Brillouin zone

The boundaries of this cell are given by planes related to points on the reciprocal lattice.

The importance of the Brillouin zone stems from the description of waves in a periodic medium given by Bloch's theorem, in which it is found that the solutions can be completely characterized by their behavior in a single Brillouin zone.

Another definition is as the set of points in k-space that can be reached from the origin without crossing any Bragg plane.

There are also second, third, etc., Brillouin zones, corresponding to a sequence of disjoint regions (all with the same volume) at increasing distances from the origin, but these are used less frequently.

In general, the n-th Brillouin zone consists of the set of points that can be reached from the origin by crossing exactly n − 1 distinct Bragg planes.

The reciprocal lattices (dots) and corresponding first Brillouin zones of (a) square lattice and (b) hexagonal lattice .
k -vectors exceeding the first Brillouin zone (red) do not carry any more information than their counterparts (black) in the first Brillouin zone. k at the Brillouin zone edge is the spatial Nyquist frequency of waves in the lattice, because it corresponds to a half-wavelength equal to the inter-atomic lattice spacing a . [ 1 ] See also Aliasing § Sampling sinusoidal functions for more on the equivalence of k -vectors.
The Brillouin zone (purple) and the irreducible Brillouin zone (red) for a hexagonal lattice .
First Brillouin zone of FCC lattice , a truncated octahedron , showing symmetry labels for high symmetry lines and points
Brillouin-zone construction by selected area diffraction , using 300 keV electrons.