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, Σ6i=1ωᵢ²(q)=1β18M̄r₀, where ωᵢ(q)=angularfrequency of the ith mode for a given wave vector q, M̄=m₊m_-(m₊+m_-), m₊=massofpositiveion, m_-=massofthenegativeion, r₀=interionicdistance, and β 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 Σ3i=1ωᵢ²(q)=ωₚₗ², where ωₚₗ²=4πne²m, which is the classical plasma frequency. (This last relation was first derived by Kohn.)