Limiting Optical Frequencies in Alkali Halide Crystals

It recently has been pointed out by Rosenstock on the basis of symmetry arguments that the frequencies of the three optical branches of an ionic crystal in the limit of infinite wavelengths are all equal. This result is in contrast with the relation $\left(\frac{{\omega }_l}{{\omega }_t}\right)={\left(\frac{{\epsilon }_0}{{\epsilon }_{\infty }}\right)}^{\frac{1}{2}}$ due to Lyddane, Sachs, and Teller, where ${\omega }_l$ and ${\omega }_t$ are the limiting longitudinal and transverse frequencies and ${\epsilon }_0$ and ${\epsilon }_{\infty }$ are the static and high-frequency dielectric constants, respectively. By use of Kellermann's model for NaCl we have obtained the small-k expansions of the elements of the dynamical matrix for a finite spherical crystal of radius $R$. It is found that, if the limit k→0 is taken before the limit $R\to \infty$, the three optical frequencies are all equal, while if the order of taking limits is reversed the result of Lyddane, Sachs, and Teller is obtained. These conclusions are in agreement with Rosenstock's result, and with remarks of Fröhlich, and provide an explicit expression for the infrared frequency in the finite-crystal case. A similar calculation for Wigner's low-density electron crystal yields the result that in a finite spherical crystal the limiting frequencies of the two transverse branches and the one "longitudinal" branch are all equal. The possibility of the experimental observation of these effects is discussed.