Under appropriate conditions, the thermal expansion of a solid is closely related to the Grüneisen parameter $\gamma$. 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 $\gamma$ 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 Grüneisen $\gamma \text{'}\mathrm{s}$. The three-dimensional $\gamma$ 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.

• Received 26 August 1952

DOI:https://doi.org/10.1103/PhysRev.89.832