Phonon frequency spectrum distribution

[sirwan]

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

 

[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 L_x,L_y,L_z the allowable k_x values are 0, 2π/L_x, 4π/L_x, ... Similar relation holds for k_y and k_z. 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 V_k(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
V_k(ω) = (4π/3) k³ = (4π/3c³)ω³
Now, it's easy to differentiate wrt ω to get
g(ω) = (4π/3) k³ = (4π/c³)ω²
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

 

[sirwan]

thank you , I got some idea.

 

[sirwan]

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

 

[Henryk]

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

 

[radium]

The photon spectrum is something you can calculate using density functional theory. A package that can do this is quantum espresso.