Let us set l = 1, and neglect the higher-order terms. Considering (4.12) and (4.13) we have then the expression:

ф(г, Е,П) = ^ф(г, Е) + ^Ш(г, Е) (4.19)

and for the scattering cross-sections

2S (£’->£, ft’ft) = SoiE’^E)+sdE’^E)p0

= -^lso(E’^E) + -^Zsl(E’^E)p0 (4.20)

where ju0 = ft ■ ft’.

The time-independent Boltzmann equation with isotropic sources is [see eqn. (4.4)]

ПЧф(г, E, ft) + 2,,ф(г, E, ft) = J Xs(E’ ->Е, П’-> И)ф(г, E’, ft’) dE’ dft’

+ _LxjE) Г jf(£^(r)£.i0.)y(£.)(j£. dft’ (4.21)

Keff 47Г J

+ jLs(r. E).

Substituting (4.19) and (4.20) in (4.21) and integrating over ft, once directly and once after multiplication by ft, we have the set of two P, equations

div J(r, E) + 2, (Е)ф(г, Е)= [ Xs0(E’ -»Е)ф(г, E’) dE’

Jo

+ -£-x(E)( V(E’)ME’№(r, E’)dE’+S(r, E), (4.22)

Keff JO

5grad ф(г, E) + 2,(E)J(r, E) = [ XS,(E’ ->E)J(r, E’) dE’.

Jo

A multi-group formulation of the Pi equations can be obtained starting from the multi-group form of the transport equation (4.6) with the group constants defined by

(4.7) and (4.8).

It is also possible to obtain the multi-group P, equations integrating directly eqn. (4.22) over the energy range of each group.

In this case it is not necessary to assume the separability

ф(г, E, ft) = f(r, Sl)(pi(E) within each group і

and it can be easily seen that some cross-sections (e. g. 2s0) will be weighted on the flux and some on the current (e. g. 2.,i). Furthermore, the 2,(1?) appearing in the first eqn. (4.22) will be weighted by the flux

while the 2,(i?) appearing in the second equation will be weighted by the current

In practice these equations are always used in codes with a very high number of groups so that this difference is unimportant.