International Tables for Crystallography, Symmetry Relations between Space Groups / Edition 1

International Tables for Crystallography, Symmetry Relations between Space Groups / Edition 1

by Hans Wondratschek, A. J. Wilson
     
 

ISBN-10: 1402023553

ISBN-13: 9781402023552

Pub. Date: 09/30/2004

Publisher: Wiley

International Tables for Crystallography is the definitive resource and reference work for crystallography and structural science.

Each of the eight volumes in the series contains articles and tables of data relevant to crystallographic research and to applications of crystallographic methods in all sciences concerned with the structure and properties of

Overview

International Tables for Crystallography is the definitive resource and reference work for crystallography and structural science.

Each of the eight volumes in the series contains articles and tables of data relevant to crystallographic research and to applications of crystallographic methods in all sciences concerned with the structure and properties of materials. Emphasis is given to symmetry, diffraction methods and techniques of crystal-structure determination, and the physical and chemical properties of crystals. The data are accompanied by discussions of theory, practical explanations and examples, all of which are useful for teaching.

This volume presents a systematic treatment of the maximal subgroups and minimal supergroups of the crystallographic plane groups and space groups. It is an extension of and a supplement to Volume A, Space-group symmetry, in which only basic data for sub- and supergroups are provided.

Group-subgroup relations, apart from their theoretical interest, are the basis of a number of important applications in crystallographic research:

(1) In solid-state phase transitions there often exists a group-subgroup relation between the symmetry groups of the two phases. According to Landau theory, this is in fact mandatory for displacive (continuous, second-order) phase transitions. Group-subgroup relations are also indispensable in cases where the symmetry groups of the two phases are not directly related but share a common subgroup or supergroup.

(2) Group-subgroup relations provide a concise and powerful tool for revealing and elucidating relations between crystal structures. They can thus help to keep up with the ever-increasing amount of crystal-structure data. Their application requires knowledge of the relations of the Wyckoff positions of group-subgroup related structures.

(3) Group-subgroup relations are of great importance in the study of twinned crystals, domain structures and domain boundaries.

(4) These relations can even help to identify errors in space-group assignment and crystal-structure determination.

(5) Subgroups of space groups provide a valuable approach to teaching crystallographic symmetry.

Volume A1 consists of three parts:

Part 1 presents an introduction to the theory of space groups at various levels and with many examples. It includes a chapter on the mathematical theory of subgroups.

Part 2 gives for each plane group and space group a complete listing of all maximal subgroups and minimal supergroups. The treatment includes the generators of each subgroup as well as any necessary changes of the coordinate system. Maximal isomorphic subgroups are given in parameterized form as infinite series because of the infinite number for each group. A special feature of the presentation is graphs that illustrate the group-subgroup relations.

Part 3 lists the relations between the Wyckoff positions of every space group and its subgroups. Again, the infinite number of maximal isomorphic subgroups of each space group are covered by parameterized series. These data for Wyckoff positions are presented here for the first time.

Audience: The volume is a valuable addition to the library of scientists engaged in crystal-structure determination, crystal physics or crystal chemistry. It is essential for those interested in phase transitions, the systematic compilation of crystal structures, twinning phenomena and related fields of crystallographic research.

Product Details

ISBN-13:
9781402023552
Publisher:
Wiley
Publication date:
09/30/2004
Series:
IUCr Series. International Tables of Crystallography Series, #2
Pages:
744
Product dimensions:
8.94(w) x 11.99(h) x 2.04(d)

Related Subjects

Table of Contents

Foreword.

Scope of this Volume.

Computer Production of Parts 2 and 3.

List of Symbols and Abbreviations used in this Volume.

PART 1 SPACE GROUPS AND THEIR SUBGROUPS.

1.1 Historical Introduction (M.I. Aroyo, U. Müller and H. Wondratschek).

1.1.1 The Fundamental Laws of Cystallography.

1.1.2 Symmetry and Crystal-Structure Determination.

1.1.3 Development of the Theory of Group–Subgroup Relations.

1.1.4 Applications of Group–Subgroup Relations.

1.2 General Introduction to the Subgroups of Space Groups (H. Wondratschek).

1.2.1 General Remarks.

1.2.2 Mappings and Matrices.

1.2.3 Groups.

1.2.4 Subgroups.

1.2.5 Space Groups.

1.2.6 Types of Subgroups of Space Groups.

1.2.7 Application to Domain Structures.

1.2.8 Lemmata on Subgroups of Space Groups.

1.3 Remarks on Wyckoff Positions (U. Müller).

1.3.1 Introduction.

1.3.2 Crystallographic Orbits and Wyckoff Positions.

1.3.3 Derivative Structures and Phase Transitions.

1.3.4 Relations Between the Positions in Group–Subgroup Relations.

1.4 Computer Checking of the Subgroup Data (F. Gähler).

1.4.1 Introduction.

1.4.2 Basic Capabilities of the Cryst package.

1.4.3 Computing Maximal Subgroups.

1.4.4 Description of the Checks.

1.5 The Mathematical Background of the Subgroup Tables (G. Nebe).

1.5.1 Introduction.

1.5.2 The Affine Space.

1.5.3 Groups.

1.5.4 Space Groups.

1.5.5 Maximal Subgroups.

1.5.6 Quantitative Results.

1.5.7 Qualitative Results.

PART 2 MAXIMAL SUBGROUPS OF THE PLANE GROUPS AND SPACE GROUPS.

2.1 Guide to the Subgroup Tables and Graphs (H. Wondratschek and M.I. Aroyo).

2.1.1 Contents and Arrangement of the Subgroup Tables.

2.1.2 Structure of the Subgroup Tables.

2.1.3 I Maximal Translationengleiche Subgroups (t-Subgroups).

2.1.4 II Maximal Klassengleiche Subgroups (k-Subgroups).

2.1.5 Series of Maximal Isomorphic Subgroups (Y. Billiet).

2.1.6 Minimal Supergroups.

2.1.7 The Subgroup Graphs.

2.2 Tables of Maximal Subgroups of the Plane Groups (Y. Billiet, M.I. Aroyo and H. Wondratschek).

2.3 Tables of Maximal Subgroups of the Space Groups (Y. Billiet, M.I. Aroyo and H. Wondratschek).

2.4 Graphs for Translationengleiche Subgroups (V. Gramlich and H. Wondratschek).

2.4.1 Graphs of the Translationengleiche Subgroups with a Cubic Summit.

2.4.2 Graphs of the Translationengleiche Subgroups with a Tetragonal Summit.

2.4.3 Graphs of the Translationengleiche Subgroups with a Hexagonal Summit.

2.4.4 Graphs of the Translationengleiche Subgroups with an Orthorhombic Summit.

2.5 Graphs for Klassengleiche Subgroups (V. Gramlich and H. Wondratschek).

2.5.1 Graphs of the Klassengleiche Subgroups of Monoclinic and Orthorhombic Space Groups.

2.5.2 Graphs of the Klassengleiche Subgroups of Tetragonal Space Groups.

2.5.3 Graphs of the Klassengleiche Subgroups of Trigonal Space Groups.

2.5.4 Graphs of the Klassengleiche Subgroups of Hexagonal Space Groups.

2.5.5 Graphs of the Klassengleiche Subgroups of Cubic Space Groups.

PART 3 RELATIONS BETWEEN THE WYCKOFF POSITIONS.

3.1 Guide to the Tables (U. Müller).

3.1.1 Arrangement of the Entries.

3.1.2 Cell Transformations.

3.1.3 Origin Shifts.

3.1.4 Nonconventional Settings of Orthorhombic Space Groups.

3.1.5 Conjugate Subgroups.

3.1.6 Monoclinic and Triclinic Subgroups.

3.2 Tables of the Relations of the Wyckoff Positions (U. Müller).

Appendix. Differences in the Presentation of Parts 2 and 3 (U. Müller and H. Wondratschek).

References.

Subject Index.

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