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Incrystallography,acentrosymmetricpoint groupcontains aninversion centeras one of itssymmetryelements.[1]In such apoint group,for every point (x, y, z) in theunit cellthere is an indistinguishable point (-x, -y, -z). Such point groups are also said to haveinversionsymmetry.[2]Point reflectionis a similar term used in geometry. Crystals with an inversion center cannot display certain properties, such as thepiezoelectric effectand the frequency doubling effect (second-harmonic generation). In addition, in such crystals, one-photon absorption (OPA) andtwo-photon absorption(TPA) processes are mutually exclusive, i.e., they do not occur simultaneously, and provide complementary information.
The followingspace groupshave 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.[3]
Point groups lacking an inversion center (non-centrosymmetric) can bepolar,chiral,both, or neither.
Apolarpoint groupis 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. Polarcrystallographic point groupsinclude 1, 2, 3, 4, 6, m, mm2, 3m, 4mm, and 6mm.
Achiral(often also called enantiomorphic)point groupis one containing only proper (often called "pure" ) rotation symmetry. No inversion, reflection, roto-inversion or roto-reflection (i.e., improper rotation)symmetryexists in such point group. Chiral crystallographic point groups include 1, 2, 3, 4, 6, 222, 422, 622, 32, 23, and 432.Chiral moleculessuch asproteinscrystallize in chiralpoint groups.
The remaining non-centrosymmetric crystallographic point groups4,42m,6,6m2,43m are neither polar nor chiral.
See also
editReferences
edit- ^Tilley, Richard (2006). "4".Crystals and Crystal Structures.John Wiley. pp.80–83.ISBN978-0-470-01821-7.
- ^Fu, Liang; Kane, C. (2007). "Topological insulators with inversion symmetry".Physical Review B.76(4): 045302.arXiv:cond-mat/0611341.Bibcode:2007PhRvB..76d5302F.doi:10.1103/PhysRevB.76.045302.S2CID15011491.
- ^Cockcroft, Jeremy Karl."The 230 3-Dimensional Space Groups".Birkbeck College, University of London.Retrieved18 August2014.