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Average number and energy of prompt neutrons from fission
Volume 27B, number 2
AVERAGE
PHYSICS LETTERS
NUMBER
AND
ENERGY
OF
10June 1968
PROMPT
NEUTRONS
FROM
FISSION
Gy. KLUGE and A. L A J T A I Central Research Institute f o r Physics, Budapest, Hungary
Received 15 April 1968
The initial temperatures corresponding to the excitation energies of individual fission fragments are predicted to a fair accuracy from a simple statistical model. These values are then used for the evaluation of the average number and energy of the neutrons emitted per fragment.
quently a p p e a r s a s excitation i s bound up in the d e f o r m a t i o n . T h i s is in c o n t r a d i c t i o n with an a s s u m e d t h e r m a l e q u i l i b r i u m . In this work an a t t e m p t i s m a d e to explain this a p p a r e n t c o n t r a d i c tion. A s s u m i n g the o n s e t of t h e r m a l e q u i l i b r i u m j u s t b e f o r e s c i s s i o n , the d i s t r i b u t i o n of the e x c i tation e n e r g y p e r f r a g m e n t p a i r is such that the l e v e l density dependent n u c l e a r t e m p e r a t u r e s of the two f r a g m e n t s [1] a r e equal and can be d e scribed as
The n u c l e a r t e m p e r a t u r e of the f r a g m e n t s in low e n e r g y f i s s i o n r e a c t i o n s , as d e t e r m i n e d e x p e r i m e n t a l l y f r o m the e m i s s i o n e n e r g i e s of n e u t r o n s , s e e m s to be s i m i l a r in a g iv e n f r a g m e n t p a i r (figs. 1 and 2). T h i s s u g g e s t s that the two f r a g m e n t s a r e in t h e r m a l e q u i l i b r i u m at the s c i s sion point. H o w e v e r , on the i n t e r p r e t a t i o n s of the m e a s u r e d d e p e n d e n c e of the a v e r a g e n u m b e r of p r o m p t n e u t r o n s on the f r a g m e n t m a s s (figs. 3 and 4) the f r a g m e n t s a r e a s s u m e d to be ' c o l d ' at s c i s s i o n , while m o s t of the e n e r g y that s u b s e J,(;2Cf
~- cA/. CULA TED (w/,,h ~ m f ~ [ (~2.D • CALCULATEO
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F~AGMENT MA5$ NfJNBER Fig. 1. A v e r a g e c e n t r e - o f - m a s s neutron kinetic energy f o r spontaneous fission of 252Cf as a function of fragment mass.
65
Volume27B, number 2
PHYSICS LETTERS
10 June 1968
2.35,, + tl~herm= / 2,4 o EXPERIMENTAL [I~] z~ CALCULATED (Mth Z)o/o f,.om [ ~,S~J)
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Fig. 2. Average centre-of-mass neutron kinetic energy for thermal-neutron-induced fission of 235U as a function of fragment mass.
Tll = dud l°gWl(U)lu=Ull
a p p r o x i m a t i o n to the e n e r g y d i s t r i b u t i o n of c a s cade n e u t r o n s in a compound p r o c e s s is u s e d which r e a d s
= = d logw2(U) dU
U=U2,- 1
T2 '
w h e r e Ui = E i - Pi(Z) - Pi(N), i = 1, 2, while E i is the total excitation energy, Pi(Z) and Pi(N) the c o r r e s p o n d i n g p a i r i n g e n e r g y f o r a f r a g m e n t with Z p r o t o n s a n d N n e u t r o n s . T1, T2, Wl(U) , w2(U) a r e the n u c l e a r t e m p e r a t u r e s and l e v e l d e n s i t i e s , r e s p e c t i v e l y . The s u m of the excitation e n e r g y of the two f r a g m e n t s , E 1 + E 2 i s given by Q - Ek w h e r e Q is the e n e r g y r e l e a s e d f o r a given f r a g ment p a i r a s s u m i n g h i g h e s t p r o b a b il it y c h a r g e d i s t r i b u t i o n and Ek i s the a v e r a g e total k in e t ic energy of the two f r a g m e n t s . In this way the e x citation e n e r g i e s E l , E 2 and the i n i ti a l f r a g m e n t t e m p e r a t u r e s can be d e t e r m i n e d . F o r the evaluation of the a v e r a g e energy of the e m i t t e d n e u t r o n s Lang and Le C o u t e u r ' s [2] 66
~p(c) ~ (~c(~m-1/q m) e x p ( - ~ / q ) , w h e r e c is the k i n et i c energy of the n e u t r o n s , while m = ~ a n d q = ~ T i. Ti i s the i n i t i a l nuc l e a r t e m p e r a t u r e . A s s u m i n g constant n eu t r o n c a p t u r e c r o s s section, we get the Maxwellian distribution 1
3
(p(E) ~ (E2/T~) exp ( - E / T i) , hence the a v e r a g e n eu t r o n e n e r g y 77 = ~- = ~ T i. Th e v a l u e s of Q, E k and the l e v e l d e n s i t i e s u s e d in the evaluation of the a v e r a g e excitation e n e r g y p e r f r a g m e n t in the spontaneous f i s s i o n of 252Cf and the t h e r m a l neutron induced f i s s i o n of 235U w e r e taken f r o m r e f s . 3-5, 6-7, and 1,8,9, r e s p e c t i v e l y . Th e c o n s i d e r a b l e d i f f e r e n c e s b e tween the r e p o r t e d data do not c a u s e a p p r e c i a b l e d e v i a t i o n s in the p r e d i c t i o n s of the n u c l e a r t e m -
10 June 1968
PHYSICS LETTERS
Volume 27B, number 2
~U~Cf 5
•
EXP£RIMENTAL
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A CALCULATfD
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Fig. 3. Neutron yields as functions of fragment mass, for spontaneous fission of 252Cf.
0 3-
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Fig. 4. Neutron yields as functions of fragment mass, for thermal-neutron-induced fission of 235U. p e r a t u r e s . Th e p r e d i c t e d v a l u e s of the a v e r a g e ne ut ro n e n e r g y shown in figs. 1 and 2 a r e c o n s i s t e n t with the m e a s u r e d v a l u e s within the e x p e r i m e n t a l a c c u r a c y of the data r e p o r t e d [10-11]. Th e a v e r a g e n u m b e r v(A) of n e u t r o n s p e r f r a g m e n t is e v a l u a t e d f r o m the s i m p l e e n e r g y balance E(A)
v(A) =
-
E¥(A)
TI(A)+'~A
(2) '
w h e r e v(A), ~?(A) and B-A a r e the a v e r a g e n u m b e r , kine t i c e n e r g y and binding e n e r g y of the n e u t r o n s
e m i t t e d f r o m the f r a g m e n t with m a s s n u m b e r A , r e s p e c t i v e l y . E ( A ) i s the total excitation e n e r g y of and E~(A) the y - e n e r g y e m i t t e d f r o m this f r a g I -ment to be equal to ~ B A . Th e v a l u e s p r e d i c t e d f r o m this f o r m u l a a r e shown in figs. 3 and 4. T h e s e v a l u e s a r e much r a t h e r influenced by the c o n c r e t e v a l u e s of l e v e l density p a r a m e t e r s , than the n u c l e a r t e m p e r a t u r e s . F o r e x a m p l e , in the c a s e of 252Cf the ap p l i cat i o n of data f r o m ref. 8 does not give so good a g r e e m e n t with the e x p e r i m e n t a s it does f o r the a v e r a g e n eu t r o n energies. 67
Volume27B, number 2
PHYSICS
T h e r e s u l t s s e e m t o i n d i c a t e t h a t , c o n t r a r y to the earlier interpretations [10,11,14-19], the r o l e of t h e d e f o r m a t i o n e n e r g y c e a s e s t o b e i m portant immediately before scission. It is reasona b l e t o b e l i e v e t h a t t h e d i s t r i b u t i o n of t h e d e f o r m a t i o n e n e r g y o v e r t h e two f r a g m e n t s i s a l s o d e t e r m i n e d by t h e l e v e l d e n s i t y of e a c h of t h e f r a g m e n t s i n v o l v e d , e v e n if w e c a n a t a l l k e e p on t h e i d e a of t h e d e f o r m a t i o n e n e r g i e s a t t h i s s t a g e of t h e f i s s i o n p r o c e s s . T h e f a c t of d e f o r m a t i o n must be reflected by the level-densities. In c o n c l u s i o n , t h e r e s u l t s of t h e p r e s e n t i n vestigation show that the non-adiabatic processes can be more important than was believed. This i n v e s t i g a t i o n w i l l b e e x t e n d e d to o t h e r n u c l i d e s , w i t h f u r t h e r i n v e s t i g a t i o n s of t h e v a l i d i t y of t h e model.
References 1. A . G i l b e r t and A . G . W . C a m e r o n , (1965) 1446. 2. K . J . L e Couteur and D.W. Lang, (1959) 32. 3. J . C. D. Milton, UCRL-9883-Rev. 4. J . W i n g and J . V a r l e y , ANL 6886
68
Can. J. P h y s . 43 Nucl. Phys. 13 (1966). (1964)
LETTERS
10 June 1968
5. U . O . M y e r s and W.J.Swiatecki, Nuel. Phys. 81 (1966) 1. 6. J . C . D . M i l t o n and J . S . F r a s e r , Can. J . Phys. 40 (1962) 1626. 7. H.W. Schmitt, I.H. Neiler and F . J . Walter, Phys. Rev. 141 (1966) 1146. 8. Yu. N. Shubin, A .U. Malyshev and V. S. Stavinsky, P r o c . Conf. on Nuclear data, P a r i s , Vol. I (1966) 163. 9. E. E r b a , U. Facchini and E. Saetta-Menichella, Nucl. Phys. 84 (1966) 595. 10. H.R. Bowman, J . C . D . Milton, S.G. Thompson and W.J.Swiatecki, Phys. Rev. 129 (1963) 2133. 11. J . C . D . Milton and J . S . F r a s e r , P r o c . Syrup. on P h y s i c s and c h e m i s t r y of fission, Salzburg, Vol. II (1965) p. 39. 12. W.E. Stein, P r o c . Syrup. on P h y s i c s and c h e m i s t r y of fission, Salzburg, Vol. I (1965) p. 491. 13. V . F . A p a l i n , J u . P . D o b r y n i n , V . P . Z a k h a r o v a , I . E . Kutikov and L . A . Mikaelyan, Atomnaja Energiya 8 (1960) 15. 14. J. T e r r e l l , Phys. Rev. 127 (1962) 880. 15. M . V . B l i n o v and V . P . E i s m o n t , Z. Eksp. i T e o r . Fiz. 42 (1962) 180. 16. R.Vandenbosch, Nucl. P h y s . 46 (1963) 129. 17. P. Fong, Phys. Rev. L e t t e r s 11 (1963) 375. 18. T.Honda, J. P h y s . Soc. Japan 18 (1963) 1697. 19. T.D. Thomas, V . M . G i b s o n and G. J. Safford, P r o c . Syrup. on P h y s i c s and c h e m i s t r y of fission, Salzburg, Vol. I (1965) p. 467.
AVERAGE
PHYSICS LETTERS
NUMBER
AND
ENERGY
OF
10June 1968
PROMPT
NEUTRONS
FROM
FISSION
Gy. KLUGE and A. L A J T A I Central Research Institute f o r Physics, Budapest, Hungary
Received 15 April 1968
The initial temperatures corresponding to the excitation energies of individual fission fragments are predicted to a fair accuracy from a simple statistical model. These values are then used for the evaluation of the average number and energy of the neutrons emitted per fragment.
quently a p p e a r s a s excitation i s bound up in the d e f o r m a t i o n . T h i s is in c o n t r a d i c t i o n with an a s s u m e d t h e r m a l e q u i l i b r i u m . In this work an a t t e m p t i s m a d e to explain this a p p a r e n t c o n t r a d i c tion. A s s u m i n g the o n s e t of t h e r m a l e q u i l i b r i u m j u s t b e f o r e s c i s s i o n , the d i s t r i b u t i o n of the e x c i tation e n e r g y p e r f r a g m e n t p a i r is such that the l e v e l density dependent n u c l e a r t e m p e r a t u r e s of the two f r a g m e n t s [1] a r e equal and can be d e scribed as
The n u c l e a r t e m p e r a t u r e of the f r a g m e n t s in low e n e r g y f i s s i o n r e a c t i o n s , as d e t e r m i n e d e x p e r i m e n t a l l y f r o m the e m i s s i o n e n e r g i e s of n e u t r o n s , s e e m s to be s i m i l a r in a g iv e n f r a g m e n t p a i r (figs. 1 and 2). T h i s s u g g e s t s that the two f r a g m e n t s a r e in t h e r m a l e q u i l i b r i u m at the s c i s sion point. H o w e v e r , on the i n t e r p r e t a t i o n s of the m e a s u r e d d e p e n d e n c e of the a v e r a g e n u m b e r of p r o m p t n e u t r o n s on the f r a g m e n t m a s s (figs. 3 and 4) the f r a g m e n t s a r e a s s u m e d to be ' c o l d ' at s c i s s i o n , while m o s t of the e n e r g y that s u b s e J,(;2Cf
~- cA/. CULA TED (w/,,h ~ m f ~ [ (~2.D • CALCULATEO
A&AA
2.2o EXPERIMENTAL CALCULATEL)
++t"+
Lt03 Z~
2,0 A
•
÷A
+
ku
+
Z~
•
A A
A
4-
+
÷A
¢.6
•
A
a,o
÷ +
fl
g
ee
/%
•
}
o4.
A
4-
4-
•
•
,
Q 0 0 •0~••
+
4.0
F~AGMENT MA5$ NfJNBER Fig. 1. A v e r a g e c e n t r e - o f - m a s s neutron kinetic energy f o r spontaneous fission of 252Cf as a function of fragment mass.
65
Volume27B, number 2
PHYSICS LETTERS
10 June 1968
2.35,, + tl~herm= / 2,4 o EXPERIMENTAL [I~] z~ CALCULATED (Mth Z)o/o f,.om [ ~,S~J)
2.0
/.8
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t.4
/,2
n o~ d A
4,0
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A
A
A
L
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~RAGMENT
MA~$
NUMBER
Fig. 2. Average centre-of-mass neutron kinetic energy for thermal-neutron-induced fission of 235U as a function of fragment mass.
Tll = dud l°gWl(U)lu=Ull
a p p r o x i m a t i o n to the e n e r g y d i s t r i b u t i o n of c a s cade n e u t r o n s in a compound p r o c e s s is u s e d which r e a d s
= = d logw2(U) dU
U=U2,- 1
T2 '
w h e r e Ui = E i - Pi(Z) - Pi(N), i = 1, 2, while E i is the total excitation energy, Pi(Z) and Pi(N) the c o r r e s p o n d i n g p a i r i n g e n e r g y f o r a f r a g m e n t with Z p r o t o n s a n d N n e u t r o n s . T1, T2, Wl(U) , w2(U) a r e the n u c l e a r t e m p e r a t u r e s and l e v e l d e n s i t i e s , r e s p e c t i v e l y . The s u m of the excitation e n e r g y of the two f r a g m e n t s , E 1 + E 2 i s given by Q - Ek w h e r e Q is the e n e r g y r e l e a s e d f o r a given f r a g ment p a i r a s s u m i n g h i g h e s t p r o b a b il it y c h a r g e d i s t r i b u t i o n and Ek i s the a v e r a g e total k in e t ic energy of the two f r a g m e n t s . In this way the e x citation e n e r g i e s E l , E 2 and the i n i ti a l f r a g m e n t t e m p e r a t u r e s can be d e t e r m i n e d . F o r the evaluation of the a v e r a g e energy of the e m i t t e d n e u t r o n s Lang and Le C o u t e u r ' s [2] 66
~p(c) ~ (~c(~m-1/q m) e x p ( - ~ / q ) , w h e r e c is the k i n et i c energy of the n e u t r o n s , while m = ~ a n d q = ~ T i. Ti i s the i n i t i a l nuc l e a r t e m p e r a t u r e . A s s u m i n g constant n eu t r o n c a p t u r e c r o s s section, we get the Maxwellian distribution 1
3
(p(E) ~ (E2/T~) exp ( - E / T i) , hence the a v e r a g e n eu t r o n e n e r g y 77 = ~- = ~ T i. Th e v a l u e s of Q, E k and the l e v e l d e n s i t i e s u s e d in the evaluation of the a v e r a g e excitation e n e r g y p e r f r a g m e n t in the spontaneous f i s s i o n of 252Cf and the t h e r m a l neutron induced f i s s i o n of 235U w e r e taken f r o m r e f s . 3-5, 6-7, and 1,8,9, r e s p e c t i v e l y . Th e c o n s i d e r a b l e d i f f e r e n c e s b e tween the r e p o r t e d data do not c a u s e a p p r e c i a b l e d e v i a t i o n s in the p r e d i c t i o n s of the n u c l e a r t e m -
10 June 1968
PHYSICS LETTERS
Volume 27B, number 2
~U~Cf 5
•
EXP£RIMENTAL
x
-.
f.42]
-
CI01
A CALCULATfD
..x
2
,t 8o
~
~o
,~o
,~o
,~o
/:RA6MENT
~o
,~o
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MA,S~ NUMBER
Fig. 3. Neutron yields as functions of fragment mass, for spontaneous fission of 252Cf.
0 3-
~lr[t(. ~ o
- - - $o
oo
IOO
~io
a~o
~Jo
,,6o
s~o
~o
.CRAGMEN T ,~,4 S$ NUMBER
Fig. 4. Neutron yields as functions of fragment mass, for thermal-neutron-induced fission of 235U. p e r a t u r e s . Th e p r e d i c t e d v a l u e s of the a v e r a g e ne ut ro n e n e r g y shown in figs. 1 and 2 a r e c o n s i s t e n t with the m e a s u r e d v a l u e s within the e x p e r i m e n t a l a c c u r a c y of the data r e p o r t e d [10-11]. Th e a v e r a g e n u m b e r v(A) of n e u t r o n s p e r f r a g m e n t is e v a l u a t e d f r o m the s i m p l e e n e r g y balance E(A)
v(A) =
-
E¥(A)
TI(A)+'~A
(2) '
w h e r e v(A), ~?(A) and B-A a r e the a v e r a g e n u m b e r , kine t i c e n e r g y and binding e n e r g y of the n e u t r o n s
e m i t t e d f r o m the f r a g m e n t with m a s s n u m b e r A , r e s p e c t i v e l y . E ( A ) i s the total excitation e n e r g y of and E~(A) the y - e n e r g y e m i t t e d f r o m this f r a g I -ment to be equal to ~ B A . Th e v a l u e s p r e d i c t e d f r o m this f o r m u l a a r e shown in figs. 3 and 4. T h e s e v a l u e s a r e much r a t h e r influenced by the c o n c r e t e v a l u e s of l e v e l density p a r a m e t e r s , than the n u c l e a r t e m p e r a t u r e s . F o r e x a m p l e , in the c a s e of 252Cf the ap p l i cat i o n of data f r o m ref. 8 does not give so good a g r e e m e n t with the e x p e r i m e n t a s it does f o r the a v e r a g e n eu t r o n energies. 67
Volume27B, number 2
PHYSICS
T h e r e s u l t s s e e m t o i n d i c a t e t h a t , c o n t r a r y to the earlier interpretations [10,11,14-19], the r o l e of t h e d e f o r m a t i o n e n e r g y c e a s e s t o b e i m portant immediately before scission. It is reasona b l e t o b e l i e v e t h a t t h e d i s t r i b u t i o n of t h e d e f o r m a t i o n e n e r g y o v e r t h e two f r a g m e n t s i s a l s o d e t e r m i n e d by t h e l e v e l d e n s i t y of e a c h of t h e f r a g m e n t s i n v o l v e d , e v e n if w e c a n a t a l l k e e p on t h e i d e a of t h e d e f o r m a t i o n e n e r g i e s a t t h i s s t a g e of t h e f i s s i o n p r o c e s s . T h e f a c t of d e f o r m a t i o n must be reflected by the level-densities. In c o n c l u s i o n , t h e r e s u l t s of t h e p r e s e n t i n vestigation show that the non-adiabatic processes can be more important than was believed. This i n v e s t i g a t i o n w i l l b e e x t e n d e d to o t h e r n u c l i d e s , w i t h f u r t h e r i n v e s t i g a t i o n s of t h e v a l i d i t y of t h e model.
References 1. A . G i l b e r t and A . G . W . C a m e r o n , (1965) 1446. 2. K . J . L e Couteur and D.W. Lang, (1959) 32. 3. J . C. D. Milton, UCRL-9883-Rev. 4. J . W i n g and J . V a r l e y , ANL 6886
68
Can. J. P h y s . 43 Nucl. Phys. 13 (1966). (1964)
LETTERS
10 June 1968
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