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. 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.

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.

Triclinic lattice system TRI(4)

File:TRI1a.PNG Triclinic Lattice type 1a (TRI1a) BZ.

File:TRI1b.PNG Triclinic Lattice type 1b (TRI1b) BZ.

File:TRI2a.PNG Triclinic Lattice type 2a (TRI2a) BZ.

File:TRI2b.PNG Triclinic Lattice type 2b (TRI2b) BZ.

See below for the aflowlib.org standard.

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

File:MCL.PNG Monoclinic Lattice (MCL) BZ.

File:MCLC1.PNG Base Centered Monoclinic Lattice type 1 (MCLC1) BZ.

File:Base-Centered Monoclinic Lattice type 2 (Brillouin zone).png Base Centered Monoclinic Lattice type 2 (MCLC2) BZ.

File:MCLC3.PNG Base Centered Monoclinic Lattice type 3 (MCLC3) BZ.

File:MCLC4.PNG Base Centered Monoclinic Lattice type 4 (MCLC4) BZ.

File:MCLC5.PNG 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)

File:Simple Orthorhombic Lattice (Brillouin zone).png Simple Orthorhombic Lattice (ORC) BZ.

File:Base-Centered Orthorhombic Lattice (Brillouin zone).png Base Centered Orthorhombic Lattice (ORCC) BZ.

File:Body-Centered Orthorhombic Lattice (Brillouin zone).png Body Centered Orthorhombic Lattice (ORCI) BZ.

File:Face-Centered Orthorhombic Lattice type 1 (Brillouin zone).png Face Centered Orthorhombic Lattice type 1 (ORCF1) BZ.

File:Face-Centered Orthorhombic Lattice type 2 (Brillouin zone).png Face Centered Orthorhombic Lattice type 2 (ORCF2) BZ.

File:Face-Centered Orthorhombic Lattice type 3 (Brillouin zone).png Face Centered Orthorhombic Lattice type 3 (ORCF3) BZ.

See below for the aflowlib.org standard.

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

File:TET.PNG Simple Tetragonal Lattice (TET) BZ.

File:BCT1.PNG Body Centered Tetragonal Lattice type 1 (BCT1) BZ.

File:BCT2.PNG Body Centered Tetragonal Lattice type 2 (BCT2) BZ.

See below for the aflowlib.org standard.

Rhombohedral lattice system RHL(2)

File:RHL1.PNG Rhombohedral Lattice type 1 (RHL1) BZ.

File:RHL2.PNG Rhombohedral Lattice type 2 (RHL2) BZ.

See below for the aflowlib.org standard.

Hexagonal lattice system HEX(1)

File:HEX.PNG 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)

File:Simple Cubic Lattice (Brillouin zone).png Simple Cubic Lattice (CUB) BZ.

File:BCC.PNG Body Centered Cubic Lattice (BCC) BZ.

File:FCC.PNG Face Centered Cubic Lattice (FCC) BZ.

See below for the aflowlib.org standard.

See also

File:Brillouin-zone construction by 300keV electrons.jpg

Brillouin-zone construction by 300-keV electrons.

• Fundamental pair of periods
• Fundamental domain

External links

• Brillouin Zone simple lattice diagrams by Thayer Watkins
• Brillouin Zone 3d lattice diagrams by Technion.
• DoITPoMS Teaching and Learning Package- "Brillouin Zones"
• Aflowlib.org consortium database (Duke University)
• AFLOW Standardization of VASP/QUANTUM ESPRESSO input files (Duke University)