In thermodynamics and solid-state physics, the Debye model is a method developed by Peter Debye in 1912 for estimating the phonon contribution to the specific heat (Heat capacity) in a solid. It treats the vibrations of the atomic lattice (heat) as phonons in a box, in contrast to the Einstein photoelectron model, which treats the solid as many individual, non-interacting quantum harmonic oscillators. The Debye model correctly predicts the low-temperature dependence of the heat capacity of solids, which is proportional to – the Debye T 3 law. Similarly to the Einstein photoelectron model, it recovers the Dulong–Petit law at high temperatures. Due to simplifying assumptions, its accuracy suffers at intermediate temperatures.
Derivation
The Debye model is a solid-state equivalent of Planck's law of black body radiation, where one treats electromagnetic radiation as a photon gas confined in a vacuum space. The Debye model treats atomic vibrations as phonons confined in the solid’s volume. Most of the calculation steps are identical as both are examples of a massless Bose gas with linear dispersion relation.
Consider a cube of side . From the particle in a box article, the resonating modes of the sonic disturbances inside the box (considering for now only those aligned with one axis) have wavelengths given by
where is an integer. The energy of a phonon is
where is the Planck constant and is the frequency of the phonon. Making the approximation that the frequency is inversely proportional to the wavelength, we have
in which is the speed of sound inside the solid. In three dimensions we will use
in which is the magnitude of the three-dimensional momentum of the phonon.
The approximation that the frequency is inversely proportional to the wavelength (giving a constant speed of sound) is good for low-energy phonons but not for high-energy phonons (see the article on phonons). This disagreement is one of the limitations of the Debye model. It produces incorrect results at intermediate temperatures, whereas the results are exact at the low and high temperatures limits.
Let's now compute the total energy in the box,
where is the number of phonons in the box with energy . In other words, the total energy is equal to the sum of energy multiplied by the number of phonons with that energy (in one dimension). In 3 dimensions we have:
Here, the Debye model and Planck's law of black body radiation differ. Unlike electromagnetic photon radiation in a box, there is a finite number of phonon energy states because a phonon cannot have arbitrarily high frequencies. Its frequency is bounded by the medium of its propagation—the atomic lattice of the solid. Consider an illustration of a transverse phonon below.
It is reasonable to assume that the minimum wavelength of a phonon is twice the atom separation, as shown in the lower figure. There are atoms in a solid. Our solid is a cube, which means there are atoms per edge. Atom separation is then given by , and the minimum wavelength is
making the maximum mode number (infinite for photons)
This number bounds the upper limit of the triple energy sum
For slowly varying, well-behaved functions, a sum can be replaced with an integral (also known as Thomas–Fermi approximation)
So far, there has been no mention of , the number of phonons with energy Phonons obey Bose–Einstein statistics. Their distribution is given by the famous Bose–Einstein statistics formula
Because a phonon has three possible polarization states (one longitudinal, and two transverse which approximately do not affect its energy) the formula above must be multiplied by 3,
Actually one uses an effective sonic velocity , i.e. the Debye temperature (see below) is proportional to , more precisely , where one distinguishes longitudinal and transversal sound-wave velocities (contributions 1/3 and 2/3, respectively). The Debye temperature or the effective sonic velocity is a measure of the hardness of the crystal.
Substituting into the energy integral yields
The ease with which these integrals are evaluated for photons is due to the fact that light's frequency, at least semi-classically, is unbound. As the figure above illustrates, this is not true for phonons. In order to approximate this triple integral, Debye used spherical coordinates.
and approximated the cube by an eighth of a sphere
where is the radius of this sphere, which is found by conserving the number of particles in the cube and in the eighth of a sphere. The volume of the cube is unit-cell volumes,