WALL DISSIPATION AND THE FUSION OF 28Si + 28Si and 40Ca + 40Ca

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WALL DISSIPATION AND THE FUSION OF 28Si + 28Si and 40Ca + 40Ca

D. Sperber, J. Stryjewski, M. Zielínska-Pfabé

To cite this version:

D. Sperber, J. Stryjewski, M. Zielínska-Pfabé. WALL DISSIPATION AND THE FUSION OF 28Si + 28Si and 40Ca + 40Ca. Journal de Physique Colloques, 1987, 48 (C2), pp.C2-275-C2-278.

�10.1051/jphyscol:1987242�. �jpa-00226510�

WALL DISSIPATION AND THE FUSION OF 2 8 ~ i + 2 8 ~ i and 4 0 ~ a + *Oca

D. SPERBER, J . STRYJEWSKI and M. ZIEL~NSKA-PFAB~*

Department of Physics, Rensselaer Polytechnic Institute, Troy, NY 12180, U.S.A.

" ~ e p a r t m e n t of Physics, Smith College, Northampton, M A 01063, U.S.A.

ABSTRACT

The fusion excitation functions for 2 8 ~ i + 2 8 ~ i and 4 0 ~ a + 4 0 ~ a are determined using a fully dynamical calculation. This calculation includes a careful treatment of entrance channel deformations and the deformation dependant inertias in both the one-body and two-body regimes. It is shown that deformation, deformation dependent inertias and wall friction affect the fusion excitation function in a fundamental way. We also perform the calculation with a weaker wall friction (surface

friction) and show that only the strong wall friction is consistant with the experimental data.

INTRODUCTION

The first full dynamical treatment of the role of deformations and their damping in heavy ion Scattering is found in references 1-3. However that model suffers from a number of shortcomings. First, the parametrization describing the shapes of the deformed nuclei in the amalgamated system is not realistic. Second, the nuclei are assumed to be spherical up to touching, at which time a neck connecting the two suddenly opens. In other words, the time evolution of the neck is not determined dynamically. The use of a nondynamical treatment of neck

formation has detrimental consequences: i) One has to make assumptions on the amount of deformation that the nuclei undergo before they touch; ii) Since

deformations are not treated dynamically in the entrance channel the amount of wall dissipation which occurs during this time can not be determined; iii) The amount of wall friction which occurs during this time can only be infered by making

assumptions as to the how much wall friction there is and as to whether or not the rolling limit is reached; iv) Finally the way in which the incident energy of the nuclei is partitioned between the relavent degrees of freedom can only be

determined by solving the equations of motion. Thus, in order to properly study heavy ion fusion it is necessary to undertake a full dynamical calculation which includes deformations and one body dissipation in both the one- and two-body regimes.

In this paper we present a complete, parameter free dynamical calculation. We solve the equations of motion numerically, incorporating all the ingredients such as: i) all inertias and their dependence on the relavent degrees of freedom, ii) the nuclear forces in both the entrance and exit channels and, iii) a proper treatment of nuclear dissipation.

(')on leave at G M I L . Caen. France

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987242

C2-276 JOURNAL DE PHYSIQUE

R e c e n t l y [ 4 , 5 ] i t h a s been s u g g e s t e d t h a t t h e s t r e n g t h of t h e w a l l f r i c t i o n s h o u l d b e r e d u c e d by a f a c t o r o f b e t w e e n t h r e e and f o u r . Hence, we compare t h e r e s u l t s of a c a l c u l a t i o n w i t h f u l l w a l l f r i c t i o n w i t h t h e r e s u l t s of a c a l c u l a t i o n w h e r e t h e s t r e n g t h o f t h e w a l l f r i c t i o n h a s b e e n r e d u c e d by a f a c t o r of t h r e e . W e show t h a t t h e r e d u c t i o n of t h e w a l l d i s s i p a t i o n a f f e c t s t h e f u s i o n c r o s s s e c t i o n s i g n i f i c a n t l y a t h i g h e r bombarding e n e r g i e s f o r h e a v i e r s y s t e m s .

THE DYNAMICAL MODEL

The c a l c u l a t i o n s p r e s e n t e d i n t h i s p a p e r a r e b a s e d on s e v e r a l commonly u s e d a s s u m p t i o n s . The m a c r o s c o p i c a p p r o a c h i s u s e d . S i n c e t h e number of n u c l e o n s i s l a r g e , we b e l i e v e t h a t t h e i n d i v i d u a l p a r t i c l e d e g r e e s o f f r e e d o m d o n o t h a v e t o be c o n s i d e r e d e x p l i c i t l y and t h e d y n a m i c s of t h e e n t i r e s y s t e m c a n be e s s e n t i a l l y d e s c r i b e d by a f e w , c l e v e r l y c h o s e n c o l l e c t i v e v a r i a b l e s .

N u c l e i c a n b e c o n s i d e r e d a s l e p t o d e r m o u s ( t h i n s k i n n e d ) which i m p l i e s t h a t t h e v o l u m e of t h e d i f f u s e s u r f a c e r e g i o n i s a f r a t i o n of t h e t o t a l volume.

The e x c i t a t i o n e n e r g i e s a r e low, s o t h a t t h e t e m p e r a t u r e of t h e s y s t e m i s much s m a l l e r t h a n t h e F e r m i e n e r g y . T h e r e f o r e , t h e n u c l e o n s c a n be t r e a t e d a s a c o l d F e r m i g a s . At t h e s e l o w t e m p e r a t u r e s , t h e P a u l i p r i n c i p l e s t r o n g l y s u p p r e s s e s n u c l e o n - n u c l e o n c o l l i s i o n s . T h e r e f o r e , t h e mean f r e e p a t h i s l o n g and t h e n u c l e o n s move i n a mean f i e l d .

Based on t h e s e a s s u m p t i o n s , t h e m a c r o s c o p i c b e h a v i o r of t h e s y s t e m of two c o l l i d i n g h e a v y i o n s i s d e s c r i b e d by t h e d y n a m i c a l e v o l u t i o n of a f a m i l y of n u c l e a r s h a p e s w i t h s h a r p s u r f a c e and t h e u n i f o r m a n d c o n s t a n t c h a r g e and mass d e n s i t y .

We r e s t r i c t t h e s h a p e s t o b e a x i a l l y s y m m e t r i c s h a p e s and u s e t h e p a r a m e t r i z a t i o n s u g g e s t e d by B l o c k 1 and S w i a t e c k i 161. The Coulomb e n e r g y i s d e t e r m i n e d by t h e method of B e r i n g e r [7] and f o r t h e n u c l e a r f o r c e we u s e t h e p r o x i m i t y f o r m a l i s m o f Kandrup [8]. W e c a l c u l a t e t h e i n e r t i a s w i t h t h e

Werner-Wheeler a p p r o x i m a t i o n t o i r r o t a t i o n a l , i n c o m p r e s s a b l e f l o w of n u c l e a r m a t t e r [Y,10]. The w a l l a n d window f r i c t i o n are c a l c u l a t e d a f t e r t h e method of B l o c k i e t . a l . [ l l ] .

DYNAMICS

When d i s c u s s i n g damped c o l l i s i o n s and f u s i o n i t i s c o n v e n i e n t t o d i v i d e t h e t r a j e c t o r y i n t o f o u r p a r t s o r " p h a s e s " . The f i r s t p h a s e i s t h e " a p p r o a c h p h a s e " , t h i s i s t h e t i m e b e f o r e t h e n u c l e i become o n e body. The n e x t p h a s e i s t h e "damping p h a s e " , w h i c h i s t h e t i m e a f t e r t h e a p p r o a c h p h a s e u n t i l t h e n u c l e i r e a c h t h e d e e p e s t p o i n t f o r t h a t t r a j e c t o r y . At c h i s p o i n t t h e m o s t compact s h a p e ( l a r g e s t n e c k ) i s r e a c h e d f o r t h i s t r a j e c t o r y . The t h i r d p h a s e i s t h e " d r i f t p h a s e " , and i s t h e t i m e a f t e r t h e damping p h a s e w h i l e t h e s y s t e m i s s t i l l i n t h e one-body

c o n f i g u r a t i o n . F i n a l l y t h e r e i s t h e " e x i t p h a s e " , w h i c h i s t h e t i m e a f t e r s c i s s i o n i n t h e e x i t c h a n n e l . The breakdown of t h e t r a j e c t o r y i n t o t h e s e f o u r p h a s e s a l l o w u s t o d i s c u s s i n d e t a i l t h e d y n a m i c s of c o l l i d i n g n u c l e i .

APPROACH PHASE AND "NOSE" FORMATION

I n t h e a p p r o a c h p h a s e t h e s y s t e m f o l l o w s a R u t h e r f o r d t r a j e c t o r y u n t i l t h e n u c l e i a r e a b o u t 3fm f r o m t o u c h i n g , t h e n t h e a t t r a c t i v e n u c l e a r p r o x i m i t y f o r c e becomes i m p o r t a n t . At t h i s t i m e we b e g i n o u r d y n a m i c a l c a l c u l a t i o n w h i c h i n c l u d e s a r i g o r o u s t r e a t m e n t of e n t r a n c e c h a n n e l d e f o r m a t i o n s .

As m e n t i o n e d e a r l i e r i t was s u g g e s t e d t h a t t h e w a l l f r i c t i o n c o e f f i c i e n t i s

t o o l a r g e by a f a c t o r of a b o u t t h r e e . Our c a l c u l a t i o n s show t h a t a r e d u c t i o n i n

t h e w a l l f r i c t i o n h a s o n l y a m i n i m a l e f f e c t o n t h e d y n a m i c s of t h e a p p r o a c h p h a s e .

T h i s b e h a v i o r i s o b s e r v e d f o r a l l s y s t e m s w h i c h w e r e s t u d i e d and i s d u e t o t h e f a c t

t h a t t h e d e f o r m a t i o n d u r i n g t h e a p p r o a c h p h a s e i s s m a l l . T h e r e f o r e , i t i s a p p a r e n t

t h a t a n y d i f f e r e n c e s i n 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 d u e t o t h e u s e of d i f f e r e n t

w a l l f r i c t i o n c a n n o t b e a t t r i b u t e d t o t h e a p p r o a c h p h a s e .

nuclei form one body and the deformation degree of freedom couples strongly to the separation degree of freedom. This causes the off diagonal terms of the friction tensors to be of approximately the same magnitude as the diagonal terms. Thus, the wall friction, which opposes deformations, has a larger component which resists the relative motion.

During the damping phase a potential energy barrier must be surmounted before the system can enter the potential pocket and fuse. If the wall friction is decreased then the ability of the system to surmount the barrier and enter the pocket is enhanced. At low incident energies very little dissipation is required to make the system fuse once the pocket is entered. Thus, at low energy a reduction in wall friction leads to anenhancementof the excitation function. At high energies the ability of the system to fuse is actually decreased when the friction is reduced because the dissipation is not sufficient to trap the system in the pocket. This dynamical result is readily visible in figure 1 where the dotted curve represents results for normal wall friction and the dashed curve represents results for the reduced wall friction (the solid line is the maximum possible excitation function which can be achieved with the current model based on very general dynamical arguments).

Figure 1 - Fusion excitation functions for 2 8 ~ i + 2 8 ~ i and 4 0 ~ a + 40~a.

The solid line represents the maximum possible function as discussed in the text. The dotted line is for full friction and the dashed line is for the reduced friction. [Data are from references 12-16]

EXIT AND DRIFT PHASES

We do not explicitly discuss the damping phase and exit phase of heavy ion interactions in this paper since these phases do not affect fusion excitation functions. These phases will be discussed in the future where their importance to the deflection function in inelastic scattering will be explored.

SUMMARY

In our dynamical calculations we used the Blocki-Swiatecki shape

parametrization, with a rigorous treatment of deformations, deformation dependent

C2-278 JOURNAL DE PHYSIQUE

inertias, conservative forces and frictions in both the one- and two-body regimes.

We found that:

i) The wall friction has little effect on the formation of "noses" in the entrance channel of heavy ion collisions.

ii) The fusion excitation function is very sensitive to the strength of the one-body wall friction at high bombarding energies.

This sensitivity of the excitation function allows us to conclude that the excitation functions for Si and Ca are reproduced reasonably well with the full one-body wall friction and that decreasing the strength of the wall dissipation results in too low a cross section at high energies.

ACKNOWLEDGMENTS

We would like to acknowledge our gratitude to Professor Paul F. Yergin and the nuclear experimental group at Rensselaer Polytechnic Institute for the use of their computing facilities.

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