(Wyckoff) positions, site symmetries and crystallographic orbits

Wondratschek, H.

doi:10.1107/97809553602060000516

International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by Th. Hahn

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International Tables for Crystallography (2006). Vol. A. ch. 8.3, pp. 732-734

Section 8.3.2. (Wyckoff) positions, site symmetries and crystallographic orbits

H. Wondratschek^a ^*

^a Institut für Kristallographie, Universität, D-76128 Karlsruhe, Germany
Correspondence e-mail: hans.wondratschek@physik.uni-karlsruhe.de

8.3.2. (Wyckoff) positions, site symmetries and crystallographic orbits

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The concept of positions and their site symmetries is fundamental for the determination and description of crystal structures. Let, for instance, $[P\bar{1}]$ be the space group of a crystal structure with tetrahedral $[AX_{4}]$ and triangular $[BY_{3}]$ groups. Then the atoms A and B cannot be located at centres of inversion, as the symmetry of tetrahedra and triangles is incompatible with site symmetry $[\bar{1}]$ . If the space group is [P2/m] , again the points with site symmetry [2/m] cannot be the loci of A or B, but points with site symmetries 2, m or 1 can.

The relations between `site symmetry' and `positions' can be formulated in a rather general way.

Definition: The set of all symmetry operations of a space group $[{\cal G}]$ that leave a point X invariant forms a finite group, the site-symmetry group $[{\cal S}(X)]$ of X with respect to $[{\cal G}]$ .²

With regard to the symmetry operations of a space group $[{\cal G}]$ , two kinds of points are to be distinguished. A point X is called a point of general position with respect to a space group $[{\cal G}]$ if there is no symmetry operation of $[{\cal G}]$ (apart from the identity operation) that leaves X fixed, i.e. if $[{\cal S}(X) = {\cal I}]$ . A point X is called a point of special position with respect to a space group $[{\cal G}]$ if there is at least one other symmetry operation of $[{\cal G}]$ , in addition to the identity operation, that leaves X fixed, i.e. if $[{\cal S}(X) \gt {\cal I}]$ .

The subdivision of the set of all points into two classes, those of general and those of special position with respect to a space group $[{\cal G}]$ , constitutes only a very coarse classification. A finer classification is obtained as follows.

Definition: A Wyckoff position $[\hbox{\sf W}_{{\cal G}}]$ (for short, position; in German, Punktlage) consists of all points X for which the site-symmetry groups $[{\cal S}(X)]$ are conjugate subgroups³ of $[{\cal G}]$ .

For practical purposes, each Wyckoff position of a space group is labelled by a letter which is called the Wyckoff letter (Wyckoff notation in earlier editions of these Tables). Wyckoff positions without variable parameters (e.g. $[0, 0, 0; 0, 0, {1 \over 2}; \ldots]$ ) and with variable parameters (e.g. $[x, y, z; x, 0, {1 \over 4}; \ldots]$ ) have to be distinguished.

The number of different Wyckoff positions of each space group is finite, the maximal numbers being nine for plane groups (realized in p2mm) and 27 for space groups (realized in Pmmm).

A finer classification of the points of $[E^{n}]$ with respect to $[{\cal G}]$ , which always results in an infinite number of classes, is the subdivision of all points into sets of symmetrically equivalent points. In the following, these sets will be called crystallographic orbits according to the following definition.

Definition: The set of all points that are symmetrically equivalent to a point X with respect to a space group $[{\cal G}]$ is called the crystallographic orbit of X with respect to $[{\cal G}]$ .

Example

Described in a conventional coordinate system, the crystallographic orbit of a point X of general position with respect to a plane group p2 consists of the points [x, y] ; $[\bar{x},\bar{y}]$ ; [x + 1, y] ; $[\bar{x} + 1, \bar{y}]$ ; [x, y + 1] ; $[\bar{x},\bar{y} + 1]$ ; [x - 1, y] ; $[\bar{x} - 1,\bar{y}]$ ; [x, y - 1] ; $[\bar{x},\bar{y} - 1]$ ; [x + 1,y + 1] ; $[\ldots]$ etc.

Crystallographic orbits are infinite sets of points due to the infinite number of translations in each space group. Any one of its points may represent the whole crystallographic orbit, i.e. may be the generating point X of a crystallographic orbit.⁴

Because the site-symmetry groups of different points of the same crystallographic orbit are conjugate subgroups of $[{\cal G}]$ , a crystallographic orbit consists either of points of general position or of points of special position only. Therefore, one can speak of `crystallographic orbits of general position' or general crystallographic orbits and of `crystallographic orbits of special position' or special crystallographic orbits with respect to $[{\cal G}]$ . Because all points of a crystallographic orbit belong to the same Wyckoff position of $[{\cal G}]$ , one also can speak of Wyckoff positions of crystallographic orbits.⁵

The points of each general crystallographic orbit of a space group $[{\cal G}]$ are in a one-to-one correspondence with the symmetry operations of $[{\cal G}]$ . Starting with the generating point X (to which the identity operation corresponds), to each point $[\tilde{X}]$ of the crystallographic orbit belongs exactly one symmetry operation $[\hbox{\sf W}]$ of $[{\cal G}]$ such that $[\tilde{X}]$ is the image of X under $[\hbox{\sf W}]$ . This one-to-one correspondence is the reason why the `coordinates' listed for the general position in the space-group tables may be interpreted in two different ways, either as the coordinates of the image points of X under $[{\cal G}]$ or as a short-hand notation for the pairs (W, w) of the symmetry operations $[\hbox{\sf W}]$ of $[{\cal G}]$ ; cf. Sections 8.1.6 and 11.1.1 . Such a one-to-one correspondence does not exist for the special crystallographic orbits, where each point corresponds to a complete coset of a left coset decomposition of $[{\cal G}]$ with respect to the site-symmetry group $[{\cal S}(X)]$ of X. Thus, the data listed for the special positions are to be understood only as the coordinates of the image points of X under $[{\cal G}]$ .

Space groups with no special crystallographic orbits are called fixed-point-free space groups. The following types of fixed-point-free space groups occur: p1 and pg in $[E^{2}]$ ; $[P_{1} \equiv C_{1}^{1}]$ (No. 1), $[P2_{1} \equiv C_{2}^{2}]$ (No. 4), $[Pc \equiv C_{s}^{2}]$ (No. 7), $[Cc \equiv C_{s}^{4}]$ (No. 9), $[P2_{1}2_{1}2_{1} \equiv D_{2}^{4}]$ (No. 19), $[Pca2_{1} \equiv C_{2v}^{5}]$ (No. 29), $[Pna2_{1} \equiv C_{2v}^{9}]$ (No. 33), $[P4_{1} \equiv C_{4}^{2}]$ (No. 76), $[P4_{3} \equiv C_{4}^{4}]$ (No. 78), $[P3_{1} \equiv C_{3}^{2}]$ (No. 144), $[P3_{2} \equiv C_{3}^{3}]$ (No. 145), $[P6_{1} \equiv C_{6}^{2}]$ (No. 169) and $[P6_{5} \equiv C_{6}^{3}]$ (No. 170) in $[E^{3}]$ .

Though the classification of the points of space $[E^{n}]$ into Wyckoff positions $[\hbox{\sf W}_{{\cal G}}]$ of a space group $[{\cal G}]$ is unique, the labelling of the Wyckoff positions by Wyckoff letters (Wyckoff notation) is not.

Example

In a space group $[P\bar{1}]$ there are eight classes of centres of inversion $[\bar{1}]$ , represented in the space-group tables by [0,0,0] ; $[0,0,{1 \over 2}]$ ; $[0,{1 \over 2},0; \ldots; {1 \over 2},{1 \over 2},{1 \over 2}]$ . The site-symmetry groups $[\{\hbox{\sf I}, \bar{1}\}]$ within each class are `symmetrically equivalent', i.e. they are conjugate subgroups of $[P\bar{1}]$ . The groups $[\{\hbox{\sf I}, \bar{1}\}]$ of different classes, however, are not `symmetrically equivalent' with respect to $[P\bar{1}]$ . Each class is labelled by one of the Wyckoff letters $[a,b,\ldots, h]$ . This letter depends on the choice of origin and on the choice of coordinate axes. Cyclic permutation of the labels of the basis vectors a, b, c, for instance, induces a cyclic permutation of Wyckoff positions b–c–d and e–f–g; origin shift from 0, 0, 0 to the point $[{1 \over 2},0,0]$ results in an exchange of Wyckoff letters in the pairs a–d, b–f, c–e and g–h. Even if the coordinate axes are determined by some extra condition, e.g. $[a \leq b \leq c]$ , there exist no rules for fixing the origin in $[P\bar{1}]$ when describing a crystal structure. The eight classes of centres of inversion of $[P\bar{1}]$ are well established but none of them is inherently distinguished from the others.

The example shows that the different Wyckoff positions of a space group $[{\cal G}]$ may permute under an isomorphic mapping of $[{\cal G}]$ onto itself, i.e. under an automorphism of $[{\cal G}]$ . Accordingly, it is useful to collect into one set all those Wyckoff positions of a space group $[{\cal G}]$ that may be permuted by automorphisms of $[{\cal G}]$ . These sets are called `Wyckoff sets'. The Wyckoff letters belonging to the different Wyckoff positions of the same Wyckoff set are listed by Koch & Fischer (1975); changes in Wyckoff letters caused by changes of the coordinate system have been listed by Boyle & Lawrenson (1973, 1978).

To introduce `Wyckoff sets' more formally, it is advantageous to use the concept of normalizers; cf. Ledermann (1976). The affine normalizer $[{\cal N}]$ ⁶ of a space group $[{\cal G}]$ in the group $[{\cal A}]$ of all affine mappings is the set of those affine mappings which map $[{\cal G}]$ onto itself. The space group $[{\cal G}]$ is a normal subgroup of $[{\cal N}]$ , $[{\cal N}]$ itself is a subgroup of $[{\cal A}]$ . The mappings of $[{\cal N}]$ which are not symmetry operations of $[{\cal G}]$ may transfer one Wyckoff position of $[{\cal G}]$ onto another Wyckoff position.

Definition: Let $[{\cal N}]$ be the normalizer of a space group $[{\cal G}]$ in the group of all affine mappings. A Wyckoff set with respect to $[{\cal G}]$ consists of all points X for which the site-symmetry groups are conjugate subgroups of $[{\cal N}]$ .

The difference between Wyckoff positions and Wyckoff sets of $[{\cal G}]$ may be explained as follows. Any Wyckoff position of $[{\cal G}]$ is transformed onto itself by all elements of $[{\cal G}]$ , but not necessarily by the elements of the (larger) group $[{\cal N}]$ . Any Wyckoff set, however, is transformed onto itself even by those elements of $[{\cal N}]$ which are not contained in $[{\cal G}]$ .

Remark: A Wyckoff set of $[{\cal G}]$ is a set of points. Obviously, with each point X it contains all points of the crystallographic orbit of X and all points of the Wyckoff position of X. Accordingly, one can speak not only of `Wyckoff sets of points', but also of `Wyckoff sets of crystallographic orbits' and `Wyckoff sets of Wyckoff positions' of $[{\cal G}]$ . Wyckoff sets of crystallographic orbits have been used in the definition of lattice complexes (Gitterkomplexe), under the name Konfigurationslage, by Fischer & Koch (1974); cf. Part 14 .

The concepts `crystallographic orbit', `Wyckoff position' and `Wyckoff set' have so far been defined for individual space groups only. It is no problem, but is of little practical interest, to transfer the concept of `crystallographic orbit' to space-group types. It would be, on the other hand, of great interest to transfer `Wyckoff positions' from individual space groups to space-group types. As mentioned above, however, such a step is not unique. For this reason, the concept of `Wyckoff set' has been introduced to replace `Wyckoff positions'. Different space groups of the same space-group type have corresponding Wyckoff sets, and one can define `types of Wyckoff sets' (consisting of individual Wyckoff sets) in the same way that `types of space groups' (consisting of individual space groups) were defined in Section 8.2.2 .

Definition: Let the space groups $[{\cal G}]$ and $[{\cal G}']$ belong to the same space-group type. The Wyckoff sets $[\hbox{\sf K}]$ of $[{\cal G}]$ and $[\hbox{\sf K}']$ of $[{\cal G}']$ belong to the same type of Wyckoff sets if the affine mappings which transform $[{\cal G}]$ onto $[{\cal G}']$ also transform $[\hbox{\sf K}]$ onto $[\hbox{\sf K}']$ .

Types of Wyckoff sets have been used by Fischer & Koch (1974), under the name Klasse von Konfigurationslagen, when defining lattice complexes. There are 1128 types of Wyckoff sets of the 219 (affine) space-group types and 51 types of Wyckoff sets of the 17 plane-group types [Koch & Fischer (1975) and Chapter 14.1 ].

References

Boyle, L. L. & Lawrenson, J. E. (1973). The origin dependence of the Wyckoff site description of a crystal structure. Acta Cryst. A29, 353–357.Google Scholar

Boyle, L. L. & Lawrenson, J. E. (1978). The dependence of the Wyckoff site description of a crystal structure on the labelling of the axes. Comm. R. Soc. Edinburgh (Phys. Sci.), 1, 169–175.Google Scholar

Fischer, W. & Koch, E. (1974). Eine Definition des Begriffs `Gitterkomplex'. Z. Kristallogr. 139, 268–278.Google Scholar

Koch, E. & Fischer, W. (1975). Automorphismengruppen von Raumgruppen und die Zuordnung von Punktlagen zu Konfigurationslagen. Acta Cryst. A31, 88–95.Google Scholar

Ledermann, W. (1976). Introduction to group theory. London: Longman.Google Scholar

International Tables for Crystallography (2006). Vol. A. ch. 8.3, pp. 732-734