Geometrical Frustration 9780511599934, 0511599935

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English Pages [319] Year 1999

Geometrical Frustration
9780511599934, 0511599935

Author / Uploaded
J. F. Sadoc
Rémy Mosseri

Table of contents :
Contents
Preface
1. Introduction to geometrical frustration
1.1 From cubism to icosahedrism
1.2 Geometry
1.3 Geometrical frustration
2. Ideal models
2.1 A unified approach to very different materials
2.2 Simple two-dimensional examples
2.3 Metals
2.4 The {3, 3, 5} polytope: an ideal template for amorphous metals
2.5 Covalent tetracoordinated structures
2.6 Frustration in lamellar liquid crystals and amphiphiles
2.7 Lamellar structures in curved spaces
2.8 Frustration and curved space structure for blue phases
2.9 Frustration in polymers
3 Finite structures
3.1 Finite clusters
3.2 Toroidal vesicles with phospholipid membranes
4. Decurving and disclinations
4.1 Disclinations
4.2 Wedge and screw disclinations
4.3 Coordination number, disclination density and Regge calculus
4.4 Conservation laws
4.5 Defects and topology
5. Hierarchical polytopes
5.1 Hierarchical polytopes and symmetry groups
5.2 Hierarchy and scaling
5.3 Matrix formulation of the hierarchical structures
5.4 Disorder and non-commutative defects
5.5 Deflation of the orthoscheme
6. Some physical properties
6.1 Structure factor of polytopes and orientational order
6.2 Specific volume variation in disordered solids: a simple model
6.3 Landau theory of frustrated systems
6.4 Two-level systems
6.5 Frustration-limited domain theory
6.6 Excitation spectrum
7. Periodic structures with large cells
7.1 Frustration and large cell crystals
7.2 Complex structures in metals
7.3 Melting of model structures
7.4 Tetracoordinated structures
7.5 Liquid crystal structures
8. Quasiperiodic order and frustration
8.1 Quasicrystals: the spectacular appearance of quasiperiodic order in solid state physics
8.2 Hierarchical clusters in quasicrystals
8.3 Random tilings
8.4 Random tilings in one dimension
8.5 Two-dimensional tilings
8.6 Three-dimensional rhombohedral tilings
8.7 Glass-like properties in quasicrystals
A1. Spaces with constant curvature
A1.1 The three geometries
A1.2 Spherical spaces
A1.3 Two- and three-dimensional cylindrical spaces
A1.4 Intrinsic curvature
A2. Quaternions and related groups
A2.1 Quaternions
A2.2 Some continuous groups acting on spheres
A2.3 Discrete groups
A3. Hopf fibration
A3.1 Fibrations
A3.2 Hopf fibration
A4. Polytopes and honeycombs
A4.1 Symmetries and orthoscheme tetrahedra
A4.2 Polytopes and honeycombs
A5. Polytope {3, 3, 5}
A5.1 The geometry of polytope {3, 3, 5}
A5.2 Description in terms of toroidal shells
A6. Frank and Kasper coordination polyhedra
A6.1 Frank and Kasper polyhedra
A6.2 Positive and negative disclinations
A7. Quasiperiodic tilings: cut and projection
A7.1 Cut and projection algorithm
A7.2 Codimension 1 approximants
A7.3 Approximants of the octagonal tiling
A7.4 An almost octagonal quasiperiodic tiling: the labyrinth
A8. Differential geometry and parallel transport
A8.1 Manifold and tangent space
A8.2 Geodesic equation
A8.3 Parallel transport and curvature
A9. Icosahedral quasicrystals and the ES lattice
A9.1 Introduction
A9.2 The E8 lattice
A9.3 A discrete Hopf fibration on the Gosset polytope
A9.4 Shelling the quasicrystal
A9.5 The 2d-ld aspect of the shell-by-shell construction of the quasicrystal
A9.6 Quasicrystals of lower dimension
Bibliography
Index