 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 isThe 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 areThe 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 $k_{B}T$ 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,For a three-dimensional free-electron gas, $g(E_F)=3N_e/(2E_F)$ and $C_{\mathrm{el}}=({\pi }^2/2)N_e{k}_B(T/T_F)$. 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 $T\to 0$ in a normal gapless 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.