the Brillouin zone
/ bree-LWAN /
The Brillouin zone is the unit cell of reciprocal space, chosen in the most symmetric possible way. Think of it as the territory of genuinely distinct wavevectors: every possible electron or phonon state in a crystal can be labelled by a k inside this one region, and any k lying outside is just a disguised copy of one inside. It is where all the interesting physics of a crystal is drawn.
The first Brillouin zone is the Wigner-Seitz cell of the reciprocal lattice: the set of all k-points that are closer to a chosen reciprocal lattice point (the origin) than to any other. You construct it by drawing the perpendicular bisector planes of the vectors from the origin to neighbouring reciprocal lattice points; the smallest volume enclosed is the zone. Its boundaries are Bragg planes, and any wavevector outside can be folded back inside by subtracting a reciprocal lattice vector G, since k and k + G describe the same Bloch state. This is why band structures E_n(k) are plotted only over the first zone.
The zone boundary is where the action happens: it is exactly there that Bragg reflection of electron waves opens energy gaps, splitting a free-electron band into the allowed bands of a solid. A frequent slip: the Brillouin zone is a region in k-space (inverse length), not a region of real space; its corners and symmetry points (labelled Gamma, X, L, and so on) are special wavevectors, not places in the crystal.
For a simple cubic lattice of spacing a, the first Brillouin zone is a cube of side 2 pi / a centred on the origin; its centre is the Gamma point (k = 0) and its face centres are the X points.
The first zone contains one k-state per allowed value per band, no more and no less.
Because k and k + G are physically the same state, restricting to the first Brillouin zone loses no information; it merely removes redundant copies.