Abstract:
A more than 75 years old, still unresolved problem to determine subgroups of a (N-dimensional) space group G happened to find, in course of time, various crystallographic and physical applications: (a) Isotropy subgroups of a space group G specify possible low symmetries of a structural phase transition where G is the prototypic symmetry; (b) Site-point subgroups, yielding Wyckoff positions of G, are applicable in structural analysis of quasicrystals (N>3); (c) Subperiodic subgroups are used in superspace approach to description of extensive family of modulated structures (N=4,5,6).
All subgroups of any space group G form an (algebraic) lattice L(G) with two binary operations of (set-)intersection and (group-)union. The subgroups are divided into N+1 classes according to the number M=0,...N of independent translations involved. Previously, we dealt with the problem of site-point subgroups (M=0), and wrote a C-program for deriving the Wyckoff positions for any finite N. Now we focus ourselves to the class of subgroups which themselves are space groups (M=N), and do form an infinite lattice. We have developed a computer software to compute, construct and visualize finite quotient L(G : F) of L(G), which consists of all subgroups of G containing a given space group F⊂ G as a lower bound. The underlying algorithm is based on the concept of lattice of crystallographic pairs (P,T), where P is a crystallographic point group, and T - a discrete translational group invariant under P. We demonstrate program capabilities on illustrative examples.