Debye Model of Solids
Quantized lattice vibrations (phonons) explain the heat capacity of solids — bridging classical and quantum regimes.
Debye Model (1912): Replace lattice vibrations with a phonon gas with linear dispersion ω = v_s·k, up to a cutoff (Debye frequency ω_D). The density of states is g(ω) ∝ ω² (parabolic), truncated at ω_D.
Heat capacity: C_V = 9Nk_B(T/θ_D)³∫₀^(θ_D/T) x⁴eˣ/(eˣ−1)² dx
— At high T: C_V → 3Nk_B (Dulong-Petit) — At low T: C_V ∝ T³ (Debye T³ law)