Impurity Scattering in Metallic Alloys

529 Pages·2002·16.14 MB·English

by  Joginder Singh Galsin (auth.)

Impurity
Scattering in
Metallic Alloys
PHYSICS OF SOLIDS AND LlaUIDS
Editorial Board: Jozef T. Devreese • University of Antwerp, Belgium
Roger P. Evrary • University of Liege, Belgium
Stig Lundqvist • Chalmers University of Technology, Sweden
Gerald D. Mahan • University of Tennessee, USA
Norman H. March • University of Oxford, England
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Impurity
Scattering in
Metallic Alloys
Joginder Singh Galsin
Punjab Agricultural University
Punjab, India
Springer Science+Business Media, LLC
Library of Congress Cataloging-in-Publication Data
Galsin, Joginder Singh, 1947-
Impurity scattering in metallic alloys/Joginder Singh Galsin.
p. cm. - (Physics of solids and liquids)
Includes bibliographical references and index.
ISBN 978-1-4613-5457-4 ISBN 978-1-4615-1241-7 (eBook)
DOI 10.1007/978-1-4615-1241-7
1. Electronic structure. 2. Alloys. 1. Title. Il. Series.
QCI76.8.E4 035 2001
530.4'1l-dc21
2001029748
ISBN 978-1-4613-5457-4
©2002, Springer Science+Business Media New York
Originally published by Kluwer AcademiclPlenum Publishers in 2002
Softcover reprint of the hardcover l st edition 2002
http://www.wkap.nl
10987654321
A c.I.P. record for this book is available fram the Library of Congress
AII rights reserved
No part of this book may be reproduced, stQred in a retrieval system, or transmitted in any form or by any
means, electronic, mechanical, photocopying, microfilming, recording, ar otherwise, without written permission
from the Publisher
To Mr. H. S. Mattewal
Advocate General, Punjab State
Dean of Faculty of Law, Guru Nanak Dev University, Amritsar
Chairman of Bar Association of Punjab and Haryana High Court, Chandigarh
Preface
Since the introduction of quantum mechanics, the general theory of solid state physics has
developed very rapidly. To date, a number of good textbooks on general solid state physics
have been written. However, research in solid state physics has become highly specialized
and undertaken in narrow fields. There is thus a great need for elementary textbooks that
deal in detail with the study of solids in a particular field in order to give students basic
knowledge in that field.
Metallic solids with an impurity, generally called alloys, are of immense importance
from both fundamental and technological points of view. The pioneering work of Bloember
gen and Rowland (1953) gave considerable impetus to the study of the electronic structure
of metallic alloys. Serious theoretical study in this field started in 1960 and, during the last
two decades, considerable success in understanding the electronic structure of simple metal
alloys has been achieved. Nonetheless the theoretical study of dilute alloys of transition
metals is still in its infancy. At present there are few review articles and original research
papers that examine the role of an impurity with respect to the electronic structure and
properties of metallic alloys. Because of the absence of an elementary textbook that presents
a comprehensive account of different aspects of the electronic structure of metallic alloys, I
have written this elementary textbook on the theory of the electronic structure of metallic
alloys. Here I have given more emphasis to the study of conduction electron distribution,
impurity screening (fundamental quantities), electric field gradients, and other electronic
properties in the real space for metallic alloys. Theoretical study of metallic alloys covers a
vast field and cannot be treated in a single book. In the present book I have chosen alloys
that are dilute and possess cubic symmetry for the following reasons: First, in dilute alloys
impurity-impurity interactions are negligible and hence the problem reduces to scattering
from a single impurity. Second, the cubic systems possess high symmetry, which simplifies
the mathematical algebra.
This book deals with theory and is designed for senior undergraduate students of
physics, chemistry, and materials science and can be prescribed for a one-semester course.
Introductory courses on quantum mechanics and general solid state physics can be consid
ered as a prerequisite. I have cited original references so that a student can consult them for
vii
viii Preface
further study. I have used Dirac's notation, which spotlights the physics contained in the
mathematics in an elegant and compact manner. I have tried my best to incorporate as many
diagrams and geometrical constructions as possible so that the reader can visualize the
physical processes. Problems have been inserted at appropriate places in each chapter. I have
also tried to list available experimental data and theoretical results in the form of tables
pertaining to dilute cubic alloys that may be of immense value to readers.
The book is organized systematically to develop the theory of electronic structure of
dilute cubic alloys. The first 10 chapters are basic in nature. Chapter 1 provides a general
introduction into metals and alloys. Chapter 2 describes crystal structure and geometry
around various types of impurities. Chapter 3 gives an account of the thermodynamic
treatment of equilibrium phase diagrams in binary metallic alloys. Chapter 4 examines the
electronic structure of pure metals. The concept of pseudopotentials is also introduced in this
chapter. Chapter 5 treats physical effects produced by an impurity in an otherwise pure solid.
A brief account of change in various electronic properties is given. The elastic continuum
theory applied to pure cubic metals and those with impurities is discussed in Chapter 6.
Chapters 7 and 8, respectively, are devoted to theoretical and experimental studies of
quadrupole interactions in dilute alloys. Chapters 9 and 10 deal with partial wave analysis
for the evaluation of impurity-induced phase shifts, conduction electron density, and impurity
scattering potential due to excess impurity valency. Chapter 10 also presents dielectric
screening theory for studying the electronic structure of dilute alloys.
Chapters 11 and 12 describe density functional theory and Green's function method, ab
initio methods, for the evaluation of conduction electron density and these include many
body effects explicitly. Chapter 13 presents lattice static methods for evaluating impurity
induced strain field in dilute metallic alloys. Starting from fundamentals, Chapter 14 gives
a detailed account of impurity-induced electric field gradients in dilute metallic alloys. It
also describes the evaluation of electric field gradients due to excess impurity valency using
different methods. Chapter 15 explains how the additional electron redistribution and electric
field gradients arise from the size difference between the impurity and host atoms. Transition
metal impurities change the electronic properties of the host metal drastically and therefore
constitute an interesting class of metallic systems. Chapter 16 examines the electronic
structure of dilute metallic alloys with transition metal impurities using the LCAO and model
band structure methods in addition to those discussed in Chapters 9-12. Finally, Chapter 17
gives experimental data and possible factors contributing toward temperature variation in
the impurity scattering potential and electric field gradients, which is an almost unexplored
aspect of the impurity problem in cubic metals.
It is a pleasure to acknowledge my indebtedness to my teacher Professor S. Prakash,
who stimulated my interest in the field of theoretical solid state physics. I also thank students,
the late Dr. S.D. Raj, Dr. S. Mahajan, Dr. B. Pal, Dr. S.K. Rattan, Dr. P. Singh, Ranjjan Verma,
P.S. Purl, Servesh Kumar, P.K. Bassi, B. Malhotra, N. Bhatia, Sanjeev Kumar, and J.P. Singh
with whom I have loved working and who have always been a source of inspiration to me.
I am grateful to Professor S.K. Sinha, Professor W.A. Harrison, Professor Nathi Singh, and
Professor S.K. Joshi for fruitful discussions at different stages of the work reported in this
book. I am very grateful to my wife, who checked the technical part of English text and also
inspired me to devote time generously for completion of this project. I am thankful to Mr.
Gagandeep Singh for typing the manuscript and Mr. M.S. Dhall and Dr. G.S. Miglani for
Preface ix
doing the artwork. The finished product has not been read by subject experts, therefore any
omission or error is my responsibility. I would appreciate that readers write to me about any
such errors.
REFERENCE
BLOEMBERGEN, N., and ROWLAND, T.J., 1953, Acta Metall. I, 731.
Contents
1. Introduction 1
1.1. Metals . 1
1.2. Alloys . 7
2. Crystal Structure and Point Defects 11
2.1. Crystal Lattice and Basis . . . . . 11
2.2. Structural Periodicity . . . . . . . 13
2.2.1. Translational Symmetry 13
2.2.1.1. Near Neighbors 14
2.2.1.2. Primitive Cell 15
2.2.2. Rotational Symmetry . 17
2.3. Electrostatic Periodicity 20
2.4. Reciprocal Lattice . . . 21
2.5. Point Defects in Solids . 25
2.5.1. Solid Solutions 25
2.5.2. Types of Solid Solutions. 26
2.6. Types of Point Defects . . . . . . . 27
2.6.1. Substitutional Point Defects 27
2.6.2. Vacancies........ 28
2.6.3. Interstitial Point Defects 30
2.6.4. Other Point Defects . 33
Suggested Reading 35
References .. . . . . . . . . . . . . 35
3. Phase Equilibrium in Binary Alloys . 37
3.1. Free Energy of a System ..... 37
xi
xii Contents
3.2. Partial Pressure . . . . . . . . . . . . . . . . . . . 38
3.3. Gibbs Free Energy of Binary System ....... . 39
3.4. Change in Gibbs Free Energy in the Formation of a
Binary System. . . . . . . . . . . . . . . . . . .40
3.5. Free Energy of a Binary Alloy with Two Phases · 45
3.6. Equilibrium Phase Diagrams · 50
3.7. Phase Rule ........ . 51
3.8. One-Component Systems .. 51
3.9. Two-Component Systems .. 53
3.9.1. Isomorphous Alloys 53
3.9.2. Eutectic Alloys . 56
3.9.3. EutectoidAlloys. 58
3.9.4. Peritectic Alloys . · 58
Suggested Reading .60
References . . . . . . . . . . . . . .60
4. Electronic Structure of Pure Metallic Solids 61
4.1. Simple Metals . . . . . . . . . . . 61
4.2. The One-Electron Approximation 63
4.3. Rartree Theory ......... . .64
4.4. The Free-Electron Approximation · 68
4.5. d-Band Metals .......... . · 71
4.6. The Friedel Model of d-Band Metals . .77
4.7. Pseudopotentials in Metallic Solids .. · 79
4.7.1. Plane Wave Representation . 80
4.7.2. Orthogonalized Plane Wave Representation 81
4.7.3. The Concept of Pseudop otential .. 82
4.7.4. Nonuniqueness in Pseudopotentials .. 83
4.7.5. The Concept of Model Potential ... 84
4.7.6. Pseudopotentials in Transition Metals 87
Suggested Reading · 91
References . . . . . . . . . . . . . . . . . . . . . . . . .92
5. Physical Effects of Impurities in Metals . 93
5.1. Physical Effects 93
5.2. Valence Effect . 95
5.3. Size Effect . . . 97
5.3.1. Blatt Correction 97
5.3.2. Direct Size Effect 98
5.4. Charge Transfer . . . . . . 100
5.5. Properties of Dilute Metallic Alloys 103
5.5.1. Entropy in Metallic Alloys 103