Watching Quasicrystals Grow

Abstract

The initial discovery in 1984 [1] that certain metallic alloys can possess rotational symmetries forbidden in crystals, such as the symmetry of the regular icosahedron, raised a host of questions about these so-called quasicrystalline materials, or quasicrystals for short. Can the forbidden symmetries of quasicrystals be reconciled with their ability to form x-ray diffraction patterns with sharp spots? Can quasicrystals form faceted polyhedral shapes? Are they thermodynamically stable? These questions and many more have largely been answered over the subsequent decades, and in 2011 Daniel Shechtman was awarded the Nobel Prize in chemistry for the discovery of quasicrystals [2]. However, because quasicrystals have long-range but nonperiodic order, one of the most perplexing questions about them continues to be, “How do quasicrystals grow?” Theoretical models and computer simulations have made important advances on this front [3–6], but experimental answers to this question have been elusive. Using in situ highresolution transmission electron microscopy (HRTEM), Keiichi Edagawa and colleagues [7] at the University of Tokyo and Tohoku University have now successfully observed—at the atomic scale—quasicrystals in the act of growing. For centuries, crystals have been understood, indeed defined, to be composed of a single building block, or “unit cell,” that is copied and stacked together to form the entire three-dimensional crystal structure. The unit cell has been the foundational concept of crystallography from which models and theoretical frameworks have been developed to understand crystals and all of their properties. In particular, our comprehension of the ability of crystals to diffract x rays has—since the pioneering work of Walter Friedrich, Paul Knipping, and Max von Laue in 1912 [8]—been dependent on the long-range, periodic translational symmetry generated by the stacking of unit cells. The structures of quasicrystals, on the other hand, can be modeled based on the famous Penrose tilings—nonperiodic arrangements that have not just one but two unit cells. But there is a dilemma in our understanding of quasicrystal formation. Crystals are built from the repetition of a single unit cell, so only local interactions between atoms are necessary to understand how they might grow. The long-range but nonperiodic order of perfect quasicrystals seems, however, to be significantly more difficult to build, tile-by-tile, using only local interactions. Consider, for example, the tile attachment in a quasiperiodic direction in the Penrose tiling (Fig. 1). Along the quasicrystal growth direction, the ideal tiling has a sequence of wide (W) and narrow (N) rows of tiles, where W and N are arranged in the predictable but nonperiodic fashion of a so-called Fibonacci sequence. To create a Fibonacci sequence, start with the short sequence WN and generate a new sequence following the replacement rules W→WN and N→W, yielding WNW. Subsequent application of these replacement rules to a prior sequence systematically generates a still longer Fibonacci sequence, such as WNWWN and WNWWNWNW (Fig. 1 illustrates part of this latter sequence). But what mechanism ensures that new layers of tiles attach in this specific way during growth? Therein lies the dilemma.