Chapter 1.5. Magnetic properties¹

A crystal placed in a magnetic field is magnetized. The magnetized state is characterized by two vectors, the magnetization (the magnetic moment per unit volume) and the magnetic induction . The Gaussian system of units is used in this chapter (see Table 1.5.10.1 at the end of the chapter for a list of conversions from Gaussian to SI units). The magnetic induction is given by This equation shows that the dimensions of B, H and M are the same in the Gaussian system. The unit for B, the gauss (G), and for H, the oersted (Oe), also coincide in magnitude, whereas the unit for M, usually called emu cm⁻³, is 4π times larger. These units are related to the corresponding SI units as follows: 1 G = 10⁻⁴ tesla (T), 1 Oe = 10³/(4π) A m⁻¹, 1 emu cm⁻³ = 10³ A m⁻¹.

In disordered magnetics, the vectors B and M are linear in the magnetic field. Using a Cartesian coordinate system, this can be expressed as where is the dimensionless magnetic susceptibility per unit volume and is the magnetic permeability. The susceptibility is frequently referred to 1 g or to one mole of substance. The mass susceptibility is written as , the molar susceptibility as .

All three vectors H, M and B are axial vectors (see Section 1.1.4.5.3 ), the symmetry of which is . Accordingly, the components of these vectors perpendicular to a mirror plane do not change sign on being reflected by this plane, whereas the components parallel to the plane do change sign. Consequently, these three vectors are invariant with respect to inversion. The quantities and are components of second-rank polar tensors. In principal axes, the tensors become diagonal and both the magnetic susceptibility and permeability of a crystal are characterized by the three values of the principal susceptibilities and principal permeabilities, respectively.

All disordered magnetics are divided into two types: diamagnets (χ < 0) and paramagnets (χ > 0).

Diamagnetism is a universal property of all materials. It is associated with the tendency of all the electrons to screen the applied external field according to the Lenz law. For materials in which the electron orbits are spherically symmetric, the relation for the diamagnetic susceptibility was calculated by Langevin. For monoatomic substances he obtainedwhere N is the number of atoms per unit volume, Z is the number of electrons per atom, e and m are the charge and the mass of the electron, respectively, and are the mean squares of the radii of the electron orbits. In polyatomic substances, the summation must be done over all types of atoms. In most chemical compounds, the orbits are not spherical and the calculation of the diamagnetic susceptibility becomes more complicated. In metals, the conduction electrons contribute significantly to the diamagnetic susceptibility. The diamagnetic susceptibility of most substances is very small () and isotropic. Rare exceptions are bismuth and some organic compounds, in which the diamagnetism is strongly anisotropic.

Most paramagnetic materials contain ions (or free atoms) with a partly filled inner electronic shell. Examples are the transition metals and the rare-earth and actinide elements. Atoms, molecules and point defects possessing an odd number of electrons are also paramagnetic. Ions with a partly filled inner electronic shell possess orbital L and spin S angular momenta, which determine the total angular momentum J if the spin–orbit interaction is strong compared with the crystal field.

The magnetic susceptibility of paramagnets follows the Curie–Weiss law in low magnetic fields (): where N is the number of magnetic ions (or atoms) per cm³, is the Bohr magneton, p is the effective number of Bohr magnetons, is the Boltzmann factor and is the Weiss constant. The Weiss constant is related to the interaction between the magnetic moments (mostly exchange interaction) and to the effect of the splitting of electron levels of the paramagnetic ion in the crystalline electric field. Many paramagnets that obey the Curie–Weiss law transform into ordered magnetics at a temperature , which is of the order of . The sign of depends on the sign of the exchange constant J [see relation (1.5.1.7)]. For the substances that at low temperatures become ferromagnets, we have , for antiferromagnets , and for ferrimagnets the temperature dependence of is more complicated (see Fig. 1.5.1.1). For those paramagnets that do not go over into an ordered state, is close to zero and equation (1.5.1.4) changes to the Curie law.

The value of the effective number of Bohr magnetons p depends strongly on the type of the magnetic ions and their environment. For most rare-earth compounds at room temperature, the number p has the same value as for free ions: where g is the Landé g-factor or the spectroscopic splitting factor (). In this case, the paramagnetic susceptibility is practically isotropic. Some anisotropy can arise from the anisotropy of the Weiss constant .

The behaviour of the transition-metal ions is very different. In contrast to the rare-earth ions, the electrons of the partly filled shell in transition metals interact strongly with the electric field of the crystal. As a result, their energy levels are split and the orbital moments can be `quenched'. This means that relation (1.5.1.5) transforms to Here the value of the effective spin S represents the degeneration of the lowest electronic energy level produced by the splitting in the crystalline field; differs from the usual Landé g-factor. The values of its components lie between 0 and 10–20. The tensor becomes diagonal in the principal axes. According to relation (1.5.1.6), the magnetic susceptibility also becomes a tensor. The anisotropy of can be studied using electron paramagnetic resonance (EPR) techniques.

The Curie–Weiss law describes the behaviour of those paramagnets in which the magnetization results from the competition of two forces. One is connected with the reduction of the magnetic energy by orientation of the magnetic moments of ions in the applied magnetic field; the other arises from thermal fluctuations, which resist the tendency of the field to orient these moments. At low temperatures and in strong magnetic fields, the linear dependence of the magnetization versus magnetic field breaks down and the magnetization can be saturated in a sufficiently strong magnetic field. Most of the paramagnetic substances that obey the Curie–Weiss law ultimately transform to an ordered magnetic as the temperature is decreased.

The conduction electrons in metals possess paramagnetism in addition to diamagnetism. The paramagnetic susceptibility of the conduction electrons is small (of the same order of magnitude as the diamagnetic susceptibility) and does not depend on temperature. This is due to the fact that the conduction electrons are governed by the laws of Fermi–Dirac statistics.

Magnetic point groups may contain rotations, reflections, the element R and their combinations. A set of such elements that satisfies the group properties is called a magnetic point group. It is obvious that there are 32 trivial magnetic point groups; these are the ordinary crystallographic point groups supplemented by the element R. Each of these point groups contains all the elements of the ordinary point group and also all the elements of this group multiplied by R. This type of magnetic point group can be represented byThese groups are sometimes called `grey' magnetic point groups. As pointed out above, all dia- and paramagnets belong to this type of point group. To this type belong also antiferromagnets with a magnetic space group that contains translations multiplied by R (space groups of type III^b).

The second type of magnetic point group, which is also trivial in some sense, contains all the 32 crystallographic point groups without the element R in any form. For this type . Thirteen of these point groups allow ferromagnetic spontaneous magnetization (ferromagnetism, ferrimagnetism, weak ferromagnetism). They are listed in Table 1.5.2.4. The remaining 19 point groups describe antiferromagnets. The groups are often called `white' magnetic point groups.

The third type of magnetic point group , `black and white' groups (which are the only nontrivial ones), contains those point groups in which R enters only in combination with rotations or reflections. There are 58 point groups of this type. Eighteen of them describe different types of ferromagnetism (see Table 1.5.2.4) and the others represent antiferromagnets.

Replacing R by the identity element E in the magnetic point groups of the third type does not change the number of elements in the point group. Thus each group of the third type is isomorphic to a group of the second type.

The method of derivation of the nontrivial magnetic groups given below was proposed by Indenbom (1959). Let denote the set of those elements of the group which enter into the associated magnetic group not multiplied by R. The set contains the identity element E, for each element H also its inverse , and for each pair , also its products and . Thus the set forms a group. It is a subgroup of the crystallographic group . Let denote an element of . All these elements enter in the form of products because and . Multiplying the elements of by a fixed element corresponds to a permutation of the elements of . This permutation maps each element of the subgroup on an element of that does not belong to and vice versa. It follows that one half of the elements of are elements of multiplied by R and the other half belong to . The relation for the magnetic point groups of the third type may therefore be written as is therefore a subgroup of index 2 of . The subgroups of index 2 of can easily be found using the tables of irreducible representations of the point groups. Every real non-unit one-dimensional representation of contains equal numbers of characters and . In the corresponding magnetic point group , the elements of with character are multiplied by R and those with character remain unchanged. The latter form the subgroup . This rule can be stated as a theorem: every real non-unit one-dimensional representation of a point group of symmetry produces an isomorphic mapping of this group upon a magnetic group (Indenbom, 1959). This concept will be developed in Section 1.5.3.

Using the Schoenflies symbols and the method described above, the point groups of magnetic symmetry (magnetic point groups) can be denoted by , where is the symbol of the original crystallographic point group and is the symbol of that subgroup the elements of which are not multiplied by R. This notation is often used in the physics literature. In the crystallographic literature, the magnetic groups are defined by Hermann–Mauguin or Shubnikov symbols. In this type of designation, the symbols of elements multiplied by R are primed or underlined. The primed symbols are used in most of the recent publications. The Hermann–Mauguin and Shubnikov definitions differ slightly, as in the case of crystallographic groups. In Table 1.5.2.1, different symbols of magnetic point groups (trivial and nontrivial ones) are compared. This is done for the family that belongs to the crystallographic point group . The symbols of the symmetry elements of these four magnetic point groups are compared in Table 1.5.2.2.

Table 1.5.2.3 gives a list of the 90 magnetic point groups belonging to types 2 and 3. The Schoenflies, Shubnikov and Hermann–Mauguin symbols of the point groups are given in the table. The entries in the Hermann–Mauguin symbol refer to symmetry directions, as explained in Section 2.2.4 of International Tables for Crystallography, Vol. A (2005). The elements of symmetry of each point group are displayed using the Hermann–Mauguin symbols. The symbol denotes N 180° rotations with axes perpendicular to the principal symmetry axis; denotes N mirror planes with normals perpendicular to the principal symmetry axis. Similar definitions hold for the primed symbols and . The point groups are arranged in families. The part of the Schoenflies symbol before the bracket is the same for each member of a family. Each family begins with a trivial magnetic point group. It contains the same elements as the corresponding crystallographic point group; its Schoenflies symbol contains no brackets. For each nontrivial point group, the list of the elements of symmetry begins with the non-primed elements, which belong to a subgroup of the head of the family . The number of the primed elements is equal to the number of non-primed ones and the total number of the elements is the same for all point groups of one family.

The overall number of the magnetic point groups of all three types is 122. There are two general statements concerning the magnetic point groups. The element does not appear in any of the magnetic point groups of type 3. Only trivial magnetic point groups (of both first and second type) belong to the families containing the point groups , and .

Only 31 magnetic point groups allow ferromagnetism. The different types of ferromagnetism (one-sublattice ferromagnet, ferrimagnet, weak ferromagnet, any magnetic order with nonzero magnetization) cannot be distinguished by their magnetic symmetry. Ferromagnetism is not admitted in any point group of type 1. For the magnetic point groups of the second type, ferromagnetism is not allowed if the point group contains more than one symmetry axis, more than one mirror plane or a mirror plane that is parallel to the axis. The same restrictions are valid for the point groups of type 3 (if the corresponding elements are not multiplied by R). If the point group contains , ferromagnetic order is also forbidden. There are the following rules for the orientation of the axial vector of ferromagnetic magnetization : . Table 1.5.2.4 lists those magnetic point groups that admit ferromagnetic order (Tavger, 1958). The allowed direction of the magnetization vector is given for every point group. Ferromagnetic order is allowed in 13 point groups of the second type and 18 point groups of the third type.

All 31 point groups of magnetic symmetry allowing ferromagnetism are subgroups of the infinite noncrystallographic group The transition from a paramagnetic to a ferromagnetic state is always accompanied by a change of the magnetic symmetry.

If the point group of symmetry describes the macroscopic properties of a crystal, its microscopic structure is determined by the space group, which contains the group of translations as a subgroup. The elements of are defined by the following relation: where , , are basic primitive translation vectors and , , are arbitrary integers. The set of points obtained by applying all the translations of the group to any point defines a lattice. All sites of the crystallographic lattice are equivalent.

The structure of the ordered magnetics is described by the magnetic lattices and corresponding magnetic translation groups . In the magnetic translation groups , some of the elements may be multiplied by R (we shall call them primed translations). The magnetic lattices then have two types of sites, which are not equivalent. One set is obtained by non-primed translations and the other set by the primed ones. The magnetic translation group is isometric to the crystallographic one that is obtained by replacing R by E in .

There are trivial magnetic translation groups, in which none of the translation elements is multiplied by R. The magnetic lattices of these groups coincide with crystallographic lattices.

Nontrivial magnetic translation groups can be constructed in analogy to relation (1.5.2.2). Zamorzaev (1957) showed that every translation group has seven subgroups of index 2. If the basic primitive translations of the group are , , , then the basic primitive translations of the seven subgroups can be chosen as follows (see also Opechowski & Guccione, 1965)

As an example, let us consider the case (1.5.2.5). In this case, the subgroup consists of the following translations: Therefore the elements of [which corresponds to in relation (1.5.2.2)] must have the following form: The corresponding magnetic translation group consists of the elements (1.5.2.12) multiplied by R and the elements (1.5.2.11).

The crystallographic lattices are classified into Bravais types or Bravais lattices. The magnetic lattices are classified into Bravais types of magnetic lattices. It turns out that there are 22 nontrivial magnetic Bravais types. Together with the trivial ones, there are 36 magnetic Bravais lattices.

Two types of smallest translation-invariant cells are in common use for the description of magnetically ordered structures: the crystallographic cell obtained if the magnetic order is neglected and the magnetic cell, which takes the magnetic order into account. The list of the basic translations of all the magnetic Bravais lattices was given by Zamorzaev (1957). The diagrams of the magnetic unit cells were obtained by Belov et al. (1957).

In Figs. 1.5.2.1–1.5.2.7, the diagrams of the magnetic unit cells of all 36 Bravais types are sketched in such a way that it is clear to which family the given cell belongs. All the cells of one family are displayed in one row. Such a row begins with the cell of the trivial magnetic lattice. All nontrivial cells of a family change into the trivial one of this family if R is replaced by E (to draw these diagrams we used those published by Opechowski & Guccione, 1965). Open and full circles are used to show the primed and unprimed translations. A line connecting two circles of the same type is an unprimed translation; a line connecting two circles of different types is a primed translation. The arrows in the trivial magnetic cell represent the primitive (primed or unprimed) translations for all the magnetic lattices of the family. The arrows in the nontrivial cells are primitive translations of the magnetic unit cell. The magnetic unit cell of a nontrivial magnetic lattice is generated by unprimed translations only. Its volume is twice the volume of the smallest cell generated by all (primed and unprimed) translations. The reason for this is that one of the primitive translations of the magnetic cell is twice a primitive primed translation. The crystallographic cell of many simple collinear or weakly non-collinear structures coincides with the smallest cell generated by the primed and unprimed translations. However, there are also magnetic structures with more complicated transformations from the crystallographic to the magnetic unit cell. The second line after each part of Figs. 1.5.2.1–1.5.2.7 gives, between braces, an extended vector basis of the magnetic translation group (Shubnikov & Koptsik, 1972). The first line gives two symbols for each Bravais type: the symbol to the right was introduced by Opechowski & Guccione (1965). The symbol to the left starts with a lower-case letter giving the crystal system followed by a capital letter giving the centring type of the cell defined by the unprimed translations (P: primitive; C, A, B: C-, A-, B-centred; I: body-centred; F: all-face-centred). The subscript, which appears for the nontrivial Bravais types, indicates the translations that are multiplied by time inversion R.

Ferromagnetism is allowed only in trivial magnetic Bravais lattices. All nontrivial magnetic lattices represent antiferromagnetic order. There are only two magnetic sublattices in the simplest antiferromagnetic structures; one sublattice consists of the magnetic ions located in the black sites and the other of the ions located in the white sites. All the magnetic moments of one sublattice are oriented in one direction and those of the other sublattice in the opposite direction. However, antiferromagnetism is allowed also in trivial lattices if the (trivial) magnetic cell contains more than one magnetic ion. The magnetic point group must be nontrivial in this case. The situation is more complicated in case of strongly non-collinear structures. In such structures (triangle, 90° etc.), the magnetic lattice can differ from the crystallographic one despite the fact that none of the translations is multiplied by R. The magnetic elementary cell will possess three or four magnetic ions although the crystallographic cell possesses only one. An example of such a situation is shown in Fig. 1.5.1.3(c). More complicated structures in which the magnetic lattice is incommensurate with the crystallographic one also exist. We shall not discuss the problems of such systems in this chapter.

There are 1651 magnetic space groups , which can be divided into three types. Type I, , consists of the 230 crystallographic space groups to which R is added. Crystals belonging to these trivial magnetic space groups show no magnetic order; they are para- or diamagnetic.

Type II, , consists of the same 230 crystallographic groups which do not include R in any form. In the ordered magnetics, which belong to the magnetic space groups of this type, the magnetic unit cell coincides with the classical one. Forty-four groups of type II describe different ferromagnetic crystals; the remaining antiferromagnets.

The nontrivial magnetic space groups belong to type III, . This consists of 1191 groups, in which R enters only in combination with rotations, reflections or translations. These groups have the structure described by relation (1.5.2.2). The magnetic space groups of this type are divided into two subtypes.

Subtype III^a contains those magnetic space groups in which R is not combined with translations. In these groups, the magnetic translation group is trivial. To these space groups correspond magnetic point groups of type . There are 674 magnetic space groups of subtype III^a; 231 of them admit ferromagnetism, the remaining 443 describe antiferromagnets.

In the magnetic space groups of the subtype III^b, R is combined with translations and the corresponding point groups are of type . They have a nontrivial magnetic Bravais lattice. There are 517 magnetic space groups of this subtype; they describe antiferromagnets.

In summary, the 230 magnetic space groups that describe dia- and paramagnets are of type I, the 275 that admit spontaneous magnetization are of types II and III^a; the remaining 1146 magnetic space groups (types II, III^a and III^b) describe antiferromagnets.

The classification of magnetic structures on the basis of the magnetic (point and space) groups is an exact classification. However, it neglects the fundamental role of the exchange energy, which is responsible for the magnetic order (see Sections 1.5.1.2 and 1.5.3.2). To describe the symmetry of the magnetically ordered crystals only by the magnetic space groups means the loss of significant information concerning those properties of these materials that are connected with the higher symmetry of the exchange forces. Andreev & Marchenko (1976, 1980) have introduced the concept of exchange symmetry.

The exchange forces do not depend on the directions of the spins (magnetic moments) of the ions relative to the crystallographic axes and planes. They depend only on the relative directions of the spins. Thus the exchange group contains an infinite number of rotations U of spin space, i.e. rotations of all the spins (magnetic moments) through the same angle about the same axis. The components of the magnetic moment density transform like scalars under all rotations of spin space. The exchange symmetry group contains those combinations of the space transformation elements, the rotations U of spin space and the element R with respect to which the values are invariant. Setting all the elements U and R equal to the identity transformation, we obtain one of the ordinary crystallographic space groups . This space group defines the symmetry of the charge density and of all the magnetic scalars in the crystal. However, the vectors may not be invariant with respect to .

The concept of exchange symmetry makes it possible to classify all the magnetic structures (including the incommensurate ones) with the help of not more than three orthogonal magnetic vectors. We shall discuss this in more detail in Section 1.5.3.3.

More information about magnetic symmetry can be found in Birss (1964), Cracknell (1975), Joshua (1991), Koptsik (1966), Landau & Lifshitz (1957), Opechowski & Guccione (1965), and in Sirotin & Shaskol'skaya (1979).

Following Dzyaloshinskii (1957a), we consider crystals belonging to the crystallographic space group . To this group belong -Fe₂O₃ and the carbonates of Mn²⁺, Co²⁺ and Ni²⁺. Weak ferromagnetism was first observed in these materials. Cr₂O₃, in which the magnetoelectric effect was discovered, also belongs to this group. The magnetic ordering in these materials occurs without change of the unit cell.

The representatives of the cosets form the group . Its symmetry operations are shown in Fig. 1.5.3.1. Directed along the z axis is the threefold axis and the sixfold roto-inversion axis . Three twofold axes run through the points at right angles to the z axis. One of these axes is directed along the x axis. Arranged normal to each of the axes are three glide planes . The y axis is directed along one of these planes. The centre of inversion is located at the point , lying on the z axis halfway between two points . The sign ~ means that the corresponding operation is accompanied by a translation along the z axis through half the period of the crystal ( means that the inversion centre is shifted from the point to the point ). In Fig. 1.5.3.1, the elementary period of translation along the z axis is marked by . Thus the crystallographic group has the following elements:

In two types of crystals, considered below, the magnetic ions are arranged on the z axis. If we place the magnetic ion at point 1 located between points and (see Fig. 1.5.3.2), then using symmetry operations (1.5.3.1) we obtain three additional positions for other magnetic ions (points 2, 3, 4). Thus, the elementary cell will contain four magnetic ions. This is the structure of oxides of trivalent ions of iron and chromium (Fe₂O₃, Cr₂O₃). The structure of these oxides is shown in Fig. 1.5.3.2. If the positions of the magnetic ions coincide with the positions of the inversion centre , we obtain the structure of the carbonates of the transition metals (MnCO₃, CoCO₃, NiCO₃, FeCO₃), which is shown in Fig. 1.5.3.3.

Evidently, the formation of a magnetic structure in the crystal does not result in the appearance of new elements of symmetry. The magnetic groups of magnetically ordered crystals may lack some elements contained in the crystallographic group and some of the remaining elements may happen to be multiplied by R (primed). Let us find the groups of symmetry that correspond to all possible collinear magnetic structures in rhombohedral crystals with four magnetic ions in the elementary cell. We shall assume that the magnetic moments are located at the points of the ion positions 1–4; they will be marked . The symmetry transformations cannot change the length of the vectors of the magnetic moments but they can change the direction of these vectors and interchange the positions of the sites 1 2, 3 4 and 1 3, 2 4. This interchange of the vectors , , , means that these vectors form a basis of a reducible representation of the group . The following linear combinations of form irreducible representations² of : Vectors characterize the antiferromagnetic states and are called antiferromagnetic vectors. The ferromagnetic vector gives the total magnetic moment of the elementary cell. These vectors describe the four possible collinear magnetic structures. Three are antiferromagnetic structures: (, ), (, ), (, ) and one is a ferromagnetic structure, (). All these types are presented schematically in Fig. 1.5.3.4.

In the description of the structures of orthoferrites, other symbols were introduced to define the linear combinations of and to denote the antiferromagnetic structures under consideration (see Bertaut, 1963). The two types of symbols are compared in Table 1.5.3.1.

It should be borne in mind that in each of these types of magnetic ordering the respective vectors and may be directed along any direction. There are 12 types of such structures in which or are directed along one of the axes or planes of symmetry. To find out to which group of magnetic symmetry each of these structures belongs, one needs to investigate how each element of the crystallographic symmetry transforms the Cartesian components of the four vectors. This is shown in Table 1.5.3.2 for the group . If the component keeps its direction, it is marked by the sign; the − sign corresponds to reversal of the component direction. In some cases, the transformation results in a change of the direction of the components or through an angle other than 0 or . This is marked by 0. With the help of Table 1.5.3.2, we can easily describe all the elements of symmetry of the magnetic group that corresponds to each structure ( or ) with the aid of the following rule. All the elements that yield the sign are included in the magnetic group as they stand, while the elements yielding the − sign must be multiplied by R; the elements which are marked by the sign 0 are not included in the magnetic group.³ With the aid of this rule, Table 1.5.3.3 of the elements of the magnetic groups for the structures under consideration was compiled. In Table 1.5.3.4, the symbols of the magnetic point groups of all the 12 magnetic structures considered are listed. The crystals with two ions in the elementary cell have only two sublattices and their antiferromagnetic structures belong to the same groups as the structures .

One can see from Tables 1.5.3.3 and 1.5.3.4 that, in accordance with general theory, the magnetic point groups of the crystals under consideration are subgroups of the trivial magnetic point group to which they belong in the paramagnetic state. In the example considered, the translation group does not change in going from the paramagnetic to the ordered state. Thus the same statement made for the point groups is also true for the space groups. Putting gives a subgroup of the crystallographic group of the crystal. For the magnetic structures with the ferromagnetic or antiferromagnetic vector directed along the z axis, it turns out that the magnetic group is isomorphic to the crystallographic group. This rule is obeyed by all (optically) uniaxial crystals if the transition occurs without change of the elementary cell. (Optically uniaxial are the non-cubic crystals with a point group possessing a threefold, fourfold or sixfold axis.)

Tables 1.5.3.3 and 1.5.3.4 show that different types of collinear structures may belong to the same point group (and also to the same space group). For the antiferromagnetic structure and the ferromagnetic the group is , and for the structures and it is . Thus the symmetry allows a phase to be simultaneously ferromagnetic and antiferromagnetic. That is not ferrimagnetic order because all the ions in the four sublattices are identical and their numbers are equal. The ferromagnetic vector and the antiferromagnetic one are perpendicular and . This phenomenon is called weak ferromagnetism and will be discussed in detail in Section 1.5.5.1. Like weak ferromagnetism, the symmetry also allows the coexistence of two orthogonal antiferromagnetic structures and . This gives rise to weakly non-collinear antiferromagnetic structures.

The strongly non-collinear structures are described by another set of basis vectors for the irreducible representations of the group . If the magnetic ions in the crystal form triangular planes one gets instead of (1.5.3.2) the relations for the basis vectors:

It is pertinent to compare the different kinds of interactions that are responsible for magnetic ordering. In general, all these interactions are much smaller than the electrostatic interactions between the atoms that determine the chemical bonds in the material. Therefore, if a crystal undergoes a transition into a magnetically ordered state, the deformations of the crystal that give rise to the change of its crystallographic symmetry are comparatively small. It means that most of the non-magnetic properties do not change drastically. As an example, the anisotropic deformation of the crystal that accompanies the transition into the ordered state (see Section 1.5.9.1) is mostly not larger than 10⁻⁴.

The formation of the ordered magnetic structures is due mainly to the exchange interaction between the spins (and corresponding magnetic moments of the atoms or ions). The expression for the exchange energy can contain the following terms [see formula (1.5.1.7)]: The exchange interaction decreases rapidly as the distance between the atoms rises. Thus, it is usually sufficient to consider the interaction only between nearest neighbours. The exchange interaction depends only on the relative alignment of the spin moments and does not depend on their alignment relative to the crystal lattice. Therefore, being responsible for the magnetic ordering in the crystal, it cannot define the direction of the spontaneous magnetization in ferromagnets or of the antiferromagnetic vector. This direction is determined by the spin–orbit and magnetic spin–spin interactions, which are often called relativistic interactions as they are small, of the order of , where v is the velocity of atomic electrons and c is the speed of light. The relativistic interactions are responsible for the magnetic anisotropy energy, which depends on the direction of the magnetic moments of the ions with regard to the crystal lattice. The value of the exchange energy can be represented by the effective exchange field H_e. For an ordered magnetic with a transition temperature of 100 K, H_e ≃ 1000 kOe. Thus the external magnetic field hardly changes the value of the magnetization or of the antiferromagnetic vector ; they are conserved quantities to a good approximation. The effective anisotropy field in cubic crystals is very small: 1–10 Oe. In most non-cubic materials, is not larger than 1–10 kOe. This means that by applying an external magnetic field we can change only the direction of , or sometimes of , but not their magnitudes.

The magnetic anisotropy energy can be represented as an expansion in the powers of the components of the vectors or . The dependence of on the direction of the magnetization is essential. Therefore, one usually considers the expansion of the spontaneous magnetization or antiferromagnetic vector in powers of the unit vector . The anisotropy energy is invariant under time reversal. Therefore, the general expression for this energy has the form where , , are tensors, the components of which have the dimension of an energy density. The forms of the tensors depend on the symmetry of the crystal. There are at most two independent components in . For a uniaxial crystal, the second-order term in the anisotropy energy expansion is determined by one anisotropy constant, K. Instead of using the components of the unit vector , its direction can be described by two angles: polar and azimuthal . Correspondingly, the anisotropy energy for a uniaxial crystal can be written as This relation is equivalent to

The direction of the magnetization vector in a ferromagnet or of the antiferromagnetic vector in an antiferromagnet is called the direction or the axis of easy magnetization. The crystals in which this axis is aligned with a threefold, fourfold or sixfold axis of the magnetic point group are called easy-axis magnetics. The magnetic crystals with the main axis higher than twofold in the paramagnetic state in which, in the ordered state, (or ) is perpendicular to this axis are often called easy-plane magnetics. The anisotropy in this plane is usually extremely small. In this case, the crystal possesses more then one axis of easy magnetization and the crystal is usually in a multidomain state (see Section 1.5.4).

If the anisotropy constant K is positive, then the vector is aligned along the z axis, and such a magnetic is an easy-axis one. For an easy-plane magnetic, K is negative. It is convenient to use equation (1.5.3.6) for easy-axis magnetics and equation (1.5.3.7) for easy-plane magnetics. In the latter case, the quantity K is included in the isotropic part of the thermodynamic potential , and (1.5.3.7) becomes . Instead we shall write in the following, so that K becomes positive for easy-plane ferromagnetics as well.

Apart from the second-order term, terms of higher order must be taken into account. For tetragonal crystals, the symmetry allows the following invariant terms in the anisotropy energy: the azimuthal angle is measured from the twofold axis x in the basal plane and the constant determines the anisotropy in the basal plane.

Trigonal symmetry also allows second- and fourth-order invariants: where is measured from the x axis, which is chosen parallel to one of the twofold axes. For easy-plane magnetics and , the vector is directed along one of the twofold axes in the basal plane. If is negative, then lies in a vertical mirror plane directed at a small angle to the basal plane. For the complete solution of this problem, the sixth-order term must be taken into account. This term is similar to the one that characterizes the anisotropy of hexagonal crystals. The expression for the latter is of the following form: where x and have the same meaning as in (1.5.3.9).

The symmetry of cubic crystals does not allow any second-order terms in the expansion of the anisotropy energy. The expression for the anisotropy energy of cubic crystals contains the following invariants:

In considering the anisotropy energy, one has to take into account spontaneous magnetostriction and magnetoelastic energy (see Section 1.5.9). This is especially important in cubic crystals. Any collinear cubic magnetic (being brought into a single domain state) ceases to possess cubic crystallochemical symmetry as a result of spontaneous magnetostriction. If is positive, the easy axis is aligned along one of the edges of the cube and the crystal becomes tetragonal (like Fe). If is negative, the crystal becomes rhombohedral and can be an easy-axis magnetic with vector parallel to one of the spatial diagonals (like Ni) or an easy-plane magnetic with perpendicular to a spatial diagonal. We shall discuss this topic in more detail in Section 1.5.9.3.

The considerations presented above can be applied to all crystals belonging in the paramagnetic state to the tetragonal, trigonal or hexagonal system that become easy-plane magnetics in the ordered state. All of them, including the cubic crystals, may possess more than one allowed direction of easy magnetization. In the example considered in the previous section, these directions can be aligned along the three twofold axes for the structures and can be parallel to the three mirror planes for .

It is worth noting that in some applications it is more convenient to use an expansion of the anisotropy energy in terms of surface spherical harmonics. This problem has been considered in detail by Birss (1964).

According to Landau (1937) (see also Landau & Lifshitz, 1951), a phase transition of the second kind can be described by an order parameter , which varies smoothly in the neighbourhood of the transition temperature . The order parameter when and rises continuously as the temperature is decreased below , but the symmetry of the crystal changes suddenly. The order parameter can be a scalar, a vector or a tensor.

Consider a crystal with known space group in the paramagnetic state. In this section, we show how the Landau theory allows us to determine the magnetic space groups that are possible after a second-kind phase transition into an ordered state. The application of the Landau theory to the magnetic transitions into different types of antiferromagnets was made by Dzyaloshinskii (1957a,c; 1964). In these cases, the order parameter is the magnetic moment density . To determine the equilibrium form of this function, it is necessary to find the minimum of the thermodynamic potential , which is a functional of . Since the transition is continuous and for , the value of must be very small in the neighbourhood below the transition point. In this region, the thermodynamic potential will be expanded into a power series of . To find the proper form of this expansion, it is convenient to represent as a linear combination of functions that form bases of the irreducible representations of the space group of the paramagnetic phase : where are functions that transform under the representation n ( is the number of the function in the representation) and . In this expansion, the quantities are independent of and transform with respect to i as the components of an axial vector. The functions are transformed into combinations of one another by the elements of the group . Instead, these elements can be regarded as transforming the coefficients and leaving the functions invariant. In this case, the quantities transform according to the direct product of the representation n of and the representation formed by the components of the pseudovector. This representation is reducible in the general case. Irreducible representations can be obtained by forming linear combinations of the . Let us denote these combinations by . These variables can be considered as components of the order parameter, and the thermodynamic potential can be expanded into a power series of . The terms of this expansion must be invariant under the transformations of the magnetic space group of the crystal in the paramagnetic state . This group possesses R as a separate element. Therefore the expansion can contain only even terms. For each irreducible representation, there is only one invariant of second order – the sum of the squares. Consequently, retaining only the square terms, the expansion of the thermodynamic potential has the form: To minimize , it is necessary to add the terms of the fourth power. All the coefficients in the relation (1.5.3.13) depend on the temperature. At all . This solution corresponds to the minimum of if all are positive. The transition into the ordered state occurs if one of the quantities changes its sign. This means that the transition temperature is the temperature at which one of the coefficients . This coefficient has the form: Accordingly, the corresponding magnetic structure is defined by the order parameters and belongs to the representation p.

The representation of the space group is realized by a set of functions of the following type: where the values of the vectors are confined to the Brillouin zone in the reciprocal lattice and the function is periodic in the real lattice. The irreducible representation defined by the vector contains the functions with all the vectors that belong to the same star. The star is the set of the vectors obtained by applying all the transformations of the corresponding point group to any vector of the star (see also Section 1.2.3.3 ). If we denote it as , then the set of the vectors of the star consists of all inequivalent vectors of the form .

There are three types of transition we have to consider: (1) the magnetic lattice is commensurate with the crystallographic one and ; (2) the magnetic lattice is incommensurate with the crystallographic one; (3) and the magnetic lattice coincides with the crystallographic lattice. Below we shall discuss in detail only the first and the third type of transition.

Neither symmetry nor energy considerations can determine the alignment of the magnetization vector in a non-chiral easy-axis magnetic (of ferro- or antiferromagnetic type). The vector may be aligned parallel or antiparallel to the positive direction of the z axis. Therefore, specimens of any magnetic are usually split into separate regions called domains. In each domain of an easy-axis magnetic, the vector has one of its two possible directions. Such domains are called 180° domains. Adjacent domains are separated by a domain wall, in which the magnetic moments are no longer strictly parallel (or antiparallel). As a result of this, both the exchange and the anisotropy energy rise inside the volume of the domain wall.

In ferromagnets (and ferrimagnets), the loss in the exchange and anisotropy energy in a multidomain sample is compensated by the gain in the magnetostatic energy. The existence of the domain structure is responsible for the behaviour of a ferromagnet in an applied magnetic field. There are two kinds of magnetization processes that one has to distinguish: the displacement of the domain walls and the rotation of the spontaneous magnetization vector from the easy direction to the direction of the applied magnetic field. The magnetization process will first be considered without taking the demagnetizing field into account. If the magnetic field is applied parallel to the axis of an easy-axis ferromagnet, the displacement of the domain wall will completely determine the magnetization process. If the sample contains no impurities and crystal defects, such a displacement must take place in an infinitely small magnetic field [see curve (1) in Fig. 1.5.4.1 and Fig. 1.5.4.3a]. If the magnetic field is applied perpendicular to the easy axis, the size of the domains does not change but their magnetization vectors rotate. Let us denote the spontaneous magnetization by . Then the sample magnetization M rises linearly with respect to the applied magnetic field: where is defined by relations (1.5.3.8)–(1.5.3.10). Some nonlinearity in H can arise from the fourth-order term with [see curve (2) in Fig. 1.5.4.1 and Fig. 1.5.4.3c]. When , the magnetizations of all the domains are rotated by 90° and the magnetization of the sample becomes oriented along the magnetic field; its value is saturated and is equal to the spontaneous magnetization . If  K, there is an additional rise in magnetization with the magnetic field. This rise, which is called true magnetization, is relatively very small at all temperatures except for the temperature region close to the transition temperature. If the magnetic field is applied at an arbitrary angle to the easy axis, the magnetization process occurs in two steps [see curves (2) in Fig. 1.5.4.2 and Fig. 1.5.4.3b]. First, as a result of the wall displacement, the magnetization jumps to the value in a small magnetic field. Next, the rotation process follows and at the sample becomes saturated [see curves (2) in Fig. 1.5.4.2]. It is essential to take the shape of the sample into account in considering the problem of the magnetization processes in ferromagnets, as the demagnetizing field can be up to . In real materials, the displacement process is partly (at low fields) reversible and partly (at higher fields) irreversible. Therefore, complicated hysteresis processes arise in magnetizing ferromagnets.

The problem of 180° domains in antiferromagnets is not as clear. These domains differ in the sign of the antiferromagnetic vector . This vector was defined as the difference of the vectors of sublattice magnetizations in a two-sublattice antiferromagnet, i.e. . Thus two such antiferromagnetic domains differ only by the numbering of the sites in the sublattices. Antiferromagnetic 180° domains are also called S-domains. The wall between two S-domains is schematically represented in Fig. 1.5.4.4.

The origin of the antiferromagnetic S-domains cannot be explained from the point of view of energy balance as in a ferromagnet. These domains give rise to additional exchange and anisotropy energies which are not compensated by a decrease of any other kind of energy. Thus the S-domain structure is thermodynamically not stable. However, experiments show that S-domains exist in most easy-axis antiferromagnets.

The formation of S-domains can be explained by assuming that when the material is cooled down to the Néel temperature, antiferromagnetic ordering arises in different independent regions. The direction of the vector in these regions is accidental. When growing regions with different directions of meet, the regular alternation of the directions of magnetic moments of the ions is broken on the border between these regions. Domain walls are created on such borders. Such domain structures can be metastable.

The existence of S-domains in easy-axis antiferromagnets was first proved in experiments in which effects that depend on the sign of were investigated. These are piezomagnetism, linear magnetostriction and the linear magnetoelectric effect. The sign of these effects depends on the sign of . We shall discuss this problem in detail in Sections 1.5.7 and 1.5.8. Later, 180° domain walls were observed in neutron scattering experiments (Schlenker & Baruchel, 1978), and the domains themselves in magneto-optical experiments (see Kharchenko et al., 1979; Kharchenko & Gnatchenko, 1981).

As pointed out in Section 1.5.3, in tetragonal non-easy-axis magnetics, in easy-plane hexagonal and trigonal and in cubic magnetics there is more than one easy magnetization direction (3, 4 or 6). As a result, domains arise in which vectors or are directed to each other at 120, 109.5, 90, 70.5 and 60°. Such domains are called twin or T-domains. The formation of magnetic T-domains is accompanied by the formation of crystallographic domains as a result of spontaneous magnetostriction. But mostly this is very small. Each of the T-domains may split into 180° domains.

The magnetization process in ferromagnets possessing T-domains is similar to the previously described magnetization of an easy-axis ferromagnet in a magnetic field directed at an oblique angle. First the displacement process allows those 180° domains that are directed unfavourably in each T-domain to disappear, and then the rotation process follows.

In easy-plane antiferromagnets, the T-domain structure is destroyed by a small magnetic field and the antiferromagnetic vector in the whole specimen becomes directed perpendicular to the applied magnetic field, as was explained in Section 1.5.3.

There are four kinds of T-domains in cubic antiferromagnets, in which the vectors are directed parallel or perpendicular to the four axes. Such a T-domain structure can be destroyed only when the applied magnetic field is so strong that the antiferromagnetic order is destroyed at a spin-flip transition.

Aizu (1970) gave a classification of domain formation when a crystal undergoes a transition from an unordered to a magnetically ordered state that has a lower point-group symmetry (see also Section 3.1.1 ). The unordered state (called the prototype phase) has a grey point group. The number of elements in this group is equal to the product of the number of elements in the point group of the ordered state (called the ferroic state) times the number of domains. Aizu found that there are 773 possible combinations of the point-group symmetries of the prototype and the ferroic state, if crystallographically inequivalent orientations of the subgroup in the group of the prototype are distinguished. These 773 combinations are called ferroic species and are characterized by a symbol giving first the point group of the prototype, then the letter F, then the point group of the ferroic state and finally a letter between parentheses if different orientations are possible. As an example, the axis of the ferroic state is parallel to the fourfold axis of the prototype in and perpendicular to it in .

Let us discuss the ferroic states of rhombohedral transition-metal oxides given in Table 1.5.3.4. The paramagnetic prototype has point group . The four monoclinic ferroic species have six domains (`orientation states') each, which form three pairs of 180° domains (`time-conjugate orientation states'). All four species are `fully ferroelastic', i.e. the three pairs show different orientations of the spontaneous strain; two of the four species ( and ) are also `fully ferromagnetic' because all six domains have different orientations of the spontaneous magnetization. Switching a domain into another with a different orientation of the spontaneous strain can be achieved by applying mechanical stress. If the domain was spontaneously magnetized, the orientation of the magnetization is changed simultaneously. Similarly, a domain can be switched into another with a different orientation of the spontaneous magnetization by means of a magnetic field. If the two spontaneous magnetizations have different directions (not just opposite sign), the direction of the spontaneous strain will change at the same time.

The Aizu classification is of interest for technological applications because it gives an overall view not only of domain formation but also of the possibilities for domain switching.

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 two-sublattice antiferromagnets MnCO₃, CoCO₃ and NiCO₃, discussed in Section 1.5.3.1. The following analysis can be applied also to the four-sublattice anti­ferromagnet -Fe₂O₃ (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 two-sublattice 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 high-temperature modification of haematite, -Fe₂O₃ (Townsend Smith, 1916; Néel & Pauthenet, 1952), MnCO₃ (Borovik-Romanov & Orlova, 1956) and later also in CoCO₃, NiCO₃ and FeBO₃. In accordance with theory, weak ferromagnetism does not occur in trigonal crystals with a positive anisotropy coefficient a. Such crystals become easy-axis antiferromagnets. Of this type are FeCO₃ and the low-temperature modification of -Fe₂O₃. For four-sublattice 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 easy-plane 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).

It is worth mentioning that 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 second-order phase transition. Such transitions are allowed because is a subgroup of both groups and .

NiF₂ 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 NiF₂ 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 (MnF₂, FeF₂ and CoF₂) and their magnetic structure is described by the magnetic point group . They are easy-axis antiferromagnets without weak ferromagnetism.

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 NiF₂) 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 . Turov (1963) determined for all crystallographic space groups the form of the invariants of lowest order that allow for collinear or weakly non-collinear 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, however, we have restricted our consideration to the phenomenological approach to this problem.

A large number of orthorhombic orthoferrites and orthochromites with the formula RMO₃ (where R is a trivalent rare-earth ion and M is Fe³⁺ or Cr³⁺) 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 transition-metal 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 four-sublattice 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.

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 M₃B₇O₁₃X (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 no-one has observed such a phenomenon.

The thermodynamic potential of a four-sublattice 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 relation (1.5.3.2)]. Such a structure may occur in Cr₂O₃. In most orthoferrites discussed above, such non-collinear 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 non-collinear 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).