Lattice Vibrations in Polar Crystals

Abstract

This paper investigates the free vibrations of a polar crystal of the sodium chloride type. The main purpose is the calculation of the Coulomb force, which has been done by an extension of the Madelung method. For numerical evaluation, a short range repulsion is assumed, and the van der Waals forces are neglected. The results are: For long waves one finds in the acoustical branch three waves whose frequencies are inversely proportional to the wave-length, in agreement with Born. The optical branch has two transverse waves (Reststrahlen) with a Coulomb force of $\frac{4\ensuremath{\pi}}{3}$ times the polarization, as Born has calculated, and one longitudinal one with a Coulomb force of $\ensuremath{-}\frac{8\ensuremath{\pi}}{3}$ times the polarization. For short waves the vibrations are in general not exactly transverse or longitudinal, but even in the optical branch there remain separate modes of vibration with different frequencies for a given wave-length. In addition, the neutral simple cubic lattice is considered. It is shown that the free vibrations are characterized by $D=0$. The influence of the shape of the crystal is discussed.