Crystal system
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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. A major application is in crystallography, to categorize crystals, but by itself the topic is one of 3D Euclidean geometry.
Contents 
[edit] Overview
There are 7 crystal systems:
 Triclinic, all cases not satisfying the requirements of any other system. There is no necessary symmetry other than translational symmetry, although inversion is possible.
 Monoclinic, requires either 1 twofold axis of rotation or 1 mirror plane.
 Orthorhombic, requires either 3 twofold axes of rotation or 1 twofold axis of rotation and two mirror planes.
 Tetragonal, requires 1 fourfold axis of rotation.
 Rhombohedral, also called trigonal, requires 1 threefold axis of rotation.
 Hexagonal, requires 1 sixfold axis of rotation.
 Isometric or cubic, requires 4 threefold axes of rotation.
There are 2, 13, 59, 68, 25, 27, and 36 space groups per crystal system, respectively, for a total of 230. The following table gives a brief characterization of the various crystal systems:
Crystal system  No. of point groups  No. of bravais lattices  No. of space groups 
Triclinic  2  1  2 
Monoclinic  3  2  13 
Orthorhombic  3  4  59 
Tetragonal  7  2  68 
Rhombohedral  5  1  25 
Hexagonal  7  1  27 
Cubic  5  3  36 
Total  32  14  230 
Within a crystal system there are two ways of categorizing space groups:
 by the linear parts of symmetries, i.e. by crystal class, also called crystallographic point group; each of the 32 crystal classes applies for one of the 7 crystal systems
 by the symmetries in the translation lattice, i.e. by Bravais lattice; each of the 14 Bravais lattices applies for one of the 7 crystal systems.
The 73 symmorphic space groups (see space group) are largely combinations, within each crystal system, of each applicable point group with each applicable Bravais lattice: there are 2, 6, 12, 14, 5, 7, and 15 combinations, respectively, together 61.
[edit] Crystallographic point groups
A symmetry group consists of isometric affine transformations; each is given by an orthogonal matrix and a translation vector (which may be the zero vector). Space groups can be grouped by the matrices involved, i.e. ignoring the translation vectors (see also Euclidean group). This corresponds to discrete symmetry groups with a fixed point. There are infinitely many of these point groups in three dimensions. However, only part of these are compatible with translational symmetry: the crystallographic point groups. This is expressed in the crystallographic restriction theorem. (In spite of these names, this is a geometric limitation, not just a physical one.)
The point group of a crystal, among other things, determines the symmetry of the crystal's optical properties. For instance, one knows whether it is birefringent, or whether it shows the Pockels effect, by simply knowing its point group.
[edit] Overview of point groups by crystal system
crystal system  point group / crystal class  Schönflies  HermannMauguin  orbifold  Type 

triclinic  triclinicpedial  C_{1}  11  enantiomorphic polar  
triclinicpinacoidal  C_{i}  1x  centrosymmetric  
monoclinic  monoclinicsphenoidal  C_{2}  22  enantiomorphic polar  
monoclinicdomatic  C_{s}  1*  polar  
monoclinicprismatic  C_{2h}  2*  centrosymmetric  
orthorhombic  orthorhombicsphenoidal  D_{2}  222  enantiomorphic  
orthorhombicpyramidal  C_{2v}  *22  polar  
orthorhombicbipyramidal  D_{2h}  *222  centrosymmetric  
tetragonal  tetragonalpyramidal  C_{4}  44  enantiomorphic polar  
tetragonaldisphenoidal  S_{4}  2x  
tetragonaldipyramidal  C_{4h}  4*  centrosymmetric  
tetragonaltrapezoidal  D_{4}  422  enantiomorphic  
ditetragonalpyramidal  C_{4v}  *44  polar  
tetragonalscalenoidal  D_{2d}  or  2*2  
ditetragonaldipyramidal  D_{4h}  *422  centrosymmetric  
rhombohedral (trigonal)  trigonalpyramidal  C_{3}  33  enantiomorphic polar  
rhombohedral  S_{6} (C_{3i})  3x  centrosymmetric  
trigonaltrapezoidal  D_{3}  or or  322  enantiomorphic  
ditrigonalpyramidal  C_{3v}  or or  *33  polar  
ditrigonalscalahedral  D_{3d}  or or  2*3  centrosymmetric  
hexagonal  hexagonalpyramidal  C_{6}  66  enantiomorphic polar  
trigonaldipyramidal  C_{3h}  3*  
hexagonaldipyramidal  C_{6h}  6*  centrosymmetric  
hexagonaltrapezoidal  D_{6}  622  enantiomorphic  
dihexagonalpyramidal  C_{6v}  *66  polar  
ditrigonaldipyramidal  D_{3h}  or  *322  
dihexagonaldipyramidal  D_{6h}  *622  centrosymmetric  
cubic  tetartohedral  T  332  enantiomorphic  
diploidal  T_{h}  3*2  centrosymmetric  
gyroidal  O  432  enantiomorphic  
tetrahedral  T_{d}  *332  
hexoctahedral  O_{h}  *432  centrosymmetric 
The crystal structures of biological molecules (such as protein structures) can only occur in the 11 enantiomorphic point groups, as biological molecules are invariably chiral. The protein assemblies themselves may have symmetries other than those given above, because they are not intrinsically restricted by the Crystallographic restriction theorem. For example the Rad52 DNA binding protein has an 11fold rotational symmetry (in human), however, it must form crystals in one of the 11 enantiomorphic point groups given above.
[edit] Classification of lattices
The 7 Crystal systems  The 14 Bravais Lattices  
triclinic (parallelepiped)  
monoclinic (right prism with parallelogram base; here seen from above)  simple  centered  
orthorhombic (cuboid)  simple  basecentered  bodycentered  facecentered 
tetragonal (square cuboid)  simple  bodycentered  
rhombohedral or trigonal (trigonal trapezohedron) 

hexagonal (centered regular hexagon)  
cubic (isometric; cube) 
simple  bodycentered  facecentered  
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 n_{1}, n_{2}, and n_{3} are integers and a_{1}, a_{2}, and a_{3} are three noncoplanar 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 crystal 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 (18011869), in 1842, who found that there were 15 Bravais lattices. This was corrected to 14 by A. Bravais in 1848.
[edit] See also
[edit] External links
 Overview of the 32 groups
 Mineral galleries  Symmetry
 all cubic crystal classes, forms and stereographic projections (interactive java applet)
