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The term centrosymmetric, as generally used in crystallography, refers to a point group which contains an inversion center as one of its symmetry elements. In such a point group, for every point (x, y, z) in the unit cell there is an indistinguishable point (-x, -y, -z). Crystals with an inversion center cannot display certain properties, such as the piezoelectric effect.
Point groups lacking an inversion center (non-centrosymmetric) are further divided into polar and chiral types. A chiral point group is one without any rotoinversion symmetry elements. Rotoinversion (also called an 'inversion axis') is rotation followed by inversion; for example, a mirror reflection corresponds to a twofold rotoinversion. Chiral point groups must therefore only contain (purely) rotational symmetry. These arise from the crystal point groups 1, 2, 3, 4, 6, 222, 422, 622, 32, 23, and 432. Chiral molecules such as proteins crystallize in chiral point groups.
The term polar is often used for those point groups which are neither centrosymmetric nor chiral. However, the term is more correctly used for any point group containing a unique anisotropic axis. These occur in crystal point groups 1, 2, 3, 4, 6, m, mm2, 3m, 4mm, and 6mm. Thus some chiral space groups are also polar.