Brillouin zoneThe 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 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 boundaries of this cell are given by planes related to points on the reciprocal lattice. In the same way the Bravais lattice is divided up into Wigner–Seitz cells in the real lattice. The first Brillouin zone is the locus of points in reciprocal space that are closer .defined primitive cell in reciprocal space. the reciprocal lattice is broken up into Brillouin zones. Equivalently.to the origin of the reciprocal lattice than they are to any other reciprocal lattice points (see the derivation of the Wigner- Seitz cell). this is the Voronoi cell around the origin of the reciprocal lattice. Another definition is as the set of points in k-space that can be reached from the origin without crossing any Bragg plane. etc. third. the first Brillouin zone is often called simply the Brillouin .. Brillouin zones. There are also second. As a result. corresponding to a sequence of disjoint regions (all with the same volume) at increasing distances from the origin. but these are used less frequently. Critical points .[1] The concept of a Brillouin zone was developed by Léon Brillouin (1889– 1969).) A related concept is that of the irreducible Brillouin zone.zone. 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 French physicist. which is the first Brillouin zone reduced by all of the symmetries in the point group of the lattice (point group of the crystal). (In general. showing symmetry labels for high symmetry lines and points Several points of high symmetry are of special interest – these are called critical points.[2] .First Brillouin zone of FCC lattice. a truncated octahedron. . They can be found in the illustrations below.Symbol Description Γ Center of the Brillouin zone Simple cube M Center of an edge R Corner point X Center of a face Face-centered 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 Body-centered 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 high-symmetry points. BZ. MCLC(5) . See below for the aflowlib. Monoclinic lattice system MCL(1).Triclinic lattice system TRI(4) Triclinic Triclinic Triclinic Triclinic Lattice type Lattice type Lattice type Lattice type 1a (TRI1a) 1b (TRI1b) 2a (TRI2a) 2b (TRI2b) BZ. BZ.org standard. BZ. (MCLC3) BZ. Base Centered Monoclinic Base Centered Monoclinic Lattice type 5 (MCLC5) Lattice type 4 (MCLC4) BZ. Base Base Monoclinic Base Centered Centered Lattice Centered Monoclinic Monoclinic (MCL) BZ. Monoclinic Lattice type 1 Lattice type 3 Lattice type 2 (MCLC1) BZ. (MCLC2) BZ. . BZ. Orthorhombic lattice system ORC(1). Lattice (ORCI) BZ. ORCI(1). ORCF(3) Simple Base Centered Orthorhombic Body Centered Orthorhombic Lattice (ORCC) Orthorhombic Lattice (ORC) BZ.org standard. . ORCC(1). BZ.See below for the aflowlib. Tetragonal lattice system TET(1). BCT(2) .Face Centered Face Centered Face Centered Orthorhombic Orthorhombic Orthorhombic Lattice type 1 Lattice type 2 Lattice type 3 (ORCF1) BZ. See below for the aflowlib. (ORCF3) BZ.org standard. (ORCF2) BZ. See below for the aflowlib.org standard. Rhombohedral lattice system RHL(2) . BZ.Simple Body Centered Tetragonal Body Centered Tetragonal Lattice Lattice (TET) Tetragonal Lattice type 2 (BCT2) BZ. type 1 (BCT1) BZ. See below for the aflowlib. Hexagonal lattice system HEX(1) .Rhombohedral Lattice Rhombohedral Lattice type 1 (RHL1) BZ.org standard. type 2 (RHL2) BZ. FCC(1) .Hexagonal Lattice (HEX) BZ. More details in links below. BCC(1). Cubic lattice system CUB(1).org standard. See below for the aflowlib. (BCC) BZ. See also .org standard. See below for the aflowlib. (FCC) BZ.Simple Cubic Body Centered Face Centered Lattice (CUB) Cubic Lattice Cubic Lattice BZ. io. Thompson. . Fundamental pair of periods Fundamental domain References 1. Ibach. Hans (1996). Kittel. An Introduction to Principles of Materials Science (2nd ed. Springer-Verlag. Nick. ISBN 0-471-14286-7. "Irreducible Brillouin Zones and Band Structures" . Introduction to Solid State Physics. bandgap. Solid- State Physics. Charles (1996).Brillouin-zone construction by selected area diffraction. Lüth.). 2. Harald. ISBN 3-540-58573-7. using 300 keV electrons. Retrieved 13 December 2017. New York City: Wiley. David (1976). Wahyu. Solid State Physics.01 0. Setyawan. Curtarolo. . N. Sci. Orlando: Harcourt. "High-throughput electronic band structure calculations: Challenges and tools". Comp.commatsci. Mat.2010. Mermin. "Les électrons dans les métaux et le classement des ondes de de Broglie correspondantes" . ISBN 0-03-049346-3. Neil W. 191 (292).Ashcroft. doi:10. 49 (2): 299–312. . Léon (1930).05. Stefano (2010). Brillouin. Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences..1016/j. org/w/index.wikipedia. DoITPoMS Teaching and Learning Package."Brillouin Zones" Aflowlib.External links Brillouin Zone simple lattice diagrams by Thayer Watkins Brillouin Zone 3d lattice diagrams by Technion.php? title=Brillouin_zone&oldid=815253042" .org consortium database (Duke University) AFLOW Standardization of VASP/QUANTUM ESPRESSO input files (Duke University) Retrieved from "https://en. .Last edited 1 month ago by Natho… Content is available under CC BY-SA 3.0 unless otherwise noted.