Interpretation of the Relaxation Time for the Electrical Conductivity of Elemental Metals Using the Fluctuation Dissipation Theorem

TL;DR

This paper links the relaxation time for electrical conductivity in elemental metals to quantum fluctuation principles, providing a theoretical basis for an empirical formula that matches experimental data within 20%.

Contribution

It derives the relaxation time from fluctuation dissipation theorem and quantum uncertainty, offering a theoretical explanation for the empirical conductivity formula.

Findings

Relaxation time matches quantum fluctuation predictions.

Electron density equals atomic density for most metals.

Temperature dependence can be expressed using ion plasma frequency.

Abstract

We proposed in an earlier paper [arXiv:1108.6141] an empirical formula of the electrical conductivity which agrees with experiments within 20 percent for the most of pure elemental metals at room temperature ranges. This is obtained, in Drude formula, by replacing the electron density with the number density of atoms n(atom), and the electron effective mass with true electron mass multiplied by G. Here the relaxation time is assumed to be (h/2pi)/(kT) for all metals (h is the Planck constant and k is the Boltzmann constant). The single free parameter G is summed electron numbers in each atomic shell, e.g. G=5+1=6 for chromium(five 3d electrons and one 4s electron). In this paper, we find that the above relaxation time can be reproduced if the autocorrelation time of electron fluctuating velocity in a simple fluctuation dissipation theorem is converted to 2delta(energy)delta(time)/kT, and if this delta(energy)delta(time) is assumed equal to (h/2pi)/2 of the Heisenberg's minimum uncertainty. This corresponds to the closest approach, or head-on colliisions. Independent from this, we find that n(electron)=n(atom), is appropriate for the conductivity in most elemental metals. In discussing the 5th power temperature dependence of resistivity, we find that, besides the use of the Debye temperature unit, the temperature unit defined by the ion plasma frequency times (h/2pi)/k is equally acceptable. Here the ion plasma frequency is only depending on n(electron) unlike somewhat ambiguous Debye temperature.