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Energy and chemical potential of a Fermi gas

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Alpha in the equation should be replaced by minus alpha. This is called the fugacity, which is related to the chemical potential according to:

alpha = - beta mu

So, the equation for the occupation number can be written as:

n_s = g_s/{1 + exp[beta (e_s - mu)]} (1)

Here g_s is the degeneracy of the energy level, beta is the temperature parameter beta = 1/(kT) and mu is the chemical potential.

The distribution at zero temperature can be obtained by letting beta go to infinity. If e_s < mu, then the exponential will be zero so n_s = g_s, while for e_s > mu, the exponential will tend to infinity, and therefore n_s = 0. So, all the states with energy below an energy of mu are occupied. The chemical potential mu at zero temperature is also called the Fermi energy

In case of electrons g_s = 2 because the electron has two spin states. In a volume of V in configuration space and volume Vp in momentum space there are:

2 V Vp/h^3

states available for electrons. The factor 2 comes from the spin degeneracy (one usually does ...

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