Sum Rule for Lattice Vibrations in Ionic Crystals

It is shown that in lattices of tetrahedral symmetry with two ions to a unit cell, in the approximation of nearest neighbor repulsive interactions, for a given wave vector q, 6i=1{₈}^2 (q) =118Mr₀, where ₈ (q) =angularfrequency of the ith mode for a given wave vector q, M=m+{m-} ({m++m-) }, m+=massofpositiveion, m-=massofthenegativeion, r₀=interionicdistance, and is the coefficient of compressibility. This theorem serves as a useful check on numerical work as well as a relation for the downward curvature of the optical modes at small q in terms of the speed of sound. In the limit of small q, this relation becomes the first Szigeti relation. A similar theorem is true for low-density electron gases where the electrons localize on a lattice. Here one can show that 3i=1{₈}^2 (q) ={₏₋}^2, where {₏₋}^2=4{e^2}m, which is the classical plasma frequency. (This last relation was first derived by Kohn. )