Volume 36B, number 1 P H Y S I C S L E T T E R S 9 August 1971

C O N F I G U R A T I O N M I X I N G A N D M Y S T E R I O U S S E C O N D Z E R O S T A T E S

J . M. IRVINE D e p a r t m e n t o f The or e t i c a l P h y s i c s , Un iver s i t y o f M a n c h e s t e r , UK

Received 4 June 1971

A s imple schemat ic model is used to highlight the difficulty of obtaining a low lying excited s tate in doubly closed shel l nuclei having the same quantum numbers as the ground state in a nuclear she l l - model configuration mixing calculation.

M a n y e v e n - e v e n n u c l e i e x h i b i t a low ly ing e x - c i t e d s t a t e h a v i n g the s a m e q u a n t u m n u m b e r s a s the g r o u n d s t a t e , i . e . , T = 0, J = 0, 7r = +. Such ' m y s t e r i o u s s e c o n d z e r o s t a t e s ' a r e d i f f i c u l t to e x p l a i n in t e r m s of the s i m p l e s h e l l - m o d e l , p a r - t i c u l a r l y in the c a s e of c l o s e d s h e l l n u c l e i . T he s u g g e s t i o n h a s b e e n m a d e [1] and g e n e r a l l y a c - c e p t e d [2], t h a t the s t r u c t u r e of s u c h s t a t e s i s d o m i n a t e d by m u l t i p a r t i c l e - m u l t i h o l e c o n f i g u r a - t i o n s , e . g . , t he 6.04 M e V s t a t e in 1 6 0 i s t hough t to be p r i n c i p a l l y a f o u r p a r t i c l e - f o u r ho le s t a t e . M i c r o s c o p i c m o d e l s of the s e c o n d z e r o s t a t e b a s e d upon the B l o c h - H o r o w i t z p r o c e d u r e [3] and B r u e c k n e r - H a r t r e e - F o c k c a l c u l a t i o n s [4] f a i l to o b t a i n s u c h s t a t e s low enough in e n e r g y . S h e l l - m o d e l c a l c u l a t i o n s s u c c e e d in e x p l a i n i n g t h e s e s t a t e s only if the m a t r i x e l e m e n t s and s i n g l e p a r t i c l e e n e r g i e s a r e t r e a t e d a s £ i t t ing p a r a m e - t e r s [5]. T h e v a l u e s of the p a r a m e t e r s r e q u i r e d to f i t t he o b s e r v e d e x c i t a t i o n s p e c t r a a r e no t c o n s i s t e n t wi th the u s u a l s h e l l - m o d e l a s s u m p - t i o n s . A s i s u s u a l in s h e l l - m o d e l c a l c u l a t i o n s , any f a i l u r e to e x p l a i n the d a t a h a s b e e n a t t r i b - u t ed to the u s e of a too r e s t r i c t e d c o n f i g u r a t i o n

s p a c e • We now s e e k to i n v e s t i g a t e t h i s a s s e r t i o n t h r o u g h the u s e of a s i m p l e s c h e m a t i c m o d e l .

We s h a l l c o n s i d e r a c o n f i g u r a t i o n s p a c e c o n - t a i n i n g a c l o s e d s h e l l s t a t e , n two p a r t i c l e - t w o ho l e s t a t e s and rn f o u r p a r t i c l e - f o u r ho le s t a t e s . We s h a l l t ake the u n p e r t u r b e d e n e r g i e s to b e z e r o f o r the c l o s e d s h e l l c o n f i g u r a t i o n , 2hw f o r e a c h of the two p a r t i c l e - t w o ho le c o n f i g u r a t i o n s a n d 4t&o f o r e a c h of the f o u r p a r t i c l e - f o u r ho le c o n f i g u r a t i o n s . We s h a l l r e s t r i c t o u r s e l v e s to t w o - b o d y i n t e r a c t i o n s so t h a t t h e r e can be no m a t r i x e l e m e n t s b e t w e e n the c l o s e d s h e l l s t a t e and any of the f o u r p a r t i c l e - f o u r ho le s t a t e s . F o r the p u r p o s e s of t h i s s c h e m a t i c c a l c u l a t i o n we s h a l l u s e a v e r a g e d m a t r i x e l e m e n t s

<closed s h e l l IV] 2p - 2h} = A ( l a )

(2p - 2h I v [ 2p - 2h} = B (Ib)

(2p - 2 h IV] 4p - 4 h } = A ( l c )

<4p - 4h IV} 4p - 4h} = C ( ld)

T h u s we o b t a i n the e n e r g y m a t r i x i l l u s t r a t e d in f ig. 1.

- E A A A 2 fiw -E B A B 2/~¢6 -E A B B

A 0 A o A 0 A

DI

A 0 0 O 0 B A A A A B A A A

2fi[O - E A A A A 4hco-E C C A C 4 hco-E C A C C 4h w -E

C

Fig. 1.

4h w -E

21

Volume 36B, number 1 P H Y S I C S L E T T E R S 9 August 1971

It may be argued that this is an extremely oversimplified model of the energy matrix, however, we are looking for collective coherent effects which will produce the mysterious second zero. These effects are at a maximum in our model and the model sets upper bounds on the collectivity. A little consideration makes it clear that the probability of obtaining a myste- rious second zero state will be lower if the ' real ' energy matrix is employed.

We a r e l ook ing f o r c o n d i t i o n s on ~'~, m , A , B and C w h i c h wi l l p r o d u c e a low ly ing e x c i t e d

1 s t a t e a t ~ 2 ~w r e l a t i v e to the i n t e r a c t i n g g r o u n d s t a t e . T h e d e t e r m i n a n t of the m a t r i x in f ig. 1 f a c t o r i s e s and h a s (n - 1) r o o t s a t an e n e r g y

E 5 = 2]~w- B . (2)

T h e r e a r e (m - 1) r o o t s at an e n e r g y

E 4 = 4]/w - C. (3)

And the t h r e e r e m a i n i n g r o o t s a r e the s o l u t i o n s of the cub ic e q u a t i o n

E 3 - { 6 h w + ( n - 1 ) B + ( m - 1 ) C } E 2

+ {(2~w + ( n - 1) B ) ( 4 ~ w + (m - 1 )C)

- n ( m + l )A2} E + nA2(41~iw + ( m - 1 ) C ) = 0 . (4)

T h e g r o u n d s t a t e is c l e a r l y d e p r e s s e d b e l o w z e r o u n l e s s

4 ~ w + ( m - 1 ) C ~ - 0 < 2h-w+ ( n - l ) B (5)

C o n s i d e r i n g two n e i g h b o u r i n g m a j o r o s c i l l a t o r s h e l l s m i s a l w a y s c o n s i d e r a b l y l a r g e r t han n, T h e n fo r no r e a s o n a b l e v a l u e s of B and C [6-8] i s the c o n d i t i o n (5) s a t i s f i e d . We s h a l l t h e r e f o r e a s s u m e tha t the g r o u n d s t a t e a l w a y s l i e s b e l o w the u n p e r t u r b e d c l o s e d s h e l l c o n f i g u r a t i o n . It f o l l o w s t ha t f o r no r e a s o n a b l e v a l u e s of B and C can the s o l u t i o n s (2) or (3) p r o v i d e us w i th an e x p l a n a t i o n of the m y s t e r i o u s s e c o n d z e r o s t a t e .

We s e e k an e x p l a n a t i o n fo r the s e c o n d z e r o s t a t e s in the s o l u t i o n s of eq. (4). F i r s t l e t us o m i t the 4 p - 4 h s t a t e s , i . e . , m = 0 . ( m - l ) = 0 and c o n s i d e r t h r e e c a s e s :

(i) A = 0: A low ly ing s t a t e a p p e a r s a t ½~w p r o - v ided

( n - 1 ) B ~ - } ~ w (6)

(ii: IAI ~ IBI : No low ly ing s t a t e a p p e a r s u n - l e s s

n ~ 60 (7)

(i i i) IAI ~ 2 [ B i : No low ly ing s t a t e a p p e a r s u n - l e s s

n ~ 250 (8)

The c o n d i t i o n (6) i s j u s t p o s s i b l e f o r r e a l i s t i c i n t e r a c t i o n s and r e a l i s t i c v a l u e s of n p r o v i d e d a l l 2 p - 2 h s t a t e s b e t w e e n two m a j o r o s c i l l a t o r s h e l l s a r e i nc luded . H o w e v e r , any r e a l i s t i c i n t e r a c t i o n g i v e s IAI >~ iBl and the c o n d i t i o n s (7) and (8) t hen y i e ld u n r e a l i s t i c v a l u e s of n , i . e . , in 1 6 0 t h e r e a r e a t m o s t 40 2 p - 2 h s t a t e s of the c o r r e c t s y m m e t r y .

F o r r e a l i s t i c i n t e r a c t i o n s [6 -8 ] , we f ind 1 1 ~ e . g . ~ IA! ~ 21B I ~ 41C I and ( -B/ t iw) '~ to too

the strength of the pairing interaction is usually taken to be ~ 20/A [9] and the oscillator size parameter yields //¢o L 40A-1/3 [I0] these are consistent with B/'l~w ~ -½A-2/3 where A is the m a s s n u m b e r . A s s u m i n g IA[ ~ ~ ~w, B - ~ / ~ w and C = - ~ ~w and n = 40 c o r r e S ) o n d i n g

,£o

1 6 0 t h e n we f ind t h a t the gap b e t w e e n the l o w e s t s t a t e and the f i r s t e x c i t e d s t a t e i s an i n c r e a s i n g f u n c t i o n of m. Wi th m = 1 i t i s p o s s i b l e to h a v e

3 a gap ~ 4 gw bu t t h i s i n c r e a s e s v e r y r a p i d l y a s m i n c r e a s e s . S ince t h e r e a r e c l e a r l y m a n y f o u r p a r t i c l e - f o u r ho le c o n f i g u r a t i o n s ( the s i n - g le d e f o r m e d 4 p - 4 h s t a t e i s , of c o u r s e , a m i x - t u r e of m a n y s h e l l - m o d e l 4 p - 4 h c o n f i g u r a t i o n s ) we r e a c h the c o n c l u s i o n t h a t t h e r e a r e no s e n s i - b l e v a l u e s of the p a r a m e t e r s n, m, A, B and C

1 f o r w h i c h the f i r s t e x c i t e d s t a t e i s a t ~ 5 ~¢o w i t h i n t h i s c o n f i g u r a t i o n s p a c e .

Our s c h e m a t i c m o d e l d o e s no t a l l ow us to s ay v e r y m u c h a b o u t s p u r i o u s c e n t r e of m a s s m o t i o n . H o w e v e r , i t i s we l l known tha t if one d i a g o n a l i s e s in the s p a c e of the two p a r t i c l e - two ho le s t a t e s t h e r e i s a s i n g l e c o l l e c t i v e l y d e - p r e s s e d s t a t e and t ha t t h i s s t a t e i s a l m o s t e n - t i r e l y s p u r i o u s [11] and the only c a n d i d a t e f o r a low ly ing s e c o n d z e r o s t a t e in o u r s c h e m a t i c m o d e l i s a s i n g l e c o l l e c t i v e l y d e p r e s s e d s t a t e a r i s i n g f r o m the 2 p - 2 h s p a c e (e .g . , eq. (6)).

T h e c o n c l u s i o n t h a t we would d r a w f r o m o u r s c h e m a t i c m o d e l i s t ha t the low ly ing s e c o n d z e r o s t a t e s c a n n o t be d e s c r i b e d as s h e l l - m o d e l s t a t e s w i th in the c o n f i g u r a t i o n s p a c e w h i c h we h a v e d e s c r i b e d and t h a t they a r e m u c h m o r e l i ke ly to f ind t h e i r o p t i m u m d e s c r i p t i o n as m o - l e c u l a r s t a t e s in the a l p h a c l u s t e r m o d e l ( s ee [12]) r e q u i r i n g f o r t h e i r m i c r o s c o p i c d e s c r i p t i o n an e v e n l a r g e r c o n f i g u r a t i o n s p a c e .

R e f e r e n c e s [1] G. E. Brown, Compt. Rend. Congr. Intern. de

Phys. Nucl6aire, Vo[. 1, Centre National de la Recherche Seientifique, Pa r i s , 1964); T. EngeIand, Nucl. Phys. 72 (1965) 68.

[2] W. H. Bass ich is and G. Ripka, Phys. Le t te r s 15 (1965) 320;

22

Volume 36B, number 1 PHYSICS LETTERS 9 August 1971

P. F e d e r m a n and I. T a l m i , P h y s . L e t t e r s 15 (1965) [8] 165; G. B e n s o n and J . M . I rv ine , P r o c . P h y s . Soc, 89 [91 (1966) 249; G . E . B r o w n and A . M . G r e e n , Nucl. P h y s . 75 (1966) [10] 401.

[3] C. Bloch and J. Horowi tz , Nucl . P h y s . 8 (1958) 91. [11] [4] B .H . Brandow. Rev. Mod. P h y s . 39 (1967) 771. [51 A . P . Zucke r . B . B u c k a n d J . B. M c G r o r y . P h y s . [12]

Rev. L e t t e r s 21 (1968) 39. [6] J . P . El l io t t et a l . . Nuel. P h y s . A121 (1968) 241. [7] J . M . I rv ine and V . F . E . Puekne l l . Nucl . P h y s .

A159 (1970) 513.

T . T . S . Kuo and G . E . Brown. Nucl. P h y s . 85 (1966) 40; A l l 4 (1968} 241. L. S. K i s s l i n g e r and R. A. So rensen , Rev. Mod. P h y s . 35 (1963) 853. R. Hof s t ad t e r , F. B u m i l l e r and M . R . Yea r i an . Rev. Mod. P h y s . 30 {1958) 482. P . J . E l l i s and L. Zamick , Ann. of P h y s . 55 (1969) 61. D. M. Br ink . The a l p h a - p a r t i c l e mode l of l ight nuc le i , In te rn . School of P h y s i c s ' E n r i c o F e r m i ' , C o u r s e XXXVI.

23