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Symmetry in Crystals

Wai-Kee Li, Gong-Du Zhou, and Thomas Chung Wai Mak

in Advanced Structural Inorganic Chemistry

Published in print:
2008
Published Online:
May 2008
ISBN:
9780199216949
eISBN:
9780191711992
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199216949.003.0009
Subject:
Physics, Crystallography

The symmetry properties of molecules were discussed in Chapter 6, and the corresponding properties of crystals are presented here. In contrast to discrete molecules, crystals have a lattice structure ... More


SYMMETRY OF MODULATED CRYSTALS

Sander Van Smaalen

in Incommensurate Crystallography

Published in print:
2007
Published Online:
January 2008
ISBN:
9780198570820
eISBN:
9780191718762
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198570820.003.0003
Subject:
Physics, Crystallography

This chapter gives a comprehensive account of the symmetry of incommensurately modulated crystals. Diffraction symmetry is shown to be given by a crystallographic point group as it is known from the ... More


Crystal structures

Erich H. Kisi and Christopher J. Howard

in Applications of Neutron Powder Diffraction

Published in print:
2008
Published Online:
January 2009
ISBN:
9780198515944
eISBN:
9780191705663
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198515944.003.0005
Subject:
Physics, Condensed Matter Physics / Materials

Neutron powder diffraction finds extensive application in the solution, and more particularly, the refinement, of crystal structures. This chapter reviews the tools (space group symmetry, ... More


Modularity at crystal scale — Twinning

Giovanni Ferraris, Emil Makovicky, and Stefano Merlino

in Crystallography of Modular Materials

Published in print:
2008
Published Online:
September 2008
ISBN:
9780199545698
eISBN:
9780191712111
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199545698.003.0005
Subject:
Physics, Crystallography

The chapter develops the reticular theory of twinning and some related worked examples. Twinning leads to oriented associations of crystals whose main types are: twin by merohedry, where the twin ... More


SUPERSTRUCTURES AND THE COMMENSURATE APPROXIMATION

Sander Van Smaalen

in Incommensurate Crystallography

Published in print:
2007
Published Online:
January 2008
ISBN:
9780198570820
eISBN:
9780191718762
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198570820.003.0005
Subject:
Physics, Crystallography

This chapter develops the relation between crystal structures of aperiodic crystals and superstructures. Superstructures are described as commensurately modulated structures, and it is shown that ... More


Fundamental crystallography

Xiaodong Zou, Sven Hovmöller, and Peter Oleynikov

in Electron Crystallography: Electron Microscopy and Electron Diffraction

Published in print:
2011
Published Online:
January 2012
ISBN:
9780199580200
eISBN:
9780191731211
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199580200.003.0002
Subject:
Physics, Crystallography

The basic concepts of crystallography, including unit cell, symmetry, crystal systems and origin specification are introduced. Bravais lattice, point groups, plane groups and space groups are ... More


Modelling inorganic structures

I. David Brown

in The Chemical Bond in Inorganic Chemistry: The Bond Valence Model

Published in print:
2006
Published Online:
January 2010
ISBN:
9780199298815
eISBN:
9780191708879
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199298815.003.0011
Subject:
Physics, Crystallography

Bond networks must obey the various bond valence rules, but these are insufficient to predict a structure. The problem of modelling inorganic structures has been approached in different ways. Some ... More


Introduction to symmetry and diffraction

William Clegg

in Crystal Structure Analysis: Principles and Practice

Published in print:
2009
Published Online:
September 2009
ISBN:
9780199219469
eISBN:
9780191722516
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199219469.003.0002
Subject:
Physics, Crystallography

This chapter explores how a crystal structure and its diffraction pattern are related, with reference to aspects of symmetry. Pure translation symmetry is characteristic of crystal structures, ... More


Space-group determination

William Clegg

in Crystal Structure Analysis: Principles and Practice

Published in print:
2009
Published Online:
September 2009
ISBN:
9780199219469
eISBN:
9780191722516
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199219469.003.0004
Subject:
Physics, Crystallography

This chapter describes the factors involved in deciding which of the 230 space groups is correct for a particular crystal structure, based on crystallographic experiments and chemical knowledge. ... More


Crystal structure factors and symmetry

Xiaodong Zou, Sven Hovmöller, and Peter Oleynikov

in Electron Crystallography: Electron Microscopy and Electron Diffraction

Published in print:
2011
Published Online:
January 2012
ISBN:
9780199580200
eISBN:
9780191731211
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199580200.003.0003
Subject:
Physics, Crystallography

The structure factors, with their hkl indices, amplitudes and phases are dealt with in great detail. With figures, text and formulas the physical meaning of amplitudes and phases is explained. ... More


Space groups

Mark Ladd

in Symmetry of Crystals and Molecules

Published in print:
2014
Published Online:
April 2014
ISBN:
9780199670888
eISBN:
9780191781124
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199670888.003.0005
Subject:
Physics, Crystallography

Space groups in one, two and three dimensions are studied, with copious illustrations, including stereoscopic views. Limiting conditions on X-ray reflection from crystals are introduced first ... More


Crystal symmetry and X-ray diffraction

ANGELO GAVEZZOTTI

in Molecular Aggregation: Structure analysis and molecular simulation of crystals and liquids

Published in print:
2006
Published Online:
January 2010
ISBN:
9780198570806
eISBN:
9780191718779
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198570806.003.0005
Subject:
Physics, Atomic, Laser, and Optical Physics

The study of crystal symmetry started as a study of the symmetry of macroscopic crystalline objects, mostly minerals. When it was realised that external symmetry reflected the internal symmetry of ... More


Basics of crystallography, part 3

Ulrich Müller

in Symmetry Relationships between Crystal Structures: Applications of Crystallographic Group Theory in Crystal Chemistry

Published in print:
2013
Published Online:
December 2013
ISBN:
9780199669950
eISBN:
9780191775086
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199669950.003.0006
Subject:
Physics, Crystallography, Condensed Matter Physics / Materials

The molecular symmetry is called the point group of the molecule. A point group can be represented by a set of matrices W. The point group of a crystal is the symmetry group of the bundle of normals ... More


Space and space groups

I. David Brown

in The Chemical Bond in Inorganic Chemistry: The Bond Valence Model

Published in print:
2006
Published Online:
January 2010
ISBN:
9780199298815
eISBN:
9780191708879
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199298815.003.0010
Subject:
Physics, Crystallography

Applying the model to solids requires an appreciation of the restricting role of symmetry in defining what crystal structures are possible. This chapter reviews those items of space group theory ... More


Extracting integrated intensities from powder diffraction patterns

William I.F. David and Devinderjit S.Sivia

in Structure Determination from Powder Diffraction Data

Published in print:
2006
Published Online:
January 2010
ISBN:
9780199205530
eISBN:
9780191718076
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199205530.003.0008
Subject:
Physics, Condensed Matter Physics / Materials

This chapter deals with the theoretical aspects of extracting integrated intensities from a powder diffraction pattern and with the ancillary issue of space group determination. Two principal methods ... More


The group-theoretical presentation of crystal-chemical relationships

Ulrich Müller

in Symmetry Relationships between Crystal Structures: Applications of Crystallographic Group Theory in Crystal Chemistry

Published in print:
2013
Published Online:
December 2013
ISBN:
9780199669950
eISBN:
9780191775086
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199669950.003.0010
Subject:
Physics, Crystallography, Condensed Matter Physics / Materials

A Bärnighausen tree is set up according to a given scheme by combining steps of maximal group-subgroup relations. Translationengleiche, klassengleiche and isomorphic subgroups are distinguished, and ... More


Conjugate subgroups, normalizers and equivalent descriptions of crystal structures

Ulrich Müller

in Symmetry Relationships between Crystal Structures: Applications of Crystallographic Group Theory in Crystal Chemistry

Published in print:
2013
Published Online:
December 2013
ISBN:
9780199669950
eISBN:
9780191775086
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199669950.003.0008
Subject:
Physics, Crystallography, Condensed Matter Physics / Materials

Subgroups conjugate in a space group 𝒢 are subgroups that are equivalent by symmetry operations of 𝒢. This can be orientational or translational conjugation. The normalizer of a subgroup ℋ in the ... More


Symmetry in Crystallography: Understanding the International Tables

Paolo Radaelli

Published in print:
2011
Published Online:
December 2013
ISBN:
9780199550654
eISBN:
9780191775093
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199550654.001.0001
Subject:
Physics, Crystallography, Condensed Matter Physics / Materials

This book provides a simple but rigorous introduction to crystallographic symmetry, up to the point where understanding and using the ‘International Tables for Crystallography’ (ITC) becomes ... More


How to handle space groups

Ulrich Müller

in Symmetry Relationships between Crystal Structures: Applications of Crystallographic Group Theory in Crystal Chemistry

Published in print:
2013
Published Online:
December 2013
ISBN:
9780199669950
eISBN:
9780191775086
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199669950.003.0009
Subject:
Physics, Crystallography, Condensed Matter Physics / Materials

Among the Wyckoff positions of a space group there often are different positions that are mutually equivalent in the Euclidean normalizer of the space group. Origin shifts or basis transformations ... More


Symmetry Relationships between Crystal Structures: Applications of Crystallographic Group Theory in Crystal Chemistry

Ulrich Müller

Published in print:
2013
Published Online:
December 2013
ISBN:
9780199669950
eISBN:
9780191775086
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199669950.001.0001
Subject:
Physics, Crystallography, Condensed Matter Physics / Materials

In crystal chemistry and crystal physics, the relations between the symmetry groups (space groups) of crystalline solids are of special importance. Part 1 of this book presents the necessary ... More


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