Hideo Toraya

X-ray powder diffraction analysis of silver behenate, a possible low-angle diffraction standard

Silver behenate, a possible low-angle diffraction standard, was characterized using the powder diffraction technique. Diffraction patterns obtained with 1.54 A synchrotron and Cu Kα radiations showed thirteen regularly spaced (00l) peaks in the range 1.5–20.0°2θ. With the National Institute of Standards and Technologys standard reference material silicon as an internal standard, the long spacing of silver behenate was accurately determined from the profile-fitted synchrotron diffraction peaks, with d001 = 58.380 (3) A. This result was in agreement with that obtained from the Cu Kα pattern. The profile widths of the silver behenate peaks were found to be consistently larger than those of the silicon peaks, indicating significant line broadening for silver behenate. The average crystallite size along the long-spacing direction of silver behenate was estimated using the Scherrer equation, giving Davg = 900 (50) A. Because silver behenate has a large number of well defined diffraction peaks distributed evenly in the 1.5–20.0°2θ range, it is suitable for use as an angle-calibration standard for low-angle diffraction. However, care must be taken if silver behenate is to be used as a peak-profile calibration standard because of line broadening.

Extended pseudo-Voigt function for approximating the Voigt profile

The formula of the pseudo-Voigt function expressed by a weighted sum of Gaussian and Lorentzian functions is extended by adding two other types of peak functions in order to improve the accuracy when approximating the Voigt profile. The full width at half-maximum (FWHM) values and mixing parameters of the Gaussian, the Lorentzian and the other two component functions in the extended formula can be approximated by polynomials of a parameter ρ = ΓL/(ΓG + ΓL), where ΓG and ΓL are the FWHM values of the deconvoluted Gaussian and Lorentzian functions, respectively. The maximum deviation of the extended pseudo-Voigt function from the Voigt profile is within 0.12% relative to the peak height when sixth-order polynomial expansions are used. The systematic errors of the integrated intensity ΓG and ΓL, estimated by fitting the extended formula to Voigt profiles, are typically less than 1/10 of the errors arising from the application of the original formula of the pseudo-Voigt approximation proposed by Thompson et al. [J. Appl. Cryst. (1987), 20, 79–83], while the time required for computation of the extended formula is only about 2.5 relative to the computation time required for the original formula.

A new powder diffractometer for synchrotron radiation with a multiple-detector system.

A new powder diffractometer for synchrotron radiation with six detector arms has been constructed. Five detector arms are attached radially at intervals of 25 degrees to the 2theta axis and form a multiple-detector system. Five scintillation counters coupled with flat Ge(111) crystal analyzers on the respective arms can simultaneously record the whole powder pattern divided into five segments, each with an equal 2theta span. The optics design is based on flat-specimen reflection geometry using a parallel beam. The intensity data are collected using a 2theta step-scan technique in asymmetric diffraction at a fixed incident angle. A sixth multi-purpose detector arm can be used in the conventional single-arm scan mode. It can be equipped with various kinds of analyzers such as long horizontal parallel slits, a fiat or channel-cut crystal analyzer, a receiving slit and a solid-state detector. Test operations of the multiple-detector system, conducted at the Photon Factory in Tsukuba, recorded a full width at half maximum of 0.022 degrees and a peak maximum intensity of more than 40000 counts s(-1) for the (111) reflection from Si powder. The whole powder pattern of Mg(2)SiO(4) over a 2theta range of 130 degrees could be step-scanned at a step interval of 0.004 degrees (2theta) in just 4 h. Results of whole-powder-pattern decomposition and Rietveld refinement of the Mg(2)SiO(4) pattern are given.

Estimation of statistical uncertainties in quantitative phase analysis using the Rietveld method and the whole-powder-pattern decomposition method

Formulae for estimating statistical uncertainties in quantitative phase analysis using the Rietveld method and the whole-powder-pattern decomposition method have been derived. The relative magnitude of statistical uncertainty for a derived weight fraction of a component in a mixture is given by σ(Wm)/Wm = (1/Wm − 1)1/2F(D\textstyle\sum_{i = 1}^NYoi)−1/2, where Wm is the weight fraction of the mth component, F is the goodness-of-fit index, D (≤1) is a factor depending on the degree of peak overlap, and ∑Yoi is the total sum of profile intensities in the 2θ range used for whole-powder-pattern fitting. If the step width Δ2θ in step scanning is halved, ∑Yoi is almost doubled; on the other hand, ∑Yoi is proportional to the fixed counting time T. Therefore, σ(Wm)/Wm ∝ (Δ2θ/T)1/2. Extension of the 2θ range for whole-powder-pattern fitting towards the high-angle region is not effective for improving the precision of the derived weight fractions if the profile intensities in that region are weak. The formulae provide guidelines for optimizing experimental parameters in order to obtain a required precision.

Deconvolution of the instrumental functions in powder X-ray diffractometry

A novel method to deconvolute the instrumental aberration functions from the experimental powder X-ray data has been developed. The method is based on the combination of scale transformation, interpolation of data and fast Fourier transformation. The effects of axial divergence, flat specimen, sample transparency and spectroscopic profile of the source X-ray are eliminated from the entire observed diffraction pattern in three-step operations. The errors in the deconvoluted data propagated from the statistical uncertainty in the source data are approximated by the reciprocal of the square root of the correlation between the reciprocal of the variance in the source data and the squared instrumental function. The deconvolution of the instrumental aberration functions enables automatic correction of peak shift and line broadening, and supplies narrow and symmetric peak profiles for a well crystallized sample, which can be fitted by a simple model function. It will be useful in preparatory data processing for precise line profile analysis, accurate determination of lattice parameters and whole pattern fitting for crystal structure analysis.

Intensity Enhancement in Asymmetric Diffraction with Parallel-Beam Synchrotron Radiation

An intensity enhancement obtained from asymmetric diffraction with a fixed incident angle α has been studied. Parallel-beam synchrotron radiation with λ = 1.54 A (Stanford Synchrotron Radiation Laboratory) and λ = 1.53 A (Photon Factory) was used to collect powder diffraction patterns of Si, CeO2 (α = 5 and 10°) and monoclinic ZrO2 (α = 10°). The synchrotron-radiation data were analyzed using single-reflection profile fitting and whole-powder-pattern fitting techniques. The integrated intensities in the asymmetric diffraction were compared with those of symmetric diffraction obtained by the conventional θ–2θ scanning technique. An intensity, after correction for a limited height of counter aperture, was enhanced by factors of 1.8 (α = 5°) and 1.7 (α = 10°) at the maximum in asymmetric diffraction and its magnitudes agreed well with those calculated from theory.

Weighting Scheme for the Minimization Function in Rietveld Refinement

A new weighting scheme for the minimization function used in the Rietveld method for structure refinement using powder diffraction data is proposed. It has the form w = 1/Yeo with e ≃ 2 (Yo = observed profile intensity) and gives relatively heavier weights on weak intensities than weighting schemes currently used in the form w = 1/Yo in Rietveld refinement. The weight function was tested by using X-ray diffraction data-sets of Ca5(PO4)3F, α-SiO2, Mg2SiO4 and monoclinic ZrO2 powders measured with laboratory sources and synchrotron radiation. By using the new weighting scheme, the accuracy of positional parameters of the test sample was significantly improved relative to the weight function 1/Yo, which weights the medium and strong intensities more heavily, is in accordance with statistical theory and gives a better overall fit between the observed and calculated powder patterns. Plots of (w(Yo − Yc)2) (Yc = calculated profile intensity) against the groups of Yo give a uniform distribution for the present weighting scheme. Weighted difference plots w1/2(Yo − Yc) can be used for examining the correctness of the model instead of conventional (Yo − Yc) plots. The use of high-resolution powder diffraction data and the proper weighting of profile intensity in the least-squares refinement are important for obtaining accurate structural parameters in Rietveld refinement.

Crystal structure refinement of α-Si3N4 using synchrotron radiation powder diffraction data: unbiased refinement strategy

The crystal structure of α-silicon nitride (Si3N4) was refined by the Rietveld method using synchrotron radiation powder diffraction data (wavelength = 1.2 A) collected at station BL-4B2 in the Photon Factory. A refinement procedure that adopted a new weight function, w = 1/Y_o^e (Yo is the observed profile intensity and e ≃ 2), for the least-squares fitting [Toraya (1998). J. Appl. Cryst. 31, 333–343] was studied. The most reasonable structural parameters were obtained with e = 1.7. Crystal data of α-Si3N4: trigonal, P31c, a = 7.75193 (3), c = 5.61949 (4) A, V = 292.447 (3) A3, Z = 4; Rp = 5.08, Rwp = 6.50, RB = 3.36, RF = 2.26%. The following five factors are considered equally important for deriving accurate structural parameters from powder diffraction data: (i) sufficiently large sin θ/λ range of >0.8 A−1; (ii) adequate counting statistics; (iii) correct profile model; (iv) proper weighting on observations to give a uniform distribution of the mean weighted squared residuals; (v) high-angular-resolution powder diffraction data.

Simulated annealing structure solution of a new phase of dicalcium silicate Ca2SiO4 and the mechanism of structural changes from α-dicalcium silicate hydrate to αL′-dicalcium silicate via the new phase

A new phase of dicalcium silicate (Ca2SiO4) was formed by heating α-dicalcium silicate hydrate [α-Ca2(SiO4H)OH = α-C2SH] at temperatures of ∼663–763 K, and it was transformed into \alpha _{\rm L} ^\prime-Ca2SiO4 (= \alpha _{\rm L} ^\prime-C2S) above ∼1193 K. The crystal structure of the new phase (hereafter called x-C2S) has been determined by simulated annealing and refined by the Rietveld method using synchrotron radiation powder diffraction data. The structure consists of isolated SiO4 tetrahedra and a three-dimensional CaOn polyhedral network, forming a new structural type of dicalcium silicate. A structural change from α-C2SH to x-C2S is compelled by large displacements of SiO4 tetrahedra, accompanied by dehydration, in the direction perpendicular to the two-dimensional Ca(O,OH)n polyhedral network in α-C2SH. With increasing temperature, sizes of CaOn polyhedra in x-C2S become too large to confine Ca atoms at the sixfold to eightfold coordination sites. Then the structure of x-C2S is transformed into \alpha _{\rm L} ^\prime-C2S, having eightfold to tenfold coordination sites for the Ca atoms.

Precision Lattice-Parameter Determination of (Mg,Fe)SiO3 Tetragonal Garnets

The tetragonal garnet (Mg,Fe)SiO3 is a high-pressure phase of pyroxene that is thought to be a major constituent of the earths upper mantle. Its crystal structure is similar to that of cubic garnet, but it is slightly distorted to tetragonal symmetry so that its x-ray powder diffraction pattern shows a very small line splitting. A suite of tetragonal garnets with different compositions in the MgSiO3-rich portion of the MgSiO3-FeSiO3 system was synthesized at about 20 gigapascals and 2000�C. The lattice parameters a and c of quenched samples were determined by whole-powder-pattern decomposition analysis of Fe Kα x-ray powder diffraction data, which has the capacity to resolve to a high degree heavily overlapping reflections. It was found that the lattice parameters can be obtained from the following equations; a (in angstroms) = 11.516 + 0.088x and c (in angstroms) = 11.428 + 0.157x, where x, teh mole fraction of FeSiO3, is 0.0 ≤ x ≤ 0.2.