  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$_2$.
