Two-dimensional indirect Fourier transformation for evaluation of small-angle scattering data of oriented samples

2013 ◽  
Vol 46 (5) ◽  
pp. 1447-1454 ◽  
Author(s):  
Gerhard Fritz-Popovski
Keyword(s):  
Small Angle ◽  
Fourier Transformation ◽  
Real Space ◽  
Small Angle Scattering ◽  
Basis Functions ◽  
Scattering Data ◽  
Normal Wood ◽  
Two Dimensional ◽  
Model Free ◽  
Angle Scattering

An extension of the indirect Fourier transformation method for two-dimensional small-angle scattering patterns is presented. This allows for a model-free investigation of real-space functions of oriented structures. The real-space function is built from two-dimensional basis functions. The Fourier transformed basis functions are approximated to the scattering pattern. The solution to this problem in reciprocal space can be used to compute the corresponding real-space functions. These real-space functions contain information on size, shape, internal structure and orientation of the structures studied. Information on structures that are oriented in different distinct directions can be partly separated. The applicability of the technique is demonstrated on simulated data of oriented cuboids and on two experimental data sets based on the nanostructure of spruce normal wood.

Interpretation of two-dimensional real-space functions obtained from small-angle scattering data of oriented microstructures

2015 ◽  
Vol 48 (1) ◽  
pp. 44-51 ◽  
Author(s):  
Gerhard Fritz-Popovski
Keyword(s):  
Small Angle ◽  
Fourier Transformation ◽  
Real Space ◽  
Small Angle Scattering ◽  
Scattering Data ◽  
Two Dimensional ◽  
Spruce Wood ◽  
Angle Scattering ◽  
Nanostructured Silica ◽  
Internal Architecture

The new two-dimensional indirect Fourier transformation converts small-angle scattering patterns obtained by means of area detectors into two-dimensional real-space functions. These functions contain identical information to the scattering patterns, but many parameters related to the microstructure can be obtained directly from them. The size and shape of the microstructures are mainly reflected in the contours of the real-space functions. Their height can be used to get information on the internal architecture of the microstructures. The principles are demonstrated on nanostructured silica biotemplated by spruce wood.

Small-angle scattering of interacting particles. II. Generalized indirect Fourier transformation under consideration of the effective structure factor for polydisperse systems

1999 ◽  
Vol 32 (2) ◽  
pp. 197-209 ◽  
Author(s):  
B. Weyerich ◽  
J. Brunner-Popela ◽  
O. Glatter
Keyword(s):  
Form Factor ◽  
Small Angle ◽  
Structure Factor ◽  
Fourier Transformation ◽  
Hard Spheres ◽  
Small Angle Scattering ◽  
Scattering Data ◽  
Model Free ◽  
Angle Scattering ◽  
The Gift

The indirect Fourier transformation (IFT) is the method of choice for the model-free evaluation of small-angle scattering data. Unfortunately, this technique is only useful for dilute solutions because, for higher concentrations, particle interactions can no longer be neglected. Thus an advanced technique was developed as a generalized version, the so-called generalized indirect Fourier transformation (GIFT). It is based on the simultaneous determination of the form factor, representing the intraparticle contributions, and the structure factor, describing the interparticle contributions. The former can be determined absolutely free from model assumptions, whereas the latter has to be calculated according to an adequate model. In this paper, various models for the structure factor are compared,e.g.the effective structure factor for polydisperse hard spheres, the averaged structure factor, the local monodisperse approximation and the decoupling approximation. Furthermore, the structure factor for polydisperse rod-like particles is presented. As the model-free evaluation of small-angle scattering data is an essential point of the GIFT technique, the use of a structure factor without any influence of the form amplitude is advisable, at least during the first evaluation procedure. Therefore, a series of simulations are performed to check the possibility of the representation of various structure factors (such as the effective structure factor for hard spheres or the structure factor for rod-like particles) by the less exact but much simpler averaged structure factor. In all the observed cases, it was possible to recover the exact form factor with a free determined parameter set for the structure factor. The resulting parameters of the averaged structure factor have to be understood as apparent model parameters and therefore have only limited physical relevance. Thus the GIFT represents a technique for the model independent evaluation of scattering data with a minimum ofa prioriinformation.

SLADS: a parallel code for direct simulations of scattering of large anisotropic dense nanoparticle systems

2017 ◽  
Vol 50 (3) ◽  
pp. 951-958 ◽  
Author(s):  
Sen Chen ◽  
Juncheng E ◽  
Sheng-Nian Luo
Keyword(s):  
Analytical Solutions ◽  
Small Angle ◽  
Real Space ◽  
Small Angle Scattering ◽  
Two Dimensional ◽  
X Ray ◽  
X Ray Scattering ◽  
Angle Scattering ◽  
Parallel Code ◽  
Diffraction Patterns

SLADS(http://www.pims.ac.cn/Resources.html), a parallel code for direct simulations of X-ray scattering of large anisotropic dense nanoparticle systems of arbitrary species and atomic configurations, is presented. Particles can be of arbitrary shapes and dispersities, and interactions between particles are considered. Parallelization is achieved in real space for the sake of memory limitation. The system sizes attempted are up to one billion atoms, and particle concentrations in dense systems up to 0.36. Anisotropy is explored in terms of superlattices. One- and two-dimensional small-angle scattering or diffraction patterns are obtained.SLADSis validated self-consistently or against cases with analytical solutions.

Bayesian estimation of hyperparameters for indirect Fourier transformation in small-angle scattering

2000 ◽  
Vol 33 (6) ◽  
pp. 1415-1421 ◽  
Author(s):  
Steen Hansen
Keyword(s):  
Bayesian Methods ◽  
Small Angle ◽  
Fourier Transformation ◽  
Posterior Probability ◽  
Small Angle Scattering ◽  
Scattering Data ◽  
Distance Distribution Function ◽  
Posterior Probability Distribution ◽  
Angle Scattering ◽  
Real Experimental Data

Bayesian analysis is applied to the problem of estimation of hyperparameters, which are necessary for indirect Fourier transformation of small-angle scattering data. The hyperparameters most frequently needed are the overall noise level of the experiment and the maximum dimension of the scatterer. Bayesian methods allow the posterior probability distribution for the hyperparameters to be determined, making it possible to calculate the distance distribution function of interest as the weighted mean of all possible solutions to the indirect transformation problem. Consequently no choice of hyperparameters has to be made. The applicability of the method is demonstrated using simulated as well as real experimental data.

Real space functions from experimental small angle scattering data

2011 ◽  
Vol 13 (13) ◽  
pp. 5872 ◽  
Author(s):  
Gerhard Fritz-Popovski ◽  
Alexander Bergmann ◽  
Otto Glatter
Keyword(s):  
Small Angle ◽  
Real Space ◽  
Small Angle Scattering ◽  
Scattering Data ◽  
Angle Scattering

An iterative method to extract the size distribution of non-interacting polydisperse spherical particles from small-angle scattering data

2014 ◽  
Vol 47 (2) ◽  
pp. 712-718 ◽  
Author(s):  
D. Sen ◽  
Avik Das ◽  
S. Mazumder
Keyword(s):  
Iterative Method ◽  
Size Distribution ◽  
Small Angle ◽  
Real Space ◽  
Colloidal Dispersions ◽  
Small Angle Scattering ◽  
Spherical Particles ◽  
Scattering Data ◽  
X Ray Scattering ◽  
Angle Scattering

In this article, an iterative method for estimating the size distribution of non-interacting polydisperse spherical particles from small-angle scattering data is presented. It utilizes the iterative addition of relevant contributions to an instantaneous size distribution, as obtained from the fractional difference between the experimental data and the simulated profile. An inverse relation between scattering vector and real space is assumed. This method does not demand the consideration of any basis function set together with an imposed constraint such as a Lagrange multiplier, nor does it depend on the Titchmarsh transform. It is demonstrated that the method works quite well in extracting several forms of distribution. The robustness of the present method is examined through the successful retrieval of several forms of distribution, namely monomodal, bimodal, trimodal, triangular and bitriangular distributions. Finally, the method has also been employed to extract the particle size distribution from experimental small-angle X-ray scattering data obtained from colloidal dispersions of silica.

SASET: a program for series analysis of small-angle scattering data

2013 ◽  
Vol 46 (4) ◽  
pp. 1187-1195 ◽  
Author(s):  
Michael Muthig ◽  
Sylvain Prévost ◽  
Reinhold Orglmeister ◽  
Michael Gradzielski
Keyword(s):  
Small Angle ◽  
Evaluation Process ◽  
Small Angle Scattering ◽  
Scattering Data ◽  
Data Series ◽  
Two Dimensional ◽  
Evaluation Tool ◽  
Fitting Procedure ◽  
Angle Scattering ◽  
Anisotropy Measurement

This article presents a new program that allows highly automatized analyses of series of, especially, anisotropic two-dimensional neutron and X-ray small-angle scattering data as well as one-dimensional data series. The main aim of this work was to reduce the effort of the analysis of complex scattering systems, which remains an essential burden in the evaluation process of complex systems. The program is built in a modular manner to support a stepwise analysis of small-angle scattering data. For example, from a two-dimensional data series, features such as anisotropy or changes of the preferred scattering direction or intensities along the radial or azimuthal directions as well as along the series axis (e.g. time axis) can quickly be extracted. Different anisotropy measurement methods are available, which are described herein. In a second step, physical scattering models can be fitted to the extracted data. More complex models can be easily added. The fitting procedure can be applied with nearly every possible constraint and works automatically on whole scattering data series. Furthermore, simultaneous fitting can be used to analyze coupled series, and parallel working methods are implemented to speed up the code execution. Finally, results can be easily visualized. The name of the program isSASET, which is an acronym standing for small-angle scattering evaluation tool.SASETis based on MATLAB.

Structure and order in lamellar phases determined by small-angle scattering

2004 ◽  
Vol 37 (5) ◽  
pp. 703-710 ◽  
Author(s):  
Thomas Frühwirth ◽  
Gerhard Fritz ◽  
Norbert Freiberger ◽  
Otto Glatter
Keyword(s):  
Form Factor ◽  
Small Angle ◽  
Structure Factor ◽  
Scattering Length ◽  
Small Angle Scattering ◽  
Scattering Data ◽  
X Rays ◽  
Model Free ◽  
Angle Scattering

Multilamellar phases can be identified and characterized by small-angle scattering of X-rays (SAXS) or neutrons (SANS). Equidistant peaks are the typical signature and their spacing allows the fast determination of the repeat distance,i.e.the mean distance between the midplane of neighbouring bilayers. The scattering function can be described as the product of a structure factor and a form factor. The structure factor is related to the ordering of the bilayers and is responsible for the typical equidistant peaks, but it also contains information about the bilayer flexibility and the number of coherently scattering bilayers. The form factor depends on the thickness and the internal structure (scattering length density distribution) of a single bilayer. The recently developed generalized indirect Fourier transformation (GIFT) method is extended to such systems. This method allows the simultaneous determination of the structure factor and the form factor of the system, including the correction of instrumental broadening effects. A few-parameter model is used for the structure factor, while the determination of the form factor is completely model-free. The method has been tested successfully with simulated scattering data and by application to experimental data sets.

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