= Solution
The <Debye model> treats a solid's low-frequency lattice vibrations as acoustic <phonons> in an elastic continuum. There are three polarizations per <wavevector>, one longitudinal and two transverse, with approximately linear <dispersion relations>. A cutoff is chosen to preserve the total $3N$ vibrational modes of $N$ atoms. The <phonon> number itself is not fixed; mode occupations follow the <Bose-Einstein distribution> with zero <chemical potential>.
For one polarization of speed $v_s$, counting <wavevector> states in a spherical shell gives $g_s(\omega)=V\omega^2/(2\pi^2v_s^3)$. Define an average Debye speed by $3/v_D^3=1/v_L^3+2/v_T^3$. The total <density of states> can then be written
$$
g(\omega)=\frac{9N\omega^2}{\omega_D^3},\quad0\leq\omega\leq\omega_D,\qquad
\int_0^{\omega_D}g(\omega)d\omega=3N.
$$
The <Debye wavevector> is $k_D=(6\pi^2N/V)^{1/3}$, the <Debye frequency> is $\omega_D=v_Dk_D$, and the <Debye temperature> is $\Theta_D=\hbar\omega_D/k_B$.
Omitting the temperature-independent <zero-point energy>, the vibrational <energy> and resulting <Debye heat capacity> are
$$
U=\int_0^{\omega_D}\frac{g(\omega)\hbar\omega}{e^{\hbar\omega/(k_BT)}-1}\,d\omega,\qquad
\boxed{C_V=9Nk_B\left(\frac T{\Theta_D}\right)^3\int_0^{\Theta_D/T}\frac{x^4e^x}{(e^x-1)^2}\,dx}.
$$
At low <temperature> the upper limit can be extended to infinity. The integral is $4\pi^4/15$, giving
$$
\boxed{C_V\simeq\frac{12\pi^4}{5}Nk_B\left(\frac T{\Theta_D}\right)^3},\qquad T\ll\Theta_D.
$$
Only the long-wavelength acoustic modes are appreciably populated, explaining the <Debye T-cubed test>. At high <temperature> every mode contributes $k_B$ by <equipartition theorem>, so $\boxed{C_V\to3Nk_B}$, the <Dulong-Petit law>.
\Image[/past-exam-of-the-physics-a-course-of-the-university-of-cambridge/2017/paper-2-d13-debye-heat-capacity.png]
{title=Debye heat capacity approaching its classical limit and its low-temperature cubic law}
The model succeeds especially at low <temperature>, where long-wavelength elasticity is reliable. It replaces real anisotropic branches by a simple continuum and cannot reproduce detailed <optical phonon branches> or every intermediate-temperature feature. Metals also have an <electronic heat capacity> approximately proportional to $T$; defects, magnetic excitations or a <phase> transition can add other terms. Experiments often measure <heat capacity at constant pressure>, which need not equal the harmonic model's $C_V$ exactly.
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