The neutron flux in heterogeneous assemblies is usually strongly space dependent and this fact complicates the calculation of resonance absorption in nuclear reactors.

It is usually possible to simplify the problem assuming a cell structure of the core, in which two regions are defined—a moderator and a fuel region. The fuel region consists in general of a mixture of resonance absorber and moderator and may have a complicated geometry (it does not need to be a single isolated body, but it may consist of a cluster of pins embedded in the moderator region, etc.). It is useful to use in this case the collision probability form of the transport equation

The index 0 refers to the fuel region, 1 to the moderator regions. P0 and P, are the escape probabilities from region 0 and 1: probability that a neutron originating in the region considered has its next collision outside of this region.

The terms P0 and Pi are in general functions of the neutron energy, and of the flux distribution in the regions considered.

The other quantities are:

Vo Vi volumes of fuel and moderator regions,

2to total cross-section of fuel region,

2s0 scattering cross-section of absorber,

2smo scattering cross-section of moderator in the fuel region 2*i scattering cross-section of moderator in the moderator region.

In practice the flat flux approximation is made for calculating the escape probabilities Pо and Pi for each region.

This assumption is made in most resonance-absorption calculations and works well.

Pо and Pi are related by the reciprocity relation (see ref. 3, p. 112):

2*i ViP, = 2t„ V0Po. Using this relation we can eliminate Pi and obtain

(7.13)

Another assumption which is ordinarily made is that of 1 IE asymptotic flux in the moderator region. Equation (7.10) becomes then

7.2. Narrow resonance approximation for heterogeneous assemblies

As in the homogeneous case a 1 IE flux is substituted in the integrals at the right-hand side of eqn. (7.13) obtaining

2,„4>o = (1 — Po) Xs0+^sm0 + Po 2,0

Using this approximation it is possible to express the resonance integral as the sum of a volume and a surface component / = /„+/* (see ref. 4) but this formulation is now seldom used.