High-temperature heat capacity of an anharmonic crystal to orderλ4

The thermal-expansion contribution to the high-temperature heat capacity of an anharmonic crystal to $O\left({\lambda }^4\right)$ is obtained. It is found that an additional term of $O\left({\lambda }^4\right)$ contributes to the $T^2$ coefficient of heat capacity at constant volume. The heat capacity at constant pressure has also been calculated to the same order and the difference $c_p-c_v$ is shown to be consistent with that obtained from the exact thermodynamic identity.