Quasicrystal

Quasicrystals are structural forms that are both ordered and nonperiodic. The term and the concept were introduced originally to denote a specific arrangement observed in solids which can be said to be in a state intermediary between crystal and glass. Producing Bragg diffraction, they share a defining property with crystals, but differ from them by lacking a simply repeating structure. Mathematical artefacts, known as aperiodic tiling, were invented in the early 1960s, but some twenty years later 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]

An ordering is nonperiodic 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 existence 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.

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A brief history of quasicrystals

For physicists, 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) arrangements 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.

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. The same year Ishimasa and coauthors sent for publishing a paper entitled ' New ordered state between crystalline and amorphous in Ni-Cr particles' in which a case twelvefold symmetry was reported. ^[3]. Soon another equally challenging case presented a sample which gave a sharp eightfold diffraction picture.^[4] Over the years hundreds of quasicrystals with various composition and different symmetries have been discovered. The first quasicrystalline materials were thermodynamically unstable. When heated, they formed regular crystals. But 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 1972 de Wolf and van Aalst^[5] reported in print that the diffraction pattern produced by a crystal sodium carbonate cannot be 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.^[6] Nowadays 'quasicrystalline' is an adjective applied to any pattern with unusual symmetry ^[7] Incommensurate structures, which have been a topic in solid state physics since the 1960's, today are acknowledged to be a legitimate kind of quasicrystals.

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 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 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.

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.^[8] Since 2004 different research groups have reported evidence for quasicrystal ordering in liquids and polymers. Such occurrences have come to be known as 'liquid' or, more generally, 'soft' quasicrystals.^[9]

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. ^ T. Ishimasa, H.-U. Nissen and Y. Fukano, New ordered state between crystalline and amorphous in Ni-Cr particles, Phys Rev Lett 55(1985)511
4. ^ N. Wang, H. Chen, and K. H. Kuo, Two-dimensional quasicrystal with eightfold rotational symmetry, Phys. Rev. Lett. 59, 1010-1013 (1987)[2]
5. ^ de Wolf R.M. and van Aalst, The four dimensional group of γ-Na₂CO₃, Acta. Cryst. Sect.A 28(1972) 111
6. ^ D. Levine and P.J. Steinhardt, "Quasicrystals: A New Class of Ordered Structures," Phys. Rev. Lett. 53 (1984) 2477 - 2480.
7. ^ W. S. Edwards and S. Fauve, Parametrically excited quasicrystalline surface waves, Phys. Rev. E 47, (1993)R788 - R791 ; Peter J. Lu and Paul J. Steinhardt (2007). "Decagonal and Quasi-crystalline Tilings in Medieval Islamic Architecture". Science 315: 1106-1110.
8. ^ E. Macia, The role of aperiodic order in science and technology, Rep. Prog. Phys. 69(2006)397-441
9. ^ R.Lifshitz, Soft Quasicrystals, cond-mat/0611115 [3]

See also

• Aperiodic tiling
• Crystal
• Penrose tiling
• Tessellation
• Girih tiles

Bibliography

• D. P. DiVincenzo and P. J. Steinhardt, eds. Quasicrystals: The State of the Art. Directions in Condensed Matter Physics, Vol 11. ISBN 981-02-0522-8, 1991.
• M. Senechal, Quasicrystals and Geometry, Cambridge University Press, 1995.
• 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", W. H. Freeman & Company, NY, 1988.
• An |online bibliography (1996 - today).