RELAXATION MODEL FOR HEAT CONDUCTION IN METALS

A time-dependent relaxation model for the heat flux in metals is derived from the quantum mechanical form of the Boltzmann transport equation. In the derivation, the manipulation of the nonlinear integral term of the Boltzmann equation is simplified by assuming that the phonons are in thermal equilibrium at all times and by using the Lorentz approximation to treat free electron-phonon interactions. The relaxation model for the heat flux is found to yield a damped wave equation for the temperature and as a result, the speed of propagation for heat is shown to be finite instead of infinite as implied by the Fourier model for the heat flux. Approximate expressions for the thermal conductivity and the isothermal electrical conductivity are derived in the appendices in order to estimate the magnitude of the relaxation times and to obtain an expression for the Lorentz number. In general it is found that the thermal and electrical relaxation times are not equal although they are estimated to be at the same order-of-magnitude, 10-14 sec for the common monovalent metals. The Lorentz number is found to be a function of the ratio of the relaxation times and as a consequence the difference in the relaxation times may account, at least in part, for the derivation of the experimental Lorentz number from the usual theoretical value based upon equal relaxation times. © 1970 The American Institute of Physics.