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1982 | Book

Space Lattice Symmetry Read chapter Read first chapter

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

Author: H. F. Franzen

Publisher: Springer Berlin Heidelberg

Book Series : Lecture Notes in Chemistry

Included in: Professional Book Archive

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About this book

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.

Table of Contents

Frontmatter
Space Lattice Symmetry
Abstract
Some properties of crystalline solids, such as the directions and symmetry of diffraction (X-ray, neutron, electron), the anisotropy of transport phenomena (electrical, thermal conduction, matter diffusion), and the anisotropy of thermal expansion and interactions of crystals with optical radiation are directly related to the underlying three-dimensional periodicities that characterize many crystalline materials. It is therefore important to understand such periodicity in order to better understand the phenomenology of the interactions of crystalline solids. It is also necessary to understand three-dimensional periodicity as a basis for development of the abstract theory of symmetry in crystalline solids (the theory of space groups and their representations). The three-dimensional periodicities are conveniently discussed in terms of space lattices, which are three-dimensional spatial arrays of discrete points which correspond to the translational symmetries of solids. The theory of space lattices, developed below, is the theory of the allowed symmetries of the periodicity of three-dimensionally crystalline solids.
H. F. Franzen
Space Group Symmetry
Abstract
Thus far the treatment of symmetry has been restricted to the proper rotational and reflection symmetries of space lattices. The discussion of the symmetry of crystalline solids does not end with the presentation of the 14 Bravais lattice types because the symmetry of a solid is the symmetry of its three-dimensionally periodic particle density, and there are more symmetries available to such periodic patterns than to the lattices which characterize their translational symmetries. This is in part because the pattern need not be centrosymmetric while the lattice must be, and in part because of the existence of symmetry operations appropriate to such patterns but not to their lattices.
H. F. Franzen
Reciprocal Space and Irreducible Representations of Space Groups
Abstract
The set of pure translational symmetry operations { ε |≠i} is a subgroup of the space group of a three dimensional crystalline solid. It is therefore meaningful to seek irreducible representations (irr. reps) and basis functions for this pure translational subgroup, and these play an important role in the theory of crystalline solids. For the benefit of the reader unfamiliar with group theory, a set of basis functions for a representation is made up of functions which transform into each other, or linear combinations thereof, under symmetry operations of the group. The irr. rep. is then the set of matrices which transform the functions under the symmetry operations (a representation) when the set cannot be reduced in the sense that a coordinate transformation results in splitting the basis functions into two or more sets of new functions each of which is a set of basis functions for a representation.
H. F. Franzen
Second-Order Phase Transitions
Abstract
This chapter treats the thermodynamic and group theoretical techniques useful in the consideration of phase transitions which occur without the coexistence of two phases, i.e., without the nucleation and growth of a new phase. The transitions under consideration occur with a change of symmetry at a certain thermodynamic state during a continuous structure change through that state. Such structure changes may be of three types: order-disorder, displacive and a combination of order-disorder and displacive. In the latter case the ordering is incommensurate with the translational periodicity of the lattice, as will be shown in a later section.
H. F. Franzen
Backmatter
Metadata
Title
Second-Order Phase Transitions and the Irreducible Representation of Space Groups
Author
H. F. Franzen
Copyright Year
1982
Publisher
Springer Berlin Heidelberg
Electronic ISBN
978-3-642-48947-1
Print ISBN
978-3-540-11958-6
DOI
https://doi.org/10.1007/978-3-642-48947-1