Crystallography in Spaces E2, E3, E4, E5 ...N0I Isomorphism Classes: Properties and Applications to the Study of Incommensurate Phase Structures, Molecular Symmetry Groups and Crystal Families of Space E5</sup

R. Veysseyre; D. Weigel; T. Phan; H. Veysseyre

doi:10.4236/apm.2015.54017

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Advances in Pure Mathematics > Vol.5 No.4, March 2015

Crystallography in Spaces E², E³, E⁴, E⁵ ...N⁰I Isomorphism Classes: Properties and Applications to the Study of Incommensurate Phase Structures, Molecular Symmetry Groups and Crystal Families of Space E^{5 ()}

R. Veysseyre^1*, D. Weigel¹, T. Phan¹, H. Veysseyre²

¹Laboratoire Mathématiques appliquées aux Systèmes, Ecole Centrale Paris, Paris, France.
²Institut Supérieur de Mécanique de Paris, Paris, France.

DOI: 10.4236/apm.2015.54017 PDF HTML 2,233 Downloads 2,635 Views Citations

Abstract

This paper mainly consists of the classification of all crystallographic point groups of n-dimensional space with n ≤ 6 into different isomorphism classes. An isomorphism class is defined by a type of finite mathematic group; for instance, the different types of mathematic groups have been well defined and studied by Coxeter. This classification may be used in the investigation of several domains of crystallography such as the study of the incommensurate phases, the quasi crystals … Indeed, each mathematic substitution group characterizes an isomorphism class of crystallographic point groups (spaces E² or E³), of point groups of super crystals (spaces E⁴ or E⁵), and of molecular symmetry groups (spaces E² or E³). This mathematic group gives interesting information about: 1) the incommensurate phase structures and their phase transitions according to the Landau’s theory in their super spaces E⁴, E⁵, E⁶, ···; 2) the molecular symmetry group of chemisorbed molecules in space E² (paragraph 2) or of the molecular crystal or solution in view of studying the molecule structure or its rotations or vibrationsin space E³; 3) the geometric polyhedron symmetry groups as the regular rhombohedron in space E³, the rhombotope in space E⁴ or the rhombotope in space E⁵. Then, thanks to the isomorphism classes, we shall give properties of some crystal families that we have not published up to now. This formalism may be used to study crystal families in n-dimensional space with n > 6.

Keywords

Crystallographic Point Groups, Isomorphism Classes, Incommensurate Phase Structures, WPV (Weigel Phan Veysseyre) Symbols of Point Groups

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Veysseyre, R. , Weigel, D. , Phan, T. and Veysseyre, H. (2015) Crystallography in Spaces E², E³, E⁴, E⁵ ...N⁰I Isomorphism Classes: Properties and Applications to the Study of Incommensurate Phase Structures, Molecular Symmetry Groups and Crystal Families of Space E^{5Advances in Pure Mathematics, 5, 137-149. doi: 10.4236/apm.2015.54017.}

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1] International Tables for Crystallography (1996) Vol. A Edited by T. Hahn. Kluwer Academic Publishers, Dordrecht/ Boston/London.

[2] Veysseyre, R. and Veysseyre, H. (2002) Crystallographic Point Groups of Five-Dimensional Space 1. Their Elements and Their Subgroups. Acta Crystallographica Section A, 58, 429-433.

[3] Weigel, D., Phan, T. and Veysseyre, R. (1987) Crystallography, Geometry and Physics in Higher Dimensions. III. Geometrical Symbols for the 227 Crystallographic Point Groups in Four-Dimensional Space. Acta Crystallographica Section A, 43, 294-304.
http://dx.doi.org/10.1107/S0108767387099367

[4] Janssen, T., Birman, J.L., Koptsik, V.A., Senechal, M., Weigel, D., Yamamoto, A., Abrahams, S.C. and Hahn, T. (1999) Symmetry Elements in Space Groups and Point Groups. Acta Crystallographica Section A, 55, 761-782.

[5] Weigel, D., Phan, T. and Veysseyre, R. (2008) Crystal Families and Systems in Higher Dimensions, and Geometrical Symbols of Their Point Groups. II. Cubic Families in Five- and n-Dimensional Spaces. Acta Crystallographica Section A, 64, 687-697.
http://dx.doi.org/10.1107/S0108767308028766

[6] Brown, H., Bul?w, R., Neubüser, J., Wondratschek, H. and Zassenhaus, H.J. (1978) Crystallographic Groups of Four-Dimensional Space. Wiley, New York.

[7] Plesken, W. (1981) Bravais Groups in Low Dimensions. Communications in Mathematical Chemistry, 97-119.

[8] Plesken, W. and Hanrath, W. (1984) The Lattices of Six-Dimensional Euclidean Space. Mathematics of Computation, 43, 573-587.
http://dx.doi.org/10.1090/S0025-5718-1984-0758205-5

[9] Plesken, W. and Schulz, T. (2000) Counting Crystallographic Groups in Low Dimensions. Experimental Mathematics, 9, 407-411.

[10] Coxeter, H.S.M. and Moser, W.O.J. (1965) Generators and Relations for Discrete Groups. Springer-Verlag, Berlin, G?ttingen, Heidelberg, New York, 117-133.

[11] Van Smaalen, S., Bronsema, K.D. and Mahy, J. (1986) The Determination of the Incommensurately Modulated Structure of Niobium Tetratelluride. Acta Crystallographica B, 42, 43-50.

[12] Chen, Z.Y. and Walker, M.B. (1989) Superspace Symmetry Modes and Incommensurate-to-Incommensurate Phase Transition in NbTe4. Physical Review B, 40, 8983-8994.
http://dx.doi.org/10.1103/PhysRevB.40.8983

[13] Sch?enflies, A. (1984) Krystallsysteme und Krystallstructur. Springer, Berlin.
http://dx.doi.org/10.1007/978-3-642-61740-9

[14] Fedorov, E.S. (1891) The Symmetry of Regular Systems of Figures (in Russian). Proceedings of the Imperial St. Petersburg Mineralogical Society, 28, 1-146.

[15] Shubnikov, A.V. and Belov, N.V. (1964) Colored Symmetry. Pergamon Press, Oxford.

[16] Bertaut, E.F. (1950) Raies de Debye-Scherrer et repartition des dimensions des domaines de Bragg dans les poudres polycristallines. Acta Crystallographica, 3, 14-18.
http://dx.doi.org/10.1107/S0365110X50000045

[17] Bertaut, E.F. (1952) Sur la correction de la transformée de Fourier d’une raie de Debye-Scherrer dans la mesure de dimensions cristallines. Acta Crystallographica, 5, 117-121.
http://dx.doi.org/10.1107/S0365110X5200023X

[18] de Wolff, P.M. (1974) The Pseudo-Symmetry of Modulated Crystal Structures. Acta Crystallographica, A30, 777-785.
http://dx.doi.org/10.1107/S0567739474010710

[19] Janner, A. and Janssen, T. (1977) Symmetry of Periodically Distorted Crystals. Physical Review B, 15, 643-651.
http://dx.doi.org/10.1103/PhysRevB.15.643

[20] Grébille, D., Weigel, D., Veysseyre, R. and Phan, T. (1990) Crystallography, Geometry and Physics in Higher Dimensions. VII. The Different Types of Symbols of the 371 Mono-Incommesurate Superspace Groups. Acta Crystallographica Section A, 46, 234-240.

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[1]	International Tables for Crystallography (1996) Vol. A Edited by T. Hahn. Kluwer Academic Publishers, Dordrecht/ Boston/London.
[2]	Veysseyre, R. and Veysseyre, H. (2002) Crystallographic Point Groups of Five-Dimensional Space 1. Their Elements and Their Subgroups. Acta Crystallographica Section A, 58, 429-433.
[3]	Weigel, D., Phan, T. and Veysseyre, R. (1987) Crystallography, Geometry and Physics in Higher Dimensions. III. Geometrical Symbols for the 227 Crystallographic Point Groups in Four-Dimensional Space. Acta Crystallographica Section A, 43, 294-304. http://dx.doi.org/10.1107/S0108767387099367
[4]	Janssen, T., Birman, J.L., Koptsik, V.A., Senechal, M., Weigel, D., Yamamoto, A., Abrahams, S.C. and Hahn, T. (1999) Symmetry Elements in Space Groups and Point Groups. Acta Crystallographica Section A, 55, 761-782.
[5]	Weigel, D., Phan, T. and Veysseyre, R. (2008) Crystal Families and Systems in Higher Dimensions, and Geometrical Symbols of Their Point Groups. II. Cubic Families in Five- and n-Dimensional Spaces. Acta Crystallographica Section A, 64, 687-697. http://dx.doi.org/10.1107/S0108767308028766
[6]	Brown, H., Bul?w, R., Neubüser, J., Wondratschek, H. and Zassenhaus, H.J. (1978) Crystallographic Groups of Four-Dimensional Space. Wiley, New York.
[7]	Plesken, W. (1981) Bravais Groups in Low Dimensions. Communications in Mathematical Chemistry, 97-119.
[8]	Plesken, W. and Hanrath, W. (1984) The Lattices of Six-Dimensional Euclidean Space. Mathematics of Computation, 43, 573-587. http://dx.doi.org/10.1090/S0025-5718-1984-0758205-5
[9]	Plesken, W. and Schulz, T. (2000) Counting Crystallographic Groups in Low Dimensions. Experimental Mathematics, 9, 407-411.
[10]	Coxeter, H.S.M. and Moser, W.O.J. (1965) Generators and Relations for Discrete Groups. Springer-Verlag, Berlin, G?ttingen, Heidelberg, New York, 117-133.
[11]	Van Smaalen, S., Bronsema, K.D. and Mahy, J. (1986) The Determination of the Incommensurately Modulated Structure of Niobium Tetratelluride. Acta Crystallographica B, 42, 43-50.
[12]	Chen, Z.Y. and Walker, M.B. (1989) Superspace Symmetry Modes and Incommensurate-to-Incommensurate Phase Transition in NbTe4. Physical Review B, 40, 8983-8994. http://dx.doi.org/10.1103/PhysRevB.40.8983
[13]	Sch?enflies, A. (1984) Krystallsysteme und Krystallstructur. Springer, Berlin. http://dx.doi.org/10.1007/978-3-642-61740-9
[14]	Fedorov, E.S. (1891) The Symmetry of Regular Systems of Figures (in Russian). Proceedings of the Imperial St. Petersburg Mineralogical Society, 28, 1-146.
[15]	Shubnikov, A.V. and Belov, N.V. (1964) Colored Symmetry. Pergamon Press, Oxford.
[16]	Bertaut, E.F. (1950) Raies de Debye-Scherrer et repartition des dimensions des domaines de Bragg dans les poudres polycristallines. Acta Crystallographica, 3, 14-18. http://dx.doi.org/10.1107/S0365110X50000045
[17]	Bertaut, E.F. (1952) Sur la correction de la transformée de Fourier d’une raie de Debye-Scherrer dans la mesure de dimensions cristallines. Acta Crystallographica, 5, 117-121. http://dx.doi.org/10.1107/S0365110X5200023X
[18]	de Wolff, P.M. (1974) The Pseudo-Symmetry of Modulated Crystal Structures. Acta Crystallographica, A30, 777-785. http://dx.doi.org/10.1107/S0567739474010710
[19]	Janner, A. and Janssen, T. (1977) Symmetry of Periodically Distorted Crystals. Physical Review B, 15, 643-651. http://dx.doi.org/10.1103/PhysRevB.15.643
[20]	Grébille, D., Weigel, D., Veysseyre, R. and Phan, T. (1990) Crystallography, Geometry and Physics in Higher Dimensions. VII. The Different Types of Symbols of the 371 Mono-Incommesurate Superspace Groups. Acta Crystallographica Section A, 46, 234-240.