STRETCHED EXCITATIONS AS A MEANS TO STUDY THE SPIN MODES OF THE NUCLEUS WITH ELECTROMAGNETIC AND HADRONIC PROBES

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STRETCHED EXCITATIONS AS A MEANS TO STUDY THE SPIN MODES OF THE NUCLEUS WITH ELECTROMAGNETIC AND HADRONIC

PROBES

R. Lindgren

To cite this version:

R. Lindgren. STRETCHED EXCITATIONS AS A MEANS TO STUDY THE SPIN MODES OF

THE NUCLEUS WITH ELECTROMAGNETIC AND HADRONIC PROBES. Journal de Physique

Colloques, 1984, 45 (C4), pp.C4-433-C4-451. �10.1051/jphyscol:1984433�. �jpa-00224099�

JOURNAL DE PHYSIQUE

Colloque C4, suppl6ment a u n03, Tome 45, m a r s 1984 page C4-433

STRETCHED EXCITATIONS AS A MEANS TO STUDY THE S P I N MODES OF THE NUCLEUS WITH ELECTROMAGNETIC AND HADRONIC PROBES

R.A. L i n d g r e n

Department of Physics and Astronomy, University of Massachusetts, Amherst MA, U.S.A.

~ & s- ~Nous a v o n s 6 t u d i C l ' e x c i t a t i o n d e s t r a n s i t i o n s d e moment a n g u l a i r e e n nous ~ &

s e r v a n t d e s o n d e s & l e c t r o m a g n & t i q u e s e t h a d r o n i q u e s . L 1 i n t e n s i t 6 s p e c t r o s c o p i q u e s e d g d u i t d e s e t u d e s d e ( e , e l ) , ( p , p l ) , ( I T , I T ' ) e t ( p , n ) d e s & t a t s a l l o n g e s

s 6 l e c t i o n n 6 s . Dans n o s e t u d e s il y a d e l l i n t e n s i t & a b s e n t e d e s e x c i t a t i o n s d e moment a n g u l a i r e e l e v e . Ce f a i t i m p l i q u e que d ' a u t r e s mgcanismes i m p o r t a n t s d ' e x t i n c t i o n e x i s t e n t p a r c e q u e l e mecanisme A - N - ~ n e c o n t r i b u e n i a u x e x c i t a t i o n s d e moment a n g u l a i r e & l e v 6 n i aux t r a n s i t i o n s i s o s c a l a i r e s . Les & t a t s a l l o n g g s s e r v e n t d ' g t a l o n pour v g r i f i e r 3 l a f o i s 1 1 i n t e n s i t 6 t e n s e u r nuclgon-nucl&on d a n s l e phgnom&ne d e d i f f u s i o n d e s p r o t o n s , e t l ' i n t e n s i t c o r b i t a l e d e moment a n g u l a i r e e n t r e l e p i o n e t l e n u c l g o n d a n s l e ph6nomene d e d i f f u s i o n d e s p i o n s . L e s

r e s u l t a t s s ' a c c o r d e n t d ' u n e f a c o n r e m a r q u a b l e a v e c l e s p r 6 d i c t i o n s d u m o d ~ l e D W I A .

A b s t r a c t - E x c i t a t i o n o f s p i n t r a n s i t i o n s u s i n g e l e c t r o m a g n e t i c and h a d r o n i c p r o b e s i s d i s c u s s e d . S p e c t r o s c o p i c s t r e n g t h s a r e deduced from s t u d i e s o f ( e , e l ) , ( p , p l ) , ( " , I T ' ) and ( p , n ) f o r s e l e c t e d s t a t e s . The m i s s i n g i s o v e c t o r a n d i s o s c a l a r s t r e n g t h s imply t h e e x i s t e n c e of o t h e r i m p o r t a n t quenching mechanisms s i n c e t h e A - N - ~

mechanism d o e s n o t c o n t r i b u t e t o h i g h s p i n e x c i t a t i o n s o r t o i s o s c a l a r t r a n s i t i o n s . The s t r e t c h e d s t a t e s a r e used a s benchmarks t o t e s t t h e nucleon-nucleon t e n s o r f o r c e i n p r o t o n s c a t t e r i n g and t h e pion-nucleon s p i n o r b i t f o r c e i n p i o n s c a t t e r i n g . The r e s u l t s a r e i n s t r i k i n g agreement w i t h p r e d i c t i o n s of t h e DWIA model.

INTRODUCTION

I n t h i s s e s s i o n o n s p i n e x c i t a t i o n s i n n u c l e i we a r e i n t e r e s t e d i n t h e s t u d y o f t h e n u c l e a r r e s p o n s e t o n u c l e o n o p e r a t o r s of t h e o~ t y p e . It h a s b e e n shown1 t h a t e f f e c t s o n t h e s p i n - i s o s p i n o p e r a t o r s d u e t o p i o n i c o l a r i z a t i o n o f t h e n u c l e a r medium a r e s t r o n g l y momentum d e p e n d e n t . S t ~ d i e s ~ , ~ . ~ . ~ of M1 and GT r e s o n a n c e s have

p r o v i d e d i m p o r t a n t and f u n d a m e n t a l i n f o r m a t i o n o n q u e n c h i n g mechanisms i n t h e s t a t i c l i m i t of q = 0 . S t u d i e s of h i g h m u l t i p o l e m a g n e t i c r e s o n a n c e s s h o u l d p r o v i d e v a l u a - b l e i n f o r m a t i o n o n w h e t h e r medium e f f e c t s a r e i m p o r t a n t a t h i g h q . Y e t , quenching e f f e c t s s u c h a s d e l t a p a r t i c l e - n u c l e o n h o l e a d m i x t u r e s f o r t h e m i s s i n g GT s t r e n g t h a r e n o t e x p e c t e d t o s t r o n g l y c o u p l e t o t h e h i g h e r m u l t i p o l e m a g n e t i c r e s o n a n c e s . 6 I n t h i s t a l k i n t h i s s e s s i o n I would l i k e t o p r e s e n t r e s u l t s from a s t u d y of t h e h i g h e r m a g n e t i c m u l t i p o l e r e s o n a n c e s o b s e r v e d i n ( e , e l ) e x p e r i m e n t s a t t h e Bates-MIT L i n e a r A c c e l e r a t o r C e n t e r , ( p , p f ) a n d ( p , n ) e x p e r i m e n t s a t t h e I n d i a n a U n i v e r s i t y C y c l o t r o n , and ( n , ~ ' ) e x p e r i m e n t s a t t h e Los Alamos Meson P h y s i c s F a c i l i t y t h a t show h i g h s p i n " s t r e t c h e d " s t a t e s a r e a l s o quenched compared t o n u c l e a r s t r u c t u r e c a l c u - l a t i o n s by amounts s i m i l a r t o t h a t o b s e r v e d i n Y1 and GT r e s o n a n c e s . The a b s e n c e of a n e x p l a n a t i o n of t h e m i s s i n g s t r e n g t h h e i g h t e n s t h e i n t e r e s t i n t h e s t u d y o f s t r e t c h e d e x c i t a t i o n s , p a r t i c u l a r l y i n v i e w of t h e i r e x p e c t e d s i m p l e s t r u c t u r e . For example, o n l y o n e n u c l e o n s p i n m a t r i x e l e m e n t i s e x p e c t e d t o c o n t r i b u t e t o t h e e l e c t r o m a g n e t i c and h a d r o n i c c r o s s s e c t i o n , ' no o r b i t a l c u r r e n t c o n t r i b u t i o n s a r e e x p e c t e d i n ( e , e l ) ,7 a n d two-body meson-exchange c u r r e n t s a r e s m a l l and c a l c u l - a b l e . 8 P e r h a p s we h a v e t h e o p p o r t u n i t y h e r e t o s t u d y t r a n s i t i o n s where t h e

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984433

C4-434 JOURNAL DE PHYSIQUE

p r i n c i p a l quenching mechanism i s m o s t l y d u e t o c o n v e n t i o n a l n u c l e a r s t r u c t u r e . S i n c e t h e s u b n u c l e o n i c e f f e c t s a r e e x p e c t e d t o b e weak, a t t e n t i o n c a n b e f o c u s s e d on t h e n u c l e o n - n u c l e o n c o r r e l a t i o n s and c o r e p o l a r i z a t i o n e f f e c t s . The p r e s e n c e of u n e x p e c t e d l y l a r g e ground s t a t e c o r r e l a t i o n s w i l l h a v e g e n e r a l i m p l i c a t i o n s r e g a r d - i n g most n u c l e a r s t r u c t u r e p r o p e r t i e s i n c l u d i n g M l t r a n s i t i o n s . The s t r e t c h e d e x c i t a t i o n s a r e o f f u r t h e r i n t e r e s t . Because o f t h e i r s i m p l i c i t y t h e y c a n b e u s e d a s benchmarks t o t e s t t h e t e n s o r component V: ( q ) o f t h e n u c l e o n - n u c l e o n f o r c e and t h e s p i n o r b i t component, t:S(q), of t h e p i o n - n u c l e o n f o r c e i n t h e n ~ c l e u s . ~ And u l t i m a t e l y we c a n u s e t h e h a d r o n i c r e a c t i o n s t o d e d u c e n u c l e a r s t r u c t u r e inform- a t i o n on t h e i s o s c a l a r m a g n e t i c s t a t e s which i s u n o b t a i n a b l e from ( e , e l ) d a t a a l o n e . Examples u t i l i z i n g t h e complementary a s p e c t s o f e l e c t r o m a g n e t i c and h a d r o n i c

r e a c t i o n s w i l l b e shown i l l u s t r a t i n g t h e s e f e a t u r e s . The d i s c u s s i o n w i l l b e i n t h e s p i r i t of m a t e r i a l p r e s e n t e d i n E e f . 1 0 .

S c h e m a t i c a l l y , t h e r e l a t i o n s b e t w e e n t h e weak, s t r o n g , and e l e c t r o m a g n e t i c o p e r a t o r s t h a t a r e i m p o r t a n t i n c o u p l i n g t h e p r o j e c t i l e t o t h e n u c l e u s i n r e g i o n s of momentum t r a n s 5 e r t h a t f a v o r 1' (q=O) a n d s t r e t c h e d h i g h s p i n e x c i t a t i o n s (q22 fm-l) a r e g i v e n i n T a b l e 1.

T a b l e 1. A t a b u l a t i o n of t h e weak, s t r o n g , and e l e c t r o m a g n e t i c o p e r a t o r s r e l e v a n t f o r e x c i t a t i o n of u n n a t u r a l p a r i t y s t a t e s of low a n d h i g h s p i n .

--

P r o b e 5' = 1+, q=O J' = 4 - , 6-, 8-,

. . .

The d i f f e r e n t s e n s i t i v i t i e s t o d i f f e r e n t r e g i o n s o f q a r i s e s b e c a u s e t h e h i g h e r o r d e r s p h e r i c a l B e s s e l f u n c t i o n j ( q r ) f o r J > l , when f o l d e d i n w i t h t h e n u c l e a r wave f u n c t i o n and t h e i n t e r a c t i o g ; l s a m P l e s t h e n u c l e u s a t h i g h e r r e g i o n s of q .

Another i m p o r t a n t o b s e r v a t i o n t h a t d i s t i n g u i s h e s t h e s t r e t c h e d s t a t e s from t h e mag- n e t i c d i p o l e s t a t e s i s t h a t t h e f o r m e r o n l y i n v o l v e t h e n u c l e a r s p i n f o r t h e p r o b e s i n T a b l e 1, w h i l e t h e l a t t e r a l s o i n c l u d e s t h e o r b i t a l momentum i n e l e c t r o n s c a t - t e r i n g . To i l l u s t r a t e t h e d i f f e r e n t s e n s i t i v i t i e s i n ( e , e r ) and ( p , p t ) f o r ex- c i t a t i o n of 1' s t a t e s , F i g . 1 shows a c o m p a r i s o n of B(EIl)+ s t r e n g t h o b t a i n e d from

( e , e l ) w i t h t h e m i r r o r image c r o s s s e c t i o n s o b t a i n e d from ( p , p l ) t a k e n a t 8 = 4' Lab

- 5 0 ~ 1 (e.eS) ^" " ~ r (c.eS)

'-

' * ~ e (e.e.1 N =

-

- -

r f ^0.5-

-

7 8 9 10 11 12 7 8 9 10 11 12 7 8 9 10 11 12 Ex ( MeV 1

F i g . 1 - A comparison of B(Ml)+ s t r e n g t h o b t a i n e d from ( e e ' ) w i t h m i r r o r image c r o s s s e c t i o n s o b t a i n e d from ( p , p 1 ) f o r 5 0 ~ i , 5 2 ~ r , and 5 Z ~ e .

f o r s e v e r a l d i f f e r e n t even-even n u c l e i i n t h e f p s h e l l ? ' l l ~ h e d i f f e r e n c e s a r e most s t r i k i n g f o r t h e s t a t e s i n 5 2 ~ r . Although some of t h e s e d i f f e r e n c e s i n e x t r a c t e d s t r e n g t h s a r i s e b e c a u s e of d i f f e r e n t s e n s i t i v i t i e s of t h e s p i n o p e r a t o r t o n e u t r o n and p r o t o n components ( o r i s o s c a l a r and i s o v e c t o r ) , t h e a d d i t i o n a l o r b i t a l compone t f o r proton e x . t a t i o n s i n ( e , e t ) c a n n o t b e n e g l e c t e d . A most d r a m a t i c c o m p a r i s o n

19

i s shown f o r "V i n F i g . 2. The s t r u c t u r e n e a r 10 MeV e x c i t a t i o n o b s e r v e d i n ( p , p l )

F i g . 2 - A c o m p a r i s o n of ( e , e l ) and

2 2.0 51 ( p , p l ) s p e 5 f y a t low q f o r t h e odd

c V ( P . P ' ) A n u c l e u s

.

3 E.: 2 0 0 K . s ~ f 1

E x c i t a t i o n Energy (MeV)

a t 3 " i s n o t a t a l l v i s i b l e i n t h e e l e c t r o n s p e c t r a . A p o s s i b l e ex- p l a n a t i o n i s t h a t t h e weaker e x c i t a t i o n s i n ( e , e l ) a r i s e s from p o s s i b l e s p i n and o r b i t a l d e s t r u c t i v e i n t e r f e r e n c e . Such a p o s s i b i l i t y becomes i n c r e a s i n g l y more i m p o r t a n t w i t h i n c r e a s i n g &. The s i n g l e p a r t - i c l e m a g n e t i c d i p o l e m a t r i x e l e m e n t f o r t h e s p i n and o r b i t a l o p e r a t o r s a r e p l o t t e d v e r s u s 9. i n F i g . 3 . At 9. = 4 t h e p r o t o n o r b i t a l r e c o u p l i n g m a t r i x e l e m e n t < ~ + 1 / 2

1 1

I I

1 1

1 I

J O U R N A L DE PHYSIQUE

I- Z F i g . 3

-

m a t r i x e l e m e n t s f o r s p i n and o r b i t a l

;

o p e r a t o r s p l o t t e d a s a f u n c t i o n of t h e _I w o r b i t a l quantum number.

X

H t h e r e would b e s i g n i f i c a n t ground s t a t e

c o r r e l a t i o n s b r e a k i n g t h e s i m p l e s h e l l

'I

model s t r u c t u r e i n o r d e r f o r t h i s _{&0 i I 0}

mechanism t o b e i m p o r t a n t . Calcu- n

l a t i o n s of t h i s t y p e s h o u l d b e performed

i n o r d e r t o t e s t t h e v a l i d i t y of t h i s 2

p o s s i b i l i t y . I- k!

L 0

S i n c e t h e i n t e r e s t i n t h i s s e s s i o n i s 4

f o c u s s e d o n t h e spin modes of n u c l e a r E e x c i t a t i o n s , we s h a l l a v o i d d i s c u s s i n g

m u l t i p o l e magnetic t r a n s i t i o n s t h a t

knowingly c o n t a i n o r b i t a l c o n t r i b u t i o n s ⁰_{0 1 2 3 4 5 6}

t o t h e c r o s s s e c t i o n s , which w i l l e l i m i n a t e t h e a m b i g u i t i e s d i s c u s s e d above.

R

ELECTRON, PROTON, AND PION INELASTIC SCATTERING CROSS SECTIONS TO STATES OF STRETCHED CONFIGURATION

I n e l a s t i c e l e c t r o n , p r o t o n , and p i o n s c a t t e r i n g c r o s s s e c t i o n s f o r u n n a t u r a l p a r i t y e x c i t a t i o n s i n v o l v e t h e m a g n e t i c m u l t i p o l e s o f t h e e l e c t r o m a g n e t i c i n t e r a c t i o n , t h e s p i n dependent c e n t r a l , t e n s o r , and s p i n - o r b i t components o f t h e nucleon-nucleon i n t e r a c t i o n , and t h e s p i n - o r b i t component of t h e pion-nucleon i n t e r a c t i o n and t h u s depend on t h e i s o s c a l a r and i s o v e c t o r s p i n and o r b i t a l t r a n s i t i o n d e n s i t i e s o f t h e t a r g e t n u c l e u s . The g e n e r a l e x p r e s s i o n s f o r t h e c r o s s s e c t i o n have been g i v e n i n Ref. 10 and 1 4 . IIere, we summarize i n p l a n e wave Born a p p r o x i m a t i o n (PWBA) t h e e x p r e s s i o n s f o r t h e i n e l a s t i c d i f f e r e n t i a l c r o s s s e c t i o n f o r 0+ -t J' " s t r e t c h e d "

e x c i t a t i o n s i n e l e c t r o n , p r o t o n , and pion-nucleus s c a t t e r i n g . If is assumed t h a t t h e " s t r e t c h e d " p a r t i c l e - h o l e c o n f i g u r a t i o n is of t h e form ( j a j i L ) ~ , where J

,

, = ja

+

+

+

c o n f i g u r a t i o n s c a n mix w i t h m u l t i - n u c l e o n - m u l t i - h o l e c o n f i g u r a t i o n s w i t h i n t h e same s h e l l . Such mixing p r o d u c e s p h y s i c a l ground and e x c i t e d s t a t e s t h a t a r e n o t p u r e c l o s e d s h e l l -ahd p a r t i c l e - h o l e wave f u n c t i o n s , r e s p e c t i v e l y ; however, t h e a d d i t i o n a l components i n t h e wave f u n c t i o n s w i l l n o t b e connected by t h e one body s p i n and' o r b i t a l c u r r e n t o p e r a t o r s . The o n l y e f f e c t i s t h e n a r e d u c t i o n of t h e t r a n s f t i o n s t r e n g t h w i t h t h e c r o s s s e c t i o n s r e m a i n i n g p r o p o r t i o n a l t o a s i n g l e p a r t i c l e m a t r i x element c o r r e s p o n d i n g t o t h e s t r e t c h e d c o n f i g u r a t i o n . An a d d i t i o n a l s i m p l i f i c a t i o n t h a t a r i s e s i s t h a t t h e o r b i t a l t r a n s i t i o n d e n s i t y and t h e s p i n t r a n s i t i o n d e n s i t y a s s o c i a t e d w i t h t h e J+1 o r d e r B e s s e l f u n c t i o n v a n i s h a s a r e s u l t of a n g u l a r momentum r e s t r i c t i o n s on t h e s i n g l e p a r t i c l e m a t r i x e l e m e n t .

For i n e l a s t i c m a g n e t i c e l e c t r o n s c a t t e r i n g 1 t h e PWBA e x p r e s s i o n f o r t h e d i f f e r e n t i a l c r o s s s e c t i o n i s

w h e r e oM i s t h e Mott c r o s s s e c t i o n , n i s a r e c o i l f a c t o r , 8 i s t h e s c a t t e r i n g a n g l e , q i s t h e momentum t r a n s f e r , F~ i s t h e t r a n s v e r s e form f a c t o r , which i s g i v e n by

T

f o r t r a n s i t i o n s t o p u r e i s o v e c t o r o r i s o s c a l a r s t r e t c h e d s t a t e s w h e r e T = 1 o r 0 , r e s p e c t i v e l y and

gs

where C =

0

+

+

where b i s t h e o s c i l l a t o r l e n g t h p a r a m e t e r . A u s e f u l a n a l y t i c e x p r e s s i o n f o r t h e t r a n s v e r s e e l e c t r o n s c a t t e r i n g form f a c t o r i s o b t a i n e d by s u b s t i t u t i n g e q . ( 3 ) and e q . (4) i n t o e q . ( 2 ) , o b t a i n i n g

a A 2 2

1 1 ( b ~ ) ~ - ' exp(-b 4 14)

j a j b c j a j b

2

7

I

[ ( J - l ) ! ! ( J + l ) ! ! ]

The t e r m s p cm and p N S a r e c e n t e r of mass and n u c l e o n f i n i t e s i z e c o r r e c t i o n f a c t o r s , r e s p e c t i v e l y . The c o e f f i c i e n t s ZJT a r e s p e c t r o s c o p i c a m p l i t u d e s p r o p o r t i o n a l t o t h e o n e body d e n s i t y m a t r i x e l e m e n t s 1 5

where T . , T f , and T a r e t h e i n i t i a l s t a t e , f i n a l s t a t e , and t r a n s i t i o n i s o s p i n , r e s - p e c t i v e i y .

For i n e l a s t i c p r o t o n n u c l e u s s c a t t e r i n g t h e c o r r e s p o n d i n g PWBA e x p r e s s i o n 1 0 , 1 4 , 1 6 , 1 7 f o r t h e d i f f e r e n t i a l c r o s s s e c t i o n t o s t r e t c h e d s t a t e s i n t h e s p i r i t of t h e i m p u l s e a p p r o x i m a t i o n i s g i v e n by

where % ( k ) i s t h e r e d u c e d e n e r g y (wave number) o f t h e i n c i d e n t n u c l e o n , t h e f a c t o r a a r i s e s Yrom t h e d e c o m p o s i t i o n o f t h e r e l a t i v e n u c l e o n - n u c l e o n momentum

-p - - -LS

p =nppp - a t p t , vT ( q ) r e f e r s t o t h e s p i n o r b i t component o f t h e e f f e c t i v e n u c l e o n - n u c l e o n i n t e r a c t i o n , and

C4-438 JOURNAL DE PHYSIQUE

a r e t h e l o n g i t u d i n a l and t r a n s v e r s e c o m b i n a t i o n s o f t h e c e n t r a l and t e n s o r i n t e r - a c t i o n components. The b a r s o n t h e i n t e r a c t i o n components a r e t o i n d i c a t e t h a t t h e y a r e d e f i n e d t o i n c l u d e , a p p r o x i m a t e l y , c o n t r i b u t i o n s a s s o c i a t e d w i t h knockout ex- change.16-'a The component

VT

where M and K a r e t h e r e d u c e d e n e r g y and i n c i d e n t wave number of t h e i n c i d e n t p i o n , a' i s a s T a t i n e q . ( 7 ) and t+S i s t h e i s o v e c t o r ( ~ = 1 ) o r i s o s c a l a r (T=O) s p i n - o r b i t c:mponent of t h e e f f e c t i v e p i o n - n u c l e o n i n t e r a c t i o n . I n t h i s framework pion- n u c l e u s s c a t t e r i n g d e t e r m i n e s u n i q u e l y t h e t r a n s v e r s e s p i n d e n s i t y . I n t h e i m p o r t - a n t i n c i d e n t e n e r g y r e g i o n of t h e ( 3 , 3 ) r e s o n a n c e , t h e pion-nucleon i n t e r a c t i o n i s v e r y n o n - l o c a l . T h i s g i v e s r i s e t o c o r r e c t i o n s i n e q . which p r o d u c e c o u p l i n g t o o r b i t a l c u r r e n t and s p i n - c u r r e n t t r a n s i t i o n d e n s i t i e s . "413 2 2 These t e r m s a r e n o t e x p e c t e d t o b e i m p o r t a n t f o r t h e t r a n s i t i o n of s t r e t c h e d c o n f i g u r a t i o n s n e a r t h e peak of t h e a n g u l a r d i s t r i b u t i o n s .

I n t h e c a l c u l a t i o n of t h e i n e l a s t i c p r o t o n and p i o n c r o s s s e c t i o n s t h e c e n t e r of mass and n u c l e o n f i n i t e s i z e c o r r e c t i o n f a c t o r s a r e n o t i n c l u d e d i n t h e s p i n d e n s i t y P:JT1(q). I n s t e a d t h e c e n t e r of mass c o r r e c t i o n i s made by m o d i f y i n g 2 3 t h e h a r - monlc o s c i l l a t o r p a r a m e t e r b by ( A / A - ~ ) ~ / ( ~ and t h e o v e r a l l n o r m a l i z a t i o n f a c t o r by [ (A-1) /A]

I n e l e c t r o n s c a t t e r i n g

gy>>gz

J J - 1

o b s e r v i n g h i g h s p l n e x c i t a t i o n s , vl f s dominant ,I4 3"-l8 s o n u c l e o n - n u c l e u s s c a t - t e r i n g i s a l s o p r e f e r e n t i a l l y s e n s l t l v e t o i s o v e c t o r s p i n - f l i p s t a t e s of h i g h s p i n . T h i s r e a c t i o n a l s o p r o v i d e s i n f o r m a t i o p o: i s o s c a l a r h ' g h s p i n s t a t e s b e c a u s e b o t h

-I7

G~ and 3LS a r e a p p r e c i a b l e a t q 2-2 f m w l t h most of vo.coming from knockout ex- cgange.lQ*16-18 For t h e i m p o r t a n t i n c i d e n t e n e r g y r e g l o n c o r r e s p o n d i n g t o t h e ( 3 , 3 ) r e s o n a n c e , tLS = 2 t k S s o p i o n - n u c l e u s s c a t t e r i n g f a v o r s i s o s c a l a r t r a n s i t i o n s . STRETCHED TRANSITIONS VIA ELECTRON SCATTERING

The h i g h s p i n s t r e t c h e d m a g n e t i c t r a n s i t i o n s a r e r e a d i l y o b s e r v e d i n e l e c t r o n s c a t - t e r i n g a t h i g h q-2 fm-' and a t a backward a n g l e 8>140° where non-magnetic t r a n s - i t i o n s a r e r e t a r d e d . I n l i g h t p s h e l l n u c l e i s t r e t c h e d 114 (d5/2p3/2! t r a n s i t i o n s h a v e b e e n o b s e r v e d i n a l l s t l e n u c l e i from 12c t h r o u g h ''0. The distribution o f M4 s t r e n g t h 2 4 y 2 5 , 8 , 2 6 , 2 7 i n "C, 13c, 14c, and 1 4 ~ a s o b s e r v e d i n e l e c t r o n s c a t - t e r i n g a t 0 = 1 8 0 ° i s shown i n F i g . 4 . The d e c o m p o s i t i o n of t h e d i s t r i b u t i o n o f s t r e n g t h i n terms of s p i n and i s o s p i n i s i m p o r t a n t t o s t u d y b e c a u s e i t p r o v i d e s i n f o r m a t i o n o n t h e s p i n and i s o s p i n d e p e n d e n t p a r t s of t h e e f f e c t i v e i n t e r a c t i o n i n n u c l e i by p r o v i d i n g well-def i n f g s y s t e m a t i c s t h a t s h o u l d b e r e p r o d u c e d by any r e a l i s t i c model. The n u c l e u s C illustrates t h e f e a t u r e t h a t o n l y o n e s t r o n g i s o v e c t o r s t r e t c h e d t r a n s i t i o n i s o b ~ e r v e d * ~ * ~ ~ i n n u c l e i whose ground s t a t e s h a v e z e r o i s o s p i n and z e r o a n g u l a r momentum. T h i s a l s o t r u e f o r M4 t r a n s i t i o n s 2 9 7 3 0 i n 1 6 0 and M6 t r a s i t i o n s 3 1 ~ 3 2 ~ 3 3 i n 2 4 ~ g and i . A second i m b o r t a n t f e a t u r e i l l u s t r a t e d 8 by i s t h a t i n n e u t r o n e x c e s s n u c l e i w i t h ground s t a t e i s o s p i n To, t h e s t r e n g t h i s f r a g m e n t e d i n t o s e v e r a l 4- s t a t e s w i t h i s o s p i n To and one s t r o n g 4-

4 -

2

F i g . 4

-

12713'14C and 14iY a t q v 2 fm and 8 = 180"

i l l u s t r a t i n g p r e f e r e n t i a l e x c i t a t i o n of M4 t r a n s i t i o n s .

I 2 c ( e , e 4 )

E o

196.5 MeV

- 8

180"

-

-

I I

cn

=

0

0

-

W

-

I3c ( e , e 4 )

E o

1 9 9 . 5 M e V

8

180"

I

M 4 1

C4-440 JOURNAL DE PHYSIQUE

s t a t e w i t h T

+

+

+

A t h i r d f e a t u r e i n which 1 4 ~ s e r v e s a s t h e o n l y example 2 6 7 2 7 s o f a r i s t h a t f o r ground s t a t e s w i t h z e r o i s o s p i n and s p i n 1, t h e i s o v e c t o r M4 s t r e n g t h i s f r a g m e n t e d i n t o t h e e x p e c t e d 3-, 4-, and 5- s t a t e s . And i n 13c where we h a v e n e h e r z e r o a n g u l a r momentum n o r z e r o i s o s p i n i n t h e ground s t a t e t h e L*4 s t r e n g t h i S a p p e a r s t o f r a g m e n t i n t o J = 712, 912 and T = 1 1 2 , 3 / 2 s t a t e s which p o s e s a s t r o n g e r theo- r e t i c a l c h a l l e n g e .

Because t h e s e s t r e t c h e d l e v e l s a r e l o c a t e d i n t h e continuum, and d i f f i c u l t t o un- r a v e l from u n r e s o l v e d n e i g h b o r i n g l e v e l s , some c a r e must b e t a k e n i n i d e n t i f y i n g m u l t i p o l a r i t i e s and d e d u c i n g s t r e n g t h s . T h e r e i s a l w a y s t h e p o s s i b i l i t y t h a ~ more

t h a n o n e l e v e l is b e i n g o b s e r v e d . I n a d d i t i o n , many o f t h e l e v e l s a r e unbound t o e i t h e r p r o t o n o r n e u t r o n e m i s s i o n a n d , t h e r e f o r e , continuum wave f u n c t i o n s 3 8 .in a r e a l i s t i c f i n i t e p o t e n t i a l w e l l s u c h a s t h e Wood-Saxon form (WSWF) s h o u l d b e con- s i d e r e d . With e x c e p t i o n o f t e s t i n g t h e s e n s i t i v i t y of t h e c r o s s s e c t i o n t o Wood- Saxon r a d i a l s h a p e s f o r r e l a t e d examples, t h e q u e s t i o n of r e a l i s t i c w e l l s i s i g n o r e d h e r e . I n s t e a d , t h e r e s u l t s o f a s y s t e m a t i c s t u d y o f most of t h e s t r o n g i s o v e c t o r l e v e l s o b s e r v e d i n ( e , e l ) b a s e d o n t h e u s e of harmonic o s c i l l a t o r r a d i a l wave f u n c t i o n s (HOWF) i s p r e s e n t e d . With t h i s a s s u m p t i o n t h e form f a c t o r f o r t h e 1 H w s t r e t c h e d p a r t i c l e - h o l e c o n f i g u r a t i o n s h a s t h e s i m p l e q dependence, i n d i c a t e d i n eq. ( 5 ) ,

which peaks a t q = =/b. T h i s s i m p l i c i t y i s a n i m p o r t a n t a s s e t i n t h e i d e n t i f i - c a t i o n o f t h e m u l t i p o l a r i t y . The s q u a r e o f t h e t r a n s v e r s e form f a c t o r F$ i s shown v e r s u s q e f f f o r s e v e r a l t r a n s i t i o n s i n v a r i o u s n u c l e i i n F i g . 5 . D i s t o r t i o n e f f e c t s due t o t h e e l e c t r o n - n u c l e u s coulomb a t t r a c t i o n a r e t a k e n i n t o a c c o u n t by c o r r e c t i n g t h e momentum t r a n s f e r u s i n g t h e r e l a t i o n

The c u r v e s t h r o u g h t h e d a t a p o i n t s i n F i g . 5 were d e t e r m i n e d by a l e a s t s q u a r e f i t of t h e form f a c t o r c a l c u l a t e d assuming t h e e x t r e m e s i n g l e p a r t i c l e - h o l e (ESPHM) model. The o s c i l l a t o r p a r a m e t e r , b , and a n o r m a l i z a t i o n f a c t o r ,

were u s e d a s f i t t i n g p a r a m e t e r s . The r e s u l t i n g v a l u e s o f b and t h e n o r m a l i z i n g f a c t o r S2 a r e summarized f o r s e v e r a l i s o v e c t o r t r a n s i i o n s i n T a b l e 2. The v a l u e s o b t a i n e d f o r b a r e c l o s e t o t h a t e x p e c t e d from t h e AlT6 r u l e . It i s no s u r p r i s e t h a t t h e f a c t o r s S2(ESpHM) a r e c o n s i s t e n t l y much l e s s t h a n one. These f a c t o r s a r e i n t e n t e d t o s e r v e a s a u n i t i n which t o c o n v e n i e n t l y compare deduced s p e c t r o s c o p i c s t r e n g t h s . S ~ ( E S P H M ) = 1 c o r r e s p o n d s t o t h e maximum i s o v e c t o r s t r e n g t h e x p e c t e d i n a s i n g l e l e v e l i n t h e ESPHM. For example i n 160 t h e M4 (d5/2P3/2) s t r e n g t h i n t h e 18.98 MeV l e v e l i s S 2 ( ~ s p H M ) = 0.41 o r 41% of t h e ESPHM p r e d i c t i o n .

To t e s t t h e s e n s i t i v i t y of t h e e x t r a t e d S2 t o s h a p e s of t h e r a d i a l wave f u n c t ' o n s , c a l c u $ g t i o n s o f 5 i h e form f a c t o r s were performed u s i n g bound s t a t e WSWF. F o r 14C,

1 4 ~ , S i , and Fe, S ~ ( E S P H M ) was found t o b e 0 . 5 1 , 0.61, 0.37, and 0.57 f o r t h e c o r r e s o n d i n g t r a n s i t i n s l i s t e d i n T a b l e 2 . To t e s t t h e q e f f a p p r o x i m a t i o n i n n u c l e i

160, 2'Mg, 2 8 ~ i , and 5zFe, DWBA c a l c u l a t i o n s , u s i n g t h e c o d e HEIMAG, 43 m o d i f i e d t o u s e HOWF, were performed by f i t t i n g t h e c r o s s s e c t i o n s i n s t e a d o f t h e form f a c t o r s . The n o r m a l i z a t i o n f a c t o r s were t h e same a s t h o s e deduced i n PWBA f i t s t o w i t h i n 3 % , b u t t h e s i z e p a r a m e t e r b was s y s t e m a t i c a l l y l a r g e r by 2-4%.

F i g . 5

-

d i f f e r e n t m u l t i p o l a r i t y i n v a r i o u s n u c l e i .

T h e r e is ample e v i d e n c e t h a t t h e ground s t a t e o r b i t s a r e n o t f u l l y o c c u p i e d a s assumed i n ESPHM. The column l a b e l e d S'(THY) c o n t a i n s t h e r e s u l t s w i t h Z ~ ( T H Y ) de- duced from v a r i o u s t h e o r e t i c a l models t h a t t a k e i n t o a c c o u n t c o r e p o l a r i z a t i o n e f f e c t s and c o r r e l a t i o n s i n t h e ground s t a t e . For example, M i l l e er-Kurath wave functionii6(MKtJf) a r e used t o c a l c u l a t e t h e M4 s t r e n g t h i n "c, "C, '*N, and "0.

I n 12c, ^{C , and} 1 4 ~ t h e c o m p l e t e p s h e l l s p a c e i s u s e d t o c a l c u l a t e t h e ground s t a t e wave f u n c t i o n . The s p a c e i s f u r t h e r e n l a r g e d t o a l l o w one n u c l e o n i n t h e sd- s h e l l t o c a l c u l a t e t h e odd p a r i t y s t a t e s . Some p r o v i s i o n i s a l s o made t o a l l o w f g r 2$w e x c i t a t i o n s from t h e I s t o l p s h e l l , b u t n o t from t h e l p t o t h e 2 s l d s h e l l . The r e s u l t s 8 of t h i s c a l c u l a t i o n f o r t h e t h r e e s t r o n g 4- s t a t e s i n 14c a r e shown i n T a b l e 3 .

S i g n i f i c a n t improvement i s o b t a i n e d f o r t h e T = l s t a t e s a t 1 1 . 7 and 1 7 . 3 MeV u s i n g t h e MKWF. S t i l l , however, t h e c a l c u l a t e d s t r e n g t h i s a b o u t 40% t o o l a r g e . Very l i t t l e improvement i s o b t a i n e d f o r t h e T=2 s t a t e a t 24.3 MeV. When two-body meson- exchange c u r r e n t s a r e i n c l u d e d i n t h e c a l c u l a t i o n of t h e M4 form f a c t o r , t h e c a l c u - l a t e d c r o s s s e c t i o n i n c r e a s e s by a b o u t 1 5 % a n d , t h e r e f o r e , i n c l u s i o n of MEC i n - c r e a s e s t h e s u b s e q u e n t d i s a g r e e m e n t between e x p e r i m e n t and t h e o r y by a b o u t 1 5 % . The i n a b i l i t y t o f i t t h e M4 s t r e n g t h s c l e a r l y i n d i c a t e s a more r e a l i s t i c t r e a t m e n t of t h e e x c i t e d s t a c e and ground s t a t e i s r e q u i r e d , f o r example, a l l o w i n g f o r 3Hw c o n f i g u r a t i o n a d m i x t u r e s i n t h e e x c i t e d s t a t e a s w e l l a s 2p-4h ground s t a t e admix- t u r e s . I n 1 6 0 where b e t t e r a g r e e m e n t i s o b t a i n e d , multi-particle-multi-h~le con-

C4-442 JOURNAL DE PHYSIQUE

T a b l e 2. Deduced s p e c t r o s c o p i c s t r e n g t h S2 from ( e , e l ) f o r s t r e t c h e d s t a t e s w i t h s t r o n g i s o v e c t o r components.

b s S2 R e f . Z 2

Nucleus

Ex ^.TT, T CONF

(ESPHM) " (THY) (ESPHM)

(MeV)

tHarmonic o s c i l l a t o r p a r a m e t e r b and n o r m a l i z a t i o n ~ 2 were d e t e r m i n e d by a l e a s t s q u a r e f i t of t h e DWBA c a l c u l a t e d c r o s s s e c t i o n t o t h e d a t a . The c e n t e r of mass and f i n i t e s i z e e f f e c t s h a v e b e e n i n c l u d e d i n t h e form f a c t o r d u r i n g t h e f i t t i n g procedu e .

t t S 2 = Z'(EXP)/Z2(ESPHM) where Z 2 (ESPHM) i s d e f i n e d i n e q s . (3) and ( 6 ) . For p u r e p r o t o n o r n e u t r o n c o n f i g u r a t i o n s ,

z2=1.

T a b l e 3 . A comparison of t h e deduced i s o v e c t o r (d ) M4 s t r e n g t h w i t h t h e p r e d i c t i o n s of t h e ESPEM and a t r ~ n c a t e d ~ ~ ~ ~ ? ~ ~ m o d e l u s i n g MKWF.

-- -- -- -

Ex J*

,

w i t h o u t w i t h MEC w i t h o u t w i t h MEC (MeV)

2 4 f i g u r a t i o n s a r e a l l o w e d b o t h i n t h e 8f;ound s t a t e and t h e e x c i t e d s t a t e . 3 6 I n Mg and 2 8 ~ i open s h e l l RPA c a l c u l a t i o n s p r e d i c t t o o much M 6 s t r e n g t h by a f a c t o r of two. Again t h e ground s t a t e i s t r e a t e d r e a l i s t i c a l l y a l l o w i n g f o r l d 5 / 2 . 2 ~ ~ / ~ . l d 3 1 2 a d m i x t u r e s w h i l e t h e e x c i t e d s t a t e o n l y a l l o w s f o r one n u c l e o n i n t h e f712 o r b i t . R e c e n t s h e l l model c a l c u l a t i o n s by Amusa and ~ a w s o n ~ ~ which u s e a s p a c e of

I I I !

- 8

-

E~

-

- BATES

E x ( M e V )

F i g . 6 - E l e c t r o n s p e c t r a from t a r g e t s of 5 4 9 5 6 ~ e and 5 8 ' 6 0 ~ i i l l u s t r a t i n g p r e f e r e n - t i a l e x c i t a t i o n of M 8 t r a n s i t i o n s .

c4-444 JOURNAL DE PHYSIQUE

t h e t y p e ( d 5 / 2 ~ 1 / 2 ) 1 1 f 7 1 2 y i e l d s s i m i l a r quenching f a c t o r s . I n t h e e x t r e m e deformed l i m i t of t h e c o l l e c t i v e model, t h e M6 s t r e n g t h i s fragmented o v e r s e v e n r o t a t i o n a l bands from K=O t o K=6. Each K#O r o t a t i o n a l band would g e t 2 / 1 3 of t h e e x t r e m e s i n g l e p t i c l e - h o l e model p r e d i c t i o n s . Although t h i s deformed l i m i t i s u n r e a l i s t i c f o r "Si, i t does s u g g e s t a mechanism f o r p r o d u c i n g f r a g r n e n t a t i o n . 4 6 , 4 7 More r e a l i s t i c f r a g m e n t a t i o n h a s b e e n p r e d i c t e d by ~ m eby i n t r o d u c i n g C o r i o l i s r ~ ~ ~ c o u p l i n g between bands b u t p r e d i c t s t o o much s t r e n g t h i n a s i n g l e s t a t e f o r r e a l i s - t i c d e f o r m a t i o n s .

I n t h e f p s h e l l t h e f r a g m e n t a t i o n of s t r e t c h e d M8 s t r e n g t h i s b e i n g s t u d i e d a s a f u n c t i o n of p r o t o n and n e u t r o n number. E l e c t r o n s c a t t e r i n g measurements have been t a k e n on 5 4 ~ e , 5 6 ~ e , 5 8 ~ i , 6 0 ~ i , a t B a t e s and a l s o on 5 2 ~ r a t NIKHEF.~' A comparison of s p e c t r a a r e shown i n F i g . 6 i n d i c a t i n g t h e s t r i k i n g s e l e c t i v i w i t h whi h t h e 8- l e v e l s a r e e x c i t e d . Form f a c t o r s 5 p e been e x t r a c t e d f o r 5 4 ~ e , " N i , and 6 S N i , and a n a l y s i s is s t i l l i n p r o g r e s s on Fe. Two c h a r a c t e r i s t i c f e a t u r e s of t h e d a t a a r e :

(1) t h e t o t a l M8 s t r e n g t h i s a b o u t 1 1 3 of t h e ESPHM p r e d i c t i o n f o r e a c h n u c l e u s and (2) t h e M8 s t r e n g t h i s s p l i t i n t o two g r o u p s w i t h i s o s p i n To and To

+

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( g g / 2 p 3 / 2 f j j 2 ) 8 - u s i n g t h e m o d i f i e d s u r f a c e d e l t a i n t e r a c t i o n (MSDI) y i e l d

S ~ ( T H Y ) = 0.72 f o r t h e same t r a n s i t i o n . However t h e comparison between e x p e r i m e n t and t h e o r y i s s t i l l poor f o r t h e T = l s t a t e s i n 5 2 ~ e a s i s shown i n F i g . 7 . Even when t h e 2p-4h c o n f i g u r a t i o n s a r e i n c l u d e d , o n l y two T = l 8- s t a t e s a r e p r e d i c t e d and a r e much t o o s t r o n g . Only t h o s e c a l c u l a t e 8- s t a t e s t h a t o n t a i n more t h a n 1% of t h e M8 s t r e n g t h a r e shown i n F i g . 7. I n 5 8 N i

calculation^^^

by t h e ESPHM a s shown i n T a b l e 2. s t r u c t u r e of e a c h l e v e l i s g i v e n by t h e I n d e d u c i n g t h e s t r e n g t h , v j 15/2i1?j$2, ^{~ i ~ ~}

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F i g . 7 - Comparison of e x p e r i m e n t a l and t h e o r e t i c a l H8 s t r e n g t h i n 5 4 ~ e f o r v a r i o u s models.

8 10 12 14 16

E x ( M e V )

c o n f i g u r a t i o n s . T h i s a s s u m p t i o n i s confirmed by r e c e n t c o r e p o l a r i z a t i o n and p a r t i c l e - v i b r a t i o n c a l c u l a t i o n s . The c o r e p o l a r i z a t i o n c a l c u l a t i o n s by Hammamoto e t a1.51 which i n c l u d e a sum of lp-lh,lHw, 2Mw, 3%w, e t c . , con£ i g u r a t i o n s a l s o p r e d i c t t h e c o r r e c t s t r e n g t h . However, t h e c a l c u l a t i o n s employ a z e r o r a n g e d e l t a f o r c e (Vo = 170 ~ e v - f r n ~ ) , which i s known t o b e t o o s t r o n g a t l a r g e a i n c o m ~ a r i s o n t o r e a l i s t i c e f f e c t i v e i n t e r a c t i o n s . Recent c o r e p o l a r i z a t i o n c a l ~ u l a t i o n s > ~ in- c l u d i n g t h e e f f e c t s of meson and r h o exchange o f t h e 12- and 14- e l e c t r o n s c a t t e r i n g form f a c t o r s by S u z u k i p r e d i c t t o o much s t r e n g t h by f a c t o r s of two and do n o t f i t t h e s h a p e of t h e form f a c t o r . On t h e o t h e r hand p a r t i c l e - v i b r a t i o n c o u p l i n g c a l c u - l a t i o n s by Krewald and ~ ~ e t h 5 ~ show a 50% r e d u c t i o n i n M12 and M14 s t r e n g t h t o t h e E = 6 . 4 3 , 7.02, and 6.74 MeV l e v e l s i n agreement w i t h e x p e r i m e n t . However, t h e m?ssing s t r e n g t h i n t h e s e s t a t e s a p p e a r s a s a d d i t i o n a l M12 and M14 s t r e n g t h i n a g r o u p of s t a t e s a t 8.0-9.0 MeV e x c i t a t i n , which h a v e n o t b e e n i d e n t i f i e d e x p e r i - m e n t a l l y . C l e a r l y , t h e s i t u a t i o n i n "'Pb needs f u r t h e r e x p e r i m e n t a l and

t h e o r e t i c a l e f f o r t . It d o e s a p p e a r , however, t h a t t h e b e s t agreement between t h e o r y and e x p e r i m e n t o b t a i n e d t o d a t e f o r t h e s e h i g h s p i n s t a t e s o c c u r s f o r '08pb.

For most of t h e To + 1 e x c i t a t i o n s i n T a b l e 2 t h e r e i s improved agreement between t h e o r y and experiment when t h e model s p a c e i s e n l a r g e d t o i n c l u d e more r e a l i s t i c ground s t a t e wave f u n c t i o n s . Using t h e ESPHM, t h e a v e r a g e s2 = 0.38 and f o r t h e more r e a l i s t i c t h e o r i e s

s2

.

t h e i n e l a s t i c and e l a s t i c m a g n e t i c s t r e n g t h s sugges?s t h a t ground s t a t e and e x c i t e d s t a t e c o r r e l a t i o n s a r e v e r y i m p o r t a n t , s i n c e t h e i n t e r f e r i n g backward going ampli- t u d e i n RPA c a l c u l a t i o n s f o r i n e l a s t i c t r a n s i t i o n s would n o t b e p r e s e n t i n e l a s t i c s c a t t e r i n g . 5 5 For t h e To s t r e t c h e d m a g n e t i c e x c i t a t i o n s i n n o n - s e l f - c o n j u g a t e n u c l e i , t h e s i t u a t i o n a p p e a r s more complex. For example, t h e r e i s even g r e a t e r f r a g m e n t a t i o n and l e s s s t r e n g t h found e x p e r i m e n t a l l y r e d i c t e d b c u r r e n t t h e o r y ( s e e R e f s . 8 , 1 4 , 35, and 36) r e g a r d i n g 14c.

''K:

HIGH SPIN STRETCHED STATES AS BENCHMARKS: COMPARISON WITH HADRONIC PROBES

Although t h e s e p u r e s p i n t r a n s f e r , s t r e t c h e d t r a n s i t i o n s a r e n o t y e t c o m p l e t e l y u n d e r s t o o d i n terms of c u r r e n t n u c l e a r s t r u c t u r e models, t h e y a r e s t i l l v e r y u s e f u l a s "benchmarks" i n t e s t i n g d i r e c t , one s t e p i n e l a s t i c s c a t t e r i n g models f o r h a d r o n i c r e a c t i o n s s u c h a s ( p , p ' ) , ( p , n ) , and ( n - , r ) . The r e a s o n f o r t h i s i s t h a t w i t h i n t h e s p i r i t of t h e DWIA model, t h e

same

( 1 0 ) . D i s t o r t i o n e f f e c t s a r e t a k e n i n t o a c c o u n t t h r o u g h t h e u s e of DWBA i n s t e a d of PWBA and t h e exchange a m p l i t u d e s f o r t h e nucleon-nucleus r e a c t i o n s a r e i n c l u d e d e x a c t l y t h r o u g h t h e u s e of t h e code DWBA70. The comparisons of i n t e r e s t h e r e p r o v i d e i n f o r m a t i o n on t h e i s o v e c t o r t e n s o r p a r t of t h e e f f e c t i v e nucleon-nucleon i n t e r - a c t i o n and t h e s p i n - o r b i t component of t h e e f f e c t i v e pion-nucleon i n t e r a c t i o n . TO i l l u s t r a t e t h e f o r m e r , t h e modulus of v : ( ~ ) , ~ k ~and ( ~ ) f o r 140 MeV p r o t o n s a s deduced from t h e f r e e nucleon-nucleon a m p l i t u d e s i n R e f . 16 i s shown i n F i g . 8 . An i m p o r t a n t f e a t u r e of t h e i n t e r a c t i o n of Ref. 1 6 i s t h a t t h e r a d i a l dependence i s g i v e n a s sums of Yukawa t e r m s w i t h t h e l o n g r a n g e p a r t s of t h e c e n t r a l and t e n s o r