Displacive transformations for the octagonal and dodecagonal phases

In this Letter we present a new geometric transformation to connect in a continuous way the octagonal phase with the related crystalline ones. This transformation avoids the tiles vanishing and hence it allows us to define a simple atomic decoration which preserves the ‘hard core’ condition. In the model presented here, the quasicrystal-related crystal density ratio is so close to 1 that it implies very short atomic displacementa and allows phase transitions to be physically possible in large enough clusters. A Shannon entropy is defined to estimate physically this evolutionary model. The strategy is extended to study the connection of the dodecagonal phase with two hexagonal phases rotated 30° from one to another.