Past exam of the condensed matter physics course of the University of Oxford / 2019 / 2 / a

The Einstein model of a solid replaces the vibrations of $N$ identical atoms by $3N$ independent quantum harmonic oscillators with a common frequency ${\omega }_E$. Each atom has three displacement directions; the restoring force is harmonic, coupling between different atoms is replaced by a fixed local restoring potential, and anharmonicity and frequency dispersion are neglected. The crystal volume and ${\omega }_E$ are held fixed when calculating the vibrational heat capacity at constant volume.

The one-oscillator partition function isConsequently, with $T_E=\hslash {\omega }_E/k_B$,Differentiating at fixed volume givesThe zero-point energy does not contribute to this derivative. The result tends to the Dulong-Petit law $3N{k}_B$ at high temperature and vanishes exponentially at low temperature; the latter is a limitation of the single-frequency model for a real acoustic spectrum.