Space groups rare for organic structures. I. Triclinic, monoclinic and orthorhombic crystal classes

Abstract

Mirror planes and rotation axes lead to rarity of space groups for organic structures. Their inhibiting effect is mitigated by the simultaneous presence of glide planes or screw axes or both. To a first approximation the number of structures in each space group of a given crystal class is given by Nsg = Acc exp (- Bcc[2]sg - Ccc[m]sg), where [2]sg is the number of twofold axes and [m]sg is the number of reflexion planes in the cell, Bcc and Ccc are parameters characteristic of the crystal class in question, and Acc is a normalizing factor, proportional to the total number of structures in the crystal class. If the cell is centred, it will contain twice or four times as many symmetry elements as a primitive cell in the same crystal class, but for a given asymmetric unit it will have (approximately) twice or four times the volume, so that the density of symmetry elements is (approximately) the same. Centred cells thus fall approximately into line with primitive cells if the actual numbers of symmetry elements are divided by two or four to give the number in a 'volume-equivalent' cell. In the first approximation no separate provision is needed for [21 ]sg and [g]sg or other glides, since ([2]sg + [21]sg) and ([m]sg + [g]sg) are constants for the volume-equivalent cell within each crystal class in these systems. In a second approximation coincidences of axes and planes and a residual effect of centring can be allowed for, and the representation becomes quantitative (R2 ≤ 0.01 for 2, m, 2/m, 222, and ≤ 0.04 for mm2, mmm).