Attenuating Wave Analysis of Heat Flow in Crystal Lattices

A direct approach to the detailed analysis of the thermal relaxation and conduction processes in crystal lattices in the classical-temperature range is presented in terms of the mechanical energy transported by attenuating lattice waves. A second-order classical perturbation procedure, formulated in terms of time- and space-dependent normal coordinates, is used to solve for the dynamics of a slightly imperfect, nonlinear general crystal lattice model under the influence of an applied temperature gradient. Only the use of a random-phase assumption for initial wave amplitudes at $t=0\mathrm{}$ and statistical averaging of the subsequent dynamical response are required for the direct determination of the accepted lattice relaxation times from the time dependence of the stored mechanical-energy density (for first- and second-order perturbation terms). In addition, the well-known anharmonic, mass-fluctuation, and force-fluctuation components of the high-temperature thermal conductivity are found directly from the steady-state mechanical-power density within the lattice. No use is made of the Boltzmann transport equation or standard phonon scattering theory, although the results obtained are wholly consistent with their use. Finally, a brief discussion is given on the extension of this attenuating-wave technique to the corresponding quantum treatment of low-temperature heat flow in crystal lattices.