Brillouin zone

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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 of the reciprocal lattice in the frequency domain. 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 the 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 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.

The concept of a Brillouin zone was developed by Leon Brillouin (1889-1969), a French physicist.

1 Critical points
2 See also
3 References
4 External links

[edit] 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]

Symbol	Description
Γ	Center of the Brillouin zone
Simple cube
M	Center of an edge
R	Corner point
X	Center of a face
Face-centered cubic
K	Middle of an edge joining two hexagonal faces
L	Center of a hexagonal face
U	Middle of an edge joining a hexagonal and a square face
W	Corner point
X	Center of a square face
Body-centered cubic
H	Corner point joining four edges
N	Center of a face
P	Corner point joining three edges
Hexagonal
A	Center of a hexagonal face
H	Corner point
K	Middle of an edge joining two rectangular faces
L	Middle of an edge joining a hexagonal and a rectangular face
M	Center of a rectangular face

[edit] See also

[edit] References

^ 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).
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). (original article)