Fourier Transforms of Tubular Objects with Spiral Structures

Abstract

Crystal structures of several naturally occurring minerals are known to contain various deformities such as cones, cylinders, and tapered hollow cylinders with different apex angles, which have been described as solid and hollow cones, “cups”, “lampshades” as well as rolled cylindrical planes. The present study was undertaken to determine how these different shapes within a crystal structure can be explained. Since the usual method of observing them is by either X-ray and electron diffraction or electron microscopy, we investigated Fourier transforms of these forms, which were considered in terms of spirals with varying radii. Three types of spirals were considered, namely: 1) Archimedean spiral; 2) Involute of a circle or power spiral and 3) Logarithmic spiral. Spiraling caused the radius r to be modified by a factor f(θ) , so that r becomes rf(θ) , where f(θ) = θ for Archimedean helix, θ n for power helices like θ 1/2 for Fermat’s helix, θ -1 for hyperbolic helix and e θ or e -θ for logarithmic helix, r and θ being co-ordinates of the map on which the helix has to be drawn, f(θ) is unaffected by the magnitude of r . Expressions have been derived that explain the diffraction of structures containing the distortions described above, and bring all of these phenomena under one “umbrella” of a comprehensive theory.