To represent such a property, each lattice point is colored black or white,[1] 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.
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.
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.
[8] The concept was more fully explored by Shubnikov in terms of color symmetry.
There are two common conventions for giving names to the magnetic space groups.
They are Opechowski-Guccione (named after Wladyslaw Opechowski and Rosalia Guccione)[15] and Belov-Neronova-Smirnova.
A full list of the magnetic space groups (in both conventions) can be found both in the original papers, and in several places online.
[21] The magnetic point groups which are compatible with ferromagnetism are colored cyan, the magnetic point groups which are compatible with ferroelectricity are colored red, and the magnetic point groups which are compatible with both ferromagnetism and ferroelectricity are purple.
Similar symmetry arguments have been extended to other electromagnetic material properties such as magnetoelectricity or piezoelectricity.
[23] The following diagrams show the stereographic projection of most of the magnetic point groups onto a flat surface.
For black-white Bravais lattices, the number of black and white sites is always equal.
, transitions to the magnetic space group of the ordered phase,
This can be tracked numerically by evolution of the order parameter, which belongs to a single irreducible representation of
Differences in the magnetic phase transitions explain why Fe2O3, MnCO3, and CoCO3 are weakly ferromagnetic, whereas the structurally similar Cr2O3 and FeCO3 are purely antiferromagnetic.
A related scheme is the classification of Aizu species which consist of a prototypical non-ferroic magnetic point group, the letter "F" for ferroic, and a ferromagnetic or ferroelectric point group which is a subgroup of the prototypical group which can be reached by continuous motion of the atoms in the crystal structure.
More abstractly, the magnetic space groups are often thought of as representing time reversal symmetry.
In the most general form, magnetic space groups can represent symmetries of any two valued lattice point property, such as positive/negative electrical charge or the alignment of electric dipole moments.
The magnetic space groups place restrictions on the electronic band structure of materials.
Specifically, they place restrictions on the connectivity of the different electron bands, which in turn defines whether material has symmetry-protected topological order.
[35][36][37] Experimentally, the main source of information about magnetic space groups is neutron diffraction experiments.
The resulting experimental profile can be matched to theoretical structures by Rietveld refinement[38] or simulated annealing.
[39] Adding the two-valued symmetry is also a useful concept for frieze groups which are often used to classify artistic patterns.