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>
HAL Id: jpa-00215107
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Submitted on 1 Jan 1972
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Colloque C5, supplement au no 8-9, Tome 33, ofi it-Septembre 1972, page C5-45
JOURNAL DE PHYSIQUE
W. J. Swiatecki
Lawrence Berkeley L a b o r a t o r y , University of California
Berkeley, California 94720, USA
RBsume5 - Une approche macroscopique du phenomene de l a f i s s i o n e t de l a fusion 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 qualitativement. Les propriktes grossieres de l'energie 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 ~ liquide.
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 approach to fission and fusion physics i s formulated.
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 . The
problem of nuclear viscosity i s mentioned and c o m p a r i s o n s with liquid He3 a r e
made. Some effects of l a r g e angular momenta a r c d e s c r i b e d .
I
- INTRODUCTION
two hundred.
We a r e entering a new stage i n nuclear
physics, c h a r a c t e r i z e d by the availability of v e r y
heavy n u c l e a r projectiles.
It i s a good t i m e t o r e -
f l e c t on the place of the developing field of heavy
ion physics i n r e l a t i o n t o nuclear f i s s i o n and, m o r e
f o r c e s c r e a t e d in off-center collisions of heavy
nuclei will be sufficient to d e f o r m a nuclear 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 stretched-out configurations, s o m e t i m e s resembling even a dumb-bell.
generally, i n relation to nuclear physics a s a
whole.
Second, the e n o r m o u s centrifugal
The extension of atomic n u m b e r s b e yond a hundred m a y r e s u l t in the d i s c o v e r y of
During the 60 y e a r s of i t s h i s t o r y nuc l e a r physics h a s had t o contend with two l i m i t a tions and the h i s t o r i c role of heavy ion physics
will be to r e l a x them.
T h e s e limitations have been
superheavy e l e m e n t s in thc: vicinity of Z
Z
=
154
-
" 114,
164, and p e r h a p s in s o m e other regions.
The consequences of t h e s e anticipated
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 felt in
s o p e r v a s i v e that we have a l m o s t stopped being
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.
a w a r e of them.
even m o r e fundamental consequence of t h e extension of the l i m i t of nuclear s y s t e m s f r o m atomic
T h e s e limitations a r e :
1.
The r e s t r i c t i o n of atomic n u m b e r s t o l e s s than
n u r n b e r s . n e a r 100 to atomic n u m b e r s n e a r 200 has
to do with the c i r c u m s t a n c e that the m o s t intense
about 100.
2.
An
The a l m o s t exclusive r e s t r i c t i o n of nuclear
e l e c t r i c fields o c c u r r i n g anywhere in t h e u n i v e r s e
s h a p e s t o those c l o s e to a s p h e r e .
a r e to be found i n the 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 to 200 i n The introduction 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 limitations.
First,
i t will be possible to study a t l e a s t t r a n s i e n t nuc l e a r s y s t e m s with atomic n u m b e r s up t o about
(')Supported by the U. S. Atomic Energy
Commission.
c r e a s e s t h i s highest field only m o d e r a t e l y , but i t
s o happens that i t i s in this range of atomic numb e r s that an atomic e l e c t r o n becomes highly r e l a tivistic and the atomic 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
SWIATECKI
W.J.
F i r s t then, what a r e the relevant d e -
The advent of v e r y heavy- on a c c e l e r a 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 turn.
I1
-
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 scopic approach i s that collective r a t h e r than
single-particle d e g r e e s of f r e e d o m become con-
THE MACROSCOPIC APPROACH
To 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.
T h i s i s saying no m o r e than
venient and relevant.
that if you have tens o r hundreds of nucleons you
For
will t r y t o formulate your problem in t e r m s of a
the f i r s t t i m e we wlil have nuclear reactions where
few intelligently chosen groupings of the nucleon
both the t a r g e t and projectile s a t i s f y well the c r i -
coordinates r a t h e r than of the whole s e t .
t e r i o n f o r a m a c r o s c o p i c approximation, namely
a g r e a t simplification.
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
This i s
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
come into i t s own.
nuclear s u r f a c e (the
This i s t o be contrasted with conventiona l n u c l e a r reaction theory which, hxstorically, h a s
m o s t nuclei).
" leptodermous"
c h a r a c t e r of
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 roots
s
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.
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 -
On the other hand it i s
Thus one will t r y t o d e s c r i b e the s t a t e of
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.
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.
unsolved problems ?
which suggests that f o r the f i s s i o n of a nucleus in-
T h e r e i s a rule
t o n p a r t s , o r in the simultaneous fusion of n
In trying to answer t h e s e questions I
-
4 collective d e g r e e s of f r e e d o m
would like to remind you that many problems in
nuclei, about 9n
physics in g e n e r a l (not just nuclear physics) a r e
should be adequate f o r many purposes.
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 e
(If each
TABLE 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) Inertial Terms.
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
...
coordinates i t s location in space, and t h r e e E u l e r
1.
A separation coordinate, s a y a 2 .
angles i t s orientation.
2.
A necking o r neck-healing coordinate, say
3.
"4'
A m a s s a s y m m e t r y coordinate, s a y
This gives
freedom for n fragments.
9n d e g r e e s of
The minus four con-
s t r a i n s the total volume of the s y s t e m to a standa 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 binary p r o c e s s of fission o r fusion
(The a ' s may be related to but a r e not t o be thought
of a s identical with the coefficients in a Legendre
Polynomial expansion of the shape. )
(n = 2 ) this rule gives about 14 d e g r e e s of freedom.
One may 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 minimum
n e c e s s a r y to give the roughest qualitatively adc quate description of nuclear shapes relevant f o r
binary 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 -
@3.
The nuclear s h a p e s described by these
d e g r e e s of f r e e d o m can 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 section at constant
a3.
The a s y m m e t r y coordinate a 3 would stick
out of the plane of the paper and would correspond
to changing the m a s s ratio of the left and right
sides of the shapes.
thing like this
Fig. 1. An illustration of f i s s i o n and fusion s h a p e s d e s c r i b e d by 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 lines f o r binary and t e r n a r y ivlsions 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
1 l ( a 2 a 3 ~ 4 I)
repre -
t w e e n two h e a v y i o n s would be r e p r e s e n t e d b y a
the a 2 ~ 3 ~
s p4
a c e , with
path 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 -
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 -
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 , followed
f i g u r a t i o n s p e c i f i e d by a 2 a 3 a 4 , and
e i t h e r by 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 .
Conversely, in nuclear fission
one 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 into t h e t w o - f r a g m c n t v a l l e y , but s o m e t i m e s , p e r h a p s , into t h e
c h a n i c a l l y then i n s t e a d of p a t h s we s h a l l be d i s -
of the 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
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 -
"
This brings u s to the second
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 down
t h e E q u a t i o n s of Motion.
T h e r e will be t h r e c types
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 with
time derivatives
of the d e g r e e s of
f rcedom
Terms
I.
Potcntial Energy T e r m s
ZEROTH
2.
F r i c t i o n , Damping o r
Dissipative T e r m s
FIRST
3.
Inertial T e r m s
SECOND
Relevant Quantities
Rayleigh' s D i s s i p a t i o n
Function o r iW(a a a . . . )
2 3 4
Inertia Tensor
(ff2a3ff4.
.. )
1
Ma
[ The number
three of
d i f f e r c n t t e r m s i s no a c c i -
+.
S c h r B d i n g c r e q u a t i o n , we s h a l l n e e d i n f o r m a t i o n
item in the
Lf thc 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 -
zH9
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
t e m s considered.
t h r e e -fragment v a l l e y .
**
i dt
giving 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 function
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
d e n t : i t i s a s s o c i a t e d with the f a c t t h a t ( m a c r o -
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
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
theory:
and s e c o n d t i m e d e r i v a t i v e s of the d e g r c c s of f r e e -
1.
Potential
d o m , but no h i g h e r . ]
2.
Damping
3.
Incrtia.
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 be 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
Raylcigh dissipation function (a generalized f r i c tion).
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
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, and v c r y l i t t l e of 2.
I believe
and the 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
t h a t i n t h c f u t u r e we will h a v e t o c o n c e n t r a t e on
a c o m p l e x p o t e n t i a l V (a a a ) t i W ( a a a ). T h e
2 3 4
2 3 4
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 well a s q u a n t u m m e -
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
chanics give r i s e to a so-called inertia m a t r i x o r
Potential Energy.
t e n s o r Ma,&,(a2a3a4),which d e s c r i b e s the i n e r t i a l
1 1
r e s p o n s e of the 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
pulling 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 Damping
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 nuc l e a r potential energy, g r e a t p r o g r e s s h a s been
F I S S I O N AND H E A W I O N S
...
C5-49
m a d e i n the l a s t few y e a r s , principally a s a r e s u l t
nuclei with widely differing N/Z
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 -
brought into contact, a r e - d i s t r i b u t i o n of neutron
bining m a c r o s c o p i c and m i c r o s c o p i c theories.
We
a r e today i n a position where we can calculate the
ratios a r e
and proton d e n s i t i e s will take place such that an
approximately uniform value of N/Z
will obtain
potential e n e r g y of a nuclear s y s t e m a s a function
throughout the s y s t e m .
of N, Z and the nuclear shape, with an a c c u r a c y
tions f r o m uniformity, and slight fluctuations
of about *I MeV.
around i t , but it i s a f a i r approximation to d i s r c -
This i s 1 MeV out of a total
binding e n e r g y of s o m e 2000 MeV.
What have we l e a r n e d ?
e n e r g y a s a function of cu2cu3cr4
gard these at first.
The potential
i s given by a pock-
T h e r e will be slight dcvia-
F o r example, f o r tangent
s p h e r e s of radii
one m a y work out i n a
R1, RZ
closed f o r m an e x p r e s s i o n f o r the Zl/Ai r a t i o
m a r k e d s u r f a c e , consisting of a smooth p a r t and
of one of the nuclei divided by the
s h e l l effect pock-marks.
the wholc s y s t e m :
The c h a r a c t e r i s t i c un-
Z/A
r a t i o of
dulations of the smooth p a r t a r e generally of the
o r d e r of t e n s of MeV, the 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
smooth p a r t i s well understood.
The p o c k - m a r k s ,
though not s o well understood, a r e a l s o beginning
to b e r e l a t e d to s i m p l e f e a t u r e s of the nuclcar
shape.
In p a r t i c u l a r the l a s t y e a r o r two have
where e i s the proton c h a r g e , r o i s the n u c l c a r
2
radius constant (with e / r o equal to about
1.2 MeV), "coeff" i s the nuclear s y m m e t r y energy
brought the realization that m a j o r shell effects in
coefficient (about 30 MeV), and F i s the following
the nuclear potential e n e r g y a r e closely r e l a t e d to
function
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
that the f i r s t s t e p i n building up a t h e o r y of heavyion reactions i s quite c l e a r : to c o n s t r u c t the pot e n t i a l - e n e r g y s u r f a c e a s a function of suitable d e f -
According to this f o r m u l a the s m a l l e r of
two tangent nuclei will have a slightly higher N/:<
ormation coordinates describing colliding and
r a t i o , but only by a t m o s t a few percent.
fusing nuclei, using the Strutinsky method ( i m -
g r e a t o s t deviation of
proved and refined where n e c e s s a r y ) .
near
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 the resulting potential e n e r g y m a p s ,
especially i n the p o c k - m a r k s .
L e t m e just point
(The
ZL/$
f r o m unity o c c u r s
A1
R1/K2 = 0.3 and i s about 6 % f o r A = 216. )
F o r s o m e p u r p o s e s this charge 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
out s o m e of the m o s t primitive f e a t u r e s t o be e x -
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
pected i n the g r o s s s t r u c t u r e of the m a p s .
p r o p e r t i e s I will not s a y m o r e about it.
T h e r e a r e four i m p o r t a n t f e a t u r e s I wish
t o mention:
1.
Existence 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 the g r o s s
The equilibration of the neutron-proton
ratio.
potential e n e r g y on the a s y m m e t r y coordinate
u3,
the m o s t important thing to b e a r in mind i s that
2.
The existence of a c r i t i c a l m a s s a s y m m e t r y .
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
3.
The existence of two misaligned valleys.
and projectile.
4.
The effect of angular momentum.
t r e m e than the c r i t i c a l (i. e. f o r a relatively light
Equilibration of N n
F o r m a s s asymmetries more ex-
heavy ion and a heavy t a r g e t ) the t a r g e t nucleus
ratio
tends to suck up the projectile.
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 dispose of i t quickly.
It i s that if two
For asymmetries
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 heavy ions
W. J. SWIATECKI
C5-50
m o r e n e a r l y comparable with the t a r g e t ) the p r o -
m a t t e r out of the s m a l l nucleus into the big one. If the
jectile will tend to g r o w towards equality with 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
t a r g e t ( s e e Fig. 2).
be the case: the l a r g e r p a r t n e r would tend t o suck up
Most heavy ion e x p e r i m e n t s
Low Z ~ / A
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 tendency 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
2
qualitative change n e a r Z /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 Nuc 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
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 configurations 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 configurations tend toward s y m m e t r y .
shown.
sphere.
One valley s t a r t s f r o m the vicinity of the
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.
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 metry.
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,
( 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
much more. )
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
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.
I hope you c a n s e e the f i s s i o n valley with
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 (idealized 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.
F o r a sufficiently s m a l l nucleus in contact with a
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 -
ridge running f r o m A t o C . R e m e m b e r a l s o that
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
marks.
Between the two i s a
(One of t h e s e i s shown: the magic hole H,
FISSION AND HEAVY IONS
...
Fig. 3a. A sketch of the potential e n e r g y landscape i n ( ~ -2 ( ~ 4s p a c e , f o r a super-heavy nucleus. A few
shapes a r e shown for orientation. The portion BC of the fission valley i s separatcd f r o m thc portion
AD of the binary valley by a ridge running f r o m A t o C.
This vibration c o r r e s p o n d s t o
responsible f o r the stability of a super-heavy nu-
able damping).
cleus. )
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.
With this potential energy s u r f a c e a s
background we c a n now sketch in a f i s s i o n o r a
fusion path corresponding t o a dividing o r fusing
system.
In f i s s i o n the nucleus d e f o r m s , goes over
the saddle and r o l l s down the fission valley.
In the
f r a g m e n t excitation.
roughly the difference in energy between points C
and D.
necking-in i s l o s t and the s y s t e m i s injected into
Because of the misalign-
m e n t of the valleys the injection i s off-axis and the
Experimentally i t i s typically 20 - 4 0 MeV
and i s eventually dissipated in neutron evaporation
f r o m the f i s s i o n f r a g m e n t s .
Now about fusion.
neighborhood of point C equilibrium against
the two-fragment valley.
, to
The excitation energy i s
ogous.
The situation i s anal-
We proceed up the Two F r a g m e n t Valley
corresponding t o approaching nuclei.
At the point
A , corresponding to tangency, equilibrium against
r e p r e s e n t a t i v e point will vibrate around the a x i s
an increasing 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 lost
a s i t descends the two-fragment valley ( o r c r e e p s
toward the bottom of the valley if t h e r e i s a p p r e c i -
and the s y s t e m i s injected into the fission valley.
Because of the off-center injection t h e r e i s
W. J. SWIATECKI
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, especially when one i s trying to make super-heavy nuclei.
(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 ~ (which
e
i s one of the most
promising candidates for making superheavie s).
These estimates a r e , however, extremely uncertain because of a crucial missing piece of infarmation-namely,
how large i s nuclear viscosity.
In
other words, how strong i s the coupling of the collective degrees of freedom to the single particle
degrees of freedom that a r e not displayed explicitly
in a macroscopic treatment.
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 injection into the binary valley takes place.
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
vibration about (or creep toward) the axis of the
fission valley, which would eventually lead to excitation of the fused system.
The amount of excita-
tion i s roughly the difference between the energy a t
A and at B.
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
-
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 system over the saddle at S . If there were no viscosity-no conversion of collective into internal endisintegration.
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.
cause re-disintegration. In t e r m s of the Potential
energy map in Fig. 3 this means that in the vicin-
Each segment has radial focusing (e. g. ,
quadrupole lenses ).
A short pre -accelerator (the
ergy-an
additional 15 MeV might be enough.
But
FISSION AND HEAVY IONS
if the viscosity i s large-if
s a y nine tenths of the
...
proportional to A- 'I6. =or water a t oOC one
40
~f wc put A = 1oZ4 molecules
~y= ~ 1 / 6.
e x t r a energy goes into i n t e r n a l excitation, then
finds
one might have t o go to bombarding e n e r g i e s
T1
(R = 1.93 c m ) , we find T:, = 0.004, and f o r such
" 150 MeV above contact energy (coulomb b a r r i e r ) .
l a r g e d r o p s viscosity i s negligible.
It would be like trying to make two charged d r o p s
A = 300,
of honey coalesce by banging them together with
viscous o r c r e e p y limit.
This s e e m s to be the c a s e
for 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 ,
Most of the collision velocity would
high energy.
go into h e a t , and after making p a r t i a l contact the
-
--
But f o r
1 5 and now we a r e i n the e x t r e m e
T2
again without e v e r reaching the s p h e r i c a l configura-
and i s considerably l e s s for water at iOO°C
TI
(= 2.5), but we have h e r e a somewhat ominous
T2
result: a s i s well known in hydrodynamics v i s c o s -
tion.
ity b e c o m e s dominant f o r sufficiently s m a l l s y s -
hot, charged honey d r o p s would be torn a p a r t
So h e r e we come a c r o s s a c e n t r a l unanswered problem in heavy-ion physics.
A r e nuclei
t e m s , and f o r a l l ordinary liquids A = 300 i s i n deed srnall in this sense.
T h e r e would be g r e a t
viscous like honey o r mobile like water o r m e r -
difficulties in the way of making a superheavy ( ! )
cury?
d r o p of w a t e r with A
In hydrodynamics the distinction between
e x t r e m e l y viscous and e x t r e m e l y nonviscous types
of flow m a y be made quantitative i n t e r m s of the
relative magnitude of the second and third t e r m s
in the equations of motion.
300 out of two d r o p l e t s with
A = 232 and A = 76 respectively (simulating the
+
2 3 2 ~ h 7 6 ~ reaction).
e
Viscosity would i n a l l
likelihood prevent fusion of the d r o p s before r e disintegration caused by the coulomb repulsion.
Now water may be an entirely misleading
F o r l a r g e viscosity the
dissipative t e r m s dominate over the i n e r t i a l t e r m s
guide t o the p r o p e r t i e s of a quantum fluid like nu-.
(which m a y then be neglected).
clear matter.
F o r s m a l l viscosity
the damping t e r m s a r e neglected compared to the
inertial terms.
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
radius R, density p , s u r f a c e tension y
efficient of viscosity q.
and c o -
Such a drop, 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
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 considering another
3
. Inserting the
3
density, s u r f a c e tension and radius c o ~ s t a n of
t He
quantum fluid, namely liquid He
at 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 in poises. The v i s c o s 2
ity of ~e~ i n c r e a s e s rapidly with decreasing t e m -
c i r c u l a r frequency w determined by y and p (if
viscosity i s negligible), o r c r e e p back t o the
p e r a t u r e , and one finds that T ~ / Ti s ~120, 24,
s p h e r i c a l shape with a n e-folding t i m e determined
tively.
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
is
1.4 a t t e m p e r a t u r e s of 0.04"K. 0.I0K, IoK, r e s p e c 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 correspond to the lower range (0.04OK to
0.i0K) and we again come out with the indication that
viscosity would be dominant f o r s y s t e m s with AZ300.
given by
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 .
-+
e-foldin
T2 =
19
time =
" creep
-
4-7T-E
5 0
This r a t i o (or the
q
.
p a r a m e t e r " q/
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.
R = r 0A'/)
(with r o
If f o r R we put
1.2 X
c m f o r nuclei.
is ~
o r 1.93 X 10-' c m f o r w a t e r ) we find that T ~ / T
I
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 .
i t s e l f ) m a y be used a s a m e a s u r e of the relative
importance of viscosity.
These e s t i m a t e s suggest e x t r e m e l y high
viscosity f o r nuclear m a t t e r (except at v c r y high
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 like ~e~ would become superfluid, with very
low viscosity,
-
and the above e s t i m a t e s should a t
b e s t be used a s a guide at not too low t e m p e r a t u r e s
(corresponding t o nuclear excitations a t which
pairing effects a r e destroyed).
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
W. J. SWIATECKI
C5-54
of electromagnetic decays) the system would go on
TABLE 11
vibrating indefinitely in that state, with no damping
a t all.
Thus for small systems a t low temper-
Ratios of e -folding Time to Vibrational Times for
Drops of Water and ~ e 3 .
atures, where level spacings a r e large and indi-
Water
vidual levels stand out, the damping i s no longer
described by a viscosity coefficient and, in particular, 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
L
=2.5
T,
5=10OoC:q=0.00284 poise,
droplets with A-values of tens o r hundred of molecule 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 mhas recently discussed the question of
damping in relation to the presence o r absence of
0=0.04"K (corresponding to 0.26 o r 0.8 M~v)":
q=0.002 poise,
with a spontaneously fissioning isomeric state).
TI
- = 120
There will also soon be many heavy-ion experi-
T2
ments which in one way o r another will depend on
%=O.l°K(corresponding to 0.65 o r 2 M~v)":
q=0.0004 poise,
vibrational levels i n heavy nuclei (especially those
1
= 24
T,
6'=I0K (corresponding to 6.5 o r 20 M ~ v ) ~ ' :
q=0.000023 poise,
T1
=1.4
T2
r ~ h conversion
e
of the Kelvin temperatures for
33e3 to equivalent nuclear temperatures i s not unambiguous. 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 quasiparticle with an experimental effective m a s s equal
to 3.08 times the m a s s of ~ e 3 .
viscosity, and from these we shall gradually unravel 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, especially in their dependence on z'/A.
F o r the heavier
nuclei in particular (i.e. , for high z'/A) the saddle point shape for fission i s cylinder-like, o r even
spheroidal, and this means that there i s a considerable 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.
He
3
refer to very large systems, in which the
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 inadequate.
An extreme case that illustrates this point
is the damping of a f i r s t vibrational level of a nucleus.
If this level happens to be the f i r s t excited
state in the energy spectrum, then (in the absence
Why then hasn't the theory
of fission already provided us with the answer a s
regards viscosity? Because the relevant calculathe viscous descent f r o m saddle to s c i s -
tions for
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 experiment.
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
The calculation of the viscous descent
C5-55
dynamical calculations a t all bcfore this question
i s c l e a r e d up?
standing problem of d i r e c t relevance t o heavy ion
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
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
problem of a steady rotation of a s y s t e m without
In this c a s e the only
intrinsic change of shape.
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
n r Z ( a neck o r "window" ), lasting f o r a time of
object, s a y a fluid m a s s , it tends to flatten, and if
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
gases.
Not really-but
t h e r e i s a s o r t of
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-
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 / ~ ,h and t h i s leads t o a
4 3
d r a g f o r c e of ( n P /h ) Av p e r unit a r e a of contact
4
between the nuclei, o r t o T P /h
3
a s the
It
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 shapelike a n oval piece of soap-for
momenta before fission.
a range of angular
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
faster?
To d i s c u s s the equilibrium s h a p e s of a
d r a g co-
efficient" p e r unit a r e a , per unit velocity differen-
rotating nucleus in a m a c r o s c o p i c theory we s e t up
tial.
),
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
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)
2r
1
np4
X ( t i m e ) = - Av. r r 2 . - = 4T (
,3 P. where
h
T h e s e simple predictions dxsregard
kF = %/P.
$
..
potential cne rgy stationary:
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
= 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 s
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 hydrodynamics, i s the standard initial condition f o r
heavy ion collisions.
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).
In the meantime the problem of nuclear
viscosity r e m a i n s v e r y unclear.
where
Can one do any
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
the moment of i n e r t i a Q i s taken a s
rot
the rigid body moment.
where i n E
I would like to give you a b i r d ' s eye view
of what happens to this 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 this p u r -
pose i t i s convenient to introduce two dimensionless
numbers specifying the relative s i z e s of the t h r e e
energy components: coulomb, s u r f a c e and r o t a tional.
We pick the surface e n e r g y of the s p h e r i -
a s a unit and specify the
c a l shape E ( O )
surface
amount of c h a r g e on the nucleus by the usual f i s sility p a r a m e t e r x
where E ( O ) i s the coulomb e n e r g y of the spherical
shape.
We specify the amount of angular m o -
Fig. 4. A classification 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 shapes
disappear altogether a t x = 0.81, but a r e a l m o s t
gone a t x = 0.6.
mentum by y
E(o)
L e t m e i l l u s t r a t e the practical p r e d i c -
rot
Y = (0)Es
tions that follow f r o m t h i s kind of calculation.
Consider a collision of a heavy ion of
where E'O)
i s the rotational energy of a rigid
rot
s p h e r e with the given angular 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.
This d i a g r a m says the following
things: If you take any nucleus in the periodic
m a s s Mi and a nucleus of m a s s M2 a t impact par 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
cm'
F r o m conservation of e n e r g y and momentum i t
readily follows that the c l o s e s t distance of approach
of projectile and t a r g e t i s given by
r min where
table then, if t h e r e w e r e no shell effects and if the
moment of i n e r t i a w e r e rigid, the nucleus would a t
f i r s t deform into a f l a t shape.
The f i s s i o n b a r r i e r
d e c r e a s e s with increasing angular momentum and
vanishes along the upper c u r v e in Fig. 4.
The
H e r e V ( r ) i s the interaction potential between the
nuclei.
F o r a given value of
middle c u r v e shows the c r i t i c a l angular momentum
a t which the flat ground s t a t e shape goes over into
hyperbola when b
a t r i a x i a l shape.
plots nb
(Some of these shapes a r e like
2
2
r
min
this i s a
.
i s plotted vs E
c m ' (In such
i s proportional to a c r o s s section. ) If,
.
the right the saddle point shape f o r the fission of
r
i s chosen to be t h e s u m of the
m
radii of the two nuclei, r
= R.
R 2 , the c o r r e m
2
sponding hyperbola divides the b v s Ecm plane
the rotating d r o p i s stable against a s y m m e t r y , to
into two regions: d i s t a n t collisions where the nu-
the left i t i s unstable.
c l e i p a s s e a c h other without appreciable nuclear
slightly indented flattened dumb-bellsJ
The dashed
curve divides the x - y plane into two regions.
To
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 o r example,
+
i n t e r a c t i o n s , and close collisions where nuclear
interactions take place (the corresponding nbL
FISSION 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. )
,..
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 vanishes) the s y s t e m h a s too much angular
f
momentum to s t a y together and collisions c o r r e 2
sponding to those values of b and E
would lead
cm
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 nucleus 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,
h
5.
A classification of nuclear collisions in
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.
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
T h e bL vs Ecm plane c a n be f u r t h e r subdivided by c u r v e s corresponding t o loci of .fixed
angular momentum L.
(Here L = h L . ) Since L
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
i s given by
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 -
we have
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 ~ +e2 7 ~ 1the
)
curve ABC does indeed predict the approximate
energy-dependence of the c r o s s section f o r thc
F o r a given L this i s again a hyperbola in a plot
. If now f r o m the upper curve in
of b2 v s E
cm
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 2
sulting hyperbola will divide the b vs E c m plane
into two regions, a s shown in Fig. 5.
T o the right
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 considerations on which Fig. 5 i s based.
There is,
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
2
the b v s Ecm plot, yet to be worked out on the
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 -
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
be 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 ,
w i l l r e l a x two age-old l i m i t a t i o n s : atomic
Only i f t h i s c r i t i c a l c u r v e happens t o be e n t i r e l y
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
above t h e c a r v e ABC f i . e . ,
n u c l e a r shapes.
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 be expected t o r e -
3 , Nuclear macro-physics based on e x p l o i t i n g
A >> 1
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
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 :
undetermined 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
t h e working o u t of p o t e n t i a l energy s u r f a c e s
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-
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
low t h e
o r more i n number).
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 ,
2 0 ~ e +2 7 ~,
1i t seems t o l i e above t h e
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-
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 -
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 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 remains, as f a r
t i v e i n e r t i a s i n dynamical problems.
such a s
a s I know, an unsolved problem i n f u s i o n dynamics.
I have been a b l e t o mention o n l y a few o f
L e t me summarize t h e main p o i n t s of my
talk
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 phys i c s . I have b i a s e d my t a l k toward macroscopic a s -
1. Heavy i o n r e s e a r c h i s expected t o have a n
im-
p a c t o n c h e m i s t r y , atomic 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 physics.
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
approach 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
fundamental one,
REFERENCES
An e x h a u s t i v e list of r e f e r e n c e s on re-
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
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 ,
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,
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,
F. P l a s i l and W . J .
Ann. Rev. Nucl. S c i e n c e 22, 1972. The f o u n d a t i o n s
T h i r d Conference on R e a c t i o n s between Complex Nu-
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. Brack et
a l . , Rev. Mod. Phys. 44, 1972, 320
, where
the for-
m u l a t i o n of t h e dynamics i s a l s o cfiscussed.
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
c l e i a t Asilomar, 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
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
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
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
R. B a s i l e e t a i , , J o u r n a l d e Physique 33, 1972, 9.
Symposium a t E b e l t o f t , May 1971, i s due t o be pu-
DISCUSSION
t h e c r i t i c a l p o i n t C i s predominantly d e t e r -
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
mined by t h e behaviour of t h e l i q u i d d r o p e n e r -
i t e m s of your t a l k :
gy o r do you expect t h e s h e l l c o r r e c t i o n s t o
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
be important i n t h i s r e s p e c t ?
which corresponded t o a b e g i n n i n g of i n s t a b i l i t y
2 ) From a n a l v e p o i n t of view I would t h i n k
w i t h r e g a r d t o t h e "necking-in
i s s u r e l y of g r e a t importance f o r t h e l a s t sta-
t h a t t h e d a t a on t h e k i n e t i c energy r e l e a s e
i n f i s s i o n indicate t h a t t h e viscosity should
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
n o t be overwhelmingly high.
mode".
T h i s mode
I a s k you t o com-
...
FISSION AND HEAVY IONS
C5-59
even b e e a s i e r t o c a l c u l a t e t h e e f f e c t i n t h i s
ment on t h i s .
frame.
J. SWIATECKI ( B e r k e l e y )
( T r a n s i t i o n of e n e r g y from c o l l e c t i v e t o
m i c r o s c o p i c v a r i a b l e s . ) I n view of t y p i c a l exam-
1 ) The p o t e n t i a l e n e r g y s u r f a c e s I showed a r e
p l e s ( r a d i a t i o n damping) i t is,however,
meant t o r e p r e s e n t a v e r a g e s , a p a r t from s h e l l
sugges-
t e d t h a t t h e c o r r e s p o n d i n g term i n t h e Schrii-
e f f e c t s , which may i n t r o d u c e i m p o r t a n t modi-
dinger equation could contain higher d e r i v a t i -
f i c a t i o n s i n p a r t i c u l a r n u c l e i . These averages
ves (e.g.
ought t o g i v e a v e r a g e t m n d s c o r r e c t l y . They
3 r d o r d e r ) of t h e c o l l e c t i v e v a r i a b l e s .
We might have even t o d o w i t h a n " i n t e g r a l - e q u a -
s h o u l d a l s o be r e l e v a n t a t e x c i t a t i o n s -0
t i o n " i n a n a l o g y w i t h well-known c l a s s i c a l c a -
MeV) a t which s h e l l e f f e c t s h a v e been l a r g e l y
s e s (e.g.
e l a s t i c material).
washed o u t .
2 ) I t h i n k t h e s t u d y of t h e k i n e t i c e n e r g y r e -
J . SWIATECKI ( B e r k e l e y )
Thank you.
l e a s e d i n f i s s i o n i s very relevant t o t h e
q u e s t i o n of v i s c o s i t y .
F o r heavy n u c l e i i n
R.
QIASTEL (Bordeaux)
p a r t i c u l a r t h e r e i s a very considerabbe saddle-
A r e i n your o p i n i o n l i g h t p a r t i c l e s e m i t t e d i n
t o - s c i s s i o n s t a g e and i n t h e a b s e n c e of v i s c o -
f i s s i ~ ? r. e l a t e d t o a p a r t i c u l a r l o c u s i n t h e
0
s i t y a c o n s i d e r a b l e k i n e t i c e n e r g y ( ~ 4 MeV)
ternary f i s s i o n valley ?
i s p i c k e d up by t h e f r a g m e n t s d u r i n g t h i s
s t a g e . One would g u e s s t h a t a l a r g e v i s c o s i t y
J. SWIATECKI ( B e r k e l e y )
I t h i n k so. I n o r d e r t o f i t i n l i g h t p a r t i c l e s
would t e n d t o c u t down t h i s p a r t of t h e e n e r -
e m i t t e d i n f i s s i o n i n t o t h e scheme I d i s c u s s e d
gy, g i v i n g much lower f i n a l k i n e t i c e n e r g i e s .
one would have t o g e n e r a l i z e t h e c o n f i g u r a t i o n
N i x ' s c a l c u l a t i o n s on t h e k i n e t i c e n e r g y r e -
space
l e a s e i n t h e non-viscous
d e s c r i b e s t h e mass r a t i o of t h e m i d d l e fragment
f i s s i o n of a l i q u i d
aZ,a3,a4
t o a f o u r t h dimension which
drop a g r e e f a i r l y well w i t h experiment, but
t o t h e s i d e f r a g m e n t s . Then a n e x t e n s i o n of t h e
one c a n n o t b e s u r e i f t h i s i n d i c a t e s t h a t v i s -
t e r n a r y v a l l e y I d i s c u s s e d i n t h e d i r e c t i o n of
c o s i t y is low b e c a u s e w e d o n ' t know how s e n s i -
t h e new d e g r e e of freedom would c o r r e s p o h d t o
t i v e t h e c a l c u l a t i o n s a r e t o v i s c o s i t y . The
t h e e m i s s i o n of l i g h t p a r t i c l e s i n f i s s i o n .
c a l c u l a t i o n of t h e v i s c o u s d i v i s i o n of a nu-
would b e an i n t e r e s t i n g problem t o f o l l o w up.
It
c l e u s i s a n o u t s t a n d i n g t h e o r e t i c a l problem.
Mrs. G. SCHARFF-GOLDIABER (Brookhaven N a t l . Lab. )
F. PLASIL ( Oak Ridge Natl. Lab.)
You may h a v e answered my q u e s t i o n a l r e a d y i n
I t a p p e a r s t h a t v i s c o s i t y shows s t r o n g s h e l l
e f f e c t s . Do you a g r e e w i t h t h i s view ?
r e s p o n s e t o Dr. D i e t r i c h ' s q u e s t i o n , b u t what
I would l i k e t o go back t o i s t h e problem of
J. SWIATECKI ( B e r k e l e y )
Certainly.
e x t r a c t i n g i n f o r m a t i o n on v i s c o s i t y from d a t a
which g i v e s t h e dependence of f i s s i o n f r a g m e n t
k i n e t i c e n e r g i e s on t h e e x c i t a t i o n e n e r g y of
t h e f i s s i o n i n g nucleus.
T h e r e i s o f a i r amount
of d a t a a v a i l a b l e . I s i t a m a t t e r 09 t h e o r e t i c a l developments t h a t one needed o r s h o u l d
some more e x p e r i m e n t s b e performed ?
Just a s there a r e strong s h e l l ef-
f e c t s i n t h e p o t e n t i a l e n e r g y , one e x p e c t s t o
f i n d s t r o n g s h e l l e f f e c t s i n t h e v i s c o s i t y and
i n t h e i n e r t i a tensor (one already
knows
that
c a l c u l a t i o n s show t h i s f o r t h e i n e r t i a t e n s o r ) .
What I t h i n k one s h o u l d d o i n a l l t h r e e c a s e s
i s t o u s e a S t r u t i n s k y t y p e a p p r o a c h of f i r s t
c a l c u l a t i n g smooth v i s c o s i t i e s and i n e r t i a s ,
J.
SWIATECKI ( B e r k e l e y )
I t h i n k i t i s d e f i n i t e l y a m a t t e r of t h e o r y .
The e x p e r i m e n t s h a v e been a v a i l a b l e f o r a l o n g
time.
K.
smooth F e r m i g a s , and t h e n a d d i n g on s h e l l e f f e c t s . We h a v e l e a r n t t h a t t h i s i s t h e i n t e l l i g e n t way t o p r o c e e d i n t h e c a s e of t h e p o t e n -
t i a l e n e r g y and I hope a s i m i l a r method w i l l
BLEULER (Bonn)
Viscosity
p r e t e n d i n g t h e l e v e l s p e c t r u m i s t h a t of a
-
i n c o n t r a s t t o a n e a r l i e r comment*
b u t i n p e r f e c t agreement w i t h t h e s p e a k e r - is8
w e l l - d e f i n e d e f f e c t i n Quantum Theory. I t might
work f o r t h e o t h e r two quantities.
' The
A.
o r i g i n of t h i s comment was a q u e s t i o n by
ZUKER ( S a c l a y ) i n which
h e had c a s t d o u b t s
W. J. SWIATECKI
on t h e u s e f u l n e s s of t h e ( c l a s s i c a 1 ) concept of
skipped from t h e proceedings a f t e r a l o n g p r i -
v i s c o s i t y i n n u c l e a r physics. The term " c l a s s i -
v a t e d i s c u s s i o n between W.J.
cal" may be considered o b j e c t i o n a b l e and could
A.
simply be replaced by "macroscopic".
d i s a g r e e on what i s a v a l i d quantum mechanical
The t h r u s t
of t h e q u e s t i o n and i t s answer was not however
a simple m a t t e r of terminology but t h e y were
SWIATECKI and
Z'JKER d u r i n g which they amicably agreed t o
d e s c r i p t i o n of n u c l e a r dynamics.