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

For anisotropic particulate samples with scattering contrast (Δ n){}^2, the leading asymptotic term of the scattering intensity, along a direction q̂ (= { q}/q) of reciprocal space, is [4π^2 (Δ n){}^2/q^4]∑_j [1/|κ_{{{ G},j}}(± q̂)|]. Here, κ_{{{ G},j}} (± q̂) denotes the Gaussian curvature value at the points (labelled by j) of the interphase surface where the normal is either parallel or antiparallel to q̂. If the Gaussian curvature vanishes at, say, the { j}th of these points, the corresponding contribution takes the form { C}_{{ j}}/q^ {αⱼ} with 2≤ αⱼ \,<\, 4, { C}_{{ j}} and αⱼ being determined by the local behaviour of the surface. However, the intensity detected by a counter pixel, with opening solid angle Δ Ω( q̂_0) along (mean) direction q̂_0, asymptotically still behaves as 4π^2 (Δ n){}^2 {S}(ΔΩ( q̂_0))/q^4, where {S}(Δ Ω( q̂_0)) is the area of that part of the interface that has its normals inside Δ Ω( q̂_0).