Second-Order Phase Transitions and the Irreducible Representation of Space Groups


Book Description

The lecture notes presented in this volume were developed over a period of time that originated with the investigation of a research problem, the distortion from NiAs-type to MnP-type, the group-theoretical implications of which were investigated in collaboration with Professors F. Jellinek and C. Haas of the Laboratory for Inorganic Chemistry at the University of Groningen during the 1973-1974 year. This distortion provides the major example that is worked through in the notes. The subject matter of the notes has been incorporated in part in the lectures of a course in Solid State Chemistry taught several times at Iowa State University, and formed the basis of a series of lectures presented at the Max-Planck Institute for Solid State Research in Stuttgart during 1981- 19821 and as part of a Solid State Chemistry course taught during the spring of 1982 at Arizona State University in Tempe. I wish here to express my gratitude to the Max-Planck Institute for Solid State Research and to Arizona State University for the opportunity and support they provided during the time I was developing and writing the lecture notes of this volume. I wish also to thank the many colleagues and students who have offered comments and suggestions that have improved the accuracy and readability of the notes, and who have provided stimulation through discussion of the ideas presented here. am especially indebted to Professors C. Haas and F.




Space Group Representations


Book Description

This book is devoted to the construction of space group representations, their tabulation, and illustration of their use. Representation theory of space groups has a wide range of applications in modern physics and chemistry, including studies of electron and phonon spectra, structural and magnetic phase transitions, spectroscopy, neutron scattering, and superconductivity. The book presents a clear and practical method of deducing the matrices of all irreducible representations, including double-valued, and tabulates the matrices of irreducible projective representations for all 32 crystallographic point groups. One obtains the irreducible representations of all 230 space groups by multiplying the matrices presented in these compact and convenient to use tables by easily computed factors. A number of applications to the electronic band structure calculations are illustrated through real-life examples of different crystal structures. The book's content is accessible to both graduate and advanced undergraduate students with elementary knowledge of group theory and is useful to a wide range of experimentalists and theorists in materials and solid-state physics.




Isotropy Subgroups Of The 230 Crystallographic Space Groups


Book Description

This book gives a rather exhaustive list of isotropy subgroups of the 230 crystallographic space groups. The symmetry changes for the vast majority of observed phase transitions in crystalline solids can be found in the list. With each entry, information is given concerning both physical and abstract characteristics of the phase transitions.




Phase Transitions and Crystal Symmetry


Book Description

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.







Group Theory in Solid State Physics and Photonics


Book Description

While group theory and its application to solid state physics is well established, this textbook raises two completely new aspects. First, it provides a better understanding by focusing on problem solving and making extensive use of Mathematica tools to visualize the concepts. Second, it offers a new tool for the photonics community by transferring the concepts of group theory and its application to photonic crystals. Clearly divided into three parts, the first provides the basics of group theory. Even at this stage, the authors go beyond the widely used standard examples to show the broad field of applications. Part II is devoted to applications in condensed matter physics, i.e. the electronic structure of materials. Combining the application of the computer algebra system Mathematica with pen and paper derivations leads to a better and faster understanding. The exhaustive discussion shows that the basics of group theory can also be applied to a totally different field, as seen in Part III. Here, photonic applications are discussed in parallel to the electronic case, with the focus on photonic crystals in two and three dimensions, as well as being partially expanded to other problems in the field of photonics. The authors have developed Mathematica package GTPack which is available for download from the book's homepage. Analytic considerations, numerical calculations and visualization are carried out using the same software. While the use of the Mathematica tools are demonstrated on elementary examples, they can equally be applied to more complicated tasks resulting from the reader's own research.




Crystal Symmetries


Book Description

Crystal Symmetries is a timely account of the progress in the most diverse fields of crystallography. It presents a broad overview of the theory of symmetry and contains state of the art reports of its modern directions and applications to crystal physics and crystal properties. Geometry takes a special place in this treatise. Structural aspects of phase transitions, correlation of structure and properties, polytypism, modulated structures, and other topics are discussed. Applications of important techniques, such as X-ray crystallography, biophysical studies, EPR spectroscopy, crystal optics, and nuclear solid state physics, are represented. Contains 30 research and review papers.




Group Theory and Quantum Mechanics


Book Description

This graduate-level text develops the aspects of group theory most relevant to physics and chemistry (such as the theory of representations) and illustrates their applications to quantum mechanics. The first five chapters focus chiefly on the introduction of methods, illustrated by physical examples, and the final three chapters offer a systematic treatment of the quantum theory of atoms, molecules, and solids. The formal theory of finite groups and their representation is developed in Chapters 1 through 4 and illustrated by examples from the crystallographic point groups basic to solid-state and molecular theory. Chapter 5 is devoted to the theory of systems with full rotational symmetry, Chapter 6 to the systematic presentation of atomic structure, and Chapter 7 to molecular quantum mechanics. Chapter 8, which deals with solid-state physics, treats electronic energy band theory and magnetic crystal symmetry. A compact and worthwhile compilation of the scattered material on standard methods, this volume presumes a basic understanding of quantum theory.




Electron-Phonon Interactions and Phase Transitions


Book Description

This NATO Advanced Study Institute was the fourth in a series devoted to the subject of phase transitions and instabilities with particular attention to structural phase transforma~ions. Beginning wi th the first Geilo institute in 19'(1 we have seen the emphasis evolve from the simple quasiharmonic soft mode description within the Landau theory, through the unexpected spectral structure re presented by the "central peak" (1973), to such subjects as melting, turbulence and hydrodynamic instabilities (1975). Sophisticated theoretical techniques such as scaling laws and renormalization group theory developed over the same period have brought to this wide range of subjects a pleasing unity. These institutes have been instrumental in placing structural transformations clearly in the mainstream of statistical physics and critical phenomena. The present Geilo institute retains some of the counter cul tural flavour of the first one by insisting whenever possible upon peeking under the skirts of even the most successful phenomenology to catch a glimpse of the underlying microscopic processes. Of course the soft mode remains a useful concept, but the major em phasis of this institute is the microscopic cause of the mode softening. The discussions given here illustrate that for certain important classes of solids the cause lies in the electron phonon interaction. Three major types of structural transitions are considered. In the case of metals and semimetals, the electron phonon interaction relie6 heavily on the topology of the Fermi surface.




Intramolecular Diels-Alder and Alder Ene Reactions


Book Description

The Diels-Alder reaction has long been a powerful tool in organic synthesis. In recent years, the Alder ene reaction has also achieved some prominence. From the beginning, it was apparent that the intramolecular variants of these reactions would be feasible. Many such have been reported, but the results are widely scattered in the chemical literature. This volume is an attempt to synthesize results observed to date, and to suggest directions for future development. One of the limiting factors in the application of the intramolecular Diels Alder reaction has been the development of methods for the preparation of the requisite trienes. The fIrst chapter of this volume summarizes methods for the preparation of dienes and dienophiles. Examples representative of every general approach to 1,3-dienes and to dienophilic functional group combinations have been included. There are two questions one might ask in considering the prospective cyclization of a given triene: what are the factors that govern the rate of cyclization? and, for cyclizations that lead to the creation of one or more new chiral centers, what are the factors that govern diastereoselectivity? These questions are addressed in Chapter Two. The third chapter is devoted to the all-carbon intramolecular Alder ene reaction. The tables in that chapter summarize all examples that could be found in the literature through 1981, with several additional examples from 1982. Leading references to heterocyclic ene reactions are also included in this chapter.