Everipedia is now IQ.wiki - Join the IQ Brainlist and our Discord for early access to editing on the new platform and to participate in the beta testing.

# Brillouin zone

k-vectors exceeding the first Brillouin zone (red) do not carry any more information than their counterparts (black) in the first 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).[1]

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

## Critical points

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.[2]

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[[CITE|3|//doi.org/10.1016%2Fj.commatsci.2010.05.010]]
Lattice systemBravais lattice (Abbreviation)
TriclinicPrimitive triclinic (TRI)Triclinic Lattice type 1a (TRI1a)Triclinic Lattice type 1b (TRI1b)Triclinic Lattice type 2a (TRI2a)Triclinic Lattice type 2b (TRI2b)
MonoclinicPrimitive 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)
OrthorhombicPrimitive 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)
TetragonalPrimitive tetragonal (TET)Simple Tetragonal Lattice (TET)
Body-centered Tetragonal (BCT)Body Centered Tetragonal Lattice type 1 (BCT1)Body Centered Tetragonal Lattice type 2 (BCT2)
RhombohedralPrimitive rhombohederal (RHL)Rhombohedral Lattice type 1 (RHL1)Rhombohedral Lattice type 2 (RHL2)
HexagonalPrimitive hexagonal (HEX)Hexagonal Lattice (HEX)
CubicPrimitive cubic (CUB)Simple Cubic Lattice (CUB)
Body-centered cubic (BCC)Body Centered Cubic Lattice (BCC)
Face-centered cubic (FCC)Face Centered Cubic Lattice (FCC)

Brillouin-zone construction by selected area diffraction, using 300 keV electrons.

• Fundamental pair of periods

• Fundamental domain

## References

[1]
Citation Linkbandgap.ioThompson, Nick. "Irreducible Brillouin Zones and Band Structures". bandgap.io. Retrieved 13 December 2017.
Sep 23, 2019, 8:39 AM
[2]
Citation Linkopenlibrary.orgIbach, Harald; Lüth, Hans (1996). Solid-State Physics, An Introduction to Principles of Materials Science (2nd ed.). Springer-Verlag. ISBN 978-3-540-58573-2.
Sep 23, 2019, 8:39 AM
[3]
Citation Link//doi.org/10.1016%2Fj.commatsci.2010.05.010Setyawan, Wahyu; Curtarolo, Stefano (2010). "High-throughput 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.
Sep 23, 2019, 8:39 AM
[4]
Citation Linkgallica.bnf.fr"Les électrons dans les métaux et le classement des ondes de de Broglie correspondantes"
Sep 23, 2019, 8:39 AM
[5]
Citation Linkwww2.sjsu.eduBrillouin Zone simple lattice diagrams by Thayer Watkins
Sep 23, 2019, 8:39 AM
[6]
Citation Linkphycomp.technion.ac.ilBrillouin Zone 3d lattice diagrams by Technion.
Sep 23, 2019, 8:39 AM
[7]
Citation Linkwww.doitpoms.ac.ukDoITPoMS Teaching and Learning Package- "Brillouin Zones"
Sep 23, 2019, 8:39 AM
[8]
Citation Linkwww.aflowlib.orgAflowlib.org consortium database (Duke University)
Sep 23, 2019, 8:39 AM
[9]
Citation Linkmaterials.duke.eduAFLOW Standardization of VASP/QUANTUM ESPRESSO input files (Duke University)
Sep 23, 2019, 8:39 AM
[10]
Citation Linkbandgap.io"Irreducible Brillouin Zones and Band Structures"
Sep 23, 2019, 8:39 AM
[11]
Sep 23, 2019, 8:39 AM
[12]
Sep 23, 2019, 8:39 AM
[13]
Sep 23, 2019, 8:39 AM
[14]
Citation Linkgallica.bnf.fr"Les électrons dans les métaux et le classement des ondes de de Broglie correspondantes"
Sep 23, 2019, 8:39 AM
[15]
Citation Linkwww2.sjsu.eduBrillouin Zone simple lattice diagrams by Thayer Watkins
Sep 23, 2019, 8:39 AM
[16]
Citation Linkphycomp.technion.ac.ilBrillouin Zone 3d lattice diagrams by Technion.
Sep 23, 2019, 8:39 AM
[17]
Citation Linkwww.doitpoms.ac.ukDoITPoMS Teaching and Learning Package- "Brillouin Zones"
Sep 23, 2019, 8:39 AM
[18]
Citation Linkwww.aflowlib.orgAflowlib.org consortium database (Duke University)
Sep 23, 2019, 8:39 AM
[19]
Citation Linkmaterials.duke.eduAFLOW Standardization of VASP/QUANTUM ESPRESSO input files (Duke University)
Sep 23, 2019, 8:39 AM
[20]
Citation Linken.wikipedia.orgThe original version of this page is from Wikipedia, you can edit the page right here on Everipedia.Text is available under the Creative Commons Attribution-ShareAlike License.Additional terms may apply.See everipedia.org/everipedia-termsfor further details.Images/media credited individually (click the icon for details).
Sep 23, 2019, 8:39 AM