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# Phase Transitions and Crystal Symmetry

## byYurii Aleksandrovich Izyumov, V.N. Syromyatnikov

### Paperback | October 7, 2011

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About half a century ago Landau formulated the central principles of the phe nomenological second-order phase transition theory which is based on the idea of spontaneous symmetry breaking at phase transition. By means of this ap proach it has been possible to treat phase transitions of different nature in altogether distinct systems from a unified viewpoint, to embrace the aforemen tioned transitions by a unified body of mathematics and to show that, in a certain sense, physical systems in the vicinity of second-order phase transitions exhibit universal behavior. For several decades the Landau method has been extensively used to an alyze specific phase transitions in systems and has been providing a basis for interpreting experimental data on the behavior of physical characteristics near the phase transition, including the behavior of these characteristics in systems subject to various external effects such as pressure, electric and magnetic fields, deformation, etc. The symmetry aspects of Landau's theory are perhaps most effective in analyzing phase transitions in crystals because the relevant body of mathemat ics for this symmetry, namely, the crystal space group representation, has been worked out in great detail. Since particular phase transitions in crystals often call for a subtle symmetry analysis, the Landau method has been continually refined and developed over the past ten or fifteen years.

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Title:Phase Transitions and Crystal SymmetryFormat:PaperbackPublished:October 7, 2011Publisher:Springer NetherlandsLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:9401073570

ISBN - 13:9789401073578

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Table of Contents

1. Introduction to Phenomenological Phase Transition Theory.- 1. Fundamentals of Landau's Thermodynamic Theory.- Elementary thermodynamic analysis.- Spontaneous symmetry breaking at a continuous phase transition.- The Landau condition for a second-order phase transition.- Further development of Landau's theory.- 2. Prerequisites on Space-group Representations.- Space-group irreducible representations.- Irreducible representations and their decomposition.- References.- 2. Physical Realization of the Order Parameters at a Microscopic Level of Description.- 3. Tensor Representation of the Space Group on a Basis of Localized Atomic Functions.- Constructing crystal space group reducible representations.- The stabilizer method.- Constructing basis functions for star arms.- 4. Permutational Representation and its Basis.- A summary of formulas.- The OP for ordering in AB type alloys.- The OP for ordering in Nb-H and Ta-H hydrides.- 5. Vector Representation and its Basis.- A summary of formulas.- The OP at a structural phase transition in A-15 compounds.- The OP at a structural phase transition in C-15 compounds.- 6. Pseudovector Representation and its Basis.- A description of the magnetically ordered state.- The OP at a magnetic phase transition in a garnet.- References.- 3. Symmetry Change at Phase Transitions.- 7. Change in Translational Symmetry.- The Brillouin zone and the symmetric points in it.- Arm mixing and the transition channel.- Magnetic lattices.- 8. The Total Symmetry Change.- Principles for finding the symmetry group of a new phase.- An example of a group-theoretic method of searching for dissymmetric phases.- 9. Domains.- Domains as a consequence of the Curie principle.- A symmetry classification of domains.- Arm, orientational and antiphase domains.- Examples of an analysis of the domain structure.- 10. The Paraphase.- The initial phase and the paraphase.- Major criteria for paraphase search.- An example of choosing the paraphase.- References.- 4. Analysis of the Thermodynamic Potential.- Invariant Expressions of the Thermodynamic Potential.- A straightforward (direct) method of constructing polynomial invariants.- Constructing the ? for the structural phase transition in C-15 compounds.- Constructing the ? for the structural phase transition in A-15 compounds.- 12. Integral Rational Basis of Invariants.- General remarks.- The IRBI construction algorithm.- Solvability of the group and the minimal IRBI.- 13. Examples of the Construction of an IRBI.- Constructing the IRBI for the structural transition in C-15 compounds.- Construction of the IRBI for the structural transition in A-15 compounds.- Constructing the IRBI for the structural transition in MnAs.- 14. Irreducible Representation Images and Thermodynamic Potential Types.- General information on the I groups.- Two- and three-component order parameters.- A multicomponent order parameter.- I groups and rotation groups in multidimensional space.- Universal classes.- 14. Irreducible Representation Images and Thermodynamic Potential Types.- 5. Phase Diagrams in the Space of Thermodynamic Potential Parameters.- 15. Theoretical Fundamentals of the Phase Diagram Construction Method.- Major physical principles.- Requisite theorems from the algebra of polynomials.- 16. The One-Component Order Parameter.- The form ofthe thermodynamic potential.- The ?6 model.- The ?8 model.- Succession of solutions to equations of state.- 17. The Two-Component Order Parameter.- The ?4 model.- The ?6 model.- Cubic invariants in the ?4 model.- Cubic invariants in the ?6 model.- 19. The Role of the IRBI in the Construction of Phase Diagrams.- The two-component order parameter.- The three-component order parameter.- 20. Coupling Order Parameters.- The interplay of two one-component order parameters.- Phase transitions in MnAs.- Phase transitions in KMnF3.- Orientation transitions.- References.- 6. Macroscopic Order Parameters.- 21. Transformation Properties of the Order Parameters.- Physical realization of the macroparameters.- Construction of basis functions.- Constructing the thermodynamic potential.- 22. Interplay of Micro- and Macroparameters.- Constructing the thermodynamic potential.- Improper transitions.- Examples of structural transitions in perovskite-typecrystals.- 23. Ferroics.- Classification of dissymmetric phases according to macroproperties.- Ferroelectrics.- Ferroelastics.- Ferrobielectrics and ferrobimagnetics.- Higher-order ferroics.- 24. Non-ferroics.- Crystal-class-preserving phase transitions.- Examples of non-ferroics.- References.- 7. Phase Transitions in an External Field.- 25. Phase Diagrams.- Constructing a potential for a system in an external field.- Phase diagram for the ?4 model.- Phase diagram for the ?6 model.- Singular points on the phase diagram.- Multi-component order parameter.- Splitting of a phase transition described by a microparameter in an external field.- Features Peculiar to the Temperature Behavior of Susceptibility in the Vicinity of the Second-Order Phase Transition.- Classification of the singularities by the Aizu indices.- Catastrophe indices.- Calculation of Susceptibilities for Second-Order Phase Transitions.- Proper phase transitions.- Improper phase transitions.- Pseudoproper phase transitions.- Singularities of the Susceptibility in the Vicinity of the First-Order Phase Transition.- Classification of the first-order transitions.- Classifying the singularities of susceptibilities.- Calculating the susceptibilities.- 29. Domains in an External Field.- A thermodynamic description of the domains.- Effect of an external field on domains.- References.- 8. Martensite Transformations.- 30. Reconstructive Structural Transitions.- Transitions without group-subgroup relation.- Geometric relation of direct lattices.- The b.c.c.-f.c.c. transition.- The b.c.c.-h.c.p. transition.- The f.c.c.-h.c.p. transition.- Orientation relations.- Interrelationship of reciprocal lattices.- 31. Thermodynamic Analysis of the Homogeneous State.- Describing the martensitic transition in terms of deformation.- Thermodynamic potential and phase diagram.- Behavior in an external field.- Shape memory effect.- 32. Inhomogeneous States in the Vicinity of the Phase Transition.- Thermodynamic potential with gradient terms.- Equations of motion.- Tetragonal-strain phase transition.- Phase transition with shear strain.- Square lattice.- 33. The Omega Phase.- Interrelationship between the lattices at the b.c.c.-?-phase transformation.- Thermodynamic potential.- Inhomogeneous states.- References.- 9. Incommensurate Periodicity Phases.- 34. General Approach to the Problem.- Commensurate and incommensurate phases.- Expansion of a thermodynamic potential with continuous order parameters.- 35. Phases without Linear Gradient Terms in Free Energy.- One-component order parameter.- Two-component order parameter.- Three-component order parameter.- Lifschitz point.- 36. Phases with Linear Gradient Terms.- The soliton lattice.- Soliton lattice stability.- The devil's staircase.- Stochastic regime.- 37. Multi-?-structures.- Conditions for many-arm structures to be thermodynamically favorable.- Multi-?-structure in CeAl2.- Multi-?-structure in Nd.- 38. Incommensurate Phases in External Fields.- Helical structure in an external field.- Effect of external fields on the wave-vector of incommensurate phases.- 39. The Thermodynamics of Phase Transitions to Incommensurate Phases.- Constructing a thermodynamic potential for non-Lifschitz stars.- Phase transition from incommensurate to commensurate phase.- Concomitant order parameters and incommensurate-phase symmetry.- Peculiarities of susceptibilities in incommensurate phases.- References.- 10. Color Symmetry and its Role in Phase Transition Theory.- 40. Color Symmetry in the Theory of Magnetic Structures.- Magnetic structures and their symmetries. Geometric aspect.- Color symmetry and the thermodynamic-potential model.- The hierarchy of approximations for describing the magnetic structure of FeGe2.- 41. Supersymmetry of Incommensurate Structures.- Incommensurate structures and the paradox of the Cheshire cat.- Phase symmetry of the thermodynamic potential and the symmetry of incommensurate structures.- 42. Icosahedral Symmetry of Crystals. Quasicrystals.- A new type of atomic ordering.- The geometric aspect.- The thermodynamic aspect.- The symmetry of quasicrystal structures.- Al86Mn14: A Fibonacci structure or a quasicrystal?.- Conclusion.- 43. Color Groups in the Theory of Systems with a Quantum Mechanical Order Parameter.- The quantum mechanical order parameter.- Classification of ordered phases on the basis of color sub-groups of the normal-phase symmetry group.- OP symmetry and the behavior of the gap in ? space.- Thermodynamic potentials.- References.- 11. Fluctuations and Symmetry.- 44. Fundamentals of the Fluctuation Phase Transitions Theory.- Critical indices.- The renormalization-group and ?-expansion method.- The isotropic model.- 45. Critical Behavior of Anisotropic Systems.- Universal classes.- Cubic anisotropy.- Examples of systems with multicomponent order parameters.- 46. Fluctuation-Induced Break-Down to First-Order Phase Transitions.- The absence of stable fixed points.- First-order transitions in magnetic systems.- 47. Fluctuations in the Vicinity of Multicritical Points.- Systems with coupled order parameters. Bicritical and tetracritical points.- Lifschitz point.- References.