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In geometry and crystallography, a Bravais lattice, studied by Auguste Bravais (1850),^[1] is an infinite set of 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 $n i$ are any integers and $\mathbf{a}_{i}$ are known as the primitive vectors which lie in different planes and span the lattice. For any choice of position vector $\mathbf{R}$ , the lattice looks exactly the same.

A crystal is made up of one or more atoms (the basis) which is repeated at each lattice point. The crystal then looks the same when viewed from any of the lattice points.

Two Bravais lattices are often considered to be 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.

[edit] 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. They are oblique, rectangular, centered rectangular, hexagonal, and square.^[2] There are 4 lattice systems, as the centered rectangular and rectangular lattices are in the same lattice system.

The five fundamental two-dimensional Bravais lattices: 1 oblique, 2 rectangular, 3 centered rectangular, 4 hexagonal, and 5 square

[edit] Bravais lattices in 3 dimensions

The 14 Bravais lattices in 3 dimensions are arrived at by combining one of the seven 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 centering (P): lattice points on the cell corners only
Body centered (I): one additional lattice point at the center of the cell
Face centered (F): one additional lattice point at center of each of the faces of the cell
Centered on a single face (A, B or C centering): one additional lattice point at the center of one 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-centered 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
triclinic
monoclinic	P	C
monoclinic
orthorhombic	P	C	I	F
orthorhombic
tetragonal	P	I
tetragonal
rhombohedral	P
rhombohedral
hexagonal	P
hexagonal
cubic	P (pcc)	I (bcc)	F (fcc)
cubic

The volume of the unit cell can be calculated by evaluating $\mathbf{a} \cdot \mathbf{b} \times \mathbf{c}$ where $\mathbf{a}, \mathbf{b}$ , and $\mathbf{c}$ 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	$abc ~ \sin\alpha$
Orthorhombic	$a b c$
Tetragonal	$a 2 c$
rhombohedral	$a^3 \sqrt{1 - 3\cos^2\alpha + 2\cos^3\alpha}$
Hexagonal	$\frac{3\sqrt{3\,}\, a^2c}{2}$
Cubic	$a 3$

[edit] Bravais lattices in 4 dimensions

In four dimensions, there are 52 Bravais lattices. Of these, 21 are primitive and 31 are centered.^[3]

[edit] See also

[edit] References

^ Aroyo, Mois I.; Ulrich Müller and Hans Wondratschek (2006). "Historical Introduction". International Tables for Crystallography (Springer) A1 (1.1): 2–5. doi:10.1107/97809553602060000537. http://it.iucr.org/A1a/ch1o1v0001/sec1o1o1/. Retrieved 2008-04-21.
^ Kittel, Charles (1996) [1953]. "Chapter 1". Introduction to Solid State Physics (Seventh ed.). New York: John Wiley & Sons. pp. 10. ISBN 0-471-11181-3. http://www.wiley.com/WileyCDA/WileyTitle/productCd-047141526X.html. Retrieved 2008-04-21.
^ Mackay AL and Pawley GS (1963). "Bravais Lattices in Four-dimensional Space". Acta. Cryst. 16: 11–19. doi:10.1107/S0365110X63000037.

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 (5th ed.), Berlin, New York: Springer-Verlag, doi:10.1107/97809553602060000100, ISBN 978-0-7923-6590-7, http://it.iucr.org/A/

[edit] External links

Smith, Walter Fox (2002), The Bravais Lattices Song, http://www.haverford.edu/physics-astro/songs/bravais.htm

[0] Aroyo, Mois I.; Ulrich Müller and Hans Wondratschek (2006). "Historical Introduction". International Tables for Crystallography (Springer) A1 (1.1): 2–5. doi:10.1107/97809553602060000537. http://it.iucr.org/A1a/ch1o1v0001/sec1o1o1/. Retrieved 2008-04-21.

[1] Kittel, Charles (1996) [1953]. "Chapter 1". Introduction to Solid State Physics (Seventh ed.). New York: John Wiley & Sons. pp. 10. ISBN 0-471-11181-3. http://www.wiley.com/WileyCDA/WileyTitle/productCd-047141526X.html. Retrieved 2008-04-21.

[2] Mackay AL and Pawley GS (1963). "Bravais Lattices in Four-dimensional Space". Acta. Cryst. 16: 11–19. doi:10.1107/S0365110X63000037.

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