International
Tables for Crystallography Volume C Mathematical, physical and chemical tables Edited by E. Prince © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. C, ch. 1.4, pp. 1522
https://doi.org/10.1107/97809553602060000575 Chapter 1.4. Arithmetic crystal classes and symmorphic space groupsArithmetic crystal classes have four main applications in practical crystallography: in the classification of space groups; in forming symbols for certain space groups in higher dimensions; in modelling the frequency of occurrence of space groups; and in establishing `equivalent origins'. Simple descriptions and complete enumerations of the arithmetic crystal classes in one, two and three dimensions are given in this chapter. Symmorphic space groups and the effect of dispersion on diffraction symmetry are also discussed. 
Arithmetic crystal classes are of great importance in theoretical crystallography, and are treated from that point of view in Volume A of International Tables for Crystallography (2005), Section 8.2.3 . They have, however, at least four applications in practical crystallography:
The tabulation of arithmetic crystal classes in Volume A is incomplete, and the relation of the notation used in complete tabulations found elsewhere (for example, in Brown, Bülow, Neubüser, Wondratschek & Zassenhaus, 1978) to that of International Tables is not immediately obvious. Simple descriptions and complete enumerations of the arithmetic crystal classes in one, two and three dimensions are therefore given here.
The 32 geometric crystal classes and the 14 Bravais lattices are familiar in threedimensional crystallography. The threedimensional arithmetic crystal classes are easily derived in an elementary fashion by enumerating the compatible combinations of geometric crystal class and Bravais lattice; the symbol adopted by the International Union of Crystallography for an arithmetic crystal class is simply the juxtaposition of the symbol for the geometric crystal class and the symbol for the Bravais lattice (de Wolff et al., 1985). For example, in the monoclinic system the geometric crystal classes are 2, m, and 2/m, and the Bravais lattices are monoclinic P and monoclinic C. The six arithmetic crystal classes in the monoclinic system are thus 2P, 2C, mP, mC, 2/mP, and 2/mC. In certain cases (loosely, when the geometric crystal class and the Bravais lattice have unique directions that are not necessarily parallel), the crystal class and the lattice can be combined in two different orientations. The simplest example is the combination of the orthorhombic crystal class^{1} mm with the endcentred lattice C. The intersection of the mirror planes of the crystal class defines one unique direction, the C centring of the lattice another. If these directions are placed parallel to one another, the arithmetic class mm2C is obtained; if they are placed perpendicular to one another, a different arithmetic class^{2} 2mmC is obtained. The other combinations exhibiting this phenomenon are lattice P with geometric classes 32, 3m, , , and . By consideration of all possible combinations of geometric class and lattice, one obtains the 73 arithmetic classes listed in Table 1.4.2.1.
In one dimension, there are two geometric crystal classes, 1 and m, and a single Bravais lattice, . Two arithmetic crystal classes result, and . In two dimensions, there are ten geometric crystal classes, and two Bravais lattices, p and c; 13 arithmetic crystal classes result. The twodimensional geometric and arithmetic crystal classes are listed in Table 1.4.1.1.

The number of arithmetic crystal classes increases rapidly with increasing dimensionality; there are 710 (plus 70 enantiomorphs) in four dimensions (Brown, Bülow, Neubüser, Wondratschek & Zassenhaus, 1978), but those in dimensions higher than three are not needed in this volume.
Arithmetic crystal classes may be used to classify space groups on a scale somewhat finer than that given by the geometric crystal classes. Space groups are members of the same arithmetic crystal class if they belong to the same geometric crystal class, have the same Bravais lattice, and (when relevant) have the same orientation of the lattice relative to the point group. Each onedimensional arithmetic crystal class contains a single space group, symbolized by and , respectively. Most twodimensional arithmetic crystal classes contain only a single space group; only 2mmp has as many as three.
The space groups belonging to each geometric and arithmetic crystal class in two and three dimensions are indicated in Tables 1.4.1.1 and 1.4.2.1, and some statistics for the threedimensional classes are given in Table 1.4.3.1. 12 threedimensional classes contain only a single space group, whereas two contain 16 each. Certain arithmetic crystal classes (3P, 312P, 321P, 422P, 6P, 622P, 432P) contain enantiomorphous pairs of space groups, so that the number of members of these classes depends on whether the enantiomorphs are combined or distinguished. Such classes occur twice in Table 1.4.3.1, as indicated by the footnotes.


The space groups in Table 1.4.2.1 are listed in the order of the arithmetic crystal class to which they belong. It will be noticed that arrangement according to the conventional spacegroup numbering would separate members of the same arithmetic crystal class in the geometric classes 2/m, 3m, 23, , 432, and . This point is discussed in detail in Volume A of International Tables, Section 8.3.4 . The symbols of five space groups [ (Aem2), C2ce (Aea2), Cmce, Cmme, Ccce] have been conformed to those recommended in the fourth, revised edition of Volume A of International Tables.
The 73 space groups known as `symmorphic' are in onetoone correspondence with the arithmetic crystal classes, and their standard `short' symbols (Bertaut, 2005) are obtained by interchanging the order of the geometric crystal class and the Bravais cell in the symbol for the arithmetic space group. In fact, conventional crystallographic symbolism did not distinguish between arithmetic crystal classes and symmorphic space groups until recently (de Wolff et al., 1985); the symbol of the symmorphic group was used also for the arithmetic class.
This relationship between the symbols, and the equivalent ruleofthumb symmorphic space groups are those whose standard (short) symbols do not contain glide planes or screw axes, reveal nothing fundamental about the nature of symmorphism; they are simply a consequence of the conventions governing the construction of symbols in International Tables for Crystallography.^{3}
Although the standard symbols of the symmorphic space groups do not contain screw axes or glide planes, this is a result of the manner in which the spacegroup symbols have been devised. Most symmorphic space groups do in fact contain screw axes and/or glide planes. This is immediately obvious for the symmorphic space groups based on centred cells; C2 contains equal numbers of diad rotation axes and diad screw axes, and Cm contains equal numbers of reflection planes and glide planes. This is recognized in the `extended' spacegroup symbols (Bertaut, 2005), but these are clumsy and not commonly used; those for C2 and Cm are and , respectively. In the more symmetric crystal systems, even symmorphic space groups with primitive cells contain screw axes and/or glide planes; () contains many diad screw axes and P4/mmm () contains both screw axes and glide planes.
The balance of symmetry elements within the symmorphic space groups is discussed in more detail in Subsection 9.7.1.2 .
In the absence of dispersion (`anomalous scattering'), the intensities of the reflections hkl and are equal (Friedel's law), and statements about the symmetry of the weighted reciprocal lattice and quantities derived from it often rest on the tacit or explicit assumption of this law – the condition underlying it being forgotten. In particular, if dispersion is appreciable, the symmetry of the Patterson synthesis and the `Laue' symmetry are altered.
In Volume A of International Tables, the symmetry of the Patterson synthesis is derived in two stages. First, any glide planes and screw axes are replaced by mirror planes and the corresponding rotation axes, giving a symmorphic space group (Subsection 1.4.2.1). Second, a centre of symmetry is added. This second step involves the tacit assumption of Friedel's law, and should not be taken if any atomic scattering factors have appreciable imaginary components. In such cases, the symmetry of the Patterson synthesis will not be that of one of the 24 centrosymmetric symmorphic space groups, as given in Volume A, but will be that of the symmorphic space group belonging to the arithmetic crystal class to which the space group of the structure belongs. There are thus 73 possible Patterson symmetries.
An equivalent description of such symmetries, in terms of 73 of the 1651 dichromatic colour groups, has been given by Fischer & Knop (1987); see also Wilson (1993).
Similarly, the eleven conventional `Laue' symmetries [International Tables for Crystallography (2005), Volume A, Section 3.1.2 and elsewhere] involve the explicit assumption of Friedel's law. If dispersion is appreciable, the `Laue' symmetry may be that of any of the 32 point groups. The point group, in correct orientation, is obtained by dropping the Bravaislattice symbol from the symbol of the arithmetic crystal class or of the Patterson symmetry.
References
Bertaut, E. F. (2005). Synoptic tables of spacegroup symbols. International tables for crystallography, Vol. A, edited by Th. Hahn, Part 4. Heidelberg: Springer.Brown, H., Bülow, R., Neubüser, J., Wondratschek, H. & Zassenhaus, H. (1978). Crystallographic groups of fourdimensional space. New York: Wiley.
Engel, P. (1986). Geometric crystallography. Dordrecht: Reidel. (Present distributor Kluwer Academic Publishers, Dordrecht.)
Fischer, K. F. & Knop, W. E. (1987). Space groups for imaginary Patterson and for difference Patterson functions in the lambda technique. Z. Kristallogr. 180, 237–242.
International Tables for Crystallography (2005). Vol. A. Spacegroup symmetry, fifth ed., edited by Th. Hahn. Heidelberg: Springer.
Opechowski, W. (1986). Crystallographic and metacrystallographic groups. Amsterdam: North Holland.
Wilson, A. J. C. (1993). Laue and Patterson symmetry in the complex case. Z. Kristallogr. 208, 199–206.
Wolff, P. M. de, Belov, N. V., Bertaut, E. F., Buerger, M. J., Donnay, J. D. H., Fischer, W., Hahn, Th., Koptsik, V. A., Mackay, A. L., Wondratschek, H., Wilson, A. J. C. & Abrahams, S. C. (1985). Nomenclature for crystal families, Bravaislattice types and arithmetic classes. Acta Cryst. A41, 278–280.
Wondratschek, H. (2005). Introduction to spacegroup symmetry. International tables for crystallography, Vol. A, edited by Th. Hahn, Part 8. Heidelberg: Springer.