Magnetic representations and magnetic space groups
As is well known nowadays, there are two streams for describing magnetic structures. (In fact, there are more, but widely used ones are the following two.) One way is to use the “magnetic representations”. Any axial vector configurations in periodic crystals should be classified by the symmetry of the underlying crystal lattice, so that they should be described as a certain linear combination of basis vectors of irreducible (axial) representations of the crystal’s space group. Indeed, in many real magnets, one (or a few) irreducible representations are enough for describing their magnetic structure, and hence, using “magnetic representation” sounds quite reasonable for practical magnetic structure description. One advantage of this method is that it can deal with incommensurate magnetic structures; simply, the irreducible representations for the (incommensurate) k-group can be used for the incommensurate structures, and hence there is no essential difference in formulation between commensurate and incommensurate structures.
The other way to go is to use the “magnetic space group”. The ordinary crystal space group contains symmetry operations such as translations, rotations, inversions etc… The “magnetic space group” further include time-reversal symmetry operation which just simply flips the magnetic moment direction (classical axial vectors). Inclusion of the time reversal symmetry enlarges the number of the groups from 230 to 1651. An advantage of using this approach is that we can see the symmetry operations (either globally or around a specific site) rather clearly. Quite often, the symmetry determines physical properties, either globally or locally, so this apparent visibility of (either global or local) magnetic symmetry in crystalline magnets is of great help for understanding and classifying their physical properties.
There are quite a few existing tools for exploring magnetic representations and magnetic space groups, which you can quickly find by google them. That said, I would like to list two of mine in the following.
One is to calculate magnetic representation basis vectors for any given k-vectors and for any given 230 space groups. The good point of this code is that by selecting basis vectors (and giving their coefficients), it will draw the magnetic structure! The drawing is done with Web3D technology, so you can just save html file to see it at hand later. The code behind it indeed calculates the irreducible representations from scratch (not using any database), so it takes a little long time; please be patient if you calculate high-symmetry crystal such as Fd-3m.
The other is to visualize a magnetic structure of a given magnetic space group. Sometimes, people like me with lack of imagination cannot imagine magnetic structures solely from the magnetic space group symbols. For such a case, this software may be of help for understanding them visually. The code behind this uses the “magnetic space group table” compiled by Harold T. Stokes and Branton J. Campbell. Without this table I am sure my life gets much complicated. (Please note that the code below uses “OG” setting only.)
Note: Please regard the above two codes as experimental. I can not ensure the results provided by the codes, not by any means. It is always a good practice to double check the results using other (much established) software. I am very happy, however, if you could let me know erroneous behavior you may encounter.