Nuclei Off the Line of Stability - American Chemical Society

11 Shell-Model Calculations near Effective Interaction 1

1

Sn Using a Realistic,

132

2

2

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C. A. Stone, W. B. Walters , S. D. Bloom , and G. J. Mathews 1

Department of Chemistry, University of Maryland, College Park, MD 20742 Lawrence Livermore National Laboratory, University of California, Livermore, CA 94550

2

We have performed shell model calculations on positive parity states of the one, two, and three quasiparticle nuclei near Sn using the Kallio-Kolltveit two-body interaction. A weak quadrupole and a weak pairing potential were added to the two-body interaction as core polarization corrections. We have found that the addition of a quadrupole potential improves the level spacing in Sn and Te. For Te and I (two and three proton systems) a pairing potential has to be included while Sn and Sb (zero and one proton systems) can not tolerate this addition. There i s evidence that the h interactions are overestimated and lead to some level misordering in Sn and 131Sb. 132

130

134

133,134

135

129,130

131

11/2

129,130

I.

INTRODUCTION There has been some success in developing effective inter actions for finite nuclei using r e a l i s t i c two-body interactions. Much of this work has been done on the lighter nuclei. Kuo and Brown [KUO66] developed a g-matrix interaction for the sd and fp shells. They used the Hamada-Johnson nucleon-nucleon interaction in their calculations with a core polarization correction. This work was extended to the region near Ca and [KU068] and near P b [KU072]. Baldridge and Vary [BAL76] have also per formed calculations in the 2uopb g i o n . The spectra of 204,206pb calculated using the Reid soft-core potential with core polarization corrections. Lane [LAN79] used the Petrovich, McManus, and Madsen two-body interaction in a limited study of calculations near 132s . Calculations in the heavy nuclei unfor tunately been limited by the large model spaces needed and also by the quality of the available experimental data. 40

2 0 8

re

w e r e

n

1

The 32sn region has recently shown i t s e l f to be a very good region in which to develop an effective interaction. This i s a region with a strong double-shell closure, stronger than a l l shell closures beyond ^ 0 . There i s also a large set of experi0097-6156/86/0324-0070$06.00/0 © 1986 American Chemical Society

Meyer and Brenner; Nuclei Off the Line of Stability ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

11.

STONE ET AL.

71

Shell-Model Calculations near Sn 132

mental data a v a i l a b l e on the n u c l e i near ^^ Sn. A l l f i v e of the n e u t r o n - h o l e s t a t e s i n '3 sn have been seen [F0G84] and f o u r o f the f i v e p r o t o n s t a t e s i n '33sb have been found [BLO83] ( o n l y the 1/2+ s t a t e i n '33$b i s m i s s i n g ) . Oyr knowledge o f the twoq u a i s p a r t i c l e n u c l i d e s , ^ O s n [F0G81], 3 S b [ST086a], and *Te [KER72] i s not as complete as the w i t h o n e - q u a s i p a r t i c l e n u c l i d e s but much o f the l o w - l y i n g s t r u c t u r e has been i d e n t i f i e d . The f o u r 3 - q u a s i p a r t i c l e n u c l i d e s , * S n [DEG80], ^ S b [ST086b], 3 3 τ β [LAN80], and [SAM85], have been s t u d i e d e x t e n s i v e l y : a l a r g e number o f l e v e l s has been seen i n each n u c l e u s . 2

T

,

2

13i

1 2

1

Our g o a l i s t o develop a semi-phenomenological e f f e c t i v e i n t e r a c t i o n f o r use near *3 sn u s i n g a s y s t e m a t i c approach. We began the study by u s i n g the s i n g l e - q u a s i p a r t i c l e n u c l i d e s t o f i x the one-body p o r t i o n o f the H a m i l t o n i a n . T b i s i s done e m p i r i c a l l y by f i t t i n g the e x c i t a t i o n s t a t e s i n '31sn and '33sb. S i n g l e p a r t i c l e e n e r g i e s (SPE's) a r e determined by f i t t i n g the e x p e r i m e n t a l s e p a r a t i o n o f the 7/2+ s t a t e from o t h e r e x c i t a t i o n s t a t e s . As c o r r e c t i o n s a r e made t o a g i v e n two-body i n t e r a c t i o n SPE's a r e r e d e t e r m i n e d . 2

A f t e r f i x i n g the one-body H a m i l t o n i a n , the next s t e p i s t o e s t the two-body H a m i l t o n i a n i n the t w o - q u a s i p a r t i c l e systems, 30sn and '34χ ^ t r i a l two-body i n t e r a c t i o n we have used the K a l l i o - K o l l t v e i t (KK) i n t e r a c t i o n [KAL64]· The KK i n t e r a c t i o n i s a G-matrix i n t e r a c t i o n based on the Scott-Moskowski c u t o f f procedure. Only even components a r e e x p l i c i t l y i n c l u d e d and they have the form o f an e x p o n e n t i a l w i t h a h a r d - c o r e . G-matrix i n t e r a c t i o n s do not take i n t o account core p o l a r i z a t i o n but guidance f o r i n c l u d i n g such c o r r e c t i o n s has been g i v e n by Brown and Kuo [BR067]. These c o r r e c t i o n s have the form o f a quadrupole p o t e n t i a l and a p a i r i n g p o t e n t i a l . We have added these poten t i a l s as c o r r e c t i o n s t o the KK i n t e r a c t i o n , u s i n g the twoquasiparticle nuclides to f i x t h e i r strengths. ββ

II.

s

o u r

Calculations

C a l c u l a t i o n s were performed a t LLNL u s i n g the v e c t o r i z e d s h e l l model code, VLADIMIR [HAU76]· The model space f o r these c a l c u l a t i o n s i n c l u d e d f i v e o r b i t s : 1g7/2» 5 / 2 » 3/2> 3 *\/2> 1 11/2· There are 64 s i n g l e p a r t i c l e o r b i t a l s i n t h i s model space which poses some problems f o r even the s i m p l e s t o f the c a l c u l a t i o n s . The VLADIMIR s h e l l model code uses an i n t e r n a l o c c u p a t i o n number r e p r e s e n t a t i o n t o d e s c r i b e a n u c l e a r con f i g u r a t i o n . The s t r i n g o f b i t s w i t h i n a word d e f i n e s a S l a t e r d e t e r m i n a n t . When a b i t i s s e t t o one the s i n g l e p a r t i c l e o r b i t a l which i t r e p r e s e n t s i s o c c u p i e d and the s i n g l e p a r t i c l e o r b i t a l i s unoccupied when t h a t b i t i s z e r o . The Cray computer has a word s i z e o f 64 b i t s . S i n c e 1 b i t i s r e s e r v e d as a s i g n b i t , o n l y 63 s i n g l e p a r t i c l e o r b i t a l s can be r e p r e s e n t e d i n a s i n g l e word. VLADIMIR does have the a b i l i t y o f r e p r e s e n t i n g a S l a t e r determinant i n m u l t i p l e words but t h i s p o r t i o n o f the code i s c u r r e n t l y w r i t t e n i n F o r t r a n , not Assembly. Two-word c a l c u l a t i o n s a r e p r e s e n t l y much s l o w e r than the one-word c a l c u l a t i o n s . 2d

2d

η


s

NUCLEI OFF THE LINE OF STABILITY

72

We have made f u r t h e r t r u n c a t i o n s t o our model space i n order t o l i m i t the r e p r e s e n t a t i o n t o one word. F o r c a l c u l a t i o n s on 129 ,130,131sn t h e i r h i i / 2 o r b i t a l s were removed. T h i s had no e f f e c t on t h e Sn i s o t o p e s . When c a l c u l a t i o n s were performed on '33sb, 133,134χ ©r » 3 5 i e x c i t a t i o n s were r e s t r i c t e d from t h e 1g7/2 o r b i t a l s . T h i s e f f e c t i v e l y p l a c e d most o f t h e 1g7/2 neutrons i n t o t h e c o r e . These t r u n c a t i o n s had t h e e f f e c t of b r e a k i n g i s o s p i n c o n s e r v a t i o n . Low-lying e x c i t a t i o n s t a t e s (below - 3 MeV) were not a f f e c t e d s i g n i f i c a n t l y but h i g h e r e x c i t a t i o n s t a t e s d i d not have a w e l l - d e f i n e d energy. 8ΐ

III.

Results A.

One-quasiparticle

nuclides

F i g u r e one shows t h e r e s u l t s o f c a l c u l a t i o n s o f 3 3 $ b and •31sn, o n e - q u a s i p a r t i c l e n u c l i d e s . The c a l c u l a t e d l e v e l s agree very w e l l w i t h e x p e r i m e n t a l e x c i t a t i o n s t a t e s . F i g u r e 1 a l s o shows r e s u l t s u s i n g t h e bare KK i n t e r a c t i o n and w i t h t h e a d d i t i o n of a weak QQ p o t e n t i a l determined i n t h e t w o - q u a s i p a r t i c l e n u c l i 1

2800

2/21

U2L. 1ZZL.

2600

mi

2400 2200 2000

2/21.

VV

5/2*

800 600 400 200

F i g u r e 1. nuclides. Calculated effective

U2L11/21

i/2*

1/2*

UIZL 201. 2121JSn,, KK KK'QQ

R e s u l t s f o r c a l c u l a t i o n s on t h e one q u a s i p a r t i c l e Experimental s t a t e s are labeled with the nucleus. s t a t e s a r e shown t o t h e r i g h t , l a b e l e d w i t h t h e i n t e r a c t i o n t h a t was used.


11.

STONE ET AL.

Shell-Model Calculations near Sn

73

132

des. T h e p o o r e s t f i t was t h e 1 1 / 2 - s t a t e a n d t h e t o t a l d e v i a t i o n was o n l y a b o u t 2 0 0 k e V . T a b l e 1 g i v e s o u r b e s t SPE's from t h e s e studies. SPE's f o r 3 3 s b and 3 1 $ n a r e n o t v e r y d i f f e r e n t and s u g g e s t t h a t a s i n g l e s e t o f S P E ' s c a n be u s e d f o r b o t h p r o t o n s y s t e m s and n e u t r o n - h o l e s y s t e m s . The S P E s f o r t h e K K + Q Q + P a i r i n g i n t e r a c t i o n a r e t h e same a s t h o s e w i t h t h e KK+QQ interaction. This i s e a s i l y understood since the p a i r i n g poten t i a l w i l l p u l l down t h e l o w e s t 0 c o n f i g u r a t i o n i n a n e v e n - e v e n nucleus. I n a n odd-A n u c l e u s , a l l s t a t e s w i t h t h e o d d p a r t i c l e (hole) coupled to t h i s 0 s t a t e w i l l be p u l l e d down i n e n e r g y . 1

1

f

+

+

T a b l e 1. S i n g l e ' p a r t i c l e energies determined for the three e f f e c t i v e i n t e r a c t i o n s .

Single Particle Energies Sn (Mell)

, 3 ,

Orbit

I33

S D

(Mell)

fluerage (MeU)

Oeuiation (Mell)

KK Potentiel -4.5000 -3.9208 -2.9674 -2.6703 -2.2358

7/2* 5/2* 11/2" 1/2* 3/2*

-4.5000 -4.0097 -3.2153 -2.2672

-4.5000 -3.9652 -3.0914 -2.6703 -2.2515

0.0000 0.0444 0.1240 0.0000 0.0157

-4.5000 -3.9802 -3.0987 -2.7799 -2.3164

0.0000 0.0296 0.1166 0.0000 0.0492

-4.5.000 -3.9802 -3.0987 -2.7799 -2.3164

0.0000 0.0296 0.1 166 0.0000 0.0492

KK«QQ P o t e n t i a l s 7/2* 5/2* 11/2" 1/2* 3/2*

I I I 1 I

-4.5000 -3.9506 -2.9821 -2.7799 -2.3656

-4.5000 -4.0097 -3.2153 -2.2672

KK*QQ*Peiring Potentials 7/2* 5/2* H/2' 1/2* 3/2*

B.

1 1 I 1 I

-4.5000 -3.9506 -2.9821 -2.7799 -2.3656

-4.5000 -4.0097 -3.2153 -2.2672

T w o - q u a s i p a r t i c l e systems

The b a r e KK i n t e r a c t i o n g i v e s f a i r l y g o o d r e s u l t s i n c a l c u l a t i o n s o n H^Te excitation states. T h e l e v e l s a r e somewhat c o m p r e s s e d and a p p e a r t o l i e a t t o o low an e n e r g y . The b e s t f i t o c c u r s when t h e KK i n t e r a c t i o n h a s a QQ c o r r e c t i o n o f V Q Q = - 0 . 0 0 0 4 1 MeV. A T=1 p a i r i n g p o t e n t i a l c a n t h e n be u s e d t o depress the ground s t a t e , r e l a t i v e t o t h e o t h e r e x c i t a t i o n s t a tes. The optimum s t r e n g t h f o r t h e p a i r i n g p o t e n t i a l i s V i = - 0 . 0 4 9 MeV. These r e s u l t s a r e summarized i n f i g u r e 2a. p a

r

R e s u l t s f o r c a l c u l a t i o n s on 1 3 0 $ w e r e s i m i l a r t o t h o s e o f 34χ . T h e b a r e KK i n t e r a c t i o n g i v e s a l e v e l o r d e r i n g w h i c h i s approximately correct. A d d i t i o n o f a weak QQ p o t e n t i a l d o e s g i v e η

1

Θ



74

an improvement, but o n l y because the s e p a r a t i o n o f the second 2 and t h e 4 i s becoming s m a l l e r . E v e n t u a l l y the f i t t o t h e e x p e r i m e n t a l 6 , 8 + , and 1 0 s t a t e s degrades g i v i n g a s h a l l o w χ minimum a t about the same v a l u e as found i n *3^Te. F o r •30sn, the a d d i t i o n of a p a i r i n g p o t e n t i a l can not be t o l e r a t e d . These r e s u l t s a r e summarized i n f i g u r e 2 b . +

+

+

+

sooo 2800 4;

2600

6*

2400 2200

6*

*4*ê

MOO 1200 4*

400

(a)

200

(b) Χ KK • QQ

• pairing

Ι 30,

90

58-|S η go

KK

KK • Q-Q

• Miring

F i g u r e 2 a , b . C a l c u l a t i o n s on ^^Te and 1 3 0 s showing r e s u l t s w i t h the bare KK p o t e n t i a l and the KK w i t h core polarization corrections. Only the p o s i t i v e p a r i t y s t a t e s w i t h even a n g u l a r momentum a r e shown. n

C.

Three-quasiparticle

systems

We have used the t h r e e - q u a s i p a r t i c l e n u c l e i t o t e s t whether our e f f e c t i v e i n t e r a c t i o n , d e r i v e d from the t w o - q u a s i p a r t i c l e n u c l e i , i s s u f f i c i e n t or whether a l a r g e r core p o l a r i z a t i o n c o r r e c t i o n i s r e q u i r e d . We a l s o were i n t e r e s t e d i n d e t e r m i n i n g i f the p a i r i n g p o t e n t i a l must be v a r i e d as the number o f the p r o t o n s was i n c r e a s e d . I n the two-neutron h o l e ' 3 0 β saw t h a t a p a i r i n g p o t e n t i a l could not be t o l e r a t e d but i n t h e two-proton '34χ p a i r i n g p o t e n t i a l was needed. The t h r e e - q u a s i p a r t i c l e n u c l e i would p r o v i d e us a w i d e r range o f n u c l e i t o t e s t t h i s behavior. η

θ

w e

a


11.

STONE ET AL.

Shell-Model Calculations near Sn

Figures 3a-3d. calculated states particle nuclei.

132

75

These f i g u r e s show the dependence of the on the p a i r i n g s t r e n g t h f o r the t h r e e q u a s i Only e x p e r i m e n t a l l y a c c e s s i b l e s t a t e s are shown.


76


I t i s not c l e a r whether the s t r e n g t h o f the QQ p o t e n t i a l needs t o be v a r i e d as we move from a t h r e e n e u t r o n - h o l e n u c l e u s t o the t h r e e proton n u c l e u s . There are a l a r g e number o f p o s s i b l e e x c i t a t i o n s t a t e s , w i t h s i m i l a r s p i n s , and t h i s makes comparison w i t h e x p e r i m e n t a l data d i f f i c u l t . F i g u r e s 3 a through 3 d show the r e s u l t s o f the c a l c u l a t i o n s on the t h r e e q u a s i p a r t i c l e n u c l e i . One f e a t u r e t h a t i s apparent i n the r e s u l t s i s the need f o r a p a i r i n g p o t e n t i a l i n ' 3 3 χ and 3 5 χ In ^Sn and '3'Sb the p a i r i n g p o t e n t i a l does not seem t o be n e c e s s a r y . Comparison o f the r e s u l t s from c a l c u l a t i o n s on ' 3 3 χ j T35j i t h those o f 9 $ n and ' 3 1 s b shows another i n t e r e s t i n g f e a t u r e . The l e v e l - o r d e r i n g i s good i n ' 3 3 χ and ' 3 5 χ t i n 9 s n and ' 3 » s b the l e v e l - o r d e r i n g i s not c o r r e c t . The primary d i f f e r e n c e b e t ween the two s e t s o f n u c l e i i s t h a t i n ^ S n and ^ I s b the lowl y i n g s t a t e s w i l l have a l a r g e h i 1/2 c h a r a c t e r t o them. I n ' 3 3 χ and T 3 5 j the h i i / 2 c h a r a c t e r w i l l c>e low. Another i n t e r e s t i n g p o i n t i s t h a t some s t a t e s have markedly d i f f e r e n t s l o p e s as the p a i r i n g p o t e n t i a l i s i n c r e a s e d . These are predominantly s i n g l e p a r t i c l e s t a t e s , s i n g l e p a r t i c l e e x c i t a t i o n s above ground s t a t e . 12

Ί

Θ

β

β

a

n

(

W

1 2

1

θ

D

2

U

12

β

IV.

Conclusions/Discussion

We have found t h a t the n u c l e a r s t r u c t u r e near ^^Sn can be r e a s o n a b l y w e l l d e s c r i b e d by an ab i n i t i o e f f e c t i v e i n t e r a c t i o n w i t h core p o l a r i z a t i o n c o r r e c t i o n s . A d d i t i o n o f a s m a l l amount of a quadrupole f o r c e improves the s e p a r a t i o n o f the c a l c u l a t e d s t a t e s i n ' 3 0 s and ^ X e . y found t h a t the p a i r i n g p o t e n t i a l was needed when t h e r e are a c t i v e p r o t o n s . The p a i r i n g c o r r e c t i o n may be an i n d i c a t i o n o f the r o l e o f the 1go/2 p r o t o n e x c i t a t i o n s . For the Sn i s o t o p e s i t does not appear t h a t the e x c l u s i o n o f the 1go/2 protons has any a f f e c t on the e x c i t a t i o n s t a t e s . In ^Tsb the optimum p a i r i n g p o t e n t i a l s t r e n g t h may be somewhat l a r g e r but t h i s c o u l d not be determined. P a i r i n g i s important i n the twop r o t o n and t h r e e - p r o t o n systems. I t i s s a t i s f y i n g t o note t h a t on the whole the p a i r i n g c o r r e c t i o n t o the KK i n t e r a c t i o n can account f o r much o f the t r u n c a t i o n e f f e c t s . n

e

1

R e s u l t s a l s o seem t o i n d i c a t e t h a t the KK p o t e n t i a l o v e r e s t i mates two-body m a t r i x elements i n v o l v i n g h i 1/2 o r b i t a l s . T h i s may have been seen i n the o n e - q u a s i p a r t i c l e n u c l i d e s ; the l a r g e s t d e v i a t i o n , through o n l y 2 0 0 keV, was y i f h the s e p a r a t i o n o f the 11/2" s t a t e from the 7 / 2 + s t a t e . I n 3 4 χ results were q u i t e good. For 3 0 s , however, the f i t was not as s a t i s f y i n g : t h e r e was some l e v e l m i s o r d e r i n g and a QQ c o r r e c t i o n d i d not account f o r t h i s . These s t a t e s w i l l be l a r g e l y h-ji/2 i n c h a r a c t e r . The problems w i t h l e v e l - o r d e r i n g were most severe i n • 9 S n and '31sb. A g a i n , many o f the l o w - l y i n g s t a t e s w i l l have a l a r g e I111/2 c h a r a c t e r t o them. 1

1

β

w

e

s

a

w t

n

e

n

2


11.

STONE E T AL.

Shell-Model Calculations near Sn 132

77

F u r t h e r developments o f a g e n e r a l e f f e c t i v e i n t e r a c t i o n f o r the '32s r e g i o n w i l l have t o f o c u s on i m p r o v i n g t h e c h a r a c t e r i z a t i o n o f the h-|i/2 i n t e r a c t i o n s . T h i s w i l l be e s s e n t i a l i f we a r e t o attempt t o c a l c u l a t e t h e n e g a t i v e p a r i t y s t a t e s and t h e isomerism w i t h i n t h i s r e g i o n . We w i l l a l s o have t o c o n s i d e r o t h e r c o r r e c t i o n s t o the two-body i n t e r a c t i o n such as t h e a d d i t i o n o f density-dependent p o t e n t i a l s , t e n s o r p o t e n t i a l s and oddcomponents t o t h e two-body p o t e n t i a l . n

References [BAL76] [BLO83] [BRO67] [DEG80] [FOG81] [FOG84] [HAU76] [KAL64] [KER72] [KUO66] [KUO68] [KUO72] [LAN79] [LAN80] [SAM85] [STO86a] [STO86b] RECEIVED

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August 12, 1986


Nuclei Off the Line of Stability - American Chemical Society

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