International
Tables for Crystallography Volume G Definition and exchange of crystallographic data Edited by S. R. Hall and B. McMahon © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. G, ch. 3.8, pp. 206-207
Section 3.8.2. Dictionary design considerations^{a}Brockhouse Institute for Materials Research, McMaster University, Hamilton, Ontario, Canada L8S 4M1 |
In 1995, a working group was established to draw up a CIF symmetry dictionary. The group quickly encountered two problems. Firstly, scientists in different fields have different expectations of space-group theory and conflicting requirements for CIF definitions, and secondly the familiar presentation of space groups that works well in a printed book such as IT A is not necessarily the best way of presenting the material in a computer-based system. These problems are illustrated by the difficulties that the working group encountered in defining the space-group symbols. Each space group can be fully described only within a particular coordinate system or setting, but most space groups can be represented in many different settings and no one setting is uniquely fitted for this role. Ideally, one would like to have access to representations of the symmetry operations in all possible settings, but this is impractical in a printed book. IT A restricts itself to representing the space groups in a small number, often only one, of the many possible settings. However, computers are not limited in the same way, since it is possible to calculate the operations for any desired setting without additional overhead. This is an advantage for experimental crystallographers who like to choose the setting to suit the crystal, but theorists prefer to use the same fixed reference setting for all instances of a space group because it leads to a consistent description of group–subgroup and other relationships between the space groups.
This difficulty is resolved in the CIF symmetry dictionary by recognizing that there are two types of space-group symbol: those such as the Schoenflies symbol and the International Tables number that identify the space group without specifying the setting; and those such as the Hall symbol [see Appendix 1.4.2.3 of Chapter 1.4 in International Tables for Crystallography Volume B (Hall & Grosse-Kunstleve, 2001)] that are symmetry generators and therefore define the space group within a particular setting. The weakness of the setting-independent symbols is that they are labels that carry at best only a limited amount of information about the symmetry. Full details of the symmetry operations can only be found by consulting a concordance that lists the operations in some arbitrarily chosen setting. The weakness of the symmetry generators is that different settings of the same space group require different symbols, making it difficult to recognize when two symbols refer to the same space group. Unfortunately, the widely used Hermann–Mauguin symbols suffer from both weaknesses. They can be used as space-group generators, but they do not cover all possible settings because they contain no information about the choice of origin. This makes them inappropriate for use when all possible settings are needed. But equally they cannot be used as unique space-group identifiers, since many space groups can be described by more than one Hermann–Mauguin symbol. They are not therefore well adapted to computer use, but they are popular because they are easy to interpret and they indicate in a simple way the relationship between the symmetry operations and the axis system.
Because CIF requires a unique and precise definition for any symbol that a program is required to interpret, the working group found it necessary to define different versions of the Hermann–Mauguin symbol. There is a tightly defined version, _space_group.name_H-M_ref, which can be used to generate the symmetry operations of the reference setting, but only one form of this symbol (that given in the list of values specified by the dictionary) can be used for a given space group. For consistency with earlier work, the reference settings in this list have all been chosen from among those listed in IT A. Where there is a choice of settings in IT A, the b-axis setting is chosen for monoclinic, and the hexagonal setting is chosen for rhombohedral space groups. Where two different origin choices are given, the second (origin at a centre of symmetry) is chosen. A second more loosely defined Hermann–Mauguin symbol, _space_group.name_H-M_alt, is defined for the benefit of those who like to use the Hermann–Mauguin symbol to indicate the axis system adopted, but as this item does not have a precise definition, it should not be interpreted by a software application. It should be treated as a text string that can be displayed for the benefit of the user.
The symmetry CIF dictionary differs in an important way from the other CIF dictionaries. While the other dictionaries define items that give the results of an experimental measurement, all the items in the symmetry dictionary are either theoretically derivable or, like the reference settings and Wyckoff letters, are arbitrarily assigned. The addition of algorithmic methods in future versions of CIF dictionaries will allow many of the symmetry items to be calculated as needed from expressions included in the dictionary as long as a space-group generator, such as the Hall symbol, is present in the CIF.
The symmetry CIF dictionary is written in DDL2 (Chapter 2.6 ) as this allows the relationships between the items to be given explicitly. Version 1.0 of the dictionary contains CIF definitions for all the basic symmetry items needed to describe a three-dimensional space group as described in more detail in Section 3.8.3. Some of these items have been incorporated in the core CIF dictionary, in which they have been converted to DDL1 (Chapter 2.5 ) to match the dictionary definition language of the core CIF dictionary. While the items in the symmetry CIF dictionary are designed in part to replace those in the symmetry category of the original core CIF dictionary, they are defined in a way that allows a CIF to include descriptions of several space groups in several settings. A single CIF could, in principle, include the symmetry operations and Wyckoff positions of all possible settings of the 230 space groups.
References
Hall, S. R. & Grosse-Kunstleve, R. W. (2001). International tables for crystallography, Vol. B, Reciprocal space, edited by U. Shmueli, 2nd ed., Appendix A1.4.2.3. Dordrecht: Kluwer Academic Publishers.