Displacive Modulation Research Articles

Unified (3 + 1)-dimensional superspace description of the 2212-type stair-like [Bi2Sr3Fe2O9] m [Bi4Sr6Fe2O16] family of compounds

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 [Bi(2)Sr(3)Fe(2)O(9)](m)[Bi(4)Sr(6)Fe(2)O(16)] 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.

A computationally efficient TLM thermal model in the beam displacement modulation regime

The beam displacement modulation (BDM) method is increasingly being used for examination of the uniformity or detection of non-uniformities in relatively large objects such as high power thyristor structures. The main feature of this method is that during the experiment the examined object is moving in relation to the source of excitation and in relation to the device which measures the disturbance of temperature field. Such method is fast, but its modeling is complicated. The complexity of this method lies in a fact that the source of excitation, e.g., optical radiation, moves over the surface of the object—also in a model. Used by the authors, well known and often used TLM thermal model is time consuming by the fact of possible multidimensional description of the real object. However this model enables the easy way to optimize the spectrum of excitation signal due to maximization of the temperature contrast of the observed physical properties of the object. Such possibility is particularly important in BDM regime as well as in using one device to examination objects with different thermal properties. To reduce the time of calculations in case of two- and three-dimensional models, the authors propose a novel method of replacing the moving excitation source over the surface of the object by a uniform excitation source that excite the entire object's surface at once. This method gives the same results as in the case of modeling the real (moving) source of excitation. Such approach in the case of usage 3D cylindrical model presented in the paper significantly speeds up the calculations.

Structure of (Ga2O3)2(ZnO)13 and a unified description of the homologous series (Ga2O3)2(ZnO)2 n + 1

The structure of (Ga(2)O(3))(2)(ZnO)(13) has been determined by a single-crystal X-ray diffraction technique. In the monoclinic structure of the space group C2/m with cell parameters a = 19.66 (4), b = 3.2487 (5), c = 27.31 (2) Å, and β = 105.9 (1)°, a unit cell is constructed by combining the halves of the unit cell of Ga(2)O(3)(ZnO)(6) and Ga(2)O(3)(ZnO)(7) in the homologous series Ga(2)O(3)(ZnO)(m). The homologous series (Ga(2)O(3))(2)(ZnO)(2n + 1) is derived and a unified description for structures in the series is presented using the (3+1)-dimensional superspace formalism. The phases are treated as compositely modulated structures consisting of two subsystems. One is constructed by metal ions and another is by O ions. In the (3 + 1)-dimensional model, displacive modulations of ions are described by the asymmetric zigzag function with large amplitudes, which was replaced by a combination of the sawtooth function in refinements. Similarities and differences between the two homologous series (Ga(2)O(3))(2)(ZnO)(2n + 1) and Ga(2)O(3)(ZnO)(m) are clarified in (3 + 1)-dimensional superspace. The validity of the (3 + 1)-dimensional model is confirmed by the refinements of (Ga(2)O(3))(2)(ZnO)(13), while a few complex phenomena in the real structure are taken into account by modifying the model.

Crystal Chemistry of Melilite [CaLa]2[Ga]2[Ga2O7]2: a Five Dimensional Solid Electrolyte

Melilite-type [A(2)](2)[B(I)](2)[B(II)(2)O(7)](2) gallates are promising ion conducting electrolytes for deployment in solid oxide fuel cells. Single crystals of [CaLa](2)[Ga](2)[Ga(2)O(7)](2), grown in an optical floating zone furnace, were investigated using a combination of transmission electron microscopy and single crystal X-ray diffraction. Strong anisotropic displacements of oxygen arise from the structural misfit between the interlayer Ca/La cations and the [Ga]-[Ga(2)O(7)] tetrahedral layers. A model employing two-dimensional modulation achieves bond lengths and bond angles that preserve satisfactory bond valence sums throughout the structure. The melilite belongs to the tetragonal superspace group P42(1)m(α, α, 0)00s(α, α, 0)000, α = 0.2160(5), with a subcell metric of a = 7.9383(2) Å, c = 5.2641(3) Å, onto which modulation vectors are superimposed: q(1) = α (a* + b*), q(2) = α (-a* + b*). Both displacive (cation and anion) and occupational (cation) modulations contribute to incommensuration. The analysis of structural adjustments that accompany changes in temperature and composition provides assurance that the crystal chemical model is correct. By better understanding the flexibility of this modulated structure a rational approach toward crystallochemical optimization of electrolyte performance by enhancing oxygen mobility becomes feasible.

Incommensurate modulation of calcium barium niobate (CBN28 and Ce:CBN28)

The incommensurately modulated crystal structures of Ca(0.28)Ba(0.72)Nb(2)O(6) (CBN28) and Ce(0.02)Ca(0.25)Ba(0.72)Nb(2)O(6) (Ce:CBN28) were refined in the supercentred setting X4bm(AA0,-AA0) of the 3 + 2-dimensional superspace group P4bm(aa½,-aa½). Both compounds are isostructural with a tetragonal tungsten bronze-type structure. The modulation of CBN28 consists of a wavy distribution of Ba and Ca atoms as well as vacancies on the incompletely occupied Me2 site with 15-fold oxygen coordination. The occupational modulation is coupled with a modulation of the atomic displacement parameters and a very weak modulation of the positional parameters of Me2. The surrounding O atoms show strong displacive modulations with amplitudes up to ca 0.2 Å owing to the cooperative tilting of the rigid NbO(6) octahedra. The Me1 site with 12-fold coordination and Nb atoms are hardly affected by the modulations. Only first-order satellites were observed and the modulations are described by first-order harmonics. In Ce:CBN28 cerium appears to be located on both the Me2 and Me1 sites. Wavevectors and structural modulations are only weakly modified upon substitutional incorporation of 0.02 cerium per formula unit of calcium.

Evidence of oxygen-dependent modulation in LuFe2O4

A polycrystalline sample of LuFe${}_{2}$O${}_{4}$ has been investigated by means of powder synchrotron x-ray and neutron diffraction and transmission electron microscopy (TEM), along with M\"ossbauer spectroscopy and transport and magnetic properties. A monoclinic distortion is unambiguously evidenced, and the crystal structure is refined in the monoclinic $C$2/$m$ space group [${a}_{M}$ $=$ 5.9563(1) \AA{}, ${b}_{M}$ $=$ 3.4372(1) \AA{}, ${c}_{M}$ $=$ 8.6431(1) \AA{}, \ensuremath{\beta} $=$ 103.24(1)\ifmmode^\circ\else\textdegree\fi{}]. Along with the previously reported modulations distinctive of the charge-ordering (CO) of the iron species, a new type of incommensurate order is observed, characterized by a vector $\stackrel{P\vec}{q}$${}_{1}$ $=$ ${\ensuremath{\alpha}}_{1}$${\stackrel{P\vec}{\mathrm{a}}}_{M}^{*}$ + ${\ensuremath{\gamma}}_{1}$${\stackrel{P\vec}{c}}_{M}^{*}$ (with ${\ensuremath{\alpha}}_{1}$ \ensuremath{\cong} 0.55, ${\ensuremath{\gamma}}_{1}$ \ensuremath{\cong} 0.13). In situ heating TEM observations from 300 to 773 K confirm that the satellites associated with $\stackrel{P\vec}{q}$${}_{1}$ vanish completely, only at a temperature significantly higher than the CO temperature. This incommensurate modulation has a displacive character and corresponds primarily to a transverse displacive modulation wave of the Lu cations position, as revealed by the high resolution, high angle annular dark field scanning TEM images and in agreement with synchrotron data refinements. Analyses of vacuum-annealed samples converge toward the hypothesis of a new ordering mechanism, associated with a tiny oxygen deviation from the O${}_{4}$ stoichiometry.

Incommensurate Modulated Structures in the Ta2O5 - Al2O3 System

Members of the (1 – x)Ta2O5·xAl2O3 series were synthesized, and the structures investigated using synchrotron X-ray powder diffraction and neutron powder diffraction data for the first time. Structural models were developed and refined using the Rietveld method and a [3 + 1] dimensional incommensurately modulated composite structure approach with a composition dependent modulation vector q, and superspace group Xmmm(0β0)s00. Displacive atomic modulation functions across the (1 – x)Ta2O5·xAl2O3 series were found to be very similar, and strongly resemble those for the Ta2O5–WO3 system, in line with the notion that there are structure types in higher dimensional space just as there are in 3D space. Bond valence sum calculations and bond distance plots showed that the introduction of the modulation to the structural model generally led to more favourable bond valence sum values and bond distances. Fourier difference plots were examined, and the occupational modulation of aluminium refined to determine that the aluminium atoms preferentially occupy the octahedral sites.

Five-Dimensional Incommensurate Structure of the Melilite Electrolyte [CaNd]2[Ga]2[Ga2O7]2

Melilite-type gallium oxides are potential intermediate temperature electrolytes for solid oxide fuel cells. Single crystals of [CaNd](2)[Ga](2)[Ga(2)O(7)](2) grown using an optical floating zone furnace have been investigated using transmission electron microscopy and powder and single-crystal X-ray diffraction. The anion array topologically conforms to a [(3.5.4.5)(2), 3.5.3.5] network that contains distorted pentagonal tunnels. The distortion is necessary to achieve space filling and accommodate structural misfit between the layers. Satisfactory bond lengths and angles are obtained through two-dimensional modulation in the tetragonal based plane, leading to five-dimensional symmetry in the superspace group P(4⁻)2(1)m(α,α,0)00s((a⁻)a,0)000, α = 0.2319(2), with modulation vectors q(1) = α(a* + b*) and q(2) = α(-a* + b*). Both displacive and occupational modulations are found. Through this mechanism, melilites are primed to accommodate mobile oxygen interstitials, suggesting a rational approach to crystallochemical tailoring that will enhance ionic diffusion and optimize electrolyte performance.

Determination of Displacive Modulation of Thermoelectric Ca3Co4O9 by Simultaneous Acquisitions of HAADF and MAADF Images

Extended abstract of a paper presented at Microscopy and Microanalysis 2011 in Nashville, Tennessee, USA, August 7–August 11, 2011.

Modulated anharmonic ADPs are intrinsic to aperiodic crystals: a case study on incommensurate Rb2ZnCl4.

A combination of structure refinements, analysis of the superspace MEM density and interpretation of difference-Fourier maps has been used to characterize the incommensurate modulation of rubidium tetrachlorozincate, Rb(2)ZnCl(4), at a temperature of T = 196 K, close to the lock-in transition at T(lock-in) = 192 K. The modulation is found to consist of a combination of displacement modulation functions, modulated atomic displacement parameters (ADPs) and modulated third-order anharmonic ADPs. Up to fifth-order Fourier coefficients could be refined against diffraction data containing up to fifth-order satellite reflections. The center-of-charge of the atomic basins of the MEM density and the displacive modulation functions of the structure model provide equivalent descriptions of the displacive modulation. Modulations of the ADPs and anharmonic ADPs are visible in the MEM density, but extracting quantitative information about these modulations appears to be difficult. In the structure refinements the modulation parameters of the ADPs form a dependent set, and ad hoc restrictions had to be introduced in the refinements. It is suggested that modulated harmonic ADPs and modulated third-order anharmonic ADPs form an intrinsic part, however small, of incommensurately modulated structures in general. Refinements of alternate models with and without parameters for modulated ADPs lead to significant differences between the parameters of the displacement modulation in these two types of models, thus showing the modulation of ADPs to be important for a correct description of the displacive modulation. The resulting functions do not provide evidence for an interpretation of the modulation by a soliton model.

Optimization of Recording of Thermal-Wave Images in BDM Method

The beam displacement modulation (BDM) method is very difficult to implement and has a lot of features and parameters which require optimization to improve the effectiveness of detection of thermal non-uniformities. The most important parameters that have to be taken under consideration during the optimization process are the width of the excitation source, the displacement speed of the excitation source, and the distance between the excitation and measuring spots. The width of the excitation source (e.g., diameter of laser beam) and speed of beam dislocation, in relation to the surface of the object, influence the spectra of excitation’s energy. It allows examination and visualization of thermal properties of the object at varied depths. This article presents the general scheme of a thermal-wave microscope with the implemented BDM mode and describes the process of optimization for which the assignment is to find the microscope’s settings that guarantee the highest temperature contrast of non-uniformities at the surface of the examined object.

Modulated structure of nepheline

The incommensurately modulated structure of a natural nepheline of composition K(0.54)Na(3.24)Ca(0.03)Al(3.84)Si(4.16)O(16) has been determined in superspace. The compound crystallizes in the trigonal centered superspace group X3(00γ)0 with γ = 0.2048 (10), X = (0, 0, 0, 0), (1/3, 2/3, 0, 2/3), (2/3, 1/3, 0, 1/3), a = 17.2889 (8) and c = 8.3622 (10) Å. The structure is characterized by a framework of corner-connected (Al,Si)O(4) tetrahedra. The additional cations are incorporated in two different types of channels of the framework. All atoms in the structure are displacively modulated with amplitudes below 0.1 Å. The modulation can be well described taking into account harmonics of first order only. Atomic positions in the smaller channels of the framework are fully occupied by Na(+). Cationic positions in the larger channel are occupationally modulated, yet the variation of electron density as a function of the internal coordinate t is very small and indicates that the incorporation of different types of cations (K(+), Na(+), Ca(2+)) and vacancies is realised in a highly disordered way. Average T-O distances indicate a nearly complete Al/Si ordering in the tetrahedral framework. A large part of the O atoms are approximated by split-atom positions, which are additionally affected by occupational modulation resulting in a high degree of disorder in the modulated structure. Occupational probabilities for the split-atom positions are complementary. Occupational modulations of the cations in the larger channels and the O atoms of the tetrahedral framework are coupled and correlations between occupational and displacive modulations exist.

Stable diurnal growth rhythms modulate root elongation of Arabidopsis thaliana

Arabidopsis root growth kinetics were investigated with high temporal and spatial resolution in combination with detailed statistical analysis to resolve presence of diurnal modulation of root tip displacement. In particular, high resolution video imaging was used to monitor root-tip displacement of Arabidopsis thaliana seedlings over several days. Root growth kinetics were sampled and statistically analyzed in two different photoperiods: long day LD (16 h-8 h) and equal LD (12 h-12 h) light. Diurnal root elongation kinetics exhibited five highly reproducible phases, one of these being a maximum of root growth rate displayed 1-2 h after the light on phase. Then, during the later part of the light period, root growth rate decreased. Several hours before darkening root elongation rates started to increase, with a profound decrease immediately after darkness. Subsequent to this dark-induced reduction in root growth rate the remaining part of the night was characterized by increasing growth activity. Together, tip elongation, in Arabidopsis roots is modulated by strong diurnal rhythms that are maintained in both photoperiods used and also in continuous illumination.

Structure of incommensurately modulated chromium pyrophosphate studied by Maximum Entropy Method (MEM)

A general discussion is given of the Maximum Entropy Method (MEM) as a method to determine the electron density in superspace from X-ray diffraction data. The MEM has been used to determine the electron density in superspace of incommensurately modulated chromium pyrophosphate from X-ray diffraction data published by Palatinus et al. [Acta Crystallogr. B 62, 556]. Chromium pyrophosphate, Cr2P2O7, contains ordered regions (83% of the volume) and regions with disorder. Analysis of the MEM density has allowed us to determine the displacive modulation functions within the ordered regions. The disordered regions can be described as the alternate occupation of two conformations of the pyrophosphate unit and two positions of the chromium atom. Both the conformations and the occupational probabilities of them depend continuously on the phase of modulation t.

Modulation functions of incommensurately modulated Cr2P2O7studied by the maximum entropy method (MEM)

The maximum entropy method (MEM) has been used to determine electron density in superspace of incommensurately modulated chromium pyrophosphate from X-ray diffraction data measured by Palatinus et al. [(2006), Acta Cryst. B62, 556-566]. Chromium pyrophosphate, Cr(2)P(2)O(7), contains ordered regions (83% of the volume) and regions with disorder. Analysis of the MEM density has allowed the determination of the displacive modulation functions within ordered regions. The disordered regions can be described as the alternate occupation of two conformations of the pyrophosphate group and two positions of the chromium atom, with occupational probabilities that depend continuously on the phase of modulation t. A structure model based on the interpretation of the MEM density provides a fit to the diffraction data of the same quality as the model given by Palatinus et al. (2006). The failure to find a model that better fits the data is attributed to the intrinsic inaccuracy of approximately 0.01 A for positions derived from the MEM and to the difficulties in constructing an appropriate model for the anharmonic ADPs and their modulation functions from electron densities.

Superspace description of the homologous series Ga2O3(ZnO)m

A unified description for the structures of the homologous series Ga(2)O(3)(ZnO)(m), gallium zinc oxide, is presented using the superspace formalism. The structures were treated as a compositely modulated structure consisting of two subsystems. One is constructed with metal ions and the other with O ions. The ideal model is given, in which the displacive modulations of ions are well described by the zigzag function with large amplitudes. Alternative settings are also proposed which are analogous to the so-called modular structures. The validity of the model has been confirmed by refinements for phases with m = 6 and m = 9 in the homologous series. A few complex phenomena in real structures are taken into account by modifying the ideal model.

Structural investigation of composite phases Ba1 + x [(NaxMn1 – x)O3] with x approx. 2/7, 5/17 and 1/3; exotic Mn4.5+ valence

Structural models are proposed for three composite compounds of chemical formula Ba1 + x [(NaxMn1 – x)O3] (x approx. 2/7, 5/17 and 1/3) by single crystal X-ray diffraction; superspace formalism is used to obtain an unified description of the three phases. The modulation affecting Ba atoms can be easily designed but the competition “occupational/displacive” modulations relating to the Mn/Na metallic columns were particularly difficult to modelize. However, the large amplitude of the displacive modulation affecting the oxygen atoms (approx. ±0.7 Å) in comparison with that observed for related compounds (approx. ±0.3 Å) makes it a direct consequence of the Na+ alkali insertion inside the trigonal prisms. Owing to this insertion, the Mn atoms exhibit, in the three phases, an “exotic” oxidation state of about +4.5. In addition, even if the sequence between face sharing MnO6 octahedra and NaO6 trigonal prisms can be analytically deduced from the x value, it is clear that the Na/Mn contrasts play in favour of its accurate determination through the XRD single crystal refinement.

Modulated Lanthanum Chains in the Crystal Structure of La3.65[Ru(C2)3

AbstractLanthanum ruthenium carbide La3.65[Ru(C2)3] crystallizes hexagonal (P63/m) with a = 887.9(2) pm and c = 1606.8(6) pm, V = 1096.8(4)·106 nm3, and Z = 6. The crystal structure contains trigonal‐planar Ru(C2)3 groups and lanthanum atoms, which form a partial structure of the chemical composition La3Ru(C2)3. Additional lanthanum atoms occupy hexagonal channels built by those carbon atoms of the C2 ligands, which are non‐bonded to the central ruthenium atoms. The distribution of these lanthanum atoms along the c axis results in a threefold superstructure, which was refined as a commensurate modulated structure (P63/m (00g)s0, q = (0, 0, {}^1/{}_3)). Harmonic occupational and displacive modulation functions were used for the structure refinement. The structure converged to R1 = 4.9 % for all data (R1 = 2.6 % for the main reflections, R1 = 11.1 % for the 1st order satellite reflections). A small homogeneity range for this phase is predicted by total energy calculations using the PAW‐GGA method (VASP). Chemical bonding analysis reveals that the C2 pairs are best described as acetylide C22– units.

LaSe1.85, CeSe1.83, NdSe1.83 and SmSe1.84 – four new rare earth metal polyselenides with incommensurate site occupancy and displacive modulation

Abstract Crystals of the four new binary rare earth metal polyselenides LaSe1.85, CeSe1.83, NdSe1.83 and SmSe1.84 were obtained by flux reactions. Their two-dimensionally incommensurately modulated structures were solved from X-ray diffraction data. Structure models were established in super space group X4/n(–αβ0)(βα0)00 with modulation wave vectors q 1 = –α a * + β b * and q 2 = β a * + α b * and the centering vector (0, 0, 1/2, 1/2, 1/2). The unit cell parameters at T = 293 K are a = 4.216(1) Å and c = 17.048(2) Å for LaSe1.85, a = 4.178(1) Å and c = 16.942(3) Å for CeSe1.83, a = 4.123(1) Å and c = 16.792(2) Å for NdSe1.83 and a = 4.080(1) Å and c = 16.672(2) Å for SmSe1.84; the components of the modulation wave vectors are α = β = 0.292(1) for LaSe1.85 and CeSe1.83, and α = β = 0.293(1) for NdSe1.83 and SmSe1.84, respectively. The modulation originates from a site occupancy wave caused by defects in planar selenium layers coupled to a displacive modulation.

Incommensurately modulated LT″-Ni 1+δSn ( δ=0.60, 0.63): Rietveld refinement, line-broadening analysis and structural relation with LT- and LT′-Ni 1+δSn

X-ray powder diffraction data of NiAs/Ni 2In-type Ni 1.60Sn and Ni 1.63Sn alloys annealed at or below about 573 K reveal the development of an incommensurately ordered phase called LT″. In this phase Ni(2) atoms occupy partially the trigonal–bipyramidal interstices formed by five Sn within an NiAs-type arrangement Ni(1)Sn. The modulated occupational ordering of Ni(2) in the LT″ phase can be described in the superspace group Cmcm( α00)0 s 0, and the parameters describing this occupational modulation were refined together with atomic displacement modulations using the Rietveld method. The structure parameters revealed close structural analogies of the LT″ phase with the previously reported commensurate LT-Ni 1+ δ Sn and incommensurate LT′-Ni 1+ δ Sn phases (A. Leineweber, J. Solid State Chem. 177 (2004) 1197–1212), which both occur for lower Ni contents than the LT″ phase. The 1st-order satellite reflections visible in the powder-diffraction patterns exhibit, with respect to the fundamental reflections, a considerable diffraction-line broadening, caused by a small size of the particularly ordered domains. This small-domain-size broadening was successfully described by a recently developed reflection-index ( hklm) dependent (anisotropic) line-broadening model (A. Leineweber, V. Petricek, J. Appl. Crystallogr. 40 (2007) 1027–1034) designed to consider the effect of fluctuations of the lattice metrics on the peak widths in powder diffraction patterns of incommensurately modulated crystal structures. The small domain sizes encountered for the LT″ phase indicate that domain coarsening is much more difficult than for the LT and LT′ phases. This special feature of the LT″ phase goes along with a compared to the LT and LT′ phases absent orthorhombic distortion and the low ordering temperature, which are discussed as a consequence of the ordering patterns due to the Ni(2) atoms.