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Geometry practice b answers
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Abstract: Brillouin zones were introduced by Brillouin [Br] in the thirties to describe quantum mechanical properties of crystals, that is, in a lattice in Rn. They play an important role in solid-state physics. It was shown by Bieberbach [Bi] that Brillouin zones tile the underlying space and that each zone has the same area. is a reciprocal lattice vector = ~k ¡~k0 from which we conclude that the periodic potential V(~r) only connects wave vectors ~k and ~k0 separated by a reciprocal lattice vector. We note that this is the same relation that determines the Brillouin zone boundary. The matrix element is then h~k0jV(~r)j~ki = N › Z ›0 eiG~¢~r0V(~r0)d3r0– ~k0 ... Pack sampling grid of the Brillouin zone. These calculations showed that DE H are converged and that different approxi-mations of the exchange-correlation energy introduce sys-tematic errors of about 0.1-0.2eV in DE H, while leaving almost unchanged ( 0.05eV) the relative energy values of H in different fcc interstitial positions.