Quasicrystal
Encyclopedia
A quasiperiodic crystal, or, in short, quasicrystal, is a structure
Structure
Structure is a fundamental, tangible or intangible notion referring to the recognition, observation, nature, and permanence of patterns and relationships of entities. This notion may itself be an object, such as a built structure, or an attribute, such as the structure of society...

that is ordered
Order and disorder (physics)
In physics, the terms order and disorder designate the presence or absence of some symmetry or correlation in a many-particle system.In condensed matter physics, systems typically are ordered at low temperatures; upon heating, they undergo one or several phase transitions into less ordered...

but not periodic. A quasicrystalline pattern can continuously fill all available space, but it lacks translational symmetry
Translational symmetry
In geometry, a translation "slides" an object by a a: Ta = p + a.In physics and mathematics, continuous translational symmetry is the invariance of a system of equations under any translation...

. While crystals, according to the classical crystallographic restriction theorem
Crystallographic restriction theorem
The crystallographic restriction theorem in its basic form was based on the observation that the rotational symmetries of a crystal are usually limited to 2-fold, 3-fold, 4-fold, and 6-fold...

, 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
A Paradigm shift is, according to Thomas Kuhn in his influential book The Structure of Scientific Revolutions , a change in the basic assumptions, or paradigms, within the ruling theory of science...

in the fields of crystallography
Crystallography
Crystallography is the experimental science of the arrangement of atoms in solids. The word "crystallography" derives from the Greek words crystallon = cold drop / frozen drop, with its meaning extending to all solids with some degree of transparency, and grapho = write.Before the development of...

. Quasicrystals had been investigated and observed earlier, but, until the 1980s, they were disregarded in favor of the prevailing views about the atomic structure of matter.

Roughly, an ordering is non-periodic if it lacks translational symmetry
Translational symmetry
In geometry, a translation "slides" an object by a a: Ta = p + a.In physics and mathematics, continuous translational symmetry is the invariance of a system of equations under any translation...

, 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
Linear independence
In linear algebra, a family of vectors is linearly independent if none of them can be written as a linear combination of finitely many other vectors in the collection. A family of vectors which is not linearly independent is called linearly dependent...

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.
The first experimental observation of what came to be known as quasicrystals was made by Dan Shechtman
Dan Shechtman
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 Ames Laboratory, and Professor of Materials Science at Iowa State University. On April 8, 1982, while on sabbatical at the U.S...

and coworkers in 1982 and it was reported in print two years later. Shechtman received the Nobel Prize in Chemistry
Nobel Prize in Chemistry
The Nobel Prize in Chemistry is awarded annually by the Royal Swedish Academy of Sciences to scientists in the various fields of chemistry. It is one of the five Nobel Prizes established by the will of Alfred Nobel in 1895, awarded for outstanding contributions in chemistry, physics, literature,...

in 2011 for his findings.

## History

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
Robert Berger (mathematician)
Robert Berger is known for inventing the first aperiodic tiling using a set of 20,426 distinct tile shapes.The unexpected existence of aperiodic tilings, although not Berger's explicit construction of them, follows from another result proved by Berger: that the so-called domino problem is...

, 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
Roger Penrose
Sir Roger Penrose OM FRS is an English mathematical physicist and Emeritus Rouse Ball Professor of Mathematics at the Mathematical Institute, University of Oxford and Emeritus Fellow of Wadham College...

proposed a set of just two tiles, up to rotation, (referred to as Penrose tiles
Penrose tiling
A Penrose tiling is a non-periodic tiling generated by an aperiodic set of prototiles named after Sir Roger Penrose, who investigated these sets in the 1970s. The aperiodicity of the Penrose prototiles implies that a shifted copy of a Penrose tiling will never match the original...

) that produced only non-periodic tilings of the plane. These tilings displayed instances of fivefold symmetry. One year later, Alan Mackay
Alan Mackay
Alan Mackay was a long-serving reporter for BBC Scotland's Reporting Scotland. He is the brother of the High Court judge Donald Mackay, Baron Mackay of Drumadoon. He left the progamme in 2007 after over twenty-five years in television journalism....

showed experimentally that the diffraction pattern from the Penrose tiling had a two-dimensional Fourier transform
Fourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...

consisting of sharp 'delta' peaks arranged in a fivefold symmetric pattern. Around the same time, Robert Ammann
Robert Ammann
Robert Ammann was an amateur mathematician who made several significant and groundbreaking contributions to the theory of quasicrystals and aperiodic tilings....

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
Peter Kramer (physicist)
- Personal Life :Kramer studied physics in Münster, Tübingen, Bristol und Marburg. He obtained his PhD in 1964 in Marburg and his Habilitation in 1968in Tübingen. He was a postdoc at the UNAM in Mexico City, where he collaborated with Marcos Moshinsky....

and Roberto Neri in 1984. 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
Dan Shechtman
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 Ames Laboratory, and Professor of Materials Science at Iowa State University. On April 8, 1982, while on sabbatical at the U.S...

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. Next year, Ishimasa et al. reported twelvefold symmetry in Ni-Cr particles. Soon, eightfold diffraction patterns were recorded in V-Ni-Si and Cr-Ni-Si alloys. 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, a mineralogical finding offered evidence that quasicrystals might form naturally under suitable geological conditions. Scientists first reported the naturally occurring quasicrystals in a new kind of mineral found in the Khatyrka
Khatyrka

River in eastern Russia
Russia
Russia or , officially known as both Russia and the Russian Federation , is a country in northern Eurasia. It is a federal semi-presidential republic, comprising 83 federal subjects...

. A sample from the Russian holotype
Holotype
A holotype is a single physical example of an organism, known to have been used when the species was formally described. It is either the single such physical example or one of several such, but explicitly designated as the holotype...

specimen was found to be an agglomerate of grains up to 0.1 millimeter in size of various phases, mostly khatyrkite
Khatyrkite
Khatyrkite is a rare mineral which is mostly composed of copper and aluminium, but might contain up to about 15% of zinc or iron; its chemical structure is described by an approximate formula Al2 or Al2. It was discovered in 1985 in placers derived from serpentine, in association with another...

, cupalite
Cupalite
Cupalite is a rare mineral which is mostly composed of copper and aluminium, but might contain up to several percents of zinc or iron; its chemical structure is therefore described by an approximate formula Al or Al. It was discovered in 1985 in placers derived from serpentine, in association with...

(zinc or iron containing), some yet unidentified Al-Cu-Fe minerals and the quasicrystal phase. The quasicrystal grains were of high crystalline quality equal to that of the best laboratory specimens.

In 1972, de Wolf and van Aalst reported that the diffraction pattern produced by a crystal of sodium carbonate
Sodium carbonate
Sodium carbonate , Na2CO3 is a sodium salt of carbonic acid. It most commonly occurs as a crystalline heptahydrate, which readily effloresces to form a white powder, the monohydrate. Sodium carbonate is domestically well-known for its everyday use as a water softener. It can be extracted from the...

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, but, until the concept of quasicrystal came to be established, they were explained away or denied. However, at the end of the 1980s, the idea became acceptable, and in 1992 the International Union of Crystallography
International Union of Crystallography
The International Union of Crystallography is a member of the International Council for Science and exists to serve the world community of crystallographers....

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. Now, the symmetries compatible with translations are defined as "crystallographic", leaving room for other "non-crystallographic" symmetries. Therefore, aperiodic or quasiperiodic
Quasiperiodic function
In mathematics, a function is said to be quasiperiodic when it has some similarity to a periodic function but does not meet the strict definition.A simple case is if the function obeys the equation:...

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.

The term quasicrystal was first used in print shortly after the announcement of Shechtman's discovery, in a paper by Steinhardt
Paul Steinhardt
Paul J. Steinhardt is the Albert Einstein Professor of Science at Princeton University and a professor of theoretical physics. He received his B.S. at the California Institute of Technology and his Ph.D. in Physics at Harvard University...

and Levine. However, the adjective quasicrystalline has been loosely applied to any pattern with unusual symmetry. Quasiperiodical structures were claimed to be observed in some decorative tilings devised by medieval Islamic architects. For example, tiles in a medieval Islamic mosque in Isfahan, Iran, are arranged in a two-dimensional quasicrystalline pattern. These claims have, however, been under some debate, as of late.

Originally, the new form of matter from quasicrystals was dubbed "Shechtmanite" to honor Shechtman in his discovery—one that took years to gain scientific legitimacy. Shechtman won the Nobel Prize in Chemistry
Nobel Prize in Chemistry
The Nobel Prize in Chemistry is awarded annually by the Royal Swedish Academy of Sciences to scientists in the various fields of chemistry. It is one of the five Nobel Prizes established by the will of Alfred Nobel in 1895, awarded for outstanding contributions in chemistry, physics, literature,...

in 2011 for his work on quasicrystals. “His discovery of quasicrystals revealed a new principle for packing of atoms and molecules,” said Lars Thelander, who leads the Nobel Committee for Chemistry at academy. “This led to a paradigm shift within chemistry.”

## Mathematical description

There are several ways to mathematically define quasicrystalline patterns. One definition, the "cut and project" construction, is based on the work of Harald Bohr
Harald Bohr
Harald August Bohr was a Danish mathematician and football player. After receiving his doctorate in 1910, Bohr became an eminent mathematician, founding the field of almost periodic functions. His brother was the Nobel Prize-winning physicist Niels Bohr...

.
Bohr showed that quasiperiodic functions arise as restrictions of high-dimensional periodic functions to an irrational slice (an intersection with one or more hyperplane
Hyperplane
A hyperplane is a concept in geometry. It is a generalization of the plane into a different number of dimensions.A hyperplane of an n-dimensional space is a flat subset with dimension n − 1...

s), and discussed their Fourier point spectrum. In order that the quasicrystal itself be aperiodic, this slice must avoid any lattice plane
Lattice plane
In crystallography, a lattice plane of a given Bravais lattice is a plane whose intersections with the lattice are periodic and intersect the Bravais lattice; equivalently, a lattice plane is any plane containing at least three noncollinear Bravais lattice points...

of the higher-dimensional lattice. De Bruijn showed that Penrose tilings can be viewed as two-dimensional slices of five-dimensional hypercubic structures. Equivalently, the Fourier transform
Fourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...

of such a quasicrystal is nonzero only at a dense set of points spanned
Linear span
In the mathematical subfield of linear algebra, the linear span of a set of vectors in a vector space is the intersection of all subspaces containing that set...

by integer multiples of a finite set of basis vectors (the projections of the primitive reciprocal lattice
Reciprocal lattice
In physics, the reciprocal lattice of a lattice is the lattice in which the Fourier transform of the spatial function of the original lattice is represented. This space is also known as momentum space or less commonly k-space, due to the relationship between the Pontryagin duals momentum and...

vectors of the higher-dimensional lattice).

The intuitive considerations obtained from simple model aperiodic tilings are formally expressed in the concepts of Meyer and Delone
Delaunay triangulation
In mathematics and computational geometry, a Delaunay triangulation for a set P of points in a plane is a triangulation DT such that no point in P is inside the circumcircle of any triangle in DT. Delaunay triangulations maximize the minimum angle of all the angles of the triangles in the...

sets. The mathematical counterpart of physical diffraction is the Fourier transform
Fourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...

and the qualitative description of a diffraction picture as 'clear cut' or 'sharp' means that singularities are present in the Fourier spectrum
Spectrum
A spectrum is a condition that is not limited to a specific set of values but can vary infinitely within a continuum. The word saw its first scientific use within the field of optics to describe the rainbow of colors in visible light when separated using a prism; it has since been applied by...

. 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
Substitution tiling
In geometry, a tile substitution is a useful method for constructing highly ordered tilings. Most importantly, some tile substitutions generate aperiodic tilings, which are tilings whose prototiles do not admit any tiling with translational symmetry...

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
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

. 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
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...

, is the appropriate tool for studying quasicrystals.

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 allowing one to construct, visualize and analyze quasicrystal structures and their diffraction patterns.

Interacting spins
Spin (physics)
In quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...

were also analyzed in quasicrystals: AKLT Model
AKLT Model
The AKLT model is an extension of the one-dimensional quantum Heisenberg spin model. The proposal and exact solution of this model by Affleck, Lieb, Kennedy and Tasaki provided crucial insight into the physics of the spin 1 Heisenberg chain...

and 8 vertex model were solved in quasicrystals analytically

## Materials science of quasicrystals

Since the original discovery of Dan Shechtman
Dan Shechtman
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 Ames Laboratory, and Professor of Materials Science at Iowa State University. On April 8, 1982, while on sabbatical at the U.S...

, 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.).

In theory, there are two types of quasicrystals. 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:
• Stable quasicrystals grown by slow cooling or casting with subsequent annealing
Annealing (metallurgy)
Annealing, in metallurgy and materials science, is a heat treatment wherein a material is altered, causing changes in its properties such as strength and hardness. It is a process that produces conditions by heating to above the recrystallization temperature, maintaining a suitable temperature, and...

,
• Metastable quasicrystals prepared by melt spinning
Melt spinning
Melt spinning is a technique used for rapid cooling of liquids. A wheel is cooled internally, usually by water or liquid nitrogen, and rotated. A thin stream of liquid is then dripped onto the wheel and cooled, causing rapid solidification. This technique is used to develop materials that require...

, and
• Metastable quasicrystals formed by the crystallization
Crystallization
Crystallization is the process of formation of solid crystals precipitating from a solution, melt or more rarely deposited directly from a gas. Crystallization is also a chemical solid–liquid separation technique, in which mass transfer of a solute from the liquid solution to a pure solid...

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
Electron diffraction
Electron diffraction refers to the wave nature of electrons. However, from a technical or practical point of view, it may be regarded as a technique used to study matter by firing electrons at a sample and observing the resulting interference pattern...

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.

• Aperiodic tiling
Aperiodic tiling
An aperiodic tiling is a tiling obtained from an aperiodic set of tiles. Properly speaking, aperiodicity is a property of particular sets of tiles; any given finite tiling is either periodic or non-periodic...

• Archimedean solid
Archimedean solid
In geometry an Archimedean solid is a highly symmetric, semi-regular convex polyhedron composed of two or more types of regular polygons meeting in identical vertices...

• Girih tiles
Girih tiles
Girih tiles are a set of five tiles that were used in the creation of tiling patterns for decoration of buildings in Islamic architecture...

• Golden ratio
Golden ratio
In mathematics and the arts, two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. The golden ratio is an irrational mathematical constant, approximately 1.61803398874989...

• Phason
Phason
Phason is a quasiparticle existing in quasicrystals due to their specific, quasiperiodic lattice structure. Similar to phonon, phason is associated with atomic motion. However, whereas phonons are related to translation of atoms, phasons are associated with atomic rearrangements...

• Tessellation
Tessellation
A tessellation or tiling of the plane is a pattern of plane figures that fills the plane with no overlaps and no gaps. One may also speak of tessellations of parts of the plane or of other surfaces. Generalizations to higher dimensions are also possible. Tessellations frequently appeared in the art...

• 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
Birkhäuser Verlag
Birkhäuser Verlag is the leading European publisher for architecture, landscaping and design. Since 2010 it is part of the ActarBirkhäuser Group.-Scientific publishing:Founded in 1879 by Emil Birkhäuser the publishers program focused on regional literature...

(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
World Scientific
World Scientific Publishing is a publisher of scientific, technical, and medical books and journals. The company was founded in 1981 and now employs more than 200 staff at its headquarters in Singapore, with offices worldwide in New Jersey, California, London, New Delhi, Tianjin, Sydney, Hong...

, 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
Philosophical Magazine
The Philosophical Magazine is one of the oldest scientific journals published in English. Initiated by Alexander Tilloch in 1798, in 1822 Richard Taylor became joint editor and it has been published continuously by Taylor & Francis ever since; it was the journal of choice for such luminaries as...

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.