International
Tables for
Crystallography
Volume A1
Symmetry relations between space groups
Edited by Hans Wondratschek and Ulrich Müller

International Tables for Crystallography (2011). Vol. A1, ch. 1.7, pp. 66-68

Section 1.7.4. Relations of Wyckoff positions for a group–subgroup pair of space groups

Mois I. Aroyo,a* J. Manuel Perez-Mato,a Cesar Capillasa and Hans Wondratschekb

aDepartamento de Física de la Materia Condensada, Facultad de Ciencia y Tecnología, Universidad del País Vasco, Apartado 644, E-48080 Bilbao, Spain, and bInstitut für Kristallographie, Universität, D-76128 Karlsruhe, Germany
Correspondence e-mail:  wmpararm@lg.ehu.es

1.7.4. Relations of Wyckoff positions for a group–subgroup pair of space groups

| top | pdf |

Consider two group–subgroup-related space groups [{\cal G}>{\cal H}]. Atoms that are symmetrically equivalent under [{\cal G}], i.e. belong to the same orbit of [{\cal G}], may become non-equivalent under [{\cal H}], (i.e. the orbit splits) and/or their site symmetries may be reduced. The orbit relations induced by the symmetry reduction are the same for all orbits belonging to a Wyckoff position, so one can speak of Wyckoff-position relations or splitting of Wyckoff positions. Theoretical aspects of the relations of the Wyckoff positions for a group–subgroup pair of space groups [{\cal G}>{\cal H}] have been treated in detail by Wondratschek (1993[link]) (see also Section 1.5.3[link] ). A compilation of the Wyckoff-position splittings for all space groups and all their maximal subgroups is published as Part 3[link] of this volume. However, for certain applications it is easier to have the appropriate computer tools for the calculations of the Wyckoff-position splittings for [{\cal G} > {\cal H}]: for example, when [{\cal H}] is not a maximal subgroup of [{\cal G}], or when the space groups [{\cal G} > {\cal H}] are related by transformation matrices different from those listed in the tables of Part 3[link] . The program WYCKSPLIT (Kroumova, Perez-Mato & Aroyo, 1998[link]) calculates the Wyckoff-position splittings for any group–subgroup pair. In addition, the program provides further information on Wyckoff-position splittings that is not listed in Part 3[link] , namely the relations between the representatives of the orbit of [{\cal G}] and the corresponding representatives of the suborbits of [{\cal H}].

1.7.4.1. The program WYCKSPLIT

| top | pdf |

To simplify the notation, we assume in the following that the group [{\cal G}], its Wyckoff-position representatives and the points of the orbits are referred to the basis of the subgroup [{\cal H}].

  • (1) Splitting of the general position. Consider the group–subgroup chain of space groups [{\cal G}>{\cal H}] of an index [i]. The general-position orbits [{\cal{O}}_{{\cal G}}(X_0)] have unique splitting schemes: they are split into [i] suborbits [{\cal{O}}_{{\cal H}}(X_{0,j})] of the general position of the subgroup, i.e. they all are of the same multiplicity: [ {\cal{O}}_{{\cal G}}(X_0) = {\cal{O}}_{{\cal H}}(X_{0,1})\cup\ldots\cup{\cal{O}}_{{\cal H}}(X_{0,i}). \eqno(1.7.4.1)]This property is a direct corollary of the relation between the index [i] and the so-called reduction factors of the site-symmetry groups [{{\cal S}}_{{\cal G}}(X)] and [{{\cal S}}_{{\cal H}}(X)] of a point [X] in [{\cal G}] and [{\cal H}] (Wondratschek, 1993[link]; see also Section 1.5.3[link] ).

    The determination of the splitting of the general-position orbit [{\cal{O}}_{{\cal G}}(X_0)] is then reduced to the selection of the [i] points [(X_{0,j})] belonging to the [i] independent suborbits [{\cal{O}}_{{\cal H}}(X_{0,j})] of [{\cal H}], equation (1.7.4.1)[link]. Owing to the one-to-one mapping between the general-position points of [{\cal{O}}_{{\cal G}}(X_0)] and the elements [\ispecialfonts{\sfi g}] of [{\cal G}], the right cosets [\ispecialfonts{{\cal H}}{\sfi g}_j] of the decomposition of [{\cal G}] with respect to [{{\cal H}}] (cf. Definition 1.2.4.2.1[link] ) correspond to the suborbits [{\cal{O}}_{{\cal H}}(X_{0,j})]. In this way, the representatives of these cosets can be chosen as the [i] points [X_{0,j}] in the decomposition of [{\cal{O}}_{{\cal G}}(X_0)].

  • (2) Splitting of a special position. The calculation of the splitting of a special Wyckoff position [{\cal W}_{\cal G}] involves the following steps:

    • (i) the determination of the suborbits [{\cal{O}}_{{\cal H}}(X_j)] into which the special-Wyckoff-position orbit [{\cal{O}}_{{\cal G}}(X)] has split;

    • (ii) the assignment of the orbits [{\cal{O}}_{{\cal H}}(X_j)] to the Wyckoff positions [{\cal W}^{\,l}_{\cal H}] of [{{\cal H}}];

    • (iii) the determination of the correspondence between the points [X_{j}^m] of the suborbits [{\cal{O}}_{{\cal H}}(X_j)] and the representatives of [{\cal W}^{\,l}_{\cal H}].

    The direct determination of the suborbits [{\cal{O}}_{{\cal H}}(X_j)] is not an easy task. The restrictions on the site-symmetry groups [{{\cal S}}_{{\cal H}}(X_j)] which follow from the reduction-factor lemma (cf. Section 1.5.3 ) are helpful but in many cases not sufficient for the determination of the suborbits. The solution used in our approach is based on the general-position decomposition, equation (1.7.4.1)[link]. It is important to note that each of the suborbits of the general position gives exactly one suborbit [{\cal{O}}_{{\cal H}}(X_j)] when the variable parameters of [{\cal{O}}_{{\cal H}}(X_{0,j})] are substituted by the corresponding parameters (fixed or variable) of the special position. The assignment of the suborbits to the Wyckoff positions of [{{\cal H}}] is done by comparing the multiplicities of the orbits, the number of the variable parameters [the number of the variable parameters of [{\cal W}^{\, l}_{{\cal H}}] is equal to or greater than that of [{\cal{O}}_{{\cal H}}(X_j)]] and the values of the fixed parameters. If there is more than one Wyckoff position of [{{\cal H}}] satisfying these conditions, then the assignment is done by a direct comparison of the points of the suborbit [{\cal{O}}_{{\cal H}}(X_j)] with those of a special [{\cal W}^{\,l}_{{\cal H}}] orbit obtained by substitution of the variable parameters by arbitrary numbers. The determination of the explicit correspondences between the points of [{\cal{O}}_{{\cal H}}(X_j)] and the representatives of [{\cal W}^{\,l}_{{\cal H}}] is done by comparing the values of the fixed parameters and the variable-parameter relations in both sets.

The program WYCKSPLIT calculates the splitting of the Wyckoff positions for a group–subgroup pair [{\cal G} > {\cal H}] given the corresponding transformation relating the coordinate systems of [{\cal G}] and [{\cal H}].

Input to WYCKSPLIT:

The program needs as input the following information:

  • (i) The specification of the space-group types [{\cal G}] and [{\cal H}] by their IT A numbers.

  • (ii) The transformation matrix–column pair [({\bi P}, {\bi p})] that relates the basis of [{\cal G}] to that of [{\cal H}]. The user can input a specific transformation or follow a link to the IT A1 database for the maximal subgroups of [{\cal G}]. In the case of a non-maximal sub­group, the program SUBGROUPGRAPH provides the transformation matrix (or matrices) for a specified index of [{\cal H}] in [{\cal G}]. The transformations are checked for consistency with the default settings of [{\cal G}] and [{\cal H}] used by the program.

  • The Wyckoff positions [{\cal W}_{\cal G}] to be split can be selected from a list. In addition, it is possible to calculate the splitting of any orbit [O_{\cal G}(X)] specified by the coordinate triplet of one of its points.

Output of WYCKSPLIT:

  • (i) Splittings of the selected Wyckoff positions [{\cal W}_{\cal G}] into Wyckoff positions [{\cal W}^{\,l}_{\cal H}] of the subgroup, specified by their multiplicities and Wyckoff letters.

  • (ii) The correspondence between the representatives of the Wyckoff position and the representatives of its suborbits is presented in a table where the coordinate triplets of the representatives of [{\cal W}_{\cal G}] are referred to the bases of the group and of the subgroup.

WYCKSPLIT can treat group or subgroup data in unconventional settings if the transformation matrices to the corresponding conventional settings are given.

Example 1.7.4.1.1

To illustrate the calculation of the Wyckoff-position splitting we consider the group–subgroup pair [{P}{4_2/mnm}] (No. 136) > [{C}{mmm}] (No. 65) of index 2, see Fig. 1.7.4.1[link]. The relation between the conventional bases [({\bf a},{\bf b},{\bf c})] of the group and of the subgroup [({\bf a}',{\bf b}',{\bf c}')] is retrieved by the program MAXSUB and is given by [{\bf a}' = {\bf a}-{\bf b}], [{\bf b}' = {\bf a}+{\bf b}, {\bf c'}={\bf c}]. The general position of [{P}{4_2/mnm}] splits into two suborbits of the general position of [{C}{mmm}]: [{16k}\ {1}\ (x,y,z) \rightarrow {16r}\ {1}\ (x_1,y_1,z_1) \cup {16r}\ {1}\ (x_2,y_2,z_2).]This splitting is directly related to the coset decomposition of [{P}{4_2/mnm}] with respect to [{C}{mmm}]. As coset rep­resen­tatives, i.e. as points which determine the splitting of the general position, one can choose [X_{0,1}= (x_1,y_1,z_1)] and [X_{0,2}=] [ (x_2,y_2,z_2)=(y, x+\textstyle{1\over 2},z+\textstyle{1\over 2}))] (referred to the basis of the subgroup).

[Figure 1.7.4.1]

Figure 1.7.4.1 | top | pdf |

Sequence of calculations of WYCKSPLIT for the splitting of the Wyckoff positions [{2a}\ {m.mm}\ (0,0,0)] and [{4d}\ {\bar 4..}\ (\textstyle{1\over 2},0,\textstyle{3\over 4})] of [{P}{4_2/mnm}], No. 136, with respect to its subgroup [{C}{mmm}], No. 65, of index 2. [({\cal{O}}_{{\cal G}})_{{\cal H}}] are the orbits of [{P}{4_2/mnm}] in the basis of [{C}{mmm}].

The splitting of any special Wyckoff position is obtained from the splitting of the general position. The consecutive steps of the splittings of the special positions [{4d}\ {\bar{4}..}\ (\textstyle{1\over 2},0,\textstyle{3\over 4})] and [{2a}\ {m.mm}\ (0,0,0)] are shown in Fig. 1.7.4.1[link]. First it is necessary to transform the representatives of [{\cal W}_{\cal G}] to the basis of [{\cal H}], which gives the orbits [({\cal O}_{\cal G})_{{\cal H}}(0,0,0)] and [({\cal O}_{\cal G})_{\cal H}(\textstyle{1\over 4},\textstyle{1\over 4},\textstyle{3\over 4})]. The substitution of the values x = 0, y = 0, z = 0 in the coordinate triplets of the decomposed general position of [{\cal G}] (cf. the corresponding output of WYCKSPLIT) gives two suborbits of multiplicity 2 for the [2a] position: [{\cal O}^{2a,1}_{\cal H}(0,0,0)] and [{\cal O}^{2a,2}_{\cal H}(0,\textstyle{1\over 2},\textstyle{1\over 2})]. The assignment of the suborbits [{\cal O}^{2a,j}_{\cal H}] to the Wyckoff positions of [{\cal H}] (cf. Table 1.7.4.1[link]) is straightforward. Summarizing: the Wyckoff position [{2a}\ {m.mm}\ (0,0,0)] splits into two independent positions of [{C}{mmm}] with no site-symmetry reduction:[{2a}\ {m.mm}\ (0,0,0) \rightarrow {2a}\ {mmm}\ (0,0,0) \cup {2c}\ {mmm}\ (0,\textstyle{1\over 2},\textstyle{1\over 2}).]

Table 1.7.4.1| top | pdf |
Wyckoff positions of [{C}{mmm}] (No. 65) with multiplicities 2 and 8

Each Wyckoff position is specified by its multiplicity and Wyckoff letter, site symmetry and a coordinate triplet of a representative element.

Wyckoff multiplicity and letterSite symmetryRepresentative element
2d mmm [(0,0,\textstyle{1\over 2})]
2c mmm [(\textstyle{1\over 2},0,\textstyle{1\over 2})]
2b mmm [(\textstyle{1\over 2},0,0)]
2a mmm [(0,0,0)]
8q ..m [(x,y,\textstyle{1\over 2})]
8p ..m [(x,y,0)]
8o .m. [(x,0,z)]
8n m.. [(0,y,z)]
8m ..2 [(\textstyle{1\over 4},\textstyle{1\over 4},z)]

No splitting occurs for the case of the special [4d] position orbit: the result is one orbit of multiplicity 8, [{\cal O}^{4d,1}_{\cal H}(\textstyle{1\over 4},\textstyle{1\over 4},\textstyle{3\over 4})]. The assignment of [{\cal O}^{4d,1}_{\cal H}(\textstyle{1\over 4},\textstyle{1\over 4},\textstyle{3\over 4})] is also obvious: there are five Wyckoff positions of [{C}{mmm}] of multiplicity 8 but four of them are discarded as they have fixed parameters 0 or [\textstyle{1\over 2}] (Table 1.7.4.1[link]). The orbit [{\cal O}^{4d,1}_{\cal H}] belongs to the Wyckoff position [{8m}\ {..2}\ (\textstyle{1\over 4},\textstyle{1\over 4},z)].

As expected, the sum of the site-symmetry reduction factors equals the index of [{C}{mmm}] in [{P}{4_2/mnm}] for both cases (cf. Section 1.5.3[link] ). The loss of the fourfold inversion axis results in the appearance of an additional degree of freedom corresponding to the variable parameter of [{8m}\ {..2}\ (\textstyle{1\over 4},\textstyle{1\over 4},z)].

References

Kroumova, E., Perez-Mato, J. M. & Aroyo, M. I. (1998). WYCKSPLIT: a computer program for determination of the relations of Wyckoff positions for a group–subgroup pair. J. Appl. Cryst. 31, 646.
Wondratschek, H. (1993). Splitting of Wyckoff positions (orbits). Mineral. Petrol. 48, 87–96.








































to end of page
to top of page