Brillouin zone

The reciprocal lattices (dots) and corresponding 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. 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 (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. 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 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 (point group of the crystal).

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

Critical points

First Brillouin zone of FCC lattice, a truncated octahedron, showing symmetry labels for high symmetry lines and points

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

SymbolDescription
ΓCenter of the Brillouin zone
Simple cube
MCenter of an edge
RCorner point
XCenter of a face
Face-centered cubic
KMiddle of an edge joining two hexagonal faces
LCenter of a hexagonal face
UMiddle of an edge joining a hexagonal and a square face
WCorner point
XCenter of a square face
Body-centered cubic
HCorner point joining four edges
NCenter of a face
PCorner point joining three edges
Hexagonal
ACenter of a hexagonal face
HCorner point
KMiddle of an edge joining two rectangular faces
LMiddle of an edge joining a hexagonal and a rectangular face
MCenter of a rectangular face

Other lattices have different types of high-symmetry points. They can be found in the illustrations below.

Triclinic lattice system TRI(4)

Triclinic Lattice type 1a (TRI1a) BZ.
Triclinic Lattice type 1b (TRI1b) BZ.
Triclinic Lattice type 2a (TRI2a) BZ.
Triclinic Lattice type 2b (TRI2b) BZ.

See below for the aflowlib.org standard.

Monoclinic lattice system MCL(1), MCLC(5)

Monoclinic Lattice (MCL) BZ.
Base Centered Monoclinic Lattice type 1 (MCLC1) BZ.
Base Centered Monoclinic Lattice type 2 (MCLC2) BZ.
Base Centered Monoclinic Lattice type 3 (MCLC3) BZ.
Base Centered Monoclinic Lattice type 4 (MCLC4) BZ.
Base Centered Monoclinic Lattice type 5 (MCLC5) BZ.

See below for the aflowlib.org standard.

Orthorhombic lattice system ORC(1), ORCC(1), ORCI(1), ORCF(3)

Simple Orthorhombic Lattice (ORC) BZ.
Base Centered Orthorhombic Lattice (ORCC) BZ.
Body Centered Orthorhombic Lattice (ORCI) BZ.
Face Centered Orthorhombic Lattice type 1 (ORCF1) BZ.
Face Centered Orthorhombic Lattice type 2 (ORCF2) BZ.
Face Centered Orthorhombic Lattice type 3 (ORCF3) BZ.

See below for the aflowlib.org standard.

Tetragonal lattice system TET(1), BCT(2)

Simple Tetragonal Lattice (TET) BZ.
Body Centered Tetragonal Lattice type 1 (BCT1) BZ.
Body Centered Tetragonal Lattice type 2 (BCT2) BZ.

See below for the aflowlib.org standard.

Rhombohedral lattice system RHL(2)

Rhombohedral Lattice type 1 (RHL1) BZ.
Rhombohedral Lattice type 2 (RHL2) BZ.

See below for the aflowlib.org standard.

Hexagonal lattice system HEX(1)

Hexagonal Lattice (HEX) BZ.

See below for the aflowlib.org standard. More details in links below.

Cubic lattice system CUB(1), BCC(1), FCC(1)

Simple Cubic Lattice (CUB) BZ.
Body Centered Cubic Lattice (BCC) BZ.
Face Centered Cubic Lattice (FCC) BZ.

See below for the aflowlib.org standard.

See also

Brillouin-zone construction by 300 keV electrons.

References

  1. ↑ Ibach, Harald; Lüth, Hans (1996). Solid-State Physics, An Introduction to Principles of Materials Science (2nd ed.). Springer-Verlag. ISBN 3-540-58573-7.

External links

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