# Bravais lattice

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

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

where ni are any integers and ai are known as the primitive vectors which lie in different directions 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.

A crystal is made up of a periodic arrangement of one or more atoms (the basis) 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 (the motive).

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.

## Bravais lattices in at most 2 dimensions

In each of 0-dimensional and 1-dimensional space there is just one type of Bravais lattice.

In two dimensions, there are five Bravais lattices, called oblique, rectangular, centered rectangular (rhombic), hexagonal, and square.[2]

The five fundamental two-dimensional Bravais lattices: 1 oblique, 2 rectangular, 3 centered rectangular (rhombic), 4 hexagonal, and 5 square. In addition to the stated conditions, the centered rectangular lattice fulfills $2\mathbf{a}_2-\mathbf{a}_1\perp \mathbf{a}_1$. This orthogonality condition leads to the rectangular pattern indicated and implies $\varphi \neq 90^\circ$.

## Bravais lattices in 3 dimensions

The 14 Bravais lattices in 3 dimensions are obtained by coupling one of the 7 lattice systems (or axial systems) with one of the lattice centerings. Each Bravais lattice refers to a distinct lattice type.

The lattice centerings are:

• Primitive (P): lattice points on the cell corners only.
• Body (I): one additional lattice point at the center of the cell.
• Face (F): one additional lattice point at the center of each of the faces of the cell.
• Base (A, B or C): one additional lattice point at the center of each of one pair of the cell faces.

Not all combinations of the crystal systems and lattice centerings are needed to describe the possible lattices. There are in total 7 × 6 = 42 combinations, but 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.

The 7 lattice systems The 14 Bravais lattices
Triclinic P
Monoclinic P C
Orthorhombic P C I F
Tetragonal P I
Rhombohedral P
Hexagonal P
Cubic P (pcc) I (bcc) F (fcc)

The volume of the unit cell can be calculated by evaluating a · b × c where a, b, and c are the lattice vectors. The volumes of the Bravais lattices are given below:

Lattice system Volume
Triclinic $abc \sqrt{1-\cos^2\alpha-\cos^2\beta-\cos^2\gamma+2\cos\alpha \cos\beta \cos\gamma}$
Monoclinic $abc ~ \sin\beta$
Orthorhombic $abc$
Tetragonal $a^2c$
Rhombohedral $a^3 \sqrt{1 - 3\cos^2\alpha + 2\cos^3\alpha}$
Hexagonal $\frac{\sqrt{3\,}\, a^2c}{2}$
Cubic $a^3$

Centred Unit Cells :

Lattice System Possible Variations Axial Distances (edge lengths) Axial Angles Examples
Cubic Primitive, Body-centred, Face-centred a = b = c α = β = γ = 90° NaCl, Zinc Blende, Cu
Tetragonal Primitive, Body-centred a = b ≠ c α = β = γ = 90° White tin, SnO2, TiO2, CaSO4
Orthorhombic Primitive, Body-centred, Face-centred, Base-centred a ≠ b ≠ c α = β = γ = 90° Rhombic sulphur, KNO3, BaSO4
Hexagonal Primitive a = b ≠ c α = β = 90°, γ = 120° Graphite, ZnO, CdS
Rhombohedral Primitive a = b = c α = β = γ ≠ 90° Calcite (CaCO3), Cinnabar (HgS)
Monoclinic Primitive, Base-centred a ≠ b ≠ c α = γ = 90°, β ≠ 90° Monoclinic sulphur, Na2SO4.10H2O
Triclinic Primitive a ≠ b ≠ c α ≠ β ≠ γ ≠ 90° K2Cr2O7, CuSO4.5H2O, H3BO3

## Bravais lattices 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.[3]

## References

1. ^ Aroyo, Mois I.; Ulrich Müller; Hans Wondratschek (2006). "Historical Introduction". International Tables for Crystallography (Springer) A1 (1.1): 2–5. doi:10.1107/97809553602060000537. Retrieved 2008-04-21.
2. ^ Kittel, Charles (1996) [1953]. "Chapter 1". Introduction to Solid State Physics (Seventh ed.). New York: John Wiley & Sons. p. 10. ISBN 0-471-11181-3. Retrieved 2008-04-21.
3. ^ Brown, Harold; Bülow, Rolf; Neubüser, Joachim; Wondratschek, Hans; Zassenhaus, Hans (1978), Crystallographic groups of four-dimensional space, New York: Wiley-Interscience [John Wiley & Sons], ISBN 978-0-471-03095-9, MR 0484179