After integration over all directions, some manipulation, and the discretisation of energy into G groups, the steady-state version of equation 1.1 for a homogeneous region of space gives G linked equa­tions, one for each of the group fluxes fg,

g-1 1 G

0 = DgV2fg — Vrgfg + Esg^gfg + kXg Vg Efgfg+ Sg. (1.9)

g’=1 g’=1

Equations 1.9 are the multigroup diffusion equations. The terms on the right-hand side of equations 1.1 and 1.9 correspond to each other. The groups are numbered in reverse order of energy, so that group

1 contains the neutrons of highest energy and group G those of the lowest. Thus the limits of summation in the third term of equation 1.9 correspond to the absence of up-scattering. The diffusion coefficients Dg are given by Dg = 1/3Esg(1 — ~fig) where д is the mean cosine of the angle of scattering in the laboratory system of coordinates. The boundary conditions at an interface between regions with different properties are that фЕ and Jg = Dg grad фЕ should be continuous.

For a reactor driven by a source (Sg ф 0) к is made equal to 1 and the фg represent the resulting flux distribution. If no solution is possible the reactor assembly is supercritical. For a critical reactor with Sg = 0 к can be thought of in two ways. The straightforward interpretation is that it is the effective multiplication constant, keff, of the reactor. But if к ф 1 the reactor cannot be steady and the фg cannot be constant, so that under this interpretation equations 1.9 have a meaning for к = 1 only. Alternatively к can be thought of as the highest eigenvalue of the set of equations that can be solved to find к and the ф^ although in this case the фg have no physical meaning unless к = 1. The composition of the reactor has to be altered to make к = 1, and when this is achieved the фg are the group fluxes in the critical reactor.

If the neutrons are treated in only one group — that is, if differences of energy can be ignored — the third term in equation 1.9 disappears and it becomes

0 = DV2ф + (vVf /к — Va )p + S. (1.10)

J2a = J2c + f has replaced r because neutrons are removed only by capture or fission.

Before the multigroup equations can be solved of course values have to be given to the constants Dg, Vrg, Vsg^g, xgvg and Vfg that depend on the microscopic cross-sections for the various materials in the reactor. Because the cross-sections vary with neutron energy the group constants involve average values over the energy range covered by each group.

In principle it is possible to make the groups so narrow that the variation of each cross-section within each group is small. However, in practice this cannot be done because the very complex fine structure of the variation of the cross-sections with energy, especially in the resonance region, would necessitate an impossibly large number of
groups. A method has to be found for calculating suitable group cross­sections to facilitate accurate calculations with 20 or 30 groups.