The asymptotic leading term of anisotropic small-angle scattering intensities. II. Non-convex particles

Abstract

For anisotropic particulate samples with scattering contrast (\Delta n){}^2, the leading asymptotic term of the scattering intensity, along a direction \hat{\bf q} (= {\bf q}/q) of reciprocal space, is [4\pi^2 (\Delta n){}^2/q^4]\sum_j [1/|\kappa_{{{\rm G},j}}(\pm\hat{\bf q})|]. Here, \kappa_{{{\rm G},j}} (\pm\hat{\bf q}) denotes the Gaussian curvature value at the points (labelled by j) of the interphase surface where the normal is either parallel or antiparallel to \hat{\bf q}. If the Gaussian curvature vanishes at, say, the {\bar j}th of these points, the corresponding contribution takes the form {\cal C}_{{\bar j}}/q^ {\alpha_{\bar j}} with 2\le \alpha_{\bar j} \,\lt\, 4, {\cal C}_{{\bar j}} and \alpha_{\bar j} being determined by the local behaviour of the surface. However, the intensity detected by a counter pixel, with opening solid angle \Delta \Omega(\hat{\bf q}_0) along (mean) direction \hat{\bf q}_0, asymptotically still behaves as 4\pi^2 (\Delta n){}^2 {S}(\Delta\Omega(\hat{\bf q}_0))/q^4, where {S}(\Delta \Omega(\hat{\bf q}_0)) is the area of that part of the interface that has its normals inside \Delta \Omega(\hat{\bf q}_0).