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Einstein's solid is a model of solid in which there are N atoms each of which is assumed to be noninteracting, identical and localized quantum mechanical oscillators. The canonical partition function is given by $$Z=\Bigg(\frac{e^{-\beta\hbar\omega/2}}{1-e^{-\beta\hbar\omega}}\Bigg)^N$$ which depends only on the $\beta$, and the Z doesn't contain any information about the volume of the solid. Therefore, the pressure formula gives $$P=\beta^{-1}\frac{\partial}{\partial V}(\ln Z)=0!$$ What is the simplest way to amend the model in a realistic way (and hence the partition function) so to obtain a nonzero pressure?

I understand this model was devised to find the temperature dependence of the specific heat. But unless one can find the pressure this doesn't seem to be a realistic model of solid.

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Of course the model is not realistic. It does not even come close to reality and is just a historic remnant. Note, that the heat capacity of solids at low temperatures is typically dominated by the acoustic phonons which have a linear spectrum, while the Einstein model describes optical phonons with a band gap. The low energy characteristic of acoustic phonons is even accessible as the speed of sound – so we can connect our model with some experimentally accessible quantity. Using an isotropic linear spectrum is known as the Debye model, which is still bad but much better than the Einstein model.

However, if you want to include some volume dependence you will have to change the energy of the phonon band in dependence of a volume (which does happen in reality, if we press a solid together we "change the spring constants between the atoms" since we now develop around another equilibrium point). This change corresponds to the non-harmonic corrections to the effective potential of the atoms in the solid.

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  • $\begingroup$ That's a great answer, However I'd just wanted to point out that, while it is true that the model is not realistic and has been superseded by more advanced models, the Einstein solid is still used as the starting point of the so-called thermodynamic integration procedure applied to crystals, which is a simulation technique used to compute the free energy of "real" solids (see here for instance) $\endgroup$
    – lr1985
    May 23, 2018 at 8:22

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