Attenuating Wave Analysis of Heat Flow in Crystal Lattices
Abstract
A direct approach to the detailed analysis of the thermal relaxation and conduction processes in crystal lattices in the classicaltemperature range is presented in terms of the mechanical energy transported by attenuating lattice waves. A secondorder classical perturbation procedure, formulated in terms of time and spacedependent 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 randomphase assumption for initial wave amplitudes at t=0 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 mechanicalenergy density (for first and secondorder perturbation terms). In addition, the wellknown anharmonic, massfluctuation, and forcefluctuation components of the hightemperature thermal conductivity are found directly from the steadystate mechanicalpower 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 attenuatingwave technique to the corresponding quantum treatment of lowtemperature heat flow in crystal lattices.
 Publication:

Physical Review
 Pub Date:
 April 1964
 DOI:
 10.1103/PhysRev.134.A163
 Bibcode:
 1964PhRv..134..163M