Discrete Relaxation Times in Neutron Thermalization

On the basis of the Van Hove theory, an analysis is given of those properties of the neutron-scattering law which influence the decay constants (the reciprocal relaxation times) in neutron thermalization. It is shown how the known differences in the long-time behavior of the correlation functions for a gas, a liquid, and a solid are reflected in the different behavior of the differential cross sections at low energies. This behavior ultimately determines the general character of the spectrum of decay constants for a uniform neutron distribution in an infinite medium. For a gaseous moderator, either there is an infinite set of discrete decay constants, strongly accumulating towards the lower limit ${\lambda }^{*}=min\left[v\Sigma \mathrm{}\right(v\left)\right]$ of the continuous spectrum, or else the spectrum below ${\lambda }^{*}$ is empty altogether. On the other hand, for a solid, the set of discrete decay constants is always finite, or possibly empty. The liquid appears to allow, in principle, all three possibilities, but normally the spectrum is expected to resemble that of a solid. In any case, the existence of the lowest decay constant ${\lambda }_0$ (and hence the existence of an infinite set for a gas) is trivial whenever absorption is absent or of the $\frac{1}{v}$ kind. Only an absorption rate $v{\Sigma }_a\mathrm{}\left(v\right)\mathrm{}$ which strongly increased in the small-$v$ region could cause the complete disappearance of the discrete spectrum. In this event any initial neutron distribution slowly evolves towards a singular distribution containing a $\delta \mathrm{}\left(v\right)\mathrm{}$ term, or some weaker singularity, and the decay rate approaches ${\lambda }^{*}$.