I think TNR did a fairly good job of clarifying (and I'm not entirely sure what the question was) but let me add a few comments which might make a few things more crisp:
(1) In fig 13.5 there are no fermi surfaces drawn -- just the brillouin zones shaded in extended zone scheme. You could translate things by G in which case each BZ would occupy the same k-states as the first BZ.
(2) If we were to have electrons in infinitely weak periodic potential, we would be tempted to just draw a parabolic dispersion E proportional to kx^2 + ky^2 at all k, oblivious to any zone boundaries, and you wouldn't think of anything periodic and there would be a single eigenstate at each value of (kx,ky). This does not show the fact that the spectrum remains unchanged when we shift momentum by a reciprocal lattice vector G. The other way to show things would be to fold everything back into a single brillouin zone, in which case we realize that, for example, G = 2pi/a is actually the same point in reciprocal space as G=0. This may seem strange, but there are now many eigenstates at this G, one of them with energy 0 and the other with energy hbar^2 (2 pi/a)^2/(2m), for example.
« Back to index