Compare Einstein and Debye models of lattice heat capacity. The Debye model explains the T³ low-temperature regime. Visualize phonon density of states, dispersion, and C_v(T) for different materials.
Debye (1912): C_v = 9Nk_B(T/T_D)³∫₀^{T_D/T} x⁴eˣ/(eˣ-1)² dx. Low-T: C_v = (12π⁴/5)Nk_B(T/T_D)³ (Debye T³ law). High-T: C_v = 3Nk_B (Dulong-Petit). Einstein model C_v = 3Nk_B(T_E/T)²eˣ/(eˣ-1)² with x=T_E/T. Debye superior at low T because it includes acoustic phonons (linear dispersion ω=vk).