Intermediate-temperature specific heat of solids and the rationale behind the Maier-Kelley empirical formula

TL;DR

This paper develops a first-principles model for the specific heat of solids at intermediate temperatures, providing a theoretical basis for the empirical Maier-Kelley formula and validating it with experimental data.

Contribution

It introduces a semi-harmonic oscillator model that explains the Maier-Kelley empirical formula using physical parameters without fitting.

Findings

Model accurately predicts specific heat for various solids.

Provides physical interpretation of Maier-Kelley coefficients.

No fitting parameters used in the theoretical predictions.

Abstract

The heat capacity of solids at intermediate-to-high temperatures is of fundamental importance to several fields ranging from geology to material science. It depends on a variety of factors, with anharmonicity and, ultimately, melting playing a pivotal role. In this work we develop a first-principles model from an analytically tractable semi-harmonic oscillator Hamiltonian. The resulting specific heat expression depends not only on the Einstein temperature of the material but also on other physical parameters. We compare our predictions with experimental data for copper, aluminum, lead, silicon, and germanium with rather satisfactory results, especially considering that there are no fitting parameters in our theory. We finish this work by showing that our results formally justify the otherwise purely empirical formula by Maier and Kelley, also providing its coefficients in terms of elementary physical quantities.