Die Symmetrie von Spiralketten

Abstract

In crystals, polymeric chain molecules often adopt helical structures. Neglecting small distortions possibly caused by an anisotropic environment within the crystal, the symmetry of the single helix can be described by a rod group, which has translational symmetry in one dimension. The rod groups have Hermann–Mauguin symbols similar to space groups, beginning with a script style \scr p followed by a screw-axis symbol; the order of the screw axis can adopt any value. In a crystal, the rod-site symmetry, the so-called penetration rod group, must be a common crystallographic rod subgroup of the molecular rod group and the space group. Instructions are given for the derivation of the rod subgroups in question for a molecular helical rod group of any order. In polymer chemistry, a helix is designated by a (chemical) symbol like 7/2, which means 7 repeating units in 2 coil turns of covalent bonds per translational period. The corresponding Hermann–Mauguin screw-axis symbol is easily derived with a simple formula from this chemical symbol; for a 7/2 helix it is 73or 74, depending on chirality. However, it is not possible to deduce the chemical symbol from the Hermann–Mauguin symbol, because it depends on where the covalent bonds are assumed to exist. Covalent bonds are irrelevant for symmetry considerations; a symmetry symbol does not depend on them. A chemically right-handed helix can have a left-handed screw axis. The derivation of the Hermann–Mauguin symbol of a multiple helix is not that easy, as it depends on the mutual position of the interlocked helices; conversion formulae for simpler cases are presented. Instead of covalent bonds, other kinds of linking can serve to define the chemical helix, for example, edge- or face-sharing coordination polyhedra.