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 symmetry group. A major application is in crystallography, to categorize crystals, but by itself the topic is one of 3D Euclidean geometry.

There are 7 crystal systems:

Triclinic, all cases not satisfying the requirements of any other system; thus there is no other symmetry than translational symmetry, or the only extra kind is inversion.
Monoclinic, requires either 1 two-fold axis of rotation or 1 mirror plane.
Orthorhombic, requires either 3 two-fold axes of rotation or 1 two fold axis of rotation and two mirror planes.
Tetragonal, requires 1 four-fold axis of rotation.
Rhombohedral, also called trigonal, requires 1 three-fold axis of rotation.
Hexagonal, requires 1 six-fold axis of rotation.
Isometric or cubic, requires 4 three-fold axes of rotation.

There are 2, 13, 59, 68, 25, 27, and 36 space groups per crystal system, respectively, together 230. The following mini-table gives a breakdown of the various different things per crystal system;

Crystal system         :   1   2   3   4   5   6   7   : Total
No. of point groups    :   2   3   3   7   5   7   5   :    32
No. of Bravais lattices:   1   2   4   2   1   1   3   :    14
No. of space groups    :   2  13  59  68  25  27  36   :   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.

1 Crystallographic point group
2 Overview of point groups by crystal system
3 Classification of lattices
4 See also
5 External links

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Crystallographic point group

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, simply by knowing its point group.

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Overview of point groups by crystal system

crystal system	point group / crystal class	Schönflies	Hermann-Mauguin	orbifold	Type
triclinic	triclinic-pedial	C₁	<math>1\ </math>	11	enantiomorphic polar
triclinic	triclinic-pinacoidal	C_i	<math>\bar{1}</math>	1x	centrosymmetric
monoclinic	monoclinic-sphenoidal	C₂	<math>2\ </math>	22	enantiomorphic polar
	monoclinic-domatic	C_s	<math>m\ </math>	1*	polar
	monoclinic-prismatic	C_2h	<math>2/m\ </math>	2*	centrosymmetric
orthorhombic	orthorhombic-sphenoidal	D₂	<math>222\ </math>	222	enantiomorphic
	orthorhombic-pyramidal	C_2v	<math>mm2\ </math>	*22	polar
	orthorhombic-bipyramidal	D_2h	<math>2/m\ 2/m\ 2/m</math>	*222	centrosymmetric
tetragonal	tetragonal-pyramidal	C₄	<math>4\ </math>	44	enantiomorphic polar
	tetragonal-disphenoidial	S₄	<math>\bar{4}</math>	2x
	tetragonal-dipyramidal	C_4h	<math>4/m\ </math>	4*	centrosymmetric
	tetragonal-trapezoidal	D₄	<math>422\ </math>	422	enantiomorphic
	ditetragonal-pyramidal	C_4v	<math>4mm\ </math>	*44	polar
	tetragonal-scalenoidal	D_2d	<math>\bar{4}2m\ </math> or <math>\bar{4}m2</math>	2*2
	ditetragonal-dipyramidal	D_4h	<math>4/m\ 2/m\ 2/m</math>	*422	centrosymmetric
rhombohedral (trigonal)	trigonal-pyramidal	C₃	<math>3 \!</math>	33	enantiomorphic polar
	rhombohedral	S₆ (C_3i)	<math>\bar{3}</math>	3x	centrosymmetric
	trigonal-trapezoidal	D₃	<math>32\ </math> or <math>321\ </math> or <math>312\ </math>	322	enantiomorphic
	ditrigonal-pyramidal	C_3v	<math>3m\ </math>or <math> 3m1\ </math> or <math>31m\ </math>	*33	polar
	ditrigonal-scalahedral	D_3d	<math>\bar{3} 2/m</math> or <math>\bar{3} 2/m 1</math> or <math>\bar{3} 1 2/m </math>	2*3	centrosymmetric
hexagonal	hexagonal-pyramidal	C₆	<math>6\ </math>	66	enantiomorphic polar
	trigonal-dipyramidal	C_3h	<math>\bar{6}</math>	3*
	hexagonal-dipyramidal	C_6h	<math>6/m\ </math>	6*	centrosymmetric
	hexagonal-trapezoidal	D₆	<math>622\ </math>	622	enantiomorphic
	dihexagonal-pyramidal	C_6v	<math>6mm\ </math>	*66	polar
	ditrigonal-dipyramidal	D_3h	<math>\bar{6}m2</math> or <math>\bar{6}2m</math>	*322
	dihexagonal-dipyramidal	D_6h	<math>6/m\ 2/m\ 2/m\ </math>	*622	centrosymmetric
cubic	tetartoidal	T	<math>23\ </math>	332	enantiomorphic
	diploidal	T_h	<math>2/m\ \bar{3}</math>	3*2	centrosymmetric
	gyroidal	O	<math>432\ </math>	432	enantiomorphic
	tetrahedral	T_d	<math>\bar{4}3m</math>	*332
	hexoctahedral	O_h	<math>4/m\ \bar{3}\ 2/m</math>	*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 11-fold rotational symmetry (in human), however, it must form crystals in one of the 11 enantiomorphic point groups given above.

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Classification of lattices

Crystal system	Lattices
triclinic (parallelepiped)	Image:Triclinic.png
monoclinic (right prism with parallelogram base; here seen from above)	simple	centered
	Image:Monoclinic.png	Image:Monoclinic-base-centered.png
orthorhombic (cuboid)	simple	base-centered	body-centered	face-centered
orthorhombic (cuboid)	Image:Orthorhombic.png	Image:Orthorhombic-base-centered.png	Image:Orthorhombic-body-centered.png	Image:Orthorhombic-face-centered.png
tetragonal (square cuboid)	simple	body-centered
tetragonal (square cuboid)	Image:Tetragonal.png	Image:Tetragonal-body-centered.png
rhombohedral (trigonal) (3-sided trapezohedron)	Image:Rhombohedral.png
hexagonal (centered regular hexagon)	Image:Hexagonal.png
cubic (isometric; cube)	simple	body-centered	face-centered
cubic (isometric; cube)	Image:Cubic crystal shape.png	Image:Cubic-body-centered.png	Image:Cubic-face-centered.png

Template:- 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

<math>\mathbf{R} = n_1 \mathbf{a}_1 + n_2 \mathbf{a}_2 + n_3 \mathbf{a}_3,</math>

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 crystal system only. They represent the maximum symmetry a structure with the translational symmetry concerned can have.

All crystalline materials recognised till now (not including quasicrystals) fit in one of these arrangements.

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.

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