Table Of ContentCrystal Defects and Crystalline Interfaces
W. Bollmann
Crystal Defects
and Crystalline Interfaces
With 158 Figures
and a Set of Moire-Models
Springer-Verlag Berlin Heidelberg GmbH
w. BOLLMANN, Dr. SC. nat., Dipl. Phys., Battelle Institute, Advanced
Studies Center, Geneva, and Privatdozent, Eidgen6ssische Technische
Hochschule, Ziirich, Switzerland
ISBN 978-3-642-49175-7 ISBN 978-3-642-49173-3 (eBook)
DOI 10.1007/978-3-642-49173-3
Illustration on the dust cover: Dislocations and grain boundaries in stainless steel, taken with the
1500 kV electron microscope at Toulouse, France, by G. Dupouy and F. Perrier.
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<1:> by Springer-Verlag Berlin Heidelberg 1970. Library of Congress Catalog Card Number 77-124069.
Originally published by Springer-Verlag Berlin Heidelberg in 1970
Softcover reprint of the hardcover 1s t edition 1970
The use of general descriptive names, trade names, trade marks, etc. in this publications, even if the fonner
are not especially identified, is not be taken as a sign that such names, as understood by the Trade Marks
and Merchandise Marks Act, may accordingly be used freely by anyone. Printed by Universitătsdruckerei
H. Stiirtz AG, Wiirzburg. Title No. 1700.
To my Mother and my Wife
Preface
It is nonnal for the preface to explain the motivation behind the
writing of the book. Since many good books dealing with the general
theory of crystal defects already exist, a new book has to be especially
justified, and here its main justification lies in its treatment of crystal
line interfaces. About 1961, the work of the author, essentially based on
the fundamental work of Professor F. C. Frank, started to branch away
from the main flow of thought in this field and eventually led to a
general geometrical theory which is presented as a whole for the first
time in this book. Although nearly all that is presented has already
been published in different journals and symposia, it might be difficult
for the reader to follow that literature, as a new terminology and new
methods of analysis had to be developed.
Special emphasis is given to discussion and many diagrams are
included in order that a clear view of the basic concepts be obtained.
Intennediate summaries try to bring out the main points of the chapters.
Instead of specific exercises, general suggestions for them are given. The
part up to chapter 9 is considered more or less as introductory, so that
the book can be studied without specific knowledge of crystals and
crystal defects. The presentation of that part developed out of lectures
given by the author at the Swiss Federal Institute of Technology (ETH)
in Zurich. As metallurgists and mineralogists are often not familiar with
linear algebra, which is the mathematical basis of the interface theory,
the procedures involved in this technique are collected in the Appendix.
Many of the ideas contained in the book originated from observations
made on moire models and the subsequent fonnulation of these observa
tions in mathematical terms. In order that the reader may participate
in this experience, a set of such models is presented with the book.
Now the work is finished, it is my great pleasure to express my
gratitude first to the Battelle Institute (BI) which by its financial sup
port made this work possible, and in particular to Dr. B. D. Thomas,
fonner president of the Battelle Memorial Institute, Dr. H. Thiemann,
General Director of the Battelle Laboratories, Geneva and Dr. F. I. Mil
ford, Director of Physical Sciences of BI, for their moral support of this
work. Much inspiration came from discussions with my friend and fonner
colleague Prof. D. G. Brandon, and I would like to express my special
VIII Preface
thanks to him. I should also like to thank Mr. P. Fontaine, my col
laborator for the last fourteen years, who prepared many of the dia
grams appearing in this book, including the precision drawings of the
moire models, Dr. and Mrs. E. Anderson for their contribution to the
translation of the original German manuscript and Mrs. R. Anderson
for the typing of this manuscript. Finally, I should like to thank the
many others, not mentioned by name here, without whose contributions
the book would not have become what it is, especially the publisher and
his collaborators.
Geneva, July 1970 w. Bollmann
Contents
Chapter 1. Introduction ......... . 1
1.1. The Significance of Crystal Defects . 1
1.2. The Nature of Crystal Defects ... 1
Chapter 2. General Aspects of the Structure of Crystals 3
2.1. Outline . . . . . . . . . . . . . . . 3
2.2. Mathematical Description of the Crystal 5
2.3. Classification of Crystal Structures 11
Chapter 3. Some Special Crystal Structures 13
3.1. Outline. . . . . . . . . . . . 13
3.2. Close-Packed Structures . . . . 13
3.3. The Face-Centred Cubic Structure 17
3.4. The NaCl-Structure. . . . . . . 19
3.5. The Diamond Structure. . . . . 20
3.6. The Body-Centred Cubic Structure. 22
3.7. The Close-Packed Hexagonal Structure 25
3.8. The Graphite Structure. . . . . . . 27
3.9. Irregularities in the Stacking Sequence 28
3.10. Energy of Stacking Faults . . . . . 33
3·11. Fibrous Structures. . . . . . . . . 34
Chapters 2-3. Summary and Discussion. 35
Chapter 4. Point Defects . . . . . . . . . . . . 37
4.1. General Considerations of the Field of Poi!1t Defects 37
Chapter 5. The Individual Dislocation: Geometrical Basis. . 41
5.1. Outline . . . . . . . . . . . . . . . . . . . . 41
5.2. Definition of the Dislocation Line and the Burgers
Vector . . . . . . . . . . . 41
5.3. Relative Orientation of b and I 45
5.4. Motion of Dislocations . . . . 47
5.5. Dislocations and Moire-Figures . 49
Chapter 6. Interaction Between a Dislocation and its Sur-
roundings .......... . 51
6.1. Outline . . . . . . . . . . . . 51
6.2. Representation of the Stress Field 51
x Contents
6.3. Forces Acting on a Dislocation 53
6.4. Energy of a Dislocation. . . . 55
6.5. The Line Tension ...... . 58
6.6. The Strain Field of a Dislocation. 61
6.7. The Stress Field of a Dislocation . 63
Chapters 5-6. Summary and Discussion . 67
Chapter 7. The Interaction of Dislocations . . . . 70
7.1. Outline .............. . 70
7.2. Interaction Between two Parallel Screw Dislocations 70
7·3· Interaction Between two Parallel Edge Dislocations 71
7.4. Dislocation Reactions. . . . . . . . . . . . . . 72
7.5. Dislocation Reactions in Special Crystal Structures. 75
Chapter 8. Partial Dislocations . . . . . . . . . . . . . . 78
8.1. Outline . . . . . . . . . . . . . . . . . . . . 78
8.2. Partial Dislocations in the Face-Centred Cubic Struc-
ture . . . . . . . . . . . . . . . . . . . . . 78
8.3. The Sequence of Partial Dislocations . . . . . . . 81
8.4. Transformation of the Face-Centred Cubic - into the
Close-Packed Hexagonal Structure. . . . . . . . 87
Chapter 9. Dualistic Representation of Dislocation Reactions 98
9.1. Introduction. . . . . . . . . . . . . 98
9.2. The Significance of the b-Net .......... 101
9·3· The Prediction of Dislocation Reactions. . . . . . 102
9.4. Dislocation Reactions Involving Partial Dislocations 107
9·5. Thompson's Notation of the Burgers Vectors .... 109
Chapters 7-9. Summary and Discussion. . . . . . 110
Chapter 10. Short Description of the Relations Between Dis
locations and Point Defects, and Dislocation Dy-
namics . . . . . . . . . . . 113
10.1. Dislocations and Point Defects ...... " 113
10.2. Dislocation Dynamics. . . . . . . . . . .. 117
Chapter 11. Dislocation Networks - Subgrain Boundaries 118
11.1. Introduction. . . . . . . . . . . . 118
11.2. Frank's Formula. . . . . . . . . . 119
11. 3. Classification of Subgrain Boundaries . 124
11.4. "Special" Subgrain Boundaries 125
11.5. Foreign Dislocations . . . . . . . 129
11.6. "General" Subgrain Boundaries . . 130
11.7. The Energy of a Subgrain Boundary 131
11.8. Some Considerations of Recrystallization 133
Contents XI
11.9. Limiting Cases . . . . . . . . . . . . .. 138
11.10. Dualistic Representation in Three Dimensions. 139
Chapter 11. Summary and Discussion . . .. 141
Chapter 12. General Geometrical Theory of Crystalline Inter-
faces . . . . . . . . . . 143
12.1. Introduction . . . . . . . 143
12.2. The Coincidence-Site Lattice 143
12.3. The O-Lattice. . . . . . . 148
12.4. The O-Lattice and Frank's Formula. 153
12.5. Theb-Lattice. . . . . . . . . . . 154
12.6. Various Aspects of the O-Points. . . 155
12.7. The Actual Formulation of the Transformation A . 158
12.8. Choice Between Various Possible Transformations A 160
12.9. Solutions of the Basic Equation. . . . . . . . . 169
12.10. The Behaviour of the O-Lattice upon Translation of
Lattice 2. . . . . . . . . . . . . . . . . . . 175
12.11. Translation of the O-Lattice for the Case of
Rank (T) < 3. . . . . . . . . . . . . 177
12.12. Subdivision of the Crystal Space into Cells 182
Chapter 12. Summary and Discussion . 183
Chapter 13. Applications of the O-Lattice Theory. . 186
13·1. Outline . . . . . . . . . . . . . . . 186
13·2. Example of an Optimum Phase Boundary 188
13·3· The Periodicity of the Pattern of Lattice Points. 196
13.4· Displacement of the Pattern . . . . . . . .. 206
13·5. Dislocation Networks in General Crystalline Inter-
faces .................... 209
Chapter 14. Completion of the Linear O-Lattice Theory and
Extension to Non-Linear Problems ...... 215
14.1. Outline ................... 215
14.2. Dislocations and Burgers Vectors in Crystalline
Interfaces . . . . . . . . . . . 215
14.3. Non-Linear Problems . . . . . . 221
Chapters 13-14. Summary and Discussion 226
Appendix • 229
Ai Matrix Calculation Procedures . . . . . 229
A2 Moire-Models ............ . 243
A3 The Direct Observation of Crystal Defects 245
References . . . . . . . 247
Sources of the Illustrations 250
Subject Index . . . . . . 251
1. Introduction
1.1. The Significance of Crystal Defects
On opening a book about crystal defects the reader may wonder why
this field should be investigated at all. It is well known that most of the
solid materials exist in crystalline form, whether this fact is directly
evident as in minerals or whether it becomes apparent only after pre
paration (polishing, etching) as in metals. Crystals are distinguished by
their periodic, highly symmetric structures which possess a strong
aesthetic appeal. The word" defect", however, indicates a disturbance,
an imperfection, which, at first sight, appears to be neither aesthetic
nor attractive.
The final aim of research is to attain an understanding of the objective,
in this case of solid matter. Here it was seen quite early that the as
sumption of a perfect crystal led to results differing greatly from the
measured data, especially with respect to flow stress, where the cal
culated results were higher than the measured data by a factor 100.
Such observations led to the assumption that defects were already
present in crystals. An understanding of the" real crystal" (in contrast
to the" ideal crystal ") is unthinkable without the knowledge of crystal
defects. Crystal defects were investigated theoretically long before they
could be made directly visible. We do not enter the history of the research
on crystal defects as this is given in the books of Nabarro (1967) and
Hirth and Lothe (1968).
1.2. The Nature of Crystal Defects
In order to become acquainted with the nature of crystal defects we
first have to consider their environment, i.e. the crystal itself (Chap
ters 2-3).
By neglecting lattice oscillations, the crystal defects can be divided
into the following groups, according to their dimensional nature:
a) Point defects: Vacancies, interstitial atoms, impurity atoms, and
combinations of these;
b) line defects: i.e. dislocations;
c) two-dimensional defects: stacking faults, grain boundaries, etc.;
d) three-dimensional defects: precipitates, inclusions, holes, etc.
I Bollmann, Crystal Defects
