Debye model

The Debye model approximates a crystal's vibrational spectrum by long-wavelength acoustic modes extended to a cutoff chosen to preserve the total number of vibrational degrees of freedom. In the elementary three-dimensional isotropic treatment, linear dispersion gives a phonon density of states proportional to ${\omega }^2$ below that cutoff.

Integrating the oscillator thermal energies then gives low-temperature lattice heat capacity proportional to $T^3$. At high temperature, the same correctly counted modes approach the classical value $3N{k}_B$ for $N$ atoms. The Debye frequency fixes the cutoff and the Debye temperature expresses its energy scale thermally. Real crystals have multiple branches, anisotropy and optical modes, so a fitted Debye parameter summarizes a spectral approximation rather than reconstructing every phonon frequency.

The Debye model is a theoretical framework used to describe the specific heat capacity of solids, particularly at low temperatures. Developed by physicist Peter Debye in 1912, it extends the classical Einstein model of specific heat by considering acoustic phonons, or quantized lattice vibrations, in a more comprehensive manner.