Long-wavelength soft modes, central peaks, and the Lyddane-Sachs-Teller relation

A derivation of the Lyddane-Sachs-Teller relation is presented which depends only on the broad requirements of statistical mechanics. Mode frequencies are introduced as peaks in the dielectric response to avoid the introduction of a Hamiltonian. Using this approach the modes may be very anharmonic, may be coupled, and may have a central-peak character and yet be precisely entered into the Lyddane-Sachs-Teller relation. The number of such modes need not conform to the number predicted by the usual group-theoretic methods applied to the symmetry of the solid. Examples are given of several types of spectra and the requirements of mode softening are discussed.