Coordinates

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 3.1.1.6.4. Coordinates

Ulrich Müller^a ^*

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

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.

References

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