To use all functions of this page, please activate cookies in your browser.

my.chemeurope.com

With an accout for my.chemeurope.com you can always see everything at a glance – and you can configure your own website and individual newsletter.

- My watch list
- My saved searches
- My saved topics
- My newsletter

## Quasicrystal
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. ## Additional recommended knowledge
## A brief history of quasicrystalsFor 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. In 1972 de Wolf and van Aalst ## Mathematical descriptionThe 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 quasicrystalsReal 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. ## References**^**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.**^**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]**^**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**^**N. Wang, H. Chen, and K. H. Kuo, Two-dimensional quasicrystal with eightfold rotational symmetry, Phys. Rev. Lett. 59, 1010-1013 (1987)[2]**^**de Wolf R.M. and van Aalst, The four dimensional group of γ-Na_{2}CO_{3}, Acta. Cryst. Sect.A 28(1972) 111**^**D. Levine and P.J. Steinhardt, "Quasicrystals: A New Class of Ordered Structures," Phys. Rev. Lett. 53 (1984) 2477 - 2480.**^**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.**^**E. Macia, The role of aperiodic order in science and technology, Rep. Prog. Phys. 69(2006)397-441**^**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).
Categories: Crystallography | Condensed matter physics |
|||

This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Quasicrystal". A list of authors is available in Wikipedia. |