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# Brillouin zone

In mathematics and solid state physics, the first Brillouin zone is a uniquely defined primitive cell of the reciprocal lattice in the frequency domain. It is found by the same method as for the Wigner-Seitz cell in the Bravais lattice. The importance of the Brillouin zone stems from the Bloch wave description of waves in a periodic medium, in which it is found that the solutions can be completely characterized by their behavior in a single Brillouin zone.

Taking the surfaces at the same distance from one element of the lattice and its neighbours, the volume included is the first 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 more rarely. As a result, the first Brillouin zone is often called simply the Brillouin zone. (In general, the n-th Brillouin zone consist of the set of points that can be reached from the origin by crossing n − 1 Bragg planes.)

A related concept is that of the irreducible Brillouin zone, which is the first Brillouin zone reduced by all of the symmetries in the point group of the lattice.

The concept of a Brillouin zone was developed by Leon Brillouin (1889-1969), a French physicist.

## Critical points

Several points of high symmetry are of special interest – these are called critical points.[1]

SymbolDescription
ΓCenter of the Brillouin zone
Simple cube
MCenter of an edge
RCorner point
XCenter of a face
Face-centered cubic
KMiddle of an edge joining two hexagonal faces
LCenter of a hexagonal face
UMiddle of an edge joining a hexagonal and a square face
WCorner point
XCenter of a square face
Body-centered cubic
HCorner point joining four edges
NCenter of a face
PCorner point joining three edges
Hexagonal
ACenter of a hexagonal face
HCorner point
KMiddle of an edge joining two rectangular faces
LMiddle of an edge joining a hexagonal and a rectangular face
MCenter of a rectangular face