Layer symmetries in crystal structures

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. 410-411 | 1 | 2 |

Section 5.2.5.1. Layer symmetries in crystal structures

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

5.2.5.1. Layer symmetries in crystal structures

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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, CdI₂. The space group of this crystal is $[P{\bar 3}m1]$ , $[D_{3d}^{3}]$ (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 CdI₂ structure is: $[\matrix{{A}&{C}&{B}&{C}\cr {\rm I}&{\rm Cd}&{\rm I}&{\rm void}}]$

Figure 5.2.5.1 | top | pdf |

The structure of cadmium iodide, CdI₂. The section planes of two orbits in special positions are distinguished by shading. The figure is drastically elongated in the c direction to exhibit the layer symmetries.

From the scanning tables, we obtain for planes with the (0001) orientation and at heights 0c or $[{{1}\over{2}}{\bf c}]$ a sectional layer symmetry $[p{\bar 3}m1]$ (L72), and for planes of this orientation at any other height a sectional layer symmetry [p3m1] (L69).

The plane at height 0c contains cadmium ions. This plane defines the orbit of planes of orientation (0001) located at points $[P+n{\bf c}]$ , where $[n \in Z]$ (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 $[{{1}\over{2}}{\bf c}]$ defines the orbit of planes of orientation (0001) located at points $[P+(n+{{1}\over{2}}){\bf c}]$ , where $[n \in Z]$ . 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 $[{{1}\over{4}}{\bf c}]$ and $[{{3}\over{4}}{\bf c}]$ contain B and A layers of iodine ions, respectively. These planes and all planes produced by translations $[n{\bf c}]$ from them belong to the same orbit because the operations $[{\bar 3}]$ exchange the A and B layers.

Example 2

The space group of cadmium chloride, CdCl₂, is $[R{\bar 3}m]$ , $[D_{3d}^{5}]$ (No. 166). Fig. 5.2.5.2 shows the structure of CdCl₂ 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 CdCl₂ structure is: $[\matrix{{A} &{C} &{B} &{A} &{C} &{B} &{A} &{C} &{B} &{A} &{C} &{B}\cr {\rm Cl} &{\rm Cd} &{\rm Cl} &{\rm void} &{\rm Cl} &{\rm Cd} &{\rm Cl} &{\rm void} &{\rm Cl} &{\rm Cd} &{\rm Cl} &{\rm void}}]$

Figure 5.2.5.2 | top | pdf |

The structure of cadmium chloride, CdCl₂. The section planes of two orbits in special positions are distinguished by shading. Notice the different location of the sectional layer groups on different levels for the same orbit. The figure is drastically elongated in the c direction to exhibit the layer symmetries.

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, $[{{1}\over{6}}{\bf c}]$ , $[{{1}\over{3}}{\bf c}]$ , $[{{1}\over{2}}{\bf c}]$ , $[{{2}\over{3}}{\bf c}]$ and $[{{5}\over{6}}{\bf c}]$ have a sectional layer group of the type $[p{\bar 3}m1]$ (L72) and at any other height have a sectional layer group of the type [p3m1] (L69).

The scanning table also specifies the location of the sectional layer groups. The position along the c axis, where the basis vector $[{\bf c} = {\bf d}]$ specifies the scanning direction, is given by fractions of d or by $[s{\bf d}]$ in the case of a general position. At the heights 0c and $[{{1}\over{2}}{\bf c}]$ , the sectional layer group is the group $[p{\bar 3}m1]$ (L72), while at the heights $[{{1}\over{3}}{\bf c}]$ and $[{{5}\over{6}}{\bf c}]$ it is the group $[p{\bar 3}m1]$ $[[({\bf a} +2{\bf b})/3]]$ (L72), and at the heights $[{{2}\over{3}}{\bf c}]$ and $[{{1}\over{6}}{\bf c}]$ it is the group $[p{\bar 3}m1]$ $[[(2{\bf a} + {\bf b})/3]]$ , (L72), where the vectors in brackets mean the shift of the group $[p{\bar 3}m1]$ in space. The planes at the heights 0d, $[{{1}\over{3}}{\bf d}]$ and $[{{2}\over{3}}{\bf d}]$ belong to one translation orbit and the layers contain cadmium ions which are shifted relative to each other by the vectors $[({\bf a}+2{\bf b})/3]$ and $[(2{\bf a}+{\bf b})/3]$ . The planes at the heights $[{{1}\over{2}}{\bf d}]$ , $[{{5}\over{6}}{\bf d}]$ and $[{{1}\over{6}}{\bf d}]$ 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 $[({\bf a}+2{\bf b})/3]$ and $[(2{\bf a}+{\bf b})/3]$ .

References

International Tables for Crystallography (2006). Vol. E. ch. 5.2, pp. 410-411