Application of the method of moments to X-ray diffraction line profiles from paracrystals: Native cellulose fibres

Mitra, G. B.; Mukherjee, P.

doi:10.1107/s0021889881009692

Cited by 15 publications

(3 citation statements)

References 26 publications

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“…The refinement procedure is continued until a good agreement between the calculated and measured profiles is achieved. Afterward, determination of crystallite size and dislocation density was performed using the Williamson–Hall (WH) (Williamson and Hall, 1953), Warren–Averbach (WA) (Warren and Averbach, 1950), modified Williamson–Hall (MWH), modified Warren–Averbach (MWA) (Ungar and Borbely, 1996), and Variance (Wilson, 1962; Mitra and Mukherjee, 1981) methods. Finally, thermal analysis of the milled powders was performed in a Metler Toledo scanning calorimeter (DSC1); with a slow heating rate of 5 K min −1 , up to 700°C.…”

Section: Methodsmentioning

confidence: 99%

An investigation on the microstructure and defects in the mechanically milled Cu and Fe powders

et al. 2014

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The microstructural characteristics of mechanically milled (MM) iron (Fe) and copper (Cu) powders are investigated by means of various X-ray crystallography analysis methods. The conventional Williamson-Hall and Warren-Averbach methods are used besides the modified Williamson-Hall, the modified Warren-Averbach, and the Variance approaches, in proper cases. Afterward, the obtained crystallite size and dislocation density are used to calculate the stored energy in the nanostructured powders. For this aim, a new geometrical approach is developed which can consider three-dimensional crystallites and the thickness of boundaries between them. Moreover, the released energy during annealing of MM Cu and Fe powders is measured using differential scanning calorimetry. The results of line broadening analysis and geometrical modelling are combined to the calorimetry of a room temperature aged Cu powder. In this way, the thickness of grain boundary in the nanostructured Cu is calculated to be 1.6 nm.

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Section: Methodsmentioning

confidence: 99%

An investigation on the microstructure and defects in the mechanically milled Cu and Fe powders

et al. 2014

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“…There are several methods for extracting information on the microstructure of a sample from X-ray data (Stokes, 1948; Warren and Averbach, 1950). One of the traditional techniques is the variance method, developed by Wilson (1962a), Langford (1968a, 1968b), and Mitra and Mukherjee (1981). The variance of a diffraction line in real space can be defined as where 2 θ 0 is the centroid position of the line intensity distribution and σ is the range, symmetrically measured from the centroid.…”

Section: Introductionmentioning

confidence: 99%

Comparison methods of variance and line profile analysis for the evaluation of microstructures of materials

Soleimanian

Aghdaee

2008

Powder Diffr.

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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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“…In general, these lattice imperfections are classified into two types: nanometresized crystals and lattice defects (classified in turn into point, line, or planar defects) (Klug & Alexander, 1974;Warren, 1990;Snyder et al, 1999;Mittemeijer & Scardi, 2004;Ungá r, 2004). One method of linebroadening analysis is the variance method (Tournarie, 1956;Wilson, 1962a,b;Langford, 1968a,b;Klug & Alexander, 1974;Mitra & Mukherjee, 1981;Sá nchez-Bajo & Cumbrera, 1997), wherein the mean values of the crystal sizes and lattice microstrains are computed from the variance coefficients of the line profiles. The variance of the line profiles [W()] is defined as the second central moment of the distribution of diffracted intensities [I(2)] (Wilson, 1962a,b), and is therefore a measure of the line broadening.…”

Section: Introductionmentioning

confidence: 99%

Analytical formulation of the variance method of line-broadening analysis for Voigtian X-ray diffraction peaks

2006

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An alternative formulation of the variance method for the line-broadening analysis of polycrystalline materials is presented. It maintains the theoretical basis of the earlier formulations of the variance method, but differs in the manner of calculating the variance coefficients of the line profiles. In the proposed formulation, these are evaluated analytically in terms of the shape parameters of Voigt functions fitted to the X-ray diffraction data. Explicit expressions are thus derived for calculating the (surface-weighted) crystal sizes and (root-mean-square) lattice microstrains from the integral breadths of the Gauss and Lorentz components of the Voigt functions that model the experimental and instrumental line profiles.

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Application of the method of moments to X-ray diffraction line profiles from paracrystals: Native cellulose fibres

Cited by 15 publications

References 26 publications

An investigation on the microstructure and defects in the mechanically milled Cu and Fe powders

An investigation on the microstructure and defects in the mechanically milled Cu and Fe powders

Comparison methods of variance and line profile analysis for the evaluation of microstructures of materials

Analytical formulation of the variance method of line-broadening analysis for Voigtian X-ray diffraction peaks

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