Orientation 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

pdf | chapter contents | chapter index | related articles

International Tables for Crystallography (2006). Vol. E. ch. 5.2, pp. 396-397 | 1 | 2 |

Section 5.2.2.5. Orientation 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

5.2.2.5. Orientation orbits

| top | pdf |

The point group G of the scanned group $[\cal G]$ acts on the orientations defined by Miller indices [(hkl)] or Bravais–Miller indices [(hkil)] . The set of all orientations $[V({\bf a}_{i}', {\bf b}_{i}')]$ obtained from a given orientation $[V({\bf a}_{1}', {\bf b}_{1}')]$ by the action of the elements of the group G is called the orientation orbit. The point group $[H_{1} \subseteq G]$ which leaves the orientation $[V({\bf a}_{1}', {\bf b}_{1}')]$ invariant is the point group of the scanning group $[{\cal H}_{1}]$ for this orientation. From the coset resolution $[G = H_{1} \cup g_{2}H_{1} \cup \ldots \cup g_{p}H_{1} \eqno(5.2.2.2)]$ we obtain orientations of the orbit by the action of cosets on the first orientation: $[V({\bf a}_{i}',{\bf b}_{i}') = g_{i}H_{1}V({\bf a}_{1}',{\bf b}_{1}') = g_{i}V({\bf a}_{1}',{\bf b}_{1}')]$ . In general, the number of orientations in the orbit is equal to the index $[p = [G:H_{1}]]$ of the subgroup $[H_{1}]$ in G. The point group $[H_{i} \subset G]$ which leaves the orientation $[V({\bf a}_{i}',{\bf b}_{i}')]$ invariant is the conjugate subgroup $[H_{i} = g_{i}H_{1}g_{i}^{-1}]$ of the point group $[H_{1}]$ in the group G. If $[H_{1} = H = G]$ , then the scanning group $[\cal H]$ is identical with the scanned group $[\cal G]$ and the orientation orbit contains just one orientation.

In the general case, to each orientation $[V({\bf a}_{i}',{\bf b}_{i}')]$ there corresponds a scanning group $[{\cal H}_{i}]$ , conjugate to the scanning group $[{\cal H}_{1}]$ . The elements of a coset $[g_{i}H_{1}]$ send the scanning vector $[{\bf d}_{1}]$ for the first orientation into scanning vectors $[{\bf d}_{i} = g_{i}H_{1}{\bf d}_{1} = g_{i}{\bf d}_{1}]$ for orientations $[V({\bf a}_{i}',{\bf b}_{i}')]$ .

The set of the conjugate scanning groups $[{\cal H}_{i}]$ is obtained from the coset resolution of the space group, which corresponds to the coset resolution (5.2.2.2) of the point group: $[{\cal G} = {\cal H}_{1} \cup \{g_{2}|{\bf s}_{2}\}{\cal H}_{1} \cup \ldots \cup \{g_{p}|{\bf s}_{p}\}{\cal H}_{1}. \eqno(5.2.2.3)]$ The scanning groups $[{\cal H}_{i} = \{g_{i}|{\bf s}_{i}\}{\cal H}_{1}\{g_{i}^{-1}|-g_{i}^{-1}{\bf s}_{i}\}]$ are related in the same way to the respective conventional bases $[({\bf a}_{i}',{\bf b}_{i}',{\bf d}_{i}) = (g_{i}{\bf a}_{1}',g_{i}{\bf b}_{1}',g_{i}{\bf d}_{1})]$ and hence they are expressed by the same Hermann–Mauguin symbols. However, the operations in the three-dimensional Euclidean space, which correspond to operations $[g_{i}]$ on the vector space, often contain additional translations $[{\bf s}_{i}]$ . Quite generally, the scanning for an orientation $[V({\bf a}_{i}',{\bf b}_{i}')]$ is described in the same manner with reference to the coordinate system $[(P+{\bf s}_{i};V({\bf a}_{i}',{\bf b}_{i}',{\bf d}_{i}))]$ as the scanning for the orientation $[V({\bf a}_{1}',{\bf b}_{1}')]$ is described with reference to a coordinate system $[(P; V({\bf a}_{1}',{\bf b}_{1}',{\bf d}_{1}))]$ .

In analogy with Wyckoff positions, see Section 8.3.2 of IT A, we distinguish three types of orientations and of orientation orbits:

(1) special orientations and special orientation orbits with fixed parameters;
(2) special orientations and special orientation orbits with variable parameter; and
(3) general orientations and general orientation orbits.

The type of the orbit is the same as the type of each of its orientations. Orientations and orientation orbits have the following characteristic properties:

(1) An orientation $[V({\bf a}', {\bf b}')]$ is a special orientation with fixed parameters if its symmetry H is either at least orthorhombic or if it is monoclinic with the vector of its unique axis orthogonal to the orientation.
(2) An orientation $[V({\bf a}', {\bf b}')]$ is a special orientation with variable parameter if its symmetry H is monoclinic and if it contains the vector of the unique axis.
(3) An orientation $[V({\bf a}', {\bf b}')]$ is a general orientation if its symmetry H is triclinic.

Example 1

Orientations defined by the Miller indices (001) are special orientations with fixed parameters for monoclinic groups with unique axis c as well as for orthorhombic and tetragonal groups. Bravais–Miller indices (0001) also define special orientation with fixed parameters. In each of these cases, the orientation orbit contains just one orientation.

Orientations (010) and (100) are special orientations with fixed parameters for all orthorhombic groups and each such orientation constitutes the orientation orbit.

Orientations (001), (010) and (100) are special orientations with fixed parameters for cubic groups and they belong to the same orientation orbit.

Example 2

Orientations [(mn0)] are special orientations with variable parameter for monoclinic groups with unique axis c. Each such orientation constitutes an orientation orbit.

For cubic groups, the orientations [(mn0)] [with the exclusion of cases $[m = \pm 1]$ , [n = 0] and [m = 0] , $[n = \pm 1]$ in groups of Laue class $[m{\bar 3}]$ ( $[T_{h}]$ ) and also of cases $[m = \pm 1]$ , $[n = \pm 1]$ in groups of Laue class $[m{\bar 3}m]$ ( $[O_{h}]$ )] are special orientations with variable parameter. The orientation orbits contain six equivalent orientations in groups of Laue class $[m{\bar 3}]$ ( $[T_{h}]$ ) and 12 in groups of Laue class $[m{\bar 3}m]$ ( $[O_{h}]$ ), see Section 5.2.4.6.

Orientation orbits are correlated with orbits of crystal faces, see Part 10 of IT A. If the group H does not contain elements that change the sign of the normal, then the orientation orbit is characterized by the same set of Miller indices as the set of equivalent crystal faces. Generally, the group H contains a halving subgroup $[H_{o}]$ whose elements leave the normal to the orientation $[V({\bf a}', {\bf b}')]$ invariant while elements of the coset change its sign. In this case, the number of equivalent crystal faces is twice the number of orientations in the orbit. The group $[H_{o}]$ is identical with the point symmetry of a crystal face of orientation $[V({\bf a}', {\bf b}')]$ . Such a face located at a point $[P+s{\bf d}]$ is sent to a face of the same orientation located at a point $[P-s{\bf d}]$ by those elements of H which are not contained in $[H_{o}]$ . These are the same elements which change the direction of the scanning.

References

International Tables for Crystallography (2006). Vol. E. ch. 5.2, pp. 396-397