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Density of Energy Levels - Strange Summation

  1. Nov 24, 2015 #1
    1. The problem statement, all variables and given/known data
    *This is not my whole problem, I am only stuck on how to interpret one part of the question. Put simply, I want to find the expression for the density of energy levels in a given energy band per unit volume (in some crystal structure). Say I have an infinitesimal interval of energy levels ## \Delta##, I would like to find the number of energy levels per ##\Delta## and unit volume.

    2. Relevant equations
    I am given the expression $$ g_n(E) = \frac{1}{ V \Delta } \sum_{\mathbf{k}, E < E_{n\mathbf{k}} < E + \Delta} 1 $$ which I write as
    $$ g_n(E) = \frac{1}{ V \Delta } \sum_{\mathbf{k}} \sum_{E < E_{n\mathbf{k}} < E + \Delta} $$

    3. The attempt at a solution
    Using the periodic boundary conditions I can convert the sum over ##\mathbf{k}## into an integral in reciprocal space,
    $$\sum_{\mathbf{k}} \rightarrow \frac{V}{(2 \pi)^3} \int d^3k $$
    at which point I am stuck on how to interpret the second sum with index ## E < E_{n\mathbf{k}} < E + \Delta ##. I was thinking about taking the limit that ## \Delta \rightarrow 0 ## the sum would become an integral,
    $$ \frac{1}{\Delta} \sum_{E < E_{n\mathbf{k}} < E + \Delta} \rightarrow \int dE $$
    but this doesn't seem quite right. I am stuck on how to deal with that sum, and what the bounds of the integral should be.
    Any help with that step would be much appreciated!
     
  2. jcsd
  3. Nov 29, 2015 #2
    Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
     
  4. Nov 30, 2015 #3

    mfb

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    Staff: Mentor

    E and k are related, I don't think you can separate them like that. The number of energy levels within that infinitesimal energy interval will depend on k, and it is some external input you'll need.
     
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