International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2013 
International Tables for Crystallography (2013). Vol. D, ch. 1.5, pp. 132133
Section 1.5.6. Reorientation transitions^{a}P. L. Kapitza Institute for Physical Problems, Russian Academy of Sciences, Kosygin Street 2, 119334 Moscow, Russia,^{b}Labor für Neutronenstreuung, ETH Zurich, Switzerland, and ^{c}Paul Scherrer Institute, CH5232 Villigen PSI, Switzerland 
In many materials, the anisotropy constants change sign at some temperature below the critical temperature. As a result, the direction of the vector (or ) changes relative to the crystallographic axes. Correspondingly, the magnetic symmetry of the material also changes. Such phase transitions are called reorientation transitions.
Cobalt is a typical ferromagnet and experiences two such reorientation transitions. It is a hexagonal crystal, which at low temperatures behaves as an easyaxis ferromagnet; its magnetic point group is . If the anisotropy energy were described by the relations (1.5.3.6) and (1.5.3.7) with only one anisotropy constant , the change of the sign of this constant would give rise to a firstorder transition from an easyaxis to an easyplane ferromagnet. This transition would occur at the temperature at which . In fact, the polar angle that determines the direction of the spontaneous magnetization increases progressively over a finite temperature interval. The behaviour of during the process of this reorientation may be obtained by minimizing the expression of the anisotropy energy (1.5.3.10), which contains two anisotropy coefficients and . If , the minimum of corresponds to three magnetic phases, which belong to the following magnetic point groups:
The lowtemperature phase is of the easyaxis type and the hightemperature phase is of the easyplane type. The intermediate phase is called the angular phase. The two secondorder phase transitions occur at temperatures which are the roots of the two equationsThe chain of these transitions (including the transition to the paramagnetic state at ) may be represented by the following chain of the corresponding magnetic point groups:
In Co and most of the other ferromagnets, the rotation of the spontaneous magnetization described above can be obtained by applying an external magnetic field in an appropriate direction. In many antiferromagnets, there occur similar reorientation transitions, which cannot be achieved by means of a magnetic field.
The first reorientation transition in antiferromagnets was observed in haematite ( Fe_{2}O_{3}), which at room temperature is a weak ferromagnet with magnetic structure or (see Tables 1.5.3.3 and 1.5.3.4 in Section 1.5.3.1). Morin (1950) found that the weak ferromagnetism in haematite disappears below K. At low temperature, haematite becomes an easyaxis antiferromagnet with the structure . Unlike in cobalt, the transition at is a firstorder transition in haematite. This is so because the anisotropy constant is negative in haematite. As a result, there are only two solutions for the angle that lead to a minimum of the anisotropy energy [(1.5.3.9)], if and if . The transition temperature is defined byThere is the following change in the magnetic space groups at this transition:
Which of the two groups is realized at high temperatures depends on the sign of the anisotropy constant in equation (1.5.3.9). Neither of the hightemperature magnetic space groups is a subgroup of the lowtemperature group. Therefore the transition under consideration cannot be a secondorder transition.
Reorientation transitions have been observed in many orthoferrites and orthochromites. Orthoferrites of Ho, Er, Tm, Nd, Sm and Dy possess the structure [see (1.5.5.8)] at room temperature. The first five of them undergo reorientation transitions to the structure at lower temperatures. This reorientation occurs gradually, as in Co. Both vectors and rotate simultaneously, as shown in Fig. 1.5.6.1. These vectors remain perpendicular to each other, but the value of varies from for to for . The coefficients and belong to the terms and , respectively. The following magnetic point groups are observed when these transitions occur:

Schematic representation of the rotation of the vectors and (in the xz plane) at a reorientation transition in orthoferrites. 
Anomalies typical for secondorder transitions were observed at the temperatures and . The interval varies from 10 to 100 K.
At low temperatures, DyFeO_{3} is an easyaxis antiferromagnet without weak ferromagnetism – . It belongs to the trivial magnetic point group . At T_{M} = 40 K, DyFeO_{3} transforms into a weak ferromagnet . This is a firstorder reorientation transition of the type
Reorientation transitions in antiferromagnets occur not only as a result of a sign change of the anisotropy constant. They can be governed by the applied magnetic field. In Section 1.5.3.3.2, we described the spinflop firstorder reorientation transition in an easyaxis antiferromagnet. This transition splits into two secondorder transitions if the magnetic field is not strictly parallel to the axis of the crystal. There is a specific type of reorientation transition, which occurs in antiferromagnets that do not exhibit weak ferromagnetism, but would become weak ferromagnets if the antiferromagnetic vector was directed along another crystallographic direction. As an example, let us consider such a transition in CoF_{2}. It is a tetragonal crystal with crystallographic space group . Below , CoF_{2} becomes an easyaxis antiferromagnet. The magnetic structure of this crystal is shown in Fig. 1.5.5.3. Its magnetic point group is = . Let us apply the magnetic field H parallel to the twofold axis x (see Fig. 1.5.6.2). In a typical antiferromagnet, the field stimulates a magnetization . The structure allows weak ferromagnetism if is perpendicular to the z axis. As a result, if the vector is deflected from the z axis by an angle in the plane yz perpendicular to the x axis, the magnetization will rise according to the relationwhere [see (1.5.5.3) and (1.5.5.4)]. As a result, there is a gain in the magnetic energy, which compensates the loss in the anisotropy energy. The beginning of the deflection is a secondorder transition. The balance of both energies determines the value of : The second secondorder transition occurs when becomes equal to at the critical field : After the reorientation transition, CoF_{2} has the same magnetic point group as the weak ferromagnet NiF_{2}, i.e. .
References
Morin, F. J. (1950). Magnetic susceptibility of αFe_{2}O_{3} and αFe_{2}O_{3} with added titanium. Phys. Rev. 78, 819–820.