International
Tables for Crystallography Volume A1 Symmetry relations between space groups Edited by Hans Wondratschek and Ulrich Müller © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. A1. ch. 2.1, p. 51
Section 2.1.5.3. Origin shift
Y. Billietc
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Each of the sublattices discussed in Section 2.1.4.3.2 is common to a conjugacy class or belongs to a normal subgroup of a given series. The subgroups in a conjugacy class differ by the positions of their conventional origins relative to the origin of the space group . To define the origin of the conventional unit cell of each subgroup in a conjugacy class, one, two or three integers, called u, v or w in these tables, are necessary. For a series of subgroups of index p, or there are p, or conjugate subgroups, respectively. The positions of their origins are defined by the p or or permitted values of u or u, v or u, v, w, respectively.
Example 2.1.5.3.1
The space group , , No. 112, has two series of maximal isomorphic subgroups . For one of them the lattice relations are , listed as for the transformation matrix. The index is . For each value of p there exist exactly conjugate subgroups with origins in the points , where the parameters u and v run independently: and .
In another type of series there is exactly one (normal) subgroup for each index p; the location of its origin is always chosen at the origin of and is thus not indicated as an origin shift.