I i Nuclear Physics A146 (1970) 1--14; (~) North-Holland Publishing Co., Amsterdam 2.E I

Not to be reproduced by photoprint or microfilm without written permission from the publisher

DERIVATION OF THE RADIAL DEPENDENCE OF THE

QUADRUPOLE FORCE FROM A WOODS-SAXON POTENTIAL

K. K U M A R *

The Niels Bohr Institute ** University o/ Copenhagen, Copenhagen, Denmark

and

B. SORENSEN ti t

Lawrence Radiation Laboratory, Department of Chemistry, University of California, Berkeley, California 94720

Received 22 September 1969

Abstract: For use in calculation of collective quadrupole phenomena we construct a modified quadrupole force, whose radial dependence is related to that of a Woods-Saxon well. Matrix elements and self-consistent interaction strengths are compared to those of the usual quadrupole force, which is based on a harmonic oscillator potential.

1. Introduction

The radial dependence of the usual quadrupole force, which has been widely applied to studies of collective motion in nuclei, is obviously unrealistic in its qua- dratic raise with radius, as pointed out, e.g., by Baranger and Kumar 2, 7).

A more realistic evaluation of the radial form of a multipole force was some time ago suggested by Bohr and Mottelson (unpublished lecture notes and ref. 4)), and by Rowe 9). This method is based on a self-consistency argument relating the radial form factor of an arbitrary multipole interaction to a realistic average potential, say of Woods-Saxon type.

We here extend the method to include also a spin-orbit term in the average potential and make a detailed comparison between matrix elements of this modified quadrupole interaction and those of the usual one.

It is finally pointed out that the renormalization of the interaction due to truncation of the configuration space is likely to take a form different from the one assumed in connection with the usual quadrupole force, which corresponds to harmonic oscil- lator single-particle wave functions, since the modified interaction requires Woods- Saxon wave functions both for bound and continuum levels.

* Present address: Oak Ridge National Laboratory, Oak Ridge, Tennessee, USA. t* Work performed under the auspices of the US Atomic Energy Commission.

tet On leave from the Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark (present address).

1

May 1970

2 K. KUMAR AND B. SORENSEN

2. Self-consistent multipole force and average potential

Let us first consider a separable multipole interaction in general,

V(rl z, , r2 22) = -zP,,(r,)P,2(r2) ~ Y*,(O, d?,)Yxu(02 d?2), (2.1) ~t

where (rOd?)are position coordinates and z is an isospin label giving the three-component (z = - 1 for protons). Assuming the nuclear average potential in which a proton or a neutron will be moving to arise entirely from the two-body interaction, it has to satisfy

V~(r) = ~ V(rz, r'v')p~,(r')dr', (2.2)

where p+ 1 is the neutron (proton) part of the density function

p(r) = ~2 p~(r). (2.3 t

Note that, although we do not assume identity between the proton and neutron average fields, we have by (2.1) in fact assumed that the two-body interaction is isospin-independent (which is the conventional choice in the type of application we are thinking about) 1- z).

We now impose a deformation of multipole order 2 on the system, assuming the surface to be of the form

r = r'(1 + Z ccau Y~(Od?)), (2.4) /t

with r ' equal to the spherical radius Ro. Provided now that the interaction (2.1) is of sufficiently short range, the surface will adjust itself completely to the shape of the average field, enabling us to relate the equipotential surfaces of the deformed average potential V~ to the spherical average potential V~ ph which was present before the interaction was switched on (assuming 2 # 0),

V~:(l'Od?) = vsPh(r'), (2.5)

where the relation between (rOd?) and r' must be given by (2.4). If the deformation implied by applying the residual interaction is small, we may retain only the leading order term in a Taylor expansion of V~Ph(r ') around the actual radius r at a given direction (0d?):

vsph( r ' ) = g s p h ( r ) - - F 0vsph(? ') Z (X~.# Yff~t(Od?). (2.6) dr u

Comparing (2.6) to (2.1) and (2.2), we get for 2 # 0

~ vsPh(r) P,(r) = r , (2.7)

Dr

z f ~ e~'(r')Yzu(O'd?')P~,(r')dr' = ~ . , (2.8)

RADIAL DEPENDENCE OF QUADRUPOLE FORCE 3

which is unique up to the choice of a scale factor. Eq. (2.7) tells that P,(r ) does not depend on 2.

When the deformation parameters ~-a, correspond to equilibrium, there is pro- portionality between the average field V~ and the density distribution p~,

V~(r) = c~p~(r), (2.9) implying, in analogy to (2.6),

p~(r) = psPh(r)-- r 63pSph(r) 2 0~;., Y~(O(o). (2.10) 0r u

Inserting (2.10) into (2.8), we get ]_1 Z = -- r 40v~Ph(r) t~pS*Ph(r) dr , (2.11)

0r 8r

which is called the self-consistent value of the coupling strength X [refs. 4, 9)]. The proportionality factors c~ can be found by integrating (2.9),

cn = - V~Ph(r)r2 dr,

r V~Ph(r) rE dr. (2.12) ¢P ~" Z

2.1. THE USUAL QUADRUPOLE FORCE (UQF)

The standard choice for the spherical average potential 1) has been the harmonic oscillator potential (rl

VS'h(r) = ½ho~ o ~ , (2.13)

which leads to a quadrupole force (2.1) with 2 = 2 and the radial form factor (2.7) given by

(J P(r) = hogo (2.14)

As it is, however, nonsense to assume the proportionality (2.9) between (2.14) and the density, one usually rewrites (2.1 l) as a sum over all the particles in the system

X = 4--~ k r2k Or---kk Or k ]

and assumes

which implies

( 2 r2 ) = yzx3"l~'2.0, (2.16) k

X - l - 3 ho9 or 2A~ ' (2.17) 4n b 2

where R o = r o A + is the nuclear radius.

K. KUMAR AND B. S~RENSEN

2.2. THE MODIFIED QUADRUPOLE FORCE (MQF)

The modificat ion of the quadrupole force which we are suggesting assumes the average potent ial to be of Woods-Saxon type 5), including a C o u l o m b par t and a spin-orbit part . The isospin dependence is chosen in accordance with the symmetry energy contr ibut ion to the nuclear binding energy

v~Ph(r) = ½(1 - - z )Hc( r )+W,{ f ( r ) - -½ v .... 2 2 ( p x s ) • V/(r )} , (2.18)

where

( f ( r ) = l + e x p , (2.19)

He(r) = zZe 2 6(r __> Ro)+ ~ ~-½ ~(~ < Ro) , (2.2O)

t ' T w~ = Vo + v1 - - , (2.21)

A

;~ = ~ ( 1 + A - % (2.22 t Mc

Here, t is the isospin vector of the nucleon and T that of the remaining A nucleons which give rise to the average potential . The m o m e n t u m and spin vectors of the nucleon are denoted p and stt, and the characteristic radii of the nuclear mass and charge distr ibutions are denoted Ro and Re. The factor (p x s) • V in (2.18) reduces to ( l / r ) / • s (~ /&) as long as the system remains spherical. The presence of the spin-orbit te rm makes the fo rm factor (2.7) state-dependent, and the derivat ion given above in the main sect. 2 is no longer valid.

We shall for a while restrict ourselves to the case of axially symmetr ic deforma- tions, following the a rgumenta t ion of Chepurnov and Nemirovksy 6), who point out that the extra componen t s of the spin-orbit te rm in (2.18)t , which in a non-axially symmetr ic case would contr ibute to single-particle matr ix elements, p resumably are small. In terms of the unit vector n o = (cos 0 cos ~b, cos 0 sin qS, - sin 0) perpendicular to r, we can write the deformed field as

Vz(r) = vsph(F) --/~ }12 0(0~)) F 0 ~ s P h ( r ) -Jr- Wr ~) .... ~2fl c~r

E × ½Y2o(Od?)O--!f(-r)-l.s - cosOsinO(no×p) .sO-f~(r)q (2.23) ~r z 2 4-~ Or 3 '

where 17~ ph is the spin-orbit independent par t of V sph. The structure of the last term in (2.23) may better be unders tood by rewriting it in the fo rm

cos 0 sin O(n o x p ) . s = Y2o(040+1 - - 3r

t The three-components in a decomposition along the axes n,, n 0 and n4~. tt Both in units of h.

RADIAL DEPENDENCE OF QUADRUPOLE FORCE 5

The three-axis is chosen along the symmetry axis, and the appearance in (2.24) of a non-invariant term just reflects the fact that with the assumption of axial symmetry also the three-component of p x s will be a constant of motion.

The procedure for extracting the quadrupole form factor analogous to (2.7) is now to pick up all terms proportional to Y20(0~b), which gives

_ _ /~32f (r) 1 Of(r)~ t (2.25) P,(r) = ½(1-r ) r ~Hc + W, Ir ~ f ( r ) _½v~ ° 221. s _ . c3r t c~r "" \ Or z r Or ])

As only terms which are totally invariant remain, we shall assume that (2.25) is correct also for non-axial symmetric systems, i.e., P,(r) can be used in the residual interaction (2.1). It should be noted that one would also arrive at the expression (2.25) if, in (2.18), one replaced (p × s) . V by (1 / r ) l . s (d /Or) and treated ( l . s ) as an r-independent constant. Because of the presence of the (l • s) term, P,(r) is no longer a state-independent form factor, and the significant quantities will be the reduced single-particle matrix elements

<illV,(r) Yz(04')ll j ) , (2.26)

evaluated between the Woods-Saxon eigenstates of the spherical average potential (2.18). These matrix elements, which are the basic ingredients of a microscopic calcula- tion using an interaction of the type (2.1), will in sect. 3 be compared to those of the U Q F considered in subsect. 2.1, and the importance of using consistent wave func- tions in the evaluation of the matrix elements (2.26) will be stressed.

3. Comparison between UQF and M Q F

Expressing the single-particle wave functions* in the helicity representation 4)

I.ljm> = 1- u.,j(r) (2j+ y. (3.1) r \ 16n 2] T

one easily finds the reduced matrix elements of the quadrupole field (2.26)

(nl j l lP, Yz l ln ' l~ j ' )= V ~ ( - ) J - ~ J ) ' ( I + ( - - ) t-r) (~' _½J 20)(nljlP~lnlj), (3.2)

where ) = (2j+ 1)~ and r,bc~ ~av) is a 3-j symbol. Fig. 1 gives, for comparison, the harmonic oscillator potential and the spin-orbit independent part of the Woods-Saxon potentials which are used in this section to illustrate the differences. They correspond to pure neutron (v = 1) wells for 12°Sn, using for the Woods-Saxon potential the parameters of table 1, assuming Ro = Re = roA ~. The parameters of the harmonic oscillator

t Phase convention: Condon and Shortley, u, tl positive for r --~ 0.

6 K. KUMAR AND B. SORENSEN

potentials are then also fixed, since

b -~- F 0 ~ 5 ( 3 ) ]

h O) 0 =

M b z

(3.3)

(3.4)

>

>

I0

- I 0

- 2 0

- 3 0

- 4 0 :

- 5 0

I J i r I 1 I t I I I /

/ /

/ /

/

® //

0)- VI -

(!) ///

/ w s

o , / - - - f

IR 0

, i , i , I I I I I I I I 0 2 4 6 8 I 0 I

r ( f m )

Fig. 1. The spin-orbit independent neutron Woods-Saxon well for 12°Sn compared with the cor- responding (cf. sect. 3) harmonic oscillator well.~The numbering of the Woods-Saxon single-particl e states will be used in the following figures. Their spins will be given in the caption to fig. 4.

TABLE 1

Parameters of the Woods-Saxon well

ro a Vo Vt v ....

1.27 fm 0.67 fm --51.0 MeV 132.4 MeV 32.0

We have used Ro = Re = ro A4e.

R A D I A L D E P E N D E N C E O F Q U A D R U P O L E F O R C E 7

-0.4 I - 0 . 6 I 0

Z

J I t I i I l t J I l I l I l I

0.4 °iI - 0 . 2

\ \L//

I I I I I l I J l I l l I r I

2 4 6 8 I 0 12 14 16 8

r (fro) Fig. 2. Examples of s.p. wave functions for Woods-Saxon (full) and harmonic oscillator (dashed)

wells. The circled numbers correspond to fig. 1.

/ i ~ - ! -

/ 1 4 - /

/c

12 /1

IO

n

6

4

"!'..~ o - ~ f , , - , , ' ' + - - ' + ' - ; . + ' . - " I . . . . . . . |

- I " ' ° " " R o

- 2 t I i I i I i 1 i I i I =

0 2 4 6 8 I 0 12 14

r (fro) Fig. 3. The spin-orbit independent part of the radial function P(r) of the modified quadrupole force (MQF) is shown as a solid line (a). The same for the usual quadrupole force (UQF) is shown by a dashed line (c), and the dotted line (b) gives the spin-orbit contribution to P(r) in MQF divided by

l" s (i.e., the factor multiplying this operator).

8 K . K U M A R A N D B. $~IRENSEN

Fig. 2 gives a few examples of the differences in single-particle wave funct ions for

the two potentials, in particular, the difference in range for wave funct ions of levels

close to the top of the Woods-Saxon potential . The problem of u n b o u n d states will

be considered later in this section.

In fig. 3, the radial funct ions P(r) are given, for the M Q F case only the spin-orbi t

independent part. However, the radial funct ion mult iplying the state-dependent

matrix element of I • s is also indicated. It is seen that the M Q F concentrates its effect

at the surface of the nucleus *, and that the spin-orbit part tends to let the interact ion

take place a little outside or inside the surface, according to which one of the two

spin-orbi t par tner levels is involved.

I0 8 6 4

% 2

o

~-2 /x -4

"O-

~ -I0 n = j~

Force Wove fct,st 8 0

:]4

-10 4 6 2 3 8 9 4 5 ill? 612 T B 8 9 9 0 18 121:3 I~.-~516-IZ-18- "~

12~ ?~ 18 6 15 15A 9~ I0,? 171R 1213 14~-Ib I¢ I~ 19

171819 1415 i I 2 3 4 5 S 7 8 9 ~ i

15 Fig. 4. Reduced single-particle matrix elements of the MQF or UQF quadrupole fields using harmonic oscillator (HO) or Woods-Saxon (SW) wave functions. The relevant s.p. levels are identified by numbers corresponding to the following orbits: 1 = ls~r, 2 = lp~r, 3 ~ lp~r, 4 = ld~r, 5 = 2s~, 6 = ld~, 7 - lf~, 8 = 2p~, 9 ~ lf~, 10 = 2p~, 11 = lg~_, 12 ~ 2d}, 13 = lgk, 14 ~ 3s~,

15 ~2dk , 16 = lh~t, 17 =2f~, 18 ~ 3pk, 19 = 3p_}.

In fig. 4, we compare reduced matrix elements of the type (3.2) for M Q F and UQF,

using in bo th cases either Woods-Saxon or ha rmonic oscillator wave funct ions in

the evaluat ion. We can thereby test whether a restricted improvement of either only

the wave funct ions or only the force would produce the main difference between the

two "na tu r a l " types of matrix elements, which use consistent potent ial and wave

funct ion. This is seen not to be the case. In fact, the two steps of improvement go in

t In this respect, the MQF is much closer to the surface deRa interaction s) than the UQF. Actually, the radial dependence of the MQF equals that of the surface delta interaction for the diffuseness a going to zero and no spin-orbit potential. Under these approximations the surface delta interaction simply equals the sum of the self-consistent multipole forces over all multipole orders.

R A D I A L D E P E N D E N C E OF Q U A D R U P O L E FORCE 9

opposite directions, and it is remarkable that the consistent use of MQF and Woods- Saxon wave functions gives matrix elements which are much closer to those of UQF with harmonic oscillator wave functions than one might expect looking at the major differences in the potentials and wave functions exhibited in figs. 1-3. In most cases, the differences in magnitude are only a few percent, and only for one significant matrix element there is a difference of 35 ~o. One important feature is that the matrix elements, which were zero because of the selection rules associated with the harmonic oscillator are non-zero for the MQF, for instance the matrix element between Ip~ and 2f~, is 12.8 MeV t. One observes through the shell structure an oscillatory be- haviour of the difference between the MQF and the UQF matrix elements. For in- stance, the matrix elements between levels just above the Fermi level (2d~) are largest for the MQF, while those between levels just below the Fermi level are largest for the UQF. The gain by using MQF for levels close to the binding limit can directly be understood from figs. 1 and 3.

>~o

_~{-i " 7 oo

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 l i

t

I I

2 8

2 2

. , . . . r . l . r .

I I I I ( I I I I I I I I I , I I 1

18 4 12 6 15 7 17 8 18 9 II 12 I$ 15 16 17 18 j

2 Zl" 4 6 6 7 7 8 B 9 II 12 15 15 16 17 18 i

Fig. 5. The differences between the full M Q F ma t r ix e lements and those with the spin-orbi t con t r ibu- t ion left out , calculated wi th the same Woods -Saxon wave functions.

Fig. 5 is concerned with the effect of the spin-orbit part in the MQF, comparing the relevant matrix elements of fig. 4 with matrix elements calculated with the l . s part left out of the MQF (yet the Woods-Saxon wave functions were not altered). The total difference conforms with the sign change of <l • s> going from one spin-orbit partner to the other, but the absolute differences are extremely small, supporting the argumentation for the approximate treatment of the deformed spin-orbit potential presented in sect. 2.

3.1. T H E S E L F - C O N S I S T E N T C O U P L I N G S T R E N G T H

Formally, the self-consistent strength of the MQF may be expressed in analogy to (2.II) -1

1 (3.5)

t This is, however , still a ra ther small mat r ix element. As there was a free choice o f scale for P ( r ) , the significant n u m b e r is Z t imes the squared single-particle mat r ix e lement , which in the case m e n - t ioned is only 4 ~o o f the same quant i ty for the s t rongest t ransi t ion.

10 K. KUMAR AND B. SORENSEN

where the c, is obtained by inserting (2.18) into (2.12). However, due to the spin- dependence of P~ (2.25), it is convenient to use the equivalent form (2.15) and second

quantization

E 1 Z = 4re ~ <X[r 5 r3P,(r))ltc><a+aK> , (3.6) K,V

where the expectation value of a+a has to be taken in a state with density distribu- tions p~Ph(r) corresponding to the spherical Woods-Saxon potential. From the deriva- tion of the MQF, this is the state of the system when the quadrupole force is absent, i.e., the state in which particles occupy the lowest available Woods-Saxon orbits. Assuming these to be the states labelled 1," = (nljm) in (3.6), we obtain

x=4,, F y~ <~cl 1 ~ 1-1 ~-,.K . . . . o,e, r ~ ~r (rze'(r))lx>

F 1' = 4re Z d,(il (r3e~(r))li>,,a,a, , (3.7) L- i ~ i F , r:

where di is the degree of filling for orbit i = (nlj)(d i = 2.] i + 1 except for i = iF) and the radial matrix elements are to be calculated with the radial Woods-Saxon wave functions u,u(r ) of (3.1). The label i on P~ signifies that l • s has to be replaced by

0.5 [Ji(J~ + 1)-l~(l~ + 1 ) -0 .75 ] in accordance with the discussion in sect. 2. In order to understand the importance of various contributions to Z, we have also

considered two further approximations. One is to neglect the spin-orbit part of P , in which case (3.5) is a simple integral. I f we further neglect the Coulomb energy and the symmetry energy, we obtain a strength )(, which only depends on A and thus may be compared with XUQF of eq. (2.17). Carrying the integration only up to R 0, as it was done in (2.16), we can evaluate the integral explicitly, obtaining

, 16~za A_ ~ (3.8) ZMQF - ~ 0.174 A -~ MeV- t 3Vo ro

The numerical value corresponds to the parameters listed in table 1. Using the same parameters (which imply b = A t- and hOgo = 59.76 % 2 A-¢ MeV) we get t f rom (2.17)

XuQv = 0.0702 A - ~ MeV- ~ (3.9)

The two self-consistent strengths Z~tov and ZUQF are compared in fig. 6. Further the full ZMQF has been calculated in a number of cases, using eq. (3.7). The figure shows the general trends for nuclei close to the ~-stability line. The inclusion of Cou- lomb effects and spin-orbit terms affects ZMOF only by a few percent, and the main reason for the difference between g~tQv and Z~QF is the symmetry energy, which for regions with large neutron excess makes ZMQF increase more slowly than A -~. For

t Note that the present definition of Z differs from the usual one by a factor of (gmo)2.

RADIAL DEPENDENCE OF QUADRUPOLE FORCE l l

the same reason, ;~MOF is almost constant through a series of isotopes. The gross picture is thus a rough A -~ dependence for nuclei at the B-stability line and no A-de- pendence for fixed Z, in contrast to the overall A- ~ dependence of XUQF-

It is not very easy to test the predicted A-dependence of X against experimental evidence. This is due to the renormalization effects which have to be taken into ac- count when truncating the single-particle basis. The UQF requires the discrete but infinite harmonic oscillator basis, and the MQF requires continuum states in addition to the Woods-Saxon bound states. If sufficient levels are included to make the effect

10 -z I i i I I I I I l l i ' l l l S l l l [ l l l l I [ i

10 -3

I0 -4 I ~ , I , X i

10 20 50 100 150 200 300

A Fig. 6. Self-consistent quadrupole strength for UQF and MQF, the latter in two versions. One, (Z'Mo~), neglects Coulomb, symmetry and spin-orbit terms in order to obtain a simple A-dependence, the other, (ZMQF), includes these effects and is thus dependent on both N and Z, for which reason we have only given the average dependence for nuclei at the fl-stability line together with a few

crossing lines indicating the N-dependence for fixed Z.

of the remaining levels structureless, a single renormalization factor will multiply the self-consistent Z. This factor will, however, be A-dependent for convenient choices of the truncation, and a further approximation in the type of many-body technique employed may be more or less justified for different nuclei. For instance, a hypotheti- cal RPA calculation using untruncated bases and the self-consistent X will presumably give reasonable fits to 2 + energies in regions with small quadrupole matrix elements and large gaps in the single-particle spectrum, but poor fits when leaving these regions.

Another possible calculation aimed at deformed nuclei would use truncated bases and calculate the static deformation as function of Z by a Hartree-Fock or Hartree-

12 K. KUMAR AND B. SORENSEN

Bogoliubov method, iterating until the potential and wave function deformation become equal. This procedure will give the correct self-consistent Z corresponding to the truncated configuration space, provided that dynamical effects can be neglected. This condition is similar to the one we used to find the unrenormalized strength, but still not equivalent, since the proportionality between field and density may hold in the complete configuration space but not in the truncated one.

We have performed such calculations for 15OSm and 192Os, including all bound Woods-Saxon states above 2p~ for protons and above 2d~ for neutrons. We thereby have obtained quadrupole strengths related to the self-consistent Z of eq. (2.7) by Z = 1.77 Zs.~. for 15°Sm and Z = 1.60 Zs.c. for 192Os .

The MQF self-consistent z-values are supported by the comments of Baranger and Kumar 7) concerning the inappropriateness of the A- ~ dependence and in particular favouring a slower A-dependence within major shells (or for a series of isotopes), which they had to introduce in their UQF calculations 2).

A comparison of the self-consistent Z with values which reproduce certain experi- mental data is of course further affected by the possible insufficiency of the quadrupole force.

3.2. T H E R O L E OF T H E C O N T I N U U M

In conventional calculations with pairing and quadrupole forces, one confines the number of single-particle levels employed to those close to the Fermi levels (usually not more than three major shells for each kind of particle). With the UQF one can hardly go further, and it may well be that even the calculations which do include three shells are unreliable 7) because of the obvious unrealistic features of the inter- action implied between distant levels. It is also in the spirit of the pairing force with constant matrix elements that the non-zero strength must only apply within a certain subspace, and that all other matrix elements are zero. On the other hand, an enlarge- ment of the number of configurations is significant in connection with the UQF, which has infinite range. It has been suggested that already the A N = 2 matrix ele- ments are questionable, and the type of correlations implied by the inclusion of more than two major shells seems to be quite unrealistic 7). Since the MQF has a radial dependence peaked around the nuclear surface, one would guess that the inclusion of more levels, either far below the Fermi level or above it, reaching up in the continuum, would not change the correlations considerably, so that a truncation of the configura- tion space would be justified. However, the Woods-Saxon wave functions do have tails reaching large radii, and further the peaking of the interaction at the surface does not apply to the Coulomb part, which in fact has a very long range. It is seen from fig. 4 that matrix elements involving levels far below the Fermi level are in general not smaller than those of the UQF, and although the matrix elements involv- ing levels close to the continuum are considerably reduced, it is not clear whether this tendency will not be compensated by the increased level density in the continuum. In order to estimate the magnitudes of matrix elements of the MQF involving contin-

RADIAL DEPENDENCE OF QUADRUPOLE FORCE 13

uum states, let us evaluate

5 p ( r ) 2 l , ) = }-, <,lP(r)Y2M(~)[x) 2

M=ml --m~e

= y' ( 2 j , + l ) - ' ( n , l , j , llP(r)Y2(~)llnlj) 2 (nl j ) bound

+ ~ (2j ,+l)- l (n , l , j , l lP(r)¥z(~) l lnl j ) 2, (3.10) (ntj) unbound

for a level t close to the Fermi level, leaving as unknown the sum involving unbound states. For the case of neutrons in 12°Sn considered earlier, we find for L = ( lh÷)

that (t15/4n p21t) = 33.7, Zbound = 14.1 and thus ~unbouna = 19.6, which is 1.4 times the bound state contribution. This may be compared to the UQF, for which we get ~.unbnund = 2 1 - 10 ----- 1 1, i.e., 1.1 times the "bound" state contribution. The quoted numbers are of course not directly related to the renormalization factors multiplying ): ..... but the indication is that the renormalization will be larger when using MQF than UQF. The relatively larger importance of the continuum states for the MQF as compared to the UQF is, in the example given here, due to the selection rules valid for harmonic oscillator states and not for Woods-Saxon states. Considering a more physical sum rule than (3.10), one which contains an energy weighting similar to that of a strength function (the energy being in the denominator), it appears reason- able that the relative importance of unbound levels in MQF will be increased further since such levels are abundant at low unbound energies, whereas the corresponding levels in an oscillator well are separated by distances comparable to those between the lower levels.

We thus conclude that, although the Woods-Saxon basis and the MQF matrix elements appear much more physical than the harmonic oscillator basis and the UQF matrix elements, the problems connected with renormalization are just as pertinent and one will, in each concrete calculation, have to consider whether the influence of the continuum can be simulated by a mere change of force strength ~ or not.

Although the magnitudes of bound state matrix elements are not very different in MQF and UQF, there can hardly be any similarity for matrix elements involving continuum (high-lying harmonic oscillator) states; so should an investigation of such contributions become necessary, only the MQF provides a physically acceptable basis.

4. Conclusions

Whereas many-body calculations performed within small configuration spaces (e.g., one major shell) cannot aim at more than a qualitative description of spectra and transition rates and therefore may not require much refinement of the interaction, then the present-day availability of many-body techniques allowing approximate

14 K. K U M A R A N D B. SORENSEN

solu t ions with large conf igura t ion spaces will most certainly lead to awareness of any

systematic deficiency in the in teract ion. In o rder to decide whether a quadrupo le type

in terac t ion is a t all useful in such calculat ions, we feel that one should at least remove

the conceptua l inconsis tency o f the usual radia l dependence.

M a n y - b o d y calcula t ions with the modif ied quadrupo le in terac t ion have been lo)

or are being pe r fo rmed by the authors , but since pre l iminary results with only bound-

state conf igurat ions included do not p rovide any conclusive evidence of improvement ,

and since the p rob l ems of including the con t inuum have not yet been sat isfactori ly

solved, we have here tr ied to exhibi t some of the basic features of the p r o p o s e d inter-

ac t ion in the hope o f creat ing a wider interest in the persuance of these problems.

References

1) L. Kisslinger and R. Sorensen, Rev. Mod. Phys. 35 (1963) 853 2) K. Kumar and M. Baranger, Nucl. Phys. Al10 0968) 529; A122 (1968) 273 3) B. S~rensen, Nucl. Phys. A97 (1967) l 4) A. Bohr and B. Mottelsoa, Nuclear structure, vol. 2, (Benjamin) to be published 5) R. Woods and D. Saxon, Phys. Rev. 95 (1954) 577 6) V. Chepurnov and P. Nemirovsky, Nucl. Phys. 49 (1963) 90;

Gareev et al., Acta Phys. Pol. 32 (1967) 461 7) M. Baranger and K. Kumar, Nucl. Phys. Al l0 (1968) 490; A122 (1968) 241 8) R. Arvieu and S. Moszkowski, Phys. Rev. 145 (1966) 830;

A. Plastino et al., Phys. Rev. 145 (1966) 837 9) D. Rowe, Phys. Rev. 162 (1967) 866

10) B. Sorensen, Nucl. Phys. A142 (1970) 411