Radiation detection and measurement

A finite element method for neutron transport If there is no upscattering, i.e. o,. = 0 for m > n the multigroup equations can be solved sequentially starting with n = N. Then the source term becomes

o~(i, f l ' . l ~ ) =

,~ a , ( i , f¥ •f~) ~bo,(r, fy) d.Q'.

(7)

re+n+ 1

By defining even O~,(r, £1) and odd Oo.(r, 11) parity fluxes; the even parity equations can be generalized (Ackroyd, 1978):

= S,+(r,11) - 11. VG, S;-(r,~),

v . 1 1 , L ( r , a ) + ~.(i)4,~;.(r, f~) = f n o,~(i, fl'.l~)~.(r,f¥)d.Q' + S~(r, fl),

(8)

+ fR d11' n-1 ~ o.(i,11"fl)~0.(r, f l ' ) d ~ ' . '

C~(r, ~ ) = o,(i)u(r, 1"1)

- fa,

a'. a)u(r, a') da',

~b~-(r, 11)--- ½{~,(r, 11) - ~n(r, - fl)},

= ½{o.n(i, a ' . a ) (9)

where, in general, on(i, 1"1"11') is the solution of

l'L V *-(~o ,(r,l ))4 -

o.(i)2

fT.(i)

C (~n(r, ~) = S~ + (r, ~),

a . V* ÷ ~;, (r, a ) - G- 1 ~ -

(r, a ) = s~ - (,, a).

(16) Note that the sign of the leakage operator has been changed. From the above equations it is possible to derive the even parity adjoint equation.

~;,(~, a . a')

fn g'(i'trt~') o;,(t~ • 11") d ~ " ,

a.v{G,(a.v~,+(r,a)} (11)

+ c, ~,+ (,, a)

-- S*+(r,a) + f L V G , S~',(r, I)). (17)

Table 1. Positive definite leakage and removal operators in plane geometry (where o~., is the Ith Legendre polynomial expansion of the scattering cross section and P ~ ) are the Legendre polynomials) ],/ N

G.

c.{,

(15)

and using similar definitions for S~n+(v,~) and Sn*-(v,l~); the even parity adjoint equations are obtained.

G.u(r, 11) = u(r, ~_~) Jr fn on(i, ~" 11')u(r, 1~') d£~', ~,(i) ... (1o)

,,

(14)

m=O

Defining even and odd adjoint parity fluxes

+ o,,(i,a'.~)},

+

(13)

S*,(r, 11) = S~(r, 11)

follows.

g~(i, [~. f¥) =

(12)

The adjoint multi-group equations can be written as

where the operators C. and G. are defined as

a.a')

- ~r,(i, - 11'.a)}.

where

a - V{G,fl. V4~'.(r, 11)} + C , ~ , ( r , fl)

~:n(i,

½{o,(i, f l ' . a )

2.2 The even parity adjoint equations

N

-

with,

To illustrate the theory Table 1 lists operators of the transport equation for infinite slabs.

S.(r, ¢~) = S,,(r, t~) + fa, dfl' x

337

-°'- 1 _ ~41+ 1 (o..-1

)

l

x

}

t ~ P21+l(#') d~'