Today I gave a lecture to a Solid State Physics class on "Magnetic quantum oscillations and mapping out the Fermi surface." I basically follow chapter 14 of Ashcroft and Mermin.
The central result is Onsager's 1952 equation that the period of the magnetic oscillations is related to extremal areas of the Fermi surface perpendicular to the magnetic field.
[Aside: This is an amazing result because it only involves fundamental constants and so the interpretation of the experiments is not "theory laden", a rare thing in condensed matter].
There is one point I struggle to explain: why extremal areas?
Ashcroft and Mermin have a figure to justify this. It is lost on me, no matter how many times I read it and stare at the pictures.
Does anyone know a clear and convincing way to demonstrate this?
The only way I know how to get this result of extremal areas is to do a very fancy calculation (Lifshitz-Kosevich) which evaluates the magnetisation (or thermodynamic potential or partition function), summing over all the Landau levels and integrating over the momentum direction parallel to field. One then evaluates the last integral using the method of steepest descent (saddle point approximation), which then picks out the extremal areas of the Fermi surface.
However, this is way beyond what one should be doing at this level (final year undergraduate).
Furthermore, the students said they have never encountered the method of steepest descents before.
That is reasonable since neither did I as an undergraduate in Australia.
I only learnt it in graduate school after learning how to evaluate Feynman path integrals by this method. Only later did I learn it also applied simple one-dimensional integrals!
Should undergraduate students learn the method of steepest descent?
In mathematics and/or physics?