MOMENTUM DISTRIBUTIONS IN DEFORMED NUCLEI

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MOMENTUM DISTRIBUTIONS IN DEFORMED NUCLEI

E. Moya de Guerra, J. Caballero, P. Sarriguren

To cite this version:

E. Moya de Guerra, J. Caballero, P. Sarriguren. MOMENTUM DISTRIBUTIONS IN DEFORMED NUCLEI. Journal de Physique Colloques, 1987, 48 (C2), pp.C2-295-C2-299.

�10.1051/jphyscol:1987245�. �jpa-00226514�

JOURNAL DE PHYSIQUE

C o l l o q u e C 2 , supplgment a u n o 6, Tome 48, j u i n 1987

E. MOYA DE GUERRA, J.A. CABALLERO and P. SARRIGUREN I.E.M., C.S.I.C., Serrano 119, SP-28006 Madrid, Spain

A b s t r a c t . We d i s c u s s some d i s t i n g u i s h i n g f e a t u r e s of momentum d i s t r i - b u t i o n s i n deformed n u c l e i .

Q u a s i e l a s t i c e l e c t r o n s c a t t e r i n g p r o v i d e s a v e r y powerful t o o l t o ( 1

i n v e s t i g a t e momentum d i s t r i b u t i o n s of p r o t o n s bound i n n u c l e i

.

U n t i l 1 now t h e e x p e r i m e n t a l i n f o r m a t i o n a v a i l a b l e h a s been a n a l y z e d i n t e r m s o f t h e i n d e p e n d e n t p a r t i c l e model assuming n u c l e i t o b e

s p h e r i c a l l y symmetric, i n which c a s e t h e s p e c t r a l f u n c t i o n t a k e s t h e s i m p l e form

where a s t a n d s f o r t h e quantum numbers of t h e s i n g l e p a r t i c l e o r b i t a l ( n l j ) ; and n a ( p ) , Ear Na a r e , r e s p e c t i v e l y , t h e momentum d i s t r i b u t i o n ( n o r m a l i z e d t o I ) , t h e b i n d i n g e n e r g y and t h e o c c u p a t i o n number o f said o r b i t a l .

I t i s w e l l known t h a t many n u c l e i a d q u i r e a deformed a x i a l l y symmetric shape i n t h e i r ground s t a t e . For t h e s e n u c l e i t h e ground s t a t e c h a r g e d e n s i t y i n t h e i n t r i n s i c frame i s g i v e n by

s i m i l a r l y t h e momentum d i s t r i b u t i o n o f t h e s i n g l e p a r t i c l e o r b i t a l s w i l l have t h e form

While t h e e f f e c t s of n u c l e a r d e f o r m a t i o n i n e l a s t i c and i n e l a s t i c . s c a t t e r i h g have been e x t e n s i v e l y d i s c u s s e d ( 2 ' , t h o s e i n q u a s i - e l a s t i c

uppor ported ^{in part} by CAICYT under grant No 1179-84

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987245

C2-296 JOURNAL DE PHYSIQUE

e l e c t r o n s c a t t e r i n g have, t o o u r knowledge, n e v e r b e f o r e been c o n s i d e r e d .

F o r t h e p r o c e s s A ( e , e ' p V ) B t h e dependence of t h e n u c l e a r s t r u c - t u r e c a n be f a c t o r i z e d o u t ( i n p l a n e wave impulse a p r o x i m a t i o n

(PWIA)) i n t h e s p e c t r a l f u n c t i o n ( ' )

where :Y J A , r e p r e s e n t s t h e wave f u n c t i o n of t h e ground s t a t e of t h e t a r g e t , w i t h a n g u l a r momentum J A and t o t a l b i n d i n g energy E,: and YB J t h e wave f u n c t i o n of t h e r e s i d u a l n u c l e u s B , w i t h b i n d i n g e n e r g y

B

E ~ . I n t h i s p a p e r we d i s c u s s t h e form of t h e s p e c t r a l f u n c t i o n f o r a x i a l l y symmetric deformed n u c l e i . To s i m p l i f y t h e d i s c u s s i o n we u s e t h e B o h r - M o t t e l ~ o n ' ~ ) f a c t o r i z a t i o n a p p r o x i m a t i o n and N i l s s o n model ( 4 ) s i n g l e p a r t i c l e wave f u n c t i o n s .

For a x i a l l y symmetric deformed n u c l e i we w r i t e t h e wave f u n c t i o n i n t h e l a b o r a t o r y system i n t e r m s of t h e r e l a t i v e o r i e n t a t i o n of t h e body f i x e d system and t h e i n t r i n s i c wave f u n c t i o n . For a g i v e n band, KA, i n t h e A n u c l e u s we w r i t e (same c o n v e n t i o n s and n o t a t i o n s as t h o s e i n r e f . ( 5 ) a r e used h e r e ) ,

and s i m i l a r l y f o r t h e s t a t e s of t h e r e s i d u a l n u c l e u s B. With t h i s approximation t h e s p e c t r a l f u n c t i o n t a k e s t h e form

where, i n p a r t i c u l a r , i f t h e t a r g e t A i s even-even, we have f o r t h e t r a n s i t i o n from t h e ground s t a t e (JA=KA=VA=O) t o a o n e q u a s i p a r t i c l e s t a t e (VB=l,KB) i n t h e r e s i d u a l n u c l e u s

where vK 2 i s t h e p r o b a b i l i t y f o r f i n d i n g t h e (deformed) s i n g l e p a r t i c l e B ( s . p . ) s t a t e KB o c c u p i e d i n t h e ground s t a t e of t h e t a r g e t n u c l e u s , and C Z j a r e t h e a m p l i t u d e s o f t h e s.p. s t a t e KB

i n t h e s p h e r i c a l b a s i s

Hence, one can s e e t h a t t h e s p e c t r a l f u n c t i o n s i n t h e deformed and s p h e r i c a l c a s e s a r e q u i t e d i f f e r e n t . F i r s t of a l l , t h e s i n g l e p a r t i c l e b i n d i n g e n e r g i e s Ea a p p e a r i n g i n eq. ( 1 ) a r e r e p l a c e d i n eq. ( 4 ) by

r o t 1 2 J B + l /2

where EJ ( = %(JB(JB+l)-K +6 (-1)

K ~ , 1 / 2

i s t h e r o t a t i o n a l e n e r g y and E t h e q u a s i p a r t i c l e e n e r g y . S i n c e t h i s s p e c t r u m i s i n g e n e r a l much more d e n s e t h a n t h a t c o r r e s p o n d i n g t o t h e s . p . e n e r g i e s E f o r a g i v e n r a n g e of m i s s i n g e n e r g i e s E t h e s p e c t r a l

a t

f u n c t i o n w i l l have many more and c l o s e r p e a k s t h a n i n t h e s p h e r i c a l c a s e . With t h e e x p e r i m e n t a l r e s o l u t i o n a v a i l a b l e a t p r e s e n t , t h e r o t a t i o n a l l e v e l s i n a g i v e n K band may n o t i n g e n e r a l be r e s o l v e d , s e v e r a l p e a k s t h u s r e s u l t i n g i n a s i n g l e b r o a d e r one. On t h e o t h e r hand f o r a narrow r a n g e o f m i s s i n g e n e r g i e s i n t h e neighbourhood of a g i v e n q u a s i p a r t i c l e ( o r h o l e ) energy t h e momentum d i s t r i b u t i o n i n t h e deformed c a s e w i l l i n g e n e r a l be q u i t e d i f f e r e n t . While i n t h e s p h e r i c a l c a s e t h e momentum d i s t r i b u t i o n w i l l be t h a t of a s i n g l e o r b i t a l nu, i n t h e deformed c a s e it w i l l be a l i n e a r combination of many o r b i t a l s .

To i l l u s t r a t e t h i s p o i n t we c o n s i d e r i n what f o l l o w s t h e momentum d i s t r i b u t i o n s o f t h e l a s t o c c u p i e d p r o t o n s t a t e s i n 2 8 ~ i , u s i n g t h e N i l s s o n model and n e g l e c t i n g p a i r i n g c o r r e l a t i o n s . I n t h e N i l s s o n model, w i t h d e f o r m a t i o n p a r a m e t e r i n t h e r a n g e -,256,$.2, t h e l a s t o c a u p i e d s t a t e s i n 2 8 ~ i a r e t h e lower K' =

P I t',

momentum d i s t r i b u t i o n , n ( p ) , o b t a i n e d by i n t e g r a t i o n o f t h e s p e c t r a l f u n c t i o n o v e r a narrow energy r a n g e (c 1 MeV) around t h e minimal removal e n e r g y , we c a n r e s t r i c t o u r a t t e n t i o n t o j u s t t h e s e t h r e e

2 7 ( 6 ) K s.p. s t a t e s . Moreover t a k i n g i n t o a c c o u n t t h e s p e c t r u m of A x one s e e s t h a t o n l y t h e l o w e s t J B K B = 7

'* , - I-'

2 2 '

c o n t r i b u t e i n t h i s e n e r g y r a n g e . I n f i g . 1 w e show t h e momentum d i s t r i b u t i o n , n ( p ) =nSl2 5 / 2 + n 1 / 2 / 2 ( i n u n i t s of (b/JfE) 3 ) a s a

JOURNAL DE PHYSIQUE

Fig. 1 - Momentum d i s t r i b u t i o n f o r 6=-.2 (-.-)

,

. .

f u n c t i o n of P ( i n u n i t s of b-' )

,

1

.

Fig. 2

-

experimental d a t a and f i t

.

0

2 W

... -

.., _C: 2 7

2 8 ~ i ( e , e l p ) AI

30

-

E <

I I

50 100 IS0 200 250 P ( U e w

The f i t i n r e f . ( 7 ) was obtained by using q. (1)

,

5/2 1/2

i n d i s t o r t e d wave Born a p p r o x i m a t i o n ( D W I A ) w i t h wave f u n c t i o n s o b t a i n e d from a Woods-Saxon p o t e n t i a l . Our r e s u l t shown by t h e

d a s h e d l i n e i n f i g . 2 c o n t a i n s a r e n o r m a l i z a t i o n f a c t o r Q = . 5 t o t a k e i n t o a c c o u n t t h e o v e r a l l r e d u c t i o n i n g o i n g f r o m PWIA t o D W I A ' ~ ) . A l s o t h e b v a l u e u s e d (b = 1.97 fm) was c h o s e n l a r g e t o t a k e i n t o a c c o u n t t h e d i s p l a c e m e n t t o l o w e r momenta i n g o i n g from PWIA t o DWIA")

.

R e f e r e n c e s

( 1 ) S. F r u l l a n i and J . Mongey, Adv. i n Nucl. Phys. V.14 ( 1 9 8 5 ) , J . W . Negele a n d E.W. Vogt, Eds.

( 2 ) W. B e r t o z z i , Nucl. Phys.

A374

E. Moya d e G u e r r a a n d A.E.L. D i e p e r i n k , Phys. Rev. (1978) 1596.

( 3 ) A. Bohr a n d B. M o t t e l s o n , N u c l e a r S t r u c t u r e , Vol. I1 (Benjamin, N . Y . , 1 9 7 5 ) .

( 4 ) S.G. N i l s s o n , Mat. Fys. Medd. Dan. Vid. S e l s k .

2,

138

( 6 ) C.M. L e d e r e r a n d V.S. S h i r l e y , Eds., T a b l e of L s o t o p e s 7th e d . ( J o h n W i l l e y & S o n s , N . Y . , 1 9 7 8 ) .

( 7 ) J . Mougey e t a l , Nucl. Phys.

A262