Questions on Statistical Physics

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Hi everyone, I have two questions from my latest homework set that are driving me nuts, so here goes:

1) "Recalling that the Fermi-Dirac distribution function applies to all fermions, including protons and neutrons, each of which have spin 1/2, consider a nucleus of 22Ne consisting of 10 protons and 12 neutrons. Protons are distinguishable from neutrons, so two of each particle (spin up, spin down) can be put into each energy state. Assuming that the radius of the 22Ne nucleus is 3.1X10^-15 m, estimate the Fermi energy and the average energy of the nucleus in 22Ne. Express your results in MeV. Do the results seem reasonable?"

For this problem the best I can come up with is to use the fermi energy (Ef) equation for electrons derived in the book. It seems like it should work for protons and neutrons as well since they are also fermions with spin 1/2. The equation is:

Ef=(h^2/2m)(3N/8piV)^(2/3)

I know the Ef has to be calculated separately for protons and neutrons, so I've been taking N/V to be the number of protons or neutrons divided by the volume of a sphere with the given radius. I've been getting 36.9 MeV for protons and 41.68 MeV for neutrons.

The answer is in the back of the book as:
Ef(protons)=516MeV <E>=310MeV
Ef(neutrons)=742MeV <E>=445MeV

I can't for the life of me figure out where those numbers come from!


2) Consider a system of N particles which has only two possible energy states, E1=0 and E2=epsilon. The distribution function is f_i=Ce^(-E_i*kT)
a)What is C for this case?
b) Compute the average energy and show that <E>-->0 as T-->0 and <E>-->epsilon/2 as T-->infinity.
c) show that the heat capacity is

C_v=Nk(epsilon/kT)^2*(e^(-epsilon/kT)/(1+e^(-epsilon/kT))^2)

d) Sketch Cv versus T.

This one seems like the sort where once the first step is correct the rest should fall into place. My best guess as to how to find C is to use the condition that the sum over the probabilities for each energy state must equal 1, so:

f=C(e^0 + e^(-epsilon*kT))=1

Which gives C=1/(1+e^(-epsilon*kT))

If that's right, which it may well not be, then the most applicable equation I can find for <E> is:

<E>=(1/N) integral(0 to infinity) E*n(E)dE

Problem is, n(E) is g(E)*f(E) and I don't know how to find g(E)!

I'm also a bit worried by the fact that the heat capacity equation has epsilon/kT and the original has epsilon*kT. Not sure how that gets switched around...

Sorry for the long post, just though I should say what I've tried so far.
Any help on either of these problems would be much appreciated!

Thanks in advance,
Ryan
 

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