Information for every subgroup

Müller, U.

doi:10.1107/97809553602060000547

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

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International Tables for Crystallography (2006). Vol. A1. ch. 3.1, pp. 429-430 | 1 | 2 |

Section 3.1.1.6. Information for every subgroup

Ulrich Müller^a ^*

^a Fachbereich Chemie, Philipps-Universität, D-35032 Marburg, Germany
Correspondence e-mail: mueller@chemie.uni-marburg.de

3.1.1.6. Information for every subgroup

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3.1.1.6.1. Index

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The entry for every subgroup begins with the index in brackets, for example [2] or [p] or [ [p^2] ] (p = prime number).

The index for any of the infinite number of maximal isomorphic subgroups must be either a prime number p, or, in certain cases of tetragonal, trigonal and hexagonal space groups, a square of a prime number [p^2] ; for isomorphic subgroups of cubic space groups the index may only be the cube of a prime number [p^3] . In many instances only certain prime numbers are allowed (Bertaut & Billiet, 1979; Billiet & Bertaut, 2005; Müller & Brelle, 1995). If restrictions exist, the prime numbers allowed are given under the axes transformations by formulae such as ` $[p = {\rm prime} = 3n-1]$ '.

3.1.1.6.2. Subgroup symbol

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The index is followed by the Hermann–Mauguin symbol (short symbol) and the space-group number of the subgroup. If a nonconventional setting has been chosen, then the space-group symbol of the conventional setting is also mentioned in the following line after the symbol $[\widehat{=}]$ .

In some cases of nonconventional settings, the space-group symbol does not show uniquely in which manner it deviates from the conventional setting. For example, the nonconventional setting [P22_12] of the space group [P222_1] can result from cyclic exchange of the axes, $[({\bf b},\,{\bf c},\,{\bf a})]$ or by interchange of b with c $[({\bf a},\,-{\bf c},\,{\bf b})]$ . As a consequence, the relations between the Wyckoff positions can be different. In such cases, cyclic exchange has always been chosen.

3.1.1.6.3. Basis vectors

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The column `Axes' shows how the basis vectors of the unit cell of a subgroup result from the basis vectors a, b and c of the space group considered. This information is omitted if there is no change of basis vectors.

A formula such as ` $[q{\bf a}-r{\bf b},\,r{\bf a}\,+\,q{\bf b},\,{\bf c}]$ ' together with the restrictions ` $[p=q^2+r^2 = {\rm prime} =4n+1]$ ' and ` $[q=2n+1\geq 1]$ ; $[r=\pm 2n'\neq 0]$ ' means that for a given index p there exist several subgroups with different lattices depending on the values of the integer parameters q (odd) and r (even) within the limits of the restriction. In this example, the prime number p must be $[p \equiv 1]$ modulo 4 (i.e. 5, 13, 17, …); if it is, say, $[p=13 =3^2+(\pm 2)^2]$ , the values of q and r may be $[q=3, \,r=2]$ and $[q=3,\,r=-2]$ .¹

3.1.1.6.4. Coordinates

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The column `Coordinates' shows how the atomic coordinates of the subgroups are calculated from the coordinates x, y and z of the starting unit cell. This includes coordinate shifts whenever a shift of the origin is required (cf. Section 3.1.3). If the cell of the subgroup is enlarged, the coordinate triplet is followed by a semicolon; then follow fractional numbers in parentheses. This means that in addition to the coordinates given before the semicolon, further coordinates have to be considered; they result from adding the numbers in the parentheses. However, if the subgroup has a centring, the values to be added due to this centring are not mentioned. If no transformation of coordinates is necessary, the entry is omitted.

Example 3.1.1.6.1

The entry $[\displaylines{{\quad\textstyle{{1}\over{3}}} x\,+\,{\textstyle{{1}\over{4}}},y+{\textstyle{{1}\over{4}}},z ;\, \pm({\textstyle{{1}\over{3}}},0,0)\hfill\cr}]$ means: starting from the coordinates of, say, 0.63, 0.12, 0.0, sites with the following coordinates result in the subgroup: $[\displaylines{\quad0.46, 0.37, 0.0\semi\quad 0.793333, 0.37, 0.0\semi\hfill\cr\quad 0.1266667, 0.37, 0.0.\hfill\cr}]$

Example 3.1.1.6.2

The entry of an I-centred subgroup $[\displaylines{\quad{\textstyle{{1}\over{2}}} x,\, {\textstyle{{1}\over{2}}}y,\, {\textstyle{{1}\over{2}}}z; \, +({\textstyle{{1}\over{2}}},0,0);\, +(0,{\textstyle{{1}\over{2}}},0); \, +(0,0,{\textstyle{{1}\over{2}}})\hfill}]$ means: starting from the coordinates of, say, 0.08, 0.14, 0.20, sites with the following coordinates result in the subgroup: $[\displaylines{\quad0.04, 0.07, 0.10\semi\quad 0.54, 0.07, 0.10\semi\hfill\cr\quad0.04, 0.57, 0.10\semi\quad 0.04, 0.07, 0.60\semi\hfill\cr}]$ in addition, there are all coordinates with $[+({{1}\over{2}},{{1}\over{2}},{{1}\over{2}})]$ due to the I-centring: $[\displaylines{\quad0.54, 0.57, 0.60\semi\quad 0.04, 0.57, 0.60\semi\hfill\cr\quad 0.54, 0.07, 0.60\semi\quad 0.54, 0.57, 0.10.\hfill}]$

For the infinite series of isomorphic subgroups, coordinate formulae are, for example, in the form $[x,\,y,\,{{1}\over{p}}z]$ ; $[+(0,0,{{u}\over{p}})]$ with $[u=1,\dots, p-1]$ . Then there are p coordinate values running from $[x,\,y,\,{{1}\over{p}}z]$ to $[x,\,y,\,{{1}\over{p}}z+{{p-1}\over{p}}]$ .

Example 3.1.1.6.3

For a subgroup with index [p^2=25] ( [p=5] ) the entry $[\displaylines{\quad {\textstyle{{1}\over{p}}}x,{\textstyle{{1}\over{p}}}y,z\semi\, +({\textstyle{{u}\over{p}}},{\textstyle{{v}\over{p}}},0)\semi\, u,v=1,\dots, p-1\hfill}]$ means: starting from the coordinates of, say, 0.10, 0.35, 0.0, sites with the following coordinates result in the subgroup: $[\displaylines{\quad 0.02, 0.07, 0.0\semi\quad 0.02, 0.27, 0.0\semi\quad 0.02, 0.47, 0.0\semi\hfill\cr\quad 0.02, 0.67, 0.0\semi\quad 0.02, 0.87, 0.0\semi\hfill\cr\quad0.22, 0.07, 0.0\semi\quad 0.22, 0.27, 0.0\semi\quad 0.22, 0.47, 0.0\semi\hfill\cr\quad0.22, 0.67, 0.0\semi\quad 0.22, 0.87, 0.0\semi\hfill\cr\quad0.42, 0.07, 0.0\semi\quad 0.42, 0.27, 0.0\semi\quad 0.42, 0.47, 0.0\semi\hfill\cr\quad0.42, 0.67, 0.0\semi\quad 0.42, 0.87, 0.0\semi\hfill\cr\quad0.62, 0.07, 0.0\semi\quad 0.62, 0.27, 0.0\semi\quad 0.62, 0.47, 0.0\semi\hfill\cr\quad0.62, 0.67, 0.0\semi\quad 0.62, 0.87, 0.0\semi\hfill\cr\quad0.82, 0.07, 0.0\semi\quad 0.82, 0.27, 0.0\semi\quad 0.82, 0.47, 0.0\semi \hfill\cr\quad0.82, 0.67, 0.0\semi\quad 0.82, 0.87, 0.0.\hfill}]$

If Volume A allows two choices for the origin, coordinate transformations for both are listed in separate columns with the headings `origin 1' and `origin 2'. If two origin choices are allowed for both the group as well as the subgroup, then it is understood that the origin choices of the group and the subgroup are the same (either origin choice 1 for both groups or origin choice 2 for both). If the space group has only one origin choice, but the subgroup has two choices, the coordinate transformations are given for both choices on separate lines.

3.1.1.6.5. Wyckoff positions

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The columns under the heading `Wyckoff positions' contain the Wyckoff symbols of all sites of the subgroups that result therefrom. They are given in the same sequence as in the top line(s). If the symbols at the top run over more than one line, then the symbols for the subgroups take a corresponding number of lines.

When an orbit splits into several independent orbits, the corresponding Wyckoff symbols are separated by semicolons, i.e. [1b;4h;4k] . An entry such as $[3\times 8j]$ means that a splitting into three orbits takes place, all of which are of the same kind 8j; they differ in the values of their free parameters.

For the infinite series of isomorphic subgroups general formulae are given. They allow the calculation of the Wyckoff-position relations for any index in a simple manner.

Example 3.1.1.6.4

The entry $[{{p(p-1)}\over{2}}\times 24k]$ means that for a given prime number p, say [p=5] , there are $[{{5(5-1)}\over{2}} = 10]$ orbits of the kind 24k.

In some cases of splittings, there is not enough space to enter all Wyckoff symbols on one line; this requires them to be listed one below the other over two or more lines. Whenever a Wyckoff symbol is followed by a semicolon, another symbol follows.

Example 3.1.1.6.5

The last subgroup listed for space group $[I{\bar 4}m2]$ , No. 119, is $[I{\bar 4}m2]$ with basis vectors $[p{\bf a},\,p{\bf b},\,{\bf c}]$ . The entry for the Wyckoff position 2a is: $[\left | \matrix { 2a;{{p-1}\over{2}}\,\times 8g; \cr \phantom{2a;}{{p-1}\over{2}}\,\times 8i; \cr {{(p-1)(p-3)}\over{8}}\,\times 16j } \right |]$ If [p=5] , it shows the splitting of an orbit of position 2a into one orbit 2a, two $[({{5-1}\over{2}}=2)]$ orbits 8g, two orbits 8i and one $[(\,{{(5-1)(5-3)}\over{8}}=1\,)]$ orbit 16j.

Sometimes a Wyckoff label is followed by another Wyckoff label in parentheses together with a footnote marker. In this case, the Wyckoff label in parentheses is to be taken for the cases specified in the footnote.

Example 3.1.1.6.6

The entry [2c(d^*)] together with the footnote $[\,^*\,p=4n-1]$ means that the Wyckoff position is 2c, but it is 2d if the index is $[p\equiv 3]$ modulo 4 (i.e. $[p=3,\,7,\,11,\,\dots]$ ).

The Wyckoff positions of an isomorphic subgroup of a space group with two choices for the origin are only identical for the two choices if certain origin shifts are taken into account. Since origin shifts have been avoided as far as possible, in some cases some Wyckoff positions differ for the two origin choices.

Example 3.1.1.6.7

The isomorphic subgroups of the space group $[P\,4_2/\!n]$ , No. 86, with cell enlargements $[{\bf a},\,{\bf b},\,p{\bf c}]$ and [p=4n-1] result in identical Wyckoff positions for the two origin choices only if there is no origin shift for choice 1, but an origin shift of $[0,0,{{1}\over{2}}]$ for choice 2. The origin shift for choice 2 has been avoided, but as a consequence some of the Wyckoff labels differ for the two choices. For the Wyckoff position 2a of the space group, the entry for these isomorphic subgroups is $[2a(b^\dagger);\,{{p-1}\over{2}}\,\times 4f]$ . The footnote reads ` $[\,^\dagger]$ origin 2 and [p=4n-1] '. Therefore, 2a is (aside from 4f) the resulting Wyckoff position for origin choice 1 and any value of p; for origin choice 2 it is also 2a if [p=4n+1] , but it is 2b if (the permitted values for p are $[p=4n\pm 1]$ ).

Warning: The listed Wyckoff positions of the subgroups apply only to the transformations given in the column `Coordinates'. If other cell transformations or origin shifts are used, this may result in an interchange of Wyckoff positions within each Wyckoff set of the subgroup.

References

Bertaut, E. F. & Billiet, Y. (1979). On equivalent subgroups and supergroups of the space groups. Acta Cryst. A35, 733–745.Google Scholar

Billiet, Y. & Bertaut, E. F. (2005). Isomorphic subgroups of space groups. International Tables for Crystallography, Vol. A, Space-group symmetry, edited by Th. Hahn, Part 13. Heidelberg: Springer.Google Scholar

Müller, U. & Brelle, A. (1995). Über isomorphe Untergruppen von Raumgruppen der Kristallklassen , $[{\bar 4}]$ , , , $[{\bar 3}]$ , , $[{\bar 6}]$ und . Acta Cryst. A51, 300–304.Google Scholar

International Tables for Crystallography (2006). Vol. A1. ch. 3.1, pp. 429-430