Mean-field-theory study of the energetics of icosahedral, decagonal, and dodecagonal quasicrystals

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Abstract
Most quasicrystals are found in alloys which have Frank-Kasper phases. These Frank-Kasper phases are characterized by a network of interpenetrating polyhedra, with coordination numbers (CN) 12, 14, 15, and 16. While the CN-12 (icosahedral) polyhedra cause icosahedral and decagonal quasicrystalline order, the CN-14 polyhedra can lead to dodecagonal order. Motivated by the structure of these polyhedra, we have studied the energetics of a number of quasicrystalline structures, which have icosahedral, decagonal, and dodecagonal symmetries, using a modified model of Kalugin, Kitaev, and Levitov. We find that in a region of low to moderate stiffness in parameter space (parametrized by elastic stiffness and temperature), the dodecagonal structure is energetically more stable, while the decagonal structure wins in the high-stiffness regime. The decagonal structure is always lower in energy than the icosahedral structure. The dodecagonal structure, however, has a higher energy barrier with respect to the isotropic phase than the decagonal and icosahedral structures, which have comparable barriers.