Icosahedral quasicrystal decoration models. I. Geometrical principles.

Abstract

It is proposed that quasicrystal structure determination should include the calculation of cohesive energies using realistic potentials. A class of atomic decoration models for $i$-AlMn is then presented, adopting the "canonical-cell" tiling geometry, with "Mackay icosahedron" clusters placed on all its nodes. The remaining atomic positions are based, as far as possible, on the known structure of $\ensuremath{\alpha}$-AlMnSi. These models guarantee good local packing of the atoms, whose displacements away from "ideal" positions are specified by only a moderate number of parameters. Certain atomic sites are uncertain as regards their occupancy and/or chemistry; variations of the decoration rules on these sites must be compared, in order to discover the correct one. Our models are well adapted to be relaxed under an effective Hamiltonian to optimize the cohesive energy; we show how the energies found in such relaxations can be used to extract an effective tile-tile Hamiltonian, as would be needed for future studies of phason elasticity and the development of long-range order. In addition, we clarify concepts needed for decoration models in general (in particular, the ways in which elaborate, more realistic decorations may be evolved from simpler ones). We also show that these decoration models are closely related, but not identical, to quasiperiodic structures defined using six-dimensional formalism.