The Electrical Resistance of Dilute Solid Solutions

TLDR

Adding a second metal to form a solid solution generally increases a metal's electrical resistance, with the resistivity rise largely temperature‑independent per Matthiessen's rule, though exceptions exist such as Cr in Au. Nordheim’s quantum‑mechanical model relates resistivity to the relaxation time τ and free‑electron density N, explaining the temperature‑independent resistivity increase in dilute solid solutions. Reference 1.

Abstract

1. As is well known, the electrical resistance of a metal is very greatly in-creased by the addition of a second metal with which it forms a solid solution. The increase Δρ in the resistivity due to the addition of a small percentage of the second metal is in general independent of the temperature (Matthiessen's rule), though there are oertain exceptions (e.g. Cr in Au). The quaritum-mechanical explanation of both these facts was first given by Nordheim, and may be expressed as follows: the electrical conductivity of any metal may be written in the form where τ is the “time of relaxation”, equal to half the time between collisions, and N is the effective number of free electrons per unit volume: hence, for the resistivity, we have