User:JTiago/Sandbox
(sandbox)
Overview
[edit]A lattice system is a class of lattices with the same point group. In three dimensions there are seven lattice systems: triclinic, monoclinic, orthorhombic, tetragonal, rhombohedral, hexagonal, and cubic. The lattice system of a crystal or space group is determined by its lattice but not always by its point group.
A crystal system is a class of point groups. Two point groups are placed in the same crystal system if the sets of possible lattice systems of their space groups are the same. For many point groups there is only one possible lattice system, and in these cases the crystal system corresponds to a lattice system and is given the same name. However, for the five point groups in the trigonal crystal class there are two possible lattice systems for their point groups: rhombohedral or hexagonal. In three dimensions there are seven crystal systems: triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic. The crystal system of a crystal or space group is determined by its point group but not always by its lattice.
A crystal family also consists of point groups and is formed by combining crystal systems whenever two crystal systems have space groups with the same lattice. In three dimensions a crystal family is almost the same as a crystal system (or lattice system), except that the hexagonal and trigonal crystal systems are combined into one hexagonal family. In three dimensions there are six crystal families: triclinic, monoclinic, orthorhombic, tetragonal, hexagonal, and cubic. The crystal family of a crystal or space group is determined by either its point group or its lattice, and crystal families are the smallest collections of point groups with this property.
In dimensions less than three there is no essential difference between crystal systems, crystal families, and lattice systems. There are 1 in dimension 0, 1 in dimension 1, and 4 in dimension 2, called oblique, rectangular, square, and hexagonal.
The relation between three-dimensional crystal families, crystal systems, and lattice systems is shown in the following table:
Crystal family | Crystal system | Required symmetries of point group | point groups | space groups | bravais lattices | Lattice system |
---|---|---|---|---|---|---|
Triclinic | None | 2 | 2 | 1 | Triclinic | |
Monoclinic | 1 twofold axis of rotation or 1 mirror plane | 3 | 13 | 2 | Monoclinic | |
Orthorhombic | 3 twofold axes of rotation or 1 twofold axis of rotation and two mirror planes. | 3 | 59 | 4 | Orthorhombic | |
Tetragonal | 1 fourfold axis of rotation | 7 | 68 | 2 | Tetragonal | |
Hexagonal | Trigonal | 1 threefold axis of rotation | 5 | 7 | 1 | Rhombohedral |
18 | 1 | Hexagonal | ||||
Hexagonal | 1 sixfold axis of rotation | 7 | 27 | |||
Cubic | 4 threefold axes of rotation | 5 | 36 | 3 | Cubic | |
Total: 6 | 7 | 32 | 230 | 14 | 7 |
Crystal systems
[edit]The distribution of the 32 point groups into the 7 crystal systems is given in the following table.
crystal family | crystal system | point group / crystal class | Schönflies | Hermann-Mauguin | orbifold | Type | order | structure |
---|---|---|---|---|---|---|---|---|
triclinic | triclinic-pedial | C1 | 1 | 11 | enantiomorphic polar | 1 | trivial | |
triclinic-pinacoidal | Ci | 1 | 1x | centrosymmetric | 2 | cyclic | ||
monoclinic | monoclinic-sphenoidal | C2 | 2 | 22 | enantiomorphic polar | 2 | cyclic | |
monoclinic-domatic | Cs | m | 1* | polar | 2 | cyclic | ||
monoclinic-prismatic | C2h | 2/m | 2* | centrosymmetric | 4 | 2×cyclic | ||
orthorhombic | orthorhombic-sphenoidal | D2 | 222 | 222 | enantiomorphic | 4 | dihedral | |
orthorhombic-pyramidal | C2v | mm2 | *22 | polar | 4 | dihedral | ||
orthorhombic-bipyramidal | D2h | mmm | *222 | centrosymmetric | 8 | 2×dihedral | ||
tetragonal | tetragonal-pyramidal | C4 | 4 | 44 | enantiomorphic polar | 4 | Cyclic | |
tetragonal-disphenoidal | S4 | 4 | 2x | 4 | cyclic | |||
tetragonal-dipyramidal | C4h | 4/m | 4* | centrosymmetric | 8 | 2×cyclic | ||
tetragonal-trapezoidal | D4 | 422 | 422 | enantiomorphic | 8 | dihedral | ||
ditetragonal-pyramidal | C4v | 4mm | *44 | polar | 8 | dihedral | ||
tetragonal-scalenoidal | D2d | 42m or 4m2 | 2*2 | 8 | dihedral | |||
ditetragonal-dipyramidal | D4h | 4/mmm | *422 | centrosymmetric | 16 | 2×dihedral | ||
hexagonal | trigonal | trigonal-pyramidal | C3 | 3 | 33 | enantiomorphic polar | 3 | cyclic |
rhombohedral | S6 (C3i) | 3 | 3x | centrosymmetric | 6 | cyclic | ||
trigonal-trapezoidal | D3 | 32 or 321 or 312 | 322 | enantiomorphic | 6 | dihedral | ||
ditrigonal-pyramidal | C3v | 3m or 3m1 or 31m | *33 | polar | 6 | dihedral | ||
ditrigonal-scalahedral | D3d | 3m or 3m1 or 31m | 2*3 | centrosymmetric | 12 | dihedral | ||
hexagonal | hexagonal-pyramidal | C6 | 6 | 66 | enantiomorphic polar | 6 | cyclic | |
trigonal-dipyramidal | C3h | 6 | 3* | 6 | cyclic | |||
hexagonal-dipyramidal | C6h | 6/m | 6* | centrosymmetric | 12 | 2×cyclic | ||
hexagonal-trapezoidal | D6 | 622 | 622 | enantiomorphic | 12 | dihedral | ||
dihexagonal-pyramidal | C6v | 6mm | *66 | polar | 12 | dihedral | ||
ditrigonal-dipyramidal | D3h | 6m2 or 62m | *322 | 12 | dihedral | |||
dihexagonal-dipyramidal | D6h | 6/mmm | *622 | centrosymmetric | 24 | 2×dihedral | ||
gyroidal | O | 432 | 432 | enantiomorphic | 24 | symmetric | ||
hexoctahedral | Oh | m3m | *432 | centrosymmetric | 48 | 2×symmetric |
Lattice systems
[edit]The distribution of the 14 Bravais lattice types into 7 lattice systems is given in the following table.
The 7 lattice systems | The 14 Bravais Lattices | |||
triclinic (parallelepiped) | ![]() | |||
monoclinic (right prism with parallelogram base; here seen from above) | simple | base-centered | ||
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orthorhombic (cuboid) | simple | base-centered | body-centered | face-centered |
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tetragonal (square cuboid) | simple | body-centered | ||
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rhombohedral (trigonal trapezohedron) |
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hexagonal (centered regular hexagon) | ![]() | |||
cubic (isometric; cube) |
simple | body-centered | face-centered | |
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In geometry and crystallography, a Bravais lattice is a category of symmetry groups for translational symmetry in three directions, or correspondingly, a category of translation lattices.
Such symmetry groups consist of translations by vectors of the form
where n1, n2, and n3 are integers and a1, a2, and a3 are three non-coplanar vectors, called primitive vectors.
These lattices are classified by space group of the translation lattice itself; there are 14 Bravais lattices in three dimensions; each can apply in one lattice system only. They represent the maximum symmetry a structure with the translational symmetry concerned can have.
All crystalline materials must, by definition fit in one of these arrangements (not including quasicrystals).
For convenience a Bravais lattice is depicted by a unit cell which is a factor 1, 2, 3 or 4 larger than the primitive cell. Depending on the symmetry of a crystal or other pattern, the fundamental domain is again smaller, up to a factor 48.
The Bravais lattices were studied by Moritz Ludwig Frankenheim (1801-1869), in 1842, who found that there were 15 Bravais lattices. This was corrected to 14 by A. Bravais in 1848.
See also
[edit]References
[edit]- Hahn, Theo, ed. (2002), International Tables for Crystallography, Volume A: Space Group Symmetry, vol. A (5th ed.), Berlin, New York: Springer-Verlag, doi:10.1107/97809553602060000100, ISBN 978-0-7923-6590-7
External links
[edit]- Overview of the 32 groups
- Mineral galleries - Symmetry
- all cubic crystal classes, forms and stereographic projections (interactive java applet)
- Crystal system at the Online Dictionary of Crystallography
- Crystal family at the Online Dictionary of Crystallography
- Lattice system at the Online Dictionary of Crystallography
- Conversion Primitive to Standard Conventional for VASP input files
- Learning Crystallography