Crystallographic point group

The point groups are named according to their component symmetries. There are several standard notations used by crystallographers, mineralogists, and physicists. For the correspondence of the two systems below, see crystal system. Schoenflies notation In Schoenflies notation, point groups are denoted by a letter symbol with a subscript. The symbols used in crystallography mean the following: • Cn (for cyclic) indicates that the group has an n-fold rotation axis. Cnh is Cn with the addition of a mirror (reflection) plane perpendicular to the axis of rotation. Cnv is Cn with the addition of n mirror planes parallel to the axis of rotation. • S2n (for Spiegel, German for mirror) denotes a group with only a 2n-fold rotation-reflection axis. • Dn (for dihedral, or two-sided) indicates that the group has an n-fold rotation axis plus n twofold axes perpendicular to that axis. Dnh has, in addition, a mirror plane perpendicular to the n-fold axis. Dnd has, in addition to the elements of Dn, mirror planes parallel to the n-fold axis. • The letter T (for tetrahedron) indicates that the group has the symmetry of a tetrahedron. Td includes improper rotation operations, T excludes improper rotation operations, and Th is T with the addition of an inversion. • The letter O (for octahedron) indicates that the group has the symmetry of an octahedron, with (Oh) 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. D4d and D6d are actually forbidden because they contain improper rotations with n=8 and 12 respectively. The 27 point groups in the table plus T, Td, Th, O and Oh constitute 32 crystallographic point groups. Hermann–Mauguin notation An abbreviated form of the Hermann–Mauguin notation commonly used for space groups also serves to describe crystallographic point groups. Group names are The correspondence between different notations ==Graphical representations==

as a guide to the eye. It is common to display point groups graphically to develop an intuitive understanding of their symmetry. Typically stereographic projections are used as they preserve angular relations. Two types of projections can be made. The first, shown here, is a projection of the symmetry elements, to display their angular relations with respect to one another. In this case, each symmetry element is represented by a symbol shown in the table below. Thin lines are used to demarcate the sphere of the stereographic projection, and axes of rotation or rotoinversion where they do not intersect with a mirror plane. Mirror planes are represented by bold lines. The second type of projection, is of a general point, and all the additional points generated from that initial point using the symmetry elements of the point group. == Isomorphisms ==

Two groups are said to be isomorphic if there exists a bijective and homomorphic mapping between the two groups. That is to say that simply renaming the elements of one group with the elements of the second group in the right manner will give you the second group and visa versa. Many of the crystallographic point groups share the same internal structure in this sense. 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 C2. All isomorphic groups are of the same order, 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 cyclic groups (C1, C2, C3, C4, C6), dihedral groups (D2, D3, D4, D6), one of the alternating groups (A4), and one of the symmetric groups (S4). Here the symbol " × " indicates a direct product. ==Deriving the crystallographic point group (crystal class) from the space group==

• Leave out the Bravais lattice 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. ==Symmetry in understanding crystal properties==

The symmetry of a material can have profound effects on what properties are 'allowed' to be displayed by that crystal. These influences are summarized in Von Neumann's principle and more generally by the Curie Law's. These laws state that the symmetry of a crystals physical properties must be at least as symmetric as the crystal itself. A common example given is in piezoelectricity and pyroelectricity. These properties generate an electric dipole in a crystal under strain or thermal changes. An electric dipole is directional and as such cannot exist in crystals with inversion symmetry. The converse, however, is not true. Crystals without inversion symmetry do not necessarily display piezo- or pyro-electricity. Symmetry in Polycrystals Polycrystals contain many small crystals of different orientations. In an idealized case, with sufficiently small crystals in a sufficiently large sample, every orientation is represented, leading to an approximately isotropic material. In effect, this allows a polycrystal to display properties of a higher symmetry than the individual crystals composing it. For example, an ideal polycrystal or pyroelectric crystallites will display electric dipoles in every direction, which cancel out, giving the polycrystal a net zero polarization. An extension of the crystallographic point groups known as the Curie groups allows us to describe the symmetry of the polycrystal by the symmetry of the orientations of each crystallite. In an ideal polycrystal, every orientation is identical, which may be though of as having infinite rotational symmetry in all direction given by ∞∞, representing two perpendicular rotational axis. This gives two groups ∞∞ if the polycrystal has a net chirality, and ∞∞m if there is no net chirality. The remaining curie groups may be derived from the ideal symmetries using the Curie's principle. ==See also==