Feature article: A non–mathematical introduction to the superspace description of modulated structures

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  electronic reprint Acta Crystallographica Section B StructuralScience ISSN 0108-7681 Editor: Carolyn P. Brock A non-mathematical introduction to the superspacedescription of modulated structures Trixie Wagner and Andreas Sch ¨onleber Acta Cryst. (2009). B 65 , 249–268 Copyrightc  International Union of CrystallographyAuthor(s) of this paper may load this reprint on their own web site or institutional repository provided thatthis cover page is retained. Republication of this article or its storage in electronic databases other than asspecified above is not permitted without prior permission in writing from the IUCr.For further information seehttp://journals.iucr.org/services/authorrights.html Acta Crystallographica Section B: Structural Science  publishes papers in structural chem-istry and solid-state physics in which structure is the primary focus of the work reported.The central themes are the acquisition of structural knowledge from novel experimentalobservations or from existing data, the correlation of structural knowledge with physico-chemical and other properties, and the application of this knowledge to solve problemsin the structural domain. The journal covers metals and alloys, inorganics and minerals,metal-organics and purely organic compounds. Crystallography JournalsOnline is available from journals.iucr.org Acta Cryst. (2009). B 65 , 249–268 Trixie Wagner et al. · Superspace description of modulated structures  feature articles Acta Cryst. (2009). B 65 , 249–268 doi:10.1107/S0108768109015614 249 Acta Crystallographica Section B StructuralScience ISSN 0108-7681 A non-mathematical introduction to the superspacedescription of modulated structures Trixie Wagner a * and AndreasScho¨nleber b a Novartis Institutes for BioMedical Research,4002 Basel, Switzerland, and b Laboratory of Crystallography, University of Bayreuth, 95440Bayreuth, GermanyCorrespondence e-mail:trixie.wagner@novartis.com # 2009 International Union of CrystallographyPrinted in Singapore – all rights reserved The X-ray analysis of (6 R ,7a S )-6-( tert  -butyl-dimethylsilanyl-oxy)-1-hydroxy-2-phenyl-5,6,7,7a-tetrahydropyrrolizin-3-one,C 19 H 27 NO 3 Si, revealed a diffraction pattern which is typicalfor modulated structures: strong Bragg peaks surrounded byweaker reflections which cannot be indexed with the samethree reciprocal lattice vectors that are used to describe thestrong peaks. For this class of crystal structures the concept of superspace has been developed which, however, for manycrystallographers still constitutes a Gordian Knot. As apossible tool to cut this knot the crystal structure of theabove-mentioned tetrahydropyrrolizinone derivative ispresented as an illustrative example for handling anddescribing the modulated structure of a typical pharmaceutical( i.e. molecular) compound. Having established a workingknowledge of the concepts and terminology of the superspaceapproach a concise and detailed description of the completeprocess of peak indexing, data processing, structure solutionand structure interpretation is presented for the incommen-surately modulated crystal structure of the above-mentionedcompound. The superspace symmetry applied is P 2 1 (  0   )0;the (incommensurate) q vector components at 100 K are  =0.1422 (2) and   = 0.3839 (8). Received 13 February 2009Accepted 27 April 2009 1. Introduction Together with quasicrystals and composite crystals incom-mensurately modulated structures constitute the category of aperiodic crystals. Since modulated structures will bediscussed in detail over the course of this paper the interestedreader is referred to ‘An elementary introduction to super-space crystallography’ by van Smaalen (2004) for a compre-hensible description of quasicrystals and composite crystals.The diffraction patterns of modulated structures are char-acterized by the existence of additional Bragg reflections sothat an integer indexing with three indices hkl  is not possible;instead, 3 + d indices are required ( d = 1, 2 or 3). The necessityfor using four or more indices must be understood as a loss of periodicity in three dimensions. To restore the periodicity, theconcept of higher-dimensional superspace was developed (deWolff, 1974, 1977). Atoms are no longer points in space, butare envisioned as d -dimensional atomic domains (Janner &Janssen, 1977).The theoretical foundation for the superspace approach isnow well established (van Smaalen, 2007). Furthermore,considerable effort has been put into making JANA 2006, acrystallographic computing system that can convenientlyhandle the data and structures of modulated compounds(Petricek et al. , 2006), available and user-friendly.However, for the majority of the crystallographic commu-nity many of the concepts and terms used in this context are electronic reprint  still not easily accessible because superspace descriptions tendto be rather mathematical and based on inorganic examples.In this manuscript a more practical approach to this topic ispresented, based on an illustrative example from a pharma-ceutical crystallographic service lab. The manuscript is dividedinto three main sections: in the first section a workingknowledge of modulated structures will be established andkey terms such as modulation vector, atomic modulationfunction, atomic domain etc. will be introduced and explained.After a brief excursion into the historical and recent devel-opments in the field of aperiodic structures this workingknowledge will be illustrated in the third section where thecomplete process of peak indexing, data processing, structuresolution and structure interpretation for an incommensuratelymodulated structure is presented for the crystal structureof (6 R ,7a S )-6-( tert  -butyl-dimethyl-silanyloxy)-1-hydroxy-2-phenyl-5,6,7,7a-tetrahydropyrrolizin-3-one, C 19 H 27 NO 3 Si. 1.1. Modulated structures – a first definition Most crystal structures are periodic in three dimensions andshow a diffraction pattern that can be indexed with threeinteger numbers (Fig. 1). Modulated structures can be derivedfrom those structures: here atoms, groups of atoms or evenwhole molecules are shifted or rotated with respect to theirneighbours such that the three-dimensional translationalsymmetry, often considered as the characteristic feature of acrystal structure, is destroyed. These shifts and rotations inmodulated structures, however, are not arbitrary; they followdistinct rules and within these distortions (or better: modu-lations) there is additional periodicity which can mathemati-cally be described by so-called atomic modulation functions (AMFs). AMFs can be harmonic (continuous) and thereforebe expressed by a sine/cosine combination or they may bediscontinuous, in which case crenel or sawtooth functions areneeded for their description (Fig. 2; Petricek et al. , 1995).Based on the periodicity of the modulation wave a distinctioncan be made between commensurately modulated structures(periodicity matches an integral number of lattice translationsof the basic cell) and incommensurately modulated structures(periodicity does not match an integral number of lattice feature articles 250 Trixie Wagner et al.  Superspace description of modulated structures Acta Cryst. (2009). B 65 , 249–268 Figure 1 Two-dimensional schematic representation of a hypothetical crystalstructure that is periodic in three dimensions and its schematic diffractionpattern, indexed with three integer numbers based on the reciprocallattice vectors a *, b * (not shown) and c *. Lattice/unit cells in light grey. Figure 2 Schematic representation of three manifestations of a modulatedstructure with lost translational symmetry along the a axis. All threedrawings in ( a ) are derived from the periodic structure shown in Fig. 1 byshifting or rotating the molecules. The atomic modulation functionssuitable for the description of the modulated atomic positions are shownas an overlay: in the top drawing the molecules are shifted up and downparallel to c in a continuous harmonic (sinusoidal) way (red curve); in themiddle drawing the molecules are rotated around an axis parallel to a , therotation angle can be described using a sawtooth function (blue) with adiscontinuity between molecules 8 and 1; in the bottom drawing themolecule adopts two different orientations which can be described by astep-like crenel function (green). Note that the modulation proceeds onlyalong a , the c direction is not affected. ( b ) In the associated schematicdiffraction pattern additional reflections (grey circles) appear along a *.The number of satellite reflections and their intensity distribution dependon the strength and nature of the modulation. For simplicity, only onediffraction scheme was drawn. electronic reprint  translations of the basic cell and is therefore incommensuratewith the periodic basic structure).Due to the periodic character of the modulation, additionalsharp peaks appear in the diffraction pattern, just as the Braggreflections produced by a three-dimensional crystal are aresult of the periodic terms in the structure factor equation.These additional peaks are referred to as satellite reflections and usually they are weaker than the main reflections. Thesatellite reflections can lie parallel to one reciprocal axis, butthey do not have to. It should be emphasized that modulatedstructures are not disordered but have long-range order whichis reflected in those discrete additional peaks rather than indiffuse streaks. For displacive modulations as described above( i.e. the atomic positions are affected), the number of satellitereflections that can be observed depends on the amplitude of the AMFs or, in other words, on the degree of distortion/modulation present in the three-dimensional crystal structure.The stronger the modulation, the stronger the satellitereflections and also the higher the order of the satellitereflections that have measurable intensity (see x 1.2.3).Even though the distribution of satellite reflections inreciprocal space is not arbitrary, they cannot be described withthe same three reciprocal lattice vectors that allow an integerindexing of the main reflections. Any attempt to do so resultsin non-integer values for h and/or k and/or l  (Fig. 3). For anincommensurately modulated structure there is not any set of three vectors that allows an integer indexing of all (main andsatellite) reflections. 1.2. Handling the diffraction pattern of modulated structures The diffraction pattern of a modulated structure (strongmain reflections surrounded by weaker satellite reflections)can be approached in three different ways, two of which arebased on a description in the traditional three-dimensionalspace, and one that is not. Since the structure, however, is notperiodic in three dimensions (except for commensurate cases)any three-dimensional approach can only be regarded as anapproximation. The three-dimensional approaches shall bementioned nonetheless since the results may contain valuableinformation that can be used at a later stage (space-groupdetermination in superspace, identification of groups withstrongest modulation, type of AMF to describe the modula-tion etc. ). 1.2.1. Basic cell and average structure . The first option is toignore the weak satellite reflections altogether and concen-trate on the main reflections. The indexing procedure thendelivers a so-called basic cell (sometimes also referred to as a subcell ) with a volume that can accommodate a realisticnumber of the molecule under study (1, 2, 3 etc. ). In the three-dimensional crystal structure using the main reflections onlycorresponds to averaging the contents of several unit cells andlooking at a structure which is commonly referred to as the average structure . It is usually characterized by large aniso-tropic atomic displacement parameters (ADPs) as well asunrealistic bond lengths and bond angles. In the hypotheticalstructure of Fig. 2, for example, the averaging would includeeight unit cells (Fig. 4). The inherent disadvantage of thisapproach is that a certain amount of information provided bythe diffraction pattern, i.e. satellite intensity, is neglected. 1.2.2. Supercell and superstructure . The second way of handling the diffraction pattern is to drop the distinctionbetween main and satellite reflections and to use all reflectionsequivalently for indexing (Fig. 5). This results in a smallerreciprocal unit cell and a reciprocal lattice where, dependingon the maximum order of the satellite reflections that areobserved, many reciprocal lattice points may be empty, i.e. have no observable intensity. In direct space the unit cell isaccordingly larger and is commonly referred to as a supercell .The resulting crystal structure is called a superstructure and ischaracterized by a large number of independent molecules inthe asymmetric unit ( Z  0 ; Z  0 = 8 inthe example of Fig. 5). Inincommensurate cases where thepositions of the satellite reflec-tions do not perfectly fit the gridof the supercell lattice a defor-mation of reciprocal space has tobe accepted which usuallymanifests itself in poor agree-ment factors, large standarddeviations of refined parameters,split atoms, large ADPs etc. In feature articles Acta Cryst. (2009). B 65 , 249–268 Trixie Wagner et al.  Superspace description of modulated structures 251 Figure 3 Indexed schematic diffraction pattern of the modulated structure of Fig.2,main reflections black, satellite reflections grey. Indices for mainreflections are based on the lattice vectors a *, b * (not shown) and c *.Note that an integer indexing for the satellite reflections is not possibleusing those vectors. The deviations from the indices of the closest mainreflection are ( À 0.25, 0, 0) for the marked satellite reflection at the topand (+0.25, 0, 0) for that at the bottom. Figure 4 Neglecting satellite reflections and solving the structure in the basic cell  corresponds to averaging thecontents of the eight unit cells shown in either of the top rows of Fig. 2( a ) by shifting the molecules parallel to a into one cell (H atoms omitted for clarity; the molecules from Fig. 2 have been moved towards each otheralready). Consequently, the resulting average structure is characterized by large displacement ellipsoids and,in the case of a strong modulation, also by unrealistic geometric parameters. electronic reprint  such cases the term superstructure approximation would bemore appropriate (see also x 1.4).For commensurate cases the description as a (three-dimensional) superstructure constitutes a valid approach ( e.g. Hao et al. , 2005; Siegler et al. , 2008) and can even be used incombination with a (higher-dimensional) superspace treat-ment (Schmid & Wagner, 2005). 1.2.3. Moving into superspace . In the third approach, too,all reflections are used, but the distinction between main andsatellite reflections is maintained. In a first step the reciprocalunit cell is established using the main reflections ( cf  x 1.2.1).The second step makes use of the fact that the distribution of satellite reflections in reciprocal space is not arbitrary; theycan be divided into groups in which they are equidistant notonly from each other but also from the main reflection towhich they belong. This systematic distribution allows thedefinition of a fourth vector, the so-called modulation wave-vector q , which describes the satellite reflections with respectto their main reflection. Now every satellite peak can beuniquely identified as being n Á q ( n = Æ 1, Æ 2, . . . ) away fromits main reflection, and one speaks of the n th-order satellitereflection of this main reflection. This manuscript will berestricted to four-dimensional cases, i.e. one modulationwavevector q . The theory for five- or six-dimensional cases,where a second or even a third set of satellite reflectionsrequires the use of one or two additional vectors, can bederived accordingly. Since five- and six-dimensional cases,however, occur mainly with higher symmetry (hexagonal,cubic) and a standardized treatment is still in the process of being developed, they are beyond the scope of this intro-ductory guide.Within the framework of the three reciprocal lattice vectors a *, b * and c *, which describe the basic cell, the modulationwavevector q can be expressed only with the help of fractionalcomponents: q =  a * +  b * +   c *. In other words, thecomponents of  q are given with respect to the basis vectors of the reciprocal lattice of the basic cell or average structure. 1 Please note that the number of non-zero components of  q isnot related to the dimensionality of the modulation, which isdefined by the number of  q vectors necessary for thedescription of reciprocal space. Depending on the rationalityof   ,  and   there is now a second way ( cf. x 1.1) to classifymodulated structures as commensurate (all componentsrational) or incommensurate (at least one of the componentsirrational). In direct space commensurate cases are char-acterized by a supercell in which all lattice parameters areinteger multiples (1, 2, . . . , n ) of the lattice parameters of thebasic cell. In practice, however, it is often not easy to distin-guish between commensurate and incommensurate cases,especially when the multipliers involve larger integers and thedifferences between a commensurate and an incommensurateapproach disappear.Working with one modulation wavevector q implies atransition into four-dimensional space ( cf. x 1.2.4) and everyreflection of the diffraction pattern can now be described in aunique way by four integer indices h, k, l  and m . This higher-dimensional concept affects all subsequent steps of structureanalysis and therefore will be discussed in detail in connectionwith the sample structure in x 3. A schematic illustration of therelation between superspace and three-dimensional space isgiven in Fig. S1 of the supplementary material. 2 1.2.4. The definition of reciprocal superspace . It is difficultto envision the fourth dimension introduced in the preceedingsection by looking at the q vector alone because q itself is onlypart of the definition of reciprocal superspace. To illustrate the feature articles 252 Trixie Wagner et al.  Superspace description of modulated structures Acta Cryst. (2009). B 65 , 249–268 Figure 5 Using main and satellite reflections equivalently results in a large unit cell(light grey) with a 0 = 8 Á a or, in reciprocal space, a 0 * = 1/8 Á a *. Note thatunless satellite reflections up to higher orders are observed, numerousreciprocal lattice points are empty ( b ). This holds especially if the satellitereflections are not along one reciprocal axis but are located in a plane( e.g. the a * c * plane, cf. x 3). 1 The q vector components  ,  and   have to be distinguished from the unit-cell parameters  ,  and   . Alternative notations such as   1 ,   2 ,   3 can befound in the literature (van Smaalen, 2007). In this manuscript, however, thenomenclature used in Vol. C of the International Tables of Crystallography (  ,  and   ) will be followed (Janssen et al. , 2006). 2 Supplementary data for this paper are available from the IUCr electronicarchives (Reference: BK5084). Services for accessing these data are describedat the back of the journal. electronic reprint
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