HAL Id: jpa-00215107
https://hal.archives-ouvertes.fr/jpa-00215107
Submitted on 1 Jan 1972
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
COMMON FEATURES AND DIFFERENCES BETWEEN FISSION AND HEAVY ION PHYSICS
W. Swiatecki
To cite this version:
W. Swiatecki. COMMON FEATURES AND DIFFERENCES BETWEEN FISSION AND HEAVY ION PHYSICS. Journal de Physique Colloques, 1972, 33 (C5), pp.C5-45-C5-60.
�10.1051/jphyscol:1972505�. �jpa-00215107�
JOURNAL DE PHYSIQUE Colloque C5, supplement au no 8-9, Tome 33, ofi it-Septembre 1972, page C5-45
W. J. Swiatecki
L a w r e n c e B e r k e l e y L a b o r a t o r y , U n i v e r s i t y of C a l i f o r n i a B e r k e l e y , C a l i f o r n i a 94720, USA
RBsume5
-
Une approche macroscopique du phenomene de l a f i s s i o n e t de l a f u s i o n e s t formul6e. Un ensemble comportant au minimum t r o i s degrks de l i b e r t k e s t d e e r i t quali- tativement. Les propriktes g r o s s i e r e s de l ' e n e r g i e p o t e n t i e l l e dans c e t espace de configurations sont discut6es. On touche au probleme de l a v i s c o s i t 6 nuclkaire e t des comparaisons sont f a i t e s avec 1' 3 ~ e l i q u i d e . Quelques e f f e t s des moments cindtiques eleves sont discut8.s.A b s t r a c t - The m a c r o s c o p i c a p p r o a c h to f i s s i o n and fusion p h y s i c s i s f o r m u l a t e d . A m i n i m u m s e t of t h r e e d e g r e e s of f r e e d o m i s d e s c r i b e d qualitatively. The g r o s s f e a t u r e s of the potential e n e r g y i n t h i s configuration s p a c e a r e d i s c u s s e d . T h e p r o b l e m of n u c l e a r v i s c o s i t y i s mentioned and c o m p a r i s o n s with liquid He3 a r e m a d e . Some e f f e c t s of l a r g e a n g u l a r m o m e n t a a r c d e s c r i b e d .
I - INTRODUCTION
We a r e e n t e r i n g a new s t a g e i n n u c l e a r p h y s i c s , c h a r a c t e r i z e d by t h e availability of v e r y h e a v y n u c l e a r p r o j e c t i l e s . It i s a good t i m e t o r e - f l e c t on t h e place of the developing field of heavy ion p h y s i c s i n r e l a t i o n t o n u c l e a r f i s s i o n and, m o r e g e n e r a l l y , i n r e l a t i o n to n u c l e a r p h y s i c s a s a whole.
During the 60 y e a r s of i t s h i s t o r y n u - c l e a r p h y s i c s h a s had t o contend with two l i m i t a - t i o n s and t h e h i s t o r i c r o l e of heavy ion physics will be t o r e l a x them. T h e s e l i m i t a t i o n s have b e e n s o p e r v a s i v e t h a t we have a l m o s t stopped being a w a r e of them.
T h e s e l i m i t a t i o n s a r e :
1. The r e s t r i c t i o n of a t o m i c n u m b e r s t o l e s s than about 100.
2. The a l m o s t e x c l u s i v e r e s t r i c t i o n of n u c l e a r s h a p e s t o t h o s e c l o s e t o a s p h e r e .
The i n t r o d u c t i o n of a c c e l e r a t o r s f o r v e r y heavy ions will r e l a x both l i m i t a t i o n s . F i r s t , i t will be p o s s i b l e to study a t l e a s t t r a n s i e n t nu- c l e a r s y s t e m s with a t o m i c n u m b e r s up t o about (')Supported by the U. S. Atomic Energy C o m m i s s i o n .
two hundred. Second, the e n o r m o u s c e n t r i f u g a l f o r c e s c r e a t e d in o f f - c e n t e r c o l l i s i o n s of heavy nuclei will be sufficient to d e f o r m a n u c l e a r s y s - t e m away f r o m i t s c u s t o m a r y n e a r - s p h e r i c a l shape into m o r e o r l e s s s t r e t c h e d - o u t c o n f i g u r a - t i o n s , s o m e t i m e s r e s e m b l i n g e v e n a dumb-bell.
The e x t e n s i o n of a t o m i c n u m b e r s b e - yond a h u n d r e d m a y r e s u l t in the d i s c o v e r y of s u p e r h e a v y e l e m e n t s in thc: vicinity of Z
"
114,Z
=
154-
164, and p e r h a p s in s o m e o t h e r regions.The c o n s e q u e n c e s of t h e s e a n t i c i p a t e d d i s c o v e r i e s a r c a l r e a d y beginning t o be f e l t in t h e o r e t i c a l c h e m i s t r y and a t o m i c physics. An e v e n m o r e fundamental consequence of t h e e x t e n - s i o n of t h e l i m i t of n u c l e a r s y s t e m s f r o m a t o m i c n u r n b e r s . n e a r 100 t o a t o m i c n u m b e r s n e a r 200 h a s t o d o with the c i r c u m s t a n c e t h a t the m o s t i n t e n s e e l e c t r i c f i e l d s o c c u r r i n g anywhere in t h e u n i v e r s e a r e to be found i n t h e vicinity of heavy nuclei. The i n c r e a s e i n a t o m i c . n u m b e r f r o m 100 t o 200 i n - c r e a s e s t h i s h i g h e s t field only m o d e r a t e l y , but i t s o h a p p e n s t h a t i t i s in t h i s r a n g e of a t o m i c n u m - b e r s t h a t a n a t o m i c e l e c t r o n b e c o m e s highly r e l a - t i v i s t i c and the a t o m i c p r o p e r t i e s of v e r y heavy nuclei will t e s t the l i m i t s of quantum e l e c t r o d y - n a m i c s under unusual conditions.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972505
W.J. SWIATECKI
The advent of v e r y heavy- on a c c e l e r a - F i r s t then, what a r e the relevant d e - t o r s will thus have a n effect on c h e m i s t r y , atomic g r e e s of f r e e d o m in a m a c r o s c o p i c t r e a t m e n t of physics and quantum electrodynamics, a s well a s heavy ion and f i s s i o n p h y s i c s ?
on n u c l e a r physics i t s e l f , t o which I will now r e - The c h a r a c t e r i s t i c f e a t u r e of a m a c r o -
turn. scopic approach i s that collective r a t h e r than
I1
-
THE MACROSCOPIC APPROACHTo m e the distinguishing f e a t u r e of heavy ion physics i s i t s m a c r o s c o p i c nature. F o r the f i r s t t i m e we wlil have nuclear reactions where both the t a r g e t and projectile s a t i s f y well the c r i - t e r i o n f o r a m a c r o s c o p i c approximation, namely A > 1. 1. A kind of nuclear m a c r o - p h y s i c s , based on this approximation a s a s t a r t i n g point, will come into i t s own.
This i s t o be contrasted with convention- a l n u c l e a r reaction theory which, hxstorically, h a s
single-particle d e g r e e s of f r e e d o m become con- venient and relevant. T h i s i s saying no m o r e than that if you have tens o r hundreds of nucleons you will t r y t o formulate your problem in t e r m s of a few intelligently chosen groupings of the nucleon coordinates r a t h e r than of the whole s e t . This i s a g r e a t simplification.
A f u r t h e r simplification immediately sug g e s t s itself in virtue of the relative thinness of the nuclear s u r f a c e (the
"
leptodermous" c h a r a c t e r of m o s t nuclei). Insofar a s a nuclear systerxl h a s a well-defined s u r f a c e , that grouping of the nucleon~ t s roots in an idealization where the projectile i s coordinates which c o r r e s p o n d s to the nuclear s u r - a s t r u c t u r e l e s s m a s s point. On the other hand it i s f a c e i s the m o s t relevant s e t of d e g r e e s of f r e e - the s a m e n u c l e a r m a c r o - p h y s i c s that i s used i n the dom. Thus one will t r y t o d e s c r i b e the s t a t e of theory of n u c l e a r f i s s i o n , and this is why n u c l e a r the s y s t e m in t e r m s of the shape of i t s s u r f a c e fission m a y be used as a guide in formulating the and the development in time of t h i s shape.
f r a m e w o r k of heavy ion physics.
In g e n e r a l many d e g r e e s of f r e e d o m a r e What a r e the c e n t r a l f e a t u r e s of the needed t o specify a c c u r a t e l y the shape of a d i - m a c r o s c o p i c approach and what a r e the principal viding o r fusing nuclear system. T h e r e i s a rule
unsolved problems ? which suggests that f o r the f i s s i o n of a nucleus in-
In trying to answer t h e s e questions I t o n p a r t s , o r in the simultaneous fusion of n would like to remind you that many problems in nuclei, about 9n
-
4 collective d e g r e e s of f r e e d o m physics in g e n e r a l (not just nuclear physics) a r e should be adequate f o r many purposes. (If each solved according to a s t a n d a r d canonical s c h e m e , f r a g m e n t is thought of a s a n approximate ellipsoid, which goes something like this: t h r e e a x e s d e s c r i b e i t s s i z e and shape, t h r e eTABLE I
Guidelines f o r Solving Many P r o b l e m s in Applied P h y s i c s : i. Isolate relevant d e g r e e s of f r e e d o m to be retained explicitly.
2. Write down Equations of Motion (Schrodinger Equation) f o r t h e s e ciegrees of freedom.
( a ) Potential Energy T e r m s .
(b) Dissipative T e r m s (representing the coupling to d e g r e e s of f r e e d o m not retained explicitly).
( c ) I n e r t i a l T e r m s .
3. Solve the Equations of Motion, using whatever techniques a r e applicable (e. g. s t a t i s t i c a l , s e m i - c l a s s i c a l , o r what have you).
4. Compare with Experiment and re-cycle.
FISSION AND HEAVY IONS
...
c o o r d i n a t e s i t s location i n s p a c e , and t h r e e E u l e r angles i t s orientation. T h i s gives 9n d e g r e e s of f r e e d o m f o r n f r a g m e n t s . The m i n u s f o u r con- s t r a i n s the t o t a l volume of the s y s t e m to a s t a n d - a r d value, and t h e c e n t e r of m a s s of the s y s t e m t o a s t a n d a r d location. )
F o r a b i n a r y p r o c e s s of f i s s i o n o r fusion (n = 2 ) t h i s r u l e gives about 14 d e g r e e s of f r e e d o m . One m a y s u r e l y go below this number without making g r o s s qualitative e r r o r s , but I believe t h r e e d e g r e e s of f r e e d o m i s the b a r e s t m i n i m u m n e c e s s a r y to give the roughest qualitatively adc - quate d e s c r i p t i o n of n u c l e a r s h a p e s r e l e v a n t f o r b i n a r y f i s s i o n o r two-ion fusion.
T h e s e d e g r e e s of f r e e d o m a r e s o m e - thing like this
1. A s e p a r a t i o n c o o r d i n a t e , s a y a 2 .
2 . A necking o r neck-healing coordinate, say
"4'
3 . A m a s s a s y m m e t r y c o o r d i n a t e , s a y @ 3 . (The a ' s may be r e l a t e d to but a r e not t o be thought of a s identical with the coefficients in a L e g e n d r e Polynomial expansion of the shape. )
The n u c l e a r s h a p e s d e s c r i b e d by t h e s e d e g r e e s of f r e e d o m c a n be displayed in a t h r e e - dimensional s p a c e , of which Fig. I attempts to r e p r e s e n t a two-dimensional s e c t i o n a t constant a 3 . The a s y m m e t r y coordinate a 3 would s t i c k out of the plane of the paper and would c o r r e s p o n d to changing the m a s s r a t i o of the left and right s i d e s of the s h a p e s .
Fig. 1. An i l l u s t r a t i o n of f i s s i o n and f u s i o n s h a p e s d e s c r i b e d b y a n elongation coordinate a 2 and a necking o r neck-healing coordinate a4. The a s y m m e t r y coordinate ad {which would point into the plane of the p a p e r ) i s held fixed. The s c i s s i o n l i n e s f o r b i n a r y and t e r n a r y i v l s i o n s a r e indicated.
W. J. SWIATECKI
T h e t i m e d e v e l o p m e n t of a c o l l i s i o n b e - c u s s i n g t h e s o l u t i o n of a S c h r i i d i n g e r e q u a t i o n i n t w e e n t w o h e a v y i o n s would b e r e p r e s e n t e d b y a
p a t h s t a r t i n g s o m e w h e r e o n t h e r i g h t and p r o - c e e d i n g t o t h e c o n t a c t ( o r s c i s s i o n ) l i n e , f o l l o w e d e i t h e r b y r e - s e p a r a t i o n , f u s i o n , o r t e r n a r y d i - v i s i o n , d c p c n d i n g on t h e c o n d i t i o n s of t h e c o l l i s i o n and o t h e r f a c t o r s . C o n v e r s e l y , i n n u c l e a r f i s s i o n o n e s t a r t s s o m e w h e r e in t h e v i c i n i t y of t h e s p h e r e and g o e s t o t h e r i g h t , u s u a l l y i n t o t h e t w o - f r a g - m c n t v a l l e y , b u t s o m e t i m e s , p e r h a p s , i n t o t h e t h r e e -fragment v a l l e y .
Lf t h c p r o b l e m i s t r e a t e d q u a n t u m m e - c h a n i c a l l y t h e n i n s t e a d of p a t h s we s h a l l b e d i s -
t h e a 2 ~ 3 ~ 4 s p a c e , w i t h
1
l ( a 2 a 3 ~ 4 )I
r e p r e-
s e n t i n g t h e p r o b a b i l i t y of t h e s y s t e m b e i n g i n a c o n - f i g u r a t i o n s p e c i f i e d b y a 2 a 3 a 4 , and
**
i d t z H 9 g i v i n g t h e t i m e d e v e l o p m e n t of t h e w a v e f u n c t i o n+.
In o r d e r t o c o n s t r u c t a d y n a m i c a l t h e o r y of t h e p a t h s i n a2a3a4 s p a c c , o r t o s o l v e t h e S c h r B d i n g c r e q u a t i o n , w e s h a l l n e e d i n f o r m a t i o n a b o u t c e r t a i n b a s i c p r o p e r t i e s of t h e n u c l e a r s y s - t e m s c o n s i d e r e d . T h i s b r i n g s u s t o t h e s e c o n d i t e m i n t h e " G u i d e l i n e s " i n T a b l e I: w r i t i n g d o w n t h e E q u a t i o n s of Motion. T h e r e w i l l b e t h r e c t y p e s of t e r m s t o c o n s i d e r :
A s s o c i a t e d w i t h t i m e d e r i v a t i v e s of t h e d e g r e e s of
T e r m s f r c e d o m R e l e v a n t Q u a n t i t i e s
I . P o t c n t i a l E n e r g y T e r m s
2. F r i c t i o n , D a m p i n g o r D i s s i p a t i v e T e r m s
Z E R O T H
F I R S T R a y l e i g h ' s D i s s i p a t i o n F u n c t i o n o r i W ( a a 2 3 4 a . . . )
3. I n e r t i a l T e r m s SECOND I n e r t i a T e n s o r
M a ( f f 2 a 3 f f 4 .
. .
)'
1[ T h e n u m b e r
three
of d i f f e r c n t t e r m s i s no a c c i - d e n t : i t i s a s s o c i a t e d with t h e f a c t t h a t ( m a c r o - s c o p i c ) e q u a t i o n s of m o t i o n c o n t a i n z e r o t h , f i r s t and s e c o n d t i m e d e r i v a t i v e s of t h e d e g r c c s of f r e e - d o m , b u t no h i g h e r . ]In c l a s s i c a l m e c h a n i c s t h e d i s s i p a t i v e t e r m s m a y b e d e s c r i b e d by a q u a n t i t y c a l l e d t h c R a y l c i g h d i s s i p a t i o n f u n c t i o n ( a g e n e r a l i z e d f r i c - t i o n ) . In q u a n t u m m e c h a n i c s the potential c n c r g y and t h e d a m p i n g t e r m s a r c s o m e t i m e s c o m b i n e d i n a c o m p l e x p o t e n t i a l V (a 2 3 4 a a ) t i W ( a a 2 3 4 a ). T h e i n e r t i a l t e r m s i n c l a s s i c a l a s w e l l a s q u a n t u m m e - c h a n i c s g i v e r i s e t o a s o - c a l l e d i n e r t i a m a t r i x o r t e n s o r M a , & ,(a2a3a4), w h i c h d e s c r i b e s t h e i n e r t i a l
1 1
r e s p o n s e of t h e s y s t e m to t i m e v a r i a t i o n s of t h e
d e g r e c s of f r e e d o m . In any c a s e t h e r e a r e
three
p i e c e s of p h y s i c s t o c o n s i d e r i n m a k i n g a d y n a m i c a l t h e o r y :
1. P o t e n t i a l 2. D a m p i n g 3. I n c r t i a .
In t h e c a s e of n u c l e a r m a c r o - p h y s i c s t h e s i t u a t i o n t o d a y i s t h a t we h a v e a good u n d e r s t a n d i n g of 1 , a l i t t l e of 3, a n d v c r y l i t t l e of 2. I b e l i e v e t h a t i n t h c f u t u r e we w i l l h a v e t o c o n c e n t r a t e on p u l l i n g u p t h e i n f o r m a t i o n i n I n e r t i a a n d D a m p i n g t o a l e v e l t h a t m a t c h e s o u r u n d c r s t a n d i n g of t h e P o t e n t i a l E n e r g y .
A s r c g a r d s o u r u n d c r s t a n d i n g of t h e n u - c l e a r p o t e n t i a l e n e r g y , g r e a t p r o g r e s s h a s b e e n
F I S S I O N AND H E A W I O N S
. . .
C5-49m a d e i n the l a s t few y e a r s , p r i n c i p a l l y a s a r e s u l t of the s u c c e s s of S t r u t i n s k y ' s p r e s c r i p t i o n f o r c o m - bining m a c r o s c o p i c and m i c r o s c o p i c t h e o r i e s . We a r e today i n a position w h e r e we c a n c a l c u l a t e the potential e n e r g y of a n u c l e a r s y s t e m a s a function of N , Z and the nuclear shape, with an a c c u r a c y of about *I MeV. T h i s i s 1 MeV out of a t o t a l binding e n e r g y of s o m e 2000 MeV.
What have we l e a r n e d ? The potential e n e r g y a s a function of cu2cu3cr4 i s given by a pock- m a r k e d s u r f a c e , consisting of a smooth p a r t and
s h e l l effect p o c k - m a r k s . The c h a r a c t e r i s t i c un- dulations of the smooth p a r t a r e g e n e r a l l y of the o r d e r of t e n s of MeV, t h e p o c k - m a r k s a r e of the o r d e r of a few MeV. The theory underlying the s m o o t h p a r t i s well understood. The p o c k - m a r k s , though not s o well u n d e r s t o o d , a r e a l s o beginning t o b e r e l a t e d t o s i m p l e f e a t u r e s of the n u c l c a r shape. In p a r t i c u l a r t h e l a s t y e a r o r two have brought the r e a l i z a t i o n t h a t m a j o r s h e l l effects in the n u c l e a r potential e n e r g y a r e c l o s e l y r e l a t e d t o c e r t a i n f e a t u r e s of c l a s s i c a l o r b i t s in a potential well. F o r example if the well i s such that a c l a s s i - c a l o r b i t c l o s e s up on i t s e l f , then m a j o r m a g i c n u m b e r s a r e t o be expected. In any c a s e I f e e l t h a t the f i r s t s t e p i n building up a t h e o r y of h e a v y - ion r e a c t i o n s i s quite c l e a r : to c o n s t r u c t the po- t e n t i a l - e n e r g y s u r f a c e a s a function of s u i t a b l e d e f - o r m a t i o n c o o r d i n a t e s d e s c r i b i n g colliding and fusing nuclei, using the S t r u t i n s k y method ( i m - proved and refined w h e r e n e c e s s a r y ) .
T h e r e will, of c o u r s e , be a wealth of s t r u c t u r e in t h e r e s u l t i n g potential e n e r g y m a p s , e s p e c i a l l y i n the p o c k - m a r k s . L e t m e just point out s o m e of the m o s t p r i m i t i v e f e a t u r e s t o be e x - pected i n t h e g r o s s s t r u c t u r e of t h e m a p s .
T h e r e a r e f o u r i m p o r t a n t f e a t u r e s I wish t o mention:
1 . The equilibration of the neutron-proton r a t i o .
2. The e x i s t e n c e of a c r i t i c a l m a s s a s y m m e t r y . 3. The e x i s t e n c e of two m i s a l i g n e d v a l l e y s . 4. The e f f e c t of a n g u l a r momentum.
E q u i l i b r a t i o n of N n r a t i o
The f i r s t point i s r a t h e r t r i v i a l and I want to d i s p o s e of i t quickly. It i s that if two
nuclei with widely differing N/Z r a t i o s a r e brought i n t o c o n t a c t , a r e - d i s t r i b u t i o n of n e u t r o n and proton d e n s i t i e s will t a k e place s u c h t h a t an a p p r o x i m a t e l y u n i f o r m value of N/Z will obtain throughout the s y s t e m . T h e r e will be slight d c v i a - tions f r o m u n i f o r m i t y , and s l i g h t fluctuations around i t , but i t i s a f a i r a p p r o x i m a t i o n t o d i s r c - g a r d t h e s e a t f i r s t . F o r e x a m p l e , f o r tangent s p h e r e s of r a d i i
R1, RZ one m a y work out i n a closed f o r m a n e x p r e s s i o n f o r the Zl/Ai r a t i o of one of t h e nuclei divided by the Z/A r a t i o of the wholc s y s t e m :
w h e r e e i s the proton c h a r g e , r o i s the n u c l c a r r a d i u s constant (with e / r o equal t o about 2
1.2 MeV), "coeff" i s the n u c l e a r s y m m e t r y e n e r g y coefficient (about 30 MeV), and F i s the following function
According t o t h i s f o r m u l a the s m a l l e r of two tangent nuclei will have a slightly h i g h e r N/:<
r a t i o , but only by a t m o s t a few p e r c e n t . (The g r e a t o s t deviation of
ZL/$
f r o m unity o c c u r s n e a r R1/K2 = 0.3 and i s about 6 % f o r A = A 1 216. ) F o r s o m e p u r p o s e s t h i s c h a r g e r e - d i s t r i b u t i o n m a y be of i m p o r t a n c e , and t h e r e i s e x p e r i m e n t a l evidence f o r it both i n f i s s i o n and p e r h a p s i n heavy ion t r a n s f e r r e a c t i o n s , but i n m y s u r v e y of g r o s s p r o p e r t i e s I will not s a y m o r e about it.E x i s t e n c e of a c r i t i c a l m a s s a s y m m e t r y .
As r e g a r d s the dependence of t h e g r o s s potential e n e r g y on the a s y m m e t r y c o o r d i n a t e
u3, the m o s t i m p o r t a n t thing to b e a r in mind i s t h a t t h e r e e x i s t s a c r i t i c a l r a t i o of m a s s e s of t a r g e t and p r o j e c t i l e . F o r m a s s a s y m m e t r i e s m o r e e x - t r e m e t h a n the c r i t i c a l (i. e. f o r a r e l a t i v e l y light heavy ion and a heavy t a r g e t ) the t a r g e t nucleus t e n d s t o s u c k up the p r o j e c t i l e . F o r a s y m m e t r i e s l e s s e x t r e m e than the c r i t i c a l (i. e . f o r h e a v y ions
C5-50 W. J. SWIATECKI
m o r e n e a r l y comparable with the t a r g e t ) the p r o - jectile will tend to g r o w towards equality with the t a r g e t ( s e e Fig. 2). Most heavy ion e x p e r i m e n t s
Low Z ~ / A
Fig. 2. An illustration of the dependence of the relative deformation e n e r g y on a s y m m e t r y . F o r light s y s t e m s (low z ~ / A ) a s y m m e t r i c configura- tions tend to become even m o r e a s y m m e t r i c . F o r high Z ~ / A this i s s t i l l t r u e f o r very a s y m m e t r i c configurations, but moderately a s y m m e t r i c con- figurations tend toward s y m m e t r y .
done t o date l i e on one s i d e of the c r i t i c a l a s y m - m e t r y . Most heavy ion e x p e r i m e n t s of the future.
in p a r t i c u l a r those aiming a t super-heavy nuclei, will lie on the other side of the c r i t i c a l a s y m m e t r y . The c r i t i c a l a s y m m e t r y i s t h e r e f o r e an important f e a t u r e t o b e a r in mind when extrapolating f r o m past experience t o future e x p e r i m e n t s with heavy ions. T h e existence of a c r i t i c a l m a s s a s y m m e t r y i s a r e s u l t of the competition between the .electric
m a t t e r out of the s m a l l nucleus into the big one. If the e l e c t r o s t a t i c e n e r g y w e r e negligible t h i s would always be the case: the l a r g e r p a r t n e r would tend t o suck up up the s m a l l e r one. F o r heavy s y s t e m s , however, when the e l e c t r o s t a t i c energy i s appreciable, the tend- ency i s r e v e r s e d , except for e x t r e m e a s y m m e t r i e s , when the p r e s s u r e f r o m the s u r f a c e e n e r g y eventually begins to dominate.
T h e r e i s m o r e to this problem than I have indicated, in p a r t i c u l a r a n inadequately understood qualitative change n e a r Z 2 /A=40, but I will now go o n to the th'rd item.
Misaligned Valleys
The third important f e a t u r e of the Nu- c l e a r Potential Energy m a p s in a2a3a4 space i s the existence of two (or m o r e ) valleys, s i m i l a r l y oriented but misaligned. L e t m e explain. Again think of a fixed m a s s - a s y m m e t r y , i. e . , a section through the a2a3a4 s p a c e along a fixed a3. The nuclear s h a p e s a s functions of a2 and a4 a r e shown i n Fig. 1.
The s i m p l e s t way t o s u m m a r i z e the findings of many people who have investigated the potential e n e r g y in s p a c e s like the a2a4 space i s to say that t h e r e a r e two principal valleys, a s shown. One valley s t a r t s f r o m the vicinity of the sphere. After a saddle, the energy goes down.
but t h e r e i s stability against changes of the necking coordinate f o r a fixed elongation coordinate. Be- low t h i s valley is a roughly p a r a l l e l two-fragment valley corresponding to approaching o r separating fragments.
( F a r t h e r up t h e r e i s a t h i r d valley, t h e T h r e e - F r a g m e n t Valley, about which I will not s a y m u c h m o r e . )
How do the valleys fit t o g e t h e r ? I have shown an oversimplified sketch t o give you a hint of what the situation looks like. A plan and an end view of the potential e n e r g y s u r f a c e a s functions of a2 and a4 a r e shown i n Figs. 3 a and 3b.
f o r c e s and the s h o r t - r a n g e n u c l e a r f o r c e s (ideal- I hope you c a n s e e the f i s s i o n valley with ized a s a s u r f a c e energy), and m a y be understood i t s saddle and stable s p h e r i c a l shape and the m i s - with r e f e r e n c e t o configurations of tangent nuclei. aligned two-fragment valley. Between the two i s a F o r a sufficiently s m a l l nucleus in contact with a ridge running f r o m A t o C . R e m e m b e r a l s o that l a r g e r one the l a r g e p r e s s u r e caused by the s u r - on top of what I d e s c r i b e d a r e shell-effect pock- face energy of the s m a l l e r nucleus tends to s q u i r t m a r k s . (One of t h e s e i s shown: the magic hole H,
FISSION AND HEAVY IONS
...
Fig. 3a. A s k e t c h of the potential e n e r g y landscape i n ( ~ 2 - ( ~ 4 s p a c e , f o r a s u p e r - h e a v y nucleus. A few s h a p e s a r e shown f o r orientation. The portion BC of the f i s s i o n valley i s s e p a r a t c d f r o m thc portion AD of the b i n a r y valley by a ridge running f r o m A t o C.
r e s p o n s i b l e f o r the s t a b i l i t y of a s u p e r - h e a v y nu- cleus. )
With this potential e n e r g y s u r f a c e a s background we c a n now s k e t c h in a f i s s i o n o r a f u s i o n path c o r r e s p o n d i n g t o a dividing o r fusing
s y s t e m . In f i s s i o n the nucleus d e f o r m s , goes o v e r the saddle and r o l l s down the f i s s i o n valley. In the neighborhood of point C equilibrium against necking-in i s l o s t and the s y s t e m i s injected into the two-fragment valley. Because of the misalign- m e n t of the v a l l e y s the injection i s off-axis and the r e p r e s e n t a t i v e point will v i b r a t e around the a x i s a s i t d e s c e n d s the two-fragment valley ( o r c r e e p s toward the bottom of the v a l l e y if t h e r e i s a p p r e c i -
able damping). T h i s v i b r a t i o n c o r r e s p o n d s t o changes i n e c c e n t r i c i t y of the f r a g m e n t s , i. e.
,
t o f r a g m e n t excitation. The excitation e n e r g y i s roughly the d i f f e r e n c e i n e n e r g y between points C and D. E x p e r i m e n t a l l y i t i s typically 2 0 - 4 0 MeV and i s eventually d i s s i p a t e d in n e u t r o n e v a p o r a t i o n f r o m the f i s s i o n f r a g m e n t s .Now about fusion. The situation i s a n a l - ogous. We p r o c e e d u p the Two F r a g m e n t Valley c o r r e s p o n d i n g t o approaching nuclei. At the point A , c o r r e s p o n d i n g to tangency, e q u i l i b r i u m a g a i n s t an i n c r e a s i n g e c c e n t r i c i t y of the f r a g m e n t s i s l o s t and the s y s t e m i s injected into the f i s s i o n valley.
B e c a u s e of the o f f - c e n t e r injection t h e r e i s
W. J. SWIATECKI
Fig. 3b. A view of the potential energy landscape in a 2
-
a4 space looking up the three valleys in the direction of the spherical system in the hollow H.Coming out of H one goes over the saddle S and rolls down the fission valley along BC. At C in- jection into the binary valley takes place.
vibration about (or creep toward) the axis of the fission valley, which would eventually lead to ex- citation of the fused system. The amount of excita- tion i s roughly the difference between the energy a t A and at B.
An analogy to these fission and fusion paths may be constructed i n t e r m s of the path of a beam particle in a linear accelerator. Imagine a linear accelerator consisting of two misaligned segments. Each segment has radial focusing (e. g.
,
quadrupole lenses ). A short pre -accelerator (the
fission valley) injects a particle into the main accelerating tube, which, however, i s misaligned.
Conversely, in fusion, a particle i s sent back up the main accelerator and i s then injected up-hill into the pre-accelerator. Because of the misalign- ment, transverse oscillations a r e set up in the beam at injection. These oscillations correspond to fission fragment excitations in the case of f i s - sion, o r to the excitation of the fused nucleus in the case of fusion.
The question of estimating the amount of excitation following a fusion reaction i s one of the outstanding problems of heavy ion physics, especi- ally when one i s trying to make super-heavy nu- clei. (If there i s too much excitation one will not be able to make them. ) Using the picture of m i s - aligned valleys one estimates excitations ranging f r o m perhaps 20 to 60 MeV for a typical case like that of 2 3 2 ~ h
+
7 6 ~ e (which i s one of the most promising candidates for making superheavie s).These estimates a r e , however, extremely uncer- tain because of a crucial missing piece of infarma- tion-namely, how large i s nuclear viscosity. In other words, how strong i s the coupling of the col- lective degrees of freedom to the single particle degrees of freedom that a r e not displayed explicitly in a macroscopic treatment. You can probably see at once that too much damping, too much viscosity, will make fusion difficult or impossible. This i s because two nuclei like Th and Ge, when brought into contact, do not in general feel a driving force tending to fuse them into a spherical shape. On the contrary, even though the nuclear forces tend to fill in the neck region, the strong electric forces tend to push the bulks of the two nuclei apart and cause re-disintegration. In t e r m s of the Potential energy map in Fig. 3 this means that in the vicin- ity of point 3 one i s still some 20-15 MeV below the saddle at S and one i s on a sloping part of the landscape, with a slope to the right, towards r e
-
disintegration. In order to achieve fusion one would increase the bombarding energy above the contact energy (the coulomb b a r r i e r ) hoping that this additional collective energy will c a r r y the sys- tem over the saddle at S . If there were no viscos- ity-no conversion of collective into internal en- ergy-an additional 15 MeV might be enough. But
FISSION AND HEAVY IONS
. ..
if the v i s c o s i t y i s large-if s a y nine tenths of the e x t r a e n e r g y goes into i n t e r n a l excitation, then one might have t o go to bombarding e n e r g i e s
" 150 MeV above contact e n e r g y (coulomb b a r r i e r ) . I t would be like trying to m a k e two c h a r g e d d r o p s of honey c o a l e s c e by banging t h e m t o g e t h e r with high energy. M o s t of the collision velocity would go into h e a t , and a f t e r making p a r t i a l contact the h o t , c h a r g e d honey d r o p s would be t o r n a p a r t again without e v e r reaching the s p h e r i c a l configura- tion.
So h e r e we come a c r o s s a c e n t r a l unan- s w e r e d p r o b l e m i n heavy-ion physics. A r e nuclei v i s c o u s like honey o r mobile like w a t e r o r m e r - c u r y ?
In hydrodynamics the d i s t i n c t i o n between e x t r e m e l y v i s c o u s and e x t r e m e l y nonviscous types of flow m a y be m a d e quantitative i n t e r m s of the r e l a t i v e magnitude of t h e second and t h i r d t e r m s i n t h e equations of motion. F o r l a r g e v i s c o s i t y the d i s s i p a t i v e t e r m s dominate o v e r the i n e r t i a l t e r m s (which m a y then be neglected). F o r s m a l l v i s c o s i t y the damping t e r m s a r e neglected c o m p a r e d to the i n e r t i a l t e r m s . A textbook i l l u s t r a t i o n i s the c a s e of s m a l l oscillations of a viscous liquid d r o p of r a d i u s R, d e n s i t y p , s u r f a c e t e n s i o n y and c o - efficient of v i s c o s i t y q. Such a d r o p , if d i s t o r t e d into a s p h e r i c a l shape, will e i t h e r v i b r a t e with a c i r c u l a r f r e q u e n c y w d e t e r m i n e d by y and p (if v i s c o s i t y i s negligible), o r c r e e p back t o the s p h e r i c a l s h a p e with a n e-folding t i m e d e t e r m i n e d by y and q (if i n e r t i a i s negligible). The r a t i o of the e-folding t i m e to the c i r c u l a r period l / w i s given by
e-foldin t i m e =
19
q -. T =
2-+
5 04-7T-E
T h i s r a t i o ( o r the
"
c r e e p p a r a m e t e r " q/i t s e l f ) m a y be used a s a m e a s u r e of the r e l a t i v e i m p o r t a n c e of v i s c o s i t y . H e r e q , y , p a r e given n u m b e r s , and the dependence on the s i z e of the s y s t e m e n t e r s through R. If f o r R we put R = r A'/) 0 (with r o I 1.2 X c m f o r nuclei.
o r 1.93 X 10-' c m f o r w a t e r ) we find t h a t T ~ / T ~ i s
p r o p o r t i o n a l to A-
'I6.
= o r w a t e r a t oOC one 40 ~f wc put A =1oZ4
m o l e c u l e s finds~y
=~ 1 / 6 .
(R = 1.93 c m ) , we find T:, T1
-
= 0.004, and f o r s u c h l a r g e d r o p s v i s c o s i t y i s negligible. But f o r A = 300,-
- - 1 5 and now we a r e i n the e x t r e m e v i s c o u s o r c r e e p y l i m i t . T h i s s e e m s to be the c a s e T2 f o r a l l o r d i n a r y liquids. The r a t i o i s l e s s f o r e t h e r , and i s c o n s i d e r a b l y l e s s for w a t e r a t iOO°CT I
(- = 2 . 5 ) , but we have h e r e a somewhat ominous r e s u l t : a s i s well known in hydrodynamics v i s c o s - T2 ity b e c o m e s dominant f o r sufficiently s m a l l s y s - t e m s , and f o r a l l o r d i n a r y liquids A = 300 i s i n - deed srnall i n t h i s s e n s e . T h e r e would be g r e a t difficulties i n the way of making a s u p e r h e a v y ( ! ) d r o p of w a t e r with A 300 out of two d r o p l e t s with A = 232 and A = 76 r e s p e c t i v e l y (simulating the 2 3 2 ~ h
+
7 6 ~ e r e a c t i o n ) . Viscosity would i n a l l likelihood p r e v e n t fusion of the d r o p s before r e - d i s i n t e g r a t i o n caused by the coulomb repulsion.Now w a t e r m a y be an e n t i r e l y m i s l e a d i n g guide t o the p r o p e r t i e s of a quantum fluid like nu-.
c l e a r m a t t e r . We shou1.d be able t o get a b e t t e r o r d e r of magnitude e s t i m a t e by c o n s i d e r i n g another quantum fluid, namely liquid He 3
.
I n s e r t i n g the d e n s i t y , s u r f a c e t e n s i o n and r a d i u s c o ~ s t a n t of He 3 a t low t e m p e r a t u r e s we find, f o r A = 300,T1
= 60,000 q, where q i s i n p o i s e s . The v i s c o s - i t y of ~e~ i n c r e a s e s rapidly with 2 decreasing t e m - p e r a t u r e , and one finds t h a t T ~ / T ~ i s 120, 24, 1.4 a t t e m p e r a t u r e s of 0.04"K. 0 . I 0 K , IoK, r e s p e c - tively. Nuclear t e m p e r a t u r e s of 1 o r 2 MeV a r e e x - pected to c o r r e s p o n d to the lower range (0.04OK to 0.i0K) and we a g a i n come out with the indication t h a t v i s c o s i t y would be dominant f o r s y s t e m s with AZ300.Table II s u m m a r i z e s t h e s e e s t i m a t e s .
These e s t i m a t e s s u g g e s t e x t r e m e l y high v i s c o s i t y f o r n u c l e a r m a t t e r (except a t v c r y high t e m p e r a t u r e s of many MeV), but they have to be taken with r e s e r v a t i o n s . F i r s t of a l l i t i s possible that a t a sufficiently low t e m p e r a t u r e a F e r m i liquid l i k e ~e~ would become s u p e r f l u i d , with v e r y low v i s c o s i t y , and the above e s t i m a t e s should a t
-
b e s t b e u s e d a s a guide a t not too low t e m p e r a t u r e s ( c o r r e s p o n d i n g t o n u c l e a r e x c i t a t i o n s a t which p a i r i n g effects a r e d e s t r o y e d ) . Secondly the v i s - c o s i t i e s o r d i n a r i l y calculated and m e a s u r e d f o r
C5-54 W. J. SWIATECKI
of electromagnetic decays) the system would go on
TABLE 11 vibrating indefinitely in that state, with no damping
Ratios of e -folding Time to Vibrational Times for Drops of Water and ~ e 3 .
Water
L
5=10OoC: q=0.00284 poise,
-
T, = 2 . 50=0.04"K (corresponding to 0.26 o r 0.8 M ~ v ) " : q=0.002 poise,
-
TI = 120T2
%=O.l°K (corresponding to 0.65 o r 2 M~v)":
q=0.0004 poise,
-
T, 1 = 246'=I0K (corresponding to 6.5 o r 20 M ~ v ) ~ ' : q=0.000023 poise,
-
T1 =1.4T 2
r ~ h e conversion of the Kelvin temperatures for 33e3 to equivalent nuclear temperatures i s not un- ambiguous. The f i r s t figure uses the m a s s of 33e3 a s m a s s of the Fermion in the F e r m i fluid r e p r e - senting ~ e 3 , the second uses the m a s s of a quasi- particle with an experimental effective m a s s equal to 3.08 times the m a s s of ~ e 3 .
He refer to very large systems, in which the 3 mean f r e e path of the particles (or quasi-particles) i s small compared to the dimensions of the system.
F o r small droplets, when this condition i s not satisfied, a discussion of damping in t e r m s of the usual viscosity coefficient may be grossly inade- quate. An extreme case that illustrates this point is the damping of a f i r s t vibrational level of a nu- cleus. If this level happens to be the f i r s t excited state in the energy spectrum, then (in the absence
a t all. Thus for small systems a t low temper- atures, where level spacings a r e large and indi- vidual levels stand out, the damping i s no longer described by a viscosity coefficient and, in partic- ular, may be very small. I wonder if one could throw some more light on the question of damping in small F e r m i fluid systems by experiments on the properties of a m i s t of 33e3, consisting of droplets with A-values of tens o r hundred of mole- cule s.
What else can we do to fill in the gaps in our understanding of the viscosity problem?
~ j d r n h o l m has recently discussed the question of damping in relation to the presence o r absence of vibrational levels i n heavy nuclei (especially those with a spontaneously fissioning isomeric state).
There will also soon be many heavy-ion experi- ments which in one way o r another will depend on viscosity, and from these we shall gradually un- ravel the answer. However, there exists already a m a s s of relevant experimental data in the allied field of fission, which could be used to estimate the nuclear viscosity. These data a r e measure- ments of fission fragment kinetic energies, especi- ally in their dependence on z'/A. F o r the heavier nuclei in particular (i. e.
,
for high z'/A) the sad- dle point shape for fission i s cylinder-like, o r even spheroidal, and this means that there i s a consider- able saddle -to-scission stage for which the dynamics will surely d-pend on the size of the viscosity. Thus if the descent from saddle to scission i s creepy, little kinetic energy will be accumulated by the f r a g - ments during this stage, and the observed fragment kinetic energies ought to be l e s s than if the descent were f r e e and mobile. Why then hasn't the theoryof fission already provided us with the answer a s regards viscosity? Because the relevant calcula-
tions
for the viscous descent f r o m saddle to s c i s -sion have not been made. Calculations for a
9-
viscous descent of an idealized drop a r e available, and the results do agree fairly well with experi- ment. One might take this a s an indication that viscosity i s small, but one cannot be s u r e , since one doesn't know how much viscosity the theory could stand before a discrepancy with experiment would begin to emerge.
F I S S I O N AND HEAW IONS
...
C5-55The calculation of the viscous descent f r o m saddle t o s c i s s i o n r e m a i n s thus a n out-
standing problem of d i r e c t relevance t o heavy ion physics.
Before leaving the subject of viscosity l e t m e mention a related problem on which p r o g r e s s a p p e a r s to be possible. This i s the question of the d r a g o r f r i c t i o n experienced by two nuclei passing each other in a grazing collision. Imagine two F c r m i g a s e s passing each other with a relative velocity Av. Imagine t h e r e i s a n a r e a of contact n r Z ( a neck o r "window" ), lasting f o r a time of the o r d e r of 2r/Av.
During the time this window i s open p a r - t i c l e s will move t o and f r o between the two F e r m i g a s e s . Because t h e r e i s a vclocity m i s m a t c h t h e r e will be a t r a n s f e r of l i n e a r momentum equal to MAv e v e r y time a p a r t i c l e of m a s s M f r o m one nucleus i s captured by the other. T h e flux of p a r - t i c l e s a c r o s s unit a r e a in a F e r m i g a s with F e r m i
4 3
momentum P i s a P / ~
,
hand t h i s leads t o a 4 3d r a g f o r c e of ( n P /h ) Av p e r unit a r e a of contact 4 3
between the nuclei, o r t o T P /h a s the I t d r a g co- efficient" p e r unit a r e a , per unit velocity differen- tial. The change in l i n e a r momentum of one of the nuclei (equal t o the change i n the o t h e r ) i s (force)
2 r 1 n p 4 Av. r r 2 .
-
=-
X ( t i m e ) = -
h 4T (
$
,3 P. wherekF = %/P. T h e s e simple predictions dxsregard the f a c t that the exclusion principle m a y inhibit thc t r a n s f e r of nucleons f r o m one nucleus t o the other.
It would be interesting t o compare the properly generalized formulae f o r the grazing d r a g with suitable experiments. The relation of this t o the problem of viscosity i s that, a s in the calculation of viscosity, t h i s i s a problem in the t r a n s p o r t of momentum a c r o s s a plane in a (nuclear) fluid. The difference i s that the velocity field, instead of being c h a r a c t e r i z e d by a uniform velocity gradient ( a s i n standard viscosity p r o b l e m s ) is c h a r a c t e r i z e d by a velocity discontinuity. But i t i s the l a t t e r velocity field which, though r a r e in o r d i n a r y hydro- dynamics, i s the standard initial condition f o r heavy ion collisions.
In the meantime the problem of nuclear viscosity r e m a i n s v e r y unclear. Can one do any
dynamical calculations a t all bcfore this question i s c l e a r e d up? Not really-but t h e r e i s a s o r t of half-dynamic, h a l f - s t a t i r stage which d o e j n ' t r c - q u i r e a knowledge of damping t e r m s . This i s the problem of a steady rotation of a s y s t e m without intrinsic change of shape. In this c a s e the only dynamical t e r m , the only kinetic e n e r g y , i s the e n - e r g y of rotation. T h i s brings m e to the fourth item I wanted t o d i s c u s s , that of the effect of angular momentum on the potential e n e r g y surface.
Effect of Angular Momentum.
We a l l know that if you spin a deformable object, s a y a fluid m a s s , it tends to flatten, and if you spin i t too h a r d it will fly a p a r t (fission). Some of you m a y be a w a r e of a f i n e r point, namcly that usually, a s you i n c r e a s e the angular momentum, a fluid m a s s will go f r o m a f l a t , axially s y m m e t r i c equilibrium shape t o a t r i a x i a l equilibrium shape- like a n oval piece of soap-for a range of angular momenta before fission. What do we expect t o happen to a nucleus a s it i s made to spin f a s t e r and f a s t e r ?
To d i s c u s s the equilibrium s h a p e s of a rotating nucleus in a m a c r o s c o p i c theory we s e t up an effective potential e n e r g y (PE) (a a a cff 2 3 4 .
. .
),and look f o r configurations that make the effective potential cne rgy stationary:
= Ecoulomb
'
Esurface'
Erotation t shells.T h e rotational energy Erot (a2a3a4+
.
) that i s added t o the usual P E may be written a swhere td is s o m e effective moment of i n e r t i a of the shape in question. As usual the problem of exploring (PE)eff m a y be split into f i r s t looking a t a smooth background and then adding shell s t r u c - t u r e wiggles. Little i s known a s yet about the full problem with shells. On the other hand the smooth background h a s been explored m o r e thoroughly
(although many of the r e s u l t s r e m a i n unpublished).
I s h a l l p r e s e n t a few of the r e s u l t s of making s t a - tionary the smooth p a r t
W. J. SWIATECKX
w h e r e i n E the m o m e n t of i n e r t i a Q i s taken a s r o t
t h e rigid body moment.
I would like to give you a b i r d ' s eye view of what happens to t h i s smooth p a r t of the e n e r g y f o r a nucleus anywhere i n the periodic table and f o r any amount of angular momentum. F o r t h i s p u r - pose i t i s convenient to introduce two d i m e n s i o n l e s s n u m b e r s specifying the r e l a t i v e s i z e s of the t h r e e e n e r g y components: coulomb, s u r f a c e and r o t a - tional. We pick the s u r f a c e e n e r g y of the s p h e r i - c a l shape E ( O ) s u r f a c e a s a unit and specify the amount of c h a r g e on the nucleus by the u s u a l f i s - s i l i t y p a r a m e t e r x
w h e r e E ( O ) i s the coulomb e n e r g y of the spherical shape.
We specify the amount of angular m o - m e n t u m by y
E(o)
r o t
Y = (0)-
E s
w h e r e E'O) i s the rotational e n e r g y of a rigid rot
s p h e r e with the given a n g u l a r momentum.
Now I s h a l l d i s c u s s the r e s u l t s in an x-y d i a g r a m . Fig. 4. T h i s d i a g r a m s a y s the following things: If you take any nucleus i n the p e r i o d i c table then, if t h e r e w e r e no s h e l l effects and if the m o m e n t of i n e r t i a w e r e r i g i d , the nucleus would a t f i r s t d e f o r m into a f l a t shape. T h e f i s s i o n b a r r i e r d e c r e a s e s with i n c r e a s i n g angular m o m e n t u m and v a n i s h e s along t h e upper c u r v e in Fig. 4. T h e middle c u r v e shows the c r i t i c a l angular m o m e n t u m a t which the f l a t ground s t a t e shape goes o v e r into a t r i a x i a l shape. (Some of these shapes a r e like slightly indented flattened dumb-bellsJ The dashed c u r v e divides the x - y plane into two regions. To the right the saddle point s h a p e f o r the f i s s i o n of the rotating d r o p i s stable a g a i n s t a s y m m e t r y , to the left i t i s unstable. Note that beyond x = 0.81 t h e r e a r c no t r i a x i a l shapes.
F i g . 4. A c l a s s i f i c a t i o n of rotating s y s t e m s a c - cording to the f i s s i l i t y p a r a m e t e r x and the r o t a - tion p a r a m e t e r y. T r i a x i a l equilibrium s h a p e s d i s a p p e a r altogether a t x = 0.81, but a r e a l m o s t gone a t x = 0.6.
L e t m e i l l u s t r a t e the p r a c t i c a l p r e d i c - tions t h a t follow f r o m t h i s kind of calculation.
Consider a collision of a heavy ion of m a s s Mi and a nucleus of m a s s M2 a t impact pa- r a m e t e r b and c e n t e r - o f - m a s s e n e r g y E c m ' F r o m c o n s e r v a t i o n of e n e r g y and m o m e n t u m i t readily follows that the c l o s e s t d i s t a n c e of a p p r o a c h of p r o j e c t i l e and t a r g e t i s given by r min where
H e r e V ( r ) i s the i n t e r a c t i o n potential between the nuclei. F o r a given value of r t h i s i s a
m i n 2 .
hyperbola when b i s plotted v s E
c m ' (In s u c h 2 .
plots nb i s proportional to a c r o s s section. ) I f , f o r e x a m p l e , r i s c h o s e n to be t h e s u m of the
m r a d i i of the two nuclei, r
m = R.
+
R 2 , the c o r r e - sponding hyperbola divides the b 2 v s E c m plane into two regions: d i s t a n t collisions w h e r e the nu- c l e i p a s s e a c h o t h e r without a p p r e c i a b l e n u c l e a r i n t e r a c t i o n s , and c l o s e collisions w h e r e n u c l e a r i n t e r a c t i o n s take place (the c o r r e s p o n d i n g nbLFISSION AND HEAVY IONS
,..
gives then the reaction c r o s s section). Because of the diffuseness of the nuclear s u r f a c e and the finite range of nuclear f o r c e s , t h e r e i s an i n t e r - mediate diffuse transition region of grazing c o l - lisions. (See Fig. 5. )
5. A classification of nuclear collisions in
h
v s EGW plane. Distant collisions give place to close collisions along a diffuse region of grazing collisions. Close collisions a r e subdivided into t h r e e regions, a s shown.T h e bL vs Ecm plane c a n be f u r t h e r sub- divided by c u r v e s corresponding t o loci of .fixed angular momentum L. (Here L = h L . ) Since L i s given by
we have
F o r a given L this i s again a hyperbola in a plot of b2 v s E
.
If now f r o m the upper curve inc m
Fig. 4 we read off the value of ycrit ( o r L c r i t ) a t which the fission b a r r i e r h a s vanished, and i n - s e r t this in the above equation f o r b 2 , the r e - sulting hyperbola will divide the b vs E c m plane 2 into two regions, a s shown in Fig. 5. T o the right
of the line m a r k e d Bf = 0 (where the f i s s i o n b a r - r i e r B f vanishes) the s y s t e m h a s too much angular momentum to s t a y together and collisions c o r r e -
sponding to those values of b and 2 E would lead c m
to redisintegration without the possibility of c o m - pound nucleus formation.
To the left of the hyperbola m a r k e d B = 0 t h e r e e x i s t s a fission b a r r i e r and a c o m -
f
pound nucleus i s in principle possible. This i s because a finite f i s s i o n b a r r i e r e n s u r e s the e x - istence i n configuration s p a c e of a potential energy hollow, which c a n keep a n excited s y s t e m confined i n i t s neighborhood (for t i m e s which d e c r e a s e e x - ponentially with d e c r e a s i n g height of the b a r r i e r ) . Thus
if
the s y s t e m g e t s captured in the hollow, a compound nucleus will be formed. (Its lifetime i s an exponentially d e c r e a s i n g fuction of the distance f r o m the c r i t i c a l hyperbola m a r k e d Bf = 0. ) But whether a m o r e o r l e s s short-lived compound nu- cleus would, i n fact, be f o r m e d f o r collisions t o the left of Bf = 0 i s a different m a t t e r . It i s a dy- namical question of whcthc r the colliding nuclei, starting off at the moment of tangency in s o m e p a r t of configuration s p a c c , would be capturcd in the potential e n e r g y hollow, o r , on the c o n t r a r y , whether they would m i s s i t altogether o r perhaps only p a s s through i t without being captured.Also shown in Fig. 5 i s the hyperbola corresponding t o the angular momentum a t which the f i s s i o n b a r r i e r h a s become equal t o the binding e n e r g y of a neutron (or proton, whichever i s lower).
In this g e n e r a l neighborhood the de-excitation mode of a compound nucleus (if one w e r e f o r m e d ) would changc f r o m fission t o particle e m i s s i o n and the compound nucleus, having survived the r i s k of f i s - sion, could be detected a s such. T h e r e a r c indica- tions that in some c a s e s (c. g.
,
2 0 ~ e+
2 7 ~ 1 ) the curve ABC does indeed predict the approximate energy-dependence of the c r o s s section f o r thc formation and s u r v i v a l of a compound nucleus.One should, however, r e m e m b e r that, a s pointed out above, the prediction of the formation of a compound s y s t e m i s outside the scope of the con- siderations on which Fig. 5 i s based. T h e r e i s , in fact, a f u r t h e r c u r v e , o r family of c u r v e s , in the b v s Ecm plot, yet to be worked out on the 2
C5-58 W.J. SWIATECKI
b a s i s of t h e dynamics of f u s i o n , which w i l l d e s c r i - b e t h e compound n u c l e u s f a r m a t i o n c r o s s s e c t i o n , Only i f t h i s c r i t i c a l c u r v e h a p p e n s t o b e e n t i r e l y above t h e c a r v e ABC f i . e . , i f f o r m a t i o n imposes n o l i m i t a t i o n ) c a n t h e l a t t e r c u r v e b e e x p e c t e d t o r e - p r e s e n t t h e c r o s s s e c t i o n f o r t h e f o r m a t i o n and s u r v i v a l of a compound n u c l e u s . About t h e a s y e t u n d e t e r m i n e d c r i t i c a l c u r v e ( o r c u r v e s ) f o r compound n u c l e u s f o r m a t i o n we o n l y know t h a t i t must l i e be- l o w t h e B f z O h y p e r b o l a ( a n d t h a t i n some c a s e s , s u c h a s 2 0 ~ e + 2 7 ~ 1
,
i t seems t o l i e a b o v e t h e B f s B n h y p e r b o l a ) . But t h e c a l c u l a t i o n o f t h e c r i - t i c a l c o n d i t i o n i n a g e n e r a l c a s e r e m a i n s , as f a r a s I know, an u n s o l v e d problem i n f u s i o n dynamics.L e t me summarize t h e main p o i n t s of my t a l k
1. Heavy i o n r e s e a r c h i s e x p e c t e d t o h a v e a n i m - p a c t o n c h e m i s t r y , a t o m i c p h y s i c s and quantum e l e c t r o d y n a m i c s , a s w e l l a s on n u c l e a r p h y s i c s .
2. W i t h i n n u c l e a r p h y s i c s heavy i o n e x p e r i m e n t s w i l l r e l a x two a g e - o l d l i m i t a t i o n s : a t o m i c numbers less t h a n a b o u t 100, and n e a r - s p h e r i c a l n u c l e a r shapes.
3 , N u c l e a r macro-physics based on e x p l o i t i n g A >> 1 s h o u l d c a n e i n t o i t s own.
4. The f i r s t s t e p i n t h e new f i e l d i s o b v i o u s : t h e working o u t of p o t e n t i a l e n e r g y s u r f a c e s a s f u n c t i o n s o f t h e d e g r e e s o f freedom ( t h r e e o r more i n number).
5. More d i f f i c u l t b u t e s s e n t i a l s t e p s a r e : under- s t a n d i n g damping e f f e c t s ( v i s c o s i t y ) and e f f e c - t i v e i n e r t i a s i n dynamical problems.
I h a v e been a b l e t o mention o n l y a few o f t h e o u t s t a n d i n g problems i n f i s s i o n and f u s i o n phy- s i c s . I h a v e b i a s e d my t a l k t o w a r d m a c r o s c o p i c a s - p e c t s which I b e l i e v e a r e t h e d i s t i n g u i s h i n g f e a t u - r b s of heavy i o n p h y s i c s . I hope o t h e r s p e a k e r s w i l l complement t h e p i c t u r e by s t r e s s i n g t h e m i c r o s c o p i c a p p r o a c h which i s , i n p r i n c i p l e a t l e a s t , t h e more f u n d a m e n t a l one,
REFERENCES
An e x h a u s t i v e l i s t of r e f e r e n c e s on re- c e n t c a l c u l a t i o n s of P o t e n t i a l Energy s u r f a c e s , u s i n g t h e S t r u t i n s k y method, may be found i n J.R.Nix, Ann. Rev. Nucl. S c i e n c e 22, 1972. The f o u n d a t i o n s o f ~ t r u t i n s k y ' s m e t h o d a r e d e s c r i b e d i n M. B r a c k et a l . , Rev. Mod. Phys. 44, 1972, 3 2 0
,
where t h e f o r - m u l a t i o n of t h e dynamics i s a l s o cfiscussed.S. ~ j d r n h o l m ' s d i s c u s s i o n o f dynamics i n f i s s i o n and f u s i o n , g i v e n a t t h e Nordic-Dutch A c c e l e r a t o r Symposium a t E b e l t o f t , May 1971, i s due t o b e pu-
b l i s h e d soon. P r e l i m i n a r y r e s u l t s on e q u i l i b r i u m s h a p e s of r o t a t i n g s y s t e m s were g i v e n by S. Cohen, F. P l a s i l and W . J . S w i a t e c k i i n P r o c e e d i n g s of t h e T h i r d C o n f e r e n c e on R e a c t i o n s between Complex Nu- c l e i a t A s i l o m a r , e d i t e d by G h i r s o , Diamond and C o n z e t t , U n i v e r s i t y of C a l i f o r n i a P r e s s , 1963. A f u l l a c c o u n t i s i n p r e p a r a t i o n . A'. r e c e n t p a p e r which d i s c u s s e s g r a z i n g c o l l i s i o n s of heavy i o n s is R. B a s i l e e t a i , , J o u r n a l d e P h y s i q u e
33,
1972, 9.DISCUSSION
K, DIETRICH (Munich)
Would you p l e a s e comment on t h e f o l l o w i n g two i t e m s of y o u r t a l k :
I f I n one o f t h e s l i d e s a p o i n t C was marked which c o r r e s p o n d e d t o a b e g i n n i n g of i n s t a b i l i t y w i t h r e g a r d t o t h e "necking-in mode". T h i s mode i s s u r e l y of g r e a t i m p o r t a n c e f o r t h e l a s t sta- g e o f t h e f i s s i o n p r o c e s s . Do you t h i n k t h a t
t h e c r i t i c a l p o i n t C i s p r e d o m i n a n t l y d e t e r - mined by t h e b e h a v i o u r of t h e l i q u i d d r o p e n e r - gy o r do you e x p e c t t h e s h e l l c o r r e c t i o n s t o b e i m p o r t a n t i n t h i s r e s p e c t ?
2 ) From a n a l v e p o i n t of view I would t h i n k t h a t t h e d a t a on t h e k i n e t i c e n e r g y r e l e a s e i n f i s s i o n i n d i c a t e t h a t t h e v i s c o s i t y s h o u l d n o t b e overwhelmingly high. I a s k you t o com-
