Papadopolos and Kasner Model


This bulk model of icosahedral Al–Pd–Mn (with positions also applicable to Al–Cu–Fe) [1,2,3] is a decoration of an F-phase icosahedral tiling *(2F) [4] by Bergman (and therefore automatically Mackay) polyhedra. This model is deterministic and in agreement with inflation rules [3,5].


The deterministic atomic positions of this model are defined uniquely by the polyhedral windows Wq , Wa, and Wb in the coding space E. Each window defines a quasilattice from a corresponding copy of an icosahedrally projected D6 module. These windows correspond to the windows Wn', Wbc, and Wn respectively, for the Katz and Gratias Model. The model, describes both i-Al–Pd–Mn and i-Al–Cu–Fe.

Wq has edge lengths of τ-1 and where = 2/(τ + 2)1/2 and τ = (1 + √5)/2. Wa is a triacontahedraon with an edge length of τ-1. Wb is obtained by taking the tetrahedra, marked Δ, away from the triacontahedron of edge length τ.

is the standard length parallel to a 5-fold axis and is the standard length parallel to a 2-fold axis.



The patch of tiling *(2F) is generated using inflation rules [5]. 10 steps of inflation are performed and a patch of size 760 Å x 760 Å x 260 Å is obtained. The downloadable file is only 100 Å x 100 Å x 100 Å.

The Al, Pd and Mn atoms (in the file "model_M_chem01_cube.dat" labeled by (1), (2) and (3) respectively) are distributed on the windows Wq, Wb, and Wa in such a way that they roughly agree to the Boudard's atomic surfaces Wn0, Wn1, Wbc1, respectively, see Fig 1b in reference [6]. Whereas the window Wa contains only Pd atoms, the windows Wq and Wb contain more than one type of atom. The distribution of the elements was further slightly changed by taking quantum chemical calculations from Quandt and Elser [7] into account [8].

Whereas the model of atomic positions was tested through the work on clean surfaces, see references [2,9], the chemistry of the model (how the atoms of Al, Pd and Mn are distributed at the atomic positions) have not yet been tested.

The 2.3 MB file available for downloading is in ASCII format. Specific information about the format can be found at the top of the file.

Appropriate normals to 2-fold, 5-fold and 3-fold planes in "model_M_chem01_cube.dat" are:

(1, 0, 0) is 2-fold
(0, 0, 1) is 5-fold
(0, 2
τ, τ-1) is a 3-fold direction next to y = (0, 1, 0)


If you need instructions on how to determine atomic 2-fold, 3-fold, and 5-fold planes orthogonal to the given directions, click HERE.


[1] 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.
[2] G. Kasner, Z. Papadopolos, P. Kramer and D. E. Bürgler Phys. Rev. B, 1999, 60, 3899.
[3] Z. Papadopolos, P. Kramer, G. Kasner and D. E. Bürgler Mat. Res. Soc. Symp., 1999, 553, 231.
[4] 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.
[5] Z. Papadopolos, C. Hohnecker and P. Kramer Discrete Math., 2000, 221, 101.
[6] M. de Boissieu, P. Stephens, M. Boudard, C. Janot, D. L. Chapman and M. Audier J. Phys.:Condens. Matter, 1994, 6, 10725.
[7] A. Quandt and V. Elser Phys. Rev. B, 2000, 61, 9336.
[8] G. Kasner and Z. Papadopolos Mat. Res. Soc. Symp., 2001, 643, K9.7.1.
[9] Z. Papadopolos, G. Kasner, J. Ledieu, E. J. Cox, N. V. Richardson, Q. Chen, R. D. Diehl, T. A. Lograsso, A. R. Ross and R. McGrath Phys. Rev. B, 2002, 66, 184207.