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Quantitative Phase Analysis Using the Rietveld Method
Journal of Applied Crystallography
Abstract
Quantitative phase analysis of multicomponent mixtures using X-ray powder diffraction data has been approached with a modified version of the Rietveld computer program of Wiles & Young [J. Appl. Cryst. (1981), 14, 149–151]. This new method does not require measurement of calibration data nor the use of an internal standard; however, the approximate crystal structure of each phase of interest in a mixture is necessary. The use of an internal standard will allow the determination of total amorphous phase content in a mixture. Analysis of synthetic mixtures yielded high-precision results, with errors generally less than 1.0% absolute. Since this technique fits the complete diffraction pattern, it is less susceptible to primary extinction effects and minor amounts of preferred orientation. Additional benefits of this technique over traditional quantitative analysis methods include the determination of precise cell parameters and approximate chemical compositions, and the potential for the correction of preferred orientation and microabsorption effects.
... Comparative application of UNI standards on Alkali-Aggregate reaction by the Whole Powder Pattern Fitting (WPPF) using the Rietveld method (Bish and Howard, 1988;Young, 1993). Crystallographic structures were selected on purpose from the literature. ...
The study focused on natural aggregates used in concrete, sourced from six different production sites in Italy, and assessed their potential alkali-silica reactivity through two national standard methods. The first method, UNI 11530, involved a detailed petrographic examination, utilizing modal analysis for identifying and quantifying potentially alkali-reactive lithotypes. The second test method, UNI 11604, involved an accelerated expansion test at 38°C and high relative humidity on concrete including an established combination of size fractions of aggregates. This method allows the assessment of aggregate reactivity based on concrete expansion after 1 year of testing and an expansion limit value of 0.04%. A correlation analysis was done on the results from both test methods aimed to establish a relationship between the type and content of potentially reactive lithotypes of aggregates and concrete expansion. Chert and Chalcedony were identified as the primary contributors to concrete expansion. The sum of Chert and Chalcedony contents in the combined aggregates was found to vary between 1.8 vol% and 10.1 vol%. A critical content of approximately 5.5 % vol for the sum of Chert and Chalcedony was identified for deleterious expansion development in concrete subjected to UNI 11604 test method. This limit proves valuable in predicting the expansive behavior of aggregates where petrographic analysis has identified the presence of lithotypes potentially reactive to alkalis.
... The Rietveld refinement procedure refines the background, profile coefficients, lattice parameters, linear absorption coefficients, and other variables in a least squares procedure to ensure the calculated pattern best fits the experimental data. Each phase's data is calculated separately, and their relations are used to calculate the phase fractions, which are mainly directly related to the calculated scale factor (proportionality correction) [54,55]. Phase fraction calculations are only performed for bulk samples that are randomly oriented due to the significant uncertainties introduced by preferred orientation. ...
At the interface of thin film development and powder metallurgy technologies, this study aims to characterise the mechanical properties, lattice constants and phase formation of Ti-Nb alloys (8–30 at.%) produced by different manufacturing methods, including conventional powder metallurgy (PM), mechanical alloying (MA) and high power impulse magnetron sputtering (HiPIMS). A central aspect of this research was to investigate the different energy states achievable by each synthesis method. The findings revealed that as the Nb content increased, both the hardness and Young’s modulus of the PM samples decreased (from 4 to 1.5 and 125 to 85 GPa, respectively). For the MA alloys, the hardness and Young’s modulus varied between 3.2 and 3.9 and 100 to 116 GPa, respectively, with the lowest values recorded for 20% Nb (3.2 and 96 GPa). The Young’s modulus of the HiPIMS thin film samples did not follow a specific trend and varied between 110 and 138 GPa. However, an increase in hardness (from 3.6 to 4.8 GPa) coincided with an increase in the β2 phase contribution for films with the same chemical composition (23 at.% of Nb). This study highlights the potential of using HiPIMS gradient films for high throughput analysis for PM and MA techniques. This discovery is important as it provides a way to reduce the development time for complex alloy systems in biomaterials as well as other areas of materials engineering.
Graphical abstract
... A Bruker D8 DISCOVER (Bruker AXS, Frankfurt am Main, Germany) X-ray diffractometer with CuKα radiation was used for the phase analysis of the synthesized layers. The content of the phases in the surface layer was determined according to the X-ray diffraction patterns using the Rietveld method, using the FullProf software [72][73][74]. ...
Oxide layers on the surface of the aluminum alloys D16 and AMg6 and on arc coatings sprayed with electrode wires made of the alloys D16 and AMg6 were synthesized using plasma electrolytic oxidation (PEO). The microstructure, phase composition and micro-hardness of the PEO layers were studied. In addition to the two main phases (α-Al2O3 and γ-Al2O3), a small amount of a metastable crystalized Al2.427O3.64 phase was found in their structure. A comparison was made of the wear resistance and friction coefficients of the synthesized PEO layers during friction tests in pairs with other PEO layers, a galvanic chromium coating, cast iron, steels and bronze of the BrC30 type. The results of the friction tests for the various PEO layers on aluminum alloys in tribo-contact with high-hardness elements made of hardened or chrome-plated steel justify the possibility of their use in a friction pair. We experimentally showed the influence of glycerin additive in motor oil 15W30 on the change in the friction coefficient of the PEO layers synthesized on the aluminum alloys and on the arc-sprayed coatings on their surfaces in tribocouples with hardened steel.
... Mineral phase identification was conducted using the DIFFRAC.EVA XRD phase analysis software (Bruker) with reference to the International Center for Diffraction Data Powder Diffraction File 4+ database (ICDD PDF4+). Rietveld refinements (Rietveld, 1969;Hill and Howard, 1987;Bish and Howard, 1988) with XRD data were used to determine mineral abundances using the TOPAS 5 software (Bruker). Fundamental-parameters peak fitting (Cheary and Coelho, 1992) was used for all phases. ...
For more than a decade, the CheMin X-ray diffraction instrument on the Mars Science Laboratory rover, Curiosity, has been returning definitive and quantitative mineralogical and mineral–chemistry data from ~3.5-billion-year-old (Ga) sediments in Gale crater, Mars. To date, 40 drilled rock samples and three scooped soil samples have been analyzed during the rover’s 30+ km transit. These samples document the mineralogy of over 800 m of flat-lying fluvial, lacustrine, and aeolian sedimentary rocks that comprise the lower strata of the central mound of Gale crater (Aeolis Mons, informally known as Mt. Sharp) and the surrounding plains (Aeolis Palus, informally known as the Bradbury Rise). The principal mineralogy of the sedimentary rocks is of basaltic composition, with evidence of post-depositional diagenetic overprinting. The rocks in many cases preserve much of their primary mineralogy and sedimentary features, suggesting that they were never strongly heated or deformed. Using aeolian soil composition as a proxy for the composition of the deposited and lithified sediment, it appears that, in many cases, the diagenetic changes observed are principally isochemical. Exceptions to this trend include secondary nodules, calcium sulfate veining, and rare Si-rich alteration halos. A surprising and yet poorly understood observation is that nearly all of the ~3.5 Ga sedimentary rocks analyzed to date contain 15–70 wt.% of X-ray amorphous material. Overall, this >800 m section of sedimentary rock explored in lower Mt. Sharp documents a perennial shallow lake environment grading upward into alternating lacustrine/fluvial and aeolian environments, many of which would have been habitable to microbial life.
Diffracted intensities from axially symmetric flat-plate or capillary specimens, composed of effectively rod- or disk-shaped crystallites, can be corrected for preferred orientation with a single pole-density profile. A convenient procedure is to approximate this profile with a function whose variable parameters are fit during least-squares structure refinement. Several functions have previously been suggested but without theoretical justification. The present study reviews the derivation of this method and examines its assumptions and applications. The several proposed functions are compared with each other and with the March function which describes the pole-density distribution produced by rigid-body rotation of inequant crystallites (i.e. crystallites with unequal sides) upon axially symmetric volume-conserving compression or expansion. For its basis, ease of use, single variable parameter, direct interpretability and good refinement test results, the March distribution is proposed as an advantageous pole-density profile function for general use.
The Rietveld [J. Appl. Cryst. (1969), 2, 65–71] method of structure refinement from powder diffraction patterns is widely used with neutron data and increasingly so with X-ray data. A computer program for the application of the method with X-ray data, or with neutron nuclear scattering data, has been written ab initio in an effort to make it versatile, user oriented, portable, convenient, and largely self-contained yet reasonably comprehensive. It is of modular construction to facilitate user-desired modifications, one of which may be additions to the choice of four profile functions now included. The program accommodates either one or two wavelengths (e.g.α1 and α2 or λ and λ/2). It permits simultaneous refinement of the structures of two phases and the background.
The whole-pattern least-squares fitting method (herein called the pattern-fitting structure-refinement, PFSR, method) developed by Rietveld and used by many with neutron powder data has been extended for use with X-ray powder diffractometer data. Examples of PFSR for specimens in five different space groups are given. Structural parameters generally agree, within 2σ, with those from single-crystal results where available. Atom positions and site occupancies were determined for LaPO4 Ca5PO4)3F, and quartz. R(pattern) values generally ranged from 9 to 20% while R(Bragg) values were a few percent lower. Reflection profiles were represented by either Gaussian or Cauchy functions: better representations are shown to be needed with an expected result of improved sensitivity accompanying lower R values. Possibilities for improved pattern-stripping for minor phase study via PFSR of the major phase are exhibited. Remarkably precise lattice parameters, e.g. 1 part in 35000 from data all below 2θ < 77° in the fluorapatite case, and parametric characterization of the diffraction broadening of the Bragg profiles are obtained as additional benefits.
A matrix-flushing method for quantitative multicomponent analysis by X-ray diffraction is reported. It is simpler and faster than, yet as reliable as, the conventional internal-standard method. In this new method, the calibration-curve procedure is shunted; a more fundamental `matrix-flushing' concept is introduced. The matrix-flushing theory gives an exact relationship between intensity and concentration free from matrix effect. Contrary to most theoretical methods the working equation is very simple, no complicated calculations are involved. The matrix-flushing theory and the analytical procedure are presented. Eight illustrative examples are drawn to demonstrate how this theory is applied to multicomponent analysis and amorphous-content determination. A novel `auto-flushing' phenomenon of binary systems was observed which appears to make the analysis of any binary system a simple matter.
A structure refinement method is described which does not use integrated neutron powder intensities, single or overlapping, but employs directly the profile intensities obtained from step-scanning measurements of the powder diagram. Nuclear as well as magnetic structures can be refined, the latter only when their magnetic unit cell is equal to, or a multiple of, the nuclear cell. The least-squares refinement procedure allows, with a simple code, the introduction of linear or quadratic constraints between the parameters.
All the information relating to the quantitative composition of a mixture is coded and stored in its X-ray diffraction pattern. It has been the goal of X-ray diffraction analysts since the discovery of X-rays to retrieve and decode this information directly from the X-ray diffraction pattern rather than resort to calibration curves or internal standards. This goal appears to be attained by the application of the `matrix-flushing theory' and the now-proposed `adiabatic principle' in applied X-ray diffraction analysis. The matrix-flushing theory offers a simple intensity-concentration equation free from matrix effects which degenerates to `auto-flushing' for binary systems. The adiabatic principle establishes that the intensity–concentration relationship between each and every pair of components in a multi component system is not perturbed by the presence or absence of other components. A key equation is derived which conducts the decoding process. Both the matrix-flushing theory and the adiabatic principle are experimentally verified.
The refinement of crystal structures using X-ray powder data in a two-stage method is described. (1) The integrated intensities of the individual reflections are derived by a profile fitting method in which the profile shapes are accurately defined using an experimentally determined instrument function and the sum of Lorentzian curves. (2) These values are then used in a powder least-squares refinement for structure determination. The results obtained with three simple structures (silicon, quartz and corundum) gave R (Bragg) values of 0.7 to 2.5%. The necessity of correcting for preferred orientation and the importance of proper specimen preparation are also discussed.