SEMICLASSICAL AND QUANTUM MECHANICAL TREATMENTS OF POSITIVE AND NEGATIVE ANGLE POLARIZED HEAVY ION SCATTERING

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TREATMENTS OF POSITIVE AND NEGATIVE ANGLE POLARIZED HEAVY ION SCATTERING

D. Mukhopadhyay, G. Grawert

To cite this version:

D. Mukhopadhyay, G. Grawert. SEMICLASSICAL AND QUANTUM MECHANICAL TREAT- MENTS OF POSITIVE AND NEGATIVE ANGLE POLARIZED HEAVY ION SCATTERING.

Journal de Physique Colloques, 1984, 45 (C6), pp.C6-435-C6-444. �10.1051/jphyscol:1984652�. �jpa-

00224255�

JOURNAL DE PHYSIQUE

Colloque C6, supplément au n°6, Tome Ï5, juin 198^ page C6-435

SEMICLASSICAL AND QUANTUM MECHANICAL TREATMENTS OF POSITIVE AND NEGATIVE ANGLE POLARIZED HEAVY ION SCATTERING

D. Mukhopadhyay and G. Grawert*

Institut ftir Theoretisohe Physik, Universit&t Heidelberg, and

Max-Ptanok-Institut filr Kernphysik, Heidelberg, F.R.G.

*Faahbereiah Physik, Vniversit'dt Marburg, F.R.G.

Résumé - Nous utilisons une théorie semiclassique pour calculer les sections efficaces et les pouvoirs d'analyse à" tout ordre en diffusion élastique d'ions lourds polarisés. Cette approche a pour but d'expliquer les déviations par rapport à l'absence de structure observée normalement dans les sections efficaces. Ces déviations sont dues aux interférences entre diffraction et réfraction. Nous montrons que la diffusion rêfractive à angle négatif ("far- side scattering") peut être interprétée comme un effet de taille de la cible

à El a b * 20 MeV pour Li + C. Il est montré que le très grand nombre

d'intégrales à calculer numériquement limite le champ d'application de la méthode semiclassique. Finalement nous présentons une approche quantique pour traiter la diffusion à angles positifs et négatifs.

Abstract - A unified semiclassical theory for calculating cross section and analysing powers of all ranks for polarized heavy ion elastic scattering is applied to explain deviations from the structureless pattern normally en- countered in the cross section, which are interpreted as due to interference between diffraction and refraction. Refractive, negative angle or far-side scattering is then discussed and is shown to be a target-size effect at E-, . ~ 20 MeV for Li + C. The limitations of the semiclassical method are then shown in terms of the necessity of performing a huge number of integrals numerically. Lastly, a quantum mechanical approach for handling positive and negative angle scattering is presented.

I - INTRODUCTION

Experimental data of polarization observables along with elastic scattering cross sec tions can most assuredly be expected to provide a more stringent handle on studying interference effects arising from negative angle scattering compared to the situation without analysing power measurements. This is because one would then utilise an ad- ditional degree of freedom, namely spin, over and above unpolarized cross section measurement, provided one can experimentally study effects of spin on cross section,

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984652

i . e . analysing powers. Thus a f u l l e r s e t o f measurements composed o f analysing pow- e r s o f as many ranks as p o s s i b l e f o r t h e same s c a t t e r i n g process would p r o v i d e a ground f o r t e s t i n g the v a l i d i t y o f any model assumptions o r r e s o l v i n g o f any o p t i c a l p o t e n t i a l a m b i g u i t i e s w i t h much l e s s u n c e r t a i n t y . One such f i e l d which has r e c e n t l y received much a t t e n t i o n , a l b e i t w i t h o u t accompanying p o l a r i z a t i o n data, i s t h e study o f r e f r a c t i v e e f f e c t s i n terms o f t h e normal, d i f f r a c t i v e p o s i t i v e a n g l e s c a t t e r i n g from t h e near s i d e o f t h e t a r g e t and the n e g a t i v e angle s c a t t e r i n g from t h e f a r side.

I n t h i s t a l k we present how p o s i t i v e and n e g a t i v e angle s c a t t e r i n g and i n t e r f e r e n c e s t h e r e o f can be d h e o r e t i c a l l y s t u d i e d using p o l a r i z a t i o n data f i r s t l y by adopting a t r a j e c t o r y p i c t u r e , t h a t i s , i n a semiclassical manner and secondly by t a k i n g the quantum mechanical approach of s p l i t t i n g t h e s c a t t e r i n g process i n t o i t s near and f a r - s i d e c o n t r i b u t i o n s inasmuch as t h e Legendre f u n c t i o n s , which occur i n p a r t i a l wave expansion w i t h i n o p t i c a l model, a r e rearranged accordingly.

A s e r i e s o f t e c h n i c a l l i m i t a t i o n s i s r e s p o n s i b l e f o r t h e f a c t t h a t u p t i l now a com-

6- rS:

p l e t e s e t o f analysing power data i s a v a i l a b l e o n l y f o r some L i ( s p i n 1) and LI

12 58

( s p i n 3/2) e l a s t i c s c a t t e r i n g from l i g h t ( C) and heavier ( N i say) t a r g e t s a t tandem energies. I n view o f t h i s f a c t and due t o t h e t h e o r e t i c a l c o n s i d e r a t i o n s i n - troduced above, i n t h i s work we concentrate on developing a t h e o r e t i c a l approach which i s n o t o n l y capable o f simul taneously accounting f o r t h e observed cross s e c t i o n as w e l l as analysing powers o f a l l ranks, b u t a l s o can i n c o r p o r a t e a s p e c i f i c e f f e c t a r i s i n g o u t o f t h e s i z e o f t h e t a r g e t .

To understand t h e t a r g e t - s i z e e f f e c t , we present t h e f o l l o w i n g argument f o r v i s u a l i s i n g e l a s t i c s c a t t e r i n g o f a l i g h t heavy-ion ( H I ) p r o j e c t i l e l i k e t h e l i t h i u m isotopes.

P o s i t i v e angle s c a t t e r i n g i s then t h e dominant process f o r s c a t t e r i n g from a heavy t a r g e t (say 5 8 ~ i o r heavier) since t h e negative angle c o n t r i b u t i o n s a r e h e a v i l y suppressed as p r o j e c t i l e s which go around t h e t a r g e t g e t more s t r o n g l y absorbed. T h i s i s how we e x p l a i n the r e l a t i v e l y f e a t u r e l e s s Fresnel-type d i f f r a c t i o n p a t t e r n f o r a/aR o f

+

5 8 ~ i e l a s t i c s c a t t e r i n g a t Elab = 20.3 MeV [I1 and t h e corresponding

second rank tensor analysing power i s then we1 1 explained by t a k i n g recourse t o the shape e f f e c t model [2]. This c l a s s i c a l model associates changes i n o v e r l a p be- tween p r o j e c t i l e and t a r g e t a t t h e p o i n t o f c l o s e s t approach t o a l i g n i n g quadrupole-

7 2

deformed p r o j e c t i l e s (here L i w i t h Q-moment = -3.9 6m ) w i t h respect t o v a r i o u s s p i n p r o j e c t i o n s , since i n a s p i n - p o l a r i z e d beam quadrupole moment and s p i n a r e co- a x i a l . T h s h e shape-effect model n o t o n l y g i v e s the c o r r e c t ( n e g a t i v e ) s i g n o f

f o r L i 7

+

5 8 ~ i beyond t h e g r a z i n g angle b u t a l s o reproduces t h e magnitude f o r a reasonable choice o f L i quadrupole moment. 7

I n c o n t r a s t , f o r e l a s t i c s c a t t e r i n g o f l i t h i u m w i t h the l i g h t e r t a r g e t 12C a t com- parable l a b o r a t o r y energies [31, t h e angular d i s t r i b u t i o n o f u/uR shows wiggles f o r angles beyond the g r a z i n g angle and t h e simple shape e f f e c t model f o r o b t a i n i n g t h e

a n a l y s i n g powers i s a u t o m a t i c a l l y doomed t o f a i l u r e . Here, one can argue t h a t t h e w i g g l e s a r e due t o i n t e r f e r e n c e e f f e c t s between d i f f r a c t i o n and n e g a t i v e a n g l e s c a t - t e r i n g and t h e r e f o r e t h e s i m p l i f i e d v e r s i o n o f t h e shape e f f e c t model can be bidden good-bye. I n t h i s work we show how t h e ' n u c l e a r m a t t e r o v e r l a p ' i d e o l o g y can be ex- tended i n a h e a v i l y m o d i f i e d f o r m t o n e g a t i v e a n g l e s c a t t e r i n g . Here we a g a i n r e - i t e r a t e t h e n o t i o n t h a t t r e a t i n g "/aR and T20 on t h e same f o o t i n g p r o v i d e s a n ex- T t r e m e l y p o w e r f u l methodology o f o b t a i n i n g a handle on n e g a t i v e a n g l e s c a t t e r i n g .

T h i s r e p o r t i s d i v i d e d i n t o f o u r s e c t i o n s . P o l a r i z a t i o n o b s e r v a b l e s a r e b r i e f l y i n - t r o d u c e d and d e f i n e d i n sect,. 2. Then we summarily d i s c u s s t h e s e m i c l a s s i c a l v e r s i o n o f t r e a t i n g n e g a t i v e a n g l e s c a t t e r i n g i n sect. 3 where t h e r e a d e r w i l l be r e f e r r e d t o o u r work on a general f o r m a l i s m o f s e m i c l a s s i c a l t h e o r y o f a n a l y s i n g powers. Next, i n s e c t . 4 t h e quantum mechanical approach o f s e p a r a t i n g t h e s c a t t e r i n g process i n t o po- s i t i v e and n e g a t i v e a n g l e c o n t r i b u t i o n s i s d e a l t w i t h .

I 1

-

DEFINITION OF ANALYSING POWERS

S t a r t i n g w i t h t h e Schroedinger e q u a t i o n H+ = EQ, t h e s c a t t e r i n g a m p l i t u d e m a t r i x M i s d e f i n e d by t h e a s y m p t o t i c expansion o f t h e t o t a l wave f u n c t i o n Y as a sum o f an incoming p l a n e wave i n a p u r e s p i n s t a t e mo and an o u t g o i n g s p h e r i c a l wave composed o f a l l p o s s i b l e s p i n s t a t e s . Thus

where K i s t h e wave number.

The centre-of-mass (c.m.) c r o s s s e c t i o n do (B~.,.

,

pin) i s o b t a i n e d by t r a n s f o r m i n g f r o m t h e measured l a b o r a t o r y a n g l e

elab

t o t h e c o r r e s p o n d i n g c.m. angle. Thus,

where t h e s p i n s t a t e o f t h e beam i s d e s c r i b e d b y t h e d e n s i t y m a t r i x pin i n s p i n space.

Expanding pin =

0sis2s

^{T t* / ( 2 s}

⁺

^{I ) ,} where s i s t h e p r o j e c t i l e s p i n , k and q a r e kq kq

t h e r a n k o f t h e t e n s o r and i t s p r o j e c t i o n and < m' I T k q ] m > =

Jm

c (sks; mqm' ) a r e t h e i r r e d u c i b l e s p h e r i c a l m a t r i c e s i n s p i n space, one o b t a i n s

The t k q l s denote t h e ' p r e p a r e d ' beam p o l a r i z a t i o n and t h e T ' s a r e t h e measured a n a l y s i n g powers. kq

S p e c i a l i s a t i o n

Rank 1 ( k = 1 ) : F o r t h e rank 1 case t h e beam i s ' p r e p a r e d ' i n a way such t h a t t h e m a t r i x p has t h e f o l l o w i n g elements:

<

milp

jm > = 6 lim/s

i n mm' 2si;l

t h a t i s , t h e o c c u p a t i o n number i s a l i n e a r f u n c t i o n o f t h e magnetic s u b s t a t e s . C l e a r - l y then. pin = 1 (1 i

<.;I

and i n t h i s c o n s t r u c t i o n t h e o n l y non-zero t i s

1 q t l O -

Such a beam i s c a l l e d a ' v e c t o r p o l a r i z e d ' beam. If now one measures t h e t r a n s v e r s e v e c t o r ( o r f i r s t r a n k ) a n a l y s i n g power, c a l l e d TlT O, b y t a k i n g t h e q u a n t i s a t i o n a x i s such t h a t e[ l t j x t f , t h e n t h i s ' t r a n s v e r s e ' q u a n t i t y i s

du p o l .

-

( x ) u n p o l .

u (&)unpol.

-

For t h e t r a n s v e r s i t y frame t h e c h o i c e o f a x i s i s

?;i

x

tf 1 I

ez and f o r t h e Madison frame 141 one chooses

?;i

x

Tf 1 1

^e For r a n k 1 t h e ' t r a n s v e r s e ' TI0 i s s i m p l y p r o -

Y' M

p o r t i o n a l t o 'Madisont iTll ( a s a whole a r e a l q u a n t i t y ! ) , i.e., T ~ l o a i Tll. F u r - t h e r d e t a i l s on t h i s s u b j e c t and i n t e r r e l a t i o n s among t h e v a r i o u s c o n v e n t i o n s of de- f i n i n g p o l a r i z a t i o n o b s e r v a b l e s can be o b t a i n e d from r e f . 141.

Rank 2 ( k = 1 ) : Here we s h a l l s i m p l i f y t h e d i s c u s s i o n by d i r e c t l y c o n s i d e r i n g t h e s i t u a t i o n f o r s p i n 3 / 2 . I n t h i s case t h e beam i s ' p r e p a r e d ' such t h a t o n l y tZ0 = 1 and a l l o t h e r t ' s a r e zero. T h i s i s achieved, say, b y h a v i n g a q u a d r a t i c ( p a r a b o l i c ) b e h a v i o u r o f t h e o c c u p a t i o n number as a f u n c t i o n o f t h e magnetic p r o j e c t i o n as f o l l o w s : kq

f o r m1 = m = 3/2

< m l j p [m > = 1

n

0 o t h e r w i s e

Such a beam i s c a l l e d ' a l i g n e d ' and t h e t r a n s v e r s e t e n s o r ( o r second r a n k ) ana- l y s i n g power i s

I11

-

SEMICLASSICAL TREATMENT OF POSITIVE AND NEGATIVE ANGLE SCATTERING

In t h e following we only s h a l l b r i e f l y i n d i c a t e those s t e p s i n t h e formal ism of t r e a t i n g negative angle s c a t t e r i n g e s s e n t i a l f o r our c a l c u l a t i o n s . Detailed t r e a t - ment of p o s i t i v e angle s c a t t e r i n g and p o l a r i z a t i o n observables within t h e semiclas- s i c a l framework i s given in our paper, vide r e f . [ 5 ] .

We s t a r t with taking t h e r e a l c e n t r a l p o t e n t i a l vc t o be t h e dominant i n t e r a c t i o n and t r e a t vc i n a c l a s s i c a l approximation t o obtain t r a j e c t o r i e s of t h e p r o j e c t i l e . All t h e r e s t of t h e p o t e n t i a l s , be they a b s o r p t i v e c e n t r a l (wc) o r complex s p i n - o r b i t ( U ) a r e t r e a t e d p e r t u r b a t i v e l y . The a b s o r p t i o n , governed by wc, t h e r e f o r e , t a k e s place simply while t h e p r o j e c t i l e moves on i t s t r a j e c t o r y and t h e spin-dependent i n t e r a c - t i o n s , caused by U, generate a time-evolution of t h e p r o j e c t i l e spin s t a t e again without a f f e c t i n g the t r a j e c t o r i e s . As a sequel t o t h e s e assumptions t h e S-matrix f a c t o r i z e s i n t o a product of two terms, t h e f i r s t f a c t o r being t h e S-matrix f o r c e n t r a l p o t e n t i a l s and t h e o t h e r being a matrix B i n spin space, which n a r r a t e s t h e e f f e c t s of spin-dependent i n t e r a c t i o n s on t h e spin s t a t e of t h e p r o j e c t i l e . Next, t h e S-matrix so obtained i s i n s e r t e d i n t o a s e m i c l a s s i c a l expression [6] f o r t h e matrix M of s c a t t e r i n g amplitudes. The usual p a r t i a l wave sum f o r t h e s c a t t e r i n g amplitude i s transformed i n t o an i n t e g r a t i o n over t r a j e c t o r i e s and t h e M-matrix i s evaluated by t h e method of s t a t i o n a r y phases 171. F i n a l l y , from t h e M-matrix t h e analysing power f o r e l a s t i c polarized HI s c a t t e r i n g a r e c a l c u l a t e d applying standard d e f i n i t i o n s [41 used f o r d e s c r i b i n g p o l a r i z a t i o n phenomena.

S t a r t i n g with t h e f u l l time-dependent Schroedinger equation f o r t h e r e l a t i v e motion of p a r t i c l e s w i t h a r e a l c e n t r a l i n t e r a c t i o n , we make t h e following s i m p l i f i c a t i o n s a s regards t h e t r a j e c t o r i e s . F i r s t l y , t h e wave packet i s a narrow one and secondly, i t s c e n t r e moves along t h e t r a j e c t o r y , The Morse saddle point equations f o r a b s o l u t e values of L a r e

(E = il)

and t h e general behaviour of t h e c l a s s i c a l deflexion f u n c t i o n

i s sketched i n f i g . 1 of r e f . [5], where i t can be seen t h a t O c i s determined essen- t i a l l y by Coulomb repulsion f o r l a r g e L values and i t goes through t h e rainbow angle O R a t L = L R and becomes i n f i n i t e f o r grazing angular momentum L In t a b l e 1 of r e f .

9'

[51 t h e rainbow parameters a r e presented f o r t h e c u r r e n t l y a v a i l a b l e d a t a s e t s f o r t h e two p r o j e c t i l e s and Li.

7-+

For t h e sake of methodological consistency within t h e s e m i c l a s s i c a l framework, 6c, the r e a l c e n t r a l phase s h i f t i s taken t o be t h e

W K B

one corresponding t o t h e r e a l c e n t r a l p o t e n t i a l vc and t h e c e n t r a l a b s o r p t i v e c o e f f i c i e n t , a c , i s defined by

Two graphical s o l u t i o n s of eq. @ f o r a s p e c i f i c 0 > O R a r e a l s o depicted i n f i g . 1 of r e f . [5]. I f we have 0 <

e R ,

t h e r e e x i s t a l s o two s o l u t i o n s of eq. @ f o r v = 0 ,

E = -1, i , e . , f o r p o s i t i v e deflexion angles 0. These l a s t two s o l u t i o n s become com- plex when varying 0 from values below O R t o those above of i t . As i n r e f . [8], ap- proximate complex s o l u t i o n s a r e c a l c u l a t e d by expanding B c around

L R

and solving

8 '

; being t h e second d e r i v a t i v e of 8, a t L = L R . Thus, f o r 0 < OR, Lo which r e p r e s e n t s a c t u a l l y two values of angular momenta, one on t h e ascending and t h e o t h e r on t h e descending s i d e of t h e deflexion f u n c t i o n , corresponds t o a p a i r of p o s i t i v e angle c o n t r i b u t i o n s denoted by go. Also f o r 0 < O R , L1 lying a s i t does between 0 and -T

generates t h e negative angle c o n t r i b u t i o n gl and L2 lying between -T and - 2 ~ gives the negative angle c o n t r i b u t i o n f o r a s i n g l e o r b i t i n g around t h e t a r g e t c a l l e d g2 e t c . For 0 > B R one of t h e p a i r of complex s o l u t i o n s Lo must be taken i n t o consider- a t i o n a s i d e from t h e r e a l s o l u t i o n s of eq.

@,

i . e . , L1, L2 e t c . The o t h e r complex s o l u t i o n i s t o be discarded because t h i s would have an e x p n e n t i a l l y increasing c o n t r i b u t i o n t o t h e s c a t t e r i n g amplitude. In o t h e r words, f o r a Fresnel d i f f r a c t i o n p a t t e r n of t h e c r o s s s e c t i o n , t h e d i f f r a c t i o n amplitude go dominates and i f one has a wiggly u / u R , t h i s i s caused by i n t e r f e r e n c e between go and gl.

The next s t e p c o n s i s t s of introducing spin-dependent p o t e n t i a l s i n t o t h e above-dis- cussed procedure f o r c e n t r a l p o t e n t i a l s and w r i t i n g down t h e time-dependent Schroe- dinger equation f o r t h e s p i n - s t a t e evolution matrix

B.

This generates s p i n - o r b i t phase s h i f t s given by

The physical meaning of t h i s r e s u l t i s t h a t p r o j e c t i l e s i n p a r t i c u l a r e i g e n s t a t e of + -?

s-L w i l l s t a y i n t h e same one while moving along a t r a j e c t o r y . The r e a l s p i n - o r b i t phase s h i f t 2m6,, and t h e s p i n - o r b i t absorption c o e f f i c i e n t a S o 3 f o r wso

+

^{0, depend}

-& +

on m , t h e eigenvalue of S.L. Needless t o say t h a t t h e i n t r o d u c t i o n of spin-dependent p o t e n t i a l s modifies t h e saddle point e q u a t i o n @ a c c o r d i n g l y , encompassing spin-de- pendent f o r c e s and s p i n - f l i p processes. As a r e s u l t of t h e f a c t o r i z a t i o n property mentioned above, we now have, a s s o c i a t e d w i t h each g i , a corresponding m u l t i p l i c a t i v e f a c t o r h i due t o t h e spin-dependent p o t e n t i a l s .

When d i f f r a c t i o n s c a t t e r i n g dominates, say, f o r a heavy t a r g e t , i.e., o/aR shows Fresnel p a t t e r n , we know from above t h a t then o n l y go c o n t r i b u t e s t o fc. C a l c u l a t i o n s were done f o r iTll f o r such a s i t u a t i o n and the r e s u l t i s shown i n f i g . 4 o f r e f . [5]. The agreement w i t h data i s e x c e l l e n t inasmuch as the c a l c u l a t i o n i s parameter- f r e e

.

We now present t h e theory f o r c a l c u l a t i n g gl, i.e., consider a s c a t t e r i n g process l i k e

+ '*c

where t h e d i f f r a c t i o n term n o n - o r b i t i n g negative angle term and i n t e r - ferences t h e r e o f a l l p l a y a r o l e due t o t h e smaller t a r g e t size. The i n p u t p o t e n t i a l s

131

obtained by o p t i c a l model f i t a r e t h e r e a l and t h e imaginary c e n t r a l p o t e n t i a l s and t h e r e a l and t h e imaginary s p i n - o r b i t p o t e n t i a l s .

The procedure i s d i v i d e d i n t o t h r e e steps.

a ) As f o r t h e p o s i t i v e angle case, one determines t h e d e f l e x i o n f u n c t i o n 8 and ob- t a i n s therefrom t h e rainbow parameters OR, LR and

BE,

where t h e double dash i n - d i c a t e s t h e second d e r i v a t i v e w i t h r e s p e c t t o L and t h e corresponding rainbow a b s o r p t i o n c o e f f i c i e n t s ac(LR) and aso(LR) and t h e rainbow s p i n - o r b i t phase s h i f t 26 ( L ) and i t s d e r i v a t i v e 26so'.

so R

b ) Then one determines t h e range R o f Ro values, Ro being t h e d i s t a n c e o f c l o s e s t approach, f o r which -IT 4 8

<

0, i .e., one i s i n t h e negative angle range.

c ) D i v i d e t h e range R i n t o p a r t s so t h a t o s c i l l a t i o n s i n t h e experimental data can be reproduced. Note t h a t t h e r e l a t i o n s h i p between Ro and

ec.,.

i s n o t l i n e a r . Con- c r e t e l y speaking, i f experimental data e x i s t between, say, 30' and 130°, i .e., w i t h i n an angular range o f 100' and o s c i l l a t i o n s need t o be reproduced t o w i t h i n O.LiO, one then d i v i d e s R i n t o approximately 200 p a r t s , t a k i n g c a r e o f s l i g h t non- l i n e a r i t y . Then, f o r each o f these values o f Ro, one c a l c u l a t e s L and 8 and the a b s o r p t i o n c o e f f i c i e n t s a c and as, along w i t h 26,,. I n a d d i t i o n , the f o l l o w i n g i n t e g r a l i s needed, namely,

and t h e d e r i v a t i v e s 8 ' , a c c

,

26so', 8 " and 26s0c' w i t h respect t o L.

A f t e r having n u m e r i c a l l y obtained these many q u a n t i t i e s , we can now w r i t e down the l a s t s e t o f formulae t o o b t a i n o/oR and iTll. Thus, t h e d i f f r a c t i o n and t h e non- o r b i t i n g n e g a t i v e angle amp1 i t u d e s are, r e s p e c t i v e l y ,

w i t h x = C- 1 ~ / ~ ( 1 8 1

-

eR)

leRi' I

and lgl[ = F r " c

,

A i being t h e A i r y f u n c t i o n , and the phase

q,

⁼

+ ^(L +

L ~ ) [ B I

- $

^{+; ( I}

-

S i g n A i ( x ) ) The c o r r e s p o n d i n g spin-dependent f u n c t i o n s , hi, a r e

Thus, t h e u n p o l a r i z e d c r o s s s e c t i o n i s

and t h e f i r s t r a n k v e c t o r a n a l y s i n g power i s

F o r each o f t h e numerous Ro v a l u e s , one t h u s has t o c a l c u l a t e s e v e r a l i n t e g r a l s and d i f f e r e n t i a l c o e f f i c i e n t s a l o n g t h e t r a j e c t o r y . I n t h i s process, extreme c a r e has t o be g i v e n t o a v o i d convergence problems, p a r t i c u l a r l y f o r i n t e g r a l s n e a r Ro. Compu- t a t i o n a l work i s i n p r o g r e s s and t h e t y p i c a l t i m e f o r one f u l l c a l c u l a t i o n i s a b o u t 10

CPU

m i n u t e s on a CRAY computer. T h i s c l e a r l y demonstrates t h a t one has reached t h e 1 im i t o f v i a b i 1 i t y o f such s e m i c l a s s i c a l c a l c u l a t i o n s n u m e r i c a l l y and t h e r e f o r e i n t h e n e x t s e c t i o n we s h a l l p r e s e n t how t o go a b o u t t a c k l i n g t h i s problem by a method w h i c h s h o u l d be more s u i t e d f o r i t , namely t h e quantum mechanical procedure.

QUANTUM APPROACH TO POSITIVE AND NEGATIVE ANGLE SCATTERING

I n o r d e r t o s i m p l i f y t h e d i s c u s s i o n , we c o n s i d e r f o r t h e t i m e b e i n g o n l y c e n t r a l po- t e n t i a l s . S t a r t i n g f r o m t h e p a r t i a l wave sum f o r t h e s c a t t e r i n g a m p l i t u d e

we decompose t h e Legendre polynomials as f o l l o w s

where

Q,

a r e t h e Legendre f u n c t i o n s o f t h e second k i n d 191.

p u t Q1(-) = (PI + j 2 Ql) q1(+) = (p1

-

i

7

2 Q ~ )

.

Then, f ( 8 ) = f ( - ) + f ( + )

and

C

E

1 f 1

=

1

f(-)1

+ /

^f(+)1

+

i n t e r f e r e n c e terms.

O R

S e m i c l a s s i c a l l y , t h e amp1 i t u d e f ( - ) corresponds t o t h e d i f f r a c t i v e / p o s i t i v e angle c o n t r i b u t i o n and f ( + ) t o the r e f r a c t i v e / n e g a t i v e angle c o n t r i b u t i o n .

S a t c h l e r and coworkers [ l o ] have coined t h e phrase ' n e a r - s i d e ' and ' f a r - s i d e ' f o r the two cases r e s p e c t i v e l y . This a s s o c i a t i o n i s obvious, s i n c e

i C + i ( l

+

p)e 1

-

i

f o r l a r g e 1.

The extension t o i n c l u d e spin-dependent p o t e n t i a l s i s r e l a t i v e l y s t r a i g h t f o r w a r d , a l b e i t cumbersome. For example, we now need the associated Legendre f u n c t i o n s o f t h e second k i n d i n s t e a d o f t h e o r d i n a r y ones s i n c e magnetic p r o j e c t i o n i s o f paramount importance now. I n p a r t i c u l a r , we n o t e t h a t i n t h e case o f L i

+

Ni a t Elab "0 MeV s c a t t e r i n g i s e s s e n t i a l l y near-side. C o n t r a r i l y , i n t h e case o f A i

+

C b o t h near-side and f a r - s i d e amplitudes c o n t r i b u t e and i n t e r f e r e . C l e a r l y , t h e Li-T 7 near-side can be expected t o behave as t h e shape e f f e c t model p r e d i c t s , w h i l e t h e f a r - s i d e 2q T ' s do

2q not. Work i s i n progress t o i n t r o d u c e these changes i n t o t h e s p i n 3/2 o p t i c a l model code LINA.

REFERENCES

[I] FICK, D., Ann. Rev. Nucl. S c i . 31 (1981) 53

[2] ZUPRANSKI, P., DREVES, W., EGELHOF, P., MOEBIUS, K.-H., STEFFENS, E., TUNGATE, G. and FICK, D., Phys. L e t t . 91B (1980) 358;

MOROZ, Z., ZUPRANSKI, P., BOETTGER, R., EGELHOF, P., MOEBIUS, K.-H., TUNGATE, G., STEFFENS, E., DREVES, W., KOENIG, I., and FICK, D., Nucl. Phys. A381 (1982) 294 [3] RUSEK, K., MOROZ, Z., CAPLAR, R., EGELHOF, P . , MOEBIUS, K.-H., STEFFENS, E.,

KOENIG, I . , WELLER, A. and FICK, D., Nucl. Phys. A407 (1983) 208

[ 4 ] SIMONIUS, M. i n Lecture notes i n physics, v o l . 30, ed. FICK, 0. (Springer, Ber- l i n , 1974) p . 38

151 GRAWERT, G. and MUKHOPADHYAY, D., Nucl. Phys. A415 (1984) 304 [ 6 ] CROWLEY, B.J.B. and HILL, T.F., Z. Phys. A300 (1981) 299 [ 7 ] FORD, K.W. and WHEELER, J.A., Ann. of Phys. 7 (1959) 259

[ 8 ] BROGLIA, R.A. and WINTHER, A., Heavy i o n r e a c t i o n s , v o l . 1 (Benjamin Cummings, London, 1981)

[ 9 ] ABRAMOWITZ, M. and STEGUN, I.A., Handbook o f mathematical f u n c t i o n s (Dover, New York)

[ l o ]

SATCHLER, G.R., FULMER, C.B., AUBLE, R.L., BALL, J.B., BERTRAND, F.E., ERB, K.A., GROSS, E.E. and HENSLEY, D.C., Phys. L e t t . 1288 (1983) 147