Exchange symmetry

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, p. 116

Section 1.5.2.4. Exchange symmetry

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

1.5.2.4. Exchange symmetry

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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 $[{\cal G}_{\rm ex}]$ 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 $[{\bf m}({\bf r})]$ transform like scalars under all rotations of spin space. The exchange symmetry group $[{\cal G}_{\rm ex}]$ contains those combinations of the space transformation elements, the rotations U of spin space and the element R with respect to which the values $[m({\bf r})]$ are invariant. Setting all the elements U and R equal to the identity transformation, we obtain one of the ordinary crystallographic space groups $[\cal G]$ . This space group defines the symmetry of the charge density $[\rho ({\bf r})]$ and of all the magnetic scalars in the crystal. However, the vectors $[{\bf m}({\bf r})]$ may not be invariant with respect to $[\cal G]$ .

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).

References

Andreev, A. F. & Marchenko, V. I. (1976). Macroscopic theory of spin waves. (In Russian.) Zh. Eksp. Teor. Fiz. 70, 1522–1538. (English translation: Sov. Phys. JETP, 43, 794–803.)Google Scholar

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

Birss, R. R. (1964). Symmetry and magnetism. Amsterdam: North-Holland.Google Scholar

Cracknell, A. P. (1975). Magnetism in crystalline materials. Oxford: Pergamon.Google Scholar

Joshua, S. J. (1991). Symmetry principles and magnetic symmetry in solid state physics. Graduate Student Series in Physics. Bristol: Hilger.Google Scholar

Koptsik, V. A. (1966). Shubnikov groups. (In Russian.) Moscow: Izd. MGU.Google Scholar

Landau, L. D. & Lifshitz, E. M. (1957). Electrodynamics of continuous media. (In Russian.) Moscow: Gostekhizdat. [English translation (1960): London: Pergamon.]Google Scholar

Opechowski, W. & Guccione, R. (1965). Magnetic symmetry. In Magnetism, Vol. IIA, edited by G. T. Rado & H. Suhl, pp. 105–165. New York: Academic Press.Google Scholar

Sirotin, Y. I. & Shaskol'skaya, M. P. (1979). Fundamentals of crystal physics. (In Russian.) Moscow: Nauka. [English translation (1982): Moscow: Mir.]Google Scholar

International Tables for Crystallography (2006). Vol. D. ch. 1.5, p. 116