= Solution
{c}
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> is
$$
z=\sum_{n=0}^\infty e^{-\beta\hbar\omega_E(n+1/2)}=\frac{e^{-\beta\hbar\omega_E/2}}{1-e^{-\beta\hbar\omega_E}},\qquad Z=z^{3N}.
$$
Consequently, with $T_E=\hbar\omega_E/k_B$,
$$
U=-\frac{\partial\log Z}{\partial\beta}=3N\hbar\omega_E\left[\frac12+\frac1{e^{T_E/T}-1}\right].
$$
Differentiating at fixed volume gives
$$
\boxed{C_V=3Nk_B\left(\frac{T_E}{T}\right)^2\frac{e^{T_E/T}}{(e^{T_E/T}-1)^2}.}
$$
The <zero-point energy> does not contribute to this derivative. The result tends to the <Dulong-Petit law> $3Nk_B$ at high temperature and vanishes exponentially at low temperature; the latter is a limitation of the single-frequency model for a real acoustic spectrum.
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