The first Brillouin zone — a region of reciprocal space (see Reciprocal lattice) centred on the origin of coordinates and defined as follows: if planes are drawn through the midpoints of the vectors joining the origin to the nearest points of the reciprocal lattice, the polyhedron they form is the first Brillouin zone. To every crystal lattice (direct lattice) there corresponds a reciprocal lattice, which in turn defines the Brillouin zone. For example, the Brillouin zone of a simple cubic lattice has the shape of a cube. For a face-centred cubic lattice the reciprocal lattice is body-centred, and the first Brillouin zone has the shape of a truncated octahedron (Fig.). The first Brillouin zone has the same symmetry properties with respect to rotations, mirror reflection and inversion as the Bravais lattice of the crystal in question (see Symmetry of crystals). The volume of reciprocal space enclosed within the first Brillouin zone is (2π)3/V0, where V0is the volume of the unit cell of the Bravais lattice.
The first Brillouin zone for a body-centred crystal lattice.
The Brillouin zone plays an important part in the theory of the propagation of waves in crystals, in particular it is used in the band theory of solids, where the waves are electron waves, elastic waves and others. The energy of any quasiparticle in a crystal (a conduction electron, a phonon and others) is a periodic function of its quasi-momentum p. The dispersion law Ε(p) of the quasiparticles is a single-valued and continuous function within the first Brillouin zone.
If a polyhedron is built in the same way for the vectors and the lattice points that come next after the nearest ones, and the polyhedron corresponding to the first Brillouin zone is subtracted from it, the second Brillouin zone is obtained, and so on. The second Brillouin zone, unlike the first, always consists of several disconnected regions. The introduction of the higher Brillouin zones plays an important part in determining the Fermi surface of metals.