Statistical attenuation of shell effects in the fission probability of excited superheavy nuclei

Volume 34B, number 3.

PHYSICS LETTERS

STATISTICAL IN T H E

FISSION

ATTENUATION

PROBABILITY

OF

OF

15 February 1971

SHELL

EXCITED

EFFECTS

SUPERHEAVY

NUCLEI

L. G. MORETTO

Centro di Radiochimica e Analisi Attivazione, Istituto di Chimica Generale, Universitlz di Pavia, Italy Received 5 November 1970

Statistical calculations on superheavy nuclei based on shell model and BCS Hamiltonian are presented. The relevance of the shell effect damping due to excitation energ3, on the location of the transition state and on the fission width is discussed. First-chance fission probabilities are calculated.

Some authors [1,2] have i n v e s t i g a t e d the s t a bility of s u p e r h e a v y nuclei c l o s e to the p r e d i c t e d m a g i c n u m b e r s Z = 114, N = 184 by c a l c u la ti n g the n u c l e a r potential e n e r g y s u r f a c e as a function of a se t of d e f o r m a t i o n p a r a m e t e r s . Such an inv e s t i g a t i o n has been m a d e p o s s i b l e by the c o m b i nation of the s h el l m o d e l and the liquid drop m o d e l as p r o p o s e d by Strutinski [3, 4]. In this way, f i s s i o n b a r r i e r s , alpha and beta decay ene r g i e s and the c o r r e s p o n d i n g half li v e s f o r spontaneous f i s s i o n , alpha and b e t a e m i s s i o n have been predicted. A n a t u r a l c o m p l e m e n t a r y p a r t to the above inv e s t i g a t i o n is the study of the s u p e r h e a v y c o m pound nucleus s t ab i l i ty a g a i n s t f i s s i o n : such s t a t i s t i c a l c a l c u l a t i o n s can be based upon the s a m e s h e l l m o d el and BCS H a m i lt o n ia n as the Strutins k i - l i k e c a l c u l a t i o n s in such a way to i n s u r e similarly reliable results. In o r d e r to a c h i e v e such a goal one needs a t h e o r y which d e s c r i b e s , on the b a s i s of the s h e ll ~nodel, with i n c l u s i o n of p a i r in g and Coulomb eff e c t s , the b e h a v i o u r of the nucleus as a function of e x c i t a t i o n e n e r g y and deformation. In a p r e v i ous c o m m u n i c a t i o n , the author [5] has d e s c r i b e d a method which m a k e s p o s s i b l e to a s s o c i a t e a s t a t i s t i c a l weight to a nucleus with a g i v e n d e f o r m a t i o n and e x c i t a t i o n e n e r g y , on the b a s i s of an a r b i t r a r y s h e l l model. In such f r a m e w o r k the pr oba bi l i t y of finding a nucleus with total e n e r g y E at a d e f o r m a t i o n e has been shown to be:

P(E,e)d¢ :~

1

2~d~cA

~1/2

P(ET,¢)d¢,

w h e r e E T : E - V(¢), V(¢) is the potential e n e r gy at the d e f o r m a t i o n ~, P(ET, ¢) is the density

of l e v e l s , rnC is the i n e r t i a l m a s s a s s o c i a t e d with the motion along E, A = [ d l n p ( x ) / d X ] x : E T Such an approach takes c a r e a u t o m a t i c a l l y of the long d i s c u s s e d p r o b l e m of the washing out of sh el l ef f ect s with i n c r e a s i n g e x c i t a t i o n energy. However, in o r d e r to gain a b e t t e r insight into this m a t t e r , let us make a few c o n s i d e r a t i o n s . The v a r i a t i o n of the n u c l e a r potential e n e r g y p r o f i l e as a function of d e f o r m a t i o n is due to m o dulations of the single p a r t i c l e l e v e l density: the potential e n e r g y fluctuations which o c c u r within a s h o r t r an g e of d e f o r m a t i o n (shell effects) a r e g e n e r a t e d by a s h o r t w a v e - l e n g t h d e p a r t u r e f r o m u n i f o r m l ev el density (the w a v e - l e n g t h being equal to, o r s m a l l e r than, a m a j o r o s c i l l a t o r shell), while the long r a n g e potential e n e r g y v ar i a t i o n s a r e g e n e r a t e d by a long w a v e - l e n g t h dep a r t u r e f r o m u n i f o r m level density (the w av elength is of the o r d e r of the F e r m i energy). Such s t r u c t u r e s in the s i n g l e - p a r t i c l e level density do influence the l ev el density (or the entropy) of a nucleus e x c i t e d at a given energy. The h i g h e r the e x c i t a t i o n e n e r g y , the s m a l l e r is the effect: m o r e p r e c i s e l y a s t a t i s t i c a l c a l c u l a t i o n shows that the washing out of such effect with inc r e a s i n g e n e r g y has the a s y m p t o t i c b e h a v i o u r : S cc e x p ( _ T n 2 / ~ ) , w h e r e S is what we may call the sh el l effect in s o m e r e l e v a n t s t a t i s t i c a l quantity, ~t is the wavelength of the p e r t u r b a t i o n in the singVe p a r t i c l e l ev el density and T is the n u c l e a r t e m p e r a t u r e . An a n a l y s i s along these lines has been made by G i l b e r t [6] in a p ap er c o n c e r n i n g l e v e l d e n s i t i e s f o r a p e r i o d i c a l l y bunched single p a r t i c l e s p e c trum. 191

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It is to b e s t r e s s e d t h a t t h e m a g n i t u d e of a p o t e n t i a l e n e r g y f l u c t u a t i o n d o e s not t e l l i m m e d i a t e l y how d e e p l y it i s r o o t e d in t h e s i n g l e p a r t i c l e s p e c t r u m : t h e r e f o r e two e q u a l l y l a r g e s h e l l e f f e c t s in the g r o u n d s t a t e m a y d i e off at q u i t e a different rate with increasing excitation energy. C o n s i d e r i n g a g a i n t h e s u p e r h e a v y n u c l e i , we m a y r e c a l l t h a t t h e i r f i s s i o n b a r r i e r is e s s e n t i a l ly due to s h e l l e f f e c t s w h i c h d e t e r m i n e not o n l y its height but also its shape. The excitation energy will t e n d to w a s h out the s h e l l e f f e c t s b u t s u c h w a s h i n g out c a n t a k e p l a c e a t a d i f f e r e n t r a t e f o r d i f f e r e n t d e f o r m a t i o n s . In t h i s c a s e , w h i c h i s t h e g e n e r a l one, t h e t r a n s i t i o n s t a t e w h i c h c o n t r o l s t h e f i s s i o n r a t e a n d w h i c h is u s u a l l y c o n s i d e r e d to b e a s s o c i a t e d w i t h t h e s a d d l e p o i n t in t h e p o tential energy, may assume a different deformat i o n t h a n t h e s a d d l e p o i n t i t s e l f . It f o l l o w s t h a t , in o r d e r to c a l c u l a t e t h e f i s s i o n w i d t h a t a g i v e n e x c i t a t i o n e n e r g y , it i s n e c e s s a r y to f i n d t h e l o c a t i o n of the t r a n s i t i o n s t a t e a t t h a t e n e r g y . Qualitative considerations along these lines have b e e n m a d e a l s o by R a m a m u r t h y e t al. [7]. T h e c a l c u l a t i o n s w h i c h we p r e s e n t h e r e w e r e b a s e d on t h e B o l s t e r l i , F i s e t a n d Nix [8] s i n g l e p a r t i c l e s p e c t r u m a s a f u n c t i o n of d e f o r m a t i o n . T h e p o t e n t i a l e n e r g y p r o f i l e w a s o b t a i n e d by a m o d i f i e d S t r u t i n s k i p r o c e d u r e [9]. T h e l e v e l d e n s i t i e s b a s e d u p o n the a b o v e m e n t i o n e d s i n g l e particle levels, are generated from a grand part i t i o n f u n c t i o n b u i l t u p o n t h e BCS H a m i l t o n i a n in a way s i m i l a r to t h a t d e s c r i b e d b y S a n o a n d Y a m a s a k i [10] a n d by D e c o w s k i et al. [ 1 1 ] * . T h e d e f o r m a t i o n p r o b a b i l i t y a s a f u n c t i o n of e x c i t a t i o n e n e r g y w a s c a l c u l a t e d in a f a s h i o n s i m i l a r to t h a t d e s c r i b e d in a p r e v i o u s c o m m u n i c a tion. T h e t r a n s i t i o n s t a t e of f i s s i o n h a s b e e n l o c a t e d by s e a r c h i n g f o r t h e d e f o r m a t i o n at w h i c h t h e s t a t i s t i c a l p r o b a b i l i t y i s at a m i n i m u m a t constant energy: the minimum statistical probab i l i t y i s t h e n c o n v e r t e d into t h e f i s s i o n w i d t h F F. The neutron width FN is also computed with the s a m e d e g r e e of a c c u r a c y b y u s i n g t h e l e v e l d e n s i t y of t h e r e s i d u a l n u c l e u s c a l c u l a t e d f r o m t h e s a m e s e t of s i n g l e p a r t i c l e l e v e l s a n d i n c l u d i n g pairing. The neutron binding energies which have b e e n u s e d in the e v a l u a t i o n of F N h a v e b e e n o b tained from the masses calculated by Nilsson et al. [2]. It is p o s s i b l e t h e n to c a l c u l a t e t h e f i r s t c h a n c e f i s s i o n p r o b a b i l i t y F F / ( F F + FN). * Note: Our level density f o r m a l i s m differs somewhat from the two mentioned above: it differs from that of Sano and Yamasaki in the n u m b e r of Lagrange multip l i e r s and from that of Decowski et al. only in the level density denominator in which there is an unexpected tack of agreement. A description of such f o r m a l i s m will be p r e s e n t e d in a forthcoming paper. 192

LETTERS

15 F e b r u a r y 1971

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Fig. 1. F i s s i o n b a r r i e r shapes for some superheavy nuclei: the potential e n e r g i e s are relative to a s p h e r i cal liquid drop. Sphericity is at y = 0. In fig. 1 the f i s s i o n b a r r i e r s f o r t h e n u c l e i 296 290 298 296 108 X, 110 x , 114 x , 116 x a r e p r e s e n t e d : t h e d e formation coordinate and the shapes associated w i t h i t a r e d e s c r i b e d b y Nix [12] a n d b y B o l s t e r l i e t al. [8]. In f i g s . 2a, 2b, 2c a n d 2 d t h e d e f o r m a tion probabilities are shown for the same nuc!ei a s a f u n c t i o n of e x c i t a t i o n e n e r g y . T h e p e r s i s t e n c e of s h e l l e f f e c t s c a n b e o b s e r v e d f o r i n s t a n c e b y l o o k i n g a t t h e h e i g h t of t h e p r o b a b i l i t y p e a k a t s p h e r i c i t y a s a f u n c t i o n of e x c i t a t i o n e n e r g y . T h e a t t e n u a t i o n of a l l t h e f l u c t u a t i o n s of p r o b a b i l i t y d u e to e x c i t a t i o n e n e r g y i s r e a d i l y o b s e r v e d . T h e t r a n s i t i o n s t a t e c o r r e s p o n d s to t h e d e f o r m a t i o n w h e r e t h e p r o b a b i l i t y i s a t a m i n i m u m . It s h o u l d b e p o i n t e d o u t t h a t t h e p o r t i o n s of c u r v e s b e y o n d the transition state are physically meaningless b e c a u s e no e q u i l i b r i u m is e x p e c t e d to b e a t t a i n e d in t h e d e s c e n t t o w a r d s s c i s s i o n . In fig. 3 t h e f i r s t chance fission probabilities for the nuclei ment i o n e d a b o v e a r e p r e s e n t e d . T h e d r a m a t i c r i s e of 9 8 v a n d 296v f i s s i o n p r o b a b i l i t y of 2114,~ 116 ~ w i t h i n c r e a s ing e n e r g y c o n t r a s t s w i t h the m o r e u n i f o r m b e 296. h a v i o r of t h e s a m e q u a n t i t y f o r t h e n u c l e i 108,x, 2110 9 6"~, y. for the latter nuclei the fission probability a s s u m e s v e r y h i g h v a l u e s a t low e n e r g i e s , t h e n it d e c r e a s e s s o m e w h a t , r e a c h e s a m i n i m u m a n d increases again with increasing excitation energy. S u c h a p e c u l i a r f e a t u r e , s l i g h t l y e v i d e n t a l s o in 2 9 6 y is m a i n l y d u e to t h e d i f f e r e n t r a t e of i n 116", c r e a s e of l e v e l d e n s i t i e s f o r t h e f i s s i o n i n g n u c l e us and the residual nucleus after neutron emission a n d to t h e c l o s e n e s s of t h e f i s s i o n b a r r i e r a n d t h e neutron binding energy. It i s i n t e r e s t i n g to n o t i c e t h a t t h e n u c l e i w h i c h we h a v e c o n s i d e r e d , a l t h o u g h r a t h e r p r o n e to undergo fission, still present reasonably wide excitation energy intervals where fission does occ u r w i t h m o d e r a t e p r o b a b i l i t y . T h e i n c r e a s e of

Votume 34B, n u m b e r 3

PHYSICS

LETTERS

15 F e b r u a r y 1971

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Fig. 2. Statistical probability as a function of nuclear deformation and excitation energy. The excitation energies in MeV are written at the left of each curve. The quantity P(E, y) h/(27Tmy)I/2 has the dimensions MeV -I/2 . 193

Volume 34B, n u m b e r 3

PHYSICS

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h e l p f o r t h e c h o i c e of t h e c o m p o u n d n u c l e i to b e p r o d u c e d a n d t h e i r r a n g e of e x c i t a t i o n e n e r g y w h i c h o p t i m i z e s t h e p r o b a b i l i t y of s u r v i v a l i n t h e competition between fission and neutron emission decays. Similar and more extensive calculations based b o t h on t h e s i n g l e p a r t i c l e l e v e l s u s e d h e r e a n d on the Nilsson diagram are in progress. We a r e i n d e b t e d to Dr. F i s e t f o r p r o v i d i n g u s with the single particle levels and potential energy c u r v e s . We t h a n k D r s . N i l s s o n a n d Nix f o r their comments on the subject.

• io8X ~6 A 110X290

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Fig. 3. F i r s t - c h a n c e fission probabilities as a function of excitation energy. e x c i t a t i o n e n e r g y d o e s not m e a n n e c e s s a r i l y a n i n c r e a s e in f i r s t - c h a n c e f i s s i o n p r o b a b i l i t y t h o u g h it m a y m e a n a n i n c r e a s e i n t o t a l f i s s i o n p r o b a b i l i t y . Of c o u r s e it i s the t o t a l f i s s i o n p r o b a b i l i t y a n d not t h e f i r s t - c h a n c e f i s s i o n p r o b a b i l i t y w h i c h is t h e q u a n t i t y t h a t d e t e r m i n e s t h e a m o u n t of n u c l e i s u r v i v i n g t h e e v a p o r a t i o n c a s cade; therefore calculations along this line s h o u l d b e p e r f o r m e d i n o r d e r to o b t a i n a r e s u l t of m o r e d i r e c t i n t e r e s t to e x p e r i m e n t a l i s t s . It s e e m s p o s s i b l e to c o n c l u d e t h a t c a l c u l a t i o n s such as those presented here may be a valuable

194

LETTERS

References

[1] Yu. A. Muzychka, V.V. Pashkewieh and V• M. Strutinski, Yad. Fis. 8 (1960) 716. [2] S. G. Nilsson, C . F . Tsang, A. Sobiczewski, Z. Szymanski, S.Wycech, G. Gustafson, I.L. Lamm, P. M~Uer and B. Niisson, Nucl. Phys. A131 (1969) 1. [3] V. M. Strutinski, Yad• Fiz. 3 (1966) 614; Soy. J. Nucl• Phys. 3 (1966) 449. [4] V. M. Strutinski, Arkiv. Fysik 36 (1967) 629. [5] L.G. Moretto and R. Stella. Phys. L e t t e r s 32B (1970) 558. [6] A . G i l b e r t , Lawrence Radiation Laboratory, Report UCRL-18095 (1968). [7] V. S. Ramamurthy, S.S. Kapoor and S. K. Kataria, Phys. Rev. L e t t e r s 25 {1970) 386. [8] M• Bolsterli, E.O. F i s e t and J• R. Nix, Proe. of the second IAEA Syrup. on Physics and Chemistry of Fission, Vienna, 1969 (International Atomic Energy Agency, Vienna, 1969) p• 183. [9] C. F. Tsang, University of California, P h . D . T h e sis (1969); Lawrence Radiation Laboratory, Report UCRL-18899 (1969)• [10] M. Sano and S. Yamasaki, P r o g r . Theor. Phys. 29 {1963) 397. [11] P. Decowski, W. Grochulski, A. Marcinkowski, K. Siwek and Z. Wilhelmi, Nue[. Phys. A l l 0 (1968) 129. [12] J . R . Nix, Nucl. Phys. A130 (1968) 241•