Principles of Condensed matter physics

Transcription

1 Principles of Condensed matter physics P. M. CHAIKIN Princeton University T. C. LUBENSKY University of Pennsylvania CAMBRIDGE UNIVERSITY PRESS

2 Preface xvii 1 Overview Condensed matter physics An example - H^O 3 1 Gaseous and liquid states 3 2 The liquid-gas phase transition 4 3 Spatial correlations in the liquid State 5 4 Ice - crystallized water 8 5 Broken symmetry and rigidity 10 6 Dislocations - topological defects 12 7 Universality of the water example 13 8 Fluctuations and spatial dimension 15 9 Overview of book Energies and potentials 17 1 Energy scales 17 2 Van der Waals attraction 18 3 Molecular hydrogen - the Heitler-London approach 20 4 Hard-sphere repulsion 22 5 Exchange interaction and magnetism 24 6 The hydrogen molecule, molecular orbitals, and bands in metals 25 Bibliography 28 References 28 2 Structure and scattering Elementary scattering theory - Bragg's law Photons, neutrons, or electrons The density Operator and its correlation functions Liquids and gases 38 1 Hard-sphere liquids Crystalline solids 43 1 Unit cells and the direct lattice 43 2 The reciprocal lattice 45 Vll

3 viii 3 Periodic functions 46 4 Bragg scattering Symmetry and crystal structure 49 1 Two-dimensional Bravais lattices 50 2 Three-dimensional Bravais lattices 53 3 Close packed structures 56 4 Space groups Liquid crystals 58 1 Isotropie, nematic and cholesteric phases 58 2 Smectics-A and -C 61 3 Hexatic phases 65 4 Discotic phases 68 5 Lyotropic liquid crystals and microemulsions One- and two-dimensional order in three-dimensional materials Incommensurate structures Quasicrystals Magnetic order Random isotropic fractals 90 Appendix 2A Fourier transforms 97 1 One dimension 97 2 d dimensions 99 3 Transforms on a lattice 100 Bibliography 101 References 102 Problems Thermodynamics and Statistical mechanics Thermodynamics of homogeneous fluids The first law of thermodynamics The second law of thermodynamics The third law of thermodynamics Thermodynamic potentials Stability criteria Homogeneous functions Equations of State Statistical mechanics: phase space and ensembles The ideal gas Spatial correlations in classical Systems Ordered Systems Symmetry, order parameters, and modeis Discrete symmetries Continuous symmetries 137

4 ix 3 Models 139 Appendix 3A Functional derivatives 140 Bibliography 142 References 142 Problems Mean-field theory Bragg-Williams theory Landau theory The Ising and n-vector modeis The nonlocal susceptibility and the correlation length O n symmetry Some mean-field transitions The liquid-gas transition The critical point and the critical isochore The coexistence curve The first-order nematic-to-isotropic transition Multicritical points Tricritical points Metamagnets and FeCl He 3 He 4 mixtures and the Blume-Emery-Griffiths model Bicritical and tetracritical points Lifshitz points The liquid-solid transition Are all crystals BCC? Criterion for freezing Improvements of the theory Changes in density Density functional theory Variational mean-field theory Two inequalities The mean-field approximation The s-state Potts model The 0 classical Heisenberg model Debye-Hückel theory 204 Bibliography 208 References 209 Problems Field theories, critical phenomena, and the renormalization group Breakdown of mean-field theory 214

5 1 Mean-field transitions revisited Construction of a field theory Coarse graining Lattice field theories and their continuum limit Gaussian integrals Mean-field theory from functional integrals Breakdown of mean-field theory revisited The self-consistent field approximation The n-vector model in the limit n > oo Critical exponents, universality, and scaling Exponents and scaling relations Scaled equation of state Multicritical points Amplitude ratios Theoretical calculations of critical exponents and amplitude ratios The Kadanoff construction The one-dimensional Ising model Exact Solution Decimation and renormalization The Migdal-Kadanoff procedure The Ising model on a hypercubic lattice General properties of recursion relations The Potts lattice gas and krypton on graphite Momentum shell renormalization group Thinning of degrees of freedom and rescaling Correlation functions The Gaussian model The e-expansion w-vector model with cubic anisotropy Quadratic anisotropy Crossover Dangerous irrelevant variables The Utility of the e-expansion 275 Appendix 5A The Hubbard-Stratonovich transformation 276 Appendix 5B Diagrammatic perturbation theory 277 Bibliography 283 References 283 Problems Generalized elasticity The xy-mom The elastic free energy 289

6 XI 2 Boundary conditions and external fields The Josephson scaling relation Fluctuations Long-range order, quasi-long-range order, and disorder Resistance of a conducting medium symmetry and nematic liquid crystals n-vector elastic energy The Frank free energy of nematic liquid crystals Cells with non-uniform n The Freedericksz transition The twisted nematic display Fluctuations and light scattering Smectic liquid crystals The elastic free energy Fluctuations Nonlinearities The nematic-to-smectic-^4 transition Elasticity of solids: strain and elastic energy The strain tensor The elastic free energy Isotropie and eubie solids Fluctuations Mercury chain salts - one-dimensional crystals Xenon on graphite - a two-dimensional crystal Vacancies and interstitials Bond-angle order and rotational and translational elasticity Elastic constants from density functional theory Lagrangian elasticity Classical theory of elasticity Elasticity of classical harmonic lattices Elasticity of solids: the stress tensor The Lagrangian stress tensor Stress-strain relations The Eulerian stress tensor The nonlinear sigma model 341 Bibliography 347 References 347 Problems Dynamics: correlation and response Dynamic correlation and response funetions Correlation funetions 354

7 2 Response functions 7.2 The harmonic oscillator 1 The undamped oscillator 2 The damped oscillator 3 The response function 4 Dissipation 7.3 Elastic waves and phonons 1 Sound waves in an elastic continuum 2 Acoustic phonons in a harmonic lattice 7.4 Diffusion 1 Fick's law 2 The Green function and dynamic response 3 The response function 4 External potentials and the Einstein relation 5 Brownian motion 6 Cooperative diffusion versus self-diffusion 7 Master equation for diffusion on a lattice 7.5 Langevin theory 1 Random forces and thermal equilibrium 2 Correlation functions for diffusion 3 Short-time behavior 4 Fluctuation-dissipation theorem for the harmonic oscillator The Fokker-Planck and Smoluchowski equations Formal properties of response functions Response to external fields Symmetry properties of response functions Dissipation Spectral representations of x'., The fluctuation-dissipation theorem Sum rules and moment expansions Inelastic scattering Scattering geometry and partial cross-sections Fermi golden rule and neutron scattering The Fermi pseudopotential Coherent and incoherent scattering Cross-sections and correlation functions Neutron scattering from crystals Magnetic scattering How neutron scattering experiments are actually done Scattering of charged particles and photons 410 Bibliography 411 References 411

8 xni Problems Hydrodynamics All 8.1 Conserved and broken-symmetry variables A tutorial example - rigid rotors on a lattice Description of the model The disordered phase The ordered phase Excitations from the classical ground State The Goldstone theorem Kubo formulae Summary Spin Systems Spin dynamics Generalized Heisenberg modeis The planar magnet The isotropic antiferromagnet Isotropic ferromagnets Hydrodynamics of simple fluids Conservation laws Thermodynamics with mass motion The entropy production equation Dissipationless hydrodynamics Dissipation The Navier-Stokes equations Hydrodynamic modes Light scattering Two-component fluids Liquid crystals, crystalline solids, and superfluid helium Nematic liquid crystals Smectic-^4 liquid crystals Crystalline solids Superfluid helium Stochastic modeis and dynamic critical phenomena Critical slowing down and the conventional theory Dissipative dynamics Dynamic scaling Poisson bracket terms Models with Poisson brackets Mode-mode coupling Nucleation and spinodal decomposition Nucleation with a nonconserved order parameter Symmetrie unstable queneh with model A dynamics 483

9 xiv 3 Conserved order parameters and spinodal decomposition Bibliography References Problems Topological defects Characterization of topological defects 1 Vortex pairs 2 Order parameters with more than two components 3 Order parameter Spaces and homotopy Examples of topological defects 1 Vortices in xy-models 2 Dislocations in smectic liquid crystals 3 Periodic solids 4 Volterra construction 5 Hexagonal and close-packed lattices 6 Disclinations in crystals 7 Strength of crystals 8 Crystal growth 9 Grain boundaries 10 Nematic and hexatic liquid crystals Energies of vortices and dislocations 1 Simple calculation of xy-vortex energies 2 Analogy with magnetism 3 Energies of dislocations in crystals 4 Dislocations in smectic liquid crystals Vortex unbinding and the Kosterlitz-Thouless transition 1 Vortices and the spin-wave stiffness 2 Vortex unbinding in two dimensions - the Kosterlitz- Thouless transition 3 Superfluid helium films Dislocation mediated melting 1 Effects of a Substrate 2 Experiments and numerical Simulation The twist-grain-boundary phase 1 Structure of the TGB phase 2 The thermodynamic critical field 3 The lower critical field 4 The upper critical field 5 X-ray scattering 6 Analogy with superconductivity Appi endix 9A Notes on the Kosterlitz-Thouless transition

10 xv 1 Integration of the KT recursion relations Longitudinal and transverse response The spin correlation function 577 Appendix 9B Duality and the Villain model Potts modeis The xy-, Villain, and lattice Coulomb-gas modeis 582 Bibliography 584 References 584 Problems Walls, kinks and solitons Some simple examples Domain walls in mean-field theory The< 4 kink The sine-gordon soliton Dynamics The Frenkel-Kontorowa model Introduction Discommensurations Devil's staircases and the FK phase diagram The continuum approximation Nature of Solutions The minimum energy Solution Repulsive interaction between discommensurations X-ray diffraction Compressional elastic constants 614 lophasons Pinned phasons Extension to two dimensions Fluctuating walls Differential geometry and the total surface area Curvature Energy of a surface Fluctuations in the harmonic approximation Nonlinearities and renormalization in fluid membranes Polymerized membranes Arrays of fluctuating walls Fluctuating walls and steric entropy Honeycomb lattice of walls Elasticity of sterically stabilized phases Dislocations and the CI transition Roughening and faceting The solid-on-solid and discrete Gaussian modeis 643

11 xvi 2 The roughening transition Faceting 648 Bibliography 655 References 656 Problems 656 Glossary 662 Index 685