crystallography Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids. Crystallography is a fundamental subject in the fields of materials science and solid-state physics ( condensed matter physics). The wor ...

, a crystallographic point group is a set of symmetry operations, corresponding to one of the

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 of a sphere. It is a subgroup of the orthogonal group O(3), the group of all isometrie ...

, such that each operation (perhaps followed by a

translation Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transla ...

) would leave the structure of a crystal unchanged i.e. the same kinds of atoms would be placed in similar positions as before the transformation. For example, in many crystals in the

cubic crystal system In crystallography, the cubic (or isometric) crystal system is a crystal system where the unit cell is in the shape of a cube. This is one of the most common and simplest shapes found in crystals and minerals. There are three main varieties ...

, a rotation of the unit cell by 90 degrees around an axis that is perpendicular to one of the faces of the cube is a symmetry operation that moves each atom to the location of another atom of the same kind, leaving the overall structure of the crystal unaffected. In the classification of crystals, each point group defines a so-called (geometric) crystal class. There are infinitely many three-dimensional

point group In geometry, a point group is a mathematical group of symmetry operations ( isometries in a Euclidean space) that have a fixed point in common. The coordinate origin of the Euclidean space is conventionally taken to be a fixed point, and every ...

s. However, the crystallographic restriction on the general point groups results in there being only 32 crystallographic point groups. These 32 point groups are one-and-the-same as the 32 types of morphological (external) crystalline symmetries derived in 1830 by Johann Friedrich Christian Hessel from a consideration of observed crystal forms. The point group of a crystal determines, among other things, the directional variation of physical properties that arise from its structure, including

optical properties The optical properties of a material define how it interacts with light. The optical properties of matter are studied in optical physics, a subfield of optics. The optical properties of matter include: * Refractive index *Dispersion * Transmittan ...

such as birefringency, or electro-optical features such as the Pockels effect. For a periodic crystal (as opposed to a quasicrystal), the group must maintain the three-dimensional

translational symmetry In geometry, to translate a geometric figure is to move it from one place to another without rotating it. A translation "slides" a thing by . In physics and mathematics, continuous translational symmetry is the invariance of a system of equati ...

that defines crystallinity.

Notation

The point groups are named according to their component symmetries. There are several standard notations used by crystallographers,

mineralogist Mineralogy is a subject of geology specializing in the scientific study of the chemistry, crystal structure, and physical (including optical) properties of minerals and mineralized artifacts. Specific studies within mineralogy include the proce ...

s, and physicists. For the correspondence of the two systems below, see crystal system.

Schoenflies notation

Schoenflies Arthur Moritz Schoenflies (; 17 April 1853 – 27 May 1928), sometimes written as Schönflies, was a German mathematician, known for his contributions to the application of group theory to crystallography, and for work in topology. Schoenflies ...

notation, point groups are denoted by a letter symbol with a subscript. The symbols used in crystallography mean the following: *''C_n'' (for

cyclic Cycle, cycles, or cyclic may refer to: Anthropology and social sciences * Cyclic history, a theory of history * Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr. * Social cycle, various cycles in so ...

) indicates that the group has an ''n''-fold rotation axis. ''C_nh'' is ''C_n'' with the addition of a mirror (reflection) plane perpendicular to the

axis of rotation Rotation around a fixed axis is a special case of rotational motion. The fixed- axis hypothesis excludes the possibility of an axis changing its orientation and cannot describe such phenomena as wobbling or precession. According to Euler's r ...

. ''C_nv'' is ''C_n'' with the addition of n mirror planes parallel to the axis of rotation. *''S_2n'' (for ''Spiegel'', German for

mirror A mirror or looking glass is an object that reflects an image. Light that bounces off a mirror will show an image of whatever is in front of it, when focused through the lens of the eye or a camera. Mirrors reverse the direction of the im ...

) denotes a group with only a ''2n''-fold rotation-reflection axis. *''D_n'' (for dihedral, or two-sided) indicates that the group has an ''n''-fold rotation axis plus ''n'' twofold axes perpendicular to that axis. ''D_nh'' has, in addition, a mirror plane perpendicular to the ''n''-fold axis. ''D_nd'' has, in addition to the elements of ''D_n'', mirror planes parallel to the ''n''-fold axis. *The letter ''T'' (for

tetrahedron In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all ...

) indicates that the group has the symmetry of a tetrahedron. ''T_d'' includes

improper rotation In geometry, an improper rotation,. also called rotation-reflection, rotoreflection, rotary reflection,. or rotoinversion is an isometry in Euclidean space that is a combination of a rotation about an axis and a reflection in a plane perpendicul ...

operations, ''T'' excludes improper rotation operations, and ''T_h'' is ''T'' with the addition of an inversion. *The letter ''O'' (for

octahedron In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at ea ...

) indicates that the group has the symmetry of an octahedron (or

cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only ...

), with (''O_h'') or without (''O'') improper operations (those that change handedness). Due to the crystallographic restriction theorem, ''n'' = 1, 2, 3, 4, or 6 in 2- or 3-dimensional space. ''D_4d'' and ''D_6d'' are actually forbidden because they contain

s with n=8 and 12 respectively. The 27 point groups in the table plus ''T'', ''T_d'', ''T_h'', ''O'' and ''O_h'' constitute 32 crystallographic point groups.

Hermann–Mauguin notation

An abbreviated form of the

Hermann–Mauguin notation In geometry, Hermann–Mauguin notation is used to represent the symmetry elements in point groups, plane groups and space groups. It is named after the German crystallographer Carl Hermann (who introduced it in 1928) and the French mineralogis ...

commonly used for

space group In mathematics, physics and chemistry, a space group is the symmetry group of an object in space, usually in three dimensions. The elements of a space group (its symmetry operations) are the rigid transformations of an object that leave it uncha ...

s also serves to describe crystallographic point groups. Group names are

The correspondence between different notations

Isomorphisms

Many of the crystallographic point groups share the same internal structure. For example, the point groups , 2, and m contain different geometric symmetry operations, (inversion, rotation, and reflection, respectively) but all share the structure of the

cyclic group In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...

C₂. All

isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...

groups are of the same

order Order, ORDER or Orders may refer to: * Categorization, the process in which ideas and objects are recognized, differentiated, and understood * Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of ...

, but not all groups of the same order are isomorphic. The point groups which are isomorphic are shown in the following table: This table makes use of

s (C₁, C₂, C₃, C₄, C₆),

dihedral group In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, ...

s (D₂, D₃, D₄, D₆), one of the

alternating group In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on a set of elements is called the alternating group of degree , or the alternating group on letters and denoted by or Basic pr ...

s (A₄), and one of the

symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group ...

s (S₄). Here the symbol " × " indicates a

direct product In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one t ...

Deriving the crystallographic point group (crystal class) from the space group

# Leave out the

Bravais lattice In geometry and crystallography, a Bravais lattice, named after , is an infinite array of discrete points generated by a set of discrete translation operations described in three dimensional space by : \mathbf = n_1 \mathbf_1 + n_2 \mathbf_2 + n ...

type. # Convert all symmetry elements with translational components into their respective symmetry elements without translation symmetry. (Glide planes are converted into simple mirror planes; screw axes are converted into simple axes of rotation.) # Axes of rotation, rotoinversion axes, and mirror planes remain unchanged.

References

External links

Point-group symbols in International Tables for Crystallography (2006). Vol. A, ch. 12.1, pp. 818-820Names and symbols of the 32 crystal classes in International Tables for Crystallography (2006). Vol. A, ch. 10.1, p. 794
{{Crystal systems Symmetry Crystallography Discrete groups