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Prompt gamma decay of fission fragments
Volume 33B, number 4
PROMPT
PHYSICS LETTERS
GAMMA
DECAY
OF
FISSION
26 October 1970
FRAGMENTS
R. SARKAR Saha Institute of Nuclear Physics, Calcutta - 9, India
and A. CHATTERJEE Calcutta University and Saha Institute of Nuclear Physics, Calcutta - 9, India Recieved 8 September 1970
Prompt ~-ray energies and yields from fission fragments for the thermal neutron fission of 235U and spontaneous fission of 252Cf are calculated from an improved renormalised Fermi gas model and the collective potential energy surface concept. The predictions agree fairly well with the experimental observations.
An a t t e m p t was made [1,2] to t r e a t the f i s s i o n e n e r g y k i n e t i c s at the s c i s s i o n p o i n t in a s e l f - c o n s i s t e n t way without the help of a conventional m a s s f o r m u l a by c o m b i n i n g two m i c r o s c o p i c m o d e l s of n u c l e a r s t r u c t u r e : (a) the collective potential e n ergy s u r f a c e concept of Mosel and G r e i n e r [3] and (b) the i n t e r a c t i n g r e n o r m a l i s e d F e r m i gas model developed by us [1, 2]. The p r o v i n g grounds of the ideas were the t h e r m a l n e u t r o n induced f i s s i o n of 233U and spontaneous f i s s i o n of 252Cf [1], the t h e r m a l n e u t r o n f i s s i o n of 235U and 239Pu [2] and the fast n e u t r o n f i s s i o n of 238U [4]. F a i r a g r e e m e n t was found in all c a s e s with the m a s s - f o r m u l a e s t i m a t e s of the total e n e r g y r e l e a s e E R , the o b s e r v e d f r a g m e n t excitation e n e r g y UF and the m e a s u r e d f r a g m e n t kinetic e n e r g y T F. The o v e r a l l a g r e e m e n t was found to be b e t t e r than p r e v i o u s a t t e m p t s b a s e d on the s t a t i s t i c a l model [5], the liquid drop model [6] and with shell c o r r e c t i o n s on the liquid drop model [7, 8]. In the work of D i c k m a n n and D i e t r i c h [7], the s h e l l and BCS p a r a m e t e r s a r e e s s e n t i a l l y the s a m e as ours. They also u s e d two liquid drop model p a r a m e t e r s , the s u r f a c e t e n s i o n ~- and the S t r u t i n s k i i type s h e l l - w i d t h ~; we have used the M o s e l - G r e i n e t collective coefficients C O and C' instead. However, the a g r e e m e n t in the f r a g m e n t e x c i t a tion e n e r g i e s with o b s e r v a t i o n s for 235U + n f i s s i o n is found to be much b e t t e r in our case [7,fig. 5; 2, fig. 2]. S i m i l a r c o m m e n t s and conc l u s i o n s apply to the work of Schmitt [8]. After a b r i e f s u m m a r y of our p r e v i o u s p r o cedure [1, 2], we suggest a few i m p r o v e m e n t s
in our t r e a t m e n t of the stiffness coefficients C o and C'. The c o n t r i b u t i o n of the r e c i p r o c a l monop o l e - q u a d r u p o l e Coulomb i n t e r a c t i o n to the p r o m p t v - d e c a y p r o c e s s e s in the f r a g m e n t s is d i s c u s s e d . C o n s i d e r a t i o n s of the f r a g m e n t deexcitation lead p r o c e s s and the nature of n u c l e a r levels in cascade E2 t r a n s i t i o n s lead to the c o r r e c t evaluation of the v - r a y e n e r g y available in, and the m e a n n u m b e r of y - r a y s e m i t t e d from, each fragment. We b r i e f l y r e c o u n t our approach in [1] arid [2]. The total i n t r i n s i c e n e r g y (BCS plus Coulomb e n e r g i e s ) of a f i s s i o n i n g nucleus A c o n s i s t i n g of Z p r o t o n s and N n e u t r o n s EA(°t) = Z~y(~" ,
-t z + A 2 / G ) + E C A (ol)
(1)
is m i n i m i s e d with r e s p e c t to the shape d e f o r m a tion ot at the s a d d l e s c i s s i o n point. The n u c l e a r p a r t in (1) is composed of (i) a static s t r u c t u r a l p a r t at the ground state d e f o r m a t i o n fl, evaluated from the r e n o r m a l i s e d F e r m i gas model and (ii) a s h a p e - d e p e n d e n t p a r t e s t i m a t e d from the axially s y m m e t r i c potential e n e r g y s u r f a c e exp a n s i o n in a power s e r i e s in the shape deviation upto (0t-/3)3 with a p p r o p r i a t e stiffness coefficients C O and C'; the Coulomb e n e r g y E C A (~) is given a s i m i l a r well-known expansion upto (o~3 _/~3). The i n t r i n s i c e n e r g y changes of the f i s s i o n i n g nucleus I were b a l a n c e d with those of the m u t u a l ly i n t e r a c t i n g p r o m p t f i s s i o n f r a g m e n t s (L and H) at the s a d d l e - s c i s s i o n point:
263
Volume 33B, number 4 E I ( ~ I)
=
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F=L, H
{EF(%)+½COF(~F-flF) 2 +
+ C~(OZF-~F )3 + E C F ( ~ F) -
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ECF(flF)}+Er-Ed+E q
c o n n e c t i o n b e t w e e n the r e n o r m a l i s e d F e r m i g a s m o d e l and p o t e n t i a l e n e r g y s u r f a c e i s o b t a i n e d by a s s u m i n g t h e m to be f u n c t i o n s of e x t r a c o r e nuc l e o n s n in a n u c l e u s AF(n/A F =-5 E F / E o ) . T h e o c c u p a t i o n d e p e n d e n c e of the c o e f f i c i e n t s a r e a s s u m e d h e r e to be of the f o r m
(3)
w h e r e Coo and C~ a r e the s t i f f n e s s c o e f f i c i e n t s of a f r e e F e r m i g a s ( r e n o r m a l i s e d F e r m i gas m o d e l c o r r e c t i o n 8E F = 0). W e h a v e c h o s e n Coo= = 320 MeV and C o = 160 M e V t h r o u g h t r i a l and e r r o r f o r b e t t e r e n e r g y f i t s of eq. (2) w i t h o b s e r v a t i o n s . T h i s s m a l l i m p r o v e m e n t (3) d o e s not c h a n g e the q u a l i t a t i v e b e h a v i o u r in the p u b l i s h e d e n e r g y c u r v e s in r e f s . [1] and [2] and in f a c t g i v e s b e t t e r s h a p e f i t s in s o m e m a s s r e g i o n s . T h e s e c h o s e n v a l u e s of C o o and C~ a g r e e w i t h the a v e r a g e v a l u e s of M o s e t and G r e i n e r [3]. It is p o s s i b l e to d e d u c e a v a l u e of C o o ~ 350 M e V f r o m e l e m e n t a r y c o n s i d e r a t i o n s of the r e n o r m a l i s e d F e r m i gas m o d e l [10]. T h e f i s s i o n e n e r g y r e l e a s e E R and the c o m p o n e n t s of i n t e r n a l e n e r g y (ER, E d , Eq and the C o u l o m b e n e r g y d i f f e r e n c e ) of the p r o m p t f r a g m e n t F h a v e b e e n r e c a l c u l a t e d by m i n i m i s i n g the t o t a l e n e r g y at the s a d d l e - s c i s s i o n p o i n t a s b e f o r e [1, 2]. T h e f i s s i o n b a r r i e r f o r 235U + n is c o r r e c t l y r e p r o d u c e d (5.64 MeV; e x p e r i m e n t a l v a l u e is 5.75 M e V ; o u r p r e v i o u s v a l u e w i t h o u t u s i n g eq. (3) w a s 4.4 MeV). T h e f r a g m e n t d e f o r m a t i o n s a r e now m o r e s t r o n g l y f l u c t u a t i n g than in r e f s . [1, 2] due to the r e l a t i v e s t r u c t u r a l d i f f e r e n c e s of the f r a g m e n t n u c t e i d e s . T h e s e c a l c u 264
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T h e i n t e r a c t i o n t e r m s Er, Ed and E a h a v e b e e n d i s c u s s e d and d e d u c e d by W i l e t s [9] f o r t e r m s l i n e a r in ~ [1, eq. (5); 2, eq. (13)]. T h e W i l e t s t e r m s E r and E d w e r e a s s u m e d in r e f s . [1, 2] to go into the t r a n s l a t i o n a l e n e r g y of the f r a g m e n t s T F ; the i n t e r a c t i o n e n e r g y E q w a s p r e d i c t e d [1, 2] to g i v e r i s e to E2 t y p e p r o m p t v - r a y s at the final s t a g e s of the g r a g m e n t d e e x c i t a t i o n process. Our p r e v i o u s t r e a t m e n t [1, 2] a s s u m e d a l l n u c l e i to h a v e the s a m e a v e r a g e s t i f f n e s s c o e f f i c i e n t s Co and C~ in eq. (2). One w o u l d e x F p e c t , h o w e v e r , the n u c l e a r s t r u c t u r e e f f e c t s to show up in C o F and C ~ . A m o r e d e t a i l e d i n t e r -
C~ = Co(t-0~F/%)
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CoF = %o(1-0~F/%),
26 October
PHYSICS LETTERS
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Fig. 1. Calculated quanhhes OZF, E y and my for the 235U + fission reaction with thermal neutrons, plotted against the prompt fragment mass number A F. 1 (a) : The scission point deformation parameter otF estimated from eq. (2) ; 1 (b) : The available y-decay energy E~nax calculated by solving eq. (2) are shown as G) and are compared with the data {@) of ref. [15]; the calculated energies are slightly lower for heavy fragments and higher for tight ones; 1(c) : The mean photon number no/shown as A are compared with the data of ref. [15]. l a t e d d e f o r m a t i o n s a r e shown at the top s e c t i o n s m a r k e d (a) in f i g s . 1 and 2 f o r the t h e r m a l n e u t r o n i n d u c e d f i s s i o n of 235U and s p o n t a n e o u s f i s s i o n of 252Cf r e s p e c t i v e l y . T h e r e a r e no l i q u i d d r o p m o d e l c a l c u l a t i o n s f o r the f r a g m e n t d e f o r m a t i o n s ~ F f o r 252Cf, but F e r g u s o n and R e a d [6] h a v e e s t i m a t e d ~ F f o r a few f r a g m e n t p a i r s f o r the 235U + n f i s s i o n f r o m the l i q u i d d r o p m o d e l ;
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Fig. 3. The s t a t i s t i c a l level density p a r a m e t e r (the a - p a r a m e t e r ) of the p r o m p t fission f r a g m e n t s as a function of the p r o m p t f r a g m e n t m a s s number. The method of calculation of ref. [18]. The e x p e r i m e n t a l data compilation of ref. [19] is also shown for c o m parison.
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Fig. 2. Calculations of oz- E max and n.. for the spon~' y -.Y taneous fission of 252Cf. F o r details (a), (b) and (c), see the r e s p e c t i v e captions of fig. 1. No e x p e r i m e n t a l c o m p a r i s o n is possible in fig. 2(b). The e x p e r i m e n t a l data of fig. 2(c) are from ref. [14]. The r e l a t i v e scales of the quantities on the right in both figs. 1 and 2 a r e those used by the e x p e r i m e n t o r s [14, 15]. their estimates for a F are in fair agreement with our calculations. The quadrupole-monopole interaction energy between the fragment pairs E a is identified as the v-emission energy EZax.-The physical situation here is that a free vibrating quadrupole q is held in constraint by a charge monopole at a finite d i s t a n c e r = ROL + R o l l a n d h e n c e t h e p o t e n t i a l e n e r g y of t h e q u a d r u p o l e i n c r e a s e s [11]; r e m o v a l of t h e i n t e r a c t i o n a l l o w e s t h e s t o r e d p o t e n t i a l e n e r g y of t h e q u a d r u p o l e to b e r e l e a s e d . S i n c e t h e p o s t s c i s s i o n c o n f i g u r a t i o n of t h e r e c o i l i n g fragments may be taken as the snapping process of t h i s i n t e r a c t i o n , t h e e x t r a s t o r e d e n e r g i e s i n t h e q u a d r u p o l e s E q L a n d EqH l e f t to t h e m s e l v e s ,
neutrons uF '
gammas
AF,* ny
•
AF'
(2)
w h e r e u F i s t h e n u m b e r of p r o m p t n e u t r o n s f r o m A F , a n d ny i s t h e n u m b e r of p r o m p t y - r a y s e m i t ted from the fragment AF, , (AF, -u F = AF,,) • T h e h i g h e s t p o s s i b l e v a l u e of t h e r e s i d u a l e x c i tation (~ neutron separation energy Sn) is 7 M e V f o r 2 5 2 C f a n d ~ 5.5 M e V f o r 2 3 5 U + n, a s p r e d i c t e d i n f i g s . l ( b ) a n d 2(b) a n d a s f o u n d max e x p e r i m e n t a l l y [12]; t h i s Ey corresponds to a single direct v-transition from near the neutron t h r e s h o l d t o t h e g r o u n d s t a t e of A F , . On t h e a v e r a g e , h o w e v e r , ( E ~ nax) a v ~ ~1 S n ~ 3.5 MeV, a n d is the average energy available for prompt y-dec a y of t h e f r a g m e n t s [13]. W e now c o r r e l a t e t h e t o t a l a v a i l a b l e y - e n e r g y Em Ya x w i t h t h e a v e r a g e y - d e c a y e n e r g y ( p h o t o n e n e r g y ) E y a n d w i t h t h e t o t a l n u m b e r of p h o t o n s ny in a fragment F'* from the renormalised Fermi gas model. Here we make the crucial a s s u m p t i o n s t h a t (1) o n l y a c a s c a d e d e e x c i t a t i o n of m e a n e n e r g y Ey i s p e r m i t t e d , (2) a p r e s c r i b e d s t a t i s t i c a l l e v e l s t r u c t u r e of t h e f r a g m e n t s e x i s t a n d (3) t h e d e e x c i t a t i o n t a k e s p l a c e b y E 2 t r a n sitions alone, i.e.,
E ~ ax = ny • Ey There are the following good experimental
rea265
Volume 33B, number 4
PHYSICS LETTERS
sons to b e l i e v e that the p r o m p t v - d e e x c i t a t i o n p r o c e s s p r o c e e d s through a s e r i e s of c a s c a d e y - r a y s ; (i) the m e a s u r e d y i e l d of y - r a y s with e n e r g y (E~naX)av = ½ E max is ~ 100 t i m e s g r e a t e r [12] than the yield with' e n e r g y E~ nax, (if) the m e a s u r e d e n e r g y - s e l e c t e d y i e l d of e n e r g y < 1 MeV is ~ 105 t i m e s g r e a t e r than of e n e r g y E~rnax [14] and (iii) it has been s t a t e d that on the a v e r a g e about 3 to 5 y - r a y s of m e a n e n e r g y < 600 keV a r e o b s e r v e d [15]. A c a s c a d e d e e x c i t a t i o n of m e a n e n e r g y E v is a l s o f a v o u r e d f r o m the following c o n s i d e r a t i o n s : (a) f i s s i o n f r a g m e n t s have to get a r i d of the l a r g e a n g u l a r m o m e n t u m l I of the f i s s i o n i n g nucleus I; s i n c e p r o m p t neutrons c a r r y away only a s m a l l f r a c t i o n of l I, the r e l i e f can c o m e only through a y - c a s c a d e p r o c e s s , and (b) the q u a d r u p o l e - m o n o p o l e type of mutual coulomb i n t e r a c t i o n E q as d i s c u s s e d e a r l i e r , is e x p e c t e d to give r i s e to the E2 type c a s c a d e y - r a y s ; s o m e d i s c u s s i o n s on this point e x i s t in the U t e r a t u r e [16]. The w e l l known s t a t i s t i c a l p r o p e r t y of the l e v e l s t r u c t u r e of the f r e e gas is that the single p a r t i c l e l e v e l density g o n e a r the F e r m i s u r f a c e eo g e n e r a t e s a n u c l e a r l e v e l density Po at an e x citation Uo [17]. T h e s e quantities a r e i n t e r a c t i o n m o d i f i e d in a r e a l n u c l e a r c a s e . The n e c e s s a r y m o d i f i c a t i o n s m ay be e v a l u a t e d f r o m the r e n o r m a l i s e d F e r m i gas model. It can be shown that the l e v e l spacing d r , and the single p a r t i c l e l e v e l density g F ' of the nucleus F ' is r e l a t e d to the f r e e gas density g o [18] by
g F ' = 1/dF' = gn +gp,
gn o r p ~ go(l+6e/eo)(4)
including the s t at e mixing e f f e c t s and e f f e c t i v e occupation f a c t o r s in a d e f o r m e d open s h e l l (coulomb contribution added). A plot of the l e v e l density p a r a m e t e r a F ( = ~ n 2 g F) f o r the p r o m p t f i s s i o n f r a g m e n t s a r e shown in fig. 3 and a r e c o m p a r e d with the e x p e r i m e n t a l data c o m p i l a t i o n of the a - p a r a m e t e r [19]; we note f a i r a g r e e m e n t between c a l c u l a t e d and m e a s u r e d v a lu e s . A c o r r e l a t e d s et of n u c l e a r l e v e l s t r u c t u r e coupled to the ground s t a t e (e. g., 0 +, 2 +, 4 +, . . , in an e v e n - e v e n and jrr + ~ k 2 k s e r i e s with/e = 0, 1, 2 , . . . , in an o d d - e v e n nucleus) is often known to p e r s i s t upto f a i r l y high e x c i t a t i o n s (~ 3 - 4 MeV). A s s u m i n g this s p e c i a l s e t of c o r r e l a t e d l e v e l s to be a s e t of equidistantly s p a c e d s t a t i s t i c a l l e v e l s of spacing def f above the r e n o r m a l i s e d gas m o d el F e r m i s u r f a c e e, we note that (i) upto about E~nax, l e v e l s of only one kind of p a r i t y a r e e f f e c t i v e for E2 t r a n s i t i o n s and (if) an upper l i m i t can be s e t at N ~E ymax (not E vmax ), as d i s c u s s e d below. Condition (i) gives 266
aeff -- 2 a F, = 2 / e r
26 October 1970
(5)
= 1/eef f "ev
R e g a r d i n g (if), it is w e l l known that h i g h e r r o t a t i o n a l - v i b r a t i o n a l groups of l e v e l s a p p e a r at N 3 MeV and that our c o r r e l a t e d s t a t i s c i t c a l l e v e l s (5) d e g e n e r a t e into an exponentially v a r y i n g l e v e l density Po above ~ ½E~nax ; si n ce the n u m b e r of l e v e l s between ½E~max and E ~nax i n c r e a s e s f r o m tens to h u n d r e d - t h o u s a n d s , the a v e r a g e d e f f is so s m a l l , and the c o m p e t i t i o n b e t w e e n E2 t r a n s i t i o n - f a v o u r i n g l e v e l s is so l a r g e , that it is i m p o s s i b l e to o b s e r v e the ef f ect s of s p e c i a l l e v e l s any m o r e . M o r e o v e r , the e l e c t r o n i c s and the d et ect i n g co u n t er e n e r g y s e l e c t i v i t y a r e often u su al l y b i a s e d ag ai n st counting t h ese low e n e r g y y - r a y s in typical e x p e r i m e n t s [12, 14, 15]. The c o r r e l a t i o n (3) thus m o d i f i e s to m ax ~E m a x (ET )av ~ ~ V ~ %.
aeff ~ E v " %
.
(6)
The n u m b e r of p r o m p t y - r a y s n 9, a r e c a l c u l a t e d f r o m the c o r r e l a t i o n (6) and a r e shown in figs. l(c) and 2(c). E x p e r i m e n t a l data a r e a l s o shown in figs. 1 and 2 when a v a i l a b l e . We note that f o r figs. l(b) and l(c), the e x p e r i m e n t a l e r r o r s have been i n c r e a s e d by a f a c t o r of 3 on the b a s i s of s o m e c o m m e n t s of the w o r k e r s [15] on 235U + n. Our absolute y i e l d s n3, a g r e e f a i r l y w e l l with the m e a s u r e d r e l a t i v e y - r a y y i e l d s [14, 15]. S e v e r a l p h en o m en a c o m p e t e with (6) to p r o v i d e a l t e r n a t i v e paths of d e e x e i t a t i o n of the p r o m p t f i s s i o n f r a g m e n t s , e . g . , the d el ay ed neutron em i s. sion, the d i r e c t E 7max t r a n s i t i o n to the ground st at e and o t h er c r o s s - o v e r y - r a y s of d i f f e r e n t m u l t i p o l a r i t i e s , e t c . . Such delayed, weak and r e t a r d e d p h e n o m e n a have b e e n a s s u m e d in the p r e s e n t w o r k to make a negligible contribution to the m a i n s t r e a m of d e e x c i t a t i o n (6). The e x p e r i m e n t a l t i m e r e s o l u t i o n of ~ 10 -9 to 10 -11 s [14, 15] ' d e f i n e s ' the p r o m p t p r o c e s s e s in y - d e c a y . This t i m e s c a l e is a p p r o p r i a t e f o r allowed E2 t r a n s i t i o n s .
Reference s [1] R. Sarkar and A. Chatterjee, Phys. Letters 30B (1969) 313. [2] R. Sarkar and A. Chatterjee, Phys. Rev. C1 (1970) 619. [3] U. Mosel and W. Greiner. Z.Physik 217 (1968) 256; 222 (1969) 261. [4] R. Sarkar, to be published. [5] Gy. Kiuge and A. Lajtai, Phys. Letters 27B (1968) 65; 30B (1969) 311. [6] J.M. Ferguson and P.A Read, Phys. Rev. 150 (1966) 1018.
Volume 33B, n u m b e r 4
PHYSICS
[7] F. Dickmann and K. Dietrich, Nucl. Phys. A129 (1969) 241. [8] H. W. Schmitt, P r o c . Intern. Sym. on Why and how shlould we investigate nucleides f a r off the stability line, Lykesil, Sweden (1966); Arkiv Phys. 36 (1966) 633; P r o c . Intern. Conf. on the P h y s i c s and c h e m i s t r y of f i s s i o n , Vienna, 1969 (L A. E. A., Vienna, 1969) p. 67. [9] L.Wilets, T h e o r i e s of nuclear fission (Oxford University, Oxford, 1964). [10] S. Chatterjee, unpublished private communication. [11] W. K. H. Panofsky and M. Phillips, Classical e l e c tricity and magnetism (Addison-Wesley, Reading, Mass. 1962) Fig. 1-3, p. 16 and Ex. 6, p. 116 [12] E. C. Maienschein, R.W. Peele, W. Zobel and T. A. Law, Second Intern. Conf. on Peaceful u s e s of atomic energy, Geneva, 1958 (United Nations, Geneva) Vol. 15, p a p e r No. P/670; F . E . W . Rau, Ann. Physique 10 (1963) 252.
LETTERS
26 October 1970
[13] V . F . Apalin, Yu.N.Gritsiuk, I.E.Kutikov, V.L Lebedev and L. A. Mikailian, Nucl, Phys. 71 (1965) 553. [14] S . A . E . Johanson, Nuel. Phys. 60 (1964) 378; 64 (1965) 147. [15] H. M a i e r - L e i b n i t z , H.W. Schmitt and P. A r m b r u s t e r Intern. Conf. on the P h y s i c s and c h e m i s t r y of f i s sion, Salzburg, 1965 (I. A. E . A . , Vienna, 1965) Vol. 2, p. 143. [16] R. B. Leaehman, Second Intern. Conf. on Peaceful uses of atomic energy, Geneva, 1958 (United Nations, Geneva) Vol. 15, p a p e r No. P/665; V, M. Strutinskii, Soviet Phys. J E T P 10 (1960) 613. [17] T . E . O . E r i c s o n Advances in Phys. 9 (1960) 425. [18] S.K. Chosh, S. Chatterjee and A. Chatterjee, to be published. [19] E. Erba, U. Facchini and E. Seatta-Menichella, Nuovo Cim. 22 (1961) 1237; U. Facchini and E. Saetta-Menichella, Energia Nucleare 15 (1968) 54.
267
PROMPT
PHYSICS LETTERS
GAMMA
DECAY
OF
FISSION
26 October 1970
FRAGMENTS
R. SARKAR Saha Institute of Nuclear Physics, Calcutta - 9, India
and A. CHATTERJEE Calcutta University and Saha Institute of Nuclear Physics, Calcutta - 9, India Recieved 8 September 1970
Prompt ~-ray energies and yields from fission fragments for the thermal neutron fission of 235U and spontaneous fission of 252Cf are calculated from an improved renormalised Fermi gas model and the collective potential energy surface concept. The predictions agree fairly well with the experimental observations.
An a t t e m p t was made [1,2] to t r e a t the f i s s i o n e n e r g y k i n e t i c s at the s c i s s i o n p o i n t in a s e l f - c o n s i s t e n t way without the help of a conventional m a s s f o r m u l a by c o m b i n i n g two m i c r o s c o p i c m o d e l s of n u c l e a r s t r u c t u r e : (a) the collective potential e n ergy s u r f a c e concept of Mosel and G r e i n e r [3] and (b) the i n t e r a c t i n g r e n o r m a l i s e d F e r m i gas model developed by us [1, 2]. The p r o v i n g grounds of the ideas were the t h e r m a l n e u t r o n induced f i s s i o n of 233U and spontaneous f i s s i o n of 252Cf [1], the t h e r m a l n e u t r o n f i s s i o n of 235U and 239Pu [2] and the fast n e u t r o n f i s s i o n of 238U [4]. F a i r a g r e e m e n t was found in all c a s e s with the m a s s - f o r m u l a e s t i m a t e s of the total e n e r g y r e l e a s e E R , the o b s e r v e d f r a g m e n t excitation e n e r g y UF and the m e a s u r e d f r a g m e n t kinetic e n e r g y T F. The o v e r a l l a g r e e m e n t was found to be b e t t e r than p r e v i o u s a t t e m p t s b a s e d on the s t a t i s t i c a l model [5], the liquid drop model [6] and with shell c o r r e c t i o n s on the liquid drop model [7, 8]. In the work of D i c k m a n n and D i e t r i c h [7], the s h e l l and BCS p a r a m e t e r s a r e e s s e n t i a l l y the s a m e as ours. They also u s e d two liquid drop model p a r a m e t e r s , the s u r f a c e t e n s i o n ~- and the S t r u t i n s k i i type s h e l l - w i d t h ~; we have used the M o s e l - G r e i n e t collective coefficients C O and C' instead. However, the a g r e e m e n t in the f r a g m e n t e x c i t a tion e n e r g i e s with o b s e r v a t i o n s for 235U + n f i s s i o n is found to be much b e t t e r in our case [7,fig. 5; 2, fig. 2]. S i m i l a r c o m m e n t s and conc l u s i o n s apply to the work of Schmitt [8]. After a b r i e f s u m m a r y of our p r e v i o u s p r o cedure [1, 2], we suggest a few i m p r o v e m e n t s
in our t r e a t m e n t of the stiffness coefficients C o and C'. The c o n t r i b u t i o n of the r e c i p r o c a l monop o l e - q u a d r u p o l e Coulomb i n t e r a c t i o n to the p r o m p t v - d e c a y p r o c e s s e s in the f r a g m e n t s is d i s c u s s e d . C o n s i d e r a t i o n s of the f r a g m e n t deexcitation lead p r o c e s s and the nature of n u c l e a r levels in cascade E2 t r a n s i t i o n s lead to the c o r r e c t evaluation of the v - r a y e n e r g y available in, and the m e a n n u m b e r of y - r a y s e m i t t e d from, each fragment. We b r i e f l y r e c o u n t our approach in [1] arid [2]. The total i n t r i n s i c e n e r g y (BCS plus Coulomb e n e r g i e s ) of a f i s s i o n i n g nucleus A c o n s i s t i n g of Z p r o t o n s and N n e u t r o n s EA(°t) = Z~y(~" ,
-t z + A 2 / G ) + E C A (ol)
(1)
is m i n i m i s e d with r e s p e c t to the shape d e f o r m a tion ot at the s a d d l e s c i s s i o n point. The n u c l e a r p a r t in (1) is composed of (i) a static s t r u c t u r a l p a r t at the ground state d e f o r m a t i o n fl, evaluated from the r e n o r m a l i s e d F e r m i gas model and (ii) a s h a p e - d e p e n d e n t p a r t e s t i m a t e d from the axially s y m m e t r i c potential e n e r g y s u r f a c e exp a n s i o n in a power s e r i e s in the shape deviation upto (0t-/3)3 with a p p r o p r i a t e stiffness coefficients C O and C'; the Coulomb e n e r g y E C A (~) is given a s i m i l a r well-known expansion upto (o~3 _/~3). The i n t r i n s i c e n e r g y changes of the f i s s i o n i n g nucleus I were b a l a n c e d with those of the m u t u a l ly i n t e r a c t i n g p r o m p t f i s s i o n f r a g m e n t s (L and H) at the s a d d l e - s c i s s i o n point:
263
Volume 33B, number 4 E I ( ~ I)
=
~
F=L, H
{EF(%)+½COF(~F-flF) 2 +
+ C~(OZF-~F )3 + E C F ( ~ F) -
Bo
ECF(flF)}+Er-Ed+E q
c o n n e c t i o n b e t w e e n the r e n o r m a l i s e d F e r m i g a s m o d e l and p o t e n t i a l e n e r g y s u r f a c e i s o b t a i n e d by a s s u m i n g t h e m to be f u n c t i o n s of e x t r a c o r e nuc l e o n s n in a n u c l e u s AF(n/A F =-5 E F / E o ) . T h e o c c u p a t i o n d e p e n d e n c e of the c o e f f i c i e n t s a r e a s s u m e d h e r e to be of the f o r m
(3)
w h e r e Coo and C~ a r e the s t i f f n e s s c o e f f i c i e n t s of a f r e e F e r m i g a s ( r e n o r m a l i s e d F e r m i gas m o d e l c o r r e c t i o n 8E F = 0). W e h a v e c h o s e n Coo= = 320 MeV and C o = 160 M e V t h r o u g h t r i a l and e r r o r f o r b e t t e r e n e r g y f i t s of eq. (2) w i t h o b s e r v a t i o n s . T h i s s m a l l i m p r o v e m e n t (3) d o e s not c h a n g e the q u a l i t a t i v e b e h a v i o u r in the p u b l i s h e d e n e r g y c u r v e s in r e f s . [1] and [2] and in f a c t g i v e s b e t t e r s h a p e f i t s in s o m e m a s s r e g i o n s . T h e s e c h o s e n v a l u e s of C o o and C~ a g r e e w i t h the a v e r a g e v a l u e s of M o s e t and G r e i n e r [3]. It is p o s s i b l e to d e d u c e a v a l u e of C o o ~ 350 M e V f r o m e l e m e n t a r y c o n s i d e r a t i o n s of the r e n o r m a l i s e d F e r m i gas m o d e l [10]. T h e f i s s i o n e n e r g y r e l e a s e E R and the c o m p o n e n t s of i n t e r n a l e n e r g y (ER, E d , Eq and the C o u l o m b e n e r g y d i f f e r e n c e ) of the p r o m p t f r a g m e n t F h a v e b e e n r e c a l c u l a t e d by m i n i m i s i n g the t o t a l e n e r g y at the s a d d l e - s c i s s i o n p o i n t a s b e f o r e [1, 2]. T h e f i s s i o n b a r r i e r f o r 235U + n is c o r r e c t l y r e p r o d u c e d (5.64 MeV; e x p e r i m e n t a l v a l u e is 5.75 M e V ; o u r p r e v i o u s v a l u e w i t h o u t u s i n g eq. (3) w a s 4.4 MeV). T h e f r a g m e n t d e f o r m a t i o n s a r e now m o r e s t r o n g l y f l u c t u a t i n g than in r e f s . [1, 2] due to the r e l a t i v e s t r u c t u r a l d i f f e r e n c e s of the f r a g m e n t n u c t e i d e s . T h e s e c a l c u 264
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T h e i n t e r a c t i o n t e r m s Er, Ed and E a h a v e b e e n d i s c u s s e d and d e d u c e d by W i l e t s [9] f o r t e r m s l i n e a r in ~ [1, eq. (5); 2, eq. (13)]. T h e W i l e t s t e r m s E r and E d w e r e a s s u m e d in r e f s . [1, 2] to go into the t r a n s l a t i o n a l e n e r g y of the f r a g m e n t s T F ; the i n t e r a c t i o n e n e r g y E q w a s p r e d i c t e d [1, 2] to g i v e r i s e to E2 t y p e p r o m p t v - r a y s at the final s t a g e s of the g r a g m e n t d e e x c i t a t i o n process. Our p r e v i o u s t r e a t m e n t [1, 2] a s s u m e d a l l n u c l e i to h a v e the s a m e a v e r a g e s t i f f n e s s c o e f f i c i e n t s Co and C~ in eq. (2). One w o u l d e x F p e c t , h o w e v e r , the n u c l e a r s t r u c t u r e e f f e c t s to show up in C o F and C ~ . A m o r e d e t a i l e d i n t e r -
C~ = Co(t-0~F/%)
1970
(2) o~
CoF = %o(1-0~F/%),
26 October
PHYSICS LETTERS
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Fig. 1. Calculated quanhhes OZF, E y and my for the 235U + fission reaction with thermal neutrons, plotted against the prompt fragment mass number A F. 1 (a) : The scission point deformation parameter otF estimated from eq. (2) ; 1 (b) : The available y-decay energy E~nax calculated by solving eq. (2) are shown as G) and are compared with the data {@) of ref. [15]; the calculated energies are slightly lower for heavy fragments and higher for tight ones; 1(c) : The mean photon number no/shown as A are compared with the data of ref. [15]. l a t e d d e f o r m a t i o n s a r e shown at the top s e c t i o n s m a r k e d (a) in f i g s . 1 and 2 f o r the t h e r m a l n e u t r o n i n d u c e d f i s s i o n of 235U and s p o n t a n e o u s f i s s i o n of 252Cf r e s p e c t i v e l y . T h e r e a r e no l i q u i d d r o p m o d e l c a l c u l a t i o n s f o r the f r a g m e n t d e f o r m a t i o n s ~ F f o r 252Cf, but F e r g u s o n and R e a d [6] h a v e e s t i m a t e d ~ F f o r a few f r a g m e n t p a i r s f o r the 235U + n f i s s i o n f r o m the l i q u i d d r o p m o d e l ;
PHYSICS L E T T E R S
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m u s t b e s p e n t i n t h e f o r m of q u a d r u p o l e v i b r a t i o n d e e x c i t a t i o n s (E2 t r a n s i t i o n s ) . The entire prompt fragment deexcitation process is visualised as
AF*
4
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Fig. 3. The s t a t i s t i c a l level density p a r a m e t e r (the a - p a r a m e t e r ) of the p r o m p t fission f r a g m e n t s as a function of the p r o m p t f r a g m e n t m a s s number. The method of calculation of ref. [18]. The e x p e r i m e n t a l data compilation of ref. [19] is also shown for c o m parison.
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Fig. 2. Calculations of oz- E max and n.. for the spon~' y -.Y taneous fission of 252Cf. F o r details (a), (b) and (c), see the r e s p e c t i v e captions of fig. 1. No e x p e r i m e n t a l c o m p a r i s o n is possible in fig. 2(b). The e x p e r i m e n t a l data of fig. 2(c) are from ref. [14]. The r e l a t i v e scales of the quantities on the right in both figs. 1 and 2 a r e those used by the e x p e r i m e n t o r s [14, 15]. their estimates for a F are in fair agreement with our calculations. The quadrupole-monopole interaction energy between the fragment pairs E a is identified as the v-emission energy EZax.-The physical situation here is that a free vibrating quadrupole q is held in constraint by a charge monopole at a finite d i s t a n c e r = ROL + R o l l a n d h e n c e t h e p o t e n t i a l e n e r g y of t h e q u a d r u p o l e i n c r e a s e s [11]; r e m o v a l of t h e i n t e r a c t i o n a l l o w e s t h e s t o r e d p o t e n t i a l e n e r g y of t h e q u a d r u p o l e to b e r e l e a s e d . S i n c e t h e p o s t s c i s s i o n c o n f i g u r a t i o n of t h e r e c o i l i n g fragments may be taken as the snapping process of t h i s i n t e r a c t i o n , t h e e x t r a s t o r e d e n e r g i e s i n t h e q u a d r u p o l e s E q L a n d EqH l e f t to t h e m s e l v e s ,
neutrons uF '
gammas
AF,* ny
•
AF'
(2)
w h e r e u F i s t h e n u m b e r of p r o m p t n e u t r o n s f r o m A F , a n d ny i s t h e n u m b e r of p r o m p t y - r a y s e m i t ted from the fragment AF, , (AF, -u F = AF,,) • T h e h i g h e s t p o s s i b l e v a l u e of t h e r e s i d u a l e x c i tation (~ neutron separation energy Sn) is 7 M e V f o r 2 5 2 C f a n d ~ 5.5 M e V f o r 2 3 5 U + n, a s p r e d i c t e d i n f i g s . l ( b ) a n d 2(b) a n d a s f o u n d max e x p e r i m e n t a l l y [12]; t h i s Ey corresponds to a single direct v-transition from near the neutron t h r e s h o l d t o t h e g r o u n d s t a t e of A F , . On t h e a v e r a g e , h o w e v e r , ( E ~ nax) a v ~ ~1 S n ~ 3.5 MeV, a n d is the average energy available for prompt y-dec a y of t h e f r a g m e n t s [13]. W e now c o r r e l a t e t h e t o t a l a v a i l a b l e y - e n e r g y Em Ya x w i t h t h e a v e r a g e y - d e c a y e n e r g y ( p h o t o n e n e r g y ) E y a n d w i t h t h e t o t a l n u m b e r of p h o t o n s ny in a fragment F'* from the renormalised Fermi gas model. Here we make the crucial a s s u m p t i o n s t h a t (1) o n l y a c a s c a d e d e e x c i t a t i o n of m e a n e n e r g y Ey i s p e r m i t t e d , (2) a p r e s c r i b e d s t a t i s t i c a l l e v e l s t r u c t u r e of t h e f r a g m e n t s e x i s t a n d (3) t h e d e e x c i t a t i o n t a k e s p l a c e b y E 2 t r a n sitions alone, i.e.,
E ~ ax = ny • Ey There are the following good experimental
rea265
Volume 33B, number 4
PHYSICS LETTERS
sons to b e l i e v e that the p r o m p t v - d e e x c i t a t i o n p r o c e s s p r o c e e d s through a s e r i e s of c a s c a d e y - r a y s ; (i) the m e a s u r e d y i e l d of y - r a y s with e n e r g y (E~naX)av = ½ E max is ~ 100 t i m e s g r e a t e r [12] than the yield with' e n e r g y E~ nax, (if) the m e a s u r e d e n e r g y - s e l e c t e d y i e l d of e n e r g y < 1 MeV is ~ 105 t i m e s g r e a t e r than of e n e r g y E~rnax [14] and (iii) it has been s t a t e d that on the a v e r a g e about 3 to 5 y - r a y s of m e a n e n e r g y < 600 keV a r e o b s e r v e d [15]. A c a s c a d e d e e x c i t a t i o n of m e a n e n e r g y E v is a l s o f a v o u r e d f r o m the following c o n s i d e r a t i o n s : (a) f i s s i o n f r a g m e n t s have to get a r i d of the l a r g e a n g u l a r m o m e n t u m l I of the f i s s i o n i n g nucleus I; s i n c e p r o m p t neutrons c a r r y away only a s m a l l f r a c t i o n of l I, the r e l i e f can c o m e only through a y - c a s c a d e p r o c e s s , and (b) the q u a d r u p o l e - m o n o p o l e type of mutual coulomb i n t e r a c t i o n E q as d i s c u s s e d e a r l i e r , is e x p e c t e d to give r i s e to the E2 type c a s c a d e y - r a y s ; s o m e d i s c u s s i o n s on this point e x i s t in the U t e r a t u r e [16]. The w e l l known s t a t i s t i c a l p r o p e r t y of the l e v e l s t r u c t u r e of the f r e e gas is that the single p a r t i c l e l e v e l density g o n e a r the F e r m i s u r f a c e eo g e n e r a t e s a n u c l e a r l e v e l density Po at an e x citation Uo [17]. T h e s e quantities a r e i n t e r a c t i o n m o d i f i e d in a r e a l n u c l e a r c a s e . The n e c e s s a r y m o d i f i c a t i o n s m ay be e v a l u a t e d f r o m the r e n o r m a l i s e d F e r m i gas model. It can be shown that the l e v e l spacing d r , and the single p a r t i c l e l e v e l density g F ' of the nucleus F ' is r e l a t e d to the f r e e gas density g o [18] by
g F ' = 1/dF' = gn +gp,
gn o r p ~ go(l+6e/eo)(4)
including the s t at e mixing e f f e c t s and e f f e c t i v e occupation f a c t o r s in a d e f o r m e d open s h e l l (coulomb contribution added). A plot of the l e v e l density p a r a m e t e r a F ( = ~ n 2 g F) f o r the p r o m p t f i s s i o n f r a g m e n t s a r e shown in fig. 3 and a r e c o m p a r e d with the e x p e r i m e n t a l data c o m p i l a t i o n of the a - p a r a m e t e r [19]; we note f a i r a g r e e m e n t between c a l c u l a t e d and m e a s u r e d v a lu e s . A c o r r e l a t e d s et of n u c l e a r l e v e l s t r u c t u r e coupled to the ground s t a t e (e. g., 0 +, 2 +, 4 +, . . , in an e v e n - e v e n and jrr + ~ k 2 k s e r i e s with/e = 0, 1, 2 , . . . , in an o d d - e v e n nucleus) is often known to p e r s i s t upto f a i r l y high e x c i t a t i o n s (~ 3 - 4 MeV). A s s u m i n g this s p e c i a l s e t of c o r r e l a t e d l e v e l s to be a s e t of equidistantly s p a c e d s t a t i s t i c a l l e v e l s of spacing def f above the r e n o r m a l i s e d gas m o d el F e r m i s u r f a c e e, we note that (i) upto about E~nax, l e v e l s of only one kind of p a r i t y a r e e f f e c t i v e for E2 t r a n s i t i o n s and (if) an upper l i m i t can be s e t at N ~E ymax (not E vmax ), as d i s c u s s e d below. Condition (i) gives 266
aeff -- 2 a F, = 2 / e r
26 October 1970
(5)
= 1/eef f "ev
R e g a r d i n g (if), it is w e l l known that h i g h e r r o t a t i o n a l - v i b r a t i o n a l groups of l e v e l s a p p e a r at N 3 MeV and that our c o r r e l a t e d s t a t i s c i t c a l l e v e l s (5) d e g e n e r a t e into an exponentially v a r y i n g l e v e l density Po above ~ ½E~nax ; si n ce the n u m b e r of l e v e l s between ½E~max and E ~nax i n c r e a s e s f r o m tens to h u n d r e d - t h o u s a n d s , the a v e r a g e d e f f is so s m a l l , and the c o m p e t i t i o n b e t w e e n E2 t r a n s i t i o n - f a v o u r i n g l e v e l s is so l a r g e , that it is i m p o s s i b l e to o b s e r v e the ef f ect s of s p e c i a l l e v e l s any m o r e . M o r e o v e r , the e l e c t r o n i c s and the d et ect i n g co u n t er e n e r g y s e l e c t i v i t y a r e often u su al l y b i a s e d ag ai n st counting t h ese low e n e r g y y - r a y s in typical e x p e r i m e n t s [12, 14, 15]. The c o r r e l a t i o n (3) thus m o d i f i e s to m ax ~E m a x (ET )av ~ ~ V ~ %.
aeff ~ E v " %
.
(6)
The n u m b e r of p r o m p t y - r a y s n 9, a r e c a l c u l a t e d f r o m the c o r r e l a t i o n (6) and a r e shown in figs. l(c) and 2(c). E x p e r i m e n t a l data a r e a l s o shown in figs. 1 and 2 when a v a i l a b l e . We note that f o r figs. l(b) and l(c), the e x p e r i m e n t a l e r r o r s have been i n c r e a s e d by a f a c t o r of 3 on the b a s i s of s o m e c o m m e n t s of the w o r k e r s [15] on 235U + n. Our absolute y i e l d s n3, a g r e e f a i r l y w e l l with the m e a s u r e d r e l a t i v e y - r a y y i e l d s [14, 15]. S e v e r a l p h en o m en a c o m p e t e with (6) to p r o v i d e a l t e r n a t i v e paths of d e e x e i t a t i o n of the p r o m p t f i s s i o n f r a g m e n t s , e . g . , the d el ay ed neutron em i s. sion, the d i r e c t E 7max t r a n s i t i o n to the ground st at e and o t h er c r o s s - o v e r y - r a y s of d i f f e r e n t m u l t i p o l a r i t i e s , e t c . . Such delayed, weak and r e t a r d e d p h e n o m e n a have b e e n a s s u m e d in the p r e s e n t w o r k to make a negligible contribution to the m a i n s t r e a m of d e e x c i t a t i o n (6). The e x p e r i m e n t a l t i m e r e s o l u t i o n of ~ 10 -9 to 10 -11 s [14, 15] ' d e f i n e s ' the p r o m p t p r o c e s s e s in y - d e c a y . This t i m e s c a l e is a p p r o p r i a t e f o r allowed E2 t r a n s i t i o n s .
Reference s [1] R. Sarkar and A. Chatterjee, Phys. Letters 30B (1969) 313. [2] R. Sarkar and A. Chatterjee, Phys. Rev. C1 (1970) 619. [3] U. Mosel and W. Greiner. Z.Physik 217 (1968) 256; 222 (1969) 261. [4] R. Sarkar, to be published. [5] Gy. Kiuge and A. Lajtai, Phys. Letters 27B (1968) 65; 30B (1969) 311. [6] J.M. Ferguson and P.A Read, Phys. Rev. 150 (1966) 1018.
Volume 33B, n u m b e r 4
PHYSICS
[7] F. Dickmann and K. Dietrich, Nucl. Phys. A129 (1969) 241. [8] H. W. Schmitt, P r o c . Intern. Sym. on Why and how shlould we investigate nucleides f a r off the stability line, Lykesil, Sweden (1966); Arkiv Phys. 36 (1966) 633; P r o c . Intern. Conf. on the P h y s i c s and c h e m i s t r y of f i s s i o n , Vienna, 1969 (L A. E. A., Vienna, 1969) p. 67. [9] L.Wilets, T h e o r i e s of nuclear fission (Oxford University, Oxford, 1964). [10] S. Chatterjee, unpublished private communication. [11] W. K. H. Panofsky and M. Phillips, Classical e l e c tricity and magnetism (Addison-Wesley, Reading, Mass. 1962) Fig. 1-3, p. 16 and Ex. 6, p. 116 [12] E. C. Maienschein, R.W. Peele, W. Zobel and T. A. Law, Second Intern. Conf. on Peaceful u s e s of atomic energy, Geneva, 1958 (United Nations, Geneva) Vol. 15, p a p e r No. P/670; F . E . W . Rau, Ann. Physique 10 (1963) 252.
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[13] V . F . Apalin, Yu.N.Gritsiuk, I.E.Kutikov, V.L Lebedev and L. A. Mikailian, Nucl, Phys. 71 (1965) 553. [14] S . A . E . Johanson, Nuel. Phys. 60 (1964) 378; 64 (1965) 147. [15] H. M a i e r - L e i b n i t z , H.W. Schmitt and P. A r m b r u s t e r Intern. Conf. on the P h y s i c s and c h e m i s t r y of f i s sion, Salzburg, 1965 (I. A. E . A . , Vienna, 1965) Vol. 2, p. 143. [16] R. B. Leaehman, Second Intern. Conf. on Peaceful uses of atomic energy, Geneva, 1958 (United Nations, Geneva) Vol. 15, p a p e r No. P/665; V, M. Strutinskii, Soviet Phys. J E T P 10 (1960) 613. [17] T . E . O . E r i c s o n Advances in Phys. 9 (1960) 425. [18] S.K. Chosh, S. Chatterjee and A. Chatterjee, to be published. [19] E. Erba, U. Facchini and E. Seatta-Menichella, Nuovo Cim. 22 (1961) 1237; U. Facchini and E. Saetta-Menichella, Energia Nucleare 15 (1968) 54.
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