Introduction to Phase Transitions and Critical Phenomena

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ISBN: 9780195053166 | 0195053168
Cover: Paperback
Copyright: 7/16/1987

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First published in 1971, this highly popular text is devoted to the interdisciplinary area of critical phenomena, with an emphasis on liquid-gas and ferromagnetic transitions. Advanced undergraduate and graduate students in thermodynamics, statistical mechanics, and solid state physics, as well as researchers in physics, mathematics, chemistry, and materials science, will welcome this paperback edition of Stanley's acclaimed text.

Notation Guide

PART I INTRODUCTION

What are the Critical Phenomena? A Survey of Some Basic Results

(21)

Classical era of critical phenomena

(8)

Modern era of critical phenomena

(9)

Phase transitions in other systems

(4)

Useful Thermodynamic Relations for Fluid and Magnetic Systems

(17)

The thermodynamic state functions U, E, G, and A

(1)

Differential relations for the state functions: the thermodynamic square

(2)

Two basic response functions: the specific heat and the compressibility

(3)

Stability and convexity relations

(2)

Geometrical interpretation of the Gibbs and Helmholtz potentials

(2)

Analogies between fluids and magnets

(1)

The thermodynamic state functions for a magnetic system

(1)

Differential relations and the thermodynamic square for magnetic state functions

(2)

magnetic response functions: specific heat and susceptibility

(1)

Convexity relations for magnetic systems

(1)

Geometrical interpretations of the thermodynamic potentials for a magnetic system

(2)

PART II CRITICAL-POINT EXPONENTS AND RIGOROUS RELATIONS AMONG THEM

Critical-Point Exponents

(28)

Definition of a critical-point exponent

(3)

The critical-point exponents α,β,γ,δ,ν and η

(4)

Numerical values of critical-point exponents

(3)

The exponents Δ and θ

(2)

Useful relations among critical-point exponents

(2)

Exponent Inequalities

The Rushbrooke and Coopersmith inequalities

(3)

The Griffiths inequality

(4)

More inequalities

(7)

PART III CLASSICAL THEORIES OF COOPERATIVE PHENOMENA

The van Der Waals Theory of Liquid-Gas Phase Transitions

(12)

Heuristic derivation of the van der Waals equation of state

(2)

van der Waals isotherms and the Maxwell construction

(2)

The van der Waals critical point

(1)

The law of corresponding states

(2)

Critical-point exponents for the van der Waals theory

(2)

The van der Waals equation of state as a mean field theory

(3)

The Mean Field Theory of Magnetic Phase Transitions

(15)

The non-interacting magnetic system

(2)

The assumption of an effective molecular field

(2)

Critical-point exponents for the mean field theory

(5)

The mean field theory as an approximation for the Heisenberg model

(2)

Equivalence of the mean field theory and an infinite interaction range

(3)

The Pair Correlation Function and the Ornstein-zernike Theory

(15)

The density---density correlation function for a fluid system

(2)

Relation between density fluctuations, the isothermal compressibility, and the density---density correlation function

(2)

The structure factor: relation between the pair correlation function and the scattering of electromagnetic radiation

(2)

Ornstein-Zernike theory of the scattering amplitude

100

(6)

Further developments of the Ornstein-Zernike theory

106

(3)

PART IV MODELS OF FLUID AND MAGNETIC PHASE TRANSITIONS

Results Provided by Exact Solution of Model Systems

109

(25)

A model Hamiltonian for a classical spin system: the generalized Heisenberg model

110

(5)

Exact solution of the case d = 1, D = 1, H = 0: The linear chain Ising model in the absence of an external magnetic field

115

(9)

The linear chain of arbitrary dimensional spins in zero magnetic field

124

(4)

The spherical model as the limit of infinite spin dimensionality

128

(3)

The transfer matrix method: application to the d = 1 Ising model in a magnetic field

131

(3)

Results Obtained from Model Systems by Approximation Methods

134

(33)

Successive approximation concept

135

(1)

Series expansion methods

136

(2)

Calculation of the coefficients in the high-temperature expansion of the partition function

138

(7)

Calculation of the coefficients in the high-temperature expansion of the two-spin correlation function

145

(3)

Physical interpretation of the terms in the high-temperature expansion of the correlation function

148

(4)

Extrapolation procedures for estimating the limiting behaviour of a power series from the behaviour of its first few terms

152

(6)

Pade approximants and transformation methods

158

(6)

Conclusions

164

(3)

PART V PHENOMENOLOGICAL THEORIES OF PHASE TRANSITIONS

Landau's Classic Theory of Exponents

167

(8)

Expansions about the critical point

167

(1)

Assumptions of the Landau theory

168

(2)

Critical-point predictions of the Landau theory

170

(2)

Critique of the Landau theory

172

(3)

Scaling Law Hypothesis for Thermodynamic Functions

175

(16)

Homogeneous functions of one or more variables

176

(5)

Static scaling hypothesis

181

(1)

Predicted relations among the critical-point exponents

182

(3)

Magnetic equation of state: scaled magnetization and scaled magnetic field

185

(6)

Scaling of the Static Correlation Functions

191

(11)

The Kadanoff construction

191

(3)

Application to the pair correlation function

194

(3)

Alternative methods of obtaining the correlation function scaling relations

197

(2)

Comparison with model calculations

199

(3)

PART VI DYNAMIC ASPECTS OF CRITICAL PHENOMENA

Introduction to Dynamic Critical Phenomena in Fluid Systems

202

(15)

Critical-point exponents for transport coefficients

203

(1)

Time-dependent correlation functions and the dynamic structure factor F(q, ω)

204

(1)

Relation between the dynamic structure factor and light-scattering experiments

205

(5)

Predictions of hydrodynamics for the spectrum of the scattered radiation

210

(4)

Predictions of hydrodynamics near the critical point

214

(3)

Measurements of the Dynamic Structure Factor for Fluid Systems

217

(16)

Time-dependent density fluctuations

217

(1)

Optical mixing spectroscopy

218

(1)

Measurements of the Rayleigh linewidth

219

(6)

Corrections to the hydrodynamic theory of the Rayleigh linewidth

225

(3)

Measurements of the Brillouin peak: velocity and attenuation of hypersonic sound waves

228

(5)

Dynamic Scaling Law and the Mode-Mode Coupling Approximation

233

(27)

Dynamic scaling hypothesis

233

(3)

Predictions of the restricted dynamic scaling hypothesis for fluid systems

236

(2)

Predictions of extended dynamic scaling for fluid systems

238

(2)

Evidence supporting the dynamic scaling hypothesis provided by magnetic systems

240

(6)

Spirit of the mode-mode coupling approach

246

(3)

Predictions of the mode-mode coupling approximation

249

(5)

Application of the mode-mode coupling approach to the interpretation of experimental results on fluids

254

(5)

Applications of the mode-mode coupling approach to other systems

259

(1)

Appendix A. The lattice-gas model of a fluid system

260

(5)

Appendix B. Exact solution of the zero-field Ising model for a two-dimensional lattice

265

(7)

Appendix C. Geometric interpretation of the static scaling hypothesis for thermodynamic potentials

272

(3)

Appendix D. The dynamic structure factor in the hydrodynamic limit

275

(5)

Appendix E. Model systems useful in the study of time-dependent cooperative phenomena: the Glauber model

280

(7)

Appendix F. Two-dimensional ferroelectric and antiferroelectric models

287

(3)

Bibliography

290

(11)

Author Index

301

(4)

Subject Index

305

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Introduction to Phase Transitions and Critical Phenomena

Summary

Table of Contents

Supplemental Materials