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I need help solving an integral over density of states for electrons.

  1. Aug 9, 2011 #1
    1. The problem statement, all variables and given/known data
    I am trying to retake an old course in statistical mechanics but run into integrals that i simply have forgotten how to solve.

    Given an denstiry of states such that
    [itex]f(\epsilon)= \frac{1}{|\epsilon |}[/itex] for [itex]\epsilon_{min} \leq \epsilon < 0 [/itex] and 0 elsewhere

    Using the mean occupation number for a fermi-dirac distribution, I am supposed to find the fermi energy for N electrons.


    2. Relevant equations

    I assume integrating

    [itex]dN(\epsilon)=\bar{n}(\epsilon)f(\epsilon)d\epsilon[/itex]

    using
    [itex]\bar{n
    }=\frac{1}{e^{-\beta(\epsilon-\mu)}+1}[/itex]

    and the above

    [itex]f(\epsilon)=\frac{1}{|\epsilon|}[/itex]
    is the way to proceed.

    3. The attempt at a solution

    The integral I seek to solve is

    N=[itex]\int^{\epsilon_{min}}_{0}\frac{1}{|\epsilon|}\frac{1}{e^{-\beta(\epsilon-\mu)}+1}d\epsilon[/itex]

    and I simply can't figure out if I need to do a subtitution of integration variables or if i am missing some other nifty technique.

    All help appreciated

    Sincerely
    Mathias Kristoffersson
     
    Last edited: Aug 9, 2011
  2. jcsd
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