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# Fermions Question - Statistics! Watch

1. Here's the question -

What is the chemical potential, mu (in units of hbar.w), at T = 0 K of 46 6Li atoms with spin ½ contained in a 3D harmonic potential well with energy levels, Ei = (i + 3/2)hbar.w and degeneracies gi = 1/2(i +1)(i + 2).

Basically, I know that the chemical potential should be equal to the fermi energy at T=0K but I really don't know how to use Ei and gi to get there. Literally no idea. Plus, we haven't discussed calculating the fermi energy in class much yet (we did last year but not everyone took that course). Help please!!?

So far in the question I have derived the f(E) distribution (the occupation probabilities) for fermions, and we are given the statistical weight of a fermi gas.

Thanks! I will rep! x
2. BUMP
3. I'm not sure if this is the neatest way to do it, but if you know the energy and degeneracy of each level you can find the partition function, and hence the average energy, and then you can use the relation between the average energy and the Fermi energy for a 3D case.

more hints
Spoiler:
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Partition function Z = sum over i (gi x exp(-beta * Ei))
Average energy = - partial derivative of ln Z w.r.t beta
and you should be able to derive the average energy in terms of the Fermi energy by considering the Fermi sphere in k-space.

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Updated: December 4, 2010
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