Change of Mean Phonon Occupation Numbers in a Cubic Crystal During Adiabatic Compression

It is shown that during an infinitesimal adiabatic compression the average number of phonons ${\overline{n}}_j$ in a normal mode of a solid changes by an amount $d{\overline{n}}_j=\left\{\frac{({\gamma }_j-\gamma )c_{j}T}{\hslash {\omega }_j}\right\}d\ln V,$ where ${\omega }_j$ is the angular frequency of the mode, $c_j$ its contribution to $C_V$, ${\gamma }_j=-\frac{d\ln {\omega }_j}{d\ln V}$, and $\gamma$ is the Grüneisen function $\frac{\beta V}{{\chi }_S{C}_P}$ (derived from the volume coefficient of expansion $\beta$, the adiabatic compressibility ${\chi }_S$, and the heat capacity at constant pressure $C_P$).