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 WignerSeitz cell). Another definition is as the set of points in kspace 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 nth 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]
Symbol  Description 

Γ  Center of the Brillouin zone 
Simple cube  
M  Center of an edge 
R  Corner point 
X  Center of a face 
Facecentered cubic  
K  Middle of an edge joining two hexagonal faces 
L  Center of a hexagonal face 
U  Middle of an edge joining a hexagonal and a square face 
W  Corner point 
X  Center of a square face 
Bodycentered cubic  
H  Corner point joining four edges 
N  Center of a face 
P  Corner point joining three edges 
Hexagonal  
A  Center of a hexagonal face 
H  Corner point 
K  Middle of an edge joining two rectangular faces 
L  Middle of an edge joining a hexagonal and a rectangular face 
M  Center of a rectangular face 
Other lattices have different types of highsymmetry points. They can be found in the illustrations below.
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)
 
Basecentered 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)
 
Basecentered orthorhombic
(ORCC) 
Base Centered Orthorhombic Lattice (ORCC)
 
Bodycentered orthorhombic
(ORCI) 
Body Centered Orthorhombic Lattice (ORCI)
 
Facecentered 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)
 
Bodycentered 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)
 
Bodycentered cubic
(BCC) 
Body Centered Cubic Lattice (BCC)
 
Facecentered cubic
(FCC) 
Face Centered Cubic Lattice (FCC)

References
 "Topic 52: Nyquist Frequency and Group Velocity" (PDF). Solid State Physics in a Nutshell. Colorado School of Mines.
 Thompson, Nick. "Irreducible Brillouin Zones and Band Structures". bandgap.io. Retrieved 13 December 2017.
 Ibach, Harald; Lüth, Hans (1996). SolidState Physics, An Introduction to Principles of Materials Science (2nd ed.). SpringerVerlag. ISBN 9783540585732.
 Setyawan, Wahyu; Curtarolo, Stefano (2010). "Highthroughput electronic band structure calculations: Challenges and tools". Computational Materials Science. 49 (2): 299–312. arXiv:1004.2974. Bibcode:2010arXiv1004.2974S. doi:10.1016/j.commatsci.2010.05.010.
Bibliography
 Kittel, Charles (1996). Introduction to Solid State Physics. New York City: Wiley. ISBN 9780471142867.
 Ashcroft, Neil W.; Mermin, N. David (1976). Solid State Physics. Orlando: Harcourt. ISBN 9780030493461.
 Brillouin, Léon (1930). "Les électrons dans les métaux et le classement des ondes de de Broglie correspondantes". Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences. 191 (292).