Bravais lattice 1

Bravais latticeIn geometry and crystallography, a Bravais lattice, studied by Auguste Bravais(1850), is an infinite array ofdiscrete points generated by a set of discrete translation operations described by:

where ni are any integers and ai are known as the primitive vectors which lie in different directions and span thelattice. This discrete set of vectors must be closed under vector addition and subtraction. For any choice of positionvector R, the lattice looks exactly the same.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 bythe translation of one unit cell (the motive).Two Bravais lattices are often considered equivalent if they have isomorphic symmetry groups. In this sense, thereare 14 possible Bravais lattices in three-dimensional space. The 14 possible symmetry groups of Bravais lattices are14 of the 230 space groups.

Bravais lattices in at most 2 dimensionsIn each of 0-dimensional and 1-dimensional space there is just one type of Bravais lattice.In two dimensions, there are five Bravais lattices, called oblique, rectangular, centered rectangular (rhombic),hexagonal, and square.

The five fundamental two-dimensional Bravais lattices: 1 oblique, 2 rectangular, 3 centered rectangular (rhombic), 4 hexagonal, and 5 square. Inaddition to the stated conditions, the centered rectangular lattice fulfills . This orthogonality condition leads to the rectangular

pattern indicated and implies .

Bravais lattice 2

Bravais lattices in 3 dimensionsThe 14 Bravais lattices in 3 dimensions are obtained by coupling one of the 7 lattice systems (or axial systems) withone of the lattice centerings. Each Bravais lattice refers to a distinct lattice type.The lattice centerings are: Primitive (P): lattice points on the cell corners only. Body (I): one additional lattice point at the center of the cell. Face (F): one additional lattice point at the center of each of the faces of the cell. Base (A, B or C): one additional lattice point at the center of each of one pair of the cell faces.Not all combinations of the crystal systems and lattice centerings are needed to describe the possible lattices. Thereare in total 76 = 42 combinations, but it can be shown that several of these are in fact equivalent to each other. Forexample, 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 ofcombinations to 14 conventional Bravais lattices, shown in the table below.

The 7 lattice systems The 14 Bravais lattices

Triclinic P

Monoclinic P C

Orthorhombic P C I F

Tetragonal P I

Rhombohedral P

Hexagonal P

Bravais lattice 3

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. Thevolumes of the Bravais lattices are given below:

Lattice system Volume

Triclinic

Monoclinic

Orthorhombic

Tetragonal

Rhombohedral

Hexagonal

Cubic

Centred Unit Cells :

LatticeSystem

Possible Variations Axial Distances (edgelengths)

Axial Angles Examples

Cubic Primitive, Body-centred, Face-centred a = b = c = = = 90 NaCl, Zinc Blende, Cu

Tetragonal Primitive, Body-centred a = b c = = = 90 White tin, SnO2, TiO2, CaSO4Orthorhombic Primitive, Body-centred, Face-centred,

Base-centreda b c = = = 90 Rhombic sulphur, KNO3,

BaSO4Hexagonal Primitive a = b c = = 90, =

120Graphite, ZnO, CdS

Rhombohedral Primitive a = b = c = = 90 Calcite (CaCO3), Cinnabar(HgS)

Monoclinic Primitive, Base-centred a b c = = 90, 90

Monoclinic sulphur,Na2SO4.10H2O

Triclinic Primitive a b c 90 K2Cr2O7, CuSO4.5H2O, H3BO3

Bravais lattice 4

Bravais lattices in 4 dimensionsIn four dimensions, there are 64 Bravais lattices. Of these, 23 are primitive and 41 are centered. Ten Bravais latticessplit into enantiomorphic pairs.

References

Further reading Bravais, A. (1850). "Mmoire sur les systmes forms par les points distribus rgulirement sur un plan ou dans

l'espace". J. Ecole Polytech. 19: 1128. (English: Memoir 1, Crystallographic Society of America, 1949.) Hahn, Theo, ed. (2002). International Tables for Crystallography, Volume A: Space Group Symmetry (http:/ / it.

iucr. org/ A/ ) A (5th ed.). Berlin, New York: Springer-Verlag. doi: 10.1107/97809553602060000100 (http:/ / dx.doi. org/ 10. 1107/ 97809553602060000100). ISBN978-0-7923-6590-7.

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

bravais. htm).

Article Sources and Contributors 5

Article Sources and ContributorsBravais lattice Source: http://en.wikipedia.org/w/index.php?oldid=618089715 Contributors: .:Ajvol:., AlexStamm, Andre Engels, Avihu, Bor75, Brane.Blokar, Clipjoint, DVdm,Danieljamesscott, David Eppstein, DhananSekhar, DragonflySixtyseven, Dvorak729, El C, Eruantalon, Fittold27, GJaxon, Giftlite, Gogo Dodo, Guillom, Hard Raspy Sci, Headbomb, Hhhippo,Jamesmcguigan, Joule36e5, Klicketyclack, LMB, Larryisgood, LewisEE, Materialscientist, Michael Hardy, Mikko Paananen, Mjgardes, Mnmngb, Mrsmime, Neeya The Great, Niceguyedc,Oysteinp, Patrick, Pfalstad, Pinethicket, Plasticup, Profjohn, Prolineserver, R.e.b., Ricardogpn, Rifleman 82, Sanalampady, Sibwarra, SietskeEN, Snigbrook, Soren.harward, Specopsgamer,Stannered, Stikonas, Tantalate, Tedickey, Tomruen, Trappist the monk, Truelight, User A1, Vsmith, WilliamDParker, Xanthe-fr, Zaslav, 93 anonymous edits

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Bravais latticeBravais lattices in at most 2 dimensionsBravais lattices in 3 dimensionsBravais lattices in 4 dimensionsReferencesFurther readingExternal links

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