Small Angle X-Ray Scattering from Randomly Oriented Cylinders of Arbitrary Cross Section

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Abstract
At the larger angles of the small‐angle region, asymptotic expansions are the appropriate means of calculating the intensity of the small angle x‐ray scattering from a sample of identical particles with uniform electron density and random position and orientation. Since other workers have previously shown that this intensity can be expressed as a Fourier integral transform involving a function characteristic of the size and shape of the particles, standard techniques for asymptotic expansion of Fourier integrals can be employed. In order to make a more general study than would be possible by consideration of particles with a single shape, right cylinders of arbitrary cross section have been chosen for this investigation. These generalized cylinders permit a fairly wide range of choice of particle shape and are well adapted to the study of both elongated and flattened particles. An integral has been developed which gives the characteristic function for the generalized cylinder in terms of the characteristic function of the two‐dimensional cross section of the cylinder. As the asymptotic expressions for the scattered intensity depend quite critically on the discontinuities in derivatives of the characteristic function, a study has been made of these derivatives. An expression for the scattered intensity from rectangular parallepipeds has been calculated. A treatment of the limiting case of highly elongated cylinders is given.

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