Rend. Fis. Acc. Lincei DOI 10.1007/s12210-012-0203-3 SI: XR DIFFRACTION Quasicrystals: a brief history of the impossible Paul J. Steinhardt Received: 25 July 2012 / Accepted: 27 August 2012 Ó Accademia Nazionale dei Lincei 2012 Abstract The 30-year history of quasicrystals is one in as a brief personal perspective rather than a formal history. which, time after time, the conventional scientific view Over 11,000 papers have been written on this subject by about what is possible has been proven wrong. First, other authors whose important contributions to the field quasicrystals were thought to be mathematically impossi- could not be included in these few pages. I strongly com- ble; then, physically impossible; then, impossible unless mend the reader to reviews and other reminiscences to synthesised in the laboratory under carefully controlled obtain a complete account. conditions. One by one, these strongly held views have been disproven, the last only recently as the result of the discovery of a natural quasicrystal found in a meteorite 1 Mathematically impossible dating back to the formation of the solar system. This paper is a brief personal perspective on this history of misun- When the concept of quasicrystals was first introduced in derstanding and discovery. 1984 (Levine and Steinhardt 1984), it met resistance because it appeared to contradict mathematical theorems Keywords Quasicrystals Á Icosahedral symmetry Á that had been proven over the last two centuries and that Diffraction Á Icosahedrite had been adopted as essential truths in crystallography and solid state physics. Solids were thought to come in one of two forms, ordered and disordered, where ordered meant This pre´cis is based on a presentation delivered at the periodic and disordered meant amorphous or glassy. Accademia dei Lincei for a meeting commemorating the (Incommensurate crystals were known, but they are subject 100th anniversary of X-ray diffraction. The topic was to the same symmetry restrictions as periodic crystals.) The quasicrystals, a subject that has captivated me for over classic theorems of Schoenflies and Fyodorov proven in the 30 years with its never-ending ability to surprise. The 19th century established that periodic solids are only recent awarding of the Nobel Prize in Chemistry to Dan allowed two-, three-, four-, or sixfold symmetry axes, Shechtman for his discovery of the first icosahedral solid limiting the number of possibilities in three dimensions to and the centenary of X-ray diffraction made this a timely 32 point groups. Five-, seven- and all higher-order sym- occasion for a historical reflection. This essay is intended metry axes—an infinite number of possibilities—are incompatible with periodicity. Most forbidden of all is icosahedral symmetry, because it contains six independent This contribution is the written, peer-reviewed version of a paper presented at the conference ‘‘The Centennial of X-Ray Diffraction disallowed fivefold symmetry axes, the maximum number (1912–2012)’’, held at Accademia Nazionale dei Lincei in Roma on possible in three dimensions. The historic work of von May 8 and 9, 2012. Laue and the Braggs showed that periodic solids exhibit point-like (Bragg) diffraction patterns with the same sym- & P. J. Steinhardt ( ) metry restrictions. By contrast, amorphous solids and Department of Physics, Princeton Center for Theoretical Science, Princeton University, Princeton, NJ 08544, USA glasses have isotropic diffuse diffraction patterns. All this e-mail: [email protected] is rigorously true. 123 Rend. Fis. Acc. Lincei Somehow, after several decades, point-like diffraction, be some analogy for solids [Conway (unpublished); Mac- order, and crystal became incorrectly conflated as mathe- kay 1981, 1982; Kramer and Neri 1984; Elser 1986], matically equivalent. This was the conventional belief in though in most cases it was not made clear what this meant. 1981 when quasicrystals began to emerge as a hypothetical The approach in Levine and Steinhardt 1984 was system- idea. Now, after quasicrystals, it is recognized that the atic in first identifying the precise symmetries and order terms are inequivalent, though the field has not yet parameters that underlie the Penrose tiling and can be used broadened its view as much as it needs to, as discussed in to define a new phase of matter, namely quasiperiodicity the concluding remarks. and bond orientational order, and then showing that they A small crack in the establishment view came with the generalize to arbitrary non-crystallographic symmetries in theory of dislocation-mediated two-dimensional melting by two and three dimensions. Nelson and Halperin (1979), in which they argued for a Icosahedral symmetry became the first target. A three- two-stage melting process from crystal to liquid punctuated dimensional quasicrystal was constructed using polyhedral by an intermediate ‘‘hexatic phase’’. Hexatic diffraction units and certain ‘‘face-to-face matching’’ rules that force a is neither Bragg nor simply diffuse; rather the pattern quasiperiodic arrangement with perfect long-range icosa- consists of a hexagonal arrangement of algebraically hedral orientational order. The existence of matching rules decaying finite-width peaks. The rotational symmetry is not was an important advance because it meant that, conceiv- remarkable, since it is crystallographically allowed, but the ably, local interactions (e.g., between atoms or molecules) introduction of a ‘‘bond orientational order parameter’’ is, could be sufficient to make the new phase a ground state. The since it opens the possibility of a new type of ordered phase hypothetical phase was ultimately dubbed quasicrystal, that had not been considered previously for monatomic short for quasiperiodic crystal (Levine and Steinhardt 1984). systems. The mathematical properties are subtle. In real space, In a study with David Nelson and Marco Ronchetti the lack of periodicity means that there is no translation of (Steinhardt et al. 1981), we examined whether a similar a quasicrystal pattern that exactly overlays the original; effect occurs in three-dimensional supercooled liquids and consequently, each atom or cluster of atoms in a quasi- glasses composed of a single type atom with Lennard– crystal has a distinct global arrangement of atoms sur- Jones interactions. Earlier, Nelson and Toner (1981) had rounding it. On the other hand, there is a sequence of proposed the possibility of a cubatic bond orientationally translations of the pattern that almost overlay the original ordered phase, and the original intent of the investigation except for a small density of misalignments; and that was to search for it. The effort failed, but something density can be made arbitrarily small by going to ever curious turned up instead: icosahedral bond orientational larger translations. This is the closest to translational order extending many interatomic spacings. This was not a symmetry without being periodic. Similarly, there is new phase, since the icosahedral correlations only spanned sequence of points such that a rotation by 2p/5 about the a finite distance, but it introduced a new element: crystal- point almost overlaps the original (see Steinhardt and Bindi lographically forbidden symmetry. 2012 for illustrations of these properties). In a colloquium given in fall 1981, shortly after joining In reciprocal space, the almost translational symmetry the faculty at the University of Pennsylvania, the hypoth- guarantees a diffraction pattern consisting only of true esis was floated that, by including two or more atomic point-like peaks just like the Bragg peaks in a periodic species, it may be possible to have a truly new phase of crystal. The peaks are arranged in a reciprocal space lattice matter with infinite-range icosahedral bond orientational of points described by an integer sum of basis vectors set order that evades the crystallographic theorems. In the by the rotational symmetry, just as for periodic crystals, but audience was a young graduate student, Dov Levine, who with a symmetry that is forbidden to crystals. However, approached afterward to ask if he could join in pursuing unlike the case for crystals, the number of integer linearly this idea. Warnings that the project was highly risky and independent basis vectors D in d-dimensions exceeds d, may have not physical relevance did not deter him, and so resulting in a reciprocal lattice with a dense arrangement of the quest started. peaks labeled by D quasi-Miller indices. For example, the Within a matter of months, the concept of quasicrystals reciprocal lattice for three-dimensional icosahedral quasi- developed. An inspiration was the two-dimensional crystals has D = 6 quasi-Miller indices. The result is a Penrose tiling invented by Sir Roger Penrose (Penrose diffraction pattern that is distinctive and instantaneously 1974) 7 years earlier. Penrose had identified a pair of tile recognizable. shapes that can only fit together non-periodically, forming An important corollary is that the real space structure can a self-similar pattern full of fivefold symmetric clusters of be viewed as a projection or cut through a subset of points in a tiles. Penrose’s tiling attracted the attention of diverse six-dimensional hypercubic lattice; and the diffraction theorists; several independently speculated that there might pattern can be computed as the convolution of the 123 Rend. Fis. Acc. Lincei six-dimensional crystal diffraction pattern with the window paper appeared in Physical Review Letters a month after