mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

and

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

, the first Brillouin zone (named after

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 Brilloui ...

) 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 necessaril ...

reciprocal space Reciprocal lattice is a concept associated with solids with translational symmetry which plays a major role in many areas such as X-ray diffraction, X-ray and Electron diffraction, electron diffraction as well as the Electronic band structure, e ...

. 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 ...

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 cell, Voronoi decomposition to a crystal lattice. It is used in the study of crystalline materials in crystal ...

s in the real lattice, the

reciprocal lattice Reciprocal lattice is a concept associated with solids with translational symmetry which plays a major role in many areas such as X-ray and electron diffraction as well as the energies of electrons in a solid. It emerges from the Fourier tran ...

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 can be expressed as plane waves modulated by periodic functions. The theorem is named after the Swiss physicist Felix Bloch, ...

, 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: Mathematics and science * Locus (mathematics), the set of points satisfying a particular condition, often forming a curve * Root locus analysis, a diagram visualizing the position of r ...

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 (geometry), 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 Interference (wave prop ...

. 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. It can be classified also as a tessellation. In the simplest case, these objects are just finitely many points in the plane (calle ...

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 group (mathematics), mathematical group of symmetry operations (isometry, isometries in a Euclidean space) that have a Fixed point (mathematics), fixed point in common. The Origin (mathematics), coordinate origin o ...

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

(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 electron states from unoccupied electron states at zero temperature. The shape of the Fermi surface is derived from the periodicity and sym ...

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 Watkins

DoITPoMS 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