IC/68/4streaming.ictp.it/preprints/P/68/004.pdfeffective interactions in nuclei from the realistic...
Transcript of IC/68/4streaming.ictp.it/preprints/P/68/004.pdfeffective interactions in nuclei from the realistic...
£ 2 5-JAN ^33
IC/68/4
INTERNATIONAL ATOMIC ENERGY AGENCY
INTERNATIONAL CENTRE FOR THEORETICAL
PHYSICS
ENERGY SPECTRUM OF 116Sn
AND EFFECTIVE NUCLEAR FORCES DERIVED
FROM THE REALISTIC NUCLEON-NUCLEON
POTENTIALS OF YALE AND OF TABAKIN
M. GMITROAND
J. SAWICKI
1968
PIAZZA OBERDAN
TRIESTE
IC/68/4
INTERNATIONAL ATOMIC ENERGY AGENCY
INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS
11 fiENERGY SPECTRUM OF Sn AND EFFECTIVE NUCLEAR FORCES DERIVED
FROM THE REALISTIC NUCLEON-NUCLEON POTENTIALS OF YALE
AND OF TABAKIN *
U. GMITRO * *
and
j . SAWICK1
TRIESTE
January 1968
* To be submitted for publication.
** On leave of absence from the Nuclear Research Institute, Rez (Prague), Czechoslovakia.
ABSTRACT
The effective nuclear forces are obtained by the core polarization
renormalization from a realistic local static (Yale) and from a non-
local (Tabakin) nucleon-nucleon potential. These forces are applied to the
11 Rquasiparticle Tamm-Dancoff theory of the spectrum of Sn. While
the respective bare (unrenormalized) forces lead to different
predictions, the renormalized Yale and Tabakin effective forces yield
almost identical results. While the second-order core polarization
corrections are quite large, the successive inclusion of all the higher-
order corrections (iteration) leads to only negligible modifications.
ENERGY SPECTRUM OF 116Sn AND EFFECTIVE NUCLEAR FORCES DERIVED
FROM THE REALISTIC NUCLEON-NUCLEON POTENTIALS OF YALE
AND OF TABAKIN
KUO and BROWN have proposed a method for reducing the
effective interactions in nuclei from the realistic nucleon-nucleon
potentials. The method is based on treating excited configurations of
the "inert" core nucleons through a renormali zation of the matrix
elements of the interaction between the "active" valence (open shell)
nucleons by second- and higher-order terms of the particle—core-
particle scattering type.
This "core polarization" correction of the nucleon-nucleon
interaction responsible for the shell model configuration mixing and
related effects has been shown to be most important and when applied to
realistic nucleon-nucleon potentials it has met with a remarkable
quantitative success. The first numerical results referred to the
1) 2)Hamacla- Johnston potential and to light nuclei ' and to the nickel
isotopes
4) 5)Most recently, Hendekovic and the present authors ' have
applied the above idea to nuclei of the vibrational region where the BCS
pairing effect appears to be most important. At the same time,the
Hilbert spaces for mixing configurations of the exact shell model have
completely prohibitive dimensions in that region, while the quasi-
particle Tamm-Dancoff methods are feasible and reasonable. The
realistic nucleon-nucleon potential was that of TABAKIN . Ref. 4)
has shown that while the energy spectra of the even tin isotopes calculated
with the "bare" matrix elements of the Tabakin potential are in a
complete qualitative disagreement with the data, quite satisfactory
results are obtained when the core polarization renormalization is
included.
In order to see whether the conclusions of ref. 4) were not due to
any peculiar (specific) properties of the non-local (separable and non-
singular) potential of Tabakin we have chosen to compare our previous
results supplemented by the odd-parity states of the spectrum with the
7)corresponding results obtained with the best Yale potentials . This
is a typical, essentially static and local potential with hard core parts.
The Yale potential also contains tensor parts. The basic effective
many-body Hamiltonian is defined in the Brueckner-Hartree-Fock
sense. The Brueckner reaction matrix for the Yale potential is taken
8)in the approximation given by SHAKIN et al. In calculating our first-
and higher-order reduced matrix elements of the reaction matrix
operator we have used the numerical tables of the Yale nucleon-nucleon
potential of the corresponding radial matrix elements in the relative
motion of Appendix of ref. 8). In a quasiparticle Ta mm -Dane off (QTD)
approximation we encounter reduced matrix elements both of the particle-
particle and of the particle- hole type coupling. The second-order core
polarization corrections to the reduced matrix elements are defined
3)explicitly in eq. (1) of KUO and in eq. (5) of ref. 5). Generalization of
these formulae to an arbitrary higher order is quite straightforward.
One can perform the summation of such polarization corrections to all
orders by using an obvious iteration procedure (cf. eq. (6) of ref. 5)).
- 3 -
In this way one can then include all the RPA bubble diagrams and related
exchange diagrams of the core nucleons. Below, we compare results,
based on only second-order corrections includedfwith those involving
al! the higher-order core polarization corrections.
The energy denominators of the particle-hole propagators for the core
nucleons can be approximated by E -E, (p denotes particle and h denotes
hole). It has been shown in ref. 5) that in calculating the second- and
higher-order corrections in the tin isotopes it is enough to assume an
average 50% occupation of the neutron valence subshells (approximation
116referred to as S2 in ref. 4)). In our calculations of Sn we have assumed
the set of the unperturbed single-particle energies -I E j- (in MeV) given in
Table I.
In Table II we compare the single qp-energies of the five valence
neutron subshells for the bare, the second-order renormalized and to-all-
order renormalized effective forces of the Yale potential (Shakin) and of
the Tabakin potential.
We note almost negligible differences between the second-order
renormalized force and the iteration-renormalized force both for the
potentials of Yale (Shakin) and of Tabakin.
In Fig. 1 we compare the calculated QTD energy levels of several
116even-parity states of Sn for the Yale and Tabakin potentials in the
three cases:
1) bare matrix elements of the reaction matrix (Shakin et al .)—l
for Yale and of the potential itself for Tabakin
2) second-order core polarization corrections included ',
3) core polarization corrections are included to all orders .
- 4 -
't Z .'- A. ••-.!•
We see that in the case 1) both for the Yale and for the Tabakin
potential all the levels considered (0o, 2 , 2 , A., 4 ) lie quite low
and the level density is high. The differences between the two potentials
considered are marked.
When the second-order core polarization corrections are included
the spectrum is spread out quite appreciably in the direction of a much
better agreement with the experimental data. At the same time we
observe the remarkable feature that the differences between the QTD
predictions for the two potentials become practically negligible.
In the case 3), where all the higher-order corrections are
included, we recover almost exactly all the results of the previous case
2). This permits the conclusion that, at least in our problem,the second-
order core polarization renormalization is quite sufficient, i. e. ,
contains all the most important information about the contributions of
the core nucleons.
A similar analysis was performed and the same situation found
for the odd-parity states 5 , 6~ and 7 . All our results for the QTD
energy levels are summarized in Table III.
The electromagnetic reduced transition probabilities B(E2) when
calculated for both the Yale and the Tabakin potentials with the core
polarization are typically increased by about 15-20% with respect to
the "bare" forces. The differences between the case of the second-order
renormalized force and that of the iteration renormalized force are
completely negligible both for the Yale and the Tabakin potentials. All
the corresponding predictions based on these two potentials are almost
identical. Although only a four-qp Tamm-Dancoff theory can explain
-5-
the observed value of the quadrupole moment Q(21) of the state 2 t ,
we have applied the QTD approximation for similar comparisons. The
calculated Q(2.) exhibits trends quite similar to those described above
for the B{E2).
It may be interesting to observe a reduction of the average nucleon-
number fluctuations (non-conservation), <( N )> - N , due to the core
polarization renormalization of the force. An example of this effect is
given in Table IV for several QTD eigenvectors calculated for the
Tabakin potential.
In Fig. 2 we compare the largest in absolute value 18 reduced
matrix elements G(abcd, J = 2) in the notation of BARANGER for the
Tabakin potential with core polarization to all orders with the corresponding
elements of the standard (Wigner) quadrupole-quadrupole (Po) force.
Our notations are: 1 = 2d5/2>
2 = l g?/2 3 = 3 s l / 2 ' 4 " 2d3/2 a n d
5 = Ih , . One observes a similar trend of the matrix elements of the
theoretical and of the phenomenological nuclear forces. A similar
situation has been observed by Brown and Kuo for the Hamada-Johnston
10)potential in the Ni isotopes . We have not found any striking
differences in the effects of the core polarization on the even J and the
odd J reduced matrix elements.
After the present Letter was completed a Letter by KUO came
to our attention in which higher-order core polarization corrections are
studied. Although Kuo's procedure is somewhat different from ours his
conclusions about the higher-order effects are quite consistent with
those reported above.
- 6 -
ACKNOWLEDGMENTS
We are indebted to J. Hendekovic for many useful discussions,
his help in some of our numerical computations and for bringing to
our attention the effect of core renormalization on the nucleon number
fluctuation. We are happy to express our thanks to Professors
Abdus Salam and P. Budini and the IAEA for their kind hospitality
at the International Centre for Theoretical Physics in Trieste.
Financial support from UNESCO to one of us (MG) is gratefully
acknowledged.
- 7 -
REFERENCES
1) T . T . S . KUO and G. E. BROWN, Nucl. Phys. 8j3, 4tK(9fi6);
A92, 481 (1967).
2) R. P . LYNCH and T . T . S. KUO, Nucl. Phys. A95, 561 (1967).
3) T . T . S . KUO, Nucl. Phys. A9Q 199 (1967).
4) M. GMITRO, J. HENDEKOVIC and J. SAWICKI, Phys. Letters ,
in press .
5) M. GMITRO, J. HENDEKOVIC* and J. SAWICKI, Trieste preprint
IC/67/75 (to be published in Phys. Rev).
6) F . TABAKIN, Ann. Phys. (NY) 30, 51 (1964).
7) K. E. LASSILA, et a l . . Phys, Rev. m , 881 (1962).
8) C M . SHAKIN, et al. . Phys. Rev. 1£1, 1006 (1967);
161, 1015 (1967).
9) M. BARANGER, Phys. Rev. 120_, 957 (1960).
10) G.E. BROWN, Proceedings of the International Conference on Nuclear
Structure, Gatlinburg (1966).
11) T . T . S . KUO, Phys. Letters 26B, 63 (1967).
- 8 -
TABLE CAPTIONS
Table I
Set of -{ E j* of Sn used in the present calculations (energies
in MeV).
Table II
Comparison of the single-qua siparticle energies (in MeV) calculated
for the Yale (Shakin) and Tabakin potentials (bare and renormalized).
Table III
itQTD level energies (in MeV) calculated for the J excited
116states of Sn for the cases defined in Table II.
Table IV
Comparison of the nucleon-number fluctuation for several QTD
116states of Sn calculated with the bare and renormalized Tabakin
potential.
FIGURE CAPTIONS
Fig. 1
11 fiComparison of (QTD) spectra of Sn for Yale (Shakin) and
Tabakin bare and renormalized (second-order and iterated) potentials.
Fig, 2
Comparison of the 18 largest reduced matrix elements
G(abcd J = 2) of the renormalized (to all orders) Tabakin potential
with those of the usual P -force.
-10-
TABLE I
State
nlj
<
Open or valence subshells
2d5/2
0.0
l g7/2
0.40
3 S l /2
1.90
2d3/2
2.20
l h l l / 2
2.40
core subshells
l g 9 / 2
-4.0
2pl/2
-12.0
l f 5 / 2
-12.0
2 p 3 /2
-12.0
TABLE II
Force
Yale
Tabakin
nlj
bare
2nd-order
iterated
bare
2nd-order
iterated
2 d 5/2
1. 94
2.09
2.08
1.91
2. 03
2. 02
l g 7 / 2
1.07
I. 71
1. 77
1.21
1.68
1. 72
3 S l / 2
1.03
1. 34
1.37
1.10
1. 32
1.34
2 d 3 /2
0.99
1. 56
1.60
1. 08
1.48
1.50
lhn/2
0. 97
1. 27
1.29
1.03
1.28
1.30
TABLE III
Force ^"""X^^
Yale
Tabakin
bare
2nd-order
iterated
bare
2nd-order
iterated
Experimenta]
1. 352. 05
2.063.01
2. 123. 05
1. 542. 21
2. 062. 90
2, 092. 92
1. 762.02
»;.•
1.011. 77
1. 222. 62
1. 232. 68
1. 272.00
1.442. 62
1.462. 65
1. 292. 11
< . =
1. 691. 99
2.152. 83
2. 192.89
1.882.20
2.272.81
2. 292.85
2. 392. 53
5 i . 21. 771. 98
2. 352.80
2..402.85
1.912. 13
2. 372. 75
2.402. 79
2. 364
V2
1.922. 04
2.612. 80
2. 672. 8fi
2. 082. 15
2. 612. 75
2. 642. 79
2. 774
\ , 2
1. 612. 02
2.472. 90
2.542. 98
1. 782. 20
2.432.88
2.462. 92
2.909
TABLE IV
Force ^ ^ ^ ^
bare
2nd-order
renormalized
0
0.
+
2
. 64
40
0
0
0.
+
3
50
30
2?1
0. 52
0.30
?,
0.
0.
+
2
73
56
4
0
0
+
1
. 6 4
. 60
4
0.
0.
+
2
13
05
- ) 2 -
MeVFig. i
2 —
1 —
/
—f=
/t
i
t
a
11 •• • • • - H
•••••• • f
— Oj
, ..,,. 2
/J
i
1
,20
—~4
2
—i,
0
. 1
•i. 2.• 0
H
2
, V
0
t
0 —
Yale Tabakin
bai-e foroe
Yale Tabakin
+2nd orderoorreotions
Yale Tabakin
•fhigher orderoorreotions
-13-
Fig. 2
WeV
0.3
0.2
0 . 1
0 . 0
0 . 1
0 .2
ab0
d
—
—
—
-
-
—
llll
iii
ifi
t
lll3
J\\
V*'
i21
r1
i313
\
1324
1 JL 11
\ !
\
VIV1343
1
1344
\ .ft J
\\ /
wIf
i443
i
!//
f
t
\ 1
\
\
2222
yi
J2224
(
1//
K' V
2244
*
\\]
2424
//
\\
i
2443
f
2444
\
\
\
y
\
2525
4122
+i
if
i
i i
1J4144
—
—
—
-
-
;
5555
0.3
0 .2
0 . 1
- o.o
-.0.1
- 0 . 2
- 1 4 -
Available from the Office of the Scientific Information and Documentation Officer,
International Centre for Theoretical Physics. Piazza Oberdan 6, TRIESTE, Italy
026