# Phonon frequency spectrum distribution

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1. Aug 26, 2015

### sirwan

hello every one , I want to know how we get phonon frequency spectrum theoretically by using three modes and dispersion relation, can any one explain it. for example it is phonon energy correspond to density of state how it is obtain?

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2. Aug 26, 2015

### Henryk

I assume that you can calculate ω(k) where k is a vector in reciprocal space. The magnitude of the k vector is 2π/λ where λ is the wavelength of the phonon and the direction is the direction of propagation of the phonon. For a solid of dimension Lx,Ly,Lz the allowable kx values are 0, 2π/Lx, 4π/Lx, ... Similar relation holds for ky and kz. Thus, the number of states is proportional to the volume in reciprocal space.
Therefore, to find phonon spectrum, you have to construct a constant ω surface in reciprocal space for all the values of ω, calculate the volume enclosed by the volume to get Vk(ω). Once you've done it, differentiate it with respect to ω and you have your spectrum.
To illustrate the point, let's consider anisotropic solid and low frequency phonons. The phonon frequency ω is related to the magnitude of the k vector by the relation
ω = c|k|
where c is the sound velocity. Constant ω surfaces will be spheres in reciprocal space and the volume given by
Vk(ω) = (4π/3) k3 = (4π/3c33
Now, it's easy to differentiate wrt ω to get
g(ω) = (4π/3) k3 = (4π/c32
Of course, you would need to normalize the expression, but these are details.
Two things to note:
g(ω) is proportional to ω2
and inversely proportional to c3

3. Aug 27, 2015

### sirwan

thank you , I got some idea.

4. Aug 27, 2015

### sirwan

what do you need to know about any element if you want to plot the spectrum of any element by that method.thanks.

5. Aug 27, 2015

### Henryk

You need to know the crystal structure, the forces between atoms and things like masses, distances between atoms.

6. Aug 27, 2015