Crystallographic group-subgroup relations and their use in crystal chemistry

Family trees of crystallographic group-subgroup relations are especially suited to display relationships among crystal structure types. Starting from an aristotype, a simple, highly symmetrical structure type, more and more complicated structures are obtained by reducing the space group symmetry. In doing so, it is important to keep trace of how the occupied Wyckoff positions develop from a space group to its subgroups and which coordinate transformations and origin shifts occur. The neccessary information can by obtained from the new volume A I of International Tables for Crystallography v. Possible error sources are demonstrated with the aid of examples. When a Wyckoff position is split into different symmetry-independent Wyckoff positions, substitution derivatives become possible. If the position does not split, its site symmetry must be reduced. thus rendering possible distorted derivative structures. Such distortions are of special importance in phase transitions, as shown with the examples K-2[TeBr6], (NO)(2)[TiCl6] and SrCu2(BO3)(2). Phase transitions involving a symmetry reduction are likely to produce twinned crystals. This can be the cause of errors in crystal structure determinations, as shown with the example of CaMnF5. The possible ways to occupy voids in sphere packings can be followed and calculated systematically, including the possibility to predict crystal structures. Sometimes the indices of the symmetry reduction take values that seem curious, such as 13 for CZr6I12 and 37 for PtCl3. The systematic application is also possible to molecular packings, such as to the modifications of P4S3. Tetraphenylphosphonium salts often crystallize in the space group P4/n or its subgroups, retaining pseudotetragonal symmetry even in monoclinic and triclinic structures.