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Incommensurate Crystallography$

Sander van Smaalen

Print publication date: 2007

Print ISBN-13: 9780198570820

Published to Oxford Scholarship Online: January 2008

DOI: 10.1093/acprof:oso/9780198570820.001.0001

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Incommensurate Crystallography

Sander Van Smaalen

Oxford University Press

Abstract and Keywords

This chapter presents the steps that are required to determine the superspace group of an aperiodic crystal from its diffraction pattern. This includes the analysis of the metric of the reciprocal lattice in superspace, the point symmetry of the diffraction pattern, and the reflection conditions.

Keywords:   aperiodic crystal, diffraction, point symmetry, reciprocal lattice, reflection conditions

9.1 Indexing of the diffraction pattern

The first step in analysing the diffraction of crystals is finding an integer indexing of the Bragg reflections. Experimentally each Bragg reflection is characterized by a specific orientation of the crystal and the direction of the diffracted beam. Four-circle diffractometers with point detectors produce a list of Bragg reflections in a random search procedure. Each reflection is described by its values for three setting angles describing the crystal orientation, and by its value for the diffraction angle 2θ corresponding to the detector position. Diffractometers with area detectors produce two-dimensional diffraction images. Intensity maxima define Bragg reflections at positions given by a pair of coordinates (x, y) that define the direction of the diffracted beam. To each image belongs a range of crystal orientations, characterized by the values of one to three setting angles and by a range parameter, Δω or Δφ, defining the range of rotation about a selected axis, during which the scattering has been collected onto the detector. The orientation of a crystal is defined by an orientation matrix A. This 3 x 3 matrix gives the coordinates of the three basis vectors of the reciprocal lattice of the crystal with respect to a Cartesian coordinate system fixed to the diffractometer, for an orientation of the crystal with all setting angles equal to zero. For periodic crystals A refers to the reciprocal lattice, for modulated crystals it refers to the reciprocal lattice of the basic structure, while for composite crystals the basic-structure reciprocal lattice of any of the subsystems is a suitable choice for the definition of the orientation matrix.

Robust methods exist for periodic crystals, that determine the orientation matrix along with the integer reflection indices (Duisenberg, 1992; Steller et al., 1997). Setting angles of Bragg reflections can be accurately measured with a point detector. For each orientation matrix A they can be used to compute accurate values of the reflection indices, ( h 1 exp h 2 exp h 3 exp ) . Reflections belonging to the reciprocal lattice defined by A will have values h i exp close to integers, typically differing from integers by less than 0.01, and this property can be used to distinguish them from non-matching reflections. Subsequent inspection or automated analysis of the real-valued indices ( h 1 exp h 2 exp h 3 exp ) of non-matching reflections will reveal possible modulation wave vectors or the second lattice of a composite crystal [eqn (2.5)], or it might reveal secondary lattices for twinned and multi-phase crystals (Duisenberg, 1992).

Accurate values are not available for the setting angles of Bragg reflections measured by area detectors, because scattered radiation is collected onto a single (p.194) image for a range of values of one of the angles describing rotation of the crystal, e.g. with Δω between 0.1° and 1.0°, while steps in centring procedures with point detectors are chosen as small as 0.001° to 0.01°. For images with Δω ∼ 0.5°, the centre of the scan can be assigned to each reflection on this image as its setting angle ω. The relatively large errors in setting angles causes experimental reflection indices to deviate from integers by considerable amounts, e.g. by 0.1, even if these reflections belong to the reciprocal lattice. Successful indexing procedures rely on the assumption that all reflections belong to a single reciprocal lattice (Steller et al., 1997). Integer indices (h 1 h 2 h 3) are assigned to the reflections, where hi is equal to the integer nearest to h i exp . This procedure fails for incommensurate crystals, because it will assign integer indices to satellite reflections, and it thus will fail to distinguish between main reflections and satellite reflections. The fine slicing technique (Δω = 0.05°−0.1°) allows a more accurate determination of the setting angles of reflections. Extended automated procedures then can distinguish between main reflections and satellite reflections, if indexing procedures are combined with refinements of the orientation matrix and refinements of the components of the modulation wave vectors (Schönleber et al., 2001; Pilz et al., 2002). In an alternative approach, the orientation matrix for the strong, presumably main, reflections can be used to compute an undistorted diffraction image in reciprocal space (Estermann and Steurer, 1998). Visual inspection of these images may reveal satellite reflections or secondary reciprocal lattices.

The result of a successful indexing is an orientation matrix and a minimal set of reciprocal vectors M = {a 1*, ⋯, a 3+d*} that is necessary to obtain an integer indexing of all observed Bragg reflections (Section 2.2).

9.2 Point symmetry

Point symmetry of the diffraction pattern of aperiodic crystals is given by a three-dimensional point group (Chapter 3). For incommensurately modulated crystals this is a crystallographic point group that is compatible with the lattice of the basic structure. All possible symmetry elements can be tested by comparing intensities of related reflections, whereby groups of equivalent reflections exclusively contain main reflections or exclusively contain satellite reflections of a single satellite order m. The result is the appropriate three-dimensional point group describing the diffraction symmetry.

Tests for symmetry are often performed through consideration of the internal R index (compare Section 7.1):

R int = j =1 N ref | I ( H j ) < I ( j ) > | j = 1 N ref < I ( j ) > ,
where 〈I(j)〉 is the average intensity computed as the average over all reflections equivalent to reflection H j. For modulated crystals it may occur that deviations (p.195) from certain symmetries appear only in the modulation and not in the basic-structure parameters. The partial R int index for main reflections [R int(0)] will have a low value in accordance with this pseudo symmetry, but R int for all reflections might be small too, if modulation amplitudes are small and satellite reflections are weak. It is thus of utmost importance to consider the partial index R int(m) for satellite reflections, in order to reveal the true point symmetry. As discussed in Section 6.3, twinning might prevent the correct point symmetry to be determined. Several examples of incommensurate crystals are known, for which pseudo-merohedral twinning has resulted in diffraction patterns with symmetries higher than the symmetries of the crystal structures.

Once the three-dimensional point group has been obtained, point symmetry operators in superspace can be determined from the consideration of the action of each operator R on the additional reciprocal vectors a 3+1* through a 3+d*.

9.3 Reflection conditions

Intrinsic translations of screw and glide operators in superspace groups can be found from the analysis of possible reflection conditions on the subgroup of the reflections with scattering vectors that are left invariant by the point symmetry element under consideration. This analysis is feasible only if the basic-structure unit cell is transformed towards a supercell, {A 1, A 2, A 3}, such that transformed modulation wave vectors, q i, have their rational component equal to zero (Section 3.9.1). Of course, a transformation is not required if q r = 0 from the outset [eqn (3.4)]. The set of invariant reciprocal points is most easily defined in physical space: invariant points are reciprocal points that lie in the mirror plane or on the rotation axis of the physical-space part of the symmetry operator. They always include a reciprocal lattice plane or reciprocal lattice line of main reflections. Depending on the components of the modulation wave vectors, satellite reflections may or may not be part of the set of invariant points. For example, satellite reflections are never invariant points for operators with ε = −1, while they may fail to be invariant points for operators with ε = 1, if the modulation wave vector contains rational components (Section 3.4).

The choice of a supercell for the basic structure introduces a centring of the superspace lattice. Furthermore, superspace lattices may already be centred for the standard choice of basic-structure unit cell. Any lattice centring can be analysed within the framework of reflection conditions, if they are considered as intrinsic translations of the unit operator E. All reflections are invariant points for this operator. For example, a reflection condition

( H 1 H 2 H 3 H 4 ) : H 2 + H 3 + H 4 = 2 n
might have been observed. This condition indicates the presence of an A′ centring ( 0 , 1 2 , 1 2 , 1 2 ) (Table 3.9), as it might be found for a non-standard setting of the superspace group Ammm1 1 0)1¯00 (No. 14 in Table 3.11).

In general, reflection conditions for reflections (H 1 H 2 H 3 H 4) are of the form [eqn (3.57)] (p.196)

p 1 H 1 + p 2 H 2 + p 3 H 3 + p 4 H 4 = p 0 n ,
with pi and n integers. The quantity pi is set to zero for all i = 1, ⋯, 4 for which Hi = 0 in the set of invariant points. Reflections may have non-zero intensities, if they fulfil the condition eqn (9.2) for some integer n, while reflections that violate eqn (9.2) must have zero intensity. The condition eqn (9.2) then corresponds to the intrinsic translation [eqn (3.58)]
v s I = ( p 1 p 0 , p 2 p 0 , p 3 p 0 , p 4 p 0 ) .

For example consider a sixfold rotation on the hexagonal lattice with modulation wave vector q = (0, 0, σ3). Invariant reciprocal lattice points are (0 0 h 3 h 4), and a possible reflection condition is

( 0 0 h 3 h 4 ) : h 3 + 3 h 4 = 6 n ,
i.e. p 1 = p 2 = 0, p 3 = 1, p 4 = 3 and p 0 = 6. This condition corresponds to the intrinsic translation ( 0 , 0 , 1 6 , 1 2 ) . The symmetry operator is (61, s), but a different choice for the modulation wave vector would lead to the standard setting (61, 0) of this sixfold screw operator, with the alternate reflection condition (Table 3.12)
( 0 0 h 3 h 4 ) : h 3 = 6 n .

Intrinsic translations obtained from indexed diffraction patterns may correspond to non-standard settings of superspace groups, and they do not necessarily appear in the list of superspace groups in the International Tables for Crystallography Vol. C (Janssen et al., 1995). A non-standard setting is often preferred, and one should not hesitate to keep it in the further structural analysis.