Subperiodic classes of reducible space groups

Abstract

Classification of reducible space groups into pairs of complementary subperiodic classes with respect to various reductions is introduced and analysed. This classification is finer than the classification into geometric classes and it intersects with the classification into arithmetic classes. It is proved that an intersection theorem holds for those classes which correspond to Z decomposition of the translation subgroups of the reducible space groups and then symmorphic representatives of subperiodic classes of reducible space groups are introduced in analogy with the ordinary concept of symmorphic space groups. In particular, it is shown that the symmorphic space group is a symmorphic representative of subperiodic classes, defined by complementary symmorphic subperiodic groups. In cases of Z reductions it is shown that the pair of complementary subperiodic classes may define none, one or several space groups; if one such group belongs to these classes, then also a set of groups which differ by shifts in space does. These shifts are determined with translation normalizers. Further ramifications and possible use of the theory are discussed.