Wikipedia:Crystal system

From Global Warming Art

In crystallography, a crystal system or crystal family or lattice system is one of six or seven classes of space groups, lattices, point groups, or crystals. Informally, two crystals tend to be in the same crystal system if they have similar symmetries, though there are many exceptions to this.

Crystal systems, crystal families, and lattice systems are similar but slightly different, and there is widespread confusion between them: in particular the trigonal crystal system is often confused with the rhombohedral lattice system, and the term "crystal system" is sometimes used to mean "lattice system" or "crystal family".

Space groups and crystals are divided into 7 crystal systems according to their point groups, and into 7 lattice systems according to their Bravais lattices. Five of the crystal systems are essentially the same as five of the lattice systems, but the hexagonal and trigonal crystal systems differ from the hexagonal and rhombohedral lattice systems. The six crystal families are formed by combining the hexagonal and trigonal crystal systems into one hexagonal family, in order to eliminate this confusion.

Contents 1 Overview 2 Crystal systems 3 Lattice systems 4 See also 5 References 6 External links

Overview

A lattice system is a class of lattices with the same point group. There are 7 lattice systems: triclinic, monoclinic, orthorhombic, tetragonal, rhombohedral, hexagonal, and cubic. A space group is assigned to a lattice system according to its underlying lattice.

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 most point groups there is only one possible lattice system, so in these cases the crystal system corresponds to a lattice system and is given the same name. However for the 5 point groups in the trigonal crystal class there are two possible lattice systems for their point groups: rhombohedral or hexagonal. There are 7 crystal systems: triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic. Space groups are assigned to crystal systems according to their point groups.

A crystal family is almost the same as a crystal system or lattice system, except that the hexagonal and trigonal crystal systems, and the hexagonal and rhombohedral lattice systems, are combined into one hexagonal family in order to eliminate the difference between crystal systems and lattice systems. There are 6 crystal families: triclinic, monoclinic, orthorhombic, tetragonal, hexagonal, and cubic..

The relation between 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

The distribution of the 32 point groups into the 7 crystal systems is given in the following table.

crystal system	point group / crystal class	Schönflies	Hermann-Mauguin	orbifold	Type	order	structure
triclinic	triclinic-pedial	C1		11	enantiomorphic polar	1	trivial
	triclinic-pinacoidal	Ci		1x	centrosymmetric	2	cyclic
monoclinic	monoclinic-sphenoidal	C2		22	enantiomorphic polar	2	cyclic
	monoclinic-domatic	Cs		1*	polar	2	cyclic
	monoclinic-prismatic	C2h		2*	centrosymmetric	4	2×cyclic
orthorhombic	orthorhombic-sphenoidal	D2		222	enantiomorphic	4	dihedral
	orthorhombic-pyramidal	C2v		*22	polar	4	dihedral
	orthorhombic-bipyramidal	D2h		*222	centrosymmetric	8	2×dihedral
tetragonal	tetragonal-pyramidal	C4		44	enantiomorphic polar	4	Cyclic
	tetragonal-disphenoidal	S4		2x		4	cyclic
	tetragonal-dipyramidal	C4h		4*	centrosymmetric	8	2×cyclic
	tetragonal-trapezoidal	D4		422	enantiomorphic	8	dihedral
	ditetragonal-pyramidal	C4v		*44	polar	8	dihedral
	tetragonal-scalenoidal	D2d	or	2*2		8	dihedral
	ditetragonal-dipyramidal	D4h		*422	centrosymmetric	16	2×dihedral
trigonal (rhombohedral)	trigonal-pyramidal	C3		33	enantiomorphic polar	3	cyclic
	rhombohedral	S6 (C3i)		3x	centrosymmetric	6	cyclic
	trigonal-trapezoidal	D3	or or	322	enantiomorphic	6	dihedral
	ditrigonal-pyramidal	C3v	or or	*33	polar	6	dihedral
	ditrigonal-scalahedral	D3d	or or	2*3	centrosymmetric	12	dihedral
hexagonal	hexagonal-pyramidal	C6		66	enantiomorphic polar	6	cyclic
	trigonal-dipyramidal	C3h		3*		6	cyclic
	hexagonal-dipyramidal	C6h		6*	centrosymmetric	12	2×cyclic
	hexagonal-trapezoidal	D6		622	enantiomorphic	12	dihedral
	dihexagonal-pyramidal	C6v		*66	polar	12	dihedral
	ditrigonal-dipyramidal	D3h	or	*322		12	dihedral
	dihexagonal-dipyramidal	D6h		*622	centrosymmetric	24	2×dihedral
cubic	tetartohedral	T		332	enantiomorphic	12	Alternating
	diploidal	Th		3*2	centrosymmetric	24	2×alternating
	gyroidal	O		432	enantiomorphic	24	symmetric
	tetrahedral	Td		*332		24	symmetric
	hexoctahedral	Oh		*432	centrosymmetric	48	2×symmetric

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.

Lattice systems

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	centered
orthorhombic (cuboid)	simple	base-centered	body-centered	face-centered
tetragonal (square cuboid)	simple	body-centered
rhombohedral (trigonal trapezohedron)	α=β=γ
hexagonal (centered regular hexagon)
cubic (isometric; cube)	simple	body-centered	face-centered

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

• Crystal structure
• Point group
• List of the 230 crystallographic 3D space groups

References

• Hahn, Theo, ed. (2002), International Tables for Crystallography, Volume A: Space Group Symmetry (5th ed.), Berlin, New York: Springer-Verlag, doi:10.1107/97809553602060000100, ISBN 978-0-7923-6590-7, http://it.iucr.org/A/ 

External links

• Overview of the 32 groups
• Mineral galleries - Symmetry
• all cubic crystal classes, forms and stereographic projections (interactive java applet)

v • d • e Crystal systems triclinic · monoclinic · orthorhombic · tetragonal · trigonal · hexagonal · cubic (isometric)

W	The content appearing above is a local copy of the Crystal system article from Wikipedia, the free encyclopedia. It was written by various Wikipedians (contributor history) and is distributed under the Creative Commons CC-BY-SA 3.0 License (license text). The version shown above has Revision ID: 310557752 and was last updated August 28, 2009, 3:33 pm.
	Current version on Wikipedia: view / edit	Read Wikipedia's disclaimers.