Powder Diffraction Intensities Research Articles

Powder diffraction intensities of α-Al2O3 calculated by traditional and DFT calculations

Powder diffraction intensities of α-Al₂O₃ calculated by traditional and DFT calculations

A noise-robust data assimilation method for crystal structure determination using powder diffraction intensity.

Crystal structure prediction for a given chemical composition has long been a challenge in condensed-matter science. We have recently shown that experimental powder x-ray diffraction (XRD) data are helpful in a crystal structure search using simulated annealing, even when they are insufficient for structure determination by themselves [Tsujimoto et al., Phys. Rev. Mater. 2, 053801 (2018)]. In the method, the XRD data are assimilated into the simulation by adding a penalty function to the physical potential energy, where a crystallinity-type penalty function, defined by the difference between experimental and simulated diffraction angles was used. To improve the success rate and noise robustness, we introduce a correlation-coefficient-type penalty function adaptable to XRD data with significant experimental noise. We apply the new penalty function to SiO2 coesite and ɛ-Zn(OH)2 to determine its effectiveness in the data assimilation method.

Convolution and deconvolutional treatment on sample transparency aberration in Bragg–Brentano geometry

Exact and approximate mathematical models for the effects of sample transparency on the powder diffraction intensity data are examined. Application of the formula based on the first-order approximation about the deviation angle is justified for realistic measurement and computing systems. The effects of sample transparency are expressed by double convolution formulas applying two different scale transforms, including three parameters, goniometer radius R, penetration depth μ−1, and thickness of the sample t. The deconvolutional treatment automatically recovers the lost intensity and corrects the peak shift and asymmetric deformation of peak profile caused by the sample transparency.

Quantitative phase analysis of amorphous components in mixtures by using the direct-derivation method

The direct-derivation (DD) method is a new technique for quantitative phase analysis (QPA) [Toraya (2016). J. Appl. Cryst. 49, 1508–1516]. A simple equation, called the intensity–composition (IC) formula, is used to derive weight fractions of individual components (w k ; k = 1–K) in a mixture. Two kinds of parameters are required as input data of the formula. One is the parameter S k , which is the sum of observed powder diffraction intensities for each component, measured in a wide 2θ range and corrected for the Lorentz–polarization factor. The other is the parameter a k −1, defined by a k −1 = M k −1∑nik 2, where M k is the chemical formula weight and n ik is the number of electrons belonging to the ith atom in the chemical formula unit. The parameter a k −1 was originally derived by using the relationship between the peak height and the integrated value of the peak at the origin of the Patterson function, implicitly assuming the presence of periodic structures like crystals. In this study, the formula has been derived theoretically from a general assemblage of atoms resembling amorphous material, and the same expression as the original formula has been obtained. The physical meaning of a k −1, which represents `the total scattering power per chemical formula weight', has been reconfirmed in the present formulation. The IC formula has been tested experimentally by using two-, three- and four-component mixtures containing SiO2 or GeO2 glass powder. In the whole-powder-pattern fitting (WPPF) procedure, incorporated into the DD method, a background-subtracted halo pattern is directly fitted as one of the components in the mixture, together with profile models for crystalline components. In the WPPF, an interaction was observed between the parameters of the background function (BGF) and the parameter for scaling the halo pattern, and this resulted in systematic deviations of w k from weighed values. The deviations were ≤0.7% in the case of binary mixtures when the BGF was fixed at the correct background height, supporting the hypothesis that the DD method is applicable to the QPA of amorphous components.

PDFgetN3: atomic pair distribution functions from neutron powder diffraction data using ad hoc corrections

PDFgetN3, a new software tool for the extraction of pair distribution functions (PDFs) from neutron powder diffraction intensity data, is described. Its use is demonstrated with constant-wavelength neutron data measured at the new powder diffractometer PEARL at the Delft University of Technology. PDFgetN3 uses an ad hoc data collection protocol similar to PDFgetX3. The quality of the resulting PDFs is assessed by structure refinement and by comparison with established results from synchrotron X-ray scattering.

Deconvolution–convolution treatment on powder diffraction data collected with CuKα X-ray and NiKβ filter

A method to remove small CuKβ peaks and step structures caused by NiK-edge absorption as well as CuKα2 sub-peaks from powder diffraction intensity data measured with Cu-target X-ray source and Ni-foil filter is proposed. The method is based on deconvolution–convolution treatment applying scale transform of abscissa, Fourier transform, and a realistic spectroscopic model for the source X-ray. The validity of the method has been tested by analysis of the powder diffraction data of a standard LaB6 powder (NIST SRM660a) sample, collected with the combination of CuKα X-ray source, Ni-foil Kβ filter, flat powder specimen and one-dimensional Si strip detector. The diffraction intensity data treated with the method have certainly shown background intensity profile without CuKβ peaks and NiK-edge step structures.

Analysis of powder diffraction data collected with synchrotron X-ray and multiple 2D X-ray detectors applying a beta-distribution peak profile model

A powder diffraction measurement system constructed on a beam-line BL5S2 at Aichi Synchrotron Radiation Center in Seto, Japan, has been modified for extensive use of two-dimensional (2D) X-ray detectors. Four flat 2D detectors are currently mounted on the movable stages on supporting rods radially attached to the 2Θ-wheel of the goniometer with the interval of 25°. The 2D powder diffraction intensity data are reduced to conventional 1D format of powder diffraction data by the method based on averaging of the pixel intensities with geometrical corrections, which also enables evaluation of standard uncertainties about the reduced intensity data. The 1D powder diffraction data of a 0.1 mm-capillary LaB6 (NIST SRM660b) sample obtained at the camera length of 340 mm have shown almost symmetric peak profile with slight asymmetry simulated by a beta-distribution profile function.

Experimental estimation of uncertainties in powder diffraction intensities with a two-dimensional X-ray detector

A method to obtain both one-dimensional powder diffraction intensities I(2θ) and statistical uncertainties σ(2θ) from the data collected with a flat two-dimensional X-ray detector is proposed. The method has been applied to analysis of the diffraction data of fine quartz powder recorded with synchrotron X-ray. The profile and magnitude of the estimated uncertainties σ(2θ) have shown that the effects of propagation of the errors in 2θ are dominant as the uncertainties about the observed intensity values I(2θ). The powder diffraction intensity data I(2θ), including nine reflection peaks have been analyzed by the Rietveld method incorporating the experimentally estimated uncertainties σ(2θ). The observed I(2θ) data have been reproduced with a symmetric peak profile function (Rwp = 0.84 %), and no significant peak shifts from calculated locations have been detected as compared with the experimental errors. The optimized values of the lattice constants of the quartz sample have nominally been estimated at a = 4.9131(4) Å and c = 5.4043(2) Å, where the uncertainties in parentheses are evaluated by the Rietveld optimization based on the estimated uncertainties σ(2θ) for intensities I(2θ). It is likely that reliability of error estimation about unit-cell dimensions has been improved by this analytical method.

Particle statistics in powder diffraction method

The statistical variation in observed powder diffraction intensity is often dominated by the limited number of crystallites that satisfy the diffraction condition. This effect has been termed with particle statistics in powder diffraction method [1]. We have reported experimental validation of the theory for particle statistics about symmetric reflection mode measurements, and proposed a new method to evaluate the crystallite size and its distribution by statistical analysis based on a generalized theoretical framework [2]. We have also demostrated improvement of structure refinement by application of the maximum likelihood estimation to statistical model explicitly taking the errors caused by the particle statistics into account [3]. As the formula suitable for the particle statistics about a continuously spinning flat specimen is still unclear, three kinds of diffraction intensity measurements, (1) in-plane (Φ)-scan, rocking angle (Ω)-scans for (2) stationary and (3) spinning flat specimens, have been conducted for standard powder samples of Si (NIST SRM640c) and lanthanum hexaboride (NIST SRM660a) at a synchrotron powder diffraction beamline BL-4B2 at the Photon Factory, KEK in Tsukuba. The statistical variance observed inΦ-scan and Ω-scan measurements for stationary specimens have shown cosecant dependence of the effective number of diffracting crystallites, just as predicted by the traditional theory. The statistical variance in Ω-scan measurements for spinning specimens has also shown a cosecant behaivior rather than the squared cosecant dependence suggested by de Wolff [1].

Thermoreversible crosslinking of maleic anhydride grafted butyl rubber with multiple hydrogen bonding arrays

Thermoreversible crosslinking supramolecular rubber with multiple hydrogen bonds are prepared via reaction of maleic anhydride grafted butyl rubber (MAn‐g‐IIR) with 3‐amino‐1,2,4‐triazole catalyzed by tetrabutyl ammonium bromide. Then, the hydrogen bonding arrays are introduced via the reaction between the obtained amide triazole‐carboxylic acid groups. Microphase‐separated aggregates that acts as physical crosslinks in the MAn‐g‐IIR is significantly tightened and strengthened after introducing multiple hydrogen bonds, resulting in increasing of X‐ray powder diffraction intensity and drastically improvement of tensile strength. After heating‐cooling treatment up to three cycles, no evidence of chemical change is observed from Fourier transform infrared spectra, and the tensile strength of thermoreversible crosslinking IIR was still much higher than that of the MAn‐g‐IIR. The reprocessability of this material was due to the thermoreversibility of hydrogen bond at elevated temperature, while the maintenance of tensile properties is due to the recovery of hydrogen bond during cooling. POLYM. ENG. SCI., 54:1783–1790, 2014. © 2013 Society of Plastics Engineers

Powder X-Ray Structure Refinement Applying a Theory for Particle Statistics

A new method for analysis of powder diffraction intensity data recently developed by the author has been modified to include the effects of possible statistical errors in the goniometer angle 2Θ. The analytical method is based on the maximum-likelihood estimation. Structure parameters refined by the method for fluorapatite Ca5(PO4)3F, anglesite PbSO4and barite BaSO4 have become closer to those obtained by single-crystal structure analyses than the results obtained by applications of a conventional Rietveld refinement to the same powder diffraction data, similarly to the previous analyses, where the errors in 2Θ are not included. The statistical errors about 2Θ are estimated at Δ2Θ = 0.0030º, 0.00099º and 0.0036º from the powder diffraction data sets of fluoroapatite, anglesite and barite, respectively.

Analytical method for observed powder diffraction intensity data based on maximum likelihood estimation

A new methodology based on maximum likelihood estimation for structure refinement using powder diffraction data is proposed. The method can not only optimize the parameters adjusted in Rietveld refinement but also parameters to specify errors in a model for statistical properties of the observed intensity. The results of structure refinements with relation to fluorapatite Ca5(PO4)3F, anglesite PbSO4, and barite BaSO4 are demonstrated. The structure parameters of fluorapatite and barite optimized by the new method are closer to single-crystal data than those optimized by the Rietveld method, while the structure parameters of anglesite, whose values optimized by the Rietveld method are already in good agreement with the single-crystal data, are almost unchanged by the application of the new method.

Generation and applications of structure envelopes for porous metal–organic frameworks

The synthesis of polycrystalline, as opposed to single-crystalline, porous materials, such as zeolites and metal–organic frameworks (MOFs), is usually beneficial because the former have shorter synthesis times and higher yields. However, the structural determination of these materials using powder X-ray diffraction (PXRD) data is usually complicated. Recently, several methods for the structural investigation of zeolite polycrystalline materials have been developed, taking advantage of the structural characteristics of zeolites. Nevertheless, these techniques have rarely been applied in the structure determination of a MOF even though, with the electron-density contrast between the metal-containing units and pore regions, the construction of a structure envelope, the surface between high- and low-electron-density regions, should be straightforward for a MOF. Herein an example of such structure solution of MOFs based on PXRD data is presented. To start, a Patterson map was generated from powder diffraction intensities. From this map, structure factor phases for several of the strongest reflections were extracted and a structure envelope (SE) of a MOF was subsequently constructed. This envelope, together with all extracted reflection intensities, was used as input to theSUPERFLIPsoftware and a charge-flipping (CF) structure solution was performed. This structure solution method has been tested on the PXRD data of both activated (solvent removed from the pores;dmin= 0.78 Å) and as-synthesized (dmin= 1.20 Å) samples of HKUST-1. In both cases, our method has led to structure solutions. In fact, charge-flipping calculations using SE provided correct solutions in minutes (6 min for activated and 3 min for as-synthesized samples), while regular charge flipping or charge flipping with histogram matching calculation provided meaningful solutions only after several hours. To confirm the applicability of structure envelopes to low-symmetry MOFs, the structure of monoclinic PCN-200 has been solvedviaCF+SE calculations.

Particle statistics in ϕ- and ω-scan powder diffraction intensity measurements

Particle statistics in ϕ- and ω-scan powder diffraction intensity measurements

Unfolding a two-dimensional powder diffraction image: conformal mapping

A new procedure aimed at unfolding a two-dimensional powder diffraction image into both a one-dimensional azimuthal and a radial scan is presented. In this approach, the sample-to-detector distance is the only parameter that must be adjusted in a separate step by using a standard sample. The technique consists of three steps: tracking the beam centre as the local maximum of the self-convolution of the original two-dimensional map, detector tilt and rotation determination by an intensity-tensor diagonalization, and azimuthal/radial intensity integration by a conformal mapping of the original two-dimensional powder diffraction image. The X-ray powder diffraction (XRPD) intensity profile of the NIST Si 640c standard sample is used to test the performance. The results show the robustness of the method and its capability of efficiently tagging the pixels in a two-dimensional readout system by matching the ideal geometry of the detector to the real beam–sample–detector frame. The technique is a fast, versatile and user-friendly tool for the simultaneous analysis of both azimuthal and radial spectra of two-dimensional XRPD images.

X-ray diffraction studies on (NaCl) x(NaBr) y−x(NaI) 1−y crystals

We have grown (NaCl) x (NaBr) y − x (NaI) 1− y crystals by the melt method for various values of x and y. Thermal parameters like Debye–Waller factor, Debye temperature, mean square amplitude of vibration and Debye frequency were determined using the X-ray powder diffraction intensity data. The results obtained are reported here.

Studies on multiphased mixed crystals grown from NaBr and KCl

We have grown multiphased binary and ternary mixed crystals by the melt method using the miscible alkali halides, viz. NaBr and KCl and physically characterized. Thermal parameters like Debye–Waller factor, Debye temperature, Debye frequency and mean square amplitude of vibration were determined using the X-ray powder diffraction intensity data. DC and AC electrical measurements were carried out by using the parallel plate capacitor method at various temperatures. Activation energies (DC and AC), mean jump frequency, compressibility and mean sound velocity were also determined. The results obtained are reported here.

KNd(MoO4)2: A New Incommensurate Modulated Structure in the Scheelite Family.

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Folding a two-dimensional powder diffraction image into a one-dimensional scan: a new procedure

A new procedure aimed at folding a two-dimensional powder diffraction pattern into a one-dimensional scan is presented. The sample-to-detector distance is the only parameter that, in this approach, needs to be adjusted in a separate step by using the `standard' sample. The technique consists of three steps: tracking the beam centre by means of simulated annealing of the diffraction rings along the same axis, detector tilt and rotation determination by a Hankel Lanczos singular value decomposition and intensity integration by an adaptive binning algorithm. The X-ray powder diffraction (XRPD) intensity profile of the standard NIST Si 640c sample is used to test the performance. Results show the robustness of the method and its capability of efficiently tagging the pixels in a two-dimensional readout system by matching the ideal geometry of the detector to the real beam–sample–detector frame. The technique is a versatile and user-friendly tool for the one-dimensional folding of two-dimensional XRPD images.

KNd(MoO4)2: A New Incommensurate Modulated Structure in the Scheelite Family

Potassium neodymium molybdenum oxide, KNd(MoO4)2, has been synthesized by the solid-state method. The structure and microstructure have been studied by X-ray powder diffraction and transmission electron microscopy (TEM). TEM revealed that the KNd(MoO4)2 structure is incommensurately modulated. The scheelite-like structure of KNd(MoO4)2 has been refined from X-ray powder diffraction intensities in the (3 + 1)D superspace group I2/b(αβ0)00 with a = 5.5202(2) Å, b = 5.33376(5) Å, c = 11.8977(3) Å, γ = 90.9591(7)°, and modulation vector q = 0.57789(4)a* − 0.14748(6)b* (RP = 3.09%, RwP = 4.04%). The ordering of the K and Nd cations appears to be the primary parameter of the structure modulations. The compositional wave of the {KMoO4} and {NdMoO4} distribution is observed in the ab structure projection. The incommensurability of the compositional wave direction with respect to the a and b lattice constants is regarded as the origin of the incommensurate structure modulations.