Basis vectors

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. 429 | 1 | 2 |

Section 3.1.1.6.3. Basis vectors

Ulrich Müller^a ^*

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

3.1.1.6.3. Basis vectors

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The column `Axes' shows how the basis vectors of the unit cell of a subgroup result from the basis vectors a, b and c of the space group considered. This information is omitted if there is no change of basis vectors.

A formula such as ` $[q{\bf a}-r{\bf b},\,r{\bf a}\,+\,q{\bf b},\,{\bf c}]$ ' together with the restrictions ` $[p=q^2+r^2 = {\rm prime} =4n+1]$ ' and ` $[q=2n+1\geq 1]$ ; $[r=\pm 2n'\neq 0]$ ' means that for a given index p there exist several subgroups with different lattices depending on the values of the integer parameters q (odd) and r (even) within the limits of the restriction. In this example, the prime number p must be $[p \equiv 1]$ modulo 4 (i.e. 5, 13, 17, …); if it is, say, $[p=13 =3^2+(\pm 2)^2]$ , the values of q and r may be $[q=3, \,r=2]$ and $[q=3,\,r=-2]$ .¹

References

International Tables for Crystallography (2006). Vol. A1. ch. 3.1, p. 429