Brillouin zone

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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. 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[edit]

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]

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. They can be found in the illustrations below.

Brillouin zone types[3]
Lattice system Bravais lattice

(Abbreviation)

Triclinic Primitive triclinic

(TRI)

Triclinic Lattice type 1a (TRI1a)

Triclinic Lattice type 1a (Brillouin zone).png

Triclinic Lattice type 1b (TRI1b)

Triclinic Lattice type 1b (Brillouin zone).png

Triclinic Lattice type 2a (TRI2a)

Triclinic Lattice type 2a (Brillouin zone).png

Triclinic Lattice type 2b (TRI2b)

Triclinic Lattice type 2b (Brillouin zone).png

Monoclinic Primitive monoclinic

(MCL)

Monoclinic Lattice (MCL)

Monoclinic Lattice (Brillouin zone).png

Base-centered monoclinic

(MCLC)

Base Centered Monoclinic Lattice type 1 (MCLC1)

Base-Centered Monoclinic Lattice type 1 (Brillouin zone).png

Base Centered Monoclinic Lattice type 2 (MCLC2)

Base-Centered Monoclinic Lattice type 2 (Brillouin zone).png

Base Centered Monoclinic Lattice type 3 (MCLC3)

Base-Centered Monoclinic Lattice type 3 (Brillouin zone).png

Base Centered Monoclinic Lattice type 4 (MCLC4)

Base-Centered Monoclinic Lattice type 4 (Brillouin zone).png

Base Centered Monoclinic Lattice type 5 (MCLC5)

Base-Centered Monoclinic Lattice type 5 (Brillouin zone).png

Orthorhombic Primitive orthorhombic

(ORC)

Simple Orthorhombic Lattice (ORC)

Simple Orthorhombic Lattice (Brillouin zone).png

Base-centered orthorhobmic

(ORCC)

Base Centered Orthorhombic Lattice (ORCC)

Base-Centered Orthorhombic Lattice (Brillouin zone).png

Body-centered orthorhombic

(ORCI)

Body Centered Orthorhombic Lattice (ORCI)

Body-Centered Orthorhombic Lattice (Brillouin zone).png

Face-centered orthorhombic

(ORCF)

Face Centered Orthorhombic Lattice type 1 (ORCF1)

Face-Centered Orthorhombic Lattice type 1 (Brillouin zone).png

Face Centered Orthorhombic Lattice type 2 (ORCF2)

Face-Centered Orthorhombic Lattice type 2 (Brillouin zone).png

Face Centered Orthorhombic Lattice type 3 (ORCF3)

Face-Centered Orthorhombic Lattice type 3 (Brillouin zone).png

Tetragonal Primitive tetragonal

(TET)

Simple Tetragonal Lattice (TET)

Simple Tetragonal Lattice (Brillouin zone).png

Body-centered Tetragonal

(BCT)

Body Centered Tetragonal Lattice type 1 (BCT1)

Body-Centered Tetragonal Lattice type 1 (Brillouin zone).png

Body Centered Tetragonal Lattice type 2 (BCT2)

Body-Centered Tetragonal Lattice type 2 (Brillouin zone).png

Rhombohedral Primitive rhombohederal

(RHL)

Rhombohedral Lattice type 1 (RHL1)

Rhombohedral Lattice type 1 (Brillouin zone).png

Rhombohedral Lattice type 2 (RHL2)

Rhombohedral Lattice type 2 (Brillouin zone).png

Hexagonal Primitive hexagonal

(HEX)

Hexagonal Lattice (HEX)

Brillouin zone in hexagonal lattice.png

Cubic Primitive cubic

(CUB)

Simple Cubic Lattice (CUB)

Simple Cubic Lattice (Brillouin zone).png

Body-centered cubic

(BCC)

Body Centered Cubic Lattice (BCC)

Body-Centered Cubic Lattice (Brillouin zone).png

Face-centered cubic

(FCC)

Face Centered Cubic Lattice (FCC)

Face-Centered Cubic Lattice (Brillouin zone).png

See also[edit]

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

References[edit]

  1. ^ Thompson, Nick. "Irreducible Brillouin Zones and Band Structures". bandgap.io. Retrieved 13 December 2017. 
  2. ^ Ibach, Harald; Lüth, Hans (1996). Solid-State Physics, An Introduction to Principles of Materials Science (2nd ed.). Springer-Verlag. ISBN 3-540-58573-7. 
  3. ^ Setyawan, Wahyu; Curtarolo, Stefano (2010). "High-throughput electronic band structure calculations: Challenges and tools". Computational Materials Science. 49 (2): 299–312. arXiv:1004.2974Freely accessible. Bibcode:2010arXiv1004.2974S. doi:10.1016/j.commatsci.2010.05.010. 

External links[edit]