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Quasicrystals

How a Crystal Can Be Ordered Without Repeating

Big Question: How can atoms produce sharp diffraction peaks and long-range order if their arrangement never repeats periodically?

Quick Answer

A periodic crystal repeats a basic unit after fixed translations in space. A quasicrystal does not have that ordinary translational periodicity, yet its atoms are not random. Their arrangement follows deterministic ordering rules that create long-range correlations. Because those correlations extend across large distances, diffraction still produces sharp peaks.

Dan Shechtman’s 1982 electron-diffraction observations revealed ten-fold symmetry in an aluminium–manganese alloy, contributing to a redefinition of what crystallinity means. He received the 2011 Nobel Prize in Chemistry for the discovery of quasicrystals.

Nobel Prize — Dan Shechtman and the discovery of quasicrystals →

What You Will Learn

• What periodicity means in a crystal.
• Why rotational symmetry is restricted in ordinary periodic lattices.
• Why five-fold symmetry was historically considered incompatible with periodic crystals.
• What long-range order means.
• Why sharp diffraction peaks do not require ordinary translational repetition.
• How Penrose-like tilings help visualise quasiperiodic order.
• Why a quasicrystal is not simply a glass.
• How electron and X-ray diffraction provide evidence.
• What phason rearrangements are.
• How quasicrystals can be described using higher-dimensional periodic structures.
• Why some quasicrystals have unusual friction, hardness or thermal properties.
• Why “non-repeating” must not be confused with “random.”

Part 1 — What Makes an Ordinary Crystal Periodic?

In an ordinary crystal, a basic arrangement of atoms repeats after translation by certain lattice vectors.

If you move the whole ideal structure by one lattice spacing in an allowed direction, the pattern lines up with itself again.

periodic order = a pattern that matches itself after specific translations.

Part 2 — Symmetry Has Rules

Periodic tilings of a flat plane can support two-fold, three-fold, four-fold and six-fold rotational symmetry. Five-fold rotational symmetry does not fill periodic space without gaps or overlaps.

The same mathematical restriction shaped classical crystallography. Certain rotational symmetries were considered incompatible with periodic atomic lattices.

That did not mean five-fold geometric shapes could not exist. It meant a three-dimensional periodic crystal lattice could not possess five-fold crystallographic symmetry.

Part 3 — The Diffraction Pattern That Changed the Definition

When a beam of electrons interacts with an ordered atomic structure, scattered waves interfere. The resulting diffraction pattern contains information about the spatial order.

Shechtman’s aluminium–manganese alloy produced sharp diffraction spots arranged with symmetry that ordinary periodic crystallography could not explain.

Sharp spots mattered because they showed the material was not merely amorphous or randomly disordered.

sharp diffraction + forbidden periodic symmetry → a new kind of long-range order.

Part 4 — Long-Range Order Without Translational Periodicity

A quasicrystal follows a structured arrangement extending across the material, but there is no finite translation that makes the complete pattern coincide with itself in the usual way.

The pattern is aperiodic yet highly constrained.

This distinction is crucial:

Structure	Long-range order?	Periodic repetition?
Ordinary crystal	Yes	Yes
Quasicrystal	Yes	No
Ideal amorphous glass	No ordinary long-range translational order	No

Part 5 — Penrose Tilings Give a Two-Dimensional Analogy

A Penrose tiling covers a plane using a small set of shapes according to matching rules. The pattern never repeats periodically, yet it is not random.

Local motifs recur, and the whole pattern can show five-fold symmetry and long-range mathematical organisation.

Real atomic quasicrystals are more complicated, but Penrose tilings provide a useful model for understanding how deterministic order can exist without translation symmetry.

Part 6 — Why Sharp Diffraction Peaks Still Appear

Diffraction peaks arise when scattered waves from many atoms add coherently at particular wavevectors.

Periodic order is one way to create those correlations, but it is not the only way.

Quasiperiodic order generates a discrete set of reciprocal-space peaks because atomic positions remain correlated over long distances.

This is a powerful scientific lesson: the measurement reveals order, but the model of that order must still be inferred correctly.

Part 7 — Icosahedral Symmetry

Many famous quasicrystals show icosahedral symmetry. An icosahedron has twenty triangular faces and symmetry axes involving five-fold rotations.

Icosahedral symmetry is natural for isolated clusters but incompatible with ordinary three-dimensional translational periodicity.

Quasicrystals make such symmetry compatible with long-range solid-state order by abandoning periodic translation.

Part 8 — Higher-Dimensional Descriptions

One elegant mathematical approach describes a quasicrystal as a projection or slice of a periodic lattice in a space with more dimensions.

The higher-dimensional structure can be perfectly periodic. When projected into ordinary three-dimensional space, the result becomes quasiperiodic.

This does not mean the physical alloy literally extends into extra everyday spatial dimensions. It is a mathematical representation that makes the ordering rules easier to express.

Part 9 — Phasons: A New Kind of Rearrangement

Periodic crystals have familiar vibrational excitations called phonons. Quasicrystals add another type of degree of freedom associated with how quasiperiodic patterns can rearrange locally while preserving their global ordering rules.

These rearrangements are called phasons.

Phason-related dynamics can influence diffusion, defects and thermal behaviour.

Part 10 — Quasicrystals Are Not Perfectly Defect-Free

Real materials contain vacancies, impurities, dislocations and other defects. Quasicrystals are no exception.

The existence of defects does not erase the underlying quasiperiodic order, just as defects do not prevent an ordinary crystal from being crystalline.

Scientists distinguish the ideal structural model from the imperfect real sample.

Part 11 — Why Their Properties Can Be Unusual

Atomic order affects how electrons and phonons move through a solid. Because quasicrystals have unusual reciprocal-space structure and local environments, some show:

• low thermal conductivity;
• unusual electrical conductivity;
• high hardness;
• low friction or low surface adhesion in some compositions;
• distinctive electronic density-of-states features.

These are material-specific properties, not universal consequences of being a quasicrystal.

Part 12 — Natural Quasicrystals Exist

Quasicrystals were first made and studied in laboratories, but natural quasicrystalline minerals have also been identified in meteorite material.

This demonstrates that quasiperiodic atomic order is not merely an artificial curiosity. Under suitable conditions, nature can produce it too.

Part 13 — Why Definitions Changed

Science often teaches definitions after they have stabilised. But definitions are sometimes revised when observations no longer fit them.

Before quasicrystals, periodicity was deeply embedded in the concept of a crystal. After quasicrystals, crystallinity became tied more broadly to ordered diffraction rather than simple translational repetition alone.

good scientific definitions must survive contact with evidence.

Part 14 — Follow the Evidence

1. An alloy sample is prepared.
2. An electron beam passes through a thin region.
3. Scattered electron waves interfere.
4. Sharp diffraction spots appear.
5. The spot arrangement shows symmetry incompatible with ordinary periodic crystallography.
6. Repeat measurements rule out simple experimental error.
7. Additional diffraction and composition studies confirm a distinct phase.
8. Mathematical quasiperiodic models reproduce the symmetry.
9. Atomic-scale imaging and structure refinement provide further support.
10. The definition of crystal is broadened to accommodate the evidence.

Think Like a Scientist: What Would Distinguish a Quasicrystal?

• Sharp diffraction peaks indicate long-range order.
• Non-crystallographic rotational symmetry challenges ordinary periodic models.
• No finite translation reproduces the full structure.
• High-resolution microscopy tests local atomic arrangements.
• Structure models must predict observed peak positions and intensities.
• Independent laboratories should reproduce the phase.

Observation vs Inference

• Observation: sharp diffraction spots occur.
• Observation: the pattern has five-fold, ten-fold or icosahedral-type symmetry in appropriate samples.
• Inference: the material has long-range quasiperiodic order.
• Further test: atomistic models must reproduce diffraction and microscopy.
• Boundary: unusual diffraction does not automatically prove every proposed atomic model.

Common Misconceptions and Better Models

Misconception	Better model
Non-repeating means random.	Quasicrystals are aperiodic but highly ordered.
All crystals must repeat periodically.	Quasicrystals show long-range order without ordinary translational periodicity.
Five-fold symmetry means the measurement is wrong.	It is a genuine signature possible in quasiperiodic order.
A quasicrystal is just glass.	Glasses lack the same long-range quasiperiodic order and sharp Bragg-like diffraction.
Higher-dimensional models mean hidden physical dimensions were discovered.	They are mathematical constructions used to describe order.
Every quasicrystal has the same properties.	Properties depend strongly on composition and structure.

Checkpoint Questions

1. What is translational periodicity?
2. Why was five-fold symmetry problematic for classical crystallography?
3. Why do sharp diffraction peaks matter?
4. How is a quasicrystal different from an amorphous solid?
5. What does a Penrose tiling illustrate?
6. Why can a quasiperiodic structure still diffract sharply?
7. What is icosahedral symmetry?
8. Why are higher-dimensional descriptions useful?
9. What is a phason?
10. What scientific lesson came from the discovery?

Answer Key

Open after attempting the questions

1. A structure matches itself after fixed spatial translations.
2. Five-fold rotational symmetry cannot tile ordinary periodic space in the required crystallographic way.
3. They show long-range coherent order rather than random disorder.
4. A quasicrystal has long-range quasiperiodic order; an amorphous solid does not.
5. Deterministic non-periodic order with recurring local motifs.
6. Long-range correlations still produce constructive interference at discrete wavevectors.
7. Symmetry based on an icosahedron, including five-fold axes.
8. They convert quasiperiodic order into periodic order in a higher-dimensional mathematical representation.
9. A quasiperiodic rearrangement degree of freedom distinct from ordinary phonon motion.
10. Definitions and models must change when reproducible evidence demands it.

Primary Science Bridge

• patterns can be regular without being simple;
• materials have internal structures;
• observations can challenge a model;
• classification rules may have exceptions;
• evidence should decide between explanations.

Secondary and JC Bridge

Core idea	Higher-resolution route
Patterns	Periodicity and quasiperiodicity
Symmetry	Crystallographic restriction
Waves	Reciprocal space and diffraction
Solids	Long-range atomic order
Mathematics	Projection from higher dimensions
Dynamics	Phonons and phasons

Deep Science Window — Reciprocal Space

Diffraction is naturally described in reciprocal space. Periodic crystals generate reciprocal lattices with a finite set of basis vectors. Quasicrystals can require more independent reciprocal basis vectors than the physical dimension of the sample, reflecting their quasiperiodic order.

Deep Science Window — Local Rules and Global Order

One of the deepest questions in quasicrystal physics is how local atomic interactions select a globally quasiperiodic state. Matching rules, energetic stability and entropy can all matter, depending on the material.

Evidence Boundaries

• Aperiodic ≠ random.
• Sharp diffraction ≠ ordinary periodicity.
• Five-fold symmetry ≠ experimental error by default.
• Mathematical higher dimensions ≠ extra physical spatial dimensions.
• Quasicrystal ≠ amorphous glass.
• One composition’s property ≠ universal quasicrystal property.

Manual Summary — KNOW → CONNECT → EXPLAIN → APPLY → CHECK

KNOW: periodicity, quasiperiodicity, diffraction, rotational symmetry, long-range order, phason.

CONNECT: atomic order to diffraction and forbidden periodic symmetry to quasiperiodic models.

EXPLAIN: how a non-repeating structure can still be highly ordered.

APPLY: distinguish periodic crystal, quasicrystal and amorphous solid from evidence.

CHECK: do not equate repetition with order.

Research Sources and Further Reading

• Nobel Prize — Dan Shechtman
• Royal Swedish Academy of Sciences — Scientific Background on Quasicrystals