The Landau Theory of Phase Transitions

The following sections are included:

• FOREWORD

• THE LANDAU THEORY OF PHASE TRANSITIONS

• TABLE OF CONTENTS

The following sections are included:

• INTRODUCTION

• MODEL EXAMPLE OF PHASE TRANSITION IN A CRYSTAL
  • The system and its degrees of freedom
  • Symmetry of the two phases
  • Variational free-energy associated to the system
  • Symmetry property of F and form of its 2nd degree Taylor expansion
  • Decoupling of the (p_x, p_y) and p_z degrees of freedom. Order-parameter
  • Need for an expansion of degree higher than 2, below T_c
  • Simple physical consequences of the 4th degree order-parameter expansion
    • Order-parameter coinciding with p_z
    • Order-parameter coinciding with (p_x, p_y)
  • Symmetry considerations
    • Relationship between equally-stable phases
  • Secondary order-parameters

• CONCLUSIONS

The following sections are included:

• INTRODUCTION

• BASIC CONCEPTS OF GROUP REPRESENTATION THEORY
  • Irreducible representations of a group G_o
    • The example of inversion
    • Generalization
    • “Physically irreducible” representations
  • Group theoretical properties used in the formulation of Landau's theory
    • Decomposition of f(r) into a sum of irreducible functions
    • Invariant polynomials
    • 1st and 2nd degree invariants for non-totally symmetric IR's
    • Standard form of the second degree invariant for a reducible set of degrees of freedom

• LANDAU'S THEORY
  • Variational and equilibrium particle density and free-energy
    • Symmetry of the system
    • Single phase of a system. Continuous phase transition
  • Decomposition of the density increment into irreducible parts
    • Definition of a small density increment
    • Symmetry properties of
  • Scalar variational degrees of freedom
  • Second degree Taylor expansion of F(η_kr)
    • Invariance property of the Taylor expansion of F
    • Form of the second-degree expansion
  • Definition of the order-parameter
    • Expression of the occurence of a symmetry change at T_c
    • Order-parameter, and form of α(T, P)
  • Order-parameter expansion and nature of the symmetry change
    • Incompatibility of f₃ (n_r) with a continuous transition
    • Role of the fourth-degree term f₄(n_r)

• TECHNICAL GROUP-THEORETICAL ASPECTS OF LANDAU'S THEORY
  • Reduction of an invariant space into irreducible spaces
  • Tensorial products of invariant spaces. n^th power of a representation
  • Symmetrized n^th power of a representation
    • Definition
    • Polynomials of n^th degree ; reduction into irreducible sets
    • Example of irreducible polynomial spaces
  • The Landau condition
    • Compliance of one-dimensional order-parameters with Landau's condition
    • Non-compliance when G is a subgroup of index 3 of G_o
  • Invariant polynomials and image of G_o in the OP-representation
    • Definition of the image
    • Application to the OP-space
    • Images of G_o for n < 3, and irreducible subgroups of O(n).[6]
    • Invariant polynomials of p-th degree. Integrity bases
    • Images and invariant polynomials for n = 4
  • Minimization of the free-energy,and symmetry breaking
    • Standard minimization of F
    • Isotropy groups in the OP-space [6,7,13,19]
    • Absolute minimum of F and maximal isotropy-groups
    • Minima of F and Morse theory [7, 27]
    • Exclusion of orbits with minimal symmetries [7]
    • Determination of the minima of F by means of the decomposition into orbits and of the integrity basis of I_o

• SECONDARY ORDER PARAMETERS
  • Coupling between secondary and primary OP
  • Possible symmetries of secondary OP, and form of φ(ζ_s,η_r)
  • Equilibrium value of the secondary OP, nearby T_c
  • Irrelevance of the secondary—OP , to the symmetry below T_c

• CONCLUSIONS

• REFERENCES

The following sections are included:

• INTRODUCTION

• CRYSTALLOGRAPHIC SPACE-GROUPS AND THEIR REPRESENTATIONS
  • Crystallographic space-groups
    • Geometrical elements composing a apace-group
    • Bravais lattice and point-group Ĝ of the crystal
    • Restrictions on non–primitive translations. Enumeration of space-groups. Notations
    • Unit-cells
  • Irreducible representations of the space-groups
    • Irreducible representations of the translation-group, Brillouin-zone
    • Group of ; Star of ; high-symmetry points
    • Small representation of ; weighted representation of
    • Construction of the Irreducible representation of G

• THE LIFSCHITZ CRITERION
  • Derivation of the Lifschitz-criterion
    • Existence of an extremum of
    • Functional form of the Landau free-energy
    • Form of
    • Form of the density increment
    • Functional form of
    • Antisymmetrized square of a representation
    • Group-theoretical expression of the Lifschitz-criterion
  • Procedure of application of the Lifachitz-criterion
    • Stars containing
    • Stars containing and
    • Stars not containing
  • Selection of vectors resulting from the Lifschitz-criterion
    • Proper symmetry of . Polar groups
    • Allowed vectors
  • Lifschitz-criterion and periodicity of the low-symmetry phases

• IMAGES AND FREE-ENERGIES FOR ACTIVE REPRESENTATIONS
  • Active representations of a space-group
  • Images of G_o for active representations
    • Finite character of the images, for active representations
    • Enumeration of the images for active representations [17]
  • Free-energies for active representations

• DETERMINATION OF THE LOW-SYMMETRY SPACE-GROUP G

• FERROIC CLASSIFICATION OF STRUCTURAL PHASE TRANSITIONS
  • Ferroicity and point-symmetry-change at T_c
    • Point-symmetry and macroscopic tensors
    • Ferroic and non-ferroic transitions
    • Classification of ferroics
    • Use of the ferroic classification
  • Proper and improper ferroics [30, 31, 13, 14]
    • Characteristics of the OP of improper ferroics
    • Physical behaviour of improper ferroics. Faintness index
  • Pseudo-proper ferroics

• EXAMPLE: THE STRUCTURAL TRANSITION IN GADOLINIUM MOLYBDATE
  • Symmetry of the primary-order parameter
    • Images, kernels, and isotropy groups
    • Low-symmetry space groups
    • Compliance with the Landau and Lifschitz conditions
  • Secondary OP and free-energy expansion
    • Improper ferroelectricity and ferroelasticity in GMO
    • Form of the Landau free-energy
  • Phenomenological theory of GMO
    • Low symmetry phase
    • Temperature dependence of the physical quantities

• CONNECTIONS TO THE EXPERIMENTAL SITUATION
  • Physical nature of the order-parameter
    • Different types of order-parameters
    • -vector of the OP-representation and superlattice reflections below T_c
    • OP-symmetry and modifications of the vibrational spectrum across T_c
  • Physical realization of different OP symmetries

• CONCLUSION

• REFERENCES

The following sections are included:

• INTRODUCTION

• FIRST-ORDER TRANSITIONS ASSOCIATED WITH HIGH-DEGREE EXPANSIONS OF THE ORDER-PARAMETER
  • Case with n = 1, and d = 6 : general features of first-order transitions
  • General features of phase diagrams associated with one-component order-parameter expansions
  • Phase diagrams with two-component order-parameters
  • Phase diagrams with multicomponent order-parameter
  • Experimental examples

• TRANSITIONS PREDICTED TO BE DISCONTINUOUS BY THE LANDAU CONDITION
  • Role of third order invariants of the order-parameter in the phenomenological description of structural transitions
  • Experimental examples

• TRANSITIONS PREDICTED TO BE DISCONTINUOUS BY THE LIFSHITZ CONDITION

• TRANSITIONS ASSOCIATED WITH MORE THAN ONE ORDER-PARAMETER
  • General properties of phase transitions associated with several order-parameters
  • Examples of phase diagrams associated with two order-parameters
    • Case of a biquadratic coupling between the order-parameters
    • Case of linear-quadratic coupling between the order-parameters
    • Case of a linear-cubic coupling between the order-parameters

• PHENOMENOLOGICAL THEORY OF PHASE TRANSITIONS WHICH HAVE NO GROUP-SUBGROUP RELATIONSHIP BETWEEN THE PHASES
  • Reconstructive transitions for which the order-parameter is a sinusoīdal function of the atomic shifts
  • Reconstructive transitions for which the variational parameter is the degree of occupation of a latent unit-cell

• SINGULAR POINTS IN PHASE DIAGRAMS

• REFERENCES

The following sections are included:

• INTRODUCTION
  • Standard experimental scheme for INC systems
  • Basic ideas for the adaptation of Landau's theory

• SYMMETRY PROPERTIES OF THE ORDER PARAMETER AND OF THE FREE ENERGY
  • Order-parameter and free-energy for an incommensurate wavevector
    • Symmetry characteristics of the INC order-parameter
    • Number of modulation directions in the INC phase
    • Partial independence of the INC free-energy on the “angular” components (phases) of the OP
  • Order-parameter and free-energy for the lock-in vector
    • Relationship between the OP at and at
    • Degrees of the lock-in terms
    • Similarity of INC phases and “high-order” commensurate phases
    • Enumeration of possible lock-in wavevectors on the basis of symmetry considerations
  • Free-energy for a modulated order-parameter ; Lifshitz invariant
    • Symmetry properties of the modulated order-parameter
    • Contribution of the OP derivatives to . Lifschitz invariant
  • Three standard examples
    • Rubidium-zinc-chloride (Rb₂ZnCl₄)
    • Barium sodium niobate
    • Thiourea, SC(NH₂)₂

• QUALITATIVE INTERPRETATION OF THE EXPERIMENTAL SITUATION
  • Simplified formulation of the first theoretical scheme
    • INC free-energy
    • Commensurate free-energy
    • Lock-in transition
    • Free-sliding of the INC distortion
  • Refinement of the first theoretical scheme
  • Second theoretical scheme, in the presence of a Lifschitz invariant
    • Role of the Lisfschitz invariant and of the squared gradient
    • Role of the lock-in term
    • Lifschitz invariants and modulation directions

• PHENOMENOLOGICAL THEORY FOR A TWO-COMPONENT (OP) IN THE (PMA) APPROXIMATION [11]
  • The θ-equation
  • The sinusoidal limit (μ ≈ O ; T < T_I)
  • General form of the free-energy for μ² ≠ 0
  • Multisoliton limit (μ² ≈ 1 ; T ≈ T_L2)
    • The single soliton solution (μ² = 1)
    • Multisoliton regime, in the “imposed ρ (T)” approximation
    • Lock-in transition with variational amplitude ρ(T)
  • Macroscopic quantities and anomalies of the susceptibilities
    • Qualitative discussion of the susceptibilities
    • Quantitative derivation of the susceptibility relative to P ∝ ρ². cos (2θ) [15]

• EXTENSIONS
  • Lifting of the PMA
  • Phenomenological theory in the absence of a Lifschitz invariant
    • Simplified theory | 19 |
    • Refined theories
    • Origin of the negative sign of σ
  • Effect of additional spatially dispersive terms
  • Higher number of OP components and of modulation directions
    • Barium sodium niobate
    • Other systems

• CONCLUSIONS

• REFERENCES

The following sections are included:

• INTRODUCTION

• MAGNETIC SYMMETRY
  • Magnetic point-groups
  • Identification of the type of magnetic ordering associated with a given magnetic group
  • Magnetic lattices and magnetic space-groups

• IRREDUCIBLE COREPRESENTATIONS OF THE MAGNETIC GROUPS

• SPECIFIC FORMULATION OF THE LANDAU THEORY FOR MAGNETIC SYSTEMS AND EXCHANGE SYMMETRIES
  • Formulation of the Landau theory
  • Exchange symmetry
    • Exchange and magnetic anisotropy energies
    • Classification of magnetic structures in terms of exchange symmetry

• PRACTICAL APPLICATION OF THE LANDAU THEORY TO MAGNETIC TRANSITIONS : SOME EXAMPLES
  • Introduction
  • Phase transitions from the paramagnetic group Pca2₁1' at k = o
    • Symmetries of the magnetically ordered phases
    • Exchange and relativistic energies : nature of the magnetic order below T_c
  • Phase transitions in α − Fe₂O₃
    • Symmetry of the ordered phases
    • Exchange antiferromagnetism and weak ferromagnetism in α−Fe₂O₃
    • Phase diagram of α− Fe₂ O₃
    • Behaviour ofα− Fe₂ O₃ in a magnetic field : magnetic susceptibilities, induced magnetization and hysteresis
  • Latent antiferromagnetism in nickel-iodine boracite
    • Identification of the order-parameter symmetry
    • Tentative model for the ordered phase of Ni–I boracite : latent antiferromagnetism

• APPLICABILITY OF THE LANDAU THEORY TO MAGNETIC SYSTEMS : ORDER-PARAMETER SYMMETRIES, FIRST-ORDER TRANSITIONS AND INCOMMENSURATE MAGNETIC PHASES
  • Order-parameter symmetries for second and first order transitions in magnetic systems
  • Specific features of the Landau theory of incommensurate magnetic systems
    • Prediction for incommensurate structures in magnetic systems
    • Examples of incommensurate magnetic phases
    • Dielectric and metallic magnetic systems : non-localized spin density

• COUPLING OF THE MAGNETIC ORDER-PARAMETER TO NON-MAGNETIC PHYSICAL QUANTITIES : SPONTANEOUS MAGNETOSTRUCTURAL EFFECTS
  • Introduction
  • Spontaneous magnetostriction
  • Spontaneous piezomagnetism
  • Spontaneous magnetoelectricty
  • Coupling between structural and magnetic transitions

• REFERENCES

The following sections are included:

• INTRODUCTION

• SYMMETRY GROUPS OF LIQUID CRYSTALS
  • An introductory example
  • The macroscopic space groups of liquid crystals

• THE IRREDUCIBLE REPRESENTATIONS OF LIQUID CRYSTAL GROUPS
  • Irreducible representations of the G point groups
  • Irreducible representations of the space groups S

• PROBABILITY DENSITIES, ORDER-PARAMETERS AND THERMODYNAMIC POTENTIALS FOR LIQUID CRYSTAL TRANSITIONS
  • Probability densities
  • Primary and secondary order-parameters
  • Thermodynamic potentials
  • Practical procedure for the determination of the symmetry changes which take place at Liquid Crystal transitions
    • Transitions from the smectic group S = Z Λ C _∞V
    • Transitions from the nematic group S = R Λ D_∞h

• APPLICABILITY OF THE LANDAU THEORY TO PHASE-TRANSITIONS IN LIQUID CRYSTALS
  • 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

• MODULATED LIQUID CRYSTAL PHASES

• TILTING ORDER AND BOND-ORIENTATIONAL ORDER IN SMECTIC PHASES

• FERROELECTRIC, FLEXOELECTRIC AND PIEZOELECTRIC EFFECTS

• TRANSITIONS FROM THE ISOTROPIC PHASE

• REFERENCES

The following sections are included:

• INTRODUCTION

• STABILITY OF ICOSAHEDRAL QUASI-CRYSTALLINE PHASES
  • Landau's model of the liquid-solid phase transition
  • Alexander and Mc Tague's extension of the model
  • Application of the model to the icosahedral phase
    • Influence of a fifth-degree term. “Edge Model”
    • Consideration of edge and vertex models
    • Triggering mechanisms

• INFLUENCE OF DEFECTS ON PHASE TRANSITIONS
  • Classification of defects
  • Phenomenological theory for symmetry-breaking point defects
    • Description of the model
    • Calculation of η_eq(r) for a single defect
    • Physical properties of the defective system
    • Extension of the theory

• VALIDITY OF LANDAU'S THEORY
  • Validity of the theory and statistical fluctuations
  • Preservation of the validity of the symmetry aspects of Landau's theory
  • Cases in which the thermodynamical results of the theory are preserved
  • Behaviour of physical quantities in the critical range
    • Wilson's theory and the Landau free-energy
    • Behaviour in the critical range

• REFERENCES