Strain and Grain Size Determination of CeO2 and TiO2 Nanoparticles: Comparing Integral Breadth Methods versus Rietveld, μ-Raman, and TEM

Various crystallite size estimation methods were used to analyze X-ray diffractograms of spherical cerium dioxide and titanium dioxide anatase nanoparticles aiming to evaluate their reliability and limitations. The microstructural parameters were estimated from several integral breadth methods such as Scherrer, Monshi, Williamson–Hall, and their variants: (i) uniform deformation model, (ii) uniform strain deformation model, and (iii) uniform deformation energy density model. We also employed the size–strain plot and Halder–Wagner method. For this purpose, an instrumental resolution function of an Al2O3 standard was used to subtract the instrumental broadening to estimate the crystallite sizes and strain, and the linear regression analysis was used to compare all the models based on the coefficient of determination. The Rietveld whole powder pattern decomposition method was introduced for comparison purposes, being the best candidate to fit the X-ray diffraction data of metal-oxide nanoparticles. Refined microstructural parameters were obtained using the anisotropic spherical harmonic size approach and correlated with the above estimation methods and transmission electron microscopy images. In addition, μ-Raman spectra were recorded for each material, estimating the mean crystallite size for comparison by means of a phonon confinement model.

X-ray diffraction analysis of kaolin M1 and M2 via the Williamson–Hall and Warren–Averbach methods

Kaolin M1 and M2 studied by X-ray diffraction focus on the mullite phase, which is the main phase present in both products. The Williamson–Hall and Warren–Averbach methods for determining the crystallite size and microstrains of integral breadth β are calculated by the FullProf program. The integral breadth ( β) is a mixture resulting from the microstrains and size effect, so this should be taken into account during the calculation. The Williamson–Hall chart determines whether the sample is affected by grain size or microstrain. It appears very clearly that the principal phase of the various sintered kaolins, mullite, is free from internal microstrains. It is the case of the mixtures fritted at low temperature (1200 °C) during 1 h and also the case of the mixtures of the type chamotte cooks with 1350 °C during very long times (several weeks). This result is very significant as it gives an element of explanation to a very significant quality of mullite: its mechanical resistance during uses at high temperature remains.

IndexCub: a ready-to-use set of routines for X-ray diffraction line profile analysis

The growing interest in the use of powder X-ray diffractometry for materials’ characterization has led to the introduction of relevant concepts (e.g. microstructure, strain, anisotropy, texture) to undergraduate teaching in engineering and science. In this concern, the study of polycrystalline materials underlays the use of appropriate software: free, licensed, proprietary, or commercial to assist research on structure determination, structure refinement, and microstructure characterization. Today with the easy access to personal computers, routines for powder diffractometry also becomes feasible to use for non-specialist. Therefore, it would be relevant that students with computing knowledge may decide to improve routines on such three tasks incorporating their own computational approaches. In this study, we show the development of a ready-to-use and open source program written in GNU-Octave (v4.2.1) focused on X-ray diffraction line-profile analysis. The programing language platform was chosen mainly because of two reasons: (1) there is no requirement for commercial licenses, meaning that both programing language and routines can be downloaded online, facilitating collaborative efforts between students, instructors, and developers, and (2) easy re-coding of evaluation strategies is always allowed through fast implementation of modules into the code. The code, IndexCub, features routines for background subtraction, whole profile smoothing, and Kα2 radiation removal, location of diffraction peaks positions, indexing for cubic specimens, multi peak separation of individual peaks, and evaluation of full-width at half-maximum and integral breadth values. Microstructure properties are characterized through the use of integral breadth methods (e.g. Williamson–Hall) and Fourier analysis (e.g. Warren–Averbach), and the anisotropy effects are incorporated introducing calculations of contrast factors. In terms of diffraction domain sizes, size distribution, and the lattice microstrain, the analysis of the microstructure is discussed along with examples for polycrystalline coarse-grained materials (NaCl), epitaxial film (Si), and thin-films (Au) specimens. The code facilitates the understanding of microstructure analysis by using theoretical approaches well established and in state-steady level.

Diffraction line profiles of spherical hollow nanocrystals

An analytical expression of diffraction line profiles of spherical hollow nanocrystals (NCs) is derived. The particular features of the profile lines, enhanced peak tail intensity, are analyzed and discussed as a function of the NC size parameters (outer and inner radius, shell thickness). The explicit formula for the integral breadth, the Fourier particle size, and the Scherrer constants are also obtained and discussed in detail. The diffraction line profiles of hollow CdS NCs of zincblende and wurtzite crystallographic structure are calculated and compared with Debye scattering profiles. The diffraction profiles of both approaches exhibit an enhanced peak tail intensity that can be considered as a fingerprint of the hollow NC structure.

Evaluation of Ductile Damage Progress of Aluminum Single Crystal with Prior Activity of Single Slip System under Tensile Loading by Using Synchrotron White X-Ray

A ductile damage progress of an aluminum single crystal with the prior activity of the single slip system under tensile loading was verified by a profile analysis using white X-ray obtained in BL28B2 beam line of SPring-8. In this study, the aluminum single crystal of the purity 6N was used as a specimen prepared in I-type geometry for tensile test. A notch was introduced into one side of the center of a parallel part of the specimen by the wire electric discharge machining. White X-ray beam, which has 50 μm in both height and width, was incident into the specimen on the Bragg angle θ of 3 degrees using energy dispersive X-ray diffraction technique. The specimen was deformed by elongation in the direction of 45°to [11 and [11 crystal orientations, respectively, and a diffraction profile of the white X-ray from Al220 plane was analyzed. In profile analysis, an instrumental function was defined in consideration both of a divergence by a slit and a response function peculiar to the energy dispersive method. The Gauss component of integral breadth related to non-uniform strain and the Cauchy component of integral breadth related to crystallite size were determined by eliminating the broadening by the instrumental function from the diffraction profile of white X-ray. As a result, the characteristics of ductile damage progress near the notch of the aluminum single crystal were inspected from the distribution of both non-uniform strain and dislocation density.

Study on Ductile Damage Progress of Aluminum Single Crystal Using Synchrotron White X-Ray

A ductile damage progress of FCC single crystal was verified by a profile analysis using white X-ray obtained in BL28B2 beam line of SPring-8. In this study, an aluminum single crystal of the purity 6N was used as a specimen prepared in I-type geometry for tensile test. A notch was introduced into one side of the center of a parallel part of the specimen by the wire electric discharge machining. White X-ray, which has 100 microns in height and 200 microns in width, was incident into the specimen on the Bragg angle θ of 3 degrees using energy dispersive X-ray diffraction technique. The specimen was deformed by elongation along crystal orientation [001], and a diffraction profile of the white X-ray which penetrated it was analyzed. In profile analysis, an instrumental function was defined in consideration both of a divergence by a slit and a response function peculiar to the energy dispersive method. The Gauss component of integral breadth related to non-uniform strain and the Cauchy component of integral breadth related to crystallite size were determined by eliminating the broadening by the instrumental function from the diffraction profile of white X-ray. As a result, the direction of progress and the characteristics of ductile damage near the notch of the aluminum single crystal were clarified from the Gauss component and the Cauchy component of integral width of the single diffraction profile.

Notes on the order-of-reflection dependence of microstrain broadening

In order to obtain systematic insight into the different manifestations of microstrain broadening in powder diffraction patterns, the consequences of the breakdown of the Stokes–Wilson approximation (negligible strain gradient in a stack of lattice planes) were investigated. To this end, a phenomenological approach for the decay of the variance of the microstrain with increasing correlation distanceL, 〈∊L2〉, was adopted, as well as a Gaussian microstrain distribution for eachL. For the case of anL-independent 〈∊L2〉 (i.e.the Stokes–Wilson approximation) the (Gaussian) microstrain distribution directly shows up (is affinely mapped) on the diffraction angle scale as well as on the length of the diffraction vector scale. Furthermore, the integral breadth (on the length of the diffraction vector scale) then increases linearly with the order of reflection or, expressed another way, with the length of the diffraction vector (i.e.the integral breadth increases with tanθ on the diffraction angle 2θ scale). For the case of a decay of 〈∊L2〉 withL, deviations from such simple behaviour of the integral breadth occur. In particular, Lorentzian line broadening and nonlinear dependence of the integral breadth on the length of the diffraction vector scale (i.e.non-tanθ dependence on the 2θ scale) are induced. It is argued that the approaches used for the description of microstrain broadening in many procedures (integral breadth and Rietveld refinement) are of limited validity,i.e.they do not warrant general unverified application.

X-Ray Diffraction Analysis on Crystallite Development of Nanostructured Aluminium

This study is to investigate the crystallite development in nanostructured aluminium using x-ray line broadening analysis. Nanostructured aluminium was produced by equal channel angular extrusion at room temperature to a total deformation strain of ~17. Samples of the extruded metal were then heat treated at temperatures up to 300oC. High order diffraction peaks were obtained using Mo radiation and the integral breadth was determined. It was found that as the annealing temperature increased, the integral breadth of the peak reflections decreased. By establishing the modified Williamson-Hall plots (integral breadth vs contract factor) after instrumental correction, it was determined that the crystallite size of the metal was maintained ~80 nm at 100oC. As the annealing temperature increased to 200oC, the crystallite size increased to ~118 nm. With increasing annealing temperature, the hardness of the metal decreased from ~60 HV to ~45 HV.

Line broadening analysis using integral breadth methods: a critical review

Integral breadth methods for line profile analysis are reviewed, including modifications of the Williamson–Hall method recently proposed for the specific case of dislocation strain broadening. Two cases of study, supported by the results of a TEM investigation, are considered in detail: nanocrystalline ceria crystallized from amorphous precursors and highly deformed nickel powder produced by extensive ball milling. A further application concerns a series of Fe–Mo powder specimens that were ball milled for increasing time. Traditional and modified Williamson–Hall methods confirm their merits for a rapid overview of the line broadening effects and possible understanding of the main causes. However, quantitative results are generally not reliable. Limits in the applicability of integral breadth methods and reliability of the results are discussed in detail.