Journal articles: 'Pseudo-Voigt function'

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Ida, T., M. Ando, and H. Toraya. "Extended pseudo-Voigt function for approximating the Voigt profile." Journal of Applied Crystallography 33, no. 6 (2000): 1311–16. http://dx.doi.org/10.1107/s0021889800010219.

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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 ΓGand ΓLare 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 ΓGand 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 Thompsonet 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.

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Huang, T. C., and G. Lim. "Resolution of Overlapping X-Ray Fluorescence Peaks With the Pseudo-Voigt Function." Advances in X-ray Analysis 29 (1985): 461–68. http://dx.doi.org/10.1154/s0376030800010582.

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AbstractA method for resolving overlapping X-ray fluorescence spectra by curve fitting is described. The profile shape of an experimental fluorescence line obtained by wavelength dispersive method is represented by a simple pseudo-Voigt function, i.e. a sum of an asymmetric Gaussian and Lorentzian, each of equal width. Results showed that the pseudo-Voigt function matched the experimental profiles with high reliability. The relative Gaussian and Lorentzian contents and the asymmetry of the profiles depended upon the analyzing crystal, coliimating system and the 2θ peak position. For fixed crystal and collimator the smaller the 2θ, the larger the Gaussian content and the lower the asymmetry. The original Gaussian and Loretzian components of the exact Voigt function calculated from the parameters of the fitted pseudo-Voigt function explain the broadening effects of the X-ray emission lines and the instrumental aberrations on observed spectra. Curve fitting method with the psuedo- Voigt function has been used successfully to analyze overlapping fluorescence spectra. Examples and applications include a thin film sample where the Kα and the Kβ lines of adjacent transition elements overlap, and a strontium zirconium oxide specimen where the Zr Kα and the Sr Kβ lines strongly interfere. Concentrations obtained from the resolved individual peak intensities of Zr and Sr Kα lines are within ±1% of the true values.

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David, W. I. F. "Powder diffraction peak shapes. Parameterization of the pseudo-Voigt as a Voigt function." Journal of Applied Crystallography 19, no. 1 (1986): 63–64. http://dx.doi.org/10.1107/s0021889886089999.

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David, W. I. F., and J. C. Matthewman. "Profile refinement of powder diffraction patterns using the Voigt function." Journal of Applied Crystallography 18, no. 6 (1985): 461–66. http://dx.doi.org/10.1107/s0021889885010718.

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The Voigt function has been incorporated as a peak-shape description into a program for the refinement of constant-wavelength X-ray and neutron diffraction patterns. The results obtained for neutron diffraction are encouraging and indicate that the Voigt function describes the symmetrical component of the profile peak shape to high accuracy even in the presence of substantial line broadening from particle-size effects. In contrast with approximations to the Voigt function, such as the pseudo-Voigt and Pearson VII functions, the present treatment allows the angular dependences of line-broadening effects resulting from particle-size and instrumental contributions to be coded independently from each other in the Rietveld technique. The present treatment, which details improvements to the symmetrical component of the peak shape, does not offer a fully rigorous description of the peak shape as asymmetry corrections such as those given by Howard [J. Appl. Cryst. (1982), 15, 615–620] are not included.

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Mani, Deepak, Andreas Kupsch, Bernd R. Müller, and Giovanni Bruno. "Diffraction Enhanced Imaging Analysis with Pseudo-Voigt Fit Function." Journal of Imaging 8, no. 8 (2022): 206. http://dx.doi.org/10.3390/jimaging8080206.

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Diffraction enhanced imaging (DEI) is an advanced digital radiographic imaging technique employing the refraction of X-rays to contrast internal interfaces. This study aims to qualitatively and quantitatively evaluate images acquired using this technique and to assess how different fitting functions to the typical rocking curves (RCs) influence the quality of the images. RCs are obtained for every image pixel. This allows the separate determination of the absorption and the refraction properties of the material in a position-sensitive manner. Comparison of various types of fitting functions reveals that the Pseudo-Voigt (PsdV) function is best suited to fit typical RCs. A robust algorithm was developed in the Python programming language, which reliably extracts the physically meaningful information from each pixel of the image. We demonstrate the potential of the algorithm with two specimens: a silicone gel specimen that has well-defined interfaces, and an additively manufactured polycarbonate specimen.

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Baizar, Davor, and Hassel Ledbetter. "Accurate Modeling of Size and Strain Broadening in the Rietveld Refinement: The “Double-Voigt” Approach." Advances in X-ray Analysis 38 (1994): 397–404. http://dx.doi.org/10.1154/s0376030800018048.

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In the “double-Voigt” approach, an exact Voigt function describes both size- and strainbroadened profiles. The lattice strain is defined in terms of physically credible mean-square strain averageid over a distance in the diffracting domains. Analysis of Fourier coefficients in a harmonic approximation for strain coefficients leads to the Warren-Averbach method for the separation of size and strain contributions to diffraction line broadening. The model is introduced in the Rietveld refinement program in the foliowing way: Line widths are modeled with only four parameters in the isotropic case. Varied parameters are both surface- and volumeweighted domain sizes and root-mean-square strains averaged over two distances. Refined parameters determine the physically broadened Voigt line profile. Instrumental Voigt line profile parameters are added to obtain the observed (Voigt) line profile. To speed computation, the corresponding pseudo-Voigt function is calculated and used as a fitting function in refinement. This approach allows for both fast computer code and accurate modeling in terms of physically identifiable parameters.

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Soleimanian, V., and S. R. Aghdaee. "Comparison methods of variance and line profile analysis for the evaluation of microstructures of materials." Powder Diffraction 23, no. 1 (2008): 41–51. http://dx.doi.org/10.1154/1.2888763.

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A comparison of different methods of X-ray diffraction analysis for the determination of crystallite size and microstrain; namely, line profile analysis, Rietveld refinement, and three approaches based on the variance method, is presented. The analyses have been applied to data collected on a ceria sample prepared by the IUCr Commission on Powder Diffraction. In the variance method, split Pearson VII, the Voigt function, and its approximation pseudo-Voigt function were fitted to X-ray diffraction line profiles. Based on the fitting results, the variances of line profiles were calculated and then the crystallite size and root mean square strain were obtained from variance coefficients. A SS plot of Langford as well as a Fourier analysis and Rietveld refinement have been carried out. The average crystallite size and microstrain were determined. The values of area-weighted domain size determined from the variance method are in agreement with those obtained from line profile analysis within a single (largest) standard uncertainty, and the volume-weighted domain sizes derived from the SS plot, Fourier size distribution, and Rietveld refinement agree within a single standard uncertainty. The results of rms strain calculated from variance and Pearson VII shape function and those from Rietveld refinements fall within a single esd. However, the variance method in conjunction with pseudo-Voigt and Voigt functions produce rms strains substantially larger than those determined from line profile analysis and Rietveld refinements.

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Avdeev, Maxim, James Jorgensen, Simine Short, and Robert B. Von Dreele. "On the numerical corrections of time-of-flight neutron powder diffraction data." Journal of Applied Crystallography 40, no. 4 (2007): 710–15. http://dx.doi.org/10.1107/s0021889807030014.

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Time-of-flight neutron powder diffraction data for NIST Standard Reference Materials have been used to study the adequacy of the peak profile model obtained from a convolution of back-to-back exponentials with a pseudo-Voigt function that is widely used in Rietveld refinement. It is shown that, while the empirical models ford-spacing (wavelength) dependence of Gaussian and Lorentzian components of the pseudo-Voigt function and rise exponent are satisfactory, the behavior of the decay exponent and peak positions demonstrate significant deviations, which can be corrected by numerical methods. The practical side of this process as implemented inGSASandFULLPROFand the effect of the corrections on the Rietveld analysis results are discussed.

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Ida, Takashi. "New measures of sharpness for symmetric powder diffraction peak profiles." Journal of Applied Crystallography 41, no. 2 (2008): 393–401. http://dx.doi.org/10.1107/s0021889807067659.

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New measures of sharpness for symmetric powder diffraction peak profiles are proposed. The sharpness parameter is defined through the \nuth-order moment of the Fourier transform of the profile function. Analytical expressions for the sharpness parameter for empirical model profile functions, namely the Gaussian, logistic distribution, hyperbolic secant, Lorentzian, Voigt, Pearson VII and pseudo-Voigt functions, and theoretical size-broadening profiles with statistical size distribution are presented. Theoretical diffraction profiles with complicated formulae can be approximated by empirical model functions assuming equivalent values of the sharpness parameter. The concept of the sharpness parameter provides a simple way to define an approximation for a theoretical diffraction peak profile with empirical model functions.

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Kawamura, Yuki, and Yoshiaki Akiniwa. "Measurement of the X-ray Elastic Constants of Amorphous Polycarbonate." Quantum Beam Science 4, no. 4 (2020): 35. http://dx.doi.org/10.3390/qubs4040035.

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In polymer materials, residual stress introduced during injection molding affects yield reduction due to deformation during molding and delayed fracture during operation, so the establishment of nondestructive stress evaluation of polymer products is desirable. The X-ray elastic constants of polycarbonate were measured for the purpose of obtaining fundamental data for X-ray stress measurement of amorphous polymer materials. The structural function was obtained from the diffraction data, and the strain measured by X-ray was determined from the shift of the first peak by the Q-space method. The peak position was determined using the pseudo-Voigt function approximation method and the diffraction line width method. The Young’s modulus measured by X-ray obtained by the diffraction line width method was close to the mechanical value. Although these values varied widely, they changed depending on the peak ratio. A simple and practical measurement method directly using the raw profile data was also discussed. The Young’s modulus determined by the diffraction line width method decreased with increasing peak ratio. On the other hand, the values determined by the pseudo-Voigt method were almost constant, irrespective of the peak ratio. The strain calculated by the line width method was determined more accurately than that by the pseudo-Voigt method.

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Pain, Jean-Christophe. "Extracting Physical Information from the Voigt Profile Using the Lambert W Function." Plasma 7, no. 2 (2024): 427–45. http://dx.doi.org/10.3390/plasma7020023.

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Spectral line shapes are a key ingredient of hot-plasma opacity calculations. Since resorting to elaborate line-shape models remains prohibitive for intensive opacity calculations involving ions in different excitation states, with L, M, etc., shells are populated, and Voigt profiles often represent a reliable alternative. The corresponding profiles result from the convolution of a Gaussian function (for Doppler and sometimes ionic Stark broadening) and a Lorentzian function, for radiative decay (sometimes referred to as “natural” width) and electron-impact broadening. However, their far-wing behavior is incorrect, which can lead to an overestimation of the opacity. The main goal of the present work was to determine the energy (or frequency) at which the Lorentz wings of a Voigt profile intersect with the underlying Gaussian part of the profile. It turns out that such an energy cut-off, which provides us information about the dominant line-broadening process in a given energy range, can be expressed in terms of the Lambert W function, which finds many applications in physics. We also review a number of representations of the Voigt profile, with an emphasis on the pseudo-Voigt decomposition, which lends itself particularly well to cut-off determination.

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Schmid, Martin, Hans-Peter Steinrück, and J. Michael Gottfried. "A new asymmetric Pseudo-Voigt function for more efficient fitting of XPS lines." Surface and Interface Analysis 46, no. 8 (2014): 505–11. http://dx.doi.org/10.1002/sia.5521.

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Schmid, Martin, Hans‐Peter Steinrück, and J. Michael Gottfried. "A new asymmetric Pseudo‐Voigt function for more efficient fitting of XPS lines." Surface and Interface Analysis 47, no. 11 (2015): 1080. http://dx.doi.org/10.1002/sia.5847.

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Angel, R. J. "Automated profile analysis for single-crystal diffraction data." Journal of Applied Crystallography 36, no. 2 (2003): 295–300. http://dx.doi.org/10.1107/s0021889803001134.

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An integration method for step-scanned single-crystal intensity data based upon fitting of the individual diffraction profiles by a pseudo-Voigt function is presented. Algorithms for both the recovery of weak intensities from data sets and the rejection of aberrant peak profiles are discussed. The ideas presented in this paper have been implemented in a software package for Microsoft Windows,WinIntegrStp, which is available at http://www.crystal.vt.edu/.

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15

Wang, H. "Step size, scanning speed and shape of X-ray diffraction peak." Journal of Applied Crystallography 27, no. 5 (1994): 716–22. http://dx.doi.org/10.1107/s002188989400186x.

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The influences of step size and scanning speed on the shape of a single X-ray diffraction (XRD) peak are analyzed quantitatively. For this purpose, it is assumed that XRD peak shapes are a mixture of Cauchy and Gauss curves. Six equations are established for the calculation of position, maximum intensity and full width at half-maximum (FWHM) errors caused by step size and two for the FWHM errors caused by counting statistics. The ratio of step size to FWHM is proposed as the shape-perfect coefficient of the XRD peak. From these equations and the relationship between the FWHM and the integral width of a peak based on the pseudo-Voigt function or Voigt function, three basic elements of a single symmetric XRD peak (peak position, maximum intensity and FWHM) can be refined. The optimum step size and scanning time can also be set from them.

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Berti, Giovanni. "Modeling and Optimization Algorithm to Analyse Xrpd Data Via Modulation and Pseudo-Voigt Functions." Advances in X-ray Analysis 39 (1995): 465–71. http://dx.doi.org/10.1154/s0376030800022886.

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The composition of the sample related-effects with those from the instrument, has been carried out by convolution of involved terms. The sample-related effects have been modelled via the modulation function accounting for packed parallel planes, named here “layers”, and other structure-related parameters. The instrumental parameters have been determined using DISVAR93, a softawre package allowing the diffraction profiles to be represented in the more convenient shape of a pseudo-Voigt function. A constrained optimization process is the basis for determining these instrumental parameters; their calibration is typically carried out by adopting an external standard sample. The comparison of the modeled intensity distribution with the experimental one is performed via minimization of χ2 where the convolutor and minimizator are embedded engines of the algorithm.

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Sánchez-Bajo, F., and F. L. Cumbrera. "The Use of the Pseudo-Voigt Function in the Variance Method of X-ray Line-Broadening Analysis." Journal of Applied Crystallography 30, no. 4 (1997): 427–30. http://dx.doi.org/10.1107/s0021889896015464.

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A modified application of the variance method, using the pseudo-Voigt function as a good approximation to the X-ray diffraction profiles, is proposed in order to obtain microstructural quantities such as the mean crystallite size and root-mean-square (r.m.s.) strain. Whereas the variance method in its original form is applicable only to well separated reflections, this technique can be employed in the cases where there is line-profile overlap. Determination of the mean crystallite size and r.m.s. strain for several crystallographic directions in a nanocrystalline cubic sample of 9-YSZ (yttria-stabilized zirconia) has been performed by means of this procedure.

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Riello, P., P. Canton, and G. Fagherazzi. "A semi-empirical asymmetry function for X-ray diffraction peak profiles." Powder Diffraction 10, no. 3 (1995): 204–6. http://dx.doi.org/10.1017/s0885715600014743.

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A new semi-empirical approximation for the asymmetry function to be used in the X-ray Rietveld analysis has resulted in lower values of the so-called goodness-of-fit index, defined as S = Rwp/Rexp, where Rwp is the R-weighted pattern and Rexp is the R-expected [R. A. Young, The Rietveld Method (Oxford U.P., Oxford, 1993)], with respect to the corresponding values obtained with the classical approximation used by Rietveld in his fundamental paper. A comparing test of the two asymmetry functions was carried out for the cubic Y2O3 and for αAl2O3 using either pseudo-Voigt or Pearson VII symmetrical functions and two diffractometers. As in the case of the Rietveld approximation, the present one, which employs an exponential function, is optimized using only one fitting parameter. Experimentally, the asymmetry can be considerably diminished by using Soller slits with a small opening angle (≤2°).

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Ida, T., and K. Kimura. "Flat-specimen effect as a convolution in powder diffractometry with Bragg–Brentano geometry." Journal of Applied Crystallography 32, no. 4 (1999): 634–40. http://dx.doi.org/10.1107/s0021889899003222.

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The flat-specimen effect on the peak profile in powder diffractometry with Bragg–Brentano geometry can quantitatively be treated as a convolution with an asymmetric window function. The formula of the window function and a practical method for numerical calculation of the convolution are proposed. By applying the model function given by the convolution of the pseudo-Voigt function with asymmetric window functions based on vertical (axial) divergence and the flat-specimen effect, experimental profiles measured with a conventional diffractometer are fairly well reproduced with only a few parameters. When this function is applied to the analysis of experimental diffraction profiles of a lanthanum hexaboride (LaB6) standard sample measured with a conventional powder X-ray diffractometer, the lattice constant is estimated asa= 4.15685 (2)–4.15693 (2) Å at 299 (1) K, without the use of any other standard material or preceding correction of systematic instrumental errors, while the specified value isa= 4.15695 (6) Å.

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Dasgupta, Prabal, and Endale Abeba Gudeta. "On the use of pseudo-Voigt function in the variance method of size-strain analysis." Acta Crystallographica Section A Foundations and Advances 78, a1 (2022): a326. http://dx.doi.org/10.1107/s2053273322096735.

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Toraya, H. "Array-type universal profile function for powder pattern fitting." Journal of Applied Crystallography 23, no. 6 (1990): 485–91. http://dx.doi.org/10.1107/s002188989000704x.

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An array-type universal profile function for powder-pattern fitting is described. It is defined as an array of unknown parameters representing the profile heights at discrete angular steps. The unknown parameters are determined during least-squares fitting together with the integrated intensity, the peak position, the peak width and background parameters. The function has been applied to profile fitting in a small 20 range, and has better R-factors than the pseudo-Voigt and Pearson VII functions. This new function has some important advantages: it can model any complex profile shape, which would be difficult to describe by a smoothly varying analytical-type function; it generally gives a better fit and thus more precise and accurate analysis than currently used analytical-type functions; and it can be applied to the data of all radiation sources including conventional X-rays, synchrotron radiation and time-of-flight and fixed-wavelength neutrons without changing the mathematical form. The optimized use of the function and the accuracy of deduced parameters are also discussed.

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Ermurat, Yakup. "Phenomenal Simulation Modelling of X-Ray Diffraction Patterns of Pyrite Ash Bioprocessed in Acetobacter aceti Contained Media." Kemija u industriji 70, no. 1-2 (2021): 29–38. http://dx.doi.org/10.15255/kui.2020.032.

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Phenomenal profiling functions were applied for the simulation of an X-ray diffraction (XRD) pattern of pyrite ash bioprocessed in Acetobacter aceti contained growth media. The full width at half maximum (FWHM) values were estimated for each pattern of pyrite ash, and the Bragg equation was used to determine the atomic layer spacing, as well crystallite size and strain broadening were accomplished by applications of Scherrer and Williamson-Hall equations. Errors in line broadening were correlated with the FWHM equations of the generalised profiling functions, and the microstrain (&lt;i&gt;ε&lt;/i&gt;) was estimated with the relation between the integral breadth (&lt;i&gt;β&lt;/i&gt;&lt;sub&gt;i&lt;/sub&gt;) and cos&lt;i&gt;θ&lt;/i&gt;. Furthermore, the XRD pattern of a discrete peak was 3D simulated using Gaussian, Lorentzian, Pearson VII, pseudo-Voigt, and Voigt functions. The Gaussian function with a round top peak was designated the most suitable profile simulation by offering the maximum peak height, &lt;i&gt;I&lt;/i&gt;&lt;sub&gt;max&lt;/sub&gt; and tails comparable to the experimental peak.

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Cruz-Gandarilla, Francisco, A. M. Salcedo-Garrido, Thierry Baudin, H. Mendoza-León, and Richard Penelle. "Generalized Pole Figures and Stored Energy Distribution Function Obtained by X-Ray Diffraction." Materials Science Forum 702-703 (December 2011): 519–22. http://dx.doi.org/10.4028/www.scientific.net/msf.702-703.519.

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Abstract. Anisotropy of physical and mechanical properties of textured polycrystalline materials strongly depends on microstructural characteristics, such as subgrain sizes, lattice deformations, etc. Generalized Pole Figures (GPF) are an attempt to estimate the anisotropy of these properties; so, the energy stored during plastic deformation is a key parameter in primary recrystallization. In this work, the technique to measure GPF (measurements and software) was implemented for X-Ray diffraction and applied to study of property anisotropy of a Fe50%Ni alloy. GPF’s of texture, crystallite size, stored energy and diffraction peak shift, (among others) have been characterized. The Full Width at Half Maximum (FWHM) of obtained instrumental functions shows that defocusing is significant for polar angle higher than 50°. The mixing parameter of the pseudo-Voigt function using in fitting, presents important dispersions.

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Bianchi, A. E., L. Montenegro, R. Viña, and G. Punte. "Microstructure anisotropy in CuO powders." Powder Diffraction 23, S1 (2008): S81—S86. http://dx.doi.org/10.1154/1.2903494.

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An anisotropic line broadening study of CuO is reported. X-ray powder diffraction line width modifications observed are modeled when comparing data coming from (1) commercial analytical grade CuO, (2) energetic ball milling sample for 1 h, and (3) samples prepared by thermally annealing the ball milled sample at various temperatures. X-ray powder diffraction data from commercial and produced samples were analyzed by the Rietveld method using a pseudo-Voigt function. Different assumptions including size and strain anisotropy were tried to improve pattern fitting. An anisotropic strain broadening, modeled using Stephens’ approximation, yielded the best fit, thus indicating that strain anisotropy is the main source of the departure from a smooth function of line broadening as a function of 2θ observed in all samples.

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Ida, T., and K. Kimura. "Effect of sample transparency in powder diffractometry with Bragg–Brentano geometry as a convolution." Journal of Applied Crystallography 32, no. 5 (1999): 982–91. http://dx.doi.org/10.1107/s0021889899008894.

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The effect of transparency and finite thickness of the sample on the peak profile in powder diffractometry with Bragg–Brentano geometry is treated as a convolution with an asymmetric aberration function. The formula of the aberration function and a practical method for numerical calculation of the convolution are proposed. The model function for the total profile is given by triple convolution of the pseudo-Voigt function with asymmetric functions based on vertical (axial) divergence, horizontal (equatorial) divergence and sample transparency. The validity of the model profile function is examined by fitting the CuKα X-ray diffraction profile of diluted lanthanum hexaboride (LaB6) standard samples (SRM660) of 0.5 mm thickness, the transparency of which is varied by mixing with different amounts of starch; all the parameters that may affect the shape of the profile function are treated as fixed parameters. Even in the case of heavily distorted profiles of highly transparent samples with the inverse linear absorption coefficients μ−1= 0.34 and 1.19 mm, experimental profiles are fairly well reproduced by the model function, typically within several % inRfactor. It is also demonstrated that the lattice constants of the transparent samples are precisely evaluated by applying this model function.

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Tashmetov, Mannab, Bekhzodjon Abdurakhimov, Matlab N. Mirzayev, and To Xuan Thang. "The effect of electron beam on nanocrystallites size, strain and structural parameters of the silicon carbide nanopowder." International Journal of Modern Physics B 33, no. 20 (2019): 1950223. http://dx.doi.org/10.1142/s0217979219502230.

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The effect of high-energy electron beam on the silicon carbide nanopowder’s structural parameters, strain and powder size was studied. The sample was irradiated with [Formula: see text]2-MeV electron beam energy under different fluencies such as 1.13 ⋅ 10[Formula: see text], 1.89 ⋅ 10[Formula: see text], 2.79 ⋅ 10[Formula: see text] and 3.69 ⋅ 10[Formula: see text] cm[Formula: see text] at the linear electronic accelerator. Initial and irradiated samples at various doses have been analyzed in the XRD. The nanostructural effects within FullProf are treated using the pseudo-Voigt profile function. The dependences of maximum strain and nanocrystallite size on irradiation dose were obtained and analyzed.

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Dasgupta, Prabal, and Girija Mitra. "Characterization of a diffraction profile using the fourth cumulant." Acta Crystallographica Section A Foundations and Advances 70, a1 (2014): C953. http://dx.doi.org/10.1107/s2053273314090469.

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In this study, line broadening in a diffraction intensity profile of powder crystalline materials due to particle size and lattice strain has been characterized by a new function- the fourth cumulant of the diffraction profile. Diffraction intensity profile is usually characterized in terms of half intensity width (FWHM), Fourier Transforms, second and fourth restricted moments etc. The cumulants, with addtional property of additivity[1], can be used to de-convolute the contribution of several effects such as crystallite size, lattice strain, stacking fault, dislocation, etc., to the line broadening. Since the first three cumulants are the same as the corresponding moments, we investigate the fourth cumulant, which is a function of fourth and second moments. Here, line broadening has only been due to particle size and lattice strain. Previous reports [1] showed that fourth cumulant of a Gaussian was zero, where as we now show that for Cauchy, Voigt and pseudo-Voigt distributions, the fourth cumulant is non-zero. Hence, the fourth cumulant of functions describing particle size and lattice strain for the latter types of distribution has been derived, as well as those for crystalline and para-crystalline materials. For crystalline materials, it was shown that fourth cumulant for particle size and strain (jointly) is the simple sum of the 4th cumulant for particle size and that of strain individually- thus proving the additivity of the fourth cumulant. This work illustrates the advantages of considering the fourth cumulant for characterizing line broadening in terms of particle size and strain.

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Denney, Jonathan J., Gerard S. Mattei, Marcus H. Mendenhall, James P. Cline, Peter G. Khalifah, and Brian H. Toby. "Determination of physically based pseudo-Voigt powder diffraction profile terms from the fundamental parameters approach." Journal of Applied Crystallography 55, no. 2 (2022): 289–95. http://dx.doi.org/10.1107/s1600576722001169.

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A methodology is developed where a fundamental parameters approach (FPA) description of a laboratory powder diffraction instrument (configured in divergent-beam Bragg–Brentano geometry) is used to determine GSAS-II profile parameters for peak asymmetry and instrumental peak widths. This allows the instrumental contribution to peak shapes to be robustly determined directly from a physical description of the instrument, even though GSAS-II does not directly implement FPA for peak shape computation. The FPA-derived parameters can be used as the starting point for instrument characterization, or to characterize sample broadening without the use of a standard to determine the instrument profile function. This new method can facilitate generation of training sets for machine learning. A plot is generated that shows the differences between the two approaches, demonstrating upper bounds for the accuracy of the GSAS-II profile model for a particular instrumental configuration.

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Smith, R. I., A. R. West, I. Abrahams, and P. G. Bruce. "Rietveld Structure Refinement of Metastable Lithium Disilicate Using Synchrotron X-Ray Powder Diffraction Data From the Daresbury SRS 8.3 Diffractometer." Powder Diffraction 5, no. 3 (1990): 137–43. http://dx.doi.org/10.1017/s0885715600015566.

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AbstractThe crystal structure of metastable Li2Si2O5, Fw = 150.05, has been refined by the Rietveld method using high resolution X-ray powder diffraction data recorded at the Daresbury Synchrotron Radiation Source on the new 8.3 diffractometer. Li2Si2O5, in keeping with many compounds of interest to the materials scientist, exhibits relatively broad diffraction peaks. It is important to establish the quality of crystal structure data that may be obtained from such materials on this new instrument. Various functions were used to model the peak shape from this instrument; a split-Pearson VII function appeared to be marginally superior to Pearson VII or Pseudo-Voigt functions. Refinement was carried out using the split-Pearson VII in the space group Pbcn (60) and terminated with a = 5.6871(6), b = 4.7846(5), c = 14.645(1) Å, V = 398.50 Å3, Z=4, Dc= 2.502 gcm−3, Rwp = 17.06, Rex = 14.48 and Χ2 = 1.39. The refined parameters are compared with those obtained from a previous single crystal X-ray determination.

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Ohno, Katsumi, Hiroshi Harada, Toshihiro Yamagata, and Michio Yamazaki. "Numerical Resolution Enhancement of X-ray Diffraction Patterns." Advances in X-ray Analysis 33 (1989): 409–16. http://dx.doi.org/10.1154/s0376030800019832.

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AbstractA numerical resolution-enhancement method was developed for x-ray diffraction data measured with a conventional x-ray diffractometer. This method removes the instrumental broadening due to x-ray optics, including the spectral distribution of the x-ray source such as the CuKα doublet. The advantages of this method are to separate the cluster of peaks in x-ray powder patterns into individual peaks without previous knowledge of the number of peaks, and to remove CuKα2 reflection peaks automatically.The instrumental window function, which was approximated by a modified pseudo-Voigt function, was calculated from the measured diffraction pattern of NBS Standard Reference Materials (640B) by a non-linear least squares method. The simple diffraction patterns, including no CuKα2 peaks, were obtained from the diffraction patterns measured with the conventional x-ray diffractometer by using the window function mentioned above.The application of the method of determination of the lattice misfit between γ and γ phases in Ni-base superalloys was also described.

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Allegra, Giuseppe, and Sergio Brückner. "Crystallite-size distributions and diffraction line profiles near the peak maximum." Powder Diffraction 8, no. 2 (1993): 102–6. http://dx.doi.org/10.1017/s0885715600017917.

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The effect of crystallite-size distribution on the shape of X-ray diffraction peaks from powder samples is investigated focusing the attention on the region within the top half of intensity. It is shown that, unlike profile tails, this central region can markedly depart from the Lorentzian shape for crystallite-size distributions that are quite acceptable from the physical point of view. Goal of this paper is to correlate the well-known m parameter in the Pearson VII function or the η weight in the pseudo-Voigt function with a number of different distributions of crystallite dimensions ranging from the δ function of all-equal-sized crystals to a very broad distribution. The suitably normalized curvature at the peak is a possible new parameter; its correlation with m and η is shown. Also, a procedure is suggested to derive the volume-average crystal thickness 〈Tw〉 from the FWHM and the knowledge of either m or η.

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Ohno, Katsumi, Tadaharu Yokokawa, Toshihiro Yamagata, Hiroshi Harada, Michio Yamazaki та Kazumasa Ohsumi. "Determination of the Coherency Strain of γ and γ′ Phases in Nickel-Base Superalloys at High Temperatures". Advances in X-ray Analysis 36 (1992): 515–26. http://dx.doi.org/10.1154/s0376030800019145.

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AbstractA method for using syncrotoron-radiation parallel-beam X-ray diffractometry for precise lattice parameters and strains of γ-γ′ type Nickel base superalloys at elevated temperature is described. The superalloys have γ′ precipitates which are an ordered L12 structure based on Ni3Al, in y-matrices having a disordered FCC structure. Lattice misfit between γ and γ′ phases was very small and peaks reflected from γ and γ′ phases made unresolved clusters of peaks.Profile fitting with a pseudo-Voigt function is used to resolve overlapping peaks. Instrumental broadening of the peak profile was removed using a deconvolution method. The standard errors of the calculated peak angle were less than 0.002°. The elastic strain of the γ′ precipitates in the alloys were smaller than those of γ-matrices.

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Sánchez-Bajo, F., and F. L. Cumbrera. "The use of the pseudo-Voigt function in the variance method of X-ray line-broadening analysis. Erratum." Journal of Applied Crystallography 30, no. 5 (1997): 550. http://dx.doi.org/10.1107/s0021889897098956.

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Ganiev, O. K. "Application of the pseudo-Voigt function to the analysis of single-particle tunneling spectra of d-wave superconductors." Physica C: Superconductivity and its Applications 610 (July 2023): 1354280. http://dx.doi.org/10.1016/j.physc.2023.1354280.

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Xu, Zhiniu, and Lijuan Zhao. "Highly accurate slope-assisted distributed fiber strain and temperature sensor based on pseudo-Voigt profile and quartic function." Optical Fiber Technology 82 (January 2024): 103645. http://dx.doi.org/10.1016/j.yofte.2023.103645.

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Collet, J. L., Françoise Bley, Alexis Deschamps, and Colin Scott. "Study of the Deformation Mechanisms of TWIP Steels (Fe-Mn-C) by X-Ray Diffraction." Advanced Materials Research 15-17 (February 2006): 822–27. http://dx.doi.org/10.4028/www.scientific.net/amr.15-17.822.

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The deformation mechanisms of an Fe-Mn-C TWIP steel have been investigated as a function of deformation and deformation temperature, using synchrotron X-ray diffraction at the European synchrotron radiation facility. Using the Warren theory, it is possible to reach a good qualitative understanding of the deformation mechanisms. We have confirmed that the deformation mechanisms shifted from the formation of martensite at very low temperature, to twinning around room temperature and dislocations at higher temperatures. Although some quantification of the density of crystalline defects can be reached using simple parameters such as peak shift and broadening, the complexity of defects present in this material require the development of more advanced data interpretation models. First results are shown, using shift and broadening of the peak and fit of intensity by a pseudo-voigt function, as well as the study of the asymptotic behavior of the intensity.

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Ida, T., and H. Hibino. "Symmetrization of diffraction peak profiles measured with a high-resolution synchrotron X-ray powder diffractometer." Journal of Applied Crystallography 39, no. 1 (2006): 90–100. http://dx.doi.org/10.1107/s0021889805040318.

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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.

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de Andrade, Mônica C., Geysa N. Carneiro, Elizabeth L. Moreira, Jorge C. Araújo, and Valéria C. A. Moraes. "Synthesis and Characterization of Barium Titanate by Solid-State Reaction." Materials Science Forum 802 (December 2014): 285–90. http://dx.doi.org/10.4028/www.scientific.net/msf.802.285.

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Solid-state reactions were used to synthesize pure and doped barium titanate powder. Barium titanate formation with tetragonal perovskite structure was detected by X-ray diffraction and occurred at a temperature above 700°C for pure powder and 500°C for doped powder. However, quite crystalline samples were observed only at 800oC and 600°C for pure and doped barium titanate, respectively, what made the refinement of the synthesized powders possible. They were characterized by X-ray diffraction and Fourier transform infrared spectroscopy and scanning electron microscopy. X-ray diffraction data was analyzed by using the Fullprof Rietveld refinement approach, Thompson-Cox-Hastings pseudo-Voigt with function. The refinement method was effective in the study of the temperature influence on the microstructure of the analysis of pure and doped barium titanate.

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Balzar, D., N. Audebrand, M. R. Daymond, et al. "Size–strain line-broadening analysis of the ceria round-robin sample." Journal of Applied Crystallography 37, no. 6 (2004): 911–24. http://dx.doi.org/10.1107/s0021889804022551.

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The results of both a line-broadening study on a ceria sample and a size–strain round robin on diffraction line-broadening methods, which was sponsored by the Commission on Powder Diffraction of the International Union of Crystallography, are presented. The sample was prepared by heating hydrated ceria at 923 K for 45 h. Another ceria sample was prepared to correct for the effects of instrumental broadening by annealing commercially obtained ceria at 1573 K for 3 h and slowly cooling it in the furnace. The diffraction measurements were carried out with two laboratory and two synchrotron X-ray sources, two constant-wavelength neutron and a time-of-flight (TOF) neutron source. Diffraction measurements were analyzed by three methods: the model assuming a lognormal size distribution of spherical crystallites, Warren–Averbach analysis and Rietveld refinement. The last two methods detected a relatively small strain in the sample, as opposed to the first method. Assuming a strain-free sample, the results from all three methods agree well. The average real crystallite size, on the assumption of a spherical crystallite shape, is 191 (5) Å. The scatter of results given by different instruments is relatively small, although significantly larger than the estimated standard uncertainties. The Rietveld refinement results for this ceria sample indicate that the diffraction peaks can be successfully approximated with a pseudo-Voigt function. In a common approximation used in Rietveld refinement programs, this implies that the size-broadened profile cannot be approximated by a Lorentzian but by a full Voigt or pseudo-Voigt function. In the second part of this paper, the results of the round robin on the size–strain line-broadening analysis methods are presented, which was conducted through the participation of 18 groups from 12 countries. Participants have reported results obtained by analyzing data that were collected on the two ceria samples at seven instruments. The analysis of results received in terms of coherently diffracting, both volume-weighted and area-weighted apparent domain size are reported. Although there is a reasonable agreement, the reported results on the volume-weighted domain size show significantly higher scatter than those on the area-weighted domain size. This is most likely due to a significant number of results reporting a high value of strain. Most of those results were obtained by Rietveld refinement in which the Gaussian size parameter was not refined, thus erroneously assigning size-related broadening to other effects. A comparison of results with the average of the three-way comparative analysis from the first part shows a good agreement.

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Parrish, W., and M. Hart. "Parallel Beam and Focusing X-ray Powder Diffractometry." Advances in X-ray Analysis 32 (1988): 481–88. http://dx.doi.org/10.1154/s0376030800020802.

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AbstractComparison of results using synchrotron radiation and X-ray tubes requires a knowledge of the fundamentally different profile shapes inherent in the methods. The varying asymmetric shapes and peak shifts in focusing geometry limit the accuracy and applications of the method and their origins are reviewed. Most o f the focusing aberrations such as specimen displacement, flat specimen and θ-2θ mis-setting do not occur in the parallel beam geometry. The X-ray optics used in synchrotron parallel beam methods produces narrow, symmetrical profiles which can be accurately fit with a pseudo-Voigt function, They have the same shape in the entire pattern. Only the width increases as tanθ due to wavelength dispersion but with higher resolution systems dispersion can be eliminated. The constant instrument function contribution to the experimental profile shape is an important advantage in studies involving profile shapes, e.g., small particle sizes and microstrains, and accurate integrated intensities. The absence of systematic errors leads to more precise lattice parameter determinations.

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Takata, Masaki, Manabu Yamada, Yoshiki Kubota, and Makoto Sakata. "High Resolution X-Ray Powder Diffraction by the Combination of Synchrotron Radiation and Imaging Plate to Observe Electron Distribution by the Maximum Entropy Method." Advances in X-ray Analysis 35, A (1991): 85–90. http://dx.doi.org/10.1154/s0376030800008703.

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AbstractA Debye-Scherrer camera (radius 572 mm) was designed for the use at the Photon Factory BL-6A2. It uses 4 pieces of the Imaging Plate to cover 0° ∼ 160° in 2θ. The profiles measured by the camera had 0.083° full width at half maximum (FWHM) and were rather symmetric compared with conventional X-ray source. The observed profiles were well represented by either the split Pearson VII or pseudo-Voigt function. In order to examine the performance of the new camera, the whole powder data of LiF was analyzed by the Maximum Entropy Method(MEM). The MEM analysis of LiF was successfully accomplished using 20 measured structure factors and gave R(Bragg) 0.25%. The density distribution obtained represented a characteristic feature of ionic crystals and was consistent with the theoretical result of the self-consistent LCAO method.

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Králik, Martin, Stanislav Jurečka, and Emil Pinčík. "Thickness and tensile stress determination of black silicon layers by spectral reflectance and Raman scattering." Journal of Electrical Engineering 70, no. 7 (2019): 51–57. http://dx.doi.org/10.2478/jee-2019-0041.

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Abstract In this work black silicon (b-Si) samples were prepared by anodic (electrochemical) etching of p-type silicon substrate in solution of hydrofluoric acid (HF). We studied influence of anodic etching conditions (etching time, electrical potential and current) on the spectral reflectance and Raman scattering spectra. Optical properties of b-Si structures were experimentally studied by UV-VIS (AvaSpec-2048) and Raman (Thermo DXR Raman) spectrometers. B-Si layer thickness of formed substrate were determined by using SCOUT software. Effective medium approximation theory (Looyenga) was used in construction of the reflectance model. Influence of the deformation of crystal lattice introduced during the substrate etching was studied by Raman scattering method. Teoretical model of the 1st order Raman scattering profile was constructed by using pseudo-Voigt function and the profile parameters were extracted. The values of biaxial tensile stress were estimated by using optimized Raman profile parameters.

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Branoiu, Gheorghe, and Ibrahim Ramadan. "Rietveld Structure Refinement of the Stilbite Crystals from Deccan Traps (India) Using X-ray Powder Diffraction Data." Revista de Chimie 70, no. 7 (2019): 2379–84. http://dx.doi.org/10.37358/rc.19.7.7344.

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The crystal structure of a spectacular sample of stilbite from Pune region located in the Deccan Traps (western India) has been refined using X-Ray powder diffraction data and the Rietveld method. The Rietveld refinement was carried out using the computer program Diffracplus TOPAS 4.1. The pseudo-Voigt (pV) profile function was used for the fit of the peaks. The Rietveld refinement of the analyzed sample in the space group C2/m (No.12): a=13.606 �, b=18.260 �, c=11.253 �, b=127.432�, Z=8, confirm the basic stilbite structure. The chemical composition of the stilbite crystals from Pune region (India) was determined by EDX analysis. The paper presents a new set of the unit cell parameters and fractional coordinates that define the stilbite crystal structure. The quality of the sample analyzed was pristine, the sample being collected from an association of apophyllite-stilbite crystals of centimetric dimensions.

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Chatterjee, Partha, S. P. Sen Gupta, and Suchitra Sen. "Microstructural transformation of vanadium pentoxide powder obtained by high-energy vibrational ball-milling." Journal of Applied Crystallography 34, no. 3 (2001): 381–85. http://dx.doi.org/10.1107/s0021889801004009.

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An X-ray powder profile analysis of vanadium pentoxide powder milled in a high-energy vibrational ball-mill is presented. The strain- and size-induced broadening of the Bragg reflection for two different crystallographic directions ([001] and [100]) were determined by Warren–Averbach (WA) analysis, using a pattern-decomposition method assuming a pseudo-Voigt function. The deformation process causes a decrease in the domain size, and a near isotropic saturation value of ∼10 nm is reached after severe milling. The initial stages of milling produce a propensity for size broadening, whereas, with increasing milling time, microstrain broadening is predominant. WA analysis indicated significant plastic strain along with spatial confinement of the internal strain fields, probably in the crystallite interfaces. Transmission electron microscopy revealed a drastic change in particle shape after 64 h of milling, indicating the existence of size anisotropy which, however, decreased with further increase of the milling time.

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Dong, Y. H., and P. Scardi. "MarqX: a new program for whole-powder-pattern fitting." Journal of Applied Crystallography 33, no. 1 (2000): 184–89. http://dx.doi.org/10.1107/s002188989901434x.

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MarqXis a computer program for the modelling of powder diffraction data. It can be used for an unconstrained profile fitting (pattern decomposition, PD) or constrained modelling of the whole powder pattern (Pawley method, PM), for single- as well as multiple-phase samples. The program output includes: lattice parameters or peak positions (for PM and PD, respectively), width and shape of the diffraction peak (in terms of half width at half-maximum and mixing parameter of a pseudo-Voigt function), corrected for the instrumental broadening component, intensity, peak area and profile asymmetry. In addition, errors on the goniometer zero and shift in sample position with respect to the goniometric axis can also be modelled, together with distance and relative intensity of the spectral components of the X-ray beam (e.g.Kα1andKα2). Specific output files are provided for line-profile analysis, including the Williamson–Hall plot and Warren–Averbach method.

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Chaves, A. C., Hélio Lucena Lira, G. A. Neves, L. L. M. Sales, D. M. A. Melo, and Bráulio Silva Barros. "Study Morphologic and Microstructural of the System CeO₂-NiO Obtained by Polymeric Precursor Method." Materials Science Forum 727-728 (August 2012): 491–96. http://dx.doi.org/10.4028/www.scientific.net/msf.727-728.491.

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Cerium oxide has a high potential to be apply in removing pollutants combustion powders, organic materials from waste water and in the technology of fuel cells. Nickel oxide is an attractive material due to its excellent chemical stability, and their optical, electrical and magnetic proprieties. In this work, CeO2-NiO systems on molar ratios of 1:1, 1:2 and 1:3 citric metal-acid were synthesized by using Pechinis method. The change in compositions and calcined temperature were studied by different techniques such as: XDR and SEM. It was also done a microstructural study by Rietvelds method, whose routine was designed to run in the Fullprof Suite program, and analyzed by pseudo-Voigt function. It has noticed that the molar ratio between metal-citric acid in the system CeO2-NiO 1:1, 1:2 and 1:3 has a strong influence on microstructural proprieties, crystallite size and cell micro-deformations and can be used to control these proprieties.

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Oliveira, Rafael P., and Carlos Driemeier. "CRAFS: a model to analyze two-dimensional X-ray diffraction patterns of plant cellulose." Journal of Applied Crystallography 46, no. 4 (2013): 1196–210. http://dx.doi.org/10.1107/s0021889813014805.

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Cellulose from higher plants is a vast renewable resource organized as crystals. Analysis of these crystals by X-ray diffraction poses very specific challenges, including ubiquitous crystallite texture and substantial overlapping of diffraction peaks. In this article, a tailor-made model named Cellulose Rietveld Analysis for Fine Structure (CRAFS) is developed to analyze two-dimensional X-ray diffraction patterns from raw and processed plant cellulose. One-dimensional powder diffractograms are analyzable as a particular case. The CRAFS model considers cellulose Iβ crystal structure, fibrillar crystal shape, paracrystalline peak broadening, pseudo-Voigt peak profiles, harmonic crystallite orientation distribution function and diffraction in fiber geometry. Formulated on the basis of the Rietveld method, CRAFS is presently written in the MATLAB computing language. A set of meaningful coefficients are output from each analyzed pattern. To exemplify model applicability, representative samples are analyzed, bringing some general insights and evidencing the model's potential for systematic parameterization of the fine structure of raw and processed plant celluloses.

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Zhang, Yanheng, Ning Lu, and Wei Qiu. "Optimal Data Processing Method for the Application of Eu3+ Photoluminescence Piezospectroscopy in Thermal Barrier Coatings." Coatings 11, no. 6 (2021): 678. http://dx.doi.org/10.3390/coatings11060678.

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Thermal barrier coatings (TBCs) are widely used to protect gas turbine blades but internal stress near the interface in TBCs is one of the main causes of thermal barrier failure under thermal cycling. A non-destructive inspection technique based on Eu3+ photoluminescence piezospectroscopy has been successfully used to analyze the residual stress in TBCs, but systematic and quantitative evaluation of data processing is still needed, especially with respect to the identification of peak positions. In this work, processing methods for Eu3+ photoluminescence spectroscopy data were studied to characterize TBC internal stress. Both physical and numerical experiments were carried out where Eu3+ luminescence spectra were obtained from a sample of europium-doped yttria-stabilized zirconia (YSZ:Eu3+) under step-by-step uniaxial loading, and the simulated spectra were numerically deduced from the measured spectra. The peak shifts were then obtained by processing the spectral data in different ways (Gaussian, Lorentzian, pseudo-Voigt fitting, and the barycenter method), and comparing the results. We found that the Gaussian function, rather than the commonly used Lorentzian function, is the most appropriate method for the application of Eu3+ photoluminescence piezospectroscopy in TBCs because it provides sufficient sensitivity, stability and confidence for quantitative stress analysis.

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Ida, T., H. Hibino, and H. Toraya. "Deconvolution of instrumental aberrations for synchrotron powder X-ray diffractometry." Journal of Applied Crystallography 36, no. 2 (2003): 181–87. http://dx.doi.org/10.1107/s0021889802021131.

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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-4B2at the Photon Factory in Tsukuba have been analysed by the method. The formula of the scale transformation for the diffractometer aberration hasa prioribeen 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.

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Parrish, W., M. Hart, T. C. Huang, and M. Bellotto. "Lattice Parameter Determination Using Synchrotron Powder Data." Advances in X-ray Analysis 30 (1986): 373–82. http://dx.doi.org/10.1154/s0376030800021510.

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AbstractA method for using synchrotron-radiation parallel-beam X-ray diffractometry for precision measurement of scattering angles and lattice parameters is described. The important advantages of the method are the high P/B made possible by wavelength selection and high source intensity, the symmetrical profiles and the absence of most systematic errors making it unnecessary to use standards. Profile fitting with a pseudo-Voigt function is used to determine 2θ to 0.0001º. The zero-angle correction and lattice parameter were determined from least-squares refinement and the average accuracy of observed-calculated 2θs was 0.0020°. Average values of ∆d/d = ∆a/a directly calculated from the individual hkl measurements ranged from 2x 10-5 to 5.7 x 10-5. The precision estimated from the standard deviation of the mean is in the 10-6 range and 1 ppm precision was obtained for Si. The determination of the exact wavelength selected remains to be solved, but ratios of lattice spacings to standards such as NBS SRM 640a can be determined.

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Journal articles on the topic 'Pseudo-Voigt function'

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