International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2013 
International Tables for Crystallography (2013). Vol. D, ch. 1.5, pp. 128132
Section 1.5.5. Weakly noncollinear magnetic structures^{a}P. L. Kapitza Institute for Physical Problems, Russian Academy of Sciences, Kosygin Street 2, 119334 Moscow, Russia,^{b}Labor für Neutronenstreuung, ETH Zurich, Switzerland, and ^{c}Paul Scherrer Institute, CH5232 Villigen PSI, Switzerland 
As was indicated above (see Tables 1.5.3.3 and 1.5.3.6), certain magnetic space groups allow the coexistence of two different types of magnetic ordering. Some magnetic structures can be described as a superposition of two antiferromagnetic structures with perpendicular antiferromagnetic vectors . Such structures may be called weakly noncollinear antiferromagnets. There can also be a superposition of an antiferromagnetic structure with a ferromagnetic one (with ). This phenomenon is called weak ferromagnetism. We shall demonstrate in this section why one of the magnetic vectors has a much smaller value than the other in such mixed structures.
The theory of weak ferromagnetism was developed by Dzyaloshinskii (1957a). He showed that the expansion of the thermodynamic potential may contain terms of the following type: (). Such terms are invariant with respect to the transformations of many crystallographic space groups (see Section 1.5.3.3). If there is an antiferromagnetic ordering in the material () and the thermodynamic potential of the material contains such a term, the minimum of the potential will be obtained only if as well. The term is a relativistic one. Therefore this effect must be small.
We shall consider as an example the origin of weak ferromagnetism in the twosublattice antiferromagnets MnCO_{3}, CoCO_{3} and NiCO_{3}, discussed in Section 1.5.3.1. The following analysis can be applied also to the foursublattice antiferromagnet Fe_{2}O_{3} (assuming , ). All these rhombohedral crystals belong to the crystallographic space group . The thermodynamic potential for these crystals was derived in Section 1.5.3.3. For the case of a twosublattice antiferromagnet, one has to add to the expression (1.5.3.26) the invariant (1.5.3.24): The coefficients of the isotropic terms (A and B) are of exchange origin. They are much larger than the coefficients of the relativistic terms (). Minimization of for a fixed value of and gives two solutions:
Weak ferromagnetism was first observed in the following trigonal crystals: the hightemperature modification of haematite, Fe_{2}O_{3} (Townsend Smith, 1916; Néel & Pauthenet, 1952), MnCO_{3} (BorovikRomanov & Orlova, 1956) and later also in CoCO_{3}, NiCO_{3} and FeBO_{3}. In accordance with theory, weak ferromagnetism does not occur in trigonal crystals with a positive anisotropy coefficient a. Such crystals become easyaxis antiferromagnets. Of this type are FeCO_{3} and the lowtemperature modification of Fe_{2}O_{3}. For foursublattice antiferromagnets, the sequence of the directions of the magnetic moments of the sublattices is also essential. For example, the structures of the types and (see Fig. 1.5.3.4 and Table 1.5.3.3) do not exhibit weak ferromagnetism.
The behaviour of weak ferromagnets in magnetic fields applied perpendicular and parallel to the trigonal axis is described by the following relations: where
An external magnetic field can freely rotate the ferromagnetic moment in the basal plane of the easyplane weak ferromagnets under consideration because their anisotropy in the basal plane is extremely small. During such a rotation, both vectors and move simultaneously as a rigid structure. On the other hand, it is impossible to deflect the vector out of the basal plane, as this is forbidden by symmetry. This is illustrated by the magnetization curves plotted in Fig. 1.5.5.2, which confirm the relations (1.5.5.3).
When the weakly ferromagnetic structure is rotated in the basal plane, a change of the magnetic space groups occurs in the following order: . Each of these symmetry transformations corresponds to a secondorder phase transition. Such transitions are allowed because is a subgroup of both groups and .
NiF_{2} was one of the first weak ferromagnets to be discovered (Matarrese & Stout, 1954). In the paramagnetic state, it is a tetragonal crystal. Its crystallographic space group is = . In the ordered state its magnetic point group is and the vectors and are directed along two twofold axes (one of which is primed) in the plane perpendicular to the former fourfold axis (see Fig. 1.5.5.3a). The invariant term responsible for the weak ferromagnetism in tetragonal fluorides has the formThe anisotropy of the crystals of NiF_{2} and the relation given above for the invariant lead to the same dependence on the magnetic field as for trigonal crystals. However, the anisotropy of the magnetic behaviour in the basal plane is much more complicated than for rhombohedral crystals (see Bazhan & Bazan, 1975). The anisotropy constant is positive for most other fluorides (MnF_{2}, FeF_{2} and CoF_{2}) and their magnetic structure is described by the magnetic point group = . They are easyaxis antiferromagnets without weak ferromagnetism.

Magnetic structures of fluorides of transition metals. (a) The weak ferromagnet NiF_{2}; (b) the easyaxis antiferromagnets MnF_{2}, FeF_{2} and CoF_{2}. 
The interaction described by the invariant in equation (1.5.5.1) is called Dzyaloshinskii–Moriya interaction. It corresponds to the interaction between the spins of neighbouring ions, which can be represented in the formwhere the vector has the components (). Terms of such type are allowed by symmetry for crystals that in the paramagnetic state belong to certain space groups of the trigonal, tetragonal and hexagonal systems. In some groups of the tetragonal system, weak ferromagnetism is governed by the term (as for NiF_{2}) and in the orthorhombic system by . In the monoclinic system, the latter sum contains four terms. The weak ferromagnetism in most groups of the hexagonal and cubic systems is governed by invariants of fourth and sixth order of . For all crystallographic space groups, Turov (1963) determined the form of the invariants of lowest order that allow, for collinear or weakly noncollinear antiferromagnetic structures, a phase transition into a state with weak ferromagnetism. The corresponding list of the numbers of the space groups that allow the transition into an antiferromagnetic state with weak ferromagnetism is given in Table 1.5.5.1. The form of the invariant responsible for weak ferromagnetism is also displayed in the table. Turov (1963) showed that weak ferromagnetism is forbidden for the triclinic system, for the six trigonal groups with point groups or , and the 12 cubic groups with point groups or .

The microscopic theory of the origin of weak ferromagnetism was given by Moriya (1960a,b, 1963). In this chapter, we have restricted our consideration to the phenomenological approach to this problem.
A large number of orthorhombic orthoferrites and orthochromites with the formula RMO_{3} (where R is a trivalent rareearth ion and M is Fe^{3+} or Cr^{3+}) have been investigated in many laboratories (cf. Wijn, 1994). Some of them exhibit weak ferromagnetism. The space group of these compounds is in the paramagnetic state. The primitive cell is the same in the paramagnetic and magnetically ordered states. It contains four magnetic transitionmetal ions (see Fig. 1.5.5.4). They determine to a large extent the properties of orthoferrites (outside the region of very low temperatures). For a foursublattice antiferromagnet, there are four possible linear combinations of the sublattice vectors, which define three types of antiferromagnetic vectors and one ferromagnetic vector [see relations (1.5.3.2) and Table 1.5.3.1]. The exchange interaction in these compounds governs magnetic structures, which to a first approximation are described by the following antiferromagnetic vector (which is usually denoted by the symbol ): In the case of orthoferrites, the other two antiferromagnetic vectors and [see relations (1.5.3.2)] are named and , respectively.

Magnetic structures of orthoferrites and orthochromites RMO_{3}. (Only the transitionmetal ions are shown; the setting Pbnm is used.) (a) weakly ferromagnetic state; (b) weakly ferromagnetic state. 
The magnetic structure of the compounds under consideration is usually called the or state. Depending on the signs and the values of the anisotropy constants, there are three possible magnetic states: The magnetic structures (I) and (III) are weak ferromagnets. They are displayed schematically in Fig. 1.5.5.4. Both are described by the same magnetic point group yet in different orientations: (i.e. ) for structure (I) and (i.e. ) for structure (III). The magnetic point group of structure (II) is .
Weak ferromagnetism is observed in boracites with chemical formula M_{3}B_{7}O_{13}X (where M = Co, Ni and X = Br, Cl, I). These compounds are unique, being simultaneously antiferromagnets, weak ferromagnets and ferroelectrics. Section 1.5.8.3 is devoted to these ferromagnetoelectrics.
Concerning the magnetic groups that allow weak ferromagnetism, it should be noted that, as for any ferromagnetism, weak ferromagnetism is allowed only in those space groups that have a trivial magnetic Bravais lattice. There must be at least two magnetic ions in the primitive cell to get antiferromagnetic order. Among the 31 magnetic point groups that admit ferromagnetism (see Table 1.5.2.4), weak ferromagnetism is forbidden in the magnetic groups belonging to the tetragonal, trigonal and hexagonal systems. Twelve magnetic point groups that allow weak ferromagnetism remain. These groups are listed in Table 1.5.5.2.

A material that becomes a weak ferromagnet below the Néel temperature differs from a collinear antiferromagnet in its behaviour above . A magnetic field applied to such a material above gives rise to an ordered antiferromagnetic state with vector directed perpendicular and magnetization parallel to the field. Thus, as in usual ferromagnets, the magnetic symmetry of a weak ferromagnet in a magnetic field is the same above and below . As a result, the magnetic susceptibility has a maximum at [like the relations (1.5.3.34) and (1.5.3.35)]. This is true only if the magnetic field is aligned along the easy axis for weak ferromagnetism. Fig. 1.5.5.5 shows the anomalous anisotropy of the temperature dependence of the magnetic susceptibility in the neighbourhood of for weak ferromagnets.
Similar anomalies in the neighbourhood of are observed in materials with a symmetry allowing a transition into a weakly ferromagnetic state for which the sign of the anisotropy constant causes their transition into purely antiferromagnetic states.
A thermodynamic potential of the form (1.5.5.1) may give rise not only to the weak ferromagnetism considered above but also to the reverse phenomenon. If the coefficient B (instead of A) changes its sign and , the material will undergo a transition into a slightly canted ferromagnetic structure, in which and the expression for isExperimental detection of such structures is a difficult problem and to date noone has observed such a phenomenon.
The thermodynamic potential of a foursublattice antiferromagnet may contain the mixed invariant [see (1.5.3.24)]Such a term gives rise to a structure in which all four vectors of sublattice magnetization form a star, as shown in Fig. 1.5.5.6 (see also Fig. 1.5.1.3b). The angle between the vectors and (or and ) is equal to if the main antiferromagnetic structure is defined by the vector [see the caption for Fig. 1.5.1.3(b)]. Such a structure may occur in Cr_{2}O_{3}. In most orthoferrites discussed above, such noncollinear structures are observed for all three cases: purely antiferromagnetic () and weakly ferromagnetic ( and ). The structure is not coplanar. Apart from the main antiferromagnetic vector aligned along the y axis, it possesses two other antiferromagnetic vectors: (aligned along the x axis) and (aligned along the z axis). The weakly ferromagnetic structure has an admixture of the antiferromagnetic structure.
The helical (or spiral) structure described in Section 1.5.1.2.3 and depicted in Fig. 1.5.1.4 is also a weakly noncollinear antiferromagnetic structure. As mentioned above, this structure consists of atomic layers in which all the magnetic moments are parallel to each other and parallel to the layer. The magnetizations of neighbouring layers are antiparallel to a first approximation; but, more specifically, there is a small deviation from a strictly antiparallel alignment. The layers are perpendicular to a vector , which is parallel to the axis of the helix. The two mutually perpendicular antiferromagnetic vectors are both perpendicular to . These vectors define the helical structure by the following relation for the density of the magnetization in the layer with the coordinate (Dzyaloshinskii, 1964; Andreev & Marchenko, 1980): Most helical structures are incommensurate, which means that the representation defined by the vector does not satisfy the Lifshitz condition (see Section 1.5.3.3).
References
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