Beam broadening is discussed in the context of the standard formalism for multiple small-angle scattering, in which coherent single-particle scattering is incoherently, or stochastically, compounded by a random system of spherical particles in a uniform matrix. Bethe's analysis of scattering when sample thickness greatly exceeds the scattering mean free path is combined with the dynamical analysis of singleparticle scattering to obtain a new scaling relation between the multiple-scattering intensity at arbitrary phase shift and the multiple-scattering intensity in the diffraction or small-phase-shift regime. A formula is derived for the curvature of the scattering at Q = 0 which expresses this scaling and which can be used in particle size determinations. It is shown that strong multiple scattering, as in very thick samples, tends to render beam broadening insensitive to the cross over from diffractive to refractive single-particle scattering.
Microstructural parameters of high‐purity alumina powder are determined quantitatively throughout the bulk of the material using small‐angle neutron scattering techniques. A unified theoretical and experimental approach for analyzing multiple scattering data is developed to obtain values for particle size, volume fraction and surface area. It is shown how particle size and volume fraction can be measured in a practical way from SANS data totally dominated by incoherent multiple scattering (`beam broadening'). The general phase‐shift dependence of single‐particle scattering is incorporated into the multiple scattering formalism, and it is also shown that the diffractive limit (small phase shift) applies even for phase shifts as large as unity (particle radii of order 1 μm). The stability of the Porod law against multiple scattering and the phase‐shift scale are described, a useful empirical formula for analysis of beam broadening data is exhibited, and the applicability of the formulations to polydispersed systems is discussed.
Small-angle neutron scattering techniques developed in the preceding paper [Berk & Hardman-Rhyne (1985). J. Appl. Cryst. 18,[467][468][469][470][471][472] are used to obtain microstructural parameters of high-purity alumina powder. The values of the particle size, volume fraction and surface area have been obtained and are compared to data from techniques such as laser light scattering, X-ray sedigraph and scanning electron microscopy. The particles exhibit a log-normal distribution and are spherical in shape with a mean particle size of 342 nm determined from SANS analyses of both beam broadening and Porod regions.
The use of small angle neutron scattering (SANS) techniques for ceramic materials is discussed. Two areas are emphasized: 1) diffraction for microstructural phenomena of less than 100 nm, and 2) beam broadening for microstructural phenomena greater than 90 nm.
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