Generalization of Porod's law of small-angle scattering to anisotropic samples

Abstract

Small-angle scattering from anisotropic samples, consisting of homogeneous particles inside a homogeneous medium with a scattering contrast (∆n) 2 , is considered. Along any direction q ≡ q/|q| of reciprocal space, at large q ( ≡ |q|) the Porod plot of the scattering intensity (i.e. q 4 I(q) vs. q) shows a plateau whose height depends on q and reads 4π 2 (∆n) 2 j,l (1/|κ G,j,l (± q)|). Here, the sum runs over all the points (labeled by (j, l)) of the surface of the j-th particle of the sample where the normal is either parallel or antiparallel to q, and κ G,j,l (± q) is the corresponding Gaussian curvature value.

Crystal shapes and crystallization in continuum modeling

“…First, the form factor goes to unity for small q. Second, in the Porod limit [50][51][52][53] P͑q͒ is proportional to the particle surface area divided by the square of the particle volume for isotropic systems. Third, the integral of P͑q͒ over all q is inversely proportional to the particle volume.…”

Fourier, Gauss, Fraunhofer, Porod and the shape from moments problem

“…Section 3 briefly discusses how the solution for two dimensional polyhedra appears when Fourier transforming three dimensional polyhedrons before showing how the surface integral form of the three dimensional Fourier transform can be used to derive not only the standard Porods law [3] for X-ray scattering from spherical particles but also the extension of this law to anisotropic particles. This extension has been discussed recently in a series of papers by Ciccariello and colleagues [12]. The derivation presented here is somewhat more direct than that used by Ciccariello and colleagues.…”

“…Porods law [12] follows from this in the case where the magnitude of the scattering wavevector k = k = 2π/λ is large, i.e, λ is small and so Porods law is often associated with X-ray scattering.…”

“…In general for an arbitrary shaped particle there will be multiple stationary phase points, i.e., σ 1 , σ 2 , · · · which must be summed to get the complete amplitude. There are also issues with positive and negative curvature which must be carefully considered as well along with how the handle the case when the curvature vanishes [12].…”

Reconstruction of the form of a particle from its three-dimensional asymptotic form factor

“…To prove this statement, referred to as property I in the following, it is recalled that, according to equation (11) of Ciccariello et al (2000), the asymptotic behaviour of e q q reads…”