Wyckoff positions

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, p. 430 | 1 | 2 |

Section 3.1.1.6.5. Wyckoff positions

Ulrich Müller^a ^*

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

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

International Tables for Crystallography (2006). Vol. A1. ch. 3.1, p. 430