Chapter 1．Introduction to Phenomenological Phase Transition Theory 1
1．Fundamentals of Landau's Thermodynamic Theory 1
Elementary thermodynamic analysis 1
Spontaneous symmetry breaking at a continuous phase transition 7
The Landau condi-tion for a second-order phase transition 9
Further development of Landau's theory 10
2．Prerequisites on Space-group Representations 12
Space-group irreducible representations 12
Irreducible representa-tions and their decomposition 15
References 17
Chapter 2．Physical Realization of the Order Parameters at a Micro-scopic Level of Description 19
3．Tensor Representation of the Space Group on a Basis of Localized Atomic Functions 19
Constructing crystal space group reducible representations 20
The stabilizer method 24
Constructing basis functions for star arms 27
4．Permutational Representation and its Basis 28
A summary of formulas 28
The OP for ordering in AB type alloys 29
The OP for ordering in Nb-H and Ta-H hydrides 30
5．Vector Representation and its Basis 36
A summary of formulas 36
The OP at a structural phase transition in A-15 compounds 38
The OP at a structural phase transition in C-15 compounds 43
6．Pseudovector Representation and its Basis 45
A description of the magnetically ordered state 45
The OP at a magnetic phase transition in a garnet 47
References 51
Chapter 3．Symmetry Change at Phase Transitions 52
7．Change in Translational Symmetry 52
The Brillouin zone and the symmetric points in it 52
Arm mixing and the transition channel 56
Magnetic lattices 72
8．The Total Symmetry Change 74
Principles for finding the symmetry group of a new phase 74
An example of a group-theoretic method of searching for dissymmetric phases 77
9．Domains 82
Domains as a consequence of the Curie principle 82
A symme-try classification of domains 83
Arm,orientational and antiphase domains 85
Examples of an analysis of the domain structure 87
10．The Paraphase 91
The initial phase and the paraphase 91
Major criteria for para-phase search 93
An example of choosing the paraphase 94
References 98
Chapter 4．Analysis of the Thermodynamic Potential 100
11．Invariant Expressions of the Thermodynamic Potential 100
A straightforward(direct)method of constructing polynomial invari-ants 100
Constructing the Ф for the structural phase transition in C-15 compounds 101
Constructing the Ф for the structural phase transition in A-15 compounds 102
12．Integral Rational Basis of Invariants 104
General remarks 104
The IRBI construction algorithm 105
Solv-ability of the group and the minimal IRBI 110
13．Examples of the Construction of an IRBI 112
Constructing the IRBI for the structural transition in C-15 com-pounds 112
Construction of the IRBI for the structural transition in A-15 compounds 114
Constructing the IRBI for the structural transition in MnAs 117
14．Irreducible Representation Images and Thermodynamic Potential Types 118
General information on the I groups 118
Two-and three-compon-ent order parameters 119
A multicomponent order parameter 121
I groups and rotation groups in multidimensional space 121
Uni-versal classes 123
References 130
Chapter 5．Phase Diagrams in the Space of Thermodynamic Potential Parameters 131
15．Theoretical Fundamentals of the Phase Diagram Construction Method 131
Major physical principles 131
Requisite theorems from the algebra of polynomials 134
16．The One-Component Order Parameter 137
The form of the thermodynamic potential 137
The η6 model 137
The η8 model 139
Succession of solutions to equations of state 146
17．The Two-Component Order Parameter 148
The η4 model 148
The η6 model 152
Cubic invariants in the η4 model 156
Cubic invariants in the η6 model 159
18．The Three-Component Order Parameter 161
The phases and the stability conditions 161
The phase diagram in the η4 model 164
19．The Role of the IRBI in the Construction of Phase Diagrams 165
The two-component order parameter 166
The three-component order parameter 167
20．Coupling Order Parameters 168
The interplay of two one-component order parameters 168
Phase transitions in MnAs 170
Phase transitions in KMnF3 173
Ori-entation transitions 178
References 186
Chapter 6．Macroscopic Order Parameters 188
21．Transformation Properties of the Order Parameters 188
Physical realization of the macroparameters 188
Construction of basis functions 191
Constructing the thermodynamic potential 192
22．Interplay of Micro-and Macroparameters 197
Constructing the thermodynamic potential 197
Improper transi-tions 198
Examples of structural transitions in perovskite-type crystals 199
23．Ferroics 202
Classification of dissymmetric phases according to macroproperties 202
Ferroelectrics 204
Ferroelastics 205
Ferrobielectrics and ferrobimagnetics 205
Higher-order ferroics 206
24．Non-ferroics 211
Crystal-class-preserving phase transitions 211
Examples of non-ferroics 212
References 214
Chapter 7．Phase Transitions in an External Field 215
25．Phase Diagrams 215
Constructing a potential for a system in an external field 215
Phase diagram for the η4 model 216
Phase diagram for the η6 model 217
Singular points on the phase diagram 222
Multi-component order parameter 222
Splitting of a phase transition described by a microparameter in an external field 226
26．Features Peculiar to the Temperature Behavior of Susceptibility in the Vicinity of the Second-Order Phase Transition 227
Classification of the singularities by the Aizu indices 227
Catas-trophe indices 231
27．Calculation of Susceptibilities for Second-Order Phase Transitions 233
Proper phase transitions 233
Improper phase transitions 234
Pseudoproper phase transitions 238
28．Singularities of the Susceptibility in the Vicinity of the First-Order Phase Transition 239
Classification of the first-order transitions 239
Classifying the sin-gularities of susceptibilities 240
Calculating the susceptibilities 243
29．Domains in an External Field 244
A thermodynamic description of the domains 244
Effect of an external field on domains 247
References 249
Chapter 8．Martensite Transformations 250
30．Reconstructive Structural Transitions 250
Transitions without group-subgroup relation 250
Geometric rela-tion of direct lattices 253
The b.c.c.-f.c.c.transition 255
The b.c.c.-h.c.p.transition 256
The f.c.c.-h.c.p.transition 257
Ori-entation relations 258
Interrelationship of reciprocal lattices 261
31．Thermodynamic Analysis of the Homogeneous State 266
Describing the martensitic transition in terms of deformation 266
Thermodynamic potential and phase diagram 268
Behavior in an external field 270
Shape memory effect 274
32．Inhomogeneous States in the Vicinity of the Phase Transition 275
Thermodynamic potential with gradient terms 275
Equations of motion 277
Tetragonal-strain phase transition 278
Phase tran-sition with shear strain 282
Square lattice 288
33．The Omega Phase 292
Interrelationship between the lattices at the b.c.c.-w-phase trans-formation 292
Thermodynamic potential 295
Inhomogeneous states 296
References 298
Chapter 9．Incommensurate Periodicity Phases 301
34．General Approach to the Problem 301
Commensurate and incommensurate phases 301
Expansion of a thermodynamic potential with continuous order parameters 303
35．Phases without Linear Gradient Terms in Free Energy 304
One-component order parameter 304
Two-component order pa-rameter 309
Three-component order parameter 311
Lifschitz point 316
36．Phases with Linear Gradient Terms 318
The soliton lattice 318
Soliton lattice stability 323
The devil's staircase 328
Stochastic regime 331
37．Multi-κ-structures 332
Conditions for many-arm structures to be thermodynamically favor-able 332
Multi-κ-structure in CeAl2 334
Multi-κ-structure in Nd 336
38．Incommensurate Phases in External Fields 340
Helical structure in an external field 340
Effect of external fields on the wave-vector of incommensurate phases 343
39．The Thermodynamics of Phase Transitions to Incommensurate Phases 347
Constructing a thermodynamic potential for non-Lifschitz stars 347
Phase transition from incommensurate to commensurate phase 350
Concomitant order parameters and incommensurate-phase symmetry 351
Peculiarities of susceptibilities in incommensurate phases 353
References 355
Chapter 10．Color Symmetry and its Role in Phase Transition Theory 358
40．Color Symmetry in the Theory of Magnetic Structures 358
Magnetic structures and their symmetries.Geometric aspect 358
Color symmetry and the thermodynamic-potential model 360
The hierarchy of approximations for describing the magnetic structure of FeGe2 362
41．Supersymmetry of Incommensurate Structures 365
Incommensurate structures and the paradox of the Cheshire cat 365
Phase symmetry of the thermodynamic potential and the sym-metry of incommensurate structures 367
42．Icosahedral Symmetry of CrystalsQuasicrystals 372
A new type of atomic ordering 372
The geometric aspect 374
The thermodynamic aspect 379
The symmetry of quasicrystal structures 385
Al86Mn14:A Fibonacci structure or a quasicrys-tal? 387
Conclusion 388
43．Color Groups in the Theory of Systems with a Quantum Mechanical Order Parameter 390
The quantum mechanical order parameter 390
Classification of ordered phases on the basis of color sub-groups of the normal-phase symmetry group 391
OP symmetry and the behavior of the gap in κ space 394
Thermodynamic potentials 399
References 401
Chapter 11．Fluctuations and Symmetry 404
44．Fundamentals of the Fluctuation Phase Transitions Theory 404
Critical indices 404
The renormalization-group and ε-expansion method 406
The isotropic model 413
45．Critical Behavior of Anisotropic Systems 415
Universal classes 415
Cubic anisotropy 417
Examples of systems with multicomponent order parameters 419
46．Fluctuation-Induced Break-Down to First-Order Phase Transitions 423
The absence of stable fixed points 423
First-order transitions in magnetic systems 424
47．Fluctuations in the Vicinity of Multicritical Points 426
Systems with coupled order parameters.Bicritical and tetracritical points 426
Lifschitz point 434
References 437
Appendix 439
Index 441