Scattering from cylindrically symmetric systems

Abstract

Expressions for the intensity of scattering by systems composed of infinitely long cylindrical particles are derived. Four types of independent scattering regions are considered: (1) isolated cylindrical rods, with and without internal radial structure; (2) aggregates of parallel rods with fixed locations; (3) systems of parallel rods with variable locations; (4) two-dimensional crystals. The effect of random and specific orientation of independent scattering regions in discussed. The results are applicable to low-angle X-ray scattering from fibres and from macromolecular or micellar solutions.In a previous paper (Oster & Riley, 1952) we considered the scattering of X-rays and visible light by isotropic systems and confined our treatment to those cases which possess spherical symmetry. We shall now extend our examination to assemblies which possess cylindrical symmetry and in which the fundamental particles are very long compared with the wavelength ), of the radiation used. On the macroscopic scale such systems may be isotropic or anisotropic. In the former case they are collections of cylindrically-symmetric domains in random orientation; in the latter, the domains themselves have preferred orientation. Micellar solutions are an example of the first type and completely oriented fibres of the second. We shall show that there is a close analogy between the expressions for the intensity of scattering by cylindrically symmetric systems and those for the equivalent spherically symmetric cases discussed in our first paper. The results obtained in the present paper are not only applicable to X-ray scattering and diffraction by certain macromolecular and colloidal systems but also to visible-light scattering and diffraction by macroscopic systems having cylindrical symmetry.A peculiar feature of systems of long rod-shaped particles is the correlation in orientation which sets in if the concentration exceeds a certain critical value, depending on the particle length and the forces of interaction (Onsager, 1949; 0ster, 1949). Very long particles will, even in fairly dilute solution, show correlation in orientation, the rods being nearly parallel over rather large domains. Fibres in bundles can usually be rendered parallel, at least in local regions, by stretching and rolling the sample. These properties of elongated particles considerably simplify the mathematics of the problem as it then reduces to a two-dimensional calculation.In this paper we shall calculate the angular distribution of scattered intensity for (1) isolated * Present address: Polytechnic Institute of Brooklyn, Brooklyn 2, New York, U.S.A. cylindrical rods, with and without internal radial structure, (2) aggregates of parallel rods with fixed locations, (3) systems of parallel rods with variable locations as expressed generally by a radial distribution function, and (4) two-dimensional crystals consisting of an infinite periodic array of parallel rods. As in the previous paper, we shall eliminate the scale factor by writing all expressions in terms of ...

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