Z. Phys. A

Atomic Nuclei 337, 237-238 (1990)

Zeitsehrift far PhysikA

Atomic Nuclei 9 Springer-Verlag 1990

Short note

Fission neutron multiplicity versus fragment mass and kinetic energy A. Ruben and H. Miirten Technische Universit/~t Dresden, Mommsenstrasse 13, DDR-8027 Dresden, German Democratic Republic Received July 2, 1990

Abstract: N e ~ t r o n m u [ t ~ p ~ c s s~ope ~ers~s / r a 9msnts" total ~ n e t t C e n e r g y ts s t u d ~ e d as f u r o r ( o n o/ m~tss ntunber ~n &he /~'cm'~)ol~ o / a p A e n o m e n o [ o @fC~s S C Z S S Z O n ~ s CE?. ,

s163

mentct[

dczt~ / o r

z52

~del

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C/Cs/D

are

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~)e[~ ~'eprod~ced.

Average neutron m u l t i p l i c i t y {)(A,TKE) as function of fragment mass number A and total k ~ e t i c energy TKE has been measured in the case of Cf(sf) by S e V eral " " authors * - - - . D a t a ~ndlcate a strong influence of shell effects giving rise to the well-known saw-tooth like behaviour of [;(A). However, a d e t a i l e d t h e o r e t i c a l understanding of the f u l l - s c a l e energy partition in fission is still crucial. In the present work, neutron multiplicity as a, measure of the average excitation energy E* of f r a g m e n t s is studied in dependence of both A and TKE defining mass a s y m m e t r y and e l o n g a t i o n of the f i s s i o n i n g n u c l e u s at scission point. According to energy balance consideration, total neutron multiplicity decreases with increasing TKE. However, shell effects d e p e n d i n g on d e f o r m a t i o n and temperature at scission have much influence on the slope O u / O T K E for individual fragments. Thus, the energy p a r t i t i o n p r o b l e m has to be solved as function of A and TKE. We make use of an energy-conservation c o n s i s t e n t scission p o ~ t model s t a r t i n g with the general energy b a l a n c e "

I

I

I

I

A1

I

I

I

]

4,

4,

:

E

- average energy release in fission, - p r e - s c i s s i o n kinetic energy,

E o~t - Coulomb

potential

energy

Ea, f

fragment

EdL ~

- dissipative

energy

E~ t

intrinsic

excitation

F

deformation

- part of s c i s s i o n energy d e p e n d i n g

at scission,

energy, (nuclear

friction),

at scission,

point p o t e n t i a l on deformation.

The average neutron m u l t i p l i c i t y is given by an energy bal~nce equation concerning fragment deexcitation including average neutron binding energy B", average c e n t r e - o f - m a s s emission energy of n e u t r o n s 6, and average total z-ray emission energy E as Y E~(A t) =

[Bn(A~)+~(A~)]5(At)

+ Ey(At).

(2)

Based on a simple t w o - s p h e r o i d representation of scission point c o n f i g u r a t i o n ( i n c l u d i n g a d i s t a n c e d between the tips of the fragments), F can e a s i l y be m i n i m i z e d with the constraint that TKE is fixed, i.e. d e f i n i n g a ridge line of p o t e n t i a l in d e f o r m a t i o n space. The following simplifications have been introduced to make the model tractable: (i)

description of deformation energy as q u a d r a t i c in change of major spheroid axis with a linear deformability coefficient a d e p e n d i n g on shell effects; (ii) d e s c r i p t i o n of C o u l o m b p o t e n t i a l at scission for two point charges e f f e c t i v e l y located in the spheroid centres; (iii) c a l c u l a t i o n of energy release AE until scission point using Eq. (I) and a s s u m p t i o n that AE=Ep~+Eat 8 , where the ratio Epro/Ea~ ~ is 9

(iv)

.

.

iO

a p p r o x i m a t e d by e x p e r i m e n t a l xndlcatlons ; d e f i n i t i o n of scission point temperature (assumed as equal . in 9 both fragments) by statistical assumptlons and corresponding p a r t i t i o n of EdLs.

D e f o r m a b i l i t y ~ can be d e s c r i b e d in the framework of liquid drop model. Its relation to shell energy ~W has been taken from Ref. Ii. In Ref. 8, effective shell c o r r e c t i o n energies for deformed fragments at scission have been d e d u c e d within the model d i s c u s s e d above on the basis of e x p e r i m e n t a l data on >(A) and TKE(A/Az). These data are r e p r e s e n t a t i v e for average fragmeDt deformations. The d e p e n d e n c e 6W(A) can be well understood in comparison with m i c r o s c o p i c c a l c u l a t i o n s of she~l energies for deformed fragments at scission . F i r s t - o r d e r a p p r o x i m a t i o n of a u / a T K E as function of A can be based on the average semi-empirical &W(A) data, i.e. the 6W d e p e n d e n c e on deformation is neglected. The minor influence of this dependence around most p r o b a b l e d e f o r m a t i o n is verified by the e x p e r i m e n t a l indication that L(TKE) for given A is nearly linear. In a second calculation, the d e p e n d e n c e of shell effects on d e f o r m a t i o n has been considered within the a p p r o x i m a t i v e description of d e f o r m a t i o n e n e r g y (6W defines deforma-

238

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]igh~ fragment heavy fragment light fragment heavy fragment

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,..~ 0.10

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"-~2.01~ 1,0

/t/r/r t/r ~.ix,Aw . ~r~r,k~t/rvrit~

6,0

........

i .........

~0.08

,~

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~ 0.06

102-150

~'3.0

,~ ~t t,

o 0 o o o o o o 0 o o o +

~

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I

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0.02

u .........

i .........

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i

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o.oq~ d. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

140

Heavy 108-144

4.o

150

Fragment

%60

170

Mass

[aS/aTKE](A),~orZ~ZCf(sf)

Fig.2 Calculated slope in comparison with experimental

data'

2,0 I;~ LO

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=========================================================================== 9.o~

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120-132

at scission influenced by shell structure. The linear dependence found ~n all experiments is well described within an energy-conservation-consistent scission point model. However, deviations from this behaviour might be due to the competition of different fission modes 9 as well as due to the h ~ o t h e t i c cold-deformed fission (i.e. at low TKE)--. More detailed investigations will be possible in the framework of t~e macroscopicmicroscopic scission point modelenabling the description of fragment~occurence probability in dependence on A, TKE, E-, and fragment charge. References

I~10~ "F

0000000000 O0 ~

,

TKE [MeV] Fig. 1 D(TKE) for selected mass splits in 252-Cf(sf) ( * / o light/heavy fragments, firstorder approximation, cf. text; lines - calculation considering ~W dependence on deformation for the mass split 108/144).

bility). The results obtained within the two procedures do not differ very much. Fig. 1 represents calculated data for some typical mass splits. The slope aS/aTKE versus A is shown in Fig. ,_~. The calculations reproduce experimental data . In summary, the conclusion can be drawn that the variation of average neutron multiplicity as function of the total kinetic energy for individual fission fragments from a given mass split can be understood as an effect of the fragment stiffness

I. H. Nifenecker et al., Proc. IAEA Symp. on Phys. and Chem. of Fission, Rochester, 1973 (IAEA, Vienna, 1974), Vol. II, p.l17 2. V.P. Zakharova, D.K. Ryazanov, Yad. Fiz. 3o, 38 (1979) 3 Liu Zuhua et al., Chin. J. Nucl. Phys. 6, 1 (1984) 4 I.D. Alkhazov et al., Yad. Fiz. .48, 1835 (1988).- Soy. J. Nucl. Phys. 48, 978 (1988) 5 R. Schmid-Fabian, Thesis, Ruprecht-KarlsUniversit~t Heidelberg, 1988 6 C. Budtz-Jorgenson and H.-H. Knitter, Nucl. Phys. "A4gO, 307 (1988) 7 B.D. Wilkins et al., Phys. Rev. C14, 1832 (1976) 8 A. Ruben et al., "Energy Partition in Nuclear Fission", submitted to Z. Phys. A 9 H. M~rten, Prcc. Int. Conf. on 50-th Anniversary of Nucl. Fiss., Leningrad, 1989, in print I0 F. G6nnenwein, Yad. Const. I, 14 (1988) Ii H. Kildir, K.K. Aras, Phys. Rev. ces, 365 (1982) 12 R.W. Hasse, Yad. Const. I, 3 (1988)