A PROPOSAL FOR A UNIFIED DYNAMICAL MODEL FOR FISSION, FUSION AND DAMPED COLLISIONS

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A PROPOSAL FOR A UNIFIED DYNAMICAL MODEL FOR FISSION, FUSION AND DAMPED

COLLISIONS

D. Sperber

To cite this version:

D. Sperber. A PROPOSAL FOR A UNIFIED DYNAMICAL MODEL FOR FISSION, FUSION AND DAMPED COLLISIONS. Journal de Physique Colloques, 1984, 45 (C6), pp.C6-415-C6-424.

�10.1051/jphyscol:1984650�. �jpa-00224252�

JOURNAL DE PHYSIQUE

Colloque C6, suppl6ment a u n06, Tome 45, juin 1984 page C6-415

A PROPOSAL FOR A UNIFIED DYNAMICAL MODEL FOR FISSION, FUSION AND DAMPED COLLISIONS

D . Sperber

Department of Physics, RensseZaer PoZytechnic Institute, Troy, fiY 12281, U.S.A.

R6sum6

-

A b s t r a c t - A u n i f i e d c l a s s i c a l dynamical model f o r f i s s i o n , f u s i o n and damped c o l l i s i o n s i s developed. The e q u a t i o n s of motion a r e s o l v e d i n - c l u d i n g d e f o r m a t i o n s , i n e r t i a s , c o n s e r v a t i v e and d i s s i p a t i v e f o r c e s .

I. INTRODUCTION

We propose a u n i f i e d c l a s s i c a l dynamical model f o r f i s s i o n , f u s i o n and damped heavy i o n c o l l i s i o n s . We e v a l u a t e t h e time dependence of t h e t r a j e c t o r y dynamically. The d e g r e e s of freedom d e s c r i b i n g t h e t r a j e c t o r y , i n c l u d e i n a d d i t i o n t o r e l a t i v e motion c o l l e c t i v e d e g r e e s of freedom d e s c r i b i n g deformations. To demonstrate t h e v a l i d i t y o f t h e model we f i r s t l i m i t o u r d i s c u s s i o n t o head on c o l l i s i o n s between i d e n t i c a l i o n s . T h i s a l l o w s u s t o s t u d y t h e time e v o l u t i o n of t h e most prominent d e g r e e of freedom, t h e neck. We s o l v e t h e Euler-Lagrange e q u a t i o n s w i t h d i s s i p a t i v e f o r c e s . The i n e r t i a s a r e c a l c u l a t e d u s i n g a hydrodynamical model. Both Coulomb and n u c l e a r f o r c e s c o n t r i b u t e t o t h e c o n s e r v a t i v e f o r c e . One body d i s s i p a t i o n i s assumed and c o n t r i b u t i o n s due t o b o t h w a l l and window f r i c t i o n a r e i n c l u d e d . We c a l c u l a t e a l l i n e r t i a s and f o r c e s from f i r s t p r i n c i p l e s , and no a d j u s t a b l e parameters a r e i n t r o d u c e d . T h e r e f o r e , an u l t i m a t e comparison w i t h experiment tests t h e l i m i t s of t h e v a l i d i t y of t h e c l a s s i c a l approximations.

To be s u r e t h e r e were o t h e r a t t e m p t s t o p r e s e n t a c l a s s i c a l dynamical model f o r heavy i o n c o l l i s i o n s which a l l o w s f o r d e f o r m a t i o n s /1,2/. However, t h e s e models s u f f e r e d from c o n s i d e r a b l e r e s t r i c t i o n s . Some models depended on r a t h e r s t r o n g a s s u m p t i o n s a s t o what o c c u r s when t h e i o n s touch. I n o t h e r words, t h e r e were a number of a d j u s t a b l e parameters t o be f i t t e d . I n o t h e r models some of t h e i n e r t i a s were n e g l e c t e d l e a d i n g t o e r r o n e o u s r e s u l t s a s d i s c u s s e d l a t e r .

The p r e s e n t model t a k e s a d v a n t a g e of a p a r a m e t e r i z a t i o n which u s e s t h e same a n a l y t i c form f o r two s e p a r a t e i o n s , o r a m u l t i p l y connected system (MCS) and one d i n u c l e a r s y s t e m o r a s i n g l y connected system (SCS). T h i s f a c i l i t a t e s t h e dynamical c a l c u l a - t i o n .

At p r e s e n t we focus on heavy i o n c o l l i s i o n s . The model is e q u a l l y a p p l i c a b l e t o f i s s i o n . Heavy i o n r e a c t i o n s and f i s s i o n cannot be d e s c r i b e d a s time i n v e r s e of one a n o t h e r . Due t o d i s s i p a t i o n t h e e q u a t i o n s of motion a r e n o t i n v a r i a n t under time r e v e r s a l and t h e r e i s an asymmetry between t h e e n t r a n c e and e x i t channel. i n a heavy i o n c o l l i s i o n , and f i s s i o n cannot be c o n s i d e r e d a s t h e time r e v e r s a l heavy i o n c o l l i s i o n .

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984650

'26-416 JOURNAL DE PHYSIQUE

In the present communication we don't dwell on a detailed comparison with experiment to be reported later. Rather, we focus on the mechanism of the interaction between two heavy ions. We show that this first calculation, limited in scope, but general in nature, can explain many of the properties of heavy ion collision at energies of a few MeV per nucleon.

11. PARAMETERIZATION

As mentioned, the present treatment is hydrodynamical. To facilitate the treatment it is customary to represent the shapes of the approaching ions by a finite number of parameters, describing their shapes. One assumes that the collective degrees of freedom associated with these shape parameters correspond to the lowest normal modes of the system. The most important degrees of freedom in heavy ion collisions and fission are (a) the distance between the center of the fragments, (b) the neck and (c) mass asymmetry. As two limiting cases one can consider two parameterizations suggested by Sobel et al. / I / and by Blocki and Swiatecki. The first parameteriza- tion maximizes the role of the nuclear force, the second may minimize it. In the present we use the Blocki Swiatecki parameterization, which is very useful when the system has axial symmetry. In our case the ions are "chopped" spheres connected smoothly to hyperholoids of revolution. In the case of a singly connected system the two spheres are connected by hyperholoid of revolution of one sheet. For a multiply connected system it is a hyperholoid of two sheets. In both cases the total volume is kept constant. Let z be the axis of axial symmetry. For a SCS the equation of the shape is given by

Here a ₌ 1 _{[ l} - (1 - ^A) P I 11 -

-

p(l-~)]

(R + R )

For a MCS the equations describing the shapes are

2 2 2

Y = R2 - (2 - ^Y) z 2 6 z 6 R 2 - r Here

a 1 = [-(-)PI [ ( I + A? I

(R, + R2l2

and

&2

+ ,

^-

^-

In our approach p, A and A are the collective variables. We, therefore, express all dynamical quantities in terms of p, A, A, 5 ,

. .

111. INERTIAS

The inertias were calculated using a hydrodynamical model. We approximate the inertias for incompressible irrotational flow using the Werner-Wheeler method.

The Werner-Wheeler approximation is based on the assumption mass points which are parts of a disk will remain parts of the same disk at all times although the disk may change in radius and move along the axis. This approximation has been mis- understood in the literature /3/ and the resulting inertias are not always correct.

C6-4 18 JOURNAL DE PHYSIQUE

Let the local fluid velocity be 5 ^then

and T = q 1 pm

I

aur auz

- - = - =

According to the Werner Wheeler approximation - ^a+ - ar a+ O' Uz = Uz(z)

Due to incompressibility div $ = 0 or

Let the surface of revolution be defined by y = y (q,z) [q stands for the collective degrees of freedom p and A 1

The functions +(z) and Bi(z) can be determined and an

Z mar

Ai(z) = 2 2 dzcy(q,z') - ^{a y (q,zc)}

Y (992) = , aqi

a a

Bi = - 1 ^{(Ai(.) Y (q.2) Y ('1.') (lob)}

This allows us to determine the components of the inertia tensor Tij (q) such that

IV. CONSERVATIVE FORCES

The Conservative Coulomb force is calculated numerically by dividing the system into thin cylinders. This force is obtained explicitly as a function of p and A.

The other conservative force is the nuclear force. There are two contributions to the nuclear potential energy. One term which can be considered as self energy is proportional to total surface of both ions or the dinuclear systems and depends explicitly on the collective variables. The interaction term is due to differences of the two interacting systems. Such an interaction can be best described using the proximity formalism. The proximity method yields simple equation for the nuclear potential and force for two interacting frozen spheres. However, in our case the approaching ions facing one another are not spherical. We therefore had to integrate the interaction energy per unit area e(s) explicitly to obtain the proximity potential Vp as

Vp = y e(s) cos B(s) do gap

Here y is the surface tension constant and we used for e(s) e(s) = - 1.0 + 0.188912222s2 otsc 1.2310668

e(s) = - 6.1445 exp (- s/0.7176) 2.74tsc8 V. DISSIPATIVE FORCES

In addition to the conservative forces there are dissipative forces. In the present treatment we assume one body dissipation. We include contributions from both wall and window frictions.

The wall formula accounts for dissipation within a singly connected domain due to shape deformation. The nuclear system is treated as a gas of non interacting particles constrained to a container with perfectly elastic moving walls.

The energy loss due to collisions with the wall can be written to lowest order as

pm = mass density <V> = average speed of nucleons W = normal speed of surface in rest frame of gas

n = normal speed of surface

To determine W we need first the velocity of the collective motion of the gas: the

"drift" velocity. A somewhat lengthy calculation leads to

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And in the present parameterization

-

Similarly to the inertias, the energy loss due to collisions with the walls reduces t 0

The previous equations allow us to determine the elements of the friction tensor Fij (q).

Randrup /5/ showed that in addition to wall friction there is one body dissipation due to the exchange of particle between the approaching ions. This dissipation is significant when the ions are close to one anocher. The energy loss can be

determined in a similar way to the proximity force hence this friction is sometimes referred to as proxf-mity friction.

VI. EQUATIONS OF MOTION

So far we showed how all the elements entering the equation of motion are

determined. We now comment briefly on the structure of the equations of motion and their solution.

Here q stands for p and A. We replace the two second order equations by four first order equations introducing pi = qi and thus

Gi

VII. RESULTS AND CONCLUSION

We show that the lighter systems fuse and the heavier systems do not. For the heavier systems we observe that the larger the bombarding energy, the larger the damping. Also, for higher bombarding energies more compact configurations are reached for a short period of time. We show that there is a considerable asymmetry between the entrance and exit channel. Since the system passes through different shapes in the exit and entrance channel, the functional relation between p A is different in the entrance and exit channel. The conservative nuclear potential is different in the two channels. This has been suggested long ago using phenomenological arguments 1 6 1 . The energy losses and interaction times are in excellent agreement with what one expects. We find that most of the energy is dissipated via the wall friction and only a smaller fraction via the window

friction. This means that the collective states can be considered as doorway states which finally are damped out. The collective states are populated due to the strong coupling between the A and p degrees of freedom. We find that the results are sensitive to the choice of inertial parameters. In particular, the inertial para- meters are considerably near the scission point. It turns out that calculations with inertial parameters twice as big or half as big yield very unreasonable results.

Figure 1

(A) Shapes of 4 0 (top) ~ ~and (B) 2 3 8 ~ as Function of Time (bottom)

0

132 MeV

00

a

(3

103 MeV

0

( 7

a

00

00

C3000000

~ ~ ~ 0 0

a

88 MeV

00

a

00

00 0

00

m o o

( 3 0 0 0 0 0

59 MeV

684 MeV

~ ~ 00 ( = 3 0 0

~ ~ ~ 0 0

000

I 0 0,o 0,o ⁰

00 0

0000 0

1539 MeV 1197 MeV 1026 MeV

JOURNAL DE _PHYSIQUE

i

WALL D I S S I P A T I O N ; U

. . . - . . .

_ _ - - - - _

- 1539 M e V . . 1197 M e V -- - 1026 M e V . - . 664 M e V

r

0.0 0.4 0.8 1.2 1.6 2.0 2.4

S E C O N D S / C ( ~

lo-=)

F i g u r e 2

Wall d i s s i p a t i o n f o r c o l l i d i n g 2 3 8 ~ s a s a f u n c t i o n of t i

0.0

0.0 0.4 0.8 1.2 1.6 2.0 2.4

SECONDS/C(x 10-9

F i g u r e 3

IJindow d i s s i p a t i o n f o r c o l l i d i n g 2 3 8 ~ a s a f u n c t i o n of time.

2

WINDOW D I S S I P A T I O N ; U

-

. . ¹¹⁹⁷ M e V

-

~ ~ _ . ~ _ _ . . _ _ . _ . . . _ . _ . _ . . . _ _

C_^___--___---

1

-1.2

0.0 0.4 0.8 1.2 1.6 2.0 2.4

SECONDS/C(x

F i g u r e 4

Proximity p o t e n t i a l a s a f u n c t i o n of time f o r a s t r o n g l y damped c o l l i s i o n between 2 3 8 ~ i o n s .

4

PROXIMITY ENERGY; U ^2a8 . . .

-

'1 +

1.1 1.3 1.4 1.6 i.7 1.9 2.0

RHO

J.

F i g u r e 5

The c o l l e c t i v e v a r i a b l e A a s a f u n c t i o n of p a l o n g a t r a j e c t o r y f o r c o l l i d i n g 130xe i o n s .

C6-424 JOURNAL DE PHYSIQUE

C o n t r a r y t o what is b e l i e v e d t h e motion i s not overdamped a l l t h e time. I n f a c t , i n t h e e n t r a n c e c h a n n e l which i s v e r y i m p o r t a n t , t h e motion i s n o t overdamped. I n t h e e x i t c h a n n e l t h e motion is overdamped. The p r e s e n t r e s u l t s g i v e us c o n f i d e n c e i n t h e method t o s t u d y p e r i p h e r a l c o l l i s i o n s as w e l l as c o l l i s i o n s between non i d e n t i c a l i o n s and f i s s i o n . U l t i m a t e l y s t a t i s t i c a l e f f e c t s w i l l be i n t r o d u c e d by i n c l u d i n g t h e t h e r m a l f l u c t u a t i o n s s t u d i e d by means of a Fokker-Planck e q u a t i o n . When a l l t h i s i s done one w i l l be a b l e t o d e t e r m i n e t h e l i m i t a t i o n of t h e c l a s s i c a l model.

R e f e r e n c e s

/1/ S.K. Samaddar, M.I. S o b e l , J.N. De, S.1 A. Garpman, D. S p e r b e r , M. Z i e l i n s k a - P f a b e and S. H o l l e r , Nucl. Phys. A332 (1979), 210.

/ 2 / S.K. Samaddar, D. S p e r b e r , M. Z i e l i n s k a - P f a h e , M.I. Sobel and S.I.A. Garpman, Phys. Rev. C23 (1981), 760.

/ 3 / I. Kelson, Phys. Rev. 136 (19641, B1667.

/ 4 / J. B l o c k i , J. Randrup, W.J. S w i a t e c k i and C.F. Tsang, Ann. Phys (N.Y.) 105

(1977), 427.

/ 5 / J. Randrup, Ann. Phys. (NY) 2 (19781, 356.

/ 6 / J.N. D e , Phys. L e t t . 66B (19771, 1977.