Eigen-System Analysis of X-ray Diffraction Profile Deconvolution Methods Explains Ill-Conditioning

Armstrong, N.; Kalceff, W.

doi:10.1107/s0021889897019638

Cited by 10 publications

(6 citation statements)

References 17 publications

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“…The ill-posedness is manifested in the oscillatory character of the computed solutions (e.g. Armstrong & Kalceff, 1998), well known in crystallography, unless regularizing algorithms (e.g. Tikhonov et al, 1995) are used, as in this work (Kojdecki, 2001).…”

Section: Aim Of Investigationmentioning

confidence: 99%

Crystalline microstructure of sepiolite influenced by grinding

et al. 2005

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The crystalline microstructure of ground sepiolite has been investigated. A reference sample of sepiolite and products of its comminution by dry grinding were studied through X-ray diffraction pattern analysis, specific surface measurements by nitrogen adsorption and complementary analysis of field emission scanning electron microscope images. A statistical model of polycrystals was applied to describe and determine the crystalline microstructure of the studied specimens. The model parameters characterizing the microstructure were prevalent crystallite shape, volume-weighted crystallite size distribution and second-order crystalline lattice strain distribution, and they were determined for each sample by modelling a selected part of the X-ray diffraction pattern and fitting the simulated pattern to a measured one. A strict correlation of microstructure parameters with grinding time and with specific surface magnitudes was observed. A parallelepiped with edge-length ratios almost independent of grinding time (for longer times) was found to be the predominant crystallite shape. The crystallite size distributions were found to be close to logarithmic normal ones, with the mean values decreasing with increasing grinding time and the standard-deviation-to-mean-value ratios approximately constant. The second-order crystalline lattice strain distributions were found to be close to some simple function with the mean value equal to zero, the mean deviation increasing with increasing grinding time and the standard-to-mean-deviation ratios approximately constant. It was demonstrated that the specific surface can be calculated on the basis of the microstructure characteristics. Some details of the relation between crystallites and crystalline grains were explained by comparing the results of analyses via X-ray diffraction and scanning electron microscopy.

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Section: Aim Of Investigationmentioning

confidence: 99%

Crystalline microstructure of sepiolite influenced by grinding

et al. 2005

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“…Bayesian/maximum entropy (MaxEnt) methods have been used with considerable success for data analysis in a range of inverse problems, including the restoration of astronomical images (Skilling & Bryan, 1984), particlesize distribution calculations from small-angle X-ray scattering (Mu È ller & Hansen, 1994;Mu È ller et al, 1996) and deconvolution of neutron diffraction pro®les, neutron re¯ectivity analysis, and structure-factor determination from powder data (Sivia, 1990;Sivia et al, 1993;Sivia & David, 1994;Geoghegan et al, 1996).³ The advantages of MaxEnt deconvolution over more established techniques in X-ray data analysis have been shown by Kalceff et al (1995), while the eigen-system analysis of Armstrong & Kalceff (1998) has explained the causes of ill-conditioning in the established techniques.…”

Section: Introductionmentioning

confidence: 99%

A maximum entropy method for determining column-length distributions from size-broadened X-ray diffraction profiles

Armstrong

Kalceff

1999

J Appl Cryst

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We present a novel application of the maximum entropy (MaxEnt) method for solving the double-inverse problems of removing instrument broadening from Xray diffraction pro®les and calculating the columnlength distribution of the crystallites. The MaxEnt approach is shown to have compelling advantages over the conventional methods it replaces: it is stable and robust, incorporates noise and a priori information into the solution, preserves positivity of the solution, and can be applied successively. We also show how uncertainties in the derived pro®les and column distributions can be determined and used in subsequent calculations, including integral breadth, Fourier coef®cients, column-length distributions and apparent particle sizes. Calculations are performed on simulated X-ray diffraction pro®les for a range of particle sizes, with a detailed study of the sensitivity of the results to background-level estimates and the use of an incorrect instrument response function. Theory Crystallite size broadening of pro®lesThe models for size broadening of X-ray diffraction pro®les in polycrystalline materials described by Bertaut 600

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“…This implies that the column vectors of K are (nearly all) linearly dependent, which has dire consequences, as any attempt to determine P ( D ) (given g ( s ), K ( s , D ), σ and b ( s )), produces a set of solutions { P ( D )} rather than a unique solution. The presence of statistical noise in the data simply worsens the situation, in that the ill-conditioning of K ( s , D ) amplifies the noise and the solution is swamped by spurious and unphysical oscillations [ 11 ]. Faced with this situation, the following question arises: How do we develop a method to extract a unique P( D ) from g(s), given our knowledge of K(s, D ), b(s) and σ 2 ?…”

Section: Bayesian and Maximum Entropy Methodsmentioning

confidence: 99%

“…amplifies the noise and the solution is swamped by spurious and unphysical oscillations (Armstrong & Kalceff 1998). Faced with this situation, the following question arises:…”

Section: The Uniqueness Of P (D)mentioning

confidence: 99%

Bayesian inference of nanoparticle-broadened X-ray line profiles

Armstrong¹,

Kalceff²,

Cline³

et al. 2004

J. Res. Natl. Inst. Stand. Technol.

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The analysis of x-ray line profile broadening can be considered as solving a series of inverse problems. There are usually two steps-removing the instrumental contribution (deconvolution), and determining the broadening contribution in terms of crystallite size and microstrain. Here we are concerned with quantifying only the size broadening, in terms of the shape and size distributions of the crystallites. We present a method that removes the instrumental broadening and determines the particle size distribution in a single step. The general theoretical framework developed makes it possible to determine the crystallite shape and average dimensions, and to fully quantify these results by also assigning uncertainties to them.In general, there are two approaches that can be adopted. The first assumes functional forms for the size distribution and shape of the crystallites, and applies least squares fitting to determine the parameters defining the size distribution [1,2]. For pragmatic reasons, this approach is often used to ensure numerical stability; however, it is based on an explicit assumption for the crystallite size distribution and does not take into account the non-uniqueness of the solution.The second approach takes into account the nonuniqueness of the problem of determining the size distribution P(D) from the experimental data, by assigning a probability to the solutions and enabling an average solution to be determined from the set of solutions; moreover, it also allows any a priori information and assumptions to be included and tested. This approach is

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Eigen-System Analysis of X-ray Diffraction Profile Deconvolution Methods Explains Ill-Conditioning

Cited by 10 publications

References 17 publications

Crystalline microstructure of sepiolite influenced by grinding

Crystalline microstructure of sepiolite influenced by grinding

A maximum entropy method for determining column-length distributions from size-broadened X-ray diffraction profiles

Bayesian inference of nanoparticle-broadened X-ray line profiles

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