Chapter 20: Metallic Structures IV: Quasicrystals

Quasicrystals represent a revolutionary departure from classical crystallographic understanding, combining long-range orientational order with sharp diffraction peaks while fundamentally lacking the three-dimensional translational periodicity that defines conventional crystals. This chapter traces the historical breakthrough of discovering icosahedral symmetry in aluminum-manganese alloys, a rotational symmetry previously thought mathematically impossible in periodic structures, and establishes the theoretical foundation by defining quasi-periodic arrangements through Fourier analysis that produces discrete delta functions indexed by a finite set of characteristic lengths. The mathematical framework relies heavily on the irrational golden mean and Fibonacci sequence to construct one-dimensional quasi-periodic lattices using recursive substitution algorithms and the cut-and-project method, which conceptually lifts structures into higher-dimensional space and projects them back to generate aperiodic order in lower dimensions. The chapter extends these principles to two-dimensional systems, examining Penrose tilings composed of kites and darts or rhombi that fill space aperiodically while maintaining strict five-fold rotational symmetry, and explains how inflation and deflation transformations coupled with matching rules enforce this geometric organization. Higher-dimensional crystallography receives detailed treatment, demonstrating how three-dimensional icosahedral quasicrystals emerge from arrangements of prolate and oblate rhombohedra and can be indexed using six integers derived from projecting basis vectors from six-dimensional hypercubic lattices. The chapter concludes by evaluating alternative structural models, including the multiple-twinning hypothesis and the icosahedral glass model, against the quasi-periodic framework, and presents microscopic evidence such as characteristic dodecahedral and triacontahedral crystal faceting to substantiate the existence of genuine quasicrystalline phases rather than approximant structures or disordered phases.