Estimation of the density distribution from small-angle scattering data

Abstract

The one-dimensional density distribution for symmetrical scatterers is estimated from small-angle scattering data. The symmetry of the scatterers may be one dimensional (lamellar), two dimensional (cylindrical) or three dimensional (spherical). Previously this problem has been treated either by a two-step approach with the distance distribution as an intermediate [Glatter (1981).J. Appl. Cryst.14, 101–108] or in a single step using spherical harmonics [Svergun, Feigin & Schedrin (1982).Acta Cryst.A38, 827–835]. A combination of these two methods is presented here, where the density distribution is estimated using constraints without the explicit use of an intermediate distribution. A maximum entropy constraint is introduced for this problem and the results are compared with the results of the conventional smoothness constraint. Bayesian methods are used for estimation of the overall noise level of the experimental data and for the maximum dimension of the density distribution. The method described is tested on both simulated and experimental data and shown to provide reliable estimates for the Guinier radius and maximum dimension. In both cases the effects of minor deviations from the assumed symmetry as well as incorrect background subtraction are studied.

Determination of Internal Density Profiles of Smart Acrylamide-Based Microgels by Small-Angle Neutron Scattering: A Multishell Reverse Monte Carlo Approach

“…As already mentioned in the introduction, a similar form-free algorithm has been proposed 62 , but not applied to spherically symmetric systems, and in particular not to microgels. 61,63 For the sake of comparison, all model predictions have been calculated using the same relative particle polydispersity (see Eq. (1) 39 .…”

“…60 Recently, Hansen has revisited the estimation of density distributions from small-angle scattering, analyzing the influence of the remaining background and different regularizations, and applied it to typical soft aggregates like micelles and proteins. 61 In some cases, the density distributions can be defined by a small set of parameters like cross-section density profiles in lamellar systems 62 or radial density profiles in globular systems. 63 The densities are described as a set of Gaussian functions, and the technique is termed Gaussian deconvolution.…”

“…There are numerous ways to find such solutions, including random search schemes running under the name of reverse Monte Carlo (RMC). , These methods are usually applied to liquid-state measurements but have been extended to soft condensed matter, namely, colloids or micelles . Recently, Hansen has revisited the estimation of density distributions from small-angle scattering, analyzing the influence of the remaining background and different regularizations, and applied it to typical soft aggregates like micelles and proteins . In some cases, the density distributions can be defined by a small set of parameters like cross-section density profiles in lamellar systems or radial density profiles in globular systems .…”

Small-angle neutron scattering correlation functions of bulk magnetic materials

“…In most situations, SANS data are analyzed in reciprocal space, by fitting a particular model to the experimental SANS cross section. An alternative real-space approach to analyzing SANS data is the computation of the (auto)correlation function of the system, for instance by means of the indirect Fourier transformation technique (Glatter, 1977;Hansen, 2000;Fritz & Glatter, 2006;Hansen, 2012), which has recently been extended to allow for the analysis of two-dimensional small-angle scattering patterns of oriented samples (Fritz-Popovski, 2013;Fritz-Popovski, 2015). For dilute, monodisperse and uniform particle-matrix systems, several analytical expressions for the density-density autocorrelation function ðrÞ or, likewise, for the distance distribution function pðrÞ ¼ r 2 ðrÞ have been derived (see e.g.…”

Evidence for the formation of nanoprecipitates with magnetically disordered regions in bulk $\mathrm{Ni}_{50}\mathrm{Mn}_{45}\mathrm{In}_{5}$ Heusler alloys

“…where j 0 (x) = sin(x)/x denotes the zeroth-order spherical Bessel function. The correlation function C(r) and the corresponding distance distribution function p(r) = r 2 C(r) can be extracted by either a direct [27][28][29][30] or an indirect [31][32][33][34][35][36] Fourier transform of dΣ +− /dΩ. Figure 6(b) shows the computed p(r).…”