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FRAGMENTATION DYNAMICS IN NUCLEAR FISSION AND IN HEAVY ION COLLISIONS
Rajiv Gupta, D. Saroha, N. Malhotra
To cite this version:
Rajiv Gupta, D. Saroha, N. Malhotra. FRAGMENTATION DYNAMICS IN NUCLEAR FISSION AND IN HEAVY ION COLLISIONS. Journal de Physique Colloques, 1984, 45 (C6), pp.C6-477-C6- 488. �10.1051/jphyscol:1984657�. �jpa-00224260�
J O U R N A L DE PHYSIQUE
Colloque C6, suppl6ment a u n06, Tome 45, juin 1984 page C6-477
F R A G M E N T A T I O N D Y N A M I C S I N N U C L E A R F I S S I O N A N D I N H E A V Y I O N C O L L I S I O N S
R.K. G u p t a , D.R. S a r o h a and N. M a l h o t r a
Department of Physics, Panjab University, Chandigarh-160014, India
Rgsumg - Dans cet article, nous avons montrg que la thgorie de la fragmentation bas6e sur le modsle en couches de deux centres donne une image unifige 2. la fois de la fission et des collisi- ons des ions lourds. Dans le cas de la fission, on montre, que la dynamique se produit d un point juste aprss la barrisre de la selle. On parvient h des solutions numgriques des gquations de Schrodinger stationnaires et d6pendant du temps, dans des coordonnges de masse ou de charge d'asymgtrie. De msme, des modsles sont propos6s pour les solutions analytiques de l'gqua- tion de Schrodinger dgpendant du temps. Le mouvement relatif est trait6 de manisre classique. La thgorie est illustrge en appliquant 2. plusieurs phgnomsnes d la fois la fission et les collisions des ions lourds.
Abstract - In this review, we have shown that the Fragmentation theory based on two centre shell model gives a unified picture of both fission and heavy ion collisions. In the case of fissi- on, the dynamics are shown to occur at a point just past the saddle barrier. Numerical solutions of the stationary and the time dependent Schrodinger equations, in mass or charge asymme- try coordinates, are carried out. Also, models are advanced for the analytical solutions of time dependent SchrBdinger equation.
The relative motion is treated classically. The theory is illu- strated by applying to various phenomena in both fission and heavy ion collisions.
I - INTRODUCTION
The fragmentation theory is a unified, quantum mechanical description of the otherwise two completely different(inverse) nuclear processes:
the nuclear fission and the nucleus-nucleus collision. The unifying aspect of these processes is that in both cases the given nuclear system transforms (disintegrates) into other nuclear systems--only the way of achieving this transformation is different. In case of fission, the system exists as a single (compound) nucleus which disintegrates spontaneously or when some enerqy is added to it. On the other hand, in nucleus-nucleus collision, some kind of an intermediate system is first formed, which then leads to a stable compound nucleus or transf- orms, depending on what energy, angular momentum and the reaction partners, etc. are involved.
A complete description of both these processes is achieved in fragmen- tation theory by introducing /I-4/ two dynamical collective coordinates
(which are subject to quantization) of mass and charge asymmetries,
nA= (A1 -A2) / (A1 +A2) and nz= (Z1 -Z2) / (Z1 + Z 2 ) ( 1 ) respectively, in the asymmetric two centre shell model (ATCSM) nuclear Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984657
C6-478 J O U R N A L D E PHYSIQUE
shape. The other commonly used coordinates are: the relative distance R (or, equivalently, the length 1 1 , the deformation coordinates a,
and B,, and the neck coordinate E (see Fig. 1). One can similarly defi- ne a neutron asymmetry coordinate TIN= (NI -N2) / (N1+N2) , but only two are sufficient as dynamical
coordinates, since they are related/2/. p l = o , l b , p z = a 2 / b 2
Here, A=A1+A2, Z=Z +Z2 and N=N1+N2 ~ ~ ~ t ~ o n s
1 P r o t o n s
with Ai,Zi and Ni (i=1,2) referring to the incoming nuclei or the outgoing fragments,depending on the phenomenon under study.
The fragmentation theory is applied /I-22/ with great success in both fission' and heavy ion collisions (HIC) and is reviewed in many articles /19-22/.
In this presentation, we consider only the dynamics of the theory in a few specific examples and give the new resu- lts obtained very recently, specifically, the analytically solvable models /23-26/.
It is gratifying that today enough expe- rimental evidence is available to treat both the fission and HIC on a completly quantum mechanical basis. In fission we have the recently observed odd-even prot-
on and neutron effects /27,28/, and in / \
HIC are the phenomena of cold fragmenta-
tion / 2 9 / , cold fusion below the barrier 1 2 Z /30/ and the a-cluster transfer /31/.
All these results support the use of Fig. 1 cold potentials /6,7,15,32/ and mass
narameters. In any case, we shall also study the role of nuclear temp- erature, though it is a statistical concept.
Statistical theories are also given (in terms of the Fokker-Planck equations) for both the fission and HIC and have met with rather celebrative success /33-35/. However, in addition to many difficulties, the above noted phenomena could not be reconcilled on these theories.
I1 - THE FRAGMENTATION DYNAMICS
The collective Hamiltonian can be written as
4 . . .
H = ~ ( ~ , ~ ~ , n ~ , n ~ , R , B ~ , n ~ , n ~ ) + V(R,8irnArnZ) (2) where the potential V is obtained in Strutinsky way (based on an appro- priate liquid drop model (LDM) /37/ and the ATCSM / 5 / ) by minimizing it in B1,B2 and E. This means adiabaticity in P ,a2 and E and the dyn- amical coordinates are R and any two of the nA?q and q . The mass
parameters B . . , defining the kinetic energy part 8 in
ar are
the cons-istently cal&alated adiabatic Cranking masses. So far we have quantiz- ed the motions in mass and charge asymmetries and treated the relative motion classically. The great advantage of talking of a theory in terms of nA!nZ or nN is that these coordinates have a direct relevance to ex~erlmentally measured quantities, like the yields and cross-sections.
Two possibilities have been considered for the R-motion :(i) assuming that both the q A and q Z motions are fast compared to the R-motion, R
i s t a k e n a s a t i m e independent p a r a m e t e r ; ( i i ) t h e r e l a t i v e motion i s s o l v e d e x p l i c i t l y t h r o u g h known c l a s s i c a l e q u a t i o n s o f motion. Both c a s e s a r e d i s c u s s e d i n t h e f o l l o w i n g s u b - s e c t i o n s . Case ( i ) i s appare- n t l y s i m p l e b u t t u r n s o u t t o be r a t h e r v e r y good f o r t h e g e n e r a l desc- r i p t i o n . T h i s happens b e c a u s e t h e dynamics a r e given j u s t p a s t t h e s a d d l e b a r r i e r . T h i s r e s u l t and many o t h e r s a r e g i v e n e x p l i c i t l y by t h e a n a l y t i c a l models based on c a s e ( i i ) .
Assuming t h a t t h e c o u p l i n g between qA and n Z i s weak, i n f i r s t approx- i m a t i o n , we can c o n s i d e r t h e a u a n t i z a t l o n i n each of t h e s e c o o r d i n a t e s
e q u a t i o n
[ - - ( 3
where q=qA o r q Z . The v a l u e o f R i s f i x e d a t t h e p o s t s a d d l e p o i n t . T h e n , t h e mass o r c h a r g e d i s t r i b u t i o n y i e l d s Y ( A i ) and Y ( Z . ) a r e g i v e n , r e s ~ e c t i v e l y , by 1 $ n A 1 or 1 Ji ( n Z ) 1 a f t e r a p r o p e t s c a l i n g .
R q ~
S i n c e $R(Y) a r e t h e v i b r a t i o n a l s t a t e s i n t h e p o t e n t i a l V , i n a d d i t i o n t o v=O, h l g h e r s t a t e s w i t h v = 1 , 2 , 3 , ... c o u l d a l s o c o n t r i b u t e . We cons- i d e r t h e s e e f f e c t s throucrh a Boltzmann l i k e f u n c t i o n
I $,I =
"r,
m I C , ' ~ ' I e x p ( - ~ ~ ( ~ ) / O ) ( 4 a )where O i s t h e n u c l e a r t e m p e r a t u r e , r e l a t e d t o t h e e x c i t a t i o n e n e r g y
W e i l l u s t r a t e t h e s u c c e s s o f t h i s approach i n t h e f o l l o w i n g s u b - s e c t - i o n s . For o t h e r a p p l i c a t i o n s o f t h e method, s e e R e f s . / 1 , 2 , 8 , 1 2 / .
11. I . 1 - ODD-EVEN PXOTON EFFECTS I N FISSION OF 236i3 AND 2 5 2 ~ f Using t h e c a l c u l a t e d c h a r g e d i s p e r s i o n p o t e n t i a l s V ( q Z ) , w i t h n e u t r o n e v a p o r a t i o n i n c l u d e d / 3 8 / , and t h e Cranking mass p a r a m e t e r s BnZqZ ( n Z )
f o r h=1.8, t h e p e r c e n t a e y i e l d s a r e c a l c u l a t e d f o r v=O f o r mass
p r o d u c t s A2=97-104 o f 236U and a r e compared w i t h t h e e x p e r i m e n t a l d a t a /27/ i n F i g . 2 . A p p a r e n t l y , b o t h t h e symmet-
r i c a s w e l l a s asymmetric d i s t r i b u t i o n s a r e o b t a i n e d . The r o l e o f mass p a r a m e t e r s is
s e e n i n g i v i n g t h e f i n e s t r u c t u r e o f t h e t EKPI
y i e l d s , r e s u l t i n g i n odd-even e f f e c t . The c a l c u l a t e d mean v a l u e s
moment), i n t e r m s o f t h e i r
t h e unchanged c h a r g e d e n s i t y v a l u e ZUCD t h e n e i g h b o u r i n g even i n t e g e r v a l u e Z , and t h e w i d t h s a Z ( s q u a r e r o o t o f t h e secgnd moment) a r e a l s o found t o compare n i c e l y w i t h t h e c o r r e s p o n d i n g e x p e r i m e n t a l d a t a . '
A s a measure o f t h e odd-even c h a r g e e f f e c t , we have e s t i m a t e d : ( i ) a q u a n t i t y which g i v e s t h e d e v i a t i o n o f even-Z and odd-Z y i e l d s from a Gaussian f u n c t i o n /27,28/:
where L = l o q { ~ ( Z + n ) } ; n = 0 , 1 , 2 , 3 a n d Y ( Z + n ) 0 3 7 3 8 38 19 LO ~1 rnm6c WUIIBER z2
a r e t h e n i s o t o n i c y i e l d s m u l t i p l i e d by t h e mass y i e l d s and n o r m a l i z e d f o r e a c h i s o t o n e ;
F i g . 2
C6-480 J O U R N A L DE PHYSIQUE
(ii) the difference between the sum of even-Z and odd-Z yields:
A p = Ye - Yo
N = 5 9
Fig.3 shows our calculated 6 ( % ) for 2 3 6 ~ in good agreement with empirical data /27/. Also, our compa?es well with the experimental /27/ value A = 17.2 % ,calculated for 37<Z2<43, which
-
(23.7+0.7)%. Effect of temperature,in complete washing away of shell effects BU /36/, is shown
in reducing A p to lo%, which remains due to the fragmentation dynamics alone. The odd-even pro- ton effect due to 6 U is thus 17.2-10.0=7.2%.
N = 6 1
The role of dynamics and the existance of odd-
I I I I I I I I
- - C r o n h ~ n g M o r r e r B,, z 1 252 Cf .----.
='% nz 3 0 -
3 0
10
39 40 41 L2
2 2
Fig. 3
- even proton e cts is also clearly shown in Fig.4 for '"Cf, where an average mass
- is used. The A p reduces to zero (the form) from 16%, though Zp and o Z do not change significantly.
51 53 5 5 5 7
C H A R G E NUMBER z 2 The scission-point models /39,40/,without
or with pairing and shell effects, are Fig. 4 shown /28/ not to account for this effect.
11.1.2 - MASS FRAGMENTATION IN 2 0 8 ~ b + 5 0 ~ i AND 1 6 0 + 4 0 ~ a REACTIONS a) 2 0 8 ~ b + 5 0 ~ i Reaction: For this and many other reactions /41/, the fusion cross sections are measured at 4.8-8 MeV/A and the fused system is also observed to disintegrate, giving symmetric mass distributions.
Within fragmentation theory, we show /42/ that the fused system is formed by crossing over of the adiabatic interaction barrier which is low enough that symmetric mass fragmentat398 is obtained by the simple adiabatic fission of the compound system 104.
Fig.5 gives the cross sections vs. Ecm. We use the sharp cut-off model o = Tr R; ( 1 - VI/Ecm)
with interaction barrier VI defined as the energy required above the entrance channel frozen configuration (the conditional saddle of Swia- tecki /43/) and RI its position (see Fig.7). The agreement between the calculations and experiments is good.
Fiq.6 gives the mass distribution, obtained by solving Eq.(3) in nA,
for V ( n A ) obtained from ATCSM and the Cranking masses. Temperature
effects are included,as per Eq.(4) for fission from excited states, and in the potential through the relation /36/:
V=VLDM+ 6U exp(-EI2/0:); Oc=1.5 MeV The symmetric fragmentation is apparently qiven. The oriqin of the shoulders, seen in experiments; is not certain /41/. The nuclear shapes also indicate neck forma-
P
tions only for symmetric and nearly symm- etric fragments.Fig. 5
The above process can be understood by loo- king at the scattering potentials V(R) ,cal- culated with the angular momentum R effects (=(fi2/21) R(R+l)) included (Fig.7) . The mom-
ent of inertia g is calculated for the nec- ked shape by usinq the generating function of the ATCSM nuclear shape /44/. We notice that for R=O, the barrier exists for the interacting system to be cantured in the pocket to form a compound system. However, this barrier is low enough that the compo- und system can fission. Also, the effective
say that for R ~ R ~ ~ S the probability of forming a compound system 1s zero. Since the capture cross
- sections are measured /4l/,using the correspondi- ng sharp cut-off approximation, we calculate in
- Table I, the R carried by the compound system formed at eachCEcm:
2 3 0
barrier against fission
2 5 0 I I I I Fig. 6
Table I
l 1 5 - This means thatrat the Elab c' r c'
given E , the com~ound 4.77 192 g
- - system ggserved with the 5.00 202 29
5.25 211 43 5.50 222 62 5.90 238 71 6.50 263 89
190 - 8.00 323 117
1.20 1 3 0 1.LO 1.50 1.60 175
A -
In Fiq.7 we have shown V(R) for R-values that Fig. 7 are close to each of these Rc and also for Rcr.
258 104
decreases as R increa-
ses. For some R=R_-., the barrier vanishes and we
C6-482 JOURNAL DE PHYSIQUE
We n o t i c e t h a t f o r each o f R < R c , t h e b a r r i e r e x i s t s which i s low eno- ugh f o r t h e formed compound sysEem t o f i s s i o n , a s i s o b s e r v e d u p t o 8 MeV/A t h a t r e f e r t o Rc=117. N o t i c e t h a t i n o u r c a l c u l a t i o n o f o ( E ) , t h e b a r r i e r f o r R=O i s used. A p p a r e n t l y , f o r a b e t t e r f i t a t eachcPfcm, t h e b a r r i e r w i t h R = R c s h o u l d b e used.
b ) 1 6 0 + 4 0 ~ a R e a c t i o n : B e t t s /31/ h a s r e c e n t l y measured t h e mass s p e c t - rum o f t h i s r e a c t i o n a t 7 5 MeV, i n d i c a t i n g an a - ? a r t i c l e t r a n s f e r . Following a s u g g e s t i o n by G r e i n e r / 4 5 / , we have c a l c u l a t e d f o r t h i s system t h e p o t e n t i a l V ( n ) , i n s t e p s o f AqA=2/A, by u s i n g t h e e x p e r i - mental b i n d i n g e n e r g i e s ,Aminimized i n q Z :
V ( n A ) = (Z1Z2e 2 / R ) - B1 (A1 , Z 1 ) - B 2 ( A 2 , Z 2 )
11.2 - NUMERICAL SOLUTION OF TIME DEPENDENT SCHRODINGER EQUATION I N nA
Q u a n t i z i n g n A motion i n t h e c l a s s i c a l c o l l e c t i v e Hamiltonian
w i t h R=R1+R2. T h i s i s shown i n F i g . 8 , i n d i c a t i n g t h e minima a t a - c l u s - t e r t r a n s f e r . Using t h i s p o t e n t i a l and mass la
p a r a m e t e r s B =v ( n A ) , t h e reduced mass
n n
f o r each poss$b!ke nA-value, F i g . 9 g i v e s t h e c a l c u l a t e d mass f r a g m e n t a t i o n y i e l d a t d i f - f e r e n t 0 - v a l u e s . The g r o s s s t r u c t u r e of t h e measured y i e l d s i s o b t a i n e d , p r o v i n g t h a t 161
t h e c o l l e c t i v e a - p a r t i c l e t r a n s f e r i s g i v e n w i t h i n t h e f r a g m e n t a t i o n t h e o r y .
we g e t t h e t i m e dependent SchrBdinger e q u a t i o n
- 15
4 1 0 -
- >
*
0 : 5 -
W :
2
5 ,- z 0
B
- 5
nrss *srnncrRu T, '' 16 20 24 1 8 3'2 36 40
FRAGMENT MASS
F i g . 8 F i g . 9
I I I I I I I I I
S 18
- '3 + :",a -:g6~f W
V) V)
I
1 6 l
I
*,
%
-'
16'
- 28
- 1 0 1 1 1 1 ~ 1 ~ ~ 1
-0.4 -03 -0.2 -0.1 0 0.1 0.2 0.3 0.4
We solve this equation by the finite difference method with R(t) and pR(t) obtained by solving Hamilton's equations:
Solution of coupled Eq. (8) also give classical estimate of n(t). The initial conditions are: at t=O, R=R for collisions and R=R. for fiss- ion,with the initial mass distributyon of Gaussian function?il form. Rc is the touching radius of the two nuclei.Thus the collective mass tra- nsfer begins at R=R and continues until1 finally the two fragments separate again at R ~ R and t=T. Then, at t=T, the quantum mechanical probability is C
dW(nA,t)= $(",t) l 2 (R(t)) dog ( 9 ) A A
This m&hod29gs been applied to the collision of 238 2 3 8 ~ and 7-8 MeV/A Kr+ U. It is found that ig4the25gse of 238:+498~1 a large mass transfer is given whereas the Kr+ U reasxion a binary process, involving only a few nucleon transfer. Kr+2i'U is sh_qyn to be a neripheral collision with reaction time of the order 6x10 sec, which is characteristic of deep inelastic collisions. For more details, we refer to /8-lo/.
I11 - ANALYTICAL MODELS BASED ON FRAGMENTATION THEORY
After having calculated the V (nA) , V(n Z) and V(R) , in the ATCSM, it is found possible to functionalize these potentials and solve the time dependent problem analytically. Two such models have been worked out so far /24-26/, which are discussed in the following sub-sections.
111.1 - ANALYTICAL MODEL OF QZ COUPLED WITH R MOTION
The time de~endent Schrodinger equation in these two coordinates is H ( ~ ~ , R ) Y(nZI~,t) = ih - a at Y(nZrRlt) (10) where, on using the separation of variable method, we get
and - %2 a I a a
+ v I @(R.t) = ih @(R.t) with
Y(nZ,R,t) = $(nZlt) @(Rtt) (13)
For the nZ motion, the smoothed V (17 ) , for fixed nA and R,, can always be approximated by a harmonic oscil?ator, with minimum at nZmln:
1 min 2
V(llz) = 2 k (nz-7lz )
Then, Eq.(ll) is solved analytically /24/,for the initial condition of a very narrow Gaussian distr_ibution,which gives the deviation from the UCD-value, and the averaged Bnvn- Cranking mass. Expanding the $(qZ,t)
X X
in terms of the stationary harmonic oscillator wavefunctions,we get
JOURNAL DE PHYSIQUE
w i t h h a l f w i d t h , i n terms o f t h e i n i t i a l w i d t h Ti,
4 4
T ( t ) = Ti(l+B - ( I - B ) cos2wt)/2B 4
( 1 6 ) and t h e mean v a l u e n o r e x p l i c i t l y Z p f o r heavy and l i g h t fragments:
z p r
ZP = - 2 I z ( I * (nzmin- (nzmin-nzUCD) coswt) 1 (1 7) w i t h 8 = (riwB / f r ) I I 2 and w = (k/g ) 1 / 2
"zqz 'Iznz
For t h e r e l a t i v e motion, g i v e n by E q . ( 1 2 ) , t h e t o p p a r t o f t h e b a r r i e r , w i t h VB and R d e f i n i n g i t s h e i g h t and p o s i t i o n , can b e approximated by an I n v e r t e 8 harmonic o s c i l l a t o r p o t e n t i a l
V ( R ) = VB - 7 1 K (R-RB) 2
f o r RSR. ( 1 9 )
f o r which H i l l and Wheeler /46/ have g i v e n a t r a n s m i s s i o n p r o b a b i l i t y
I @ ( R l t ) I = l / { l ~ x ~ [ - z T ( v ( R ~ ) - v ~ ) / ~ o ~ ] 1 ( 2 0 ) where w R = J ~ T , w i t h y a s t h e reduced mass.
The t o t a l c h a r g e d i s t r i b u t i o n y i e l d , a t any t i m e t , i s t h e n
For d e t e r m i n i n g t h e f i s s i o n t i m e t = T from R . t o R=R , t h e s c i s s i o n p o i n t , we approximate V ( R ) f o r R3Ri by an e 4 u a t i o n g? s t r a i g h t l i n e w i t h s l o p e s . We o b t a i n ,
The c o n d i t i o n o f c o n s e r v a t i o n o f e n e r g y a t t h e t o p o f t h e b a r r i e r
i s u s e d , which g i v e s t h e i n i t i a l v e l o c i t y v . = k ( t = O ) . A p p l i c a t i o n o f
t h e model i s made
[i5$ 3H o f i s s i o n U . F i q s . 1 0 and 11 show t h e f i t t o V ( n Z ) and V ( R ) , r e s p e c t i v e - l y , f o r d i f f e r e n t 'I v a l u e s . Using tfie c o r r e s p o n d i n g p o s i t i o n s o f R B I R and R , a s i % i c a t e d i i n F i g . 11, and p=56.76
( i n u n i t s o f p r o t o n m a s s ) , t h e f i s s i o n t i m e s T f o r d i f f e r e n t q Z = 0 . 1 5 0 9 , ~ 0 . 1 7 2 9 , 0.1949, 0.2174 and 0.2391 a r e , r e . s g e c t i v e l y , 4.27
Charge Asymmetry
, I I
Y) 6 55.2 59.8
Charge Number Z H
F i g . 1 0
~ ~ ~ ~ ~ ~ ~ ~ ' ~ ~ ~ ~ ~ ~ ~ ' ~ " ' ~ " ~ ~ ' ~ ~
10%- ErO.!'Y9 ----. a o r L
2032 -
20% -
10%
2018
1011-
1021
2011
1020- 2018 -
X11e -
2011 -
ZOU ll.I .I .I, I, I. I. I. I. I.I .
5 6 7 8 9 10 11 13 14 15
RYm)
F i g . 1 I
4.23,4.22,4.20 and 4 . 2 6 x 1 0 - ~ ~ sec. The ,oo?
final charge dis~ersion yields, calculat- . 296"
ed by using Eq. (21), for t=T(q ) are plotted in Fig. I2 (solid lines?. The charge distribution without barrier pene- tration is also calculated for the avera- ge T = 4 . 2 3 6 ~ 1 0 - ~ ~ sec., by using Eq. (15) .
This is of a Gaussian functional form, shown as dotted line in Fig.12, since only smoothed potential V(11 Z) and the averaged mass parameter is used. In other words, the odd-even proton effects due to both the shell correction and structure in masses, is reduced to zero.
The experimental data is from Ref./38/, for A1=141 Ind 142. The model is shown to fit the data nicely,with the barrier penetration making the charge distribu- tion non-Gaussian. This gives arise to odd-even proton effects: 6 (Z=54.5) =
29.04% and 6 (2=55.5) =30.98% , which are
apparently due to the q and R coupling 1 2 53 I 54 I 5 5 I 56 I 57 I
alone. The mean value Zp and width o Z Z H are 54.83, 0.81 and 54.89, 0.75, respec-
tively, for the Gaussian and non- Gauss- Fig. I2 ian distributions.
111.2 - ANALYTICAL MODEL OF COUPLED PROTON AND NEUTRON ASYMMETRIES Usinq the
Hamiltonian for the H ( Q ~ ~ ' ~ ~ ) = -
with k =B w and k =B w 2
z nZqZ z q ~ Q ~
The last term gives the coupling of two harmonic oscillators withIgl&l.
The cou~ling in kinetic energy is neglected, since B =O.The harmon- ic oscillator approximation is good for the notentia?~'~calculated in LDM, for which / g / = l . This stresses the relevance of an exact treatme- nt of the couplina term, compared to any perturbation method /47/.
The time dependent Schrodinger equation and the initial conditions are /26/:
H(nzf~N)'f'(nzl~Nf t) = i* a Y(nz,qNrt) (26)
with
Here 'zof 'Nof o ~ o f o ~ o f a ~ o ~ o refer to initial asymmetries, variances and covariance. For obtaining y (qZ ,qN, t) , the Hamiltonian (24) is first transformed to normal coordinates QZ and ON, by introducing a transfo- rmation matrix, such that H(QZ,QN) 1s a sum of two uncoupled oscilla- tors :
C6-486 JOURNAL DE PHYSIQUE
where 2 2 2
Qf = w +a z and QN = .,-a with 1 2 2 2 2 2 2 2 2
a = -iJ;w -w ) +4g uZuN -u +w 3 for w >u
2 Z N Z N Z N
Since $(Q,t) for a single oscillator is known, in terms of its Green s function solution, the corresponding Y(Q ,Q ,t) can be calculated by using a product of the Green functions o? tRe two uncoupled oscillat- ors. A lengthy but straightforward algebra gives the probability density, which on transforming back to original coordinates is
6 2 2 2 2 2
= ( F ) 'I2 exp[-26{oN(nZ-TZ)-2azN('7Z-%Z) (rlN-TN)+uZ (qN-TN) 11
with 2 2 4 -1
F =I4 (oZaN-azN) 3 (30)
where the time dependence of the mean values TZ and 5 the variances o2 and a2 and the covariance a2 is obtained explicity; /26/. Knowing pfnZ ! nN , p) , the conditional orgk?abilities are also calculated. For detalls, see Ref./26/.
Applications of this model are made to the reactions I2'xe on 6 ~ n and 124Sn1 for 0; =O and the LDM approximation for V('7 z) and V(rlN) .
Fig.13 gives the Pegult of the calculation in comparison with the experimental data /48/. xZ =a2 /aZaN is
the correlation coef f icienp. ?!Re scale -1110 =..cI of El ss/time = ~ O X I O ~ ~ M ~ V / S ~ C . and
a2 /aq is fixed by an extrapolation of No Zo
the data to E1oss=O. The calculated results are very satisfactory for the underestlmaged. However, a$ =O is ratios oi/a2 though the correlations are
Gross Horfmonn
considered, though the data0 O
indicate non-zero value as well. Also, the shell effects are important. The resent fits are comparable with those 0 2
of Gross and Hartmann /33/ for the
statistical theory. Thus, the saturation
of charge widths in the initial stages,
-
EL.,, IYVIwithin EIQss of 80MeV, is shown to be obtained In the simple model of cou~led
charge and neutron asymmetries, without Fig. I3 introducing any additional mechanism
for dampinq.
However, for any dissipative nhenomenon, a mechanism for loosing the energy has to be considered. In the above quantum mechanical model, it is shown to be required for larue reaction times, giving an exten- ded saturation effect. In view of this result, a model has been work- ed out /49/, describing two collective degrees of freedom interacting with an environment representing the intrinsic degrees of freedom. The environment is considered to be in a pure quantum state or in statis- tical equilibrium and a perturbation series expansion of the coordin- ate operator is made. It is found that, under certain conditions, the
~erturbation series can be summed up to all orders, thereby yielding an exact solution. This exact solution gives an exponential decay of
the mean values, interpreted as damping due to friction. Since no app- lication to any experimental data is as yet made, we do not go into the details of this model here. Such models (upto second order pertur- bations) for a statistically described environment, based on the werturbed density matrix formulation, have been worked out before by Hofmann and Siemens /50/ and by Hasse /51/.
IV - CONCLUSION
We have shown that the fragmentation dynamics, based on the two centre shell model picture, plays a rather significant role in the descript- ion of both the nuclear fission and heavy ion collision processes.
This work is supported by the University Grants Commission, New Delhi, India.
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