From Hamiltonians to Phase Diagrams: The Electronic and Statistical-Mechanical Theory of sp-Bonded Metals and Alloys

415 Pages·1987·30.2 MB·English

by  Professor Dr. Jürgen Hafner (auth.)

70
Springer Series in Solid-State Sciences
Edited by Peter Fulde
Springer Series in Solid-State Sciences
Editors: M. Cardona P. Fulde K. von Klitzing H.-J. Queisser
40 Semiconductor Physics An Introduction 58 The Recursion Method and Its Applications
3rd Edition By K. Seeger Editors: D. Pettifor and D. Weaire
41 The LMTO Method 59 Dynamical Processes and
Muffin-Tin Orbitals and Electronic Structure Ordering on Solid Surfaces
By H.L. Skriver Editors: A. Yoshirnori and M. Tsukada
42 Crystal Optics with Spatial Dispersion, 60 Excitonic Processes in Solids
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43 Resonant Nonlinear Interactions of 61 LocaIization, Interaction, and
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44 Elastic Media with Microstructure n 62 Theory of Heavy Fermions
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45 Electronic Properties of Doped Semiconductors 63 Electronic Properties of
By B.1. Shklovskii and A. L. Efros Polymers and Related Compounds
Editors: H. Kuzmany, M. Mehring, and S. Roth
46 Topological Disorder in Condensed Matter
Editors: F. Yonezawa and T. Ninomiya 64 Symmetries in Physics: Group Theory
Applied to Physical Problems
47 Statics and Dynamics of Nonlinear Systems
By W. Ludwig and C. Falter
Editors: G. Benedek, H. Bilz, and R. Zeyher
65 Phonons: Theory and Experiments II
48 Magnetic Phase Transitions
Experiments and Interpretation of
Editors: M. Ausloos and R. J. Elliott
Experimental Results By P. Briiesch
49 Organic Molecular Aggregates, Electronic 66 Phonons: Theory and Experiments III
Excitation and Interaction Processes Phenomena Related to Phonons
Editors: P. Reineker, H. Haken, and H.C. Wolf By P. Briiesch
50 Multiple Diffraction of X-Rays in Crystals 67 Two-Dimensional Systems: Physics
By Shih-Lin Chang and New Devices
51 Phonon Scattering in Condensed Matter Editors: G. Bauer, F. Kuchar, and H. Heinrich
Editors: W. Eisenmenger, K. LaBmann, 68 Phonon Scattering in Condensed Matter V
and S. Dottinger Editors: A. C. Anderson and J. P. Wolfe
52 Superconductivity in Magnetic and Exotic 69 Nonlinearity in Condensed Matter
Materials Editors: A.R. Bishop, D.K. Campbell,
Editors: T. Matsubara and A. Kotani P. Kumar and S. E. Trullinger
53 Two-Dimensional Systems, Heterostructures, 70 From Hamiltonians to Phase Diagrams
and Superlattices The Electronic and Statistical-Mechanical
Editors: G. Bauer, F. Kuchar, and H. Heinrich Theory of sp-Bonded Metals and Alloys
54 Magnetic Excitations and Fluctuations By J. Hafner
Editors: S. Lovesey, V. Balucani, F. Borsa, 71 High Magnetic Fields in Semiconductor Physics
and V. Tognetti Editor: G. Landwehr
55 The Theory of Magnetism II 72 One-Dimensional Conductors
Thermodynamics and Statistical Mechanics By S. Kagoshima, T. Sambongi,
By D. C. Mattis and H. Nagasawa
56 Spin Fluctuations in Itinerant Electron 73 Quantum Solid-State Physics
Magnetism By T. Moriya Editors: S. V. Vonsovsky and M.1. Katsnelson
57 PolycrystaIline Semiconductors, 74 Quantum Monte Carlo Methods in Equilibrium
Physical Properties and Applications and Nonequilibrium Systems
Editor: G. Harbeke Editor: M. Suzuki
Volumes 1-39 are listed on the back inside cover
Jiirgen Hafner
From Hamiltonians
to Phase Diagrams
The Electronic and Statistical-Mechanical Theory
of sp-Bonded Metals and Alloys
With 147 Figures
Springer-Verlag Berlin Heidelberg New York
London Paris Tokyo
Professor Dr. Jiirgen Hafner
Institut fUr Theoretische Physik, Techn. Universitat Wien, Karlsplatz 13,
A-1040 Wien, Austria
Series Editors:
Professor Dr., Dr. h. c. Manuel Cardona
Professor Dr., Dr. h. c. Peter Fulde
Professor Dr. Klaus von Klitzing
Professor Dr. Hans-Joachim Queisser
Max-Planck-Institut fUr Festkorperforschung, Heisenbergstrasse 1
D-7000 Stuttgart 80, Fed. Rep. of Germany
ISBN -13 :978-3-642-83060-0 e-ISBN- 13: 978-3-642-83058-7
DOl: 10.1007/978-3-642-83058-7
Library of Congress Cataloging·in·Publication Data. Hafner, J. (Jiirgen), 1945-. From Hamiltonians to
phase diagrams. (Springer series in solid-state sciences; 70). Bibliography: p. Includes index. 1. Metal-metal
bonds. 2. Alloys. 3. Phase transformations (Statistical physics) 4. Phase diagrams. I. Title. II. Series.
QD461.H16 1987 530.4'1 87-9846
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Preface
The development of the modern theory of metals and alloys has coincided
with great advances in quantum-mechanical many-body theory, in electronic
structure calculations, in theories of lattice dynamics and of the configura
tional thermodynamics of crystals, in liquid-state theory, and in the theory
of phase transformations. For a long time all these different fields expanded
quite independently, but now their overlap has become sufficiently large that
they are beginning to form the basis of a comprehensive first-principles the
ory of the cohesive, structural, and thermodynamical properties of metals
and alloys in the crystalline as well as in the liquid state. Today, we can set
out from the quantum-mechanical many-body Hamiltonian of the system of
electrons and ions, and, following the path laid out by generations of the
oreticians, we can progress far enough to calculate a pressure-temperature
phase diagram of a metal or a composition-temperature phase diagram of a
binary alloy by methods which are essentially rigorous and from first prin
ciples.
This book was written with the intention of confronting the materials
scientist, the metallurgist, the physical chemist, but also the experimen
tal and theoretical condensed-matter physicist, with this new and exciting
possibility. Of course there are limitations to such a vast undertaking as
this. The selection of the theories and techniques to be discussed, as well
as the way in which they are presented, are necessarily biased by personal
inclination and personal expertise. The book begins with a discussion of
the fundamental concepts that guide our way, starting with things that are
as basic as the adiabatic approximation or the idea of the self-consistent
field. To the condensed-matter theorist this is of course all familiar and he
or she might just glance rapidly over the first chapter, but I hope that the
non-specialist will find this introduction helpful. The introduction also sets
the framework for the remaining chapters which should provide the reader
with a working knowledge of the electronic theory of interatomic interac
tions, the theory of cohesion and structure, the thermodynamics of crystals,
liquid-state theory, etc., as far as is necessary to reach the final goal. These
chapters also present a necessarily very personal view of the state of the art
in each field. The presentation is as self-contained as possible; no familiar
ity with advanced theoretical concepts beyond the level of basic quantum
mechanics, statistical mechanics and solid-state theory is assumed.
v
Most of what is presented in this book resides upon the concept of
effective interatomic potentials. To date, the most reliable technique for
calculating these potentials in metals and alloys is based on reponse theory,
which is valid only in systems whose ions scatter the electrons weakly and
are describable by pseudopotentials. These are the sp-bonded or simple
metals. For this reason most of the book is devoted to this particular class
of metals and their alloys, with only short excursions into the theory of
d-band or transition metals to sketch the essential differences between sp
and d-bonding and to draw the reader's attention to some interesting new
developments. But here, as everywhere else in this book, I could not aim
for encyclopaedic completeness.
The book reflects the long-time research interests ofthe author. Several
chapters grew out of lectures given at the Technical University of Vienna and
at the Laboratoire de Thermodynamique et Physico-Chimie Metallurgiques
of the Ecole Normale Superieure d'Electrochimie et d'Electrometallurgie de
Grenoble. The ideas presented here have been shaped in many discussions
with friends and colleagues all over the world. I do not attempt to name
them all, as such a list could never be complete. However, I could not forget
to mention Volker Heine and the late Heinz Bilz - it has always been a
great privilege to work with them and I greatly enjoyed their hospitality at
the Cavendish Laboratory of the University of Cambridge and at the Max
Planck-Institut fUr Festkorperforschung. I should also not forget to mention
my friends Pierre Hicter, Ferencz Igloi, Sitaram Jaswal, Gerhard Kahl, Alain
Pasturel, Gottfried Punz, and Werner Weber, whose cooperation I deeply
appreciate. To all others who might have been cited but have'not, 'I extend
my apologies, and I sincerely hope that t'hey will accept my gratitude.
I thank my colleagues at the Institute of Theoretical Physics under the
direction of Prof. Otto Hittmair for creating a congenial athmosphere for
my scientific work.
I thank Prof. Peter Fulde who suggested that I should write this book.
I am grateful to Dr. II.K.V. Lotsch for his patience and to Dr. A.M. Lahee
for her efficient cooperation and for making several excellent suggestions
concerning the presentation.
Finally, I should not forget to thank my wife Anneliese and my sons
Thomas and Michael. They certainly had to bear an absentminded husband
and father over many evenings and weekends and they have tolerated it with
good humour.
The book is dedicated to my parents.
Vienna" January 1987 Jurgen Hafner
VI
Contents
1. Introduction................................................. 1
1.1 Why This Book Was Written and What It Contains..... .. 1
1.2 The Thermodynamic Origin of Phase Diagrams. . . . . . . . . . . 4
1.2.1 First-Order Transitions Without Compositional
Change - The pT Phase Diagram. . . . . . . . . . . . . . . . . . 5
1.2.2 First-Order Transitions with Compositional Change
- The Alloy Phase Diagram. . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.3 Second-Order Transitions. . . . . . . . . . . . . . .. . . . . . . . . . . 8
1.3 Adiabatic Decoupling of the Ionic and Electronic Degrees
of Freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4 The One-Electron Approximation........ ................. 10
1.5 Tightly-Bound and Nearly-Free Electrons; Potentials and
Pseudopotentials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.6 Response Theory and Interatomic Interactions. . . . . . . . . . . . . 18
1.7 The Statistical Mechanics of a Vibrating Lattice. . . . . . . . . . . 20
1.8 Periodic, Aperiodic and Quasi-Periodic Structures. . . . . . . . . 22
1.9 Elementary Excitations in Aperiodic Structures. . . . . . . . . . . 24
1.9.1 Effective Medium Approximations. . . . . . . . . . . . . . . . . . 24
1.9.2 The Recursion Method and Related Techniques. . . . . 27
1.10 Configurational Thermodynamics of Solids and Liquids. . . . 28
1.10.1 Order-Disorder Transitions........................ 29
1.10.2 From the Interatomic Force Law to the Structure
of a Liquid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.10.3 Order-Parameter Approach to Freezing and Melting 31
2. Interatomic Forces in Metals and Alloys.................. 34
2.1 Pseudopotentials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.1.1 The Operator Approach............................ 35
2.1.2 The Scattering Approach. . . . . . . . . . . . . . . . . . .. . . . . . . . 37
2.1.3 Model Potentials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.2 Response Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.2.1 Nonlocality........................................ 45
2.2.2 Screening Beyond the Random Phase Approximation 46
2.3 Effective Pair Potentials in Pure Metals. . . . . . . . . . . . . . . . . . . 48
VII
2.4 Effective Pair Potentials in Binary Alloys. . . . . . . . . . . . . . . . . . 54
2.4.1 Chemical Compression............................. 58
2.4.2 Chemical Ordering.. . . ... . .. . . .. . . . . . . .. . . . . . .. . . . . 59
2.5 Interatomic Forces in Non-Simple Metals and Alloys. . . . . . . 63
2.6 Beyond Perturbation Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3. Phase Stability of Crystalline Metals. . . . . . . . . . . . . . . . . . . . . . 67
3.1 Simple-Metal Cohesion.................................... 67
3.1.1 An Excursion into Transition-Metal Cohesion....... 72
3.2 Structural Stability.. .. .. . .. . .. . .. .. .. . . . . .. .. .. .. .. .. . . . . 77
3.3 Trends in Crystal Structures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.3.1 A Brief Remark on Transition Metal Structures. . . . 82
3.3.2 The Crystal Structures of the Lanthanides. . . . . . . . . . 84
3.3.3 Charge-Density Analysis of Bonding .... " . . . . . . . . . . 85
3.4 Pressure-Induced Phase Changes. . . . . . . .. . . . . . . . . . . . . . . . . . 88
3.5 Thermodynamics of Crystalline Metals. . . . . . . . . . . . . . . . . . . . 90
3.5.1 Harmonic Lattice Dynamics. ...... ................. 90
3.5.2 Anharmonicity.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.5.3 Variational Method for Calculating Thermodynamic
Properties (Gibbs-Bogoljubov Method)............ . 96
3.6 Temperature-Induced Phase Changes. . . . . . . . . . . .. . . . . . . . . . 98
4. Structure and Thermodynamics of Liquid Metals.... . . . . 102
4.1 Computer Simulations. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . 103
4.2 Integral Equation Approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.3 Thermodynamic Perturbation Theories.................... 107
4.3.1 Thermodynamic Variational Method for Liquids
(Gibbs-Bogoljubov Method)..................... ... 107
4.3.2 Repulsive Forces: Weeks-Chandler-Andersen Theory 110
4.3.3 Attractive Forces: Random Phase Approximation,
Optimized Random Phase Approximation and Mean
Spherical Approximation. . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.4 Trends in Liquid Structures. . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . 121
4.5 Expanded Fluid Metals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
4.6 Structure and Thermodynamics of Liquid Transition and
Rare-Earth Metals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
4.7 Atomic Motion in Liquid Metals.......................... 128
5. The pT Phase Diagram of Pure Metals. . . . . . . . . . . . . . . . . . . 133
5.1 Solid-Liquid Trnasitions: The Total Energy Approach. . . . . 133
5.2 Microscopic Theories of Melting and Freezing. . . . . . . . . . . . . 137
5.3 The Liquid-Vapour Transition............................. 142
VIII
6. Alloy Formation and Stability............................. 145
6.1 Nearly-Free-Electron Approach to Alloy Formation. . . . . . . . 146
6.1.1 Criteria for Solubility in Homovalent Systems. . . . . . . 146
6.1.2 Variations of the Atomic Volume in Heterovalent
Alloy Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
6.1.3 The Chemical Potential Model for the Heat of
Formation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
6.1.4 Band Picture for Simple-Metal Alloys.............. 157
6.1.5 Real-Space Picture for Alloy Formation... ......... 159
6.1.6 A Brief Summary..................... ...... ....... 162
6.2 Miedema's Semiempirical Theory of Alloy Formation...... 163
6.2.1 Microscopic Interpretation of Miedema's Alloying
Rules: Simple Metals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
6.2.2 Microscopic Interpretation of Miedema's Alloying
Rules: Transition Metals. . . . . . . . . . . . . . . . . . . . . . . . . . . 165
7. Solid Substitutional Alloys................................. 169
7.1 Primary Solid Solutions..... ........ ...................... 171
7.1.1 The Homovalent Case.... .. .. .. .. .. .. .. .. .. .. .. .. .. 171
7.1.2 The Heterovalent Case.. .. .. .. .. .. .. .. .... .. .. .. .. . 177
7.2 Hume-Rothery Phases...... ..................... ......... 180
7.3 Static Lattice Distortions. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . 184
7.4 Ordering in Substitutional Alloys. . . . . . . . . . . . . . . . . . . . . . . . . . 188
7.4.1 Long-Range Order................................. 190
7.4.2 Short-Range Order. . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . 195
7.4.3 Ordering in Substitutional Transition-Metal Alloys. 196
7.5 Thermodynamics of Alloys.......... ...... ................ 197
7.5.1 Lattice Dynamics of Substitutional Alloys.......... 197
7.5.2 Thermodynamic Perturbation Theory.............. 202
7.5.3 Vibrational Dynamics and the Ordering Transition. 205
8. Intermetallic Compounds...... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
8.1 Structure Maps........... ........ ........ ................ 207
8.2 Empirical Pair-Potential Analysis of Intermetallic Phases. . 211
8.3 Classification of Intermetallic Phases According to Building
Principles and Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
8.4 Topologically Close-Packed Intermetallic Compounds
(Frank-Kasper Phases).................. .................. 215
8.4.1 Heat and Volume of Formation of Laves Phases..... 215
8.4.2 Structural Stability of Laves Phases. . . . . . . . . . . . . . . .. 221
8.4.3 Other Topologically Close-Packed Compounds. . . . . . 227
8.4.4 Charge-Density Analysis of Bonding. . . . . . . . . . . . . . . . 230
8.4.5 Lattice Dynamics of Topologically Close-Packed
Compounds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
IX
8.5 Intermetallic Phases with Large Band-Structure
Stabilization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
8.5.1 Charge-Density Analysis of Bonding in Zintl Phases 238
8.5.2 Lattice Dynamics of Zintl Phases. . . . . . . . . . . . . . . . . . . 239
9. Liquid Alloys................................................ 241
9.1 Computer Simulations of Binary Liquid Alloys............ 243
9.2 Thermodynamic Variational Calculations. . . . . . . . . . . . . . . . . . 246
9.2.1 Systems with a Nearly Ideal Mixing Behaviour..... 247
9.2.2 Liquid Alloys with Strong Chemical Short-Range
Order.............................................. 254
9.2.3 Liquid Alloys with a Miscibility Gap. . . . . . . . . . . . . . . 262
9.3 Thermodynamic Perturbation Theory. . . . . . . . . . . . . . . . . . . . . 267
9.3.1 Repulsive Forces: Weeks-Chandler-Andersen
Theory. ............. .. ............................ 267
9.3.2 Long-Range Forces: Optimized-Random-Phase
Approximation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
9.4 Structure and Thermodynamics of Liquid Transition-Metal
Alloys.................................................... 275
9.5 Collective Excitations in Liquid Alloys. . . . . . . . . . . . . . . . . . . . 278
10. Alloy Phase Diagrams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
10.1 First Principles Calculations of Alloy Phase Diagrams. . . . . 282
10.2 Chemical Short-Range Order and Alloy Phase Diagrams... 285
10.2.1 Melting Extrema. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286
10.2.2 Eutectic Diagrams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
10.2.3 Compound Formation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
10.2.4 Phase Separation in the Liquid State. . . . . . . . . . . . . 289
10.3 Molecular Theory of the Freezing of Liquid Alloys. . . . . . . . 289
11. Beyond the Phase Diagram: The Formation and
Properties of Metastable Phases. . . . . . . . . . . . . . . . . . . . . . . . . . . 292
11.1 Amorphous Alloys - Metallic Glasses.... .................. 292
11.1.1 Glass-Forming Ability. . . . . . . . . . . . . . . . . . . ... . . . .. . . 293
11.1.2 Atomic Structure of Metallic Glasses.............. 298
11.1.3 Elementary Excitations in Metallic Glasses.. . . . . . . 307
11.2 Quasi-Crystals............................................ 309
12. Conclusions and Outlook................................... 313
Appendices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315
A. Density-Functional Pseudopotentials . . . . . . . . . . . . . . . . . . . . . . . . . . . 315
A.1 Optimized Pseudopotentials - the Operator Approach.... 319
A.2 Norm-Conserving Pseudopotentials - the Scattering
Approach. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . .. . . . . . 323
x