Ultrasonic Study of Three-Phonon Interactions. I. Theory

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Abstract
The nonlinear interaction of two ultrasonic waves in a homogeneous, isotropic medium is investigated by using the first-order time-dependent perturbation theory of quantum mechanics to calculate transition probabilities between available phonon states. Excluding collinear interactions, it is shown that there are two general types of possible interactions, depending on whether the zeros of the scattered wave displacement amplitude do or do not depend on the third-order elastic constants. Using correspondences between phonon densities and classical displacement amplitudes, and between generated phonons and Huygens wave sources, the theoretical displacement amplitudes for the scattered waves are derived. The amplitudes agree exactly with those derived from classical theory and are plotted for various materials and interaction geometries.

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