Breakdown of the Concentration Expansion for the Zero-Temperature Impurity Resistivity

Transport coefficients for classical gases do not possess simple expansions in powers of the density. A similar phenomenon occurs in the problem of expressing the zero-temperature resistivity of a metal as a function of nᵢ, the density of impurities. Both semiclassical and quantum-mechanical treatments reveal that the resistivity ρ breaks down at order nᵢ³. The leading term of the breakdown is of the form nᵢ³ lnnᵢ.