Brillouin zone

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. The boundaries of this cell are given by planes related to points on the reciprocal lattice. It is found by the same method as for the Wigner–Seitz cell in the Bravais 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.

Taking surfaces at the same distance from one element of the lattice and its neighbours, the volume included is the first Brillouin zone. 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 more rarely. 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.

The concept of a Brillouin zone was developed by Léon Brillouin (1889-1969), 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

• Fundamental pair of periods
• Fundamental domain

References

1. ^ Ibach, Harald; Hans Lüth (1996). Solid-State Physics, An Introduction to Principles of Materials Science (Second ed.). Springer-Verlag. ISBN 3-540-58573-7. 

• Kittel, Charles (1996). Introduction to Solid State Physics. New York City: Wiley. 
• Ashcroft, Neil W.; N. David Mermin (1976). Solid State Physics. Orlando: Harcourt. 
• Brillouin, Léon (1930). "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). http://gallica.bnf.fr/ark:/12148/bpt6k31445. 

External links

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