International
Tables for Crystallography Volume E Subperiodic groups Edited by V. Kopský and D. B. Litvin © International Union of Crystallography 2010 |
International Tables for Crystallography (2010). Vol. E, ch. 5.2, p. 414
Section 5.2.5.1. Layer symmetries in crystal structures^{a}Freelance research scientist, Bajkalská 1170/28, 100 00 Prague 10, Czech Republic, and ^{b}Department of Physics, The Eberly College of Science, Penn State – Berks Campus, The Pennsylvania State University, PO Box 7009, Reading, PA 19610–6009, USA |
The following two examples show the use of layer symmetries in the description of crystal structures.
Example 1
Fig. 5.2.5.1 shows the crystal structure of cadmium iodide, CdI_{2}. The space group of this crystal is , (No. 164). The anions form a hexagonal close packing of spheres and the cations occupy half of the octahedral holes, filling one of the alternate layers. In close-packing notation, the CdI_{2} structure is:
From the scanning tables, we obtain for planes with the (0001) orientation and at heights 0c or a sectional layer symmetry (L72), and for planes of this orientation at any other height a sectional layer symmetry (L69).
The plane at height 0c contains cadmium ions. This plane defines the orbit of planes of orientation (0001) located at points , where (Z is the set of all integers). All these planes contain cadmium ions in the same arrangement (C layer filled with Cd).
The plane at height defines the orbit of planes of orientation (0001) located at points , where . All these planes lie midway between A and B layers of iodine ions with the B layer below, the A layer above the plane. They contain only voids.
The planes at levels and contain B and A layers of iodine ions, respectively. These planes and all planes produced by translations from them belong to the same orbit because the operations exchange the A and B layers.
Example 2
The space group of cadmium chloride, CdCl_{2}, is , (No. 166). Fig. 5.2.5.2 shows the structure of CdCl_{2} in its triple hexagonal cell. The anions form a cubic close packing of spheres and the cations occupy half of the octahedral holes of each alternate layer. In close-packing notation, the CdCl_{2} structure is:
We choose the origin at a cadmium ion and the hexagonal basis vectors a, b as shown in Fig. 5.2.5.2. This corresponds to the obverse setting for which the scanning table is given in Part 6. The planes with the (0001) orientation at the heights 0c, , , , and have a sectional layer group of the type (L72) and at any other height have a sectional layer group of the type (L69).
The scanning table also specifies the location of the sectional layer groups. The position along the c axis, where the basis vector specifies the scanning direction, is given by fractions of d or by in the case of a general position. At the heights 0c and , the sectional layer group is the group (L72), while at the heights and it is the group (L72), and at the heights and it is the group , (L72), where the vectors in brackets mean the shift of the group in space. The planes at the heights 0d, and belong to one translation orbit and the layers contain cadmium ions which are shifted relative to each other by the vectors and . The planes at the heights , and contain the voids and are located midway between layers of chlorine ions; they belong to another linear orbit and again are shifted relative to each other by the vectors and .