Debye model: Encyclopedia II - Debye model - Derivation

Debye model - Derivation

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.

Consider a cube of side L. 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 n is an integer. The energy of a phonon is

where h is Planck's constant and ν_n is the frequency of the phonon. We make the approximation that the frequency is inversely proportional to the wavelength, giving:

in which c_s is the speed of sound inside the solid. In three dimensions we will use:

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 is one of the limitations of the Debye model.

Let's now compute the total energy in the box

where is the number of phonons in the box with energy E_n. 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:

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. 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 N atoms in a solid. Our solid is a cube, which means there are atoms per side. Atom separation is then given by , and the minimum wavelength is

making the maximum mode number n (infinite for photons)

This is the upper limit of the triple energy sum

For slowly-varying, well-behaved functions, a sum can be replaced with an integral (a.k.a Thomas-Fermi approximation)

So far, there has been no mention of , the number of phonons with energy E. Phonons obey Bose-Einstein statistics. Their distribution is given by the famous Bose-Einstein formula

Because a phonon has three possible polarization states (one longitudinal and two transverse) which do not affect its energy, the formula above must be multiplied by 3

Substituting this 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 boldly approximated the cube by an eighth of a sphere

where R 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 N unit-cell volumes,

so we get:

The substitution of integration over a sphere for the correct integral introduces another source of inaccuracy into the model.

The energy integral becomes

Changing the integration variable to ,

To simplify the look of this expression, define the Debye temperature T_D -- a shorthand for some constants and material-dependent variables.

We then have the specific internal energy:

where D₃(x) is the (third) Debye function.

Differentiating with respect to T we get the dimensionless heat capacity:

These formulae give the Debye model at all temperatures. The more elementary formulae given further down give the asymptotic behavior in the limit of low and high temperatures.

Adapted from the Wikipedia article "Derivation", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki