Pressure of a Solid

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Homework Statement


Show that the pressure of a solid is given by
[tex] P = -\frac{\partial\Phi_0}{\partial V} + \gamma \frac{U}{V}[/tex]

[tex]\Phi_0(V) [/tex] is the potential energy of the solid when all atoms are at rest in their equilibrium positions and V is the volume of the solid.
[tex]U[/tex] is the internal energy arsing from the vibrations of the atoms.

[tex]\gamma = -\frac{\partial ln\omega}{\partial ln V} \approx 1/3[/tex] is the Gruneisen parameter
[tex]\omega_i (V)[/tex] where (i = 1,2,...,3N-6) are the normal frequencies of vibration

Homework Equations



[tex]E = \Phi_0 + \sum_i(n_i + 1/2) \hbar \omega_i [/tex]

[tex]U = \sum_i \hbar\omega_i + \sum_i \frac {\hbar\omega_i}{e^{\frac{\hbar\omega_i}{k T}}-1}[/tex]

The Attempt at a Solution


I guess I need to find how the frequencies w depend on volume?
 

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