International
Tables for Crystallography Volume A Spacegroup symmetry Edited by M. I. Aroyo © International Union of Crystallography 2016 
International Tables for Crystallography (2016). Vol. A, ch. 3.5, pp. 838850

The Euclidean and the affine normalizer of a space group form the appropriate tool to define equivalence relationships on sets of objects that are not symmetryequivalent in this space group but `play the same role' with respect to this group. Two such objects referring to the same space group will be called Euclidean or affineequivalent if there exists a Euclidean or affine mapping that maps the two objects onto one another and, in addition, maps the space group onto itself.
3.5.3.2. Equivalent point configurations, equivalent Wyckoff positions and equivalent descriptions of crystal structures
In the crystal structure of copper, all atoms are symmetryequivalent with respect to space group . The pattern of Cu atoms may be described equally well by Wyckoff position or . The Euclidean normalizer of gives the relation between the two descriptions.
Two point configurations (crystallographic orbits)^{3} of a space group are called Euclidean or equivalent (affine or equivalent) if they are mapped onto each other by the Euclidean (affine) normalizer of .
Affineequivalent point configurations play the same role with respect to the spacegroup symmetry, i.e. their points are embedded in the pattern of symmetry elements in the same way. Euclideanequivalent point configurations are congruent and may be interchanged when passing from one description of a crystal structure to another.
Starting from any given point configuration of a space group , one may derive all Euclideanequivalent point configurations and – except for monoclinic and triclinic space groups – all affineequivalent ones by successive application of the `additional generators' of the normalizer as given in Tables 3.5.2.3, 3.5.2.4 and 3.5.2.5.
Examples
The difference between the two examples is the following: The reference point of the first example does not change its site symmetry .3m when passing from to . Point of the second example, however, has site symmetry in , but in .
The following rule holds without exception: The number of point configurations equivalent to a given one is equal to the quotient , with i being the subgroup index of in its Euclidean or affine normalizer and the subgroup index between the corresponding two sitesymmetry groups of any point in the original point configuration.
When referring to the Euclidean normalizer, i_{s} may be higher than 1 only if the eigensymmetry of the point configuration under consideration (i.e. the group of all motions that maps the point configuration onto itself, cf. Sections 1.4.4.4 and 3.4.1.3 ) is a proper supergroup of . If designates the intersection group of with the eigensymmetry group of the point configuration, the number of Euclideanequivalent point configurations equals the index of in .
Example
The Euclidean and affine normalizer of is with index 8. Point configuration with forms a facecentred cubic lattice with eigensymmetry . The reference point has site symmetry .3. in but in . The number of equivalent point configurations, therefore, is . One additional point configuration is generated by the translation : with , the two others by applying the dglide reflection , , to the first two point configurations: with and . The intersection group of the eigensymmetry with the normalizer is . Its index 4 in gives again the number of equivalent point configurations.
The set of equivalent point configurations is always infinite if the normalizer contains continuous translations, but this set may be described by a finite number of subsets due to noncontinuous translations.
Example
The Euclidean and affine normalizer of is (a, b, c). With the aid of the `additional generators' given in Table 3.5.2.5, one can calculate two subsets of point configurations that are equivalent to a given general point configuration with , , : with and . If, however, the coordinates for the original point configuration are specialized, e.g. to , or to , , only one subset exists, namely or , respectively. The reduction of the number of subsets is a consequence of the enhancement of the site symmetry in the normalizer (.2. or 622, respectively), but the index , as introduced above, does not necessarily give the reduction factor for the number of subsets.
It has to be noticed that for most space groups with a Euclidean normalizer containing continuous translations the index is larger than 1 for all point configurations, i.e. the number of subsets of equivalent point configurations is necessarily reduced. The general Wyckoff position of such a space group does not belong to a characteristic type of Wyckoff sets (cf. Sections 1.4.4 and 3.4.1.3 ) and the eigensymmetry of all corresponding point configurations is enhanced.
Example
The Euclidean and affine normalizer of P6 is (a, b, c). As a consequence of the continuous translations, the site symmetry of any point is at least m.. in . With the aid of the `additional generators', one calculates four subsets of point configurations that are equivalent to a given general point configuration with , , : ; ; ; . The first two and the second two subsets coincide, however.
According to the above examples, Euclidean (affine) equivalent point configurations may or may not belong to the same Wyckoff position. Consequently, normalizers also define equivalence relations on Wyckoff positions:
Two Wyckoff positions of a space group are called Euclidean or equivalent (affine or equivalent) if their point configurations are mapped onto each other by the Euclidean (affine) normalizer of .
Euclideanequivalent Wyckoff positions are important for the description or comparison of crystal structures in terms of atomic coordinates. Affineequivalent Wyckoff positions result in Wyckoff sets (cf. Sections 1.4.4 and 3.4.1.2 ) and form the necessary basis for the definition of lattice complexes. All sitesymmetry groups corresponding to equivalent Wyckoff positions are conjugate in the respective normalizer.
Examples
The Euclidean and affine normalizer of is . It maps the point configurations , , and (bodycentred tetragonal lattices) onto each other. Accordingly, Wyckoff positions a to d are affineequivalent and together form a Wyckoff set. Analogous point configurations exist in subgroup of (again Wyckoff positions a to d). The Euclidean and affine normalizer of , however, is , not containing . Therefore, Wyckoff positions a and b form one Wyckoff set, c and d a different one. This is also reflected in the sitesymmetry groups .. and 2.22.
The existence of Euclideanequivalent point configurations results in different but equivalent descriptions of crystal structures (exception: crystal structures with symmetry or ). All such equivalent descriptions are derived by applying the additional generators of the Euclidean normalizer of the space group to all point configurations of the original description. Since an adequate description of a crystal structure always displays the full symmetry group of that structure, the number of equivalent descriptions must equal the index of in .
Example
Ag_{3}PO_{4} crystallizes with symmetry (cf. Masse et al., 1976): P at , Ag at and O at with . with index 4 gives rise to three additional equivalent descriptions: yields P at , Ag at and O at with ; inversion through the origin results in P at , Ag at , O at with and in P at , Ag at and O at with . Although the phosphorus configuration is the same for all descriptions and the silver and oxygen atoms refer to only two configurations each, their combinations result in a total of four different equivalent descriptions of the structure.
If the Euclidean normalizer of a space group contains continuous translations, each crystal structure with that symmetry refers to an infinite set of equivalent descriptions. This set may be subdivided into a finite number of subsets in such a way that the descriptions of each subset vary according to the continuous translations. The number of these subsets is given by the product of the finite factors listed in the last column of Tables 3.5.2.3, 3.5.2.4 and 3.5.2.5.
Example
The tetragonal form of BaTiO_{3} has been described in space group P4mm (cf. e.g. Buttner & Maslen, 1992): Ba at with , Ti at with , O1 at with , and O2 at with . gives rise to equivalent descriptions of this structure. The continuous translation with vector yields a first infinite subset of equivalent descriptions: Ba at with , Ti at with , O1 at with , and O2 at with . The translation with vector generates a second infinite subset: Ba at with , Ti at with , O1 at with , and O2 at with . Inversion through the origin causes two further infinite subsets of equivalent coordinate descriptions of BaTiO_{3}: first, Ba at with , Ti at with , O1 at with , and O2 at with ; second, Ba at with , Ti at with , O1 at with , and O2 at with .
For any chiral crystal structure two variants with opposite handedness exist and the corresponding symmetry group is a Sohncke space group, i.e. a group without improper motions. Two cases should be distinguished:
Example for case (1)
The phosphorus atoms of NaP form helical chains with (slightly distorted) symmetry (rod group of the molecules, not imposed by the space group; Schnering & Hönle, 1979). The chains wind round axes parallel to b in the space group . The Euclidean normalizer is with halved basis vectors (Table 3.5.2.4); the chiralitypreserving Euclidean normalizer is its noncentrosymmetric subgroup . The index of in is 16 and shows the existence of 16 equivalent descriptions, namely of 8 enantiomorphic pairs. The index of in is 8. This corresponds to eight equivalent coordinate sets which can be obtained by application of the translations ; ; ; ; ; ; and . The remaining eight, with the opposite chirality, result from the same translations in addition to an inversion. The inversion converts the lefthanded helices to the enantiomorphic righthanded helices. The chirality is a property of the polymeric ions. The space group itself is not chiral, but it contains no symmetry operations that interconvert the enantiomeric ions; it is a Sohncke space group. The nonchiral space group is compatible with either of the enantiomorphic forms of NaP.
Example for case (2)
Lowtemperature quartz crystallizes either with symmetry (lefthanded quartz) or (righthanded), which is a pair of enantiomorphic (chiral) space groups. Their Euclidean normalizers also form a pair of enantiomorphic space groups, namely = = and with halved c vectors; they consist only of proper motions and thus are chirality preserving. The index of in its Euclidean normalizer is 4, which means that there are four equivalent descriptions for lefthanded quartz. They are mapped onto each other by the translation and the rotation (Table 3.5.2.5). The same applies to righthanded quartz. Lefthanded quartz cannot be mapped onto righthanded quartz by the Euclidean normalizer. The four equivalent descriptions retain the chirality.
More details on Euclideanequivalent point configurations and descriptions of crystal structures have been given by Fischer & Koch (1983) and Koch & Fischer (2006).
All the different but equivalent descriptions of a crystal structure refer to different but equivalent lists of structure factors. These lists contain the same moduli of the structure factors , but they differ in their indices and phases .
In the previous section, the unit cell (basis and origin) of a space group has been considered fixed, whereas the crystal structure or its enantiomorph was embedded into the pattern of symmetry elements at different but equivalent locations. In the present context, however, it is advantageous to regard the crystal structure as being fixed and to let transform the basis and the origin with respect to which the crystal structure is described. This matches the usual approach to resolve the ambiguities in direct methods by fixing the origin and the absolute structure.
Each matrix–column pair (P, p) representing an element of describes a unitcell transformation of . According to Section 1.5.2 the following equations hold: As a consequence, the phase of a given structure factor also changes into .
Similar to equivalent descriptions of a crystal structure, it is possible to derive all equivalent lists of structure factors: The additional generators of are pure translations that leave the indices h of all structure factors unchanged but transform their phases according to . Therefore, the origin for the description of the crystal structure may be fixed by appropriate restrictions of some phases. The number of these phases equals the number of additional generators of , given in Tables 3.5.2.3, 3.5.2.4 or 3.5.2.5. These generators [together with the inversion that generates , if present] also determine the parity classes of the structure factors and the ranges for the phase restrictions.
The inversion that generates changes the handedness of the coordinate system in direct space and in reciprocal space and, therefore, gives rise to different absolute crystal structures. The indices of a given structure factor change from h to , whereas the phase is influenced only if the symmetry centre is not located at .
If no anomalous scattering is observed, Friedel's rule holds and the moduli of any two structure factors with indices h and −h are equal. As a consequence, different absolute crystal structures result in lists of structure factors and indices that differ only in their phases. Therefore, one phase may be restricted to an appropriate range of length π to fix the absolute structure. This is not possible if anomalous scattering has been observed.
If differs from , i.e. if and belong to different Laue classes, the further generators of always change the orientation of the basis in direct and in reciprocal space. Therefore, the indices of the structure factors are permuted, but their phases are transformed only if . The choice between these equivalent descriptions of the crystal structure is made when indexing the reflection pattern. In the case of anomalous scattering, the similar choice between the absolute structures is also combined with the indexing procedure.
Example
According to Table 3.5.2.4, eight equivalent descriptions exist for each crystal structure with symmetry F222. Four of them differ only by an origin shift and the other four are enantiomorphic to the first four. transforms all phases according to , which gives rise to four parity classes of structure factors: , , and (n integer). As generates all additional translations of , restriction of one phase to a range of length fixes the origin. Restriction of a second phase to an appropriately chosen range of length π discriminates between pairs of enantiomorphic descriptions in the absence of anomalous scattering. For inversion through the origin , the corresponding change of phases is . Table 3.5.3.1 shows, for structure factors from all parity classes, how their phases depend on the chosen description of the crystal structure. Only phases from parity classes or determine the origin in a unique way. The phase that fixes the absolute structure may be chosen from any parity class but the appropriate range for its restriction depends on the parity classes of and and, moreover, on the range chosen for . If, for instance, with is restricted to , one of the following restrictions may be chosen for : for ; for ; for ; for . If, however, the phase of the same first reflection was restricted to , the possible restrictions for the second phase change to: for or ; for or (for further details, cf. Koch, 1986).

The Euclidean or affine normalizer of a space group maps any subgroup or supergroup of either onto itself or onto another subgroup or supergroup of . Accordingly, these normalizers define equivalence relationships on the sets of subgroups and supergroups of (Koch, 1984b):
Two subgroups or supergroups of a space group are called Euclidean or equivalent (affine or equivalent) if they are mapped onto each other by an element of the Euclidean (affine) normalizer of , i.e. if they are conjugate subgroups of the Euclidean (affine) normalizer.
In the following, the term `equivalent subgroups (supergroups)' is used if a statement is true for Euclideanequivalent and affineequivalent subgroups (supergroups), and is used to designate the Euclidean as well as the affine normalizer.
The knowledge of Euclideanequivalent subgroups is necessary in connection with the possible deformations of a crystal structure due to symmetry reduction. Affineequivalent subgroups play an important role for the derivation and classification of blackandwhite groups (magnetic groups) and of colour groups (cf. for example Schwarzenberger, 1984). Information on equivalent supergroups is useful for the determination of the idealized type of a crystal structure.
For any pair of space groups and with , the relation between the two normalizers and controls the subgroups of that are equivalent to and the supergroups of equivalent to . The intersection group of both normalizers, may or may not coincide with and/or with . The following two statements hold generally:
Equivalent subgroups are conjugate in if and only if . In this case, contains elements not belonging to and the cosets of in refer to the different conjugate subgroups.
Examples
3.5.3.5. Reduction of the parameter regions to be considered for geometrical studies of point configurations
Each point configuration with spacegroup symmetry may be described by its metrical and coordinate parameters. To cover all point configurations belonging to a certain spacegroup type exactly once, the metrical parameters of have to be varied without restrictions, whereas the coordinate parameters x, y and z must be restricted to one asymmetric unit of . For the study of the geometrical properties of point configurations (e.g. spherepacking conditions or types of Dirichlet domains etc.), the Euclidean normalizers (cf. e.g. Laves, 1931; Fischer, 1971, 1991; Koch, 1984a) as well as the affine normalizers (cf. Fischer, 1968) of the space groups allow a further reduction of the parameter regions that have to be considered.
Examples
References
Buttner, R. H. & Maslen, E. N. (1992). Structural parameters and electron difference density in BaTiO_{3}. Acta Cryst. B48, 764–769.Fischer, W. (1968). Kreispackungsbedingungen in der Ebene. Acta Cryst. A24, 67–81.
Fischer, W. (1971). Existenzbedingungen homogener Kugelpackungen in Raumgruppen tetragonaler Symmetrie. Z. Kristallogr. 133, 18–42.
Fischer, W. (1991). Tetragonal sphere packings II. Lattice complexes with two degrees of freedom. Z. Kristallogr. 194, 87–110.
Fischer, W. & Koch, E. (1983). On the equivalence of point configurations due to Euclidean normalizers (Cheshire groups) of space groups. Acta Cryst. A39, 907–915.
Koch, E. (1984a). A geometrical classification of cubic point configurations. Z. Kristallogr. 166, 23–52.
Koch, E. (1984b). The implications of normalizers on group–subgroup relations between space groups. Acta Cryst. A40, 593–600.
Koch, E. (1986). Implications of Euclidean normalizers of space groups in reciprocal space. Cryst. Res. Technol. 21, 1213–1219.
Koch, E. & Fischer, W. (2006). Normalizers of space groups, a useful tool in crystal description, comparison and determination. Z. Kristallogr. 221, 1–14.
Laves, F. (1931). Ebenenteilung in Wirkungsbereiche. Z. Kristallogr. 76, 277–284.
Masse, R., Tordjman, I. & Durif, A. (1976). Affinement de la structure cristalline du monophosphate d'argent Ag_{3}PO_{4}. Existence d'une forme haute température. Z. Kristallogr. 144, 76–81.
Schnering, H. G. v. & Hönle, W. (1979). Zur Chemie und Strukturchemie der Phosphide und Polyphosphide. 20. Darstellung, Struktur und Eigenschaften der Alkalimetallmonophosphide NaP und KP. Z. Anorg. Allg. Chem. 456, 194–206.
Schwarzenberger, R. L. E. (1984). Colour symmetry. Bull. London Math. Soc. 16, 209–240.