EXPERIMENTAL EVIDENCE AND PHYSICAL IMPLICATIONS OF THE TIME EVOLUTION ALONG THE MASS ASYMMETRY MODE IN HEAVY ION REACTIONS

HAL Id: jpa-00216587

https://hal.archives-ouvertes.fr/jpa-00216587

Submitted on 1 Jan 1976

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of

sci-entific research documents, whether they are

pub-lished or not. The documents may come from

teaching and research institutions in France or

abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est

destinée au dépôt et à la diffusion de documents

scientifiques de niveau recherche, publiés ou non,

émanant des établissements d’enseignement et de

recherche français ou étrangers, des laboratoires

publics ou privés.

EXPERIMENTAL EVIDENCE AND PHYSICAL

IMPLICATIONS OF THE TIME EVOLUTION

ALONG THE MASS ASYMMETRY MODE IN

HEAVY ION REACTIONS

L. Moretto, R. Schmitt

To cite this version:

J01JRNAL DE PHYSIQUE Colloque C5, supplément au n° 1 1 , Tome 3 7 , Novembre 1976, page C5-109

EXPERIMENTAL EVIDENCE AND PHYSICAL IMPLICATIONS OF THE TIME EVOLUTION ALONG THE MASS

ASYMMETRY MODE IN HEAVY ION REACTIONS*

L. G. Moretto and R. Schmitt

Nuclear Science Division, Lawrence Berkeley Laboratory,

and Department of Chemistry, University of California,

Berkeley, California 94720

Résumé : Des caractéristiques expérimentales complexes sont associées .aux distribu-tions de .masse ou charge et aux distribudistribu-tions angulaires en fonction de la masse ou de la charge du fragment. Elles prouvent l'existence d'une structure intermédiaire, ou complexe intermédiaire, dont l'asymétrie de masse évolue en fonction du temps. De fortes preuves indirectes suggèrent que cette évolution avec le temps correspond à une diffusion et peut être décrite en termes d'Equation Maîtresse ou d'Equation de Pokker-Planûk. Il est établi expérimentalement que la distribution de masse est lar-ge pour les grands rapports E/B - où E est l'énergie dans le centre de masse et B la barrière d'interaction - et étroite avec un maximum S la masse du projectile (et de la cible) pour les petits rapports E/B. Ceci est interprété comme étant dû à une aug-mentation du temps de vie du complexe avec E. Pour les temps de vie courts, le

sys-tème a peu de temps pour modifier son asymétrie de masse et donne donc naissance à des distributions de masse assez étroites et centrées au voisinage de la masse du pro-jectile (et.de la cible). Pour les temps de vie longs, le degré de liberté asymétrie de masse est en voie d'équilibre, ce qui conduit à de très larges distributions de masse. De la même façon, le maximum des distributions angulaires semble se déplacer depuis un angle éloigné de 0° (pic sur le côté) jusqu'à un angle très proche de 0°

(pic en avant) lorsque E/B croît. Ceci est interprété comme étant dû à une transition depuis un régime où le temps de vie court et la vitesse angulaire faible ne permettent pas au système de tourner plus loin que 0 ° , jusqu'à un régime à temps de vie long et vitesse angulaire élevée, qui conduit à une rotation importante du système sur lui-même et à une émission au-delà de 0°. Lorsque la charge nucléaire du produit s'éloi-gne de celle du projectile, l'évolution depuis le pic sur le côté jusqu'au pic en avant est due au décalage de temps introduit par la diffusion dans la population des fragments les plus éloignés en Z du projectile. La variation des distributions an-gulaires et de charge avec l'énergie cinétique du fragment permet de relier l'amor-tissement en énergie à l'amorl'amor-tissement en asymétrie de masse. Des calculs théori-ques .basés sur les modèles de diffusion permettent de bien rendre compte des distri-butions angulaires et de masse, et aussi d'obtenir des probabilités de transition et des coefficients de Fokker-Planck. La validité des différentes méthodes d'analyse est discutée.

Abstract: The complex experimental features associated with the mass or charge distributions, and with the angular distributions as a function of fragment mass or charge, are interpreted as evidence of an intermediate structure, or intermediate complex, evolving in time along the mass asymmetry mode. Strong circumstantial evidence suggests that this time evolution is diffusive in nature and can be described in terms of the Master Equation or the Fokker-Planck Equation. The experimental evidence of broad mass distributions for large ratios E/B, where E is the center of mass energy and B is the interaction barrier, and narrow mass dis-tributions peaked at the projectile and target mass for small ratios E/B, is interpreted as due to an increasing lifetime of the complex with energy. For short lifetimes, the system has little time to evolve in mass asymmetry and gives rise to rather narrow distributions centered about the target and projectile mass. For long lifetimes the system undergoes extensive relaxation in mass asymmetry and gives rise to very broad mass distributions. Similarly the angular distributions seem to evolve from side peaked to forward peaked with increasing E/B. This is interpreted as due to a transition from a short lifetime-slow angular velocity regime which does not allow for orbiting beyond 0°, to a long lifetime-large angular velocity regime which produces orbiting past 0°. The evolution from side peaking to forward peaking in the same reaction as one moves away in Z from the projectile is interpreted as

L.G. MORETTO and R . SCHHITT

due t o t h e time lag introduced by diffusion i n the population of fragments f a r t h e r

removed i n

Z

from t h e p r o j e c t i l e . The variation of charge and angular d i s t r i b u t i o n

with the fragment k i n e t i c energy allows one t o connect t h e energy lcelaxation t o

t h e mass asymmetry relaxation. Theoretical c a l c u l a t i o n s based on diffusion models

allow one t o f i t mass and angular d i s t r i b u t i o n s as well a s t o e x t r a c t t r a n s i t i o n

p r o b a b i l i t i e s and Fokker-Planck c o e f f i c i e n t s . The r e l i a b i l i t y of various methods of

analysis i s discussed.

INTRODUCTION

I t appears t h a t the accelerator development has

occurred along a sound pedagogical l i n e , well

suited t o our education i n nuclear physics.

Early

machines provided us with simple p r o j e c t i l e s which

had t h e merit of mainly inducing two kinds of

nuclear reaction: d i r e c t ' r e a c t i o n s on the one

hand, and compound nuclear reactions on t h e other.

Both kinds were d'i'ligently studied by our parents

in science.

The d i r e c t reactions, i t was learned, portray

a strong dynamical coupling between entrance and

e x i t channels, t h e i r degree of i n e l a s t i c i t y

is

minimal, and very few degrees of freedom of the

t a r g e t a r e excited. Then the g r e a t chapter of

p a r t i c l e spectroscopy was w r i t t e n , and the s i n g l e

p a r t i c l e s t r u c t u r e of nuclei (she1

1

s t r u c t u r e ) ,

'U

with t h e added refinement of residual i n t e r a c t i o n s ,

was revealed.

The compound nucleus reactions, on t h e

contrary, t o l d q u i t e a d i f f e r e n t story.

A

com-

p l e t e decoupling between entrance and e x i t channels

was observed, together w i t h an extreme degree of

i n e l a s t i c i t y and the involvement of a l l of t h e

nuclear degrees of freedom.

The compound nucleus

was then postulated a s a long-lived intermediate

i n which a l l the degrees of freedom a t t a i n

s t a t i s t i c a l equilibrium.

The chapter of t h e

s t a t i s t i c a l nuclear properties was then written

w i t h

a l l the c o r o l l a r i e s of s t a t i s t i c a l d i s t r i -

butions, s t a t i s t i c a l decay, evaporation, e t c .

I t seems n a G r a l t h a t , a f t e r having received

such primers i n nuclear physics, we should e n t e r a

new, more comprehensive f i e l d which bridges the

simplicity of the s t a t e s explored by d i r e c t

reactions t o t h e complexity of t h e compound s t a t e s .

Such a c c h e c t i o n involves t h e great absent-in

e a r l y nuclear physics, t h e time. Heavy ion

reactions do in f a c t reveal a sequence of patterns

L

whose connection i s unmistakably the time.

In t h i s

sequence, the-re1 a t i v e l y simple entrance channel

conf'igurations appear t o evolve i n t o more and more

'complex configurations, approaching, with variable

degree, t h e ultimate s t a t i s t i c a l d i s t r i b u t i o n s .

Three c o l l e c t i v e degrees of freedom a r e

p a r t i c u l a r l y noticeable in heavy ion reactions f o r

t h e i r various stages of relaxation: the re1 a t i v e

-distance of the fragments, t h e neutron-to-proton

r a t i o , and t h e mass asymmetry. In t h e f i r s t degree

of freedom, a l l the i n i t i a l k i n e t i c energy appears

t o relax i n t o the internal degree of freedom

through t h e action of viscous forces. Such a

d i s s i p a t i v e process i s so immediately v i s i b l e , and-

involves such a l a r g e f r a c t i o n of t h e t o t a l cross

s e c t i o n , t h a t special names have been created f o r

i t , l i k e deep i n e l a s t i c , strongly damped or

re1 axed [l

-41,

even though differences a r e

frequently seen in the degree of relaxation

associated with t h e other degrees of freedom.

The

energy d i s s i p a t i o n i s a r e l a t i v e l y f a s t process,

and has r e l a t i v e l y l i t t l e overlap with t h e slower

relaxation processes associated with other degrees

of freedom.

The second degree of freedom, t h e neutron-to-

proton r a t i o , a l s o appears t o e q u i l i b r a t e very f a s t ,

so t h a t i t s relaxation seems t o proceed e s s e n t i a l l y

a t constant mass asymmetry

[5-71.

The relaxation processes associated w i t h t h e

f i r s t two degrees of freedom have been t r e a t e d a t

length i n Galin's t a l k . So we s h a l l concentrate

e s s e n t i a l l y on t h e f e a t u r e s associated with the

t h i r d degree of freedom, the mass asymmetry. This

degree of freedom i s r a t h e r slow in i t s time

evolution, so t h a t i n many cases i t s relaxation

occurs,while other degrees of freedom have already

attained t h e i r equilibrium d i s t r i b u t i o n s :

i n

other words, t h e k i n e t i c energy has been mostly

dissipated i n t o the internal degrees of freedom,

giving r i s e t o a "warm" system; t h e neutron-to-

proton r a t i o has been balanced; e t c .

T I M E EVOLUTION ALONG THE MASS ASYMMETRY MODE C5-l l l

mass asymmetry degree o f freedom, we l i k e t o c a l l " i n t e r m e d i a t e complex" i n analogy w i t h t h e i n t e r - mediate complex o f chemical r e a c t i o n s 18-1 l].

Various f a c t s seem t o support t h e e x i s t e n c e o f t h e i n t e r m e d i a t e complex. For instance, t h e g r e a t amount o f c r o s s s e c t i o n c o n c e n t r a t e d a t l o w k i n e t i c energies shows t h a t i n general t h e r e i s enough t i m e f o r t h e k i n e t i c energy t o r e l a x . Furthermore, t h e spreading o f t h e Z d i s t r i b u t i o n seems t o occur m o s t l y w h i l e t h e k i n e t i c energy i s v e r y low, i f n o t completely thermalized. T h i s means t h a t t h e mass asymmetry mode r e l a x e s s l o w l y .

T h i s general p i c t u r e suggests t h a t t h e e v o l u t i o n along t h e mass asymmetry i s "creepy", o r dominated by t h e viscous f o r c e s . Apparently a continuous readjustment i n t h e e q u i l i b r i u m c o n d i t i o n s i s p o s s i b l e as t h e system moves along t h e mass asymmetry c o o r d i n a t e . I n t h i s case memory e f f e c t s a r e unimportant, and t h e t i m e e v o l u t i o n can w e l l be described i n terms o f a d i f f u s i o n mechanism [8,9,11-131.

An i m p o r t a n t aspect o f t h e i n t e r m e d i a t e complex i s i t s decay time, which c o n t r o l s t h e e x t e n t o f r e l a x a t i o n observed f o r t h e v a r i o u s degrees of freedom. I n t h e l i g h t e r systems i t i s n o t c l e a r whether t h e decay o f t h e complex i s s t a t i s t i c a l l y o r dynamically c o n t r o l l e d . I n t h e h e a v i e r systems t h e evidence seems t o i n d i c a t e t h a t t h e decay t i m e i s more a dynamical q u a n t i t y than a s t a t i s t i c a l q u a n t i t y . Therefore, i t

appears t h a t t h e p r e v a i l i n g regime i s t h a t o f n o n e q u i l i b r i u m s t a t i s t i c a l mechanics f o r some degrees o f f r e e d o q a n d almost p u r e l y dynamical f o r o t h e r s .

We s h a l l d i v i d e what f o l l o w s i n t o f o u r s e c t i o n s . I n t h e f i r s t s e c t i o n we s h a l l g i v e a q u a l i t a t i v e j u s t i f i c a t i o n f o r t h e use o f a d i f f u s i o n model a p p l i e d t o t h e e v o l u t i o n i n mass asymmetry. The Master Equation w i l l be e x p l i c i t l y w r i t t e n down f o r t h i s process, t h e corresponding Fokker-Planck Equation w i l l be considered, and t h e d r i f t and spread c o e f f i c i e n t s w i l l be r e l a t e d t o t h e t r a n s i t i o n p r o b a k i l i t i e s . The p o s s i b i l i t y o f a d i r e c t use o f t h e Fokker-Planck Equation i n o r d e r t o analyze t h e mean displacement and t h e w i d t h s o f t h e experimental d i s t r i b u t i o n s w i l l be discussed. The a l t e r n a t e and more r i g o r o u s procedure o f c o u p l i n g t h e Master Equation o r t h e Fokker-Planck Equation t o t h e dynamics o f o t h e r degrees o f freedom w i l l be presented.

I n t h e second section, t h e experimental data

on t h e mass ( o r charge) d i s t r i b u t i o n w i l l be - presented. The dependence o f these d i s t r i b u t i o n s on angle and energy windows w i l l be shown and t h e p h y s i c a l i m p l i c a t i o n s w i l l be discussed.

S i m i l a r l y i n t h e t h i r d s e c t i o n , t h e a n g u l a r d i s t r i b u t i o n s and t h e i r dependence upon Z and energy windows w i l l be presented. The t i m e f a c t o r c o n t r o l l i n g t h e f e a t u r e s o f t h e angular d i s - t r i b u t i o n s w i l l be discussed.

I n t h e b r i e f f o u r t h s e c t i o n , examples o f t h e o r e t i c a l c a l c u l a t i o n s w i l l be compared w i t h experiment, and numerical values f o r t h e decay times and d i f f u s i o n c o e f f i c i e n t s o b t a i n e d from v a r i o u s sources w i l l be discussed.

SECTION I. THEORETICAL CONSIDERATIONS Lagrangian and D i f f u s i v e Approaches t o t h e D e s c r i p t i o n o f Time Dependent Processes

The f i s s i o n process has been one o f t h e f i r s t n u c l e a r processes t o be t r e a t e d i n a time-

dependent f a s h i o n . I n a couple o f b r i l l i a n t papers,

N i x [74,15] d e s c r i b e d t h e t i m e e v o l u t i o n f r o m saddle t o s c i s s i o n p o i n t by i n t r o d u c i n g a

Lagrangian i n t h e c o l l e c t i v e v a r i a b l e s . The l i q u i d drop model was used f o r t h e p o t e n t i a l energy, and an i r r o t a t i o n a l f l o w was assumed f o r t h e i n e r t i a tensor. I n p r i n c i p l e t h e e x t e n s i o n o f these c a l c u l a t i o n s t o heavy i o n r e a c t i o n s i s t r i v i a l . I t i s n o t c l e a r , however, i f t h i s approach i s s u f f i c i e n t l y general.

The Lagrangian approach e s t a b l i s h e s a p o i n t - t o - p o i n t correspondence between t h e i n i t i a l and t h e f i n a l phase space and thus i s completely d e t e r m i n i s t i c . More c l e a r l y , t h e t r a j e c t o r y , i n a Lagrangian f o r m u l a t i o n , i s a w e l l d e f i n e d e n t i t y , and f o r a g i v e n i n i t i a l c o n d i t i o n , o r p o i n t i n phase space, t h e r e i s one and o n l y one t r a j e c t o r y . The f i n a l d i s t r i b u t i o n s depend e x c l u s i v e l y upon t h e d i s t r i b u t i o n s o f i n i t i a l c o n d i t i o n s .

While such an approach, g e n e r a l i z e d by t h e i n t r o d u c t i o n o f t h e R a y l e i g h d i s s i p a t i o n f u n c t i o n t o handle viscous f o r c e s , may be a p p l i c a b l e under c e r t a i n circumstances [l 6-24], i t a c t u a l l y has s e r i o u s d e f i c i e n c i e s which may p r e v e n t i t s success i n d e s c r i b i n g t h e o v e r a l l e v o l u t i o n o f t h e shape parameters i n heavy i o n r e a c t i o n s . The s h o r t - comings o f t h e Lagrangian approach t o t h e d e s c r i p t i o n o f a manybody system a r i s e from t h e n e g l e c t o f t h e i n t e r n a l degrees o f freedom [ll].

C 5 - 1 1 2 L.G. MORETTO AND R . SCHMITT

d i v e r g i n g s e t o f t r a j e c t o r i e s , r a t h e r than by a where h i s t h e microscopic t r a n s i t i o n z z '

s i n g l e t r a j e c t o r y , because o f t h e u n s p e c i f i e d i n i t i a l c o n d i t i o n s f o r t h e i n t e r n a l degrees o f freedom. Therefore, an a c c u r a t e d e s c r i p t i o n of t h e t i m e e v o l u t i o n o f t h e ensemble cannot be com- p l e t e l y d e t e r m i n i s t i c , b u t must a l s o c o n t a i n t h e s t a t i s t i c a l i n f l u e n c e o f t h e i n t e r n a l degrees o f freedom i n d e t e r m i n i n g t h e d i s t r i b u t i o n o f t h e elements o f t h e ensemble i n c o l l e c t i v e phase space.

One can l o o k a t t h i s problem m r e c o n c r e t e l y as fo71ows. A f t e r t h e k i n e t i c energy i s d i s s i p a t e d , t h e i n t e r m e d i a t e complex has a temperature t h a t may range, t y p i c a l l y , between 1 and 4 MeV. While t h i s system f o l l o w s a Lagrangian t r a j e c t o r y i n c o l l e c t i v e phase space w i t h a few tens o f MeV k i n e t i c energy, i t i s subjected t o random Brownian impulses which a r e comparable t o t h e momentum o f t h e system along t h e c o l l e c t i v e c o o r d i n a t e . As a consequence t h e Lagrangian t r a j e c t o r y i s s e r i o u s l y perturbed, .so t h a t t h e a c t u a l t r a j e c t o r i e s o f t h e v a r i o u s elements o f t h e ensemble tend t o diverge.

Norenberg w i l l show i n g r e a t e r d e t a i l under which c o n d i t i o n s t h e use o f t h e Master Equation can be j u s t i f i e d . We s h a l l assume t h a t t h e Master Equation i s indeed a s u i t a b l e t o o l t o d e s c r i b e t h e e v o l u t i o n i n t i m e o f t h e mass d i s t r i b u t i o n and we s h a l l apply i t d i r e c t l y t o o u r problem.

A p p l i c a t i o n o f t h e Master Equation t o t h e D i f f u s i o n Along t h e Mass Asymmetry Coordinate

L e t us l a b e l t h e asymmetry o f t h e i n t e r m e d i a t e complex by means o f t h e atomic number Z o f one o f t h e two fragments i n c o n t a c t . Furthermore, l e t us assume t h a t t h e complex evolves i n t i m e through c o n f i g u r a t i o n s o f d i f f e r e n t asynunetries by means o f a s t o c h a s t i c process, as r e q u i r e d by t h e Master Equation. Then, t h e t i m e e v o l u t i o n o f t h e popu- l a t i o n @(Z,t) can be w r i t t e n as [8]:

where

6

i s t h e t i m e - d e r i v a t i v e o f t h e p o p u l a t i o n and AZ,,,, Azlz a r e the macroscopic t r a n s i t i o n p r o b a b i l i t i e s c o u p l i n g t h e c o n f i g u r a t i o n s Z ' and Z.

The form o f AZZl and A,,, and t h e range o f Z ' s over which one must extend t h e sum, must be described. Without any l o s s o f g e n e r a l i t y we can r e w r i t e :

p r o b a b i l i t y (which i s symmetric because o f micro- scopic r e v e r s i b i l i t y ) ; and pZ, P,, a r e t h e s t a t i s t i c a l weights o f t h e macroscopic c o n f i g u r a - t i o n s . The l a t t e r q u a n t i t i e s can be i d e n t i f i e d w i t h t h e l e v e l d e n s i t i e s o f t h e complex:

where E i s t h e t o t a l energy o f t h e system; and V, i s i t s p o t e n t i a l energy ( i n c l u d i n g r o t a t i o n a l energy). F o r small V, one can expand t h e l e v e l d e n s i t y as f o l l o w s [8]:

P(E

-

VZ) = P(E) exp-Vz/T where

The q u a n t i t y T can be i d e n t i f i e d w i t h t h e thermodynamic temperature.

The q u a n t i t y hZZ, can be w r i t t e n as [8]:

where K i s a v e l o c i t y o f t h e o r d e r o f t h e Fermi

v e l o c i t y , and f i s a form f a c t o r which we t a k e t o be equal t o t h e window open between t h e two fragments:

N o t i c e how t h e mean l e v e l d e n s i t y contained i n t h e denominator o f h,,, a l l o w s t h e macroscopic t r a n s i t i o n p r o b a b i l i t y /LZZl t o remain o f t h e o r d e r o f ~ f . The sum can be r e s t r i c t e d t o values o f

Z = Z c 1 i n t h e s p i r i t o f t h e independent p a r t i c l e model.

The master equation can now be w r i t t e n as: ~f

vz

+ V,'

z ' = Z ? l exp

-

2T

TIME EVOLUTION ALONG THE MASS ASYMMETRY MODE C5-113

The Fokker-Planck Approximation c o o r d i n a t e Z and does n o t depend on any i n i t i a l The i m p l i c a t i o n o f t h e Master Equation as v e l o c i t y . Therefore, i t can be i d e n t i f i e d w i t h t h e w r i t t e n above, can be b e t t e r appreciated i f we l i m i t i n g v e l o c i t y ( t + m ) V =

K

C associated w i t h t h e c o n s i d e r i t i n i t s approximate Fokker-Planck form: d i f f e r e n t i a l equation:

where m i s t h e mass, K i s t h e v i s c o s i t y c o e f f i c i e n t , 1

a2

+

7

- [ u 2 ( z ) @ ( z , t ) l

.

and c i s t h e f o r c e . The v i s c o s i t y c o e f f i c i e n t i s

az2

_{then :}

The q u a n t i t i e s p1 (Z) and p2(Z) a r e g i v e n by: _3T

K = - K f ' z+l p2 =

1

( Z 1

-

z ) ~ AzOzdZ8 z - l and can be c a l c u l a t e d e x p l i c i t l y .

The t r a n s i t i o n p r o b a b i l i t y AZZ1 can be w r i t t e n as f o l l o w s :

N o t i c e t h a t K i s n e a r l y independent o f temperature

f o r a h i g h l y degenerate Fermi gas:

The p o t e n t i a l V Z , can be expanded as:

and we have: V'h Azz, = ~f exp

(- +-)

.

F i n a l l y , we can c a l c u l a t e p, and p2:

v;

I n t h e f a i r l y common l i m i t o f small

2~

,

we o b t a i n :

From t h e d e f i n i t i o n o f p1 one sees t h a t i t corresponds t o t h e average displacement i n Z p e r u n i t time. I n o t h e r words, i t represents t h e average v e l o c i t y along Z. Also one can n o t i c e t h a t such a v e l o c i t y depends o n l y upon t h e

where i s t h e Fermi energy. It f o l l o w s t h a t t h e v i s c o s i t y c o e f f i c i e n t i s p r o p o r t i o n a l t o T f o r T cF and p r o p o r t i o n a l t o T"* f o r T >> cF. The

-7

L I

c o e f f i c i e n t u2 can be r e w r i t t e n as p 2 =

.

I n t h e case T << E ~ , p2 does n o t depend on temperature.

L e t us now c o n s i d e r two simple cases o f p r a c t i c a l importance. The f i r s t case i s t h a t o f a c o n s t a n t f o r c e , which corresponds a t o c o n s t a n t slope i n t h e d r i v i n g p o t e n t i a l , and c o n s t a n t temperature. For t h e i n i t i a l c o n d i t i o n

@(Zo,O) = 6(Z

-

Zo), t h e s o l u t i o n o f t h e Fokker- Planck equation i s :

T h i s i s a Gaussian whose c e n t r o i d moves w i t h v e l o c i t y p1 and whose second moment i s :

A conceivable experimental t e s t o f t h e a p p l i c a b i l i t y o f t h e above equation i s a p l o t o f t h e f i r s t moment o f t h e experimental d i s t r i b u t i o n vs. t h e second moment. Since b o t h a r e p r o p o r t i o n a l t o t h e t i m e t, such p l o t should be l i n e a r . From such an a n a l y s i s the r a t i o p1/u2 be obtained. 2 5

C 5 - 1 1 4 L .G. MORETTO AND R. SCHMITT

w h e r e g is the moment of inertia; and

I

the angular

momentum. Since both

eo

a n d 2

depend on

I,

only

a detailed knowledge of the deflection function can

lead to an accurate determination OF the time.

A1 ternatively, one can rely upon the energy

loss in order to obtain the time. For a viscous

dissipation in the relative motion of target and

projectile one obtains:

or, in order to account for Coulomb and rotational

energies Ec and ER,

This expression can be of some use in relating

the energy dissipation along the relative motion

coordinate to the diffusion along.

the mass

asymmetry coordinate. However, the strong I

dependence of this expression and the uncertainty

about

K

and m make its use doubtful.

The second case of interest is that of the

diffusion driven by a parabolic potential:

In many practical cases the potential energy

along the mass asymmetry coordinate is nearly

harmonic close to the symmetric value Zs. If the

initial condition is:

@(ho,O)

=

6(h

-

ho)

,

the Fokker-Planck Equation gives:

@(h,t)

=

d

C

2~rT(1

-

exp

(-

F))

2

c(h

-

hoexp

(-

t

)

)

x

exp

--

ZT(1

-

exp(-

F))

Notice that the solution is a Gaussian whose

centroid moves following the equation:

i ; + K ~ + C h = O

m

m

(26)

which, in the limit

-

C >>

1 has the solution:

m

A limitation of this formalism is associated with

the constant temperature T. Even when the potential

is parabolic, the energy difference between the

injection point and symmetry may be so large that

the temperature along the parabola changes

substantialiy.

Application of the Master Equation to the Evaluation

of Cross Sections and Angular Distributions

Moretto and Sventek [8,11,26) have performed

diffusion calculations for some of the reactions

studied experimentally. A most important quantity

for this calculation is the potential energy of

the intermediate complex as a function of Z for

each partial R wave. The potential energy has

been calculated assuming the shape of two touching

spheres for the complex. The energies are computed

by means of the liquid drop model:

where the first two terms are the liquid drop

masses of the two fragments; VCoul is the Coulomb

interaction of the two fragments; and ERot is the

rotational energy of the complex. Examples of the

potential energies and of the populations pro-

babilities as a function of time are shown in

Fig. 1. The drift of the distributions from the

injection point towards low potential energies is

well illustrated. Perhaps more impressive is the

spread of the distribution which increases very

dramatically with time. This illustrates how a

Lagrangian approach might miss a most important

feature, namely the spreading of the distribution

which, for large times, dominates the picture. It

is also important to notice how the angular

momentum shifts the Businaro-Gallone mountain with

respect to the injection point and how this affects

the direction of.the drift. This feature is

particularly visible in the Ag

+

Ne case.

At this point it is possible to calculate the

Z

distribution integrated over angle, provided one

knows the distribution of lifetimes as a function

of impact parameter b,lT(t,b)

:

One can go one step farther and evaluate the

cross section as a function of Z and

e

directly.

The differential cross section can be written as

follows [8,26]:

azan

sine

-

(b,t) Il(t,b)/

,

(29)

TIME EVOLUTION ALONG THE MASS ASYMMETRY MODE C5-115

where P(b) is the probability that a collision at

SECTION 11. THE MASS OR CHARGE DISTRIBUTIONS

impact parameter b leads to a deep inelastic

collision. The sum is carried over all the impact

parameters b which result in a particle

Z

being

emitted at the angle

0

after a time t. The

quantity II(t,b) is the probability that the inter-

mediate complex characterized by an impact

parameter b will live a time t. This expression

is very general and it implies a complicated fold-

ing over unknown deflection functions. It is

possible to obtain reasonable results if, as done

by Moretto and Sventek [8,11] and by Sventek and

Moretto [26],

one assumes rigid rotation of the

complex with a moment of inertia suitably averaged

between the entrance channel and the exit channel

asymmetry. In the former paper, where the cal-

culation was performed for lighter systems, a

statistical time distribution was used:

independent of b, where

T is the average lifetime

of the system. An example of this calculation

is shown in Figs. 32 and 33. In the latter paper,

the calculation was performed for a heavier system

where the experiment suggests a much narrower time

distribution which was assumed to be of the form:

where N(b) is a normalization constant,

These forms for the first and the second moment

of the time distribution have a linear dependence

on b suggested by trajectory calculations similar

to those performed by Tsang [20].

The former and

the latter expression for II(b,t), quite different

for a heavy system like Kr

+

Au, give nonetheless

the same result when applied to the lighter system

studied in Ref.

8

because of the relatively narrow

window in impact parameters leading to deep in-

elastic collisions in that reaction. Examples of

the calculation for Kr

+

Au are shown in Figs. 34

and 35. Further discussion on the agreement

between experiment and theory will be given in

Section IV.

An Experimental Note

A

very powerful tool in studying heavy ion

reactions is the AE, E telescope which allows one

to measure the atomic number of the reaction

products. This method, for its great simplicity

and versatility has been widely used. In particular,

in our group, we have developed a gas AE

detector

C611

that enables us to resolve individual

atomic numbers up and above

Z

=

60

(Fig. 2). A1

l

of our data presented in this paper have b"en

taken with such a device. As a result, we have

obtained

Z

distributions, without any information

on the masses. Because of the rapid equilibration

between target and projectile insofar as the

N/Z

ratio is concerned, we may loosely use the work

mass asymmetry when we have actually measured the

charge asymmetry, etc.

The Two Regimes in the Mass Distributions

The mass distributions obtained in heavy ion

induced reactions can be divided in two classes.

The first class includes very broad distributions,

without a well defined peak in the vicinity. of

the projectile and of the target. The second

class includes relatively narrow distributions

peaked at the target and at the projectile masses.

The latter class is associated with the so called

"quasi-fission" reactions, while the former is

associated with the so called "deep inelastic"

reactions.

C5-1 16 L.G. MORETTO AND R. SCHMITT

u n c e r t a i n t y due t o t h e s t i l l sketchy experimental s i t u a t i o n , one can t e n t a t i v e l y conclude t h a t , f o r each r e a c t i o n , t h e l i f e t i m e o f t h e i n t e r m e d i a t e complex increases w i t h i n c r e a s i n g E/B. I f t h i s i s t h e case, one must conclude t h a t t h e l i f e t i m e i s a dynamical r a t h e r than a s t a t i s t i c a l q u a n t i t y , and can be presumably associated w i t h t h e t i m e o f c o n t a c t o f t h e two n u c l e i moving i n and o u t along a r a d i a l coordinate. T h i s p o i n t w i l l be taken up again i n t h e d i s c u s s i o n about t h e angular d i s t r i b u t i o n s .

The l o n g L i f e t i m e Regime o r t h e Regime o f Broad Charge D i s t r i b u t i o n s

The study o f r e a c t i o n s induced by l i g h t p r o j e c t i l e s [l ,2,5-7,27-391 (up t o Ar) on a v a r i e t y o f t a r g e t s a t f a i r l y l a r g e energies above t h e i n t e r a c t i o n b a r r i e r showed t h e presence o f two f a i r l y w e l l i d e n t i f i a b l e components: t h e quasi- e l a s t i c component, v i s i b l e i n

a

narrow angular range about t h e g r a z i n g a n g l e and i n a narrow Z range above t h e p r o j e c t i l e Z; and t h e r e l a x e d com- ponent, p r e s e n t a t a l l angles and e s s e n t i a l l y f o r a l l atomic numbers. I n c o n t r a s t t o t h e narrow Z d i s t r i b u t i o n o f t h e q u a s i - e l a s t i c component, t h e Z d i s t r i b u t i o n o f t h e r e l a x e d component i s v e r y broad, reminding one o f f i s s i o n . I t was n o t immediately c l e a r whether these d i s t r i b u t i o n s were due t o compound nucleus f i s s i o n , p o s s i b l y enhanced by t h e l o w e r i n g o f t h e b a r r i e r due t o angular momentum, o r e l s e a new noncompound nucleus mechanism was i n v o l v e d . The f i r s t p o s s i b i l i t y was favored by t h e n e a r l y t h e r m a l i z e d k i n e t i c energy d i s t r i b u t i o n s associated w i t h t h e r e l a x e d o r deep i n e l a s t i c component o f t h e c r o s s s e c t i o n . F u r t h e r - more t h e general shape o f t h e d i s t r i b u t i o n s Y(Z) q u a l i t a t i v e l y respected t h e s t a t i s t i c a l

p r e d i c t i o n [27-311:

where VZ i s t h e p o t e n t i a l energy o f t h e system a t t h e r i d g e p o i n t w i t h t h e r e q u i r e d mass asymmetry; and T i s - t h e temperature. I n o t h e r words, t h e c r o s s s e c t i o n appeared t o be h i g h where t h e p o t e n t i a l energy i s low, and v i c e versa. T h i s can be seen i n some o f t h e Z d i s t r i b u t i o n s shown i n F i g . 3 t o F i g . 7. Furthermore, a general i n c r e a s e i n w i d t h o f t h e d i s t r i b u t i o n s , o r a f l a t t e n i n g o f t h e slopes w i t h i n c r e a s i n g e x c i t a t i o n energy, seemed t o i n d i c a t e a L I T e f f e c t , very much s t a t i s t i c a l i n n a t u r e [27-311.

However, a c a r e f u l i n s p e c t i o n showed unmistakable entrance channel e f f e c t s i n t h e Z d i s t r i b u t i o n s , e s p e c i a l l y when comparisons o f r e a c t i o n s expected t o l e a d t o s i m i l a r compound n u c l e i , b u t w i t h s u b s t a n t i a l l y d i f f e r e n t entrance channel mass asymmetries, were made [g]. For instance, i n t h e r e a c t i o n s 1 0 7 y 1 0 9 ~ g

+

(Ref. 30) and 1 0 7 ' 1 0 g ~ g

+

4 0 ~ r (Ref. 29) shown i n Figs. 3 and 4, t h e general p a t t e r n s o f t h e Z d i s t r i b u t i o n s appear t o be reversed. I n t h e 1 0 7 y 1 0 9 ~ g

+

4 0 ~ r r e a c t i o n , one observes an i n c r e a s e o f c r o s s s e c t i o n w i t h i n c r e a s i n g Z; i n t h e

1 0 7 y 1 0 9 ~ g

+

_{r e a c t i o n one observes a minimum}

i n c r o s s s e c t i o n a t about Z = 15, a sharp i n c r e a s e o f t h e c r o s s s e c t i o n a t l o w e r 2 ' s and a weak i n c r e a s e o f t h e c r o s s s e c t i o n a t h i g h e r Z's. These f e a t u r e s seem t o be d e f i n i t e l y r e l a t e d t o entrance channel e f f e c t s . I n f a c t i n t h e case o f 1 0 7 ' 1 0 9 ~ g

+

_{t h e c r o s s s e c t i o n i s l a r g e s t a t} l o w Z ' s i n t h e general v i c i n i t y o f Z = 10, w h i l e , i n t h e case o f 1 0 7 s 1 0 9 ~ g

+

4 0 ~ r , t h e c r o s s s e c t i o n i s l a r g e s t a t h i g h Z ' s i n t h e general v i c i n i t y o f Z = 18. These s u b s t a n t i a l changes a r e n o t expected from a compound nucleus decay, s i n c e t h e r i d g e l i n e , c o n t r o l l i n g t h e s t a t i s t i c a l emission o f fragments o f d i f f e r e n t masses o r changes, should be very s i m i l a r f o r t h e two r e a c t i o n s .

A more p l a u s i b l e assumption, c o n s i s t e n t w i t h t h e observed entrance channel e f f e c t s , i s t h a t t h e experimental Z d i s t r i b u t i o n s a r e generated by a d i f f u s i o n process a l o n g t h e mass asymmetry c o o r d i n a t e . I f t h i s i s t h e case, one should be a b l e t o observe t h e e f f e c t s o f t h e p o t e n t i a l energy '

a l o n g t h e mass asymmetry c o o r d i n a t e ( r i d g e l i n e ) upon t h e d i f f u s i o n process. A comparison w i t h t h e r i d g e l i n e p o t e n t i a l energies and w i t h t h e d i f f u s i o n c a l c u l a t i o n s i s a c t u a l l y v e r y i n s t r u c t i v e ( F i g . 1 ).

I n t h e case o f Ag

+

Ne, f o r many o f t h e 8 waves, t h e i n j e c t i o n p o i n t i s s l i g h t l y t o t h e l e f t o f t h e Businaro-Gallone mountain, l e a d i n g t o a r a p i d d r i f t towards s m a l l e r atomic numbers, as e x p e r i - m e n t a l l y observed. I n t h e case o f t h e

1 0 7 y 1 0 9 ~ g

+

4 0 ~ r r e a c t i o n , t h e i n j e c t i o n p o i n t i s s l i g h t l y t o t h e r i g h t o f t h e Businarb-Gallone mountain, l e a d i n g t o a d r i f t i n t h e d i s t r i b u t i o n towards l a r g e r atomic numbers, a l s o as observed. For t h e r e a c t i o n 5 8 ~ i

+

4 0 ~ r [5], shown i n F i g . 5, a s i t u a t i o n s i m i l a r t o t h a t o f 107,l OgAg + 40Ar

TIME EVOLUTION ALONG THE MASS ASYMMETRY XODE

I n t h e case o f t h e r e a c t i o n lg7Au

+

4 0 ~ r shown i n F i g . 6 t h e r e i s a peaking i n t h e v i c i n i t y o f t h e p r o j e c t i l e a t t h e most forward angles on a back- ground r i s i n g w i t h Z [31]. T h i s i s c o n s i s t e n t w i t h t h e f a c t t h a t t h e i n j e c t i o n p o i n t i s indeed t o t h e r i g h t o f t h e Businaro-Gallone p o i n t . How- ever, one should n o t f o r g e t t h a t more o r l e s s o r d i n a r y f i s s i o n should be present.

For t h e r e a c t i o n 1 0 7 y 1 0 9 ~ g

+

8 6 ~ r shown i n F i g . 7, t h e i n j e c t i o n p o i n t i s so c l o s e t o t h e bottom o f t h e symmetry minimum t h a t t h e Z d i s t r i b u t i o n a t s u f f i c i e n t l y backward angles i s completely symmetric 1401. I n t h i s case, o n l y t h e forward peaking angular d i s t r i b u t i o n s suggest t h a t one i s n o t d e a l i n g w i t h compound nucleus r e a c t i o n s . S i m i l a r l y f o r a l l o t h e r r e a c t i o n s mentioned above, t h e angular d i s t r i b u t i o n s a r e t h e most d e c i s i v e evidence a g a i n s t a compound nucleus mechanism.

I n conclusion, the Z d i s t r i b u t i o n s considered so f a r do n o t e a s i l y b e t r a y t h e i r non-compound nucleus o r i g i n . The degree o f r e l a x a t i o n a l o n g t h e mass asymmetry c o o r d i n a t e i s such t h a t o n l y r e l a t i v e l y weak signs o f t h e entrance channel asymmetry a r e v i s i b l e . The d i s t r i b u t i o n s a r e so broad t h a t , as p r e d i c t e d by t h e f i s s i o n model, phenomena r e f l e c t i n g t h e r a t i o VZ/T becomes dominant. I n t h i s r e s p e c t i t becomes q u i t e d i f f i c u l t t o d i s t i n g u i s h and r e s o l v e c o n t r i b u t i o n s coming from t r u e f i s s i o n from those coming from deep i n e l a s t i c r e a c t i o n s .

The Short L i f e t i m e Regime, o r t h e Regime o f Sharp Charge D i s t r i b u t i o n s

The f i r s t r e a c t i o n s s t u d i e d w i t h p r o j e c t i l e s h e a v i e r than Ar on heavy t a r g e t s [3,4,41-451 immediately showed a mass d i s t r i b u t i o n centered about t h e t a r g e t and t h e p r o j e c t i l e . The group who discovered t h e phenomenon: l a b e l e d i t

q u a s i - f i s s i o n , i n view o f t h e f a c t t h a t t h e k i n e t i c energies associated w i t h the products were n e a r l y thermal i z e d o r f i s s i o n - l i k e ( F i g . 8 ) . The

o b s e r v a t i o n o f these f e a t u r e s removes any doubt about t h e q u a l i t a t i v e l y d i f f e r e n t n a t u r e o f these r e a c t i o n s from e i t h e r compound nucleus o r d i r e c t r e a c t i o n s . A more d e t a i l e d study o f these r e a c t i o n s has been performed by means o f the Z d e t e r m i n a t i o n o f t h e i n d i v i d u a l fragments. L e t us c o n s i d e r t h e f o l l o w i n q r e a c t i o n s f o r sake o f example:

l g 7 ~ u + 8 6 ~ r , + 8 6 ~ r , b o t h a t 620 MeV and lg7Au

+

1 3 6 ~ e , 1 5 ' ~ b

+

1 3 6 ~ e b o t h a t 980 MeV. The corresponding Z d i s t r i b u t i o n s a r e shown i n F i g s . 9-12.

L e t us c o n s i d e r f i r s t t h e r e a c t i o n

lg7Au + 8 6 ~ r [ l 1 ;46]. .The Z d i s t r i b u t i o n s shown i n F i g . 9 show a v a r i o u s degree o f peaking a t t h e Z o f t h e p r o j e c t i l e , depending upon t h e a n g l e o f measurement. Sharper Z d i s t r i b u t i o n s a r e observed a t i n t e r m e d i a t e a n g l e s w h e r e t h e angular d i s t r i b u t i o n s a r e peaking. Broader d i s t r i b u t i o n s a r e seen a t more forward angles, and even broader d i s t r i b u t i o n s a r e seen a t more backward angles. I n t h e l a t t e r case i t i s very d i f f i c u l t t o say where t h e d i s t r i b u t i o n s a r e a c t u a l l y peaking, s i n c e they a r e n o t symmetric and t h e i r maxima a r e so broad t h a t t h e c r o s s s e c t i o n i s about c o n s t a n t over more than t e n Z u n i t s .

Sharp d i s t r i b u t i o n s , i n d i f f u s i o n language, a r e young d i s t r i b u t i o n s t h a t have n o t had t i m e t o spread. Therefore, moving from forward t o backward angle, we have t h e sequence: m i d d l e age, young, o l d d i s t r i b u t i o n s . We have commented e l s e - where [11,46] t h a t t h i s i s s t r o n g l y suggestive o f a l i f e t i m e decreasing w i t h i n c r e a s i n g a n g u l a r momen tum.

For t h e same v e l o c i t y v. a t t h e i n t e r a c t i o n r a d i u s , small impact parameters have a l a r g e r a d i a l v e l o c i t y (which may mean l a r g e r a d i a l i n t e r p e n e t r a t i o n and l o n g l i f e t i m e ) and a slow angular v e l o c i t y , w h i l e t h e l a r g e s t impact para- meters have a small r a d i a l v e l o c i t y and thus a s h o r t l i f e t i m e , and a l a r g e angular v e l o c i t y . Therefore, i t seems p o s s i b l e t o a s s o c i a t e t h e backward a n g l e d i s t r i b u t i o n s w i t h l a r g e impact parameters, t h e i n t e r m e d i a t e angle d i s t r i b u t i o n s w i t h small impact parameters and t h e forward angle d i s t r i b u t i o n s w i t h i n t e r m e d i a t e impact parameters.

T h i s a s s o c i a t i o n i s a l s o j u s t i f i e d i n terms o f angular v e l o c i t y . Small impact parameter systems l i v e l o n g b u t r o t a t e s l o w l y and cannot reach very forward angles. Large impact parameter systems r o t a t e much f a s t e r , b u t decay so soon t h a t they cannot r o t a t e t o o forward. The i n t e r m e d i a t e impact parameters have o p t i m a l angular v e l o c i t y and l i f e t i m e t o reach t h e most forward angles.

I n support o f what has been s a i d above, i t can be observed t h a t t h e young d i s t r i b u t i o n s a r e n o t v e r y w e l l r e l a x e d i n k i n e t i c energy, w h i l e t h e m i d d l e age and t h e o l d d i s t r i b u t i o n s have pro- g r e s s i v e l y more r e l a x e d k i n e t i c energy d i s t r i b u t i o n s . Comments a l o n g t h e same l i n e have been made a l s o by Wolf and Roche [47].

C5-118 L.G. MORETTO AND R. SCHMITT

picture, however, is that of a more extensive

relaxation. In particular, the most backward

distributions are not at all peaked in the neighbor-

hood of the projectile, nor are the intermediate

angle

Z

distributions if their higher energy com-

ponent (quasi-elastic) is removed. In both

reactions one would like to see a drift of the

mass distribution peak towards symmetry. However,

while there is an excess cross section at larger

Z's, it is hard to see a well defined peak in the

backward angle distributions. In many respects

the Z distributions at backward angles, expecially

for

+

86~r,

begin to resemble those observed

in the 107y109Ag

+

8 6 ~ r

reaction [40],

illustrated

above.

In the reactions lg7Au

+

1 3 6 ~ e

[49] and

1 5 9 ~ b

+

13%e

[50]

at 979 MeV (Figs. 1 1 and

12), the short decay time features are even more

enhanced. The peaking at the projectile seems

to persist over a broader angular range than in

the previous reactions.

In the lg7Au

+

1 3 6 ~ e

reaction, a strong

fission component is observed. It can be separated

from the deep inelastic component because the

kinetic energy spectrum shows two peaks. This

component, arises from the fission of the quasi

Au fragment. The fission of the quasi target

is essentially absent in the reaction

+

1 3 6 ~ e

because of the much higher fission barriers

invol ved.

As a final example of mass distributions we

consider those arising from light target-projectile

combinations studied in the previous subsection,

but at much smaller values of

E/B.

The reaction Ig7Au

+

40~r,

which, already

at 288 MeV bombarding energy shows a peak in the

mass distribution in the vicinity of the projectile,

and other quasi-fission features [31] (Figs.

6

and

23), shows at 220 MeV a more dramatic quasi-fission

pattern in the mass distribution [51].

Similarly

the reaction 6 3 ~ u

+

9 3 ~ b

at 280 MeV shows a mass

distribution centered about the projectile and the

target [52].

Even the reaction 107y109~g

+

4 0 ~ r

changes its

charge distribution 1533 when the energy is lowered

at 170 MeV, showing two peaks. A first sharp peak

is centered at the projectile and a broader peak

seems to be centered at symmetry (see Fig. 13).

This represents the evidence indicating that the

mass distributions should be classified in terms

of the ratio E/B rather than in terms of the

target-projectile combinations.

1

Dissipation

In many reactions, especially in those of the

quasi-fission type, the relaxed and the quasi-

elastic components are, at times, bridged by a

partially relaxed component. It is then desirable

to find out the dependence of the charge or mass

distribution on the degree of damping.

The mass distributions, when observed for

bins of decreasing fragment kinetic

energy [11,54,55] start out very narrow,nnd become

progressively broader. Their shape is approximately

Gaussian and the centroid at times seems to drift

with decreasing kinetic energies, at other times

it seems to remain fixed at the

Z

of the projectile.

Examples of such distributions are given in

Fig. 14 and Fig. 15. In certain cases these dis-

tributions are given for a fixed lab. angle [Ill,

in other cases, like in Fig. 14 and Fig. 15, the

distributions are integrated over angle. The Z

distributions shown in Figs. 14 and 15 have been

obtained by defining the energy bins by means of

E~ragment

vs. Z lines parallel to the experimentaj

EFragment

VS.

Z

line corresponding to the relaxed

kinetic energy centroids as determined from

measurements at backward angle. This fancy

procedure has the advantage of defining the energy

bins for constant energies above the completely

relaxed energy line.

The qualitative meaning of these distributions

is clear. At small degrees of energy relaxation,

one observes narrow

Z

distributions, while at

greater degrees of energy damping, the

Z

distri-

bution become substantially larger. This shows

that the diffusion in mass asymmetry occurs also

before complete energy relaxation, which is not

too unexpected.

T I M E EVOLUTION ALONG THE MASS ASYMMETRY MODE _C5-119

seems reasonable t o conclude t h a t t h e analysis of

these data without unfolding t h e

R

d i s t r i b u t i o n

can y i e l d o"ly

o r a t best semiquantita-

t i v e r e s u l t s .

This point has been touched i n Section I and

will be discussed again in Section IV.

SECTION 111. THE ANGULAR DISTRIBUTIONS

Lifetime and Rotational Period, o r t h e Two

Angular Distribution Regimes

As i n t h e case of t h e charge d i s t r i b u t i o n s

discussed i n t h e previous section, two kinds of

angular d i s t r i b u t i o n s a r e observed in deep

i n e l a s t i c reactions. Forward peaked angular

d i s t r i b u t i o n s a r e observed, frequently, but not

necessarily associated w i t h broad

Z d i s t r i b u t i o n s ,

and side-peaked angular d i s t r i b u t i o n s a r e observed

usually associated with narrow

Z d i s t r i b u t i o n s ,

peaked a t t h e Z of the p r o j e c t i l e .

As will be seen l a t e r on i n t h i s section,

there i s a continuous evolution from one kind of

angular d i s t r i b u t i o n t o t h e other. The physical

quantity which determines t h e prevailing angular

d i s t r i b u t i o n regime i s the r a t i o between t h e

l i f e t i m e of t h e complex and i t s mean rotational

period.

I f t h i s r a t i o i s small, t h e system does

not have a chance t o r o t a t e enough t o reach

0 ° ,

and consequently t h e products a r e emitted a t wide

angles, on t h e same s i d e of t h e impact.

I f t h i s

r a t i o i s l a r g e and approaches one, t h e system

r o t a t e s past

O0

decaying in t h e meantime. This

generates a forward-peaked angular d i s t r i b u t i o n .

Provided t h a t the Coulomb b a r r i e r i s c l o s e t o t h e

i n t e r a c t i o n b a r r i e r , t h e above quantity can be

written a s follows:

where

i s t h e reduced mass of t h e t a r g e t - p r o j e c t i l e

a t t h e i n t e r a c t i o n radius R; v.

i s t h e center of

mass velocity a t t h e same radius; and Z , , Z2 a r e

the t a r g e t and p r o j e c t i l e atomic numbers.

Let us now consider an impact parameter such

t h a t the radial and t a n g e n t i l e v e l o c i t i e s a r e

the same.

A t the interaction radius these veloci-

v

t i e s a r e

2

.

Then t h e f i r s t square bracket i n

Ji-

the l a s t expression represents t h e time i t takes

the system, subject t o a

f o r c e equal t o t h e

Coulomb force a t t h e i n t e r a c t i o n radius, t o move

radially. i n and out.

Insofar a s v i s c o s i t y i s

neglected (which may be dangerous); insofar a s the

Coulomb force i s representative of t h e forces a t

the i n t e r a c t i o n radius (which i s not a s bad a s

i t sounds because t h e centrifugal force a t

s u f f i c i e n t l y l a r g e r

R

values may p a r t l y compensate

the nuclear force); and f o r a moderate value of

v,, t h e f i r s t square bracket can be i d e n t i f i e d

w i t h

the l i f e t i m e :

I f t h i s r a t i o i s very l a r g e , much l a r g e r than one,

T Y V

v

t h e system has a chance t o r o t a t e several times

zNmax

lifetime

=

2

2.

2

0

p

=

-

C

a

z1z2

B

3

(35)

before i t decays and, i f t h e l i f e t i m e d i s t r i b u t i o n

i s s u f f i c i e n t l y broad, t h e angular d i s t r i b u t i o n

will become symmetric about

90"

and will tend,

f o r l a r g e angular momenta t o the l / s i n 6 limit.

In order t o evaluate such a r a t i o , one needs

a r a t h e r d e t a i l e d and sophisticated model.

However,

i t has been empirically observed t h a t t h e r a t i o

E/B

i s a good predictor of t h e angular d i s t r i b u t i o n

regimes.

A t E/B values below 1.6, one observes

s i d e peaking, while f o r values of E/B above

1.6,

one observes a forward peaking.

The possible reason why

E/B

i s a good empirical

parameter can be seen a s follows. Let us consider

the quantity:

where v i s t h e velocity; c i s t h e force;

-

R,,,

i s

the maximum

R

wave associated with t h e reaction.

The second square bracket i s j u s t t h e angular

velocity (assuming no s t i c k i n g and no d i s s i p a t i o n ) .

The product of t h e two brackets i s t h e angle

of r o t a t i o n of t h e complex, p r i o r t o decay:

A r o t a t i o n of 1 radian may be a good c r i t e r i o n f o r

discriminating between o r b i t i n g past

O0

and s i d e

decay. We then obtain:

E

E - B

- -

1

= -

B

B

In

conclusion 2

(i

-

1) represents t h e product of

C5- 120 L.G. MORETTO AND R . SCHMITT

The rotation of a radian i s obtained with E/B

=

1.5.

For those who do not l i k e t h i s a posteriori

j u s t i f i c a t i o n , Table I shows t h a t indeed f o r E/B

around 1.6 or lower, the s i d e peaking in the angular

d i s t r i b u t i o n i s we1 l established while f o r values

l a r g e r than 1.6 t h e angular d i s t r i b u t i o n s a r e

compl e t e l y forward peaked.

The Regime of Long Relative Lifetimes

( E / B

>

1.6)

A s

i t was seen i n the previous section,

reactions induced by f a i r l y l i g h t p r o j e c t i l e s

(up t o

Ar) a t moderately high energies, a r e

characterized by broad mass d i s t r i b u t i o n s . The

nearly thermalized kinetic energy spectra associated

with such products, together with the broad Z

d i s t r i b u t i o n s a r e very reminiscent of compound

nucleus f i s s i o n . On the other hand, various

f e a t u r e s in the Z d i s t r i b u t i o n suggested a non-

compound nucleus mechanism.

Table

I .

However, remarkable features observed in t h e

angular d i s t r i b u t i o n s [ l ,

2 ,

5-7, 27-39],

l i k e forward peaking i n excess of l / s i n e and i t s

dependence upon the Z difference between t h e actual

fragment and t h e p r o j e c t i l e a r e crucial in ruling

out compound nucleus reactions.

I t i s worth

repeating t h a t t h i s forward peaking i s associated

with the relaxed component of the cross section.

While i n many cases the forward peaking i s

associated w i t h s l i g h t l y higher k i n e t i c energies,

e s p e c i a l l y a t angles smaller than the grazing

angle, s t i l l such increases a r e minimal when com-

pared with the bombarding energy (Fig. 23). The

s l i g h t increase in k i n e t i c energy with angle

appears t o be associated with an increase in impact

parameter leading t o a decreased degree of energy

d i s s i p a t i o n .

An overall picture of the features associated

with the angular d i s t r i b u t i o n a.s a function of t h e

Reaction

ELab(MeV)

ECM(MeV)

B(MeV)

E/B

*

TIME EVOLUTION ALONG THE MASS ASYMMETRY MODE

Z o f t h e fragment can be o b t a i n e d from Fig. 16 t o F i g . 21. I n a l l o f these r e a c t i o n s t h e most remarkable f e a t u r e i s t h e excess f o r w a r d peaking i n t h e c e n t e r o f mass angular d i s t r i b u t i o n s , e s p e c i a l l y i n t h e v i c i n i t y o f t h e p r o j e c t i l e .

The forward peaking i m p l i e s t h e e x i s t e n c e o f an i n t e r m e d i a t e complex w i t h a l i f e t i m e s h o r t e r t h a n t h e mean r o t a t i o n a l p e r i o d . I t a l s o suggests t h a t such an i n t e r m e d i a t e complex r e t a i n s i n i t s shape a memory o f t h e i n i t i a l t a r g e t - p r o j e c t i l e combination. T h i s i s an i m p o r t a n t o b s e r v a t i o n : a completely randomized shape, even i f a s s o c i a t e d w i t h a very s h o r t l i f e t i m e , would n o t l e a d t o a backward-forward asymmetry. A f u r t h e r o b s e r v a t i o n i s t h a t t h e k i n e t i c energy d i s s i p a t i o n occurs on a t i m e s c a l e s h o r t e r than b o t h t h e r o t a t i o n a l p e r i o d and t h e mean l i f e t i m e .

The extend o f r o t a t i o n p r i o r t o t h e decay of t h e i n t e r m e d i a t e complex can be e s t a b l i s h e d approximately by t h e f o l l o w i n g observations: t h e forward peaking i m p l i e s o r b i t i n g p a s t 0°, w h i l e t h e presence o f f a i r l y l a r g e cross s e c t i o n s , and o c c a s i o n a l l y even o f a minor peaking i n t h e back- ward d i r e c t i o n suggests t h a t , a t l e a s t a t times, o r b i t i n g extends t o almost one complete r o t a t i o n , as f a r back as 180'.

By f a r t h e most i m p o r t a n t f e a t u r e , observed i n a l l these reactions, i s t h e dependence o f t h e a n g u l a r d i s t r i b u t i o n upon t h e d i f f e r e n c e i n Z between t h e observed fragment and t h e p r o j e c t i l e . More s p e c i f i c a l l y , t h e forward peaking i s more pronounced i n t h e v i c i n i t y o f t h e p r o j e c t i l e , and fades away toward t h e l / s i n e l i m i t f o r fragments w i t h Z ' s f a r removed from t h e p r o j e c t i l e .

T h i s phenomenon f i n d s i t s q u a l i t a t i v e e x p l a n a t i o n [8,9,11] i n an i n c r e a s i n g t i m e l a g associated w i t h t h e p o p u l a t i o n o f c o n f i g u r a t i o n s f a r t h e r removed i n mass asymmetry from t h e i n j e c t i o n asymmetry. C o n f i g u r a t i o n s w i t h fragments c l o s e i n Z t o t h e p r o j e c t i l e a r e q u i c k l y populated and can q u i c k l y decay, thus g e n e r a t i n g a s u b s t a n t i a l forward peaking. Fragments f a r t h e r removed from t h e p r o j e c t i l e a r e populated on a l a r g e r t i m e s c a l e and decay over a l o n g e r t i m e period, thus g e n e r a t i n g angular d i s t r i b u t i o n s more symmetric about 90".

I t was such a p e c u l i a r combination o f k i n e t i c energy t h e r m a l i z a t i o n , associated w i t h a t i m e e v o l u t i o n a l o n g t h e mass asymmetry c o o r d i n a t e t h a t l e d us t o p o s t u l a t e t h e e x i s t e n c e o f an i n t e r m e d i a t e complex, n e a r l y e q u i l i b r a t e d i n a l l t h e c o l l e c t i v e degrees o f freedom and e v o l v i n g along t h e mass

asymmetry mode by means o f t h e d i f f u s i o n mechanism [8,9,11].

The apparent e f f e c t o f t h e r i d g e - l i n e p o t e n t i a l upon t h e t i m e e v o l u t i o n o f t h e system a l s o seems t o suggest a d i f f u s i o n process. For instance, t h e disappearence o f t h e excess forward peaking as one moves away i n Z from t h e p r o j e c t i l e , appears t o be asymmetric a t times. I n

14N

+

7 0 7 y 1 0 9 ~ g r e a c t i o n [28], and a l s o i n 2 0 ~ e

+

1 0 7 y 1 0 9 ~ g r e a c t i o n 1301 shown i n F i g . 16, t h e forward peaking i s more pronoynced and p e r s i s t e n t f o r fragments lower i n Z than t h e p r o j e c t i l e , w h i l e , f o r h i g h e r Z fragments, t h e excess forward peaking r a p i d l y disappears. A check o f t h e p o t e n t i a l energy vs. mass asymmetry shows t h a t f o r b o t h o f these r e a c t i o n s , t h e i n j e c t i o n p o i n t l i e s more o r l e s s t o t h e l e f t o f t h e Businaro-Gal l o n e mountain (Fig. 1 )

.

Con- sequently,the p o t e n t i a l energy appears t o d r i v e t h e d i f f u s i o n process r a p i d l y t o t h e l e f t and s l o w l y t o t h e r i g h t o f t h e i n j e c t i o n p o i n t . Con- f i g u r a t i o n s associated w i t h fragments l o w e r i n Z than t h e p r o j e c t i l e a r e q u i c k l y populated and t h e y can q u i c k l y decay w i t h t h e r e s u l t i n g sharp f o r w a r d peaking. Instead, t h e fragments h i g h e r i n Z than t h e p r o j e c t i l e a r e populated on a slower t i m e s c a l e and consequently t h e excess f o r w a r d peaking r a p i d l y disappears. An i n v e r s i o n o f these phenomena can be observed i n t h e r e a c t i o n lg7Au

+

4 0 ~ r a t b o t h 288 and 340 MeV bombarding energy L311 ( F i g . 17). I n t h i s case, t h e i n j e c t i o n p o i n t i s t o t h e r i g h t o f t h e Businaro-Gallone mountain and t h e p o t e n t i a l d r i v e s t h e d i f f u s i o n towards symmetry. The experimental data shows t h a t , c o n t r a r y t o t h e

1 0 7 ' 1 0 9 ~ g

+

2 0 ~ e case, t h e excess f o r w a r d peaking i s r e t a i n e d from Z = 18 t o about Z = 29 o r 11 atomic numbers above t h e p r o j e c t i l e . A d i s t a n c e o f o n l y 4 t o 5 atomic numbers was s u f f i c i e n t t o reduce t h e angular d i s t r i b u t i o n s t o t h e l / s i n 8 form i n t h e .case o f 'ONe

+

107,109 Ag

An i n t e r m e d i a t e s i t u a t i o n occurs f o r t h e r e a c t i o n s 1 0 7 y 1 0 9 ~ g

+

4 0 ~ r ( F i g . 18) 1291 and