pseudo-Voigt Function Research Articles

Numerical conversion between the Pearson VII and pseudo-Voigt functions

The shape coefficient of a curve, which is defined as the ratio of its full width at half-maximum intensity to integral width, is a measure of variation in shape. From the definitions of the pseudo-Voigt, Pearson VII and Voigt functions used in X-ray diffractometry, three shape-converting formulae are deduced. Numerical solution helps to establish a numerical model for interconverting the shapes of the pseudo-Voigt, Pearson VII and Voigt functions. Assessment of the error level and some comments on the shape of the pseudo-Voigt and Pearson VII functions are thereby made.

Ab initioand experimental pre-edge investigations of the MnK-edge XANES in oxide-type materials

Mn $K$ edge ab initio FEFF8.2 calculations of the pre-edge features of the x-ray-absorption near-edge structure (XANES) region were undertaken for a series of Mn-bearing oxide-type compounds. The aim of the study is to provide a reliable method for determining quantitative and accurate redox and symmetry information for manganese. In agreement with multiplet calculations by Glatzel and co-workers, FEFF8.2 predicts a doublet and a triplet for Mn(II) and Mn(III) in octahedral symmetry, respectively, in agreement with high-resolution XANES experiments. Site distortion increases notably the contribution from dipolar transitions and, consequently, the pre-edge feature integrated area. An even more intense pre-edge feature is calculated and measured for the ${T}_{d}$ symmetry (singletlike). For Mn(IV), a triplet is predicted and measured for the ${O}_{h}$ symmetry. However, additional transitions are found in Mn(IV)-rich compounds, that are related to metal-metal transitions. These transitions overlap strongly with the ``true pre-edge,'' making extraction of redox and symmetry information for Mn(IV) more challenging. However, a model of the pre-edge with pseudo-Voigt functions of fixed calculated width (based on core-hole lifetime and experimental resolution) helps to separate the contributions related to first-neighbor symmetry from those of the metal-metal pairs. Application to multivalent defective manganese oxide materials suggests that the pre-edge information varies linearly as a function of Mn redox state or symmetry but varies nonlinearly as a function of both parameters. Finally, the polymerization of the manganese networks can be estimated from the metal-metal transitions found in the pre-edge region.

Size driven phase transition of barium titanate nanoparticles prepared by plasma chemical vapor deposition

Size driven phase transition of barium titanate nanoparticles prepared by plasma chemical vapor deposition

Size–strain line-broadening analysis of the ceria round-robin sample

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.

Study of different type of dislocations in GaN thin films

High-resolution X-ray diffraction was used to analyze the type of dislocations in GaN epitaxial thin films. Rocking curves of five planes were investigated, (0 0 0 2) , (1 0 1 ̄ 3) , (1 0 1 ̄ 2) , (1 0 1 ̄ 1) , and (2 0 2 ̄ 1) , respectively. Pseudo-Voigt function was used to simulate the rocking-curve of every plane. Every extension of the rocking-curve was regarded as the effect of the interaction of the twist and tilt fractions of the dislocations. From the result, it is found that carrier mobility is more sensitive to substrate normal tilt dislocation than to in-plane twist and the interaction of the twist and tilt fractions also affect the carrier mobility.

Structural study of Bi4Ti3O12using neutron powder diffraction data

Recently, a thin film of Bi4Ti3O12 (BTO) has been widely studied as one of candidate materials for applications of ferroelectric random access memory due to its large remnant polarization at room temperature. The crystal structure of BTO consists of pervoskite type units, Bi2Ti3O10 (An−1BnO3n+1 for n = 3), and interleaved Bi2O2 layers along the c-axis. In the Bi2Ti3O10 units, Ti ions are surrounded by oxygen octahedrons, and Bi ions occupy the outside of TiO6 octahedrons [1, 2]. Although the crystal structural information of BTO is essential to understand the remnant polarization at room temperature, the structural study of BTO has not been performed sufficiently because the difference between two kinds of crystal systems, monoclinic and orthorhombic, is subtle. This present work is focused on the structural refinement of BTO at room temperature using neutron powder diffraction data. The Rietveld refinement of Bi4Ti3O12 was carried out using neutron powder diffraction data, which provide a better refined results for a number of oxygen atoms in the unit cell, though Ti atom has a negative scattering length (−0.3438 × 10−12 m) for neutron radiation. The sample of BTO was prepared by a normal solidstate reaction using Bi2O3 and TiO2 as a starting material. The mixture was annealed at 1050 ◦C for 17 h. The second annealing process was carried out at 1150 ◦C for 48 h followed by intermediate regrinding. Neutron powder diffraction data were collected over scattering angles 0 ◦–160 ◦ using 1.8348 A neutron on the High Resolution Powder Diffractometer (HRPD) at Hanaro Center of Korea Atomic Energy Research Institute in Korea. The General Structure Analysis System (GSAS) program was used to do the Rietveld refinement [3]. The pseudo-Voigt function was used as a profile function among profile ones in GSAS program [4]. According to the report of the single crystal X-ray experiment [5], the phase of BTO at room temperature belongs to the monoclinic system with B1a1 space group. However, Hervoches et al. reported that BTO at room temperature has orthorhombic crystal system from the Rietveld refinement using neutron powder diffraction. Also, they mentioned that the deviation between orthorhombic and monoclinic system was too small to be detectable with neutron diffraction data [6]. When the reflections by the b-glide in the orthorhombic system with B2cb space group are not observed, it is difficult to discern the difference between the two crystal systems quantitatively because the final diffraction patterns for both crystal systems are almost identical. In order to start the Rietveld refinement, a reasonably approximated model of BTO for the actual structure based on the monoclinic and orthorhombic crystal system is required. As a starting model of BTO, two kinds of models based on the monoclinic and orthorhombic systems were used. The model based on the B1a1 space group (M-model) was built with crystallographic data taken from the single crystal X-ray diffraction [5]. The crystallographic data reported by Hervoches et al., were used to make the other one (O-model) [6]. The Rietveld refinement of BTO was performed by neutron powder diffraction data, which is powerful to determine atomic positions and occupancies factors of oxygen atoms. Because BTO contains a large number of oxygen atoms in the unit cell. The neutron scattering lengths for Bi, Ti and O atoms are 0.853 × 10−12 m, −0.3438 × 10−12 m and 0.5805 × 10−12 m, respectively. The initial refinement for the O-model was done by the unit-cell, the zero-point shift and background parameters. The zero-point shift was correct by neutron powder diffraction data of α-Al2O3 as a standard sample. A good match of the peak positions was achieved from the preliminary refinement, and then the peak profile parameters including the peak asymmetry were refined. A pseudo-Voigt function in this study was used as a profile function. It assumes that there is no defect for oxygen sites because the samples were prepared under air atmosphere. The final weighted and profile R-factors, Rwp and Rp, obtained from the O-model through the Rietveld refinement procedure showed 8.61% and 6.31%, respectively. Fig. 1 illustrates the Rietveld refinement patterns for the O-model. In the case of the same Rietveld refinement procedure applied to the M-model, all R-factors for M-model were lower than those for O-model. Among R-factors, the weighted R-factor, Rwp, was considerably decreased from 8.61% to 5.59%. The profile factor, Rp, which is a true quantity based on the discrepancies between observed and calculated intensities, was 4.17%. Also, the goodness-of-fit indicator, S(= Rwp/Re), was decreased from 2.15 to 1.40. The Rietveld refinement patterns of the M-model are shown

Strain-induced martensite in ADI alloys

The remarkable mechanical properties of austempered ductile irons arise from their microstructure, which is composed of a high-carbon stabilized austenite matrix filled with small isles of ferrite platelets, appropriately called ausferrite. Cold work imposed on the material yields a partial transformation of retained austenite into martensite, which can strongly influence the mechanical behavior of the alloy. The present study deals with the determination of transformed austenite upon plastic deformation in the range from 0 to 25% thickness reduction of the samples. The analyzed material was a Cu–Mo–Ni-alloyed ADI, obtained by austempering at 400 °C for 90 min after austenization at 900 °C for 80 min under Ar-protecting atmosphere. The determination of retained austenite and induced martensite made use of the classical X-ray diffraction technique and Rietveld refinement of the intensity data, with all experimental profile modeled by non-linear least-squares fitting, using pseudo-Voigt functions. The overall results showed that the starting contents of retained austenite in three groups of samples, i.e. 43.20, 35.79 and 44.05 vol.%, gradually evolved to 18.23, 13.02 and 18.59%, respectively, after 20% cold work, while formation of martensite increased from nearly zero to 24.95, 24.01 and 25.52% under the same deformation process. These results are further analyzed on the basis of some classical transformation theories.

Automated profile analysis for single-crystal diffraction data

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/.

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-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.

Microstructural Characterization of Rhenium Powder Obtained at Various Temperatures

This work deals with the measurement of particle-size distribution of rhenium powder obtained by reduction of ammonium perrhenate under hydrogen flux. Depending upon the reduction temperature, variable amounts of hydrogen, nitrogen and oxygen remain interstitially located in the material, causing large X-ray diffraction line broadening, which accounts for particle-size domains in the microstructure of the metallic particles. The present study was carried out in the range of 400 to 1000 °C, where the oxidized species can be reduced to rhenium powder. All calculations regarding the line-broadening effects made use of the powerfull Warren-Averbach method, while the experimental diffraction profiles were modeled by non-linear least-squares fitting, using pseudo-Voigt functions. The average particle size increased from 8.7 to 120 nm with increasing temperature from 400 to 750 °C, with no further change above this last temperature. On the other side, strain distribution within the same temperature range varied from 15 to 735x10 -5 .

Effects of calcination temperature on phase and microstructure evolution of BaTi4O9 powders

Calcining of the mixture of BaCO3 and TiO2 with a ratio 1: 4 at different temperatures was carried out to synthesize BaTi4O9 powders. Phase evolution of the samples was studied using the differential thermal analysis (DTA) and X-ray diffractometry (XRD). Both techniques confirmed that the formation of BaTi4O9 started around 1000 °C. The XRD peaks showed that BaTi4O9 was most pronounced at 1250 °C. X-ray line broadening methods were employed to study the variation of particle size and microstrain of the BaTi4O9 powders. The Voigt function in a single line and the pseudo-Voigt function in the variance methods were used in our case. We found that both functions resulted in the same trends, i.e., the particle size increased with the temperature with the biggest size of 180 and 160 nm, whilst the microstrain yielded the opposite trend with the lowest values of 6.2 × 10−3 and 1.1 x 10−3. The scanning electron microscopy (SEM) study revealed the size of the large particles formed, due to agglomeration, to be about 0.5–1.9 μm. Furthermore, it was shown that irregular shapes of BaTi4O9 powders necked to each other appeared at 1000 °C and grew into ellipse and rod shapes at 1250 °C.

Microstructural transformation of vanadium pentoxide powder obtained by high-energy vibrational ball-milling

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.

Particle fracture and plastic deformation in vanadium pentoxide powders induced by high energy vibrational ball-mill

An X-ray powder profile analysis in vanadium pentoxide powder milled in a high energy vibrational ball-mill for different lengths of time (0–250 h), is presented. The strain and size induced broadening of the Bragg reflection for two different crystallographic directions ([001] and [100]) was determined by Warren-Averbach analysis using a pattern-decomposition method assuming a Pseudo-Voigt function. The deformation process caused a decrease in the crystallite size and a saturation of crystallite size of ∼ 10 nm was reached after severe milling. The initial stages of milling indicated a propensity of size-broadening due to fracture of the powder particles caused by repeated ball-to-powder impact whereas with increasing milling time microstrain broadening was predominant. WA analysis indicated significant plastic strain along with spatial confinement of the internal strain fields in the crystallite interfaces. Significant strain anisotropy was noticed in the different crystallographic directions. A near-isotropy in the crystallite size value was noticed for materials milled for 200 h and beyond. The column-length distribution function obtained from the size Fourier coefficients progressively narrowed down with the milling time.

Extended pseudo-Voigt function for approximating the Voigt profile

The formula of the pseudo-Voigt function expressed by a weighted sum of Gaussian and Lorentzian functions is extended by adding two other types of peak functions in order to improve the accuracy when approximating the Voigt profile. The full width at half-maximum (FWHM) values and mixing parameters of the Gaussian, the Lorentzian and the other two component functions in the extended formula can be approximated by polynomials of a parameter ρ = ΓL/(ΓG+ ΓL), where 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.

X-ray study of the crystallographic structure of BaTiO3 powder prepared by solid state reaction

The structure of barium titanate powder prepared by conventional solid state reaction has been investigated by x-ray diffractometry. The pseudo-Voigt function fitted to the diffractographs to discriminate between cubic and tetragonal peaks. The Scherrer formula, with correction for instrument broadening, was employed to estimate the particle size. The method of internal standards was applied to calibrate the amount of cubic phase in the particles. The fraction of cubic shell, covering tetragonal core, was found to increase with decreasing particle size.

Structural studies and influence of the structure on the electrical and optical properties of microcrystalline silicon thin films produced by RF sputtering

Microcrystalline silicon thin films were produced by reactive magnetron sputtering on glass substrates under several different conditions (RF power and gas mixture composition). The film structure was studied by X-ray diffractometry (XRD), transmission electron microscopy (TEM) and Raman spectroscopy, allowing the determination of crystal sizes, crystallinity and mechanical strain. These parameters were evaluated by fitting a pseudo-Voigt function to the X-ray data, and by the application of the strong phonon confinement model to the Raman spectra. The degree of crystallinity and the presence of single crystals or crystal agglomerates, which was confirmed by TEM, depends on the preparation conditions, and strongly affects the optical spectra and the electrical transport properties.

MarqX: a new program for whole-powder-pattern fitting

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.

Effect of sample transparency in powder diffractometry with Bragg–Brentano geometry as a convolution

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.

Flat-specimen effect as a convolution in powder diffractometry with Bragg–Brentano geometry

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.

Relationships among carbonated apatite solubility, crystallite size, and microstrain parameters.

The use of the metastable equilibrium solubility (MES) concept to describe the solubility properties of carbonated apatites (CAPs) and human dental enamel (HE) has been well established in previous studies using a range of CAPs with varying carbonate contents and crystallinities. It was shown in these studies that the mean value of the CAP MES is directly related to the broadening parameter full width at half maximum (FWHM) of the 002 reflection of the X-ray diffraction profile. The apparent solubility of the CAPs increased monotonically with an increase in the broadening of the diffraction peaks, and when this peak broadening was taken into account, carbonate had no additional effect upon the MES. The broadening of the diffraction peaks has been used as an indicator of crystallinity, and is generally influenced by both crystallite size and microstrain. The purpose of the present study was to extract the crystallite size and microstrain parameters separately from the X-ray diffraction peaks and then to determine their relationships to the corresponding MES values. The samples studied were CAPs synthesized by precipitation from Ca(NO3)2 and NaH2PO4 solutions in carbonate containing media at temperatures of 95, 80, and 70 degrees C, and powdered HE. The crystallite size and microstrain parameters were determined simultaneously with the refinement of the structural parameters with the Rietveld method of whole-pattern-fitting structure-refinement. A modified pseudo-Voigt function was used to model the observed peak profiles. The MES distributions for the CAPs and HE were determined by a previously described method. The results of this study showed that the CAPs possessed an MES distribution and therefore provided further support that MES distribution is a common phenomenon, regardless of the method of CAP synthesis. The crystallite size decreased and the microstrain increased with increasing carbonate content and decreasing temperature of synthesis of the CAPs. A plot of the mean of the MES distribution versus the microstrain parameter showed that the apparent solubility of the CAPs and HE correlated very well with the microstrain parameter. On the other hand, a plot of the mean of the MES distribution versus the crystallite size parameter showed a poor correlation between MES and crystallite size. These findings support a view that microstrain, rather than crystallite size, is the dominant factor governing the effective solubility of the CAPs and dental enamel.