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. 3.1, pp. 4454
doi: 10.1107/97809553602060000506 Chapter 3.1. Spacegroup determination and diffraction symbolsIn this chapter, the determination of space groups from the Laue symmetry and the reflection conditions, as obtained from diffraction patterns, is discussed. Apart from a small section where differences between reflections hkl and due to anomalous dispersion are discussed, it is assumed that Friedel's rule holds, i.e. that . This implies that the reciprocal lattice weighted by has an inversion centre, even if this is not the case for the crystal under consideration. Accordingly, the symmetry of the weighted reciprocal lattice belongs, as was discovered by Friedel, to one of the eleven Laue classes. Laue class plus reflection conditions in most cases do not uniquely specify the space group. A summary is given of methods that help to overcome these ambiguities, especially with respect to the presence or absence of an inversion centre in the crystal. 
In this chapter, the determination of space groups from the Laue symmetry and the reflection conditions, as obtained from diffraction patterns, is discussed. Apart from Section 3.1.6.5, where differences between reflections hkl and due to anomalous dispersion are discussed, it is assumed that Friedel's rule holds, i.e. that . This implies that the reciprocal lattice weighted by has an inversion centre, even if this is not the case for the crystal under consideration. Accordingly, the symmetry of the weighted reciprocal lattice belongs, as was discovered by Friedel (1913), to one of the eleven Laue classes of Table 3.1.2.1. As described in Section 3.1.5, Laue class plus reflection conditions in most cases do not uniquely specify the space group. Methods that help to overcome these ambiguities, especially with respect to the presence or absence of an inversion centre in the crystal, are summarized in Section 3.1.6.
Spacegroup determination starts with the assignment of the Laue class to the weighted reciprocal lattice and the determination of the cell geometry. The conventional cell (except for the case of a primitive rhombohedral cell) is chosen such that the basis vectors coincide as much as possible with directions of highest symmetry (cf. Chapters 2.1 and 9.1 ).
The axial system should be taken righthanded. For the different crystal systems, the symmetry directions (blickrichtungen) are listed in Table 2.2.4.1 . The symmetry directions and the convention that, within the above restrictions, the cell should be taken as small as possible determine the axes and their labels uniquely for crystal systems with symmetry higher than orthorhombic. For orthorhombic crystals, three directions are fixed by symmetry, but any of the three may be called a, b or c. For monoclinic crystals, there is one unique direction. It has to be decided whether this direction is called b, c or a. If there are no special reasons (physical properties, relations with other structures) to decide otherwise, the standard choice b is preferred. For triclinic crystals, usually the reduced cell is taken (cf. Chapter 9.2 ), but the labelling of the axes remains a matter of choice, as in the orthorhombic system.
If the lattice type turns out to be centred, which reveals itself by systematic absences in the general reflections hkl (Section 2.2.13 ), examination should be made to see whether the smallest cell has been selected, within the conventions appropriate to the crystal system. This is necessary since Table 3.1.4.1 for spacegroup determination is based on such a selection of the cell. Note, however, that for rhombohedral space groups two cells are considered, the triple hexagonal cell and the primitive rhombohedral cell.
The Laue class determines the crystal system. This is listed in Table 3.1.2.1. Note the conditions imposed on the lengths and the directions of the cell axes as well as the fact that there are crystal systems to which two Laue classes belong.

In Section 2.2.13 , it has been shown that `extinctions' (sets of reflections that are systematically absent) point to the presence of a centred cell or the presence of symmetry elements with glide or screw components. Reflection conditions and Laue class together are expressed by the Diffraction symbol, introduced by Buerger (1935, 1942, 1969); it consists of the Laueclass symbol, followed by the extinction symbol representing the observed reflection conditions. Donnay & Harker (1940) have used the concept of extinctions under the name of `morphological aspect' (or aspect for short) in their studies of crystal habit (cf. Crystal Data, 1972). Although the concept of aspect applies to diffraction as well as to morphology (Donnay & Kennard, 1964), for the present tables the expression `extinction symbol' has been chosen because of the morphological connotation of the word aspect.
The Extinction symbols are arranged as follows. First, a capital letter is given representing the centring type of the cell (Section 1.2.1 ). Thereafter, the reflection conditions for the successive symmetry directions are symbolized. Symmetry directions not having reflection conditions are represented by a dash. A symmetry direction with reflection conditions is represented by the symbol for the corresponding glide plane and/or screw axis. The symbols applied are the same as those used in the Hermann–Mauguin spacegroup symbols (Section 1.3.1 ). If a symmetry direction has more than one kind of glide plane, for the diffraction symbol the same letter is used as in the corresponding spacegroup symbol. An exception is made for some centred orthorhombic space groups where two glideplane symbols are given (between parentheses) for one of the symmetry directions, in order to stress the relation between the diffraction symbol and the symbols of the `possible space groups'. For the various orthorhombic settings, treated in Table 3.1.4.1, the top lines of the twoline spacegroup symbols in Table 4.3.2.1 are used. In the monoclinic system, dummy numbers `1' are inserted for two directions even though they are not symmetry directions, to bring out the differences between the diffraction symbols for the b, c and a settings.
Example
Laue class:
Reflection conditions: As there are both c and n glide planes perpendicular to b, the diffraction symbol may be given as or as . In analogy to the symbols of the possible space groups, C1c1 (9) and , the diffraction symbol is called .
For another cell choice, the reflection conditions are: For this second cell choice, the glide planes perpendicular to b are n and a. The diffraction symbol is given as , in analogy to the symbols A1n1 (9) and adopted for the possible space groups.
Reflection conditions, diffraction symbols, and possible space groups are listed in Table 3.1.4.1. For each crystal system, a different table is provided. The monoclinic system contains different entries for the settings with b, c and a unique. For monoclinic and orthorhombic crystals, all possible settings and cell choices are treated. In contradistinction to Table 4.3.2.1 , which lists the spacegroup symbols for different settings and cell choices in a systematic way, the present table is designed with the aim to make spacegroup determination as easy as possible.
^{†}Pair of space groups with common point group and symmetry elements but differing in the relative location of these elements.
^{‡}Pair of enantiomorphic space groups, cf. Section 3.1.5. ^{§}Condition: . ^{¶}For obverse and reverse settings cf. Section 1.2.1 . The obverse setting is standard in these tables. The transformation reverse obverse is given by , , . ^{††}For No. 205, only cyclic permutations are permitted. Conditions are 0kl: ; h0l: ; hk0: . 
The lefthand side of the table contains the Reflection conditions. Conditions of the type or are abbreviated as h or . Conditions like are quoted as h, k; in this case, the condition is not listed as it follows directly from . Conditions with , , or more complicated expressions are listed explicitly.
From left to right, the table contains the integral, zonal and serial conditions. From top to bottom, the entries are ordered such that left columns are kept empty as long as possible. The leftmost column that contains an entry is considered as the `leading column'. In this column, entries are listed according to increasing complexity. This also holds for the subsequent columns within the restrictions imposed by previous columns on the left. The makeup of the table is such that observed reflection conditions should be matched against the table by considering, within each crystal system, the columns from left to right.
The centre column contains the Extinction symbol. To obtain the complete diffraction symbol, the Laueclass symbol has to be added in front of it. Be sure that the correct Laueclass symbol is used if the crystal system contains two Laue classes. Particular care is needed for Laue class in the trigonal system, because there are two possible orientations of this Laue symmetry with respect to the crystal lattice, and . The correct orientation can be obtained directly from the diffraction record.
The righthand side of the table gives the Possible space groups which obey the reflection conditions. For crystal systems with two Laue classes, a subdivision is made according to the Laue symmetry. The entries in each Laue class are ordered according to their point groups. All space groups that match both the reflection conditions and the Laue symmetry, found in a diffraction experiment, are possible space groups of the crystal.
The space groups are given by their short Hermann–Mauguin symbols, followed by their number between parentheses, except for the monoclinic system, where full symbols are given (cf. Section 2.2.4 ). In the monoclinic and orthorhombic sections of Table 3.1.4.1, which contain entries for the different settings and cell choices, the `standard' spacegroup symbols (cf. Table 4.3.2.1 ) are printed in bold face. Only these standard representations are treated in full in the spacegroup tables.
Example
The diffraction pattern of a compound has Laue class mmm. The crystal system is thus orthorhombic. The diffraction spots are indexed such that the reflection conditions are ; ; ; . Table 3.1.4.1 shows that the diffraction symbol is mmmPcn–. Possible space groups are Pcn2 (30) and Pcnm (53). For neither space group does the axial choice correspond to that of the standard setting. For No. 30, the standard symbol is Pnc2, for No. 53 it is Pmna. The transformation from the basis vectors , used in the experiment, to the basis vectors of the standard setting is given by for No. 30 and by for No. 53.
Possible pitfalls
Errors in the spacegroup determination may occur because of several reasons.
Table 3.1.4.1 contains 219 extinction symbols which, when combined with the Laue classes, lead to 242 different diffraction symbols. If, however, for the monoclinic and orthorhombic systems (as well as for the R space groups of the trigonal system), the different cell choices and settings of one space group are disregarded, 101 extinction symbols^{1} and 122 diffraction symbols for the 230 spacegroup types result.
Only in 50 cases does a diffraction symbol uniquely identify just one space group, thus leaving 72 diffraction symbols that correspond to more than one space group. The 50 unique cases can be easily recognized in Table 3.1.4.1 because the line for the possible space groups in the particular Laue class contains just one entry.
The nonuniqueness of the spacegroup determination has two reasons:
In some cases, chemical information determines whether or not the space group is centrosymmetric. For instance, all proteins crystallize in noncentrosymmetric space groups as they are constituted of Lamino acids only. Less certain indications may be obtained by considering the number of molecules per cell and the possible spacegroup symmetry. For instance, if experiment shows that there are two molecules of formula per cell in either space group or and if the molecule cannot possibly have either a mirror plane or an inversion centre, then there is a strong indication that the correct space group is . Crystallization of in with random disorder of the molecules cannot be excluded, however. In a similar way, multiplicities of Wyckoff positions and the number of formula units per cell may be used to distinguish between space groups.
This is discussed in Chapter 10.2 . In favourable cases, suitably chosen methods can prove the absence of an inversion centre or a mirror plane.
Xray data can give a strong clue to the presence or absence of an inversion centre if not only the symmetry of the diffraction pattern but also the distribution of the intensities of the reflection spots is taken into account. Methods have been developed by Wilson and others that involve a statistical examination of certain groups of reflections. For a textbook description, see Lipson & Cochran (1966) and Wilson (1970). In this way, the presence of an inversion centre in a threedimensional structure or in certain projections can be tested. Usually it is difficult, however, to obtain reliable conclusions from projection data. The same applies to crystals possessing pseudosymmetry, such as a centrosymmetric arrangement of heavy atoms in a noncentrosymmetric structure. Several computer programs performing the statistical analysis of the diffraction intensities are available.
The application of Patterson syntheses for spacegroup determination is described by Buerger (1950, 1959).
Friedel's rule, , does not hold for noncentrosymmetric crystals containing atoms showing anomalous dispersion. The difference between these intensities becomes particularly strong when use is made of a wavelength near the resonance level (absorption edge) of a particular atom in the crystal. Synchrotron radiation, from which a wide variety of wavelengths can be chosen, may be used for this purpose. In such cases, the diffraction pattern reveals the symmetry of the actual point group of the crystal (including the orientation of the point group with respect to the lattice).
One or more of the methods discussed above may reveal whether or not the point group of the crystal has an inversion centre. With this information, in addition to the diffraction symbol, 192 space groups can be uniquely identified. The rest consist of the eleven pairs of enantiomorphic space groups, the two `special pairs' and six further ambiguities: 3 in the orthorhombic system (Nos. 26 & 28, 35 & 38, 36 & 40), 2 in the tetragonal system (Nos. 111 & 115, 119 & 121), and 1 in the hexagonal system (Nos. 187 & 189). If not only the point group but also its orientation with respect to the lattice can be determined, the six ambiguities can be resolved. This implies that 204 space groups can be uniquely identified, the only exceptions being the eleven pairs of enantiomorphic space groups and the two `special pairs'.
References
Buerger, M. J. (1935). The application of plane groups to the interpretation of Weissenberg photographs. Z. Kristallogr. 91, 255–289.Buerger, M. J. (1942). Xray crystallography, Chap. 22. New York: Wiley.
Buerger, M. J. (1950). The crystallographic symmetries determinable by Xray diffraction. Proc. Natl Acad. Sci. USA, 36, 324–329.
Buerger, M. J. (1959). Vector space, pp. 167–168. New York: Wiley.
Buerger, M. J. (1960). Crystalstructure analysis, Chap. 5. New York: Wiley.
Buerger, M. J. (1969). Diffraction symbols. Chap. 3 of Physics of the solid state, edited by S. Balakrishna, pp. 27–42. London: Academic Press.
Catti, M. & Ferraris, G. (1976). Twinning by merohedry and Xray crystal structure determination. Acta Cryst. A32, 163–165.
Crystal Data (1972). Vol. I, General Editors J. D. H. Donnay & H. M. Ondik, Supplement II, pp. S41–52. Washington: National Bureau of Standards.
Donnay, J. D. H. & Harker, D. (1940). Nouvelles tables d'extinctions pour les 230 groupes de recouvrements cristallographiques. Nat. Can. 67, 33–69, 160.
Donnay, J. D. H. & Kennard, O. (1964). Diffraction symbols. Acta Cryst. 17, 1337–1340.
Friedel, M. G. (1913). Sur les symétries cristallines que peut révéler la diffraction des rayons Röntgen. C. R. Acad. Sci. Paris, 157, 1533–1536.
International Tables for Xray Crystallography (1952; 1969). Vol. I, edited by N. F. M. Henry & K. Lonsdale. Birmingham: Kynoch Press. [Abbreviated as IT (1952) and IT (1969).]
Koch, E. (2004). Twinning. International Tables for Crystallography Vol. C, 3rd ed., edited by E. Prince, ch. 1.3. Dordrecht: Kluwer Academic Publishers.
Lipson, H. & Cochran, W. (1966). The determination of crystal structures, Chaps. 3 and 4.4. London: Bell.
PerezMato, J. M. & Iglesias, J. E. (1977). On simple and double diffraction enhancement of symmetry. Acta Cryst. A33, 466–474.
Wilson, A. J. C. (1970). Elements of Xray crystallography, Chap. 8. Reading, MA: Addison Wesley.