What is the Definition and Mathematical Explanation of Density of States?

[Greg Bernhardt]

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

<br /> g(E)=\sum_{s}\delta(E-E_s)<br />
<br /> N=\int dE g(E)<br />

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 \mathcal{V}, the density of states for a given wavevector \mathbf{k} (rather than energy) is a constant:
<br /> g_{\mathbf{k}}=\frac{\mathcal{V}}{{(2\pi)}^3}.<br />
The above equation is the basis for the well-known replacement
<br /> \sum_{\mathbf{k}}(\ldots)\to\int \mathcal{V}\frac{d^3 k}{{(2\pi)}^3}(\ldots)<br />

Extended explanation

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

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

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

* 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!

 

[AaronS]

Thanks for the overview of density states!