Thermoelectricity and Resistivity in Metal Alloys at Low Temperatures

A simple model is proposed for the "resonant scattering" of electrons from foreign atoms in a crystal lattice. The model assumes the existence of highly selective scattering mechanisms characterized by widths of the order of ${10}^{-4\mathrm{}}$ electron volts and larger; in this sense it is similar to the model of Korringa and Gerritsen (1953), although the present model makes use of only the general analytical characteristics of the relaxation time and does not specify the details of the scattering mechanism. Mott's well-known approximation formula $S=\left(\frac{{\pi }^2{k}^{2}T}{3e}\right){\left[\frac{d\ln \sigma \mathrm{}\left(E\right)\mathrm{}}{\mathrm{dE}}\right]}_{E=E_F}$ for the absolute thermoelectric power of a metal alloy is strictly valid and physically meaningful only at temperatures $\mathrm{kT}\ll a$, where $2a$ is the width of the resonance. At temperatures $\mathrm{kT}\gg a$ the formula leads to useful and valuable information on the thermoelectric properties of alloys, but the formula in this temperature region has only a rather artificial physical meaning. In the intermediate temperature region where $\mathrm{kT}$ is comparable to $a$, the Mott formula is entirely invalid. But it is in precisely this intermediate temperature range that the resistance and thermoelectric anomalies occur, so that Mott's approximation cannot be used for the treatment of these anomalies. The model satisfactorily explains many of the details of this anomalous behavior. It is suggested that the solvent metals used in the experimental studies of these effects will have to be much purer than those presently available before we shall be able to specify unambiguously the effects of a given kind of impurity.