The Thermal Expansion of Solids

TLDR: In this article, the dependence on volume of the velocity of plane waves in one-, two-, and three-dimensional lattices is obtained, and hence the corresponding Gruneisen parameter is predicted for a body with a purely harmonic atomic potential.

Abstract: Under appropriate conditions, the thermal expansion of a solid is closely related to the Gr\"uneisen parameter $\ensuremath{\gamma}$. This in turn may be derived from the variation of the characteristic frequency of the lattice with volume. If, however, this variation is calculated from the usual expressions for the velocity of sound in solids at zero pressure, the $\ensuremath{\gamma}$ does not agree with that predicted from lattice theory, and an anomalous thermal expansion is predicted for a solid with a purely harmonic atomic potential.General expressions for the dependence on volume of the velocity of plane waves in one-, two-, and three-dimensional lattices are obtained, and hence the corresponding Gr\"uneisen $\ensuremath{\gamma}'\mathrm{s}$. The three-dimensional $\ensuremath{\gamma}$ differs by a numerical constant from that used by Slater. All three expressions are now consistent when applied to a body with a purely harmonic atomic potential and predict no thermal expansion.