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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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