Particle shapes and oscillatory deviations from the Porod law: The hyperbolic contact point case

The leading term in the asymptotic expansion of the isotropic component of the X-ray intensity scattered by a homogeneous right circular cylindrical particle (of height H and diameter D) can be written as: Ch-4[ST-2SB cos (hH)-SL sin (hD)]. Here SL, SB and ST denote respectively the lateral, the base and the total surface area of the cylinder, while C is a normalization factor. From this expression one is led to conjecture that the oscillatory deviations from the Porod law, originated by the two interface subsets which have the same normal and are separated by δ, are proportional to cos (hδ) or to sin (hδ) depending on whether the tangency points between one of the interphase subsets and the spheres of radius δ centred on the opposite subset are elliptical or hyperbolic respectively. It is proved that the conjecture is true and general expressions for the coefficients in front of the two oscillatory functions are obtained. (orig.)