Reorientation transitions

Borovik-Romanov, A. S.; Grimmer, H.

doi:10.1107/97809553602060000632

International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

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International Tables for Crystallography (2006). Vol. D. ch. 1.5, pp. 131-132

Section 1.5.6. Reorientation transitions

A. S. Borovik-Romanov^a ^‡ and H. Grimmer^b ^*

^a P. L. Kapitza Institute for Physical Problems, Russian Academy of Sciences, Kosygin Street 2, 119334 Moscow, Russia, and ^bLabor für Neutronenstreuung, ETH Zurich, and Paul Scherrer Institute, CH-5234 Villigen PSI, Switzerland
Correspondence e-mail: hans.grimmer@psi.ch

1.5.6. Reorientation transitions

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In many materials, the anisotropy constants change sign at some temperature below the critical temperature. As a result, the direction of the vector $[\bf L]$ (or $[{\bf M}_{s}]$ ) 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 easy-axis ferromagnet; its magnetic point group is $[{\bi D}_{6h}({\bi C}_{6h}) = 6/mm'm']$ . If the anisotropy energy were described by the relations (1.5.3.6) and (1.5.3.7) with only one anisotropy constant $[K_{1}]$ , the change of the sign of this constant would give rise to a first-order transition from an easy-axis to an easy-plane antiferromagnet . This transition would occur at the temperature [T_c] at which [K_1(T) = 0] . In fact, the polar angle $[\theta]$ which determines the direction of the spontaneous magnetization increases progressively over a finite temperature interval. The behaviour of $[\theta]$ 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 $[K_{1}]$ and $[K_{2}]$ . If $[K_{2}\,\gt\, 0]$ , the minimum of $[U_{a}]$ corresponds to three magnetic phases, which belong to the following magnetic point groups:

(1) $[{\bi D}_{6h}({\bi C}_{6h}) = 6/mm'm']$ ; for this phase $[\theta = 0, \pi]$ . It is realized at temperatures $[T \,\lt\, T_1= 520]$ K, where $[K_{1}\,\gt\, 0]$ .
(2) $[{\bi C}_{2h}({\bi C}_{i}) = 2'/m']$ ; for this phase $[\sin\theta = \pm(-K_{1}/2K_{2})^{1/2}]$ . It is realized at temperatures $[T_1 = 520 \,\lt\, T \,\lt\, T_{2} = 580]$ K, where $[-2K_{2} \,\lt\, K_{1} \,\lt\, 0]$ .
(3) $[{\bi D}_{2h}({\bi C}_{2h}) = mm'm']$ ; for this phase $[\theta = \pi/2]$ . It is realized at temperatures $[T_{2} = 580 \,\lt\, T \,\lt\, T_c = 690]$ K, where $[K_{1} \,\lt\, -2K_{2}]$ .

The low-temperature phase is of the easy-axis type and the high-temperature phase is of the easy-plane type. The intermediate phase is called the angular phase. The two second-order phase transitions occur at temperatures which are the roots of the two equations $[K_{1}(T_{1}) = 0; \quad K_{1}(T_{2}) + 2K_{2}(T_{2}) = 0.\eqno(1.5.6.1) ]$ The chain of these transitions (including the transition to the paramagnetic state at [T = T_c] ) may be represented by the following chain of the corresponding magnetic point groups: $[\eqalign{{\bi D}_{6h}({\bi C}_{6h}) = 6/mm'm' &\,\longleftrightarrow \,{\bi C}_{2h}({\bi C}_{i}) = 2'/m'\cr& \,\longleftrightarrow \,{\bi D}_{2h}({\bi C}_{2h}) = mm'm' \cr&\,\longleftrightarrow \,({\bi D}_{6h} + {\bi RD}_{6h}) = 6/mmm1'. }]$

In Co and most of the other ferromagnets, the rotation of the spontaneous magnetization described above may 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 ( $[\alpha]$ - Fe₂O₃), which at room temperature is a weak ferromagnet with magnetic structure $[{\bi A}_{3x}]$ or $[{\bi A}_{3y}]$ (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 $[T_{M} \simeq 260]$ K. At low temperature, haematite becomes an easy-axis antiferromagnet with the structure $[{\bi A}_{3z}]$ . Unlike in cobalt, the transition at [T_M] is a first-order transition in haematite. This is so because the anisotropy constant $[K_{2}]$ is negative in haematite. As a result, there are only two solutions for the angle $[\theta]$ that lead to a minimum of the anisotropy energy $[U_{a}(3)]$ [(1.5.3.9)], $[\theta = 0]$ if $[K_1\,\gt\, -K_2]$ and $[\theta = \pi/2]$ if $[K_1 \,\lt\, -K_2]$ . The transition temperature $[T_{M}]$ is defined by $[K_1(T_M) + K_2(T_M) = 0. \eqno(1.5.6.2)]$ There is the following change in the magnetic space groups at this transition: $[\displaylines{\hfill{R\overline{3}c' {\buildrel{T_M}\over {\longleftrightarrow}} \left\{ \matrix{ P2/c \hfill \cr P2'/c'\hfill}\right.} \hfill\matrix{(1.5.6.3)\cr(1.5.6.4)}}%fd1.5.6.4]$

Which of the two groups is realized at high temperatures depends on the sign of the anisotropy constant $[K'_{\perp}]$ in equation (1.5.3.9). Neither of the high-temperature magnetic space groups is a subgroup of the low-temperature group. Therefore the transition under consideration cannot be a second-order transition.

Reorientation transitions have been observed in many orthoferrites and orthochromites. Orthoferrites of Ho, Er, Tm, Nd, Sm and Dy possess the structure $[{\bi G}_x{\bi F}_z]$ [see (1.5.5.8)] at room temperature. The first five of them undergo reorientation transitions to the structure $[{\bi G}_z{\bi F}_x]$ at lower temperatures. This reorientation occurs gradually, as in Co. Both vectors $[\bf L]$ and $[{\bf M}_{D}]$ rotate simultaneously, as shown in Fig. 1.5.6.1. These vectors remain perpendicular to each other, but the value of $[{\bf M}_{D}]$ varies from $[(d_{1}/B)L]$ for $[M_{Dz}]$ to $[(d_{2}/B)L]$ for $[M_{Dx}]$ . The coefficients $[d_{1}]$ and $[d_{2}]$ belong to the terms $[L_{x}M_{z}]$ and $[L_{z}M_{x}]$ , respectively. The following magnetic point groups are observed when these transitions occur: $[{{2'_x}\over{m'_x}} {{2'_y}\over{m'_y}} {{2_z}\over{m_z}} \; {\buildrel{T_1}\over{\longleftrightarrow}} \; {{2'_y}\over{m'_y}}\; {\buildrel{T_2}\over{\longleftrightarrow}} \; {{2_x}\over{m_x}} {{2'_y}\over{m'_y}} {{2'_z}\over{m'_z}}. \eqno(1.5.6.5)]$

Figure 1.5.6.1 | top | pdf |

Schematic representation of the rotation of the vectors $[\bf G]$ and $[\bf F]$ (in the xz plane) at a reorientation transition in orthoferrites.

Anomalies typical for second-order transitions were observed at the temperatures [T_1] and [T_2] . The interval $[T_{2} - T_{1}]$ varies from 10 to 100 K.

At low temperatures, DyFeO₃ is an easy-axis antiferromagnet without weak ferromagnetism – $[{\bi G}_y]$ . It belongs to the trivial magnetic point group $[{\bi D}_{2h} = mmm]$ . At T_M = 40 K, DyFeO₃ transforms into a weak ferromagnet $[{\bi G}_x{\bf F}_z]$ . This is a first-order reorientation transition of the type $[{\bi D}_{2h}=mmm\; {\buildrel{T_M}\over {\longleftrightarrow}} \;{\bi D}_{2h}({\bi C}_{2h})=m'm'm. \eqno(1.5.6.6)]$

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 spin-flop first-order reorientation transition in an easy-axis antiferromagnet. This transition splits into two second-order 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 CoF₂. It is a tetragonal crystal with crystallographic space group $[{\bi D}^{14}_{4h} = P4_2/mnm]$ . Below [T_N] , CoF₂ becomes an easy-axis antiferromagnet. The magnetic structure of this crystal is shown in Fig. 1.5.5.3. Its magnetic point group is $[{\bi D}_{4h}({\bi D}_{2h}) =4'/mmm']$ . 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 $[M = \chi_{\perp}H]$ . The structure $[{\bi D}^{14}_{4h} = P4_2/mnm]$ allows weak ferromagnetism if $[\bf L]$ is perpendicular to the z axis. As a result, if the vector $[\bf L]$ is deflected from the z axis by an angle $[\theta]$ in the plane yz perpendicular to the x axis, the magnetization will rise according to the relation $[M = \chi_{\perp}(H + H_{D}\sin \theta), \eqno(1.5.6.7)]$ where $[H_D=M_D/\chi_{\perp}]$ [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 second-order transition. The balance of both energies determines the value of $[\theta]$ : $[\sin \theta = (H_{e}/H_{a}H_{D})H. \eqno(1.5.6.8)]$ The second second-order transition occurs when $[\theta]$ becomes equal to $[\pi /2]$ at the critical field $[H_{c}]$ : $[H_{c} = H_{D}H_{a}/H_{e}. \eqno(1.5.6.9)]$ After the reorientation transition, CoF₂ has the same magnetic point group as the weak ferromagnet NiF₂, i.e. $[{\bi D}_{2h}({\bi C}_{2h}) = mm'm']$ .

Figure 1.5.6.2 | top | pdf |

Schematic representation of the rotation of the vector $[\bf L]$ under the action of a magnetic field applied to CoF₂ perpendicular to the fourfold axis z (reorientation transition) (see Figs. 1.5.5.3a and b).

References

Morin, F. J. (1950). Magnetic susceptibility of α-Fe₂O₃ and α-Fe₂O₃ with added titanium. Phys. Rev. 78, 819–820.Google Scholar

International Tables for Crystallography (2006). Vol. D. ch. 1.5, pp. 131-132