Geometrical Modeling of Crystal Structures with Use of Space of Elliptic Riemannian Geometry

ABSTRACT

The space of internal geometry of a model of a real crystal is supposed to be finite, closed, and with a constant Gaussian curvature equal to unity, permitting the realization of lattice systems in accordance with Fedorov groups of transformations. For visualizing computations, the interpretation of geometrical objects on a Clifford surface (SK) in Riemannian geometry with the help of a 2D torus in a Euclidean space is used. The F-algorithm ensures a computation of 2D sections of models of point systems arranged perpendicularly to the symmetry axes l3, l4, and l6. The results of modeling can be used for calculations of geometrical sizes of crystal structures, nanostructures, parameters of the cluster organization of oxides, as well as for the development of practical applications connected with improving the structural characteristics of crystalline materials.

The space of internal geometry of a model of a real crystal is supposed to be finite, closed, and with a constant Gaussian curvature equal to unity, permitting the realization of lattice systems in accordance with Fedorov groups of transformations. For visualizing computations, the interpretation of geometrical objects on a Clifford surface (SK) in Riemannian geometry with the help of a 2D torus in a Euclidean space is used. The F-algorithm ensures a computation of 2D sections of models of point systems arranged perpendicularly to the symmetry axes l3, l4, and l6. The results of modeling can be used for calculations of geometrical sizes of crystal structures, nanostructures, parameters of the cluster organization of oxides, as well as for the development of practical applications connected with improving the structural characteristics of crystalline materials.

KEYWORDS

F-Algorithm, Crystal Lattice Systems, Microstructure, Riemannian Geometry, Space of Interpretation

F-Algorithm, Crystal Lattice Systems, Microstructure, Riemannian Geometry, Space of Interpretation

Cite this paper

nullS. Rudnev, B. Semukhin and A. Klishin, "Geometrical Modeling of Crystal Structures with Use of Space of Elliptic Riemannian Geometry,"*Materials Sciences and Applications*, Vol. 2 No. 6, 2011, pp. 526-536. doi: 10.4236/msa.2011.26071.

nullS. Rudnev, B. Semukhin and A. Klishin, "Geometrical Modeling of Crystal Structures with Use of Space of Elliptic Riemannian Geometry,"

References

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[2] S. Van Aert, K. J. Batenburg, M. D. Rossell, R.Erni and G. Van Tendeloo, “Three-dimensional atomic imaging of crystalline nanoparticles,” Nature, Vol. 470, February 2011, pp. 374-377.(doi:10.1038/nature09741)

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[4] S.V. Rudnev, “Application of Elliptic Riemannian Geometry to Problems Crystallography,” Computers and Mathematics with Applications, Vol. 16, No. 5-8, 1988, pp. 597-616.

[5] B.S. Semukhin, A.N. Sergeev and S.V. Rudnev, “Non-Euclidean Interpretation of the Structures of Crystalline Materials,” Crystallography Reports, Vol. 4, No. 5, 1999, pp. 738-741.

[6] S.V. Rudnev and V.A Ermolaev, “On Application of Elliptical Riemannian Geometry to Study of Crystalline Structure,” Proc. Geometry, TGU Tomsk State University, Tomsk, Vol. 25, 1985, pp.113-121.

[7] B.S. Semukhin, S.V. Rudnev and R.V. Galiulin, “Application of Riemann Geometry to Structures of Nano- and Macrocrystals,” Crystallography Reports, Vol. 53, No. 4 2008, pp. 541-544.

[8] S.V. Rudnev, A.N. Sergeev, V.G. Bamburov and G.P. Shvykin, “Geometrical Modeling of Superplastic Constructional Ceramics Structure,” Doklady Chemistry, Vol. 341, No. 4, 1995, pp.589-501.

[9] S. A. Bogomolov, “An Introduction to Non-Euclidian Geometry,” GTTI, Moscow-Leningrad, 1934.

[10] W.K. Clifford, “Preliminary Sketch of Biquaternions,” Proc. Lond. Math. Soc. IV, 1873, pp. 381-395.

[11] H. A. Rahnamaye Aliabad and S. R. Ghorbani, “Structural and Spin Polarization Effects of Cr, Fe and Ti Elements on Electronical Properties of α–Al2O3 by First Principle Calculations,” Journal of Modern Physics, Vol. 2, 2011, pp. 158-161.

[12] Jizhong Sun, T. Stirner and A. Matthews, “Structure and electronic properties calculation of ultrathin α-Al2O3 films on (0001) α-Cr2O3 templates,” Surface Science, Vol. 601, November 2007, pp. 5050-5056. (doi:10.1016/j.susc.2007.09.004)

[13] M. M. Zhong, X. Y. Kuang, H.Q. Wang; H. F. Li, Y. R. Zhao, “Density functional study of the structural and electronic properties of tetra-aluminum oxide (3 ≤ n ≤ 8, λ = 0, –1) clusters,“ Molecular Physics: An International Journal at the Interface Between Chemistry and Physics, Vol. 109, Issue 4, 2011, pp. 603-612. (doi: 10.1080/00268976.2010.542777)

[14] R Ahuja, J. M. R Ahuja, J. M. Osorio-Guillen, J. Souza de Almeida, B. Holm, W. Y. Ching and B Johansson, “Electronic and optical properties of γ-Al2O3 from ab initio theory, ” Journal of Physics: Condensed Matter, Vol.15, N16, 2004, pp.2891-2898. (doi: 10.1088/0953-8984/16/16/013)

[15] S.V. Rudnev and А.N. Sergeev, “ICS Simulation of Crystal Mineral Grouth and Deformation,” TGU Tomsk State University, Tomsk, 1994.

[16] B.A. Dubrovin, A.T. Fomenko and S.P. Novikov, “Modern Geometry-Methods and Applications: Part I: The Geometry of Surfaces, Transformation Groups, and Fields,” Springer-Verlag, 1991.

[17] А. P. Klishin, A.N. Zakutaev, S.V. Rudnev, V.A. Ermolaev and T.A. Habas, “Modeling Process of Structural Transformations in Al2O3 at Magnetothermal Treatment,” Constructions From Composite Materials, Vol. 1, 2008, pp. 12-17.

[18] А. P. Klishin, S.V. Rudnev and V.I. Vereshagin, “New Technological Approaches of Reception Alumina Materials with Usage of Magnetothermal Treatment,” Refractories and Technical Ceramics, No. 6, 2009, pp. 30-34.

[19] N.V. Belov, “Structural Crystallography,” USSR Academy of Sciences, 1952.

[1] R.V. Galiulin, “The Geometrical Theory of Crystallization,” Crystallography Reports, Vol. 43, No. 2, 1998, pp. 366-347.

[2] S. Van Aert, K. J. Batenburg, M. D. Rossell, R.Erni and G. Van Tendeloo, “Three-dimensional atomic imaging of crystalline nanoparticles,” Nature, Vol. 470, February 2011, pp. 374-377.(doi:10.1038/nature09741)

[3] T. Janssen. International Tables for Crystallography, Vol. D, Physical Properties of Crystals, eduted by A. Authier, ch. 1.2, Representations of crystallographic groups. Dordrecht: Kluwer Academic Publishers, 2003.

[4] S.V. Rudnev, “Application of Elliptic Riemannian Geometry to Problems Crystallography,” Computers and Mathematics with Applications, Vol. 16, No. 5-8, 1988, pp. 597-616.

[5] B.S. Semukhin, A.N. Sergeev and S.V. Rudnev, “Non-Euclidean Interpretation of the Structures of Crystalline Materials,” Crystallography Reports, Vol. 4, No. 5, 1999, pp. 738-741.

[6] S.V. Rudnev and V.A Ermolaev, “On Application of Elliptical Riemannian Geometry to Study of Crystalline Structure,” Proc. Geometry, TGU Tomsk State University, Tomsk, Vol. 25, 1985, pp.113-121.

[7] B.S. Semukhin, S.V. Rudnev and R.V. Galiulin, “Application of Riemann Geometry to Structures of Nano- and Macrocrystals,” Crystallography Reports, Vol. 53, No. 4 2008, pp. 541-544.

[8] S.V. Rudnev, A.N. Sergeev, V.G. Bamburov and G.P. Shvykin, “Geometrical Modeling of Superplastic Constructional Ceramics Structure,” Doklady Chemistry, Vol. 341, No. 4, 1995, pp.589-501.

[9] S. A. Bogomolov, “An Introduction to Non-Euclidian Geometry,” GTTI, Moscow-Leningrad, 1934.

[10] W.K. Clifford, “Preliminary Sketch of Biquaternions,” Proc. Lond. Math. Soc. IV, 1873, pp. 381-395.

[11] H. A. Rahnamaye Aliabad and S. R. Ghorbani, “Structural and Spin Polarization Effects of Cr, Fe and Ti Elements on Electronical Properties of α–Al2O3 by First Principle Calculations,” Journal of Modern Physics, Vol. 2, 2011, pp. 158-161.

[12] Jizhong Sun, T. Stirner and A. Matthews, “Structure and electronic properties calculation of ultrathin α-Al2O3 films on (0001) α-Cr2O3 templates,” Surface Science, Vol. 601, November 2007, pp. 5050-5056. (doi:10.1016/j.susc.2007.09.004)

[13] M. M. Zhong, X. Y. Kuang, H.Q. Wang; H. F. Li, Y. R. Zhao, “Density functional study of the structural and electronic properties of tetra-aluminum oxide (3 ≤ n ≤ 8, λ = 0, –1) clusters,“ Molecular Physics: An International Journal at the Interface Between Chemistry and Physics, Vol. 109, Issue 4, 2011, pp. 603-612. (doi: 10.1080/00268976.2010.542777)

[14] R Ahuja, J. M. R Ahuja, J. M. Osorio-Guillen, J. Souza de Almeida, B. Holm, W. Y. Ching and B Johansson, “Electronic and optical properties of γ-Al2O3 from ab initio theory, ” Journal of Physics: Condensed Matter, Vol.15, N16, 2004, pp.2891-2898. (doi: 10.1088/0953-8984/16/16/013)

[15] S.V. Rudnev and А.N. Sergeev, “ICS Simulation of Crystal Mineral Grouth and Deformation,” TGU Tomsk State University, Tomsk, 1994.

[16] B.A. Dubrovin, A.T. Fomenko and S.P. Novikov, “Modern Geometry-Methods and Applications: Part I: The Geometry of Surfaces, Transformation Groups, and Fields,” Springer-Verlag, 1991.

[17] А. P. Klishin, A.N. Zakutaev, S.V. Rudnev, V.A. Ermolaev and T.A. Habas, “Modeling Process of Structural Transformations in Al2O3 at Magnetothermal Treatment,” Constructions From Composite Materials, Vol. 1, 2008, pp. 12-17.

[18] А. P. Klishin, S.V. Rudnev and V.I. Vereshagin, “New Technological Approaches of Reception Alumina Materials with Usage of Magnetothermal Treatment,” Refractories and Technical Ceramics, No. 6, 2009, pp. 30-34.

[19] N.V. Belov, “Structural Crystallography,” USSR Academy of Sciences, 1952.