Fourier Transforms of Tubular Objects with Spiral Structures
Author(s)
G. B. Mitra*
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, θn for power helices like θ1/2 for Fermat’s helix, θ-1 for hyperbolic helix and eθ 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.
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, θn for power helices like θ1/2 for Fermat’s helix, θ-1 for hyperbolic helix and eθ 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.
Cite this paper
G. Mitra, "Fourier Transforms of Tubular Objects with Spiral Structures," Journal of Crystallization Process and Technology, Vol. 2 No. 4, 2012, pp. 161-166. doi: 10.4236/jcpt.2012.24024.
G. Mitra, "Fourier Transforms of Tubular Objects with Spiral Structures," Journal of Crystallization Process and Technology, Vol. 2 No. 4, 2012, pp. 161-166. doi: 10.4236/jcpt.2012.24024.
References
[1] M. S. Taggart Jr., W. O. Milligan and H. P. Studer, “Electron Micrographic Studies of Clays,” Clays and Clay Minerals, Vol. 3, No. 1, 1954, pp. 31-95. doi:10.1346/CCMN.1954.0030104 [2] H. Jagodzinski and G. Kunze, “The Rolled Structure of Chrysotile. II. Far Winklen Interferences,” Neues Jahrb. Mineral Monatsh, Vol. 6, 1954, pp. 113-130. [3] H. Jagodzinski and G. Kunze, “The Rolled Structure of Chrysotile. III. The Manner of Growth of the Rolls,” Neues Jahrb. Mineral Monatsh, Vol. 7, 1954, pp. 137150. [4] H. Jagodzinski and G. Kunze, “Die Rcillchenstruktur des Chrysotils. I. Allgemeine Beugungtheorie und Kleinwinkelstreung,” Neues Jahrb. Mineral Monatsh, Vol. 10, 1954, pp. 219-240. [5] G. Honzo and K. Mihama, “A Study of Clay Minerals by Electron-Diffraction Diagrams Due to Individual Crystallites,” Acta Crystallographica, Vol. 7, No. 6-7, 1954, pp. 511-513. [6] J. Waser, “Fourier Transforms and Scattering Intensities of Tubular Objects,” Acta Crystallographica, Vol. 8, No. 3, 1955, pp. 142-150. doi:10.1107/S0365110X55000583 [7] E. Whittaker, “A Classification of Cylindrical Lattices,” Acta Crystallographica, Vol. 8, No. 9, 1955, pp. 571-574. doi:10.1107/S0365110X55001771 [8] H. B. Haanstra, W. F. Knippenberg and G. Verspui, “Columnar Growth of Carbon,” Journal of Crystal Growth, Vol. 16, No. 1, 1972, pp. 71-79. doi:10.1016/0022-0248(72)90091-7 [9] S. Iijima, “Helical Microtubules of Graphitic Carbon,” Nature, Vol. 354, No. 6348, 1991, pp. 56-58. doi:10.1038/354056a0 [10] M. Ge and K. Sattler, “Observation of Fullerene Cones,” Chemical Physics Letters, Vol. 220, No. 3-5, 1994, pp. 192-196. doi:10.1016/0009-2614(94)00167-7 [11] A. Krishnan, E. Dujardin, M. M. J. Treacy, et al., “Graphitic cones and the Nucleation of Curved Carbon Surfaces,” Nature, Vol. 388, No. 6641, 1997, pp. 451454. doi:10.1038/41284 [12] S. Iijima, M. Yudasaka, R. Yamada, et al., “Nano-Aggregates of Single-Walled Graphitic Carbon Nano-Horns,” Chemical Physics Letters, Vol. 309, No. 3-4, 1999, pp. 165-170. doi:10.1016/S0009-2614(99)00642-9 [13] L. Bourgeois, Y. Bando, W. Q. Han, et al., “Structure of Boron Nitride Nanoscale Cones: Ordered Stacking of 240 and 300 Disclinations,” Physical Review B, Vol. 61, No. 11, 2000, pp. 7686-7691. doi:10.1103/PhysRevB.61.7686 [14] Y. Gogotsi, S. Dimovski and J. A. Libera, “Conical Crystals of Graphite,” Carbon, Vol. 40, No. 12, 2002, pp. 2263-2267. doi:10.1016/S0008-6223(02)00067-2 [15] G. Zhang, X. Jiang and E. Wang, “Tubular Graphite Cones,” Science, Vol. 300, No. 5618, 2003, pp. 472-474. doi:10.1126/science.1082264 [16] G. B. Mitra and S. Bhattacherjee, “The Structure of Halloysite,” Acta Crystallographica Section B, Vol. 31, No. 12, 1975, pp. 2851-2857. doi:10.1107/S0567740875009041 [17] S. Wang and P. R. Buseck, “Cylindrite: The Relation between Its Cylindrical Shape and Modulated Structure,” American Mineralogist, Vol. 77, No. 7-8, 1992, pp. 758764. [18] M. E. Zolensky and I. D. R. Mackinnon, “Microstructures of Cylindrical Tochilinites,” American Mineralogist, Vol. 71, 1986, pp. 1201-1209. [19] J. L. Jambor, “New Occurences of the Hybrid Sulphide Tochilinite,” Geological Survey of Canada Paper, Vol. 76, No. 1B, 1976, pp. 65-69. [20] M. Vigodsky, “Mathematical Handbook—Higher Mathematics,” 1975, Mir Publishers, Moscow. [21] T. L. J. Ferris, A. Nafalski and M. Saghafifar, “Matching Observed Spiral Form Curves to Equations of Spirals in 2-D Images,” In: H. Tsuboi and I. Sebestyen, Eds., A pplied Electromagnetics and Computational Technology, IOS Press, Amsterdam, 2001, pp. 151-158. [22] G. Oster and D. P. Riley, “Scattering from Cylindrically Symmetric Systems,” Acta Crystallographica, Vol. 5, No. 2, 1952, pp. 272-276. doi:10.1107/S0365110X5200071X [23] M. E. Essington, “Soil and Water Chemistry: An Integrative Approach,” CRC Press, Boca Raton, 2004, p. 440. [24] B. Eksioglu and A. Nadarajah, “Structural Analysis of Conical Carbon Nanofibers,” Carbon, Vol. 44, No. 2, 2006, pp. 360-373. doi:10.1016/j.carbon.2005.07.007 [25] G. Mitra, “Diffraction Intensities from a Cluster of Curved Crystallites. I. General Theory for Oneand TwoDimensional Cases,” Acta Crystallographica, Vol. 18, No. 3, 1965, pp. 464-467. doi:10.1107/S0365110X65001020 [26] G. B. Mitra and S. Bhattacherjee, “Diffraction Intensities from a Cluster of Curved Crystallites. II. The Effect of Curvature,” Acta Crystallographica Section A, Vol. 24, No. 2, 1968, pp. 266-269. doi:10.1107/S0567739468000422 [27] G. B. Mitra and S. Bhattacherjee, “Diffraction Intensities from a Cluster of Curved Crystallites. III. The ThreeDimensional Case,” Acta Crystallographica Section A, Vol. 27, No. 1, 1971, pp. 22-28. doi:10.1107/S0567739471000056 [28] G. Mitra, “Low-angle scattering by cylindrical structures,” Acta Crystallographica Section A, Vol. 66, No. 1, 2010, pp. 93-97. doi:10.1107/S0108767309044791
[1] M. S. Taggart Jr., W. O. Milligan and H. P. Studer, “Electron Micrographic Studies of Clays,” Clays and Clay Minerals, Vol. 3, No. 1, 1954, pp. 31-95. doi:10.1346/CCMN.1954.0030104 [2] H. Jagodzinski and G. Kunze, “The Rolled Structure of Chrysotile. II. Far Winklen Interferences,” Neues Jahrb. Mineral Monatsh, Vol. 6, 1954, pp. 113-130. [3] H. Jagodzinski and G. Kunze, “The Rolled Structure of Chrysotile. III. The Manner of Growth of the Rolls,” Neues Jahrb. Mineral Monatsh, Vol. 7, 1954, pp. 137150. [4] H. Jagodzinski and G. Kunze, “Die Rcillchenstruktur des Chrysotils. I. Allgemeine Beugungtheorie und Kleinwinkelstreung,” Neues Jahrb. Mineral Monatsh, Vol. 10, 1954, pp. 219-240. [5] G. Honzo and K. Mihama, “A Study of Clay Minerals by Electron-Diffraction Diagrams Due to Individual Crystallites,” Acta Crystallographica, Vol. 7, No. 6-7, 1954, pp. 511-513. [6] J. Waser, “Fourier Transforms and Scattering Intensities of Tubular Objects,” Acta Crystallographica, Vol. 8, No. 3, 1955, pp. 142-150. doi:10.1107/S0365110X55000583 [7] E. Whittaker, “A Classification of Cylindrical Lattices,” Acta Crystallographica, Vol. 8, No. 9, 1955, pp. 571-574. doi:10.1107/S0365110X55001771 [8] H. B. Haanstra, W. F. Knippenberg and G. Verspui, “Columnar Growth of Carbon,” Journal of Crystal Growth, Vol. 16, No. 1, 1972, pp. 71-79. doi:10.1016/0022-0248(72)90091-7 [9] S. Iijima, “Helical Microtubules of Graphitic Carbon,” Nature, Vol. 354, No. 6348, 1991, pp. 56-58. doi:10.1038/354056a0 [10] M. Ge and K. Sattler, “Observation of Fullerene Cones,” Chemical Physics Letters, Vol. 220, No. 3-5, 1994, pp. 192-196. doi:10.1016/0009-2614(94)00167-7 [11] A. Krishnan, E. Dujardin, M. M. J. Treacy, et al., “Graphitic cones and the Nucleation of Curved Carbon Surfaces,” Nature, Vol. 388, No. 6641, 1997, pp. 451454. doi:10.1038/41284 [12] S. Iijima, M. Yudasaka, R. Yamada, et al., “Nano-Aggregates of Single-Walled Graphitic Carbon Nano-Horns,” Chemical Physics Letters, Vol. 309, No. 3-4, 1999, pp. 165-170. doi:10.1016/S0009-2614(99)00642-9 [13] L. Bourgeois, Y. Bando, W. Q. Han, et al., “Structure of Boron Nitride Nanoscale Cones: Ordered Stacking of 240 and 300 Disclinations,” Physical Review B, Vol. 61, No. 11, 2000, pp. 7686-7691. doi:10.1103/PhysRevB.61.7686 [14] Y. Gogotsi, S. Dimovski and J. A. Libera, “Conical Crystals of Graphite,” Carbon, Vol. 40, No. 12, 2002, pp. 2263-2267. doi:10.1016/S0008-6223(02)00067-2 [15] G. Zhang, X. Jiang and E. Wang, “Tubular Graphite Cones,” Science, Vol. 300, No. 5618, 2003, pp. 472-474. doi:10.1126/science.1082264 [16] G. B. Mitra and S. Bhattacherjee, “The Structure of Halloysite,” Acta Crystallographica Section B, Vol. 31, No. 12, 1975, pp. 2851-2857. doi:10.1107/S0567740875009041 [17] S. Wang and P. R. Buseck, “Cylindrite: The Relation between Its Cylindrical Shape and Modulated Structure,” American Mineralogist, Vol. 77, No. 7-8, 1992, pp. 758764. [18] M. E. Zolensky and I. D. R. Mackinnon, “Microstructures of Cylindrical Tochilinites,” American Mineralogist, Vol. 71, 1986, pp. 1201-1209. [19] J. L. Jambor, “New Occurences of the Hybrid Sulphide Tochilinite,” Geological Survey of Canada Paper, Vol. 76, No. 1B, 1976, pp. 65-69. [20] M. Vigodsky, “Mathematical Handbook—Higher Mathematics,” 1975, Mir Publishers, Moscow. [21] T. L. J. Ferris, A. Nafalski and M. Saghafifar, “Matching Observed Spiral Form Curves to Equations of Spirals in 2-D Images,” In: H. Tsuboi and I. Sebestyen, Eds., A pplied Electromagnetics and Computational Technology, IOS Press, Amsterdam, 2001, pp. 151-158. [22] G. Oster and D. P. Riley, “Scattering from Cylindrically Symmetric Systems,” Acta Crystallographica, Vol. 5, No. 2, 1952, pp. 272-276. doi:10.1107/S0365110X5200071X [23] M. E. Essington, “Soil and Water Chemistry: An Integrative Approach,” CRC Press, Boca Raton, 2004, p. 440. [24] B. Eksioglu and A. Nadarajah, “Structural Analysis of Conical Carbon Nanofibers,” Carbon, Vol. 44, No. 2, 2006, pp. 360-373. doi:10.1016/j.carbon.2005.07.007 [25] G. Mitra, “Diffraction Intensities from a Cluster of Curved Crystallites. I. General Theory for Oneand TwoDimensional Cases,” Acta Crystallographica, Vol. 18, No. 3, 1965, pp. 464-467. doi:10.1107/S0365110X65001020 [26] G. B. Mitra and S. Bhattacherjee, “Diffraction Intensities from a Cluster of Curved Crystallites. II. The Effect of Curvature,” Acta Crystallographica Section A, Vol. 24, No. 2, 1968, pp. 266-269. doi:10.1107/S0567739468000422 [27] G. B. Mitra and S. Bhattacherjee, “Diffraction Intensities from a Cluster of Curved Crystallites. III. The ThreeDimensional Case,” Acta Crystallographica Section A, Vol. 27, No. 1, 1971, pp. 22-28. doi:10.1107/S0567739471000056 [28] G. Mitra, “Low-angle scattering by cylindrical structures,” Acta Crystallographica Section A, Vol. 66, No. 1, 2010, pp. 93-97. doi:10.1107/S0108767309044791