About the role of time and space inversion in chirality

Abstract

Because certain ambiguities in the conceptual definition of magnetic chirality, in this contribution the concept of chirality will be revised from both crystallographic and magnetic points of view. The term chirality, in a crystallographic sense, refers to the symmetry restriction of the absence of improper rotations in a crystal or in a molecule. A chiral space group can then only consist of pure rotations, pure translations, and screw rotations. Consequently, chiral crystals can occur in two different enantiomer forms, which are related by a difference in their handedness. These two enantiomer forms of a molecule or a crystal are mirror-related and not super-imposable (non congruent). Conceming magnetic structures a great variety of types have been described. The first classification regards translation properties of the magnetic structure via the propagation vector. Both commensurate, i.e. the magnetic lattice contains Q-rational times the nuclear lattice, and incommensurate structures can be found. If only the structural motif is considered then ferromagnetic, antiferromagnetic, collinear, non collinear, helical, fan, squared, modulated, etc., structures are observed. A magnetic structure is formed by an ordered arrangement of magnetic moments which transform as pseudo-vectors under symmetry operations. The mathematical tool necessary to describe such structure should be based on the symmetry transformations of the moments including the time-inversion as a new symmetry operation. Then, the 230 Space Groups, or Fedorov groups, are no longer valid to classify magnetic structures, as they do not include time-inversion as symmetry operation. Inclusion of such operation in the Fedorov groups leads to the 1651 Shubnikov groups that help to classify most magnetic structures. There are however some magnetic structures like the helical ones that can not be described by Shubnikov groups and further generalizations are required, e.g., magnetic superspace groups, etc. If the conditions leading to chirality in a nuclear lattice are weli defined, it does not seem to he such in the case of magnetic structures. In this contrihution we generalize the crystallographic definition to the case of magnetic space groups, including the time inversion symmetry.