Peeter’s lecture notes from class. May not be entirely coherent.
We’ve considered a momentum sphere as in fig. 1.1, and performed various appromations of the occupation sums fig. 1.2.
The physics of Fermi gases has an extremely wide range of applicability. Illustrating some of this range, here are some examples of Fermi temperatures (from )
- Electrons in copper:
- Neutrons in neutron star:
- Ultracold atomic gases:
We’d like to work with a fixed number of particles, but the calculations are hard, so we move to the grand canonical ensemble
Again, we’ll consider free particles with energy as in fig. 1.3, or
Again introducing fugacity , we have
We’ll consider systems for which
Observe that at large energies we have
For small energies
Observe that we require (or ) so that the number distribution is strictly positive for all energies. This tells us that the fugacity is a function of temperature, but there will be a point at which it must saturate. This is illustrated in fig. 1.4.
Let’s calculate this density (assumed fixed for all temperatures)
With the substitution
This implicitly defines a relationship for the fugacity as a function of temperature .
It can be shown that
As we end up with a zeta function, for which we can look up the value
where the Riemann zeta function is defined as
At high temperatures we have
(as does down, goes up)
Looking at leads to
How do I satisfy number conservation?
We have a problem here since as the term in above drops to zero, yet cannot keep increasing without bounds to compensate and keep the density fixed. The way to deal with this was worked out by
- Bose (1924) for photons (examining statistics for symmetric wave functions).
- Einstein (1925) for conserved particles.
To deal with this issue, we (somewhat arbitrarily, because we need to) introduce a non-zero density for . This is an adjustment of the approximation so that we have
as in fig. 1.5, so that
Given this, we have
We can illustrate this as in fig. 1.6.
At we have , whereas at we must introduce a non-zero density if we want to be able to keep a constant density constraint.