Debye and Einstein heat-capacity models

A positive-frequency harmonic mode has thermal energy $\hbar\omega/(e^{\hbar\omega/(k_BT)}-1)$ above its zero-point energy. Differentiate at fixed frequency to obtain its constant-volume heat capacity. Summing over modes, or integrating against a density of states g, gives the harmonic lattice contribution. The density must count every physical mode once, including polarization. Its integral is three times the atom count in a bulk three-dimensional model, up to the separately treated overall zero modes whose fraction vanishes in the thermodynamic limit.

The Einstein model places all those modes at one angular frequency. It captures quantum freeze-out and the classical high-temperature limit, but its low-temperature heat capacity is exponentially small. A real three-dimensional crystal has acoustic modes at arbitrarily low frequency in the thermodynamic limit, so a single nonzero Einstein frequency misses their power-law contribution. The model can still be useful for a narrow optical band or as one component of a fitted vibrational spectrum. It is an approximation to the frequency distribution, not an alternative statistical rule for phonons.

For a monatomic three-dimensional bulk solid, replace the three acoustic branches by isotropic linear dispersions. Counting wavevectors in a sphere gives a density proportional to frequency squared. A cutoff omega D is chosen so the integrated count is 3N. In the simplest one-speed version the cutoff wavevector is $(6\pi^2N/V)^{1/3}$. Real longitudinal and transverse speeds differ; a suitable inverse-cube average can preserve the low-frequency density without claiming that every polarization travels at the same speed.

At temperatures well below the Debye temperature, thermally occupied frequencies lie far below the cutoff. Extending the dimensionless integral to infinity produces the cubic temperature law. At high temperature every harmonic oscillator contributes Boltzmann's constant, giving the Dulong–Petit value. The zero-point energy contributes no temperature derivative when the frequencies are held fixed. A multi-atom cell has optical modes beyond its three acoustic branches; replacing all 3N modes by the Debye density is then a surrogate model, rather than literal acoustic branch counting across the entire spectrum.

The continuum low-temperature law requires many modes below the thermal frequency scale. For a finite sample, the lowest nonzero acoustic frequency is of order sound speed divided by sample length. Below that discrete spacing, the thermodynamic-limit cubic law need not hold. One therefore takes a large-system limit before extrapolating the bulk low-temperature power law indefinitely. Two-dimensional membranes and anisotropic or nonlinearly dispersing low modes also have different densities of states, so their temperature powers require separate derivations.

These formulas describe vibrational heat capacity at constant volume within a harmonic model. Experiments often measure constant-pressure heat capacity and can include electrons, magnetic excitations, defects, or structural changes. Anharmonic thermal expansion changes frequencies and helps distinguish the two thermodynamic constraints. A fitted Debye temperature can summarize a particular temperature range without being a unique sharp maximum of a real dispersion. Use low-temperature scaling, high-temperature mode count, and known additional excitations as independent checks before treating a one-parameter fit as a complete microscopic spectrum.