Principles of Condensed Matter Physics

by P. M. Chaikin , T. C. Lubensky

• ISBN13: 9780521432245
• ISBN10: 0521432243
• Format: Hardcover
• Copyright: 1995-06-30
• Publisher: Cambridge University Press

Summary

Now in paperback, this book provides an overview of the physics of condensed matter systems. Assuming a familiarity with the basics of quantum mechanics and statistical mechanics, the book establishes a general framework for describing condensed phases of matter based on symmetries and conservation laws. After surveying the structure and properties of materials with different symmetries, it explores the role of spatial dimensionality and microscopic interactions in determining the nature of phase transitions. Particular attention is given to critical phenomena and renormalization group methods. The properties of liquids, liquid crystals, quasicrystals, crystalline solids, magnetically ordered systems and amorphous solids are investigated in terms of their symmetry, generalized rigidity, hydrodynamics and topological defect structure. In addition to serving as a course text, this book is an essential reference for students and researchers in physics, applied physics, chemistry, materials science and engineering, who are interested in modern condensed matter physics.

Table of Contents

Preface	xvii
Overview	1	(28)
Condensed matter physics	1	(2)
An example - H2O	3	(14)
Gaseous and liquid states	3	(1)
The liquid-gas phase transition	4	(1)
Spatial correlations in the liquid state	5	(3)
Ice - crystallized water	8	(2)
Broken symmetry and rigidity	10	(2)
Dislocations - topological defects	12	(1)
Universality of the water example	13	(2)
Fluctuations and spatial dimension	15	(1)
Overview of book	16	(1)
Energies and potentials	17	(12)
Energy scales	17	(1)
Van der Waals attraction	18	(2)
Molecular hydrogen - the Heitler-London approach	20	(2)
Hard-sphere repulsion	22	(2)
Exchange interaction and magnetism	24	(1)
The hydrogen molecule, molecular orbitals, and bands in metals	25	(3)
Bibliography	28	(1)
References	28	(1)
Structure and scattering	29	(79)
Elementary scattering theory - Bragg's law	29	(4)
Photons, neutrons, or electrons	33	(1)
The density operator and its correlation functions	34	(4)
Liquids and gases	38	(5)
hard-sphere liquids	40	(3)
Crystalline solids	43	(6)
Unit cells and the direct lattice	43	(2)
The reciprocal lattice	45	(1)
Periodic functions	46	(1)
Bragg scattering	47	(2)
Symmetry and crystal structure	49	(9)
Two-dimensional Bravais lattices	50	(3)
Three-dimensional Bravais lattices	53	(3)
Close packed structures	56	(1)
Space groups	57	(1)
Liquid crystals	58	(13)
Isotropic, nematic and cholesteric phases	58	(3)
Smectics-A and -C	61	(4)
Hexatic phases	65	(3)
Discotic phases	68	(1)
Lyotropic liquid crystals and microemulsions	68	(3)
One-and two-dimensional order in three-dimensional materials	71	(6)
Incommensurate structures	77	(5)
Quasicrystals	82	(3)
Magnetic order	85	(5)
Random isotropic fractals	90	(18)
Appendix 2A Fourier transforms	97	(1)
One dimension	97	(2)
dimensions	99	(1)
Transforms on a lattice	100	(1)
Bibliography	101	(1)
References	102	(1)
Problems	103	(5)
Thermodynamics and statistical mechanics	108	(36)
Thermodynamics of homogeneous fluids	108	(9)
The first law of thermodynamics	109	(2)
The second law of thermodynamics	111	(1)
The third law of thermodynamics	111	(1)
Thermodynamic potentials	112	(1)
Stability criteria	113	(2)
Homogeneous functions	115	(1)
Equations of state	116	(1)
Statistical mechanics: phase space and ensembles	117	(5)
The ideal gas	122	(1)
Spatial correlations in classical systems	123	(4)
Ordered systems	127	(5)
Symmetry, order parameters, and models	132	(12)
Discrete symmetries	135	(2)
Continuous symmetries	137	(2)
Models	139	(1)
Appendix 3A Functional derivatives	140	(2)
Bibliography	142	(1)
References	142	(1)
Problems	142	(2)
Mean-field theory	144	(69)
Bragg-Williams theory	146	(5)
Landau theory	151	(1)
The Ising and n-vector models	152	(7)
The nonlocal susceptibility and the correlation length	154	(2)
On symmetry	156	(1)
Some mean-field transitions	157	(2)
The liquid-gas transition	159	(9)
The critical point and the critical isochore	162	(3)
The coexistence curve	165	(3)
The first-order nematic-to-isotropic transition	168	(4)
Multicritical points	172	(16)
Tricritical points	173	(2)
Metamagnets and FeCl2	175	(4)
He3 - He4 mixtures and the Blume-Emery-Griffiths model	179	(2)
Bicritical and tetracritical points	181	(3)
Lifshitz points	184	(4)
The liquid-solid transition	188	(10)
Are all crystals BCC?	189	(3)
Criterion for freezing	192	(1)
Improvements of the theory	192	(2)
Changes in density	194	(1)
Density functional theory	195	(3)
Variational mean-field theory	198	(15)
Two inequalities	198	(2)
The mean-field approximation	200	(1)
The s-state Potts model	201	(1)
The On classical Heisenberg model	202	(2)
Debye-Huckel theory	204	(4)
Bibliography	208	(1)
References	209	(1)
Problems	209	(4)
Field theories, critical phenomena, and the renormalization group	213	(75)
Breakdown of mean-field theory	214	(3)
Mean-field transitions revisited	216	(1)
Construction of a field theory	217	(9)
Coarse graining	217	(2)
Lattice field theories and their continuum limit	219	(2)
Gaussian integrals	221	(2)
Mean-field theory from functional integrals	223	(2)
Breakdown of mean-field theory revisited	225	(1)
The self-consistent field approximation	226	(4)
The n-vector model in the limit n → ∞	229	(1)
Critical exponents, universality, and scaling	230	(7)
Exponents and scaling relations	230	(4)
Scaled equation of state	234	(1)
Multicritical points	235	(1)
Amplitude ratios	236	(1)
Theoretical calculations of critical exponents and amplitude ratios	237	(1)
The Kadanoff construction	237	(5)
The one-dimensional Ising model	242	(6)
Exact solution	242	(3)
Decimation and renormalization	245	(3)
The Migdal-Kadanoff procedure	248	(8)
The Ising model on a hypercubic lattice	248	(4)
General properties of recursion relations	252	(1)
The Potts lattice gas and krypton on graphite	253	(3)
Momentum shell renormalization group	256	(32)
Thinning of degrees of freedom and rescaling	256	(4)
Correlation functions	260	(1)
The Gaussian model	261	(2)
The &epsis;-expansion	263	(4)
n-vector model with cubic anisotropy	267	(2)
Quadratic anisotropy	269	(1)
Crossover	270	(3)
Dangerous irrelevant variables	273	(2)
The utility of the &epsis;-expansion	275	(1)
Appendix 5A The Hubbard-Stratonovich transformation	276	(1)
Appendix 5B Diagrammatic perturbation theory	277	(6)
Bibliography	283	(1)
References	283	(1)
Problems	283	(5)
Generalized elasticity	288	(65)
The xy-model	289	(9)
The elastic free energy	289	(1)
Boundary conditions and external fields	290	(2)
The Josephson scaling relation	292	(1)
Fluctuations	293	(2)
Long-range order, quasi-long-range order, and disorder	295	(2)
Resistance of a conducting medium	297	(1)
On symmetry and nematic liquid crystals	298	(10)
n-vector elastic energy	298	(1)
The Frank free energy of nematic liquid crystals	298	(2)
Cells with non-uniform n	300	(2)
The Freedericksz transition	302	(2)
The twisted nematic display	304	(2)
Fluctuations and light scattering	306	(2)
Smectic liquid crystals	308	(8)
The elastic free energy	309	(3)
Fluctuations	312	(2)
Nonlinearities	314	(1)
The nematic-to-smectic-A transition	315	(1)
Elasticity of solids: strain and elastic energy	316	(14)
The strain tensor	316	(2)
The elastic free energy	318	(1)
Isotropic and cubic solids	319	(2)
Fluctuations	321	(1)
Mercury chain salts - one-dimensional crystals	322	(2)
Xenon on graphite - a two-dimensional crystal	324	(1)
Vacancies and interstitials	325	(3)
Bond-angle order and rotational and translational elasticity	328	(1)
Elastic constants from density functional theory	329	(1)
Lagrangian elasticity	330	(4)
Classical theory of elasticity	330	(2)
Elasticity of classical harmonic lattices	332	(2)
Elasticity of solids: the stress tensor	334	(7)
The Lagrangian stress tensor	334	(3)
Stress-strain relations	337	(1)
The Eulerian stress tensor	338	(3)
The nonlinear sigma model	341	(12)
Bibliography	347	(1)
References	347	(1)
Problems	347	(6)
Dynamics: correlation and response	353	(64)
Dynamic correlation and response functions	354	(5)
Correlation functions	354	(1)
Response functions	355	(4)
The harmonic oscillator	359	(7)
The undamped oscillator	359	(1)
The damped oscillator	360	(2)
The response function	362	(3)
Dissipation	365	(1)
Elastic waves and phonons	366	(3)
Sound waves in an elastic continuum	366	(1)
Acoustic phonons in a harmonic lattice	367	(2)
Diffusion	369	(12)
Fick's law	369	(1)
The Green function and dynamic response	370	(1)
The response function	371	(2)
External potentials and the Einstein relation	373	(2)
Brownian motion	375	(1)
Cooperative diffusion versus self-diffusion	376	(2)
Master equation for diffusion on a lattice	378	(3)
Langevin theory	381	(9)
Random forces and thermal equilibrium	381	(2)
Correlation functions for diffusion	383	(2)
Short-time behavior	385	(2)
Fluctuation-dissipation theorem for the harmonic oscillator	387	(1)
The Fokker-Planck and Smoluchowski equations	388	(2)
Formal properties of response functions	390	(9)
Response to external fields	390	(2)
Symmetry properties of response functions	392	(2)
Dissipation	394	(1)
Spectral representations of χ'' ϕ	395	(2)
The fluctuation-dissipation theorem	397	(1)
Sum rules and moment expansions	398	(1)
Inelastic scattering	399	(18)
Scattering geometry and partial cross-sections	399	(1)
Fermi golden rule and neutron scattering	400	(2)
The Fermi pseudopotential	402	(2)
Coherent and incoherent scattering	404	(1)
Cross-sections and correlation functions	405	(1)
Neutron scattering from crystals	406	(1)
Magnetic scattering	407	(1)
How neutron scattering experiments are actually done	408	(2)
Scattering of charged particles and photons	410	(1)
Bibliography	411	(1)
References	411	(6)
Hydrodynamics	417	(78)
Conserved and broken-symmetry variables	417	(2)
A tutorial example - rigid rotors on a lattice	419	(15)
Description of the model	420	(1)
The disordered phase	421	(5)
The ordered phase	426	(4)
Excitations from the classical ground state	430	(2)
The Goldstone therem	432	(1)
Kubo formulae	432	(1)
Summary	433	(1)
Spin systems	434	(6)
Spin dynamics	434	(1)
Generalized Heisenberg models	435	(1)
The planar magnet	436	(2)
The isotropic antiferromagnet	438	(1)
Isotropic ferromagnets	439	(1)
Hydrodynamics of simple fluids	440	(14)
Conservation laws	441	(2)
Thermodynamics with mass motion	443	(1)
The entropy production equation	444	(1)
Dissipationless hydrodynamics	445	(1)
Dissipation	446	(2)
The Navier-Stokes equations	448	(1)
Hydrodynamic modes	449	(3)
Light scattering	452	(1)
Two-component fluids	453	(1)
Liquid crystals, crystalline solids, and superfluid helium	454	(10)
Nematic liquid crystals	454	(2)
Smectic-A liquid crystals	456	(3)
Crystalline solids	459	(1)
Superfluid helium	460	(4)
Stochastic models and dynamic critical phenomena	464	(15)
Critical slowing down and the conventional theory	464	(2)
Dissipative dynamics	466	(3)
Dynamic scaling	469	(3)
Poisson bracket terms	472	(3)
Models with Poisson brackets	475	(2)
Mode-mode coupling	477	(2)
Nucleation and spinodal decomposition	479	(16)
Nucleation with a nonconserved order parameter	480	(3)
Symmetric unstable quench with model A dynamics	483	(1)
Conserved order parameters and spinodal decomposition	484	(7)
Bibliography	491	(1)
References	491	(1)
Problems	492	(3)
Topological defects	495	(95)
Characterization of topological defects	495	(11)
Vortex pairs	499	(1)
Order parameters with more than two components	499	(2)
Order parameter spaces and homotopy	501	(5)
Examples of topological defects	506	(20)
Vortices in xy-models	506	(1)
Dislocations in smectic liquid crystals	507	(5)
Periodic solids	512	(3)
Volterra construction	515	(1)
Hexagonal and close-packed lattices	515	(2)
Disclinations in crystals	517	(1)
Strength of crystals	518	(4)
Crystal growth	522	(1)
Grain boundaries	522	(2)
Nematic and hexatic liquid crystals	524	(2)
Energies of vortices and dislocations	526	(16)
Simple calculation of xy-vortex energies	526	(4)
Analogy with magnetism	530	(1)
Energies of dislocations in crystals	531	(5)
Dislocations in smectic liquid crystals	536	(6)
Vortex unbinding and the Kosterlitz-Thouless transition	542	(13)
Vortices and the spin-wave stiffness	542	(2)
Vortex unbinding in two dimensions - the Kosterlitz-Thouless transition	544	(7)
Superfluid helium films	551	(4)
Dislocation mediated melting	555	(6)
Effects of a substrate	558	(1)
Experiments and numerical simulation	559	(2)
The twist-grain-boundary phase	561	(29)
Structure of the TGB phase	561	(3)
The thermodynamic critical field	564	(1)
The lower critical field	565	(1)
The upper critical field	566	(2)
X-ray scattering	568	(3)
Analogy with superconductivity	571	(2)
Appendix 9A Notes on the Kosterlitz-Thouless transition	573	(1)
Integration of the KT recursion relations	573	(2)
Longitudinal and transverse response	575	(2)
The spin correlation function	577	(1)
Appendix 9B Duality and the Villain model	578	(1)
Potts models	579	(3)
The xy-, Villain, and lattice Coulomb-gas models	582	(2)
Bibliography	584	(1)
References	584	(1)
Problems	585	(5)
Walls, kinks and solitons	590	(72)
Some simple examples	591	(4)
Domain walls in mean-field theory	595	(6)
The ϕ4 kink	597	(2)
The sine-Gordon soliton	599	(1)
Dynamics	599	(2)
The Frenkel-Kontorowa model	601	(19)
Introduction	601	(1)
Discommensurations	602	(1)
Devil's staircases and the FK phase diagram	603	(2)
The continuum approximation	605	(3)
Nature of solutions	608	(2)
The minimum energy solution	610	(3)
Repulsive interaction between discommensurations	613	(1)
X-ray diffraction	613	(1)
Compressional elastic constants	614	(1)
Phasons	615	(2)
Pinned phasons	617	(1)
Extension to two dimensions	618	(2)
Fluctuating walls	620	(15)
Differential geometry and the total surface area	620	(3)
Curvature	623	(2)
Energy of a surface	625	(1)
Fluctuations in the harmonic approximation	626	(3)
Nonlinearities and renormalization in fluid membranes	629	(1)
Polymerized membranes	630	(5)
Arrays of fluctuating walls	635	(8)
Fluctuating walls and steric entropy	635	(3)
Honeycomb lattice of walls	638	(1)
Elasticity of sterically stabilized phases	638	(2)
Dislocations and the CI transition	640	(3)
Roughening and faceting	643	(19)
The solid-on-solid and discrete Gaussian models	643	(3)
The roughening transition	646	(2)
Faceting	648	(7)
Bibliography	655	(1)
References	656	(1)
Problems	656	(6)
Glossary	662	(23)
Index	685