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Axial asymmetric fission-barrier distortions
Volume 34B, n u m b e r 4
AXIAL
PHYSICS
ASYMMETRIC
LETTERS
1 March 1971
FISSION-BARRIER
DISTORTIONS
*
H i l d e g a r d S C H U L T H E I S a n d R. S C H U L T H E I S Sektion Physik der Uni~,ersittit Miinchen, Theoretische Physik, M~nchen, Germany Received 15 D e c e m b e r 1970 P o t e n t i a l - e n e r g y s u r f a c e s of even actinides are calculated according to t r i - a x i a l ellipsoids by a modified Myers-Swiatecki energy formula. The curve of m i n i m u m energy connecting the ground state and the second hump of the fission b a r r i e r c o r r e s p o n d s to non-axial distortions.
On t h e p a t h f r o m t h e n u c l e a r g r o u n d s t a t e to fission the liquid-drop model energetically fav o r s a x i a l s y m m e t r i c d e f o r m a t i o n s of t h e n u c l e u s . On t h e o t h e r h a n d t h e r e a r e s o m e i n d i c a t i o n s [1], t h a t s h e l l e f f e c t s p r e f e r n u c l e a r s h a p e s w i t h o u t a n a x i s of s y m m e t r y . T h e r e f o r e i n a r e a l i s t i c m o d e l of n u c l e a r f i s s i o n n o n - a x i a l t e n d e n c i e s of t h e s h e l l e n e r g y m i g h t p o s s i b l y p r e d o m i n a t e o v e r t h e s t a b i l i z i n g i n f l u e n c e of t h e liquid-drop energy against non-axial distortions. By u s e of S t r u t i n s k y ' s c o m p u t e r r o u t i n e , a l t e r e d f o r t r i - a x i a l e l l i p s o i d s , P a s h k e v i c h [2] i n deed has found for some actinides an axial asymm e t r i c of t h e n u c l e a r s h a p e a t t h e s a d d l e p o i n t between ground state and isomeric state. But the computation is not quite satisfactory, because the results, as far as they concern axial-symmetric distortions, indicate significant disagreem e n t s w i t h m o r e r e f i n e d c a l c u l a t i o n s in t h i s f i e l d [3]. So t h e / 3 - v a l u e s of t h e s e c o n d a r y m i n i m a g e n e r a l l y a r e t o o h i g h , a n d f o r s o m e of t h e nuclei calculated the secondary minima and the s e c o n d h u m p s of t h e f i s s i o n b a r r i e r a r e l a c k i n g a t all. T h e r e f o r e we h a v e c o m p u t e d t h e e n e r g y s u r f a c e of s o m e e v e n a c t i n i d e s a c c o r d i n g to a n other method. As our results for axial nuclear s h a p e s a r e i n good a g r e e m e n t w i t h t h o s e of r e f . [3], w e h o p e t h e y a r e r e a l i s t i c f o r a x i a l - a s y m m e t r i c d e f o r m a t i o n s too. In t h e p r e s e n t p a p e r t h e s h e l l e n e r g y i s e x p r e s s e d by a c u r v a t u r e t e r m b a s e d on t h e M y e r s S w i a t e c k i s h e l l f u n c t i o n [4]. It i s c a l c u l a t e d by a w e i g h t e d m e a n o v e r t h e s h e l l e n e r g i e s of s m a l l s p h e r i c a l s e c t i o n s i n s c r i b e d to t h e n u c l e a s , t h a t h a s b e e n a l r e a d y a p p l i e d to a b o u t 50 n u c l e i , a s suming axial-symmetric s h a p e s [5]. F o r m a l l y it * This work has been supported in p a r t by the Bundesm i n i s t e r i u m flit Bildung und Wissenschaft.
consists of an integral over the nuclear surface, the integrand of which is taken from the above shell function for spherical nuclei. Now the nuclear shape is approximated by a tri-axial ellipsoid. It is described by two degrees of freedom, one for deviations from spherical symmetry (for example /3 or •), and one for deviations from axial symmetry (for ins t a n c e ~). T h e f o l l o w i n g f i g u r e s s h o w l e v e l l i n e s of t h e p o t e n t i a l - e n e r g y s u r f a c e a s a f u n c t i o n of t h e n u c l e a r s h a p e . T h e a x e s of t h e c o n t o u r m a p c o r r e s p o n d to a x i a l s y m m e t r i y : t h e o r i g i n to a s p h e r e , t h e a b s c i s s a to p r o l a t e (~ = 0°), t h e o r d i n a t e to o b l a t e s p h e r e s (), = 60o). W e u s e s h a p e p a r a m e t e r s p a n d q. d e f i n e d a s t h e d i f f e r e n c e s of t h e s e m i - a x e s of t h e ( t r i - a x i a l ) e l l i p s o i d in u n i t s of the nuclear charge radius Rc: p=
(z o - ro)/R
c ,
q = (Yo - X o ) / R c
,
where Xo, Yo, Zo are the semi-axes lipsoid, Z o being the major one,
of t h e e l -
Xo, Yo < Zo T h e r e f o r e q = 0 a n d p > 0 c o r r e s p o n d s to p r o l a t e , P = 0 a n d q > 0 to o b l a t e d e f o r m a t i o n s , P = q = 0 t o t h e s p h e r e , p > 0 a n d q > 0 to s h a p e s w i t h o u t a n a x i s of s y m m e t r y . F o r a x i a l - s y m m e t r i c s h a p e s o u r p a r a m e t e r s a r e r e l a t e d to t h e u s u a l o n e s by p ~ •, q = 0 f o r y = 0 ° I} 3/-~-fi4~ p=O, q~-• for V=60 ° •~2 4nn = 0 . 9 5 f i if • << 1. T h e g e n e r a l r e l a t i o n f o r ~ = 0 i s p = exp
(~•)-exp(-~•)•+~•2+~•3+ =
....
245
O0
I
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U 238
01
TH22B
4
1
5
0
0:2
2
6
7
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0 T O T A L ENERGY
5
T O T A L ENERGY
0.'4
05
o:s
06
11
0;~
16
p
p
~
oo
0.2
O.
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-2
CF2Sg
01
-I.5
-2.5
-2
02
0
0
2
0'4
04 T O T A L ENERGY
1
TOTAL_ ENERGY
0'.3
1
o:s
.
06
5
06
,
6
+
p
P = (z o - yo)/R
Fig. 1. Contour map of the potential e n e r g y of 228Th, 238U, 250Cf and 254Fm. The s c a l e is 0.5 MeV. The shape p a r a m e t e r s a r e the d i f f e r e n c e s c , and q = (Yo - %o)/Rc of the ellipsoidal s e m i - a x e s in units of the n u c l e a r c h a r g e r a d i u s R. Oblate s p h e r o i d s lay on the ordinate, p r o l a t e on the a b s c i s s a . F o r s m a l l d e f o r m a t i o n s p e q u a l s N i l s s o n ' s ~. Saddle points a r e m a r k e d by c r o s s e s . The c u r v e connecting the m i n i m a a c c o r d i n g to r e l a t i o n (4) is plotted for 238U, 250Cf and 25~4Fm.
02!
oo
0.0
0.2
3
p
0
~o~
C~ c/3
c~
~
¢-
Volume 34B, number 4
PHYSICS L E T T E R S
Calculations of the potential energy s u r f a c e have been p e r f o r m e d for 228Th, 232Th, 234U, 236U, 238U, 240pu ' 242pu, 244Cm, 250Cf, 254Fm, 256Fm and 260Fm. The ground state (g) of all nuclei c o n s i d e r e d is found to have the shape of a p r o l a t e spheroid, the coordinates of which can be a p p r o x i m a t e d within 0.01 by the r e lation /~ = 0.002Z - 0.006N+ 0.98,
qg = 0,
(1)
Z and N being the proton or n e u t r o n n u m b e r , respectively. Expect for 228Th all oblate m i n i m a have t u r n e d out to be i n s t a b l e a g a i n s t n o n - a x i a l d e f o r m a t i o n s . They a r e connected with the p r o l a t e m i n i m u m by a monotonically descending valley, and their energy exceeds that of the ground state by some MeV. Only for 228Th, the lightest n u cleus inquired, the oblate m i n i m u m is s e p e r a t e d from the ground state by a 0.5 MeV b a r r i e r . M o r e o v e r , t h e r e is another flat m i n i m u m for the s p h e r i c a l shape. But these two local m i n i m a p r e s u m a b l y do not p e r m i t of stable states b e c a u s e of the z e r o - p o i n t motion. In all nuclei c o n s i d e r e d the n o n - a x i a l connection between p r o l a t e and oblate m i n i m u m is, a c c o r d i n g to our c a l c u l a t i o n s , m o r e d i s t i n c t i v e than in P a s h k e v i c h ' s r e s u l t s [2]. Since the valley from p r o l a t e to oblate s p h e r e s g e n e r a l l y is s e v e r a l MeV deeper than the path via the sphere, it is of some i n t e r e s t for collective motions and shall be i n v e s t i g a t e d by s i m i l a r c a l c u l a t i o n s in the r a r e - e a r t h region. Both, the p r o l a t e and the oblate m i n i m u m a r e r a t h e r soft with r e s p e c t to n o n - a x i a l d i s t o r t i o n s . The ground state m i n i m u m b r o a d e n s m o r e and m o r e in the a s y m m e t r i c region with i n c r e a s i n g m a s s n u m b e r , and the o b l a t e - p r o l a t e difference extends from 1.7 MeV (228Th) up to 4.9 MeV (254Fm). The p o t e n t i a l - e n e r g y m i n i m u m c o r r e s p o n d i n g to the f i s s i o n i s o m e r i c state (f) is g e n e r a l l y s i t u ated on the a b s c i s s a , a p p r o x i m a t e l y at Pf = 0 . 0 1 Z - 0 . 0 1 N + 1.04 ,
qf = 0
(2)
The stiffness of y - v i b r a t i o n a p p e a r s to be s m a l l e r in the s e c o n d a r y m i n i m u m than in the ground state. This kind of shape dependence is c o n t r a r y to P a s h k e v i c h ' s r e s u l t [2], that does not differ f r o m the p r e d i c t i o n s of the pure l i q u i d - d r o p model [6]. This may be of some i m p o r t a n c e in a d y n a m i c f i s s i o n theory. F o r two nuclei (238U and 250Cf) the s e c o n dary m i n i m u m s e e m s to be somewhat n o n - a x i a l , but this region i s v e r y flat, and the energy diff e r e n c e to the p r o l a t e spheroid is l e s s than 0.1 MeV.
i March 1971
The energy of the i s o m e r i c state above the ground state is too high c o m p a r e d with ref. [3], b e c a u s e in the p r e s e n t paper the l i q u i d - d r o p energy, being calculated for ellipsoids only, tends to infinity with i n c r e a s i n g deformation, c o n t r a r y to the energy of a m o r e r e a l i s t i c (constricting} n u c l e a r shape. F o r none of the nuclei i n v e s t i g a t e d the saddle point between the f i r s t and the second m i n i m u m is situated on the a b s c i s s a . The a s y m m e t r y f l u c tuates between q = 0.04 (228Th)and q = 0.14 (254Fm), that m e a n s , the difference between the axes of the spheroid a m o u n t s up to 28% of the n u c l e a r charge r a d i u s . By ignorging the n o n - a x i a l d e g r e e of f r e e d o m the s a d d l e - p o i n t e n e r g y is i n c r e a s e d by 0.1 MeV (228Th) up to 0.6MeY (250Cf). In the l a t t e r case the f i r s t hump of the f i s s i o n b a r r i e r would be o v e r e s t i m a t e d by about 10% of its height above the ground state and by about 20% r e l a t i v e to the s e c o n d a r y m i n i m u m . M o r e o v e r , the a s y m m e t r i c saddle point is shifted to s m a l l e r fl-values by Aft ~ A~ ~ Ap ~0.05 c o m p a r e d with the s p u r i o u s saddle point on the axis. F o r all nuclei other than 228Th the c o o r d i n a t e s of the saddle points (s) a r e , c o n t r a r y to [2], slowly v a r y i n g functions of the n e u t r o n and proton n u m b e r s N and Z. The following e m p i r i c a l r e l a t i o n is valid within an e r r o r of 0.02: Ps = -0.006Z - 0.0025N + 1.335, qs = 0.01Z - 0.0025N - 0.475. In addition, the whole curve following the g r a dient between the saddle point and the m i n i m a takes its c o u r s e in the a s s y m m e t r i c region. As a s i m p l e a p p r o x i m a t i o n of this lowest path b e tween both the m i n i m a , a parabola is plotted in the f i g u r e s , which a p p r o x i m a t e l y p a s s e s through the t h r e e points. It is defined by the relation: p g p f _ p ( p g + pf) + p2 q : qs
PgPf - P's(Pg + P f ) + Ps 2
(4)
C o n t r a r y to the shapes g, f and s, which a r e i n v a r i a n t s with r e s p e c t to p a r a m e t e r t r a n s f o r m a tion, the "lowest path" depends somewhat on the choice of the d e f o r m a t i o n p a r a m e t e r s . Just as the f i r s t hump of the f i s s i o n b a r r i e r the second hump is g e n e r a l l y reduced, if one a l lows for axial a s y m m e t r y . But the effect is s m a l l e r c o m p a r e d with the f i r s t hump. The m a x i m u m energy difference is 0.4 MeV (236U). Again, the second hump is shifted towards the s p h e r e like the f i r s t one by the n o n - a x i a l d e g r e e 247
Volume 34B, n u m b e r 4
PHYSICS
of f r e e d o m . In t h e h e a v i e s t n u c l e u s ( 2 6 0 F m ) no a s y m m e t r y of the s e c o n d h u m p i s f o u n d , but in this case the second minimum might be situated on t h e e n e r g y s l o p e t o w a r d s t h e s c i d d i o n p o i n t , if the nuclear shape would be more realistic than spheroidal ones. A p e c u l i a r i t y o c c u r s in t h e s e c o n d h u m p of t h e f i s s i o n b a r r i e r of 250Cf. T h e r e a r e two s a d d l e p o i n t s w i t h e q u a l e n e r g y , s e p a r a t e d by a n e n e r gy h i l l of 0.2 M e V . One m a y s p e c u l a t e t h a t two p a t h s to f i s s i o n a r e p o s s i b l e , c o r r e s p o n d i n g to d i f f e r e n t d e f o r m a t i o n s . T h e y would c a u s e d i f f e r ent k i n e t i c e n e r g i e s of t h e f i s s i o n f r a g m e n t s a n d p e r h a p s a n e x t r a a s y m m e t r y in t h e m a s s d i s t r i bution. W e a r e g r a t e f u l to P r o f e s s o r G. S ~ l s s m a n n f o r stimulating discussions and valuable suggestions.
248
LETTERS
1 March 1971
References [1] D.A. Zaikin, Soviet P h y s i c s J E T P 8 (1959) 365; T.D. Newton, Can. J. Phys. 38 (1960) 700: K. Kumar. M. B a r a n g e r , Nuc[.Phys. A l l 0 (1968) 529; E.Nadjakov. Y. Piperova, D. Karadjov, I. Sivriev. Comptes rendus de l'Acad~mie bulgare des Sciences 22 (1969) 639. [2] V.V. Pashkevich. NucI. Phys. A133 (1969) 400. [3] S.G. Ni[sson. C . F . T s a n g , A.Sobiczewski. Z. Szymafiski. S. Wvcech. C.Gustafson, I.-L. Lamm. P. MSller and B. Nilsson, Nucl. Phys. A131 (1969) 1. [4} W. D. Myers and W. J. Swiateeki. University of California, UCRL 17070 (1969). [5] H. Schultheis, R. Schultheis. G. Siissmann. Nuel. Phys. A144 (1970) 545. [6] M. Zie|ifiska-PfabS. Phys. L e t t e r s 29B (1969) 280.
AXIAL
PHYSICS
ASYMMETRIC
LETTERS
1 March 1971
FISSION-BARRIER
DISTORTIONS
*
H i l d e g a r d S C H U L T H E I S a n d R. S C H U L T H E I S Sektion Physik der Uni~,ersittit Miinchen, Theoretische Physik, M~nchen, Germany Received 15 D e c e m b e r 1970 P o t e n t i a l - e n e r g y s u r f a c e s of even actinides are calculated according to t r i - a x i a l ellipsoids by a modified Myers-Swiatecki energy formula. The curve of m i n i m u m energy connecting the ground state and the second hump of the fission b a r r i e r c o r r e s p o n d s to non-axial distortions.
On t h e p a t h f r o m t h e n u c l e a r g r o u n d s t a t e to fission the liquid-drop model energetically fav o r s a x i a l s y m m e t r i c d e f o r m a t i o n s of t h e n u c l e u s . On t h e o t h e r h a n d t h e r e a r e s o m e i n d i c a t i o n s [1], t h a t s h e l l e f f e c t s p r e f e r n u c l e a r s h a p e s w i t h o u t a n a x i s of s y m m e t r y . T h e r e f o r e i n a r e a l i s t i c m o d e l of n u c l e a r f i s s i o n n o n - a x i a l t e n d e n c i e s of t h e s h e l l e n e r g y m i g h t p o s s i b l y p r e d o m i n a t e o v e r t h e s t a b i l i z i n g i n f l u e n c e of t h e liquid-drop energy against non-axial distortions. By u s e of S t r u t i n s k y ' s c o m p u t e r r o u t i n e , a l t e r e d f o r t r i - a x i a l e l l i p s o i d s , P a s h k e v i c h [2] i n deed has found for some actinides an axial asymm e t r i c of t h e n u c l e a r s h a p e a t t h e s a d d l e p o i n t between ground state and isomeric state. But the computation is not quite satisfactory, because the results, as far as they concern axial-symmetric distortions, indicate significant disagreem e n t s w i t h m o r e r e f i n e d c a l c u l a t i o n s in t h i s f i e l d [3]. So t h e / 3 - v a l u e s of t h e s e c o n d a r y m i n i m a g e n e r a l l y a r e t o o h i g h , a n d f o r s o m e of t h e nuclei calculated the secondary minima and the s e c o n d h u m p s of t h e f i s s i o n b a r r i e r a r e l a c k i n g a t all. T h e r e f o r e we h a v e c o m p u t e d t h e e n e r g y s u r f a c e of s o m e e v e n a c t i n i d e s a c c o r d i n g to a n other method. As our results for axial nuclear s h a p e s a r e i n good a g r e e m e n t w i t h t h o s e of r e f . [3], w e h o p e t h e y a r e r e a l i s t i c f o r a x i a l - a s y m m e t r i c d e f o r m a t i o n s too. In t h e p r e s e n t p a p e r t h e s h e l l e n e r g y i s e x p r e s s e d by a c u r v a t u r e t e r m b a s e d on t h e M y e r s S w i a t e c k i s h e l l f u n c t i o n [4]. It i s c a l c u l a t e d by a w e i g h t e d m e a n o v e r t h e s h e l l e n e r g i e s of s m a l l s p h e r i c a l s e c t i o n s i n s c r i b e d to t h e n u c l e a s , t h a t h a s b e e n a l r e a d y a p p l i e d to a b o u t 50 n u c l e i , a s suming axial-symmetric s h a p e s [5]. F o r m a l l y it * This work has been supported in p a r t by the Bundesm i n i s t e r i u m flit Bildung und Wissenschaft.
consists of an integral over the nuclear surface, the integrand of which is taken from the above shell function for spherical nuclei. Now the nuclear shape is approximated by a tri-axial ellipsoid. It is described by two degrees of freedom, one for deviations from spherical symmetry (for example /3 or •), and one for deviations from axial symmetry (for ins t a n c e ~). T h e f o l l o w i n g f i g u r e s s h o w l e v e l l i n e s of t h e p o t e n t i a l - e n e r g y s u r f a c e a s a f u n c t i o n of t h e n u c l e a r s h a p e . T h e a x e s of t h e c o n t o u r m a p c o r r e s p o n d to a x i a l s y m m e t r i y : t h e o r i g i n to a s p h e r e , t h e a b s c i s s a to p r o l a t e (~ = 0°), t h e o r d i n a t e to o b l a t e s p h e r e s (), = 60o). W e u s e s h a p e p a r a m e t e r s p a n d q. d e f i n e d a s t h e d i f f e r e n c e s of t h e s e m i - a x e s of t h e ( t r i - a x i a l ) e l l i p s o i d in u n i t s of the nuclear charge radius Rc: p=
(z o - ro)/R
c ,
q = (Yo - X o ) / R c
,
where Xo, Yo, Zo are the semi-axes lipsoid, Z o being the major one,
of t h e e l -
Xo, Yo < Zo T h e r e f o r e q = 0 a n d p > 0 c o r r e s p o n d s to p r o l a t e , P = 0 a n d q > 0 to o b l a t e d e f o r m a t i o n s , P = q = 0 t o t h e s p h e r e , p > 0 a n d q > 0 to s h a p e s w i t h o u t a n a x i s of s y m m e t r y . F o r a x i a l - s y m m e t r i c s h a p e s o u r p a r a m e t e r s a r e r e l a t e d to t h e u s u a l o n e s by p ~ •, q = 0 f o r y = 0 ° I} 3/-~-fi4~ p=O, q~-• for V=60 ° •~2 4nn = 0 . 9 5 f i if • << 1. T h e g e n e r a l r e l a t i o n f o r ~ = 0 i s p = exp
(~•)-exp(-~•)•+~•2+~•3+ =
....
245
O0
I
01
U 238
01
TH22B
4
1
5
0
0:2
2
6
7
0
o:a
0 T O T A L ENERGY
5
T O T A L ENERGY
0.'4
05
o:s
06
11
0;~
16
p
p
~
oo
0.2
O.
FM?___54
-2
CF2Sg
01
-I.5
-2.5
-2
02
0
0
2
0'4
04 T O T A L ENERGY
1
TOTAL_ ENERGY
0'.3
1
o:s
.
06
5
06
,
6
+
p
P = (z o - yo)/R
Fig. 1. Contour map of the potential e n e r g y of 228Th, 238U, 250Cf and 254Fm. The s c a l e is 0.5 MeV. The shape p a r a m e t e r s a r e the d i f f e r e n c e s c , and q = (Yo - %o)/Rc of the ellipsoidal s e m i - a x e s in units of the n u c l e a r c h a r g e r a d i u s R. Oblate s p h e r o i d s lay on the ordinate, p r o l a t e on the a b s c i s s a . F o r s m a l l d e f o r m a t i o n s p e q u a l s N i l s s o n ' s ~. Saddle points a r e m a r k e d by c r o s s e s . The c u r v e connecting the m i n i m a a c c o r d i n g to r e l a t i o n (4) is plotted for 238U, 250Cf and 25~4Fm.
02!
oo
0.0
0.2
3
p
0
~o~
C~ c/3
c~
~
¢-
Volume 34B, number 4
PHYSICS L E T T E R S
Calculations of the potential energy s u r f a c e have been p e r f o r m e d for 228Th, 232Th, 234U, 236U, 238U, 240pu ' 242pu, 244Cm, 250Cf, 254Fm, 256Fm and 260Fm. The ground state (g) of all nuclei c o n s i d e r e d is found to have the shape of a p r o l a t e spheroid, the coordinates of which can be a p p r o x i m a t e d within 0.01 by the r e lation /~ = 0.002Z - 0.006N+ 0.98,
qg = 0,
(1)
Z and N being the proton or n e u t r o n n u m b e r , respectively. Expect for 228Th all oblate m i n i m a have t u r n e d out to be i n s t a b l e a g a i n s t n o n - a x i a l d e f o r m a t i o n s . They a r e connected with the p r o l a t e m i n i m u m by a monotonically descending valley, and their energy exceeds that of the ground state by some MeV. Only for 228Th, the lightest n u cleus inquired, the oblate m i n i m u m is s e p e r a t e d from the ground state by a 0.5 MeV b a r r i e r . M o r e o v e r , t h e r e is another flat m i n i m u m for the s p h e r i c a l shape. But these two local m i n i m a p r e s u m a b l y do not p e r m i t of stable states b e c a u s e of the z e r o - p o i n t motion. In all nuclei c o n s i d e r e d the n o n - a x i a l connection between p r o l a t e and oblate m i n i m u m is, a c c o r d i n g to our c a l c u l a t i o n s , m o r e d i s t i n c t i v e than in P a s h k e v i c h ' s r e s u l t s [2]. Since the valley from p r o l a t e to oblate s p h e r e s g e n e r a l l y is s e v e r a l MeV deeper than the path via the sphere, it is of some i n t e r e s t for collective motions and shall be i n v e s t i g a t e d by s i m i l a r c a l c u l a t i o n s in the r a r e - e a r t h region. Both, the p r o l a t e and the oblate m i n i m u m a r e r a t h e r soft with r e s p e c t to n o n - a x i a l d i s t o r t i o n s . The ground state m i n i m u m b r o a d e n s m o r e and m o r e in the a s y m m e t r i c region with i n c r e a s i n g m a s s n u m b e r , and the o b l a t e - p r o l a t e difference extends from 1.7 MeV (228Th) up to 4.9 MeV (254Fm). The p o t e n t i a l - e n e r g y m i n i m u m c o r r e s p o n d i n g to the f i s s i o n i s o m e r i c state (f) is g e n e r a l l y s i t u ated on the a b s c i s s a , a p p r o x i m a t e l y at Pf = 0 . 0 1 Z - 0 . 0 1 N + 1.04 ,
qf = 0
(2)
The stiffness of y - v i b r a t i o n a p p e a r s to be s m a l l e r in the s e c o n d a r y m i n i m u m than in the ground state. This kind of shape dependence is c o n t r a r y to P a s h k e v i c h ' s r e s u l t [2], that does not differ f r o m the p r e d i c t i o n s of the pure l i q u i d - d r o p model [6]. This may be of some i m p o r t a n c e in a d y n a m i c f i s s i o n theory. F o r two nuclei (238U and 250Cf) the s e c o n dary m i n i m u m s e e m s to be somewhat n o n - a x i a l , but this region i s v e r y flat, and the energy diff e r e n c e to the p r o l a t e spheroid is l e s s than 0.1 MeV.
i March 1971
The energy of the i s o m e r i c state above the ground state is too high c o m p a r e d with ref. [3], b e c a u s e in the p r e s e n t paper the l i q u i d - d r o p energy, being calculated for ellipsoids only, tends to infinity with i n c r e a s i n g deformation, c o n t r a r y to the energy of a m o r e r e a l i s t i c (constricting} n u c l e a r shape. F o r none of the nuclei i n v e s t i g a t e d the saddle point between the f i r s t and the second m i n i m u m is situated on the a b s c i s s a . The a s y m m e t r y f l u c tuates between q = 0.04 (228Th)and q = 0.14 (254Fm), that m e a n s , the difference between the axes of the spheroid a m o u n t s up to 28% of the n u c l e a r charge r a d i u s . By ignorging the n o n - a x i a l d e g r e e of f r e e d o m the s a d d l e - p o i n t e n e r g y is i n c r e a s e d by 0.1 MeV (228Th) up to 0.6MeY (250Cf). In the l a t t e r case the f i r s t hump of the f i s s i o n b a r r i e r would be o v e r e s t i m a t e d by about 10% of its height above the ground state and by about 20% r e l a t i v e to the s e c o n d a r y m i n i m u m . M o r e o v e r , the a s y m m e t r i c saddle point is shifted to s m a l l e r fl-values by Aft ~ A~ ~ Ap ~0.05 c o m p a r e d with the s p u r i o u s saddle point on the axis. F o r all nuclei other than 228Th the c o o r d i n a t e s of the saddle points (s) a r e , c o n t r a r y to [2], slowly v a r y i n g functions of the n e u t r o n and proton n u m b e r s N and Z. The following e m p i r i c a l r e l a t i o n is valid within an e r r o r of 0.02: Ps = -0.006Z - 0.0025N + 1.335, qs = 0.01Z - 0.0025N - 0.475. In addition, the whole curve following the g r a dient between the saddle point and the m i n i m a takes its c o u r s e in the a s s y m m e t r i c region. As a s i m p l e a p p r o x i m a t i o n of this lowest path b e tween both the m i n i m a , a parabola is plotted in the f i g u r e s , which a p p r o x i m a t e l y p a s s e s through the t h r e e points. It is defined by the relation: p g p f _ p ( p g + pf) + p2 q : qs
PgPf - P's(Pg + P f ) + Ps 2
(4)
C o n t r a r y to the shapes g, f and s, which a r e i n v a r i a n t s with r e s p e c t to p a r a m e t e r t r a n s f o r m a tion, the "lowest path" depends somewhat on the choice of the d e f o r m a t i o n p a r a m e t e r s . Just as the f i r s t hump of the f i s s i o n b a r r i e r the second hump is g e n e r a l l y reduced, if one a l lows for axial a s y m m e t r y . But the effect is s m a l l e r c o m p a r e d with the f i r s t hump. The m a x i m u m energy difference is 0.4 MeV (236U). Again, the second hump is shifted towards the s p h e r e like the f i r s t one by the n o n - a x i a l d e g r e e 247
Volume 34B, n u m b e r 4
PHYSICS
of f r e e d o m . In t h e h e a v i e s t n u c l e u s ( 2 6 0 F m ) no a s y m m e t r y of the s e c o n d h u m p i s f o u n d , but in this case the second minimum might be situated on t h e e n e r g y s l o p e t o w a r d s t h e s c i d d i o n p o i n t , if the nuclear shape would be more realistic than spheroidal ones. A p e c u l i a r i t y o c c u r s in t h e s e c o n d h u m p of t h e f i s s i o n b a r r i e r of 250Cf. T h e r e a r e two s a d d l e p o i n t s w i t h e q u a l e n e r g y , s e p a r a t e d by a n e n e r gy h i l l of 0.2 M e V . One m a y s p e c u l a t e t h a t two p a t h s to f i s s i o n a r e p o s s i b l e , c o r r e s p o n d i n g to d i f f e r e n t d e f o r m a t i o n s . T h e y would c a u s e d i f f e r ent k i n e t i c e n e r g i e s of t h e f i s s i o n f r a g m e n t s a n d p e r h a p s a n e x t r a a s y m m e t r y in t h e m a s s d i s t r i bution. W e a r e g r a t e f u l to P r o f e s s o r G. S ~ l s s m a n n f o r stimulating discussions and valuable suggestions.
248
LETTERS
1 March 1971
References [1] D.A. Zaikin, Soviet P h y s i c s J E T P 8 (1959) 365; T.D. Newton, Can. J. Phys. 38 (1960) 700: K. Kumar. M. B a r a n g e r , Nuc[.Phys. A l l 0 (1968) 529; E.Nadjakov. Y. Piperova, D. Karadjov, I. Sivriev. Comptes rendus de l'Acad~mie bulgare des Sciences 22 (1969) 639. [2] V.V. Pashkevich. NucI. Phys. A133 (1969) 400. [3] S.G. Ni[sson. C . F . T s a n g , A.Sobiczewski. Z. Szymafiski. S. Wvcech. C.Gustafson, I.-L. Lamm. P. MSller and B. Nilsson, Nucl. Phys. A131 (1969) 1. [4} W. D. Myers and W. J. Swiateeki. University of California, UCRL 17070 (1969). [5] H. Schultheis, R. Schultheis. G. Siissmann. Nuel. Phys. A144 (1970) 545. [6] M. Zie|ifiska-PfabS. Phys. L e t t e r s 29B (1969) 280.