Generators

Billiet, Y.

doi:10.1107/97809553602060000542

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. 2.1, p. 52 | 1 | 2 |

Section 2.1.5.4. Generators

Y. Billiet^c

2.1.5.4. Generators

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The generators of the p (or [p^2] or [p^3] ) conjugate isomorphic subgroups $[{\cal H}]$ are obtained from those of $[{\cal G}]$ by adding translational components. These components are determined by the parameters p (or q and r, if relevant) and u (and v and w, if relevant).

Example 2.1.5.4.1

Space group [P2_13] , No. 198.

In the series defined by the lattice relations $[p{\bf a},p{\bf b}, p{\bf c}]$ and the origin shift $[u,\,v,\,w]$ there exist exactly [p^3] conjugate subgroups for each value of p. The generators of each subgroup are defined by the parameter p and the triplet $[u,\,v,\,w]$ in combination with the generators (2), (3) and (5) of $[{\cal G}]$ . Consider the subgroup characterized by the basis $[7{\bf a}, 7{\bf b},7{\bf c}]$ and by the origin shift $[u=3,\ v=4,\ w=6]$ . One obtains from the generator (2) $[\overline{x}+{{1}\over{2}},\overline{y},z+{{1}\over{2}}]$ of $[{\cal G}]$ the corresponding generator of $[{\cal H}]$ by adding the translation vector $[({{p}\over{2}}-{{1}\over{2}}+2u){\bf a}+2v{\bf b}+({{p}\over{2}}- {{1}\over{2}}){\bf c}]$ to the translation vector $[{{1}\over{2}}{\bf a}+ {{1}\over{2}}{\bf c}]$ of the generator (2) of $[{\cal G}]$ and obtains $[{{19}\over{2}}\,{\bf a}+8{\bf b}+{{7}\over{2}}\,{\bf c}]$ , so that this generator of $[{\cal H}]$ is written $[\overline{x}+{{19}\over{2}},\overline{y}+8,z+{{7}\over{2}}]$ .

References

International Tables for Crystallography (2006). Vol. A1. ch. 2.1, p. 52