Magnetic lattices

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. 112-115

Section 1.5.2.2. Magnetic lattices

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

1.5.2.2. Magnetic lattices

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If the point group of symmetry describes the macroscopic properties of a crystal, its microscopic structure is determined by the space group, which contains the group of translations $[\cal T]$ as a subgroup. The elements $[\bf t]$ of $[\cal T]$ are defined by the following relation: $[{\bf t} = n_1{\bf a}_1 + n_2{\bf a}_2 + n_3{\bf a}_3, \eqno(1.5.2.3)]$ where $[{\bf a}_1]$ , $[{\bf a}_2]$ , $[{\bf a}_3]$ are basic primitive translation vectors and [n_1] , [n_2] , [n_3] are arbitrary integers. The set of points $[{\bf r}']$ obtained by applying all the translations of the group $[\cal T]$ to any point $[\bf r]$ defines a lattice. All sites of the crystallographic lattice are equivalent.

The structure of the ordered magnetics is described by the magnetic lattices and corresponding magnetic translation groups $[{\cal M}_T]$ . In the magnetic translation groups $[{\cal M}_T]$ , some of the elements $[\bf t]$ may be multiplied by R (we shall call them primed translations). The magnetic lattices then have two types of sites, which are not equivalent. One set is obtained by non-primed translations and the other set by the primed ones. The magnetic translation group $[{\cal M}_T]$ is isometric to the crystallographic one $[{\cal G}_0]$ that is obtained by replacing R by E in $[{\cal M}_T]$ .

There are trivial magnetic translation groups, in which none of the translation elements is multiplied by R. The magnetic lattices of these groups coincide with crystallographic lattices.

Nontrivial magnetic translation groups can be constructed in analogy to relation (1.5.2.2). Zamorzaev (1957) showed that every translation group $[\cal T]$ has seven subgroups of index 2. If the basic primitive translations of the group $[\cal T]$ are $[{\bf a}_1]$ , $[{\bf a}_2]$ , $[{\bf a}_3]$ , then the basic primitive translations of the seven subgroups $[\cal H]$ can be chosen as follows (see also Opechowski & Guccione, 1965) $[\eqalignno{{\cal H}_{1}&: 2{\bf a}_1, {\bf a}_2, {\bf a}_3 &(1.5.2.4)\cr {\cal H}_{2}&: {\bf a}_1, 2{\bf a}_2, {\bf a}_3 &(1.5.2.5)\cr {\cal H}_{3}&: {\bf a}_1, {\bf a}_2, 2{\bf a}_3 &(1.5.2.6)\cr {\cal H}_{4}&: 2{\bf a}_1, {\bf a}_1 + {\bf a}_2, {\bf a}_3 &(1.5.2.7)\cr {\cal H}_{5}&: 2{\bf a}_2, {\bf a}_2 + {\bf a}_3, {\bf a}_1 &(1.5.2.8)\cr {\cal H}_{6}&: 2{\bf a}_3, {\bf a}_3 + {\bf a}_1, {\bf a}_2 &(1.5.2.9)\cr {\cal H}_{7}&: 2{\bf a}_1, {\bf a}_1 + {\bf a}_2, {\bf a}_1 + {\bf a}_3.&(1.5.2.10)}%fd1.5.2.10]$

As an example, let us consider the case (1.5.2.5). In this case, the subgroup $[\cal H]$ consists of the following translations: $[{\bf t}({\cal H}) = n_1{\bf a}_1 + 2n_2{\bf a}_2 + n_3{\bf a}_3. \eqno(1.5.2.11)]$ Therefore the elements [G_i] of $[\cal(T-H)]$ [which corresponds to $[\cal(P-H)]$ in relation (1.5.2.2)] must have the following form: $[{\bf t}(G_i) = n_1{\bf a}_1 + (2n_2+1){\bf a}_2 + n_3{\bf a}_3. \eqno(1.5.2.12)]$ The corresponding magnetic translation group consists of the elements (1.5.2.12) multiplied by R and the elements (1.5.2.11).

The crystallographic lattices are classified into Bravais types or Bravais lattices. The magnetic lattices are classified into Bravais types of magnetic lattices . It turns out that there are 22 nontrivial magnetic Bravais types. Together with the trivial ones, there are 36 magnetic Bravais lattices.

Two types of smallest translation-invariant cells are in common use for the description of magnetically ordered structures: the crystallographic cell obtained if the magnetic order is neglected and the magnetic cell, which takes the magnetic order into account. The list of the basic translations of all the magnetic Bravais lattices was given by Zamorzaev (1957). The diagrams of the magnetic unit cells were obtained by Belov et al. (1957).

In Figs. 1.5.2.1–1.5.2.7 , the diagrams of the magnetic unit cells of all 36 Bravais types are sketched in such a way that it is clear to which family the given cell belongs. All the cells of one family are displayed in one row. Such a row begins with the cell of the trivial magnetic lattice. All nontrivial cells of a family change into the trivial one of this family if R is replaced by E (to draw these diagrams we used those published by Opechowski & Guccione, 1965). Open and full circles are used to show the primed and unprimed translations. A line connecting two circles of the same type is an unprimed translation; a line connecting two circles of different types is a primed translation. The arrows in the trivial magnetic cell represent the primitive (primed or unprimed) translations for all the magnetic lattices of the family. The arrows in the nontrivial cells are primitive translations of the magnetic unit cell. The magnetic unit cell of a nontrivial magnetic lattice is generated by unprimed translations only. Its volume is twice the volume of the smallest cell generated by all (primed and unprimed) translations. The reason for this is that one of the primitive translations of the magnetic cell is twice a primitive primed translation. The crystallographic cell of many simple collinear or weakly non-collinear structures coincides with the smallest cell generated by the primed and unprimed translations. However, there are also magnetic structures with more complicated transformations from the crystallographic to the magnetic unit cell. The second line after each part of Figs. 1.5.2.1–1.5.2.7 gives, between braces, an extended vector basis of the magnetic translation group (Shubnikov & Koptsik, 1972). The first line gives two symbols for each Bravais type: the symbol to the right was introduced by Opechowski & Guccione (1965). The symbol to the left starts with a lower-case letter giving the crystal system followed by a capital letter giving the centring type of the cell defined by the unprimed translations (P: primitive; C, A, B: C-, A-, B-centred; I: body-centred; F: all-face-centred). The subscript, which appears for the nontrivial Bravais types, indicates the translations that are multiplied by time inversion R.

Figure 1.5.2.1 | top | pdf |

Magnetic lattices of the triclinic system.

Figure 1.5.2.2 | top | pdf |

Magnetic lattices of the monoclinic system (the y axis is the twofold axis).

Figure 1.5.2.3 | top | pdf |

Magnetic lattices of the orthorhombic system.

Figure 1.5.2.4 | top | pdf |

Magnetic lattices of the tetragonal system.

Figure 1.5.2.5 | top | pdf |

Magnetic lattices of the rhombohedral system.

Figure 1.5.2.6 | top | pdf |

Magnetic lattices of the hexagonal system.

Figure 1.5.2.7 | top | pdf |

Magnetic lattices of the cubic system.

Ferromagnetism is allowed only in trivial magnetic Bravais lattices . All nontrivial magnetic lattices represent antiferromagnetic order. There are only two magnetic sublattices in the simplest antiferromagnetic structures; one sublattice consists of the magnetic ions located in the black sites and the other of the ions located in the white sites. All the magnetic moments of one sublattice are oriented in one direction and those of the other sublattice in the opposite direction. However, antiferromagnetism is allowed also in trivial lattices if the (trivial) magnetic cell contains more than one magnetic ion. The magnetic point group must be nontrivial in this case. The situation is more complicated in case of strongly non-collinear structures. In such structures (triangle, 90° etc.), the magnetic lattice can differ from the crystallographic one despite the fact that none of the translations is multiplied by R. The magnetic elementary cell will possess three or four magnetic ions although the crystallographic cell possesses only one. An example of such a situation is shown in Fig. 1.5.1.3(c). More complicated structures in which the magnetic lattice is incommensurate with the crystallographic one also exist. We shall not discuss the problems of such systems in this chapter.

References

Belov, N. V., Neronova, N. N. & Smirnova, T. S. (1957). Shubnikov groups. (In Russian.) Kristallografiya, 2, 315–325. (English translation: Sov. Phys. Crystallogr. 2, 311–322.)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

Shubnikov, A. V. & Koptsik, V. A. (1972). Symmetry in science and art. (In Russian.) Moscow: Nauka. [English translation (1974): New York: Plenum.]Google Scholar

Zamorzaev, A. M. (1957). Generalization of Fedorov groups. (In Russian.) Kristallografiya, 2, 15–20. (English translation: Sov. Phys. Crystallogr. 2, 10–15.)Google Scholar

International Tables for Crystallography (2006). Vol. D. ch. 1.5, pp. 112-115