Sample transparency aberration in Bragg–Brentano geometry affected by interference with opaque and translucent sample holders has been formulated. The formulation for an opaque sample holder should be classified to 5 cases, depending on the apparent diffraction angle, beam width, specimen width, and specimen thickness. The cumulants of the aberration function for a translucent sample holder with an arbitrary linear attenuation coefficient can numerically be evaluated by a Gauss–Legendre quadrature. The use of a function defined by the convolution of truncated exponential and rectangular functions has been tested as the model for the aberration function. A double deconvolutional treatment (DCT) designed to cancel the effects of the first and third order cumulants of the aberration function has been applied to the XRD data of Si standard powder, NIST SRM640d. The diffraction peak profile in the data treated by the DCT method certainly shows improved symmetry. The main features of the symmetrized peak profile in the DCT data have been simulated by instrumental and specimen parameters. It is suggested that the current analytical method could be utilized for texture analysis, if the manufacturer of an XRD instrument should supply a more accurate information about the instrument.

Least-squares analysis on the diffraction intensity values certified for NIST SRM676a and SRM1976c α-Al2O3 (corundum) have shown that the intensities of SRM1976c can be simulated by the March-Dollase preferred orientation model along the (001)-direction. Diffraction intensities of randomly oriented corundum crystallites have been calculated from electron density data obtained by conventional and density functional theory (DFT) calculations, on the assumption of independent and similar atomic displacements for Al and O atoms. The results of DFT calculations support that the strongest peak of randomly oriented α-Al2O3 crystalline powder should be 113-reflection, though the intensities simulated by DFT calculations are not closer to NIST SRM676a intensities than those expected for a fully ionized model ${\rm Al}_2^{3 + } {\rm O}_3^{2-}$. Diffraction data of two types of relatively fine (nominally 2–3 μm and ca 0.3 μm) α-Al2O3 powder have been collected and processed by a deconvolutional treatment (DCT). Integrated peak intensities extracted from the DCT data by an individual peak profile fitting method also support that the 113-reflection is the strongest reflection of randomly oriented α-Al2O3 crystalline powder.

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.

A mathematical system for modeling the effects of symmetrized instrumental aberrations has been developed. The system is composed of the truncated Gaussian, sheared Gaussian, and Rosin-Rammler-type functions. The shape of the function can uniquely be determined by the standard deviation and kurtosis. A practical method to evaluate the convolution with the Lorentzian function and results of application to the analysis of experimental powder diffraction data are briefly described.

The application of continuous-scan integration to collect X-ray diffraction data with a Si strip X-ray detector (CSI-SSXD) introduces additional effects on the peak shift and deformation of peak shape caused by the equatorial aberration. A deconvolutional method to correct the effects of equatorial aberration in CSI-SSXD data is proposed in this study. There are four critical angles related to the effects of spillover of the incident X-ray beam from the specimen face in the CSI-SSXD data. Exact values of cumulants of the equatorial aberration function are efficiently evaluated by 4 × 4 point two-dimensional Gauss–Legendre integral. A naïve two-step deconvolutional method has been applied to remove the effects of the first and third-order cumulants of the equatorial aberration function from the observed CSI-SSXD data. The performance of the algorithm has been tested by analyses of CSI-SSXD data of three LaB6 powder specimens with the widths of 20, 10, and 5 mm, collected with a diffractometer with the goniometer radius of 150 mm.

A deconvolutional method for preprocessing powder diffraction data has been improved. The cumulants of instrumental aberration functions of Bragg-Brentano (Parrish) diffractometer calculated up to the fourth order are presented. The treatments of axial-divergence aberration and the effective spectroscopic profile of the source X-ray have been simplified from those used in previous methods. The current method has been applied to powder diffraction data collected with a Cu-target X-ray tube, used over 20 years, and a Ni-foil Kβ 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.

Four series of small parasite peaks observed in powder diffraction data recorded with a Cu-target X-ray tube and a Ni filter on the diffracted beam side in Bragg–Brentano geometry are investigated. One series of the parasite peaks is assigned to the tungsten Lα-emission. Other three types of the parasite peak series are likely to be caused by the K-emissions of Ni, but the peak locations are deviated from those predicted by the Bragg's law. An empirical formula to locate the parasite peaks and a method to remove them from observed powder diffraction data are proposed. The method is based on the whole-pattern deconvolution–convolution treatment on the transformed scale of abscissa. The parameters optimized for the diffraction data measured for Si powder has been applied on treatment of the data of LaB6 powder recorded under the same experimental conditions. It has been confirmed that the parasite peaks in the observed data can effectively be removed by the deconvolution treatment with parameters determined by a reference measurement.

An improved method to correct observed shift and asymmetric deformation of diffraction peak profile caused by the axial-divergence aberration in Bragg–Brentano geometry is proposed. The method is based on deconvolution–convolution treatment applying scale transform of abscissa, Fourier transform, and cumulant analysis of an analytical model for the axial-divergence aberration. The method has been applied to the powder diffraction data of a standard LaB6 powder (NIST SRM660a) sample, collected with a one-dimensional Si strip detector. The locations, widths and shape of the peaks in the deconvolved–convolved powder diffraction data have been analyzed. The finally obtained whole powder diffraction pattern ranging from 10° to 145° in diffraction angle has been simulated by the Pawley method applying a symmetric Pearson VII peak profile model to each peak with ten background, two peak-shift, three line-width, and two peak-shape parameters, and the Rp value of the best fit has been estimated at 4.4%.

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.

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 (R wp = 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.

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.

The Monte Carlo method is applied to finding missing atoms in solving inorganic crystal structures without applying a rigid-body approximation. Whole powder patterns of α-SiO2 and Mg2SiO4 were used for testing a procedure. Four atoms among the six in the asymmetric unit of Mg2SiO4 could be found in the present analysis. The use of well-refined profile parameters enhanced the frequency of correct structure configurations in the Monte Carlo search. Utilizing structural information available for constructing a trial configuration is also considered to be important for efficiently searching the structure solution. A procedure for assignment of equivalent positions to respective atoms is presented. The method can be used as a powerful tool for finding missing atoms in a partially solved structure. A histogram of weighted reliability index in Monte Carlo calculations is very informative for evaluating the performance of the method. ©