Origin shift

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. 51 | 1 | 2 |

Section 2.1.5.3. Origin shift

Y. Billiet^c

2.1.5.3. Origin shift

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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 $[{\cal G}]$ . 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, [p^2] or [p^3] there are p, [p^2] or [p^3] conjugate subgroups , respectively. The positions of their origins are defined by the p or [p^2] or [p^3] permitted values of u or u, v or u, v, w, respectively.

Example 2.1.5.3.1

The space group $[{\cal G}]$ , $[P\overline{4}2c]$ , No. 112, has two series of maximal isomorphic subgroups $[{\cal H}]$ . For one of them the lattice relations are $[[p^2]\, {\bf a}'=p{\bf a},{\bf b}'=p{\bf b}]$ , listed as $[p{\bf a},p{\bf b},{\bf c}]$ for the transformation matrix. The index is [p^2] . For each value of p there exist exactly [p^2] conjugate subgroups with origins in the points $[u,\,v,\,0]$ , where the parameters u and v run independently: $[0\leq u \,\lt\, p]$ and $[0\leq v \,\lt\, p]$ .

In another type of series there is exactly one (normal) subgroup $[{\cal H}]$ for each index p; the location of its origin is always chosen at the origin $[0,\,0,\,0]$ of $[{\cal G}]$ and is thus not indicated as an origin shift.

Example 2.1.5.3.2

Consider the space group [Pca2_1] , No. 29. Only one subgroup exists for each value of p in the series $[{\bf a}, {\bf b},p{\bf c}]$ . This is indicated in the tables by the statement `no conjugate subgroups'.

References

International Tables for Crystallography (2006). Vol. A1. ch. 2.1, p. 51