Inelastic Electron Scattering and the Lorenz Ratio of Liquid Metals

Abstract

Recent experimental work has led to the surprising conclusion that the Lorenz ratios of some of the liquid metals exhibit substantial negative deviations from the ideal Sommerfeld value, hence indicating that inelastic electron scattering effects may be an important factor in some liquid metals. Motivated by these observations, we present in this paper a calculation of the electronic transport properties of a liquid metal, which takes into account the effects of inelastic electron scattering from the ionic density fluctuations to leading order in the small dimensionless parameter $\frac{{E}_{0}}{{k}_{B}T}$. Here ${E}_{0}$ denotes a typical inelastic energy transfer and ${k}_{B}T$ the Boltzmann factor. This calculation, based on the nearly-free-electron model, is carried through by use of the exact sum rules of Placzek and of de Gennes on the dynamic structure factor $S(q,\ensuremath{\omega})$ of the classical ionic liquid component of the metal. The effects of inelastic electron scattering on the electrical resistivity are found to be negligible. Non-negligible corrections, however, associated with small-angle inelastic processes, are found to enter the electronic thermal resistivity. The corresponding depression in the Lorenz number is expressed in terms of the electron-ion pseudopotential, the static liquid structure factor, and the collective-mode frequencies associated with the density fluctuations of the liquid component. Inspection of the theoretical expression for the deviation in the Lorenz ratio reveals, however, that it is too small to account for the experimentally observed deviations. It is concluded that, within the framework of the nearly-free-electron theory, the effects of inelastic electron scattering are not the dominant cause of the rather large anomalies in the observed Lorenz ratios.