Advances in Natural Quasicrystals and Quasicrystal Tilings

Abstract

The first part of this dissertation reports recent progress on natural quasicrystals. We present new evidence from a fragment of the quasicrystal-bearing CV3 carbonaceous chondritic meteorite Khatyrka that shows cross-cutting relationships and redox reaction between Al-Cu-bearing alloys and silicate phases. The new evidence establishes that the Al-Cu-bearing alloys (including quasicrystals) formed in outer space during a complex, multi-stage process. Some Al-bearing grains (including some quasicrystals) formed as a direct result of an impact in space a few 100 Ma. Most other Al-bearing grains (including quasicrystals) existed prior to the impact and thus formed in space at an earlier time. We also present the discovery of two new quasicrystals, including a second distinct Al-Cu-Fe icosahedral phase in Khatyrka---the first quasicrystal found in nature prior to discovery in the lab---and a synthetic Al-Fe-Cu-Cr-Ni icosahedral phase---the first quasicrystal to be synthesized in a laboratory shock experiment.

In the second part of this dissertation, we explore how different local isomorphism (LI) classes of quasicrystals vary in their structural and physical properties. We examine the continuum of LI classes of pentagonal quasicrystal tilings obtained by direct projection from a five-dimensional hypercubic lattice.

Our initial focus is on hyperuniformity, the suppression of long-wavelength density fluctuations relative to typical structurally disordered systems. We study how the degree of hyperuniformity depends on LI class. The results show that the degree of hyperuniformity is dominantly determined by the local distribution of vertex environments, and also exhibits a non-negligible dependence on the restorability. Among the pentagonal quasicrystal tilings, the Penrose tiling is the most hyperuniform. The difference in the degree of hyperuniformity is expected to affect physical characteristics, such as transport properties.

We then turn to a study of photonic quasicrystal heterostructures derived from the continuum of pentagonal quasicrystal tilings. We demonstrate that, with the exception of the Penrose LI class, all other LI classes result in degenerate, effectively localized states, with precisely predictable and tunable properties (frequencies, frequency splittings, spatial configurations). A relationship between the degeneracy of these states and the number of certain vertex environments is discussed, and potential applications for these states are described.