# Phonon Energy and Density of States

1. Mar 10, 2015

### Karim Habashy

Hi all,

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

U (Total Phonon Energy ) = Σkp((ħ*ωk,p)/((exp(ħ*ωk,p/τ))-1))

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

U (Total Phonon Energy ) = ∑p∫dω*Dp(ω)*((ħ*ωk,p)/((exp(ħ*ωk,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/Vg

so it be :

U (Total Phonon Energy ) = ∑p∫dω*(1/Vg)*((ħ*ωk,p)/((exp(ħ*ωk,p/τ))-1))

Last edited: Mar 10, 2015
2. Mar 11, 2015

### Staff: Mentor

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.

3. Mar 11, 2015

### Karim Habashy

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

Thanks