The conventional basis of the scanning group

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. 395-396 | 1 | 2 |

Section 5.2.2.3. The conventional basis of the scanning group

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

5.2.2.3. The conventional basis of the scanning group

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In the Scanning tables of Part 6 , we follow the usual crystallographic practice to define the orientation of planes by their Miller indices (Bravais–Miller indices in hexagonal cases). This itself already guarantees that the orientations considered are crystallographic. The choice of vectors $[{\bf a}']$ , $[{\bf b}']$ and $[{\bf d}]$ is governed by a convention in which we distinguish the cases of orthogonal and inclined scanning.

Convention : Given the orientation of planes by Miller or Bravais–Miller indices, we choose vectors $[{\bf a}']$ , $[{\bf b}']$ and the vector d of the scanning direction according to the following rules:

(i) Orthogonal scanning: If the scanning group $[\cal H]$ is of orthorhombic or higher symmetry, or if it is monoclinic with the direction of its unique axis orthogonal to the orientation of the planes, we call the scanning orthogonal and the vectors $[{\bf a}']$ , $[{\bf b}']$ , $[{\bf d}]$ are chosen in such a way that the triplet $[({\bf a}',{\bf b}',{\bf d})]$ constitutes a conventional right-handed basis of the scanning group $[\cal H]$ .
(ii) Inclined scanning: If the scanning group is either triclinic or monoclinic with its unique axis parallel to the section planes, we call the scanning inclined. In this case we choose vectors $[{\bf a}']$ , $[{\bf b}']$ in such a way that they constitute a conventional basis of the vector lattice $[T({\bf a}',{\bf b}')]$ , common to all sectional layer groups, while the scanning vector d is chosen as the shortest complementary vector.

Note that, in cases of orthogonal scanning, the first two vectors $[{\bf a}']$ , $[{\bf b}']$ of the conventional basis of the scanning group $[\cal H]$ automatically constitute a conventional basis of the lattice $[T({\bf a}',{\bf b}')]$ and d is orthogonal to the orientation $[V({\bf a}',{\bf b}')]$ . In cases of inclined scanning it is always possible to choose the vectors $[{\bf a}']$ , $[{\bf b}']$ so that they constitute a conventional basis of the vector lattice $[T({\bf a}',{\bf b}')]$ . However, it is generally impossible to choose all three vectors $[{\bf a}']$ , $[{\bf b}']$ and d as a strictly conventional basis of the scanning group because the first two vectors must lie in the space defined by Miller (Bravais–Miller) indices, which usually leads to a clash with the metric conditions as they are given, for example, in Section 9.1.4 (vi ) and (vii ) of IT A (2005).

The choice of the scanning direction d as that of a vector of the basis of the scanning group guarantees the periodicity d of the scanning. As a result, it is sufficient to describe the scanning for a given orientation, i.e. the sectional layer groups and orbits of planes, only in the interval with $[0 \leq s \,\lt\, 1]$ on the scanning line $[P+s{\bf d}]$ . Indeed, the crystal structure of symmetry $[\cal G]$ is periodically repeated with periodicity d in the scanning direction. The sectional layer groups are, however, repeated in the scanning direction with the periodicity of the translation normalizer of $[\cal G]$ . This is identical with the periodicity of the translation normalizer of the scanning group $[\cal H]$ (see the examples in Section 5.2.5.1). We recall that the translation normalizer of the space group $[\cal G]$ , as defined by Kopský (1993b,c), is the translation subgroup of the Cheshire group (Euclidean normalizer) of $[\cal G]$ [see Hirschfeld (1968) and Koch & Fischer in Part 15 of IT A, 1987 edition or later].

In the application of the convention we note the following:

Item 1 . If for a certain orientation of planes so that this orientation is invariant under all elements of the point group G of the space group $[\cal G]$ , then $[{\cal G} = {\cal H}]$ , i.e. the scanning group $[\cal H]$ coincides with the original space group $[\cal G]$ .

The typical cases of this relationship are orientations (001) for the monoclinic, orthorhombic and tetragonal groups and the orientations (0001) for the trigonal and hexagonal groups. In these cases, the conventional basis of the original space group $[\cal G]$ also coincides with the conventional basis of the scanning group $[\cal H]$ and the group $[\cal H]$ is therefore represented by the same Hermann–Mauguin symbol as the group $[\cal G]$ .
Item 2 . The conventional basis of the scanning group $[\cal H]$ may differ from the conventional basis of the original group $[\cal G]$ even if these groups are identical. In this case the group is generally denoted by different Hermann–Mauguin symbols. This always happens in the cases of monoclinic and very frequently in cases of orthorhombic groups for other orientations than (001) because the conventional vectors $[{\bf a}']$ , $[{\bf b}']$ , d of the scanning group $[\cal H]$ cannot be made identical with the conventional basis vectors a, b, c of the group $[\cal G]$ .

Example

Consider the space group $[{\cal G} = Pmmm]$ ( $[D_{2h}^{1}]$ ) and the orientations described by the Miller indices (001), (100), (010). The scanning group $[{\cal H} = {\cal G}]$ is identical with the scanned group and its Hermann–Mauguin symbol is the same for all three orientations.

If, however, the scanned group is the group $[{\cal G} = Pmma]$ ( $[D_{2h}^{5}]$ ), then again the scanning group $[\cal H]$ is identical with the scanned group $[\cal G]$ for the three orientations, but the Hermann–Mauguin symbols of the scanning group are now different: they are , and for the orientations (001), (100) and (010), respectively.
Item 3 . If $[H \subset G]$ , so that the point group H is a proper subgroup of the point group G, then the conventional basis of the scanning group $[\cal H]$ is usually different from the conventional basis of the original group $[\cal G]$ , although the groups are equitranslational, i.e. have the same translation subgroup. The conventional basis of the scanning group $[\cal H]$ in the case when $[H \subset G]$ actually coincides with the conventional basis of the space group $[\cal G]$ only in the cases of the orientations (001), (100) and/or (010) if $[\cal G]$ is cubic of lattice type P or I and hence $[\cal H]$ is tetragonal of the same lattice type. The centring type of the scanning group $[\cal H]$ is also frequently different from the centring type of the original group $[\cal G]$ .

References

Hirschfeld, F. L. (1968). Symmetry in the generation of trial structures. Acta Cryst. A24, 301–311.Google Scholar

International Tables for Crystallography (2005). Vol. A. Space-group symmetry, edited by Th. Hahn. Heidelberg: Springer. [Previous editions: 1983, 1987, 1992, 1995 and 2002. Abbreviated as IT A (2005).]Google Scholar

Kopský, V. (1993b). Translation normalizers of Euclidean motion groups. I. Elementary theory. J. Math. Phys. 34, 1548–1556.Google Scholar

Kopský, V. (1993c). Translation normalizers of Euclidean motion groups. II. Systematic calculation. J. Math. Phys. 34, 1557–1576.Google Scholar

International Tables for Crystallography (2006). Vol. E. ch. 5.2, pp. 395-396

Section 5.2.2.3. The conventional basis of the scanning group

5.2.2.3. The conventional basis of the scanning group

Example

References