Solving Fermi Nuclear Model Homework: A & B

[Sebastian]

Homework Statement

This question is about the Fermi nuclear model (see Das & Ferbel page 56 or http://www.e12.physik.tu-muenchen.de/stud/vorlesungen/fabbietti/WS2007/Lecture3.pdf" ). Let the nucleon mass be $m$, the number of nucleons be $A$ and the number of protons be $Z$. Assume a hypothetical situation where the only difference between the proton and the neutron is that the proton has magnetic moment $\mu_p$ and the neutron has magnetic moment $\mu_n$. The nucleus is placed in a magnetic field $B$. Assume also that the protons and neutrons may align the directions of their magnetic moments such that energy is minimized.

A. Find a formula connecting the Fermi energy of the protons with that of the neutrons.
B. Find a formula connecting the Fermi momentum of the protons with that of the neutrons.

Homework Equations

See link above.

The Attempt at a Solution

I think the binding energy of a nucleon should be $-\mu B$. However, I don't see how to continue from here.

PS: I will appreciate a direct answer instead of hints. This is from the material for an exam I'm taking tomorrow, and I'm quite stressed about not being able to solve it. Thanks in advance! :)

 

[mark726]

Thank you for your question. I understand that you are having difficulty solving this problem for your exam. Let me try to help you by providing a direct answer.

A. The Fermi energy of a proton and a neutron can be expressed as:

Ep = (p^2/2m) + \mu_pB
En = (p^2/2m) + \mu_nB

Where p is the momentum of the nucleon, m is the nucleon mass, \mu_p and \mu_n are the magnetic moments of the proton and neutron respectively, and B is the magnetic field.

Since we are assuming that the protons and neutrons can align their magnetic moments to minimize energy, we can equate the Fermi energies of the two particles:

Ep = En
(p^2/2m) + \mu_pB = (p^2/2m) + \mu_nB

Solving for p, we get:

p = \sqrt{\frac{2m(\mu_p-\mu_n)B}{1/m}}

Substituting the values for \mu_p and \mu_n from the given information, we get:

p = \sqrt{\frac{2m(\mu_p-\mu_n)B}{1/m}} = \sqrt{\frac{2m(\mu_p-\mu_n)B}{1/m}} = \sqrt{2m(A-Z)B}

Thus, the formula connecting the Fermi energy of the protons with that of the neutrons is:

Ep = En = \frac{p^2}{2m} = \frac{2m(A-Z)B}{2m} = (A-Z)B

B. The Fermi momentum of a proton and a neutron can be expressed as:

Pp = \sqrt{2mEp}
Pn = \sqrt{2mEn}

Substituting the values of Ep and En from part A, we get:

Pp = Pn = \sqrt{2m(A-Z)B}

Thus, the formula connecting the Fermi momentum of the protons with that of the neutrons is:

Pp = Pn = \sqrt{2m(A-Z)B}

I hope this helps. Best of luck with your exam!