Solid State Chaikin, Lubensky, Principles of Condensed Matter Physics

Contents......Page 7
Preface......Page 17
1.1 Condensed Matter Physics......Page 21
1 Gaseous and Liquid States......Page 23
2 The Liquid-Gas Phase Transition......Page 24
3 Spatial Correlations in the Liquid State......Page 25
4 Ice - Crystallized Water......Page 28
5 Broken Symmetry and Rigidity......Page 30
6 Dislocations - Topological Defects......Page 32
7 Universality of the Water Example......Page 33
8 Fluctuations and Spatial Dimensions......Page 35
9 Overview of Book......Page 36
1 Energy Scales......Page 37
2 Van der Waals Attraction......Page 38
3 Molecular Hydrogen - the Heitler-London Approach......Page 40
4 Hard-Sphere Repulsion......Page 42
5 Exchange Interaction and Magnetism......Page 44
6 The Hydrogen Molecule, Molecular Orbitals, and Bands in Metals......Page 45
2.1 Elementary Scattering Theory - Bragg's Law......Page 49
2.2 Photons, Neutrons, or Electrons......Page 53
2.3 The Density Operator and its Correlation Functions......Page 54
2.4 Liquids and Gases......Page 58
1 Hard-Sphere Liquids......Page 60
1 Unit Cells and the Direct Lattice......Page 63
2 The Reciprocal Lattice......Page 65
3 Periodic Functions......Page 66
4 Bragg Scattering......Page 67
2.6 Symmetry and Crystal Structure......Page 69
1 Two-Dimensional Bravais Lattices......Page 70
2 Three-Dimensional Bravais Lattices......Page 73
3 Close Packed Structures......Page 76
4 Space Groups......Page 77
1 Isotropic, Nematic and Cholesteric Phases......Page 78
2 Smectics -A and -C......Page 81
3 Hexatic Phases......Page 85
5 Lyotropic Liquid Crystals and Microemulsions......Page 88
2.8 One- and Two-Dimensional Order in Three-Dimensional Materials......Page 91
2.9 Incommensurate Structures......Page 97
2.10 Quasicrystals......Page 102
2.11 Magnetic Order......Page 105
2.12 Random Isotropic Fractals......Page 110
1 One Dimension......Page 117
2 d Dimensions......Page 119
3 Transforms on a Lattice......Page 120
3.1 Thermodynamics of Homogeneous Fluids......Page 128
1 The First Law of Thermodynamics......Page 129
3 The Third Law of Thermodynamics......Page 131
4 Thermodynamic Potentials......Page 132
5 Stability Criteria......Page 133
6 Homogeneous Functions......Page 135
7 Equations of State......Page 136
3.2 Statistical Mechanics: Phase Space and Ensembles......Page 137
3.3 The Ideal Gas......Page 142
3.4 Spatial Correlations in Classical Systems......Page 143
3.5 Ordered Systems......Page 147
3.6 Symmetry, Order Parameters, and Models......Page 152
1 Discrete Symmetries......Page 155
2 Continuous Symmetries......Page 157
3 Models......Page 159
Appendix 3A: Functional Derivatives......Page 160
4 Mean-Field Theory......Page 164
4.1 Bragg-Williams Theory......Page 166
4.2 Landau Theory......Page 171
4.3 The Ising and n-Vector Models......Page 172
1 The Nonlocal Susceptibility and the Correlation Length......Page 174
2 O(n) Symmetry......Page 176
3 Some Mean-Field Transitions......Page 177
4.4 The Liquid-Gas Transition......Page 179
1 The Critical Point and the Critical Isochore......Page 182
2 The Coexistence Curve......Page 185
4.5 The First-Order Nematic-to-Isotropic Transition......Page 188
4.6 Multicritical Points......Page 192
1 Tricritical Points......Page 193
2 Matemagnets and FeCl2......Page 195
3 He3-He4 Mixtures and the Blume-Emery-Griffiths Model......Page 199
4 Bicritical and Tetracritical Points......Page 201
5 Lifshitz Points......Page 204
4.7 The Liquid-Solid Transition......Page 208
1 Are All Crystals BCC?......Page 209
3 Improvements of the Theory......Page 212
4 Changes in Density......Page 214
5 Density Functional Theory......Page 215
1 Two Inequalities......Page 218
2 The Mean-Field Approximation......Page 220
3 The S-State Potts Model......Page 221
4 The Omega(n) Classical Heisenberg Model......Page 222
5 Debye-Huckel Theory......Page 224
5 Field Theories, Critical Phenomena, and the Renormalization Group......Page 233
5.1 Breakdown of Mean-Field Theory......Page 234
1 Mean-Field Transitions Revisited......Page 236
1 Coarse Graining......Page 237
2 Lattice Field Theories and their Continuum Limit......Page 239
3 Gaussian Integrals......Page 241
4 Mean-Field Theory from Functional Integrals......Page 243
5 Breakdown of Mean-Field Theory Revisited......Page 245
5.3 The Self-Consistent Field Approximation......Page 246
1 The n-Vector Model in the Limit n->Infinity......Page 249
1 Exponents and Scaling Relations......Page 250
2 Scaled Equation of State......Page 254
3 Multicritical Points......Page 255
4 Amplitude Ratios......Page 256
5.5 The Kadanoff Construction......Page 257
1 Exact Solution......Page 262
2 Decimation and Renormalization......Page 265
1 The Ising Model on a Hypercube Lattice......Page 268
2 General Properties of Recursion Relations......Page 272
3 The Potts Lattice Gas and Krypton on Graphite......Page 273
1 Thinning of Degrees of Freedom and Rescaling......Page 276
2 Correlation Functions......Page 280
3 The Gaussian Model......Page 281
4 The e-Expansion......Page 283
5 n-Vector Model with Cubic Anisotropy......Page 287
6 Quadratic Anisotropy......Page 289
7 Crossover......Page 290
8 Dangerous Irrelevant Variables......Page 293
9 The Utility of the e-Expansion......Page 295
Appendix 5A: The Hubbard-Stratonovich Transformation......Page 296
Appendix 5B: Diagrammatic Perturbation Theory......Page 297
6 Generalized Elasticity......Page 308
1 The Elastic Free Energy......Page 309
2 Boundary Conditions and External Fields......Page 310
3 The Josephson Scaling Relation......Page 312
4 Fluctuations......Page 313
5 Long-Range Order, Quasi-Long-Range Order, and Disorder......Page 315
6 Resistance of a Conducting Medium......Page 317
2 The Frank Free Energy of Nematic Liquid Crystals......Page 318
3 Cells with Non-Uniform n......Page 320
4 The Freedericksz Transition......Page 322
5 The Twisted Nematic Display......Page 324
6 Fluctuations and Light Scattering......Page 326
6.3 Smectic Liquid Crystals......Page 328
1 The Elastic Free Energy......Page 329
2 Fluctuations......Page 332
3 Nonlinearities......Page 334
4 The Nematic-to-Smectic-A Transition......Page 335
1 The Strain Tensor......Page 336
2 The Elastic Free Energy......Page 338
3 Isotropic ad Cubic Solids......Page 339
4 Fluctuations......Page 341
5 Mercury Chain Salts - One-Dimensional Crystals......Page 342
6 Xenon on Graphite - a Two-Dimensional Crystal......Page 344
7 Vacancies and Interstitials......Page 345
8 Bond-Angle Order and Rotational and Transitional Elasticity......Page 348
9 Elastic Constants from Density Function Theory......Page 349
1 Classical Theory of Elasticity......Page 350
2 Elasticity of Classical Harmonic Lattices......Page 352
1 The Lagrangian Stress Tensor......Page 354
2 Stress-Strain Relations......Page 357
3 The Eluerian Stress Tensor......Page 358
6.7 The Nonlinear Sigma Model......Page 361
7 Dynamics: Correlation and Response......Page 373
1 Correlation Functions......Page 374
2 Response Functions......Page 375
1 The Undamped Oscillator......Page 379
2 The Damped Oscillator......Page 380
3 The Response Function......Page 382
4 Dissipation......Page 385
1 Sound Waves in and Elastic Continuum......Page 386
2 Acoustic Phonons in a Harmonic Lattice......Page 387
1 Fick's Law......Page 389
2 The Green Function and Dynamic Response......Page 390
3 The Response Function......Page 391
4 External Potentials and the Einstein Relation......Page 393
5 Brownian Motion......Page 395
6 Cooperative Diffusion Versus Self-Diffusion......Page 396
7 Master Equation for Diffusion on a Lattice......Page 398
1 Random Forces and Thermal Equilibrium......Page 401
2 Correlation Functions for Diffusion......Page 403
3 Short-Time Behavior......Page 405
4 Fluctuation-Dissipation Theorem for the Harmonic Oscillator......Page 407
5 The Fokker-Planck and Smoluchowski Equations......Page 408
1 Response to External Fields......Page 410
2 SYmmetry Properties of Response Functions......Page 412
3 Dissipation......Page 414
4 Spectral Representations of X......Page 415
5 The Fluctuation-Dissipation Theorem......Page 417
6 Sum Rules and Moment Expansions......Page 418
1 Scattering Geometry and Partial Cross-Sections......Page 419
2 Fermi Golden Rule and Neutron Scattering......Page 420
3 The Fermi Pseudopotential......Page 422
4 Coherent and Incoherent Scattering......Page 424
5 Cross-Sections and Correlation Functions......Page 425
6 Neutron Scattering from Crystals......Page 426
7 Magnetic Scattering......Page 427
8 How Neutron Scattering Experiments are Actually Done......Page 428
9 Scattering of Charged Particles and Photons......Page 430
8.1 Conserved and Broken-Symmetry Variables......Page 437
8.2 A Tutorial Example - Rigid Rotors on a Lattice......Page 439
1 Description of the Model......Page 440
2 The Disordered Phase......Page 441
3 The Ordered Phase......Page 446
4 Excitations from the Classical Ground State......Page 450
6 Kubo Formulae......Page 452
7 Summary......Page 453
1 Spin Dynamics......Page 454
2 Generalized Heisenberg Models......Page 455
3 The Planar Magnet......Page 456
4 The Isotropic Antiferromagnet......Page 458
5 Isotropic Ferromagnets......Page 459
8.4 Hydrodynamics of Simple Fluids......Page 460
1 Conservation Laws......Page 461
2 Thermodynamics with Mass Motion......Page 463
3 The Entropy Production Equation......Page 464
4 Dissipationless Hydrodynamics......Page 465
5 Dissipation......Page 466
6 The Navier-Stokes Equations......Page 468
7 Hydrodynamic Modes......Page 469
8 Light Scattering......Page 472
9 Two-Component Fluids......Page 473
1 Nematic Liquid Crystals......Page 474
2 Smectic-A Liquid Crystals......Page 476
3 Crystalline Solids......Page 479
4 Superfluid Helium......Page 480
1 Critical Slowing Down and the Conventional Theory......Page 484
2 Dissipative Dynamics......Page 486
3 Dynamic Scaling......Page 489
4 Poisson Bracket Terms......Page 492
5 Models with Poisson Brackets......Page 495
6 Mode-Mode Coupling......Page 497
8.7 Nucleation and Spinodal Decomposition......Page 499
1 Nucleattion with a Nonconserved Order Parameter......Page 500
2 Symmetric Unstable Quench with Model A Dynamics......Page 503
3 Conserved Order Parameters and Spinodal Decomposition......Page 504
9.1 Characterization of Topological Defects......Page 515
2 Order Parameters with More than Two Components......Page 519
3 Order Parameters Spaces and Homotopy......Page 521
1 Vortices in XY-Models......Page 526
2 Dislocations in Smectic Liquid Crystals......Page 527
3 Periodic Solids......Page 532
5 Hexagonal and Close-Packed Lattices......Page 535
6 Disclinations in Crystals......Page 537
7 Strength of Crystals......Page 538
9 Grain Boundaries......Page 542
10 Nematic and Hexatic Liquid Crystals......Page 544
1 Simple Calculation of XY-Vortex Energies......Page 546
2 Analogy with Magnetism......Page 550
3 Energies of Dislocations in Crystals......Page 551
4 Dislocations in Smectic Liquid Crystals......Page 556
1 Vortices and the Spin-Wave Stiffness......Page 562
2 Vortex Unbinding in Two Dimsions: The Kosterlitz-Thouless Transition......Page 564
3 Superfluid Helium Films......Page 571
9.5 Dislocation Mediated Melting......Page 575
1 Effects of a Substrate......Page 578
2 Experiments and Numerical Simulation......Page 579
1 Structure of the TGB Phase......Page 581
2 The Thermodynamic Critical Field......Page 584
3 The Lower Critical Field......Page 585
4 The Upper Critical Field......Page 586
5 X-Ray Scattering......Page 588
6 Analogy with Superconductivity......Page 591
1 Integration of the KT Recursion Relations......Page 593
2 Longitudinal and Transverse Response......Page 595
3 The Spin Correlation Function......Page 597
Appendix 9B: Duality and the Villain Model......Page 598
1 Potts Models......Page 599
2 The XY-, Villain, and Lattice Coulomb-Gas Models......Page 602
10 Walls, Kinks and Solitons......Page 610
10.1 Some Simple Examples......Page 611
10.2 Domain Walls in Mean-Field Theory......Page 615
1 The Phi4 Kink......Page 617
3 Dynamics......Page 619
1 Introduction......Page 621
2 Discommensurations......Page 622
3 Devil's Staircases and the FK Phase Diagram......Page 623
4 The Continuum Approximation......Page 625
5 Nature of Solutions......Page 628
6 The Minimum Energy Solution......Page 630
8 X-Ray Diffraction......Page 633
9 Compressional Elastic Constants......Page 634
10 Phonons......Page 635
11 Pinned Phasons......Page 637
12 Extension to Two Dimensions......Page 638
1 Differential Geometry and the Total Surface Area......Page 640
2 Curvature......Page 643
3 Energy of a Surface......Page 645
4 Fluctuations in the Harmonic Approximation......Page 646
5 Nonlinearities and Renormalization in Fluid Membranes......Page 649
6 Polymerized Membranes......Page 650
1 Fluctuating Walls and Steric Entropy......Page 655
3 Elasticity of Sterically Stabilized Phases......Page 658
4 Dislocations and the CI transition......Page 660
1 The Solid-on-Solid and Discrete Gaussian Models......Page 663
2 The Roughening Transition......Page 666
3 Faceting......Page 668
Glossary......Page 682
Index......Page 705