Simple derivation of magnetic space groups (I)

Abstract

The magnetic translation lattices can be described by invariant wave vectors k. Advantages of the wave vector notation over the notations used by Belov et al. and Opechowski et al. are pointed out. In a one-dimensional real representation a space group element (α | τα) has either the character + 1 (symmetry element) or — 1 (antisymmetry element). Thus the square of any space group operation must have the character + 1 in a one-dimensional real representation. This simple « square criterion » is used to limit the admissible k-vectors and to derive the family of magnetic space groups, i. e. the set of all possible magnetic space groups, belonging to the same crystallographic space group. In the discussion some useful side results are obtained. Not only the real one-dimensional representations of point groups are connected to real one-dimensional representations of space groups, but a direct connection is shown to exist between one-dimensional complex representations of the point groups 3, 4 and 6 and one-dimensional real representations, belonging to [math] = P2c (Pc )-lattices with screw axes 31 , 32 , 42 , 62 and 64 .Finally we derive rules for finding the Belov symbol when the Opechowski-Guccione symbol of the magnetic space group is known and use this opportunity for correcting errors in the Opechowski-Guccione tables.