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Encyclopedia > Crystal system

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. 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. ... A discrete point group in 3D is a finite symmetry group in 3D that leaves the origin fixed. ... 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 Synthetic bismuth crystal Insulin crystals Gallium, a metal that easily forms large single crystals A huge monocrystal of potassium dihydrogen phosphate grown from solution by Saint-Gobain for the megajoule laser of CEA. In chemistry and mineralogy, a crystal is a solid in which the constituent atoms, molecules... 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 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, together 230. The following mini-table gives a breakdown of the various different things per crystal system - In crystallography, the triclinic crystal system is one of the 7 lattice point groups. ... A translation slides an object by a vector a: Ta(p) = p + a. ... In crystallography, the monoclinic crystal system is one of the 7 lattice point groups. ... The triskelion appearing on the Isle of Man flag. ... Figures with the axes of symmetry drawn in. ... 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 seven lattice point groups. ... In crystallography, the hexagonal crystal system is one of the 7 lattice point groups. ... The cubic crystal system is a crystal system where the unit cell is in the shape of a cube. ...

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, named after Auguste Bravais, 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. ... In crystallography, the hexagonal crystal system is one of the 7 lattice point groups. ... The cubic crystal system is a crystal system where the unit cell is in the shape of a cube. ...

  • 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. ...

Contents

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.) 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 (strictly speaking, two affine 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... 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, by simply 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-pinacoidal Ci bar{1} 1x centrosymmetric
monoclinic monoclinic-sphenoidal C2 2 22 enantiomorphic polar
monoclinic-domatic Cs m 1* polar
monoclinic-prismatic C2h 2/m 2* centrosymmetric
orthorhombic orthorhombic-sphenoidal D2 222 222 enantiomorphic
orthorhombic-pyramidal C2v mm2 *22 polar
orthorhombic-bipyramidal D2h mmm *222 centrosymmetric
tetragonal tetragonal-pyramidal C4 4 44 enantiomorphic polar
tetragonal-disphenoidial S4 bar{4} 2x
tetragonal-dipyramidal C4h 4/m 4* centrosymmetric
tetragonal-trapezoidal D4 422 422 enantiomorphic
ditetragonal-pyramidal C4v 4mm *44 polar
tetragonal-scalenoidal D2d bar{4}2m or bar{4}m2 2*2
ditetragonal-dipyramidal D4h 4/mmm *422 centrosymmetric
rhombohedral (trigonal) trigonal-pyramidal C3 3 ! 33 enantiomorphic polar
rhombohedral S6 (C3i) bar{3} 3x centrosymmetric
trigonal-trapezoidal D3 32 or 321 or 312 322 enantiomorphic
ditrigonal-pyramidal C3v 3mor 3m1 or 31m *33 polar
ditrigonal-scalahedral D3d 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 C3h bar{6} 3*
hexagonal-dipyramidal C6h 6/m 6* centrosymmetric
hexagonal-trapezoidal D6 622 622 enantiomorphic
dihexagonal-pyramidal C6v 6mm *66 polar
ditrigonal-dipyramidal D3h bar{6}m2 or bar{6}2m *322
dihexagonal-dipyramidal D6h 6/mmm *622 centrosymmetric
cubic tetartoidal T 23 332 enantiomorphic
diploidal Th mbar{3} 3*2 centrosymmetric
gyroidal O 432 432 enantiomorphic
tetrahedral Td bar{4}3m *332
hexoctahedral Oh 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 Schönflies (April 17, 1853 Landsberg an der Warthe(Gorzów) – 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 17–1961 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 – 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. ... In crystallography, the triclinic crystal system is one of the 7 lattice point groups. ... The term chiral (pronounced ) is used to describe an object which is non-superimposable on its mirror image. ... Polar may refer to: Polsk Ost med KATING WAPOOOOW CHING CHING WOWOWOWOW/Gling, Oink oink. ... 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. ... The term chiral (pronounced ) is used to describe an object which is non-superimposable on its mirror image. ... Polar may refer to: Polsk Ost med KATING WAPOOOOW CHING CHING WOWOWOWOW/Gling, Oink oink. ... Polar may refer to: Polsk Ost med KATING WAPOOOOW CHING CHING WOWOWOWOW/Gling, Oink oink. ... In geometry, an n-sided prism is a polyhedron made of an n-sided polygonal base, a translated copy, and n faces joining corresponding sides. ... 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 7 lattice point groups. ... The term chiral (pronounced ) is used to describe an object which is non-superimposable on its mirror image. ... This article is about the polyhedron pyramid (a 3-dimensional shape); for other versions including architectural Pyramids, see Pyramid (disambiguation). ... Polar may refer to: Polsk Ost med KATING WAPOOOOW CHING CHING WOWOWOWOW/Gling, Oink oink. ... A bipyramid is a polyhedron formed by joining two identical pyramids base-to-base. ... 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. ... The term chiral (pronounced ) is used to describe an object which is non-superimposable on its mirror image. ... Polar may refer to: Polsk Ost med KATING WAPOOOOW CHING CHING WOWOWOWOW/Gling, Oink oink. ... 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 ) is used to describe an object which is non-superimposable on its mirror image. ... Polar may refer to: Polsk Ost med KATING WAPOOOOW CHING CHING WOWOWOWOW/Gling, Oink oink. ... 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. ... The term chiral (pronounced ) is used to describe an object which is non-superimposable on its mirror image. ... Polar may refer to: Polsk Ost med KATING WAPOOOOW CHING CHING WOWOWOWOW/Gling, Oink oink. ... 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 ) is used to describe an object which is non-superimposable on its mirror image. ... Polar may refer to: Polsk Ost med KATING WAPOOOOW CHING CHING WOWOWOWOW/Gling, Oink oink. ... 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 crystal system is one of the 7 lattice point groups. ... The term chiral (pronounced ) is used to describe an object which is non-superimposable on its mirror image. ... Polar may refer to: Polsk Ost med KATING WAPOOOOW CHING CHING WOWOWOWOW/Gling, Oink oink. ... 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 ) is used to describe an object which is non-superimposable on its mirror image. ... Polar may refer to: Polsk Ost med KATING WAPOOOOW CHING CHING WOWOWOWOW/Gling, Oink oink. ... 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 is a crystal system where the unit cell is in the shape of a cube. ... The term chiral (pronounced ) is used to describe an object which 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 ) is used to describe an object which 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. ... A representation of the 3D structure of myoglobin, showing coloured alpha helices. ... The term chiral (pronounced ) is used to describe an object which is non-superimposable on its mirror image. ... The term chiral (pronounced ) is used to describe an object which is non-superimposable on its 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. ... The term chiral (pronounced ) is used to describe an object which is non-superimposable on its mirror image. ...


Classification of lattices

Crystal system Lattices
triclinic (parallelepiped) Triclinic
monoclinic (right prism with parallelogram base; here seen from above) simple centered
Monoclinic, simple Monoclinic, centered
orthorhombic (cuboid) simple base-centered body-centered face-centered
Orthohombic, simple Orthohombic, base-centered Orthohombic, body-centered Orthohombic, face-centered
tetragonal (square cuboid) simple body-centered
Tetragonal, simple Tetragonal, body-centered
rhombohedral
(trigonal) (3-sided trapezohedron)
Rhombohedral
hexagonal (centered regular hexagon) Hexagonal
cubic
(isometric; cube)
simple body-centered face-centered
Cubic, simple Cubic, body-centered Cubic, 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. In crystallography, the triclinic crystal system is one of the 7 lattice point groups. ... In geometry, a parallelepiped (now usually pronounced , traditionally[1] in accordance with its etymology in Greek παραλληλ-επίπεδον, a body having parallel planes) is a three-dimensional figure like a cube, except that its faces are not squares but parallelograms. ... Image File history File links No higher resolution available. ... In crystallography, the monoclinic crystal system is one of the 7 lattice point groups. ... In geometry, an n-sided prism is a polyhedron made of an n-sided polygonal base, a translated copy, and n faces joining corresponding sides. ... This article or section is not written in the formal tone expected of an encyclopedia article. ... Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... In crystallography, the orthorhombic crystal system is one of the 7 lattice point groups. ... In anatomy, the cuboid bone is a bone in the foot. ... Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... 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. ... Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... In crystallography, the rhombohedral (or trigonal) crystal system is one of the seven lattice point groups. ... The trapezohedron is the dual polyhedron of the corresponding antiprism. ... Image File history File links No higher resolution available. ... In crystallography, the hexagonal crystal system is one of the 7 lattice point groups. ... A regular hexagon. ... Image File history File links No higher resolution available. ... The cubic crystal system is a crystal system where the unit cell is in the shape of a cube. ... A cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. ... Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... Calabi-Yau manifold Geometry (Greek γεωμετρία; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. ... 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 are commonly denoted by the above symbol. ...


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 must, by definition fit in one of these arrangements (not including quasicrystals). Quasicrystals are aperiodic structures which produce diffraction. ...


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. ... Year 1848 (MDCCCXLVIII) was a leap year starting on Saturday (link will display the full calendar) of the Gregorian Calendar (or a leap year starting on Monday of the 12-day slower Julian calendar). ...


See also

Enargite crystals In mineralogy and crystallography, a crystal structure is a unique arrangement of atoms in a crystal. ... In mathematics, point group is a group of geometric symmetries (isometries) leaving a point fixed. ...

External links


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