Small-angle scattering behavior of thread-like and film-like systems

A film-like or a thread-like system is a system such that one of its constituting homogeneous phases has a constant thickness $\delta$ or a constant normal section of largest diameter $\delta$. The stick probability function of this phase, in the limit $\delta \to 0$, naturally leads to the definition of the correlation function (CF) of a surface or a curve. This CF fairly approximates the generating stick probability function in the range of distances larger than $\delta$. The surface and the curve CFs respectively behave as $1/r$ and $1/r^2$ as $r \to 0$. In the two cases, this result implies that small-angle scattering intensities of the relevant samples respectively behave as $1/q^2$ and $1/q$ in an intermediate range of the scattering vector $q$ and as $1/q^4$ in the outermost one. One reports the analytic expressions of the pre-factors of these behaviors. It may happen that a sample looks thread-like at large scale resolution and film-like at smaller one. The surface and the curve CFs have explicitly been evaluated for some simple geometrical shapes. Besides, it is also reported the algebraic expression of the circular cylinder CF in terms of two elliptic integral functions, and it is shown that the limits of this CF, as the height or the radius of the cylinder approaches to zero, coincide with the CF of a disk or a linear segment, respectively.