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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.

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) 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 3 dimensions

In three-dimensional space, there are 14 Bravais lattices. These are obtained by combining one of the seven lattice systems (or axial systems) with one of the seven lattice types (or lattice centerings). In general, the lattice systems can be characterized by their shapes according to the relative lengths of the cell edges (a, b, c) and the angles between them (α, β, γ). The lattice 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)
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
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)
Rhombohedral (R): lattice points on the cell corners only where a = b = c and α = β = γ ≠ 90° (special case for the rhombohedral lattice system)

Not all combinations of lattice systems and lattice types are needed to describe all of the possible lattices. If we consider R equivalent to P, then 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 rhombohedral lattice is officially assigned as type R in order to distinguish it from the hexagonal lattice in the trigonal crystal system. However, for simplicity this lattice is often shown as type P.

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 :
L

Lattice System	Possible Variations	Axial Distances (edge lengths)	Axial Angles	Corresponding 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]
See also

Translational symmetry
Lattice (group)
classification of lattices
Miller Index
Translation operator (quantum mechanics)

References

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.
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.

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