= Solution
The <Debye model> treats the crystal as an isotropic elastic continuum with independent harmonic vibrational modes. There are three acoustic <polarizations>, approximated by linear dispersions and a common effective speed. A cutoff is chosen to retain $3N$ modes for $N$ atoms. The model neglects anharmonic interactions and the detailed high-wavevector dispersion. In a multi-atom primitive cell, optical branches require additional treatment; copper has one atom per primitive cell even though its conventional cubic cell contains four atoms.
A <frequency> interval contains, per <volume>,
$$
g_{\rm ph}(\omega)\,d\omega=3\frac{4\pi q^2dq}{(2\pi)^3}
=\frac{3\omega^2}{2\pi^2c^3}\,d\omega
=\frac{9n\omega^2}{\omega_D^3}\,d\omega.
$$
Each <phonon> mode has <Bose-Einstein distribution> occupation $(e^{\hbar\omega/(k_BT)}-1)^{-1}$. The fixed <zero-point energy> contributes no <heat capacity>. Its <thermal energy> density is
$$
u_{\rm ph}=\frac{9n\hbar}{\omega_D^3}\int_0^{\omega_D}\frac{\omega^3\,d\omega}{e^{\hbar\omega/(k_BT)}-1}
=9nk_BT\left(\frac{T}{\theta_D}\right)^3\int_0^{\theta_D/T}\frac{x^3\,dx}{e^x-1}.
$$
For $T\ll\theta_D$, extending the upper limit to infinity gives $u_{\rm ph}=(3\pi^4/5)nk_BT^4/\theta_D^3$. Therefore
$$
\boxed{C_{\rm ph}=\frac{12\pi^4}{5}nk_B\left(\frac{T}{\theta_D}\right)^3
=\frac{2\pi^2k_B^4}{5\hbar^3c^3}T^3}.
$$
The cubic law reflects the three-dimensional acoustic mode density proportional to $\omega^2$. It is a low-temperature limit, not a valid extrapolation through the classical high-temperature regime.
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