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The Debye model treats lattice vibrations in a crystal as a gas of noninteracting phonons with a density of states proportional to ω². A cutoff frequency (Debye frequency ω_D) ensures the correct number of modes. The model predicts C_V → 12π⁴/5 (Nk) at T ≪ T_D (T³ law) and C_V → 3Nk at T ≫ T_D (Dulong-Petit), in good agreement with experiment.
From your study of the Planck distribution and blackbody radiation, you already know how to treat a system of quantum harmonic oscillators in thermal equilibrium: each mode of frequency ω carries an average energy ℏω/(eℏω/kT − 1), the Planck function. Photons in a cavity are exactly this — a collection of oscillators with frequencies spanning a continuous spectrum. The key difference in a crystal is that the "photons" are now phonons: quantized lattice vibrations. Instead of an electromagnetic field, the oscillating objects are atoms in a crystal lattice, and the normal modes of their collective motion are the vibrational modes. The Debye model asks: what is the spectrum of frequencies these modes span, and how does their thermal energy depend on temperature?
The Einstein model (which this topic builds toward) took a crude guess: all modes have the same frequency ω_E. This captured the quantum suppression at low temperature but failed quantitatively because real crystals have modes at many frequencies. Debye's improvement was to model the crystal as an elastic continuum — a 3D solid where sound waves propagate at a speed v_s. For sound waves in 3D, the number of modes with frequency below ω is proportional to ω³ (from the volume of a sphere in k-space), so the density of states g(ω) ∝ ω². Unlike photons, however, a crystal with N atoms has exactly 3N vibrational modes — not an infinite number. Debye imposed a hard cutoff at the Debye frequency ω_D chosen so that ∫₀^{ω_D} g(ω)dω = 3N. This cutoff defines the Debye temperature T_D = ℏω_D/k, a material-specific scale separating quantum from classical behavior.
With this density of states, the total energy is U = ∫₀^{ω_D} g(ω) · ℏω/(eℏω/kT − 1) dω. Taking the temperature derivative gives the heat capacity C_V = ∂U/∂T. In the two limiting regimes, the math simplifies beautifully. At high temperature (T ≫ T_D), every mode has kT ≫ ℏω, so the Planck function reduces to kT and each mode gets exactly kT of energy — the classical Dulong-Petit result C_V = 3Nk. At low temperature (T ≪ T_D), most modes are frozen out because kT ≪ ℏω_D. Only the lowest-frequency modes (long-wavelength acoustic phonons) are thermally excited, and their contribution scales as T³. The ω² density of states is essential here: it gives just the right weighting to produce the T³ law, which is experimentally confirmed for virtually all insulators at low temperature and is one of the landmark successes of quantum statistical mechanics.
The Debye model is approximate — it assumes a linear dispersion relation (ω ∝ k) that breaks down at short wavelengths and it ignores anharmonic effects — but its predictions match experiment far better than the Einstein model. More importantly, it provides the conceptual template for treating any quantum many-body system as a gas of bosonic excitations (phonons, magnons, plasmons) with a given density of states. The density of states function g(ω) is the central object: once you know it, thermodynamic quantities follow from the same Planck-distribution integrals you already know.