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# Disclaimer

Peeter’s lecture notes from class. May not be entirely coherent.

# Fermi gas

**Review**

Continuing a discussion of [1] section 8.1 content.

We found

With no spin

Fig 1.1: Occupancy at low temperature limit

Fig 1.2: Volume integral over momentum up to Fermi energy limit

gives

This is for periodic boundary conditions \footnote{I filled in details in the last lecture using a particle in a box, whereas this periodic condition was intended. We see that both achieve the same result}, where

**Moving on**

with

this gives

Over all dimensions

so that

Again

## Example: Spin considerations

{example:basicStatMechLecture16:1}{

This gives us

and again

}

**High Temperatures**

Now we want to look at the at higher temperature range, where the occupancy may look like fig. 1.3

Fig 1.3: Occupancy at higher temperatures

so that for large we have

Mathematica (or integration by parts) tells us that

so we have

Introducing for the *thermal de Broglie wavelength*,

we have

Does it make any sense to have density as a function of temperature? An inappropriately extended to low temperatures plot of the density is found in fig. 1.4 for a few arbitrarily chosen numerical values of the chemical potential , where we see that it drops to zero with temperature. I suppose that makes sense if we are not holding volume constant.

Fig 1.4: Density as a function of temperature

We can write

or (taking (and/or volume?) as a constant) we have for large temperatures

The chemical potential is plotted in fig. 1.5, whereas this function is plotted in fig. 1.6. The contributions to from the term are dropped for the high temperature approximation.

Fig 1.5: Chemical potential over degenerate to classical range

Fig 1.6: High temp approximation of chemical potential, extended back to T = 0

**Pressure**

For a classical ideal gas as in fig. 1.7 we have

Fig 1.7: Ideal gas pressure vs volume

For a Fermi gas at we have

Specifically,

or

so that

We see that the pressure ends up deviating from the classical result at low temperatures, as sketched in fig. 1.8. This low temperature limit for the pressure is called the *degeneracy pressure*.

Fig 1.8: Fermi degeneracy pressure

# References

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