Bravais lattice

In geometry and crystallography, a Bravais lattice, studied by Auguste Bravais (1850),^[1] is an infinite array of discrete points in three dimensional space 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 n_i are any integers and a_i 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 at most two dimensions

In zero-dimensional and one-dimensional space, there is only one type of Bravais lattice.

In two-dimensional space, there are five Bravais lattices: oblique, rectangular, centered rectangular, hexagonal (rhombic), and square.^[2]

The five fundamental two-dimensional Bravais lattices: 1 – oblique, 2 – rectangular, 3 – centered rectangular, 4 – hexagonal (rhombic), 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

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 7 lattice systems	The 14 Bravais lattices
Triclinic	P
Monoclinic	P	C
Orthorhombic	P	C	I	F
Tetragonal	P	I
Rhombohedral	R or 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	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 sulfur, 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 sulfur, 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

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

Further reading

• Bravais, A. (1850). "Mémoire sur les systèmes formés par les points distribués régulièrement sur un plan ou dans l'espace". J. Ecole Polytech. 19: 1–128. (English: Memoir 1, Crystallographic Society of America, 1949.)
• Hahn, Theo, ed. (2002). International Tables for Crystallography, Volume A: Space Group Symmetry A (5th ed.). Berlin, New York: Springer-Verlag. doi:10.1107/97809553602060000100. ISBN 978-0-7923-6590-7.

External links

• Catalogue of Lattices (by Nebe and Sloane)
• Smith, Walter Fox (2002). "The Bravais Lattices Song".