= Solution
A solid stores thermal <energy> in <lattice> vibrations and, when available, electronic and other excitations. For a nonmagnetic insulating <crystal> at sufficiently low <temperature>, the gap suppresses mobile electronic excitations and the leading bulk contribution normally comes from acoustic <phonons>. The three-dimensional <Debye heat capacity> is
$$
C_{\rm ph}=9Nk_B\left(\frac{T}{\theta_D}\right)^3\int_0^{\theta_D/T}\frac{x^4e^x}{(e^x-1)^2}\,dx.
$$
The vibrational <density of states> scales as $\omega^2$, so only a fraction proportional to $T^3$ of the modes is thermally active at low <temperature>. The two limiting forms are
$$
\boxed{C_{\rm ph}\simeq\beta T^3,\quad\beta=\frac{12\pi^4Nk_B}{5\theta_D^3}\quad(T\ll\theta_D),\qquad C_{\rm ph}\to3Nk_B\quad(T\gg\theta_D).}
$$
The high-temperature value is the <Dulong-Petit law>, or $3R$ per mole of atoms. Optical <phonons> add thermally activated contributions as their characteristic energies become accessible. A multiatom cell has more branches, but still three vibrational degrees of freedom per atom.
A normal <metal> has an additional low-temperature <electronic heat capacity>. The <Pauli exclusion principle> restricts excitations to a shell of width about $k_BT$ around the <Fermi energy>. The number of thermally active <Electrons> is therefore proportional to $T$, rather than the total <Electron> number contributing a classical <heat capacity>. With spin-summed electronic <density of states> $g(E_F)$ for the whole specimen,
$$
\boxed{C_{\rm el}=\gamma T,\qquad \gamma=\frac{\pi^2}{3}k_B^2g(E_F),\qquad C_{\rm metal}\simeq\gamma T+\beta T^3.}
$$
For a three-dimensional free-electron gas, $g(E_F)=3N_e/(2E_F)$ and $C_{\rm el}=(\pi^2/2)N_ek_B(T/T_F)$. Only a small fraction of the classical electronic value is accessible when $T\ll T_F$, even though the electronic term dominates the <phonon> term as $T\to0$ in a normal gapless <metal>.
\Image[/past-exam-of-the-quantum-condensed-matter-course-of-the-university-of-cambridge/2013-heat-capacity.png]
{title=Debye lattice heat capacity and the low-temperature electronic intercept in a normal metal}
Plotting $C/T$ against $T^2$ gives a straight line with intercept $\gamma$ and slope $\beta$ in the appropriate low-temperature range. This separates the electronic density-of-states information from the acoustic-phonon scale. In a <Fermi liquid>, the interacting <quasiparticle> <density of states> replaces the free value, so an enhanced <effective mass> enhances $\gamma$.
The qualifications matter experimentally. Superconductivity can gap electronic excitations, magnetic order adds magnetic excitations, defects can generate additional low-temperature terms, and low-dimensional or very small specimens need not obey the bulk cubic law. The Debye expression is for constant-volume <heat capacity>; constant-pressure <heat capacity> differs through thermal expansion, although the difference is often small at low <temperature>. Thus the normal-metal and ordinary insulating limits are useful leading behaviors, not universal formulas for every solid in every phase.
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