Abstract
Quasiperiodic structures (QSs), incommensurately modulated structures (IMSs) and composite structures (CSs) are the main types of aperiodic crystal structures (Cummins 1990,van Smaalen 1995,Axel and Gratias 1995,Yamamoto 1996,Steurer and Haibach 1998). Diffraction patterns of aperiodic crystals consist of sharp Bragg reflections like those of regular periodic crystals. In contrast, aperiodic crystal structures lack lattice periodicity (Burzlaff et al. 1992, Wondratschek 1992) in d-dimensional (dD) physical space. It was first shown by de Wolff (1974) that diffraction patterns of IMSs can be described as projections of appropriate n-dimensional (nD) reciprocal lattices onto physical space. Consequently, the direct space IMSs result from cutting nD periodic hypercrystals with physical space. All types of aperiodic crystals (excluding almost periodic structures) can be described entirely by the same nD approach. Crystallography of quasicrystals (QCs), therefore, corresponds mainly to crystallography extended to n dimensions. Hermann (1949) was the first to study the number of dimensions necessary for the existence of m-fold rotational symmetry preserving translation symmetry. For instance, a lattice being invariant under five-, eight-or twelvefold rotational symmetry has to be at least four-dimensional. In general, any aperiodic structure with a Fourier module corresponding to a ℤ-module of rank n, can be embedded in n dimensions to achieve a lattice periodic structure. Symmetry restrictions of the dD physical space drastically limit the number of symmetry groups needed in the corresponding nD symmetry analysis. A comprehensive description of the symmetry operations for nD periodic and quasiperiodic structures has been given, e.g., by Janssen (1992a).
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Axel, F., Gratias, D. (1995): Beyond Quasicrystals. Springer-Verlag, Berlin
Bancel, P.A., Heiney, P.A., Stephens, P.W., Goldman, A.I., Horn, P.M. (1985): Phys. Rev. Lett. 54, 2422
Burzlaff, H., Zimmermann, H., de Wolff, P.M. (1992): in International Tables for Crystallography, Vol. A, Hahn, T. (ed). Kluwer Academic Publishers, Dordrecht, p 736
Cervellino, A., Haibach, T., Steurer, W. (1998): Phys. Rev. B 57, 11 223
Cummins, H.Z. (1990): Phys. Rep. 185, 211
de Wolff, P.M. (1974): Acta Crystallogr. A 30, 777
Edagawa, K., Suzuki, K., Ichihara, M., Takeuchi, S., Shibuya, T. (1991): Philos. Mag. B 64, 629
Elser, V. (1985): Phys. Rev. B 32, 4892
Goldman, A.I., Kelton, R.F. (1993): Rev. Mod. Phys. 65, 213
Gratias, D., Katz, A., Quiquandon, M. (1995): J. Phys. Condens. Matter 7, 9101
Hermann, C. (1949): Acta Crystallogr. 2, 139
Janner, A. (1992): Acta Crystallogr. A 48, 884
Janssen, T. (1988): Phys. Rep. 168, 55
Janssen, T. (1992a): Z. Kristallogr. 198, 17
Janssen, T. (1992b): Philos. Mag. B 66, 125
Janssen, T., Janner, A., Looijenga-Vos, A, de Wolff, P.M. (1992): in International Tables for Crystallography, Vol. C, Wilson, A.J.C. (ed). Kluwer Academic Publishers, Dordrecht, p 797
Kalning, M., Kek, S., Krane, H.G., Dorna, V., Press, W., Steurer, W. (1997): Phys. Rev. B 55, 187
Koch, E. (1992): in International Tables for Crystallography, Vol. C, Wilson, A.J.C. (ed). Kluwer Academic Publishers, Dordrecht, p 10
Levine, D., Steinhardt, P.J. (1986): Phys. Rev. B 34, 596
Levitov, L.S., Rhyner, J. (1988): J. Phys. (Paris) 49, 1835
Luck, J.M., Godrèche, C., Janner, A., Janssen, T. (1997): J. Phys. A 26, 1951
Niizeki, K. (1991): J. Phys. A 24, 3641
Niizeki, K. (1992): J. Phys. A 25, 1843
Pavlovitch, A., Kléman, M. (1987): J. Phys. A 20, 687
Penrose, R. (1974): Bull. Inst. Math. Appl. 10, 266
Rabson, D.A., Merrain, N.D., Rokhsar, D.S., Wright, D.C. (1991): Rev. Mod. Phys. 63, 699
Rokshar, D.S., Merrain, N.D., Wright, D.C. (1988): Acta Crystallogr. A 44, 197
Socolar, J.E.S., Steinhardt, P.J. (1986): Phys. Rev. B 34, 617
Steurer, W. (1996): in Physical Metallurgy, Vol. 1, Cahn, R.W., Haasen, P. (eds). Elsevier Science BV, Amsterdam, p 371
Steurer, W., Haibach, T. (1998): in International Tables for Crystallography, Vol. B, Shmueli, U. (ed). Kluwer Academic Publishers, Dordrecht, in press
Steurer, W., Haibach, T. (1998): Acta Crystallogr. A, in press
van Smaalen, S. (1995): Cryst. Rev. 4, 79
Wang, R., Wenge, Y., Hu, C., Ding, D. (1997): J. Phys. Condens. Matter 9, 2411
Willis, B.T.M., Pryor, A.W. (1975): Thermal Vibrations in Crystallography. Cambridge University Press. Cambridge
Wondratschek, H. (1992): in International Tables for Crystallography, Vol. A, Hahn, T. (ed). Kluwer Academic Publishers, Dordrecht, p 711
Yamamoto, A. (1992): Phys. Rev. B 45, 5217
Yamamoto, A. (1996): Acta Crystallogr. A 52, 509
Zhang, H., Kuo, K. H. (1990): Phys. Rev. B 42, 8907
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1999 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Steurer, W., Haibach, T. (1999). Crystallography of Quasicrystals. In: Stadnik, Z.M. (eds) Physical Properties of Quasicrystals. Springer Series in Solid-State Sciences, vol 126. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-58434-3_3
Download citation
DOI: https://doi.org/10.1007/978-3-642-58434-3_3
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-63593-9
Online ISBN: 978-3-642-58434-3
eBook Packages: Springer Book Archive