The Debye Model
Field note — quantized lattice vibrations and the low-temperature heat capacity of solids.
Why the Einstein model fails
Einstein modeled a solid as independent quantum harmonic oscillators, all at the same frequency ω_E. It got the right qualitative picture — heat capacity drops at low T — but the wrong functional form. The experimental data shows C_V ∝ T³ as T → 0, while Einstein predicts an exponential drop, C_V ∝ exp(−Θ_E/T). The discrepancy was a known problem by the early 1930s. Debye solved it by abandoning the single-frequency assumption entirely.
Debye's insight
Debye treated the solid as a continuous elastic medium supporting phonon modes — collective lattice vibrations with a linear dispersion ω = v_s·k, where v_s is the speed of sound. There are three polarizations: two transverse and one longitudinal. In a crystal with N atoms, there are 3N normal modes total. Debye imposed a cutoff frequency ω_D such that the total number of modes is exactly 3N:
∫₀^{ω_D} g(ω) dω = 3N
This gives the Debye frequency and the corresponding Debye temperature:
Θ_D = ℏω_D/k_B
The cutoff is the key physical idea: a crystal is not an infinite elastic continuum, so there must be a maximum frequency. The lattice spacing sets the shortest wavelength (λ_min ≈ 2a), and therefore the maximum wavevector k_max ≈ π/a, and therefore ω_max. Debye compressed all of this into a single parameter Θ_D, which varies from ~100 K for lead to ~2200 K for diamond.
The heat capacity
The total vibrational energy is:
U = ∫₀^{ω_D} ℏω · [1/(e^{ℏω/k_BT} − 1) + 1/2] · g(ω) dω
where g(ω) = (9N/ω_D³)·ω² is the Debye density of states (quadratic, unlike the flat Einstein spectrum).
High temperature (T ≫ Θ_D)
All 3N modes are excited. U ≈ 3Nk_BT (ignoring zero-point energy). Therefore C_V → 3Nk_B = 3R per mole — the Dulong-Petit law. Every degree of freedom contributes k_B/2 for kinetic and k_B/2 for potential energy.
Low temperature (T ≪ Θ_D)
Only the lowest-frequency modes matter. The upper cutoff becomes irrelevant; you can extend the integral to ∞ with negligible error:
C_V ≈ (12π⁴/5) · Nk_B · (T/Θ_D)³
This is the T³ law. The T³ comes from the ω² density of states combined with the Bose-Einstein occupation factor, which for small ω becomes linear in ω, and the k_BT energy scale. The result is exact and universally observed at sufficiently low T.
What Θ_D really is
Debye temperature is not just a fitting parameter. It is directly related to the sound velocity and the atomic density:
Θ_D = (ℏv_s/k_B) · (6π²N/V)^{1/3}
where v_s is an appropriate average over longitudinal and transverse sound velocities. You can compute Θ_D from the elastic moduli of the crystal, or measure it from low-temperature heat capacity data, or extract it from neutron scattering. The consistency between these methods is one of the quiet successes of condensed matter physics.
Beyond the Debye model
The Debye model is a harmonic approximation in a continuous medium. Real crystals have:
- Optical phonons (not captured — Debye only has acoustic modes).
- Non-linear dispersion at high k (the ω = v_s·k relation breaks down near the Brillouin zone boundary).
- Anisotropy (v_s depends on direction).
- Anharmonicity (thermal expansion, phonon-phonon scattering).
Still, the Debye model gives the correct low-T behavior, captures the crossover from T³ to constant heat capacity, and provides a single parameter that encodes the stiffness of an entire solid. It is, in the physicist's sense, the right model.
Summary
The Debye model treats lattice vibrations as quantized acoustic phonons with a linear dispersion and a mode-counting cutoff. It predicts the observed T³ low-temperature heat capacity and the Dulong-Petit limit at high temperature. One parameter, Θ_D, controls everything.