DYNAMICS OF NUCLEAR FUSION

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Submitted on 1 Jan 1984

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J. B_ocki

To cite this version:

J. B_ocki. DYNAMICS OF NUCLEAR FUSION. Journal de Physique Colloques, 1984, 45 (C6),

pp.C6-489-C6-499. �10.1051/jphyscol:1984658�. �jpa-00224261�

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JOURNAL DE PHYSIQUE

Colloque C6, supplkment au n06, T o m e 45, juin 1984 page c6-489

D Y N A M I C S O F NUCLEAR F U S I O N

*

I n s t i t u t e o f NucZear S t u d i e s , 05-400 Zwierkk, Poland

R6sum6

-

Le calcul de la dynamique classique de processus de fusion nuclgaire basg sur le modSle 2 particules indgpendantes de dissipation est prgsente. Les sections efficaces et le temps d'interactions ont 6t6 calculgs pour diffgrentes rgactions.

Abstract - The c l a s s i c a l dynamical c a l c u l a t i o n s based on the one body d i s s i p a t i o n a r e presented. Results of the fusion c r o s s s e c t i o n s and i n t e r a c t i o n times f o r d i f f e r e n t r e a c t i o n s a r e shown.

1, Introduction

A

l a r g e amount of the experimental data concerning heavy ion c o l l i - s i o n s was cumulated during l a s t ten years. However i t seems we a r e s t i l l f a r away from t h e f u l l understanding of a l l t h e phenomena which take place i n these c o l l i s i o n e ,

As 8

very simple example we can take binary c o l l i s i o n s . Very many d i f f e r e n t models can describe them i n a r e l a t i v e l y good way, ao i t i s impossible t o decide which one of them i s more r e a l i s t i c , Moreover t h e r e a r e a l s o a l o t of fundamental questions which a r e s t i l l not answered. Let me present few of them which a r e c l o s e l y r e l a t e d t o the s u b j e c t of the present paper:

/i/ how much energy above the p o t e n t i a l b a r r i e r i s needed f o r fusion t o take place. Connected with t h i s i s another question:

i s t h e r e any acaling i n heavy ion c o l l i s i o n e ; /ii/ what a r e the i n t e r a c t i o n times

;

/iii/ what i s the main mechanism of f r i c t i o n ; is i t based on one body o r two body damping.

We would l i k e t o present here some r e s u l t s based on the c l a s s i c a l dynamical c a l c u l a t i o n s using one body type of damping and l i q u i d drop i d e a l i z a t i o n of t h e p o t e n t i a l energy. For t h e mass t e n s o r i n t h e k i n e t i c energy we u a hydrodynamlcal approximation; s p e c i f i -

'j9

c a l l y a Werner-Wheeler one. The d e t a i l s of t h e o a l c u l a t i o n s together with t h e a e s u e d shape parametrization a r e ppesented below.

2. Calculations

The idea of the c a l c u l a t i o n s i e the following: f o r a given parame- t r i z a t i o n of the nuclear ahape we solve a classical system of t h e d y ~ a m i c a l equatione of motiom. Altogether t h e r e a r e s i x degrees of

* Work done i n c o l l a b o r a t i o n w i t h H. Peldmeier, Gesellschaft f u r Schwerionenforschung, Darmstadt, West Germany and W.J, Swiqtecki, Lawrence Berkeley Laboratory, Berkeley, USA.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984658

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freedom t h r e e of them a r e degrees of freedom defining the shape of the r w l e u s aad t h r e e a r e r o t a t i o n a l variables. So we have a s e t of s i x secomd order d i f f e r e n t i a l equations which we need t o solve.

where L

XI

T - ^V

⁸

a lagrangia~a / k i n e t i c energy T milsus p o t e n t i a l energy V/, am3 gf i s a Rayleigh disaipatiom f u a c t i o ~ . Equations /I/

a r e solved above s c i s s i o a l i a e g we s t a r t from two touching spheres and f i n i s h e i t h e r a t the s c i s s i on l i n e o r a compound raucleue.

The first s t e p i n a l l the dynamical c a l c u l a t i o n 6 i a t o decide what f a m i l i e a of shapes we would l i k e t o describe aad which shape para- meters i n heavy i o n collisioms we would l i k e t take. This we have already defined is one of the previous papers 31. He chose shapes which a r e a l l a x i a l l y symmetric and correspond t o two spheres joined smoothly by a p o r t i o a of a t h i r d quadratic surface of revolution /Fig. I/.

A

convenieat s e t of t h r e e degrees of freedom d e s c r i b i n g nuclear shapes i s t h e followiagr

Distaace v a r i a b l e

t

_? - - ^R,+R, ⁵

L i *

L r

Neck variable: = Rl+R,

Fig. 1

Fig. 1 - ^A parametrization of the nuclear shapes

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Let u s w r i t e the equation f o r Y a s follows:

R: - ^r

2

^{- R i 4} ^{2 4} R, a + b r + c z 2 Z i 4 2 4 2,

R: - ^{( r -} ^sI2 ^z2< ^z ⁴ ^siR2

The c o n t i n u i t y of Y and its first d e r i v a t i v e a t t h e junction points

e

and

z2

l e a d t o the following s o l u t i o n s f o r t h e parameters of t h e a d c l e a r shape:

where u i s a s c a l i n g f a c t o r which can e a s i l y be obtained from the volume c o ~ s e r v a t i o m cowdition:

Equation f o r the s$corad degree conic surface /eq. 2/ has one solu- t i o n Y = 0, when b - ^4ac ⁼ 0. I n such a case t h e aeck i s a p a r t of a cone and t h i s happens -he=:

which corresponds t o the s c i s s i o n surface, independent of the

a sgmme t ry

_"

which i s t h e locue of comes capped with g o r t f o r s of spheres.

When A = ¹ /see

eq.

3/, c = 0, which meaaos t h a t t h e neck becomes

a paraboloid. So we have h ⁼ 1 a s the locus of paraboloids capped

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with p o r t i o r s of spheres. If i a a d d i t i o n A-0 , ^then

^b^-P

0 and

ha 1 i s i n t h i s case the l o c u s of c y l i n d e r s capped with hemispheres.

S n a l l e r sphere i s swallowed by a l a r g e r ome when s = R, - ^R2 ^i,e.

y = A

Pure i n t e r s e c t i n g spheres /z2 - ^zl ⁼ ^O/ correspond to: (?I h = F - p

which i s asymmetry indeperdeat.

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The smaller sphere i s swallowed up by the spheroidal neck when z2

^a

s + R2. !Phis l e a d s t o ;

A t y p i c a l s e c t i o n of t h e cornfiguration space a t = 0.2, say, i s presented i n Pig. 2.

SPHEROID CAPPED WITH SPHERE h = 2 - h - ~

S

(CONVEX)

(BINARY

1 2 3

P

Pig. 2. A aection of the (PJ A,A) configuratioa space a t 0 = 0.2

2 -1. Energies

a / Kiaetic energy

The e q u a t i o ~ l f o r a k i n e t i c energy can be w r i t t e n i n a followirg nays

7

where g; = (;>k3n) i s

a

s e t of our c o l l e o t i r e parameters. Rotation

of t h e system i s taken i ~ t o accourmt i n such a nay: two spheres can

r o t a t e iadependeatly with angular v e l o c i t i e s miand rA2and the whole

system r o t a t e s with angular v e l c i t y

W-L.

The mass tensor M

i s

c a l c u l a t e d ia a Weraer-Wheeler 91 approximatim of a hgdrodya&ioal

mass, I n e r t i a s of t h e two spheres I1 and I2 a r e taken a s r i g i d bodies

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t o t

and Irel = Irigid body - ^I, - ^I2 i s a i n e r t i a of t h e t o t a l system.

b /

D i s s i p a t i o n

For t h e r a t e of t h e d i s s i p a t e d c o l l e c t i v e energy we take a l i n e a r combination of w a l l amd w a l l p l u s window formulae 3/

window

+ waU

I n t h e l i m i t of t h e macroscopic dynamics c h a r a c t e r i z e d

by

t h e Chaotic Regime Dynamics one can w r i t e :

where 3 i s t h e mass d e ~ s i t y of t p e mnucleue, - V i s t h e mean speed of t h e nucleons i n t h e rnucleug aad n i s t h e normal v e l o c i t y of a n e l e m e n t d b of t h e ~ u c l e a r s u r f a c e . !he q u a n t i t y D is a normal compo- n e n t taken a t db of a d r i f t v e l o c i t y impressed on t h e nucleolsa by t h e presence of $he t r a a s l a t i o n a l and r o t a t i o n a l components i n t h e v e l o c i t y f i e l d . For window p l u s w a l l p a r t of t h e d i s s i p a t i o n we can w r i t e

:

The f i r s t two terms r e p r e s e n t t h e w a l l formula d i s s i p a t i o n s f o r two n u c l e i . The l a s t two terms 4 / a r e t h e window formulas w i t h

U r

and

UT

being t h e r a d i a l and t a n g e n t i a l components of t h e r e l a t i v e velo- c i t y . The q u a n t i t y 6' i s t h e a r e a of t h e window and \ji i s t h e time d e r i v a t i v e of t h e volume of one of t h e n u c l e i .

The formfactor c i n eq. 19 i s t h e only f r e e parameter of t h e theory.

I t d e c i d e s when one should switch from t h e d i n u c l e a r t o mononuclear regimes.

A s

a f o r m f a c t o r we chose:

I + I&)

-

where ( 3 ~ 4 - i+ p and = 1 corresponds t o our upper boundary when s m a l l e r of t h e spheres i s swallowed up completely by a neck; do and

a A a r e parameters t o be e s t a b l i s h e d . Additionally we t r i e d t o d e s c r i -

be t h e following idea: t h e window i s completely gone when i t s r a d i u s

i s equal t o t h e r a d i u s of s m a l l e r of t h e spheres. This i s e q u i v a l e n t

t o

²²

= s and t h e f ormfactor

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c / P o t e n t i a l energy

For the potelstial eaergy we t a k e a l i q u i d drop model energy, i n which t h i s energy i s a sun of surface d Coulomb terms, with para- meters taken from Myera a ~ d Swiqtecki 37. The use of a l i q u i d drop model energy l e a d s t o a f a m i l i a r kink a t two touching spheres, which r e f l e c t s t h e f a c t t h a t the nuclear a t t r a c t i v e p a r t of the energy was not taken i n t o account. Ira o r d e r t o mock up t h i s e f f e c t our s t a r t i n g point waa chosen somewhere oa the s c i s s i o n l i n e s l i g h t l y below two touching spheres. We have checked t h a t i n our overdamped dyaamics t h i s does not i n f l u e a c e the r e s u l t s . For t h e potezatial energy one can w r i t e ax expresaiomr

where B s = E s / ~ s

CO)

i s t h e r a t i o of t h e urface energy iE, of t h e shape s u r f a c e emergg €5 of the cornporurd sphere. The i s t h e same r a t i o of the e l e c t r o s t a t i c energiee f i e s i l i t g parameter.

3. R e s u l t s

A

s e t of eq. 1 was solved numericaly using a predictor-corrector method. d i r e c t i o n the change of t h e k i n e t i c energy T i a m a l l

ao i a t h i s d i r e c t i o n we decided t o use creeping l i m i t . This causes q u i t e an i n c r e a s e i n the t i n e s t e p eapecially i n t h e regions where mass tensor i n d i r e c t i a a i s big. Ia t h i s way t h e o v e r a l l time

evolution takes muoh l e s s of t h e computer time. We have checked the above approximation i n few c a s e s sad i t does not a f f e c t the f i n a l r e s u l t s .

The l a s t term i n formula /13/ irnfluence very much the r e s u l t s a s one can s e e i n Fig. 3 where we presemt t h e average f i a a l charge

Pig. 3.

A

comparison of the experimemtal poimts with the r e s u l t s

of c a l c u l a t i o a a f o r t h e mean value of t h e p r o j e c t i l e

charge number

2

vs t o t a l k i n e t i c eaergy.

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number

Z

of one of the fragments f o r d i f f e r e n t energy losses. One can s e e t h a t i n t h e experiment t h e r e is ao d r i f t toward synunetrg evela f o r very high energy losses. The broken curve i s m outoome of t h e t r a j e c t o r y c a l c u l e t i o n s where the l a s t term i n formula /13/ wars n o t included. !The curve i a completely off the experimental p o l n t s with a s t r o n g d r i f t toward symmetry.

One

should mention t h a t t h e curve s t a r t s about looMeV below the e l a s t i c peak, which r e f l e c t s t h e f a c t t h a t our s t a r t i n g point f o r the t r a j e o t o r y c a l c u l a t i o n s i s a t two touching spheres, s o i t misses e l a s t i c and q u a s i e l a @ t i c components i n the reaction, The f u l l curve correeponds t o the r e s u l b of the c a l c u l a t i o n s where t h e last terra i n formula /13/ was taken i n t o account. One can see t h a t t h i a curve follows n i c e l y t h e experi- mental points,

Let us s t a r t the discussion of t h e r e s u l t a by presenting Table 1 which shows the i n t e r a c t i o n times i.e. times i t t a k e s systems t o go from two touching spheres t o s c i s s i o n . For two combinations: A$+A2=

80+80 and 90+90 systems reach the upper boundary which meana they end up a t t h e spheroidal shapes c l o s e t o t h e tompound ephere. A s f a r a s the i s t e r a c t i o n times a r e concerned they i n c r e a s e with decreasing nears of the compound nucleus and with i n c r e a s i n g asymmetry, In co- lumns 5 and 6 of t h e Table we show t h e Coulomb repulaive energy a t s c i s s i o n and k i n e t i c t r a n s l a t i o n a l energy a t s c i s s i o n , too. In co- lumns 7 and 8vpresent the value of the f i n a l asymmetry and corres- ponding t o i t the combination of A1 and A2. From the Table one can s e e that f o r the systems which could be i n t e r e s t i n g f o r experimenta- lists the i n t e r a c t i o n times a r e c e r t a i n l y too s h o r t t o allow X-ray mea ements t o be performed a s they r e q u i r e times of the order of

i o - f v s e c .

In Fig.

f

r e present experimental 6/ fueion c r o s s s e c t i o n s f o r 2 0 8 ~ b + O T i together with the r e s u l t s of the calculations. I n expe- riments f u s i o n c r o s s sectiona were e s t a b l i s h e d by taking a l l events with a mass t r a n s f e r toward symmetry g r e a t e r than 40 nucleons. There- f o r e i n t h e c a l c u l a t i o n s f o r fusion we decided t o take a s lmx the v a l u e f o r which t h e r e i s a t r a n s f e r of 40 nucleons, In Fig. 4 t h e

FUSION CROSS SECTIONS

Fig. 4. Fusion c r o s s s e c t i o n s f o r

2 0 8 ~ b

+ 5 0 ~ i reaction,

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Table 1 r--- I--- r-,,---

I

*I I A2

I

1 E~~~~ 1 Ekin /final asym./ : At , f

C

i----,+---i---+---

---m,,----,--L---t-t--t---

80

1

80 0

t

115.221

I

/ system reached the upper boundary / 80 f 160 0 1 74.123 ' 146.19

1

7.78 .08864 , 1'151.26, 88.74/

I I

10

1

118.783 1149.12

f

7.99 -07483 ! h46.54, 93.46/

I

1 20 1 134.878 1 150.00 1 8.45 ; .07115 /145.27, 94*73/

1 I

1 50 166.206 !150.27 1 8.89 .06496 /143.12, 96.88/ 80

f

240 1 0 j 77.251

I

150.24 1 8.05 -13959 1 /223.71, 96.29/

I I

10

I

103.775 1 157.53

r

8.34

8

.I2223

I

! /216.43,103.57/

I

1 20 1 110.143 f 158.72 1 9.23 1 .I1897

7

/215.04,104.96/

I I

s 50

I

121.447 162.55 1 9.86 1 .I1488

I I

f /213.28,106.72/ 160 f 160 f 0 : 23.811 1 244.56 f 37.00 1 0 1

I

s 10

I

37.911 r 229.08

1

34.96

8

0

L I I

1 20 ' 43.415 230.13 f 32.50 1 0

I I 8 I

50 1 54.039 1 226.69 31.85 1 0 s

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C6-497

Tabl e 1 /continuation /

54

C r

4

°A r

86

K r

124

S n

100

M o

209

B i 209

M

206

pb 160

G d

92

M o

110

P d

124

S n

124

S n

124

S n

0 0 0 0 0 0 0 0 0 0 0 0 0

148.99 5 170.30 3 196.92 7 330.87 9 107.77 5 83.21 1 58.89 1 328.93 6 102.40 3 81.66 7 57.35 5 122.11 5 94.98 7

137.3 9 139.1 8 142.0 4 144.7 2 138.3 1 145.2 4 154.1 8 128.3 5 143.9 3 158.4 2 145.1 5 142.4 5

4.7 5 4.2 4 3.7 7 4.9 8 6.6 1 7.4 4 9.7 2 5.0 4 5.7 0 8.6 4 4.7 9 4.6 9

.1194 4 .1170 1 .1139 6 .0896 7 .0933 2 .0793 2 .0297 9 0 0 0 .0295 8 .0269 7

/179.59 , 87.41 / /176.04 , 86.96 / /172.30 , 86.70 / /155.40 , 90.60 / /156.65 , 89.35 / /151.79 , 94.21 / /133.97,112.03 / /119.74,100.26 / /121.04,102.96 /

I i *

syste m reache d th e uppe r boundar y

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two curves correspond t o two d i f f e r e n t the formfactor c i n the d i s s i p t i o n tensor. For c = s i n 2 we varied parame- t e r s d o a n d

dtl

t o f i t the data. This with the values

do = 0 and

O(

= 0.75. For c = (f,;,/~,)' we obtained a broken curve which i s too h i g h a s compared t o t h n t a l data. Fig, 5 p r e m n t s t h e sane f o r the r e a c t i o n ~ ~ ~ " P b P e s ~ ~ Fe. This time the same values of do and a s before give a s o l i d curve which i s lower than experimental p o i n t s and a broken curve i s again higher than these points, Summarizing i t seems possible t o have a formfactor c which i s d i f f e r e n t than t h e two used here and s t i l l w i l l reproduce q u i t e well t h e experimental data but we d i d n o t check t h i s get.

Pig. 5. Fusion c r o s s s e c t i o n s f o r 2 0 8 ~ b + 5 8 ~ e reaction.

I n Fig, 6 we present o e of t h e r e s u l t s of the c a l c u l a t i o n s with a Uranium p r o j e c t i l e on 37Al t a r g e t . For t h e value of the i n i t i a l angu- l a r momentum t i n = 30 .k one can see i n ($,A1 plane a b e a u t i f u l examp- l e of the fusion-fission t r a j e c t o r y . The t r a j e c t o r y was c a l c u l a t e d with a formfactor c i n the d i s s i p a t i o n tensor equal ( ~ w ; ) ~ ~ / R ~ 12. ^{h e}

can s e e t h a t a t f i r s t neck i s growing very f a s t and system s a r t s t o go toward compound sphere and then i t takes q u i t e some time t o rese- p a r a t e by taking advantage of the asymmetry degree of freedom, The system i s f i s s i g j i n g aymmetricaly and the o v e r a l l time of separa- t i o n i s 5 10' sea,

4. Conclusions

We have presented h e r e j u e t t h e f i r s t s t a g e of our c a l c u l a t i o n s f o r t e s t i n g t h e one body dynamics. The model is a b l e t o reproduce t h e experimental d a t a , however a t i l l f u r t h e r i n v e s t i g a t i o n s a r e required i n order t o e s t a b l i s h when and how f a e t one should switch from a di- nuclear t o , a mononuclear regime, Once t h i s i s e s t a b l i s h e d i t w i l l be possible t o reproduce fusion c r o s s s e c t i o n s f o r the r e a c t i o n s 208pb+

d i f f e r e n t t a r g e t s and 2 3 8 ~ + d i f f e r e n t t a r g e t s , It a l s o w i l l be pos-

s i b l e t o c a l c u l a t e an e x t r a and extra-extra pushes and see i f t h e r e