Generalisation of the density states of phonons Is it possible to generalize de density of states for phonons $\left( \left(\frac{L}{2\pi} \right )^3  \int \frac{dS_\omega}{v_g}\right)$ to a density of states which is also applicable to Bloch electrons?
I thought about changing $v_g$ to $v_f$ (the Fermi velocity), but that did not agree with de density of states for electrons.
 A: The following equations for the density of states are applicable in all (3D) situations:*
$$g\left(E\right) = \int \frac{d\mathbf{k}}{4\pi^3} \delta\left(E-E\left(\mathbf{k}\right)\right) = \int_{S\left(E\right)} \frac{dS}{4\pi^3}\frac{1}{\left|\nabla E\left(\mathbf{k}\right)\right|}$$
See Ashcroft and Mermin equations 8.57 and 8.63 (and the surrounding section). You're basically quoting the right-most part of the equation; $\left|\nabla E\left(\mathbf{k}\right)\right|$ is proportional to the group velocity: $\mathbf{v}\left(E\right) = \frac{1}{\hbar} \nabla E\left(\mathbf{k}\right)$ (Ashcroft and Mermin equation 8.51), which follows from the fact that $E = \hbar \omega$.
However, the group velocity is not constant. The Fermi velocity is the group velocity only for certain $\mathbf{k}$ --- not for all $\mathbf{k}$. So, if you want to use the equation, you need to have the group velocity be a function of $\mathbf{k}$.
Note that for phonons, you can sometimes approximate the group velocity as being constant because its dispersion relation is roughly linear near $\mathbf{k} = 0$. Except in special situations (e.g. graphene), Bloch electrons have mass, so their dispersion relation will not be linear, and their group velocity will not be constant.
EDIT:
* Those equations are for electrons, and the most common convention for electrons is to include an extra factor of 2 because the electron states are (normally) two-fold degenerate. Sometimes this degeneracy factor is kept separate from the density of states because the degeneracy is not set in stone. E.g. you can get rid of it by applying a magnetic field.
A: I thought about this : 
Density of states =  $\left( \left(\frac{L}{2\pi} \right )^3  \int \frac{dS_\omega}{v_g}\right)$ with $v_g = \frac{d\epsilon}{dk}$, with $\epsilon = \hbar^2k^2/2m$ then is $v_g = \hbar^2k/2 = \hbar^2/m \sqrt{2\epsilon m /\hbar^2}$ and thus for the density of states (with $\int d_S = 4\pi k^2$ ) I get to $v/2\pi^2 * 2m/\hbar^2 \sqrt{\epsilon}* \sqrt{m/2\epsilon\hbar^2} $. Which is the result without a factor 2 wrong, is there anyone who sees my fault?
