= Solution
For three-dimensional acoustic modes, $\omega\propto q$ and the mode count in a spherical shell is proportional to $q^2dq$, giving <phonon> DOS proportional to $\omega^2$. Thermally occupied modes have $\hbar\omega$ of order $k_BT$, so $U_{la}-U_0\propto\int\omega^3n_B(\omega)\,d\omega\propto T^4$. Differentiation gives \b[$C_{la}\propto T^3$]. At $T\gg\theta_D$, all $3N$ harmonic modes are classically populated; each has mean <energy> $k_BT$ by equipartition. Thus the <molar heat capacity> tends to \b[$3R$], the <Dulong-Petit law>. Anharmonic expansion and electronic contributions are not included in that lattice limit.
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