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A quantitative comparison of similar crystal structures is often convenient to cross-check different experimental and/or theoretical structural models of the same phase coming from different sources. It is also important for the identification of different phases with the same symmetry, and it is fundamental for the still open problem of the classification of structures into structure types. In most cases, even if the setting of its space group is fixed, there is more than one equivalent description for a given structure. The existence of various equivalent structure descriptions makes the comparison of different structural models a non-trivial task in general. To deal with it, the programCOMPSTRUhas been developed, available at the Bilbao Crystallographic Server (http://www.cryst.ehu.es). The program measures the similarity between two structures having the same space-group symmetry (or space groups that form an enantiomorphic pair) with the same or different compositions, and under the condition that the sequence of the occupied Wyckoff positions is the same in both structures (isopointal structures). The efficiency and utility of the program are demonstrated by a number of illustrative examples. It is also shown how the program can be used to outline different structure types within a set of isopointal structures.

We report the synthesis, single crystal X-ray diffraction, and thermal, dielectric, Raman and infrared studies of a novel heterometallic formate [C2H5NH3][Na0.5Fe0.5(HCOO)3] (EtANaFe). The thermal studies show that EtANaFe undergoes a second-order phase transition at about 360 K. X-ray diffraction data revealed that the high-temperature structure is monoclinic, space group P2(1)/n, with dynamically disordered ethylammonium (EtA(+)) cations. EtANaFe possesses a polar low-temperature structure with the space group Pn and, in principle, is ferroelectric below 360 K. Dielectric data show that the reciprocal of the real part of dielectric permittivity above and below the phase transition temperature follows the Curie-Weiss, as expected for a ferroelectric phase transition. Based on theoretical calculations, we estimated the polarization as (0.2, 0, 0.8) μC cm(-2), i.e., lying within the ac plane. The obtained data also indicate that the driving force of the phase transition is ordering of EtA(+) cations. However, this ordering is accompanied by significant distortion of the metal formate framework.

Two new programs, MTENSOR and TENSOR, hosted on the open-access website known as the Bilbao Crystallographic Server, are presented. The programs provide automatically the symmetry-adapted form of tensor properties for any magnetic or non-magnetic point group or space group. The tensor is chosen from a list of 144 known tensor properties gathered from the scientific literature or, alternatively, the user can also build a tensor that possesses an arbitrary intrinsic symmetry. Four different tensor types are considered: equilibrium, transport, optical and nonlinear optical susceptibility tensors. For magnetically ordered structures, special attention is devoted to a detailed discussion of the transformation rules of the tensors under the time-reversal operation 1′. It is emphasized that for non-equilibrium properties it is the Onsager theorem, and not the constitutive relationships, that indicates how these tensors transform under 1′. In this way it is not necessary to restrict the validity of Neumann's principle. New Jahn symbols describing the intrinsic symmetry of the tensors are introduced for several transport and optical properties. In the case of some nonlinear optical susceptibilities of practical interest, an intuitive method is proposed based on simple diagrams, which allows easy deduction of the action of 1′ on the susceptibilities. This topic has not received sufficient attention in the literature and, in fact, it is usual to find published results where the symmetry restrictions for such tensors are incomplete.

In this article we comment on the results published by Thompson et al. (, J. Solid State Chem. 219 (2014) 173–178) on the crystal structure of SrFeO 2 F, who claim the compound to crystallize in the cubic space group Pm -3 m . We give a more detailed explanation of the determination of our previously reported structural model with Imma symmetry (Clemens et al., J. Solid State Chem. 206 (2013) 158–169), with addition of variable temperature XRD measurements with high counting time to provide unambiguous evidence for the Imma model being correct for our sample.

The numerous structures that have been reported for the different phases of the lead zirconate titanate system, PbZr1−x Ti x O3 (PZT), are analysed by means of a systematic symmetry-mode analysis. The distortion corresponding to the order parameter has been separated out and expressed in all phases in a comparable form. The fact that the physical origin of the PZT phases is an unstable threefold degenerate polar mode, plus in some cases an unstable octahedral tilting mode, produces structural correlations between the different phases. These correlations had remained unnoticed until now but are directly observable in a mode parameterization. They can be used both to characterize the evolution of the order parameters through the phase diagram and as a stringent test of the reported structural models. It is further shown that the activity of a single polar mode yields a specific feature in the mode decomposition of the monoclinic phases. This single-mode signature can be observed in the majority of the monoclinic structures proposed, making the others questionable. In fact, this internal constraint is satisfied by PZT to such a high degree that it drastically reduces the number of effective structural degrees of freedom. It is conjectured that this type of structural constraint beyond space-group symmetry can be a rather general property of low-symmetry distorted structures. As shown here, its existence can be detected and assessed by a symmetry-mode analysis, if considered in relation to the single underlying multidimensional order parameter.

NaFePO4 is known as a promising intercalation cathode material for sodium-ion batteries. During the electrochemical reaction, an intermediate phase forms with a composition close to Na≈2/3FePO4, whose crystal structure is defined with a supercell that results from both Na/vacancy and charge ordering. In this work, we present a detailed study of this superstructure through synchrotron powder X-ray diffraction and electron microscopy studies.

The pavonites is a homologous series of sulfosalts with galena-like modules of variable size. A survey of their unit cells reveals that they are severely constrained by the metrics of an underlying common average sublattice. The unit cell of any compound of the series accommodates with high precision an integer number of approximately equal subcells. This explains a peculiar non-uniform variation of the unit cell parameters within the series and evidences that the interface between the galena-like modules, despite having a very different topology, approximately maintains the subperiodicity of the modules, and must therefore be subject to strong steric restrictions. It also implies that cations and anions occupy the nodes of the observed common underlying average sublattice according to a striking alternate cation/anion occupational modulation. This is the starting point for a description of these materials as modulated structures, which can make a proficient use of the approximate atomic positional non-crystallographic correlations caused by their modular character. Under this approach only four parameters suffice to define a realistic approximate model of any member of the series. A full structural characterization of any of the compounds only requires the determination of additional small/smooth modulations. As an example, the case of the P 7 pavonite Ag3Bi6.2Pb0.8S12, is analyzed.

A symmetry based framework is used to describe the complex phase diagrams observed in the multiferroic delafossite compounds. A free energy Landau functional is derived from the analysis of the transformation properties of the most general incommensurate magnetic spin order parameter. A principle of maximal symmetry is invoked and the stability of each of the different higher symmetry phases considered. The competition between different potential ground states is analysed within the scope of a simplified model, which emphasizes the role of the symmetry allowed phase dependent biquadratic couplings. The cross-over between the different competing states is also discussed. The results show that the diverse set of phase diagrams that are experimentally observed in this class of triangular lattice antiferromagnets and, in particular, the stabilization of magnetically induced ferroelectric states, can be well interpreted and described within this integrated phenomenological approximation., This work has been supported by FEDER (through the COMPETE Program and project MAT2012-34740), by the Portuguese Foundation for Science and Technology (FCT) in the framework of the Strategic Project PESTC/ FIS/UI607/2011, by the Spanish Ministry of Economy and Competitiveness, and by the Government of the Basque Country (project IT779-13).

International audience; The investigation of the phase transition sequence in SrBi2Ta2O9 (SBT) and SrBi2Nb2O9 (SBN) is reported using single-crystal X-ray diffraction. By monitoring specific reflections as a function of temperature, sensitive either to the superstructure formation or to polar displacements, it was possible to check the existence or not of an intermediate phase. This latter was confirmed in SBT, but within experimental accuracy could not be detected in SBN.

The (3 + 1)-dimensional superspace approach is applied to describe and refine a series of sheared compounds related to layered high T c superconducting oxides. Two commensurate members (m = 4, 5) of the 2212 stair-like [Bi2Sr3Fe2O9] m [Bi4Sr6Fe2O16] family of compounds, previously studied using single-crystal diffraction data, are analyzed. A common average unit cell has been identified and a composition-dependent modulation wavevector is proposed. The model is built using only three independent atomic domains, one for the metal atoms and two for the O atoms. The three Sr, Bi and Fe species are described using close-connected crenel-like functions forming a continuous atomic domain along the internal space. The two oxygen domains are represented by crenel functions and the displacive modulation functions are built up by Legendre polynomials recently implemented in the program JANA2006. Surprisingly, the results of the refinements show a striking similarity of the displacive modulations for the two compounds analyzed, indicating that a unique model can be used to describe the correlations between the atomic displacements of the 2212 stair-like series. This final model is then applied to predict the structure of new members of the family.

Crystal structures of low temperature phases III, IV and V of Rubidium Zinc Bromide have been determined by single crystal X-ray diffraction at 90 K, 85 K, and 30 K, respectively. The analysis of the structures in terms of symmetry-adapted modes of the high temperature prototype phase permits the identification of the modes responsible for each phase and to assess its thermal evolution. The primary mode responsible for the incommensurate and lock-in phases remains dominant throughout the whole transition sequence, maintaining its structure and slightly increasing its amplitude as temperature decreases. The two monoclinic phases IV and V are presumably due to the instability of a second mode associated with a single active irreducible representation. However, the presence with similar weights of several symmetry-adapted distortions of different symmetry indicates a strong hybridization in phase III of the softer normal modes. A symmetry-mode description of the structures of phase IV and V fully consistent with a simple Landau model, can then only be obtained if phase III is taken as effective parent phase. Besides the insight on the microscopic origin of the different phases, the symmetry modes were used as crystallographic coordinates in the refinement of the structure of phase V. The result shows the advantages of using these non-conventional collective structural parameters instead of individual atomic coordinates. The refinable parameters can be reduced to the amplitudes of those symmetry modes with significant weight in the distortions.

New ferroelectrics can be predicted by considering the existence of pseudosymmetry with respect to a higher symmetry structure using the so-called atomic displacement method and investigating the minimal super-groups of the given structure's space group. This analysis can be performed with the new version of the computer program PSEUDO, located at the Bilbao Crystallographic Server. After defining the procedures for the detection and quantification of pseudosymmetry, we present the new program, illustrating its use with worked cases of polar structures from the literature which are either known or reported as possible ferroelectrics.

Some Aurivillius compounds exhibit features beyond the single soft-mode picture valid for other ferroelectrics. In SrBi2Ta2O9 (SBT) two primary distortions are needed to reach the ferroelectric phase, causing the presence of an intermediate phase. However, no intermediate phase has been detected in SrBi2Nb2O9 (SBN), and an avalanche transition seems to happen, with the two order parameters condensing simultaneously. According to ab-initio calculations, the usually assumed triggering mechanism through biquadratic coupling of the two order parameters can be discarded, and we have looked for other possible causes of these avalanche transitions. Using a simple microscopic Hamiltonian (an extended model), mean-field techniques (independent-site approximation) and Monte Carlo simulations, we show that a strong trilinear coupling between the two order parameters and an additional secondary mode could explain the existence of a single first-order transition with the simultaneous condensation of two order parameters.

The description of displacive distorted structures in terms of symmetry-adapted modes is reviewed. A specific parameterization of the symmetry-mode decomposition of these pseudosymmetric structures defined on the setting of the experimental space group is proposed. This approach closely follows crystallographic conventions and permits a straightforward transformation between symmetry-mode and conventional descriptions of the structures. Multiple examples are presented showing the insight provided by the symmetry-mode approach. The methodology is shown at work, illustrating its various possibilities for improving the characterization of distorted structures, for example: detection of hidden structural correlations, identification of fundamental and marginal degrees of freedom, reduction of the effective number of atomic positional parameters, quantitative comparison of structures with the same or different space group, detection of false refinement minima, systematic characterization of thermal behavior, rationalization of phase diagrams and various symmetries in families of compounds etc. The close relation of the symmetry-mode description with the superspace formalism applied to commensurate superstructures is also discussed. Finally, the application of this methodology in the field of ab initio or first-principles calculations is outlined. At present, there are several freely available user-friendly computer tools for performing automatic symmetry-mode analyses. The use of these programs does not require a deep knowledge of group theory and can be applied either a posteriori to analyze a given distorted structure or a priori to parameterize the structure to be determined. It is hoped that this article will encourage the use of these tools. All the examples presented here have been worked out using the program AMPLIMODES [Orobengoa et al. (2009). J. Appl. Cryst. 42, 820-833].

The structure of pyrrhotite (Fe1 − x S with 0.05 ≤ x ≤ 0.125) has been reinvestigated in the framework of the superspace formalism. A common model with a centrosymmetric superspace group is proposed for the whole family. The atomic domains in the internal space representing the Fe atoms are parametrized as crenel functions that fulfil the closeness condition. The proposed model explains the x-dependent space groups observed and the basic features of the structures reported up to now. Our model yields for any x value a well defined ordered distribution of Fe vacancies in contrast to some of the structural models proposed in the literature. A new (3 + 1)-dimensional refinement of Fe0.91S using the deposited dataset [Yamamoto & Nakazawa (1982). Acta Cryst. A38, 79–86] has been performed as a benchmark of the model. The consistency of the proposed superspace symmetry and its validity for other compositions has been further checked by means of ab initio calculations of both atomic forces and equilibrium atomic positions in non-relaxed and relaxed structures, respectively.

The Bilbao Crystallographic Server is a web site with crystallographic databases and programs available on-line at www.cryst.ehu.es. It has been operating for about six years and new applications are being added regularly. The programs available on the server do not need a local installation and can be used free of charge. The only requirement is an Internet connection and a web browser. The server is built on a core of databases, and contains different shells. The innermost one is formed by simple retrieval tools which serve as an interface to the databases and permit to obtain the stored symmetry information for space groups and layer groups. The k-vector database includes the Brillouin zones and the wave-vector types for all space groups. As a part of the server one can find also the database of incommensurate structures. The second shell contains applications which are essential for prob lems involving group-subgroup relations between space groups (e.g. subgroups and supergroups of space groups, splittings of Wyckoff positions), while the third shell contains more sophisticated programs for the computation of space-group representations and their correlations for group-subgroup related space groups. There are also programs for calculations focused on specific problems of solid-state physics. The aim of the article is to report on the current state of the server and to provide a brief description of the accessible databases and crystallographic computing programs. The use of the programs is demonstrated by illustrative examples.

Structural pseudosymmetry is a fundamental property to be considered when trying to predict structural phase transitions at high temperatures. In the present work, two methods for detecting and characterizing pseudosymmetry, the here-called atomic displacements and electron density methods are briefly introduced, analyzed and compared. Conceptually both approaches are similar: the pseudosymmetry check is carried out with respect to a supergroup of the crystal space group. However, due to the different way of estimating the deviations between the initial and the possible high-symmetry structure, the two approaches study distinct and, to a certain extent, complementary aspects of the pseudosymmetry problem. This conclusion is also confirmed by the lack of functional relationship between the corresponding quantitative parameters used to estimate the structural pseudosymmetry. Several examples illustrate the main features of the two methods. The temperature dependences of the two parameters and their empirical relations with the transition temperature in ferroelectric transitions are discussed. Algorithmic procedures based on the two approaches have been implemented in two computer programs. They form part of the Bilbao Crystallographic Server as free web tools.

International audience; A solid solution (1 - x)LiCdVO4 - xLi1/3Cd 1/3□1/3CdVO4 has been observed for 0 ≤ x ≤ 1. From the cell parameters evolution, two domains of the solid solution have been distinguished with a discontinuity at x ≈ 0.6. The first one for 0 ≤ x ≤ 0.6 corresponds to the LiCdVO4 substructure of the Na2CrO4-type. For 0.75 ≤ x ≤ 1 satellite peaks are observed which correspond to a modulated structure. The first end-member LiCdVO4 (x = 0) crystallizes in Cmcm space group with a = 5.911, b = 8.975, and c = 6.513 Å. The structure consists of infinite edge-sharing chains of [CdO6] octahedra linked together by [VO4] tetrahedra and [LiO4] tetrahedra. According to the general formula Li1-2x/3Cdx/3□x/3CdVO4 of the solid solution, when x increases cadmium replaces lithium in the [LiO 4] tetrahedra leading to [(Li/Cd/□)O4] disordered tetrahedra. All the phases in the domain 0.7 ≤ x ≤ 1 are one-dimensionally modulated with wavevector q = γc* and γ values of 0.75, 0.733, and 0.722 for x = 3/4, 6/7, and 9/10, respectively. The superspace group is Xmcm(00γ) where X stands for (1/2, 1/2, 0, 1/2) centering. The main structural result is a strong occupation modulation of lithium and cadmium in the tetrahedral site.

The structure of Tl2SeO4 at 293, 100 and 30 K has been determined. Space group at 293 and 100 K is Pmcn, Z=4, lattice parameters at 293 K are a=6.0838(12)(100 K : 6.0336(19)), b=10.965(3) ( 100 K : 10.903(5)) and c=7.9385(15) A (100 K : 7.921(2)). At 30 K space group is P212121, Z=4, with lattice parameters a=6.2333(17), b=10.533(3) and c=7.828(2). Measurement was carried out with a Stoe IPDS ( 293 K ) and a Nonius CCD using an Oxford He-cryostat for cooling (100 and 30 K ). Final agreement factors at 293 K after refinement with the program SHELXL97 are R(F)=0.061 ( 100 K : 0.097), w R(F ∗∗ 2)=0.051 ( 100 K : 0.156) for 566(100 K : 678) unique reflections and R(F)=0.039, w R(F ∗∗ 2)=0.097 for 1162 unique reflections at 30 K . The high-temperature phase is isotypical to the β-K2SO4 structure, characterized by isolated selenate tetrahedra and two different cation sites with coordination number 9 and 11. In the low-temperature phase, the tetrahedra are rotated and the coordination numbers of the Tl+ ions are reduced to 8 and 10, respectively. Structural changes are similar to the ones observed in high- and low-temperature phases of Tl2MoO4. A comparison to other Tl2TX4 compounds shows that the structural instabilities are closely connected to shortest Tl–O distances.

Two different pseudosymmetry-search methods are compared and applied in the search for new ferroelectrics among the binary, ternary and quaternary compounds of Pna21 symmetry. Among the 322 compounds with this symmetry, 18 candidates for ferroelectrics are detected.

The stability of SrBi2Ta2O9 has been studied by means of ab-initio calculations. The main instability of the tetragonal configuration concerns a non-polar mode at the Brillouin zone border. It can be related through Brillouin zone folding to the well-known soft rigid-unit mode R25 in perovskites. This implies a complex scenario for the whole Aurivillius family with two competing order parameters, which can result in a sequence of two continuous phase transitions or a single discontinuous one. A third hard-mode is shown to play an essential role in the stabilization of the ferroelectric phase.

The recently proposed superspace description of the Aurivillius compounds in their ferroelectric phase is here extended to their high-temperature phases. The probable existence of an intermediate phase between the tetragonal high-symmetry and the ferroelectric phases is stressed. The relevant superspace groups for both the tetragonal and the intermediate phases are identified. Consequently, the 3D space groups for the intermediate phases as a function of composition are predicted.

Following the general model proposed for the series of compounds Sr n (Nb,Ti) n O3n + 2 , the recently determined structure of the related compound NaCa 4 Nb 5 O 17 has been re-analyzed in superspace and refined using a set of crenel occupational functions. The atomic positions in real space can be described using eight independent atomic domains with smooth modulations under the superpace monoclinic symmetry X112 1 /b(α β 0) (X standing for the centring translations generated from (3/4 1/2 0 1/4)), and with modulation wave vector q = 1/12 b* (α = 0, β = 1/12). A refinement performed with the JANA2000 program using the same experimental data as the standard three-dimensional refinement gave indeed equivalent results. The superspace approach, however, reduced the number of structural variables.

A general structural model of B -site deficient hexagonal layer perovskites AB 1− x X 3 having no tilting of anion octahedra in consecutive layers is presented. They are described as modulated layered structures using a (3+1)-dimensional superspace approach. The presence or not of a given atom along the stacking direction is described in superspace through step-like (crenel) occupational modulation functions. Being essentially composition independent, the model is applicable to any compounds related to this structural type. The component γ of the primary modulation vector q =γ c * is connected to the chemical composition by the relation γ =(1+ x )/3. The model is confirmed and the efficiency of the superspace approach is demonstrated by the structural analysis of some of the hexagonal perovskite found in the system Ba 5 Nb 4 O 15 –BaTiO 3 , where long-period intergrowth compounds have been recently identified.

Bilbao Crystallographic Server is a web site with crystallographic programs and databases available on-line. The programs give access to general information related to space groups (generators, general positions, Wyckoff positions, irreducible representations), group-subgroup or group-supergroup pairs of space groups, and/or results on specific crystal structures. The utility of the programs is illustrated by treating phase-transition problems related to structural pseudosymmetry.

We report the structure determination on a single crystal of Ba 12 Co 11 O 33 , which belongs to the family of 1-dimensional structures related to the 2H hexagonal perovskite. The structure can be derived from the hcp stacking of [Ba 8 Co 2 O 18 ] and [Ba 8 O 24 ] layers. Assuming a similitude in the building principle with that present in the homologue rhombohedral series based on [A 3 A'O 6 ] and [A 3 O 9 ] layers, a generic model within the superspace formalism of the whole family is proposed and used successfully for Ba 12 Co 11 O 33 . Two different strategies are presented. The first one considers the structure as a commensurate modulated composite containing two subsystems [CoO 3 ] and [Ba]. With the [CoO 3 ] subsystem chosen as the reference, the superspace group is Fddd(00γ c )0s0 with a = 11.4129(2) A, b = 19.7677(2) A, c 1 = 2.4722(1) A, and q = (6/11)c 1 * (Z = 8 for the formula Ba 12/11 CoO 3 ). The final global R value is 4.46% for 4625 independent reflections (at a I/σ(I) > 3 level) and only 123 refineable parameters. The second model follows the natural description of the structure as formed by the uniform hcp stacking of the two types of layers [Ba 8 Co 2 O 18 ] and [Ba 8 O 24 ]. The structure is formulated as BaCo 11/12 O 33/12 (Z = 16) and considered to be a commensurate modulated phase with crenel occupational modulations. The superspace group can be denoted as Xdcd(00γ L )qq0 with a = 11.4129(2) A, b = 19.7677(2) A, c = 4.5324(1) A, and q = (1/6)c*. The final global R value is 4.36% for the same number of independent reflections and 151 refineable parameters. Both approaches yield similar results, demonstrating for the first time the feasibility of refining these structures as layerlike modulated structures, instead of composites. The chains consist of polyhedra sharing faces formed by 10 consecutive CoO 6 octahedra followed by one trigonal prism. Previous models based on simulated and high-resolution electron microscopy images are refuted.

A generalized structural model of Aurivillius-type compounds is presented using a 4D superspace group analysis where Aurivillius structures are considered as cation-deficient perovskites with the general formula AB 1− x O 3 . Being essentially composition independent, the model is valid for any Aurivillius-type compounds where x is the only composition-dependent parameter. The atomic domains representing the atoms in superspace are described by means of crenel functions. For any composition, the conventional space groups can be easily derived from a unique superspace group. This work is supported by a TEM investigation where the continuously variable character of the diffraction diagram indicates that the various stacking sequences can be interpreted in terms of a structural modulation over a common average structure.

Three-axes elastic neutron scattering measurements demonstrate that the five-fold modulated phase (phase 1/5) of BCCD exhibits under electric field a phase transition without change of superlattice periodicity. Through the monitoring of high-order satellite diffraction peaks as a function of electric field and temperature, the competition between this phase and neighboring polar phases with other periods has been characterized. At a threshold electric field of about 20 kV/cm, a rather abrupt redistribution of the satellite intensities of phase 1/5 is observed, without change of the corresponding primary modulation wave vector ( ⅕ ). A quantitative analysis of these intensity variations confirms the earlier conjecture based on dielectric experiments that the modulation essentially changes from a non-polar sequence 5up5down ( ) of polarized z-perpendicular layers of basic semicells, to a polar sequence 6up4down ( ). The transition is caused by the flip of the average polarization of one of the interface layers, and can then be described as a bounded discrete motion of the wall separating positive and negative microdomains within the five-fold unit cell. This type of polarization-flip phase transition had been detected and characterized in one-dimensional theoretical models as generalized Frenkel-Kontorova models or spin chains with elastic couplings, but had not been anticipated in theoretical analyses of BCCD, for which other phenomenological or microscopic models (as the ANNNI model) have been considered adequate. Only recently and in view of the experimental results reported here, we demonstrated, using a general phenomenological displacive model, the possibility of this type of transition in systems as BCCD [Phys. Rev. B 62, 11418 (2000)]. Phase diagrams with spin-flip phase transitions yield very peculiar phase diagrams with a checkerboard topological structure and self-similar features. In particular, they may present special critical points as the so-called upsilon points [J. Statistical Phys. 62, 45 (1991)]. BCCD may be then the first experimental system where they could be observed.

Coh et al. presented in [Phys. Rev. B 88, 121106(R) (2013)] a systematic search of the simplest so-called ``canonical'' structures allowing isotropic magnetoelectric response, and reported a total of 30 such magnetic configurations. Using magnetic symmetry we show in this Comment that this listing is severely incomplete, and 14 additional distinct cases satisfying the same conditions should be added. The complete list of these elementary magnetic arrangements is then presented in a short and efficient form as distinct Wyckoff positions of some cubic magnetic space groups.

The triangular based antiferromagnet $\ensuremath{\alpha}$-${\mathrm{NaFeO}}_{2}$ has been studied by magnetization, dielectric, and neutron diffraction measurements as a function of temperature and magnetic field. The appropriate $(H\ensuremath{-}T)$ phase diagram was constructed revealing a complex behavior due to a competition between several magnetic phases. In zero field, the system undergoes a sequence of magnetostructural transitions; initially from paramagnetic $R\overline{3}m{1}^{\ensuremath{'}}$ phase to the incommensurate spin density wave (ICM1) at ${T}_{\mathrm{N}1}=11$ K with the nonpolar (3+1) magnetic superspace group $C2/m{1}^{\ensuremath{'}}(0,\ensuremath{\beta},\frac{1}{2})s0s$, then, below ${T}_{\mathrm{N}2}=7.5$ K, the ICM1 phase coexists with the polar cycloidal ordering (ICM2) possessing the $Cm{1}^{\ensuremath{'}}(0,\ensuremath{\beta},\frac{1}{2})0s$ superspace symmetry and finally the commensurate collinear ordering (CM) with the nonpolar magnetic space group ${P}_{a}{2}_{1}/m$ develops below ${T}_{\mathrm{N}3}=5.5$ K as the ground state of the system. A small amount of ICM2 coexists with the ICM1 and CM phases resulting in a nonzero measured polarization below ${T}_{\mathrm{N}2}$. Magnetic field destabilizes the collinear ground state and promotes the polar ICM2 phase resulting in a drastic increase of the polarization. The symmetry of the zero field cycloidal structure allows the two orthogonal components ${\mathbit{p}}_{1}\ensuremath{\propto}{\mathbit{r}}_{ij}\ifmmode\times\else\texttimes\fi{}({\mathbit{S}}_{i}\ifmmode\times\else\texttimes\fi{}{\mathbit{S}}_{j})$ and ${\mathbit{p}}_{2}\ensuremath{\propto}{\mathbit{S}}_{i}\ifmmode\times\else\texttimes\fi{}{\mathbit{S}}_{j}$ to contribute to the macroscopic polarization through the inverse DM effect. The applied magnetic field reduces the symmetry of the ICM2 phase down to the triclinic $P1(\ensuremath{\alpha},\ensuremath{\beta},\ensuremath{\gamma})0$, resulting in admixture of another cycloidal and helical components both generating magnetic field switchable polarization ${\mathbit{p}}_{3}$ perpendicular to ${\mathbit{p}}_{1}$ and ${\mathbit{p}}_{2}$.

Quantum saturation of displacive order parameters may be observed at temperatures higher than expected in a simple soft-mode model. The experimental results have been explained sometimes by coupling of the softphonon branch with hard mode branches. We analyse the effect of this coupling in the limit of coupled Einstein oscillators. It is shown that a significant change through this mechanism of the low-temperature quantum saturation of the order parameter requires abnormally large coupling terms. In any case, under favourable circumstances, the coupled hard-phonon spectrum can be derived from the second temperature derivative of the square of the measured order parameter.

The structure of Ba 1.1064 CoO 3 has been solved in the (3+1)-dimensional formalism. The structure is described as a modulated chain composite with two subsystems, [CoO 3 ] and [Ba], respectively. The superspace group is R -3 m (00 γ )0 s with a =9.8842(20) A, c =2.4785(12) A, and q =0.5532(4) c * ( Z =3). A saw-tooth function was used to model both the occupational and displacive modulations. Each atomic saw-tooth function is defined by its center ○ 4 along the fourth dimension, its width (Δ), and the maximum amplitude of the displacive modulation ( δ ). The paper describes how, as a first approximation, the columns (CoO 3 ) can be mainly described by a single free parameter, based on the height difference of the trigonal prisms and octahedra that constitute the transition metal chains. As a result, this superspace formalism requires only a small number of variables to be refined, compared to the conventional superstructure description.

A phenomenological description of the quasicrystal-to-crystal transformation is presented in terms of Landau theory. The basic idea is to describe the structures of quasicrystals as incommensurately modulated structures and to use the tools developed for phase transitions of this kind. The usefulness of this description is demonstrated on quasiperiodic model structures: the one-dimensional (1D) Fibonacci sequence and the 2D Penrose tiling. The transformation to periodic structures is discussed generally for rational approximants.

Eu8/7TiS3 and Sr8/7TiS3, two members in the hexagonal perovskite-like A x TiS3 family, have been obtained as single crystals and their structure solved from twinned crystal X-ray data within the (3+1)-dimensional [(3+1)D] formalism. Eu8/7TiS3 and Sr8/7TiS3 crystallizes in the trigonal symmetry, R3̅m(00γ)0s superspace group with the following lattice parameters: Eu8/7TiS3, as = 11.3892(13) Å, cs = 2.989(2) Å, q = 0.5741(4) c* and Vs = 335.8(3) Å3; Sr8/7TiS3, as = 11.480(5) Å, cs = 2.9940(13) Å, q = 0.5716(2) c* and Vs = 341.7(5) Å3. The structures were considered as commensurate (P three-dimensional (3D) spacegroup) but refined within the (3+1)D formalism (t = 0 section) to a residual factor: Eu8/7TiS3, R = 3.84% for 128 parameters and 4500 independent reflections; Sr8/7TiS3, R = 4.92% for 125 parameters and 2719 independent reflections. The two structures are isotypic and resemble that of Sr9/8TiS3. They differ from the structures of the oxide counterparts by the presence of original PO [TiS6] units, intermediate between octahedra (Oh) and trigonal prisms (TP), and replacing the TP units found in the oxides. In addition, they are different from the Sr9/8TiS3 structure by the occurrence of a particular twinning and the presence of two different [TiS3] chain types, with two consecutive PO units in one of them.

A general nomenclature applicable to the phases that form in any sequence of transitions in the solid state has been recommended by an IUCr Working Group [Acta Cryst. (1998). A54, 1028–1033]. The six-field notation of the first Report, hereafter I, was applied to the case of structural phase transitions, i.e. to transformations resulting from temperature and/or pressure changes between two crystalline (strictly periodic) phases involving modifications to the atomic arrangement. Extensive examples that illustrate the recommendations were provided. This second Report considers, within the framework of a similar six-field notation, the more complex nomenclature of transitions involving magnetic phases, incommensurate phases and transitions that occur as a function of composition change. Extension of the nomenclature to the case of phases with less clearly established relevance to standard schemes of transition in equilibrium systems, namely polytype phases, radiation-induced and other transient phases, quasicrystalline phases and their transitions is recommended more tentatively. A uniform notation for the translational periodicity, propagation vector or wavevector for magnetic and/or incommensurate substances is specified. The notation adopted for incommensurate phases, relying partly on the existence of an average structure, is also consistent with that for commensurate phases in a sequence. The sixth field of the nomenclature is used to emphasize the special features of polytypes and transient phases. As in I, illustrative examples are provided for each category of phase sequence.

A new, unified superspace approach to the structural characterization of the perovskite-related Sr n (Nb,Ti) n O3n + 2 compound series, strontium niobium/titanium oxide, is presented. To a first approximation, the structure of any member of this compound series can be described in terms of the stacking of (110)-bounded perovskite slabs, the number of atomic layers in a single perovskite slab varying systematically with composition. The various composition-dependent layer-stacking sequences can be interpreted in terms of the structural modulation of a common underlying average structure. The average interlayer separation distance is directly related to the average structure periodicity along the layer stacking direction, while an inherent modulation thereof is produced by the presence of different types of layers (particularly vacant layers) along this stacking direction. The fundamental atomic modulation is therefore occupational and can be described by means of crenel (step-like) functions which define occupational atomic domains in the superspace, similarly to what occurs for quasicrystals. While in a standard crystallographic approach, one must describe each structure (in particular the space group and cell parameters) separately for each composition, the proposed superspace model is essentially common to the whole compound series. The superspace symmetry group is unique, while the primary modulation wavevector and the width of some occupation domains vary linearly with composition. For each rational composition, the corresponding conventional three-dimensional space group can be derived from the common superspace group. The resultant possible three-dimensional space groups are in agreement with all the symmetries reported for members of the series. The symmetry-breaking phase transitions with temperature observed in many compounds can be explained in terms of a change in superspace group, again in common for the whole compound series. Inclusion of the incommensurate phases, present in many compounds of the series, lifts the analysis into a five-dimensional superspace. The various four-dimensional superspace groups reported for this incommensurate phase at different compositions are shown to be predictable from a proposed five-dimensional superspace group apparently common to the whole compound series. A comparison with the scarce number of refined structures in this system and the homologous (Nb,Ca)6Ti6O20 compound demonstrates the suitability of the proposed formalism.

Commensurate and incommensurate structures, which can be described as composites but do not fit into the usual picture of a composite are briefly reviewed. The modulation is an essential part of these systems and cannot be considered as a perturbation. In many cases, the superspace phason sliding mode does not have a physical meaning.

A new unified superspace approach to the structural characterization of two compositionally flexible compound series, La n Ti n-δ O 3n and Sr n (Nb,Ti) n O 3n+2 , is presented. To a first approximation, the structure of any compound within either compound series can be described in terms of the stacking of layers derived from the ideal perovskite structure. A unique superspace model for each compound series is shown to predict the space group and layer stacking sequence for any particular composition.

New Rietveld refinements of powder diffraction data of the cation deficient perovskite-related compounds mentioned in the title were performed in 4D using a recently proposed superspace model with discrete atomic occupation domains. The results confirm the validity and efficiency of the superspace description of these layered systems. The resulting 3D structure for La5Ti4O15 corrects a previously reported model.

A new, essentially composition-independent structural model is proposed for the perovskite-related compound series LanTin-δO3n, with δ ≤ n/4, based on superspace formalism. In the standard 3-dimensional crystallographic approach, one must specify space group and cell parameters separately for each composition. In the superspace approach, the superspace group is invariant while the cell parameters and primary modulation wave-vector vary slowly and continuously with composition. To a first approximation, the structure at any particular composition can be described in terms of mixed cubic and hexagonal stacking of close-packed LaO3 (and Ti) layers along the z-axis. The actual layer stacking sequence varies abruptly with composition, and the titanium vacancies occur as entire layers between neighbouring hexagonal close-packed LaO3 layers. The continuously variable, composition-dependent character of reciprocal space indicates that the various layer stacking sequences must be intimately correlated and shows that all the compounds can be interpreted in terms of a structural modulation over a common average structure. It is shown that a very simple model in superspace is sufficient to describe the layer stacking sequences for any particular composition. The atomic surfaces representing the atoms in superspace are shown to be well described by means of crenel functions. The cation displacive modulations are approximately described by sawtooth functions. The only variable parameters are the composition-dependent (but smoothly varying) primary modulation wave-vector and associated Ti occupation domain width. Conventional space groups can be easily determined from the superspace group and the composition. From a practical point of view, this means that we have a very good single starting point to refine the actual atomic positions for any composition, including incommensurate or long period structures. Even more importantly, the superspace approach has the flexibility to represent in a simple manner the symmetry-allowed deviations from the ideal model. A comparison with the scarce number of refined structures in this system and in the homologous (Ba,La)nTin-1O3n system, demonstrates the suitability of the formalism for the analysis of these compound series and for the analysis of compositionally flexible systems in general.

The structure of Ba 1+x Cu x Ir 1− x O 3 , x =0.2708 has been solved using the superspace formalism. The superspace group is R 3 m(00 γ )0s with a =10.1490(8) A, c =2.7164(3) A and q =0.6354(2) c 1 * ( Z =3). The structure can be described as a modulated columnar composite containing two subsystems [(Cu, Ir)O 3 ] and [Ba]. A sawtooth function was used to model the z-component of the atomic modulation functions. Each atomic sawtooth function is defined by its center x 4 along the fourth dimension, its width ( Δ ) and the maximum amplitude of the displacive modulation ( δ ). The paper describes how, as a first approximation, the columns (Cu x Ir 1− x )O 3 can be simply described by a single free parameter, based on the height difference of the prisms and octahedra that make up the transition metal chains. As a result, this superspace approach requires only a very small number of variables to be refined, compared to the conventional superstructure description. In addition, this paper presents a structural comparison of this structure with other members of this family and outlines how the structure determination is completely general and can be applied to other modulated composite structures of this family. The structural analysis presented in this paper is another demonstration of the efficiency of the 4-dimensional superspace formalism for describing the structures of so-called ‘composition flexible’ systems.

As an alternative to the usual Landau-Ginzburg-type continuous field approximation, a free energy featuring two anticrossed phonon branches and defined in a discrete lattice is proposed for dealing with modulated phases with inherent discrete lattice effects. Using this approach, it is shown that, at some threshold electric (or conjugate) field, phases may appear in the phase diagram of solids having several modulated phases of different periods. These phases are the result of the sign-reversal of a particular polar local mode in a spinlike modulation of the order parameter, and have indeed been recently observed in the modulated ferroelectric betaine calcium chloride dihydrate (BCCD). This type of field-driven polarization-flip transition had not been anticipated by previous theoretical approaches, including microscopic ones such as the ANNNI model. The model proposed also explains the quite peculiar topological features of the phase diagram under the electric field observed in BCCD.

We have performed detailed first-principles calculations to determine the eigenvectors of the zone-center modes of hexagonal BaTiO3 and shown that the experimentally relevant low-energy modes (including the non-polar instability) can be represented as suitable combinations of basic local polar distortions associated with the instability of the cubic perovskite phase. The hexagonal structure provides a testing ground for the analysis of the influence of the stacking of TiO6 octahedra: the occurrence of relatively high-energy chains of dipoles highlights the importance of local effects related to the coherent hybridization enhancement between Ti and O ions. Our results provide simple heuristic rules that could be useful for the analysis of related compounds., Comment: 4 pages, 3 eps figures

Polar structures with pseudosymmetry related to a hypothetical non-polar configuration are considered as good candidates for displacive ferroelectrics. Recently we have developed a procedure for the systematic search of pseudosymmetric structures in the available databases. The aim of the present contribution is the generalization of the pseudosymmetry search procedure to include possible displacive ferroelectric structures which implies an additional optimization of the continuous parameter(s) describing the location of the non-polar symmetry elements. Here we present the results of the search for possible displacive ferroelectrics among the compounds with space group Pba 2, listed in the Inorganic Crystal Structure Database.

We present a new scheme for the construction of highly localized lattice Wannier functions. The approach is based on a heuristic criterion for localization and takes the symmetry constraints into account from the start. We compare the local modes thus obtained with those generated by other schemes and find that they also provide a better description of the relevant vibrational subspace., Comment: 6 pages, ReVTeX, plus four postscript files for figures