# Brillouin zone

In mathematics and solid state physics, the first Brillouin zone is a uniquely defined primitive cell in reciprocal space. In the same way the Bravais lattice is divided up into Wigner–Seitz cells in the real lattice, the reciprocal lattice is broken up into Brillouin zones. 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 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.

The first Brillouin zone is the locus of points in reciprocal space that are closer to the origin of the reciprocal lattice than they are to any other reciprocal lattice points (see the derivation of the Wigner-Seitz cell). Another definition is as the set of points in k-space that can be reached from the origin without crossing any Bragg plane. Equivalently, this is the Voronoi cell around the origin of the reciprocal lattice.

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. As a result, the first Brillouin zone is often called simply the Brillouin zone. 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.

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 (point group of the crystal).[2]

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

## Critical points

Several points of high symmetry are of special interest – these are called critical points.[3]

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

Other lattices have different types of high-symmetry points. They can be found in the illustrations below.

Brillouin zone types[4]
Lattice system Bravais lattice

(Abbreviation)

Triclinic Primitive triclinic

(TRI)

Triclinic Lattice type 1a (TRI1a)

Triclinic Lattice type 1b (TRI1b)

Triclinic Lattice type 2a (TRI2a)

Triclinic Lattice type 2b (TRI2b)

Monoclinic Primitive monoclinic

(MCL)

Monoclinic Lattice (MCL)

Base-centered monoclinic

(MCLC)

Base Centered Monoclinic Lattice type 1 (MCLC1)

Base Centered Monoclinic Lattice type 2 (MCLC2)

Base Centered Monoclinic Lattice type 3 (MCLC3)

Base Centered Monoclinic Lattice type 4 (MCLC4)

Base Centered Monoclinic Lattice type 5 (MCLC5)

Orthorhombic Primitive orthorhombic

(ORC)

Simple Orthorhombic Lattice (ORC)

Base-centered orthorhombic

(ORCC)

Base Centered Orthorhombic Lattice (ORCC)

Body-centered orthorhombic

(ORCI)

Body Centered Orthorhombic Lattice (ORCI)

Face-centered orthorhombic

(ORCF)

Face Centered Orthorhombic Lattice type 1 (ORCF1)

Face Centered Orthorhombic Lattice type 2 (ORCF2)

Face Centered Orthorhombic Lattice type 3 (ORCF3)

Tetragonal Primitive tetragonal

(TET)

Simple Tetragonal Lattice (TET)

Body-centered Tetragonal

(BCT)

Body Centered Tetragonal Lattice type 1 (BCT1)

Body Centered Tetragonal Lattice type 2 (BCT2)

Rhombohedral Primitive rhombohederal

(RHL)

Rhombohedral Lattice type 1 (RHL1)

Rhombohedral Lattice type 2 (RHL2)

Hexagonal Primitive hexagonal

(HEX)

Hexagonal Lattice (HEX)

Cubic Primitive cubic

(CUB)

Simple Cubic Lattice (CUB)

Body-centered cubic

(BCC)

Body Centered Cubic Lattice (BCC)

Face-centered cubic

(FCC)

Face Centered Cubic Lattice (FCC)