Bravais lattice

In geometry and crystallography, a Bravais lattice, named after Auguste Bravais (1850),^[1] is an infinite array (or a finite array, if we consider the edges, obviously) of discrete points generated by a set of discrete translation operations described in three dimensional space 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 (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) 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 motif).

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 2 dimensions

1 – oblique (monoclinic), 2 – rectangular (orthorhombic), 3 – centered rectangular (orthorhombic), 4 – hexagonal, and 5 – square (tetragonal).

In two-dimensional space, there are 5 Bravais lattices,^[2] grouped into four crystal families.

Crystal family	Point group (Schönflies notation)	5 Bravais lattices
Crystal family	Point group (Schönflies notation)	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	$ab\,\sin \theta$	a ≠ b	θ ≠ 90°
Orthorhombic	$ab$	a ≠ b	θ = 90°
Hexagonal	${\frac {\sqrt {3}}{2}}\,a^{2}$	a = b	θ = 120°
Tetragonal	$a^{2}$	a = b	θ = 90°

In 3 dimensions

2×2×2 unit cells of a diamond cubic lattice

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.^[3]

Crystal Family	Lattice System	Schönflies	14 Bravais Lattices
Crystal Family	Lattice System	Schönflies	Primitive (P)	Base-centered (C)	Body-centered (I)	Face-centered (F)
Triclinic		C_i
Monoclinic		C_2h
Orthorhombic		D_2h
Tetragonal		D_4h
Hexagonal	Rhombohedral	D_3d
Hexagonal	Hexagonal	D_6h
Cubic		O_h

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)^[4]	Axial angles^[4]	Corresponding examples
Triclinic		$abc \sqrt{1-\cos^2\alpha-\cos^2\beta-\cos^2\gamma+2\cos\alpha \cos\beta \cos\gamma}$	(All remaining cases)		K₂Cr₂O₇, CuSO₄·5H₂O, H₃BO₃
Monoclinic		$abc\,\sin \beta$	a ≠ c	α = γ = 90°, β ≠ 90°	Monoclinic sulphur, Na₂SO₄·10H₂O, PbCrO₃
Orthorhombic		$abc$	a ≠ b ≠ c	α = β = γ = 90°	Rhombic sulphur, KNO₃, BaSO₄
Tetragonal		$a^2c$	a = b ≠ c	α = β = γ = 90°	White tin, SnO₂, TiO₂, CaSO₄
Hexagonal	Rhombohedral	$a^3 \sqrt{1 - 3\cos^2\alpha + 2\cos^3\alpha}$	a = b = c	α = β = γ ≠ 90°	Calcite (CaCO₃), cinnabar (HgS)
Hexagonal	Hexagonal	${\frac {\sqrt {3}}{2}}\,a^{2}c$	a = b	α = β = 90°, γ = 120°	Graphite, ZnO, CdS
Cubic		$a^3$	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.^[5]

References

↑ Aroyo, Mois I.; Müller, Ulrich; Wondratschek, Hans (2006). "Historical Introduction". International Tables for Crystallography. A1 (1.1): 2–5. CiteSeerX 10.1.1.471.4170. doi:10.1107/97809553602060000537. Archived from the original on 2013-07-04. 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 978-0-471-11181-8. Retrieved 2008-04-21.
↑ Based on the list of conventional cells found in Hahn (2002), p. 744
1 2 Hahn (2002), p. 758
↑ 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

External links

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

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[1] Aroyo, Mois I.; Müller, Ulrich; Wondratschek, Hans (2006). "Historical Introduction". International Tables for Crystallography. A1 (1.1): 2–5. CiteSeerX 10.1.1.471.4170. doi:10.1107/97809553602060000537. Archived from the original on 2013-07-04. 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 978-0-471-11181-8. Retrieved 2008-04-21.

[3] Based on the list of conventional cells found in Hahn (2002), p. 744

[axial_dimensions-4] 1 2 Hahn (2002), p. 758

[5] 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