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Bravais lattice
 
 
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In geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
 and crystallography
Crystallography

Crystallography is the experimental science of determining the arrangement of atoms in solids. In older usage, it is the scientific study of crystals....
, a Bravais lattice, named after Auguste Bravais, is an infinite set of points generated by a set of discrete translation
Translation (geometry)

In Euclidean geometry, a translation is moving every point a constant distance in a specified direction. It is one of the Euclidean groups . A translation can also be interpreted as the addition of a constant vector space to every point, or as shifting the Origin of the coordinate system....
 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.






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In geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
 and crystallography
Crystallography

Crystallography is the experimental science of determining the arrangement of atoms in solids. In older usage, it is the scientific study of crystals....
, a Bravais lattice, named after Auguste Bravais, is an infinite set of points generated by a set of discrete translation
Translation (geometry)

In Euclidean geometry, a translation is moving every point a constant distance in a specified direction. It is one of the Euclidean groups . A translation can also be interpreted as the addition of a constant vector space to every point, or as shifting the Origin of the coordinate system....
 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.

Development of the Bravais lattices

The 14 Bravais lattices are arrived at by combining one of the seven crystal system
Crystal system

A crystal system is a category of space groups, which characterize symmetry of structures in three dimensions with translational symmetry in three directions, having a discrete class of Point groups in three dimensions....
s (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 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. The volumes of the Bravais lattices are given below:

Crystal system Volume
Triclinic
Monoclinic
Orthorhombic
Tetragonal
Rhombohedral
Hexagonal
Hexagonal crystal system

In crystallography, the hexagonal is one of the 7 crystal systems. It contains 7 point groups . It has the same symmetry as a right prism with a hexagonal base....
Cubic
Cubic crystal system

The cubic crystal system is a crystal system where the unit cell is in the shape of a cube. This is one of the most common and simplest shapes found in crystals and minerals....


Bravais lattices in 2D


In two dimensions, there are five Bravais lattices. They are oblique, rectangular, centered rectangular, hexagon
Hexagon

In geometry, a hexagon is a polygon with six edges and six Vertex . A regular hexagon has Schl?fli symbol ....
al, and square.

Bravais lattices in 4D


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

See also

  • translational symmetry
    Translational symmetry

    In geometry, a translation "slides" an object by a a: Ta = p + a.In physics and mathematics, continuous translational symmetry is the invariance of a system of equations under any translation....
  • lattice (group)
    Lattice (group)

    In mathematics, especially in geometry and group theory, a lattice in Rn is a discrete subgroup of Rn which linear span the real number vector space Rn....
  • classification of lattices
    Crystal system

    A crystal system is a category of space groups, which characterize symmetry of structures in three dimensions with translational symmetry in three directions, having a discrete class of Point groups in three dimensions....
  • Miller Index
    Miller index

    Miller indices are a notation system in crystallography for planes and directions in Bravais lattices.In particular, a family of lattice planes is determined by three integers , , and , the Miller indices....


External links

  • A witty written and performed by Walter Fox Smith