I'm studying from "Solid State Physics" by Ashcroft-Mermin. In particular, in chapter 12 it talks about the semiclassical model and tries to reason about the Hall effect in the limiting case of high magnetic field.
The first case the book considers is when, in a band, either all occupied or all unoccupied levels give rise to closed orbits in the reciprocal lattice (in presence of perpendicular uniform and constant Electric and magnetic field) according to the semiclassical equations of motion. In footnote 36, page 235, the book says that, in this case, since the band energy $$\varepsilon(\mathbf{k})$$ is periodic in the reciprocal lattice, it can't be that all levels, both occupied and unoccupied, give rise to closed orbits. Either all occupied levels give rise to closed orbits and some unoccupied levels give rise to open orbits or viceversa.
Can someone explain why is that or how can I prove that? I can't see why this would be true; I can immagine a band where the surface corresponding to the maximum of the energy band is enclosed in the first Brillouin zone, and that would be periodic and have only closed obits, wouldn't it?
