Bravais lattice
In geometry and crystallography, a Bravais lattice, stud-ied by
Auguste Bravais (1850),[1] is an innite array ofdiscrete points
generated by a set of discrete translationoperations described
by:
R = n1a1 + n2a2 + n3a3where ni are any integers and a are known
as the primi-tive vectors which lie in dierent directions and span
thelattice. This discrete set of vectors must be closed undervector
addition and subtraction. For any choice of posi-tion vector R, the
lattice looks exactly the same.When the discrete points are atoms,
ions, or polymerstrings of solid matter, the Bravais lattice
concept is usedto formally dene a crystalline arrangement and its
(nite)frontiers. A crystal is made up of a periodic arrangementof
one or more atoms (the basis) repeated at each latticepoint.
Consequently, the crystal looks the same whenviewed from any
equivalent lattice point, namely thoseseparated by the translation
of one unit cell (the motive).Two Bravais lattices are often
considered equivalent ifthey have isomorphic symmetry groups. In
this sense,there are 14 possible Bravais lattices in
three-dimensionalspace. The 14 possible symmetry groups of Bravais
lat-tices are 14 of the 230 space groups.
1 Bravais lattices in at most 2 di-mensions
In zero-dimensional and one-dimensional space, there isonly one
type of Bravais lattice.In two-dimensional space, there are ve
Bravais lattices:oblique, rectangular, centered rectangular,
hexagonal(rhombic), and square.[2]
2 Bravais lattices in 3 dimensionsIn three-dimensional space,
there are 14 Bravais lattices.These are obtained by combining one
of the seven latticesystems (or axial systems) with one of the
seven latticetypes (or lattice centerings). In general, the lattice
sys-tems can be characterized by their shapes according tothe
relative lengths of the cell edges (a, b, c) and the an-gles
between them (, , ). The lattice types identify thelocations of the
lattice points in the unit cell as follows:
|a | = |a |, = 901 2|a | = |a |, = 1201 2
a1
|a | |a |, = 901 2 |a | |a |, 901 2|a | |a |, 901 21 2 3
54
a1
a2 a2 a1
a2
a1
a2
a1
a2
The ve fundamental two-dimensional Bravais lattices: 1oblique, 2
rectangular, 3 centered rectangular, 4 hexagonal(rhombic), and 5
square. In addition to the stated conditions,the centered
rectangular lattice fullls 2a2 a1 ? a1 . This or-thogonality
condition leads to the rectangular pattern indicatedand implies '
6= 90 .
Primitive (P): lattice points on the cell corners only(sometimes
called simple)
Body-Centered (I): lattice points on the cell cornerswith one
additional point at the center of the cell
Face-Centered (F): lattice points on the cell cornerswith one
additional point at the center of each of thefaces of the cell
Base-Centered (A, B, or C): lattice points on thecell corners
with one additional point at the centerof each face of one pair of
parallel faces of the cell(sometimes called end-centered)
Rhombohedral (R): lattice points on the cell cornersonly where a
= b = c and = = 90 (specialcase for the rhombohedral lattice
system)
Not all combinations of lattice systems and lattice typesare
needed to describe all of the possible lattices. If weconsider R
equivalent to P, then there are in total 7 6= 42 combinations, but
it can be shown that several ofthese are in fact equivalent to each
other. For example,the monoclinic I lattice can be described by a
monoclinicC lattice by dierent choice of crystal axes.
Similarly,all A- or B-centred lattices can be described either by
aC- or P-centering. This reduces the number of combina-tions to 14
conventional Bravais lattices, shown in the ta-ble below. The
rhombohedral lattice is ocially assignedas type R in order to
distinguish it from the hexagonal lat-
1
2 7 EXTERNAL LINKS
tice in the trigonal crystal system. However, for simplicitythis
lattice is often shown as type P.The volume of the unit cell can be
calculated by evaluat-ing a b c where a, b, and c are the lattice
vectors. Thevolumes of the Bravais lattices are given below:Centred
Unit Cells :
3 Bravais lattices in 4 dimensionsIn four dimensions, there are
64 Bravais lattices. Ofthese, 23 are primitive and 41 are centered.
Ten Bravaislattices split into enantiomorphic pairs.[3]
4 See also Translational symmetry Lattice (group) classication
of lattices Miller Index Translation operator (quantum
mechanics)
5 References[1] Aroyo, Mois I.; Ulrich Mller; Hans
Wondratschek
(2006). Historical Introduction. International Ta-bles for
Crystallography (Springer) A1 (1.1):
25.doi:10.1107/97809553602060000537. Retrieved 2008-04-21.
[2] Kittel, Charles (1996) [1953]. Chapter 1. Introductionto
Solid State Physics (Seventh ed.). New York: John Wi-ley &
Sons. p. 10. ISBN 0-471-11181-3. Retrieved2008-04-21.
[3] Brown, Harold; Blow, Rolf; Neubser, Joachim; Won-dratschek,
Hans; Zassenhaus, Hans (1978), Crystallo-graphic groups of
four-dimensional space, New York:Wiley-Interscience [John Wiley
& Sons], ISBN 978-0-471-03095-9, MR 0484179
6 Further reading Bravais, A. (1850). Mmoire sur les systmes
for-ms par les points distribus rgulirement sur unplan ou dans
l'espace. J. Ecole Polytech. 19: 1128. (English: Memoir 1,
Crystallographic Societyof America, 1949.)
Hahn, Theo, ed. (2002). International Tables forCrystallography,
Volume A: Space Group SymmetryA (5th ed.). Berlin, New York:
Springer-Verlag.doi:10.1107/97809553602060000100. ISBN
978-0-7923-6590-7.
7 External links Catalogue of Lattices (by Nebe and Sloane)
Smith, Walter Fox (2002). The Bravais LatticesSong.
38 Text and image sources, contributors, and licenses8.1
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Bravais lattices in at most 2 dimensionsBravais lattices in 3
dimensionsBravais lattices in 4 dimensionsSee alsoReferencesFurther
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