£ 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

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0.

0.

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2

73

56

4

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1

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4

0.

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+

2

13

05

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MeVFig. i

2 —

1 —

/

—f=

/t

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2

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0

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Yale Tabakin

bai-e foroe

Yale Tabakin

+2nd orderoorreotions

Yale Tabakin

•fhigher orderoorreotions

-13-

Fig. 2

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—

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—

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5555

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0 . 1

- o.o

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- 0 . 2

- 1 4 -

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