Posted by TNR on April 10, 2014, 3:49 pm, in reply to "Confusion about Brillouin Zones"
I'll give a shot at answering this, but I'm only an undergraduate so take my supposed understanding with a bucket of salt (thank you internet anonymity...) The periodicness in 2D isn't actually represented in diagram 16.5 or 16.3 because we don't see the ENERGIES occupied by the states that the Fermi sea shows. If you imagine a z axis coming out of that picture on which we plotted the surface for the energies (i.e. for every (k_x, k_y) state we plotted it's energy on the z axis) we would see that the energy surface is periodic in k space. Just like how in 1D, when the omega or energy function went into the 2nd Brillouin zone, the function itself remained periodic in the sense that if you cut it off as it crossed over into k = (+/)pi/a, it's periodic at the boundaries. That's what you would see if you could see the energy surface plotted above diagram 16.6, but in 2D instead of 1D. And yeah, in figure 16.6, I think the extended zone scheme is what you would get if you could just see which states were occupied as though we were looking at figure 13.5 (i.e. if 13.5 didn't already have colours for the different Brillouin zones and you coloured in the occupied states, you would get the extended zone in figure 16.6). Ok, I looked at it some more, and the reason that the surface of occupied states in k space intersects the boundary in a periodic way in 16.3 is because for a strong potential, the dispersion of the electrons depends on more than just the magnitude of k; the energy as a function of K itself is such that the surface is periodic in k space because it depends on the exact combination of k_x and k_y (for the tight binding, the energy goes something like E = cos(k_x) + cos(k_y)). However, for no period potential, the energy is a function of only the magnitude of k, hence why the occupied states are spherical, so they are periodic in k, but only the actual "Energy" surface projected above our 2D map. Man I hope that makes any kind of sense 
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