Phonon Energy and Density of States

[Karim Habashy]

Hi all,

In Charles Kittel (Introduction to Solid State Physics) He writes :

U (Total Phonon Energy ) = Σ_k∑_p((ħ*ω_k,p)/((exp(ħ*ω_k,p/τ))-1))

I understand this, but then he integrate over k and multiply by density of states :

U (Total Phonon Energy ) = ∑_p∫dω*D_p(ω)*((ħ*ω_k,p)/((exp(ħ*ω_k,p/τ))-1))

I understand the Integration, but why multiply by density of states, if he wants to change the variable dk to dω why not just use the dispersion relation i.e k=g(ω) so dk=(the first dervative g(ω))*dω , dk=dω/V_g

so it be :

U (Total Phonon Energy ) = ∑_p∫dω*(1/V_g)*((ħ*ω_k,p)/((exp(ħ*ω_k,p/τ))-1))

Thanks in Advance.

 

[DrClaude]

As soon as you go from a sum to an integral, you need to introduce the density of states. The number of states in the range $(k,k+dk)$ or $(\omega,\omega+d\omega)$ depends on $k$ and $\omega$, respectively.

 

[Karim Habashy]

Ok, that makes senses, but what's the physical meaning that at ω=0, we have a the Density of States g(ω) = (N/π)*√(M/K).

Thanks