Triclinic system

Kopskyacute, V.; Litvin, D. B.

doi:10.1107/97809553602060000652

International
Tables for
Crystallography
Volume E
Subperiodic groups
Edited by V. Kopský and D. B. Litvin

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International Tables for Crystallography (2006). Vol. E. ch. 5.2, p. 402 | 1 | 2 |

Section 5.2.4.1. Triclinic system

V. Kopský^a ^* and D. B. Litvin^b

^a Department of Physics, University of the South Pacific, Suva, Fiji, and Institute of Physics, The Academy of Sciences of the Czech Republic, Na Slovance 2, PO Box 24, 180 40 Prague 8, Czech Republic, and ^bDepartment of Physics, Penn State Berks Campus, The Pennsylvania State University, PO Box 7009, Reading, PA 19610–6009, USA
Correspondence e-mail: kopsky@fzu.cz

5.2.4.1. Triclinic system

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The triclinic groups are trivial even from the viewpoint of scanning but it is non-trivial to express the vectors $[{\bf a}']$ , $[{\bf b}']$ and d in terms of vectors a, b, c and of Miller indices ( [hkl] ). Since the groups are related in the same way with respect to any given basis, we do not identify bases in the two tables. The specification Any admissible choice for the scanning group means that the vectors $[{\bf a}']$ , $[{\bf b}']$ have to be chosen as a basis of the translation group in the subspace defined by Miller indices and d should be the vector that completes the basis of the translation group in the whole space.

The scanned groups are identical with the scanning group for all orientations in the triclinic groups [P1] , $[C_{1}^{1}]$ (No. 1) and $[P{\bar 1}]$ , $[C_{i}^{1}]$ (No. 2). There is only one orientation in each orientation orbit. In the case of the group [P1] , $[C_{1}^{1}]$ (No. 1), there is one type of linear orbit consisting of planes generated by translations d from either one of the set and the respective layer symmetries are the trivial groups [p1] (L01). In the case of the group $[P{\bar 1}]$ , $[C_{i}^{1}]$ (No. 2), the orbit with a general location consists of a pair of planes, located symmetrically from a symmetry centre at distances $[\pm s]$ in the scanning direction d, which is then periodically repeated with periodicity d; the sectional layer symmetry of these planes is [p1] (L01). Furthermore, there are two linear orbits corresponding to positions $[0{\bf d}]$ and $[{{1}\over{2}} {\bf d}]$ , each of which consists of a periodic set of planes with periodicity d; the sectional symmetry in each of these cases is $[p{\bar 1}]$ (L02).

The triclinic scanning also applies to general orientation orbits of all space groups of higher symmetry than triclinic. If the space group $[\cal G]$ is noncentrosymmetric, then the number of orientations in the orientation orbit is the order [|G|] of the point group G and the linear orbits are described for each orientation as in the case of the group [P1] , $[C_{1}^{1}]$ (No. 1). If the space group $[\cal G]$ is centrosymmetric, then the number of orientations in the orientation orbit is [|G|/2] and the linear orbits are described for each orientation as in the case of the group $[P{\bar 1}]$ , $[C_{i}^{1}]$ (No. 2).

References

International Tables for Crystallography (2006). Vol. E. ch. 5.2, p. 402