I N D E P E N D E N T F I S S I O N - P R O D U C T Y I E L D S A N D

F A S T - N E U T R O N E M I S S I O N BY F I S S I O N P R O D U C T S

A . ]3. K o l d o b s k t t , V . M. K o l o b a s h k t n , a n d A . A . G u s e v

UDC 539.173.8:539.173.84

A method ts descr tbed for calculatmg the mdependent yields of h s s ton products on the basts of the quantitative cha rac te r t s t t c s of the fas t -neut ron emiss ion . The formulation and solution of the mver se problem are discussed for fission by neutrons at ~14.5 MeV.

1. The charge distribution of the hss ton products of heavy nuclei ts of interest for several r easons : f i rs t , for solving several pract ica l problems m radiation physics (e .g . , calculating the acttvity of fission products); second, for studying the fission p rocess itself, on the basts of its final products; third, for plannmg and ca r ry ing out exper iments tn fission physics . The only complete experimental work tn ths field (measurements of the independent yields of the fission products) has been cs r r i ed out for the fission of U 235 and (partially) for the hss ton of U ~33 and Pu 239 by thermal neut rons . * There ts ex t remely ht t le m- formation avatlable on other types of fission, and no theory for nuclear f ission ts avatlable which gives accura te values of the mdependent f i ss ion-product y te lds . Consequently, empir ical and semiemptr tca l methods a re used to determine these independent y ie lds . The baste c r i te r ion for the vahdity of a par t icular calculation method is the agreement between its resu l t s and experiment .

2. Measurements of the independent yields show that the isobaric charge distribution can be approx- Lmated by a Gausstan function:

*For brevi ty below we specify the type of f ission by ZEA, where Z ts the f issi le element, A ts [ts mass number , and E is the energy of the neutron causing f ission. Thus the fission of U 235 by thermal neutrons is descr ibed by U~d 5, and the f ission of Th 232 by neutrons with an energy of about 14.5 MeV is described by ThZl~ 2 .

TABLE 1. Charge Distribution of the I sobar i c Chain with A = 93 (F is - sion of U 2~5 by Thermal Neutrons)

Z . Relatzve independent Zp ~ yzeld

expermaent calculated

37,26 0,551 35

36 0,075-t- O,010 0,002 37 0,59~0,08 38 0,32=]:0,08 39 0,0t6~0,004 40 6--5 41

Not__._~e Here and below we use the notatzon "-n" instead of -10-n- for brevzty (e g, "8.54-4" instead of "8.54-10"4").

8,54--4

0,0811

0,586 0,320 0,0132 4,12--5 9,73--9

0 o,9 2

O,a .~i tl

0.7 ar

" : I " ~ O,O ~ ~" o,, O:o2 o

o

a,J 80 tO0

o

_ ~ o

o " c ~ 0 A

".'"c ~ %

~ o 120 140

Fig. 1. The ~(A) dependence. �9 P resen t study; A) data of [5].

Moscow Engineer ing- -Phys ics Inst t tute . Translated f rom Izves t iya Vysshikh Uchebnykh Zavedenh, Fiztka, No. 7, pp. 7-13, July, 1974. Ortgmal ar t ic le submitted May 10, 1973.

�9 76 Plenum Pubhshmg Corporation, 22 7 West 1 7th Street, New York, N Y 10011 No part o f this pubheatlon mat be reproduced, stored m a retrieval system, or transmitted, tn any form or by any means, electromc, rnechamcal, photocopying, mtcrofilmzng, recordmg or otherwise, without written permzsston o f the pubhsher A copy o f this artwle ts avadable from the pubhsher for $15 O0

905

TABLE 2. By Thermal Neutrons

Type of flssmn

Comparative Resul ts of the Yteld Calculattons for Ftssion

Ill

Flsslon product

Relative independent (cumulative) ymld

expernnental value th~s study

(0,00024::J:0,00006) Kr95

reuolmahz- [ mon ] [~l y h% I

(o,oa3) (o,ooolss) 23q Putn (0,000268)

U233 ! [,20 1,3~0,2--3 1,34--3 6,57--4 I 1,41--3 th ! 1,08:J:0,06--3

u~ 3 sn~- ' (0,025+0,005) (0,029) (0,1~1) (0,0400)

(0,0165~0.0012 (0,018) (0,09347) (0,0534) p ~ Xe,~ o,ooo7)

UZ~3 Xe,~O (0,225~0,06) (0,266) (0,597) (0,379) th

p239 RbS~ 1,1--4 1,29--4 1,5 '3--5 3,75--4 Uth

y(A, Z)=(c~)-I/2exp[ - (Z-~ZP)2]. (i)

Here y(A, Z) is the relative independent yield of the fission product of mass A and charge Z, Zp is the "most probable charge" (which need not be an integer and usually is not), c = 2(02 + 1/12), and 02 is the dispersion of the Gaussian distribution.

With the experimental mformation available on the independent yields for the case of T,235 it is Uth, possible to determine the optimum values of the parameters Z_ and o, with a good reproduction of the p initial data, tn several cases (specifically, when at least two independent yields are available for the i sobar ic chain with the given value of A). Table 1 i l lus t ra tes the situation with the initial experimental values [1-3] and the calculated values of the independent yields for A = 93.

If the ,ndependent yield ts known for only one isotope for a given isobaric chain, to calculate the other independent yields we must assume one of the pa rame te r s m distribution (1), namely Z_, to be p known. We choose Zp because it Ls possible to predtct the value of this pa rame te r on the barns of the genera lna tu re of the Zp(A) dependence for the given type of fission [4]. By way of contrast , the a(A) depen- dence constructed f rom the "experimental" values of a is very i r r egu la r (1), as can also be seen f rom the

4 !

, ~ o p 2 b~2o2" I

8Q [00 120 /4g A

Fig. 2

Fig. 2. Construct ion of the neu t ron -emi s s ion func- tion for U2~ 5. �9 Values of y(A) obtained f rom Eq. (11); dotted curve) approximate v ' (A)dependence for U235 dashed curve) neu t ron-emiss ion function for f i s - t4 , siou of U 233 by protons at N 12 MeV [23]; solid curve) final v ( A ) curve for TT235 normal ized to V. ~ 1 4 ,

))

4.' / ' ~ >2/ I

80 10Y /go 140 ,4

Fig. 3.

Fig. 3. Calculated neutron-emission function for n.~.232 (solid curve) and U **Li4

(dashed curve). The points were ob- tained f rom Eq. (11); O) for ~h232 [20]; .,. t*14

A) for r~2~8 [14]. u14

906

T A B L E 3. C o m p a r a t i v e R e s u l t s of the Y ie ld C a l c u l a t i o n s fo r F t s -

s ton b y 14-MeV N e u t r o n s

Type of fission

U235 14

U235 14

U235 14

U235 14

U•a8 14

Th232 "14

p 23~ UI4

Fission

product

Krg3

p3~-

Xe135

Xet35

Xe138

Xei35

ReLative independent (cumuLanve) yzeld

expermlental value

0,039+0, 05 --0,002 )

0,15~0,0l

0,43-]-0,02

0,26•

0,0560+0,037

0,083q=0,005

0,47•

0,46q_.=0,03

thas study

(0,032)

0 0851

0,430

0,268

0,0520

0,055

0,460

0,375

t renormallz- atlon method

(0,106)

[8]

(0,0177)

0,0t04 0,0286

0,125 0,39l

0 0447

0,0447

0,0447

0,435

0,0447

0,149

0,9248

0,0240

0,366

0,283

da ta of [5]. We do not b e l i e v e tha t the va lue of the p a r a m e t e r (7 can be p r e d i c t e d by m t e r p o l a t i n g b e tween known v a l u e s [6], and ff thLs p a r a m e t e r ts se t equa l to a c o n s t a n t ( e . g . , 0 .56 [4]) fo r a l l i s o b a r i c c h a i n s we run into a s i g n i f i c a n t d i s c r e p a n c y with e x p e r i m e n t (by an o r d e r of m a g n i t u d e o r m o r e ) m s o m e c a s e s .

I f , on the o t h e r hand , e x p e r i m e n t a l v a l u e s of the m d e p e n d e n t y i e l d s a r e not a v a i l a b l e f o r the e l e m e n t s of the t s o b a r t c cha in , the p a r a m e t e r a m th i s s tudy i s a s s u m e d to equa l t he va lue (r = 0.575 a v e r a g e d o v e r the " e x p e r i m e n t a l " v a l u e s fo r o t h e r i s o b a r i c c h a m s ; th i s v a l u e a g r e e s we l l wi th the v a l u e s cr = 0 .56 [4] and ~ = 0 .57 [6].

3. In t h i s m a n n e r we ob ta ined the p a r a m e t e r s Z_ and g of the c h a r g e d t s t r t b u t i o n f o r A = 72--161 fo r the p r o c e s s UI~ 5. S m c e much l e s s e x p e r i m e n t a l inform~atmn ts a v a t l a b l e fo r o t h e r t y p e s of f i s s i o n , i t t s n a t u r a l to adop t t h i s p r o c e s s a s the " b a s t s " p r o c e s s .

,T23s to a & f f e r e n t p r o c e s s only t h r o u g h a r e n o r m a h z a - In [7] the t r a n s i t i o n was m a d e f r o m the p r o c e s s ~ th t ton of d i s t r t b u t t o n (1) to the o t h e r m a s s - n u m b e r y i e l d . In th i s p r o c e d u r e the v a l u e s of Zp, cr, and thus y r e m a m the s a m e a s f o r TT235 Al though thLS p r o c e d u r e i s e x t r e m e l y r a p i d , i t f r e q u e n t l y l e a d s to e x t r e m e l y Uth �9 l a r g e e r r o r s m the v a l u e s d e t e r m i n e d fo r the i ndependen t y i e l d s (T a b l e s 2 and 3).

C l e a r l y , to ge t a m o r e a c c u r a t e p r e d i c t i o n of the independen t y i e l d s we m u s t change the p a r a m e t e r s of the c h a r g e d t s t r i b u t t o n .

At the ou t s e t we s t t p u l a t e tha t a l l the equatLons u sed to c a l c u l a t e c ha nge s of thLs type invo lve only the p a r a m e t e r Z Wi th r e g a r d to the p a r a m e t e r g, on the o t h e r hand , we a s s u m e it to r e m a i n c o n s t a n t p . f o r each i s o b a m c chain f o r any type of f i s m o n (but tt i s d i f f e r e n t fo r d i f f e r e n t c h a i n s ) . The o p t i m u m v a l u e s of Zp w e r e sought m [8] on the b a s i s of

[Zp]~: (A) = [Zp] u~; (A) + [-~ Zp] ~, (2)

[-~Zp]~=O-5(Zx--92)--O,19(Ax--236)+O,19(~x vu~). (3)

H e r e Z x and A x a r e the c h a r g e and m a s s of the compound n u c l e u s fo r whose f i s s i o n the c o r r e c t i o n ts beLng sought , and Px ts the a v e r a g e n u m b e r of f a s t n e u t r o n s e m i t t e d d u r m g the f i s s i o n of th, s n u c l e u s .

E q u a t i o n s (2) and (3) i m p r o v e the a g r e e m e n t wi th e x p e r i m e n t , but t hey s t i l l do not t ake mto accoun t an e f fec t whtch t s e x t r e m e l y i m p o r t a n t fo r th t s p r o b l e m - the d i f f e r e n c e b e t w e e n the f a s t - n e u t r o n emiss iorLs by d i f f e r e n t i s o b a r i c c h a m s .

907

In the p r e se n t s tudy we take this effect into account by modify ing (3); spec i f ica l ly , we wr i te

[AZp (A)L , = 0,5 (Z x -- 92) -- 0,19 (A x - - 236) + + 0.38 {[, (A)]+ -- [~ (A)]u~ } (4~

H e r e v(A) is the n u m b e r of fas t neu t rons dur ing whose e m i s s i o n by f i s s ion f r a g m e n t s the f i s s ion p roduc t with m a s s n u m b e r A is f o r m e d . A f ac to r of two a p p e a r s in the las t t e r m because of double f i s s ion .

By defmit ton, the quant i t ies v(A) and V a r e r e l a t ed by

2Y~ (A) YA - - A ~ = (5)

ZY~ A

H e r e YA is the cumula t ive m a s s - n u m b e r yield (determLned f r o m the F e r m i cu rve fo r the f i s s ion p roduc t s ) . A l t e rna t ive ly , we can wr i te

- 1 �9 ~ = - ~ , ~ . ~ v (A) Ya (6)

(taking into account the no rm a hz a t L on of the cumula t ive y i e lds to 200~r).

As a ru le , e x p e r i m e n t s on f a s t - n e u t r o n e m i s s i o n yie ld the dependence of the n u m b e r of emLtted n e u t r o n s not on A -- the m a s s of the p roduc t f o r m e d a s a r e s u l t of the neu t ron emissLon, but on A ' -- the m a s s of the f r a g m e n t e m i t t m g the n e u t r o n s . Us ing the obvious relatLon

A = A' -- ~ (A'), (7)

we can ea s i l y t r a n s f o r m f r o m v(A') to v(A) and ca lcu la te AZp f r o m Eq . (4).

Le t us c o n s i d e r some p a r t i c u l a r app l ica t ions of this me thod .

1 . F i s s i o n o f U 233 a n d P u 23a b y T h e r m a l N e u t r o n s

tT235 (the ba si s function), "~th The v(A') dependence fo r the p r o c e s s e s " t h TT233 and P u ~ a a r e taken f r o m [9-11]. rT235 The v(A') dependence fo r '~th ts a l s o used in s tudying f i s s ion by 14-MeV n e u t r o n s . The r e su l t i ng v(A)

dependences w e r e checked by ca l cu la t ions on the bas i s of Eq . (5). The va lues of ~ a r e taken f r o m the r e v i e w [12], while the values of YA a r e taken f r o m [13].

TT235 to ca lcu la te The r e su l t i ng va lues of AZp(A) a r e used a long with Z_~, and ~ for the b a s t s p r o c e s s ~th the mdependent y i e lds on the b a s t s of E q s . (2) and (1). The r e s u l t s f o r certaLn chains a r e shown in Table 2; the expe r imen t a l va lues of the y ie lds a r e taken f r o m [14-17]. We see f r o m Table 2 that , a l though c a l - cu la t ions based on Eq . (4) r e p r o d u c e the expe r imen ta l r e s u l t s m o r e a c c u r a t e l y than ca lcu la t ions based on Eq . (3), the d i f f e rence between the r e s u l t s fo r t h e r m a l - - n e u t r o n f i s s ion ts s m a l l . The a g r e e m e n t can be a t t r ibu ted to the s i m i l a r i t i e s in the behav io r of the funct ions v(A) fo r t he se f i s s ion p r o c e s s e s .

In fact if the d i f f e rence be tween iv(A)] x and iv(A)] is cons tan t and equal to Utah,

['~ (A)] x = ['~ (A)]ut~s -t- to, then the las t t e r m Ln (4) b e c o m e s

0.38 ([~ (A)]~ -- Iv (A)]u~p~} = 0.38 to. (8)

On the o the r hand, UsLng (3), we find

U s m g the app rox ima t ion

0.19 (~x__ ~L~as) = 0,19 1

x [ r A ] + + ~ [ r A i . - [, (A)luf~3. [ r,i~p@. (9)

we find 0.19(Vx--~U~]~)= 0 .38 K, in a g r e e m e n t with (8).

2 . F i s s i o n o f T h T M , U T M , U 2 ~ U 238, a n d P u 23s b y

_ N e u t r o n s a t A b o u t 1 4 . 5 M e V

If the n e u t r o n - - e m i s s i o n funct ions f o r t h e r m a l - n e u t r o n f i s s ion have been studied quite thoroughly , t h e r e is a l m o s t no tn fo rmat ton ava i lab le on f i s s ion by 14-MeV n e u t r o n s . T h e r e a r e only two p a p e r s [18, 19]

908

T~23~ in [18] a g r e e wi th i n f o r m a t i o n on t h i s p r o c e s s and even h e r e the func t ions v(A') fo r the p r o c e s s -14 ~t.232 TT238 p o o r l y * with the f u n c t m n s f o r IL,14 and u16 tn [19] and e i t h e r do not sa t , ss a check of the n o r m a h z a t i o n

to ~ [18] o r c a n n o t b e c h e c k e d m t t n s m a n n e r [19]. U s i n g t h e s e r e s u l t s f o r c a l c u l a t i o n s on the b a s i s os E q . (4), we f ind v a l u e s wh, ch do not a g r e e s a t t s f a c t o r i l y wi th the a v a i l a b l e e x p e r i m e n t a l da t a on the i ndependen t

TT238 [14]. y i e l d s fo r ~LI 1~t'2326 [20], "TT235 [ 1 4 , 2 1 ] ' 1 6 and u16

T h i s c h e c k t s c a r r i e d out in the f o l l o w m g m a n n e r . R e w r , t m g (4), we have

['~ (A)]~ = [~ (A)]u2~ + I {[-~ZRI~--O5(Z,--92)+O19(A~--236)}. (10) th ~ -

U s i n g the v a l u e s of Zp fo r the g iven p r o c e s s , found f r o m an a n a l y s t s of the e x p e r i m e n t a l da t a on the i n d e - penden t y i e l d s , and u s , n g the v a l u e s of Zp and v(A) fo r the b a s t s p r o c e s s U ~ 5, we c a l c u l a t e the quan t i t y iv(A)] x , which i s c o m p a r e d wi th the c o r r e s p o n d i n g va lue of the n e u t r o n - - e m i s s i o n func t ion .

The fo l lowing me thod has been w o r k e d out f o r f ind ing the n e u t r o n - - e m i s s i o n funct ion:

(1) The e x p e r i m e n t da t a on the m d e p e n d e n t y i e l d s and thus on the p a r a m e t e r s of the c h a r g e d t s t r i - but , on fo r the p r o c e s s ,T235 [14.21] a r e u s e d a long with Eq . (10) to c a l c u l a t e the v a l u e s of [v(A)]TT~35 fo r "14 ~16

v a r i o u s v a l u e s os A . The r e s u l t i n g po in t s a t v a r i o u s po in t s a long the a x i s c o r r e s p o n d i n g to the f i s s i o n - p r o d u c t m a s s n u m b e r a r e u s e d to d r a w an a p p r o x i m a t e c u r v e fo r the n e u t r o n - - e m i s s i o n func t ion .

Th i s c u r v e h a s much m c o m m o n with the n e u t r o n - - e m t s s , o n funct , ons d e t e r m i n e d m [22-24] fo r f i s s , on of U 233 and U 238 by p r o t o n s at c o m p o u n d - - n u c l e u s e x c i t a t i o n e n e r g i e s a p p r o x i m a t e l y equal to t h o s e o b s e r v e d in f i s s i o n by 1 4 - M e V n e u t r o n s . The da ta f r o m t h e s e s t u d i e s a r e u s e d to c o n s t r u c t an a p p r o x - i m a t e n e u t r o n - - e m i s s i o n c u r v e fo r TT235 ~16 o v e r t h o s e i n t e r v a l s of the m a s s a x i s fo r which no e x p e r i m e n t a l r e s u l t s a r e a v a i l a b l e on the i ndependen t y i e l d s and w h e r e Eq . (10) cannot be u s e d . We deno te the r e s u l - t i ng c u r v e by v' (A) (F ig . 2) .

(2) The s u m

S - 1 JO0 ~ ~' (A) YA (11) A

i s c a l c u l a t e d fo r the p r o c e s s TT235 "14 on the b a s i s of the da ta of [13] and the new n e u t r o n - - e m i s s i o n c u r v e p ' (A) .

(3) The q u a n t i t y

R = S -~ ",~,~,~ i s c a l c u l a t e d ; the v a l u e s of ~TT235 a r e t aken f r o m [12].

'J14

(4) The f ina l f o r m of the n e u t r o n - e m i s s i o n f tmct lon f o r t h e TT235 p r o c e s s (F ig . 2) t s found: w14

( d ) = ~' ( A ) ; ~

The e x t r e m e l y s l , gh t d i f f e r e n c e b e t w e e n the [ m t t a l n e u t r o n - - e m i s s i o n funct ion and the f ina l func t ion , n o r m a h z e d to P (F ig . 2), can be t a k e n a s e v i d e n c e m f a v o r of the v a l , d t t y of t h i s c a l c u l a t i o n p r o c e d u r e .

(5) I t , s then a s s u m e d tha t the f o r m of the n e u t r o n - - e m i s s i o n func t ions f o r the g iven p r o c e s s e s of f l s s , o n by n e u t r o n s at ~ 14 .5 MeV i s g e n e r a l l y s x m f l a r .

The q u a n t i t y

S ~ - - 1 a00 A

/~,232 VT233 TT238 T~,239~ t s c a l c u l a t e d . H e r e the s u b s c r i p t "x" r e f e r s to one of the fou r f i s s i o n p r o c e s s e s ~• u14 , u 1 4 , ru l4 ! and v(A) i s the n e w l y c a l c u l a t e d n e u t r o n - - e m i s s i o n func t ion s TT235 w14 �9

(6) The quan t i t y R x = (Sx)-lfi x ~s c a l c u l a t e d .

(7) The n e u t r o n - - e m i s s i o n func t ion f o r the g iven f i s s i o n p r o c e s s t s found: iv(A)] x = Rxv(A ).

F i g u r e 3 shows the p(A) d e p e n d e n c e fo r .~t.232 U~38 ~,,t4 and c a l c u l a t e d b y the me thod d e s c r i b e d a b o v e .

235|~ (8) F o r each t ype of f i s s i o n the u(A) d e p e n c e n c e fo r Uth .~ u s e d a long with E q s . (4) and (2) to f ind Zp . The known v a l u e s os the p a r a m e t e r a a r e used in Eq . (1) to c a l c u l a t e the r e l a t i v e , ndependen t y i e l d s .

�9 I f we a s s u m e s i m , l a r d e p e n d e n c e s fo r the f i s s i o n p r o c e s s e s wi th s i m i l a r c o m p o u n d - - n u c l e u s e x c , t a t i o n e n e r g i e s .

909

The product yields for 5ssLon by neutrons at ~ 14.5 MeV calculated by the three procedures discussed above are compared in Table 3. The experimental data on the yields are taken from [14, 20, 21, 25, 26).

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