 Quasicrystal

A quasiperiodic crystal, or, in short, quasicrystal, is a structure that is ordered but not periodic. A quasicrystalline pattern can continuously fill all available space, but it lacks translational symmetry. While crystals, according to the classical crystallographic restriction theorem, can possess only two, three, four, and sixfold rotational symmetries, the Bragg diffraction pattern of quasicrystals shows sharp peaks with other symmetry orders, for instance fivefold.
Aperiodic tilings were discovered by mathematicians in the early 1960s, and, some twenty years later, they were found to apply to the study of quasicrystals. The discovery of these aperiodic forms in nature has produced a paradigm shift in the fields of crystallography. Quasicrystals had been investigated and observed earlier,^{[2]} but, until the 1980s, they were disregarded in favor of the prevailing views about the atomic structure of matter.
Roughly, an ordering is nonperiodic if it lacks translational symmetry, which means that a shifted copy will never match exactly with its original. The more precise mathematical definition is that there is never translational symmetry in more than n – 1 linearly independent directions, where n is the dimension of the space filled; i.e. the threedimensional tiling displayed in a quasicrystal may have translational symmetry in two dimensions. The ability to diffract comes from the existence of an indefinitely large number of elements with a regular spacing, a property loosely described as longrange order. Experimentally, the aperiodicity is revealed in the unusual symmetry of the diffraction pattern, that is, symmetry of orders other than two, three, four, or six. For materials which are intermediate between liquids and crystals (the socalled called liquid crystals), the existence of a phase with fivefold symmetry (also with symmetries of all other Platonic solids) was proposed in 1981 by H. Kleinert and K. Maki.^{[3]}</ref>^{[4]} The first experimental observation of what came to be known as quasicrystals was made by Dan Shechtman and coworkers in 1982 and it was reported in print two years later.^{[5]} Shechtman received the Nobel Prize in Chemistry in 2011 for his findings.^{[6]}
Contents
History
In 1961, Hao Wang asked whether determining if a set of tiles admits a tiling of the plane is an algorithmically unsolvable problem or not. He conjectured that it is solvable, relying on the hypothesis that any set of tiles, which can tile the plane can do it periodically (hence, it would suffice to try to tile bigger and bigger patterns until obtaining one that tiles periodically). Nevertheless, two years later, his student, Robert Berger, constructed a set of some 20,000 square tiles (now called Wang tiles), which can tile the plane but not in a periodic fashion. As the number of known aperiodic sets of tiles grew, each set seemed to contain even fewer tiles than the previous one. In particular, in 1976, Roger Penrose proposed a set of just two tiles, up to rotation, (referred to as Penrose tiles) that produced only nonperiodic tilings of the plane. These tilings displayed instances of fivefold symmetry. One year later, Alan Mackay showed experimentally that the diffraction pattern from the Penrose tiling had a twodimensional Fourier transform consisting of sharp 'delta' peaks arranged in a fivefold symmetric pattern.^{[7]} Around the same time, Robert Ammann had created a set of aperiodic tiles that produced eightfold symmetry.
Mathematically, quasicrystals have been shown to be derivable from a general method, which treats them as projections of a higherdimensional lattice. Just as the simple curves in the plane can be obtained as sections from a threedimensional double cone, so too various (aperiodic or periodic) arrangements in two and three dimensions can be obtained from postulated hyperlattices with four or more dimensions. Icosahedral quasicrystals in three dimensions were projected from a sixdimensional hypercubic lattice by Peter Kramer and Roberto Neri in 1984.^{[8]} The tiling is formed by two tiles with rhombohedral shape.
Shechtman first observed tenfold electron diffraction patterns in 1982, as described in his notebook. These results were not published until two years later when Ilan Blech, using computer simulation, suggested that the diffraction patterns resulted from an aperiodic structure. Blech simulations as well as the diffraction patterns were published in 1984 in a joint paper with Shechtman entitled “The Microstructure of Rapidly Solidified Al6Mn”, Metallurgical Transactions A, 16A (1984) 1005. Later on, a second paper entitled, "Metallic Phase with LongRange Orientational Order and No Translational Symmetry" was submitted for publication, Dan Shechtman et al. both papers demonstrated a clear diffraction pattern with a fivefold symmetry. The pattern was recorded from an AlMn alloy which had been rapidly cooled after melting.^{[5]} Next year, Ishimasa et al. reported twelvefold symmetry in NiCr particles.^{[9]} Soon, eightfold diffraction patterns were recorded in VNiSi and CrNiSi alloys.^{[10]} Over the years, hundreds of quasicrystals with various compositions and different symmetries have been discovered. The first quasicrystalline materials were thermodynamically unstable—when heated, they formed regular crystals. However, 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. In 2009, a mineralogical finding offered evidence that quasicrystals might form naturally under suitable geological conditions. Scientists first reported the naturally occurring quasicrystals in a new kind of mineral found in the Khatyrka River in eastern Russia.^{[11]} A sample from the Russian holotype specimen was found to be an agglomerate of grains up to 0.1 millimeter in size of various phases, mostly khatyrkite, cupalite (zinc or iron containing), some yet unidentified AlCuFe minerals and the Al_{63}Cu_{24}Fe_{13} quasicrystal phase. The quasicrystal grains were of high crystalline quality equal to that of the best laboratory specimens.^{[12]}
In 1972, de Wolf and van Aalst^{[13]} reported that the diffraction pattern produced by a crystal of sodium carbonate cannot be labeled with three indices but needed one more, which implied that the underlying structure had four dimensions in reciprocal space. Other puzzling cases have been reported^{[3]}, but, until the concept of quasicrystal came to be established, they were explained away or denied.^{[14]}^{[15]} However, at the end of the 1980s, the idea became acceptable, and in 1992 the International Union of Crystallography altered its definition of a crystal, broadening it as a result of Shechtman’s findings, reducing it to the ability to produce a clearcut diffraction pattern and acknowledging the possibility of the ordering to be either periodic or aperiodic.^{[16]} Now, the symmetries compatible with translations are defined as "crystallographic", leaving room for other "noncrystallographic" symmetries. Therefore, aperiodic or quasiperiodic structures can be divided into two main classes: those with crystallographic pointgroup symmetry, to which the incommensurately modulated structures and composite structures belong, and those with noncrystallographic pointgroup symmetry, to which quasicrystal structures belong.
The term quasicrystal was first used in print shortly after the announcement of Shechtman's discovery, in a paper by Steinhardt and Levine.^{[17]} However, the adjective quasicrystalline has been loosely applied to any pattern with unusual symmetry.^{[18]} Quasiperiodical structures were claimed to be observed in some decorative tilings devised by medieval Islamic architects.^{[19]}^{[20]} For example, tiles in a medieval Islamic mosque in Isfahan, Iran, are arranged in a twodimensional quasicrystalline pattern.^{[21]} These claims have, however, been under some debate, as of late.^{[22]}
Originally, the new form of matter from quasicrystals was dubbed "Shechtmanite"^{[23]} to honor Shechtman in his discovery—one that took years to gain scientific legitimacy. Shechtman won the Nobel Prize in Chemistry in 2011 for his work on quasicrystals. “His discovery of quasicrystals revealed a new principle for packing of atoms and molecules,” said Lars Thelander, who leads the Nobel Committee for Chemistry at academy. “This led to a paradigm shift within chemistry.” ^{[24]}^{[25]}
Mathematical description
There are several ways to mathematically define quasicrystalline patterns. One definition, the "cut and project" construction, is based on the work of Harald Bohr.^{[26]} Bohr showed that quasiperiodic functions arise as restrictions of highdimensional periodic functions to an irrational slice (an intersection with one or more hyperplanes), and discussed their Fourier point spectrum. In order that the quasicrystal itself be aperiodic, this slice must avoid any lattice plane of the higherdimensional lattice. De Bruijn showed that Penrose tilings can be viewed as twodimensional slices of fivedimensional hypercubic structures.^{[27]} Equivalently, the Fourier transform of such a quasicrystal is nonzero only at a dense set of points spanned by integer multiples of a finite set of basis vectors (the projections of the primitive reciprocal lattice vectors of the higherdimensional lattice).^{[28]}
The intuitive considerations obtained from simple model aperiodic tilings are formally expressed in the concepts of Meyer and Delone sets. The mathematical counterpart of physical diffraction is the Fourier transform and the qualitative description of a diffraction picture as 'clear cut' or 'sharp' means that singularities are present in the Fourier spectrum. There are different methods to construct model quasicrystals. These are the same methods that produce aperiodic tilings with the additional constraint for the diffractive property. Thus, for a substitution tiling the eigenvalues of the substitution matrix should be Pisot numbers. The aperiodic structures obtained by the cutandproject method are made diffractive by choosing a suitable orientation for the construction. This is indeed a geometric approach which has also a great appeal for physicists.
Classical theory of crystals reduces crystals to point lattices where each point is the center of mass of one of the identical units of the crystal. The structure of crystals can be analyzed by defining an associated group. Quasicrystals, on the other hand, are composed of more than one type of unit, so, instead of lattices, quasilattices must be used. Instead of groups, groupoids, the mathematical generalization of groups in category theory, is the appropriate tool for studying quasicrystals.^{[29]}
Using mathematics for construction and analysis of quasicrystal structures is a difficult task for most experimentalists. Computer modeling, based on the existing theories of quasicrystals, however, greatly facilitated this task. Advanced programs have been developed^{[30]} allowing one to construct, visualize and analyze quasicrystal structures and their diffraction patterns.
Interacting spins were also analyzed in quasicrystals: AKLT Model and 8 vertex model were solved in quasicrystals analytically ^{[31]}
Materials science of quasicrystals
Since the original discovery of Dan Shechtman, hundreds of quasicrystals have been reported and confirmed. Undoubtedly, the quasicrystals are no longer a unique form of solid; they exist universally in many metallic alloys and some polymers. Quasicrystals are found most often in aluminium alloys (AlLiCu, AlMnSi, AlNiCo, AlPdMn, AlCuFe, AlCuV, etc.), but numerous other compositions are also known (CdYb, TiZrNi, ZnMgHo, ZnMgSc, InAgYb, PdUSi, etc.).^{[32]}
In theory, there are two types of quasicrystals.^{[30]} The first type, polygonal (dihedral) quasicrystals, have an axis of eight, ten, or 12fold local symmetry (octagonal, decagonal, or dodecagonal quasicrystals, respectively). They are periodic along this axis and quasiperiodic in planes normal to it. The second type, icosahedral quasicrystals, are aperiodic in all directions.
Regarding thermal stability, three types of quasicrystals are distinguished:^{[33]}
 Stable quasicrystals grown by slow cooling or casting with subsequent annealing,
 Metastable quasicrystals prepared by melt spinning, and
 Metastable quasicrystals formed by the crystallization of the amorphous phase.
Except for the Al–Li–Cu system, all the stable quasicrystals are almost free of defects and disorder, as evidenced by xray and electron diffraction revealing peak widths as sharp as those of perfect crystals such as Si. Diffraction patterns exhibit fivefold, threefold, and twofold symmetries, and reflections are arranged quasiperiodically in three dimensions.
The origin of the stabilization mechanism is different for the stable and metastable quasicrystals. Nevertheless, there is a common feature observed in most quasicrystalforming liquid alloys or their undercooled liquids: a local icosahedral order. The icosahedral order is in equilibrium in the liquid state for the stable quasicrystals, whereas the icosahedral order prevails in the undercooled liquid state for the metastable quasicrystals.
See also
References
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 ^ Steurer W. (2004). "Twenty years of structure research on quasicrystals. Part I. Pentagonal, octagonal, decagonal and dodecagonal quasicrystals". Z. Kristallogr. 219 (7–2004): 391–446. doi:10.1524/zkri.219.7.391.35643.
 ^ ^{a} ^{b} Kleinert H. and Maki K. (1981). "Lattice Textures in Cholesteric Liquid Crystals". Fortschritte der Physik 29 (5): 219–259. doi:10.1002/prop.19810290503. http://www.physik.fuberlin.de/~kleinert/75/75.pdf.
 ^ T Seideman (1990). "The liquidcrystalline blue phases". Rep. Prog. Phys. 53 (6): 659–705. Bibcode 1990RPPh...53..659S. doi:10.1088/00344885/53/6/001. http://chemgroups.northwestern.edu/seideman/Publications/The%20liquidcrystalline%20blue%20phases.pdf.
 ^ ^{a} ^{b} Shechtman, D.; Blech, I.; Gratias, D.; Cahn, J. (1984). "Metallic Phase with LongRange Orientational Order and No Translational Symmetry". Physical Review Letters 53 (20): 1951. Bibcode 1984PhRvL..53.1951S. doi:10.1103/PhysRevLett.53.1951.
 ^ "The Nobel Prize in Chemistry 2011". Nobelprize.org. http://www.nobelprize.org/nobel_prizes/chemistry/laureates/2011/. Retrieved 20111006.
 ^ Mackay, A.L. (1982). "Crystallography and the Penrose Pattern". Physica A 114: 609. Bibcode 1982PhyA..114..609M. doi:10.1016/03784371(82)903594.
 ^ Kramer, P.; Neri, R. (1984). "On periodic and nonperiodic space fillings of E(m) obtained by projection". Acta Crystallographica A40 (5): 580. doi:10.1107/S0108767384001203.
 ^ Ishimasa, T.; Nissen, H.U.; Fukano, Y. (1985). "New ordered state between crystalline and amorphous in NiCr particles". Physical Review Letters 55 (5): 511–513. Bibcode 1985PhRvL..55..511I. doi:10.1103/PhysRevLett.55.511. PMID 10032372.
 ^ Wang, N.; Chen, H.; Kuo, K. (1987). "Twodimensional quasicrystal with eightfold rotational symmetry". Physical Review Letters 59 (9): 1010–1013. Bibcode 1987PhRvL..59.1010W. doi:10.1103/PhysRevLett.59.1010. PMID 10035936.
 ^ Bindi, L.; Steinhardt, P. J.; Yao, N.; Lu, P. J. (2009). "Natural Quasicrystals". Science 324 (5932): 1306–9. Bibcode 2009Sci...324.1306B. doi:10.1126/science.1170827. PMID 19498165.
 ^ Steinhardt, Paul; Bindi, Luca (2010). "Once upon a time in Kamchatka: the search for natural quasicrystals". Philosophical Magazine 91 (19–21): 1. Bibcode 2011PMag...91.2421S. doi:10.1080/14786435.2010.510457. http://www.physics.princeton.edu/~steinh/naturalquasicrystals.html.
 ^ de Wolf, R.M. and van Aalst, W. (1972). "The four dimensional group of γNa_{2}CO_{3}". Acta. Cryst. A 28: S111.
 ^ Pauling, L (1987 Jan 26). "Socalled icosahedral and decagonal quasicrystals are twins of an 820atom cubic crystal. Pauling L". Phys Rev Lett. 58 (4): 365–368. doi:10.1103/PhysRevLett.58.365. PMID 10034915.
 ^ Kenneth Chang (October 5, 2011). "Israeli Scientist Wins Nobel Prize for Chemistry". NY Times. http://www.nytimes.com/2011/10/06/science/06nobel.html.
 ^ The concept of aperiodic crystal was coined by Erwin Schrödinger in another context with a somewhat different meaning. In his popular book What is life? in 1944, Schrödinger sought to explain how hereditary information is stored: molecules were deemed too small, amorphous solids were plainly chaotic, so it had to be a kind of crystal; as a periodic structure could not encode information, it had to be aperiodic. DNA was later discovered, and, although not crystalline, it possesses properties predicted by Schrödinger—it is a regular but aperiodic molecule.
 ^ Levine, Dov; Steinhardt, Paul (1984). "Quasicrystals: A New Class of Ordered Structures". Physical Review Letters 53 (26): 2477. Bibcode 1984PhRvL..53.2477L. doi:10.1103/PhysRevLett.53.2477.
 ^ The use of the adjective 'quasicrystalline' for qualifying a structure can be traced back at least to the mid 194050s, e.g. in Kratky, O; Porod, G (1949). "Diffuse smallangle scattering of xrays in colloid systems". Journal of Colloid Science 4 (1): 35–70. doi:10.1016/00958522(49)90032X. PMID 18110601.; Gunn, R (1955). "The statistical electrification of aerosols by ionic diffusion". Journal of Colloid Science 10: 107. doi:10.1016/00958522(55)900817.
 ^ E. Makovicky (1992), 800yearold pentagonal tiling from Maragha, Iran, and the new varieties of aperiodic tiling it inspired. In: I. Hargittai, editor: Fivefold Symmetry, pp. 67–86. World Scientific, SingaporeLondon
 ^ Peter J. Lu and Paul J. Steinhardt (2007). "Decagonal and Quasicrystalline Tilings in Medieval Islamic Architecture". Science 315 (5815): 1106–1110. Bibcode 2007Sci...315.1106L. doi:10.1126/science.1135491. PMID 17322056. http://peterlu.org/sites/peterlu.org/files/Science_315_1106_2007.pdf.
 ^ Lu, P. J.; Steinhardt, P. J. (2007). "Decagonal and QuasiCrystalline Tilings in Medieval Islamic Architecture". Science 315 (5815): 1106–1110. Bibcode 2007Sci...315.1106L. doi:10.1126/science.1135491. PMID 17322056.
 ^ Makovicky, Emil (2007). "Comment on "Decagonal and QuasiCrystalline Tilings in Medieval Islamic Architecture"". Science 318 (5855): 1383–1383. Bibcode 2007Sci...318.1383M. doi:10.1126/science.1146262.
 ^ "Impossible' Form of Matter Takes Spotlight In Study of Solids". New York Times. http://www.nytimes.com/1989/09/05/science/impossibleformofmattertakesspotlightinstudyofsolids.html?pagewanted=all&src=pm.
 ^ "Nobel win for crystal discovery". BBC News. http://www.bbc.co.uk/news/scienceenvironment15181187. Retrieved 20111005.
 ^ Andrea Gerlin (5 October 2011). "Tecnion’s Shechtman Wins Nobel in Chemistry for Quasicrystals Discovery". http://www.bloomberg.com/news/20111005/technionsshechtmanwinschemistrynobelfordiscoveryofquasicrystals.html.
 ^ Bohr, H. (1925). "Zur Theorie fastperiodischer Funktionen I". Acta Mathematicae 45: 580. doi:10.1007/BF02395468.
 ^ de Bruijn, N. (1981). "Algebraic theory of Penrose's nonperiodic tilings of the plane". Nederl. Akad. Wetensch. Proc A84: 39.
 ^ J. B. Suck, M. Schreiber, and P. Häussler, Quasicrystals: An Introduction to Structure, Physical Properties, and Applications (Springer: Berlin, 2004)
 ^ Paterson, Alan L. T. (1999). Groupoids, inverse semigroups, and their operator algebras. Springer. p. 164. ISBN 0817640517.
 ^ ^{a} ^{b} Yamamoto, Akiji (2008). "Software package for structure analysis of quasicrystals" (freedownload review). Science and Technology of Advanced Materials 9: 013001. Bibcode 2008STAdM...9a3001Y. doi:10.1088/14686996/9/3/013001.
 ^ V.E. Korepin Completely integrable models in quasicrystals ( Comm. Math. Phys. Volume 110, Number 1 (1987), 157–171.)
 ^ MacIá, Enrique (2006). "The role of aperiodic order in science and technology". Reports on Progress in Physics 69 (2): 397. Bibcode 2006RPPh...69..397M. doi:10.1088/00344885/69/2/R03.
 ^ Tsai, An Pang (2008). "Icosahedral clusters, icosaheral order and stability of quasicrystals—a view of metallurgy" (freedownload review). Science and Technology of Advanced Materials 9: 013008. Bibcode 2008STAdM...9a3008T. doi:10.1088/14686996/9/1/013008.
Further reading
 V.I. Arnold, Huygens and Barrow, Newton and Hooke: Pioneers in mathematical analysis and catastrophe theory from evolvents to quasicrystals, Eric J.F. Primrose translator, Birkhäuser Verlag (1990) ISBN 3764323833 .
 Christian Janot, Quasicrystals – a primer, 2nd ed. Oxford UP 1997.
 HansRainer Trebin (editor), Quasicrystals, WileyVCH. Weinheim 2003.
 Marjorie Senechal, Quasicrystals and geometry, Cambridge UP 1995.
 JeanMarie Dubois, Useful quasicrystals, World Scientific, Singapore 2005.
 Walter Steurer, Sofia Deloudi, Crystallography of quasicrystals, Springer, Heidelberg 2009.
 Ron Lifshitz, Dan Shechtman, Shelomo I. BenAbraham (editors), Quasicrystals: The Silver Jubilee, Philosophical Magazine Special Issue 88/1315 (2008).
 Peter Kramer and Zorka Papadopolos (editors), Coverings of discrete quasiperiodic sets: theory and applications to quasicrystals, Springer. Berlin 2003.
 Barber, Enrique Macia (2010). Aperiodic Structures in Condensed Matter: Fundamentals and Applications. Taylor & Francis. ISBN 9781420068276.
External links
 A Partial Bibliography of Literature on Quasicrystals (1996–2008).
 BBC webpage showing pictures of Quasicrystals
 What is... a Quasicrystal?, Notices of the AMS 2006, Volume 53, Number 8
 Gateways towards quasicrystals: a short history by P. Kramer
 Quasicrystals: an introduction by R. Lifshitz
 Quasicrystals: an introduction by S. Weber
 Steinhardt's proposal
 Quasicrystal Research – Documentary 2011 on the research of the University of Stuttgart
 Thiel, P.A. (2008). "Quasicrystal Surfaces". Annual Review of Physical Chemistry 59: 129–152. Bibcode 2008ARPC...59..129T. doi:10.1146/annurev.physchem.59.032607.093736. PMID 17988201.
 Foundations of Crystallography .
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