Orthogonal scanning, standard tables

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. 403 | 1 | 2 |

Section 5.2.4.3.1. Orthogonal scanning, standard tables

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

5.2.4.3.1. Orthogonal scanning, standard tables

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Orientation orbits [(001)] , [(100)] and [(010)] : These three orientation orbits represent all orbits with fixed parameters in the orthorhombic system. Each of these consists of a single orientation. Hence the scanning group $[\cal H]$ for each of these orientations and for any orthorhombic group $[\cal G]$ coincides with the group $[{\cal G} = {\cal H}]$ itself. The Hermann–Mauguin symbols of the scanning groups are, however, generally different for the three orientations because they refer to different bases $[{\bf a}']$ , $[{\bf b}']$ , $[{\bf c}' = {\bf d}]$ . For the orientation (001) they always coincide with the Hermann–Mauguin symbol used in IT A.

The scanning groups for groups of geometric classes [222] ( $[D_{2}]$ ) and [mmm] ( $[D_{2h}]$ ) are not only the same (identical with the scanned group) for all three orientations, but in a few cases they also have the same Hermann–Mauguin symbols, so the entries in the columns of the scanning group and of the sectional layer groups coincide. The orbits are separated by horizontal lines in the first column and further through the column with the scanning group, orbits and sectional layer groups, if they are different; when the Hermann–Mauguin symbol of the scanning group and hence the two remaining columns are identical, we give them as a common row for all the three orbits, which are then separated only in the first two columns. In the tables for groups of geometric class [mm2] ( $[C_{2v}]$ ), the orbit (001) is separated by double lines across the table from the remaining orbits (100) and (010), which are separated by single lines across the tables.

The bases for the scanning groups and for the sectional layer groups associated with these orbits are chosen in a standard manner for all orthorhombic groups:

(1) For the orientation (001), it is natural to choose $[{\bf a}' = {\bf a}]$ , $[{\bf b}' = {\bf b}]$ and $[{\bf c}' = {\bf d} = {\bf c}]$ . The symbol of the scanning group then coincides with the symbol of the space group itself, i.e. its symbol in the () setting.
(2) The scanning direction for orientations (100) and (010) are along $[{\bf d} = {\bf a}]$ and $[{\bf d} = {\bf b}]$ , respectively. We choose the remaining vectors $[{\bf a}']$ , $[{\bf b}']$ in such a way that ( $[{\bf a}']$ , $[{\bf b}']$ , d) is a right-handed basis, hence $[{\bf a}' = {\bf b}]$ , $[{\bf b}' = {\bf c}]$ for the orientation (100) and $[{\bf a}' = {\bf c}]$ , $[{\bf b}' = {\bf a}]$ for the orientation (010). Accordingly, the Hermann–Mauguin symbols for the scanning groups are the symbols which correspond to the settings () and ( $[{\bar{c}\bar{a}}b]$ ), respectively.

References

International Tables for Crystallography (2006). Vol. E. ch. 5.2, p. 403