Phonon density of states and density of states of free electrons

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SUMMARY

The discussion focuses on calculating the density of states (DOS) for free electrons and phonons. The free electron DOS is established as proportional to (energy)^(-1/2) in one dimension and constant in two dimensions. The participant expresses uncertainty about representing phonon DOS in terms of energy rather than frequency, questioning the validity of using D(E) versus D(w). References to Kittel and Ashcroft and Mermin are suggested for further understanding of phonon DOS calculations, specifically noting that in 1D and 2D, phonon DOS is expressed as 1/pi*vg and k/2pivg, respectively, both lacking direct energy relationships.

PREREQUISITES
  • Understanding of density of states (DOS) concepts
  • Familiarity with phonon dispersion relations
  • Knowledge of group velocity (vg) in solid-state physics
  • Basic principles of quantum mechanics related to energy and frequency
NEXT STEPS
  • Research the relationship between phonon density of states and energy using the equation D(E) = D(w) * dw/dE
  • Study the dispersion relations for phonons in Kittel's "Introduction to Solid State Physics"
  • Examine Ashcroft and Mermin's treatment of phonon DOS in solid materials
  • Explore advanced topics in solid-state physics regarding the implications of DOS on material properties
USEFUL FOR

Students and researchers in solid-state physics, particularly those studying the properties of materials and the behavior of electrons and phonons in various dimensions.

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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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