Abstract In an earlier paper we extracted an asymptotic series solution in half-integral powers of ( kT E ) from the Boltzmann transport equation for neutrons coming into thermal equilibrium with a moderating material. In this paper we consider the series in detail for moderators having simple crystalline structure, or in which vibrational modes predominate, and give the coefficients of the series explicitly in terms of moments of the frequency spectrum of crystal vibrations. Our calculations indicate that the neutron density in the asymptotic region increases as one hardens the spectrum of crystal vibrations, whence the “hardness” of the neutron spectrum increases, too. We also discuss some aspects of the mass-expansion for these systems, and compare our calculations with those of other workers, and with typical experimental data for thermalization in water. Finally, we discuss the asymptotic expansion for moments of the scattering kernel and for ξσ s , the slowing down power.