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, $\Sigma \stackrel{6}{i=1\mathrm{}}{{\omega }_i}^2\left(\mathrm{q}\right)=\frac{1}{\beta }\frac{18}{\overline{M}}{r}_0,$ where ${\omega }_i\left(\mathrm{q}\right)=\mathrm{angular}\mathrm{frequency}$ of the $i\mathrm{th}$ mode for a given wave vector q, $\overline{M}=\frac{m_{+}{m}_{-}}{(m_{+}+m_{-})}$, $m_{+}=\mathrm{mass}\mathrm{of}\mathrm{positive}\mathrm{ion}$, $m_{-}=\mathrm{mass}\mathrm{of}\mathrm{the}\mathrm{negative}\mathrm{ion}$, $r_0=\mathrm{interionic}\mathrm{distance}$, and $\beta$ 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 $\Sigma \stackrel{3}{i=1\mathrm{}}{{\omega }_i}^2\left(\mathrm{q}\right)={{\omega }_{\mathrm{pl}}}^2,$ where ${{\omega }_{\mathrm{pl}}}^2=\frac{4\pi n{e}^2}{m}$, which is the classical plasma frequency. (This last relation was first derived by Kohn.)