An unusual case of OD-allotwinning: 9,9′-(2,5-dibromo-1,4-phenylene)bis[9H-carbazole]

The title compound crystallizes as allotwins made up of two OD-polytypes with a maximum degree of order.


Introduction
A twin is a heterogeneous edifice made up of homogeneous crystals of the same phase in different orientations related by an operation that does not belong to the point group of the individual (Friedel, 1904). The crystalline domains diffract independently and the orientation relationship is well defined (Hahn & Klapper, 2006). Allotwinning (Nespolo et al., 1999) is a generalization of twinning, where the domains are different polytypes of the same compound (Greek o& = different) composed of equivalent layers. In contrast to the structural characterization of classical twins, the concurrent refinement of two or more structural models against the same data set is not yet routine. It has nevertheless been performed successfully in a few cases (e.g. Friese et al., 2003;Krü ger et al., 2009;Jahangiri et al., 2013;Stö ger et al., 2015).
The importance of applying such refinement strategies was shown in recent work by our group on the allotwinning of potassium silver carbonate (KAgCO 3 ; Hans et al., 2015). Therein, we argued that not only the frequency of allotwinning, but also the volume fraction of the minor domain in allotwins tend to be significantly underestimated.
In a continuation of this work, we present an unusual case of allotwinning, which we serendipitously discovered during routine structural analysis of organic intermediates. 9,9 0 -(2,5-Dibromo-1,4-phenylene)bis [9H-carbazole] (1) was synthesized as an intermediate for further functionalization towards functional organic materials for potential applications in organic electronics (Fig. 1). The two bromine substituents allow for chemical modification via Pd catalyzed crosscoupling reactions. Crystals of (1) turned out to be allotwins made up of two members of an order-disorder (OD) polytype family (Dornberger-Schiff & Grell-Niemann, 1961). They can, therefore, be designated as OD-allotwins in analogy to ODtwins, which are made up of different orientations of the same OD-polytype. A non-OD allotwin would be composed of polytypes that do not follow the principles of OD theory. ISSN 2052-5206 To encourage future structural characterization of allotwins, a detailed account of data treatment and refinement will be given. Furthermore, an interpretation according to the OD theory is presented, to plausibly explain the polytypism of the compound. Finally, the crystals of (1) are compared with those of KAgCO 3 , to highlight the diversity of the phenomenon.

Data collection
Crystals were selected under a polarizing microscope, embedded in perfluorinated oil and attached to Kapton1 micromounts. They were subjected to short scans at 100 K in a dry stream of nitrogen on a Bruker KAPPA APEX II CCD diffractometer system (Bruker, 2014) using graphite-monochromated Mo K radiation. All crystals featured weak diffraction, with a sharp intensity drop off at higher diffraction angles and distinct streaking. A full data set up to 2 = 55 of the crystal giving the best diffraction pattern was collected with long exposure. A 0.7 mm collimator ensured a full immersion of the platy (0.36 Â 0.24 Â 0.02 mm) crystal in the X-ray beam. Data were integrated using SAINT-Plus (Bruker, 2014). An adequate correction for absorption effects was performed by using the multi-scan approach implemented in SADABS (Bruker, 2014).
After a first successful refinement, data of four more crystals were collected. Unfortunately, all of them featured even poorer reflection quality and were unfit for a proper structural characterization.

Structure solution and refinement
The automatic unit-cell determination suggested an orthorhombic base-centered (oS) Bravais lattice (Burzlaff & Zimmermann, 2006). In the conventional C-centered setting ða C ; b C ; c C Þ it had the cell parameters a ' 9.1 Å , b ' 14.5 Å , c ' 16.9 Å . The rather large R int = 0.082 value for the mmm Laue class was attributed to the poor reflection quality. No structure solution was successful even in the lowest symmetry orthorhombic space groups. Likewise, attempts using any space group belonging to the monoclinic C-centered (2=mC) Bravais flock (R int = 0.068) were unsuccessful. Thus, the lattice basis was transformed into the primitive setting b ' 16.9 Å ,c ' 8.58 Å ,' 116 (Fig. 2). Assuming a space group of the monoclinic primitive (2=mP) Bravais flock, systematic absences suggested the space group P2 1 . Owing to streaking in the ½010 direction, relying on absences is treacherous, because the streaks are interpreted by the integration software as positive Bragg intensity. A structure solution in P2 1 with the novel dual space method implemented in SHELXT (Sheldrick, 2015) did not converge to a usable result, as is common for the intensity data of twins. The direct methods implemented in SHELXS (Bruker, 2014), on the other hand, successfully located the Br atoms and most of the remainder of the molecule. Having succeeded in obtaining a preliminary structural model, no further attempts in triclinic space groups were undertaken.  The lattice of the crystal under investigation projected along the monoclinic axis of the major polytype and the bases used during refinement. The horizontal line is the twin plane.
Being aware of the orthorhombic metrics, the first refinements were performed under consideration of twinning by metric merohedry. The twin plane is indicated in Fig. 2. In the ða P ; b P ; c P Þ oP setting it is parallel to ð101Þ; in the ða C ; b C ; c C Þ oC setting parallel to (100). H atoms were placed at calculated positions and henceforth refined as riding on the parent C atoms. Visual inspection of the structure suggested the higher P2 1 =n symmetry [the (1) molecule being located on a center of inversion], which was confirmed by the ADDSYM routine of PLATON (Spek, 2009;100% fit). The correctness of the higher symmetry, which was used for subsequent refinements, was proven by reasonable anisotropic atomic displacement parameters (ADPs) of all non-H atoms, in contrast to the P2 1 model.
Nevertheless, the residuals were disappointing (R obs ' 0:09) and a prominent peak in the difference Fourier map ( = 7.68 e Å À3 , charge = 2.36 e) was observed close to the carbazole moiety. The coordinates of this peak were approximately ðx þ 1 2 ; y; z þ 1 2 Þ with respect to the coordinates ðx; y; zÞ of the Br atom. Since Br is the only heavy atom in the structure, this suggested a spurious 'phantom' or 'shadow' molecule, which is characteristically observed for structures with improperly resolved stacking faults. The Br atom was refined as positionally disordered between the original position and the 'phantom' (refined ratio $ 92:8), resulting in a significant drop of the residuals (R obs ' 0:06).
In many, if not most, cases polytypism is plausibly explained by pseudo-symmetry of distinct layers, called order-disorder (OD) layers (Grell, 1984). Indeed, the structure is built of such layers, which are pseudo-symmetric by reflection at ð101Þ with respect to the oP setting (the deviation from perfect symmetry is quantified in x3.4). The linear part of this reflection relates the orientation of the twin individuals. Application of this operation to an adjacent layer maps the positions of the Br and the phantom-Br in that layer.
The Bravais lattice of the OD layers was rectangular basecentered (os, note the lower case Bravais letter), with the nonprimitive basis ða B ; c B Þ ¼ ða P À c P ; a P þ c P Þ ¼ ðb C ; a C Þ (Fig. 2). To simplify the OD description, the basis was therefore transformed into the corresponding B-centered setting ða B ; b B ; c B Þ. In this setting the symbol of the space group is B2 1 =d, because the intrinsic translation vector of the glide reflection is a quarter of the ð010Þ face-diagonal of the unit cell. The twin plane is parallel to ð100Þ. It has to be noted that the Hermann-Mauguin symbol B2 1 =d is ambiguous with respect to the orientation of intrinsic translations. Here it designates the group with the d glide reflection with intrinsic translation vector ða þ cÞ=4, but not ða À cÞ=4.
To explain the 'shadow atoms', an interpretation according to OD theory (Dornberger-Schiff & Grell-Niemann, 1961) was performed (for details, see x3.2). The most probable minor polytype was determined to possess the orthorhombic F2dd symmetry with a doubled y-axis, i.e. the lattice basis ða B ; 2b B ; c B Þ. The corresponding reflections, albeit weak and diffuse, indeed exist and were missed by the automatic unit cell determination. Thus, data reduction and absorption correction were repeated with respect to the B-centered cell with the basis ða B ; 2b B ; c B Þ. The corresponding reciprocal basis defines the smallest unit cell in which the reflections of both polytypes can be indexed with integral hkl values (x3.5).
The occupational disorder of the Br atom was removed from the model of the B2 1 =d polytype. A model of the F2dd polytype was generated from the model of the B2 1 =d polytype by moving the origin to ð0; 0; 1 4 Þ, halving the y-coordinates and applying the F2dd symmetry. This model featured a chemically implausibly bent dibromobenzene fragment. To achieve a reasonable model, the atoms of this fragment were all placed onto the x ¼ 0 d glide plane (but not fixed there in subsequent refinements).
The models of both polytypes were then combined and refined against the same data set using the JANA2006 (Petříček et al., 2014) software. As previously, the B2 1 =d model was refined with two orientations related by reflection at (100). The F2dd model was refined as two orientations related by inversion. Thus, in total the intensity data was calculated as originating from four independently diffracting domains. The C atoms in the minor F2dd polytype were refined with isotropic ADPs. The refinements converged quickly to satisfying residuals (R obs ¼ 0:059).
In diffraction patterns of crystals with a high stacking fault probability, a systematic misevaluation of intensities owing to different shape and backgrounds of different reflection classes may occur. It can lead to erroneous volume fractions of the polytypes (Ď urovič effect; Nespolo & Ferraris, 2001). In the crystal under investigation, reflections hkl with h ¼ 2n, n 2 Z (or equivalently l ¼ 2n) are distinctly elongated and located on rods of diffuse scattering (x3.5). To at least partially counteract this effect, for reflections h ¼ 2n and h ¼ 2n þ 1 different scale factors were used. Although the refined volume ratios changed only within the estimated standard uncertainty (e.s.u.) range, the residuals improved distinctly (R obs ¼ 0:055). Moreover, the highest peaks in the difference Fourier maps of both polytypes were less pronounced (B2 1 =d: ¼ 1:95 versus 1.47 e Å À3 , charge = 1.40 versus 0.90 e; F2dd: 2.16 versus 1.76 e Å À3 , charge = 0.05 versus 0.04 e). More details on data collection and refinement are compiled in Tables 1 and 2.  Molecules of (1) (Fig. 3) in both polytypes possess 2=m pseudo-symmetry (actual symmetry 1 and 2, respectively). The carbazole moieties are virtually planar [maximum distance to leastsquares (LS) plane in the major B2 1 =d polytype: 0.045 (7) Å (N1)]. Remarkably, the C-N bond connecting the benzene and the carbazole is distinctly tilted with respect to the plane of the carbazole [angle of N(carbazole)-C(benzene) to the LS plane of carbazole: 15.5 (4) ].

OD description
The crystal under investigation was made up of two polytypes (Fig. 4), each in two orientation states. The layers of the polytypes extend in the ð010Þ plane [here and in the following discussion all directions are given with respect to the oB setting ða B ; b B ; c B Þ of x2.3], and will be designated as A n , where n is a sequential number. The benzene rings are located at the center of these layers, the carbazoles at the interfaces.
An OD interpretation of polytypism is based on identifying pseudo-symmetry of these layers. The operations of these layers are called -partial operations (POs) (Dornberger-Schiff & Grell-Niemann, 1961). They are partial operations, because their domain of definition is a strict subset of Euclidean space E 3 . The operations relating different layers are likewise POs and are called -POs.

Figure 3
Geometry of molecule (1). C, N, Br and H atoms are represented by gray, blue, orange and white spheres of arbitrary radius. Coordinates taken from the B2 1 =d polytype.

Figure 5
An A n layer viewed down the stacking direction [010]. Color codes as in Fig. 3. H atoms are omitted for clarity. (Pseudo)-symmetry operations are indicated using the usual graphical symbols (Hahn, 2006), the unit cell by a black rectangle. Coordinates taken from the B2 1 =d polytype.
tradition of OD literature parentheses are used to mark the direction lacking translational symmetry (Dornberger-Schiff & Grell-Niemann, 1961)]. Adjacent layers are related by a d glide reflection at a plane parallel to ð010Þ.
Because the reflection planes of adjacent layers do not overlap, the structure belongs to an OD family of layers of one kind. Since the layers are non-polar with respect to the stacking direction, the OD family belongs to category I (Dornberger-Schiff & Grell-Niemann, 1961). The OD groupoid family symbol reads in the oB setting as B 2=m ð1Þ 1 f À 2 2 =n r;s 2 r =n s;2 g ð1Þ according to the notation of Dornberger-Schiff & Grell-Niemann (1961). The first line of the symbol indicates the B2=mð1Þ1 layer symmetry of the A n layers, the -POs. Below are listed the operations relating A n to A nþ1 in one possible arrangement (-POs). 2 2 is a generalization of the 2 1 notation and describes a screw rotation with an intrinsic translation vector b 0 , which is the vector perpendicular to the layer planes and the length of one layer width. In analogy, n r;s is a glide reflection with the intrinsic translation vector ðsa þ rcÞ=2 etc.
In the actual OD family of the title compound, the metric parameters describing the intrinsic translations of the -POs are ðr; sÞ ¼ ð 1 2 ; 1 2 Þ. This can be written by the symbol B 2=m ð1Þ Here, the n r;s glide reflection becomes a n1 2 ; 1 2 glide reflection, which is a d glide reflection with an intrinsic translation vector a=4 þ c=4, corresponding to a quarter of the face diagonal of the unit cell. Application of the B centering produces a second n À 1 2 ;À 1 2 operation which is likewise a d glide reflection. On the other hand, no d glide reflections n1 2 ;À 1 2 or n À 1 2 ; 1 2 exist in this case. The alternative stacking possibilities are determined by application of the NFZ relationship (Ď urovič, 1997). To choose the correct form of the NFZ relationship, it is crucial to realise that the n1 2 ; 1 2 PO has reverse continuations (Dornberger-Schiff & Grell-Niemann, 1961), meaning that it maps A n onto A nþ1 , but also A nþ1 onto A n . In other words, n1 2 ; 1 2 is a symmetry operations of the ðA n ; A nþ1 Þ pairs of layers.
In such a case the NFZ relationship reads as Z ¼ N=F ¼ ½G n : G n \ G nþ1 , where G n is the group of those operations of the A n layer that do not reverse the orientation with respect to the stacking direction (--POs in the OD literature). G n \ G nþ1 is the subgroup of those operations that are also valid for A nþ1 . Thus, given a fixed A n layer, the A nþ1 layer can appear in ½Bmð1Þ1 : B1ð1Þ1 ¼ 2 positions, which are related by the m ½100 operation of the A n layer. In the alternative stacking possibility, the n1 2 ;À 1 2 and n À 1 2 ; 1 2 glide reflections are realised.
By these stacking rules, the A n layers can be arranged to an infinity of different polytypes, which are all locally equivalent up to at least two layer widths. Of these, two polytypes are of a maximum degree of order (MDO), i.e. they cannot be decomposed into simpler polytypes (Dornberger-Schiff, 1982). Fig. 6a) is generated by repeated application of n1 2 ; 1 2 glide reflections; MDO 2 (F2dd, b ¼ 4b 0 , Fig. 6b) by alternating application of n1 2 ; 1 2 and n1 2 ;À 1 2 glide reflections. All other polytypes consist of fragments of the MDO 1 and MDO 2 .
These two MDO polytypes make up the crystal under investigation. Indeed, experience shows that in the vast majority of observed cases, OD polytypes are of the MDO type. Identification of the MDO polytypes is therefore a crucial part of the interpretation of OD structures (Ď urovič, 1997).

Twinning and allotwinning
To determine the possible orientation states that each polytype can adopt, the point group of the OD family (Fichtner, 1977), which is the point group generated by the  linear parts of all POs of a member, is determined. In the title compound it is mmm. The point group of any member of the family is a subgroup of this point group.
The possible orientation states of a polytype are then determined by the coset decomposition of the point group of the member in the point group of the family. For the B2 1 =d MDO 1 polytype ½mmm : 2=m ¼ 2 and there are therefore two orientation states. The coset decomposition is f1; 1; 2 ½010 ; m ½010 g and f2 ½100 ; m ½100 ; 2 ½001 ; m ½001 g. The latter is the twin law (Hahn & Klapper, 2006) relating the two orientations.
In analogy, for the F2dd MDO 2 polytype ½mmm : 2mm ¼ 2 and there are again two orientation states described by the cosets f1; 2 ½100 ; m ½010 ; m ½001 g and f1; m ½100 ; 2 ½010 ; 2 ½001 g. This twinning is by inversion and therefore determination of the twin volume ratio was only possible owing to the resonant scatterer Br.

Desymmetrization
As is characteristic for ordered polytypes, the actual layers in both MDO polytypes are desymmetrized (Ď urovič, 1979) compared with the idealized OD description (the prototype layers). A too extreme desymmetrization casts doubt on an OD interpretation, because it means that the presumption of geometrically, and therefore energetically, equivalent layers is not valid.
In the MDO 1 polytype, the symmetry of the A n layers is related to the prototype layers by a translationengleiche symmetry reduction of index 2 from B2=mð1Þ1 to B1ð1Þ1. The point symmetry of the (1) molecules is reduced from 2=m to 1. Of the -POs relating adjacent layers only the n1 2 ; 1 2 glide reflections and the 2 2 screw rotations are retained as symmetry operations of the whole polytype.
In the MDO 2 polytype, the symmetry of the A n layers is related to the prototype layers by a translationengleiche symmetry reduction of index 2 from B2=mð1Þ1 to B2ð1Þ1. The symmetry of molecules (1) is 2. Of the -POs relating adjacent layers, the n1 2 ; 1 2 and the n1 2 ;2 glide reflections are retained (the latter is likewise written as d in the F2dd symbol owing to the doubled b-axis).
To quantify the desymmetrization, in both polytypes the lost symmetry was applied to an actual A n layer (in the MDO 1 polytype the m ½100 reflection; in the MDO 2 polytype the inversion). An overlap of the actual A n layers and their images are given in Fig. 7. The distance of the atoms mapped by pseudo-symmetry are compiled in Table 3.
The most striking difference in the desymmetrization of both polytypes pertains to the Br atom (MDO 1 : 0.435 Å , MDO 2 : 0.010 Å ). In MDO 1 the large deviation is due to a distinct inclination of the central dibromobenzene fragment of (1) with respect to the m ½100 pseudo-reflection plane. In MDO 2 on the other hand, the Br atom is located virtually on the pseudo-reflection plane (a d glide plane in the actual polytype) for chemical reasons. An inclination with respect to the reflection plane would result in a bent benzene ring, owing to the twofold rotation axis normal to the plane. The benzene rings are indeed apparently bent, though it has to be noted that the localization of the atoms in the MDO 2 polytype is not exact owing to poor diffraction data (x3.5). The generally small deviations demonstrate the validity of the OD interpretation.

Diffraction pattern
The family structure of an OD family is the fictitious structure in which all stacking possibilities of the family are realised (Ď urovič et al., 2006). It plays a crucial role in the interpretation of the diffraction pattern of OD structures. The reflections of the family structure (family reflections) are always sharp (supposing that desymmetrization plays only a minor role). All polytypes contribute equally (proportional to the volume ratio) to the family reflections. The remaining reflections are only generated by certain polytypes and are therefore called characteristic reflections. They can be sharp if the polytypes are highly ordered,  but may also be broadened in the case of frequent stacking faults.
The family structure of the OD family under investigation is derived by application of the m ½100 operation of the A n layers onto the A nþ1 layers (Fig. 6c). The symmetry of the family structure, the superposition group (Fichtner, 1977), is Pmnn with lattice basis ða=2; 2b 0 ; c=2Þ. The symmetry of any stacking arrangement, ordered or disordered, is a subgroup of the superposition group.
In the reciprocal basis of the MDO 1 polytype ða Ã ; b Ã ; c Ã Þ ¼ ða=a 2 ; b 0 =ð2b 2 0 Þ; c=c 2 Þ the family reflections are located at h ¼ 2n h ; k ¼ n k ; l ¼ 2n l with n h ; n k ; n l 2 Z. It can be shown (Ferraris et al., 2008;Hans et al., 2015) that if the layers are translationally equivalent, non-zero intensity on the rods with the family reflections is only observed for the family reflections, even for long-period polytypes and disordered stackings. Here, adjacent layers are not equivalent, but an analoguous result is obtained by decomposing the structure into A 2n and A 2nþ1 layers. Thus, characteristic reflections and diffuse scattering are only expected on rods h ¼ 2n h þ 1, l ¼ 2n l þ 1. The characteristic reflections of both MDO polytypes are distinct. A summary of the reflection conditions is given in Table 4.
Indeed, in the actual diffraction patterns family reflections are sharp (Fig. 8a), whereas the characteristic reflections ( Fig. 8b) are located on distinct streaks, indicative of disordered domains. The characteristic reflections are elongated in the stacking direction, i.e. the ordered domains are small. The intensities of the MDO 2 characteristic reflections are weak and in the same order of magnitude as the intensities of the streaks.
A careful inspection of the family reflections reveals that they are likewise located on distinct, albeit very faint, streaks of diffuse scattering. This is due to the layers in the structure not being translationally equivalent as a consequence of desymmetrization (x3.4). Notably, the strongly diffracting Br atoms alternate between two positions in the MDO 1 polytype and adopt a third orientation in the MDO 2 polytype, where they are located close to the idealized positions. It can therefore be expected that in the disordered parts of the crystals the Br atoms feature a distinct modulation of their position, resulting in faint streaks.

Determination of the volume ratio
The volume ratio of the four domains (MDO 1 :M-DO 1 :MDO 2 : MDO 2 ) was determined by a concurrent refinement (x2.3) as 36.6 (2):32.4 (2):14.9 (13):16.1 (12). Thus, the MDO 1 :MDO 2 ratio is 69.0:31.0. As expected for a crystal with a high stacking fault probability, the volume fractions of the two orientations of each polytype are approximately equal. The ESU of the fractions of the MDO 2 domains is significantly higher, because it is derived from the anomalous scattering of the Br atoms.
The refinement was performed under the assumption of ideal (allo)twinning. In such a case the individual domains diffract independently and the intensities are calculated as the weighted sums of the intensities of the individuals. Given the signs of disorder and small ordered domains (x3.5), this assumption is only valid to a certain degree. Moreover, the intensities of the MDO 2 characteristic reflections are in the order of magnitude of the intensities of the streaks and therefore the determined volume ratio has to be considered as inexact.
For comparison, the polytype fractions were also indirectly determined by a refinement using only the MDO 1 reflections. The twin volume fraction, which describes the ratio of both orientations of the MDO 1 polytype, refined to 53.1:46.9 (2). Indeed, owing to the laws of probability, volume fractions in a polysynthetic twin with a high stacking fault probability are expected to be close to equal. The Br position was refined as occupationally disordered (x2.3). As we have shown for KAgCO 3 (Hans et al., 2015), the actual fraction of the major polytype is  Table 4 Integral reflection conditions of both MDO polytypes and the family structure with respect to the reciprocal base (a Ã ; b Ã ; c Ã ) = ða=a 2 ; b 0 =ð2b 2 0 Þ; c=c 2 Þ; n h ; n k ; n l 2 Z.
Polytype Symmetry, lattice basis Family structure Pmmn; ða=2; 2b 0 ; c=2Þ k ¼ n k - Figure 8 The (a) 2kl and (b) 3kl planes of reciprocal space of the crystal under investigation reconstructed from CCD data.
domains diffract independently and secondly the contributions of both polytypes to the family reflections are equal. As noted above, we cannot ascertain the former and also the latter is certainly not perfectly fulfilled owing to the strong desymmetrization of the Br position. Nevertheless, the refined occupancy ratio of the Br atoms is 92.25:7.75 (14), which corresponds to an MDO 1 :MDO 2 ratio of 72.16:27.84. The similarity of the ratios derived from both refinements indicates that the assumption of independent diffraction is mostly valid.

Comparison to KAgCO 3
In two instances the (1) allotwins differ notably from the KAgCO 3 allotwins that we have described previously (Hans et al., 2015). Firstly, KAgCO 3 featured a high degree of order and large domains, as witnessed by sharp reflections and an absence of diffuse scattering. Secondly, KAgCO 3 featured virtually no desymmetrization and therefore contributions to the family reflections can be assumed to be equal for all polytypes. These two points are seemingly paradoxical, because one would expect that distinct polytypes are stabilized by desymmetrization. Hence, it is shown that crystallization is a complex dynamic process, which cannot be broken down to such simple rules.

Conclusion and outlook
The crystals of (1) are a further addition to the growing set of known allotwins and here it should be stressed again that the phenomenon is more common than one might expect. It is therefore crucial to recognize and pursue the, often subtle, signs of allotwinning like peaks in the difference Fourier maps and weak additional reflections. Moreover, we want to emphasize the diversity of the phenomenon, making it necessary to address every allotwin as a unique problem.
In a final, and longer, remark it is noted that the OD structure of (1) is particularly unusual owing to the metric parameters ðr; sÞ ¼ ð 1 2 ; 1 2 Þ. Only for this particular pair of parameters the MDO polytypes possess B2 1 =d and F2dd symmetry.
OD groupoid families, which are the only accepted symmetry-classification system of OD families, disregard these metric parameters, despite their fundamental importance. Indeed, to the OD groupoid family of (1) [equation (1)] also belong the groupoids of fully ordered OD structures (i.e. with only one structure in the family). It therefore seems necessary to develop a finer classification system.
On the example of the OD groupoid family of (1), a short overview of the special values that the metric parameters can adopt will be developed. First, one has to realise that for a given A n layer different ðr; sÞ pairs describe the same OD family. Notably, the intrinsic translation of the n r;s glide reflection need only be considered modulo the rectangular centered (ob) lattice of the layers. Moreover, owing to the group of --POs [Bmð1Þ1], if n r;s is a -PO in an alternative stacking arrangement n r;Às is also a -PO. Therefore, for a given A n layer, the pairs ðr; sÞ and ðr; ÀsÞ describe the same family of structures. In total, for this OD groupoid family the intrinsic translation vectors of the n r;s glide reflection can be limited to the asymmetric unit of the A n layers. The corresponding metric parameters are fðr; sÞj0 r<2, 0 s< 1 2 g [ fðr; 1 2 Þj0 r<1g (Fig. 9). Note that in other OD families conjugation of -POs with --POs may alter the linear part, leading to more complex situations.
More complexities arise when considering the symmetry of the family structure. Here, one has to differentiate between rational and irrational metric parameters. If irrational, the lattice of the family structure may become dense in one or two directions.
Certainly, devising a proper definition of the different classes of metric parameters and their tabulation will be a difficult task. Nevertheless, given the frequent occurrence of OD structures, it is overdue.