Long-Wave Optical Vibrations in Simple Ionic Crystals

Abstract

This paper resolves some of the difficulties recently pointed out by Rosenstock in the theory of the lattice vibrations of ionic crystals of NaCl or CsCl structure. Long optical waves in these crystals (but not in crystals with only short-range forces) are seen to be purely longitudinal and transverse. With the electrostatic approximation, the Lyddane-Sachs-Teller equation ${\left(\frac{{\omega }_l}{{\omega }_t}\right)}^2=\frac{{\epsilon }_0}{{\epsilon }_{\infty }}$ holds independently of the boundary conditions for modes in which the phase of the ionic vibrations is effectively constant over distances which are large compared with the lattice spacing; such modes must be either curl-free or divergence-free, with respective frequencies ${\omega }_l$ and ${\omega }_t$. If both curl and divergence vanish, the frequency is indeterminate unless boundary conditions are specified, so that there is a singularity but no inconsistency as the wave number k→0. When Coulomb retardation is taken into account this singularity disappears, and the frequency at k=0 is found to be ${\omega }_l$; the range of validity of the Lyddane-Sachs-Teller equation is then ${10}^{-4\mathrm{}}\gtrsim k^{-1\mathrm{}}\gtrsim {10}^{-6\mathrm{}}$ cm.