Low-angle scattering by cylindrical structures

A theory of diffraction at small angles of scattering by a cylindrically constructed structure has been developed and general expressions for two- and three-dimensional cylindrically curved crystallites have been obtained. Modified expressions have been derived for special cases of (a) a single arc of equiangularly spaced identical atoms, (b) several coaxial arcs each having an equal number of equiangularly spaced identical atoms and (c) such arcs arranged at equal distances along the common axis of the arc, including single cylinders and multilayered cylindrical structures. From the expressions obtained, it is possible to measure the magnitudes of the radius of the cylinder, the length of the cylinder, the angular distance between the scattering atoms on each cylindrical arc, their atomic number and the radius of the hole inside each cylindrical structure when a particular tube is considered for a particular purpose or even when predicting the properties required by a nanotube for a particular purpose.