Chapter 3.4. Classification and use of modulated and composite structures data

Aperiodic structures do not have lattice periodicity, but do show long-range order. Their diffraction patterns exhibit sharp Bragg peaks that have to be indexed using more than three Miller indices. All aperiodic structures can be classified into one of three groups: incommensurately modulated structures, incommensurate composite structures and quasicrystals. It is the purpose of the modulated and composite structures CIF dictionary (msCIF dictionary), presented in Chapter 4.3 , to provide machine-readable descriptions of the first two of these groups.

Modulated structures represent the simplest case and are described by periodic distortions of an underlying (reference) crystal structure. The distortions involve one or several atomic parameters – positions, occupation probability or thermal parameters (the term displacement parameters is ambiguous in this context) – and their periodicity may be commensurate or incommensurate with the lattice of the reference structure. The first case corresponds to a three-dimensionally periodic crystalline phase, whereas the second case defines an incommensurate structure. In both cases, the diffracted intensities can be divided into two groups: the prominent main reflections, which are located in a three-dimensional reciprocal lattice (that of the reference structure), and additional (generally) weaker satellite reflections situated at points determined by the wave vectors of the modulations. Strictly speaking, the number of parameters needed to describe the atomic modulations of an incommensurate structure is infinite. In practice, however, the number of structural parameters is often drastically reduced owing to the strong hierarchy (demonstrated by the discreteness of the diffraction diagrams) that exists among them.

Composite structures can be thought of as being built of two or more periodic subsystems whose lattices are mutually incommensurate. Therefore, the set of main reflections no longer defines a unique three-dimensional reciprocal lattice. Moreover, the interaction between the different subsystems provokes modulations and, as a consequence, the presence of satellite reflections in the diffraction diagram.

Quasicrystals not only lack three-dimensional lattice periodicity but also show noncrystallographic symmetry.

The methodology for solving aperiodic crystal structures has been well developed since the introduction of the concept of superspace (de Wolff, 1974; de Wolff et al., 1981). Superspace allows the recovery of the periodicity and a simple description of the symmetry of quasicrystalline structures in a higher-dimensional space. The real aperiodic structure is recovered from the superspace through appropriate three-dimensional sections. The dimension of the superspace (3 + d) is equal to the number of reciprocal vectors needed to index the whole diffraction pattern of the quasicrystalline structure. More information on the superspace approach can be found, for example, in van Smaalen (1995) and Janssen et al. (2004).

The success of this concept relies above all on the systematic description of the symmetry of the aperiodic materials using superspace groups. Superspace groups for the simplest (but most common) case of one-dimensional modulated structures are tabulated in Janssen et al. (2004).

Within the superspace approach, all the aperiodic atom positions are embedded in dense sets (atomic domains) in the (3 + d)-dimensional unit cell of the associated periodic structure in superspace. They are parallel on average to the `internal' (or `perpendicular') space, which is a d-dimensional subspace chosen to be orthogonal to the real (physical or parallel) space. The three-dimensional structure is then a section of the (3 + d)-dimensional structure parallel to the real space. The atomic domains are distorted along the internal space by the modulation functions. In many incommensurate structures and composites, the atomic domains are continuous periodic functions along the internal subspace and are parameterized by Fourier series. Some compounds, however, need to be described using discrete atomic domains whose parameterization is more complicated and will be discussed in Section 3.4.2. Note that commensurate structures can also be included within the superspace approach. The difference between incommensurate and commensurate modulations is that for commensurate modulations only a finite number of values of the atomic modulation functions are relevant.

The number of modulated and composite structures solved with software that uses the superspace formalism has grown rapidly and in many cases the determination of the structures of such systems is now almost routine. A standard for the description of incommensurate modulated structures has been established by the Commission on Aperiodic Crystals of the International Union of Crystallography (Chapuis et al., 1997) using a checklist that is easily extensible to composite crystals. However, there was until recently no standard way to represent these structures electronically. Structural databases tend to contain only a brief reference to the modulated character of the structure and it has not been possible to transmit, archive and retrieve information about modulated and composite structures as efficiently as for normal crystal structures. Extending the core CIF dictionary to form a modulated and composite structures CIF (msCIF) dictionary seemed the appropriate way to deal with these problems, and has the additional benefits derived from the use of a well tested standard for which several tools have been developed.

In the case of quasicrystals, however, although the theoretical foundations seem to be well established, the determination of accurate models requires a combination of different strategies and techniques. The parameterization of the atomic domains with physical meaning is far from being an automated procedure (Cervellino et al., 2002) and some of the existing models are now disputed. Major problems arise from sample quality, intrinsic disorder and rather low data-to-parameter ratios (Haibach et al., 2000). Quasicrystals are not covered by the msCIF dictionary detailed in Chapter 4.3 .

The CIF dictionary for modulated and composite structures (msCIF dictionary) is an extension of the core CIF dictionary (Chapter 4.1 ). This means that the dictionary defines items that are basically related to single-crystal data. The close relationship between the msCIF and core dictionaries has led to synergies between and benefits for both dictionaries.

The design of the msCIF dictionary had two objectives: (i) it should be as functional as possible, i.e. as little information in an msCIF as possible should be given as unstructured text; (ii) it should be possible to include even the oldest modulated and composite structures in an msCIF, even if the way they were reported did not follow the guidelines used now.

There were two major difficulties in implementing the msCIF dictionary. Firstly, the number of additional wave vectors used to index a diffraction diagram is theoretically not limited. Secondly, a CIF containing information about a modulated or composite structure should, in general, be composed of several (related) data blocks. As CIF definitions do not at present include vectors or matrices as distinct types of data values, an arbitrary upper limit of 11 was assigned to the dimension of superspace to limit the number of new data names. Linking between data blocks is handled by using recommended values for items in the AUDIT and AUDIT_LINK categories, like those used in the powder CIF (pdCIF) dictionary (Chapters 3.3 and 4.2 ).

An additional problem arises when special (ideal) modulation functions are considered. Although periodic modulations are normally parameterized by Fourier series, in certain cases it is convenient to use discontinuous functions which lead to a severe reduction in the number of structural parameters. The shape of these functions is not restricted and new materials could require new functions. Given that it is not possible at this moment to define logical or mathematical relations between data values within a CIF [although an initiative for including algorithms in the definitions of CIF dictionaries has been proposed by Spadaccini et al. (2000)], general functions cannot be defined and therefore the type of special functions included in the msCIF dictionary are those implemented in the most widely used program, JANA2000 (Petříček & Dušek, 2000). They only apply to one-dimensional modulations and are sawtooth displacive functions and occupational crenel functions. Both functions define discontinuous occupational atomic domains and are normally combined with (smoother) atomic modulation functions (involving atom positions and/or thermal parameters) that are expressed by Fourier series. Because of the discreteness of the atomic domains, members of the set of harmonic functions used to expand these series are no longer mutually orthogonal as they are only defined within each atomic domain and not in the (internal space) interval [0, 1]. As a consequence, severe correlation effects among the coefficients of the Fourier series are expected. A solution for this problem lies in the selection and orthogonalization of a set of basic functions (Petříček et al., 1995). The atomic modulation functions are then expressed as linear combinations of an orthogonal basis whose elements are specific combinations of harmonic functions. Discontinuous atomic domains are being increasingly used in composite materials, in some cases revealing that considering such materials as composites or modulated structures is a matter of convenience (Elcoro et al., 2003; Pérez-Mato et al., 2003).

CIFs that conform to the msCIF dictionary are highly itemized for the human reader, but have a strong relational structure even though the dictionary itself is written in DDL1.

The major drawbacks of the dictionary are:

(i) Items describing the superspace symmetry should be reconsidered and perhaps included within the symmetry CIF (symCIF) dictionary (Chapters 3.8 and 4.7 ).

(ii) There is still some information in an msCIF that cannot be interpreted by a computer [e.g. rigid rotations and translations around (along) noncrystallographic axes cannot be parsed, since the description of the axes is textual].

(iii) A full description of the modulation in terms of orthogonalized functions (used when the atomic domains are discrete) is not supported yet.

The msCIF dictionary detailed in Chapter 4.3 includes 19 new categories. Another 18 already exist in the core CIF dictionary, but include new items (16) or revised definitions (2). The category structure of the msCIF dictionary is summarized in Table 3.4.2.1 and is listed in full in Appendix 3.4.1. The appendix also lists for each category the section of this chapter in which the category is described.

Table 3.4.2.1 | top | pdf | Category groups defined in the msCIF dictionary The groups are listed in the order in which they are described in this chapter. Section Category group Subject covered (a) Experimental measurements 3.4.3.1.1 CELL The unit cell, especially cell subsystems for composite structures 3.4.3.1.2 DIFFRN Multi-dimensional diffraction pattern 3.4.3.1.3 EXPTL Details of the experiment specific to modulated and composite structures       (b) Analysis 3.4.3.2.1 REFINE Refinement procedures 3.4.3.2.2 REFLN Reflection measurements indexed in higher-dimensional space       (c) Atomicity, chemistry and structure 3.4.3.3.1 to 3.4.3.3.3 ATOM Atom sites in a modulated structure 3.4.3.3.4 GEOM Geometry of a structure in superspace 3.4.3.3.5 SYMMETRY Symmetry information       (d) File metadata 3.4.3.4 AUDIT The structure of the CIF

Many of the modifications to categories that already exist in the core CIF dictionary result from the need to use more than three integer indices to label the diffracted intensities (in the cases of CELL, DIFFRN_REFLN, DIFFRN_REFLNS, DIFFRN_STANDARD_REFLN and EXPTL_CRYSTAL_FACE) or the need to use superspace symmetry (in the cases of GEOM_ANGLE, GEOM_BOND, GEOM_CONTACT, GEOM_TORSION, SPACE_GROUP and SPACE_GROUP_SYMOP). Apart from the categories that describe the atomic modulation functions, there are two that are specific to composite structures (CELL_SUBSYSTEM and CELL_SUBSYSTEMS).

The rest of this section summarizes the contents of the dictionary, organized by categories within the functional groups outlined in Table 3.1.10.1 . As in the other chapters in this part of the volume, the classification is under the headings Experimental measurements (Section 3.4.3.1), Analysis (Section 3.4.3.2), Atomicity, chemistry and structure (Section 3.4.3.3) and File metadata (Section 3.4.3.4). The msCIF dictionary adds no new data items concerned with the publication or reporting of structures to those already present in the core CIF dictionary.

The data items within each category are listed in the detailed commentary below. Where relevant, data items that represent a unique identifier for a looped list (`category keys') are listed first and are marked by a bullet (). The remaining data items in each category are listed alphabetically.

3.4.3.2. Analysis

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The categories related to refinement that have been extended in this dictionary are as follows:

Refinement techniques and results (§3.4.3.2.1)  REFINE ¶ The reflections used in the refinement (§3.4.3.2.2)  REFLN ¶  REFLNS ¶

Categories marked with ¶ are already defined in the core CIF dictionary.

3.4.3.2.1. Refinement techniques and results

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New data items in this category are as follows:

REFINE

During the early stages of the development of the msCIF dictionary, several sets of data items were defined to accommodate the need to specify residual R factors for the different sets of main reflections and satellite reflections. It was then recognized that the binning of reflection classes had more general application, and these new data items were transferred to the core CIF dictionary, where, of course, they are still available for use in an msCIF.

The new items in the REFINE category in the msCIF dictionary are specific to the refinement of modulated structures. _refine_ls_mod_func_description allows a free-text description of the types of modulation present in the structural model and how they are handled. The treatment of hydrogen-atom modulation parameters is specified by _refine_ls_mod_hydrogen_treatment. Information on an overall phason correction (the use of which should in general be discouraged) may be given using the _refine_ls_mod_overall_phason_* items.

3.4.3.2.2. The reflections used in the refinement

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New data items in these categories are as follows:

(a) REFLN

(b) REFLNS

As with the _diffrn_refln_* and _diffrn_reflns_* items (Section 3.4.3.1.2), these data names extend the corresponding core data items into the higher-dimensional space used in the treatment of modulated structures and composites. They apply to the list of reflections used in the refinement, as distinct from the experimentally collected set of intensities described by the _diffrn_* data items.

3.4.3.3. Atomicity, chemistry and structure

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The categories relevant to the description of the structural model are as follows:

ATOM group Atom sites (§3.4.3.3.1)  ATOM_SITE ¶  ATOM_SITE_PHASON Modulation functions as Fourier series (§3.4.3.3.2)  ATOM_SITE_DISPLACE_FOURIER  ATOM_SITE_DISPLACE_FOURIER_PARAM  ATOM_SITE_FOURIER_WAVE_VECTOR  ATOM_SITE_OCC_FOURIER  ATOM_SITE_OCC_FOURIER_PARAM  ATOM_SITE_ROT_FOURIER  ATOM_SITE_ROT_FOURIER_PARAM  ATOM_SITE_U_FOURIER  ATOM_SITE_U_FOURIER_PARAM  ATOM_SITES_DISPLACE_FOURIER  ATOM_SITES_MODULATION  ATOM_SITES_ROT_FOURIER Special modulation functions (§3.4.3.3.3)  ATOM_SITE_DISPLACE_SPECIAL_FUNC  ATOM_SITE_OCC_SPECIAL_FUNC Molecular or packing geometry (§3.4.3.3.4) GEOM group  GEOM_ANGLE ¶  GEOM_BOND ¶  GEOM_CONTACT ¶  GEOM_TORSION ¶ Symmetry information (§3.4.3.3.5) SYMMETRY group  SPACE_GROUP ¶  SPACE_GROUP_SYMOP ¶

Categories marked with ¶ are already defined in the core CIF dictionary.

Most of the new categories introduced to the msCIF dictionary appear here, since their function is to describe in great detail the modulation of the atom-site properties. They fall naturally into families describing the modulation of atomic displacement, of site occupation or of thermal parameters.

New data items are added to several categories in the core CIF dictionary that describe molecular or packing geometry. There are also new data items to describe superspace-group symmetry.

3.4.3.3.1. Atom sites

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Data items in these categories are as follows:

(a) ATOM_SITE

(b) ATOM_SITE_PHASON

The bullet () indicates a category key. The arrow () is a reference to a parent data item.

The ATOM_SITEcategory is extended in the msCIF dictionary by the addition of a small number of items that may appear in the main looped list of atom-site information (see Section 3.2.4.1.1 ). The *_flag items indicate whether each individual atom site has been modelled through modulation of atomic displacement, site occupation or thermal parameters. In each case, the default value of the item is no, so that any or all of the flags may be omitted when that particular type of modulation has not been applied to the structural model.

_atom_site_subsystem_code identifies the cell subsystem to which the atom site must be assigned in the description of composite structures. Each value of _atom_site_subsystem_code must match one of the values of _cell_subsystem_code in the overall description of the subsystems defined for a composite.

The ATOM_SITE_PHASON category allow details of an atom-dependent phason correction, as implemented in JANA2000, to be given. The use of these phason corrections is discouraged.

3.4.3.3.2. Modulation functions as Fourier series

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Data items in these categories are as follows:

(a) ATOM_SITE_DISPLACE_FOURIER

(b) ATOM_SITE_DISPLACE_FOURIER_PARAM

(c) ATOM_SITE_FOURIER_WAVE_VECTOR

(d) ATOM_SITE_OCC_FOURIER

(e) ATOM_SITE_OCC_FOURIER_PARAM

(f) ATOM_SITE_ROT_FOURIER

(g) ATOM_SITE_ROT_FOURIER_PARAM

(h) ATOM_SITE_U_FOURIER

(i) ATOM_SITE_U_FOURIER_PARAM

(j) ATOM_SITES_DISPLACE_FOURIER

(k) ATOM_SITES_MODULATION

(l) ATOM_SITES_ROT_FOURIER

The bullet () indicates a category key. The arrow () is a reference to a parent data item.

It is common to represent a modulated structure using a reference periodic structure on which are superimposed atomic modulation functions expanded as Fourier series. (A full discussion of this is given in Section 3.4.4.3.) The msCIF dictionary provides separate categories for listing the modulated parameters that apply to atom positions, site occupancies and thermal parameters. The structuring of the data items within each of these categories follows a similar pattern.

For example, consider the modulation of the atomic displacements. The ATOM_SITE_DISPLACE_FOURIER category allows a listing of the axis along which the displacement occurs (*_axis) and the wave vectors contributing to that displacement component (*_wave_vector_seq_id) for each relevant atom site (labelled by _atom_site_displace_Fourier_atom_site_label). *_wave_vector_seq_id is a pointer to the description of the separate modulation wave vectors and must match one of the identifiers _atom_site_Fourier_wave_vector_seq_id listed separately in the ATOM_SITE_FOURIER_WAVE_VECTOR category. Likewise, the *_atom_site_label data item must match a value of _atom_site_label in the main list of atom positions. This is how the modulation is linked to the atom list. The item _atom_site_displace_Fourier_id is the formal key for the ATOM_SITE_DISPLACE_FOURIER category. It is used to locate the matching Fourier coefficients in the ATOM_SITE_DISPLACE_FOURIER_PARAM category. The coefficients may be reported in a sine–cosine ( _atom_site_displace_Fourier_param_sin, *_cos) or modulus–argument (*_mod, *_phase) representation.

Where a group of atoms is treated as a rigid group, the categories above describe only the translational part of the positional distortion. ATOM_SITE_ROT_FOURIER and ATOM_SITE_ROT_FOURIER_PARAM are used to describe the rotational components.

ATOM_SITE_OCC_FOURIER and ATOM_SITE_U_FOURIER, and their associated *_PARAM categories, are the analogous categories for the modulation of site occupation and thermal parameters.

All the categories above describe the properties of individual atom sites. Larger-scale descriptions of the displacive modulation or of the rotational component of a rigid group are covered by the categories ATOM_SITES_DISPLACE_FOURIER and ATOM_SITES_ROT_FOURIER, each of which at present contains one descriptive data item.

The ATOM_SITES_MODULATION category contains data items describing the initial phases of the modulation waves, which are essential for determining the space group of the commensurate superstructure. More details are given in the dictionary.

3.4.3.3.3. Special modulation functions

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Data items in these categories are as follows:

(a) ATOM_SITE_DISPLACE_SPECIAL_FUNC

(b) ATOM_SITE_OCC_SPECIAL_FUNC

The bullet () indicates a category key. The arrow () is a reference to a parent data item.

Several data items cover modulation functions that are not expressed as Fourier expansions. The examples in the current msCIF dictionary are restricted to the one-dimensional modulations (sawtooth displacive and occupational crenel functions) implemented in the program JANA2000 (see Section 3.4.2).

3.4.3.3.4. Molecular or packing geometry

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New data items in these categories are as follows:

(a) GEOM_ANGLE

(b) GEOM_BOND

(c) GEOM_CONTACT

(d) GEOM_TORSION

For each of the geometry categories, there are two groups of extensions. One set covers maximum, minimum and average values of bonds, contact distances, angles and torsion angles. The other extends the symmetry-operation code used in geometry listings in the core CIF dictionary (see Section 3.2.4.3.2 ) to the higher-dimensional superspace form.

3.4.3.3.5. Symmetry information

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New data items in these categories are as follows:

(a) SPACE_GROUP

(b) SPACE_GROUP_SYMOP

At present, the msCIF dictionary extends the core CIF dictionary symmetry categories to describe superspace groups for one-dimensional modulated structures in four ways: as the superspace-group number in Janssen et al. (2004) ( _space_group_ssg_IT_number), as the International Tables superspace-group symbol (*_ssg_name_IT), as one of the notations from de Wolff et al. (1981) (*_ssg_name_WJJ, *_ssg_WJJ_code), or in some other formalism (*_ssg_name). At present, superspace-group names for higher dimensions can only be indicated using _space_group_ssg_name.

Symmetry operations in the superspace group are specified in the SPACE_GROUP_SYMOP category by an obvious extension to the method used in the core dictionary. These items must always be present in a CIF corresponding to a modulated or composite structure.

In this section, some of the capabilities of the dictionary will be demonstrated using simple examples. More detailed examples can be found at http://www.iucr.org/iucr-top/cif/ms and on the CD-ROM accompanying this volume.