The Physics of Quasicrystals

The following sections are included:

• PREFACE

• CONTENTS

The development of the concept of quasicrystals has challenged many widely held assumptions of crystallography and solid state physics. It had been generally assumed that pentagonal symmetry is impossible in solids, that Bragg diffraction peaks imply periodic translational order, that all pure solids are either crystalline or glassy, and that the (classically) stable state of a solid is crystalline. The discovery of icosahedral phases in metallic alloys has compelled theorists to reconsider these assumptions and to confront many new problems. Under what conditions will a material solidify in the quasicrystalline state? Can a real material be found or “designed” such that the quasicrystal phase is the classically stable state? How does one uniquely determine the atomic structure of a quasicrystal from diffraction data? What new thermal, structural, elastic, phonon, and electronic properties should quasicrystalline solids exhibit? Can the growth and physical properties of crystal, quasicrystal, and glass be related in a unified theoretical picture? These are are just a few of the questions that can be raised; few, if any, have been fully answered. Clearly, quasicrystals present a fundamental challenge to theoretical physicists to re-examine traditional concepts and devise new analytical methods for determining their physical properties…

This section consists of an introduction by the author(s) and the following reprint articles are included:

• Quasicrystals: A New Class of Ordered Structures

• Comment on Quasicrystals: A New Class of Ordered Structures

• On Periodic and Non-periodic Space Fillings of E^m Obtained by Projection

• Al_0.86 Mn_0.14: A 6-Dimensional Crystal

• The Diffraction Pattern of Projected Structures

• Quasiperiodic Patterns and Icosahedral Symmetry

• Icosahedral Crystals from Cuts in Six-Dimensional Space

• Quasicrystals I: Definition and Structure

• Quasicrystals II: Unit-cell Configurations

This section consists of an introduction by the author(s) and the following reprint articles are included:

• Crystallography of Quasi-Crystals

• Scale Invariance and the Group Structure of Quasicrystals

• Rudimentary quasi-crystallography: The icosahedral and decagonal reciprocal lattices

• Beware of 46-Fold Symmetry: The Classification of Two-Dimensional Quasicrystallographic Lattices

This section consists of an introduction by the author(s) and the following reprint articles are included:

• Metallic Phase with Long-Range Orientational Order and No Translational Symmetry

• Quasicrystal with One-Dimensional Translational Symmetry and a Ten-fold Rotation Axis

• Icosahedral Quasicrystals of a Melt-Quenched Al-Mn Alloy Observed by High-Resolution Microscopy

• Structure of Rapidly Quenched Al-Mn

• Indexing Problems in Quasicrystal Diffraction

• Indexing of Icosahedral Quasiperiodic Crystals

• Diffraction from quasicrystals: Geometric structure factor

• Distortion and Peak Broadening in Quasicrystal Diffraction Patterns

This section consists of an introduction by the author(s) and the following reprint articles are included:

• Al₄Mn Quasicrystal Atomic Structure, Diffraction Data and Penrose Tiling

• A Structural Determination of the Al-Mn Icosahedral Phase

• Crystal and Quasicrystal Structures in Al-Mn-Si Alloys

• Sphere Packings and Local Environments in Penrose Tilings

• Icosahedral Crystals: Where Are the Atoms?

• Supercooling of Liquids

• Complex Alloy Structures Regarded as Sphere Packings, Part I. Definitions and Basic Principles.

• A Dense Non-crystallographic Packing of Equal Spheres

This section consists of an introduction by the author(s) and the following reprint articles are included:

• Apparent Icosahedral Symmetry is due to Directed Multiple Twinning of Cubic Crystals

• Pauling's Model not Universally Accepted

• So-called Icosahedral and Decagonal Quasicrystals Are Twins of an 820-Atom Cubic Crystal

• The Microstructure of Rapidly Solidified Al₆Mn

• Sharp Diffraction Maxima from an Icosahedral Glass

• Perfect and Imperfect Icosahedral Solids and the Projection Method

• Systematics of Disorder in Quasiperiodic Material

• The Growth or Quasicrystals

The second part of this book focusses on the physical properties of quasicrystals. Although there is a powerful theoretical machinery for analyzing the physical properties of periodic crystals, it has proven very difficult to extrapolate those techniques to the case of quasicrystals. Significant progress has been made in understanding the stability, elasticity, hydrodynamics, and defect properties of quasicrystals, which is the focus of Chapter 6. On the other hand, comparatively little progress has been made in understanding electronic and phonon properties. Chapter 7 discusses the attempts thus far to make progress on these issues and describes the more substantial results that have been made in studying the analogous problem in a one-dimensional quasiperiodic (Fibonacci) lattice.

This section consists of an introduction by the author(s) and the following reprint articles are included:

• Phenomenological Theory of Icosahedral Incommensurate ('Quasiperiodic') Order in Mn-Al Alloys

• Mean-field Theory of Quasi-crystalline Order

• Elasticity and Dislocations in Pentagonal and Icosahedral Quasicrystals

• 6-Dimensional Properties of Al_0.86 Mn_0.14 Alloy

• Periodic Quasicrystal

• Long-Range Icosahedral Orientational Order and Quasicrystals

• Order in Metallic Glasses and Icosahedral Crystals

• 'Martensitic' Instability of an Icosahedral Quasicrystal

• Symmetry and Stability of Icosahedral and other Quasicrystalline Phases

• A Quasicrystalline Structure for the Cholesteric Fog Phase

• Quasicrystalline Textures of Cholesteric Liquid Crystals: Blue Phase III?

• Hydrodynamics of Icosahedral Quasicrystals

• Symmetry, Stability, and Elastic Properties of Icosahedral Incommensurate Crystals

• Phonons, Phasons, and Dislocations in Quasicrystals

• Topological Constraints on Quasicrystal Transformations

This section consists of an introduction by the author(s) and the following reprint articles are included:

• One-Dimensional Schrödinger Equation with an Almost Periodic Potential

• Electronic States on a Penrose Lattice

• Localization Problem in One Dimension: Mapping and Escape

• Energy Levels and Wave Functions of Bloch Electrons in Rational and Irrational Magnetic Fields

• Analyticity Breaking and Anderson Localization in Incommensurate Lattices

• Bandwidths for a Quasiperiodic Tight-binding Model

• Wavefunction Scaling in a Quasi-periodic Potential

• Chaotic States of Almost Periodic Schrödinger Operators

• A Soluble Quasi-Crystalline Magnetic Model: the XY Quantum Spin Chain

• Oscillatory Length-Dependent Conductivity in Periodic and Almost Periodic Crystals

• Electronic and Vibrational Spectra of Two-dimensional Quasicrystals

• Electronic and Vibrational Spectra of Two-dimensional Quasicrystals

• Quasiperiodic Lattice: Electronic Properties, Phonon Properties, and Diffusion

• Eigenstates in 2-Dimensional Penrose Tiling

• Density of States for a Two-Dimensional Penrose Lattice: Evidence of a Strong Van Hove Singularity

This section consists of an introduction by the author(s) and the following reprint articles are included:

• APPENDIX I
  • The Role of Aesthetics in Pure and Applied Mathematical Research
  • Sequences of Zeros and Ones Generated by Special Production Rules
  • Algebraic Theory of Penrose's Non-periodic Tilings of the Place
  • De Nive Quinquangula: On the Pentagonal Snowflake
  • Crystallography and the Penrose Pattern
  • Non-periodic Central Space Filling with Icosahedral Symmetry using Copies of Seven Elementary Cells
  • Two and Three Dimensional Non-periodic Networks Obtained from Self-similar Tiling
  • Should All Crystals Be bcc? Landau Theory of Solidification and Crystal Nucleation

• APPENDIX II
  • Additional References