NUCLEAR STRUCTURE WITH HEAVY-ION TRANSFER REACTIONS

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NUCLEAR STRUCTURE WITH HEAVY-ION TRANSFER REACTIONS

G. Morrison

To cite this version:

G. Morrison. NUCLEAR STRUCTURE WITH HEAVY-ION TRANSFER REACTIONS. Journal de

Physique Colloques, 1972, 33 (C5), pp.C5-111-C5-126. �10.1051/jphyscol:1972509�. �jpa-00215111�

JOURNAL DE PHYSIQUE C o l l o q u e C 5 , s u p p l e m e n t au n o 8-9, Tome 33, AoGt-Septembrc 1972, p a g e C 5 - i l l

G.C. MORRISON

Argonne N a t i o n a l L a b o r a t o r y , U.S,A.,and C.E.N. S a c l a y , F r a n c e .

A b s t r a c t . - A t t h e p r e s e n t t i m e t h e s t u d y of n u c l e a r s t r u c t u r e w i t h heavy-ion t r a n s f e r re- a c t i o n s i s t h e f o c u s of i n c r e a s e d a t t e n t i o n , b o t h e x p e r i m e n t a l a n d t h e o r e t i c a l . The g e n e r a l f e a t u r e s e n c o u n t e r e d i n t h e m a j o r i t y o f s u c h r e a c t i o n s a r e s u r v e y e d i n t e r m s o f t h e w e l l - known s e m i - c l a s s i c a l d e s c r i p t i o n . T h i s i s f o l l o w e d by a d e s c r i p t i o n o f c u r r e n t DWBA ap- p r o a c h e s and t h e i r u s e t o e x t r a c t q u a n t i t a t i v e s p e c t r o s c o p i c i n f o r m a t i o n from s i n g l e and m u l t i - n u c l e o n t r a n s f e r r e a c t i o n s . F i n a l l y t h e s t r i k i n g s e l e c t i v i t y i n l e v e l s e x c i t e d a n d o t h e r f e a t u r e s r e v e a l e d i n r e c e n t e x p e r i m e n t a l r e s u l t s a t h i g h e n e r g i e s a r e d i s c u s s e d .

R6sum6,

-

^A l ' h e u r e a c t u e l l e 1 1 6 t u d e de l a s t r u c t u r e n u c l s a i r e 3 l ' a i d e d e s r i a c t i o n s de t r a n s f e r t p a r i o n s l o u r d s e s t un c e n t r e d ' i n t i r z t c r o i s s a n t a u s s i b i e n s u r l e p l a n e x p i r i - men,tal q u e t h s o r i q u e . Les a s p e c t s gdn6raux r e n c o n t r h s d a n s l a m a j o r i t i d e c e s r i a c t i o n s s o n t e x p o s i s e n t e r m e s d e l a d e s c r i p t i o n s e m i - c l a s s i q u e b i e n connue. S u i t une d e s c r i p t i o n d e s a p - p r o c h e s c o u r a n t e s e n DWBA e t d e l e u r u t i l i s a t i o n pour e x t r a i r e l e s i n f o r m a t i o n s s p e c t r o s c o - p i q u e s q u a n t i t a t i v e s 1 p a r t i r d e s r g a c t i o n s 1 un ou p l u s i e u r s n u c l 8 o n s . F i n a l e m e n t l a s 6 l e c - t i v i t i f r a p p a n t e d a n s l c s n i v e a u x e x c i t i s e t l e s a u t r e s a s p e c t s r i v 6 l i s p a r l e s r s s u l t a t s e x p i r i m e n t a u x 3 h a u t e 6 n e r g i e s o n t d i s c u t e s .

I

.

I n t r o d u c t i o n .

-

D u r i n g t h e p a s t few y e a r s an i n d i c a t i o n b o t h o f p r e s e n t t r e n d s and t h e d i r e c - i n c r e a s i n g f r a c t i o n of n u c l e a r r e a c t i o n s t u d i e s h a s t i o n o f f u t u r e d e v e l o p m e n t s . But by way of i n t r o - i n v o l v e d t h e u s e o f heavy i o n s a s p r o j e c t i l e s . Such d u c t i o n i t would seem a p p r o p r i a t e t o b r i e f l y s u r - i n v e s t i g a t i o n s o f heavy-ion t r a n s f e r r e a c t i o n s have vey some f e a t u r e s which a r e e n c o u n t e r e d i n t h e ma- b e e n s t i m u l a t e d by t h r e e r e c e n t developments : ( a ) j o r i t y o f heavy-ion r e a c t i o n s , The d i s c u s s i o n w i l l

t h e a v a i l a b i l i t y o f a v a r i e t y o f h i g h - q u a l i t y be b a s e d on t h e well-known s e m i - c l a s s i c a l d e s c r i p - heavy-ion beams of s u f f i c i e n r e n e r g y and i n t e n s i t y , t i o n

[I:,

o r i g i n a l l y d e v e l o p e d f o r sub-Coulomb r e -

t o g e t h e r w i t h improved p a r t i c l e - i d e n t i f i c a t i o n a n d a c t i o n s . T h i s d e s c r i p t i o n h a s t h e a d v a n t a g e s o f d a t a - h a n d l i n g s y s t e m s ; (b) t h e i n t e n s i v e work on b e i n g t r a n s p a r e n t , of i n d i c a t i n g t h e main f e a t u r e s r e a c t i o n t h e o r i e s and c o d e s b a s e d on s e m i c l a s s i c a l of t h e r e a c t i o n mechanism, a n d o f b e i n g a p p l i c a b l e and d i f f r a c t i o n models a s w e l l a s on t h e DWBA, and e v e n a t e n e r g i e s above t h e Coulomb b a r r i e r .

( c ) t h e i n t e r e s t i n few-nucleon c o r r e l a t i o n s , e s p e -

2 . S e m i - c l a s s i c a l D z s c r i p t i o n . - a )

Cenemz feu-

c i a l l y t h e p o s s i b i l i t y o f q u a r t e t t i n g phenomena i n

t w e s .

-

Because o f t h e i r l a r g e c h a r g e p r o d u c t f p - s h e l l n u c l e i .

Z t Z 2 _. and s m a l l v e l o c i t y v , heavy i o n r e a c t i o n s a r e

-

The u s e o f complex n u c l e i a s p r o j e c t i l e s l e a d s t o c h a r a c t e r i z e d by l a r g e v a l u e s o f t h e Sommerfeld b o t h q u a l i t a t i v e a n d even q u a n t i t a t i v e d i f f e r e n c e s p a r a m e t e r ri = z 1 z 2 e 2 / 6 v = b/2K, where b i s t h e

from t h e f a m i l i a r t r a n s f e r r e a c t i o n s produced by d i s t a n c e o f c l o s e s t a p p r o a c h i n a head-on c o l l i - l i g h t i o n s , More t h a n i n l i g h t i o n t r a n s f e r t h e s i o n . A l a r g e n i m p l i e s a small. X a n d h e n c e a w e l l s t r u c t u r e o f t h e p r o j e c t i l e a n d t h e dynamics of t h e l o c a l i z e d R u t h e r f o r d t r a j e c t o r y f o r t h e s c a t t e r i n g r e a c t i o n a r e i n t e r l i n k e d . T h u s t h e s p e c i f i c f e a t u r e s process. ( o f course,

x

may be small e v e n for o f t h e r e a c t i o n p r o c e s s h a v e t o b e w e l l u n d e r s t o o d

n

s m a l l . ) F o r a f i x e d s c a t t e r i n g a n g l e 8 , t h e d i s - t o hope t o r e a c h t h e l e v e l where n u c l e a r s t r u c t u r e t a n c e of c l o s e s t a p p r o a c h i n s u c h a t r a j e c t o r y p r o p e r t i e s c a n be d e t e r m i n e d . Much o f t h e i n i t i a l ( F i g . 1 ) i s g i v e n by :

s t u d i e s h a v e i n v o l v e d t h e t r a n s f e r o f s i n g l e nu-

c l e o n s whereby t h e n o v e l f e a t u r e s o f heavy-ion r e - n f

Rmin = ( 1 + C S C

1

8) = ( 1 + C S E 8 ) . ( 1 )

a c t i o n s may b e d e t e r m i n e d by comparison w i t h c o r - 2 2Ec.m.

r e s p o n d i n g l i g h t - i o n r e a c t i o n s .

T h i s r e l a t i o n a l s o g i v e s t h e d i s t a n c e o f c l o s e s t I n t h i s p a p e r , I c a n n o t o f n e c e s s i t y r e v i e w ex- a p p r o a c h i n t h e i n i t i a l and f i n a l t r a j e c t o r i e s , R;

t e n s i v e l y t h e f i e l d o f heavy-ion r e a c t i o n s , b u t and Rf f o r a t r a n s f e r r e a c t i o n a t a n a n g l e 8.

r a t h e r s h a l l c o v e r a few t o p i c s i n some d e t a i l a s a n Semi-c-assically t h e r e a c t i o n w i l l o n l y t a k e p l a c e

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972509

G.C. MORRISON

which t h e a n g u l a r d i s t r i b u t i o n s peak is mainly de- termined by t h e e l a s t i c s c a t t e r i n g b e f o r e and a f t e r the t r a n s f e r , even f o r 2 t r a n s f e r o t h e r than optimum.

The matching requirements on t h e i n g o i n g and outgoing o r b i t s i n c o o r d i n a t e o r a n g u l a r momentum space l e a d t o an optimum Q v a l u e [2] f o r the t r a n s - f e r r e a c t i o n . The value can be found from t h e semi- c l a s s i c a l c o n d i t i o n f o r an u n d i s t u r b e d t r a j e c t o r y a t t h e n u c l e a r s u r f a c e , i . e . Bi

2

B E

(2

Ograzing), and ROi

2 Ref.

Hence :

F i g . 1.

-

C l a s s i c a l Rutherford t r a j e c t o r y .

-2.

1 1 1

( 1 + c s c - 8 . ) =

2Ec.m. 2 1

w i t h a p p r e c i a b l e p r o b a b i l i t y i f t h e i r d i f f e r e n c e does n o t exceed t h e o r d e r of A.

The f u r t h e r c h a r a c t e r i s t i c f e a t u r e of t h e heavy i o n r e a c t i o n i s t h e s t r o n g a b s o r p t i o n a t small i n t e r - n u c l e a r d i s t a n c e s . A s the o v e r l a p between t h e two n u c l e i i n c r e a s e s the p r o b a b i l i t y f o r t r a n s f e r i n c r e a s e s . However, i f the minimum d i s t a n c e becomes s m a l l e r than a c e r t a i n r a d i u s - t h e s t r o n g i n t e r a c - t i o n r a d i u s , Ro

-

t h e p r o b a b i l i t y f o r a d i r e c t re- a c t i o n d e c r e a s e s d r a s t i c a l l y and t h e incoming f l u x i s absorbed i n t o a compound n u c l e u s . Thus t h e s t r o n g a b s o r p t i o n o p t i m i z e s the t r a n s f e r r e a c t i o n t o t r a - j e c t o r i e s matched a t t h e n u c l e a r s u r f a c e .

The l o c a l i z a t i o n i n c o o r d i n a t e space has an equiv- a l e n t l o c a l i z a t i o n i n a n g u l a r momentum space

-

which has t h e advantage t h a t t h e a n g u l a r momentum t r a n s f e r i s made e x p l i c i t . Associated w i t h a p a r t i c u l a r t r a - j e c t o r y i s a w e l l d e f i n e d a n g u l a r momentum L where :

Because of t h e s h o r t wavelength ?C of t h e heavy-ion, c a u s i n g both l a r g e L v a l u e s and r a p i d changes i n b a r r i e r p e n e t r a b i l i t y , only a narrow band of L v a l - ues marks the n u c l e a r s u r f a c e . For t h e t r a n s f e r re- a c t i o n t o proceed s t r o n g l y , t h e few v a l u e s of L a s s o c i a t e d w i t h t h e n u c l e a r s u r f a c e i n t h e i n g o i n g and outgoing c h a n n e l s must be matched a p p r o p r i a t e l y

-

which d e f i n e s an optimum a n g u l a r momentum t r a n s f e r 9, =

l~~ - ~ ~ 1 .

^{I t} can a l s o be understood t h a t , s i n c e t h e momentum t r a n s f e r t o t h e t a r g e t is much l e s s than t h e momentum of t h e heavy-ion p r o j e c t i l e ( a t l e a s t f o r few nucleons t r a n s f e r r e d ) , t h e a n g l e a t

which f o r €Ii = 0 f l e a d s t o an optimum r e a c t i o n Q value :

For e n e r g i e s below t h e Coulomb b a r r i e r ,

8. = 8 = 180°, and t h e matching c o n d i t i o n i s inde- 1 f

pendent of

P

t r a n s f e r ; above t h e b a r r i e r , Q i = O f i m p l i e s Li # Lf (equ. 2) and hence t h e c a l c u l a t e d Qopt d e s c r i b e s a p a r t i c u l a r L _OP =

I

^L~

-

^L

I .

^Devi-

t f

a t i o n s from the optimum Q v a l u e r e s u l t i n a de- c r e a s e i n t h e t r a n s f e r c r o s s s e c t i o n . However, t h e heavy-ion r e a c t i o n i s c h a r a c t e r i z e d by a s e v e r i t y of Q dependence not found i n l i g h t - i o n r e a c t i o n s . The r e a c t i o n i s e i t h e r a t t e n u a t e d by t h e Coulomb b a r r i e r (Q Q ) o r is f o r c e d t o t a k e p l a c e f a r

OP t

out with l i t t l e o v e r l a p (Q Q )

.

The semi-clas- OP t

s i c a l c o n s i d e r a t i o n s can a l s o be extended t o o b t a i n t h e optimum Q v a l u e f o r o t h e r .P t r a n s f e r s - o r t h e optimum k t r a n s f e r f o r a p a r t i c u l a r Q v a l u e . How- e v e r , f o r % ' c

kept,

t h e optimum c r o s s s e c t i o n s a r e s m a l l e r i n magnitude and t h e i r dependence on Q value i s l e s s s h a r p .

The e x p r e s s i o n f o r Qopt (equ. 4) a l s o d i s t i n - g u i s h e s t h e s p e c i f i c t r a n s f e r p r o c e s s which o c c u r s ( F i g . 2 ) . For n e u t r o n t r a n s f e r o r t h e t r a n s f e r o f charge between n u c l e i w i t h s i m i l a r c h a r g e s , Qopt

2

0 . For n u c l e i w i t h d i f f e r e n t c h a r g e s , t h e s i g n of Q i s n e g a t i v e o r p o s i t i v e , depending on

OP t

whether charge is t r a n s f e r r e d i n a s t r i p p i n g o r a pick-up r e a c t i o n . Since t h e Q v a l u e s of mcst heavy-

HEAVY-ION TRANSFER REACTIONS

F i g . 2.

-

Q dependence f o r d i f f e r e n t t r a n s f e r r e a c t i o n s .

i o n r e a c t i o n s a r e n e g a t i v e , s t r i p p i n g r e a c t i o n s a r e s t r o n g l y f a v o r e d . S e m i - c l a s s i c a l l y t h e w i d t h of t h e d i s t r i b u t i o n a r o u n d Q a l s o depends on t h e t y p e of

o p t

t r a n s f e r r e a c t i o n , b e i n g l e a s t f o r n e u t r o n t r a n s f e r and low i n c i d e n t e n e r g i e s .

b) Comparison w i t h experiment. - The v a l i d i t y o f t h e f o r e g o i n g s e m i - c l a s s i c a l c o n s i d e r a t i o n s may b e e x e m p l i f i e d w i t h r e s u l t s t a k e n from a s u r v e y [3] of

l p t r a n s f e r i n d u c e d by 160 p r o j e c t i l e s i n t h e f p s h e l l . For e n e r g i e s above t h e Coulomb b a r r i e r

(Q & 1 5 ) , one o b s e r v e s t h e c h a r a c t e r i s t i c b e l l - s h a p e d a n g u l a r d i s t r i b u t i o n s shown i n f i g u r e 3.

T h e i r s i m p l e s h a p e h a s a n a t u r a l i n t e r p r e t a t i o n b a s e d on t h e c l a s s i c a l r e l a t i o n ( I ) . The t r a n s f e r c r o s s s e c t i o n i n c r e a s e s w i t h a n g l e b e c a u s e of t h e i n c r e a s i n g o v e r l a p between t h e two n u c l e i , r e a c h e s a maximum a t a n a n g l e

-

t h e g r a z i n g a n g l e

-

^which

d e f i n e s an i n t e r a c t i o n r a d i u s , Ro f o r t h e c o n t a c t c o l l i s i o n , t h e n f a l l s a t backward a n g l e s a s a b s o r p - t i o n p r o c e s s p r e d o m i n a t e o v e r t h e s i m p l e t r a n s f e r p r o c e s s . Moreover, t h e g r a z i n g a n g l e i s found t o be r e l a t i v e l y i n s e n s i t i v e t o t r a n s f e r r e d .t.

E q u a t i o n ( I ) a l s o p r e d i c t s t h a r t h e t r a n s f e r r e - a c t i o n s h o u l d peak a t more backward a n g l e s f o r l o v e r bombarding e n e r g i e s , more n c g a t i v e Q v a l u e s , and i n c r e a s e d Z of t h e t a r g e t - a s i s i n d e e d f o u n d . F o r E d EB, t h e g r a z i n g a n g l e i s 180°, a s i t u a -

c.m.

t i o n c h a r a c t e r i s t i c of t h e sub-Coulomb domain. Kela- t i o n ( I ) a l s o i m p l i e s f o r t h e t r a n s f e r r e a c t i o n t h e e q u i v a l e n c e o f e x c i t a t i o n f u n c t i o n s and a n g u l a r d i s - t r i b u t i o n s ; i n b o t h c a s e s t h e peak i n t h e c r o s s s e c t i o n d e f i n e s R

.

F i g . 3 .

-

4 8 ~ a ( 1 6 0 , 1 5 ~ ) 4 9 ~ c a n g u l a r d i s t r i b u t i o n a t Elab = 42, 48 and 56 MeV f o r t r a n s i t i o n s t o t h e 712- ground s t a t e and t h e l o w e s t 312- e x c i t e d

s t a t e . The L v a l u e s d e n o t e t h e t o t a l a n g u l a r momentum t r a n s f e r . The s o l i d c u r v e s a r e f i n i t e -

r a n g e DWBA c a l c u l a t i o n s .

Q V A L U E , M e V

F i g . 4 . - E x p e r i m e n t a l peak c r o s s s e c t i o n s d i v i d e d by t h e c o r r e s p o n d i n g ( 3 ~ e , d ) s p e c t r o s c o p i c f a c t o r s f o r t h e f 7 / 2 ground s t a t e and f c r p r o m i n e n t ~ 3 1 2

s t a t e s . 'The s o l i d c u r v e s a r e DlJM peak c r o s s s e c t i o n s ( m u l t i p l i e d by a n 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 N = 1 . 5 ) . The s i n g l e n u c l e o n t r a n s f e r d a t a a l s o d e m o n s t r a t e

C5-i 14 G .C. MORRISON

t h e s e v e r e e f f e c t o f Q v a l u e on t h e m a g n i t u d e of t h e peak c r o s s - s e c t i o n . T h i s i s d i s p l a y e d i n a p l o t o f t h e "reduced" p e a k c r o s s s e c t i o n s - t h e e x p e r i - m e n t a l v a l u e s d i v i d e d by t h e p r o t o n s p e c t r o s c o p i c s t r e n g t h (25 + 1 ) ~ ' s from ( 3 He,d) e x p e r i m e n t s

-

a s a f u n c t i o n o f t h e Q v a l u e ( F i g . 4 ) . The optimum Q v a l u e d e m o n s t r a t e d i n t h i s way i s a p p r o x i m a t e l y t h a t g i v e n by e q u . 4 , namely Q

?,

- 2 . 9 MeV and

OP t

II I . m e c r o s s s e c t i o n f o r low II t r a n s f e r i s OP t

a l s o s e e n t o b e more r a p i d l y a f f e c t e d a s Q d e v i a t e s from t h e optimum v a l u e

-

a consequence of t h e t i g h t e r a n g u l a r m o m e n t u ~ m a t c h i n g r e s t r i c t i o n s .

Such a Q window r e p r e s e n t s b o t h an a d v a n t a g e and d i s a d v a n t a g e f o r t h e heavy-ion r e a c t i o n . I t h a s t h e o b v i o u s a d v a n t a g e o f any window e f f e c t f o r t h e f a - v o r e d r e g i o n o f e x c i t a t i o n , b u t t h e d i s a d v a n t a g e t h a t Q may be s u c h a s t o s e v e r e l y l i m i t t h e r a n g e

"t

of e x c ~ t a t i o n e n e r g i e s . However, t h e wide v a r i e t y o f heavy-ion p r o j e c t i l e s now a v a i l a b l e f a c i l i t a t e s t h e s e l e c t i o n of p a r t i c u l a r e x c i t a t i o n e n e r g i e s . F i g u r e 5 shows t h e Ip pick-up r e a c t i o n on 4 0 ~ a i n i t i a t e d by two d i f f e r e n t p r o j e c t i l e s . F o r t h e ( 1 8 0 , 1 9 F ) and ( I 5 ~ , l 6 0 ) r e a c t i o n s t h e optimum Q v a l u e s a r e + 2 . 3 and + 3 . 0 MeV, r e s p e c t i v e l y . Thus Q f a l l s w e l l

OP t

beyond t h e ground s t a t e i n t h e f o r m e r r e a c t i o n and a t a b o u t E = 0 . 8 MeV i n t h e l a t t e r . The e f f e c t s on t h e Q window a r e o b v i o u s .

The s y s t e m a t i c f e a t u r e s a s have been p r e s e n t e d h e r e have been f o u n d f o r a v a r i e t y of heavy-ion r e - a c t i o n s on medium t o heavy n u c l e i f o r e n e r g i e s a t and a b o v e t h e Coulomb b a r r i e r (n >> I ) . Thus t h e i r c o n s i s t e n t e x p l a n a t i o n w i t h a s e m i - c l a s s i c a l de- s c r i p t i o n s e r v e s t o c o n f i r m t h e i r i n t e r p r e t a t i o n a s a d i r e c t r e a c t i o n

-

even f o r t h o s e r e a c t i o n s i n - v o l v i n g t h e t r a n s f e r of s e v e r a l n u c l e o n s , where a d i r e c t comparison w i t h l i g h t i o n r e a c t i o n s i s n o t p o s s i b l e . However, e v i d e n c e now e x i s t s [3] t h a t t h e s i m p l e s e m i - c l a s s i c a l p i c t u r e based on R u t h e r f o r d t r a j e c t o r i e s a p p e a r s i n a d e q u a t e t o i n t e r p r e t " i n t e r -

F i g . 5. - Energy s p e c t r a of 3 9 ~ p o p u l a t e d i n two d i f f e r e n t p r o t o n pick-up r e a c t i o n s on 4 0 ~ a .

3 . DWBA A n a l y s i s . - A l l c u r r e n t a p p r o a c h e s t o t h e e x t r a c t i o n o f s p e c t r o s c o p i c i n f o r m a t i o n from heavy i o n t r a n s f e r r e a c t i o n s a r e based on o r d e r i v e from DWBA a n a l y s i s . Although s u c h a framework i s w e l l a c c e p t e d f o r l i g h t i o n s , i t s a p p l i c a t i o n t o heavy i o n s r a i s e s c o n s i d e r a b l e problems i n p r a c t i c e (and i n p r i n c i p l e ) . These p r i m a r i l y a r i s e from t h e need t o t r e a t t h e f i n i t e e x t e n s i o n o f t h e heavy

i o n which makes d i f f i c u l t b o t h t h e c o r r e c t t r e a t - ment of t h e k i n e m a t i c s and t h e c o m p l e t e d e s c r i p t i o n of t h e form f a c t o r .

a c t i o n r a d i i " e t c .

-

p a r t i c u l a r l y f o r r e a c t i o n s i n - F u r t h e r d i s c u s s i o n r e q u i r e s a b r i e f s k e t c h of t h e v o l v i n g more t h a n one n u c l e o n t r a n s f e r . Thus t h e DWBA f o r m a l i s m . L e t u s c o n s i d e r a t r a n s f e r r e a c t i o n f o r e g o i n g d e s c r i p t i o n i s m a i n l y u s e f u l a s a f i r s t A ( a , b ) B , where a = b

+

x and B = a + x. R e f e r r i n g

A d e t a i l e d i n t e r p r e t a t i o n o f the angu- t o t h e f a m i l i a r " t r i a n g l e " d i a g r a m f o r a t r a n s f e r l a r d i s t r i b u t i o n and t h e e x t r a c t i o n o f q u a n t i t a t i v e r e a c t i o n ( F i g . 6 ) , t h e DWBA a m p l i t u d e can be w r i t - s p e c t r o s c o p i c i n f o r m a t i o n r e q u i r e a c o m p l e t e and t e n i n t h e p o s t a p p r o x i m a t i o n :

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

H E 4 W-ION TRRNSFER REACT IONS _{C 5 - i 19}

This e x p r e s s i o n e x p l i c i t l y r e v e a l s t h e i n t e r l i n k e d dependence on t h e c o o r d i n a t e s of t h e system. Since f o r heavy i o n s the s i m p l i f y i n g zero-range approxima- t i o n ( r x b = 0) i s c l e a r l y i n j u s t i f i e d , t h e evalua- t i o n of the DWBA t r a n s i t i o n amplitude i s formidable.

The c o r r e c t t r e a t m e n t of t h e kinematics r e q u i r e s e i t h e r t h e c a l c u l a t i o n of t h e f u l l six-dimensional i n t e g r a l o r i t s r e d u c t i o n t o two three-dimensional i n t e g r a l s by r e a s o n a b l e approximations wich permit a s e p a r a t i o n of v a r i a b l e s .

F i g . 6 .

-

R e l a t i v e c o o r d i n a t e system f o r t r a n s f e r r e a c t i o n .

A common approximation i s t h e n e g l e c t of s o - c a l l e d r e c o i l terms, which assumes t h a t the mass of t h e t r a n s f e r r e d p a r t i c l e i s s m a l l compared t o t h e c o r e s A and b , and hence t h a t a and b have t h e same cen-

t r o i d . This approach has been used by Tobocman [4]

and e a r l i e r by B u t t l e and Goldfarb [5] t o a c h i e v e a s e p a r a t i o n of v a r i a b l e s . However, t h e s e t r e a t m e n t s s t i l l take f i n i t e range i n t o account i n t h e form- f a c t o r i n t e g r a t i o n . They should be most a p p l i c a b l e f o r single-nucleon t r a n s f e r where t h e r e c o i l terms

(%

I/ma) w i l l be s m a l l e s t .

More r e c e n t approximations attempt t o allow cor- r e c t l y f o r t h e t r a n s f e r of mass by a p p e a l i n g t o phy- s i c a l i n s i g h t t o determine from where t h e maximum c o n t r i b u t i o n t o t h e i n t e g r a l a r i s e s and t o determine t h e r e l a t i o n between the c o o r d i n a t e s a t t h a t p o i n t . The u s u a l assumption i s t h a t t h e p a r t i c l e i s on t h e i n t e r c o n n e c t i n g l i n e of t h e two n u c l e i . (The "fixed range" approximation of Piihlhofer 161 i s an e a r l y example of such a method a l t h o u g h i t reduces t o a zero-range c a l c u l a t i o n ) . When i t i s f u r t h e r assumed

t h a t t h e t r a n s f e r r e d p a r t i c l e i s on t h e s u r f a c e of a , t h i s approach [2] permits a r e d e f i n i n g of t h e r e l a t i v e c o o r d i n a t e s i n t h e i n i t i a l and f i n a l chan- n e l , namely :

-f

where r i s t h e d i s t a n c e between t h e two c o r e s . While a n a l y s i s i n v o l v i n g t h e use of t h e s e " s c a l i n g "

f a c t o r s has been mainly a t e n e r g i e s about t h e Coulomb b a r r i e r , t h e approach would appear t o o f f e r wider a p p l i c a b i l i t y . I t a l s o l e a d s t o an e x p r e s s i o n

f o r Q which i n c l u d e s r e c o i l c o r r e c t i o n s ( t h e OP t

term zfZf/ziZi i n equ. 4 b e i n g m u l t i p l i e d by u l v ) . F i n a l l y t h e r e now e x i s t v a r i o u s DWBA codes [7]

which perform the f u l l six-dimensional i n t e g r a l (equ. 5 ) . T h e i r wide spread a p p l i c a b i l i t y i s l i m - i t e d by t h e s i z e of t h e computer r e q u i r e d and by the time involved i n t h e c a l c u l a t i o n

-

t h e i r use becomes u n f e a s i b l e f o r e l a s t i c channels i n v o l v i n g a l a r g e number of p a r t i a l waves (>50-80). Neverthe- l e s s , t h e i r e x i s t e n c e i s comforting, p a r t i c u l a r l y i n view of t h e need t o t h e t e s t t h e r e l i a b i l i t y of programmes u t i l i z i n g v a r i o u s k i n e m a t i c approxima- t i o n s . I n p a r t i c u l a r t h e approximate methods d i s - cussed above obey the u s u a l f a r i t y r e s t r i c t i o n s on

1+L2+L

t h e it t r a n s f e r , namely ( - I ) should be even, where i t I and it2 r e f e r t o t h e a n g u l a r momentum i n

t h e p r o j e c t i l e and f i n a l n u c l e u s , r e s p e c t i v e l y . The f u l l f i n i t e range c a l c u l a t i o n r e l a x e s t h i s r e s t r i c - t i o n

-

p o s s i b l e L t r a n s f e r being determined by t h e more g e n e r a l s e l e c t i o n r u l e ,

I

j l

-

j 2 1 4 i < j l + j 2 . Such r e c o i l e f f e c t s a r e expected t o be of i n c r e a s - i n g importance a t h i g h e r e n e r g i e s and f o r h e a v i e r t a r g e t s

.

A d i f f e r e n t approach t o the heavy-ion r e a c t i o n i s the s e m i - c l a s s i c a l t r e a t m e n t of B r o g l i a and

Winther r8]. Although i n s p i r e d by an a n a l y s i s of t h e s e m i - c l a s s i c a l l i m i t of DWBA i91, i t i s formu- l a t e d i n t h e framework of the coupled-channel theory of Coulomb e x c i t a t i o n

-

extended i o i n c l u d e nucleon t r a n s f e r r e a c t i o n s . I t s advantages should become i n c r e a s i n g l y more important when thq number of p a r t i a l waves becomes very l a r g e and when 2nd o r d e r e f f e c t s i n v o l v i n g i n e l a s t i c e x c i t a t i o n s have

C.C. MORRISON

t o be c o n s i d e r e d .

Even i f t h e k i n e m a t i c e f f e c t s a r e t r e a t e d proper- l y , t h e r e s t i l l a r i s e s t h e problem of c a l c u l a t i n g form f a c t o r s f o r more than one nucleon t r a n s f e r

-

a problem whose complexity i n c r e a s e s r a p i d l y f o r t h e

t r a n s f e r of s e v e r a l nucleons. I n p a r t i c u l a r f o r heavy i o n s , t h e convenient assunrption of a r e l a t i v e

Is c o n f i g u r a t i o n f o r t h e i n t e r n a l motion of t h e t r a n s f e r r e d c l u s t e r , i s n o t v a l i d [gal. Thus t h e t r a n s i t i o n a m p l i t u d e has t o be considered a s a co- h e r e n t sum o v e r a l l p o s s i b l e i n t e r n r e d i a t e s t a t e s of

t h e t r a n s f e r r e d nucleons, and ltence a l l p o s s i b l e quantum numbers of t h e c l u s t e r r e l a t i v e ntotion.

T h i s f a c t may l e a d t o c o n s i d e r a b l e d i f f i c u l t i e s f o r t h e c o n v e n t i o n a l t r e a t m e n t .

4. Conrparison w i t h E x p e r i m n t .

-

^a) Oyla-nuoleon tmsfe'ep.

-

The s u c c e s s of t h e DWBA d e s c r i p t i o n of one-nucleon t r a n s f e r t o g i v e q u a n t i t a t i v e s p e c t r o - s c o p i c i n f o r m a t i o n i s now w e l l e s t a b l i s h e d f o r en- e r g i e s a t and above t h e Coulomb b a r r i e r [3,10].

5-

p i c a l l y , i t h a s been demonstrated i n a comparison of t h e ( l 6 0 , I 5 ~ ) r e a c t i o n on n u c l e i i n t h e f p s h e l l

133 w i t h the p r e d i c t i o n s of t h e RDXC program of Tobocman [4]. Examples of t h e goodness of t h e f i t s both f o r a n g u l a r d i s t r i b u t i o n s and f o r t h e v a r i a - t i o n o f peak c r o s s s e c t i o n w i t h Q v a l u e a r e shown i n F i g s . 3 and 4. The t h e o r e t i c a l c u r v e s were c a l - c u l a t e d with s t r o n g l y a b s o r b i n g p o t e n t i a l s [I l]

which o b v i a t e t h e need f o r a cut-off r a d i u s . The p a r a m t e r s f o r t h e i n g o i n g 160 c h a n ~ % e l g i v e a good d e s c r i p t i o n of t h e e l a s t i c s c a t t e r i n g d a t a ; t h e parameters f o r t h e outgoing 15N c h a n n e l a r e t h e same e x c e p t t h a t t h e d i f f u s e n e s s was s l i g h t l y in- c r e a s e d t o y i e l d a b e s t f i t t o t h e 48 MeV d a t a of f i g u r e 3 , A t p r e s e n t no "N e l a s t i c s c a t t e r i n g d a t a e x i s t i n t h i s mass region.

I n t h i s work a somewhat unexpected J dependence of t h e ( 1 6 0 , 1 5 ~ ) c r o s s s e c t i o n was found. The e f - f e c t i s demonstrated i n the s p e c t r a of t h e 6 2 7 6 4 ~ i ( 1 6 ~ , 1 5 ~ ) 6 3 7 6 5 ~ ~ r e a c t i o n s (Fig. 7 ) . For example t h e 0.67 KeV 112- s t a t e i n 6 2 ~ i i s consid- e r a b l y weaker than t h e 312- ground s t a t e , whereas t h e measured 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 comparable CC 2 S = 0.66 and 0.70 f o r t h e 312- and 112- s t a t e s , r e s p e c t i v e l y ) . The observed r e d u c t i o n i s w e l l de- s c r i b e d by

DWBA

c a l c u l a t i o n s (Fig. 8). It can be

F i g . 7.

-

Energy s p e c t r a of t h e ( 1 6 ~ , i S ~ ) r e a c t i o n on 6 2 3 6 4 N i t a r g e t s a t ~ ( 1 6 0 )

-

^{48 MeV,}

F i g , 8.

-

C a l c u l a t e d DWBG d i s t r i b u t i o n s f o r L = 0, 2, 4 t r a n s i t i o n s l e a d i n t o various f i n a l

s t a t e s i n t h e 62Ni(i60, 15N)68Cu r e a c t i o n a t

~ ( ~ = ~48 MeV. 0 )

HEkW-ION TRANSFER REACTIONS C5-117

e x p e c t e d t o be a g e n e r a l f e a t u r e o f heavy-ion t r a n s f e r from a j I = R 1

-

112 o r b i t a l t o

j 2 = E2

-

112 o r b i t a l , f o r which t h e s e m i - c l a s s i - c a l l y f a v o r e d !L t r a n s f e r (= L I + 2, ) i s n o t a l -

2

lowed. The J d e p e n d e n t f a v o r i n g of j 2 = i 2 + 112 m y be e x p l o i t e d by comparing t h e r e l a t i v e i n t e n s i -

160 15

t i e s i n t h e (

,

N) r e a c t i o n o f t h o s e s t a t e s which have t h e same 2 t r a n s f e r i n t h e ( 3 He,d) r e - a c t i o n . Moreover, d i f f e r e n t i n t e n s i t y r a t i o s s h o u l d be e x p e c t e d i n a c o m p a r i s o n o f j 2 = R 2 ^f 112 s t a t e s p o p u l a t e d by two heavy i o n r e a c t i o n s s u c h a s

( I 6 0 , l 5 N ) a n d ( 1 2 ~ , 1 1 ~ ) , f o r which t h e t r a n s f e r r e d n u c l e o n comes from a p I l 2 and p31Z o r b i t a l , r e s p e c -

t i v e l y .

P e r h a p s a p a r t i a l f a i l u r e o f t h e DWBA a n a l y s i s of t h e ( 1 6 ~ , 1 5 ~ ) r e a c t i o n was an i n a b i l i t y i n some c a s e s t o r e p r o d u c e t h e d a t a a t t h e most f o r d a r d an- g l e s , e s p e c i a l l y a t v e r y n e g a t i v e Q v a l u e s ( F i g . 9, s o l i d l i n e s ) . T h i s c a u s e s some c o n c e r n i n view of t h e s e m i - c l a s s i c a l p i c t u r e t h a t a t t h e s e a n g l e s o n l y Coulomb d i s t o r t i o n s s h o u l d be e f f e c t i v e . However, a b e t t e r f i t t o t h e d a t a c a n be o b t a i n e d ( F i g . 9, d a s h e d l i n e s ) w i t h a s l i g h t l y m o d i f i e d s e t of p a r a m e t e r s l123, b u t one which d o e s n o t now r e - p r o d u c e t h e e l a s t i c s c a t t e r i n g . (The same s i t u a t i o n w i l l b e a g a i n found i n t h e a n a l y s i s o f s e v e r a l nu- c l e o n t r a n s f e r ) .

160 15 F i g . 9. - A n g u l a r d i s t r i b u t i o n s of t h e (

,

N) r e a c t i o n on even-A Ca i s o t o p e s a t 48 MeV l e a d i n g

t o t h e f i n a l s t a t e s shown. DWBA d i s t r i b u t i o n s a r e d i s c u s s e d i n t h e t e x t .

The j u s t i f i c a t i o n f o r u s i n g p o t e n t i a l s from e l a s - t i c s c a t t e r i n g t o d e s c r i b e t h e t r a n s f e r r e a c t i o n i s l e s s c l e a r f o r heavy-ion t h a n f o r l i g h t - i o n t r a n s - f e r r e a c t i o n s owing t o t h e i n c r e a s e d p o s s i b i l i t y o f c o r e - c o r e e x c i t a t i o n s . Even t h e u s e o f " s t a n d a r d "

o p t i c a l p o t e n t i a l s f o r heavy-ion s c a t t e r i n g h a s been r e c e n t l y q u e s t i o n e d [13], and i t has been sug- g e s t e d t h a t c o r r e c t p r o c e d u r e s h o u l d t a k e p r o p e r a c c o u n t of t h e s i z e o f t h e p r o j e c t i l e . However, i t can be n o t e d a t t h i s p o i n t t h a t a v e r y s e n s i t i v e t e s t of o p t i c a l p a r a m e t e r s now e x i s t s a s a r e s u l t o f t h e r e c e n t d i s c o v e r y o f t h e d e s t r u c t i v e i n t e r - f e r e n c e between Coulomb a n d n u c l e a r e x c i t a t i o n i n heavy-ion i n e l a s t i c s c a t t e r i n g [14]. The experimen- t a l e f f e c t i s shown i n F i g . 10 f o r e x c i t a t i o n o f t h e 2+ s t a t e i n 5 8 ~ i by 160 ; t h e c u r v e s a r e p r e - d i c t i o n s o f t h e a n g u l a r d i s t r i b u t i o n s by s e m i - c l a s - s i c a l and q u a n t a 1 c a l c u l a t i o n s D 5 j .

1 ^{F m i}

3.L 1.56 0.60 ^1fiO+5BNi

imaginary 6.0 1.27 0.5L E~~~ = LLMeV

pl

quantum mechanical

P

( qne~astic / ~ e ~ a s t i c ) e x p t .

F i g . 10. - E x p e r i m e n t a l and t h e o r e t i c a l a n g u l a r d i s t r i b u t i o n s f o r 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 n g

of 160 on 5 8 ~ i a t ~ ( ~ = ~44MeV. 0 )

(25-1 18 G . C . MORRISON

F i n a l l y , i t may be remarked t h a t t h e r e s u l t s of t h e ( 1 6 ~ , 1 5 ~ ) experiments i n t h e f-p s h e l l g i v e no evidence t h a t the r e a c t i o n has any p e c u l i a r a s p e c t unique t o heavy-ion r e a c t i o n s , c o n t r a r y t o e a r l i e r

"?

b e l i e f . The s p e c t r o s c o p i c s t r e n g t h s e x t r a c t e d from 75 - ^r.

-

^{"? - 1}

-

t h e ( 1 6 0 , 1 5 ~ ) r e a c t i o n a r e found t o be c l o s e l y the same a s those e x t r a c t e d from ( 3 He,d)

-

s o t h a t two- s t e p p r o c e s s e s appear unimportant. The s i t u a t i o n ³

U

should be very d i f f e r e n t f o r h e a v i e r , deformed t a r -

25 -

g e t n u c l e i , and even t h e p o s s i b i l i t y of a s t r o n g c o n t r i b u t i o n of two-step p r o c e s s e s i n t h e mass-90 r e g i o n cannot be excluded [16].

500 550 600 650 700 750 B i l l

b) MuZti-nuoZeo?~ t r a n s f e ~ .

-

^{A s} d i s c u s s e d e a r l i e r t h e DWBA a n a l y s i s of m u l t i - n u c l e o n t r a n s f e r is com- p l i c a t e d by two f e a t u r e s : t h e need t o t a k e account of r e c o i l e f f e c t s and t h e need to c o n s i d e r i n t e r n a l c l u s t e r c o n f i g u r a t i o n s . Ln connection w i t h t h e l a t -

t e r problem, a new approach by Bonche and Giraud [17] appears of i n t e r e s t . T h e i r a n a l y s i s i s based on a generator-coordinate d e s c r i p t i o n of t h e s c a t - t e r i n g waves. It u s e s d i r e c t l y s h e l l model wave f u n c t i o n s , w i t h o u t e x t r a c t i n g t h e c e n t r e of mass motion, t o d e s c r i b e the t r a n s f e r r e d nucleons i n the i n i t i a l and f i n a l n u c l e i , and thereby t a k e s ac- count e x p l i c i t l y of a l l o r d e r s of i n t e r n a l configu- r a t i o n s of t h e t r a n s f e r r e d c l u s t e r , With such a m i - c r o s c o p i c d e s c r i p t i o n of the t r a n s f e r form f a c t o r , s p e c t r o s c o p i c wave f u n c t i o n s can be d i r e c t l y ana- l y s e d . I n t h e o p t i c a l waves t h e c e n t e r of mass of t h e t r a n s f e r r e d p a r t i c l e s i s assumed to be a t t h e c e n t e r of mass of t h e two c o r e s . Thus t h e t r e a t m e n t of r e c o i l e f f e c t s may s t i l l be inadequate f o r c o r e s of unequal masses.

The ' * 8 ~ a ( 1 6 ~ , 1 4 ~ ) 5 0 ~ i r e a c t i o n i s a good t e s t of t h i s new approach f o r the c a s e of two-proton t r a n s - f e r . Fig. 1 1 shows a spectrum o b t a i n e d a t 59.5 MeV.

The f i r s t f o u r s t a t e s i n 5 0 ~ i a r e known t o have s p i n s 0+, 2 + , 4+ and 6+ r e s p e c t i v e l y based on a r e l a t i v e l y p u r e (n f 7,Z)2 c o n f i g u r a t i o n . Thus t h e p r e d i c t e d shape and magnitude of 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 s can be compared with experiment f o r d i f f e r e n t L t r a n s f e r , d i f f e r e n t e x c i t a t i o n e n e r g i e s and d i f f e r e n t bombarding e n e r g i e s .

Angular d i s t r i b u t i o n s o b t a i n e d a t 48 MeV f o r t h e f i r s t f o u r s t a t e s i n " ~ i a r e shown i n F i g . 12.

They a r e s i m i l a r i n shape t o those observed i n Ip t r a n s f e r e x c e p t , a s has been n o t e d p r e v i o u s l y [3],

CHANNEL NUMBER

F i g . 1 1 .

-

Energy s p e c t r a of t h e 4 8 ~ a ( 1 6 0 , 1 4 ~ ) 5 0 ~ i r e a c t i o n a t ~ ( ' ~ 0 ) = 59.5 MeV.

-

B o n c h e I

----

_E.FR.

E= L8'M eV

F i g . 12.

-

Experimental and t h e o r e t i c a l a n g u l a r d i s t r i b u t i o n s of t h e 4 8 ~ a ( 1 b 0 , 4 ~ ) 5 0 ~ i r e a c t i o n l e a d i n g t o t h e f i n a l s t a t e s shown. ~ ( ~ = ~48 MeV. 0 )

they peak much forward of t h e "grazing" angle pre- d i c t e d by t h e customary s t r o n g i n t e r a c t i o n r a d i u s . This e f f e c t i s not accounted f o r i n DWBR c a l c u l a - t i o n s i n c l u d i n g r e c o i l w i t h parameters which f i t t h e (160, I5N) r e a c t i o n . The f i t s o b t a i n e d i n both c a l c u l a t i o n s (Bonche and e x a c t f i n i t e range, E.F.R.

[ ~ e v r i e s ] ) r e q u i r e a l a r g e r v a l u e of d i f f u s e n e s s a (= 0.7) i n e n t r a n c e and e x i t channels [or equiv- a l e n t l y a l a r g e r v a l u e of r a d i u s parameter r o (- 1.3513

.

Such changes e s s e n t i a l l y f o r c e t h e r e a c t i o n t o t a k e p l a c e f a r o u t and hence reproduce t h e observed forward peaking. However, with t h i s choice of parameters, t h e observed q u a l i t y of f i t

HEA W-ION TRANSFER REACI'IONS C5-119

was m a i n t a i n e d a t 4 2 , 56 and 59.5 MeV bombarding e n e r g y .

T a ~ l e I g i v e s t h e e x p e r i m e n t a l peak c r o s s s e c t i o n a t 48 MeV a n d t h e t h e o r e t i c a l p r e d i c t i o n s assuming t h a t t h e two p r o t o n s i n ' O T ~ occupy e i t h e r I f

712 O r

2p3,2 s u b s h e l l s . The f i n a l column g i v e s t h e re- q u i r e d n o r m a l i z a t i o n f o r a p u r e ( I f ) 2 c o n f i g u r a -

7 / $

t i o n . I t i s s e e n t h a t a s m a l l ( 2 ~ a d m i x t u r e i n ~ ~ ~ ) t h e 0' and 2' wave f u n c t i o n s c o u l d l e a d t o an over- a l l improvement i n n o r m a l i z a t i o n . A s i m i l a r en- hancement of t h e 0 + and 2+ c r o s s s e c t i o n s i s found c o n s i s t e n t l y a t e a c h of t h e o t h e r e n e r g i e s s t u d i e d .

T a b l e I

EjperimentaZ and calcuZated peak cross sections for the ( 1 8 ~ a f i G ~ ,

i 4 ~ . J 5 0 ~ i

~ e a c t i o n a t 40 MeV.

Ex J~ o 2

e x p u t h ( f 7 / 2 ) ' t h ( ~ 3 i 2 ) ~ 'exp ^'

(MeV) s s ( ~ b l s r ) o t h ( f 7 , 2 ) ²

0

of

200 2.8 16.8 72

1.55 2' 290 5.7 32.3 5 1

2.68 4' 190 4.3

-

⁴³

3.20 6+ 8 5 2.1

-

³⁹

I t i s beyond t h e scope o f t h i s r e v i e w t o d i s c u s s t h e l a r g e n o r m a l i z a t i o n f a c t o r s r e q u i r e d - o n e i s reminded o f s i m i l a r problems i n t h e a n a l y s i s of ( t , p ) r e a c t i o n s . F u r t h e r m o r e , t h e s i t u a t i o n i s ex- a c e r b a t e d by t h e e x t r e m e o p t i c a l p a r a m e t e r s r e - q u i r e d t o f i t t h e a n g u l a r d i s t r i b u t i o n s , Neverthe- l e s s , i t s h o u l d b e r e n a r k e d t h a t t h e Bonche a n a l y - s i s s t i l l d o e s n o t g i v e a good d e s c r i p t i o n of t h e e n e r g y dependence o f t h e c r o s s s e c t i o n

-

^{i n}

c o n t r a s t t o t h e E.F.R. c a l c u l a t i o n s

-

p e r h a p s i n d i c a t i n g some inadequacy of t h e r e c o i l approxima- t i o n .

The s e n s i t i v i t y of 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 s t o s m a l l a d m i x t u r e s o f 2p312 i n I f 7 l 2 c o n f i g u r a - t i o n s h a s been remarked ( T a b l e 1 ) . S i n c e t h e heavy-

i o n r e a c t i o n i s s t r o n g l y l o c a l i z e d a t t h e nucleon s u r f a c e , t h e d i f f e r e n t magnitude of t h e p u r e con- f i g u r a t i o n s stems from d i f f e r e n c e s i n t h e form f a c - t o r of t h e t r a n s f e r r e d n u c l e o n s t h e r e , I n t u r n t h i s depends on t h e s i n g l e n u c l e o n b o u n d - s t a t e wave f u n c t i o n s , t h e 2p3,2 p e a k i n g f u r t h e r o u t t h a n t h e I f l f 2 . ( I t can depend i n a d d i t i o n on t h e m a t c h i n g of t h e t r a n s f e r r e d n u c l e o n c o n f i g u r a r i o n s i n t h e p r o j e c t i l e a n d f i n a l n u c l e u s ) . The e f f e c t s h o u l d i n c r e a s e a s t h e p r o d u c t o f t h e number o f n u c l e o n s t r a n s f e r r e d , a s i s i n d e e d a p p r o x i m a t e l y o b s e r v e d

a l p h a t r a n s f e r r e a c t i o n s h o u l d b e p a r t i c u l a r l y s a n - s i t i v e t o 2p3/2 a d m i x t u r e s .

The DWBA a n a l y s i s of t h e f o u r n u c l e o n t r a n s f e r by means of t h e Bonche code i s a l s o p r o c e e d i n g a t S a c l a y , i n d e e d h i s t o r i c a l l y i t was f i r s t u s e d i n a r e a c t i o n mechanism s t u d y of t h e (160, "c) r e a c t i o n [18]

.

I n a p p a r e n t c o n t r a s t t o t h e ( I 6 0 , l 4 c ) r e - a c t i o n , i t i s found t h a t t h e peak p o s i t i o n i n t h e a n g u l a r d i s t r i b u t i o n a p p e a r s t o 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 o p t i c a l p a r a m e t e r s d e r i v e d from t h e e l a s t i c s c a t t e r i n g

-

a t l e a s t f o r t h e

5 4 ~ e ( 1 4 0 , 1 2 ~ ) 5 8 ~ i r e a c t i o n . Moreover t h e a n a l y s i s p e r m i t s t h e i n t e n s i t y v a r i a t i o n i n t h e

I2c

s p e c t r a due t o r e a c t i o n mechanism - t h e Q dependence

-

t o be e s t i m a t e d .

T h i s i n i t i a l s t u d y was b a s e d , n o t u n r e a s o n a b l y , on t e s t i n g p u r e s t r e t c h - s c h e m e f o u r - p a r t i c l e wave f u n c t i o n s . I-lowever, t h e extreme s e n s i t i v i t y t o t h e p r e s e n c e of ( 2 ~ c o n f i g u r a t i o n presumably a l s o ~ ~ ~ ) i m p l i e s a s e n s i t i v i t y t o c o n f i g u r a t i o n s c o n t a i n i n g ( 2 ~components. I n t h i s c o n n e c t i o n , a r e c e n t ~ ~ ~ ) ~ comparison

[lq

^{o f} ( a , u t ) and a l p h a t r a n s f e r r e a c - t i o n s t o 5 8 ~ i s u g g e s t t h e i m p o r t a n c e o f t r a n s f e r t o 3- s t a t e s . The s t r o n g e x c i t a t i o n o f s t a t e s b a s e d on s u c h o b v i o u s l y mixed components i m p l i e s t h e n e e d t o c o n s i d e r t h e c o n t r i b u t i o n of mixed components i n o t h e r c a s e s ( e . g . even p a r i t y s t a t e s ) . However, then t h e l a r g e number of p o s s i b l e c o n f i g u r a t i o n s based on f o u r p a r t i c l e e x c i t a t i o n s , p r e v e n t s a d i - r e c t i n t e r p r e t a t i o n from t h e d a t a

-

e v e n f o r s t a t e s of known s p i n . One is l e f t i n t h e p o s i t i o n o f t e s t i n g t h e o r e t i c a l wave f u n c t i o n s

-

a r e a l i s t i c p r o p o s i t i o n o n l y f o r low-lying s t a t e s .

5. T r a n s f e r K e a a t i o n s a t High E n e r g i e s . I would l i k e now t o c o n c l u d e t h i s r e v i e w by d i s c u s -

s i o n o f some r e c e n t t r a n s f e r e x p e r i m e n t s performed a t h i g h e r e n c r g i e s (5-10 MeV p e r n u c l e o n ) . These a r e mainly c y c l o t r o n e x p e r i e n t s i n which a v a r i e t y of p r o j e c t i l e s have been used t o bombard mainly l i g h t and heavy t a r g e t s . T h e i r d i s c u s s i o n h a s t h e a d v a n t a g e o f e n d i n g w i t h r c s u l t s which, i n t h e i r r e m a r k a b l e s e l e c t i v i t y , r e v e a l new a s p e c t s o f t h e heavy i o n t r a n s f e r r e a c t i o n and t h u s a r e a d i r e c t encouragement t o t h e more w i d e s p r e a d achievement o f h i g h e r e n e r g y heavy i o n s .

i n t h e c a l c u l a t i o n s of a l p h a t r a n s f e r . Thus t h e

C5-120 G . C . MORRISON

The b a s i c f e a t u r e of t h e heavy-ion t r a n s f e r r e a c - t i o n a t h i g h e n e r g i e s a p p e a r s t o be t h e f a v o r i n g of l a r g e a n g u l a r momentum t r a n s f e r [20]. T h i s s e l e c - t i v i t y i s shown i n F i g s . 13 and 14 f o r one a n d t h r e e - n u c l e o n t r a n s f e r s [21] on r e l a t i v e l y l i g h t n u c l e i . I t i s most c l e a r l y d e m o n s t r a t e d i n t h e l a t -

t e r example i n which t h e s t a t e s of maximum p o s s i b l e a l i g n m e n t a r e most s t r o n g l y p o p u l a t e d . T h i s r e s u l t a p p e a r s t o b e a n a t u r a l consequence of t h e l a r g e l i n e a r momentum o f t h e t r a n s f e r r e d n u c l e o n o r c l u s - t e r of n u c l e a r which h a s t o be c o n v e r t e d i n t o an- g u l a r momentum. Thus, i f t h e r e a c t i o n t a k e s p l a c e a t t h e n u c l e a r s u r f a c e , e a c h n u c l e o n c a n b e t h o u g h t

t o t r a n s f e r a n g u l a r momentum L

%

k R , where ko i s t h e wave number of t h e r e l a t i v e n u c l e o n v e l o c i t y and R is t h e r a d i u s of t h e t a r g e t . Moreover, t h e s e l e c t i v i t y w i l l be g r e a t e s t and t h e c r o s s s e c t i o n l a r g e s t f o r r e a c t i o n s whose Q v a l u e s f a v o r t h e l a r g e momentum t r a n s f e r . T h i s c o n t r a s t s w i t h t h e s i t u a t i o n a t e n e r g i e s n e a r t h e Coulomb b a r r i e r where k

2

^{0 .}

F i g . 13.

-

S i n g l e - p r o t o n t r a n s f e r r e a c t i o n induced by 1 1 4 MeV

I2c

i o n s on

I ~ c ,

160 and , 4 0 ~ a .

12Fi8. 14.

-

Three-nucleon t r a n s f e r r e a c t i o n ( C Be) i n d u c e d by 1 1 4 MeV 1% i o n s ( a ) on

and (b) on I ~ c , 160 and 4 0 ~ a ,

These q u a l i t a t i v e c o n s i d e r a t i o n s have been ex- tended by B r i n k 1223 t o d e r i v e q u a n t i t a t i v e k i n e - m a t i c a l c o n d i t i o n s on t h e t r a n s f e r r e a c t i o n a t en- e r g i e s w e l l above t h e Coulomb b a r r i e r . These r e - l a t e t h e Q v a l u e of t h e r e a c t i o n t o t h e a n g u l a r momentum o f t h e t r a n s f e r r e d n u c l e o n s i n t h e i n i - t i a l and f i n a l n u c l e u s . S i n c e l a r g e c r o s s s e c t i o n s a r e o n l y p o s s i b l e i f t h e s e k i n e m a t i c a l c o n d i t i o n s a r e s a t i s f i e d , one can t h e r e b y o b t a i n d i r e c t i n - f o r m a t i o n on t h e s t r u c t u r e o f t h e s t a t e s s e l e c - t i v e l y p o p u l a t e d .

The Oxford group [23] h a s a l s o u s e d t h e b a s i c o n e - n u c l e o n s e l e c t i v i t y a s a b u i l d i n g b l o c k t o s t u d y t h e same f i n a l n u c l e u s formed by r e a c t i o n s which t r a n s f e r a d i f f e r e n t number of n u c l e o n s . F o r example t h e two-neutron t r a n s f e r r e a c t i o n on 1 2 c may be compared w i t h t h e one-neutron t r a n s f e r re- a c t i o n on

I3c

( F i g . 15) .One f i n d s t h a t o n l y t h e 3- s t a t e i n

14c,

based on t h e diI2 @ p I ,? c o n f i g c r a - t i o n , i s p o p u l a t e d i n t h e 2n ( o r 2p) t r a n s f e r whereas t h e r e a c t i o n on 1 3 c p o p u l a t e s b o t h t h e 3- and 2 - s t a t e s ( T a b l e 1 1 ) . The i n h i b i t i o n o f u n n a t - u r a l p a r i t y s t a t e s i n 2n ( o r 2p) t r a n s f e r a p p e a r s a's a n a t u r a l consequence i f t h e two i d e n t i c a l nu- c l e o n s a r e t r a n s f e r r e d i n a 0' r e l a t i v e I s c o n f i g - u r a t i o n

-

t h e c o r r e s p o n d i n g 3- and 2- s t a t e s a r e b o t h p o p u l a t e d i n np c r a n s f e r l e a d i n g t o I4F4. How- e v e r . i t i s a l s o found t h a t t h e 2n and 22 t r a n s f e r

F i g . 15.

-

P o p u l a t i o n of s t a t e s of

I4c

by one and two-neutron t r a n s f e r r e a c t i o n s on 13c and

I2c,

r e s p e c t i v e l y . Einc

2

1 0 MeV/nucleon,

HEAW-ION TRANSFER REACTIONS

Table 11

States select.tvely eazited i n sin Ze-and two-nsutrm stripping reactions on 2% and 2 3 ~ .

~ e a c t i o n Ex J" C o n f i g u r a t i o n ( a ) 1 2 ~ ( 1 1 ~ , 1 0 ~ ) 1 3 ~ 0.0 112- VP1/2

3.85 512' vd

512

is much reduced compared with the np t r a n s f e r

-

^and

even w i t h t h r e e nucleon ( t o r 3 ~ e ) t r a n s f e r . This r e s u l t , which i s n o t an e f f e c t of Q v a l u e , i s n o t understood a t p r e s e n t .

The e f f e c t of t h e " s p e c t a t o r " nucleon has been c a r r i e d t o h i g h e r 0rde.r f o r 160 by comparing t h e s t a t e s e x c i t e d i n three-nucleon t r a n s f e r r e a c t i o n s on

12c

with those populated i n two-nucleon t r a n s f e r r e a c t i o n s on

I3c,

Such s t u d i e s make use of t h e wide v a r i e t y of a v a i l a b l e p r o j e c t i l e s so t h a t problems of d i s t i n g u i s h i n g between e x c i t e d s t a t e s of t h e o u t g o i n g p a r t i c l e and t h e new s t a t e s of i n t e r e s t i n t h e f i n a l n u c l e u s can be overcome. This combination of t h e t r a d i t i o n a l methods used t o examine weak c o u p l i n g w i t h t h e high s e l e c t i v i t y of t h e heavy-ion r e a c t i o n appears t o have g r e a t p o t e n t i a l .

A f u r t h e r s o u r c e of s e l e c t i v i t y i n r e a c t i o n s on l i g h t n u c l e i has been e a r l i e r observed by von Oertzen [24], nameiy, t h a t t h e t r a n s f e r of s e v e r a l nucleons i s s t r o n g l y favored when t h e i r i n t e r n a l s t r u c t u r e i s n o t changed i n the i n i t i a l and f i n a l n u c l e u s .

This i s demonstrated i n F i g u r e 16 which shows t h a t t h e 1 2 ~ ( 1 9 ~ , 60) 1 5 ~ r e a c t i o n p o p u l a t e s most s t r o n g l y t h e 1/2+ s t a t e a t 5.3 MeV i n 15N which was t h e same c o n f i g u r a t i o n a s the p r o j e c t i l e , namely (Pl12)-4(sd)3. The d i f f e r e n t s e l e c t i v i t i e s of t h e ( 1 60, 'N) and ( 160, r e a c t i o n s a r e a l s o a p p a r e n t .

A p r e f e r e n t i a l e x c i t a t i o n of s p e c i f i c s t a t e s i s observed i n t h e one-nucleon t r a n s f e r r e a c t i o n s on Pb [23], [25] a t high e n e r g i e s

-

a f a v o r i n g of high s p i n s t a t e s f o r n e g a t i v e Q v a l u e s is a l s o found 1261 i n two-nucleon t r a n s f e r r e a c t i o n s on some f p - s h e l l n u c l e i . S p e c t r a o f t h e 2 0 8 ~ b (

I2c,

I3c) and

-excitat~on energy

channel number-

F i g . 16.

-

S e l e c t i v e p o p u l a t i o n of s t a t e s of I5N by d i f f e r e n t t r a n s f e r r e a c t i o n s .

2 0 8 ~ b ( 1 2 ~ , "B) r e a c t i o n s a t 78 MeV [252 a r e shown i n F i g u r e 17. (They have been o b t a i n e d w i t h t h e Berkeley magnetic s p e c t r o m e t e r system u s i n g a fo- c a l p l a n e d e t e c t o r and r e p r e s e n t t h e c u r r e n t o p t i - mum i n p a r t i c l e i d e n t i f i c a t i o n and energy r e s o l u -

t i o n f o r heavy i o n s . ) C l e a r l y seen i s the wide range of j,R v a l u e s of t h e s i n g l e p a r t i c l e s t a t e s . More than i n t h e r e a c t i o n s on l i g h t n u c l e i , t h e s p e c i f i c dependence of a n g u l a r momentum t r a n s f e r on t h e p a r t i c u l a r p r o j e c t i l e and r e a c t i o n i s r e v e a l e d . The main f e a t u r e s a g a i n seem t o have t h e i r i n t e r p r e - t a t i o n i n t h e k i n e m a t i c s e l e c t i o n r u l e s [22]relating Q v a l u e and a n g u l a r momentum t r a n s f e r , One may a l s o a n t i c i p a t e i n multi-nucleon t r a n s f e r r e a c t i o n s an i n c r e a s i n g l y important r o l e of r e a c t i o n s such a s ("c, lone) and ( 1 3 c , ' ~ e ) i t high e n e r g i e s i n o r d e r t o use t h e i r more n e g a t i v e Q v a l u e s t o enhance high

C.C. MORRISON

"

"

b

("c. "c)

' O ' P ~

60' lab

:%a- - - -. .- -. - - - -.

I

''21. 1oaPb(s2C, " 8 ) '098i

60' lob

1

1

' R

i

C H A N N E L

F i g . 17.

-

Single-nucleon t r a n s f e r r e a c t i o n s induced by 7 8 MeV

12c

i o n s on '08pb.

a n g u l a r rnomentum s t a t e s .

The Berkeley d a t a a l s o shows c l e a r evidence f o r t h e J dependent e f f e c t s d i s c u s s e d e a r l i e r . A com- p a r i s o n of t h e r e a c t i o n s (160, 1 5 ~ ) and ( 1 2 c , 1 1 ~ ) on 2 0 8 ~ b ( F i g . 18) shows the r e d u c t i o n i n i n t e n s i t y of

160 15

t h e j2 = i2

-

^{1 / 2} s t a t e s i n t h e (

,

N ) r e a c t i o n . However, t h e r e d u c t i o n i s l e s s than p r e d i c t e d by DWBA c a l c u l a t i o n s [4] which n e g l e c t r e c o i l . Since a t high e n e r g i e s , r e c o i l e f f e c t s a r e n o t n e g l i g i b l e , t h i s d i s c r e p a n c y may be evidence f o r t h e p a r i t y - v i o l a t i n g & t r a n s f e r (= t 1 + E2

-

1) c o n t r i b u t i n g t o t h e r e a c t i o n ; p a r t i c u l a r l y a s t h i s would p e r m i t a l a r g e r a n g u l a r momentum t r a n s f e r than t h a t a l - lowed by p a r i t y . I f confirmed by e x a c t f i n i t e range c a l c u l a t i o n s , i t would c o n s t i t u t e d i r e c t evidence f o r t h e importance of such r e c o i l e f f e c t s i n heavy- ion t r a n s f e r .

A t s t i l l h i g h e r e n e r g i e s a new f e a t u r e of t h e heavy-ion r e a c t i o n i s r e v e a l e d . This is t h e p o s s i - b i l i t y t h a t t h e heavy i o n beam may c a r r y i n a n g u l a r momenta above t h e c r i t i c a l a n g u l a r momentum of the compound n u c l e u s [ 2 7 ] . Thus t h e formation of the

compound nucleus w i l l be r e s t r i c t e d t o small i m - p a c t parameters and t h e c r o s s s e c t i o n f o r d i r e c t r e a c t i o n enhanced. While t h e experimental conse- quences a r e n o t completely c h a r t e d , i t would ap- p e a r most l i k e l y t h a t the phenomenon m a n i f e s t s i t - s e l f i n an i n c r e a s e d p r o b a b i l i t y of very i n e l a s t i c s c a t t e r i n g c o l l i s i o n s showing up a t very forward a n g l e s . W e v e r r e c e n t experiments 1281 a l s o sug- g e s t an e f f e c t on t h e t r a n s f e r t o low l y i n g s t a t e s

-

a l t h o u g h t h e o b s e r v a t i o n s do n o t exclude an in- t e r p r e t a t i o n i n terms of r e c o i l e f f e c t s 1203.

This c l e a r l y i s an a r e a of much f u t u r e i n t e r e s t

-

and one whose u n d e r s t a n d i n g may p r o v i d e t h e neces- s a r y l i n k between heavy-ion t r a n s f e r used t o o b t a i n t r a d i t i o n a l microscopic n u c l e a r s t r u c t u r e informa- t i o n , a s d i s c u s s e d i n t h i s review, and the heavy- i o n r e a c t i o n used t o e x p l o r e macroscopic p r o p e r t i e s of t h e nucleus. One r e a l i z e s a l s o t h a t our c u r r e n t p r o j e c t i l e s a r e mainly a t t h e l i g h t end of the heavy-ion r e g i s t e r , C l e a r l y we a r e o n l y s e e i n g t h e b e g i n n i n g of t h e heavy-ion t r a n s f e r r e a c t i o n a s a n e x c i t i n g , many-f a c e t e d field of r e s e a r c h .

HEA ^W-ION TRANSFER REACTIONS

P ~ ~ ~ ( C " , 6 " ) Bi EC12 =

78 MeV

e, = 60 Deg

C H A N N E L NUMBER

F i g . 18.

-

S i n g l e - p r o t o n s t r i p p i n g r e a c t i o n s induced by 104 MeV 160 and 7 8 MeV

'*c

i o n s on *08pb.

Acknowledgements. - I am p l e a s e d t o acknowledge t o spend a y e a r s a b b a t i c a l i n F r a n c e

-

and i n p a r - h e l p f u l d i s c u s s i o n s w i t h D r s . P. Bonche, R. DeVries t i c u l a r t o acknowledge t h e h o s p i t a l i t y e x t e n d e d and R , ' S c h a e f f e r . I a l s o w i s h t o t a k e t h i s o p p o r t u - t h r o u g h o u t by Mme H. F a r a g g i and D r . E. C o t t o n . n i t y t o t h a n k C.E.N. S a c l a y f o r making i t p o s s i b l e

R e f e r e n c e s

(#)work performed i n p a r t under t h e a u s p i c e s o f t h e P r e s s , New York, t o be p u b l i s h e d . U.S. Atomic Energy Commission.

[I]

BREIT (G.) and EBEL (M.), Phys. Rev., 1956, The P r o c e e d i n g s of t h e Symposium on Heavy I o n

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103, 679.

R e a c t i o n s and Many P a r t i c l e E x c i t a t i o n s , S a c l a y BREIT (G.), Phys. Rev., 1964,

135,

B1323.

(19711, Suppl. J. P h y s i q u e , 1971,

2,

C6

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a b b r e - FRAHN (W.E.) and VENTER (R.H.)

,

Nucl, Phys,

,

v i a t e d t o S a c l a y Conf. - g i v e a good a c c o u n t o f 1964,

2,

651.

r e c e n t work.

[2] BUTTLE (P.J.A.) a n d GOLDFARB (L.J.B.), Nucl.

A r e c e n t comprehensive r e v i e w i s " S i n g l e a n d Phys., 1971,

-,

299.

M u l t i n u c l e o n T r a n s f e r Reactions", Von O e r t z e n (W.),

i n N u c l e a r S p e c t r o s c o p y ( e d . J, Cerny)

,

Academic [3] MORRISON (G.C.), S a c l a y Conf., C6-69.