Dimensional Superspace Group Research Articles

Occupational modulation in the (3+1)-dimensional incommensurate structure of (2S,3S)-2-amino-3-hydroxy-3-methyl-4-phenoxybutanoic acid dihydrate.

The incommensurately modulated structure of (2S,3S)-2-amino-3-hydroxy-3-methyl-4-phenoxybutanoic acid dihydrate (C11H15NO4·2H2O or I·2H2O) is described in the (3+1)-dimensional superspace group P212121(0β0)000 (β= 0.357). The loss of the three-dimensional periodicity is ascribed to the occupational modulation of one positionally disordered solvent water molecule, where the two positions are related by a small translation [ca 0.666 (9) Å] and ∼168 (5)° rotation about one of its O-H bonds, with an average 0.624 (3):0.376 (3) occupancy ratio. The occupational modulation of this molecule arises due to the competition between the different hydrogen-bonding motifs associated with each position. The structure can be very well refined in the average approximation (all satellite reflections disregarded) in the space group P212121, with the water molecule refined as disordered over two positions in a 0.625 (16):0.375 (16) ratio. The refinement in the commensurate threefold supercell approximation in the space group P1121 is also of high quality, with the six corresponding water molecules exhibiting three different occupancy ratios averaging 0.635:0.365.

Topological magnetic structures in MnGe: Neutron diffraction and symmetry analysis

From new neutron powder diffraction experiments on the chiral cubic ($P2{_1}3$) magnet manganese germanide MnGe, we analyse all of the possible crystal symmetry-allowed magnetic superstructures that are determined successfully from the data. The incommensurate propagation vectors $k$ of the magnetic structure are found to be aligned with the [100] cubic axes, and correspond to a magnetic periodicity of about 30 $\r{A}$ at 1.8 K. Several maximal crystallographic symmetry magnetic structures are found to fit the data equally well and are presented. These include topologically non-trivial magnetic hedgehog and "skyrmion'' structures in multi-$k$ cubic or orthorhombic 3+3 and orthorhombic 3+2 dimensional magnetic superspace groups respectively, with either potentially responsible for topological Hall effect. The presence of orthorhombic distortions in the space group $P2_12_12_1$ caused by the transition to the magnetically ordered state does not favour the cubic magnetic hedgehog structure, and leave both orthorhombic hedgehog and "skyrmion'' models as equal candidates for the magnetic structures. We also report on a new combined mechanochemical and solid-state chemical route to synthesise MnGe at ambient pressures and moderate temperatures, and compare with samples obtained by the traditional high pressure synthesis.

Enumeration and tabulation of magnetic (3+d)-dimensional superspace groups.

A magnetic superspace group (MSSG) simultaneously constrains both the magnetic and non-magnetic (e.g. displacive, occupational, rotation and strain) degrees of freedom of an incommensurately modulated magnetic crystal. We present the first enumeration and tabulation of all non-equivalent (3+d)-dimensional magnetic superspace groups for d = 1, 2 and 3 independent incommensurate modulations, along with a number, symbol and reference setting for each group. We explain the process for generating an exhaustive set of inequivalent magnetic superspace groups, describe several examples, and show how the tables can be accessed via the ISO(3+d)D interface within the ISOTROPY Software Suite. We recommend that published incommensurate magnetic structures indicate a magnetic superspace-group number and symbol from these tables, as well as the transformation matrix from the published group setting to the reference setting used in these tables.

A Revision of the Sb-Te Binary Phase Diagram and Crystal Structure of the Modulated γ-Phase Field

A new experimental study of Sb-Te, based on long term annealed samples homogenized by intermediate powdering and re-annealing of the pressed powder pellets was performed. More than 25 samples were prepared, covering the entire composition range between 0 and 60 at.% Te. This part of the phase diagram was disputed for many years, as the number of phases and their crystal structures were uncertain. Samples were characterized by optical microscopy, SEM/EDX, powder XRD and DTA. Our results show the existence of a continuous single phase field (γ) ranging from 11.4 to 56.9 at.% Te at 520 °C. A comparison of all obtained diffraction patterns showed clear evidence for a continuously modulated crystal structure. It was possible to refine all powder patterns by applying a structural model in the (3 + 1)-dimensional super-space group R­3m(0,0,γ). Lattice parameters of the basic cell as well as the modulation vector are varying continuously throughout the phase field and the γq-component ({mathbf{q}} = gamma_{q} cdot {mathbf{c}}^{user2{*}}) of the incommensurate q-vector shows a linear extrapolation to pure Sb (γq = 3/2) and Sb2Te3 (γq = 6/5). In agreement with previous authors, the single-phase field γ was found to have a melting point minimum (30 at.% Te, 535 °C). Furthermore, two peritectic reactions were observed and the course of liquidus lines was determined in the whole investigated composition range. The new phase diagram differs considerably from all previous versions, and, for the first time, gives a complete rationalization of the crystal structures observed in the debated composition area.

Crystal structure of incommensurate ηʺ-Cu1.235Sn intermetallic

Abstract The crystallographic parameters of the incommensurately ordered phase ηʺ of the composition Cu1.235Sn are reported. This phase belongs to the group of ordered Ni2In/NiAs-type phases, with a NiAs-type arrangement Cu(1)Sn and additional Cu(2) atoms partially occupying trigonal-bipyramidal interstices in an ordered fashion, leading to the formula Cu(1)Cu(2)0.235Sn = Cu1.235Sn. The structure model, afterward refined on the basis of powder X-ray diffraction data, has been derived on the basis of the slightly Cu-poorer commensurately ordered η′-Cu6Sn5 (=Cu1.2Sn) phase but also on previously reported commensurate structure models η8-Cu1.25Sn and η4+1-Cu1.243Sn derived from selected area electron diffraction data. In line with a recent work (Leineweber, Wieser & Hügel, Scr. Mater. 2020, 183, 66–70), the incommensurate ηʺ phase is regarded as a metastable phase formed upon partitionless ordering of the η high-temperature phase with absent long-range ordering of the Cu(2) atoms. The previously described η8 and η4+1 superstructure are actually of the same phase, and the corresponding superstructure models can be regarded as approximant structures of the ηʺ phase. The refined structure model is described in 3+1 dimensional superspace group symmetry C2/c(q 10-q 3)00 with a unit cell of the average structure with lattice parameters of a av = 4.21866(3) Å, b av = 7.31425(5) Å, c av = 5.11137(3) Å and b av = 90.2205(5)° and a unit cell volume V = 157.717(2) Å3. The modulation vector is with q 1 = 0.76390(4), q 3 = 1.51135(5), and governs the spatial modulation of the occupancy of the Cu(2) atoms described by a Crenel function. The occupational ordering is accompanied by displacive modulations of the atoms constituting the crystal structure, ensuring reasonable interatomic distances on a local level. In particular, the spatial requirements of pairs of edge-sharing Cu(2)Sn5 trigonal bipyramids (Cu(2)2Sn8) lead to a measurable splitting of some fundamental reflections in the powder diffraction data. This splitting is considerable smaller in η′-Cu1.20Sn, which lacks such pairs due to the lower Cu content.

Antiferromagnetic Semiconductor BaFMn0.5Te with Unique Mn Ordering and Red Photoluminescence.

Semiconductors possessing both magnetic and optoelectronic properties are rare and promise applications in opto-spintronics. Here we report the mixed-anion semiconductor BaFMn0.5Te with a band gap of 1.76 eV and a work function of 5.08 eV, harboring both antiferromagnetism (AFM) and strong red photoluminescence (PL). The synthesis of BaFMn0.5Te in quantitative yield was accomplished using the "panoramic synthesis" technique and synchrotron radiation to obtain the full reaction map, from which we determined that the compound forms upon heating at 850 °C via an intermediate unknown phase. The structure refinement required the use of a (3+1)-dimensional superspace group Cmme(α01/2)0ss. The material crystallizes into a ZrCuSiAs-like structure with alternating [BaF]+ and [Mn0.5Te]- layers and has a commensurately modulated structure with the q-vector of 1/6a* + 1/6b* + 1/2c* at room temperature arising from the unique ordering pattern of Mn2+ cations. Long-range AFM order emerges below 90 K, with two-dimensional short-range AFM correlations above the transition temperature. First-principles calculations indicate that BaFMn0.5Te is an indirect band gap semiconductor with the gap opening between Te 5p and Mn 3d orbitals, and the magnetic interactions between nearest-neighbor Mn2+ atoms are antiferromagnetic. Steady-state PL spectra show a broad strong emission centered at ∼700 nm, which we believe originates from the energy manifolds of the modulated Mn2+ sublattice and its defects. Time-resolved PL measurements reveal an increase in excited-state lifetimes with longer probe wavelengths, from 93 ns (at 650 nm) to 345 ns (at 800 nm), and a delayed growth (6.5 ± 0.3 ns) in the kinetics at 800 nm with a concomitant decay (4.1 ± 0.1 ns) at 675 nm. Together, these observations suggest that there are multiple emissive states, with higher energy states populating lower energy states by energy transfer.

Reply to comment on Couzi et al. (2018): a phenomenological model for structural phase transitions in incommensurate alkane/urea inclusion compounds.

In a recent paper (Couzi et al. 2018 R. Soc. open sci. 5, 180058. (doi:10.1098/rsos.180058)), we proposed a new phenomenological model to account for the I↔II↔“III” phase sequence in incommensurate n-alkane/urea inclusion compounds, which represents an alternative interpretation to that proposed in work of Toudic et al. In a Comment (Toudic et al. 2019 R. Soc. open sci. 6, 182073. (doi:10.1098/rsos.182073)), Toudic et al. have questioned our assignment of the superspace group of phase II of n-nonadecane/urea, which they have previously assigned, based on a (3 + 2)-dimensional superspace, as C2221(00γ)(10δ). In this Reply, we present new results from a comprehensive synchrotron single-crystal X-ray diffraction study of n-nonadecane/urea, involving measurements as a detailed function of temperature across the I↔II↔“III” phase transition sequence. Our results demonstrate conclusively that “main reflections” (h, k, l, 0) with h+k odd are observed in phase II of n-nonadecane/urea (including temperatures in phase II that are just below the transition from phase I to phase II), in full support of our assignment of the (3+1)-dimensional superspace group P212121(00γ) to phase II. As our phenomenological model is based on phase II and phase “III” of this incommensurate material having the same (3+1)-dimensional superspace group P212121(00γ), it follows that the new X-ray diffraction results are in full support of our phenomenological model.

Na3Sm(PO4)2: (3+1)-dimensional commensurately modulated structure model and photoluminescence properties

Abstract For the first time, we determined the detailed crystal structure of Na3Sm(PO4)2 using single crystal X-ray diffraction and established the commensurately modulated structure model using the superspace formalism. The structure adopts a (3+1)-dimensional superspace group Pca21(0β0)000 with the modulation vector q=1/3b*. At the same time, the photoluminescent properties of Na3Sm(PO4)2 sintered at different temperatures between 800 and 1200°C were studied. Under near-UV excitation (402 nm), Na3Sm(PO4)2 shows intense characteristic emission bands of Sm3+ (561, 596, 642 nm) with the CIE coordinate of (0.5709, 0.4282), corresponding to orange color. The excitation spectrum covers a wide range from 350 nm to 470 nm, which indicates that Na3Sm(PO4)2 can be efficiently activated by near-UV or visuable LED ship.

A phenomenological model for structural phase transitions in incommensurate alkane/urea inclusion compounds

n-Alkane/urea inclusion compounds are crystalline materials in which n-alkane ‘guest’ molecules are located within parallel one-dimensional ‘host’ tunnels formed by a helical hydrogen-bonded arrangement of urea molecules. The periodic repeat distance of the guest molecules along the host tunnels is incommensurate with the periodic repeat distance of the host substructure. The structural properties of the high-temperature phase of these materials (phase I), which exist at ambient temperature, are described by a (3 + 1)-dimensional superspace. Recent publications have suggested that, in the prototypical incommensurate composite systems, n-nonadecane/urea and n-hexadecane/urea, two low-temperature phases II and ‘III’ exist and that one or both of these phases are described by a (3 + 2)-dimensional superspace. We present a phenomenological model based on symmetry considerations and developed in the frame of a pseudo-spin–phonon coupling mechanism, which accounts for the mechanisms responsible for the I ↔ II ↔ ‘III’ phase sequence. With reference to published experimental data, we demonstrate that, in all phases of these incommensurate materials, the structural properties are described by (3 + 1)-dimensional superspace groups. Around the temperature of the II ↔ ‘III’ transition, the macroscopic properties of the material are not actually associated with a phase transition, but instead represent a ‘crossover’ between two regimes involving different couplings between relevant order parameters.

How to assign a (3 + 1)-dimensional superspace group to an incommensurately modulated biological macromolecular crystal.

Periodic crystal diffraction is described using a three-dimensional (3D) unit cell and 3D space-group symmetry. Incommensurately modulated crystals are a subset of aperiodic crystals that need four to six dimensions to describe the observed diffraction pattern, and they have characteristic satellite reflections that are offset from the main reflections. These satellites have a non-integral relationship to the primary lattice and require q vectors for processing. Incommensurately modulated biological macromolecular crystals have been frequently observed but so far have not been solved. The authors of this article have been spearheading an initiative to determine this type of crystal structure. The first step toward structure solution is to collect the diffraction data making sure that the satellite reflections are well separated from the main reflections. Once collected they can be integrated and then scaled with appropriate software. Then the assignment of the superspace group is needed. The most common form of modulation is in only one extra direction and can be described with a (3 + 1)D superspace group. The (3 + 1)D superspace groups for chemical crystallographers are fully described in Volume C of International Tables for Crystallography. This text includes all types of crystallographic symmetry elements found in small-molecule crystals and can be difficult for structural biologists to understand and apply to their crystals. This article provides an explanation for structural biologists that includes only the subset of biological symmetry elements and demonstrates the application to a real-life example of an incommensurately modulated protein crystal.

How to assign a (3+1)-dimensional superspace group to an incommensurately modulated biological macromolecular crystal

How to assign a (3+1)-dimensional superspace group to an incommensurately modulated biological macromolecular crystal

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.

Superspace description of trimethyltin hydroxide at T = 100 K

Abstract At low temperatures the metalorganic compound trimethyltin hydroxide, (CH3)3SnOH, possesses a commensurately modulated crystal structure, the modulation wave vector can be described as q = 1 2 c ∗ . ${\bf{q}}\; = \;{1 \over 2}{{\bf{c}}^ * }.$ The crystal structure is studied by analysing single-crystal X-ray diffraction data within the (3+1)-dimensional superspace approach and superspace group P21212(00γ)00s. The corresponding twofold superstructure has space group symmetry P212121. The structure is characterised by polymeric chains running along c-axis, generated by Sn–O–Sn bridges between neighbouring Sn atoms and packed in a distorted hexagonal pattern and linked via C–H···O interstrand hydrogen bonds along the (orthorhombic) directions [110] and [11̅0], but not along [100].

High-resolution synchrotron x-ray powder diffraction study of the incommensurate modulation in the martensite phase ofNi2MnGa: Evidence for nearly 7M modulation and phason broadening

The modulated structure of the martensite phase of Ni2MnGa is revisited using high resolution synchrotron x-ray powder diffraction (SXRPD) measurements, which reveals higher order satellite reflections up to the 3rd order and phason broadening of the satellite peaks. The structure refinement, using the (3+1) dimensional superspace group approach, shows that the modulated structure of Ni2MnGa can be described by orthorhombic superspace group Immm(00$\gamma$)s00 with lattice parameters a= 4.21861(2) {\AA}, b= 5.54696(3) {\AA}, and c= 4.18763(2) {\AA} and an incommensurate modulation wave vector q= 0.43160(3)c*= (3/7+$\delta$)c*, where $\delta$=0.00303(3) is the degree of incommensuration of the modulated structure. Additional satellite peak broadening, which could not be accounted for in terms of the anisotropic strain broadening based on a lattice parameter distribution , has been modeled in terms of phasons using fourth rank covariant strain tensor representation for incommensurate structures. The simulation of single crystal diffraction patterns from the refined structural parameters unambiguously reveals a rational approximant structure with 7M modulation. The inhomogeneous displacement of different atomic sites on account of incommensurate modulation and the presence of phason broadening clearly rule out the adaptive phase model proposed recently by Kaufmann et al.[1] and suggests that the modulation in Ni2MnGa originates from soft-mode phonons.

(3 + 1)D superspace description of the incommensurate modulation in the premartensite phase of Ni2MnGa: a high resolution synchrotron x-ray powder diffraction study

Le Bail and Rietveld analysis of high resolution synchrotron x-ray powder diffraction (SXRPD) data shows unambiguous signatures of the failure of the commensurate 3M modulation model. Using (3 + 1) dimensional superspace group formalism, we have not only confirmed the incommensurate modulation in the premartensite phase with a modulation wavevector of q = 0.337 61(5)c* but also determined the superspace group (Immm(00γ)s00), atomic positions and amplitude of modulations for the incommensurate premartensite phase of Ni2MnGa for the first time. Our results may have important implications in the understanding of the martensitic transition and hence the magnetic field induced strains.

Crystal Structure and Thermoelectric Properties of Misfit-Layered Sulfides [Ln2S2] p NbS2 (Ln = Lanthanides)

Polycrystalline samples of misfit-layered sulfides [Ln2S2] p NbS2 (Ln = lanthanides) have been prepared using the CS2 sulfidation method. Except for Ln = Eu and Lu, single-phase samples were prepared. The compounds consist of an alternate stacking of a trigonal prism-type [NbS2] layer and a double-layered NaCl [rock salt (RS)]-type [Ln2S2] block, parallel to the c-axis. Three types of stacking modes of the [NbS2] layer and the RS-type block have been confirmed with a (3 + 1)-dimensional superspace group description: two face-centered orthorhombic structures and one face-centered monoclinic structure. Slightly cation-deficient samples with Ln = Yb exhibit the highest Seebeck coefficient S ≈ 59 μV/K. With electrical resistivity of $$\rho = 1.4 \,\hbox{m}{\Upomega}$$ cm and thermal conductivity of κ = 0.69 W/Km, this yields a dimensionless figure of merit of ZT = 0.11 at 300 K.

Equivalence of superspace groups.

An algorithm is presented which determines the equivalence of two settings of a (3 + d)-dimensional superspace group (d = 1, 2, 3). The algorithm has been implemented as a web tool findssg on SSG(3+d)D, providing the transformation of any user-given superspace group to the standard setting of this superspace group in SSG(3+d)D. It is shown how the standard setting of a superspace group can be directly obtained by an appropriate transformation of the external-space lattice vectors (the basic structure unit cell) and a transformation of the internal-space lattice vectors (new modulation wavevectors are linear combinations of old modulation wavevectors plus a three-dimensional reciprocal-lattice vector). The need for non-standard settings in some cases and the desirability of employing standard settings of superspace groups in other cases are illustrated by an analysis of the symmetries of a series of compounds, comparing published and standard settings and the transformations between them. A compilation is provided of standard settings of compounds with two- and three-dimensional modulations. The problem of settings of superspace groups is discussed for incommensurate composite crystals and for chiral superspace groups.

Thermally induced structural changes in incommensurate calcium barium niobate Ca0.28Ba0.72Nb2O6 (CBN28)

The incommensurately modulated crystal structure of relaxor ferroelectric CBN28 was refined at elevated temperatures up to 330°C in the 3+2 dimensional superspace group P4bm(αα1/2,−αα1/2). The structural modulations mainly consisting of cooperative tilting of NbO6 octahedra and an occupational modulation of the large cation site Me2 persist beyond the diffuse ferroelectric transition with slightly reduced amplitudes. A change of symmetry was not observed. Both symmetrically non-equivalent NbO6 octahedra are distorted by off center shifts of the Nb atoms in the same direction along the tetragonal c-axis. The displacements of the Nb atoms are gradually reduced with increasing temperatures until one of the Nb atoms crosses the center of its coordination polyhedron near the transition temperature TM adopting an uncompensated anti-ferroelectric configuration. This change is accompanied by enhanced thermal motions of the Nb atoms along the c-axis. Structural distortions and electric polarization do not completely vanish at TM but may persist in fluctuating polar nanodomains.

Structural phase transitions in SrRh2As2

SrRh2As2 exhibits structural phase transitions reminiscent to those of BaFe2As2, but crystallizes with three polymorphs derived from the tetragonal ThCr2Si2-type structure. The structure of \alpha-SrRh2As2 is monoclinic with a = 421.2(1) pm, b = 1105.6(2) pm, c = 843.0(1) pm and \beta = 95{\deg} and was refined as a partially pseudo meroedric twin in the space group P21/c with R1 = 0.0928. \beta-SrRh2As2 crystallizes with a modulated structure in the (3+1) dimensional superspace group Fmmm(10\gamma)\sigma 00 with the unit cell parameters a = 1114.4(3) pm, b = 574.4(2) pm and c = 611.5(2) pm and an incommensurable modulation vector q = (1, 0, 0.3311(4)). High temperature single crystal diffraction experiments confirm the tetragonal ThCr2Si2-type structure for \gamma-SrRh2As2 above 350{\deg}C. Electronic band structure calculations indicate that the structural distortion in alpha-SrRh2As2 is caused by strong Rh-Rh bonding interactions and has no magnetic origin as suggested for isotypic BaFe2As2.

Generation of (3 +d)-dimensional superspace groups for describing the symmetry of modulated crystalline structures

A complete table of (3 + 1)D, (3 + 2)D and (3 + 3)D superspace groups (SSGs) has been enumerated that corrects omissions and duplicate entries in previous tables of superspace groups and Bravais classes. The theoretical methods employed are not new, though the implementation is both novel and robust. The paper also describes conventions for assigning a unique one-line symbol for each group in the table. Finally, a new online data repository is introduced that delivers more complete information about each SSG than has been presented previously.