Fermi energy for a Fermion gas with a multiplicity function

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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Fermi energy for arbitrary multiplicity
I ran across the following problem :

Statement:

Consider a gas of fermions and suppose that each energy level has a multiplicity of . What is the Fermi energy and the average energy of this gas when ?

My attempt:

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



Usually if the system has a fixed degeneracy, say only the spin degeneracy , one can write the total number of particles as an integral over :



One can than find the Fermi energy in the limit .

But this is not the case when ... Any hints on how to do this ?
 
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  • #2
The generic equation for the total number of fermions is

where

is the Fermi-Dirac distribution and is the density of states. The degeneracy factor is part of the density of states, so it will stay inside the integral if is dependent on (so dependent on ).

You should however be looking at the equation for the average energy. In the limit , the energy levels can be considered continuous and an integral similar to the one above is obtained.
 
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