International
Tables for
Crystallography
Volume G
Definition and exchange of crystallographic data
Edited by S. R. Hall and B. McMahon

International Tables for Crystallography (2006). Vol. G, ch. 3.4, pp. 131-140
https://doi.org/10.1107/97809553602060000736

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

G. Madariagaa*

aDepartamento de Física de la Materia Condensada, Facultad de Ciencia y Tecnología, Universidad del País Vasco, Apartado 644, 48080 Bilbao, Spain
Correspondence e-mail: gotzon.madariaga@ehu.es

The core CIF dictionary has been extended to provide a formalism for the description in software of certain types of aperiodic crystal structures. Aperiodic structures do not have lattice periodicity, but show long-range order; their diffraction patterns exhibit sharp Bragg peaks that have to be indexed using more than three Miller indices; and they can be classified in one of the three groups: incommensurately modulated structures, incommensurate composite structures or quasicrystals. Modulated structures are described by periodic distortions of an underlying (reference) crystal structure. Composite structures can be thought of as being built of two or more periodic subsystems whose lattices are mutually incommensurate. Both may be described using a CIF dictionary of terms (msCIF) that express the modulation or superposition of structural models according to the standard established by the IUCr Commission on Aperiodic Crystals. Quasicrystals not only lack three-dimensional lattice periodicity but also show noncrystallographic symmetry. Theoretical descriptions of quasicrystal structures are still relatively poorly developed, and are not described by the terms in the msCIF dictionary.

3.4.1. Introduction

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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[link] , 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[link]; de Wolff et al., 1981[link]). 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[link]) and Janssen et al. (2004[link]).

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[link]).

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[link]. 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[link]) 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[link]) 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[link]). Quasicrystals are not covered by the msCIF dictionary detailed in Chapter 4.3[link] .

3.4.2. Dictionary design considerations

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The CIF dictionary for modulated and composite structures (msCIF dictionary) is an extension of the core CIF dictionary (Chapter 4.1[link] ). 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[link] and 4.2[link] ).

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[link])], 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[link]). 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[link]). 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[link]; Pérez-Mato et al., 2003[link]).

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[link] and 4.7[link] ).

(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.

3.4.3. Arrangement of the dictionary

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The msCIF dictionary detailed in Chapter 4.3[link] 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[link] and is listed in full in Appendix 3.4.1[link]. The appendix also lists for each category the section of this chapter in which the category is described.

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Category groups defined in the msCIF dictionary

The groups are listed in the order in which they are described in this chapter.

SectionCategory groupSubject covered
(a) Experimental measurements
3.4.3.1.1[link] CELL The unit cell, especially cell subsystems for composite structures
3.4.3.1.2[link] DIFFRN Multi-dimensional diffraction pattern
3.4.3.1.3[link] EXPTL Details of the experiment specific to modulated and composite structures
     
(b) Analysis
3.4.3.2.1[link] REFINE Refinement procedures
3.4.3.2.2[link] REFLN Reflection measurements indexed in higher-dimensional space
     
(c) Atomicity, chemistry and structure
3.4.3.3.1[link] to 3.4.3.3.3[link] ATOM Atom sites in a modulated structure
3.4.3.3.4[link] GEOM Geometry of a structure in superspace
3.4.3.3.5[link] SYMMETRY Symmetry information
     
(d) File metadata
3.4.3.4[link] 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[link] . As in the other chapters in this part of the volume, the classification is under the headings Experimental measurements (Section 3.4.3.1[link]), Analysis (Section 3.4.3.2[link]), Atomicity, chemistry and structure (Section 3.4.3.3[link]) and File metadata (Section 3.4.3.4[link]). 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 ([\bullet]). The remaining data items in each category are listed alphabetically.

3.4.3.1. Experimental measurements

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The categories relevant to a structure determination experiment are in the CELL, DIFFRN and EXPTL groups. Most of the items extend the existing core CIF categories, but the categories describing cell subsystems are new.

3.4.3.1.1. Cell and modulation wave vectors

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The categories describing the unit cell (or cells for composite structures), the wave vectors of the modulations and, for composites, the cell subsystems are as follows:

CELL group
 CELL
 CELL_SUBSYSTEM
 CELL_SUBSYSTEMS
 CELL_WAVE_VECTOR
 CELL_WAVE_VECTORS

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

The data items in these categories are as follows:

(a) CELL [Scheme scheme1]

(b) CELL_SUBSYSTEM [Scheme scheme2] [\vdots]

(and _cell_subsystem_matrix_W_m_n for all combinations [1 \le m,n \le 11])

[\vdots] [Scheme scheme3]

(c) CELL_SUBSYSTEMS [Scheme scheme4]

(d) CELL_WAVE_VECTOR [Scheme scheme5]

(e) CELL_WAVE_VECTORS [Scheme scheme6]

As explained in Section 3.4.2[link], the msCIF dictionary arbitrarily allows an upper limit of 11 for the dimension of superspace for which data names are defined. _cell_modulation_dimension specifies the number of additional reciprocal vectors needed to index the whole diffraction pattern and has values d ranging from 1 to 8 to express the dimensionality (3 + d) of the superspace. _cell_reciprocal_basis_description is a text field allowing a free description of the higher-dimensional basis chosen.

For a composite structure, different cell subsystems may be specified. Each such subsystem is identified and characterized by the data items in the CELL_SUBSYSTEM category (see Section 3.4.4.1[link]). _cell_subsystems_number gives the number of such subsystems as an independent check of the completeness of the description.

Data items in the CELL_WAVE_VECTOR category specify the wave vectors of the modulation (see Section 3.4.4.1[link]). In accordance with the limits on dimensionality of the current version of the msCIF dictionary, no more than eight additional modulation wave vectors may be specified. The number used must agree with the value of _cell_modulation_dimension.

The data items in the CELL_WAVE_VECTORS category describe the experimental conditions during the determination of the independent modulation wave vectors.

3.4.3.1.2. Data collection

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The categories describing data collection are as follows:

DIFFRN group
 DIFFRN_REFLN
 DIFFRN_REFLNS
 DIFFRN_STANDARD_REFLN

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

New data items in these categories are as follows:

(a) DIFFRN_REFLN [Scheme scheme7]

(b) DIFFRN_REFLNS [Scheme scheme8]

(c) DIFFRN_STANDARD_REFLN [Scheme scheme9]

The data items in these categories are straightforward extensions of the core CIF dictionary definitions to the indexing of diffraction intensities by higher-dimensional components. The _diffrn_refln_index_m_* items are the additional Miller indices [m_i] indexing the modulation wave vectors [{\bf q}_i] when the diffraction wave vector is written as [{\bf H}=h {\bf a}^{*}+k {\bf b}^{*}+ l{\bf c}^{*}+ \textstyle\sum_i m_{i} {\bf q}_{i}].

The upper limit of 8 was chosen arbitrarily to limit the number of data items defined in the msCIF dictionary.

_diffrn_reflns_limit_index_m_* items provide independent checks on the range of values recorded for each of the additional Miller indices [m_i]. _diffrn_standard_refln_index_m_* items allow the higher-dimensional Miller indices of standard reflections to be recorded.

3.4.3.1.3. Experimental measurements on the crystal

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The categories describing measurements on the crystal or crystals used in the experiment are as follows:

EXPTL group
 EXPTL_CRYSTAL
 EXPTL_CRYSTAL_FACE

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

New data items in these categories are as follows:

(a) EXPTL_CRYSTAL [Scheme scheme10]

(b) EXPTL_CRYSTAL_FACE [Scheme scheme11]

_exptl_crystal_type_of_structure specifies the structure type as cryst (crystalline), mod (modulated) or comp (composite). These are the only three types of structure handled at present by the msCIF dictionary.

The extensions to the EXPTL_CRYSTAL_FACE category permit the indexing of crystal faces using the higher-dimensional Miller indices introduced for aperiodic structures.

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[link])
 REFINE
The reflections used in the refinement (§3.4.3.2.2[link])
 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 [Scheme scheme12]

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 [Scheme scheme13]

(b) REFLNS [Scheme scheme14]

As with the _diffrn_refln_* and _diffrn_reflns_* items (Section 3.4.3.1.2[link]), 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[link])
 ATOM_SITE
 ATOM_SITE_PHASON
Modulation functions as Fourier series (§3.4.3.3.2[link])
 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[link])
 ATOM_SITE_DISPLACE_SPECIAL_FUNC
 ATOM_SITE_OCC_SPECIAL_FUNC
Molecular or packing geometry (§3.4.3.3.4[link])
GEOM group
 GEOM_ANGLE
 GEOM_BOND
 GEOM_CONTACT
 GEOM_TORSION
Symmetry information (§3.4.3.3.5[link])
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 [Scheme scheme15]

(b) ATOM_SITE_PHASON [Scheme scheme16]

The bullet ([\bullet]) indicates a category key. The arrow ([\rightarrow]) 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[link] ). 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 [Scheme scheme17]

(b) ATOM_SITE_DISPLACE_FOURIER_PARAM [Scheme scheme18]

(c) ATOM_SITE_FOURIER_WAVE_VECTOR [Scheme scheme19]

(d) ATOM_SITE_OCC_FOURIER [Scheme scheme20]

(e) ATOM_SITE_OCC_FOURIER_PARAM [Scheme scheme21]

(f) ATOM_SITE_ROT_FOURIER [Scheme scheme22]

(g) ATOM_SITE_ROT_FOURIER_PARAM [Scheme scheme23]

(h) ATOM_SITE_U_FOURIER [Scheme scheme24]

(i) ATOM_SITE_U_FOURIER_PARAM [Scheme scheme25]

(j) ATOM_SITES_DISPLACE_FOURIER [Scheme scheme26]

(k) ATOM_SITES_MODULATION [Scheme scheme27]

(l) ATOM_SITES_ROT_FOURIER [Scheme scheme28]

The bullet ([\bullet]) indicates a category key. The arrow ([\rightarrow]) 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[link].) 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 [Scheme scheme29]

(b) ATOM_SITE_OCC_SPECIAL_FUNC [Scheme scheme30]

The bullet ([\bullet]) indicates a category key. The arrow ([\rightarrow]) 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[link]).

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 [Scheme scheme31]

(b) GEOM_BOND [Scheme scheme32]

(c) GEOM_CONTACT [Scheme scheme33]

(d) GEOM_TORSION [Scheme scheme34]

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[link] ) 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 [Scheme scheme35]

(b) SPACE_GROUP_SYMOP [Scheme scheme36]

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[link]) ( _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[link]) (*_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.

3.4.3.4. File metadata

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The categories modified in the msCIF dictionary to formalize the construction of a multi-block description of modulated or composite structures are as follows:

AUDIT group
 AUDIT
 AUDIT_LINK

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

Data items revised in these categories are as follows:

(a) AUDIT [Scheme scheme37]

(b) AUDIT_LINK [Scheme scheme38]

The core dictionary definitions of these items are revised in order to formalize the relationships between multiple data blocks representing reference and modulated structures. Guidance is provided in the msCIF dictionary on how to label data blocks in a way that makes their mutual relationships clear.

3.4.4. Use of the msCIF dictionary

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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.

3.4.4.1. Description of reciprocal space

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Modulated and composite structures need more than three reciprocal vectors in order to index the whole set of reflections with integer numbers. Hence a diffraction vector is written as[{\bf H}=h {\bf a}^{*}+k {\bf b}^{*}+l {\bf c}^{*}+ m_{1} {\bf q}_{1}+\ldots+m_{d} {\bf q}_{d},\eqno(3.4.4.1)]where the notation has been chosen according to the core CIF dictionary. In the case of a modulated structure, [{\bf a}^{*}], [{\bf b}^{*}] and [{\bf c}^{*}] are the reciprocal vectors of the reference structure (and therefore h, k and l index the main reflections). [{\bf q}_{1}, \ldots, {\bf q}_{d}] are the modulation wave vectors. They are three-dimensional vectors with some irrational component (if the modulated structure is incommensurate) in the lattice spanned by [{\bf a}^{*}], [{\bf b}^{*}] and [{\bf c}^{*}]. d is the dimension of the modulation. In the case of composite structures, the diffraction pattern can be indexed using 3 + d (arbitrarily selected) vectors [{\bf a}^{*}_{k}] [(k=1,\ldots,3+d)]. [{\bf a}^{*}_{1}] [(\equiv {\bf a}^{*})], [{\bf a}^{*}_{2}] [(\equiv {\bf b}^{*})] and [{\bf a}^{*}_{3}] [(\equiv {\bf c}^{*})] normally span the reciprocal lattice of the main reflections of one of the substructures (notice that this is only one particular, but highly intuitive, choice). The remaining d vectors with [k = 4, \ldots, d] are the wave vectors of the modulation [[{\bf q}_{1}, \ldots, {\bf q}_{d}] in equation (3.4.4.1[link])].

In a composite structure, the (3 + d)-dimensional reciprocal basis of the subsystem [\nu] is determined by a (3 + d) × (3 + d) matrix [W^{\nu}] [see van Smaalen (1995[link]) and references therein]:[{{\bf a}}^{* \nu}_{i} = \textstyle\sum\limits_{k=1}^{3+d} W^{\nu}_{ik} {{\bf a}}^{*}_{k}, i=1,\ldots,3+d, \eqno(3.4.4.2)]where the subscripts i = 1, 2 and 3 label the reciprocal vectors [{\bf a}^{* \nu}], [{\bf b}^{* \nu}] and [{\bf c}^{* \nu}], and [i = 4, \ldots, d ] label the wave vectors of the modulation expressed as linear combinations of [{\bf a}^{* \nu}], [{\bf b}^{* \nu}] and [{\bf c}^{* \nu}].

The simplest case corresponds to a one-dimensional (d = 1) modulated structure. Consider for example the incommensurate phase of K2SeO4. The wave vector of the modulation can be chosen to be [{\bf q}_{1}=\alpha {\bf a}^{*}]. Relevant information about the diffraction pattern of this compound is expressed using both the core CIF and msCIF dictionaries as shown in Example 3.4.4.1[link].

Example 3.4.4.1. msCIF description of the diffraction pattern of a one-dimensional modulated structure.

[Scheme scheme39]

A more complicated example is the composite structure (LaS)1.14NbS2. The two mutually incommensurate subsystems (along the a axis) are (van Smaalen, 1991[link]) NbS2 ([\nu =1]) and LaS ([\nu =2]). The reciprocal basis can be chosen to be [{\bf a}^{*}_{1}={\bf a}^{*1}], [{\bf a}^{*}_{2}={\bf b}^{*1}], [{\bf a}^{*}_{3}={\bf c}^{*1}] and [{\bf a}^{*}_{4}={\bf a}^{*2}]. For this particular choice, the two W matrices [see equation (3.4.4.2[link])] are[W^{1}=\pmatrix{1 & 0 & 0 & 0 \cr 0 & 1 & 0 & 0 \cr 0 & 0 & 1 & 0 \cr 0 & 0 & 0 & 1 }, \quad W^{2}=\pmatrix{0 & 0 & 0 & 1 \cr 0 & 1 & 0 & 0 \cr 0 & 0 & 1 & 0 \cr 1 & 0 & 0 & 0 }.]

This information is transcribed to CIF format as shown in Example 3.4.4.2[link]. (Note that the default values for the wave vector components and the elements of W are 0.)

Example 3.4.4.2. Representation of two mutually incommensurate subsystems.

[Scheme scheme40]

3.4.4.2. Description of symmetry

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The symmetry of a modulated or composite structure is described by a superspace group which leaves the (3 + d)-dimensional embedding of the structure invariant. Superspace is built of two orthogonal subspaces and both of them are kept invariant separately by the superspace symmetry operations. (In reciprocal space this means that, for these structures, main reflections and satellite reflections are never transformed into one another by superspace symmetry operations.) Consequently, superspace groups are not general (3 + d)-dimensional space groups. The standard notation for superspace groups only covers the one-dimensional superspace groups, which are listed in Janssen et al. (2004[link]). As a consequence, msCIFs must include a list of all the symmetry operations in an (x, y, z) format (using as symbols [x_{1}\ldots x_{3+d}]) similar to that used in the core CIF dictionary. Superspace-group names for one-dimensional structures can be expressed either according to Janssen et al. (2004[link]) or according to the original notation of de Wolff et al. (1981[link]). Alternative names or higher-dimensional superspace groups can also be included, but not parsed.

In the particular case of K2SeO4, the superspace group is [P^{Pnma}_{\;\overline{1}ss}] (de Wolff et al., 1981[link]) or [Pnma(\alpha00)0ss] (Janssen et al., 2004[link]). This information would appear in a CIF as shown in Example 3.4.4.3[link].

Example 3.4.4.3. Symmetry description of a superspace group.

[Scheme scheme41]

3.4.4.3. Description of the structure

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A modulated structure is described by a reference periodic structure and the atomic modulation functions. Such functions are periodic and are normally expanded as Fourier series. The modulated parameters may apply to the atom positions (displacive modulation), the site occupancies (occupational modulation) and/or the temperature factors. In composite structures, each substructure is referred to the crystallographic basis defined by the W matrices [see equation (3.4.4.2[link])]. The simplest case corresponds to a one-dimensional displacive modulated structure. In this case, the atomic modulation functions are given by[u^{\mu}_{\alpha}=\textstyle\sum\limits_{n=1}^{\infty} U^{\mu}_{n \alpha} \cos ( 2 \pi n {{\bf q} \cdot {\bf r}} + \varphi ^{\mu}_{n \alpha} ),\eqno(3.4.4.3)]where [( U^{\mu}_{n \alpha},\varphi ^{\mu}_{n \alpha})] is the complex amplitude of each Fourier term; [\mu] labels the atoms; [\alpha=x, y, z]; r is the average atom position; and [n{\bf q}] represents the successive harmonics of the modulation. At present, displacive modulations along axes other than a, b and c can be calculated with the restriction stated in Section 3.4.2[link].

Atomic displacements can also be expressed as rigid rotations and translations. The incommensurate phase of K2SeO4 is one-dimensional displacive and sinusoidal (at least over a wide range of temperature), i.e. n = 1 in equation (3.4.4.3[link]). The tetrahedral SeO4 groups behave as rigid bodies. Symmetry considerations restrict the possible translations to occur along c where the only allowed rigid rotations are around the a and b axes. The incommensurate structure is then expressed as shown in Example 3.4.4.4[link].

Example 3.4.4.4. Atomic displacements as translations and rigid rotations.

[Scheme scheme42]

Alternatively, equation (3.4.4.3[link]) can be expressed as a Fourier series with real amplitudes. This form is also covered by the msCIF dictionary. Note that there is a global phase, which is irrelevant in the incommensurate case but fixes the space group if the modulated structure is commensurate. Global phases are also defined in the dictionary.

3.4.4.4. Block pointers

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CIFs for modulated and (in particular) composite structures often need several linked data blocks. For example, the average structure of a modulated phase refined using only the main reflections could be reported in one data block and then the results of the refinement of the complete structure using all the reflections could be added as a separate data block at a later stage. Similarly, each substructure in a composite structure can be fully described in a separate data block. It is therefore essential to keep all these blocks together, since all of them together describe the composite structure. Block pointers are described in the core CIF dictionary, but they need to be used with particular care in msCIFs. Therefore, the msCIF dictionary gives additional guidelines for naming the data blocks, as the aim (as in pdCIF) is to define unique names. Example 3.4.4.5[link] shows how the structure of (LaS)1.14NbS2 can be represented using several data blocks. In this example, the block named 1997–07-24|LaSNbS2|G.M.| is linked to those in the loop below.

Example 3.4.4.5. Relationship between data blocks in an msCIF.

[Scheme scheme43]

3.4.4.5. Other information

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The msCIF and core CIF dictionaries also include items for R factors for different reflection classes (for example, for the main reflections and for the satellite reflections; see Section 3.2.3[link] ) and for describing the variation of interatomic distances due to the modulation. msCIF can be used to create a complete record of the determination of a modulated or composite structure suitable for submitting as an article to an academic journal or for archiving in a database. The obvious next step in the development of msCIF is to adapt or write software that can filter and manage msCIF data and that can generate msCIFs automatically.

Appendix A3.4.1

A3.4.1. Category structure of the msCIF dictionary

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Table A3.4.1.1[link] shows all categories that are represented in the msCIF dictionary.

Table A3.4.1.1| top | pdf |
Categories in the modulated structures CIF dictionary

Numbers in parentheses refer to the section of this chapter in which each category is described in detail.

ATOM group (§3.4.3.3[link]) AUDIT group (§3.4.3.4[link])
 ATOM_SITE (§3.4.3.3.1[link](a)) ¶  AUDIT (§3.4.3.4[link](a)) [\ddag]
 ATOM_SITE_DISPLACE_FOURIER  AUDIT_LINK (§3.4.3.4[link](b)) [\ddag]
  (§3.4.3.3.2[link](a)) CELL group (§3.4.3.1.1[link])
 ATOM_SITE_DISPLACE_FOURIER_PARAM  CELL (§3.4.3.1.1[link](a)) ¶
  (§3.4.3.3.2[link](b))  CELL_SUBSYSTEM (§3.4.3.1.1[link](b))
 ATOM_SITE_DISPLACE_SPECIAL_FUNC  CELL_SUBSYSTEMS (§3.4.3.1.1[link](c))
  (§3.4.3.3.3[link](a))  CELL_WAVE_VECTOR (§3.4.3.1.1[link](d))
 ATOM_SITE_FOURIER_WAVE_VECTOR  CELL_WAVE_VECTORS (§3.4.3.1.1[link](e))
  (§3.4.3.3.2[link](c)) DIFFRN group (§3.4.3.1.2[link])
 ATOM_SITE_OCC_FOURIER  DIFFRN_REFLN (§3.4.3.1.2[link](a)) ¶
  (§3.4.3.3.2[link](d))  DIFFRN_REFLNS (§3.4.3.1.2[link](b)) ¶
 ATOM_SITE_OCC_FOURIER_PARAM  DIFFRN_STANDARD_REFLN
  (§3.4.3.3.2[link](e))   (§3.4.3.1.2[link](c)) ¶
 ATOM_SITE_OCC_SPECIAL_FUNC EXPTL group (§3.4.3.1.3[link])
  (§3.4.3.3.3[link](b))  EXPTL_CRYSTAL (§3.4.3.1.3[link](a)) ¶
 ATOM_SITE_PHASON (§3.4.3.3.1[link](b))  EXPTL_CRYSTAL_FACE
 ATOM_SITE_ROT_FOURIER   (§3.4.3.1.3[link](b)) ¶
  (§3.4.3.3.2[link](f)) GEOM group (§3.4.3.3.4[link])
 ATOM_SITE_ROT_FOURIER_PARAM  GEOM_ANGLE (§3.4.3.3.4[link](a)) ¶
  (§3.4.3.3.2[link](g))  GEOM_BOND (§3.4.3.3.4[link](b)) ¶
 ATOM_SITE_U_FOURIER (§3.4.3.3.2[link](h))  GEOM_CONTACT (§3.4.3.3.4[link](c)) ¶
 ATOM_SITE_U_FOURIER_PARAM  GEOM_TORSION (§3.4.3.3.4[link](d)) ¶
  (§3.4.3.3.2[link](i)) REFINE group (§3.4.3.2.1[link])
 ATOM_SITES_DISPLACE_FOURIER  REFINE (§3.4.3.2.1[link]) ¶
  (§3.4.3.3.2[link](j)) REFLN group (§3.4.3.2.2[link])
 ATOM_SITES_MODULATION  REFLN (§3.4.3.2.2[link](a)) ¶
  (§3.4.3.3.2[link](k))  REFLNS (§3.4.3.2.2[link](b)) ¶
 ATOM_SITES_ROT_FOURIER SYMMETRY group (§3.4.3.3.5[link])
  (§3.4.3.3.2[link](l))  SPACE_GROUP (§3.4.3.3.5[link](a)) ¶
   SPACE_GROUP_SYMOP
    (§3.4.3.3.5[link](b)) ¶

¶ Categories already defined in the core CIF dictionary but containing new items.
[\ddag] A stricter definition than that given in the core CIF dictionary.

Acknowledgements

The msCIF dictionary exists thanks to the continuous help and support received from I. David Brown and Brian McMahon. The suggestions and criticisms of Sander van Smaalen, Howard Flack, Juan Manuel Pérez-Mato and, especially, Vaclav Petříček (who has implemented a CIF input/output routine in the program JANA2000) were important.

References

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Elcoro, L., Pérez-Mato, J. M., Darriet, J. & El Abed, A. (2003). Superspace description of trigonal and orthorhombic A1+xA′xB1−xO3 compounds as modulated layered structures; application to the refinement of trigonal Sr6Rh5O15. Acta Cryst. B59, 217–233.
Haibach, T., Cervellino, A., Estermann, M. A. & Steurer, W. (2000). X-ray structure determination of quasicrystals – limits and potentiality. Z. Kristallogr. 215, 569–583.
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Smaalen, S. van (1995). Incommensurate crystal structures. Crystallogr. Rev. 4, 79–202.
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Wolff, P. M. de, Janssen, T. & Janner, A. (1981). The superspace groups for incommensurate crystal structures with a one-dimensional modulation. Acta Cryst. A37, 625–636.








































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