International
Tables for Crystallography Volume A Spacegroup symmetry Edited by Th. Hahn © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. A, ch. 14.2, pp. 849872
Section 14.2.3. Descriptive symbols^{a}Institut für Mineralogie, Petrologie und Kristallographie, PhilippsUniversität, D35032 Marburg, Germany 
For the study of relations between crystal structures, latticecomplex symbols are desirable that show as many relations between point configurations as possible. To this end, Hermann (1960) derived descriptive latticecomplex symbols that were further developed by Donnay et al. (1966) and completed by Fischer et al. (1973). These symbols describe the arrangements of the points in the point configurations and refer directly to the coordinate descriptions of the Wyckoff positions. Since a lattice complex, in general, contains Wyckoff positions with different coordinate descriptions, it may be represented by several different descriptive symbols. The symbols are further affected by the settings of the space group. The present section is restricted to the fundamental features of the descriptive symbols. Details have been described by Fischer et al. (1973). Tables 14.2.3.1 and 14.2.3.2 give for each Wyckoff position of a plane group or a space group, respectively, the multiplicity, the Wyckoff letter, the oriented site symmetry, the reference symbol of the corresponding lattice complex and the descriptive symbol.^{1} The comparatively short descriptive symbols condense complicated verbal descriptions of the point configurations of lattice complexes.
Invariant lattice complexes in their characteristic Wyckoff position are represented by a capital letter eventually in combination with some superscript. The first column of Table 14.2.3.3 gives a complete list of these symbols in alphabetical order. The characteristic Wyckoff positions are shown in column 3. Lattice complexes from different crystal families but with the same coordinate description for their characteristic Wyckoff positions receive the same descriptive symbol. If necessary, the crystal family may be stated explicitly by a small letter (column 2) preceding the latticecomplex symbol: c cubic, t tetragonal, h hexagonal, o orthorhombic, m monoclinic, a anorthic (triclinic).

Example
D is the descriptive symbol of the invariant cubic lattice complex a as well as of the orthorhombic lattice complex Fddd a. The cubic lattice complex cD contains – among others – the point configurations corresponding to the arrangement of carbon atoms in diamond and of silicon atoms in βcristobalite. The orthorhombic complex oD is a comprehensive complex of cD. It consists of all those point configurations that may be produced by orthorhombic deformations of the point configurations of cD.
The descriptive symbol of a noncharacteristic Wyckoff position depends on the difference between the coordinate descriptions of the respective characteristic Wyckoff position and the position under consideration. Three cases may be distinguished, which may also occur in combinations.
In noncharacteristic Wyckoff positions, the descriptive symbol P may be replaced by C, I by F (tetragonal system), C by A or B (orthorhombic system), and C by A, B, I or F (monoclinic system). If the lattice complexes of rhombohedral space groups are described in rhombohedral coordinate systems, the symbols R, , M and of the hexagonal description are replaced by P, I, J and , respectively (preceded by the letter r, if necessary, to distinguish them from the analogous cubic invariant lattice complexes).
The descriptive symbols of lattice complexes with degrees of freedom consist, in general, of four parts: shift vector, distribution symmetry, central part and siteset symbol. Either of the first two parts may be absent.
Example
..2 C4xxz is the descriptive symbol of the lattice complex in its characteristic position: is the shift vector, ..2 the distribution symmetry, C the central part and 4xxz the siteset symbol.
Normally, the central part is the symbol of an invariant lattice complex. Shift vector and central part together should be interpreted as described in Section 14.2.3.2. The point configurations of the regarded Wyckoff position can be derived from that described by the central part by replacing each point by a finite set of points, the site set. All points of a site set are symmetrically equivalent under the sitesymmetry group of the point that they replace. A site set is symbolized by a string of numbers and letters. The product of the numbers gives the number of points in the site set, whereas the letters supply information on the pattern formed by these points. Site sets replacing different points may be differently oriented. In this case, the distributionsymmetry part of the reference symbol shows symmetry operations that relate such site sets to one another. The orientation of the corresponding symmetry elements is indicated as in the oriented sitesymmetry symbols (cf. Section 2.2.12 ). If all site sets have the same orientation, no distribution symmetry is given.
Examples

In the case of a Weissenberg complex, the central part of the descriptive symbol always consists of two (or more) symbols of invariant lattice complexes belonging to the same crystal family and forming limiting complexes of the regarded Weissenberg complex. The shift vector then refers to the first limiting complex. The corresponding siteset symbols are distinguished by containing the number 1 as the only number, i.e. each site set consists of only one point.
Example
In , each of the two points and , represented by , is replaced by a site set containing only one point 1z, i.e. the points are shifted along the z axis. The shifts of the two points are related by a twofold rotation .2., i.e. are running in opposite directions. The point configurations of the two limiting complexes and B refer to the special parameter values and , respectively.
The central parts of some lattice complexes with two or three degrees of freedom are formed by the descriptive symbol of a univariant Weissenberg complex instead of that of an invariant lattice complex. This is the case only if the corresponding characteristic spacegroup type does not refer to a suitable invariant lattice complex.
Example
In , each of the two points and , represented by , is replaced by a site set 2y of two points forming a dumbbell. These dumbbells are oriented parallel to the y axis.
The symbol of a noncharacteristic Wyckoff position is deduced from that of the characteristic position. The four parts of the descriptive symbol are subjected to the transformation necessary to map the characteristic Wyckoff position onto the Wyckoff position under consideration.
Example
The lattice complex with characteristic Wyckoff position Imma 8h 0yz has the descriptive symbol for this position. Another Wyckoff position of this lattice complex is . The corresponding point configurations are mapped onto each other by interchanging positive x and negative y directions and shifting by . Therefore, the descriptive symbol for Wyckoff position Imma i is .
In some cases, the Wyckoff position described by a latticecomplex symbol has more degrees of freedom than the lattice complex (see Section 14.2.2.1). In such a case, a letter (or a string of letters) in brackets is added to the symbol.
Different kinds of relations between lattice complexes are brought out.
In many cases, limitingcomplex relations can be deduced from the symbols. This applies to limiting complexes due either to special metrical parameters (e.g. etc.) or to special values of coordinates (e.g. both P4x and P4xx are limiting complexes of P4xy). If the site set consists of only one point, the central part of the symbol specifies all corresponding limiting complexes without degrees of freedom that are due to special values of the coordinates (e.g. . for the general position of ).
References
Donnay, J. D. H., Hellner, E. & Niggli, A. (1966). Symbolism for lattice complexes, revised by a Kiel symposium. Z. Kristallogr. 123, 255–262.Fischer, W., Burzlaff, H., Hellner, E. & Donnay, J. D. H. (1973). Space groups and lattice complexes. NBS Monograph No. 134. Washington: National Bureau of Standards.
Hermann, C. (1960). Zur Nomenklatur der Gitterkomplexe. Z. Kristallogr. 113, 142–154.