I solving an integral over density of states for electrons.

[mattek1979]

Homework Statement

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
$f(\epsilon)= \frac{1}{|\epsilon |}$ for $\epsilon_{min} \leq \epsilon < 0$ and 0 elsewhere

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

Homework Equations

I assume integrating

$dN(\epsilon)=\bar{n}(\epsilon)f(\epsilon)d\epsilon$

using
$\bar{n }=\frac{1}{e^{-\beta(\epsilon-\mu)}+1}$

and the above

$f(\epsilon)=\frac{1}{|\epsilon|}$
is the way to proceed.

The Attempt at a Solution

The integral I seek to solve is

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

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

 

[mark726]

Hello Mathias,

It looks like you are on the right track with your approach. To solve this integral, you may want to consider using a substitution of variables. Let u = e^(-beta(epsilon-mu)) and see if that helps you simplify the integral. You may also want to try using integration by parts to see if that leads you to a solution. Additionally, you can try looking up tables of integrals or using software such as Mathematica to help you with the integration. I hope this helps and good luck with your studies!