DISSIPATIVE COLLISION AND COMPOUND-NUCLEUS FORMATION

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DISSIPATIVE COLLISION AND

COMPOUND-NUCLEUS FORMATION

W. Nörenberg

To cite this version:

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JOURNAL DE PHYSIQUE _CoLZoque_CIO, _{suppL6ment au nolZ, Tome 41, de'cembre}_1980,_page _C10-167

DISSIPATIVE COLLISiON

AND

COMPOUND-NUCLEUS FORMATION W. NGrenberg.

Gese Z l s c h a f t fiir Seherionenforschung (GSII 0-61 00 Darmstadt, F . R. Germany.

Abstract.- The successes, limitations and generalizations of transport theories for dissipative heavy- ion collisions are reviewed. The evolution of the collision complex can be divided roughly into three stages : nutual approach of the nuclei, local equilibration and slow relaxation of macroscopic degrees of freedom. In particular, the last stage has been thoroughly investigated within the framework of transport theories. The main steps in the derivation of transport equations is discussed and applications of Fokker-Planck equations are presented. Recent developments try to bridge the gap between the initial stage and the third stage of the process. An important feature of the mutual approach is due to the initial correlations imposed by the ground-state configurations (doorway) of the colliding nuclei. These correlations give rise to an explicity time'-dependent dynamical potential in the relative motion. The resulting modified Fokker-Planck equation is applied to compound-nucleus formation and dissipative collisions. The possible existence and some characteristic features of a slow component of dissipative collisions (besides the familiar fast component) are studied.

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

We consider c o l l i s i o n s between heavy n u c l e i w i t h energies o f t y p i c a l l y a few MeV p e r nucleon above t h e i n t e r a c t i o n b a r r i e r (Coulomb b a r r i e r ) . An im- p o r t a n t f e a t u r e o f t h e i n t e r a c t i n g composite system a r e the l a r g e angular momenta which a r e i n v o l v e d i n t h e bombardment w i t h heavy n u c l e i . Depending essen- t i a l l y on t h e i n i t i a l value o f t h e r e l a t i v e angular momentum t h e c o l l i d i n g n u c l e i may f u s e and form a compound nucleus or break a p a r t w i t h o u t going through

-

t h e i n t e r m e d i a t e stage o f a compound nucleus. The l a t t e r case i s r e f e r r e d t o as d i s s i p a t i v e heavy-ion c o l l i s i o n and i s c h a r a c t e r i z e d by a p a r t i a l memory t o t h e i n c i d e n t channel. The most e x c i t i n g discov- e r i e s i n these c o l l i s i o n s a r e t h e continous l o s s o f r e l a t i v e k i n e t i c energy and t h e broadening o f t h e fragment-mass d i s t r i b u t i o n w i t h i n c r e a s i n g k i n e t i c - energy l o s s [ l - 4 1 . W i t h i n t h e c l a s s i c a l equation o f motion t h e l o s s o f k i n e t i c energy has been a s c r i b e d t o f r i c t i o n f o r c e s a c t i n g between t h e n u c l e i [5,6]. The mass d i s t r i b u t i o n was subsequently i n t e r p r e t e d as a r e s u l t of a s t a t i s t i c a l t r a n s f e r ( d i f f u s i o n ) o f nucleons between t h e i n t e r a c t i n g n u c l e i and hence,

has been described by master equations and Fokker- Planck equations [7-91. The d i s s i p a t i v e c h a r a c t e r i s common t o t h e t y p i c a l phenomena observed i n these processes. Here, t h e word d i s s i p a t i o n i s n o t r e - s t r i c t e d t o the d i s s i p a t i o n o f r e l a t i v e k i n e t i c energy and angular momentum. For example, a l s o t h e d i f f u s i o n o f nucleons between t h e n u c l e i i s a d i s s i - p a t i v e process. D i s s i p a t i v e c o l l i s i o n s cover t h e range between d i r e c t r e a c t i o n s and compound-nucleus formation. Whereas o n l y a few degrees o f freedom p a r t i c i p a t e i n d i r e c t r e a c t i o n s , a l l degrees o f freedom a r e i n v o l v e d i n compound-nucleus formation. Therefore, d i s s i p a t i v e c o l l i s i o n s c a r r y i n f o r m a t i o n about t h e d i s s i p a t i v e processes which l e a d from simple nuclear s t a t e s t o more complex c o n f i g u r a t i o n s .

D i s s i p a t i v e heavy-ion c o l l i s i o n s show t h e f o l - l k i n g w e l l - e s t a b l i s h e d c h a r a c t e r i s t i c f e a t u r e s 110-141-.

(1) P r o j e c t i l e and t a r g e t a r e m o s t l y r a t h e r heavy n u c l e i w i t h mass numbers A 40.

(2) The i n c i d e n t energy a t t h e Coulomb b a r r i e r i s t y p i c a l l y 1 t o 3 MeV/u (MeV p e r nucleon) which

12

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C10-168 JOURNAL DE PHYSIQUE

correspond t o maximal angular momenta of a few hundred 5.

(3) The i d e n t i t y of p r o j e c t i l e and t a r g e t i s essen- t i a l l y preserved throughout, although

( 4 ) a considerable amount of mass (AA 6 30) can be transferred during the c o l l i s i o n .

(5) The angular d i s t r i b u t i o n i s strongly non-iso- tropic which means t h a t the time r i n t of the nuclear interaction i s s i g n i f i c a n t l y smaller

=

.

~ O - ~ ~ S ) than the time needed , f o r a complete rotation of the composite system

(6) A large amount of r e l a t i v e k i n e t i c energy ( t y - pical l y more than 100 MeV) and r e l a t i v e angular momentum (up t o = 50 u n i t s of 5 ) i s dissipated, i . e . transferred from r e l a t i v e motion i n t o in- t r i n s i c excitation. The excitation energy i s d i s t r i b u t e d on the fragments proportional t o the fragment masses.

( 7 ) Dissipative c o l l i s i o n s cover the t o t a l range between d i r e c t reactions and compound-nucleus formation. Their share in the t o t a l cross-section

increases with increasing bombarding energy and masses of the colliding nuclei. For heavy nuclei typical values f o r the d i s s i p a t i v e cross-section a r e 1 t o 2 barns.

(8) The measured cross-sections a r e inclusive cross- sections because not a l l reaction products can be observed. Usually only s c a t t e r i n g angle, energy and charge of one fragment a r e detected. Since the excitation energies a r e l a r g e , the channels cannot be resolved and therefore, only averaged q u a n t i t i e s a r e observed.

Several reviews on theories of d i s s i p a t i v e col- l i s i o n s ' c a n be found i n the l i t e r a t u r e [11,13-151. An' a t t r a c t i v e approach i s t h e time-dependent Hartree- Fock theory (TDHF) which t r e a t s the single-particle motion in the time-dependent mean f i e l d generated by

the nucleons C161. W i t h respect t o the c o l l e c t i v e degrees, however, the r e s u l t s of TDHF calculations show many c l a s s i c a l features which a r e imposed by the mean f i e l d . This i s particularly apparent f o r the r e l a t i v e motion. For a given impact parameter the TDHF c o l l i s i o n leads t o a d e f i n i t e deflection angle. Therefore, whereas mean q u a n t i t i e s l i k e mean scat- t e r i n g angle, mean mass s p l i t and mean t o t a l kinetic energy come out reasonably we1 l

,

the corresponding fluctuations a r e severely underestimated in TDHF calculations. Theoccurrence of large fluctuations even f o r small interaction times (=10-*ls) indicate t h a t more complex configurations than those described by a S l a t e r determinant a r e excited already e a r l y in the col l i s i on process.

As i l l u s t r a t e d in f i g . 1 , the t o t a l process can be roughly divided i n t o three stages.

(1) The i n i t i a l stage i s characterized by the mutual approach of the nuclei in t h e i r ground s t a t e s . The motion of the nucleons during t h i s f i r s t stage of

the c o l l i s i o n i s expected t o be governed by t h e i r self-consistent s i n g l e - p a r t i c l e potential which e- volves slowly i n time. This stage should therefore be well described by the time-dependent Hartree- Fock theory.

( 2 ) By residual interactions the S l a t e r determinant of time-dependent single-particle s t a t e s decays t o more complex configurations. This decay leads t o a local s t a t i s t i c a l equilibrium. A t the end of t h i s decay the system occupies t h e t o t a l phase space

( t o t a l configuration space) which i s available f o r fixed values of the macroscopic ( c o l l e c t i v e ) degrees of freedom.

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plied by t h e local s t a t i s t i c a l equilibrium. This d e n s i t i e s , precompound reactions). Furthermore i t leads naturally t o the formulation of transport seems t o be most appropriate t o further extensions

theories. of transport theories ( c f . section 3 ) .

2.1 Derivation of transport equations

There a r e three essential steps in the derivation

1

.

I N I T I A L of transport equations. For simplicity we t r e a t th?

D

1 dI,

we define the macroscopi.~ occupation p r o b a b i l i t i e s

CORRELATIONS A N I ,

fl&i

A 2

.

LOCAL EQUILIBRATION 3

.

RELAXATION OF McROSCOPIC QUANTITIES (TRANSPORT THEORIES) = r o t f ( t ) 2

1

P Pm", ( t ) (1)

"

=mass

which denote the t o t a l occupation probabilities of the r e l a t i v e motion c l a s s i c a l l y and assume t h a t the t r a -

-

trad

jectory F ( t ) and the coupl ing V(t) between excited s t a t e s of the interacting nuclei a r e known. This

-27 " teoc

-- 70 S

= r

semiclassical treatment [8,13,211 of r e l a t i v e motion

a"9

has been generalized by including the r e l a t i v e vari- = E ables e x p l i c i t l y in the equation of motion [l91 ( c f .

de f

2.3).

--

165

( 1 ) By dividing the t o t a l channel space i n t o subsets

t

i6f3s

s u b s e t s z

.

Rewriting the Liouville equation f o r the

U

density matrix p ( t )

,

yields the general ized master Fig. 1. Evolution of the c o l l i s i o n complex (double- equation [81

nucleus s t r u c t u r e [12]) i n a d i s s i p a t i v e t-to

c o l l i s i o n .

f U ( t ) = ~ ~ ( t , t ~ ) +

I

j d ~ K U v ( t . r ) ~ f v ( t - r ) . ( 2 )

v 0

2. Transport theory

Transport theories of d i s s i p a t i v e heavy-ion col- l i s i o n s have been formulated by several groups 18, 17-20]. All derivations s t a r t from an assumed sepa- ration of the degrees of freedom i n t o Slow macroscopic ( c o l l e c t i v e ) and f a s t equilibrating " i n t r i n - s i c " variables. The corresponding s t a t i s t i c a l equilibrium is e i t h e r represented by a heat bath of gi- ven temperature [17,20] or by t h e random properties of the coupl ing matrix elements [8,18,19]. Since the c l a s s i c a l k i n e t i c theory f201 and the l i n e a r response theory [l71 a r e well established i n s t a t i s - t i c a l mechanics we l i m i t our discussion t o t h e non- perturbative quantum-stati s t i c a l approach C8,18,291. This approach i s most closely related t o s t a t i s t i c a l theories of other nuclear properties (e.g. level

The f i r s t term on the r.h.s. represents the contri- bution from those p a r t s of the i n i t i a l density mat- r i x p ( t O ) which a r e not contained in f u ( t ) . The second term on t h e r.h.s. describes the coupling between d i f f e r e n t subsets. In analogy t o the Boltzmann equation i t i s referred t o as the c o l l i s i o n term. The col1 ision term consists of gain terms ( v

+

U) which correspond t o t r a n s i t i o n s from a l l subsets V $ P t o the subset U , and another term V = U which describes the loss from the subset U. The kernel K ( t , ~ ) i s non-local in time. The change of the

U v

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c10-170 JOURNAL DE PHYSIQUE

(2) The second step in the derivation of a transport equation is the introduction of random properties of the coupling matrix V(t). This leads to the following structure of the memory kernel,

KUv(t ,r) = ti-2<~mn(t)~nm(t-T);N~V!J(t,t-r) + C.C. (3) The mean correlation function <V (t)Vnm(t-T)>

mn - vp

decays as function of T within a characteristic time

v11

rCOr (correlation time). This correlation time is related to a correlation length o by T~~~ = G /

i ? ~

.

The correlation length G is typically given by the

mean distance between the nodes of the single-particle wave functions. For strong overlap we have

o = 1 fm, while in the nuclear surface a = (3.. .4)fm [221. Therefore, rCor 2 10-'~s. The mean propagator Gvp(t,t-~) for a state n td4,and a state m E $ ' ~ has

a lifetime denoted by -r&. The relative values of rV' _corand *:Ec determine the two limits of weak and strong coup1 ing. The weak-coupling limit is defined

by

! J = V' <<

mem - cor dec ( 4 )

and the strong-coupling limit by

TV" E T;;C << =;Er.

mem 15)

Calculations [231 of the decay time for small over-. lap give valuer T~~~ .', 5 - 1 0 - ~ ~ s . Thus, in the basis

of asymptotic eigenstates rdec << lor, such that the strong-coupling limit is applicable for dissipative col l isions. The initial correlations IUv(t ,to) have the same structure as the memory kernel (3). It is usually assumed that the rest11 ting transport equation is applied for t-tO>zri&, where I (t,to) can

!J v

be neglected. The memory time characterizes the time interval during which the phase correlations are lost. (3) As discussed at the beginning of this section, the macroscopic variables have to be chosen as the slow modes of the system. This implies that the relaxation times for the macroscopic variables satis- fy rrelax

>>';Ern

qnd hence, justifies the Markov

approximation. It consists in replacing

fv(t-T) by fv(t) in eq. (2) and yields the master equation for t-to >> ,,:,r;

d

f,,(t) =

1

wuv(t) [dufv(t)-dvf,(t)1 (6)

vs !J m

with the transition probability wuV(t)

-

jd~K~~(t,~).

0

Such a master equation was originally introduced by Paul i 1241 in order to justify thelf-theorem from the quantum-mechanical point of view. Moretto and Sventek L93 have directly applied such a master equation to the diffusion of nucleons between the colliding nuclei.

2.2 Fokker-Planck equation, transport coefficients and re1 axation phenomena

A particularly convenient method of describing the solutions of the master equation (6) has been introduced by transforming the integral equation into a second-order differential equation, the Fok- ker Planck equation

Here we have replaced the discrete variables !J by

the continuous variable

?

with g components. The transport coefficients vi (drift coefficients) and Dij (diffusion coefficients) are completely determined by the second moments of the transition probabilities and the density of states. Drift and diffusion coefficients are related by (generalized) Einstein relations [7,13,14,21,251.

In the further evaluation of the microscopic expressions for the transport coefficients the method of spectral distributions is used which has been successful 1y applied to the calculation of mean

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of averages in such a shell-model space. The evaluation of the level densities d($), the memory or de-

-b +

cay time T~~~ (y,yl) and the transition probabili-

ties W($,$') are reduced to the calculation of such averages.

The transport coefficients have been calculated for three macroscopic variables [25]: the mass A1 of one fragment (mass-asymmetry variable), the z-component M of the total intrinsic angular momentum and the total excitation energy E*. Drift and diffusion coefficients are simple analytical expressions depending on the macroscopic variables, on the total mass, on the relative angular momentum t and on three characteristic parameters A j ,

A and y of the interaction matrix elements. The allowed change A j of angular momentum in the transfer or excitation of a single nucleon is determined by the size of the overlap region. For touching nuclei we estimate 1141 A j = 1.5 (A1

+

A~)"~. The corresponding energy change is determined by

a

= (R~/$&~) (t

-

M) A j where

Tel

is the moment of inertia for the relative motion. The third parameter y determines the strength of the coupling matrix elements.

A considerable number of mass distributions measured in dissipative col1 isions have been analyzed [26,271 using a Fokker-Planck equation (7) for the single variable Al. Figure 2 shows as an example the fit (solid curve) of the mass-transport coefficients vA and DAA to the data for 8 6 ~ r (5.99 MeV/u) .

+

166~r. Using the theoretical values for the transport coefficients the dashed line is obtained.

Figure 3 compares experimental and theoretical values for the diffusion constant for various reactions 1141. In calculating the theoretical values we use y = 3 MeV and the dynamical values Aj and

Fig. 2. Calculated element distributions for the reaction 86Kr (5.99 MeV/u) + 166Er compared with experimental data 1291. The dashed curve corresponds to theoretical values of drift and diffusion coefficients, the solid curve to a fit by adjusting only the drift coefficient. From ref. [301.

Fig. 3. Comparison of experimental and theoretical values for the mass diffusion coefficient. The reactions are: (0) 86Kr (515, 619,

703 MeV) + 166Er, (a) 136Xe (900, 1130 MeV)

+

209Bi, (A)

208Pb (1456, 1560 MeV)

+

2@*Pb,

E4Kr (712 MeV)

+

Z09Bi, (A)13'Xe (779 MeV) !.12oSn, ) (. 20aPb (1560 MeV)

+

238U, ( X )

20Ne (168 MeV)

+

63Cu.

A as given above with $

=T

= ( 2 / 3 ) t and M = Mst gr

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JOURNAL DE PHYSIQUE

Fig. 4. The t o t a l angular momentum Itot (Z1) o f t h e fragments as f u n c t i o n o f t h e fragment charge number and corresponding y - m u l t i p l i c i t y data from r e f . L291 f o r t h e r e a c t i o n 86Kr (5.99 MeV/u)

+

l'j6Er. The dashed curve i s obtained by n e g l e c t i n g f l u c t u a t i o n s . From r e f . [30].

s t r e n g t h parameter i s i n agreement w i t h estimates f o r t h e touching c o n f i g u r a t i o n . Comparing e x p e r i - mental and t h e o r e t i c a l values we r e a l i z e t h a t t h e dependence of t h e mean d i f f u s i o n c o e f f i c i e n t on bombarding energy, t o t a l mass and mass asymmetry i s w e l l reproduced. Large e f f e c t s from s h e l l s t r u c - t u r e i n t h e mass d i f f u s i o n c o e f f i c i e n t a r e n o t apparent from t h e experimental values, c f . a l s o r e f . L281. Note t h a t these values correspond t o mean d i f f u s i o n c o e f f i c i e n t s a t r a t h e r h i g h e x c i - t a t i o n energies. The good agreement w i t h t h e experimental values show t h a t p r e d i c t i o n s o f mass d i s t r i b u t i o n s on t h e b a s i s o f these t h e o r e t i c a l d i f f u s i o n c o e f f i c i e n t s a r e r a t h e r r e l i a b l e .

F i g u r e 4 compares t h e c a l c u l a t e d angular momentum o f t h e fragments w i t h r e s u l t s from y - m u l t i p l i - c i t y measurements. The d i p around t h e p r o j e c t i l e charge i s understood as f o l l o w s : The fragments c l o s e t o t h e p r o j e c t i l e a r e predominantly produced i n c o l l i s i o n s w i t h l a r g e a-values where t h e i n t e r - a c t i o n time and hence, t h e d i s s i p a t e d angular momentum i s small. S u f f i c i e n t l y f a r away from t h e

p r o j e c t i l e-charge number, t h e d i s s i p a t e d angular momentum saturates. These fragments a r e m a i n l y populated i n c o l l i s i o n s w i t h a-values where t h e i n t e r a c t i o n t i m e i s l a r g e enough t o r e a c h s t i c k i n g . An important f e a t u r e o f angular-momentum d i s s i

-

p a t i o n i s t h e occurrence o f l a r g e f l u c t u a t i o n s which a f f e c t even t h e mean values shown i n f i g . 4. These f l u c t u a t i o n s i n angular momentum t o g e t h e r w i t h f l u c t u a t i o n s i n t h e t o t a l k i n e t i c energy l o s s o r Q-value imply t h a t t h e r e i s no sharp c o r r e l a t i o n between impact angular momentum'& and t h e Q-value. T h i s becomes i m p o r t a n t i n p a r t i c u l a r f o r small

&-values. Corresponding e f f e c t s have been c l e a r l y demonstrated i n s e q u e n t i a l - f i s s i o n measurements i n bombardments o f U and Pb on v a r i o u s t a r g e t s by von Harrach e t a l . [311.

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grazing c o l l i s i o n s up t o several 10-~'s f o r close c o l l i s i o n s (with small parameters b ss b ). They

g r

have been used in the analyses of mass d i s t r i b u t i o n s and angular momentum dissipation. From a f i t t o experimental energy spectra the relaxation times f o r the loss of radial k i n e t i c energy and f o r

the evolution of fragment deformations ( T ~ ~ ~ ) have been determined [271. In t h i s analysis the theore- t i c a l relaxation time T f o r t h e dissipation of

ang

r e l a t i v e angular momenta has been used ( c f . f i g , 4 ) . The values f o r the relaxation times

(Trad = 0.3

.

1 0 - ~ l s , Tang = 1.0

.

1 0 - ~ l s , -21

T~~~ = 4 10 S) imply t h a t the f a s t loss of radial k i n e t i c energy i s followed by t h e dissipation of r e l a t i v e angular momentum and f i n a l l y by t h e evolution of fragment deformations. The analysis of mass d i s t r i b u t i o n s ( c f . f i g . 2) show t h a t no equilibrium i s reached i n the mass-asymmetry coordinate f o r typical interaction times. The corresponding equilibration time (rmass = 2 10-~Os) i s larger by an order of magnitude and t y p i c a l l y of the same order of magnitude as the mean time rrOt necessary f o r a complete rotation of the c o l l i s i o n complex.

2 . 3 E x p l i c i t treatment of r e l a t i v e motion So f a r we have focused our a t t e n t i o n t o the relaxation phenomena with respect t o observable pro- p e r t i e s of the fragments. In doing t h i s we have assumed t h a t the c l a s s i c a l t r a j e c t o r y i s known. This treatment has the advantage t o bring out the main ingrediences and f e a t u r e s of transport theory, but of course, i s not completely s a t i s f y i n g . General transport equations which include the e x p l i c i t t r e a t - ment of r e l a t i v e motion, have been given by Agassi e t a l . 1181 i n a s t a t i o n a r y formulation and by Ayik and Norenberg L191 f o r the time-dependent problem.

Instead of t h e macroscopic occupation probabili- t i e s f , ( t ) defined by eq. (1) we consider the macro-

scopic Wigner functions f (F,F;t) which in t h e v

semi-classical l i m i t a r e t h e j o i n t probability dis- t r i b u t i o n s f o r finding the system a t time t in the subset p a t the r e l a t i v e distance

?

with the re- l a t i v e momentum

5.

The general transport equation

is given by

in the semi-classical l i m i t [19,321. The left-hand side of (8) describes the change of the probability d i s t r i b u t i o n fv(?,$;t) due t o t h e velocity

$/M\,

and force -ifruV(?) within the subset v. The q u a n t i t i e s Mv and uV(;) denote the corresponding mean reduced mass and the mean p o t e n t i a l , respectively. The col-

l i s i o n term on the right-hand side of (8) describes the coup1 ing t o other subsets. Expl i c i t expressions

+ + +

f o r the memory kernels KvU(r,p;r1 ,$l ; T ) a r e given i n

r e f . [l91 and a r e not discussed here. KO e t a l . 1 3 3 have studied the off-shell behaviour of the collision term within a stationary transport equation f o r a one-dimensional model. Their r e s u l t s can be i n t e r - preted within t h e time-dependent formulation a s f o l - lows. The decay of the memory kernel i s character-. ized by the memory time T&. Therefore, t r a n s i t i o n s

a r e allowed which can go off the energy s h e l l u p t o the order f / r i L m . The resulting off-she1 l probabil i - t i e s a r e not allowed to survive in the asymptotic region t + m , r + m. When the nuclei s t a r t t o separ-

a t e , t h e memory time begins t o increase and tends t o asymptotically. As a consequence, a l l off-shell

l

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C10-174 J O U R N A L DE PHYSIQUE

In addition, due t o the density-of-state factor in the c o l l i s i o n term, the mean value of the i n t r i n s i c excitation energy Ev d r i f t s towards higher d e n s i t i e s of s t a t e s . Since the off-shell t r a n s i t i o n s a r e limited t o %/T;&, the ;-distribution has t o fol low (with some delay) the drag which i s enforced t o i t by the Ev-distribution. According t o eq. ( 4 ) the memory time in the weak-coupling l i m i t i s given by the correlation time. The energy change f o r on-shell t r a n s i t i o n s i s a l s o limited by the correlation time. As-a r e s u l t of t h i s equality the d i s t r i b u t i o n s in Ev and

;

remain close t o on-shell d i s t r i b u t i o n s and hence, do not change s i g n i f i c a n t l y by the feed-back from off-she1 l t o on-she1 l probabi l i t i e s when the nuclei separate. In contrast t o t h i s

-iiEm

= << rVIJ in the strong-coupling l i m i t and hence, the

cor

system can go f a r off-shell such t h a t the mean value of the excitation-energy d i s t r i b u t i o n i s f a r from the on-shell value. Therefore, in the strong-coupling l i m i t the feed-back mechanism i s in general an e s s e n t i a l stage i n the reaction process.

A stationary transport equation i s obtained from eq. (8) by considering the solution fv(?,$;t) f o r an impinging plane wave. For t- the time dependence induced by the wave f r o n t has died out. Thus we ob- t a i n the corresponding s t a t i o n a r y transport equation

+- +

by taking fV(f,$;t-.m) f v ( r , p ) t o be independent of t. By t h i s procedure the i n i t i a l - v a l u e problem i s transformed t o a boundary-value problem with a plane wave of sharp energy impinging from i n f i n i t y . The s t a t i o n a r y l i m i t of t h e time-dependent transport equation (8) i s quite similar t o the transport equation of Agassi e t a l . [181. The left-hand sides a r e i d e n t i c a l . The right-hand s i d e s both contain gain and loss terms which carry information about a t r a n s i t i o n being on s h e l l or off s h e l l . In d e t a i l there a r e differences, which a r e due t o some d i f f e -

r e n t approximations concerning the coupling between i n t r i n s i c and r e l a t i v e angular momentum and the evaluation of the intermediate propagator i n the memory kernel S.

A1 though the general transport equation (8) i s considerably simpler than the Liouville o r Schrodin- ger equation, numerical solutions of t h i s integro- d i f f e r e n t i a l equation a r e hard t o obtain. For the practical application i t i s therefore important t o look f o r f u r t h e r approximations. These a r e obtained a t the additional dispense of a detailed knowledge of the macroscopic Wigner functions f (?,$;t) by

lJ

considering the macroscopic occupation p r o b a b i l i t i e s , c f . eq. ( l ) ,

f v ( t ) r /d3r d3p fv(?,$;t) ₍₉₎

of the subsets V and/or the mean phase-space d i s t r i -

bution

f(?,$;t)

1

fv(F,i;;t). ₍₁₀₎

v

For these q u a n t i t i e s the coupled equations a

-

a t f v ( t ) =

C

wv,,(t) { d y f , ( t )

-

d l J f v ( t ) l y (11) U

{D, (?,Et; t ) f (?,Et; t )

1

a r e obtained [34]. The master equation (11) and the Fokker-Planck equation (12) a r e coup1 ed through the d e f i n i t i o n s of the t r a n s i t i o n p r o b a b i l i t i e s W _{( t )}

VlJ and the transport c o e f f i c i e n t s v i ( f ,$it) and Di

(?,ff;t). Whereas the t r a n s i t i o n p r o b a b i l i t i e s a r e averages taken with f(Sf,$;t)

,

the transport coef- f i c i e n t s a r e determined from averages over f ( t ) .

lJ

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recovered i f only the on-shell t r a n s i t i o n s are considered.

A numerical treatment of the transport equations discussed here has been given by Agassi e t a l . [351. Whereas the dissipation of momentum i s treated by a microscopic input, t h e mass t r a n s f e r i s taken i n t o account by f i t t i n g a diffusion c o e f f i c i e n t . The strong-coupling l i m i t i s replaced by a weak-coupling l i m i t with an e f f e c t i v e coupling strength. KO 1361 includes i n t h i s treatment shape deformations in a phenomenologica? way a s introduced by Siwek-Wilczyn- ska and Wilczynski [37]. Figure 5 shows the r e s u l t s f o r 1 3 6 ~ e (1130 MeV)

+

' 0 9 ~ i . A rather good agreement with the experimental data i s obtained.

ELab=1130

MeV

25"5Ocrn<-75"

l0-'

'i0

'i8

'

i 6 ' i - 4 '

i 2 '

;O'

;8'

ii

Z

(ATOMIC

NUMBER)

2

Fig. 5. Double-differential cross-section d o/dEdZ (integrated over c.m. s c a t t e r i n g angle as indicated) f o r the reaction 1 3 6 ~ e ( E l a b = 1130 MeV)

+

2 0 9 ~ i . The dashed curves a r e the r e s u l t s of the calculation. From r e f . 1361.

2.4 Limitations and insufficiencies

We may summarize the applications of transport equations by the f o l lowing statements. The microscopic theories a r e able t o account quantitatively f o r various relaxation phenomena, i n many cases even without f i t t i n g any parameter. We understand best

the slow processes which a r e connected with the t r a n s f e r of nucleons and the dissipation of angular momentum. Despite of thissuccess the transport theo- r i e s , a s formulated up t o now, a r e subject t o four major r e s t r i c t i o n s .

(1) Restriction due t o the choice of c o l l e c t i v e and macroscopic variables. For the sake of simplicity one takes into account only a few c o l l e c t i v e degrees of freedom ( f o r example, the r e l a t i v e distance between the centers of the fragments) and some macroscopic variables which characterize the observable i n t r i n s i c properties of the fragments ( f o r example, excitation energy, mass asymmetry and i n t r i n s i c angular momentum). I t i s c l e a r t h a t f u r t h e r collec- t i v e coordinates l i k e the deformations of the fragments, and maybe a l s o additional macroscopic variables a r e necessary f o r a more complete understand-

ing of the process.

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c10- 176 JOURNAL DE PHYSIQUE

domness assumption w i t h i n a p a r t i c u l a r b a s i s ( f o r approximations can be found such t h a t numerical example t h e e i g e n s t a t e s o f t h e separated n u c l e i ) a r - treatments become p o s s i b l e . Note, t h a t i n o r d e r t o t i f i c i a l l y e l i m i n a t e s c o r r e l a t i o n s which m i g h t be account f o r t h e f l u c t u a t i o n s i t i s necessary t o con- important. s i d e r n o t o n l y t h e s i n g l e - p a r t i c l e d e n s i t y b u t a l s o ( 4 ) I n t h e a p p l i c a t i o n t o experimental data addi- t h e c o r r e l a t e d p a r t o f t h e t w o - p a r t i c l e d e n s i t y . t i o n a l assumptions a r e made. Since t h e treatment I n view o f t h e numerical e f f o r t s which a r e necess- o f o f f - s h e l l c o n t r i b u t i o n s i s d i f f i c u l t , Agassi e t a r y i n a l l extended TDHF c a l c u l a t i o n s , i t i s advan- a l . C351 use an e f f e c t i v e weak-coupling l i m i t there- tageous t o f o l l o w a s i m p l e r although n o t so r i g o r o u s by r e s t r i c t i n g a1 l tr a n s i t i o n s t o be on the energy approach which i s s t r o n g l y r e l a t e d t o t h e t r a n s p o r t s h e l l . T h i s i s s i m i l a r t o t h e assumption [81 t h a t t h e o r i e s t r e a t e d i n s e c t i o n 2.

t h e c o u p l i n g m a t r i x elements themselves a r e l i m i t e d

t o be on s h e l l . T h i s r e s t r i c t i o n t o on-shell t r a n s i - 3.1 I n i t i a l c o r r e l a t i o n s and l o c a l e q u i l i b r a t i o n tions introduces f o r b o t h approaches e f f e c t i v e During t h e f a s t approach o f t h e n u c l e i t h e i n d i - s t r e n g t h s . E f f e c t s from o f f - s h e l l c o n t r i b u t i o n s have v i d u a l nucleons cannot f o l l o w t h e lowest p o s s i b l e been s t u d i e d r e c e n t l y by Saloner and Weidenmiil l e r 081. ( a d i a b a t i c ) l e v e l S. The nucleonic wave f u n c t i o n s

s t a y e s s e n t i a l l y unchanged. Such a ' d i a b a t i c ' (i.e., 3. Extension o f t r a n s p o r t t h e o r i e s . I n i t i a l c o r r e - non-adiabatic) behaviour i s w e l l known f o r e l e c t r o n s

l a t i o n s

.

i n atom-atom c o l l i s i o n s [43]. S i m i l i a r non-adiabatic

The r e s t r i c t i o n t o l o c a l s t a t i s t i c a l e q u i l i b r i u m processes around quasi-crossings o f a d i a b a t i c l e v e l s l i m i t s t h e a p p l i c a b i l i t y o f present f o r m u l a t i o n s o f have been considered i n t h e f i s s i o n process [44-471 t r a n s p o r t theory e s s e n t i a l l y t o t h e t h i r d stage o f and i n heavy-ion c o l l i s i o n s [48]. The motion o f nuc- t h e t o t a l process (cf. f i g . l ) . The apparent success leons on d i a b a t i c l e v e l s g i v e s r i s e t o a l a r g e po- o f t r a n s p o r t t h e o r i e s i n t h e i n t e r p r e t a t i o n o f ex- t e n t i a l energy i n a d d i t i o n t o t h e a d i a b a t i c p o t e n t i a l . perimental data can be a t t r i b u t e d t o t h e s h o r t du- This a d d i t i o n a l p o t e n t i a l can be estimated from t h e r a t i o n o f t h e i n i t i a l stages o f t h e process as com- i n s p e c t i o n o f two-center shell-model c a l c u l a t i o n s pared t o t h e t h i r d stage. I t i s a c h a l l e n g i n g theo- [44,49,50]. From a more schematic c o n s i d e r a t i o n o f r e t i c a l problem t o extend t h e present f o r m u l a t i o n s c o r r e l a t i o n diagrams we o b t a i n

o 1/3

o f t r a n s p o r t theory towards t h e i n i t i a l stage o f ( A U ) ~ ~ ~ ~ 30 A1 MeV (13) t h e process.

As mentioned i n t h e i n t r o d u c t i o n t h e time-dependent ~ a r t r e e - ~ o c k t h e o r y (TDHF) seems t o be p a r t i c u -

l a r l y s u i t e d f o r d e s c r i b i n g t h e approach phase o f t h e c o l l i s i o n . I n o r d e r t o extend TDHF towards t h e f o l l o w i n g stages i t i s necessary t o take t h e r e s i - dual i n t e r a c t i o n s i n t o account 139-421. A l l extensions o f TDHF conserve t h e s e l f c o n s i s t e n c y and hence, l e a d t o h i g h l y n o n - l i n e a r i n t e g r o - d i f f e r e n t i a l equations. A t present i t i s n o t c l e a r i f reasonable

f o r t h e a d d i t i o n a l r e p u l s i v e p o t e n t i a l a t t h e compound-nucleus shape f o r A1 = A2. T h e o c c u r r e n c e o f an a d d i t i o n a l r e p u l s i v e p o t e n t i a l has been recognized a l s o from TDHF c a l c u l a t i o n s 1511.

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decay via residual interactions. The time which i s necessary t o obtain local s t a t i s t i c a l equilibrium between a l l excited s t a t e s a t a given shape, i s denoted by T ~ Estimates f o r t h i s local equilibra- ~ ~ .

tion time can be obtained from the decay time of one of the particle-hole s t a t e s o r from the corresponding time i n precompound reactions. These con-

-21 siderations lead t o T~~~ = 10 S .

3 .'2 Correlation parameter and consequences f o r transport theories

For describing the e f f e c t of the doorway configuration and i t s decay, we introduce an order parameter which measures the degree of correlation with respect t o the entrance channel. I t i s defined t o be one i n i t i a l l y and approaches zero f o r

t >> T ~ This parameter i s an additional macro- ~ ~ .

scopic variable which has to be taken into account e x p l i c i t l y in the transport theory [191. Only i f

T would t u r n out t o be much smaller than a l l

l oc

other c h a r a c t e r i s t i c times of the c o l l i s i o n process we could neglect X . As a consequence the r e l a t i v e motion of the nuclei would be determined by the adiabatic potential. Since we expect T~~~ t o be of

t h e order of 1 0 - ~ l s , such an approximation seems not t o be j u s t i f i e d .

We assume in the following t h a t the correlation parameter ~(t') i s given by

x ( t ) = exp L

-

1 j

f ( r ( t ' ) ) d t 1 1

loc to (14)

where we regard T~~~ a s an unknown parameter. The

integral over the form f a c t o r f ( r ) smoothly switches on the equilibration. The q u a n t i t i e s r ( t ) and to denote respectively the mean t r a j e c t o r y and a time well before the c o l l i s i o n . As compared to the trans-

port theory formulated so f a r C17-201, the e s s e n t i a l new feature i s the introduction of an e x p l i c i t l y time-dependent dynamical potential

,

where Uad and Udiab denote t h e adiabatic and diaba- t i c potentials

,

respectively. The r e l a t i v e motion of the nuclei and the transfer of nucleons is-described by f i v e variables: position r and momentum

p of radial motion, angle 9 and angular momentum e

of rotational motion, and mass asymmetry

a E (A1-A2)/(A1+A2) without a corresponding momentum ( c f . [7,251). With these variables the Fokker- Planck equation reads

with the reduced mass U and the r e l a t i v e moment

Cy

of i n e r t i a f r e l . The potential U includes be- dyn

sides the dynamical potential U of eq. (15) the dyn

rotational energy. The transport c o e f f i c i e n t s a r e calculated from the expressions of r e f . [251 with modifications a r i s i n g from the treatment of the radial motion. In p a r t i c u l a r , a form f a c t o r i s introduced in accordance with the numerical calcu- l a t i o n s of r e f . [22]. For the adiabatic potential we use the r e s u l t s of f i l l e r and Nix [521. The dia- b a t i c potential i s given by adding t o the adiabatic potential the central value (13) with an adequate form f a c t o r . The proximity form f o r t h e mass- asymmetry dependence i s used f o r calculating (hU)Odiab f o r d i f f e r e n t projecti l e - t a r g e t combinations,

1/3 1/3

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c10-178 JOURNAL DE PHYSIQUE

Here, the dynamical potential e n t e r s via the effec- t i v e excitation energy

which i s available as heat. Whereas the transport c o e f f i c i e n t s v p $ DPP. v a , D R R , DRa, D a r e only

PP affected via the e f f e c t i v e temperature

*

Teff (Eef f / a )

'I2

with the l eve1 -densi t y parameter a , the mass-drift c o e f f i c i e n t va becomes e x p l i c i t l y

N

proportional t o t h e dynamical force -aUdyn/aa and hence, should d i r e c t l y show the local e q u i l i b r a t i o n . An indication of such an e f f e c t i s shown in f i g . 6 where the mean values of the element d i s t r i b u t i o n f o r 8 6 ~ r (5.99 MeV/u, 8.18 MeV/uj + 1 6 6 ~ r a r e plotted as functions of the t o t a l kinetic energy loss AE and the interaction time T ~ , , ~ , respectively.

A t both bombarding energies a local equilibration time of the order 1oe21s i s indicated. This re- tardation in the d r i f t c o e f f i c i e n t explains a l s o the discrepancy of the experimental and theoretical element d i s t r i b u t i o n s shown in f i g . 2.

Fig. 6. Correlation of the mean values <Z1> of the element d i s t r i b u t i o n s with the t o t a l kine-

t i c energy loss AE and the interaction time Tint, respectively, f o r the reactions 86Kr (5.99 and 8.18 MeV/u)

+

l G 6 E r . From r e f . L-271.

3.3 Numerical r e s u l t s f o r the compound nucleus

2 4 8 ~ m

For the numerical treatment of the Fokker-Planck equation we introduce a moment expansion. Figure 7

shows three d i f f e r e n t t r a j e c t o r i e s f o r 4 0 ~ r

(400 MeV)

+

2 0 8 ~ b in the (r,a)-plane f o r a local equilibration time 0.75 10-~'s. The t r a j e c t o r y with the smallest %-value (a=51) leads t o the formation of a compound nucleus.

The l a r g e s t %-value (a=104) corresponds t o a f a s t process known a s d i s s i p a t i v e (or deeply i n e l a s t i c ) c o l l i s i o n . The t r a j e c t o r y with a = 102, although s i m i l a r t o t h a t with = 104 f o r small times, i s captured. In c o n t r a s t t o the t r a j e c t o r y with a = 51 i t does not lead to a compound nucleus because L > "rjt 70 ( l a r g e s t E-value f o r the existence of the compound nucleus 248~m). Instead, the system develops towards mass symmetry (a=O) and i s expected t o s p l i t i n t o two fragments i f deformations of the fragments a r e allowed. This long-living (slow) component of d i s s i p a t i v e c o l l i s i o n s r e s u l t s a f t e r the system has rotated several times and hence, exhibits

Fig. 7. Trajectories f o r three L-values i n the

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e s s e n t i a l l y a l / s i n e angular d i s t r i b u t i o n l i k e the fragments from compound-nucleus f i s s i o n . In contradistinction t o compound-nucleus f i s s i o n , the mass d i s t r i b u t i o n of t h i s long-living component of d i s s i p a t i v e c o l l i s i o n s i s expected t o be broader because i t i s not limited by the saddle-point shape.

W can divide the reaction cross-section schema- e t i c a l l y according t o the a-values. For 0 < L c gCri we have compound-nucleus formation i f the c r i t i c a l L-value a f o r capture (here = 103) i s

cap cap

l a r g e r than ecrit (here acrit = 70). For

a

<a< cap

xcrit

compound-nucleus formation i s limited by The slow component o f d i s s i p a t i v e c o l l i s i o n s i s obtained f o r t c r i t

< L < L c a ~ . Calculated r e s u l t s f o r the f a s t component f o r Kr (8.18 MeV/u)

+

Er a r e compared in f i g . 8 with experimental data.

Fig. 8. Angular d i s t r i b u t i o n , element d i s t r i b u t i o n and energy loss A E vs. variance U$ f o r the

f a s t component of Kr (8.18 MeV/u)

+

Er. Note t h a t t h e erperimental points E541 in do/de correspond t o a l l events whereas only events with AE 5 160 MeV have been considered i n da/dZ1. From r e f . [55].

Since no deformations of the fragments a r e included in the model, energy losses AE l a r g e r than 190 MeV cannot be described. Because of t h i s we probably miss some cross section f o r

e

30'. The element d i s t r i -

2

bution and the c o r r e l a t i o n A E V S . oZ a r e we1 l re-

produced f o r AE 5 190 MeV. A value 1 0 - ~ l s i s used f o r T ~ The variation of the value within a f a c t o r ~ ~ .

two does not s i g n i f i c a n t l y a f f e c t the r e s u l t s . Note t h a t no f u r t h e r parameter i s adjusted in these c a l - culations.

Fig. 9. Excitation functions f o r capture in Ar

+

Pb calculated f o r three d i f f e r e n t values of the local equilibration time. The q u a n t i t i e s VC

and A, J., denote the interaction b a r r i e r and

the resuced mass, respectively. From r e f . 1551.

Figure 9 i l l u s t r a t e s the calculated e x c i t a t i o n functions f o r the capture cross-section o i n

cap u d i t s of the reaction cross-section oR. For o

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c10- 180 JOURNAL I)E PHYSIQUE

fhe dashed line which corresponds to cCri t. Above long-living dissipative component have been reported this cross-over a long-living (slow) dissipative for 238~ (5.7 MeV/u)

+

48~a 1561

,

13'xe (5.9 MeV/u) collision occurs. The threshold for this component + 5 6 ~ e l571,

ON^

+

n a t ~ e and 4 0 ~ r

+

165~o 1581. is well above the interactidn barrier VC. A sig- Thresholds somewhat above the reaction barrier as nificant dependence of the capture cross-section on indicated in figs, 9 and 10 have been observed. the local equilibration time is obtained only for

(ECM-VC)/Ared > 3 MeV. In this region the capture 4. Concluding remarks cross-section is roughly determined by an upper

limit !Lcap of relative angular momentum which only depends on T ~ As a result of this the capture ~ ~ .

cross-section exhibits a rather pronounced m a x i m as function of the bombarding energy.

Fig. 10. Excitation functions for capture in the collisions Ar

+

Pb, Kr

+

Er and Xe

+

Sn calculated for T~~~ = 10-~ls. From ref. 1551.

Figure 10 illustrates the dependence of the capture cross-section on the projectile-target com- bination, The depth of the pocket in the relative potential is reduced for more symmetric projectile- target combinations and hence, the capture cross- section goes down corresponding1 y.

Experimental evidences for the existence of the

Dissipative collisions play a dominant role in heavy-ion reactions and reveal new nuclear properties which are connected Nith mass transfer, kinetic energy loss and angular momentum dissipation. They represent an interesting many-body problem,at rather high excitation energies. In contrast to nuclear spectroscopy dissipative heavy-ion collisions supply information about relaxation phenomena in nuclei. Such phenomena have also been observed in precompound reactions and in fission, but only in heavy-ion collisions a rich variety of such phenomena have been discovered. In this respect the study of dissipative collisions has opened a new field of nuclear research.

Transport theories have been successfully applied in describing the dissipative feature: of heavy-ion collisions. Transport coefficients calculated from microscopic theories account quantitatively for the gross properties of mass transfer, angular momentum dissipation and kinetic energy loss. Still many questions concerning a detailed account of the data remain open, in particular the dynamics of fragment deformations.

Present formulations of microscopic transport theories encounter two essential problems, the strong-coup1 ing limit and the assumption of local equilibrium. The restriction to local statistical equilibrium does not apply to the first stages of the collision process. In order to overcome these problems two ways can be considered :

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from r e s i d u a l i n t e r a c t i o n s and

( i i ) t h e e x t e n s i o n o f t h e microscopic t r a n s p o r t t h e o r i e s by the. expl i t i t treatment o f l o c a l e q u i l i- b r a t i o n . A p a r t l y phenomenological model i n v o l v i n g an e x p l i c i t l y time-dependent dynamical p o t e n t i a l i n t h e r e l a t i v e coordinate, has been i n t r o d u c e d as a f i r s t step. Then the mechanism o f energy l o s s becomes two-fold. P a r t o f t h e k i n e t i c energy i s l o s t d i r e c t l y by f r i c t i o n . Another p a r t i s f i r s t s t o r e d as p o t e n t i a l energy (dynamical p o t e n t i a l ) and then transformed i n t o heat v i a r e s i d u a l i n t e r a c t i o n s . Thus we expect t h e r e l a t i v e motion t o be q u i t e d i f - f e r e n t from t h e treatment by f r i c t i o n f o r c e s and t h e a d i a b a t i c p o t e n t i a l .

This concept has been a p p l i e d t o heavy-ion c o l - l i s i o n s d e s c r i b i n g b o t h compound-nucleus f o r m a t i o n and d i s s i p a t i v e c o l l ~ i s i o n s . A l o n g - l i v i n g (slow) component o f d i s s i p a t i v e c o l l i s i o n s i s found t o e x i s t besides t h e w e l l e s t a b l i s h e d f a s t component The l o n g - l i v i n g component i s c h a r a c t e r i z e d by s i m i l a r i t i e s t o compound-nucleus f i s s i o n . I t d i f f e r s from compound-nucleus f i s s i o n by a broader mass d i s - t r i b u t i o n and by a t h r e s h o l d i n bombarding energy which l i e s w e l l above t h e i n t e r a c t i o n b a r r i e r . Cal- c u l a t i o n s o f mass d i s t r i b u t i o n s and angular d i s t r i - b u t i o n s i n c l u d i n g a dynamical treatment o f fragment deformations a r e i n progress 1591.

References :

[l] Artukh, A.G., G.F. Gridnev, V.L. Mikheev, V.V. Volkov and J. Wilczynski, Nucl. Phys. A215 (1973) 91

C23 L e f o r t , M., C. Ng6, J. Peter and B. Tamain, Nucl. Phys. A216 (1973) 166

Hanappe, F. ,Ivl.Lefort, C. Ng6, J . P'eter and B. Tamain, Phys. Rev. L e t t .

32

(1974) 738 131 Moretto, L.G., D. Heunemann, R.C. Jared,

R.C. G a t t i and S.G. Thomson, i n "Physics and Chemistry o f F i s s i o n 1973" ( I n t e r n a t i o n a l Atomic Energy Agency, Vienna, 1974) Vol. 11, p. 351

141 Wolf, K.L., J.P. Unik, J.R. Huizenga, J. B i r - kelund, H. F r e i e s l e b e n and V.E. V i o l a , Phys. Rev. L e t t .

3

(1974) 1105

151 Beck, R., and D.H.E. Gross, Phys. L e t t .

478

(1973) 143

[ 6 ] Wilczynski

,

J.

,

Phys. L e t t .

478

(1973) 484 [7] Norenberg, W., Phys. L e t t .

528

(1974) 289 [81 Norenberg, W,, Z. Physik A

274

(1975) 241 and

276 (1976) 84

193 Moretto, L.G., and J.S. Sventek, Phys. L e t t . 588 (1975) 26

-

[ l 0 1 Schroder, W.U., and J.R. Huizenga, Ann. Rev Nucl. Sci,

-

27 (1977) 465

Cl11 L e f o r t , M - , and C. Ng8, Ann. Phys. ( P a r i s )

V

3

(1978) 5

[ l 2 1 Volkov, V.V., Phys. Rep.

44

(1978) 93 1131 Norenberg, W,, and H.A. Weidenmiil l e r , I n t r o -

d u c t i o n t o t h e Theory o f Heavy-Ion C o l l i s i o n s , Lecture Notes i n Physics, vcl. 51, second e d i t i o n (Springer-Verlag, B e r l in-Heidelberg- New York, 1980)

[ l 4 1 Gobbi, A., and W. Norenberg, i n "Heavy-Ion C o l l i s i o n s " , ed. by R. Bock, v o l . I I . ( N o r t h - Holland, Amsterdam, 1980), GSI-Preprlnt 80-6 [ l 5 1 Weidenmuller, H.A.,Progress i n Nuclear and

P a r t i c l e Physics ( i n press)

1161 Negele, J.W., Proceedings o f the NATO Advanced Study I n s t i t u t e o f T h e o r e t i c a l Methods i n Me- dium Energy and Heavy-Ion Physics, Madison, Wisconsin, USA

,

1978, ed. by

K.W.

MC Voy and W.A. Friedmann (Plenum Press, New York and London, 1978) p. 235

L171 Hofmann, H., and P.J. Siemens, Nucl. Phys. A257 (1976) 165 and

g

(1977) 464

-

[ l 8 1 Agassi, D., C.M. KO and H.A. Weidenmuller, Ann. Phys. (N.Y.)

107

(1977) 140

L191 Ayik, S., and W. Norenberg, Z. Physik A 288 (1978) 401

1201 Randrup, J., Nucl. Phys.

A307

(1978) 319 and A327 (1979) 490

[211 Norenberg, W., Proceedings of t h e Predeal I n t e r n a t i o n a l School on Heavy-Ion Physics, Predeal

,

Romania, 1978, ed. by A . Berinde e t a l . (Central I n s t i t u t e o f Physics, Bucha- . ' r e s t , 1978) p . 825, GSI-Report 79-5

[22] B a r r e t t , B.R., S. Shlomo and H.A. WeidenmUller, Phys. Rev. C

17

(1978) 544

c231 Ayik, S., B. Schurmann and W. Norenberg, Z. Physi k A

277

(1976) 299

[24] P a u l i , W., i n "Probleme d e r modernen Physik", F e s t s c h r i f t zum 60. Ceburtstag A. Sommerfelds, ed. by P. Debye (Hirzel-Verlag, L e i p z i g , 1928) p. 30

[25] Ayik, S., G . Wolschin and W. Norenberg, 2 .

Physik A

286

(1978) 271

(17)

C10- 182 JOURNAL DE PHYSIQUE

L271 Riedel, C., G. Wolschin and W. Norenberg, 2. 1431 L i c h t e n , W., Phys. Rev. 164 (1967) 131

Physik A -- 290 (1979) 47 Smith, F.T., Phys. Rev.

f7V

(1969) 111

[28] Schurmann, B., W. Norenberg and M. Simbel, [44] H i l l , D.L., and J.A. Wheeler, Phys. Rev.

3

Z. Physik A 286 (1978) 263 (1953) 1102

[291 Olmi, A., H. Sann, D. P e l t e , Y. Eyal, A. Gobbi, [45] Wilets, L., Theories o f Nuclear F i s s i o n W. Kohl, U. Lynen, G. Rudolf, H. S t e l z e r and (Clarendon Press, Oxford, 1964)

5

5.2 R. Bock, Phys. Rev. L e t t .

5

(1978) 688

[46] G r i f f i n , J.J., Phys. Rev. L e t t . - 2 1 (1968) 826 [301 Wolschin, G., and W. Narenberg, Phys. Rev.

L e t t . 41 (1978) 691 [47] Schutte, G., and L. Wilets, Nucl. Phys.

A252

(1975) 21 [311 von Harrach, D., P. Glassel

,

Y. Civelekoglu,

R. Manner and H.J. Specht, Phys. Rev. L e t t . 1481 Glas, D., and U. Mosel, Nucl. Phys. A264 (1976) 42 (1979) 1728

- 268 and Phys. L e t t . 498 (1974) 301

C321 Norenberg, W., Proceedings o f t h e European Conference on Nuclear Physics w i t h Heavy Ions, Caen, France, 1976, J. Physique

2

(1976) C5

-

141

[331 KO, C.M., D. Agassi and H.A. Weidenmuller, Ann. Phys. (N.Y.)

-

117 (1979) 237

[34] Ayik, S., and W. Norenberg, t o be p u b l i s h e d l351 Agassi, D., C.M. KO and H.A. Weidenmuller,

Phys. Rev. C18 (1978) 223 and Ann. Phys. (N.Y.)

117

m 7 9 ) 407

[36] KO, C.M., Phys. L e t t . 81B (1979) 299 C371 Siwek-Wilczynska, K., and J . Wilczynski,

Nucl. Phys. A264 (1976) 115

1381 Saloner, D., and H.A. Weidenmiiller, 2. Phys. A 294 (1980) 207

_-

C391 Wong, C.Y., and H.F. Tang, Phys. Rev. L e t t . 40 (1978) 1070

-

L401 Orland, H., and R. Schaeffer, 2 . Physik A 290 (1979) 191

1411 Grangk, P., G. Wolschin and H.A. Weidenmuller, Proceedings o f t h e I n t e r n a t i o n a l Workshop V111 on Gross P r o p e r t i e s o f N u c l e i and Nuclear E x c i t a t i o n s , Hirschegg, K l e i n w a l s e r t a l

,

A u s t r i a 1980

1421 A y i k , S., Proceedings o f t h e I n t e r n a t i o n a l Workshop V I I I on Gross P r o p e r t i e s o f N u c l e i and Nuclear E x c i t a t i o n s , Hirschegg, K l e i n - w a l s e r t a l , A u s t r i a , 1980; t o be p u b l i s h e d

[491 Scharnweber, D., W. Greiner and U. Mosel

,

Nucl

.

Phys. A164 (1971) 257

[50] Norenberg, W., Phys. Rev. C5 (1972) 2020 and Z. Physik

197

(1966) 246

-

1511 Flocard. H.. S.E. Koonin and M.S. Weiss, Phus.

- - ~- . -

Rev. C i 7 ( i 9 7 8 ) 1682

Bonche,T., B. Grammaticos and S.E. Koonin, Phys. Rev. C 17 (1978) 1700

Dhar, A.K., p x v a t e communication (March 1979) C521 M o l l e r , P., and J.R. Nix, Nucl. Phys. A

281

(1977) 354

C531 Norenberg, W., and C. Riedel, 2. Physik A

290

(1979) 335

C541 Rudolf, G., A. Gobbi, H. S t e l z e r , U. Lynen, A. Olmi, H. Sann, R. Stokstad and D. P e l t e , Nucl. Phys. A ( i n press)

1551 Riedel, C., and W. Norenberg, t o be published C561 Sann, H., U. Lynen, V . Metag, A. Olmi, S. B j o r n -

holm, D. Habs, H.J. Specht, R. Bock, A. Gobbi and H. S t e l z e r , GSI Annual Report, GSI-J-1, p. 30

C571 Heusch, B., C. Volant, H. Freiesleben, R.P. Chestnut, K.D. Hildenbrand, F. Puhl hofer, W.F.W. Schneider, B. Kohlmeyer and W. P f e f f e r , Z. Physi k A 288 '(1978) 391

L581 Lebrun, C., F. Hanappe, J.F. L e c o l l e y , F. Le- febvres, C. Ng5, J. P'eter and B. Tamain, Nucl. Phys.

A321

(1979) 207