NationMaster - Encyclopedia: Crystal systems

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. The space group of a crystal is a mathematical description of the symmetry inherent in the structure. ... Sphere symmetry group o. ... A translation slides an object by a vector a: Ta(p) = p + a. ... The symmetry group of an object (e. ... Crystallography (from the Greek words crystallon = cold drop / frozen drop, with its meaning extending to all solids with some degree of transparency, and graphein = write) is the experimental science of determining the arrangement of atoms in solids. ... Quartz crystal In chemistry and mineralogy, a crystal is a solid in which the constituent atoms, molecules, or ions are packed in a regularly ordered, repeating pattern extending in all three spatial dimensions. ... Euclid Euclidean geometry is a mathematical system attributed to the Greek mathematician Euclid of Alexandria. ...

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; A translation slides an object by a vector a: Ta(p) = p + a. ... Rotational symmetry is symmetry with respect to some or all rotations in m-dimensional Euclidean space. ... Figures with the axes of symmetry drawn in. ... In crystallography, the tetragonal crystal system is one of the 7 lattice point groups. ... To meet Wikipedias quality standards and make it easier to understand, this article or section may require cleanup. ...

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(Trigonal)	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: In crystallography, a crystallographic point group or crystal class is a set of symmetry operations that leave a point fixed, like rotations or reflections, which leave the crystal unchanged. ... In geometry and crystallography, a Bravais lattice is an infinite set of points generated by a set of discrete translation operations. ... The space group of a crystal is a mathematical description of the symmetry inherent in the structure. ... In crystallography, the triclinic crystal system is one of the 7 lattice point groups. ... In crystallography, the monoclinic crystal system is one of the 7 lattice point groups. ... In crystallography, the orthorhombic crystal system is one of the 7 lattice point groups. ... In crystallography, the tetragonal crystal system is one of the 7 lattice point groups. ... In crystallography, the rhombohedral (or trigonal) crystal system is one of the 7 lattice point groups. ... To meet Wikipedias quality standards and make it easier to understand, this article or section may require cleanup. ...

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. See lattice for other meanings of this term, both within and without mathematics. ... The space group of a crystal is a mathematical description of the symmetry inherent in the structure. ...

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

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.) In crystallography, a crystallographic point group or crystal class is a set of symmetry operations that leave a point fixed, like rotations or reflections, which leave the crystal unchanged. ... The symmetry group of an object (e. ... In geometry, an affine transformation or affine map (from the Latin, affinis, connected with) between two vector spaces consists of a linear transformation followed by a translation: In the finite-dimensional case each affine transformation is given by a matrix A and a vector b, which can be written as... In matrix theory, a real orthogonal matrix is a square matrix Q whose transpose is its inverse: // Overview An orthogonal matrix is the real specialization of a unitary matrix, and thus always a normal matrix. ... In mathematics, the Euclidean group is the symmetry group associated with Euclidean geometry. ... A discrete point group in 3D is a finite symmetry group in 3D that leaves the origin fixed. ... The crystallographic restriction theorem in its basic form is the observation that the rotational symmetries of a crystal are limited to 2-fold, 3-fold, 4-fold, and 6-fold. ...

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. Crystal optics is the branch of optics that describes the behaviour of light in anisotropic media, that is, media (such as crystals) in which light behaves differently depending on which direction the light is propagating. ... A calcite crystal laid upon a paper with some letters showing the double refraction Birefringence, or double refraction, is the decomposition of a ray of light into two rays (the ordinary ray and the extraordinary ray) when it passes through certain types of material, such as calcite crystals, depending on... The Pockels effect, or Pockels electro-optic effect, is the production of birefringence in an optical medium induced by a constant or varying electric field. ...

Overview of point groups by crystal system

crystal system	point group / crystal class	Schönflies	Hermann-Mauguin	orbifold	Type
triclinic	triclinic-pedial	C₁	$1$	11	enantiomorphic polar
triclinic	triclinic-pinacoidal	C_i	$bar{1}$	1x	centrosymmetric
monoclinic	monoclinic-sphenoidal	C₂	$2$	22	enantiomorphic polar
	monoclinic-domatic	C_s	$m$	1*	polar
	monoclinic-prismatic	C_2h	$2/m$	2*	centrosymmetric
orthorhombic	orthorhombic-sphenoidal	D₂	$222$	222	enantiomorphic
	orthorhombic-pyramidal	C_2v	$mm2$	*22	polar
	orthorhombic-bipyramidal	D_2h	$mmm$	*222	centrosymmetric
tetragonal	tetragonal-pyramidal	C₄	$4$	44	enantiomorphic polar
	tetragonal-disphenoidial	S₄	$bar{4}$	2x
	tetragonal-dipyramidal	C_4h	$4/m$	4*	centrosymmetric
	tetragonal-trapezoidal	D₄	$422$	422	enantiomorphic
	ditetragonal-pyramidal	C_4v	$4mm$	*44	polar
	tetragonal-scalenoidal	D_2d	$bar{4}2m$ or $bar{4}m2$	2*2
	ditetragonal-dipyramidal	D_4h	$4/mmm$	*422	centrosymmetric
rhombohedral (trigonal)	trigonal-pyramidal	C₃	$3 !$	33	enantiomorphic polar
	rhombohedral	S₆ (C_3i)	$bar{3}$	3x	centrosymmetric
	trigonal-trapezoidal	D₃	$32$ or $321$ or $312$	322	enantiomorphic
	ditrigonal-pyramidal	C_3v	$3m$ or $3m1$ or $31m$	*33	polar
	ditrigonal-scalahedral	D_3d	$bar{3} m$ or $bar{3} m 1$ or $bar{3} 1 m$	2*3	centrosymmetric
hexagonal	hexagonal-pyramidal	C₆	$6$	66	enantiomorphic polar
	trigonal-dipyramidal	C_3h	$bar{6}$	3*
	hexagonal-dipyramidal	C_6h	$6/m$	6*	centrosymmetric
	hexagonal-trapezoidal	D₆	$622$	622	enantiomorphic
	dihexagonal-pyramidal	C_6v	$6mm$	*66	polar
	ditrigonal-dipyramidal	D_3h	$bar{6}m2$ or $bar{6}2m$	*322
	dihexagonal-dipyramidal	D_6h	$6/mmm$	*622	centrosymmetric
cubic	tetartoidal	T	$23$	332	enantiomorphic
	diploidal	T_h	$mbar{3}$	3*2	centrosymmetric
	gyroidal	O	$432$	432	enantiomorphic
	tetrahedral	T_d	$bar{4}3m$	*332
	hexoctahedral	O_h	$mbar{3}m$	*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. In mathematics, point group is a group of geometric symmetries (isometries) leaving a point fixed. ... Arthur Moritz SchÃ¶nflies (April 17, 1853 Landsberg an der Warthe(GorzÃ³w) â€“ May 27, 1928) was a German mathematician, known for his contributions to the application of group theory to crystallography, and for work in topology. ... German professor of crystallography Carl Hermann (1898 June 17â€“1961 September 12) was an inventor (with Charles-Victor Mauguin) of an international standard notation for crystallographic groups. ... French professor of mineralogy Charles-Victor Mauguin (July 19, 1878 â€“ April 25, 1958) was a founder of the International Union of Crystallography (IUCr), and inventor (with Carl Hermann) of an international standard notation for crystallographic groups. ... In topology and group theory, an orbifold (for orbit-manifold) is a generalization of a manifold. ... Chirality is a manga by Satoshi Urushihara Chirality (Greek handedness, derived from the word stem Ï‡ÎµÎ¹Ï~, ch[e]ir~ - hand~) is an asymmetry property important in several branches of science. ... ... ... ... In geometry, a prism is a polyhedron made of two parallel copies of some polygonal base joined by faces that are rectangles or parallelograms. ... This article or section does not cite its references or sources. ... ... A bipyramid is a polyhedron formed by joining two identical pyramids base-to-base. ... In crystallography, the tetragonal crystal system is one of the 7 lattice point groups. ... ... ... ... ... ... ... To meet Wikipedias quality standards and make it easier to understand, this article or section may require cleanup. ... A representation of the 3D structure of myoglobin, showing coloured alpha helices. ... Chirality refers to several phenomena, all having to do with objects that differ from their mirror image. ... The crystallographic restriction theorem in its basic form is the observation that the rotational symmetries of a crystal are limited to 2-fold, 3-fold, 4-fold, and 6-fold. ...

Classification of lattices

Crystal system	Lattices
triclinic (parallelepiped)
monoclinic (right prism with parallelogram base; here seen from above)	simple	centered

orthorhombic (cuboid)	simple	base-centered	body-centered	face-centered
orthorhombic (cuboid)
tetragonal (square cuboid)	simple	body-centered
tetragonal (square cuboid)
rhombohedral (trigonal) (3-sided trapezohedron)
hexagonal (centered regular hexagon)
cubic (isometric; cube)	simple	body-centered	face-centered
cubic (isometric; cube)

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. In geometry, a parallelepiped (pronounced ; meaning of parallel planes) or parallelopipedon is a three-dimensional figure like a cube, except that its faces are not squares but parallelograms. ... Triclinic crystal structure File links The following pages link to this file: User:DrBob/Figures Crystal structure Triclinic Categories: GFDL images ... In geometry, a prism is a polyhedron made of two parallel copies of some polygonal base joined by faces that are rectangles or parallelograms. ... A parallelogram. ... Monoclinic crystal structure. ... Monoclinic base-centred crystal structure File links The following pages link to this file: User:DrBob/Figures Crystal structure Monoclinic Categories: GFDL images ... In anatomy, the cuboid bone is a bone in the foot. ... Orthorhombic crystal structure. ... Orthorhombic base-centred crystal File links The following pages link to this file: User:DrBob/Figures Crystal structure Orthorhombic Categories: GFDL images ... Orthorhombic body-centred crystal File links The following pages link to this file: User:DrBob/Figures Crystal structure Orthorhombic Categories: GFDL images ... Orthorhombic face-centred crystal File links The following pages link to this file: User:DrBob/Figures Crystal structure Orthorhombic Categories: GFDL images ... In crystallography, the tetragonal crystal system is one of the 7 lattice point groups. ... In anatomy, the cuboid bone is a bone in the foot. ... Tetragonal crystal structure. ... Tetragonal body-centred crystal File links The following pages link to this file: User:DrBob/Figures Crystal structure Tetragonal Categories: GFDL images ... The trapezohedron is the dual polyhedron of the corresponding antiprism. ... Rhombohedral crystal structure. ... A regular hexagon In geometry, a hexagon is a polygon with six edges and six vertices. ... To meet Wikipedias quality standards and make it easier to understand, this article or section may require cleanup. ... A cube [1] (or regular hexahedron) is a three-dimensional Platonic solid composed of six square faces, facets or sides, with three meeting at each vertex. ... Cubic crystal structure. ... Image File history File links Cubic-body-centered. ... Image File history File links Cubic-face-centered. ... Table of Geometry, from the 1728 Cyclopaedia. ... Crystallography (from the Greek words crystallon = cold drop / frozen drop, with its meaning extending to all solids with some degree of transparency, and graphein = write) is the experimental science of determining the arrangement of atoms in solids. ... The symmetry group of an object (e. ... A translation slides an object by a vector a: Ta(p) = p + a. ... See lattice for other meanings of this term, both within and without mathematics. ...

Such symmetry groups consist of translations by vectors of the form

$mathbf{R} = n_1 mathbf{a}_1 + n_2 mathbf{a}_2 + n_3 mathbf{a}_3,$

where n₁, n₂, and n₃ are integers and a₁, a₂, and a₃ are three non-coplanar vectors, called primitive vectors. The integers consist of the positive natural numbers (1, 2, 3, â€¦), their negatives (âˆ’1, âˆ’2, âˆ’3, ...) and the number zero. ...

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. The space group of a crystal is a mathematical description of the symmetry inherent in the structure. ...

All crystalline materials recognised till now (not including quasicrystals) fit in one of these arrangements. This article or section is in need of attention from an expert on the subject. ...

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. In solid state physics and mineralogy, particularly in describing crystal structure, a primitive cell is a minimum volume cell corresponding to a single lattice point. ... In mathematics, given a lattice Γ in a Lie group G, a fundamental domain is a set D of representatives for the cosets G/Γ, that is also a well-behaved set topologically, in a sense that can be made precise in one of several ways. ...

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. 1842 was a common year starting on Saturday (see link for calendar). ... Auguste Bravais (c. ... 1848 is a leap year starting on Saturday of the Gregorian calendar. ...

External links

Overview of the 32 groups
Ditto - uses a different notation
Mineral galleries - Symmetry

Categories: Symmetry | Euclidean geometry | Crystallography | Mineralogy

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Contents

Overview of point groups by crystal system

Classification of lattices

See also

External links