Superspace Symmetry Research Articles

The incommensurate composite YxOs4B4 (x = 1.161).

YxOs4B4 (x = 1.161) crystallizes as a tetragonal incommensurate composite of columns of Y atoms extending along [001] in an Os4B4 framework. The structure was refined using the superspace approach. The basic structure of the Y subsystem can be idealized as having I4/mmm symmetry, with a crystallographically unique Y atom located on the 4/mmm position. The actual superspace symmetry is P42/nmc(00σ3)s0s0. The Y atoms feature only subtle positional modulation in the [001] direction. The Os4B4 subsystem [P42/ncm(00σ3)00ss superspace symmetry] is built of columns of edge-sharing Os4 tetrahedra extending along [001] and B2 dumbbells. The Os4 tetrahedra feature pronounced positional modulation with a distinct variation of the Os-Os bond lengths. Modulation of the B2 dumbbells is best described as a rotation about the [001] axis.

Analysis of magnetic structures in JANA2020.

JANA2020 is a program developed for the solution and refinement of regular, twinned, modulated, and composite crystal structures. In addition, JANA2020 also includes a magnetic option for solving magnetic structures from powder and single-crystal neutron diffraction data. This tool uses magnetic space and superspace symmetry to describe commensurate and incommensurate magnetic structures. The basics of the underlying formulation of magnetic structure factors and the use of magnetic symmetry for handling modulated and non-modulated magnetic structures are presented here, together with the general features of the magnetic tool. Examples of structures solved in the magnetic option of JANA2020 are given to illustrate the operation and capabilities of the program.

Methylhydrazinium]2PbCl4, a Two-Dimensional Perovskite with Polar and Modulated Phases

Two-dimensional (2D) lead halide perovskites are a family of materials at the heart of solar cell, light-emitting diode, and photodetector technologies. This perspective leads to a number of synthetic efforts toward materials of this class, including those with prescribed polar architectures. The methylhydrazinium (MHy+) cation was recently presumed to have an unusual capacity to generate non-centrosymmetric perovskite phases, despite its intrinsically nonchiral structure. Here, we witness this effect once again in the case of the Ruddlesden–Popper perovskite phase of formula MHy2PbCl4. MHy2PbCl4 features three temperature-dependent crystal phases, with two first-order phase transitions at T1 = 338.2 K (331.8 K) and T2 = 224.0 K (205.2 K) observed in the heating (cooling) modes, respectively. Observed transitions involve a transformation from high-temperature orthorhombic phase I, with the centrosymmetric space group Pmmn, through the room-temperature modulated phase II, with the average structure being isostructural to I, to the low-temperature monoclinic phase III, with non-centrosymmetric space group P21. The intermediate phase II is a rare example of a modulated structure in 2D perovskites, with Pmmn(00γ)s00 superspace symmetry and modulation vector q ≅ 0.25c*. MHy2PbCl4 beats the previous record of MHy2PbBr4 in terms of the shortest inorganic interlayer distance in 2D perovskites (8.79 Å at 350 K vs 8.66 Å at 295 K, respectively). The characteristics of phase transitions are explored with differential scanning calorimetry, dielectric, and Raman spectroscopies. The non-centrosymmetry of phase III is confirmed with second harmonic generation (SHG) measurements, and polarity is demonstrated by the pyroelectric effect. MHy2PbCl4 also exhibits thermochromism, with the photoluminescence (PL) color changing from purplish-blue at 80 K to bluish-green at 230 K. The demonstration of polar characteristics for one more member of the methylhydrazinium perovskites settles a debate about whether this approach can present value for the crystal engineering of acentric solids similar to that which was recently adopted by a so-called fluorine substitution effect.

Interpretation of diffuse scattering using superspace crystallography

Single crystal diffuse scattering is generally interpreted using correlation parameters that describe probabilities for certain configurations on a local scale. In this paper we present a novel interpretation of diffuse maxima using a disordered superspace approach. In (D+d)-dimensional superspace two modulation functions are disordered along the superspace axis $\vec{a}_{s,i}$ for i = 1,..,D, while the periodicity along the internal dimensions is maintained. This simple approach allows the generation of diffuse maxima of any width at any position in reciprocal space. The extinction rules that are introduced by superspace symmetry are also fulfilled by the diffuse maxima from structures generated using the disordered superspace approach. In this manuscript we demonstrate the disordered superspace approach using a simple a simple two-dimensional binary disordered system. The extension of the approach to (3+D)-dimensional superspace is trivial. The treatment of displacement and magnetic disorder in a similar approach is straight forward.

Description and the basic spectrum of type crystals Cu<sub>7</sub>GeS<sub>5</sub>I

The crystalline structure of the Cu 7 GeS 5 I type is considered and its supra-spatial description is described, considering the latter as a natural superlattice. The complete (3 + 3) -mean basis, a set of modulation vectors and mass modulation functions are given. The dispersion of the phonon spectrum is calculated, and the dispersion dependences of the phonon spectrum in highly symmetric directions (G - X - M - R - Г - M) of the Brillouin zone of the given crystal. Key words: argygoid, crystalline structure, superposition, over spatial symmetry, phonon spectrum, Maple. Introduction : The compound Cu 7 GeS 5 I relates to a large family of complex chalcogenides, which crystallize in the structures of argillaceites. At ambient temperature, the crystals under investigation are characterized by high symmetry: cubic sinongion, spatial group F - 43m. The main feature of copper-bearing argillites is the high solid-state ion conductivity of Cu + ions, which makes it possible to use these argilloid crystals as functional electronic materials. Purpose : The main goal of this work was to develop a program suitable for theoretical calculation of the phonon spectra of Cu 7 GeS 5 I argillites and to carry out appropriate calculations (within the concept of superspace symmetry). Methods : In the present work, the crystalline structure of Cu 7 GeS 5 I argillites was analyzed and described using the superspace symmetry concept. Results : A program working in the Maple environment is developed and is suitable for theoretical calculation of the phonon spectra of Cu 7 GeS 5 I argillate crystals. Phonon spectra were calculated and presented for the model phases of Cu 7 GeS 5 I. The eigenvalues of the generalized dynamical matrix are found and the dispersion dependences for the directions of the Brightouin zone G - X- M - R - G - M are constructed.

Incommensurate atomic and magnetic modulations in the spin-frustrated β−NaMnO2 triangular lattice

The layered {\beta}-NaMnO2, a promising Na-ion energy-storage material has been investigated for its triangular lattice capability to promote complex magnetic configurations that may release symmetry restrictions for the coexistence of ferroelectric and magnetic orders. The complexity of the neutron powder diffraction patterns underlines that the routinely adopted commensurate structural models are inadequate. Instead, a single-phase superspace symmetry description is necessary, demonstrating that the material crystallizes in a compositionally modulated q= (0.077(1), 0, 0) structure. Here, Mn3+ Jahn-Teller distorted MnO6 octahedra form corrugated layer stacking sequences of the {\beta}-NaMnO2 type, which are interrupted by flat sheets of the {\alpha}-like oxygen topology. Spontaneous long-range collinear antiferromagnetic order, defined by the propagation vector k= (1/2, 1/2, 1/2), appears below TN1= 200 K. Moreover, a second transition into a spatially modulated proper-screw magnetic state (k+-q) is established at TN2= 95 K, with an antiferromagnetic order parameter resembling that of a two-dimensional (2D) system. The evolution of 23Na NMR spin-lattice relaxation identifies a magnetically inhomogene-ous state in the intermediate T-region (TN2 <T< TN1), while its strong suppression below TN2 indicates that a spin-gap opens in the excitation spectrum. High-resolution neutron inelastic scattering confirms that the magnetic dynamics are indeed gapped ({\Delta}~5 meV) in the low-temperature magnetic phase, while simulations on the basis of the single-mode approximation suggest that Mn-spins residing on ad-jacent antiferromagnetic chains, establish sizable 2D correlations. Our analysis points that novel struc-tural degrees of freedom promote, cooperative magnetism and emerging dielectric properties in this non-perovskite-type of manganite.

Magnetic structure of the swedenborgiteCaBa(Co3Fe)O7derived by unpolarized neutron diffraction and spherical neutron polarimetry

We present a study that combines polarized and unpolarized neutrons to derive the magnetic structure of the swedenborgite compound $\mathrm{CaBa}({\mathrm{Co}}_{3}\mathrm{Fe}){\mathrm{O}}_{7}$. Integrated intensities from a standard neutron diffraction experiment and polarization matrices from spherical neutron polarimetry have been simultaneously analyzed revealing a complex order, which differs from the usual spin configurations on a kagome lattice. We find that the magnetic structure is well described by a combination of two one-dimensional representations corresponding to the magnetic superspace symmetry $P{2}_{1}^{\ensuremath{'}}$, and it consists of spins rotating around an axis close to the [110] direction. Due to the propagation vector $\mathbf{q}=(\frac{1}{3}00)$, this modulation has cycloidal and helicoidal character rendering this system a potential multiferroic. The resulting spin configuration can be mapped onto the classical $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ structure of a kagome lattice, and it indicates an important interplay between the kagome and the triangular layers of the crystal structure.

The true structural periodicities and superspace group descriptions of the prototypical incommensurate composite materials: Alkane/urea inclusion compounds

The prototypical family of incommensurate composite materials are the n-alkane/urea inclusion compounds, in which n-alkane guest molecules are arranged in a periodic manner along one-dimensional tunnels in a urea host structure, with an incommensurate relationship between the periodicities of the host and guest substructures along the tunnel. We develop interpretations of the structural periodicities, superspace group descriptions and symmetry properties of the low-temperature phases of n-alkane/urea inclusion compounds, based in part on a high-resolution synchrotron single-crystal X-ray diffraction study of n-nonadecane/urea. Specifically, we prove that, on passing from phase I to phase II, the C-centering of the orthohexagonal unit cell is lost for both the host and guest substructures, and that the symmetries of all phases I, II and III are described completely by (3 + 1)-dimensional superspace groups.

MAGNDATA: towards a database of magnetic structures. II. The incommensurate case

A free web page under the name MAGNDATA, which provides detailed quantitative information on more than 400 published magnetic structures, has been made available at the Bilbao Crystallographic Server (http://www.cryst.ehu.es). It includes both commensurate and incommensurate structures. In the first article in this series, the information available on commensurate magnetic structures was presented [Gallego, Perez-Mato, Elcoro, Tasci, Hanson, Momma, Aroyo &amp; Madariaga (2016). J. Appl. Cryst. 49, 1750–1776]. In this second article, the subset of the database devoted to incommensurate magnetic structures is discussed. These structures are described using magnetic superspace groups, i.e. a direct extension of the non-magnetic superspace groups, which is the standard approach in the description of aperiodic crystals. The use of magnetic superspace symmetry ensures a robust and unambiguous description of both atomic positions and magnetic moments within a common unique formalism. The point-group symmetry of each structure is derived from its magnetic superspace group, and any macroscopic tensor property of interest governed by this point-group symmetry can be retrieved through direct links to other programs of the Bilbao Crystallographic Server. The fact that incommensurate magnetic structures are often reported with ambiguous or incomplete information has made it impossible to include in this collection a good number of the published structures which were initially considered. However, as a proof of concept, the published data of about 30 structures have been re-interpreted and transformed, and together with ten structures where the superspace formalism was directly employed, they form this section of MAGNDATA. The relevant symmetry of most of the structures could be identified with an epikernel or isotropy subgroup of one irreducible representation of the space group of the parent phase, but in some cases several irreducible representations are active. Any entry of the collection can be visualized using the online tools available on the Bilbao server or can be retrieved as a magCIF file, a file format under development by the International Union of Crystallography. These CIF-like files are supported by visualization programs like Jmol and by analysis programs like JANA and ISODISTORT.

Combining powder diffraction with TEM for solving modulated structures

In many materials competing interactions of different nature may give rise to incommensurate modulations causing extreme structure complexity. Ab initio solution of the modulated structures even with using high quality synchrotron X-ray and/or neutron powder diffraction data appears to be a very challenging problem due to weakness of the satellite reflections, ambiguity in the determination of the modulation vector(s) and superspace symmetry and difficulties in building the initial model for further Rietveld refinement. These problems can be resolved or, at least, mitigated if the diffraction, imaging and spectroscopic advanced transmission electron microscopy techniques are combined with the analysis of powder diffraction data. Complete reconstruction of the reciprocal space, structure solution using quasi-kinematical electron diffraction data, mapping projected scattering density in the unit cell, visualization of the light atoms, displacive and occupational ordering, mapping chemical composition and coordination number can be utilized to reveal the nature of incommensurate modulations and construct the reliable model for the refinement from powder diffraction data. The benefit of the strategy of combining the powder diffraction data with the reciprocal and real space information obtained using aberration-corrected scanning transmission electron microscopy will be illustrated on the examples of the transition metal oxides: Li3xNd2/3-xTiO3 perovskites with frustrated incommensurately modulated octahedral tilting pattern [1]; perovskites (Bi,Pb)1-xFe1+xO3-y, modulated by crystallographic shear planes [2]; CaGd2(1-x)Eu2x(MoO4)4(1-y)(WO4)4y scheelites with incommensurately modulated ordering of cation vacancies [3].

Superspace symmetry and phenomenology of incommensurate type-II multiferroics

The recent upsurge of research work on multiferroic materials has made clear the need for a comprehensive knowledge of the constraints imposed by symmetry on the magnetic structure and tensor properties of incommensurate magnetic phases. That knowledge can be provided by the magnetic superspace formalism in a simple and robust manner. In this communication I will briefly review some of the basic concepts of this formalism pertaining to magnetic systems. Several practical examples will be given in order to illustrate how the assignment of a magnetic superspace group to a given phase is, in general , more restrictive and wide-ranging than the simple assignment of the order parameter irreducible representation. The direct connection between symmetry and phenomenology, which naturally arises via Landau theory, will then be used in order to illustrate how several complex phase diagrams of type-II magneto-electric multiferroics can be interpreted, and how the magnetically induced ferroelectric states can be anticipated on the basis of general symmetry considerations.

Visualizing Structural Modulation in Melilite Electrolytes by X-ray and Electron

Nowadays novel melilite-type oxides are of growing interest as promising new oxide-ion conductors with potential use in clean energy devices such as solid oxide fuel cells (SOFCs).[1] These oxides have a general formula of A2BC2O7, where A = large divalent alkali earth metals like Ca, Sr, Ba or trivalent rare earth metals like La, Nd, Gd; B and C = small cations like Mg, Si, Ga. In addition to the conventional tetragonal symmetry with space group P¯421m, these melilites also show a five-dimensional superspace symmetry P¯421m(α, α, 0)00s(¯α, α, 0)000 with modulation vectors q1=α(a*+b*) and q2=α(-a*+b*). Both displacive (cation and anion) and occupational (cation) modulations have been found.[2-3] Here a series of melilite single crystals, such as Ca2MgSi2O7, SrLaGa3O7, SrGdGa3O7, BaLaGa3O7 and CaNdGa3O7 have been successfully obtained using the optical floating zone and Czochralski method. Further investigations were made by a combination of temperature-variant X-ray diffraction and electron microscopy. The mechanism between substitutions in A- and B- sites and structural modulations in melilites is clarified, which paves the way for future crystallographic optimization of electrolyte performances via tailoring interstitial oxygen concentration and enhancing ion mobility.

Sheding light on the multiferroicity in Mn1-xCoxWO4using superspace formalism

The use of the superspace symmetry formalism allows to rationalize the physical properties induced by an incommensurate magnetic ordering [1]. The incommensurate magnetic structures (ICMS) in Mn1-xCoxWO4 have been studied in the light of this formalism. MnWO4 is a multiferroic material in which the magnetic order of one of its magnetic phases induces ferroelectricity. Like most multiferroic materials MnWO4 is extremely sensitive to small perturbations such as chemical substitution. It turned out that doping with Co2+ is particularly interesting since it strongly stabilizes the multiferroic phase at low temperatures, and moreover, by increasing the cobalt amount in the crystals the orientation of the electric polarization flops from the b axis to the ac plane [2]. This change of orientation is linked to a symmetry change. The ICMS of the x = 0 and x = 0.10 compounds, which exhibit completely different behavior, have been studied thoroughly using superspace formalism. We have found, not only the symmetry of the magnetic structures and their intrinsic restrictions, but also information about the tensor properties of each incommensurate phase, such as ferroelectric and magnetostructural properties [3]: in the paramagnetic state, there is a unique independent magnetic atom in the crystallographic unit cell in both cases, but when the system enters into the multiferroic phase, this is no-longer true. In the multiferroic phase of MnWO4, the two Mn atoms in the unit cell become symmetry independent, whereas in the x = 0.10 substituted compound, they are still symmetry-related. This difference is related to the change of the electric polarization.

THE CONDITION OF THE COMBINATION FOR THE CRYSTAL BaTiO&lt;sub&gt;3&lt;/sub&gt;

For crystals of BaTiO 3 type due to superspace symmetry conception over all vectors k i of Brillouin zone full vibrational images were obtained and the analysis of correlation combination was carried out. Within the framework of proposed methods the vibrational branches of phonon spectrum and contribution to symmetry of each crystal atom were determined.

A (3 + 3)-Dimensional “Hypercubic” Oxide-Ionic Conductor: Type II Bi2O3–Nb2O5

The high-temperature cubic form of bismuth oxide, δ-Bi2O3, is the best intermediate-temperature oxide-ionic conductor known. The most elegant way of stabilizing δ-Bi2O3 to room temperature, while preserving a large part of its conductivity, is by doping with higher valent transition metals to create wide solid-solutions fields with exceedingly rare and complex (3 + 3)-dimensional incommensurately modulated "hypercubic" structures. These materials remain poorly understood because no such structure has ever been quantitatively solved and refined, due to both the complexity of the problem and a lack of adequate experimental data. We have addressed this by growing a large (centimeter scale) crystal using a novel refluxing floating-zone method, collecting high-quality single-crystal neutron diffraction data, and treating its structure together with X-ray diffraction data within the superspace symmetry formalism. The structure can be understood as an "inflated" pyrochlore, in which corner-connected NbO6 octahedral chains move smoothly apart to accommodate the solid solution. While some oxide vacancies are ordered into these chains, the rest are distributed throughout a continuous three-dimensional network of wide δ-Bi2O3-like channels, explaining the high oxide-ionic conductivity compared to commensurately modulated phases in the same pseudobinary system.

Incommensurate magnetic structures of multiferroic MnWO4studied within the superspace formalism

The magnetic structures of the incommensurate phases of the prototypic multiferroic MnWO${}_{4}$ are analyzed using superspace symmetry formalism. This approach shows in a very simple way the relation between the magnetic symmetry of the incommensurate phases and any tensorial property, such as ferroelectricity. The magnetic structures have been redetermined by single-crystal neutron diffraction experiments, which revealed fine details of the AF3 and AF2 magnetic structures. The spin modulations of the two manganese ions of the basic unit cell are symmetry-related in the paraelectric incommensurate-magnetic structure (AF3)---they are not forced to be collinear but they must have opposite chirality---while in the multiferroic phase (AF2) they become symmetry independent. It is due to this symmetry break that the electric polarization arises along the $b$ axis.

ChemInform Abstract: Magnetic Superspace Groups and Symmetry Constraints in Incommensurate Magnetic Phases

Superspace symmetry has been for many years the standard approach for the analysis of non-magnetic modulated crystals because of its robust and efficient treatment of the structural constraints present in incommensurate phases. For incommensurate magnetic phases, this generalized symmetry formalism can play a similar role. In this context we review from a practical viewpoint the superspace formalism particularized to magnetic incommensurate phases. We analyse in detail the relation between the description using superspace symmetry and the representation method. Important general rules on the symmetry of magnetic incommensurate modulations with a single propagation vector are derived. The power and efficiency of the method is illustrated with various examples, including some multiferroic materials. We show that the concept of superspace symmetry provides a simple, efficient and systematic way to characterize the symmetry and rationalize the structural and physical properties of incommensurate magnetic materials. This is especially relevant when the properties of incommensurate multiferroics are investigated.

Symmetry and magnetic field driven transitions in the 2D triangular lattice compound RbFe(MoO4)2

The temperature versus magnetic field phase diagram of the 2D triangular lattice and multiferroic compound RbFe(MoO4)2 is analysed from the point of view of symmetry. The paramagnetic space group and its irreducible representations are used in order to obtain the magnetic symmetry of the possible modulated phases and characterize the restrictions imposed by this symmetry on the corresponding magnetic structures. Superspace symmetry is considered in the case of incommensurate phases. It is shown that the experimentally observed phases correspond to different isotropy subgroups originating in the same irreducible representation of the paramagnetic symmetry group. The relevant couplings between the primary transverse spin modulation and the electric polarization, the in-plane magnetization and the secondary longitudinal magnetic modulation are discussed. The mechanisms for the destabilization of the improper ferroelectric chiral phase and the origin of the different orientation of the spins with respect to the external field in the two collinear phases are analysed from a symmetry based perspective.

Spiral ground state against ferroelectricity in the frustrated magnet BiMnFe2O6

The spiral magnetic structure and underlying spin lattice of BiMnFe${}_{2}$O${}_{6}$ are investigated by low-temperature neutron powder diffraction and density functional theory band structure calculations. In spite of the random distribution of the Mn${}^{3+}$ and Fe${}^{3+}$ cations, this centrosymmetric compound undergoes a transition into an incommensurate antiferromagnetically ordered state below ${T}_{N}\ensuremath{\simeq}220$ K. The magnetic structure is characterized by the propagation vector $\mathbf{k}=[0,\ensuremath{\beta},0]$ with $\ensuremath{\beta}\ensuremath{\simeq}0.14$ and the $P{22}_{1}{2}_{1}{1}^{\ensuremath{'}}(0\ensuremath{\beta}0)0s0s$ magnetic superspace symmetry. It comprises antiferromagnetic helixes propagating along the $b$ axis. The magnetic moments lie in the $\mathit{ac}$ plane and rotate about $\ensuremath{\pi}(1+\ensuremath{\beta})\ensuremath{\simeq}204.8$-deg angle between the adjacent magnetic atoms along $b$. The spiral magnetic structure arises from the peculiar frustrated arrangement of exchange couplings in the $\mathit{ab}$ plane. The antiferromagnetic coupling along the $c$ axis cancels the possible electric polarization and prevents ferroelectricity in BiMnFe${}_{2}$O${}_{6}$.

Orthogonality of Hermite polynomials in superspace and Mehler type formulae

In this paper, Hermite polynomials related to quantum systems with orthogonal O(m)-symmetry, finite reflection group symmetry 𝒢<O(m), symplectic symmetry Sp(2n) and superspace symmetry O(m)×Sp(2n) are considered. After an overview of the results for O(m) and 𝒢, the orthogonality of the Hermite polynomials related to Sp(2n) is obtained with respect to the Berezin integral. As a consequence, an extension of the Mehler formula for the classical Hermite polynomials to Grassmann algebras is proved. Next, Hermite polynomials in a full superspace with O(m)×Sp(2n)-symmetry are considered. It is shown that they are not orthogonal with respect to the canonically defined inner product. However, a new inner product is introduced, which behaves correctly with respect to the structure of harmonic polynomials on superspace. This inner product allows to restore the orthogonality of the Hermite polynomials and also restores the hermiticity of a class of Schrödinger operators in superspace. Subsequently, a Mehler formula for the full superspace is obtained, thus yielding an eigenfunction decomposition of the super Fourier transform. Finally, the new results for the Sp(2n)- and O(m)×Sp(2n)-symmetry are compared with the results in the different types of symmetry.