Debyemodell

Debye model is a theoretical framework for the vibrational contribution to the heat capacity of crystalline solids. Proposed by Peter Debye in 1912, it extends the Einstein model by treating lattice vibrations as a continuum of acoustic phonon modes with a finite maximum frequency. In the Debye approximation, the solid is assumed to have 3N vibrational modes for N atoms, with a density of states proportional to ω^2 up to a maximum frequency ω_D and zero beyond. The Debye temperature θ_D is defined as θ_D = ħ ω_D / k_B.

The internal energy and the heat capacity at constant volume can be derived from the Bose–Einstein statistics for the vibrational modes. In the Debye model this yields the relation C_V = 9 N k_B (T/θ_D)^3 ∫_0^{θ_D/T} x^4 e^x/(e^x − 1)^2 dx, or equivalently C_V = 9 N k_B (T/θ_D)^3 D_3(θ_D/T), where D_3 is the Debye function.

Behavior with temperature: at high temperatures (T >> θ_D), C_V tends toward the classical Dulong–Petit value 3N k_B. At low temperatures (T << θ_D), C_V varies as T^3, in agreement with many experimental observations for crystalline solids.

Applications and limitations: the Debye model captures the essential low-temperature T^3 behavior and links θ_D to elastic properties, but it neglects optical phonons, assumes isotropy and a simple lattice, and does not describe all materials, especially those with complex unit cells or strong anharmonic effects. Nevertheless, it remains a foundational model for understanding lattice heat capacity and for estimating the Debye temperature from calorimetric data or elastic constants.