  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.
