Ground state energy of free electron fermi gas

In summary, the ground state energy of a free electron fermi gas is not just 2 times the integral of \frac{\hbar^2 k^2}{2m} 3k^2 dk, as there are a few errors in the formula. The correct result can be obtained by taking into account the volume of k-space each state occupies and converting the 3D integration into a 1D integration only in the radial direction. The total particle number can also be found in a similar way. Ultimately, the ground state energy can be written in terms of the number of particles and the fermi energy.
  • #1
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Can someone explain to me why the ground state energy of a free electron fermi gas is not just:

[tex]
E = 2 \int_0^{k_f} \frac{\hbar^2 k^2}{2m} 3k^2 dk
[/tex]

Where the factor of two is due to the fact that there are two electron states for each value of k. The idea is to add up all the energies of all states within the fermi sphere, but it does not give the correct result which is:

[tex]
E = \frac{3}{5} N k_f
[/tex]

Where N is the number of electrons, and [tex]k_f[/tex] is the radius of the fermi sphere. What am I doing wrong? If you need more info please let me know.

Thanks in advance
René
 
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  • #2
What is the integral defining N? (ie, the integral over the fermi sphere of dN)
 
  • #3
There are a few things wrong with your formulas:

Your total energy integral should look like this:

[tex]

E = 2 \int_0^{k_f} \frac{\hbar^2 k^2}{2m} \frac{V}{(2\pi)^3} 4 \pi k^2 dk

[/tex]

The [tex]\frac{V}{(2\pi)^3}[/tex] is necessary because you need to account for the volume of k-space each state occupies. The [tex] 4\pi [/tex] comes from converting the 3D integration in spherical coordinates into a 1D integration only in the radial direction (dk).

You'll see that you can also find the total particle number the same way, without the [tex] \frac{\hbar^2 k^2}{2m}[/tex]:

[tex]

N = 2 \int_0^{k_f} \frac{V}{(2\pi)^3} 4 \pi k^2 dk

[/tex]

After performing the integrations, you can write E in terms of N, and you'll find that:

[tex]

E = \frac{3}{5} N \frac{\hbar^2}{2m} (k_f)^2 = \frac{3}{5} N E_f

[/tex]

Hope that helps.
 
  • #4
Thanks very much for this solution.
 
  • #5


The ground state energy of a free electron fermi gas is not simply the sum of all the energies of the states within the fermi sphere because of the Pauli exclusion principle. This principle states that no two electrons can occupy the same quantum state simultaneously. Therefore, in a fermi gas, as the electron states fill up, the energy levels become increasingly crowded and it becomes more difficult for electrons to occupy higher energy states. This results in a lowering of the overall energy of the system.

In your calculation, you are assuming that all the electrons are at the same energy level, which is not the case due to the Pauli exclusion principle. The correct expression for the ground state energy takes into account the distribution of electrons among energy levels, which is given by the Fermi-Dirac distribution. This distribution shows that the energy levels are not evenly spaced, with more electrons occupying lower energy levels and fewer electrons occupying higher energy levels.

Therefore, the correct expression for the ground state energy of a free electron fermi gas is:

E = \frac{3}{5} N \epsilon_f

Where N is the number of electrons and \epsilon_f is the fermi energy, which is related to the fermi wavevector k_f by:

\epsilon_f = \frac{\hbar^2 k_f^2}{2m}

This expression takes into account the distribution of electrons among energy levels and gives the correct result for the ground state energy. I hope this helps to clarify your understanding.
 

1. What is the ground state energy of a free electron fermi gas?

The ground state energy of a free electron fermi gas is the lowest possible energy state that a collection of free electrons can have at absolute zero temperature. It is determined by the Fermi energy, which is the maximum energy that an electron in the gas can have.

2. How is the ground state energy of a free electron fermi gas calculated?

The ground state energy of a free electron fermi gas can be calculated using the Fermi-Dirac distribution, which takes into account the number of energy levels available to the electrons and the probability of an electron occupying each level. The equation for the ground state energy is E = (3/5)EF, where EF is the Fermi energy.

3. What factors affect the ground state energy of a free electron fermi gas?

The ground state energy of a free electron fermi gas is primarily affected by the density of the gas and the mass of the electrons. As the density of the gas increases, the Fermi energy also increases, resulting in a higher ground state energy. Similarly, a decrease in electron mass leads to a decrease in the ground state energy.

4. How does the ground state energy of a free electron fermi gas relate to other properties of the gas?

The ground state energy of a free electron fermi gas is closely related to other properties of the gas, such as the heat capacity and electrical conductivity. As the ground state energy increases, the heat capacity and electrical conductivity also increase, as more energy is available for the electrons to move and transfer heat.

5. Why is the ground state energy of a free electron fermi gas important in materials science?

The ground state energy of a free electron fermi gas is an important concept in materials science because it helps to explain the behavior of metals and other conductive materials. It also plays a role in understanding phenomena such as superconductivity, where the ground state energy is crucial in determining the properties of the material at low temperatures.

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