Quasicrystals

Lead Summary

Quasicrystals are solid materials that possess long-range atomic order without translational periodicity. They produce sharp Bragg diffraction peaks — the signature of crystalline order — yet their atomic arrangements never repeat identically through any translation. This combination had been considered impossible: classical crystallography held that order required periodicity, and that certain rotational symmetries (five-fold, eight-fold, ten-fold, twelve-fold) were mathematically forbidden in any periodic lattice.

When Dan Shechtman observed a ten-fold diffraction pattern in an aluminum-manganese alloy on April 8, 1982, he discovered that nature had found a third way — neither the periodic order of crystals nor the disorder of glasses, but something structurally richer than either. The discovery was so disruptive that it took over two years to publish, prompted a fierce decade-long dispute with Linus Pauling, and ultimately forced the International Union of Crystallography to rewrite its definition of "crystal." Shechtman received the unshared Nobel Prize in Chemistry in 2011.

Today quasicrystals are found in hundreds of synthetic alloys, in at least two minerals inside a 4.5-billion-year-old meteorite, inside trinitite formed by the first nuclear detonation, and in engineered colloidal systems assembled from DNA-coated nanoparticles. Their unusual electronic, thermal, and tribological properties are being translated into practical coatings, photonic devices, and composite materials.

Definition & Scope

A quasicrystal is distinguished from both conventional crystals and amorphous solids by two simultaneous properties: it has long-range atomic order (evidenced by sharp, discrete Bragg peaks in diffraction experiments) and it lacks translational periodicity (no lattice vector maps the structure onto itself). This combination is the defining feature.

The crystallographic restriction theorem, a mathematical theorem established over a century before Shechtman's discovery, proved that no periodic lattice in two or three dimensions can possess five-fold rotational symmetry. The allowed rotational axes in a periodic crystal are strictly 2-fold, 3-fold, 4-fold, and 6-fold. Five-fold, eight-fold, ten-fold, and twelve-fold axes are forbidden because periodic lattice points must map to integer combinations of basis vectors under rotation — a constraint that five-fold symmetry cannot satisfy.

Quasicrystals evade this theorem because aperiodicity is the mechanism by which forbidden symmetries become possible. The restriction theorem applies only to periodic structures. A quasiperiodic structure — one that never exactly repeats — is simply not bound by its constraints, and so five-fold or ten-fold rotation operations can produce valid structures in real materials.

In 1992, the International Union of Crystallography (IUCr) formalized this understanding by expanding the definition of "crystal" from "a material with a periodic atomic arrangement" to "any solid with an essentially discrete diffraction diagram." This redefinition officially recognized quasicrystals as legitimate crystal structures and ended a century of crystallographic orthodoxy.

Historical Development

Mathematical precursors

Roger Penrose published his aperiodic tiling work in 1974 in the paper "The rôle of aesthetics in pure and applied mathematical research". His tilings — plane-filling arrangements of two rhombic tile types assembled according to specific matching rules — exhibit five-fold rotational symmetry while never repeating under any translation. The ratio of "fat" to "skinny" rhombi in any Penrose tiling is the golden ratio (φ ≈ 1.618), an irrational number that emerges directly from the aperiodic matching constraints. Penrose tilings also exhibit hierarchical self-similarity under inflation and deflation transformations: inflating a tiling increases tile edge lengths by a factor of φ, producing a valid tiling at a larger scale — a property that explains why translational symmetry is impossible at any hierarchical level.

Also in 1984, mathematically preceding the experimental publication, Peter Kramer and Roberto Neri showed that icosahedral quasicrystals in three dimensions can be constructed as projections from a six-dimensional hypercubic lattice — the lowest-dimensional parent lattice in which icosahedral symmetry is mathematically possible. This theoretical work provided crucial foundations for interpreting the experimental results.

Shechtman's discovery

On April 8, 1982, Dan Shechtman was studying a rapidly solidified aluminum-manganese alloy (86% aluminum, 14% manganese) at the National Bureau of Standards in Gaithersburg, Maryland, using electron diffraction. The diffraction pattern he observed showed ten-fold symmetry — the pattern repeated after a 36-degree rotation — which was physically impossible for any periodic crystal.

His initial manuscript was rejected by the Journal of Applied Physics. The paper was eventually published on November 12, 1984, in Physical Review Letters — more than two and a half years after the discovery. It was coauthored with Ilan Blech (Technion), John Cahn (NIST), and Denis Gratias (CNRS, Paris).

Controversy and acceptance

"There is no such thing as quasicrystals, only quasi-scientists." — Linus Pauling, American Chemical Society conference at Stanford

Linus Pauling, a two-time Nobel laureate, became the most prominent and persistent opponent of the quasicrystal interpretation. His alternative explanation was that the observed diffraction patterns were artifacts of icosahedral twinning — multiple conventional cubic crystals intergrown at shared boundaries — producing composite diffraction patterns that mimicked forbidden symmetries. He published numerous rebuttal articles, including in Nature (October 10, 1985) and the Proceedings of the National Academy of Sciences. Shechtman was asked to leave his research group as a result of the hostility his discovery provoked.

The tide turned when internationally known materials scientist John Cahn publicly endorsed Shechtman's interpretation following the 1984 publication. A community of researchers rapidly synthesized hundreds of new quasicrystals in alloys of aluminum with other elements, and improved characterization techniques confirmed the structures. Pauling maintained his opposition until his death in 1994, never publicly accepting the quasicrystal interpretation.

Dan Shechtman received the unshared Nobel Prize in Chemistry in 2011 "for the discovery of quasicrystals."

Core Concepts

Aperiodic order

The central conceptual achievement of quasicrystal science is demonstrating that long-range order does not require periodicity. Quasicrystals produce sharp Bragg peaks — the hallmark of crystalline long-range order — yet no lattice vector maps one part of the structure identically onto another. The combination distinguishes them from periodic crystals (which have both) and from amorphous glasses (which have neither).

The cut-and-project formalism

The mathematical explanation for why quasicrystals can exist rests on the cut-and-project method: a quasicrystalline structure in three dimensions can be generated by selecting and projecting a slice of a higher-dimensional periodic lattice into physical space. The cutting plane must be positioned at an irrational angle relative to the parent lattice directions. This irrational angle is what produces aperiodicity in the physical structure while preserving the long-range order inherited from the parent lattice's periodicity — explaining why Bragg peaks remain sharp.

For icosahedral quasicrystals, the parent lattice lives in six dimensions. This framework was established by Harald Bohr's earlier mathematical work showing that quasiperiodic functions arise as restrictions of high-dimensional periodic functions to an irrational slice.

Phasons

Beyond the ordinary phonon (acoustic vibration) modes of conventional crystals, quasicrystals possess additional elastic degrees of freedom called phasons. A phason flip is a local atomic rearrangement that restores nearest-neighbor configurations while preserving the overall quasicrystalline structure — corresponding, in the cut-and-project picture, to a shift in the position of the cutting plane in the perpendicular (internal) space. Phason modes scatter phonons and are responsible for much of the anomalously low thermal conductivity of quasicrystals. They also play a central role in growth, defect dynamics, and plastic deformation.

Classification & Taxonomy

Quasicrystals fall into two main structural classes based on how many dimensions are quasiperiodic:

Icosahedral quasicrystals are aperiodic in all three spatial dimensions. Their symmetry group is that of an icosahedron: fifteen 2-fold rotation axes, ten 3-fold axes, and six 5-fold axes. The presence of six 5-fold axes in the same three-dimensional structure violates the crystallographic restriction theorem for any periodic arrangement. These are the most extensively studied and industrially relevant class.

Polygonal (dihedral) quasicrystals are quasiperiodic in a plane but retain translational periodicity in one perpendicular direction. They are further divided by their local symmetry:

• Decagonal (10-fold symmetry) — quasiperiodic in the plane, periodic along the stacking axis
• Octagonal (8-fold symmetry)
• Dodecagonal (12-fold symmetry)

Decagonal quasicrystals exhibit friction anisotropy correlated directly with their symmetry: high friction along periodic directions, low friction along aperiodic directions — a structural property with no analog in icosahedral phases.

Mechanism & Process

Stability and formation

Quasicrystal stability is governed by competing mechanisms. In many metallic systems, the Hume-Rothery electronic mechanism stabilizes the quasiperiodic phase: specific valence-electron-to-atom ratios (e/a) produce pseudogaps or band gaps at the Fermi level through Fermi-surface–Brillouin-zone interactions. First-principles calculations reveal sharp pseudogaps near the chemical potential in Al-Ni-Co and Al-Cu-Fe systems, predicting which alloy compositions favor icosahedral phases.

Entropy also plays a stabilizing role. In Al-Cu-Fe, the icosahedral phase is not the ground state at zero temperature but becomes thermodynamically favored at elevated temperatures (600–800 K and above) through contributions from vibrational entropy and phason fluctuations. At low temperatures, periodic approximant structures are lower in enthalpy by approximately 3 meV/atom, but the quasicrystal's larger configurational entropy tips the balance above a critical temperature. Recent DFT calculations on nanoparticle scaling have shown that icosahedral quasicrystals ScZn₇.₃₃ and YbCd₅.₇ are genuine ground-state phases at zero temperature, proving that translational symmetry is not a necessary condition for thermodynamic stability in inorganic solids.

Growth and phason dynamics

In-situ TEM observations of Al-Ni-Co decagonal quasicrystal growth reveal frequent local errors in tiling patterns followed by phason-mediated restructuring that restores long-range order. When multiple quasicrystalline grains collide during solidification, phason dynamics facilitate coalescence at small misorientations through rigid rotation, allowing the formation of single monocrystalline quasicrystals from initially misaligned seeds. This error-and-repair mechanism has no direct counterpart in periodic crystals.

Shock synthesis

Shock compression at pressures exceeding 5 GPa and temperatures above 1200°C can synthesize quasicrystalline phases directly. The Khatyrka meteorite provides natural evidence: hypervelocity impact generated heterogeneous pressure-temperature fields sufficient to melt Al-Cu-Fe mineral precursors, which then rapidly solidified into icosahedrite. Shock-synthesized quasicrystals exhibit surprisingly low phason strain, suggesting that shock is an effective pathway for producing essentially strain-free, highly perfect quasicrystals. Phason shifts during shock modulate local atomic symmetry without altering bulk thermodynamics, enabling the system to explore multiple nucleation pathways simultaneously.

Entropy-driven soft-matter assembly

Quasicrystalline phases also emerge in colloidal systems through purely entropic mechanisms, without any energetic stabilization. Binary mixtures of spheres with different sizes self-assemble into quasicrystalline phases that optimize sphere-packing entropy rather than specific interparticle interactions — demonstrating that quasiperiodic order is a general packing principle. DNA-programmable assembly of gold nanoparticle decahedra with sequence-specific bonding can produce icosahedral and dodecagonal quasicrystalline superlattices, bridging bottom-up design with quasicrystal structure.

Physical Properties

Electronic and electrical properties

Despite their metallic composition, quasicrystals behave electrically like semiconductors rather than metals. They display low electrical conductivity (0.1–1 Ω⁻¹cm⁻¹) and the counterintuitive property that resistivity decreases with increasing temperature — behavior opposite to conventional metals. Resistivity also increases with structural perfection — again the reverse of metals. This arises from a pseudogap in the electronic density of states at the Fermi level, related to the Hume-Rothery stabilization mechanism.

Thermal properties

Quasicrystals exhibit anomalously low thermal conductivity for metallic systems, typically 1–2 W/mK at room temperature — more than two orders of magnitude lower than pure aluminum and comparable to steel. Their thermal behavior is glass-like: at low temperatures, thermal conductivity follows a T² dependence typical of glasses, and a phonon saturation plateau appears at high temperatures. This glassy thermal transport coexists with metallic electronic structure — a hybrid combination unique to quasicrystals. The low conductivity is driven by phason scattering, structural complexity, and phonon-phonon interactions peculiar to the quasiperiodic lattice.

Mechanical properties

Quasicrystals are hard and brittle at room temperature, with mechanical behavior resembling ceramics or semiconductors despite being composed of metallic elements. This brittleness — a fundamental limitation for bulk applications — is attributed to the difficulty of dislocation motion in the complex quasiperiodic structure. At elevated temperatures, icosahedral quasicrystals deform plastically primarily through dislocation climb (diffusion-assisted non-conservative motion) rather than the glide mechanism dominant in conventional crystals, confirmed by in-situ TEM experiments.

Tribological properties

Quasicrystals possess exceptionally low surface energy (approximately 28 mJ/m²), comparable to PTFE, derived from the deep pseudogap at the Fermi level. This translates to friction coefficients around 0.16 — significantly below most metals and alloys. Their high hardness and superior wear resistance make them effective tribological materials: quasicrystal-reinforced fluorinated ethylene propylene composites demonstrate up to 50 times better wear resistance than unfilled polymer. Al-Cu-Fe and related systems also show high corrosion resistance in acidic and alkaline solutions, with protective aluminum oxy-hydroxide surface films.

Natural Quasicrystals

Until 2009, every known quasicrystal had been synthesized in laboratories. The existence of natural quasicrystals was uncertain.

The Khatyrka meteorite

The Khatyrka meteorite, classified as a CV3 carbonaceous chondrite approximately 4.5 billion years old, was collected from the Koryak Mountains of far eastern Russia. In 2009, Paul Steinhardt and Luca Bindi (a mineralogist at the Museum of Natural History in Florence) identified an unusual sample in the museum collection originally mislabeled "khatyrkite". This led to the discovery of icosahedrite (Al₆₃Cu₂₄Fe₁₃), the first naturally occurring quasicrystal, with icosahedral symmetry.

A 2011 expedition to the Koryak Mountains recovered additional meteorite fragments. Analysis of these new specimens identified a second natural quasicrystal: decagonite (Al₇₁Ni₂₄Fe₅), with decagonal (10-fold) symmetry and a two-layer structure based on the pentagonal Penrose tiling, formally recognized by the IMA-NMNC Commission (IMA2015-017).

The quasicrystals in Khatyrka formed during a hypervelocity asteroid collision in the early solar system. Impact exposed precursor Al-Cu-Fe alloys to pressures exceeding 5 GPa and temperatures above 1200°C. The presence of stishovite (a high-pressure silica polymorph forming only above several GPa) among the coexisting mineral phases in the meteorite independently confirms extreme shock conditions. Laboratory shock-recovery experiments have reproduced this formation mechanism, and high-pressure studies show that pressure strongly stabilizes the icosahedral structure at temperatures far above its ambient-pressure stability range.

Trinitite

An icosahedral quasicrystal with composition Si₆₁Cu₃₀Ca₇Fe₂ was discovered in trinitite — the glass created by the first nuclear detonation at Alamogordo, New Mexico, on July 16, 1945 (the Trinity test). This is the oldest known anthropogenic quasicrystal with a precisely documented place and moment of origin. The extreme transient conditions of a nuclear explosion, broadly comparable to those of astrophysical impacts, are sufficient to produce quasicrystalline phases.

Applications

Coatings and composites

The room-temperature brittleness of bulk quasicrystals limits their use as structural materials, but their surface properties make them valuable as thin coatings and reinforcing particles. Plasma-sprayed Al-Cu-Fe coatings provide low-friction, wear-resistant surfaces for cookware and medical devices. Quasicrystalline particles incorporated into polymer matrix composites improve wear resistance by more than a factor of 2 compared to conventional reinforcement. Quasicrystals have also been processed via selective laser sintering and stereolithography into composite components with complex geometries.

Photonic applications

Photonic quasicrystals — structures with quasiperiodic dielectric patterns — exploit the ability to realize rotational symmetries of any order, including those forbidden in periodic photonic crystals. High rotational symmetry (8-fold, 10-fold, 12-fold) enables nearly isotropic photonic band gaps even with modest refractive-index contrast — a regime where periodic photonic crystals struggle. Key demonstrated applications include:

• Lasers: Photonic quasicrystals support stimulated emission and lasing through controlled optical feedback from their band-gap properties.
• Solar cells: 12-fold symmetric quasicrystal patterning on silicon solar cells achieves superior broadband antireflectance compared to conventional periodic photonic crystal patterns.
• Optical fibers: Quasiperiodic fiber cladding produces more spherically symmetric photonic band gaps than periodic cladding, with applications in dispersion control and polarization maintenance.
• Nonlinear optics: The wider tunability of quasicrystalline configurations enables phase-matching of multiple nonlinear processes simultaneously in a single nanostructure.
• Phoxonic structures: Simultaneous photonic and phononic band gaps in quasicrystalline structures enable joint control of light and sound.
• Topological photonics: Photonic quasicrystals host higher-order topological corner and edge states characterized by the Bott index (since conventional momentum-space methods cannot be applied to aperiodic systems).

Light localization

Three-dimensional icosahedral photonic quasicrystals support wave localization analogous to Anderson localization even though the structure is deterministic and ordered — without any random disorder. This represents a departure from conventional Anderson localization theory. One-dimensional photonic quasicrystals also exhibit reentrant delocalization transitions: light initially localizes, then delocalizes again as quasiperiodic modulation strength increases — a transport phenomenon more complex than simple Anderson localization.

Cultural Significance

The quasicrystal controversy stands as a case study in how paradigm-challenging discoveries encounter institutional resistance. Shechtman's original notebook entry from April 8, 1982, reads "10 fold???" — a question that took a decade of controversy and twenty-nine years of accumulated evidence to settle with a Nobel Prize. The episode illustrates both the conservatism of mature scientific fields and the mechanisms by which genuine anomalies eventually overturn established frameworks.