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 $\ensuremath{\Sigma}{i}^{}\ensuremath{\omega}_{i}^{}{}_{}{}^{2}(\mathbf{q})$ 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.