The relation between the particle shape and the outer part of the small-angle X-ray scattering curve

The relation between the form of the scattering particles and the outer part of the small-angle X-ray scattering curve has been studied. The particles are assumed to be independent, identical, and randomly oriented and to have a uniform electron density and a smooth, strictly convex boundary surface. The electron density of the solvent is also assumed to be uniform. As earlier calculations by the authors and others have shown, the effects of the particle shape on the scattered intensity can often be conveniently described by a function called the chord, or intersect, distribution G(M). A chord, or intersect, is a straight line that has both ends on the particle boundary surface, and G(M) is defined to have the property that G(M)dM is the probability that the chord length will lie between M and M+ dM. The outer part of the scattering curve is shown to depend on the form of G(M) only in the neighborhoods of M = 0 and of any M values at which G(M) or G′(M) are discontinuous. Methods are developed for finding where these discontinuities occur and for calculating the form of G(M) in the neighborhood of these M values. In the outer part of the scattering curve, the intensity I(h) is shown to have the limiting form

where h = 4πλ ⁻¹ sin (θ/2), 2 is the X-ray wavelength, θ is the scattering angle, I_e is the intensity scattered by a single electron, A is the particle surface area, the D_i are the values of M at which G(M) or G′(M) is discontinuous, and j₋₂ and the j_i, φ_i, and μ_i are quantities which can be calculated from the principal curvatures and other properties of the surface at the two points where it contacts the chord with length D_i. The values of the μ_i are shown to lie in the interval 0 ≤ μ_i ≤ 1. In this equation the assumption is made that only the term or terms which vanish least rapidly as h increases are to be retained. In addition to the assumptions which conventionally are made in the analysis of the small-angle X-ray scattering from dilute suspensions, the limiting expression for the intensity for large h requires only that the particle boundary be smooth and strictly convex. This approximation is useful for determining the effect of the particle shape on the outer part of the scattering curve. In addition, the equation can be employed for numerical calculations for large h, where other methods of computation often are unwieldy or inapplicable.