## PHY452H1S Basic Statistical Mechanics. Lecture 17: Fermi gas thermodynamics. Taught by Prof. Arun Paramekanti

Posted by peeterjoot on March 26, 2013

# Disclaimer

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?**

We had

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

Using

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 [1] appendix section E. For large it can be shown that this is

so that

Assuming a quadratic form for the chemical potential at low temperature as in fig. 1.5, we have

or

We have used a Taylor expansion for small , for an end result of

# References

[1] RK Pathria. *Statistical mechanics*. Butterworth Heinemann, Oxford, UK, 1996.

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