research papers
Equivalence of
groups^{a}Laboratory of Crystallography, University of Bayreuth, Bayreuth, Germany, and ^{b}Department of Physics and Astronomy, Brigham Young University, Provo, Utah 84602, USA
^{*}Correspondence email: smash@unibayreuth.de
An algorithm is presented which determines the equivalence of two settings of a (3 + d)dimensional (d = 1, 2, 3). The algorithm has been implemented as a web tool on , providing the transformation of any usergiven to the standard setting of this in . It is shown how the standard setting of a can be directly obtained by an appropriate transformation of the externalspace lattice vectors (the basic structure unit cell) and a transformation of the internalspace lattice vectors (new modulation wavevectors are linear combinations of old modulation wavevectors plus a threedimensional reciprocallattice vector). The need for nonstandard settings in some cases and the desirability of employing standard settings of groups in other cases are illustrated by an analysis of the symmetries of a series of compounds, comparing published and standard settings and the transformations between them. A compilation is provided of standard settings of compounds with two and threedimensional modulations. The problem of settings of groups is discussed for incommensurate composite crystals and for chiral groups.
Keywords: symmetry; superspace groups; twodimensionally modulated crystals; threedimensionally modulated crystals.
1. Introduction
Symmetry is one of the most important concepts in the solidstate sciences. Knowledge of the symmetry of a crystalline compound allows the understanding of many aspects of its physical behavior, including degeneracies, the possibility of possessing nonlinear properties and the anisotropy of the response to external fields. A change in symmetry at different temperatures, pressures or compositions is used as the key parameter for characterizing phase transitions of a compound. Symmetry is used for the description of phonon and electron bands and thus allows the interpretation of spectroscopic measurements on materials. Not least, symmetry restrictions on structural parameters are essential for successful refinements of crystal structures.
Theoretically, the classification of symmetry is solved. The 230 space groups give the 230 possibilities for the symmetry of a periodic structure (Hahn, 2002). Aperiodic crystals lack threedimensional (3D) translational symmetry (Janssen et al., 2006, 2007; van Smaalen, 2007). The structures of incommensurately modulated crystals are characterized by a threedimensional lattice for the average structure together with d modulation waves () describing deviations from the latticeperiodic structure. Their symmetries are given by (3+d)dimensional [(3+d)D] groups (de Wolff et al., 1981). The latter are space groups of (3+d)D space, which have to obey particular conditions in order to qualify as symmetry groups for the symmetries of aperiodic crystals. Incommensurate composite crystals are described by the same groups as modulated crystals (van Smaalen, 2007), while quasicrystals require a slightly modified treatment (Janssen et al., 2007; Steurer & Deloudi, 2009).
We have recently generated a complete list of (3+d) for d = 1, 2 and 3 (Stokes et al., 2011a). The list agrees with previous information on (3+1)D groups (Janssen et al., 2006), but it contains numerous corrections for groups of dimensions d = 2 and 3 (Yamamoto, 2005) and even some corrections to the Bravais classes of dimensions 2 and 3 (Janssen et al., 2006). The results of Stokes et al. (2011a) are compiled in the form of the webbased data repository (Stokes et al., 2011b). provides several types of information for each including the the list of symmetry operators and in both standard and supercentered settings.
groups and their Bravais classes of dimensionsIt is noticed that Stokes et al. (2011a) have defined the standard settings and their symbols by a set of judiciously chosen rules, which, however, include subjective choices. The standard setting thus is defined as the setting included in the list of groups on SSG(3+d)D.
The use of alternate settings of space groups is a well known feature for threedimensional space groups. Volume A of the International Tables for Crystallography (Hahn, 2002) provides several settings for monoclinic space groups, thus showing, for example, that C2/c, A2/n and I2/a denote different settings of No. 15.^{1} Alternate settings of threedimensional space groups arise owing to different choices of the Trivial transformations include a simple relabeling of the axes a, b and c. For monoclinic space groups this implies the freedom to select the unique axis as , or . For the example of No. 15 the transformation
takes the setting C2/c into A2/n, while both settings refer to a unique axis . Applying the transformation of equation (1) to the setting A2/n results in the third setting I2/a. Other notorious pairs of equivalent settings include the I and F settings for space groups based on the Icentered tetragonal lattice, primitive and centered hexagonal settings as well as obverse versus reverse settings for rhombohedral space groups, and the Hcentered setting as an alternative to the primitive setting for trigonal space groups (Hahn, 2002).
Different diffraction experiments independently lead to any of the possible settings of the
It is then of high practical importance to find the transformation between these settings or to establish that two different space groups have indeed been obtained. The latter situation implies different compounds or different phases of one compound, while different settings of one imply that the same compound has been studied. In other experiments it is important that a previously defined orientation of a crystalline material is reestablished, thus requiring the relation to be found between the newly found setting and a standard setting.As a start, (3+d)D groups are based on a basic structure lattice and in threedimensional space. In addition, groups may appear in many more different settings, owing to the ambiguity in the choice of the modulation wavevectors characterizing the structure and the diffraction pattern. The equivalence of different settings of a is not always obvious. In some cases, establishing an actual equivalence can be a computationally prohibitive task unless appropriate algorithms are used. Here we present such an algorithm, which was used but not described in detail in our previous publication Stokes et al. (2011a). It is available within SSG(3+d)D as a tool with which to determine the transformation between a userprovided setting of a (3+d)D and the standard setting defined by SSG(3+d)D (Stokes et al., 2011b). It thus can be used to establish or disprove the equivalence of settings.
groups for incommensurately modulated compounds and incommensurate composite crystals exhibit the same variation of settings as threedimensional space groups do, sinceCoordinate transformations between different settings of threedimensional space groups are discussed in Volume A of International Tables for Crystallography (Hahn, 2002). For (3+1)D groups, typical transformations are presented in van Smaalen (2007). The possibility to combine two modulation wavevectors into an equivalent but different set of two wavevectors leads to new types of transformations for d = 2 and 3.
One goal of this paper is to present an overview of typical coordinate transformations that may occur between settings of (3+d)D space, as given on SSG(3+d)D (Stokes et al., 2011a), and an experimentally related description in terms of a rotation and an origin shift in three dimensions (van Smaalen, 2007). Where available, we use substances published in the literature to illustrate important transformation types.
groups. Particular attention is given to the relation between the formal description in2. Equivalence of groups
2.1. Definitions
The following definitions are used by Stokes et al. (2011a), van Smaalen (2007) and Janssen et al. (1995). A ddimensionally modulated structure is characterized by d rationally independent modulation wavevectors with components compiled in a d ×3 matrix according to
For an aperiodic structure at least one component in each row of is an irrational number. The reciprocal vectors in physical space correspond to the reciprocal basis vectors in
The basis vectors of the in are and the coordinates of a point in are .Note that the SSG(3+d)D data repository, and also the webbased findssg and transformssg tools described herein, presently use the {x,y,z,t} notation to indicate coordinates, though the same notation is also commonly used to indicate physical and internalspace coordinates.
An operator g of a (3+d)D G consists of a rotation R_{s}(g) and a translation given in matrix form as
where R is a 3 ×3 integer matrix and is a threedimensional column vector, together defining the operator in physical space. is a d ×d integer matrix, and det(R) = . The d ×3 integer matrix M is defined as [equation (2)]
M has nonzero components only in the case that at least one of the modulation wavevectors incorporates nonzero rational components. Following Stokes et al. (2011a), each operator g can be written as an augmented (4+d) ×(4+d) matrix
that simultaneously treats the point and translational parts of the operation. The action of operator g on a point in then is given by the matrix product
In
a coordinate transformation can be accomplished by the augmented affine transformation matrixThe components of S_{R}, S_{M} and are required to be integers. Also, det(S_{R}) = 1 and . The transformation S can be interpreted as a rotation S_{s} in followed by a change of origin S_{v}. The effect of this transformation in physical space can be described in terms of a rotation of the reciprocal basis and the choice of an alternate set of modulation wavevectors, according to
Two (3+d)D groups, G^{1} and G^{2}, are equivalent if a single transformation S can be found, such that for every g^{2} G^{2},
for some g^{1} G^{1}. Note that these definitions imply that a primitive setting is used for the where all lattice translations are represented by integers, even those which are centering translations in a conventional setting (Stokes et al., 2011a).
It is sufficient that the relation of equivalence [equation (9)] is tested for corresponding pairs of nontranslational generators from the two groups; the generators of the translation need not be considered. Furthermore, a transformation of the type of equation (9) can only be found between operators g^{1l} and g^{2l} if [equation (3)]
where g^{1l} is the lth generator of G^{1} and g^{2l} is the corresponding generator of G^{2}. The appropriate pairs of generators are obtained by consideration of the basic structure implied by the Other, trivial properties that need to be fulfilled for equivalence and that are easily tested include the number of operators in the of the which must be equal for G^{1} and G^{2}.
2.2. The algorithm determining equivalence
The goal of testing for equivalence of two S with which the operators of G^{2} are transformed into corresponding operators of G^{1}, or to establish that a matrix S that solves equation (9) simultaneously for all pairs of generators does not exist. Equation (9) is quadratic in S but can be recast in linear form as SA(g^{2}) = A(g^{1})S. Given that a pairing has been established for the n_{gen} generators of G^{1} and G^{2}, this results in (3+d)^{2}n_{gen} equations for (3+d)^{2} variables S_{s}_{ik},
groups is to find the augmented matrixand (3+d)n_{gen} equations for additional (3+d) variables S_{v}_{i},
The translational parts of the operators g^{1l} and g^{2l} are only known up to a lattice translation, which is taken into account by the mod 1 in equation (12).
Employing the special structure of S [equation (7)], the variables S_{sik} can be ordered in a column vector as
where, for example, is obtained by juxtaposition of the columns of S_{M} into a single column matrix. This procedure eliminates the 3d variables that are zero according to equation (7) and results in [(3+d)^{2}3d]n_{gen} equations in [(3+d)^{2}3d] variables ,
The [(3+d)^{2}3d]n_{gen}×[(3+d)^{2}3d] matrix B_{ij} is obtained by rearrangement of equation (11), followed by linear row operations that bring it into row echelon form. In this form, the first nonzero element in each row occurs in a column where it is the only nonzero element. If B_{ij} is such an element, then B_{kj} = 0 for all and B_{i+k,j+m} = 0 for all and . This equation relates the `dependent' variable to `independent' variables according to
The number of independent equations is smaller than or equal to [(3+d)^{2}3d]n_{gen}. If the number of independent equations is larger than [(3+d)^{2}3d], a solution does not exist for , and the two groups are shown to be inequivalent. Alternatively, the number of independent equations can be equal to [(3+d)^{2}3d], then defining a unique solution for . Finally, the number of independent equations can be smaller than [(3+d)^{2}3d], resulting in more than one solution to equation (15). Once values for the independent variables have been chosen, equation (15) can be used to compute the remaining variables of and S_{s}. For each solution of equation (15), equation (12) may or may not provide a solution for the translational parts of the transformation.
The strategy for finding the transformation S is now as follows. For each trial set of integers for the independent variables , check that all dependent variables compute to have integer values and that det(R) = 1 and . If not, discard the trial set. If so, use the values (both dependent and independent) in equation (12) and explore trial integer sets of variables S_{vi} in search of a modulo 1 solution. If a solution is found, then equations (11) and (12) are both satisfied and the two sets of superspacegroup operators, G^{1} and G^{2}, represent distinct but equivalent settings of the same If no solution is found, then we can assume that the groups are not equivalent, provided that we have a robust algorithm that searches a sufficiently wide range of trial values for each independent variable as to guarantee a solution to equations (11) and (12) provided one exists. The method of choosing these variable exploration ranges is described in the supplementary material.^{2}
The number of variables, and therefore the computational complexity of the search, increases with the dimension d of the modulation. Furthermore, the goal of the proposed analysis is to determine which in the tables a usergiven is equivalent to. Since the number of groups strongly increases with d, the number of candidate equivalencies that need to be tested increases dramatically with increasing dimension of the easily reaching several hundreds of groups in the worst case (orthorhombic symmetry). Thus we need an algorithm for evaluating the possible equivalence of two groups that is not only robust but also efficient. The efficiency of the algorithm boils down to finding the most restrictive number of trial sets of integers for which equivalence needs to be tested (see the supplementary material).
An algorithm based on these rules has been implemented in the software (Stokes et al., 2011a). For any usergiven set of operators, the web tool determines the complete list of operators (modulo lattice translations) of the that they generate, as well as a minimal list of generators, identifies the equivalent in the tables and provides the coordinate transformation S to the standard setting [equation (7)].
3. Alternate settings of (3 + 1)dimensional groups
3.1. The basic structure space group
An important reason for the occurrence of nonstandard settings of ^{3}
groups is the common use of different standard settings for groups and threedimensional space groups. Structural analysis of modulated crystals often proceeds by the initial determination from the main reflections of the periodic basic structure along with its threedimensional (the basic structure BSG). Subsequently, satellite reflections are considered and modulation functions and the are determined. For other substances the incommensurate phase is the result of a so that the threedimensional of the unmodulated structure at ambient conditions is known independently. This or one of its subgroups, is preserved as the BSG of the incommensurate phase.In all these cases the BSG is specified before the symmetry of the modulation is considered. It is then a matter of chance that the Pnma (Hahn, 2002). In this setting, TaSe_{0.36}Te_{2} is modulated with and [equation (2); van der Lee et al. (1994)], so that the (3+1)D superspacegroup symbol is , which is the standard setting for No. 62.1.9.1 in .
thus obtained will or will not be in its standard setting. These points can be illustrated by No. 62 with standard settingThiourea has a latticeperiodic structure with Pnma at ambient conditions. Below T_{i} = 202 K it develops an incommensurate modulation with in the Pnma setting (Gao et al., 1988; Zuñiga et al., 1989). Combining the BSG and modulation wavevector leads to the (3+1)D . shows that this is an alternate setting of No. 62.1.9.3, for which the standard symbol is . The augmented matrix S that transforms coordinates from the original (unprimed) to the standard (primed) settings [equation (6)] is given in as
According to equation (8) the new basis vectors of the basic structure are obtained as the upperleft 3 ×3 part of the transpose of S^{1}. Inspection of equation (16) shows that the basis vectors of the basic structure in the standard (primed) setting are obtained by a transformation of the basis vectors in the original (unprimed) setting as
The fourth row of S shows that the modulation wavevector remains the same, but its components with respect to the transformed basic structure reciprocal basis vectors are obtained by equation (8),
in accordance with the standard setting of
No. 62.1.9.3.3.2. Choice of the modulation wavevector
A second source of variation of settings is the freedom in the choice of the modulation wavevector. Given a modulation with modulation wavevector , any reciprocal vector
where n_{i} are integers is an appropriate choice for the modulation wavevector. A common choice is to select within the first Brillouin zone of the basic structure, i.e. to choose the shortest possible vector [equation (19)]. This choice does not necessarily correspond to the standard setting of the group.
Transformations that change settings have been extensively discussed in van Smaalen (2007). The principles are illustrated by the symmetries of A_{2}BX_{4} ferroelectric compounds with the K_{2}SO_{4} structure type and orthorhombic symmetry according to No. 62 (Hahn, 2002). Basic structures have been described in the standard setting Pnma for some compounds, but the most frequently employed settings are Pnam and Pmcn (Cummins, 1990).
K_{2}SeO_{4} develops an incommensurate modulation below T_{i} = 129.5 K with in the Pnam setting (Yamada & Ikeda, 1984). The incommensurate component is with equal to a small positive number that depends on temperature. The is . shows that this is an alternate setting of No. 62.1.9.6 , involving a transformation of basis vectors and the selection of an alternative modulation wavevector according to
With respect to the transformed reciprocal basis vectors, the components of the modulation wavevector are [equation (8)]
The transformed modulation wavevector has a negative component and a length larger than , which might be considered an unfavorable situation. contains the tool transformssg, with which any userspecified transformation can be applied to the reciprocal basis vectors and modulation wavevectors. Employing this tool with [equation (20)]
shows that a modulation wavevector with components [equation (8)]
again represents the standard setting of
No. 62.1.9.6 . Alternatively, the transformationleads to a nonstandard setting of
No. 62.1.9.6.The analysis of symmetry alone does not consider numerical values of lattice parameters or modulation wavevectors. Therefore, does not employ this information. Accordingly, it is impossible to give preference to one of the transformations of equation (20) or equation (22). Instead, the tool transformssg can be used for transformation to the desired values.
Rb_{2}ZnCl_{4} is incommensurately modulated below T_{i} = 375 K with in the Pmcn setting (Hogervorst, 1986). The incommensurate component is with at room temperature. The is . shows that this is another alternate setting of No. 62.1.9.6 . The transformation now only involves the choice of a different modulation wavevector:
Like in the previous example, the transformation given by does not lead to the setting with the shortest possible modulation wavevector for the case of Rb_{2}ZnCl_{4}. Employing transformssg shows that the standard setting of No. 62.1.9.6 can also be obtained by the transformation of modulation wavevector
As discussed in van Smaalen (2007), replacement of by may change the apparently intrinsic translational component along the fourth coordinate for symmetry operators that possess a nonzero intrinsic translational component in the direction corresponding to the incommensurate component of the modulation wavevector. In the present example that is (c,s) [mirror operation with intrinsic translation ] being replaced by (c,0) [mirror operation with intrinsic translation ]. These two settings of the correlate with different normalmode descriptions of the same for which it has been established that the modulation wavevector with describes a distortion in terms of a soft optical phonon, while leads to the preferred description of the distortion in terms of a soft optical phonon (Axe et al., 1986). It is well known that a change of setting will sometimes change the irreducible representation that contributes to a distortion without changing the physical distortion itself.
3.3. The supercentered setting
Aperiodic crystals are characterized by d modulation wavevectors, each of which possesses at least one irrational component. According to symmetry, the values of these components can be viewed as variable rather than having specific irrational values. The other components are either zero or may assume rational values as allowed by the It is easily checked against the list of Bravais classes of (3+d)D groups that the allowed rational components of modulation wavevectors are and in the standard superspacegroup settings, as well as nonzero integers in the case of BSGs based on a centered lattice of the basic structure. The modulation wavevector is usually separated into a rational part, with zeros and rational numbers as components, and an irrational part, with zeros and the variable components, according to
A modulation wavevector with nonzero rational components may naturally occur when a diffraction pattern of a modulated crystal is indexed, first determining the e.g. the shortest possible vector. However, rational components of imply that the lattice in is a centered lattice with the special feature that centering vectors contain nonzero components both along the fourth coordinate and along at least one of the three physical coordinates. Employing centered unit cells for centered lattices is common practice. It has several advantages in crystallographic analysis, facilitating the description of and the analysis of For groups it has been denoted as the supercentered setting as opposed to the BSG setting, where a modulation wavevector with rational components is combined with the standard centered setting of the BSG (Stokes et al., 2011a). provides symmetry operators for both the BSG and supercentered settings (Stokes et al., 2011b).
– centered if required – of the basic lattice, and then selecting an appropriate modulation wavevector,As an example, consider blue bronze K_{0.3}MoO_{3}. (Despite the decimal subscript in the usual form of the chemical formula, it possesses a fully ordered crystal structure with two formula units K_{3}Mo_{10}O_{30} per unit cell.) Blue bronze develops an incommensurate chargedensity wave (CDW) below T_{CDW} = 183 K with a modulation wavevector
at T = 100 K in the setting of the hightemperature Schutte & de Boer (1993) have determined the crystal structure of the incommensurate phase. With the modulation wavevector of equation (28), they obtained the (3+1)D . This mixed setting contains the center , which possesses a nonzero component along the fourth coordinate (van Smaalen, 2007), and which is different from the BSG setting comprising the C center . shows that is an alternate setting of No. 12.1.8.5 (Table 1 and Fig. 1). In physical space, the transformation from the setting of Schutte & de Boer (1993) to the standard BSG setting involves a permutation of the unitcell axes and the choice of a new modulation wavevector according to
Furthermore, an origin shift of is required in order to bring the origin onto the operator (2,0) instead of (2,s). The components of with respect to the transformed (primed) reciprocal basis vectors follow from equation (8) as [compare equation (21)]
The standard supercentered setting is obtained from the standard BSG setting by the transformation of
basis vectors as given in , and corresponds to the following transformation of the physicalspace basis vectors and modulation wavevector (capital letters indicate the supercentered setting):Observe that the modulation wavevector is purely irrational in the supercentered setting, as expected [equation (27)].

 Figure 1 to No. 12.1.8.5 . 
4. A plethora of settings
4.1. (3 + 2)Dimensional groups
4.1.1. General features
Different settings of (3+d)D groups are obtained when different settings of the BSG are chosen (§3.1). For d = 1, further settings result from the ambiguity in the choice of the modulation wavevector: all modulation wavevectors that differ by a reciprocallattice vector of the basic structure are equally valid [equation (19)] and may define different settings of a (§3.2). For , additional coordinate transformations involve the replacement of the modulation wavevectors by linear combinations of them.
Generalizing equation (7) shows that any set of reciprocal vectors ()
is an appropriate choice for the set of d modulation wavevectors, where S_{Mji} and are integers and . Such linear combinations of the modulation vectors of the standard setting are often necessary to make the description of experimental diffraction data simpler and more intuitive, but can also have the opposite effect if applied arbitrarily.
In analyzing d = 2, it is useful to distinguish between (3+2)D and (3+1+1)D groups, where the latter refer to incommensurate crystals with two independent modulation waves, while the (3+2)D groups refer to crystals with two symmetryrelated modulation waves, such as those in No. 2.57 .
groups withAs an example of a (3+1+1)D consider . shows that this is an alternate setting of No. 10.2.5.6 . The transformation which brings the original setting into the standard BSG setting is
The difference between the two settings is that the operator (2,ss) in one setting is equivalent to the operator (2,s0) in the standard setting. The transformation modifies the intrinsic translation of a twofold axis along one of the coordinates. This is a feature specific to transformations of the type of equation (33), while the modification of the modulation wavevector by a reciprocallattice vector of the basic structure [equation (19)] can only affect the intrinsic translations of symmetry operators that are screw axes or glide planes in three dimensions. More complicated linear combinations of modulation wavevectors may be required, as in the transformation between and the standard setting of No. 79.2.62.3 [equation (32)]:
The same concept can be applied for reducing the number of rational components of the modulation wavevectors. shows that
is an alternate setting of No. 16.2.19.3 with and . The transformation between these settings involves a linear combination of the two modulation wavevectors as well as a change of the setting of the BSG according towhere again the primed vectors refer to the standard BSG setting.
4.1.2. NbSe_{3}
Several of the features discussed here are illustrated by the example of NbSe_{3}. NbSe_{3} develops an incommensurate CDW below T_{CDW1} = 145 K. A second, independent CDW develops below T_{CDW2} = 59 K, then resulting in an incommensurately modulated structure with two independent modulation waves, and with symmetry given by the (3+2)D No. 11.2.6.4 from . Inspection of the list of groups shows that No. 11.2.6.4 is the only in its that has BSG P2_{1}/m. This implies that any possible combinations of nonzero intrinsic translations along the fourth and fifth coordinate axes are equivalent to the setting by a suitable transformation.
The modulated, lowtemperature crystal structure of NbSe_{3} has been described in (unique axis) with (van Smaalen et al., 1992)
This setting naturally arises for the following choices:
(i) The BSG is equal to the
of the periodic structure at ambient conditions, which has a unique axis that is the preferred setting for monoclinic threedimensional space groups.(ii) The choice of axes and is that of the previously determined periodic crystal structure at ambient conditions.
(iii) Modulation wavevectors are chosen within the first Brillouin zone.
(iv) The first modulation wavevector is that of the first CDW and the second modulation wavevector applies to the second CDW.
All four choices need to be adapted, in order to arrive at the standard setting of this
group:(i) The standard setting of the
has incommensurate components of the modulation wavevectors along , thus requiring a reordering of the basic structure axes.(ii) Transforming the second modulation wavevector into a wavevector with one nonzero rational component requires a basic structure monoclinic ]. Notice that this transformation does not affect the symbol of the BSG.
that involves linear combinations of the axes and [compare to equation (1)(iii) The transformation of (2_{1}, s 0) into (2_{1}, 0 0) requires the transformation [equation (32)].
(iv) The standard setting requires interchanging the two modulation wavevectors.
Altogether, the transformation from the published setting to the standard BSG setting of
No. 11.2.6.4 is achieved bywhich implies a transformation of reciprocal basis vectors of the basic structure as
The components of the modulation wavevectors with respect to the transformed reciprocal basis vectors follow from equation (8) or by inspection of equations (37) and (38):
The tool transformssg can be used to demonstrate that an alternate transformation, defined by a different choice of the second modulation wavevector, also leads to the standard BSG setting of No. 11.2.6.4:
Choices (i), (ii) and (iv) are arbitrary – there does not appear to be a compelling reason to adhere to the standard setting except to establish the equivalence of different crystal structures. The choice (iii) of the modulation wavevector is related to the important question about the real wavevectors of the CDWs, which is not obvious because the incommensurate components of the modulation wavevectors can either be or , depending on the setting. This is most easily analyzed with the help of the supercentered setting, which follows from the standard BSG setting by the transformation
Structural analysis has shown that the first CDW ( in the standard setting) is located on a pair of chains of niobium atoms, denoted as the Nb3 atoms, while the second CDW ( = ) is located on a pair of chains of Nb1 atoms (Fig. 2). provides the explicit form of the symmetry operators in the supercentered setting. Employing these operators, one finds that the double chain of Nb1 atoms centered on of the supercentered is located on the screw axis .^{4} This is a screw operator (2_{1},s0) as it is generated by the combination of the screw (2_{1},00) and the centering translation . The pair of chains of Nb3 atoms is related by the operator , which is a screw operator (2_{1},00). We judge that = = (0,0,0.759) is the real wavevector of the Nb3 modulation, because an additional phase shift is not involved on application of this symmetry. On the other hand, the second wave with = (0,0,0.260) [equations (39) and (40)] implies symmetry for the pair of Nb1 chains involving a phase shift of one half. The real wavevector thus is = (0,0,0.740), resulting in the setting of No. 11.2.6.4. shows that the standard setting can be restored by a shift of the origin of along . With this final transformation, the symmetry of NbSe_{3} is described in the standard setting, and the components of the modulation wavevectors show that both CDWs are waves with wavevectors of on their respective double chains of niobium atoms.
4.1.3. Centerings in internal space
Table 2 compiles groups for a series of compounds with twodimensional modulations. Symbols for the groups from the original publications encompass a disparate set of notations, including symbols based on the online database of (3+d)D groups (d = 1,2,3) of Yamamoto (2005), as in the case of Ca_{2}CoSi_{2}O_{7}, symbols based on Janner et al. (1983), as in the case of Mo_{2}S_{3}, and symbols derived from these notations, such as replacing p by in the case of Sm_{2/3}Cr_{2}S_{4}, as well as other ad hoc symbols.
While for several compounds a permutation is required of the basis vectors of the basic structure
in order to transform the published setting into the standard setting, other, less trivial transformations occur too. shows that the symmetry of tetrathiafulvalene tetracyanoquinodimethane (TTF TCNQ), , is based on a supercentered lattice, where the centering exclusively involves the two internal coordinates. The supercentered setting has modulation wavevectors = and = withresulting in the reflection condition and corresponding centering translation:
Monoclinic symmetry is the lowest symmetry where this kind of _{2}CoSi_{2}O_{7}, on the other hand, the does not possess a centered lattice, despite the seemingly simpler modulation wavevectors = and = , which are related to and as in equation (42). The reason is that and are related by symmetry in the same way as and are, and the wouldbe supercentering does not have an advantage over the primitive lattice from the point of view of symmetry. Examples of supercentered lattices with higher symmetries are given for threedimensional modulations in §4.2.
lattice centering can occur. For CaA peculiar feature of the modulation of TTF TCNQ is that one of the unrestricted components is not experimentally distinguishable from zero (). The explanation probably lies in the optimal phase relations between the CDWs on neighboring stacks of TTF or TCNQ molecules, as it is governed by the physics of CDW formation. However, in this case the phase relation is not reflected in the symmetry of the crystal structure. Similar observations can be made for Mo_{2}S_{3} ( and ), (Bi,Pb)_{2}(Sr,Bi,Pb,Ca)_{2}CuO () and LaSe_{1.85} (). For these compounds, the special values of the components of the modulation wavevectors are reminiscent of the higher symmetries at high temperatures [monoclinic for Mo_{2}S_{3} and orthorhombic for the highT_{c} superconductor (Bi,Pb)_{2}(Sr,Bi,Pb,Ca)_{2}CuO] or the higher symmetry of a hypothetical basic structure (Laue symmetry 4/mmm for LaSe_{1.85}).
The lattice type (primitive, centered BSG or supercentered) is the same for all (3+2)D groups. Since all these Bravais classes contain groups with acentric trigonal symmetry (Table 3), it is necessary to choose a pair of modulation wavevectors that enclose an angle of 120° and not 60° (Fig. 3). With the exception of the recent study on Cu_{3+x}Si, this condition has not been obeyed in studies of the compounds with trigonal or hexagonal symmetries listed in Table 2, where the angle between and was chosen as 60°. While not wrong in these cases, it is highly preferable to describe these structures using an angle of 120° between the modulation wavevectors so as to be consistent with the settings of their Bravais classes.
groups belonging to a Likewise, the choice of modulation wavevectors should be the same for all groups within a single the is defined by the of the lattice together with the modulation wavevectors. A further requirement on the modulation wavevectors is that they must transform according to the threedimensional of the These requirements become important for the selection of modulation wavevectors in the case of trigonal and hexagonal Bravais classes of

4.2. (3 + 3)Dimensional groups
Different settings of (3+3)D groups are obtained by means of the same that apply to (3+2)D groups. That is, the setting of a (3+3)D depends on the choice of basic structure basis vectors and on the freedom in the choice of modulation wavevectors, including the possibility to replace the modulation wavevectors by linear combinations of them [equation (32)].
4.2.1. Supercentered setting of (TaSe_{4})_{2}I
(TaSe_{4})_{2}I has a periodic structure with I422 at ambient conditions. A CDW develops below T_{CDW} = 263 K. It is expressed in the diffraction by the presence of eight incommensurate satellite reflections around each main reflection, which can be indexed as firstorder satellite reflections according to the four modulation wavevectors
van Smaalen et al. (2001) incorrectly reported this as a fourdimensional modulation (Table 4), but then continued to show that the is accompanied by a lowering of the and the formation of a multiply twinned crystal with a onedimensional incommensurate modulation in each domain. Nevertheless, for the purpose of illustrating a fundamental issue of symmetry, we will proceed as though all modulation wavevectors would originate in a single domain, where the number of symmetryequivalent modulation wavevectors is larger than the dimension of the modulation. The modulation in equation (44) is actually threedimensional, because = . Despite this relationship between the modulation wavevectors, the tetragonal symmetry requires that modulation wavefunctions are symmetric in the four arguments
This symmetry becomes obvious in the supercentered setting, where shows that there are two symmetryequivalent modulation wavevectors, and , in addition to a third wavevector, , parallel to the tetragonal axis:
The four pairs of satellite reflections as well as the four equivalent arguments of the modulation wavefunctions then follow as all four equivalent linear combinations of with or :
In accordance with the centering of the (j = 1,2,3), and modulation functions do not contain harmonics involving arguments , but only contain linear combinations like [equation (47)].
lattice, diffracted satellite reflections do not appear at

4.2.2. symmetry with BSG
All known compounds with a threedimensional modulation possess cubic symmetry. Wustite, Fe_{1x}O, is based on an Fcentered cubic lattice with BSG and the simple modulation with = (0.398, 0, 0) (Table 4). The is symmorphic and centerings other than the Fcentering,
of the BSG do not occur.
Three compounds have been reported with symmetry according to _{4})_{2}I (§4.2.1), four symmetryequivalent modulation wavevectors exist. The supercentered setting clearly reveals the threedimensional nature of the modulation with = (Table 4), and
No. 225.3.215.7 . They have the same BSG as wustite but different modulation wavevectors. As with (TaSeFor Bi_{0.85}Mo_{0.16}O_{1.74} the modulation wavevectors of the supercentered setting are
The centering translations of the supercentered setting combine the Fcenter of the basic structure [equation (48)] with a socalled `Fcenter' among the internal coordinates, the latter being defined as
This can be compared with ], but now combines the Fcenter of the basic structure [equation (48)] with an `Icenter' among the internal dimensions with centering translation .
No. 225.3.212.5, based on modulation wavevectors of the type , where the supercentered setting again involves modulation wavevectors of the type = [equation (50)Interestingly, replacing threevalent molybdenum atoms by fivevalent niobium or tantalum atoms leads to a similar, but different structure involving mirror planes with nonzero intrinsic translational components along the internal ).
dimensions (Table 44.2.3. Modulation in the Icentered lattice of V_{6}Ni_{16}Si_{7}
V_{6}Ni_{16}Si_{7} is a threedimensionally modulated crystal with symmetry based on the cubic Icentered lattice and BSG (Table 4). Withers et al. (1990) report an indexing of the electron diffraction based on the modulation wavevectors
where and = = .
Withers et al. (1990) also report the observed but then provide an analysis based on the theory of irreducible representations (normalmode analysis). Yamamoto (1993) has assigned to V_{6}Ni_{16}Si_{7} the (3+3)D with the tentative symbol .
shows that such a et al. (1990) or Yamamoto (1993), we could not use the tool on for computing the transformation to the standard setting. However, does show that the only possible modulation wavevectors for threedimensional modulations with BSG are , and (Bravais classes 3.208, 3.211 and 3.214, respectively). Indeed, the modulation wavevectors can be rewritten as
does not exist. Since symmetry operators are not provided by Witherswhere = = . Notice that we cannot add the basic structure reciprocal vector to the modulation wavevectors [equation (19)], because this is a forbidden reciprocal vector for the Icentered lattice. Instead, we have added the vector (1,1,0) to in order to arrive at a reciprocal vector along the diagonal of the cubic Of course, this goes at the expense of a considerably increased length for the modulation wavevectors. Nevertheless, a description that respects the symmetry of the problem requires these long modulation wavevectors. With the new indexing, the nonsymmorphic is obtained, which corresponds to No. 229.3.214.8 in (Table 4).
5. Incommensurate composite crystals
Incommensurate composite crystals comprise two or more subsystems, each of which has an incommensurately modulated structure. The basic structures of the subsystems are mutually incommensurate, but for all known compounds, any pair of subsystems share a common reciprocallattice plane of their basic structures. The third reciprocal basis vector of one subsystem then acts as modulation wavevector for the other subsystem, and the other way around. The symmetry of a composite crystal is given by a (3+d)D while the symmetry of each subsystem is also given by a (3+d)D These socalled subsystem groups often are different (inequivalent) groups according to the definition of employed in de Wolff et al. (1981) and Stokes et al. (2011a).
The various aspects of the structures and symmetries of composite crystals are illustrated by the example of [Sr]_{x}[TiS_{3}] (Onoda et al., 1993), where square brackets indicate the subsystems. The seemingly nonstoichiometric composition with x = 1.132 reflects the incommensurate ratio of the volumes of the basic structure unit cells of subsystem 1 (TiS_{3}) and subsystem 2 (Sr). [Sr]_{x}[TiS_{3}] is a composite crystal of the columnar type, where chains of Sr atoms and columns of TiS_{3} are alternatingly arranged on a twodimensional hexagonal lattice (Fig. 4). The basic structure reciprocal lattices share the basis vectors in the basal plane, while the third direction (parallel to the chains) is the incommensurate direction:
where , for example, denotes the third reciprocal basis vector of the first subsystem and is the first (and in this example only) modulation wavevector of the first subsystem with = .
An indexing of all reflections with four integers is obtained with the four reciprocal basis vectors M^{*} = . Along with its modulation wavevector, the reciprocal basis vectors of subsystem () are obtained from the four reciprocal vectors M^{*} by a (3+d) ×(3+d) integer matrix (d = 1 in the present example) according to
The matrices extract the basic structure reciprocal basis vectors and modulation wavevectors of subsystem from the basis vectors used for indexing. In this sense, represents a coordinate transformation in M^{*} and the natural subsystem which is specific to each subsystem. Operators of the subsystem follow as (van Smaalen, 1991)
between the arbitrarily chosen representationBecause reciprocal basis vectors of one subsystem act as modulation wavevectors of the other subsystem, must be a coordinate transformation that mixes the first three dimensions and the additional dimensions for at least some of the subsystems. This coordinate transformation is a forbidden transformation when establishing the equivalence of et al., 2011a). Therefore, the subsystem groups are generally inequivalent, unless they are equivalent by chance, as is the case for the mineral levyclaudite which possesses triclinic symmetry (Evain et al., 2006).
groups (StokesFor [Sr]_{x}[TiS_{3}] equation (54) shows that W^{1} is the identity matrix. This choice of M^{*} has become a de facto standard for composite crystals. It implies a setting where the symmetry of [Sr]_{x}[TiS_{3}] and the symmetry of the first subsystem are described by the same Onoda et al. (1993) give the , which is found to be an alternate symbol for No. 166.1.22.2, on . Apart from the Rcentering of the hexagonal basic structure other centerings in do not exist for this lattice.
Equation (54) leads for the second subsystem to
The Rcentering of the original setting transforms by W^{2} [equation (56)] into the centering vectors and , which represent an Htype centering of the BSG and which has been denoted as the centering of the lattice (van Smaalen, 2007). SSG(3+d)D shows that the transformation by W^{2} [equation (57)] leads to the supercentered setting of the (3+1)D No. 163.1.23.1, .
The subsystem _{x}[TiS_{3}] turn out to be inequivalent (3+1)D groups, although they are of course equivalent as 4D space groups as governed by the coordinate transformation W^{2}. The case of [Sr]_{x}[TiS_{3}] is special as it combines different Bravais lattices of the BSG for the subsystems. The rhombohedral lattice with an Rcentering and the primitive trigonal described with an Hcentered can be considered as different centerings of the hexagonal which share a common reciprocallattice plane perpendicular to the trigonal axis (Fig. 5).
groups of [Sr]Other incommensurate composite crystals, including misfit layer ), misfit layer cobalt oxides (Isobe et al., 2007) and urea inclusion compounds (van Smaalen & Harris, 1996), also exhibit a pairing of two inequivalent groups. A detailed analysis of this feature is outside the scope of the present overview and will not be discussed further here.
(Wiegers, 1996Standard settings and alternate settings of M^{*} of reciprocal basis vectors is preferably chosen to contain reciprocal basis vectors of the basic structures of the subsystems. For example, for the case of [PbS]_{1.18}[TiS_{2}] and isostructural [Ca_{0.85}OH]_{1.256}[CoO_{2}] (van Smaalen et al., 1991; Isobe et al., 2007), M^{*} has been chosen as
groups occur for incommensurate composite crystals by means of the same kinds of coordinate transformations as have been discussed for modulated crystals. One difference is the stronger inclination for employing nonstandard settings in the case of composite crystals, because the setThis results in a mixed setting of the (3+1)D as with . shows that this is an alternate setting of No. 12.1.7.4 . Apart from the trivial transformation of the setting of the BSG, the transformation toward the standard BSG setting involves the choice of an alternate modulation wavevector:
The interpretation of as a reciprocal basis vector of the second subsystem is lost in this representation [equation (59)]. Therefore, the mixed setting (i.e. not BSG setting or supercentered setting) with centering translation is preferred over the standard setting in the case of these composite crystals.
6. Chiral groups
Chiral space groups are space groups that may be the symmetry of crystals containing chiral molecules. They are of particular importance in the life sciences, because all proteins and et al., 2008).
are molecules of this type (LovelaceChiral space groups are those space groups of which the ). Chiral groups are then defined as the groups for which the threedimensional of the BSG contains rotations only (Souvignier, 2003). A list of (3+d)D groups (d = 1,2,3) has been generated with this criterion and is available on . It is noticed that the fraction of groups that is chiral strongly decreases on increasing dimension d (Table 5). We did not find a compelling theoretical reason for this feature. But we do observe that the number of ways to combine the intrinsic translations of the BSG with the intrinsic translations along the additional dimensions, or with supercentering translations, increases with d. So it appears that the intrinsic translations of chiral BSG operations are more restricted in the combinations in which they can participate.
contains rotations only (Blow, 2002

det(S_{R}) = 1; equation (7)], i.e. that preserve This definition leads to pairs of enantiomorphic groups in cases where the BSG is an enantiomorphic like the (3+2)D groups No. 76.2.60.2 and No. 78.2.60.2 . Intrinsic translations along the additional dimensions do not give rise to enantiomorphic groups. For example, No. 75.2.60.4 is not enantiomorphic (q stands for the fractional translation ). Instead of being an is an alternative setting of No. 75.2.60.4, and is transformed into the standard setting by the choice of a different modulation wavevector: [equation (19)].
groups are defined on the basis of equivalence relations that only allow coordinate transformations that preserve the handedness of the coordinate axes in threedimensional space [7. Conclusions
The computational complexity of finding the transformation between two settings of a (3+d)D is surprisingly high, especially for d = 2,3. Here an efficient algorithm is presented, which either establishes two groups to be different groups or determines them to be different settings of the same and then provides the transformation between these settings. The algorithm has been implemented as an internetbased utility called `', which identifies any usergiven (3+d)D (d = 1,2,3) based on the superspacegroup operators provided, and displays the transformation to the standard setting of this in the tables.
The algorithm considers coordinate transformations in
It is shown that in general such a transformation corresponds to one, or a combination, of the following three types of transformations in physical space:(i) A transformation of the basic structure unit cell.
(ii) Adding any reciprocallattice vector of the basic structure to the modulation wavevector [equation (19)].
(iii) Replacing originally chosen modulation wavevectors by linear combinations of the same [only for ; equation (32)].
These transformations are illustrated by the analysis of the symmetries of a series of compounds with d = 1,2,3, comparing published and standard settings and discussing the transformations between them. It is argued that nonstandard settings are needed in some cases, while standard settings of groups are desirable in other cases. A compilation is provided of standard settings of compounds with two and threedimensional modulations (Tables 2 and 4). It appears that several ad hoc notations have been used in the literature for (3+d)D groups, especially for d = 2 and 3.
For d = 2 groups with trigonal/hexagonal symmetry, an angle of 120° between the two modulation wavevectors is preferred and is the only correct choice for acentric trigonal cases (Table 3). This is the standard setting for all relevant Bravais classes in , in contrast to the use of a 60° angle in most published structures (§4.1.3 and Table 2).
The problem of superspacegroup settings, including the choice of origin, is subtle. Therefore, we strongly advise authors to explicitly document for each structure the list of symmetry operators (or at least the generators) of the . It would also be useful to include the number and symbol of the standard setting on for each structure, because this will make it easier to check the equivalences of structures and symmetries in future studies.
along with the explicit form of the modulation wavevectors as in equation (2)Supporting information
Output of SSG(3+d)D for different settings of https://doi.org//10.1107/S0108767312041657/pc5018sup1.zip
groups. DOI:Algorithm for determining the transformation S between two settings of (3+d)dimensional https://doi.org//10.1107/S0108767312041657/pc5018sup2.pdf
groups (d=1, 2, 3). DOI:Footnotes
^{1}The mathematically correct designation is that of equivalent space groups providing different settings of spacegroup type No. 15.
^{2}Supplementary material for this paper is available from the IUCr electronic archives (Reference: PC5018). Services for accessing this material are described at the back of the journal.
^{3}The modulated phase often appears as a twinned crystal if the BSG is a of the threedimensional of the periodic phase (van Smaalen, 2007).
^{4}The symbol indicates an operator with a twofold rotation 2_{z} of the BSG along combined with a unit 2 ×2 matrix , and followed by a translation .
Acknowledgements
We thank Gloria Borgstahl for pointing out the importance of chiral
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