Historical Overview

V. Elser in reference [1] stated that the Katz and Gratias model [2,3] of icosahedral Al–Cu–Fe and the Boudard and de Boissieu model [4,5] of icosahedral Al–Pd–Mn can both be explained as a decoration of the primitive icosahedral tiling P [6] by the Bergman and therefore automatically by Mackay polyhedra. Elser introduced in [1] a unifying model that is partly deterministic and partly probabilistic, i.e. some atomic positions are fixed and another occupied with certain probabilities. Z. Papadopolos introduced an analogous unifying model as a decoration by Bergman polyhedra of an F-phase tiling *(2F) [7,8], using the fact that the tiling P is locally derivable from the tiling *(2F) [7]. She constructed the windows in [8] for the unifying model formulated in [1]. The Papadopolos and Kasner model [9,10] is a deterministic model with polyhedral windows, somewhat different from the Katz and Gratias windows.


[1] V. Elser Phil. Mag. B, 1996, 73, 641.
[2] A. Katz and D. Gratias J. Non-Cryst. Solid, 1993, 153&154, 187.
[3] A. Katz and D. Gratias in Proceedings of the 5th International Conference on Quasicrystals, Ed. by C. Janot and R. Mosseri, World Scientific, Singapore, 1995, 164.
[4] M. Boudard, M. de Boissieu, C. Janot, G. Heger, C. Beeli, H.-U. Nissen, H. Vincent, R. Ibberson, M. Audier and J. M. Dubois J. Phys.: Condens. Matter, 1992, 4, 10149.
[5] M. de Boissieu, P. Stephens, M. Boudard, C. Janot, D. L. Chapman and M. Audier J. Phys.:Condens. Matter, 1994, 6, 10725.
[6] P. Kramer and R. Neri Acta Cryst. A, 1984, 40, 580.
[7] P. Kramer, Z. Papadopolos and D. Zeidler in Symmetries in Science V: Algebraic Structures, their Representions, Realizations and Physical Applications, Ed. by B. Gruber, L. C. Biedenharn and H. D. Döbner, Plenum Press, New York, 1991, 395.
[8] Z. Papadopolos, P. Kramer and W. Liebermeister in Aperiodic '97, Ed. by M. de Boissieu, J.-L. Verger-Gaugry and R. Currat, World Scientific, Singapore, 1998, 173.
[9] G. Kasner, Z. Papadopolos, P. Kramer and D. E. Bürgler Phys. Rev. B, 1999, 60, 3899.
[10] Z. Papadopolos, P. Kramer, G. Kasner and D. E. Bürgler Mat. Res. Soc. Symp., 1999, 553, 231.

General Comments


The shortest interatomic distances in the model, along the 5-fold, 2-fold and 3-fold axes are τ-1, τ-1 and τ-1, respectively. The standard distances , , and are related by /√3 = /(τ+2)1/2 = /2, where τ = (1 + √5)/2. The standard distance = 1 / √2 in E|| is set to be 4.561 Å for i-Al–Pd–Mn and 4.465 Å for i-Al–Cu–Fe.

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