ON THE APPLICATION OF SEMI-CLASSICAL CONCEPTS TO HEAVY ION SCATTERING

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ON THE APPLICATION OF SEMI-CLASSICAL

CONCEPTS TO HEAVY ION SCATTERING

C. Marty

To cite this version:

JOURNAL DE PHYSIQUE

Colloque C6, supplément au n ' é , Tome 45 juin 1984 page C6-121

ON THE APPLICATION OF SEMI"CLASSICAL CONCEPTS TO HEAVY ION SCATTERING

C. Marty

Division de Physique Théorique*, Institut de Physique Nucléaire, B.P. 1, 91406 Orsay Cedex, France

Résumé - Les méthodes semi-classiques s ' a p p l i q u e n t d ' o r d i n a i r e bien aux c o l l i s i o n s d ' i o n s l o u r d s . T o u t e f o i s i l e x i s t e des contre-exemples. L'un e s t discuté i c i .

A b s t r a c t - S e m i - c l a s s i c a l methods work q u i t e w e l l usually f o r heavy-iQn c o l l i s i o n s . However counter examples e x i s t and one o f them i s discussed here.

Even a t low e n e r g i e s , due to t h e i r large masses, c o l l i s i o n s between heavy ions are always associated w i t h small de B r o g l i e wave-lengths ( w i t h respect to the geometri-cal c h a r a c t e r i s t i c s o f d i f f u s i n g i n t e r a c t i o n s ) . I t i s t h e r e f o r e n a t u r a l t o f i n d s e m i - c l a s s i c a l ( s . c . ) ideas l a r g e l y used i n t h a t branch o f p h y s i c s . One i s close t o an " o p t i c a l " s i t u a t i o n and the words, i f not q u i t e the concepts, o f d i f f r a c t i o n , r e f r a c t i o n , d e f l e c t i o n f u n c t i o n s , rainbows, g l o r i e s . . .are c u r r e n t l y encountered. The s . c . methods work sometimes remarkably w e l l . A c l a s s i c a l example i s the c a l c u l a -t i o n of -the pure Coulomb s c a -t -t e r i n g ampli-tude fc(e)between p o i n t charges under an

angle 9. The WKB approximation associated w i t h saddle p o i n t or s t a t i o n a r y phase techniques give the exact r e s u l t f o r fc. Though s u r p r i s i n g t h i s can be p a r t l y

understood / l / . What f o l l o w s i s r e l a t e d t o a counter example where the a p p l i c a t i o n o f s . c . approximations y i e l d s to a completely wrong r e s u l t .

I t has been recently asserted / 2 / t h a t f o r charged p a r t i c l e s a forward g l o r y i s p r e -sent i n the e l a s t i c channel. This has t o be understood i n the f o l l o w i n g way. The usual o p t i c a l theorem should be m o d i f i e d f o r long range i n t e r a c t i o n s such as the Coulomb one. For spinless p a r t i c l e s i n the i n c i d e n t channel a new i n t e g r a t e d e l a s t i c c r o s s - s e c t i o n a ' , ( 9 o ) i s defined as

IT

( ! ) ae l ^9° ^ = 21T ( d o / d f i - d q _ / d n ) s i n e . d e -'eo

where do/dfi i s the d i f f e r e n t i a l e l a s t i c c r o s s - s e c t i o n and d0c/dfi the Rutherford

one. In the l i m i t e0- * - 0 , a " f o r m a l " c a l c u l a t i o n using the p r o p e r t i e s o f Legendre

polynomials y i e l d s :

( I I ) o | = a ^1 +aR = 4 T T l m fN( 0 ) / k

In ( I I ) , ol i s a new t o t a l c r o s s - s e c t i o n , sum o f the standard r e a c t i o n one aR plus

the l i m i t a^ f o r 60^ 0 o f a ' ^ 6 0 ) . fN i s the s o - c a l l e d short-range amplitude

and k the asymptotic wave number o f the c o l l i d i n g p a r t n e r s . Evaluating f|\|(0) by the s t a t i o n a r y phase approximation i t i s asserted i n / 2 / t h a t aj o s c i l l a t e s when k i n c r e a s e s . These o s c i l l a t i o n s are the s i g n a t u r e o f a forward g l o r y since they are r e l a t e d t o p r o p e r t i e s o f the d e f l e c t i o n f u n c t i o n .

* laboratoire Associé au C.N.R.S.

C6-122 JOURNAL DE PHYSIQUE

As i t has been demonstrated elsewhere

/ 3 / ,

equation (11) i s indeed i n c o r r e c t . The e x a c t r e s u l t t o s t a r t w i t h i s ( O < B O G ) :

a l l ( 0 0 + O R = 41r

I

f N ( 0 )

I

s i n ( A r g f N ( 0 )

+

2n l o g ( ~ i n 8 ~ / 2 ) )

/

k

(111) 0 0

- I r

I,

{

1

fN12

+

2 ~ e f Z [ f ~ - f ~ ( 0 ) 1 } s i n e .dB

I n (111) a l l the q u a n t i t i e s ave been defined w i t h the exception o f n which i s the

₉

Sommerfeld parameter Z1Z2e / ( i i v ) between two charges Zle, Z2e moving w i t h a r e l a t i v e v e l o a i t y v. I t has t o be noted how p a r t o f the Coulgmb-nuclear i n t e r f e r e n c e aFnears i n the RHS of (111). I n s t e a d o f the usual i n t e g r a n d fc f one has

f c [ f N - f N ( 0 )

l

.

The o r i g i n o f t h i s p a r t i c u l a r f e a t u r e w i l l be s f e t c h e d below. I n the l i m i t

e 0

-t 0 the i n t e g r a l o f (111) can be shown t o vanish whereas the f i r s t term o f the RHS s t a r t s t o o s c i l l a t e more and more r a p i d l y , b e i n g a t the l i m i t undefined b u t bounded. I f one f o l l o w s the authors o f /2/ one would, f o r the forward d i r e c t i o n , average o u t t h i s term. What i s obtained i s simply

a r e s u l t t o t a l l y independent o f the energy. One concludes t h a t the very existence o f a forward n u c l e a r rainbow i s a pure a r t e f a c t .

The o r i g i n o f the e r r o r i n r e f . / 2 / i s n o t merely due t o a S . C . approximation, b u t t o t h e "formal" d e r i v a t i o n o f (11), r e a r r a n g i n g terms i n s c a l a r p r o d u c t o f Legendre expansions o f f and f

.

Such a method i n u n c o n t r o l l e d since f c i s a d i s t r i b u t i o n / I / . I n s c a h r pro!ucts where f c . i s involved, i t has always t o be m u l t i p l i e d by " t e s t - f u n c t i o n s " . f N - f N ( 0 ) , which vanishes f o r B = 0 , has t h a t property, b u t n o t f~ i n general and i t i s the reason o f the appearence o f f N - f N ( 0 ) i n ( I I I ) . For a complete discussion, see /3/.

The above remarks are r a t h e r academical s i n c e i t i s impossible i n p r a c t i c e t o make any measurement i n the forward d i r e c t i o n due t o the enormous f l u x i n the i n c i d e n t channel. The i n t e r e s t o f (111) i s f o r angles B o i n the r e g i o n o f the Coulomb n u c l e a r i n t e r f e r e n c e (Fresnel zone). I n t h i s s i t u a t i o n i t i s p o s s i b l e t o give model independent upper and lower bounds t o DR. T h i s w i l l be p u b l i s h e d elsewhere.

References :

/l/ ROWLEY N., J. Phys. A l l (1978) 1545.

/ 2 / HUSSEIN M.S. e t a l . , P h y s . L e t t . 1146 (1982) 1. /3/ MARTY C., Zei t. Phys

.

A309 ( 1 9 8 3 ) 2 6 1 .