Point groups in crystallography

Abstract

Two dual spaces are extensively used in crystallography: the space E n , hosting the crystal pattern; and the vector space V n , where face normals and reciprocal-lattice vectors are defined. The term point is used in crystallography to indicate four different types of groups in these two spaces. 1) Morphological groups in V n ; they can be obtained by determining subgroups of maximal holohedries (holohedries not in group-subgroup relation): this gives 21 and 136 groups in V 2 and V 3 , respectively, which are classified into 10 and 32 point-group types (on the basis of which geometric crystal classes are defined) falling into 9 and 18 abstract isomorphism classes. 2) Symmetry groups of atomic groups and coordination polyhedra in E n ; they coincide with molecular groups, which are infinite in number because the symmetry operations forming these groups are not subject to the crystallographic restriction. 3) Site-symmetry groups in E n ; they are finite groups but infinite in number due to conjugation by the translation subgroups of the space groups. They are classified in geometric crystal classes exactly like groups in V n . A finer classification of site-symmetry groups into species is however introduced that takes into account their orientation in space: species of site-symmetry groups in E n uniquely correspond to groups in V n . 4) Groups of matrices representing the linear parts of space group operations in E n ; they are isomorphic to the groups in V n and are also isomorphic to the factor groups G/T, where G is a space group and T its translation subgroup.