For an electron in a periodic potential the Schrödinger equation has solutions for which there are large gaps in the energy. This is used to explain properties relating to the electric conduction in solids. In my book the formation of energy bands is explained using the Bragg diffraction in a crystal with reciprocal lattice vectors G. As far as I can understand the idea is that we consider the electrons as free electrons, i.e. waves in the solid, which then bounce off the potenial walls formed by the different nuclei in the crystal. The bragg diffraction is: k' = k + G which are fulfilled by wavevectors on the boundary of the Brillouin zone. I guess some of that makes sense. But what about the k-vectors which do no lie near the zone boundaries. My teacher told me nothing happens to these electron waves. WHY is that? Given the nature of the model we should also expect these to be reflected at the potential walls - whether or not they fulfill the Bragg condition. Maybe I am wrong in assuming that we see the electrons as free waves, which scatter of the periodic potential?
What do you mean by: "Given the nature of the model" ? The electrons only scatter if their wavenumber fulfills the Bragg-condition. In that case, the electron wave is diffracted, we have constructive interference and the electron is scattered. Now, if the Bragg-condition is not fulfilled, then we always have destructive interference. There will be always a diffracted sub-wave and a second sub-wave that is in opposite phase so that we get destructive interference. But I do not understand well what you mean by:"reflected at the potential walls"..
Isnt it the idea that we view the electrons as free electrons moving in free space but with periodic potential peaks, where the waves are reflected and transmitted. If there is destructive interference for electron waves in the region of k space, where the bragg condition is not fulfilled, why is it that nothing happens to the energy of the solutions in this area? My teacher said these solutions are just what you would expect if the electron where indeed completely free to move.