Long‐Wavelength Longitudinal Phonons of Multi‐Mode Crystals

TLDR

A rigorous method identifies longitudinal optic mode frequencies by locating minima of the complex dielectric constant modulus, enabling individual LO determination for any n‑mode crystal when oscillator parameters are known. The study demonstrates that Drude's rule fails for multi‑resonance systems, that LO frequencies correspond to minima of |ε| while TO frequencies correspond to its maxima, and that the method reproduces known relations and experimental data.

Abstract

Abstract It is shown that the application of the so‐called Drude's rule in obtaining the k ≈︁ 0 longitudinal optic mode frequencies (LO's) of multi‐resonance damped oscillators is not correct, although it may be used as an approximate method only when a few oscillators, all of appreciable strength, are present. A rigorous procedure suitable for all cases, single or multi‐resonance, is shown to be to identify the frequencies at which the minima of the modulus of the complex dielectric constant (|ϵ|) occur with the LO modes. The frequencies of the maxima of |ϵ|, on the other hand, approximately represent the k ≈︁ 0 transverse optic modes. It is also shown that for a general n ‐mode crystal it is possible to obtain the LO frequencies individually provided that all the parameters of the Lorentz oscillators needed to represent an observed optical property are known. In the case of a single resonance oscillator the equations reduce to the well‐known Lyddane‐Sachs‐Teller relation. Examples are given where LO's are obtained by the above methods using reflectivity data and compared with whatever available directly measured experimental LO values, e.g., from Raman or neutron scattering measurements.