Diffraction and modeling solve the structure of ytterbium-cadmium quasicrystals

Where are the atoms? Per Bak posed this question about the positions of atoms in icosahedral quasicrystals ¹ two years after their discovery in 1984. But despite thousands of papers published during the subsequent two decades, not one quasicrystal structure is known with the detail and accuracy that crystallographers can claim for normal crystals. The strange nature of quasicrystals accounts for the difficulty. ² Judging by the sharp Bragg peaks in their diffraction patterns, quasicrystals can possess a long-range order comparable to the most perfectly crystalline material. But unlike normal crystals, quasicrystals also possess a rotational symmetry that forbids packing into a repeated array of unit cells. Consequently, they lack periodic translational order and a straightforward path from Bragg peaks to structure.

The sensitive interaction of x rays with atomic electrons makes x-ray diffraction foremost among the methods available to determine the arrangement of atoms in a crystal. X rays reflected from the lattice planes produce a set of Bragg peaks of various intensities, with each peak characterized by a vector that measures the momentum transfer. For normal crystals, it takes just three integers—Miller indices—to label the reflections, from which a map of the triply periodic electron density can be reconstructed, provided one can also infer the phases of the reflected waves.

The lack of periodic order in quasicrystals complicates the analysis of their diffraction data. Calculations are done in the framework of hyperspace crystallography, a mathematical approach that treats a quasicrystal as a three-dimensional slice through a structure that is periodic in a higher-dimensional space. In that framework, a quasicrystal structure is defined by arrangements of 3D hypersurfaces in 6D space—in contrast to a normal crystal, which is defined by arrangements of 0D points, the atoms, in 3D space.

To determine the physical structure, the crystallographer must go through the arduous process of determining the positions, sizes, and detailed shapes of those hypersurfaces, sometimes called occupation domains, through a complete 6D analysis of the diffraction spectrum. Strictly speaking, the occupation domains are regions where electron densities are concentrated in the 6D unit cell. To recover a 3D perspective, one then cuts a slice through that 6D cell.

Based on that kind of procedure, researchers led by Hiroyuki Takakura from Japan’s Hokkaido University propose the most complete structural model to date for a type of icosahedral quasicrystal alloy made of ytterbium and cadmium. ³ Takakura and company observed 5024 unique x-ray diffraction peaks, an order of magnitude larger set of reflections than typically used to analyze intermetallic alloys such as brass or bronze. “Considering the quasicrystal’s high symmetry, that’s a quantity of data in reciprocal space matched only by the very largest biomolecules,” comments Cornell’s Veit Elser.

With just two elements, Yb–Cd offers the advantage of chemical simplicity. Until a few years ago, most researchers thought that only ternary quasicrystals could be stably formed. In 2000 Takakura and coauthor An Pang Tsai, then at the National Research Institute for Metals in Tsukuba, Japan, identified YbCd_5.7 as quasicrystalline, and it remains one of only two known stable binary quasicrystals (see Physics Today, February 2001, page 17 ). Moreover, the large difference in atomic number between the two elements ideally suits the system for x-ray diffraction.

X-ray diffraction, however, captures the spatially averaged structure. Some atomic sites appear to be only partially occupied—that is, a site might be occupied at some locations in the sample, while equivalent sites remain vacant elsewhere. The effect is to blur the electron-density distribution in the 6D unit cell. The intrinsic disorder creates ambiguities that, despite the huge data set, prevent the unique assignment of every atom to its own specific site.

Takakura and company therefore had to interpret their recent diffraction data in concert with a model based on what are called approximants, crystalline phases whose compositions vary ever so slightly from the quasicrystal but whose structures are periodic and well known. Comparing the structures of the cubic approximants YbCd₆ and Yb₁₃Cd₇₆ with the reconstructed electron density of the quasicrystal, the team realized that the largest common building block for the approximants, a rhombic triacontahedron (RTH) decorated by 92 Cd atoms on its edges and vertices, also makes up the bulk of YbCd_5.7 (see the figure on page 24).

Like a nested set of Russian dolls, four subshells fit concentrically inside the RTH. The oddball among them is the four-sided tetrahedron, whose symmetry breaks the otherwise fivefold icosahedral symmetry of the rest of the lattice. The tetrahedron’s Cd atoms are likely to rotate among several symmetry-equivalent orientations, as they do in the YbCd₆ approximant when temperatures rise above 100 K.

Only two other geometrical motifs are required to reconstruct the quasicrystal lattice: acute and obtuse rhombohedra that link the RTH clusters together. One can liken the researchers’ approach to assembling the pieces of a puzzle in which every atom is assigned a plausible position. The x-ray data confirm the model’s prescription of where Yb and Cd reside within the various clusters and reveal the packing and distribution of those clusters in the lattice.

The 3D slab pictured here represents a finite part of their solution. The cogwheel complex, the stellate polyhedron, and their fragments illustrate the rich variety of shapes that can form locally from particular arrangements of acute and obtuse rhombohedra.

A strength of the model, according to Carnegie Mellon University’s Michael Widom, lies in its flexibility: Though deterministic, the model accounts for certain degrees of freedom in the arrangement of building blocks while the linkages between clusters are preserved. Clusters may dynamically and subtly flip from one configuration to another. It remains unclear to what extent quasicrystals are stabilized by adopting an ideal, deterministic arrangement of clusters that minimizes the system’s energy or by adopting an ensemble of arrangements that maximizes its entropy. But one can now ask whether the new structural elements found in the quasiperiodic phase provide any energetic dividend. That would be a step toward answering an even more fundamental question than Bak’s: Why should quasicrystals form at all?