Statistical approach in cluster analysis of two-dimensional quasicrystals

An analytical formula for the structure factor of Penrose tiling in the cluster approach was derived and tested. Probability distributions obtained for each Penrose position allow the number of different atoms that can decorate the cluster to be found. Calculations were performed in the average-unit-cell approach for Gummelt's cluster of 33 atoms, divided into three independent groups of atoms, and a kite cluster of 17 atoms, divided into seven independent groups.