Symmetries and Semiclassical Features of Nuclear Dynamics: Invited Lectures of the 1986...
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A. SEMICLASSlCAL FEATURES OF NUCLEAR MOTION
Transcript of Symmetries and Semiclassical Features of Nuclear Dynamics: Invited Lectures of the 1986...
A. SEMICLASSlCAL FEATURES OF NUCLEAR MOTION
QUANTUM FOUNDATIONS FOR A THEORY OF COLLECTIVE MOTION.
S EMICLASSICAL APPROXIMATIONS AND THE THEORY OF LARGE AMPLITUDE COLLECTIVE MOTION.
Abraham Klein Department of Physics, University of Pennsylvania
Philadelphia, Pennsylvania 19104 USA
I. INTRODUCTION
In 1962 A. Kerman and the writer applied Heisenberg's equations of motion (matrix
mechanics) in a new way to the study of nuclear collective motion. We showed how to
preserve the symmetry properties of the underlying Hamiltonian in a fully quantum al-
beit approximate form of dynamics 1'2) The main consequences of this early work were
that all mean field approximations such as Hartree-Fock, Hartree-Fock-Bogolyubov, BCS,
RPA and self-consistent cranking were seen to be contained within the new framework.
This original formulation, variously termed the generalized Hartree-Fock approximation
or self-conslstent core-particle coupling method, can be characterized as a generaliz-
ed quasi-particle theory which requires a simultaneous self-consistent determination
of the properties of states in both even and neighboring odd nuclei. In the long run,
this most ambitious form of the theory has been found wanting both theoretically and
practically (.see ref. 12, 18). It has been revived, however, in a linearized form re-
quiring externally supplied models for the even nuclei 3'4) .- In this form, it is still
the most complete and fundamental semi-phenomenological core-partlcle coupling method
extant.
Beginning in 19685) , it has become increasingly clear that the problems associat-
ed with the study of collective motion in even nuclei could be attacked on their own
merits, without dipping into the underlying single fermion structure, by utilizing
the Lie algebras of Fermion pair and multipole operators 6-II). Both the single-parti-
cle and Lie algebraic approaches have been thoroughly reviewed recently 12) . Shorter
recent reviews 13'14) may also be of interest.
Of the algebraic method, it may be said that it has proved to be a useful approach
to numerous models with collective properties and at the same time that its potential
has hardly been tapped. Until recently all applications have been to two types of
models. In the first, the Lie algebra is sufficiently simple that group representa-
tion theory may be applied without approximation. (We shall illustrate this type of
model below). In general such models are not very realistic (toy models). In the
second class of models, we deal with realistic shell structure where the Lie algebra
is already too complicated to be dealt with routinely but simplification is achieved
by using separable interactions such as pairing and quadrupole forces ("specificity"
forces), Under these circumstances it is also not difficult to derive a closed compu-
tational scheme which characterizes a set of states (band) called the collective sub-
space,
In the toy models the collective subspace is synonymous with the states of an
irreducible representation (irrep) of the Lie Algebra. In the realistic models the
collective suhspace is identified by reference to experiment. In an idealization it
may further be identified with a Hilbert space of small dimension ("geometric" model)
or with a finite dimensional space ("algebraic" model).
The main goal of the current lectures is to extend the algebraic method for even
nuclei to interactions which are not separable and to propose several new applications.
It goes without saying that many older applications can be "revisited", but we have
chosen not to look backwards because some of these subjects are out of fashion.
After these preliminary remarks, we turn to the substance of this section which
is to smooth the path from the toy models to the fully realistic ones by discussion
of an example. Consider the problem of N identical fermions in two N-fold degenerate
levels separated by the single-particle energy E. Let @t(po) create a fermion in the
magnetic sublevel p of the level ~ = ±i (for upper and lower respectively) and let
~(po) be the corresponding annihilation operator. A general shell-model Hamilton±an
for this system is
i H = ~8 r O @+(pU)$(p~)
i + t + ~V (Pl~l,P2~2;P3o3,P4~4)@ (Pl~l)~ (P202)$(P4O4)@(P3~3) , (i.i)
where the matrix elements V are antisymmetrized.
Let us first consider the widely known "Lipkin" model 15), which we regain when
the interaction takes the especially simple form
V(PlOl,P2~2;P3O3,P404) = V[~(p2,P4)~(o2,-~4)~(pl,P3)~(~l,-~3)
- (i+-+2)]~(01,o2)~(o3,o4) (1.2)
In this case (i.i) reduces to
H = gJ + i 2 2) o ~V(J+ + J_ ,
where j+ = (j_)t = I St(p+)$(p_) ,
P
i l~[ ~#(p+)~(p+)-~#(p-)@(p-) ] Jo = ~p
are generators of SU(2).
be applied to this model.
tions of SU(2)
[ J+,J_ ] = 2J °
[Jo,i± ] = ±i±
the "equations of motion"
[J±,H ]= • ~J± - VJ~ + 2VJoJ $
(1.3)
(i.4)
(1.5)
Many years ago, we showed 6) how the algebraic method could
The equations which are utilized are the commutation rela-
, (1.6)
, (1.7)
(1.8)
[Jo,H] , _ V(j+2 _ j_2) , (1.9)
and the equation which identifies the irrep of interest, [N] , giving the value of
the Caslmir invariant,
i i 2 1 1 J+J- + ~ J-J+ + Jo = ~(~N + i) (i.i0)
We do not wish to review this old work in detail, but only to retrieve a few
salient features. From the equations (1.6)-(1.10) (or from a subset of them) we ob-
tain a set of non-linear algebraic equations which characterize a collective subspace
of states. These statesconstitute a subspace of the irrep [~] which has N+I states i
(IJo I ~ ~ N) in all. We choose this subspace as follows: (i) it consists of eigen-
states of the full Hamiltonian, which we label In>, n = 0,1,2,... in order of ascend-
ing energy. (ii) Matrix elements of products are evaluated by sum rules, e.g.
<n]JoJ+In'> = <nlJoln"><n"IJ+In'> (I.ii)
Thus matrix elements of (1.6)-(1.9) yield, by use of equations such as (i.ii), non-
linear algebraic equations where the variables are matrix elements of J+ and J and - o
energy differences (En-En,). (iii) An essential element is that viable and accurate
approximation schemes may be obtained without having to include the entire space of
[N] because of the fact that in general, that is f_or all strengths of q0pp!ing, the
matrix elements <nlJ±In'> and <nlJoln'> decrease rapidly with In-n'[. Thus we may
start from the ground state and formulate schemes involving successively higher states.
(iv) Of course details may differ sharply according to the value of V. We even go
through a "phase transition" in this model: For small V the model is equivalent to
a one-dlmensional anharmonle oscillator, but for sufficiently large V we go over into
a double well oscillator with its characteristic almost degenerate doublet structure.
The number of levels utilized and the choice of suitable input for self-consistent
algebralc solution must reflect some knowledge of the regime in which one finds one-
self. (v) In general the information contained in the totality of commutation rela-
tions and equations of motion is redundant. For example, notice that
2 IJo'H] " [[J+'J-]'H]
= - [r~,J+],J ] - [[J_,H],J~ (1.12)
in consequence of the Jacobi identity, so that we do not utilize the equations of mo-
tion (1.9). But the scheme must include the information contained in Eq. (i.i0) be-
cause it expresses the Pauli principle for this problem.
For further details of this calculation and related topics - for example construc-
tion of a boson representation of the results - we refer to the previous literature
6,16) What we wish to extract most particularly from the above is the feature which
permits the formulation of a collective scheme within a relatively restricted subspace
of states. At first sight one is tempted to say that this is the group theoretical
structure of the model which restricts one to a single irrep. But even in this most
trivial of models, this is only a partial answer. A more profound reason is dynamical.
If there is collective motion, then there is at least one collective operator (in this
case a generator as well) which prefers or enforces transitions dominantly between
neighboring levels. In the Lipkin model, for small V, it Is J+, since Jo is almost
a good quantum number, but for large V, it turns out to be J = (J+ + J ). x
To move closer tc the study of a realistic model, let us still consider the
Hamiltonian (1.3), but study the equation of motion for a general density fluctuation
operator, for example
[~%(pl-)~(p2+),H] - e ~%(pl-)~(p2 +)
+ ½V {[**(Pl-l*(P2-)- *%(pl+)*(P2+l],J+} (i.13)
Even though the operator ~%~ can connect different representations of SU(2), because
H has eigenstates within a given irrep, the equation of motion can once again be con-
fined to a given irrep. This is because a sum rule such as (i.ii) requires only one
of the two factors to be a generator. In fact, as long as we restrict ourselves to
matrix elements within a given irrep, we can replace (1.13) by the previous equations
of motion.
Finally, let us investigate the most general case represented by the Hamiltonlan
Eq. (i.i). We introduce a condensed notation (p@) + ~. We have (with p~8=~Bt~=) of
^ h A -
1 1 + ~ V y6~ cBtCyt¢~ - ~ Vy~e~yt~t¢~ ~ , (1.14)
where
h~8 = ~8 ~ (SO,+ - SO,-) (1.15)
We ask the following question: Can we specify a sufficient set of conditions under
which at least a partial set of consequences of (1.14) is equivalent to the Lipkin
model. To put it in more physical terms, let us imagine that (i.i) refers to a real
physical system and that a breathing mode is observed experimentally. Let us assume
that this breathlng mode is associated with an approximately decoupled SU(2) symmetry.
Then the following conditions suffice: (1) A subset of all matrix elements V~¥~
have, approximately at least, the coherence properties specified by Eq. (1.2). (ii)
The remaining matrix elements do not necessarily vanish, or are they necessarily de-
void of all coherence properties, but we do assume that, e.g.,
z' <n[¢S% ~,a¢~ where In> and In'> are nearby members of the monopole spectrum and the prime means the
sum excludes the coherent subset. (This is a "random phase" approximation.) Then for
matrix elements within the collective subspaee, we can replace the full model with the
"toy" model.
Of most importance for our future efforts and the final point of this section is
that there is a way of combining the two requSKements (i) and (ii) above. It is to
assume a generalized factorization of matrix elements of the two body operators that
occur in (1.15):
- (c~B) - (6+-~y) + (~+8,~+-~y)} (1.17)
This approximate ($enerallzed Hartree-Fock) factorization has a number of attractive
properties which will be discussed more fully in the next section. Here we remark
that for an interaction of the form (1.2), it becomes exact when averaged with the
interaction in forming either a matrix element of the Hamiltonian or a contribution to
the equations of motion. Thus it selects the part of the Hamiltonian effective in the
collective subspace. The factorization (1.17) will play a fundamental role in the
further development of the concepts in these lectures.
II. REVISED GENERALIZED DENSIR~I ~PclXMETHOD
A. Derivation. We study next a general non-relativlstic Hamiltonian of the
form i
~a*~b*~d~c , (2.1) H = hab ~a*~b + ~ Vabcd
where we use summation convention whenever possible, and the indices a, b,... on the
nucleon creation (~T) and annihilation (~) operators may refer, according to the
application intended, either to space, spin, and Isospin, a = (~,O,T) or to the quantum
numbers of a slngle-partlcle orbit, and satisfy the usual Fermion anticommutation rela-
tions. We take h and V to be Hermitian matrices, hab = hba , Vabcd = Vcdab and V to
describe antisymmetrized matrix elements, Vabcd - - Vbacd = - Vabdc. We suppose fur-
thermore, if H is a "realistic" Hamiltonian with h the kinetic energy operator, that
V is invarlant under translations, rotations and Galilean transformations.
Our initial aim is to derive from (2.1) an approximately closed set of equations
of motion for matrix elements of the one-body density operator, equations which charac-
terize a collective subspace. In the example discussed in the introductory section,
the collective subspace, here labelled as IA>, IB>, IC> ..... had a group theoretical
significance which made its choice more or less obvious. This simplification was, in l)
fact, absent from our original conception , which was based on the idea that the re-
lationships among the members of the collective subspace was enforced by coherence
properties of sums of products of matrix elements of the two body interaction with
matrix elements of the density (or pairing) operator. (This conception also entered
the considerations of See. I.) It is to this initial conception that we shall return
here in order to effect an improvement in its implementation.
We study the equation of motion for the density operator
Oab= *b**a ' (2.2)
namely (Cf. (1.14))
i pa b = [0ab,H] = hacPcd - 0achcb
i i + 2 Vacde ~b~ct~e~d - 2 Vcdbe ~ct~dt~e~a (2.3)
By taking matrix elements of (2.3) between states of the collective subspace, we are
led to the study of the generalized density matrix (GDM) elements of the one and two
particle density operators,
p(aAlbB ) = <BI~bT~aIA> (2.4)
o(abAlcdB ) : <Bl~cT0d#0b~alA>
= 0(cdBlabA)* , (2.5)
where the latter also have certain obvious antisymmetry properties. Thus a matrix
element of (2.3), (E A .... are eigenvalues of H)
(EA-EB)P(aAIDB) = haC0(cAlbB) - o(aAlcB)hcb
1 0(deAlbcB) - ½ Vcdbe p(aeAIcdB) , (2.6) + ~ Vacde
can lead to closed equations only if the elements (2.5) can be expressed in terms of
the simpler objects (2.4). AS is well-known, this can never be the case if pairing
correlations are significant. We exclude such correlations from the present study,
since we have enough substance to communicate which does not depend in an essential
way on these currelations.
The goal of closing the set (2.6) is accomplished by means of the fundamental
assumption of our work, the generalized factorlzation hypothesis first encountered in
(I.17),
i (<Bl,c*~alC><Ci,d**bl~ > <Bl~c*~dt~b0a IA>
- (e+-+d) - (a+-+b) + (c+-+d,a+-+b)} (2.7)
This factorizatlon 17'18) which generalizes the proposal found in our earliest work has
the following desirable features: (1) It preserves the antlsymmetry and hermiticity
properties of the two-body GDM. (ll) The approximate equations of motion to which it
leads satisfy all the conservation laws inherent in the original equations of motion.
It is thus a conserving approximation. (lii) In the semi-classical limit it reduces
to tlme-dependent Hartree-Fock theory. (iv) Though the previous criteria provide ex-
cellent recommendations for the decomposition (2.7), to look more deeply into its
possible origin and significance, we must return to the physical arguments contained
in our original work. There it was reasoned that it was asking t o o much to expect
(2.7) to hold for an arbitrary choice of single particle indices. All that can Be
expected is that selected averages of (2.7) lead to coherent sums which favor transi-
tions within the collective subspace. Where these sums form generators of Lie algebras
as in the model studied in Sac. I, the matter is especially clear. But if we wish to
deal with realistic interactions, we cannot insist on such mathematical clarity and
must hope that physical intuition will pull us through. We view this mode of reason-
ing as the moxt extensive use of the random phase argument encountered in the theory
of collective motion: Certain coherent subsets of matrix elements favor transitions
within the collective subspace whereas the overwhelming majority of the remaining
elements cancel out because of "random phases".
If we accept (2.7) in the sense described, the equation of motion (2.6) can be
rewritten with the help of several convenient definitions. These include a collective
Hamiltonian in which energies are referred to the ground state energy,
~c(aAlbB) = ~ab6AB(EA-Eo) E ~ab~AB~A (2.8)
and a generalized Hartree-Fock Hamiltonian H,
4H(aAIbB) = hab ~AB + v(aAlbB) ' (2.9)
v(aAIbB) = Vacbd p(dAIcB) (2.10)
We then find that (2.6) may be written in operator form as
1 i [~c '~ = ~ [~'~ ]+ ~'~]e (2.11)
and, e.g. (Hp) is a special matrix product e
~P)e(aAlbB) =~(aC[cB)o(cA[bC)
= 0(cA[bC~H(aC[cB) . (2.12)
According to the order in (2.12), it is thus seen that the symbol e may refer either to
exchange of the collective coordinates or to the exchange of the single particle in-
dices.
If we replace the time-independent operators ~a by time-dependent operators,
~a(t) = exp(iHt)~a exp(-iHt), we may replace (2.11) by an equation which is properly
termed "quantized" TDHF, namely
dp i i i ~ = ~ [~,p] + ~ [~,0] e (2.13)
From the form (2.13), which is that of an initial value problem, it is especially
clear that we need a kinematical constraint to set the scale of p. This can again be
derived from (2.7) by setting b=d, sunning over b and using number conservation in
the forms
N]A> = N]A>. N *alA> " (N-I),alA> , (2.14)
where
= E *a** ( 2 . 1 5 ) a a
We find (replacing p2 . 0 of Hartree Fock theory)
1 2 i( 2. P = ~ P + ~ P )e " (2.16)
Equations (2.11), (2.13), and (2.16) will provide the starting points for the re-
mainder of these lectures. Here Eq. (2.16) replaces the Casimir invariant of the SU(2)
model of Sec. I. Its derivation from (2.7) represents an extended application of that
factorization. To include the analogue of all the elements of the simple model we
should also take note of the algebra of the density operators. This we do only in
passing, since these relations will not play any direct role in the applications to
be discussed.
B. ConserVation Laws' and Sum Rules. Suppose that we have an exact conservation
law associated with the Hamiltonlan (2.1), of the form
d /dx ~(X) ~(x) = 0 (2.17) dt
where 8 can, for example, be the single-particle linear or angular momentum operator.
The invar~ance properties of (2.1) which result in (2.17) can be w~Itten in the form
(Xl[e, Id IY) = exhxy " hxy0y = O (2.18)
and
(xy][ (81449 2) ,V] ]zW) = (Sx+Sy)Vxyzw - Vxyzw(SZ+Ow) = 0 (2 .19)
We now show that in consequence of (2.18) and (2.19) and the approximate equations of
motion (2.13), Eq. (2.17) is satisfied within the collective subspace. We calculate
Id/f /dx <A' l * * (x )0* (x ) IA> (2.20)
Since the contributions from the single-particle and two-particle parts of H vanish
separately, we consider only the interaction part, which is
!2/0x[ Vxz,~, P(wAIzS)0(YSIxA') - 0(xA]yS)VyzxwO(wSlzA,)
+ Vxzy, w 0(wSlzA')0(yAIxB) - p ( x B I Y A ' ) V y z x w 0(wAlzS)
= 21--/ (xyI[01-~2,V] Izw)0(wAlzB)p(ySlxA') = 0 , ( 2 .21 )
requiring only a reshuffling of variables.
Next we show that our method satisfies both energy weighted and energy squared
weighted sum rules. Let F be an arbitrary one-partlcle ~ermitian operator ^ ^
= feb Pba = Tr f0 , (2.22)
and consider the ground state expectation values
^ k ^
Sk(F) ~ <O[F(H-E o) FIn> (2.23)
For k = I, two exact versions of Sl~f) are
I <0I [~, tH,~] ] i . > (2.24)
fOA " <O[FIA> (2.2S)
We prove that this sum rule continues to be satisfies in the GDM approximation, pro-
vided the left hand side is evaluated from the solution of our equations and provld=,/
FI0> is a state in the collective subspace. For then the right hand side of (2.24) is
10
equal to
1 i <O[EH,F] IA> (2.26) foA <AI[H'FI lO> - 7 fAO
But according to the GDM approximation, we evaluate
i [0,~] (cOIdA) <A[[H,F][O> = fdc {~
i * + 2 [0'~]e(COldA) = fdc[O'~4c ](cOIdA) = ~AfOA (2.27)
Combined with a similar evaluation for the second term, we easily reach the required
identity,
The method described works not only for the S I sum rule, but also the S 2
which takes the form
S2(F) ~ Z(~A)mlfoA 12 = <OI[F,H][H,F]I0> (2.28)
The proof is immediate if one follows the outline of the argument given above.
III. SEMI-CLASSICAL LIMITS
In this section we shall describe several different problems whose central ele-
ments can be understood by starting with the GDM equations and passing to the semi-
classical limit in which the theory reduces to a version of time-dependent Hartree-
Fock theory (TDHF).
A. Periodic Motion. 19) " For illustrative purposes only, we restrict our study
to a one-dlmensional collective mode, a breathing mode, for example, so that IA>+In>,
with n an integer, n = 0 referring to the ground state. We study the matrix element
0(xltn[x2tn') = <n'l*t(x2t)*(Xl t) In>
= exp [i(En,-Bn)t]<n'I~+(x2)~(xl)In > (3.1)
One standard method of approaching the semi-classlcal limit is as follows: Introduce
new variables
= ½(n+n'), ~ = n'-n , (3.2)
and approximate the energy difference
En,-E n ~ (dE/dn)~ E ~(n)9 , (3.3)
If the correspondence principle approximation (3.3) is Justified, then (3.1) becomes
. . i~(~)~t 0(xltnlx2tn' ) ~ p,,~XlX2)e , (3.4)
and the Fourier sum
O~(XlX21t ) =~O(xltnlx2tn' ) (3,51
defines a periodic function of t, period T(n) = 2z/~(n). From its definition, the
Fourier coefficients of 0~(xlx21t) are the various transition matrix elements of the
density matrix.
Provided 0~,v(XlX2) is a slowly varying function of n and a function peaked in V,
sum rule)
11
we shall prove that pfi(XlX21t) is a solution of the TDHF equation. The peaking in
is required for the correspondence principle approximation (3.3) to be valid. The
necessity for the postulated behavior in n will become evident below.
To prove the assertion made above, form the sum on 9 in Eq. (2.13). We also form
a similar sum in the factorlzation (2.7). A typical term (with slngle-particle indices
suppressed) contributing to the latter at time t is
ei~(n)~tp(nln")p(n"In ') (3.6)
We i n t ~ o d n c e ~" = ~ ( n ~ " ) , V' ~ n " - n , ~ ' = ~ ( n ' + ~ " ) , ~ ' = n ' - n " (~ Then t h e
Sum ( 3 . 6 ) c a n be w r i t t e n a p g r o x i m a t e l 2 a s
i ~ ( G ) ~ ' t . . ~ i ~ ( ~ ) ~ " t . (Z e p~,M,)~o,,e p~,~ , , ) ( 3 . 7 )
provided we can set
p~,,,~ ~ p~,~ , p~,,~ ~p~,~ , (3.8)
which requires weak dependence on the average of the quantum numbers, as has been
assumed.
We have thus been able to transform a double sum of products into a product of
single sums. In addition to the previous physical assumptions concerning the depend-
dence of the transition amplitudes on their variables, the composition property of
the exponential function has played a central role. Therefore we expect to be able
to prove semi-classical limits (in general) only for Fourier series or integrals of
transition amplitudes.
The major consequence of the transition from (3.6) to (3.7) is that the factoriza-
tlon (2.7) turns into the standard Hartree-Fock factorization,
P~(XlX21X3X 4) = p~(xlx3)P~(x2x 4) - p~ (x2x3)P~(XlX4), (3.9)
(the time variable has been suppressed). This transforms Eq. (2.13) into the TDHF
equation
i~ = ~,p ] (3.10)
and Eq. (2.16) into
2 p = p , (3.11)
which guarantees that p describes a Slater determinant. Here e.g.
4H +~(XlX2) = h(XlX2) + V(XlX2X3X4) pff(x4x3) , (3.12)
is the standard HF Hamiltonlan, only the subscript n reminding us that we started from
the GDM formalism. In the weak coupling limit, we regain the RPA via the usual argu-
ments, namely that only the Fourier components Pff,o and P~,±I occur, that the former
is of zero order and the latter of first order.
The search for periodic solutions for physically identifiable e~citatlons which
incorporate the full non-llnearity of TDHF is a new and exciting problem. We mention
several possible approaches. As one possibility Umar and Straayer 20) have studied
12
solutions of TDHF which contain (inltlally) all frequencies and found that in the
course of time TDHF amplifies some collective modes. These can he filtered out (they
contain the P~,v Fourier amplitudes-) and recomblned to form an initial density matrix
which might generate a truly periodic solution, at least after further filtering and
iteratlon~
As another method we propose active use of the semi-classlcal quantization condi-
tion
f (n)dt b) (3.13)
where the sum is over the occupied orbitals and ~h(Xl t) is one of the occupied orbltals
at time t,
p~(XlX21 t) = Z ~h(Xlt)~h*(X2t) (3.14). h
We shall include a "new" proof of (3.13) below, but let us consider how this condition
can be utilized. At t = 0, we might choose a density matrix representing an RPA period-
ic solution or one constructed from the theory of large amplitude collective motion
(next subsection). We then integrate the TDHF equation forward in time until the in-
tegral (13.13) achleve~ the value 27 (for example). A function Q~(XlX21t) is thus de-
fined over 0 < t < T(n) and can be taken as defining a period function. The Fourier
coefficients of this function can be assembled into a new initial condition which can
again be integrated forward in time and the previous procedure repeated. Hopefully,
after a sufficient number of integrations the Fourier coefficients will converge.
The application of the above procedure lles in the future. We complete the pre-
sent discussion with a derivation of (3.13) which we believe is more direct than 21-24)
those previously available in the literature . The occupied orhitals ~h(xt) are
the solutions which evolve in time from an initially prescribed set ~h(XO ) according
to the time dependent equation
! !
i8 t ¢h(xt) = ~ ( x , x It)~h(X t) . (3.15)
These equations (and thelr complex conjugates) can 5e derived from the usual Hamilton's
principle
~I n = 6 / T(n) <#(t) l(i~t-HlI*(t)> = 0 , (3.16) o
where #(t) is the Slater determinant composed of the orbltals #h(Xt), and the varia-
tlon is with respect to ~h and ~h subject to the boundary conditions
~h(XO) = ~h(XT) - 6T(n) = 0 (3,17)
It is more convenient for our purposes to utilize the principle of least action,
~S = ~ fT(n) <~(t) li~ t].. ~ (t)> - 0 (3.18) n o
Here the variations satisfy the conditions
IS
~(xO) = l~¢(XTn) + ~t~h(XTn)~Tnl
= ~ <$(t) IHi#(t)> = 0 (3.19)
The consistency of the last condition follows from the constancy in time of the Har-
tree-Fock energy. The proof of (3.18) with the conditions (3.19) is the same as that
to be found in any mechanics textbook 25) and will not be repeated. Utilizing the
definition (3.16) of In, we may rewrite (3.18) as
AS n = A(I n + EnT n) = 0 (3.20)
We apply this principle to the problem at hand by computing (dSn/dn) . This can
he written symbolically as a sum of two terms
dS AS ~S ~S n n ~ = n (3.21~
~f-n = "Tn + ~n ~n '
where the first term (which is then dropped) is the contrlbution to the derivative
from those terms which enter into the establishment of the variational principle
ASn = 0 and for this reason doeano:tcontribute. The quantity (BSn/Bn) is then computed
By varying only those quantities which both depend on n and remain unvaried in deriving
(3.20), In the light of these remarks we note that (~In/~n) = 0 and also (~Tn/~n) = 0.
From (3.20) we then have
dS dE n n = =
dn d~ Tn w(n)Tn 2n (3.22)
Thus we obtain the well-known condition
s = {Tnd~li~tl*> n 0
I Tndt E(*21i~ti*h) = 2zn (3.23) 0 h "
B. Large Amplitude Collective Motion, Adiabatic Limit. 26-28) The major portion
of these lectures will be devoted to the detailed theoretical development of this sub-
Ject. The purpose of this section is to reach the starting point fDr the later develop-
ment and to illustrate a second way in which a seml-classlcal limit of the GDM equa-
tions may he taken.
As with the argument given in the previous section, it may be applied to any num-
ber of degrees of freedom, but for illustrative pruposes, we continue to discuss only
a monopole spectrum of states In>. Referring to Eq. (2.8), we suppose - this is called
the large amplitude adiabatic hypothesis - that the excitation energies ~n are those
of an operator
1 ~c(p,q) = ~ {p, {p,~(q)}} + ~(q) , (3.24)
With
HC ~n (q) = ~n ~n (q) (3.25)
We must comment, before continuing, on the form chosen for the kinetic energy. This
14
form is singled out (see below) because it has the simplest Wiener transform among all
possible competitors. It is an elementary exercise (sometimes called a theorem) to
show that any other Hermitian form differs from that used by contributions of "order
h 2" to the potential energy.
The complete set ~n(q) can be thought of as the wave functions of a phenomeno-
logical model. Since we have introduced a Hilbert space, in this case, it is a "geo-
metrical" model. This is the approach which we shall follow in these lectures, though
presumably one could develop a purely algehralc scheme as well.
With the help of the complete set ~n(q), we change the representation in which
we study the GDM equation (2.11) (and (2.13)), namely we define
p (aq l bq ' ) = < q ' I~ht~a I q>
= ~n,(q')<n'I~%b~In> ~n*(q) (3.26)
In this collective coordinate representation, the passage to the semi-classlcal limit
is carried through by means of the Wigner transform, which is defined as the Fourier
transform wlth respect to the difference coordinate y = (q'-q),
0(ablQP) = fexp(-lyP) 0(ablQy) (3.27)
where Q = ~ (q+q') and 0(aqlbq')E0(abIQy) .
What is the physical basis for the application of the Wigner transform to the GDM
equations? The analogue (but not the precise equivalent) of the physical assumptions
made in the previous section is that the density matrix o(abIQy) is a slowly varying
function of the average coordinate Q and strongly peaked in y about y = 0. Therefore
in the Wiener representation, 0(abIQP ) would be a slowly varying function of Q and P.
These assumptions are utilized in the GDM equations (2.11) and (2.13) by the
application of the convolution theorem for Wigner transforms: If A,B,C, are operators
and C = AB, i.e.,
<q'JCIq> = /<q'IAlq">dq"<q"IBIq> , (3.28)
then
c(Q ) - r Ars(QF) sr(Q )' (3 .29 ) r=O a=O
where, e.g.
A = (~/~Q)r (~/~p)S A(QP) (3.30) rs
Thus to first order (3.29) yields a famous result,
3A 3B ~A 3B C(QP) g A(QP)B(QP) + i ( ~ 8P 3Q ) (3.31)
It is evident from (3.27) that we are utilizing units in which the product (QP) is
dimensionless. Our assumptions correspond to a physical situation where this quantity
is of the order of the number of degrees of freedom ("~") which contribute to
the collective motion and successive terms of (3.29) stand to each other in the ratio ~"
15
Under these circumstances we evaluate the products and commutators which appear
in (2.11) and (2,13) by using the first non-vanlshlng terms of (3.29). Several observa-
tions are necessary: A first essential ingredient of the reduction of the GDM equa-
tion is the Wigner tranmform of the generalized factorization (2.7), namely (using only
the first term of (3.31))
p(abcdlQP ) = p(acIQP)p(bdIQP )
.. p(bc IQp) p(ad IQp ) , (3.32)
which is once again the standard HF factorization.
Let us next notice (without proof) that the exact Wigner transform of the collec-
tive Hamiltonian (3.24) is
1 p%(Q) +q(Q) (3.33) (4 c (QP) =
Putting these results together, we can combine (2.11) and (2.13) into equations expres-
Sing the semi-classlcal limit,
6(ab) = ~Hc,P(ab) ]pB (3 .34a )
= - i [ ~ , p ] (ab) (3 .34b)
Here the dependence on Q and P has been suppressed, PB means Polsson bracket with res-
pect to the variables Q,P and H is the usual HF Hamiltonlan corresponding to the den-
sity matrix p(abIQp) ' w~th p2 = P. It should be apparent that the PB term of (3.34)
arose from a commutator in the collective space and therefore the second term of (3.31)
contributed whereas the other term involved a commutator in the single-particle space
only which was unaffected by passage to the seml-class~cal limit.
The net result of our efforts is a pair of equations: The first is a classical
Liouville equation and the second a TDHF equation. These will form the starting point
of a theory of collective motion whose aim is to calculate the classical Hamiltonlan
H e , Eq. (3.33), i.e., to find the functions ~(Q) and ~(Q). By inverse Wither trans-
form, this gives us the quantum Hamiltonian (3.29) from which spectra and other pro-
perties of the collective states can be determined. This study will be pursued in
See. IV,
i 29) C. Heavy Ion Scatter ng. This example is in principle and practice more
complicated than the ones discussed in the preceding sections. To derive TDHF for
the appropriate amplitude, we need all the arguments given previously plus some addi-
tional ones. Our first task is to specify the states IA> to which the generalized
factorization (2.7) is to apply, We deal once more with a schematic model which,
however, contains all the essential physical elements. We shall allow for momentum
transfer, particle transfer, and excitation of collective states, the latter schematiz-
ed by a monopole degree of freedom. Thus
IA>~ INl,~l,nl; N2,~2,n2> (3.35)
is an exact scattering eigenstate of a "nucleus" NI, momentum PI' state of collective
16
excitation n I by a nucleus N 2 with corresponding quantum numbers. Here we shall assume
that the collective excitation goes mainly through modes, like giant resonances whose
properties vary slowly with nucleon number and do not depend in an essential way on
the details of shell structure.
We therefore study the density matrix element
<NI',Pl ,nl';N2',P2',n2'J~#(x2t)~(xlt)JNl,Pl,nl;N2,P2,n2 >
To construct from this quantity one that satisfies TDHF theory, various transformations
are necessary. For example, we replace mi by the collective variable ql (remember
(3.26) and (3.27)) and then pass to Wigner transform varlables QIPI , Q2P2 . To perform
a similar trick for the particle-transfer properties, we first define NI, ~2,~ and ~'
by
= NI- ~ ' N2 = N2 + ~ N I
!
NI - ~' ' N2' = N2 + ~' ' (3.26) N I
and (displaying only relevant variables) set
ip~l~2 ' I~0 <~i_~,~2+~,i~#$1~i_~,~2+~> (3.27) <8' 18> ~ E (2~) -I e -i~'0 e
We then treat 8 and ~' Just llke q and q' for a transformation to Wigner variables,
O = ~(8+8') and its canonically conjugate variable ~. Finally, just integrating over
momentum transfer (which is, after all, a special case of the procedure for other
variables), we write
0NIN2(Xlt,X2~,QIPI,Q2P 2)
= f d(8'-e)exp -i~(8'-8)"fd~id~ 2 ~dYldY2ex p (-iPlYl-iP2Y 2)
<Ni,0',Pl-kl,ql';N2,8',P2-k2,q2 J~#(x2t)~(xlt)INi,0,~l,ql;N2,8,p2,q2 >. (3.28)
If the momentum transfer variables were absent, the arguments f~t deriving TDHF theory
would now be the same as those used in Subsection B, which can be made independently for !
each degree of freedom (where e.g. Yl = ql -ql )"
It remains therefore only to discuss the integrals over momentum transfer. In
practice we would actually replace the individual momenta by total momentum and rela-
tive momentum and further decompose the latter into relative energy and impact para-
meter variables. Furthermore we could have introduced parameters additional to ~ ~,
QI,PI,Q2,P2 via the momentum integrals. For any of these and also for the restricted
integral displayed in (3.28), it has long been known 30) that it is possible to derive
the Hartree-Fock factorization from the generalized factorization provided
l~il << l~iI , (3.29)
which states that the maximum momentum transfer that can be supp.orted by the density
matrix is small compared to the incident momentum. At the same time we must expect
the mean field picture to break down if Ipll becomes so large that particle emission
~7
becomes important. A reasonable upper bound is l~il Z N i PF' where ~F is the Fermi momentum. Since the form factor (3.28) will not support momentum transfer larger than
PF' I~iI ~ < PF' we see that (3.29) can be satisfied.
Following these arguments through, we reach the conclusion that the multi-para-
meter amplitude defined by (3.28) is the density matrix of a Slater determinant and
satisfies the TDHF equation. It is thus the prototype of a reasonable candidate for
the density matrix describing Heavy Ion (HI) scattering, and gives substance to the
co,anon understanding that the TDHF approach to HI reactions provides an inclusive
description.
We can, however, go further and ask whether it is possible to extract exclusive
scattering information from this approach. We illustrate how a positive answer to this
question may be obtained. Consider only the question of particle transfer. For the
initlal condition (t +-~), we expect (and can prove) that the density matrix decomposes o
into a sum of contributions from projectile and target; number conservation then re-
quires ~ = ~,. It follows that (cf. (3.27))
t°~'~" -i
<8'I0~i~21~> ~ (2~) ~Zexp [i~(@-e')~
[<NI-VJ011NI-V> + <N2+~[02JN2+v>] (3.30)
This shows, first of all, that Ois an ignorable coordinate (conservation of particles).
Referring to (3.27) we have furthermore that
0~l~2(Xlto,X2toI~...) + <NI-~I011NI-~> + <N2+~IP21N2+~> , (3.31)
so that ~ is a particle transfer variable.
The conclusion is that if we study TDHF as a (smooth) function of the nucleon
number of the individual incident ions (for a fixed total number) we can disentangle
(in seml-classlcal accuracy) the particle transfer behavior. Similar arguments can
be made for any variable for which the full Wigner transform description has been intro-
duced. In general terms we must solve TDHF for a sufficient range of values of the
classical canonical pairs and then calculate the inverse transformations necessary to
Pick out the individual or subset of individual channels of interest. Though hardly
trivial from a technical or an economic point of view, it is important that the prin-
ciple be established. Moreover, it doesn't seem unreasonable to choose simplified
examples on which some start can be made on the extension of existing technology.
IV. LARGE AMPLITUDE COLLECTIVE MOTION. CLASSICAL LIMIT.
We return to the principle detailed subject of these lectures, large amplitude
collective motion. This subject has been studied for fifteen years with many beautiful
and many difficult papers to be found in the literature. We include a representative list 31-53) We are a relative newcomer in this field, and our published work to date
contains only a piecemeal account of our results. 26"28'54) In these lectures, we shall
attempt a coherent account of our current approach, which certainly involves mainly
18
a restatement of older ideas, but also, we believe, some new results. It turns out
that the major points of the theory can be understood within the framework of classi-
cal Hamiltonlan mechanics. Therefore we shall first study this aspect and later tie
it to TDHF theory for fermlons.
A. Classical Hamiltonlan Mechanlcs o fDecoupled Large.Amplltude Collective Motion.
We study a Hamiltonlan of the form
1 H(£,~) = ~ p~ B~S(xl...xN)p8 + V(x I ..x N) (4.1)
containing N pairs of "slngle-particle" coordinates and momenta (xl...x N) = (~),
(pl...pN) ~ (£). We pose the following question: Can we find a (locally invertible)
point canonical transformation to a new set (~) and ([)
x = ~(QI...Q N) , (4.2)
p~ = (~Q~/bx=)p (4.3)
such that the transformed Hamiltonlan can be separated in at least one way into two
non-interactlng parts,
I~(e(R,£), ~(R)) = ~([,R)
. Hc(PI...PK;Q1...Q K) + HNc(PK+I...PN;QK+I...Q N) . (4.4)
Here the subscripts C and NC stand for collective and non-colleetive~ respectively.
Since our initial definition of collectivity renders it synonymous with separability,
the distinction between the two parts of (4.4) is, at this stage, artificial. The
distinction will become meaningful only when we turn to applications of the mathema-
tical theory. It will be based both on well-known physical criteria and on the fact
that we shall want to adapt the mathematical theory to the usual realistic situation
where exact decoupling doesn't occur.
Under the transformations (4.2), (4.3), H (for example) will have the form C
(i, J ' l . . . . K)
1 . . He = 2 Pi~iJ(Ql" .QK)pj + ~(QI ..QK), (4.5)
~lJ , (~Qi/~x~) B~8(SQJ/Sx ~) . (4.6)
HNC will possess a similar form in terms of the variables Qa Pa' a ~ K+I.,.N. A set
of conditions for decoupling is then easily lis:ted: (1) V separates into two parts,
one depending on the Q i, i ~ I.,.K, the other on the Qa a = K+I...N. (ii) ~ia =
~al = 0, i.e., the mass tensor has block diagonal form. (ill) ~lJ does not depend on
Q~ and does not depend on ~i i.e., ~ab
(3~lJ/~Qa) = (~ab/~Qi) ~ 0 (4.7)
Let us consider _ 4 specifically conditions for the determination of HG, i.e. of ~(QI...Q K ) and ~ij(QI...QK). For example, from (14.1) - (4,5) we have
Vc(QI...QK) ~(QI...Q K) = V(~(~)) Qa=0 ~ v(~(~)IZ, , (4.8)
19
i.e. V is the value of V on a hypersurface Z, defined by
x u = ~e(QI...QK QK+I = ...QN = 0) ~ ~U(Qi) . (4.9)
Thus to determine HC, we have to find the functions (4.9) for the hypersurface Z as
well as the quantities (~Qi/3xe) (cf. (4.6). From (4.8) we have on Z
(I) V ~ (~V/~x ~) = (~q/~Qi)(~Qi/Bx@ ) . (4.10)
This equation is a convenient expression of condition (i) for separability (together
with a similar equation for the complementary space).
To express condition (ll), let us consider the kinematical constraints
(~Q~/~Q~) = (~Q~/3x~)(B~/~Q ~) = ~ (4.11a)
relating to the complete transformation (4.2), (4.3). The form of (4.6) for such a
transformation is obtained from (4.6) by replacing (lj)+(~). Utilizing (4.11) we
then derive
~9(D¢a/~x 9) = B~(gq~/~x8 ) (4.12)
On Z, in the separable case, this becomes in virtue of condition (ii),
(II) ~ij(~/~Q~ = B~8(~Qi/~x s) (4.13)
To Conditions (I) and (II) we add those equations (4.11a) which refer to ~, namely
(IIl) (~Qi/~QJ) = (~Qi/~x~)(~,~QJ) = ~iJ (4.11b)
and the first of Eqs. (4.7) (which we call (IV)). If we can find a surface ~(Qi).
and a set of K vectors (BQi/~x ~) satisfying conditions (I)-(IV) (plus a complementary
set characterizing ~C ) , we will have succeeded in decoupling the original Hamiltonian
into two-non-lnteractlng parts.
Though there are rare (mainly trivial) examples where one can solve conditions
(1)-(IV) directly in most cases further development of the equations is both necessary
and desirable. Let us, for instance, consider the determination of the hypersurface
Z, x ~ = $~(Qi). This will be done by specification of the tangent plane at every point
of the surface. Note that the vectors (~/~Qi), i = I...K form a basis for the tan-
gent plane. Next note that by combining (I) and (If), (4.10) and (4.13), we have
V e H B~BV 8 = [(~/~Qi)BiJ ](~/~Qj) E %1 ~e/~Qi (4.143
This equation identifies V ~ as a vector in the tangent plane, a result depending only
on Conditions (I) and (II).
What follows is our most important new result, namely by combining conditions
(!~, (II) and (IV1, we can identify an indefinite number of tangent vectors to Z.
Define a sequence of point functions,
(1)V - V , (4.15)
i(o) VeBeB(~)VB (°+l)v = y (4.16)
We Prove that provided the gradient of (~)V (V(~)~ is in the tangent plane, so is
20
v(~+l)v. v e n t o r y o f t a n g e n t v e c t o r s .
P r o o f : We a s s u m e ( c f . ( 4 . 1 0 ) )
(g)V~ = (~)%i(~Qi/~x ~) ,
where (g)%i = (g)%i (QI'''Qk) is a point function on ~.
and remembering (4.6), we find
~ij (o+i) v = 1 (~)
(° )x i xj Now compute the gradient
(~+I)v ~ = [(~(~)%j/~Qi)(o)% k~jk] (DQi/~x~)
+ [l (~) (o) %j %k(~Jk/~Qi) ] (~Qi/$x~)
Since we already know this to be true for o = i, we can thus Build an in-
(4.17)
Substituting (4.17) into (4.16)
(4.18)
(°+l)~i(~Qi/~x~) , (4.19)
where in the second term, we have utilized condition (IV). Combining (4.19) and (4.12),
we obtain the result sought (including a definition of (o+l)~i)
(~+l)v~ = (~+l)li (~/~Qi) (4.20)
The consequence of the theorem just proved is that it provides a fully determined
constructive procedure for finding the collective Hamiltonian, defined on the hypersur-
face Z. For suppose that Z is K dimensional. It follows that the K + 1 vectors
(~=I...K+I)
(~)v ~ = (o)~i(3~/3Qi) (4.21)
are linearly dependent at any point. Viewed as a system of equations for the computa-
tion of the tangent vectors (3~/~Qi), they are overdetermined. To see the consequenc¢~
let us consider a special case N=3, K=2. Without loss of generality we choose
i ~i x = i = 1,2
x3 = ~(QIQ2) = ~(xl x 2) (4.22)
We are in effect seeking a two-dimensional surface in a three-dimensional space and
have chosen a convenient parametrization, which simplifies the equations (4.21). For
= i = 1,2, we have simply from (4.21) and (4.22)
(~)V i = (~)~i (i = 1,2) (4.23)
This leaves the equations (O = 1,2,3)
(O)V3 = (~)vl(~/~x I) + (~)V(2)(~/~x 2) (4.24)
These equations are consistent if and only if the determinant
detI(O)V~l = 0 . (~,~ = 1,2,3) (4.25)
21
Provided (4.25) has at least one solution, that solution has the form (4.22). It is
then guaranteed that (4.24) can be solved for the tangent vectors (~/~xi). (A numeri-
cal example (of sorts) will be given in the next section.) A corresponding procedure
can be fashioned for any values of K and N. Since, as shown later, we can construct
the mass matrix~IJ from the tangent vectors, we have, in principle, completed our
task.
There remain a number of theoretical points to discuss, but the (temporarily)
Saturated reader may prefer to go on to the next section on applications and return to
developments given below at a later time.
Before going on, the relation of the conditions (1)-(IV) to Hamilton's equations
needs clarification. Condition (III) is the affirmation that we are dealing with a
point canonical transformation. The other conditions have a dynamical origin. We
Study Hamilton's equations for points on I. These can be written in two ways.
~ = (~H/~p~)l ~ = [x~,H§]pB , (4.26)
p~ =~(aH/Dx~)[ z = [p~,Hc]pB , (4.27)
the Poisson bracket (with respect to the Qi and P.) taking explicit note of the fact i
that on E, H ~ H . Upon substitution of (4.2) and (4.3), each side of (4.26) and c
(4.27) becomes a polynomial of degree two in Pi" We thus obtain conditions of order
Zero, one and two by equating coefficients. The first two are conditions (I) and (II)
respectively. The condition of order two looks rather formidable:
(~Qi/~x~)(~BB~/~x~)($QJ/~xY) = (~Qk/~x~)(~iJ/~Qk)
- (D2Qi/~x~ ~Qk)~kj_(~2QJ/~x~ ~Qk)~ki (4.28)
It is not difficult to show, however, that provided conditions (II), (III) and (IV)
are satisfied, (4.28) is not an independent equation. Conversely we can (and do) re-
Place (4.28) by the more transparent condition (IV), which played an essential role in
developing the constructive procedure described previously.
Condition (I) is often given in the form
(~V/aQ a) ~ = 0 , (4.29)
Which is the statement that for points on E the gradient of V has no component ortho-
gonal to ~. But this is the same as the statement that the gradient of V lies in the
tangent plane at ~, which is the more useful form for us. Formally, utilizing (I)
(aV/~Qa) E = (~V/~x ~) E(~/aQa)
= (~V/~Qi).(~Qi/~x~)(~/~Q a) , (4.30)
which indeed vanishes because of (4.11a).
Next we consider a geometrical interpretation of the theory which yields the same
determinantal conditions as found from the previous consideration (e.g. Eq. (4.25)).
22
Let us consider the case K = 2 again. If ~I and ~2 are Lagrange multiplier functions,
the necessary conditions for the vanishing of the first variation (in a space with
N> 3)
6(1)V - £~i ~(2)V - 22 ~(3)V = 0 , (4.31)
i.e.
(1)V~ - ~I(2)V ~ - ~2(3)¥ ~ = 0 (~=I...N) (4.32)
are consistent only if certain three by three determinants vanish. For N = 3, it is
Just the single determinant (4.25) and for N > 3, the same set of determinants occur
as in the theory of the tangent plane described above.
To explore the meaning of the variational principle (4.31), first consider ~ = }
(~2 = 0). Remember that (i)V is the potential , (2)V the (square of the) magnitude
of VV, (3)V the (square of the) magnitude of the gradient of the (square of the) mag-
nitude of VV, etc. For K = i, the variational principle is a form of the definition
of stationary paths 43) (including valleys) on the potential energy surface V with
respect to the metric tensor B ~. This is usually expressed in a different form,
namely (A Lagrange multiplier)
~(2)V -A~(1)V = 0 , (4.33)
i.e., we seek the minimum slope of the potential among all points on an equipotential.
(In the form (4.31) for K = 1 we seek the maximum of the potential among points of
equal magnitude of the gradient of the potential.) For ~ = 2, which has not appeared
in the physics literature, we have a two-dimenslonal generalization of the concept of
stationary path. By means of our family of point functions (~)V, we can introduce a
variational definition of a stationary hypersurface, Z, of arbitrary dimension K in a
space of N dimensions. As we have shown, this surface is our candidate for decoupled
collective subspace.
It is "obvious" from the variational formulation (4.31) that stationary hypersur-
faces of dimension K contain all stationary hypersurfaces of dimension < K.
We must finally face the really difficult problem. How can we use the theory de-
veloped in a case of "approximate" decoupling? How can we measure approximate decoupl-
ing in an intrinsic way? In order to understand our viewpoint, let us write the basic
equations (4.21) in the form
(O)v~ (o)li = Yi (4.34)
where we have introduced a set of quantities Yi' which are the components of K vectors
Yi" According to the previous discussion, provided the appropriate determinental con-
ditions have a solution, Eqs. (4.34) determine a hypersurfaee Z, described by the equa-
tions x ~ = ~(Qi) and at each point a ~ determined by the K vectors y~. For exact ~
ly decoup]ed motion y~ = (~/~Qi) and the plane determined is the tangent phase.
Before continuing our main considerations, let us notice that whereas the functio~ ~
a ~a(Qi) x = determine the collective potential energy, the collective mass matrix can
23
be calculated either from the formula (4.6) or from the equivalent formula
~ij = (~/~Qi)B~B(~/~QJ) ' (4.35)
where ~iJ is the matrix reciprocal to ~ij and B B is reciprocal to B ~B. Thus we have
a decision to make, in the event of approximate decoupling, concerning how to evaluate
(4.35). We would appear to have two choices: Use either the actual tangent vectors
(to the approximate surface X~) or use the solutions y~ of Eqs. (4.34). This also
Suggests a "natural" measure of error of the decoupling at each point. Let
i = Yi - (~/~Qi) ~ Yi - ~i (4.36)
Then a reasonable measure of "error" is
= 6~ / ~ ~) , (4.37) &(QI...Qk) (~ ~ij J) (~i~iJ
and we should require A<< 1 (on the average) for "reasonable" decoupling.
It would be a falsification of the existing literature to pretend that the dis-
CUssion just given exhausts all the possibilities. Actually some of the discussions
found in the literature have distorted the situation because of undue emphasis on the
One-dimensional case - the collective path; here attention has been concentrated on
physical p~oblems where tile potential energy surface possesses an absolute minimum
and a saddle point, and the decoupling is not exact. There is a valley running up
from the minimum to the saddle point which the "exact" theory chooses as the "surface"
~. In practice (so far) we have used the actual tangent to the valley path. The
alternative, YI' is equivalent in this case to a local normal mode (as discussed below).
But there is at least one more choice. There is a fall line (line of force)
which passes through the two critical points. One group 40'47) has chosen this
Collective path and the tangent to it and used the resulting theory in some impressive
applications. For exact decoupling this does coincide with the valley and for approxi-
mate decoupling it should be a reasonable choice. Its main defects are that it doesn't
Work if there is no saddle point and it doesn't generalize to more than one dimension.
On the other hand, there is another general veiwpoint in the literature 32'41'43'28)
Which merits our attention, that of the local harmonic approximation (LHA). We shall
approach this idea by means of an "end run". We ask once again to what extent we can
satisfy directly conditions (I) - (IV). For this purpose condition (IV) (the absence
of centripetal forces between collective and non-collective space) is not (as previous-
ly remarked) really a constructive equation, so that we put it aside temporarily and
ask: How many equations do the conditions (I) - (Ill) (often called Villars' equa- 33)~ tion s j provide for the unknowns ~, (~/~Qi) and (~Qi/~x~). In Tab]e I we list and
COunt the unknowns. The only non-trivial point concerning this enumeration is the num-
ber of independent functions ~. Because of the freedom of point transformations on
the collective hypersurface, K of the N functions ~ may be chosen arbitrarily. For
instance, in some cases, as already illustrated, it is even convenient to choose
i( Qi i ~) = = x , i = I...K (4.38)
Table I.
24
Unknowns in Villars' Equations
Variable Number
¢m ~N-K)
(~Qi/~xa) (N) (K)
~q/~Qi K
~iJ K 2
(~¢~/~QJ) (N-K)(K)
If we study the table, a useful viewpoint that emerges is that the number of unknowns
exceeds the number of equations by the number of elements (~¢~/$Qi).
It is a tenable view therefore that the system can be completed by providing equa"
tlons for the tangent vectors. This is what we have done previously for the case of
exact decoupling, in a way which incorporates the additional condition (IV). But if
decoupling is not exact, the equations derived are not consistent and have to be "in-
terpreted". In such an uncomfortable situation, it might be just as acceptable to in-
vent convenient~ but approximate, alternative equations for the tangent vectors. This
is the role of the LHA.
There are several ways to derive LHA. Following the discussion Just given, we
choose the most straightforward (and least elegant). To simplify the algebra a bit,
let the original metric tensor B ~B be the unit matrix. Differentiating condition I
respect to Qi, we have (V B=(~2V/Dx~$xB), etc.) with
fJ &B (4 39)
where (~QJ/~x a) = fJ and (~2QJ/~xm~x~) = fJ We further compute with repeated help ~ " from condition (II),
= (i) + (ii) (4.40)
But (using (4.35))
(i) ~i%(~J ~/3Qm ) ~mk = ~k
where we have also used condition (IV).
general, (m <__ K, a > K)
~ ~ ~mf ~ ~ a
B mk ~ ab. = ¢~m ~ *k + (~a ~ @b (4.42)
Here recall that the indices a and b refer to the non-collective space and thus it
appears that the equations do not define vectors tangent to Z.
An exception occurs when there is only a single-collective coordinate. In that
, (4.41)
Notice however that in the second term,in
25
Case term (li) becomes
=i ~ - 1 • (d~B/dQ)~ (d~8/dQ) ~ ~ ~
4 - i ~ dQ 2 d o d--Q." ' (4.43)
and (4.39) reduces to ((4.41) and (4.43) combining)
d~ B 2(Q) d~1 (4.44) V~ dQ L-= NQ '
where
2(Q) = (d2V/dQ2) + ~(dV/dQ)(H~/dQ) (4.45)
is not necessarily positive. In any event (4.44) is a local RPA equation which deter-
mines a tangent vector, it being understood that we choose the lowest frequency solu-
tion. This is the LHA.
There are two other circumstances, one exact and one approximate and imposed where
We Obtain an LHA. From Eq. (4.42), if the transformation ~(Qi) is linear, then the
term displayed there is zero and we find exactly
V 8(~+ /$Q ) = eik(8~/~Q k) , (4.46) Where
Lik = ~ij ~jh +~'3 ~iA (~jA/~Qn)~mk (4.47)
It is consistent with the assumption already made concerning @e(Q) to drop the second
term of (4.47). Then by going to a coordinate system (different at each point) in
whlch~B jk is the unit matrix, we see that Lik is equivalent to a (real) symmetric
matrix which can further be dlagonalized by a local orthogonal transformation. We end
up With a different choice of collective coordinates, - call them ql _ but ones in
which the tangent vectors are determined by the equations of the LHA,
2 ~/BqJ (J=l ..K) (4.48) v~ ~B/~qJ = ~j . .
Recall that since V ~ = V B(#Y), a procedure must and can be given (but not here) in
Which (4.48) are solved in conjunction with conditions (I) - (Ill). We emphasize,
however, that the quantities called (~/Bq j) in (4.48) will not be the tangent vectors
to ~(~) unliess (i) the decoupling is exact and (ii) either K = 1 or the transforma-
tion ~(q) is linear. Thus for K = l, this theory is equivalent to the theory given
PreViously. For K > i, it is different and, in general, is not exact even when the
decoupling is exact: This does not imply, however, that this approach has no value,
Since in the cases of interest physically, where decoupling is not exact, this method
may supply results of comparable worth to those from the tangent plane method. As
discUSsed in the next section, this appears to be the situation for one of the examples
we have studied.
The theoretical conclusions we have reached concerning the LHA can be verified
using the equations of the generalized valley, but this will. not be done here.
An extremely important and interesting problem not discussed so far is how to
26
include approximately the effects of the non-collectlve degrees of freedom. Instead
of describing a formal theory, we will furnish an illustration, in the next subsection.
B. Examples. We have barely begun our program of applications. Before describ-
ing out initial efforts, we should credit the impressive applications to fusion reac- 40,475
tions and inelastic scattering already in existence . The method used in that
work cannot, as far as we know, he extended beyong K = i and therefore we are hopeful
that we are taking the first tentative steps toward new applications to Nuclear physicS,
i. Generalized Lipkin Model. Our most eomplete numerieal results have been obtained
for a potential model derived from the generalized (multi-level) Lipkln model 55-57) .
We shall record the Ramiltonian actually studied. Let (xk,Pk) be canonical pairs,
k = l...n-l, where n is the number of original slngle-particle levels. In this model
£ measures the overall scale of the single particle energy; thus if ~k C is the energy
of the (k+l)st level relative to the lowest level, ENk = i. 2J is the degeneracy of
each level and f measures the strength of the interaction. A reduced Hamiltonian h
is studied,
h = (H/EJ) = t + v , (4.49)
= 1 n-i 1 ~? _ 1 t ~ k~ 1 (~k+l) pk 2 + ~ ~ f(£2) 2 (4.50)
i (nk+l)Xk2 1 ~ x 2 i v = - (i + ~) + ~ ~ - ~ _ + 8 f( 2)2 , (4.51)
2 x 2 n where, e.g. x = E k ' ~ = fll+ ~), and k
[xk,P£] = (i/J) ~k% (4.52)
The procedure leading from the original model to (4.49) - (4.52) has been described in
the literature 56) . For present purposes we are confronted with a system of coupled
"oscillators".
Limiting ourselves to n=3 (2 coordinates), the interest of this problem, first
treated as an example in the theory of large amplitude collective motion some time
ago 31) is that the nature of the potential energy surface depends on the value of f. i
We describe our calculations 57) for f > f2 = ~ (2-NI)" This is the strong coupling
regime in which V(Xl,X2) has the following critical points: (i) A local maximum at 1
x I = x 2 = 0. (2) Symmetric minima at x = O, x12 = 2[f - ~ (l-~lS]/f (3) Symmetric 12
saddle points at x I = 0, x22 = 2 If - (2-nl)]/f.
We apply the tangent plane theory to this example. We have B 12 = B 21 = O,
i (Bii/EJ) = ~ (l+~i) + f , (i = 1,2) (4.53)
= I B~vB IV E v (Eq. 4.515 and (2)V E u ~ v~ (4.54)
The theory described in Sec. (IVA) yields a two by two determinantal condition for the
valley path. Selected results for NI = 0.i, 2J = 28, and various values of f are
shown in Fig. i. Except near f ~ i, these curves, passing through the minima and
saddle points are well-approximated by ellipses. A convenient choice of collective
coordinate for present purposes is the polar angle ~,
1.50 ,,
27
1.25 -
! .00 -
x ~ 0.75 -
0 .50 -
0.25 -
0.00 0.00 0125 I .25 l .50 0'.50 0.75 l .00
X~
Fig. i. The collective path x2(xl) in the quadrant (x2>O, Xl>0) for various values
of the coupling strength f. The values 2J = 28 and ql = 0.i are held fixed.
= tan-l(~)"2t~ll (4.55)
2 We can numerically invert this equation and the equation x = x2(x I) to find the
COllective path in the form
x ~ = ~(@) , ~ = 1,2. (4.56)
Using (3.14) and (3.16), we can thus calculate
q(@) = V(#I(#),¢2(@)) , (4.57)
and ~-i(~) = M(@), the collective mass. These are periodic functions of ~ with period
2~. In Fig. 2 one period of V is shown and in Fig. 3, a corresponding range for M(~).
If we shift the origin of @ to the first maximum of V (a minimum of M), and consider
an interval - z to w about this point, we see that V has a double well structure.
MOreOver the difference in energy between maximum and minimum of the potential is now
- 22 .5
28
- 23 .0
-23 .5 -
-24.
- 2 4 . 5
-25.0
-25.5
-26.
°26.5
-27.0 'I , =' I
0.0 l.O 2'.0 3'.0 4.0 5.0 6.0 7.0 CD
Fig. 2. The collective potential V(~, with 2J = 28, H I
tion of the polar angle %.
= 0.i and f = 2.5 as a func-
much larger than a characteristic excitation energy (see below), so that we anticipate
a doublet structure for the quantum levels, which becomes more pronounced as f increas~
In this example, we shall-illustrate the applicability of the method used by "qua¢
tlzing" the classical Hamiltonian and comparing results with e~act diagonalization of
the original model Hamiltoniano We thus need a"recipe"for quantization of the klne
energy. Because we started with a quantum theory, the question can actually be settled
in a correct mathematical way 26'27) , It is, however, unnecessary to pursue this ques-
tion here: For the parameters used in this paper, particularly the choice 2J = 28, the
effect on the energy of changing the prescription is negligible. We shall see this by
presenting results for two prescriptions
p2~(~) + p~p (4.58)
and
P%(~) ~ ¼{P,{P,~}} , (4.59)
0.47
29
0.'16
0.45
0
0.44
0.43
0,42 ' ' ' ' ' 5' 6' 0 . 0 1 .0 2 . 0 3 .0 4 .0 .0 .0 7 . 0
Fig. 3. The collective mass M(#) with 2J = 28, QI = 0"i' and f = 2.5 as a function of the polar angle ~.
which differ from each other and from other prescriptions by contributions of relative
Order j-2 to the potential energy, as one easily verifies by commutation after setting
P * - ( i / J ) ( d / d e ) .
If we use the choice (4.58), for example, we have the Schrodlnger equation
t - i + = E , ( ¢ ) (4 .6o) 2j2
Which we solve numerically by expansion in a finite fourier series
nH .in~ ~(¢) = E C e ( 4 . 6 1 )
n n=-n M
leading to a matrix eigenvalue problem, In Fig. 4 we compare the exact eigenvalues
(Column (a)) with the solutions of (4.60) (column (b)) displaced so that ground states
Coincide and a third set of eigenvalues (column (c~) obtained with the prescription
30
-IB
-|7
-18
-J9
"~. -20 5_I
-21
-22
-23
-24
................ 2 ........2
............... 2 ........ 2
-2 ........ 2 ~ 2 ............
~ 2 . . . . . . . . . . . . . . . . 2 ........ 2
~ 2 ................ 2 ........
z ill,ill [iii[ :':::.''.."
2 . . . . . . . . . . . . . . . . 2 ........ 2
~ 2 ............... 2 ........ 2 la [hi [c]
Fig. 4. Comparison of the exact eigenvalues shown in column (a) with the elgenvalues of the collec- tive Hamiltonian (Eq. (4.60)) shown in column (b). Parameter values are 2J = 28, nl = 0.3,
f ~ 2.5. Column (c) are the results when the kinetic energy operator of Eq. (4.59) is utilized.
The potential minima correspond to QI = QI0
v I
(4.59). It is seen that on the
scale displayed there is no appre-
ciable difference between the re-
sults obtained by the two methods.
We now discuss the approxim-
ate inclusion of the "non-collec-
tive" degree of freedom in particu-
lar its contribution to the zero-
point energy. It is necessary to
find a complete point transforma-
tion which reduces to the collec-
tive path when the non-collectlve
coordinate has some fixed value
(conventionally zero). We consider
values of f where this can be done
analytically. For such values of
f the collective path can be fitted
to a curve of the form (as is evi-
dent from Fig. i)
(xl)2 (x l) 4 (x2~ 2 + a~ + +
i "'" b~ + (x2) 4
+ ... = i. (4.62) 4
b 2
For f ~ 2 we find that the terms
other than the quadratic ones are
negligible, and thus we can fit
the collective path to an ellipse.
This suggests the introduction
of hyperbolic coordinates, a pair
of orthogonal curvilinear coordin-
ates defined by the equations,
x I = c coshQ 2 cosQ I,
2 x = c sinhQ 2 sinQ I . (4.63)
For a given set of parameters, the
collective path corresponds to a
fixed value of c and Q2 = Q20.
= 0 or ~. In these coordinates we write
= ~(QI) = V(x l(Ql,q20),x 2(QI,Q20)) (4.64)
31
and
~2(Q 2) ~ V(xI(QI0,Q2),x2(QI0,Q2)) - V(Q I0) (4.65)
This is to be viewed as a prescription for separation. We approximate V as the (separ-
able) sum of (4.64) and (4.65).
The collective mass parameters M i -i = ~li are given by the expressions
= (~x ~) 2 _i_l Mi(QI'Q2) ~ ~Qi B~ (4.66)
In practice we use
MI(Q I) E MI(QI,Q 20) , (4.67)
M2(Q 2) ~ M2(QIO,Q 2) (4.68)
The coordinate QI, though cyclic, differs from the polar coordinate used in (4.55)
by a one-dimensional canonical transformation. Because of the imposed approximate
separability we shall not have to construct this transformation, since we already have
the energies we need. The additional non-collective Hamiltonian is defined by the
functions (cf. (4.65))
~2 = ~ el 2 [~(i + HI) - f](cosh2Q2-cosh2Q 20) + ~I c4f(cosh4Q2 - cosh4Q20) (4.69)
and (cf. (4.68))
~22)-i = M2(Q2) = c sinhQ2~ II (4.70)
Finally we solve the Schrodinger equation
1 d B22(Q 2) d ~ d--~ d~ +V2(Q2)~(Q2) = (4.71) E~(Q 2)
for 0 ~ Q2 ~ ~, (which is only an approximate range). In Table 2 we tabulate some
representative results. We may conclude that we have good approximate separability,
this desirable property improving as the value of the coupling strength f increases.
Table 2. Ground state energies for 2J = 28, H I = 0.i and two values
of f (in units of e). The column named E denotes the exact ex
energy, and the column E is the sum of the energies in the cor
first two columns, which are the contributions of the collective
and non-collective degrees of freedom.
2.
f E 1 E 2 Eex Ecor % Err.
3.0 -30.142 2.348 -27.909 -27.794 0.4
5.0 -45.160 3.648 -41.576 -41.512 0.15
G~.enera l ized L a n d s c a p e Mode l . We p l a n to e x t e n d t h e p r e v i o u s work t o t h e c a s e o f
three degrees of freedom and two collective coordinates. In the meantime we can de-
scribe a preliminary calculation involving the decoupling of two degrees of freedom
from a three-dimensional system 58) . The model studied is described by the Hamiltonian
32
1 2 + P32) H(x,[) = 2(Pl + P22 + V(XlX2X3)
1 ~ ~(x12 (4.72) V(Xl'X2'X3)= 2 (~i 2 xi2 )- + x22)x 3 i=l
we suppose that el 2 < e22 < ~32. The corresponding~ two-dimensional model, Here studied
previously, has been labeled the landscape model 4~j . The reason for the name becomes
apparent when we observe that the potential energy surface (4.72) has a minimum at the~
origin and four saddle points:
x 2 = 0, x 1 = ± e3 (x3/8)½' x3 = (el 2/28) ' (4.73)
x I = 0, x 2 = ± ~3 (x3/8)½' x3 = (~22/28) (4.74)
In the following, for illustration, we shall focus on points in the positive (Xl,X2)
and (x2,x3) planes.
We apply the tangent plane theory of the previous subsection to this model focus-
ing on the collective hypersurface and the collective potential energy. For K = i, we
find valleys connecting the minimum to each of the saddle points. For K = 2, we ex-
pect and find a surface x 3 = ~(XlX2) connecting the two one-dimensional paths. This
is shown in Fig. 5. Here we have something new to report: This figure was obtained
in three different ways. First it was found by use of a version of the LHA (we do not
describe the mathematical details). Second it was found by solving the determlnental
condition with a choice of (C)V, ~ = 1,2,3 as given in the text, in particular (remem-
ber the unit metric)
(3)V ~ (2)V~(2)V~ (4.75)
Thirdly we tried
(3)9 = (1)V (2)V ~
which is also an allowable choice, but has not previously been discussed. We found
that each method gave essentially indistinguishable surfaces and therefore indlstln-
guishable collective potential surfaces, the result exhibited in Fig. 6. However,
the approximate non-selfconslstent tangent planes given by each can differ substan-
tially. The one closest to the "actual" tangent plane in this case is the LHA. We
are trying to understand this result.
We have also examined some analytically soluble models 59) but shall omit them
for lack of space. Instead analogous models involving Fermlons will be considered.
V. LARGE AMPLITUDE COLLECTIVE MOTION, TDHF THEORY.
A. TranscriPtion of TDHF Theory intqHami!ton/an.Form . We finally confront the ques-
tion of what the classical mechanics studied in the previous section has to do with
nuclear physics, The answer (.now well-known) is that TDHF is a disguised form of
Hamilton's equations! We consider the equation
33
>
X I
Fig. 5.
003= 3
" ' Z
Collective hypersurface x 3 = $(Xl,X2) for the generalized landscape model
with parameters of Eq. (4.72) as shown (~ = ~I=~2).
i Pab = [~' p] ab (5.1)
in a representation in which p is diagonal,
According to a we l l -known theorem, p i s c o m p l e t e l y c h a r a c t e r i z e d by i t s p a r t i c l e - h o l e
(Ph) .and Chp) m a t r i x e l e m e n t s i n an a r b i t r a r y bas i s , of s i n g l e p a r t i c l e s t a t e s . We
34
XI 0.00.0 1.0 2.0 3.0
0.5
X2 1.o
1.5
2.0
Fig. 6. Representation of the collective potential energy corresponding to the surface of Fig. 5.
therefore study such matrix elements of (5.1) in the diagonal representation. We
have
i 0ph = ~ph = (6WRF~p) E (6H/~Php) , (5.3a)
i 0h p =- ~hp = (~WHF/80pN) ~ (6H/~0pN) ' (5.3b)
forms which resemble Hamilton's equations, with
Hip] = WHF[O]
1 habPba + ~ VabcdPcaPdb (5.4)
However, pp h, Php are matrix elements of generators of a Lie algebra and are thus
not canonical variables. We shall solve this problem by exhibiting a non-linear
transformation which introduces such variables 60-61) (and is nothing other than a
classical version of a Holstein-Primakoff mapping from a representation of a unitary
algebra onto a Heisenberg-Weyl algebra, i.e. a boson mapping.) The canonical
35
Variables ~ph and ~ph' satisfying
~ph = hp
nph = -~hp
are determined by the equations 27)
* ~ 1 2 ] ½}ph Oph = Ohp = {(i + i~)[l - ~(! + 2) ,
and their inverses
~ph =~{(ry-~)p h + (ry-½)hp} ,
~ph = -i ~ { (r_y_-½)p h - (ry-½)hp}
Here ! is a matrix such that
(5.5)
(5.6)
~.7)
(5.8)
(5.9)
rph = 0p h, rhp = 0h p, rpp, = rhh , = o , (5.10)
=~½- [i + (i + 4r%r) ½]__ , (5.11)
and it is understood that ~ and ~ have the same support as ~, namely ph and hp indices Only.
To show that the (~,~) are canonical variables, we note that in the representa-
tion in which O is diagonal we have 0p h = ~ph = ~ph = O, similarly for the (hp) ele-
ments; as a consequence, we deduce from (5.7) - (5.9) that
'h') (~0ph/~p =~pp'~hh' (5.12)
(~Pph/~p,h,)= i~pp,~hh, (5.13)
(~Ph/~Op'h') =~--~ ~pp'~hh' ' (5.14)
(~ph/~Dp,h,) = -i~½ ~pp,~hh, (5.15)
By applying the chain rule to (5.3a) and utilizing (5.12) - (5.15) together with the
established properties of ~, ~ and 0, we find that this equation becomes
i~Ph -~ph = (~H/~Ph) + i(~H/~nph) (5.16)
Equation (5.3b) yields the same equation except for a change in the sign of i. Thus
We arrive at Hamilton's equations
%ph = (~H/~ph) , (5.17a)
nph = - (~H/$~Ph) (5.17b)
There are two ways in which this classical problem differs from that considered
in Sec. IV. First, if we started with a coordinate space nuclear Hamiltonian (as
OpPosed to a shell model Hamiltonian), we have an infinite number of equations in
(5.17). Of more i~ediate impact for us is the fact that the Hamiltonian, H, is
generally not quadratic in the nph' so that we cannot apply the adiabatic theory given
36
in Sec. IV. It is possible to formulate an approximate theory in the general case, 32)
in a form first proposed by Rowe and Basserman , which is a generalized version of
what we have called LHA. This theory has so far never been applied (though it could
be) and therefore we refer to the literature for further discussion. 28)
Instead we describe briefly how to expand the Hartree-Fock energy to second order
in ~. Even though questions may be raised concerning the convergence of the resulting
expansion, if we limit further transformations to point transformations3 as we do in
the adiabatic theory, it is the only part of WHy which contributes to large amplitude
collective motion. Actually we have carried the expansion to fourth order in z28),
but we restrl¢~ the present account to second order terms. We have explicitly, as a
starting point, the equation
m(~,~) = WHF(P(~,E)) (5.18)
To carry out our task, we expand p(~,~) to second order terms
1 P(i,E) = P(O)(1) + P(l~)(1)~ +~p(2~s)(1)~ + " ' " (5.19)
Here, of course, @,8 run over all (ph) indices in som9 representation. We shall per-
form the present calculations using a representation in which p(O)(!) is diagonal.
This simplifies the ensuing formulas.
2 From (5.19) and the conditions p = p , we have, first of all, the constraints
p(O) p(O) 2 = , (5.20)
p(l~) = p(O)p(im) + p(la) p(O) (5.21)
p(2~) = p(O) p(2e~) + p(2~B)p(O) + p(l~)p(iB) + p(18)p(l~) (5.22)
These equations have the following well-known consequences: In the representation
in which 0 (01 is diagonal, p(1) has only (ph) and (hp) elements~ and the (pp'1 and
(hh') elements of p(2) are determined by the elements O (0) and 0 (I) . Next, if we
carry out a formal expansion of (5.7) (which then refers only to the (ph) and (hp)
parts of p), we have to second order
1 2.½ 1 2.½ 1 1 2.--~ /~p = ~(i- ~ ) + i~(l - ~ ) - ~ ~ 2 (i - ~ ) (5.23)
The consequences of (.5.23) are: First~ in the representation in which p(01 is dia-
gonal, we have
and ~ph = hp = 0.
(p(0))ph = [~(1- ½~2~½]ph = 0 (5.24)
It follows that in the representation studied (5.23) reduces to
= i + 0(~31 (5.25) ~"Z Pph ~ph
Upon comparison with (5.19), we conclude that
(i p'h'1 = _ 0(I p'h') = i Dph hp
Pph (2p'h'p''h'') = 0 ,
i ~pp~hh' ' (5.26)
(5,271
37
If the expansion (5.19) is substituted into (5.18) and the results (5.24) -
(5.27) utilized, we find the classical form
i H = V(~) + ~ zen8 B~8(~) , (5.28)
where
V(~) = WHF(O (0)(%)) , (5.29)
a well-known result, and
BPhp'h' = ~2 6hh' [~pp, +Up,p] _ 21 ~pp, [~hh' +~h'h ]
i +I +i i - ~ Vhh,pp, ~ Vhp,ph, ~ Vph,hp, - ~ Vpp,hh, (5.30)
Now the beauty of these results is that if we wish to calculate the parameters of
a collective Hamiltonian (such as in Eq. (3.24)), we do not have to go hack to the
TDHF equations. We go directly to the classical mechanics of See. IV~ Thus we
translate Eqs. (4.2) and (4.3) into
~ph = ~ph(Qi) , (5.31)
Zph = (~Qi/~ph) Pi ' (5.32)
understanding, however, that we have already projected to the collective hypersurface,
Z . Let us examine condition (I), Eq. (4.10). We have
V +Vph = ~WHF/~ph = (~/~Qi)(~Qi/~ph) (5.33)
In fact from (5.23), it follows easily in the p(O) diagonal representation that
1 (~/~ph) =~ [(~/~Pph ) + (~/~Php)] , (5.34)
and (5.33) can be replaced by the equation (recall (5.3))
(1) ~ph = (8~/8Qi) (SQi/~Php) (5.35)
and its transpose. (We should really write (~Qi/~P)hp, but this will be understood.)
It is also understood or taken for granted that in the nuclear physics problem the
natural definition of the collective coordinate Qi will he in terms of the density
matrix, P. Similarly condition (II) becomes
(II) B ph P'h'(~Qi/~0h,p,) = ~iJ(~Oph/~QJ ) (5.36)
Besides (5.34), this also involves the first order expansion of (5.8), namely
/~ ~ph ~ + (5.37) Pph Php
Finally, with the help of both (5.34) and (5.35), the condition (~Qi/~QJ) = ~lJ is turned into
(IIl) 1 (5.38) (~Qi/~Pph).(~0ph/~QJ) = ~ 61j
Only one more technical point is necessary to be able to proceed with the appll-
= 1 VaB~VB ' cation of the theory. Consider the quantity (1)V ~ which currently turn
out to be (assuming ~ph = ~hp )
38
p'h' (I~v =~ph ~ph 4{p,h, (5.39)
How do we calculate the gradient of (1)V or what is the same thing (~(1)V/~Pph).
This can be done in two ways. In the first (1)V can be exhibited explicitly as a
function of P, and the derivative taken directly. Previously we followed another
method 27) which we describe without proof Let a,b,c be orbltals in the p(O) dia-
gonal representation and let ~,~,y be orbitals in an arbitrary representation. Let
ga = f~ U~a ' (5.40)
with ga and f the wave functions of the corresponding representations. For single-
particle and two particle operators, we have
~ab =~ U~U~b ' (5.41)
= = U t t -abcd EaBy~ as UbB Uyc U~d (5.42)
we then prove the formulas
(~Uaa/~Pph) ~ ~p~ab ' (5.43a)
(~Uaa/30hp) = -6~h ~ap ' (5.43b)
(3U~y/~0ph) = - ~cp~yh , (5.43c)
(~U~y/~0hp) = ~ch~yp (5.43d)
Since (5.39) is a linear combination of products of terms like (5.41) and (5.42),
the application of (5.43) allows all required derivatives to be computed.
Before considering applications, it is interesting to add some general remarks
concerning the Fermlon significance of the geometrical theory developed in Sec. IV.
Let us, for example consider the covariant forms of Eqs. (4.21) and use the notation
~a ~ ~(0), (~+l)va ~ (~) ~ . Thus these equations take the form (i=I...K,~=O,I...K)
~(~) = (~(~) .(d) ~Qi/~P)ph 0 (5.44) p h - A i =
This is a set of coupled HF cranking problems each with a different HF Hamiltonian,
.(d) and a single set of cranking ~(a) a different set of Lagrange multipliers h i
operators (~Qi/]p). Actually we must discover that set of cranking operators for
which Eqs. (5.44) can be simultaneously satisfied. Equivalently we must find a den- *
slty matrix p 0 = Z~h~h such that the ~h are the simultaneous eigenfunctions of H (a)
Another way of viewing this theory is in terms of the generalized valley picture.
Then we have
~(0) K ~ (a) - Z : 0 (5.45)
~ph i Aq ph
In this form, the theory appears in the guise of a single cranking equation where
the cranking operators are the gradients of the point functions " (G)V, d = 2,...K+I,
39
which are al_l positive function s. In this form it is readily apparent that we are
dealing with a single density matrix. Below we shall examine by means of the most
elementary of examples how to solve one or the other form of the theory.
B. A p_~!ication" We describe the beginnings of a program of application of the
methods developed to nuclear physics. We deal first with a model with two collective
COordinates which can be treated analytically. In "first quantized" form, we study
a system of N spinless fermions in one-spatial dimension, with
N i 2 i ~o2X 2) i FI 2 i F22 H = Z + + KI + <2 (5.55) ~i (2 P~ 2 ~ 2 '
where ~I and K 2 are coupling constants and
F 1 : ~ x ~ , F 2 ~ ~ ( x ~ ) 2 ( 5 , 5 6 )
are dipole and monopole moments respectively. For the model to be treated analytical-
ly we have to make a Hartree rather than a HF approximation. The Hartree Hamiltonian
is
= 1 p2 i ~(O) 2 + ~ ~o2X 2 + ~IXQ 1 + K2x2Q2 , (5.57)
where (tr = trace)
Q1 = tr p(0)x, Q2 = tr pox2 , (5.58)
a notation which is strictly consistent with the previous developments only if Q1
and Q2 turn out to he the self-consistent collective coordinates. This will be
seen to be the (exceptional) case where the "obvious" cranking coordinates are also
the self-conslstent ones. (This is what makes the problem so easy.) We use the
following method to establish a solution: We obtain a solution of conditions (I)-
(Ill) and then verify that condition (IV) is satisfied. It will also be clear that
we Will have satisfied the equations of the theory of the tangent plane.
Starting with (I), Eq. (5.35) or better (5.44), we write, with l~/~Qi),
~(0~pn ll(X)ph - 12(X2)ph = ~(0)ph = 0 (5.59)
By trivial manipulation ~(0) is recognized as the Hamiltonian of a displaced frequency
altered harmonic oscillator,
i 2 2 ½(2/2) :~2+~y _ , (5.60)
where
y = x + (~/~2), ~ = ml Q1 - 11 , (5.61)
~2 = ~2o + 2 (<2 Q2 - 12 ) (5.62)
The N lowest energy eigenfunctions of ~ ~h (energy En), will provide us with a den-
Slty matrix 0 (0) and a representation in which (5.59) is satisfied.
As the next step it is convenient to calculate the potential energy surface
~(QI Q2) associated with p(O) Because of the simple character of ~(0) and ~, this
can be done explicitly with Just a little help from the virial theorem for harmonic
40
oscillators.
for q, namely
Putting back the definlt~on of the %i' we obtain a differential equatio~
2q= 4Q2 + 2 2 QI [KIQI (~/~QI)]
i _ I 2 + ~ [~IQI-(~/~QI) ]2 + ~2[QI(~v/~QI) ~ KI(Q I)
i 2 2 + Q2(~V/~Q2) ~ ~2(Q ) ] (5.63)
The solution of this equation can be given by inspection,
=i 1 2 1 1 2 V(QI,Q2) ~KI( Q ) + 2 ~o2Q2 + 2 ~2(Q2) + (~/Q2) , (5.64)
where B is so far undetermined. It can be fixed by the following equilibrium consider"
ations: We impose the two requirements
=I 2 2 2 0 = (~y/~Q2) 2 ~o + <2Qo 2 - (8/(Qo) ) ' (5.65)
2 p(o) 21 ~ ½(N2/~) (5.66) Qo = tr IQ 1 = (~VlaQ2) =0
In the first we get a relationship between the unknown constant B and the equilibrium
"radius" represented by Q2. In the second we have (a straightforward harmonic oscil-
lator result) evaluated Q~. To complete an evaluation we note that at equilibrium,
(5.62) becomes
e + (~o 2 + 2~ 2 Qo2) ½ (5.67)
Combining (5.65) - (5.67) we find 8 = (I/8)N 4. Let us emphasize that up to this point
there is no proof that the potential energy surface (5.64) is self-conslstent.
We turn next to the calculation of the collective mass matrix. In the case where
the collective coordinates are written Qi = tr p(0)q i, qi real, it is easy to find a
simple general formula for ~iJ in the Hartree approximation with separable two-body
interactions. This is because the last four terms of (5.30) give contributions
which cancel out. We thus find
~i iJ = tr~p0qi(l_pO)~(0)(l_p0)qJ
(l_pO)qipO B(O) pO qJ] (5.68)
Using the properties of~ (0) andS, we evaluate these formulas and find
~ii : N[I+(%2/ 2) ] , (5.69)
~12 :~21 : 2QI + (fIN/e2) + (4%2QI/e2) , (5.70)
~22 : 4Q2 + (41lQI/ 2) + (6X2(QI)2/Ne2) + (%2N/e3) (5.71)
We have now satisfied condition (II) as well as (I) and our choice of Qi automatically
satisfied condition (III). But in fact, since the ~iJ ~xhibited in (5.69) - (5.71)
are explicit functions of Q1 and Q2 and no other variables, we have also satisfied
condition (IV) and thus proved that for this model cranking theory is a self-
41
Consistent solution of the theory of large amplitude collective motion.
It should he clear finally that we have satisfied not only the first of Eqs.
(5.44) but all of them. Consider for example the quantity
(I) v = ½(O)vaB~(O)vB
p'h' ~(0) i ~iJ%j =~(0) BPh ÷ h i (5,72) ph p'h' 2 '
where we have used (5.59) and the definition of ~ij i.e., we confine ourselves to
the domain of solutions of (5.59). We now ask whether this domain is a generalized
valley. Because we have found that %i and ~ij depend only on QI and Q2 we have
~(1)V/~0h p = (~(1)V/~Ql)~ph + (~(1)V/~Q2)(X2)ph , (5.73)
which is equivalent to the equation~ ) = 0. This argument can be continued and
shown to be correct, in this case, forM"any (~)V, o ~ i.
It is worth remarking that what we have just studied can be viewed as a one-
dimensional version of the Nilsson model, and that it can be generalized to any number
of Spatial dimensions including the correct number as long as the cranking coordinates
are linear and quadratic functions of the slngle-particle coordinates.
We conclude these lectures with some remarks which indicate the direction in
which we are moving and also to show the reader how to proceed when the obvious
Cranking choice of collective coordinate is not self-consistent. We will also explain
where we lose the simple character of the solution found above.
and
Instead of (5.55), we consider the Hamiltonian
1 2 2. 1 1 H = Z(2 p 2 + ~o X~ ) + ~ K2F22 + ~ K4F42 , (5.74)
F 4 = Z x 4 (5.75)
This is the one-dimensional analogue of a system with quadrupole and hexadecupole
Separable interactions. We shall not supply any details, but assert that the steps
leading from (5.59) to (5.64) can (with the help only of the virial theorem) all be
duplicated and an explicit solution found for ~(Q2,Q4), namely
I 1 2 2 = ~ eo2Q 2 + ~ <2 (Q) + (82/Q 2)
1 4 2 (~4~Q4)½) +~<4( Q ) + , (5.v6)
where the values of ~2 and B 4 can be determined from equilibrium considerations.
Where this example differs essentially from the previous one is in what happens
When we set out to evaluate the formulas (5.68) for the mass matrix. It is not diffi-
i x q Cult to show that if q ex p and then ~iJ will contain terms depending on Qp+q-2
Where
Qp+q-2 = tr 0 (0) x p+q-2 (5.77)
This explains why only Q1 and Q2 occur in the first example studied and why cranking
42
is therefore self-conslstent for this case. In the present example Q6 will occur in ~22
(Q2+QI Q4+Q2) and thus the original choice of collective coordinate is not
self-consistent. Under these conditions we are "cranking up" to apply the full theory
either in the form (5.44) or (5.45), but are only in the initial stages of this
enterprise.
Most of the current research described in these lectures is being carried out in
collaboration with G. Do Dang and A. S. Umar. This work was supported in part under
U. S. Department of Energy grant number 40264-5-20441.
REFERENCES
I. A. Kerman and A. Klein, Phys. Left. ~, 185 (1962); Phys. Rev. 132, 1326 (1963). 2. A. Klein, L. Celenza, and A. K. Kerman, Phys. Rev. 140, B245 (1965). 3. F. D~nau and S. Frauendorf, Phys. Lett. 71B, 263 (1977). 4. P. B. Semmes, G. A. Leander, D. Lewellen, and F. D~nau, Phys. Rev. C33, 1476
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Suppl. Extra Number, 42, 179 (1968); Sov. J. Nuel. Phys. 9, 287 (1969). 6. G. J. Drelss and A. Klein, Nucl. Phys. A139, 81 (1969). 7. C. Dasso, A. Klein, C. Y. Wang-Kelser, and G. J. Dreiss, Nucl. Phys. A205,
200 (1973). 8. M. Vallieres, A. Klein, and R. M. Dreizler, Phys. Rev. C7, 2188 (1973). 9. C. T. Li and A. Klein, Phys. Rev. C19, 2023 (1979). i0. S. T. Belyaev and V. G. Zelevlnsky, Soy. J. of Nucl. Phys. l_!l, 416 (1970);
16, 657 (1973); ibld, 17, 269 (1974). II. M. I. Shtokman, Soy. J. of Nucl. Phys. 22, 247 (1975). 12. A. Klein in Pro~.[ess in Particle and Nuclear Physics, edited by D. Wilkinson
(Pergamon, London, 1983), Vol. i0, p. 39. 13. A. Klein, Prog. Theor. Phys. Suppl. Nos. 75 and 76, 237 (1983). 14. V. G. Zelevinsky, Prog. Theor. Phys. Suppl. Nos. 75 and 76, 251 (1983). 15. H. J. Lipkin, N. Meshkov, and A. J. Glick, Nucl. Phys. 62, 188 (1965). 16. A. Klein, in Dynamical Structure of Nuclear States , edited by D. J. Rowe
(University of Toronto, Toronto, 1972), p. 38. 17. A. Friederich, W. Gerllng, and K. Bleuler, Z. Naturforsch. 309, 142 (1975). 18. A. Klein, Phys. Rev. C3-0, 1680 (1984). 19. A. Klein and A. S. Umar, U. of Pennsylvania report UPR-O022NT (1986) submitted
to Phys. Rev. C. 20. A.. Umar and M. R. Straayer, Phys. Letters BI71, 353
(1986). 21. K.-K. Kan, J. J. Griffin, P. C. Lichtner, and M. Dworzecka, Nucl. Phys. A332,
109 (1979). 22. S. Levit, J. W. Negele, and Z. Paltlel, Phys. Rev. C21, 1603 (1980). 23. J. P. Blaizot, Phys. Rev. 107B, 331 (1981). 24. M. Baranger and I. Zahed, Phys. Rev. C29, 1005 (1984). 25. H. C. Corben and Ph. Sthele, Classical Mechanics, 2nd edition (Kreiger Publishing
Co., Huntington, New York), p. 169. 26. A. Klein, Nucl. Phys. A410, 74 (1983). 27. A. Klein, Nuel. Phys. A431, 90 (1984). 28. G. Do Dang and A. Klein, Nucl. Phys. A441, 271 (1985). 29. A. Klein and A. S. Umar, U. of Pennsylvania report 0025NT (1986), submitted
to Phys. Rev. C. 30. A. Klein and F. R. KreJs, Phys. Rev. All8, 1343 (1978). 31. G. Holtzwarth and T. Yukawa, Nucl. Phys. A219, 125 (1974). 32. D. J. Rowe and R. Basserman, Can. J. Phys. 54, 1941 (1976). 33. F. Villars, Nucl. Phys. 6285, 269 (1977). 34. K. Goeke and P. G. Reinhard, Ann. of Phys. 12, 328 (1978). 35. M. Baranger and M. Veneroni, Ann. of Phys. 114, 123 (1978). 36. D. M. Brink, M. T. Giannonl, and M. Veneroni, Nucl. Phys. A258, 237 (1976).
43
37. T. Marumorl, Prog. Theoret. Phys. 57, 112 (1977). 38. V. G. Zelevlnsky, Nuel. Phys. A344,~ 109 (1980). 39. A. Klein, T. Marumori, and T. Une, Phys. Rev. C29, 246 (1984). 40. P.-G. Relnhard, J. Marun, and K. Goeke, Phys. ~. Lett. 44, 1740 (1980). 41. K. Goeke, P.-G. Reinhard and D. J. Rowe, Nucl. Phys. A3597-408 (1981). 42. A. K. Mukher0e e and M. K. Pal, Phys. Lett. IOOB, 457 (1981); Nucl. Phys. A!73 , 289 (1982). 43. D. J. Rowe and A. Ryman, J. Math. Phys. 2_~3, 732 (1982). 44. L. P. Brito and C. A. Sousa, J. of Phys. AI4, 2239 (1981). 45. j. p. Blaizot and B. Grammaticos, Phys. Lett. B53~ 231 (1974). 46. M. J. Giannonl and P. 0uentin, Phys. Rev. C21,--~60 (1980), C21, 2076 (1980). 47 D. Provoost, F. Grurmmer, K. Goeke, and P.-G. Reinhard, Nuel. Phy ~ A431, 139
(1984). 48. M. Iwasaki, Prog. Theor. Phys. 65,' 2042 (1981). 49. A. Kuriyama and M. Yamamura, Pro---g. Theor. Phys. 71, 973 (1984). 50. M. 51. S. 52. F.
51 53. E. 54. G. 55. S. 56. A. 57. A.
to 58. G. 59. G. 60. E. 61. j.
Yamamura, A. Kurlyama, and $. llda, Prog. Theol. Phys. 7_~2, 513 (1984). lida, Prog. Theor. Phys. 73, 374 (1985). Sakata, T. Marumorl, K. Muramatsu, and Y. Hashimoto, Prog. Theor. Phys. 74, (1985). J. V. de Passos and F. F. de Souza Guz, to be published in J. of Phys. A. Do Dang and A. Klein, Phys. Rev. Lett. 55, 2265 (1985). Y. L~, A. Klein, and R. M. Dreizler, J.-Math. Phys. Ii, 975 (1970). Klein and C. T. Li, Phys. Rev. Lett. 46, 895 (1981). S. Umar and A. Klein, U. of Pennsylvania report UPR-0021NT (1986), be published in Nuel. Phys. A. Do Dang and A. Klein, U. of Pennsylvania report UPR-O04NT (1985). Do Dang and A. Klein, U. of Pennsylvania report UPR-005NT (1985). R. Marshalek and J. Weneser, Phys. Rev. C2, 1682 (1970). P. Blalzot and Y. Orland, Phys. Rev. C2__~4, 1740 (1981).
Classical Image of Many-Fermion System
and Its Canonical Quantization
Masatoshi YAMAMURA and Atsushi KURIYAMA
Faculty of Engineering, Kansai University, Suits Osaka 564/JAPAN
§i. Introduction
One of the most fundamental problems in many-body theory may be to give a general
scheme, with the aid of which not only collective motion but also its coupling with
intrinsic degrees of freedom can be treated. Any many-fermlon system is essentially
quantal. Therefore, in principle, the general scheme should be given in the frame-
work of the quantum mechanics. However, in many cases, it is almost impossible to
give it directly in the quantum mechanics. The next best scheme may be as follows:
we transcribe the quantal system under investigation in the classical mechanics and
express the classical system in terms of the collective and the intrinsic degrees of
freedom. Then, we requantize the system in an appropriate method. If the classical
image can be derived in terms of canonical form, we can use various technique of the
analytical mechanics. Further, canonical quantization can be expected.
If we adopt the above-mentloned scheme, the first problem is to find a method
to get the classical image of many-fermion system in the canonical form. For this
aim, we notice the time-dependent Hartree-Fock (TDHF) theory. This method has played
a central role for investigating various nuclear properties, especially, large ampli -
rude collective motion. I)'2) Originally, the TDHF theory was proposed as a method
for obtaining a Slater determinant which satisfies the tlme-dependent Schroedinger
equation approximately. With the use of a certain condition which will be later call ~
ed canonicity condition, we can formulate the TDHF theory in the Hamilton's canonical
form. We can prove that this form gives us a classical image of the original many-
fermion system and, by an appropriate canonical quantization, the image goes back to
the original quantal system in disguise. Boson expansion is one of the examples. 3)-5)
For studying the collective motions, a viewpoint which attracts the attention
is the introduction of extra degrees of freedom. This means that the total number
of the degrees of freedom is excess of the original ones and, then, there should exlS~
constraints. For the system with constraints, the Dirac's canonical theory is avail ~
able. 6) In this sense, it is an interesting problem for the study of the cl~smical
image of many-fermlon system to weave the Dirac's canonical theory into the TDHF
theory.
The TDHF theory is valid only for describing the collective and the intrinsic
motions only in even-fermion system. The both types of the motions are of the parti ~
cle-hole pair type. If we intend to give a classical image for odd-fermion system,
45
the TDHF theory should be extended. For such systems, we must express the intrinsic
degrees of freedom in terms of the independent-particle picture. As was formulated
by Casalbuonl,7) a classical image of fermlons can he given by the use of the Grass-
mann variables. Therefore, it becomes interesting to investigate how the TDHF theory
is extended so as to describe the collective and the independent-partlcle motions in
terms of the collective and the Grassmann variables.
Main aim of this talk is to sketch a possible idea for the above-mentioned prob-
lems. The TDBF theory in the Hamilton's canonical form was formulated by Marumori,
Maskawa, Sakata and Kuriyama. 2) The present authors proved that the formulation gives
us a classical image of many-fermion system 8) and the requantizatlon 3) becomes the
hOSon expansion given by Marshalek. 9) Obeying the true spirit of the TDHF theory in
the Hamilton's canonical form, the second and the third problem have been completely
Originated by the present authors. With the aid of the TDHF theory in the Dirac's
Canonical form, I0) we can obtain equation of collective submanlfold and it gives us
the separation of the whole degrees of freedom into the collective and the intrinsic Ones.ll),12)
In the third problem, the fermion coherent state constructed on the
Slater determinant plays a central role. 13) In this case, also, the Dirac's canonical
theory is indispensable. Through the quantization, 14) we can arrive at the boson=
fermion expansion given by Marshalek. 15) Further, we can obtain equation of collec-
tive submanifold in which the effect of the independent-particle excitations is in- Cluded.13),16)
In the next section, the classical image of many-fermion system based on the
TDHF theory is given. Section 3 is devoted to proposing the use of the Dirac's
theory in the TDHF theory. In §4, with the use of the Grassmann variables, the ex-
tension of the TDHF theory is given. In Appendix, an example is discussed.
§2. Classical image of many-fermion system in the Hamilton's canonical form
2-I A possible form of TDHF theory parametrlzed in terms of canonical variables
The TDHF theory gives us a Slater determinant satisfying the tlme-dependent
SCroedlnger equation in the variational sense. The Slater determinant Ic0> can be
expressed a s
le0 :Gl0>. (G+G-GG Here, G is related to one-body operator and the most popular example is
= exPZi~(albiPi~ -biaLPi~) (2.2) ^
The Symbols ax and bi stand for the particle and the hole operator in the state E and
i given in a static Hartree-Fock field, respectively, and the state I0> is the vacuum *
for a% and hi" The quantities FiE and FiE are time-dependnet parameters, the number
of which is 2NM (~2f0). Here, N and M denote total numbers of the hole and the par-
tlCle States, respectively. In any form, a Slater determinant is characterized by
2NM parameters and we denote them as F collectively.
46
From the following viewpoint, the time-dependence of F can be determined: We in-
0 p0 ( r=l,2,---,f0 ) which can be regarded as caninical troduce new parameters Qr and r
variables (hereafter, denoted as (Q0, p0) collectively). After determining the func-
tional forms of F for (Q0, p0), we investigate the time-dependence of (Q0 p0). This
viewpoint was initiated by Marumori, Maskawa, Sakata and Kuriyama.
The TDHF theory starts from the following variation:
t2 Ldt 0 , L ~ <c01i~t -HIc0> . (2.3) ~ftl
The variational parameters are (Q0, p0). First, with the use of S o = S(Q°,P°), we
set up
~S ° . ~ ~S ° < c o l i Ico> = P° + , <col --Ico> = (2 .4 )
~Q~ r ~Q~ 3P°r ~P~ Then, L can be expressed as
0"0 +~0 H ~ (2.5) e = ~rPrQr -H , <c01Hlc0>
The variation (2.3) gives us
"0 = ~H ~0 =_ ~H (2.6) Qr ~p0 ' r ~QO
r r
The basic relations of the present formulation are Eqs.(2.4) and (2.6). The condition
(2.4) gives us F expressed in terms of (Q0 p0). Solving Eqs.(2.6), the time-depen-
dences of (Q0, p0) are determined. Thus, [c0> is obtained as a function of t. From
the form (2.5), we can regard (Q0, p0) as canonical variables. Further, L and H are
Lagrangian and Hamiltonian, respectively, and Eqs.(2.6) are Hamilton's canonical equa"
tions. We call the set of Eqs.(2.4) as canonicity condition. In this way, we formu-
lated the TDHF theory in the Hamilton's canonical form. The dimension of the phase
space is 2f0. Hereafter, we abbreviate it as [f0]-space.
Let Ic0> be parametrlzed in terms of another set of variables (QZ, pl). We have
the relations replaced the index 0 with 1 for Eqs.(2.4), (2.5) and (2.6). Since the
Lagrangian for both cases should be identical, we have
0 " 0 I*I Zr(PrQ r -PrQr) = d(sl -S O ) (2.7)
The above relation means that the transformation from (Q0 p0) to (QI pl) is canoni-
cal. This fact can be also shown from the following argument: From the canonicity
conditions for the indices 0 and I, we have the relations of the Lagrange bracket
(A, B)L
Qs)L s L " ' Ps)L rs
Therefore, for parametrizing the Slater determinant, there exist infinite possibil-
ities, which are canonically equivalent to one another.
2-2 Classlcal image obtained in the framework of the TDHF theory
The TDHF theory gives us a classical image of many-fermlon system. We define
47
the following operators:
~r = • ~ = (2.9) ~pOr ~Q:
^ n ~o are one-body operators. With the use of the canon- It can be proved that ~r and r
lelty Condltlon (2.4) and the definitions (2,9), we have the following relations for
the commutators [A, B]_:
<co I ^ o ^ ^ tar, 5.~I_Ico> - <colt~:, ~I Ioo> - 0 , <Co { ^ 0 ^
[e r, , T ~ ] _ [ c o > : Jars ( 2 . 1 0 )
A set of one-body operators {On } forms a closed algebra with the structure con- Stants C £ .
nm
[8 n, 8# . ~£C~m 8£ . (2.11)
Taking the expectation values of It0>, we have
<c°[[O n, Om]_Ico> = ~C~m 0£ , O~ ~ <co[0£{co> . (2.12)
0he-body operator 8 n can be generally expressed as ^ ^
8n " Z r % r a ~ + + n r ~ ) +6(v) +~(°) (2.13) n n Here O(v) • S(v) Ic0> = <co 6 (v) = 0 and 6 (c) is a c= n is a one-body operator satisfying O n n n
number. The definition (2.13) gives us
<c0118 n, Om]_Ic0> = iEr(~nr@mr -@nrTmr ) (2.14)
With the use of the definition (2.9), we have
@O ^ ^0 n ^ ~On = i<c01[O n, ~ r ] _ [ c o > = ~ (2.15) : - i<col 9=~] I CO> - - nr ~Q0 [On' - = ~nr' ~p0
r r Therefore, Eqs.(2.13) can be rewritten, in the following forms, as the relations of
the POisson brackets (A, B)p:
i(o n, Om) P = Z£C~m O£ . (2.16)
The algebra of {O } gives in the Poisson brackets is completely identical to that of n} n
{6 given in terms of the commutators. Therefore, 0 can be regarded as a classical image of 8 : n
n
6 ~ O ( 2 . 1 7 ) n
From the above correspondence, a canonical quantizatlon from {O n } to {On } may be pos- ^0 p0 0 and p0 with q-numbers Qr and satisfying Slble by replacing c-numbers Qr r r
^Thus, we arrive at the following conclusion: The expectation value of any opera-
tor O as a function of {O n} for Ic0>, O, may be regarded as a classical image of 8.
For example, H is a classical correspondence of H. Our discussion started from re-
gardlng the TDHF theory as a method for obtaining a Slater determinant in the varia-
48
tional sense. However, at the present stage, we can see that the TDHF theory gives
us a classical image of the original many-fermion system and, by an appropriate re-
quantlzatlon, the classical image may go back to the original quantal system in dis-
guise.
2-3 Boson expansion derived from canonical quantizatlon through the Poisson bracket
A concrete example of the quantization of the classical image is boson expansiom
Let us show this fact. For the form (2.2), the expectation values of the fermion =
pairs with respect to It0> is given by
<col~' , i ; 'x lco, ~ (1,¢-aK-Ao co)iX- (/I -Aro cT)xl . (2.19)
<co laxau lco> . (C~Co)x~ , <co l^* ^ * T ^*^ bibj[c0> = (CoCo)Ij , (2.20)
*T (A0)x~ - (C%0C0)x~ , (A0)Ij -- (CoC0)]i . (2.21)
Here Co is a matrix, the i-I element of which is defined by
(co)ix = [r(/rirl-lsin/r+rllx = [sln/rr¢(/rr+l-lrlix (2.22)
We adopt boson type variables
i (Q0 +ipo) x* - (Qo _i o) Xr =72 r ' r
i(Xr, Xs) P = i(Xr, Xs) P = 0 , i(Xr' Xs)p = ~rs (2.24)
As for the canonlcity condition, the following form derived for S 0 o o = -½ZrQrPr is use~:
~X r = 2rr~u°~ r ~Xr = ~xr '
1_ .~tBCo ~C~ _! X <c01 ~,Ic0> = ~rtu~ ~ ~ Co) = 2 r "
BX BX BX r r r
(2.25)
A possible solution of Eqs.(2.25) is as follows:
(C0)IX = X r , (C0)IX = X r ( r = iX ) (2.26)
Here, the index r is identified to iX and, hereafter, we will use the index (iX).
Substituting the result (2.26) into Eqs.(2.19), (2.20) and (2.21), the expectation
values of the fermlon-palrs can be expressed in terms of the canonical variables Xi%
and XiX.
The quantization can be performed by the following replacement:
^ . ^*
Xix --+ Xix , Xi % --+ Xix , (2.27) ^, ^ ^,
[~ix' ~ju l- = [i[x' xj~]_ = 0 , [xlx, xj~] = ~i16xu (2.2s)
Then, we have
^ ^
49
^*^ (X*X)A~ ^ * ^ ^ * ^ T (ala~) B = , (blbj) B = (X X )ij ' (2.30) ^
(yo)x~ = (~t~)X~ (~o)ij ^*^T , = (X X )Ji (2.31)
The above result is completely identical to the boson expansion for the particle-hole
Pairs given by Marshalek. 9)
§3, Classical image of many-fermion system in the Dirac's canonical form
3-I Extra canonical variables for collective motion and constraints governing the
whole degrees of freedom
Main interest of this section is how to separate a many-body system into collec-
tive and intrinsic degrees of freedom. For simplicity, the case of one kind of col-
lective mode is treated. We start from the classical image given in §2. In order to
regard the collective mode as an apparently extra degree of freedom, we prepare a
2(f° +1)-dimensional phase space (hereafter, abbreviated as [f0 +1J-space). The ca-
~Onlcal variables (Q, p) and (qn' Pn; n=0,1 .--,f0-1) are introduced in the If0 +I]= Space:
(Q, P)p = I , (qm' Pn)P = dmn " ( the others )p = 0 (3.1)
A certain 2f0-dimensional subspace in the [fo+l]-space corresponds to the original
[f0]-Space. The subspace may be specified by two constraints governing the whole var- iables.
Let Ic0) and (Qo p0) in the [f0 +l]-space correspond to Ic0> and (QO pO) in
the [f0]-space, respectively. Then, they should obey the same relations as those giv-
en in Eqs.(2.4):
(coli-A_ico) . po + ~s ° (=oli~ ~s° r , - - ; I c o ) r . . . . . . (3.2)
~ere, So is obtained by replacing (Q0, p0) in S O = S(Q0 p0) with (Q0, p0). Our start-
ing point is the picture that (Q0 p0) consist of simple sum of collective and intrin-
Slc COmponents:
Q~ = ~r(Q,P) +~r(q,p) , (ar(00) = 0 ) (3.3)
po ~ ~r(Q,p ) +~r (q'p) . ( ~ (00) = 0 ) f r r
The first terms (~,,p) come from the collective degree of freedom and the second
Ones (a, ~) represent the intrinsic fluctuations around the collective parts. There-
fore, the collective and the intrinsic modes are described in terms of the variables
(Q' Y) and (q, p), respectively. Wlth the use of the canonlcity condition (3.2) and
the relations (3.3), we have
(Col Ic0) = p +~+×p
(Coli~ ~w Ic0) = Pn +--
~qn
, (co co) = ~ -XQ , (3.4)
(co c0) .... • (3.5) ' ~Pn
50
W = S(Q °, F °) -2j(og~,~D ) +Er~r4D r , \ (3,6)
J w = S(~ 0, F 0) -s(q, p) +Erar~_..r
For the above derivation, the following relations were used:
×Q = Er (e r ~ -~r ) " ×P = Er(~r " - ~ -~ r ) , (3.7)
~qr ~s ~2r ~s Er~D r ~Q P ~Q , Er~D r @~ ~p , (3.8)
lr~r ----~r = Pn - ~__~_s • Er~r r ~s , (3.9) 3qn ~qn ~Pn 3Pn
In relation to specifying the suhspace which corresponds to the [f0]-space, we
set up
XQ - 0 , Xp = 0 (3.10)
Then, the expectation value (c01iB/~t -Hlc0) ( ~ L ) is given by
L = PQ +EnPnq n -~ +W , (3.11)
= (colH[co) =O~(~(~ °', F °) = ~ ( ~ + a , ~ + r ) , (3.12)
W = S -~-s +Er~r~D r +QXp -PXQ (3.13)
We can see that L is a Lagrangian in the [f0 +l]-space and (Q, P) and (q, p) are ca-
nonical. However,, it should be noted that L was given under the conditions (3.10).
From the relations (3.10), XQ and Xp can be regarded as constants of motion and, by
the assumption, we set up
×Q ~ 0 , Xp = 0 (3.14)
The relations (3.14) play a role of the constraints to specify the subspace which
corresponds to the original [f0]-space.
For the variation of L, we impose two requirements. First is the requirement
that the collective mode is stationary for the intrinsic fluctuations. The Hamilton-
Jan can be decomposed to
" ~(~+~,J=+~) = ~e +~d (3.15)
Here, ~d is linear with respect to (a, ~):
~d = Zr(er B~(41,~) +~ @~4~(~Z,/m) ) B~r r 8~r
The first requirement tells us that ~ d should vanish.
following form:
~d = IQXQ +%pXp (3.17)
Through the constraints (3.14), ~d vanishes. Second is the requirement that the
variation of L should be performed under the constraints (3.14): tl
~ft0( L +~QXQ +~pXp) at = 0 (3.18)
(3.16)
We can make ~q~d vanish in the
51
Here, ~Q and Ap denote the Lagrange multiliers.
The variation (3,18) with the relations (3.]0) gives us the following equations:
(Xq,~)p (Xp,~)p Cp -Ap = (XQ, Xp)e ' @Q -IQ = (Xp, xQ)p ' (3.19)
ffi (Q'~)D ' P ffi (P'~)D " (3.20)
qn ~ (qn,~e) D , Pn = (Pn'~)D (3.21)
Here, (A, B)D represents the Dirac bracket:
1 (A, B)D = (A, B)p (XQ, Xp)p [(A' Xp)p(xQ, B)p -(A, XQ)p(Xp, B)p] (3.22)
The relations (3.20) and (3.21) are nothing but the Dirac's canonical equations. 6)
If We know (~, ~) and (a, ~) as functions of (Q, P) and (q, p), respectively,
can be determined. Then, we can solve Eqs.(3.20) and (3.21).
3-2 Canonically invariant form of equation of collective submanifold and specification
of collective-lntrinsic coordinate system
Let us start from finding equation to determine (~,~) in terms of (Q, P).
For this aim, we go back to Eq.(3.17). The definitions of (×Q, ×p) and ~ d given in
Rqs'(3-71 and (3.16), respectively, lead us to
A ~r " .~(o0~, ~]m ) A ~ ~(~' $~ ) (3.23)
a (3.241
The above is called equation of collective submanifold. We can determine (2, 9)
as functions of (Q, P), together with (~Q, ~p). Needless to say, the relations (3.8)
or the following relation in which S is eliminated from Eqs.(3.8) are used:
r ~Q .... ~p- - 3 ~ ~
In order to find the solution, let us investigate the property of Eqs.(3.23) and
(3.25). With the use of Eq.(3.25), we can derive the following relations from
Eq-(3.23) :
~Q • = ~p .....
Then, we can see that Eqs.(3.23) and (3.25) are symbolically of the form
~A ~B ~A ~B (3.27)
~ere, A, B and C are functions of (~,~). We introduce another set of canonical
Variables (Q. p.) which connects to (Q, P) through
B~Q~'~ " ~-~-'3P" ~ ' ~ = ~p i (3.28)
Then, we have
~'~A ~.-~ B ~A ~B (3.291 ~q" 29" " ~p-'--;"~q~ f f i C
52
The above fact tells us that Eqs.(3.23) and (3.25) are canonically invariant. This
is a quite natural fact, because the collective submanifold does not depend on the
choice of its coordinate system. However, it is necessary to fix the coordinate sys ~
tem in order to express the Hamiltonlan in a concrete form.
Let us discuss a typical example. We note ~( ~, J D). This is a function of
only (Q, P) through ~r(Q,P) and ~r(Q,P). Therefore, we call@~(~, J D ) the collec"
tlve Hamiltonian and, hereafter, we denote it as Hcoll. Our task is to determine
Hcoll concretely. A practical method is to obtain it successively from the lower to
higher terms in the framework of power series expansion for P:
p2 Hcoll = V(Q) + 2-~(Q) +Z~= 3 hn(Q)P n (3.30)
Here, we assumed that Hcoll is stationary at the point Q = P - O. Practically, we
have to stop the expansion at a finite power. Therefore, it is undesirable that the
power, at which we stop the expansion, changes if we view from another coordinate
system. Of course, it connects to the original by a canonical transformation. The
point transformation, which is shown in the following, satisfies this condition:
P Q" - f(Q) , P" = df(~-----7) , (3.31)
dq
where f(Q) is an arbitrary function.
Keeping the invariance property of expansion (3.30) in mind, let us investigate
how to solve Eqs.(3.23) and (3.25). As was already mentioned, they are invariant lot
any canonical transformation. Therefore, in the framework of Eqs.(3.23) and (3.25),
it is impossible to specify the coordinate system in one special type. Instead of
Eq.(3.25), we adopt Eqs.(3.8), from which Eq.(3.25) is derived. In this sense,
Eqs.(3.23) and (3.8) with S = 0 are not invarlant for any canonical transformation,
but for the point transformation (3.31). Therefore, in the framework of Eqs.(3.23)
and (3.8) with S - 0, we can specify the coordinate system except the choice of the
function f(Q) in Eqs.(3.31). In order to fix f(Q), we set up M(Q) = I, where M(Q)
appears in the expansion (3.30). In order to solve Eqs.(3.23) and (3.25), we further,
investigate the boundary condition, which is given in the small amplitude limit. We
assume that if Q and P vanish, then, (~,~) also vanish. In this limit, the colleC'
rive Hamiltonian is given in the following form:
.~ -I Loll 2 rs rs rPs +½ rsKrs r s
-I = -, = ~ ) (~-rs ~q'sr ' K r s s r
Then, the equation of conective submanifold becomes
3~r ~r -I ~J~r A p " ~ - A Q ~p " Zs~rs~s Ip - - ~ -~Q ~Dr • ' ~--V-
~ r 3~r Zr.~/Dr ~----~- P , ZrJ3r ~--7-= 0
A s o l u t i o n of Eqs . (3 .33 ) and (3 .34) i s g iven by
(3.32)
' f f i -Z s ~ r s ~ s , (3.33)
(3.34)
53
~r ~ ~r Q , ~r ffi #r p ' IP = P " IQ ffi CQ (3.35)
~ere, Vr' ~r and C areglven by identifying them with the solution showing the high-
est collectivity, ~rO' ~r0 and ~0 in the following equation:
Es(~?~rs -~rs)~s ffi 0 , Ersq?[rs~r@s ffi I , #r = ZsT?trs~s (3.36)
With the use of the above boundary condition, we can determine (~,~D) as functions of (q, p).
Our final problem is to determine (~, ~) in terms of functions of (q, p). We
note that they obey Eqs.(3.9). By adopting s = 0, let us diagonalize the following
Hamiltonlan:
Hintr " ½ rs' ;IVs + r s K rs'r°s This Hamiltonlan is obtained from ~e(~,~) for small limit of (~, ~).
(3.37) is diagonalized in the form
=r " En Arnqn ' ~r ffi EnDrnPn (3.38)
It is easily verified that Arn and Drn are given as the n-th solutions of Eqs.(3.36):
A rn ~ ~rn ' Drn = ~rn (3.39)
The above is a possible choice of the intrinsic coordinate system.
Finally, we will give a comment concerning the small limit of (Q, P) and (~, n).
In this limit, the Hamiltonlan is given in the sum of Hcoll and Hintr shown in
Eqs'(3.32) and (3.37), respectively:
~(~ffi ½p~ ~oQ2 I f-I 2 I f-1 2 -~Znffi 0 pn-~Zn=O ~nq n (3.40)
It should be noted that the mode with n = 0 is double-counted. However, XQ and Xp
are given in the forms
XQ ~ q0 ' Xp = P0 (3.41)
Therefore, the intrinsic mode with n = 0 is rejected from~ given in Eq.(3.40).
(3.37)
The Hamiltonian
§4. Classical image of many-fermlon system in a canonical form including Grassmann
variables
4"i An extension of TDHF theory based on the use of Grassmann variables for independ-
ent particle motions
As was shown in §3, the TDHF theory gives us a method how to describe collective
a~d intrinsic motions only for even-particle system. In this case, both kinds of
motions are of the partlcle-hole pair type. However, the description of odd-particle
system is inevitable for the understanding of collective and independent-partlcle
mOtions. In this case~ the independent-particle motion is a kind of intrinsic motion,
hut it is not of the partlcle-hole pair type. Therefore, in order to treat the col-
lective and the Independent-partlcle motions, we must extend the TDHF theory. Of
54
course, we regard the collective motion as of the partlcle-hole pair type. In this
section, we investigate this problem.
We note that a classical image of fermlon can be given in terms of Grassmann
variables. 7) Then, first, let us investigate a method how to weave the Grassmann
variables into the TDHF theory. The particle and the hole operator, ~% and n i for
the vacuum Ic0> ( ~Ic0> - ~ilc0> = 0 ), are, in the matrix form, given by ^
I" "I( 1 ~* = C0* , D0* ~, , ( DO = ¢I -A0 ) (4.1)
Here, A0 and Co are defined in Eqs.(2.21) and (2.22), respectively. Following Casal"
buoni, we introduce the following fermion-coherent state [c> for ~% and ~i:
Ic>- Ico>, (4.21 ^, ,^ ^, ,^
- exp[Zl(~lf I -fl~l) +El(Dig i -glni)] (4.3)
The state Ic0 TM is given for the case (2.2). The coherent state Ic> satisfies
E Ic> * ql=> , Ailc> = gll c> (4.4) • *
gi and gi^are Grassmann numbers, which cau be regarded as classical im" Here, fAl,fAl ~,
ages of ~A' ~A' ~i and ~i' respectively. For the exchange of the order of the pro-
duct and the complex conjugation, they are, for example, governed by ^
gifA = -fAg i • gi~% = -~lg i , ]
• * * ~ * ^* * I (4.5) (glfA ~ = flgi ' (gi l ) = ~Agl
The state le> contains the parameters Ci%, Cil, fA, fk, gi and gi"
Basic idea of our extension of the TDHF theory is in the variation of the folloW'
ing quantity which is similar to Eq.(2.3):
• <cli Ic> (46) In the present case, we cannot regard the parameters contained in [c> as canonical.
Further, they are not independent of one another. In the case of even-partlcle sys-
tem, the number of the independent parameters is 2f0. Therefore, there exist certaiO ^ ^
constraints governing these parameters. We consider the expectation values of bia l ^.^,
and aAb i for }c>, which play a special role in the study of collective motion. The ^ ^
expectation value of bla % is expressed as
<clbla%Ic> = (bla%) C +(bia%) p , (4.7)
(bias) C = ~Ep~(Do)~l(6 -2f~f )(C0)i +½Ejk(D0)ij(6jk -2gkgj)(C0)k~ , (4.7a)
(biaA) P = Zj~(D0)ij(Do)BkVju -(C0)lu(Co)jkYj~) , (4.7b)
YJM = glf~ ' YJu = f~gJ (4.8) ^ ^
Relations (4.7) tell us that the pair mode <c[biallc> can be shared between the two
components (bial) C and (biaA) P. In the TDHF theory, the collective motion is associ"
55
ated with the tlme-variation of the self-consistent field, which is treated in terms of * (C0)IA and (C0)i%. The independent-partlcle motions are described with the aid of
the particles and the holes referring to the tlme-dependent self-conslstent field.
Classically, the motions are treated in terms of (f%, f~) and (gl' g:)" The second component (bia%) P contains ¥jp and ¥jp which represent the pair modes associated with
the, independent-partlcle motions. On the other hand, the first contains (C0)il and
(C0)i% which are associate d with the collective motion. The total number of <clbla%ic>
and <el ̂ *^* * * akbilc> is 2f0 and the total number of (C0)ll, (C0)i% , Yil and yi%,is 4f0. *
This means that they should be governed by 2f0 constraints. The terms fufp and gkg j
are expected to play a role of disturbing the collective motion as a result of the
Independent-particle excitations. In our present system, we consider the case where
f Collective modes are contained explicitly. Then, (C0)i ~ and (C:)i% are expressed
as functions of 2f collective variables. This means that all (C0)i% and (C0)il are
implicitly restricted by 2(f0 -f) constraints. Therefore, it is enough to find 2f
constraints Xr = 0 and X r = 0 ( r=l,2,''',f ), which govern all Yi% and Yi%, As is
suggested from Eqs.(4.7), they are of the linear combinations for Yi% and Yi%"
For the variation of6~ , we introduce boson type collective variables (X r, X* ;
~'l,2,"-,f) and the Grassmann variables, (y%, yl; l=I,2,...,M) and (z i, zl; i=1,2,..,
N). For the even and the odd power for the Grassmann variables, E and O, the modified
POlason brackets {A, B}p are defined by
r (8EI 8E2 8E 2 ~E,) .3~E 8E2 8E2 8EI~ = -Ze(Bx ~ 8x ,, , (4.9a) {El, E2}p -r'BXr ~X* 8Xr 3X e 3x ~ 3x
r r ~
+ (0~, o~p = ~r%-~ 3X~ 3Xr 3X r 3x 3x 3" ~= '
{0, E}p . Zr(~xOr 3E BE 3%) +l (~ ~ + BE 3__O_O. (4.9c) 3X* 3Xr 3X = 3x 8xa 3X *) '
r r C~ (l = Z (BE 80 BO BE) BE 80 BE)
% . . . . * 3x r + " (4.9d) 8X r BX r 3XcL 3x 3x~
Here, xe and x*e denote, (yl,,zi) and (y~,,z~), respectively. For example, the combi-
nations of (Xr,
{X r, X:}p =
Xr)' (Yl' Yl ) and (z i, zi), we have
*} = * 6rs ' {Y~' YP P ~U ' {z i, Zj}p = ~ij ' (4.10)
{ the others }p = 0
In order to get~, first, we must calculate <clB/BXrlC>, <clB/By%Ic> and
<cl~/BZilc>. The results are given by
= 1 * +K(r)* (4.11) <cl~Ic> ~K r K* = T-.~*~cor ~ C ~ . * -D 3Do 3Do D ~ ,+3Co 3 C ~ . r L~u03X ~ ~0J -Zl~flf~( 0-~ ~ 0 +C0-~-- - 8X u0)~ l (4.12)
r r r r r r
3Co~T~ *3gi 3gi , LiJgig~U~r 8X r- ~°~ r 3XrU°JiJ +El(fl~r ~'fr ) +Ei(gi~r 3Xr gi)'
56
K(r)* . .. (r) * . (r)* . '= ~ll~Lil Yil -rail YII ) '
L(r) 1.3Co o 3Do 3Co ~D0o ,
r r r r
= _ coax - E ~' ~ - ' ~ '° +Do3x ~X C°)il |
r r r r
~y! I * _K(I)*
<CI~ilC > = 1 * _ K ( i ) * -~K i
• * 3fl ~fl * ~gi 3gi m = Z~(f I ~ + ~x fl ) +Zi(gi ~ + ~-- gl )
+Tr(C~ 3c~ _ 3c~ Co) 3x 3x
ct ct
- Z ~ ) f * f ~ "¢(Do 3x aDO - '3"D'03x DO +C#0 3x3C° 3x3C%0 C°)VU
• T 3~o 3 ~ o +c~ 3¢° 3 d c 0 ) ~ -Zjkgkg J(D0 3x 3x 3x 3x '
C~ C~ O~ Ot
K ( ~ ) * : - , - ( a ) * + M ~ l * y i l ) ~il thil Yil
L(~) 1.3Co 3Do 3Co 3Do Co)i 1 il = ~(,~--~--- Do -Co ~ +Do 3x 3x "
• * ~ o +~'[ ~Co 3D'o ~ c* ) i~ . (a)* 1,3Co D~O ' -Co (X ~ Ct r~
(4.13)
(4.13s)
(4.13b)
(4.14)
(4.15)
(4.16)
(4.17)
(4.17~)
* K(~)* * By replacing x with yl and z i in Eqs.(4.16) and (4.17), we obtain K%, K i and
K (i)*. As was already suggested, 2f constraints Xr = 0 and ×r = 0 are of the linear
combination for Yil and Vil. Then, we set up
* = K(r) * Xr = K (r) ( = 0 ), Xr ( = 0 ). ( r = I, 2,,.., f ) (4.18)
Further, as the canoniclty conditions, we set up
~s, * ~s K r = X r -21 , K = X r +2i " - , (4.19a)
~X r ~X r r
* 3S 3S
H e r e , S i s a f u n c t i o n o f (X r , X r ) , (y~ , y~) and ( z j , z j ) . Then , we h a v e
i *" "* *" "* +Zr(X;~r "* <cll ~tlc> = ~[Z (ypyu +y~y~) +Zj(zjzj +zjzj) -XrXr)] +K d -S , (4.20)
K d Z "K (d) * +K (d)* " (4.21) = il ~ il ~il il Yil ) '
• (~) . "*..(p), -iEj(~jL~ ) +-'*"(J) -iZ (y Lil ~Tumil j zjv~il ) . (4.21s)
(4.17b)
57
The expectation value <c[HIc> is given by
<clHlc> = H +H d , (4.22)
.. (d) * .. (d)* . (4.23) H d ffi Zilt~i% Ti% tni% Yi~)
Here, We impose the following requirement: The coupling between the collective and
the independent-partlcle motions should be minimal. This means that the linear terms for *
Yi% and Ti % should not exist in the Hamiltonlan, i.e., H d should vanish. The
elimination of H d can be performed in the following way: We make K d and H d coming
from <cli~/~tlc> and <c]HIc>, respectively, cancel and set up
K d -H d = Er(¢rXr +¢rXr ) (4.24)
Thus, we get ~ in the following form:
i *. .* *. .* +Zr(X~r -* ¢~ ~[~ (y~y~ +y~y~) +Ej(zjzj +zjzj) m _XrXr)]
-H +Er(¢rXr +¢rXr ) -S (4.25)
Here, Xr and Xr are given in Eqs.(4.18). We can see that ~ is of the form of the
CSnventlonal Lagrangian. Then, we perform the variation under the constraints Xr
X r " O:
~S~{l:~ +Zr(ArX: +A:Xr)}dt = 0 , (4,26)
where Ar and A* " "* r are the lagrange multipliers. If we note Xr ffi Xr ffi O, the present
sYStem can be treated in the Dirac's canonical form. The above is our basic idea for
an extension of the TDHF theory.
4-2 Equation of collective submanifold for the case of one collective mode
In this subsection, we investigate one extreme case f ffi I, that is, the system
contains one collective mode. In this case, the variation (4 26) with XI "* • = XI = 0
gives us
iXl = {Xl, H} D , iXx ffi {XI, H} D , (4.27a)
• m • ** ~ { * lYu {Yu' H)D iy~ y~, H} D , (4.275)
i z i ffi {z i , H} D , i z : ffi { z : , H} D • ( 4 . 2 7 c )
Here, (A, B} D denotes the modified Dirac bracket:
{A, B}D . {A, B}p -{A, X~}p 1 * {XI, B}p {Xl, ×1}p
I {×~, B}p (4 .28 ) -{A, Xl}p , (Xl, Xl}p
With the use of Eqs.(4.18), (4.21) and (4.23) for r = I, the relation (4.24)
leads us to the following equation:
58
. (1 ) * ±~ '1 . (1) , +[{XI, H} D [{XI H} D -~I]LiX ~ijr*il
.~ .(U) Nl L (j) +{z~, "~ M (j)1 = H (d) "~ " (~) +{Y~' nTDnil ] -ZJ[{ZJ ' -'D il n;D il j il
-[{Xl. H} D -*l]M~ )* -[{X~, H} D ~ij~il±~*1"(1)
+X [{y , HI M (~)* +(y~, (~)* "D-il H}DLil ] +Zj[{zj,
, (4.29a)
HI~DM(J)*il +{z~, H}DL(J)*]il J
= H (d)* (4.29b) iE '
are expressed as functions of (Cil, Here, it should be noted that H, H (d) and H (d)* , , , i~ ik
Ci~), (f , f ) and (gj, gl).~ The set of Eqs.(4.29) can be called equation of collec"
tive submanifold. By solving it, together with the canonicity conditions (4.19) for
r ffi I, we can determine (Cil, Cil), (fp, f), (gj, gj) and (¢I, ¢i) as functions of
(XI, Xl), (yg, y~) and (z j , z j ) .
It can be proved that equation of collective submanifold is invariant under the
Sp(2R)- and the SO(2(N+M))-transformation for the collective and the Grassmann vari-
ables, respectively. Therefore, for the canonicity conditions (4.19), we should
choose S ffi 0. Then, the conditions (4.19) are invariant under the above-mentioned
two transformation. The Sp(2R)- and the SO(2(N+M))-transformation contain 3 and
(N+M)(2(N+M) -I) free parameters, respectively, Which must he fixed. This problem
can be solved by requiring that H C and Hp at the small amplitude limit are of the
form
ffi 1 p2 +<Q2 Hp * * H C ~ , = E cuy~y ~ -Zi~iziz i (4.30)
Under the above-mentloned requirement, we can determine (Cil, Cil), (fu" fu ) and
(gj, gj) as functions of (Xl, Xl), (Yu' YP ) and (zj, zj). Then, Hamiltonian and
other physical quantities can be expressed as functions of the canonical variables.
4-3 Boson-fermion expansion given as a result of canonical quantization through DirsC
bracket
The purpose of this subsection is to investigate the other extreme case ( f ffi
f0 ). The result is reduced to the hoson-fermion expansion given by Marshalek.
First, we define the following matrices:
ffi COBp Dp = DOBp , cp p ' p
C h = ~l -w h BhC0, D h = ~I -~ BhD0.
Here, Bp and B h are certain unitary matrices and Wp
B B% - B~B ffi 1BhB~ ffi B~Bh ffi 1 , PP PP '
(Wp)l~ " f~fl ' (Wh)iJ ffi gJgi "
The matrices Bp and B h are determined under the conditions
(4.31s)
(4.31b)
and w h are hermite matrices:
(4.32)
(4.33)
59
D %p = Dp , D h% = D h • (4.34)
Then, we have
DZp +CtCp P = I -Jp , D Tz +ChC h* = I -w hT , ChD p -DhC p = 0 (4.35)
From the relation (4.35), Dp and D h can be expressed as
D -- ~I -A , A = C%C +w T , (4.36a) P p P P P P
*TT D h = ~I -A h , A h = (ChC h) +w h (4.36b)
^ ^
With the use of the above-deflned matrices, the expectation value <clbia X[c> etc.
can be expressed as ^ ^
<clbia%Ic> = Ei~ +Fi~ , (3.37a)
^*^ E l ^* ̂ <cla%aHlc> = ~ +F%H ' <clbibjle> = Eji +F , (4.37b)
<cla/Ic> = E l , <clbilc> = E i , (4.37c)
Ell -- (A -A T T +(/i -(~/I -A~ C0)i% P Cp) ~i -Ah Ch) i%
= (/I -A T cr)%i +(/I -~ Ch)i% -( I~--~-Ao cT)ki , (4.38a) P P
i * T * T * T I (4.38b) Ej . (CpCp)ij +(ChCh)i j -(CoCo)ij +(wT)ij ,
E ~ (/I -A T 1 I = _ _ f)% +(g% Ch)A ,
P •i' -Wp ~I -w h I
J Ei ~ _(f+ 1 ,T +(J1 -~ I g)i " Cp) i /i -w h
P
(4.38e)
e°nstraints:
Yi~ ~ 0 , Yil = 0
From the relations (4.30), we can see that Fil,
conditions (4.19) can be expressed as
~C 3C % ~C h ~C~ Tr(c * ~ p - P +Tr(C~ -Tr(C~ 8C0 p ~X ~X Cp) ~X ~X Ch) 3X
r r r r r
3f~ ~f% * ~gi ~gl * +Z%(f~ ~X ~X f%) +Zi(gl ~X ~X gi ) = Xr '
r r r r
F ~ and i vanish. Fj
~C,~ Co) 3X
r
(4.39)
The canonicity
(4.40a)
F % and i * The quantities Fi% , P Fj are expressed as linear combinations for Yi% and yi %
and, in the later discussion, we will show that these terms vanish.
Next, we contact with the constraints (4.18). Since we consider the case f - f0,
the relations (4.18) with the definitions (4.13) give us the following forms for the
60
Here, we choose the following form as S:
iBp i8 h S = Tr(Sp) +Tr(B h) ( Bp - e , B h - e ) (4.41)
Now, let us search a possible solution of Eqs.(4.40). First, it is noted that
we can choose the followlng solution: Cp, C h and Co do not depend on the Grassmann
variables and f~ and gi is equal to
f~ = Y% ' gi = zi (4.42)
Then, Yi%' Yi%' (Wp)%p and (Wh)ij become
Yil = ziY~ ' Yi% ~ Y%Zi ' (4.43)
(Wp)%~ = y~yl , (Wh)ij = zjz i (4.44)
With the use of the constraints (4.43), we have
(Wp) l~'(Wh)iJ = YpYl'zjzi = Y~ZJ'ZiY% = ~J~Yi% = 0 (4.457
The above relation implies that the following three subspaces in the whole phase space
are consistent with the constraints (4.39):
w ~ 0 , w h ~ 0 ; the space p , (4.46a) P
i Wp 0 , w h ~ 0 ; the space h , (4.46b)
w ~ O , w h " 0 ; the space O . (4.46c) P
Thus, we obtain the following expression of C in terms of the canonical variable X : r
(Cp) i% = X r , (Ch) il = (C0)i X for the space p , (4.47a)
(Ch)i% = X r , (Cp)i% = (C0)i% for the space h , (4.47b)
(Cp)i~ " (Ch) i~ " (C0)i~ " X r for the space 0 . (4.47c)
Here, the index r is identified to ik and, hereafter, we will use the index (il). In
t h e t h r e e s u b s p a c e , <clbi le> e tc , are given by
^ ^
<clbla%lc> = (/I _yTp xT)k i +(41'-Yh x) i% -(~/i '--Yo X)iA
" (A _yTp xZ)% i +(/I -Yh X) i% -(/I _yT X T)%i "
^*^ T <c[a%a Ic> - (xtx ~Wp)%~ ,
<el:~Ic> = (/~ _yT 1 y)% +(z* P ¢T-Wp /i--w h
<cf~ilc> . _(yt 1 ET) i +(¢I -Yh _1 Fl'-w h
P
Here, Yp, Yh and Yo are defined by
^*^ T <clbib j [ c> - (x*x T +w h) lj '
X) ~ ,
z) i ,
(4.48a)
(4.48h)
(4.48c)
61
T = (x*xT T (Yp)x~ = (x~x +Wp)X~ ' (Yh)ij +Wh)lj ' ~ (4.49)
(Y0)I~ = (X#X)I~ , (Y0)ij = (x*xT)~i , J Thus, we have accomplished our task to give a classical image of many-fermlon system.
Clearly, if the Grassmann variables vanish, our results are reduced to those given in
the TDHF theory of §2. In this sense, we can call our present theory as an extension
of the TDHF theory.
As our final stage, we will sketch a possible quantlzatlon of our system. It
can be performed by the following replacement: ^ ^
Xil-'+ Xil ' Y% -'+ Yl ' zi--+ z i (4.50)
^ ^, The operators Xil and Xil obey the commutation relations (2.28). Further, if there
^ ^ ^, do not exist the constraints, (yl, ;~) and (zl, zl) obey the antl-commutation rela-
t ions [~, Bl+: [~, ^* ~ ^,
y~]+ = ~ip , [z i, zj]+ = 61j . [ the others ]+ = 0 (4.51)
Of Course, the commutators ~, ;1 among (Xil' ^* ^ ^* * ^* - Xi% ) and (Yl' Y%' zi' zi) vanish.
HOWever, our present system is in the Dirac canonical form. Then the quantization
should be performed by the modified Dirac bracket:
• -i {A, B}D = {A, B}p -Eijl~[{A, yiA}p($ )il,j {yj , B}p
-i * -{A, yj }p(~ )il,jp{Yil, B}p] . (4.52)
Here, ~-i is the inverse of ~, the element of which is given by
(~)il,j~ " {YiA' 7jp}p
= ~ij~Au -(~)il,ju ' (4.53)
1. * * * * (P)il,jp = 26ij(6~ -YEY~ +YpY%) +½61p(~ij -zizJ +zJzi) (4.53a)
Then, (~-l)il,jp is expressed as
( ~ - l ) i t , j ~ = ~ij~ip +Zn=~(~n)ik,j~ (4.54)
The q-number replacement of ~ is performed by
(~,(no constraint) (~)il,J~ "-+ "~)il,J~
1 ^ ^* ^*^ ^ ^* ^*^ = ~lj(~lp -YlYp +YpYA) +½61~(~lj -ziz j +zjz i)
^,^ ^ = ^*^ +6A zjz i ~ (4.55) ~ijY~Yl (~)il,jp
~Y replacing the modified Dirac bracket with the antl-commutator and using the rela-
ti°n (4.55), we have
62
" ^ * ^ * ^ - 1 ^ [Yx' Y~]+ = ~xu -zlJzi(~ )ix,juz3 '
^ ^* ^* ^-1 ^ [z i, zj]+ = ~ij -E~y~(¢ )iX,j~y ~ ,
(4.56) ^ ^* ^* ^-I ^ ^ ^* ^* ^-i ^
[z i , yp]+ ffi Ej%y%(@ ) i X , j p z j , [ y l , z j ] + = Zi zi(@ ) i%, juy p,
[ the others ]+ = 0 ,
^n ; - I = 1 + Z n = l ~ (4.57)
By replacing (Xix, Xix) ,^~yx , y l ) and (z i , z i ) in Eqs . (4 .48) and (4.49) wi th (X , ^, ^ ^, ^ ^ ^ ^ , ^ ^ , ^ i ~
Xix) , (yx, yx) and ( z i , z i ) , r e p e c t i v e l y , we can o b t a i n (biax)BF, ( a ~ a ) B F , (b ib j )BF, ^ ^
I t can be shown t h a t the r e s u l t i s comple te ly i d e n t i c a l to t h a t (aX)BF and (bI)BF. I5) g iven by Marshalek.
Acknowledgements
The authors would like to express their gratitude to Professor A.A. Raduta for
giving a chance to report the present work at the Brasov International Summer School.
One of the authors (M. Y.) is indebted to the Yamada Science Foundation for support-
ing the travel expenses between Osaka and Bucharest.
Appendix
In this Appendix~ we will give an application of the present general theory to
the case of an illustrative model, i.e., Lipkln model. This model consists of two
levels with the same degeneracy. We denote it as 2~ (= 2j +i, j; half-integer). ^ ^
The following set of three operators, S± and So, which are fundamental in the Lipkln
model, forms the SU(2)-algebra:
^* ] ^* ^ 1 ~ ^* ^ ^* ^ ;+ = S_ ffi Em= - c+1,mC_l,m , So = ~Em. - (e+1,mC+1,m -C_l,mC_l,m), (A.la)
[;o, ;±] = ±;+, [i_, ;+]_ =-2;0 (A.ib)
The indices ±I denote the upper and the lower level, respectively. The Hamiltonian
is given by
= 2e; -~(s+V ^2 +;~) (A.2)
Here,2~ and V denote the energy distance between the two levels and the strength of
the interaction, respectively. It is convenient for the later treatment to introduce
the particle and the hole operator:
C+l,m " am ' C-l,m = (-l)3-m b~m (A.3)
^ ^
With the use of these operators, S± and S0 can be rewritten as
= I ^*^ +bmbm)-n (A.4) S+ S* = E ( - I ) j-m a ' b * $o = ~Em(ama m - - m m -m
63
The above is mathematical framework of the Lipkin model.
With the uae of the following Slater determinant, the classical image of the
Lipkln model is given:
]c0> = ~Jo> . ~ = ~+r -~_r (A.5) ^
The expectation values of S± and So can be calculated in the forms
^ , ^ ,
<C°IS+Ic0> = <c01S Ic0~ = 2~v*/l -v v , <c01Solc0> " 2~v v -~ , (A.6)
v E sinlFl. ~ (A.7)
We can show three classical images based on the three sets of the parameters, (x, y),
(8, r) and (X, ~):
v = ~2(x +iy) = 4~r e -is 1 cos2~sin X +Isin2~ =~7~ (A.8) ¢i +cos2¢cosx
The canonlclty condition (2.4) for each case gives us the following relations:
x 1 QO y 1 pO ~ , = -- , (A.9)
O QO pO = , r = ~ , (A~IO)
pO × = Q0 , sin2~ = ~- (A. II)
^
Of COUrse, (Q0 p0) are canonical variables. The expectation values of S+ and S0 are,
for each case, expressed as
<c01E+Ic0> = B*/2~ -B*B , <c01E01c0> = B*B -~ , ~ (A.12)
Q0 ~ ~2(B +B*) , p0 E i~2( B -e*) J <c°IS+Ico> = J(~-J) el+ ' <c°IS°le°> = J -~ ' i
(A. 13) Q0 ~ ~ , p0 E j J
^ i ^ <c01SxlC0> = <c01:(S+ +S_)Ic0> = /~-~-~T sinQ0 ,
<colby{CO> ~ <col~(~+ -~_)[co> = -P° . (A.~4)
<co[~lco> = <col~olco> = -~-~cosQ ° ,
The flrat case is nothing but the classical Holstein-Prlmakoff representation. The
second is an expression in terms of the action and the angle variables. The third
is suitable for the adiabatic TDHF treatment, because the Hamiltonian can be express-
ed in terms of the even power for the momentum p0.17) This case corresponds to the
PresCription given by Baranger and Veneroni. I)
We apply the above third case to the coupled Lipkln model. 18)'19) This model
Was first used for the calculation of collective path by Hayashi and lwasaki 20) and
essentially SU(2)xSU(2) model. The Hamiltonlan is given by
64
^ I V 22 +S~,_)] -V3 ^ ^ +$2 (A.15) - Zr=l,2[2~rSr, 0 -~ r(br,+ (SI,+$2 _ ,+SI,_ ) ^ ^
The operators Sr, ± and Sr, 0 form the SU(2) algebra and are given by attaching the in- ^ ^
dex r (=i, 2) to those of the Lipkin model. Therefore, <c01Sr)+Ic0> and <c0]Sr,olC0>
are expressed in terms of the variables (Q$,_ P ~ ) , a s i s shown i n E q s . ( A . 1 4 ) . I f t h e
o we get the fol- classical Hamiltonlan is expressed up to the quadratic order for Pr'
lowing form:
1 o o o o H A = ~Zrr-Urr,(Qz,Q2)PrPr~ +%~(Q~,QR) • (A.16)
I +sln2Q~)]/~£ +2V~{slnQ~slnQ~/~2 Prr = [2erC°SQ~ +2Vr(~r -~)(I , (A.16a)
pI2(Q~,Q~) = ~2z(Q~,Q~) = -2V3 , (A.16D)
O~(Q~,Q~) = _Er~[2erCOSQ ~ +Vr(~ i 2 0 -~)sln Qr ] -2V3~f~slnQ~sinQ~ (A.16c)
In the above relations, the degeneracy ~r is replaced by ~ = ~r + ½' in order to take
into account the quantum fluctuation effect. The equation of collective submanifold
(3.23) for the Hamlltonlan (A.16) is solved approximately in the framework of the fol ~
lowing for~s:
~r(Q,P) : qr(Q) , ~Dr(Q,P) = pr(Q)p (A~I7)
The above approximation is consistent to the conventional adiabatic TDHF treatment.
Then, the collective Hamlltonian is given by
HC . ~pl ~ +Vc(Q) , (A.18)
Vc(Q) "qJ~(qz(Q),q2(Q)) (A. I8~)
We will show numerical results of the collective path (qz, q2) and the potential
energy Vc(Q) under the following set of values of various parameters:
~z " ~2 " 5 , cz = 1.5 , E2 = 2.0 , 1 (A.19)
VZ = 0.05X , V2 = 0.10X , V3 = 0.075X , J which is that adopted in Ref.20). Figure I, which is cited from that of Ref.18),
shows the collective path (qz, q2) and the countour plot of the potential energy
O~ (, .nJ 02 (n . . ) 02 (n . , )
la) (b) (c)
F t g . l Collective Paths
65
(Q~, Q~). Three illustative cases are given: (a) X = 1.0, (b) X =2.0, (c) X = 4.0.
Figure 2, which is also cited from that of Ref.18), shows the collective potential
energy Vc(Q ) for the cases (a), (b) and (c). From these figures, we can see that the
Smaller, the middle and the larger value of X correspond to a rather stable normal,
a transitional and a rather stable super-conductlng cases, respectively. The classi-
Cal Hamiltonian (A.16) can be quantized in the following q-number replacement:
Figure 3 cited from that of Ref.18) shows the energy spectra given under the above
quantlzatlon~ For the comparison, we give also the results of the TDHF method 3)
(the rightmost) and the exact ones (the leftmost). We can learn that the present
Prescription based on the adiabatic TDHF treatment works well in the specification
of collective submanifold and also in the reproduction of the collective energy spec-
tra.
-50
COLL. POT.
L\
-3 *3
I0 0
m
" ' " 1 - - - i
- - _ _ ~ . . - . . .
- - - . . . -
. . . . . . .
x=1 --x:2 x=4
Fig'2 Collective potential energies Fig.3 Excitation energies of collective states
The present collective submanlfold is 2-dimenslonal and is embedded in a 4-dimen-
SlOnal symplectic manifold. Hence, the motion on this submanifold becomes always pe-
riodic. Then, we have the following two questions: I) What type of orbits on the 4=
dimensional manifold do orbits on the collective submanifold simulate? 2) In what
sense, does the collective submanifold simulate the quantal collective subspace?
For the first question, we solve the Hamilton's canonical equation (2.6) for the pre-
sent model. Figure 4 shows two examples of the solutions for X = 2.0, where boundary
e°ndltions are rather arbitrarily chosen. This figure is cited from that of Ref.19).
It Seems hard to understand that the periodic orbits on the collective path can slm-
ulate such TDHF orbits shown in Fig.4. Figures 5 and 6 cited from those of Ref.19)
Show the TDHF and the adiabatic TDHF orbits with the RPA-type boundary condition.
bach orbit of this class is localized in a narrow region along a certain simple curve.
FUrther, we can see that the collective path simulates the class of orbits with the
RPA-tyPe boundary condition as a whole.
66
Fig.4 TDHF orbits in the coordinate space
p
Fig.5 TDHF orbits in the coordi- nate space with the RPA= type boundary condition
Fig.6 Adiabatic TDHF orbits in the coordinate space with the RPA-type boundary con- dition
Next, we see that the relation between the collective path and the quantal col-
lective suhspace. For this aim, we quantize the classical image in the following
form: _i ̂ _i ~
tiS = _ e+2Qr/~.2 _~2 e+2Qr , S = _~ , % rx rz r r ry r [ (A.21) ^ ^
[Qr' Ps] = i~rs
By substituting the relations (6.21) into H given in Eq.(A.15), we have H expressed ^ ^
in terms of (Qr" ^ Pr )" In the coordinate representation, the operator Pr can be glve~
as Pr = -i~/~qr (Qr = qr )" Then, we can get the wave functions of exact elgenstateS
in the coordinate representation by solving
H~(ql,q2) = E V (ql,q2) , (A.22)
- fh. ~2~ I ei(mlql +mzq2) ~(ql.q2) = ~ ~ ~ -- (A.23) ml=-~l m2=-~2 ~,mlm2 2~
Then, we can easily evaluate the probability density in the coordinate space (ql, q2)
by calculating 0~(ql,q2) = [~(ql,q2)[ 2. Figure 7 cited from that of Ref.19) shows
p~(q ,q ) of eigenstates identified to be collective. It may be clear that these
states belong to a particular class of states, where, as the excitation energy becom~
higher, the number of nodes of wave function increases one by one along a certain dl-
rection. This direction just coincides with that of the collective path. Each TDHF
orbit with the RPA-type boundary condition and the corresponding energy goes through
near tops of the probability density. The above is the answer of the two questions.
87
I Fig.7 Contour plot of probabi-
lity density in the coordinate space
x~ ~ ~
Reference s
1) D.J. Rowe and R. Bassermann, Can. J. Phys. 54(1976), 1941. F. Villars, Nucl. Phys. A285(1977), 297. K. Goeke and P.G. Reinhard, Ann. of Phys. 112(1978), 328. M. Baranger and M. Veneroni, Ann. of Phys. 114(1978), 123.
2) T. Marumori, T. Maskawa, F. Sakata and A. Kuriyama, Prog. Theor. Phys. 6-4(1980), 1294.
3) A. Kuriyama, Proceedings of the 5th Kyoto Summer Institute, Kyoto, July 12-16, 1982, ed. Y. Abe and T. Suzuki ( Prog. Theor. Phys. Suppl. Nos.74&75, 1983), 66.
4) F. Sakata, T. Marumorl, Y. Hashlmoto and T. Une, Prog. Theor. Phys. 70(1983), 424.
5) A.A. Raduta, V. Ceauseseu, A. Gheorghe and M.S. Popa, Nucl. Phys. A427(1984), i. 6) P.A.M. Dirae, Can. J. Math. 2(1950), 129. 7) R. Casalbuoni, Nuovo Cim. 33A(1976), 115, 389. 8) 9) A. Kuriyama and M. Yamamura, Prog. Theor. Phys. 71(]984), 122. I0) E.R. Marshalek, Nucl. Phys. A161(1971), 401. II) M. Yamamura and A. Kurlyama, P-~rog. Theor. Phys. (submitted) 12) M. Yamamura, A. Kuriyama and S. Iida, Prog. Theor. Phys. 71(1984), 109, 752.
M. Yamamura and A. Kuriyama, Prog. Theor. Phys. 75(1986),--272, 583. 13) M. Yamamura and A. Kuriyama, Prog. Theor. Phys. ~(1981), 550. 14) M. Yamamura and A. Kuriyama, Prog. Theor. Phys. ~(1981), 2147.
M. Yamamura, Proceedings of the 5th Kyoto Summer--~nstitute, Kyoto, July 12-16, 1982, ed. Y. Abe and T. Suzuki ( Prog. Theor. Phys. Suppl. Nos.74&75, 1983),
15) 271. ~.R. Marshalek, Nucl. Phys. A357(1981), 429.
16) - Yamamura and A. Kuriyama,--~g. Theor. Phys. (submitted) 17) ~. Kuriyama and M. Yamamura, Prog. Theor. Phys. 69(1983), 681. 18) • Kuriyama, M. Yamamura and S. Iida, Prog. Theo~?. Phys. 72(1984) 1273. 19) ~. Kurlyama, M. Yamamura and S. lida, Prog. Theor. Phys. ~(1985)[ 837. 20) • Hayashi and S. lwasaki, Prog. Theor. Phys. 6~3(1980), i~3.
68
NOTE
After (4.40a), the following equation should be inserted:
* ~f~ ~f~ * ~gi ~gi Zl(f I ~-~-- + ~--~- fl) +li(gi ~-~- + ~--~'- gi )
BC # ~C h ~C~ 3C~ Co) = x* ~Cp _ P +Tr(C~ Ch) -Tr(C~ ~Co +Tr(C ~x ~x Cp) ~x ~x ~x ~x @
(x = y~, z i) (4.40b)
CLASSICAL LIMIT AND QUANTIZATION OF HAMILTONIAN SYSTEMS
V. Ceausescu, A. Gheorghe
CENTRAL INSTITUTE OF PHYSICS, P.O. Box MG6, ROmAnIA
O. INTRODUCTION
The idea to present a lecture on dynamical (Hamiltonian) systems (with empha-
sis on co l lec t i ve motion) at a Nuclear Physics Summer School is motivated by the
fact that co l lec t ive features (such as rotat ional - v ibrat ional models) play an im-
Portant role in our understanding of the complicated nuclear system. Moreover re-
cently, many such ideas have been developed and introduced (co l lec t i ve path, coherent
States) in the descript ion of large nuclear dynamics amplitude motion with the hope
of understanding (maybe) even such processes as f i ss ion . We shal l however be more
modest in our approach and r e s t r i c t our discussion to various co l lec t ive aspects of
nuclear dynamics that manifest themselves in nuclear spectra. There are many models
that attempt such descriptions and they can be seen to be algebraic that is the Ha-
miltonian is a polynomial function of the generators of some group of symmetries
G (most have a sym~aetry group that is compact). I t has also become appearent that
lately the domain of quantum mechanics is being invaded by concepts that were speci-
f ic to classical mechanics. Indeed solitons and instantons (in f ie ld theory) are but
the top of the iceberg.
This search for a correspondence between classical and quantum systems has pro-
duced a new mixture of ideas that has affected our understanding of nuclear systems.lt
is thus natural to ask ourselves at some point whether this rendering of quantum me-
Chanics in terms of classical objects and concepts is accidental (fashionable maybe)
or necessary. We make no secret that our main point is to show that i t is no accident.
We Shall show that at least for some class of systems i t makes sense to introduce
Classical concepts and that they appear in a natural way. Thus we shall take a geome-
t r ic view - describe the geometric properties of our systems - and the main (mathema-
tical) tools w i l l be those of Symplectic Geometry and Group Representation Theory.
The relevancy of this approach for nuclear physics is, as already mentioned,
basically exhibited by the great nunber of collective models (some of them obtained
as restrictions of the many body problem,and others purely phenomenological). I t is
to be noted that no matter the source they are al l equally valid in the sense
that they do give an in tu i t ive description of some relevant features of the quantum
nuclear system. One point of interest is that the most of the models used by nucle-
ate physicists make (quite freely) use of parameters to f i t spectra(especially the so
70
called phenomenological models). We shall consider thus families of operators (and
functions) and consider i t a basic feature of our approach. Indeed the idea to
employ a corresponding classical system stems from the d i f f i cu l t i es encountered
by many-body approaches to a complete understanding of some simple collective
features of the nuclear system. In this lecture we shall point out that a corres-
pondence between quantum and classical systems exists and that i t can be employed
for a qual i tat ive understanding of some (as yet) unsolvable problems.
To be more precise we shall show that there is a compatible realization of
certain Lie algebras of observables both in the Hilbert space (e.g. square integrable
functions) and on the phase space of some "classical" system - that is there is an
embedding of the phase space into the quantum space of states and a homomorphism
of observables. In order to i l l us t ra te our approach we shall mainly res t r ic t our
discussion to that of a simple case - the SU(2) algebraic models (which are however
conceptually relevant to nuclear physics). This are introduced in Section 1 and
the symmetry condition (invariance) yields a formal description as a "classical"
Hamiltonian system. Thus for a proper choice (coherent states) the expectation
values of our operators obey classical equations of motion on a manifold (the
Riemann sphere ident i f ied with a two-sphere).
This is investigated in Section 2 where the basic concept of G-hamiltonian
systems is introduced and a connection between quantum Hamiltonian dynamics and
classical G-Hamiltonian systems (as submanifolds of the quantum space of states P(V)
is established. This is sometime (improperly) called a dequantization or a classi-
cal l im i t of the system. In Section 3 we solve the dynamical system with SU(2) symmetry. Exp l ic i t
analytic solutions are obtained but they are not investigated in detail here but
rather we discuss the phase portra i t changes with the variation of parameters.
These qual i tat ive features of the dynamical system are important in connection
with the correspondent quantum picture.
The quantization of classical phase spaces and observables (a realization of
the correspondence principle) is presented in Section 4. We also give a classifying
correspondence of quantizable elementary systems (for semisimple groups).The results
are used to find al l irreducible representations and their geometric realization for
compact groups (SU(2) - as an application) and the I~amiltonian dynamics is given on
these spaces by global d i f ferent ia l operators with an exp l i c i t realization for the
SU(2) model.
In this presentation we tr ied to give a self-consistent ( i f not self-contained)
account of the problem of the correspondence of classical and quantum systems (quanti"
zation) and also keep i t (this presentation)at an elementary level (gradually intro"
ducing new concepts). I f however, now and then, we have been forced to leave some
of the concepts and structures undefined, i t is our hope that the reader w i l l be
compelled to search for further information.
71
1. QUANTUM ALGEBRAIC SU(2) HAMILTONIANS
Some of the problems related to the quantization of classical
Hamiltonian systems with symmetry (on reduced phase space) can best
be analysed on a very simple model. We shall consider the family of Hamiltonians {~H a} defined by
2 2 2 ( I 1) H a : a l J I + a2J 2 + a3J 3 + a4J 3 ,
whose dynamica l group i s SU(2) ; t h a t i s the a l g e b r a i c m o d e l s l ) t h a t are
at most q u a d r a t i c i n i J k the g e n e r a t o r s o f SU(2) . They can
be g i ven in m a t r i x form for a 2j + i - d i m e n s i o n a l complex space V%¢ 2 j + i
of an i r r e d u c i b l e u n i t a r y r e p r e s e n t a t i o n o f SU(2) . The c a n o n i c a l bas i s
of V is denoted by { l j m > } ; m = - j , - j + l . . . . . j . W r i t i n g in the usua l manner
J+ : Jl ~ iJ2 ; Jo : J3
One has the commutation r e l a t i o n s
[Jo' J+] : ± J+ ~ [J+' J-] : 2Jo
With jo lJm> = ml jm>; J _ l j , - j > = 0 and the C a s i m i r o p e r a t o r
2 2 j2 = j + J2 + J3 : j ( j+1) ]I , j : O, i /2 , I . . .
The family of Hamiltonians (1.1) contains some of the schematic models
most extensively discussed in nuclear physics for their relevance to the Study of nuclear col lect ive motion. Indeed, these are the simplest
non-tr ivial models considered in the treatment of large amplitude mo- tions and the selection of a col lect ive subspace.
One can see easily that
i) the microscopic two level part icle-hole interaction model consisting of two single part ic le levels with the same degeneracy
(N) and the level distance E (Lipkin model)2)can be written
aS
H = ~Jo + ½ V(j2+ + j2). i r terms of the quasispin operators
N N J+ = z o p=1 ap' lap, '1 ~ Jo =s ~ ~ j < N/2 p=1 o=+_I aPo ap°
72
i i ) the semiphenomenological model (Bohr-Mottelson) 3) of a single
pa r t i c l e of constant angular momentum ( j ) in a quadrupole
deformed a x i a l l y symmetric potent ia l given by
2 H = Q J3 ~c J1 with Q = c t / j ( j + 1 )
and 4)
i i i ) the phenomenological model - the asyn~netric rotor
L 2 L 2 L 2 x z =
H = ~ x + + 7Tzz ( Ix > ly > I z,
are a l l of the form considered in (1.1) and for our purpose the ex-
p l i c i t r ea l i za t i on of d i f f e r e n t models is i r r e l evan t ( th is becomes
meaningful only for t he i r physical i n te rp re ta t i on in each case).
Given such a fami ly of Hamiltonians and a symmetry group one can
eas i l y show that there are subsets of vectors in the H i lber t space such
that an equivalent " c l ass i ca l " Hamiltonian system is obtained.
Indeed consider the fo l lowing complex parametrization of the
Bloch vectors5):
Z J IZ+> : e Z J+ l J, -J> ; IZ-> : e l J, J> , (1.2)
Z in ~; j > 0
t ions ( f i n i t e dimensional in the case of compact groups) of SU(2).
These mixtures of canonical basis vectors (wi th in a unirrep)
have certa in geometric propert ies (which we shal l invest igate in de-
t a i l l a t t e r on) and are cal led coherent vectors.
On such subsets of the H i lber t space the expectation values of
cer ta in observables exh ib i t a " c l ass i ca l " behaviour.
Indeed, consider the c lass ica l symbol, or energy funct ion, for
the Hamiltonian (1.1) - a polynomial in the generators of SU(2)
defined as the expectation value
<Z~IH~Z±> H+(Z.Z) : (1.3)
- < z ~ I z ~ >
w h e r e we d e n o t e
F(Z,~) : <Z±IZ~ : (1 + ZZ) 2j
( th i s is now a smooth function of Z and ~).
character iz ing the uni tary i r reduc ib le representa-
73
The dynamics of the system can be obtained apply ing the v a r i a - t iona l p r i n c i p l e to the ac t ion i n t e g r a l :
t " y S(y) : r t + ( Z ( t ) , Z-(t)) dt ; t + Z( t ) , (1.4)
t i - -
Where
L + ( Z ( t ) , ~ ( t ) ) = Z ~ ( Z ( t ) , Z ( t ) ) - H+(Z( t ) ,Z( t ) )_ (1.5)
is the " c l a s s i c a l "
~ , ( Z ( t ) , Z ( t ) )
(z
Lagrangian, and
aZ
being the time d e r i v a t i v e of Z) . The v a r i a t i o n a l p r i n c i p l e (T .D.V.P. ) 5 '6)
6S : 0 ( 1 . 6 )
Yields the Euler-Lagrange equat ions:
@L+ BL d (_Z_~) = _+ (1.7) d-~
Or
• BH± ( i . 8 ) i ~ (Z ,~ )Z : BZ
With
~(Z, Z) : ~21°gF 2j ( I 9 ~za~ ( i + z~) z
the symple~ ic form on the subset of coherent vec to rs . Hence, we can Obtain the Euler -Poisson equat ions:
Z = {Z, H+}
With the Poisson bracket def ined by
(i. I0
i +z~)2(~f ~g ~f { f , g } = - ~-~(! ~z ~z ~z ~z )
for any smooth f ,g of Z and Z f unc t i ons .
(i.il)
74
The phase space on which the expectation values have a classical beha-
viour is obtained as the set of al l vectors ~ - with the
constraint J~ + J~ + 1~ : j2. Thus f o r each i r r e d u c i b l e r e p r e s e n t a t i o n ( cha rac te r i zed by the
weight j ) one has
S~ : {J : ( J i , J 2 , ] 3 ) in ~31 2 2 2 .2 j J1 + J2 + J3 = J ) ~ (1.12)
where ] i ' J2' J3
J2' J3 of SU(2) given by
yC) <Z±IdkIZ±> -
<Z±[Z±> (.)2 jj+)2
+ J2 + :
are the expectation values of the generators J1'
j~-)2 + ]~_)2 + j~_)2 = j2
Thus the corresponding geometric objects are spheres of radi i ~j
(Bloch spheres).
In what follows we shall exploi t the d i f f e ren t i a l structure on S 2 _~j- which can be i l l us t ra ted in the fol lowing manner - le t ~ = j~ ;
x = (x i , x 2, x3), then:
S 2 S 2 ~3 ~ = i = {~ in Ix + x + x~ = i }
is the unit sphere. The local coordinates x I , x 2, x 3 are well defined
on the charts (given by the stereographic projections)
"~ 3 $ Z ~ c c 3 ~ X l ' N(LT,0,1)
c g, ",i, m " , a j
stereographic projection from the north pole N
stereographic projection from the south pole S
with the complex structure on the projection plane given by
75
x - i x 2 x I + i x 2 Z : i : ~ - i n ; Z ' - = g ' + i q '
i - x 3 r + x 3
In order that these char~s be compatible (holomorphically
the following condition must hold
( l . i3 )
joined)
Z Z ' = 1 f o r Z , Z ' # 0 ( 1 . 1 4 )
Up to this point we have established that there are subsets of vectors
in the Hi lber t space (chosen in a certain manner - which we shall esta-
blish l a t t e r on), so that in a formal way one can establish a corres-
POndence with classical Hamiltonian systems. The basic question we
Shall address in this lecture is that of precising this correspondence
(When certain dynamical symmetries are present) and solve the problem
completely for the simple case of SU(2). This generalizes readi ly for
the case of compact groups and can be established for certain represen-
tations of non-compact groups ~.e. those of the discrete ser ies).
Once such a corresoondence is established a possible analysis of
the dynamical system yields q u a l i t a t i v e l y (at least) the features of
the or ig inal problem considered (in the f u l l Hi lber t space). A very
important point is that the dynamics must be considered on the whole
Phase space (S~j-__ in our case) since we can not r e s t r i c t the t ra jecto-
ries on a single chart. That is we must look at objects that are glo-
bal (everywhere smoothly defined). In our case one can easi ly check that
H+(Z,Z) : H_(Z I , ~ ' )
~jZ = ~jZ'
That is a l i s t of global objects comprises:
• the phase space S 2 • ~ j ' • the energy function ~ ;
• the Poisson bracket { J i ' J j } = ~ i j k ]k
• the Poisson equations: ~k = {Jk 'H} ; k = 1,2,3
While othersare local (only defined in each chart) such as:
canonical coordinates (action - angle var iab les) ;
76
Lagrangians
• the action functional S(y)
T.D .V .P.
coherent vector sets {IZ+>}
{ IZ->} respect ively.
Since the equations of motion have been obtained from the var ia t ional
pr inc ip le this poses the question of the i r extension since the var ia-
t ional pr inc ip le is only loca l l y val id (on each chart).
We see thus that not any subset of vectors (or states) can
y ie ld a corresponding classical Hamiltonian system. For the moment we
shall leave this question open but rather t ry to describe ( b r i e f l y ) the
classical systems that are quantizable. A c lass i f i ca t ion (description)
of such systems w i l l contain the phase space we have obtained in
the SU(2) case.
In order to describe more precisely (from the mathematical point
of view) such systems we shall need a resum6 of some results of
symplectic geometry.
2. HAMILTONIAN SYSTEMS
The results of the previous section show that for a "good choice" the expectation values of certain quantum observables obey "classical" equations of motion. Also these state - vectors form a subset in the
Hi lber t space that can be formally put into correspondence with a
"classical phase space" (symplectic manifold).
Since these are not accidental properties of the algebraic
system considered (1.1) we shall t ry to b r i e f l y review some of the 7)
results of symplectic geometry. Our main task w i l l be to define what
a Hamiltonian system means and to establish the moment ma~ for such
systems which is essential in the study of co l lec t ive motionS)Later on
we shall use these results by ident i fy ing several such structures in
a chain that y ields a correspondence pr inciple between classical and
quantum systems.
We shall s ta r t by reminding that a sympl¢ctic manifold is a
2n-dimensional smooth manifold M (you may think of the phase space)
with a given symplectic structure m that is a 2-for~ which in local
coordinates (x 1 , . . . , x 2n) has the expression
77
I dx i A dx j : Z ~ i ~ i , j J
Where ~ i j is a skew symmetric tensor (you may think
element), with the fo l lowing propert ies:
(2.1)
of i t as a surface
i ) m is a closed 2-form:
amij amjk Bmki + • + ..... . = 0 ; (2.2)
a x 1 a x j
i i ) ~ is non-degenerate
det (~ i j ) # 0 (2.3)
Although the term phase space is commonly used to describe sym-
Plectic manifolds of the type { ( q 1 ' ' ' ' ' qn' Pl . . . . . pn)} (cotangent
bundles) for an n -pa r t i c l e system we shal l henceforth use i t for
SYmplectic manifolds ( in general) that do not necessari ly have an under-
lying (global) conf igurat ion space. Otherwise stated th is means that
there is no global i form 8 (symplectic po ten t ia l ) such that ~ = do
(m =Z i dPi A dqi ; o = zip i 4qi in local Darboux coordinates).
In the framework of classical mechanics the c lass ica l state of a System is a point m in M while the c lass ica l observable is a smooth real-valued function f on M, The set of al l real-valued smooth functions deno-
ted by F(M) forms an algebra with the Poisson bracket defined as:
i j af ag { f , g } = s m xi . for any f ,g in F(M) (2.4)
i , j ~ ax j
Where ( i j ) - the inverse o f ( m i j ) i s well defined since m is non-de-
generate. For any smooth function h in F(M) (the Hamiltonian is
Such an observable) one may define a Hamiltonian vector f i e l d X h of F(M) into F(M)such that
Xh(f) = { h , f } (2.5)
that is a l inear ( d i f f e r e n t i a l ) operator whose expression in local coordinates is given by
i j ah X h = z
i , j a x I a x J (2.6)
78
We s h a l l denote the se t o f a l l H a m i l t o n i a n v e c t o r f i e l d s by Ham(M)
and n o t i c e t h a t i t i s a l so a L ie a l g e b r a .
The Ham i l t on e q u a t i o n s o f mo t ion are r e a d i l y o b t a i n e d in the
form
= { f , h } : - X h ( f ) ( 2 . 7 )
We can nex t d i s c u s s the sequence of L i e a l geb ras
+ F(M) ~ Ham(M)
where ~ i s the se t of a l l c o n s t a n t s ~ n c l u d e d in F(M)~ and u a
homomorphism of L ie a l g e b r a s d e f i n e d as u ( f ) = Xf such t h a t
[ X f , Xg] = X { f , g } f o r f , g in F(M) ( 2 . 8 )
and f o r any X in Ham(M) t h e r e e x i s t s some f u n c t i o n f i n F(M) such
t h a t X = × f .
I f Xf = Xg then f - g is c o n s t a n t ( i . e . Ham(M) ~ F (M) / ~ ) .
Tha t i s the se t o f a l l H a m i l t o n i a n v e c t o r f i e l d s i s i s o m o r p h i c to the
se t o f c l a s s i c a l o b s e r v a b l e s modulo some c o n s t a n t .
We nex t c o n s i d e r the a c t i o n o f a L ie group G on the s y m p l e c t i c
m a n i f o l d (M,m) . Here G is the symmetry group of the model c o n s i d e r e d
and ac ts on M by c a n o n i c a l t r a n s f o r m a t i o n s (wh ich p rese rves m ) .
We s h a l l c a l l t h i s a H a m i l t o n i a n G - a c t i o n i f the d iagram o f L i e
a l g e b r a homomorphisms commutes
p F(M) , Ham(M)
~ - L i e a l g e b r a of G.
o r
p ( x ) , x ( x ) : ~ ( x )
x
79
The map v o f ~ into Ham(M) gives a rea l izat ion of the Lie algebra ~ of
G in terms of l inear d i f f e ren t ia l operators:
d f ( g ( t ) m ) (2 .9) ( ~ ( x ) f ) (m) = ~ 't=O
With m in MI f in F(M), x in ~ ; g ( t ) : e x p ( - t x ) in G. g ( t ) is the ca-
nonical t r a n s f o r m a t i o n corresponding to g ( t ) . Then
[ v ( x ) , ~ ( y ) ] : V [ x , y ] X,y in
Thus ~(G) is a L ie a lgebra of canonical t r a n s f o r m a t i o n s .
On the other hand
(2.10)
. ( p ( x ) ) : Xp(x) : ~(x)
(x,y in ~ ) , {p(x) , p(y)} = P[x,y] (2.11)
Hence ~(~) is the Lie a lgebra of generators fo r the Hami l ton ian G-ac t i on .
In general there may be obstructions (of cohomology type) to
the existence of the homomorphism p. However in our case (compact
groups) and in the more general case of semisimple Lie groups8)j
the conditions for the existence of p are f u l f i l l e d . We shall cal l
a Hamiltonian G- space (M,~,p) a symplectic manifold with a Hamiltonian
G-action. A Hamiltonian G-space is elementary i f the action of G is
t ransi t ive ( i f m and m' are .ooints of M, then m' = ~m for some g in G).
Since in our example the manifold M = SU(2)/U(1) ~ S 2
is a homogeneous space} the action of SU(2) on S2~is t rans i t i ve . This
also means that M is a single orb i t of G or M = ~m with m in M for an
elementary G-space. A canonical map m of symplectic manifolds)
from M into M' such that y = m(x) for x in M and y in M' changes
the symplectic form by
@Yi ~Yj (2.12~ m, kl(y) = s miJ(x) i , j ~ ~Xl
is a canonical isomorphism i f i ts inverse is canonical.
80
We shall next define the moment map which allows us to establish
a correspondence between d i f fe rent }~¢quivaZent ~ Hamiltonian systems.
I t plays a crucial role in defininq co l lect ive motion since (in Qenerall
the reduced Hamiltonian system has fewer degrees of freedom (col-
lect ive var iables).
Indeed, consider a Hamiltonian G-system (M,~,p) a~d the map ~ of
M into ~* , where ~ ~ is the dual of the Lie algebra ~ of G
( i . e . the linear space of a l l real-valued l inear functions on ~ ),
defineG by
@(m)(x) : p(x) (m) fo r m in M , x i n ~ • (2 .13)
(that is the value of the l inear functional ~(m)in ~* at the element
x of ~ is that of the corresponding function p(x ) in F(M) at the point m of M).The map ~ is called the moment map.
Thus, without going into detai ls, a colZective Hamiltonian H
is always defined as H = h o ¢ for a smooth function h on ~* (in our
case we can iden t i f y ~* with the real d-space ~d ~* ; d = dim ). Given the function h and the moment map ~ one obtains through
the homo~orphism ~ defined above the corresponding Hamiltonian vector
f i e l d X h. Thus i f m(t) is the integral curve of the Hamiltonian system
X h in M then one can see that the image through ¢ of this t ra jectory
l ies en t i re ly on the orb i t 0 (m) and i t is a solution of the Hamilto-
nian system rest r ic ted to this orb i t .
The coadjoint orb i t 0 through the point ~ of ~ ~* is defined bY
0 : {Ad*(g)~ Ig in G} (2 .14)
where the c o a d j o i n t ac t i on of G on ~* g iven by
( A d ' ( g ) ~ ) ( x ) : e ( g - I x g) f o r x in ~ • (2 .15)
Then one can de f i ne the symplec t i c formson c o a d j o i n t o r b i t s in the
f o l l o w i n g manner: the l i f t of ~ to Hami l ton ian vec to rs at the o r b i t
through B is g iven by
d~ f(ad*(g(t~B) (2 16) ( v ( x ) f ) ( B )
t=Q wi th x in G, f a f u n c t i o n in F ~ * ) and B in ~ * .The symp lec t i c form is
~ ( v ( x ) , v ( y ) ) : ~ ( [ y , x ] ) (2 .17) 9,1o)
Coad jo in t o r b i t s are e lementary Hami l ton ian G-spaces.
81
I f (M,m,p) i s an e lemen ta ry G-space f o r a s imp l y connected L ie group G,
then the moment map is a c o v e r i n g map from M to a c o a d j o i n t o r b i t 0
With r e s p e c t to the K o s t a n t - K i r i l l o v - S o u r i a u s y m p l e c t i c form ~ I i , ~ 2 )
A ve ry i m p o r t a n t r o l e in what f o l l o w s is p layed by the project ive Space which we show has a s y m p l e c t i c s t r u c t u r e and a l so has a complex
S t r u c t u r e . With the cho ice (1 .2 ) f o r the v e c t o r s we have c o n s t r u c t e d
Such a space in Sec t ion i .
Let {e i } be a complete o r thonorma l system in a sepa rab le complex
H i l b e r t space V such t h a t ~ = ~ a j e j and d e f i n e the s y m p l e c t i c form
~V : i z da k A da (2 .18 ) k k'
For e lements ~ in V\{O} c o n s i d e r an e q u i v a l e n c e r e l a t i o n such t h a t
[~] = { I U I L in { ; ~ # O} i s an e q u i v a l e n c e c l a s s , then the se t P(V)
of a l l e q u i v a l e n c e c lasses [~] i s c a l l e d the project ive space of V.
We nex t show t h a t P(V) i s a complex m a n i f o l d d e f i n i n g l o c a l coor - 13) .
d i n a t e s ( w k ) i n the f o l l o w i n g manner
l ak/a i k < i
Wk = f o r i f i x e d k > i
a k + i / a i
Then we can d e f i n e a system of cha r t s in P(V)
(2.19)
Qi = { [ @ ] l a i = < e i ' ~> # 0 } .
I f V i s a f i n i t e d imens iona l v e c t o r space, say cn+ l ,we see t h a t p ( { n + l ) =
cpn is a compact complex m a n i f o l d w i t h ~i ~ $n.
(For n= l~{p I i s the c l a s s i c a l Riemann sphere i somorph i c to $2) .
A complex s y m p l e c t i c m a n i f o l d such t h a t the s y m p l e c t i c form
is l o c a l l y g i ven by ~2 F
: i z gkk ' dZk A d~ k, ' gkk' ~ (2 .20 ) k , k ' ~Zk ~Zk'
is a K~hler m a n i f o l d . P(V) has a l so a s y m p l e c t i c s t r u c t u r e g i ven by the
( l o c a l ) K~h ler p o t e n t i a l 13):
F = I n ( l + ~ w k Wk ) (2 .21 ) k
Cons ider now a u n i t a r y i r r e d u c i b l e r e p r e s e n t a t i o n T of a L ie
group G on V.
82
We see t h a t t h e r e i s a G - a c t i o n on P(V) g i ven by
g [~ ] = [ T ( g ) ~ ]
f o r any g in G and ~ in V \ { O } . Moreove r , i f g = e x p ( t x ) w i t h x i n ~ ,
we have
exp( t~) = T(exp tx )
( t h a t i s x i s an i n f i n i t e s i m a l g e n e r a t o r o f T ( G ) ) . Then we can d e f i n e
the homomorphism p o f the L ie a l oeb raU~ i n t o F ( P ( V ) ) by the smooth
f u n c t i o n p ( x ) onP(V) such t h a t
p ( x ) ( [ ~ ] ) = < i ~ (2 22)
Then ( P ( V ) , m , p ) i s a H a m i l t o n i a n G-space. The complex and symp lec -
t i c s t r u c t u r e s are c o m p a t i b l e on P(V) . Thus the p r o j e c t i v e space has
the s t r u c t u r e o f a K~h le r m a n i f o l d .
Since P(V) i s a H a m i l t o n i a n space one can r e a d i l y o b t a i n the e q u a t i o n s
o f mo t ion i n the form
w i t h
• BH
i z gkk' Wk' - (2.23) k' ~Wk
<~IHI~> (2.24) H - <~I~ >
f o r the H a m i l t o n i a n H on V.
We r e c o g n i z e i n t h i s f o r m u l a t i o n the r e s u l t s o f S e c t i o n 1o
N o t i c e t h a t s t a r t i n g w i t h an i r r e d u c i b I e r e p r e s e n t a t i o n o f SU(2) on
V = C n (n = 2j + 1) we can r e o b t a i n the e q u a t i o n s of mo t ion ( I . 8 ) .
S ince the space phase o b t a i n e d in s e c t i o n 1 was f o r each u n i r r e p
the sphere S 2 ( w h i c h can be d e s c r i b e d us ing the c o o r d i n a t e s of the ~J 2
u n i t 2 - sphe re S ) i t i s o f i n t e r e s t to e s t a b l i s h the n a t u r a l connec-
t i o n between the p r o j e c t i v e space { p l and S 2.
I ndeed , as above (2 ,19 ) we c o n s i d e r { p I as the un i on
o f the c h a r t s ~ I and ~2 d e f i n e d by
83
~i : { [ a l ' a2] in (~p1 la 2 ~ O} ~ $
With ( a l , a 2 ) in {2 and the map ~I of ~I
a 1
Z I = ~ in ¢ ~ ~ i [ a i , a2] =
and
onto $1 g iven by
Z 1
(2 .25)
Q2 =' { [ a l , a2] in ¢pi la I # O) ~
With the map ~2 of ~2 onto ~ defined by:
a 2 Z2 = ~11 in C; ~2 [a l , a2] = Z 2 (2 .26)
Thus on i~ ! and s~2 we have the complex coo rd ina tes Z 1 and Z 2, r e s p e c t i -
Vely, and the complex a n a l y t i c (ho ]omorph ic ) t r a n s f o r m a t i o n of c o o r d i -
nates Z2 = Z ; Z I # 0 ( f o r [ a l , a 2 ] in the i n t e r s e c t i o n ~12 i and ~2}" i i o f
Th is can be i l l u s t r a t e d by the diagram
[a I , a 2 ] in ~12
in ~ \ { 0 } ,Z 2 in C\{O}
Then the canonical isomorphism between ¢p1 and S in the f o l l o w i n g way.
We l e t
and
2 can be establ ished
2ReZ I 21mZ I I - IZ112 [ Z l , I ] -~ x 1 = ( ~ , - ,
I+ZIZ I I+ZIZ 1 1 + IZ l Iz)
(2ReZ 2 21mZ 2 [ i , Z2] ~ x : ~ ,
I+Z2Z 2 I+Z2Z 2
x I in
12 (2 27) IZ 2 - 1 1 +IZ2 IZ ); x2 in IR 3 ,
84
as can be easily checked
x I in $2\{0, O, - I }
and
x2in $2\{0, 0, 1}
gives the south pole stereographic projec-
tion for Z 1 in
gives the north pole stereographic projec-
tion for Z 2 in
The fact that the map x1 o 7; I is holomorphic on ~IZ
( i . e . ZIZ 2 = i) yields the re lat ion x I = x 2 on the intersect ion of
the north and south pole charts. Thus we have established the isomorphism between $p1 ( a
K~hler manifold with symplectic structure defined above by (2.!8))and
the unit 2 - sphere S 2
Simi lar ly , one can establish d i rec t ly (checking the geometric
structure) the isomorphism between the project ive space {p1 and 0 the quotient space SU(2)/U(1) where U(1) = {(~ T) Icl = I} is
the set of diagonal matrices. One can see that
[Z 1, I ] ~ u1(Zl) U(1)
[ I , Z 2] + u2(Z2) U(1) (z.2~)
with uI(Z1), u2(Z2)in SU(2) of the form:
I+IZI12 71 i
(:) = I / 1 2 - I u2(72 )
V 1+]Z212 Z2
(2.29)
The fact that the two spaces correspond (are isomorphic) represents
only an example of the c lass i f i ca t ion of quotient spaces G/K}where G
is a Lie group and K is the centralizer of a torus (product of circle groups).
Such quotient spaces are called f lag manifolds, for a compact group G 14,15) the generic example being
U(n)/U(1)x . . . x U(1) is the manifold of flags
{0 = VoC V I c . . . ~V n = sn } in complex n-spaces,where V i are complex subspaces of dimension i .
85
Projective spaces and Grassmann manifolds are f lags of type G/C(T)
(where T is a c i r c l e subgroup and C(T) are a l l elements g in G such that
gt=tg for any t in T). A l l inequivalent unirrep of G are c l ass i f i ed by SOme G/T with T = C(T).
There is also a descr ipt ion of f lag manifolds as quotients of
the corresponding complex groups G C and the complex and symplectic
Structures are compatible such that they are homogeneous K~hler mani- folds 15 ) , then one can regard them as (complex) Hamiltonian G-spaces.
In fact f lag manifolds completely c l ass i f y the coadjoint or-
bits of G which carry a natural symplectic form (see above (2.17)) ,
that is each o rb i t is a f lag manifold and a l l f lag manifolds appear as
Coadjoint o rb i t s .
Flag manifolds can be covered by open dense neighborhoods
(Borell ce l ls ) 16! i d e n t i f i e d with complex Euclidean spaces.(The K~h-
le t phase spaces for noncompact semisimple groups are given in 16, ~7)).
We shal l now i l l u s t r a t e th is discussion by i den t i f y i ng (isomor-
Phical ly) {p1 with the complex f lag manifold F = SL((2, ¢)/P where
a b la ,b,c ,d in ¢; ad bc 1} SL(2, { ) : { ( c d ) - :
subgroup
u
P : { 0
and P is the parabolic
v _l) lU,V in C , u ~ 0 } (2.30)
u
then
[Z I , I ] + g l ( Z I ) P
[ i , Z2] ÷ g2(Z2)P
With g l ( Z l ) , g2(Z2) in SL(2, C) of the form
Z I -1 g1(Zl) : )
i 0
I 0 g2(Z2) = )
Z 2 i
and the ana ly t i c transformation of coordinates expressed as:
(~.3L)
(2 .32)
86
Z lZ 2 : i <=> g ~ l ( Z l ) g 2 ( Z 2 ) : Z 2 1
( ) in P. (2.33) - I + ZIZ 2 Z I
We shall l a t t e r on make use of this parametrization in the construc-
tion of the unitary i r reducible representations of SU(2). 2 F i n a l l y , we see t h a t S Is endowed w i t h a complex s t r u c t u r e
( c o m p a t i b l e w i t h the s y m p l e c t i c s t r u c t u r e ) and can be regarded as a
compact homogeneous K~h le r m a n i f o l d , The complex p a r a m e t r i z a t i o n o f
S 2 can be bes t v iewed on the f o l l o w i n g d i a g r a m :
unit 2-sphere M=S 2 i complex_ F=SL(2,C)lPflag monifo~
unit 3-sphere S 3 ( ~ 1 ) = SU(2)IU(1)
where i denotes he isomorphisms established above, ~ is the inclu-
sion map (of S 3 - the unit 3-dimensional sphere in the 2-dimen-
sional complex space ¢2) and p is the projection of ¢~{0} onto tP I .
We have seen that the project ive space P(V) is a K~hler manifold' There ex is t K~hler embeddings of f lag manifolds into projective spa-
ces 23). We recal l that a K~hler isomorphism is a biholomorphic canoni"
cal transformation and a K~hler embedding gives a K~hler isomorphism
onto a K~hler submanifold. The complex submanifolds of P(V) are
K~hler 13) The isomorphisms of the preceeding diagram are K~hler.
On the other hand, the projective space P(V) of an i n f i n i t e di-
mensional separable complex Hi lber t space V of quantum state-vectors
is a K~hler manifold with symplectic structure (2.20) induced by the
t rans i t ion probabi l i ty 18-21) We now r e s t r i c t our discussion to a
connected, simply connected, semisimple Lie group G.
87
We can thus show that quantum mechan ics has a H a m i l t o n i a n s t r u c -
tu re . Indeed, l e t V be the separab le complex H i l b e r t space of s t a t e -
Vectors assoc ia ted to a quantum mechanical system. Then the imaginary
Part of the i nne r product in V de f ines a symp lec t i c form m V on V 18)
As i t is known, quantum observab les correspond to s e l f - a d j o i n t
Operators on V and thus, a l i n e a r ope ra to r A on V is s e l f - a d j o i n t i f
and Only i f iA is a complete Hami l ton ian vec to r f i e l d (w i th mV -
re la ted to the d i f f e r e n t i a l of the e x p e c t a t i o n value of A in V).
MOreover a l i n e a r opera to r U which is u n i t a r y ( leaves i n v a r i a n t the
inner product of V) is symp lec t i c w i th respec t to m V.
Then the dynamics is descr ibed by a Hami l ton ian f low r e a l i z e d
by an one-parameter group of u n i t a r y opera to rs U( t ) = e x p ( i t ~ - I H ) ,
Where the s e l f - a d j o i n t opera to r H is the quantum Hami l ton ian of the
system. Consequent ly , quantum dynamics can be regarded as Hami l ton ian
dynamics f o r the Hami l ton ian system (V,~v,H) !8)
HoweverI the p h y s i c a l l y s i g n i f i c a n t space is t ha t of quantum pure
States which is the p r o j e c t i v e space P(V) .
The next step in our c o n s t r u c t i o n (or r e d u c t i o n ) is to cons ider SUbmanifolds of P(V) (which preserve the r i ch s t r u c t u r e of P(V)) and
i d e n t i f y the e lementary systems.
As we have seen a u n i r r e p T of the dynamical group G on V
induces a Hami l ton ian G-ac t ion on P(V) , such tha t the ac t ion of G on
P(V) is g iven by K~hler automorphisms{which is e q u i v a l e n t to the con- d i t i o n tha t t r a n s i t i o n p r o b a b i l i t i e s are i n v a r i a n t 19-21)}.The quantum
Observables are g iven by e x p e c t a t i o n va lues .
The ac t i on of G in t roduces an o r b i t s t r u c t u r e on P(V) w i th the o r b i t
through the po in t [~] g iven by 0 [ ~ ] : {g [~ ] I g i nG} . I f i t is a
~bmanifold of P(V) we cal l the o rb i t O[~]thc coherent state manifold 22) and denote i t by M. As we know the orb i t O[¢]is defined as the coset Space
0[~] ~ G/G[~]= {gG[~ ] Ig in G} ) (2 .34)
Where G [ ~ ] i s the i s o t r o p y group of the po in t [~] in P(V) , t ha t is the
Set of a l l elements tha t do not change [4]
G [~ ] : {glg in G such t ha t ~ [ } ] = [ } ] } (2 .35)
and t h e ( l e f t ) coset gGr~lis the set of a l l e lements of the form gg' w i th LTJ24~ 9' in G[¢](g in G f i xed) , J
The coherent state manifolds O{ ](4 in V) become Hamiltonian
j and p' = plo[~ ] , G'Spaces with m' = ~ 0[~]
88
However, i f G is compact, there is a unique orb i t 0 o which is
a K~hler submanifold of P(V) and such that (Odm',p') is an elementary
Hamiltonian G-system. This orb i t is the one that passes through [~o ] 23) in P(V) , where ~o is a maximal weight vector of representation T.
Then the coherent states sa t is fy the de f in i t ion of Perelomov 24) for ~o"
We have already encountered such elementary Hamiltonian G-spaces
when we have considered the moment map ~: M - ~ , (they are coadjoint
orbits 0~. I t is natural to look for a c lass i f i ca t i on of our systems
in terms of coadjoint orbits (or f lag manifolds).
On the(unique)orbit 0 o the natural symplectic structure (2.17) is
precisely that res t r i c t i on induced by the symplectic structure on P(V).
I t is thus a K~hler manifold (~ ' is induced by the imaginary part of the
hermitian structure on P(V)) and can be ident i f ied with a flaq mani-
fold ( f o r compact g r o u p s ) .
We can now e s t a b l i s h an e x p l i c i t c o n n e c t i o n between e l e m e n t a r y
systems in quantum and c l a s s i c a l mechan ics ,
Indeed , i f V i s the ( i n v a r i a n t ) subspace of a u n i t a r y i r r e d u c i b l e
r e p r e s e n t a t i o n T o f the L ie group G ( t h a t i s P(V) i s an e l e m e n t a r y
quantum space) than one can see t h a t t h e r e i s the f o l l o w i n g d iagram o f H a m i l t o n i a n G - spaces:
( v \ { o } . . V) g (P (V ) ,~ ) ~- (~=0[~,~ ' ,~ , ) ~ (M = 0~o ,~ ,o )
F i n a l l y , the c o r r e s p o n d i n g phase space i s the o r b i t i n ~ * t h r o u g h the
r o o t ~o " c o r r e s p o n d i n g to the maximal w e i g h t v e c t o r ~o o f G 25) .
N o t i c e t h a t the c o r r e s p o n d i n g c Z a s ~ i c a l system c a r r i e s an i n v a r i a n t
g l o b a l s t r u c t u r e ( i n f a c t has p r e s e r v e d a l l the r i c h s t r u c t u r e i n P(V))
and thus a l l i m p o r t a n t q u a l i t a t i v e f e a t u r e s of the quantum system may
be o b t a i n e d ,
We remark t h a t the V a r i a t i o n a l P r i n c i p l e on v e c t o r - s t a t e s (on V)
y i e l d s the same e q u a t i o n s on 0 o, however they are o n l y l o c a l ( t h e r e is
an o b s t r u c t i n n to the e x i s t e n c e o f a g l o b a l exac t s y m p l e c t i c form
wh ich appears as a phase o f @),
A l t h o u g h the p r e s c r i p t i o n f o r the co r respondence o f e l e m e n t a r y sY s"
tems g i v e n above seems necessary (unde r c e r t a i n c o n d i t i o n s on G and
G/ K 17) ) one must ask w h e t h e r i t r e a l l y e s t a b l i s h e s a c o r r e s p o n -
dence p r i n c i p l e , t h a t i s i f we can o b t a i n a l l e q u i v a l e n c e c l asses o f
i r r e d u c i b l e u n i t a r y r e p r e s e n t a t i o n s f o r the group G s t a r t i n g w i t h o r b i t s o f ,
T h i s we s h a l l deal w i t h i n S e c t i o n 4.
89
For now, we want to have a look on the dequan~tizaJcion diagram in the case of
the algebraic system with SU(Z)-symmetry. Since the unitary irreducible representa-
tions of SU(2) are all finite-dimensional (SU(2) is compact), the ~eme~Ya~y quantum
System is a cp2J-space and we see that the diagram becomes
~2j+1 p ~p2j ~ ~ ~ M. J
0
~ Hj ~ {pl ~, S,~j ~ SL(Z,¢)/P ~ SU(2)/U(1)
Thus the coherent state manifold F¢ is a dcfo~u~e~ sphere embedded into a projective
SPace ~p2j One can perhaps better understand the situation i f one notices that the
Projective space P(V) (as a Hamiltonian G-space) admits a moment map ~ of P(V) into
and that the Hamiltonian flow lies entirely on the orbit in ~ ' . The classical
Phase space Mj can be embedded into {p2j such that the complex structures match, i t
cannot be embedded (as a submanifold) in £2j+! since the complex structures do not
match (they are rigid). This is another manifestation of the phase obstruction and
explains why the set of coherent vectors (1.2) can not be joined by a holomorphic
transformation and thus do not form a symplectic complex manifold in V. Furthermore,
one can see why the Variational Principle (on coh~a~t vector) gives only local equations of motion.
The main results of this Section are two. First, we showed that quantum col-
lective dynamics can be expressed in an explicit Hamiltonian form on coherent state
SUbmanifolds of the quantum projective space (endowed in a natural way with the
K~hler structure induced by transition probabilities). Second, we have shown that
these manifolds of coherent states are images of certain elementary G-spaces with
respect to dequantization maps (realized by K~hler isomorphisms). According to tl~e
Kiri llov- Ko stant-Souri au construction of moment maps, the elementary G-spaces are
reduced phase spaces for classical collective models. The remarkable symplectic,
Complex and algebraic structures of these G-spaces are transported on coherent
State manifolds (for the discrete series of any semisimple group S). Then the re-
Sults of quantum collective dynamics can be precisely expressed in terms of clas-
Sical Hamiltonian dynamics (without any variational principle or functional inte-
goal). An explicit realization of the Hamiltonian dynamics for the collective
SU(2) _ models is given in the next Section.
90
3. DYNAMICS ON TWO-SPHERES
In th is sect ion we consider the dynamics of the Hamiltonian sys-
tem (1.1) on P(V) reduced to the coadjoint o r b i t s i n ~* ( that is
on two - spheres) Since, as already stated we are not p r ima r i l y
in terested in various appl icat ions of the Hamiltonian system in nuclear
physics (these are to be published elsewhere)~7~e~ " shal l only make a
short presentat ion of the q u a l i t a t i v e aspects of the dynamical problem,
emphasizing the points of i n te res t for our discussion on quant iza-
t ion of Hamiltonian systems (Section 4).
However, before passing on to the classicaZ dynamical system
we shal l s imp l i f y the Hamiltonian (1.1) by f u r t he r symmetry considera-
t i ons . Indeed, for various spec i f i c re la t i ons between the parameters
a the invar iance groups of the Hamiltonian are subgroups of
SU(2), The Hamiltonian (1.1) is t r i v i a l for a I = a 2 = a 3.
Thus fo r
la I a31 ~ la l a21% al # a3% a4(a I - a3) a D , (3 .1 )
the Hamiltonian (1.1) reduces to the form
H = (a I - a3 ) (J + + 2VoJ 3) + a3J
with
a - a
2 3
a 1 - a 3 a 4
v 0 = ~-T~ I - a3~
and the condi t ions
(3.2)
- I ~< u < 1 ; v o > 0 (3.3)
In what fo l lows we shal l consider the corresponding dynamical system
for
L(U,Vo)= J~ + uJ~ + 2VoJ 3 (3.4)
(the crancked Arnold rotor26~which although simple contains a l l the
i n t e res t i ng models mentioned in Section 1 as subsystems ( fo r spec i f i c
values of the parameters u, v) . Indeed, we have
91
i ) for v o > O, u = 0
i i ) u = -1, v o > 0
i i i ) u # O, 1~ v o = 0
i v ) u = v o = 0
v) u = i , v o = 0
v i ) u = 1, v o > 0
These cases give a model
the Bohr-Mottelson model Hamiltonian
for the par t i c le -core in te rac t ion
the Lipkin model - the asymmetric t r i a x i a l ro tor model
- a x i a l l y symmetric ro tor with oblate
deformation
- a x i a l l y symmetric rotor with prolate
deformation
the system with ax ia l symmetry.
descr ipt ion of d i f f e ren t extreme s i tua t ions
( invariance subgroups and spectral propert ies) and the i r un i f ied des-
c r i p t i on , given ~ L(U,Vo)allows for the descr ipt ion of the t r ans i t i on
between them, and thus hopeful ly for a bet ter understanding of the
mechanism responsible for d i f f e r e n t possible phase-l ike t r a n s i t i o n .
One should notice that there is a d iscrete transformation
Which changes J1' J2 into -J1' -J2 .(There is an invariance group for
L (U,Vo)" We shal l discuss the impl ica t ion of i t at the end of Section 5 ).
Making use of the resu l ts of Section 2 for the expectation
Values on coherent states of the operators JK(k=1,2,3) we can read i l y 2 Obtain the c l a s s i c a l H a m i l t o n ~ a n H (on S~j):
H = j ( j - ½)h + ~( I + u) (3.5)
?hen on the uni t sphere (using the real parametr izat ion) one has the
fol lowing Hamiltonian funct ion
h = x 2 I + ux + 2vx 3 (3.6)
and the condit ion
2 x I + x~ + x~ : I (3.7)
Which describes the two-dimensional surface S 2 in ~3.
The Hamiltonian (3.6) defined on S 2 (the phase space) depends
On 2 control parameters u and v which (due to somesymmetry conside-
rations) are res t r i c ted to a subset K of ~2:
K = { (u ,v ) in {R21 -1 < u < I , v ~ 0 }
92
Given the (global) symplectic st ructure on S 2 one obtains a Hamiltonian flow (Xh)
on the phase space y ie ld ing the non-l inear equations of motion:
I Xl = 2x2(ux3 - v) Xk : {x k, h (u,v)} (:=~ xz =-2x1(x3 - v) (3.8)
k = %,2 ,3 ~3 = 2 ( ± - u ) XlX 2
(Notice that is no need to use a Variational Principle in order to obtain (3.0)).
Since a Hamiltonian system is conservative(h is a constant of motion)one may
picture the fol iat ion on S 2 induced by theHamiltonian flow in the following way:
the trajectories are curves obtained by intersecting the unit 2-sphere S 2 wi th
the Z - dimensional surface
x~ + ux 2 +2vx 3 = E (3.9)
and E is the energy forthe resulting solution curve.
With conditions (3.7) and (3.9) the set of nonlinear equations becomes a
{one-dimensional) completely integrable system. Integrating the system one obtains 27)
the integral curves x = f ( t ,E ,u , v ) as analyt ic expressions in terms of Jacobian e l l i p t i c funct ions 28) and the period spectrum T(E,u,v) is given by e x p l i c i t formulae
( fami l ies of periods T I and T2). The analysis of how the integral curves of the Hamil"
tonian system change with control parameters is b r i e f l y explained below in %his case.
[See Appendix for e x p l i c i t formulae).
The c r i t i c a l s t ructure of the Hamiltonian system gives an algebraic s t r a t i f i -
cation of the parameter set , Strata are sometimes interpreted as def ining d i f f e ren t
phases. This re f lec ts us the existence of s ingu la r i t ies of the phase por t re t (the
space of integral t ra jec to r ies ) . The phase space is thus topolog ica l ly characterized
by domains D i which are manifolds of per iodic orb i ts (Appendix) separated by
separatr ices. Cr i t i ca l behaviour ( i . e . Morse theory) is extremely useful in f inding
some per iodic solut ions since around c r i t i c a l points l inear s t a b i l i t y assures
the i r existence. Notice that ~ propert ies established for the system (3.8) can
be generalized to other integrable systems of the same type (~geO~Le.aZly comple- t ~ y imvteg~blel. Indeed, by Arnold-L iouv i l le theorem, the connected compact inva- r ian t manifolds are real tor i for most values of the energy and control parameters.
In our case the stat ionary ( c r i t i ca l ) po in t s of h(x ,u,v) are given by the
set of equations (3.7) and
x2(ux 3 - v) : O; Xl(V - x3) : 0 ~ ( I - U)XlX 2 : 0 (3.10)
A simple analysis of the c r i t i c a l s t ructure shows thus that there are
I) ( iso lated) non-degenerate c r i t i c a l points (maxima,minima, sadd le p o i n t s ) .
- l i near iza t ion and existence of per iodic solut ions is assured.
2) degenerate c r i t i c a l points (give b i furcat ion and catastrophe sets).
93
3) manifolds ( c i r c l e s ) o f degenerate c r i t i c a l points
~ith va r ia t i on of parameters u and v in the control space K the character of a c r i t i c a l point changes. C r i t i c a l energies are
Values of h at c r i t i c a l points.
C r i t i c a l points and c r i t i c a l energies for the Hamiltonian fami- ly {h} are represented on the diagram on parameter space. The
b i furcat ion set ( tha t is a l l points in K for which degenerate c r i t i -
cal points ex is t ) is represented by dashed l ines .
lul ~< 1 "u" 't.Y >~ 0
~ = 1
\
~0~ E_ \
m
E 2 E_ E+ m s s " " ~3' E 2 E_
\
E 1 \,, M " • • %.
0
E1 Q E1 m s M I..~
E_ E+ I m M
E ~ s M ," ~ r"
/
/ / E_ E+ E 2 E 1 /
/ " m ITI S M • • • e o •
E_ E 2 E t
E_ E+ @
u=1
U E_@ , M
M
v 2 Cr i t i ca l energies: E+ =~2v; E 1 = I + v 2. E 2 = u + ~--
" minimum; s - saddle; M - maximum. I i
non-degenerate c r i t i c a l energy; I I
degenerate c r i t i c a l energy for two c r i t i c a l points;
0 degenerate c r i t i c a l energy for a c r i t i c a l c i r c l e ;
the b i fu rca t ion set
"( created c r i t i c a l points;
)" destroyed c r i t i c a l points.
94
To be p r e c i s e , we present the f o l l o w i n g c r i t i c a l po in ts and c r i -
t i c a l energ ies :
i ) The south pole x = ( 0 , 0 , - i ) is a c r i t i c a l p o i n t of h w i th
the c r i t i c a l energy E_ = h ( x _ , u , v ) =-2v, f o r (u ,v ) # ( 0 , 0 ) . x is a
minimum f o r u + v ~ 0 and a saddle f o r u + v < O. x is a degenerate
c r i t i c a l po i n t of h f o r u + v = O.
2) The north pole x+ = ( 0 , 0 , 1 ) is a c r i t i c a l po i n t o f h w i th
the c r i t i c a l energy E+ = h ( x+ ,u , v ) = 2v, f o r (u ,v ) # ( 0 , 0 ) . x+ is a
maximum f o r v ~ I , a saddle f o r u ~ v < i , (u ,v ) # ( 0 , 0 ) , and a m in i -
mum f o r u > v. x+ is a degenerate c r i t i c a l po in t of h f o r u = v and
v = i .
+ ( ~ v 2, O, v) w i th u,v < 1 are c r i t i - 3) The po in ts x~ = +
cal po in ts of h w i th the c r i t i c a l energy E I = h(x T, u ,v) = h (u ,v ) =
= i + v 2. x ; and Xl are maxima, x~ and Xl are degenerate c r i t i -
cal po in ts of h f o r u = v and u = 1. _
± 1 ~ _ v 2 v 4) The po in ts x 2 = (0, ~ ~ , ~) w i th v < l u l ; u < 1
+
are c r i t i c a l po in ts of h w i th the c r i t i c a l energy E 2 = h(x~, u, v) = 2
= u + v x~ ~ and x2 are minima fo r u + v < 0 and saddle U
po in ts f o r v < u. x~ and x2 are nondegenerate c r i t i c a l po in ts of h.
5) The c i r c l e S 1 c o n s i s t i n g of the po in ts x = (O,x2,x3) w i th
x 2 + x = i f o r u = v = O. Each po in t x of S I is a degenerate minimum
f o r h w i th the c r i t i c a l energy E = h ( x , u , v ) = O.
i 6)~ The c i r c l e S v c o n s i s t i n g of the po in ts x ~ (X l ,X2,V) w i t h 2 V 2 i is a degenerate x I + x = 1 - f o r u = i ; v < i . Each po in t S v 2
maximum f o r h w i th the c r i t i c a l energy E I = h ( x , u , v ) = I + v
e E,,,o,,~e÷ E~Phose /tonslt/on
X ~lu/
tl,~O ~ phuse ~ronxlfion
zi>O
Groundstate phase
t r a n s i t i o n s
95
Notice that ~et rans i t ions to d i f fe ren t phases occur passing from
one domain to another through the bi furcat ion set. This qua l i ta t ive
behaviour can be fur ther observed on the diagram of periodic orbits in parameter space (the orb i t manifolds D i are given in Appendix):
' \
\ \
\ \
\
E l . . \
D/+ T 2 O 0 \ , ,
D 1 T 1 0 E _ o ~
D 3 T 2 O 0 I E 2 . ,
E I . -
T 2 O 0
E_ CE)
T2 O 0
E2 " ' I
DI
E- TI • 0
\ \
\ \
\ \
\
~7 E_ I
_I
E+
/ /
/
0
E I 0
T 2 O 0
DI
Ti E + 0 "
T2 O 0
/ /
/ /
/
/ /
/
E1 / / g O /
/
/ E l . " /
/ T 2 O 0 / / D 5
E 2 (]0
D 3 T 1 O 0 t E+ 0
?
E I . ,
T2 O 0 l
E 2 (I)
T 1 O 0
E__ • •
E 1 0
T 1 O 0 l
E+ ®
T1 0 E - •
-LI El
T1 O 0 I E-
= v 2 Cr i t ical energy E+ = ~ 2v; E 2 u + ~-- ; E I = 1 + v
Period T1,T 2 ; non-degenerate point • ;
degenerate c i rc le for EI O
Separatrix ~ C]E)
All phase t ransi t ions with variat ion of parameters (that is
Change in spectral propert ies)are represented.
We present the fol lowing qua l i ta t ive argument to explain why
the Study of completely integrable systems seems important in this COntext:
96
Periodic (quasi-periodic}solut ions are restrict~dQ)~d to an open
subset of the phase space (and in this sense are stable). This seems
to indicate why such solutions have been mostly used in the Bohr
quantization rule . I t makes sense thus to further investigate the
re lat ion between s t a b i l i t y and quantization for simple Hamiltonian
systems since for Hamiltonian dynamics s t a b i l i t y is not the rule but
almost the exception (at best quasi -per iodic i ty and randomness
coexist)~ 0)~ This strongly indicates that ouantization of periodic orbits
(or manifolds) is an exceptional case.
Thus the invest igat ion of completely integrable systems (which
have periodic solutions) is of paramount importance.
A f ina l point we want to make with respect to the Hamiltonian
system considered(see (3.6)) is that the exact solutions obtained on S 2 are those of the Hartree-Fock approximation (which can be trans-
ported on the projective space without any loss of information) and
the analysis of the degree of approximation is very important for other (more complex) systems of the same type,
4. QUANTIZATION OF HAMILTONIAN SYSTEMS
The theory of (geometric) quantization is the answer to the
question of how to get the quantum picture of a classical dynamical
(Hamiltonian) system.
Recall that in Section 2 we have given d i f fe ren t real izat ions of
dynamical systems and in par t icu lar have shown how a co l lec t ive model
( " c la s s i ca l " ) can be obtained by dequant i za t ion . This was an elementa"
ry G-hamiltonian system (M,w,p, H) (as an example we used the SU(2)
elementary system) and in what follows we show that the manner in
which such a co l lec t ive model has been obtained is not accidental, but
represents a unique choice for a class of systems, and is compatible
with geometric quantization. (This is sometimes called requantization)'
Thus we shall spec i f i ca l l y consider the quantization of G-elementa-
ry systems (M,m,p, h) where M ~ G/K (c lass i f ied by coadjoint o rb i ts ) .
Thus we need to explain what a quantizable system means and shall
s ta r t by giving a br ie f account of what quantization is a l l about.
In one sentence we can state i t as:
f inding a correspondent V to the phase space (M,m) such that V i5
a complex Hi lbert space (representation space) together
with a correspondence between classical and quantum observableS.
97
For Hamiltonian G-systemsthis means to f ind a g e o m e t r i c constr(~c.~ion
for a~l unitary equivalence classes of i r reducible unitary representa-
tions of G. Notice that quantization has a global character
(that is i t is a transport map of global invar iant structures).
Recall that a classical observable is a real-valued smooth func-
tion f on the symplectic manifold M, whose dynamics is given by the
classical equation of motion
i t = { f t , h}
Where f t : f ( ~ ( t~ and ¢ is the f low generated by the Hamil tonian vector f i e l d Xh.
On the other hand,the time evo lu t i on (dynamics) of a quantum observable (a s e l f - a d j o i n t operator A on the H i l b e r t space V) is given by the quantum equat ion of motion
i~A t : [A t , HI
Where [ , ] denotes the commutator of operators and
A t : U(t) - I A U(t) , U(t) = e x p ~ - I t H )
This analogy between the classical and quantum dynamical sys-
tems Suggests the Dirac rule of associating classical observables to
quantum ones such that the Poisson bracket becomes a "correspondent" COmmutator7,8, 31)
Then a Dirac q~antization is a map Q of a set Fo(M ) of real-va-
lued smooth functions on M onto a set B of se l f -ad jo in t operators on
V Such that the map f ÷ Q(f) sa t is f ies the following conditions:
i) Q(f+g) : Q(f) + Q(g) ; Q(~f) = ~Q(f) , (4.1)
[Q(f ) , Q(g)] = i~ Q({f ,g})
for any real number ~ and any functions f ,g belonging to Fo(M ).
This COndition ensures that the quantization Q is a homomorphism
Of the Lie algebras (Fo(M), { , } ) and {B, (i~) -1 [ , ] ) } .
ll) Q ( 1 ) = I
where I is the ident i ty operator on V corresponding to the
constant I on M.
98
This condition ensures that a pair of classical observables that
has a constant Poisson bracket shall have the commutator of the
corresponding quantum observables required by the uncertainty p r i n c i p l e .
i i i ) The operators Q(f) (with f in Fo(M)) act i r reduc ib ly on V.
This condition is needed to construct the Hi lber t space V of a
quantum elementary system and means that Fo(M ) must contain
p(~) (that is there are enough generators to generate a group
of canonical transformations acting t r a n s i t i v e l y on M).
The r i g i d i t y of F(M) implies that there are few Dirac quanti-
zations.
Indeed, the theorems of Gr~enewald and van Hove show that a
canonical quantization - a Dirac quantization for a canonical phase
space { (q l . . . . . qn" Pl . . . . . Pn )} (or cotangent bundle)~ - of a l l the classical observables is , in general, impossible. 31'32)
However)the algebra of quadratic polynomials in the posit ion
(q) and momentum (p) functions admits a canonical quantization.
This is given by the osc i l l a to r representation and an example
of this quantization is that of the (5-dimensional) harmonic Bohr-
Mottelson Hamiltonian .
The class of elementary systems we consider does not, however,
posses this nice property, that is we can not give a global descrip-
t ion in terms of configuration space ( recal l that there is no glo6al symplec t i c p o t e n t i a l e such that ~ : de for compact phase spaces).
We shall next analyse the geometric structure of V (as a cer-
tain construct - l ine bundle on M) and shall for the moment drop 10,11,32)
condition i i i ) . Then we get a prequantization map Q'
Q'( f ) = f - i~Vxf
which maps F(M) onto a set of se l f -ad jo in t operators on a complex
Hi lber t space.
As we know that any symplectic manifold M allows a local
description in canonical (Darboux) coordinates ql . . . . . qn' Pl . . . . . Pn
the covariant der ivat ive has ( l oca l l y ) the expression
: Xf - i / ~ ~ Pk ~f VXf k ~Tk '
where e = E Pk dq i.s a local symplectic potential k k
99
The i dea o f q u a n t i z a t i o n i s t hus to ~ i v e a r e p r e s e n t a t i o n o f
c l a s s i c a l o b s e r v a b l e s and the space on w h i c h t h e y ac t . The { g l o b a l )
g e o m e t r i c o b j e c t on w h i c h t he p r e q u a n t i z a t i o n Q ' ( f ) ac t s i s t h a t o f
sec t ion o f a complex l i n e b~ndle L ~ M c a l l e d a quantum l i n e bundle~ ~)
Such a l i n e b u n d l e 34) I s a smooth m a n l f o l d ( thetotalapace) w~th a
Smooth p ro jec t ion ~ onto M (the base smace).For each po in t m of M, there ~s an open
neighborhood Q of m in M together w i th a smooth isomorphism m of Q x ~ onto - I ( ~ ) SUch tha t the diagram
~-l ~)... <o x C
p r o j e c t i o n
commutes and the induced maOmm, of {m'} x C onto ~- l (m ') is S - l i nea r f o r each
Point m' o f ~ . The pa i r (~,m) is ca l led a local ty~vializa~Lon. .~ I te rna t i ve ly M can be reqarded as a oarametr iza t ion space f o r the fibres
~'l(m) which are one dimensional t - v e c t o r sgaces.Consider two local t r i v i a l i z a -
tions ( ~ i ' mi I and (~ j , mj) o f L. Then on the i n te r sec t i on ~ i j of .9 i and ~j we have a smooth func t ion c i j such tha t
mi(m,w ) = c i j ( m ) ~i(m,w! (4. 2~
for any m in ~ i j and any comolex number w. c i j is ca l led a /~y~W.Y~Lon fune.Y~Lon. A ( l oca l ) sect ion s def ined on a neighborhood ~ of m is a mao of ~ i n t o L
Such tha t ~(s(m)) = m. That is we can regard s.(,e) as a subset of - I ( Q ) . The set
r(Q) of a l l sect ions over Q is a comolex vector soace (de f i n i no add i t ion and sca la r
~.~multiolicationh_ ' oo in twise. ) . I t s elements can be m u l t i n l i e d bv smooth func t ions on ~.
~s a c o v e r i n g { ~ i } where each Qi is an open neighborhood which admits a symolect ic
POtential ( ~ i is a gauge neighborhood). Thus L can be r e g a r d e d as a u n i o n o f
OPen neighborhoods ~ ' l ( Q i ) which admits a loca l t r i v i a l i z a t i o n mi of the p r o d u c t
e i x ~ onto ~ - l ( Q i ) .
We see tha t V' = r (M) ( the soace of a l l smooth sect ions over M)is a geometr i-
cal const ruct such tha t the smooth global orooer t ies are car r ied over and also
SUch that local domains on which the svmolect ic no ten t i a l is def ined are car r ied
Over as sect ions ( l o c a l l y re la ted by un i ta ry t ransformat ions o f smooth func t ions (~ ) ) .
Thus the wave vectors are sect io~ of a complex l i ne bundle w i th the c lass ica l
Phase space as base and a connection VX 33) . When the local svmplect ic oo ten t ia l
is replaced by o + du the wave function ¢' is reolaced by exn ( i~ ' l u ) \~ .
The ex is tence of such a l i ne bundle w i th connection (havind 2~ i~ ' I~ as curva-
ture form) imposes a topo log ica l cond i t ion on the svmolect ic form m - tha t is the_
~ I of the svmelect ic form m over any closed or iented surface in M must be an
100
We c a l l a s y m p l e c t i c m a n i f o l d M quantizable i f the W e i l ' s i n - t e g r a l i t y c o n d i t i o n i s s a t i s f i e d 31,33)
This c o n d i t i o n i s c l o s e l y r e l a t e d to the Bohr -Wi l son-Sommer fe ld
q u a n t i z a t i o n r u l e in the o ld quantum t h e o r y . Cons ider a c o a d j o i n t
o r b i t M ~ G/K and ~ in M such t h a t K is the s t a b i l i z e r of ~. The i s o -
t r opy a l g e b r a ~ o f K c o n s i s t s of a l l e lements x of the L ie a l geb ra
of G such t h a t ~ ( [ x , y ] ) = 0 f o r any e lement y o f 3 . Then the map
2 ~ i f o f ~ onto i ~ de f i ned by x ~ 2 ~ i : ( x ) i s a hemomerphism of
L ie a l g e b r a s . A c o a d j o i n t o r b i t i s i n t e g r a l i f 2~i~ induces a homo- i morphism x~ o f K o onto the c i r c l e group S (K o i s the i d e n t i t y connec-
ted component o f K). Since G is a connected s imp l y connected L ie group,
a c o a d j o i n t o r b i t i s q u a n t i z a b l e i f and on ly i f i s integral (2~ i~
i n t e g r a t e s to a u n i t a r y c h a r a c t e r x a of Ko). The prequantum H i l b e r t
space V' f o r the q u a n t i z a b l e o r b i t M can be i d e n t i f i e d w i th the space
Ind ( G , K , x ' ) c o n s i s t i n g of a l l comp lex -va lued f u n c t i o n s ~ on G which
are square i n t e g r a b l e on M and s a t i s f y the r e l a t i o n
~(gk) = x ' ( k - l ) ~ ( g ) (4. 3 )
f o r any g in G and k in K. Here x ' i s an e x t e n s i o n of ×~ to a u n i t a r y
c h a r a c t e r of K ( the se t of a l l p o s s i b l e e x t e n s i o n s i s p a r a m e t r i z e d by
the c h a r a c t e r s o f the group K/Ko) . The u n i t a r y r e p r e s e n t a t i o n U' o f G
induced by x' on I n d ( G , K , x ' ) is g iven by
U ' ( g ) ~ ( g ' ) = ~ ( g - l g , ) (4. 4 )
f o r any g, g' in G and m in I n d ( G , K , x ' ) .
U n f o r t u n a t e l y , the prequantum H i l b e r t space V' i s too b ig to re -
p resen t the space of s t a t e - v e c t o r s o f a p h y s i c a l l y reasonab le quantum
system. Moreover , the r e p r e s e n t a t i o n U' induced by p r e q u a n t i ~ a t i o n i s
r e d u c i b l e .
To f i x a r e p r e s e n t a t i o n ( i n the sense of D i r a c ) we need to cons-
t r u c t a g loba l geomet r i c ob j ec t t h a t cor responds to the idea of a maxi"
mal set of commuting c l a s s i c a l o b s e r v a b l e s . This ( f o r reasons to become
t r a n s p a r e n t l a t e r ) s h a l l be a complex suba lgebra o f the c o m p l e x i f i c a "
t i o n ~ c of our symmetry algebra ~ w i t h c e r t a i n c o n d i t i o n s which we now
proceed to d e s c r i b e . Cons ider ~ in a q u a n t i z a b l e c o a d j o i n t o r b i t M~G/K,
where K is the s t a b i l i z e r of a. Let ~c and ~ c be c o m p l e x i f i c a t i o n s
o f the L ie a lgeb ras ~ and ~ f o r the L ie groups G and K, r e s p e c t i -
v e l y . A complex p o l a r i z a t i o n at ~ i s a complex suba lgebra o f ~c
such t h a t 3 5 ) :
101
1) ]~c is contained in ~b;
2) ~ ( [ x , y ] ) = 0 for any elements x,y of 9 ;
3) dime ~c + dim~ ~c = 2dim c 9 ;
4) ~ is stable under ~ ;
5) ~ + ~ is a Lie subalgebra of ~c '
Where dim{ denotes complex dimension and ~ + ~ cons is ts of a l l elements X + ~ with x,y in ~. The quantum H i l b e r t space V is the space of al l Vectors ~ of V' such that
~Xm : 2~i ~(X)m ( 4 . 5 )
for any x in ~'~, where the Lie de r i va t i ve ~x is defined by
d t=O (~x ~)(g) - dt m(g exp ( - t x ) ) ( 4 . 6 )
for any element g of G.
U : U, I is a unitary representation of G. Q Q'IV V = is a geome-
t r i c quantization. Then x + (ii~) -1 Q(p(x)) is a unitary irreducible representation of ~ and the unitary representation U is irreducible.
Thus the complex polarization uniquely determines (for speci- f ic cases) the real izat ion of the quantum Hilbert space as a space of
global geometric objects (functions, sections, k-forms) of a homoge- neous holomorph~e l i ne L over the quantizab]e classical phase space
x M×. That is Mx,Lxare complex manifolds and the projection ~ and transi-
tion functions are holomorphico 34) x is a holomorphic extension of ×'
The quantization construction described above and the coadjoint orbit structure of elementary Hamiltonian systems considered (such
that the t r a n s i t i v i t y of the G-action should be equivalent to the
I r reduc ib i l i t y of unitary representations) can be summarized as fo l -
lows(for square integrable rep~sentations of semisimple groups):
M has a natural G-invariant symplectic structure ~ (2.17) x
and admits a unique prequantization given by the integral i ty of 2~i~ to a unitary character x~ • The complex polarization gives a holomorphic structure to L x and the sections are holomorphic. The
quantization Q yields a unitary irreducible representation on V (SUch that representations arising from di f ferent choices of~Xare
uni tar i ly equivalent). Every irreducible unitary representation of G
arises from a~adjointorbit of G in . As one can observe we take the
f°11OWing position regarding the quantization of Hamiltonian systems:
102
This is given by symp~e~c ~b~ng~ of classical Hamiltonian spaces
into quantum projective spaces (with the K~hler structure given by the
t rans i t ion p robab i l i t y ) .
We shall r e s t r i c t our discussion to K~hler embeddings.
For a large class of elementary spaces there exist K~hler s~[u~- tures with K~hler complex polarizations (realized by holomorp~ic vector fields) ~ ,3 .
One may establish an equivalence between a given unitary i r r e -
ducible representation and the quantized representation constructed
such that the geometric structures are transported.
There is thus a global, e x p l i c i t correspondence between clas-
sical and quantum Hamiltonian systems which is given by the following
correspondence diagram:
^ / ,
M~ = M~
The right-hand side of the diagram is given in Section 2. P(V) is
the project ive space of the quantum Hi lbert space V. M is the coherent
state manifold. ~ is the moment map and ~ the dequantization map. M X is
a quantizable classical phase space, V x is the Hi lber t space obtained
by quantization and n is an isomorphism. P(V×) is the projective space
of the Hilbert space V x. ~ is an embedding of M x with the image Mx" P'~ are projections, i , i X , i are inclusion maps. The diag~a~ is classi~ing for e l e ~ ta~ classical sys~(as coadjoint orbits)and the corresponding elementa~ quantum sys~ms. Here M is a quantizable phase space realized as a K~hler integral co"
x 6, adjoint orb i t G/K of a connected simply connected semisimple Lie group
The unitary character × of the isotropy group K is determined by
Weil's i n t e g r a l i t y condition (prequantization). Let U X be the irredu-
cible unitary representation of G on the complex Hi lber t space V x
constructed by the geometric quantization procedure and related to the
103
homogeneous holomorphic l ine bundle L ÷ M . Let ~ be a K~hler embed- x x
ding of M x into the project ive space P(V ), which is the base space of x
the canonical K~hler bundle with the total space V {0} and projec- × tion Px"
Consider m o in M x and define the vector s o in P~(E(mo)) of
V, ~x is defined by ~x(g ) = UX(g)s o for g in G.
Let V be a Hi lber t space and U an i r reducible unitary represen-
tation of G on V, such that U is un i t a r i l y equivalent to U X and M the
Coherent state manifold (as a submanifold of the K~hler manifold P(V)
With notation introduced in Section 2).Then there is a unique K~hler embed-
ding and a K~hler isomorphism v such that the diagram is commutative. , . , . . . E
In this way we give an e x p l i c i t geometric rea l izat ion of global
Objects on the classical quantizable phase space M x such that the sym-
Plectic,complex,algebraic structures are transported (without any loss)
on Submanifolds of the quantum project ive space P(Vx) ~6~hen on the Hilbert space V x together with the rea l iza t ion of quantum observa- bles as global d i f f e ren t i a l operators. 37)
The quantizable phase space M can be (isomorphically) iden t i f i ed .X
to the manifold of coherent state M (with the K~hler structure of M ). x
Thus the dequantization given by coherent states (as submanifolds of
P(V)) is compatible with geometric quantization and is e x p l i c i t l y
given by the qeometric construction of square-integrable represen-
tations of G 3B ,3g ) . A v e c t o r ~ of V\{O} is coheren t i f [4 ]= p (~ ) .The co-
he~nt vectors give the to ta l space of a ho lomorph ic bundle on M e q u i v a -
len t With L X ÷ M x such t h a t they are g l oba l s e c t i o n s and admi t r e a l i -
za t ions by ho lomorph ic f u n c t i o n s as l o c a l s e c t i o n s on ho lomorph ic gauge Charts of M x ' o n l y . T h e y , i n our case ( e l e m e n t a r y systems) d e f i n e
natural c o l l e c t i v e coordinates (give uniquely a complex po lar i - zation).
The quantization procedure and the commutativity of the corres-
POndence diagram are e x p l i c i t for al l the discrete series represen-
tations of semi-simple Lie groups 37~. O ~ i n the case when M = G/K is a
bounded symmetric domain 39) in {~ the quantum space consists of
h°Iomorphic functions on M (in this case the boson real izat ions con- Structed by the geometric quantization procedure are global) .
An e x p l i c i t rea l iza t ion is given for the SU(2) (compact groups) in the next section.
104
5. QUANTIZATION OF COLLECTIVE SU(2)-MODELS
Let us consider a classical co l lec t ive model, the symmetry group
of which is a connected~simply connected cQmpact group Lie G. The classical
phase space of this model is precisely a flag manifold M ~ G/K ~ Gc/P,
where P is a parabolic subgroup of the complexification G c of G
(see Section 2 ). Assume that M is quantizable and le t X ~ be a uni-
tary character of K determined by Weil's i n t e g r a l i t y condition (prequan °
t i za t i on ) . Suppose that we are given a complex polar izat ion ~, where
~ is the Lie algebra of P.
The quantum phase space is represented by the complex Hi lber t X
space VBW of a l l holomorphic functions F on Gc, square integrable on
G, such that
F(gp) = x - l (p ) F(g) ( 5 . 1 )
for any g in G c and p in P. The holomorphic character X is a holo-
morphic function on P characterized by
( i ) x ( p ) x ( p ' ) =
( i i ) x(k) = ×'(k)
x(pp')
( 5 . 2 )
for any k in K and p, p' in P. Then there is an isomorphism of the
Borel-Weil space V~W onto the space V x (constructed by the geome-
t r i c quantization procedure in Section 4 ) defined by F ÷
with m(g) = F(g) for any g in G Consider the representation U x of • C
× defined by G c on VB~!
×(g) F(g') F(g - I ' u c : g ) ( 5 . 3 )
, V X for any g and g in G c. Let UXbe the representation of G on BW
defined by UX(g) = uX(g) for any g in G. The unitary subrepresenta- c
t ion U X of U X is i rreducible23). C
40,4i ) . Remark. The space V x is f ini te-dimensional
BW Indeed, since a uniformly bounded sequence of holomorphic functions
has a subsequence converging uniformly on compacts, V~; (topologized
by uniform convergence on the compact manifold M×)is a loca l ly compact
Banach space, hence f in i te-dimensional . We now prove the Borel-Weil
theorem. Since G c is semisimple, V~WD is a direct sum of the subspaceS
105
S k where G c acts irreducibly. Let F k be a highest weight vector in S k. There is a
SUbgroup N of G c such that NP is open and dense in G c and the intersection of N and
P is the identi ty element e of Gc.(As we know K:C(T) (Section 2) and let H be a ma-
Ximal torus with T contained in H. The intersection of H and G/K in consists of
a f in i te number of points m i . The Lie a lgebra~of !i is the orthogonal complement
° f ~ i n ~Jc" The orbit of N through m i is ~i which is open and dense in M X. Local
coordinates in ~i are given by the complex exponential parametrization of N. The
Bore] ceils ~i are holomorphic gauge neighborhoods). By (5.2) and (5.3) we have
Fi(nP) = x-1(p)Fi (n ) for any n in ~ and p in P. Then the set of al l highest weight
Vectdrs is contained in an one-dimensional complex vector subspace of JBM" Hence
the holomorphic representation UcXxiS i r reduc ib le . Since G is a maximal compact sub-
group of Gc, the representation U is i r reduc ib le on r(M}. Here ?(H) is the space
of global holomorphic sections over M.
÷ M . We now construct the homogeneous holomorphic line bundle L× x
The base space is the n-dimensional complex manifold Mx~ Gc/P. This manifold
is covered by the Borel cells h i ~ {n. Each point m of ~i has a local coordinate
System Zi such that m = gi(Zi)P with gi(Zi) in G c. The matrix elements of gi are
Polynomials on cn. (Indeed, N is nilpotent and m = ~(Zi)m i with n=exp(skZikXk) in
N, where {Xk} is a basis for ~ and Z i = {Zik} ).
The tota l space L× consists of al l equivalence classes (g; w) of elements
(g,w) in G c x ¢ under the equivalence relation (g, x(P)W) ~ (gP, w) with p in the
Parabolic subgroup P of the complex Lie group G c. Therefore, i f g is an element of
G c and w is a complex number, then the point (g; w) of L× is the set of all pairs
(g', w') where g' = gp and w' = x(p'l)w with p in P.
The projection ~ of L x onto the flag manifold M • Gc/P is defined by
~(g;w) = gp. Then the f ibre over gP (consisting of all (g; w) with w in {) and the
Complex line ¢ can be identif ied by (g;w) ÷ w.
The action of G c on L× is defined by g'(g; w) = (g'g; w) for any g' in G c and (g;w) in L× .
Local t h i v i a l i z a t i o ~ and . ~ n s i t i o n functions
The total space L× is covered by the open neighborhoods I~!i = ~ '1(~ i ) such
that there is a biholomorphic map ~i of ~i x ~ onto h'.1 for each Borel cel l h i of
the f lag manifold M .Thus the 7oca] coordinate system (Z i , w) in ~ consists of the X
local coordinate system Z i in h i and the complex coordinate w in { . Each point of ~' l can be wr i t ten as (g i (Z i ) ,w) where the matrix elements of gi are holomorphic
functions of the local coordinates Z i for the point ~ (g i (Z i ) ; w) of M .Thus, L × x admits a n a t u r a l s t r u c t u r e o f complex m a n i f o l d such t h a t the
106
project ion ~ is holomorphic. Moreover, each (~i,mi) is a l oca l t r i v i a - l i z a t i o n and the t r a n s i t i o n funct ions c i j are holomorphic on the
in tersect ion Rij of ~i and ~j:
mj(m,w) = cij(m)mi(m,w ) = mi(m, t i j (Z j )w) , ( 5 . 4 )
where t i j are holomorphic functions of the local coordinates Zj and m
is a point belonging to ~ i j with the local coordinate system Zi and
Zj in Ri and ~j, respect ively. Then we have
(g j (Z j ) ; w) : ( g i ( Z i ) ; t i j ( Z j ) w)
t i j ( Z j ) : X-~(giZi) ) x (g j (Z j ) ) ( 5 . 5 )
A 91obal ho~omorphic s ec t i on s of the holomorphic l ine bundle
L X + Mxis a map s of M into L X such that s(m) = ( g i ( Z i ) ; f s i ( Z i ) ) where fs i is a holomorphic function of the local coordinates Z i for the point
m of ~i" Then sl~ " = fs i oi where each uni t scction o i over ~i is de- fined by s(m) = I g i ( Z i ) ; I ) .
According to ( 5 . 4 ) , we have
f s j (Z j ) : t j i (Z i ) f s i ( Z i ) ( 5 . 6 )
for any m in ~ i j "
We now apply the resul ts above to a c lassical co l lec t i ve model with the symmetry group SU(2). The c lassical phase space is the 2-sphere of radius ~ j :
S 2 ~ {p1 ~ SL(2,£)/P ~ SU(2)/U(1) ~ S0(3)/S0(2) ~j
a) Prequant iza t ion
By Weil's i n t e g r a l i t y condi t ion, we have
I ~ j : i ~ f 2 j ( 1 + ZZ) -2 dZ A dZ : 2~n~. (5.7 )
5 2 ¢ ~j
C l e a r l y , S 2 is q u a n t i z a b l e w i t h r e s p e c t to the s y m p l e c t i c form ~mj ~j i f and on l y i f j = n/2 f o r some p o s i t i v e i n t e g e r n. Assume t h a t
S 2 is q u a n t i z a b l e Then the re i s a prequantum H i l b e r t space V' ~j f o r the u n i t a r y c h a r a c t e r x I g i ven by
107
e- i ra/2 0 1 x ' ( k ) = e - i j m ; k = 0 e ira/2 ( 5 . 8 )
the space of for any k in K = U(1) . Then Ind (SU(2), U(1) , x ' ) i s / a u n i t a r y rep re -
Sentat ion of G = SU(2) on V ' ( ( s e e Sect ion 4) .
b) P o l a r i z a t i o n and q u a n t i z a t i o n
A complex polar izat ion is given by the Lie subalgebra ~ of the Parabolic group P° Here ~ and P consist of a l l matrices x and p, respect ively, given by
x : 0 -p ; P = 0 u - I ( 5 . 9 )
where u , ~ , u, v are complex numbers and u # O. Then the subrepresen-
t a t i on of Ind (SU(2) , U(1) , x I) on the (2 j + 1) - d imensional quan-
tum H i l b e r t space V× is u n i t a r y and i r r e d u c i b l e .
Moreover, t h i s s u b r e p r e s e n t a t i o n is e q u i v a l e n t to the u n i t a r y i r r e d u -
Cible r e p r e s e n t a t i o n U x on the Bore l -Wei l space V X The holomorphic BW' charac ter x ( the unique holomorphic ex tens ion of x ' ) is de f ined by
X(P) : u "2j (5.10)
for any ma t r i x p g iven by ( 5 . 9 ) .
+ S 2 .The t o t a l We now c o n s t r u c t the holomorphic l i n e bundle L X ~ j
SPace LX cons i s t s of a l l po in ts (g; w) where g is a ma t r i x be long ing
to Gc = SL(2, { ) and w a complex number. Each (g; w) is the equ iva-
lence c lass of a l l pa i rs (gp, x (p -1) w) w i th p in P. The loca l t r i v i a -
l i z a t i o n s are (~1' °1) and (R2' m2 ) ' where the Borel c e l l s ~1 and R2
are the nor th pole cha r t and the south pole char t of the Riemann Spher e ~pl ~ S 2 ~j.Here mi(m,w) = ( g i ( z i ) ; w) with m = g i (z i )P in the
2 Borel cel l ~i of the f lag manifold SL(2,¢)/P ~ ~j , S . ( i = I 2). The matri-
Ces g1(z1 ) and g2(Z2) are given in Section 2 By (2.32) and
( 5 " 5 ) , we obtain the t rans i t i on functions t12 and t21 given by
2j t12(Z2) = Z I ; t 2 1 ( Z l ) : Z j
Where ZI ' Z 2 are complex coord ina tes and ZIZ 2 = i .
(5.11)
108
A global holomorphic section s of the holomorphic l ine bundle S 2 is a map of S 2 L X ~j ~ j i n t o L X such that s(m) = ( g i ( z i ) ; f s i ( Z i ) )
where fs i is a holomorbhic function of Z i and m = gi(Zi)P is a point of ~i ( i = 1,2). Denote ~i(s) = f s i " According to ( 5 . 6 ) and (5.11), we obtain
fs2(Z2) : Z~ j f s i ( Z l ) (5.12)
for any complex numbers Z 1 and Z 2 with ZIZ 2 = 1. Then fs l and fs2 are polynomials of degree at most~ 2j. Hence the
of a l l global holomorphic sections Fj = Fx(S~j)is a (2j + 1) space dimensional complex Hi lbert space with respect to the scalar product
( s , s ' ) : $ h (s ,s ' )m j = < f s i ' f s ' i > S 2 i~j
2j+i~ I Tsi(Z ) f s , i ( Z ) ( l + Z Z ) "2 j -2 d(ReZ)d(ImZ); i =1,2. (5.13) m 2
Consider the complex Hi lber t space F~ i ) of a l l polynomials fs i with s in r j , the scalar product of which is given by (5.13). I f A is a l i - near operator on r j , the l inear operator A ( i ) on r~ i ) is defined by
A ( i )~ i ( s ) : ~i(As) for any s in r j ( i=1,2) .
Let Tj be the Borel-Weil uni tary i r reduc ib le representation of SU(2) on r j . The in f i n i tes ima l generators of Tj are denoted by
- i J k (k=1,2,3). The canonical basis of r j consists of the sections Sjm ~ : . Then Tj is determined
(m=-j, - j + l . . . . . j ) . Denote f ) ~i(Sjm) T! 1) and T~ 2) of SU(2) on by the uni tary i r reduc ib le representations J r~ I) and r~ 2), respec t i ve ly . To be precise:
I) the north pole chart I
T ~ l ) ( g ) f s l ( Z )
j ~ i ) d = ~ Z
(~Z + m)2j - - ~) fs l ( ~ + m
j (1) -2jZ + Z 2 d : ~ - ~ ,
C 5 . 1 4 )
d +(i) ~i) (i) Jo(1) : j _ ZE:Z ., J = J _+ i J2 ' ( 5. ;.5)
109
f ( 1 ) j+m . ( 2 j ) ! ~1/2 zj-m jm (Z) = ( 11 ) ( ( j - m ) ! ( j + m ~ '
2) the south pole chart ~..
: ( ~ z + ~) TJ 2) (g) fs2 (Z) (~ " 6z)2J fs2 -BZ
j~2) -2 jZ + Z 2 d j ( 2 )= d = ~ Z ; - d--Z '
j 2) = _ j + Z ~ ; j~2) = j~2)+_ i j~2)
(5. 16)
(5-17)
, (5 .18)
12~)! 1/2 zj+m f!2)(Z)Jm : ( - l )J+m ( ( j -m ! ( j+m) ! )
Here f = si ~ i ( s ) fo r some s in r j ( i = 1,2)
matrix of SU(2).
We remark tha t r ! I ) and F! 2) J 3
funct ions based on coherent s tates:
• ( 5 . 1 9 )
and g : ( ) is a
are Hi lber t spaces of holomorphic
f!]~) (Z) : ( - I ) 2j <jmlexp(-Z j ) [ j j > (5.20) jm - ,
f~m 2) (Z) = <jmiexp(-Z j + ) l j j > . (5.21)
The Subspaces of local holomorphic ~ections over Borel cells are precisely the spaces z~.~n.olomorphic functions based on coherent states and the corresponding Dyson reali- :~lons are given by the local trivializations of quantized linear Hamiltonians.
cJ ~uantum Hamil tonLans
Af te r the preceding p repara t ions , we consider the fo l l ow ing family of Hamil tonians on the space of sec t ionsF j :
Ha = al J + a2 J + a3 J3 + a4
Where a : ( a l , a2 ' a3 ' a4 ) is a system of cont ro l parameters and
la I - a3I a la I a2{ ; a4(a I - a3) a 0 (5.23)
Any Hamiltonian H on ?j expressed as polynomial of degree at most two in j
i ' J2' J3 has a nont r iv ia l invariance group i f and only i f H : Tj(g) Ha T j (g -1 ) , where g is a matrix of SU(2) and a is a system of Control parameters.
I f al = a3 ' then H a has the eigenvalues a l j ( j + l ) + a4m corres- POnding to the eigensections Sjm.
110
Assume now that a I # a 3. Then H a can be wr i t ten as
H a = (a I a3) L(U,Vo) + a 3 j ( j + l ) I (5 .24 )
where I is the i den t i t y operator on r j and
L(U,Vo) : J~ + uJ~ + 2VoJ 3
a 2 a 3 a 4 = ~ V 0 = U al " a3 ~-~-1 a3~
(5.25)
(5.26)
By (5.23), we have v o > 0 and - 1 < u ~ i .
The local t r i v i a l i z a t i o n s L(Z)(U,Vo ) of L(u, Vo) are given by
d 2
+-12 [ (2 j i ) ( i + u) z I -(2j - i ) ( i - o) z~ + 4v~ z] ~ +
+ ½ (i u) j (2j i) z~ + ½ (i + u)j - 2vzj ( 5 . 2 7 )
on rj(Z) (Z = 1,2), where v I =-v o and v 2 = v o. Putt ing Z~1 =t in 2~I (5.27) we obtain the algebraic form of the Heun d i f f e r e n t i a l operator
(the Heun equation is res t r i c ted to polynomial so lu t ions) . Modern
l i t e r a t u r e uses the Jacobian form as a basis. Then we obtain the f o l -
lowing operators:
d2 ~ L(Z)(u,v£) : - ~ - 2v I s + k 2 j , j + l , s 2 ( ~ + 2vzjcd , (5.28)
where s = sn(q,k) ; c = cn(q,k) ; d = dn(q,k) 28) (see Appendix) and
c+d c-d k 2 ~/~ k 2 Z I = ~ ; Z 2 = ~ ; : u ; k ' : - .
Then can be w r i t t e n as:
c+d £ ) (Z) : j c d - s ~ j Z) : i ~ ( j l ~ J ); Jo
the i n f i n i t es ima l generators of SU(2) and the canonical basis
(z) fjm + mjm
2j ] 112
j+m
j-m
(_ik,s)J-m(c+d) m ; m ~ 0
( 5 . 2 9 )
( - i k ' s ) O + m ( c - d ) m ~ ; m < 0
111
I f 2j is an even in teger , then @jm are homogeneous polynomials in s,
c, d of degree j . On the other hand, i f 2j is an odd in teger , we have
mjm = Pjm~l + Qjm~2 (5 .30 )
Where Pjm' Qjm are homogeneous polynomials of degree j - 1/2 in s, c, d and
2 2 ~I : d + c , ~2 = d - c , E I~ 2 : k ' s (5 . 3 1 )
The operator given by (5.28)(wi th v ° = 0 and u # O, i ) is the Lam~
d i f f e r e n t i a l operator (the asymmetric top Hamiltonian) 28)
I f Vo : 0 and k = 0 (or k = I ) , then (5.28) becomes a symmetric top
Hamiltonian. In th is case or when k = i the d i f f e r e n t i a l operator
(5.28) has e x p l i c i t eigenvectors in terms of Legendre polynomials.
R ~ . Consider 0 < k < i (0 < u < I) and v o = O. Then we have the
fol lowing asymptotic expression for the eigenvalues of L(u,O) :
+ l+k2 (m 2 + 1) ( I -k2)2 m(m2+3) E~ = k( j + ½)m - ~ - 32k(2 j+ l )
. _(1+k21(l_k 2 2 256k2(2 j+ l ) I (5m 4 + 34m 2 + 9) + . . . + O+(e - c j ) _ (5.32)
Where m = 1, 3, 5, . . . and kj >> I . I f kj is f ixed and j ÷
then the l im i t i ng Lam~ operator is a Mathieu d i f f e r e n t i a l operator .
The asymptotic ground-eigenvector of L(u,v o) is given by the action of
an one-parameter subgroup of SL(2, t ) on Sjo
sB = exp(iBJ3) exp(- IZ J2) Sjo
Where
, (5.33)
- 1 5k 4 - 2k 2 + 5 -2B o = 1+k 5. e 2(B-Bo) = I - T~ - 16j2 R2 + . . . ; e T:-k' ( 34)
I f ~'u = 2v o , then SBo is prec ise ly the ground-eigenvector of
L(u'v o) for any j (an exact large-amplitude so lu t ion ) . Moreover
J+~lim[(sB° ' L(u, Vo) SBo ) - E~ ] : 0 (5.35)
112
Wi thout going i n t o d e t a i l s , us ing the d i f f e r e n t i a l o p e r a t o r s which
have been e x p l i c i t l y ob ta ined here by geomet r i c q u a n t i z a t i o n and sym-
met ry c o n s t r a i n t s ( d e f i n e d on q u o t i e n t l i n e bundles w i t h respec t to
i n v a r i a n c e subgroups) we o b t a i n the exac t forms of the ground sec t i ons
f o r some va lues of parameters (U,Vo) in the case of n o n - a x i a l systems
( w i t h d i s c r e t e i n v a r i a n c e subg roups ) .Mo reove r , we o b t a i n e x p l i c i t asym-
p t o t i c s in the l i m i t j + ~ f o r ( l o w - l y i n g ) energy l e v e l s (5 ,23 ) and
the co r respond ing wave v e c t o r s . One must remark t h a t in the a s y m p t o t i c
f o rmu lae the re i s a c o m p e t i t i o n between the c o n t r o l parameters u, v o
and the we igh t of the r e p r e s e n t a t i o n j ( " d e f o r m a t i o n - r o t a t i o n " compe-
t i t i o n ) . Th is s t r a t i f i c a t i o n of the c o n t r o l parameter space (g i ven by
a s y m p t o t i c s ) i s even f i n e r t h a t the ( a l r e a d y c o m p l i c a t e d ) s t r a t i f i c a t i o n
ob ta ined in the c l a s s i c a l case. Th is m i x t u r e of domains where the asym-
p t o t i c s e x i s t w i t h those where they are des t royed suggests a c r i t e r i o n
of "paramete r s t a b i l i t y " f o r some quantum c o l l e c t i v e models ( e . g . the
a s y m p t o t i c s around u = v o : 0 - f o r the symmetr ic top - are des t royed
by p e r t u r b a t i o n s ) .
6. CONCLUSIONS
Let us summarize b r i e f l y the above r e s u l t s . The main r e s u l t o f
t h i s l e c t u r e is the cor respondence diagram presented in Sec t i on 4. The
geomet r i c q u a n t i z a t i o n procedure can be expressed in a s imp le and e l e -
gant H a m i l t o n i a n form in terms of s y m p l e c t i c embeddings of c l a s s i c a l
H a m i l t o n i a n spaces i n t o p r o j e c t i v e spaces of quantum pure s t a t e s (en-
dowed w i t h the K~hler s t r u c t u r e induced by the t r a n s i t i o n p r o b a b i l i t y ) "
We r e s t r i c t our d i s c u s s i o n to the geomet r i c c o n s t r i c t i o n f o r the square"
i n t e g r a b l e i r r e d u c i b l e u n i t a r y r e p r e s e n t a t i o n s of semis imp le L ie group s.
Then the phase spaces of the c l a s s i c a l c o l l e c t i v e models are K~hler
m a n i f o l d s . The g loba l p r o p e r t i e s and the s y m p l e c t i c , complex and a lge -
b r a i c s t r u c t u r e s of these spaces are t r a n s p o r t e d onto the coheren t
s t a t e m a n i f o l d s ( subman i f o l ds of the quantum p r o j e c t i v e spaces ) . Thus
the coheren t s t a t e m a n i f o l d s are e l emen ta r y H a m i l t o n i a n spaces. More-
ove r , the geomet r i c p r o p e r t i e s of the coheren t v e c t o r sets are coded
in c e r t a i n homogeneous ho lomorph ic l i n e bundles on the c l a s s i c a l phase
spaces. The quantum H i l b e r t spaces, r e s u l t i n g from geomet r i c q u a n t i z a "
t i o n , c o n s i s t o f some g loba l s e c t i o n s over the c l a s s i c a l phase space.
Then the coheren t v e c t o r s are g l oba l s e c t i o n s ( r e a l i z e d as g l oba l holO"
morphic f u n c t i o n s on ly in the case of a bounded symmetr ic domain) .
The quan t i zed obse rvab les are g l oba l d i f f e r e n t i a l o p e r a t o r s ( r e l a t e d
to c e r t a i n e l l i p t i c o p e r a t o r s ) . Moreover , the l o c a l t r i v i a l i z a t i o n s of
113
these operators can be exp l ic i te ly identi f ied to the boson realizations42).
In Section 3 we obtained the Hartree-Fock phase portrai ts (or- bits oriented with arrowheads) on two-spheres considered as classical
Phase spaces for the family of Hamiltonians quadratic in the i n f i n i t e - Simal generators of the group SU(2) and with nontr ivial invariance sub-
groups. The manifolds of periodic orbits and the period spectrum are
given by exp l i c i t analytic expressions (in terms of Jacobian e l l i p t i c
functions). We obtained an exp l i c i t c r i t i ca l structure of the control- led Hamiltonian system (al l c r i t i ca l points, c r i t i ca l manifolds, c r i -
tical energies, separatrices, bifurcation and catastrophe sets, phase
transit ions). The control parameter space is s t ra t i f ied with respect
to the c r i t i ca l elements and invariance subgroups(which completely or-
ganize the qual i tat ive global properties of the Hamiltonians). The
B°hr-~ilson-Sommerfeld rule on the manifolds of periodic orbits gives the leading terms of the quantum excitation energies for some strata
of control parameters (the f i r s t nontr ivial correction is obtained by the geometric quantization procedure). The preceeding results are gene- ric for a large class of algebraically completely integrable col lect ive
Hamiltonians (e.g. the crancked Arnold rotors) Unfortunately, this Class of nontr iv ial Hamiltonians is singular in the space of col lect ive Hamiltonians.
As emphasized in Section 5, the considered family of algebraically COmpletely integrable Hamiltonians admits an exp l ic i t e l l i p t i c real iza-
tion ( i l lus t ra ted by exact and asymptotic analytical expressions for
ground-energy, excitation energies and the corresponding eigenvectors).
Final ly, we recall that the correspondence diagram provides a natural framework in which to understand the exp l ic i t relationship between classical and quantum collective dynamicsj and the competition between the classical l im i t of large quantum numbers and deformations
in the control parameter space.
The diagram is classifying for ~e classical collective models and the cor-
responding quantum collective models with semisimple Lie groups of symmetries. The results can be extended to the square integrable irreducible unitary representa-
tions of nilpotentjalgebraic solvable, exponential solvable Lie groups and some semi-
di~ct products. By transport of structure, the geo~tric construction of group re-
Presentations, symplectic geomet~, K~hler geo~t~, algebraic geomet~, global ana-
lysis, Hamiltonian dynamics and geometric quantization can be applied to the des-
cription of the complex collective nuclear motion.
114
A P P E N D I X :
DI: E < E < E+
x I :~-I - ~ c~
x 2 : R - r ( l -~1 )sd~
Man i f o l ds of p e r i o d i c o r b i t s Per iod spectrum
~i-~3 ~ = ( I - ~ us2) - I
: 2/R--RTt
T1 = ~T F (.~2_E)
D2: E+ < E < E2;
( 1 - u ) ( I - ~ ) _. s2~
x3 =~4 + R( lep )
X 1 = V ~ - ~ cd~
x2 ~ R ' ( I + p ) ' s~
( 1 - u ) ( 1 - ~ ) s2 C x3 =~1 + R ' ( I + p )
V < U < I
( l - u ) ( l - ~ ) 2 -1 ~=[I-2RR,,,(, I+O,)-- us ]
T=/2RR'(I+p) t
2~ FtP-l~ T I /2RR' ( l+p )
(I-u)(I-~) s2 ]_ I = [ 1+ -Z'RR' ( l+p )
• : /2RR'( I+p) t
2~T F / P - l , ~ p +--TT) T I - ~ 2 R R ' ( I + p )
D3: E 2 < E < E_;
Xl :+-~T~-~ (I-~)sc~
( 1 - u ) L 1- 4 2~ x3 =~4 + - - R ( I - p ) s
V < "U
~3-~4 - I ~=[1- s 2 ] ~3-~1
~: V2RR'(1-p) t
~T T2 = , 2 f2RR (1-p~ F(-T~S-P)
115
D4: E+ < E < m i n ( E 1 , E2)
Xl : 2 V I - ~ d~
~ 2R(1-e~) ×2 : ± - R - ~ T sc
( 1 - u ) ( 1 - ~ ) 2 x3 = ~1 + ~ ' ~ ' ~ s
~ = [ I m2 -e l
~2-~4 s 2 ] - I
: vr2RR'(1-~ t
IT 2
F (T~-) T2=V2RR' ( l -p )
D5: E 2 < E < E 1
Xl = + 2 ~ - ~ d~
x2 : ' ~ ,
~ 'E ~ - 4v ~ ( l - u )
x3 = ~ I + 2R(1+c )~
: [I+c+ ( I -c )C] -1
I E2_4v
= 2/1-~u ~/E 2 - 4v 2 t
F(k 2 ) .....
k 2 : ½ 1 1 + P ]
Vp~+ I
~ : R : ~ I + v 2- E ; R' = ~/u 2 + v z Eu ; ~ i : v - R ; c~ 2 : v + R; - l 2(u~ + ~) - E(u + I)
3 " ~(V - R'); e4 = ~(v + R'); P = - 2RR'
F(z) = zFI(½, ½; i; z) (hypergeometric function).
Jacobian elliptic functions:
~P 1 0 < k < I T = f dco' = F ( e , k ) ; 0 ~ - k 2 s i n 2 p ,
~T k 2 s = s n ( ~ , k ) = s imp ; T = 4K ; K = ~- F( ) ,
c = cn ( -~ ,k ) : cos~p ; T = 4K ; - - = -
d = d n ( T , k ) = ~/ i - k2sin2~p ; T : 2K
116
R E F E R E N C E S
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1 1 7
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t ive SU(2) Models, - t o be published. 2B)A.Erd~lyi: Higher Transcedental Functions, vol.3, New York: McGraw-
Hi l l , 1955. 29Jj.Moser:' Stable and random motions in dynamical systems. Ann.Math.
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eds. G.Casati, J.Ford. Lecture Notes in Physics 93, Berlin: Sprin- 31xger Verlag, 1979.
JN.Woodhouse: Geometric Quantization, Oxford: Oxford Univ. Press, 1980. 32)
V.Guillemin, S.Sternberg: Geometric Asymptotics. Mathematical Sur- veys, No.14, Amer.Math.Soc., Providence, R.I . , 1977.
33)D.J.Simms, N.Woodhouse Lectures on Geometric Quantization,Lecture Notes in Physics 5__33, New York: Springer Verlag, 1976.
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J.H.Rawnsley: Coherent States and K~hler Manifolds, Quart.J.Math. 28, 403 (1977).
37)M.F.Atiyah,-- W.Schmid: A Geometric Construction of the Discrete Series. Invent.math. 42, I (1977).
3B)Harish-Chandra: Representations of Semisimple Lie groups IV-VI.Amer. J.Math. 77, 743(1955); 78, 1 (1956); 78, 564 (1956).
39) - - - - - - VoBargmann: On a Hilbert Space of Analytic Functions and an Associ- ated Integral Transform I. Comm. Pure Appl.Math. i_44, 187 (1961).
4Q)H.Cartan, J.-P.Serre: Un th~or~me de finitude concernant les vari~t~s analytiques compactes. C.R.Acad.Sci.Paris 237, 128-130 (1953).
41)S.Helgason: A Duality for Symmetric Spaces with Applications to Group Representations I. Advan.Math. 5, 1-154 (1970). ~2) J.Dobaczewski A Unification of Boson Expansion Theories I. Nucl PhJ~s. A36921~ ) (1981 .
B. SYMMETRIES AND SUPERSYMMETRIES IN NUCLEI
DYNAMICAL SYMMETRY IN NUCLEAR COLLECTIVE AND REACTION PHYSICS
P. Kramer Institut fHr Theoretische Physik der Universit~t THbingen, F.R.G.
In this report a geometric approach is taken to nuclear collective and
reaction phenomena. As a starting point we take the linear canonical
transformations of the many-body coordinate and momentum observables iN
state space. In section I we develop this scheme as a basis for appli-
cations. The coherent states for the various groups of linear canonical
transformations will play an important part. In section 2 we describe
some advances in nuclear collective theory. In section 3 we discuss
molecular resonances and introduce the concept of unitary reaction
channels. In section 4 we extend the analysis of reaction channels from
the unitary to the symplectic group. Notes and references are given in
section 5.
I. The nuclear many-body system, oscillator coherent states, and the
symplectic scheme of canonical transformations
We consider observables and states of a system of A= n+1 particles 3
labelled by s = I,.., n+1 in ~ labeled by i = 1,2,3. Instead of the
usual Schr6dinger observables ~is and corresponding momenta Zis we may +
use oscillator crea~on and annihilation operators ais, ais. In what
follows we shall remove the c.m. coordinate and work with n relative
Jacobi coordinates. Now we use the overall oscillator ground state I0 >
with the property
IO> : aislO> = O, <0;O> = I
for the
1.1 Def. The oscillator coherent states are given by
+ e(x) = exp( ~ Xis ais)I0>
i,s This definition leads to the well-known Bargmann /I/ space with the
properties
1.2 Prop. The oscillator coherent states yields aHilbert space H of
analytic functions f(x) with the properties (HI) : H has a scalar prodUCt
121
<flq > = /f-~) qlx) dp(x),
d~(x) = P(x,~) ~ d Re(Xis) dIm(xis) i,s
p(x,x) = Po exp (- [ Xis Xis ) i,s
(@2): H has a reproducing kernel I(x,x'),
I(x,x') = exp( [ Xis X~s )- i,s
(G): the oscillator operators act in H according to
+ ais : f(x) ÷ (Xisf) (x) = Xis f(x)
ais : f(x) + (Disf) (x) = (~/~Xis)f(x)
~t should be clear that the oscillator coherent states are convenient
in particular when dealing with oscillator states. That their use is
not restricted to oscillator shell theory will be seen in the applica-
tion to collective excitations and to reaction theory.
The oscillator coherent states allow for a very explicit introduction
0f the linear transformations and their unitary representations in
State space. For the integral operators we refer to Bargmann.
1"3 Prop. The linear canonical transformations are generated by the
sYrnplectic group Sp(6n,]R) whose generators are of the form
I Cis jt = ~ (Xis Djt + Djt Xis)
I Kis it,+ = ~ Xis Xjt
1 Kis jt,- = 2 Dis Djt
~o introduce what we call the symplectic scheme we construct generators
of collective groups by contracting over the particle index, and gener-
ators of intrinsic groups by contracting over the IR 3 index. As collec-
tive groups we then generate the symplectic group Sp(6,~R) with genera-
tors Cij, Kij, + and Kij,_ and the unitary group U(3) with generators
Cij. Other collective groups could be generated by linear combinations
of these sets of generators. The rotation group associated with the
total orbital angular momentum is generated by C..- C... As intrinsic ij 3 l
grOUps we find a symplectic group Sp(2n,]R), a unitary group U(n) gene-
rated by Cst, and an orthogonal group O(n,~) generated by Cst- Cts.
~he orthogonal group O(n,IR) contains as a finite subgroup the symmetric
grOUp S(n+1). The complete scheme of groups is given by
1"4 ProD. The collective and intrinsic groups of linear canonical
transformations have the group/subgroup scheme
122
Sp(6,1R)
U 3) ~ ~
O(3,~)
collec£ive
Sp(2n,~)
1 u(n)
O (n,]R)
1 S (n+1)
intrinsic
The subgroups connected by broken lines commute with one another.
In state space the collective and intrinsic groups give rise to unitary
representations. The full n-body Hilbert space contains two irreducible
representations of the group Sp(6n,IR) , both together form what is
called the metaplectic representation. If this metaplectic representa-
tion is reduced according to the scheme given in Prop. 1.4, we obtain
1.5 Prop. The irreducible representations of collective and intrinsic
groups in the subduction from Sp(6n,IR) are related by complementarit~
The irreducible representations of pairs of commuting collective and
intrinsic qroups occur once and only once in correlated pairs.
We turn now to some aspects of the nuclear many-body system in the
light of this symplectic scheme. The fermion properties of nucleons
may be introduced in the symplectic scheme via the supermultiplet
classification which leads to the specification of the spin-isospin
symmetry. This symmetry determines the irreducible representation f
of the symmetric group S(n+1) of orbital permutations. The oscillator
shell theory arises if we assume the intrinsic group U(n) as a synmetry
group of the system. This assumption implies that the irreducible re-
presentation [hlh2h 30 n-3] of this group be fixed. By complementarity,
Prop. 1.5, one gets that the collective group U(3) is a dynamical sym-
metry group of the model. So the most general operator diagonal in the
oscillator shell model is a function of the generators Cij of U(3),
acting in a representation space of the IR [hlh2h3]. The symplectic
shell model arises if instead of U(n) we take O(n,IR) as a symmetry
group. Then by complementarity the symplectic group Sp(6,IR) becomes
the dynamical group of the model. The irreducible representation space
of the group is infinite-dimensional but it possesses a lowest state
labeled by the subgroup U(3). This extremal state obeys the equations
123
I{jlJ2J3}>: Kij,_ l{jlJ2J3}> = O
Ci< j l{jlJ2J3}> = O
Cii ]{JlJ2J3 }> = I{JlJ2J3}> Ji
Or in other words, it is characterized by a lowest IR of U(3). The
choice of this IR is restricted by the requirement of a minimal repre-
sentation of S(n+1) which induces a minimal representation of O(n,IR).
So in the symplectic shell model we consider an infinite-dimensional
representation space of Sp(6,]R) based on a single shell model base
State. We refer to Draayer, Weeks and Rosensteel /2,3/ for computations
in this model.
2. Collective theory and coherent states
We refer to the review on advances in the theory of collective motion
in nuclei /4/ and to the review of collective classical dynamics given
at the 1982 Brasov School /5/. We start with a brief historical intro-
duction /4/.
The theory of collective motion in nuclei has as its geometric origin
the comparison of certain nuclear phenomena with properties of a liquid
drop. Bohr and Mottelson (1952,53) /7,8/ introduced the idea of a irro-
tational flow and explained a variety of collective phenomena by the
deformations and vibrations of a nuclear fluid. The nuclear shell theo-
rY succeeded later on in the representation of collective excitation by
Coherent superpositions of many single-particle excitations. Elliott
(1958) /9/ showed that collective levels in the shell theory can be
Connected to a group SU(3). Independently of the shell theory, various
attempts were made to develop the geometric ideas implicit in the Bohr-
Mottelson model. Weaver, Biedenharn and Cusson /10,11,12/ introduced
the group SL(3,1R) of volume-preserving deformations into the collective
theory. With this group, they connected kinematical transformations of
the system of A nucleons, the vortex spin, and a spectrum generating
algebra. Inclusion of the mass quadrupole tensor leads to a natural ex-
tension of this group which was also studied by Rowe, Rosensteel and
COllaborators /13,14/. In the geometric models, it is the final goal to
eXPlain collective phenomena from the point of view of many-body dyn-
•ics. Therefore one has to link the collective coordinates to the sin-
gle-particle coordinates. This program was already started by Lipkin
(1955) /15/ and by Villars (1957) /16/. Whereas these authors tried to
keep the single-particle coordinates, new viewpoints were developed
124
later by Zickendraht /17/, by Dzyublik et al. /18/, and by Buck Bieden-
harn and Cusson /19/ by use of the orthogonal intrinsic group SO(n,]R)
acting on the particle indices and commuting with S0(3,IR). Rowe and
Rosensteel (1980) /20/ were the first to analyze this scheme through
an orbit analysis in configuration space. Vanagas (1977) /21/ pointed
out the close relation of the group SO(n,IR) to the symmetric group of
orbital permutatfons and proposed the qroup SO(n,]R) as a symmetry
group of the collective hamiltonian. The symplectic group Sp(2r,IR)
(r = 1,2,3 is the dimension of the space) as a phenomenological group
for collective motion was proposed by Goshen and Lipkin for r = I /22/
and 2 /23/ and studied by Rosensteel and Rowe /24/ for r = 3.
The main tool for a systematic study of both quantum and classical
collective dynamics is provided by coherent states. In section I we in-
troduced the oscillator coherent states which are known to belong to
the Weyl group of translations in phase space /6/. For collective theO-
ry we require coherent states associated with the symplectic group.
These coherent states may be obtained but are technically much more
difficult to handle than the oscillator states.
The essential idea for coherent states is to generate a continuous set
of states by acting with the group operators on an extremal state in a
representation space. This extremal state may have a stability subgroup
given by
H = {hIU-1(h) Iextr> = lextr>l(h)}
Then one should decompose the group G into cosets w.r.t. H and genera-
ted by representatives c and define the coherent states by
U -1 (c) l extr > : = I c >
For the application to physics one would like to have an interpretation
of the coset parameters contained in c in terms of observables.
We apply these ideas to the symplectic group Sp(6,IR). Its extremal
states are the lowest U(3) states characterized by a highest weight
{jlJ2J3 }. The stability group H becomes
Table 1
{JlJ2J 3 }
J1=J2=J3
J1>J2=J3
Jl>J2>J3
H
U(3)
U(1)×U(2)
U(1)×U(1)×U(1)
dim (Sp (6, ]~) /H)
21-9 = 12
21-5 = 16
21-3 = 18
~25
The first one of these three cases corresponds to collective excitations
of nuclei with a base state of a closed harmonic oscillator state. In
/25,26,27/ we employed the Iwasawa decomposition of the non-compact
group Sp(6,~R) to obtain a characterization of the cosets Sp(6,]R~U(3).
To describe it we require the real form of the generators of Sp(6,]R)
given by
I I = ~ (P +Q)ij + i ~ (A-tA)ij Cij
I I Kij'± = 4 (Q-P)ij ~ i ~ (A +tA) ij
2.1 Prop. The unitary coherent states for the case H = U(3) become
I [ 0kl(Alk +Akl)] l{JoJo jO }> Is,Z> = exp[i ~ I Zkl Qlk]exp[ -i I
k,l k,l
@t = @, s = exp@>O, tz = Z
The expectation values of the generators of Sp(6,1R) in this case
become
Qik (s,Z) = Jo(S2)ik ,
Aik (s,Z) = Jo(S2Z)ik ,
Pik (s,Z) = jo(ZS2Z +s-2) ik .
The dequantization procedure shows that the generalized Poisson brac-
kets become
{(s2)ij, Zrs} = jo1(~ir~js + 6is~jr)
{(s2)ij, (S2)rs} = O
{Zij, Zrs} = O
so that the quantities Zij play the role of generalized momenta asso-
Ciated with the coordinate quadrupole.
An alternative form of the same coset in terms of analytic parameters
Was given in /28/. It leads to a Hilbert space of analytic functions in
Six complex variables. In the meantime we succeeded in getting for all
eases of Table I general analytic and non-unitary coherent states /29/.
2'2 Prop Let B = (bij) denote a complex symmetric 3 x3 matrix res-
tricted to a Siegel domain, I-B+B>O, and let k be a complex upper tri-
angular matrix with all diagonal elements equal to one. There exists
a Hilbert space H of analytic functions f(k,B) with the properties
(HI) H has a scalar product
<flq> = f f(k,S) q(k,B) d~(k,B),
126
d~(k,B) = p(k,B) n dRe(kij)dIM(kij) ~ dRe(brs) d In(brs) i<j r<s
2 j3-J2-2(A123)-w 3 +6 p(k,B) = PO (A~)J2-Jl-2(AI2) 23
with the A's being subdeterminants of the 3x3 matrix
k(l -B+B) -I k +
(H2) H has the reproducing kernel
<k m 1 k'B'> = (A 11)j1-j2 (A12)12 j2-J3 (At 2323 )j3
with the £'s being subdeterminants of the 3x3 matrix
k(I -(B')+B)-I(k') +
(G) The symplectic group Sp(6,1R) acts in H through an explicit inte-
gral operator. We shall not discuss here the rather technical aspects
of this result. So far it has been used in cooperation with Moshinsky
/30/ to derive boson realizations of symplectic algebras, a topic which
we cannot cover here.
3. Molecular resonances and unitary reaction channels
The reaction theory of composite particles may be formulated effective-
ly in terms of the Bargmann Hilbert space described in section 1. It
now becomes essential to use Jacobi coordinates adapted to the split-
ting of a reaction channels into two fragments and the relative motion-
We follow /31/ in the analysis and refer to the reviews by Cindro /32,
33/ for the experiments and theories of nuclear molecular resonances.
To motivate the unitary analysis, consider two nuclear reaction part-
ners 1,2 and their relative motion (r.m.). If j1 j2 and jr.m. denote r
the corresponding angular momenta, the total angular momentum is given
by j = j1 + j2 + jr.m.
The corresponding eigenstates of a reaction channel are characterized,
with respect to angular momentum, by the coupling
j' x j,' x jr-m" + j
Suppose now that we can characterize the reaction partners by oscilla-
tor shell model states labeled by irreducible representations or parti"
tions h' = [h~ h 2' hl] and h" = [h'~ h~ h 3''] respectively of the corres-
ponding unitary groups U(3). For an excitation N = O,1,2.. the relative
motion carries with respect ot U(3) a partition [N] = [NOO]. A U(3)
channel state may now be characterized by the eigenstates of the summed
127
U(3) generators and by the coupling
h' × h" x [N] + h = [h I h 2 h3].
This coupling is governed by Littlewood's rules. To describe a general
state of the relative motion we must admit any N in this coupling. If
in a channel the overall partition h has been f~xed we speak of a U(3)
Channel.
The coupling rules for U(3) channels depend only on the Wigner-Racah
algebra of this group, but the occurrence of possible partitions h',
h" and H is related to the underlying system of A = n+1 nucleons. The
relations can be inferred from a comparison of the U(3) channel states
of partition h = [h I h 2 h 3] with the compound nucleus states of parti-
tion h c = [hlCh~h~ ]. One obtains the following types of configura-
tions /31/
Type partner I partner 2 compound system
la s... s... s... 4
b s p...
4 4 2a s... s p... s p...
412 b s p (sd)...
4 4 4 3a s p... s p... s p...
412 b s p (sd) ...
An analysis of the U(3) channels shows the following features:
(a) Only for collisions of type 3b, where two sp partners are colliding,
the compound states s4p12(sd).., are strictly excluded in the U(3)
Channels. More precisely, for
h = [h I h 2 h3]: h I - h 3 > 12
the lowest shell configuration cannot be formed.
(h) We assume without loss of generality h{ - h½ > h~' - h½' and
denote the quantum excitation of the compound nucleus by m. For
h~ + h½' + 2(h i + hi' ) > 16 + 2m
the formation of a compound state with a quantum excitation m is for-
hidden.
(c) If n is the lowest value of m which does not fulfill the inequa~ty,
the formation of an excited compound state requires the transfer of m
nUCleons from the sd to the pf shell.
Now we apply the following U(3) principle for nuclear molecular reso-
nances: Nuclear molecular resonances occur in U(3) channels which do
128
not match the lowest shell model compound levels. A detailed account of
the fragment configurations and of the numbers n is given in Table 2 of
/311.
A complete analysis of the channels 160 + 12+bx, o ~b~ 4 is given in
/31/. The orbital partitions f in these channels combine according to
f, x f,, ÷ f
[44 ] x [43b] + [47b]
With respect to the U(3) partitions, we have
[4 4 4] × [4 4 b] x [N O O] = [8 8 4 + b] × IN O O]
4 = ~ [8 + N + W - 4 8 8 + b - W]
W=b
We call W the mode quantum number. The compound nucleus configurations
admit those U(3) partitions which would correspond to N > 16+b. In con"
trast, the U(3) channel states require N > 16+2b. This means that for
4 ~ b > I the U(3) channels cannot match the compound states for frag-
ment partitions h' = [4 4 4], h" = [4 4 b]. Moreover it turns out that
for b < 3 and N = 16 + 2b, only the U(3) coupling mode W= b with the
most compact partition is open while all other modes are forbidden.
Generally one finds that the mode W only exists if the relative motion
is excited by at least n = W additional oscillator quanta. These features
then single out candidates for nuclear molecular resonances.
The U(3) channel states may be written as
l[h I h 2 h3]f ) = cfl [h , x h" x [N]]h>
f where c is the Young operator for the symmetric group S(n+1) and the
square brackets indicate U(3) coupling of fragment and relative motion
states. The numbers
q = ([h I h 2 h3]fl[h I h 2 h3]f) # O
are obtained in /31/ as the eigenvalues of the normalization operator
/6/ from the analytic expressions in Bargmann space. For 160 + 160,
these numbers are non-zero only for the even values N = 24,26.. and
hence display the extreme even-odd effect. This effect extends qualita"
tively to the whole configuration. In the U(3) theory, the partition
h = [h I h 2 h 3] is often interpreted in terms of deformation. In all
cases considered, we have h I - h 2 >> I, indicating a strong deformation
of the system in one space dimension, in accordance with the simple
quasimolecular picture.
The interaction in the U(3) channel configuration has for 160 + 160
been studied in /31/ with Brink-Boeker, Volkov and Coulomb forces by
Tab
le 2
: S
ymp
lect
ic
de
com
po
sitio
n
in
the
160
+ 16
0 ch
an
ne
l. Th
e u
nit
ary
p
art
itio
n
is
[32
+ 2k
8
8],
k =
0,i
,...
an
d th
e sy
mp
lect
ic
pa
rtit
ion
is
{3
2 +
2~
8 8
},
p =
0,1
,...
2 p:
0
I 2
3 4
5 6
k:
6 0.
1720
2809
1 0.
3781
2115
0 0.
3096
7811
9 0.
1175
6066
7 0.
0210
6909
2 0.
0015
1913
6 0.
0000
2374
4
5 0.
2237
1098
2 0.
4138
6531
7 0.
2738
4661
0 0.
0787
3307
3 0.
0094
9829
9 0.
0003
4571
8
4 0.
2942
7953
8 0.
4399
3304
7 0.
2205
0455
1 0.
0426
8299
9 0.
0025
9986
4
3 0.
3918
0022
6 0.
4437
7372
6 0.
1497
8423
5 0.
0146
4181
3
0.52
8276
931
0.72
1804
511
0.40
3015
391
0.27
8195
489
0.06
8707
678
0 1.
0000
0000
0
130
Table 3: Symplectic decomposition in the 160 + 13X channel. The uni tary
pa r t i t i on is
a) [29 + 2k 8 5], k = 0 , I , . . and the symplectic pa r t i t i on is
{29 + 2~ 8 5}, U : 0 , i , . . (odd channel par i ty )
b) [30 + 2k 8 5], k = 0 , i , . . and the symplectic pa r t i t i on is
{30 + 2u 8 5} ~ = 0 , i , . . (even channel par i ty ) a)
2 ~ 0 1 2 3 4
k: 4 0.121272128 0.408018088 0.336916458 0.117750836 0.016042490
3 0.189169002 0.487719007 0.274198210 0.048913781
2 0.306593787 0.538687772 0.154718441
0.526739937 0.473260063
0 1.000000000
b)
2 ~! 0 I 2 3 4
k: 4 0.253613667 0.405889143 0.254414354 0.076633253 0.009449583
3 0.353464135 0.433296099 0.184834287 0.028405479
2 0.495813026 0.414005513 0.090181461
0.700879165 0.299120835
0 1.000000000
131
analytic methods. The model space is chosen with 24 < N < 40 and
O~L! 40. It is found that a good fit to the experimentally observed
resonances is obtained only upon inclusion of many U(3) channel states.
The good reproduction of the experimental resonances in the low angular
momentum region by a quasi-bound calculation suggests that a quasi-
molecule is formed and shows that the phenomenon of molecular resonan-
ces can be linked to states in the entrance channel which are highly
Selective with respect to the U(3) and supermultiplet quantum numbers.
4. Symplectic properties and selection rules for unitary reaction
channels.
The unitary group used in the coupling of reaction channels is the one
Which appeared as the collective group U(3)in the symplectic scheme
discussed in section I. Now we want to consider reaction channels in
the wider frame of the symplectic group Sp(6,~).
A general state extremal in U(3) may be written as
I{91 J2 J3 } [hl h2 h3]>
in the Sp(6,~)> U(3) chain of groups. We have studied the symplectic
decomposition of U(3) channel states wrt the symplectic group. We de-
fine the normalized channel state
- I / 2 l [ h I h 2 h 3 ] f > = l [ h I h 2 h 3] f )
a~d consider for fixed f the expansion
I[hl h2 h3]f> = ~ I{91 J2 93} [hl h2 h3] f > {Jij2J3 }
{Jl J2 93}' [hl h2 h3])
The coefficients a have been obtained by solving recursion relations,
and examples are given in Tables 2,3, for the configurations discussed
already in section 3. We describe here the main features of the ex-
Pansion:
(a) the U(3) channel states of lowest excitation belong to a single
syrnplectic representation identical to this U(3) representation.
(b) the U(3) channel states of higher excitation contain superpositions
of symplectic representations. All these states are orthogonal to the
SYmplectic compound states.
For the physics of these states, the extension from the unitary to the
Symplectic group has important consequences: Not only the overlap, but
132
also the matrix elements of the symplectic generators between channel
and compound states vanish. Among them there are the kinetic energy,
the collective potential energy and the (mass) quadrupole operator. In
the U(3) analysis, a change of frequency between the channel and com-
pound states removes in part the orthogonality due to this group. In
the symplectic analysis, a change of frequency cannot mix different ir-
reducible representations, and the orthogonality of channel and com-
pound states persists. The interpretation of the symplectic partition
{Jl J2 J3 } admits an interpretation in terms of intrinsic deformation.
In this interpretation, the symplectic channel states are linearly
strongly deformed in agreement with the quasimolecular picture. The
irreducible representation of the symplectic group admit rotational
states of a more general form and involving strong U(3) mixtures.
We therefore propose for nuclear molecular resonances the following
symplectic principle: Nuclear molecular resonances occur in symplectiC
reaction channels which do not match the lowest symplectic shell model
compound levels. The results of the calculation on 160 + 160 which lead
to strong U(3) mixtures support this broader interpretation.
From this point of view of nuclear reaction physics, the channel ana-
lysis presented here implies that the channel states build up, for
sufficient high mass number, selective symplectic many-body quantum
numbers and differ from the usual collective states of the shell model
analysis. The dependence of these selective channel states on the mass
number should have other relevant consequences for heavier nuclei
which should be explored.
5. Notes and References
The derivation of the Hilbert space of analytic functions given by
Bargmann /I/ is generalized in /32/ and applied in particular to the
group Sp(6,~). The complementarity relation of Prop. 1.5 is given by
Moshinsky and Quesne /35/. The harmonic oscillator states for nucleons
are treated by Kramer and Moshinsky /36/. The model based on the
O(n,~) - invariance was introduced by Vanagas /21/.
For examples of coherent states we refer to Perelomov /37/. The syste-
matic exploration of cosets of the symplectic group for collective
theory is developed in /38,28,39,25,26,27,29/. ADplications of the co"
herent states of Prop. 2.1 are given in /28/. The general dequantiza-
tion scheme is treated by Kramer and Saraceno /43/. The branching of
133
symplectic states into reaction channels is treated by Arickxs/44/, by
Vasilewski and Filippov /42/, Hecht and Braunschweig /43/ and by
Suzuki /44/. In section 4 we treat the inverse problem.
Acknowledgment
The work reported here was supported by the Deutsche Forschungs-
gemeinschaft
References
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/3/ G. Rosensteel, J.P. Draayer and K.J.Weeks, Nucl.Phys.A 419(1984)I-12
/4/ p. Kramer, Advances in the theory of collective motion in nuclei, Lecture Notes in Physics (1984) 343-351
/5/ p. Kramer and M. Saraceno, Geometry and Application of the Time- Dependent Variational Principle in Quantum Mechanics, in: Nuclear Collective Dynamics, ed. by D. Bucuresti, V. Ceausescu and N.Y. Zamfir, World Scientific (1983) 46-73
/6/ p. Kramer, G. John and D. Schenzle, Group Theory and the Inter- action of Composite Nucleon Systems, Vieweg, Braunschweig (1980) 75-99
/7/ A. Bohr, Dan.Mat.Fys.Medd. 26 (1952) 14
/8/ A. Bohr and B. Mottelson, Dan.Mat. Fys.Medd. 27 (1953) No. 16
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/12/ O.L. Weaver, R.J. Cusson and L.C. Biedenharn, Ann. Phys. 102 (1976) 493
/13/ p. Gulshani and D.J. Rowe, Canad.J.Phys. 54 (1976) 970
/14/ G. Rosensteel and D.J. Rowe, Ann. Phys. 123 (1979) 36
/15/ H.J. Lipkin, A. De Shalit and I. Talmi, Nuovo Cimento II(1955)773
/16/ F. Villars, Nucl. Phys. 3 (1957) 240
/17/ W. Zickendraht, J.Math. Phys. 12 (1971) 1663
/18/ A.Ya. Dzyublik, V.I. Ovcharenko, A.I. Steshenko and G.F. Filippov, Sov.J.Nucl. Phys. 15 (1972) 487
/19/ B. Buck, L.C. Biedenharn and R.Y. Cusson, Nucl. Phys. A317(1979)205
/20/ D.J. Rowe and G. Rosensteel, Ann. Phys. 126 (1980) 198
/21/ V. Vanagas, The microscopic theory, Lecture notes, University of Toronto, 1977
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/22/ S. Goshen and H.J. Lipkin, Ann. Phys. (N.Y.) 6 (1959) 301
/23/ S. Goshen and H.J. Lipkin in Spectroscopic and Group Theoretical Methods in Physics, ed. F. Block, North-liolland, Amsterdam 1968
/24/ G. Rosensteel and D.J. Rowe, Ann. Phys. (N.Y.) 126 (1980) 343
/25/ P. Kramer, Z. Papadopolos and W. Schweizer, Nucl.Phys.A 431 (1984) 75-89
/26/ P. Kramer, Z. Papadopolos and W. Schweizer, Nucl. Phys. A 441 (1984) 461-476
/27/ P. Kramer, Z. Papadopolos and W. Schweizer, J.Math. Phys. 27 (1986) 24-28
/28/ P. Kramer, Ann. Phys. 141 (1982) 254-268 and 269-282
/29/ P. Kramer and Z. Papadopolos, J.Phys. A19(1986)1083-1092
/30/ O. Castanos, P. Kramer and M. Moshinsky, J.Phys.A Math. Gen. 18 (1985) L493-L498 0. Castanos, P. Kramer and M. Moshinsky, J.Math. Phys. 27 (1986) 924-935
/31/ R. Bader and P. Kramer, Nucl. Phys. A 441 1985) 174
/32/ N. Cindro, The Resonant Behaviour of Heavy-Ion Systems. Rivista del Nuovo Cim. 6 (1981) 1-64
/33/ N. Cindro, Nuclear Molecules: The Present Status, in: Nuclear Collective Dynamics, compare ref /5/ (1983)397-420
/34/ P. Kramer, R. Bader, Z. Papadopolos and W. Schweizer, Contr. Int. Conf. on Symmetries in Nuclear Structure and Reactions, Dubrovnik (1986)
/35/ M. Moshinsky and C. Quesne, J.Math. Phys. 11 (1970) 1631
/36/ P. Kramer and M. Moshinsky, Group Theory of Harmonic Oscillators and Nuclear Structure, in: Group Theory and its Applications, ed. by E.M. Loebl (1968)
/37/ Perelomov, A.M. Comm.Math. Phys. 261 (1972) 222
/38/ P. Kramer, Kinam 4 (1982) 279-292
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/40/ P. Kramer and M. Saraceno, Geometry of the Time-Dependent Varia- tional Principle in Quantum Mechanics, Lecture Notes in Physics 140 (1981) 1-98
/41/ F. Arickxs, Nucl. Phys.A 284 (1977) 264
/42/ V.S. Vasilewski and G.F. FiliDpov, Sov. J.Nucl. Phys.33(1981)500-506
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/44/ Y. Suzuki, Nucl. Phys. A 448 (1986) 395
THE RESTRICTED DYNAMICS NUCLEAd ~ODELS:
CONCEPTIONS ~D APPLICATIONS
V.Vanagas
Institute of Physics, Academy of Sciences of the Lithuanian SSE
232600,Vilnius, USSR
Table of Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . .
2. ~ain ideas and definitions . . . . . . . . . . . . . . . .
3. The constant non-abelian fields H~niltonian H~ .....
4. The decomposition of J'~G in terms of the restricted I •
dynamics operators . . . . . . . . . ...... • • . . .
5. The microscopic collective model and its simpli£ied versions
6. Anticollective Hamiltonians . . . . . . . . . . . . . . .
7. Spectra of microscopic collective and anticollective
Hamil tonians . . . . . . . . . . . . . . . . . . . . . . .
8. Concluding remarks . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . • • . . . . . . • .
i. Introduction
A lot of publications ~ books, review articles, lecture notes and
Papers ~ have been devoted to the nuclear models with algebraic struc-
ture, developed throughout the last two decades. Different terminology
and mathematical methods were used, which makes it difficult to estab-
lish the relationship between them, even in cases when they are equi-
Valent or very similar. One of the aims of our lecture notes is to
SYstematize various algebraic approaches to the many-body problem in
nuclear physics, beginningwith most general microscopic models and
endin~ with simplest phenomenological ones.
We will make use of the circumstace that at the bottom of many al~
gebraic models lies the unifying restricted dynamics idea, which makes
it possible to construct a lot of nuclear models both traditional and
non-traditional.Our aim is also to present a large scale overall review
of the restricted dynamics models, outlining the conceptions and gene-
ral features. In order not to be bogged down with details, instead of
Scrutinizing models, and commenting results, following from them, we
t36
will meke use of the obvious fact, that all the features of a given
model are represented by its Hamiltonian, thus two models are equiva-
lent if their B,m~ltonians are identic. ;~e will also state, that the j~l j~ I.I'/IZ/QIz,/" m@del is a submodel of if the Hamiltonian ,,O~,~]for~c is
a particular case of the Hamiltonian H 0 ~v~)for v~ . Additionally
if H 0 (~) is microscopically__ derived and ~/e(~') is the phenomeno-
logicalj,. --,Hamilt°nian' then ~/0 (J~) gives the microscopic derivation
of H oct/), i.e. the microscopic justification of the phenomenolo-
gical model o
A very specific feature of the restricted dynamics is associated
with characteristics of the spaces in which the Hamiltonians act. In
the restricted dynamics models both Hamiltonians and spaces they act
in are equally important. W.e are not going to use approximate expres-
sions for the operators, ususally used, when some non-diagonal ele~
merits are neglected. In order to introduce the restricted dynamics
Hamiltonians there is no need to neglect some non-diagonal matrix ele-
ments. Those Hamiltonians conserve the additional quantum members, io
e. possess the additional integrals of motion, which characterize the
spaces involved. Thus we may say, that the restricteddynamics model
is defined by t-.the Hamiltonian~, ~O and by the space ~O where it
acts, i.e. by l~o~ J(ol" In order to explain properl~ this main re-
stricted dynamics feature we included in our lecture notes the recent ~
ly developed example of the application of the restricted dynamics de-
composition to the ~U~l)constant non-abelian gluon field Hanuiltonian+
Limited space prevents the possibility to include many topics re-
lated to the title of lecture notes. Being devoted to very definite
questions, namely to general description of the restricted dynamics
idea and models, this course must be treated as some part of a large
mosaic, presenting picture of the restricted dynamics models and their
applications° Other fragments of this mosaic, not always completed, ~re
scattered over many publications~ some of them can be found in the
lecture notes [1-4] and in review articles [5-8J.
2. ~ain ideas and definitions
The general method of constructing microscopic models of nucleus
in the framework of the restricted dynamics rests on the following
idea proposed and developed in L2 ] (see also [ 6 ] ). Let us consider
the quantum many-particle system, described by the Schrodinger equa-
tion H~=~ . Not able to solve this equation because of its lJ ~,. Q.,
complexity we substitute it by equation ~I0~o=6o ~0 for the restric-
137
ted dynamics Hamiltonian ~0 °
This Hamiltonian is introduced using the operatorial decomposi-
tion
H = IWo -~ l-l~ * l-l~ -~. . . (2. l ) With the first term, acting within some subspace ~(A ~O)of the space
~(A) , presented as the direct sum
(2.2) In (2.2) A denote all the integrals of motion of the original Hamil-
tertian ~I , and ~O - characteristics of the subspace ~(A K~) •
From the definition of ~o it follows, that the matrix represen-
tation of H in the basis ~[A K-P)( ~ denotes some basis indices
in <k~ ~ oJ) satisfies the condition
< a Ko r the I/l'l<~ £'>= 2(A A') ~(~o k'~) <Cl FIo~4 ~> I F'>. (2.3)
In (2.3) ~(AA I) reflects the conservation laws for the integrals of motion of H while ~(~oK~) guarantees the restriction of ~o with- i~ ~C~o). '
There are two methods of restriction of H to the subspace
~(A Me) , based on the algebraic technique. In the first one - we Shall call it the first kind of restriction- the decomposition (2.1)
is performed by means of the irreducible analysis of H with respect
to some group ~ . In this case
H=/.icon÷ y, ;~ :0 ~ (2.4)
where Ho~ H (0) is the G -scalar term and ~'/ (~) with ~ (0) deno- tes the rest terms of H .
The second method of restriction of ~/ - we shall call it the
Second kind of restriction - rests on properties of the infinitesi-
mal operators. It is well known that the infinitesimal operators of
the group G in the G -irreducible basis are diagonal with respect
to the ~ -irreducible representations. Thus the decomposition of I~
la terms of the multilinear form of the infinitesimal operators ~. &
Of the group ~ guarantees the restriction H to ~(~Ko). Let us
Present this decomposition in the form of H = Herb ~ , where ~% 11 /%
~ " ~k (2.5)
In (2.5) O~t~... ~k - coefficients depending on ~/o . If ~o stand
138
g for -irreducible representations, then H o satisfies the condi-
tion (2.3), thus (2.5) is restricted dynamics He~iltonian.
In both the first and the second kinds of restriction the opera- ||
tot H 0 is projectedH from H , and the Schrodinger equation is int-
roduced for H o . ,~e say, that the physical picture of the nucleus,
described by the eigenfunctions ~o of H 0 , is the model of nucle-
us, originated from the Hamiltonian 14 , restricted to the subspace
~(A" Ko) , or, briefly - the nuclear model with restricted dynanlics.
V~e have stated already in the previous section, that in this sense
"--~~o)~(AK°)~ gives the nuclear model with the restricted dynamics.The
constructive ex~nples of both kinds of restrictions will be discussed
in two next sections.
From the definition (2.3) it follows, that in the restricted dy-
n~nics models the additional quantum nun~bers are conserved. In this
respect the~ essentially differ from traditional nuclear models, in
the latter the total H~iltonian H and some approxin~te characteris-
tics of the ~1odel functions are used, for example - the configurations
in the shell-model, Nilssonorbits in deformed nuclei, ~/5(~-ff) -irre-
ducible representations in the method of h2perspherical functions,etc.
Those ch~acteristics restrict the space, where the total Hamiltoni~uu
H H acts. However this tEpe of restriction is artificial, because
does not conserve those characteristics. Thus in traditional models it
is taken for granted that the ei~envalues and eigenstates of ~/ ,found
in this restricted space, reflect the properties of the solutions of
the Schrodinger equation for ~/ well enough. Strictly speaking, that
is not true, thus the results depend considerably on the basis and t~-
pe of the restriction used.
In non-traditional models, based on the ~ -scalar Ha~iltonian
~ ~(e) in (2.4) (the first kind of restriction), or on the ~ -infini -
tesimal type Hamiltonian (2.5) (the second kind of restriction) addi-
tional characteristics Ko are conserved. We have Got a simpler Ha -
~itonian ~0 and its integrals of motion K O . This accounts for
the essential difference between the quality of results, obtained in
non-traditional models in comparison with those in traditional ones
(more detailed discussion see in [8] ).
3. The constant non-abelian fields Hamiltonian
We will return to the general desription of the nuclear Hanuilto~
nians in section 5. Now we are going to illustrate the first and the
second kind of restrictions by means of an example, related to the
gluon field model, resting on the s~plified quantum chromodinamics
139
equations. In Ha~niltonian form with temporary gauge those equations
are equivalent to the functional Schrodinger equation (see [9] and
~eferences in this paper). In its turn this equation in [lO] has been
Simplified and reduced to the usual time-independent Schrodinger equa-
tion with tile following constant non-abelian gluon fields Ilamiltonian:
~o Ze
~=~ ~i~ (3.1) I~ (3.1) [ , ] - the co~r~nutator, Z6 gives the space dimension, ~
denotes following linear combinations of the infin_itesin~l operators
~ . of the group G :
o _7.
g=~ ~ (3.2)
Where ~ (~_ number of ~he operators 7 i and 3<2 -variables for
I G , running values-oo < ~ <+ o ° . In case of ~ang-~dills fields
i~ a ~peoial ~ita=j g U(~) ~roup ~ith J< : 2, 3, .... ~f ~e
onsmder ~: aetlng in an arbitrar) S U ~ J ~ ) - i r r e d u c i b l e space ~(),
the dependence of ~@ on ~ is relatively smmple and can be taken
into account b3 substituting C k and £p in (3.1) with ne~ cons-
~t~ ~) ~nd c,',(~) depe~din~ o~ ~ (see fo~ ~etails / ' i l l ). Thus H7 does not ~essentially change if we take either the simplest
g U(*)-irreduoible represent.~tlon [~3.~[~0"-. 4] or She ~djoint gU(~)-irreducible representation [{'q3 W =- [~ 0... 0 -{3 as A •
~or ~e first o~se gU(~)oon~t=t ~ields ~{~iltoni~n ~/~ ,~it~ tO = 2, 3, obtained in [12] , has the expression
Zo ~:o =- I ~ I £ f , ,
I
-0=~ .~<~ =~ (3.3) ~he re V and ~ - some positive constants and o~9~, - the 8/igle
ue~ween tAree dimensional vectors Z and t ~ (
e ~(~ ~-~3 , and similarly - for ~9' ).
The potential term of HG can be presented in few different
~Orms, follo~ving from the general explicit expression of the~U(YV)Ha-
~iltoni~- obt=ined in rll]- If we introduce dimensionless variables
9,'~ (V@} 3)(; (eigenvalue of l-l~ is a number,mul~iplied by q'/3V 13)
then ~ for'3 U{.[) field and arbitrary "~> "~ has the fodrm of
140
where
(3.4)
~o 3
~o
= T - p . ( 3 . 5 )
In (3.5) Z o = min( ~o ,3). The first expression in (3.5) gives
in terms of the Carthesian variables. From the second expression in
(3.5) follows (3.3); this expression shows, that (3-4) is a scalar
operator with respect to the transformations of the orthogonal group
03 • Analogically from the next expression in (3.5) it follows,that
(3.4) is also O~ =scalar operator. Thus we conclude, that Hr. in
case of ~T~ '~ i~ ~ xOz O - sca la r operator, Consequently ~/~ con- serves both O~ -irreducible representation LO and U, -irreducib-
le representation ~ . The eigenvalues 6_ of t'l_ depend on ,
The las t two microscopic expressions in (3-5), obtained in.~ ~ I l l ] , rgiVe H G in terms of the collective variables ~(4o. int oduced iN r "~
L13J or in terms of the global radius ~ and collective ~ variable, I
used in nuclear theorb~, generalized to the ZO -dimensiofial case (see
details [6, n] ) . The Hamiltonian (3-4) depends on ~ Z 0 Carthesian variables, and
l I
the Schrodinger equation for ~ , having the form
formally can be treated as an equation for the ~o+~ equivalent par-
ticle system (with the excluded centrum-of-mass motion), interacting
with a specific collective potential ~dZ~-~)/~Zo-- --~&/~- . In case
of Z O = 2, (3.6) is equivalent to the three-body Schrodinger equa-
tion; if Z~ = 3, we have the analogue of the equation for the four-
body system. In both cases (3.6) can be solved numerically, using com ~
141
puters. Our aim, however, is to study the features of ~]G and its
Spectrum employing analytical methods.
4. The decomposition of ~/~ in terms of the restricted d3~arr~ics operators
Let us perform the decomposition (2.1) of ~/G • In order to re-
strict this operator to some subspace ~O of ~ we must choose a de-
finite chain of groups, containing the symmetry group q ~ OZo of ~/C
as a subgroup. The algebraic methods, developed in the nuclear theor 2
(see [14, i~ 5] ) suggest following chains [ii] :
U U ~ ~ U
Which are equivalent by means of the realization of the Hilbert space
metric U,, -irreducible representation as E O-[&O 0... and both
U~ - an~ ° U~o -irreducible representations, contained in E O , as
~[~I"" ~ 0... O] ; note, that irreducible representations for
both "~3 an~ ~ are given by the same partition E . Only two
of the ~p(6Z0,~,) °-irreducible representations are needed in order to provide the complete basis for ~G . One representation with space
Parity ~--- + ~ consists of all even ~O , another with parit~ ~= -~ -
all odd ~o ( ~ is uniquely defined by ~J , thus~G(uJ)~)also has a definite parity).
The matrix representation of bl~ in the basis of the chain (4.1) has the expression
(H IH r.E,r,, P c ' Where ~ - additional characteristics needed for complete labelling
of the basis (see for details [14- 1. 5] ). Usin~ (4.2) we can restrict
~Ir by means of (2.3) either to the space ~(o) , where E~ is
U~ -irreducible representation, or to the space ~( j , with t ~O TY" I F teated as ~3 - or U~0 -irreducible representation. ~'fe can perform
the first or the second kind of restriction; thus there are six possi-
bilities. A part of the restricted dynamics Hamiltonians, obtained
this way overlap, therefore it is convenient to join some of them.
Let us present H G in the form
o
H G = H G -, H -, H ÷ t-I¢' , (43)
142 o
where H~ denotes the U.X L)'. -scalar part of (the first kind
of restriction to q<'~ , with ~ treated as the irreducible repre-
sentation for both U~ and Uz ). The operator ~ gives a part
of H~ , not include~ in H; o, expressible in terms of both ~ - and ~Z -infinitesimal operators (the second kind of restriction to
o ) 0 I ~ / I I J<" ). Analogically FI~ denotes a part of m~ , not included
/0 in terms of U~zo -infinitesimal H #--GH , expressible ~ operators
(the~second kind of restriction to ~(E0) ); H~ gives the rest p~rt
of PI~ , not included in first three ter~s.
The explicit construction of the decomposition (4.3) has been p er-
formed in [15J • In the following text of this section we will use the
results of this paper adopting them for the illustration of the rest-
ricted dynamics method. A
The potential term in (3-4) is a polynomial in ~/ which makes
it eas2 to obtain the decomposition (4.3) in a coordinate representa-
tion using the well known expressions of ~? and derivatives with
respect to them in terms of the creation ~ and annihilation ~? ~c" g , .I operators. Using the technique and the results described in Oh.IV Ll~,
formula has been obtained ~15] the following
7Zo÷3- / , (# ~'_ 3~(%o.+I)" ~K3"~o )
32,(%+0 K % K 3 j 7Erl o
+
(4o4) I i o
In (4.4) the square brackets separate the expressions for ~G and
HG/o ," ~(0, ~(~) and ~(~)denote the Casimir operators, defined
in (18.22), (18.23) and (18.33) [14] •
The third term in (4.3) is U 3ZO -, ~3 - and UZo -irreducible
operator with the following structure:
a (4.5) The definition and f~tures of T~. are desribed in 0h.IV [14 3 ~the
characteristics of l~ in the brackets indicate its irreducible pr o~
143
Parties.
The last term in (4.3) depends on the ~ (6~ ~) -infinitesimal
Operators. In El5] it has been obtained, that
H =V
where ~ , ~)~.~" - the operators conjugated with 3 , Bej", ZEj=
Values -9 = I, 2, ..., Z^ i b = i, 2, 3. It should be noticed,
that (4.4) is valid in ~&-~ spaces only; the general expression,
suitable for the arbitrary realization of the Hilbert space, is ob-
tained in [15] •
Allthe operators in (4.4) are diagonal in the basis of the chains
(4.1). Substituting ~<(4), K(~) and ~(R) in (4.4) by their eigenvalu-
es given in (18.25), (18.26) and (18.35) [14] , and also taking into
account (18.29) [14] we derive following explicit expressions for the
" ° 14' ° eigenvalues ~ and ~ 6 of
o
E° ÷ 6 Zo(~o'~4)
:c(~.o÷I) %- zfzo+i) 3 ~ Zo÷1 zo z(~o*~) $
~(~o,~) .
In (4.7) ~o takes only two values Z 0 = 2, 3 and
R __ I ~o t I i (EcEz) (E ' -Ez÷ ' t )> i~ ~o=I, .
~rom (4.7) we see, that both and
8Pect to ~ .
(4.7)
if Zo=3~
(4.8) are degenerated with re-
144
As a matter of fact, we have solved the following Schrodinger
equ.tion for H7 ~ H~ ° o o /o
The eigenfunctions '(/2~ / are identical with the 3 ~o-dimensional iso-
tropic harmonic oscillator functions in the basis of chains (4,1).The o /O
eigenvalue gG @ ~ G is given '~ O in (4.7). lhe spectrum ~2 of the
~ X ~Z -scalar operator f12 is degenerated with respect to 6~)
and i~ . ~he spectrum 6G of HG O consists of rotational-ripe fi-
nite Lo - and i -bands built up for a definite ~ ~ ~ -irredu- o ciblerepresentation [6~ . E%].
The Schrodinger equation can be exactl2 solved even in case of
the Hamiltonian ~ / ~ 0 /. HG , acting within the finite dimension
space ~(E-o)i this equation differs from (4.9) bj the additional term /
H~ , which is not diagonal w i t h respect to E and p . T h e mat- /
fix representation of HG can be obtained using digner-Eckart theo-
rem with respect to the group ~3Z 0 . Taking into account (4.5) we
get :
/ / l /
-{ V ; fzo,~,) ~ x
(4.10) where C - the Clebsch-Gordan coefficients of the group ~3~ ° for
the basis of chains (4.1). Both (4.7) and (4.10) give the matrix rep- o / O l" /
resentation for /-]6 -~/~/~ /-/~ . Performing diagonalization of the finite matrix for given ~0 we obtain the spectrum and eigenfunctionS
H,e~ / o f /
~n order to est imate the e f fec t s , o r i g i n a t i n g from H e , l e t us consider a p a r t i c u l a r case w i th ~o = 2 and L =o, lo For those L v6 ~ lues ~ is absent. In [15 3 it has been found that in this case the diagonal with respect to E matrix element of H~ is expressed ~s
follows :
1
a ~=<Eo[E~c,]~jL- -o, I IHGI~ 'o [E ,£ ;J~ , )L=Oj f>=
(Eo-a +4 L,S)(Eo, 4-4 L~ 3 )
(4.11)
145
where Z~ = 0 if ~o - even and /% = 1 if E o - odd.
The last term of (4.3), given in (4.6), has non-diagonal elements
with respect to E e , ~ , ~ • Considering it we have complete
Hamiltonian HG ; its eigenfunctions can be decomposed onlx in terms
of the infinite number of basis functions labelled by chains (4ol).
~urther we will discuss the results of such calculations, obtained in
[ 15] for the states with L = o.
The analysis of the spectrum of the constant 5~(~) non-abelian
fields Hamiltonian HG for g~s= 2 and L = 0 is presented in Fig.
l, with 4 given in OZ/3 ~/-v3 units. In the left side of Fig.1
; 6 z L,o 2 ,.11[~2,o]
[6o] o [~o,o] ' o ~ I - - , ~ N t r - 8 ~ ]
.--r ~ - , [ 6 o ] ] o _ z,+a_.._ 2 °
' :--4 , . .- :- [ ~ o ]
c [oo3 o ,.0 [~o]
8 ~2 16 20 24
a ~ c d e
~Ig.l. The spectrum of the restricted d~uamics and total constant
~U(1) non-abelian fields Hamiltonians
Six-dimensional isotropic oscillator levels with energy E~+3 are shown.
The levels, indicated by OL , give the spectrum ~-~)/3~labelled by
~3 ~ ~2~. -irreducible representation [[4 E~] ; we see that H Qhifts amd in case of E~: 0 splits the harmonic oscillator levels.
The spectrum of ~G ~HG with the eigenvalues ~ G given in
146
(4.7) is not shown because of its u2 -band structure~ needed~o~to be
improved by an additional term (4.11). The spectrum 8G @ ~ g ~ 6
consisting of ~ levels possesses the inversed ~D -band structure.
The OO -splitting of the ~X [f~-levels [f4~3 is large enough ,~ ~r
arid the inversed ~.j -bands do not overlap only in the lowest ~SXUZ
states~ The levels C give the spectrum of the total Hamiltonian
H G , calculated by computer using the matrix of ~/G given by the
elements (4.2) with the number of oscillator quanta ~0 ~ 8 . Some le-
vels have shifted considerably and the spectrum obtained has ~he ir-
regular structure° The fragment ~ shows how the levels change their
position when the number of quanta ~o used in the matrix represen-
tations is increased from 8 up to 24. The spectrum ~ is calculated
using ~o % 24; in it we see the semi-fine 649 -splitting of the~3X ~
states with regular inversed CO -bands, the distances in which dec-
rease with uO increasing. The comparison of ~ and ~ shows that
diagonalization compresses the 0.) -bands and changes relative dis-
tances within them.
The features of the spectrum discussed are informative because
of the possibility to compare the analiticall~ derived results for
the restricted dynamics Hamiltonians with those obtained numerically
for the total Hamiltonian H G . This example outlines characteristic
features of the restricted d~namics method. In particular we see how
crucially important it is to choose the proper realization of the Hil ~
bert space. ~e did it using chain (4.1) and as a consequence of the
realization used we have obtained the decomposition (4.3) with the
H first term ..~ the of the spectrum.
It was not enough J~ take into account the next term in (4.3); only
adding (4.11) to ~ ÷ g/o Q we obtain the spectrum, reasonably,, imi-
tating the energ~ of the ground and low-lying states of ~I G . Se h a v e
not discussed the exact solutions for ~/ e__~ *.. 6 (in order to
do that some special technique is needed for the calculation of the
matrix elements (4.10)), but from the results obtained the conclusioD
can be made that the second kind of restriction ''~ to~(e°)gives" =
the term, needed to build up the realistic dynamic model describing
the states of the constant &U(~) non-abelian fields Ha~iltonian ~/~-''
5- The microscopic collective model and its simplified verslons
Let us return to the nuclear structure problems. The ~ -particle
nucleus H~niltonian H is very complicated and only experience and
intuition help to choose the proper realization of the space ~ and
subspaces ~O , suitable to construct the restricted dynanics modelS"
147
Those realizations have been already described in [14, I, 2, 5-8~ and
their discussion is not our aim, thus we continue with systematization
and general description of the }~riltonians.
In order to construct realistic models of nucleus, at least the
Collective and anticollective (pairing-like) effects must be conside-
red (for details see [2, 6, 8J ). Other terms of the restricted dyna-
mics Hamiltonian, taking into account coupling between the space and
Spin degrees of freedom can also be needed (see the very end of [ 2~ ),
but there are many states, especially in the even nuclei, where those
effects can be disregarded. This is the reason, why we are not going
to discuss those questions here. The anticollective terms of H will
be described in the next section~ now let us begin with the collective o~es®
The most general micrscopic collective model with isospin quantum
nUmbers used, based on the space collective degrees of freedom has been
introduced in [2, 6J (see also [16J ). The Hamiltonian ~/~ of this
model is obtained by means of the first kind of restriction /-/ to the
O~- I -irreducible space ~(A~) ( ~ - the number of nucleons,
tO - the ~ ~ -irreducible representation). In other words /~a~
la the O~_, -%calar term of ~/ •
The explicit expression of / ~ both in coordinate and m~trix
representations has been obtained in ~16-18~ (see also [2-4, 6J ). Va-
rious kinds of collective models can be interpreted microscopically
and generalized starting with the Schrodinger equation for ~@~ .Fig.
illustrates the relationship and hierarchy between the H~miltonians
of most common collective models, as well as some new ones, naturally
following from simplified versions of the general microscopic collec- tive Hamiltonian ~/~ .
The subscripts used in }'ig.2 for /~ read as follows: the first
letters ~ or ~E in subscripts mean Ndcroscopic or ~icroscopic
~atended, 0 indicates the collective Hamiltonians with strictly re-
Stricted dynamics, ~ stands for "Generalized", ~ , ~-~ , I~I~,
VPS and S~ correspondingly denote the Hamiltonians of Elliott,
Bohr-~ottelson, Interacting BoSCh Model (IBM), Violated Permutation
SYmmetry and Symplectic.
In the first line of Fig.2 the general microscopic collective Ha-
~lltonian ~ is shown. Its simplified versions are indicated b2
a~rows. There are three main branches, begininning with ~ • The
first of them ends with Elliott Hamiltonian ~O~ , while the rest
two _ with IB~ and Bohr-~ottelson Hamiltonians ~ZS~ and H~.IW -
The first branch, begininning with HO~ consists of the strictly
~eneral ~[icroscopic Collective
~c=o
~oop
±c ~
o~-I
_ l~±=o~copio Generalizedl_ I~licroscopic V-~o
late
d Permu-
ot,
e oo
n ,
~3oh
r-N
ott
elso
~
, !
k
..
..
.
_~
..
..
•
' ~icroscopic Genera I
[iiicroscopic ~x--t~n-~ !S
ymplectic Collec- I
lize
d
IB~I
H ..
...
~-~,
ded
IB~'
,I P]
....
...
~ti
ve
F-Ig
6
Olt~l
bP1
i 'l
r'lC,J~
e] ]
[ p
,.~-~.
/ !Bgl \
\ ..
....
/ Pig. 2. The
collective Han~ltonians
~trictl$ Restricted
] ',ollective /
-/0 ¢ 0
~
I ;ene ralize
d~]
O G [~
Elli°t t '
s t
~lliot
tC~]
i I
[ ~'lo
etf _
j
I Symplectic Ant
icol
lect
ivei
~|Ge
nera
l ~c
rosc
opic
Anticollective
L fo
"7
~trictly Restricted Sj
~lec
tic.
Strictl~
il Anticollective ~O~p(nOA~g6
~ collect~
L i
~ llomentum Independent Anti-[
collective H~IA~ ~
mi
)
y ilesti~icted An
ti-
~ollective ~]0~ ~
~ig. 3. The
anticollective Hamiltonians
149
restricted Hamiltonians (by the definition, introduced in [19] , Ho~ is the ~n-q -scalar part of H ~ ), namely the generalized Elli-
ott ~O@~ and Elliot HOE6~ Hamiltonianso
The second branch starts with the symplectic collective Hamilto-
nian Hgp 6 , which is a particular case of Hc~ (we will return
to this question in the last section). This branch is continued with
the Hamiltonian I~IMEISM giving tile microscopic extention of the
IB~ }[amiltonian to both even and odd nuclei. The next Hamiltonian of
this branch gives the nuicroscopic generalization for ~/16M suitable
Only for even nuclei. This branch ends with ~/I~M "
The third branch is very specific; the Hamiltonians shown in it
act in the space with the violated permutation synm~etry, i.e. in the
Space not allov;ed for fermions by the Pauli principle. This t3pe o£
the restricted d3namics Hamiltonians has been introduced in [2] in
Order to explain the microscopic meaning of the collective Bohr-[~ot-
telson and IB~ models. This branch begins with the ~icroscopic viola-
ted permutation symmetry Hamiltonian HMVPS , suitable for both even
and odd nuclei. It is continued with the z~icroscopic generalized Bohr-
I~Ottelson Hamiltonian I-/~4g~_ M , then tile microscopic Bohr-l~1ottelson
Hamiltonian I-IM~.~ 4 follows and finally the phenomenological Bohr-
~ottelson Hamiltonian ~/~-M " Two arrows connect the second and third
branches. The first arrow shows the possibility to obtain ~MEZBM
from ~I~4Vp S. The second arrow indicates the relationship between
~I~G/5_ M and H~G_r~] ~ . i, rom ~ig.2 we see, that ~MG~-~4 is the
l a s t "cross-ro d" on the fro to both H .jV M . ~here is no arrow between H~_/~ and ~]/TBM , thus the relationship
between the Bohr-l~ottelson and interacting boson models can be estab-
lished onl2 by means of their generalization via HMB. M and ~-[MGZG/W (see for details [20, 4] ).
The arbitrary nucleon-nucleon interaction potentials can be used
in the }~miltonians, listed in the first two lines of Fig.2 . This
DrOperty is indicated by solid rectangular lines. Some of them, namely
Hce~[ , HMvps , H~$B-M and H~48-~ are defined for both con-
tinuous and discrete spectra, thus their eigenfunctions not necessari-
ly belong to the square-integrable functions. The strictly restricted
°~erat°r /"/0~-~ and its simplified versions HO~" ~ and / " /0~ '~ DOSsess only the discrete spectrum and their most suitable form for
aDPlications is the matrix representation given in the isotropic os- Cillator basis.
The Hamiltonians listed in the third line of Fig.2 indicated by
dotted squire rectangles, are based on the decompostions in terms of
150
the infinitesimal operators of the symplectic algebra ~p(6,~)and its
subalgebras F(6, R)-i initesimal operators are scalars consequently ~$p6 and its simplified versions are also O~_ I -sca-
lars, i.e. the collective Hamiltonians.
In the last line of Figo2 well-known collective Ha~ltonians are
shown; their phenomenological nature is indicated by dotted round line.
The Bohr-Mottelson Han~ltonian ~IB_jW can possess both continuous and
discrete spectra, while I~rS~ - onl~ a discrete spectrum.
6. Anticollective Hamiltonians
It has been already mentioned, that collective effects, represen-
ted by I~miltonians of various degrees of complexity, exposed in Fig -2
and discussed in the previous section, do not exhaust many-sided fea-
tures of the nuclei. In the realistic model Hamiltonians the anticol-
lective term is present. The variables used allow to notice the es-
sential difference between the collective and anticollective effectS,
The collective degrees of freedom are described dy the microscopic c ol~
lective ~_variables ~(4), £(~), 9 (3) introduced in ~13J and Euler ~g~
les ~/~/ ~ . Those variables are scalars with respect to the perm~"
tation of the nucleon space indices (see details in ~l, 5J ). In order
to expose the anticollective effects, variables sensitive to the r~nY ~
particle structure of the nucleus must be used, for example, Jacobi A
variables 9~ ( L = i, 2, ..., ~-1; O = i, 2, 3) which are tranS"
lational invariant linear combinations of the components of one- par ~
ticle vectors Z I , Z~ , ..., • But/,that is insufficient. 11 ~
order to separate the anticollective term k~A~ from /7 we must
supress the collective and stress the anticollective features. The
latter are caused by the correlations between nucleons. They depend
on both general structure of the microscopic Hamiltonian /~ (for
example on the exchange interactions), and on nucleon-nucleon inter-
action potentials.
Taking into account only the space correlation and seeking for
symmetr~ of rotations in the 3-dimensional space, where real motion
of particles takes place and rotations in n-l-dimensional space, cl o~
sely related with the separation of the sollective degrees of freedom,
the general microscopic.~ anticollective,, Hamiltonian ~ A ~ can be de"
fined as the ~ -scalar part of ~ , from which collective compO"
nents are excluded. Due to this definition the most general anticol-
lective operator for space correlations has the form
151
In case if a small difference of the proton and neutron mass is neg- lected, Hk = ~ .
The definition (6.1) is too general for practical applications.
In order to introduce the constructive anticollective Hamiltonian,furt-
her simplifications are needed. They can be achieved in different ways. ~n particular, H ~ can be simplified using the restricted dynamics technique.
Let us discuss various versions of anticollective Hamiltonians,
Presented in Fig.3, where the subscript A indicates Antico!lective,
I means Momentum Independent, 0 denotes strictly restricted Ha-
~iltonians, and ~ stands for Symplectic. Considerable si~liolification
of ~I~ can be carried out separating from the nucleon-nucleon in-
teraction potential V(~{~) its term, independent on the relative mo-
tion momentum ~ . In the matrix representation this can be easily
Performed using the following substitution for two-particle matrix ele- ments:
(6.2)
QUa with arbitrary characteristics ~ ~nd Y~t - spherical harmo-
aics. Let us introduce the Hamiltonian ~IM~ jb , obtained from (6.1)
by SUbstituting C~)~/ according to (6.2) ~orC~(V)~/~. We will
~efer to this Hamiltonian as to the anticollective Hamiltonian with
the momentun~independent interaction. If ~ denotes the irreducible
~eDresentation of some group ~ , H M ~ i s the g -scalar term
Of HAC~ , thus in this case ~I~Z~ is the Hantiltonian, obtained
h~ means of the first kind of restriction.
Future simplifications of [~IA~ *~Y be achieved usin~ the
~i~st kind of restriction to the ~ -irreducible space ~(~). In
thls case ~ means the ~ -irreducible representation i~f~ OOJ
for the three-dimensional isotropie oscillator basis functions of the
~elative motion of nucleons. The anticollective H~iltonian obtained
la Such a way is the U~ -scalar part of H~c~4 ~ • To this Hamilto-
~i~n we will refer as t~ a strictly restricted anticollective Hamilto-
nian~ we will denote it HOIWf A~ • In both ~ / ~ ~ and ~ 0M[A~ as well as in ~IA ~ the arbitrary nucleon-nucleon in-
e~action potentials can be used; this feature is indicated in Pig.3 by solid rectangular lines°
Another kind of restriction of ~ I ~ may be carried out b~
152 If
changing microscopic variables in kJA~ by their expressions in
terms of the creation and annihilation operators and separating the
terms, depending on the £p(~(n-1))g)-infinitesimal oper~tors.(In
section 4 we have already discussed an ex~r, ple of the symplectic-type
decomposition). As a result we get the anticollective H~ltonian
- .H~p(~-f)~ , shown in the l.h.s, of theLl first line in ~ig.3. 7~_Ca~
rying out the first~.~{kindE)of restriction of FlSpfn.1~A.~Yg~. ~-~ to the oU'-
irreducible space ~ we obtained the strictl~ restricted anti-
collective Hamiltonian ~I05~(H.1)~which can also be derived from
HoN[Ac~.~ . Both ~p(~-1)A~ and Hogp(~.1)~ depend on the
infinitesimal-type interaction, which is indicated in Fig.3 by dotted
lines.
At this point we end the general description of restricted dyns ~
mics collective and anticollective Hamiltonians. New let us discuss
some featuyes of their spectra.
7- Spectra of microscopic collective and anticollective Hamilton_tans
The restricted dynamics model Hamiltonian, taking into account
both collective and anticollective effects)has an expression
In (7.1) instead of H6e ~ and /-J~,~ we can take various Hamiltonl ~
ans, listed in f'ig.2 and 3. Depending on the operators used, (7.1) g i-
yes a great variety of the restricted d~amics Hamiltonians, starting
from very sophisticated, in which realistic potentials are employed,
and finishing with very simple ones, such as r~croscopic versions of
the IBM Hamiltonians with additional anticollective terms.
The limited volume of those lecture notes prevents from descri -
bing many applications of the models, based on the Hamiltonians (7.1)'
Therefore we will discuss only some new effects, following from the
restricted dynamics models, illustrating them with simple examples.
Let us begin with the structure of H.-## • The total H~miltoni ~
H I!
an of the nucleus consists of the kinetic ~k~Coulomb ~I~ ccitt ~l
C , vectorial H6- and tensorial H t interaction ter~, i.e.
H : * H e • H t . (7.2) Every term in (7.2) has its ~N-4 -scalar part, thus the general m ic~
roscopic collective Hamiltonian has the form
(7.3)
153
Instead of Hd~ , in (7-3) we expose its expression in terms of
the collective part of ~igner H~ , ~jorana H~ , Bartlett A/~ and Heisenberg H H interactions. The last three terms corresponding-
(~) (r) P~) and isospin ~ i~ depend on the spin-isospin P£% Pg; , spin
exchange operators, i~rom (7.3) we~see,~that the general collective
Hamiltonian ~IO~ ~ consists of eight terms, originating from the cor-
responding interactions in fl . 3ome of them give a verj specific
contribution, which we are going to discuss further.
The phenomenological interpretation of the collective excitations
in even nuclei is based on a ~ell known geometrical picture, treating
the atomic nucleus as a deformed system with specific modes of rotati-
Onal-vibrational forms of motion. Large values of electronmgnetic
transitions as well as static electric quadrupole moments help to se-
lect the collective states, which are interpreted as sta~es of the
Phenomenological collective Hamiltonian.
S~arting premises of the microscopic theory of the collective mo-
tion in nuclei essentially differ from those in the phenomenological
aPProaches. In the ~icroscopic theorj collective states and their pro-
Perties are uniquely defined by H ~ without referring to the phe-
~Omenological geometrical picture. In particular there is no need to
SUpPose, that the ~gnitude of electromagnetic transitions gives cri-
teria helping to identify collective states. ~e can calculate them for
states of given ~ and find out, that their values depend on addi-
tional integrals of motion H ~ possesses, on the shape of the nuc-
leon-nucleon interaction potentials used, etc.
The qualitative analysis of the ~6~ spectrum can be carried
out on grounds of the integrals of motion, possessed by fl~ .Let
isouss for ter s (7.3). spectr= of Hk * i~haracterized by the parity ~=~ ~ , orbital angular momentum L
and O~.@ -irreducible representation ~J (in case of ~ > 6, ~ de-
~Otes three numbers ( uJ I ~ UJ 3 ); ~=T~, if ~I¢~$+~3 is even
and 2T=-~ , if ~u~ ~u)~ +bO 3 is odd). Sets of UD , ailo~ed by the
Pauli principle, are specific for every nucleus with a definite n~mber of Protons and neutrons, i.e. QJ = 60 (~> ~T), where ~T is the
total isospin T projection. For every 6£) , starting with the lo-
West one, the spectrum of H k ~ @ H ~ eozmists of the 6~ -bands
With levels labelled b~ L . Depending on the shape of the potential
V W (Z~) , ~0 -bands consist of either infinite series of the dis-
~rete spectrum, or a finite number of discrete levels and the continu-
OUs Spectrum. In the lowest states of the even nuclei, when ~ is gi-
Ven b~ the partition (LO 4 0()~ 5U~ ) with all ~ , ~)~ , t4]~ - even
154
( tnu~ JT= . 1 ) , t h e s p e o t r ~ , o r Hk~-~H~, ~ o ~ e . p o s s e s s e s f ea - ture s s i m i l a r t o t h o s e oi7 t h e w e l l - k n o w n c o l l e c t i v e b a n d s i n p h e n o m e -
n o l o g i c a l a p p r o a c h e s . I n s p e c i a l m a t r i x r e p r e s e n t a t i o n s t h e u p p r o x i r ~ "
te quantum ntunber ~ can 0e introduced ( K is analogous to / projection in the intrinsic ~ -axis), i.e. ~ -bands ,~ithin a definite bO -band can be obtained° The ~r~gnitude of mixing the ~- b a n d s a s well as back-banding e f f e c t s d e p e n d on t h e s h a p e o.f V w ( t L ] ' ] °
Besides even parity states with J~--+ ~ , the Hamiltonian flk~ ~ +Hw~2 has odd parity states, with J~---~)building up negative parity co-bands. This is a new feature oi' the microscopic collective theorj, having no analogue in phenomenological models with quadrupole variables. The new quantum number UJ , with td1+~O2j+bO 3 - odd, miS- sing in the phenomenological approaches, provides this possibility, without introducing octupole degrees of freedom, necessary in order to get negative parity states in phenomenological models.
I I
Thus we arrive at the conclusion that ~k~-k~Wt~__ has well developed spectra in both even an odd nuclei and presents a far tea - ching generalization of the rotational-vibrational model. However the spectrum of Hk~2T~w ~ is still highly degenerated and this de ~ generation can be taken off by the rest terms in (7.3) as well as the anticollective term in (7.1).
Let us take into account the interaction ''~ in F/ , compose~ from the ~,lajorana potential multiplied bj spin-isospin exchange oper ~
have !!i~ ii!iiite Is ~i~iil Jr i~~iil)~ definite I ! i~!iiinii!th! ii!!i! ~~i i~° ii~! i~ii~ ! LJ 4- a o p e r a t o r tUIw,,J#.w~ ' t h u s t h e t l a m i l t o n i a n I , k ~
+ H w , ~ - t H H ~ with the exchange term included. Adding to it H/3 ~ and t~,,,/# originated from Barttlet and Heisenberg interactions, co m~
. . . . (~, ( ¢ ) Cr) p o s e d from V ~_") R ' . " and V~I~Z~ ' I ' ) P . , . , a s w e l l a s the
, , ~ ~ -~ -" e '-,,I I I e t term ~ c ~ , fo l lowi~lg from the Coulomb i n t e r a c t i o n r ] ~ , we g a collective ~miltonian
(7.4) I I I ,
f rom ( 7 . 3 ! , b y the o p e r a t o , . w h i c h d i f f e r s L /
A l l the terms of f ~ always contribute~ i n case~of both eve~ and odd to the in te rac t ions , , ' energ~ of c o l l e c t i v e s ta tes fo r ~ depends on the t o t a l spin and i sosp in T which are the i n t e g r a l s of motion f o r H ~ . The term ~ . . ~ of tha t operator takes i n to account the c o l l e c t i v e par t
155
Of the Coulomb interaction which causes the dependence of the collec-
tive spectrum of the isospin projection ~T • /
From this analysis it follows, that uO -bands of ~ with
Parity ~ , labelled by the orbital momentum L , are built up
from definite values of ~ , T . In their turn in the supermulti-
Plet basis values of ~ and T are defined by the space symmetry
( ~ denotes the irreducible representation of the/s~mmetric C~) .
group ~n ). Ihus the spectrum of the Hamiltonian H ~ depends
ca LO ~ ~ 5 T ~r and all the quantum numbers listed are the in-
tegrals of motion for H ~ . Similarly to the case of H&~,~,Ew~,
Within the Od -band, it is possible to build-up approximate ~ -bands
and to study ~ -band mixing.
Now let us turn to the last two terms in (7.3). The vectorial
~@u~ and tensorial ~ collective interactions couple the space
and spin-isospin degrees of freedom and violate the well known reflec-
tion symmetry of the Bohr-Nottelson collective Hamiltonian [18J . The
contributions of and essenti ll depend on the
of the total spin ~ which is the reason, why the collective excita-
tion spectr~n differs so much in even and odd nuclei. The terms ~@~
~nd H ~ are often suppressed in even nuclei by the following rea-
son. For a definite ~ , for example A = f~... @3 in light even
nUolei or A=~'''~5~'''~ ~3 in heavy even nuclei with r=~, S
takes the zeroth value, thus the contribution of the vectorial and
teasorial terms is also zero; in this case (7.3) is identic to (7-4)
and we obtain the collective spectrum already discussed.
Both in odd nuclei and in the states of even nuclei with S >0 the term of contributes. his interaction
couples ~ and L to ~ , and as a consequence the integrals of
action for (7.3) consist of a set of quantum numbers ~)AT IWTq /~ 4 @'
~or ~ i v e n I C I T M T , c~ -b~nds a re l a b e l l e d by t he t o t a l momentum
~r which takes values for odd Y[ and integer values half-integer rl. • even ~ with S > O . Snls t~pe of the collectlve spectrum in
Odd nuclei naturall~ following from the 1~croscopic theory is unknov~n
ia traditional collective nuclear models, treating the odd nucleus as
deformed core and extra particles or quasiparticles moving on l~ils-
son Orbits or in a self-consistent field.
Now let us briefly discuss spectra of anticollective Hamiltonians.
~°mentum independent anticollective ~miltonians have the most clear-
ly expressed features. The spectrum of I ~ ~ consists of groups
~f levels, labelled b~ ~ f , which are the integrals of motion for
~I~ A ~ ~ the strictly restricted Hamiltonian ~IOMIA~ posses-
156
ses the additional integral of motion E.
The schematic illustration of the spectrum for the strictly re-
stricted Hamiltonian ~/Ok~/-/O~/-~ONIAc~C~__ __ with collective
and anticollective terms, obtained using the quadrupole t~pe interac-
tion)and a simple operator, taking off the degeneration with respect
to ~ is shown in Fig.4. In the right top corner of Pig.4 we see
finite (because of the strict restriction) ~t) -bands, additionally
characterized by ~=[~..-~ and ~ =[~... ~ ~f3with T=~ :0. On
the left side the anticollective spectrum is exposed, in which we see
the levels of c6~£HoMIA ~ and a clearly expressed gap between the
levels of the most symmetric [~..- ~] and the less symmetric~...~3~
and [~..- q ~jpartitions. A fragment of the spectrum of the Hamil-
tonian with collective and anticollective terms is shown in the midd-
le of Fig.4. We see three CO -bands built up on definite states of
the anticollective Hamiltonian, characterized by ~ , ~ and 7- ,
common for both collective and anticollective spectra. ~ore detailed
discussion of this spectrum including the splitting caused by ~I0~ ~
as well as the theoretical description of the level density observed
in nuclei with ~ = 20 is presented in [21] •
8. Concluding remarks
In these lecture notes we have described only following topics:
principal ideas and definitions, the restricted dynamics decompositi ~
on of the constant SU(~)non-abelian fields Hamiltonian, general de~
scription of collective and anticollective Hamiltonians, and some fe ~
tures of their spectra. ~e have omitted many questions related with
both grounds and applications. Let us list some of them with referen"
ces to look up for details.
What is the physical reason causing collective and anticollecti ~
ve effects, so important in order to understand the structure of n u°~
lei? In L 2, 6, 8] it has been stressed, that the crucial condition is
the equivalence of the particles composing the nucleus. This statem e~
has been checked up in [22, 23] by means of the following "computer
experiment". The three particle Hamiltonian has been presented in the
form allowing to break up the equivalence of the particles. The dir e°~
comparison of the spectrum of H k ~ H 6 with the spectrum of the r e~
stricted Hamiltonian H~yTHa~@~-/oMrA~.~ has clearly de~
monstrated that only in case of the equivalent particle system the
spectrum of the restricted IIamiltonian reasonably imitates the spec t~
r u m o f
O~
o
l:b
¢~
o
O 0 0 0 ~r
b ~
0
0 ~0 ~+
£El
Lib
0 u~
O
~
I
(O
~4...
~]
[#...
~-22
]
-- IIIil
lllill
~,.-s
~ruc
±ure
of
the
[
II I
lllllll
Ill
(~,6~4J- band
wi{h
X=
E4.
.-q22
]~ w =
S=O
I#O
1 ~
7 Z+/
-- ]
Darld
~ I0
~1 ~
~th
7L= E¢,,.4]11-= S-- 0
F it
h ~-- [4...43, T=S
~D
o ~+
~-.
- I
o~,
I-J
r4r
Ub
0
H Ce
l l II l
M 11t
M 1
158
,.hat is the main idea and proper technique, underljing the mic-
roscopic foundation of the phenomenological collective models'? In [23
it has been proved that the essential difference between .~Ita2F ~nd ~ m
I-I depend= on chara teri =ic o: =he sp ,oes H N g ~Jc~ is defined for LKJ allo,ved by Pauli principle. ,-~hile
acts in the violated permutation sjmmetry space ~[A~ ~={#)) !fhe
specific structure of this space) nam~elb' its ~/~-I -scalar property.
gives a "deep freezing" of microscopical collective features of /~g~
and m~kes ,hem similar to those in the Boh~-i~]o%telson theory. If we
span -- ~ (4~ ~ = ( O ) ) ~ ~ o,~ t h e s q u a r e - i n t e ~ r a b l e b a s i s r ~ . c t i o n s , i n
p~rticular, on the ~,(]'~(~-4~-irreducible basis nith &On(O) and use
the s21~.plectie tjpe decomposition of H M ~ ~-]W , ~Je obtain n-~icros-
topically the interacting boson model ~.amiltonian and izs "deep col- lectivity" interpretation [20, 4], without referring to active let-
O[ mions coupled to $ and bosoms. The generalization of Hr~ ~
for odd nuclei as well as for odd parity' states of even nuclei can b,
carried out introducing the microscopic violated permutation sb'rr~?.et-._~ [! ~DA . . . ~-
r E model "with the }-h, m i l t o n i a n j , r-I ~ t i P S a c t ~ n t : ! n ~he s p a c e d(. ( ) ~ ]
I n t h i s c a s e t h e t e r m s k l ~ c ~ a n d ~ ¢ ~ c o n t r i b u t e , w h i c h G i v e s
s p e c i f i c f e a t u r e d o f t h e spectru~..~ a l r e a d j d i s o u s s e d .
Le t us p u t t h e l a s t q u e s t i o n we a r e g o i n g ~o d i s c u s s i n t h o s e
l e c t u r e n o t e s . . , ~ h a t t i e s g e n e r a l m i c r o s c o p i c a nd s j m p l e c t i c m i c r o s -
c o p i c c o l l e c t i v e m o d e l s ? Their" c o n n e c t i o n c a n be e x p l a i n e d on t h e ex-
amp le o f t h e Ha: r~Ll tan i~n ( 3 . 4 ) , i f t a k e i n s t e a d of ( 3 . 5 ) a more comp-
l i c a t e d potential, for example the Gaussian or Yukawa-t~pe interactZ-
on. For those interactions it is very difficult to separate from H ter , depe din on p(6, )i ,finitesim&l operators. r ctica i
we can take into account only a few terms in the Ta~lor expansion
of the potentials. Therefore Hsp 6 obtained is much simpler tha~
~-{tz~.~l~ o This explains, wh.y the s~m~piectic collective model is o n l y
a simplified version of the general collective model (for details see
[ 7 ] ). Often the term "s3mplectic collective model" is used for no~ completely microscopic approaches, for example, those, described in
[24] , with a phenomenological potential not related to microscopic
collective potentials, introduced in [16] and explicitly obtained in
[
159
~eferences
i. Vanagas V. ~ethods of the Theory of Group Representations and Se- paration of the Collective Degrees oY ;'reedom of the NucleusoLec- ture Notes at the 1974 ~osco# Engineering 2hFsios Institute School, ~IPI, ~oscow, 1974.
2. Vanagas V. The Microscopic Nuclear Theory within the Fr~nework of the Eestricted Dynamics. Lecture Notes. Univ. Toronto, 1977.
3. Vanmgas V. The licroscopic Theory of the Collective ~otion in Nuclei. Lecture Notes.in: Group Theory and its Applications in Physics - 1980. Latin American dchool Phys., AIP Conf. Proc., 1981, No. T1, p. 220-293.
4. Vanagas V. Introduction to the Zicroscopic Theory of the Collec- tive ~otlon in Nuclei. Lecture Notes.in: Proc. V--th Int. School On Nuclear ~nd ~eutron Physics ~ud Nuclear Energy, Varna - 198!, Bulgarian Acad. Sc., ~ofia, 1982, po 185-229.
5. Vanagas Vo Soy. J. Part. Nucl., 1976, ~, 309.
6. V~uagas V. Soy. J. Part. Nucl., 1980, l_~l, 454.
7. Vanagas V. The Symplectlc ~odels of Rucleus. In: Group-Theoreti- cal iethods in Physics. Proc. 1982 Zvenigorod int. Sem., Chur, Paris, London, Har~ood Academic publ., 1985, Vol. l, p. 259-282.
8. Vanagt~s V. The J<epresentations of the Orthogonal and Unitary Groups and Nuclear Models. itogi Nauki i Techniki, Set. mat. ana- lisis, iosco~, VINITI, 1984, Volo 22, p. 3-35.
9. Badalyan A. Soy. J. Nucl. Phys., 1983, ~, 779-
I0. Simonov ~u. A. Phys. Left., 1984, l~6B, 105.
ii. Vanagas V. Soy. J. Nucl. Phys., 1985, 41, 1474.
12. Badalyan A. Coy. J. Nucl. Phys., 1984, ~, 947-
13. Dzyublik A. ~a. Preprint ITF-71-122~. Inst. Theoret. Phys., Kiev, 1971.
14. Vanagas V. Algebraic ~ethods in Nuclear Theory, ~tintis, Vilnius, 1971 (in lussian).
15. Vanagas V., KatkeviOius O. The StI'ucture and Spectrum of the Cons- tant SU(2) Non-Abelian Fields Hamiltonian (to be published).
16. Vanagas V. Soy. J. Nucl. Phys., 1976, 2/, 950.
17. Katkevi6ius O., Vanagas V. Liet. fiz. rinkinys, 1981, 21, No.2, 3o
16. Vanagas V. Liet. fiz. rinkin~s, 1982, 22_, No. 2, 3o
19. V~uuagas V. The Restricted Dynamic Nuclear ~odel. In: Proc. int. Conf. on Selected Topics in Nuclear Structure. Dubna, 1976, Vol. l, p. 46.
20. Vanagas V. Bulg. J. Phys., 1982, ~, No 3, 231.
~l. Vanagas V., Sabaliauskas L., Eriksonas K. Liet. fiz. rinkinys, 1982, 22, No. 6, 12.
22. Katkevi6ius 0., Vanagas V. Liet. fiz. rinkinys, 1985, 25, No. 2, I0.
23. Katkevi6ius 0., Van~gas V. Liet. fiz. rinkinys, 1985, 25, No.3, 3-
24" }{osensteel G., Rowe D. J. Ann. Phys. (N.Y.), 1980, 126, 343.
AN INTRODUCTION TO THE USE OF GROUP THEORETIC TECHNIQUES IN SCATTERING
F. Iachel]o A.W.Wright Nuclear Structure Laboratory
Yale University New Haven, Connecticut 06511
Abstract
An introduction to the use of group theOretic techniques in scattering is presented. In particular, the role of dynamic symmetries in the analysis of scattering data is discussed.
]. INTRODUCTION
Group theoretic techniques have been used extensively in the analysis of bound state problems (spectra] properties) in many fields of physics. Notable examples are the quark model (Gell-Mann-Ne'eman SU(3) [I]) in elementary particle physics, the interacting bosun model in nuclear physics [2], the exploitation of the symmetries of the Coulomb problem in atomic physics [3] and the vJbron model in molecular physics [4]. In recent years, there have been several attempts to extend the use of these techniques to include scattering.
The collision of two objects with structure is described in terms of three sets of coordinates, Fig.l: (i) the internal coordinates of the first object, denoted in Fig. 1 by ~]. ; (ii) the internal coordin"
P ares of the second ob]ect, denoted by (2p and (iii) the relative coordinates, r.. Group theoretical techniques may help in the treat" ment of any of these three sets of coordlnates.
The first use of group theoretic techniques is in the treatment of the internal coordinates. This use is an extension of that done pre- viously in the treatment of bound state problems and will be dis- cussed in Sect.2. The second use is in the description of the colli" sion process itself (algebraic treatment of the relative variables), Sect.3. Finally, it may be possible to combine both descriptions into a single framework.
2. GROUP THEORETIC TREATMENT OF THE INTERNAL DEGREES OF FREEDOM
I shall consider for simplicity non-relativistic scattering of a structureless particle from an object with structure. The correspond" ing Hamiltonian can be written schematically as
H = T + H(~) + V(r,~), (2.1)
where T is the kinetic energy of the incomlng particle, fl(~) the Hami]tonian of the particle with structure and V(r, ~) their interaC" tion. If the internal structure of the particle can be described by a group G, then one can replace both H(() and V(r, ~) by algebraic vari"
161
Fig.l. Collision of two ob3ects with structure (two deformed nuclei).
ables. This replacement produces significant simplifications in the evaluation of the corresponding S-mattlces ( and thus of the cross- Sections). There are four cases that have been analyzed so far, listed in Table I.
Table I. Algebraic targets
Target Algebraic model Group structure, G
NUcleus Interacting Boson Model-1 NUcleus Interacting Bosch Model-2 Diatomic molecule Vibron Model-I Triatomic molecule Vibron Model-2
U(6) U(6)×U(6)
U(4) U(4)×U(4)
D e s p i t e t h e c o n s i d e r a b l e s i m p l i f i c a t i o n s i n t r o d u c e d by t h e use o f a l g e b r a i c t e c h n i q u e s , t h e c a l c u l a t i o n o t t h e S - m a t r i x when t h e p a r - t i c l e with structure has many bound states is still a formidable PrOblem. Further simplifications ace obtained in eases in which the Salutlon of the scattering problem can be treated approximately but in Closed form. Such is the case of high-energy scattering. Here one can neglect H(~) compared with T and use the eikonal (or Glauber) aPProximation, Fig.2, which is a representation of the S-matrix in the space of impact parameter,b. In the eikonal approximation, the Scattering matrix from an i n i t i a l state, i, to a final state, f, can be written as
Sfi(b) = <fl eli(b) li > , (2,2)
and the scattering amplitude Afi as
P F i g . 2 . S c a t t e r i n g in
t h e e i k o n a l a p p r o x i m a t i o n ,
162
All(q) = d2b e iq'b <fJ e ix(b) -1 Ji>, (2,3)
where q = k ' - k Is the momentum t rans fe r . The phase x(b) depends on the interaction between the incoming particle and the target, V(r,~). In cases in which the target can be described by an algebraic model, the interaction V(r, ~) can be written in terms of the algebraic form of the operators exciting the target. Quite often, these are genera- tors of the same group G that describes the bound states. Thus one finds oneself in the situation in which the 5-matrix elements (2.2) are schematically given by
<Irrep of GJ e iaG JIrrep of G>, (2.4)
where Irrep of G denotes an irreducible representation of the group G. But (2.4) is nothing but the group element of G, a generalization of the familiar Wigner D-matrices for 5U(2),
(J) iZuBuJ u D (01,02,03) = <J, MJe IJ, M'> (2.5) M M'
The quan t i t i es 0 u are the group parameters (Euler angles), whi le the operators Ju are the group generators (angular momenta). Thus algebraic models for the target when combined with the eikonal approximation lead to a definite mathematical problem which in fact can be soleved exactly. This implies that by using the group technique one can solve the multiohannel coupled problem to all orders, a significant result.
2.1. Scattering of high-energy protons off nuclei
As first application, consider the scattering of 800 MeV protons off nuclei [5,6]. In so far as the nucleus can be described by the interacting bosch model, the algebraic-eikonal approach can be inple- mented. The interacting boson model [7] in its sinplest version (IBM- i) describes the nucleus in terms of the group U(6) spanned by monopole (Jr0) and quadrupole (Jr2) bosons, denoted respectively by s and d. (~=0,±I,±2) or, altogheter, by b a (a=l ..... 6). The generators of th~ group U(6) are the bilinear products of creation and annihila- tion operators
GaB = baths , a,B=l ..... 6. (2.6)
There are 6x6=36 such generators. It is convenient to rewrite the 36 generators in terms of coupled tensors. For the applications one has in mind here the important tensors are the quadrupole (E2) tensorS since one expects that the excitation goes predominatly via a quad" rupole mode. The corresponding excitation operator is of the form
T(2) = a2(stxB + dtxs) (2) + B2(dtxB) (2) (2.7)
where I have used the same notation as in tel.r7]. The target- pro]~g~ile interaction can be obtained by taking the scalar product of T'~'u with the quadrupole operator of the incoming particle. This leads t6 a phase ,(b) given by [6]
163
x(b) = [o2(b) (stxa+dtxE) (2) + ~2(b) (dtxd) (2)] " y(2)(~) , (2.8)
Where a2{b) and Bg(b) are given in terms of the boson form factors Q2(t) and B2(r) [8~ by
O2(b) = -- dz o21 ) •
k (2.9)
~2(b) = -- dz ~2(2, /~+z 21 k
The evaluation of the S-matrix elements (2.2) is done, in general, by expanding the initial and final state wave function of the target in terms of representations of U(6) amd then evaluating the group ele- aents (2.4). A particularly simple situation arises when the initial and final states have a dynamic symmetry. In the interacting boson mOdel-i there are three such cases, corresponding to the chain decom- Positions
U(5) D 0(5) D 0(3) 3 0(2) , (I) U(6) -~ SO(3) D 0(3) D 0(2) , (II) (2.10)
"~ 0(6) D 0(5) D 0(3) D 0(2) (Ill)
In these cases, the evaluat ion can be done in closed form. Otherwise One must expand the wave functions in an appropriate basis, for example that of chain (1), and evaluate the group elements in this basis,
[N] D (e u ) =
n d, v, n A, L, M; nd' , v ' , n A', L ' , M'
iEuSuOu , , , <[N],nd, v, nA, L,M I • I[N3, n d , v ' , n A ,L ' ,M >, (2.11)
Where n~,v,n.,L,M are the quantum numbers labelling the basis. In ~eneral, u the~e are 36 groun parameters e., and generators G~ However, the r ~i limits the number of im- _ physics of the problem at hand usua y , _ POrtant group parameters. For example, in (2.8) there are only two ~ arameters a 2 and Bg. An example of application [5] of this technique O a nucleus described by the 5U(3) symmetry is shown in Fig.3.
It has been known for some time that the description of bound ~tates of nuclei can be improved if one introduces explicitly proton- neutron degrees of freedom. This is done algebraically by introducing DrOton t, A ~ ,^~ ,..~^~ i, d ) bosons (interacting boson model- ~/,The corresponding group s~ructure becomes that ol _~ne ~Ir?c~ ~T°duct U,(6)XUv(6). Within the elkonal-algebrazc approac 9 t.ne inc~u_ ~on of proton-neutron degrees of freedom presents .no..lurtne~ a l l . C lty. T,e expression to evaluate i, 9ul( e qu fuP c ;~OZtation operator contains generators G" m~ aria u e~ ........
ase z(b) then becomes
164
IO 6
I0 5
~ P
10 4
E .-. i0 3
10 2 b
c I0 I o
4 1 ,
u I0 0 o
( / )
, , , 161 R o 162
id 3
m
i
I I I I I I
i i I l l j V, "¢L4 0 . 5 1.0 1.5 2 . 0 2 . 5 3 . 0 3.5 4 .0
Momentum Transfer q ( f m " l )
Fig.3.Calculated elastic cross section of an 800 MeV proton from a nucleus with SU(3) symmetry[5].
x(b) = [a2T(b) (sTtxd~+d~txs~) (2) + P2~(b) (d~tx~) (2) +
+a2v(b) (SvtXdv+dvtxsv) (2) + B2v(b) (dvtxav)(2)]'Y(2)(~), (2.12)
The use of the proton-neutron formalism is particularly important if one wants to understand differences between scattering of high-energY electrons and hlgh-enecgy protons off nuclei. In fact, these two projectiles have different interactions with the protons and neutrons contained in the target nucleus. This leads to different transition densities, i.e. different functions ~2T(r), B2T(r), a2v(r), B2v(r). A comparison between the two processes thus provides information on proton-neutron effects in nuclei. While electrons interact in nuclei in a single step, protons experience multistep processes. A detailed treatment of the multistep processes (as provided by the elkonal- algebraic approach) is thus necessary if one wants to understand these differences.
2.2.Scattering of high-energy electrons off molecules
The second application of the group technique decsribed above is to scattering of high-energy (10 eV) electrons off molecules [9]. Here again the eikonal approximation is a good approximation since the kinetic energy of the projectile is much larger than the energy associated with the taro-vibrations of the molecule. Rotation-vibra- tion states of molecules can be decsribed algebraically by introduc- ing scalar and vector bosons, o, TU(u=O,±I) (vibron model) [4]. The associated group structure is that of U(4). It is interesting to con- trash this situation with that occurring in nuclei, Fig.4. The classical collective variables ~p in nuclei are the componenents of a quadrupole tensor while in diatomic molecules are the components of a vectot,t u. Similarly, while the excitation occurs in nuclei ptedomin" antly through quadrupole transitions, it occurs in molecules
la)
165
z
[9, 7, 8, ,8z,8 3
(b)
z
r,@,•
Fig.4. Internal degrees of free- dom in nuclei (a) and molecules(b).
Predominantly via dipole (El) transitions. In the vibron model, the dipole transition operator can be written as
T(1)~ = a (otx~+wtx~) (1) (2 13)
This produces an eikonal phase of the type
x(b) = a(b) ( o t x ~ + , t x ~ ) ( 1 ) . y ( 1 ) ( ~ ) , ( 2 .14 )
where the quantity a(b) is given in terms of the radial form factor a ( r ) by
a(b) = - ~2 k ~ dz a ( 2 , t ~ + z 2) ~ 2 + z 2 (2 .15)
Once more, one can note t h a t the phase x(b) is g iven in terms of the genera to rs of the group U(4) of the v i b r o n model. Fur themore, the initial and final states i and f of the molecule are given in terms of representations of the same group U(4). Th evaluation of the scattering matrix thus amounts to an evaluation of the group elements of O(4). In the two dynamic symmetries of the vibron model,
, ~ U ( 3 ) D 0 (3 ) D 0 (2 ) • (2 .16)
U(4) ,,4,0(4 ) D 0(3) D 0(2) ,
the evaluation can be done analytically. In other cases, it must be done numerically.
The same approach can be used to describe scattering off more cOmplex molecules. Triatomic molecules, for example, can be described algebraically by two sets of vibron operators, o1,1 and a2,, 2. [4]. This situation (vibron model-2) is similar to that ~ the interacting bOson model-2. The appropriate group here is UI(4)xU2(4), and for dipole excitations, the phase operator is
x(b) = [al(b) (oltx~l+11tx~l)(I) +
a2(b) (o2tx~2+,2tx~2)(1)].Y(1)(~) (2.17)
166
Again it is in the treatment of these complex situations that algebraic techniques are particularly useful.
2.3.Coulomb excitation of nuclei
In addition to the two cases described above and based on the eikonal approximation, another application of the group techniques has been done by Wanes, Yoshinaga and Dieperink [10] to the study of Coulomb excitation of nuclei. Here the incoming ion follows a classi- cal Coulomb trajectory rather then a straight line. However, if the excitation occurs trhough the guadrupole operator it can be computed using the same technique described above.
3.GROUP THEORETIC TREATMENT OF THE RELATIVE REGREEES OF FREEDOM
A more ambitious program is to treat also the relative degrees of freedom by group theoretic techniques. This is a difficult task since the calculation of the scattering amplitude requires the knowledge of both the Hamiltonian describing the system and of the boundary condi- tions at infinity. Some progress ahs been made recently towards the solution of this problem [Ii] which I shall now discuss.
I shall consider the case of scattering of two structureless par- ticles, with Hamiltonian
H = T + V(r) (3.1)
In the traditional approach, the cross section is obtained by first solving the differential equation associated with (3.1)
M2 [- -- V 2 + V(~)] ~(r) = E ~(r) (3.2)
2p
with appropriate boundary conditions. For example, for a potential decreasing faster than i/r,
eikr ~(r) ~ • ik'r + a(~) (3.3)
r + ~ r
This method produces the scattering amplitude a(~). For non-relativ- istic problems it is convenient to introduce a partial wave expansion of (3.2). The S-matrix for partial wave £ can then be written as
2i6£(k) St(k) = e , (3.4)
and the scattering amplitude as
a(k,8) = E (2t+l) [S£(k)-l] P£(cosO) , (3.5) 2ik t=0
The Glauber (or eikonal) approximation used in the preceding section amounts in the present case to approximating (3.5) by
k[ a(q) = d2b e i q ' b [ e i x ( b ) - l ] ( 3 . 6 ) 21 i
167
The cross section is now a function only of momentum transfer and the phase ,(b) is obtained by integrating the potential V(r),
x(b) = - -- dz V(r) (3.8) ~2k
3. l.S-matrices from group theory
It has been found [Ii] that explicit forms of S-matrices can be obtained from a purely abstract formulation of the scattering problem using group theory. This occurs whenever the problem has a dynamic symmetry described by a group G. The process by means of which the S- matrix can be computed consists in an expansion of the representa- tions of G describing the system in the presence of interactions into representations of a group F (called the asymptotic group) describing the system in the absence of interactions, schematically written as
I Representations of G = I (3.9)
= (S-matrix) x Representations of F
It is important to note that, contrary to the case of bound state problems where the knowledge of the dynamic group G is sufficient to determine the properties of the system, for scattering problems we need the knowledge of both the dynamic group G and the asymptotic group F. When considered in detail, this is a complex mathematical problem [12], but for obtaining the results presented here it is sufficient to consider a simpler form. For non-relativistic problems, in a partial wave formalism, the asymptotic group F must contain the Euclidean group E(3). The relation (3.9) then corresponds to the Josh expansion
ikr e ~(k,r,~) • ~(-)(£,k) Y£m(~) +
r ~ r
-ikr e
+ @(+)(~,k) Y(m(~) , (3.10) r
as it will become apparent in the sub-sections below.. Group theory Provides algebraic relations for the Josh functions ~(±)(L,k) which can be solved explicitly.
3.2.The Coulomb problem and SO(3,1) amplitudes
The simplest illustration of the group method is provided by the non-relativistic Coulomb problem. Here it has been known for years [13] that the dynamic group G is SO(3,1), with six generators, the angular momentum, L, and the Runge-Lenz vector,A. The asymptotic group is E(3). with six generators, the angular momentum, L, and the mo- mentum, P. The representations of G are labelled by
(3.11)
Those of F by
168
I E(3) D S0(3) D S0(2) 1 F : # ~ # (3.12)
(±k,0) ~ m
In (3.11) the continuous unitary representations of SO(3,1) have been used with
<C2> = w(w+2) ; w= -l+if(k) ;
<C'2> = 0 • (3.13)
where C2=L2-K2, C'2=L.K, K:WA/k. TSe representations of E(3) are characterized by the eigenvalues of P ,
<p2> = k 2 (3,14)
and labelled by ±k (Note the doubling and the fact that in configura- tion space the representations (3.12) do not exist at the origin, r÷0). Also here <P.L>=0. Eq.(3.9) can now be written ex- plicitly in the form
Iw,£,m> = A£(k)l-k,£,m> + B~(k)l+k,£,m> , (3.15)
where l-k,~,m> and l+k,£,m> are incoming and outgoing waves. Eq. (3.15) is formally equivalent to the Jost expansion (3.10). The Jost functions #(±)(~,k) are replaced by A~(k) and B£(k) in ((~{15). In order to obtain recursion relations for the functions A£ and B£(k) one writes the generators of G in terms of those of F. This technique, called Euclidean connection, plays an important role in the final derivation, since it provides recursion relations for A[(k), BR(k) and their ratios R~(k)=B~(k)/A£(k),
£+l+if(k) R£+l(k) - R£(k) (3.16)
~+l-if(k)
The recv {7 r la ions can Sg(k)=e 1~z Rg(k), i.e.,
F(~+l+if(k)) S~(k) :
F(£+l-if(k))
be solved to yield the S-matrix
(3.17)
and the cross section
do f2(k) - - =
d~ 4k2sin4(e/2) (3.18)
The particular functional form (3.17) is due to the dynamic group 50(3,1). All problems with 50(3,1) dynamic symmetry have S-matrices of this form. The function f(k) is determined by the relation between the Hamiltonian and the Casimir invariants of SO(3,1), if this rela- tion is known, or can be obtained by fitting the experimental data. For pure Coulomb scattering,
169
~B2
H - , B = ZlZ2 e2 , (3.19) 2(C2+I)
gives
f(k) : ~Blk (3.20)
The 50(3,1) scattering amplitude (3.17) can be generalized by making f(k) a function of &. Since ~ is related to the eigenvalues of the Casimir invariant of $0(3), this generalization maintains the 50(3,1) symmetry but makes the relation between the Hamiltonian H and the Casimir invariants more complex. Making f(k) a function of ~, i.e. f£(k), adds slightly to the numerical complexity of the problem. The S-matrix is still given by an explicit expression. The only dif- ficulty now is that the partial wave expansion cannot be summed in closed form as in (3.18). S-matrices of the type
F(£+l+if~(k)) St(k) : (3.21)
F(~+l-if~(k))
can be used to analyze scattering by potential more complex than the Coulomb potential, for example by the Yukawa potential, V(r)=-ne-vr/r. M~ller and Schilcher [14] several years ago considered S-matrlces of the form (3.21) for the Yukawa potential. ~owever the function f (k) was obtained as a series aexpansion in i/k ~ and it is not very ~ell suited for practical applications. It would be more convenient here to find an explicit functional form of f~(k).
An interesting property of the SO(3,~) amplitudes (3.21) is that its values can be computed easily by making use of the relations
- y Y arg F(x+iy) = y~(x) + Z (-- - arctan )
n=0 x+n x+n
¢(I) = -~ ,
( x + i y ¢ O , - 1 , - 2 . . . . ) ( 3 . 2 2 )
n-lt_ 1 ~(n) =-~ + ?. (n~2), ~ = .57721 ,
t=l
where ~(z) is the digamma function. The phase shifts 6~(k) corre- sponding to (3.21) can be obtained directly as
28~(k) = f~(k) ~(~+I) + ® f~(k) f~(k) Z [ - arctan
n=0 £+l+n ~+l+n
For large ~, one has
26£(k) : f£(k) *(£+i)
] (3.23)
(3.24)
The SO(3,1) amplitude is thus well suited for analysis of experi- mental data.
3.2. The modified Coulomb problem and SO(3,2) amplitudes
This is the problem of particular interest in nuclear physics since the interaction between two ions is dominated at large dis-
170
tances by their Coulomb repulsion while at short distances this re- pulsion is modified by the short range nuclear interaction. While, in principle, one could use the SO(3,1) amplitude (3.21) to analyze this case, it is more convenient to use a more general form of scattering amplitudes which separates explicitly the short range from the long range behavior. This is acheived by introducing a more general con- struct based on the larger dynamic group GISO(3,2). The asymptotic group F corresponding to this case is FmE(3)xE(2) composed of a part, E(3), decsrlblng the asymptotic space group and a part, E(2), describing the asymptotic internal group. The representations of G are labelled now by
S0(3,2) D 50(3)x$0(2) D SO(2)xSO(2)> O : + ~ ~ ~ (3.25)
(w,O) ~ m v
while those of F are labelled by
I E(3)xE(2) D $0(3)xS0(2) D SO(2)xSO(2)> F : ~ + + ~ ~ , (3.26)
(±k,O) ( i ) ~ m v
where again I have used the continuous unitary representatlons of S0(3,2) with
3 <C2> : w(w+3) , w : - - + if(k) ,
2
<C'2> = 0 (3.27)
While the quantun numbers w,~,m, tk have the same meaning as in the Coulomb problem, the quantum number v has the meaning of the added interaction as it will become clearer later on.
The Just expansion is in this case
l~,£,m,v> = A~v(k)l-k,&,m,v> + B£v(k)I+k,£,m,v> (3.28)
Using the Euclidean connection one can obtain the S-matrix in the form
S~(k) = x
x e i(21n2)f(k) (3.29)
It is interesting to observe that if one lets in (3.29) v=I/2, one obtains back the Coulomb S-matrix (3.17) as one can see by using the duplication formula
F(z) F(z+l/2) = 2 l-2z w I/2 F(2z) (3.30)
I t i s thus more convenient to r e w r i t e (3.29) by i n t r o d u c i n g the
171
quantity u = v - I / 2 ,
if(k) (£-u+l+if (k))
S ~ ) k ) =
x e i(21n2)f(k) , (3,31)
where now u has precisely the meaning of the interaction added to the Coulomb interaction.
The amplitude (3.31) can now be further generalized by making u and f functions of both quantum numbers k and ~. Since in heavy ion collisions we want to have the long range part of the interaction to be purely the Coulomb interaction we shall, for applications to this Problem, retain
~B2
H - • ~ = ZIZ2 e2 • (3.32) 2(C2+i)
and
f(k) = Up/k (3 ,33 )
The quantity u will instead be taken to be a function of ~ and k. This function contains all the physical information on the scattering Process. The generalized S-matrix which has been suggested [15] should be used to analyze elastic collisions between heavy ions is
S~(k) =
F( ~+u(£'k)+2+iuB/k ) 2 F(£-u(£'k)+l+iu~/k)2 " '
{.+o,. k,+. i 0jk I (. F F
2 2
× e i(21n2)p~/k (3.34)
I shall call this the S0(3,2) model of heavy ion collisions. One can note at this stage that the S-matrix (3.29) is manifestly
unitary. However, if one wants to describe elastic collisions between heavy ions in which many channels are open, one needs, in the Schr~d- Inger approach, a complex potential which takes into account absorp- tion into the channels not included. This effect can be taken into account in (3.29) by making v complex. The associated S-matrlx is no longer unitary. In order to satisfy the unitarity bound,
]S~(k ) l ~ I , ( 3 .35 )
one must have
Im(v 2) ~ 0 (3,36)
172
If (3.31) is used, the unitarity bound is satisfied if
Im[(u+l/2) 2] ~ 0 (3.37)
o- or R i0-I
16 2
1(#3 ~ 0
I 1 I I ! I I I
ECM =15 MeV
40 80 120 160
OCM
Fig.5. Schematic representation of the angular distribution for the collision of two heavy ions [11].
O"
Id'
,6 2
I [ I I ! I I I 1 I I I 1 I
=
I I I I I I I I I I I I 12 16 20 24 28 32 36
ECM(MeV)
Fi~.6. Schematic representation of the exc i ta t ion function at eCM=90 for the co l l i s ion of two heavy ions [11].
173
A form of the "potential" u particularly suited for heavy ion colli- sions is
u(£,k) = (uR-iu I)
with
l+exp[(£-£O)/A] (3.38)
£0 = £0 (k) (3.39)
Fig.5 shows the angular distribution that one obtains in this case. This angular distribution appears to have all the features observed in heavy ion collisions. The quantity £0 has the meaning of a grazing angular momentum. Its relation to k, ~q. (3.39), will determine the excitation function (i.e. the cross section as a function of energy at a fixed angle). For example, if one assumes the form suggested by semiclassical arguments
£0 = /R2(k2-kO 2) ' (3.40)
where R is a measure of the radius at which the nuclear interaction sets in (barrier top) and
k02/2~ = B/r =V c (3.41)
is the Coulomb barrier at r=R, one obtains the excitatlon_function shown in Fig.6. The expression (3.41) is only valid for k2>k0 2. In general, a better procedure would be to determine the function ~0(k) by fitting the angular distribution at different values of k. Fig.7 shows the results of this procedure [15].
The S-matrix (3.34) can be viewed as a convenient parametrization of the scattering amplitude in the space of ~ and k. An important Problem is the relation between the function u(£,k] and the corre- sponding function U(r,k) in configuration space. Amado and Sparrow [16] have studied this relation in the peripheral region where the eikonal approximation can be used. In this region, £=kb~l. By invert- ing the eikonal approximation, one obtains
1 [~ t'((z2+r2) I/2)
g(r,k) = - --, dz (z2+r2)i/2
d~ h 2 ~'(b) = -- , ~(b) =
db ~kb 2 u(kb)f(k) (3.42)
The analysis of Amado and Sparrow applied to the form (3.38) shows that it 5~rresponds to a potential that for large r goes like exp(-vr)/r which is considerably faster than pure exponential fall off. The preliminary results shown in Fig.7 appear to indicate that the S0(3,2) S-matrix model may provide a convenient framework for the analysis of heavy ion collisions.
174
4. COMBINING THE GROUP THEORETIC TREATMENT OF INTERNAL AND RELATIVE DEGREES OF FREEDOM
In the previous sections, I have discussed a group theoretic tre- atment of the internal degrees of freedom . This treatment is well developed and already proven to be useful. I have also discussed
28Si ( 160, 160)28Si
I0 O ,c m : 18.67 MeV' ' , ~ ~ , ' "/"J47~
I0 o L ~ " ~ ~ V I
" Ec m "20J2 MeV ~N, , .~ " / io ° - - : - - . - , - . ~ , ~ "q"
&: ~ L ~ lo~ 89 '~ Ecru:211MeV "',
b ~Eo~-2,S6M,V "X_.. ~o-~o .; o°I I00 " _ Ecru :22.30 MeV ~-~ l^: 109
I0 -I ~ ~ " h / o : I I.?/
i O 40 80 120 160
8c.m. (deg)
Fig.7.Elastic angular d~tri~tion for the O~V+Si~v scattering at several energies above the Coulomb barrier, The solid lines are calcu- lated by using (3.34) with u -iui=6-17i and A=I. T~e values of £0 are specified in each case.
several attempts to treat the relative degrees of freedom by group theoretic techniques. The final goal of reaction theory would be to combine both approaches, This sub)ect is still at an early stage. The idea here is as follows. The main ingredient in the construction of the analytic S-matrices discussed above is the use of the Euclidean connection. This is a set of relations in which step operators act on both sides of the Jost expansion. For example, for the Coulomb problem, the action of the operators that shift £ up or down by one can be w r i t t e n as
X+l~'~'m> =[ (2~+l)((£+l)2-m2)((£+l)2+f2(k))(2£+3) ]
x (-I) lu.~.m>
1/2
(4.1a)
175
X-l~'£'m> =I (2£+l)(£2-m2)(~2+f2(k))(2£-1) ] x (1) lu,~,m>
112
(4.1b)
when ac t i ng on S0(3,1) r e p r e s e n t a t i o n s and as
X+I±k'£'m> = _ II 1/2 (2£+3) J
x(£+l±if(k))(tl)Itk,£+l,m> • (4.2a)
X-ltk'£'m> = [(2£+I)(£2-m2) ] I/2(2£+I)
x(£¥if(k)) (¥I) Irk, £,m> , (4.2b)
When acting on representations of E(3). These algebraic relations replace the SchrSdinger equation since they provide the [ecursion relations that determine the Jost functions A~(k) and B£(k). For example, by comparing (4.1a) and (4•2a) one obtains
-i[(£+l)2+f2(k)]I/2 A[+l(k) = (£+l-if(k)) A~(k) , (4.3)
-i[(£+l)2+f2(k)]i/2 B~+l(k) = (~+l+if(k)) B~(k)
One can use these relations to construct a purely algebraic theory of scattering. In a multichannel problem, the Jost functions become an array and the action of the shift operators becomes a matrix. The Euclidean connection produces matrix recursion relations. If these can be solved in closed form, one can solve the corresponding coupled channel problem in closed form. Otherwise, one can use numerical methods to obtain the solution• If one denotes the Jost arrays by the Column vectors
A£(1)(ki>
At(k) = A£(2)(k B£(k) =
B£(1)(ki>
B£(2)(k , (4.4)
the S-matrix is the trasformation matrix from the column vector A to the column vector B,
A = 5 B (4.5)
The solution of the coupled channel problem can be simplified by the use of some formal relations satisfied by the r-functions. For example, in a two channel problem, one must introduce 2x2 matrices. The space of 2x2 matrices is spanned by the unit matrix I and by the three Pauli matrices Ox, Oy, O z. By using the integral representation
176
F(z) = [~ t z-I e -t dt , (4.6)
one can define a V-function over matrices which satisfies the rela- tion
1 1 F ( a l + ~ , o u ) = - [ F ( a + ~ , ) + F ( a - ~ , ) ] + - a u [ F ( a + l ~ ) - F ( a - ~ . ) ] • ( 4 . 7 )
2 2
where c u is any arbitrary Pauli matrix, Ou=U.O. Eq.(4.7) is obtained with the help of the relations
Ou 2n = I , Ou 2n+l - c u , (4.8)
and is similar to the relation satisfied by the exponential function (rotation operator)
-i0o u e = cos~ - iousin0 (4.9)
A particularly simple case of practical interest is provided by scattering by a modified Coulomb interaction when the short range potential contains a spin orbit term. Since the interaction is diagonal in £,s and ], the Euclidean connection is diagonal and the corresponding recursion relations can be solved yielding for spin 1/2,
S~,]=£±i/2(k) =
F(~+u(±) (£'2k)+2+i~/k )
[~+u(-+) (£, k)+2-i~/kl F
2
V(£-u(-+)(~' 2k)+l+ip~/kl
x e i(21n2)H~Ik (4.10}
F(,-u (+)(,,k)+l-ip~/k)
2
Here
u(+)(~,k) = Uc(~,k) + ~Us(~,k)
u(-)(£,k) = Uc(~,k) - (~+l)Us(~,k) (4.11)
The functional form of u_(~,k) is expected in general to be different from that of Uc(~,k). D1fferentlal cross sections and polarizations can be obtained in the ususal way in terms of the amplitudes
177
a(k,e) = F.
2 i k ~ = 0 {(£+i)[S£,£+1/2(k)-i] +
+£[S£,~_i/2(k)-I]} P~(cosS),
1 b(k,8) = Z (S~,£=i/2(k)-Z~,~_I/2(k)}
2ik £=I
x p~l(cose)
(4.12)
5.CONCLUSIONS
The use of group theoretic techniques appears to provide a new tool for the analysis of nuclear reactions. This tool has been applied in considerable detail in the analysis of inelastic excita- tions of nuclei by proton impact in the eikonal approximation. These applications make use of an algebraic description of the internal coordinates and are a direct extension of the techniques used pre- viously in the treatment of bound state problems.
A more ambitious program is that of analyzing the collision process itself by means of algebraic tecnhigues, A preliminary study of elastic collisions between heavy ions appears very promising. These applications imply a major step forward in the use of algebraic techniques in physics. They make use of non-compact groups and of their continuous unitary representations.
Finally, one may attempt to combine the algebraic treatment of the internal and relative degrees of freedom into a single treatment.
The main trust of the algebraic approach to scattering is not so much for dealing with non-relativistic situations but to attack re- lativistic problems. Some preliminary studies have been done but further work is needed i n order to understand fully the implications in the analysis of actual experiments. As far as nuclear reactions are concerned, the major area of application of the algebraic techniques may be in the study of heavy ion ~eactions. The S0(3,2) S- matrix model, if further developed, may provide here a framework as useful as that provided by the interacting bosch model in the description of collective states of nuclei.
AKNOWLEDGEMENT5
This work has been performed in part under Department of Energy Contract No. DE-AC-02-76 ER 03074.
REFERENCES
[I] M.Gell-Mann, Phys. Rev. 125, 1067 (1962); Y.Ne'eman, Nucl. Phys. 26, 222 (1961).
[2] A.Arima and F.lachello, Ann. Phys. (N.Y.) 99, 253 (1974); Iii, 201 (1978); 123, 468 (1979).
[3] H.Pauli, Z. Physik 36, 336 (1926); V. Fock, Z. Physik 98, 145 (1935); V. Bargmann, Z. Physik 99, 576 (1936).
[4] F.lachello, Chem. Phys. Lett. 78, 581 (1981); F.Iachello and R.D.Levine, J. Chem. Phys. 77, 3046 (1982); O.S.vanRoosmalen, F.lachello, R.D.Levine and A.E.L. Dieperink, J. Chem. Phys. 79, 2515 (1983).
[5] J--.N.Ginocchio, T.Otsuka, R.Amado and D.Sparrow, Phys. Rev. C33, 247 (1986).
178
[6] J,N.Ginocchio and G. Wenes, Proc, Int, Conf. on Nuclear Struc£ure, Reactions and Symmetries, Dubrovnik, Yugoslavia (1986).
[7] For a review, see A.Arima and F.Iachello, Adv. in Nucl. Phys. 13, 139 (1984).
[8] F. Iaohello, Nucl. Phys. A358, 890 (1981). [9] R.Bi]ker,R.D.Amado and D.A, Sparrow, Phys. Rev. A33, 871 (1986);
R.Bi]ker and R.D.Amado, Phys. Rev. A, to be published. [i0] G.Wenes,N.Yoshinaga and A.E.L.Dieperink, Nucl. Phys. A, to be
published, [II] Y.Alhassid, F.Gursey and F.lachello, Phys, Rev. Left, 5__O0, 873
(1983); Ann. Phys. (N.Y.) 148, 356 (1983); A. Frank and K,B.Holf, Phys. Reu. Lett. 52, 1737 (1984); J. Math. Phys. 26, 1973 (1985); J.Alhassid,J.Engel and J.Wu, Phys. Rev. Lett. 5__33, 17 (1984); Y.Alhassid, F.Gursey and F. Iachello, Ann. Phys. N.Y.) 16 Z, 181(1986); Y.Alhassid, F.Iachello and J.Wu, Phys. Rev. Lett 56, 271 (1986); J.Wu, F.lachello and Y.Alhassid, Ann. Phys. (N.Y), to be published; A. Frank,Y.Alhassid and F. Iachello, Phys. Rev. A, to be published.
[12] L,C.Biedenharn and A.Stahlhofen, Proc. Int, Conf, on Nuclear Structure, Reactions and Symmetries, Dubrovnik, Yugoslavia, (1986).
[13] D.Zwanzige[, J. Math. Phys. 8, 1858 (1967), [14] H.J.W,M~Iler and K.Schileher, J. Math. Phys. 9, 255 (1968). [15] Y.Alhassid and F. lachello, to be published. [16] R.D.Amado and D.A. Sparrow, Proc. Int. Conf. on Nuclear
Structure, Reactions and Symmetries, Dubrovnik, Yugoslavia, (1986).
ALGEBRAIC AND SUPERSYMMETRIC TREATMENT OF ODD-ODD NUCLEI
V.Paar,S.Brant,D.Vretenar and D.K.Sunko Department of Theoretical Physics, University of Zagreb, 41000 Zagreb,
Yugoslavia
A.B.Balantekin Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA
T.H~bsch University of Maryland, College Park, ;~D 20742, USA
ABSTRACT Four related algebraic approaches to odd-odd nuclei are described. Results of recent calculations are given, and possible extensions and current problems are discussed
I. INTRODUCTION
In this lecture several interrelated alaebraic approaches to odd-
Odd nuclei will be described. They may all be viewed as various
extensions of the IBFM/PTQM model of odd-even nuclei I'2, which is
itself based on the SU(6) model of even-even nuclei first introduced 3
by Janssen,Jolos and D~nau and later much investigated 4 by Arima and
lachello, in a different representation.
Here we shall use the more widely known IBM notation, although concre-
te calculations have been done in TQM-based codes.
~he IBFM Hamiltonian of the odd-even nucleus is
HIBFM = HIB M + H F + HIN T (i)
COnsisting of the SU(6) boson core term, single particle fermion term
and a boson-fermion interaction.
The most straightforward extension of (I) to the odd-odd case is to
make the replacements
.(12) HF + HF I) + H~ 2) + "F
• (I) + H(2) (2) HINT ÷ hINT INT
In other words, an additional fermion is attached to the core, and
the two fermions interact through the term H (12) This model is - --F "
Called 5 IBFFM/OTQM, and has only recently been tested in a concrete
Calculation 6 .
A~ important refinement of this approach comes under the heading
180
of bose-fermi symmetry 7'8. It is a way to constrain the parameters
of the full IBFFM/OTQM Hamiltonian bv the same method devised by
Arima and Iachello 4 for IBM: subgroup chains} 5
Bose-fermi symmetry was originally introduced for odd-even nuclei
by Iachello 9 and subsequently used in the context of an even more
severe restriction, the one imposed by supersymmetry I0. Here the
complete subgroup chain is required to be imbedded in a representa-
tion of a supergroup. This restricts the subgroup representations
that may appear further down the chain. It also introduces correla-
tions between neighboring nuclei, originally odd-even and even- I0
even , but now also odd-odd II'19
There is another approach to suDersymmetry, which does not expli-
citly introduce any supergroup restriction 12. It is based on the
fact that for a certain special limit of IBFM an exact analytical
solution for the wave functions is known 2. The solution exhibits
certain supersymmetric properties, and has close analogues in the
even-even and odd-odd systems. It is interesting to note that
this approach seems to have recently received independent corrobo-
rationl3: in two model systems unrelated to IBFM, containing a
boson and boson-fermion term, the existence of exact solutions was
explicitly linked to supersymmetry.
2. THE ODD-ODD MODEL IBFFM/OTQM
There are two kinds of interaction terms in IBFFM, the boson-fermion
interaction inherited from IBFM and the residual interaction between
the two fermions. The first of these, (see (2))
H(i) + . + i=1,2 (3) (i) (i) (i) INT = HMON nDYN HEXC'
contains a monopole-monopole interaction, the auadrupole dynamic
interaction
i { (C~l~j 2) O2 } (4) (i) = ~ Fjlj 2 2 o HDYN J132
where Q2 is the quadrupole operator of the bosons, and the exchange
interaction which takes into account the fermionic structure of the
boson core. The residual fermion interaction consists of three
terms: h(12) + + (5) F = HSDI HSSI HSTI
the first two of which are central (the surface-delta interaction
and the spin-spin interaction), and the third is the surface tensor;
181
interaction. All of these are constrained to act on the nuclear sur-
face only.
Obviously, the full model has a large number of free parameters. Some
rather elegant ways to constrain them will be considered later. Here
we consider an ad hoc, practical constraint: to describe a realistic
nucleus with as few parameters as possible. Such a calculation has
recently been done 6 for 198. 79Aui19 •
To begin with, one should select the symmetry of the core, in this 200+
Case 80ng120. It is reasonable to assume that a pure limit of IBM
Should suffice, since the low lying spectra of odd-odd nuclei are
dominated by the extra nucleons, and putting too much detail into the
Core is not expected to yield enough for the investment of additional
free parameters. The limit chosen was 0(6), consistent with previous
IBM-2 calculations 14 of 200Hg. The parameters of the core were then
adjusted to a best fit for mercury, thus reducing the free parame-
ters actually fitted to gold to those of the various interactions.
These were further constrained by assuming the dynamical interaction
Parameters to be the same for the proton and neutron, and the exchan-
ge interaction was taken to be zero; As for the residual proton-
neutron interaction (5), only the shin-spin part was taken into
account, the other two being set to zero.
0.6"
>.
~J
3 +
---, --'I".2"
0.2"41 +&. . . . . 1-
0 + - - O"
1 + - - 1"
Oi 2 + 2- T h e E x p
0+7
3 +
0.5' 7 ~ , - - 6. +
0+"
..........................
_ _ 16 +)
5* . . . . . . . . . . . . . . 5* The £ x p
Fig.l a) negative and b) positive parity states of
198Au (from ref.6)
With these restrictions a satisfactory fit was produced, shown in
Pig.l. The overall agreement with experiment is seen to be very good,
especially for the negative-parity levels. However, some qualitative
182
features remain to be resolved. Most important, the model predicts
four 0 states below 500 keV, while only one is at present observed.
3. BOSE-FErmi SY~9~ETRIES
The concept of bose-fermi symmetry is best explained in the context
of odd-even nuclei, where it was first introduced 9. The full number
-conserving symmetry of a fermion of spin j is UF(2j+I). If a fer-
mion is coupled to a boson system having itself a dynamic symmetry
(such as an IBM core), the full symmetry of the combined system is
uB(N) ~ uF(M), where B(F) is just to remind that the concrete rea-
lization of the generators is bosonic (fermionic) . Now subgroup
chains of both dynamical groups are taken. Bose-fermi s vmmetry occurs
if at some point the same group appears in both chains (in practice,
it is two groups which are isomorphic). The direct product of a group
with itself is reducible:
G®G ~ G (6)
and so at this point the two subgroup chains merge into one. The sub-
group chain of G is then followed down to the angular momentum group
SU (2).
It should be borne in mind that (6) is true only for the diagonal sub-
group of G O G, i.e. the one in which the two group elements multinli-
ed directly are parametrized by the same parameters. In this way bose-
fermi symmetry not only constrains parameters by the choice of parti-
cular subgroup chains in the boson and fermion sectors, but also
specifies the interaction between the two.
A specific example for the odd-odd case is the Spin (6) symmetry 7, an
extension of the odd-even results of Iachello 9. The relevant subgroup
chain is
uB(6) ® u F"(4) ® uF"(4)
s u "(4) ® su ~(4)
soB(6) ® su "'(4)
sp.~ (6) I BF
sp~ "'(s)
Spin "(3) I BF
Spin "'(2).
(7)
183
The derivation of the energy formula for (7) is given in ref.7.Fig.2
is taken from there, and shows a typical spectrum. The four lowest
bands are connected by dotted lines.
E (M eV)
f" i /
I,,r~, "xbg ,i i
0 1 2 3 4 5 6 7
Fig.2 Typical spectrum for chain (7}
(from Ref. 7)
15 ~hree other subgroup chains for the same system have been discovered
One is similar to (7), but the other two are quite different. They
are based on the fact that several Sp(4) representations describing
a neutron-proton state may be taken together to span a single repre-
sentation of a larger group, namely UF(5) and UF(6) in this case. This
"reversal of direction" in the subgroup chain may alternatively be
Viewed as a standard subgroup decomposition, but with uF(16) taking
the place of uF(4) ~ uF(4).
Generally speaking, if the neutron (proton) spin has nn(n p) components,
the total system has n n angular momentum components, and the maximal pn
SYmmetry group describing them is UF(npnn). The new decompositions
a~°unt to considering the neutron-proton system as quasibound, which
is then coupled to the core. The various chains correspond to different
binding energies.
It is finally worthwile to note that it is possible to realize the
Parabolic rule for proton-neutron multiDlets in the framework of
b°se-fermi symmetryl6.The narabolic rule was previously calculated 17
in the collective model. The leading order contributions from the
184
exchange of quadrupole (2 + ) and spin-vibratlonal (I +) phonons between
the proton and neutron were found to give the energy dependence of the
multiplet as parabolic in I(I+l). The same parabolic dependence is
also a leading order effect in OTO>I, where the spin-spln interaction
plays the role of the 1 + phonon 18. In the bose-fermi approach, the
rule arises at the level of representations 16. If the odd fermion is
partlcle-like, it is placed in the fundamental representation of SU(n);
if it is hole-like, it is placed in the adjoint representation. The
energy formula arisinq from the reduction of SU(n) ~ SU(n) is the same
for both cases; but since the quantum numbers (representation labels)
are different, the formula gives different energies for the two cases,
and two mutually inverted parabolas arise, as they should 17.
4. SUPERSYMMETRY: STANDARD APPROACH
The standard approach to supersymmetry in nucleil0("dynamical super-
symmetry") is to embed a subgroup chain into a supergroup. It is our
intention in this section to do that for chains describing odd-odd
nuclei.
Instead of taking the proton-neutron system as quasibound, the whole
fermlon space is considered as having np+n n components. This is a more
flexible approach, since it allows the individual particle to be dis-
tr~m/ted over several levels, np(n n) then being the number of all the
projection states of these levels. The supergroup to be decomnosed is
then
U (6/np+n n) (8)
Since the number np+ n n gives no clues as to its origin, the physics of
the problem is specified by the next step in the decomposition. For
odd-odd nuclei, we define the canonical decomposition to be 19
U(6/np+n n) ~ uB(6) x UF(np+ nn)=UB(6) x uF(np) x uF(n n) (9)
which is then continued in var$ous ways. The point is that now several
neighboring nuclei are correlated, and a single supermultiplet contains
even-even, odd-even and odd-odd nuclei.
For a specific example, take the supermultiplet based on 194pt. The
proton space is the jp = 3/2 level and the neutron space consists of
= 1/2,3/2,5/2. Thus np=4 and nn= 12. The ~= 7 representation of Jn U(6/16) iS reduced according to (~):
185
+
+ ( [5], {12})
%u .o,
(~) (4) X UF(V) (12) U(6/16)=~(6) X UF(16) = UB(6) X U F
{ 7 } : :
194pt
: ( [ 6 ] , { I } , { 0 } ) + ( [ 6 ] , { 0 } , { 1 } )
195Au 195pt
: (Is], {12}, {0})+( [5] ,{1}, {1})+( [5] ,{0), {12}) 196Hg* 196Au 196pt*
(1o)
Under each representation the corresponding nucleus is given, with
an asterisk denoting excited states.
In the case when np= n n = n , an alternative to (9) is possiblell:
U(6/2n) ~ uB(6) ~ uF(2n) ~ uB(6) x spF(2n) (ii)
This is in fact the only supersymmetric approach known to have a
Chance of describing the odd-odd nucleus 196Au; previous attempts 8'20
COuld not account for the ground state spin. It may be shown II that
for the odd-odd members of the supermultiplet, (Ii) is equivalent
to a bose-fermi symmetry of the type
uB(6) ~ uF(n) ~ uF(n) ~ uB(6) ~ uF(n) (12)
With the bar denoting that conjugate representations should be taken.
This restriction follows from supersymmetry, and it means that (ii)
Will apply to those supermultiplets where the odd-odd members have one
fermion particle-like and one hole-like. Such is the case of 196Au:
the odd neutron is mostly in the levels Pl/2,P3/2,f5/2, the odd nro-
ton in Sl/2,d3/2,d5/2[land d~/2(P~/2 ) is hole-like (particle-like). A
Particular prediction xs given in Fig.3.
The ground state is 2-, which is correct. Since not enough is known
of the 196Au levels, an educated guess about the performance of (12)
for the excited states may be made by comparing Fig. 3 with the levels
of 198Au (Fig.l). If there is some similarity between the spectra of
the two gold isotopes, one would be quite satisfied with the results
of this approach. It is perhaps not rec%~ndant to note that the four
lOWest levels of Fig. 3 lie on a parabola in the E vs. I(I+l) plane.
~PERSYM~4ETRY: NONSTANDARD APPROACH
This is a different line of attack than the dynamical supersymmetry
described in the previous section. It makes no assumptions about the
e~heddings of the Hamiltonian. Instead, it is shown that a particular
li~it of IBFFM/OTQM has supersymmetric properties. It is defined like
186
ORNL" OWL; 85 - 16~77
0.50
0 .25
O
Fig.
3,O)[ - -
In + I. 1,0) (n * 2, 0,0)
6"
4"
3" (2,01{ 4- - - - 0 " ~ T
5" [ ° _ ( 2 . t ) 3- . . 2 "
4" 12 ,01{ _ _ 2 -
(1,O) 2"
3" (0 ,0 ) O"
( t , 4 ) { _ _ '1"
SU B (6) x Sp F (24)
- - ( t ,0) 2"
) for 196Au Prediction of scheme (I
(from ref. ii)
the various limits of IBM 4, by conditions on the parameters. It
relies heavily on the existence of a particular analytic solution 2'12
of the (odd-even) IBFM/PTOM, which has close counterparts both in the
odd-odd and even-even systems.
The limit is obtained 12 by taking only the dynamic Dart of the intera-
ction, with an SU(3) core term. Specifically,
B IB HIB M = 6I (13a)
F HIBFM = ~I B • I B + F(Q~ - 02 ) (13b)
F F HIBFF M = fi B • I B + Fp(Q~ • Q2 p) + Fn(0~- O2 n) (13c)
where the Casimir term is dropped since only states based on the
ground-state band (GSB) of the core are considered. The solutions 12
mentioned above are, respectively
2N 1 (14a) Ic> = [ ]IGsB>
I=0,2,..
IK:j JM >: PJ {IjK > Ic > }
MK=j <14b) 1
= [ ~II < j j IOIJK > I (J'IGsB)JM >
187
J
IK=jp+ Jn JM >= PMK=jp+Jn { 13p~p> ljn~ n > Ic > }
14c) 1 . . . . . . . .
= ~ B~I <3p3p3n3nl 3pn3pn ~ <3pn3pnIO IJK >
I (JpJn) Jpn'IGSB ; JM
Where (14a) is the coherent state of the core 2. The wave function
(14b) is a solution of (13b) for a particular ratio (F/~)=(F/~.~USy(j) ,
Which fixes the limit. The ratio depends only on the spin j of the
Odd particle. The odd-odd wave function (14c) is an exact solution of
(]3c) for (FP/~) = (F/6)SUSy(jp) and (Fn/~) = (F/~)SUSy(jn). The
SUSy values are taken from the odd-even model for the appropriate
Particle spins. Thus the critical ratios are the same for (13b) and
(13c). Even more remarkably, the moments of inertia of the respecti-
ve bands (K=j and K=3p' + jn ) are the same, and equal to that of the
ground state band of the core:
EGS B = EK= j = EK=jp+j n = 6J(J+l) (15)
This property has been explicitly predicted 8 as a consequence of
SUpersymmetry.
Working independently, one of us has recently shown ~3 that the exi-
Stence of exact solutions in a model Hamiltonian containing bosonic
and boson-fermion interaction terms is linked with supers_vmmetry. As
°POsed to dynamic supersymmetry, the approach was called spectrum-
generating supersymmetry. It is conjectured that the results pre-
Sented here will turn out to be another concrete examnle of spectrum-
generating supersymmetry, and work along these lines is in progress.
REFERENCES
I) F.lachello and S.Kuyueak, Ann.Phys. (N.y.)f36(1981)19
2) V.Paar,S.Brant,L.F.Canto,G.Leander and M.Vouk, Nucl.Phys. A378(1982)
41
3) D.Janssen,R.V.Jolos and F.D~nau, Nucl. Phys. A224(1974)93
4) A.Arima and F.Iachello, Ann. Phys. (N.Y.)99(1976)253; 1!1(1978)201;
1.~.323 (1979) 468 5) S.Brant,V.Paar and D.Vretenar, Z.Phys. A319(1984)355
6) V.Lopac,S.Brant,V.Paar,O.W.B.Schult,H.Seyfarth and A.B.Balantekin,
Z.Phys. A323(1986)491
7) T.Hubsch,V.Paar and D.Vretenar,Phys. Lett. 151B(1985)320
8) P.Van Isacker, J.Jolie,K.Heyde and A.Frank,Phys.Lett. 54(1985)653
188
9) F.Iachello,Phys.Rev. Lett. 44(1980)772
i0) A.B.Balantekin, I.Bars and F.Iachello, Nucl. Phys. A370(1981)284
i]) A.B.Balantekin and V.Paar, ORNL preprint, to be published
12) D.K.Sunko,S.Brant,D.Vretenar and V.Paar, invited talk in "Proceed-
ings of the International conference on Nuclear Structure,Symme-
tries and Reactions, Dubrovnik 1986", eds. R.A.Meyer and V.Paar,
World Scientific, in print
D.K.Sunko and V.Paar, Phys. Lett. 146B(1984)279
13) A.B.Balantekin, Ann. Phys. (N.Y.) 164(1985)277
14) A.F.Barfield et al., Z.Phys. 311(1983)205
15) T.H~bsch and V.Paar, Z.Phys. A319(1984)III
16) A.B.Balantekin and V.Paar, Phys.Lett. 169B(1986)9
17) V.Paar, Nucl.Phys. A331(1979)16
18) V°Paar, In-Beam Nuclear Spectroscopy, Ed.Z. Dombradi and T.F6nyes
(Akademiao Kiado, Budapest 1984) Vol.2,p.675
19) A.B.Balantekin, T.H~bsch and V. Paar, submitted to Ann. Phys. (N.Y.)
20) I.Bars, in "Bosons in Nuclei",ed. D.H.Feng,S.Pittel and M.ValliereS,
World Scientific (1984)]55
A.B.Balantekin,I.Bars and F.Iachello, unpublished
Proton-Neutron Sy~netry ~ Low Lying Collective Modes
P. yon Brentano
Institut fiir Kernphysik der Universitat zu Koln
D-5000 K61~ 41
Dedicated to Prof. G.E. Brown on the occasion of his sixtieth birthday.
Abstract:
Properties of collective states which depend on the proton and neutron content of
the nucleus are investigated in the frame work of the F-spin concept. Experimental
evidence for F-spin as well as data on F-spin multiplets is given. "Global"
descriptions of groups of nuclei in the Interacting Boson Model are discussed.
F-spin forbidden M1 transitions are discussed and values for F-spin purity of the
gamma bar~ in 128Xe and 168Er are given.
I~ntroduction
The proton-neutron symmetry of low-lying collective modes in nuclei is a very
interesting subject. The copenhagen school formulated the Dogma that the collective
properties of the low-lying modes depend only on the overall shape of the nucleus.
Today we certainly have to modify this Dogma. Actually, attempts for ir~provements
of the Dogma go back to work by Greiner and Faessler in the sixties (1-3). In the
following, two methods will be discussed by which one can obtain information on the
proton-neutron s~mmetry of collective states. These are:
I) global plots of collective properties versus N and Z, and
2) M1 transitions and g-factors of collective levels.
The global plots of collective properties versus proton and neutron numbers are an
obvious tool to find the dependence of the collective Hamiltonian on the proton and
neutron variables. Such plots have been made for the energies using the concept of
the F-spin multiplets and the Np.N n scaling. Furthermore, such plots have been
made for electromagnetic transition moments and g-factors.
M1 transitions between collective levels are a particular interesting
phenomenon. This subject has been stimulated by the suggestion of a low-lying
proton-neutron vibration mode in deformed nuclei by Lo Iudice and Palumbo (7,9) and
its subsequent discovery by Bohle and Richter (13-15) in Darmstadt. This
very successful experiment was also strongly stimulated by predictions from the
I~-2 model (10-12) alxl from geolaetric models (13). The M1 giant state raixes with
190
low-lying collective levels and thus induces M1 transitions among these levels.
There is a large body of experimental data on these M1 transitions, which can be
found in the nuclear data tables of isotopes. Until recently little use of this
data has been made. see however: (13-15). Before we discuss these phenomena in
detail, it should be mentioned, that there were various theoretical approaches
proposed for the study of these questions. In particular the Hatree-Fock shell
model approach of the Mdnchen group, the MONSTER model of the Tubingen group (16),
and various collective models. Already in the sixties Faessler and Greiner (1-3)
suggested the use of independent mean proton and neutron fields in the geometric
models and these ideas have been taken up again in the two-rotor model of Lo Iudice
and Palumbo (7-9) and also recently by the T~bingen and Frankfurt groups (13-15).
In the algebraic models, Arima, Iachello, Otsuka and Talmi (10-12) introduced in
1977 the Interacting Boson Model (II~-2) with separate proton and neutron bosons.
Also the approach by the Bucharest group (22) must be mentioned. In the following
the Interacting Boson Model will be used, since it is particularly well-suited for
a global description of collective phenomena.
The Proton-Neutron Interaction Boson Model (I~M-2) and F-spin:
As we will use the Interacting Boson Model a lot in the follc~ing we give
a short reminder. (10-12, 17-21)
Interactinq Proton-Neutron Boson Model (I~-2) Primer
i) Bosons: Pairs of valence particles (holes)
2) Types of bosons in IHM-2: (A > 50)
Proton bosons ~=(p,p) and neutron bosons bn=(n,n ) , no deuteron
boson bd=(P,n ) (for A < 50:I~4-4 with 2 extra deuteron bosons)
3) Interactions: 1 and 2 boson interactions; recently also 3 boson interactions.
4) Spins and parities of bosons:
sd-model: JP (s) =O + JP (d) =2 +
extended model: JP(g)=4 +, JP(p)=l-, JP(f)=3-
5) Mass difference: m(d)-m(s)=ed, (fitparameter)
6) g-factors: gp=g(~), gn=g(dn ) (fitparameters)
schematic model: g(~)=l, g(dn)=O ,
7) Transitions: d .... > s + ganm%a, d .... > s + meson
8) Effective quadrupole charges of bosons:
q (~) =qp q(dn) =qn (fitparameters)
Schematic model: A = 160: qp = qn = 13 efm 2
9) Electric charge of Bosons: schematic model: ep=2e, en=O
191
i0) Boson counting: Particle bosons from beginning to middle of major
shell, hole bosons beyond middle of major shell.
II) Quadrupole and magnetic dipole operators are the most general one body
operators.
The Interacting Boson Model I~M-2 with proton and neutron bosons contains many more
states than the Interacting Boson Model II~4-l,which has only one kind of bosons.
Arima, Iachello, Otsuka and Talmi introduced the notion of F-spin (10-12) from the
beginning, in order to classify these states and to give a relation between the two
~Odels. F-spin is the isospin in a system of bosons. The new name F-spin (F for
France) was introduced to distinguish F-spin from the isospin of the underlying
Fermion system. Formally, F-spin and T-spin are the same, as is shown in the
following table:
Isospin
Fe~iens
P T 1/2
T Z 1/2
F-spin
Bosons
n b n
1/2 F 1/2 1/2
-1/2 F 0 1/2 -1/2
T+ n -- p, T_ p = n F+ b n = ~, F_ ~ = b n
From these operators for the individual bosons one obtains the F-spin operators for
a system of s and d bosons. These operators have been given by Otsuka et al. In
Particular, one finds the F-spin operators FO, F+, F_ for the whole nucleus,
from which the operators F = (Fx, Fy, Fz) and F 2 are obtained by the
relations: F O = Fz, F+ = F x + i Fy and F_ = F x - i Fy. In real
nuclei the operator F O = 1/2 (Np - Nn), is found to be diagonal, and the
lowest states in nuclei have a rather pure F-Spin, F = Fma x = 1/2 (Np + Nn),
where Np and N n are the proton and neutron boson numbers.
R__elation of I~M-2 and I~-I (Projection Method)
The notion of F-spin gives a very clear relation between the two beson models IHM-I
a/nd I~-2 as shown by Iachello (23). Namely the states of the IBm4-1 correspond to
the states with maximum F-spin, F=-Fmax, in the IH~4-2. This relation has been
strengthened by Harter and others (25-27) to construct for each I~4-2 Hamiltonian
HIH~4_ 2 a unique corresponding IHM-I Hamiltonian HIHM_ I.
192
This projection method is shown schematically below:
model : IH~-I I~M-2
in a special nucleus with boson numbers ~, N n
states: (a) (Fmax, Fo,a )
Hamiltonian: HII~M_ 1 (~,N) HI~4_ 2
The two Hamiltonians are chosen to have equal matrix elements in the
F = Fma x subspace of IHM-2 and the IHM-I space
(Fmax,Fo,a/ HIH~4_2/ Fmax, Fo,a ) = (a/ HIHM_ 1 (Np, N)/ a) Thus the energies of the original 1~4-2 and the projected IH~-I Hamiltonian agree
exactly for those states which have a pure F-spin, F = Fma x (25).
F-spin multiplets
The most direct way of observing spin is to look at the splitting of a level in a
magnetic field. Similarly the most direct way to observe isospin and F-spin is to
look for isospin and F-spin m~Itiplets. An F-spin multiplet is a series of nuclei
with the same character (particle or hole), constant total boson number N = Np +
N n and constant maximum F-spin, F = Fma x = 1/2 (Np + Nn), but different
values of numbers of proton bosons ~ and different values of F O = (Fnk%x -
Np) (24,28). If the I~M-2 Hamiltonian is F-spin invariant, i.e. if HF+= F+ H,
all levels in the F-spin multiplet have exactly the same energy. But normally the
situation is more complex, because the IHM-2 Hamiltonian contains terms like the
proton-neutron quadrupole interaction Qp'Qn, which mix F-spin. Thus it is not a
priori clear whether F-spin is useful and pure in real nuclei. The most direct
method of investigating this question is to look at F-spin multiplets in real
nuclei. The observation of a complete F-spin multiplet is a much easier task than
the observation of complete isospin multiplets, because the low-lying levels in
nuclei have F = Fma x and therefore belong to F-spin multiplets, whereas some of
the members of an isospin multiplet are in general highly excited states and
difficult to observe. In figs. (1-3) we show examples of F-spin multiplets. Fig.
(i) shows an F-spin multiplet with N = 13 and Fma x = 6.5 in the rare earth
region, from reference (24). It is remarkable how constant the excitation energies
of the ground band and the ~ band are in this F-spin multiplet, even though
158Dy and 182pt differ by 24 mass units. One has to stress that without the
Interacting Boson Model (I~M-2) and the F-spin concept there would have been no
reason to examine this particular series of nuclei, which form the F-spin
multiplet. In order to test the significance of the observed constancy of the
energies of the ground band within the multiplet, several series of isobaric
nuclei, for which the total boson number, N = ~ + Nn, and the F-spin, Fma x =
1/2 N, vary strongly are shown in fig. 2. One notes that, indeed, th~ energies in
193
>
5- 1,0,
O5.
8 + _@+ . 6 +
z ... . 2" --,:2, .-" 2 + 0"
~* ...6+ 6" . . . @. ..
4 + L ÷ L + L*
2* 2* 2 + 2*
B + ... @* .- L÷
4 ÷ 4+
3" . -3+ • Z"
4 + _. 4" Z~+ . . , - -
r __- 2* . . 2~.-Z~
O. Q* ~ O* O* O~ o ÷ O ÷
158 162 165 1"70 174 178 182 Dy E r Yb Hf W Os Pf
Np, N n B,5 7,6 6,7 5,8 L 9 3,10 2,11
Figure i: F-spin multiplet with F=13/2 o Np and N n are the proton and neutron
boson numbers. We show the levels of the ground band, the quasi gamma band and the
quasi beta band up to an excitation energy around 1 MeV.
P, von Brentano et al (24).
1.5.
> (9 5-
1,0.
0,5~
8"
B,
V
.. 6t ~* .,,- 4÷
r Z . . . . z 0 + Cr' Q÷ 174 1% 174
15.
/
.¢. >
1.0.
~+ ,
. ~+ /
0.5. ~" ' 4"
~+ ,'"
r r "2 - - - - -
Q* O. Q* o ~ o ~
174 162Dy 162Ep 162 Yb Hf W Os Yb
Np,N~ 6,11 5,10 4,9 3.8 8.7 7,6 6.5
Figure 2: Levels of the ground band up to the 8 + levels for several chains of
isobaric nuclei with A=174 and A=162. Np and N n are the proton and neutron
boson numbers. We note that N=~+N n varies in these c2~ins of isobars.
P. von Brentano et al (24).
194
i i
6 ÷ / / / 4.
2" - - - 2 luJ 2 + Z "
0" 0 + 0" 9" 0+ 124 128 1:32 1:36 140
Te Xe B e C e Nd 1 p" 2p + 3p* 4p + 5D÷ 5n - An- 3n- 2n- l n -
Figure 3 : The figure shows a part of an F-spin multiplet for 124Te-140 Nd.
Nuclei with Z=50 and N=82, which are not described by the boson model, are deleted.
All known positive parity levels up Ex=2 MeV and the ground and ga~na bands up to
the energy of the 81+ level are shown. From Harter et al (28).
t . 2
~..0
O.B
0 .6
0 .4
0 . 2
0 .0
Fmax - t 3 / 2 m u l t i p l e t
B "~ . + ~-...
X ~... .X X X ~ ~ - ' x '---~--'X- . . . . "
. . . . I ' ' ' ' I . . . . I ' ' ' " I ' ' ' ' 1 " ' ' ' l ' ' ' ' l ' ' ' '
64
x + 4~ex 2g. +~
2 ; m ex
o; ex
+ IBM 2g
÷ IBM 4q
2 y IBM
O; IBM
65 58 70 72 74 76 78 BO
Figure 4: In the figure the energies of the 2~ 4~, 2~ ar~ 05 states frc~ the
P--Fmax=13/2 multiplet of fig. 1 are compared to the predictions from a global II~-2
fit with 5 constant par-dmeters. Sala et al (31).
195
these series of nuclei vary strongly with N. The 6 + energy of 162yb is about 70
% higher than the 6+ energy in 162Dy, although the two nuclei have the same mass
number. The observed constancy of the energies in the F-spin multiplets is
therefore a real effect. Similar tests can be done by plotting g-factors and
qtladrupole charges vs N and Z (32-34).
"G_~lobal,, fits with the Interactinq Boson Model
What are the consequences of the observation of rather constant energies in F-spin
multiplets? The most simple explanation is to assume that the I~4-2 Hamiltonian is
F~spin invariant and has constant parameters for all nuclei in the multiplet (28).
An elegant way in which this concept can be strenc/thened is to postulate a dynamic
U(12) symmetry, as has been suggested by Frank and van Isacker (29,30). The problem
With this approach is that it implies rigorously constant energies within the
rm/itiplet. As shown in fig. 3 the energies in the multiplets vary with FO,
however, indicating the presence of F-vector and F-tensor components in the IBM-2
Hamiltonian (28). ~us the phenomenological boson interaction must contain not only
F~scalar terms, as e.g. terms like (Qp(X) + Qn(X)) 2 , but also F-vector and
F-tensor operators, which for example are contained in the proton neutron
quadrupole interaction Qp'Qn which is usually assumed to be the dominant
interaction in the IH~4-2. However, if we use this interaction it becomes a problem
to understand why the energies are nearly constant in the rare earth multiplet
shown in fig. (3). A solution of this puzzle has been given by Sala et al (31) who
have obtained a fit to spectra of 22 rare earth nuclei including the multiplets
with the following IH~4-2 Hamiltonian H = e (~ + ~) + k Qp(X).Qn(X) +
the fit of all 22 nuclei the following 5 constant parameters where used: e = 0.48
MeV, k = -O.i MeV, a = -0.02 MeV, b = 0.0026 MeV, Xp = X n = -0.4 = X, C - 20
MeV. The actual calculations where done with the projection method (25) to save on
computer time. Examples of these fits are shown in figs. (4-6). In the discussed
Work the energies of the ground band and the gamma band are well reproduced,
Whereas the beta band is not so well reproduced. I also want to mention a similiar
Work by Casten et al (35) in which a "global" fit to spectra of about iOO nuclei is
achieved in the II~4-1 by using 6 constant parameters and a parametrized functional
dependence of the boson energy on the Casten scaling parameter N~n. Finally I
want to show a recent fit of many A = 130 nuclei by Novoselsky and Talmi (36) in
fig. 7. The success of these "global" fits clearly shows the basic validity of the
Interacting Boson Model.
196
1 . 2
1 .0
0 . 8
0 . 6
0 . 4
0 . 2
0 . 0
Fmax = 6 multlplet
.... ''"......
/ /
x x x _ _ . _ ~ . . . . x . . . . . ~- . . . . ~ / g
L-- --.X--------X. ---.-~W. . . . . -X- .... -X-"~'~"/~
" ' ' ' 1 ' ' ' ' 1 ' ' ' " I ' ' ' ' 1 ' ' ' z t ' ' ' ' I .... I ' ' ' ' 1 ' ' ' "
6 4
X
2~. 4rex • + 9 .
÷ e x 2 y e
2~ IBM
; IBM
2y IBM
O; IBM
56 6B 70 72 74 76 7B BO 82
Figure 5: Same as figure 4, but for the F=Fmax=6 multiplet (31).
182Pt
-I MeV
MeV
+ _ _ + _ _
+
s + - -
4 + _ _ 4 +
0 + .. 0 + _ _
EXP.
5 + 4 + - - 4 + - -
4 + - - 3 + 2+
4 + 2 + _ _
3+- - 2 + _ _ o +
2+ _ _ 0 +
IBM-I EXP. IBM-I EXP. IBM-I
Figure 6: Conioarison of the low lying levels of 182pt with the predictious of the
global IE~-2 fit by Sala et al. The calculation was done using the projection
technique with the IBM-I code (31).
197
1
°7 I
H e V
2 '
c
] , j J f I 2 4 6 8 i0
5eB~
b
I I l l l ~
61 _
5j
31
2a
d
I I I i ] I
n
2 4 6 8 i0
NO. OF N0uCr0n B0s0n$
Figure 7: The f i g u r e ~ e s the energies o f l e v e l s from the ground band and the
gan~na band of several Ba and Xe nuclei, with an IHM-2 calculation by Novoselsky and
Tal4qti (36).
8 O O
T 6OO
o_) 2~
%~400 t.u
200
Np- Nn
0 ~,4 1~ 192
v, ,, Ba
I°. \Z s~ /; \~, Ge 1¼ ~ Dy
\\ Er
~\" S\ ÷ \
Z ~ 64 ~
6 I% 32 48 N~Nv
Figure 8: The figure gives the energies of the 2~ levels of nuclei around A=I50
Dlotted vs Np N n. Casten et al (40).
198
NpN n Scaling
The F-spin multiplets connect collective levels in a series of nuclei. A comparison
of collective properties for an even larger set of nuclei can be made by the use of
the Np N n scaling parameter, as has been discovered by Casten (37-40). In a
series of papers he has shown that various nuclear properties, e.g. energies and
B(E2) values, depend strongly on the product of proton and neutron bosons Np.N n
and only weakly on the relative proton concentration N~ (Np+Nn) . We find e.g.
for the 2~ energies
E(2~; Np, Nn) = E(2~, N~n , N~(Np+Nn) = E'(2~, (Np. Nn) )
In fig. (8,9) examples of the success of the Casten scaling method are shown, but a
word of caution is needed. The method depends crucially on the knowledge of the
proper boson numbers ~ and Nn, which depend strongly on various subshells
(e.g. the Z=64 shell) and are somewhat uncertain for nuclei in the vicinity of
these subshells. Conversely, however, it is possible in these cases to determine
the proper boson numbers from the Np N n scaling method. The ~ N n scaling
is an ~irical finding, with up to now no detailed theoretical basis (see however
the references in ref. (41). However, it is related to an idea proposed many times
by Igal Talmi, that the proton-neutron force is the dcmd/nant force in the
collective model. Assuming this the collective energies should depend on the total
number of (p.n) valenc~ pairs, i.e on Np.N n. Another rather interesting
observation has been made recently by Theuerkauf et al (41), who realized that the
energies of the global II~M-2 Hamiltonian of Sala et al (33) follow the Np N n
scaling to a surprising degree fig (i0) . The astounding fact is that the
calculations follow the Np Nn-SCaling even in the vicinity of the magic shells,
where calculations and the data differ. So the ~ N n scaling seems to be an
inherent property of an global IE~4-2 model with constant parameters.
Extended F-spin multiplets,
particle-hole syr~netry for bosons and NpN n scalinq
It seems useful to introduce the notion of an extended F-spin multiplet as a group
of nuclei in a major shell which have constant boson number N=(Np+Nn) , but
without distinction of particle and hole bosons (40). Such groups of nuclei are
shown in fig. (Ii). One notes in the figure that an extended F-spin multiplet has a
diamond shape for nuclei in the middle of a major shell. The levels in an extended
F-spin multiplet have the same energy, if an F-spin invariant I~M-2 Hamiltonian
with constant parameters is used in the whole nuclear region. In the following
199
Ba Ce Nd Sm Gd Dy Er Yb
. . . . . . . . ir 3.2 ~ 96
3.0 ELI g2 - - 28
E2~ 2.6
2.2
2.0
18 5"6 s'~ 6B 6 & 6'6 ~'8 7; z
Figure 9: The figure shows the energy ratio
isotones vs Z and vs Np N n. Casten (37).
Np- N n 24 ~8 72 95 12O ]~4 168 192
°
6 ~'2 ~'8 2'~ 3'0 3'6 4'z ~8 N~" N v
E(4~) / E(2~) plotted for chains of
.~. o.4
LU
0,2
| . . . . I • , , • I . . . . I , •
0 ~ . . . . ! . . . . I . . . . I • ' '
0 100 200 NDNn
' Z . . . . l , ; , , I . . . . I . . . . I , ,
:: B O ' ~
: 70-
i 50-
50 ..., .... , . . . . , . . . . i ' '
90 100 110 t20 N
~, 1.0z
U.I
0 . 5
I . . . . I . . . . I . . . . I , . .
\,
I . . . . I . . . . I . . . . I ' ' •
0 100 200 NDNn
7 0 -
6 0 -
50 ..., .... , .... , . . . . , . r
90 100 t ~ 0 %20 N
Figure I0: The figure gives a contour plot of the 2~ and 4~ energies vs N and Z
Calculated for the rare earth nuclei with a global I~-2 model with constant
parameters. Also shown is a plot of these energies vs ~ N n. Note that the
Calculation fits the data only in the middle of the shell. It shows, however, that
the ~ N n scaling is an inherent property of the I~M-2 model with constant
parameters. Theuerkauf et al (41) .
2 0 0
N n
= Np
Z=66 Z
N=82 N=104 N=126 N=82 N~104 N=126
Figure ii: Comparison of isodeformation contours for three collective models with
the e2apirical results for the nuclei in the Z=50-82 and N=82-I06 regions. The
contours (and their labels) are those of constant deformation for the collective
model obtained with the Strutinski method by the Lund group, those of constant
NpNn, of constant total boson number (Fl~ix) and of the experimental excitation
energies E(2 D of the 2~ states. The diamond lines of constant N are the
extended F-spin multiplets discussed in the text. Casten et al (40).
201
J.2
~.0
.IB
.6
.4
,2
O.
8+
6+
S+
E+
4+ d+
8+ - - B + B+ 8+ - - 5 + ' '8+
6+ ~ 6 + ,,, 6+ 6 + 6 + 6+
..... 8+
4÷
- - 6 +
B + B, +
6+ ,.. 6+
- - 4 + - - 4 + d+ 4+ 4 + - - 4 + '4+ 4+ _ _ ~ + 4+ 4+
2+ 2+ 2÷ 2+ 2+ 2+ 2+ 2+ 2+ 2+
1 ~ t 5 6 1 ~ 160 164 1 ~ 172 176 180 184 184 184 1 8 4 5 2 6 4 ~ 6 8 70 7 2 7 4 7 6 7 8 80 7 8 7 5 7 4 9 4 9 ; ' 9 0 9;2 9 4 9 6 9 8 t 0 0 102 "104 106 108 % t 0 S m G d D y E P Y b H f W O s P t ~ P t O s W
~.,2
t . O
. 6
. 6
. 4
.2
O.
t . 2
1.0
. B
.6
,4
.2
O.
m s +
8+ o+
o+
8~ 6+ 6+ 6+
A + ' ' 4 + 4 +
4 + 4 + 4 + 4 + ~ 4 + ,. 4 + 4 + 4 +
2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 *
158 15/9 1 5 8 162 ~66 1 7 0 174 178 182 182 182 182 6 2 6 4 ~ 6S 70 7 2 7 4 7 6 7 8 7 6 7 4 7 2
9 4 9 2 94 9 6 9 8 200 202 ~04 ~06 ~08 ~ 0 S m G d O y Em Y D H f W O S P t O s W H f
B +
6+
- ' 9 + 8 + ' 8 + IB+
6+ G+ ' 6 + 6 +
- - 8 +
6+
8+
8+
m 6 +
6+
--4+ 4+ .... A÷
4+ 4÷ &÷ 4÷ 4+
' 2 + 2 + ..... 2+ 2 + 2 + 2 ÷ 2 + 2 +
16o leo 16'4 i48 i72 176 leo ~eo ~ o 6 4 5 6 6 8 7 0 7 2 7 4 7 5 7 4 7 2 9 6 9 4 9 6 9 8 1 0 0 1 0 2 1 0 4 1 0 6 t 0 8 G d D y E P Y D H f W O s W H f
8+
6+
4+
2+
Figure 12 : Extended F-spin multiplets in the region of the rare earth nuclei.
C. Wesselborg et al (66).
202
figures (12) experimental energies for several extended F-spin multiplets are
shown. One finds that the energies are reasonably constant even after the crossing
of the middle of the shell, in some extended multiplets (66). This shows that the
assumptions of particle-hole symmetry for bosons and constant parameters of the
IBM-2 Han~iltonian are approximately valid in the middle of the shell for the rare
earth nuclei. The same idea can also be used in the NpN n scheme and the
corresponding plots of constant Np N n contours are shown in figure (Ii) , where
contour plots of the experimental 2~ energies are also given. It is amazing
that the extended F-spin multiplet and the extended N~n scheme are
representing the qualitative features of the data. This is a clear proof that the
proton-neutron force is the origin of the nuclear deformation as suggested by
Talmi.
Breaking of F-spin
After one notes that F-spin is a useful quantum number, one wonders wether it is
also a good quantum number. So it is necessary to discuss the mechanism of F-spin
breaking and the experimental methods by which the purity of F-spin can be
obtained. In this context it is necessary to cxmpare F-spin and T-spin which is done
in the following table.
comparison between F-spin and Isospin (T-spin] bands in nuclei:
Fmax-2 Train +2
Vmax-i ....................................... Tmin+l
Fmax Tmin
E (Fma x - i, Fmax) E (Tmin + i, T) =
= 3 MeV 15 MeV (A=I60)
mixing term
Qp(~) • Qn (Xn) Vcoul
203
F-spin allowed Ml
gamma transition
( O+ Fmax) .... >(I +, Fmax-l)
F-spin forbidden Ml
transition
(Ii Frnax) .... >(12, Fma x)
isospin allowed
beta transition
(0 +, T)---->(O +, T)
isospin forbidden
beta transition
(O +, T) .... >(O +, T+I)
~-spin multiplets Isospin multiplets
Looking at this comparison we note that one expects the F-spin mixing in heavy
nuclei, to be stronger than the isospin mixing. There are two reasons: I) the
energy difference between the mixing bands is much smaller for the F-spin bands
than for the isospin bands, and 2) the F-spin mixing force is a part of the
dominant proton-neutron quadrupole force, whereas isospin is only mixed by the weak
nondiagonal part of the Coulomb force.
~ y of F-spin
The above figure shows that one can study F-spin admixtures by the measurement of
electromagnetic M1 transitions in a similar way in which one studies isospin
admixtures from isospin forbidden 0+---0 + Fermi beta transitions. Thus one
needs to investigate electromagnetic M1 transitions among collective levels in
g~eat detail.
As mentioned above, progress has been made in the discovery of the lowest (i +,
Fr~x-I ) state in Darmstadt (4-6). Subsequently, much experi/ne~tal and theoretical
Work in th~s field has been done, which has been reviewed by Richter (4), Van
Isacker et al (64), Iachello (43), Dieperink (42) Arima et al (18) and Palta~bo
(7-9). In discussing M1 transitions in the II~-2 one has to find the proper M1
transition operator T(MI). The M1 operator in a Fermion system is:
This Fermion operator has the following boson image (42-64):
T(M1) = ( 3 / 4 pi)1/2 ~K(gpI ~ + gnLn) '
~aere ~ = ( lo)l /2(d~dk)l
204
The M1 transition operator T(MI) can also be written in the following
form (42,64) :
T(MI) : (3/4pi)1/2 ~k(gs L + gv(~-Ln)), where L = ~ +L n and
2gs = gp + gn and 2 gv = gp-gn"
This form is convenient because the first part of T(MI) is a diagonal operator, so
only the second part contributes to transitions between different states: The M1
transitions between the 0 + gs. and the 1 + state therefore depend only on the F
vector g-factor gv" In order to determine gv, B(MI) values to the giant M1
state found in (e,e') and (garma,gamma') reactions can be used. The scalar g-factor
can be obtained from the g-factor of the 2~ state as shown in fig. 13 .For
SU(3) nuclei, the following expression of the B(MI) for the g.s transition to the
(i +, Fmax-i ) giant state has been obtained (46,42,64).
B(m,o .... >i) = 3/4 pi (gp~)2 s N~W (2 (~+~) -i) ~{
An alternative value for gv is gv=l from the schematic model:
gp=l, gn=O.
This value seems reasonable, because (p,p') experiments have shown that the M1
giant state is essentially an orbital mode, corroborating the schematic model.
Another way to obtain gs and gv are plots of g(2[)vs N and Z (4).
Comparison of F-spin forbidden magnetic transitions in 168Er and
128Xe with IPIM-2
Once the M1 transition operator T(MI) is determined, one can calculate the magnetic
transitions using an IE~4-2 Hamiltonian. The parameters of this Hamiltonian are not
uniquely determined. Calculations by Harter et al. in K61n (50) were done with the
following IH~4-2 Hamiltonian, which projects aproximately to the consistent-Q I~M-I
Hamiltenian by Casten and Warner (51,52).
H = e d n d + k' L 2 + k Qp(Xp)Qn(X n ) - c F 2
The parameters ed, k', k and (~+Xn) were determined by fitting the projected
Hamiltonian (25) to the energies and B(E2) 's. The parameter X'=(~ - Y~) is
used to break the F-spin and to induce M1 transitions without much change in the
energies (fig. 14) and B(E2) 's. The strength parameter c of the Majorana force
cF 2 was adjusted to the energy of the giant (i +, Fmax-i ) state. The ~
205
g-factor and B(MI, 0-i) 168Er
I, K IBA-2 pure F-spin g-expt
2,0 0.457 0.437 0.315(10)
4,0 0.455 0.437 0.303(5)
B(MI--> 1 +) 3.41 1.75
Figure 13: The figure compares the
experimental g-factors g(2~) and the
B(MI, 0[--->i[) with I~M-2 predictions
using the g-factors for the bosons
gl~=l gn--0 from the schematic model. The
8Er data are from Metzger et al.
C. Wesselborg (56).
168Er: E2/MI mixing ratio
Ii,Ki If,Kf IBA-2(a) exp
3,2 2,0 8.512 "- 4 +2 ±J" -0.8
3,2 2,2 1.401 1.418~0.042
5,2 4,2 1.356 1.565~0.080
o~,+0.96 6,2 5,2 1.295 ±.ox,_0.48
Figure 15: Same as figure 14, but
for absolute values of the
E2/MI mixing ratios (53).
168Er: energies ground-, gamma and beta band, Star~iation: 0.022 MeV
I, K exp exp-IHM-2
2,0 0.0798 -0.002 4,0 0.2641 -0.010 6,0 0.5487 -0.028 2,2 0.8212 0.012 3,2 0.8958 -0.000 4,2 0.9947 -0.016 5,2 1.1176 -0.037 6,2 1.2639 -0.060 0,0' 1.2172 0.001 2,0' 1.2763 -0.029 4,0' 1.4111 -0.i01 6,0' 1.6168 -0.215 i,i 3.4 O.i
Figure 14: The figure cor~0ares 168Er
data with an IH~4-2 calculation. This
figure compares energies.
Harter et al (50).
branching ratios from gamma band in 168Er g(v)=l g(v)=O
Ii If,Kf expt expt/ expt/ 168Er IBA IBA
0 2 4 2 4 2 2 4 6 2 2 4 0 6 0 3 2 4 2 4 0 6 0 4 2 5 ,2
0 90.2 0.99 0.98 0 I00 i. 00 i. 00 0 1.63 0.93 0.91 0 i00 i. 00 i. 00 0 17.9 1.02 1.23 2 0.037 1.23 1.85 0 60.5 I.i0 1.04 0 i00 1.00 1.00 0 1.2 1.00 0.92
0.94 1.22 1.15 i00 1.00 1.00 16.4 0.96 1.01 4.1 1.24 1.21 0.37 1.48 2.18
62.0 1.22 1.09 i00 1.00 1.00 19.8 1.37 1.22 0.83 1.34 1.89
Figure 16: Same as figure 14 but for
the branching ratios from the ganm~
band of 168Er. IH~: Hatter et al(51).
Data: Davidson et al (54).
206
calculation by Harter was done for the 168Er data (53-55) because this is a
particular// cor~lete and beautiful set of data. The 168Er data was obtained with
the ARC method in Brookhaven and from (n, ganm~) data by the ILL-Laboratory in
Grenoble. This nucleus has a rather low lying ganm~ band beginning at 800 keV.
Therefore very accurate values of the branching ratios within this band are known,
which have rather large B(MI) matrix el~ts. Furthermore 168Er is
well-descr~ by the consistent Q IIIM-I formalism (51-52). In the following only
the ground and the gamma band will be discussed. In figure 15 the E2/MI mixing
ratios are compared to the II~-2. The strong M1 mixing ratios in the gamma band are
reproduced nearly within the error. In figure 16 the results for 18 branching
ratios from the gamma band are shown. The E2/MI mixing ratios and the branching
ratios are found to be very sensitive to the parameter X' = (~ - Xn) for which
in 168Er a value X'= - 1.2 was found. This value determines the F-spin mixing in
the wavefunctions, given in figure 17. It must be stressed that the value of the
parameter X' obtained from the fit of F-spin forbidden M1 transitions depends on
the choice of the F-vector g-factor gv, because the data essentially determines
only the product gv.X'. One finds that the F-spin impurity of the gs and gamma
band is about 2 % for gv = 1 and about 4 % for gv = 0.7, which gives the B(MI
O + .... > 1 +) obtained from Metzger's data (56). In order to test the F-spin in
another nuclear region, the Xenon and Barium nuclei will be discussed. These nuclei
have been rather well studied in Koln, Kl-akow and Yvascylla and are very good
examples of the dynamic 0(6) sy~netry of I~M-I (57-60). In Koln 128Xe was
investigated in order to get complete complete spectroscopic data and accurate
branching ratios for this nucleus. There are also new data from a Coulomb
excitation experiment done at the Upsalla Lab by Srebrny et al (62), which give new
accurate B(E2) values. An IHM-2 Hamiltonian, H = e d n d + k.Qp(~).Qn(Xn) - c F 2,
which is projected onto a consistent Q I~-i Hamiltonian, was chosen and the
parameters were determined as discussed above for the 168Er case. This
Hamiltonian did not reproduce the energy staggering in the ganm~ band very well
(see fig. 18). It has been shown, however, that energy staggering can be fitted
with a single three-boson interaction term without much change in the wave
functions. In figure 19 the results of Harter et al (63) for E2/MI mixing ratios
and some branching ratios are shown. The dependence of the M1 matrix elements on
the para/neter X' was calculated. The optimum value is X' = -0.22. This value leads
to an F-spin mixing of the wave functions for 128Xe, given in fig. 20. This
analysis of the magnetic properties of the 128Xe nucleus is a little bit less
reliable than the corresponding analysis of 168Er because the energy and the
F-vector g-factor gv of the (i +, Fmax-i ) state in 128Xe is not known, so
the values Ex(l +) = 2,5 MeV and gv = 1 were used.
The results shown seem to indicate that the F-spin in heavy nuclei may indeed
be a good quantum number for the low-lying collective levels. But before drawing a
207
168Er anE01itudes
I,K F-spin Fr~ix-i Fmax-2 FrmgzK-3
0,0 -.9926 .i093 -.0531 .0054
2,0 -.9926 .1089 -.0529 .0054
2,2 -.9900 .1307 -.0520 .0072
3,2 -.9900 .1310 -.0517 .0072
i,I -.0000 .9910 -.1107 .0747
Figure 17: F-spin an~plitude for low lying states in 168Er. From an I~-2 calculation
by Harter et al (50).
O.
E Mev]
5--
i_
6 O 128 6_.
Xe O_ 5_. O_ 5_.
8 8 I __,4--" I __o i __o
6_ 2 3--- 6_ 3 6_ (1) 5_ 21=. -.
1 -- 1 ----'4_ 5 _ 2 - -"4 - - 2__ 2_.
5-- O_ 2 - Z'- 2-- 0- - 2 - 6 _ 6 _
4_0_ 4_0_ 4_ 3_ 3-- 3--
0_ _
- 2 _ 4_ 4 _
Exp 2_ A 2 _ B
2m 2
0_ 0_
Figure 18: Co~%0arison of data on low lying collective levels in 128Xe from refs
(57-60) with an I~-2 calculation by Halter et al (63). The figure compares to
IE~I-2 calculations done with a Qp Qn interaction (A) and with a (Qp Qn)2
interaction (B).
208
128xe: %=-0 22, ~-~22, %=1 I, K --> I, K branching branching
exp. Harter
3,2 --> 4,0
3,2 --> 2,2
4,2 --> 4,0
4,2 M> 2,2
5,2 --> 4,2
5,2 --> 3,2
5,2 M> 6,0
5,2 --> 3,2
0.178(3) 0.219084
0.786(15) 0.641
0.138(11) 0.153743
< 0.041(4) 0.041764
128Xe : E2/MI mixing ratio
I,K--> I,K Hatter exp.
3,2 --> 4,0 i. 86 6.3 (+3.2-1.8)or
-0.45 (+0.08,-0.05)
4,2 --> 4,0 2. 24 1.48 (+0.40,-0.29) or
0.25 (+0.09,-0.09)
2,2--> 2,0 4. 71 6.1+/- 0.5
Figure 19: The figure compares
branching ratios and mixing
ratios for the gamma band in
128Xe from refs (57-60,62)
with results from an I~M-2
calculation by
Harter et al (63).
F-spinamplitudes for 128Xe ampli~x~es Harter
I,K Fma x Fmax-I Fmax-2 Fma x Fmax-i Fmax-2
2,0 .9687 -.2245 .1062 .9827 -.1313 .1306
4,0 .9715 -.2144 .i010 .9819 -.1461 .1209
2,2 .8389 -.5301 .1232 .9806 -.1547 .1207
3,2 .6439 -.7534 .1333 .9786 -.1776 .1049
4,2 .7932 -.5891 .1542 .9794 -.1727 .1043
5,2 .5376 -.8308 .1441 .9746 -.2090 .0808
Figure 20: F-spin a~plitude in 128Xe obtained from I19~-2 calculations by Novoselsky
and Talmi (36) and by Hatter (63).
209
final conclusion regarding the question of F-spin purity, one has to discuss the
impressive I~M-2 calculations from the Weizmann group (36), which imply a much
stronger F-spin mixing than the above values. The energies of the IH~4-2
Calculations by the Wei~ group (36) are actually better than the corresponding
energies from the Koln group. On the other hand the Weizmann group did not include.
magnetic transitions in their ~rison to experimental data. This means that
their calculations were done with gv=O. If a value gv=l is used within their
Calculations the magnetic transitions are overpredicted by a factor of about 5-10
i65), Clearly the question of the F-spin breaking is intimately connected with the
finding of the proper M1 transition operator T(MI) and the proper value of gv"
S~_ummary_
It has been shown that the F-spin concept is a useful concept in the Proton-
Neutron Interacting Boson Model, which describes quantitatively the proton neutron
symmetry and which unifies many different phenomena in collective models. In
Particular this concept explains the rather constant energies in F-spin multiplets
and it suggests the possibility of "global" fits with the II~M-2 model. It also
gives a unified description of the new 1 + states from the Darmstadt group.
Furthermore, it seems to lead to a description of the forbidden M1 transitions in
Collective nuclei in the frame of the IIIM-2 model. From this description a value of
the F-spin purity in the ground and gaa~na band below 4 % is obtained and so this
Work on the forbidden M1 transitions in the frame of the I194-2 model is only the
beginning of a new interesting subject.
A_ccknowledqements
The author wantstothankA. Gelberg and H. Halter, with whommost of the work from
the Koln group reported here was done. He furthermore thanks R.F. Casten and B.
Barrett, A. Barfield, A. Dewald, W. Frank, W. }<rips, W. Lieberz, I. Morrison, T.
Otsuka, J. Theuerkauf, C. Wesselborg and K.O. Zell for stimulating dis-
cussions. Supported by Bundesministerium fOr Forschung und Technologie.
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Wesselborg, U. Neuneyer, M. Vohl, P. von Brentano: to be published
C. GIANT RESONANCE STATES
EXCITATION AND DECAY OF ELECTRIC GIANT RESONANCES - ESPECIALLY THE ISOSCALAR GIANT MONOPOLE RESONANCE
A. van der Woude Kernfysisch Versneller Instituut
Zernikelaan 25, 9747 AA Groningen, The Netherlands
Abstract
In this paper recent progress in the excitation and decay of electric giant
resonances, especially the isoscalar giant monopole resonance will be discussed. The
compressibility of nuclear matter Kn.m. will be determined from the newest
experimental data on the GMR over the mass range 24 < A < 208 using a semi-
phenomenological expression for the nuclear compressibility K A in terms of a volume,
surface, asymmetry and Coulomb contribution. The resulting value, Kn.m. ~ 260 MeV is
in good agreement with nuclear matter calculations.
Recently new data for the neutron decay of the GMR in 208pb and 124Sn have been
obtained. These data turn out to be compatible with a nearly 100% statistical decay
mode if in the statistical model calculations a realistic level scheme for the
residual nuclei is used. A similar analysis applied to existing data on the GDR in
208pb shows that also for this resonance the data are compatible with a nearly 100%
statistical decay mode.
i. Review of electric (AS = 0) giant resonances
A survey of what is known experimentally on electric (AS = O) giant resonances
is given in table 1 I). It shows that the AT = 0 and i, i hm and 2 n~ resonances are
pretty well established, the only exception being the isovector giant quadrupole
resonance (IVGQR). The main difference with a previous, 1981 review 2), has been the
discovery of the isovector giant monopole resonance (IVGMR) (see references 4 and 5
and references therein). This was made possible by the development of the (~±,~°)
facility at Los Alamos. As it turns out, the spinless isovector n-meson is an ideal
tool to study AS = 0, AT = 1 isovector giant resonances, better than the (p,p') or
(p,n) reactions which because of the nature of the nucleon-nucleon force will excite
much stronger the AS = i, AT = 1 magnetic resonances. The obvious problem is that the
pion beam is only available as a secondary beam which implies a reduced beam intensi-
ty and/or beam quality, but nevertheless various experiments using this reaction have
already been performed 4'5). Most experiments are done at an energy E ~ 165 MeV, in
215
the region of the A-resonance, where the pions are strongly absorbed resulting in a
characteristic forward angular distribution (see section 2). This feature has been
Used to identify the IVGMR.
Higher lying 3,4,...hm strength is difficult to detect. The strength dis-
tributions are predicted to become broader causing the various multipolarity strength
distributions to overlap while simultaneously the strength/MeV becomes less 3). This
makes it difficult to distinguish them from each other and from the underlying
COntinuum, not only because the uncertainties involved in separating resonance and
COntinuum become larger but also because the angular distributions of the observed
Structures, now being a mixture of several multipolarities, become increasingly
featureless. The only higher lying resonance identified as such by various groups
Using different probes is the 3n~ isoscalar octupole resonance. Table 2 shows some
recent results on this resonance in 208pb. These data indicate that there is a
resonance around E ~ 19.5 MeV but also that there is a considerable uncertainty with x
Table i.
Survey of giant resonances
EO
E1
E2
~3
E4
0 0 2 rico
0 i 2hco
1 0 3 hco
I i lh~
2 0 2he0
2 1 2 hw
3 0 i he
31~
3 i lh~
3 he0
4 0 2 llm
4nm
4 i 2nm
4 n ~
Heavy nuclei A > 90
status Ex(MeV ) S(%) a)
*** ~80 A -I/3 50-100
** ~59 A -1/6b) 50-i00 c)
*** 32 A -I/3 + ~i00
20.6 A -I/6
*** 65 A -1/3 50-100
* 130 A -1/3 50-100
*** ~30 A -I/3 10-20
** (IIO-120)A -I/3 ~50?
(60-80)A I/3 ~20
Deformed heory Exp.
v" ***
v ~ ,
Light nuclei A < 4(
Status S(%)
** 50-100
* 50~100
*** 50-100
*** 40-80
** 10-20
* Resonance/effect probably observed in some nuclei
** Resonance/effect observed but information not yet complete
*** Resonance/effect well established
a) Percentage of relevant energy-weighted-sumrule
h) Obtained from interpolating the (T±I) components
e) Ratio between experimental and calculated cross section
216
Table 2.
Overview of some recent experimental results on the AT = O, L = 3 resonance
(HEOR) in 208pb by means of inelastic hadron scattering.
Probe/Energy (MeV) Ex(MeV ) F(MeV) S(%) Reference
(3He,3He') 130 20.5±1.0 78±15 6
(a,a') 172 18.7*1.5 5.010.9 60,15 7
(p,p') 200 20.9±1.0 5.9±1.0 36±12 8
(p,p') 800 19.8,0.2 7.1 50,12 9
respect to the details, reflecting the inherent problems in studying high lying
resonances. In this connection also the (a,~') work at E = 340 and 480 MeV by Bonin
et al. I0) should be mentioned in which they treated the continuum up to about 60 MeV
excitation energy in a systematical phenomenological way. By doing so they claim to
observe broad structures all the way up to E ~ 50 MeV on top of a smooth continuum. x
Specifically they claim the presence of AL = 3 strength broadly distributed over the +50
interval 18 ~ E x ~ 35 MeV exhausting 150_60% of the sum rule. Although one might
worry whether their analysis can be carried as far as was actually done, it certainly
emphasizes the point that high quality data are a necessary condition. But even then
the unambiguous identification of such broadly distributed strength remains a tricky
business.
2. The isoscalar giant monopole resonance
Next to the isovector GDR, which is selectively excited in 7-absorptlon experi-
ments, probably the best known resonance from the experimental point of view is the
isoscalar giant monopole resonance (GMR). This is due to the fact that this resonance
is selectively excited in small angle scattering. For particles llke a-partlcles and
pions, strong absorption results in a nice diffractive angular pattern. This is illu-
strated in figure i for the 208pb (a,a') reaction. The excitation of L = 0 and L = 1
strength results in a characteristic angular distribution but for L > 2 strength it
becomes increasingly difficult to assign multipolarities based on angular
distributions.
The characteristic feature of the GMR angular distribution at small angles has
been used to identify both, the AT=0 GMR by (a,a') and (3He,3He') reactions and
the AT~I GMR by means of the (~*,~°) reaction. Small angle, including 0 ° (a,a,)ll,12)
and (3He,3He')13) scattering experiments require a very careful experimental set-up.
Examples of spectra obtained at the KVI are shown in figures 2 and 312'14).
Figures 2a, b and e show (~,e') spectra at E = 120 MeV on 208pb for 0°~O~,~3 °,
217
io ~ 208pb [~,~') [E r~ -120 MeV [- E~=IO MeV ~ IO0% EWSR
iO 2
E
• ~ . i o ~ b
I00
T -t - L = O
L:I, AT=O L=2 L=4
h I ' ~ I I I r I ~ / ~ ~ ' ' i
\ ~ ~ "\ ; \ ".. ...." .,... "-'~ l~' ~./ ' l '>, / - ' , '..
/',..& ..,,,....,,'",. x " / \ ' / ~,
IO -I I J 0 5 I0 15
Ocm (de(j)
Fig. i. DWBA predictions for the angular distributions of various multipolarities exhausting the full EWSR.
k 2OSpb (~'a') ZOepb I ° Ee.120 MeV i, Ilhj~
80000
40000L " "" "~'~, I
[
4 i '/
3
°r: ~ ' 4
E~ {MeV)
F i g . 2. S i n g l e s spec t ra a t E~=120 MeV (a ) 0" < O a, < 3 °, (b) 0" < e~, < 1.5 °, (c) 1.5" < O , <3 °, (d) difference between (b) and (c). The full curves are the con- tributions of the GQR and the GMR to the spectra. The error bars include all sta- tistical errors arising from background subtraction etc.
0°~Oa,~l.5 ° and 1.5°<@ ,(3 °, respectively. It is clear from figure 2 that the GMR
Strength distribution shows up considerably enhanced over those of other multipolari-
ties and over the continuum by considering the difference between the spectra of
figure 2 b (0 ° _< O , < 1.5 ° ) and 2 c (1.5 ° < 8 , __< 3 ° ) as is shown in figure 2 d. In
figure 2 d the small bump at Ea ~ 10.8 MeV is the remnance of the GQR - its magnitude
agrees with what is expected from DWBA calculations. A small amount of continuum
218
remains after the subtraction. From these single data it is not yet clear whether
thls is due to an instrumental background or to truly continuum excitation, but the
neutron decay coincidence data show that most of it is due to genuine continuum
excitation 12). For future reference it is important to note that the observed
strength for 12.5 ~ E x ~ 14.5 MeV is overwhelmingly due to monopole excitation. Also,
by subtracting from figure 2 b or 2 c an appropriate amount of the GMR spectrum of
figure 2 a one if left with a rather pure continuum spectrum.
Until recently one of the big remaining problems with respect to the GMR was its
apparent lack of strength in light nuclei 2'13). This situation has recently changed
due to the fact that an appreciable amount of isoscalar GMR strength has now been
located in 24Mg12'14) and in 28Si15). For 24Mg this is illustrated in figure 3 which
shows spectra for various small-angle intervals. By comparing these spectra it turns
out that there is a lot of L = 0 strength for i0 ~ E x ~ 20 MeV. For instance the peak
at E = 13.9 MeV and the broad structures around E ~ 18 MeV show a pronounced AL = 0 x x
behaviour. A careful analysis of the angular distributions over the interval 0 ° - 8 °
60
4 0
20
0
j 2 o
o
'24Mg (a,a') 24Mg E e - 120 MeV
O°(8a, d5 o
15°~Oa, e3 °
1600
800
0 600
3 0 0 5
o
4020 3° ~ ~)a"45° ~ ~ k l ~ ~ ~ ~ " ~ 600300
r ~ - "~i 0 20 18 16 14 , 12 I0
E X (MeV)
Fig. 3. Inelastic ~-spectra for 24Mg for three different scattering angle intervals. From references 12, 14.
>~
~2o rr
0 Io QJ
o
9 M I0
0
> m
- . 7 ' ~" "°°r
24Mg (eoa') 24Mg FI i Ee=120 MeV I I I EO 51rength I~ ~L] L--~ I
E I, AT=O slrength
E2 sfrenglh
IoL
o~_. 12 16 20
E X (MeV)
Fig. 4. Strength distribution for iso- scalar monopole, dipole and quadrupole strength distributions. From referen- ces 12, 14.
219
resulted in the strength distribution shown in figure 4. Not only has nearly all of
the GMR strength now been located in 24Mg, but also an appreciable amount of AT =
O, AL = i strength has been found. The GMR strength is centered around E ~ 17.2 MeV x
and has a width (FWHM) of about 6 MeV. Its strenght distribution is very fragmented
like all known multipolarity strength in light nuclei 1'2). The recent (a,a')
experiment of 28Si by Liu et al. 15) located similarly a large amount of GMR strength
at E ~ 17.9 MeV with a width (FWHM) ~ 4.8 MeV. x For both nuclei, 24Mg and 28Si, it then turns out that the isoscalar GMR and GQR
Strenghts are strongly intermingled. This also explains why it has been so difficult
up till now to locate substantial amounts of GMR strength in A < 90 nuclei 13). It is
Only from high quality inelastic scattering data at very forward angles including 0 °
that monopole strength can be really extracted.
3. The compressibility of nuclei and nuclear matter
Intuitively the excitation energy of the GMR should be related to the
COmpressibility K A of the nucleus defined by:
2 d2(E/A) (i)
KA = X0 dX 2 -I X=X 0
Where X is a collective variable and (E/A) the total energy per nucleon. In the
following we will limit ourselves to the scaling model for which R=<r2> ~ plays the
role of X 16).
In a macroscopic hydrodynamic model:
h 2 Ex(GMR) = [~m KA~½ (2)
<r2> J
Where m is the nucleon mass. A similar expression can be derived using sumrules and
the monopole operator 00 = Z r~: k
k <fl001i>2 m k = Z Ef
f
The quantities m k can be calculated once the strength distribution is known
experimentally. Various mean energies E ~k can be defined as:
Ek = [mk/mk-2 l~
In the scaling model it can be shown 16) that:
h2 K S
220
which is identical to the hydrodynamical model expression. For a Gaussian strength
distribution with excitation energy E x and width F:
-~2 E 2 E3 = x + 3(F/2"35)2
which amounts to a correction of only a few percent for all nuclei. Using these
relations the KA-values can be calculated.
A quantity of great interest is the compressibility Kn.m. of nuclear matter. The
S the data for real nuclei, to Kn.m. is not at all extrapolation from KA,
straightforward since surface, Coulomb and asymmetry effects considerably modify the
restoring force from its nuclear matter value to the finite nuclei value.
Various procedures have been suggested. One method is to perform self-consistent
RPA calculations on the response function of a few finite nuclei to the monopole
operator lkr k using various phenomenological interactions 17). The calculated reso-
nance energies can then be compared to the experimental ones. The same calculations
give also the ground state properties of nuclear matter like the energy per nucleon
(E/A), density Pn.m. and the compression modules Kn.m. defined as:
2 d2(E/A) (4) Kn.m. = 9 Pn.m. do2
n.m.
Using the best experimental value of E x ~ 13.8 MeV for 208pb one finds 17) that
Kn.m. ~ 220 MeV. This method has some obvious disadvantages. The RPA calculations can
be only performed for a few selected nuclei and in fact the whole extrapolation to
Kn.m. hinges strongly on the experimental Ex-value for 208pb. Also it is not yet
clear how reliable such RPA caluclatlons actually are 3'17) •
Another method, suggested by the liquid drop formula, is to assume that the
compression modulus K A can be written asl6'17):
_ Z 2
KA = KV + KsurfA 1/3 + Ksymm(~)2 + KC ~ (5)
In the scaling model Kn.m. = K V 16).
Using the scaling model the various coefficients can be further analysed. In
particular it can be shown 16'17)"
K C - 3/5 (e2/ro) (12.5 - 1215 / Kn.m.)
Using r 0 ~ 1.18 fm this Coulomb term amounts to ~ 30 MeV and ~ 18 MeV for 208pb
and 40Ca, respectively. Also Ksurf and Ksymm are slightly modified to K' and K' s symm'
respectively. One can then perform a three-parameter fit to the data on K A. Such an
analysis has been recentLy performed by various groups 12'13'15'18). The results are
summarized in table 3.
The value obtained for Kn.m. turns out to be remarkable constant, although
different data sets were used in the analysis. For instance the data sets used in the
221
analysis of Texas A&M and KVI are quite different, yet the only probably significant
difference in the final results is in the value of the asymmetry term K • This term
is mainly determined by the behaviour of K A (or Ex) in an isotopic chain. The
difference can be traced back to the fact that recent measurements on the GMR in
i12'i14'i16'120'124Sn at the KV118) showed less of a (N-Z) dependence of E x than
previous measurements as summarized in reference 13, had indicated.
The various parameters in table 3 are strongly correlated and thus a least-
Square fit may have a series of minima. However, it has been shown 12'18) that by
Using different initial values, the final result of the search was within the quoted
Uncertainties approximately independent of the initial choice.
Table 3
Extraction of the various compressibility parameters as defined in relation 5 by
different groups using different sets of input data. The quoted values are in MeV.
Group Kn.m. K's KZ' Reference Input data
(volume) (surface) (asymmetry)
Grenoble 278~18 -591±20 -432± 85 13 a)
Texas A&M 270513 -607543 -5505195 15 b)
KVI i 253±16 -488±56 -285,448 12 c)
KVI 2 273~12 -551±50 -3025118 18 d)
Theory 356 -461 -372 16 e)
216 -237 -272 16 f)
a) Set of 33 nuclei of which the data are summarized in reference 13.
b) Data for 28Si, 64'66Zr, I12'I16'118'120'124Sn, ll51n, 142Nd, 144Sm, 197Au
and 208pb.
c) Data for 24Mg, 28Si and 25 nuclei with A > 90 for which the observed GMR nearly
completely exhausts the E-O sum rule.
d) KVl-data for 24Mg, 28Si, i12'i14'i16'120'124Sn, 144'148Sm and 208pb.
e) Using the SK III interaction.
f) Using the SK M interaction.
For determining the values of K' the surface coefficient it is very important
to have data for light nuclei as is the case for the Texas A&M and KVI 1,2 sets, sin-
Ce then the range of the A -I/3 coefficient is considerable enhanced. In the Grenoble
analysis also data for light nuclei (A < 90) were used, but since in these experl-
ments only a small fraction of the isoscalar E-O sum rule was observed, it was quite
reasonable to assume that more monopole strength would be located at higher excita-
tion energiesl3). This would then increase the value of K A in these light nuclei and
thus decrease the value of K' and consequently also of Kn.m.. Thus the new data for s
222
24Mg and 28Si have considerably increased the reliability of extracting the Kn.m.
value from the data using the seml-phenomenological relation given in formula 5.
Of course this klnd of analysis hinges on the validity of the relation given in
formula 5 for extrapolating from finite nuclei to infinite nuclear matter. This
problem has been studied in some detail by Treiner et al. 16) who concluded that for
the scaling model such an extrapolation is justified indeed.
4. The decay of giant resonances
As is clear from the examples shown in the previous sections, the strength
distribution of the various giant resonances in spherical nuclei can be qualitatively
described by a Gausslan or Lorentzlan strength distribution with a width F, which is
llke the excitation energy Ex, a smooth function of the mass number A. As a first
approximation the total width P can be written as:
F = ~F + F~ + F+
Here AF, the Landau damping, is due to the fact that not all collective Ip-lh
strength which acts as a doorway for the giant resonance, is necessarily concentrated
in one single state as would follow from a simple schematic model, but can already be
appreciably fragmented. This is especially true for the higher multipole resonances
and for the s-d shell nuclei.
Giant resonances are located in general at excitation energies above the
particle binding energy so that they will decay predominantly by particle emission,
charged particles and neutrons in light nuclei and because of the Coulomb barrier,
neutrons only in heavier nuclei, y-decay is also possible but the relative decay-
branch is small: Py/Ftota I = 10 -3 to 10 -5 • In the actinide nuclei also the fission
process is an important decay mode. In general one might hope to achieve various
goals by performing decay-experiments:
(i) Whenever there is a "background" of physical or instrumental origin,
which is not due to the excitation of the nucleus itself, coincidence experiments
will clean up the spectra. A good example are the 28Si (e,e',~) experlments 20) and
the 208pb (e,e',n) 21) experiments where one got rid of the radiation tails which
dominated the singles (e,e')-spectra. Another example is the 238U (e,~'f) experl-
ment 22) where in the flssion-coincident q-spectrum the instrumental background due
to oxygen and carbon contaminants in the target, was eliminated.
(ii) From the angular correlations of the decay particles one sometimes has
been able to extract information on the spin-parlty of the decaying state. However,
in most cases, the fact that the various resonances overlap and that for one speci-
fic spin-parity of the decaying state more than one combination of spin and orbital
angular momentum combination can contribute to the decay, makes the analysis of the
angular correlation functions ambiguous. The only case where it turns out to be
223
possible to perform an unambiguous analysis is for the ~o-decay of resonances in the
s-d shell nuclei to a J~ = 0 + groundstate in the residual nucleus 23).
A recent suggestion has been to determine the isovector GQR strength in
208pb by virtue of its large (~ 10 -3 ) y-decay branch to the ground and first excited
State in 208pb 24).
(iii) Most important, decay experiments can tell something about the decay
mode of the giant resonance.
Particle (neutron) decay can occur through various processes - see figure 5. The
i ~
F~' direct
(e,e') ~; : ~ r~' A-1
A
KV~ 3934
/ 2p-Zh
A-I
F ,It
l 3p-3h Cornpound]~Nucleus
A-I
Fig. 5. Schematic representation of the various nucleon decay modes of a giant r e s o n a n c e .
COupling of the (Ip-lh) state to the continuum gives rise to semidirect decay into
the hole states of the (A-I) nucleus with a partial width F+, the escape width. If
this process is the only one populating the hole state, then the relative population
of these states would reflect the microscopic structure of the giant resonance. The
Other component, the spreading width F+ arises from the fact that the (lp-lh) doorway
State is mixed through the residual interaction with the more complicated and numer-
ous (2p-2h) states which are present in the vicinity of the resonance. One assumption
often made is that the (2p-2h) states couple again to (3p-3h) .... (np-nh) states
till finally a completely equilibrated system is reached. At each intermediate level
Particle decay can occur. For a fully equilibrated system the decay will be similar
to that of a compound nucleus with the same excitation energy, spin and parity as the
giant resonance. The total spreading width F+ = F+~ + r++where F++ is the partial
Width for pre-equillbrium and F++ for statistical decay.
Several microscopic calculations on the width r have been performed in the
framework of RPA without and with coupling of the (ip-lh) states to (2p-2h) states.
For heavy nuclei they have been restricted to closed shell nuclei. Especially 208pb
has been studied extensively.
Most calculations agree on the fact that for heavy nuclei the escape width
F~ << F 25,26,27). An exception is the calculation of Van Giai and Sagawa 28) on the
224
width of the GMR in 208pb. They calculate in a self consistent (ip-lh) calculation
without truncation of the (Ip-lh) model space, and not including any mixing with (2p-
2h) states, that the escape width F+ ~ 2 MeV, only slightly lower than the
experimental width P = (2.5±0.3) MeV.
In a full (Ip-lh) + (2p-2h) calculation using empirical requirements for the
single particle spectrum and residual interaction, it was found that including mixing
with (2p-2h) states appreciably increases the width p25). Of special interest in the
damping process is the excitation of low-lying surface vibrations by the inelastic
scattering of the particle (hole) from the nuclear surface. The same process is also
important in the damping of single particle and hole states 29), but for the coherent
(ip-lh) states the process is more complicated because of destructive interference
between the particle and hole contributions tO the damping width. This destructive
interference is especially strong for the GMR and in fact the predicted GMR width
from coupling to these surface vibrations is for 208pb only about i MeV, much smaller
than the experimental width. With respect to the GQR it was calculated that for a
range of closed shell nuclei between 40Ca and 208pb damping into the doorway states
(particle-hole + vibration) should account for about 60% of the experimental
width26).
Experimentally the various decay modes can be distinguished at least in
principle by measuring the residual (A-l) spectrum. Semi-dlrect decay will populate
(neutron) hole states which can be located by means of a nucleon pick-up reaction on
the nucleus A.
Statistical decay can be inferred from a comparison of the experimental spectrum
of the (A-l) nucleus with the one calculated by means of the Hauser-Feshbach
formalism in which the essential input parameters are the transmission coefficients
of the decaying particle and the level density of the residual nucleus as a function
of excitation energy and spin-parity.
The transmission coefficients are obtained from an optical model description of
elastic scattering. Usually one takes a global set of optical model parameters but
the overall behaviour of the transmission coefficients turns out to be not very
sensitive to the actual cholce 12'30).
In the past one often used an extrapolated level-density formula to calculate
pf(Ef,Jf). However it has recently been pointed out 12'30) that such a prescription
may well lead to wrong conclusions concerning the population of Iow-lylng hole
states, especially if the available energy for decay is not very large. A better
procedure 12) is to use in such cases for the lower excitation energies the actual
level scheme and for the higher excitation energies a level density formula based on
the back-shifted Fermi gas model with empirically determined parameters.
The result of pre-equilibrium decay, the P+~ component, is difficult to predict
except for the special case of the coupling to surface vibrations. This component in
nucleon decay will reveal itself by enhanced population of those states in the final
225
(A-I) nucleus arising from the coupling of a hole to a phonon of the nucleus A.
Otherwise the spectrum resulting from pre-equilibrium decay might well resemble the
statistical decay spectrum so that it will be very difficult if not impossible to
distinguish experimentally pre-equilibrium and statistical decay.
A special case is the fission decay of giant resonances in actinide nuclei.
Fission is assumed to be the result of a complicated process from an equilibrated
System and as such it can be identified with statistical decay.
5. Neutron-decay of giant resonances in medium and heavy nuclei (A > 90)
The experimental situation with respect to the isoscalar GQR and GMR resonances
has been summarized in table 4. For all nuclei studied the statistical decay mode is
the dominant one. GMR decay in 208pb is compatible with pure statistical decay while
the existing data for medium heavy nuclei are not incompatible with the presence of a
Semi-direct decay branch of 10-20%. As an illustration some of the results of the re-
Cent neutron decay experiments on the GMR of 208pb12) will be shown. In this experi-
ment the GMR was excited by the 208pb (a,~') 208pb* reaction at E = 120 b~V at scat-
tering angles 0 ° ~ O , ~ 3 ° . The inelastically scattered a-particles were measured in
COincidence with neutrons of which the energy was determined by a time-of-flight mea-
SUrement. In order to avoid effects due to knock-out processes only neutrons emitted
in backward directions were used for determining the GMR decay properties. Similar to
what has been discussed in section 3 for determining the GMR strength distribution
from singles (a,a') measurements the neutron decay spectrum of the GMR proper was iso-
lated from that of the underlying continuum by comparing the neutron decay spectrum
in the GMR excitation energy range 12.5 <_Ea <__ 15.5 MeV for O" < 0a' < 1"5° (GMR +
Continuum) and 1.5 ° < 0 , < 3" (mainly continuum). The results are shown in figure 6.
The data in figure 6 a show the neutron decay of the continuum underneath the GMR
while those in figure 6 b are of the monopole proper. The histograms in figures 6 a'b
are the results of a statistical model calculation. In both calculations the known
207pb level scheme was used up till an excitation energy E x ~ 3.7 MeV. For E x > 3.7
MeV the level density was determined according to the back-shifted Fermi-gas distri-
bution of which the parameters were determined by a fit of the known level densities
at lower excitation energies E x ~ 3 MeV and those derived from neutron capture reso-
nances (E x ~ 7 MeV). The calculations for the continuum, figure 6 a, were made under
the assumption that the decaying states in 208pb had J~ = 2 + , while for the calcula-
tions shown in figure 6b J~ = 0 +. For both spectra the calculations were normalized
to the total number of counts in the experimental spectrum.
The overall agreement between the measured and calculated 207pb spectrum con-
Cerning the decay of the continuum underneath the GMR, figure 6 a, is quite good. The
main discrepancies at E x ~ 1.6 MeV and between 3 and 4 MeV are due to an excessive
population of J~ = 13/2 + states and a cluster of states with J~ = 9/2 + , 11/2 + respee-
226
I r r
Z400~" 2 0 a p b (cl,cl 'n)
8,~,-(3 8 . ~ , ~ _
125~£
115500
I k t
Oi,,,
°oo I!!;
[ ,
E = . 120 M e V
i
b L i
; TT ¸ • I i
, tli/', ii ii+
L I 8 15 4 2 0
E x (MeV)
Fig. 6. Comparison of neutron decay data for the excitation energy region 12.5 < E x < 15.5 MeV in ~ Pb with statistical model predictions. (a) Q-value spectrum for the background. The calculation (histogram) has been normalized to the total number of counts in the experimental spectra. (b) Q-value spectrum for the decay of the GMR. The calculation has been normalized to the same number of counts as the experimental data. (c) Same as (b), except for the calculation which has been normalized to the plateau in the experimental spectrum at about 4 MeV excitation energy (= 60% of total number of counts in experimental spectrum). The arrows indicate the unperturbed excitation energies of the multiplets due to coupling of the various neutron holes to the J" = 3- surface vibration.
tively. This is an indication for the presence of L > 4 strength in the lower part of
the GMR excitation energy range, since in this energy region the branching ratios for
such L > 4 strength to 9/2 + , 11/2 + and 13/2 + final states are considerably larger
than for J~ = 2 + strength. Thus the decay of the continuum is compatible with purely
statistical decay. A similar good overall flt is obtained for the GMR, figure 6 b,
thus showing that the GMR decay is also compatible with pure statistical decay.
An interesting question is to what extent these data might show any evidence for
a non-statistical decay mode like the surface coupling mode 26). Especially the sudden
jump in the data at E x ~ 4.3 MeV which is not reproduced in the calculations and the
227
Table 4
SUmmary of experiments on neutron decay of isoscalar giant resonances in A >
90 nuclei
Nucleus Resonance Decay mode Remarks Reference
Statlsti- Non- Pre-
cal Statis- Equili-
tical brlum
90Zr GMR dominant (10-15)% possible L>4 strength 31
in GQR region
92Zr GQR dominant ~16% 32
llgsn GQR ~20% 33
124Sn GMR dominant 34
208pb GQR L>4 strength 35
in GQR region
compatible with
pure statistical
decay 30)
GMR dominant ~15% -5% 36
GMR dominant <10% <30% 12
POssible structure between 4 and 6 MeV might be an indication of such processes. The
COupling of the GMR to surface vibrations would lead to population of those states in
207pb which can be described as a neutron hole coupled to a surface vibration.
Calculations indicate that coupling to the J~ = 3- octupole vibration at E x = 2.61
MeV in 208pb is the most important one. The approximate location of the multiplets
which contain low spin members has been indicated by the arrows in figure 6 c.
One can then try to decompose the observed spectrum into various components: a
Statistical one chosen in such a way that the data between 3 and 4 MeV are fitted, a
direct component which is then the difference between the observed and calculated
POpulation of low-lying hole states and a pre-equilibrium component presumably due to
GMR-surface vibration coupling. Such a decomposition, which in its details is of
COurse quite uncertain is shown in figure 6 c. Here the statistical component amounts
to ~ 60%, the semi-dlrect one to ~ 10% and the pre-equilibrium one to ~30% of the
tOtal spectrum. However given the accuracy of the data such a decomposition should be
COnsidered to be rather speculative. Anyhow the semi-direct component is < 10%, much
Smaller than suggested by various theories 26'28).
A similar analysis can be applied the existing data on the GDR in 208pb, indica-
ting that the non-statistical branch is at most 5%. This is in contrast with earlier
analysis 37) where one fitted the experimental spectrum with an exponential one, N(En)
E n exp(-En/T) , and concluded to a i0 to 15% direct decay branch 37).
228
Summarizing the Pb-data, we conclude that the decay of both the GDR and the GMR
is compatible with statistical decay. The (semi-)direct decay branch, if at all
present, is <__ 10%, while there is no clear-cut evidence for any pre-equilibrium type
of decay like the surface-coupllng mode.
A preliminary analysis of recent data taken for the neutron-decay of the GMR of
124Sn also shows that the data are compatible with pure statistical decay 34). A
similar conclusion was reached for 238U where a study of the fission decay of the GMR
showed that by assuming a fisslon-decay probability for the GMR similar to that of
the GDR or of a compound nucleus, essential 100% of the GMR sumrule could be
recovered 22). Thus, in an indirect way, also in this case it can be shown that the
GMR decay is compatible with pure statistical decay.
The work described in this paper has been part of the thesiswork of S.
Brandenburg. It has been performed in close collaboration with M.N. Harakeh - now at
the Free University in Amsterdam - and as far as the neutron decay work is concerned
with our Swedish colleagues L. Nilsson, P. EkstrDm, A. Hakanson, A. Lindholm, N.
Olsson and Italian colleagues R. de Leo and M. Pignanelli.
References
i) A. van der Woude, Progr. in Part. and Nucl. Phys. (ed. A. Faessler) to be published 1986.
2) J. Speth and A. van der Woude, Rep. Progr. Phys. 44 (1981) 719. 3) K. Coeke and J. Speth, Ann. Rev. Nucl. Part. Sci 32 (1982) 65. 4) J.D. Bowman et al., Phys. Rev. Lett. 50 (1983) 1195; Journ. de Physique C4 (1984)
351. 5) A. Erell et al., Phys. Rev. Lett. 52 (1984) 2134; preprint 1985. 6) T. Yamagata et al., Phys. Rev. C23 (1981) 937. 7) H.P. Morsch, Journ. de Physique 45 (1984) C4-185. 8) D.K. McDaniels et al., Phys. Rev. C33 (1986) 1943. 9) C.S. Adams et al., Phys. Rev. C33 (1986) 2054. i0) B. Bonln et al., Nucl. Phys. A430 (1984) 349. ii) D.H. Youngbloud et al., Phys. Rev. C23 (1981) 1997.
C.M. Rozsa et al., Phys. Rev. C21 (1980) 1252. 12) S. Brandenburg et al., to be published and Thesis, Groningen 1985. 13) M. Buenerd, Journ. de Physique 45 (1984) C4-I15. 14) H.J. Lu et al., Phys. Rev. C33 (1986) 1116. 15) Y.-W. Lul, Phys. Rev. C31 (1985) 1643. 16) J. Treiner et al., Nucl. Phys. A371 (1981) 253. 17) J.P. Blaizot and B. Grammatlcos, Nucl. Phys. A355 (1981) 115. 18) M.M. Sharma et al., private communication, KVI, 1986. 19) B. Ter Haar, Thesis, Gronlngen 1986. 20) Th. Klhm et al., Phys. Rev. Lett. 56 (1986) 2789. 21) G. Bolme, Ph.D. Thesis, Illinois 1983. 22) S. Brandenburg et al., Phys. Rev. Lett. 49 (1982) 1687.
R. de Leo et al., Nucl. Phys. A441 (1985~---591. 23) F. Zwarts et al., Nucl. Phys. A439 (1985) 117. 24) J. Speth et al., Phys. Rev. C31 (1985) 2310. 25) J. Wambach and B. Schweslnger, Journ. de Physique 45 C4-281
229
26) G.F. Bertsch et al., Rev. Mod. Phys. 55 (1983) 287. 27) P.F. Bortignon et al., Phys. Lett. 14-8-B- (1984) 20. 28) N. Van Giai and H. Sagawa, Nucl. Phys. A371 (1981) i. 29) F. Azaiez et al., Nucl. Phys. A444 (1985) 373. 30) H. Dias and E. Wolynec, Phys. Rev. C30 (1984) 1164.
H. Dias et al., Phys. Rev. C33 (1986) 1955. 31) K. Fuchs et al., Phys. Rev. C3---2 (1985) 418. 32) H° Ohsumi et al., Phys. Rev. C-32 (1985) 1789. 33) K. Okada et al., Phys. Rev. Le~-t. 48 (1982) 1382. 34) W.T.A. Borghols et al., KVI Annual---Report 1985. 35) H. Steuer et al., Phys. Rev. Lett. 47 (1981) 1702. 36) W. Eyrich et al., Phys. Rev. C29 (i-~4) 418. 37) L.S. Cardman, Nucl. Phys. A354--~1981) 173c.
TIlE SCISSORS MODE
F. Palumbo 1NFN - Laboratori Nazionali di Frascati, P.O.Box 13, 00044 Frascati (Italy)
ABSTRACT
A dipole magnetic resonance has been predicted in deformed nuclei by a two-rotor model, as a state in which the nucleus rotates while the proton neutron symmetry axes stay a fixed angle in a scissors-like configuration.
This resonance has been also predicted in the VPM and in the IBA, and has been experimentally confirmed by high resolution electron scattering in three regions of the periodic table, i.e. the deformed rare earth nuclei, the f7/2-shell nuclei 46,48Ti and the actinides.
I will report the theoretical description in terms of the two-rotor model and the RPA, and 1 will give a survey of the experimental status.
l. - INTRODUCTION
As it is well known the E1 giant resonance is a collettive mode existing in all nuclei. It has a simple
semiclassical description in terms of translational oscillations of protons against neutrons(l).
This two-fluid picture suggests the existence of another collective mode which can occur only in
deformed nuclei. Protons and neutrons can be assumed to form two separate rigid bodies of ellipsoidal
shape, free to rotate independently around a common axis with opposite velocities. In such a two-rotor
model (TRM) the restoring force generated by the displacement of protons against neutrons gives rise
classically either to relative rotational oscillations or to a configuration in wl~ich the nucleus rotates while the
proton-neutron symmetry axes stay at a fixed angle in a scissors-like configuration (Fig. 1). This latter state
is strongly excited by Ml-radiation through the coupling to the convection current(2).
The existence of this magnetic resonance, which has been predicted also in the VPM(3) and in the
IBA(4), has been proved for the first time by a high resolution electron scattering experiment(5) in t56Gd,
and it is by now confirmed in three regions of the periodic table, i.e. the deformed rare earth nuclei(6), tt~e
fT/2-shell nuclei(7) 46,48Ti mid the actinldies(8).
While there is little doubt that the magnetic resonances found experimentally correspond to the scissors
mode, a detailed understanding of the related phenomenology is still lacking. The present situation is in fact
more complex than that of the E1 mode.
Unlike the E1 state, the M1 state is bound, so that instead of reproducing a broad pick the theoretical
models have to reproduce the pattern of fragmentation, which is remarkably modest. Although the
231
investigation of the excitation spectrum at higher energy has not so far been completed, it seems that with a
few exceptions the M1 strength is concentrated in few fragments of much smaller intensity than the main
level and very close to it(9). In the few exceptions the M1 state rather than fragmented, is splitted into two
fragments of comparable strengh. This has suggested(t0) that also in these cases one cannot talk of
fragmentation in the ordinary sense and that the observed splitting might be related to a triaxial deformation.
Now there exists no systematic theoretical description of this pattern of fragmentation, although the
Spectrum of 156Gd is fairly reproduced in a HFB calculation(1 l).
a)
/ \ FIG. 1 - Classical motions in the TRM: (a) rotation around the
axis; (b) rotational oscillation around the ~ axis.
Other theoretical features specific to the Ml -mode which complicate its description are
SUperconductivity, deformation and strong coupling of the scissors mode with quadrupole oscillations(3,t2).
The mentioned reasons of complexity are, however, also the reasons why the Ml-mode is the potential
SOurce of a lot of information about nuclear structure.
In these lectures I will report the theoretical description in terms of the TRM and the RPA, and I will
give a survey of the experimental status. I will not discuss the important subject of the IBA approach, apart
from reporting that it yields the same eigenstate equation for the Ml-mode as the TRM, and for mentioning
its success in some quantitative predictions.
I will be detailed in the description of the semiclassical TRM in spite of the existence of sophisticated
Calculations, because it is very transparent and leads to an understanding of the physics of the scissors-
mode which is then conceptually simple to describe in terms of microscopic models.
232
2. - T I l E T W O R O T O R M O D E L
If relative translational motion is neglected, the classical tlamiltonian of file TRM is
1 (p)2+ 1 H = I I (n)2 + V,
2,/p 2J n (2.1)
where I(P), 1(n), Jp and Jn are the angular momenta and moments of inertia of protons and neutrons, while
V is the potential energy.
It is now convenient to introduce the total angular momentum I
I = I(P) + I(n),
S = I(P) - I(n),
and rewrite H as
H = , , 1 02+ $ 2 ) + 2./
where
4Jn Jp J = - -
Jp+Jn
Jn -Jp I ' S + V , 4Jn Jp
(2.2)
(2.3)
(2.4)
We assume the potential to depend on the angle 20 between the symmetry axes ~(p), ~(n) of the proton
and neutron ellipsoids
cos(20) = ~(p).~(n) (2.5)
It is therefore natural to introduce this variable along with a set of other variables necessary to identify
~(p), ~(n). These variables may be the Euler angles or, 13, 7 of the intrinsic frame defined by
~(P) X ~(n)
sin(2e)
~) _~(n) r l -
2sin0
~(P) + 4(n)
2cos0
(2.6)
The correspondence {~(P), ~(n)} = {~, ~], 7, 0} is one to one and regular for 0 <0 < 7t/2.
The variables (~(P), ~(n)) = {0t, [3, y, 0} are not sufficient to describe the configurations of the classical
system. However, they describe uniquely the quantized system owing to the constraints
I (p) = 1 (n) = 0, (2 .7 )
233
appropriate to rigid bodies with axial symmetry. These constraints are automatically satisfied if we take wave functions depending on ~(P), ~(n) only.
We quantize by replacing I and S by their Cartesian operator realizations and then perform the change of variable (~(p), ~(n)) -4 (oq [3, y, 0). This change of variables is a unitary transformation provided the scalar
product in the new variables is defined by
2:~ ~t 2re ~/2 / , P / , / ,
< "q/Iv'> = J dc~ J dl3sin~ J dy J d(20)sin (20) V * ( o ~ 0 ) V '(0~'~0). (2.8) 0 0 0 0
The properties of this transformation are given in the second of the refs. (2), where it is shown that the
transformed S operator and Hamiltonian are
S t = i 3/00, Sq = - cot Ol~, S~ = - tan 0171, (2.9)
H = l/(2J ) 12 + HI,
H I =l/(2J ) [cot20 I ; 2 + tan 2 0 Ir12 - 02 / 002 - 2cot(20) 0 / 00 ]+ (Jn-Jp)/(4JnJp) (2.10)
I - t an0 I;Ilq- cot0 14 I ; + iI~ 0/00 ] + V.
We assume an harmonic approximation for the potential which, due to the geometry of the system must
have the form
1/2 C02, 0 g 0 ~ ~/4, V(0) = (2.11)
1 / 2 C [ ~ / 2 - 0 1 2 , ~/4 S O S ~ / 2 .
3. . T H E E I G E N V A L U E P R O B L E M
The general expression for the eigenfunctions is
VIMa = [(2I+1)/(87z2)] I/2 k~ D IMK (ot~Y)~lKa(0),
where (5 stands for all necessary quantum numbers.
These eigenfunctions must satisfy the constraint
R~(P) 0z) VIMa = R~ (n) (g) VlMa = VlMa,
(3.1)
(3.2)
234
owing to the fact that configurations of the system differing by a rotation of protons or neutrons of
around the ~ axis are indistinguishable. The symmetry of the system imposes two sets of relations.
The f'~st one is
tl~lK O. (0) = (-1) I tI)l_Ka (0). (3.3)
Using this relation we can rewrite the eigenfunctions as
hUlMa = [(2I+1)/( 16r~2)11/2 Y+ 1/ ' /I+SK0 [ DIMK + (-)IDIM_K] q)lKa(0). (3.4)
K2.0
The second set of constraints relates the values of the O's in the regions 0_<0_<~/4 and 7t/4<0<~/2
O00 a (~/2-0) = O00 a (0),
OIK o (~/2-0) = - OIK ~ (0),
O22a ( ~ 2 - 0 ) = 1/2 O22a (0) - 112 ~312 O20a (0),
0 2 t a (~/2-0) = 021 a (0),
020 a (7~/2-0) = - 43/2 ~22a (0) - 1/2 ~20a (0).
(3.5)
It is therefore sufficient to solve the eigenvalue problem for 0 < 0 < n/4. The solution of the eigenvalue
problem is simplified by the following transformation, which eliminates the first term linear in the 0
derivative from the Hamittonian (2.10),
def (UdPlKcr)(0) = ~sin(20)OiKa(0) = ~iKo(0), (3.6)
H' =UHIU -1 = 1/(2J) {cot 2 0 I2g+ tan20 12TI - ~2/~02-[2+cot20]}
+ (Jn-Jp)/(4JaJp)[ 1/2 (cot0 + tan0)(Iqlrl+Iqlq)-iI~ " 0/~0 ] + V(0).
(3.7)
Consistently with the harmonic approximation for V we expand H' in powers of 0 up to second order
H' = l /(2J) [ 1/02 (12q-1/4)--02/~02-2 ] + 1/2 C02+1/(2J) 0212.q+ (Jn-Jp)/(4JnJp)
[1/2(1/0+0)(Iqlq+I.qlg)-iI{ 0/~0], 0 < 0 < ~/4. (3.8)
Let us introduce the definitions
03 = "~C/J, 00 = (JC) -1/4 , x = 0/0 0. (3.9)
Omitting the constant term - l / J , H' can be rewritten
H ' = 1/2 o~ [-b2/~x 2 + l /x 2 (I2q-l14)+x 2) ]+ 1/2 03040 x 2 12rl+ 0 O 0 (Jn-Jp)/(Jn+Jp) (3.10)
[ 1/2 (x + 020 (l /x)) (Iqlrl+Irllq) - i I ~ O/Ox ].
235
According to the estimates made in Section (6) for a heavy nucleus 00-__0.05, so that ( Jn-Jp)
/ (Jn +Jp) 00 is of order of 1%, and we can neglect the last term in the Ilamiltonian
H' = 1/2 o [ -32/3x 2 + 1/x 2 (12q-1/4) +x 2) ] , 0 < x < •/(4 00). (3.11)
This Hamiltonian does not contain any coupling between states with different K. In the region rt/4 < 0
-< re/2 states with different K are instead coupled, because in that region, writing [r~/2-0]/00 =y, we have
t t ' = 1/2 o [-02/Oy 2 + l/y 2 (I211-1/4) + y2) ] . (3.12)
Constraints (3.5) can be shown to be in agreement with the above approximate expression of the
Iqamiltonian. In this approximation the nucleus still has axial symmetry.
Note, however, that the terms we have neglected in H' , while small with respect to the intrinsic
excitation energies, are comparable to the rotational energy, so that deviation from pure rotational spectrum
might be expected.
The eigenfunctions of H ' are
tPlK n (0 )= tPKn (0 )= [ nl /((n+K+l)00) ]1/2 (0/00)K+l/2
x e -1/2(°/°°)2 L~(02/020), (3.13)
with eigenvalues
eKn = C0 (2n + K + I). (3.14)
The total eigenfunctions are
~IMKn (0) = [ (2I+l)/(16n 2 (l+Sko)) ] t/2 [DIMK + (_ 1)1 DIM_ K ] tpK, ~/'420 (3.15)
We remark that states with n=l , K=0, correspond to the classical vibrations shown in Fig. la, while
states with n=0, K=I , correspond to the classical rotations shown in Fig. lb.
4, . ELECTROMAGNETIC TRANSITION PROBABILITIES AND FORM FACTORS
The electromagnetic strengths are evaluated to leading order in the deformation parameter & To first
Order only the B(M1) and B(E2) are nonvanishing, while the B(M3) turns out to be of order 1~513. The
Strengths of higher multipoles are much smal4er than the single particle value and will not be reported.
While the Ml- t ransi t ion is the signature of the mode, the study of the E2- and M3-transitions is
important to establish whether a rotational band exists as predicted by the model.
236
A - M a g n e t i c T r a n s i t i o n s
The general expression for the magnetic transition operator is
M(M)~,I.t) = -1/(~+1) ~ dr (j(r).(r xV)(r Z-Yx~t)
= -1/(~,+1) I dr pp(r )Vp(r ).(r xV)(r ~'Yxg) =
= -1/(m()~+l)) I dr pp(r )lp.V(r XYxl.t),
(4.1)
where m is the proton mass, 1 the angular momentum and pp the charge density normalized to the total
number of protons.
The transition operator can be conveniently rewritten by using the classical relations
1 = m r(r- ~2)- m r2 f2
~p = l/Jp lp. (4,2)
After some straightforward algebra we obtain
M(MX,p.) = -1/(20~+1) ) Sx/J p f dr pp(r )(r 2Ox-x r.V) (r ~'Y~.l.t)" (4.3)
F o r L = l w e g e t
M(M 1,tx) = ~ , _+ I ~3/(32~) S x laN (4.4)
and the corresponding strength is
B(M1)']'= 3/(16r~) Jr0 g2 N . (4.5)
For ~, > 1 the integral in Eq. (4.3) vanishes unless the density deformation is taken into account. We
(4.6)
(4.7)
(4.8)
(4.9)
therefore use the full expression of the density
pp(r ) = pp[ r(1- [3 Y20) ] ,
where
[3 = "~4rc/5 2/3
and 5 is the deformation parameter
5 = (R3-R1)/R, R = 1/3 (2 Rt+R3)'
For ~, = 3 we get (third of refs.2)
M(M3,1X)= 51. t + 1 "1 7/(3r~) 1/5 I 5 1 mZ <r4> S x (l/Jp) P-N,
and the corresponding strength is
237
B(M3)I"= 14/(75n ) 82 m 2 A 2 <r 4 >2 o,)/J I.tN 2 fm 4.
For the numerical estimates we will assume
(4.1 O)
<r n > = 3/(3 + n) R n. (4.11)
II . E l e c [ r i c S l r e n g t h
The E2 operator is
M(E2, g) = e ~ dr pp (R0-1 r ) r 2 Y~.I~
= e ~dr pp ( r ) r 2 Y2g (R0 r) = e Y~ Q2v(P) <2v [ exp ( - i0 I~)l 2g> (4.12) V
Where
Q2g (p) = I dr pp(r) r 2 Y2lx = Q2o 590 = "] 5/(16g) Qo 890 (4.13)
for an ellipsoidal shape. To first order in 0 and g/2 - 0, Eq. (4.12) becomes
M(E2, Ix) = Mo(E2, Ix) + Me(E2, Ix), (4.14)
where
Mo(E2, It) = e Q20(P) [ 8g 0 S(0-rd4) + < 20 1 exp ( - i n/4 I@ 2Ix > S (n/4-0)l (4.15)
Mo(E2, g) = - i e Q20(P) ~ 3/2 15~1 [0 S (0-n/4) + (n/2-0) S (~/4-0)1 (4.16)
and 1, x < 0
S(x) =
0, x > 0 .
Using Eqs. (3.5) and (3.13) we get
<I=2, K = I , n = o I [ M ( E 2 , K = I ) [ I I = K = n = O > =
rd4
= - i e Q2o(P) 2 ~j 3 ~o dO ¢P21o (0) 0 ¢Pooo(0) -= - i ~] 3 Q2o(P)0o e.
The E2-transition probability results
B(E2)T= 3 e 2 002 (Q2o(P))) 2 = 15/(16~) e 2 1/(21 ~ ) (Qo(P))) 2 (4.17)
For the numerical estimates we will assume
238
Qo(p) = 4/3 Z l 8 l< r 2 > . (4.18)
c - Magnetic Form Factors
The gener',d expression of the magnetic operator for electron scattering is
T(m) (MX,q) = - i / ~/X (~.+l)f drjp (r)-r x V (jk(qr)Y:ku)
=-i/(2"~.(~+l) Sx/JpSdrpp(r)(j~.(qr)/rX[(y2+z2)Ox (4.19)
- x y O y - X Z O z ] ( rX YX~. ).
For )~ = 1 the only nonvanishing matrix element is
< I = K = 1, n=O II Tlg(m) (q) III = K = n=O > =
,,, (4.20) =-4rt/3 4m/J ~o d r r 3pff) j l (qr) .
In the above equation Off) is the spherical density normalized to the total number of nucleons and we
have checked that the first order correction in 8 is of the order of 1%.
For X = 3 the only nonvanishing matrix element is
< I = 3, K = 1, n=0 [I Tl~(m) (q) [[ I = K = n=0 > = (4.21)
= 4 / 1 5 ~ 2 ~ / 7 1 8 l " , / ~ / J q l d r r 4 p ( r ) j 2 ( q r ) .
5. - S P L I T T I N G OF T I l E M1 S T A T E AND T R I A X I A L D E F O R M A T I O N
If the nucleus has axial symmetry the relative rotation of protons against neutrons can occur only around
one of the axes orthogortal to the symmetry axis, but ff the nucleus has triaxial symmetry, rotations around
each of the three axes become obviously possible.
Let us assume a small deviation from axial symmetry and let us consider relative rotations around the
axes orthogonal to the axis of approximate symmetry. In such a case to first order in the deformation
parameter we can retain the results of the TRM, but we must take into account that the restoring constant C
is now different for rotations around the ~- or r I- axes, so that we have two different frequences
o)~ = 4 C~/]~ = 4C/J (R;-R~I)/R (5.1)
% = "4 Cn/J n = -qc/J ( R ; - R O / R
In the above formula J= 1/2 (J~ +J.q ).
239
Expressing R i (i = ~, 11, 4) in terms of the deformation parameters 13 and y
R~ = R + ~ 5/4g 13 R cos (y - 2nk/3 ), (5.2)
we get
I 0 . '2- col I / ( 0-'2+ COl ) = 1/"/3 tg 3' (5 .3 )
Since according to Eq. (4.5) the B(M1) is proportional to the excitation energy
B2(M1)I" / BI(M1)'[" = ( 1- 1/~3 tg y) / ( 1+ 1/~/3 tg y ) . (5.4)
Note that the above equation provides a check of the hypotesis of triaxiality, relating the strengths of the
fragments to the energy splitting. Once the triaxial shape is established Eqs. (5.3), (5.4) allow a very Precise measurement of the 3~-parameter.
6 . . NUMERICAL ESTIMATES
The TRM has been developed in analogy to the Goldhaber and Teller model for the E1 giant resonance. The analogy has been followed also in the evaluation of the parameters co and 0 o.
Untill now we have only assumed the rotors to be rigid, and this is the only assumption intrinsic to the model.
Let us now assume the rotors to have a constant density and the same deformation for protons and
neutrons. In such a case the moments of inertia are
Jp - 2/5 Z m R 2 (1+ 0.31 13) (6.1)
Jn = 2/5 N m R 2 (1+ 0.31 [3).
In order to evaluate C we observe that when 0 becomes larger than a critical value 0c, 1/2 pA~ (0)
neuron-proton pairs do not interact any longer causing an increase of the nuclear potential energy
AV N = 1/2 pAD (0) V o. (6.2)
Here p is the nuclear density, V o is the neutron-proton interaction potential and 1/2 A~o the volume
variation of the nucleus due to the neutron-proton relative rotation
A~ (0) Z 16/3 0 I R23-R21 l/R21 R33. (6.3)
The value of 0 c has been determined by requiring that for 0=0cthe relative displacement of neutrons
W.r. to protons be equal to the range of the interaction r o
ro'~ 4/(3~) 0 c [ R23-R21 1/R21 R 3 . (6.4)
240
At 0=0 c the restoring force given by AV N must be equal to that given by V
C = 32/(9~) p V o/r o I R23-R21 12 (R3/R1)4
To lowest order in the deformation parameter
8 = (R3-RI) / R
we obtain
C = 128/(9~) V o/r o 82 9 R4 •
Assuming the usual parametrization for Po
(6.5)
(6.6)
(6.7)
p o = A / ( 4 ~ / 3 R 3) , R = 1.2A l /3fm
and
V o = 4 0 M e v ; r o = 2 f m ,
we obtain
co = 42 1 8 1A -1/6 MeV; 002 = 1.7 A -3/21 81-1
(6.8)
(6.9)
(6.10)
For A=156, 8 =0.25, 0=4.7 MeV, ] 002 I = 3.10 -3, B(M1)I"= 17l.t2N ,
B(E2)']'=289 e 2 fm 4 , B(M3)I" --- 0.4 ~2 N b 2.
These values are to be compared to the orbital single-particle estimates(13,19)
B(M1)I" -- 0.1 I.t2N
B(E2)$ .7.- 270 e 2 fm 4
(6.~1)
(6.12)
It should be noted that the above estimates for o and 0 o are very crude. Let me mention some effects
which should appreciably alter them.
The first one is common to the E1 mode, and is related to the fact that the volume Au (0) lies on the
surface of the nucleus. The average density Po should therefore be replaced by the surface density. ThiS
effect, which is taken into account when the restoring force is determined by comparison with the
symmetry energy(14) would lower C and therefore energy and M1 strength. To have an idea of its
importance we can compare with the El-mode where, when it is neglected, one still gets a qualitative
agreement with experiment with the same values of the parameters given by Eqs.(6.9).
The other features which make the above estimate inadequate are specific of deformed nuclei.
241
The first one is related to the geometry. To see where the geometry makes a substantial difference with
respect to the E1 mode let us rewrite Eq. (6.4) by introducing the deformation parameter
ro -" 8/(3rt) 0c I~l R . (6.13)
For 156Gd we get 0 c -n/2, which exceeds the maximum value of 8 = n/4, not to mention that all our
formulae have been derived by retaining the dominant terms in the expansion for small 0. (There exists no
analogous limitation in the translation displacement of the E 1 mode). This does not mean, however, that the
estimate (6.7) be completely unreliable, as shown by the fact that a similar result has been obtained by a
different method(12).
The second reason of inadequacy of the estimates (6.10) is that, due to superlluidity, the moment of
inertia is half the rigid body value. Such a reduction can be accounted for in a two-fluid model(t5), in which
the moment of inertia is the rigid body moment evaluated for the normal fluid component of the nucleus.
This is the nuclear region external to the largest sphere enclosed in the nucleus, the latter being the
rotationally invariant superfluid components. (This picture is similar to that resulting in the IBA, where
Only the bosons describing Cooper pairs of valence nucleons are dynamical). Let me emphasize that it is
perfectly consistent to introduce in this way superfluidity into the TRM, whose only assumption is that of
rigid motion of protons against neutrons.
The last point to be mentioned in connection with the numerical estimates is related to the possibility that
protons and neutrons have a different shape, as suggested by the fact that the pairing strength for protons is
larger than for neutrons(16). Assuming R l to be the same for protons and neutrons, and R3(P) < R3(n) , we
Would obviously get a restoring force constant smaller than for R3(P) = R3(n) (Fig. 2). Let me mention
however, that there is experimental evidence(17) that the protons are more deformed than the neutrons in 154Sm '
FIG. 2 - Different proton-neutron deformation
While the relevance of all the above effects is under investigation(18), it is perhaps worth-while to erapasize that the procedure adopted to evaluate the parameters ~ and 0 o is not intrinsic to the TRM.
242
7. - THE SCHEMATIC RPA
The relation between TRM and RPA can be easily established in the framework of the unified theory of
collective motion.
To this purpose it is necessary to observe that a rotation of protons against neutrons around one of the
intrinsic axes, for instance the xl-axis, induces a density variation of the type
519 ,~ f(r)r 2 1/(i'J2) (Y21+Y2,_l) z30. (7.1)
One would then be tempted to follow the prescriptions of the unified theory to conclude that for axially
symmetric nuclei the TRM is equivalent to a (quasi-) degenerate RPA eigenvalue problem with a separable
interaction of the form
V(1, 2) = X, F*(1) F(2) , (7.2)
where the field F has the expression
F = x 3 r2y21. (7.3)
This however would be correct only if the particle-hole (p-h) states excited by F were all (quasi-)
degenerate, which is not the case.
Let us in fact assume that the nucleons move in an anisotropic harmonic oscillator potential of frequency
66± and e03. These obey the usual volume conserving condition 603 662& = 663o, where o) o is the frequency
for zero deformation.
The field F can be decomposed into the sum of two terms(3)
F = F o + F 2 = F01 + i F02 + F21 + i F22 (7.4)
where
F01 = -~15/(8~:) 1/(2m"~661 6o3) (a 3 a+l + a+3 al)
F21 = - x/15/(87~) 1/(2mX/66_L 663) (a3 al + a+l a+3)
(7.5)
with analogous expressions for 1::02 and F22. Here the a + and a are creation and annihilation operators
for oscillator quanta in different directions.
Now the F 0 terms excite p-h states of energy eo= [to_t.-to3[ ~ 1 5 1 too, which are the ones entering into the
M1-TRM state. The F 2 terms however excite p-h states of energy E2= to&+ 033 =__ 2660, which enter into a
state describing isovector quadrupole oscillations. In other words the scissors-mode is coupled to surface
vibrations(12).
243
Such a coupling can be accounted for in the framework of a two levels RPA schematic model, whosc
eigenvalue equation is
[ < 0 IF I ph > 12= 1/(2Z) (7.6) ~o/(~02- ~o2) Z~ I < 01 F I ph > 12 + E2/(~02 - ~22)p~ c2 phc ~0
The lowest eigenvalue is
o3 = [ B I o3o "~ ( l+2b) / ( l+b) (7.7)
where
b = Z / Co (7.8)
and
= rc/A V 1/< r 4 > , C O = 815 ~ m o~2o/(A < r 2 >), (7.9)
V1 being the symmetry potential.
The M1 su'enght is
t](M 1)'1" = 3/(16n) Jo3 (gp - gn) 2 ~t2N . (7.10)
It should be noted that the above expression coincides with that of the TRM, Eq. (4.5).
/f the coupling with the quadrupole osci/ladon is neglected one gets
o3degenerate = [ ~i [ (% ~1+ b , (7.11 )
which is the value which can be directly compared to the TRM one.
The E2 strength is (3,19)
B(E2)T = 1 / (1+ b) 2 B(E2)I" I degenerate, (7.12)
Where
]3(E2)1" [ degenerate = 5/(32rc) 82 A < r 2 > / (m co) . (7.13)
While the coupling with the quadrupole oscillations lowers the value of the energy but does not alter the
expression of the M1 strength, it drastically reduces the B(E2)l"because, as we will see, b "- 2.
The pairing has the following renormalizing effect (20)
o3"--~ ( E / e ) o)
t~(M1) ~ (UhVp-UpVh)2 B(M1) Z (Uh2-Vh 2) B(MI) ~_ (E/E)Z B(M1), (7.14)
Where E is the two quasi-partlcle (qp) energy, Vh,V p (Uh,Up) the occupation (vacancy) probabili ty
amplitudes respectively for the single-hole and-particle states entering into the p-h state of energy [ 8 [ o~ o.
244
The ratio e/E can be estimated by imposing that the moment of inertia be half the rigid body value
1/2 = d/Jrig ~ ¢/E (Uh 2 -Vh2)2(E/E)3 (7.15)
This gives e/E-- 0.79. A furter 10% reduction in the M1 strength comes from the neutron excess(20).
For numerical estimates we assume
V 1 = 130 MeV
< r 2 > = 3/5 R 2
< 1.4 > = 3/7 R 4
~o = 41 A -1/3 MeV
(7.16)
which yields b = 2.
For 156Gd we get 03degenerate = 3.4 MeV, m = 2.6 MeV. Such a reduction, due to the coupling with the
quadrupole mode, cannot obviously be accounted for in the semiclassical TRM. Taking into account
superconductivity and neutron excess we get the general formulae
03_- 66 I ~51 A -1/3 MeV
B(M1)?_ -0 .024151 A4/31.t2 N.
B(E2)I" z. 0.003 A 2 1 ~ [ e 2 fm 4 .
(7.17)
(7.18)
(7.19)
For 156Gd, t0=3.2 MeV, B(M1) ] '= 5.8 ld.2N, and B(E2)$ = 18 e 2 fm 4, to be compared with the
experimental values 03=3.1 MeV, B(M1)I"= 2.3 + 0.5 g2 N and B(E2)I" = 40 + 6 e 2 fm 4 .
8. - TRIAXIALITY IN TIlE SCItEMATIC RPA
In the triaxial case we assume that in the intrinsic frame the nucleons move in an anisotropic potential
with frequencies
03i = 030 exp (-0ti) (8.1)
where
o~ i = o~ cos (~t- i 27t/3) (8.2)
We assume as before a schematic interaction of the form
Vi = X Fi (1) F i (2) (8 .3)
where the fields F i are given by (second of Refs. 10)
245
FI = x3 r2/(iq2) (Y21 + Y2,-I)
F2 = '~3 r2/q2 (Y21 " Y2,-I) (8.4)
F3 = z3 r2/(i.42) (Y22" Y2,-2)
The x 1- and x 2- eigenmodes are
~i = cos ) '[ 1 - ( - 1 ) i 1/43 tg y] co, i = 1, 2 (8.5)
Where c0 is the RPA eigenvalue in the axial limit (y---0). The splitting between the two levels is
Aco = 2/~/3 sin y co (8.6)
The M1 strenghts are
Bi(MI)I"= I /2cos y [ 1 - ( - 1 ) i 1/'43 tg 'y] B(M1)~", i = 1, 2 (8.7)
Where B(MI)I" is the value in the axially symmetric limit y=0.
The above equations agree with Eqs. (5.3) and (5.4) of the TRM.
A third mode absent in axial nuclei emerges with an energy
t03 = 2/~3 sinT0~ (8.8)
and a M 1 strenght
B3(M1)']'= 2/'43 sin y B(M1)$ (8.9)
The two quantities vanish in the axial limit consistently with the fact that in this limit the state must disappear. For very small values of "/, the x 3- mode is very low in energy and weakly excited, so that it is
Very unliked to be observed, if it exists at all.
9 . . REALISTIC CALCULATIONS
A number of realistic calculations havo been performed for 156Gd. They all give a value of the total
Strength more than twice the experimental one. All these calculations with one exception give an M I
Strength concentrated into 2 regions.
In ref. (21) wave functions of a deformed Wood-Saxon potential with a 1-s term are used. The two-
body potential contains paring, quadrupole and spin-spin interactions (Third plot of Fig. 3). In spite of the
orbital nature of the main state, the authors question its interpretation as a scissors mode.
246
FIG. 3 - Experimental M1 strength distribution in 156Gd (upper part, where full lines correspond to collective rotational excitations) compared to the theoretical predictions of ref. (11) (second plot), ref. (21) (third plot), ref. (22) (fourth plot), ref. (23) (fifth plot).
I t -
1 _
o/,,I ii~. 2 1
2
1
0 2 -
1
0 2,0
i +
i
I I I,,~l
115 6 I I
Gd te ,e ' )
E x p e r i m e n l
LILI~I i-~ [ -
lIF8
, l l , i RPA
I I I I ,, I ~ , i h I~ ' I
Q R P A - A
t I I I
b /RpA-B
I
I
41o
I ,
,L
L 1, .5 2.5 ].0 3.5 50
Excitalion Energy (HEY) In ref. (22) wave functions of a deformed oscillator with a Skyrme interaction are used (Fourth plot of
Fig. 3), and in ref. (23) wave functions of an axially symmetric Wood-Saxon potential with a quadrupol¢
interaction (Fifth plot of Fig. 3).
The exception is ref. (11), which gives a strength of fl~e main level much higher fllan the strength of the
other fragments (second plot of Fig. 3). The distinctive feature of the calculation is the addition of a
quadrupole pairing to a quadrupote interaction. Wave functions of a Nillson potential with a 1-s term are
used. A ttam'ee-Bogoliubov plus RPA calculation shows that the quadrupole paring plays an important role
in bringing collectivity into the M 1-state, while the l-s produces a significant fragmentation with a result
similar to the experimental one.
Finally a calculation(24) on nuclei of mass around A=130 shows a pattern of fragmentation analogous to
the papers of the first group (Fig. 4). The wave functions are linear combinations of particle-number at~d
spin-projected 0 q-p and 2 q-p determinants obtained from an optimal HFB mean field. The interaction is
sligthly renormalized Bruckner G-matrix.
247
I~IG. 4 - Predicted B(M1)q" stmgth distributions for Selected Ba isotopes. Free values for the proton (glP=5.587, glP=l.0) and neutron (gsn= -3.383, gen=0.0) gyromagnetric factors are used. In addition non-weighted (YNw) and energy-weighted (~EW) sum rules are presented.
k~
2.0 .... w 1 ~ - - i
~tB a
oo ] Jltl
0.0 It
u°ez I 2 3S j~
1.0 Jl
O0 ] I I I
Excitation
I - - T - ~ T I
E M~=~a ~- tw=J, z ]
1,[ ...... 2 ~7 [.w=O 0
[ tw=t , 11
I, Ii , l i l t , i
[.~-8 8 ~-t'd=39 9
Lit, I, I I J I i l
Energy (MeV)
10.. TltE 1BA
I will only sketch how the M1 state in the IBA is related to the scissors mode of the TRM and then I will
report some IBA predictions.
Tile ttamiltonian used is
H --- ed nd+ K (Qn + Qv) " (Qn + Qv) + X]'vl (10.1)
The first term accounts for the pairing, the second one for the quadrupole and the third one for Majorana
interaction. The operators Q and M are given in second quantized form using the boson creation and
annihilation operators for bosons of angular momentum zero and two (s and b bosons) as
Qp = (s+p d o + d+ 9 Sp)(2) + Xp (dp+~O)(2) , (10.2) ~ .-. fg}
M = (s+nd+v + d+ns+v)(2) .(s vd'~ + ~vSn) ( 2 ) - 2 Y. (dr;dv)" (d+nd+v) (k) k=l,3 (10.3)
with p = (n, v ) denoting proton and neutron bosons, respectively. If the structure constants of the quadrupole operator are equal, i.e. Z=-Xn=Zv, the Itamiltonian is
synunetric under the interchange of proton and neutron variables. This symmetry is related to the boson
248
quantum number F. Bosons are assumed to have F-spin F=I/2 with projection F z = 1/2 for proton and
Fz=-I/2 for neutron bosons. With the help of this new quantum number the boson states can be labeled
according to their symmetry in the proton neutron degree of freedom. The low lying symmetric states have Fmax = (Nr~+Nv)/2 while the mixed symmetric JTt=l+ states have F=Fmax-1. The Majorana operator (10.3)
used in the Hamiltonian (10.1) reduces to a simple form in the presence of this symmetry, namely
M = Fma x (Fma x + 1) - F(F+I) (10.4)
The parameters of the pairing and quadrupole part have been fixed by the well known low energy
spectra mad the strength of the Majorana force has been determined from the excitation energy of the J 7:= 1 +
state.
By use of coherent states of the form
I Va > -- exp (z (O~sp Sp + + O~dp d+p ) [ 0 > (10.5) P
a classic',d Hamiltonian can be constructed
I-1c (O~sn:, O-sv, O~d~, Otdv) = < ~0t I 1t I ~ a >. (10.6)
A procedure analogous to the one adopted for the TRM can be followed at this stage. By transformiz~g to the intrinsic deformation parameters ~n "¢~ and [~v "/v and to Euler angles for the whole nucleus, the
intrinsic part of the energy becomes a function of the intrinsic deformation parameters and of three angles 0 t 02 03 describing the relative orientation of protons versus neutrons. If both neutrons and protons posses
axial symmetry and equal deformation ",/~ ---),v=0, [3r~ =[3v= [~ and a single angle 0 is needed to specify the
relative orientation. For small values of 0 the TRM Hamiltonian of Eq. (4.11) is obtained.
Let me now list the predictions of the model for 164Dy
B(M1)I"= 4.1 ~2 N
B(E2)q'= 102 e 2 fm 4 (10.7)
B(M3)T= 0.07 ~t2 N b 2
Tile B(M3) estimate is very uncertain, however, due to the difficulty of making the appropriate fermion"
boson mapping. The value reported above has been obtained in ref. (9) by fitting the gyromagnetic factors.
11. - EXPERIMENTAL RESULTS
A state with the properties of the jr~= 1 +, K= 1 state predicted by the TRM has been first discovered (5) ir~
a high resolution (e,e') experiment on 156Gd. It is now confirmed in three regions of the periodic table, i.e.
the deformed rare earth nuclei(5, 6) (Figs. 5, 6), the f7/2-shell nuclei(7) 46,48Ti (Fig. 7) and the actinides(S)
(Fig. 8).
249
f I I "l-- I 6 Ee=2SHeV
9=1650 *s~ Gdte.e'l
~ '~ Gdle,e'p k j
a
E
vj o , , , ,~ " ,+..,.
(71 T
- - l / I l L. l 2B 3.0 3 I 3 f,
Excitation Energy (HeV)
FI(;. 5 - I n e l a s t i c e l e c t r o n s c a t t e r i n g spec t r a . P e a k s d e n o t e d by a r r o w s are
interpreted as jrc= 1 + state
FIG. 6 - See cap t ion o f Fig. 5.
{, tsL SmKF.e') ~l [,=29HeV
t t 0 =165'
o
6 f " I~ Dyle.e')
o
. 4 - - .
4 I I?~Y HeJ") ~ ~#
2.6 28 10 J,2 ] z. 36 3.8
Excitation Energy (HEY)
250
J*~l" ~'~' l i l e , e ' ) 3 J,
r- 2 ~ [°~OHIV
t'- )I ~ e ~16S" fO
~--~ 0 E I k| Ti( le f~')
IV) J'=l" " ~ ~ J ' , l "
r - 7
0
0
Excitation Energy {HEY) F I G . 7 - h l e ] a s t i c e l e c t r o n s c a t t e r i n g s p e c t r a o n 4 6 , 4 8 T i . I n t he [ r i a x i a l n u e ] e u s 4 8 T i t w o
l + states are excited. They are separated by an energy of roughly 1.8 M e V .
V', 2o ,4,-- ¢-
4~ L J
2.0
I I I I I
2 ~ " U I I , ~ ' )
1 I t I I 21 2.2 2.} 2.~ 2.S
E~(MeV)
26
FIG. g - S e e c a p t i o n o f F ig . 5 .
Two features of these states give special support to their interpretation as tile scissors mode.
The first one is that the electron scattering form factors are in good agreement with the predicti o~s
which assume an orbital excitation mode and no deviations have been observed so far (see for instance Fig,
9).
251
The second is that the (p,p') reactions do not appreciably excite these states(7). Since the intermediate
energy proton scattering at small angles excites magnetic dipole states only through the spin part of the
nUcleon-nucleon interaction(26), this finding is consistent with the orbital nature of the Ml-states.
The comparison between (e,e') and (p,p') experiments allows a quantitative evaluation of the orbital and
spin contribution to the strength. In electron scattering one measures
B(M1) T=3/ (16n) [ 1 /2gs<f l l ]Eckx~lli>+ g l < f l l X k : x ~ l l i > l 2 K I¢
for a AT=I transition. Hence
N / B(MI) $ = I + 4 B(o) + ~ B(1) l,
Where the + accounts for the uncertainty in the relative phase between ~B(I) and ~B(o).
The (p,p') cross-section at q~0 can be written
dff/d[] [q~0 = (IM(2r0) 2 kf/ki ND [ Vcr~ [ 2 l< lql Y. OK'C ~ II i > 12 K
Where N D is a distortion factor andVox is the a-'~ nucleon-nucleon interaction. The (p,p') cross-section is
therefore proportional to B(o').
I I I I 15t,
Dyle,e'l
10-1 ~ J~ =f"
Ex=3.110HeV
10-1
I I
- - IBA-2
FIG. 9 . Transverse fo rm factors o f the ~0-, - -TR~ transition to the J~=l + state in 164Dy at o D a r r n s l a d f ~. Ex=3.11 MeV. Compared to the data are the A Amsterdam -
IBM and TRM prediction. I I I '~ I 0.5 10 1.S 20
q°,,(fm" ) The result of the comparison is reported in Table I, which shows that the orbital contribution is always
dominant with the exception 46Ti. In this connection we must note that Zamick(27) has been the first to
Point out that low lying Jn=l+ states should exist also in medium light nuclei, but with important spin effects. I te has calculated that the B(1)/B(c) ratio in the f7/2-shell should be of order 1, while the
experimental value is 3.
Having established the nature of tile state let us have a quantitative loock at energy and strength.
The energy scales rather weU according to the RPA result
252
o) = 66 16] A -1/3 MeV,
with the exception of 46Ti, while the TRM predicts a larger value. In this connection we must recall that the
coupling to the quadrupole oscillations is essential to lower the RPA energy. The experimental strength in 156Gd is B(M1) 1" = 2.3 + 0.5 IIN 2. This is less than half the schematic
RPA with pairing and the realistic calculations value, and much smaller than the TRaM prediction.
TABLH 1 - B(MI)~ s t rength and ratio r ~ / ~ - ~ f rom (e.e',) and (p,p') scattering.. b)assumlng the positive slgnl c) assuming the negative sign.
.u~l,u. s~ B(pl~t ~i ) /4~T63 (MeV) (~N) (¢angel
>12.8 3
156 b) Gd 2.18 1.3 ~ 0.2 >(1.3 - l.?)c)
3.075 >{3.3 - 3.7)
164Dy 3.11 1.5 * 0.3 >(1.8 - 2.3) b)
3.16 +1.4 • 0.4 >(3.4 4.3) c)
- 2 . 9 ~- 0.5
46Ti 4.32 1.0 • 0.2 1.2 - 2.5 b}
3.2 - 2.4 c)
We must remind, however, that the expression for the strength is exactly the same in the TRM and i,
the RPA, so that this difference is only due to the different values of co and J used in the two cases. The
agreement with the IBA is instead good.
Let us now come to the pattern of fragmentation. The investigation of the energy spectra above 4 MeV
has not yet been completed, but preliminary results show no Ml-strength in this region(9). All the strength
seems therefore concentrated very closely around the main level (Fig. 3) in fragments of very small
individual strength, with the exception of 164Dy, 157yb, 48Ti and 238U. Apart from this latter nucleus, i ,
these cases we actually have a splitting rather than a fragmentation. It was in fact this observation in 164Dy
and 174yb which suggested a relation to triaxiality(10). Now recent results with higher resolution have
shown that there are three levels in 164Dy rather than two, at energies 3.111, 3.159 and 3.173 MeV witla B(M1) 1 ̀1.3, 1.25 and 1.1 t.tN 2 respectively(28). These results are no longer compatible with a splitting due
to triaxialily. The relationship between triaxiality and splitting is instead confirmed in 48Ti (Fig. 7), where ] (o2-0)1 I
-1.8 MeV and B2(M1) 1" / BI(M1 ) 1" = 0.6 agree with Eqs. (5.4) and provide a value of'T"24*.
In conclusion the pattern of fragmentation is reproduced only by the calculation of ref. (11), which
seems and indication of the importance of the quadrupole pairing.
253
It remains only to mention that there exists a candidate for the Jn=2+ member of the band(29), with a
strength
B2(E2 ) 1" = 40 + 6 e2fm 4.
in reasonable agreement with the RPA and IBA predictions.
12. - CONCLUSION
We have seen that there is a new collective mode in deformed nuclei, which is described in essentially
the same way in different models. It is remarkable that the expression for the B(M1) in the TRM and in the
RPA coincide, as it is the case for the eigenstate equation in the TRM and in the IBA.
While the general features of this mode are well understood, there are a few points which deserve
further investigation.
From the experimental side it is necessary
i) to complete the measurement of the M1 strenght at higher energy.
ii) to study the members J = 2 +, 3 + of the band, in particular with respect to the orbital and spin
contribution to the strength. Let me remind in this connection that in the latest IBA analysis the M3
mode is not expected to be a dominantly orbital mode(9).
iii) to study other triaxial nuclei,
From the theoretical side 1 think it would be very interesting to investigate the effect of a different
proton-neutron deformation on the total strenght and fragmentation of the Ml-mode. A quantitive prediction
of these features of the mode remains in fact the main open problem.
ACKNOWLEDGEMENTS
I would like to thank N. Lo Iudice and A. Richter for many discussions of the material of these lectures.
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Phys. Rev. Lett. 16, 1661 (1985).
THE DESCRIPTION OF THE QUADRUPOLE
COLLECTIVE MOTION OF A PROTON-NEUTRON INTERACTING SYSTEM
WITHIN A GENERALIZED COHERENT STATE MODEL
A.A. R~du:~
C e n t r a l I n s t i t u t e o f P h y s i c s , P.O.Box MG6
R-76900 B u c h a r e s t , ROMANIA
ABSTRACT
The coherent state model is extended to the description of a com-
Posite system of protons and neutrons. Six rotational bands are simul-
taneously treated by means of an effective quadrupole boson Hamiltonian.
Among them there are two (p,n) asymmetric bands with K ~ = I +. Special
attention is paid to the MI state I +. All its properties are quantita-
tively described for 156'158'160Gd although the results are free of
any adjustable parameter.
I . INTRODUCTION
One o f the most e x c i t i n g d i s c o v e r y o f the n u c l e a r s p e c t r o s c o p y
r e f e r s to the low energy c o l l e c t i v e MI mode. Th i s i + s t a t e has been
i d e n t i f i e d f o r 156Gd in a h igh r e s o l u t i o n ( e , e ' ) e x p e r i m e n t a t back-
Ward ang les i ) . The r e s u l t s c o n c e r n i n g the e x c i t a t i o n energy and the
BMI va lue f o r the e x c i t a t i o n 0 + ~ I + were con f i rmed by a n u c l e a r r e - g 2)
Sonance f l u o r e s c e n c e e x p e r i m e n t In r e f . 2 the c o r r e s p o n d i n g data
f o r 158Gd and 160Gd are a l so r e p o r t e d . R e c e n t l y , the ( e , e ' ) a n a l y s i s
has been ex tended to o t h e r deformed n u c l e i o f the ra re e a r t h r e g i o n : 158Gd ' 154Sm ' 164Dy ' 168Er 3)
The mechanism o f e x c i t i n g such a s t a t e has been t h e o r e t i c a l l y
i n v e s t i g a t e d by s e v e r a l groups both f rom the phenomeno log i ca l and the
m i c r o s c o p i c p o i n t o f v i ew . In the f ramework of phenomeno log i ca l models
the i + s t a t e i s a s s o c i a t e d to an i s o v e c t o r mode d e s c r i b i n g a c o l l e c t i v e
mot ion of the p ro tons a g a i n s t the n e u t r o n s . Th i s k ind of mo t ion has
been f i r s t s t u d i e d in r e f . 4 where the l i q u i d drop model 5) was con-
S idered in an u n i f i e d f a s h i o n f o r p ro tons and n e u t r o n s ; the two sys -
tems i n t e r a c t w i t h each o t h e r by a r e s t o r i n g f o r c e wh ich p r e v e n t s the
256
dem ix ing of the two systems. The s t r e n g t h of t h i s f o r c e was de te rmined
by r e l a t i n g the a s s o c i a t e d r e s t o r i n g f o r c e to the asymmetry energy
g i ven by the mass f o r m u l a o f Weisz~cker . In r e f . 6 i t has been proved
t h a t t h i s p i c t u r e i s q u i t e s u i t a b l e f o r d e s c r i b i n g the magne t i c p rope r -
t i e s o f heavy n u c l e i .
However i t shou ld be ment ioned t h a t the above ment ioned f o rma l i sm
has been used to d e s c r i b e the i s o v e c t o r s t a t e 2 + o f s p h e r i c a l n u c l e i
and not the magne t i c s t a t e I + o f the deformed n u c l e i . Such a 2 + s t a t e
has been p r e d i c t e d 4) at an e x c i t a t i o n energy of about 15 MeV. Recent -
l y a more c a r e f u l e s t i m a t i o n o f the r e s t o r i n g f o r ce p r o v i d e d a c o n s i - 7) d e r a b l e improvement compared to the e x p e r i m e n t a l data
A s i m i l a r i d e a , bu t aimed to d e s c r i b e the i + s t a t e , is t h a t o f
two r o t o r model (TRM) 8) wh ich uses f o r the p r o t o n and n e u t r o n systems
two a x i a l l y s ymme t r i ca l r o t o r s i n t e r a c t i n g t h rough a r e s t o r i n g f o r c e
p r o p o r t i o n a l to the square of the ang le (2e) between the two symmetry
axes. The s t r e n g t h of t h i s f o r c e is e v a l u a t e d by a method s i m i l a r to 9)
t h a t used by Go ldhaber and T e l l e r f o r the g i a n t d i p o l e resonance
The s t a t e i + i s then c h a r a c t e r i s e d by s c i s s o r l i k e e o s c i l l a t i o n s of
the a x i a l l y deformed p r o t o n p a r t a g a i n s t the n e u t r o n p a r t o f the nu-
c leus around an ax i s p e r p e n d i c u l a r to the Z ax i s o f the i n t r i n s i c
f rame.
I t i s wo r th m e n t i o n i n g t h a t the TRM has the b ig m e r i t o f be ing
the f i r s t f o r m a l i s m p r e d i c t i n g the i s o v e c t o r s t a t e i +. Moreover TRM
p r o v i d e s a very e l e g a n t and i n t u i t i v e i n t e r p r e t a t i o n o f 1 + as a s c i s -
sor l i k e mode. Desp i t e o f i t s b e a u t y , the TRM is not ab le to p r e d i c t
any of the q u a n t i t a t i v e c h a r a c t e r i s t i c s of the s t a t e I +. Indeed the
t h e o r e t i c a l p r e d i c t i o n s f o r the e x c i t a t i o n energy as w e l l as f o r the
BMI va lue o f the t r a n s i t i o n I + 0 + g are much l a r g e r than the c o r r e s -
pond ing da ta . A l t h o u g h the r e s u l t s m igh t be s e n s i b l y improved by a
c a r e f u l use of the q u a n t i z a t i o n p r o c e d u r e , our b e l i e f i s t h a t the b ig
d e v i a t i o n s from the e x p e r i m e n t a l da ta are caused by the f a c t t h a t the
c o u p l i n g of t h i s mode to the o t h e r degrees o f f reedom i s i g n o r e d .
Indeed the dynamica l 8 and y d e f o r m a t i o n s are f r o s e n and the mot ion of
the e v a r i a b l e i s c r u e l y decoup led from the mot ion of the o t h e r 5 i nde"
pendent ang les .
In the i n t e r a c t i n g boson model ( IBA2) the MI s t a t e i s caused by
b r e a k i n g the F sp in degeneracy by a Majorana i n t e r a c t i o n o f the p ro ton
and n e u t r o n - l i k e bosons I 0 ) . Here the p o s i t i o n o f i + i s de te rm ined by
f i x i n g the s t r e n g t h o f the Majorana term.
Seve ra l a t t emp ts have been made to i n t e r p r e t the MI s t a t e a l so
i n the frame of a m i c r o s c o p i c p i c t u r e : a) the v i b r a t i n g p o t e n t i a l
257
method 11); b) the sum rule approach 12); c) the RPA method applied
to an isovector quadrupole quadrupole in teract ion 13,14)., d) the me-
thod of normal coordinates for a many fermion system moving in an
ax ia l l y deformed o s c i l l a t o r potent ial 15) In the RPA treatment addi-
t ional effects coming from pair ing in teract ion and neutron excess have
also been considered 13). A nice feature of the RPA approach consists
in i t s a b i l i t y to r e a l i s t i c a l l y describe the dependence of the MI form
factor of the momentum transfer 14)
Although the proposed formalisms d i f f e r in some technical detai ls
they have several common features:
i ) The protons and neutrons are dist inguishable e n t i t i e s ;
i i ) The I + state has a dominantly isovector character; the microscopic
approaches allow for a small admixture with the isoscalar compo-
nents;
i i i ) I f one associates a co l lec t i ve coordinate to th is co l lec t i ve mo-
t ion i t is mainly determined by the components L+ of the angular
momentum;
iv) The M1 t rans i t i on to the ground state is caused by the convection
part of the nuclear current;
v) The theories mentioned above consider the magnetic state 1 + in
i so la t ion .
The IBA2 formalism is an exception but as we said before i t is not
able to make predictions for I + . Indeed the strength of the Majorana
interact ion is adjusted so that the energy of I + be reproduced. Further-
more i t is not clear to me how the conclusions of such theories modify
When one wants to describe the neighbouring co l lec t i ve states. Are these
formalisms able to give a consistent descript ion for both the i + state
and the t r ad i t i ona l quadrupole co l lec t i ve states ? Treating on equal
footing the state 1 + and the co l lec t i ve quadrupole type states is the
agreement with experiment improved or i t becomes even worse. Of course
Such questions can not be answered using the actual stage of the
attempts we have enumerated before. Indeed: a) in the TRM formalism
the B and y variables are frosen. The model Hamiltonian is too simple
to account for dynamical motion of several degrees of freedom, b) The
model of two in teract ing l i qu id drops uses a Hamiltonian which is qua-
drat ic in the quadrupole co l lec t i ve variables, but the harmonic picture
is by far not sui table to describe the complex s i tua t ion of in terac t ing
COllective bands, c) The microscopic theories keep themselves at the
RPA level which should be the f i r s t approximation step for a chain of
addit ional ones aiming to a more r e a l i s t i c studies.
258
Due to these features my be l i e f is that the problem of explain-
ing the properties of the magnetic state 1 + is not yet closed. By con-
trary there are many open questions refer ing both to the quant i ta t ive
descript ion and a consistent treatment.
Having th is in mind we have started the study of the MI state I +
wi th in an extended version of the coherent state model (CSM); this mo-
del has been proposed, by myself and my collaborators 19), for the des-
c r ip t ion of three in teract ing bands.
For those to whom the CSM is not very fami l ia r I shall spend one
hour discussing the underlying ideas as well as some of i t s performan-
ces (Section 2).
In the second part of my lecture I shall adapt the CSM to the
aim of describing a composite system of protons and neutrons.
F i rs t I shal l define a res t r ic ted co l lec t i ve space which is sui-
table for accounting of the main features of s ix co l lec t i ve rotat ional
+ ~ )(Section 3). Within this model space an effec" bands (g, B, y, y, i , I +
t i ve Hamiltonian w i l l be analysed. Among the s ix co l lec t i ve bands con-
sidered here, there are two isovector K ~ = I + bands whose in teract ion
to the other bands receive a special at tent ion (Section 4). The elec- +
tromagnetic properties of the state I are in extenso discussed in Section 5. The numerical results for 156'158'160Gd w i l l be compared
with the avai lable data (Section 6).
2. SHORT REVIEW OF THE CSM
I t is well known that very important features of the co l lec t ive
bands are ref lected by the re la t i ve posi t ion of the excited bands B
and Y. Along the periodic table one dist inguishes three cathegories
of nuclei characterized by:
i ) E~+ > E2+. This is speci f ic to the Pt region. Using the group theo-
ry ~ language these nuclei obey the 06 symmetries.
2) E + > EO+. This re la t ion is sa t is f ied by the y-stable nuclei of Sm
2y. ~heir behaviour can be easi ly interpreted in terms of the reglon.
i r rep of the SU5 group.
3) Ej÷ ~ Ej+. A typ ica l example of th is kind is the isotope 232Th.
Th~ quas~degeneracy of the levels J; and j+ y suggests the presence
of a SU3 symmetry.
A steady aim of the co l lec t ive models is to provide a uni f ied
descript ion for the three classes of nuclei . A somewhat s impl i f ied ver-
sion of th is scope whould be to describe by a single theory the col lec-
259
t i r e bands of a b ig number of n u c l e i r a n g i n g from the s p h e r i c a l r e g i o n
to the ve ry deformed one. Such a t h e o r y shou ld be ab le to say how the
C O l l e c t i v e p r o p e r t i e s change a g a i n s t a smooth v a r i a t i o n of the n u c l e a r
d e f o r m a t i o n . In p a r t i c u l a r a co r respondence between the energy l e v e l s
of a deformed nuc leus and those of a s p h e r i c a l one can be e s t a b l i s h e d .
Such a r e l a t i o n has been s e m i e m p i r i c a l l y p o i n t e d out by S h e l i n e 16)
and Sakai 17) and w i l l be h e r e a f t e r r e f f e r e d to as the SS scheme.
The f u l l c o n t e n t o f the SS scheme i s : a) The s t a t e s of the ground
band o f a deformed nuc leus o r i g i n a t e f rom the quad rupo le h i g h e s t s e n i o -
r i t y s t a t e s of the v i b r a t i o n a l l i m i t ; b) The odd sp i n s t a t e s o f the
Y band are o b t a i n e d t h rough a c o n t i n u o u s d e f o r m a t i o n from the h i g h e s t
S e n i o r i t y v i b r a t i o n a l s t a t e s w h i l e the s t a t e s of even a n g u l a r momentum
cor respond to the second h i g h e s t s e n i o r i t y s t a t e s ; c) The s t a t e head ing
the B band (0~) i s a two phonon s t a t e w h i l e the o t h e r B band s t a t e s are
Obta ined from the t h i r d h i g h e s t s e n i o r i t y s t a t e s . S ince we take f o r +
gu idance the Pt i s o t o p e s , one shou ld ment ion t h a t t he re the s t a t e 0 R
has a t h r e e phonon c h a r a c t e r . Th is i s suppo r ted by the e x p e r i m e n t a l • . + + .
f a c t s a y i n g t h a t the B(E2) va lue f o r the t r a n s l t l o n O^ + 2 i s much ~ + g +
s m a l l e r 18) than t h a t c o r r e s p o n d i n g to the t r a n s i t i o n 2 ~ 2^. The num- + ~ Y
her of phonons a s s o c i a t e d to an a r b i t r a r y s t a t e JF i s J~ /2 + 3. Rep la -
c ing c) by the c o r r e s p o n d i n g s t a t emen t wh ich s u i t s the Pt r e g i o n one
Obta ins a new p i c t u r e wh ich w i l l be c o n v e n t i o n a l l y c a l l e d the m o d i f i e d
SS scheme.
The " p h y s i c a l " c o l l e c t i v e space w i l l be d e f i n e d as a p iece of
the q u a d r u p o l e boson space wh ich accounts f o r the e x p e r i m e n t a l f e a t u r e s
ment ioned b e f o r e .
Now l e t us enumerate the main ideas u n d e r l y i n g the CSM 19) : + + j+
i ) The f u n c t i o n s d e s c r i b i n g the e x p e r i m e n t a l l e v e l s Jg, J~, Y are mut-
t u a l l y o r t h o g o n a l and o b t a i n e d t h rough p r o , i e c t i o n from t h r e e o r t h o g o n a l
deformed s t a t e s , r e s p e c t i v e l y . The u n p r o j e c t e d ground s t a t e ~g is an
a x i a l l y deformed s t a t e of a c o h e r e n t t ype w i t h r e s p e c t to the quad ru -
Pole boson mode w h i l e the B and y s t a t e s are p o l y n o m i a l boson e x c i t a -
t i o n s o f Cg chosen so t h a t
i i ) i n the ext reme l i m i t o f smal l d e f o r m a t i o n s the s e m i e m p i r i c a l r u l e
of S h e l i n e and Sakai 16,17) i s s a t i s f i e d , w h i l e f o r the l a r g e deforma-
t i o n reg ime , the p r e d i c t i o n s of the t r a d i t i o n a l B o h r - M o t t e l s o n model
is r e c o v e r e d t ~ ~..= In the l a t t e r case the Alaga r u l e s c o n c e r n i n g the i n -
te rband a n d / i n t r a b a n d E 2 t r a n s i t i o n s shou ld be o b t a i n e d . ÷ .
i i i ) the E2 t r a n s i t i o n J+ B ÷ Jg is smal l f o r any d e f o r m a t i o n .
Le t us denote by m3M, mB , JM mJM the s t a t e s s a t i s f y i n g the c o n d i - t i o n s i - i i i ) . The " p h y s i c a l " c o l l e c t i v e space is spanned by
260
{m~M, m~N, m~M} J.
i v ) The nex t i m p o r t a n t s tep of CSM is the c o n s t r u c t i o n o f the model
H a m i l t o n i a n . This is w r i t t e n in terms of q u a d r u p o l e bosons and chosen
in such a way t h a t the model s t a t e s are app rox ima te e i g e n s t a t e s in the
r e s t r i c t e d c o l l e c t i v e space, Th is c o n d i t i o n is r i g o r o u s l y s a t i s f i e d in
the two ext reme l i m i t s of smal l and l a r g e d e f o r m a t i o n s w h i l e f o r t r a n s i "
t i o n a l n u c l e i t he re e x i s t a smal l o f f d i a g o n a l m a t r i x e lement r e l a t i n g
the s t a t e s J+ and J+ g The p o i n t s i - i v ) cover the main f e a t u r e s o f CSM. The numer i ca l
a p p l i c a t i o n s r e v e a l e d a very good agreement w i t h the e x p e r i m e n t a l data
both f o r e n e r g i e s and E2 t r a n s i t i o n s . The compar ison w i t h e x p e r i m e n t 20)
has been made f o r the near v i b r a t i o n a l even-even n u c l e i o f Pt r e g i o n 21)
( t he i s o t o p e s o f P t , Os, Hg) as w e l l as f o r the w e l l deformed n u c l e i l i k e 232Th, 234-238U, 172-178Hf"
By t h i s , our d e c l a r e d d e s i r e o f hav ing a t h e o r y which d e s c r i b e s
in a u n i f i e d f a s h i o n the extrems 06 and SU3 n u c l e i as w e l l as the
t r a n s i t i o n a l ones is f u l l y accomp l i shed .
I t is w o r t h w h i l e n o t i n g t h a t the CSM works e s p e c i a l l y w e l l f o r
the deformed n u c l e i and f o r s t a t e s of h igh a n g u l a r momentum 20) , t h a t
is j u s t in the area where the IBA, wha teve r would be i t s v e r s i o n , does
not work at a l l . The main reason f o r t h i s success is t h a t each s t a t e
o f the r e s t r i c t e d c o l l e c t i v e space is b u i l t up on vacuum by means o f
an i n f i n i t e s e r i e s o f quad rupo le bosons. At v a r i a n c e w i t h our model is
the IBA approach which uses a c o l l e c t i v e space o f a l a r g e r d imens ion
but each v e c t o r s t a t e has a very l i m i t e d number o f bosons. In f a c t t h i s
is the reason why IBA is not ab le to approach the s t a t e s o f h igh sp in
as w e l l as those o f the deformed n u c l e i . A lso i t is wor th m e n t i o n i n g
t h a t whenever IBA was faced to some d i f f i c u l t i e s in e x p l a i n i n g the
d a t a , the au thors have i n t r o d u c e d e i t h e r some a d d i t i o n a l degrees o f
f reedom (bosons s and i s o s p i n dependence of bosons) or some c o r r e c -
t i v e terms in the model H a m i l t o n i a n (Majorana i n t e r a c t i o n } , By con-
t r a s t our model d e s c r i b e s n i c e l y a l o t o f data bu t us ing on ly one
k ind o f bosons i . e . the quad rupo le one.
What about the SU5 n u c l e i , can they be d e s c r i b e d by CSM ?
As f a r as the e x c i t a t i o n e n e r g i e s are concerned the CSM is q u i t e f l e -
x i b l e and enab les a q u a n t i t a t i v e d e s c r i p t i o n f o r them. Th is has been
shown f o r 156Dy 19)and 1 5 ~ I ~ 22).Also ~ny aspects of the E2 t r a n s i t i o n s
are n i c e l y rep roduced . This tu rns out e s p e c i a l l y f o r the t r a n s i t i o n s
i n v o l v i n g the s t a t e s from the ground and the gamma bands. These r e s u l t s
say n o t h i n g e lse bu t t h a t the two reg ions of Sm and Pt have s e v e r a l
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common f e a t u r e s . However we have found out t h a t the measured E2 t r a n -
+ ÷ J~+ f o r the Gd i s o t o p e s can not be d e s c r i b e d by the CSM. s i t i o n s J~
In f a c t these data i n d i c a t e a s i t u a t i o n wh ich is c o n s i s t e n t w i t h the +
o r i g i n a l SS scheme, i . e . the s t a t e O~ has a two phonon c h a r a c t e r . The
above ment ioned f a c t s sugges t two p o s s i b l e ways to d e s c r i b e the t r a n -
s i t i o n from SU5 to SU3 symmetry. One would be to e n l a r g e the r e s t r i c -
ted c o l l e c t i v e space by c o n s i d e r i n g an a d d i t i o n a l se t of model s t a t e s .
f o r a band ~ headed on a two phonon v i b r a t i o n a l s t a t e s . Th is wou ld l e t
us to d e s c r i b e s i m u l t a n e o u s l y the t r a n s i t i o n s from 06 to SU3 and from
SU5 to SU3. A n o t h e r p o s s i b i l i t y i s to r e p l a c e the s t a t e mJM w i t h ano-
t he r one wh ich is c o n s i s t e n t w i t h the o r i g i n a l SS scheme. The l a t e r
S i t u a t i o n has been in ex tenso s t u d i e d i n r e f . 22.
Now l e t me enumerate some a d d i t i o n a l v i r t u e s o f CSM. D e t a i l s
about them you can f i n d in the c o r r e s p o n d i n g r e f e r e n c e s . The r o t a t i o -
nal l i m i t o f CSM was e x p l i c i t e l y s t u d i e d i n r e f . 21. T h e r e i n the asymp-
t o t i c b e h a v i o u r of the model s t a t e s as w e l l as of the e x c i t a t i o n ene r -
g ies are d e r i v e d . The e x p r e s s i o n s f o r e n e r g i e s are very s imp le f u n c -
t i o n s o f J ( j + 1 ) wh ich can be e a s i l y hand led in numer i ca l c a l c u l a t i o n s .
The reduced p r o b a b i l i t i e s f o r i n t r a b a n d and i n t e r b a n d t r a n -
s i t i o n s behave a c c o r d i n g to the A laga r u l e . Moreover t h e i r r e l a t i v e
magni tude prove a very pronounced band s t r u c t u r e i . e . the B(E2) va-
lues f o r the i n t r a b a n d t r a n s i t i o n s are much l a r g e r than those f o r the
i n t e r b a n d t r a n s i t i o n s . The q u e s t i o n i s , does t h i s s t r u c t u r e p e r s i s t
in the v i b r a t i o n a l l i m i t ? Th i s q u e s t i o n is p o s i t i v e l y answered i n
r e f . 20 where a f u l l s t udy of the v i b r a t i o n a l l i m i t i s p r e s e n t e d . More-
Over, i t i s shown t h a t f o r s t a t e s of l a r g e a n g u l a r momentum i t ho lds
a f a c t o r i s a t i o n wh ich is ana logous to t h a t g i ven by A l a g a ' s r u l e . In
t h i s l i m i t , s e v e r a l E2 t r a n s i t i o n s are f o r b i d d e n wh ich agree w i t h the
e x p e r i m e n t a l data showing ve ry smal l va lues f o r them. We have s t u d i e d
how the f o r b i d d e n t r a n s i t i o n s i n the ~ band depend on the d e f o r m a t i o n
Parameter 23) I t i s p o i n t e d out t h a t they reach a maximum va lue f o r
the t r a n s i t i o n a l r e g i o n . In the v i b r a t i o n a l l i m i t the model s t a t e s are
exac t e i g e n f u n c t i o n s of the model H a m i l t o n i a n i n the whole boson space.
In the y band the p a i r o f s t a t e s ( 3 + , 4 + ) , (5 +, 6 ) , e t c . are degene ra te .
S w i t c h i n g on the d e f o r m a t i o n the degeneracy i s removed and t h e r e
appear a d o u b l e t s t r u c t u r e wh ich i s c o n s i s t e n t w i t h the da ta . For
l a r g e d e f o r m a t i o n the s t a g g e r i n g e f f e c t i s d i f f e r e n t f rom t h a t men-
t i o n e d above. Indeed , the d o u b l e t o r g a n i z a t i o n is ( 2 + , 3 + ) , ( 4 + , 5 + ) , e t c .
and t h i s a g a i n , i s observed e x p e r i m e n t a l l y f o r the deformed n u c l e i . In
the t r a n s i t i o n a l r e g i o n o f course t he re i s no s t a g g e r i n g e f f e c t .
262
The s t u d y o f t h e model s t a t e s i n t he i n t r i n s i c f r a m e o f r e f e r e n c e
e n a b l e s one a d e t a i l e d a n a l y s i s o f t h e n u c l e a r shapes w h i c h m i g h t be
d e s c r i b e d w i t h i n t he p h y s i c a l boson s p a c e . In r e f . 24 t h e g e n e r a t i n g
i a re a n a l y t i c a l l y e x o r e s s e d in t e r m s o f t h e ~ and s t a t e s { m J M } i = g , ~ , ¥ v a r i a b l e s as w e l l as o f t he E u l e r a n g l e s n o f i x i n g t h e p o s i t i o n o f
t h e i n t r i n s i c f r a m e . By means o f t h e s e one has been s t u d i e d the p r o b a -
b i l i t y d e n s i t y f o r t he d i s t r i b u t i o n o f t h e d y n a m i c a l d e f o r m a t i o n s i ( g j ( ~ ) , i = g , ~ , ~ ) and y ( f ~ ( ~ ) , i = g , 6 , ~ ) r e s p e c t i v e l y . The n u m e r i c a l
a p p l i c a t i o n f o r Pt r e g i o n i s a l s o p r e s e n t e d . The c o n c l u s i o n s a re t h e
f o l l o w i n g ones : 1) f ~ ( y ) has a y - u n s t a b l e b e h a v i o u r f o r J=O w h i l e
f o r h i g h s p i n a f l a t maximum a p p e a r s a round y = 45 o . 2) f o ( y ) has
two e q u a l l y h i g h and v e r y p r o n o u n c e d maxima f o r y = 0 ° and y = 60 o
These maxima become f l a t and s h i f t t o y = I0 ° and y = 50 ° , r e s p e c t i -
v e l y f o r J = l O . The most u n l i k e shape i s t h a t f o r y : 30 o and t h i s i s
t r u e f o r any a n g u l a r momentum. 3) The s i t u a t i o n f o r ~ band i s some-
wha t o p o s i t e t o t h a t o f t h e ~ band . f ~ ( Y ) has a maximum a t ~ = 3O o
w h i l e f ~ o ( ~ ) has two sha rp maxima a t Y = 0 and Y = I0 ° . 4) The f u n c -
t i o n s gg ~ and g~ have a s i n g l e maximum a t B = Bg = 0 and ~ = 6y ~ O,
B has two maxima r e s p e c t i v e l y w h i l e go
B e f o r e c l o s i n g t h e summary p a r t o f my t a l k , I w o u l d l i k e t o men-
t i o n t h a t t h e CSM has been e x t e n d e d by c o n s i d e r i n g t h e c o u p l i n g o f the
q u a d r u p o l e c o l l e c t i v e modes t o t he o c t u p o l e ones 21) Thus , two o c t u -
p o l e bands o f K ~ = O" and K X = I - a re d e s c r i b e d f o r 232Th and s e v e -
r a l even i s o t o p e s o f U. A i m i n g a t d e s c r i b i n g t h e c r o s s i n g o f t h e c o l -
l e c t i v e and t h e two q u a s i p a r t i c l e s b a n d s , t he c o u p l i n g o f t h e two qua-
s i p a r t i c l e s t o a c o m p l e x c o r e d e s c r i b e d by t h e CSM was c o n s i d e r e d i n
r e f . 25, 26. A more c o m p l e x s i t u a t i o n a p p e a r s i n t h e odd mass n u c l e i 2 7 ) .
whe re one and t h r e e q u a s i p a r t i c l e s a r e i n t e r a c t i n g t o t h e CSM c o r e
So much a b o u t the CSM. I hope I have c o n v i n c e d you t h a t t h e CSM
is n o t o n l y a u s e f u l t o o l t o d e s c r i b e in a r e a l i s t i c f a s h i o n t h e d a t a
f o r t h e g, 8 and y bands o f t h e e v e n - e v e n n u c l e i , b u t a l s o a good s t a r t -
i n g p o i n t f o r s t u d y i n g t h e c o u p l i n g o f d i f f e r e n t d e g r e e s o f f r e e d o m .
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3. THE GENERALIZED VERSION OF CSM
I t is easy to unders tand t h a t the i s o v e c t o r s t a t e 1 + can not be
desc r ibed by the CSM. Indeed s ince the CSM uses on l y one k ind o f qua-
d rupo le bosons the s t a t e I + is f o r b i d d e n due to symmetry reasons .
A iming at d e s c r i b i n g the low l y i n g i s o v e c t o r s t a t e s , we s h a l l ex-
tend the CSM by l e t t i n g the q u a d r u p o l e bosons have an i s o s p i n depen-
dence i . e . the s u r f a c e v i b r a t i o n s o f the neu t ron and the p r o t o n systems
W i l l be d e s c r i b e d by two i ndependen t bosons b + and b + , r e s p e c t i v e l y . n~ p~ In c o n s t r u c t i n g the c o l l e c t i v e space we s h a l l f o l l o w the gene ra l
Scheme o f CSM which has been d e s c r i b e d in some d e t a i l s in the p r e v i o u s
l e c t u r e . However t h e r e are s e v e r a l s p e c i f i c f e a t u r e s de te rm ined by the
f a c t t h a t here we dea l w i t h two d i s t i n c t systems ( o f p ro tons and neu-
t r o n s ) . These f e a t u r e s can be summarized as f o l l o w s :
a) F i r s t we suppose t h a t the ground s t a t e is a symmet r ic f u n c t i o n w i t h
respec t to the p r o t o n and neu t ron c o o r d i n a t e s . This means t h a t the two
e l l i p s o i d s a s s o c i a t e d to the p r o t o n and neu t ron systems r e s p e c t i v e l y
are i d e n t i c a l and have the same symmetry axes. The e x c i t a t i o n s o f the
ground s t a t e w i l l be s e l e c t e d so t h a t the c o n d i t i o n s i - i i i ) r e q u i r e d by
CSM (see S e c t i o n 2) be s a t i s f i e d .
b) Concern ing the SS scheme we n o t i c e t h a t in the Gd r e g i o n , which is
taken f o r g u i d a n c e , the head s t a t e o f be ta band is d o m i n a n t l y a two
phonon s t a t e . This c o n c l u s i o n is suggested by the e x p e r i m e n t a l data + +
showing a r e l a t i v e l y l a r g e BE2 va lue f o r the t r a n s i t i o n JB ÷ (J+2)g .
But t h i s is c o n s i s t e n t w i t h the o r i g i n a l SS scheme. There are two pos-
S i b l e e x c i t a t i o n s of the ground band s t a t e s which go in the l i m i t o f
v a n i s h i n g d e f o r m a t i o n , to a two phonon s t a t e , r e s p e c t i v e l y . Here on l y
the symmetr ic f u n c t i o n s are c o n s i d e r e d f o r the B band.
c) Concern ing the y band, l e t us ana l yze the l owes t v i b r a t i o n a l
s t a t e s o f sp in 2. These are o b t a i n e d by a c t i n g on the vacuum s t a t e
+ b + b + b + I0 > w i t h the f o l l o w i n g o p e r a t o r s : (bp~ + ) ( - ) n~ ' nu p~ '
b+b - .b+b +, + + p p)2~ + { n n )2 , (bpbp)2~ - (b~b~)2 , (b+b+ ' ' n p)2u" We know t h a t the
f i r s t s t a t e be longs to the ground band w h i l e the t h i r d one is the sym-
m e t r i c y band which is o b t a i n e d by e x t e n s i o n from CSM. Here we s h a l l
a l so s tudy the band headed in the v i b r a t i o n a l l i m i t by (b + - b + ) I 0 > . pu n~
I t seems t h a t t h i s has a l l the c h a r a c t e r i s t i c s o f a ~ band. Which
band is the lower one is de te rm ined by the parameters o f the model
H a m i l t o n i a n 34 -36 ) . In our c a l c u l a t i o n s f o r Gd r e g i o n , one o b t a i n s a
very good d e s c r i p t i o n o f the e x p e r i m e n t a l data when the a c t u a l v band is a s s o c i a t e d to the asymmetr ic band 28 ,29 ) .
264
d) In the v i b r a t i o n a l l i m i t the lowes t s t a t e 1 + is a two phonon s t a t e
(bnbp)l~+ + IO>. However there are two e x c i t a t i o n s of the coherent ground
s t a t e @g which y i e l d , a f t e r p r o j e c t i o n and a f t e r qoing to the v i b r a t i o -
b+b +" iO> These are (b + - b + and nal l i m i t , the s t a t e ( n p)1~ " n! p l ) ~g • +b + • (bn p } l l ~g" The f i r s t f u n c t i o n is exc luded s ince i t does not s a t i s f y
the o r t h o g o n a l i t y c o n d i t i o n w i th the model s t a t e f o r y band. We note
t h a t the f i r s t 1 + appears to be an one boson e x c i t a t i o n of the s t a t e
2~ ( i n the v i b r a t i o n a l l i m i t ) . Of course we can b u i l d 1 + s t a t e s by
e x c i t i n g e i t h e r of the s t a t e s 2 +Y and 2 B.+ For the s i m p l i c i t y reasons we
s h a l l cons ide r here on ly the l a t e s t case,
e) Thus, the model space we propose f o r the d e s c r i p t i o n of the low
l y i n g s t a t e s of angu la r momentum J is sDanned by the f o l l o w i n g s i x
o r thogona l s t a t e s :
m(g)JM = N~ g) P~o~g " ~g : exp[d(b +po+bno)-d(bpo+bno)]IO>+ , (3.1)
• (~) : N~Y) P~2 ( b+ + JM n2 - bp2) ~g ' (3.3)
~) N Y) J + ~+ : PM2 (~y,p,2 + y,n,2) @g , (3.4)
m(1) N~I) J + + JM PMI )11 @g = (bnb p , (3.5)
~( i ) : ~ I ) J + _ b+ + JM PM1 (bn l p l ) ~ ~g (3 .6 )
where d is a rea l parameter and IO> is the common vacuum s t a t e . The pro-
j e c t i o n o p e r a t o r P~K is de f i ned as:
P~K 2J+l * : ~ f D~K(~ ) R ( ~ ) d~ ( 3 . 7 )
where R(~) is the r o t a t i o n o p e r a t o r w h i l e D~K is the r o t a t i o n m a t r i x
de f i ned accord ing to Rose's conven t ion 30) . N~ i ) stands f o r the norma-
l i z a t i o n f a c t o r s . We a lso have used the f o l l o w i n g n o t a t i o n s :
~+ . + b + . J ~ - + y , k , 2 = (Dk k )2 ,2 + d bk, 2 ; k = p,n , (3 .8 )
+ + + ÷ = ~ + R - 2Rpn ~B p n
+ + + d 2 = - - - , k = p,n ~k (bkbk)o
(3.9)
(3.i0)
265
d 2 - - ( 3 . 1 1 ) pn = ( b p b n ) o /E~
f ) F o l l o w i n g the CSM we seek an e f f e c t i v e H a m i l t o n i a n H f o r wh ich t he
f u n c t i o n s ( 3 . 1 - 6) a r e , a t l e a s t i n a good a p p r o x i m a t i o n , e i g e n s t a t e s
in the r e s t r i c t e d c o l l e c t i v e s p a c e . A p o s s i b l e s o l u t i o n f o r H i s :
H AI(N p + Nn) + A2(Npn + Nnp ) + (AI+A2)(Q + pn np
where
: ~ : N+ (3 13) = ~ b mbTm , T =p ,n ; Npn ~ b mbnm , Nnp pn ' " NT m m '
and ~2 is the square angular momentum operator. One can easi ly check
that H has only an of f -d iagonal m.e. in the basis (3.1 - 6). That is
~ ~ ( Y ) ) However ou r c o n c r e t e c a l c u l a t i o n s show t h a t t h i s a f f e c t s (~ ) I H I ~ j M . t he e n e r g i e s o f B and y bands o n l y by an amount o f a few keV. T h e r e f o r e
the e x c i t a t i o n e n e r g i e s o f the s i x bands are in a ve ry good a p p r o x i m a -
t i o n g i v e n by t he d i a g o n a l e l e m e n t .
( k ) i H ~ ( k ) > Eo k , ~ 1 ~ 1 ( 3 . 1 4 ) E k) = <mJM • JM ' = g ' Y ' ' Y ' '
where
E o < ~ ( g ) I H I (g)> = O0 ~00 (3.15)
Using a n a l y t i c a l e x p r e s s i o n s f o r the m.e. o f H we can s tudy E l k ) J as a
f u n c t i o n o f d. In t h i s way i t can be p roved t h a t the model s t a t e s ~(k)
JM a re e i g e n s t a t e s o f H i n the w h o l e boson space f o r bo th the v i -
b r a t i o n a l and the r o t a t i o n a l l i m i t s . The d e c o u p l i n g c o n d i t i o n h o l d i n g
f o r any v a l u e o f d , one may hope t h a t , a t l e a s t f o r t he low l y i n g
bands , t he i n f l u e n c e o f the c o m p l e m e n t a r y boson space is s m a l l .
From ( 3 . 1 2 ) we no te t h a t H i n v o l v e s f i v e f r e e p a r a m e t e r s ,
( A I , A 2 , A 3 , A 4 , d ) . As we s h a l l see l a t e r on , t hese p a r a m e t e r s a re de-
t e r m i n e d by f i t t i n g the e n e r g i e s o f some low l y i n g s t a t e s o f the g,
and y bands . Thus , t he e n e r g i e s o f the r e m a i n i n g s t a t e s o f the t h r e e
a d d i t i o n a l bands a re f r e e o f p a r a m e t e r s .
The s e t o f s t a t e m e n t s a - f ) d e f i n e the g e n e r a l i z e d v e r s i o n o f
the CSM.
266
4. SHORT COMMENTS ON THE EFFECTIVE HAMILTONIAN AND A FEW
ADDITIONAL REMARKS ON THE MODEL STATES
Now l e t me compare the model H a m i l t o n i a n ( 3 . 1 2 ) w i t h t h o s e used
by o t h e r p h e n o m e n o l o g i c a l a o p r o a c h e s . A s t r a i g h t r e l a t i o n b e t w e e n H
and t h e H a m i l t o n i a n used by A. F a e s s l e r i n r e f . 4 i s o b t a i n e d by u s i n g
a new boson r e p r e s e n t a t i o n
B+ i (b + + + C + : i__ (b + _ b + P = 7--~ np bpp) , ~ ~ np n~ ) ( 4 . 1 )
I n d e e d , w r i t i n g H in t e rms o f B + and C + bosons and c o m p a r i n o t h e r e s u l -
t i n g H a m i l t o n i a n w i t h t h a t o f r e f . 4 one sees t h a t H c o n t a i n s an a d d i -
t i o n a l t e r m o f f o u r t h o r d e r in boson o p e r a t o r s . Thus t h e d e m i x i n q o f
t h e p r o t o n and n e u t r o n sys tem i s p r e v e n t e d h e r e no t o n l y by a q u a d r a t i c
t e r m b u t a l s o by a q u a r t i c one .
A l t e r n a t i v e l y H can be w r i t e n in t e rms o f t he i n t r i n s i c v a r i a -
b l e s c h a r a c t e r i s i n g the p r o t o n and t h e n e u t r o n s y s t e m . In t h i s r e p r e -
s e n t a t i o n one f i n d s t h a t one p i e c e o f H i s j u s t t h e H a m i l t o n i a n d e s -
c r i b i n g two i n t e r a c t i n g r o t o r s w i t h t h e moments o f i n e r t i a d e p e n d i n g
on t h e v a r i a b l e s B~, ~ (~ = p , n ) . Thus the H a m i l t o n i a n ( 3 . 1 ) d e s c r i b e s
a s i t u a t i o n w h i c h i s by f a r more c o m p l e x t h a n t h a t c o r r e s p o n d i n g t o t he
TRM 8) whe re the d e f o r m a t i o n s # and ~ a r e f r o z e n .
I t i s w o r t h w h i l e m e n t i o n i n g a g a i n t h a t t he IBM2 model was e x -
t e n d e d t o t h e d e s c r i p t i o n o f the m a g n e t i c s t a t e s by i n t r o d u c i n g t h e
c o n c e p t o f t h e F - s p i n i 0 ) In t he new v e r s i o n the e f f e c t i v e H a m i l t o -
n i a n i n v o l v e s t h e M a j o r a n a i n t e r a c t i o n w h i c h d o e s n o t a f f e c t the
s t a t e s c h a r a c t e r i z e d by an F - s p i n o f maximum v a l u e . In t h i s way the
s t r e n g t h o f t h e M a j o r a n a i n t e r a c t i o n i s f u l l y d e t e r m i n i n g t h e e n e r g y +
o f t h e s t a t e i
To s t r e s s on the d i f f e r e n c e b e t w e e n ou r model H a m i l t o n i a n and
t h a t used by IBM2 l e t us c o n s i d e r t h e boson r e p r e s e n t a t i o n f o r t h e
g e n e r a t o r s o f t h e F - s p i n - SU2 a l q e b r a
r o = ½ (~p - i n) , F+ = Npn ' F_ : ~np ( 4 . 2 1
where
F+ = F 1 ~ iF 2 , F o = F 3 ( 4 . 3 )
and F i a re t h e componen ts o f t h e F s p i n o p e r a t o r s . S i n c e H ( 3 . 1 2 ) i s
l i n e a r i n t h e g e n e r a t o r F i i t i s m a n i f e s t t h a t i t i s n o t F s p i n i n v a -
r i a n t e . g . [ F i , H ] ~ 0 f o r i = 1 , 2 , 3 . T h e r e f o r e the e i g e n s t a t e s o f H
267
are F and F o mixed s t a t e s . However i t can be e a s i l y checked t h a t the
expected va lue o f F o c o r r e s p o n d i n g to the model s t a t e s ~ ) are equal
to ze ro . Th is happens on l y in the s i t u a t i o n when the d e f o r m a t i o n pa ra -
meters a s s o c i a t e d to t h e two k ind of bosons b + (3 = p ,n) are e q u a l . T
But in t h i s case the model s t a t e s have a d e f i n i t e p a r i t y w i t h r e s p e c t
to the p-n p e r m u t a t i o n . F i n a l l y , we conc lude t h a t the F s p i n - i n v a -
f i a n c e , which s u i t s the r e s t r i c t i o n f o r the t o t a l number of bosons in
the frame of IBM2 is broken in the p r e s e n t f o r m a l i s m .
Coming back to the model space, I would l i k e to ment ion some use- i
f u l p r o p e r t i e s o f the s t a t e s ~JM d e f i n e d by (3 .1 - 6) .
~ (y) ~(1) and (Y) are symmetr ic w h i l e ~JM JM The s t a t e s ~ M' mJM mJM ' '
°o(i) mjM are a n t i s y m m e t r i c f u n c t i o n s w i t h r e s p e c t to the p e r m u t a t i o n
t r a n s f o r m a t i o n ( p , n ) .
Deno t ing by 0 ( i ) r ,
(3.1 - 6) one o b t a i n s
r . J ' ( i ) = z CK~ 0 mjM J ' 1
With
= = + ~ >) + (~ ri Ki 2(8i,y i,~ i,l
the o p e r a t o r o f rank r i a c t i n g on ~JQ
J N~ i )
in
(4.4)
+ 6 i , ~ ) ( 4 . 5 )
The r e l a t i o n ( 4 .4 ) e s t a b l i s h e s a d i r e c t c o n n e c t i o n between the p r o j e c -
ted s t a t e s o M and ~JM w i t h i # g. Due to t h i s r e l a t i o n one may say
t h a t our method is to some e x t e n t e q u i v a l e n t to a d i a g o n a l i z a t i o n p ro -
Cedure w i t h i n the bas is [0~i). m~)]dM .
Using (4 .4 ) one can e a s i l y o b t a i n the v i b r a t i o n a l as w e l l as the
a s y m p t o t i c ( d - l a r g e ) b e h a v i o u r o f the model s t a t e s . In o rde r to save
the space we s h a l l g i ve here on l y the r e s u l t s f o r the s t a t e s I +. In I
the v i b r a t i o n a l l i m i t the s t a t e mlM goes to the two boson s t a t e
b+b - p ) iM lO>. I n c r e a s i n g the d e f o r m a t i o n parameter "d" the c o m p e t i t i o n n between v i b r a t i o n a l and r o t a t i o n a l degrees o f f reedom is p l a y i n g an
i m p o r t a n t r o l e in d e t e r m i n i n g the p r o p e r t i e s o f the model s t a t e s . Such
an i n t e r f e r i n g e f f e c t s can be v i z u a l i z e d by p l o t t i n g the average o f
" Np + N n on the model s t a t e s as a f u n c t i o n o f d 29) One f i n d s out t h a t
f o r d l a r g e the r o t a t i o n a l b e h a v i o u r is dominan t .
In t h i s r e g i o n o f d the vave f u n c t i o n s have a s p e c i f i c K-compo-
nent which p r e v a i l s over the o t h e r ones. In o r d e r to see which is t h i s
K-component we s h a l l w r i t e the f u n c t i o n s (3 .1 - 6) in the i n t r i n s i c
f rame. To do t h i s i t is use fu l to i n t r o d u c e the f o l l o w i n g n o t a t i o n s :
S lab , S S i are the l a b o r a t o r y f rame the i n t r i n s i c frame o f the T' nt '
268
T-system and the body fixed system for the whole nucleus, respectively.
The position of S T and Sin t with respect to Sla b are specif ied by the
Euler angles ~T and ~int respectively
I ) Sla b = S R( i l T ' nt ) Slab = S i n t " ( 4 . 6 )
From these d e f i n i t i o n s i t r e s u l t s :
( ~ I ~ i R nt ) Sin t = S T ( 4 . 7 )
F u r t h e r we denote
(mT, l ' ~e ' mT 3 ) = Q~I ~I. T : p , n " o = 6 - 6 ( 4 . 8 ) , l n t ' ' T,p T,n
Note t h a t the ang le between the symmetry axes o f the two systems (p and
n) i s 2e. From the f i v e v a r i a b l e s d e f i n e d by ( 4 . 8 ) , o n l y t h r e e are
i n d e p e n d e n t . H e r e a f t e r , these w i l l be denoted by ( m ' , ~ , m " ) . In t h i s
way the p o s i t i o n of the p r o t o n - n e u t r o n system can be d e s c r i b e d by
( m ' , ~ , m " , ~ i n t ) • Now, we s h a l l w r i t e the boson o p e r a t o r s i n terms of
the c o n j u g a t e c o o r d i n a t e s (~ ,~ ) of Sla b and then express these in terms
of the i n t r i n s i c c o o r d i n a t e s of S T wh ich w i l l be p a r a m e t r i z e d in the
usual way by the d e f o r m a t i o n s ~ and YT
I (b + i k ~" = k ~ T~ + (-)l~b~_~j) ' ~ - ~Z--T(')" b +~,_~_ b ~ ) , ( 4 . 9 )
T = p , n ,
R(~T)~T, m R(~ - I = ~T(~m cosy~ + I (~ ) s i ) T ) = aT,m ,o - ~ m,2+am,-2 nYT
(4 .10 )
The c a n o n i c a l t r a n s f o r m a t i o n ( 4 . 9 ) is de te rm ined up to a r ea l f a c t o r
k wh ich p lays the r o l e o f a s c a l i n g f a c t o r f o r the 6 d e f o r m a t i o n .
~ i ) can be e a s i l y P e r f o r m i n g these t r a n s f o r m a t i o n s , the s t a t e s m M
w r i t t e n in terms of the v a r i a b l e s a s s o c i a t e d w i t h Sin t . The r e s u l t s are:
= sr ( i ) (~p ~ ; ~p,¥ • ~ ' , e , ~ " ~* ~ ( i ) JM K JK ' n n ' ; d)D K ( ~ i n t ) ,
i = g , 6 , T , ~ , 1,1 ( 4 . 11 )
The a n a l y t i c a l e x p r e s s i o n s f o r the a m p l i t u d e s Fj~)fl are g i ven in r e f .
29. Now, t a k i n g the f i r s t o rde r expans ion of F I~) around yp = yn = e=0
and then us ing f o r the i n v o l v e d o v e r l a p i n t e g r a l s the a s y m p t o t i c
expans ion w i t h r e s p e c t to d, one o b t a i n s the f o l l o w i n g l e a d i n g beha-
i + v i o u r f o r the s t a t e s :
269
( i ) = N1 eG(6n,6 ,d -e ) F~ (Bn ,6p ; ~p mn " ~ i n t ) mJM ,J p ' i , 3 ' , 3 ' ' ( 4 .12 )
~(1) ~ ~ )2 ~ ~ 2 mJM = N~,j [(6p-B n + 126ngpe - 7]e G(Bn,6p,d;e )
Where F ~ I ( B n ' B p ; mp,3 ' mn,3; R in t ) ' ( 4 .13 )
2 ~ 2] -3dB8 = i_ e [ ( d - B p )2+( - d-6n) e (4 .14 )
G(Bn,6 p , d ;e ) 6 2- 3e -2 '
~T = kT ~T ' T = 6, n', 6 = ( P + Bn)/VrZ (4 .15 ) ~r
F l ( 6 n , 6 p ; ~ p , 3 ' ~n ,3 ; ~ i n t ) = (6pe ImP, 3 + 5 n e ' i m n , 3 ) D l ( ~ i n t )
eimp,3 + 6 eimn,3) ( )J+l J* + (6p n - DM _i ( ~ i n t ) . ( 4 .16 )
The f a c t o r s N i , J w i t h i = I , i are i r r e l e v a n t cons tan t s depend ing on J.
S i m i l a r f a c t o r i z a t i o n s have been a lso o b t a i n e d 29) f o r the o t h e r
S t a t e s . From the r e l a t i o n s ( 4 . 1 2 , 13) one sees t h a t m(1)jM and ~(1)mjM are
indeed s t a t e s w i t h K = i . For those va lues o f d which do not be long to
( I ) ~ ( I ) are s u p e r p o s i t i o n s of the a s y m p t o t i c r e g i o n , t h e s t a t e s mJM ' mJM COmponents of d i f f e r e n t K. However the component w i t h K = 1 is dominant .
We note t h a t the expans ion we made does not a l t e r the symmetry
P r o p e r t i e s w i t h r e s p e c t to the p,n p e r m u t a t i o n . The p-n asymmetry of the asymptotic functions ~(I) and ~ I ) is caused by their dependence JM M on the angle e. In the case of m~) this is symmetric function of e
but an asymmetric one with respect to 6 n and Bp.
The assumption we made about the axial symmetry of the two sys- tems r e f l e c t s i t s e l f in t h a t m (g) ~ ) JM and m M depend on ly on f o u r ang le
V a r i a b l e s ( ~ i n t , e ) . By c o n t r a s t , the o t h e r f u n c t i o n s i n v o l v e even in
the lower o rde r e x p a n s i o n , a l l s i x ang les . When m~,3 is smal l m M
behaves s i m i l a r l y to the c o r r e s p o n d i n g f u n c t i o n o f TRMo However t h e r e
is no p h y s i c a l i n s i g h t to suppo r t such an a p p r o x i m a t i o n .
Be fo re a n a l y z i n g the numer ica l a p p l i c a t i o n s , l e t me f i r s t des-
c r i b e the bas i c fo rmu lae a s s o c i a t e d to the e l e c t r o m a g n e t i c p r o p e r t i e s
of the s t a t e s I +.
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5. ELECTROMAGNETIC TRANSITION PROBABILITIES AND THE MI FORM FACTOR
A s e n s i t i v e t e s t f o r the assumptions of our model are the e l e c t r o -
magnet ic t r a n s i t i o n p r o b a b i l i t i e s . T h e r e f o r e we s h a l l i n v e s t i g a t e here
the reduced p r o b a b i l i t i e s f o r the MI and E2 t r a n s i t i o n s of the two i s o - ( I ) ~ (1)
v e c t o r s t a t e s I + d e s c r i b e d by mJM and "'JM ' r e s p e c t i v e l y . To t h i s pur -
pose we s h a l l assume f o r the magnet ic and e l e c t r i c t r a n s i t i o n o p e r a t o r s
the f o l l o w i n g e x p r e s s i o n s
Mlk = p ,k + gnJn,k)PN = gcJc ,k ~N' k = O,&l ( 5 . I )
O2p = Oo(b + PP + ( - )~ bp ,_~) , ( 5 . 2 )
where gp , gn' gc denote the gy romagne t i c f a c t o r s of the p r o t o n s , neu-
t rons and o f the whole sys tem, r e s p e c t i v e l y ; ~N stands f o r the n u c l e a r
magneton. JT ,k denotes the angu la r momentum c a r r i e d by the T-system
JT,k : ~ (b+bT)l,T k ' T = p,n , k = O, _+i (5 .3 )
w h i l e J c , k is the t o t a l angu la r momentum
J c , k = Jp ,k + Jn ,k ( 5 .4 )
For the e l e c t r i c quad rupo le t r a n s i t i o n o p e r a t o r , we have assumed t h a t
the e f f e c t i v e charge of the neu t ron system is equal to ze ro .
The reduced t r a n s i t i o n p r o b a b i l i t i e s are d e f i n e d by:
B(MI; J+ +' ( i ) -~ J ( i ' ) )
B(E2; J+ + J+ ' ( i ) ( i ' ) )
(5.5)
(5.6)
From (5 .1 ) one e a s i l y f i n d s the gy romagne t i c f a c t o r c h a r a c t e r i -
s ing the ground band
v/•< J J $ o gc~N = ~jj(g)iM10 ~JJ(g)>/ ' (5.7)
The t r a n s i t i o n o p e r a t o r s M1k and Q2k d e f i n e d by ( 5 . 1 ) , ( 5 .2 ) can be ob ta ined from t h e i r c l a s s i c a l c o u n t e r p a r t th rough the q u a n t i z a t i o n
p rocedure 31) . To be more conc re te we s h a l l i l l u s t r a t e t h i s fo r the MI
o p e r a t o r . Indeed , the c l a s s i c a l magnet ic moment f o r the compos i te sys-
tem is
271
1 M k = ~ f pp (~" x ~)k d~ (5 .8 )
where p is the charge d e n s i t y and ~ is the v e l o c i t y f i e l d . The i n t e - P
g r a t i o n domain is the drop assoc ia ted to the pro ton system. Taking the
second order expansion of M k in terms of the c o l l e c t i v e coo rd ina tes and t h e i r con jugate momenta one f i nds out the M k is p r o p o r t i o n a l to
(~p~p) Ik where dot denotes the t ime d e r i v a t i v e o p e r a t i o n . This expres - Sion is quan t i zed by r e p l a c i n g ~ by i t s boson r e p r e s e n t a t i o n s and by the boson exp ress ion r e s u l t i n g from the equat ion of mot ion
i ] &pk = ~ [H, ~pk (5.9)
where H and ~ are g iven by (3 .12) and (4 .9 ) r e s p e c t i v e l y . In t h i s way one o b t a i n s :
Mik = ~ Mk ; M k = /-~ # Ro E k u N , Ro : I . 2 A l / 3 [ f m ] , ( 5 . 1 0 )
where M is the nucleon mass, R the nuc lea r rad ius and o ./~- 3ZR A 3
Fk=-l'~8"~--°T2{(Al+6A4)J4~'i~ck pk + ~-- i nk + T ~ ( A 2 - A 1 ) [ (bnbp) l+ + k + ( b + b p l l k
+ ( b ; b n ) l k _ (bnbp ) l k ] + ~ A3 [_ 1 ~ ( ~ Jpk + Jpk~n)
+ Q+ b;bn) ) + ( + + pn (( l k - (bnbp lk ) ( b n b p ) i k + ( b ~ b p ) i k ) £np] } (5 .11)
1 + 0 + To the t r a n s i t i o n ~ c o n t r i b u t e the terms p r o p o r t i o n a l to
Jpk" Jnk and (b~b + p ) I k " Accord ing to our c a l c u l a t i o n s the l a t e s t term a f f e c t s the BMI value by an r e l a t i v e amount which may range from I0 to 30%
One should emphasize the f a c t t ha t a l though the magne t i za t i on e f f e c t coming from the i n t r i n s i c magnet ic moments are neg lec ted there
appears a c o n t r i b u t i o n due to the neut ron system which is determined by the equat ion of mot ion of the charged p a r t i c l e s ( 5 . 9 ) .
S i m i l a r l y , s t a r t i n g w i th the c l a s s i c a l exp ress ion f o r Q2k one f i nds t h a t the lead ing term in bosons is t ha t i nvo l ved in the r e l a t i o n
( 4 . 2 ) . Of course the next o rder terms w i l l be (b+b)2k'T (b+b+)2k + +b + T T
+ (bTbTl2k w i th , : p, n and (b~bn)Zk + (b~bp l2k , (bp n) + (bnbp l2k . But these terms g ive a smal l c o n t r i b u t i o n to the t r a n s i t i o n I + ÷ 2 + g Which is cons idered here.
272
Another test for any attempt to describe the state 1 + is the de-
pendence of the MI form factor on the momentum transfer for the (e,e ' )
process. On this purpose the fo l lowing assumptions are adopted. Since
the scatter ing process takes place at backward angles, the d i f f e r e n t i a l
cross-section is mainly determined by the transverse term of the e.m.
in te rac t ion . The contr ibut ion of the i n t r i n s i c magnetic moments of nu-
cleons to the total nuclear current is small. Therefore we have res-
t r i c ted the MI operator to i t s convection part. Also we have adopted
the plane wave Born approximation. Under these circumstances the MI 32):
t rans i t i on operator describing the i ne las t i c electron scattering is
TIM(q) : ~ I dr j l ( q r ) CIMyII ~p (5.12)
where the standard notations for the spherical Bessel funct ion J l (q r )
and the vector ia l spherical functions have been used. The proton
current ~p is defined by means of the charge density pp and the velo-
c i t y ~ of the proton flow
Jp = Pp(r) v (5.13)
The charge density is taken constant inside the nuclear surface of the
proton system which is parametrized by the quadrupole coordinate p~" Taking the second order expansion of TIM (5.12) in ~ , ~ and quantizing
the resul t by means of the re lat ions (4.9) and (5.9) one obtains
TIM(q ) : e FMJI(qRo) (5.14)
where F k is defined by (5.11) and R o
form factor IF~g(q)l 2 is defined by:
is the nuclear radius. The magnetic
2 Fig 2 {<o~ IT (q)IIi+>] 2 e I (q)1 = I I (5.15)
and w i l l be compared with the corresponding experimental data:
I~9(q112 z 2 (~+ tg2~) do d~ exp = T ~ (~ ) / (d -~ )Mot t (5.16)
where a denotes the scatter ing angle.
In order to account for the renormalization of the momentum transfer
due to the Coulomb in te rac t ion , the theoret ical form factor IFTg(q)I 2
w i l l be represented as a funct ion of the e f fec t ive momentum transfer
273
q e f f q + ~m__ 6 = Ro s i n
Where m denotes the f i n e s t r u c t u r e c o n s t a n t °
( 5 .17 )
6. NUMERICAL APPLICATIONS AND DISCUSSIONS
The f o r m a l i s m wh ich was d e s c r i b e d be fo re has been a p p l i e d to 1 5 6 ' 1 5 8 ' 1 6 0 G d . E x p e r i m e n t a l data were taken from r e f . 33 f o r g round ,
beta and gamma bands and from r e f s . 2,3 f o r the M1 s t a t e 1 + .
The e x c i t a t i o n e n e r g i e s are d e s c r i b e d by the r e l a t i o n ( 3 . 1 4 ) .
The parameters A I , A 2, A 3, A 4 and d have been de te rm ined by the f o l -
l ow ing p r o c e d u r e . For a g i ven va lue of d we have f i x e d the r e m a i n i n g
+ I0~ 2 + + f o u r parameters by f i t t i n g the e n e r g i e s o f the s t a t e s 2g, ~ ' y , O B.
S ince the energy spac ings i n the B-band are ve ry s e n s i t i v e to the
change o f d, we have f i x e d d to o b t a i n an o v e r a l l agreement i n the
band. The r e s u l t i n g parameters are g iven in Tab le 1. The e n e r g i e s f o r 156
Gd are v i s u a l i z e d in f i g s . I , 2. The agreement w i t h e x p e r i m e n t wh ich
has been o b t a i n e d f o r 158Gd and 160Gd is o f the same q u a l i t y as t h a t
f o r 156Gd. In the r i g h t upper c o r n e r o f f i g . i we have p l o t t e d the %
e n e r g i e s of the y band.
Tab le 1. The parameters A. (keV) and p i n v o l v e d in the H a m i l t o n i a n I
(3.12) and fixed by fitting the data for ~, I0 +g, 2 +Y, O~ .+
p = d v~Z A I A 2 A 3 A 4
156Gd 3.2195 915.4013 -182 .2986 768.0178 0.5706
158Gd 3.598 940.5615 -191 .7085 560.4169 2.0937
160 Gd 3.826 892.353 44.2469 425.7166 1.588
I t i s i n t e r e s t i n g to see what i s the c o n t r i b u t i o n of each term
of H on the e x c i t a t i o n e n e r g i e s of the s i x bands. The e n e r g i e s o f E~ g)
and E~ Y) are de te rm ined e x c l u s i v e l y by the terms hav i ng the s t r e n g h t s
A I , A 2 and A 4. The term m u T t i p l i e d by ~ (A 1 + A2) c o n t r i b u t e s to
274
7" ZSB02
IS6Gd 6" 21,Z2 Z
5"' 22113.1 10 ~ 2219.91( ~ 22311.6
C 21(,/' 9 ~. 206e5
2 ' - - - 1995.
6" 18~,B..2 O + 18e, Z ~ 1el,96 ~ 1797.0
g-bQnd 6+ I¢#.3.1 6"- 16243
6" ~S~0.3 6* ~5/'Z.S 5. 1507. S ~ IL,~ ~I 3
I0" I/,16 I(~ IZ' I(P' 1297a ~ 1292 2 /'~ 135S:3 4+" 134.26
3* 12/'8. 3% 123G.
I ~,5.1 ir 9611, ~ I0/'9 ~ O" 10,I, 9/' [X.~. T h .
LExp. Th., ~bQ~nn d
6" s8/',2 i" se~'6 p : 3.;~195
AI= 91S ~-013 k e V
4" 2B8.Z ~+, 289~. A z = - I B 2 2 9 8 6 k e V
A3= ~ 8 . 0 1 } B keV
z + 89 . Ig, A~: 0.5]'06 key
o* ["~-. 0"" Th,
F i g . I The e x p e r i m e n t a l ( E x p ) and T h e o r e t i c a l (Th ) e x c i t a t i o n s e n e r g i e s f o r g r o u n d , b e t a and gamma b a n d s .
41t31 9'
IS6Gd
3?99 ? l " /*. 3715.8
2" 3)039
I0~ 351a i 350 5
6 + 3398 5
32321 3" O? ]tel. S
3015 30~5 } ~" 315/' B
Exp. ;ZllBO ? 6°" 2913t
e ° S*" 28~3 6
3" 25B2 2666.9
~" ?SSl
l h
10 ~ 2172.8 Th
Fig. 2 The excitation energies corresponding to the model state , ~ ( I ) , ( I ) ( I ) ( ] e f t ) ana ~JM ~ r l g h t ) . The e n e r g y o f t h e s t a t e ~IM ~JM
is to be compared to the e x p e r i m e n t a l (Exp) data f o r I +
275
Ej I )~ by less than i00 keV i f J is odd but by about 1MeV f o r J -even
( i t s h i f t s them up) . The presence of t h i s term a f f e c t s ~ !Y) by an amount
of 300 keY. The A 3 term a f f e c t s m a i n l y the e n e r g i e s E~, ~ J l ) and E~ I ) + ~ 160Gd . (J = even ) . For example 2~ and I + i n are Dressed down due to t h i s
term by 927 and 666 keV, r e s p e c t i v e l y w h i l e the h igh sp in s t a t e s i0~
i0 + are c o r r e c t e d in the same d i r e c t i o n as the l ower sp in s t a t e s by
1018 and 718 keV,, r e s p e c t i v e l y . I t is wor th m e n t i o n i n g t h a t A 3 has no
on Ej I) w i t h J = odd but i t pushes down the 2 + s t a t e o f t h i s , i n f l u e n c e
band by about 70 keV and the 10 + s t a t e by 1800 keV. I t shou ld be men-
t i o n e d t h a t t h i s term causes a n o n v a n i s h i n g o f f - d i a g o n a l m.e.
<m MIHI~ j I n c l u d i n g t h i s , the ene rg i es of the B and the ~ bands
are a f f e c t e d by few keV (~ I 0 ) . In f a c t , t h i s j u s t i f i e s the app rox ima-
t i o n we made f o r the e n e r g i e s E~ and E Y)
C o n c l u d i n g , the major c o n t r i b u t i o n to E{ I ) ' comes from the harmo-
+ N But t h i s s u p p o r t s once aga in our cho i ce f o r the model n i c p a r t ~ n '~+
. The I band shows a s t a g g e r i n g e f f e c t ( J , J + l ) . Th is i s a s t a t e s m M
r e m i n i s c e n c e of the v i b r a t i o n a l s t r u c t u r e o f these s t a t e s . Moreover due
to the s t r o n g i n f l u e n c e of the A 3 term on the energy of ~+ t h i s s t a t e s
l i e s below the s t a t e i + .
Now l e t me d i s c u s s the p r e d i c t i o n s f o r the e.m. p r o p e r t i e s of the % s t a t e I + and I + c o l l e c t e d i n Tab le 2. The f i r s t number l i s t e d i n the
r o t second column f o r B (MI ; O~ ~ i + ) , was o b t a i n e d by u s i n g / t h e the M1 t r a n -
s i t i o n o p e r a t o r the e x p r e s s i o n ( 5 . 1 ) w i t h gp = 1 and gn = O. Th i s cho i ce
f o r the gy romagne t i c f a c t o r s gp and gn i s suggested by the rough p i c -
t u re t h a t the MI t r a n s i t i o n i s caused e x c l u s i v e l y by the mot ion of the
charged p a r t i c l e s . However the r e l a t i o n s (5 .10 ) and ( 5 .11 ) i n d i c a t e
t h a t the p r o t o n - n e u t r o n c o u p l i n g induces not o n l y a n o n v a n i s h i n g g y r o -
magnet i c f a c t o r f o r the n e u t r o n system but a l so some a d d i t i o n a l e f f e c t s
~ + + nbp) due to the terms (b b p ) i k , ( b b ) i k , ( b p b n ) i k , (b Ik as w e l l as to the terms c o u p l i n g the a n g u l a r momentum to s c a l a r o p e r a t o r s . In r e f . 2 9
we have proved t h a t i f one keeps gn = O, a c o n s i s t e n t d e s c r i p t i o n f o r
the gy romagne t i c f a c t o r o f the ground band and f o r the B(MI) va lues
2 + 2 + = 0.8 (and + i s o b t a i n e d f o r gp c h a r a c t e r i z i n g the t r a n s i t i o n Y g
not gp = i ) . Indeed us ing the r e l a t i o n ( 5 . 7 ) f o r gc one f i n d s
gc = ½(gp + gn ) . The f a c t o r ½ i s on l y o b t a i n e d when the d e f o r m a t i o n
Parameters (d) f o r the p ro ton and neu t ron systems are e q u a l . Tak ing
Z = 0 one gets gp = 0.8 Th i s p rocedure gc = ~ = 0,4 and keeping gn
0 + Y i e l d s f o r B(M1; g ~ i +) the second va lue f rom Tab le 2. Us ing these
2~) 156Gd. • ÷ f o r Values f o r gp and gn we have c a l c u l a t e d B(MI , 2 + Y
Ta
ble
2
The
p
red
icti
on
s
for
the
B
MI
va
lue
an
d th
e
rele
va
nt
MI
and
E2
bra
nc
hin
g
rati
os
a
re
com
pa
red
wit
h
the
e
xp
eri
me
nta
l d
ata
: a
) R
ef.
3
; b)
R
ef.
2
; c)
R
ef.
2
bu
t th
e
BM
lt
sum
s a
re
co
ns
i-
de
red
. T
he
ore
tic
al
res
ult
s
(Th
) fo
r B
MI~
c
orr
es
po
nd
to
d)
g
p=
l,
gn=
O;
e)
gp
=O
.8,
gn=
O;
f)
Eq
ua
tio
ns
(5
.10
) w
ith
k
giv
en
by
(6
.6);
q
) k
giv
en
by
(6
.7).
A
lso
th
e
res
ult
s
for
the
M
I
tra
ns
itio
n
I +
÷ +
0 B
, a
re
lis
ted
. T
his
an
d th
e
bra
nc
hin
g
rati
os
a
re
ca
lcu
late
d
by
mea
ns
of
(5.1
).
156G
d
158G
d
160G
d
B(MI
; 0 +
÷ 1 +)
(~)
Exp
Th
4)
1.3+
0.2
3.15
1 d)
b)
2.01
7 e)
1.5+
0.3
1.24
6 f)
2.15
5 g)
a)
4.14
4 d)
1.4+
0.3
2.65
2 e)
c)
1.45
8 f)
2.3+
0.5
3.51
4 g)
4.8
d)
3.07
e)
c)
1.80
4 f)
2.3+
0.6
5.53
4 g)
B(MI
; 1 +
+ 2g
)
B(MI ;
1
+ 0
Exp b)
0.7
+0
.6
0.45
+0.0
4
Th
0.55
6
0.54
3
0.53
8
B(M
1;
1+
÷ 0~!
B(HI
; 1 +
÷ 2$
)
9.68
4
12.6
7
B(E2
; 1 +
÷ 2
;)
B(E2
; 2 + g
+ 0g
)
0.11
34
0.0
93
6
0.71
+0.1
4
%
+
B(M
1;
1 + +
0~)
(~)
0.0
29
0
0.0
33
2
13
.96
0.084
0.0227
277
F u r t h e r , t a k i n g f o r B(E2", 2 +Y ~ 2~) i t s e x p e r i m e n t a l va lue one ob ta i ns
fo r the m i x i n 9 r a t i o the r e s u l t : ~ = - 10.1. But t h i s agrees q u i t e w e l l +2.6
With the e x p e r i m e n t a l data repo r ted fn r e f . 33, i . e . 6 .5_7 .9 . A b e t t e r
agreement is p o s s i b l e i f f o r gp - Cn a lower va lue was taken. In f a c t
these r e s u l t s might be a l so ob ta ined f o r a set (gp, gn) w i t h gn # 0
and gp + gn = 0 .8 . + I + In t a b l e 2 we a lso g i ve the r e s u l t s f o r B(MI; Og + ) ob ta ined
,by means of the r e l a t i o n ( 5 . 1 0 ) . From (5 .11 ) one notes t h a t the M1 t r a n -
s i t i o n o p e r a t o r depends on the cons tan t k d e f i n i n g the canon i ca l t r a n s -
f o rma t i on ( 4 . 9 ) . Here, I s h a l l i n d i c a t e two ways of f i x i n g k. F i r s t we
Sha l l i d e n t i f y the e x c i t a t i o n energ ies of the ground band s t a t e s g iven
by the p resen t model w i t h those co r respond ing to the r o t a t i o n a l ene r -
g ies p r e d i c t e d by the l i q u i d drop model. Since the va lues f o r the para-
meter p(=dv~Z) ob ta ined from the energy a n a l y s i s f o r Gd i so topes be-
long to the a s y m p t o t i c reg ion of the o v e r l a p i n t e g r a l s de f i ned in the
CSM, one can use the r e s u l t s f o r the r o t a t i o n a l l i m i t of the ground
band energ ies
= ' . . . . + A 4 ] J ( j + Z ) ( 6 . 1 )
On the o the r hand the l i q u i d drop p r e d i c t s
E rOt : J J ~ (6 .2 ) J 6B6 ~
where B is the mass parameter de f i ned by
B : ~ MAR2o ' RO = 1.2 A I / 3 [ fm] (6 .3 )
With M and A the nucleon mass and the atomic mass number° Equat ing the
exp ress ions ( 6 . 1 ) , (6 .2 ) f o r the r o t a t i o n a l ene rg ies one ob ta i ns f o r
the e q u i l i b r i u m de fo rma t i on
2 = ~ ~2 AI + A2 - I ~o ~ ~ A-5 /3 [ - - - - - T + A 4] ( 6 .4 )
6p
In the g e n e r a l i z e d CSM, the de fo rma t i on of the whole system is equal to
tha t of the p ro ton and neut ron systems which i s determined by:
2 : <¢g[ S ~ ~ l~g> , T : p,n (6 .5 ) ~T,O T~ Tp
The e q u i l i b r i u m de fo rma t i on 6~,o g iven by (6 .5 ) i n c o r p o r a t e s a spur ious
C o n t r i b u t i o n due to the z e r o p o i n t m o t i o n . By e l i m i n a t i n g t h i s c o n t r i -
278
b u t i o n one f i n d s
k~ ° = p , p = d~-2- (6 .6 )
C o n c l u d i n g k can be de te rm ined by means o f ( 6 . 4 ) and ( 6 . 6 ) . I n s e r t i n g
t h i s va lue of k i n ( 5 . 10 ) one o b t a i n s f o r BMI(O + I + g + ) ( 5 . 5 ) the t h i r d
number o f the second column of the Tab le 2.
I t i s w e l l known t h a t ~o g i ven by ( 6 . 4 ) does not c o i n c i d e w i t h
t h a t wh ich f i t s the e x p e r i m e n t a l data f o r B(E2; 2 + 0 + g ~ g ) . U s u a l l y , the
l a t e r one is taken as the e x p e r i m e n t a l va lue f o r the d e f o r m a t i o n para-
meter Bex p. A l t e r n a t i v e l y , we may suppose t h a t k i s j u s t the s c a l i n g
f a c t o r f o r ~exp:
kBexp = ~o (6 .7 )
To t h i s k co r respond the BMI va lues l i s t e d i n the f o u r t h p lace in
Tab le 2.
I t is i n t e r e s t i n g to see how does the BMI va lue f o r the t r a n s i -
0 + i + t i o n ÷ depend on the d e f o r m a t i o n pa ramete r . Of c o u r s e , the answer g is f u n c t i o n o f the e x p r e s s i o n one takes f o r the t r a n s i t i o n o p e r a t o r .
Tak ing the eq. ( 5 . 1 ) as a d e f i n i t i o n f o r the MI t r a n s i t i o n o p e r a t o r and
p l o t t i n g B (MI ; O~ ~ i +) as f u n c t i o n of p(=dv~2) one o b t a i n s a pa rabo la
o f the t ype Cp 2. T h e r e f o r e , i n the v i b r a t i o n a l l i m i t the MI t r a n s i t i o n
i s f o r b i d d e n .
However as we have seen be fo re the f o r m u l a (5 .10 ) i s more t r u s t -
f u l s i nce i t s d e r i v a t i o n is based on the e q u a t i o n of mot ion of the qua-
d r u p o l e degrees of freedom wh ich i n f a c t takes care of the s t r u c t u r e
of the model H a m i l t o n i a n . In the v i b r a t i o n a l l i m i t the r e l a t i o n y i e l d s
a n o n v a n i s h i n g r e s u l t :
Z2R 4 M 2 2 (a2 - A I ) 2 0 + i + o B(M1; ÷ ) =
g 1 . 0 ~4 k 4 ~N ' (6.8)
and where. A I , A 2 are the s t r e n g t h s of the harmon ic terms Np n
Npn + Nnp r e s p e c t i v e l y , i n v o l v e d in the model H a m i l t o n i a n ( 3 . 1 2 ) .
A rough e s t i m a t i o n is o b t a i n e d by t a k i n g the data f o r 156Gd. In t h i s 2
way one o b t a i n s f o r BMI the va lue 0.99 ~N i f k i s de te rm ined by means
of ( 6 .~ ) and ( 6 . 6 ) w h i l e f o r k d e f i n e d by ( 6 . 4 ) and ( 6 . 7 ) the va lue
0.56 UN is o b t a i n e d . C e r t a i n l y the r e s u l t depends s t r o n g l y on the f ac - 2
t o r (A 2 - A I ) wh ich i n the v i b r a t i o n a l l i m i t m igh t be by a f a c t o r
r a n g i n g from 4 to I0 s m a l l e r than t h a t f o r 156Gd.
279
In t a b l e 1, the b r a n c h i n g r a t i o s of the MI t r a n s i t i o n s
(I + 2 +g)/(l + ~ , i + 0 + 2¥)+ ÷ + 0 ) ( + g ) / ( 1 + ÷ c o r r e s p o n d i n g to the MI
t r a n s i t i o n o p e r a t o r ( 5 . 1 ) are g i v e n . A l so we have l i s t e d the v a l u e s f o r
• + . ~ ~ 0 + B(MI, I + 0 +) The MI t r a n s i t i o n I + i s f o r b i d d e n . Aga in t h i s i s g not t r u e i f we take f o r the t r a n s i t i o n o p e r a t o r the e x p r e s s i o n ( 5 . 1 0 ) .
C o n s i d e r i n g an harmon ic s t r u c t u r e f o r the E2 t r a n s i t i o n o p e r a t o r one
Obta ins f o r B(E2; 1 + 2g) the va l ue o f about 900 e2fm 4.
Concern ing the MI form f a c t o r ( 5 . 15 ) we have p l o t t e d i t i n f i g . 3 ,
f o r the case of 156Gd, as a f u n c t i o n o f the e f f e c t i v e momentum t r a n s f e r
de f i ned by ( 5 . 1 7 ) . A l so the e x p e r i m e n t a l form f a c t o r ( 5 . 1 6 ) and the
e r r o r bars are drawn. For a b e t t e r p r e s e n t a t i o n the t h e o r e t i c a l r e s u l t s
are m u l t i p l i e d by a f a c t o r 2. The c o n s t a n t k i n v o l v e d in ( 5 . 11 ) was de-
te rmined by the r e l a t i o n s ( 6 . 4 ) and ( 6 . 6 ) . Chos ing the o t h e r way o f
f i x i n g k i . e . by means o f ( 6 . 4 ) and ( 6 . 7 ) , one o b t a i n s an enhancement
of the form f a c t o r by a f a c t o r of 1 .74 . C o n c l u d i n g , t a k i n g f o r k de-
f i n e d by ( 4 . 9 ) , the va l ue Bo/Bexp, (where Bo i s g i ven by (6 .4 ) and Bexp
is the e x p e r i m e n t a l va l ue f o r the e q u i l i b r i u m d e f o r m a t i o n ) both the
Shape o f the MI form f a c t o r and the BMI va l ue o f the t r a n s i t i o n 0 + + i + g are ve ry w e l l d e s c r i b e d .
/~4L
156
7B B OS 10 -
# rrrfm'O
F ig . 3 The t h e o r e t i c a l HI form f a c t o r , m , iF~g(q) l 2 ve rsus the e f f e c -
t i v e momentum t r a n s f e r ( 5 . 1 7 ) . The e x p e r i m e n t a l data are
also represented by means of (5.16).
280
7. SUMMARY AND CONCLUSIONS
Let me now refresh the main information I wanted to transmit
to you by my ta l k .
A. The in t roductory part of my lecture stressed on the fact that
the subject is r e l a t i v e l y new and despite the big number of the
attempts and claims, is s t i l l far away from the stage of being closed.
B. Since the formalism I have used for the MI states is an exten-
sion of the CSM, I have spent a while for describing the underlying
ideas of CSM as well as some of i t s outstanding performances.
C. The main part of my lecture was devoted to the descr ipt ion of
the extension of the CSM to the composite system of protons and neutronS'
F i r s t a model space for s ix c o l l e c t i v e bands is constructed and
then an e f f ec t i ve quadrupole boson and F-spin noninvar iant Hamiltonian
is considered.
The parameters involved in the model Hamiltonian are determined
by f i t t i n g some exc i ta t i on eneroies from ground, beta and gamma bands.
Therefore the remaining energies are predict ions in our model. Note that
we keep the conventional names for the f i r s t two excited bands although
i t is not possible to introduce a global set of deformations B and
Special emphasis are addressed to the states I +. In the v i b ra t i o "
b + b + l nal l i m i t the state i + is a two phonon state ( n p'iM I0> while in the
ro ta t i ona l l i m i t is a K = 1 state depending on a l l the s ix angle var ia-
bles (m ' ,e ,~ " ,~ in t ) and not only on four var iables (e, mint) as the
TRM predicts.
The M1 t r ans i t i on 0 + ~ 1 + has been calculated by successively g using two expressions for the t r a n s i t i o n operator:
i ) One is given by the r e l a t i o n (5.1) . The ca lcu lat ions from Table 2
correspond to (gp, gn) equal to ( i . , 0.) and (0.8, 0.) respect ive ly .
The l a te r set induces for the gyromagnetic factor of the ground band
a value of 0.4 while for the mixing r a t i o of the t r ans i t i on 2 + ~ 2 + Y g
the value - 10.1. But these agree quite well to the experimental data.
The M1 t r ans i t i on 0 + 1 + g + has a parabol ic dependence on the deformation
parameter. That means that this t r a n s i t i o n is forbidden in the v ibra-
t i ona l l i m i t . A~so, such a structure for M I (5.1) does not allow for
the t r a n s i t i o n 1 + ÷ 0~. The predict ions for the ra t ios of the BM1 va-
- i + ÷ 2 + lues ( g ) / ( l + + Og) agree very wel l to the experimental data.
281
i i ) The expression of the M1 t rans i t i on operator should be consistent
With both the structure of the model Hamiltonian and the M1 form factor .
Having th is aim in mind we have derived a boson representation for MI
OPerator (5.10) by quantizing i t s c lassical counterpart by means of
(4.9) and (5.9). The t r ans i t i on operator Tlk describing the (e ,e ' )
Process has been treated in a s imi lar way. The MI operator involves
a parameter k def ining the canonical transformation (4.9). We have in-
dicated two ways of f i x i ng k. A very good descript ion of both the
~(M1; 0 + -~ I + g ) and the M1 form f a c t o r is ob ta ined f o r k = ~o/Bexp w i th
Bo g iven by ( 6 . 4 ) ,
The r e l a t i o n (5 .11) i n d i c a t e s t ha t the MI ope ra to r i n v o l v e s a
nonvanish ing value f o r gn and a lso a d d i t i o n a l terms l i k e (b+b+) l k,
( bnbp ) l k , (b+bp) i k , ( bpbn ) l k and some o thers coup l ing Jn and" "~Jp to a Scalar two boson o p e r a t o r . I n t e r p r e t i n g the c o e f f i c i e n t s of Jp and Jn as the gyromagnet ic f a c t o r s gp and gn r e s p e c t i v e l y one f i nds
9n % 0.17+ gp+. Due to the term (bnbp) Ik the v i b r a t i o n a l l i m i t o f B(MI; Og ÷ ) is d i f f e ren t from zero. This is a very important d i f f e -
rence between our model and the other ones. Indeed, in the v ibra t iona l
l im i t a l l of them predict vanishing values for both the energy El+
and the reduced t rans i t i on p robab i l i t y B(M1; 0 +g ÷ I +).
The state ~+ is one of several dipole states characterized by
0 + Weak (or forbidden) MI t rans i t i on to g and ly ing lower in energy than
the MI state we have investigated here. They can be described in our
formalism by enlarging the basis (3.1 - 3.6). For example, two of them
~ay be associated to the funct ions:
~JM ~ (bn l + b p l ) ] ~ g
( I") = N~I" )P~I (b~ b+ mJM p)31 ~g
But t h i s is c o n s i s t e n t w i th the data f o r 156Gd showing the presence of
Several s t a t es I + l y i n g below the MI s t a t e . But before t h i n k i n g of
a q u a n t i t a t i v e comparison w i th the exper imen t , more data concern ing the e.m. p r o p e r t i e s of these s t a t es are necessary .
As a f i n a l conc lus ion one may say t h a t the gene ra l i zed CSM PrOvides a f u l l and r e a l i s t i c d e s c r i p t i o n f o r the magnet ic s t a te i +.
A l l i t s p r o p e r t i e s are q u a n t i t a t i v e l y ob ta ined a l though there is no ad jus tab le parameter .
282
A c k n o w l e d g e m e n t : ~ o s t o f t h e r e s u l t s eoneermlng th~ e n e r g y and
t h e BMI v a l u e of I I have o b t a i n e d i n c o l l a b o r a t i o n wt~h P r o f .
V a l e n t i n Ceausescu a~d P r o f . Amend F a ~ s l e ~ Zn r ~ f . 28, 29. The MI form
f a c t o r was d e s c r i b e d i n r e f . 31 wher~ I h~ve b e n e f i t e d of t h e coopera ~
t i o n w r t h Drd. I . Ursu and Drd. D. D e l i o n . I would ~ k e to t h a n k a l~
o f them for t h e l r v a l u a b l e e o n t r l b u t i o n .
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i . D.Bohle, A .R ich te r , W.Stef fen, A .E .L .D iepe r i nk , N.Lo lud ice , F.Palumbo and O.Schol ten, Phys .Le t t . 137B (1983) 27.
2. U.E.P.Berg, C.B l~ss ing, J . D r e x l e r , R .D .He i l , U .Kne isse l , W.Naatz, R.Ratzek, S.Schennach, R.Stock, T.Weber, H.Wicker t , Phys .Le t t . 149B (1984) 59.
3. D.Bohle, G.K~chler, A .R ich te r and W.Stef fen, Phys.Lett .148B(1984)260'
4. A .Faess le r , Nucl.Phys. 85 (1966) 653. 5. A.Bohr, Mat.Fys.Medd.Dan. Vid. Slesk 26, No.14 (1952)
A.Bohr, B.R.Mot te lson, ib id 27, No. 16 (1953). 6. V.Maruhn-Rezwani, W.Greiner and J.A.Maruhn, Phys .Le t t . 57B(1975)lOg" 7. A .Faess le r , Z.Bochnacki and R.Nojarov, p r e p r i n t T~bingen, 1985. 8. N. Lo ludice and F.Palumbo, Phys.Rev.Let t . 74(1978) 1046,
Nucl.Phys. A326 (1979) 193. G. De Franceschi , F.Palumbo, N. Lo l ud i ce , Phys.Rev. C29 (1984)1496
9. M.Goldhaber and E . T e l l e r , Phys.Rev. 74 (1948) 1046.
i0. F . l a c h e l l o , Nucl.Phys.A358(1981) 89c, Phys.Rev.Let t .53(1984)1427. A E .L .D ieper ink , P rog r .Par t .Nuc l .Phys . 9(1983) 121.
11 T Suzuki and D.J. Rowe, Nucl.,Phys. A289 (1977) 461. 12 E L ippa r i n i and S . S t r i n g a r i , Phys .Le t t . 130B(1983) 139. 13 D R.Bes and R .A .Brog l i a , Phys .Le t t . 137B(1984) 141.
14 H Kurasawa, ToSuzuki, Phys .Le t t . 144B (1984) 151. 15 R R .H i l t on , Z.Phys. A316 (1984) 121.
16 R K.Shel ine, Rev.Mod.Phys. 32 (1960)1. 17 M Sakai, Nucl .Phys. A104 (1967) 301. 18 M Finger et a l . , Nucl.Phys. A188 (1972) 369.
19 A A.Raduta, V.Ceausescu, A.Gheorghe and R .M.Dre i z le r , P rep r in t FT-190 (1980)IPNE-Bucharest; Phys .Le t t . 99B (1981) 444; Nucl .Phys. A381 (1982) 253.
20. A.A.Raduta, S.Stoica and N.Sandulescu, Rev. Roum. Phys. 29, 1(1984)56'
21. A.A.Raduta and C.Sabac, Ann. Phys. 148 (1983) I .
22. A.A.Raduta and N.Sandulescu, P rep r in t FT-269 (1985)IPNE-Bucharest; Revue Roum. Phys. 31, 8 (1986) 765.
23. A.A.Raduta, Rev. Reum.Phys. 28, 3 (1983) 195.
283
24
25
26 27
28 29
30
A.A.Raduta
A.A.Raduta
A.A.Raduta
A.A.Raduta A.A.Raduta A.A.Raduta
A.Faess le r , Th.K~ppe~ and C.Lima, Z.Phys. A312 (1983)23.
C.Lima, A .Faess le r , Phys .Le t t . 121B (1983) I .
C.Lima, A .Faess le r , Z.Phys. #3!3(1983) 69.
S .S to ica , to be publ ished. V.Ceausescu, A .Faess le r , Rev.Roum.Phys. 31, 7(1986) 649. A ,Faess le r , V.Ceausescu, to appear in Phys.Rev. C.
M.E.Rose, Elementary Theory of Angular Momentum, John Wiley & Sons, New York, 1957.
31 . A.A.Raduta, l . l . U r s u , D.S.Del ion, Rev.Roum. Phys. 31, 8(1986)799. 32. T. de Forest , J r . and J.D.Walecka, Adv.Phys. 15 (1966) I .
33. P.O.Lipas, J.Kumpulainen, E.Hammaren, T.Honkaranta, M.Finger, T . l .K rakova , l .Prochazka and J .Ferenze i , Phys. Scr. 27 (1983) 8.
34. A,B~ck l in et a l . , Nucl.Phys. A380 (1982) 189.
35. S . P i t t e l , J .Duke lsky , R.P.J.Perazzo and H.M.Sof ia, Phys .Le t t . 144B (1984) 145.
36. l ,Novose lsky , communication of Dubrovnik Conference, June, 1986.
P R O T O N - N E U T R O N V I B R A T I O N S I N T H E C O L L E C T I V E M O D E L I
M. S. M. Nour El-Din, 2 S. G. Rohozirlski, s j . A. M a r u h n , 4 and W. Gre iner Ins t i tu t fiir Theoret ische Physik
Universit~/t Frankfur t D6000 Frankfur t am Main, West G e rman y
1 I n t r o d u c t i o n
The idea that a partial separation of proton and neutron shape degrees of freedom in nuclei might influence the magnetic properties came quite early in the development of the collective model. Faessler and Greiner [Fa64,Gr65] explained the deviations of g-factors from the constant value Z / A by the difference in pairing strength of protons and neutrons. Then while Greiner [Gr66] extended these ideas to the description of E 2 / M 1 mixing ratios, Faessler [Fa66] pursued the idea of excited states based on relative vibrations of the protons with respect to the neutrons, i. e. the quadrupole isovector states. Unfortunately, his estimate for the energy of these vibrations placed them much higher than what seems plausible now on the basis of recent experiments [Ri83,Bo84]. Later, V. Maruhn-Rezwani and coworkers [Ma75] extended the model to link it to the generalized collective model as developed by Gneuss and Greiner [GnT1], and Seiwert [Se81] applied it to high spin states. Only recently has the experimental evidence for quadrupole isovector states prompted renewed investigation of the description of these states within the model by Rohozifiski et al. [Ro85]
2 G e n e r a l F r a m e w o r k
Extending the usual definition of the quadrupole deformation tensor a2v to define separate surfaces for protons and neutrons,
R/(0, ¢) = Ro, (1 + ~ ~Y2"~(0, ~) ) , i E { p , n } (1) P
the collective coordinates a p and a n are introduced. Because these deformations should be coupled strongly by the symmetry energy, it is advantageous to introduce instead the average deformation a and the relative separation ~ according to
Bpa p + Bna a a B ' ( = u p - a " ' (2)
where Bp and Bn are the mass parameters for proton and neutron collective motion, respectively, and B = Bp + Bo. The associated transformation of the conjugate momenta ~" and rl (the latter conjugate to ~) is
BpTc n + Bnr p r = ~ r P + ~ r " , O (3)
B
The Hamiltonian of the system is decomposed according to
H = Hi(a) + H2(~) + Hi(a , ~). (4)
J Work supported by the Bundesministerium fiir Forschung und Technologie and by the Gesellsehaft ffir Schwerionenforse hung.
~Present address: Physics Department, Faculty of Science, Benha University, Benha, Egypt.
3Fellow of the Alexander yon Humboldt Foundation. Permanent address: Institute for Theoretical Physics, University of Warsaw, Poland.
qnvi ted speaker.
285
Computing the change in symmetry energy arising from small nonzero values of ~, it was found [Ma75a] that it can be approximated quite well by a parabola, so that a natural assumption for H2(~) appears to be that of a harmonic oscillator:
= × ° + × , I ° .
For the first applications of this model it was assumed that, according to Faessler [Fa66], the vibrational energy described by the Hamiltonian (5) would be of the order of 15MeV. In this case the eigenstates of the total Hamiltonian (4) consist of well-separated bands distinguished by the vibrational quantum number n~ and corresponding essentially to the coupling of the usual low-energy vibrational and rotational states to these ~-vibrations. The observed low-energy spectrum in particular is then produced by coupling the eigenstates of Hx(a) to the ground state in ~.
For the interaction energy also in this case the approximation of small vibrations in ~¢ can be made, leading to
g , ( ~ , ~) = C , [ ~ × ~10 + C2[[~ × ~l ~ × ~l ° + . . . (6)
If we are interested in transitions between the low-lying states only, this coupling potential can be treated in perturbation theory. It couples those states only to the n,, = l-bands, and calculating the matrix elements of the E2 and M1 operators between perturbed states I ¢ > based on the coupled states I Ja >1 n~ = 0 > yields for the magnetic moment and transition operator
< ¢ ] M11 ¢ ' > = ix/Td +C¢2Baj<Jal[axr l* lJa>
and for the electric transitions
< ¢ l E2 [ ¢, > = , 0 R ~ [ ( 1 C, Bn)
So both magnetic and electric operator can be replaced, for transitions between the low-lying levels, by effective operators which take into account the presence of the ~¢-mode. The magnetic operator of equation (7) is identical to the one originally proposed by Greiner [Gr66].
The first term of eq. (7) contributes to the g-factor, while the second term modifies the g-factor and gives rise to Ml-transitions. The theory in this form contains the three parameters G1/C,, C2/C~, and Bp/B beyond those contained in the usual low-energy Hamiltonian Hi(a).
The theory, coupled to the Gneuss-Greiner Hamiltonian [GnTl], was applied to some Osmium and Platinum isotopes [Ma75], yielding good agreement with experiment both for the mixing ratios and the g-factors. Here, however, we want to present the new calculations of M. S. M. Nour El-Din, in which the structure of the operators given in eqs. (7) and (8) is used within the rotation-vibration model.
3 Application in the Rotatlon-Vibration Model
In the spirit of the rotation-vibration model it appears appropriate to take into account effects of the ~-vibrations only to lowest order. Following the treatment of Seiwert et al. [Se81], we consider the Potential energy
The potential takes its minimum for
Thus the core is forced to oscillate with a constant ratio of ~ to a. and one can introduce a new coordinate u . by ap = uj, cos 6, ~j, = ul, sin 6-. Now the calculation proceeds as follows: the Hamiltonian is fitted in
286
the normal way to the energy levels, but u replaces a everywhere (which does not change the energies). Then trastsitioa probabilities and g-factors are obtained by expressing a t' in terms of u, which will yield expressions similar to those in eqs. (7) and (8), but with the eoet~cients functions of the parameter b'. The matrix elements of these can then be evaluated between the rotation-vibration eigenstates.
In this procedure the parameters of the Hamiltonian are fitted to the spectra only, and then as the single new parameter is used to enhance overall agreement with g-factors, mixing ratios, and B(E2)-values. Here we can present a small, but characteristic selection of results only. only.
Table I shows a comparison of E2/M1 mixing ratios with experimental data and with other models. Apparently the overall agreement is quite competitive; it is instructive that most models still have trouble with the sign in some cases. In the present model the sign appears in some cases to be in conflict with B(E2) vahles: the B(E2) values could be improved vastly by accepting a wrong sign in some ~(E2/M1) value. However, one should not forget that signs of mixing ratios are notoriously di~cult to measure and signs have changed in the past.
Isotope
152SI n
154Gd
t56Gd
Tra~lsition
4 + --+ 4 +
2 ~ ~ 2~
4 + --, 4 +
2}- 2:
4 +
~(E2 /M1) Exp. Present I PPQ DNSB -9.6 -11.9 -24.0 -2.8 -6.0 -10.0 -8 -11.3 +11.0 +8 -6.1 +4.0 -9.7 -13.4 -41.0 -16.3 -4.1 -6.5 -12 .4 -6.2
+8.3 -14.5 +4.9 -10.1 +2.9 -7.6 +2.1 +69.6 -17.2 -10.1 -41.0
-4 -9.2 +14.0 -14 -18.8 +21.0
Table 1: Mixing ratios for three isotopes computed in the present model and compared to experiment [Sa84], the dynamic pairing-plus-quadrupole model [La82] DPPQ, and the dynamic Nilsson, Strutinsky, and Belyacv model [Ku79] DNSB.
Fig. 1 shows some B(E2)-values in 'S2SnI in comparison with the standard rotation-vibration model and some other theories. The improvement is quite impressive and reproduces the structure of the experimental data faithfully. In general, of course not all isotopes can be described as well, but there is always significant inlprovement over the standard rotation-vibration mode].
4 S t r o n g C o u p l i n g M o d e l a n d 1 + S t a t e s
The discovery [Ri83,Bo84] that the quadrupole isovector excitation in 156'15SGd is located at a much lower energy than expected previously IFa66] has stimulated interest also in the excited states of ~-vibrations themselves, not only in their influence on the properties of the low-energy spectrum. Rohozinski aa~d Greiner [Ro85] have investigated the properties of a Hamiltonian like (4) in the strong coupling approximation.
They keep the approximation of a harmonic oscillator for H2(~), but use a much more general interaction Hamiltonian:
~ . ( ~ ) + E{~(]?")(~') × 1~ × ~l~"l {~ × ~]~." (11) AV -VM~
'~"'~r ( l l} , •
At* A# ApM
287
152 Sm- isotope
BlE2)Bronching rat io
/"%, ~.~ /" \
/' '\ /
~.~ / \
/
0.15
0A
0.2
\ 'b
~, Exp, 13 Th.with INCD)
o l~.~,ithout lIED)
Alone o /BA
i1 " ~ I
- - I I
-. , ,4," \ \ _ ",, ,, X./ \ \
* o / Trans i t ion 0
E F_
Fig~,re 1: B(E2) branching ratios for ~S2Sm in experiment IKo82], the present theory ("without Homogeneous Charge Distribution"), the rotation-vibration model (~with HCD'), the Alaga rule [Ko821, and the IBA [Sc781. The transitions concerned are indicated on the abscissa.
With all a-dependent terms in turn being expanded in powers of a. For example,
• l v @ ~ (12) "2~1") = x ~ + ~ 2 2 t × ~12~ + . ' .
This example also indicates the indexing used later on. The intrinsic system is defined by the average shape parameter a, so that the intrinsic coordinates
are the Euler angles t91, 02, and 03 as well as the intrinsic mass deformations a0 = ~ , a2 = ~ and the real and imaginary parts of the intrinsic relative deformations:
Now the strong coupling approximation proceeds as in the case of the pure rotation-vibration model [Ei76]. The coordinates a0 and a2 are assumed to deviate only slightly from their equilibrium values 1~ ~ d 0, respectively, while the equilibrium value for the z's and p's is determined by the interaction. The Ptlrely a-dependent part of the Hamiltonian reduces exactly to the same expression as in the rotation- vibration model, H~(r/) retains its harmonic oscillator form, and the coupling part can be written as
2
k=O
288
where h ~ 0 2 1
H0(~o) - 2~k O ~ + C0(=0 - f)2 ( i s )
and
ttk(=~,y~) = -~-~ + + c~(4+y~,), i t{I,2}. 06)
It ca~ be shown that the excitations described by eqs.(15) and (16) correspond to angular momen- tum projection K = 0 and K = k respectively. On each vibrational state the low-energy part of the Hamiltonian will build rotational bands with angular momenta L = K , K + 1 ,K + 2, . . . for K ¢ 0 or L = 0, 2 , . . . for K = 0. Thus for the lowest excitations of the proton-neutron relative vibrations we end up with three rotational bands of a defimite K ~ = 0 +, 1 +, 2 + and oscillator quantum numbers (n0, nx, n2) = (1, 0, 0), (0, 1, 0), (0, 0, 1), respectively.
In this approach the interaction is hidden in the parameters Bk, Ck (k E {1, 2}) and ~', which depend in complicated ways on k and ft. In particular, a static axially symmetric relative deformation
= C = - (~x~') 2Vl/2o~7~, (,)~ - x = ) ~Co (IT) go@ &
arises mainly owing to the interaction term [c~ x ~]o. The other interaction terms serve to shift the K x = 0 +, 1 +, 2 + bands relative to each other.
4! E[MeV]
n~ -1
4t-- . . . .
4*---'- - - - 3* '," ~ g-o 2**--" K-2 n°'1
L1K. n2-1 L ni"1
--7 + --8÷--5~
_ 4+ 2 * - - 8 * - - ~÷ K-2
----5+~1 6 * K=O
_ _ Z *
n~-O
NPD
F=~ -1
[fl,f2]-[n-1,1 ]
- - 6 * _ _ 4 + ~ 5 "
- - 8 _ _ 2+~_3" _ _ 6 +"~.00÷K'2 4. , - (2n-4,2)
~.oo" (2n,O)
(7* 6"--- , J 6 4 , - - - - - - - - - / 5 ~-- z,:_2_~__3"-- j 4~_____ 0;'-- K=2 I~,=: K-O
I*K 1 (2n-4,2) L I;-2,~1
[f132]" [n,O ]
IBM-2
Figure 2: Schematic comparison of the energy spectra in well-deformed nuclei predicted in the strong coupling approximation of the present model ( " N P D ' ) and the SU(3) limit of the IBA-2 [laSl] under the assumption of equal excitation energy of the lowest 0 +, 1 +, and 2 + states in both models. All rotational bands are, in principle, infinite in the NPD, whereas they are finite in the IBA-2.
It is worth while to mention two characteristics of the 14, state in this model. Its excitation energy is equal to
E,+-- h' ~7' (is) and the reduced transition probability to the ground state is
9 2 2 ,~2 B ( M I ' 0 + -"* 1+) ~ -~]l"NgrelP V ~ (19)
289
The experimental data for the lowest 1 + state in lSeGd [Bo84] read El+ = 3075keV and B(M1,0 + 1+) = 1.3 :t= 0.2p.~,. Taking the values fl = 0.307 for the deformation [Lo70], d = 33 .7h2 /MeV for the moment of inertia [Bu76], and gr,l = I for the gyromagnetic factor, we obtain Cx = 58 .9MeV,B1 = 6"3h2/MeV. This should be contrasted with the early calculations of Faessler [Fa66], who obtained C~ ~ 2000 - 3000MeV from the symmetry energy. Recently, however, Faessler and coworkers [Fa86] have revised this value, the reason for the difference in estimates being that the symmetry energy should be evaluated for densities near the surface and not for bulk density.
Fig. 2 shows a comparison of the level spectra in this model to that of the IBA-2 [Ia81] in the 8U(3) limit. As usual, the main practical difference between the two models is that in the IBA the rotational bands are cut off, while the geometric model in principle allows arbitrary angular momenta. The principal bands are in agreement and finer details like the ordering of the multiplet K ~ = 0 +, 1 +, 2 + can be adjusted by fitting the interaction terms that can be added in both models.
References
[Bo84]
[Bu76]
[Bi761
[Fa64} [l~an6l [Pa861 [Gn71]
[Or661 [1aS1} [~o821 [~:u791 [La821
[LoTO] [MaT~ l
[Ri83{
[rto85]
{sa841 [8c78] lSe811
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D. EXTREME STATES AND DECAY MODES IN NUCLEI
GRAND UNIFIED THEORIES AND THE DOUBLE BETA-DECAY
Amand Faessler
University of T~bingen, Institute for Theoretical Physics
D-7400 T~bingen, West-Germany
ABSTRACT
Grand Unified and the superstring theories suggest that the neutrino
is a Majorana particle identical with its antiparticle. The same theo-
ries indicate that it has a small finite mass and that it also inter-
acts by a right handed current. These properties can be tested by the
neutrinoless double beta-decay. We show that due to the virtual inter-
mediate neutrino in the neutrinoless double-beta decay the classifica-
tion according to allowed, first forbidden and second forbidden tran-
sitions does not reflect anymore the strength of the transition proba-
bility. We show that the relativistic corrections to the nucleon wave
functions contribute to the leading term to the double neutrinoless
beta-decay. This term has never been calculated before. We apply the
theory to the 76Ge to 76Se transition. We use the new generation of
microsopic many body wave function of the nucleus. This approach is
called MONSTER and VAMPIR. The starting point of these nuclear struc-
ture calculation is the Hartree-Fock-Bogoliubov approach with projec-
tion on good proton and neutron number and on good angular momentum
before the variation (VAMPIR). Short range correlations between the
nucleons and the finite extention of the nucleons have been included.
We show that the matrix elements due to the relativistic recoil correC"
tions including weak magnetism are by a factor 200 larger than the
contributions calculated up to now. This yields from the lower limit
of the lifetime measured by Avignone to an upper limit for the right
handedness of the weak interaction which is two orders of magnitude
more stringent than previous results. This work yields for the neu-
trino mass m~ 2.4 eV, for the right handedness parameter n 6.8xi0 -8
and for the right handedness of the leptonic and the hadronic interac"
tion I 4.3xi0 -6.
293
INTRODUCTION
The neutrino is the only Fermion from which we do not know that it
must be different from its antiparticle. If it is indeed different
from its antiparticle, as assumed by the standard model, we say it is
a Dirac particle. If it is identical with its antiparticle, as assumed
by most of the grand unified theories and by the superstring theories,
we speak of a Majorana particle. Already in 1955 one did believe that
one has answered this question, so that the neutrino is a Dirac par-
ticle (Davis, 1955).
Davis (1955) used the neutrinos from the fission products of a reac-
tion for the following reactions:
n p + e + v (Reactor)
37Ci + ÷ 37Ar + e
(1)
if the anti-neutrino ~ is identical with the neutrino, then Davis
argued that the inverse beta-decay of 37CI where one needs a neutrino
Should be possible and one should find 37Ar. Since this reaction did
not show up, one concluded that neutrino and anti-neutrino are diffe-
rent particles. But this conclusion was done without knowing about
the parity violation in weak interaction (Lee and Yang, 1956; Wu et
al. 1957). The fact that parity is maximally violated in the weak
interaction means that one emmits a right handed neutrino in the decay
of the neutron (positive helicity) and one has to absorb for the in-
Verse beta-decay in 37C1 a left-handed (negative helicity) neutrino.
Since helecity is a good quantum number for a neutrino with mass zero
the reaction (I) is forbidden even if the neutrino is a Majorana par-
ticle that means if the neutrino is identical with the anti-neutrino.
Thus the question is again open: Is the neutrino a Dirac- particle or
a Majorana particle?
One may ask why one is just today worrying about this problem although
it is well-known since 1957. The reason is that most grand unified
theories which combine the electroweak interaction with the color for-
ces, predict that a neutrino is a Majorana particle, has a finite mass
and interacts also with a right-handed current.
294
A detailed description of grand unified theories (Langacker 1981; Stech
1980; Buras 1978) or of the superstring theories (Schmid, Grimmer and
Faessler, 1984; Schmid, Grimmer, Hammar~n and Faessler, 1985) goes
beyond this talk. But for the people not familiar with this theories
I want to give a simplyfied impression of the basic ideas: A quark
can be red, blue or yellow. But local color gauge invariance requires
that it should not matter if I rename the color differently or more
exactly, at each place and at each time I can name the three colors
as I like without changing the physics. This means that a rotation in
the three dimensional complex color space where the rotational angles
depend on position and time is not changing the physics.
(i) <!) i(~'~(~ t)) r = e ' (2)
Here ~ are the eight color Gelman matrices and ~(r,t) are eight angles
discribing the rotation in the three-dimensional complex color space.
If one neglects for a moment the complexity which comes from the heli-
city one can extend the ideas behind eq. (2) to include to the three
colors red, blue, yellow of a quark the two dimensional degrees of
freedom "neutrino" and "electron". We now request in a five dimensio-
nal complex space invariance of physics under the rotations.
r
r iX ~(~,t) = e (3)
Here ~ are now 24 5x5 matrices and ~(r,t) are 24 angles discribing
the rotations in the complex 5 dimensional space. If one includes the
helicity degrees of freedom these ideas lead to the SU(5) unification
of Georgi and Glashow. In this theory the 15 Fermions of the first
family including the helicity degrees of freedom are grouped in a qui ~
tet and a decuplet. A neutrino is a Dirac-particle and there no right-
handed vector bosons interacting with right-handed currents. This Grand
unification predicts a lifetime for the proton of 1031 years. But re-
cent measurements indicate that the proton lives longer than 1032 years
and thus one favoures different grand unifications. One of two fa-
voured ones are SOl0 (Langacker 1981; Stech 1981) and E6 (Stech 1980;
295
Buras et al. 1978). The exceptional group E6 has a subgroup SO10 and
the same is true for the group structure E8xE8 or SO32 requested by
the superstring theories. All of them hay as subgroups SO10. SO10 pre-
dicts for one family 16 particles. The additional particle not inclu-
ded in the quintet and the decuplt of SU5 is a right-handed neutrino.
Thus the theories include always right-handed vector bosons and right-
handed weak interaction currents. They prefer that the neutrino is a
Majorana particle and they predict roughly a mass for the neutrino
between 10 -5 and 1 eV.
If the neutrino is a Majorana particle, then it is obvious that the
lepton number is not conserved in such a theory.
In the present lecture I will show you that it is possible to test in
the neutrinoless double beta-decay if indeed the neutrino is a Majorana
Particle and if it has a right-handed weak interaction and/or a finite
mass m .
To describe the nuclear matrix elements we shall use a new generation
of microscopic nuclear structure calculations which we call VAMPIR
(~ariation After Mean Field Projection In Realistic Model Spaces).
This method is based on the Hartree-Fock-Bogoliubov approach with par-
ticle number and angular momentum projection before variation. We in-
Clude also the short range Brueckner correlations and the finite exten-
Sion of the nucleons.
But the main point is that in previous calculations the leading term
for double neutrinoless beta-decay has been neglected. We show that a
term which has to be thought to be extremely small is for the matrix
element about a factor 200 times larger than all the matrix elements
Calculated previously. In connection with a lower limit measured for
the lifetime of the double neutrinoless beta-decay from 76Ge to 76Se
this gives an upper limit for the right handedness of the weak inter-
action relative to the left handedness which is by two orders of mag-
nitude more stringent than all the estimates before.
296
DOUBLE NEUTRINOLESS BETA-DECAY
I I *
o ÷
0 ÷ / P
3, 28MeV
,eCo __~a Ti : 2m'cZ "2'O<'IMeVl ~
82 Se B2Kr ?6
130.132 Te 128,130 Xe
_ 0 ÷
345e ~2
F ig . I :
Figure i: Schematic double beta-decay of 76Ge through 76As into
76Se. One would expect that the main intermediate 76As
states are the 0 + and 1 + states with allowed Fermi and Ga-
mov-Teller transitions. We will lateron see that for double
neutrinoless beta-decay the usual Fermi and Gamov-Teller
transitions give not the leading contributions. The Q-value
of this double beta-decay is 3 to 8 MeV, that means that
2.041 MeV are available to be transfered into kinetic
energy. In the left lower corner other double beta-decays
with an appreciable large Q-value are indicated.
Figure I shows the double beta-decay of 76Ge to 76Se with its q-value.
The decay from 76Ge to 76As is not possible since 176As is less bound
than the two even mass nuclei due to the reduction of the pairing cor-
relations by the odd proton and the odd neutron.
297
In the standard model the double beta-decay happens by the fact that
first one neutron and then a second neutron is decaying in a proton,
an electron and an antineutrino. Since the two protons remain bound
inside the nucleus, the kinetic energy (apart of the recoil) is dis-
tributed over the two electrons and the two antineutrinos. Since we
have only 2.041 MeV available for the kinetic energies of these four
Particles the density of final states is quite low since it is propor-
tional to the surface of four momentum spheres spaned up by the mo-
menta of the four emmited particles. If one can remove two of the four
particles one would drastically increase the phase space which one
can reach since now only the two remaining particles are now only sha-
ring the 2.041 MeV energy. With the same nuclear matrix elements this
increase in phase space would enlarge the transition probability by a
factor 106. Thus even if the matrix elements are smaller by a factor
10-3 the transition probability remains still the same for a neutrino-
less double betra-decay where only two electrons are emitted into the
continuum (see Fig. 2).
From Fig. 2 it is obvious that in the standard model with neutrinos
and antineutrinos different it is not possible to have a neutrinoless
beta-decay. But even with a Majorana neutrino the first emitted neu-
trino has a positive helicity (is right-handed) due to the left-handed
interaction at the vertex. The neutrino to be absorbed by a left-han-
ded interaction must have a negative helicity (left-handed). To allow
for the double neutrinoless beta-decay one must have either a Majorana
neutrino with a mass or/and with a right-handed interaction. But those
Properties are predicted by the most favoured grand unified and super
string theories.
298
Fig.2-
pl
J V-A
e~
Figure 2: The double neutrinoless beta-decay is in the standard
model not possible. At the first neutron decay with the
left-handed interaction L=V-A an antineutrino is emmited,
at the second neutron decay one needs to absorb a neutrino
if one has a left-handed interaction L=V-A. Even if the
neutrino is a Majorana particle the double neutrinoless
beta-decay is not possible for a purely left-handed weak
interaction with a massless neutrino.
The measurement for double beta-decay of 76Ge is in principle straight
forward but in practice extremely complicated. One uses ?6Ge as a probe
and as a counter. The two electrons emmited in the double beta-decay
with and without neutrinos transform their kinetic energy into a cur-
rent which is proportional to the total energy. In principle the events
as a function of the sum of the energy of the two electrons should
look like in Fig. 3.
299
_J UJ Z Z <[ I C)
u9
Z
O £D
2 vO# ~ ~ , ~6Ge ~Se
',dg . 2,0&IMeV
SUMENERGY Ele~) + E(e~)
Fig .3"
Figure 3: Sketch of the number of counts per channel as a function
of the sum-energy of the two emmited electrons in the
double beta-decay with and without neutrinos. The peak at
2.041 MeV would indicate a double neutrinoless beta-decay 76
in Ge.
The geochemical measurements are not useful for the double neutrino-
less beta-decay since calculations and measurements suggest that a
neutrinoless double beta-decay is much weaker than the beta-decay with
two neutrinos. Since the geochemical measure both transitions at the
Same time they mainly give information about the double beta-decay
With two neutrinos.
Laboratory experiments to look into the double neutrinoless beta-decay 76
of Ge have been performed by several groups (Belotti, 1984; Forster,
1984; Simpson, 1984; Avignone, 1985).
300
4 3
0
! keY/CHANNEL | keV/OHANNEL
(,) (~)
i 2 o
Figure 4:
o o
t
(c)
t$
LZ
0
6
3
0
! keY[CHANNEL 1 teV/CH~NNEL
(d) (c)
Counts per i keV channel near 2.041 MeV for the double beta-
decay of 76Ge to 76Se. Figure (a) are the results of
Avignone (1985). (b) and (c) are different experiments of
Belotti and co-workers (Belotti, 1984). (d) is the result
of a measurement of Forster et al. (1984) and the part (e)
has been measured by Simpson et al. (1984). Of these mea-
surements the best statistics is shown by Avignone and co-
workers (1985). He analyzed all these data and
found a lower limit for the lifetime of 76Ge with respect ~ T0vBB>I 7x10 ~3
to the double neutrinoless beta-decay ol " yr 112
TO calculate theoretically the double beta-decay, one has to obtain
the second order transition amplitude:
T = ~ k Ek-Ei
< fl H w rk ><k J H w li > (4)
The energy denominator
Ek-E i = E(el) + E(V) + Ek(AS) - Ei(Ge )
(5) E(e I) + E(v) + I0 MeV
301
is given by the energy of the first electron, the virtual neutrino
energy and the energy difference of the intermediate excited state in
76As relative to the ground state of 76Ge. A major point of this contri-
bution is now that the intermediate neutrino is virtual and that we
have to integrate over all energies and momenta of the neutrino. The
momentum of the neutrino is only limited by the momentum a nucleon
Can emmit or absorb inside the nucleus. This is twice the Fermi momen-
tum. In a beta-decay where the neutrino is real, a typical momentum
• of the neutrino is 1 to 2 MeV/c due to the Q-value of the decay. The
aVerage energy of the virtual neutrino is determined by the fact that
One gets the largest contribution in the integral over the energy if
the wavelengths of the neutrino is of the order of the distance bet-
Ween two neighbouring nucleons. If the energy gets higher and thus
the wavelength shorter, the contributions have a tendency to inter-
fere away. Thus we expect that we get the major contribution for neu-
trino energies of about 100 MeV. Thus if we replace in the energy deno-
minator eq. (4) the excitation energy of As relative to the ground-
State of Ge by an average value of 10 MeV we can even allow that we
are by a factor 2 off the right average value and we are still only
making an error of the order of 10%. Since we aim at an accuracy of a
factor 2 there are no difficulties in replacing the nuclear part of
the energy denominator by an average value which allows instead of
the sum over the excited states in 76As to use the completeness rela-
tion.
Table 1 shows various contributions to the transition operator for a
transition from a 0 + initial state to 0 + final state for a double neu-
trinoless beta-decay.
TABLE 1
302
+
Different contributions to the transition operator for a 0
to 0 + transition in a double neutrinoless beta-decay. The
second line shows the operators for a left-handed weak inter"
action vertex for the decay of the first and the second
neutron. This decay can only happen on the neutrino side
due to the finite neutrino mass since it mixes to a negative
helicity neutrino also one with a positive helicity. The
two emited electrons are both in S-states which have by far
the highest overlap with the emiting nucleon. A p-wave elec-
tron would reduce the probability by almost a factor 10 -2 .
The nuclear transitions are either at both vertices Fermi + +
transitions (llxl 2) or Gamov-Teller ransitions (a I ~2 ) . The
lower three lines show the interference between a left-handed
and a right-handed vertex which is proportional to the right-
handedness parameter n defined in eq. (67. The neutrino tran-
sition operator is either the neutrino energy Ev with posi-
tive parity or the neutrino momentum pv with negative parity.
To conserve the parity one can in the case where one has
the neutrino momentum as the transition operator use either
an s- and a p-wave for the two electrons and the usual Fermi
and Gamov-Teller transitions on the nuclear side or two
s-wave electrons and relativistic corrections (small ampli-
tudes) for the nucleon wave functions. One normally calls
them recoil corrections and the leading term is proportio-
nal to the neutrino momentum. Since the neutrino is virtual
this neutrino momentum can be as big as twice the Fermi
momentum. Thus the ratio p~/M N is not anymore small.
TABLE 1
Vertex
m ° L
L'nR
v(~)
m v (+)
E (+) V
e( ~ ) Nucleus
SS(+) 11"22; 1 2 (+)
+
SS(+) 11"12; ~i'°2 (+)
n<<l Pv (-) SP(-) 11.12 ; oi-02 (+7
SS(+) ° 1 x Pv /M N (-)
303
AS one can see from table i we are proposing to include also the small
amplitudes from the nucleon wave functions in the calculations. The
term which we include is proportional to v/c and goes with the neutrino
momentum Pv" These terms are normally called second forbidden. But
Since the neutrino is virtual the neutrino momentum Pv is only limi-
ted by twice the Fermi momentum 2kF=560 MeV/c which is not anymore
Small compare to 938 MeV/c. This term is also proportional to the ano-
malous g-factors (gp-gn)=4.7 due to the weak magnetism (Tomoda, Faess-
let, Schmid, Grimmer, 1986).
H w JLJL + KjLJ R + njRJ L
+ IjRJ R
j~(x) = e 7U(l-y5}v
(J0 L(x) , i L(x) } = Z n
+
Tn+6 (X-Xn} (gv' gAOn )
(6)
The capital letters J indicate the hadronic and lower case letters j
the leptonic currents. To come from the left-handed currents "L" to
the right-handed ones one has to change for the leptonic currents of
the sign ¥5 and for the hadronic currents the sign of the axial vector
COUpling constant gA" For the hadronic currents we immediately used
the non-relativistic reduction. The recoil term given in the last line
of table i is obtained by taking into account the small amplitudes of
the order of v/c.
The term in (6} proportional to < does not yield new selection rules.
It only corrects the double neutrinoless beta-decay going with the
neutrino mass m v by a factor (I+K). Since one expects that < is ex-
tremely small compared to one it is impossible to measure the contri-
bution due to the K
term. Thus we neglect this term. Both terms the one proportional to q
and to ~ have a right-handed leptonic interaction thus they allow the
double neutrinoless beta-decay by having at one of the vertices in
figure 2 a right-handed leptonic interaction.
The lower limit of the measured lifetime (Avignone et al., 1985)
304
~I/2 (0v 76Ge ) > 1.7 x 1023 [yr]
0v in2 w 0v
~i/2
< 1.3 x 1031 [sec -I] (7)
implies an upper limit of the transition probability for the neutrino-
less beta-decay of 76Ge. The theoretical transition probability is
given by a sum of matrix elements proportional to the neutrino mass
my, to the right-handedness parameter ~ and the right-handedness para-
meter I. If one squares this expression to get the transition probabi-
lity one obtains an elipsoid in a three dimensional parameter space
spaned by the three axis proportional to the neutrino mass m , to the v
right-handedness parameter ~ and the right-handedness parameter I.
The projection of this elipsoid into the m -~ plain for our matrix v
elements (Tomoda, Faessler, Schmid, Grimmer, 1986) including the last
line of table 1 and also for the matrix elements of Haxton, Stephen-
son and Strottman (1981) is shown in Fig. 5.
305
mv[eV]
xton
I o 1 ~. IoS q -
Fig. 5:
Figure 5: Using the lower limit for the lifetime and therefore the
upper limit of the transition probability (7) for the
double neutrinoless beta-decay of 76Ge one obtains for
given nuclear matrix elements an allowed area which corres-
ponds to an elipsoid in a three dimensional parameter space
of the neutrino mass m and the right-handedness parameters
q and I. The projection onto the m v and n plain is shown
for the matrix elements of Haxton and coworkers (1981) and
for our matrix elements (Tomoda et al. 1986) including also
the recoil terms from the relativistic corrections of the
nucleon wave functions. Haxton et al. obtain: m <3.7eV; n V
<l.2x10 -5; l<8.1x10 -6. We obtain: mv<2.6eV; n <6.8x10-8;
I<4.3xi0 -6. Thus we are able to reduce the upper limit for
the right-handedness parameter by more than a factor 10 -2.-
Therefore we have here the most stringent upper limit for
the right-handedness n.
306
SUMMARY
The double neutrinoless beta-decay is only possible if the neutrino
is identical with its antiparticle and thus violates lepton number
conservation. But even if the neutrino is a Majorana particle (v=~)
the double neutrinoless beta-decay is not yet possible. One must
assume that the neutrino has either a mass m or/and a right-handed
interaction as indicated in eq. (6).
For calculating the double neutrinoless beta-decay due to the right-
handed currents it turns out to be very important that on includes
the relativistic corrections for the nucleons which normally only con-
tribute to the second forbidden beta-decay. By this inclusion the nuc-
lear matrix elements are enlarged by about a factor 200 and by that
the upper limit of the right-handedness parameter q is reduced by more
than two orders of magnitudes.
It is very important to have reliable nuclear structure wave functions
for the calculation of the double neutrinoless beta-decay. We use the
new generations of microscopic nuclear structure wave functions based
on the Hartree-Fock-Bogoliubov approach with particle number and angu-
lar momentum projection before the variation (Schmid, Grimmer, Faess-
ler, 1984). The neutrino exchange between to two nucleons leads to a
short range two body operator for the double neutrinoless beta-decay
and thus one has to include the short range correlations and the finite
size of the nucleon.
The reduction of the upper limit of the right-handedness parameter n
by two orders of magnitudes is an important step towards tests of
grand unifications and the superstring theories. But to be able to
make a positive test one has to improve the lower limit on the life-
time by several orders of magnitude.
I would like to thank Dr.'s Tomoda, K.W. Schmid and Grimmer who colla-
borated with me on this problem.
307
REFERENCES
Avignone, F. T., R. L. Brodzinski, D. P. Brown, J. C. Evans, W. K.
Hensley, J. H. Reeves and N. A. Wogman (1985). Phys. Rev. Lett. 54,
2309 and preprint Oct. 1986.
Belotti, E., O. Cremonesi, E. Fiorini, C. Lignori, A. Pullia, P. Sver-
Zellati, L. Zanotti (1984). Phys. Lett. 146B, 450.
RUras, A. B., J. Ellis, M. K. Gailard, D. V. Nanopoulos (1978). Nucl.
Phys. B135, 66-92.
Davis, R. (1955). Phys. Rev. 97, 76.
Forster, A., H. Kwon, J. K. Markey, F. Boehm, H. E. Henrikson (1984).
Phys. Lett. 138B, 301.
Haxton, W. C., G. J. Stephenson, D. Strottman (1981). Phys. Rev. D25,
2360.
Langacker, p. (1981). Phys. Rep. 72, 185.
Lee, T. D. and C. N. Tang (1956). Phys. Rev. 104, 254.
Schmid, K. W., F. Grimmer, A. Faessler (1984). Nucl. Phys. A431, 205.
Sehmid, K. W., F. Grimmer, E. Hammar@n, A. Faessler, (1985). Nucl.
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Simpson, J. J., P. Jagam, J. L. Campbell, H. L. Mahn, B. C. Robertson
(1984). Phys. Rev. Lett. 54, 141.
8tech, B. (1980). Unification of the Fundamental Particle Interactions;
Ed. S. Ferrara, J. Ellis, P. van Nienwenhuizen; Plenum Press.
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Extrapolation of Nuclear Properties to the Region Near Z----184 1
J. F ink , L. Neise , M. Rufa , J. M a r u h n 2 a n d W. G r e i n e r I n s t i t u t fiir T h e o r e t i s c h e P h y s i k , U n i v e r s i t ~ t F r a n k f u r t , W e s t G e r m a n y
P. G. R e i n h a r d I n s t i t u t fiir T h e o r e t i s c h e P h y s i k
UniversitS.t E r l a n g e n - N i i r n b e r g , W e s t G e r m a n y
J. l, Yiedrich l n s t i t u t fiir K e r n p h y s i k , Un i ve r s i t S t M a i n z , Wes t G e r m a n y
1 M o t i v a t i o n
Extrapola t ing the properties of mlc]ei from the elements ohserved in na ture to the unknown region of hitcher mass nmnl~ers is one of the ()Her preocrupat ions of nuclear physics. Much work has gone into the prediction of lifetimes of superheavy nuclei with Z ~ 114 . . . 126. In this case the t)rincipal probh' in is the calculation of shell corrcctlons, which serve to enhance the fission barrier and thus cause longer fission halfliw's.
As we proceed to even heavier systems, it, is expected that the uncer ta inty in the other ingrediellt of a mass formula, the bulk part which is oftcii described by a liquid drop or droplet, formula, will also become more and more uncertain. It is the main aim of the various theoretical efforts collected here to examine the extral)olation to the region near Z ~ 184, f. e. a system with the slzc about double tha t of a Ilraniuin imcleus.
Since "normal ~ supcrheavy nuclei have yet to be discovered, such an effort does require .some external s t imulat ion, which in this case was provided by the nlystcrious positron lines observed at. GSI in he:wy ion collisions of systems like 23aU+23sU ([Sc83,C184,Co85], for the latest exper imental s ta tus see the contr i tmtions in [Gr86]).
These lines are sufllciently narrow to require a lifetime of tile emi t t ing sys tem of 10 2 . . . . 1()-1%, a t ime tha t is much longer than the expected sticking time of the nuclei with the normal reaction merhan ism. One seemingly natural explanation is the emission of the posi trons through the spontaneous process [Re84] by a compund system stabilized by some unexpected feature m the ion-ion interaction poienlial . In fact, one ran produce a pocket for molecular configurations lSe85] tha t produces reasonable st irking times, lint this is not in agreement with the observed energies of the lines which indicate a much smaller (]eformaiion. Now the available energy in the exper iment is just barely sufficient to bring two I i ranium nuclei into a touching configuration, and on the basis of tile s tandard liquid drop model the potential ener~w should increase steeply for closer distances, so tha t an additional energy of the order of 150 MeV is required to reach the more compact shape.
The question that arose at this point is whether the liquid drop model (or any other model) can |)e extraI)olated reliably enough tn allow the prediction of the energy of a I l ran iun , -Uranimn compomttI nucleus to within 150 MeV. Or, posing the problem in ;mother way, can one adjust the parameters in the
476 models in such a way as not to destroy agreement with the known nuclei but enhance binding for ts4X |)y 150 MeV?
This is the question tha t will addressed in the following in a nmnber of quite different, nuclear models.
'Work suppor ted |)y the Bundesminis te r imn fiir Forschung mid Technologic and by the Gesellschaft fiir
Schwerionenforsc hm~g,
2Invited speaker,
309
1.20
I.la
1.15
200
0
200 0
-20C
r°[fm]
Kv[MeV] I
+ ~+,~[MeVl .]____ J'f~l
20
0
-2600
_ ~+.10 a [~V]
I I I II I
-2800 -3DOO E 1292,184) [ MeV ]
18 -%[MeV] I
17 ~ _ _
16 I 02[NOV ! - - ~___ [
2O
1.0 - C l [ M e V }
O2
0.~ 25 (1 [MeV}
2 ' 2 +XR. e X~-ot
-2600 Z800 -300e E { 2'92,18q [~V I
-!+
160
140
120
1.70 r 0 ~
1.18
1.16 KvIl~eV] I
o t 1 2001 ~7~w7~;~ i-T 0
2500 2800 3000 E(2~2,184) [MeV ]
I
O.[M~V] I
L X: / ' I.+ ../!
~600 2800 30~ E (212,18~,) [ ~ v I
Piotre 1: The droplet model parameters a~ld X 2 as a flmction of the assumed binding energy of tile super-heavy nnclcus ~,~X. In addition to tile total X 2 also tile contributions from radii and binding energies arc given separately. The value for the binding energy at mmimunl X 2 is given together with the statistical uncertainty. Tile plot to tile right shows fits with K,,,~ I a~,d K,, kept lixed while on tlw left all parameters wcrc allowed to vary fi'eely.
2 T h e d r o p l e t m o d e l
Perhaps tl,e most widely used extension of the liquid drop idea is tile dr,)plet model [My76,My77}. A very important advaaltage compared to the liquid drot I model is the use of iron-homogeneous densities, albeit only to lowest order in the energy expressions. Tiros tile droplet model does describe a reduction of the proton density in the center of the nucleus, which would lead one to expect enhanced binding ill Very heavy systems.
In order to investigate the precision of the predictions made by the mode[, we have used the model for least squares fits to both a large number of known nuclei and to an assunled 1)indmg energy of ~X[SeS5a] . Ia this way it becomes apparent to what extent the agreement in tile know~, region is reduced by changing the desires value for ~{X. In addition the sensitivity of the results to an additional higher order term of the form a4A 2 was tested. Thus we proceed from the general expression
[ J E ( N , Z ) = (-,q + aa 2 - I K e 2 ) A + a=+Q A ++/a
+ [qA -~/a - c2A l/a - caA - t - c4Z-2/aA -l/a - es]g 2 (1)
¢- Wignerterm + e v e n - oddterm + a4A 2
For the definition of the average compression ~ and tile average asymmetry ~ see [My7G,My77]. The surface term has the form given in {My69], and all nuclei are treated as spherical.
For the nuclear incompressibility we use the following (N, Z)-dependence [Kr75,B[81]
K = Kv + K ' " ' IA-1 /a + K ' " ( ~ z - ) 2 ( + K~o,,,Z2A -413 + K,,, N ~ a A_,/3 (2) 2
310
co i
E
4 J
C (1)
0.20
~3.15
0.I0
~zJ 0.05
0.00 5 10 15
rad ius (fro)
Figure 2: Density protiles of xmclei with 300 < A < 1000. A increases by 100 from curve to curve. ~111 curve: p, dashed line: #r, dotted line: p , .
U ~ 2> 4
a) -7
C
r3 -8
X D _ c j = ~ I ~ 1 • 2 _ _ ~ - - - Z - - L ~
0 200 400 600 800 I000
nucleon number
Pigalrc 3: Binding energies per nucleon of nucM in the density functional model (full cnrve) arid the liquid drop model (dashes).
with the different terms describing tile contribution from volume, surface, symmetry, Coulomb and surface-symmetry, respectively. For the lit, however, only K,, and K,,,rI were allowed to vary and tile others were kept at tile fixed values of K,v,, = 508 MeV, Kco,,t = -5.07 MeV, and K,,~ = 1390 MeV.
For tile following discussion it will be important to note that even though this mass formula is fitted to the binding energies, the individual terms in it are still supposed to retain some physical significance. E. g. the coefficient Kt, should correspond to the incompressibility of nuclear matter , and one should worry if the fitted values were outside a range of 200-400 MeV. The same of course is trale for the other coefficients.
In the calculations the set of data in the known region consisted of the binding energies of 64 nuclei with 40 _< A _< 208 taken from [Ma651 with shell corrections subtracted according to [My761, and tl~e radii of 30 mtclei determined from electron scattering [Pr82]. As errors we take 0.5% for radii and 1 MeV for binding energies. The desired binding energy for ~7~X could then I,e added as an additional "experimental" quantity, and it turned out that it was always practically exactly reproduced by the fit.
Fig. 1 shows the results of the fit with the additional term (a4A 2) included. The behaviour of the 4 7 6 X2-wdues as a function of the assumed binding energy for t84X is quite encouraging: it seems that the
value necessary for understanding the positron experiments (indicated hy the dash-dot ted line) is only a little more than one standard deviation from the optimum prediction of the model. However, one should not neglect the variation of the model paranteters, which are shown in the other parts of the plot. All of them change systematically, but the most dramatic dependence is in the incompressibility parameters Kv and K, url. In order to explain the exthanced binding of 47n-e-Is4 ~, K~ would have to come clown to less thala
311
100 MeV and Ks,,rl would have to beconle large and positive, 1)oth of which seem to be unacceptable in the light of cnrrent beliefs about these coefficients [Kr75,Wo81]. The conclusion front this fit thus has to be tha t :m ~.nhanced binding of ~ X is compatible with the droplet model regarded a.9 a pure fittm~j formula but conflicts with its* physicM in.tcrprctation.
The other fits add to this picture. Sett ing a4 to zero does not drastically change the si tuation, and fixing K,,,¢ I at the credible value of -310 MeV makes the min imum in X 2 very narrow and exchides the
476 desired value for l s ,X completely. This lat ter fit. is also shown m fig. 1. A subsequent investigation [Ke86] checked whether replacing a4A 2 1)y sonic other simple term wonld
change the si tuation. It was shown that a l)~)lynoniial in Z ()r A of at least fifth or sixth order iv necessary, or in other words, essentially a threshold term is required. Also it was found tha t the various expansimis in the droplet model still seen), to work corrcclly fro" these heavy systems.
3 A simple density functional model
Although the investigations in the framework of the droph't model yMded a surprising consistency in the extrapolation, it would still be reassuring to have a model that allows arl)itrary distort ions of the proton and neutron distr ibutions inside the nuch, us. A simple empirical model of this type is provided by the density flmctional first developed and successfully apl,llt'd to the description of densi ty t)r(dih's 1)y Stock [St76] and later used by Heinz ct al.[H,.77].
The construct ion of the density fimctional is closely mot iwded hy the s tandard liquid drop model. It contains the same types of terms:
E(pp.?,,) - E.,o~(?p + o,,) ~ E.,,.s(pj, + e,,)
E<o,,~(/,.) + E.~,,,(ep o,,,:,p + :,,) (3)
The volume energy is a spatial integral ow'r the energy densily in nuclear ma t t e r pW(p), where W(p), the equation of state of cold mwlear mat te r is paramctr ized by a parabola which fits nuclear mat te r properties Po = 0.17 fm - s , W(9o ) = -16 .5 MeV. The resulting incompressibil i ty is 297 MeV.
For the surface energy a term proposed by Greiner and Scheid [EiT0] is used, which obviously describes Sllrface effects:
~ [[ ,:- b " - r l / . E . . , : : ( r ) - : ( , ) ) , s , ,
i i
The Coulomb energy was taken to be just the ususal direct term phis the exchange term calculated in the Slater approximation.
Ooncc:rning the sylonlelry energy, there is sonie ambiguity in the l i terature its h) how to translate the corresponding term in the Bethe-Weizsiicker formula into a local density expression. Excel)l f(ir the Wigner term, which just contains IPv - P,I, there is agreement on using (pp -- p,,)2 m,dlildied by stone function of the densi ty let70 I. We have fl)und tha t all terms of the form
E,v,,, c J (pp - Pn)2dr 2 1 . . . . . . . . . . . e {1, 0} (s)
could reproduce the valley of stability. However, examining the density distr ibutions, it turned out that for l, > 0 the enhancement of the term near the surface prevented the formation of a neut ron skin. in COUtrast to present belief. So we finally adopted l: = 0 and used a vahn" for C of 217.2 MeV fin ~ obtained from a fit to the valley of stability.
The final parameters to be fixed are those of the surface em'rgy, V0 and l*. These were fitted to the binding energies of known nuclei. It turned out that to first or(ler results only depend on the c()mbination V0#i 4 (as is expected from an expansion of the integrand into gradient terms), so tha t the adopted values of # = 0.5 fni and V0 = -1429 MeV fm are less precisely fixed. Tile fimctional is then minimized under the constraint of spherical symmet ry and for given numbers of protons and nen.trons.
For the known region of nuclei the results for the binding energies tend to agree be t ter with exper iment than the liquid drop model, the main difference being reduced values of bo th volume and surface energies (in absolute niagnitude). Tile density distr ibutions :ire quite reasonable but lack exponential tails, which also leads to r.m.s, radii slightly smaller than usual, following R = roA l/a with r0 - 0.927 f in .
312
E[MeV]! ' ' i , ! ,/~/ff ~,
2650 a LDM A=/+76
D Sk3 / / • o 5kM / / ' ,~
] /
2700 ; ' // ./
2750 / III;';' 12[ / / / h
/ /' ~.. . ~ - /
2800 / ,.,
[ . . . . . I _ , I ,
160 170 180 Z
Figure 4: Binding energies of the isobars with A : 476, for the liquid (trop model (LDM) and the Hartree-Fock calculation with the fi)rces Sk3 and S k M
For the extrapolated region the density profiles arc shown in fig. 2, which clearly exhibits the transition from central depression to bubble nuclei [Wo73 I. This would seem to iml)ly a substantial reduction of the Coulomb energy, but this is not the case. In fact, as fig. 3 shows, the total binding energy is smaller than m the liquid drop model, mad even the Coulomb energy increases because of an overall smaller radius. Some of the interesting properties for the special case of ~87~X are summarize(t in table 1 and clearly shows this behaviour that rams counter to expectations.
Table 1. Propert ies of ~ ] X in the density fimctiona[ model. "Polarization" denotes the ratio of central to maximum density.
Density Functional Liquid Drop Bindixxg energy [MeV] Coulomb energy [MeV] Rms-radius [fm] Rms-radius protons [fin] Rms-radius neutrons Ifml Proton polarization Density polarization
-2723 -2779 3088 3079 7.27 7.38 7.18 7.38 7.33 7.38 1.19 1.00 1.09 1,00
4 S k y r m e f o r c e H a r t r e e - F o c k c a l c u l a t i o n s
Thinking about tile extrapolation of binding properties, the first model that comes to mind is tile oscillator potential single-particle shell model, that was so widely employed for the prediction of the properties of "conventional" superheavy nuclei in the Z = 110. . . 126 region. Even then the necessity of finding appropriate values for the spin-orbit s trength x and the/2-coefficient/z meant a large uncertainty in tile extrapolation. Compare e. g- the mass-dependent paramctrizations for e; and ,u given by Gustafsson [Gu66], Nilsson [Ni69], or Seeger [Se67], where there is not even agreement as to whether they should be proportional to A, to A l/~, or depend oil tile shell considered. Thus it should not come as a surprise that we fmmd essentially no predictive power in the model when extrapolating to Z ~ 184. Magic numbers could be produced or destroyed ahnost ad libitum.
313
0.15
0 . 1 0
'1= Q..
0.05
0
20
0
Z ~ - 2 0
-/+0
. . . . i . . . . I • ' " G --)
- L . . . . . _ - \ / . . . . . -ne-u[r°ns
................ _'/"'. --t°tat- .
• ' " ' I ' ' ' "
- protons . . . . . . . n m ~ o n s
• /
/
I I
- ~ _ _ J t
/ /
~'~ i , , ] , ' , t L
5 qO
R[fm]
b)
15
f'4 I
tO CO
E
476 Figure 5: a) Proton, ueutron, and total densities in ls4X calculated for the force Sk i . b) Average potentials.
Also the s i tuat ion here is quite different from that of the region Z ~ 114, where there was a well- known background of liquid drop energy to which relatively small shell corrections were to be applied. Pot the present problem, however, it is the bulk liquid drop energy tha t is under investigation and it makes no sense to use a model that describes only the shell corrections. Therefore a single-particle model allowing a calculation of both bulk cncrg'y and shell effects has to bc considered, and maybe the most Widely used formulat ion of such a ntodel at present is the Skyrme force Hartree Fork model [Sk56,Sk59].
The general expression for a central Skyrme force is given by
VMr) : too + ~0/'o)b(r) + t~(l + ~ , f . ) Ik '2~(r) + 6(,)k~l +t2(1 + x2Po)k'6(r)k + iW(rrt + a2)k' x 6(r)k (6)
+(t~/G)(1 + ~3/' , ,)b(,)p"(,) ,
Where r = r~ - r2 is the relative separation and P . the spin exchange operator. The Coulontb interaction is added to the model as an additional force.
This nucleon-nucleon interaction is used in a Hartree-Fock approximation, where the advantage lies ia the fact tha t the short range nature of the force makes the average fields contain only local quant i t ies Such as the densities of protons and neutrons or the currents. The ground s tate wave fimctions for a given nucleus can then be obtained numerically, h ts tead of examining these tedious details, however, let Us look at the use of this model for a mass formula. The masses depend on the ten parameters t0-s, : ~ 0 ~ 3 , or , a n d W .
For these parameters various sets are available in the li terature, the older set with a = 1 denoted by Skl-Sk6, while ~modified" Skyrme forces are referred to as S k M and S k M °. We concentrated attention on the forces Skl [Va72,Va73], Sk3 [Be75], and SkM" [Ba82] (however, the forces of Giarmoni [qli80], Krivine [Kr80], and KShler [Ko76] were also tested)• The nuclear matter properties of these three forces are shown in table 2, from where it becomes apparent that the main advantage of the nmdified
314
[ MeV
4
2
0
-2
-4
-6
[MeV]
-8
-10
-12
PROTONS
SKM SK1 S K 3
3D 3 / 2 . . . . . . 1K1.2 :: :-,i .iiiiiii ...... 3D 5 /2 ,, . . . . . . . . . . :,-" ." '
1313/2 . . . . ~ . 2G 7 /2 .... ,t,.,... 4 -
26 9 /2 164- 164 154
,-,..
1315/2 154 154 3P 1/2
3P 3/2 154 ............
1111/2 - - 2F 5/2 - - ...... "".iiiiii ......... 2F 7 / 2 - -
2 ~ 4D 5 / 2 1L17/2 __
1M21/2 - 36 7 /2
3G 9 /2 - -
-4 2 I l l / 2
-6 2113/2 4P I / 2 4P 3 /2 1L19/2 3F 5 /2 1K15/2 3F 7 /2 - -
2H 9 / 2
NEUTRONS
SKM S K I S K 3
S K 4
- - 3D 5 / 2
1 8 6 1313/2 *'2G 7 / 2
164 2G 9 / 2
154 1315/2
3P 1/2 • - - 3P 3 /2
S K 4
3G 9 / 2
2111/2
308 308 308 2113/2
7 " .... ~:~: . . . . . . . . . . . . 308 ' ~ . " _ _ 4P 1/2 i.~ . ' i::;" ~ .... 4.~P 3 /2
......... " ; i :" : . . . . . . i . . ' - - 1 L 1 9 / 2
.... 3F 5 /2
' " 3F 7 / 2
- - IK15/2
o~
c a i
t/3 cO n r - E
Figure 6: Single particle levels for tile nucleus 4767 for different Skyrme forces. The Fermi energy is 1 8 4 ~
indicated by an arrow.
315
X21
100
50
X~ ------ ~ ms [ i I l
0 - 485 490 495 500 [1~_~]
2 ~ X~- ~ . ~ J ~-
Figure 7: The total X2-value and the contributions from binding energies, radii and surface thicknesses {~a') for the fit with the nonlinear mean field model, as a function of the scalar meson mass which was kept fixed in the fit. For comparison the best Skyrme force fit [Fr86] is given by a dashed bar.
Skyrme forces is the better value obtained for the incompressibility of nuclear matter. It seems that incompressibility is a problematic feature in many models!
Table 2 Nuclear mat ter )erties of the forces Sk i , Sk3, and S k M ' . E / A a#y,n Koo p k l rn * / rn MeV MeV McV fm -~ fm -1
Sk i -16.0 29.3 370.0 0.155 1.32 0.93 Sk3 ° -15.9 31.2 356.0 0.145 1.29 0.76 S k M -15.8 30.8 217.0 0.160 1.33 0.79
Figure 5 shows tile density distribution in 476v for S k l , the results with tlle othrr forces beiu,, very I S 4 ~ o
sindlar. There is clearly a considerable depression of the proton density in the center, albeit not as pronounced as in the other models.
In general the binding energies agree surprisingly well with the conventional liquid drop model. This is demonstrated by figure 4 showing the binding energies of a series of isobars with A = 476. The results differ only by about 40 MeV. It is interesting to note that the a-decay Q-value is 23.1 MeV [or the U-U system and 24 MeV for U-Cm. If one extrapolates the exponential dependence of the half life time on the decay energy, decay times of less than 10-2°s are thinkable, so that a-decay might take place even during the heavy-ion collision. Still, tile final word here is also no enhanced binding for ~ ]X.
With respect to the question of possible stability the single particle spectra are of considerable interest. e 476~ There are several nlagic nuntbers: In Fig. 6 they are shown for various Skyrmc forces in the case ol is4 A.
g=154, 164, and (for Sk4 only) 186; N=308 for all forces. Figs. 4 and 6 together show that the nucleus ~X30s should be the most stable one in this region w. r. t. a-decay or proton emission. On the other haIld, for systems such as U-U or U-Cm the Fermi energy is in a region of higher level density, so that it Seems unlikely that shell stnlctures could stabilize these systems appreciably.
316
-6
1- - o
~-7 Z
~ - 8 ¢- -
La.A
- 9
' I '
[] 8ca t 0 Sex p
- "~ 8Skyrrne M "M"
I I l
.['71
o? .tq
,.. - ' "
0 100 200 i I L I
300 400 A
I
E
I 5OO
Figalre 8: Binding energy systematics extrapolated up to giant nuclei. The meson field theory (~¢o1) is compared with experiment and, for some cases, with Skyrme M* calculations. The "experimental" point
4 7 6 for ls4X is shown with a question mark, since it simply denotes the energy necessary for the mentioned interpretation of the positron experiments.
5 R e l a t i v i s t i c m e a n f i e ld t h e o r y
Finally we will examine a recent addition to the theories of nuclear masses: tile relativistic mean-field theory as formulated by Waleeka [Wa74] and Chin [Ch77]. It has been applied with considerable success to the description of finite nuclei [Bo77,Br78,Ho81,Bo84], the nuclear equation of state [Bo83,Th84], and even in dynamical calculations [MuSl,Cu86]. Since this model had not been fitted explicitly for use as a mass formula, we decided to do this in a systematic way [Re86].
The starting point for the theory is the Lagrangian density
- I [ ( O . . V . - O . L . ) O " V " - m2.V~V ' '] - a~V"¢ '1 , ,¢ (7)
- ~ - l (a .R, - 0 , f t . ) . a , a ~ - , ~ R , . a" l - l ~gnR ~' . ¢%,?¢
-I[(o~,A,-I~,A,,)O"At, I-~cA"¢(1 + r0)%'.b }
The individual lines in this lengthy expression correspond in that order to the physical ingredients of the theory: the nucleon field ¢, the scalar meson field (I), tile vector meson field Vv, the isovector-vector meson field Rv, and the photon field A~. Note that the 7r-meson is not included hecanse it does not contribute in the nuclear ground state unless parity and isospin symmetry are broken. It will also be important that the scalar meson field ~ contains nonlinear self-interactions [Bo77], which are necessary for describing nuclear matter properties correctly (as may be expected already, this concerns especially the incompressibility properties).
Specializing to static solutions in the mean field approximation, the resulting equations of motion really become quite similar to standard Hartree-Fock equations, the main difference being the use of the
317
Dirac equat ion for the nucleonic single-particle s tates ¢~, and the appeara~lce of several types of densities, namely the scalar density
the vector densi ty
p,, = ~ ¢~%,¢o , (9)
the isovector-vector density
and finally the charge four-current density
l + r o
o
The sum is over all occupied s tates . In the static case only the t ime component of the vector densities will survive,and we obtain the following equations of motion:
( - -A + , , d ) ~ = --g, - t,2~ 2 - b3¢ ~ (12)
m~,)Vo = g~po ( - A + z 03)
( - A + ~ ) R o , , , = x ~g~:o,o 04)
- A A 0 = e P,,o (15)
1 A l + r 0 ~. (aCa = "Ya(--i'Y" V + m B + 9,¢P + gvVo"io + ~gRRo,oro"lo + e o ~ ' f a } { o , , (16)
From this point on it is essentially a numerical problem - - a l though a highly nontrivial one [Re86] - to obtain the single-particle s tates ¢o, the various mean meson fields, aald finally tile total energy of the nucleus. Some details still had to be added, such as pairing (in order to allow non-closed shell nuclei), and a center-of-mass correction.
The free I~arameters of the model are the masses ms, my, mR, and tile coupling cons tan t s g,, gv, gn, b2, and b3, while the nucleon mass is fixed to the experimental value of mz~ = 938 MeV. It is customary, however, to replace tile coupling cons tan ts gi by the dimensionless quant i t ies c~ = (9, 'mB/mi) 2, m~d the nonlinear term coefficients by B2 = b2/(g3, mB) and Bz = b3/g~. Thus we end up with the eight free
z c~, c,~, B2, and B3. Since it turned out tha t the fit is quite insentit ive to parameters ms~ rn, u, trtR, c#~ the value of m~, it was replaced by the mass of the corresponding meson in an OBEP [HoSla], i. e.
mr = 763 MeV. So we finally work with seven adjustable parameters . The exper imental quant i t ies used for tile fit were the binding energies, mean square radii, and surface
thicknesses of the eight different spherical nuclei 160, ~°Ca, 4sca , SSNi, 9°Zr, n6Sn, x24Sn, and a°spb. As iu the previous discussion of the droplet model, the quality of the fits is summar ized in terms of the total g2-value. The first result was that the nmdel m the linear version, i. e. with b2 = bs = 0 did not give a fit tha t in any way could compete with Skyrme force calculations. For this reasons we here concentrate on the nonlinear model; the broader discussion can be found in [Re86 I.
Fig. 7 shows tile variation of the a t ta ined X2-valne with the mass of the scalar meson. It comes down dose to the presently best Skyrme force fit [Fr861 indicated by a dashed bar. It is interest ing to obserw~ how the fit finds a compromise between the requirements of the various observables: the surface thicknesses would prefer a smaller value for ms, the binding energies a larger value, and the radii are rather indifferent. The pa ramete r values for this fit mid the associated nuclear ma t t e r propert ies are summar ized in table 3. One should not tha t such a reasonable value for the incompressibil i ty requires tile nonlinear theory. Surprising is the relatively low value for the reduced mass and the large one for the symmet ry energy coefficient, at least compared to values in the literature [Ho81,Bo84].
Finally, of course, we come to the prediction for the mass of ~ X . Fig. 8 shows the ext rapola t ion in comparison to a Skyrme force calculation. The agreement between the two is quite good and both produce binding energies tha t are much smaller titan is necessary to explain the posi t ron exper iments .
318
Table 3. Values of the Mean F M d Theory parameters for the best. fit. and associated mxclear mat te r
properties, m* refers to the effect|w, mass.
Parameter Value Error Property
c~ 6 B2 B3
m s
~ t v
373.18 245.46 149.67
- 2 . 4 6 - 1 0 3
- 3 . 4 3 - 1 0 -2 492 795
±0.42 ±0.30 ±6.31
.t_1.44 - 10 -r,
±1.73 - 1() -2 ±3.5 ±13.2
E/A[MeV] po[fm -s] m" IMeV] K[MeVI
a,um[Mev]
6 C o n c l u s i o n
/alue
16.43 .1542 i36.0 212 43.6
To get an overview of the predictions made by the diiferent m[idels, we have compiled the binding ene,'gies 476 for ls4X in the following table: Table 4. Predictions for the bmdi,~g energy of ,176 - Is4X m all the nuclear models discussed in this lecture.
Energy [MeV] 2724 2819 2723 2688 2691 2673 2713
Liquid drop model I)rolilet model Density funeti(mal model Skyrme force S k i Skyrme force Sk3 Skyrme force S k M " Relativistic mean field theory Average of all models 2719 Standard deviation 48 Needed for positron lines 2923
Note tha t we included the droplet model prediction with fixed values of the incomt)ressibility t)aram" eters I)ccause, as discussed above, the other prediction is not a.s credible.
The, from our poiut of view, surprising result is the very good agreement between the different models leading to thc small s tandard deviation. Al though the use of tile statist ical deviation to interpret these data is highly questionabh~, it does give an indication of the degree of unaninl i ty reached. Thus the re:m1 result of all this work, az~d independent of the positron litte motivation, is tha t the bindin9 energies of nuclei can be eztrapolated far outside the known region with a surprisingly ~mall uncertainty. This was not to be ext)eeted because of the very different way in which they treat e. ft. the polarization of the p r . t on distributions: yet tl,e liqui(t drop model with its quite unreal|st:it assumpt ion of homogeneous charge distr ibulions gives a result that is very cltise to the more realistic models.
On the other hand, the iml)lication for the positron line becomes quite clear: i f the interpretation of the lines as spontaneous emission by the compound system has to be maintained, it is not in agreement with prescnt knowlcdye about nuclear structure and requires the invocation of some new effect that becomes important only for nuclei abovc a threshold in mass that must lie between Z - 109 and Z = 184. Although with the observation of coincidences hetween positrons and electrons [Gr86] the explanat ion in terms of a new elementary particle that decays into an electron-positron pair seems to be favored at present , the si tuati(m is by no means clear enough to allow to dismiss alternatives.
R e f e r e n c e s
[Ba82]
(neT~l [BI81]
[Bo771
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319
[Bo83
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[13,,78
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II~i70 [Fr82
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ICuGG
[He77
[~081 [Ho81a
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[My7T
[Ni69
IRe84
[i~e86] [Sc8a] [Se67] [s,,8q [s.85.~] lSk~6] IskGgl [st761
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QUANTUM ASPECTS OF NUCLEAR DYNAMICS
A.S~ndulescu
Central Inst i tute of Physics, Bucharest, ROMANIA
and
Joint Inst i tute for Nuclear Research, Dubna,USSR
1. INTRODUCTION
The quantum aspects of the nuclear dynamics are discussed in terms
of new col lective modes as open quantum systems. The new col lective coor"
dinates are the charge asymmetry, the mass asymmetry, the neck and the
relat ive distance between the fragments in addition to the usual coor-
dinates for the deformation of the fragments or the angular coordinates.
The potential, as function of the above coordinates, is obtained by one
of the usual macroscopic models corrected for the shell effects. The
corresponding moments of inert ia are given by the cranking model. The
dissipation, i .e . the interaction with the neglected col lect ive or in-
t r ins ic degrees of freedom, is introduced by adopting the Lindblad's
axiomatic way of introducing the dissipation in quantum mechanics based
on completely positive dynamical semigroups with bounded generators.
Fi rst , we discuss the equations of motion for one or two coupled
damped col lective modes. We present the followino new results obtained
in the framework of the Lindblad theory for open quantum systems: a
generalization of the fundamental constraints on quantum mechanical
diffusion coefficients which appear in the corresponding master equa-
tions, a generalization of the Hasse pure state condition and a gene-
ralized Schr6dinger type nonlinear equation for an open system. The
Schr~dinger, Heisenberg and Weyl-~igner-Moyal representations of the
Lindblad equation are given exp l i c i t l y . Based on these representations,
one shows that various master equations for the damped quantum osci l la-
tor used in the l i terature for the description of the damped col lective
modes in deep inelast ic coll isions are part icular cases of the Lindblad
equation and that the majority of these equations are not satisfying
the constraints on quantum mechanical diffusion coeff icients.
Also, the case of two damped coupled col lect ive modes as two
damped coupled harmonic osci l lators is shortly discussed. The time de-
321
Pendence Qf the e x p e c t a t i o n va l ues and of the Wigner f u n c t i o n are ex-
P l i c i t l y o b t a i n e d f o r both cases.
Second l y , by assuming t h a t the d i s s i p a t i o n cou ld be n e g l e c t e d in
nuc l ea r decays , a u n i f i e d d e s c r i p t i o n of a lpha decay , heavy c l u s t e r ra -
d i o a c t i v i t i e s and a new type of symmet r i c f i s s i o n w i t h compact shapes
is g i ven as spontaneous f i s s i o n modes w i t h compact shapes in wh ich one
or both f ragments have magic numbers of a lmost magic numbers magic
r a d i o a c t i v i t y . Making a cu t i n the p o t e n t i a l energy s u r f a c e s f o r f i s -
§ ion a long the mass asymmetry and m i n i m i z i n g in the charae asymmetry
and the shape degrees of f reedom, we can see t h a t i n a d d i t i o n to the
V a l l e y c o r r e s p o n d i n g to mass A ~ 140 w i t h s t r o n g l y a l o n g a t e d f r a g m e n t s ,
ano the r v a l l e y appears at much l a r g e r mass-asymmetry c o r r e s p o n d i n g to
the doub le magic nuc leus 208pb w i t h n e a r l y s p h e r i c a l f r a g m e n t s . The
new v a l l e y g i ves a d d i t i o n a l peaks in the f i s s i o n mass d i s t r i b u t i o n -
SUperasymmetr ic f i s s i o n modes. A s imp le p a r a m e t r i z a t i o n f o r the des-
c r i p t i o n of these modes i n c l u d i n g a lpha decay is used two i n t e r a c -
t i n g spheres - w i t h i n t h ree macroscop ic models ( t he l i q u i d drop model ,
the f i n i t e range n u c l e a r f o r c e s model and the Y u k a w a - p l u s - e x p o n e n t i a l
model) ex tended f o r d i f f e r e n t charge d e n s i t i e s . In the frame of the
l i q u i d drop model an a n a l y t i c a l supe rasymmet r i c f i s s i o n model is q i v e n .
The f o r m a l i s m i n c l u d e s a phenomeno log i ca l s h e l l c o r r e c t i o n based on
the e x p e r i m e n t a l Q-va lues and a zero p o i n t v i b r a t i o n a l energy i n t r o -
duced to d e s c r i b e the o v e r a l l t r ends of the a b s o l u t e h a l f l i v e s va-
l ues . The model was a p p l i e d to a lpha decay and heavy c l u s t e r r a d i o a c -
t i v i t i e s .
Making a cu t i n the p o t e n t i a l energy s u r f a c e , at zero charge and
mass asymmetry a long the neck degree of f reedom, by m i n i m i z i n g i n the
Shape degrees of f reedom of the f ragments we can see t h a t f o r the very
heavy e lements around 264Fm in the p o t e n t i a l energy s u r f a c e s f o r f i s -
Sion appears , in a d d i t i o n to the above two v a l l e y s , a t h i r d v a l l e y co r -
respond ing to the symmetr ic f r a g m e n t a t i o n f o r wh ich both f ragments have
h e a r l y s p h e r i c a l shapes - the two symmetr ic f i s s i o n modes and the super -
asymmetr ic f i s s i o n mode. E x p e r i m e n t a l ev idences f o r c l u s t e r r a d i o a c t i v i -
t i e s and f o r the new f i s s i o n mode are r e v i e w e d .
We c o n c l u d e , t h a t the magic decays (a l pha decay, c l u s t e r r a d i o a c -
t i v i t i e s and the new symmetr ic f i s s i o n mode w i t h compact shapes) are
e s s e n t i a l l y quantum p rocesses . E v i d e n t l y f o r the d e s c r i p t i o n of the
new f i s s i o n mode or f o r the em iss ion of l a r g e c l u s t e r s , due to the f a c t
t ha t the t o t a l k i n e t i c energy of the f ragments i s not e x a c t l y the Q-va- 14C_ 24 lue l i k e i n or Ne - decays , we must a l so i n c l u d e the damping.
322
In §4 we d iscuss some processes appea r i ng in heavy ion c o l l i s i o n s
which may have some quantum e f f e c t s . The f i r s t process is the charge
e q u i l i b r a t i o n mode which we t r e a t as a damped harmonic o s c i l l a t o r in
the charge asymmetry. The p o t e n t i a l was apo rox ima ted w i t h a harmonic
o s c i l l a t o r w i t h the s t i f f n e s s g i ven by the l i q u i d drop. The mass para-
meter was de te rm ined by assuming t h a t the c o r r e s p n n d i n g f r e q u e n c i e s are
g iven by the f r e q u e n c i e s o f an i s o v e c t o r d i p o l e g i a n t resonance o f a
m o l e c u l a r d i - n u c l e a r system. I t is shown t h a t the charge v a r i a n c e s as
a f u n c t i o n of energy loss or t ime are very w e l l rep roduced f o r the
r e a c t i o n s f o r which the c o r r e c t i o n f o r charged p a r t i c l e e v a p o r a t i o n is
not i m p o r t a n t . We conc lude t h a t the s a t u r a t i o n va lues of the charge
v a r i a n c e s are r e l a t e d to the zero p o i n t mot ion of t h i s c o l l e c t i v e mode.
We a l s o s h o r t l y d iscuss the charge and mass e q u i l i b r a t i o n in heavy ion
c o l l i s i o n s .
For o t h e r p rocesses , as a c r i t e r i o n f o r quantum e f f e c t s we con-
s i d e r the t r a n s f e r o f a l a r g e number o f nuc leons a t very low e x c i t a -
t i o n s energy of the c o l l i d i n g p a r t n e r s . In a s i m i l a r way w i t h the c lus"
t e r r a d i o a c t i v i t e s , we expec t in the c o l l i s i o n of two n u c l e i below or
a t the top of the b a r r i e r , co ld rea r rangemen t o f a l a r g e number o f nu-
c l e o n s . Consequen t l y f o r c o l l i s i o n s of l i g h t p r o j e c t i l e s w i t h heavy
t a r g e t s , due to the f a c t t h a t the p r o j e c t i l e tends to fuse w i t h the
t a r g e t , we expec t l a r g e t r a n s f e r o f nuc leons to the t a r g e t l e a d i n g to
the p r o d u c t i o n o f h e a v i e r e lements and the emiss ion of a Dar t of the
p r o j e c t i l e at f o rwa rd ang les° In the f o l l o w i n g we ana l yse both proces-
ses w i t h i n phenomeno log i ca l models assuminq f r a g m e n t a t i o n o f the pro-
j e c t i l e .
For the emiss ion o f c l u s t e r s of i n t e r m e d i a t e mass w i t h 2 ~ Z # 4
we ana l yse the e x p e r i m e n t a l data o b t a i n e d in Dubna f o r the 181Ta + 22Ne, 232Th + 22Ne and 232Th + l IB systems at 8 MeV/nuc leon
and f o rwa rd a n g l e s . We use the Serber model m o d i f i e d to d e s c r i b e the
s e q u e n t i a l f e a t u r e o f the break-up r e a c t i o n and to e s t i m a t e the abso-
l u t e f o r m a t i o n cross s e c t i o n s of v a r i o u s i s o t o p e s . We show t h a t the
p r o j e c t i l e b reak -up is the main mechanism r e s p o n s i b l e f o r the emiss ion
of c l u s t e r s .
In o rde r to d e s c r i b e the m u l t i n u c l e o n t r a n s f e r r e a c t i o n s l e a d i n g
to heavy a c t i n i d e s we use a s imp le model based on p r o j e c t i l e f ragmen-
t a t i o n . The p r ima ry d i s t r i b u t i o n s of i s o t o p e s are o b t a i n e d by assuming
t h a t the c l u s t e r s sepa ra ted from the p r o j e c t i l e are cao tu red by the
t a r g e t as a who le . We show t h a t the p r o d u c t i o n cross s e c t i o n s of Fm-,
Md-, No- and L r - i s o t o p e s o b t a i n e d at LBL and GSI are w e l l d e s c r i b e d
by t h i s model a f t e r c o r r e c t i o n f o r neu t ron e m i s s i o n .
323
F i n a l l y , we d i s cuss the q u a s i - f i s s i o n and the p o s s i b i l i t y o f the
e x i s t e n c e of a new type of ~ u a s i - f i s s i o n i n wh ich both f ragments have
COmpact shapes, i . e . the p o s s i b i l i t y o f the ez i~t~nce of the bimoda~
~ u a s i - f i s s i o n i n comple te ana logy w i t h b imodal f i s s i o n .
We conclude that such an approach of the nuclear dynamics, based on quantum
equations of motion for open systems in few new co l lec t i ve coordinates may g i ve
a u n i f i e d p i c t u r e o f many a p p a r e n t l y d i f f e r e n t p rocesses , lie a l r e a d y
Obta ined by d e c o u p l i n 9 the c o o r d i n a t e s and making cuts i n the po ten -
t i a l a long mass asymmetry and neck deqrees of f reedom, the c l u s t e r
r a d i o a c t i v i t i e s and a new type of f i s s i o n w i t h compact shades both
Phenomena con f i rmed by the e x p e r i m e n t s . In heavy ion c o l l i s i o n s by
making a cu t a long the charge asymmetry c o o r d i n a t e we o b t a i n e d the
Zero p o i n t energy f o r t h i s degree of f reedom. By i n t r o d u c i n g the s h e l l
e f f e c t s i n t h i s c o o r d i n a t e we expec t an add~tioma~ minimum i n the
charge asymmetry p o t e n t i a l . We a l so e x p e c t , by c o n s i d e r i n g the c o u p l i n g
of these c o o r d i n a t e s , th~ p o s s i b l e ex i s tence of char q~-m~ss or neck -
~Somers i n comple te ana logy w i t h the shape i somers . We a l so showed t h a t
large co ld rea r rangemen ts of nuc leons may e x i s t i n heavy ion c o l l i s i o n s .
We conc lude t h a t a l l these phenomena are d i f f e r e n t quantum aspects o f
the n u c l e a r dynamics .
U n f o r t u n a t e l y the p r e s e n t p a r a m e t r i z a t i o n s of the l i o u i d drop mo-
del wh ich are not a l l o w i n g in a n a t u r a l way the f o r m a t i o n o f compact
Shapes and wh ich are not v a l i d at l a r g e charge and mass asymmetr ies d id
not a l l o w the use of the m i c r o s c o p i c models to the many of the above
Phenomena and f o r ced us to use d i f f e r e n t phenomeno log i ca l models f o r
d i f f e r e n t p rocesses . Much work must be done in o rde r to have a un ique
d e s c r i p t i o n o f a l l these processes on a m i c r o s c o p i c b a s i s .
2~ QUANTUM OPEN SYSTEMS
In heavy ion c o l l i s i o n s (HIC) we can s tudy e x p e r i m e n t a l l y the dy-
namics of n u c l e a r m a t t e r . In the l a s t years a l a rge body of expe r imen-
tal data has been accumula ted in t h i s f i e l d I ) wh ich a l l o w s a v i v i d d i s -
~USsion between the two ext reme t h e o r e t i c a l approaches : the t r a n s p o r t
t heo r i es wh ich v iew t h i s process as be inn due to i ndependen t p a r t i c l e
~ o p a g a t i o n thus s t r e s s i n q the s ~ o c h a s t i c ~ random wa lk na tu re of the
r e l a x a t i o n phenomenon 2) and the quantum mechan ica l c o l l e c t i v e theo -
r ies wh ich v iew t h i s process as be ing due to l a r g e sca le c o l l e c t i v e
324
modes thus s t r e s s i n g the cohe ren t na tu re of the r e l a x a t i o n pheno-
menon 3) .
I t is now w i d e l y adm i t t ed t h a t the d e s c r i p t i o n of the f r i c t i o n
in quantum mechanics is f a r from t r i v i a l . Perhaps, the g r e a t e s t d i f -
f i c u l t y a r i s e s from the f a c t t h a t in c l a s s i c a l mechanics the d i s s i p a -
t i o n o f the energy is d i r e c t l y r e l a t e d to the presence o f a nonzero
momentum ( the f r i c t i o n f o r c e is p r o p o r t i o n a l to the v e l o c i t y ) . On the
o t h e r hand i t is known t h a t such systems w i t h fo rces p r o p o r t i o n a l to
the v e l o c i t i e s cannot be d e s c r i b e d by the s tandard H a m i l t o n i a n mecha-
n ics and t h a t the L i o u v i l l e ' s theorem is not v a l i d . Such f o r c e s , which
cause a decrease in the phase space vo lume, are more s u i t a b l e descr ibed
in the frame of the t h e o r y of s t o c h a s t i c p rocesses .
In o rde r to p r e v e n t the f a l l in t ime of any f i n i t e volume in
phase space i n t o a volume s m a l l e r than (~ /2 ) n a d i f f u s i o n process
( s t o c h a s t i c p rocess) which i n c r e a s e s the volume in phase space is need"
ed. An e q u i l i b r i u m s t a t e is a s t a t e in which these two o p p o s i t e tenden"
c ies ba lance . I t f o l l o w s t h a t in o rde r to o b t a i n a Quantum t h e o r y f o r
systems w i t h f r i c t i o n f o r ces i t is necessary to unders tand how a r i s e
such quantum d i f f u s i o n processes which ba lance the f r i c t i o n f o r ces
and p r e v e n t the v i o l a t i o n of the u n c e r t a i n t y r e l a t i o n s .
In the l i t e r a t u r e t he re is an enormous number o f papers which
t r y to so l ve t h i s prob lem by i n t r o d u c t i o n , in a d d i t i o n to the f r i c t i o n
f o r c e s , of some d i f f u s i o n c o e f f i c i e n t s , us ing q u i t e d i f f e r e n t and con-
t r a d i c t o r y arguments .
We adopt the L i n d b l a d a x i o m a t i c way o f i n t r o d u c i n g d i s s i p a t i o n
in quantum mechanics 4 ' 5 ) based on c o m p l e t e l y p o s i t i v e dynamica l semi"
groups w i t h bound g e n e r a t o r s .
We succeeded to o b t a i n 6) : a g e n e r a l i z a t i o n of the fundamenta l
c o n s t r a i n t s on quantum mechan ica l d i f f u s i o n c o e f f i c i e n t s which appear
in the c o r r e s p o n d i n g master e q u a t i o n , a g e n e r a l i z a t i o n of the Hasse
pure s t a t e c o n d i t i o n and a g e n e r a l i z e d Sch r~d inge r type n o n l i n e a r and
n o n - h e r m i t i c e q u a t i o n f o r an open system. Based on the S c h r ~ d i n g e r ,
He isenberg and Weyl -Wigner-Moyal r e p r e s e n t a t i o n s of the L i n d b l a d
master e q u a t i o n s we show t h a t v a r i o u s master e q u a t i o n s f o r the damped
quantum o s c i l l a t o r used in the l i t e r a t u r e f o r the d e s c r i p t i o n o f the
damped c o l l e c t i v e modes in HIC are p a r t i c u l a r cases of the L i n d b l a d ' S
e q u a t i o n and t h a t the m a j o r i t y of these e q u a t i o n s are not s a t i s f y i n g
the c o n s t r a i n t s on quantum mechan ica l d i f f u s i o n s c o e f f i c i e n t s .
The L i n d b l a d ' s master e q u a t i o n s are o f the f o l l o w i n g form:
325
dp _ L(p(t)) = i i N , , dt - ~ - [ H , p ( t ) ] + Zl~ ~ ( [ Y j o ( t ) , V j ] + [ V j , p ( t ) V j ] ) , j=1
(I) where p ( t ) is the d e n s i t y m a t r i x and Vj the c o r r e s p o n d i n g bound ope-
r a t o r s .
For the damped quantum harmon ic o s c i l l a t o r w i t h unbound Vj o f
the form Vj = a j p + b j q , j 1 ,2 , where a j ,b i_ are complex numbers and q
and p, the usual o p e r a t o r s w i t h the commuta t ion r e l a t i o n [ q , p ] = i~
eq. ( I ) becomes
d p : L ( p ( t ) ) (2) d t
where
and
i i (X-u)_ [ p , p q + q p ] _ iX L (p ) = - y6-[Ho,P]- 241 ~ rq ,pp + pp] -
D D ( + ~-~ [q'[q'P]]- ~-~ [ p , [ p , p ] ] + Dpq Dqp) [P [ q , p ] ] 412 •
2 i 2 2 (pq +
H ° = ~ p + ~ q , H = H o qP) Z
(3)
Here we denoted by Dpp, Dqq, Dpq and Dqp the d i f f u s i o n c o e f f i -
c i e n t s and by ~ the f r i c t i o n c o n s t a n t , a l l the p a r a m e t e r s be ing r e a l and
r e l a t e d to the complex numbers a i and bj by the r e l a t i o n s
2 ~ 2 Dqq = -~ F. l a j l 2 • D = ~ Ib 12
j=1 ' PP ~- j=1 J 2 2
Dpq = Dqp = - ~ Re ~ a j b j , x = - Im s j = l j = l
~ j b j (4)
where ~ j means the complex c o n j u g a t e o f a j .
Thus, the f o l l o w i n g c o n s t r a i n t s f o r the quantum mechan i ca l d i f -
f u s i o n c o e f f i c i e n t s r e s u l t :
• D 2 X2~ 2 O" Dpp ~ O; DqqDpp Pq ~ ~ (5) Dqq
the l a s t i n e q u a l i t y be ing a consequence of the Schwar tz i n e q u a l i t y
2 )2 2 )2 2 2 2 (Re s ~ j b j + ( ~ m ~ ~ I b S S la I X j~__ I b j l 2 ( 5 ' )
j=1 j=1 J j j=1 J --i
In the He i senbe rg r e p r e s e n t a t i o n the mas te r eq. (1) i s :
326
= L(A(t)) = ~[H,A(t)]+ ~ s(V~[A(t),Vj]+[V~,A(t)]Vj). (6) d t j ,,
I f t he t ime e v o l u t i o n o f the o p e r a t o r M = ~V~Vj i s c o n s i d e r e d we d ~
o b t a i n f rom the 2 - p o s i t i v i t y o f t he d y n a m i c a l s e m i g r o u p g e n e r a t e d by
(6) t h a t
M ( t ) a Z V j ( t ) * V • j ( t ) (7) J
From t h i s r e l a t i o n an i n e q u a l i t y f o r mean va lues of M, Vj and Vj
f o l l o w s i m m e d i a t e l y i f we obse rve t h a t the map A ÷ TrpA i s a l s o 2 - p o s i -
t i v e .
TrpM(t) ~_ ~ Tr(pVp(t) ) Tr(pVj(t)). (8) J
By the d u a l i t y T r p ( A ( t ) ) = T r p ( t ) A r e l . (8 ) becomes:
T r ( p ( t ) z V~Vj ) a z T r ( p ( t ) V~ J J j ) T r ( p ( t ) V j ) (9)
T h i s i n e q u a l i t y i s a g e n e r a l i z a t i o n o f the i n e q u a l i t y (11) f rom
r e f o 7 to a l l M a r k o v i a n mas te r e q u a t i o n s .
For t he damped quantum h a r m o n i c o s c i l l a t o r , the above i n e q u a l i t y
becomes:
4i 2 Dqq~pp(t) + DppOqq( t ) 2Dp Opq(t) -> X 10)
where we have used the n o t a t i o n s
: T r ( p ( t ) q ) " u : T r ( p ( t ) p ) " q ' p , = T r ( p t ) q 2) - o ( t ) 2 qq
= T r ( p ( t ) p 2 ) -o ( t ) 2" { ( t ) : T r ( p ( t ) ( P - q - , ~ ) ) - ~ ( t ) o ( t PP P , pq P q
11)
The e q u a l i t y i n r e l . (9) i s a n e c e s s a r y and s u f f i c i e n t c o n d i t i o n
f o r p ( t ) to be a pure s t a t e f o r t he t ime t > O. I n d e e d , the c o n d i t i o n
p 2 ( t ) = p ( t ) w h i c h i s a n e c e s s a r y and s u f f i c i e n t c o n d i t i o n f o r p ( t )
to be a pure s t a t e 8 -9 ) g i v e s T r p ( t ) 2 = I f o r a l l t > O. Th i s i m D l i e s :
d )2 ~-~ T r ( p ( t ) = T r m ( t ) L ( p ( t ) ) = 0 t > 0 (12)
or by u s i n g the e x p l i c i t fo rm o f L ( p ( t ) ) g i v e n by eq. ( I )
327
i ~ ( T r ( p ( t ) V j p ( t ) V ; ) - T r ( p ( t ) 2 V ? V . T r ( p ( t ) L ( p ( t ) ) = ~ - J J 'I ) (13)
and the c o n d i t i o n s p 2 ( t ) = p ( t ) and p ( t ) A p ( t ) = T r ( p ( t ) A ) p ( t ) we have
T r i p ( t ) jzVjVj) = J~ T r ( p ( t ) V j ) T r ( p ( t ) V * j ) (14)
This e q u a l i t y is a g e n e r a l i z a t i o n of the Hasse pure s t a t e cond i - 8-9 t i on to a l l Markovian master equa t ions .
2( = The c o n d i t i o n p t ) p ( t ) imp l i es f i r s t l y t ha t p(t)@ = ( # ( t ) , @ ) • # ( t ) f o r any wave f u n c t i o n ¢ and secondly t ha t i t s d e r i v a t i v e
d p ( t ) d p ( t ) 2 dt = dt = L ( p ( t ) ) p ( t ) + p ( t ) L ( p ( t ) ) (15)
is e q u i v a l e n t w i th the f o l l o w i n g Schr~d inger type n o n l i n e a r and nonher-
m i t i c equa t ion :
= - (H + i JS ( ~ ( t ) , V i e ( t ) ) V.j -
, * V - * .V ( ~ ( t ) Sj Vj j @ ( t ) ) ½ JS Vj j ) @(t) (16)
This r e s u l t s i s g e n e r a l i z a t i o n to a l l Markovian master equat ions of the r e s u l t s ob ta ined f o r p a r t i c u l a r master equat ions in r e f s . 10 and 7,8 .
For the damped quantum harmonic o s c i l l a t o r the new "Ham i l t on i an " is
H + X(Op( t )q Oq( t )p ) +
Dqq 2 D 2 + i { x~ - T ((p - ° D ( t ) ) + ~ p p ( t ) ) - - ~ ( ( q - o q ( t ) ) + O q q ( t ) ) +
D + - ~ ( ( p - ~ q ( t ) ) ( q - o q ( t ) ) + ( q - o q ( t ) ) ( p - o p ( t ) ) + 2 O p q ( t ) ) l (17)
I t is i n t e r e s t i n g to remark tha t the mean va lue of t h i s new "Hamil t on ian " is equal to the mean value of the o r i g i n a l Hami l ton ian H i f the e q u a l i t y is v a l i d in the i n e q u a l i t y (10) . In t h i s l a s t case the new
"Ham i l t on i an " is equal to
i )2 ( q - ~ q ( t ) ) H + X(Op( t )q - Oq( t )p ) - ~ { D q q ( p - ~ p ( t ) + Dpp
+ Dpq((p - O p ( t ) ) ( q - u q ( t ) ) + ( q - e q ( t ) ) ( p - ~ p ( t ) ) - ~ - ~ ]
2 +
(18)
328
This result, from the physical point of view, is quite natural
since the average value of the new "Hamiltonian" of the nonlinear and
nonhermitic Scbr~dlnger equation descrihing the open system must give
the energy of the open system. Another possible representation of the Lindblad master equa-
tion is the Weyl-Wigner-Moyal representation. This is a phase-space
representation of the quantum mechanics. Roughly speaking such a repre-
sentation is a mapping from the Hilbert space operators to the functions
on the classical phase space in such a way that i f A is mapped into
fA(x,y) and p is mapped onto fp(X,y), then
Tr(pA) = f f fp(X,Y)fA(x,y ) dxdy (19)
This representation can be easily obtained by using Wigner map- ping of the density operators p(t) from the Hilbert space onto the
functions fp(t~(x,y), on the classical phase space
I - ® -~(xn-y{) f ( x , y , t ) = fp(t~(x,y),, (2x~)~r- -,f -~f e Tr(p(t)W({,n)d{dn
(z0) where W(~,n) is the Weyl operator.
Indeed, taking the time derivative of the Wigner function (20), using the master equation in the Heisenberg representation (6) and the expl ic i t action of the dynamical semigroup on the Weyl operators we obtain:
. _ )Bxf(x,y, t) = y @f(x,y,t) + m 2xBf(x,y,t) + (X.u + @t m @x - @y @x
~@ f x _ ~ @2f(x'y't) + D @2f(x'~ ' t ) + (~+~, @y + Dqq Bx 2 pp- @yZ +
B2f(x 'y ' t ) (21) + 2Dpq BxBy
This equation looks very classical, l ike an equation of the Fokker-Planck
type, but we must be very careful with the i n i t i a l function f(x,y,O) on
the phase space which must be a Wigner transform of a density operator in order to keep the quantum mechanical properties of the system.
Because the most frequently choice for f(x,y,O) is a Gaussian
function and because eq. (21) preserves th~s Gaussian type, i . e . ,
f ( x , y , t ) is also a Gaussian function, the differences between the quan-
tum mechanics and classical mechanics are completely lost in this repre- sentation of the master equation. This is a possible explanation for the frequently occured ambiguities on this subject in the l i terature.
329
In the following we Show that various master equations for the
damped quantum osc i l l a to r used in the l i t e ra tu re for the description of
the damped col lect ive modes in HIC are par t icu lar cases of the Lindblad
equation and that the majority of these equations are not sat isfy ing the
constraints on quantum mechanical d i f fusion coef f ic ients .
Indeed, in the form (3) a di rect comparison with eo. (1) from
refs. 7, 10, 11 is possible. I t follows that this master equation sup-
Plemented with the fundamental constraints (5) is a par t icu lar case of
eq. ( 3 ) , when ~ = ~.
A lso a p a r t i c u l a r case of eq. (3) i s the master equa t i on (12) con-
S idered in r e f . 12 f o r ~ : y(m)/2m = u; Dqq = O; Dpp = y(m)T (m); and
Dpq = O. E v i d e n t l y the c o n s t r a i n t s (5) are not s a t i s f i e d .
A n a l o g o u s l y , the master equa t i on (A.36) cons idered in r e f . 9 is
a p a r t i c u l a r case ° f eq" (3) f ° r ~ = ~ ~ ~ /2 ; Dpp = D; Dqq = O; Dpq = I 2 the c o n s t r a i n t s Dqp = -d /2 and H 0 = p2/2m + I/2m~m 2- q Again (5)
are not s a t i s f i e d .
In the form (21) a d i r e c t comparison w i t h two k inds of quantum
master e q u a t i o n s , w r i t t e n f o r the Wigner t r a n s f o r m of the d e n s i t y ma-
t r i x , ob ta ined r e c e n t l y in r e f . 13 is p o s s i b l e .
The f i r s t master equa t ion (see eq. (5 .1 ) of r e f . 13 ) i s a p a r t i c u -
l a r case of eq. (21) f o r ~ = ~ = ? /2 ; Dpp = D/2; Dqq = O; Dpq = Dqp =B/2 Amm q2 and H = H o - T + f ( t ) q . Evidently the constraints (5) are not
sat is f ied.
The second master equation (see eq. (5.6) of ref . 13) is also a
part icular case of eq. (21) for u = 0", ?lip = FIIRAI~•- ~; Dpp = -½ D lIp ",
Dqq ~ D~ I = 0 and H = H 0 - A I I ~'-m~ 2 ~ p2 = ; Dpq q - + f ( t ) q . Th is
equa t ion s a t i s f i e s the fundamenta l c o n s t r a i n t s ( 5 ) ,
The above procedure can be s t r a i g h t f o r w a r d g e n e r a l i z e d to many 14) d imensions . A l so , we can e a s i l y ob ta i n the e x p l i c i t t ime-dependence
of the e x p e c t a t i o n va lues and of the Wigner f u n c t i o n .
For example in two d imens ions , we take the ope ra to rs V j ( j = 1 , 2 , 3 , 4 )
in the form
2 2 Vj = g ajkPk + ~ bjkqk (22)
k=l k=l
Where a jk and b jk are complex numbers, and the H a m i l t o n i a n H in the form
of two coupled o s c i l l a t o r s 2
i ? mkWk 2 H = k~1( m~2~k p~ +-~-~-qk ) + k12 plP2 +
330
2 + D '3
kl,k2=l ~klk 2 (Pk2qk 1 qk2'kl) + 12qlq2
By i n t r o d u c i n g the a b b r e v i a t i o n s (k = 1,2)
2 I 2 mkWk 2
Hok = 2m k Pk + ~ qk
Dqkq p Dqpq k = ~ Re(aka ~)
kp~ =D = Re(bkb ) Dp PuPk
:
DqkPu qk = - 7
~12 = - ~21 = " Im(a la2)
B12 = - ~21 = - Im(b lb2)
Lk~ = - Im(akbu)
and denoting the vector with the four components by ~ ( t ) and the fo l lowing 4x4 matr ix by ~ ( t ) :
, O'q , ql 2 ~q3
(23)
(24)
and o q4
~( t ) :
~q °q lq l ~qlq2 IPl qlP2
d °q2ql Oq2q2 q2Pl ~q2P2
~p OPlql ~Plq2 IPl PlP2
~P2ql °P2q 2 ~P2Pl °P2P 2
we obtain via d i rec t ca l cu la t i on where
do
- ~ i I + ~ I I
-~21 + u21
2 - ml~ I
-612 - v12
-~ 12 + ~ 12
-~22 + ~22
612 " ~12
2 -m2~ 2
Y~
1/mi - ~ 1 2 + k12
~12 + k12 I/m2
-~11 "~11 "L21 - P21
-x12 -~12 -X22 - ~22
(25)
(26)
27)
331
From Eq. (26) i t fo l lows that
~ ( t ) = M(t) q~(O) = exp( tY) ~(0) (28)
where ~ 0 ) is given by the i n i t i a l cond i t i ons . The mat r i x M(t) has to f u l f i l the cond i t i on
lim M(t) = 0 (29) t~m
In order tha t th i s l i m i t e x i s t s , Y must have only eigenvalues With negat ive real par ts .
A lso, v ia d i r e c t ca l cu l a t i on we obta in
d__S_s = ~ + ~T + 2D (30) dt
Where D is the mat r i x of the d i f f u s i o n c o e f f i c i e n t s
Dqlq I
Dq2a I 6:
Dplq I
Dp2q I
Dq Dq D 1 lq2 IP l qlP2
Dq2q 2 Dq2P I Dq2P 2
D Dplq 2 PlPl DplP 2
D D D P2q2 P2Pl P2P2
(31)
and yT the transposed mat r i x of Y. The t ime-dependent so l u t i on of Eq. (30) can be w r i t t e n as
~ ( t ) : M(t ) (~(O) - ~o ) MT(t) + ~o (32)
Where M(t) is def ined in Eq. (28). The mat r i x ~o is t ime- independent and solves the s t a t i c problem of Eq. (30) (d~/d t = 0) :
Y~o + ~o ~T + 26 = 0 (33)
Now we assume tha t the f o l l ow ing l i m i t s e x i s t fo r t ~ :
~(~) = l i m ~ ( t ) t ~
In tha t case i t fo l lows from (20) wi th Eq. (29):
(34)
332
£(~) : ~o ( 3 5 )
I n s e r t i n g Eq. (35) i n t o Eq. (32) we ob ta i n the bas i c equa t ion f o r our
purpose:
~ ( t ) = M ( t ) ( ~ ( O ) - ~ (~) ) MT(t ) + ~(~) (36)
where
#~( . ) + ~(=) ~T = _ 26 (37)
The e x p l i c i t t ime-dependence of the Wigner f u n c t i o n in two dimen-
s ions i s
f ( x l , x 2 , Y l , Y 2 , t ) = ( d e t ( 2 ~ s ( t ) ) ) - I / 2 x
x exp ( - ½ (5 - ~ ( t ) ) z ( t ) - I (~ - ~ ( t ) ) )
where ~ = ( x l , x 2 , Y l , Y 2 ) w i t h the i n i t i a l c o n d i t i o n f ( x l , x 2 t=O) which i s the we l l known r e s u l t f o r Wigner f u n c t i o n s 1 5 - 1 7 ) - ' Y I ' Y 2 '
In t h i s way we have been ab le to w r i t e a system of coupled equa-
t i o n s f o r a damped quantum system in many d imensions and even more, f o r
the harmonic a p p r o x i m a t i o n of the p o t e n t i a l , we succeeded to f i n d e x p l i -
c i t exp ress ions f o r the t ime dependence of the e x p e c t a t i o n va lues and of
the Wigner f u n c t i o n , i . e . we have g iven a mathemat ica l f ramework f o r the
nuc lea r dynamics i n c l u d i n g the quantum e f f e c t s .
F i n a l l y we should l i k e to s t r e s s t h a t the quantum c o l l e c t i v e f l u c -
t u a t i o n s have not been revea led w i t h c l a r i t y by expe r imen t . Now i t i s
c l e a r t h a t , due to the s i m i l a r i t y of the equa t ions and s o l u t i o n s in
both extreme t h e o r e t i c a l approaches: t r a n s p o r t t h e o r i e s and quantum col"
l e c t i v e t h e o r i e s , the e f f e c t s are s i m i l a r . We cons ide r t h a t i t i s prema"
tu re to conc lude , l i k e the m a j o r i t y of the recen t papers i ) , t h a t the
p resen t data suggest t h a t the dynamical e v o l u t i o n of the d i n u c l e a r sys-
tem may be seen as an independent p a r t i c l e exchange process c o n s t r a i n e d
by the u n d e r l y i n g p o t e n t i a l energy su r f ace (PES).
3. MAGIC DECAYS
The purpose of t h i s chapter is to p resen t a u n i f i e d d e s c r i p t i o n of
a lpha decay, heavy c l u s t e r r a d i o a c t i v i t i e s and the new type of symmetr ic
f i s s i o n w i t h compact shapes in which one or both f ragments have magic
333
numbers or almost magic numbers magic r ad i oac t i v i t y , We consider main-
ly the s ta t i c aspects of the nuclear evolut ion based essent ia l l y on the
shel l effects in the potent ia l . Simple assumptions are made for the
ine r t ia parameters. No diss ipat ion has been introduced.
The s i m i l a r i t y between f i ss ion and alpha decay was recognized in
the early stages of the f i ss ion theory 18-20). Nevertheless, the theo-
ries of these phenomena were developed on essent ia l l y d i f f e ren t grounds.
Nuclear reaction microscopic methods have been used in the theory of
alpha decay and the phenomenological l iqu id-drop model (LDM) in nuclear
f i ss ion theory.
The asymmetric d i s t r i bu t i on of the fragment masses from the spon-
taneous or low exc i ta t ion energy induced f i ss ion was a long-standing
puzzle of the theory. The f i r s t attempt to consider both the co l lec t i ve
nature of the nucleonic motion and the single par t i c le effects by adding
the shell corrections to the LDM energy 21) lead to a good estimation
of nuclear ground state (gs) deformations. The next important step, pro-
ducing a renewed in terest for the development of the f i ss ion theory was
the idea of deformed nuclear shel ls and the microscopic shel l correc-
t ion method 22). The experimental discovery 23) of f i ss ion isomers
strongly stimulated this f i e l d . Secondary minima in the co l lec t i ve po-
t en t i a l (shape isomers) and the i r ef fect on the low-energetic co l lec- 24)
t ive nuclear structure were already considered in 1965 by Greiner
By considering the shel l effects i t was shown that in the poten-
t i a l energy surfaces for f i ss ion appears a val ley corresponding to the
mass A ~ 140 with strongly alongated fragments. In th is way i t was pos-
s ib le to explain q u a l i t a t i v e l y the f i ss ion asymmetry 25-29) as being 30-31) e s s e n t i a l l y dominated by s h e l l s t r u c t u r e . A l s o , i t was shown
t h a t f o r ve ry heavy e lements around 264Fm in the p o t e n t i a l energy s u r -
faces f o r f i s s i o n appears , in a d d i t i o n to the above v a l l e y , ano the r
V a l l e y c o r r e s p o n d i n g a l so to symmet r i c f r a g m e n t a t i o n but i n wh ich both
f ragments have n e a r l y s p h e r i c a l shapes - the b imodal symmet r i c f i s s i o n
modes ( t he usua l spontaneous f i s s i o n and the S n - d e c a y ) . S i g n i f i c a n t
p rog ress was ach ieved w i t h the deve lopment o f the two c e n t e r s h e l l mo-
del (TCSM) by the F r a n k f u r t schoo l 32-33) and i t s e x t e n s i o n to asymme-
t r i c b reak -ups (asymmet r i c two c e n t e r s h e l l model - ATCSM). I t a l l o w e d
to f o l l o w the s h e l l s t r u c t u r e a l l the way f rom the o r i g i n a l nuc leus
over the b a r r i e r i n t o the i n d i v i d u a l f r a g m e n t s . The ATCSM can be c o n s i -
dered as the bas i c s h e l l model f o r a l l f i s s i o n and heavy ion (quas imo-
l e c u l a r ) phenomena. The f r a g m e n t a t i o n t h e o r y 34) based on the two cen-
t e r s h e l l model , was s u c c e s s f u l i n d e s c r i b i n g both r e g i o n s of low and h igh mass asymmetry 35-38) I t was shown, t h a t in the p o t e n t i a l energy
334
s u r f a c e s f o r f i s s i o n i n a d d i t i o n to the above two v a l l e y s e x i s t s ano-
t h e r v a l l e y at much l a r g e r mass-asymmetry c o r r e s p o n d i n g to the doub le
magic nuc leus 208pb w i t h f ragments hav ing n e a r l y s p h e r i c a l shapes. The
new v a l l e y g i ves a d d i t i o n a l peaks in the f i s s i o n mass d i s t r i b u t i o n -
the supe rasymmet r i c f i s s i o n modes.
For the a s y m p t o t i c r e g i o n R > R c the p o t e n t i a l V(R ~ Rc,n ) can be
computed e a s i l y as the sum of the Coulomb i n t e r a c t i o n and the g round -
s t a t e b i n d i n g e n e r g i e s of the two n u c l e i , B ( A i , Z i ) 39)
As an example, we show in F i g . I the p o t e n t i a l energy s u r f a c e f o r 252
No, i n the l e n g t h 1 and the mass asymmetry n c o o r d i n a t e s m i n i m a l i z e d
r e l a t i v e to the d e f o r m a t i o n of each f ragmen t (61 ,~2 ) and the neck para-
meter c. We can see t h a t two s h e l l model v a l l e y s appear , in t h i s coo r -
d i n a t e m a i n l y due to t he double magic s t r u c t u r e of 132Sn and 208pb, the
f i r s t v a l l e y w i t h s t r o n g l y a l onga ted f ragments and the second one w i t h
n e a r l y s p h e r i c a l f ragments 36-38) The mass d i s t r i b u t i o n s r e s u l t i n g
f rom the f i s s i o n of e x c i t e d medium heavy n u c l e i have been d i scussed 4o)
r e c e n t l y
-0820 21 2,?. 28 Z eng/h i-Pro]
Fig. I Potential surface for 252No as function of length of the nucleus 1 and the mass asymmetry coordinate q, The two valleys are clearly seen on the picture.
335
We should l i k e to ment ion t h a t in the f i s s i o n process s t a r t i n g
from i n s i d e the p o t e n t i a l both mass-asymmetry v a l l e y s have been obser -
Ved e x p e r i m e n t a l l y , the most symmetr ic one through the mass d i s t r i b u -
t i o n of the f i s s i o n f ragments and the most asy i~metr ic one through the 14 C 24No and r a d i o a c t i v i t i e s
A s i m i l a r s i t u a t i o n e x i s t s f o r the shape degree of freedom i n -
c l ud i ng the neck. For ve ry heavy e lements around 264Fm, we expect on
the p o t e n t i a l energy su r f ace m i n i m a l i z e d on the shape c o o r d i n a t e s at
symmetr ic f r a g m e n t a t i o n (n=O) c lose to the s c i s s i o n c o n f i g u r a t i o n two
v a l l e y s in the shape degree of f reedom. One v a l l e y cor responds to the
usual spontaneous f i s s i o n w i t h s t r o n g l y a longa ted f ragments and the
Other one, to a new type of f i s s i o n w i t h n e a r l y s p h e r i c a l f r agmen ts .
As an example, we g ive in F ig . 2A the p o t e n t i a l energy su r f ace of 264Fm,
m i n i m a l i z e d f o r a r a t h e r u n r e s t r i c t e d p a r a m e t r i z a t i o n of the nuc lea r
su r face shape 26) as a f u n c t i o n of the hexadecapole de fo rma t i on m4
at f i x e d va lue of the parameter m= 0.98 which in the l i q u i d drop model
corresponds to the shapes f o r which s c i s s i o n takes p lace . We can see
two m in ima , t he f i r s t q u i t e deep and the second one very s h a l l o w . The
co r respond ing shapes are drawn in F ig . 2B, the deep minimum corresponds
to n e a r l y s p h e r i c a l f ragments and the sha l l ow minimum to s t r o n g l y a l l o n -
gated f ragmen ts . A r e l a t i v e l y high b a r r i e r between them a l l ows to de-
f i n e two types of f i s s i o n 31) Of course , we expec t o the r shape v a l -
leys at l a r g e r mass asymmetry va lues when one of the f ragments is c lose
to a double magic nuc leus .
The c l a s s i c a l paper of S~ndulescu, Poenaru and Gre i ne r e n t i t l e d
"New type of decay of heavy nuc l e i i n t e r m e d i a t e between f i s s i o n and
m-decay" s t a r t e d the f i e l d of c l u s t e r r a d i o a c t i v i t y of nuc l e i 38) I t
con ta ined a l l r e l e v a n t aspects of the phenomenon. Four t h e o r e t i c a l mo-
dels have been developed assuming f o r the mechanism of these new r a d i o -
a c t i v i t i e s e i t h e r emiss ion of a pre formed c l u s t e r or a very asymmetr ic
f i s s i o n p rocess . New superasymmet r i c peaks co r respond ina to 208pb and
the p a r t n e r of 208pb were p r e d i c t e d in the f i s s i o n mass d i s t r i b u t i o n s
of heavy n u c l e i , based on the f r a g m e n t a t i o n theo ry (FT) . E igh t even-even (ee) c l u s t e r s (14C, 24Ne, 28Mg, 3 2 ' 3 4 S i , 46Ar and 48'50Ca) have been
i d e n t i f i e d in the p e n e t r a b i l i t y spec t ra of 16 e-e n u c l i d e s ( v a r i o u s i s o -
topes of the e lements Ra, Th, Pu, Cm, Cf , Fm and No) as the most proba-
b le em i t t ed c l u s t e r s from these p a r e n t s . A numer ica l superasymmet r i c
f i s s i o n model (NSAFM) d e r i v e d f o r b i n a r y systems w i t h charge asymmetry
d i f f e r e n t from the mass asymmetry was deve loped , to compute the h a l f - l i f e . The model was used to desc r i be both the m-.decay process and the
new c l u s t e r decay modes.
336
r ' ' - - 4 0
u. 20
0
- ', ".-1o,. -
, i i
-0.~ -0.2
i I i I i
26'F$. ~
oc = 0. 98 .4
,..;'2" s~,a# f
e° • ##
- ~ '~ -...E(,minocs,oca~,o}
0 .0 8.2 d.~ oc L
(I
b
A B
Fig. 2 A) The deformation energy(in MeV) in the liquid-drop mo- del, ELD (dots) and taking into account the shell cor- rection, E, (dashed and solid curves) as a function of the hexadecapole deformation ~4 at the fixed deforma- tion ~ = 0.98. The solid curve with open points corres- ponds to the minimum of E with respect to ~6, ~8 and
~I0" B) The nuclear shape in the minimum with respect to ~6,
~B and ~I0 at ~ = 0.98 and ~4 = -0.093 (a) and ~4 = 0.I (b)o A half-volume sphere is shown by dots for comparison.
In a systematic search one has to take in to considerat ion a very
large number of combinations parent - emitted c lusters. For th is purpose)z
an ana ly t i ca l f i s s ion model (AFT4) was derived 4!) and i t was extended 4
to account for angular momentum and small exc i t a t i on energy e f fec ts . In
analogy with the fragmentation theory, a simple rule governing these
processes was found: for a given emitted c lus te r , the maximum value of
the emission rate (minimum h a l f - l i f e ) is obtained for the daughter ha-
ving magic neutron and proton numbers (N I = 126, Z 1 = 82) or not very
far from these. The rule was confirmed when other emitted clusters with
increasing Z 2 were considered. A comprehensive l i s t , containing more
than 140 predicted new decay modes was reported recent ly by Poenaru
et a l . 43) The branching ra t ios r e l a t i v e to m - decay, or the abso-
lute values of the pa r t i a l h a l f - l i v e s determined up to now in the expe-
riments, are in agreement (wi th in 1.5 orders of magnitude) with the estimations made in the framework of AFM. Deta i ls about the predictionS
337
based on t h i s model are g i ven in the l e c t u r e of Dr. Poenaru at t h i s
s c h o o l . Shi and S w i a t e c k i 44) deve loped a l so a supe rasymmet r i c f i s -
s i on model based on the p r o x i m i t y + Coulomb p o t e n t i a l and an a n a l y t i c a l
i n t e r p o l a t i o n of the p o t e n t i a l b a r r i e r i n the o v e r l a p p i n d r e g i o n , a l l o w -
ing to compute the b r a n c h i n g r a t i o s r e l a t i v e to the a lpha decay in
agreement w i t h the e x p e r i m e n t a l r e s u l t s .
In the p r e s e n t c h a p t e r we d i s c u s s a p o s s i b l e g e n e r a l i z a t i o n 4 5 ) . A s
we know the main r o l e i n the magic r a d i o a c t i v i t y i s p layed by the s h e l l
e f f e c t s i n the f i n a l s t a t e . C o n s e q u e n t l y , f o r heavy n u c l e i we have h igh
em iss i on r a t e s when one o f the f ragments i s c l ose to the magic numbers
N = 126 and Z = 82 and f o r medium mass n u c l e i when one of the f ragmen ts
i s c lose to the magic numbers N=82, 50, 28 and Z=50, 28, 20.
I t i s q u i t e n a t u r a l to ask i f such decay modes are not e x i s t i n g
f o r doub le magic numbers Z=50, i . e . f o r ve ry heavy Fm- iso topes or c l ose
to t h i s ext reme case, when both p a r t n e r s are s t r o n g l y bound. The problem
c o n s i s t s i f the h e i g t h of the b a r r i e r , wh ich is r a t h e r h igh f o r such
symmet r i c f r a g m e n t a t i o n s , is compensated or not by the c o r r e s p o n d i n g
l a r g e Q - v a l u e s . The m i c r o s c o p i c c a l c u l a t i o n s i n d i c a t e the p o s s i b i l i t y
o f such a decay 3 0 - 3 1 ) . I f we look as b e f o r e from the p o i n t o f v iew o f
mass d i s t r i b u t i o n s , such a decay mode w i l l be very s i m i l a r w i t h the
spontaneous f i s s i o n . F o r t u n a t e l y , t h e r e are some o t h e r c h a r a c t e r i s t i c s
of t h i s decay mode, c a l l e d in the f o l l o w i n g S n - r a d i o a c t i v i t y , due to
the f a c t t h a t at l e a s t one of the f ragments must be an i s o t o p e of Sn,
Which makes t h i s process d i f f e r e n t f rom the spontaneous f i s s i o n .
In o rde r to make them c l e a r we w i l l c o m p a r a t i v e l y r ev i ew the c r i -
t i c a l moments a s s o c i a t e d w i t h the two p rocesses : the spontaneous f i s -
s i on and the S n - r a d i o a c t i v i t y o In both processes the t o t a l a v a i l a b l e
energy i s g i ven by the d i f f e r e n c e between masses in the i n i t i a l channel
( f i s s i o n i n g n u c l e u s ) and the f i n a l channe l ( f i s s i o n f r a g m e n t s ) . In f i s -
s i on the energy ba lance i n c l u d e s the TKE o f the f ragments and t h e i r p r i -
mary e x c i t a t i o n e n e r g i e s . These p r i m a r y e x c i t a t i o n e n e r g i e s are o b t a i n -
ed on the accoun t o f the d e f o r m a t i o n of the f ragments at s c i s s i o n and
on the d i s s i p a t i v e f o r c e s t h a t ac ted up to t h i s p o i n t . Th is energy is
e v e n t u a l l y r e l e a s e d by p a r t i c l e e v a p o r a t i o n and ~ e m i s s i o n . The d e f o r -
mat ion of the f i s s i o n i n g nuc leus and the c o r r e s p o n d i n g appearence of the
neck ensures in the frame of the l i q u i d drop model the l o w e r i n g of the
b a r r i e r h e i g h t and hence the p e n e t r a t i o n o f t h i s b a r r i e r w i t h r e l a t i v e l y
low TKE.
In the o t h e r p rocess , the S n - r a d i o a c t i v i t y , the s c i s s i o n p o i n t is
at a more compact c o n f i g u r a t i o n as the f r agmen ts are c o n s i d e r e d s p h e r i -
c a l . I n c l u s i o n o f the f i n i t e range f o r c e s (Yukawa + e x p o n e n t i a l or p ro-
338
ximity forces) ensures a drastic decrease of the Coulomb forces at
short distances so that the barr ier is considerable lowered. Another
character ist ic of the new Drocess is that the fragments are almost com-
p le te ly unexcited, the dissipat ive forces could act only along the new
path with compact shapes which is much shorter than the spontaneous f i s -
sion path with alongated shapes. Consequently, the compactness of the
scission configurations together with the absence of the fragment exci-
tat ion w i l l lead to values of the k inet ic energy of the fragments much
higher than in the f iss ion process, almost approaching the reaction
Q-value. Microscopically the introduct ion of the potent ial energy sur-
faces in the neck and the re la t i ve distance between the fragments w i l l
allow to describe the TKE-distributions and the two main paths corres-
ponding to compact and alongated shapes. The dynamical treatment must
give the d is t r ibu t ion of the scission point shapes, i .e . the TKE-dis-
t r ibut ions in a s imi lar way with the treatment of the mass d is t r ibu-
tions 38)
Based on the microscopic calculations 31) and on the above con-
siderations we can apply the AFM for symmetric fragmentations by assu-
ming that not only the incorrect height of the barr ier but also the
energy loss can be renormalized with the help of the zero point vibra- 45)
tions
In Fig. 3A the systematics of the spontaneous ha l f - l iVes of even
Fm-isotopes is presented together with calculated ha l f - l i ves for Sn-
emission of the same isotopes. The s-emission hal f - l iVes are also pre-
sented. We can see that for l i gh t Fm-isotopes the probab i l i t y for Sn-
emission is many orders of maqnitudes below that for the spontaneous
f iss ion. For the heavier isotopes the two curves come closer. We expect
the shortest ha l f - l i ves for 264Fm which could decay in two double magic
nuclei I~sn82,^^ af ter which the ha l f - l i ves must increase again reach-
ing the ms region at N ~ 168-170. In Fig. 3B we can see that the
same behaviour remains val id for the other very heavy elements. Due to
the fact that we expect that the ha l f - l i ves w i l l decrease up to N=164
(double N=82 closed shells) we conclude that the ha l f - l i ves of the very
heavy elements with 158 ~ N ~ 170 are shorter than ms. That can explain
the lack of success to produce superheavy elements with longer ha l f -
l ives.
We expect a s imi lar s i tuat ion for double magic numbers N=50, for !80H9" 90 example f i s s i o n from e x c i t e d s ta te of in two 40Zr50 n u c l e i . Of
course, the same game can be played w i th o ther magic numbers.
Only four years a f t e r the p r e d i c t i o n s , exper imenta l ev idences fo r
one of the new decay modes, 14C r a d i o a c t i v i t y of 223Ra, was pub l i shed
15 I#
12
..-,,10 ~ a
2 o
-2
339
~ 18 ~
14
#
I 2
4 I 1 t
, , • , • ' " ~ ' ~ - 2 I42 146 150-Iid~ I 5 8 N~
-fm
Fig. 3 A)
B)
Logarithm of the half-lives for spontaneous fission (x), alpha decay (A) and Sn-emission (e) of Fm-iso- topes. For Sn-Emisslon only the first two combinations of clusters with shortest half-lives are given. We can see that half-liVes for all three processes become comparable for the heavier isotopes. 242Fm(120S n + 122Sn ' 104Ru + 138Ba)
244Fm(122Sn + 122Sn 120Sn + 124Sn)
246Fm(122Sn + 124Sn 120Sn + 126Sn )
248Fm(124Sn + 124Sn 122Sn + 126Sn )
(124 126Sn 122 S 128 250Fm Sn + n + Sn) 252Fm(126Sn + 126Sn ' 124 S I n + 28Sn)
254Fm(126Sn + 128Sn ' 124Sn + 130Sn )
256Fm(128Sn + 128Sn , 126Sn + 130Sn)
258Fm(128Sn + 130Sn, 126Sn + 132Sn)
Logarithm of half-lives for the new process cold fission with compact shapes or Sn emission-for even- even elements with 98 ~ Z ~ 102o Only the first com- bination of clusters with the shortest half-llfe are given. We conclude that the half-lives of the very heavy elements with 58.6 N ~ 170 are shorter than ms.
(]08Ru 242Cf(I0 Ru + 136Xe) 2q4Cf + 136Xe)
246Cf(]18Cd + 128Sn ) 248Cf(120Cd + 128Sn)
250Cf(~20Cd + ~30Sn) 252Cf(322Cd + 130Sn)
254Cf(]22Cd + J325n) 248No(J22Sn + 126Te )
250No(J22Sn + J28Te) 252No (J22 n + 33 OTe )
254No(]24Sn + ~30Te) 256No(~265n + ]30Te)
340
4c) by Rose and Oones from Oxford U n i v e r s i t y at the beginning of 1984 This experiment was confirmed in Moscow 47) Orsay48~ Berkeley- Geneva 49) and Argonne 50). Other experiments have shown that 14C is
49) 226Ra 51-52) emitted also from 222'224Ra and • Soon, using sol id
state track recording fi lms 53) the 24Ne rad ioac t i v i t y of 231pa 54 ) "
232 U 55), 233 U 56) and 230Th 57 has been discovered. Preliminary ex-
periments show 28Mg- rad ioac t i v i t y of 236pu and 234U. Recently the hi-
modal f iss ion of 258Fm, 259Mg, 260Mo and 258No, has been measured 58)
in which one of the component could be related to the Sn-radioact iv i -
ty 45) In the following we shall present short ly only the 24Ne and
Sn rad ioac t i v i t i es . Details about 14C rad ioac t i v i t y are given in the
lecture of Dr. Hourani at this School.
24Ne radioactivity. Toward the end of 1984 a second decay mode, 24Ne 231pa 54)
rad ioac t i v i t y , was discovered simultaneously in Dubna from and Berkeley from 232U 55) Both groups used polyethylene terephthalate
detectors sensit ive only to Z ~ 6, the l im i t ing alpha dose being
< 1012 cm -2 allowing to detect a branching rat io > 10 "15
In Berkeley, a hemispherical array of Cronar Film (polyethy-
lene terephthalate) was exposed for one month to the 0.5 mCi 232U
source in a vacuum chamber at pressure of 0.01 torr . From the two 2 scanned in transmitted l igh t 200 cm surface (af ter etching) detectors,
24+7 (the second detector was covered with a 15 pm absorber f i lm)
tracks of 24Ne were found using the s i l icon replicas 53). The Cronar
f i lm was calibrated with 2ONe and 180 ion beams.
In Dubna plane sources were prepared on a 0.1 mm nickel backing
in the form of the oxides of the isotopes being investigated, The spe-
cial-purpose technology used for the preparation of the sources e l imi -
nated the contamination of the detectors by the very active specimens
under study during lona exposures. For studying the emitted clusters the 175 ~m Melinex detectors
were i r radiated in a i r using Pay U, Th and Np sources at a distance of
I mm and with an Am source in vacuum at a distance of 1 cm. The detec-
tors allowed one to detect clusters with atomic numbers 6 ~ Z ~ 20 and
separate them from f iss ion fragments.
The exposure time was determined taking into account the alpha-
a c t i v i t y of each source. For accurate spectrometric measurements in
polyethylene terephthalate detectors the integral alpha par t ic le f lux
has to be lower than 5 x 1011~/cm 2
Inside the polymer material i r radiated in a i r , some processes
occur along the par t ic le t rajectory leading to a change in the material
341
structure which affected the measured geometrical parameters of the
track. I t was e_xpe~imentally shown that this process, i . e . the change
in the etched track length, in the case of 46 MeY 2ONe ions, is f i n i sh -
ed at most one month af ter the Crradiation. Moreover, the high alpha
par t ic le f lux makes this process faster. Nevertheless, in order to en-
Sure that a l l the recorded cluster tracks, in the case of a very long
exposure time, have the same etching character is t ics, the detectors
were stored 30 days in a i r pr ior to the chemical nrocessing, which was
carried out in a 20% NaOH solution at a temperature of 60°C.
Together with the detector exposed to the radioactive sources,
there were etched the cal ibrat ion samples i r radiated with 180, 2ONe and
26Mg ions with energies of 1.6-3.0 MeV/nucleon and with dip angles of
300 and 450 . I t has to be mentioned that the cal ibrat ion samples were
exposed to sources together with the detectors. The geometric e f f i c i en -
cy of cluster detection was equal to 0.66 2x.
Since the iden t i f i ca t i on method chosen is the var iat ion of the
Vt/V b with the residual range, the detectors were etched 2-3 times for
d i f fe rent time in terva ls . After a tota l time of 4 hours, the etching
reached the cluster stopping point. From the cone geometrical parameters,
measured under optical microscopes with a magnification of 16x25x40,the
track length and Vt/V b were calculated. Using the cal ibrat ion with 160,
2ONe and 26Mg ions, the Vt/V b = f(dE/dx) dependence was obtained for
the given alpha par t ic le f lux of 2x2011 ~/cm2: Vt/V b = 6.3xlO-3(dE/dx) 2"I
Where dE/dx is expressed in MeV cm2/mg.
By taking into account the dE/dx - range re lat ionship, the Vt/V b
v a r i a t i o n w i t h the r e s i d u a l range f o r d i f f e r e n t ions is o b t a i n d ( F i g .
4A) . As can be seen, the e x p e r i m e n t a l p o i n t s measured w i t h the Th source
are s i t u a t e d around the curve f o r the 24Ne i o n s . The data t r e a t m e n t
us ing the minimum method x 2 leads to the c o n c l u s i o n t h a t 24Ne is the
most p r o b a b l e type o f the decay c l u s t e r ° The Vt /V b va lues have been ob-
t a i n e d on l y f o r a p a r t o f the c l u s t e r t r a c k s r e v e a l e d , as the c o r r e s -
Ponding measurements are too t i m e - c o n s u m i n g .
The t r a c k l e n g t h d i s t r i b u t i o n f o r the decay p roduc t s o f Th, g i ven
i n F i g . 4B has a maximum at about 29 ~, wh ich is the range of the 24Ne
ions c o r r e s p o n d i n g to the c a l c u l a t e d va l ue o f the k i n e t i c energy of the
e m i t t e d heavy c l u s t e r . The d i s t r i b u t i o n of the f u l l ranges R i s r a t h e r
broad. This is due to some experimental shortcomings, namely a large
SOurce thickness and a rather high recoi l nuclei background, which have
introduced some ambiguities in the estimate of the cluster mass number
because of the d i f f i c u l t i e s either in the estimation ofthe self-absor-
Ption process in the source or in the measurements of the geometrical
342
76
~ 7 6
\ 'Z2 ~
110 ! |
3D 50 100
Rre s :tim)
NTPI /V 230~, 251;. rp //7
~0
N 20253035
Fig. 4 A) The calculated dependences of the etching selectivity VT/V B (where V T and V B are the rates of polymer etch" ing along the track and the bulk etch rate for the de" rector material, respectively) on the residual range Rre s. The dots indicate experimental data for clus- ters, the crosses give the results f measurement~ with control detectors exposed to I~0, 20Ne and 26Mg ions with energies of 1.6 to 3.0 MeV/nucleon.
B) The distribution of the full ranges of the 230Th de- cay products and those of fraaments from the neutron" induced fission of 233U in polyethylene terephthalateo
2 33U sou was equal The thickness of the 230Th and rces to 0.3 mg/cm2, respectively.
parameters of the track.
By taking into account the theoretical predictions as well as the
experimental results (Fig. 4) one can conclude that the new decay mode
of 230Th takes place most probably by the 24Ne ion.
No clusters have been detected from the Np and Am sources, this
fact being additional evidence for the absence of a background in Dubna
experiment. I t is worth mentioning that from the experimental data 51-52) as
well as from the 226Ra quantity contained in the 230Th + 232Th source
one should expect about 40 tracks of 226Ra decay by the emission of 14C'
Due to the s e n s i t i v i t y of the detector, the maximum etched track length
for 14C ions is equal to about 10 um. Since the C and 0 recoi l nuclei
have lower energies, they are etched faster than the 14C cluster trackS"
343
A f t e r a l o n g e r e t c h i n g t ime the background becomes so h igh t h a t i t is
i m p o s s i b l e to d i s c r i m i n a t e the 14C c l u s t e r t r a c k s a g a i n s t the r e c o i l
n u c l e i t r a c k s .
The e x p e r i m e n t s have not been aimed at s t u d y i n g spontaneous f i s -
s i o n . N e v e r t h e l e s s , in expe r imen t s w i t h t h o r i u m t h e r e have been de tec -
ted 25 spontaneous f i s s i o n f ragments which d i f f e r from c l u s t e r s both
in the range (see F i g . 4B) and shape. Bear ing in mind the p o s s i b l e
sources o f the backg round , f o r example , t h o r i u m f i s s i o n induced by
neu t rons from the ( ~ , n ) r e a c t i o n on the l i g h t n u c l e i o f the d e t e c t o r
' m a t e r i a l , the e f f e c t o b t a i n e d can be used on l y f o r e s t i m a t e s of the
lower l i m i t o f the spontaneous f i s s i o n h a l f - l i f e of 230Th. A c c o r d i n g
to our da ta , t h i s l i m i t i s se t at ~2xI018 y e a r s , t h a t is one o rde r o f
magn i tude h i g h e r than p r e v i o u s l y known va l ue of 1 . 5 x i 0 1 7 y e a r s . The 24 N . . • p a r t i a l h a l f - l i f e f o r e - e m i s s i o n is ( 1 . 3 + 0 . 3 ) 1017 years
232 U 233 U The r e s u l t s f o r 231pa, and are p resen ted i n the l e c -
t u res of Drs. Houran i and Poenaru at t h i s Schoo l .
Sn r a d i o a c t i v i t y . The r e c e n t measured T K E - d i s t r i b u t i o n of f o u r ve ry heavy n u c l e i 258Fm, 259Md, 260Md and 258No i n d i c a t e a compos i te
of two energy d i s t r i b u t i o n s one peaked at 200 MeV and a n o t h e r one peak-
ed at 235 MeV 58) Of course t h i s f a c t was i n t e r p r e t e d as a m i x t u r e
of l i q u i d - d r o p - l i k e and f ragmen t s h e l l d i r e c t e d symmetr ic f i s s i o n and
not as an ev idence f o r S n - e m i s s i o n . We conc lude t h a t a l r e a d y f o r t h i s
r eg i on the two processes the Sn -em iss i on and the spontaneous f i s s i o n
are comparab le . More than t h a t i t was a lso shown e x p e r i m e n t a l l y t h a t
the even ts w i t h very h igh TKE cou ld a lways be a s s o c i a t e d w i t h the sym-
m e t r i c and t h a t the asymmet r i c f i s s i o n mass f ragments are a lways asso-
c i a t e d w i t h low TKE d i s t r i b u t i o n s , f a c t s wh ich f o l l o w e v i d e n t l y f rom
Our i n t e r p r e t a t i o n (see f i g s . 5A and B) .
The o t h e r c h a r a c t e r i s t i c s o f the new p r o c e s s , the absence o f neu-
t rons and y - r a y s or l o w - e n e r g y and low m u l t i p l i c i t y y - r a y s make the new
mechanism d i s t i n g u i s h a b l e e x p e r i m e ~ t a l l y from the usual spontaneous f i s -
S ion . I ndeed , c o i n c i d e n c e measurements o f mass and k i n e t i c energy of
the f ragments a s s o c i a t e d w i t h no neu t rons and e s p e c i a l l y w i t h low ener -
gy and low m u l t i p l i c i t y y - r a y s w i l l a l l o w the d e t e r m i n a t i o n o f a smal l
component on a l a r g e background of spontaneous f i s s i o n f r a g m e n t s , due
to the f a c t t h a t i n spontaneous f i s s i o n we have a h igh m u l t i p l i c i t y
Y - r a y s .
The observed T K E ~ d i s t r i b u t i o n s o f 258Fm, 259Md, 260Mo and 258No,
Skewed upward o~ downward from the peak in each case cou ld be e a s i l y e x p l a i n e d i f we assume the e x i s t e n c e of the new decay mode the Sn- e m i s s i o n . New data c l ose to the p r e s e n t data l i k e 256,254Fm or
344
L~
~ 30
~ 20
Md-260
I '1 ' 1 ' 1 ' I 'l ' l ~ l l l I
150 160/70180 IgO 200 210 2202302~02~0250270 Total k/mehc energy { MeV)
A)
150
u~
0
'"1 ' ! ° I ' I ' d ' I ' ' I , I ' I
90 lOO ilO 12# 130 l~O 150 160 170 I80 Frogmenf Mczss (ainu)
Fig° 5 A)
B)
B)
Fission events (o) for 260Md as function of total kine- tic energy (TKE). Clearly two components are seen. Ten- tatively two gaussians, one for the high TKE component (-.-) and another one for the low TKE component (---) are fitted with the experimental data (o) •
Fission events (o) for 260Md as function of the frag- ment mass. We can see that the high TKE component cor- responds to a very narrow mass distribution and the low TKE component to a very large mass distribution.
345
256'254No cou ld c l a r i f y the s i t u a t i o n . A l so the l a r g e chan~e observed
in the T K E - d i s t r i b u t i n n s from 258Fm wh ich i s m a i n l y peaked at h igh TKE
to 259Md wh ich is m a i n l y peaked at low TKE, n u c l e i wh ich d i f f e r on l y by
one p r o t o n , cou ld be e x p l a i n e d s i m p l y by l a r g e r e f f e c t s o f the odd p ro -
ton on the f i s s i o n b a r r i e r f o r compact shapes w i t h few l e v e l s at the
Fermi s u r f a c e than f o r a l onga ted shape w i t h a l a r g e d e n s i t y of l e v e l s .
F i n a l l y we shou ld l i k e to ment ion t h a t the t h e o r e t i c a l c a l c u l a -
t i o n s f o r Sn -em iss i on are based on the s i m p l e s t model w i t h no r ea l pa-
rame te rs . A l so the mass c o e f f i c i e n t s are taken as s imp le reduced mas-
' s e s , even the o v e r l a p of the two f ragments i s c o n s i d e r a b l y l a r g e r than
f o r e m i t t e d l i g h t e r c l u s t e r s l i k e 14C and 24No. The absence of the mea-
sured b i n d i n g e n e r g i e s i n t h i s r e g i o n makes the e v a l u a t e d h a l f - l i v e s
u n c e r t a i n . Even w i t h these l i m i t a t i o n s , we c o n s i d e r t h a t the gene ra l
t r ends o f the new p rocess , the S n - e m i s s i o n are w e l l d e s c r i b e d .
In c o n c l u s i o n , we shou ld l i k e to ment ion t h a t in the f u t u r e s t u -
d ies we have to i n t r o d u c e the d i s s i p a t i o n in the d e s c r i p t i o n of a l l
these phenomena. The case of 260Md where the e x p e r i m e n t a l data show an
average t o t a l k i n e t i c energy loss of ~ 25 MeV i n d i c a t e such a need. We
hope t h a t the f i n e s t r u c t u r e i n a lpha decay and o t h e r c l u s t e r decays
(no t y e t d i s c o v e r e d ) can a lso be d e s c r i b e d by the i n t r o d u c t i o n of the
d i s s i p a t i o n .
4. HEAVY ION COLLISIONS
In t h i s chap te r we p r e s e n t some processes wh ich appear in heayy
ion c o l l i s i o n s , l i k e charge e q u i l i b r a t i o n , em iss ion of c l u s t e r s a t f o r -
Ward ang les and the c l u s t e r t r a n s f e r r e a c t i o n s l e a d i n g to heavy a c t i n i -
des, wh ich may i n d i c a t e some quantum e f f e c t s . We know from magic decays
t h a t the f r a g m e n t a t i o n of a nuc leus i s a very p robab le p r o c e s s , l i k e
the f o r m a t i o n of an a lpha p a r t i c l e , but due to the presence of a l a r g e
Coulomb b a r r i e r these decays become very r a r e . ConseQuen t l y , we expec t
t h a t a d i - n u c l e a r sys tem, i . e . two n u c l e i in c o n t a c t w i t h no Coulomb
b a r r i e r between them, w i l l tend w i t h a l a r g e p r o b a b i l i t y to r e a r r a n g e
the nuc leons in a co ld way a c c o r d i n g l y to the c o r r e s p o n d i n g f r agmen ta -
t i o n p o t e n t i a l . As a c r i t e r i o n f o r quantum e f f e c t s we c o n s i d e r the ap-
Pearence of such co ld rea r rangemen ts of a l a r g e number of nuc leons w i t h
smal l e x c i t a t i o n of the i n t r i n s i c degrees of freedom or c l u s t e r t r a n s -
f e r as a b a s i c e x p l a n a t i o n o f these p rocesses . F i r s t , we t r e a t the charge e q u i l i b r i u m process as a damped harmo-
n i c o s c i l l a t o r i n the charge asymmetry c o o r d i n a t e ~ = ( Z I - Z 2 ) / ( Z I + Z 2 ) .
346
We consider a r b i t r a r y values for the f r i c t i o n constant x,parameter and d i f f u s i o n c o e f f i c i e n t s
{mm Dqq = - ~ Pl ; Dpp = - --~- P2 ; Dpq = 7 ~3 (39)
where PI ' P2' P3 are dimensionless q u a n t i t i e s . The d i f f u s i o n c o e f f i - c ients (39) have to s a t i s f y the fundamental cons t ra in ts (5) . The vec- to r of variances
X(t)
mmOqq(t)
I Opp(t)
Opq(t)
(40)
s a t i s f y the fo l l ow ing equation
X( t ) = (T ektT) (X(O) - X(~))+ X(~) (41)
wi th two so lu t ions underdamped (~ < (~) and overdamped (~ > m). The asymptot ic values for both cases are 6).
1 2 Oqq(~) = 2X 2 ((mu~) (2~(x+p)+~ 2) Dqq 2(m~) (~2+ 2_~ )
+ m2Dpp + 2mm2(~+~)Dpq)
= __. i ((m~)2 2D ~PP(~) 2X(X2+ Z_ 2) qq
2m~2(x-1,)Dpq)
+ (2~(~-~)+~2)D PP
i + (~-p)Dp ~pq(-) = 2 m ~ ( ~ 2 ~ _ ~ ) (-(~+u)(mm)2Dqq p
+ 2m(~2 _ 2 ) Dpq)
2 2 In the underdamped case (~ < m) we have: Q =
TektT = e-2~t b11 b12 b13 1 ~ b21 b22 b23
J b31 b32 b33
2 - i J
(42)
(43)
347
w i t h
bl I = (2..~2) co52£t -. 2.pQsin 2£t -, 2
bl 2 = (2+Q2) cos2Rt - 2
b13 = 2m(~cos2~t - Qsin2~t - #)
b21 = ( 2+ 2) cos2~2t - 2
b22 ( 2 _2 ) = c o s 2 ~ t + 2 ~ s i n 2 ~ t - 2
b23 = 2m(ucos2Qt + o s i n 2 ~ t - ~)
b31 = - m ( u c o s 2 o t - Q s i n 2 o t - ~)
b32 = - m ( ~ c o s 2 ~ t ÷ £ s i n 2 Q t - ~)
2 b33 = -2 (m 2 c o s 2 ~ t - ~ )
In t he ove rdamped case (~ > ~) we have : 2 1)
2 2
(44 )
Wi th
- 2 ~ t T e k t T _ e
al i a12 a13
a21 a22 a23
a31 a32 a33
(p2+ 2 a l l = ) cosh 2~ t + 2~v s i n h 2 v t +
a12 = ( ~ 2 - v 2 ) cosh 2 v t - 2
a13 = 2m(# cosh 2~ t + v s i n h 2 v t - ~)
a21 = ( p 2 . ~ 2 ) cosh 2 v t - ~ 2
a22 = (~2 +v2) cosh 2 v t - 2 ~ v s i n h 2 v t - 2
a23 = 2m(~cosh 2 v t - v s i n h 2 v t . , ~)
a31 = - m ( p c o s h 2 v t ÷ v s i n h 2 v t - u)
a32 = - ~ ( p c o s h 2 v t - v s i n h 2 v t = 2
a33 -2 (m 2 cosh 2 v t - p )
~)
(45)
( 46 )
348
I n i t i a l and f i na l values of the variances have tO sat is fy the
fo l lowing fundamental constraints:
1 (),-~J) Oqq(,,:.) ~ Opq(~) > 0
2 (x+~) Opp(~o) + me Crpq(~) > 0
2 (oo)) 4( ) ,2+(~2- .2) ( (~qq(~)Opp(~) - dpo
i (=) + 2~ o (~)2 2 2 - (m~Z~qq(~) + ~ ~pp pq ) ~-fi
Dqq app(~) + Dpp aqq(~)
Dqq ~pp(O) + Dpp aqq(O)
~2~ 2 Dpq ~pq(~) ~ T
~2~ 2 Dpq Opq(0) a --2-- (47)
In the overdamped case the r es t r i c t i on ~ > v is necessary.
In order to compare with the experimental data the tota l energy
loss as funct ion of time is needed. I t was chosen a model parametriza-
t ion of the def lect ion funct ion 5g) in terms of relaxat ion times for ra-
dial T r , tangential T Z and deformation T d motions, The above times were
f ixed at the values T r = 1.5, T Z = 10, T d = 30 by reproducing the ridge
of the double d i f f e r e n t i a l cross section d2o/d~dE plotted as a contour
diagram in the total k inet ic energy versus scatter ing angle (Wilczynski p lot) for the 56Fe + 165Ho system 60) In th is model parametrization
there is a retardation of the charge equ i l i b ra t i on mode t o for the time
in which the neck radius develops enough to obtain the saturated value
of the k inet ic coe f f i c ien t B~. This i n i t i a l value T O was chosen around
few MeV energy loss. One can see that for short reaction times corres-
ponding to an energy loss of 70-80 MeV, the k inet ic energy loss is pro"
port ional with the reaction time ~ 30 MeV/10"22s. This period is the
most important period for the charge equ i l i b ra t ion mode since during
this period the charge variances are saturated.
The shell effects on the fragmentation potent ial are considered
neg l ig ib le and i t is assumed that th is potent ial as funct ion of the
charge asymmetry coordinate ~ = (Z1-Z2)/(ZI+Z2) can be approximated with
an harmonic osc i l l a t o r potent ia l :
V(~) = ½ K~ 2 (48)
with the s t i f fness K given by the l i qu id drop model:
349
Z2 ~I 1 ) k a 2 ( _ ~ + 1 K ~ ~ . + 8kal ( + A~2 - 8 al ~ )
+ 2c3( 1 + 1 2e 2
Where Z = Z 1 + Z 2 the to ta l charge of the dinuclear
The various coef f i c ien ts are:
system.
(49)
a I = 15.3941MeV ; a 2 = 17.9439 MeV ; c 3 = 0.7053 MeV ;
2 k = 2.53 MeV ; e = 1.44 MeV (50)
For the systems considered, the experimental ly deduced values of the
frequencies for th is co l l ec t i ve mode - the charge equ i l i b ra t i on mode -
are smaller by 15-20% than the frequencies of the isovector dipole
giant resonance of the molecular two-center system for which we have 61) roughly
70 h~DG R = A~/3 + A~/3 MeV (51)
This may be due to the deformations of the two fragments or to the for -
mation of a large neck. So we have used a frequency calculated accord- ing to:
70 with d = 2fm (52) h~DGR = A~/3 + A~/3 + d
Correspondingly, the mass parameter was f ixed so as to sa t is fy the re la- t ion:
2 K~ = B~ ~ (53)
We have chosen for comparison two systems, 56Fe + 209Bi and 56Fe +
+ 238 U 62) for which the correction for charged pa r t i c l e evaporation
is not important 63).
The i n i t i a l value for the charge variance was chosen from the ex-
Periment. Correspondingly we have ~pp(O) = ~2/4Oqq(O) and ~pq(O) = O.
For our systems, in both cases: under and overdamped, we have
Used the fol lowing parameters:
S~stem K~(lO4~leY) B~(IO~44MeVs 2) m(10~22s ~1) ~qq(O) ~mIMeV)
Fe+Bi 1,6363 16127 1.0073 0.000045 6.63
Fe+u 1.8683 19217 0.986 0.00006 6.49
350
The bes t f i t is o b t a i n e d f o r c l ose va lues of f r i c t i o n , dampinfl
and d i f f u s i o n c o e f f i c i e n t s . One can see in F i g s , 6A and B t h a t the over"
damped s o l u t i o n d e s c r i b e s e x t r e m e l y w e l l the e x p e r i m e n t a l data f o r both
systems w h i l e in the underdamped case the f i t i s not so good.
R e c e n t l y 64) we t r e a t e d the p ro ton and n e u t r o n asymmetr ies modes
in deep i n e l a s t i c c o l l i s i o n s (DIC) i n the frame of the usua l quantum
mechanics in two d imens ions w i t h no d i s s i p a t i o n assuming two coup led
harmon ic o s c i l l a t o r s i n these c o o r d i n a t e s . We s t u d i e d the t ime e v o l u -
t i o n o f the mean va lues ( c e n t r o i d s ) and the v a r i a n c e s ( w i d t h s ) , F i t s to i16 S the e x p e r i m e n t a l data are made f o r the r e a c t i o n s 129Xe on n and
124Sn. A n a l y t i c s o l u t i o n s f o r two d imens ions i n c l u d i n g the d i s s i p a t i o n
are p r e s e n t l y a v a i l a b l e 14) Up to now, we have not made a d e t a i l e d
compar ison w i t h the e x p e r i m e n t a l da ta .
S e c o n d l y , we c o n s i d e r the em iss ion of c l u s t e r s a t f o r w a r d ang les
in heavy ion co l l is ions at low energies 65-66) In the last years in-
creasing experimental evidence has shown that part ic les (clusters) of
intermediate mass with 2 ~ Z ~ 17 are emitted with quite large cross
sections at forward angles in heavy-ion reactions 67-69). We concen-
trated on the measurements performed at Dubna 68-69) for the H, He, Li and Be isotopes emitted by the 181Ta + 22Ne, 232Th + 22Ne and 232Th+IIB
systems at small emission angles (even as small as 0 °) and incident
energies of 141 and 178 MeV. The f l a t and wide energy d is t r ibut ions of
various He to Be isotopes, with a maximum located at an energy corres-
ponding to the beam ve loc i ty , suggest that p ro jec t i le fragmentation is the
{ . W "~ }
i
S~ Iql ÷ l ~ Bi
i i i t I
t (W "~r s )
A)
351
F ig . 6 A)
i
~£Fe .* 25gU
. . . . . . . . . ___J . . . . . . !
B)
i i i
B)
2 V a r i a n c e s o~ in the charge asymmetry c o o r d i n a t e w i t h fixed initial mass numbers A I and A 2 as function of the interaction time for the 56Fe + 209Bi system in the underdamped case (UD) (---) with ~ = 1.7, ~ = 0, Pl = I, P2 = 20, P3 = 4 and the overdamped case (OD)
( ) with ~ = 3.5, ~ = 3.3, Pl = I, p2=~qO, P3 = 9.
2 Variances o 6 in the charge asymmetry coordinate with fixed initial mass numbers A I a r)d A 2 as function of the interaction time for the bbFe + 238U systems in the underdamped case (UD) (---) with ~ = 1.8, u = 0, Pl = I, P2 = 20, P3 : 4 and the overdamped case (OD) (-- ) with >, = 3.1, !J = 2.3, Pl = I, P2 = 90,P3 : 9.
mechanism r e s p o n s i b l e f o r c l u s t e r e m i s s i o n . Then a model based on such
a mechanism shou ld g i v e a c o n s i s t e n t d e s c r i p t i o n o f the whole se t o f
e x p e r i m e n t a l da ta : energy and a n g u l a r d i s t r i b u t i o n s as w e l l as Droduc-
t i o n cross s e c t i o n s of v a r i o u s i s o t o p e s .
We used the Serber b reak -up model r e c e n t l y deve loped f o r heavy-
ion r e a c t i o n s 70-72) w i t h two a d d i t i o n a l f e a t u r e s : the s o e c t r o s c o p i c
f a c t o r and the assump t i on , suppo r t ed by the e x i s t i n g c o r r e l a t i o n measu-
rements at low e n e r g i e s , o f a t w o - s t e p ( s e q u e n t i a l ) p rocess in wh ich
the p r o j e c t i l e is f i r s t l y e ,xc i ted and s u b s e q u e n t l y decays i n the f i e l d
of the t a r g e t n u c l e u s . Both t h r e e - b o d y and two-body p r o c e s s e s , c o r r e s -
POnding r e s p e c t i v e l y to the observed f r a g m e n t , the unobserved one and
the t a r g e t nuc l eus i n the f i n a l s t a t e and to the f u s i o n of the unobse r -
Ved f r agmen t w i t h the t a r g e t n u c l e u s , were c o n s i d e r e d .
352
The cross section is wri t ten as
d2~l
~ F 1 : k lT l 2 ~S F (54)
where k is a n o r m a l i z a t i o n f a c t o r , T is the t r a n s i t i o n m a t r i x e lement
and @ i s the phase-space f a c t o r c a l c u l a t e d o p t i o n a l l y f o r the two-body
and t h ree -body processes . In the two-body case exp ress ion (54) was mul-
t i p l i e d by the f u s i o n cross s e c t i o n c a l c u l a t e d in the c l a s s i c a l b a r r i e r 73)
a p p r o x i m a t i o n . The s p e c t r o s c o p i c f a c t o r (SF) as de f i ned by Friedman
was i n t r o d u c e d i n t o the e x i s t i n g f o r m u l a t i o n of the Serber model in
o rder to ob ta in abso lu te va lues of the p roduc t i on cross s e c t i o n s . To
desc r i be the s e q u e n t i a l c h a r a c t e r of the break-up process an e f f e c t i v e
bombarding energy was c a l c u l a t e d by s u b t r a c t i n Q from the incoming kine"
t i c energy the energy co r respond ing to the e x c i t a t i o n of the p r o j e c t i l e
be fo re i t s f r a g m e n t a t i o n . We note t h a t i t was not necessary to change
the amount of e x c i t a t i o n energy accord ing to the d i f f e r e n t s e p a r a t i o n
energ ies of the va r i ous i s o t o g e s . Th is suggests t h a t , du r i ng the i n t e r -
ac t i on w i th the t a r g e t nuc leus , a f r a c t i o n of the k i n e t i c energy of the
p r o j e c t i l e i s t rans fo rmed i n t o o the r degrees of f reedom. An e x c i t e d
and m o r e - o r - l e s s deformed nucleus r e s u l t s , and t h i s is e a s i e r to f r a g -
ment even i f i t s e x c i t e d s t a t e is not above the cor resDonding separa-
t i o n energy of a g iven f ragment . C o r r e c t i o n s t ak i ng i n t o account the
Coulomb d i s t o r t i o n s in the f i e l d of the t a r g e t nucleus and the c u t - o f f
o f the spec t ra as a r e s u l t of the f i n i t e t h i c k n e s s of the bE d e t e c t o r
were i n t r o d u c e d .
In F igs . 7 and 8 we g ive the measured energy d i s t r i b u t i o n s of the
He(a) L i ( b ) and Be(c) i so topes em i t t ed by 181Ta + 22Ne (178 MeV) and
r e s p e c t i v e l y 6He, 7Li and 9Be i so topes em i t t ed by 181Ta + 22Ne(141 MeV)
compared w i t h t h e o r e t i c a l c a l c u l a t i o n s . The comparison of the Serber
c a l c u l a t i o n s w i th the expe r imen ta l va lues f o r the cross s e c t i o n s correS" ponding to the maximum of the energy d i s t r i b u t i o n f o r 232Th + 115 (89 MeV) 232Th + 22Ne (178 MeV), 181Th + 22Ne (141MeV) at d i f f e r e n t
angles i s g i ven in F ig . 9. We can see t h a t the energy and angu la r d i s -
t r i b u t i o n s are w e l l desc r ibed by the p resen t break-up model. Alpha par"
t i c l e s have not been ana lysed as t h e i r spec t ra i n d i c a t e a more compleX
structure, presumably due to s t a t i s t i c a l , break~up and pre-equilibrium
processes.
We conclude that the forward emission of clusters with 2 ~ Z ~ 4
at energies lower than 10 MeV/nucleon can be accounted for consistently
by the pro jec t i le break-up mechanism.
10°! 6
i
10-1i +
>= 6
# 2 ~0 3
10° % 6
2
6
2 1(~ 2
6
2 10-3
353
a
lo - l r F o +)He I { ~ .3 / SHe 10° 7Li :I~ 8L=O ° ~ 'X'~e+L=O°~ e+10 ° 6 ,. : ~0
:... ,0-2"- ,2.1 \ "~." ~ I ~o- ~. :/.__\ ~
9Be eL=20 ° lOBe Etob (HeV)
e~
10 ~
d
• • I I l I I . 1 1 I l I I I I I I
SO 70 90 110 "'+0 60 BO 100 120 EiobtHeV)
Fig. 7 Measured energy d i s t r i b u t i o n s of He (a), LI (b) and
Be (c) isotopes emitted by 18]Ta + 22Ne (178 MeV) com-
pared with the Serber model ca lcu la t ions ( f u l l curves
= two-body va r ian t ; dotted curve = three body var iant )
354
Isc: d 6 " o
x: 2
~-3 8 6 4
6 He
, . ~2
J_ " 10_ 3
7Li
\,\ 10"3
,~" 9 Be
'1% \ \
o
o
I I l I . . I I I I I ;~1 I I I I I
30 50 "20 ~) 60 40 60 ~,0 Elnb{HeV)
Fig. 8 6He, 7Li and 9Be iso- Measured energy distributions 9f ~lTa + (141 MeV) com" topes emitted at Olab=20 ° by 1 22Ne
pared with Serber model calculations (three body variant assuming that the projectile is excited at 10.62 MeV (full curves) and 15 MeV (dotted curves)).
T h i r d l y , i n o rde r to d e s c r i b e the m u l t i n u c l e o n t r a n s f e r r e a c t i o n s
l e a d i n g to heavy a c t i n i d e s we c o n s i d e r a model based on p r o j e c t i l e f r ag "
m e n t a t i o n . The p r ima ry d i s t r i b u t i o n of the i s o t o p e s is o b t a i n e d by assU"
ming t h a t the c l u s t e r s separa ted from the p r o j e c t i l e are cap tu red by
the t a r g e t as a who le . We show t h a t the p r o d u c t i o n cross s e c t i o n s of
Fm-, Md-, No- and L r - i s o t o p e s , o b t a i n e d at LBL and GSI, are w e l l des-
c r i b e d by t h i s model a f t e r c o r r e c t i o n f o r neu t r on e m i s s i o n .
In the l a s t y e a r s , m u l t i n u c l e o n t r a n s f e r r e a c t i o n s have been used
f o r the p r o d u c t i o n of heavy a c t i n i d e s 7 4 - 7 6 ) . The u n d e r s t a n d i n g of the
mechanism o c c u r i n g i n such r e a c t i o n s would be o f g r e a t i n t e r e s t f o r
allowing predictions of the production cross sections for heavier ele-
ments. The current interpretat ion of the reactions has been based so
far on the theory of strongly damped col l is ions 77) in the context of
the surviving probabi l i ty of the primary products 78)
355
10 °
5
5
"~ I(} d
/ I 10 °
/ lo .2
!
'He *He IHe
E
1,21
10" ' ~ '
.,;3~- #He'He • IV II l I i I llI I
~*He *He *Li iLl 'Be
'He ,'H,~ ,'L i ,"Be :B,~ IHe 'He 'Li ~Li 'Be
Fig. 9 Comparison of the Serber calculations (white points) and experimental values (black points) of the cross sections corresponding to the maximum of the energy distribution for the 232Th + 11B (~9 MeV), Olab=20°(a) ;
O°(c) ; =0° (b) and Blab:2 232Ta + 22Ne (178 MeV), Bla b
181Ta + 22Ne (141 MeV), Bla b =20 ° (d) .
Systemat ic measurements fo r mu l t i nuc leon t r a n s f e r reac t i ons in 254Es the case of the bombardment of a given t a r g e t as f o r example by
Various p r o j e c t i l e s 160, 180, 22Ne and 48Ca) 75,76) have revea led com-
Parable cross sec t ions fo r a (AZ,~N) t r a n s f e r , desp i t e a d i f f e r e n t i so -
tope d i s t r i b u t i o n fo r var ious elements (Fm, Md, No and L r ) . Such a pro-
J e c t i l e dependence suggests tha t the f ragmenta t ion of the p r o j e c t i l e
OCcurs producing la rge c l u s t e r s tha t are subsequent ly captured by the
ta rge t (massive t r a n s f e r , incomplete f u s i o n , two-body break-up reac-
t i o n ) . Much exper imenta l evidence has been qathered r e c e n t l y which
Shows tha t p r o j e c t i l e break-up processes are not n e g l i g i b l e even at
356
lower than 10 MeV/n incident energies 79) as happen~ to be the case
of the above mentioned reactions.
Starting with this observation we o~pose for multinucleon trans"
fer reactions a simple model based on the following physical picture:
the incoming pro jec t i le breaks up in the Coulomb and nuclear f i e ld of
the target nucleus the result ing large clusters being captured by the
target nucleus. The hypothesis of a large cluster transferred as a 8 whole is supported by experimental evidence for ~ and Be transfer ob-
tained in 12C reactions on Au and Bi 80) as well as by the recently
growing experimental evidence for the incomplete fusion reactions.
Therefore the transfer cross section for a F i (Z i ,Ni ) c ~ s t e r
is given by the fragmentation probabi l i ty of the pro jec t i le (~p1) mul-
t i p l i ed by the capture probabi l i ty Oc(Fi,T):
F. = K yp1 Oc(Fi,T) (55) o(F i )
Here P and T denote the pro jec t i le and target nuclei and K is a normali"
zation factor. The fragmentation probabi l i ty is given by Friedman model 73)
Fi e-2~Fi XoF i b YP ~ 3 (56) SFi x ( i b)
°F i where ~F, = V 2 mrE ~ (m r is the reduced mass of the two fragments
i l and E S represents the separation energy of the fragment F i from the
pro jec t i le ) and the cutoff radius of the cluster internal wave func-
tion x = 1.2 A~/3 fm. SFi is the spectroscopic factor which re- °Fi i
presents the re la t ive probabi l i ty for f i nd ing toge ther the necessary
~rotons and neutrons which must be removed from the p ro jec t i le to pro-
duce the fragment 73) In the case of unstable part icles as for example 6 8 5He, 7He, 5Li~i Be, Be a second fragmentation was taken into account
mult iplying Yp by the corresponding spectroscopic factor S~i.
The capture probabi l i ty was calculated in the semiclassical appro"
ximation by integrating over the energies of the fragment:
VFi,T ~c(Fi, T) = ~R~ ? jmax (I - ---C--- ) dE (57)
i V F ,T i
Here ~R~i- is the geometrical cross section with RFi = 1.22 (A;C3+--.
+ A~/3) • V F is the Coulomb barr ier and Ema x - the maximum energy i ,T
357
Qf the fragment as determined by the condition that the Excitation ener-
gy of the residual nucleus is equal to either the ~eparation energy of the neutron or the f iss ion barr ier, The Coulomb barrie~ was calculated
in the touching sphere approximat ion 2
~ ZF, Z T e 1
VFi,T
, . 1 /3 + 113 rct~F1 A F' )
Z T e 2
r C A~ F3 for protons
Where
r = c
I 1.81 fm for protoi lFi
2.452-0.408 lOglO .ZT) for 4He
2.0337-0.2412 log10 (ZFi-ZT)
(ZFi.Z T ~ 500)
(58)
(59)
B1) Was taken from the exist ing systematics As the transfer products result in an excited state, the primary
d is t r ibut ion is modified by neutron emission. The neutron emission pro- bab i l i ty is calculated by using the empirical formula of Sikkeland et al. 82) for Fn/r f , which is assumed constant over the whole range of excitat ion energies.
The prediction of the model have been compared to the recent mea- surements performed at GSI and LBL 75,76) for the multinucleon trans- fer reactions produced by 180, 22Ne and 48Ca on 254Es at E/A 4.5=MeV/u. By considering that 255-257Fm isotopes are produced by I-3H capture and the 253'254Fm are only the result of neutron emission the isotopic
Yields were calculated as shown above. The separation energies as well as Q-values have been calculated by using the exist ing tables 83). The normalization factor is determined for alpha transfer and is unique for a given system. Simi lar ly, 256-261Md isotopes correspond to the capture of 2-7He fragments while 254'255Md result from their deexcitation by
neutron emission. Larger clusters as 3"8Li and 4"gBe are associated With the formation of the 257-262No and 258-263Lr, whose neutron emis- sion leads to 254'255'256No and 256'257Lr. The results for the studied
Systems are shown on f igs. 10. We can see that the theoretical predic-
104~
10 3
101
358
T~¢ms~mr~d
Fm 2,~ 2S6Hd 251 259 261No 259 i~onJ~oLpe-
I
a
102
._=.=.
10 °
10-2
lO-4 !~ .~l ~- 4 o . . . . . . . . . . . . . . . "rr~,,s-~r,~ • . .l~e.lie. lie DLiSLiTLi ~BeGBellBe ¢tuater Fm 2 5 4 ~ d as 2572"~ 261 N o25425625626°2G2 L r25926~263 iF~'~'t°~e
a ) b)
10/'.J
102
~I0 °
10-2
/
#
J
i 6 ;
Trensfered i0-4" , !.H 3 H 2FI#..GH.e_ .~Li ~iL.i . ,~p~13,e clusfe[
Fm 2542~='L~f'ld 258 No~6~62601_r257259 2M Finol isotope
10 5
103 . o
Tronsfered
Fm 2~A ~'Md 25s 2~7 259 No 259 Finol - ISOTOpe
C) d)
Fig. I0 a) Comparison of measured (black points) and calculated (open Pg~nts) 25~is topic cross sections for the 98 MeV J°O on Es, Black squares represent the va- lues obtained by extrapolating the experimental distribution.
b) Same as in a) but for ]26 MeV 22Ne on 254Es.
254Es c) Same as in a) but for ]01 MeV 160 on
d) Same as in a) but for 266 MeV 48Ca 2 on 54Es.
359
t i o n s f o l l o w the genera l t r end o f the i s o t o p i c y i e l d s and g i v e va lues
q u i t e c lose to the e x p e r i m e n t a l ones.
C a l c u l a t i o n s performed f o r v a r i o u s t a r g e t - p r o j e c t i l e c o m b i n a t i o n s ,
even as heavy as 238U + 248Cm show t h a t the model succeeds to d e s c r i b e
Well the cross s e c t i o n s 84)
We conc lude t h a t a l l these p rocesses , l i k e magic decays , show a
quantum b e h a v i o u r . Much work must be done in o rde r to have a u n i f i e d
approach o f a l l these phenomena in the frame of our n u c l e a r dynamics.
E s p e c i a l l y we need a t w o - c e n t e r s h e l l model i n c l u d i n g a r e n o r m a l i z a -
t i o n p rocedure both v a l i d at l a r g e asy~letr ies w i t h the he lp o f which we
can d e f i n e the f r a g m e n t a t i o n p o t e n t i a l and the i n e r t i a l pa rame te rs .
Such s t u d i e s are in p rog ress .
5. CONCLUSIONS
F i r s t l y , based on f i s s i o n t h e o r y by making cuts on the many d i -
mens iona l p o t e n t i a l energy s u r f a c e , a u n i f i e d d e s c r i p t i o n of a lpha de-
cay, heavy c l u s t e r emiss ions and a new type o f symmetr ic f i s s i o n w i t h
compact shapes is p r e s e n t e d , a th reemoda l spontaneous f i s s i o n in which
one or both f ragments have magic numbers or a lmost magic numbers
magic decays.
C o n s e q u e n t l y , due to the f a c t t h a t on the p o t e n t i a l energy su r -
face we found t h ree minima a long the mass asymmetry c o o r d i n a t e , we
expec t by ana logy a t h i r d minimum a long the neck degree o f f reedom at
very l a r g e d e f o r m a t i o n s when o t h e r magic numbers can appear w i t h very
I o~ t o t a l k i n e t i c energy o f the f r a g m e n t s . Such a minimum cou ld be con-
nected w i t h t h e e m i s s i o n of a p~on in t h e spon taneous ~ i s s i o n ok heavy e l e m e n t s 85) , f o r which we need to c r e a t e i t s r e s t mass (m o=139,9626MeV,
m + = 139,5669MeVj. A lso we expec t t h a t the s h e l l e f f e c t s w i l l c r e a t e
f o r some systems at l e a s t a second m~nimum on the po~en t i ae emeray
a long the charge asymmetry c o o r d i n a t e . By c o n s i d e r i n g the i n t e r a c t i o n s
of the above s h e l l model v a l l e y s on the many d i m e n s i o n a l p o t e n t i a l
energy s u r f a c e we expec t mass asymmetry i somers or neak i s o m e r s .
E v i d e n t l y these magic decays must be enhanced by the e x c i t a t i o n
o f the nuc leus . For example , a f t e r n - c a p t u r e we exoec t not o n l y s - p a r -
t i c l e s l i k e in the case of (n ,~ ) r e a c t i o n but a l so emiss ion of c l u s -
t e r s ( n , H l ) and not on ly the usual f i s s i o n f ragments w i t h r e l a t i v e l y
low t o t a l k i n e t i c energy l i k e in ( n , f ) r e a c t i o n but a l so compact
f ragments w i t h l a r g e t o t a l k i n e t i c energy of the f ragments c lose to
Q + B n, i . e . t h e t h r e e m o d a l f i s s i o n a f t e r n - c a p t u r e . We shou ld l i k e
360
to s t ress tha t a l ready have been observed e x p e r i m e n t a l l y not only the
(n ,a) r eac t i ons but a lso w i th a la rge p r o b a b i l i t y f i s s i o n fragments
with compact shapes. Also we expect that the n-emission in the f i ss ion
of the very heavy elements w i l l be enhance~ by n-capture. After exci ta-
t ion of the nuclei by ~-rays or very l i g h t par t ic les p, d or ~ we
expect a large enhancement of these magic decays.
Secondly, we presented exper imenta l ev idence of a new m e c h a n i s m
i n m u l t i n u c l e a r t r a n s f e r r e a c t i o n s and f o r t h e d i r e c t e m i s s i o n of c l u S -
t e r s in heavy ion collisions, based on the fragmentation of the pro-
j e c t i l e . A complete check of the new mechanism can be done by measu-
ring simultaneously the emission of l i g h t clusters and the rezidual
products. Evident ly, th is mechanism is a general izat ion of the quantum
fragmentation of a nucleus, which gives a unique descript ion of the
magic decays, to the case of heavy ion co l l i s i ons .
F ina l l y we should l ike to mention that the present quantum
nuclear dynamics must be applied to the cold fusion of heavy and very
heavy nucle i . This approach use essent ia l ly the potent ial energy sur-
faces computed in the frame of the l i qu id drop model with shell cor-
rect ions. Consequently we consider that is much more reasonable to des-
cribe the damping of the co l lec t i ve coordinates in a se l fconsis tent
way by introducing quantum f r i c t i o n and quantum d i f fus ion coef f ic ients
than introducing the one-body f r i c t i o n which is based essent ia l l y on
the shel l model which is not able to describe the binding energies of
the nucle i . Further studies w i l l elucidate which approach is better
suitabTe for the descript ion of nuclear dynamics.
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HEAVY ION RADIOACTIVITIES,
COLD FISSION AND ALPHA DECAY IN A UNIFIED APPROACH
Central
D.N.Poenaru and M. Iva~cu
I n s t i t u t e o f Physics, P.O.Box MG-6
R-76900 Bucharest, Romania
1. INTRODUCTION
The f i r s t variant of the analytical superasymmetric f ission model (ASAFM) has
been derived in 1980. I t was described in a paper i) in which the numerical super-
asymmetric f ission model (NSAFM) was reviewed and a related new semiempirical
formula for alpha decay was developed (see also Ref. 2-3)). NSAFM and
other two models used to predict heavy ion radioact iv i t ies, namely fragmentation
theory and Gamow penetrabil i t ies ( l ike in tranditional theory of alpha decay) were
also presented in a review paper 4). I t was shown that among other new decay modes,
14C nuclei are the most probable spontaneously emitted ions from 222Ra and 224Ra
parent nuclides. Earlier works were quoted in the Ref. ip4)
ASAFM was successively improved and intensively exploited since 1980, due to
its ab i l i t y to compute a measurable quantity (the half l i f e ) in a short time, al-
lowing to take into consideration a large number of combinations parent - emitted
ions (of the order of 105), requested in a systematic search of new decay modes.
As usually we shall call A, Z the mass and atomic numbers of the parent nu-
clides, AdZ d and AeZ e the corresponding quantities for the daughter (heavy frag-
ment) and the emit ted ion ( l i g h t f ragment). The va r ious nuclear phenomena which could
be t rea ted in a un i f i ed way in the framework of ASAFM are
A A d A Z ~ Z d + ez e (1)
where A = A d + Ae; Z = Z d + Ze; A d >- A/2; Z d -> Z/2.
Each kind of heavy ion r a d i o a c t i v i t y corresponds to a given emit ted nucl ide which
could be spontaneously emit ted from nuc le i . For example A e = 14, Z e = 6 is ca l led
14C r a d i o a c t i v i t y , in analogy wi th alpha decay (A e = 4, Z e = 2) and proton rad io -
a c t i v i t y (A e = i , Z e = I ) .
A f t e r showing 1) that a l l nucl ides o f known masses 5) are s tab le r e l a t i v e to
emission o f 2'3H, 3'6-9He, 4Li , 7B, 9C but 8Be, 12C, 160, etc . are among the pre-
fer red candidates to the new decay modes, ASAFM was extended in 1983 6) to account
for the angular momentum and small e x c i t a t i o n energy e f fec ts . The systematic search
365
of new decay modes was made by increasing ZeA e. ~(e isotopes with A e = 3-10 were con-
sidered f i r s t 6), when 5He- and E-enhanced Hl-emissions were predicted. The same
zero-point v ibrat ion energy was used~or ~-decay in calculations 1"5"7) performed
before the f i r s t evidence 8 ) This experiment was made for the emit ter 223Ra, bet-
ween 222'224Ra considered in ref . 4)
Then by f i t t i n g ~ - decay and 14C rad ioac t i v i t y of 223Ra experimental ha l f -
l i ves , E v was allowed to vary with A e and i t was shown 9) that Rose and Jones measu-
red the most favourable case (maximum branching ra t io re la t i ve to alpha decay. The
f i r s t successful experiment made with a semiconductor detector telescope 8) was con- 10) 11,12) and with firmed with a s im i la r technique , with a magnetic spectrometer
solid state nuclear track detectors 13)
After the comment 14) ASAFM was improved by including shell ef fects in E v and
nDre than 140 new decay modes with Z e -< 24 have been predicted 15,16). I t was
shown 17,18) that penet rab i l i t y calculations could lead to wrong results and some
new estimations have been made 19) superseeding previously published ones. The re-
gion of heavy ion emitters is extended beyond that of alpha emit ters; even the
"stable" nuclides with Z > 40 are metastable with respect to several new radioac-
t i v i t i e s 20), but the measurable decay rates are mostly expected for parent nuclei
with Z -> 84 and daughters with N d = 126. A table with most probable emitted ions
(Z e -< 28) from parents with known masses 21) is now avai lable 22) 23) Double alpha decay and mult ip le heavy ion rad ioac t i v i t i es are predicted
A superasymmetric f iss ion model based on the proximity potential was elaborated by 24) Shi and Swiatecki
Up to now, there are 8 successful experimental results: four on 14C rad ioac t i v i t y of 222Ra 13,25) 223Ra 8,10-13), 224Ra 13) and 226Ra 25,26); four on 24Ne radioac-
t i v i t y of 232U 17), 231pa 2 8 ) 233 U 29), and 230Th 30). Upper l im i t s for the bran-
ching rat ios re la t i ve to alpha decay have been obtained for: 14radioact iv i ty of 221Fr, 221Ra and 225Ac; 24Ne rad ioac t i v i t y of 232Th; 30Mg and 32Si rad ioac t i v i t y of
237Np and 34Si rad ioac t i v i t y of 241Am (see systematics 31,26,32)).
The experiments were guided by the above mentioned theoret ical works and the
agreement between the data and the predicted ha l f l i ves is rather good. By taking into account an even-odd e f fec t in zero point v ibrat ion energy 26,33) (see also the
refs. 20,22)) the agreement is fur ther improved.
The new prescr ipt ion for zero point v ibrat ion energy 33) was used to up-date
the table 22) and to include also cold f iss ion phenomena 34), with Z e > 28, which have been considered ea r l i e r 15,35)
ASAFM was extended 36) to take into account the deformation of the parent, 37) daughter and emitted nuclei and the t ransi t ions to excited states of the daughter
The influence of deformations of parent and daughter nuclei end of shell ef fects
were considered also by Shi and Swiatecki 38) Preformation of heavy clusters have
been calculated 39,40) and a mechanism in which 12C fragment is transformed into
366
14C by picking up two neutrons during the tunnel ing was proposed 41). The Dionic 42) rad ioac t i v i t y was predicted
The review paper 43) was successively up dated 44,45). A comprehensive re-
view of spontaneous and beta-delayed emission of various par t ic les from nuclei is
~ade 46) by many of the spec ia l is ts d i r ec t l y implied in the development of th is
f i e l d.
In the f i r s t part of the lecture we shal l present b r i e f l y the new prescr ip t ion
for zero point v ibrat ion energy, the extension of ASAFM for deformed nuclei and we
shall i l l u s t r a t e the shel l e f fects . Some of the experimental works on cold f i ss ion
phenomena and theoret ical approach in the framework of ASAFM w i l l be reviewed in the
second part .
2. PARENT EVEN-ODD EFFECT OF EMISSION RATES
The zero point v ibra t ion energy, E v, is a parameter of ASAFM al lowing to cal-
culate the h a l f - l i f e re la t i ve to heavy ion spontaneous emission 1,6,9).
R b h l n 2 TI/2 = ~ exp ~- . I {2~[E(R) - Q,]}1/2 dR), (2)
R a
where h is the Planck constant, p the reduced mass, E(R) is the potent ia l in terac-
t ion energy between the two fragments separated by a distance R and E(Ra) = E(Rb) =
=Q' = Q + E v. In previous publ icat ions (see for example 15,16,22)) the fo l lowing r e l a t i o n -
ship has been used:
4 - A E v = Q ~0.056 + 0.039 exp ( - ~ ) ] ; Q > 0 , (3)
where A > 4 is the mass number of the emitted ion. This was obtained from a f i t e with experimental data on ~ decay and 14C rad ioac t i v i t y of 223Ra.
An odd-even e f fec t in the parent nuclei has been observed for ~ - decay 3)
When the 379 ~ - emitters have been grouped according to even and odd proton and
neutron numbers, i t was shown that the agreement with experimental resul ts is
improved i f Ev/Q takes d i f f e ren t values in various groups of nucl ides, instead of
only one (0.095) for a l l ~ - parents, regardless t he i r even-odd character. Now we
can use the experimental data not only for ~ - decay and 14C rad ioac t i v i t y of
223Ra, but also on other new rad ioac t i v i t i es (see the Table i ) .
367
T a b l e 1 - E x p e r i m e n t a l d a t a and new c a l c u l a t i o n s
Emi t t e d Expe r i ment
i on R e f e r e n c e P a r e n t I Q
n u c l e u s - (MeV)
Group A , ~ I
14C 222Ra I 33.0 13) 25)
e - e "224Ra 30.5
226 Ra 28 .2
221Ra 32.4 31)
24Ne
32 S
34Si
e - o 223R a 31 .8
221Fr 31 .3 o - e
225Ac 30.5
230Th 5 7 . 8
E3TTh 55.6 ~ e - e
2--~-UU-- 6 2 . 3
e - o 2 33 U 60.
o - e Pa 6 0 . 4
o - e 237 N p 88 .4
o - e 241Am 9 3 . 8
~ / 2 / T I / 2
(3.7 + 0.5)10 -10
(3.1 + 1 .0) i0 - I0
13) (4.3 + 1.1)10 -1.1
25, 26) (3.2 + 1 .6) i0 -11 F
< 4.4 x 10 -12
8) lO) 11)
13)
12)
31)
31)
3O)
31)
(8.5 + 2.5)10 -10
(7.6 + 3.0)10 -10
(5.5 + 2.0)10 -10
(6.1 + 0.8)10 "10
(4.7 + 1.3)10 -10
< 4.4 x 10 -12
< 3.16x 10 -13
(5.6 + 1.0)10 -13
< 6.3 x 10 -11
27) (1.0 +_0.25)10 -12
29) (7.5 + 2.5)10 -13
28) (4.3 + 0.9)10 -'i'2
29) < 8.5 x 10 -14
25) < 3 x 10 -12
31) < 4 x 10 -14
29) < 5 x 10 -15
Thee r)
log Tl /2(s )
11.01+0.06 - i i . 16
11.09+0.17
I 15.87+0.12 15.93 21.19+0.30 20.97
> 12.8 14.27
15.06+0.15
15.11+0.22
15.25+0.20 15.20
15.20+0.07
15.32+0.14
> 13.8 14.41
> 18.1 17.83
24.64+0.09 25.27
> 27.8 28.79
21.36 + 0.1220.81
24.82+0.18 24.82
23.38+0.10 23.38
> 27.2 27.6
> 21.6
> 23.5 24.5
> 24.4
*) Mass v a l u e o f the d a u g h t e r n u c l e u s n o t measured .
The experimental data on 14C and 24Ne r a d i o a c t i v i t i e s o f even - even (e - e) ,
odd - even (o - e) and e - o parent nucle i are obeying qu i te wel l the systematics
p lo t ted in Figure 1 33). A fami ly of smooth curves going through the experimental
points is given by the re l a t i onsh ip :
368
E v ~ ax2 + bx + c
Q l c, A e
fo r 4 ~ Ae < Aeb
-> Aeb ; x = (A e - 24)/20
where the parameters a, b, c are given in Table 2.
(4)
Ev
0.08
o.o
m
_ ~ ~
L " I I' I l I , ~ I ! L I
zO 20 3t7 Ae
~ - Variation of Ev/Q ratio with the mass number of emitted ion in various groups of parent nuclei
Table 2 - Parameters a, b, c o f the eq. (4)
Group o f parent Aeb a b c nucl ides
e - e 27 0.0178 -0.0341 0.0530
o - e 24 0.0230 -0.0261 0.0457
e - o 24 0.0452 0.0066 0.0437
24 0.0410 0.0400 o - o 0 . 0 0 8 5
From the value o f parameter c one can see tha t emission rates from odd-even parents
are somewhat hindered wi th respect to that from even-even nucle i or e q u i v a l e n t l y :
emission rates from even-even parents are enhanced. With new p resc r ip t i on given by
eq. (4) , the agreement wi th experimental data is improved (see Table 1).
Now the h a l f - l i v e s and branching ra t ios fo r d i f f e r e n t kind o f r a d i o a c t i v i -
t ies lead ing to 208pb daughter are given in Figure 2a and 2b, respec t i ve l y .
369
F
15
2 # 70
c Ne
20 30 #0 50 Ae
Fi~. 2.- The decimal logarithm of the life-times (a) and the branching ratios relative to ~ - decay (b) for emission leading to the double magic 208pb daughter nucleus.
By comparing the new prescr ip t ion with the old one (dashed curve in Fig. 1)
one can see that our calculat ions previously performed on the basis of eq. (3) are
too op t im is t i c , especia l ly for A e > 24, For example one has now log T(s) = 21
instead of 19,8 for 28Mg rad ioac t i v i t y of 236pu and log T = 23.6 instead of 21.6
for 46Ar and 48Ca emission from 252Cf (see re f . 34) for other de ta i l s ) .
3, ASAFM FOR DEFORMED NUCLEI
Spherical shapes have been assumed as the f i r s t approximation in the ASAFM
up to now. There are some processes ( fo r example emission from shape isomers or
from superdeformed nucle i ) where one has to take into consideration the nuclear
deformation. For the t rans i t ions taking place between ground states one can use
the nuclear deformations E 2, ~4 and E 6 calculated by M~ller and Nix 47)
We consider spheroidal shapes of the two fragments with semiaxes (c, a)
of the emitted ion and (c d, ad) o f the daughter nucleus. For a ~iven set o f
370
deformations (~2' ~4' c6) 47 ) 3= spheroid approximating this shape and conserving the volume a2c = R o
= (I.2249)3A fm 3.
The Coulomb energy of the two spheroids 49) at the touching point,
Rtd = c e + Cd, is given by:
ZIZ2e2 c~ - a 2 c 2 - a 2 Ecd - R F(x .y ) ; x2 = ~ d d ; y2 = e e R 2
td
one can f i nd 48,36) the semiaxes c and a of the
(5)
whe re
F(x,y) = s(x) + s(y) - I + S(x ,y) (6)
S(x,y) : Z 3 3 n=1 m=l (2n+l)(2n+3) (2m+1)(2m+3)
5 [ (x + ~ ) a r c tg (x) - I ] . c < a;
s(x) = ~ 1 , c : a
L 0 . 7 5 [2 + x I In i + x -7- I
By using the notations similar to ref. 6) one has
(2n+2m)! - x2ny2m (2n)!(2m)!
x = VT,
c > a
Eid = Ecd + E d b Q'-Q • = c - Ce; ~h2~(~+i) ; = EbO ' Rid E~d = 2~R2d
AIA 2 1/2 Kov = 0 . 2 1 9 6 ( E ~ - ) (Rtd-Rid)[qrl~--b - b In i + ~ ]
4~
(7)
(8)
(9)
The e x i t po in t Rbd is f ind e i t h e r wi th the Wegstein's numerical procedure, or wi th
an ana ly t i ca l approximation in the neighbourhood o f
Rt Ec Q,E l 1/2
by solving the equation (in which R t, E c, EC refers to spherical shape):
ZlZ2e2 Rt 2 E(Rbd) = Rbd F(Xb' Yb) + E~d " ~b = Q'
Then r = Rtd / Rbd and
( l l )
371
2 K s = ~ 2~,,/~ Rbd ~r F(x,~) r E~ d r 2 -F - . -
,F(xt,Yt) ~ Q' ~ i dt ( 12 )
One has
f : ( i - r 2 E i d l Q ' ) / ( 1 - r ) ; g : 1 - f (13)
and the i n teg ra l is f ind numer ica l ly w i th a Gauss - Leqendre quadrature, or is
approxi mated by
lg + r
I f g= F arc cos - q ~ ) - ( - ~ + 2 - f
+ ~/~ in[2vr~ ~ r ~ + 2~ + f r ] (14) r ( 2 - f )
F ina l l y K = Kov + K s and decimal logar i thm of the h a l f - l i f e , T, in seconds i s :
log T = 0.43429 K - log E v + 20.8436 (15)
where E v is the zero point v ibrat ion energy obtained from the f i t with experimental data.
:N ## , 18 So ~ .,; 3# ~ I
# E lO 15 20 25 30 R (Fin)
Fi 9. 3.- Potential barrier for very [email protected] oblate and prolate shapes compared with spherical ones for 14C radioactivity of 223Ra.
The influence of the deformation on the potent ial bar r ie r for 14C radioac-
t i v i t y of 223Ra is shown in Figure 3, by taking very deformed oblate (c/a = 0.4)
and prolate (c/a = 1.8) spheroids in comparison with the sphere (c = a). The bar r ie r
height is smaller for prolate shapes and larger for oblate ones. I f only the pa-
rent nucleus is deformed, the overl.apping part Of the bar r ie r becomes thinner
for prolate or thicker for oblate ones. When also fragments are deformed, one can
see an important reduction of the bar r ie r height for prolate shapes and an increase
for ob la te ones.
372
Ev
" 8S88
kT.4L'kN I
b, /
I 60 ?0 80 90 I00 IlO f2-O 130 I~O 150N4
Jig. 4.- Zero point vibration energy allowing to reproduce experinw~ntal half-lives for ~ - decay of even-even nuclei: a) deformed; b) spherical shapes.
In Figure 4a we have plotted E v values allowing to reproduce the experimen-
tal hal f - l ives of 125 even-even alpha emitters when the deformations calculated in
ref. 47) are used. Similar diagram for spherical nuclei is shown in Figure 4b. For
prolate deformations of the parent and daughter one need smaller E v to obtain the
same ha l f - l i f e as for the spherical shapes. A prescription for Ev/Q similar to that
given in eq. (4) was obtained 36) by f i t with experimental data on partial half-
lives for ~ - decay, 14C and 24Ne radioact iv i t ies.
4. SHELL EFFECTS
Various regions of parent nuclides are obtained when di f ferent partial hal f-
l ives T, relative to heavy ion radioact ivi t ies are allowed for the most probable
emission with Z e -< 28fromeach parent. Figure 5 shows three such regions: a small
one (501 nuclides) in which log T(s) _< 30 and bigger ones for loq T <_ 50 (941
nuclides) and log T_< 100 (1330 nuclides) respectively.
Let us count how many times appears as the most probable a given daughter
nucleus (with Z d protons and N d neutrons) in the reoion with log T <_ 100. The
frequency of finding a given daughter is shown in Figure 6. From this figure i t
is very clear that the magic number of neutrons 28, 50, 82 and 126 are prefered.
The same is true for the magic number of protons 28, 40 and 82. Nevertheless the 132Sn I00~ magic number Z d = 50 is not so often met because either 50 82 or 50~n50 are
far o f f the l ine of beta s tab i l i t y ( in spite of the shell effect, the l iquid drop
component is not favourable).
373
Z
90
80
70
60
50
_- .aSn ~
~0 50 60 70 80 90 I00 fro 12013B-~ N
Fig. 5 . " Regions o f pa ren t nuc le i w i t h d i f f e r e n t l i m i t s o f the h a l f - l i f e .
z~8 8Z P6~a6
,05, /
Z8
Fig. 6 . - Frequency of appearence o f some daughter nuc le i as the most probab le ones. Z e < 29: log T(s) -< 100.
374
S im i la r diagrams are p l o t t ed in Fig. 7 and Fig. 8 fo r the heavy fragments in
the cold f i ss ion phenomena considered heavy ion emission wi th Z e >28 when
13].,54 ~ e 82 '3Z S ~ t : ~
eeer 4oZ~o ~-.-.-..,.,.,.;;,.:-:-..;..;.-.-.---.-.-'~.-.-_:.tllllllb,~ 38 50 - ' - - - - - ' - ' - ' - ' - ' - ' - ' -" - ; ; ;~%;; - ' - ;~ ' - ' - ' - . . . . .
~ ' - - - - - " - - - - ~ " ; - - - : . ' . ' . ' ~ ' ~ ' . - , ~ - s o ~ s Z d
Nd
Fi~. 7.- Frequency of appearence of some heavy fragments as the most probable ones i~ cold fission (Z e > 28) of nuclei with known masses 2 . log T(s) ~ 50.
log T(s -< 50. Figure 7 is based on ca lcu la t ions performed by using Wapstra - Audi 21)
masses in the region 65 <_ Z _< 108, 80 -<N -< 157.
i r 135 738
VO s~ Xe e2 ,~B° e2
13z3M ~ /
.'/.;'.~c'.c~ 111111!~,:.:.;/.>;.;././7°u
78 75 ~0 85 90 @
Fi 9. 8.- The same as Fig. 7 for parent nuclei4wlth Z >-90 and masses calculated by Mol]er and Nix 7),
Masses calculated by MiJller and Nix for parent nuclides with Z> 90 are used to
obtain the results plotted in Figure 8. Again the neutron shell effects at the
375
magic numbers N d = 50 and 82 are more important than the proton shel l e f fec ts
at Z d = 40 and 50.
5. EXPERIMENTS ON COLD FISSION
Usual ly the f i s s i on fragments at sc iss ion po in t are deformed and exc i ted 50)
The to ta l k i n e t i c energy (TKE) E k is smal ler than the Q - value, Q = M - (M e + Md),
because par t of i t is transformed in to e x c i t a t i o n energy, E ~, released by neutron
evaporat ion, E v, and y - decay:
Q = E k + E v + Ey = E k + E ~
E = <v > + (16) v (Bn En)
where < ~ > is the mean number o f evaporated neutrons, B n is the neutron binding
energy and E n the neutron k i n e t i c energy.
The most probable to ta l k i n e t i c energy is given by the Vio la systematics 51)"
E k = (0.1189 + 0.0011) Z 2 A - I / 3 + 7.3 (_+ 1.5) HeY (17)
r e l a t i n g the k i n e t i c energy wi th the Coulomb repuls ion at the sc iss ion po in t .
As ea r l y as 1956 i t was observed 52) that in thermal neutron induced f i s s i on
of 233,235U and 239pu, another f i s s i on mechanism is possib le: in a small f rac t i on
of events E ~ < B n, the neutron emission is un l i ke l y and the primary fragments
could be d i r e c t l y determined.
The thermal neutron induced f i s s i on on 233'235U and 239pu targets has been
recent ly studied by using a time o f f l i g h t technique with semiconductor d e t e c -
tors 53,54) a t the Grenoble High Flux Reactor, a twin gridded i o n i z a t i o n chamber at
the ORPHEE reactor 55), the f i ss ion product separator LOHENGRIN and the f i ss ion
fragment spectrometer COSI FAN TUTTE at ILL Grenoble 56 -59 ) I t was found tha t
i per m i l l e o f a l l fragments occur w i th E ~ < 7 MeV, hence E k - Q. The mechanism
of cold f i s s i on corresponds to no e x c i t a t i o n energy and to the most compact sc is -
sion con f i gu ra t i on , the process being the inverse o f a fusion phenomenon. I t was
observed a preference for the formation o f even-even fragments. Unfor tunate ly the
low experimental accuracy does not a l low to observe whether fragments are exc i ted
or not , but very l i k e l y even the co ldest f ragmentat ion imply an e x c i t a t i o n o f
two q u a s i - p a r t i c l e s , proving that f i ss ion dynamics is viscous.
Fig. 9 (~elected from 54)) shows the l i g h t fragment con t r ibu t ion to the mass
y ie ld in the thermal neutron induced f i s s i o n o f 233'235U and 239pu. When the k ine-
t i c energy of the l i g h t fragment is high enough (exhaust ing the Q - va lue) , the
maximum y i e l d is obtained at the mass number A L = i00 fo r 234U compound nuc leus,a t
A L = 104 fo r 236U~and at A L = 106 fo r 240pu.
376
YU gE 1## AL (b) • | 10a5o ft~~ t
7# 80 90 JOB AL
118 MeV A J
80 90 lOO AL1/g
Fig. 9.- Mass yield for specified light fragment kinetic energy. Neutron induced fission of 233U (a), 235U (b) and 239Pu (c) 54).
Spontaneous f iss ion of heavy nuclei was studied extensively (see for exam-
ple 60-65)). Strong deviations from systematic act inide properties have been ob-
served in a l imi ted region of nuclides near Z = I00, N >- 158. There is a sharp
decrease of the spontaneous f iss ion h a l f - l i f e going from 254No to 258No (about
eight orders of magnitude), strongly affected by the 152 neutron subshelI. This
is a reversal trend compared with the LDM trend of isotopes of the element 104 where the h a l f - l i f e increases by a factor of 40 from 254104 to 260104. Two re-
gions of symmetric mass d is t r ibu t ions have been found: one around 81T] - 89Ac
characterized by a large width ( l i qu id drop model (LDM) l i ke ) and another for heaviest isotopes of Fm (258'259Fm), Md (259'260Md) and 258No which was called
"fragment shel I directed symmetric fi ssion". The total kinetic energy of 258'259Fm fission fragments is by about 40 MeV
larger than the corresponding value given by Viola systematics. The TKE distribu- tions in the spontaneous fission of 258Fm, 259"260Md and 258No 65) can be
decomposed into two Gaussian components which are peaked near 200 MeV (like in Viola systematics) and 240 MeV, as i t is shown in Fig. 10 66). The high
377
; 'A! r ; t~l k/nelie energy (Me V)
FicJ. I0.- Kinetic energy distribution in fission of 260Md.
TKE contribution is about 65% for 260Md, 56% for 258Fm, 15% for 259Md and 9% for
258No. This was called 65) bimodal symmetric fission. Potential energy surfaces
Calculated in the framework of the macroscopic-microscopic Strutinsky method with
the two-center shell model for 258Fm 67,68) as well as for 240pu by using density -
dependent Hartree -Fock -Bogolyubov approach 69) are showing two valleys: a
deeper one, corresponding to the usual fission path with elongated fragments and
a new valley with more compact shapes. The difference in energy at the intersec-
tion of scission line with these two fission trajectories is almost equal to 35 MeV 68)
6. COLD FISSION AS HEAVY ION EMISSION
From the calculations in the framework of ASAFM reported in 198415) of the
233U half- l ives spectrum for various fragmentations going from ~ - decay to sym-
metric sp l i t , i t is clear that there are three dist inct regions of mass asymmetry.
In each of the three regions one of the ions is the most probable to be emitted. For 233U, besides ~ - decay and 24Ne radioactivity one has also 99Zr emission.
This case with small mass asymmetry was interpreted in 1985 35) to correspond to
Cold fission as i t has compact shapes and high kinetic energy exhausting the
Q - value. A similar spectrum for 234U 35) is plotted in Fiq. 11. One can see that
the most probable heavy ion emitted at small asymmetry from 234U is 100Zr - in good
~ greement with experimental data shown in Fig. 9a. Fig. 11a shows that alpha decay,
8Mg radioactivity and cold fission can be treated in a unified way by using ASAFM.
Alpha decay is so strong that even far from the magic number of neutrons
and protons of the daughter nucleus the emission rate is very high. 28Mg is the
next heavy ion emitted from 234U because the corresponding daughter (206Hg) has
378
i 30
~0
~2J~ez 230 '
(a) 9oTb/~ e
Ioo. t3~ ~o ~F6O 5Z~ eoz
206 8Q ' ','TE
20 ~0 ~0 80 IQO IZO t~O f60 MO 200 220A
0 ' ( b )
l h"iil,,',,. . ..,.rlrll
O0 ~0 gO 100 120 NO [$0
(c~ tz9 nTg 0 '30
I0 zoo
20 ~zu
f L A , , , ' , , , , , , , , , , , , , , , , , , ,~1
- fO f3ZRn Cd) ,TO-- 82
I.... ea
3O
I 1 I * I •
#0" ZQ 11 EO 80 IQO fz# I~0 lEO 180 Z## Z'~ Z~ 2tl A t
Fig.11.- Ha)f lives spectra for emission fOrvarious heavy ions from 234U (a), IUOHg(b), 258Fm(c), and 254Fm (d).
379
13a T a magic number of neutrons. The heavy fragment 52 e82 corresponding to lOOzr has
also magic number of neutrons.
Many other nuclei are included in the two tables 34) based on measured mas-
ses 21) and on the calculated ones 47) in the region of heavy nuclides where expe-
rimental values are s t i l l missing.
From these tables, one can see that a symmetric cold fission process is associa-
ted with a fission in two magic or almost magic fragments. For l ight nuclei one has
either N = 100 (N/2 = 50) or Z = 80 (Z/2 = 40). As is shown in Fig. 11b for 180Hg
90 emission is very low fission the branching ratio relative to alpha decay of 40Zr50
- i t is not possible to measure this process at present.
The next region is at Z= 100 where the symmetric bimodal fission was observed.
Now the competition with alpha decay is gained by spontaneous fission, as one can
see from Fig. 11c. Calculated 47) mass values has been used for 258Fm and 264Fm,
which are not tabulated by Wapstra and Audi 21) The next sp l i t in two magic nuclei
132c~ is obtained as a cold f iss ion of a neutron rich 264Fm (see Fig. 11d). 50~'82
- 6
,..,, 2.
8 10 12
I'
i: C A
Y i \ I
, i , , . I , ' , , , , I J l
150 150 IV 170
Fi~. 12.- Half lives of the spontaneous fission of even - N Fm isotopes, from experiment (dotted line) and calculations of cold fission mechanism (full line).
Calculated half lives by using estimated masses 47) for the cold fission of
various neutron rich Fm even-N isotopes are compared with experimental ones in
Fig. 12. One can see that the half-lives decrease up to N = 164 and after that the
trend is reversed. I t is expected to find a steeper variation of the hal f - l ives,
closer to the experimental ones by using a s l ight ly dif ferent mass values, because
very l ike ly the contribution of the cold fission mechanism to the total sponta-
neous fission process of 258Fm is very important, as the experimental mass and
kinetic energy distributions have already shown.
380
In conclusion some important features of alpha decay, heavy ion radioact iv i t ies
and cold f ission phenomena are well explained in the framework of ASAFM, but further
theoretical and experimental e f fo r t is needed to understand better these processes
ACKNOWLEDGEMENTS
Part of this lecture is based on research work performed in cooperation with
Professor W. Greiner and Dr. K. Depta from Frankfurt University and with D. Mazilu,
Dr. I.H. Plonski and R. Gherghescu from Central Inst i tu te of Physics, Bucharest. We
are grateful to Prof. P. Armbruster, Dr. E. Hourani, Dr. M. Sch~del, Dr. K. SUmmerer
for useful comments, to Dr. K. Hulet, Dr. C. Signarbieux and Dr. F. G~Jnnenwein for
the correspondence on cold f ission processes.
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SPONTANEOUS EMISSION OF FRAGMENTS FROM NUCLEI
E. HOURANI
Institut de Physique Nucl~aire, BP n ° ]
9 ]406 ORSAY C~dex, France
] - INTRODUCTION
The spontaneous emission of f ragments from nuclei is a recent ly d iscovered
radioact iv i ty. Essent ia l ly, the d iscovery was that some radioact ive a -sou rces like Ra and U
isotopes are emit ters of f ragments like ]4C, 2 4 , 2 5 N e , . . . with branching rat ios of ]0 - ] 0 or
tess relat ive to the a-par t i c les .
The theoret ica l pred ic t ion of this radioact iv i ty was publ ished in 1980 by Sandu lescu,
Poenaru and Gre iner ] ) ,
The measurement which establ ished this radioact iv i ty was publ ished in ]984 by Rose
and Jones who repor ted the emission of 14C nuclei from 223Ra with a branching ratio of
( 8 . 5 ± 2 . 5 ) x 3 0 - ] 0 relat ive to a -pa r t i c les 2) .
Within a few months fol lowing the publ icat ion of Rose and Jones several groups
from di f ferent laborator ies using var ious techn iques conf i rmed their result 3 - 6 ) . The search
for other new cases of 14C radioact iv i ty was the log ica l next step. Such cases were found
In other isotopes of rad ium, i .e , 222,224Ra5) and 226Ra7-8) . Further effort y ie lded the
d iscovery of Ne emission from 232U9) , 2 3 3 u t ] ) , 23 ]pa10) and 230Th12)
During their p ioneer ing work, Rose and Jones met a ser ious diff iculty in the cho ice
of the decay to invest igate because they were guided only by Q-va lue and Gamow factor
cons idera t ions. The fol lowing invest igators were or iented eff ic ient ly by the real ist ic
pred ic t ions of Poenaru et al. ]3) obta ined, thereaf ter , by normal izat ion to the result of Rose
and Jones. The dif f icult ies encountered by these invest igators o r ig ina ted , mainly, in the
preparat ion of intense radioact ive sources and in the setup of select ive detect ion
techniques,
384
At p resen t , the status in the spon taneous f r agmen t emiss ion study resemb les many
o ther advanced f ie lds in nuc lea r phys ics : very spec ia l i zed techn iques are needed to add
new exper imen ta l resul ts and theore t i ca l model re f i nemen ts a re be ing es tab l i shed in o rde r
to take into accoun t the overa l l fea tu res of the exper imen ta l resul ts .
We found it conven ien t to c lass i fy the expe r imen ta l resu l ts ob ta ined so far acco rd ing
to the d i f fe ren t de tec t ion techn iques . Thus, a f ter a sec t ion for basic cons ide ra t i ons , we
devote th ree sec t ions to desc r i be the exper imen ts c o r r e s p o n d i n g to th ree de tec t ion
techn iques , i . e . the z~E x E t e l escope , the magnet i c spec t r ome te r s and the t rack reco rd ing
fo i ls . In a last sec t ion we p resen t a genera l d iscuss ion .
2 - BASIC CONSIDERATIONS
The decay of a nuc leus ( Z , A ) of mass M ( Z , A ) into a f r agmen t ( z , a ) and a
res idua l nuc leus ( Z - z , A -a ) Is ene rge t i ca l l y cha rac te r i zed by the Q-va lue de f ined by :
Q = M ( Z , A ) - M ( z , a ) - M ( Z - z , A - a )
The decay is poss ib le only when Q > 0. Using a tab le of masses , one can see that
Q is posi t ive for a la rge number of heavy nucle i decay ing into severa l types of f r agmen ts
and that the la rges t Q -va lues are ob ta ined in the decays where the res idua l nuc leus is the
doub ly magic nuc leus 208pb or a ne ighbo r i ng nuc leus . The kinet ic ene rgy of the emit ted
f r agmen t is deduced f rom Q by Q x ( A - a ) / A . In add i t i on , the main cha rac te r i s t i c of the
decay , which is the decay ra te , is very sens i t ive to Q.
The s imples t way to eva lua te the decay rate is the t rad i t iona l a - m o d e l where the
f r agmen t Is supposed to be p re fo rmed ins ide the nuc leus , and imp ing ing on a con f in ing
Cou lomb bar r ie r . The tunne l ing probab i l i t y ac ross the ba r r i e r , a lso ca l led Gamow fac to r , Is
g iven, within the WKB approx ima t ion by :
G -- exLD [ -- 2 ~ d r ( 2 / z (U ( r ) - Q) / f~2) 1"/2 ]
where U ( r ) = z ( Z - z ) e 2 / r , a = r o x ( ( A - a ) 1 / 3 + a ] / 3 ) l p , is the reduced mass and b is
de f ined by U ( b ) = Q.
385
We ca lcu la ted the f r a g m e n t / a (3amow factor rat ios for the most probable decays of
the members of the 235U natural ser ies. We present , In table 1, their values for r o = 1. 15
and ~. 20. These Gamow factor rat ios show the high sensit ivi ty of the tunnel ing to the values
of Q and r o. On the other hand, they correct ly se lect for a given parent the most probable
decay. Also, they are meaningful to compare the probabi l i t ies of the decays of two
ne ighbor ing parents. Unfortunately, they complete ly fall to predict ei ther a real ist ic value
for a given decay or the rat io of the decays of two parents aS~ar from each other
as 223Ra and 235U.
TABLE 1 : Most p robab le decays in the 235U natural ser ies
DECAY
235 U ~ 28Mg + 207Th
231pa_ ~ 24Ne + 207TI
227Th ~ 14 C + 213po
_~ 180 + 209pb
223Ra_ # 12 C + 21~b
.~ 13 C + 210pb
14 C + 2Ogpb
2zgR~ ' z% . 2ossg
Q-Value (re,v)
72.20
60.42
29.45
44.21
27.73
28.85
31.84
28 .Ii
Gamow factor ratio
to= 1.1S I to= l.aO
FRAGMENT/a
I Branching ratio ref.13 I ref.14
1.6xlO -II
1. Oxl0 -I0
4.0xl0 -16
15. OxlO -16
14. Ox10 -1S
14,0xlO -14
-9 3.2x10
6.3xlO -21
3.3XI0 -3
2.9XIO -3
5.3XIO -12
2. IW.IO -10
--12 6.9XIO
i. 3XIO -IO
--5 1.2X.IO
-16 2.6xlO
1.14
0.37
9.2XIO -II
9.8xlO -9
8.9XIO -II
-9 i. 9xlO
1,7XI0 -4
4.3XI0 -15
1 • 47XI0 -I{
9 • 4 XlO -I"
9.2 xlO -I:
7.3 xIO -I|
7.4 Y.lO -I[
i. 1 XIO -I'
--9 6.5 X10
5.'7 X3.0 -23
At present , it is the f lssionl lke descr ip t ion , even in Its s impl i f ied form as was given
by Poenaru et a1."13) and, later on, by Yi-JIn Shl and Swiateck114), which yields the best
predict ions for the spontaneous emiss ion of f ragments. Such a descr ip t ion takes into
account the nuc lear col lect ive osci l la t ions t imes the f ragment penetrabi l i ty factor through a
deformat ion energy barr ier parametr lzed to fit exper imenta l data. In table 1, are presented
the f r a g m e n t / a branching rat ios ca lcu lated with these supersymmetr lc f ission models for the
decays a l ready selected accord ing to Gamow factor rat ios. Not only do the calcu lated
branching rat ios select 223Ra as the best f ragment emit ter cand idate but also they pred ic t
real ist ic values when compared to observed results.
386
3 - THE 14C RADIOACTIVITY OF 223Ra MEASURED WITH A AExE TELESCOPE
A z~ExE SI su r face ba r r i e r de tec to r t e l escope was used by Rose and Jones 2) in
the i r d i scovery fo r the 14C decay o! 223Ra and la ter on by A texandrov et al. 4) in the i r
repet i t ion of the m e a s u r e m e n t of Rose and Jones . With this t echn ique the t e l escope is
p laced In d i rec t view of the source . In the expe r imen t of Rose and Jones , the z~E de tec to r
was 8 . 2 ~.m th ick with an a rea of 200ram 2, the E de tec to r had an a rea of 300 mm 2 and the
so l id ang le z~ was 1 / 3 s r , whe reas in the expe r imen t of A lexandrov et al. the de tec to rs were
16 p.m and 500 p.m th ick with z~3 = 0. 1 sr.
Since the h igh coun t ing rate of c~-part ic les p roduc ing mul t ip le p i l e - u p events iS the
most seve re cons t ra in t in a z~ExE te l escope m e a s u r e m e n t , the sou rce and the m e a s u r e m e n t
t ime shou ld be care fu l l y chosen .
The matn decay s e q u e n c e of 235U is :
235U(a,7.108y ) -, 231Th(~-,25.6h) -, 231pa(~ , 3.3xlO4y) -~ 227Ac(~-,22y )
-~ 227Th(us18.Td) -~ 223Ra(u,ll.4d) -P 219Rn(u,3.9s) ~ 215po(u,l. Sms)
-~ 211pb(~-,36.1m) -~ 211Bi(u,2.15m) -, 207Tl(~-,4.Sm) -* 2°7pb (stable)
Examin ing this s e q u e n c e one sees , e f fect ive ly , that 227Ac is the most conven ien t
p r e c u r s o r for 223Ra with regard to Its ha l f - l i f e of 22 y and to a m e a s u r e m e n t t ime of severa l
months .
In the expe r imen t of Rose and Jones , the Ac sou rce had an act ivi ty of 3 . 3 /~Ci in
= - p a r t i c l e s g iv ing a count ing rate of ~ 4 0 0 0 c / s and p roduc ing up to quad rup le p i l e -up ,
w h e r e a s tn the expe r imen t of A lexandrov et al. an Ac s o u r c e of ~- 85 ~Cl y ie lded a coun t ing
rate of 25 000 c / s and p roduced up tO qu in tup le p i l e -up . F igure 1 shows the resu l ts of Rose
and Jones ob ta ined in a run of 189 days. In this f i gu re , 11 events l ie a round a value of 30
MeV of the total ene rgy which c o r r e s p o n d s to the ene rgy ( d e d u c e d f rom the Q-va lue ) of the
14C nuc le i expec ted to be emi t ted by 223Ra, whe reas the i r ~E fal l ins ide the two dashed
l ines de l imi t ing the loca t ion of C i so topes as ex t rapo la ted f rom a ca l ib ra t ion with an ~-
sou rce . In f igure 2, we show the resul ts of A lexandrov et al. ob ta ined in a m e a s u r e m e n t of
30 days. The re a re 7 events lying a round the total ene rgy of the expec ted 14C nucle i
( i nd i ca ted by an a r row) and within the tocat lon of C i so topes ( l im i ted by dashed t ines) .
387
H e r e t he a r r o w and t he d a s h e d l ines r e p r e s e n t t he c a l i b r a t i o n e s t a b l i s h e d wi th a 12C b e a m
f r om the c y c l o t r o n of the K u r c h a t o v Ins t i t u te of A t o m i c E n e r g y .
120
8O
LU ~0 <3
X
t l t ~ j . . . . . . . . . . . . . . • - - ~'~'-----_\~i
t t 12C x 1~ C
2 ~ 2'8 ~o EToto L (MeV)
. . . . . . . . l E Totot (ChonneL)
Fig. 1 Results of Rose and Jones 2) with an 22-7Ac source end • ~ x E telescope. In dots, the 140 nuclei emitted by ##3RI. The arrows and the dashed lines are the results of the calibration. The lower of the two crosses represents a quadruple pile- up. The upper cross is an event which was recorded during a thunderstorm.
Fig. 2 Results of Alexandrov et el. 4) with sn 227Ac source and a Z~ExE telescope. In dots, the 14C nuclei emitted by 223Ra. The arrow end the dashed lines are the results of the calibration. The solid lines bound the region of quadruple co~'ncidences of pulses, The crosses represent quintuple co'(nci- dences.
On the bas is of t he l o c a t i o n o f t he ] ] e v e n t s in the z~ExE p lo t ( f ig . "1) a n d of a
f a v o r a b l e c a l c u l a t e d G a m o w f a c t o r ( t a b l e "1) Rose and J o n e s e s t a b l i s h e d the 14C d e c a y of
223Ra . As to the resu l t s of A l e x a n d r o v et e l . , t hey s h o w e d tha t the m e a s u r e m e n t of Rose
and J o n e s was r e p r o d u c i b l e ,
C o n s i d e r i n g the z~ExE t e l e s c o p e t e c h n i q u e I tse l f , we see tha t It I nvo l ves h e r e the
d e t e c t i o n of the v e r y h igh f lux of a - p a r t i c l e s em i t t ed by t he s o u r c e s , a fac t wh i ch no t on l y
p r o d u c e s a h i gh ra te of mu l t i p l e p i l e - u p even t s even in l ong m e a s u r e m e n t s bu t a lso
d a m a g e s the d e t e c t o r s . These f e a t u r e s led phys i c i s t s to use In the la te r e x p e r i m e n t s on
S p o n t a n e o u s f r a g m e n t e m i s s i o n m o r e s e l e c t i v e t e c h n i q u e s , i . e . the m a g n e t i c s p e c t r o m e t e r s
and t he t r a c k d e t e c t o r s .
388
4 - EXPERIMENTS ON 14C DECAY OF Ra ISOTOPES WITH MAGNETIC SPECTROMETERS
Because the decay into f ragments occurs as very low branching rat io relat ive to c~-
emission in an a - s o u r c e , an at t ract ive Idea is to select the f ragments f rom the very high
flux of co-particles and to d i rect them towards a detector . This Is accompl ished by magnetic
spect rometers . The main advantage In such a method is to use very strong sources with no
risk of damaging the detectors . The l imitat ion in the per fo rmances or ig inates in the
smal lness of the solid angle of the existing spec t rometers and the acceptab le th icknesses of
the sources which do not disturb too much the kinetic energ ies of the emit ted f ragments .
Many spec t rometers were cer ta in ly cons idered but only two led to diffused publ icat ions : the
superconduct ing soleno'l 'dal coi l SOLENO 15) at Orsay and the we l l -known type Enge spl i t -
pole of Argonne. The main charac ter is t i cs of these devices will be succint ly descr ibed and
the exper imenta l results will be given.
4. ] - Exper iments with SOLENO
The spec t romete r SOLENO is a superconduct ing solenoidal coil sur rounded by an
Iron shield and envelopping a vacuum chamber . It presents a per fect azlmutal symmetry
with a large solid angle of 100 msr and a large bandwidth in Bp of 5%, which seem to be
the sui table charac ter is t i cs for f ragment select ion, When the e lect r ic cur rent of SOLENO is
set to focus the f ragments emit ted f rom the source on the de tec tor , the doubly charged
a--part ic les, a ++, emit ted by the source ere focussed well before the detec tor and the singly
charged a + are focussed well behind the detector 3) . So both of them do not impinge on the
detector ( f ig. 3a) . The t ransmiss ion of SOLENO is usually descr ibed in terms of the
ef fect ive solid angle at Its en t rance versus the magnet ic r igidity ~p of the inc ident ions (f ig.
3b) . When the ions have a we l l -de f ined energy, e+g. in case of thin sources, the
focussing is adjusted in order to place the representat ive point of these ions at the maximum
of the t ransmiss ion curve, whereas when the Ions have a wide distr ibut ion of energy, e .g .
In case of thick sources , their representat ive points lie along the whole t ransmiss ion curve.
In the later case, it results In a reduced effect ive solid angle.
389
{mar)
lO0
50
b • , . . . . , . . . . , ,
12CS+ 13C~* l~*l/,cS+ 200S*
~+l ~ cc ÷
0.9 1.o 1.1 Bp/<Bp>
~E (HeVJ
10
AE (HEY)
o) ,,,, '~o : , ",j,.)
/ ' : i . . . . ,.~,~.
go;" .
!..~.."
,o ,5 Jo ;s ~0
10 ~5 ~0 ~5 30 3'5 E (HEY)
Fig. 3 (a) Setup inside the vacuum chamber of SOLENO : 1 -source , 2 - baffle, 3 , 4 - Z:~.xE telescope. The trajectories of focussed 14C6+ and unfocusoed a + and ¢x ++ are shown. (b) Transmis- sion curve of SOLENO established with an a-source. The arrows indicate the positions of degraded ¢¢+ and of different fragments which could be emitted from the daughters of 227Ac. From reference 3.
F,g. 4 Results of Gal~s et el.3) with an 227Ac source end a AExE telescope placed in the focussing plane of SOLENO. (a) In dashed lines are the results of the calibration. A group of events falls inside the expected location of 14C nuclei emitted by 223Ra. At low energy counts due to a-particles. (b) Background measurement where a slight contami- nation from Cf source w ~ revelled.
The d e t e c t i o n sys tem c o n s i s t e d of a Z~ExE t e l e s c o p e of s i l i con s u r f a c e b a r r i e r
d e t e c t o r s . The d i m e n s i o n s of the AE and E d e t e c t o r s w e r e 9 / .¢mx 200 mm 2 and 200 /~m x
300 mm 2 r e s p e c t i v e l y .
The t e l e s c o p e was c a l i b r a t e d wi th a 14C b e a m f r om the T a n d e m of Orsay .
Desp i t e the h igh m a g n e t i c r e j e c t i o n of SOLENO fo r the <x-par t ic les em i t t ed by the
S o u r c e s t h e r e w e r e two c a t e g o r i e s of a wh i ch cou ld r e a c h the d e t e c t o r s :
a) t he a pa r t i c l es em i t t ed by t he Rn and its d a u g h t e r s d e p o s i t e d a r o u n d t he
d e t e c t o r s ( t h e Rn as a gas e m a n a t e s f r om the s o u r c e and t r ave l s i ns i de t he v a c u u m t o w a r d s
the d e t e c t i o n a r e a ) ,
390
b) the degraded c( + part ic les, especia l ly in case of thick sources , which fall inside
the t ransmiss ion band of SOLENO.
However, all the detected c( are represented near the or igin in the z~,ExIE plane.
Only mult iple p i le -up could give events extending to the ]4C locat ion.
In f igure 4a, we show the results 3) of a run of 5 days with a source of 2:~7Ac of 2 ]0
v-Ci In 223Ra. The presence of a group of ] ] e v e n t s in the middle of the f igure and inside
the locat ion of the expected ]4C nuclei (g iven by dotted l ines) has unambiguously
conf i rmed the 14C decay of 223Ra d}scovered by Rose and Jones. In f igure 4a, the counts
at the left or ig inate in a -par t i c les and the six d ispersed events observed at the r ight are due
to a sl ight contaminat ion by a Cf source during a pre l iminary test ( revealed in the
background measurement of f igure 4b) .
In f igure 5, are shown the results 7) of the ]4C decay of 222Ra, character ized by a
str iking clar i ty, which have conf i rmed the results of Price et a l . 5 ) per formed with track
detectors .
In f igure 6, are shown the results 7) which have proven the 14C decay of 226Ra.
Four events, label led f rom 1 to 4 and located around the 14C line character iz ing a thick
source emiss ion, were identi f ied as 14C nuclei emit ted by the 226Ra source.
-10
. ~ .
10 L i
1~ C
20 E (MeV J
~E i MeV
-10 11 o o
50 -5
0
COJNCI DENT -", E x E -TYPE EVENTS
5 1o 15 MeV i i I
5'0 I~0 Chonnel N
F,g. 5 Results of HouranJ et al.7) with a 230U source and • ~:.xE telescope placed in the focussing plane of SOLENO. A group of events co'fnoide with the location of the 14C fragments expected from the daughter 222Ra. Events at low energy are clue to G- icarticles.
Fig. 6 Results of Hourani et al, 7) with a 226Rs source • n d • AFxE telescope pl•ced in the focussing plane of SOLEN0. Events labellqKI from I to 4 fall within the location of 14C given by dashed lines. They are 14C fragments emitted by 226Ra. Other events •re due to G-partlcles.
391
Final ly , It is wor thwhi le to ment ion o ther capab i l i t i es of SOLENO :
- Tak ing into accoun t the smal l s ize (4, < 16 ram) and the reduced th ickness (<
l i n g / e r a 2) requ i red fo r the sou rce , f ragment /c ( b ranch ing ra t ios of ] 0 - 1 3 or bet ter cou ld be
a t ta ined for sou rces of ha l f - l i ves sho r te r than ten years .
- P rec ise ene rgy m e a s u r e m e n t or mass ident i f i ca t ion are poss ib le , espec ia l l y when
the vers ion of SOLENO with two coi ls is used.
4 2 - Exper imen t with an Enge sp l i t - po le magnet i c s p e c t r o g r a p h
Kutschera et a l . 6 ) measu red the ene rgy and the mass of the C nuc le i emi t ted in the
decay of 2 2 3 R a A s t rong 227Th sou rce con ta in ing 223Ra as a daugh te r was used and an
Enge sp l i t - po le magnet i c spec t r og raph with a gaseous de tec to r in its foca l p lane served to
se lec t and to detec t the C nucle i .
The sou rce of 227Th, 100p-g/era 2 th lch, with a d i ame te r of 5ram, had an ave rage
activi ty of 9 . 2 mCi In 223Ra. The spec t r og raph had a sol id ang le of 4 . 3 8 x 10 - 4 of 4 rr and
it accep ted ]4C6+ Ions with ene rg ies rang ing f rom 20 to 33 MeV.
The de tec to r measu red for each focussed Ion its magne t i c r lgl t l ty #p ( d e d u c e d f rom
a pos i t ion m e a s u r e m e n t in the focal p l a n e ) , Its spec i f i c ene rgy loss z~E and its total ene rgy
E.
A m e a s u r e m e n t of 6 . 0 9 days with the 227Th s o u r c e was pe r fo rmed . The count ing
rate was of abou t 700c/s due to the same or ig ins as exp la ined with the spectrometer
SOLENO. A backg round m e a s u r e m e n t of 2 85 days was ob ta ined by tu rn ing the sou rce
ho lder by ] 80 ° . The compa r i son of the two m e a s u r e m e n t s , In e i the r ~ExE or Ex/Sp p lane ,
y ie lded a total of 24 events fa l l ing outs ide the backg round reg ion and at the 14C locat ion as
ob ta ined with a ca l ib ra t ion us ing a 14C beam f rom the Tandem of A rgonne .
There was a cons ide rab le sp read In z~E m e a s u r e m e n t resu l t ing f rom the
d e p e n d e n c e of the path lengths ac ross the ~,E de tec to r on the angu la r i nc idence at the
e n t r a n c e of the spec t rog raph . The total ene rgy s igna l was bet ter ca r r i ed out and there was
essent ia l l y no de te r i o ra t i on in the /sp s igna l . The re fo re we p resen t , in f igure 7, the 24
events p lot ted in the Ex/3p p lane. The la rge part of the events c lear ly fai ls on the 14C6+
mass l ine. Th ree of them fal l on a d i f fe ren t l ine which is cons i s ten t with the locat ion of
]4C5+"
392
The re is no doub t the resu l t s of th is e x p e r i m e n t d e m o n s t r a t e d that the d e t e c t e d Ions
had a mass n u m b e r 14. Un fo r t una te l y the s ta t i s t i cs of the da ta and the e n e r g y reso lu t i on
w e r e not su f f i c i en t to a l low e i the r the Iden t i f i ca t i on of poss ib le t r ans i t i on to exc i ted s ta tes in
209pb ( a s was a t t e m p t e d by the au tho rs in f i gu re 8) o r the ex i s tence of a pa ra l le l decay of
223Ra by 13C e m i s s i o n ( s u g g e s t e d by the o c c u r r e n c e of one even t on the 130 mass l ine in
f i gu re 7 ) .
120
o 80
uJ 40
12C6+
~ " ~ ~ ~q.A1/, C 6
-C " tC
~' ,/14C5 +
i i I 40 I 8L0 120
Bp (Channel)
2o9pb
+ +~ , +~~=.
"itl 8p ~s
°2I + 0 ! i . . . . ~ •
20 22 24 26 28 30 3'2 1/. C ENERGY (HEY)
Fig. 7 Results of Kutlmhers et el. 6) with an 227Ac source and a gaseous detector placed in the focal p l a n e of a split-pole spectrograph. The events eoTncide with the location of 14(3 nuclei as given in dashed line==, by the calibration.
Fig. 8 The events shown in figure 7 i re plotted h e r e
versus the total energy. The positions of the g.s. and of some e~m,0ted ~=tates of the daughter 209pb are indicated.
5. EXPERIMENTS WITH TRACK DETECTORS
T rack d e t e c t o r s a re Insu la t ing so l ids w h e r e paths of heav i ly ion iz ing par t i c les may
be m a d e v is ib le in a m i c r o s c o p e a f te r a c h e m i c a l e t ch ing ] 6 ) . P o l y c a r b o n a t e d e t e c t o r s have
been used so far in the de tec t i on of f r a g m e n t s . Tuf fak fo i ls wh ich a re sens i t i ve to pa r t i c l es
wi th Z ) 2 w e r e used by Pr i ce et e l . 5 ) to m e a s u r e the c h a r g e n u m b e r and the r ange of C
nuc le i . A m a x i m u m a - d o s e of 6 . '10"10a/cm 2 was t o l e r a t e d In o r d e r to con t ro l the d e g r e e of
ove r l app ing of sho r t t r acks of C or 0 Ions p r o d u c e d as reco i l nuc le i in e las t i c co l l i s i ons wi th
= - p a r t i c l e s ins ide the p o l y c a r b o n a t e . Much less sens i t i ve p o l y c a r b o n a t e , I . e . the
po l ye thy lene t e r e p h t h a l a t e , a lso ca l l ed c r o n a r , was used t h e r e a f t e r by d i f f e ren t au tho rs to
de tec t Ne f r a g m e n t s 9 - 1 2 ) . This d e t e c t o r is sens i t i ve to ions wi th Z ) 6 and the l imi t ing ¢x-
d o s e Is abou t 2 . 1 0 l ] ~ / c m 2.
393
5.'1 - Exper iments on Ra isotopes
Pr ice et al, 5) se lected rad ium and f ranc ium isotopes p r o d u c e d in spal la t ion
react ion at the C E R N synchrocyc lo t ron using the ISOLOE faci l i ty. The ions of masses 2 2 1 -
224 were implan ted in the bases of cups on the top and sides of which were dep loyed a
125p.m Tuffak sheet covered with a "lOp.m Makrofol shee t ( f ig . 9 ) .
60 keV ISOLDE beam Polycarbonate sheets
+ Fig. <3 Radium isotope collector cup used in the experiment of Price et el. 5) where polycarbonate sheets were taken as track detectors.
In f i gu re "10, they p lo t ted the ra te of the c h e m i c a l e t ch ing a long the t rack aga ins t the
res idua l r ange . B e c a u s e the e tch ing ra te a long the t r a c k d e p e n d s on the p r imary ion iza t ion
ra te , wh i ch is a f unc t i on of the ve loc i t y and the c h a r g e n u m b e r Z of the pa r t i c l e , th is p lot
has the p rope r t y of a AExE d e t e c t o r t e l e s c o p e with r e s p e c t to the i den t i f i ca t i on in Z. in
f i gu re 10, m e a s u r e m e n t s at two po in ts a long the t racks of ] 4 C emi t ted f r om 223Ra a re
10 "~,. I ~ . t |
8 ~_,O " ' ' , F " , N e %% %%% %% |
-- %%% %% %%
-- %% %%
Vr/V4 -___~ N ~.,~ ~ '~4
~, .c . . . : ~,, ~, . .
LB ~ ' . " " -
- - . . . _--..
10 2Q 30 40 Range (pro)
~ ' ;3c1~ ~zc ~ ~'~c 15
2't2R~ ~ C ÷ 2°aPb tO
o _ _ + - . I I ~ r I I
is - ,'c I l'~c "c
2~Ra ~ C + 2°gpb /
10 - " L
0 ~ 30 35 40 45 Range (~m)
1 Fig. 10 Ratio of etching rate along track to general etching rate plotted as = function of residual range. Dots represent measurements at two different points along the tra)ectory Of 14C nuclei emitted from 223Ra. Dashed lines are the calibration, From rat.
5.
Fig. 11 ResultsofPriceetal. 5) with track detectors and 222,223,224R& sources. The histograms are the measured range distributions of 14C compared with ranges calculated from Q-values.
394
represented by dots whereas cal ibrat ions obtained with heavy ions f rom the LBL Superhi lac
are given in dashed l ines. The locat ion of the dots co inc ide with the line of C. In f igure " l l ,
are compared the h is tograms of measured ranges with expected ranges cor respond ing to
the Q values of the decays. The agreement in ranges cor responds to an agreement in
kinet ic energ ies better than 0 .4 MeV.
5 . 2 - E x p e r i m e n t s on d e c a y s in to Ne f r a g m e n t s
Further exper iments at Berkeley 8) and Dubna ] 0 - ] 2 ) investigated decays into
f ragments heavier than ]4C, using Cronar as a t rack detector . The fol lowing sources were
cons idered : 232 ,233U, 2 3 ] p a , 230Th, 237Np and 24JAm. The common features to all
these exper iments are related to the very low f r a g m e n t / a branching rat ios which were
invest igated, i . e . two orders of magni tude lower than in the "14C decay case. Intense
sources with extended sizes ( in order to keep them relat ively thin) and detectors of large
area were used. The i r radiat ion t ime was tens of days. Some data are summarized in table
2. It was found that 232U, 23"lpa and 230Th are 24Ne emit ters and that 233U is 24Ne or
25Ne emit ter .
TABLE 2 :
Investigation of decays inlo fragments heavier than 14C,
Some characteristics or the experiments with track 0electors In the
Decays Ref. Strength or Slze o£ mass of the the det~c-
source tor (c m-)
0.5mCi 250
7 mg 17.5
75 mg 227
210 mg 1040
320tag 500
3.7 mg 36
Irrad. time ( d a y s )
30
13
2 8
64
122
30
232t~24Ne+208pb 9
231pa-*24Ne+207TI i0
233U-~24Ne+209pb II
~25Ne+208pb
230Th~24Ne+206ng 12
237~>~30Mg+207TI 12
~3Zsi+2OSAu
241~3%SI÷207TI 12
No o f eventB
Range ( ~ )
165
G
31 32.9,0.23
25 30±3
16 30.5"1.4
29
Beanu3 for
c a l i b r a t ion
2ONe ' 180
2ONe
2ONe
160,20Ne,26Mg
395
The number of observed f ragments t oge the r with the i r measu red ranges and the beams used
for ca l ib ra t ion a re g iven in the table. S ince no f r agmen ts were observed with 237Np and
24 ]Am sou rces , only upper l imits were set for the i r b ranch ing rat ios,
With all these f ind ings , the t rack de tec to r t echn ique a p p e a r s to be very powerfu l
and inc i tes phys ic i ts to sea rch for new decays with stil l lower b ranch ing rat ios.
6 - SUMMARY AND DISCUSSION
The d i f fe ren t decays into f ragments which have been observed or invest igated up to
now are repor ted in tab le 3 with the c o r r e s p o n d i n g expe r imen ta l and ca lcu la ted b ranch ing
rat ios re la t ive to a - d e c a y s . It is seen that decays of Ra iso topes into ]4C o c c u r with Q-
va lues of about 30 MeV and b ranch ing rat ios of abou t ]0 -`10 , whereas decays of some Th,
Pa and U iso topes are cha rac te r i zed by Q-va lues of abou t 60 MeV and by b ranch ing rat ios
of abou t "10 - ] 2 . All these decays are exp la ined within one o rde r of magn i tude by the
Superasyrnmet r ic f iss ion model of r e f e r e n c e s 13 and '14. Bet ter quant i ta t ive exp lana t ions ,
however , can be ca r r i ed out with the same models if h i n d r a n c e ef fec ts for e v e n - o d d parents
re lat ive to e v e n - e v e n parents a re taken into accoun t as it is common ly done in a - d e c a y
Work. The co lumns 7 and 8 in tab le 3 c o m p a r e the resul ts of b ranch ing rat io ca lcu la t ions
pe r fo rmed wi thout and with e v e n - o d d e f fec ts respect ive ly .
221R0
222R0
223R0
224R0
225Ac
225Ro
230Th t 231Po~-
232U ~ 233U L
Log10 ( B e x p / B t h ) i r
~.-o.111,
These resul ts a re d isp layed in f igure 12 where
the full squa res c o r r e s p o n d i n g to e v e n - o d d
ef fec ts exhib i t , wi thout doubt , an i m p r o -
vement in the exp lana t ion of the measu red
b ranch ing rat ios,
Other fea tu res of the a - d e c a y may a lso apply
here , e . g . the sensi t iv i ty of the decay rate to
the de fo rma t i on of the daugh te r nuc leus or
the poss ib le t rans i t ion 6 , 1 7 ) to exci ted states
in the later.
Fig, 12 Rstioofthemeasuredtocaloulatodbranching ratios displaying the columns 7 and 8 of table 3. In dots, for calculation without even-odd effects and in squares for calculation with oven-odd effects.
396
TABLE 3 : Decays into f r a g m e n t s obse rved up to now with the i r m e a s u r e d and ca lcu la ted
b ranch ing ra t ios re la t ive to ¢x-decay.
Decay Q I Fragment/u ( MeV ) Exp. Branch.
Ratio
22!Fr~I4c+207TI 31.28
221Ra~I4c÷207pb 32.39
222Ra~I4c+20Spb 33.05
223Ra~I4c+209pb 31.85
224Ra.~14C+210pb
225Ac_~14C+21~i
226Ra~14C+212pb
~30Th~24Ne+206Hg
~31pa_~24Ne+207Tl
~32U~24Ne+208pb
Z33U~24Ne+209pb
~25Ne+206pb
237Np_430Mg+207TI
~32Si+205Au
241Au~3%Si+207TI
( a ) Ref. 13
(b )
30.54
30.47
28.21
57.78
60.42
62.31
60.50
60.85
75.02
88.41
93.84
Re f,
-14 < 5.0xi0 8
-13 < I. 2x10 8
3.7¢0.5)10-10 5
3.1±1.0)10 -10 7
8.5z2 .5 )10-1~ 2
5.5~2.0)I0 -I0 3
?. 6~3.0 )i0-i~ 4
6.1¢0.8)10 -10 5
4.7ZI.3 )10 -I0 6
4.3±1.1)10 -11 5
<4.0XI0 -13 I 8
3.2¢1.6 )10-I~ 7
i
5.6±1.0 )10-1~ 12
4.3¢0.9)10--I~- i0 !
2.0~0.5 ~i0 -I:~" 9
ii -13
i 7.5z2.5)I0
< %x10 -14
< 3x10 -12
< 3xI0 -15
12
7
12
Bexp, Fr agment/u ( Bth
Theor. Br. Ratio Ref.13
--12 3.2XIO
--12 1.2XI0
1. OxlO -II
3.2xlO -9
i. 6xlO -12
6.3XIO -13
2. OXI0 -12
4. OXI0 -13
1. OXI0 -I0
1.3X10 -II
5. OXI0 -II
3.2xlO -II
2.5XI0 -12
2.5xlO -12
4. OxlO -13
L°glO
Re£.14 (a) (b) I
--12 7.9XI0 --0.80 -i. 39
--12 8.2X-10 --i. 02 --0.55
--9 l, 7xlO 1.53 0.53
--9 6.9xI0 -0.72 -0.32
--ii 6.2xlO 1. %3 --0.04
-12 1.6XIO --O. 20 --0.87
-Ii 3. IxlO 1.20 -0.23
3.6XI0 -13 0.15 0.52
-12 9.4xlO -i. 37 0.01
4.9xlO -II --0.'81 --0 . 41
-11 3.7xlO -i. 82 --0. I0
-I0 2.5xlO
-14 2.9XI0
-18 3.8xlO
i. Oxl0 -15
As in (a ) but with e v e n - o d d e f fea t accoun ted for via zero point v ib ra t ion e n e r g y 19) .
397
Let us, a lso, ment ion the possib le coup l ing interact ions between the f ragment and
the daughter nucleus dur ing their relat ive tunnel ing, by analogy to the ones proposed in the
inverse react ions, i .e . the low-energy heavy- ion fusion react ions. Such coupl ing inducing
transfer react ion as well as inelast ic exci tat ion could affect the decay rate to the ground
state of the daughter and also al low for t ransi t ions to f inal exci ted conf igurat ions 18),
7. CONCLUSION
Several cases of spontaneous emission of f ragments from heavy nucl ides have been
exper imenta l ly observed up to now. So, spontaneous decays in termediate between a - d e c a y
and f ission, previously pred ic ted, are def in i t ively establ ished.
Three exper imenta l techniques have been involved so far giving a str ik ing
agreement in branch ing ratio measurements , The first technique of a ~ExE sol id state
detector te lescope p laced in a d i rect view of the source as in the d iscovery of the
phenomenon seems to have no role left to play. The second technique based on a magnet ic
se lect ion of the f ragments from the very high flux of s -pa r t i c l es Is much more powerful In
f r agmen t ta branch ing ratio measurements and present a speci f ic interest In its capabi l i ty of
measur ing with preolston var ious charac ter is t ics of the f ragments, e . g . the energy, Z,A.. . It Is, however, the third technique using t rack sol id detectors which seems to yield the
smal lest f r agmen t / a branching rat ios, and consequent ly to have in the future the most
impor tant role to play.
We have shown that the predic t ions of the f ragment emiss ion were achieved by the
Use of the t radi t ional models of a - d e c a y and f ission. At present , the d iscovery of several
Oases of f ragment emission and the measurement of their f r a g m e n t / a branching rat ios offer
a new field for test ing and ref ining many theoret ica l aspects involved in a - d e c a y , in f ission
and even In l ow-energy heavy- ion fusion.
Final ly, it is Interest ing to point out that many f ragment emit ters, e . g .
223,224,226Ra, 231pa and 230Th, occur in the natural ser ies of 232Th or 235,238U.
Fragment concent ra t ion measurements In U and Th ores should be of Interest for appl ied
research.
398
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)
11)
12)
]3 )
14)
]5 )
16)
17)
18)
19)
REFERENCES
A. Sandu lescu , D .N . Poenaru and W. Gre ine r , Sov. J. Part. Nucl. ] ] ( ] 9 8 0 ) 528
H.J . Rose and G.A . Jones , Nature 307 ( ] 9 8 4 ) 245
S. Gal~s, E. Houran l , M. Hussonno is , J .P . Schap i ra , L. Stab and M. Vergnes , Phys. Rev. Lett. 53 (1984) 759
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W. Ku tschera , I. Ahmad, S .G. Armato III, A . M . F r i edman , J .E . G ind le r , W. Henn lng , T. Ish0, P. Paul and K.E. Rehm, Phys. Rev. C32 ( ] 9 8 5 ) 2036
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Relativistic Heavy Ion Collisions and the Nuclear Equation of State
R. Bock
GSI Darmstadt
Abstract
Central collisions between heavy nuclei in the relativistic regime have been systematically
investigated at the Bevalac in Berkeley by a joint GSI-LBL-collaboration during the last years
with 4~-detector devices, the LBL-Streamer Chamber and the GSI-Plastic Ball. These investi-
gations have provided strong evidence for the compression of nuclear matter by a factor of
up to 4. Moreover they have shown a collective flow of nuclear matter as predicted by
hydrodynamical models. Many features of the space-time development of these reactions
have been investigated and compared with theoretical predictions. The analysis of recent re-
sults give evidence for a rather stiff equation of state in contrast to considerations from
astrophysical models for neutron star formation.
Introduction
Heavy ion collisions at relativistic energies provide the only means to investigate nuclear
matter at densities greater than normal. One of the ultimate and most important goals of
relativistic heavy ion physics in the range up to several GeV is, therefore, the investigation
of the equation of state of nuclear matter ~'2,=. The prediction of phase transitions at high nu-
clear density and high temperature, in particular the transition of hadronic matter into quark
matter, but also the astrophysical aspects of the nuclear equation-of-state stimulated strongly
this kind of investigations.
Whether this aim could be reached was not at all clear when these investigations were
started. The amount of energy which can be deposited in the nuclear medium in a collision
of two heavy nuclei, strongly depends on the nuclear stopping power. When we entered this
field more than a decade ago, opinions of theoreticians about nuclear transparency were
vastly diverging and predominantly discouraging. All the more we have to give credit to the
pioneers in this field, most of their names being exhibited in the list of reference. Some
400
names should be ment ioned explicitely: Hermann Grunder with his col laborators of the Ac-
celerator Division of the Lawrence Berkeley Laboratory, who created the Bevalac and pro-
vided the heavy ion beams in the GeV region up to Uranium; Reinhard Stock who init iated the
exper imenta l activit ies and - with Hans Gutbrod and the GSI/LBL group at the Bevalac -
pushed this field at the exper imenta l front; Walter Greiner with his col laborators in Frankfurt
who developed the hydrodynamical model and st imulated the exper imenta l effort. I wi l l
mainly report on recent results of the GSI-LBL col laborat ion, two groups under the leadership
of Reinhard Stock and Howel Pugh (Streamer Chamber), and Hans Gutbrod and Arthur
Poskanzer (Plastic Ball).
What are the necessary condit ions for an investigation of the nuclear equation-of-state and
how can they be achieved? In a coll ision of two heavy nuclei near the speed of light, the
nucleons in the zone of contact are slowed down by the interaction and a shock compression
of nuclear matter is produced (Fig. 1). The kinetiG energy of the project i le is transfered into
compressional and thermal energy and a "nuclear f ireball "5 is formed. For a phase transit ion
three aspects are important:
1) the nuclear stopping power needs to be large enough to produce
a high energy density in nuclear matter,
2) the interaction region must be large enough, and
3) the system must l ive long enough in order to achieve equi l ibir ium.
In order to meet these condit ions, heavy projecti les are necessary and the coll ision needs to
be central. High mult ipl ici ty of reaction products has been identified as a signature for a small
impact parameter, and is used as a tr igger condition for the identif ication of central events.
Typical charged-part ic le mult ipl ici ty distr ibutions at var ious projecti le energies are shown in
Fig. 2. It became obvious very soon, that inclusive detection methods are of l imited value only.
Therefore the development of 4~-detector techniques brought the breakthrough on the exper-
iment side. Two detectors were used for the investigations to be discussed in this lecture:
the LBL Streamer Chamber ~ and the GSI Plastic Ball. ~ Fig. 3 shows a schematic v iew of the
Plastic Ball 4~-delector consisting of 815 modules of a A,E-E detector system. Not only pions,
protons, deuterons, tr i tons and a-part icles can be dist inguished by each single module over
a broad range of energies, but also elements up to about oxygen can be identif ied.
My lecture wil l be organized as follows: Before deal ing wilth my main subject, the nuclear
equation-of-state and the relevant recent experiments, I will, in the first part, briefly summa-
rize some important facts about the reaction mechanism and its t ime evolut ion and about the
observables which can tell us something about the high compression phase of the reaction.
The second part is devoted to a presentat ion of exper iments which can give information about
the equation-of-state.
1.
\ ' Compression
ReLative vel0cit'y r] ~ o.8 h ,% 10 -Z]s
2 / Zone /
• , /
/ • \
Figure 1 Two succesive stages of a high energy central colli- sion in the c.m. system (ignoring Lorentz con- traction). The hatched area is the shock compression zone of the developing fireball ~.
401
t -
Figure 2 Charged-particle multiplicity distribution for Au --* Au at various projectile energies 4.
~oILIP'~01Q("
Beom Cou~ler
PLASTIC BALL PLASTIC WALL 655 modulas ¢- 160
,.. $~L~nol } / - - . - 60 paLr~ of counters
Licjntguide
L ,~,o~ , ~ ill / ' * " / ~-~-~u t6 counters, 0.64 xSx16 cm
Figure 3 Schematic view of the Plastic Ball 4tt-detector. The 815 AE/E-modules allow the separation o f ~ ÷ , ~', p, d, t, and o-particles over a wide range of energies =.
402
1. Some Basic Facts about Relativistic Heavy Ion Reactions
Theoretical calculat ions on the space-t ime evolut ion of a reaction between two heavy nuclei
at relativist ic energies predict that the formation of an equi l ibrated f ireball can be reached. If
this is the case, it should be possible to exlract information about the nuclear equation-of-state
from such coll isions. The main question at this point is, which propert ies of the hot and com-
pressed state could be determined exper imental ly , and which observables can tell u., some-
thing about it. Many different reaction products, their spectra, their angular distr ibution and
their correlat ions can be measured: What information about the equation-of-state do they
provide?
Before answering some of these questions let us briefly sketch our expectat ions about the
equation-of-state and about the t ime evolut ion of the interacting nucleus-nucleus system at
relat ivist ic energies up to several GeV/nucleon. In Fig. 4 a schematic densi ty- temperature
phase diagram for nuclear matter is shown. ~ The solid lines indicate where, due to theoret ical
considerat ions, phase transit ions are expected. In the context of this lecture most important
is the deconf inement transit ion from hadronic matter into the quark gluon plasma which is
expected to happen at densit ies of p/po > 5 and temperatures between 150 and 250 MeV. Fig.
5 shows a "historical" schematic presentat ion of the equation-of-state, the dependence of
compressional energy W(P/po, T =0) from the nuclear matter density. At the t ime it was con-
ceived nothing was known exper imenta l ly except a small region around the nuclear matter
ground state at P = P o Information about the curvature of the equation-of-state around Ihis
point comes from the giant monopole resonance, depending on the compressibi l i ty K of nu-
clear matter. This constant K is often used to parametr ize the extrapolat ion of the equation-
of-state to high density: A large value for K means a strong increase of the compressional
energy with density, a stiff equation-of-state. From the physics .point of v iew there is no reason
for the assumption of a parabol ic form at higher densities, but it is convenient for a
parametr izat ion. Fig. 5 shows two different versions of an equation-of-slate, a stiff (K=400
MeV) and a soft one (K=200 MeV). In addition, the case of an equation-of-state with phase
transtit ion and density isomer which seemed possible at that t ime even at rather low com-
pression is schematical ly shown.
If we want to learn something about the transient high-density state of the coll ision we have
to get an idea about the t ime evolut ion of the reaction. Fig. 6 exhibits cascade model 9 calcu-
lations of a central nucleus-nucleus col l is ion) ° As an example, three typical quantit ies, the
fireball density (a), the number of coll isions per t ime unit (b) and the (~+A)-product ion (c) are
plotted for La + La at 1 GeV/nucleon as a function of t ime. By looking first at the number of
nucleon-nucleon coll isions, we can distinguish three phases" of the coll ision:
403
ZI]O
150
100
o
0
• ".''z •
• ".~'-" ..,
o L.) o ~ ,}tiOli< - ljllsl!n
. ? L H,13/Oq I~:15
/ ..~i-~ \
? ] - ~ g 7 Oensit y ( RGtio to Llormat Density I
Figure 4 T-p phase diagram for nuclear matteP. The solid line marks the region where phase transitions are expected.
0
P/po I I i I00
n%Boilisioos [--
I o 5
Figure 6
5 10 15 20 25 30
Cascade 10 GeV/u La +}La b <- 3.4 fm
,~ooOOO00000~ *r e ~ I0 "i~t S
i 1 . . . . . ~ . , ~ < L
t %
':" I c,, t I (cl ~,
<, . . . .
/:l e e I Ill" ,I- ~ f ChomiCII I gqui l ib.um
I I
= I
I 10 16 20 25 30
t (fro/c)
I00
10
Time evolution of some characteristic quantities in a collision the fireball density P/Po' the number of baryon-baryon colli- sions per t ime unit (tin/c) and the (n + A)-produclion for a central collision of La + La at t GeV/nucleon, based on internu- clear cascade calculations =.
g
z 0 =_
10 . . . . . . . , . ' ! / ' - ~ ' - T ~
/ / !
i
i/ \+, : il
V ~ith density isomer '
G~'.=.,L - I0 . ~", i , I , I , '
0 l 2 3 4 ,5
N u c l e o n N u m b e r D e n s i t y n/n o
Equation-of-state for nuclear matter, as suggested ten years ago with and without a phase transition (schematic). Plotted is the internal energy as a function of the nucleon number density.
crawl ,At
" t 7 "T~t
~.)
(~/P- ~'*~°) =~ %ha I >, T
T~Tr ~ He~u~s E. r
~le -" Me*~ve~ ~tr'¢ 5sure
Figure 7 Time evolution of various quantities which characterize the state of nuclear matter during the collision (schematic), (Stocker et al.*),
404
Phase I: Interpenetration: The interacting nucleons loose momentum, which is converted
into heated and compressed nuclear matter.
Phase I1: High density fireball: A plateau is reached after a few times 10-23s, the fireball re-
mains in a transient shock compressed state for another few 10-23s. Thermal and
chemical equil ibrium should be reached (due to the calculation).
Phase II1: Expansion: After about 10 x 10"23s the thermal freeze-out radius is reached and the
decomposit ion begins.
For the case considered, cascade calculations predict a maximum density p / p o ~ 4 in phase
II and an exponential decrease in phase Ill witl~ "Cp~10"23s, whereas the pions are produced
in phase I and II and the number of pions and deltas being only very slowly decreasing in
phase III because of a dynamical equil ibrium of pion absorption and reemission in the nucleus
before the pions have reached the surface and get lost. This is an important feature of such
a reaction because the number of pions produced is characteristic for the high-compression
phase of the reaction, a so-called primordial quantity. It carries information about the state
of highest density. On the contrary, the measured particle spectra for example do not reflect
the maximum temperature reached in the fireball state, but the much lower freeze-out tem-
perature reached after the expansion.
The features discussed so far lead us to the following conclusion: For the determination of the
equation-of-state at high density we need information about the high compression phase of the
reaction. Not all the quantities which can be measured conserve this information, so we have
to choose the "pr imordial" observables. Following a presentation of StScker et al. a the t ime
evolution of characteristic quantities is plotted in Fig. 7. It can be realized that some quantities
have reached nearly its final value at a t ime when the density is at maximum and that this
value remains nearly constant during the expansion phase, tn addition to the pion yield, which
has been already discussed, this holds for the entropy production (measured by the composite
particle yield, the d/p-ratio for example) and for the average flow angle and transverse mo-
mentum transfer. These observables provide the methods for the determination of the
equation-of-state which wil l be discussed in the second part of this lecture.
As mentioned in the beginning, it is a crucial condition for these considerations that equil ib-
rium has been reached. That this is the case can be verified experimental ly: A strong indi-
cation for dynamical equil ibration is the momentum distribution of the emitted particles which
they have obtained during the reaction. Full equil ibration leads to an isotropic momentum
distribution which is expressed by R = (2/~) x < IP.LI > / < IPlll > = 1. For the centra~ collision
of Ar + Pb at 0.8 GeV/nucleon R has been measured on an event by event basis (Fig. 8a). t2
The symmetry around the diagonal shows that isotropy is reached. This is not fully the case
model for the cascade predictions (Fig. 8b). In Fig. 8c the impact parameter dependence is
given for the same reaction for experiment and cascade calculations, showing again the ira-
405
/
/
/
<R,>~ . . . . ]
IO0
Figure 8 Contour plot of mean transverse and mean longitudinal momentum, event by event. 1.5- a) Data for Ar + Pb (Streamer Chamber datat:). b) the corresponding cascade model prediction t°, using
the code of Cugnon g. c) Contour plot of the ralio R = 2 < P J . > / n < P i l > for Ar
+ Pb events (data plotted vs. participant prblon multi- R 1 . . . . plicity). The dotted lines give the ridge of the data and of a cascade model prediction.
0.5
O t0
0,8 6"~WU At o Pb CaICQ~' (" b ~
/
/
/
I IO0
" % + Pb (c) 0.77 AGeV
Ridge of dale
(b)~2.§ fm ~
~lscede mo~el
20 30 40 50 Mp
60
3~
100
, c ~ + c . I I I ~Jm " 400 MeVIN
_ M,. >_ 3o
/ /
/ /
/ /
f I t
bNb+Nb I I I ~ -- 400 MeV/N
- - M ~ --> 55
/
/ _
,,oJY /
/ I 1 I
,~ Ip,I,/A (M~V/c) ,.~ I~111 '/A (~.~V/c)
Figure 9 Same as fig. 9 for two symmetric reactions: Experimental resulfs of central events for a liqht (Ca + Ca) and a heavy system (Nb + Nb). Only in the heavy system full equil ibralion is obtained. (Plastic Ball data) '= .
406
portance for central collisions for this kind of investigations. In light systems deviations may
occur even in central collisions because of the corona effect. The nucleons in a pheripheral
zone around the nuclei perpendicular to their flight direction do not traverse enough matter
to be completely stopped. In the "corona" the nucleons keep part of the forward momentum
as shown in Fig. 9 for Ca + Ca, in contrast to Nb + Nb. t3
The last topic in this section concerns the thermal equilibration. Thermal equil ibration leads
to a MaxwelI-Boltzmann distribution d Z a / d ~ pEe -(E/T) of the emitted particles, with an ex-
ponential slope proportional to the inverse of the temperature. This relation has been widely
used for the determination of temperatures. In the case of strongly interacting particles, like
protons and pions the temperature measured should be that of freeze-out. A comparison
between proton and pion temperatures, however, exhibited a drastic discrepancy as shown
in Fig. 10a for protons (T=118 MeV) and in Fig. 10b for pions (T=69 MeV). 1" I twas argued that
this discrepancy is due to the fact that protons freeze-out much earl ier than pions, the tem-
perature being further reduced in between by the expansion. ~ Recent calculations showed,
however, that this difference is caused by the parent-daughter effect TM of the A ---* N + ~ decay
and that the discrepancy can be explained nearly quantitatively by the A-decay kinematics, t4
Only a 5 % effect is left which may be due to higher isobars.
Our present knowledge can be summarized as follows:
1. In central coll isions of heavy nuclei full stopping is achieved up to the maximum energy
investigated (2 GeWu).
2. Thermal and chemical equil ibrium is reached during the collision time.
3. Some measurable quantities exist (Pion yield, entropy, transverse momentum transfer)
which can give information about the high-density phase of the colliding system.
4. Measured particle spectra allow the derivation of temperatures. The large difference
found between proton and pion temperatures can be explained by the A-kinematics
("parent-daughter effect") and are not a result of different freeze-out temperatures.
2. Recent Experiments on the Nuclear Equation-of-State
In the last section it has been discussed which observables may give information about the
high-compression phase of a nucleus-nucleus collision (Fig. 7). Accordingly, three types of
experiments have been carried out during the last years in order to study the equation-of-state
at high density:
407
>~
,0
Figure ~O
" ~ t .8 GeV,'~ucfeon
Proton C.M. energy (GeV)
~ \ - - - J ~ \ O e ~ 90 ~ c,m.
I i'l ~' "
"N]lli Iltt //~
Pion c .m. energy (GeV)
Thermal equilibration. The energy spectra of emitted particles give information about the temperature of the emitting source. The exponential slope of the proton spectrum shown in (a) c o r r e s p o n d s to a temperature of T=118 MeV. The pion temperature (b) seems to be much lower. Calculations, taking ca re of the parent-daughter effect of the A-decay show good agreement between p- and x-spectra&s14.
C O
r -
Z
Q/
20 0 c ¢__
. 4 - -
r--
-16
zJ ~:= 3~/o z̀3
] / @
E T/u .<T-p ,~I" "
t=O ( ~
®®
• = , ,~. f~.¢
, , I n T ~°) _ e~, 3
~.~,,,.pfo p~,.,)
3 compression P/Po
r
Figure t l Sketch of the function W(p, T=O) relating the energy per baryon ol nuclear matter to the density given in units of Po The reaction cycle is indicated by 1 (before interaction), 2 (maximum compression) and 3 (expansion towards freeze-out). During stage 2 the internal energy is shared between thermal (ET} and compressional (Ec) energy z.
408
1. The pion yield has been used for the determinat ion of the compressional fraction of the
energy deposited in the fireball, in a series of exper iments by Stock et al. 2,t7 ("Pion
Thermometer") .
2. The col lect ive flow, first observed in Plastic Ball exper iments two years ago, can be used
as a "barometer" for the compression built up in the part icipant region of the col l iding
nuclei (Gutbrod, Poskanzer, Ritter et al.). 2~
3. The d/p rat io can be used as an "entropymeter "26, the missing entropy production giving
qual i tat ive information about the equation-of-state. 27
I wil l mainly concentrate on the first two experiments, which have been developed to a state,
where they can give quanti tat ive results.
2.1 The Pion Thermometer
The basic idea of the exper iment 2't° is the following: Consider a central coll ision of two heavy
nuclei with a kinetic c.m. energy Ecru. At maximum compression this energy is converted into
internal energy of the coll iding system, full stopping assumed. This first step of the reaction
is indicated in the W(p)-diagram (Fig. 11) by a path from (1) to (2). At the point (2) the total
energy Ecm=ET + E C is deposited in the equi l ibrated fireball part ly as thermal energy E T,
part ly as potential energy E C in form of compression. As indicated in Fig. 11 the splitt ing of
the energy in the two parts is determined by the equation-of-state: More thermal energy is
avai lable for a soft equation-of-state and less thermal energy if it is stiff. What can be meas-
ured? By the initial condit ions Ecm is known. If we would know E T we could deduce E C. Con-
cerning the measurement orE T one can use the fact that only the thermal energy contr ibutes
to the pion production. The missing pion yield, therefore, can be used as a measure of the
compressional energy.
In order to determine the missing pion yield the measured pion yield has to be compared with
the calculated yield based on a thermal model. The procedure is discussed below. A second
quantity, the density, has to be determined also by a theoret ical model. In the Hugoniot-
Rankine shock compression model the relat ion between the density and pressure is given by
F= ~). ~.(e,T)= p.~(£,]'=o) p,~e I =
This differential equation which can be solved by iteration determines the absolute scale of
the abscissa in Fig. 11.
The pion yield has been measured for several symmetr ic systems as a function of energy. The
extrapolat ion to impact parameter b = 0 has been made by a careful determinat ion of the pion
409
multipl icity nn-as a function of the number of participating protons, as shown in Fig. 12 for Ar
on a KCl-target t° and in Fig. 15 for the system La + La ~9. The pion multipl icity increases lin-
early with number of participants. Fig. 13 exhibits the next step of the analysis. The data show
a smooth increase with the cm. energy. The various calculations for the n--yield without
compressional energy based on several models show good agreement with each other. The
difference between theory and experiment determines the compressional energy as shown in
the lower part of Fig. 13, still as a function of c.m. energy. Finally, the compressional energy
W(p,T=0) is given as a function of the density, which has been calculated by the procedure
discussed at the beginning of this section (Fig. 14).
New data of a much heavier symmetr ic system, La + La, TM measured at 5 different energies
between 530 and 1350 MeV/u have been analyzed recently. The pion yield shows the same
l inear behaviour as a function of participants (Fig. 15) and plotted as a function of c.m. energy
the < n n / A > is in excellent agreement with the Ar data (Fig. 16). The pion yield is increasing
l inearly with energy and it is far below the thermal prediction of a purely thermal model shown
in the same figure. The agreement between the two systems of different size confirms the
assumption that no pions are lost by absorption in nuclear matter, because in this case the
yield in La would be lower. Obviously, the pion yield is a bulk matter probe. The potential (=-
eompressional) energy and the fireball temperatures derived from these data are shown in
Fig. 17.
The equation-of-state W(p) derived from the measurements described, together with theore-
tical predictions are plotted as a function of the density p in Fig. 18. Good agreement is ob-
tained with calculations of Molitoris and St~bcker 2° based on the VUU theory (Vlasov, Uehling,
Uhlenbeck). This reaction theory combines 2-body coll isions accounting for Pauli Blocking with
a self-consistent mean field. In terms of a parabol ic approximation for the density-
dependence of the equation-of-state (which is not necessarily a reasonable extrapolation for
high compression as discussed before), the compressibi l i ty is described by K~300 MeV,
which is characteristic for a stiff equation-of-state. The equation-of-state requested for the
description of neutron star formation in a supernova explosion is much softer.
In conclusion, the following results (based on th pion yield data) have been found:
1. The nuclear equation-of-state is a smooth function increasing with the nuclear density.
No indication of a phase transition can be observed up to p /po=4 .
2. The compressibi l i ty of nuclear matter is rather high, comparison with theory favors a stiff
equation-of-state.
3. Some ideas but no clear explanations exist yet for the discrepancy to the equation-of-
state requested neutron star formation.
410
Figure 12 Dependence of the =-multiplicity of the number of participating protons in Ar ---, KCI for various projectile energies between 1.0 and 1.8 GeV/u '°. The ex- trapolation to maximum proton number corresponds to the impact parameter b = 0 .
Ar + KCI - - ~ nTr- + o
L- .~ 18
~-- - ~ - - f - - Z o • I
_ _ .
:) 4 8 12 16 20 24 28 32 36
O
0 5
0 4
c~ 03
+ <
Z 0 2
OI
ELA B (GeV/u) 0 5 1.0 15
I
A r 4- K C I - n n
• OmtR (b ~0} . . . . . . Chem,cal modal
- - - - ~ Cugnon casclcle "J Yartv+FfaenkeJ ca$c lde
~ 4 t. -4
Z r 4
,~. . .~ / - +
~'" -4, ,,~ ~,
2 0
O 0 0 1OO 200 300 400
Ec m ( M e V / u )
Figure 13
140
120
100
~ ~°
4 O
2 0
0
¢ t b
'~ C h e m i c a l m o d e l .,o
,~ C s s c s d e m o d e l ~ : ~ a
/¢y/" / a . . -
e . . . . ¢.
100 2 0 0 3 0 0
Ecru (MeV/u) 4 0 0
The multiplicity of pion plus deltas per participant is shown in (a) as a function of the energy. The data points are for Ar + KCI, extrapolated to zero impact parameter. The dashed and dotted curves show the Cugnon cascade g and the chemical model 'e results for the depend- ence of < ~ + A > / A on the fireball thermal energy content E T. Horizontal arrows represent the deduced compressional energy per baryon E C. This fraction of the fireball energy content is shown again in (b).
140
120
100
> 80
~ 8 o
Figure 14 Nuclear matter energy vs. density at zero 40 temperature. (a) The empirical result (without Fermi energy corrections) ob- tained by comparing pion yields to pred- 20 ictions of the chemical equilibrium plus shock compression model, and to cascade results. 0
I
'- Cascade model
_ ! _ t f
Zt- I
I
2 3 4
/J/po
411
25
20
\ 15 L
10
~39La + ~39La
5o ioo 16o 200 250 300
Participant nucleons A
Figure t5 Dependence of the K--multiplicity of the number of par- ticipating protons for the reaction La + La TM.
,~ 0.3
c 0 . 2
Ec=b/A (GeV} 0.5 t 1.5
0,5
• Ar + IKC! ~ /
0.4 o La + La /
Thermal
/ o., / 7 ' 'I'4>
0 . 0 • .l ~ . i ,
100 200 300 400
Ecm/A (MeV}
Figure t6 < n = > I A for La + La and Ar + KCI as a function of c m. energy TM.
160 .
(a)
100 • Ar + KCI O L8 4- Le
50
i
, t
~oo tb)
i-- 60
40 . . . . 1 0 IOO
200
7
--"F . . . . ,
200 3 ~ 0 400
Ec,,JA (MeV)
Compressional energy (upper part) and fireball temperature (lower part) as derived from the La and Ar data. The fraction of compressional en- ergy is 13 % at the low-energy end and 33 % of the total energy at the high-energy end tg.
u 1 ' ' i 4 VUU/ -
7 °
-I
150
:~m 100.
8 50
O-
-50 I I 0 2 4 6
Fj~re 18 ~/Po Nuclear equation-of-state W(~, T=0): Compar- ison with theory ~,2~. VUU: Molitoris and St{3cker, FB: Friedmann and Panaharibande, BCK: Baron et al., 4 0nd 3: Sano el al., B: Boguta and St(3cker.
412
2.2 Collective Flow and the Equation-of-State
The hydrodynamic model has predicted a col lect ive flow of nuclear matter in a coll ision be-
tween heavy nuclei, 22 As long as relat ively light projecti les were used this effect was not ob-
served exper imental ly . Only when heavier project i les became avai lable at the Bevalac some
years ago and adequate 4=-detectors were avai lable, a considerable non-symmetr ic
s idewards flow of nucleons in the mid-rapidi ty region was discovered. 23 Fig. 19 shows the or-
iginal data together with hydrodynamic and cascade predictions. The cascade model without
a mean field gives only a small asymmetry of angular distr ibutions and cannot explain the
data? The important feature with respect to the equation-of-state is that this col lect ive matter
flow measures the fireball pressure distribution at high density, as i l lustrated in Fig. 20.
During the last two years a lot of systematic data were taken, in part icular with the Plastic Ball
Detector, but also with the Streamer Chamber. Two methods have been used for the analysis:
1. In a Spherici ty Analysis the energy flow tensor is determined for each single event and
the flow angle is measured as indicated in Fig. 20.
2. An improved method is the Transverse Momentum Analysis, 24 in which auto-correlat ions
are removed and higher sensit ivity is obtained, also in the case of only a small number
of emitted particles.
The detai ls of both methods wil l be skipped, they are well documented in the l i terature. Two
examples are given for systematic reasons, showing the f low measured for three different
symmetr ic systems, Ca, Nb and Au at 400 MeV/u (Fig. 21) and for the system Au on Au at five
different energies (Fig, 22), in all cases for 5 impact parameter bins ranging from peripheral
to central coll isions. '~'z~
In Fig. 23 a Transverse Momentum Flow Analysis is shown, ~ were the average transverse
momentum per nucleon projected into the reaction plane < P x / A > is plotted over the c.m.
rapidity. The slope of this curve at mid-rapidi ty determines the strength of the flow and thus
the stiffness of the equation-of*state. As shown in Fig. 24 the flow increases strongly with in-
creasing mass; it increases also with increasing energy up to a region where saturation oc-
curs.
The last figure in this series exhibits a t ransverse momentum analysis of s t reamer chamber
data and its comparison with theory: Two sets of VUU-theory 3 with a stiff equation-of-state
(K=380 MeV) and a medium equation-of-state {K=200 MeV) and a prediction of the cascade
model (Fig. 25), The intranuclear cascade model in which no mean field is included gives only
413
Nb.Nb 400 MeV/n
}
ul
l 0 o 30 ° 60 ° 90 °
Flow Anqle 0 Figure 19 Hydrodynamic flow data first ob- served in a Plastic Ball experiment logether with theoretical predictions (hydrodynamical model and cascade model)~,2=, z~.
E / A = 4 0 0 M e V
C a + C a Nb + N b A u + A u ~_~ z
I
os ~ !
o s
Z -0 0 '
0,
0 0 ~'0 ~0 0 ~0 ~0 0 ~0 ~0 90
Figure 21 Flow angle e (degrees) ........ Systematic measurements of angular dis- tributions dN/dcos~ for the collision of light, medium and heavy nuclei at various impact parameters ~.
Projec t i le
.J~ Side splash
-:'7~.-'~/ es * - - FIo, .
Figure 20 Schematic picture ro the sidewards flow of nuclear matter in a heavy ion collision.
A u + A u e = -
150 MeV 250 MeV 400 MeV 850 MeV 800 MeV ~z z:
0.5 V
0 . . L, L ,
05 ' ~,
8 0
Z ~ - ( 3 o - , . . . . . ~ ,
0.s I i ~ o O, "T L.r)
0 .
0
0 30 60 '0 30 50 C 30 60 0 30 60 0 30 b0 90
Figure 22 F I o w a n g l e ~ ( d e g r e e s ) ,,. ........ Same as fig, 21 for Au 4- Au at various projectile energies z3. The maximum of the distribution ( ' f low angle') increases with decreasing impact parameter.
200
Figure 23 Transverse momentum per nucleon projected in the reaction plane < px./A> as a function of rapidity z~.
A c o 100
E
o
-100 m v
- 2 0 0 -1,6
200
414
I
-1 - 0 6
Y/Yp,o,
i i
0 B I 1.5
Au
Figure 24 Mass dependence of hydrodynamic flow: Transverse momentum per nucleon projected in the reaction plane < P x / A > plotted over the im- pact parameter. The decrease at small impact parameters is caused by the fact that the distribution of emitted reaction products gets more and more rotational symmetric 2s.
o o 3 E
N t
Figure 25 Transverse momentum analysis for Ar -* KCI at 1.8 GeV/u (Streamer Chamber data) (a) and comparison with theoretical predictions: Cascade model (b) and VUU approach with stiff (C) and soft (d) equations-of- state 3.
100
5O
0 - -
-50
-I00
~ I00
50
0
-50
-I00
150
tOO
60
f i I f
2E, 5 0 7 5 t O 0
Percent of maximum multiplicity
I LBL-GSt DATA ~j.
t /
YT
t Yp
t ttt
CASCADE
-I.5 "O.5 0 0.5
VUU K =380 MeV
VUU K=200 MeV
-t.5 RAPIDITYy
126
~=u,e~.6, t
• °
415
a very weak Px-dependence as a function of the rapidity. Among the two VUU predictions the
stiff equation-of-state is in better agreement with the flow data. Moreover, the two methods,
pion thermometer and hydrodynamic flow, agree with each other reasonably well.
2.3 Entropy and the Equation-of-State
It is a well-known phenr>menon of shock compression that entropy S is mainly produced dur-
ing the compression phase, whereas the increase during expansion is small. Entropy there-
fore is a signature of the high-compression phase. 26 It can be determined by measuring the
ratio of deuterons to protons, or in gerneral the ratio of clusters (or cluster-like configurations)
to the nucleon-like particles being formed at freeze-out. The dlike/Plik e ratio measured in Ca
+ Ca and Nb + Nb at 2 different energies 27 is plotted in Fig. 26.
In a comparison between the fireball prediction for the entropy per nucleon S/A and the data,
a considerable deficit of entropy ("missing entropy") is observed (Fig. 27). This discrepancy
can be qualitatively understood by the following consideration: In a thermal model without
compression the fireball temperature and hence the entropy gets much higher as compared
to the case where compression is included because in the latter case part of the total collision
energy is deposited in form of potential energy which is non-productive for entropy. This
consideration is supported by calculations including a mean field, which give much better
agreement with the data.
2.4. Conclusions
Collective flow in relativistiv heavy ion reactions has been observed in coll isions meas-
ured in 4~ by the Plastic Ball detector and the Streamer Chamber. The flow angle is in
good agreement with hydrodynamic model predictions.
A number of experiments can give information about the nuclear equation-of-state. Three
types of measurements have been investigated: The pion yield, the collective flow and the
entropy production. They give more or less quantitative results about the slope of the
equation-of-state in the density range between P/Po = 2 to 4. They favor a stiff equation-
of-state, in contrast to a much softer equation-of-state requested for supernova dynamics.
3, The results are well based on systematic data taken with adequate sets of a new gener-
ation of detectors. However, the number of nuclear systems investigated so far is very
small and it is mandatory to increase the data base for such a fundamental relation as the
416
!
06i
0.~
0.2
Ca + Ca
400 MeVlnucleon •
~ S ~ t050 MeV/nucieon
r' /
10 20 30 40 5-0-
f v 650 MeVInucleon
0 10 20 30 40 50 60 ;~0 80 90 100
Np
d iv~,/p ik,, ratio Ior Ca + Ca and Nb + Nb at two different energies as a function o f the niJ~'be~"~T participating nucleons 27. "Plike" and "dlik,=" are defined as follows: Plike = P + d + t + 2(3He + 4He); dlike = d + ~J~(t + =He) + 3"He)
~C
~ F- - - - I ¸ - - - [ - - J
5 ~ j
J , / /
/ 3
o I ~ [ ,, I 250 ~ 7"50
L__ 1000
Bornbardm, g energy (MeVlnuclecn)
~gure 27 Entropy production derived from the data of fig. 26 and comparison with theoret ical pred- ictions of St~Scker et al, The "missing entropy" is explained by the lower f ireball temperature due to the fact that part o f the coll ision energy is deposited in compressionat energy which is not contributing to the entropy production 27.
nuclear equation-of-state, to extent investigations to higher and lower energies, and to
remove existing discrepancies.
Discussions with R. Stock, H.H. Gutbrod and H. St6cker are grateful ly acknowledged. The
names of the group members are given in ref. 10 for the Streamer Chamber group and in ref.
25 and 27 for the Plastic Ball group.
417
References
1. GF. Chapline, M.H Johnson, E Teller and M.S. Weiss, Phys. Rev. D8 (1973) 4302 W. Scheid, H. MiJller and W, Greiner, Phys. Rev. Lett. 32 (1974) 741 M.I. Sobel, P.J. Siemens, J,P. Bondorf and HA. Bethe, Nucl. Phys. A251 (1975) 502
2. R. Stock, Physics Reports 135 (1986) 259
3. H. Stocker and W. Greiner, Physics Reports 137 (1986) 277
4, H H Gutbrod, private communication
5. J. Gosset et al. Phys. Rev. C16 (1977) 629
6. A. Sandoval el al., Nucl. Phys. A400 (1983) 365c
7. A. Baden and H.H. Gulbrod et at., Nucl. Instr. & Meth. 203 (1982) 189
8. J J Molitoris, H. StOcker and W. Greiner, to be published
9. J. Cugnon, D. Kinet and J. Vandermeulen, Nucl. Phys. A379 (1982) 553 Y. Yariv and Z. Fraenkel, Phys. Rev. C24 (1981) 488 J. Cugnon and D. L'HSte, Nucl. Phys. A452 (1986) 738
10. J.W. Harris, R. Stock, R, Bock, R. Brockmann, A. Sandoval, H. Stnbbele, G. Odyniec, H.G. Pugh, L.S. Schroeder, R E Renfordt, D. Scha~l, D, Bangert, W. Rauch and K.L. Wolf, Phys. Lett. 153B (1985) 377 J.W. Harris and R. Stock, LBL-Report 17054 (1984)
11, I. Montvay and J. Zimanyi, Nucl. Phys. A316 (1979) 490 J.W. Harris and R. Stock, Notas de Fisica 7 (1984) 61
12. R.E. Renfordt et al., Phys. Rev. Lett. 53 (1984) 763
13 H.A. Guslafsson et al., Phys. Lett. 142B (1984) 141
14. R. Brockmann et al., Phys. Rev. Lett. 53 (1984) 2012
15. S Nagamiya, Phys. Rev. Lett. 49 (t982) 1383
16. R. Hagedorn and J. Ranft, Suppl. Nuovo Cim. 6 (1968) 169 R. Hagedorn CERN Preprint TH3684 (1983)
17, R. Stock el al., Phys. Rev, Lett. 49 (1982) 1236 R. Stock et al. Physica Scripta T5 (1983) 130
18, J,l. Kapusta, Phys. Rev, C16 (1977) 1493
19. J.W. Harris et al., LBL-Report and Phys. Rev. Lett., to be published
20. 5.J. Molitoris and H. Stocker, Phys. Rev. C32 (1985) 346
21. J.J. Molitoris and H. St~,cker, Erice Lectures 1985 B. Friedmann and V.R. Pandharipande, Nucl. Phys. A361 (1981) 502 J. Boguta and H. Stocker, Phys. Lett. 120B (1983) 289 M. Sano, M, Gyulassi, M, Wakai and Y. Kitazoe, Phys. Lett. 156B (1985) 27 E. Baron, J. Cooperstein and S. Kahana, Nucl. Phys. A440 (1985) 744
22, H, Stocker, J.A. Maruhn and W. Greiner, Phys. Rev. Lett. 44 (1980) 725
23. H.G. Ritter et al., LBL Report 16110 and 20086; Nucl. Phys. A447 (1986) 3c H.A. Gustafsson el al., Phys. Rev. Left. 52 (1984) 1590
24. P. Danielewicz and G. Odyniec, Phys. Lett. 129B (1985) 283
25. K~G.R. Doss, H.A. Gustarsson, H.H. Gutbrod, K.H. Kampert, B. Kolb, H. Lohner, B. Ludewigt, A.M. Poskanzer, H.G Ritter, HR. Schmidt and H. Wieman, Phys. Rev. Lett. 57 (1986) 302
26. P.JSiemens, and J.I. Kapusta, Phys. Rev. Left. 43 (1979) 1486 G Bertsch and J. Cugnon, Phys. Rev. C24 (1981) 2514 H. St4bcker et al., Nucl. Phys. A400 (1983) 63c £)~ Hahn and H. SlOcker, to be published
27. K.G.R. Doss, H.A. Gustafsson, H.H. Gulbrod, B. Kolb, H. L(bhner, B. Ludewigt, A.M. Poskanzer, T. Rennet, H Riedesel, H.G Ritter, A. Warwick and H Wieman, Phys. Rev. C32 (1985) 116
ASTROPHYSICAL ASPECTS OF COULOMB BREAK-UP
OF NUCLEAR PROJECTILES
H. Rebel Kernforschungszentrum Karlsruhe GmbH
Institut fur Kernphysik P.O.B. 3640, D-7500 Karlsruhe Federal Republic of Germany
ABSTRACT
Experimental studies of the break-up of light nuclear projectiles in
the Coulomb field of a heavy nucleus, acting as a source of virtual pho -
tons, are proposed as an access to information about the reverse reac-
tion: the fusion of the fragment particles at small relative energies.
The mechanism of Coulomb dissociation is theoretically studied and the
cross section of such reactions, being potentially of astrophysical in-
terest, is estimated. The case of Coulomb break-up of 6Li is alternati-
vely considered on the basis of a DWBA approach and the feasibility of
dedicated experiments is discussed.
I. INTRODUCTION
Most of the experimental approaches to nuclear astrophysics, investi-
gating charged-particle-induced reactions in stellar burning processes,
involve the bombardment of rather thin targets by low-energy protons,
3He, ~-particles or other light ions. The cross sections are almost al-
ways needed at energies far below those for which measurement can be
performed in the laboratory, and they must therefore be obtained by ex- 1,2
trapolation from the laboratory energy region.
Tab. I presents some selected cases of interest at various astrophysic-
al sites. The 3He(4He,y)7Be radiative capture reaction which at solar
temperatures affects the solar neutrino flux and bears strongly on the
long-standing solar neutrino problem 3'4, is experimentally studied 4'5
down to the CM-energy ECM = 165 keV, while the cross section is actually
needed at 1-20 keY. A similar situation is found for the 12C(a,¥)160
reaction 6 which is important for the stellar helium-burning processes
in red giant stars. To which extent a nucleosynthesis of 7Li and 6Li
takes place 100-500 sec after the beginning of the expansion of the uni-
419
Verse is determined by the (a+t) and (a+d) radiative capture cross sec-
tions at a temperature near 109K. 7'8 The capture reaction D(a,y)6Li
has been studied in the laboratory at CM energies ECM ~ I MeV 9, and
the present statement that essentially all 6Li is produced in the ga-
lactic cosmic rays rather than just after the primeval big bang is bas-
ed on a purely theoretical estimate and extrapolation of the reaction
rate, whose uncertainty is not known 7. On the other side the production
of the L£-isotopes and the comparison with the actual abundances pro-
vide a stringent test of the assumptions of the standard bis bang model
(see also Ref. 10).
EXAMPLE Emeas~l ASTROPHYSICAL INTEREST
Hydrogen Burning
1- + 3 He ---- T Be . .i I
E o = 10 keY
Hetium Burning
1 = + ~2C - " 1 6 0 * ' 1
E o --- 300 keV
Big Bang Nuc(eosynthesis
= ", t " - ' " ? L i + l
a * d ~ 6 L i ' ' g
Eo = 100 keY
165 keV
~ I. 31, MeV
Solar Neutrino Probtem
Ashes of Red Giant
( C / O Ratio)
Li Be B Production
~>lMeV Test of the Standard
Big Bang Model
Tab. 1 Some examples of radiative nuclear capture reactions of
actual astrophysical interest.
The direct capture process is a transition from a continuum state of
the reaction partners, the relative motion of which is described by a
Coulomb distorted wave, to a bound final state with a particular angu-
lar momentum, induced by the electromagnetics interaction and with emis-
Sion of y-rays of corresponding multipolarities. This is schematically
indicated in Fig. I for the example 160(p,y)17F.
420
Direct Rodiative Copture
tf=2 ld5/2
l d
16 0 ,,, p 17F
El 2dM1 ''' ~)
1sO . p
IO (E,LJi~--Jf r~) = ~ 81"I:(L+1) k 2L*IB ( L[(2L.1)!!|2 y capt (E.L,Ji ~-Jf~) I
Fig. I Schematic scheme of direct capture transitions in the
case of 160(p,y) 17F.
The capture cross section can be expressed in terms of an electromagnet"
ic transition probability Bcapt(E,L) with the initial state being a
(Coulomb) scattering state. Therefore Bcapt is dependent from the ener-
gy in the entrance channel, dominated by the Coulomb barrier penetra-
tion which strongly suppresses the cross sections at small energies.
In cases of nonresonant direct capture reactions the energy dependence
due to the Coulomb barrier penetration is usually factored out by a Ga-
mov factor thus defining the astrophysical S-factor
S (EcM) = Ocapt
with the usual Coulomb parameter
Z 1 •
• ECM exp(2z~)
2 Z 2 e
q = ~%v
in obvious notation. This S-factor shows a smooth energy dependence and
seems to be adequate for an extrapolation to astrophysically relevant
421
energy ranges. However, in most cases the extrapolation covers several
orders of magnitude and is particularly suspect if resonances and sub-
threshold resonances are expected to be of influence.
In view of the considerable uncertainties of astrophysical considera-
tions, introduced by the experimental difficulties in measuring radia-
tive capture reactions, any alternative access to the reduced transi-
tion probabilities of the relevant transitions (between a bound state
of the two nuclear particles and low-energy continuum states), is of
interest.
In the present study we analyse a recently proposed 11-13 approach which
suggests the use of the Coulomb field of a large Z nucleus for inducing
photointegration processes of fast projectiles.
In fact, instead of studying directly the capture process
b + c + a + y (1.1)
one may consider the time reversed process (with a being in the ground-
state) y + a + b + c (1.2)
The corresponding cross sections are related by the detailed balance
theorem
(2Ja+I)2 k 2
o(b+c+a+¥) = (2Jb+1) (2Jc+i) ~ o(a+y÷b+c) (].3)
The wave number in the (b+c) channel is
k 2 = 2~bc ECM ~2 (1.4)
With ~bc the reduced mass while the photon wave number is given
Ey _ EcM+Q ky = ~c ~c (1.5)
(neglecting a small recoil correction) in terms of the Q value of the
capture reaction. Except for extreme cases very close to threshold
(k+o), the phase space favours the photodisintegration cross section as
compared to the radiative capture. However, direct measurements of the
photodisintegration near the break up threshold do hardly provide expe-
rimental advantages and seem presently impracticable (see ref. 11). On
the other hand the copious source of virtual photons acting on a fast
charged nuclear projectile when passing the Coulomb field of a (large Z)
422
nucleus offers a more promising way to study the photodisintegration
process as Coulomb dissociation. Fig. 2 indicates schematically the
d£ssociation reaction.
b _. U
ZP ')
F~g. 2 Coulomb dissociation a * b + c in the field of a target
nucleus (ZT).
At a sufficiently high projectile energy the two fragments b and c emer-
ge with rather high energies (around the beam-velocity energies) which
facilitates the detection of these particles. At the same time the
choice of adequate kinematical conditions for coincidence measurements
allows to study rather low relative energies of b and c and to ensure
that the target nucleus stays in the ground state (elastic break up).
In ad~it£on, it turns out that the large number of virtual photons seen
by the passing projectile leads to an enhancement of the cross section.
In the following, we will illustrate these features considering the ac-
tual cases mentioned above and discuss the theoretical and experimental
implications of the studies proposed.
2. THE COULOMB BREAK UP CROSS SECTION
The double-differential cross section for Coulomb excitation of a pro-
ject£1e by an electric multipole transition of the order L as given by
the first order theory of Alder and Winther 21 can be rewritten in the
form
d2o I dqEL photo = ~-- ~ - OEL (2.1)
x x
4 2 3
3 OE Lphoto = L[(2L+I) !1] 2 - (2~) (L+I) k2L-Iy B(EL;Ii+If)pf(Ey ) (2.2)
is related to the Bcapt(EL)-value and the capture cross section, respec-
tively. The function d~EL/d~ does not depend on the internal structure
of the projectile. It only depends on the excitation energy Ey and the
relative motion. We call dHEL/d~ the virtual photon number per unit
solid angle seen by the projectile, scattered by the Coulomb field. It
actually depends on the incident energy, the mass and Z of the projec-
t£1e and of the impact parameter. This factorization of the cross sec-
tion corresponds the Weizs~cker-Williams method used for deriving the
Coulomb dissociation cross section of relativistic projectiles. The vir-
tual photon spectrum has been explored more in detail by several au-
thors 14. Fig. 3 displays the electric dipole component, relevant for the
two considered examples: the dissociation of 7Be and 160 when passing
208pb with an impact parameter b = 10 fm at two different projectile
energies. The corresponding break-up thresholds are marked.
"-" 3 E 1 0 ' .- I- o , e l
I I C'J
2 "-o 1 0 \
W C "0
1 0 1 0
w=80
W=30 v .~~..
7Be +208pb ~~, l , ;60+208pb
t I I I i
Eth 5 Eth 1 0 1 5
E (MeV)
Fig. 3 E1 virtual photon spectra seen by the projectiles with
b = 10 fm at different projectile energies W (MeV/amu)
424
The most interesting feature is the high intensity of the virtual photon
spectra which actually leads to an enormous enhancement of the photodis-
sociation cross section. This is one of the main advantages of the pro-
posed method. The examples given in Tab. 2 demonstrate the effect. The
table gives the double-differential cross sections for the excitation
of the projectile to the continuum energy Ebc of the emerging fragments
when the projectile or the fragment center-of-mass, respectively, is
scattered to d~. Assuming a specific detection geometry this cross sec-
tion can be transformed into the triple differential cross section,
which we actually are going to measure. Obviously the resulting values
appear to be experimentally accessible, in contrast to the correspond-
ing acapt-values.
REACTION Ebc Ocapt
b+ c,-*, a IMeV] [ n b ]
E1
a+ 3He~7Be 0.1 = 0.5
El a.+UC ~'s0 1.0 = 0.1
E2
(:{÷d H BLi 0.5 :1.0
d 2 0 Diss
dEbt dQ [ l ib MeV ~ stera d-l]
11
2
10 ~
d 3 G Diss
dEbdQ b d~c [p,b ~,leV "~ sterad -2 ]
8U= 5 °
52 %; 7 °
Elastic Coulomb break up with 2°BPb
Eproj = 30MeV/ a inu- Impact parameter 10fm
Ethr
[ I,,leV ]
1 .58
7.162
1.Z,7
Tab. 2 Numerical values of break-up cross sections for selected
examples of astrophysical interest.
Surprisingly the quadrupole component (see 6Li case in Tab. 2) of the
virtual photon field is much stronger than the El component at that im-
pact parameter and projectile energy (see Ref. 13). For large b the El
component would dominate.
3. COULOMB BREAK-UP OF 6Li
The D(~,y)6Li radiative capture reaction has been experimentally studied
at energies down to E~d=1MeV 9. At E d=0.71MeV, there is a L = 2 reso-
* For sake of simplicity isotroplc decay of the excited projectile has
been assumed for the example given in Tab. 2.
425
nance corresponding to the first excited state E3+ = 2.185 MeV in 6Li.
The resonance strength can be derived from the inelastic scattering
cross section, e.g.
Focusing to the 6Li case, we have started a series of studies to explo-
re the feasibility of the break-up approach.
ASPECTS OF THE 6Li CASE
(1) Test of the method and the concept
(2) Test of the theory for quadrupole transitions
(3} 6Li production in BIG BANG
Robertson et ot 119811
t i O
t !
E l ONLY
CENTER,OF-i',~SS Et,~_FI~GY {MeV}
F[G. 1. Cross section for fhe reaction ~II(,,7)~LI. Open circles, MSU dnts; closed eh'elcs, CIIN[. dntrt; triangles, r'Ll(e,e'#) (Rer. 7); crosses, CRNL dnts for El oomponent. The curves ,'ire s d|rect-capture ertl-
cul.atlon.
6Li
0 + 3.56
3 + - - 2.185
1J.7
I +
a+d
Fig. 4 Cross section for the reaction D(~,y)6Li (from Robertson
et al. 9 )
Fig, 5 shows spectra of a-particles from reactions of 156 MeV 6Li ions
with 208pb. At forward angles these spectra are dominated by a bump
around the beam-velocity energy, indicating break-up processes as being
the origin. However, the bump is mainly related to nonelastic break-up
processes, where the nonobserved deuteron interacts nonelastically with
426
the target 16 (in particular by break-up fusion).
0
(D ~0
0
8
0
o 8
L9 o
I "O 0
. . . . . . . . . . . ~ - ; . , : ; ~ ; _ T _ ' ~ _ ~ . . . . 171 ........
. . . . . . ~ 22°
27 o
I 32 °
. . . . . . . . . . . i 50 100 150
[MeV]
Fig. 5 Inclusive ~-particle spectra of break-up of 6Li from 208pb at
156 MeV, observed at various emission angles.
The inclusive measurements of the break-up yields have been extended
to very forward angles (Fig. 6), using the recently in stalled spectro-
meter "Little John" at the beam of the Karlsruhe Isochronous Cyclotron.
We are now in the position to start s-d coincidence measurements in the
extreme forward hemisphere and to measure the cross section for binary
dissociation the Coulomb field.
The kinematic situation for a typical detector arrangement with a a-
particle and a deuteron detector in fixed-angle-position is displayed
in Fig. 7. The kinematics for three particles in the final state lead
to a correlation of the s-particle and deuteron energy (for a particu-
lar value of target excitation). For a heavy target this is approxima-
tely a linear relation, as shown in Fig. 7 for the case of 208pb, which
remains in the ground state. Along this kinematical line all events of
elastic break-up are distributed. Fig. 7 shows additionally the rela-
tive energy Esd plotted over the ELabs axis, and one recognizes that
the E d value appears twice (once the s-particle the slower fragment,
once the deuteron). There is a remarkably slow variation of E~d around
the Em~n-value ("magnifying glass effect") which allows a good resolu-
427
tion on the relative-energy scale. We have just to measure the coinci-
dence cross section on the kinematical curve around the minimum region.
10 £
C
10 3-
6Li ÷ 208pb at 156MeV
~ k ~ - - Elastic Scattering
Inclusive x~ ~ ~t -break up component -~ × x
X X
I .......... I I 5 ° I 0 ° 15 °
81Q b
-2
-I
-0.5
-0.2
-0.1
-0.05
X
I
20 °
Fig. 6 Elastic scattering and inclusive a particles (break-up
component) from 156 MeV 6Li collision with 208pb.
Due to considerable cancellations of various contributions to the rela-
tive energy E~d, the energy resolution dE d on the E d scale is much
better than on the scale of the laboratory energies. Since for the ve-
locities V~d, v~, v d
2 2 2 Vad = v + v d - 2 v~ v d cos Oad
then Vad dVad = (v -v d cos 0~d) dv a + (Vd-V a cos O d) dv d
4 2 8
150
>
@
100
50
10
0
7%=2 o
Z°aPb (6Li, u d ) 2°epb
\ r - 2.~8s x~
\ "
E~ooO.,q ; X I t
0 30 60 90 120 150 EL°b[MeV]
Fig. 7
Kinematic loci of
the emerging deu-
teron and u-partic-
le from a 6Li dis-
sociation on 208pb
at Ela b = 26 MeV/
amu.
As for beam-velocity particles (v ~v d)
cone (cos 0 d % I),
dEud << dE a, dE d
emerging within a narrow angle
However.the resolution is affected by the accuracy of 0 d
2/m a md Eu'E d = sin d0ud deed m a + m d 0ud
and requires a good angular accuracy of the experimental set-up.
Most recently the cross section has been alternatively considered 12 by
a DWBA approach on the basis of the Rybicki - Austern 15 formulation of
the break-up theory. For the Coulomb interaction and the special case
L = 2 due to some simplifications the cross section can be given by a
closed expression in terms of the electromagnetic transition probabili-
ty B(E2). (Fig. 8).
With a reasonable estimate of the B(E2, Ead) distribution, the triple
differential cross section for the Coulomb break-up of 6Li with 208pb
is given in Fig. 9.
Usually due to finite angular resolution and due other experimental li-
mits the observed spectra are broadened and ha~e to be analysed by Monte
429
D W B A A P P R O A C H
OF PROJECTILE B R E A K UP
Ryb i ck i and A u s t e r n :
ITf, = < Xg' (~)¢~'(r) IV~s(~ r) IX~[' {~)%(r)>I
2 Zb ZX Za Vre s = ZAe (rbA rxA
m x L m b L r L
= 4~zae2 [ { zb~-~-2 + Z x ~ I } R L*I L,M>_I
2L+I
Fig. 8
DWBA approach 12 of
Coulomb break-up of
6Li projectiles.
Quadrup.ote case L = 2
6Li --.)
d 3 o~ =
df~a dQ d dEa
a * d
90h {2If+l) ~ C2(~ 'Ead ) *
B ( E 2 ; E a i E ~ 4 ) )
Carlo methods. These methods are well worked out. Fig. 10 shows a s-d
coincidence spectrum (with poor s-particle resolution), just taken in
order to explore the feasibility of the measurements under the domi-
nance of the elastic scattering at very forward angles. (As a conse-
quence of the poor ~-particle energy resolution, there is a contribu- + 208pb ) £ion from inelastic break up with exciting the 31 state in
4. CONCLUSIONS
The proposed approach for studies of the interaction of nuclear partic-
les at small relative energies requires experiments at extreme forward
angles, in a region where Oelastic/aR=1. The elastic scattering cross
section provides, in fact, a calibration of the break-up cross sections.
The values of the estimated coincidence cross sections are rather small,
but appear to be measurable by present days' experimental techniques.
The kinematic situation with three outgoing particles provides particul-
ar advantages for studies of the excitation function i.e. the varitation
430
>o 5"
U~ - . , , . . . , .
t~ E
0.2-
0.1- /
q
0.0 80 90 130
O~x : 2 °
Od =-2 °
3 + 3 +
.10 .2 xi0-2
1 16o liO 12o
Ec~ [ MeV ]
Fig. 9
Resonant and nonreso-
nant excitation of
the ~ + d continuum
in 6Li by projectile
break-up in the Cou-
lomb field of 208pb
at ELi = 156 MeV as
calculated on the
basis of a DWBA 12
approach
with relative nergy of the emerging fragments, and of the angular dis-
tribution in the rest frame of the fragments subsystem. Investigations
of the latter aspect, however require a quite good angular resolution.
The cross sections can be interpreted in terms of electromagnetic inter-
action matrix elements which just determine the radiative capture cross
section. There are a number of problems which have to be investigated
more in detail, experimentally mainly arising from the dominance of Cou-
lomb scattering. The theory has to be refined with respect to orbital
dispersion and Coulomb distortion effects.
Very interesting and improved experimental possibilities would be pro-
vided by a dedicated set up at a synchotron-cooler ring (see ref. 18)
with suitable magnetic spectrometers (like the proposal of ref. 19) en-
abling particle coincidence studies at very forward emission directions.
The use of a storage ring seems to be indispensable when working with
radioactive beams like with 7Be. Even, if the acceleration and prepara-
tion of such a beam would be successful in a conventional approach, the
con-tamination problems arising from the accumulation of the radioactivity
(TI/2[7Be] = 53.3 d) impose serious limits. On the other side, in a sto-
rage ring even a current of 10 mA corresponds to a sufficiently small
number of stored radioactive particles. A Hg vapour jet target 20 e.g.
431
3 0
3 0 -
2 0 - -
1 0 -
0 -
2°8pb(6Li,O.d ) 208pbgr:;3 ~
e~= 5 ° ed = -2 °
ELi=156MeV
0 ' :31 {23 J~4 '6a 9~
E~ ( MeV )
Fig. 10 Experimental ~-d coincidence spectra at very forward
angles from collisions of 156 MeV 6Li ions with 208pb.
may serve as reaction target for the Coulomb break-up measurements.
I would like to thank Dr. G. Baur, Dr. C.A.Bertulani and Dr. D.K.
Srivastava for a friutful collaboration in the theoretical foundation,
and Dr. H.J. Gils and DP H. Jelitto for clarifying discussions on ex-
perimental aspects.
REFERENCES
I. W.A. Fowler, Rev. Mod. Phys. 5_66 (1984) 149
2. C. Rolfs and H.P. Trautvetter, Ann. Rev. Nucl. Sci. 28 (1978) 115
3. T. Kajino and A. Arima, Phys. Rev. Lett. 5_~2 (1984) 739
4. J.L. Osborne, C.A. Barnes, R.W. Kavanagh, R.M. Kremer,
G.J. Mathews, J.L. Zyskind, P.D. Parker, and A.J. Howard,
Phys. Rev. Lett. 4_~8 (1982) 1664 - Nucl. A419 (1984) 115
5. K. Nagatani, M.R. Dwarakanath, and D. Ashery, Nucl. Phys.
A 128 (1969) 325
6. K.U. Kettner, H.W. Becker, L. Buchmann, J. G~rres, H. Kr~hwinkel,
C. Rolfs, P. Schmalbrock, H.P. Trautvetter, and A. Vlieks,
Z. Phys. A 308 (1982) 73
K. Langanke and S.E. Koonin, Nucl. Phys. A 439 (1985) 384
Q.K.K. Liu, H. Kanada, and V.C. Tang, Phys. Rev. C 33 (1986) 1561
7. S.M. Austin, Prog. Part. Nucl. Phys. 7 (1981) I
432
8. R.V. Wagoner, Astrophys. J. 179 (1973) 343; D.N. Schramm and R.V.
Wagoner, Ann. Rev. Nucl. Sci. 27 (1977) 37
9. R.G.H. Robertson, P. Dyer, R.A. Warner, R.C. Melin,
T.J. Bowles, A.B. McDonald, G.C. Ball, W.G. Davies, and E.D. Earle,
Phys. Rev. Lett. 47 (1981) 1867
10. D.N. Schramm, Nature 317 (1985) 386
11. H. Rebel, Workshop "Nuclear Reaction Cross Sections of Astrophysic-
al Interest" unpublished report, Kernforschungszentrum Karlsruhe,
February 1985.
12. D.K. Srivastava and H. Rebel, Journ.Phys.G: Nucl.Phys. (in press)
13. G. Baur, C.A. Bertulani and H. Rebel, Nucl. Phys. A (in press)
KfK-Report 4037 (Jan. 1986); Contr. Internat. Symposium on Weak
and Electromagnetic Interactions in Nuclei, I-5 July 1986,
Heidelberg (Germany)
14. C.A. Bertulani and G. Baur, Nucl. Phys. A 442 (1985) 739;
A. Goldberg, Nucl. Phys. A 420 (1984) 636
15. F. Rybicki and N. Austern, Phys. Rev. C6 (1972) 1525
16. R. Planeta, H. Klewe-Nebenius, J. Buschmann, H.J. Gils,
H. Rebel, and S. Zagromski, T. Kozik, L. Freindl, and
K. Grotowski, Nucl. Phys. A 448 (1986) 110
17. A.C. Shotter, in Proc. 4th Int. Conference on Clustering Aspects
of Nuclear Structure and Nuclear Reactions, Chester, U.K.
23-27 July, 1984; D. Reidel Publ. Company, p. 199 ff
18. COSY, Proposal for a cooler-synchotron as a facility for nuclear
and intermediate energy physics at the KFA J~lich, JHlich 1985;
Studie zum Bau eines kombinierten KUhler-synchotron Rings an der
KFA J~lich - J~l-Spez-242 Februar 1984 ISSB 0343 - 7639
19. J.W. Sunier, K.D. Bol, M.R. Clover, R.M. De Vries, N.J. Di
Giacomo, J.S. Kapustinsky, P.L. McGaughey and W.E.Sondheim,
Nucl. Instr. and Meth. A 241 (1985) 139
20. G. Montagnoli, H. Morinaga, U. Ratzinger and P. Rostek, Jahresbe-
richt 1984 des Beschleunigerlaboratoriums der Universit~t und
Technischen Universit~t M~nchen, contr. 4.2.3. p. 109
21. K. Alder and A. Winther, Electromagnetic Excitation (North-
Holland, Amsterdam, 1975)
K. Alder and A. Winther, Coulomb Excitation (Academic
Press, New York 1966).
SUMMARY TALKS
SUMMARY TALK: THEORETICAL
Abraham Klein Physics Department, University of Pennsylvania
Philadelphia, Pennsylvania 19104-6396, USA
INTRODUCTION
My unenviable task is to summarize 38 hours of lectures in approximately one
hour. If it were possible to do this perfectly, we could have saved a lot of time
and played much more football and tennis. I therefore beg your indulgence from the
very beginning.
The order in which the lectures were presented at the school was determined hy
the schedules of the various speakers, most of whom could not spend the full two weeks
at the school. This summary is, however, arranged according to the bins of subject
matter as announced by the organizers in the poster advertising the school:
I. Semi-classical features of nuclear motion.
II. Symmetries and supersymmetries in nuclei.
III. Giant resonance states.
IV. Extreme states and decays.
I am deeply indebted not only to those speakers who gave nearly perfect lectures,
but also those, who like myself, fell short of this ideal, but were willing, neverthe-
less, to help with manuscripts, reprints and discussions. With this aid, I have man-
aged to clarify, at least in my own mind, points where my understanding was previously
deficient. Any errors that remain, however, must be attributed to my own inadequacies.
In this summary, I shall provide neither references nor figures. For this you
must refer to the original manuscript. What I have tried to do is reconstruct brief-
ly the essence of each lecture.
I. SEMI-CLASSICAL FEATURES OF NUCLEAR MOTION
A. M. Yamamura. Classical Image of Many Fermion System and its Canonical
Quantization.
Professor Yamamura's lectures consisted mainly of an account of very beautiful
formal structures, but it also included several interesting applications to which we
come below. It is well known that for a system of an even number of fermions, the
time-dependent Hartree-Fock theory (TDHF) represents the classical limit of the quan-
tum many-body problem. It is not quite so well-known that TDHF is a disguised form
435
of Hamilton's canonical equations of classical mechanics. In the former the indepen-
dent coordinates are the particle-hole matrix elements of the single-particle density
matrix. These variables can be mapped (non-linearly) onto canonical pairs of coor-
dinates and momenta, and, of course, this can be done in infinitely many ways. A
particular means is chosen so that upon requantization one obtains the exact Marshalek
mapping of the original quantum system to a system of particle-hole bosons. By the
use of ~assmanvariables, it was shown that the entire procedure can be extended to
odd systems and the requantization leads to the exact Marshalek Bose-Fermi mapping.
There is a problem of ordering in the requantization which is certainly resolved with
the help of prior knowledge of the exact solution.
These results are not directly applicable to the problem of collective motion.
Therefore Yamamura studied another method for introducing such coordinates, treating
them as redundant within the context of Dirac's generalization of classical mechanics.
It turns out that the classical theory of collective motion which emerges is identical
in its fundamentals with the theory developed by the writer. I will therefore des-
cribe the results in my language. Consider a simple problem with two coordinates x 1
and x 2. For small velocities we may describe the motion in configuration space
rather than phase space. For the given Hamiltonian H(Xl,X2,Pl,P2) , we ask if there
are motions in which the system moves (for long times or forever) along a one-dimen-
sional (curved) path called the collective path, Such a path is described by the
equations
x i = xi(Q) (i = 1,2),
= ~-~i P (i = 1,2) X i
The latter being a well-known equation of Lagrangian mechanics.
this motion, we have two equivalent forms of the Hamiltonian,
H (Q,P) ~ , pi(Q,p)) c H(xi(Q) '
and the motion can be described by two forms of Hamilton's equations
Pi = -- ~H = {Pi,He }
(IA.I)
(IA.2)
It follows that for
where, e.g.
~x. ~H ~x. ~H i C i C
{xi'Hc) aQ ~P ~P ?Q
(IA.3)
(IA.4)
, (IA.5)
(IA.6)
is a Poisson Bracket. Th___~e~oxed equations are a geometrical c haracter.iz~tion of the
collective submanifold Z.
To solve these equations (our method is different) Yamamura and his collaborator
436
Kuriyama expand in powers of P. Everyone does that[ They also expand in Q (not
accurate for large Q) but then analytically continue by the method of Pad~ approxim-
ants. They obtain excellent results in application to a model of coupled Lipkin
systems (requantization and comparison of energy levels), but Prof. Yamamura agreed
that the method is probably too cumbersome for extension to real physical systems.
These authors also displayed large amplitude classical periodic solutions which
can be built up from small amplitude periodic solutions obtained upon linearization
of the equations. This method of constructing periodic solutions may be quite use-
ful for more realistic cases. It was also shown that the periodic path passes through
maxima of the probability density of the associated quantized motions.
B. A. Klein. Quantum Foundations for the Theory of Collective Motion. Semiclassical
Approximations and the Theory of Large Amplitude Collective Motion.
I first described the most recent version of what used to be called the Kerman-
Klein theory of collective motion, which I now prefer to call the generalized density
matrix method (GDM). This is a general theory of collective motion, of the same de-
gree of generality, at least in practice, as generator coordinates and other such
methods. I then described in detail how to make the transition from the GDM to TDHF
for three important phenomena - periodic motion, large amplitude collective motion
and heavy ion scattering. The TDHF solution is seen in each case to have a physical
interpretation as a Fourier superposition, sum and/or integral of density matrix ele-
ments satisfying suitable classical conditions.
The remaining lectures zeroed in on the problem of large amplitude collective
motion treated as a problem in classical Hamiltonian mechanics, using the ideas des-
cribed in association with Eqs. (IA.4,5) above. A new method, called the method of
the "generalized valley" was derived for constructing the collective Hamiltonian, H , c
in an arbitrary number of discussions. An older method, the local harmonic approxima-
tion can be viewed as an approximation to the new method. Physically this method be-
comes important when the one-dimensional valley, usually discussed, is too flat to
expect the system to remain on or near it.
Some very simple application~ of the same type as described by Prof. Yamamura,
have been carried out, but in the future, we hope to study rotational motion of de-
formed nuclei, the potential energy surface for fission, and low energy fission and
fusion reactions, among other problems.
C. V. Ceausescu. qlas~Ical Limit of Ham iltonian - >ystems and Requantizationu
Professor Ceausescu studied the class of all SU(2) invariant Hamiltonians which
are at most quadratic in the "angular momentum" generators Ji' i = 1,2,3, as an exam-
ple of systems which are integrable with an associated group structure. Later he ap-
plied the theory to the two-parameter system
437
H = Jl 2 + uJ22 + 2v 33 (It.i)
In the formal study he considered a map from the Hilbert space onto a simplectie
manifold (phase space) provided by a set of coherent states Ip,q>, where the nota-
tion is meant to imply that p,q are to be viewed as a pair of (local) canonical co-
ordinates. We have such equations as
~(pq) = <pqIHlpq> , (I.C.2)
i(p,q ) = <pq]Jilpq> (I.C.3)
The choice of Ip,q> will be specified more closely below. Such a mapping provides a
local chart from the sphere
~ 2 +~22 +~32 = j(j+l) (I.C.4)
onto a plane. Because the poles are singular points of the sphere, we need at least
two charts to "cover" the sphere.
Actually Ceausescu used complex coordinates, the two charts (coset charts)
ZJ Ipq> + I~+> = £ *lj,~J> , ( I . C . 5 )
and solved the problem of Joining the charts analytically so as to have an analytic
global map. He established which quantities can be defined globally, namely the
generators, the Hamiltonian ~, the symplectic structure and the Poisson brackets.
After the global manifold was characterized, symmetry preserving (invariant)
subspaces were constructed. These provide irreducible representations for quantiza-
tion (or requantization) of the system. The latter task was completed by finding
realizations of the group generators on these spaces. The solution was found to be
the Dyson alternative.
The theory thus completed was applied to the Hamiltonian (IC.I): (i) All cri-
tical points and phase boundaries were identified. (ii) Exact solutions of the
classical Hamilton's equations were computed. (iii) Linearization of the equations
of motion was shown to be accurate only for values of the parameters (u,v) well
removed from phase changes. (iv) The requantization procedure was illustrated.
This very interesting approach is restricted to integrable systems, which un-
fortunately does not include most realistic cases in nuclear physics.
D. O. B ohigas. Spectral Fluctuations and Chaotic Systems.
In the first part, Professor Bohigas discussed fluctuations of "complex" quantum
systems. Let N(E) be the number of levels with energy J E (staircase function). In
particular the experimental results for the resonant scattering of slow neutrons by
166Er was ultimately analyzed. We write
N(E) = Nay(E) + Nfluc(E) (ID.I)
438
For theoretical models Nav(E ) can be defined in terms of the properties of the model
itself, but for experimental data it is defined by least squares fit of a straight
line to the observed staircase function. On the other hand, Nfluc(E) is characteriz-
ed by various moments and by p(x) = distribution of spacings between neighboring
levels.
It was shown that p(x) and the various moments were very well fitted by a Gaus-
sian Orthogonal Ensemble (GOE) which is the prediction of random matrix theory and is
parameter free. The question that naturally arises is how complex a system has to be
to exhibit this kind of behavior.
To study the deeper meaning of "complex system" a natural ground is provided by
recent advances in classical mechanics. For the most familiar (but limited) class of
(conservative) integrable systems, if there are N canonical pairs, the trajectories
are confined to an N-torus (N degrees of freedom). At the other limit systems call-
ed K-systems are defined by the property of exponential instability in time of neigh-
boring orbits. Since the latter possess only the energy integral, they have 2N-I de-
grees of freedom. Thus a distinction already exists at N = 2. The energy level dis-
tributions for the quantum mechanical systems associated with a number of N = 2 K
systems (Sinai billiard, stadium, etc.) have been shown to follow the GOE. At the
other extreme of an integrable system, the two-dimensional harmonic oscillator total
level distribution is Poisson. The Hydrogen atom in a magnetic field is a system
which goes from integrable (B ~ O) to K (B large). Experiments are under way to test
the previous conclusions for this accessible but sufficiently complex system.
E. A. A. Raduta. Semi-Classical Treatment of the Interaction Between Individual and
Quadrupole Desrees of Freedom.
An overly simple model of even (deformed) nuclei is provided by the Hamiltonian
n s II s
H = f~N b + Z ^ 1 o k=l (~k- l)Nk- ~ G ~ P Pk' (IE.I) k,k'=l
2 bj', bl, phonon number operator, and the Here ~ = quadrupole phonon energy, N b =
lJ =-2 second and third terms represent the usual pairing Hamiltonian where n s is the number
of fermion levels. Thus H describes a superconducting ground state with (harmonic) o
quadrupole excitations. Greater realism is achieved by adding the two terms
H' =r, ~s [b~ Pk + boPk#]- %j2 , (IE.2) k=l
where the first term introduces a core particle coupling but destroys the rotational
invariance, but the second term "restores" the latter. The Hamiltonian H I = Ho + H'
is then studied by the following sequence of steps: (i) The time-dependent variation-
al principle is applied to a suitable coherent state. This is an example of the tech-
nique described in Ceausescu's lecture of a mapping onto a symplectic manifold,
439
yielding a classical Hamiltonian2~ I. (ii) The mean field approximation, applied to
~l yields a set of generalized BCS equations giving J-dependent results for the
the quasi-particle energies, and the occupation probabilities. (iii)~ 1 energy gap,
is expanded to quadratic order in deviations from the mean field and the resulting
form is diagonalized (classical random phase approximation). (iV)~l is requantized
by means of the Wilson-Sommerfeld procedure. (v) The results are applied success-
fully to 188pt. An interesting new interpretation of band structure (excitation at
fixed angular momentum) is identified. (vi) A side result is the discovery of a new
way (analogous to Yamamura's result) for generating boson mappings.
The physical significance of this work is that the effect of core-particle coup-
ling is taken into account for all the energy levels of the collective model, even
the low-lying ones (variable moment of inertia effect).
II, SYMMETRIES AND SUPERSYMMETRIES IN NUCLEI
A. A. A. Raduta. Coherent State Model (CSM).
(i) This phenomenological model of collective motion belongs to the tradition of
geometric models as there is no fixed boson number, and it had its genesis in a
suggestion by Lipas in 1972.
(ii) The calculation of an energy spectrum is vastly simplified by restricting the
parts of the Hilbert space of the quadrupole phonon model which are included, in the
simplest example, to three bands representing the ground (g), y, and ~ bands.
(iii) The basis states are obtained by angular momentum projection from mutually
orthogonal coherent states and, at least up to now~ can be chosen so that the pro-
Jected states are also orthogonal.
(iv) Much of the analysis can be done analytically, greatly reducing needed computer
time.
The present status of the model can be characterized as follows: (a) The g, y
and B bands are in excellent agreement with experiment in range of nuclei in the
second half of the rare earth region (in the language of IBM, the SU(3) ~ SO(6)
range), including high spin states. (b) The SU(3) + SU(5) range has not yet been
done. In the joint opinion of the writer and Prof. Raduta, this will require a
different definition of the ~ band. (c) Therefore a "global" theory will require at
least two B bands. (d) For the inclusion of neutron-proton differences, see tbe
separate discussion in Sec. III.
B. F. lachello. Symmetries and Supersymmetries in Nuclei.
Prof. Iachello first reviewed the foundations of the standard IBM1 model and
subsequently described the use of graded Lie algebras as the foundation for the possi-
ble application of supersymmetry to the correlation of spectra of neighboring even
and odd nuclei. The best known experimental example from the Pt-Ir region was cited.
440
The bulk of his presentation concerned two developments in the application of alge-
braic methods to scattering and reaction problems.
The first considered the role played by IBM (so far IBM_I) in the analysis of
elastic and inelastic scattering at "intermediate" energies (e.g. i GeV protons).
This application involves a marriage of the eikonal approximation and the IBM. In
this approximation, valid for small momentum transfer, the S-matrlx element for tran-
sition from nuclear states If> to If> is represented as
Sfi = <flexp[iX(b)]li> , (llB.l)
where
V(r ,~) X(b) =-(~/k)Jdz ^ + (lib.2)
is the eikonal phase (operator) obtained by integrating the interaction over a +
straight-line path. Here ~ is reduced mass, k the relative wave number, r the separa-
tion of projectile and target and i the internal coordinates of the target (to be
represented later by a boson model). The eikonal formula sums all effects of coupled
channels. To use this formula, however, we need a model for V(~,~) and for the
states If> and If>. The first calculations (Amado) were done for limiting cases of
the geometrical collective model (vibrations, rotations) and agreement with experi-
ment was achieved. Much more extensive calculations have now been carried out using
an IBMI representation of the target states of interest for both limiting symmetries
and intermediate cases not only for proton-nucleus scattering but also for electron-
molecule scattering where the algebra is U(4) rather than U(6). The basic calcula-
tion involved in (ll.B.l) "reduces" to the evaluation of matrix elements of genera-
tors of the group in that we ~nd after integration over external (relative) coordin-
ates (for the nuclear case e.g.)
Sfi ~ <fI ~i~Gli> , (lib.3)
where G is a generator of the U(6) algebra and ~ is a determined function.
Prof. Iachello next described his latest ideas: concerning the application of
group theory to the relative coordinates in a collision. What is involved here is
the "accidental degeneracy" group of the scattering states, if one exists. For in-
stance, for the Coulomb problem this is the group S0(3,1). As a first major result,
it has been shown by purely algebraic methods that the element S%(k) = exp(2J~(k))
for any S0(3,1) invariant Hamiltonian, H, has the form (in terms of gamma functions)
F_i%+l+if(k~) S£(k) = F(£+l-if(k)) , (IIB.4)
where f(k) is known if H is known but otherwise must be treated phenomenologically.
To derive (lib.4) one writes
I'G'> = A~(k) l'F',-k> + B£1'F',k> , (lib.5)
441
where 'G' is a state in the irreducible representation (IR) of the symmetry group,
G, and F is the corresponding "asymptotic" group obtained by group contraction
(E(3) in this case). It is emphasized that the states 'F' include +k as a label.
In fact (lIB.5) is simply an algebraic version of the "well-known" representation
of scattering solutions in terms of Jost functions and S ~ (B%/A%). What is most
original technically about this approach is the calculation of (B%/A%) by purely
algebraic means.
To go beyond the well-known case of Coulomb scattering, a full analysis has also
been carried out for the group SO(3,2) and a formula which generalizes (lIB.4) has
been given. It involves besides f(k), a second function v~(k) which is understood to
characterize a short range potential. The resulting theory has been applied to exam-
ples of heavy ion elastic scattering where it is much simpler to carry out than the
usual partial wave optical model analysis, and yields excellent results. Further
generalization to coupled channels appears feasible.
C. P. van Brentano. Theoretical Aspects of IBM2.
Professor von Brentano discussed some recent theoretical advances in the applica-
tion of IBM2, the version which distinguishes neutrons and protons. He first explain-
ed the concept of F-spin, which can be viewed as an isospin variable for the bosons.
We have
N = N + N v , (IIC.I)
1 F ° = ~ (N -N v) (IIC.2)
1 and the max value of F, Fma x = ~N. The low-lying states which are well-described by
IBMI may be considered to be states with F = Fma x. The most important state clearly
identified as carrying F = F -i is the "scissors" mode discussed later in this re- max
port. It was explained why even if H(2), the Hamiltonian of IBM2 does not commute
with ~, the states of Fma x could lie lowest in the spectrum with a substantial split-
ting from the (Fmax-l) levels. This occurs if H(2) contains a term of the form
l[Fmax(Fmax+l) _ ~21 (IIC.3)
with D0 and large enough. This is the so-called Majorana interaction.
Computations with IBM2 are considerably more difficult then aith IBMI. There-
fore an important problem is to find the equivalent H(1) which can be applied to the
F states. In an obvious notation, H(1) can be defined by means of the requirement max
<2,FmaxlH(2) 12',Fmax > ~ <llH(1) ll'> (IIC.4)
The result for H(1) obtained by implementing this requirement was described.
Remarkable global fits with IBM2 have now been obtained using a five parameter
Hamiltonian,
442
n H(2) = e(ndP + n d ) + k QoQn
+ a ndP nd n + b(ndP + ndn)2 - c~ 2 (IIC.5)
Here the b-term may be considered to define a single particle energy depending on the
total boson number. Finally a review was given of the N N scaling concept. Its p n
remarkable success establishes the dominance of the neutron-proton interaction for
the description of deformation. Global IBM2 fits are found to have this scaling
property.
D. D. K. Sunko. Algebraic Approach to Odd-Odd Nuclei.
These lectures commenced with a description of the straightforward extension of
IBM and IBFM to such nuclei. The extended model IBFFM is described by a Hamiltonian
(in an almost obvious notation)
HIBFFM = HIBM ÷ H~ I) + H~ 2) + H(12)F + H(1)int + H(2)int ' (liD.i)
. (i) refers to fermion-boson interaction. In the sole application carried out where Hint
thus far, to 198Au, where the core is treated in the S0(6) limit and reasonable agree-
ment has been found, certain approximations were made which included dropping the ex-
change terms from H ~i)" " and including only surface spin interactions in H$12)- int
Dr. Sunko then explained clearly how Bose-Ferml symmetry works. In a parallel
reduction of GB~G F (the product symmetry), if we reach a point where (GB) ' is iso-
morphic to (GF) ' (this isomorphism is not universal, but occurs only in special
cases), then the subsequent reduction can proceed through (GBF) ' down to angular mo-
mentum quantum numbers. He emphasized that the reduction to (GBF) ' can be made only
if the parameters of (GB) ' and (GF) ' are set equal. It is this requirement which
fixes the interaction and is thus responsible for the symmetry.
Over and above Bose-Fermi symmetry we may have supersymmetry. In the canonical
approach we have
G(BIF ) ~ GB~G F (lID.2)
(graded Lie algebra)
after which the reduction proceeds as above. The importance of supersymmetry is that
now the spectra of neighboring nuclei are predicted by the same parameters. For odd-
odd nuclei, it turns out that an alternative approach to supersymmetry is possible if
= ~ = ~, i.e. equal level degeneracy for protons and neutrons. In this case we p n
can have the reduction
U(GI2~) ~ uB(6) ~ uG(2a)
) uB(6) ~ spF(2~) (liD,3)
A clearly successful application is being sought.
A new approach to supersymmetry has perhaps also been uncovered. Exact analytic
solutions have been found with the core in the SU(3) limit such that the ground state
443
band of the (ee) nucleus, the K = J band of the (eo) nucleus and the K = Jp + Jn band
of the (00) nucleus all are described by the energy formula
Ej = ~J(J + i) (lID.4)
with ~ the same constant in all cases and with other simplifying features. A proof
that this is indeed a supersymmetry is being sought.
E. V. Vanagas. The Restricted Dynamics Nuclear Model. Conceptions and Applications
As an introduction to his mode of thought Professor Vanagas first described a
derivation of both the Interacting Boson Model and of the Bohr-Mottelson model. We
consider the full complement of Jacobi (relative) coordinates of an n particle
system, pi s (i = l...n-l,s=x,y,z). A set of "deep collectivity" coordinates (quadru-
pole tensor) are defined by the formula
ss' ~ 2 n-i s s' q = (Ts') i~l qi qi (liE.l)
These describe monopole (trace) and quadrupole (traceless tensor) degrees of freedom.
A rotationally invariant Hamiltonian HGB M
= H ~ ss' ' = -ih(~/Dqss,)) (lIE.2) HGBM GBM tq ' PSS SS T pSS '
can, by going over to those linear combinations of q and which define hoson
operators~ be identified with HIB M (s and d bosons) if we drop the part which is not
U(6) invariant. By freezing the monopole degree of freedom HBM can be identified.
Though this is phenomenologically correct, in the present writer's opinion it is ss I
questionable physics because the q are reasonable operators to describe giant
quadrupole and monopole resonances. This is not the same as IBM or BM physics.
The method of symplectic decomposition (see the discussion of Kramer's lectures
below) was illuminated by application to a problem abstracted from QCD - a simplified
model of glueballso
In the main part of the lectures, the problem of decomposing a general "micro-
scopic" Hamiltonian, H, into a sum, H = Hcoll + H', where Hcoll is the collective
part~ was addressed. A definition of collectivity was offered: Collective coordin- ss v
ares are scalars in particle index space. It follows that q or equivalent are the
only collective coordinates. The remaining coordinates may be chosen as (3n-9)
"angles" in internal space, which is the space of the group 0n_ I. In the general
Hamiltonian written as H = T + V, the kinetic energy T is already a scalar in On_ I.
A general two-body interaction V is a general tensor in 0n_ 1 and Vcoll is the On_ 1
scalar part of this. Straightforward techniques for carrying out such calculations
have been developed.
Some successful applications of this program were described. Subsequently an
interesting way to include pairing correlations was discussed very briefly. The
basic idea is that these belong to the anticolleetive part of V. This is the part
444
which is obtained by averaging over the spatial variables rather than the particle
indices.
What emerges from these lectures is an original point of view toward the study
of nuclear spectra. It is fully microscopic in the sense that one starts with the
coordinates of n nucleons. However, so far the nucleon-nucleon interactions utilized
bear no relation to the two-body force in free space. Therefore Vanagas's theory is
at present no more microscopic than the conventional shell model, only a different
approach to the same problem.
F. P. Kramer. DyNamical Symme.try in Nuclear Collective and Reaction Physics.
These lectures were fully concerned with the theory and application of Sp(6,R)
as a dynamical symmetry group. The generators are written (here following Kramer,
we use i,J = x,y,z, s = l...n)
Lij = ~is~js - ~js~is
Sij = ~is~js + Sjs~is
QiJ = ~is~Js
PiJ = ~is~Js
(rotatlons)(3) ,
(surface vibrations) (6) ,
(deformations) (6) ,
(kinetic energy) (6) ,
a total of 21 generators. The quantities ~is~js , (no sums) are generators of
Sp(6n,r) (the most general linear canonical transformation of ~is and nl,s,).
(IIF.i)
The
reduction through Sp(6,R) (trace over particle indices) and through Sp(2n,R) (trace
over space indices) have a complementary structure indicated in the diagram displayed
below
Sp(6n,R)
S p ( 6 , R ) . . . ~ ~ - - ~ ~ ...- . . S p ( 2 n , R )
U(3) . . . ~ ~ U(n)
0 (3, R) "- ~ O(n, R) . (IIF.2)
The groups connected by dashed lines have complementary representations. This is
the analogue of the well-known space-spln relationship in atomic physics and the
space- (spin-isospin) relationship in nuclear physics, the latter related to Wigner's
supermultlplet theory.
This analysis is relevant when we assume the system of interest has Sp(6,R)
as an approximate dynamical symmetry. This generalizes Elliott's SU(3), an infinite
number of SU(3) IR's belonging to one IR of the non-compact Sp(6,R). Considerable
and elegant process was described in the mathematical formalism, but practical appli-
cations remain to be carried out.
A recent and potentially exciting application of Sp(6,R) to molecular resonances
445
was then described. Consider as an example the collision of two 160 nuclei, each
described by an IR of SU(3), h' = (hl'h2'h3') and h" = (hl"h2"h3") (they would be
equal in this particular case). Suppose that the relative motion is also described
by SU(3) with h ~m = (N,O,O). When the combined system is in the s-d shell, as in
the example chosen, it turns out that the IR's contained in h' x h" x h rm, sub$ect
to the Pauli principle, have no IR's in common with the SU(3) IR's of the compound
nucleus (32S in our example), provided we neglect the change in the oscillator fre-
quency. We suggest the name "group theory isomer". When the analysis is carried out
for Sp(6,R) representations, the same result is found even without the approximation
concerning the oscillator frequency. The following principle was therefore proposed:
Nuclear molecular resonances occur in symplectic reaction channels which do not match
lowest symplectic IR's of the compound system. A model calculation on 160 + 160 pro-
vides support for the hypothesis.
For the most part the model proposed is sitll in the kinematical stage. There
is a dynamical analogue found in Hartree-Fock and TDHF calculations where isomeric
states have been idantified.
III. GIANT RESONANCE STATES
A. F. Palumbo. The Scissors Mode.
An analysis of the two rotor model (TRM) was given which
leads to the picture of the neutron and proton densities
oriented at a finite angle to each other. In the lowest
mode (i +) this arrangement can rotate rigidly, but excited
vibrational states can also occur and each such state can
serve as handhead for a rotational band (not yet observed).
MI, E2 and E2 rates were computed and crude numerical estimates made for all quan-
tires.
Approaches to this excitation through the random phase approximation (RPA) were
also described. These included schematic (2 level) models as well as "realistic"
calculations. In addition to M1 strength in the neighborhood of the resonant state
(3.1 MeV in 156Gd), strength above 4 MeV has been predicted but not yet seen. The
IBM2 approach was also reviewed briefly.
In these approaches the scissors mode is predicted to be a purely orbital mode.
Experimental support for this view comes from the fit to form factors in the (e,e')
reaction and the fact that so far the mode has not been excited appreciably in (p,p')
at small angles where excitation is mainly through spin couplings.
B. J. Maruhn. Magnetic Properties of Heavy Nuclei.
In this lecture there was first a review of "ancient" history: In the mid '60's
Faessler suggested the scissors mode and he and Greiner showed that E(2)'s and
446
E(2)-M(1) mixing ratios of even low-lying spectra could be influenced by distinguish-
ing neutron and proton contributions.
In the main part of his lecture, Professor Maruhn described the development of
a BM-Frankfurt model which distinguished n's and p's. Let ~ , z be suitable aver-
of ~P~ and ~n which represent the old average quadrupole variables and let ~E~, ages
~ be corresponding difference variables representing the new degrees of freedom.
We write
H = Ho(~,-,r) + Hl(i,~) + H2(~,!,[,~) (IIIB.I)
A strong coupling solution of this Hamiltonian, suitably restricted, should be equiva-
lent to IBM2. But Maruhn reported that in contrast to IBM2, he needed a two-body M1
operator. This difference may be due to the fact that the Frankfurt people use too
restrictive a quadrupole operator.
C.A.A. Raduta. Description of the K = I + Isovector States with Generalized CSM.
Continuing the account of results with the CSM, for this problem a six band
description distinguishing n's and p's was utilized: The bands are designated g, 6,
(even under interchange of n and p) and ¥, i, 1 (antisymmetric under n+-~p). The
reason for using six bands is that experiments in the vicinity of 156Gd indicate 2
y bands and a low-lying 1 + band. The lowest state of the band designated 1 is iden-
tified with the scissors mode. A Hamiltonlan can be found which is almost diagonal
in this basis and even under the interchange of n and p. All parameters are adjusted
to the properties of g, 6, and y. Both the energy and M1 of the scissors mode agree
well with experiment and are thus predicted parameter-free:
IV. RARE DECAY MODES
A. D. Poenaru. Heavy Ion Radioactivity and Co~d Fissio N.
Professor Poenaru described and illustrated the successes of the Analytic Super-
asymmetric Fission Model (ASAFM) which covers the entire range between ~ decay and
the traditional domain of fission theory and predicts a tremendous number of new de
cays. The elements of ASAFM are (i) E(R) = the energy of two (spherical = cold) frag-
ratic function of R is chosen to interpolate.
ments as a function of R, the
separation of their centers of
mass. For R > Rt, where R t is
the touching radius E(R) is the
Coulomb energy. For R + 0 it
approaches, by choice of vertic-
cal scale, the Q value of the
decay. For 0 < R ~ R t, a quad-
(ii) The half-life T½ is given, in
447
where Q' = Q + Ev,
empirical formula
terms of elementary barrier penetration theory, in the form
T½ h2%n2Ev exp {~2 Rf~ [2~(E(R)-Q')]½dR , (IVA.I)
a
and the "zero-point vibrational energy", Ev, is represented by an
[ 4-A2]} (~VA. 2) Ev = Q {0.056 + 0.039 exp ~ 2.5~ '
Q > 0 and A 2 ~ 4. The factor Q takes into account shell effects. The formula for
E v was later refined by including additional factors [1.105 (ee), 0.947 (eo), 1,000
(oe), 0.789 (00)] for the odd-even effect.
B. A. Sandelescu. Fission Decay Modes with Compact Shapes, Deep Inelasti~
Collisions, Damping and Open Systems.
The theoretical basis for much of Prof. Sandelescu's lectures was considered to
be fragmentation theory with an underlying Hamiltonian
H = T(~,Dz,R,~) + V(~,~z,R,g )
Her e
(IVB.I)
= (AI-A2)/(AI+A 2) = mass asymmetry, (IVB.2)
Nz = (ZI-Z2)/(ZI+Z2) = charge asymmetry, (IVB.3)
R = separation of mass centers, (IVB.4)
E = neck coordinate (IVB.5)
The latter represents in a combined way the effect of all even multipoles higher than
two. In most applications friction must be included in (IVB.I).
To study this model, first consider an extreme simplification in which we de-
couple the ~ coordinate and study V(~) by the~trutinskyprocedure. In addition to
the usual fission minimum, one or more additional minima for superasymmetric fission
develop (with T½ reasonably predicted by ASAFM).
Next we fix n at a minimum and explore in the "~ direction" by investigating de-
formation parameters ~4' ~6'''" This leads to a new discovery - bimodal fission or
fission with compact shapes where both decay modes are magic or near magic~ As an
example the process
256Fm + 128Sn + 128Sn (IVB.6)
can occur either in the "hot" form with decay into elongated shapes or in the "cold"
spherical form. The probability for the latter is even larger than for the former.
Full, time-dependent solutions of fragmentation theory for deep inelastic colli-
sions were also mentioned. Here the masses are taken from cranking and frictional
forces of several types were utilized.
448
Another recent discovery by Professor Sandulescu and his coworkers was large
cluster transfer processes asia way of building heavy aetinides.
In a more formal vein a quantum mechanical theory of open (Markovian) systems
("Lindblad's master equation") was applied to a discussion of a harmonic oscillator
with linear dissipative coupling. In this theory there are constraints which, it
turns out, have often been violated in previous work (by others). It may be remarked
that one can always derive a theory of open systems from a theory of closed systems.
The result will not generally be Markovian, except approximately.
C. A. Faessler. Double Beta Decay as a Test of the Grand Unification Theory.
Double $ decay (sequential processes n ÷ p + e + ~) can occur in two ways: ~B
with the emission of 2 neutrinos (2~) is allowed in all existing models, but is severe-
ly reduced by phase space considerations. Neutrinoless B~ (~0~) in which the neu-
trino of the first step is subsequently absorbed, is possible only if the neutrino
mass, m # 0 and/or there are both left-handed (L) and right-handed (R) weak cur-
rents (L is dominant, remember, in standard electroweak theory). Both mechanisms
are operative in SO(10) grand unified theories, today's favorite class of theories.
The weak decay Hamiltonian can be written
G cos0 c [JL~ + H = ---- JL ~ ~JL~ JR ~ w 2~-
L+ }JR R + ~JR~ J~ ~J~ ] '
where 0 c
currents.
(ivc. l)
is the Cabibbo angle, J refers to the weak currents and J to the hadron
A calculation leads to the formula for (BB0~) ,
~(decay rate) = Im~ M m + nM n + AM~I 2 , (IVC.2)
whale the M's are matrix elements. This is an ellipsoid in the parameter space, A
sharp peak in the summed electron energy distribution is predicted,
Experiments performed on 76Ge have led to a bound ~< 5 x 10 -3 transitlons/see
yielding m < 1.7 eV, as well as bounds on ~, ~. These are still six orders of mag-
nitude larger than needed to test SO(10).
The contribution of the Faessler group to this problem was the recognition that
relativistic contributions from J, in contrast to usual B decay calculations, domin-
ates because the intermediate ~ energy is relativistic.
The writer is indebted to Prof. A. A. Raduta and his colleagues for the warm
hospitality extended to him during the Brasov summer school and for financial support.
He is indebted as well for support from the NSF International Programs Office and to
the U.S. Department of Energy for support under grant number 40264-5-20441.
SUMMARY TALK: EXPERIMENTAL
Pe te r von Brentano
Institut fiir Kernphysik, Universitat zu Kdln, FRg
Topics:
I. Models versus data, 2. New states, 3. New M1 r~ode, 4. Chaos,
5. Fragmentation, 6. Nuclear equation of state and reactions,
7. Fundamental Physics, 8. Ending remarks
I. Models versus data
We heard talks by Iachello on I~M and dynamic s~tries, by Raduta on CSM, by
Barfield on f bosons, by Brentano on F-spin and global IHM fits, by Sunko on
super~try in odd-odd nuclei, by Iachello on groups and scattering and by
Faessler on excited VAMPIR calculations. Furthe/TaDre, models with nearly no
comparison to data were discussed by: Ceausescu, Dumitrescu, Klein, Kramer,
Raitchev, P~tter, Vanagas and Yaraazm/l-a.
let us discuss first the common ground of these different talks. These talks
discussed various substructures of the nucleus. These substructures of the nucleus
correspond to the substructures of the human society. So we will cc~e the
nuclear and the him%an societies in the following table.
Nuclear society Human Society
mean field
208pb, 132Sn core
quartets, alpha clusters
bosons: s, d, g, p, f
proton and neutron bosons, F-spin
protons and neutrons, Isospin
blue, red and yellow quarks
co~t~
city
(2 couples)
"gay" couples, 2 male or 2 female
future family of 3: 2 parents and 1 child
blue, red and yellow people
These subjects have been discussed in the following talks. Meanfield: Ceausescu,
Faessler, Klein, Kramer, Vanagas. 14C, Pb, Sn clusters: Hourani, Poenaru,
450
Sandulescu, silisteanu and boson talks. Alpha clusters: Dumitresu. Bosons:
Barfield, Brentano, Iachello, Maruhn, Raduta, Raitchev, Sunko and Yamamura.
At the beginning we had a beautiful talk by Iachello on the Interacting Boson
Model (I~4). This model is now universally acknowledged and used in nuclear
physics, so Iachello gave a lecture on it in the way in which one lectures at a
school, using the black board and only a few transparencies. I think everybody
including the professors enjoyed his clear presentation. He discussed in particular
the application of the ooncept of dynamical ~tries to the description of
nuclear spectra. Originally this concept was used in particle physics to solve an
old problem in quantum mechanics discussed in Dirac's book. The systematic
construction of a crmplete set of commuting observables, which we need for the
solution of quantum mechanical problems. These oc~1~u~ting observables are found to
be the Casimir operators of the subgroups in a complete chain of subgroups of the
dynamical synm~try group, lachello also shewed a number of applications. For
example the description of many excited bands of 156Gd by the SU(3) dynamic
syr~etry of the I~q. Of interest are several bands in the range between i. 5 and 2
MeV. However, more data is needed to confirm his SU(3) interpretation of these
bands.
After Iachello we heard Raduta discussing his coherent state model (CSM) which
is very much related to the interacting boson model and to the TQM model. Its great
virtue is its capability of reproducing rotational bands up to rather high spins.
This model can describe a variable moment of inertia which cannot be done in the
usual IH~I model with two boson interactions. The reason for the success of the CSM
model is probably the use of a special three-boson interaction. But it is certainly
ir~pressive that Raduta can fit 3 bands in the nucleus 232Th; with 4 parameters he
fits a J=30 state with an error of 10 -3 . Unfortunately not so many data of this
kind is available because usually the backbending phenomenon already begins at much
lower spins. It would be very interesting to make a detailed ~ison of the CSM
model with the interacting boson model. It is very useful to translate the various
theoretical model languages into the language of the other models, because we can
learn a lot in this way. This is true even though the models are usually not fully
isomorphic in a mathematical sense.
The next talk by Ariel Barfield concerned the application of the interacting
boson model to odd parity levels by including f-bosons. She showed us that a number
of negative parity bands in 168Er can be described in the I~ with an f-boson.
However, there are several quasiparticle bands with negative parity in 168Er
which are outside the IHM model space. Her criterion, which bands are in the I~M
f-boson space, is that these are the bands with a large B(E3) to the ground band.
It would be interesting to add a p-boson to her work and see what happens.
Then my talk discussed the proton neutron-synm~try of collective modes and
451
F-spin quantum number. F-spin is the isospin in a boson system and I must say that
the name F instead of T has given rise to many misunderstandings. So let me remind
you F stands for Franco. The F-spin ooncept seems to be a very useful concept to
describe many different phenomena in one language. We find direct evidence for
F-spin in the form of F-spin multiplets with rather constant energies in F-spin
multiplets. The F-spin concept also leads to a global description of many nuclei
with constant parameters with an II~M-2 Hamiltonian. Furthermore it leads to an
understanding of the Np-Nn-SCaling parameter introduced by Casten. It is of
particular interest that the F-spin is not only a useful classification scheme but
seems to be a reasonably good quantum number for the ground and gan~a bands of
heavy nuclei. From Harter's calculations I have quoted an upper limit of F-spin
admixtures of less than 4 % for 168Er and 128Xe.
Next there was a talk by Sunko on supersynm~try in edd-odd nuclei. This concept
of supersy~netry is one of the fundamental ideas of the recent particle theory; yet
there is no convincing comparison to experimental data in this field. Thus the
transplantation of the supers3amaetry concept by Iachello to nuclear physics, where
there are a lot of data, was a very i~oortant step. This concept, however, remains
a little bit formal and it is difficult to understand the physical basis of
supers~try in nuclear physics. There has been an important progress in
understanding of the supersy~aetry concept by the Zagreb people. They showed that
supersyr~netry means a strong similarity of the wavefunctions of even, odd and, as
generalized by Sunko also odd-odd systems.
Now I come to Iachello's talk on group theory and scattering. In this talk he
discussed methods by which the scattering matrix could be obtained directly from
group theory. He showed us a quite general solution for elastic scattering of spin
zero particles. As an example he discussed a fit of 28Si + 160 elastic
scattering data. This method is a very exciting and flandamentally new approach. If
I were a student in the audience, I would think very hard whether not to join in
this enterprise, by fitting data or by doing proper theory.
But let us discuss a doubt: If we compare Iachello's fit of the 28si + 160
data with the superb fit shown to us by Dr. Khoa from Dubna to the extremely
accurate alpha-scattering data by Rebel and Co. from Karlsruhe, I would doubt that
Iachello can ever get a fit with such a small Chi square, but I think that's not
his aim. The aim is to decribe not only elastic data but also to get inelastic
scattering and reactions in multi-step excitations in one model, which ccmiorises
structure and reactions. This would be a marriage of nuclear structure and nuclear
reactions and it could well change the field, clearly such an approach will be most
valuable in cases where very many coupled channels need to be taken into account.
452
New states
This subject was discussed in talks by Borcea, Dragulescu, Lister and
Ionescu-Bujor. The talks on the new M1 mode and on giant resonance will be
discussed in the next section.
The first talk was by Lister and it started with the discovery of the 21 +
and 41 + states in 80Zr with the new Daresbury recoil mass seperator. This is
impressive new instrument, made a big discovery in its first use. We have got used
to hear a lot about Tessa i, Tessa 2, Tessa 3 and soon from Euro Tessa 30 and we
will get used to hear exciting new data from the Daresbury recoil mass separator. The
interest in 80Zr goes back a long time. I remember that in the late sixties
Metag, Repnow and myself did some calculations with the Strutinski method and we
were very excited because 72Kr came out to be oblate. There is still no proof for
this although many valiant efforts to prove it have been done by many people,
including us with OSIRIS. But Lister is now clearly ahead because he found a method
to observe extremely weak gamma lines with tiny cross sections of 10 -5 barn. This
enabled him to find the 21+ and 41+ states in 80Zr. He believes that 76Sr
and 72Kr could be investigated much mere easily using his trick, because they are
populated with a larger cross section. That the spectaum of 80Zr was very
difficult to observe has been known for a long time. In the early seventies Paul
Kienle created the Paul Kienle price for 80Zr consisting of a number of champagne
bottles. When I tell Lister this secret I hope he cuts me in when drinking the
champagne. Another interesting point was that the Casten scaling parameter NpN n
seems to give a proper prediction of the 80Zr spectrum from the known 78Sr
spectrum. The point is that 78Sr and 80Zr look nearly the same and they
have the same value of Np N n. Then Lister talked on light rare earth nuclei and
he shewed us that they become as deformed as the heavy rare earth nuclei. In
particular this concerns 128Ce and 128Nd. Another interesting question concerns
the Grodzins formula which relates the 2 + energies and the B(E2) values and seems
to be surprisingly successful.
The next talk was by Manuela Ionescu-Bujor on the meas~t of a quadrupole
moment of a 21/2- isomer in ll7sb. This measurement has been done at the
Bucharest cyclotron by the hyperfine group, which are long known to be experts in
their field. The interesting point of this measurement is that a while ago the
Stony Brook group found a deformed 9/2 + band in the odd Sb isotopes. Recently it
was shown by the Amsterdam group that a corresponding 0 + rotational band exists
in the nuclei ll2sn - ll8sn. There has always been the question whether these
"rotational" states are really deformed or whether they are spherical and have a
rotational spectlna~ for some strange reasons. New this question is settled through
the new exper~t by the Bucharest people. As the groundstate of the magic Sn
453
nucleus is spherical this is a beautiful case of shape coexistence and for the
deformed band in the Sn and Sb nuclei a deformation parameter beta = 0.2 is found
So we learned a lot about exciting new states in heavy nuclei. However, there
are also interesting news and progress in light nuclei. We learned from Borcea of
the new work in Dubna on the tri and tetra neutron and on 4H - 6H. He presented
a few spectra of excited states in 5He, 6He and 7He which showed us that the
first excited state in 6He can now be studied with statistics of a few i000
counts. This shows that these states can now be studied very well and we except
interesting results.
New M1 Mode:
The new collective M1 mode has ever been at the center of interest in nuclear
physics since a prediction by LD Iudice and Palumbo in 1978 and its experimental
discovery by Bohle and Richter 1983 in Darmstadt. This was an succesful experiment,
and the mode has r~any other parents: I~-2, geometric models and so on. We heard
talks by Faessler, Maruhn, Palumbo, Raduta and me on this mode. Unfortunately Dr.
Bohle from Darmstadt could not come. The picture of the new mode which is in
everybody's mind is certainly Palumbo's two rotor model, where the proton and the
neutron fluids oscillate against each other, qhis model gives the mode the shape of
a scissor thus it is called the scissor mode. Palumbo discussed in detail its
properties and compared then with the data. It is found that these states lie at an
energy around 3 MeV in many rare earth nuclei and such states have also been
identified in the actinide nuclei. A strong M1 transition of several UN 2
strength to the ground state has been observed. These states are interesting in
many respects, and there has been an enormous amount of papers since their
discovery and many theorists tried to put this 1 + mode into their bag.
The M1 mode was discussed further by Maruhn in the framework of the
proton-neutron version of the Bohr-Mottelson model which we could call H~4-2 model
or also Greiner-Gneuss model 2 (GG2) or even Marlhhn and Maruhn model 2 (MM-2).
Marnhn pointed out that several calculations on E2/MI mixing ratios and gamma
branching ratios from F = Fma x states were already done by Greiner and himself in
the sixties and seventies although not much data were available at that time. He
also presented some new calculations of this kind. If one ~ these
calculations with similiar ones done recently by Harter in the framework of the
interacting boson model which I have shown, it is surprising that the interacting
boson model uses a one boson T(MI) operator, whereas Maruhn uses a 2 boson
operator. The origin of this difference is still not clear.
Then Raduta talked about the proton-neutron generalisation of the coherent
state model which could be called CSM-2. It is remarkable that he can get the
454
energy of the mixed symmetry 1 + level from the energies of the fully symmetric
states. This is different from the IE~4-Model and we need to understand the reason.
Finally Faessler came with monsters and vampires, very fitting as we were so
near to Dracula's Bran Castle. The development of the MONSTER, VAMPIR and as the
last entry the EXCITED VAMPIR models is quite an important progress. Because it
allows calculations in the frame of a shell model approach for problems, which
where believed to be accessible only to various phenomenological collective models.
In particular the EXCITED VAMPIR model promises help in understanding the shape
coexistence phenomenon in nuclei. His model is much more comprehensive than the
collective model and Faessler used it to calculate many more 1 + states. A
particularly interesting finding of Faessler and Co. on the M1 mode is that in
46Ti the lowest 1 + state has 50% orbital and 50% spin excitation. This,
however, is in contrast to the collective models which treat the 1 + mode as a
purely orbital excitation. Inelastic proton scattering experiments done by the
Kdln, Darmstadt and ~ulich collaboration and by the Darmstadt Saclay collaboration
seem to favour, however, a dominant orbital excitation for the Ml-mode at least in
heavy nuclei. This is certainly a very crucial question and it needs more study.
More emphasis should be put on a comparison of various theories. In this
respect the interacting boson model IE~4-2 seems to be the most simple theory and it
has the concept of F-spin. So it is interesting to ask wether there is also a
quantum number like F-spin in the other theories. In this respect Faessler remarks
that F-spin can be introduced in a modified way also in his geometric model.
Now I come to the talk by Bohigas on chaotic behaviour in nuclear physics. This
subject of chaos is a new and very interesting field which has been developed the
last ten years. This field doesn't seem to require expensive equipment for its
study, but its problems are quite intricate. The basic observation of chaos physics
is that most observables are chaotic. The definition of quantum mechanical chaos is
a difficult problem. A crude definition of a chaotic system is the following one:
Describe an observable by a Hamiltonian H and some boundary values B. Now you look
at the change of the observable when you do a small change in H and B ( say a
10 -2 change ). Then an observable which shows a drastic change in its value is
called a chaotic observable. The amazing discovery is that almost all properties
and observables are chaotic. Examples are nuclear spectra at an energy around 7-10
MeV, the weather in 14 days in Poiana Brasov and quite generally our future. So the
chaos physicists say that even in classical physics God cannot know the future. And
this is quite a change in opinion when you ~ e this statement to the famous
statement by Laplace on the existence of a world spirit, who knows everything: past
and future. The word god was not considered philosophical enough at the time of
Laplace. In nuclear physics we have certainly many clear examples of chaotic
observables. A most obvious example is Ericson's fluctuations of cross sections of
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statistical nuclear reactions. Bohigas showed us that the energy spacings of
neutron resonances are also a chaotic observable. He described the probability
distribution of the normalized energy spacings between neutron resonances by random
matrix theory with the so called grand orthogonal ensemble GOE. It is remarkable
how good this theory describes the probability distribution of the spacings and its
various moments. Thus the nuclear spectrum at an excitation energy of ~7 MeV is
chaotic. The low lying collective levels are, however, clearly non chaotic, and as
Bohigas and Weidenmftller stressed there has to be a transition from order to chaos
at some energy below 7 MeV. Now the an interesting problem is the determination of
this transition energy especially for odd-odd nuclei. These were discussed in the
supersynm~try model by Shnmko, and a band masurement in 134pr was presented by
Dragulescu. Dragulescu found bands which have large connecting B(E2),s which is
clearly a collective, non chaotic behavior. But the question of chaoticity in
odd-odd nuclei is still far from beeing settled and there is still much more work
done in order to answer it.
Fraqmentation
We now cc~e to the subject of fragmentation which has been discussed by Hourani,
Sandulescu, Poenaru, silisteanu, Brancus, Petrascu and others.
At the beginning we had a talk by Hourani on experiments concerning the new
magic radioactivity as it's called by Sandulescu or more prosaic on heavy cluster
emission. Having known the alpha decay now for nearly i00 years it came quite as a
surprise when Rose and Jones published in Nature in 1984 evidence for 14C decay
from the ground state of 223Ra. Now clearly one can't just say that this heavy
cluster decay mode has been carelessly overlooked. If we compare the 14C to alpha
branching then it has a value of i0 -I0. So Rose and Jones looked for a really
small branch. They used a delta-E/E telescope and they had to collect data for 190
days and 4 times during the experiment they had to replace their Si-detectors
because they had been damaged by the alpha beam. So after their experiment there
was still some room for doubt which, however, was quickly settled by Hourani and
company at Orsay who used a magnetic spectrograph and a much stronger source
finishing the experiment in 5 days. Subsequently quite a number of other heavy
cluster emitters were discovered, I mention 222-226Ra. Recently also 24Ne decay
has been di~=movered by Price et al in Berkley and by Sandulescu et al in Dubna. Now
it is rather interesting that the theory of the heavy cluster decay was formulated
already in 1980 by Sandulescu, Poenaru and Greiner published in the soviet Journal
of Nuclear and Particle Physics. But Hourani reminded us that these authors did not
tell about the most probable decay and Rose and Jones had to go on their own.
Probably they didn't know this paper anyhow. After the discovery of Rose and Jones,
456
theorists could renormalize the theory to the 14C decay data and now the theory
has a very good predictive power. A new paper by Poenaru, Ivascu, Sandulescu and
Greiner was published in 1984, and there is also a theory from Shi and Swiateczki
in Berkeley.
The in~oact of the discovery of heavy cluster emission from the ground state has
been quite large. In particular it lead people to realize that many phenomena which
until now had been treated by different theories were all aspects of nuclear
fragmentation. This is surmnarized in the following table.
Nuclear Fraqmentation
cold fragmentation: all nuclei at T = 0 (g.s)
hot fragmentation: some nuclei at T ~ O
binary fraqmentation: A ...... > C+D
spontaneous fragmentation: T A = 0, induced fragmentation: T A ~ 0
examples: p, n decay, alpha decay, 8Be, 14C, 24Ne decay,
magic radioactivity, cold fission, fission.
binary reactions: A+B- ..... > C+D TC, T D ~ 0
examples: rearrangement reactions, deep inelastic scattering, heavy ion
fission.
multi fraqmentation reactions: A+B ...... > CI+C2...+C n
examples:
fusion reactions (xn, yp), inccm%olete fusion reactions, partition reactions.
We notice in particular that alpha decay, cold fission and magic radioactivity or
heavy cluster decay are essentially the same thing: cold nuclear fragmentation. One
further notes that if some of the fragments are excited then friction becomes
important so one needs a theory which incorporates friction, but that,s not so
easy. If we look at the predictive power of the new cold fragmentation theory by
Sandulescu, Poenaru, Greiner and Ivascu then it predicts the cluster to alpha
branching ratios within a factor of ten cc~pared with the experiments. And that's
quite good. We heard a number of interesting predictions for new 14C, 24Ne
cluster decays, on which several groups work intensively. However, these
experiments are at the limit of what can be done. Once the idea of cold
fragmentation had been formulated, it's not surprising that people realized that
the cold fission, which was found by Clerk, Armbruster and others at the ILL
Grenoble, is also an exanple of this phenomenon. A calculation was shown which
457
predicts correctly lOOzr as the most probable decay product in the cold fission
of 233U+n. Another rather interesting aspect which was discussed by Poenaru and
Sandulescu is the so called bimodal fission which has been discovered at Berkley
laboratory by Hoffmann et al. What is bimodal fission? If you look at the fragment
distribution in normal fission you get a doubly hunloed mass distribution. For
264Fm which is 132Sn + 132Sn you find a single peaked raass distribution in
fission. But the fission process wants to be doubly humped, so if the mass
distribution becomes singly humped you find the distribution of kinetic energies to
be doubly humped instead, and that is what is called b~al fission. It's
interesting that various fission theories seem to be able to describe this
phenomenon, and this is quite a progress. So alpha decay, heavy cluster decay and
cold fission are all aspects of cold fragmentation. For hot fragmentation one has
friction which conplicates the theory very much. Friction is ir~0ortant in deep
inelastic scattering, and beautiful example was shown in the talk by Iliana Brancus
who discussed an 19F + 2~Mg deep inelastic scattering experiment. Most of you
have not heard her talk as it was scheduled during the soccer matcll Romania -
Austria. Now we did not have many talks by women although the audience was 50/50
men and women. And it was clearly only sc~e/x~y who is wise enough not to be
interested in football who should talk during the game. So it was quite logical for
the organizers to schedule a woman at the time of the game and so few of you heard
the talk. She showed the variables which characterize the deep inelastic reaction;
and the mass distribution of the fragments. This is a case for hot fragmentation
theory and one has to include friction which is done at present by the
Fokker-Planck approach, very different from cold fragmentation.
Nuclear equation of state, con~oressibility (relativistic heavy ion
reactions and qiant monopole resonances)
On these subjects we heard talks by Bock, van der Woude, Rebel and Jellito.
Bock talked about the nuclear equation of state studied in central relativistic
heavy ion collisions. This subject has attracted much interest in recent times,
through the efforts of the GSI-Berkley collaboration.
If you shoot 2 heavy ions onto each other you can increase the density in the
central part of the collision and thus ccm~0ress the nuclear fluid. Some of the
total energy is converted to kinetic energy, that is temperature T, and if one
is able to measure the temperature T with a nuclear thermometer then the nuclear
equation of state can be determined. Now the relativistic heavy ion people found a
nuclear thermometer in the form of the pion multiplicity. This thermometer seems to
work quite well as was shown to us by Bock who coa~ the pion multiplicities
from various heavy ion reactions. Bock showed us examples of measurements of the
458
nuclear equation of state with nuclear densities varying from 1.7 to 4 times normal
density. Everybody was very i~pressed by this beautiful kind of work. There is,
however, a small snag. The snag is that one can also obtain information on the
nuclear compressibility from a study of the breathing mode of nuclei which is also
called the giant monopole resonance, and these numbers do not agree.
So let us first look at the experiments on giant monopole resonances, which
were discussed in the talk by van der Woude. As he told us there recently has been
an izloortant progress in this field. Namely the study of giant resonances is
usually plagued by large backgrounds, and in particular the giant monopole
resonance was barely visible as a small peak on a huge background. A new trick
ir~proved the situation. The trick is to substract spectra at an angle of 0 ° and
at an angle of 3 O, and it is then found that the giant monopole resonances become
much more visible and the background is diminished very much. This trick is used by
several laboratories, and beautiful work from Groningen was shown to us. At this
point Rebel added an i~0ortant comment. He told us that if you use a 6Li beam
instead of an alpha beam the background is reduced furthermore and you can see the
monopole mode even without the subtraction trick. So if both methods are combined
one clearly should get perfect data.
Already with the present experiments it is possible to get from the data very
detailed information on the nuclear con~oressibility comparing the energy of the
monopole resonances in various nuclei throughout the periodic system. The value
which comes out of this analysis is Knm=270_+15 Mev. These are quite
accurate numbers. Our respect for these numbers increases even more if we consider
that there are only a few numbers which characterize nuclear matter. These are the
nuclear density, the binding energy per nucleon, the nuclear compressibility and
very few others. Clearly we do not need only the nuclear compressibility but we
would like also to determine its dependence on (N-Z) and on A, which has also been
obtained as discussed by van der Woude. In order to get these dependences very
accurately one needs accurate values of the menopole resonances for many nuclei and
I hope that these experiments will be done soon. ~ i n g Knm from the giant
menopole resonance with that from the relativistic heavy ion experiments the two
values disagree as mentioned above. In principle this doesn't need to worry us too
much because the giant resonances measure the nuclear conloressibility at the normal
density, whereas the relativistic heavy ion experiments measure the density at 3
to 4 times the normal density. But this problem still needs a lot of attention. As
the analysis of the relativistic heavy ion experiments is quite involved, much more
experimental and theoretical work is needed in this exciting field. Finally I would
like to mention that van der Woude gave us a beautiful ~ of what is
experimentally known on the various electric giant resonances, and we learned that
for the El, E2 and E3 giant resonances a lot of data are available and we start to
459
get information even for the E4 giant resonances.
Fundamental Physics:
On fundamental physics we heard a talk by Faessler on tests of grand unified
theory (GUT) from double beta decay experiment. He told us that the neutrinoless
double beta decay is a very useful tool for such tests. The experiment is very
appealing. You just take a germanium detector, shield it and look at the total
electron spectrum from the 76Ge decay in the detector. That's all. The nice
aspect of the neutrinoless decay is that it gives a sharp line in the spectrum
whereas the usual double neutrino double beta decay produces a continuous
background spectrum. So one is much more sensitive to the neutrinoless double beta
decay and one gets very large lower limits on the lifetime of this decay mode. Now
from looking at those lifetimes Faessler obtained very strong limits on various
parts of the weak interaction. He got a value smaller than 1.7 electron volt for
the neutrino mass and he got an upper limit of the right handedness parameter eta =
10 -8 for the weak interaction. This is an important steps toward a test of the
grand unified theory, which predicts eta to have a value of around 10 -12 . Clearly
the present experiment is far from this and even if you could produce a germanium
detector out of pure 76Ge one probably would not reach the neccessary
sensitivity, although it would be a challenge to produce such device. But there are
some other ideas how such an experiment might be done. This is certainly a very
exciting field.
Ending Remarks
Now I come to the most pleasant part of my talk. As the last foreign speaker I
would like to speak for all foreign participants and thank at first the speakers
for clear presentations, the audience for lively questions and patience during our
14 day school and finally the organizers particulary Prof. Ivascu for making all
this possible, some of the organizing team worked extremly hard even during the
conference, I would like to mention Prof. Raduta, Dr. Zamfir, N. Sandulescu and
last not least Mrs. ~etta Uglai. To them we give our particular thanks. Personally
I want to thank the nearby mountains for giving us so welcome recreation in the
afternoons.
S E M I ! b A R S
(Published in Revue Rou.~,aine de Physique, nos.5-6,1987)
Unif ied d e s c r i p t i o n of p o s i t i v e and negat ive p a r i t y s t a t e s in the
deformed heavy nuc l e i by means of two i n t e r a c t i n g vector bosons,
P. R a i t c h e v
An IBM d e s c r i p t i o n of oc tupole bands in deformed nuc l e i ,
A.F. ~ar f ie ld
High spin spec troscopy of Pr nuc l e i ,
E.Dragulescu, M.Ivascul M.lonescu-Bujor, A.lordachescu, C.Petrache, D.Popescu, G.Pascovici, G.Semenescu, l.Gurgu, F.Baciu, R.A.Meyer, V.Paar, S.Brant, D.Vorkapic, D.Vretenar, H.Ajaraj
A deformed high spin i someric s t a t e in I17sb,
M.lonescu-Bujor, A.lordachescu, G.Pascovici and C.Stan-Sion
Experimental s tudy of mu l t ineu t ron systems and heavy i so topes of H and H 2
C.Borcea, A.V.Belozyorov, Z.Dlouhy, A.M.Kal inin
I sosca lar and i s o v e c t o r g ian t resonances in the gas -d rop l e t model
V.Yu.Denisov
S t a t e d e n s i t i e s for f i n i t e s i n g l e p a r t i c l e c o n f i g u r a t i o n s ,
K.W.Zimmer and A .Ca ' Iboreanu
A l p h a - l i k e four nucleon c o r r e l a t i o n s in the s u p e r f l u i d phases of atomic
nuc le i , M.Apostol, l .Bulboaca, F.Carsto iu, O.Dumitrescu, M.Horoi
On the ra tes of r a d i o a c t i v e decays by emission of heavy c l u s t e r s ,
M.Ivascu, A.Sandulescu and l .S i l i s t eanu
Cooperative e f f e c t s in nuc l e i and s e l f o r g a n i z a t i o n , l .Ro t te r
Approximate ampli tude for the s c a t t e r i n g with e x c i t a t i o n of the r o t a t i o -
nal s t a t e s . General ized Blair phase s h i f t ru l e , N.Grama
Exchange e f f e c t s in e l a s t i c and i n e l a s t i c alpha- and heavy-ion s c a t t e -
r ing , Dao Tien Khoa
A new c lass of r e s o n a n t s t a t e s , C.Grama, N.Grama, l .Zamfirescu
Nuclear molecular dynamics approach to the nuc leus-nuc leus p o t e n t i a l ,
E . B e t a k
Fragmentation processes in mul t inue leon t r a n s f e r r e a c t i o n s , M.T.Magda, A.Pop, A.Sandulescu
461
P r o j e c t i l e f r a g m e n t a t i o n in heavy ion r e a c t i o n s a t low energy ,
A.Pop, M.Cenja, M.Duma, R.Dumitrescu, A.Isbasescu, M.T.Magda
S p e c t r o s c o p i c a m p l i t u d e s fo~ a nuc l eon t r a n s f e r b e t ~ e n e x c i t e d
s t a t e s of Ip - s h e l l n u c l e i , L.Kwasniewierz, J .K is ie l
I n t e r m e d i a t e mechanism i n t h e 19F ÷ 59Co r e a c t i o n ,
M.Parlog, M.Duma, L.Trache
Gross p r o p e r t i e s of t h e y i e l d s of t h e charged p a r t i c l e s e m i t t e d from
t h e 19F ÷ 59Co r e a c t i o n ,
M.Parlog, M.Duma, E.lacob, D.Lazarovici, D.Moisa, L.Trache
Measurements of l i g h t p a r t i c l e e m i s s i o n a t v e ry forward ang le s i n 6Li
i nduced n u c l e a r r e a c t i o n s a t 26 MeV per n u c l e o n ,
H.Je l i t t o , J .J .G i l s , H.Rebel, S.Zagranski
Dynamical c a l c u l a t i o n o f deep i n e l a s t i c i n t e r a c t i o n i n heavy and
l i g h t c o m p o s i t e s y s t e m s , l .MoBrancus , A . C o n s t a n t i n e s c u
Pion p r o d u c t i o n in high energy n u c l e u s - n u c l e u s c o l l i s i o n s ,
C.Besliu, A.Jipa, A.Olariu, R.Topor Pop, V.Boldea, L.Popa, V.Popa
V.Topor Pop
~ependence of average c h a r a c t e r i s t i c o f 7- mesons on number of
i n t e r a c t i n g pro tons in n u c l e u s - n u c l e u s c o l l i s i o n s a t 4.2 GeV/c per
n u c l e o n , L.Simic, S.Baskovic, H.Agakshiev, V.Giskin, T.Kanarek, E.Kladnitskaye, V.Boldea, S.Dita
