Dimensional Space Group Research Articles

Experimental and computational analysis of binary Fe-Sn ferromagnetic compounds

Ferromagnetic Fe3Sn, Fe5Sn3 and Fe3Sn2 single crystals were synthesized using the reactive flux technique. Derived from single crystal x-ray diffraction and Transmission Electron Microscopy (TEM), a new structural model is proposed for the Fe5Sn3 crystals - the threefold twinning of an orthorhombic unit cell with (3 + 1) dimensional space group Pbcm(α00)0s0. The spontaneous magnetization (Ms) and the anisotropy constants K1 and K2 of Fe3Sn, Fe5Sn3 and Fe3Sn2 single crystals were determined in a wide temperature range using M(H) dependencies and a modified Sucksmith-Thompson technique. Ms and K1 were also evaluated in the framework of Density Functional Theory (DFT) and an overall good agreement was observed between the calculated and experimental results. Furthermore, a critical evaluation of different analytical models for the assessment of magnetocrystalline anisotropy was performed, which are restricted to the analysis of uniaxial magnetic domain patterns, and it is shown that such high-throughput techniques can lead to unrealistic results. Finally, a DFT high-throughput screening of the Fe-Sn phase diagram was used to identify Fe-Sn based phases with potential to be stabilized upon alloying, and their magnetization and magnetocrystalline anisotropy were evaluated. The results show that a similar strong anisotropy as observed in Fe3Sn may also be found in other Fe-Sn based phases, having higher potential to be used as hard magnetic material.

A Natural 2D Heterostructure [Pb3.1Sb0.9S4][Au xTe2- x] with Large Transverse Nonsaturating Negative Magnetoresistance and High Electron Mobility.

We report the two-dimensional (2D) natural heterostructure [Pb3.1Sb0.9S4][Au xTe2- x] ( x = 0.52-0.36) which shows anomalous, transverse nonsaturating negative magnetoresistance (MR). For x = 0.52, the material has a commensurately modulated structure with alternating [Pb3.1Sb0.9S4] rocksalt layers and atomically thin [Au xTe2- x] sheets, as determined by single-crystal X-ray diffraction using a (3 + 1)-dimensional space group; for other x compositions, the modulated structure is absent and the Au and Te atoms are disordered. The transport properties in this system at low temperature (<100 K) are dominated by an unusual 2D hopping mechanism, while at room temperature a high carrier mobility of ∼1352 cm2 V-1 s-1 is obtained ( x = 0.36). The confined electrons within the [Au xTe2- x] layers are also exposed to interlayer coupling with the insulating [Pb3.1Sb0.9S4] layers, and as a result, the properties of the heterostructures emerge not only from the constituent layers but also the interactions between them. Furthermore, the various Au and Te coordination patterns found in the [Au xTe2- x] sheets as a function of x further contribute to a unique electronic structure that leads to the anomalous nonsaturating negative MR with different field dependent behaviors. First-principles calculations indicate that the [Au xTe2- x] sheets are responsible for the unusual electrical transport properties in this 2D system.

Lanthanum-strontium cuprate: A promising cathodic material for solid oxide fuel cells

Samples of lanthanum-strontium cuprate LaSrCuO3.61 are synthesized by a solid-phase method at 1473 K, in air. Electron diffraction patterns reveal low-intensity satellites whose position is described in the reciprocal space by the (3 + 2)-dimensional space group I4/mmm(αα0, α-α0)00mg. Using a five-layer cell LaSrCuO4-δ|YSZ|LaSrCuO4-δ|YSZ|LaCuO4-δ prepared by isostatic hot pressing, the ionic component of conductivity of LaSrCuO4-δ is determined (σ1179 K = 3.8 × 10−3 S cm−1), which is commensurate with other mixed conductors based on complex oxides of cobalt and iron.

Modulated Structure of the Composite Crystal Urea/n-Heptadecane

The structure of the composite crystal urea/heptadecane was determined by single-crystal X-ray diffraction. The lattice constants of the hexagonal substructures are a = 8.218 (2), b = 8.218 (3), c h = 11.017 (3) Å for the urea (host) structure and c g = 23.713 (1) Å for the heptadecane (guest) structure, respectively. The (3+ 1)-dimensional space group is P6122(00\rho)011. Refinements on 786 reflections converged to wR = 0.0232 for 608 main host reflections, wR = 0.1506 for 10 main guest reflections, wR = 0.0328 for 49 common main reflections and wR = 0.753 for 119 pure satellite reflections. A comparison of the refinement of only the urea substructure and a refinement of the whole composite crystal, including the guest subsystem and the mutual modulations of both subsystems, shows that the main reflections, previously assigned to the urea host exclusively, are affected by a nonnegligible contribution of the heptadecane satellite scattering. The modulation of the guest structure has a maximum when the CH2 groups of the heptadecane molecule are facing the channel walls at heights corresponding to troughs in this wall. Thus, the modulation of the guest structure can be interpreted by an adaptation of the guest molecules to the host structure. The modulation of the host structure was found to be very weak, as the satellite scattering of the host structure is low. \psi-scans of the 00 l reflections revealed that the observed violation of the 61 screw axis extinction rule of the host structure and guest structure modulation can be explained by considerable umweganregung.

Resolution of the 66-Fold Superstructure of DySe1.84 by X-ray Diffraction and Second-Moment Scaled Hückel Calculations

We report the superlattice of the modulated structure DySe1.84. The modulation is of the charge density wave type, associated with electronic states on selenium atoms arranged in two-dimensional square nets. The complete modulated structure can be described in the superspace group formalism in a 3+2 dimensional space group . The basic cell observed by single-crystal X-ray diffraction is orthorhombic with a = 3.9912(3) Å, b = 3.9863(1) Å, and c = 8.206(1) Å. The modulation wave vectors are q1 = αa*+βb* + 1/2c* and q2 = αa* − βb* + 1/2c*, where α = 0.33338(16) and β = 0.27284(6). Structure refinement based on first- and second-order satellites as well as main lattice reflections produced solutions in four different space groups that could not be differentiated on the basis of agreement factors. A best solution was sought by use of second-moment scaled, tight-binding band calculations. The lowest energy model found was similar to but not identical with two of those refined for X-ray data. This lowest energy structure can be rationalized by analysis of second-nearest-neighbor interactions within the selenium square nets.

Derivation of Wyckoff positions ofN-dimensional space groups. Theoretical considerations

An algorithm to calculate Wyckoff positions of n-dimensional space groups is developed and a detailed theoretical background is supplied. The algorithm is based on concepts of symmetry support and of translational normalizer

Convergent-beam electron diffraction study of incommensurately modulated crystals. II. (3+1)-dimensional space groups

The convergent-beam electron diffraction (CBED) method for determining three-dimensional space groups is extended to the determination of the (3+1)-dimensional space groups for one-dimensional incommensurately modulated crystals. It is clarified than an approximate dynamical extinction line appears in the CBED discs of the reflections caused by an incommensurate modulation. The extinction enables the space-group determination of the (3+1)-dimensional crystals or the one-dimensional incommensurately modulated crystals. An example of the dynamical extinction line is shown using an incommensurately modulated crystal of Sr2Nb2O7. Tables of the dynamical extinction lines appearing in CBED patterns are given for all the (3 + 1)-dimensional space groups of the incommensurately modulated crystal.

Diffraction analysis of incommensurate phases in crystals of quasi-binary system Sn2P2(S1−xSex)6

Abstract X-ray analysis of Sn2P2(S1 − xSex)6 single crystals with 0 < x < 1 is performed in the temperature interval 70 to 400 K in which there exist the incommensurate modulated structures. Crystallographic characteristics of the modulation wave are obtained: the direction of a wave vector q at an angle 9° with respect to the [100] direction in the (010) plane; the polarization plane (010); the transversal wave of tin atoms displacement. A description of (3 + l)-dimensional space group for incommensurate phase is given. Temperature and concentrational dependencies for crystallographic characteristics are obtained. It is shown that the Lifshitz point on the concentrational phase diagram corresponds to a composition with X = 0.28 at T = 280 K. We have analyzed the evolution of thermal expansion tensor in the range of phase transition. The temperature boundaries for incommensurate phase range are shown to be fixed by the knees of thermal expansion and inside the temperature range the “C” parameter remains unchanged (invar effect). Tin atoms displacement direction in the modulation wave is shown to be normal to the “invar” plane (010).

収束電子回折法による４次元空間群の決定

The interrelation between the symmetries of CBED patterns and symmetry symbols for incommensurately-modulated structures given by de Wolff, Jannsen and Janner are clarified. 1) Point groups; The symmetry symbol 1 shows no symmetry in CBED patterns. The symbols other than 1 show the symmetries of the corresponding symbols for the average structure. 2) Space groups; The four dimensional glide planes and screw axes give dynamical extinction in the kinematically forbidden reflections due to the symmetry elements. Dynamical extinction rules in CBED patterns were tabulated for one example of the four dimensional space groups.

Exact random-walk models in crystallographic statistics. VI. P.d.f.'s of E for all plane groups and most space groups

An exact calculation of the probability density function (p.d.f.) of magnitude of [E], the magnitude of the normalized structure factor, can be developed in terms of Fourier and Fourier-Bessel series whose coefficients can be expressed in terms of the characteristic function. This article provides the formulae for atomic contributions to such characteristic functions. The results presented in this study are applicable to all the plane groups and to 206 three dimensional space groups. Only the space groups isomorphous to the cubic point groups 432, 43m and m3m were omitted due to the complexity of the resulting expressions and the small deviations of the corresponding densities from the central-limit-theorem approximation, which have been observed in simulations for extreme atomic heterogeneities. Representative derivations illustrating the problems and techniques of their solution are provided. All the theoretical results have been computed numerically and compared with simulated distributions. Some results of these computations are illustrated in the accompanying paper, Part VII of this series [Rabinovich, Shmueli, Stein, Shashua & Weiss (1991). Acta Cryst. A47, 336-340].

Structural investigation of the incommensurate modulated phase of BaMnF 4

The incommensurate modulated structure of BaMnF 4 has been studied at 100 K by X-Ray diffractometry and described on the concept of the four dimensional spacegroup, W p2 1nb 199 The final R-factor of the refinement is 0.062 for 3501 reflections. Atomic displacements are given and discussed. The modulation involves the rotation of the MnF 6 octahedra about the a-axis and a correlated translation of the Ba-atoms in the plane (a,b). The analogy between the anisotropic temperature factors in the high temperature phase and the directions of the atomic displacements in the modulated phase has been discussed.

Refinement of the incommensurate crystal structure of NaNO2in (3+1)-dimensional space groups

Two basic models of the incommensurate (IC) structure of NaNO2, one with purely occupational modulation and one with mixed modulation, have been tested using (3 + 1)-dimensional space groups. The refinements were performed in the two superspace groups PI m_{\bar 1} ms m1 and PI 2s ms m1. The main feature of this IC structure is the presence of the weak displacive waves superposed on the main modulating waves of the occupational type. The displacements of atoms take place only along the b axis. The amplitude of the displacive wave of the Na+ ion was found to be almost twice as large (0.077 A) as that of the NO-2 molecule (0.042 A) with the occupational amplitudes approximately the same. The best refinement yields an overall R factor of 0.063 in the PI 2s ms m1 group, revealing a small phase shift between individual waves of the Na+ ion and the NO-2 molecule. There are, however, practically no phase shifts between the two kinds of modulation waves involved. All waves were found to be harmonic.

Experimental investigation of 4-dimensional superspace crystals

The symmetry of incommensurate crystals can be described by higher dimensional space groups in the so called superspace approach. The basic ideas are explained and used for showing that superspace groups provide an adequate frame for analyzing experimental results on incommensurate crystals.

Graphical Representation of n -Dimensional Space Groups

Graphical Representation of <i>n</i> -Dimensional Space Groups

Structure factor of modulated crystal structures

A general structure-factor formula for n-dimensionally modulated structures (n = 1, 2, ...) has been derived to have both density (or substitutional) and displacive modulations. This structure-factor formula covers any modulated structure described by a (3 + n)-dimensional space group. Future papers show the applications of this formula to one- and three-dimensional modulations.

Superspace groups

Superspace groups have been introduced to describe the symmetry of n-dimensional incommensurate crystal phases. The (n + d)-dimensional Euclidean space in which these groups act as space groups has a distinguished n-dimensional subspace called position space. This implies that superspace groups are not merely (n + d)-dimensional space groups but that they have additional structure. Here the mathematical structure and properties of these groups is discussed. It is shown that the usual crystallographic concepts can be generalized. Taking into account the additional structure a definition is given for the equivalence of superspace groups. From this equivalence relation follow the equivalence relations for point groups and for lattices. A constructive method is given for the derivation of all (n + d)-dimensional superspace groups, their arithmetic and geometric crystal classes and their corresponding lattices starting from a knowledge of the n- and d-dimensional crystal classes. For the major part of the theory the fact that the crystal phase is incommensurate is not essential. At the end the consequences of the incommensurability are discussed.