Frequency Shifts of Acoustic Phonons in Heisenberg Paramagnets. III

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Abstract
The propagation of sound waves in ferromagnetic and antiferromagnetic insulators is studied within the framework of two models which describe the interaction between the spin system and the lattice. Expressions for the frequency shifts (phonon renormalizations) at high temperatures and near the transition temperatures are obtained in terms of time-dependent correlation functions. The frequency shifts for long-wavelength phonons are found to be negative, to increase rapidly in the vicinity of the transition temperature, and to be less singular than the attenuation coefficients. The ratio of the frequency shift to the unperturbed phonon frequency is shown to be independent of the phonon frequency for long wavelengths. These results agree qualitatively with present experiments.