Modelling the transformation from quasicrystal to crystal

Abstract

The transformation from quasicrystal to crystal tilings using the projection formalism can be done in three ways. The first is the rational orientation of the window, the second is the continuous reorientation of the projection plane, and the third is the continuous evolution of the hyperlattice. Using the analogy that the atoms move in a phase transformation, the suggestion that they be moved in the hyperspace too makes the third choice an attractive one. Here the window as well as the projection plane are fixed in the same orientation and then the magnitude of basis vectors describing the hyperlattice are continuously changed. Such a transformation from a quasiperiodic tiling to a periodic one will take the phase through a series of intermediate rational approximate structures as well as other quasiperiodic ones which exhibit some micro-crystalline series of intermediate rational approximate structures as well as other quasiperiodic ones which exhibit some micro-crystallline features. The final crystalline form does not contain overlapping vertices as obtained in the second alternative method. Three examples are shown consisting of the octagonal, the intermediate and the periodic tilings. The intermediate tiling is made of two different squares and a parallelogram, and the periodic tiling has a superlattice. The Ammann lines are preserved and are continuous through the entire process.