Quantum treatment of thermal expansion. Griineisen parameter

Quantum treatment of thermal expansion. Griineisen parameter 

To begin with, note that by using relations [1.152], [1.154] and [1.155], we can write  

In order to calculate the linear expansion coefficient, we first need to calculate the thermodynamic coefficient (dPIdT). For this 

We will consider our solid to be a collection of quantum oscillators, phonons, each of which has its own natural frequency of vibration 

The characteristic function with variables T, V is the Helmholtz energy. Thus, for each phonon, we can write its contribution to the Helmholtz energy in the form  

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