Random Phase Approximation Calculations of Gamow-Teller Î²

23 Random Phase Approximation Calculations of Gamow-Teller β-Strength Functions in the A = 80-100 Region with Woods-Saxon Wave Functions 1

2

3

2

K.-L. Kratz , J. Krumlinde , G. A. Leander , and P. Möller 1

Institut für Kernchemie, Universität Mainz, D-6500 Mainz, Federal Republic of Germany Department of Mathematical Physics, Lund Institute of Technology, Lund University, Box 118, S-22100 Lund, Sweden UNISOR, Oak Ridge Associated Universities, Box 117, Oak Ridge, TN 37831

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We discuss some features of a model for calculation of β-strength functions, in particular some recent improvements. An essential feature of the model is that it takes the microscopic structure of the nucleus into account. The initial version of the model used Nilsson model wave functions as the starting point for determining the wave functions of the mother and daughter nuclei, and added a pairing interaction treated in the BCS approximation and a residual GT interaction treated in the RPA-approximation. We have developed a version of the code that uses Woods-Saxon wave functions as input. We have also improved the treatment of the odd-A Δv=0 transitions, so that the singularities that occured in the old theory are now avoided. The c a l c u l a t i o n o f t h e β - s t r e n g t h f u n c t i o n i n v o l v e s e v a l u a t i n g t h e m a t r i x e l e m e n t o f t h e β t r a n s i t i o n o p e r a t o r between t h e i n i t i a l wave f u n c t i o n φ . o f t h e m o t h e r n u c l e u s and t h e wave f u n c t i o n s o f t h e f i n a l s t a t e s i n t h e daugh t e r nucleus. A model f o r t h e β - s t r e n g t h f u n c t i o n i s e s s e n t i a l l y e q u i v a l e n t t o d e v e l o p i n g a model f o r t h e wave f u n c t i o n s ψ. We s h a l l h e r e g i v e a b r i e f o u t l i n e o f one such model d e v e l o p e d by [KRU84] and d i s c u s s i n somewhat g r e a t e r d e t a i l some new f e a t u r e s we r e c e n t l y added t o t h e m o d e l . The model d e v e l o p e d by [KRU84] uses N i l s s o n wave f u n c t i o n s as t h e s t a r t i n g p o i n t f o r c o n s t r u c t i n g t h e wave f u n c t i o n φ . o f t h e g r o u n d s t a t e o f t h e m o t h e r n u c l e u s and t h e wave f u n c t i o n s

o f t h e g r o u n d and e x c i t e d s t a t e s

of

the daughter nucleus. I t i s i n s t r u c t i v e to consider the e f f e c t of the various i m p r o v e m e n t s , w h i c h have been added t o t h e model beyond t h e use o f N i l s s o n wave f u n c t i o n s , on some p a r t i c u l a r t r a n s i t i o n , f o r w h i c h a l s o an e x p e r i m e n t a l value i s a v a i l a b l e . I n r é f . [B0H75] t r a n s i t i o n s i n t h e r a r e e a r t h r e g i o n a r e d i s c u s s e d and r é f .

[KRU84] s e l e c t e d t h e t r a n s i t i o n

[523 | ]

p

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n

for

Yb. The l o g f t v a l u e f o r t h i s t r a n s i t i o n i s 4 . 8 . The c o n n e c t i o n between t h e t r a n s i t i o n m a t r i x e l e m e n t and t h e f t v a l u e i s g i v e n by t h e f o r m u l a [B0H75] 1 7 0

2

- I 4

ft(sec)

2

Thus t h e e x p e r i m e n t a l v a l u e f o r t h e above m a t r i x may be o b t a i n e d . t h a t f o r t h e above t r a n s i t i o n ^ ±

It

χ ρ

We f i n d

s 0.04

is pointed out in r é f .

[B0H75] t h a t a p u r e d e f o r m e d o s c i l l a t o r

0097-6156/ 86/ 0324-0159506.00/ 0 © 1986 American Chemical Society

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

model,

160

NUCLEI OFF THE LINE OF STABILITY

assuming good a s y m p t o t i c quantum numbers, w o u l d y i e l d t h e v a l u e 1 f o r t h e above m a t r i x e l e m e n t . W i t h N i l s s o n wave f u n c t i o n s we o b t a i n 0 . 8 6 . The s m a l l

- -

9

r e d u c t i o n i s due t o t h e m i x i n g o f o r b i t a l s due t o t h e £«s and £ t e r m s i n t h e s i n g l e - p a r t i c l e p o t e n t i a l and a s m a l l amount o f distortions. Now, w i t h t h e a d d i t i o n o f p a i r i n g , t h e i n c r e a s i n g c o m p l e x i t y o f t h e wave f u n c t i o n s r e d u c e s the c a l c u l a t e d value o f the t r a n s i t i o n m a t r i x element t o 0 . 3 7 . According t o [B0H75] one can e x p e c t a r e d u c t i o n by a b o u t a f a c t o r o f 4 due t o p a i r i n g f o r l e v e l s c l o s e t o t h e Fermi s u r f a c e , because t h e r e d u c t i o n i s u u o r ν ν and u ρ η ρ η and ν a r e ^ Fermi s u r f a c e . T h a t we o b t a i n e d a s m a l l e r r e d u c t i o n i n 1

a

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t h i s p a r t i c u l a r case i s due t o t h e f a c t t h a t t h e e n t e r i n g p a i r i n g f a c t o r s 1 2 l a r g e r than

An a d d i t i o n a l

r e d u c t i o n o f +

i n t h e model

oped by [KRU84] comes a b o u t f r o m t h e a d d i t i o n o f a r e s i d u a l s p e c i f i c t o GT d e c a y , VQ-|. = : β

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o f l o w - e n e r g y GT decay r a t e s , and as we saw i n o u r example a b o v e , t h e s t r e n g t h was r e d u c e d by a b o u t a f a c t o r o f f o u r . The c a l c u l a t e d s t r e n g t h i s s t i l l a b o u t a f a c t o r o f 2 l a r g e r than the observed s t r e n g t h . The c a l c u l a t e d s t r e n g t h c o u l d be f u r t h e r r e d u c e d by i n c r e a s i n g f u r t h e r t h e s t r e n g t h χ o f t h e V* j G

i n t e r a c t i o n , w h i c h i s s e t a t χ = 23/A MeV by [KRU84] as i s a l s o done by most other investigators. However, t h e s t r e n g t h χ i s d e t e r m i n e d by t h e r e q u i r e m e n t t h a t t h e c a l c u l a t e d p o s i t i o n o f t h e g i a n t Gamow-Teller resonance agrees w i t h the experimental r e s u l t s . T h u s , t h e mechanism b e h i n d t h e r e m a i n i n g f a c t o r - o f two d i s c r e p a n c y between t h e e x p e r i m e n t a l and c a l c u l a t e d s t r e n g t h i s t h o u g h t t o be o f a d i f f e r e n t o r i g i n . Two mechanisms t h a t a r e b e i n g i n v e s t i g a t e d a r e c o u p l i n g s t o 2p2h s t a t e s [BER82] and t h e Δ ( 1 2 3 2 ) i s o b a r [ B 0 H 8 1 ] . To a c c o u n t f o r t h e m i s s i n g s t r e n g t h i n h a l f - l i f e c a l c u l a t i o n s we d i v i d e t h e c a l c u l a t e d s t r e n g t h by 2 f o r such a p p l i c a t i o n s . The c a l c u l a t e d s t r e n g t h s shown i n t h i s c o n t r i b u t i o n a r e n o t d i v i d e d by 2 however. We have added some new f e a t u r e s t o t h e model d e s c r i b e d a b o v e , w h i c h was d e v e l o p e d by [KRU84]. I n p a r t i c u l a r we have o b s e r v e d t h a t t h e p e r t u r b a t i o n e x p r e s s i o n s f o r t h e Δν = 0 t r a n s i t i o n s f o r odd-mass and odd n u c l e i used by [KRU84] ( e q s . ( 4 3 ) - ( 4 7 ) i n t h a t p a p e r ) b r e a k down o c c a s i o n a l l y . Similar e x p r e s s i o n s have a l s o been used e a r l i e r by [HAL67] and [RAN73]. When t h e e x p r e s s i o n s b r e a k down, a s i n g l e Δν = 0 t r a n s i t i o n may have a s t r e n g t h t h a t i s many t i m e s t h e sum r u l e S" - s t = 3 ( N - Z ) f o r t h e Δν = 2 t r a n s i t i o n s . However, Ρ Ρ t h e e q u a t i o n s ( 4 3 ) - ( 4 7 ) o f r é f . [KRU84] can be m o d i f i e d somewhat t o remove this difficulty. The e q u a t i o n s f o r t h e Δν = 0 s t r e n g t h c o n t a i n sums o v e r t e r m s w i t h a m p l i t u d e s A (nu>) = l / ( E " " )· q u a n t i t i e s u> a r e t h e r o o t s o f t h e RPA e q u a t i o n s and a r e " c l o s e " t o t h e a s y m p t o t e s E + E but not E

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c l o s e enough t o cause any s i n g u l a r i t y i n t h e Δν = 2 t r a n s i t i o n s t r e n g t h s . By " a c c i d e n t " t h e y may be so c l o s e t o t h e q u a n t i t y E - E t h a t t h e p e r t u r b a t i o n p

e x p a n s i o n b r e a k s down and a s i n g u l a r i t y o c c u r s .

R

T h i s s i n g u l a r i t y can be


23.

Calculations of β-Strength Functions

KRATZ ET AL.

161

removed by i n t r o d u c i n g a w i d t h d , and m o v i n g t h e p o l e Ε plane. T h u s , we r e p l a c e Α ( η ω ) =

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I n f i g s , l a - l b we show t h e r e s u l t s f o r t h e o r i g i n a l model ( a ) w h i c h e x h i b i t s a p r o n o u n c e d s i n g u l a r i t y and f o r d = 0 . 1 MeV and d = 1.0 MeV ( b ) . In f i g . l b we have i n d i c a t e d t h e d = 0 . 1 MeV r e s u l t s w i t h dashed l i n e s . We see t h a t d i f f e r e n c e s between t h e d = 0 . 1 MeV and d = 1.0 MeV r e s u l t s a r e m i n o r and t h a t t h e r e s u l t s t h e r e f o r e a r e i n s e n s i t i v e t o t h e w i d t h d , as s h o u l d b e . We have s e l e c t e d d = 0 . 1 MeV f o r o u r c a l c u l a t i o n s . 1

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ENERGY CIWV) ENERGY (MeV) Fig. lb Fig. la I n f i g . 2 we e x h i b i t some r e s u l t s f o r a sequence o f Rb n u c l e i w i t h t h i s modification included. We have a l s o c o r r e c t e d an e r r o r t h a t o c c u r e d i n an e a r l i e r v e r s i o n o f t h e c o m p u t e r code f o r some Δν = 0 t r a n s i t i o n s i n t h e s p h e r i c a l case. The e r r o r was u s u a l l y s m a l l . Compared t o [KRU84] f i g . 2 c o n t a i n s o n l y t h e s e two m o d i f i c a t i o n s and t h e d i f f e r e n c e s r e l a t i v e t o t h e e a r l i e r r e s u l t s are s m a l l . However t h e i n t r o d u c t i o n o f t h e w i d t h d i s v e r y i m p o r t a n t i n some o t h e r cases as can be seen i n f i g . 1 . The c a l c u l a t e d r e s u l t s a r e d i s c u s s e d e x t e n s i v e l y i n [ K R U 8 4 ] , t o w h i c h we r e f e r f o r a more c o m p l e t e d i s c u s s i o n . H e r e , l e t us j u s t n o t e t h a t f o r t h e t o p f o u r s p e c t r a i n f i g . 2 we used t h e p a r a m e t e r s e t "A = 1 0 0 " and f o r t h e l o w e r f o u r f i g u r e s we used t h e p a r a m e t e r s e t "N = 6 0 " . The Ν = 60 s e t r e p r o d u c e s b e s t t h e l a r g e i n c r e a s e i n s t r e n g t h o f t h e l o w - e n e r g y peak t h a t i s seen e x p e r i mentally in Rb r e l a t i v e to Rb. The Ν = 60 s i n g l e - p a r t i c l e p a r a m e t e r s e t was a d j u s t e d t o r e p r o d u c e r e s u l t s i n t h e v i c i n i t y o f t h e Ν = 56 s u b s h e l l [ A Z U 7 8 ] , so i t i s t o be e x p e c t e d t h a t t h i s s e t g i v e s t h e b e t t e r d e s c r i p t i o n o f some f e a t u r e s i n t h i s r e g i o n . One m a j o r d i f f i c u l t y i n t h e N i l s s o n model i s t h e d e t e r m i n a t i o n o f t h e s i n g l e - p a r t i c l e p a r a m e t e r s κ and μ f o r v a r i o u s r e g i o n s o f n u c l e i . Usually t h e s e p a r a m e t e r s a r e d e t e r m i n e d by a d j u s t i n g c a l c u l a t e d s i n g l e - p a r t i c l e l e v e l s t o experimental data. S i n c e κ and μ can v a r y r a t h e r u n p r e d i c t a b l y f r o m r e g i o n t o r e g i o n , t h e model i s t h e r e f o r e somewhat u n s u i t a b l e f o r e x t r a p o l a t i o n s t o unknown r e g i o n s o f n u c l e i . W i t h t h e aim o f b e i n g a b l e t o c a l c u l a t e p r o p e r t i e s f o r such n u c l e i more r e l i a b l y we a r e now d e v e l o p i n g t h e code t o a c c e p t wave § 5

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of nuclei. See f o r example t h e s t u d i e s [ M 0 L 8 1 a ] , [M0L81b] and [BEN84] o f t h e folded-Yukawa p o t e n t i a l , which covers the e n t i r e p e r i o d i c system. In t h i s i n i t i a l s t u d y we s h a l l use Woods-Saxon wave f u n c t i o n s , m a i n l y because such a


164


code has been made a v a i l a b l e t o us by [NAZ84] i n a f o r m t h a t r u n s on VAX and NORSK DATA c o m p u t e r s . The f o l d e d - Y u k a w a code c u r r e n t l y r u n s o n l y on CDC computers. The Woods-Saxon model i s d e s c r i b e d i n [DUD82]. F i g u r e 3 shows β - s t r e n g t h f u n c t i o n s c a l c u l a t e d w i t h Woods-Saxon wave f u n c t i o n s f o r a sequence o f Rb n u c l e i . Above a l l b u t t h e l a s t o f t h e c a l c u l a t e d s t r e n g t h f u n c t i o n s , w h i c h have t h e words BETA STRENGTH a l o n g t h e v e r t i c a l a x i s , t h e r e a r e p l o t s o f e x p e r i m e n t a l r e s u l t s f r o m [KRA83] and [ K R A 8 1 ] . The e x p e r i m e n t a l r e s u l t s have t h e l a b e l B ' ( G T ) a l o n g t h e v e r t i c a l a x i s . The r e s u l t s w i t h Woods-Saxon w a v e - f u n c t i o n s a r e s i m i l a r t o t h e r e s u l t s o b t a i n e d w i t h t h e o s c i l l a t o r m o d e l , and a l s o a g r e e f a i r l y w e l l w i t h e x p e r i m e n t . Here we s h a l l j u s t comment on two a s p e c t s o f t h e r e s u l t s . F i r s t , the l a r g e i n c r e a s e i n s t r e n g t h seen e x p e r i m e n t a l l y when g o i n g f r o m R b t o R b i s s l i g h t l y l e s s w e l l r e p r o d u c e d w i t h t h e Woods-Saxon w a v e - f u n c t i o n s , t h a n w i t h t h e N i l s s o n wave f u n c t i o n s w i t h t h e Ν = 60 s e t . However, t h e s e t Ν = 60 was o p t i m i z e d t o r e p r o d u c e t h e Ν = 56 s u b s h e l l c o n d i t i o n s . The Woods-Saxon c a l c u l a t i o n was p e r f o r m e d w i t h a " u n i v e r s a l parameter s e t [DUD82], which should i n g e n e r a l be more r e l i a b l e f o r e x t r a p o l a t i o n s f a r f r o m s t a b i l i t y . Second, f o r t h e d e f o r m e d R b and R b r e s u l t s t h e r e i s v e r y l i t t l e s t r e n g t h a t low e n e r g y , i n c o n t r a s t t o t h e e x p e r i m e n t a l s i t u a t i o n and t h e m o d i f i e d o s c i l l a t o r results. However, t h e Woods-Saxon c a l c u l a t i o n was r u n f o r a d e f o r m a t i o n t h a t was somewhat s m a l l e r t h a n t h e a p p r o p r i a t e v a l u e f o r R b and R b . Due t o c i r c u m s t a n c e s beyond o u r c o n t r o l , we have o n l y been a b l e t o p r e s e n t v e r y few i n i t i a l r e s u l t s w i t h t h e Woods-Saxon wave f u n c t i o n s h e r e . We hope, h o w e v e r , t o e x p l o r e t h e model more f u l l y i n t h e n e a r f u t u r e , i n p a r t i c u l a r t o r u n R b and R b w i t h a more a p p r o p r i a t e v a l u e o f β and t o e x p l o r e 9 3

9 5

1 1

9 7

9 9

9 7

9 7

9 9

9 9

2

a d d i t i o n a l regions o f n u c l e i f a r from s t a b i l i t y , f o r which a p p l i c a t i o n s the Woods-Saxon s i n g l e - p a r t i c l e model s h o u l d be advantageous compared t o t h e Nilsson model, which i s less r e l i a b l e f o r e x t r a p o l a t i o n s . Acknowledgments We a r e g r a t e f u l t o J . Dudek and W. N a z a r e w i c z Woods-Saxon code a v a i l a b l e t o u s .

f o r making a copy o f

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References [AZU78] R.E. Azuma, G. L. Borchert, L.C. Carraz, P.G. Hansen, B. Jonson, S. Mattson, O.B. Nielsen, G. Nyman, I. Ragnarsson and H.L. Ravn, Phys. Lett. 80B 4 (1978). [BEN84] R. Bengtsson, P. Möller, and J.R. Nix, Phys. Scr. 29 402 (1984). [BER82] G.F. Bertsch, and I. Hamamoto, Phys. Rev. C26 1323 (1982). [BOH75] A. Bohr, and B.R. Mottelson, Nuclear Structure, vol. II (Benjamin, New York, 1975) pp. 306, 307. [BOH81] A. Bohr and B. R. Mottelson, Phys. Lett. 100B 10 (1981). [DUD82] J. Dudek, Z. Szymanski, T. Werner, A. Faessler, and C. Lima, Phys. Rev. C26 1712 (1982). [HAL67] J.A. Halbleib, S r . , and R.A. Sorensen, Nucl. Phys. A98 542 (1967). [KRA81] K.L. Kratz, H. Ohm, A. Schröder, H. Gabelmann, W. Ziegert, H. V. Klapdor, J . Metzinger, T. Oda, B. Pfeiffer, G. Jung, L. Alquist and G.I. Crawford, Proc. 4th Int. Conf. on nuclei far from s t a b i l i t y , Helsinger, 1981 (CERN 82-09, Geneva, 1981) p. 317. [KRA83] K.L. Kratz, Priv. comm. (1983). [KRU84] J . Krumlinde, and P. Möller, Nucl. Phys. A417 419 (1984). [MOL81a]P. Möller, and J.R. Nix, Nucl. Phys. A361 117 (1981). [MOL81b]P. Möller, and J . R. Nix, At. Data Nucl. Data Tables 26 165 (1981). [NAZ84] W. Nazarewicz, and J . Dudek, Priv. com. (1984). [RAN73] J . Randrup, Nucl. Phys. A207 209 (1973). RECEIVED August 20, 1986


Random Phase Approximation Calculations of Gamow-Teller Î²

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