What is the Fermi surface

What is the Fermi surface?

What is the Fermi surface? I hope this question is not too basic for this forum and I apologize in advance if it is.

Allow me to explain my confusion. I think I have a feel for the Fermi level. For example, I can understand it as a characteristic parameter μ in the Fermi-Dirac distribution of the energy levels for the electrons in the system:

f (ϵ) = 1e (ϵ-μ) / kT + 1
ignore other physical interpretations for the moment. Thus, it is the unique energy level that has a 1/2 chance of being occupied.

The definition of the Fermi surface, on the other hand, is usually given as "isosurface of states with an energy equal to the Fermi level" in the three-dimensional space of wave vectors k, for example in this Wikipedia article:


In other words, it is defined as such k such that

E. (k) = µ.
So far, so good. The problem is, I don't quite understand what E. (k) is.

One situation seems simple, namely a Fermigas with identical particles. Then

E. (k) = k22m
and the Fermi surface is a sphere. However, if we are in an infinite periodic potential, the usual idealized model for the Bloch theory, the solutions for the Schrödinger equation result in the form
ψkn (r) = eggrukn (r),