8.1. The spectrum calculations

It is now routine for thermal reactor calculations to perform detailed spectrum calculations in a high number of energy groups with only a rough approximation of the spatial dependence and then to perform few-group space-dependent calculations with constants calculated by averaging the fine group cross-sections on this spectrum.

Various types of methods and codes are being used to perform these multi-group spectrum calculations. All of these codes contain a library in a fine energy structure (40 to 200 groups) including cross-sections for all interesting reactions, transfer matrices (the £so.,-.k. £sl, i_k or higher-order components of the scattering cross-section described in Chapter 4), resonance parameters, fission spectra, etc.

In the thermal-energy range the transfer matrices of the moderators are stored for each material for various temperatures in order to take into account the effect of thermal motion of the moderator. Transfer matrices are sometimes stored for the moderators only, while the more sophisticated codes include transfer matrices for heavy materials in order to enable treatment of inelastic scattering in the fast energy range.

Normally the interval between fission energy and thermal energy (at which the neutron energy spectrum is in equilibrium with the thermal energy of the moderator) is subdivided into a certain number of energy ranges.

First one can distinguish between a fast and a thermal range. In the fast-energy range neutrons are slowed down by every collision. When the neutron energy has been decreased to the range of the thermal energy of the moderator it becomes possible for the neutrons to gain energy in a collision. As the Maxwell-Boltzmann distribution of thermal energy gives probabilities different from zero for the whole energy range, there is no clear cut between fast and thermal energy. The boundary between these two energy ranges is normally set at a level at which the probability for a neutron of gaining energy in a collision becomes negligibly small. This boundary depends, of course, on the moderator temperature so that it has to be higher in HTRs. Once fixed, this boundary is kept constant, even for calculations of a reactor in cold conditions.

The normal value for HTR calculations lies between 2 and 4 eV. This value is very high if compared with the traditionally used Cd cut-off energy of ~0.68 eV (below which energy a Cd foil sharply becomes opaque to neutrons).

Within the fast-energy range further subdivisions are possible, especially according to the representation of the cross-section of the heavy metals. For these nuclides

neutron absorption occurs mainly in resonances whose energies are well separated and whose parameters are well known in a lower energy range (up to 3 8 keV) while the

individual resonance levels are not resolved experimentally at higher energies. Above ~ 100 keV the resonance structure of the heavy metal cross-sections becomes unimpor­tant for reactor calculations. According to those three fast-energy ranges different calculational methods are used.

The methods used for the treatment of resonances in the resolved and unresolved energy ranges have been described in Chapter 7.