Abstract
It is argued that the prevailing definition of quasicrystals, requiring them to contain an axis of symmetry that is forbidden in periodic crystals, is inadequate. This definition is too restrictive in that it excludes an important and interesting collection of structures that exhibit all the well-known properties of quasicrystals without possessing any forbidden symmetries.
Similar content being viewed by others
REFERENCES
P. M. Chaikin and T. C. Lubensky, Principles of Condensed Matter Physics (Cambridge University Press, Cambridge, 1995), p. 680.
S. Van Smaalen, “Incommensurate crystal structures, ” Cryst. Rev. 4, 79(1995).
M. Senechal, Quasicrystals and Geometry (Cambridge University Press, Cambridge, 1996), p. 31(but see the comment on p. 33).
D. Gratias, “A brief survey of quasicrystallography, ” Ferroelectrics 250, 1(2001).
D. Levine and P. J. Steinhardt, “Quasicrystals I: Definition and structure, ” Phys. Rev. B 34, 596(1986).
D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, “Metallic phase with long-ranged orientational order and no translational symmetry, ” Phys. Rev. Lett. 53, 1951(1984).
International Union of Crystallography, Acta Cryst. A 48, 922(1992).
F. Denoyer, P. Launois, T. Motsch, and M. Lambert, “On the phase transition mechanism in Al-Cu-Fe: structural analysis of the modulated quasicrystalline and of the microcrystalline states, ” J. Non-Cryst. Solids 153–154, 595(1993).
N. Menguy, M. de Boissieu, P. Guyot, M. Audier, E. Elkaim, and J. P. Lauriat, “Modulated icosahedral Al-Fe-Cu phase: a single crystal X-ray diffraction study, ” Ibid., p. 620.
R. Lifshitz, “The symmetry of quasiperiodic crystals, ” Physica A 232, 633(1996).
T. Janssen, “Crystallography of quasicrystals, ” Acta Cryst. A 42, 261(1986).
A. Katz and D. Gratias, in Lectures on Quasicrystals, F. Hippert and D. Gratias, eds. (Les Editions de Physique, Les Ulis, 1994), p. 247.
L. Elcoro, J. M. Perez-Mato, and R. Withers, “Intergrowth polytypoids as modulated structures: The example of the cation deficient oxides LaTi1−xO3, ” Zeit. Krist. 215, 727(2000).
V. Petříček, A. van der Lee, and M. Evain, “On the use of Crenel functions for occupationally modulated structures, ” Acta Cryst. A 51, 529(1995).
B. Grünbaum and G. C. Shephard, Tilings and Patterns (Freeman, New York, 1987), Sec.10.4.
R. Paredes, J. L. Aragón, and R. A. Barrio, “Nonperiodic hexagonal square-triangle tilings, ” Phys. Rev. B 58, 11990(1998).
Z. H. Mai, L. Xu, N. Wang, K. H. Kuo, Z. C. Jin, and G. Cheng, “Effect of phason strain on the transition of an octagonal quasicrystal to a β-Mn-type structure, ” Phys. Rev. B 40, 12183(1989).
M. Baake, D. Joseph, and P. Kramer, “The Schur rotation as a simple approach to the transition between quasiperiodic and periodic phases, ” J. Phys. A: Math. Gen. 24, L961(1991).
S. Baranidharan and E. S. R. Gopal, “Quasiperiodic tilings with fourfold symmetry, ” Acta Cryst. A 48, 782(1992).
R. Lifshitz, “The square Fibonacci tiling, ” J. Alloys Comp. 342, 186(2002).
M. Hase, M. Egashira, N. Shinya, H. Miyazaki, K. M. Kojima, and S. Uchida, “Optical transmission spectra of two-dimensional quasiperiodic photonic crystals based on Penrose-tiling and octagonal-tiling systems, ” J. Alloys Comp. 342, 455(2002).
K. Fradkin-Kashi and A. Arie, “Multiple-wavelength quasi-phase-matched nonlinear interactions, ” IEEE J. Quantum Electron. 35, 1649(1999). K. Fradkin-Kashi, A. Arie, P. Urenski, and G. Rosenman, “Multiple nonlinear optical interactions with arbitrary wave vector differences, ” Phys. Rev. Lett. 88, 023903(2002).
Y. C. Feng, G. Lu, H. Q. Ye, K. H. Kuo, R. L. Withers, and G. Van Tendeloo, “Experimental evidence for and a projection model of a cubic quasicrystal, ” J. Phys.: Condens. Matter 2, 9749(1990).
P. Donnadieu, H. L. Su, A. Proult, M. Harmelin, G. Effenberg, and F. Aldinger, “From modulated phases to a quasiperiodic structure with a cubic point group and inflation symmetry, ” J. Phys. I France 6, 1153(1996).
P. Donnadieu, “The deviation of the Al6Li3Cu quasicrystal from icosahedral symmetry— A reminisence of a cubic crystal, ” J. Phys. I France 4, 791(1994).
J. Dräger, R. Lifshitz, and N. D. Mermin, “Tetrahedral quasicrystals, ” in Proceedings of the 5th International Conference on Quasicrystals, C. Janot and R. Mosseri, ed. (World Scientific, Singapore, 1996), p. 72.
H. Selke, U. Vogg, and P. L. Ryder, “New quasiperiodic phase in Al85Cr15, ” Phys. Stat. Sol. A 141, 31(1994).
N. W. Ashcroft and N. D. Mermin, Solid State Physics (Saunders College, Philadelphia, 1976).
N. D. Mermin, Boojums All the Way Through (Cambridge University Press, Cambridge, 1990).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Lifshitz, R. Quasicrystals: A Matter of Definition. Foundations of Physics 33, 1703–1711 (2003). https://doi.org/10.1023/A:1026247120031
Issue Date:
DOI: https://doi.org/10.1023/A:1026247120031