The significance of multiple scattering in the interpretation of small-angle neutron scattering experiments

Effects of multiple scattering encountered for various small-angle scattering model functions

The means by which multiple scattering contributions in experimental small-angle scattering data can be estimated, and how they can be included in the data analysis, are reviewed and discussed. The multiple scattering effects for a range of relevant model scattering functions are calculated using semi-analytically derived solutions to Hankel transforms as well as Monte Carlo simulations.

In small-angle scattering theory and data modeling, it is generally assumed that each scattered ray – photon or neutron – is only scattered once on its path through the sample. This assumption greatly simplifies the interpretation of the data and is valid in many cases. However, it breaks down under conditions of high scattering power, increasing with sample concentration, scattering contrast, sample path length and ray wavelength. For samples with a significant scattering power, disregarding multiple scattering effects can lead to erroneous conclusions on the structure of the investigated sample. In this paper, the impact of multiple scattering effects on different types of scattering pattern are determined, and methods for assessing and addressing them are discussed, including the general implementation of multiple scattering effects in structural model fits. The modification of scattering patterns by multiple scattering is determined for the sphere scattering function and the Gaussian function, as well as for different Sabine-type functions, including the Debye–Andersen–Brumberger (DAB) model and the Lorentzian scattering function. The calculations are performed using the semi-analytical convolution method developed by Schelten & Schmatz [J. Appl. Cryst. (1980 ▸). 13, 385–390], facilitated by analytical expressions for intermediate functions, and checked with Monte Carlo simulations. The results show how a difference in the shape of the scattering function plotted versus momentum transfer q results in different multiple scattering effects at low q, where information on the particle mass and radius of gyration is contained.

H2O/D2O Contrast Variation for Ultra-Small-Angle Neutron Scattering to Minimize Multiple Scattering Effects of Colloidal Particle Suspensions

This study investigated the use of solvent contrast (H2O/D2O ratio) as a means to optimize the ultra-small-angle neutron scattering (USANS) signal. By optimizing the signal, it was possible to reduce the undesirable effects of coherent multiple scattering while still maintaining a measurable scattered intensity. This result will further enable the use of USANS as a probe of the interactions between colloidal particles and their structures within concentrated suspensions as well as particle dispersion/aggregation. As a model system, we prepared silica colloidal particle suspensions at different solid concentrations. USANS curves were measured using the classical Bonse–Hart double crystal diffractometer while varying the scattering length density of the aqueous phase, thus varying the contrast to the silica particles. As a means of assessing the impact of multiple scattering effects on different q-values, we analyzed the scattered intensity at different contrasts at three different q values. The data were then used to determine the match point of the silica particle suspensions from the expected square root dependence of the scattered intensity with solvent composition, to analyze any differences associated with the solid concentration change, and to determine the optimum H2O/D2O ratio in terms of high transmission (TSAS > 80%) and high enough scattering intensity associated with the contrast of the system. Through this investigation series, we confirmed that adjusting the contrast of the solvent (H2O/D2O) is a good methodology to reduce multiple scattering while maintaining a strong enough scattering signal from a concentrated suspension of silica particles for both USANS and rheometric USANS (rheo-USANS) experiments.

Calibrating SANS data for instrument geometry and pixel sensitivity effects: access to an extended Q range

An improved data-reduction procedure is proposed and demonstrated for small-angle neutron scattering (SANS) measurements. Its main feature is the correction of geometry- and wavelength-dependent intensity variations on the detector in a separate step from the different pixel sensitivities: the geometric and wavelength effects can be corrected analytically, while pixel sensitivities have to be calibrated to a reference measurement. The geometric effects are treated for position-sensitive 3He proportional counter tubes, where they are anisotropic owing to the cylindrical geometry of the gas tubes. For the calibration of pixel sensitivities, a procedure is developed that is valid for isotropic and anisotropic signals. The proposed procedure can save a significant amount of beamtime which has hitherto been used for calibration measurements.

Everything SAXS: small-angle scattering pattern collection and correction

For obtaining reliable nanostructural details of large amounts of sample--and if it is applicable--small-angle scattering (SAS) is a prime technique to use. It promises to obtain bulk-scale, statistically sound information on the morphological details of the nanostructure, and has thus led to many a researcher investing their time in it over the last eight decades of development. Due to pressure from scientists requesting more details on increasingly complex nanostructures, as well as the ever improving instrumentation leaving less margin for ambiguity, small-angle scattering methodologies have been evolving at a high pace over the past few decades. As the quality of any results can only be as good as the data that go into these methodologies, the improvements in data collection and all imaginable data correction steps are reviewed here. This work is intended to provide a comprehensive overview of all data corrections, to aid the small-angle scatterer to decide which are relevant for their measurement and how these corrections are performed. Clear mathematical descriptions of the corrections are provided where feasible. Furthermore, as no quality data exist without a decent estimate of their precision, the error estimation and propagation through all these steps are provided alongside the corrections. With these data corrections, the collected small-angle scattering pattern can be made of the highest standard, allowing for authoritative nanostructural characterization through its analysis. A brief background of small-angle scattering, the instrumentation developments over the years, and pitfalls that may be encountered upon data interpretation are provided as well.

Learning about SANS instruments and data reduction from round robin measurements on samples of polystyrene latex

Measurements of a well-characterized `standard' sample can verify the performance of an instrument. Typically, small-angle neutron scattering instruments are used to investigate a wide range of samples and may often be used in a number of configurations. Appropriate `standard' samples are useful to test different aspects of the performance of hardware as well as that of the data reduction and analysis software. Measurements on a number of instruments with different intrinsic characteristics and designs in a round robin can not only better characterize the performance for a wider range of conditions but also, perhaps more importantly, reveal the limits of the current state of the art of small-angle scattering. The exercise, followed by detailed analysis, tests the limits of current understanding as well as uncovering often forgotten assumptions, simplifications and approximations that underpin the current practice of the technique. This paper describes measurements of polystyrene latex, radius 720 Å, with a number of instruments. Scattering from monodisperse, uniform spherical particles is simple to calculate and displays sharp minima. Such data test the calibrations of intensity, wavelength and resolution as well as the detector response. Smoothing due to resolution, multiple scattering and polydispersity has been determined. Sources of uncertainty are often related to systematic deviations and calibrations rather than random counting errors. The study has prompted development of software to treat modest multiple scattering and to better model the instrument resolution. These measurements also allow checks of data reduction algorithms and have identified how they can be improved. The reproducibility and the reliability of instruments and the accuracy of parameters derived from the data are described.