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FLUCTUATIONS IN THE NUCLEAR DYNAMICS OF FISSION
U. Brosa
To cite this version:
U. Brosa. FLUCTUATIONS IN THE NUCLEAR DYNAMICS OF FISSION. Journal de Physique Colloques, 1984, 45 (C6), pp.C6-473-C6-475. �10.1051/jphyscol:1984656�. �jpa-00224259�
JOURNAL DE PHYSIQUE
Colloque C6, supplkrnent a u n06, Tome 45, juin 1984 page C6-473
FLUCTUATIONS IN THE NUCLEAR D Y N A M I C S OF FISSION
U. Brosa
Fachbereich Physik, PhiZipps-Universitat, Renthof 6 , 0-3550 Marburg, F.R. G.
Rdsum6 - On d 6 c r i t l e s premiers pas v e r s une t h d o r i e dynamique de r u p t u r e a l d a t o i r e du c o l . On u t i l i s e l a c o u r b u r e g a u s s i e n n e a f i n d'exa- miner l e comportement dynamique du noyau en c o u r s d e f i s s i o n .
A b s t r a c t - F i r s t s t e p s towards a dynamical t h e o r y of random neck r u p t u r e a r e d e s c r i b e d . Gaussian c u r v a t u r e i s used t o examine t h e dynamical behaviour of t h e f i s s i o n i n g n u c l e u s .
The p e r h a p s most i m p o r t a n t f i n d i n g from s o many e x p e r i m e n t s on f i s s i o n and d e e p - i n e l a s t i c r e a c t i o n s i s t h e d e f i n i t e f a i l u r e o f s t a t i s t i c a l e q u i l i b r i u m and t r a n s p o r t models / 1 , 2 / . T h i s s t i m u l a t e s hope t h a t n u c l e a r p h y s i c s might o f f e r more t h a n r e f o r m u l a t i o n s o f i d e a s which have been developed f o r o t h e r b r a n c h e s of p h y s i c s . An e x t r a o r d i n a r y c h a r a c t e r i s t i c o f n u c l e a r p h y s i c s a r e t h e enormous f l u c t u a t i o n s which a r e a s l a r g e a s t h e mean v a l u e s ; it i s i m p o s s i b l e t o t r e a t t h e s e f l u c t u a t i o n s a s s m a l l p e r t u r b a t i o n s .
A p r i n c i p l e of o u r approach i s t o c o n s i d e r n u c l e a r f i s s i o n and t h e d e e p e s t i n e l a s t i c c o l l i s i o n s a s s i m i l a r p r o c e s s e s where t h e same mechanism a p p l i e s . T h i s mechanism i s random neck r u p t u r e , i . e . p r i o r t o s c i s s i o n t h e n u c l e u s forms a massive neck, which f i n a l l y r u p t u r e s . The measured f l u c t u a t i o n s a r e caused by v a r i a t i o n s of t h e r u p t u r e p o i n t . D e s p i t e of t h e s i m p l i c i t y o f i t s p r e s e n t form, random neck r u p t u r e e x p l a i n s t h r e e t y p e s of h i t h e r t o p o o r l y u n d e r s t o o d d a t a : a ) Nuclear f i s s i o n measurements show t h a t heavy f r a g m e n t s e v a p o r a t e
d i s p r o p o r t i o n a t e l y many n u c l e o n s / I / .
b ) For t h e s m a l l e s t measured t o t a l k i n e t i c e n e r g i e s t h e r e i s an ex- t e n d e d V i o l a s y s t e m a t i c s which e x c e e d s t h e r a n g e o f v a l i d i t y o f t h e f a m i l i a r s y s t e m a t i c s by a f a c t o r f i v e / 2 / .
C ) Large v a r i a n c e s of t h e mass and c h a r g e d i s t r i b u t i o n a r e c o n n e c t e d w i t h v e r y l i t t l e d r i f t / 2 / .
I want t o g i v e h e r e f i r s t i n d i c a t i o n s on t h e t r e a t m e n t of t h e dynamics.
The problem i s t o f i n d t h e r e a s o n why s m a l l t h e r m a l o r q u a n t a 1 n o i s e may i n d u c e t h o s e l a r g e f l u c t u a t i o n s of t h e r u p t u r e p o i n t ; one needs something l i k e an a m p l i f i e r o f f l u c t u a t i o n s .
I f we c o n s i d e r t h e n u c l e u s a s a mechanical system, t h e dynamics comes o u t a s a consequence o f t h e s t r u c t u r e o f p o t e n t i a l and k i n e t i c energy.
I t t u r n s o u t t h a t t h e p o t e n t i a l e n e r g y r a t h e r r e s t r i c t s f l u c t u a t i o n s , b u t t h e k i n e t i c e n e r g y d e s t a b i l i z e s t h e p o s i t i o n of t h e r u p t u r e p o i n t .
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984656
C6-474 JOURNAL DE PHYSIQUE
The concept of s t a b i l i t y we need h e r e i s k i n e m a t i c a l s t a b i l i t y : A t r a j e c t o r y x ( t ) (which i n g e n e r a l i s more t h a n a p o i n t ) i s s a i d t o be s t a b l e i f a n o t h e r t r a j e c t o r y % ( t ) remains t o be a neighbour f o r a l l t i m e s when it was a neighbour t r a j e c t o r y a t some i n i t i a l time. I n o t h e r words: Adiverging bundle of t r a j e c t o r i e s s i g n a l i z e s k i n e m a t i c a l i n s t a b i l i t y . Can one decide about k i n e m a t i c a l i n s t a b i l i t y without computing such bundles? The f a m i l i a r procedure i s t o c o n s i d e r n o t k i n e m a t i c a l b u t only s t a t i c s t a b i l i t y ( i . e . s t a b i l i t y i n t h e v i c i n i t y of a p o i n t ) , t o compute a p o t e n t i a l energy s u r f a c e , and t o look f o r i t s extrema. But t h i s does n o t s u f f i c e f o r t h e p r e s e n t problem.
Rather one has t o f i n d some measure of s t a b i l i t y which i n c l u d e s t h e k i n e t i c energy. This measure i s t h e Gaussian c u r v a t u r e , well-known i n d i f f e r e n t i a l geometry / 3 / .
Suppose we have a mechanical system with two degrees of freedom u and v , with p o t e n t i a l V ( u , v ) , k i n e t i c energy 2T=e ( u , v ) fi2 +2f ( u r v ) fi$+
g ( u , v ) C 2 and t o t a l energy H. Then one i d e n t i f i e s E ( u , v ) : = ( H - ~ ( u , v ) ) e ( u , v )
F.(u,v) := (H-V(u,v))f ( u , v ) ( 1 )
G ( U , V ) := ( H - V ( U , V ~ ~ ( U , V )
where E l F, and G denote Gaussr fundamental elements of t h e m e t r i c dsZ=Edu2+2Fdudv+Gdv2. From t h i s t h e Gaussian c u r v a t u r e K f o l l o w s a s
where s u b s c r i p t s a r e meant a s p a r t i a l d e r i v a t i v e s / 4 / . The mechanical importance of K i s expressed by J a c o b i ' s v a r i a t i o n a l e q u a t i o n
d 2 A - K A
- - - ( 3 )
d s 2
where A s t a n d s f o r t h e d i s t a n c e between two neighbouring t r a j e c t o r i e s . Thus i f D O , t h e t r a j e c t o r i e s o s c i l l a t e around each o t h e r forming a kind of p l a i t . Small i n i t i a l p e r t u r b a t i o n s a r e a m p l i f i e d only i f K<O.
The j u s t mentioned i d e a s have been c l e a r l y spoken o u t by J . L . Synge /5/ while t h e i r importance f o r modern developments i s s t r e s s e d by D . V . Anosov and V . I . Arnold, s e e e.g. / 6 / .
To make t h e a p p l i c a t i o n of Gaussian c u r v a t u r e t o n u c l e a r f i s s i o n a b i t moreprecise, it i s u s e f u l t o d i s c u s s a s c e n a r i o of n u c l e a r f i s s i o n :
1 . Due t o a l a r g e f l u c t u a t i o n i n c o l l e c t i v e motion t h e compound s t a t e i s l e f t .
2 . The nucleus surmounts t h e s a d d l e . A shape forms with two heads and a massive neck.
3. The shape e l o n g a t e s while t h e heads g e t s m a l l e r . The neck t e l e - scopes o u t and s t r a i g h t e n s more and more.
4 . The p o s i t i o n of s m a l l e s t neck diameter ( t h e p r o s p e c t i v e r u p t u r e p o i n t ) d e s t a b i l i z e s .
5. When t h e neck becomes t o o long, it snaps. I t snaps a t t h e p o s i t i o n of s m a l l e s t neck diameter.
P o i n t 5 was d i s c u s s e d i n / I / . Here q u a l i t a t i v e arguments s h a l l be given f o r t h e a s s e r t i o n s made i n p o i n t 3 and 4 .
Why d o e s t h e s h a p e e l o n g a t e by d i m i n i s h i n g i t s h e a d s w i t h o u t s t r a n g - u l a t i n g i t s n e c k ? T h i s i s a c o n s e q u e n c e o f R a y l e i g h ' s j e t i n s t a b i l i t y a s e x p l a i n e d i n / I / a n d / 2 / : S t r a n g u l a t i o n s o f t h e neck a r e s u p p r e s s e d u n t i l t h e s h a p e r e a c h e s a c e r t a i n l e n g t h . The m a t e r i a l f o r e l o n g a t i o n i s t a k e n from t h e h e a d s s i n c e t h e y move f a s t e s t i n t o t h e o u t w a r d d i r e c t i o n s . The m a t e r i a l i n t h e neck however moves v e r y l i t t l e , t h e l e s s t h e c l o s e r it i s t o t h e c e n t e r of mass. T h i s u n a v o i d a b l y s t r a i g h - t e n s t h e n e c k .
Why d o e s t h e r u p t u r e p o i n t d e s t a b i l i z e ? Suppose we c a n n e g l e c t f o r a moment t h e v a r i a b i l i t y o f t h e p o t e n t i a l , i . e . s e t V = c o n s t , and
r e s t r i c t t h e dynamics i n p h a s e 4 t o two d e g r e e s o f freedom: t o t a l l e n g t h 1 o f t h e s h a p e a n d t h e p o s i t i o n a o f s m a l l e s t neck d i a m e t e r
( " a " t o i n d i c a t e " a s y m m e t r y " ) . The k i n e t i c e n e r g y r e a d s
I t i s i m p o r t a n t t o n o t e t h a t p i s a p p r o x i m a t e l y t h e r e d u c e d mass, a c o n s t a n t , w h e r e a s v ( 1 ) d e c r e a s e s when 1 i n c r e a s e s . T h i s i s b e c a u s e t h e neck s t r a i g h t e n s when t h e s h a p e e l o n g a t e s , t h e s t r a i g h t e r t h e n e c k , t h e l e s s mass i s n e c e s s a r y t o s h i f t t h e p o s i t i o n o f s m a l l e s t neck d i a - m e t e r . F u r t h e r m o r e ~ ( 1 ) i s bounded from b e l o w , v i z . v ( 1 ) > 0 . T h e r e f o r e v ( 1 ) must be a concave f u n c t i o n , v i z . ~ . , ~ ( 1 ) > 0 a n d a l s o
" ' m l 1 > 0 . ( 5 )
I d e n t i f y i n g ( 4 ) a c c o r d i n g t o ( 1 ) a n d i n s e r t i n g i n t o ( 2 ) y i e l d s up t o a p o s i t i v e c o n s t a n t
a n d t h i s i s d e f i n i t e l y n e g a t i v e . I t f o l l o w s from J a c o b i ' s e q u a t i o n ( 3 ) t h a t t h e s t r e t c h i n g n e c k a m p l i f i e s f l u c t u a t i o n s .
Numerical c a l c u l a t i o n s of the r e s p e c t i v e p o t e n t i a l were p r e s e n t e d i n / 7 / ; it i s , a s a f u n c t i o n of a , s u f f i c i e n t l y f l a t . N e g l e c t i n g however t h e c h a r a c t e r i s t i c dependence o f t h e i n e r t i a p a r a m e t e r v ( 1 ) on t h e l e n g t h 1 would l e a d t o a p o s i t i v e G a u s s i a n c u r v a t u r e .
I w i l l i n g l y a d m i t t h a t a l l t h e s e c o n s i d e r a t i o n s a r e s t i l l o n l y q u a l i - t a t i v e o n e s a n d may f a i l by c l o s e r i n s p e c t i o n . But it i s b e t t e r t o h a v e a n i d e a o f t h e d i r e c t i o n of s e a r c h b e f o r e o n e s t a r t s t o t o r m e n t t h e computer. Some d i s c u s s i o n w i t h S. Grossmann a n d t h e s e l f -
s a c r i f y i n g h e l p o f D r . Hahn w i t h a p i c t u r e which I c o u l d show o n l y i n t h e t a l k a r e g r a t e f u l l y acknowledged.
R e f e r e n c e s
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4 . KOMMERELL V., KOMMERELL K . : T h e o r i e d e r Raumkurven und krurnmen F l a c h e n , Bd. I , B e r l i n : d e G r u y t e r 1931
5. SYNGE J . L . , P h i l . T r a n s . A 226 ( 1 9 2 6 ) 31
6 . ARNOLD V . I . : M a t h e m a t i c a l Methods o f C l a s s i c a l Mechanics, New York: S p r i n g e r 1978
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