The heat capacity of a crystal

Based on equipartition, we expect the molar heat capacity for a solid to be $3RT$ since each atom is free to vibrate in three directions. This is the law of Dulong and Petit, and it works well for a variety of solids at room temperature. It is reproduced, as might be expected, by the Einstein model for a crystal which considers each atom linked to its neighbours by six springs ($3N$ in total)--the algebra is just like that of the vibrations of a diatomic molecule giving

$$\begin{eqnarray*} \left\langle E \right\rangle \!\!\!&=&\!\!\!{\textstyle\frac 3... ...bar\omega\beta)^2 \left({\rm sinh} \hbar\omega\beta\right)^{-2} \end{eqnarray*}$$

At low temperature ( $\beta\to\infty$) the energy tends to $3 N k_{\scriptscriptstyle B}(\hbar\omega\beta)^2e^{-\hbar\omega\beta}$. Although this tends to zero, it does not agree with the observed low temperature behaviour, which is proportional to $T^3$. More sophisticated models, such as that of Debye, allow for collective vibrations of many atoms which have much lower frequency, and hence contribute to the internal energy and heat capacity at much lower temperatures.