Phonon density of states and density of states of free electrons

chikchok
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Homework Statement
Compare phonon DOS and DOS of free electrons in 1D and 2D
Relevant Equations
D(E)1d=(1/L)dN/dE
D(E)2d=1/A)dN/dE for electrons
In the following pdf I tried to calculate the density of states of free electrons and phonons. First, I found the free electron DOS in 1D, it turns to be proportional to (energy)^(-1/2) and in 2D it is constant. However, I am not sure I found the DOS for phonons in the second part of the solution. Because the homework said to compare two DOS, I thought phonon DOS needs to be in terms of energy D(E) and not frequency w D(w). But I suspect it is wrong. Can phonon density of states be in terms of energy? If so, how to find it? And if not, should I find it trough equation D(w)dw?
 

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This is a pretty broad question and I would suggest looking at Kittel or Ashcroft and Mermin for the phonon part. I believe the question wants you to see that the dispersion relation for phonons can resemble that of photons. There is a lot of good physics in this question and it is worth some effort.
 
I`ve looked up the DOS of a phonon in Kittel`s book and in 1D DOS is 1/pi*vg (vg as a group velocity dw/dk) and in 2D it`s k/2pivg . Both of them have no relationship with energy. That is why I was wondering if there is a way to calculate the DOS as a function of energy.
 
$$E=\hbar \omega$$
 

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