Rod groups and their settings as special geometric realisations of line groups

Abstract

Rod groups (monoperiodic subgroups of the 3-periodic space groups) are considered as a special case of the commensurate line groups (discrete symmetry groups of the three-dimensional objects translationally periodic along a line). Two different factorizations of line groups are considered: (1) The standard L = T(a)F used in crystallography for rod groups; F is a finite system of representatives of line-group decomposition in cosets of 1-periodic translation group T(a); (2) L = ZP used in the theory of line groups; Z is a cyclic generalized translation group and P is a finite point group. For symmorphic line groups (five line-group families of 13 families) the two factorizations are equivalent: the cyclic group Z is a monoperiodic translation group and P is the point group defining the crystal class. For each of the remaining eight families of non-symmorphic line groups the explicit correspondence between rod groups and relevant geometric realisations of the corresponding line groups is established. The settings of rod groups and line groups are taken into account. The results are presented in a table of 75 rod groups listed (in international and factorized notation) by families of the line groups according to the order of the principal axis q (q = 1, 2, 3, 4, 6) of the corresponding isogonal point group.