Basic Elements of Crystallography

207 Pages·2010·16.5 MB·English

Published by
Pan Stanford Publishing Pte. Ltd.
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8 Temasek Boulevard
Singapore 038988
Email: [email protected]
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British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
BASIC ELEMENTS OF CRYSTALLOGRAPHY
Copyright © 2010 by Pan Stanford Publishing Pte. Ltd.
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ISBN-13978-981-4241-59-5
ISBN-10981-4241-59-8
Printed in Singapore.
Rhaimie - Basic Elements of Crystallography.pm1d 2/26/2010, 9:50 AM
PREFACE

This book is intended to be a complete and clear introduction to the field
of crystallography for undergraduate and graduate students and lecturers in
physics, chemistry, biology, materials and earth sciences, or engineering. It
includes an extensive discussion of the 14 Bravais lattices and the reciprocal
to them, basic concepts of point group symmetry, the crystal structure of
elements and binary compounds, and much more. Besides that, the reader
can find up-to-date values for the lattice constants of most elements and
about 650 binary compounds (half of them containing rare earth metals).
The entire notation in this book is consistent with the International Tables
for Crystallography.
We have made all possible effort to attract the reader with high quality
illustrations showing all basic concepts in this area. Our purpose was to
show rather than describe “using many words” the structure of materials and
its basic properties. We believe that even readers who are completely not
familiar with the topic, but still want to learn how the atoms are arranged in
crystal structures, will find this book useful.
The text is organized into six chapters. Chapter I introduces basic
concepts and definitions in the field of crystallography starting with one-
and two-dimensional structures. Chapter II provides a detailed description of
the 14 Bravais lattices. Chapter III describes the most important crystal
structures of the elements with special emphasis on the close-packed
structures and the interstices present in them. Chapter IV presents the
structures of the most important binary compounds and reports the lattice
constants of about 650 of them. Chapter V is devoted to the reciprocal
lattice. Chapter VI, which is the final one, shows the relation between a
direct lattice and its reciprocal.
All chapters are accompanied by exercises designed in such a way to
encourage students to explore the crystal structures he/she is learning about.
Our goal always is to help the reader to develop spatial intuition by solving
the exercises graphically. Since computers are an essential part of today’s
education, we invite the reader to make use of crystallographic databases. In
most of the database web pages, it is possible to visualize crystal structures
in 3D either directly from the web browser or by downloading input files
with the coordinates of the structures. Some of the freely available (or with
open access options) databases are:

v
vi Preface
• ICSD for WWW-interface with free demo access
(http://icsd.ill.fr)
• American Mineralogist Crystal Structure Database (AMCSD)
(http://rruff.geo.arizona.edu/AMS/amcsd.php)
• Crystal Lattice Structures
(http://cst-www.nrl.navy.mil/lattice/index.html)
• Crystallography Open Database (COD)
(http://cod.ibt.lt).

We also encourage more advanced readers to create their own input files
with crystal structure coordinates or download them from the web. The
structures can be then viewed with, e.g., the freely available software called
Jmol (http://www.jmol.org). This program allows for the structure to be
manipulated, i.e., rotated, scaled, and translated, and allows for the
measurement of internal coordinates, e.g., bond lengths and angles.
We hope the reader will enjoy this book and will use it as a gateway for
understanding more advanced texts on this topic.

N. Gonzalez Szwacki
Houston, USA
T. Szwacka
Mérida, Venezuela
July 2009
CONTENTS

PREFACE v
ABBREVIATIONS xi
I. CRYSTAL STRUCTURE
1. Introduction 1
2. One-Dimensional Crystal Structures 1
3. Two-Dimensional Crystal Structures 4
4. Problems 13
II. THREE-DIMENSIONAL CRYSTAL LATTICE
1. Introduction 17
2. Examples of Symmetry Axes of Three-Dimensional
17
Figures
3. Symmetry Axes of a Cube 20
4. Symmetry Axes of a Set of Points 22
5. Crystal Systems 24
6. Conventional Cell for the Trigonal System 27
7. The 14 Bravais Lattices 28
7.1. Introduction 28
7.2. The Triclinic System 29
7.3. The Monoclinic System 29
7.4. The Orthorhombic System 32
7.5. The Tetragonal System 34
7.6. The Cubic System 34
7.7. The Trigonal and Hexagonal Systems 35
7.8. Symbols for Bravais Lattices 35
7.9. Conclusions 36
8. Coordination Number 38
9. Body Centered Cubic Lattice 39
10. Face Centered Cubic Lattice 42
11. Rhombohedral Unit Cell in a Cubic Lattice 46
11.1. Rhombohedral Unit Cell of the sc Lattice 46
11.2. Simple Cubic Crystal Structure 48
11.3. Interpretation of Data for As, Sb, Bi, and Hg 49
12. Trigonal Lattice 50

vii
viii Contents
13. Triple Hexagonal Cell R in a Cubic Lattice 55
14. Wigner-Seitz Cell 56
14.1. Construction of the Wigner-Seitz Cell 56
14.2. The Wigner-Seitz Cell of the bcc Lattice 56
14.3. The Wigner-Seitz Cell of the fcc Lattice 58
15. Problems 58
III. CRYSTAL STRUCTURES OF ELEMENTS
1. Introduction 67
2. Pearson Notation and Prototype Structure 68
3. The Filling Factor 69
4. Simple Cubic Structure 69
5. Body Centered Cubic Structure 71
6. Face Centered Cubic Structure 73
7. Close-Packed Structures 75
8. Double Hexagonal Close-Packed Structure 79
9. Samarium Type Close-Packed Structure 81
10. Hexagonal Close-Packed Structure 83
11. Interstices in Close-Packed Structures 87
12. Diamond Structure 92
13. Atomic Radius 98
14. Problems 102
IV. CRYSTAL STRUCTURES OF IMPORTANT BINARY

COMPOUNDS
1. Introduction 107
2. The Ionic Radius Ratio and the Coordination Number 107
3. Zinc Blende Structure 112
4. Calcium Fluoride Structure 114
4.1. Fluorite Structure 114
4.2. Anti-Fluorite Structure 117
5. Wurtzite Structure 118
6. Nickel Arsenide Related Structures 124
6.1. NiAs Structure 124
6.2. TiAs Structure 133
7. Sodium Chloride Structure 134
8. Cesium Chloride Structure 144
9. Problems 149
Contents ix
V. RECIPROCAL LATTICE
1. Introduction 153
2. The Concept of the Reciprocal Lattice 153
3. Examples of Reciprocal Lattices 162
3.1. Reciprocal of the Triclinic Lattice 162
3.2. Reciprocal of the Simple Cubic Lattice 162
3.3. Reciprocal of the Face Centered Cubic Lattice 164
4. Problems 165
VI. DIRECT AND RECIPROCAL LATTICES
1. Introduction 169
2. Miller Indices 169
3. Application of Miller Indices 175
4. Problems 180
REFERENCES 185
INDEX 187