QUASICRYSTAL

'Quasicrystals' are aperiodic structures that produce Bragg diffraction, i.e. they show pure point spectra.
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]
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 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.

Contents

A brief history of quasicrystals

Mathematical description

The physics of quasicrystals

References

See also

External links

Technical references

Bibliography

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.
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

External links

Technical references

★ What is... a Quasicrystal?, ''Notices of the AMS'' 2006, Volume 53, Number 8

★ Quasicrystals: an introduction by R. Lifshitz

★ Quasicrystals: an introduction by S. Weber

★ Steinhardt's proposal

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).