The (minimum) potential energy of a solid, when all its atoms are "at rest" at their equilibrium positions, may be denoted by the symbol \(\Phi_{0}(V)\), where \(V\) is the volume of the solid. Similarly, the normal frequencies of vibration, \(\omega_{i}(i=1,2, \ldots, 3 N-6)\), may be denoted by the symbols \(\omega_{i}(V)\). Show that the pressure of this solid is given by

\[
P=-\frac{\partial \Phi_{0}}{\partial V}+\gamma \frac{U^{\prime}}{V}
\]

where \(U^{\prime}\) is the internal energy of the solid arising from the vibrations of the atoms, while \(\gamma\) is the Grüneisen constant:

\[
\gamma=-\frac{\partial \ln \omega}{\partial \ln V} \approx \frac{1}{3}
\]

Assuming that, for \(V \simeq V_{0}\),

\[
\Phi_{0}(V)=\frac{\left(V-V_{0}ight)^{2}}{2 \kappa_{0} V_{0}}
\]

where \(\kappa_{0}\) and \(V_{0}\) are constants and \(\kappa_{0} C_{V} T \ll V_{0}\), show that the coefficient of thermal expansion (at constant pressure \(P \simeq 0\) ) is given by

\[
\alpha \equiv \frac{1}{V}\left(\frac{\partial V}{\partial T}ight)_{N, P}=\frac{\gamma \kappa_{0} C_{V}}{V_{0}}
\]

Also show that

\[
C_{P}-C_{V}=\frac{\gamma^{2} \kappa_{0} C_{V}^{2} T}{V_{0}}
\]