Debye model

1. Debye model 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 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, which is proportional to T3. The Debye model is a solid-state equivalent of Planck's law of black body radiation, where one treats electromagnetic radiation as a gas of photons in a box. The Debye model treats atomic vibrations as phonons in a box (the box being the solid). Most of the calculation steps are identical. There is a close analogy between photons and phonons: both are “unconserved” bosons. Distinctions: (a) the speed of propagation of phonons (~ the speed of sound waves) is by a factor of 105 less than that for light, (b) sound waves can be longitudinal as well as transversal, thus 3 polarizations (2 for photons), and (c) because of discreteness of matter, there is an upper limit on the wavelength of phonons – the interatomic distance. If cs is the speed of sound waves, and L is the length of the crystal: Nobel 1936 Peter Debye (1884 – 1966)

2. The total thermal energy of the crystal is: Now, this is where Debye model and Planck's law of black body radiation differ. Unlike electromagnetic radiation in a box, there is a finite number of phonon energy states because a phonon cannot have infinite frequency. Its frequency is bound by the medium of its propagation—the atomic lattice of the solid. Atoms are tied together with bonds, so they can't vibrate independently. The vibrations take the form of collective modes which propagate through the material. Such propagating lattice vibrations can be considered to be sound waves, and their propagation speed is the speed of sound in the material. three polarizations (one longitudinal, and two transverse) Modes of oscillations of a row of atoms on a crystal

3. 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 N atoms in a solid. Our solid is a cube, which means there are N1/3 atoms per edge. Atom separation is then given by L/N1/3 , and the minimum wavelength is Debye temperature The higher the sound speed and the density of ions, the higher the Debye temperature. However, the real phonon spectra are very complicated, and D is better to treat as an experimental fitting parameter.

4. Debye temperature When When