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 Fphase 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 W_{q} , W_{a}, and W_{b} in the coding space E_{⊥}. Each window defines a quasilattice from a corresponding copy of an icosahedrally projected D_{6} module. These windows correspond to the windows W_{n'}, W_{bc}, and W_{n} respectively, for the Katz and Gratias Model. The model, describes both iAl–Pd–Mn and iAl–Cu–Fe. W_{q} has edge lengths of τ^{1} and where = 2/(τ + 2)^{1/2} and τ = (1 + √5)/2. W_{a} is a triacontahedraon with an edge length of τ^{1}. W_{b} is obtained by taking the tetrahedra, marked Δ, away from the triacontahedron of edge length τ. is the standard length parallel to a 5fold axis and is the standard length parallel to a 2fold 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 W_{q}, W_{b}, and W_{a} in such a way that they roughly agree to the Boudard's atomic surfaces W_{n0}, W_{n1}, W_{bc1}, respectively, see Fig 1b in reference [6]. Whereas the window W_{a} contains only Pd atoms, the windows W_{q} and W_{b} 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 2fold, 5fold and 3fold planes in "model_M_chem01_cube.dat" are:
(1,
0, 0) is 2fold
(0, 0, 1) is 5fold
(0, 2τ, τ1)
is a 3fold direction next to y = (0, 1, 0)
If you need instructions on how to determine
atomic 2fold, 3fold, and 5fold 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. VergerGaugry 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.