Representation analysis of magnetic structures

In the analysis of spin structures a `natural' point of view looks for the set of symmetry operations which leave the magnetic structure invariant and has led to the development of magnetic or Shubnikov groups. A second point of view presented here simply asks for the transformation properties of a magnetic structure under the classical symmetry operations of the 230 conventional space groups and allows one to assign irreducible representations of the actual space group to all known magnetic structures. The superiority of representation theory over symmetry invariance under Shubnikov groups is already demonstrated by the fact proven here that the only invariant magnetic structures describable by magnetic groups belong to real one-dimensional representations of the 230 space groups. Representation theory on the other hand is richer because the number of representations is infinite, i.e. it can deal not only with magnetic structures belonging to one-dimensional real representations, but also with those belonging to one-dimensional complex and even to two-dimensional and three-dimensional representations associated with any k vector in or on the first Brillouin zone. We generate from the transformation matrices of the spins a representation Γ of the space group which is reducible. We find the basis vectors of the irreducible representations contained in Γ. The basis vectors are linear combinations of the spins and describe the structure. The method is first applied to the k = 0 case where magnetic and chemical cells are identical and then extended to the case where magnetic and chemical cells are different (k ≠ 0) with special emphasis on k vectors lying on the surface of the first Brillouin zone in non-symmorphic space groups. As a specific example we consider several methods of finding the two-dimensional irreducible representations and its basis vectors associated with k = ½ b2 = 0½0 in space group Pbnm (D162h). We illustrate the physical context of representation theory by constructing an effective spin Hamiltonian H invariant under the crystallographic space group and under spin reversal. H is even in the spins and automatically invariant under the (isomorphous) magnetic group. We show by the example of CoO that the invariants in H, formed with the help of basis vectors, give information on the nature of spin coupling as for instance isotropic (Heisenberg–Néel) coupling, vectorial (Dzialoshinski–Moriya) and anisotropic symmetric couplings. Magnetic structures, cited in the text to show the implications of the representation theory of space groups are ErFeO3, ErCrO3, TbFeO3, TbCrO3, DyCrO3, YFeO3, V2CaO4, β-CoSO4, Er2O3, CoO and RMn2O5 (R = Bi, Y or rare earth). Representation theory of magnetic groups must be considered when the Hamiltonian contains terms which are odd in the spins. The case may occur when the magnetic energy is coupled with other forms of energy as for instance in the field of magneto-electricity. Here again representation theory correctly predicts the couplings between magnetic and electric polarizations as shown on LiCoPO4 and (previously) on FeGaO3.