Questions on Statistical Physics

[RyanA1084]

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

 

[anathema]

Hello Ryan,

It seems like you've made a good start on both of these problems. I'll try to offer some guidance and hopefully help you figure out where the book's answers are coming from.

1) For the first problem, you're on the right track using the Fermi energy equation for electrons. However, since protons and neutrons are distinguishable particles, you need to take into account the fact that there are 10 protons and 12 neutrons in the nucleus. This means that the total number of particles (N) in the equation should be 22, and you'll need to calculate the Fermi energy separately for protons and neutrons.

Also, instead of using the volume of a sphere with the given radius, you should use the volume of a nucleus with that radius. This will give you different values for N/V, and ultimately different values for the Fermi energy.

For the average energy, you can use the formula <E> = (3/5)Ef, which is derived from the Fermi energy equation. This should give you the correct answers listed in the back of the book.

2) For the second problem, you're correct in using the condition that the sum of the probabilities for each energy state must equal 1. This will give you the correct value for C.

For part (b), you're on the right track using the formula for <E>. To find g(E), you can use the fact that for a two-state system, the degeneracy of the ground state (g1) is 1 and the degeneracy of the excited state (g2) is N-1. This should help you calculate n(E) and ultimately <E>.

For part (c), you'll need to use the formula for the heat capacity of a system with two energy states, which is C_v = Nk(epsilon/kT)^2*(e^(-epsilon/kT)/(1+e^(-epsilon/kT))^2). This will give you the correct answer listed in the book.

I hope this helps! Let me know if you have any other questions or if you need further clarification. Good luck!

 

[wais]

Hi Ryan,

For your first question, it seems like you are on the right track. The Fermi energy equation you are using is correct, but there are a few things to consider. First, the N/V term should be the total number of particles (protons + neutrons) divided by the volume of the nucleus. So for 22Ne, it would be 22 particles divided by the volume of a sphere with radius 3.1x10^-15 m. Additionally, the factor of (3/8pi)^(2/3) should be included in the h^2/2m term. This should give you the correct values for the Fermi energy for protons and neutrons.

For the second part of the question, the average energy of the nucleus can be found by summing the energies of each particle (protons and neutrons) in the nucleus. Since there are 10 protons and 12 neutrons, the total average energy would be the sum of 10 times the average energy for protons and 12 times the average energy for neutrons. This should give you the values in the back of the book.

For your second question, your approach for finding C is correct. For part b, you can use the equation for <E> that you mentioned, but keep in mind that the factor of n(E) is just the number of particles in each energy state, which in this case is just 1 for both E1 and E2. So you can simplify the integral to <E>=(1/N)(0*1+epsilon*1)=epsilon/N. This should give you the correct result for <E> as T-->0 and T-->infinity.

For part c, you can use the definition of heat capacity (C_v=d<E>/dT) to find the expression given in the problem. As for the epsilon/kT and epsilon*kT, it looks like a typo in the problem. It should be epsilon/kT in both cases.

I hope this helps! Let me know if you have any further questions. Good luck with your homework!