crystallography Crystallography is the branch of science devoted to the study of molecular and crystalline structure and properties. The word ''crystallography'' is derived from the Ancient Greek word (; "clear ice, rock-crystal"), and (; "to write"). In J ...

, a centrosymmetric

point group In geometry, a point group is a group (mathematics), mathematical group of symmetry operations (isometry, isometries in a Euclidean space) that have a Fixed point (mathematics), fixed point in common. The Origin (mathematics), coordinate origin o ...

contains an inversion center as one of its

symmetry Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is Invariant (mathematics), invariant und ...

elements. In such a

, for every point (x, y, z) in the unit cell there is an indistinguishable point (-x, -y, -z). Such point groups are also said to have ''inversion'' symmetry. Point reflection is a similar term used in geometry. Crystals with an inversion center cannot display certain properties, such as the piezoelectric effect and the frequency doubling effect (

second-harmonic generation Second-harmonic generation (SHG), also known as frequency doubling, is the lowest-order wave-wave nonlinear interaction that occurs in various systems, including optical, radio, atmospheric, and magnetohydrodynamic systems. As a prototype behav ...

). In addition, in such crystals, one-photon absorption (OPA) and two-photon absorption (TPA) processes are mutually exclusive, i.e., they do not occur simultaneously, and provide complementary information. The following

space groups In mathematics, physics and chemistry, a space group is the symmetry group of a repeating pattern in space, usually in three dimensions. The elements of a space group (its symmetry operations) are the rigid transformations of the pattern that ...

have inversion symmetry: the triclinic space group 2, the monoclinic 10-15, the orthorhombic 47-74, the tetragonal 83-88 and 123-142, the trigonal 147, 148 and 162-167, the hexagonal 175, 176 and 191-194, the cubic 200-206 and 221-230. Point groups lacking an inversion center (non-centrosymmetric) can be '' polar'', ''

chiral Chirality () is a property of asymmetry important in several branches of science. The word ''chirality'' is derived from the Greek language, Greek (''kheir''), "hand", a familiar chiral object. An object or a system is ''chiral'' if it is dist ...

'', both, or neither. A '' polar''

is one whose symmetry operations leave more than one common point unmoved. A polar point group has no unique origin because each of those unmoved points can be chosen as one. One or more unique polar axes could be made through two such collinear unmoved points. Polar

crystallographic point group In crystallography, a crystallographic point group is a three-dimensional point group whose symmetry operations are compatible with a three-dimensional crystallographic lattice. According to the crystallographic restriction it may only contain o ...

s include 1, 2, 3, 4, 6, m, mm2, 3m, 4mm, and 6mm. A ''

'' (often also called enantiomorphic)

is one containing only proper (often called "pure") rotation symmetry. No inversion, reflection, roto-inversion or roto-reflection (i.e., improper rotation)

exists in such point group. Chiral crystallographic point groups include 1, 2, 3, 4, 6, 222, 422, 622, 32, 23, and 432. Chiral molecules such as

proteins Proteins are large biomolecules and macromolecules that comprise one or more long chains of amino acid residues. Proteins perform a vast array of functions within organisms, including catalysing metabolic reactions, DNA replication, re ...

crystallize in chiral

s. The remaining non-centrosymmetric crystallographic point groups , 2m, , m2, 3m are neither polar nor chiral.

References

{{Reflist Symmetry ru:Центральная симметрия

See also

References