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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
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.
* Fundamental pair of periods
**
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
In 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 ...
, the first Brillouin zone is a uniquely defined primitive cell
In geometry, biology, mineralogy and solid state physics, a unit cell is a repeating unit formed by the vectors spanning the points of a lattice. Despite its suggestive name, the unit cell (unlike a unit vector, for example) does not necessari ...
in reciprocal space. 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 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 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
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 ...
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
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 ...
of the lattice (point group of the crystal).
The concept of a Brillouin zone was developed by Léon Brillouin (1889–1969), a French physicist.
Within the Brillouin zone, a ''constant-energy surface'' represents the loci of all the -points (that is, all the electron momentum values) that have the same energy. Fermi surface 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.
See also
* Fundamental pair of periods
** Fundamental domain
Given a topological space and a group acting on it, the images of a single point under the group action form an orbit of the action. A fundamental domain or fundamental region is a subset of the space which contains exactly one point from each o ...
References
Bibliography
* * *External links
Brillouin Zone simple lattice diagrams by Thayer Watkins
{{DEFAULTSORT:Brillouin Zone Crystallography Electronic band structures Vibrational spectroscopy