Bravais lattice

Overview

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 the arrangement of atoms in solids. The word "crystallography" derives from the Greek words crystallon = cold drop / frozen drop, with its meaning extending to all solids with some degree of transparency, and grapho = write.Before the development of...

, a

**Bravais lattice**, studied by , is an infinite array of discrete points generated by a set of discrete translation

Translation (geometry)

In Euclidean geometry, a translation moves every point a constant distance in a specified direction. A translation can be described as a rigid motion, other rigid motions include rotations and reflections. A translation can also be interpreted as the addition of a constant vector to every point, or...

operations described by:

where

*n*are any integers and

_{i}**a**are known as the primitive vectors which lie in different planes and span the lattice.

_{i}Unanswered Questions

Encyclopedia

In geometry

and crystallography

, a

operations described by:

where

A crystal is made up of a periodic arrangement of one or more atoms (the

Two Bravais lattices are often considered equivalent if they have isomorphic symmetry groups. In this sense, there are 14 possible Bravais lattices in three-dimensional space. The 14 possible symmetry groups of Bravais lattices are 14 of the 230 space groups.

In two dimensions, there are five Bravais lattices. They are oblique, rectangular, centered rectangular (rhombic), hexagonal, and square.

Each Bravais lattice refers to a distinct lattice type.

The lattice centerings are:

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 volume of the unit cell can be calculated by evaluating

Centred Unit Cells :

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 the arrangement of atoms in solids. The word "crystallography" derives from the Greek words crystallon = cold drop / frozen drop, with its meaning extending to all solids with some degree of transparency, and grapho = write.Before the development of...

, a

**Bravais lattice**, studied by , is an infinite array of discrete points generated by a set of discrete translationTranslation (geometry)

In Euclidean geometry, a translation moves every point a constant distance in a specified direction. A translation can be described as a rigid motion, other rigid motions include rotations and reflections. A translation can also be interpreted as the addition of a constant vector to every point, or...

operations described by:

where

*n*are any integers and_{i}**a**are known as the primitive vectors which lie in different planes and span the lattice. This discrete set of vectors must be closed under vector addition and subtraction. For any choice of position vector_{i}**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 of the lattice points.Two Bravais lattices are often considered equivalent if they have isomorphic symmetry groups. In this sense, there are 14 possible Bravais lattices in three-dimensional space. The 14 possible symmetry groups of Bravais lattices are 14 of the 230 space groups.

## Bravais lattices in at most 2 dimensions

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

## Bravais lattices in 3 dimensions

The 14 Bravais lattices in 3 dimensions are arrived at by combining one of the seven lattice systems (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.
- Base centered (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. 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 lattice systems |
The 14 Bravais lattices |
|||

triclinic | P | |||

monoclinic | P | C | ||

orthorhombic | P | C | I | F |

tetragonal | P | I | ||

rhombohedral |
P | |||

hexagonal | P | |||

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. The volumes of the Bravais lattices are given below:Lattice system |
Volume |
|||

Triclinic | ||||

Monoclinic | ||||

Orthorhombic | ||||

Tetragonal | ||||

rhombohedral | ||||

Hexagonal | ||||

Cubic Cubic crystal system In crystallography, 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.... |

Centred Unit Cells :

Crystal System | Possible Variations | Axial Distances (edge lengths) | Axial Angles | Examples |
---|---|---|---|---|

Cubic | Primitive, Body centred, Face centred | a = b = c | α = β = γ = 90° | NaCl, Zinc Blende, Cu Copper Copper is a chemical element with the symbol Cu and atomic number 29. It is a ductile metal with very high thermal and electrical conductivity. Pure copper is soft and malleable; an exposed surface has a reddish-orange tarnish... |

Tetragonal | Primitive, Body centred | a = b ≠ c | α = β = γ = 90° | White tin White tin White tin is refined, metallic tin. It contrasts with black tin, which is unrefined tin ore as extracted from the ground. The term "white tin" was historically associated with tin mining in Devon and Cornwall where it was smelted from black tin in blowing houses.... , SnO _{2}, TiO_{2}, CaSO_{4} |

Orthorhombic | Primitive, Body centred, Face centred, End centred | a ≠ b ≠ c | α = β = γ = 90° | Rhombic Sulphur Allotropes of sulfur There are a large number of allotropes of sulfur. In this respect, sulfur is second only to carbon.The most common form found in nature is yellow orthorhombic α-sulfur, which contains puckered rings of . Chemistry students may have seen "plastic sulfur"; this is not an allotrope but a mixture of... , KNO _{3}, BaSOBarium sulfate Barium sulfate is the inorganic compound with the chemical formula BaSO4. It is a white crystalline solid that is odorless and insoluble in water. It occurs as the mineral barite, which is the main commercial source of barium and materials prepared from it... _{4} |

Hexagonal | Primitive | a = b ≠ c | α = β = 90°, γ = 120° | Graphite Graphite The mineral graphite is one of the allotropes of carbon. It was named by Abraham Gottlob Werner in 1789 from the Ancient Greek γράφω , "to draw/write", for its use in pencils, where it is commonly called lead . Unlike diamond , graphite is an electrical conductor, a semimetal... , ZnO, CdS CDS -Computing and electronics:* Cadence Design Systems, American Electronic Design Automation software company* Chromatography data system, software to control chromatography instruments* Cockpit display system* Compact Discs... |

Rhombohedral (trigonal) | Primitive | a = b = c | α = β = γ ≠ 90° | Calcite Calcite Calcite is a carbonate mineral and the most stable polymorph of calcium carbonate . The other polymorphs are the minerals aragonite and vaterite. Aragonite will change to calcite at 380-470°C, and vaterite is even less stable.-Properties:... (CaCO _{3}, CinnabarCinnabar Cinnabar or cinnabarite , is the common ore of mercury.-Word origin:The name comes from κινναβαρι , a Greek word most likely applied by Theophrastus to several distinct substances... (HgS) |

Monoclinic | Primitive, End centred | a ≠ b ≠ c | α = γ = 90°, β ≠ 90° | Monoclinic Sulphur Allotropes of sulfur There are a large number of allotropes of sulfur. In this respect, sulfur is second only to carbon.The most common form found in nature is yellow orthorhombic α-sulfur, which contains puckered rings of . Chemistry students may have seen "plastic sulfur"; this is not an allotrope but a mixture of... , Na _{2}SO_{4}.10H_{2}O |

Triclinic | Primitive | a ≠ b ≠ c | α ≠ β ≠ γ ≠ 90° | K_{2}Cr_{2}O_{7}, CuSO_{4}.5H_{2}O, H_{3}BO_{3} |

## Bravais lattices in 4 dimensions

In four dimensions, there are 64 Bravais lattices. Of these, 23 are primitive and 41 are centered. Ten Bravais lattices split into enantiomorphic pairs.## See also

- Translational symmetryTranslational symmetryIn 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 spans the real vector space Rn. Every lattice in Rn can be generated from a basis for the vector space by forming all linear combinations with integer coefficients...
- classification of lattices
- Miller IndexMiller indexMiller indices form a notation system in crystallography for planes and directions in crystal lattices.In particular, a family of lattice planes is determined by three integers h, k, and ℓ, the Miller indices. They are written , and each index denotes a plane orthogonal to a direction in the...