Linear orbits

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. 397-398 | 1 | 2 |

Section 5.2.2.6. Linear orbits

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

5.2.2.6. Linear orbits

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We consider a section plane with orientation $[V({\bf a}',{\bf b}')]$ and location $[P+s{\bf d}]$ . The orbit of planes generated by the action of the scanned group $[\cal G]$ on this section plane splits into subsets of planes with the same orientation. The suborbit of planes with the same orientation is identical with the orbit under the action of the scanning group for this orientation. This suborbit is called the linear orbit of planes. If the orientation orbit contains only one orientation (scanning group = scanned group), then the linear orbit contains all planes of the orbit. If there are several orientations in the orientation orbit, then to each of these orientations there corresponds its own linear orbit. As shown in the previous section, the description of the scanning with reference to corresponding coordinate systems is identical for different orientations of the orientation orbit. The separation of planes and their sectional layer symmetries are the same in each of these orbits. In other words, the spatial distribution of layer symmetries is the same for all orientations of the orientation orbit; the scanning, however, begins generally at a point $[P+{\bf s}_{i}]$ for the orientation $[V({\bf a}_{i}',{\bf b}_{i}')]$ . We shall concentrate our attention now to one linear orbit.

The parameter s in the description of linear orbits defines the position of the section plane by its intersection $[P+s{\bf d}]$ with the scanning line. The parameter therefore specifies the distance of the section plane from the origin P in units of d and is referred to as the level at which the section plane is located. Intersections at $[P+(s+n){\bf d}]$ , $[n \in Z]$ (integer) are translationally equivalent to an intersection at $[P+s{\bf d}]$ where $[0 \leq s \,\lt\, 1]$ . The section planes at levels $[P+(s+n){\bf d}]$ form an orbit under the translation group $[T({\bf d})]$ generated by the scanning vector $[{\bf d}]$ . The set of these planes is called the translation orbit. Each translation orbit has exactly one representative plane in the interval $[0 \leq s \,\lt\, 1]$ . The linear orbit consists of one or several translation orbits.

We distinguish two types of locations and linear orbits:

(1) Special locations of section planes and special linear orbits .
(2) General locations of section planes and general linear orbits.

With reference to parameter s, the special locations always correspond to a fixed parameter, the general locations to a variable parameter. Special locations are singular in the sense that in the infinitesimal vicinity of a section plane at a special location there are only section planes of general location. The sectional layer groups corresponding to these locations have the following properties:

(1) The sectional symmetry of a plane in a special location is a layer group which contains operations changing the direction of the normal to the plane.
(2) The sectional symmetry of a plane in a general location is a layer group which does not contain operations changing the direction of the normal to the plane.
(3) The sectional symmetries of planes in special locations are always maximal layer subgroups of the space group $[\cal G]$ as well as of the scanning group $[\cal H]$ . The sectional symmetry of a plane in a general location is a common halving subgroup of all sectional layer groups for special locations. We say that such a sectional layer group is floating in the scanning direction.

Comment : If the point group H of the scanning group $[\cal H]$ does not contain elements that change the normal to section planes, then all locations are general locations and there is only one sectional layer group common to all locations of section planes. The scanning group with this property is also called floating in the scanning direction.

The number of planes in a translation orbit : The total number of planes in a translation orbit is infinite because the index of the sectional layer group in the scanning group is. We can, however, count the number of planes in a translation orbit in an interval $[0 \leq s \,\lt\, 1]$ . If the point group of the scanning group is H and the point group of the sectional layer group for a given translation orbit is L, then the number of planes in this orbit in the interval $[0 \leq s \,\lt\, 1]$ equals the index [[H:L]] when the centring of the scanning group is P or C. When the centring of the scanning group is A, B, I or F, this number is [2[H:L]] ; when the centring type of the scanning group is R, this number is [3[H:L]] .

The number f of planes in an orbit with a general parameter s per unit interval also defines the length of the fundamental region of the space group $[\cal G]$ as well as of the scanning group $[\cal H]$ in this interval. This length $[s_{o}]$ is a fraction of unit interval, $[s_{o} = {{1}\over{f}}]$ , where [f = [H:L]] , [2[H:L]] or [3[H:L]] according to the centring of the scanning group and L is the point group of sectional layer groups corresponding to a general orbit.

References

International Tables for Crystallography (2006). Vol. E. ch. 5.2, pp. 397-398