Incoherent approximation for neutron up-scattering cross sections and its corrections for slow neutrons and low crystal temperatures

The incoherent approximation (IA) is often used for calculating the one-phonon inelastic neutron scattering cross section for arbitrary solids. It is valid for thermal neutrons but for slow neutrons it requires a correction, which is significant for isotopes that are strong coherent scatterers. In this article, we present the extension of the Placzek–Van Hove corrections for slow neutrons in the limit of low temperatures using the example of solid \emph{ortho}-deuterium (sD${}_2$). Our approach yields realistic one-phonon up-scattering cross sections for sD${}_2$ and shows the IA to be a factor of 2 to 5 too high for ultracold neutron (UCN) up-scattering in sD${}_2$. Our calculations are compared with previously published Monte Carlo calculations of the one-phonon cross section based on the dynamic structure function $S(q,\omega )$ of polycrystalline \emph{ortho}-deuterium and are found to be consistent with them. Furthermore, we provide the means for easily replicable calculations of the one-phonon up-scattering cross sections of solid \emph{ortho}-deuterium for slow neutrons. These should from now on be used in calculations and simulations of UCN scattering in sD${}_2$.