Sum Rule for Lattice Vibrations in Ionic Crystals

Abstract

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, $\ensuremath{\Sigma}\stackrel{6}{i=1}{{\ensuremath{\omega}}_{i}}^{2}(\mathrm{q})=\frac{1}{\ensuremath{\beta}}\frac{18}{\overline{M}}{r}_{0},$ where ${\ensuremath{\omega}}_{i}(\mathrm{q})=\mathrm{angular}\mathrm{frequency}$ of the $i\mathrm{th}$ mode for a given wave vector q, $\overline{M}=\frac{{m}_{+}{m}_{\ensuremath{-}}}{({m}_{+}+{m}_{\ensuremath{-}})}$, ${m}_{+}=\mathrm{mass}\mathrm{of}\mathrm{positive}\mathrm{ion}$, ${m}_{\ensuremath{-}}=\mathrm{mass}\mathrm{of}\mathrm{the}\mathrm{negative}\mathrm{ion}$, ${r}_{0}=\mathrm{interionic}\mathrm{distance}$, and $\ensuremath{\beta}$ 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 $\ensuremath{\Sigma}\stackrel{3}{i=1}{{\ensuremath{\omega}}_{i}}^{2}(\mathrm{q})={{\ensuremath{\omega}}_{\mathrm{pl}}}^{2},$ where ${{\ensuremath{\omega}}_{\mathrm{pl}}}^{2}=\frac{4\ensuremath{\pi}n{e}^{2}}{m}$, which is the classical plasma frequency. (This last relation was first derived by Kohn.)