PHY452H1S Basic Statistical Mechanics. Lecture 17: Fermi gas thermodynamics. Taught by Prof. Arun Paramekanti
Posted by peeterjoot on March 26, 2013
Peeter’s lecture notes from class. May not be entirely coherent.
Fermi gas thermodynamics
- Energy was found to be
- Pressure was found to have the form fig. 1.1
- The chemical potential was found to have the form fig. 1.2.
so that the zero crossing is approximately when
That last identification provides the relation . FIXME: that bit wasn’t clear to me.
How about at other temperatures?
FIXME: references to earlier sections where these were derived.
We can define a density of states
where the liberty to informally switch the order of differentiation and integration has been used. This construction allows us to write a more general sum
This sum, evaluated using a continuum approximation, is
where the roots of are , we have
In 2D this would be
and in 1D
What happens when we have linear energy momentum relationships?
Suppose that we have a linear energy momentum relationship like
An example of such a relationship is the high velocity relation between the energy and momentum of a particle
Another example is graphene, a carbon structure of the form fig. 1.3. The energy and momentum for such a structure is related in roughly as shown in fig. 1.4, where
Continuing with the 3D case we have
FIXME: Is this (or how is this) related to the linear energy momentum relationships for Graphene like substances?
where as usual, and we write . For the low temperature asymptotic behavior see  appendix section E. For large it can be shown that this is
Assuming a quadratic form for the chemical potential at low temperature as in fig. 1.5, we have
We have used a Taylor expansion for small , for an end result of
 RK Pathria. Statistical mechanics. Butterworth Heinemann, Oxford, UK, 1996.