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The Landau Theory of Phase Transitions Downloaded from www.worldscientific.com by FREIE UNIVERSITAET BERLIN MATHEMATIK / INFORMATIK on 06/29/15. For personal use only. THE LANDAU THEORY OF PHASE TRANSITIONS The Landau Theory of Phase Transitions Downloaded from www.worldscientific.com by FREIE UNIVERSITAET BERLIN MATHEMATIK / INFORMATIK on 06/29/15. For personal use only. This page is intentionally left blank The Landau Theory of Phase Transitions Downloaded from www.worldscientific.com by FREIE UNIVERSITAET BERLIN MATHEMATIK / INFORMATIK on 06/29/15. For personal use only. THE LANDAU THEORY OF PHASE TRANSITIONS Application to Structural, Incommensurate, Magnetic, and Liquid Crystal Systems. Jean-Claude TOLEDANO Centre National d'Etudes des Telecommunications FRANCE Pierre TOLEDANO University of Amiens FRANCE V£> World Scientific wfc Singapore • New Jersey • Hong Kong Published by The Landau Theory of Phase Transitions Downloaded from www.worldscientific.com by FREIE UNIVERSITAET BERLIN MATHEMATIK / INFORMATIK on 06/29/15. For personal use only. World Scientific Publishing Co. Pte. Ltd. P.O. Box 128, Farrer Road, Singapore 9128 U.S.A. office: World Scientific Publishing Co., Inc. 687 Hartwell Street, Teaneck NJ 07666, USA Library of Congress Cataloging-in-Publication data is available. THE LANDAU THEORY OF PHASE TRANSITIONS Copyright © 1987 by World Scientific Publishing Co Pte Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. ISBN 9971-50-025-6 9971-50-026-4 (pbk) Printed in Singapore by Kim Hup Lee Printing Co. Pte. Ltd. The Landau Theory of Phase Transitions Downloaded from www.worldscientific.com by FREIE UNIVERSITAET BERLIN MATHEMATIK / INFORMATIK on 06/29/15. For personal use only. A NOS PARENTS The Landau Theory of Phase Transitions Downl; oaded from www.worldscientific.com by FREIE UNIVERSITAET BERLIN MATHEMATIK / INFORMATIK on 06/29/15. For personal use only. This page is intentionally left blank The Landau Theory of Phase Transitions Downloaded from www.worldscientific.com by FREIE UNIVERSITAET BERLIN MATHEMATIK / INFORMATIK on 06/29/15. For personal use only. FOREWORD Why a book on Landau's theory of phase transitions ? To many physicists working in the field of phase transitions, this question will appear as doubly relevant. Indeed, why describe in detail the foundations and consequences of a theory whose basic hypothesis (the absence of singularity in the transition free-energy), and whose essential physical result (the specification of a critical behaviour) have been known for 40 years to be questionable ? Why, on the other hand, restrict to this phenomenological approach, at a time when microscopic models can be handled by various theoretical and numerical methods, and provide a "royal way" to the investigation of phase transitions ? The existence of a satisfactory answer to these questions is attested by the fact that, ever since its formulation, Landau's theory has been used without interruption as the theoretical background of many studies of systems undergoing phase transitions. More strikingly, it is in the last 15 years, after the advent of the modern statistical theory of critical phenomena, that the utility of Landau's theory has been demonstrated most clearly, when it has been applied to the intricate patterns of transitions observed in the structural, magnetic, and liquidcrystalline systems, and more recently, to the investigation of the stabilities and properties of the incommensurate phases, and of the icosahedral quasi-crystalline phases. There are several reasons to this persistent use of Landau's theory. A first set of reasons is related to the fact that the objected lack of validity of the basic assumptions and results of Landau's theory is not of a clearcut nature. Thus, the symmetry aspects which constitute an Important part of the theory, are rigorous. Besides, there are classes of systems, governed by long range interactions (e.g. elastic interactions) for which the critical behaviour is expected to be correctly described by Landau's theory. More significantly, the temperature range adjacent to the transitions, in which the behaviour of a system is dominated by the fluctuations (and in which, accordingly Landau's theory fails), usually constitutes a small fraction of the temperature range of experimental Interest. In most of the latter range, the Landau theory is an adequate tool to Investigate the physical behaviour of the system. The Landau Theory of Phase Transitions Downloaded from www.worldscientific.com by FREIE UNIVERSITAET BERLIN MATHEMATIK / INFORMATIK on 06/29/15. For personal use only. VIII Another important set of reasons pertains to the possibility of manipulating, through a mathematically simple and flexible theory, the complicated degrees of freedom which describe the states of real systems (e.g. sets of collective atomic displacements, or intricate spin configurations). Likewise, one has the possibility of relating simply to eachother a variety of physical properties (mechanical, optical, lattice-dynamical, structural,...) whose microscopic description would be very difficult. The mathematical simplicity of the theory is a consequence of the clever manner by which Landau defines the order-parameter through the substitution of a small set of scalar (spatially uniform) quantities, to a set of functions having rapid variations at the atomic level. This substitution which appears as a "trick" of mere mathematical convenience, has the important consequence of permitting the description of the evolution of a complex spatial configuration of particles by means of an ordinary polynomial expansion. This trick also gives to the order-parameter a duality of meanings. As a spatially uniform quantity (or a smoothly varying one, in certain cases) it can be considered as of macroscopic nature. On the other hand, the functions it substitutes are clearly of microscopic nature. At choice, one can put the accent on one interpretation or the other (e.g. on the dielectric polarization of a ferroelectric crystal, or on the structural changes and lattice dynamical mode related to the polarization). The flexibility of the theory resides in its modular character. Aside from the primary order-parameter, additional degrees of freedom can be incorporated in the theory as measure as the acquisition of the experimental data requires Interpreting a larger set of results. For instance, in the study of a crystalline transition, once the primary order-parameter is given a sense in terms of atomic displacements, one can focus successively on the anomalies induced by the considered transition in the thermal expansion, the optical properties, the vibrational atomic spectrum, etc... In this view one will add terms in the Landau free-energy respectively corresponding to the mechanical deformations, to the dielectric polarization, to other collective atomic displacements, etc... The only rigid feature of the theory is its symmetry framework which imposes the form of the interactions between the various degrees of freedom, and the number of adjustable phenomenological coefficients. The form of the interactions determines the relationship between the laws governing the behaviour of the various physical quantities. The explanatory power of Landau's theory resides in the checking of the overall consistency of the observed laws. Finally, an additional reason of consideration of Landau's theory is its specific status in respect to the statistical theory of critical phenomena (Wilson's theory). From this standpoint, Landau's theory appears as a necessary point of passage, and also as a tool. On the one hand, it is the order-parameter defined by Landau's theory whose fluctuations give rise to a singularity at the transition point. Accordingly, it is on this set of degrees of freedom that the statistical theory operates. On the other hand, Landau's theory provides the rules for constructing the effective Hamiltonian density, function of the orderparameter and of the secondary degrees of freedom, on which the renormalization-group transformations act. The Landau Theory of Phase Transitions Downloaded from www.worldscientific.com by FREIE UNIVERSITAET BERLIN MATHEMATIK / INFORMATIK on 06/29/15. For personal use only. IX The contents of this book stems from three different objectives. In the first place, it is an introduction to the basic principles and techniques of Landau's theory, which is intended for teaching purposes. In this spirit, it includes an introduction to the peripheric group-theoretical and crystallographic concepts required to work out the theory. This part of the book is an expanded version of courses taught by the authors in various circumstances. Chapter I is a self-contained, simplified, introduction to the basic aspects of Landau's theory, which is well adapted to a teaching at the undergraduate level. The first paragraphs of chapter II constitute a complete presentation of the theory. They involve a thorough discussion of the starting assumptions and an explicit decomposition of the steps of the argumentation. The same pedagogical purpose has presided over the writing of chapter III, of the two first paragraphs of chapter IV, and of the four first paragraphs of chapters VI and VII. These chapters are respectively devoted to the applications of Landau's theory to structural transitions, to first-order transitions, and to magnetic and liquid-crystalline systems. Their contents is rooted in courses given at the graduate level. A second purpose of the book is to provide the practical "recipes" for applying Landau's theory to complex systems. In this view, each element of the method is illustrated by several examples and the intermediate steps of many calculations are explicitely reproduced. Thus, one can find the constructions of the matrices of the order-parameter representation, or corepresentation (chaps. II, III, V and V I ) , the construction of the Landau free-energy (chaps. II and V ) , the description of the procedures of its minimization, and the method of identification of the low symmetry group for structural (chap. Ill) magnetic (chap. VI) and liquid crystal (chap. VI) systems. The procedure of application of the Landau criterion (chap. II) and of the Llfschltz one (chap. Ill) are exposed in details. For incommensurate systems (chap. V ) , an extensive description is given of the construction of the Lifschitz-invariant and other spatially dispersive terms, as well as the resolution of the equations relative to the standard situation of a two-component orderparameter. In the chapter devoted to liquid crystals, we have adopted a unified description of the various types of transitions occuring in these systems while existing theories often derive from a variety of approaches (chap. V I I ) . The last objective of the book is to Incorporate the developments which have arisen in the last 15 years from the extensive application of the theory to a variety of physical systems. These developments involve several aspects. On the one hand, certain bases of the theory itself have been discussed. The meaning of the Llfschltz criterion has been analyzed by a number of authors and substantially clarified through the study of incommensurate systems. Its initially derivation by Llfschltz (chap. V ) has been replaced by other derivations, physically more transparent, and mathematically more correct, though more complex (chap. III). Conversely, it has been understood that Lifschitz's derivation provided a method for the study of incommensurate systems (chap. V ) . The second aspect of progress concerns the specification of the essential symmetries underlying the Landau theory, through the replacement of groups acting in the physical space, by groups acting in the order The Landau Theory of Phase Transitions Downloaded from www.worldscientific.com by FREIE UNIVERSITAET BERLIN MATHEMATIK / INFORMATIK on 06/29/15. For personal use only. IV parameter space. One has been able, by this means, to express In a more efficient way the Intrinsic symmetry of the order-parameter, the symmetry of the truncated free-energy expansion, and the characteristics of the symmetry breaking across the transition. New procedures of minimization of the free-energy have been based on the consideration of these essential symmetries. These methods have clarified the nature of the mathematical problem set by the Landau theory : find the absolute minimum of a m t n degree polynomial in several variables, 1) having a local maximum at the origin, 11) positive and infinite at infinity in any direction, ill) whose extrema have a symmetry-specifled degeneracy, and liii) which possesses obligatory extrema along symmetry directions in the order-parameter space (chap. II, paragraph 4). Finally, a large part of the book is devoted to the systems, already mentioned above, which have been studied, in recent years, by means of Landau's theory : continuous or discontinuous structural transitions involving coupling between several relevant degrees of freedom (chaps III and IV) magnetic transitions (chap. VI), transitions in liquidcrystals (chap. VII), incommensurate phases (chap. V ) . We have also outline the principles of the application of the theory to icosahedral phases and to defects (chap. VIII). In certain chapters, we had the choice between various distinct approaches. It is worth pointing out that the theory of first-order transitions is essentially inspired by the works of Gufan and co-workers (chap. IV). The theories of magnetic and liquid crystal systems have respectively their roots in methods elaborated by Dzyaloshinski and by Indenbom and coworkers (chaps VI and VII). The multiplicity of objectives pursued has the consequence that the different chapters or paragraphs are not treated evenly. In certain parts of the book, each statement is justified, while in others, dealing with more recently developped fields, the reader is directed, for complete justification to appropriate reference works. The latter situation will be found, in particular, in large fractions of chapters IV-VIII. In writing this book, we feel indebted to a number of colleagues. We are especially greatful to Louis Michel. From discussions with him we have learned most of the considerations pertaining to the essential symmetries of the Landau theory, which are included in the book. We have also benefitted from meeting several times Yu M. Gufan and V.P. Dimitrlev who have shared with us their deep understanding of the physical implications of the Landau theory. The enlightening explanations of E. Brezin have been very helpful to clarify our view of the situation of Landau's theory from the standpoint of statistical physics. We had stimulating discussions with several experts in the handling of Landau's theory, namely N. Boccara, V. Dvorak, and A.P. Levanyuk. We are also indebted to F.W. Ainger,R. Chaves, A. Janner.T. Janssen J.Mozrzymas, P.M. Raccah, H. Schmid, and D. Weigel,who have invited us to teach the courses which have eventually resulted in this book. We acknowledge stimulating collaborations in various fields of application of Landau's theory with M. Clin, G. Errandonea, J. Schneck, H. Schmid, P. Schobinger-Papamantelos, and R. Tekaia. It is a pleasure to thank M. Coiret, P. Durand, and M.C. Mourier for their competent realization of the huge task of typing the manuscript. The Landau Theory of Phase Transitions Downloaded from www.worldscientific.com by FREIE UNIVERSITAET BERLIN MATHEMATIK / INFORMATIK on 06/29/15. For personal use only. THE LANDAU THEORY OF PHASE TRANSITIONS The Landau Theory of Phase Transitions Downloaded from www.worldscientific.com by FREIE UNIVERSITAET BERLIN MATHEMATIK / INFORMATIK on 06/29/15. For personal use only. This page is intentionally left blank XIII TABLE OF CONTENTS The Landau Theory of Phase Transitions Downloaded from www.worldscientific.com by FREIE UNIVERSITAET BERLIN MATHEMATIK / INFORMATIK on 06/29/15. For personal use only. Chapter I : INTUITIVE APPROACH TO THE BASIC IDEAS OF LANDAU'S THEORY 1 - INTRODUCTION 1 2 - MODEL EXAMPLE OF PHASE TRANSITION IN A CRYSTAL 2 2.1 2.2 2.3 2.4 - 2.5 2.6 2.7 2.8 2.9 - The system and its degrees of freedom Symmetry of the two phases Variational free-energy associated to the system Symmetry properties of F and form of its 2 n " degree Taylor expansion Decoupling of the (p ,p ) and p degrees of freedom Order parameter Need for an expansion of degree higher than 2, below T c th Simple physical consequences of the 4 degree orderparameter expansion Symmetry considerations Secondary order-parameters 3 - CONCLUSIONS 2 3 4 5 6 8 8 15 19 23 Chapter II : FORMULATION OF LANDAU'S THEORY 1 - INTRODUCTION 25 2 - BASIC CONCEPTS OF GROUP REPRESENTATIONS 25 2.1 - Irreducible representations of a group G 2.2 - Group theoretical properties used in the formulation of Landau's theory 3 - LANDAU'S THEORY 3.1 - Variational and equilibrium particle density and free-energy 3.2 - Decomposition of the density increment into irreducible parts 3.3 - Scalar variational degrees of freedom 3.4 - Second degree Taylor expansion of F(n, ) 3.5 - Definition of the order-parameter 3.6 - Order-parameter expansion and nature of the symmetry change 4 - TECHNICAL GROUP THEORETICAL ASPECTS OF LANDAU'S THEORY 4.1 - Reduction of an invariant space into irreducible spaces , 4.2 - Tensorial products of invariant spaces n power of a representation tn 4.3 - Symmetrized n power of a representation 26 31 35 35 36 38 39 40 41 47 47 48 49 XIV 4.4 - The Landau Condition 4.5 - Invariant polynomials and image of G in the OP-representation 4.6 - Minimization of the free-energy and symmetry breaking The Landau Theory of Phase Transitions Downloaded from www.worldscientific.com by FREIE UNIVERSITAET BERLIN MATHEMATIK / INFORMATIK on 06/29/15. For personal use only. 5 - SECONDARY ORDER-PARAMETER 5.1 - Coupling between secondary and primary OP 5.2 - Possible symmetries of secondary OP, and form of 0 (?s, nr) 5.3 - Equilibrium value of the secondary OP nearby T 5.4 - Irrelevance of the secondary OP, to the symmetry below T 6 - CONCLUSIONS 53 56 66 88 88 89 91 92 93 Chapter III : CONTINUOUS STRUCTURAL TRANSITIONS BETWEEN PERIODIC PHASES 1 - INTRODUCTION 96 2 - CRYSTALLOGRAPHIC SPACE-GROUPS AND THEIR REPRESENTATIONS 97 2.1 - Crystallographic space-groups 2.2 - Irreducible representations of the space-groups 97 103 3 - THE LIFSCHITZ CRITERION 110 3.1 - Derivation of the Lifschitz criterion 3.2 - Procedure of application of the Lifschitz criterion ^ 3.3 - Selection of k -vectors resulting from the o Litschitz criterion 3.4 - Lifschitz criterion and periodicity of the lowsymmetry phases 4 - IMAGES AND FREE-ENERGIES FOR ACTIVE REPRESENTATIONS 4.1 - Active representations of a space-group 4.2 - Images of GQ for active representations 4.3 - Free-energies for active representations 111 115 118 120 122 122 123 125 5 - DETERMINATION OF THE LOW-SYMMETRY SPACE GROUP G 127 6 - FERROIC CLASSIFICATION OF STRUCTURAL PHASE TRANSITIONS 130 6.1 - Ferroicity and point-symmetry change at T c 6.2 - Proper and improper ferroics 6.3 - Pseudo proper ferroics 7 - EXAMPLE OF THE STRUCTURAL TRANSITION IN GADOLINIUM MOLYBDATE 7.1 - Symmetry of the primary order-parameter 7.2 - Secondary OP and free-energy expansion 131 137 143 144 144 147 XV 7.3 - Phenomenological theory of GMO 148 8 - CONNECTIONS TO THE EXPERIMENTAL SITUATION 153 The Landau Theory of Phase Transitions Downloaded from www.worldscientific.com by FREIE UNIVERSITAET BERLIN MATHEMATIK / INFORMATIK on 06/29/15. For personal use only. 8.1 - Physical nature of the order-parameter 8.2 - Physical realization of different OP-symmetries 9 - CONCLUSION 153 159 161 Chapter IV : FIRST ORDER TRANSITIONS 1 - INTRODUCTION 166 2 - FIRST ORDER TRANSITIONS ASSOCIATED WITH HIGH-DEGREE EXPANSIONS OF THE ORDER-PARAMETER 167 2.1 - Case with n=l and d=6 : general features of firstorder transitions 2.2 - General features of phase diagrams with onecomponent order-parameter expansions 2.3 - Phase diagrams with two-component order-parameters 2.4 - Phase diagrams with multi-component orderparameters 2.5 - Experimental examples 3 - TRANSITIONS PREDICTED TO BE DISCONTINUOUS BY THE LANDAU CONDITION 3.1 - Role of third order invariants of the orderparameter in the phenomenological description of structural transitions 3.2 - Experimental examples 168 172 176 179 183 185 185 190 4 - TRANSITIONS PREDICTED TO BE DISCONTINUOUS BY THE LIFSCHITZ CONDITION 5 - TRANSITIONS ASSOCIATED WITH MORE THAN ONE ORDER-PARAMETER 5.1 - General properties of phase transitions associated with several order-parameters 5.2 - Examples of phase diagrams associated with two order-parameters 6 - PHENOMENOLOGICAL THEORY OF PHASE TRANSITIONS WHICH HAVE NO GROUP-SUBGROUP RELATIONSHIP BETWEEN THE PHASES 6.1 - Reconstructive transitions for which the orderparameter is a sinusoidal function of the atomic shifts 6.2 - Reconstructive transitions for which the variational parameter is the degree of occupation of a latent unit cell 7 - SINGULAR POINTS IN PHASE DIAGRAMS 190 193 193 195 202 202 207 210 XVI Chapter V : LANDAU THEORY OF INCOMMENSURATE PHASES 1 - INTRODUCTION The Landau Theory of Phase Transitions Downloaded from www.worldscientific.com by FREIE UNIVERSITAET BERLIN MATHEMATIK / INFORMATIK on 06/29/15. For personal use only. 1.1 - Standard experimental scheme for INC systems 1.2 - Basic ideas for the adaptation of Landau's theory 2 - SYMMETRY PROPERTIES OF THE ORDER-PARAMETER AND OF THE FREE-ENERGY 215 215 218 220 2.1 - Order-parameter and free-energy for an incommensurate wavevector 2.2 - Order-parameter and free-energy for the lock-in vector k"L 2.3 - Free-energy density for a modulated orderparameter; Lifschitz invariant 2.4 - Three standard examples 231 3 - QUALITATIVE INTERPRETATION OF THE EXPERIMENTAL SITUATION 254 3.1 - Simplified formulation of the first theoretical scheme 3.2 - Refinement of the first theoretical scheme 3.3 - Second theoretical scheme, in the presence of a Lifschitz invariant 4 - PHENOMENOL0GICAL THEORY FOR A TWO-COMPONENT ORDERPARAMETER IN THE PMA APPROXIMATION 4.1 4.2 4.3 4.4 4.5 - The 6-equation . The sinusoidal limit (u = 0; T< Tj) General form of the free energy for u £0 Multisoliton limit (y2= 1; T= TL) Macroscopic quantities and anomalies of the susceptibilities 5 - EXTENSIONS 5.1 - Lifting of the PMA 5.2 - Phenomenological theory in the absence of a Lifschitz invariant 5.3 - Effect of additional spatially dispersive terms 5.4 - Higher number of order-parameter components and of modulation directions 6 - CONCLUSIONS 220 239 250 255 257 261 263 265 269 270 271 278 283 284 287 293 295 302 Chapter VI : THE LANDAU THEORY OF MAGNETIC TRANSITIONS 1 - INTRODUCTION 307 2 - MAGNETIC SYMMETRY 308 2.1 - Magnetic point-groups 2.2 - Identification of the type of magnetic ordering associated with a given magnetic group 308 310 The Landau Theory of Phase Transitions Downloaded from www.worldscientific.com by FREIE UNIVERSITAET BERLIN MATHEMATIK / INFORMATIK on 06/29/15. For personal use only. XVII 2.3 - Magnetic lattices and magnetic space-groups 312 3 - IRREDUCIBLE COREPRESENTATIONS OF THE MAGNETIC GROUPS 313 4 - SPECIFIC FORMULATION OF THE LANDAU THEORY FOR MAGNETIC SYSTEMS AND EXCHANGE SYMMETRIES 315 4.1 - Formulation of the Landau theory 4.2 - Exchange symmetry 5 - PRACTICAL APPLICATION OF THE LANDAU THEORY TO MAGNETIC TRANSITIONS : SOME EXAMPLES 5.1 - Introduction 5.2 - Phase transitions from the paramagnetic group Pca2 1 1' at k=0 5.3 - Phase transitions in a-Fe~0-, 5.4 - Latent antiferromagnetism in nickel-iodine boracite 6 - APPLICABILITY OF THE LANDAU THEORY TO MAGNETIC SYSTEMS : ORDER PARAMETER SYMMETRIES, FIRST-ORDER TRANSITIONS AND INCOMMENSURATE MAGNETIC PHASES 315 317 321 321 321 329 341 349 6.1 - Order-parameter symmetries for second and firstorder transitions in magnetic systems 6.2 - Specific features of the Landau theory of incommensurate magnetic systems 353 7 - COUPLING OF THE MAGNETIC ORDER-PARAMETER TO NON-MAGNETIC PHYSICAL QUANTITIES : SPONTANEOUS MAGNETOSTRUCTURAL EFFECTS 361 7.1 7.2 7.3 7.4 7.5 - Introduction Spontaneous magnetostriction Spontaneous piezomagnetism Spontaneous magnetoelectricity Coupling between structural and magnetic transitions 349 361 362 365 367 369 Chapter VII : THE LANDAU THEORY OF LIQUID CRYSTALS 1 - INTRODUCTION 374 2 - SYMMETRY GROUPS OF LIQUID-CRYSTALS 374 2.1 - An introductory example 374 2.2 - The macroscopic space-groups of liquid crystals 377 3 - THE IRREDUCIBLE REPRESENTATIONS OF LIQUID CRYSTAL GROUPS 379 3.1 - Irreducible representations of the G point-groups 3.2 - Irreducible representations of the space-groups S 4 - PROBABILITY DENSITIES, ORDER-PARAMETERS AND THERMODYNAMIC POTENTIALS FOR LIQUID CRYSTAL TRANSITIONS 379 383 4.1 - Probability densities 4.2 - Primary and secondary order-parameters 388 388 389 XVIII The Landau Theory of Phase Transitions Downloaded from www.worldscientific.com by FREIE UNIVERSITAET BERLIN MATHEMATIK / INFORMATIK on 06/29/15. For personal use only. 4.3 - Thermodynamic potentials 4.4 - Practical procedure for the determination of the symmetry changes which take place at Liquid Crystal Transitions 5 - APPLICABILITY OF THE LANDAU THEORY TO PHASE TRANSITIONS IN LIQUID-CRYSTALS 5.1 5.2 5.3 5.4 5.5 - The nematic to smectic A transition The nematic-smectic A-smectic C phase diagram Reentrant nematic phases The uniaxial to biaxial nematic transition Transitions between smectic phases 6 - MODULATED LIQUID CRYSTAL PHASES 390 391 399 400 404 407 407 408 410 7 - TILTING ORDER AND BOND-0RIENTATIONAL ORDER IN SMECTIC PHASES 413 8 - FERROELECTRIC, FLEXOELECTRIC AND PIEZOELECTRIC EFFECTS 414 9 - TRANSITIONS FROM THE ISOTROPIC PHASE 417 Chapter VIII : RECENT DEVELOPMENTS AND FIELD OF VALIDITY OF LANDAU'S THEORY 425 1 - INTRODUCTION 425 2 - STABILITY OF ICOSAHEDRAL QUASI-CRYSTALLINE PHASES 425 2.1 - Landau's model of the liquid-solid phase transition 2.2 - Alexander and Mc Tague's extension of the model 427 428 2.3 - Application of the model to icosahedral phase 430 3 - INFLUENCE OF DEFECTS ON PHASE TRANSITIONS 3.1 - Classification of defects 3.2 - Phenomenological theory for symmetry-breaking point defects 4 - VALIDITY OF LANDAU'S THEORY 4.1 - Validity of the theory and statistical fluctuations 4.2 - Preservation of the validity of the symmetry aspects of Landau's theory 4.3 - Cases in which the thermodynamical results of Landau's theory are preserved 4.4 - Behaviour of physical quantities in the critical range 433 434 435 440 440 442 443 444