M. Hargittai, I. Hargittai - Symmetry through the Eyes of a Chemist (793765), страница 76
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The physical observation is that very small aggregates neednot be crystalline, although they may nevertheless be perfectly structured. Mackay’s proposal is to apply the name crystalloid to them. Heoffered the following definitions [133]:Crystal: The unit cell, consisting of one or more atoms, or otheridentical components, is repeated a large number of times by threenoncoplanar translations. Corresponding atoms in each unit cell havealmost identical surroundings. The fraction of atoms near the surfaceis small and the effects of the surface can be neglected.Crystallite: a small crystal where the only defect is the existence ofthe external surface. The lattice may be deemed to be distorted but itis not dislocated.
Crystallites may further be associated into a mosaicblock.Crystalloid: a configuration of atoms, or other identical components, finite in one or more dimensions, in a true free energyminimum, where the units are not related to each other by three latticeoperations.The above ideas have been further developed mainly by translatingthem into more quantitative descriptions and by applying them to9.8. Quasicrystals489various structural problems. They can also be compared with similarlynew definitions mentioned in the next Section. These new attemptsof taxonomy by no means belittle the great importance of the 230three-dimensional space groups and their wide applicability.
What isreally expected is that they will eventually help in the systematizationand characterization of the less easily handled systems with varyingdegrees of regularity in their structures.The appearance of quasicrystals on the scene of materials has givena great thrust to these developments.9.8. QuasicrystalsThe term “quasicrystal” was coined by Dov Levine and Paul Steinhardt, who studied the structure of metallic glasses by theoreticalmeans and modeling [134]. Using this term, they wanted to expressthe connection between crystals on the one hand and quasiperiodiclong-range translational order, on the other. Here, long-range translational order means that the position of a unit cell far away in the latticeis determined by the position of a given unit cell. In a crystal structure there is only one unit cell, whereas in a quasiperiodic structurethere is more than just one.
The repetition of the unit cell is regular inthe crystal whereas it is not regular, nor is it random, in the quasiperiodic structure. In the two-dimensional space this is accomplished, forinstance, by a Penrose tiling [135], which was originally created moreas recreational mathematics than an extraordinarily important scientific tool that it has eventually become. A Penrose tiling is shownin the Introduction where some attempts of pentagonal tiling overthe centuries are also mentioned. There is a detailed and systematic discussion of pentagonal tilings in Grünbaum and Shephard’sbook [136].The discovery of the Penrose tilings was a breakthrough in thatpentagonal symmetry occurred in a pattern otherwise described byspace group symmetry.
Curiously, the Penrose tiling was first communicated not by its inventor but by Martin Gardner in the January 1977issue of Scientific American [137]. Mathematical physicist RogerPenrose himself published subsequently a paper in a university periodical which was then reprinted in a mathematical magazine. The title4909 Crystalsof the communication was rather telling, Pentaplexity, with a moresomber subtitle, A Class of Non-Periodic Tilings of the Plane [138].Alan Mackay made the connection with crystallography [139]. Hedesigned a pattern of circles based on a quasi-lattice to model apossible atomic structure. An optical transformation then created asimulated diffraction pattern exhibiting local tenfold symmetry (see,in the Introduction).
In this way, Mackay virtually predicted the existence of what was later to be known as quasicrystals, and issued awarning that such structures may be encountered but may stay unrecognized if unexpected!‡The unique moment of discovery came in April 1982 when DanShechtman was doing some electron diffraction experiments onalloys, produced by very rapid cooling of molten metals. In the experiments with molten aluminum with added magnesium, cooled rapidly,he observed an electron diffraction pattern with tenfold symmetry(see, the pattern in the Introduction). It was as great a surprise asit could have been imagined for any well-trained crystallographer.Shechtman’s surprise was recorded with three question marks in hislab notebook, “10-fold???” [140].Fortunately, Mackay’s fear that quasicrystals may be encounteredbut may stay unrecognized did not materialize.
Although Shechtmanwas not familiar with the Penrose tiling and its potential implicationsfor three-dimensional structures, he had what Louis Pasteur called, aprepared mind for new things [141]. He did not let himself discouraged by the seemingly well-founded disbelief of many though didnot attempt to publish his observations until he and his colleaguesfound a model that could be considered a possible origin of theexperimental observation. Ilan Blech constructed a three-dimensionalmodel of icosahedra filling space almost at random, and added restrictions to the model stipulating that the adjacent icosahedra touch eachother at edges, or, in a later version, at vertices. The model simulated a diffraction pattern, which was consistent with Shechtman’sobservations.The first report about Shechtman’s seminal experiment did notappear until two and a half years after the experiment.
The delay‡Alan L. Mackay gave two remarkable lectures on fivefold symmetry at theHungarian Academy of Sciences, Budapest, in September, 1982, where he issuedthis warning.9.8. Quasicrystals491was caused by Shechtman’s cautiousness and by some journal editors’skepticism. The paper was titled modestly Metallic Phase with LongRange Orientational Order and No Translational Symmetry [142]. Itstarts with the following sentence: “We report herein the existenceof a metallic solid which diffracts electrons like a single crystal buthas point group symmetry m3 5 (icosahedral) which is inconsistentwith lattice translations.” The three-page report was followed by anavalanche of papers, conferences, schools, special journal issues, andmonographs.Independent of Mackay’s predictions and Shechtman’s experiments, there was another line of research by Steinhardt and Levine,leading to a model encompassing all the features of shechtmanite (theoriginal quasiperiodic alloy was eventually named so) and other materials that are symmetric and icosahedral, but nonperiodic [143].
It wasa perfect timing that as soon as they built up their model and producedits simulated diffraction pattern, they could see its proof from a realexperiment.Steinhardt, like Mackay before (see, Section 9.7), felt the needfor redefinition of materials categories [144]. His suggestions nowincluded the newly discovered quasicrystals. Steinhardt has succinctlycharacterized the crystals, glassy materials, and quasicrystals asfollows:Crystal: highly ordered, with its atoms arranged inclusters which repeat periodically, at equal intervals, throughout the solid.Glassy material: highly disordered, with atomsarranged in a dense but random array.Quasicrystal: highly ordered atomic structure, yetthe clusters repeat in an extraordinarily complexnonperiodic pattern.The appearance of quasicrystals caused a minirevolution in crystallography.
The lack of periodicity was a major obstacle in applying thetraditional terms and approaches to this domain of materials. This wasan interesting development also from the point of view of Mackay’ssuggestions for generalized crystallography. He truly anticipated thebreakdown of the perfect traditional system, which he felt a little tooperfect and, certainly, too rigid.4929 CrystalsIt has been suggested to treat quasicrystals as three-dimensionalsections of materials that are periodic in more than three dimensions.
On the other hand, a new and more general formulation ofcrystallography has also been proposed by David Mermin, whichwould stay within the realm of three-dimensionality and would nothave the concept of periodicity in the focus of its foundation [145].He compared abandoning the traditional classification scheme ofcrystallography, based on periodicity, to abandoning the Ptolemaicview in astronomy and likened changing it to a new foundation toastronomy’s adopting the Copernican view. This is why he gavethe title “Copernican Crystallography” to his communication. Thesuggestion was to build the new foundation on the three-dimensionalconcept of point-group operations that would have the concept ofindistinguishable densities in its focus, rather than identical densities, to correspond to the character of quasisymmetries, describing,among others, the quasicrystals. Incidentally, the first challenge tothe periodicity paradigm of crystallography was the observation ofincommensurately modulated structures [146].
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