Brillouin zone


In mathematics and solid state physics, the first Brillouin zone is a uniquely defined primitive cell in reciprocal space. In the same way the Bravais lattice is divided up into Wigner–Seitz cells 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 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.
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. 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.
of waves in the lattice, because it corresponds to a half-wavelength equal to the inter-atomic lattice spacing a. See also for more on the equivalence of k-vectors.
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There are also second, third, etc., Brillouin zones, corresponding to a sequence of disjoint regions 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 of the lattice.
The concept of a Brillouin zone was developed by Léon Brillouin, a French physicist.

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.
Lattice systemBravais lattice
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TriclinicPrimitive triclinic
Triclinic Lattice type 1a
Triclinic Lattice type 1b
Triclinic Lattice type 2a
Triclinic Lattice type 2b
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MonoclinicPrimitive monoclinic
Monoclinic Lattice
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MonoclinicBase-centered monoclinic
Base Centered Monoclinic Lattice type 1
Base Centered Monoclinic Lattice type 2
Base Centered Monoclinic Lattice type 3
Base Centered Monoclinic Lattice type 4
Base Centered Monoclinic Lattice type 5
OrthorhombicPrimitive orthorhombic
Simple Orthorhombic Lattice
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OrthorhombicBase-centered orthorhombic
Base Centered Orthorhombic Lattice
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OrthorhombicBody-centered orthorhombic
Body Centered Orthorhombic Lattice
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OrthorhombicFace-centered orthorhombic
Face Centered Orthorhombic Lattice type 1
Face Centered Orthorhombic Lattice type 2
Face Centered Orthorhombic Lattice type 3
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TetragonalPrimitive tetragonal
Simple Tetragonal Lattice
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TetragonalBody-centered Tetragonal
Body Centered Tetragonal Lattice type 1
Body Centered Tetragonal Lattice type 2
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RhombohedralPrimitive rhombohederal
Rhombohedral Lattice type 1
Rhombohedral Lattice type 2
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HexagonalPrimitive hexagonal
Hexagonal Lattice
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CubicPrimitive cubic
Simple Cubic Lattice
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CubicBody-centered cubic
Body Centered Cubic Lattice
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CubicFace-centered cubic
Face Centered Cubic Lattice
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