Like The Cosby Show and the Macintosh computer, quasicrystals made their debut in 1984. By rapidly cooling an alloy of aluminum and manganese, a group led by Technion’s Dan Shechtman created a solid whose sharp x-ray diffraction patterns indicated a high degree of order. 1 But the patterns also exhibited symmetries that are impossible to realize in a regular repeating array in three dimensions. Neither amorphous nor crystalline in the traditional sense, quasicrystals have long-range orientational order but lack periodic translational order.
The first quasicrystalline materials were thermodynamically unstable. When heated, they formed regular crystals. But in 1987, the first of many stable quasicrystals were discovered, making it possible to produce large samples for study and opening the door to potential applications. Unlike Shechtman’s prototype, these stable quasicrystals contained three chemical elements, leading some to believe that quasicrystal stability required three elements.
That state of affairs changed two years ago when Matthias Conrad and Bernd Harbrecht of the University of Marburg and Frank Krumeich of the Swiss Federal Research Institute in Zurich succeeded in creating stable quasicrystals from two elements, tantalum and tellurium. 2 Those quasicrystals are two-dimensional in that the stacking is periodic along one axis, but along the other two it’s quasiperiodic with dodecagonal (12-fold) symmetry.
Now, a team led by An Pang Tsai from Japan’s National Research Institute for Metals in Tsukuba has discovered quasicrystals of cadmium–ytterbium that are stable and exhibit three-dimensional icosahedral symmetry. 3 The icosahedron, which has 20 identical triangular faces, has the largest finite group of symmetries in the three-dimensional world but its symmetry is off-limits to regular crystals. “Tsai’s discovery is one of the most exciting things to happen in quasicrystals in a long time,” says Iowa State University’s Pat Thiel.
Tsai didn’t stumble on Cd–Yb crystals by chance. In 1994, his team discovered stable icosadehral phases in ternary compounds of zinc and magnesium coupled with various rare earth (RE) elements. Four years ago, Tsai tried replacing Zn with Cd, which appears in the same column of the periodic table. But, not knowing the right stoichiometry in advance, he couldn’t find any trace of the quasicrystal.
Then, in March last year, Tsai revisited the Cd-Mg-RE system with a fresh approach and discovered eight to nine stable icosahedral quasicrystals. Among those alloys, Cd-Mg-Yb turned out to be particularly stable in that its formation didn’t require annealing. Curious about this property, Tsai looked up the Cd–Yb phase diagram (see figure 1). Next to the cubic Cd6Yb is a stable phase, Cd5.7Yb, which metallurgists had dubbed “unknown” and whose odd structure they had attributed to a very large lattice constant.
Figure 1. Binary phase diagram of cadmium–ytterbium alloys. The looping curved line marks the boundary between the liquid phase above and the solid phases below. Shown in red is the quasicrystalline Cd5.7Yb, whose stoichiometry is almost identical to its closest neighbor on the phase diagram, the cubic Cd6Yb.
Figure 1. Binary phase diagram of cadmium–ytterbium alloys. The looping curved line marks the boundary between the liquid phase above and the solid phases below. Shown in red is the quasicrystalline Cd5.7Yb, whose stoichiometry is almost identical to its closest neighbor on the phase diagram, the cubic Cd6Yb.
Tsai realized that Cd6Yb could be what quasicrystallographers call an approximant, a crystalline phase whose composition is very close to that of its quasicrystal relative but whose components (atomic clusters) are arranged periodically rather than quasiperiodically. If Cd6Yb really is an approximant, then the unknown phase was an excellent candidate for a quasicrystal. Inspired by this hunch, Tsai’s colleague Junqing Guo made grains of Cd5.7Yb in an induction oven. Electron microscope images confirmed the material’s quasicrystalline nature, as did x-ray diffraction patterns, one of which is reproduced on the cover.
Why should Yb, alone among the rare earths, form a stable quasicrystal with Cd? Tsai sees a clue in the size of the Yb atom, which comes in divalent and trivalent forms. Divalent Yb—the sort that forms the quasicrystal—has a radius of 1.94 Å, whereas trivalent Yb measures 1.74 Å. Europium is the only other rare earth that has both divalent and trivalent forms, but it doesn’t form binary quasicrystals with Cd. Most other rare earths have atomic radii between 1.75 and 1.85 Å; divalent Eu’s is 2.04 Å; so, reasons Tsai, Cd’s atomic partner must be bigger than 1.85 Å and smaller than 2.04 Å. Calcium, though not a rare earth, fits in that range, and, as Tsai’s team has recently discovered, also forms stable binary icosahedral quasicrystals with Cd.
Energy versus entropy
Despite 18 years of quasicrystal research written up in more than 5000 papers, not one quasicrystal structure is known with the completeness and accuracy that crystallographers take for granted. Nor has a definite explanation emerged for how and why quasicrystals form. Although the discovery of stable binary quasicrystals won’t lead directly to a solution, it does at least simplify the calculations.
The atoms in nearly all regular crystals arrange themselves in a symmetric repeating pattern because that arrangement has the lowest energy. But some regular crystals adopt the arrangement—or, rather, an ensemble of arrangements—that gives them the highest entropy. This energy-versusentropy dichotomy exists in the field of quasicrystals, too, but with the difference that some theorists believe that most, if not all, quasicrystals could be stabilized by entropy.
Advocates of both energy and entropy stabilization find evidence in Tsai’s discovery to support their cases. Cornell University’s Veit Elser of the entropy camp notes that Cd5.7Yb is a congruent melter—that is, the solid can be in equilibrium with the liquid, suggesting entropic kinship between the two phases. Says Elser: “When you take the high-temperature liquid cadmium–ytterbium and cool it down, the first thing that forms is the quasicrystal. And to get the approximant, you need a solid-state reaction involving the quasicrystal and the other crystal phase at lower temperature. That, for me, is a key point.”
Elser also finds support for the entropy picture in the structure that Tsai proposed for Cd5.7Yb’s atomic building blocks (see figure 2). At the center of the cluster is a tetrahedron of Cd atoms that breaks the cluster’s overall icosahedral symmetry. The energetic view relies on the deterministic packing of identical units, but in the entropic picture the units don’t have to be arranged so carefully. “You just don’t care how the tetrahedron is oriented. All orientations are just as likely,” says Elser.
Figure 2. A preliminary structure for the atomic cluster that forms the basis of Cd5.7Yb quasicrystal. Cadmium atoms are shown in red, ytterbium in yellow. Like a Russian doll, the successively smaller units (c, b, a) fit within the largest unit (d).
Figure 2. A preliminary structure for the atomic cluster that forms the basis of Cd5.7Yb quasicrystal. Cadmium atoms are shown in red, ytterbium in yellow. Like a Russian doll, the successively smaller units (c, b, a) fit within the largest unit (d).
In the energy camp, Princeton University’s Paul Steinhardt sees the cluster’s tetrahedron as providing the asymmetry that energy stabilization needs. In the energy picture, a repeating unit, the quasi-unit cell, finds the lowest energy state by maximizing its density. “The discovery of stable binary quasicrystals,” says Steinhardt, “is consistent with a simple relationship with crystals. Energy stabilization, local growth rules, simple repeating units—all these features, which are found in crystals, are also found in quasicrystals.”
The key difference between quasicrystals and regular crystals is that the units in quasicrystals can overlap, making it possible to realize symmetries that are forbidden to regular crystals. Though once rather complicated, the rules for overlapping atomic clusters are now simpler in the latest version of the energy-stabilization theory and don’t require the long-range collusion that was a feature of earlier versions.
Despite their differences, both camps agree that more and better structural data are needed on the Cd–Yb and Cd–Ca systems. For one thing, Tsai’s atomic cluster model is a first guess based on the Cd6Yb approximant. Also needed are measurements of the materials’ mechanical, thermal, and electrical properties. But don’t expect too many applications for Cd5.7Yb. Cadmium is poisonous. “The allure of these new materials will remain intellectual for the foreseeable future,” says Thiel.
