H. Hibino

Deconvolution of instrumental aberrations for synchrotron powder X-ray diffractometry

A method to remove the effects of instrumental aberrations from the whole powder diffraction pattern measured with a high-resolution synchrotron powder diffractometer is presented. Two types of asymmetry in the peak profiles caused by (i) the axial-divergence aberration of the diffractometer (diffractometer aberration) and (ii) the aberration of the monochromator and focusing optics on the beamline (beamline aberration) are both taken into account. The method is based on the whole-pattern deconvolution by Fourier technique combined with the abscissa-scale transformation appropriate for each instrumental aberration. The experimental powder diffraction data of LaB6 (NIST SRM660) measured on beamline BL-4B2 at the Photon Factory in Tsukuba have been analysed by the method. The formula of the scale transformation for the diffractometer aberration has a priori been derived from the instrumental function with geometric parameters of the optics. The strongly deformed experimental peak profiles at low diffraction angles have been transformed to sharp peak profiles with less asymmetry by the deconvolution of the diffractometer aberration. The peak profiles obtained by the deconvolution of the diffractometer aberration were modelled by an asymmetric model profile function synthesized by the convolution of the extended pseudo-Voigt function and an asymmetric component function with an empirical asymmetry parameter, which were linearly dependent on the diffraction angle. Fairly symmetric peak profiles have been obtained by further deconvolution of the empirically determined asymmetric component of the beamline aberration.

Peak profile function for synchrotron X-ray diffractometry

A formula of the instrumental function for a high-resolution synchrotron X-ray diffractometer, equipped with a flat crystal analyser and a set of Soller slits for limiting the axial divergence of the diffracted beam, has been derived. The formula incorporates the effects of (i) the axial divergence of the diffracted beam limited by the Soller slits, (ii) the Bragg angle of the flat crystal analyser, and (iii) the tilt angle defined as the deviation of the normal direction of the analyser face from the goniometer plane. The model profile function given by the convolution of a Lorentzian function with the instrumental function has been applied to fit the experimental diffraction peak profiles of standard Si powder (NIST SRM640b) measured with a high-resolution synchrotron X-ray diffractometer, MDS, on beamline BL4B2 at the Photon Factory in Tsukuba. The convolution has been calculated by applying an efficient algorithm for numerical integration. The profile function reproduces not only the experimental profiles measured with a well aligned crystal analyser, but also significantly distorted profiles arising from misalignment of the analyser, with Rp values within 1.4%, by varying only the instrumental parameter for the tilt angle. It is suggested that further convolution with a Gaussian distribution is practically not necessary for the model instrumental function to fit the data collected with MDS. More rapid computation can be achieved by applying an analytical formula of the profile function, when the tilt angle of the crystal analyser is within about 0.2°.

Evaluation of particle statistics in powder diffractometry by a spinner‐scan method

The uncertainty in measured diffraction intensities caused by particle statistics, which originates from the limited number of crystallites satisfying the diffraction condition, has been evaluated by a step-scan measurement about the rotation angle of a specimen-spinning attachment of a laboratory powder X-ray diffractometer. The residual statistical variance of the spinner-scan intensity data, after subtraction of periodic drift and variance caused by counting statistics, was assigned to the variance caused by particle statistics. Particle statistics for a standard Si powder (NIST SRM640c) and three size fractions (nominally 3–7, 8–12 and 18–22 µm in Stokes diameter) of quartz powder separated by a sedimentation method have been analysed by scanning electron microscopy (SEM) and the spinner-scan method using a powder X-ray diffractometer. It has been confirmed that the observed ratio of the squared diffraction-peak intensity to the variance caused by particle statistics is proportional to the multiplicity of reflections predicted by the crystal structure. The spinner-scan intensity data for the standard Si powder (NIST SRM640c), the effective particle diameter of which was estimated at 5.6 µm by SEM image analysis, was used as the standard for crystallite-size evaluation of quartz powder based on analysis of spinner-scan data. The effective crystallite diameters of the three quartz powder samples have been estimated at 6.5 (2), 11.7 (2) and 22.8 (2) µm by the analysis of the spinner-scan data, while the effective particle diameters evaluated by SEM image analysis are 7.1, 12 and 25 µm, respectively. Other possible applications of the analysis of particle statistics based on the spinner-scan method are also discussed.

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 an a priori theoretical 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 BL4B2 at 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.

Kα1–Kα2 characteristic of a parabolic graded multilayer

Line shapes of the Kα1–Kα2 doublet beam reflected from a parabolic graded multilayer (PGM) were analysed by ray tracing and rocking-curve measurements using an Si(400) flat single crystal. The integrated intensity and the intensity ratio of Kα2 to Kα1 of the reflected beam vary with the angle of incidence at the PGM. The rates of these variations are considered to increase with increasing spectral resolution of the PGM. The Kα1 and Kα2 beams are reflected from the PGM in slightly different directions. Therefore, the angular separation between the Kα1 and Kα2 peaks of the observed diffraction profile of a sample becomes smaller than that calculated from the two wavelengths for Kα1 and Kα2 when the PGM and the sample are arranged in the (+−) setting, and vice versa when they are in the (++) setting. The magnitude of the shift of the angular separation is close to the experimental uncertainty in the determination of the peak positions when the PGM consists of W/Si bilayers, whereas it is estimated to be three times as large when a PGM of high spectral resolution is used.

Quantitative basis for the rocking-curve measurement of preferred orientation in polycrystalline thin films

A quantitative basis for the rocking-curve measurement of the preferred orientation in polycrystalline thin films is presented. Gaussian functions are used for modeling the density distribution of the normals to the crystal plane around the normal to the specimen surface. An intensity formula for the rocking curve is derived from the kinematical theory applied to the case of asymmetric Bragg reflection. The density distribution is determined by the least-squares fit of a theoretical rocking curve to the observed curve, and a volume fraction of crystallites, whose normals to the crystal plane are present within a defined angular range, can be obtained from it. AlN and Au polycrystalline thin films were used for testing the present procedure. Parameter values of the model function, refined using both synchrotron radiation and laboratory X-rays, agree well with each other within the experimental errors although these intensity data sets were collected under different experimental conditions in instrumentation and wavelength. A distribution of depth-dependent preferred orientation in the AlN thin film was revealed by using double-layer and multiple-layer models. A very small degree of preferred orientation in Au thin films could also be measured. Parallel-beam optics and integrated intensities instead of peak height intensities are important for reliable rocking curve measurement.

Evaluation of crystallite size distribution by a capillary spinner-scan method in synchrotron powder diffractometry

A method to evaluate the effects of particle statistics in capillary-specimen transmission mode x-ray diffraction measurements has been developed. Average crystallite size of about several ?m and dispersion of crystallite size distribution have been evaluated by statistical analysis of spinner-scan diffraction intensity data. The method can be applied to polycrystalline materials and also multi-phase mixtures.

Statistical properties of measured X-ray intensities affected by counting loss of detection system

Asx Te100 x glasses with x ≤ 40 show single stage crystallization and those with x ≥ 40 exhibit a double stage crystallization and at x = 40, this is associated with “rigidity percolation” and “chemical stoichiometric ordering”. In the present study the effect of pressure on the thermal crystallization of Asx Te100 x, Asx Te100 x y Se y glasses has been investigated by differential thermal analyzer at high pressure (HP-DTA). For As = 40 and 50 system, in Asx Te100 x and Asx Te100 x y Sey, the first exothermic peaks are converted to endothermic under pressure and this is considered as rigidity percolation. The second exothermic peak do not converted to endothermic or no structural transformation takes place. This is considered as electron localization to delocalization. In As = 30, 40 and 50 system, as the Se content increases, the volume decreases from the initial value and the shifting of the temperature of the peaks reduces than the basic system because of less structural transformation. Thus it is concluded that the second peak is generated because of the electron localization.

A Quantitative Basis for Rocking Curve Measurement of Highly Oriented Polycrystalline Thin Films

A quantitative basis for rocking curve measurements of preferentially oriented polycrystalline thin films is presented. The Gaussian function is used for modeling the preferred orientation of crystallites around the plane normal of the specimen surface. A theoretical rocking curve is fitted to the observed curve by the least-squares method, and the degree of preferred orientation, given in volume fraction, can be derived from a refined preferred orientation parameter of the distribution function even when the preferred orientation is very small. Uses of diffractometers equipped with parallel-beam optics and the integrated intensity rather than peak intensity are important for reliable rocking curve measurement.