Fermi energy for a Fermion gas with a multiplicity function ##g_n##

In summary, the problem discusses a gas of N fermions with energy levels of varying degeneracy. The Fermi energy and average energy of the gas are sought as N approaches infinity. The equations for the average occupation number and total number of particles are given, and the Fermi energy can be found in the limit of T approaching 0. However, the degeneracy factor must be taken into consideration and the equation for average energy should be used instead.
  • #1
phos19
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TL;DR Summary
Fermi energy for arbitrary multiplicity
I ran across the following problem :

Statement:

Consider a gas of ## N ## fermions and suppose that each energy level ## \varepsilon_n## has a multiplicity of ## g_n = (n+1)^2 ##. What is the Fermi energy and the average energy of this gas when ## N \rightarrow \infty## ?

My attempt:

The average occupation number for a state of the ##n##th level is:

$$\langle N_n \rangle = \dfrac{1}{ e^{\beta(\varepsilon_n + \mu)} + 1 }$$

Usually if the system has a fixed degeneracy, say only the spin degeneracy ##g = 2s +1## , one can write the total number of particles ##N## as an integral over ##\vec{p}##:

$$
N = \sum_n \langle N_n \rangle = \dfrac{gV}{h^3} \int d^3 p \ \dfrac{1}{ e^{\beta(\varepsilon_p + \mu)} + 1 }
$$

One can than find the Fermi energy in the limit ##T \rightarrow 0##.

But this is not the case when ##g = g(n)##... Any hints on how to do this ?
 
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  • #2
The generic equation for the total number of fermions is
$$
N = \int_0^\infty f(\varepsilon) D(\varepsilon) d\varepsilon
$$
where
$$
f(\varepsilon) = \frac{1}{e^{\beta(\varepsilon + \mu)} + 1}
$$
is the Fermi-Dirac distribution and ##D(\varepsilon)## is the density of states. The degeneracy factor ##g## is part of the density of states, so it will stay inside the integral if is dependent on ##n## (so dependent on ##\varepsilon##).

You should however be looking at the equation for the average energy. In the limit ##N \rightarrow \infty##, the energy levels can be considered continuous and an integral similar to the one above is obtained.
 
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What is Fermi energy?

Fermi energy is the energy level at which all the lowest energy states of a system are filled by fermions, particles with half-integer spin. It is a key concept in understanding the behavior of fermions in a system.

What is a Fermion gas?

A Fermion gas is a system of particles that obey the Pauli exclusion principle, meaning that no two particles can occupy the same quantum state. Examples of Fermion gases include electrons in a metal and neutrons in a neutron star.

What is the multiplicity function ##g_n##?

The multiplicity function ##g_n## is a mathematical function that describes the number of ways that ##n## particles can be distributed among different energy levels. It is used to calculate the entropy of a system and plays a crucial role in understanding the behavior of Fermi gases.

How is Fermi energy related to the multiplicity function ##g_n##?

Fermi energy is directly related to the multiplicity function ##g_n##. In fact, the Fermi energy can be calculated by finding the energy level at which the multiplicity function reaches its maximum value. This energy level is known as the Fermi level.

What are the applications of studying Fermi energy for a Fermion gas with a multiplicity function ##g_n##?

Studying Fermi energy and the multiplicity function ##g_n## has many applications in physics and engineering. It helps us understand the behavior of electrons in metals, which is crucial for developing new materials and technologies. It also has applications in astrophysics, such as in understanding the properties of neutron stars.

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