Temperature Dependence of the Frequency Spectrum of a Paraelectric Material

The temperature-dependent frequency spectrum of a paraelectric material is discussed with use of an effective-harmonic-lattice equation of motion. For a given wave vector, the effective couplings between atoms are written as a sum of a temperature-independent part, arising from harmonic interactions, and a part linear in the temperature, arising from fourth- and third-order anharmonic interactions. The conditions required for the observed Curie-Weiss behavior of the dielectric constant and associated temperature dependence of certain long-wavelength transverse optical frequencies are examined. It is shown that the salient features of the temperature-dependent low-lying optical branch in SrTi${\mathrm{O}}_3$ can be understood by considering the effects of anharmonic interactions at constant volume.