= Debye heat capacity
{c}
{title2=$C_V=9Nk_B(T/\Theta_D)^3\int_0^{\Theta_D/T}x^4e^x/(e^x-1)^2\,dx$}
The three-dimensional <Debye model> normalizes its <phonon> <density of states> to $3N$ modes: $g(\omega)=9N\omega^2/\omega_D^3$ below the <Debye frequency> $\omega_D$. The thermal <energy> is $U=\int_0^{\omega_D}g(\omega)\hbar\omega/[e^{\hbar\omega/(k_BT)}-1]d\omega$. Differentiate at fixed <volume> and substitute $x=\hbar\omega/(k_BT)$ to obtain the displayed <heat capacity at constant volume>.
At $T\ll\Theta_D$, integration by parts gives $\int_0^\infty x^4e^x/(e^x-1)^2dx=4\int_0^\infty x^3/(e^x-1)dx=4\pi^4/15$. Therefore $C_V\sim(12\pi^4/5)Nk_B(T/\Theta_D)^3$. At $T\gg\Theta_D$, each mode approaches the <equipartition theorem> value $k_B$ of <heat capacity>, so $C_V\to3Nk_B$, the <Dulong-Petit law>. Fixed harmonic <zero-point energy> does not contribute to either limit.
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