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What are density of states

  1. Jul 24, 2014 #1
    Definition/Summary

    This term most commonly refers to the number of quantum states having energy within a given small energy interval divided by that interval.

    Equations

    [tex]
    g(E)=\sum_{s}\delta(E-E_s)
    [/tex]
    [tex]
    N=\int dE g(E)
    [/tex]

    The "density of states" need not (but it most often does) refer to states per energy interval. For example, for free particles in a box of volume [itex]\mathcal{V}[/itex], the density of states for a given wavevector [itex]\mathbf{k}[/itex] (rather than energy) is a constant:
    [tex]
    g_{\mathbf{k}}=\frac{\mathcal{V}}{{(2\pi)}^3}.
    [/tex]
    The above equation is the basis for the well-known replacement
    [tex]
    \sum_{\mathbf{k}}(\ldots)\to\int \mathcal{V}\frac{d^3 k}{{(2\pi)}^3}(\ldots)
    [/tex]

    Extended explanation

    The density of states
    [tex]
    g_{\mathbf{k}}=\frac{\mathcal{V}}{{(2\pi)}^3}\;,
    [/tex]
    results from applying periodic boundry conditions to free waves in a box of volume [itex]\mathcal{V}[/itex] and counting. Thus
    [tex]
    \delta N = d^3 k g_{\bf k}=d^3 k\frac{\mathcal{V}}{{(2\pi)}^3}\;.
    [/tex]

    If the energy E only depends on the magnitude of [itex]\mathbf{k}[/itex], E=E(k), then we may also write
    [tex]
    \delta N = d k k^2 \frac{4\pi \mathcal{V}}{{(2\pi)}^3}
    =
    \frac{4\pi\mathcal{V}}{{(2\pi)}^3}dE \frac{k^2}{v}\equiv dE g(E)\;,
    [/tex]
    where
    [tex]
    v=\frac{dE}{dk}\;,
    [/tex]
    is the velocity.

    For the case where momentum is carried by particles with an effective mass [itex]m^*[/itex] we have
    [tex]
    k=m^*v\;,
    [/tex]
    and
    [tex]
    g(E)=\frac{4\pi \mathcal{V}}{{(2\pi)}^3}m^*\sqrt{2 E m^*}\;.
    [/tex]

    * This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!
     
  2. jcsd
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