International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. D, ch. 1.5, pp. 116125
Section 1.5.3. Phase transitions into a magnetically ordered state^{a}P. L. Kapitza Institute for Physical Problems, Russian Academy of Sciences, Kosygin Street 2, 119334 Moscow, Russia, and ^{b}Labor für Neutronenstreuung, ETH Zurich, and Paul Scherrer Institute, CH5234 Villigen PSI, Switzerland 
Most transitions from a paramagnetic into an ordered magnetic state are secondorder phase transitions. A crystal with a given crystallographic symmetry can undergo transitions to different ordered states with different magnetic symmetry. In Section 1.5.3.3, we shall give a short review of the theory of magnetic secondorder phase transitions. As was shown by Landau (1937), such a transition causes a change in the magnetic symmetry. The magnetic symmetry group of the ordered state is a subgroup of the magnetic group of the material in the paramagnetic state. But first we shall give a simple qualitative analysis of such transitions.
To find out what ordered magnetic structures may be obtained in a given material and to which magnetic group they belong, one has to start by considering the crystallographic space group of the crystal under consideration. It is obvious that a crystal in which the unit cell contains only one magnetic ion can change only into a ferromagnetic state if the magnetic unit cell of the ordered state coincides with the crystallographic one. If a transition into an antiferromagnetic state occurs, then the magnetic cell in the ordered state will be larger than the crystallographic one if the latter contains only one magnetic ion. Such antiferromagnets usually belong to the subtype III^{b} described in Section 1.5.2.3. In Section 1.5.3.1, we shall consider crystals that transform into an antiferromagnetic state without change of the unit cell. This is possible only if the unit cell possesses two or more magnetic ions. To find the possible magnetic structures in this case, one has to consider those elements of symmetry which interchange the positions of the ions inside the unit cell (especially glide planes and rotation axes). Some of these elements displace the magnetic ion without changing its magnetic moment, and others change the moment of the ion. It is also essential to know the positions of all these elements in the unit cell. All this information is contained in the space group . If the magnetic ordering occurs without change of the unit cell, the translation group in the ordered state does not contain primed elements. Therefore, there is no need to consider the whole crystal space group . It will suffice to consider the cosets of in . Such a coset consists of all elements of that differ only by a translation. From each coset, a representative with minimum translative component is chosen. We denote a set of such representatives by ; it can be made into a group by defining () as the representative of the coset that contains . Obviously, is then isomorphic to the factor group and therefore to the point group of .
Once more, we should like to stress that to construct the magnetic structures and the magnetic groups of a given crystal it is not enough to consider only the point group of the crystal, but it is necessary to perform the analysis with the help of its space group in the paramagnetic state or the corresponding group of coset representatives. An example of such an analysis will be given in the following section.
Following Dzyaloshinskii (1957a), we consider crystals belonging to the crystallographic space group . To this group belong Fe_{2}O_{3} and the carbonates of Mn^{2+}, Co^{2+} and Ni^{2+}. Weak ferromagnetism was first observed in these materials. Cr_{2}O_{3}, 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 rotoinversion 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 (Fe_{2}O_{3}, Cr_{2}O_{3}). 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 (MnCO_{3}, CoCO_{3}, NiCO_{3}, FeCO_{3}), 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 representations^{2} 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.

Four types of magnetic structures of rhombohedral oxides of transition metals. The direction of is shown conventionally. 
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 noncubic 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 noncollinear antiferromagnetic structures.
The strongly noncollinear 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 nonmagnetic 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 noncubic 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 secondorder 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 easyaxis 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 easyplane 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 easyaxis one. For an easyplane magnetic, K is negative. It is convenient to use equation (1.5.3.6) for easyaxis magnetics and equation (1.5.3.7) for easyplane 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 easyplane ferromagnetics as well.
Apart from the secondorder 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 fourthorder invariants: where is measured from the x axis, which is chosen parallel to one of the twofold axes. For easyplane 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 sixthorder 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 secondorder 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 easyaxis magnetic with vector parallel to one of the spatial diagonals (like Ni) or an easyplane 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 easyplane 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 secondkind 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.

The temperature of transition from the paramagnetic to the ferromagnetic state is called the Curie temperature. The thermodynamic treatment of the behaviour of uniaxial ferromagnets in the neighbourhood of the Curie temperature is given below.
In the case of a ferromagnet , the thermodynamic potential (1.5.3.27) near including the magnetic energy is given by (see 1.5.3.25) where is used to designate the thermodynamic potential in variables [instead of ]; at the given field, should be a minimum. The equilibrium value of the magnetization is found by minimizing the thermodynamic potential.
First consider the ferromagnet in the absence of the external field . The system of equations has three solutions:
In the whole range of temperatures when , the minimum of the potential is determined by solution (I) (i.e. absence of a spontaneous magnetization). The realization of the second or third state depends on the sign of the coefficient b. If , then the third state is realized, the magnetization being directed along the axis. In this case, the transition from the paramagnetic into the ferromagnetic state will take place at (when ). If , the magnetization is directed perpendicular to the axis. In this case, the Curie temperature is (when ). In the absence of a magnetic field, the difference between the two values of has no physical meaning, since it only means another value of the coefficient B [see (1.5.3.25)]. In a magnetic field, both temperatures may be determined experimentally, i.e. when B becomes zero and when becomes zero.
If a magnetic field is applied parallel to the z axis and , the minimization of the thermodynamic potential leads toThis relation has been verified in many experiments and the corresponding graphical representations are known in the literature as Arrott–Belov–Kouvel plots (see Kouvel & Fisher, 1964). Putting according to (1.5.3.14), equations (1.5.3.32) and (1.5.3.33) may be used to derive expressions for the initial magnetic susceptibilities (for ): where .
The Landau theory of phase transitions does not take account of fluctuations of the order parameter. It gives qualitative predictions of all the possible magnetic structures that are allowed for a given crystal if it undergoes a secondorder transition. The theory also explains which of the coefficients in the expression for the thermodynamic potential is responsible for the corresponding magnetic structure. It describes also quantitative relations for the magnetic properties of the material if where is the coefficient in the term which describes the gradient energy. In this chapter, we shall not discuss the behaviour of the material in the fluctuation region. It should be pointed out that, in this region, in relations (1.5.3.34) and (1.5.3.35) depends on the dimensionality of the structure n and equals 1.24 for , 1.31 for and 1.39 for . Similar considerations are relevant to the relations (1.5.3.31) and (1.5.3.32), which describe the temperature dependence of spontaneous magnetization.
The relations (1.5.3.31) and (1.5.3.32) describe the behaviour of the ferromagnet in the `saturated' state when the applied magnetic field is strong enough to destroy the domain structure. The problem of the domains will be discussed later (see Section 1.5.4).
The transition from the paramagnetic to the ferromagnetic state is a secondorder transition, provided that there is no magnetic field. In the presence of a magnetic field that is parallel to the easy axis of magnetization, the magnetic symmetry of the crystal is the same () both above and below . From the point of view of symmetry, no transition occurs in this case.
Now let us proceed to the uniaxial antiferromagnet with two ions in the primitive cell. The thermodynamic potential for such an antiferromagnet is given in accordance with (1.5.3.26) and (1.5.3.27) by (Landau, 1933)
If the magnetic field is absent , then because B, D and . Then three possible magnetic states are obtained by minimizing the potential with respect to only:
When , state (II) with is thermodynamically stable. When , state (III) is stable and the antiferromagnetic vector is directed along the axis. This means that the term with the coefficient a is responsible for the anisotropy of the uniaxial antiferromagnet. We introduce the effective anisotropy field: where is the sublattice magnetization.
Formulas (1.5.3.39) and (1.5.3.14) in the form yield the expression for the temperature dependence of the sublattice magnetization: where is the Néel temperature. The assertions relating to formulas (1.5.3.34) and (1.5.3.35) concerning the fluctuation region are also valid for the temperature dependence of the sublattice magnetization.
The minimization of the potential with respect to for given when yields the following relation for the magnetization: where . Thus the magnetization of an antiferromagnet is linear with the magnetic field, as for a paramagnet, if the magnetic field is not too strong. The main difference is in the anisotropy and temperature dependence of the susceptibility. The parallel susceptibility decreases when the temperature is lowered, and does not depend on temperature () (see Fig. 1.5.3.6). The coefficient B belongs to the exchange term and defines the effective exchange field

Temperature dependence of the mass susceptibility for a uniaxial antiferromagnet along () and perpendicular () to the axis of antiferromagnetism (see Foner, 1963). 
As seen from Fig. 1.5.3.6, . Therefore, when the magnetic field applied parallel to the axis of a uniaxial antiferromagnet reaches the critical value ( is the value of L at ), a flopping of the sublattices from the direction along the axis to some direction in the plane perpendicular to the axis occurs. In this spinflop transition (which is a firstorder transition into a new magnetic structure), the magnetization jumps as shown in Fig. 1.5.3.7.

Dependence of the relative magnetization on the magnetic field at . The dashed line corresponds to , the full line to . is the field of spinflop, is the field of spinflip. 
A secondorder transition into a saturated paramagnetic state takes place in a much stronger magnetic field . This transition is called a spinflip transition. Fig. 1.5.3.7 shows the magnetic field dependence of the magnetization of a uniaxial antiferromagnet. Fig. 1.5.3.8 shows the temperature dependence of both critical fields.
The quantitative behaviour of the critical magnetic fields in the neighbourhood of for both directions of the magnetic field ( and ) can be determined from the theory of secondorder phase transitions starting from the thermodynamic potential and taking into account that L is small and close to .
In the presence of the magnetic field , is parallel to , , the coefficient A at is replaced by and the latter is zero at the new transition point. The critical field is given by the relation
If the field is applied parallel to the z axis, then remains parallel to if ( in the neighbourhood of ). Therefore, If , becomes perpendicular to the z axis and the anisotropy term has to be taken into account:
Formulas (1.5.3.46)–(1.5.3.48) show that the transition temperature is reduced by applying the magnetic field. The displacement of the transition point is directly proportional to the square of the applied field. Fig. 1.5.3.9 shows the phase diagram of an antiferromagnet in the neighbourhood of . Unlike ferromagnets, antiferromagnets maintain the secondorder phase transition when a magnetic field is applied because the symmetry of the crystal in the antiferromagnetic state differs essentially from that in the paramagnetic state also if the crystal is placed into a magnetic field.

Phase diagram for a uniaxial antiferromagnet in the proximity of , calculated for MnCl_{2}·4H_{2}O. Experimental data are taken from Gijsman et al. (1959). 
Formula (1.5.3.43) describes the magnetization process only in easyaxis antiferromagnets. For easyplane antiferromagnets, the anisotropy in the plane is usually extremely small and the antiferromagnetic vector rotates freely in the basic plane. Therefore, for any direction of the magnetic field, the vector becomes aligned perpendicular to the applied magnetic field. Correspondingly the magnetization becomes where and are unit vectors parallel and perpendicular to the axis.
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