HOW DOES THE CLASSICAL SURFACE FRICTION MODEL COMPARE WITH THE TIME-DEPENDENT HARTREE-FOCK METHOD ?

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HOW DOES THE CLASSICAL SURFACE FRICTION MODEL COMPARE WITH THE TIME-DEPENDENT

HARTREE-FOCK METHOD ?

P. Fröbrich

To cite this version:

P. Fröbrich. HOW DOES THE CLASSICAL SURFACE FRICTION MODEL COMPARE WITH

THE TIME-DEPENDENT HARTREE-FOCK METHOD ?. Journal de Physique Colloques, 1984, 45

(C6), pp.C6-425-C6-433. �10.1051/jphyscol:1984651�. �jpa-00224253�

JOURNAL DE PHYSIQUE

Colloque C6, supplément au n°6, Tome 45, juin 198* page C6-425

HOW DOES THE CLASSICAL SURFACE FRICTION MODEL COMPARE WITH THE TIME-DEPENDENT HARTREE-FOCK METHOD ?

P. Frobrich

Hahn-Meitner-Institut fur Kernforsehung, Berlin-West and

Fachbereiah Physik (WES), Freie Universitat Berlin, F.R.G.

Résumé - Le modèle phénoménologique classique de friction de surface qui décrit bien les aspects universels des collisions deep inélastiques et de fusion entre ions lourds est comparé de façon détaillée avec des calculs microscopiques de TDHF.

Abstract - The phenomenological classical surface friction model which successfully describes universal features of deep-inelastic collisions and fusion of heavy ions is compared in detail with microscopic time-dependent Hartree Fock calculations.

I - INTRODUCTION

Over the last years several phenomenological friction models have been developed in order to describe in a universal way the increasing amount of experimental data on deep-inelastic collisions, fusion and capture of heavy ions. One of these models is the so called surface friction model which extends the spherical model of Gross and Kalinowski 11/ by including dynamical deformations 12/ with the same deformation parameter for projectile and target. The essential feature of this model is that the friction acts between the juxtaposed surfaces of the collision partners. The resulting Rayleigh dissipation function is proportional to the square of the gradi- ent of the nuclear ion-ion potential times the square of the relative velocity of the two nuclear surfaces. The corresponding dissipative coupling between relative motion and deformation modes leads to enhanced oblate deformations in the entrance channel, whereas strongly prolate shapes develop in the exit channel. The surface friction model has been extended and refined in a series of papers /3-8/ by includ- ing different (quadrupole-) deformations for target and projectile, by allowing for charge and mass transfer and by taking into account statistical fluctuations in all degrees of freedom. This extended model was applied to deep-inelastic collisions in /3-5/, charge and mass transfer was included in 111, and fusion and capture reac- tions were treated in /6/. Although the model can be put on a microscopic basis / 3 / no attempt has been made to calculate the microscopic transport coefficients, in- stead phenomenological coefficients have been used. The model is described in de- tail in ref. / 5 / . Here we only mention the essential input that is necessary for the solution of the multi-dimensional Fokker-Planck equation, from which all rele- vant cross sections can be calculated. The nuclear potential is obtained by a single folding procedure including curvature corrections due to the deformaton modes. These are described within the liquid drop model. The dissipation is gov- erned by only three universal parameters: the strengths of the radial friction, of the tangential friction and of the intrinsic damping of the deformation modes. The dissipation-fluctuation theorem fixes the corresponding diffusion coefficients. For mass and charge transfer theoretical coefficients of-ref. /10/ are used. This clas- sical surface friction model is fairly successful in reproducing at least the over- all trends of experimental data on deep-inelastic collisions and in particular on fusion. In order to check the reliability of its phenomenological input it is in- teresting to compare its results and also its detailed dynamics with the outcome of a more microscopic theory, namely TDHF (for reviews see /11.12/).

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984651

JOURNAL DE PHYSIQUE

The same comparison i s made f o r '%+24126Mg i n f i g . 2.

I 1

-

^{FUSION OF HEAVY IONS}

By s o l v i n g t h e equations f o r t h e f i r s t moments o f t h e F o ~ k e r Planck equation ( c l a s - s i c a l equations o f motion) t h e f u s i o n cross s e c t i o n i s determined by those t r a j e c - t o r i e s which are trapped. I n t h e f o l l o w i n g we present a comparison o f experimental f u s i o n e x c i t a t i o n f u n c t i o n s w i t h those c a l c u l a t e d w i t h t h e surface f r i c t i o n model and t h e TDHF method.

I n f i g . 1 t h i s comparison i s made f o r t h e systems

'60, '%

and 1%+2%g, 27~l.

The surface f r i c t i o n model i s i n good agreement w i t h experiment i n t h e whole energy range. The TDHF f u s i o n cross sections agree w i t h experiment f o r small energies, whereas they are t o o l a r g e a t h i g h eneraies.

The surface f r i c t i o n model i s i n agreement w i t h t h e data except f o r t h e h i g h e s t energies, where incomplete f u s i o n events are expected t o be contained i n t h e data.

The TDHF cross s e c t i o n s are t o o l a r g e also a t low energies f o r t h e '%+'%lg case.

1 4 0 0 - 1 2 0 0 -

I)

.Elooo- q 8 0 0 -

6 6 0 0 - 4 0 0 - 2 0 0

We should mention t h a t t h e f u s i o n cross sections i n TDHF are very s e n s i t i v e t o t h e f o r c e used i n t h e c a l c u l a t i o n and also t o t h e p a r t i c u l a r approximations i n t h e c a l - c u l a t i o n s ( f r o z e n approximation, i s o s p i n degeneracy, etc., see r e f .

1121

f o r more d e t a i l s ) . The l i g h t heavy i o n systems represent a very s e n s i t i v e t e s t o f t h e models since a change o f one u n i t i n t h e c a l c u l a t e d c r i t i c a l &value f o r f u s i o n c o r r e - sponds t o a change o f about

100

mb i n t h e f u s i o n cross section.

1 2 ~ + ' 6 ~ FUSION CROSS SECTIONS

I TDHF

I

EXFiRiMENT

-

I I I I I

The surface f r i c t i o n model

g ~ i

es n t r s u l s

161

f o r t h e capture data

1161

o f heavy systems 1 ik e O'Pb+qk~,"~?]%i,

"icr,'%

e. The strong o b l a t e d e f o r - mations i n t h e entrance channel

-

a behavior q u i t e d i f f e r e n t from t h a t given by ex- t r a push dynamics

-

g i v e r i s e t o a higher b a r r i e r as compared t o c a l c u l a t i o n s w i t h

0 0 . 0 2 0 . 0 4 0 . 0 6 0.08 0.10

1 4 0 0 -

l2 c + 1 8 0 FUSION CROSS SECTIONS

~ T O H F

I

EXPERIMENT

^. 1 2 0 0 - n E TODD-

% 8 0 0 -

6

6 0 0 - 4 0 0 - 2 0 0 -

I I I I I

0 0 . 0 2 0 . 0 4 0.06 0.06 0.10 O 4 0 80 120 I60 200

( M ~ v - ' E LAB ( MeV)

Fig. 1

-

Experimental f u s i o n e x c i t a t i o n f u n c t i o n s are compared w i t h s u r f a c e f r i c - t i o n model c a l c u l a t i o n s (Lsolid l i n e s ) and w i t h TDHF c a l c u l a t i o n s f o r

"c+'~o,

I %

( r e f .

1131)

and f o r

'%+

h g , 2 7 ~ 1 ( r e f .

1 1 4 1 ) .

spherical nuclei, so t h a t i t i s harder f o r the t r a j e c t o r i e s t o pass over the saddle t h a t determines capture. Thus in the surface f r i c t i o n model the predicted capture excitation functions are

-

in agreement with experiment

-

below those of dynamical calculations with spherical nuclei.

I I I I

,

^- I I I I I I

1500- 1500 - -

12 26 C + Mg

-

D

E

4

7

- 0

", 3

by 500- -

5 0 0 - ^-

$62 ob3 o h o b s obs 0;;. oba 009 0 - I I I I L I

0 0 2 0 0 3 0 0 4 0 0 5 0 0 6 007 008 009

E;: ( M ~ V - ' I

€;A

^IM~V-'1

Fig. 2

-

Experimental fusion cross sections are compared with t h e surface f r i c t i o n model ( s o l i d l i n e s ) and TDHF calculations (bars, ref. /15/).

The s i t u a t i o n f o r TDHF f o r heavy sys- tems seems t o be controversal. A com- parison 181 of the surface f r i c t i o n model with a TDHF calculation by Dhar /17/ f o r the 20$b+7%e system (ELab (Pb) = 1600 MeV) shows a q u i t e similar behavior with respect t o energy l o s s , mean values of mass and charge trans- f e r , interaction times and s c a t t e r i n g angle a l l as a functon of i n i t i a l angu- l a r momentum (see f i g . 3 ) . However, t h e predicted capture cross section f o r t h i s system in TDHF ( F = 300 mb) 1s

V

190 Charpe

d e f i n i t e l y below t h a t calculated from t h e surface f r i c t i o n model ( F =

470 mb).

As

the l a t t e r describes the ^{7 0}

s l i g h t l y l i g h t e r systems of ref. /I61 _I

f a i r l y accurate I would favor the sur- face f r i c t i o n prediction f o r the fusion cross section of 20$b+7%e.

L L

Fig. 3

-

( a ) The t o t a l k i n e t i c energy loss; (b) the mean charge and mass of t h e p r o j e c t i l e l i k e fragment; ( c ) the interaction time; and id) the deflec- t i o n function f o r the 0 $ b + 7 k e col- 1 ision a t ELab (Pb) = 1600 MeV are plotted a l l as a function of the i n i - t i a l angular momentum L. The curves with the dots are the TDHF calculations of Ref. 17, whereas the solid lines re- present the surface f r i c t i o n r e s u l t s .

The numbers at the solid l i n e s are the _:-60

values f o r the L = 128 h t r a j e c t o r y of o r w 140 1 8 0 2 2 0 2 6 0 300

the surface f r i c t i o n model. Anpular momentum (ti)

C6-428 JOURNAL DE PHYSIQUE

On t h e o t h e r hand t h r ex\%ts a TDHF c a l c u ~ a t i o n f o r t h e ' 6 b + Ca system /18/ which gives a t o o high capture cross s e c t i o n as compared t o experiment, whereas t h e s u r f ace f r i c t i o n model i s close t o t h e data (see f i g . 4 ) .

1500. ¹

1000 Fig. 4

-

The experimental capture cross

s e c t i o n f o r 20$b+4&a ( f i l l e d c i r c l e s , r e f . /16/) i s compared w i t h a TDHF c a l - c u l a t i o n /19/ ( t r i a n g l e s , dashed l i n e i s

t o guide t h e eye) and a surface f r i c t i o n ^{E [ MeV lu} I

c a l c u l a t i o n ( s o l i d l i n e ) . 4 5 6 7 8

I 1 1

-

DEEP-INELASTIC COLLISIONS

I n f i g . 3 a comparison has already been made between d namical d e t a i l s o f t h e sur- f a c e f r i c t i o n model and a TDHF c a l c u l a t i o n f o r ~ ~ $ b + % e . I n t h e f o l l o w i n g we g i v e some more examples. We discuss m a i n l y mean values o f one-body operators and do n o t p u t much emphasis on t h e d i s p e r s i o n around these mean values, f o r which expectation values o f two-body operators are needed, which are not expected t o be described c o r r e c t l y i n TDHF. We discuss i n p a r t i c u l a r angle-energy c o r r e l a t i o n s (Wi l c z y n s k i p l o t s ) f o r d i f f e r e n t systems.

I n f i g . 5 t h e c a l c u l a t e d r i d g e l i n e s due t o TDHF /19/ and due t o t h e surface f r i c - t i o n model are entered i n t h e experimental Wilczynski p l o t f o r t h e r e l a t i v e l y

l i g h t system ' 6 ~ + 9 3 ~ b . The two branches i n t h e TDHF c a l c u l a t i o n are separated by a f u s i o n window (between a> = 78 and a< = 30) g i v i n g r i s e t o a f u s i o n cross

s e c t i o n o f ^OF =I373 mb, which compares f a v o r a b l y w i t h t h e experimental value o f 1350 mb (not c o r r e c t e d f o r incomplete f u s i o n events). The surface f r i c t i o n model (expected t o describe complete f u s i o n ) never gives a f u s i o n window; f o r t h e present case a f u s i o n cross s e c t i o n of q = 960 mb i s c a l c u l a t e d .

Fig. 5

-

Experimental E- 0 c o r r e l a t i o n f o r

'%+

9 3 ~ b compared t o t h e r i d g e 1 ines from TDHF (dashed l i n e ) and surface f r i c - t i o n ( s o l i d l i n e ) calcu- l a t i o n s . The dots and open c i r c l e s correspond t o t h e i n d i c a t e d p a r t i a l waves.

The example o f 36Kr+'3%a i s shown i n f i g . 6. The TDHF c a l c u l a t i o n 1201 does n o t g i v e s u f f i c i e n t energy loss, whereas t h e s u r f ace f r i c t i o n model r e s u l t i s compa- t i b l e w i t h t h e data / 2 1 / . From t h e 2-values i n d i c a t e d i n t h e f i g u r e i t can be seen t h a t t h e f u s i o n cross s e c t i o n i s considerably l a r g e r i n t h e surface f r i c t i o n model than i n TDHF.

Fig. 6

-

Experimental Wilczynski p l o t I 2 1 1 f o r

8%r+'3%a and t h e r i d g e l i n e s from TDHF (dashed l i n e ) and t h e surf ace f r i c t i o n r e s u l t ( s o l i d l i n e ) . The dots i n d i c a t e t h e corresponding i n i t i a1

-

^{377 MeV}

610 MeV 8 6 ~ r + 350 -

2-val ues

.

Next we show t h e r e s u l t s f o r t h e experiinentally w e l l i n v e s t i g a t e d system ' 3 6 ~ e + 2 0 % i 1221. C a l c u l a t i o n s f o r t h i s system have been made w i t h t h e surface f r i c t i o n model 1 5 1 and w i t h TDHF 123,241 we consider t h e Wilczynski diagram f o r E~~~ = 1422 MeV i n f ~ g . 7.

Fig. 7

-

Experimental W i l c z y n s k i diagram f o r

' 3 6 ~ e + 2 0 9 ~ i and t h e t h e o r e t i c a l E- 0 c o r r e - 1ations.The s o l i d l i n e w i t h dots i s t h e TDHF c a l c u l a t i o n , t h e s o l i d l i n e w i t h crosses t h e surface f r i c t i o n model r e s u l t . The dots and crosses i n d i c a t e t h e i n i t i a1 2-values.

C6-430 JOURNAL DE PHYSIQUE

Both models lead t o s u f f i c i e n t energy loss. For small p a r t i a l waves t h e TDHF c a l - c u l a t i o n leads t o a Coulomb l i k e behavior, whereas t h e surface f r i c t i o n model pre- d i c t s capture below M O O . A f u r t h e r d i f f e r e n c e l i e s i n t h e d i s t r i b u t i o n o f 8-values along t h e r i d g e l i n e . The TDHF c a l c u l a t i o n y i e l d s more energy l o s s f o r k-values i n t h e q u a s i e l a s t i c r e g i o n .

Some o f t h e dynamical p r o p e r t i e s o f t h e models can be read o f f from f i g . 8, where t h e d i s t a n c e o f c l o s e s t approach, t h e center o f mass angle, t h e energy l o s s and t h e i n t e r a c t i o n times are shown f o r t h r e e d i f f e r e n t energies a l l as a f u n c t i o n of t h e i n i t i a l 8-value. The r e s u l t s d i f f e r mainly i n t h e d i s t a n c e o f c l o s e s t approach (more than 2 fm): i t i s seen t h a t i n TDHF t h e n u c l e i overlap r a t h e r more s t r o n g l y than i n t h e s u r f a c e f r i c t i o n model. The o t h e r q u a n t i t i e s behave s i m i l a r l y except i n t h e case o f t h e most deep-inel a s t i c events (small 8-values), where t h e surface f r i c t i o n p r e d i c t s capture f o r EL, 1130, 1420 MeV and TDHF shows I o i l o m b l i k e behavior.

Fig, 8

-

For ' 3 6 ~ e + 2 0 9 ~ i a t ELab = 940, 1130, and 1422 MeV c a l c u l a t i o n s f o r t h e d i s t a n c e o f c l o s e s t approach Ro, t h e s c a t t e r i n g angle, t h e energy l o s s TKEL and t h e i n t e r a c t i o n t i m e a1 1 as a f u n c t i o n o f i n i t i a l angular mo- mentum are shown. S o l i d l i n e s w i t h dots are TDHF and f a t s o l i d l i n e s are surface f r i c t i o n r e s u l t s .

Angular

momentum

(%)

Q40 M e V 1130 M e V 1420 MeV

v, 2

0 300 0 300 0 300

Angular m o m e n t u m

('h)

I n f i g . 9 t h e experimental mean values o f t h e pro- j e c t i l e l i k e fragments are compared w i t h both mod- e l s . The s u r f ace f r i c t i o n model does not show a d r i f t t o asymmetry f o r t h e 1422 MeV case, whereas t h e experiment and TDHF do, although t h e r e i s a l a r g e s c a t t e r i n g o f t h e c a l c u l a t e d p o i n t s .

Fig. 9

-

a) Experimental and s u r f ace f r i c t i o n ( f a t s o l i d l i n e s ) mean values <z> are shown f o r t h r e e

energies f o r '36Xe+20%i. TDHF

b) TDHF ( s o l i d l i n e s w i t h dots) and t h e s u r f a c e f r i c t i o n ( f a t s o l i d l i n e s ) r e s u l t s are compared.

The surface f r i c t i o n model i s s u p e r i o r t o TDHF

/23/ i n t h e c a l c u l a t i o n of t h e d i s p e r s i o n around ₅₂

-

t h e mean values, because it takes s t a t i s t i c a l

f l u c t u a t i o n s i n t o account. This i s demonstrated i n

,,

f i g . 10, where charge variances are shown as a

f u n c t i o n o f t h e energy l o s s f o r ' 3 6 ~ e + 2 0 % i a t ^(b)

1130 MeV. The variances i n t h e surface f r i c t i o n ⁴⁸ o 100 200 300 400 500 600 ^{1 1 2 1}

model are, however, s t i l l s y s t e m a t i c a l l y t o o small _{Energy IOSS (MeV)} /5,7/.

2 0 9 g j + 1 3 6 ~ ~ 850 - 2 0 9 g i + 1 3 6 ~ ~

E L = 1422MeV

650 -

5 5 0 r

.

-d2c mb 350 -

dn,,dE sr MeV / -1

2 0 30 4 0 5 0 6 0 70 0 0 90

@,,(deg) ^{Z IATOMlC NUMBER)}

Fig. 10

-

Experimental (dots), TDHF

Isquares) and s u r f ace f r i c t io n mqdel s o l i d l i n e ) charge variances

a;

are

.

shown as a f u n c t i o n o f t h e energy l o s s f o r ' 3 6 ~ e + 2 0 9 ~ i a t 1130 MeV.

.

^Expermen!

o TDHF

' 3 s ~ e + 2 0 9 ~ t Elab=l130 MeV

The extent t o which t h e surface f r i c -

Fig. 11

-

Experimental ( s o l i d l i n e s ) cross sections 1221 d20/dEdi2 and d2ddEdz are compared w i t h surface f r i c t i o n model c a l c u l a t i o n s (dashed l i n e s )

80

The circumstance t h a t the-cross sections d2uldEdn i n t h e r e g i o n o f smaller energy losses are t o o narrow and t o o h i g h i n s t a t i s t i c a l t h e o r i e s i s u s u a l l y a t t r i b u t e d t o t h e lack o f quanta1 d i f f r a c t i o n , see e.g. t h e discussion i n r e f . 151. The compari- son w i t h TDHF c a l c u l a t o n s ( f i g s . 3, 6, 7 and o t h e r examples) shows t h a t t h e d i s t r i - b u t i o n o f a-values on t h e r i d g e l i n e o f a E - 8 p l o t i s such t h a t i n p a r t i c u l a r i n t h e q u a s i - e l a s t i c r e g i o n t h e r e are fewer &values than i s t h e surface f r i c t i o n model. Using fewer a-values would reduce t h e cross sections i n a s t a t i s t i c a l theory and can be an i n d i c a t i o n t h a t t h e phenomenological surface f r i c t i o n form f a c t o r should be changed t o one w i t h a longer range, which would lead t o more energy l o s s i n t h e grazing region. This c o n s i d e r a t i o n i s supported by a microscopic c a l c u l a t i o n i n t h e framework @f l i n e a r response t h e o r y I 2 5 1 which p r e d i c t s a f r i c t i o n form f a c - t o r o f longer range than t h a t o f t h e surface f r i c t i o n model.

t i o n model can reproduce t h e widths o f experimental cross sections i s demon- s i r a t e d i n f i g . 11 by t h e e3amples o f d oldEdQ and d2u/dEdz f o r

'

6 ~ e + 2 0 % i a t 1422 MeV.

JOURNAL DE PHYSIQUE

I V

-

CONCLUSION

The phenomenological c l a s s i c a l surface f r i c t i o n model has been compared i n d e t a i l t o t h e microscopic time-dependent Hartree Fock method and w i t h experiments f o r f u - s i o n and deep-inel a s t i c c o l l i s i o n s o f heavy ions.

The f u s i o n e x c i t a t i o n f u n c t i o n s f o r l i g h t systems are f a i r l y w e l l described by b0t.h t h e s u r f ace f r i c t i o n model and TDHF, although t h e l a t t e r g e n e r a l l y p r e d i c t s s l i g h t - l y t o o l a r g e cross sections a t h i g h e r energies. As t h e TDHF r e s u l t s are very sensi- t i v e t o t h e nuclear f o r c e and t o t h e d i f f e r e n t approximations i n h e r e n t i n t h e c a l - c u l a t i o n s no f i n a l answer can be given e.g. t o t h e question which f o r c e should be used f o r a u n i v e r s a l d e s c r i p t i o n o f t h e data throughout t h e p e r i o d i c t a b l e . For heavy systems t h e TDHF r e s u l t s appear t o be c o n t r a d i c t o r y , whereas t h e surface f r i c t i o n model describes the experiments very w e l l w i t h t h e

same

parameters as f o r l i g h t systems. The s u r f a c e f r i c t i o n model never p r e d i c t s a f u s i o n window, as TDHF o f t e n does.

Deep-inelastic c o l l i s i o n s g i v e more d e t a i l e d i n f o r m a t i o n on t h e dynamics o f t h e models. Ridge 1 ines o f Wilczynski diagrams (E- c o r r e l a t i o n s ) are simi 1 a r l y w e l l

described i n TDHF and t h e surface f r i c t i o n model; t h e r e s u l t s o f t h e l a t t e r i s somewhat c l o s e r t o experiment a t l a r g e r energy losses. A d i f f e r e n c e e x i s t s i n t h e d i s t r i b u t i o n s o f %-values along t h e r i d g e l i n e . There are fewer L v a l u e s f o r TDHF i n t h e grazing r e g i o n where i t i s expected t o be a good approximation. From t h i s one may conclude t h a t using fewer p a r t i a l waves i n t h e g r a z i n g r e g i o n might remedy t h e shortcoming o f t h e surface f r i c t i o n model, v i z . t h a t t h e cross s e c t i o n s are t o o narrow and too h i g h i n t h e q u a s i e l a s t i c region. This can be an a l t e r n a t i v e expla- n a t i o n t o t h e u s u a l l y assumed ocurrence o f quanta1 d i f f r a c t i o n , which i s n o t ac- counted f o r i n pure s t a t i s t i c a l t h e o r i e s and also not i n TDHF, where i n t e r f e r e n c e s between d i f f e r e n t %-values are n o t taken i n t o account. Thus TDHF suggests a change o f t h e phenomenological surface f r i c t i o n form f a c t o r t o one w i t h a longer range.

Since TDHF c a l c u l a t i o n s show octupole deformations and neck formation i t would be d e s i r a b l e t o i n c l u d e such degrees o f freedom i n t h e s u r f a c e f r i c t i o n model. Another d i f f e r e n c e o f t h e models i s t h a t f o r l a r g e energy losses t h e n u c l e i are coming c l o s e r by more than 2 fm i n TDHF than i n t h e surface f r i c t i o n model. It i s an open question whether t h i s d i s t a n c e o f c l o s e s t approach i s increased by going beyond t h e s i n g l e S l a t e r determinant approximation o r by i n c l u d i n g c o l l i s i o n terms i n TDHF.

The l a t t e r can also p l a y a r o l e i n e n l a r g i n g widths o f cross sections i n an ex- tended TDHF treatment. Due t o t h e i n c l u s i o n o f s t a t i s t i c a l f l u c t u a t i o n s t h e surface f r i c t i o n model reproduces experimental variances much b e t t e r than TDHF.

The fundamental attempt t o describe heavy-ion dynamics w i t h t h e microscopic TDHF method by s t a r t i n g from a nucl eon-nucl eon i n t e r a c t i o n , has l e d t o v e r y encouraging r e s u l t s , although one i s f o r c e d t o perform a number o f approximations /12/ ( f i l l i n g approximation, n e g l e c t o f s p i n - o r b i t terms, neglect o f exchange terms o f non zero- range forces, s p i n and i s o s p i n degeneracy, c l u t c h i n g approximation) i n order t o cope w i t h t h e numerical d i f f i c u l t i e s . We conclude t h a t

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i n s p i r e d by t h e good fea- t u r e s o f these c a l c u l a t i o n s

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i t might be worthwhile t o i n v e s t i g a t e whether s l i g h t m o d i f i c a t i o n s o f t h e phenomenological i n p u t o f t h e surface f r i c t i o n model can lead t o an improved r e p r o d u c t i o n o f experimental d a t a i n p a r t i c u l a r i n t h e quasi - e l a s t i c region. The phenomenological model, on t h e o t h e r hand, shows i t s v i r t u e e.g. i n t h e c a l c u l a t e d variances by a1 lowing f o r s t a t i s t i c a l f l u c t u a t i o n s , thus overcoming basic shortcomings o f TDHF. It i s hoped t h a t t h e surface f r i c t i o n model can simu- l a t e t o a l a r g e e x t e n t t h e r e s u l t s o f a more microscopic theory, so t h a t it can be used w i t h c o n s i d e r a b l y l e s s e f f o r t than TDHF t o make meaningful p r e d i c t i o n s f o r t h e outcome o f heavy i o n c o l l i s i o n s .

V

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