The vertex contribution to the Kirste–Porod term

Abstract

It is shown that, close to the origin, the correlation function [y(r)] of any N-component sample with interfaces made up of planar facets is always a third-degree polynomial in r. Hence, the only monotonically decreasing terms present in the asymptotic expansion of the relevant small-angle scattered intensity are the Porod [-2y'(0+)/h 4] and the KirstePorod [4')(3)(0 +)/h 6] contributions. The latter contribution is non-zero owing to the contributions arising from each vertex of the interphase surfaces. The general vertex contribution is evaluated in closed form and the )'(3)(0+) values relevant to the regular polyhedra are reported.

Coherent analysis of disordered mesoporous adsorbents using small angle X-ray scattering and physisorption experiments

“…g(0) is an indicator of the degree of angularity in the samples. As reported by Ciccariello et al [67][68][69][70] a value of g(0) 4 0 hints towards angular structures. A sphere's CLD is characterized by g(0) = 0, which is also the case for polydisperse spheres.…”

Diffuse Interfaces and Small-Angle Scattering Intensity Behaviour

“…Its expression is where (HZ)s and (x)s denote, respectively, the averages of the squared mean curvature and the Gaussian curvature of the interface throughout the latter. The corrections to the value of Era,, arising from the presence of vertices and sharp curvilinear edges, have been discussed recently by Sobry et al (1991) and Ciccariello & Sobry (1995). For smooth interfaces also, the contributions monotonously decreasing as h -s and h -1° have been explicitly evaluated by Wu & Schmidt (1971) and Ciccariello (1995), respectively.…”

The intersect distribution of the regular tetrahedron

“…More definitely, as r ! 0 þ , the value of the first derivative is linearly related to the area of the particle surface (Porod, 1951), the value of the second derivative to the presence of sharp edges (Porod, 1967;Mé ring & Tchoubar, 1968;Ciccariello & Benedetti, 1982), and the value of the third derivative to a surface average of the Gaussian and the mean curvatures of the particle surface (Kirste & Porod, 1962), as well as to eventual vertices (Ciccariello & Sobry, 1995) or curvilinear edges (Ciccariello, 1993) of the particle surface; for values of the higher derivatives, the reader is referred to the work of Ciccariello (1995). The existence of discontinuities in the second derivative of the CF was first pointed out by Wu & Schmidt (1974).…”