  Direct detection experiments are looking for nuclear recoils from scattering
of sub-GeV dark matter (DM) in crystals, and have thresholds as low as ~ 10 eV
or DM masses of ~ 100 MeV. Future experiments are aiming for even lower
thresholds. At such low energies, the free nuclear recoil prescription breaks
down, and the relevant final states are phonons in the crystal. Scattering
rates into single as well as multiple phonons have already been computed for a
harmonic crystal. However, crystals typically exhibit some anharmonicity, which
can significantly impact scattering rates in certain kinematic regimes. In this
work, we estimate the impact of anharmonic effects on scattering rates for DM
in the mass range ~ 1-10 MeV, where the details of multiphonon production are
most important. Using a simple model of a nucleus in a bound potential, we find
that anharmonicity can modify the scattering rates by up to two orders of
magnitude for DM masses of O(MeV). However, such effects are primarily present
at high energies where the rates are suppressed, and thus only relevant for
very large DM cross sections. We show that anharmonic effects are negligible
for masses larger than ~ 10 MeV.
