The Debye model treats the crystal as an elastic continuum with a cutoff frequency ω_D set by the total number of modes (3N). The heat capacity C_V = 9Nk_B (T/θ_D)³ ∫₀^(θ_D/T) x⁴eˣ/(eˣ−1)² dx interpolates between the T³ law at low temperatures (quantum regime) and the classical Dulong-Petit limit C_V → 3Nk_B at high temperatures. The Debye temperature θ_D = ℏω_D/k_B characterizes the material: diamond (~2230 K) is stiff; lead (~105 K) is soft. The Einstein model uses a single frequency for all modes — it matches the high-T limit but overestimates the exponential decay at low T.