Ferromagnets (including ferrimagnets)

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. 107-108

Section 1.5.1.2.1. Ferromagnets (including ferrimagnets)

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

1.5.1.2.1. Ferromagnets (including ferrimagnets)

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As stated above, all ordered magnetics that possess a spontaneous magnetization M_s different from zero (a magnetization even in zero magnetic field) are called ferromagnets. The simplest type of ferromagnet is shown in Fig. 1.5.1.2(a). This type possesses only one kind of magnetic ion or atom. All their magnetic moments are aligned parallel to each other in the same direction. This magnetic structure is characterized by one vector M. It turns out that there are very few ferromagnets of this type in which only atoms or ions are responsible for the ferromagnetic magnetization (CrBr₃, EuO etc.). The overwhelming majority of ferromagnets of this simplest type are metals, in which the magnetization is the sum of the magnetic moments of the localized ions and of the conduction electrons, which are partly polarized.

Figure 1.5.1.2 | top | pdf |

Ordered arrangements of magnetic moments $[{\boldmu}_{i}]$ in: (a) an ordinary ferromagnet $[{\bf M}_s = N{\boldmu}_1]$ ; (b) a ferrimagnet $[{\bf M}_s =]$ $[(N/3)({\boldmu}_1 +]$ $[{\boldmu}_2 +]$ $[{\boldmu}_3)]$ ; (c) a weak ferromagnet $[{\bf M} =]$ $[{\bf M}_D =]$ $[(N/2)({\boldmu}_1 +]$ $[{\boldmu}_2)]$ , $[{\bf L} =]$ $[(N/2)({\boldmu}_1]$ $[- {\boldmu}_2)]$ , ( $[L_x \gg M_y]$ ; [M_x=] [M_z=] [L_y=] [L_z=0] ). (N is the number of magnetic ions per cm³.)

More complicated is the type of ferromagnet which is called a ferrimagnet. This name is derived from the name of the oxides of the elements of the iron group. As an example, Fig. 1.5.1.2(b) schematically represents the magnetic structure of magnetite (Fe₃O₄). It contains two types of magnetic ions and the number of Fe³⁺ ions ( $[{\boldmu}_{1}]$ and $[{\boldmu}_{2}]$ ) is twice the number of Fe²⁺ ions ( $[{\boldmu}_{3}]$ ). The values of the magnetic moments of these two types of ions differ. The magnetic moments of all Fe²⁺ ions are aligned in one direction. The Fe³⁺ ions are divided into two parts: the magnetic moments of one half of these ions are aligned parallel to the magnetic moments of Fe²⁺ and the magnetic moments of the other half are aligned antiparallel. The array of all magnetic moments of identical ions oriented in one direction is called a magnetic sublattice. The magnetization vector of a given sublattice will be denoted by $[{\bf M}_{i}]$ . Hence the magnetic structure of Fe₃O₄ consists of three magnetic sublattices. The magnetizations of two of them are aligned in one direction, the magnetization of the third one is oriented in the opposite direction. The net ferromagnetic magnetization is [M_s = M_1-M_2+M_3 = M_3] .

The special feature of ferrimagnets, as well as of many antiferromagnets, is that they consist of sublattices aligned antiparallel to each other. Such a structure is governed by the nature of the main interaction responsible for the formation of the ordered magnetic structures, the exchange interaction. The energy of the exchange interaction does not depend on the direction of the interacting magnetic moments (or spins S) relative to the crystallographic axes and is represented by the following relation: $[U_{\rm ex} = - \textstyle \sum \limits_{m,n} J_{mn} {\bf S}_m {\bf S}_n. \eqno(1.5.1.7)]$ Here $[{\bf S}_{m}]$ , $[{\bf S}_{n}]$ are the spins of magnetic atoms (ions) and $[J_{mn}]$ is the exchange constant, which usually decreases fast when the distance between the atoms rises. Therefore, usually only the nearest neighbour interaction needs to be taken into account. Hence, according to (1.5.1.7), the exchange energy is a minimum for the state in which neighbouring spins are parallel (if $[ J\,\gt\, 0 ]$ ) or antiparallel (if $[ J \,\lt\, 0 ]$ ). If the nearest neighbour exchange interaction were the only interaction responsible for the magnetic ordering, only collinear magnetic structures would exist (except in triangle lattices). Together with the exchange interaction, there is also a magnetic dipole interaction between the magnetic moments of the atoms as well as an interaction of the atomic magnetic electrons with the crystalline electric field . These interactions are much smaller than the exchange interaction. They are often called relativistic interactions. The relativistic interactions and the exchange interaction between next-nearest atoms bring about the formation of non-collinear magnetic structures.

A simple non-collinear structure is the magnetic structure of a weak ferromagnet . It contains identical magnetic ions divided in equal amounts between an even number of sublattices. In the first approximation, the magnetizations of these sublattices are antiparallel, as in usual antiferromagnets. In fact, the magnetizations are not strictly antiparallel but are slightly canted, i.e. non-collinear, as shown in Fig. 1.5.1.2(c). There results a ferromagnetic moment [M_D] , which is small compared with the sublattice magnetization [M_i] . The magnetic properties of weak ferromagnets combine the properties of both ferromagnets and antiferromagnets. They will be discussed in detail in Section 1.5.5.1.

References

International Tables for Crystallography (2006). Vol. D. ch. 1.5, pp. 107-108