Generalized Penrose tiling as a quasilattice for decagonal quasicrystal structure analysis

Area of Science:

• Materials Science
• Crystallography
• Mathematical Physics

Background:

• Penrose tilings are aperiodic structures with unique properties.
• Generalized Penrose tilings introduce a tunable parameter (s) affecting long-range order.
• Conventional methods for analyzing these tilings can be computationally intensive.

Purpose of the Study:

• To derive a structure factor formula for arbitrarily decorated generalized Penrose tilings.
• To demonstrate a method for directly controlling long-range order via the shift parameter.
• To offer a more efficient alternative to higher-dimensional analysis methods.

Main Methods:

• Utilizing the average unit cell approach for structure factor derivation.
• Developing a formula that operates directly in physical space.
• Implementing the shift parameter (s ∈ 〈0; 1)) to define tiling variations.

Main Results:

• A novel structure factor formula is derived, dependent on the shift parameter (s).
• The formula allows direct modification of long-range order by adjusting 's' without altering tile decoration.
• This approach bypasses the need to rebuild the structure model for each parameter change.

Conclusions:

• The derived physical space formula offers a significant advantage for analyzing generalized Penrose tilings.
• This method simplifies the study of aperiodic structures and their tunable order.
• It provides a more efficient pathway for refining structural models compared to traditional higher-dimensional techniques.