The Thermal Expansion of Solids

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.