In geometry and crystallography, a Bravais lattice, named after Auguste Bravais,^[1] is an infinite set of points generated by a set of discrete translation operations. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position Crystallography is the experimental science of determining the arrangement of Atoms in Solids In older usage it is the scientific study of Crystals The August Bravais (23 August 1811 Annonay – 30 March 1863, Le Chesnay, France was a French physicist well known for his work in Crystallography In Euclidean geometry, a translation is moving every point a constant distance in a specified direction 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. In all, there are 14 possible Bravais lattices that fill three-dimensional space. Related to Bravais lattices are Crystallographic point groups of which there are 32 and Space groups of which there are 230. In Crystallography, a crystallographic point group is a set of Symmetry operations like rotations or reflections that leave a point fixed while moving each atom The space group of a Crystal or crystallographic group is a mathematical description of the Symmetry inherent in the structure

Development of the Bravais lattices

The 14 Bravais lattices are arrived at by combining one of the seven crystal systems (or axial systems) with one of the lattice centerings. A crystal system is a category of Space groups which characterize Symmetry of structures in three dimensions with Translational symmetry in three directions Each Bravais lattice refers 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 Crystal systems	The 14 Bravais lattices
triclinic	P
monoclinic	P	C
orthorhombic	P	C	I	F
tetragonal	P	I
rhombohedral (trigonal)	P
hexagonal	A
cubic	P (pcc)	I (bcc)	F (fcc)

The volume of the unit cell can be calculated by evaluating where , and are the lattice vectors. In Crystallography, the triclinic Crystal system is one of the 7 lattice Point groups A crystal system is described by three basis vectors In Crystallography, the monoclinic Crystal system is one of the 7 lattice Point groups A crystal system is described by three vectors. In Crystallography, the orthorhombic Crystal system is one of the seven Lattice Point groups Orthorhombic lattices result from stretching In Crystallography, the tetragonal Crystal system is one of the 7 lattice Point groups Tetragonal Crystal lattices result from stretching a cubic In Crystallography, the rhombohedral (or trigonal) Crystal system is one of the seven lattice point groups named after the two-dimensional In Crystallography, the hexagonal is one of the 7 Crystal system, it contains 7 Point groups. The cubic crystal system (or isometric) is a Crystal system where the Unit cell is in the shape of a Cube. The volumes of the Bravais lattices are given below:

Crystal system	Volume
Triclinic
Monoclinic	abcsinα
Orthorhombic	abc
Tetragonal	a2c
Rhombohedral
Hexagonal
Cubic	a3

Bravais lattices in 2D

In two dimensions, there are five Bravais lattices. In Crystallography, the triclinic Crystal system is one of the 7 lattice Point groups A crystal system is described by three basis vectors In Crystallography, the monoclinic Crystal system is one of the 7 lattice Point groups A crystal system is described by three vectors. In Crystallography, the orthorhombic Crystal system is one of the seven Lattice Point groups Orthorhombic lattices result from stretching In Crystallography, the tetragonal Crystal system is one of the 7 lattice Point groups Tetragonal Crystal lattices result from stretching a cubic In Crystallography, the rhombohedral (or trigonal) Crystal system is one of the seven lattice point groups named after the two-dimensional In Crystallography, the hexagonal is one of the 7 Crystal system, it contains 7 Point groups. The cubic crystal system (or isometric) is a Crystal system where the Unit cell is in the shape of a Cube. They are square, rectangular, centered rectangular, hexagonal, and oblique. Regular hexagon The internal Angles of a regular hexagon (one where all sides and all angles are equal are all 120 ° and the hexagon has 720 degrees ^[2]

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

Bravais lattices in 4D

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

References

1. ^ Aroyo, Mois I. ; Ulrich Müller and Hans Wondratschek (2006). "Historical Introduction". International Tables for Crystallography A1 (1. 1): 2-5. Springer. doi:10.1107/97809553602060000537. A digital object identifier ( DOI) is a permanent identifier given to an Electronic document.  
2. ^ Kittel, Charles [1953] (1996). "Chapter 1", Introduction to Solid State Physics, Seventh Edition (in English), New York: John Wiley & Sons, 10. ISBN 0-471-11181-3. Retrieved on 2008-04-21. 2008 ( MMVIII) is the current year in accordance with the Gregorian calendar, a Leap year that started on Tuesday of the Common Events 753 BC - Romulus and Remus found Rome ( traditional date)  
3. ^ Mackay AL and Pawley GS (1963). "Bravais Lattices in Four-dimensional Space". Acta. cryst. 16: 11–19. doi:10.1107/S0365110X63000037. A digital object identifier ( DOI) is a permanent identifier given to an Electronic document.  

See also

• translational symmetry
• lattice (group)
• classification of lattices
• Miller Index

External links

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