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Encyclopedia > Bravais lattice

In geometry and crystallography, a Bravais lattice, named after Auguste Bravais, is an infinite set of points generated by a set of discrete translation operations. 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. Table of Geometry, from the 1728 Cyclopaedia. ... Crystallography (from the Greek words crystallon = cold drop / frozen drop, with its meaning extending to all solids with some degree of transparency, and graphein = write) is the experimental science of determining the arrangement of atoms in solids. ... Auguste Bravais (c. ... In Euclidean geometry, translation is a transformation of Euclidean space which moves every point by a fixed distance in the same direction. ... 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 of the crystal to the position of an atom of the same kind. ... The space group of a crystal 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. In mineralogy and crystallography, a crystal structure is a unique arrangement of atoms in a crystal. ...

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.

Crystal system	Bravais lattices
triclinic	P
triclinic
monoclinic	P	C
monoclinic
orthorhombic	P	C	I	F
orthorhombic
tetragonal	P	I
tetragonal
rhombohedral (trigonal)	P
rhombohedral (trigonal)
hexagonal	P
hexagonal
cubic	P	I	F
cubic

In crystallography, the triclinic crystal system is one of the 7 lattice point groups. ... Image File history File links No higher resolution available. ... In crystallography, the monoclinic crystal system is one of the 7 lattice point groups. ... Monoclinic crystal structure. ... Monoclinic base-centred crystal structure File links The following pages link to this file: User:DrBob/Figures Crystal structure Monoclinic Categories: GFDL images ... In crystallography, the orthorhombic crystal system is one of the 7 lattice point groups. ... Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... In crystallography, the tetragonal crystal system is one of the 7 lattice point groups. ... Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... In crystallography, the rhombohedral (or trigonal) crystal system is one of the 7 lattice point groups. ... Image File history File links No higher resolution available. ... In crystallography, the hexagonal crystal system is one of the 7 lattice point groups. ... Image File history File links No higher resolution available. ... In crystallography, the cubic crystal system (or isometric crystal system) is the most symmetric of the 7 crystal systems. ... Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ...

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:

Crystal system	Volume
Triclinic	$abc sqrt{1-cos^2alpha-cos^2beta-cos^2gamma+2cosalpha cosbeta cosgamma}$
Monoclinic	$a b c sinβ$
Orthorhombic	$a b c$
Tetragonal	$a 2 c$
Rhombohedral	$a^3 sqrt{1 - 3cos^2alpha + 2cos^3alpha}$
Hexagonal	$frac{sqrt{3,}, a^2c}{2}$
Cubic	$a 3$

In crystallography, the triclinic crystal system is one of the 7 lattice point groups. ... In crystallography, the monoclinic crystal system is one of the 7 lattice point groups. ... In crystallography, the orthorhombic crystal system is one of the 7 lattice point groups. ... In crystallography, the tetragonal crystal system is one of the 7 lattice point groups. ... In crystallography, the rhombohedral (or trigonal) crystal system is one of the 7 lattice point groups. ... A regular hexagon A hexagon (also known as sexagon) is a polygon with six edges and six vertices. ... The cubic crystal system is a crystal system where the unit cell is in the shape of a cube. ...

See also