Resonance in Phonon-Phonon Scattering

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Abstract
The scattering process, two phonons → two phonons, is examined in perturbation theory. One of the contributions of second order in the cubic anharmonic potential gives rise to an energy denominator that can vanish. This difficulty is removed by noting that the intermediate phonon has a complex self-energy associated with its instability in the presence of anharmonic forces. Approximating the irreducible self-energy by its lowest-order contribution, one finds a Breit-Wigner form of width equal to the decay rate, one phonon → 2 phonons, calculated to lowest order. An experiment is proposed to observe directly the predicted resonance. This experiment appears to be somewhat beyond the capabilities of present apparatus. The effect of the resonance on heat conduction, phonon drag, and the analogous resonance expected in spin-wave scattering, are not discussed in this paper.

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