Equilibrium faceting shapes for quasicrystals

The equilibrium shapes (ES's) of icosahedral quasicrystals are analyzed for a wide class of lattice models which incorporate finite-range two-body interactions. Completely faceted shapes have been predicted for such models at temperature T=0. We prove that a number of simple shapes cannot be ES's for any model in this class for which the atomic interactions are constrained to be pure-attractive. This extends the result of Ho et al. [Phys. Rev. Lett. 59, 1116 (1987)], who showed that the pentagonal dodecahedron, a shape observed in grains of icosahedral Al-Cu-Fe and Ga-Mg-Zn, is a forbidden shape for pure-attractive models. We then introduce a lattice model for quasicrystals which incorporates mixed attractive and repulsive interactions and show that the possible ES's include the dodecahedron and other previously forbidden shapes.