Leading asymptotic term of the small-angle scattered intensity.

Abstract

It is shown that the leading asymptotic term of the small-angle intensity scattered by any amorphous sample is determined by the parallelism among subsets of the sample interfaces. Its general expression is ${\mathrm{\ensuremath{\Sigma}}}_{\mathit{i}}$[${\mathit{A}}_{\mathit{i}}$cos(${\mathrm{\ensuremath{\delta}}}_{\mathit{i}}$h)+${\mathit{B}}_{\mathit{i}}$sin(${\mathrm{\ensuremath{\delta}}}_{\mathit{i}}$h)]/${\mathit{h}}^{4}$, where the ${\mathrm{\ensuremath{\delta}}}_{\mathit{l}}$'s denote the distances between parallel surfaces and the ${\mathit{scrA}}_{\mathit{l}}$'s and the ${\mathit{scrB}}_{\mathit{l}}$'s are appropriate geometrical averages of the corresponding Gaussian curvatures. Since each surface is parallel to itself with a relative null distance, in the former expression the well-known Porod contribution comes out from the term relevant to \ensuremath{\delta}=0. The expression is specialized to the case of those three-component samples where one of the constituting phases has a constant thickness and lies in between the remaining two phases which have no common interface. Different approximations are considered and, in the most favorable cases, it appears that the average Gaussian, mean, and squared mean curvatures of the dividing film can be determined.