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Chapter 1．Introduction to Phenomenological Phase Transition Theory1

1．Fundamentals of Landau's Thermodynamic Theory1

Elementary thermodynamic analysis1

Spontaneous symmetry breaking at a continuous phase transition7

The Landau condi-tion for a second-order phase transition9

Further development of Landau's theory10

2．Prerequisites on Space-group Representations12

Space-group irreducible representations12

Irreducible representa-tions and their decomposition15

References17

Chapter 2．Physical Realization of the Order Parameters at a Micro-scopic Level of Description19

3．Tensor Representation of the Space Group on a Basis of Localized Atomic Functions19

Constructing crystal space group reducible representations20

The stabilizer method24

Constructing basis functions for star arms27

4．Permutational Representation and its Basis28

A summary of formulas28

The OP for ordering in AB type alloys29

The OP for ordering in Nb-H and Ta-H hydrides30

5．Vector Representation and its Basis36

A summary of formulas36

The OP at a structural phase transition in A-15 compounds38

The OP at a structural phase transition in C-15 compounds43

6．Pseudovector Representation and its Basis45

A description of the magnetically ordered state45

The OP at a magnetic phase transition in a garnet47

References51

Chapter 3．Symmetry Change at Phase Transitions52

7．Change in Translational Symmetry52

The Brillouin zone and the symmetric points in it52

Arm mixing and the transition channel56

Magnetic lattices72

8．The Total Symmetry Change74

Principles for finding the symmetry group of a new phase74

An example of a group-theoretic method of searching for dissymmetric phases77

9．Domains82

Domains as a consequence of the Curie principle82

A symme-try classification of domains83

Arm,orientational and antiphase domains85

Examples of an analysis of the domain structure87

10．The Paraphase91

The initial phase and the paraphase91

Major criteria for para-phase search93

An example of choosing the paraphase94

References98

Chapter 4．Analysis of the Thermodynamic Potential100

11．Invariant Expressions of the Thermodynamic Potential100

A straightforward(direct)method of constructing polynomial invari-ants100

Constructing the Ф for the structural phase transition in C-15 compounds101

Constructing the Ф for the structural phase transition in A-15 compounds102

12．Integral Rational Basis of Invariants104

General remarks104

The IRBI construction algorithm105

Solv-ability of the group and the minimal IRBI110

13．Examples of the Construction of an IRBI112

Constructing the IRBI for the structural transition in C-15 com-pounds112

Construction of the IRBI for the structural transition in A-15 compounds114

Constructing the IRBI for the structural transition in MnAs117

14．Irreducible Representation Images and Thermodynamic Potential Types118

General information on the I groups118

Two-and three-compon-ent order parameters119

A multicomponent order parameter121

I groups and rotation groups in multidimensional space121

Uni-versal classes123

References130

Chapter 5．Phase Diagrams in the Space of Thermodynamic Potential Parameters131

15．Theoretical Fundamentals of the Phase Diagram Construction Method131

Major physical principles131

Requisite theorems from the algebra of polynomials134

16．The One-Component Order Parameter137

The form of the thermodynamic potential137

The η6 model137

The η8 model139

Succession of solutions to equations of state146

17．The Two-Component Order Parameter148

The η4 model148

The η6 model152

Cubic invariants in the η4 model156

Cubic invariants in the η6 model159

18．The Three-Component Order Parameter161

The phases and the stability conditions161

The phase diagram in the η4 model164

19．The Role of the IRBI in the Construction of Phase Diagrams165

The two-component order parameter166

The three-component order parameter167

20．Coupling Order Parameters168

The interplay of two one-component order parameters168

Phase transitions in MnAs170

Phase transitions in KMnF3173

Ori-entation transitions178

References186

Chapter 6．Macroscopic Order Parameters188

21．Transformation Properties of the Order Parameters188

Physical realization of the macroparameters188

Construction of basis functions191

Constructing the thermodynamic potential192

22．Interplay of Micro-and Macroparameters197

Constructing the thermodynamic potential197

Improper transi-tions198

Examples of structural transitions in perovskite-type crystals199

23．Ferroics202

Classification of dissymmetric phases according to macroproperties202

Ferroelectrics204

Ferroelastics205

Ferrobielectrics and ferrobimagnetics205

Higher-order ferroics206

24．Non-ferroics211

Crystal-class-preserving phase transitions211

Examples of non-ferroics212

References214

Chapter 7．Phase Transitions in an External Field215

25．Phase Diagrams215

Constructing a potential for a system in an external field215

Phase diagram for the η4 model216

Phase diagram for the η6 model217

Singular points on the phase diagram222

Multi-component order parameter222

Splitting of a phase transition described by a microparameter in an external field226

26．Features Peculiar to the Temperature Behavior of Susceptibility in the Vicinity of the Second-Order Phase Transition227

Classification of the singularities by the Aizu indices227

Catas-trophe indices231

27．Calculation of Susceptibilities for Second-Order Phase Transitions233

Proper phase transitions233

Improper phase transitions234

Pseudoproper phase transitions238

28．Singularities of the Susceptibility in the Vicinity of the First-Order Phase Transition239

Classification of the first-order transitions239

Classifying the sin-gularities of susceptibilities240

Calculating the susceptibilities243

29．Domains in an External Field244

A thermodynamic description of the domains244

Effect of an external field on domains247

References249

Chapter 8．Martensite Transformations250

30．Reconstructive Structural Transitions250

Transitions without group-subgroup relation250

Geometric rela-tion of direct lattices253

The b.c.c.-f.c.c.transition255

The b.c.c.-h.c.p.transition256

The f.c.c.-h.c.p.transition257

Ori-entation relations258

Interrelationship of reciprocal lattices261

31．Thermodynamic Analysis of the Homogeneous State266

Describing the martensitic transition in terms of deformation266

Thermodynamic potential and phase diagram268

Behavior in an external field270

Shape memory effect274

32．Inhomogeneous States in the Vicinity of the Phase Transition275

Thermodynamic potential with gradient terms275

Equations of motion277

Tetragonal-strain phase transition278

Phase tran-sition with shear strain282

Square lattice288

33．The Omega Phase292

Interrelationship between the lattices at the b.c.c.-w-phase trans-formation292

Thermodynamic potential295

Inhomogeneous states296

References298

Chapter 9．Incommensurate Periodicity Phases301

34．General Approach to the Problem301

Commensurate and incommensurate phases301

Expansion of a thermodynamic potential with continuous order parameters303

35．Phases without Linear Gradient Terms in Free Energy304

One-component order parameter304

Two-component order pa-rameter309

Three-component order parameter311

Lifschitz point316

36．Phases with Linear Gradient Terms318

The soliton lattice318

Soliton lattice stability323

The devil's staircase328

Stochastic regime331

37．Multi-κ-structures332

Conditions for many-arm structures to be thermodynamically favor-able332

Multi-κ-structure in CeAl2334

Multi-κ-structure in Nd336

38．Incommensurate Phases in External Fields340

Helical structure in an external field340

Effect of external fields on the wave-vector of incommensurate phases343

39．The Thermodynamics of Phase Transitions to Incommensurate Phases347

Constructing a thermodynamic potential for non-Lifschitz stars347

Phase transition from incommensurate to commensurate phase350

Concomitant order parameters and incommensurate-phase symmetry351

Peculiarities of susceptibilities in incommensurate phases353

References355

Chapter 10．Color Symmetry and its Role in Phase Transition Theory358

40．Color Symmetry in the Theory of Magnetic Structures358

Magnetic structures and their symmetries.Geometric aspect358

Color symmetry and the thermodynamic-potential model360

The hierarchy of approximations for describing the magnetic structure of FeGe2362

41．Supersymmetry of Incommensurate Structures365

Incommensurate structures and the paradox of the Cheshire cat365

Phase symmetry of the thermodynamic potential and the sym-metry of incommensurate structures367

42．Icosahedral Symmetry of CrystalsQuasicrystals372

A new type of atomic ordering372

The geometric aspect374

The thermodynamic aspect379

The symmetry of quasicrystal structures385

Al86Mn14:A Fibonacci structure or a quasicrys-tal?387

Conclusion388

43．Color Groups in the Theory of Systems with a Quantum Mechanical Order Parameter390

The quantum mechanical order parameter390

Classification of ordered phases on the basis of color sub-groups of the normal-phase symmetry group391

OP symmetry and the behavior of the gap in κ space394

Thermodynamic potentials399

References401

Chapter 11．Fluctuations and Symmetry404

44．Fundamentals of the Fluctuation Phase Transitions Theory404

Critical indices404

The renormalization-group and ε-expansion method406

The isotropic model413

45．Critical Behavior of Anisotropic Systems415

Universal classes415

Cubic anisotropy417

Examples of systems with multicomponent order parameters419

46．Fluctuation-Induced Break-Down to First-Order Phase Transitions423

The absence of stable fixed points423

First-order transitions in magnetic systems424

47．Fluctuations in the Vicinity of Multicritical Points426

Systems with coupled order parameters.Bicritical and tetracritical points426

Lifschitz point434

References437

Appendix439

Index441