Magnetic Point Groups and Space Groups

Magnetic groups – also known as antisymmetry groups, Shubnikov groups, Heesch groups, Opechowski–Guccione (OG) groups, as well as dichromatic, 2-color, or simply black-and-white groups – are the simplest extension to standard point group and space group theory. They allow directly to describe, classify, and study the consequences of the symmetry of crystals, characterized by having a certain property, associated with each crystal site, that can take one of two possible values. This is done by introducing a single external operation of order two that interchanges the two possible values everywhere throughout the crystal. This operation can be applied to the crystal along with any of the standard point group or space group operations, and is denoted by adding a prime to the original operation. Thus, any rotation g followed by this external operation is denoted by g′. To start with, a few typical examples of this two-valued property are given, some of which are illustrated in Figure 1. In the section ‘‘Magnetic point groups’’ the notion of a magnetic point group is discussed, followed by a discussion on magnetic space groups in the section ‘‘Magnetic space groups’’. The section ‘‘Extinctions in neutron diffraction of anti-ferromagnetic crystals’’ describes one of the most (Figure presented) direct consequences of having magnetic symmetry in crystals which is the extinction of magnetic Bragg peaks in neutron diffraction patterns. A conclusion is given in the section ‘‘Generalizations of magnetic groups’’ by mentioning the generalization of magnetic groups to cases where the property associated with each crystal site....