Calculation of Energy Exchange between Lattice Oscillators

Abstract

The rate of transfer of energy between lattice oscillators due to anharmonic perturbations is computed when different portions of the frequency spectrum are not in thermal equilibrium. This calculation is needed for the preceding paper and is not to be confused with that of ordinary thermal conductivity, since the present article is concerned with the flow of energy between oscillators at the same point of space, rather than that associated with spatial temperature gradients. The mean free paths for the two problems are shown to be unrelated, particularly at low temperatures, inasmuch as spatial thermal resistivity can be created by boundary or mosaic reflections, free from energy exchange, but does not arise from energy-transferring collisions between oscillators unless they are of the abnormal "unklapp" type in which momentum is imparted to the grating framework. The force constants needed for the calculation for the alums are furnished by Bridgman's compressibility data. The rate of flow of energy between a band of low frequency oscillators and the main body of lattice vibrations when the two are at slightly different temperatures proves to be proportional to $T^4$ at low temperatures. The corresponding collision frequency between oscillators is about ${10}^5$ ${\sec .\mathrm{}}^{-1\mathrm{}}$ even at helium temperatures and thus is of such a magnitude that equilibrium between the different vibrations can be considered as secured instantaneously from the standpoint of macroscopic acoustical experiments, but not at all as far as paramagnetic dispersion is concerned.