When a neutron passing through a crystallite has insufficient energy to eject an atom from its site, excitation of a lattice vibration occurs instead. This paper deals mainly with neutrons having energy between 0.5 and 5.0 electron volts, the range where both the cross section for elastic scattering ${\ensuremath{\sigma}}_{\mathrm{el}}$, and that for inelastic scattering ${\ensuremath{\sigma}}_{\mathrm{in}}$ vary rapidly with energy. In order to deal with inelastic collisions in which many phonons are exchanged, Finkelstein replaced the Debye model of the crystal by the Einstein model and gave expressions for ${J}_{n}(E)$, the probability that a neutron having energy $E$ colliding with an atom of mass $M$ loses at a collision $n$ quanta. Computations of ${J}_{n}$ for large and small $M$ show in detail that in the slowing-down process by this mechanism a heavy element is much less efficient than a light element, as in the familiar case of fast neutrons. Since in this range of energy ${\ensuremath{\sigma}}_{\mathrm{el}}$ is inversely proportional to $E$, and since the observed total cross section is independent of energy, this implies that ${\ensuremath{\sigma}}_{\mathrm{in}}$ is of the form ${\ensuremath{\sigma}}_{\mathrm{in}}={\ensuremath{\sigma}}_{0}(1\ensuremath{-}\frac{A}{E})$. Computed values of $\ensuremath{\Sigma}{J}_{n}$ are found to lead to this form, irrespective of the particular Einstein temperature assumed for the solid.