'Quasicrystal' Investigation Is Mounting

During the last four years, a protracted debate has been evolving in an effort to explain the existence of solids with “forbidden symmetries.” The debate ensued following the 1984 discovery (D.Shechtmanetal., Physical Review Letters, 53, 195 1-3, 12 November 1984) of an aluminum-magnesium alloy whose atomic structure appears to have an icosahedral symmetry—a symmetry that is supposed to be impossible, according to a century-old dictum of solid-state physics. The icosahedron is a regular polyhedron with fivefold symmetry axes that, according to the well-known theorems of crystallography, are geometrically incompatible with crystalline. (periodic) structure. In just the few years since the initial discovery, more than one hundred other alloys with icosahedral symmetry have been found, adding further incentive for finding an explanation for the solids. Finally, in just the past few months, two break- throughs have taken place that appear to settle the issue. These advances were presented at a recent workshop held at the University of Pennsylvania in Philadelphia, sponsored by the university’s School of Arts and Sciences Natural Sciences Association. The new results strongly support the proposal that the icosahedral solids are quasicrystals, a fundamen- tally new phase of solid matter on equal par with crystals (D. Levine, PJ. Steinhardt, Physical Review Letters, 53, 2477-80, 24 December 1984). According to the quasicrystal picture, atoms can be grouped into regularly repeating clusters, but the clusters are not spaced periodically (equally), as they are in crystals. Instead, the clusters are spaced quasiperiodically: The sequence of spacings can be expressed as a sum of periodic functions in which the ratio of the periods is an irrational number. The inspirational analog is the Penrose tiling, space-filling, two-dimensional, quasiperiodic arrangement of two tile shapes. In a quasicrystalline. solid, one imagines that each tile type is replaced by some type of atomic cluster. Because they are not periodic, quasicrystals can violate the usual laws of crystallography that restrict the allowable symmetries. It has been shown that any symmetry, including icosahedral symmetry, is possible. The competing proposals represent less dramatic departures from traditional solid-state physics. According to the multiple-twinning model, the icosahedral alloys are really ordinary, penodic crystals. A given alloy forms from a collection of many crystalline grains that, for energetic reasons, join into icosahedrally symmetric arrangements. In the icosahedral glass model, the solids are intrinsically disordered atomic configurations obtained by randomly packing together icosahedral clusters. The random packing maintains icosahedral symmetry but does not produce the well-ordered spacing of clusters characteristic of the quasiciystal model. The key experimental tests that distinguish the models are diffrac tion measurements obtained by scattering electrons, X-rays, or neutrons from the samples. For a “perfect” icosahedral quasicrystal, the diffraction pattern should consist of perfectly sharp (Bragg) peks arranged in an icosahedrally symmetric pattern. Although this description matched the original diffraction measurements obtained from the icosahedral solids to first approximation, higher resolution experiments soon revealed that the mensured peaks were neither perfectly sharp nor perfectly located at icosahedrally symmetric positions. One obvious explanation is that the technique used to form alloys introduced imperfections strains, and defects, which distort the peak pattern. In particular, theorists have predicted that a particular kind of distortion, known as “phason strains,” is likely to predominate in quasiciystals. Phason strains correspond to local re-shuffling of two or more distinguishable atomic clusters without distorting the clusters themselves (like a rearrangement of tiles in a Penrose tiling). They have no analog in crystals. Later calculations showed-that the phason strains predicted by the quasicrystal model provide a qualitative and quantitative accounting of the observed peak broadenings and peak shifts. Nevertheless, despite its predictive success, the quasicrystal model has not been broadly accepted owing to two related, persistent beliefs: (1) strong theoretical arguments seemed to suggest that it is physically impossible to grow a perfect quasicrystal; and (2) according to wide-ranging experimental experience, it seemed technically impossible ever to form samples with significantly reduced phason-strain imperfections, suggesting the existence of some physical impediment to reaching the quasicrystal state. The recent breakthroughs have shown that both beliefs are myths. For several years, it had been claimed that growing quasicrystals (or any quasiperiodic structure, for that matter) is impossible unless there are long-range atomic interactions. The proviso was damning, since long-range interactions are physically unlikely in real materials. The claim was motivated partially by mathematical proof that growing perfect quasiperiodic atomic chains in one dimension requires infiniterange interactions. Perhaps the strongest argument was that there appeared to be no way of constructing perfect Penrose tilings via a local “aggregation procedure” (that is, adding one tile at a time using only local information to determine where to add the tile). In his original construction, Roger Penrose had produced a set of “matching rules” that constrain the way any two tiles may join edge-to-edge. He showed that the only space-filling tiling that can be constructed in a manner consistent with the matching rules is the quasicrystalline Penrose tiling. However, if one imagines “growing” a Penrose tiling by random additions of tiles at the surface of a cluster in accordance at each stage with Penrose’s edge-matching rules, one finds that this growth process fails to produce a perfect Penrose tiling. There is sufficient free choice that tiles added to distant points in the cluster may be misaligned, resulting in untileable regions as the growth process continues. It was believed. that global (long-range) rules were necessary to avoid such misalignment. Hence, the discovery of an alternative set of local rules leading to a successful growth algorithm for constructing perfect Penrose tilings was a significant conceptual break- through (G.Y Onoda et al., Physical Review Letters, 60, 2653-6, 20 June 1988). IBM’s David DiVincenzo reviewed the development at the recent workshop. The new local rules replace Penrose’s edge matching with vertex matching. To decide whether to add a given tile to a given vertex on the surface of a growing cluster, only local information about the other tiles attached to the same vertex is required. The proof that the new rules lead to perfect growth requires a novel application of quasicrystalline orientation and scaling (self-similarity symmetries, raising many new questions about the role of symmetry in growth. Furthermore, the research has shown that a new class of defects, termed “decapod” defects, can act as ideal seeds for rapid growth. The decapod defects play a role similar to screw dislocations in crystal growth—they ensure that there always exist growth sites on the surface where growth occurs most rapidly. Isolating such defects in the laboratory and using them as seeds in quasicrystal growth may lead to improvements in the quality of quasicrystalline materials, Kevin Ingersent of the University ol Pennsylvania reported on very recent progress showing that similar local growth rules can be con structed for a well-defined, countable set of other fivefold thingd. Confirmations Multiply By pure coincidence, just a few months after the key conceptual barrier to growing perfect quasicrystals had been eradicated, two new icosahedral alloys exhibiting a strikingly improved degree of perfection were discovered. Results by three different groups were reported in the workshop. IBM’s Peter Bancel spoke about his studies of large (1 cm radius), single-grain samples of AlFeCu. Franz Spaepen and Stephane Ebalard of Harvard University described their studies of the same alloy. Alan Goldman of Iowa State University spoke about studies of icosahedral AlRuCu, work done in collaboration with P. Stephens of the State University of New York at Stony Brook and K. Hiraga, A.P. Tsai, A. Inouc, and T. Masumoto of Tohoku University.. All groups reported near-perfect, energetically stable icosahedral solids. In particular, the phason defects that had plagued all previous icosahedral alloys were not observed. Bancel’s samples were the most impressive in that he could not resolve any peak broadening in his highest-resolution X-ray diffraction experiments. Using the instrumental resolution as a bound, he concluded that the diffraction peak widths must be more than 20 times sharper (on average) than those of any previously observed icosahedral alloys. The peak positions agree to within at least 0.03% of the ideal quasicrystal predictions. The degree of perfect icosahedral and positional atomic order indicated by these results is very difficult to explain it either the multiple-twinning or the icosahedral glass models. Goldman, who, along with P.W Stephens, D. Shechtman, and I. Blech, developed the compet-. ing icosahedral glass model, graciously conceded that “this observation indicates that the quasicrystalline model...is the only appropriate description of these alloys.” Hence, the notion that quasicrystals are impossible to grow has now been removed both theoretically and experimentally. The discoveries signal an important milestone in the field. Since the discovery of the first icosahedral alloys, the research emphasis has been on characterizing the atomic arrangement: Is it quasicrystalline or some alternative structure? Perhaps there is no best model for the earlier, more disordered solids. However now that it is clear that near perfect quasicrystals can be grown in the laboratory, the focus should turn to determining the new physical properties resulting from their unique symmetries. It has already been ob served that orientational symmetry and self-similarity can have a surprising effect on quasicrystal growth. A deeper understanding of these effects may lead to predictions of unexpected electronic, phonon, and thermal properties never seen in previously existing materials. Paul J. Steinhardt, who coined the term "quasicrystal," is a professor of physics at the University of Pennsylvania in Philadelphia.

'Quasicrystal' Investigation Is Mounting

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