SEMI CLASSICAL APPROACH TO ELASTIC COLLISIONS OF DEFORMED HEAVY-IONS

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SEMI CLASSICAL APPROACH TO ELASTIC COLLISIONS OF DEFORMED HEAVY-IONS

S. Mukherjee, L. Pandey

To cite this version:

S. Mukherjee, L. Pandey. SEMI CLASSICAL APPROACH TO ELASTIC COLLISIONS OF DEFORMED HEAVY-IONS. Journal de Physique Colloques, 1984, 45 (C6), pp.C6-445-C6-453.

�10.1051/jphyscol:1984653�. �jpa-00224256�

S E M I C L A S S I C A L APPROACH TO E L A S T I C C O L L I S I O N S OF DEFORMED HEAVY-IONS

S.N. Mukherjee and L.N. Pandey

Department o f Physics, Banaras Hindu University, Vuranasi-221005, India

REsum6 - Des donndes t r 2 s r d c e n t e s d e d i f f u s i o n d l a s t i q u e d e 160 s u r -utour d e 72 MeV s o n t a n a l y s d e s dans l e c a d r e du modSle semi-clas- s i q u e de d i f f r a c t i o n de Frahn. Une comparaison e s t f a i t e avec une a n a l y s e semblable de c o l l i s i o n d e 160 s u r 2% a u t o u r de l a mGme Gnergie. La r e c e t t e ' p o i n t q u a r t ' u t i l i s C e dans l e rnodsle donne un rayon p l u s i m p o r t a n t pour 24Mg que c e l u i de 2 8 ~ i . Afin de r d s o u d r e c e t t e anomalie des r a y o n s , des dgformations q u a d r u p o l a i r e s des noyaux s o n t u t i l i s d e s avec prudence.

A b s t r a c t - Very r e c e n t e l a s t i c s c a t t e r i n g d a t a of 160 on 2 4 ~ g around 7 2 ~ e ~ a r e a n a l y s e d i n t h e framework of E ' r a h ' s s e m i c l a s s i c a l

' d i f f r a c t i o n ' model and compared w i t h a s i m i l a r a n a l y s i s of c o l l i s i o n of 160 on 2 8 ~ i around t h e same energy. The q u a r t e r p o i n t r e c i p e used i n t h e model y i e l d s a g r e a t e r r a d i u s f o r 2 4 ~ g t h a n t h a t of 2 8 ~ i . I n o r d e r t o r e s o l v e t h i s r a d i u s anomaly, quadrupole d e f o r m a t i o n s of n u c l e i a r e used w i t h some c a r e .

I- INTRODUCT I O N

In most of t h e Heavy-Ion (HI) c o l l i s i o n s , t h e l o c a l wavelength of r e l a t i v e motion i s s m a l l a s compared t o t h e n u c l e a r i n t e r a c t i o n r e g ion. Theref o r e , t h e c l a s s i c a l approximat ions a r e u s e f u l i n d e s c r i b i n g and understanding H I r e a c t i o n s . But t h e r e a r e two d e f e c t s , namely, t h e n e g l e c t of q u a n t a l , and a b s o r p t i o n e f f e c t s , which l i m i t t h e a p p l i c a b i l i t y of c l a s s i c a l d e s c r i p t i o n of e l a s t i c s c a t t e r i n g . However, t h e s e two e f f e c t s do n o t d e s t r o y t h e

c l a s s i c a l p i c t u r e completely and hence make s e m i - c l a s s i c a l approximat ion t o HS e l a s t i c s c a t t e r i n g t h e o r y f e a s i b l e and p r a c t i c a b l e .

A t l a r g e e n e r g i e s t h e d e f l e c t ion f u n c t i o n which is c l a s s i c a l l y d e s c r i b e d remains a u s e f u l t o o l f o r d e s c r i b i n g t h e e l a s t i c

s c a t t e r i n g , though t h e y a r e s i g n i f i c a n t l g modified due t o q u a n t a l e f f e c t s . D i f f r a c t i o n model of e l a s t i c s c a t t e r i n g emphasized t h e q u a n t a l a s p e c t s a s s o c i a t e d w i t h t h e a b s o r p t i v e p r o p e r t i e s of heavy- i o n i n t e r a c t i o n due t o t h e presence of many n o n - e l a s t i c channels;

t h e s e m a n i f e s t themselves i n quantum d i f f r a c t i o n e f f e c t s and

c h a r a c t e r i s t i c p a t t e r n s ' r e s u l t i n g from q u a n t a l i n t e r f e r e n c e between s t r o n g a b s o r p t i v e s c a t t e r i n g and s t r o n g Coulomb s c a t t e r i n g .

The s e m i c l a s s i c a l a s p e c t s of t h e heavy-ion e l a s t i c c o l l i s i o n s a r e w e l l reproduced by t h e F r a h n ' s d i f f r a c t i o n model /I/. The well known f e a t u r e s i n t h e a n g u l a r d i s t r i b u t i o n s of heavy- ion e l a s t i c c o l l i s i o n s c l o s e l y resemble t h e c h a r a c t e r i s t i c s of F r e s n e l d i f f r a c - t i o n i n o p t i c s . The F r e s n e l e f f e c t s a r e a s s o c i a t e d w i t h s t r o n g Coulomb f i e l d and high p r o j e c t i l e energy. The s e m i c l a s s i c a l F r e s n e l c r o s s - s e c t i o n has t h e p r o p e r t y t h a t i t s r a t i o t o t h e Rutherford

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984653

C6-446 JOURNAL DE PHYSIQUE

c r o s s - s e c t i o n f a l l s t o one q u a r t e r a t an a n g l e Qc, corresponding t o t h e c r i t i c a l a n g u l a r momentum

A,.

This i s t h e w e l l known ' q u a r t e r p o i n t r e c i p e .

11- THERADIUS ANOMALY I N FRESNEL DIFFRACTION MODEL

The a n a l y s i s of experimental d a t a using t h e above q u a r t e r p o i n t r e c i p e of F r a h n l s d i f f r a c t i o n theory l e a d s t o an anomaly of t h e t a r g e t r a d i u s . For example, t h e a n a l y s i s of t h e s c a t t e r i n g of ' * ~ r by 232Th akd '08pb r e v e a l s t h a t t h e r a d i u s of 2 0 8 ~ b i s g r e a t e r than t h e r a d i u s of 2 3 2 ~ h . The a p p a r e n t anomaly i n t h e d e t e n i n a t i o n ef t h e t a r g e t r a d i u s was f i r s t i n d i c a t e d by Brink and Rowley/B/ and was l a t e r resolved by Rowley/3/ t a k i n g n u c l e a r d e f o r m a t i m as an a d j u s t a b l e parameter i n t h e Frahn' s s h a r p - c u t o f f d i f f r a c t i ~ n model.

However, t h e f i t t o t h e e l a s t i c s c a t t e r i n g d a t a was very poor though t h e r a d i u s anomaly was resolved. This i s q u i t e l i k e l y because the assumption of s h a r p - c u t o f f i n the a n g u l a r momentum space makes the model inadequate i n s e v e r a l r e s p e c t s . Frahn/4/ h a s r e c e n t l y

g e n e r a l i z e d t h e d i f f r a c t i g n model by c o n s i d e r i n g s t r ~ n g abeorption Md o t h e r q u a n t a a s p e c t s of the heavy-ion c o l l i s i o n s . The most i m p o r t a n t q u a n t i t i e s r e q u i r e d a r e t h e values @f c r i t i c a l angular momentum>c and t h e w i d t h A o f the A -space 'window' of p a r t i a l waves

t h a t take p a r t i n e l a s t i c and quasi - e l a s t i c c o l l i s i o n s . It was s t i l l very much an open question as t o what e x t e n t the

g e n e r a l i z e d F r e s n e l model can r e s o l v e t h e r a d i u s andmaly ? Pandey and Mukher jee/5/ have r e c e n t l y i n v e s t i g a t e d t h e above problem by

i n c l u d i n g gradual t r a n s i t i o n i n X-space of the e l a s t i c p a r t i a l wave S-matrix and the n u c l e a r deformation i n t h e F r e s n e l model. I n

a d d i t i o n t o t h i s , they have a p p l i e d t h e model t o r e s o l v e the r a d i u s anomaly, found i n the d i f f r a c t i o n model a n a l y s i s of e l a s t i c

s c a t t e r i n g d a t a of heavy deformed n u c l e i 2 0 ~ e on 2 0 8 ~ b and 2 0 ~ e on 2 3 5 ~ and demonstrated t h e g r e a t e r scope of t h e d i f f r a c t i o n model i n

e x p l a i n i n g the e l a s t i c c o l l i s i o n o f heavy deformed n u c l e i with t h e minimum of assumptions about the i n t e r a c t i o n .

The purpose of the p r e s e n t i n v e s t i g a t i o n i s t o show t h a t the r a d i u s anomaly s e e n i n Frahnf s d i f f r a c t i o n model i s n o t only confined t o heavy mass deformed n u c l e i but could a l s o be seen i n t h e c o l l i s i o n s of l i g h t mass deformed n u c l e i j60 on 2%g and 160 on 2 8 ~ i , and i n o r d e r t o r e s o l v e the anomaly i n the l a t t e r case one must use the value o f quadrupole deformation with some c a r e . One can see the

i n t e r p l a y between c l a s s i c a l and q u a n t a l dynamics i n t h e above a n a l y s i s . A d i r e c t p h y s i c a l d e s c r i p t i o n of t h e F r e s n e l type of heavy-ion

e l a s t i c c o l l i s i o n s i s provided by F r a h n l s rnodel/l/ which t a k e s t h e advantage o f an e l a s t i c s c a t t e r i n g S - m a t r i x f o r s t r o n g l y absorbed p a r t i c l e s which i s governed by h i g h i n c i d e n t energy and a s t r o n g Coulomb f i e l d .

With u s u a l approximation and s i m p l i f i c a t i a n s as given i n reference/5/, s c a t t e r i n g amplitude l e a d s t o an e x p r e s s i o n f o r c r o s s - s e c t i o n with s h a r p c u t o f f approximation a s :

where G(W) and s(W) a r e F r e s n e l t s c o s i n e and s i n e i n t e g r a l s and

6

0-0,

w

= ( 2 n / ~ ) S i n ( - r ) Cosec (Qc/2) ( 2

( q u a r t e r p o i n t r e c i p e ) .

The s h a r p - c u t o f f model i s i n c a p a b l e o f d e s e r i b i n g q u a n t i t a t i v e f e a t u r e s o f t h e e l a s t i c s c a t t e r i n g c r o s s - s e c t i o n s i n c e i t n e g e l e c t s r e f l e c t i o n of p a r t i a l waves above t h e b a r r i e r , a b s o r p t i o n o f p a r t i a l waves below t h e b a r r i e r , and d e v i a t i o n s from i i u t h e r f o r d o r b i t s c a u s e d by t h e r e a l p a r t o f t h e n u c l e a r p o t e n t i a l . The most i m p o r t a n t c o r r e c - t i o n is owing t o t h e g r a d u a l t r m s i t i o n i n a n g u l a r momenturn s p a c e

( A

⁾

o f t h e e l a s t i c p a r t i a l wave S- m a t r i x . The f i n i t e w i d t h of t h i s t r a n s i t i o n r e g i o n i s measured by a p a r m e t e r A t h a t d e f i n e s t h e s i z e of t h e

h

- s p a c e twindowt t h r o u g h which t h e e l a s t i c s c a t t e r i n g and most q u a s i - e l a s t i c r e a c t i o n s p r o c e e d . I t s e f f e c t on t h e s c a t t e r i n g ampli- t u d e i s d e s c r i b e d by a f u n c t i o n F ( D x ) , d e f i n e d a s t h e F o u r i e r t r a n s - form /6,7,8/ of t h e d e r i v a t i v e D ( h ) ⁼ d % ( h

)/a ,

F ~ A X ) = r d D(A ~ ) exp i ( A - A,) x]

-m (3

where%( ) i s t h e r e f l e c t i o n f u n c t i o n . With t h e f o l l o w i n g form of r e f l e c t i o n f u n c t i o n / 9 / .

We have

F ( O ) = 1 p r e s e r v e s t h e q u a r t e r p o i n t p r o p e r t y o f t h e s i m p l e F r e s n e l formula: U / r R =

3

a t 8=5,. The g e n e r a l i z e d F r e s n e l f o r m u l a 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 w i t h t h e a s s u m p t i o n s o f 'smooth-cut o f f i s t h e n g i v e n i n terms o f F r e s n e l i n t e g r a l s c ~ C ( [ W I ) and STS ( ( I w ] ) by

where F F P ( U ~ - G ) ] i s a r e a l f u n c t i o n .

11- CALCULATION OF

h,

FROM INTERACTICN POTENTIAL OF DEFGRMED NUCLEI The d i f f r a c t i o n model assumes t h a t t h e r e i s a c e r t a i n c r i t i c a l a n g u l a r momentum

h

f o r which t h e h e i g h t o f t h e b a r r i e r formed by Coulomb,

c e n t r i f u g a l and r e a l p a r t o f t h e n u c l e a r p o t e n t i a l i s e q u a l t o t h e c e n t r e of mass e n e r g y o f t h e r e a c t i o n i . e . ,

where,

and K d e n o t e s t h e reduced mass. The p o s i t i o n R o f t h e b a r r i e r is d e t e r m i n e d by

where t h e n u c l e a r p o t e n t i a l V N ( r ) is o b t a i n e d by f o l d i n g t h e d e n s i t y

C6-448 JOURNAL DE PHYSIQUE

d i s t r i b u t i o n of t h e p r o j e c t i l e w i t h t h e r e a l p a r t of t h e s i n g l e n u c l e o n o p t i c a l p o t e n t i a l of t h e t a r g e t / 5 / . F o r s p h e r i c a l n u c l e i t h i s p o t e n t i a l i s a p p r o x i m a t e l y g i v e n by

VN ('1 = Vo exp

[ - {

r- (Rp+RT

1 1 .

^{( 1 1 1}

F o r deformed n u c l e u s Rp and RT a r e g i v e n by

/lo/

R i = Roi (1+

pi

Y20) where i = P o r T. (12 1

d e n o t e s t h e q u a d r u p o l e d e f o r m a t i o n p r e s e n t i n t h e p r o j e c t i l e and/or t a r g e t and Roi= roi ~ ! / 3 . On account of Eq. ( l l ) , Eq. (121, n u c l e a r

~ o t e n t i a l c& be r e w r i t t e n a s

v p ( r ) =

v

^exp

[ - {

^r

- & R ~ ~ B ~

y z o 3 n ] (131 where,

v

⁼

v0

^exp

[

^{( R + ~}R~~ ^~)B]

,

and Vo is a c o n s t a n t f o r t h e system under c o n s i d e r a t i o n , From Eqs.

( 7 ) and ( 9 ) one o b t a i n s

D i f f e r e n t i a t i n g Eq. (10) and employing Eq. (15) we g e t

where V; ( R ) is t h e d e r i v a t i v e o f n u c l e a r p o t e n t i a l ,

S i n c e R

))

T , one can n e g l e c t t h e term

a

2 VN(R) f o r s i m p l i c i t y t o d e r i v e t h e r e l a t i o n between t h e p o s i t i o n of b a r r i e r s when n u c l e i are s p h e r i c a l and deformed. D i f f e r e n t i a t i n g e q u a t i o n (16) w i t h r e s p e c t t o t h e v a r i a b l e s , and s u b s t i t u t i n g

n E- V

T h e r e f o r e , 6 R

- ^{6 x}

^{= "} _{TI- -2}

' 6 ~

S i n c e T <<R and E r VC,

65 -

6 x < < 6 R , and one o b t a i n s

where X

=

x/D and D i s t h e p o s i t i o n of t h e b a r r i e r f o r X = C.

Equation (19) i m p l i e s t h a t t h e d i s t a n c e between t h e n u c l e a r s u r f a c e a t t h e b a r r i e r is p r a c t i c a l l y independent of t h e i r o r i e n t a t i o n s . I n s e r t i n g t h e v a l u e of R from Eq. ( 1 9 ) i n t o Eq. (15), we g e t a r e l a t i o n between d e f o r m a t i o n and q u a r t e r - p o i n t p r o p e r t i e s a s

h2 X ~ Y Z P

D(J-+x) \ ( E v ~ ( R ) ) (I+x)

- %zp

^{e 2 J (20)}

and hence

xc = c O t 2 ( e C / 2 )

(y-)

c o t 2 (9:/2) (1+x12 + 4 X ( l + X ) d ( 2 1 1

where 0:. is t h e c r i t i c a l a n g l e which one would o b t a i n f o r s c a t t e r i n g of s p h e r i c a l n u c l e i of r a d i i ROp and ROT,

d =

E/V; is t h e Coulomb

111- NUMERICAL CALCULAT I O N 3

Now we w i l l d e s c r i b e t h e n u m e r i c a l d e t e r m i n a t i o n of t h e q u a r t e r p o i n t f o r c o l l i s i o n s of 2 0 ~ e on 2 0 8 ~ b a t 161.2 MeV, 2 0 ~ e a t 1 7 5 MeV, 8 4 ~ r on 2 0 8 ~ b a t 500 AeV, 8 4 ~ r on 2 3 2 ~ h a t 500 PIev,160 on 24?4g a t 72.5 MeV and on 2 g ~ i 72 deV. Our aim i s t o compare t h e c a l c u l a - t e d q u a r t e r p o i n t s w i t h t h o s e o b t a i n e d d i r e c t l y from e x p e r i m e n t a l d a t a .

F i r s t we s o l v e Eq. ( 1 6 ) t o f i n d t h e p o s i t i o n o f t h e b a r r i e r , D , .by changing t h e s e p a r a t i o n d i s t a n c e between t a r g e t and p r o j e c t i l ? I n s t e p s of 0.001 fm and keeping t h e i r r a d i i f i x e d . V p j ( r ) and V N ( r ) a r e c a l c u l a t d from Eq. (14) of r e f e r e n c e / 5 / caking b o t h n u c l e i t o s p h e r i c a l ( a i = O ) and O E e q u i v a l e n t s u r f a c e t h i c k n e s s .

T = 2 n T i

/ r

^Ti.

The p a r a m e t e r s u s e d f o r c a l c u l a t i n g n u c l e u s - n u c l e u s i n t e r a c t i o n a r e t a k e n from t h e work of B r o g i l a and Winter /11/. The v a l u e

CVX(r)

3

r = D i s t h e n c a l c u l a t e d by n u m e r i c a l i n t e g r a t i o n ( 2 4 p o i n t s G a u s s i a n q u a d r a t u r e ) o f Eq. (13) o f r e f e r e n c e / 5 / . S u b s t i t u t i n g V (D), V;(D) and E i n Eq. ( 2 0 ) we g e t t h e c r i t i c a l a!lgular momentum

2

and h e n c e , t h e c r i t i c a l a n g l e @

E.

T h i s p r o c e s s i s r e p e a t e d by v a r y i n g rOT i n s t e p s o f 0,0001 fm u n t i l t h e c a l c u l a t e d c r i t i c a l a n g l e a g r e e s w i t h t h e e x p e r i m e n t a l v a l u e . The v a l u e o f

Bi

u s e d i n t h e c a l c u l a t a t i o n s a r e t a k e n from t h e work of J t e l s o n e t a l . /12/, Lobner e t a l . /13/, Moller and Nix /14/. The above p r o c e d u r e i s t h e n a g a i n r e p e a t e d t o f i n d and Bc.

I V - RBULTS k Y D DISCUSS IONS

Having f o u n d t h e t a r g e t r a d i u s from t h e q u a r t e r p o i n t a n a l y s i s w i t h and w i t h o u t n u c l e a r d e f o r m a t i o n i t is of some i n t e r e s t t o compare t h e r e s u l t a ~ t r a d i u s and t h e e l a s t i c ' c E u s s - s e c t i o n . We s e e from T a b l e 1, t h a t t h e i n c l u s i o n o f t h e n u c l e a r d e f o r n a t i o n w i l l y i e l d c o n s i s t e n t l y g r e a t e r r a d i u s f o r heavy mass n u c l e i ompared t o l i g h t mass n u c l e i . For example, t h e r a d i i of 23% and 2S2Th a r e found g r e a t e r t h a n 2 0 8 ~ b a s e x p e c t e d . However, t h i s is n o t t h e c a s e when t h e d e f o r m a t i o n i s excluded. Our p r e s e n t a n a l y s i s d i f f e r s from t h a t o f Rowley /J/ i n t h e s e n s e t h a t t h e l a t t e r h a s u s e d u n c l e a r deforma- t i o n a s a v a r i a t i o n a l parameter.The k i n e t i c e n e r g y of r e l a t i v e motion i n t h e r e g i o n o f b a r r i e r i s l a r g e compared t o t h e e n e r g i e s o f r o t a t i o n a l l e v e l s of t h e s e n u c l e i and t h e y might have no t i m e t o change t h e i r s h a p e and s i z e a p p r e c i a b l y . T h e r e f o r e , it i s p h y s i c a l l y more m e a n i n g f u l t o u s e i n t r i n s i c d e f o r m a t i o n of n u c l e i d e t e r m i n e d from e x p e r i a e n t a l v a l u e o f g r o u n d s t a t e q u a d r u p o l e moment.

T a b l e 1 s l s o shows t h e v a l u e s of €I:,which one would o b t a i n f o r s c a t t e r i n g o f s p h e r i c a l n u c l e i o f r a d i i FLOP and ROT. It i s i n t e r e s t - i n g t o n o t e t h a t t h e n u c l e a r d e f o r m a t i o n h a s pushed o u t t h e a c t u a l q u a r t e r p o i n t

e0

^{t o}

ec,

and hence e x p l a i n s t h e c a u s e of r a d i u s

C

anomaly.

C6-450 JOURNAL DE PHYSIQUE

T a b l e 1. C a l c u l a t e d t a r g e t r a d i u s w i t h o u t and w i t h quadrupole d e f o r m a t i o n from " ~ e on 2 0 8 ~ b , 2 0 ~ e on 2 3 2 ~ , 8 4 ~ r on 2 0 8 ~ b , 8 4 ~ r on 2 3 2 ~ h e l a s t i c c o l l i s i o n s

---- --- ----

--- ^--- - ---

Deformation T a r g e t Radius Experimental

RTi P ~ ~ * ( I + B 2 ~ 2 0 ) ~ ~ , ' ~ quarter point property

---

---

T a r g e t rgT (fm) ROT (fm)

@ c

No deformat- 235u 1.238 7.64 56.2 92.1

i o n (

PZ

^=O) _208pb ¹ ₁₁₁₀₉ ³⁰⁰ _6.81 ^7.70 _125.0 ^{53.1 91.0}

232Th 1.181 7.00 108.4

2 0 8 ~ b 101.0 156.6

- -- --- ---

0.259 235u 1.439 8.88 56.2 92.1

0.000 208pb 1.430 8.47 53.1

1.232 7.57 91.0

0.267 125.0

2322h 1.228 7.28 105.4

0.000 08p b 101.0 156.6

Experimental values- r e t ' e r e G Y f s T , ---

An i n t e r e s t i n g f e a t u r e o f t h e above a n a l y s i s i s r e v e a l e d when a p p l i e d f o r t h e d e s c r i p t i o n of e l a s t i c c o l l i s i o n of l i g h t mass h e a v i l y deformed n u c l e i . For example a v e r y r e c e n t e l a s t i c s c a t t e r - ing d a t a of 160 on 241yg around 72.5 MeV /15/ i s a n a l y s e d h e r e and compared w i t h a s i m i l a r c a l c u l a t i o n of c o l l i s i o n s o f f60 on 2 8 ~ i around same i n c i d e n t p r o j e c t i l e energy. The u s u a l q u a r t e r p o i n t r e c i p e used i n t h e model a g a i n y i e l d s a g r e a t e r r a d i u s f o r 2 4 ~ g than t h a t of 2 8 ~ i a s it was found i n t h e e a r l i e r c a s e s of heavy deformed n u c l e i . However, t h e a p p a r e n t r a d i u s anomaly can n o t be r e s o l v e d by

employing quadrupole d e f o r m a t i o n (

B ( M ~

⁾ =0.65 and P ( 3 i ) =0.40) o b t a i n e d from e ~ p e r i m e ~ i t a l quadrupole moment /12/ and t h e assumption of smooth c u t o f f . R e c e n t l y , t h e r e h a s been r e v i v e d i n t e r e s t i n t h e u s e o f an asymmetric r o t o r model f o r 24%, t h e m o t i v a t i o n coming from two s o u r c e s . F i r s t l y ? t h e i n c r e a s e d s o p h i s t i c a t i o n of coupled c h a n n e l a n a l y s i s o f 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 e x p e r i m e n t s h a s prompted e x t e n s i v e use o f s u c h approaches t o determine n u c l e a r shape parameter. Recent coupled-channel a n a l y s i s by Nurzynski e t a l . /15/ p r e d i c t s t h a t t h e quadrupole and hexadecapole deformation p a r a m e t e r /17,18/, needed t o f i t 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

d a t a , s h o u l d be t a k e n around 0.25 and-0,065 r e s p e c t i v e l y . Hence, when we i n c l u d e @ (Mg) = 0.25 i n s t e a d of B ( H g ) = 0.65, we could r e s o l v e t h e r a d i u s anomaly. The r e s u l t s o f o u r a n a l y s i s a r e shown i n T a b l e 2.

F u r t h e r , d i f f r a c t i o n c r o s s - s e c t i o n

6(8 )/qR

( 0 ) c a l c u l a t e d from

Eq.(6) f o r v a r i o u s v a l u e s o f A , a r e shown i n Fig.1, which i l l u s t r a t e t h e p o s s i b l e range of a n g u l a r d i s t r i b u t i o n s h a p e s covered by t h e g e n e r a l i z e d F r e s n e l model from undamped (0 =0, s h a r p - c u t o f f ) t o t h e

c l a s s i c a l l i m i t (A

= m

). The v a l u e s of

ec

and A correspond t o

C

t h e system 160 on 24Mg a t 72.5 XeV i n c i d e n t energy. A s u b s t a n t i a l improvement i n t h e d i f f r a c t i o n model f i t s of l 6 0 on 24Mg a t 72.5 MeV and 160 on 2 8 ~ i a t 72.0 MeV d a t a i s a l s o a c h i e v e d by t a k i n g P =1, which i s shown i n Fig.2.

The smooth c u t o f f model, a s i d e from l c o s m e t i c l e f f e c t of g i v i n g much

d i f f u s e n e s s parameter / 5 / .

Table 2. C a l c u l a t e d t a r g e t r a d i u s without and w i t h quadrupole deformat ion from 160 on 24Mg and 160 on 2 8 ~ i e l a s t i c c o l l i s i o n s .

- - - - - - - - - - - - - .- -

Deformation Target Ra i u s

3

Experimental

RT=ror%l (1+&yZ0) q u a r t e r p o i n t p r o p e r t y

- -

Target roT (fm) ROT(fm @ c c

No deformat- 24 1.410 3.29 27.0 29.71

i o n ( p 2 = o ) 2 8 2 1.061 3.22 30.0 31.16

Fig.1. D i f f e r e n t i a l c r o s s s e c t i o n V(8 ) /

VR

(8 ) c a l c u l a t e d

from Eq.(6) f o r v a r i o u s v a l u e s of w i d t h A , t o

i l l u s t r a t e t h e p o s s i b l e range of angular d i s t r i - bution shapes covered by t h e g e n e r a l i z e d F r e s n e l model, from t h e undamped p a t t e r n

( A

= 0 ) t o t h e

c l a s s i c a l l i m i t (A =OO ).

The v a l u e s of 8, and

A c

correspond t o system 160 on 24Mg a t 72.5 MeV i n c i d e n t energy.

JOURNAL DE PHYSIQUE

where k i s t h e wave number. P a r t i c u l a r l y i t s v a l u e s f o r h e a v i e r

p r o j e c t i l e a r e s u b s t a n t i a l l y s m a l l e r t h a n f o r t h e l i g h t e r p r o j e c t i l e s . T h i s is most l i k e l y t o be a t t r i b u t e d t o t h e q u a s i - e l a s t i c contamina- t i o n of v e r y heavy-ion d a t a .

Fig .2. Shows t h e e l a s t i c s c a t t e r i n g c r o s s - s e c t i o n

r(€I

^)/r8(0)

o f 160 on ^24Mg and 160 on 2 8 ~ i . The c i r c l e s r e p r e s e n t t h e e x p e r i m e n t a l v a l u e s t a k e n from t h e work of

Nurzynski e t a l . /15/ and Cramer e t ^a1./16/.

V- CONCLUS IONS

We make t h e following e s s e n t i a l conclusions from t h e p r e s e n t i n v e s t i - g a t ions.

The a d d i t ion of n u c l e a r deformation i n F r e s n e l d i f f r a c t i o n model r e s o l v e s t h e r a d i u s anomaly and t h e assumption of smooth c u t o f f of t h e a n g u l a r momentum l e a d s t o a b e t t e r f i t t o t h e e l a s t i c heavy-ion c o l l i s i o n d a t a . The presence o f a n g u l a r momentum space window width, a i s d e from i t s cosmetic e f f e c t s , p r o v i d e s t h e a d d i t i o n a l information about t h e n o n - e l a s t i c events. C l e a r l y by a n a l y s i n g s u b s t a n t i a l amount of d a t a one can hope t o g i v e a v a l i d d e s c r i p t i o n of c o l l i s i o n s of heavy-ion deformed n u c l e i i n terms o f s e m i - c l a s s i c a l d i f f r a c t i o n model on p a r w i t h o p t i c a l model a n a l y s i s .

The f i n a n c i a l s u p p o r t from t h e Department o f Atomic Fhergy, Government

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