In solid state physics, the magnetic space groups, or Shubnikov groups, are the symmetry groups which classify the symmetries of a crystal both in space, and in a two-valued property such as electron spin. To represent such a property, each lattice point is colored black or white, and in addition to the usual three-dimensional symmetry operations, there is a so-called "antisymmetry" operation which turns all black lattice points white and all white lattice points black. Thus, the magnetic space groups serve as an extension to the crystallographic space groups which describe spatial symmetry alone. The application of magnetic space groups to crystal structures is motivated by Curie's Principle. Compatibility with a material's symmetries, as described by the magnetic space group, is a necessary condition for a variety of material properties, including ferromagnetism, ferroelectricity, topological insulation.
History A major step was the work of Heinrich Heesch, who first rigorously established the concept of antisymmetry as part of a series of papers in 1929 and 1930. Applying this antisymmetry operation to the 32 crystallographic point groups gives a total of 122 magnetic point groups. However, although Heesch correctly laid out each of the magnetic point groups, his work remained obscure, and the point groups were later re-derived by Tavger and Zaitsev. The concept was more fully explored by Shubnikov in terms of color symmetry. When applied to space groups, the number increases from the usual 230 three dimensional space groups to 1651 magnetic space groups, as found in the 1953 thesis of Alexandr Zamorzaev. While the magnetic space groups were originally found using geometry, it was later shown the same magnetic space groups can be found using generating sets.
Description
Magnetic space groups The magnetic space groups can be placed into three categories. First, the 230 colorless groups contain only spatial symmetry, and correspond to the crystallographic space groups. Then there are 230 grey groups, which are invariant under antisymmetry. Finally are the 1191 black-white groups, which contain the more complex symmetries. There are two common conventions for giving names to the magnetic space groups. They are Opechowski-Guccione (named after Wladyslaw Opechowski and Rosalia Guccione) and Belov-Neronova-Smirnova (named after Nikolay Vasilyevich Belov, Nina Nikolaevna Neronova and Tamara Serafimovna Smirnova, then Kuntsevich). For colorless and grey groups, the conventions use the same names, but they treat the black-white groups differently. A full list of the magnetic space groups (in both conventions) can be found both in the original papers, and in several places online.
The types can be distinguished by their different construction. Type I magnetic space groups, M I {\displaystyle {\mathcal {M}}_{I}} are identical to the ordinary space groups, G {\displaystyle G} .
M I = G {\displaystyle {\mathcal {M}}_{I}=G}
Type II magnetic space groups, M I I {\displaystyle {\mathcal {M}}_{II}} , are made up of all the symmetry operations of the crystallographic space group, G {\displaystyle G} , plus the product of those operations with time reversal operation, T {\displaystyle {\mathcal {T}}} . Equivalently, this can be seen as the direct product of an ordinary space group with the point group 1 ′ {\displaystyle 1'} .
M I I = G + T G {\displaystyle {\mathcal {M}}_{II}=G+{\mathcal {T}}G}
M I I = G × 1 ′ {\displaystyle {\mathcal {M}}_{II}=G\times 1'}
Type III magnetic space groups, M I I I {\displaystyle {\mathcal {M}}_{III}} , are constructed using a group H {\displaystyle H} , which is a subgroup of G {\displaystyle G} with index 2.
M I I I = H + T ( G − H ) {\displaystyle {\mathcal {M}}_{III}=H+{\mathcal {T}}(G-H)}
Type IV magnetic space groups, M I V {\displaystyle {\mathcal {M}}_{IV}} , are constructed with the use of a pure translation, { E | t 0 } {\displaystyle \{E|t_{0}\}} , which is Seitz notation for null rotation and a translation, t 0 {\displaystyle t_{0}} . Here the t 0 {\displaystyle t_{0}} is a vector (usually given in fractional coordinates) pointing from a black colored point to a white colored point, or vice versa.
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