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Quasicrystals are aperiodic structures which produce diffraction. Thus they share a defining property with crystals, but differ from them by lacking a regular repeating structure. They were considered to be mathematical artefacts, known as aperiodic tiling, but physical experiments gave evidence of their material existence. Within the field of crystallography and solid state physics the discovery has produced a paradigm shift which is indeed a minor scientific revolution^[1]. Image File history File links Information_icon. ... Crystal (disambiguation) Insulin crystals A crystal is a solid in which the constituent atoms, molecules, or ions are packed in a regularly ordered, repeating pattern extending in all three spatial dimensions. ... An aperiodic tiling is a tiling of the plane by a set of prototiles that can only be tiled in a non-repeating (non-periodic) pattern. ... Paradigm shift is the term first used by Thomas Kuhn in his 1962 book The Structure of Scientific Revolutions to describe a change in basic assumptions within the ruling theory of science. ...

An ordering is aperiodic if it lacks translational symmetry, which means that a shifted copy will never match exactly with its original. The ability to diffract comes from the existance of an indefinitely large number of elements with a regular spacing, a property loosely described as long-range order. Experimentally the aperiodicity is revealed in the unusual symmetry of the diffraction pattern. The first officially reported case of what came to be known as quasicrystals was made by Dan Shechtman and coworkers in 1984 ^[2]. Between a mathematical model of a quasicrystal, such as the Penrose tiling, and the corresponding physical systems, the distinction is taken to be evident and usually does not have to be emphasized. In mathematics, a periodic function is a function that repeats its values, after adding some definite period to the variable. ... A translation slides an object by a vector a: Ta(p) = p + a. ... Dan Shechtman is the Philip Tobias Professor of Materials Science at the Israel Institute of Technology. ... A Penrose tiling A Penrose tiling is an aperiodic tiling of the plane discovered by Roger Penrose in 1973. ...

A Brief History of Quasicrystals

In physics, the discovery of quasicrystals was a surprise even if their mathematical description was already established. In 1961 Hao Wang proved that the tiling of the plane is an algorithmically unsolvable problem, which implied that there should be aperiodic tilings. Two years later an example involving some 20000 shapes was produced. The number of tiles which allow only aperiodic tilings was rapidly reduced and in 1976 Roger Penrose proposed a set of two tiles which produced an aperiodic tiling with fivefold symmetry when some rules were observed. Later it transpired that around the same time Robert Ammann had also discovered this solution and another one which produced the eightfold case. It was established that the Penrose tiling, as it came to be known, had a two dimensional Fourier transform consisting of sharp 'delta' peaks arranged in a fivefold symmetric pattern. These two examples of mathematical quasicrystals have been shown to be derivable from a more general method which treats them as projections of a higher dimensional lattice. Just as the simple curves in the plane can be obtained as sections from a three-dimensional double cone, various (aperiodic or periodic) arrangments in 2 and 3 dimensions can be obtained from postulated hyperlattices with 4 or more dimensions. This method explains both the arrangement and its ability to diffract. Hao Wang 王浩 (1921 – 1995) was a Chinese-American logician, philosopher and mathematician. ... Sir Roger Penrose, OM, FRS (born 8 August 1931) is an English mathematical physicist and Emeritus Rouse Ball Professor of Mathematics at the Mathematical Institute, University of Oxford and Emeritus Fellow of Wadham College. ... Robert Ammann (October 1, 1946-May, 1994) was an amateur mathematician who made several significant and groundbreaking contributions to the theory of quasicrystals and aperiodic tilings. ... A Penrose tiling A Penrose tiling is an aperiodic tiling of the plane discovered by Roger Penrose in 1973. ... In mathematics, the Fourier transform is a certain linear operator that maps functions to other functions. ...

The standard history of quasicrystals begins with the paper entitled 'Metallic Phase with Long-Range Orientational Order and No Translational Symmetry' published by D. Shechtman and others in 1984. The discovery was made nearly two years before, but their work was met with resistance inside the professional community. Shechtman and coworkers demonstrated a clear cut diffraction picture with an unusual fivefold symmetry produced by samples from an Al-Mn alloy which has been rapidly cooled after melting. Three years later another equally challenging case presented a sample which gave a sharp eightfold diffraction picture^[3]. Over the years hundreds of quasicrystals with various composition and different symmetries have been reported.

In crystallography the existence of aperiodicity in the form of incommensurate structures has been known for many years {{citation needed}}.In 1972 de Wolf and van Aalst^[4] reported in print that the diffraction pattern produced by a sample of Cobalt-sodium compound cannot labeled with three indexes but needed one more, which implied that the underlying structure had four dimensions in reciprocal space. Other puzzling cases have been reported but until the concept of quasicrystal came to be established they were explained away or simply denied. However at the end of the 1980s the idea became acceptable and in 1991 the International Union of Crystallography amended its definition of crystal, reducing it to the ability to produce a clear-cut diffraction pattern and acknowledging the possibility of the ordering to be either periodic or aperiodic. The term 'quasicrystal' was used for the first time in print shortly after the announcement of Shechtman's discovery in a paper by Steinhardt and Levine^[5]. Paul J. Steinhardt is the Albert Einstein Professor of Science at Princeton University and a professor of theoretical physics. ...

Lu and Steinhardt have argued that nearly perfect quasi-crystalline Penrose patterns can be observed in Medieval Islamic architecture dating to the 15th century^[6]. Paul J. Steinhardt is the Albert Einstein Professor of Science at Princeton University and a professor of theoretical physics. ...

Mathematical description

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 quasicrytals. 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 cut and project method are made diffractive by chosing a suitable orientation for the construction. This is indeed a geometric approach which has also a great appeal for physicists. An aperiodic tiling is a tiling of the plane by a set of prototiles that can only be tiled in a non-repeating (non-periodic) pattern. ... In mathematics, the Fourier transform is a certain linear operator that maps functions to other functions. ... In most modern usages of the word spectrum, there is a unifying theme of between extremes at either end. ... An aperiodic tiling is a tiling of the plane by a set of prototiles that can only be tiled in a non-repeating (non-periodic) pattern. ... A tile-substitution is a simple way to generate highly ordered tilings without any translational symmetry, in other words: aperiodic tilings. ... In linear algebra, a scalar λ is called an eigenvalue (in some older texts, a characteristic value) of a linear mapping A if there exists a nonzero vector x such that Ax=λx. ... In mathematics, a Pisot-Vijayaraghavan number is an algebraic integer α which is real and exceeds 1, but such that its conjugate elements are all less than 1 in absolute value. ...

The Physics of quasicrystals

Real world systems are finite and imperfect, so the distinction between quasicrystals and other structures is an always open question. Since the original discovery of Shechtman hundreds of quasicrystals have been reported and confirmed. Such structures are found most often in aluminium alloys (Al-Ni-Co, Al-Pd-Mn, Al-Cu-Fe), but other compositions are also possible (Ti-Zr-Ni, Zn-Mg-Ho, Cd-Yb). Different mechanisms have been proposed to explain the generation of quasicrystals and are still discussed. The physical properties of quasicrytals are still studied and new results are currently obtained^[7]. Since 2004 different research groups have reported evidence for quasicrystal ordering in liquids and polymers. Such occurences have come to be known as 'liquid' or, more generally, 'soft' quasicrystals^[8].

References

1. ^ J. W. Cahn, On the discovery of quasicrystals as a Kuhnian Scientific Revolution: "Epilogue", Proceedings of the 5th International Conference on Quasicrystals, Ed. C. Janot and R. Mosseri (World Scientific, Singapore 1995) 807-810.
2. ^ D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, Metallic Phase with Long-Range Orientational Order and No Translational Symmetry, Phys. Rev. Lett. 53, 1951-1953 (1984) [1]
3. ^ N. Wang, H. Chen, and K. H. Kuo, Two-dimensional quasicrystal with eightfold rotational symmetry, Phys. Rev. Lett. 59, 1010-1013 (1987)[2]
4. ^ de Wolf R.M. and van Aalst, The four dimensional group of γ-Na₂Co₃, Acta. Cryst.Sect.A 28(1972) 111
5. ^ D. Levine and P.J. Steinhardt, "Quasicrystals: A New Class of Ordered Structures," Phys. Rev. Lett. 53 (1984) 2477 - 2480.
6. ^ P. J. Lu and P. J. Steinhardt, Decagonal and Quasi-Crystalline Tilings in Medieval Islamic Architecture, Science Vol. 315 no. 5815, 1106-1110 (2007) [3]
7. ^ E. Macia, The role of aperiodic order in science and technology, Rep. Prog. Phys. 69(2006)397-441
8. ^ R.Lifshitz, Soft Quasicrystals, cond-mat/0611115 [4]

Dan Shechtman is the Philip Tobias Professor of Materials Science at the Israel Institute of Technology. ... John Cahn (1927 - ), American Scientist, winner of the National Medal of Science (1998), based at the National Institute of Standards and Technology since 1977. ...

See also

• Aperiodic tiling
• Crystal
• Penrose tiling
• Tessellation

An aperiodic tiling is a tiling of the plane by a set of prototiles that can only be tiled in a non-repeating (non-periodic) pattern. ... Quartz crystal Copper(II) sulfate and iodine crystal Synthetic bismuth crystal Insulin crystals Gallium, a metal that easily forms large single crystals A huge monocrystal of potassium dihydrogen phosphate grown from solution by Saint-Gobain for the megajoule laser of CEA. In chemistry and mineralogy, a crystal is a solid... A Penrose tiling A Penrose tiling is an aperiodic tiling of the plane discovered by Roger Penrose in 1973. ... A tessellated plane seen in street pavement. ...

External links

History

• Medieval Muslims made stunning math breakthrough
• Medieval Mosques Illuminated by Math by Nell Boyce (NPR report on Medieval quasicrystal tiling)

Technical references

• What is... a Quasicrystal?, Notices of the AMS 2006, Volume 53, Number 8
• Quasicystals: an introduction by R. Lifshitz
• Quasicystals: an introduction by S. Weber
• Steinhardt's proposal

Notices of the AMS, March 2005 issue. ...

Bibliography

• D. P. DiVincenzo and P. J. Steinhardt, eds. 1991. Quasicrystals: The State of the Art. Directions in Condensed Matter Physics, Vol 11. ISBN 981-02-0522-8.

M. Senechal, Quasicrystals and Geometry (1995), Cambridge University Press

J. Patera, Quasicrystals and Discrete Geometry , 1998

E. Belin-Ferre et al., eds. Quasicrystals,2000

Hans-Rainer Trebin ed., Quasicrystals: Structure and Physical Properties 2003

Peterson, Ivars, "The Mathematical Tourist" 1988, W. H. Freeman & Company, NY

An | online bibliography (1996 - today)