Nonconventional settings of orthorhombic space groups

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. 431-432 | 1 | 2 |

Section 3.1.4. Nonconventional settings of orthorhombic space groups

Ulrich Müller^a ^*

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

3.1.4. Nonconventional settings of orthorhombic space groups

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Orthorhombic space groups can have as many as six different settings, as listed in Chapter 4.3 of Volume A. They result from the interchange of the axes $[{\bf a}, {\bf b}, {\bf c}]$ in the following ways:

Cyclic exchange: $[{\bf b}\,{\bf c}\,{\bf a}]$ or $[{\bf c}\,{\bf a}\,{\bf b}]$ .
Exchange of two axes, combined with the reversal of the direction of one axis in order to keep a right-handed coordinate system: $[\displaylines{{\bf b}\,{\bf a}\,{\bar {\bf c}} \quad {\rm or} \quad {\bf b}\,{\bar {\bf a}}\,{\bf c} \quad {\rm or} \quad {\bar {\bf b}}\, {\bf a}\,{\bf c}\semi\cr{\bf c}\,{\bf b}\,{\bar {\bf a}} \quad {\rm or} \quad {\bf c}\,{\bar {\bf b}}\,{\bf a} \quad {\rm or} \quad {\bar {\bf c}}\, {\bf b}\,{\bf a}\semi\cr{\bf a}\,{\bf c}\,{\bar {\bf b}} \quad {\rm or} \quad {\bf a}\,{\bar {\bf c}}\,{\bf b} \quad {\rm or} \quad {\bar {\bf a}}\, {\bf c}\,{\bf b}.}]$

The exchange has two consequences for a Hermann–Mauguin symbol:

(1) the symmetry operations given in the symbol interchange their positions in the symbol;
(2) the labels of the glide directions and of the centrings are interchanged.

In the same way, the sequences and the labels and values of the coordinate triplets have to be interchanged.

Example 3.1.4.1

Take space group Pbcm, No. 57 (full symbol $[P2/b\,2_1/c2_1/m]$ ), and its Wyckoff position 4c $[(x,\,{{1}\over{4}},\,0)]$ . The positions in the symbol change as given by the arrows, and simultaneously the labels change: [Scheme scheme2]

The notation b c a means: the former b axis is now in the position of the a axis etc. or: convert b to a, c to b, and a to c.

The corresponding interchanges of positions and labels for all possible nonconventional settings are listed at the end of the table of each orthorhombic space group. They have to be applied to all subgroups.

Example 3.1.4.2

Consider the nonconventional setting Pcam of Pbcm. The entry at the bottom of the page for space group Pbcm, No. 57, shows the necessary interchanges for the setting Pcam: $[a \,\,{{\lower4pt\hbox{$\rightarrow$}}\atop{\raise4pt\hbox{$\leftarrow$}}} \,\,b]$ , $[{\bf a}\,\, {{\lower4pt\hbox{$\rightarrow$}}\atop{\raise4pt\hbox{$\leftarrow$}}}\,\,-{\bf b}]$ , and $[x\,\,{{\lower4pt\hbox{$\rightarrow$}}\atop{\raise4pt\hbox{$\leftarrow$}}}\,\, -y]$ .

For the subgroup Pbna (last entry in the block of klassengleiche non-isomorphic subgroups) this means: Pbna has to be replaced by Pnab, the axes conversion $[2{\bf a},\,{\bf b},\,{\bf c}]$ has to be replaced by $[{\bf a},-2{\bf b},\,{\bf c}]$ and the coordinate transformation $[{{1}\over{2}} x+{{1}\over{4}},\,y,\,z;\,+({{1}\over{2}},0,0)]$ has to be replaced by $[x,-{{1}\over{2}} y-{{1}\over{4}},\,z;\, +(0,-{{1}\over{2}},0)]$ .

Pbna and Pnab are nonconventional settings of Pbcn, No. 60.

The interchange of the axes does not affect the Wyckoff labels, just the corresponding coordinates.

Example 3.1.4.3

The Wyckoff position 4c $[(x,\,{{1}\over{4}},\,0)]$ of Pbcm, No. 57, retains its label for any of the other settings of this space group. In the setting Pbma, this Wyckoff position is still 4c and has the coordinates $[{{1}\over{4}},\,0,\,z]$ . In this case, no ambiguity arises because the different settings of space group Pbcm all have different Hermann–Mauguin symbols that uniquely show how the axes have to be interchanged (Pmca, Pbma, Pcam, Pmab and Pcmb).

The interchange of the axes must also be performed for those subgroups that have equivalent directions and where the Hermann–Mauguin symbol does not uniquely show the kind of setting. Otherwise, the wrong Wyckoff positions can result.

Example 3.1.4.4

Space group Cmmm, No. 65, has two klassengleiche subgroups of type Immm, No. 71, with doubled c axis. In the nonconventional setting Bmmm of Cmmm, the same subgroups Immm result from a doubling of the b axis. In the conventional setting of Immm, the Wyckoff positions 4e, 4g and 4i represent orbits with the coordinates $[(x,\,0,\,0)]$ , $[(0,\,y,\,0)\,]$ and $[(0,\,0,\,z)]$ , respectively. In the space group Cmmm, the position 4k corresponds to $[(0,\,0,\,z)]$ and upon transition to either of the subgroups Immm it splits to $[2\times 4i]$ .

If Bmmm is obtained from Cmmm by cyclic exchange of the axes ( $[{\bf a} \leftarrow {\bf b} \leftarrow {\bf c} \leftarrow {\bf a}]$ ), its Wyckoff position 4k obtains the coordinates $[(0,\,y,\,0)]$ . Upon doubling of b and transition to Immm, 4k will split to two orbits with the coordinates $[(0,\,{{1}\over{2}}y,\,0)]$ and $[(0,\,{{1}\over{2}} y\,+\,{{1}\over{2}},\,0)]$ . These are two orbits 4i of Immm, but this is only correct if the axes of [Immm] have also been interchanged in the same way. If the interchange of axes has not been performed in the subgroup Immm in the assumption that in Immm all axes are equivalent anyway, wrong results will be obtained. That is, Immm also has to be used in a nonconventional setting, although this is not apparent from the Hermann–Mauguin symbol. Of course, the Wyckoff symbols can then be relabelled so that they correspond to the conventional listings of Volume A ( $[4i \rightarrow 4g]$ etc.). It is recommended that this return to the conventional setting of Immm is performed, because using the label 4i for $[(0,\, y,\,0)]$ in Immm is likely to cause confusion if the nonconventional setting is not explicitly stressed.

References

International Tables for Crystallography (2006). Vol. A1. ch. 3.1, pp. 431-432