A quasicrystal and related approximants generally have markedly complex
atomic structures.
In fact, their structures allow a unified description in which both the
structures are represented as different cuts through a given
"hyper-crystal", namely a hypothetical periodic structure defined in a
higher-dimensional space (called the hyperspace). This is why
researchers recently call them "hypermaterials".
Whereas the higher-dimensional crystallography (developed since 1990s)
serves as a proper language to describe their symmetries, the theory of
quasiperiodic tiling plays an important role in describing atomic
arrangement.
We have recently been focusing on the structure of thermodynamically
stable icosahedral quasicrsytals and related approximants in
alminium-transition metal alloy systems. For instance, we used
single-crystal X-ray diffraction data to perform a structure analysis of
a novel high-order approximant in the Al-Pd-Cr-Fe system. It turned out
that that the atomic structure is well described using a
three-dimensional tiling decorated with atomic clusters. By generalizing
this knowledge we have proposed a new framework applicable to more
general approximants and also quasicrystals.
Apart from this particular example, several other structural
investigations are also under way, aiming to gain essential insight into
the structural complexity for understanding the structure stability and
physical properties.