Crystal system - Citizendia

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. The space group of a Crystal or crystallographic group is a mathematical description of the Symmetry inherent in the structure Symmetry generally conveys two primary meanings The first is an imprecise sense of harmonious or aesthetically-pleasing proportionality and balance such that it reflects beauty or In Geometry, a translation "slides" an object by a vector a: T a (p = p + a In Geometry, a Point group in three dimensions is an Isometry group in three dimensions that leaves the origin fixed or correspondingly an isometry group A major application is in crystallography, to categorize crystals, but by itself the topic is one of 3D Euclidean geometry. Crystallography is the experimental science of determining the arrangement of Atoms in Solids In older usage it is the scientific study of Crystals The In Materials science, a crystal is a Solid in which the constituent Atoms Molecules or Ions are packed in a regularly ordered repeating Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria.

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

Overview

There are 7 crystal systems:

Triclinic, all cases not satisfying the requirements of any other system. In Crystallography, the triclinic Crystal system is one of the 7 lattice Point groups A crystal system is described by three basis vectors There is no necessary symmetry other than translational symmetry, although inversion is possible. In Geometry, a translation "slides" an object by a vector a: T a (p = p + a
Monoclinic, requires either 1 twofold axis of rotation or 1 mirror plane. In Crystallography, the monoclinic Crystal system is one of the 7 lattice Point groups A crystal system is described by three vectors. Generally speaking an object with rotational symmetry is an object that looks the same after a certain amount of Rotation. Reflection symmetry, line symmetry, mirror symmetry, mirror-image symmetry, or bilateral symmetry is Symmetry with respect
Orthorhombic, requires either 3 twofold axes of rotation or 1 twofold axis of rotation and two mirror planes. In Crystallography, the orthorhombic Crystal system is one of the seven Lattice Point groups Orthorhombic lattices result from stretching
Tetragonal, requires 1 fourfold axis of rotation. In Crystallography, the tetragonal Crystal system is one of the 7 lattice Point groups Tetragonal Crystal lattices result from stretching a cubic
Rhombohedral, also called trigonal, requires 1 threefold axis of rotation. In Crystallography, the rhombohedral (or trigonal) Crystal system is one of the seven lattice point groups named after the two-dimensional
Hexagonal, requires 1 sixfold axis of rotation. In Crystallography, the hexagonal is one of the 7 Crystal system, it contains 7 Point groups.
Isometric or cubic, requires 4 threefold axes of rotation. The cubic crystal system (or isometric) is a Crystal system where the Unit cell is in the shape of a Cube.

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. In Crystallography, a crystallographic point group is a set of Symmetry operations like rotations or reflections that leave a point fixed while moving each atom of bravais lattices	No. In Geometry and Crystallography, a Bravais lattice, named after Auguste Bravais, is an infinite set of points generated by a set of discrete translation 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. The space group of a Crystal or crystallographic group 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 A crystal system is described by three basis vectors In Crystallography, the monoclinic Crystal system is one of the 7 lattice Point groups A crystal system is described by three vectors. In Crystallography, the orthorhombic Crystal system is one of the seven Lattice Point groups Orthorhombic lattices result from stretching In Crystallography, the tetragonal Crystal system is one of the 7 lattice Point groups Tetragonal Crystal lattices result from stretching a cubic In Crystallography, the rhombohedral (or trigonal) Crystal system is one of the seven lattice point groups named after the two-dimensional In Crystallography, the hexagonal is one of the 7 Crystal system, it contains 7 Point groups. The cubic crystal system (or isometric) is a Crystal system where the Unit cell is in the shape of a Cube. 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. In Mathematics, especially in Geometry and Group theory, a lattice in R n is a Discrete subgroup of 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. The space group of a Crystal or crystallographic group is a mathematical description of the Symmetry inherent in the structure

Crystallographic point groups

Main article: 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). In Crystallography, a crystallographic point group is a set of Symmetry operations like rotations or reflections that leave a point fixed while moving each atom The Symmetry group of an object ( Image, signal, etc eg in 1D 2D or 3D is the group of all Isometries under which it is In Geometry, an affine transformation or affine map or an affinity (from the Latin affinis, "connected with" between two Vector In Matrix theory, a real orthogonal matrix is a square matrix Q whose Transpose is its inverse: Q^T Space groups can be grouped by the matrices involved, i. e. ignoring the translation vectors (see also Euclidean group). In Mathematics, the Euclidean group E ( n) sometimes called ISO( n) or similar is the Symmetry group of n -dimensional This corresponds to discrete symmetry groups with a fixed point. There are infinitely many of these point groups in three dimensions. In Geometry, a Point group in three dimensions is an Isometry group in three dimensions that leaves the origin fixed or correspondingly an isometry group However, only part of these are compatible with translational symmetry: the crystallographic point groups. This is expressed in the crystallographic restriction theorem. The crystallographic restriction theorem in its basic form was based on the observation that the Rotational symmetries of a Crystal are usually limited to (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. Crystal optics is the branch of Optics that describes the behaviour of Light in Anisotropic media, that is media (such as Crystals For instance, one knows whether it is birefringent, or whether it shows the Pockels effect, by simply knowing its point group. Birefringence, or double refraction, is the decomposition of a ray of Light into two rays (the ordinary ray and the extraordinary ray The Pockels effect, or Pockels electro-optic effect produces 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-disphenoidal	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	$m\bar{3}\$	3*2	centrosymmetric
	gyroidal	O	$432\$	432	enantiomorphic
	tetrahedral	T_d	$\bar{4}3m$	*332
	hexoctahedral	O_h	$m\bar{3}m$	*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. In Mathematics, a point group is a group of geometric symmetries ( isometries) leaving a point fixed Arthur Moritz Schönflies ( April 17, 1853 &ndash May 27, 1928) was a German Mathematician, known for his contributions Carl Hermann ( 17 June 1898 &ndash 12 September 1961) was a German professor of Crystallography. French professor of Mineralogy Charles-Victor Mauguin ( July 19, 1878 &ndash April 25, 1958) was a founder of the In the mathematical disciplines of Topology and Geometric group theory, an orbifold (for "orbit-manifold" is a generalization of a Manifold. In Crystallography, the triclinic Crystal system is one of the 7 lattice Point groups A crystal system is described by three basis vectors The term chiral (pronounced /ˈkaɪɹ(əl̩/ is used to describe an object that is non- superimposable on its mirror image The term centrosymmetric, as generally used in Crystallography, refers to a Space group which contains an inversion center as one of its Symmetry elements In Crystallography, the monoclinic Crystal system is one of the 7 lattice Point groups A crystal system is described by three vectors. The term chiral (pronounced /ˈkaɪɹ(əl̩/ is used to describe an object that is non- superimposable on its mirror image General right and uniform prisms A right prism is a prism in which the joining edges and faces are perpendicular to the base faces The term centrosymmetric, as generally used in Crystallography, refers to a Space group which contains an inversion center as one of its Symmetry elements In Crystallography, the orthorhombic Crystal system is one of the seven Lattice Point groups Orthorhombic lattices result from stretching The term chiral (pronounced /ˈkaɪɹ(əl̩/ is used to describe an object that is non- superimposable on its mirror image Volume The Volume of a pyramid is V = \frac{1}{3} Bh where B is the area of the base and h the height from the base to the apex Equilateral triangle bipyramids Only three kinds of bipyramids can have all edges of the same length (which implies that all faces are Equilateral triangles: the The term centrosymmetric, as generally used in Crystallography, refers to a Space group which contains an inversion center as one of its Symmetry elements In Crystallography, the tetragonal Crystal system is one of the 7 lattice Point groups Tetragonal Crystal lattices result from stretching a cubic The term chiral (pronounced /ˈkaɪɹ(əl̩/ is used to describe an object that is non- superimposable on its mirror image The term centrosymmetric, as generally used in Crystallography, refers to a Space group which contains an inversion center as one of its Symmetry elements The term chiral (pronounced /ˈkaɪɹ(əl̩/ is used to describe an object that is non- superimposable on its mirror image The term centrosymmetric, as generally used in Crystallography, refers to a Space group which contains an inversion center as one of its Symmetry elements In Crystallography, the rhombohedral (or trigonal) Crystal system is one of the seven lattice point groups named after the two-dimensional The term chiral (pronounced /ˈkaɪɹ(əl̩/ is used to describe an object that is non- superimposable on its mirror image The term centrosymmetric, as generally used in Crystallography, refers to a Space group which contains an inversion center as one of its Symmetry elements The term chiral (pronounced /ˈkaɪɹ(əl̩/ is used to describe an object that is non- superimposable on its mirror image The term centrosymmetric, as generally used in Crystallography, refers to a Space group which contains an inversion center as one of its Symmetry elements In Crystallography, the hexagonal is one of the 7 Crystal system, it contains 7 Point groups. The term chiral (pronounced /ˈkaɪɹ(əl̩/ is used to describe an object that is non- superimposable on its mirror image The term centrosymmetric, as generally used in Crystallography, refers to a Space group which contains an inversion center as one of its Symmetry elements The term chiral (pronounced /ˈkaɪɹ(əl̩/ is used to describe an object that is non- superimposable on its mirror image The term centrosymmetric, as generally used in Crystallography, refers to a Space group which contains an inversion center as one of its Symmetry elements The cubic crystal system (or isometric) is a Crystal system where the Unit cell is in the shape of a Cube. The term chiral (pronounced /ˈkaɪɹ(əl̩/ is used to describe an object that is non- superimposable on its mirror image The term centrosymmetric, as generally used in Crystallography, refers to a Space group which contains an inversion center as one of its Symmetry elements The term chiral (pronounced /ˈkaɪɹ(əl̩/ is used to describe an object that is non- superimposable on its mirror image The term centrosymmetric, as generally used in Crystallography, refers to a Space group which contains an inversion center as one of its Symmetry elements In Mineralogy and Crystallography, a crystal structure is a unique arrangement of Atoms in a Crystal. Proteins are large Organic compounds made of Amino acids arranged in a linear chain and joined together by Peptide bonds between the Carboxyl The term chiral (pronounced /ˈkaɪɹ(əl̩/ is used to describe an object that is non- superimposable on its mirror image The term chiral (pronounced /ˈkaɪɹ(əl̩/ is used to describe an object that is non- superimposable on its mirror image The protein assemblies themselves may have symmetries other than those given above, because they are not intrinsically restricted by the Crystallographic restriction theorem. The crystallographic restriction theorem in its basic form was based on the observation that the Rotational symmetries of a Crystal are usually limited to 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. The term chiral (pronounced /ˈkaɪɹ(əl̩/ is used to describe an object that is non- superimposable on its mirror image

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	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 Crystallography, the triclinic Crystal system is one of the 7 lattice Point groups A crystal system is described by three basis vectors Properties Any of the three pairs of parallel faces can be viewed as the base planes of the prism In Crystallography, the monoclinic Crystal system is one of the 7 lattice Point groups A crystal system is described by three vectors. General right and uniform prisms A right prism is a prism in which the joining edges and faces are perpendicular to the base faces In Geometry, a parallelogram is a Quadrilateral with two sets of Parallel sides In Crystallography, the orthorhombic Crystal system is one of the seven Lattice Point groups Orthorhombic lattices result from stretching In anatomy the Cuboid bone is a bone in the foot See also Hyperrectangle Oblong In Crystallography, the tetragonal Crystal system is one of the 7 lattice Point groups Tetragonal Crystal lattices result from stretching a cubic In anatomy the Cuboid bone is a bone in the foot See also Hyperrectangle Oblong In Crystallography, the rhombohedral (or trigonal) Crystal system is one of the seven lattice point groups named after the two-dimensional Forms Trigonal trapezohedron - 6 (rhombic faces - dual Octahedron * A Cube is a special case trigonal trapezohedron In Crystallography, the hexagonal is one of the 7 Crystal system, it contains 7 Point groups. Regular hexagon The internal Angles of a regular hexagon (one where all sides and all angles are equal are all 120 ° and the hexagon has 720 degrees The cubic crystal system (or isometric) is a Crystal system where the Unit cell is in the shape of a Cube. A cube is a three-dimensional solid object bounded by six square faces facets or sides with three meeting at each vertex. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position Crystallography is the experimental science of determining the arrangement of Atoms in Solids In older usage it is the scientific study of Crystals The The Symmetry group of an object ( Image, signal, etc eg in 1D 2D or 3D is the group of all Isometries under which it is In Geometry, a translation "slides" an object by a vector a: T a (p = p + a In Mathematics, especially in Geometry and Group theory, a lattice in R n is a Discrete subgroup of

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 (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French

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. The space group of a Crystal or crystallographic group is a mathematical description of the Symmetry inherent in the structure 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). Quasicrystals are structural forms that are both ordered and nonperiodic

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. In Geometry, Solid state physics and Mineralogy, particularly in describing Crystal structure, a primitive cell, is a minimum cell corresponding Depending on the symmetry of a crystal or other pattern, the fundamental domain is again smaller, up to a factor 48. In Geometry, the fundamental domain of a Symmetry group of an object or pattern is a part of the pattern as small as possible which based on the Symmetry

The Bravais lattices were studied by Moritz Ludwig Frankenheim (1801-1869), in 1842, who found that there were 15 Bravais lattices. Year 1842 ( MDCCCXLII) was a Common year starting on Tuesday (link will display the full calendar of the Gregorian Calendar (or a Common This was corrected to 14 by A. Bravais in 1848. August Bravais (23 August 1811 Annonay – 30 March 1863, Le Chesnay, France was a French physicist well known for his work in Crystallography Year 1848 ( MDCCCXLVIII) was a Leap year starting on Saturday (link will display the full calendar of the Gregorian Calendar (or a Leap

External links

Overview of the 32 groups
Ditto - uses a different notation
Mineral galleries - Symmetry
all cubic crystal classes, forms and stereographic projections (interactive java applet)

In Mineralogy and Crystallography, a crystal structure is a unique arrangement of Atoms in a Crystal. In Mathematics, a point group is a group of geometric symmetries ( isometries) leaving a point fixed Triclinic system Monoclinic system Orthorhombic system Tetragonal system Rhombohedral (trigonal system

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Contents

Overview

Crystallographic point groups

Overview of point groups by crystal system

Classification of lattices

See also

External links