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CIF news 7: CIF and symmetry

[Brown] David Brown
A good definition of a crystallographer is someone who can find their way around International Tables for Crystallography Vol. A since it is symmetry that lies at the heart of crystallography and is what distinguishes it from other condensed matter sciences. It is surprising therefore to find how few symmetry items are defined in the CIF dictionaries. This is all the more strange given that the precise definitions of space group theory ideally lend themselves to handling by computer. That omission has now been corrected with the recent release of the CIF Symmetry Dictionary which can be found on the IUCr web site ftp://ftp.iucr.org/pub/cif_sym.dic.

Because the concepts of space group theory are precisely defined, one would expect it to be easy to provide them with good CIF definitions, but the working group that prepared the dictionary ran into some unexpected problems. A space group cannot be fully described without choosing a setting, and since each space group can be found in many possible settings none of which is unique, there are many ways in which it can be described. The choice of which description to use is arbitrary and depends on how the information is to be used.

There are 230 space groups and each of these is defined by the set of symmetry operations that comprise the group. However, these symmetry operations can only be properly described by expressing them in an explicit algebraic form, and this can only be done by choosing a specific coordinate system or setting. What is then the best way to represent a space group in a CIF? One method is to list the symmetry operations of the group in the setting of choice. This is the option most often found in the current CIF archives of crystal structures where the setting is chosen to give the simplest description of the structure. In theoretical studies, on the other hand, it is better to choose one invariant reference setting for each space group, even though the choice is arbitrary, since this permits a consistent description of the relationships between space groups.

[Cube] N-dimensional Rubics cube
In a printed work, such as International Tables, it not feasible to display the space group in all possible settings, so the setting chosen becomes, de facto, the reference setting needed for theoretical studies, but a computer is more flexible. The symmetry operations and special positions can be calculated on the fly in any desired setting from the Hall space group symbol (International Tables Vol B (1993) 1.4 Appendix B) using a computer application such as the Windows program WGETSPEC (www.ccp14.ac.uk/ccp/web-mirrors/lmpg-laugier-bochu).This opens up possibilities that are not available to printed tables.

Space groups are identified by their name or symbol. There are two types of symbol, those that define the space group in terms of its setting and those that are independent of the setting. Each has its advantages and disadvantages. The Schoenflies symbol and the International Tables (IT) number are unique identifiers of the space group that are independent of the setting. Their disadvantage is that the name itself, e.g., IT-62, contains no explicit information on the operations the space group. To get this information one must consult a separate listing of the symmetry operations, but any such listing immediately implies a particular setting. On the other hand, names such as the Hall symbol, which contain complete information about the symmetry operations, necessarily imply a particular setting, with the result that different symbols are needed for different settings of the same space group. Because so many different Hall symbols describe the same space group, it is not easy to identify the space group from a cursory examination of the symbol.

The Hermann-Mauguin (H-M) symbols can be decoded to give the symmetry operations, but they cannot reach all possible settings since they contain no information about the choice of origin, or the axis system for rhombohedral space groups. They are of limited use as generators of the space group operations, but, because they are setting dependent, many different H-M symbols can be used to describe the same space group.

Since the symbol cannot be precisely defined as either a space group generator or as a setting independent space group name, the CIF Symmetry Dictionary defines several different versions of the H-M symbol. One of these, _space_group.name_H-M_ref, is tightly defined. Only one form of the symbol, defined by an enumeration list, may be used for each space group. It can therefore act as a setting-independent space group name, but it can also be decoded to give the symmetry operations of the reference setting (normally the first setting given in International Tables Vol A). An alternative H-M symbol, _space_group.name_H-M_alt, is provided for people who like to use the Hermann-Mauguin symbol in other ways, such as to indicate the setting used as well as the space group. There are no rules about how this symbol is to be written or used and, and for this reason it cannot be reliably used as a space group generator. The CIF stores this symbol solely for the information of the user and it is not intended to be used as an instruction to a computer program.

Version 1.0 of the CIF Symmetry Dictionary defines the basic terms encountered in space group symmetry such as the different space group symbols, the symmetry operations, the reference setting and transformations between the reference setting and the setting adopted in the CIF. There are also items describing the special positions, their symmetry and Wyckoff letters. Future versions of the dictionary will include symmetry in higher dimensions (used for incommensurate structures and quasi-crystals) and group - subgroup relationships (useful in analysing displacive phase transitions). In the meantime the new set of _space_group items will be added to the core and macromolecular dictionaries as the old _symmetry items are phased out.

David Brown