The Boltzmann equation which is conceptually very simple can only be solved analytically in very few cases.

We will not deal here with these classical solutions although they are very instructive and important for the reactor physicist. The most important classical cases can be seen in many textbooks (e. g. refs. 1 and 2).

Numerical solutions are, of course, always possible, and these numerical methods are of fundamental interest for the reactor physicist. The problem is strongly complicated by the complex energy dependence of the parameters of the Boltzmann equation. The neutron cross-sections vary by many orders of magnitudes in the energy range of interest and most heavy isotopes exhibit sharp resonances where cross-sections have enormous variations in a very small energy interval.

During the fission reactions neutrons are emitted with an energy distribution which has an average value of approximatively 2 MeV and becomes negligibly small above 10-15 MeV so that the maximum energy of interest in nuclear reactors is of about 15 MeV. Neutrons are then slowed down by collisions with the reactor components approaching, in a thermal reactor, the energy distribution corresponding to the moderator temperature.

To describe the energy-dependent neutron spectrum in a reactor means to follow the neutron history in this energy range considering its probability of absorption, slowing down by elastic and inelastic collision, and of leakage out of the system.

The only accurate way of treating this complex problem consists of discretizing the energy dependence in a high number of energy ranges. Because of the very strong energy dependence of some neutron cross-sections this number of ranges should be extremely high. The problem has normally been solved confining the treatment to a rather limited number of energy ranges (groups). As the energy dependence of the cross-sections within these groups cannot be neglected, group average cross-sections have to be produced. The number of groups to be used varies according to the complexity of the calculation being performed.

Normally between 40 and 200 groups have been used for high-temperature neutron spectrum calculations, but in recent times codes with a much higher number of groups have been developed for calculations at resonance energies.

In an N group scheme, if the energy boundaries for group і are Е,-1 and Et, the angular flux of group і is defined as

(4.5)

A straightforward way of obtaining the multi-group equations would consist in integrating the Boltzmann equation over the energy range of each group. Unfortunately
this procedure would result in the definition of angle-dependent group cross-sections (ref. 2, p. 51).

[‘ Zx(r, E)il>(r, E,il)dE

2,. , (r, ft) = E[ with x = t, f, s, etc.

ф(г, E, ft) dE

A simple way of avoiding this problem is to assume a separation of the energy dependence from the space and angular dependence within each group and within the region of interest:

ф(г, E, SI) = f (r, ft)<p, ІЕ) within group i,

ф>(г, п)=fir, ft) /;

Integrating eqn. (4.4) over the group і we have, under the assumption of separability,

ft • Уфііг, ft) + 2u<Mr, ft) = X

k = 1

+ ТГЇ-2 *2/.* [ Фir, Sl)dSl’

#Ceff47T к = і J

where we have defined the average constants:

fj 2,ir, E)<p, iE) dE

S*,i(r) = E’~’ E|—————————- with x = t,/, s, etc.;

I <p, iE)dE

J E,_,

The fine energy dependence of <p,(.E) within the group і is not known. Approximations must be used with which we will deal later. In general, if the number of groups is sufficiently high, the cross-sections are fairly constant within the group, in which case the group cross-sections are almost independent of the assumption made for <p,(.E). This is not true in case of resonances unless the number of energy groups is extremely high (see Chapter 7).

Another possibility of obtaining a multi-group formulation while avoiding the definition of angle dependent cross-sections consists in first approximating the angular dependence by means of expansion in eigenfunctions (P, method) or numerical discretization (S„ method) and then integrate over the energy range of each group.

In the above treatment we have supposed that the groups define non-overlapping energy ranges. A treatment with overlapping groups has sometimes been used. For example, at a boundary between two regions (e. g. core and reflector) it is possible to suppose that the flux is a linear combination of the two spectra typical of the two adjacent regions. These two spectra can be considered as two overlapping groups. In order to calculate the fluxes it is necessary to have a physical model describing the transfer of neutrons from one group to another. The difficulty of finding an adequate model has until now strongly limited the utilization of this method.