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_{ph} = 9 N k_{B} (\frac{T}{θ _{D}})^{3} \int_{0}^{θ_{D} / T} \frac{x ^{4} e ^{x}}{( e ^{x} - 1 ) ^{2}} d x .

The vibrational density of states scales as

ω^{2}

, so only a fraction proportional to

T^{3}

of the modes is thermally active at low temperature. The two limiting forms are

C_{ph} ≃ β T^{3}, β = \frac{12 π ^{4} N k _{B}}{5 θ _{D}^{3}} (T ≪ θ_{D}), C_{ph} \to 3 N k_{B} (T ≫ θ_{D}) .

(2)

The high-temperature value is the Dulong-Petit law, or

3 R

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,

C_{el} = γ T, γ = \frac{π ^{2}}{3} k_{B}^{2} g (E_{F}), C_{metal} ≃ γ T + β T^{3} .

(3)

For a three-dimensional free-electron gas,

g (E_{F}) = 3 N_{e} / (2 E_{F})

and

C_{el} = (π^{2} /2) N_{e} k_{B} (T / T_{F})

. Only a small fraction of the classical electronic value is accessible when

T ≪ T_{F}

, even though the electronic term dominates the phonon term as

T \to 0

in a normal gapless metal.

Figure 1.
Debye lattice heat capacity and the low-temperature electronic intercept in a normal metal
. **Details**.

Plotting

C / T

against

T^{2}

gives a straight line with intercept

γ

and slope

β

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

γ

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.