scholarly journals Experimental noise in small-angle scattering can be assessed using the Bayesian indirect Fourier transformation

2021 ◽  
Vol 54 (5) ◽  
pp. 1281-1289 ◽  
Author(s):  
Andreas Haahr Larsen ◽  
Martin Cramer Pedersen
Keyword(s):  
Small Angle ◽  
Fourier Transformation ◽  
Soft Matter ◽  
Small Angle Scattering ◽  
Scattering Data ◽  
Data Set ◽  
Angle Scattering ◽  
Experimental Errors ◽  
Multiple Data Sets ◽  
Model Fits

Small-angle X-ray and neutron scattering are widely used to investigate soft matter and biophysical systems. The experimental errors are essential when assessing how well a hypothesized model fits the data. Likewise, they are important when weights are assigned to multiple data sets used to refine the same model. Therefore, it is problematic when experimental errors are over- or underestimated. A method is presented, using Bayesian indirect Fourier transformation for small-angle scattering data, to assess whether or not a given small-angle scattering data set has over- or underestimated experimental errors. The method is effective on both simulated and experimental data, and can be used to assess and rescale the errors accordingly. Even if the estimated experimental errors are appropriate, it is ambiguous whether or not a model fits sufficiently well, as the `true' reduced χ2 of the data is not necessarily unity. This is particularly relevant for approaches where overfitting is an inherent challenge, such as reweighting of a simulated molecular dynamics trajectory against small-angle scattering data or ab initio modelling. Using the outlined method, it is shown that one can determine what reduced χ2 to aim for when fitting a model against small-angle scattering data. The method is easily accessible via the web interface BayesApp.

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.

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.

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.

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.

scholarly journals 2017 publication guidelines for structural modelling of small-angle scattering data from biomolecules in solution: an update

2017 ◽  
Vol 73 (9) ◽  
pp. 710-728 ◽  
Author(s):  
Jill Trewhella ◽  
Anthony P. Duff ◽  
Dominique Durand ◽  
Frank Gabel ◽  
J. Mitchell Guss ◽  
...  
Keyword(s):  
Small Angle ◽  
Task Force ◽  
Data Bank ◽  
Small Angle Scattering ◽  
Scattering Data ◽  
Quality Data ◽  
Angle Scattering ◽  
Publication Guidelines ◽  
Guidelines And Recommendations

In 2012, preliminary guidelines were published addressing sample quality, data acquisition and reduction, presentation of scattering data and validation, and modelling for biomolecular small-angle scattering (SAS) experiments. Biomolecular SAS has since continued to grow and authors have increasingly adopted the preliminary guidelines. In parallel, integrative/hybrid determination of biomolecular structures is a rapidly growing field that is expanding the scope of structural biology. For SAS to contribute maximally to this field, it is essential to ensure open access to the information required for evaluation of the quality of SAS samples and data, as well as the validity of SAS-based structural models. To this end, the preliminary guidelines for data presentation in a publication are reviewed and updated, and the deposition of data and associated models in a public archive is recommended. These guidelines and recommendations have been prepared in consultation with the members of the International Union of Crystallography (IUCr) Small-Angle Scattering and Journals Commissions, the Worldwide Protein Data Bank (wwPDB) Small-Angle Scattering Validation Task Force and additional experts in the field.

A new time-of-flight small-angle scattering instrument at the Helmholtz-Zentrum Berlin: V16/VSANS

2014 ◽  
Vol 47 (1) ◽  
pp. 237-244 ◽  
Author(s):  
Karsten Vogtt ◽  
Miriam Siebenbürger ◽  
Daniel Clemens ◽  
Christian Rabe ◽  
Peter Lindner ◽  
...  
Keyword(s):  
Momentum Transfer ◽  
Small Angle ◽  
Soft Matter ◽  
Scattering Length ◽  
Time Of Flight ◽  
Small Angle Scattering ◽  
Neutron Guide ◽  
Macromolecular Assemblies ◽  
Angle Scattering ◽  
The Individual

Small-angle scattering methods have become routine techniques for the structural characterization of macromolecules and macromolecular assemblies like polymers, (block) copolymers or micelles in the spatial range from a few to hundreds of nanometres. Neutrons are valuable scattering probes, because they offer freedom with respect to scattering length density contrast and isotopic labelling of samples. In order to gain maximum benefit from the allotted experiment time, the instrumental setup must be optimized in terms of statistics of scattered intensity, resolution and accessible range in momentum transferQ. The new small-angle neutron scattering instrument V16/VSANS at the Helmholtz-Zentrum in Berlin, Germany, augments neutron guide collimation and pinhole optics with time-of-flight data recording and flexible chopper configuration. Thus, the availableQrange and the respective instrumental resolution in the intermediate and high momentum transfer regions can be adjusted and balanced to the individual experimental requirements. This renders V16/VSANS a flexible and versatile instrument for soft-matter research.

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