Sum Rule for Lattice Vibrations; Application to Forces in Diamond Structures

Abstract

It is shown that in a crystal of any number $n$ atoms per unit cell many interatomic forces—to wit, all forces which are electrostatic in nature (Coulomb, dipole,..., multipole), and all forces which act only between unlike atoms—contribute only constant terms to the trace of the dynamical matrix. Therefore, the sum $\Sigma i^{}\omega _i^{}{}_{}{}^2\left(\mathbf{q}\right)$ over the $3n$ squared frequencies when the wave vector is q is, in fact, independent of q in crystals in which all forces fall into one of these classes; conversely, if experiment shows that sum to actually be a function of q, then forces which fall into neither class—to wit, forces, not electrostatic in nature, between like atoms—must be present. Analysis of data for germanium and diamond suggests that such forces are present in the latter, but, within experimental uncertainty, absent in the former.