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The incoherent approximation (IA) is often used for calculating theone-phonon inelastic neutron scattering cross section for arbitrary solids. Itis valid for thermal neutrons but for slow neutrons it requires a correction,which is significant for isotopes that are strong coherent scatterers. In thisarticle, we present the extension of the Placzek--Van Hove corrections for slowneutrons in the limit of low temperatures using the example of solid\emph{ortho}-deuterium (sD$_2$). Our approach yields realistic one-phononup-scattering cross sections for sD$_2$ and shows the IA to be a factor of 2 to5 too high for ultracold neutron (UCN) up-scattering in sD$_2$. Ourcalculations are compared with previously published Monte Carlo calculations ofthe one-phonon cross section based on the dynamic structure function$S(q,\omega)$ of polycrystalline \emph{ortho}-deuterium and are found to beconsistent with them. Furthermore, we provide the means for easily replicablecalculations of the one-phonon up-scattering cross sections of solid\emph{ortho}-deuterium for slow neutrons. These should from now on be used incalculations and simulations of UCN scattering in sD$_2$.

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