Phonon density of states ↩

aliases: []
shorthands: {}
created: 2021-11-03 21:21:39
modified: 2022-01-11 15:59:34

Where is the lattice constant. In a sample of length , due to the periodic boundary condition:

This should be true for every mode, so for some . From this, cannot be a continuous, only discrete:

%% setting some definitions %%
This way, the density of states in the wavenumber space is:

And for more than one dimensions, let's say in dimensions:

Where is the volume of the sample.

is called the state number and it is the number of vibration modes in the crystal lattice whose frequency is smaller than . The frequency dependent state number can be defined like:

Where are the dispersion relations of the dispersion braches as seen in Harmonic approximation of lattice vibrations, indexed by . And is the Heaviside function.

The density of states is the derivative of this function:

Where is the Dirac delta function. In a system with state density, the number of states between the frequencies and is just .

If one of the dispersion branches are isotropic, then a simpler method can be utilized to determine the state density:

So in 1D in the isotropic case (when ), for an long sample:

The same in 2D, for a sample of area :

And in 3D, for a sample of volume :

The frequency dependent state density plays an important role when determining thermodynamical properties of a crystal material.

Debye model

The Debye model gives us a simple dispersion relation for the lattice vibrations. It is a good approximation of the acoustic dispersion branches for small wavenumbers.