SADDLE-TO-SCISSION LANDSCAPE IN FISSION : EXPERIMENTS AND THEORIES

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SADDLE-TO-SCISSION LANDSCAPE IN FISSION : EXPERIMENTS AND THEORIES

M. Asghar, R. Hasse

To cite this version:

M. Asghar, R. Hasse. SADDLE-TO-SCISSION LANDSCAPE IN FISSION : EXPERI- MENTS AND THEORIES. Journal de Physique Colloques, 1984, 45 (C6), pp.C6-455-C6-462.

�10.1051/jphyscol:1984654�. �jpa-00224257�

JOURNAL DE PHYSIQUE

Colloque C 6 , suppl6ment au n06, Tome 45, juin 1984 page C6-455

SADDLE-TO-SCISSION LANDSCAPE IN FISSION : EXPERIMENTS AND THEORIES

M . Asghar and R.W. Hasse*

CEN and USTHB, B.P. 1017, Alger-Gare, A l g e r i a

* ~ n s t i t u t Law-Langevin, 156 X, 38042 Grenoble Cedex, France

Re'sum6 - On e x p l o r e l e paysage de l ' 6 n e r g i e p o t e n t i e l l e du modzle de l a g o u t t e l i q u i d e (LDM) e n t r e l e p o i n t s e l l e e t l e p o i n t s c i s s i o n . Le p o i n t s o r t i e oh l e c o l j o i g n a n t l e s fragments n a i s s a n t s e s t encore 6 p a i ~ d g f i n i t l a c o n f i g u r a t i o n de s c i s s i o n ; c e c i e s t c o n f r o n t 6 aux r e s u l t a t s exp6rimen- taux. On d i s c u t e l e s donndes expdrimentales pour montrer que l e paysage e n t r e l e p o i n t s e l l e e t l e p o i n t s c i s s i o n e s t probablement p l u t d t p l a t et:

non r a i d e c o m e p r d v o i t l e LDM o r d i n a i r e .

A b s t r a c t - The LDM s a d d l e - t o - s c i s s i o n p o t e n t i a l energy landscape i s i n v e s t i - gated. The e x i t p o i n t , where t h e neck j o i n i n g t h e n a s c e n t f r a g m e n t s i s s t i l l r a t h e r t h i c k , i d e n t i f i e s t h e s c i s s i o n c o n f i g u r a t i o n ; t h i s i s c o n f r o n t e d w i t h d i f f e r e n t experimental evidence

.

Nuclear f i s s i o n i s a r a t h e r complex p r o c e s s , because i t c o n s i s t s of a large-amplitu- de motion, where a l l t h e nucleons of t h e f i s s i o n i n g nucleus a c t i v e l y . Although much t h e o r e t i c a l and experimental p r o g r e s s has been made over t h e p a s t decade i n understanding a p a r t of t h e p o t e n t i a l energy s u r f a c e of a f i s s i o n i n g system, which shows d i f f e r e n t b a r r i e r s and minima a s a f u n c t i o n of t h e s t r e t c h i n g and necking-in deformation c o o r d i n a t e s , we know l i t t l e about t h e r e g i o n from t h e o u t e r b a r r i e r down t o s c i s s i o n . For example, we do n o t even know whether t h e poten- t i a l energy landscape i n t h i s r e g i o n determined from t h e o r d i n a r y l i q u i d drop model (LDM) r e p r e s e n t s n a t u r e o r whether i t l a c k s some s i g n i f i c a n t i n g r e d i e n t s . Obviousl) i t i s important t o t r y t o understand t h i s landscape b e f o r e going on t o t h e dynamics of t h e p r o c e s s .

1 . 2

The p o t e n t i a l energy land- scape c a l c u l a t e d with t h e LDM (which i s l i m i t e d t o s u r f a c e , s u r f a c e asymmetry and Coulomb c o n t r i b u t i o n s ) of F i g . 1 shows t h a t , s l i g h t l y beyond t h e b a r r i e r , t h e configura- t i o n of s e p a r a t e d fragment shapes i n t h e f u s i o n v d l l e y becomes much lower i n energy

t h a n t h e one of continuous shapes of t h e same e l o n g a t i o n a l o n g t h e LDM v a l l e y , b u t a l s o t h a t t h e r e i s a r i d g e s e p a r a t i n g t h e s e v a l l e y s , which a c t s as a b a r r i e r f o r

t h e nucleus when necking-in and f i s s i o n i n g i n t o two fragments

.

F i g . 1 - The LDM p o t e n t i a l energy landscape f o r 2 4 0 ~ ~ i n terms of t h e {pcm,h) parameters.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984654

C6-456 JOURNAL DE PHYSIQUE

and a t a w e l l d e f i n e d v a l u e of deformation - c a l l e d t h e e x i t p o i n t - n o t only i t d i s a p p e a r s , b u t t h e LDM v a l l e y c e a s e s t o e x i s t and t h e system becomes u n s t a b l e a g a i n s t s c i s s i o n 11-31. A t t h e e x i t p o i n t , t h e c e n t e r of mass d i s t a n c e pm of t h e two h a l v e s of t h e nucleus i s approximately

where % i s the r a d i u s of t h e f i s s i o n i n g n u c l e u s . Pcm i s p r a c t i c a l l y independent of t h e f i s s i b i l i t y parameter x . The r a d i u s d of -the neck a t t h e e x i t p o i n t i s r a t h e r t h i c k

,

I t i s i n t e r e s t i n g t o n o t e t h a t t h i s value of d i s comparable t o t h e width of t h e t a i l of t h e d e n s i t y of a heavy nucleus. Beyond t h i s p o i n t , t h e l o s s of s a t u r a t i o n proper- t y of n u c l e a r m a t t e r and a s t r o n g Coulomb r e p u l s i o n between t h e two n a s c e n t fragments r e s u l t s i n a s t r o n g necking-in f o r c e t h a t should l e a d t o a very f a s t r u p t u r e of t h e neck. Therefore, i t i s q u i t e n a t u r a l t o i d e n t i f y t h e e x i t p o i n t a s t h e p h y s i c a l s c i s s i o n c o n f i g u r a t i o n . The e x i t c o n f i g u r a t i o n i s very compact and i t e x i s t s f o r both symmetric and asymmetric modes of f i s s i o n 121.

Let us nowdiscuss t h e experimental evidences s u p p o r t i n g t h e e x i s t e n c e of t h e e x i t p o i n t and of t h e s c i s s i o n c o n f i g u r a t i o n 131.

1 . V a r i a t i o n of t h e mean f i s s i o n fragment k i n e t i c energy <Ek> a s a f u n c t i o n of t h e f i s s i o n i n g nucleus.

Since pcm a t t h e e x i t p o i n t i s almost independent of t h e f i s s i b i l i t y parameter x and i f t h e neck r u p t u r e i s f a s t , t h e n <Ek> should r e s u l t mainly from Coulomb r e p u l s i o n between t h e two h a l v e s of t h e nucleus with a c e n t e r of mass d i s t a n c e pcm

,

2 113

<Ek> = k ZF / AF , (3)

where k i s a c o n s t a n t , and AF and ZF a r e t h e mass and charge of t h e f i s s i o n i n g nucleus. An e m p i r i c a l c o r r e l a t i o n of t h i s type has been known f o r a long time 141.

I f we t a k e ro = 1.2049 fm, then k = 0.1220 a s compared t o t h e e m p i r i c a l v a l u e of k = 0.1240 found by Viola /4/ from d a t a on symmetric f i s s i o n . This c o r r e l a t i o n i n d i c a t e s t h a t a t l e a s t a major p a r t of <Ek> comes from Coulomb r e p u l s i o n and t h e p r e s c i s s i o n energy should be q u i t e s m a l l .

2. Dependence of <Ek> on e x c i t a t i o n energy of t h e f i s s i o n i n g nucleus

It i s w e l l known t h a t <Ek> changes l i t t l e and remains p r a c t i c a l l y c o n s t a n t a s a f u n c t i o n of e x c i t a t i o n energy of t h e f i s s i o n i n g nucleus 151. Most of t h e e x c i t a t i o n energy i s r e l e a s e d a s prompt n e u t r o n s . A s d i s c u s s e d above, t h e e x i t p o i n t which determines <Ek>, r e s u l t s from t h e macroscopic (LDM) e n e r g e t i c s of t h e f i s s i o n i n g nucleus. This p r o p e r t y of t h e LDM parameters s h o u l d b e i n s e n s i t i v e t o t h e e x c i t a - t i o n energy of % 50-100 MeV corresponding t o a n u c l e a r temperature T % 1-2 MeV 161.

The v a r i a t i o n of <E > w i t h e x c i t a t i o n energy s e e n e x p e r i m e n t a l l y can b e accounted f o r by t h e microscopic ( s i n g l e k article) e f f e c t s .

3. Width of a l p h a - p a r t i c l e a n g u l a r d i s t r i b u t i o n s i n t e r n a r y f i s s i o n .

A t t h e e x i t p o i n t t h e c o n f i g u r a t i o n s of t h e f i s s i o n i n g n u c l e i , s a y , from Th t o 2 5 2 ~ f a r e p r a c t i c a l l y i d e n t i c a l because t h e d i s t a n c e pcm changes o n l y by about 3 %. Any phenomenon which scans t h i s s c i s s i o n c o n f i g u r a t i o n should have s i m i l a r p r o p e r t i e s f o r a l l t h e f i s s i o n i n g n u c l e i . Therefore, a l p h a - p a r t i c l e s i n t e r n a r y f i s s i o n , which a r e b e l i e v e d t o r e s u l t from t h e sudden r u p t u r e of t h e neck j o i n i n g t h e two n a s c e n t b i n a r y fragments / 7 , 8 / , should have s i m i l a r d i s t r i b u t i o n s . The b e s t e x i s t i n g r e s u l t s show t h a t , indeed, t h e width B a , ~ (with r e s p e c t t o t h e l i g h t f i s s i o n fragment L) i s

% 19' f o r both 2 3 5 ~ ( n t h , f ) and 252Cf(s,f) 19,101. These r e s u l t s a l s o confirm t h a t t h e p r e - s c i s s i o n energy i s small : l e s s t h a n 9 MeV, f o r both systems - c o n t r a r y t o t h e non-viscid dynamical LDM c a l c u l a t i o n v a l u e s of about 25 MeV and 40 HeeV f o r 2 3 5 ~

and 2 5 2 ~ f , r e s p e c t i v e l y .

4. Variance a: of t h e n u c l e a r charge d i s t r i b u t i o n i n f i s s i o n .

The experimental r e s u l t s on 2 3 5 ~ ( n t h , f ) show t h a t t h e average v a l u e of t h e charge v a r i a n c e 0; i s about 0.35 and i t i s c o n s t a n t and independent of t h e t o t a l deformation- e x c i t a t i o n energy Etotal a v a i l a b l e t o t h e f i s s i o n i n g nucleus i n t h e range of @12 t o 40 MeV 1121. This r e s u l t was shown i n Ref. 13 t o be c o n s i s t e n t w i t h t h e z e r o p o i n t o s c i l l a t i o n of a c o l l e c t i v e i s o v e c t o r g i a n t d i p o l e resonance of t h e composite system a t t h e e x i t p o i n t . I n t h i s p i c t u r e t h e charge mode i s e x c i t e d somewhere between s a d d l e and e x i t and t h e f i s s i o n i n g nucleus moves down from s a d d l e towards s c i s s i o n s o slowly t h a t t h e charge d i s t r i b u t i o n wave f u n c t i o n i s a b l e t o a d j u s t a d i a b a t i c a l l y t o t h e i n s t a n t a n e o u s e q u i l i b r i u m v a l u e . A t t h e e x i t p o i n t , however, t h e motion of t h e neck r a d i u s becomes suddenly s o r a p i d (due t o t h e p r o c e s s of r u p t u r e ) t h a t t h e charge wave f u n c t i o n can no longer a d j u s t a d i a b a t i c a l l y and a; undergoes a ' f r e e z e - o u t ' a t t h i s p o i n t and remains c o n s t a n t t h e r e a f t e r .

S i n c e t h e s c i s s i o n c o n f i g u r a t i o n s of d i f f e r e n t f i s s i o n i n g n u c l e i a r e p r a c t i c a l l y i d e n t i c a l , t h e v a r i a n c e

05

02

5. Saddle-to-scission ( e x i t ) p o t e n t i a l energy landscape and t h e cold f i s s i o n phenomenon

We g i v e i n Table I t h e s a d d l e - t o - s c i s s i o n energy d i f f e r e n c e Ess f o r a few represen- t a t i v e n u c l e i . One n o t i c e s t h a t Ess i n c r e a s e s q u i t e r a p i d l y w i t h t h e ZF of t h e f i s s i o n i n g nucleus. S i n c e t h e e x i t p o i n t i s n o t a s t a t i o n a r y p o i n t , i t i s n o t i n - v a r i a n t under c o o r d i n a t e t r a n s f o r m a t i o n . However, s i n c e t h e c e n t e r of mass d i s t a n c e

TABLE I

Some quantities relevant for the scission energy. Tile scission energies were calculated with the Pauli-Ledergerber parameters

-

Pissioning nucleus

229 Th(nth.f) 2 3 5 ~ ( n t h . ~ ) 239

P"(hth,f) 245~m(nth,f) 252~f(s,f)

Neutron bind+=

Energy (H,, (MeV)

6.79 6.54 6.52 6.45 6.62

Outer Barrier VB (MeV)

6.5 5.53 5.07 4.3 3.6 I n n e r Barrier

V, (MeV)

6. I 5.63 5.57 5.7 5.6

Calculated s e i s -

$ i o n - Energy relative to Ground state ESo (MeV)

6.24 10.77 15.27 20.05 24.94

Scission energy relative t o s ~ d d l e ESS (me") f so + V~

12.74 16.30 20.34 24.35 28.54

C6-458 JOURNAL DE PHYSIQUE

Heavy fragment m o s s

[PHI

Fig. 2 - The fragment mass distributions for 2 2 9 ~ as a function of the heavy fragment mass for different light-fragment kinetic energy windows 1 1 7 1 .

Heavy f r a g m e n t m a s s [pH ]

Fig.3 - The fragment mass distributions for 2 3 2 ~ , 2 3 3 ~ and 23% as a function of the heavy fragmentmass for different light-fragment kinetic- energy windows 1 1 7 1 .

Pcm i s a n a t u r a l dynamical v a r i a b l e , i t should minimize t h e non-diagonal elements of t h e i n e r t i a l t e n s o r and t h e dynamics of t h e f i s s i o n process may be determined mostly by t h e p o t e n t i a l energy 1/21. I n t h e f o l l o w i n g d i s c u s s i o n , we t a k e t h i s t o be t r u e .

I n F i g u r e s 2 and 3, we show t h e experimental d a t a on mass d i s t r i b u t i o n s f o r t h e t h e r - mal-neutron induced f i s s i o n of 229Th, 2 3 2 ~ , 2 3 3 ~ and 2 3 5 ~ as a f u n c t i o n b f t h e l i g h t fragment k i n e t i c energy window 1 / 7 1 EL. These r e s u l t s show t h e e x i s t e n c e of f i s s i o n e v e n t s whose k i n e t i c e n e r g i e s Ek,max approach t h e i r s p e c i f i c Q-values w i t h i n an experimental u n c e r t a i n t y of 2-3 MeV. This i s p a r t i c u l a r l y t r u e f o r e v e n t s w i t h heavy fragment masses MH around 144 f o r 2 2 9 ~ h and 2 3 2 ~ and & around 134 f o r 23%

and 2 3 5 ~ , and 2 3 8 ~ ~ , 2 3 9 ~ ~ and 2 4 1 ~ ~ ( r e f . / 1 8 / ) n o t shown h e r e . I n Fig, 4, .we show t h e r e s u l t s of S i narbieux e t a1 I191 of

EkImaX (MH) f o r 293JJ and 23511 compared w i t h the corresponding Q ( t 4 ~ ) v a l u e s . These d a t a were o b t a i n e d w i t h a r e l a t i v e - l y simple and p u r e l y e l e c t r o n i c method t h a t helped t o s e p a r a t e t h e fragments mass by mass i n t h e h i g h fragment k i n e t i c energy r e g i o n 1201. One n o t i c e s h e r e , t o o , t h a t t h e Ek of f i s s i o n e v e n t s w i t h MH around 114 reach t h e h i g h e s t p o s s i b l e Q-values. L e t us c o n s i d e r t h e MH/ML = 1341102 f r a g m e n t a t i o n of 2 3 5 ~

( n t h , f ) . I f t h e p r e s c i s s i o n energy Epre i s small a s d i s c u s s e d above and i f one t h e r e f o r e can assume t h a t t h e fragment k i n e t i c energy r e s u l t s from Coulomb -- r e p u l s i o n o n l y , then

2

190t

\

1

E~ = E = F Z 1 Z 2 e Ip,,, (4) where F i s t h e form f a c t o r , and Z 1 and 22 a r e t h e n u c l e a r charges of the f r a g - ments. We can t r a c e t h e l o c u s of t h e f u s i o n v a l l e y (fragments s e p a r a t e d ) as a f u n c t i o n of Ek f o r t h i s f r a g m e n t a t i o n . This i s shown s c h e m a t i c a l l y i n F i g . 5 a l o n g with t h e LDM v a l l e y and t h e double humped-barrier f o r 2 3 6 ~ . One knows t h a t t h e LDM p r e d i c t s only symmetric f i s s i o n , b u t i n r e a l i t y t h e asymmetric mode i s t h e dominant mode i n t h e a c t i n i d e r e g i o n . Furthermore, a l a r g e v a r i e t y of d a t a on f i s s i o n o b s e r - - v a b l e s s u g g e s t s t r o n g l y t h a t t h e poten-.

t i a l energy of t h e f i s s i o n i n g nucleus p l a y s a d e c i s i v e r o l e i n determining t h e i r p r o b a b i l i t i e s . Hence one can assume t h a t t h e t o t a l p o t e n t i a l energy, i . e . LDM p l u s microscopic e f f e c t s , should be lower than t h e LDM v a l l e y energy a l o n e . For t h e asymmetric mode of f i s s i o n t h i s has been shown by Mustafa e t a 1 /21/ t o be t h e c a s e . A s a1 r u l e , each MH/ML fragmentation has a p o t e n t i a l energy s u r f a c e . However, we l i m i t o u r s e l v e s t o t h e LDM v a l l e y . As d i s c u s s e d b e f o r e ( p o i n t l ) , t h e mean -

hence, t h e most probable f i s s i o n f o r a given M ~ / M ~ - occurs a t t h e e x i t p o i n t , where t h e b a r r i e r between t h e two v a l l e y s d i s a p p e a r s . I n t h i s p i c t u r e ,

1

120 130 140 150

HEAVY MASS (a.m.u.1 F i g . 4

-

I F i g . 5 - The s e p a r a t e d fragment shapes i n

t h e f u s i o n v a l l e y f o r M H / ~ L = 1341102 and the double-humped b a r r i e r a l o n g w i t h t h e LDM v a l l e y . The e l o n g a t i o n c o o r d i n a t e i s t h e c e n t e r of mass d i s t a n c e p cm'

C6-460 JOURNAL DE PHYSIQUE

t h e f i s s i o n e v e n t s w i t h Ek h i g h e r t h a n <Ek> r e s u l t from a p e n e t r a t i o n (when necking- i n ) through t h e b a r r i e r between t h e two v a l l e y s 1221 ; t h i s can happen a s long a s t h e f u s i o n v a l l e y i s lower t h a n t h e LDM v a l l e y . However, a s Ek i n c r e a s e s , pcm d e c r e a s e s and t h e d i f f e r e n c e between t h e two v a l l e y s d e c r e a s e s and f o r Ek = E ~ , A (Fig. 5 ) , t h e two v a l l e y s have t h e same energy. For Ek -t Q and f o r pcm % 14.5 fm, one n o t i c e s i n Fig. 5 t h a t t h e f u s i o n v a l l e y i s AE % 10 MeV h i g h e r t h a n t h e LDM v a l l e y and y e t t h e nucleus f i s s i o n s . Even by t h e u n c e r t a i n t y r e l a t i o n t h e nucleus could borrow t h i s energy only d u r i n g A t % 6.6 x 10-23 s e c t o f i s s i o n , much s h o r t e r t h a n a n u c l e o n i c time i n a nucleus. AE i s even h i g h e r f o r h e a v i e r n u c l e i l i k e 2 3 9 ~ ~ . One can invoke an involved s o l u t i o n t o t h i s problem 1171. However, t h e s i m p l e s t s o l u t i o n may b e t o c o n s i d e r t h a t t h e p o t e n t i a l energy s u r f a c e i n t h i s r e g i o n i s r a t h e r f l a t ( t h e p o i n t s B and C a r e c l o s e t o each o t h e r ) and n o t s t e e p a s p r e d i c t e d by t h e o r d i n a r y LDM.

Hence, t h e LDM, a s used g e n e r a l l y , l a c k s some e s s e n t i a l i n g r e d i e n t s . The r e c e n t r e s u l t s of c o n s t r a i n e d HFB c a l c u l a t i o n s of Berger e t a l . / 2 3 / show a r e l a t i v e l y f l a t energy s u r f a c e i n t h i s r e g i o n - s u b s t a n t i a t i n g t h i s c o n j e c t u r e . Furthermore, when t h e c u r v a t u r e and t h e i n c o m p r e s s i b i l i t y terms a r e added t o t h e LDM, t h e landscape becomes f l a t t e r s i m i l a r t o t h e HFB r e s u l t 124,251. However, t h e b a r r i e r a l s o becomes much h i g h e r t h a n t h e experimental v a l u e . I t i s argued t h a t t h e z e r o p o i n t e n e r g i e s on top of t h i s landscape should t a k e c a r e of t h e i n c r e a s e i n t h e b a r r i e r h e i g h t . This p o i n t i s p r e s e n t l y under i n v e s t i g a t i o n . P e r t i n e n t t o t h i s c o n t e x t i s t h e c a l c u l a -

t i o n of Brack a t al.1261 with a r e a l i s t i c SKMX-force using an extended Thomas-Fermi (ETF) formalism. The SKMX i s normalised t o t h e LDM b a r r i e r of 2 4 0 ~ ~ . T h e i r land- scape beyond t h e s a d d l e p o i n t i s q u i t e s i m i l a r t o t h e LDM. Hence i t would seem t h a t t h e energy of t h e b a r r i e r determines t h e shape of t h e landscape.

With such a f l a t landscape i n mind, Ess f r e e d from s a d d l e t o s c i s s i o n should b e much l e s s t h a n given i n Table 1 . Ess may be d i v i d e d i n t o d i f f e r e n t p a r t s a s

where E c o l l r e p r e s e n t s t h e p a r t going i n t o c o l l e c t i v e modes o t h e r t h a n deformation, and E ? ~ i n t o i n t r i n s i c e x c i t a t i o n . We take Epre % 0. S i n c e , a s d i s c u s s e d above, probably only t h e minimum number of c o l l e c t i v e modes g e t e x c i t e d , a few MeV may be taken up by them. The remaining energy f r e e d w i l l end up a s E ? ~ . One can g e t some i d e a of

E F ~

2 1 - 2 0 -

y i e l d s . The amplitude o f t h i s odd-even e f f e c t

jvh+z ^:

15- I O -

- -

s u g g e s t s t h a t E ? ~ should be s m a l l e r t h a n about 127,281 6-7 MeV because some p a i r s may be bro- ken d u r i n g t h e r a p i d neck r u p t u r e 1291. These r e s u l t s seem t o i n d i c a t e t h a t t h e r e i s not much 6 1 s -

-

3 up with t h e f i s s i l i t y parameter. This may ex-

\O:;

*,." ^-

r l 5 -

0

-

12 % f o r /14,30,31/ 2 3 9 ~ u ( n ~ ~ , ~ ) .

0 5 - 2s."

> ^.7

%lo- The r e l a t i v e f l a t n e s s of t h e landscape, t h e

presence of only a r e s t r i c t e d number of c o l l e c -

:

z I D - - c o n s e r v a t i o n of t h e K-quantum number, t h e pro-

a l i c

1 5 - 11)

00 LO 2 9 l i g h t e r t o h e a v i e r n u c l e i . The a n i s o t r o p y

NEUTRON ENERGY (MeV) v a r i e s s t r o n g l y w i t h energy f o r t h e l i g h t e r elements, b u t becomes r a t h e r s t r u c t u r e l e s s f o r Fig. 6

-

2 r r ~ u

-

lo

-- -

-

-

I

j e c t i o n of t h e t o t a l a n g u l a r momentum J on t h e n u c l e a r symmetry a x i s a s evidenced by t h e f r a g - ment a n g u l a r d i s t r i b u t i o n i n low energy f i s s i o n

1321. The f i s s i o n fragment a n i s o t r o p i e s of F i g . 6 show a s y s t e m a t i c change i n t h e charac-

t e r of a n g u l a r d i s t r i b u t i o n s , when going from

b a r r i e r t o change from a n e n e r g y h i g h e r t h a n t h e i n n e r b a r r i e r f o r t h e l i g h t e r e l e - ments t o a n energy lower t h a n t h e i n n e r b a r r i e r f o r t h e h e a v i e r e l e m e n t s . The l a c k of s t r u c t u r e i n t h e a n i s o t r o p y f o r t h e h e a v i e r e l e m e n t s h a s been e x p l a i n e d o n t h e assumption t h a t t h e K v a l u e s a r e n o t conserved d u r i n g t h e p a s s a g e through t h e second minimum. It was assumed t h a t t h e t i m e o f p a s s a g e through t h e r e g i o n o f t h e second minimum i s s u f f i c i e n t l y l o n g s o t h a t t h e memory of K-values a t t h e f i r s t b a r r i e r i s l o s t and r e e s t a b l i s h e d a t t h e second b a r r i e r . When t h e second b a r r i e r i s lower t h a n t h e f i r s t o n e , a q u a s i - s t a t i s t i c a l d i s t r i b u t i o n o f K-values may p r e v a i l r e s u l t i n g i n a s t r u c t u r e l e s s a n i s o t r o p y . However, we have j u s t s e e n above t h a t t h e a m p l i t u d e o f odd-even e f f e c t s d e c r e a s e s r a p i d l y a s one goes from t h e l i g h t e r e l e m e n t s t o t h e h e a v i e r ones and i t h a s been shown t h a t p a i r s a r e n o t broken a t t h e second b a r r i e r 1311. tie t h i n k , t h e r e f o r e , t h a t a l a r g e p a r t of t h e a l i g n m e n t o f t h e f i s s i o n i n g n u c l e u s

-

However, i t s h o u l d be n o t e d t h a t t h i s s a d d l e - t o - s c i s s i o n l a n d s c a p e does n o t seem t o be a b l e t o e x p l a i n t h e fragment e n e r g y d i s t r i b u t i o n which, f o r a g i v e n MH/ML r a t i o , i s about Gaussian i n s h a p e 117,221. As d i s c u s s e d above, t h e upper h a l f of t h e Gaussian, E k > < E k > , r e s u l t s from a p e n e t r a t i o n t h r o u g h t h e b a r r i e r between t h e two v a l l e y s and a s Ek goes up, t h e pcm d e c r e a s e s and t h e b a r r i e r h e i g h t i n c r e a s e s r e s u l t - i n g i n a d e c r e a s e d p r o b a b i l i t y o f f i s s i o n . The p r o d u c t i o n p r o b a b i l i t y f o r low e n e r g y e v e n t s , Ek < <Ek>, i s r a t h e r s i m i l a r y e t t h e r e i s no b a r r i e r p r e s e n t i n t h i s c a s e . F u r t h e r m o r e , one h a s t o l o o k i n t o a s t o how t h e r e l a t i v e f l a t n e s s of t h e p o t e n t i a l l a n d s c a p e beyond t h e second b a r r i e r may i n f l u e n c e t h e ground s t a t e spontaneous f i s s i o n h a l f - l i v e s .

Acknowledgements

One of u s (R.W.H.) i s i n d e b t e d t o t h e members of t h e Commissariat aux E n e r g i e s N o u v e l l e s , A l g i e r s , f o r t h e warm h o s p i t a l i t y e x t e n d e d t o him.

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