A Fermi gas is a gas consisting of fermions. We will investigate a few of its properties and applications here.
Fermi-Dirac Distribution Function
Consider a system that is a single orbital for a fermion. Remember that only 0 or 1 fermions may be in the orbital. The Gibbs sum is easy to calculate then; in fact, we already solved this problem in a previous problem set: Z G = 1 + λe -/τ . We can then directly calculate f , remembering that it is equivalent to < N > .
Notice that in the classical limit, f must be much less than 1, so the exponential must be large compared to one. An expansion yields the classical distribution function we derived before.
We are concerned with the case where the gas is dense compared to the quantum concentration. The gas is called degenerate in this regime, as opposed to in the classical regime already discussed.
The chemical potential in general is a function of the temperature. However, we define the chemical potential at a temperature of zero for a fermi gas to be μ(τ = 0) = where is called the Fermi energy.
The significance of the Fermi energy is that for τ , the orbitals of energy ≤ are completely occupied and the orbitals above the Fermi energy are completely vacant.
This should make sense. A system wants to minimize its total energy, and the fermions would pack into the lowest energy states first. If the temperature is low, there is little thermal excitation to promote any fermions to orbitals with higher energy.
The Ground State of a Fermi Gas
We can solve for the Fermi energy by setting the total number of particles below that energy equal to the total number of particles in the system. We obtain:
We use the term "ground state" to refer to the state in which no fermions are excited to higher energy states beyond the Fermi energy. We can calculate the energy of the ground state by summing up the energies of the orbitals below the Fermi energy, to obtain:
We can go through and calculate all of the other relevant quantities just as we did for the ideal gas.
The Fermi gas appears throughout any study of physics. Electrons form a Fermi gas. The electrons in a metal, the "sea of electrons", act as a Fermi gas. In astrophysics, white dwarf stars are prevented from collapsing upon themselves by the pressure of the Fermi gas and the resistance it gives to having its orbitals pushed together.
A Bose gas is a gas consisting of bosons. We will treat the topic briefly as above with the Fermi gas.
Bose-Einstein Distribution Function
An orbital can support any number of bosons, which fundamentally changes the Gibbs Sum and thus the distribution function. Instead of summing over N = 0, 1 we must sum over all N . The final result is:
Since there is no restriction on the number of particles in the ground state, a low enough temperature would deny the system of the thermal excitation required to promote very many bosons out of the lowest energy orbital.
There is, then, a transition temperature below which the lowest energy "ground" orbital possesses a large number of bosons. Above this temperature, entropy and thermal excitation render the ground orbital sparsely populated. This transition temperature is known as the Einstein condensation temperature, and the effect of bosons crowding the ground orbital is known as the Einstein condensation.
The Einstein condensation temperature is given by:
The most common condensate is liquid Helium. The crowding is so profound that one can actually see macroscopically the ground orbital of a Helium liquid with the proper equipment. Physics such as superfluidity are also outgrowths of the study of this condensation.