Drude's lesser known error of a factor of two and Lorentz's correction

Abstract

As is well known, Paul Drude put forward the very first quantitative theory of electrical conduction in metals in 1900. He could successfully account for the Wiedemann-Franz law which states that the ratio of thermal to electrical conductivity divided by temperature is a constant called the Lorenz number. As it turns out, in Drude's derivation, there is a lucky cancellation of two errors. Drude's under-estimate (by an order of 100) of the value of square of the average electron velocity compensated his over-estimate of the electronic heat capacity (by the same order of 100). This compensation or cancellation of two errors lead to a value of the Lorenz number very close to its experimental value. This is well known. There is another error of a factor of two which Drude made when he calculated two different relaxation times for heat conductivity and electrical conductivity. In this article we highlight how and why this error occurred in Drude's derivation and how it was removed 5 years later (that is in 1905) by Hendrik Lorentz when he used the Boltzmann equation and a single relaxation time. This article is of pedagogical value and may be useful to undergraduate/graduate students learning solid state physics.