Table Of ContentMODERN PROBLEMS IN
CONDENSED MATTER SCIENCES
Volume 31
Series editors
V.M. AGRANOVICH
Moscow, USSR
A.A. MARADUDIN
Irvine, California, USA
Advisory editorial board
F. Abeles, Paris, France
F. Bassani, Pisa, Italy
N. Bloembergen, Cambridge, MA, USA
E. Burstein, Philadelphia, PA, USA
I.L. Fabelinskii, Moscow, USSR
P. Fulde, Stuttgart, Germany
M.D. Galanin, Moscow, USSR
V.L. Ginzburg, Moscow, USSR
H. Haken, Stuttgart, Germany
R.M. Hochstrasser, Philadelphia, PA, USA
I.P. Ipatova, Leningrad, USSR
A.A. Kaplyanskii, Leningrad, USSR
L.V. Keldysh, Moscow, USSR
R. Kubo, Tokyo, Japan
R. Loudon, Colchester, UK
Yu.A. Ossipyan, Moscow, USSR
L.P. Pitaevskii, Moscow, USSR
A.M. Prokhorov, Moscow, USSR
K.K. Rebane, Tallinn, USSR
J.M. Rowell, Red Bank, NJ, USA
NORTH-HOLLAND
AMSTERDAM LONDON NEW YORK TOKYO
ELASTIC STRAIN FIELDS
AND DISLOCATION MOBILITY
Volume editors
V.L. INDENBOM
Moscow, USSR
J. LOTHE
Oslo, Norway
H
m
1992
NORTH-HOLLAND
AMSTERDAM LONDON · NEW YORK · TOKYO
© 1992 Elsevier Science Publishers B.V. All rights reserved
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ISBN: 0 444 88773 3
North-Holland
Elsevier Science Publishers B. V.
P.O. Box 211
1000 AE Amsterdam
The Netherlands
Library of Congress Cataloging-in-Publication Data
Elastic strain fields and dislocation mobility / volume editors, V.L.
Indenboni, J. Lothe.
p. cm. — (Modern problems in condensed matter sciences ; v.
31)
Includes bibliographical references and index.
ISBN 0-444-88773-3
1. Dislocations in crysta Is.find s strains. 2. Strains and
stresses. I. Indenbom, V. L. II. Lothe, Jens. III. Series.
QD921.E49 1992
548'.842—dc20 91-39415
CIP
Printed on acid-free paper
Printed in The Netherlands
MODERN PROBLEMS IN CONDENSED MATTER SCIENCES
Vol. 1. SURFACE POLARITONS
V.M. Agranovich and D.L. Mills, editors
Vol. 2. EXCITONS
E.I. Rashba and M.D. Sturge, editors
Vol. 3. ELECTRONIC EXCITATION ENERGY TRANSFERS IN
CONDENSED MATTER
V.M. Agranovich and M.D. Galanin
Vol. 4. SPECTROSCOPY AND EXCITATION DYNAMICS OF
CONDENSED MOLECULAR SYSTEMS
V.M. Agranovich and R.M. Hochstrasser, editors
Vol. 5. LIGHT SCATTERING NEAR PHASE TRANSITIONS
H.Z. Cummins and A.P. Levanyuk, editors
Vol. 6. ELECTRON-HOLE DROPLETS IN SEMICONDUCTORS
CD. Jeffries and L.V. Keldysh, editors
Vol. 7. THE DYNAMICAL JAHN-TELLER EFFECT IN
LOCALIZED SYSTEMS
Yu.E. Perlin and M. Wagner, editors
Vol. 8. OPTICAL ORIENTATION
F. Meier and B.P. Zakharchenya, editors
Vol. 9. SURFACE EXCITATIONS
V.M. Agranovich and R. Loudon, editors
Vol. 10. ELECTRON-ELECTRON INTERACTIONS IN
DISORDERED SYSTEMS
A.L. Efros and M. Pollak, editors
Vol. 11. MEDIUM-ENERGY ION REFLECTION FROM SOLIDS
E.S. Mashkova and V.A. Molchanov
Vol. 12. NONEQUILIBRIUM SUPERCONDUCTIVITY
D.N. Langenberg and A.I. Larkin, editors
v
MODERN PROBLEMS IN CONDENSED MATTER SCIENCES
Vol. 13. PHYSICS OF RADIATION EFFECTS IN CRYSTALS
R.A. Johnson and A.N. Orlov, editors
Vol. 14. INCOMMENSURATE PHASES IN DIELECTRICS
(Two volumes)
R. Blinc and A.P. Levanyuk, editors
Vol. 15. UNITARY TRANSFORMATIONS IN SOLID STATE
PHYSICS
M. Wagner
Vol. 16. NONEQUILIBRIUM PHONONS IN NONMETALLIC
CRYSTALS
W. Eisenmenger and A.A. Kaplyanskii, editors
Vol 17. SOLITONS
S.E. Trullinger, V.L. Pokrovskii and V.E. Zakharov, editors
Vol. 18. TRANSPORT IN PHONON SYSTEMS
V.L. Gurevich
Vol. 19. CARRIER SCATTERING IN METALS AND
SEMICONDUCTORS
V.F. Gantmakher and LB. Levinson
Vol. 20. SEMIMETALS-1. GRAPHITE AND ITS COMPOUNDS
N.B. Brandt, S.M. Chudinov and Ya.G. Ponomarev
Vol. 21. SPECTROSCOPY OF SOLIDS CONTAINING RARE
EARTH IONS
A.A. Kaplyanskii and R.M. Macfarlane, editors
Vol. 22. SPIN WAVES AND MAGNETIC EXCITATIONS
(Two volumes)
A.S. Borovik-Romanov and S.K. Sinha, editors
Vol. 23. OPTICAL PROPERTIES OF MIXED CRYSTALS
RJ. Elliott and I.P. Ipatova, editors
VI
MODERN PROBLEMS IN CONDENSED MATTER SCIENCES
Vol. 24. THE DIELECTRIC FUNCTION OF CONDENSED SYSTEMS
L.V. Keldysh, D.A. Kirzhnitz and A.A. Maradudin, editors
Vol. 25. CHARGE DENSITY WAVES IN SOLIDS
L.P. Gor'kov and G. Grüner, editors
Vol. 26. HELIUM THREE
W.P. Halperin and L.P. Pitaevskii, editors
Vol. 27. LANDAU LEVEL SPECTROSCOPY
(Two volumes)
G. Landwehr and E.I. Rashba, editors
Vol. 28. HOPPING TRANSPORT IN SOLIDS
M. Pollak and B. Shklovskii, editors
Vol. 29. NONLINEAR SURFACE ELECTROMAGNETIC
PHENOMENA
H.E. Ponath and G. Stegeman, editors
Vol. 30. MESOSCOPIC PHENOMENA IN SOLIDS
B.L. Altshuler, P.A. Lee and R.A. Webb, editors
Vol. 31. ELASTIC STRAIN FIELDS AND DISLOCATION
MOBILITY
V.L. Indenbom and J. Lothe, editors
Vll
Oh, how many of them there
are in the fields!
But each flowers in its
own way-
In this is the highest achievement
of a flower!
Matsuo Basho
1644-1694
PREFACE TO THE SERIES
Our understanding of condensed matter is developing rapidly at the present
time, and the numerous new insights gained in this field define to a significant
degree the face of contemporary science. Furthermore, discoveries made in this
area are shaping present and future technology. This being so, it is clear that the
most important results and directions for future developments can only be
covered by an international group of authors working in cooperation.
'Modern Problems in Condensed Matter Sciences' is a series of contributed
volumes and monographs on condensed matter science that is published by
Elsevier Science Publishers under the imprint of North-Holland. With the
support of a distinguished Advisory Editorial Board, areas of current interest
that have reached a maturity to be reviewed are selected for the series. Both
Soviet and Western scholars are contributing to the series, and each contributed
volume has, accordingly, two editors. Monographs, written by either Western or
Soviet authors, are also included. The complete series will provide the most
comprehensive coverage available of condensed matter science.
Another important outcome of the foundation of this series is the emergence
of a rather interesting and fruitful form of collaboration among scholars from
different countries. We are deeply convinced that such international collabora
tion in the spheres of science and art, as well as other socially useful spheres of
human activity, will assist in the establishment of a climate of confidence and
peace.
The publishing house 'Nauka' publishes the volumes in the Russian language.
This way the broadest possible readership is ensured.
The General Editors of the Series,
V.M. Agranovich A.A. Maradudin
IX
PREFACE
What should go into a book is always a matter of discussion and must also
depend on the general purpose of the book. We will not give a full account of our
deliberations on this matter, but state that among several motives a thorough
treatment of elastic anisotropy and a thorough discussion of dislocation mobil
ity have been main objectives. We believe that on these two main topics, the
book offers a coverage not found elsewhere.
In chapter 1 there is some overlap with material covered in other chapters,
and in particular chapters 2-7. Chapter 1 serves a dual purpose. It serves the
purpose of a synthesizing and generalizing overview after study of chapters 2-7,
as well as the purpose of a theoretical introduction for readers with a sufficiently
strong background. Chapter 1 also efficiently underlines, by examples, the
relevance of theory for discussion of experiments. In two addenda to chapter 1,
methods for direct observation of lattice defects and the phenomena of the
production of internal stresses and dislocations in crystal growth are reviewed
and discussed.
In chapter 2 the reader will find a summary of the solutions of important
dislocation problems in the approximation of elastic isotropy. The microscopic
(atomic) models for dislocations and cracks are presented in chapter 3. Chapters
4-6 are devoted to dislocations in infinite and semiinfinite anisotropic media.
The mathematical apparatus developed in these chapters is applied in chapter 7
to a discussion of uniformly moving dislocations and elastic surface waves. In
chapters 8-12 various aspects of dislocation motion and dislocation mobility
important for the physical behaviour of materials are reviewed: glide and climb
(chapter 10), interaction with point defects (chapter 9), motion in the periodic
lattice potential (chapter 8), the dynamical damping connected with interaction
with phonons and electrons (chapter 11), and finally, in chapter 12, the particu
lar behaviour of dislocations under conditions of irradiation which leads to such
phenomena as radiational creep and swelling.
We thank all the contributors who have helped us preparing the book. Our
special thanks are due to Professor Alshits whose efforts have been essential for
the completion of the book.
Moscow/Oslo, June 1991 V.L. Indenbom/J. Lothe
XI
CHAPTER 1
Dislocations and Internal Stresses
V.L. INDENBOM
A.V. Shubnikov Institute of Crystallography
Academy of Sciences of the USSR
Leninsky Prospect 59
Moscow, 117333, USSR
Elastic Strain Fields and Dislocation Mobility
Edited by
Elsevier Science Publishers B.V., 1992 V.L. Indenbom and J. Lothe
1
Contents
1. Introduction 5
2. Basic definitions 6
2.1. Classification of the crystal-lattice defects 6
2.2. Dislocations in crystals 7
2.2.1. Microscopic picture 7
2.2.2. Burgers circuit and vector 7
2.2.3. Macroscopic description 9
3. Integral theorems in the theory of internal stresses 11
3.1. Introduction 11
Example 1; internal stresses for twinned crystals 13
Example 2; internal stresses for a dislocation loop 13
3.2. Mean stress theorem and its corollaries 14
Example 3; Eshelby theorem 14
3.3. Colonnetti theorem and its applications 15
Example 4; bending of a plate with an edge dislocation 17
Example 5; twisting of a whisker crystal with a screw dislocation 17
Example 6; the force acting on a dislocation in an external field 19
3.4. Reciprocity theorem; influence function for intrinsic deformations 19
3.5. Reciprocity theorems and influence functions for tensors of dislocation density
and deformation incompatibility 20
Example 7; the displacement field of a single dislocation loop in an infinite
elastically isotropic medium 22
Example 8; electron microscopy imaging of dislocations under conditions of
anomalous transmission (the so-called halo contrast) 23
3.6. Influence functions for the deformation incompatibility tensor (the case of a
semi-infinite elastically isotropic medium) 26
Example 9; stresses due to sticking of particles to the crystal surface 28
4. One-dimensional problems 29
4.1. One-dimensional distributions of intrinsic deformations 29
4.2. Internal stresses in thin plates with one-dimensional intrinsic deformations 30
4.3. Three-dimensional distributions of intrinsic deformations; Fourier method 32
4.4. Stress jump at interfaces 34
2
