mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

and

solid state physics Solid-state physics is the study of rigid matter, or solids, through methods such as quantum mechanics, crystallography, electromagnetism, and metallurgy. It is the largest branch of condensed matter physics. Solid-state physics studies how the l ...

, the first Brillouin zone is a uniquely defined primitive cell in

reciprocal space In physics, the reciprocal lattice represents the Fourier transform of another lattice (usually a Bravais lattice). In normal usage, the initial lattice (whose transform is represented by the reciprocal lattice) is usually a periodic spatial fu ...

. In the same way the

Bravais lattice In geometry and crystallography, a Bravais lattice, named after , is an infinite array of discrete points generated by a set of discrete translation operations described in three dimensional space by : \mathbf = n_1 \mathbf_1 + n_2 \mathbf_2 + n ...

is divided up into

Wigner–Seitz cell The Wigner–Seitz cell, named after Eugene Wigner and Frederick Seitz, is a primitive cell which has been constructed by applying Voronoi decomposition to a crystal lattice. It is used in the study of crystalline materials in crystallography. ...

s in the real lattice, the

reciprocal lattice In physics, the reciprocal lattice represents the Fourier transform of another lattice (group) (usually a Bravais lattice). In normal usage, the initial lattice (whose transform is represented by the reciprocal lattice) is a periodic spatial fu ...

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 description of waves in a periodic medium given by

Bloch's theorem In condensed matter physics, Bloch's theorem states that solutions to the Schrödinger equation in a periodic potential take the form of a plane wave modulated by a periodic function. The theorem is named after the physicist Felix Bloch, who d ...

, 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 Locus (plural loci) is Latin for "place". It may refer to: Entertainment * Locus (comics), a Marvel Comics mutant villainess, a member of the Mutant Liberation Front * ''Locus'' (magazine), science fiction and fantasy magazine ** ''Locus Award' ...

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 In physics, a Bragg plane is a plane in reciprocal space which bisects a reciprocal lattice vector, \scriptstyle \mathbf, at right angles. The Bragg plane is defined as part of the Von Laue condition for diffraction peaks in x-ray diffraction c ...

. Equivalently, this is the

Voronoi cell In mathematics, a Voronoi diagram is a partition of a plane into regions close to each of a given set of objects. In the simplest case, these objects are just finitely many points in the plane (called seeds, sites, or generators). For each seed t ...

around the origin of the reciprocal lattice. Phonon k 3k

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 In geometry, a point group is a mathematical group of symmetry operations (isometries in a Euclidean space) that have a fixed point in common. The coordinate origin of the Euclidean space is conventionally taken to be a fixed point, and every p ...

of the lattice (point group of the crystal). The concept of a Brillouin zone was developed by

Léon Brillouin Léon Nicolas Brillouin (; August 7, 1889 – October 4, 1969) was a French physicist. He made contributions to quantum mechanics, radio wave propagation in the atmosphere, solid state physics, and information theory. Early life Brillouin ...

(1889–1969), a French physicist. Within the Brillouin zone, a ''constant-energy surface'' represents the loci of all the

\vec

-points (that is, all the electron momentum values) that have the same energy.

Fermi surface In condensed matter physics, the Fermi surface is the surface in reciprocal space which separates occupied from unoccupied electron states at zero temperature. The shape of the Fermi surface is derived from the periodicity and symmetry of the cryst ...

is a special constant-energy surface that separates the unfilled orbitals from the filled ones at zero kelvin.

Critical points

Several points of high symmetry are of special interest – these are called critical points. Other lattices have different types of high-symmetry points. They can be found in the illustrations below.

References

Bibliography

* * *

External links

Brillouin Zone simple lattice diagrams by Thayer WatkinsDoITPoMS Teaching and Learning Package- "Brillouin Zones"Aflowlib.org consortium database (Duke University)
{{DEFAULTSORT:Brillouin Zone Crystallography Electronic band structures Vibrational spectroscopy

Critical points

See also

References

Bibliography

External links