N E U T R O N M O D E R A T I N G I~OWER OF I N E L A S T I C

S C A T T E R I N G M E D I A

D . A . K o z h e v n i k o v a n d S . S . C h e k a n o v a UDC 621.039.51

In a lmost all modera tors , especial ly those containing a large quantity of nuclear fuel and cons t ruc- tional mater ia ls , inelastic scat ter ing plays a determInative role in the slowing down of neutrons with ener - gies g rea te r than 1-2 MeV.

The effect of inelastic scat ter ing on the slowing-down length of neutrons was f i r s t est imated in [1], in which there was given an approximate numerical solution of the equations for the spatial moments of the neutron distribution function.

Considerable p rogres s has been made in recent years in the study of the s p a c e - - e n e r g y distribution of neutrons slowed down as a resul t of inelastic scat ter ing. This has been due to the construction of mult i - group p rograms with grouped constants est imated f rom the stat is t ical model (see [2], for example).

The advantages of being able to direct ly calculate the moderat ing power of an inelastic scat ter ing me- in dium S M (E) are obvious. This quantity is given by

i n in - i n E SM(E)--~M (E)~M(), (1)

where ~ and ~ are respect ively the c ross section for inelastic scat ter ing and the mean variat ion of the lethargy for the inelastic scat ter ing of neutrons with a fixed energy E by a given nucleus M. Knowledge of this quantity facili tates the interpretation of existing experimental data and, with the aid of known analytical and numerical techniques, enables a quantitative allowance to be made for the slowing down brought about as a resul t of inelastic scat ter ing.

The mean lethargy ~(E) of neutrons of energy E in a modera tor of a rb i t r a ry composition is given by:

= ~M (E), (2) (M) (M)

el and in where h M h M are the part ia l probabili t ies of elastic and inelastic scat ter ing respect ively by a given nucleus (the summation is ca r r i ed cut over all mass numbers), or by

= (E) A- S~, (E)], (2') (M)

el and Sin(E) are the macroscopic moderat ing powers of the nuclei where k (E) is the total f ree path, and S M for elast ic and inelastic scat ter ing respect ively.

By definition

E

~ (E) = <In E/E'>M -= l d(o' S In (E/E') W ~ (E--> E', (o ---> o~') dE'. (3) ~q 0

In the s tat is t ical model of the nucleus, the inelastic scat ter ing indicatrix for neutrons is given by [3]:

W~ ( E --> E', o ---> co') : ~ E' e-~'/T(E) , ( 4)

where CM (E)=TM (E) -~ [t --(1 -~ E/T (E)) e--E/T(E)] -1. (4')

Transla ted f rom Atomnaya ]~nergiya, VoL26, No.4, pp.334-337, April, 1969. Original ar t ic le sub- mitted March 11, 1968.

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Here the angular distribution of the inelast ically sca t te red neutrons is taken to be isotropic; E and E' are the neutron energies before and after scat ter ing; T(E) is the final nuclear tempera ture .

Substituting (4) into formula (3), we obtain

E

~I'~(E)=C(E) I E'ln(E/E')e-~'/v(Z)dE' V--t+lnz+e-~+Et(x) - - t _ _ ( t _ _ x ) e - X ,

o

x=E/T(E) , (5)

where Et(x) is the f i r s t - o r d e r exponential integral function and y = 0.5772 (Euler 's constant). Formula (5) was f i r s t obtained in [9].

In this manner ~in (E) is determined by the rat io of the incident neutron energy to the target nuclear t empera ture in the final s tate.

A cer ta in amount of experimental data has been gathered to date on secondary neutron energy spec t ra for va r ious nuclear excitation energies [4-8]. This enables ~in (E) and S in (E) to be calculated for a number of elements di rect ly f rom the measured nuclear t empera tures .

Discrepancies exist, however, in the nuclear tempera tures reported by different authors, due to im- p rec i se est imation of the contribution of secondary neutrons f rom (n, 2n) react ions and to the fact that the nuclear tempera tures were determined f rom different portions of the spect rum. Also, for some abundant elements of the Ear th ' s core there is no experimental data at all. One must therefore r e so r t to a theore- t ical calculation of this important pa rame te r . This calculation can be ca r r i ed through with the aid of the F e r m i - g a s model.

The F e r m i - g a s model leads to the following expression for the nuclear tempera ture :

/ - ~ - 1

(6)

where a is a p a r a m e t e r charac te r iz ing the nuclear level density, ~ is the target nucleus excitation energy, and n = 5/4 for the total level density. The pa rame te r a is independent of the target nucleus excitation energy and depends only on the proper t ies of the nucleus itself.

The mos t successful express ion for the level density pa r ame te r a is that obtained by Newton [10], who took shell effects into account:

a = 2a (]-N + -]z + t) A 2/~, (7)

where J'N and J'Z are the effective neutron and proton angular momenta. For a value of a in this expression of 0.0374 (obtained by Lang [11]), the quantity a given theoret ical ly by expression (7) is in accord with ca l - culations based upon experimental ly determined nuclear t empera tures .

The excitation energy e of the residual nucleus is the difference between the energies of the incident neutron and the inelastically sca t te red neutron. The mean energy of the sca t tered neutrons E' equals 2T, i.e., the excitation energy of the target nucleus

e = E - - 2T. (8)

In the F e r m i - g a s model, upon which expression (6) is based, the nucleons are considered as a gas of weakly interacting par t ic les which occupy energy space f rom an upper boundary E F (the F e r m i surface) . On account of the pai rwise corre la t ion of nucleons in a real nucleus, the Fe rmi surface lies lower than that given by the F e r m i - g a s model. Some of the incident neutron energy goes on breaking up the nucleon pa i r s , so that the effective excitation energy of the residual nucleus equals:

U = E - - 2 T - - A e , hp = hz + AN. (9)

Here A Z and A N denote the reduction of the proton and neutron Fe rmi surfaces for an even number of the respect ive par t ic les . The total level density pa r ame te r s neglecting and allowing for pairwise nucleon c o r - relat ion (a and ap, respectively) are given by the following express ions:

a=e/T2+2,5/T, e = E--2T; (10)

379

a , = U / T ~ + 2 . 5 / T , U = E - - 2 T - - A p . (10')

F o r m u l a (5) was employed to calculate the contribution of inelastic sca t t e r ing to the modera t ing power of more than thi r ty e lements with exper imenta l ly known nuclear t e m p e r a t u r e s .

In the calculat ions on Na, K, Mg, Cr, Ti, Si, Ca, and S, the p a r a m e t e r a was de te rmined f r o m Newton's fo rmula (7) and the nuclear t e m p e r a t u r e s thereby calculated. The modera t ing powers of these e lements were conqurrent ly calculated f r o m the exper imenta l ly de te rmined [4, 7] values of the p a r a m e t e r

a. The values of t / 'ws- : ( cE :n :bd :~a~d :d~ iune:Oo~d angf : ;mc : t f o r Cr and Ti. There a 1 and C. There a re two explanations for this .

F i r s t ly , the p a r a m e t e r s a of the la t te r nuclei a re de te rmined in [4] f r o m reduced values of the nu- c lea r t e m p e r a t u r e s (from the effective t empera tu res ) , which cor responds to an increased ~in.

Secondly, these nuclei a re even-even, so that pa i rwise nucleon cor re la t ions a re important . Allow- ance was made for the la t te r fac tor by reca lcu la t ing the nuclear t e m p e r a t u r e s through the Newton p a r a - m e t e r a in t e r m s of an equation der ived f r o m express ion (10'):

,.Q

S t o

2~

TrT-I-I-C1-T1

1

O l 2 3 4 5 6 7 8 9 tl) fl 12 E~MeV

o ,I:, • ~ _ ~

4

2 cu

2 \ / A~

/ / .

2 3 4 5 6 7 8 9 lO ff t2. E, MeV

2 2 4 5 6 7 8 9 iO li 12 E~MeV 5

' ! I

r .....

7YV , ,

0 ~ 2 3 4 5 5 I0 II 12 E~MeV

Fig. 1. Macroscopic modera t ing power of some e lements with allowance made for inelast ic sca t t e r ing as function of neutron energy E. The resu l t s of calculat ions based upon exper imenta l ly de te rmined nuclear t e m p e r a t u r e s depend on data ob- tained or sys t ema t i zed in the following works : A) Buccino [6]; T) Gordeev et al. [7]; | Thomson [5]; O) Seth et al. [8]; m) Sal 'nikov et al. [4]; []) Malyshev [13, 14].

380

a~;T ~ - - 2 T - - ( E - - Ap) = 0. ( i l )

The values of Ap were taken f r o m [12].

Curves showing the calculated energy dependence of the modera t ing power of var ious e lements with inelast ic s ca t t e r ing taken into account a r e shown in Fig. l ( a , b, c, d). Some of the curves we re drawn through points calculated f r o m m e a s u r e d nucleax t e m p e r a t u r e s taken f r o m [4-8]. The inelastic sca t t e r ing c ro s s sect ions employed in the calculat ions were taken f r o m [15, 16]. Fo r pu rposes of compar i son we have included curves of the modera t ing power of hydrogen as a r e su l t of e las t ic sca t te r ing .

F o r e lements with A < 30, the quantity s in (E) does not exceed the modera t ing power SH(E ) of hydro- gen, or is comparab le with it. Fo r e lements with 30 < A < 130, the quantity sin(E) exceeds the m o d e r a t - ing power of hydrogen by a f ac to r of 3-4 for E > 5 MeV. For e lements with A > 130, sin(E) exceeds SH(E ) by a fac tor of 6-8 over a wide range of energ ies (E > 3 MeV).

L I T E R A T U R E C I T E D

1. H. Volkin, Phys . Rev., 91, 425 (1953). 2. T . A . Germogenova et al., in: P r o b l e m s in the Phys ics of Reac to r Pro tec t ion [in Russian] , Vol. 2,

Atomizdat , Moscow (1966), p .22 . 3. J . Blat t and V. Weisskopf , Theore t i ca l Nuclear Phys ic s [Russian t ranslat ion] , IL, Moscow (1954). 4. V . B . Annfrienko et al., Yadernaya Fizika, 2, 26 (1965). 5. D. Thomson, Phys . Rev., 129, No. 4, 1649 (1963). 6. S. Buccino et al. , Nucl. Phys . , 60, 17 (1964). 7. I . V . Gordeev, D. A. Kardashev , and A. V. Malyshev, Constants of Nuclear Phys ics [in Russian] ,

Gosatomizdat , Moscow (1963). 8. K. Seth et al. , Phys . Rev. Le t t e r s , 1__1.1, 301 (1964). 9. J . Chernik, P a p e r s p r e sen t ed at the 1955 Geneva Internat ional Conference on the Peaceful Uses of

Atomic Energy [Russian t ranslat ion] , Vol. 5, Izd. AN SSSR, Moscow (1958). 10. T. Newton, Canad. J . Phys . , 34, 804 (1956). 11. D. Lang, Nuc. Phys . , 2_6_6, 434 (1961). 12. P . 1~. Nemi rovsk i i and Yu. V. Adamchuk, Nuc. Phys. , 3 ! , 533 (1962). 13. A . V . Malyshev, Zhl~TF, 4_~6, 1187 (1963). 14. A . V . Malyshev, P roc . Intern. Conf. on the Study of Nuclear Structure with Neutrons, Holl. Publishing,

Antwerp (1966), p. 236. 15. Phys ic s of Fas t Reac to r s [in Russian], Atomizdat , Moscow (1965). 16. Bulletin of Informat ion Center on Nuclear Data [in Russian], Vol. 2, Atomizdat , Moscow (1965).

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