In the thermal region the neutrons achieve a thermal equilibrium with the atoms of the moderator material (in
the idealized situation where no absorption is present). That is the neutrons behave as a strongly diluted gas in thermal equilibrium. These neutrons do not all have the same energy, there is a distribution of energies, usually known as the Maxwell-Boltzmann distribution:
in which k is the Boltzmann constant (k = 8.52⋅10-5 eV/K). For the thermal neutron flux density it thus holds that:
in which n0 is the total thermal neutron density.
The most probable energy (for which the spectrum is maximum) is E = kT. At room temperature this is 0.025 eV. The velocity corresponding with this energy is 2200 m/s. This energy is of particular importance since reference data, such as nuclear cross-sections, are tabulated for a neutron velocity of 2200 m/s.
At a reactor temperature of 320°C (593 K), a value characteristic for PWRs, the most probable velocity is 3100 m/s and the corresponding energy is 0.051 eV.
But this distribution only holds for complete thermal equilibrium. Unfortunately, in a nuclear reactor, some absorption will always be present and this equilibrium will never be complete. As a result of 1/v behaviour, low energy neutrons are absorbed preferentially, which leads to a shift of the spectrum to higher energies.
On the other hand, the neutron leakage has an opposite effect. With decreasing energy the diffusion coefficient D decreases as a result of the increasing cross-sections, therefore the neutron leakage preferentially removes neutrons with higher energies. This effect strongly depends on the size of the multiplying system, but in most cases it is much less important than the presence of absorption.
Thermal reactor neutron spectrum.