scholarly journals Numerically stable form factor of any polygon and polyhedron

2021 ◽  
Vol 54 (2) ◽  
pp. 580-587
Author(s):  
Joachim Wuttke
Keyword(s):  
Form Factor ◽  
Small Angle ◽  
Form Factors ◽  
Small Angle Scattering ◽  
Important Application ◽  
Divergence Theorem ◽  
Series Expansions ◽  
Stable Form ◽  
Application Domain ◽  
Angle Scattering

Coordinate-free expressions for the form factors of arbitrary polygons and polyhedra are derived using the divergence theorem and Stokes's theorem. Apparent singularities, all removable, are discussed in detail. Cancellation near the singularities causes a loss of precision that can be avoided by using series expansions. An important application domain is small-angle scattering by nanocrystals.

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.

scholarly journals Analysis of small-angle scattering data using model fitting and Bayesian regularization

2018 ◽  
Vol 51 (4) ◽  
pp. 1151-1161 ◽  
Author(s):  
Andreas Haahr Larsen ◽  
Lise Arleth ◽  
Steen Hansen
Keyword(s):  
Prior Knowledge ◽  
Small Angle ◽  
Regularization Method ◽  
Prior Information ◽  
Form Factors ◽  
Small Angle Scattering ◽  
Scattering Data ◽  
Model Parameters ◽  
Bayesian Regularization ◽  
Angle Scattering

The structure of macromolecules can be studied by small-angle scattering (SAS), but as this is an ill-posed problem, prior knowledge about the sample must be included in the analysis. Regularization methods are used for this purpose, as already implemented in indirect Fourier transformation and bead-modeling-based analysis of SAS data, but not yet in the analysis of SAS data with analytical form factors. To fill this gap, a Bayesian regularization method was implemented, where the prior information was quantified as probability distributions for the model parameters and included via a functional S. The quantity Q = χ2 + αS was then minimized and the value of the regularization parameter α determined by probability maximization. The method was tested on small-angle X-ray scattering data from a sample of nanodiscs and a sample of micelles. The parameters refined with the Bayesian regularization method were closer to the prior values as compared with conventional χ2 minimization. Moreover, the errors on the refined parameters were generally smaller, owing to the inclusion of prior information. The Bayesian method stabilized the refined values of the fitted model upon addition of noise and can thus be used to retrieve information from data with low signal-to-noise ratio without risk of overfitting. Finally, the method provides a measure for the information content in data, N g, which represents the effective number of retrievable parameters, taking into account the imposed prior knowledge as well as the noise level in data.

scholarly journals Neither Sphere nor Cube - Analyzing the Particle Shape Using Small-Angle Scattering and the Superball Model

2021 ◽  
Author(s):  
Dominique Dresen ◽  
Asmaa Qdemat ◽  
Dominika Zákutná ◽  
Erik Wetterskog ◽  
Emmanuel Kentzinger ◽  
...  
Keyword(s):  
Form Factor ◽  
Small Angle ◽  
Particle Shape ◽  
Small Angle Scattering ◽  
Scattering Data ◽  
Monodisperse Nanoparticles ◽  
Statistical Relevance ◽  
Angle Scattering ◽  
Microscopy Data ◽  
User Friendly

<div>Accurate characterization of the nanocrystal shape with high statistical relevance is essential for exploiting the strongly shape-dependent properties of cuboidal nanoparticles towards applications. <br></div><div>This work presents the development of a new small-angle scattering form factor based on the superball geometry. The superball quantifies the characteristic rounding of corners and edges of cuboidalnanoparticles with a single parameter. Applied to small-angle scattering data of sufficiently monodisperse nanoparticles, the superball form factor enables differentiation between the effects of extended<br></div>particle size distribution and irregular particle shape. The quantitative application of the superball form factor is validated against microscopy data for a series of monodisperse nanoparticles and implemented into the user-friendly, open source software Sasview.

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.

scholarly journals Neither Sphere nor Cube - Analyzing the Particle Shape Using Small-Angle Scattering and the Superball Model

2021 ◽  
Author(s):  
Dominique Dresen ◽  
Asmaa Qdemat ◽  
Dominika Zákutná ◽  
Erik Wetterskog ◽  
Emmanuel Kentzinger ◽  
...  
Keyword(s):  
Form Factor ◽  
Small Angle ◽  
Particle Shape ◽  
Small Angle Scattering ◽  
Scattering Data ◽  
Monodisperse Nanoparticles ◽  
Statistical Relevance ◽  
Angle Scattering ◽  
Microscopy Data ◽  
User Friendly

<div>Accurate characterization of the nanocrystal shape with high statistical relevance is essential for exploiting the strongly shape-dependent properties of cuboidal nanoparticles towards applications. <br></div><div>This work presents the development of a new small-angle scattering form factor based on the superball geometry. The superball quantifies the characteristic rounding of corners and edges of cuboidalnanoparticles with a single parameter. Applied to small-angle scattering data of sufficiently monodisperse nanoparticles, the superball form factor enables differentiation between the effects of extended<br></div>particle size distribution and irregular particle shape. The quantitative application of the superball form factor is validated against microscopy data for a series of monodisperse nanoparticles and implemented into the user-friendly, open source software Sasview.

scholarly journals Simplified tube form factor for analysis of small-angle scattering data from carbon nanotube filled systems

2007 ◽  
Vol 40 (s1) ◽  
pp. s88-s92 ◽  
Author(s):  
Ryan S. Justice ◽  
David H. Wang ◽  
Loon-Seng Tan ◽  
Dale W. Schaefer
Keyword(s):  
Form Factor ◽  
Carbon Nanotube ◽  
Small Angle ◽  
Small Angle Scattering ◽  
Scattering Data ◽  
Angle Scattering

Generalizing small-angle scattering form factors with linear transformations

2020 ◽  
Vol 53 (5) ◽  
pp. 1387-1391
Author(s):  
Matt Thompson
Keyword(s):  
Small Angle ◽  
Form Factors ◽  
Large Data ◽  
Small Angle Scattering ◽  
Data Sets ◽  
High Symmetry ◽  
Linear Transformations ◽  
Angle Scattering ◽  
Anisotropic Structures ◽  
And Inversion

Nanostructure characterization using small-angle scattering is often performed by iteratively fitting a scattering model to experimental data. These scattering models are usually derived in part from the form factors of the expected shapes of the particles. Most small-angle-scattering pattern-fitting software is well equipped with form factor libraries for high-symmetry models, yet there is more limited support for distortions to these ideals that are more typically found in nature. Here, a means of generalizing high-symmetry form factors to these lower-symmetry cases via linear transformations is introduced, significantly expanding the range of form factors available to researchers. These linear transformations are composed of a series of scaling, shear, rotation and inversion operations, enabling particle distortions to be understood in a straightforward and intuitive way. This approach is expected to be especially useful for in situ studies of nanostructure growth where anisotropic structures change continuously and large data sets must be analysed.

Scatter: software for the analysis of nano- and mesoscale small-angle scattering

2010 ◽  
Vol 43 (3) ◽  
pp. 639-646 ◽  
Author(s):  
S. Förster ◽  
L. Apostol ◽  
W. Bras
Keyword(s):  
Small Angle ◽  
Calculated Data ◽  
Form Factors ◽  
Real Space ◽  
Small Angle Scattering ◽  
Scattering Data ◽  
Mathematical Treatment ◽  
Mesoscale Structures ◽  
Structure Factors ◽  
Angle Scattering

Scatteris a new software for analysis, modeling and fitting of one- and two-dimensional small-angle scattering data of non-ordered, partially ordered or fully ordered nano- and mesoscale structures. The calculations are based on closed analytical expressions for the scattering intensity, enabling efficient evaluation of form factors and structure factors. The software allows one to sequentially fit large series of scattering curves and scattering patterns automatically. It provides further tools for data loading, beam centering, calibration, zooming, binning, lattice identification, calculation of density profiles and size distributions, and visualization of real-space structures. Presentations of experimental and calculated data can be saved as is for presentations or exported for further graphical or mathematical treatment.

Small angle scattering study of silica gels and zeolites

1993 ◽  
Vol 03 (C8) ◽  
pp. C8-393-C8-396
Author(s):  
T. P.M. BEELEN ◽  
W. H. DOKTER ◽  
H. F. VAN GARDEREN ◽  
R. A. VAN SANTEN ◽  
E. PANTOS
Keyword(s):  
Small Angle ◽  
Small Angle Scattering ◽  
Silica Gels ◽  
Angle Scattering ◽  
Scattering Study
Sign in / Sign up

Export Citation Format

Share Document