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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
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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.
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is one containing only proper (often called "pure") rotation symmetry. No inversion, reflection, roto-inversion or roto-reflection (i.e., improper rotation)
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exists in such point group. Chiral crystallographic point groups include 1, 2, 3, 4, 6, 222, 422, 622, 32, 23, and 432.
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is a similar term used in geometry. Crystals with an inversion center cannot display certain properties, such as the
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there is an indistinguishable point (-x, -y, -z). Such point groups are also said to have
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Benzene is a centrosymmetric molecule having a centre of symmetry at the centre
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Fu, Liang; Kane, C. (2007). "Topological insulators with inversion symmetry".
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The remaining non-centrosymmetric crystallographic point groups
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198:Point groups lacking an inversion center (
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407:. Birkbeck College, University of London
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405:"The 230 3-Dimensional Space Groups"
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240:(often also called enantiomorphic)
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317:Tilley, Richard (2006). "4".
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181:second-harmonic generation
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298:Rule of mutual exclusion
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293:Centrosymmetric matrix
256:crystallize in chiral
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177:piezoelectric effect
159:elements. In such a
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