Auxiliary 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, pp. 401-402 | 1 | 2 |

Section 5.2.3.2. Auxiliary 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

5.2.3.2. Auxiliary tables

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The auxiliary tables describe cases of monoclinic/inclined scanning for groups of orthorhombic and higher symmetries. They are clustered together for groups of each Laue class, starting from Laue class $[D_{2h}]$ – [mmm] , after the tables of orthogonal scanning, i.e. after the standard-format tables for this Laue class.

All possible cases of monoclinic/inclined scanning reduce to cases where the scanned group $[\cal G]$ itself is monoclinic and the orientation is defined by the Miller indices [(mn0)] . These cases are described as a part of the standard-format tables for monoclinic groups. Two bases are used in this description:

(i) The conventional basis $[({\bf a},{\bf b},{\bf c})]$ of the group $[\cal G]$ in its role as the scanned group.
(ii) The conventional basis (in the sense of the convention for scanning groups, see Section 5.2.2.3) $[({\bf a}',{\bf b}',{\bf d})]$ of the group $[{\cal H} = {\cal G}]$ in its role as the scanning group.

If the scanned group $[\cal G]$ is of higher than monoclinic symmetry, then the monoclinic scanning group $[{\cal H} \subset {\cal G}]$ and we use three bases:

(i) The conventional basis $[({\bf a},{\bf b},{\bf c})]$ of the scanned group $[\cal G]$ .
(ii) The conventional basis $[({\widehat {\bf a}}, {\widehat {\bf b}}, {\widehat {\bf c}})]$ of the monoclinic scanning group $[\cal H]$ , which is further called the auxiliary basis. This basis is always chosen so that the vector $[{\widehat {\bf c}}]$ is the unique axis vector.
(iii) The conventional basis (in the sense of the convention for scanning groups, see Section 5.2.2.3) $[({\bf a}',{\bf b}',{\bf d})]$ of the scanning group $[{\cal H}]$ .

Two types of tables from which orbits of planes and sectional layer groups can be deduced are given:

(1) Tables of orientation orbits and auxiliary bases of scanning groups. These contain Miller indices of orientations in the orbit and define auxiliary bases $[({\widehat {\bf a}}, {\widehat {\bf b}}, {\widehat {\bf c}})]$ of the respective scanning groups in terms of the basis $[({\bf a}, {\bf b}, {\bf c})]$ of the scanned group $[\cal G]$ and of the Miller indices of the orientation.
(2) Reference tables. These serve to give a reference to that table of a monoclinic group from which one can read the scanning data.

In the next two sections we describe the construction of these two types of tables and their use in detail.

5.2.3.2.1. Tables of orientation orbits and auxiliary bases of scanning groups

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The cases of monoclinic/inclined scanning occur when the orientation of the section plane:

(i) contains the direction of some symmetry axis of even order [scanning group of geometric class 2 ( $[C_{2}]$ )],
(ii) is orthogonal to a symmetry plane [scanning group of geometric class m ( $[C_{s}]$ )],
(iii) contains the direction of some symmetry axis of even order and at the same time is orthogonal to a symmetry plane [scanning group of geometric class ( $[C_{2h}]$ )].

Auxiliary basis of the scanning group . In each of these cases, there is a set of orientations for which the property (i), (ii) or (iii) is common and all orientations of this set contain the vector that defines the unique axis of a monoclinic scanning group which is also common for all orientations of the set. An auxiliary basis $[({\widehat {\bf a}},{\widehat {\bf b}},{\widehat {\bf c}})]$ of this scanning group is defined with reference to that one orientation of the set which is described by Miller indices [(mn0)] .

The first column of each table describes orientations of the orbit by Miller indices with reference to the conventional basis $[({\bf a}, {\bf b}, {\bf c})]$ of the scanned group $[\cal G]$ . Various possible situations can be distinguished by three criteria:

(1) The structure of orbits.

(i) All orientations of the orbit contain the vector of the unique axis of the scanning group. This also means that there is only one scanning group for all orientations of the orbit.

This situation occurs for orientations that contain the vector of principal axis c in tetragonal and hexagonal groups. It occurs also for orientations which contain the vector of any of the orthorhombic axes c, a or b.
(ii) The orbit splits into sets of orientations where each set has its own common unique axis and scanning group.

This situation occurs for orientations that contain vectors of auxiliary axes of groups of Laue classes ( $[D_{4h}]$ ), $[{\bar 3}m]$ ( $[D_{3d}]$ ), ( $[D_{6h}]$ ), $[m{\bar 3}]$ ( $[T_{h}]$ ) and $[m{\bar 3}m]$ ( $[O_{h}]$ ).

(2) Possible increase of the symmetry for special orientations.

(i) All orientations of the set with common unique axis have the same monoclinic scanning group.

This is the case of groups of Laue classes ( $[C_{4h}]$ ) and ( $[C_{6h}]$ ), and of orientations that contain the vector c of the principal axis.
(ii) In all other cases there appear special orientations in the set which have higher symmetry than monoclinic.

(3) Auxiliary basis of the scanning group.

The auxiliary bases of scanning groups are their conventional bases corresponding to unique axis c.

(i) If the conventional basis of the scanning group can be based on the same vectors as the conventional basis of the scanned group, parameters m, n are used in the Miller indices that define the orientation.
(ii) If the conventional basis of the scanning group cannot be based on the same vectors as the conventional basis of the scanned group, parameters h, k, l are used in the Miller indices that define the orientation with reference to the conventional basis $[({\bf a}, {\bf b}, {\bf c})]$ .

In these cases, the transformation of Miller indices with reference to the conventional basis $[({\bf a}, {\bf b}, {\bf c})]$ to Miller indices with reference to auxiliary basis $[({\widehat {\bf a}},{\widehat {\bf b}},{\widehat {\bf c}})]$ is given in a row under the orientation orbit. The letters m and n are always used for Miller indices with reference to auxiliary bases.

The second column assigns to each orientation the conventional basis $[({\bf a}',{\bf b}',{\bf d})]$ of the monoclinic scanning group that is related to the auxiliary basis $[({\widehat {\bf a}},{\widehat {\bf b}},{\widehat {\bf c}})]$ given in the third column in the same way as to the standard basis $[({\bf a}, {\bf b}, {\bf c})]$ in the case of monoclinic groups.

The conventional basis $[({\bf a}',{\bf b}',{\bf d})]$ is always chosen so that its first vector $[{\bf a}']$ is the vector of the common unique axis. Vector $[{\bf b}']$ is defined by the orientation of section planes and hence by Miller indices (either directly or indirectly through transformation to a monoclinic basis). There is the same freedom in the choice of the scanning direction d as in the cases of monoclinic/inclined scanning in the case of monoclinic groups.

5.2.3.2.2. Reference tables

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Each table of orientation orbits for a certain centring type(s) is followed by reference tables which are organized by arithmetic classes belonging to this centring type(s). The scanned space groups $[\cal G]$ are given in the first row by their sequential number, Schönflies symbol and short Hermann–Mauguin symbol. They are arranged in order of their sequential numbers unless there is a clash with arithmetic classes; a preference is given to collect groups of the same arithmetic class in one table. If space allows it, groups of more than one arithmetic class are described in one table.

The first column is identical with the first column of the table of orientation orbits. On the intersection of a column which specifies the scanned group $[\cal G]$ and of a row which specifies the orientation by its Miller (Bravais–Miller) indices is found the scanning group, given by its Hermann–Mauguin symbol with reference to the auxiliary basis $[({\widehat {\bf a}},{\widehat {\bf b}},{\widehat {\bf c}})]$ . This symbol, which may also contain a shift of origin, instructs us which monoclinic scanning table to consult. The vectors $[{\bf a}']$ , $[{\bf b}']$ , d that determine the lattice of sectional layer groups and the scanning direction are those given in the table of orientation orbits. Depending on the values of parameters m, n, p, q we find the scanning group in its basis $[({\bf a}', {\bf b}', {\bf d})]$ and the respective sectional layer groups.

References

International Tables for Crystallography (2006). Vol. E. ch. 5.2, pp. 401-402