## Large volume Fermi gas density

Posted by peeterjoot on September 5, 2013

[Click here for a PDF of this post with nicer formatting]

Here’s part of a problem from our final exam. I’d intended to redo the whole exam over the summer, but focused my summer study on world events instead. Perhaps I’ll end up eventually doing this, but for now I’ll just post this first part.

## Question: Large volume Fermi gas density (2013 final exam pr 1)

Write down the expression for the grand canonical partition function of an ideal three-dimensional Fermi gas with atoms having mass at a temperature and a chemical potential (or equivalently a fugacity ). Consider the high temperature “classical limit” of this ideal gas, where and one gets an effective Boltzmann distribution, and obtain the equation for the density of the particles

by converting momentum sums into integrals. Invert this relationship to find the chemical potential as a function of the density .

Hint: In the limit of a large volume :

## Answer

Since it was specified incorrectly in the original problem, let’s start off by verifing the expression for the number of particles (and hence the number density)

Moving on to the problem, we’ve seen that the Fermion grand canonical partition function can be written

so that our density is

In the high temperature classical limit, where we have

This is

where

Inverting for we have

or

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