quasicrystal

Atomic model of an aluminium-palladium-manganese (Al-Pd-Mn) quasicrystal surface. Similar to Fig. 6 in Ref.^[1]

A quasiperiodic crystal, or, in short, quasicrystal, is a structure that is ordered but not periodic. A quasicrystalline pattern can continuously fill all available space, but it lacks translational symmetry. While crystals, according to the classical crystallographic restriction theorem, can possess only two, three, four, and six-fold rotational symmetries, the Bragg diffraction pattern of quasicrystals shows sharp peaks with other symmetry orders, for instance five-fold.

Aperiodic tilings were discovered by mathematicians in the early 1960s, and, some twenty years later, they were found to apply to the study of quasicrystals. The discovery of these aperiodic forms in nature has produced a paradigm shift in the fields of crystallography. Quasicrystals had been investigated and observed earlier,^[2] but, until the 1980s, they were disregarded in favor of the prevailing views about the atomic structure of matter. In 2009, after a dedicated search, a mineralogical finding, icosahedrite, offered evidence for the existence of natural quasicrystals.^[3]

Roughly, an ordering is non-periodic if it lacks translational symmetry, which means that a shifted copy will never match exactly with its original. The more precise mathematical definition is that there is never translational symmetry in more than n – 1 linearly independent directions, where n is the dimension of the space filled; i.e. the three-dimensional tiling displayed in a quasicrystal may have translational symmetry in two dimensions. 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, that is, symmetry of orders other than two, three, four, or six. In 1982 materials scientist Dan Shechtman observed that certain Aluminium-Manganese alloys produced the unusual diffractograms which today are seen as revelatory of quasicrystal structures. It took him two years to convince the scientific community and to publish the results^[4] for which he was awarded the Nobel Prize in Chemistry in 2011^[5].

History

A Penrose tiling

In 1961, Hao Wang asked whether determining if a set of tiles admits a tiling of the plane is an algorithmically unsolvable problem or not. He conjectured that it is solvable, relying on the hypothesis that any set of tiles, which can tile the plane can do it periodically (hence, it would suffice to try to tile bigger and bigger patterns until obtaining one that tiles periodically). Nevertheless, two years later, his student, Robert Berger, constructed a set of some 20,000 square tiles (now called Wang tiles), which can tile the plane but not in a periodic fashion. As the number of known aperiodic sets of tiles grew, each set seemed to contain even fewer tiles than the previous one. In particular, in 1976, Roger Penrose proposed a set of just two tiles, up to rotation, (referred to as Penrose tiles) that produced only non-periodic tilings of the plane. These tilings displayed instances of fivefold symmetry. One year later, Alan L. Mackay showed experimentally that the diffraction pattern from the Penrose tiling had a two-dimensional Fourier transform consisting of sharp 'delta' peaks arranged in a fivefold symmetric pattern.^[6] Around the same time, Robert Ammann had created a set of aperiodic tiles that produced eightfold symmetry.

Mathematically, quasicrystals have been shown to be derivable from a 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, so too various (aperiodic or periodic) arrangements in two and three dimensions can be obtained from postulated hyperlattices with four or more dimensions. Icosahedral quasicrystals in three dimensions were projected from a six-dimensional hypercubic lattice by Peter Kramer and Roberto Neri in 1984.^[7] The tiling is formed by two tiles with rhombohedral shape.

Shechtman first observed tenfold electron diffraction patterns in 1982, as described in his notebook. These results were not published until two years later when Ilan Blech, using computer simulation, suggested that the diffraction patterns resulted from an aperiodic structure. Blech simulations as well as the diffraction patterns were published in 1984 in a joint paper with Shechtman entitled “The Microstructure of Rapidly Solidified Al6Mn”, Metallurgical Transactions A, 16A (1984) 1005. Later on, a second paper entitled, "Metallic Phase with Long-Range Orientational Order and No Translational Symmetry" was submitted for publication, Dan Shechtman et al. both papers demonstrated a clear diffraction pattern with a fivefold symmetry. The pattern was recorded from an Al-Mn alloy which had been rapidly cooled after melting.^[4] Next year, Ishimasa et al. reported twelvefold symmetry in Ni-Cr particles.^[8] Soon, eightfold diffraction patterns were recorded in V-Ni-Si and Cr-Ni-Si alloys.^[9] Over the years, hundreds of quasicrystals with various compositions and different symmetries have been discovered. The first quasicrystalline materials were thermodynamically unstable—when heated, they formed regular crystals. However, 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 2009, following a 10 year systematic search, scientists reported the first natural quasicrystal, a mineral found in the Khatyrka River in eastern Russia.^[3] This natural quasicrystal exhibits high crystalline quality, equalling the best artificial examples.^[10] The natural quasicrystal phase, with a composition of Al₆₃Cu₂₄Fe₁₃, was named icosahedrite and it was approved by the International Mineralogical Association in 2010. Furthermore, analysis indicates it may be meteoritic in origin, possibly delivered from a carbonaceous chondrite asteroid.^[11]

Electron diffraction pattern of an icosahedral Ho-Mg-Zn quasicrystal

In 1972, de Wolf and van Aalst^[12] reported that the diffraction pattern produced by a crystal of sodium carbonate cannot be labeled with three indices but needed one more, which implied that the underlying structure had four dimensions in reciprocal space. Other puzzling cases have been reported,^[13] but until the concept of quasicrystal came to be established, they were explained away or denied.^[14][15] However, at the end of the 1980s, the idea became acceptable, and in 1992 the International Union of Crystallography altered its definition of a crystal, broadening it as a result of Shechtman’s findings, 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.^[16] Now, the symmetries compatible with translations are defined as "crystallographic", leaving room for other "non-crystallographic" symmetries. Therefore, 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.

Originally, the new form of matter was dubbed "Shechtmanite".^[17] The term "quasicrystal" was first used in print by Steinhardt and Levine^[18] shortly after Shechtman's paper was published. The adjective quasicrystalline has been already in use but now it came to be applied to any pattern with unusual symmetry.^[19] 'Quasiperiodical' structures were claimed to be observed in some decorative tilings devised by medieval Islamic architects.^[20][21] For example, Girih tiles in a medieval Islamic mosque in Isfahan, Iran, are arranged in a two-dimensional quasicrystalline pattern.^[22] These claims have, however, been under some debate.^[23]

Shechtman was awarded the Nobel Prize in Chemistry in 2011 for his work on quasicrystals. “His discovery of quasicrystals revealed a new principle for packing of atoms and molecules,” stated the Nobel Committee and pointed that “this led to a paradigm shift within chemistry.” ^[24][25]

Mathematical description

A penteract (5-cube) pattern using 5D orthographic projection to 2D using Petrie polygon basis vectors overlaid on the diffractogram from an Icosahedral Ho-Mg-Zn quasicrystal

A hexeract (6-cube) pattern using 6D orthographic projection to a 3D Perspective (visual) object (the Rhombic triacontahedron) using the Golden ratio in the basis vectors. This is used to understand the aperiodic Icosahedral structure of Quasicrystals.

There are several ways to mathematically define quasicrystalline patterns. One definition, the "cut and project" construction, is based on the work of Harald Bohr.^[26] Bohr showed that quasiperiodic functions arise as restrictions of high-dimensional periodic functions to an irrational slice (an intersection with one or more hyperplanes), and discussed their Fourier point spectrum. In order that the quasicrystal itself be aperiodic, this slice must avoid any lattice plane of the higher-dimensional lattice. De Bruijn showed that Penrose tilings can be viewed as two-dimensional slices of five-dimensional hypercubic structures.^[27] Equivalently, the Fourier transform of such a quasicrystal is nonzero only at a dense set of points spanned by integer multiples of a finite set of basis vectors (the projections of the primitive reciprocal lattice vectors of the higher-dimensional lattice).^[28] 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 choosing a suitable orientation for the construction. This is indeed a geometric approach which has also a great appeal for physicists.

Classical theory of crystals reduces crystals to point lattices where each point is the center of mass of one of the identical units of the crystal. The structure of crystals can be analyzed by defining an associated group. Quasicrystals, on the other hand, are composed of more than one type of unit, so, instead of lattices, quasilattices must be used. Instead of groups, groupoids, the mathematical generalization of groups in category theory, is the appropriate tool for studying quasicrystals.^[29]

Using mathematics for construction and analysis of quasicrystal structures is a difficult task for most experimentalists. Computer modeling, based on the existing theories of quasicrystals, however, greatly facilitated this task. Advanced programs have been developed^[30] allowing one to construct, visualize and analyze quasicrystal structures and their diffraction patterns.

Interacting spins were also analyzed in quasicrystals: AKLT Model and 8 vertex model were solved in quasicrystals analytically ^[31]

Materials science of quasicrystals

A Ho-Mg-Zn icosahedral quasicrystal formed as a dodecahedron, the dual of the icosahedron

Since the original discovery of Dan Shechtman, hundreds of quasicrystals have been reported and confirmed. Undoubtedly, the quasicrystals are no longer a unique form of solid; they exist universally in many metallic alloys and some polymers. Quasicrystals are found most often in aluminium alloys (Al-Li-Cu, Al-Mn-Si, Al-Ni-Co, Al-Pd-Mn, Al-Cu-Fe, Al-Cu-V, etc.), but numerous other compositions are also known (Cd-Yb, Ti-Zr-Ni, Zn-Mg-Ho, Zn-Mg-Sc, In-Ag-Yb, Pd-U-Si, etc.).^[32]

There are two types of known quasicrystals.^[30] The first type, polygonal (dihedral) quasicrystals, have an axis of eight, ten, or 12-fold local symmetry (octagonal, decagonal, or dodecagonal quasicrystals, respectively). They are periodic along this axis and quasiperiodic in planes normal to it. The second type, icosahedral quasicrystals, are aperiodic in all directions.

Regarding thermal stability, three types of quasicrystals are distinguished:^[33]

• Stable quasicrystals grown by slow cooling or casting with subsequent annealing,
• Metastable quasicrystals prepared by melt spinning, and
• Metastable quasicrystals formed by the crystallization of the amorphous phase.

Except for the Al–Li–Cu system, all the stable quasicrystals are almost free of defects and disorder, as evidenced by x-ray and electron diffraction revealing peak widths as sharp as those of perfect crystals such as Si. Diffraction patterns exhibit fivefold, threefold, and twofold symmetries, and reflections are arranged quasiperiodically in three dimensions.

The origin of the stabilization mechanism is different for the stable and metastable quasicrystals. Nevertheless, there is a common feature observed in most quasicrystal-forming liquid alloys or their undercooled liquids: a local icosahedral order. The icosahedral order is in equilibrium in the liquid state for the stable quasicrystals, whereas the icosahedral order prevails in the undercooled liquid state for the metastable quasicrystals.

See also

• Aperiodic tiling
• Archimedean solid
• Fibonacci quasicrystal
• Girih tiles
• Golden ratio
• Icosahedrite
• Phason
• Tessellation

References

Further reading

• V.I. Arnold, Huygens and Barrow, Newton and Hooke: Pioneers in mathematical analysis and catastrophe theory from evolvents to quasicrystals, Eric J.F. Primrose translator, Birkhäuser Verlag (1990) ISBN 3-7643-2383-3 .
• Christian Janot, Quasicrystals – a primer, 2nd ed. Oxford UP 1997.
• Hans-Rainer Trebin (editor), Quasicrystals, Wiley-VCH. Weinheim 2003.
• Marjorie Senechal, Quasicrystals and geometry, Cambridge UP 1995.
• Jean-Marie Dubois, Useful quasicrystals, World Scientific, Singapore 2005.
• Walter Steurer, Sofia Deloudi, Crystallography of quasicrystals, Springer, Heidelberg 2009.
• Ron Lifshitz, Dan Shechtman, Shelomo I. Ben-Abraham (editors), Quasicrystals: The Silver Jubilee, Philosophical Magazine Special Issue 88/13-15 (2008).
• Peter Kramer and Zorka Papadopolos (editors), Coverings of discrete quasiperiodic sets: theory and applications to quasicrystals, Springer. Berlin 2003.
• Barber, Enrique Macia (2010). Aperiodic Structures in Condensed Matter: Fundamentals and Applications. Taylor & Francis. ISBN 978-1-4200-6827-6. 

External links

• A Partial Bibliography of Literature on Quasicrystals (1996–2008).
• BBC webpage showing pictures of Quasicrystals
• What is... a Quasicrystal?, Notices of the AMS 2006, Volume 53, Number 8
• Gateways towards quasicrystals: a short history by P. Kramer
• Quasicrystals: an introduction by R. Lifshitz
• Quasicrystals: an introduction by S. Weber
• Steinhardt's proposal
• Quasicrystal Research – Documentary 2011 on the research of the University of Stuttgart
• Thiel, P.A. (2008). "Quasicrystal Surfaces". Annual Review of Physical Chemistry 59: 129–152. Bibcode 2008ARPC...59..129T. doi:10.1146/annurev.physchem.59.032607.093736. PMID 17988201. 
• Foundations of Crystallography.
• Pentagon tile by Alexander Braun based on quasicrystals.