Exploring oxide quasicrystals in internal space

Quantitative assessment of the structural quality of quasicrystals is a challenging task. Whereas diffraction techniques are a powerful tool for the coherently scattering regions, only diffuse scattering addresses structural inhomogeneities. Alternatively, local imaging techniques allow for direct inspection of the structures and for statistical evaluation of tiling elements. In both cases, inspecting the internal space characteristic for quasicrystals offers an additional, more sensitive perspective. Here, we analyze atomically resolved scanning tunneling microscopy images for three dodecagonal oxide quasicrystal (OQC) systems. Upon uplifting the 2D coordinates into a 4D hyperspace, the quasicrystal structure can be discussed in the internal space as well as in the physical space. The quasicrystal acceptance domain in internal space of all three OQCs increases logarithmically with the system size. From the acceptance domain expansion we determine the effective phason elastic constants, which reflect phason disorder within their square-triangle-rhombus tiling.