Asymptotic analysis of small-angle scattering intensities of plane columnar layers

The asymptotic behaviour, at large scattering vector q, of the small-angle scattering intensities of isotropic plane samples is similar to that of three-dimensional samples. In fact, its expression, limited to the first two leading terms, is c₁γ⁽¹⁾(0)/q³ + c₃γ⁽³⁾(0)/q⁵, where c₁ and c₃ are appropriate numerical constants, and γ⁽¹⁾(0) and γ⁽³⁾(0) the values, at the origin, of the first and third derivatives, respectively, of the two-dimensional correlation function. These values are proportional to the specific length and to the mean square reciprocal curvature radius of the interface curve. The angularity of the latter can also be determined, while the presence of oscillations in the appropriate Porod plot is related to a parallelism condition obeyed by the interface curve. These results are useful for analysing the small-angle scattering intensities collected under grazing incidence and diffused by film samples that are a collection of homogeneous cylinders of arbitrary right sections.