Construction of 6D icosahedral crystals from approximate icosahedral 3D crystalline structures and phase transitions

Abstract

From any so-called approximant 3D crystal, i.e. exhibiting approximate icosahedral units in real space and in reciprocal space, we build, using an optimal rational cut in R⁶, a 6D icosahedral crystal and the 3D i-phase associated with it. This construction becomes complete after “saturation” of the atom orbits under the action of a 6D icosahedral space group. Within the projection framework from R⁶, this predicts a splitting of the Wyckoff positions, in the 6D crystal, above each 3D Wyckoff position of the approximant crystal. Arithmetic selection rules, different from a band algorithm, are required to recover 3D approximant crystals by projection. No tiling of Euclidean physical space is considered here; only atom positions serve in the construction. Generalized order parameters can be defined in this scheme.

Key Words:

• Icosahedral phase
• approximant crystal
• 6D crystal
• 6D crystallography