# Bravais lattice

In geometry and crystallography, a Bravais lattice, named after Auguste Bravais (1850),[1] is an infinite array of discrete points generated by a set of discrete translation operations described in three dimensional space by:

${\displaystyle \mathbf {R} =n_{1}\mathbf {a} _{1}+n_{2}\mathbf {a} _{2}+n_{3}\mathbf {a} _{3}}$

where ni are any integers and ai are primitive vectors which lie in different directions (not necessarily mutually perpendicular) and span the lattice. This discrete set of vectors must be closed under vector addition and subtraction. For any choice of position vector R, the lattice looks exactly the same.

When the discrete points are atoms, ions, or polymer strings of solid matter, the Bravais lattice concept is used to formally define a crystalline arrangement and its (finite) frontiers. A crystal is made up of a periodic arrangement of one or more atoms (the basis, or motif) repeated at each lattice point. Consequently, the crystal looks the same when viewed from any equivalent lattice point, namely those separated by the translation of one unit cell.

Two Bravais lattices are often considered equivalent if they have isomorphic symmetry groups. In this sense, there are 14 possible Bravais lattices in three-dimensional space. The 14 possible symmetry groups of Bravais lattices are 14 of the 230 space groups. In the context of the space group classification, the Bravais lattices are also called Bravais classes, Bravais arithmetic classes, or Bravais flocks.[2]

## In 2 dimensions

In two-dimensional space, there are 5 Bravais lattices,[3] grouped into four crystal families.

Crystal family Point group
(Schönflies notation)
5 Bravais lattices
Primitive Centered
Monoclinic C2 Oblique
Orthorhombic D2 Rectangular Centered rectangular
Hexagonal D6 Hexagonal
Tetragonal D4 Square

The unit cells are specified according to the relative lengths of the cell edges (a and b) and the angle between them (θ). The area of the unit cell can be calculated by evaluating the norm ||a × b||, where a and b are the lattice vectors. The properties of the crystal families are given below:

Crystal family Area Axial distances (edge lengths) Axial angle
Monoclinic ${\displaystyle ab\,\sin \theta }$ ab θ ≠ 90°
Orthorhombic ${\displaystyle ab}$ ab θ = 90°
Hexagonal ${\displaystyle {\frac {3{\sqrt {3}}}{2}}\,a^{2}}$ a = b θ = 120°
Tetragonal ${\displaystyle a^{2}}$ a = b θ = 90°

## In 3 dimensions

In three-dimensional space, there are 14 Bravais lattices. These are obtained by combining one of the seven lattice systems with one of the centering types. The centering types identify the locations of the lattice points in the unit cell as follows:

• Primitive (P): lattice points on the cell corners only (sometimes called simple)
• Base-centered (A, B, or C): lattice points on the cell corners with one additional point at the center of each face of one pair of parallel faces of the cell (sometimes called end-centered)
• Body-centered (I): lattice points on the cell corners, with one additional point at the center of the cell
• Face-centered (F): lattice points on the cell corners, with one additional point at the center of each of the faces of the cell

Not all combinations of lattice systems and centering types are needed to describe all of the possible lattices, as it can be shown that several of these are in fact equivalent to each other. For example, the monoclinic I lattice can be described by a monoclinic C lattice by different choice of crystal axes. Similarly, all A- or B-centred lattices can be described either by a C- or P-centering. This reduces the number of combinations to 14 conventional Bravais lattices, shown in the table below.[4] Below each diagram is the Pearson symbol for that Bravais lattice.

Crystal family Lattice system Schönflies 14 Bravais lattices
Primitive (P) Base-centered (C) Body-centered (I) Face-centered (F)
Triclinic Ci

aP

Monoclinic C2h

mP

mS

Orthorhombic D2h

oP

oS

oI

oF

Tetragonal D4h

tP

tI

Hexagonal Rhombohedral D3d

hR

Hexagonal D6h

hP

Cubic Oh

cP

cI

cF

The unit cells are specified according to the relative lengths of the cell edges (a, b, c) and the angles between them (α, β, γ). The volume of the unit cell can be calculated by evaluating the triple product a · (b × c), where a, b, and c are the lattice vectors. The properties of the lattice systems are given below:

Crystal family Lattice system Volume Axial distances (edge lengths)[5] Axial angles[5] Corresponding examples
Triclinic ${\displaystyle abc{\sqrt {1-\cos ^{2}\alpha -\cos ^{2}\beta -\cos ^{2}\gamma +2\cos \alpha \cos \beta \cos \gamma }}}$ (All remaining cases) K2Cr2O7, CuSO4·5H2O, H3BO3
Monoclinic ${\displaystyle abc\,\sin \beta }$ ac α = γ = 90°, β ≠ 90° Monoclinic sulphur, Na2SO4·10H2O, PbCrO3
Orthorhombic ${\displaystyle abc}$ abc α = β = γ = 90° Rhombic sulphur, KNO3, BaSO4
Tetragonal ${\displaystyle a^{2}c}$ a = bc α = β = γ = 90° White tin, SnO2, TiO2, CaSO4
Hexagonal Rhombohedral ${\displaystyle a^{3}{\sqrt {1-3\cos ^{2}\alpha +2\cos ^{3}\alpha }}}$ a = b = c α = β = γ ≠ 90° Calcite (CaCO3), cinnabar (HgS)
Hexagonal ${\displaystyle {\frac {\sqrt {3}}{2}}\,a^{2}c}$ a = b α = β = 90°, γ = 120° Graphite, ZnO, CdS
Cubic ${\displaystyle a^{3}}$ a = b = c α = β = γ = 90° NaCl, zinc blende, copper metal, KCl, Diamond, Silver

## In 4 dimensions

In four dimensions, there are 64 Bravais lattices. Of these, 23 are primitive and 41 are centered. Ten Bravais lattices split into enantiomorphic pairs.[6]