# Fermi Gas Model / Fermi energy and momentum

• mmisk
In summary, the conversation is about a person seeking help with calculating the Fermi momentum and energy using experimental values. They mention that they seem to be missing something and are struggling with the calculations. They mention a possible geometrical consideration and a math error in their calculations.

#### mmisk

Hi,

Hope somebody can help - I seem to be missing the obvious and bonking

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

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!

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.

## 1. What is the Fermi gas model?

The Fermi gas model is a theoretical model used to describe the behavior of a gas of non-interacting particles at very low temperatures. It assumes that the particles have no interactions with each other and are subject to the laws of quantum mechanics.

## 2. What is Fermi energy?

Fermi energy is the highest energy level occupied by a particle at absolute zero temperature in a system described by the Fermi gas model. It is a measure of the maximum energy that a particle can have in a system and is often used to describe the energy of electrons in metals.

## 3. How is Fermi energy related to Fermi momentum?

According to the Fermi gas model, Fermi momentum is the momentum of a particle at the Fermi energy level. It is equal to the product of Fermi energy and the speed of light, divided by Planck's constant.

## 4. What is the significance of Fermi energy and momentum?

Fermi energy and momentum are important quantities in the study of condensed matter systems, particularly in understanding the behavior of electrons in metals. They help to explain phenomena such as electrical conductivity and heat capacity in these systems.

## 5. How does temperature affect Fermi energy and momentum?

In the Fermi gas model, as temperature increases, the Fermi energy level and Fermi momentum also increase. This is because at higher temperatures, more energy is available for particles to occupy higher energy levels, resulting in a larger Fermi energy and momentum.