Fermi Gas Model / Fermi energy and momentum

[mmisk]

Hi,

Hope somebody can help - I seem to be missing the obvious and bonking
my head on the wall.

In Wong and also in Feshbach/deShalit, they calculate the Fermi momentum,
Kf using the experimental value of Rho-0 and come up with Kf=1.3 fm-1

So far so good.

Where I stumble is in their calculation of the Fermi Energy, Ef. For
instance, Wong has Ef=(hbar*Kf)^2/2M, M=nucleon mass, giving
Ef about 37 Mev. But when I plug in the appropriate values I get an
Ef of 11.2 ish Mev with M = M neutron ~939.5 Mev/c^2.

I'm sure I'm missing the obvious 'adjustment' to the privously
calculated Kf, and suspect it is a geometrical consideration as I'm
off roughly pi.

Thanks

Mike

 

[mmisk]

er..ah.. stupid math error

multiplying by 1.xx ^ +15 and dividing by 1.xx ^ -15 give answers to same order of magnitude...just that one is 11.7 and the other is 37 I've been wearing my dunce hat for a few days now!

 

[sir-pinski]

Hi Mike,

The Fermi gas model is a simplified theoretical model that describes a system of non-interacting fermions (particles with half-integer spin) at low temperatures. It assumes that the fermions are confined to a finite volume and that their energy states are quantized. The Fermi energy, Ef, is defined as the energy of the highest occupied state at zero temperature. This means that all states with energy less than Ef are filled with fermions, while all states with energy greater than Ef are empty.

The Fermi momentum, Kf, is related to the Fermi energy through the following equation: Ef = (hbar*Kf)^2/2M, where hbar is the reduced Planck's constant and M is the mass of the fermion. This equation can also be written as Kf = (3*pi^2*n)^1/3, where n is the number density of fermions in the system.

In the case of a Fermi gas of nucleons (neutrons and protons), the mass M in the equation for Ef should be the average mass of a nucleon, which is approximately 939.5 MeV/c^2. Using this value, we can calculate the Fermi energy to be around 37 MeV, which is consistent with the value you obtained from Wong and Feshbach/deShalit.

It is possible that the discrepancy you are seeing is due to using the mass of a neutron (939.5 MeV/c^2) instead of the average mass of a nucleon. Another potential source of error could be in the experimental value of Rho-0 that was used to calculate Kf. However, without knowing the specific values you used in your calculation, it is difficult to pinpoint the exact cause of the discrepancy.

I hope this helps clarify the concept of the Fermi gas model and the relationship between Fermi energy and momentum. Keep in mind that this is a simplified model and may not accurately describe more complex systems, but it can provide a useful starting point for understanding the behavior of fermions in certain situations.