The reciprocal lattices (dots) and corresponding first Brillouin zones of (a) square lattice and (b) hexagonal lattice.

In mathematics and solid state physics, the first Brillouin zone is a uniquely defined primitive cell in reciprocal space. The boundaries of this cell are given by planes related to points on the reciprocal lattice. 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 surfaces at the same distance from one element of the lattice and its neighbours, the volume included is the first Brillouin zone (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.

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

Critical points[link]

First Brillouin zone of FCC lattice, a truncated octahedron, showing symmetry labels for high symmetry lines and points

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

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

[edit] Triclinic lattice system TRI(4)

See below for the aflowlib.org standard.

[edit] Monoclinic lattice system MCL(1), MCLC(5)

See below for the aflowlib.org standard.

[edit] Orthorhombic lattice system ORC(1), ORCC(1), ORCI(1), ORCF(3)

See below for the aflowlib.org standard.

[edit] Tetragonal lattice system TET(1), BCT(2)

See below for the aflowlib.org standard.

[edit] Rhombohedral lattice system RHL(2)

See below for the aflowlib.org standard.

[edit] Hexagonal lattice system HEX(1)

See below for the aflowlib.org standard.

[edit] Cubic lattice system CUB(1), BCC(1), FCC(1)

Simple Cubic Lattice (CUB) BZ.

File:BCC.PNG Body Centered Cubic Lattice (BCC) BZ.

File:FCC.PNG Face Centered Cubic Lattice (FCC) BZ.

See below for the aflowlib.org standard.

See also[link]

• Fundamental pair of periods
• Fundamental domain

References[link]

1. ^ Ibach, Harald; Hans Lüth (1996). Solid-State Physics, An Introduction to Principles of Materials Science (Second ed.). Springer-Verlag. ISBN 3-540-58573-7. 

• Kittel, Charles (1996). Introduction to Solid State Physics. New York City: Wiley. ISBN 0-471-14286-7. 
• Ashcroft, Neil W.; Mermin, N. David (1976). Solid State Physics. Orlando: Harcourt. ISBN 0-03-049346-3. 
• 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). http://gallica.bnf.fr/ark:/12148/bpt6k31445. 
• Setyawan, Wahyu; Curtarolo, Stefano (2010). "High-throughput electronic band structure calculations: Challenges and tools". Comp. Mat. Sci. 49: 299–312. DOI:10.1016/j.commatsci.2010.05.010. 

External links[link]

• Brillouin Zone simple lattice diagrams by Thayer Watkins
• Brillouin Zone 3d lattice diagrams by Technion.
• DoITPoMS Teaching and Learning Package- "Brillouin Zones"
• Aflowlib.org consortium database (Duke University)
• AFLOW Standardization of VASP/QUANTUM ESPRESSO input files (Duke University)