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Quasicrystals are structural forms that are both ordered and nonperiodic. Structure is a fundamental and sometimes Intangible notion covering the Recognition, Observation, nature, and Stability of They form patterns that fill all the space but lack translational symmetry. 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. In Materials science, a crystal is a Solid in which the constituent Atoms Molecules or Ions are packed in a regularly ordered repeating Glass in the common sense refers to a Hard, Brittle, transparent Solid, such as that used for Windows many Producing Bragg diffraction, they share a defining property with crystals, but differ from them by lacking a simple repeating structure. Bragg diffraction (also referred to as the Bragg formulation of X-ray diffraction) was first proposed by William Lawrence Bragg and William Henry Bragg

Mathematical artefacts, known as aperiodic tiling, were invented in the early 1960s, but some twenty years later physical experiments gave conclusive 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. Paradigm shift, sometimes known as extraordinary science or revolutionary science, is the term first used by Thomas Kuhn in his influential [1] It was realized that quasicrystals had been investigated and observed earlier[2]but until then the prevailing views about atomic structure of matter lead to their being explained away.

An ordering is nonperiodic if it lacks translational symmetry, which means that a shifted copy will never match exactly with its original. In Geometry, a translation "slides" an object by a vector a: T a (p = p + a 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. Dan Shechtman is the Philip Tobias Professor of Materials Science at the Technion - Israel Institute of Technology, an Associate of the US Department of Energy's [3] 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. A Penrose tiling is a nonperiodic tiling generated by an aperiodic set of Prototiles named after Roger Penrose, who investigated these sets

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A brief history of quasicrystals

For 20th century 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. Wang Hao, also Hao Wang ( 20 May 1921 &ndash 13 May 1995) was a Chinese American Logician, Philosopher Two years later an example involving some 20,000 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. Sir Roger Penrose, PhD, OM, FRS (born 8 August 1931) is an English Mathematical physicist and Emeritus With hindsight similar patterns were observed in some decorative tilings devised by medieval Islamic architects [4][5]. 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. A Penrose tiling is a nonperiodic tiling generated by an aperiodic set of Prototiles named after Roger Penrose, who investigated these sets This article specifically discusses Fourier transformation of functions on the Real line; for other kinds of Fourier transformation see Fourier analysis and Later it transpired that around the same time Robert Ammann had also discovered this solution and another one which produced the eightfold case. Robert Ammann ( October 1, 1946 &ndash May 1994 was an amateur mathematician who made several significant and groundbreaking contributions to the theory 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. [6]. Soon another equally challenging case presented a sample which gave a sharp eightfold diffraction picture. [7] 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[8] 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[9]. The International Union of Crystallography (IUCr is a member of the International Council for Science (ICSU and exists to serve the world community of crystallographers Now the symmetries compatible with translations are defined as "crystallographic", leaving room for other "non-crystallographic" symmetries. Thus aperiodic or quasiperiodic structures can be divided into two main classes: those with crystallographic point-group symmetry, to which the incommensurately modulated structures and composite structures belong, and those with non-crystallographic point-group symmetry, to which quasicrystal structures belong. In Mathematics, a function f is said to be quasiperiodic with quasiperiod (sometimes simply called the period) &omega if for certain

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. Paul J Steinhardt is the Albert Einstein Professor of Science at Princeton University and a Professor of Theoretical physics. [10] Nowadays 'quasicrystalline' is an adjective applied to any pattern with unusual symmetry [11]

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. This article specifically discusses Fourier transformation of functions on the Real line; for other kinds of Fourier transformation see Fourier analysis and A spectrum (plural spectra or spectrums) is a condition that is not limited to a specific set of values but can vary infinitely within a continuum. 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. A tile substitution is a useful method for constructing highly ordered tilings Most importantly some tile substitutions generate Aperiodic tilings which are tilings In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes In Mathematics, a Pisot-Vijayaraghavan number, also called simply a Pisot number or a PV number, is an Algebraic integer &alpha which is real 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. [12] 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. [13]

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. ^ Steurer W. , Z. Kristallogr. 219 (2004) 391–446
  3. ^ D. Shechtman, I. Dan Shechtman is the Philip Tobias Professor of Materials Science at the Technion - Israel Institute of Technology, an Associate of the US Department of Energy's Blech, D. Gratias, and J. W. Cahn, Metallic Phase with Long-Range Orientational Order and No Translational Symmetry, Phys. John Cahn (1927 -) is an American Scientist and winner of the National Medal of Science in 1998 Rev. Lett. 53, 1951-1953 (1984) [1]
  4. ^ E. Makovicky (1992), 800-year-old 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, Singapore-London
  5. ^ Peter J. Lu and Paul J. Steinhardt (2007). "Decagonal and Quasi-crystalline Tilings in Medieval Islamic Architecture". Science 315: 1106-1110. Science is the Academic journal of the American Association for the Advancement of Science and is considered one of the world's most prestigious Scientific  
  6. ^ 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
  7. ^ N. Wang, H. Chen, and K. H. Kuo, Two-dimensional quasicrystal with eightfold rotational symmetry, Phys. Rev. Lett. 59, 1010-1013 (1987)[2]
  8. ^ de Wolf R. M. and van Aalst, The four dimensional group of γ-Na2CO3, Acta. Cryst. Sect. A 28(1972) 111
  9. ^ For the record: 'aperiodic crystal' was a concept 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. What is Life? with Mind and Matter is a non-fiction book on science for the lay reader written by physicist Erwin Schrödinger. Year 1944 ( MCMXLIV) was a Leap year starting on Saturday (link will display full calendar of the Gregorian calendar. DNA was later discovered and, although not crystalline, it possesses properties predicted by Schrödinger: it is a regular but aperiodic molecule. Deoxyribonucleic acid ( DNA) is a Nucleic acid that contains the genetic instructions used in the development and functioning of all known
  10. ^ D. Levine and P. J. Steinhardt, "Quasicrystals: A New Class of Ordered Structures," Phys. Rev. Lett. 53 (1984) 2477 - 2480.
  11. ^ 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), loc. cit.
  12. ^ E. Macia, The role of aperiodic order in science and technology, Rep. Prog. Phys. 69(2006)397-441
  13. ^ R. Lifshitz, Soft Quasicrystals, cond-mat/0611115 [3]

See also

External links

Technical references

Bibliography

Dictionary

quasicrystal

-noun

  1. (physics),(chemistry) any solid with conventional crystalline properties but exhibiting a point group symmetry inconsistent with translational periodicity
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