Individual Profile Fitting Research Articles

Quantitative Phase Analysis by X-ray Diffraction—Doping Methods and Applications

X-ray powder diffraction is an ideal technique for the quantitative analysis of a multiphase sample. The intensities of diffraction lines of a phase in a multiphase sample are proportional to the phase fraction and the quantitative analysis can be obtained if the correction for the absorption of X-rays in the sample is performed. Simple procedures of quantitative X-ray diffraction phase analysis of a multiphase sample are presented. The matrix-flushing method, with the application of reference intensities, yields the relationship between the intensity and phase fraction free from the absorption effect, thus, shunting calibration curves or internal standard procedures. Special attention is paid to the doping methods: (i) simultaneous determination of the fractions of several phases using a single doping and (ii) determination of the fraction of the dominant phase. The conditions to minimize systematic errors are discussed. The problem of overlapping of diffraction lines can be overcome by combining the doping method (i) and the individual profile fitting method, thus performing the quantitative phase analysis without the reference to structural models of particular phases. Recent suggestions in quantitative phase analysis are quoted, e.g., in study of the decomposition of supersaturated solid solutions—intermetallic alloys. Round Robin on Quantitative Phase Analysis, organized by the IUCr Commission on Powder Diffraction, is discussed shortly. The doping methods have been applied in various studies, e.g., phase transitions in titanium dioxide, biomineralization processes, and phases in intermetallic oxide systems and intermetallic alloys.

A new method for quantitative phase analysis using X-ray powder diffraction: direct derivation of weight fractions from observed integrated intensities and chemical compositions of individual phases

A new method for the quantitative phase analysis of multi-component polycrystalline materials using the X-ray powder diffraction technique is proposed. A formula for calculating weight fractions of individual crystalline phases has been derived from the intensity formula for powder diffraction in Bragg–Brentano geometry. The integrated intensity of a diffraction line is proportional to the volume fraction of a relevant crystalline phase in an irradiated sample, and the volume fraction, when it is multiplied with the chemical formula weight, can be related to the weight fraction. The structure-factor-related quantity in the intensity formula, when it is summed in an adequate 2θ range, can be replaced with the sum of squared numbers of electrons belonging to composing atoms in the chemical formula. Unit-cell volumes and the number of chemical formula units are all cancelled out in the formula. Therefore, the formula requires only single-measurement integrated intensity datasets for the individual phases and their chemical compositions. No standard reference material, reference intensity ratio or crystal structure parameter is required. A new procedure for partitioning the intensities of unresolved overlapped diffraction lines has also been proposed. It is an iterative procedure which starts from derived weight fractions, converts the weight fractions to volume fractions by utilizing additional information on material densities, and then partitions the intensities in proportion to the volume fractions. Use of the Pawley pattern decomposition method provides more accurate intensity datasets than the individual profile fitting technique, and it expands the applicability of the present method to more complex powder diffraction patterns. Test results using weighed mixture samples showed that the accuracy, evaluated by the root-mean-square error, is comparable to that obtained by Rietveld quantitative phase analysis.

Symmetrization of diffraction peak profiles measured with a high-resolution synchrotron X-ray powder diffractometer

The asymmetry of diffraction peak profiles observed with a high-resolution synchrotron powder X-ray diffractometer has been successfully removed by a double deconvolution method. In the first step, the asymmetry caused by the axial divergence aberration of the diffractometer is removed by a whole-pattern deconvolution method based on ana prioritheoretical model for the aberration. In the second step, the residual asymmetry, the origin of which can be ascribed to the aberrations of the beamline optics, is also removed by a whole-pattern deconvolution method, based on an empirical model derived from the analysis of experimental diffraction peak profiles of a standard Si powder (NIST SRM640b). The beamline aberration has been modelled by the convolution of a pseudo-Voigt or Voigt function with an exponential distribution function. It has been found that the angular dependence of the asymmetry parameter in the exponential function is almost proportional to tanθ, which supports the idea that the residual asymmetry should be ascribed mainly to the intrinsic asymmetry in the spectroscopic distribution of the source X-ray supplied by the beamline optics of the synchrotron facility. Recently developed procedures of whole-pattern deconvolution have been improved to treat the singularity of the instrumental function in the measured angular range. Formulae for the whole-pattern deconvolution based on the Williamson–Hall-type dependence of the width parameter of the instrumental function have also been developed. The method was applied to the diffraction intensity data of a standard ZnO powder sample (NIST SRM674) measured with a high-resolution powder diffractometer on beamline BL4B2at the Photon Factory. The structure parameters of ZnO were refined from the integrated peak intensities, which were extracted by an individual profile fitting method applying symmetric profile models. The refined structure parameters coincide fairly well with those obtained from single-crystal data.

New Method for Estimation of the Magnesium Fraction in Magnesian Calcite

A new method for estimation of the magnesium fraction in magnesian calcite is described. It involves measuring the XRD pattern of magnesian calcite in a narrow 20 range and individual profile fitting of diffraction lines 113 and 202. The intensity ratio I113/I202 is linearly correlated with the Mg fraction. The method resulted from an XRD study of adult sea urchins Sterechinus neumayeri.

An X-ray diffraction study of the effects of heat treatment on bone mineral microstructure

A series of human cortical bone specimens has been heated to temperatures up to 1200°C and the mineral content examined in detail by X-ray diffraction. Line profile analysis of the diffraction data has been undertaken to characterise the microstructural (crystallite size and microstrain) features of the mineral at each temperature. Individual profile fitting of several maxima from each diffractogram has also provided precise lattice parameters of the apatite at each temperature. The apatite did not show any significant decomposition over the temperature range although CaO was increasingly formed at temperatures above 600°C. Both finite crystallite size and microstrain contributed significantly to the diffraction peak broadening below 600°C. When heated to >800°C, the small, rod-like mineral crystallites changed from a highly anisotropically strained state to one with significantly larger equidimensional crystals possessing little microstrain. The findings are discussed in the context of graft bone substitutes and surgical heating of bone.

Estimation of errors in the measurement of unit-cell parameters. I. Statistical uncertainties of peak positions of powder diffraction lines determined by individual profile fitting

A formula for estimating the statistical uncertainty of peak positions, determined by individual profile fitting, has been derived. The magnitude of the statistical uncertainty is given by (sum of peak profile intensities)−1/2× (full width at half-maximum) ×F×G×Q× (goodness-of-fit index). The factorFdepends on the shape of the diffraction profile; it becomes smaller by 40% when the profile shape changes from Lorentzian to Gaussian. The factorGrepresents the parameter correlation between the peak position and the parameter for profile asymmetry; it becomes unity when the profile shape is symmetric or the parameter for profile asymmetry is fixed during the least-squares fitting. The factorQdepends on the peak-to-background ratio. The formula was experimentally verified by using diffraction data sets for CeO2and Si powders. The statistical uncertainty is essentially based on the counting statistics and profile width. Therefore, the peak position of a strong and sharp peak in the low-angle region can be determined more precisely than that of a weak and broadened peak in the high-angle region. The formula provides a guideline for optimizing experimental parameters for required precision.

Anisotropic diffraction peak broadening and dislocation substructure in hydrogen-cycled LaNi5and substitutional derivatives

Anisotropic peak broadening in hydrogen-cycled hexagonal LaNi5and substitutional derivatives has been studied by means of synchrotron powder diffraction. The data have been analysed by a local lattice parameter variation method implemented in a Rietveld code and by an individual profile fitting using a dislocation peak broadening model. Two main dislocation systems, both with Burgers vector 1/3〈\bar{2}110〉, are activated by misfit of the lattice parameters between the intermetallic compounds and their hydrides. Two types of diffraction peak broadening effect were observed as a function of the substitution in LaNi5: (i) a decrease or disappearance of the broadening related to the decrease of the total dislocation density and (ii) a change in the anisotropy of the broadening related to the change of the nature of the dislocation system involved. The latter effect was attributed to a change in shape of the hydride precipitates.

多目的パターン・フィッティングシステムＲＩＥＴＡＮ‐２０００とその微細孔物質への応用

A heavy-duty system of high performance, RIETAN-2000, has recently been developed for Rietveld refinement, Le Bail refinement, whole-pattern fitting based on a maximum-entropy method (MEM), and individual profile fitting. An original feature called partial profile relaxation is applicable to (nearly) isolated reflections with relatively large residuals, which leads to better fits in these reflections. The state-of-the-art technology of MEM-based wholepattern fitting allows us to represent positional disorder, chemical bonds, and anharmonic thermal motion more adequately than the conventional Rietveld method. Four examples of applying RIETAN-2000 to microporous materials are introduced to demonstrate its high ability to analyze structural details.

A Polycrystalline Thin Film Diffractometer for Asymmetric Diffraction Using Parallel Beam and High Resolution Parallel Slits

A polycrystalline thin film diffractometer for asymmetric diffraction at a fixed incident angle using parallel beam and high-resolution-parallel slits has been developed. Well collimated CuK α1 incident beam was obtained by coupling of a rotating anode generator with a channel-cut crystal monochromator. The diffracted beam was analyzed with parallel slits with 0.17° angular aperture. The observed profiles of CeO2 standard powder sample were analyzed by using an individual profile fitting technique. The minimum full-width at half-maximum (FWHM) was 0.15° in the 20 range of 40° ∼ 70°, and the variation of FWHM with 20 was almost independent of the incident angle. Intensity in asymmetric diffraction was enhanced compared to that in the symmetric 0-20 scan, and its variation was well in accordance with the theory. The use of the parallel slits compromizes the intensity and resolution for polycrystalline thin film diffraction in the present setup, but gives the high quality of diffraction data for structural study.

特集 粉末回折法の新しい展開 １． 粉末回折法の新しい展開

A brief history of the new evolution of the powder diffraction method is described in this chapter. The advent of the Rietveld method gave a great impact on the traditional powder diffraction method. It was just a small step, according to Dr, H.M. Rietveld, to refrain from the integrated intensity and to use a profile intensity as an observation. Since then, however, the Rietveld method has been used to refine a number of crystal structures by using powder diffraction data, and it is now an indispensable technique in materials research. It also stimulated a development of the method such as the Pawley method, and induced interactions with other techniques such as the individual profile fitting. A rush of conferences held in a recent few years indicates an increasing interest of the people in powder diffraction.

特集 粉末回折法の新しい展開 ６． プロファイル関数とパターン分解法

The first half of this chapter is devoted for describing: 1) a brief history of profile functions in both neutron and X-ray powder diffractions, 2) the relation between the Voigt and pseudo-Voigt functions and that between the pseudo-Voigt and Pearson VII functions, 3) the methods of asymmetrizing the symmetric profile function, and 4) the effect of profile-function truncation in powder pattern fitting. In the latter half of this chapter, the theory and application of the pattern decomposition method are described. Examples are pattern decompositions by using the individual profile fitting method and the Pawley method, The use of convolution function in pattern fitting derives the physically meaningful information from the profile shape. A sophisticated example of whole-powder-pattern fitting using a lattice-direction-dependent profile-width-model is given.