Anharmonic effects in nuclear recoils from sub-GeV dark matter

Direct detection experiments are looking for nuclear recoils from scattering of sub-GeV dark matter (DM) in crystals, and have thresholds as low as $\sim 10\mathrm{eV}$ or DM masses of $\sim 100\mathrm{MeV}$. 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 $\sim 1–10\mathrm{MeV}$, 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 $\mathcal{O}(\mathrm{MeV})$. 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 $\sim 10\mathrm{MeV}$.