# Density of states (solid state)

calculate the density of states and average energy for an elctron gas in 1d,2d and 3d

I know the number of states is

$$N= \int_{0}^{infinity} g(e)f(e) de$$

$$and E = \int_{0}^{infinity} g(e)ef(e) de$$

$$and g(e) =dN/de$$

## Answers and Replies

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That doesn't really make sense as you've written it, since N is just a number, not a function of e. The formula you want to use for N(e) is:

$$N(\epsilon) = \int_0^\epsilon d\epsilon' g(\epsilon')$$

Then from the fundamental theorem of calculus, it's clear dN/de=g(e). Of course, this is not a helpful definition when you want to use N to compute g. But there's another interpretation of the integral on the RHS: it's just the total number of states whose energy is less than e. In other words, in 3D, it's the number of states inside the fermi-sphere corresponding to energy e. In 2D, you'd have a fermi disk, and so on. Can you see how to count the states inside such a fermi sphere?