Retardation Effects on the Limiting Phonon Frequencies of Finite-Size Ionic Crystals

The discussion, given by Maradudin and Weiss, of the limiting optical-phonon frequencies of a finite spherical ionic crystal has been reconsidered, including interactions with the radiation field. Using Kellermann's model for alkali-halide crystals, it is shown that the degeneracy of the three optical frequencies in the long wavelength limit is lifted by retardation effects. Moreover, the longitudinal and one of the transverse normal-mode frequencies of the crystal harmonically coupled to the radiation field are found to satisfy the relation $\frac{{{\omega }_L}^2}{{{\omega }_T}^2}=\frac{{\epsilon }_{\infty }}{{\epsilon }_0}$, instead of the usual Lyddane-Sachs-Teller equation $\frac{{{\omega }_L}^2}{{{\omega }_T}^2}=\frac{{\epsilon }_0}{{\epsilon }_{\infty }}$, where ${\epsilon }_0$ and ${\epsilon }_{\infty }$ are the static and high-frequency dielectric constants of the crystal, respectively.