Extension of Kirste–Porod's law in the case of angulous interfaces

A preceding paper handled, by way of application, the usefulness of Porod's law extended to the second nonoscillating term. The h⁻⁶ term allows the structure of the phases to be better characterized. This paper is mainly concerned with the setting up of the main equations used in this preceding paper. The h⁻⁶ term is analysed from the correlation function γ(r) and related to the `stick probability function'. It can be positive or negative. The positive case appears in smooth phases and has been previously analysed by Kirste & Porod. The negative case occurs in the presence of linear edges resulting from the meeting of surfaces that are planar in the vicinity of their intersection. More precisely, it is shown that the h⁻⁶ negative term results from the finite length of the edge. Its magnitude depends on the dihedral angles at the vertex defined by the limited sharp edges. The smaller the dihedral angles, the greater the h⁻⁶ term amplitude. The new concept of angulosity, θ, a pure number characterizing the geometry of the phase, is introduced. In this way, it is possible to develop similar equations for a specific surface, angularity and angulosity. Some simple-geometry examples are developed. The region where the extended Kirste–Porod law is useful in analysing small-angle scattering curves is discussed.