The Dark Matter Phonon Coupling

Abstract

Generically, the effective coupling between the dark matter and an atom scales with the number of constituents in the atom, resulting in the effective coupling being proportional to the mass of the atom. In this limit, when the momentum transfer is also small, we show that the leading term in the scattering of a particle off the optical phonons of an array of atoms, whether in a crystal or in a molecule, vanishes. Next-generation dark matter direct detection experiments with sub-eV energy thresholds will operate in a regime where this effect is important, and the suppression can be up to order $10^6$ over naive expectations. For dark matter that couples differently to protons and neutrons, the suppression is typically of order $10-100$ but can be avoided through a judicious choice of material, utilising variations in nuclear ratios $Z/A$ to break the proportionality of the coupling to mass. We provide explicit illustrations of this effect by calculating structure factors for di-molecules and for the crystals NaI and sapphire.