First Brillouin zones of (a) square lattice and (b) hexagonal lattice. In Mathematics, the square lattice is one of the five two-dimensional lattice types The hexagonal lattice or equilateral triangular lattice is one of the five 2D lattice types

In mathematics and solid state physics, the first Brillouin zone is a uniquely defined primitive cell of the reciprocal lattice in the frequency domain. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Solid-state physics, the largest branch of Condensed matter physics, is the study of rigid Matter, or Solids The bulk of solid-state physics theory and In Geometry, Solid state physics and Mineralogy, particularly in describing Crystal structure, a primitive cell, is a minimum cell corresponding In Crystallography, the reciprocal lattice of a Bravais lattice is the set of all vectors K such that e^{i\mathbf{K}\cdot\mathbf{R}}=1 Frequency domain is a term used to describe the analysis of Mathematical functions or signals with respect to frequency It is found by the same method as for the Wigner-Seitz cell in the Bravais lattice. The Wigner-Seitz cell (named after E P Wigner and Frederick Seitz) is a geometrical construction which helps in the study of Crystalline material in In Geometry and Crystallography, a Bravais lattice, named after Auguste Bravais, is an infinite set of points generated by a set of discrete translation 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. A Bloch wave or Bloch state, named after Felix Bloch, is the Wavefunction of a particle (usually an Electron) placed in a periodic potential

Taking the surfaces at the same distance from one element of the lattice and its neighbours, the volume included is the first Brillouin zone. The volume of any solid plasma vacuum or theoretical object is how much three- Dimensional space it occupies often quantified numerically 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, Bragg's law is the result of experiments into the Diffraction of X-rays or neutrons off Crystal surfaces at certain angles Equivalently, this is the Voronoi cell around the origin of the reciprocal lattice. In Mathematics, a Voronoi diagram, named after Georgy Voronoi, also called a Voronoi Tessellation, a Voronoi decomposition, or

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 more rarely. As a result, the first Brillouin zone is often called simply the Brillouin zone. (In general, the n-th Brillouin zone consist of the set of points that can be reached from the origin by crossing n − 1 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. In Mathematics, a point group is a group of geometric symmetries ( isometries) leaving a point fixed

The concept of a Brillouin zone was developed by Leon Brillouin (1889-1969), a French physicist. Léon Nicolas Brillouin ( August 7, 1889 &ndash December 1969 was a French physicist

Critical points

First Brillouin zone of FCC lattice showing symmetry labels for high symmetry lines and points

Several points of high symmetry are of special interest – these are called critical points. ^[1]

See also

• period lattice
• fundamental domain

References

1. ^ Harald Ibach & Hans Lüth, Solid-State Physics, An Introduction to Principles of Materials Science, corrected second printing of the second edition, 1996, Springer-Verlag, ISBN 3-540-58573-7

• Charles Kittel, Introduction to Solid State Physics (Wiley: New York, 1996). In Mathematics, a fundamental pair of periods is an Ordered pair of Complex numbers that define a lattice in the Complex plane. In Geometry, the fundamental domain of a Symmetry group of an object or pattern is a part of the pattern as small as possible which based on the Symmetry
• Neil W. Ashcroft and N. David Mermin, Solid State Physics (Harcourt: Orlando, 1976).
• Léon Brillouin Les électrons dans les métaux et le classement des ondes de de Broglie correspondantes Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences, 191, 292 (1930). Léon Nicolas Brillouin ( August 7, 1889 &ndash December 1969 was a French physicist (original article)

External links

• Brillouin Zone simple lattice diagrams by Thayer Watkins
• Brillouin Zone 3d lattice diagrams by Technion.