The methods we have seen up to now to treat resonance absorption require a considerable computational effort, so that it is often desirable to use more rapid approximate methods. This is usually done by establishing a library of resonance integrals in homogeneous medium, and for all practical cases by looking for the homogeneous mixture which is equivalent to the actual heterogeneous case.

Such an equivalence can be theoretically proved under the following assumptions:

1. Narrow resonance approximation for moderator collisions.

2. Rational approximation for the collision probabilities.

Under assumption 1, eqn. (7.3) for the homogeneous case becomes:

(7.30)

and eqn. (7.14) for the heterogeneous medium:

Assumption 2 implies with = 2,(1 — С) = (1 — С)Цо (the results are also valid if a Bell factor a is used in this formula) which substituted in (7.31) gives

(7.32)

A comparison between (7.32) and (7.30) shows that the heterogeneous system is equivalent to a homogeneous system in which the term has been added both to the scattering and the total cross-section. This term can be interpreted as a pseudo-cross­section representing escape from the lump (escape cross-section). The equivalence relation can also be explained in the following way. The absorber region is limited geometrically to the fuel region and energetically to the resonance region (which in this approximation is supposed to be narrow). A neutron can escape resonance absorption leaving this region either geometrically or energetically (scattering collision in the narrow resonance approximation). The equivalence relation consists in simulating the geometrical escape by the addition of a fictitious scattering cross-section (energetical escape). This result is, of course, also valid if NR or IM approximations are made for the collision with the absorber atoms.

Very often the macroscopic cross-sections 2to, 2sm, etc., are transformed into microscopic cross-sections dividing them by the atomic density N0 of the absorber in the fuel region. In this way one can speak of a scattering cross-section per absorber atom.

In the NR or IM approximations where for the resonance absorber only the potential scattering appears one speaks of a aP, potential scattering cross-section per absorber atom, and of a apctt, effective potential scattering cross-section per absorber atom

0-p eft = crP + cr e (7.33)

and

if the equivalence relation is used.

Using the equivalence relation it is possible to save a considerable amount of computer time in codes for spectrum calculations. Instead of performing a detailed calculation for every case, many computer codes have resonance integrals stored for different homogeneous mixtures and different temperatures, i. e. for each resonance absorber two-dimensional tables of resonance integrals are stored as function of crpPtt and temperature. If different moderators are considered a second type of equivalence can be established between any given moderator and hydrogen. This procedure is used in the UKAEA WIMS code<l8) where resonance integrals are stored only for homoge­neous mixtures of resonance absorbers and hydrogen. For each moderator it is possible to establish the equivalent amount of hydrogen to be used in WIMS.<,9,20)