Phonon Boltzmann Equation and Second Sound in Solids

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Abstract
It has been suggested that two types of second sound, "drifting" and "driftless," are possible in dielectric crystals. The conditions for the existence of these two types of second sound are obtained, both from a heuristic analysis of the problem and from an exact solution of the complete linearized Boltzmann equation. The exact solution is given in terms of the eigenvalues and eigenvectors of the collision matrix, with the effects of normal processes, umklapp processes, and imperfections included. It is shown that to get drifting second-sound normal-process scattering must dominate so that crystal momentum is approximately conserved; while to get driftless second sound, the scattering must be such that a uniform energy flux will decay exponentially. These conditions for the two types of second sound are not mutually exclusive. It is found that normal-process scattering need not dominate for second sound to exist; but that only when it does dominate, is second sound likely to be observable. The relaxation times for both types of second sound are shown to be the same and equal to the reciprocal of smallest nonzero eigenvalue of the collision matrix. An expression is given for a lower limit on this relaxation time.