N E U T R O N T R A N S P O R T
P R O T O N M E D I U M
A. V. M a r k o v
IN A P O L A R I Z E D
UDC 621.039.51.12
Neutron t r anspor t in polar ized proton media is considerably different f rom t ranspor t in unpolarized media because of the s t rong spin--spin dependence of the sca t ter ing cross section. For example, the neu- t r o n - p r o t o n interact ion c ross sect ion for paral le l spins is ~3b and is ~28b for antiparallel spins. Po la r - ized proton ta rge ts a re used at the present t ime to obtain beams of polar ized neutrons and the question of the use of a polar ized proton re f lec tor in a pulsed fast r eac to r has been considered [1]. It was shown that one could achieve a neutron pulse halfwidth At ~ 1 ~sec by changing the albedo of such a ref lec tor by means of a magnetic field. Because of this, there is in teres t in exact solutions of cer ta in problems involving neutron t ranspor t in polar ized proton media. Equations describing neutron t ranspor t in polar ized proton media have been obtained [1].
~VF-(P, ~)+Z- F- (P~ Q) = d~ {F4W--F-+Z+~W+-F+}; (1)
(r, ~VF+ ~)+ Z+F+ (r, ~)= ~ d~ {ZZW-+F-+~W++F+},
where F- and F + a re the neutron fluxes in the t r iplet and singlet s ta tes; Y.- and Z + a re the interaction c ros s sections for the corresponding s tates; W-- , W +-, W -+, and W ++ are the probabil i t ies of neutron spin reor ientat ions during scat ter ing.
In plane geometry , Eq. (1) can be written for isotropic sca t ter ing in the following manner :
1 0 x
."~x ~ ( , ~ )+~(~ ,~ )=~ j ~(~, ~')d~', (2) - 1
where ~(x, ~) = F+ is a vector; ~ is a mat r ix with elements Cij
i z~- w__ z~ w+- ~= 2Z+ 2Z+ (3)
-~+W-+ Z~ w++ 2;~+ A
and the mat r ix ~ is
where 5 = (Z-/~,+) > 1.
Y=[0 '
We write the solution of Eq. (2) in the form
(x,/~) = �9 (v, ~) exp (--x/v).
One can delineate th ree regions for v:
1 ~ v E ( - - l / a , 1/~);
2 ~ uE(--l, --I/c) or (l/a, I);
3 --* d i scre te values ui lying outside the interval (--1, 1).
The genera l solution of Eq. (2) can be writ ten in the following fo rm:
(4)
Trans la ted f rom Atomnaya Energiya, Vol. 38, No. 3, pp. 179-181, March, 1975. Original ar t ic le submitted August 13, 1974.
�9 19 75 Plenum Publishing Corporation, 22 7 West 17th Street, New York, N. Y. 10011. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission of the publisher. A copy o f this artiele is available from the publisher for $15.00.
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h -x/v i ~ ( x , ~ t ) : ~ [A(vi) i~(vi ,~)e + A ( - - v i ) { l ) ( - - v i , ~ ) e x/vi]
i= i
j ( j + (l) (2)
The d i s p e r s i o n funct ion fo r the de t e rmina t i on of d i s c r e t e va lues vi is
A (z)= i - -2Ci izT (i/(~z)--2C~2zT ( i /z)+4Cz2T (l/z) T (i/(~z),
where T(1/z) = a rg th (1 / z ) , C = det C and the e igerdunct ions a r e
~ (~ vl,r3 = (~ ~ t~ C2~-- 2CviT (I/(~vi) v~
vt~p, [ -- C i~ (~v-- ~)
[ Cv__~ +~(~v- -p , ) [C~- -2~CT (or)] ., *~ ' (~, ~) = |~
\ - - C~i~ (v-- ~)
[ City ) [ crv--~
*(" (v' Vt) = ~ vv (--~) +(o v (v-- }z) '
where
] (v) = Cz~-- 2vCT (I/(~v), ~ (v) = i - - 2vC~2T (v) - - 2vCiiT (I/(~v).
(5)
(6)
(7a)
(7b)
(7c)
(7d)
The o r thogona l i ty and n o r m a l i z a t i o n r e l a t ions needed fo r the d e t e r m i n a t i o n of the coeff ic ients A (v i), A 1 (v), and A2(v) have been d e m o n s t r a t e d [2]. Since the functions ~1) and ~ l ) a r e not o r thogona l fo r v = v ' , we in t roduce the fol lowing two new funct ions :
Xt (~, ~)=N~e~i~ (v, ~)--Ni2~9 ) (v, W); x~ (~, ~)=N~O7' (~, ~)--N~Oi" (v, ~). (8a)
Fo r Xt(v, /~) and X2(v, /~), the o r thogona l i ty r e l a t ions exhibit the fol lowing f o r m :
(X~, Op0 = N~5 (v-- v');
(x~, (I)(#)=0; (8b) (x~, ~ , ) : N ~ (V-~); (x~, Oi~') = O,
where by def ini t ion i
- 1
~+(v, 9) is a so lu t ion of the adjoint equat ion where
(~)(2h (1)(~9 = N28 (v-- v').
We in t roduce the e x p r e s s i o n s fo r the n o r m a l i z a t i o n funct ions :
L(crv~) 2 - [ - o'~'i i -- T (i/(~vi)]_ N i • = • 2v~ { Ci~C~t
-~[C~2--2C,,T (I/ovi)]2 [ ~ - - T(i/'vi)] } ;
N U : -- CjiV {Cii + C22-- 2vC [T (v) + T (av)]} ; IVii = v {Ct2C21 ~- [C ti - - 2vCT (~)]2 + a~C2v2; N ~ =: v { C i2C 2i q- [ C ~2 -- 2vC T (~;)]~+ ~C2v2;
N l = v2C: { i -- 2vCliT ((Iv) - - 2vC~2T (v) + v~C [4T (~) T (o~) - - n~]}~+ ~2C2va {2Cv [T (v)
+ T (av)]-- Cti-- C2~} ~; N~ = ~ {l--2vC22T (v)--2vCitT ( l /or)
+ 4"~CT (v) T ( i / ~ ) } 2 + n~va {C2 2 _ 2vCT (i/~4;) }3:
where C = det C.
We obtain a so lu t ion of Eq. po l a r i zed p ro ton m e d i um .
(9)
(10a)
(10b) (10c) (10d)
(10e)
(10f)
(1) f o r a plane i so t rop ic s o u r c e emi t t ing p o l a r i z e d neu t rons in an infinite This can be done by us ing the G r e e n ' s funct ion [3] fo r Eq. (2) in an infinite
232
medium.
F (x)= ~ 2Ct~T ( i /avi) - x / v ~ N~--~ ~ ~
i
/ • t[C~--2C'viT (l /ovi) ] viT (i/v~) J
l/o e -~/~ " ,f C2~
+ S dv 9-~t +L(C2t'tV2t+C~2Ntt) [.--C~tl o
i e 2Ct~vr (l/6v) ~ .
l/a
Assuming 100% polar izat ion of the medium, we obtain numerical values for the elements of the mat r ix C
- (t,8~5 o - k4,052 0,44S)
by using the resul ts of [1] and (r = 11.839, In this case, the d ispers ion function has two roots on the real axis, v i = ~:1.86948 [2].
We consider the case of a source emitting neutrons with a spin antiparallel to the spin of the pro-
(11)
i , 2. 3.
L I T E R A T U R E C I T E D
Yu~ N. Kazachenkov and V. V. Orlov, At. Energ. , 32, 297 (1972). C. Siewert and P. Shieh, Nucl. Energy, 2__77, 5 (1967). K. Case and P. Zweifel, Linear T ranspor t Theory [Russian translation], Mir, Moscow (1972).
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