Estimation of chord length distributions from small-angle scattering using indirect Fourier transformation

2003 ◽  
Vol 36 (5) ◽  
pp. 1190-1196 ◽  
Author(s):  
Steen Hansen
Keyword(s):  
Small Angle ◽  
Fourier Transformation ◽  
Length Distribution ◽  
Small Angle Scattering ◽  
Chord Length ◽  
Scattering Data ◽  
Additional Tool ◽  
Additional Information ◽  
Chord Length Distribution ◽  
Angle Scattering

It is shown that it is possible to estimate the chord length distribution from small-angle scattering data by indirect Fourier transformation. This is done for several examples of scatterers varying in structure from globular to elongated as well as scatterers consisting of separated parts. The presented examples suggest that the chord length distribution may give additional information about the scatterer. Therefore it may be relevant to consider estimation of the chord length distribution as an additional tool for analysis of small-angle scattering data.

Chord length distribution of pentagonal and hexagonal rods: relation to small-angle scattering

2009 ◽  
Vol 42 (2) ◽  
pp. 326-328 ◽  
Author(s):  
Wilfried Gille ◽  
Narine G. Aharonyan ◽  
Hrachya S. Harutyunyan
Keyword(s):  
Small Angle ◽  
Length Distribution ◽  
Small Angle Scattering ◽  
Chord Length ◽  
Density Functions ◽  
Explicit Formulas ◽  
Chord Length Distribution ◽  
Regular Pentagon ◽  
Angle Scattering

Based on explicit formulas of chord length density functions (CLDs) for a regular pentagon and a hexagon, the CLDs of infinitely long regular homogeneous pentagonal/hexagonal cylinders are discussed. Characteristic properties of the small-angle scattering of these cylinders are studied.

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.

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.

scholarly journals Prospects for investigating deterministic fractals: Extracting additional information from small angle scattering data

2012 ◽  
Vol 393 ◽  
pp. 012032
Author(s):  
A Yu Cherny ◽  
E M Anitas ◽  
V A Osipov ◽  
A I Kuklin
Keyword(s):  
Small Angle ◽  
Small Angle Scattering ◽  
Scattering Data ◽  
Additional Information ◽  
Angle Scattering

scholarly journals Deterministic fractals: Extracting additional information from small-angle scattering data

2011 ◽  
Vol 84 (3) ◽  
Author(s):  
A. Yu. Cherny ◽  
E. M. Anitas ◽  
V. A. Osipov ◽  
A. I. Kuklin
Keyword(s):  
Small Angle ◽  
Small Angle Scattering ◽  
Scattering Data ◽  
Additional Information ◽  
Angle Scattering

scholarly journals Chord-length distributions cannot generally be obtained from small-angle scattering

2020 ◽  
Vol 53 (1) ◽  
pp. 127-132
Author(s):  
Cedric J. Gommes ◽  
Yang Jiao ◽  
Anthony P. Roberts ◽  
Dominique Jeulin
Keyword(s):  
Porous Materials ◽  
Small Angle ◽  
Hollow Spheres ◽  
Boolean Model ◽  
Small Angle Scattering ◽  
Chord Length ◽  
Scattering Data ◽  
Angle Scattering ◽  
Dilute Suspension ◽  
Geometrically Complex

The methods used to extract chord-length distributions from small-angle scattering data assume a structure consisting of spatially uncorrelated and disconnected convex regions. These restrictive conditions are seldom met for a wide variety of materials such as porous materials and semicrystalline or phase-separated copolymers, the structures of which consist of co-continuous phases that interpenetrate each other in a geometrically complex way. The significant errors that would result from applying existing methods to such systems are discussed using three distinct models for which the chord-length distributions are known analytically. The models are a dilute suspension of hollow spheres, the Poisson mosaic and the Boolean model of spheres.

The small-angle scattering correlation function of the cuboid

1999 ◽  
Vol 32 (6) ◽  
pp. 1100-1104 ◽  
Author(s):  
Wilfried Gille
Keyword(s):  
Correlation Function ◽  
Asymptotic Behaviour ◽  
Small Angle ◽  
Length Distribution ◽  
Chord Length ◽  
Scattering Intensity ◽  
Chord Length Distribution ◽  
Angle Scattering ◽  
Higher Derivatives ◽  
Derivatives Of

The analytical expression of the correlation function γ(r) for the cuboid with edgesa,bandcis established. The calculation is based on the chord-length distribution. Details of these structure functions at essentialrpositions are analysed, including higher derivatives of the correlation function at the maximum chord length. The result was checked on closer analysis of the corresponding scattering intensityI(h) and its asymptotic behaviourI∞(h).

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 analysis of the spherical half-shell

2007 ◽  
Vol 40 (2) ◽  
pp. 302-304 ◽  
Author(s):  
Wilfried Gille
Keyword(s):  
Correlation Function ◽  
Small Angle ◽  
Pair Correlation ◽  
Length Distribution ◽  
Small Angle Scattering ◽  
Scattering Intensity ◽  
Chord Length Distribution ◽  
Angle Scattering ◽  
Analytic Expressions

For a spherical half-shell (SHS) of diameter D, analytic expressions of the small-angle scattering correlation function \gamma_0(r), the chord length distribution (CLD) and the scattering intensity are analyzed. The spherically averaged pair correlation function p_0(r)\simeq r^2\gamma_0(r) of the SHS is identical to the cap part of the CLD of a solid hemisphere of the same diameter. The surprisingly simple analytic terms in principle allow the determination of the size distribution of an isotropic diluted SHS collection from its scattering intensity.

Sign in / Sign up

Export Citation Format

Share Document