Other weakly non-collinear magnetic structures

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. 130-131

Section 1.5.5.2. Other weakly non-collinear magnetic structures

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

1.5.5.2. Other weakly non-collinear magnetic structures

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A thermodynamic potential $[\tilde {\Phi}]$ 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 $[b\,\gt\, 0]$ , the material will undergo a transition into a slightly canted ferromagnetic structure, in which $[M_{s} \gg L_{D}]$ and the expression for $[L_{D}]$ is $[L_{D} = (d/B)M_{s\perp}. \eqno(1.5.5.11)]$ Experimental detection of such structures is a difficult problem and to date no-one has observed such a phenomenon.

The thermodynamic potential $[\tilde {\Phi}]$ of a four-sublattice antiferromagnet may contain the mixed invariant [see (1.5.3.24)] $[d_{1}(L_{1x}L_{2y} - L_{1y}L_{2x}). \eqno(1.5.5.12)]$ Such a term gives rise to a structure in which all four vectors of sublattice magnetization $[{\bf M}_{\alpha}]$ form a star, as shown in Fig. 1.5.5.6 (see also Fig. 1.5.1.3b). The angle $[2\alpha]$ between the vectors $[{\boldmu}_{1}]$ and $[{\boldmu}_{3}]$ (or $[{\boldmu}_{2}]$ and $[{\boldmu}_{4}]$ ) is equal to $[d_{1}/A_{2}]$ if the main antiferromagnetic structure is defined by the vector $[{\bf l}_{2}]$ [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 ( $[{\bi G}_y]$ ) and weakly ferromagnetic ( $[{\bi G}_x{\bi F}_z]$ and $[{\bi G}_z{\bi F}_x]$ ). The structure $[{\bi G}_y]$ is not coplanar. Apart from the main antiferromagnetic vector $[\bf G]$ aligned along the y axis, it possesses two other antiferromagnetic vectors: $[\bf A]$ (aligned along the x axis) and $[\bf C]$ (aligned along the z axis). The weakly ferromagnetic structure $[{\bi G}_x{\bi F}_z]$ has an admixture of the $[{\bi A}_{y}]$ antiferromagnetic structure.

Figure 1.5.5.6 | top | pdf |

A weakly non-collinear magnetic structure corresponding to (1.5.5.12).

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 $[\bf k]$ , which is parallel to the axis of the helix. The two mutually perpendicular antiferromagnetic vectors $[{\bf L}_{\alpha}]$ are both perpendicular to $[\bf k]$ . These vectors define the helical structure by the following relation for the density of the magnetization $[{\bf M(r)}]$ in the layer with the coordinate $[\bf r]$ (Dzyaloshinskii, 1964; Andreev & Marchenko, 1980): $[{\bf M(r)} = {\bf L}_1\sin {\bf kr} - {\bf L}_2\cos {\bf kr}. \eqno(1.5.5.13)]$ Most helical structures are incommensurate, which means that the representation defined by the vector $[\bf k]$ does not satisfy the Lifshitz condition (see Section 1.5.3.3).

References

Andreev, A. F. & Marchenko, V. I. (1980). Symmetry and the macroscopic dynamics of magnetic materials. (In Russian.) Usp. Fiz. Nauk, 130, 39–63. (English translation: Sov. Phys. Usp. 23, 21–34.)Google Scholar

Dzyaloshinskii, I. E. (1964). Theory of helicoidal structures in antiferromagnets. (In Russian.) Zh. Eksp. Teor. Fiz. 46, 1420–1437, 47, 336–348 and 992–1002. [English translation: Sov. Phys. JETP, 19, 960–971, 20 (1965), 223–231 and 665–671.]Google Scholar

International Tables for Crystallography (2006). Vol. D. ch. 1.5, pp. 130-131