Commensurate and Incommensurate Phase Transitions: (Phase Transition Phenomena)

In this volume new group-theoretical methods for investigating phase transitions and symmetry problems in various types of crystals and liquid crystals are presented. New topics pertaining to the area of application of the theories of Landau and Birman, such as, transitions to modulated structures and the description of their symmetry, are discussed. Examples of applications of the group-theoretical methods are given, containing detailed calculations of symmetry changes at structural, magnetic, improper ferroelectric, commensurate-to-incommensurate, field-induced metamagnetic and liquid crystal phase transitions. This book will be invaluable to research workers at universities and scientific laboratories and postgraduate students specializing in solid state theory.

Table of Contents:
Contents. Chapter 1. Group-theoretical methods. Introduction. Group actions. The complete rational integrity basis of invariants. Crystallographic colour groups. Magnetic group corepresentations. The basis functions of tensor field representations. Superspace groups. Symmetry of modulated crystals. Chapter 2. Developments of Landau's theory. Introduction. Landau's theory and group actions. The inverse Landau problem in symmetry changes. Symmetry changes with the thermodynamic potential depending on complete rational integrity basis. Phase transitions with two order parameters. Phase transitions to modulated structures. Space-dependent basis functions and space-independent expansion coefficients in modulated structure calculations. Chapter 3. Developments of Birman's theory. Introduction. Symmetry restoration. Physical applications of crystallographic colour groups. Hermann's space-group decomposition theorem and chain subduction criterion. Symmetry restrictions implied by group-subgroup relation. Symmetry changes in two-dimensional systems. Chapter 4. Improper ferroelectrics. Introduction. The phase transition in gadolinium molybdate. Phase transitions in boracites. Improper ferroelectric-nonferroelastic phase transitions. The susceptibility tensor. Chapter 5. Symmetry changes in ferroic crystals. Introduction. Ferroic crystals. Purely ferroelastic phase transitions. Phase transitions of nonmagnetic ferroics. Nonferroic phase transitions. Chapter 6. Modulated structures. Introduction. Examples of incommensurate crystal phases. The internal dimensions of a supercrystal. Bravais classes for incommensurate crystals. Superspace-group symmetry operations. Determination of superspace groups for modulated crystals. Space-group representations and superspace symmetry. Superspace symmetry without a supercrystal. The commensurate-to-incommensurate phase transition in dysprosium crystal. The commensurate-to-incommensurate phase transition in a neodymium crystal. Chapter 7. Symmetry changes in metamagnets. Introduction. Field-induced metamagnetic phase transitions. Phase transitions at the tricritical point in metamagnets. Symmetry changes in the tricritical point in metamagnets. Symmetry changes at the tricritical point of a FeCl 2 -type crystal. Symmetry changes in the tricritical point of a metamagnet with P6 3 mmc(D) space-group symmetry and an hcp lattice. Chapter 8. Symmetries of liquid crystals. Introduction. ``Statistical'' symmetry with applications to phase transitions. Phase transitions from an isotropic liquid to liquid crystals. Symmetry of ferroelectric liquid crystals. The subgroups of the symmetry group of an A-type smectic with a symmetry centre. Critical X-ray scattering and the symmetry of liquid crystals. Appendix. Bibliography. Author index. Subject index.