PPubMed20 Sep 2018Cited 33×

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

Grethe Vestergaard Jensen, John George Barker

Abstract

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.

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Multiple scattering in small-angle scattering can be estimated and corrected using semi-analytical Hankel transforms and Monte Carlo simulations.

What’s new

i.Reviewed and discussed methods to estimate and include multiple scattering contributions in small-angle scattering analysis.
ii.Developed semi-analytical solutions using Hankel transforms for several model scattering functions.
iii.Provided analytical expressions and approximate formulas for forward scattering and radius of gyration corrections.
iv.Verified the semi-analytical results with Monte Carlo simulations.
v.Presented experimental examples illustrating transmission-based estimation of scattering power.

How they did it

The authors calculated multiple-scattering-affected scattering functions I_m(q) for representative model functions using the one-dimensional convolution method of Schelten & Schmatz, which relies on semi-analytical Hankel transforms. Scattering powers τ were obtained from transmission measurements or model integrals, and the results were cross-checked with Monte Carlo simulations of ray trajectories. Calculations were performed for spheres, Gaussian, DAB, Sabine and Lorentzian functions, and the impact on radius of gyration and forward scattering was analyzed.

What they found

Scattering powers for polystyrene spheres in D2O were measured between τ = 0.028 and τ = 3.13, matching theoretical values (Fig. 5a).
The full width at half-maximum of the double-scattering curve increased by approximately 30 % compared with the single-scattering curve (Fig. 4b).
For τ = 1 the normalized probabilities of the first three scattering orders are P₁′ = 0.582, P₂′ = 0.291, P₃′ = 0.061 (Fig. 2).
Monte Carlo simulated scattering curves agree closely with the semi-analytical Hankel-transform results for all tested functions.
Approximate power-law expressions for I(0) and R_g as functions of τ were derived for five scattering models (Table 2).

Caveats

✕Deconvolution of multiple-scattering data would require extremely low-noise intensity measurements over a wide q range, which is often impractical.
✕The semi-analytical Hankel-transform method assumes small-angle scattering and may not be valid for very large scattering angles without truncation.
✕Refraction effects can become significant for large particles (R₀ > 4.7 µm for SANS, >0.8 µm for SAXS) and are not accounted for in the presented corrections.
✕Monte Carlo simulations are time‑consuming, limiting their routine use for extensive parameter sweeps.

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