Centrosymmetry

The term centrosymmetric, as generally used in crystallography, refers to a space group which contains an inversion center as one of its symmetry elements. In such a space 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.

Space groups lacking an inversion center (non-centrosymmetric) are further divided into polar and chiral types. A chiral space 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 space groups must therefore only contain (purely) rotational and translational 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 space groups.

The term polar is often used for those space groups which are neither centrosymmetric nor chiral. However, the term is more correctly used for any space 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.

See also

Centrosymmetric matrix