On the Polar Vibrations of Alkali Halides

The long wave-length, polar lattice vibrations of alkali halide crystals are discussed without making any specific assumptions about the detailed interactions between the ions. This is made possible by the introduction of the effective charge, $e^{*}$, of an ion defined as follows: All of the positive ions in a crystal slab are displaced by an equal amount in a direction perpendicular to the faces of the slab and all of the negative ions in the opposite direction. Then $e^{*}$ is the ratio of the dipole moment per ion pair induced in the slab by this displacement to the relative displacement of the positive and the negative ions. Expressions are obtained for the frequency, ${\omega }_l$, of the longitudinal vibration and the frequency, ${\omega }_t$, of the transverse vibration in terms of the dielectric constant, $k$, of the crystal, the dielectric constant, $k_0$, obtained by extrapolating the square of the index of refraction of the crystal from high frequencies to zero frequency, and $e^{*}$. The ratio of the two frequencies is found to be independent of $e^{*}$ and given by $\frac{{\omega }_l}{{\omega }_t}={\left(\frac{k}{k_0}\right)}^{\frac{1}{2}}$.