EDITOR IN CHIEF Rudy J. M. Konings European Commission, Joint Research Centre, Institute for Transuranium Elements, Karlsruhe, Germany
SECTION EDITORS Todd R. Allen Department of Engineering Physics, University of Wisconsin, Madison, WI, USA Roger E. Stoller Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN, USA Shinsuke Yamanaka Division of Sustainable Energy and Environmental Engineering, Graduate School of Engineering, Osaka University, Osaka, Japan
Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, The Netherlands The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, UK 225 Wyman Street, Waltham, MA 02451, USA Copyright © 2012 Elsevier Ltd. All rights reserved The following articles are US Government works in the public domain and not subject to copyright: Radiation Effects in UO2 TRISO-Coated Particle Fuel Performance Composite Fuel (cermet, cercer) Metal Fuel-Cladding Interaction No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone (þ44) (0) 1865 843830; fax (þ44) (0) 1865 853333; email:
[email protected]. Alternatively you can submit your request online by visiting the Elsevier web site at http://elsevier.com/locate/permissions, and selecting Obtaining permission to use Elsevier material Notice No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein, Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Catalog Number: 2011929343 ISBN (print): 978-0-08-056027-4 For information on all Elsevier publications visit our website at books.elsevier.com Cover image courtesy of Professor David Sedmidubsky´, The Institute of Chemical Technology, Prague Printed and bound in Spain 12 13 14 15 16 10 9 8 7 6 5 4 3 2 1
Editorial : Gemma Mattingley Production: Nicky Carter
EDITORS BIOGRAPHIES Rudy Konings is currently head of the Materials Research Unit in the Institute for Transuranium Elements (ITU) of the Joint Research Centre of the European Commission. His research interests are nuclear reactor fuels and actinide materials, with particular emphasis on high temperature chemistry and thermodynamics. Before joining ITU, he worked on nuclear fuel-related issues at ECN (the Energy Research Centre of the Netherlands) and NRG (Nuclear Research and Consultancy Group) in the Netherlands. Rudy is editor of Journal of Nuclear Materials and is professor at the Delft University of Technology (Netherlands), where he holds the chair of ‘Chemistry of the nuclear fuel cycle.’
Roger Stoller is currently a Distinguished Research Staff Member in the Materials Science and Technology Division of the Oak Ridge National Laboratory and serves as the ORNL Program Manager for Fusion Reactor Materials for ORNL. He joined ORNL in 1984 and is actively involved in research on the effects of radiation on structural materials and fuels for nuclear energy systems. His primary expertise is in the area of computational modeling and simulation. He has authored or coauthored more than 100 publications and reports on the effects of radiation on materials, as well as edited the proceedings of several international conferences.
Todd Allen is an Associate Professor in the Department of Engineering Physics at the University of Wisconsin – Madison since 2003. Todd’s research expertise is in the area of materials-related issues in nuclear reactors, specifically radiation damage and corrosion. He is also the Scientific Director for the Advanced Test Reactor National Scientific User Facility as well as the Director for the Center for Material Science of Nuclear Fuel at the Idaho National Laboratory, positions he holds in conjunction with his faculty position at the University of Wisconsin.
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Editors Biographies
Shinsuke Yamanaka is a professor in Division of Sustainable Energy and Environmental Engineering, Graduate School of Engineering, Osaka University since 1998. He has studied the thermophysics and thermochemistry of nuclear fuel and materials. His research for the hydrogen behavior in LWR fuel cladding is notable among his achievements and he received the Young Scientist Awards (1980) and the Best Paper Awards (2004) from Japan Atomic Energy Society. Shinsuke is the program officer of Japan Science and Technology Agency since 2005 and the visiting professor of Fukui University since 2009, and he is also the associate dean of Graduate School of Engineering, Osaka University since 2011.
PREFACE There are essentially three primary energy sources for the billions of people living on the earth’s surface: the sun, radioactivity, and gravitation. The sun, an enormous nuclear fusion reactor, has transmitted energy to the earth for billions of years, sustaining photosynthesis, which in turn produces wood and other combustible resources (biomass), and the fossil fuels like coal, oil, and natural gas. The sun also provides the energy that steers the climate, the atmospheric circulations, and thus ‘fuelling’ wind mills, and it is at the origin of photovoltaic processes used to produce electricity. Radioactive decay of primarily uranium and thorium heats the earth underneath us and is the origin of geothermal energy. Hot springs have been used as a source of energy from the early days of humanity, although it took until the twentieth century for the potential of radioactivity by fission to be discovered. Gravitation, a non-nuclear source, has been long used to generate energy, primarily in hydropower and tidal power applications. Although nuclear processes are thus omnipresent, nuclear technology is relatively young. But from the moment scientists unraveled the secrets of the atom and its nucleus during the twentieth century, aided by developments in quantum mechanics, and obtained a fundamental understanding of nuclear fission and fusion, humanity has considered these nuclear processes as sources of almost unlimited (peaceful) energy. The first fission reactor was designed and constructed by Enrico Fermi in 1942 in Chicago, the CP1, based on the fission of uranium by neutron capture. After World War II, a rapid exploration of fission technology took place in the United States and the Union of Soviet Socialist Republics, and after the Atoms for Peace speech by Eisenhower at the United Nations Congress in 1954, also in Europe and Japan. A variety of nuclear fission reactors were explored for electricity generation and with them the fuel cycle. Moreover, the possibility of controlled fusion reactions has gained interest as a technology for producing energy from one of the most abundant elements on earth, hydrogen. The environment to which materials in nuclear reactors are exposed is one of extremes with respect to temperature and radiation. Fuel pins for nuclear reactors operate at temperatures above 1000 C in the center of the pellets, in fast reactor oxide fuels even above 2000 C, whereas the effects of the radiation (neutrons, alpha particles, recoil atoms, fission fragments) continuously damage the material. The cladding of the fuel and the structural and functional materials in the fission reactor core also operate in a strong radiation field, often in a dynamic corrosive environment of the coolant at elevated temperatures. Materials in fusion reactors are exposed to the fusion plasma and the highly energetic particles escaping from it. Furthermore, in this technology, the reactor core structures operate at high temperatures. Materials science for nuclear systems has, therefore, been strongly focussed on the development of radiation tolerant materials that can operate in a wide range of temperatures and in different chemical environments such as aqueous solutions, liquid metals, molten salts, or gases. The lifetime of the plant components is critical in many respects and thus strongly affects the safety as well as the economics of the technologies. With the need for efficiency and competitiveness in modern society, there is a strong incentive to improve reactor components or to deploy advanced materials that are continuously developed for improved performance. There are many examples of excellent achievements in this respect. For example, with the increase of the burnup of the fuel for fission reactors, motivated by improved economics and a more efficient use of resources, the Zircaloy cladding (a Zr–Sn alloy) of the fuel pins showed increased susceptibility to coolant corrosion, but within a relatively short period, a different zirconium-based alloy was developed, tested, qualified, and employed, which allowed reliable operation in the high burnup range.
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Nuclear technologies also produce waste. It is the moral obligation of the generations consuming the energy to implement an acceptable waste treatment and disposal strategy. The inherent complication of radioactivity, the decay that can span hundreds of thousands of years, amplifies the importance of extreme time periods in the issue of corrosion and radiation stability. The search for storage concepts that can guarantee the safe storage and isolation of radioactive waste is, therefore, another challenging task for materials science, requiring a close examination of natural (geological) materials and processes. The more than 50 years of research and development of fission and fusion reactors have undoubtedly demonstrated that the statement ‘technologies are enabled by materials’ is particularly true for nuclear technology. Although the nuclear field is typically known for its incremental progress, the challenges posed by the next generation of fission reactors (Generation IV) as well as the demonstration of fusion reactors will need breakthroughs to achieve their ambitious goals. This is being accompanied by an important change in materials science, with a shift of discovery through experiments to discovery through simulation. The progress in numerical simulation of the material evolution on a scientific and engineering scale is growing rapidly. Simulation techniques at the atomistic or meso scale (e.g., electronic structure calculations, molecular dynamics, kinetic Monte Carlo) are increasingly helping to unravel the complex processes occurring in materials under extreme conditions and to provide an insight into the causes and thus helping to design remedies. In this context, Comprehensive Nuclear Materials aims to provide fundamental information on the vast variety of materials employed in the broad field of nuclear technology. But to do justice to the comprehensiveness of the work, fundamental issues are also addressed in detail, as well as the basics of the emerging numerical simulation techniques. R.J.M. Konings European Commission, Joint Research Centre, Institute for Transuranium Elements, Karlsruhe, Germany T.R. Allen Department of Engineering Physics, Wisconsin University, Madison, WI, USA R. Stoller Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN, USA S. Yamanaka Division of Sustainable Energy and Environmental Engineering, Graduate School of Engineering, Osaka University, Osaka, Japan
FOREWORD ‘Nuclear materials’ denotes a field of great breadth and depth, whose topics address applications and facilities that depend upon nuclear reactions. The major topics within the field are devoted to the materials science and engineering surrounding fission and fusion reactions in energy conversion reactors. Most of the rest of the field is formed of the closely related materials science needed for the effects of energetic particles on the targets and other radiation areas of charged particle accelerators and plasma devices. A more complete but also more cumbersome descriptor thus would be ‘the science and engineering of materials for fission reactors, fusion reactors, and closely related topics.’ In these areas, the very existence of such technologies turns upon our capabilities to understand the physical behavior of materials. Performance of facilities and components to the demanding limits required is dictated by the capabilities of materials to withstand unique and aggressive environments. The unifying concept that runs through all aspects is the effect of radiation on materials. In this way, the main feature is somewhat analogous to the unifying concept of elevated temperature in that part of materials science and engineering termed ‘high-temperature materials.’ Nuclear materials came into existence in the 1950s and began to grow as an internationally recognized field of endeavor late in that decade. The beginning in this field has been attributed to presentations and discussions that occurred at the First and Second International Conferences on the Peaceful Uses of Atomic Energy, held in Geneva in 1955 and 1958. Journal of Nuclear Materials, which is the home journal for this area of materials science, was founded in 1959. The development of nuclear materials science and engineering took place in the same rapid growth time period as the parent field of materials science and engineering. And similarly to the parent field, nuclear materials draws together the formerly separate disciplines of metallurgy, solid-state physics, ceramics, and materials chemistry that were early devoted to nuclear applications. The small priesthood of first researchers in half a dozen countries has now grown to a cohort of thousands, whose home institutions are anchored in more than 40 nations. The prodigious work, ‘Comprehensive Nuclear Materials,’ captures the essence and the extensive scope of the field. It provides authoritative chapters that review the full range of endeavor. In the present day of glance and click ‘reading’ of short snippets from the internet, this is an old-fashioned book in the best sense of the word, which will be available in both electronic and printed form. All of the main segments of the field are covered, as well as most of the specialized areas and subtopics. With well over 100 chapters, the reader finds thorough coverage on topics ranging from fundamentals of atom movements after displacement by energetic particles to testing and engineering analysis methods of large components. All the materials classes that have main application in nuclear technologies are visited, and the most important of them are covered in exhaustive fashion. Authors of the chapters are practitioners who are at the highest level of achievement and knowledge in their respective areas. Many of these authors not only have lived through a substantial part of the history sketched above, but they themselves are the architects. Without those represented here in the author list, the field would certainly be a weaker reflection of itself. It is no small feat that so many of my distinguished colleagues could have been persuaded to join this collective endeavor and to make the real sacrifices entailed in such time-consuming work. I congratulate the Editor, Rudy Konings, and
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the Associate Editors, Roger Stoller, Todd Allen, and Shinsuke Yamanaka. This book will be an important asset to young researchers entering the field as well as a valuable resource to workers engaged in the enterprise at present. Dr. Louis K. Mansur Oak Ridge, Tennessee, USA
Permission Acknowledgments The following material is reproduced with kind permission of Cambridge University Press Figure 15 of Oxide Dispersion Strengthened Steels Figure 15 of Minerals and Natural Analogues Table 10 of Spent Fuel as Waste Material Figure 21b of Radiation-Induced Effects on Microstructure www.cambridge.org The following material is reproduced with kind permission of American Chemical Society Figure 2 of Molten Salt Reactor Fuel and Coolant Figure 22 of Molten Salt Reactor Fuel and Coolant Table 9 of Molten Salt Reactor Fuel and Coolant Figure 6 of Thermodynamic and Thermophysical Properties of the Actinide Nitrides www.acs.org The following material is reproduced with kind permission of Wiley Table 3 of Properties and Characteristics of SiC and SiC/SiC Composites Table 4 of Properties and Characteristics of SiC and SiC/SiC Composites Table 5 of Properties and Characteristics of SiC and SiC/SiC Composites Figure 5 of Advanced Concepts in TRISO Fuel Figure 6 of Advanced Concepts in TRISO Fuel Figure 30 of Material Performance in Supercritical Water Figure 32 of Material Performance in Supercritical Water Figure 19 of Tritium Barriers and Tritium Diffusion in Fusion Reactors Figure 9 of Waste Containers Figure 13 of Waste Containers Figure 21 of Waste Containers Figure 11 of Carbide Fuel Figure 12 of Carbide Fuel Figure 13 of Carbide Fuel Figure 4 of Thermodynamic and Thermophysical Properties of the Actinide Nitrides Figure 2 of The U–F system Figure 18 of Fundamental Point Defect Properties in Ceramics Table 1 of Fundamental Point Defect Properties in Ceramics Figure 17 of Radiation Effects in SiC and SiC-SiC Figure 21 of Radiation Effects in SiC and SiC-SiC Figure 6 of Radiation Damage in Austenitic Steels Figure 7 of Radiation Damage in Austenitic Steels Figure 17 of Ceramic Breeder Materials Figure 33a of Carbon as a Fusion Plasma-Facing Material Figure 34 of Carbon as a Fusion Plasma-Facing Material i
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Figure 39 of Carbon as a Fusion Plasma-Facing Material Figure 40 of Carbon as a Fusion Plasma-Facing Material Table 5 of Carbon as a Fusion Plasma-Facing Material www.wiley.com The following material is reproduced with kind permission of Springer Figure 4 of Neutron Reflector Materials (Be, Hydrides) Figure 6 of Neutron Reflector Materials (Be, Hydrides) Figure 1 of Properties and Characteristics of SiC and SiC/SiC Composites Figure 3 of Properties and Characteristics of SiC and SiC/SiC Composites Figure 4 of Properties and Characteristics of SiC and SiC/SiC Composites Figure 5 of Properties and Characteristics of SiC and SiC/SiC Composites Figure 6 of Properties and Characteristics of SiC and SiC/SiC Composites Figure 7 of Properties and Characteristics of SiC and SiC/SiC Composites Figure 8 of Properties and Characteristics of SiC and SiC/SiC Composites Figure 9 of Properties and Characteristics of SiC and SiC/SiC Composites Figure 10 of Properties and Characteristics of SiC and SiC/SiC Composites Figure 11 of Properties and Characteristics of SiC and SiC/SiC Composites Figure 12 of Properties and Characteristics of SiC and SiC/SiC Composites Figure 22d of Fission Product Chemistry in Oxide Fuels Figure 3 of Behavior of LWR Fuel During Loss-of-Coolant Accidents Figure 14a of Irradiation Assisted Stress Corrosion Cracking Figure 14b of Irradiation Assisted Stress Corrosion Cracking Figure 14c of Irradiation Assisted Stress Corrosion Cracking Figure 25a of Irradiation Assisted Stress Corrosion Cracking Figure 25b of Irradiation Assisted Stress Corrosion Cracking Figure 1 of Properties of Liquid Metal Coolants Figure 5b of Fast Spectrum Control Rod Materials Figure 3 of Oxide Fuel Performance Modeling and Simulations Figure 8 of Oxide Fuel Performance Modeling and Simulations Figure 10 of Oxide Fuel Performance Modeling and Simulations Figure 11 of Oxide Fuel Performance Modeling and Simulations Figure 14 of Oxide Fuel Performance Modeling and Simulations Figure 5 of Thermodynamic and Thermophysical Properties of the Actinide Nitrides Figure 51 of Phase Diagrams of Actinide Alloys Figure 6 of Thermodynamic and Thermophysical Properties of the Actinide Oxides Figure 7b of Thermodynamic and Thermophysical Properties of the Actinide Oxides Figure 9b of Thermodynamic and Thermophysical Properties of the Actinide Oxides Figure 35 of Thermodynamic and Thermophysical Properties of the Actinide Oxides Table 11 of Thermodynamic and Thermophysical Properties of the Actinide Oxides Table 13 of Thermodynamic and Thermophysical Properties of the Actinide Oxides Table 17 of Thermodynamic and Thermophysical Properties of the Actinide Oxides Figure 18 of Radiation Damage of Reactor Pressure Vessel Steels Figure 7 of Radiation Damage Using Ion Beams Figure 9b of Radiation Damage Using Ion Beams Figure 28 of Radiation Damage Using Ion Beams Figure 34 of Radiation Damage Using Ion Beams Figure 35 of Radiation Damage Using Ion Beams Figure 36d of Radiation Damage Using Ion Beams Figure 37 of Radiation Damage Using Ion Beams Table 3 of Radiation Damage Using Ion Beams
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Figure 5 of Radiation Effects in UO2 Figure 9a of Ab Initio Electronic Structure Calculations for Nuclear Materials Figure 9b of Ab Initio Electronic Structure Calculations for Nuclear Materials Figure 9c of Ab Initio Electronic Structure Calculations for Nuclear Materials Figure 10a of Ab Initio Electronic Structure Calculations for Nuclear Materials Figure 23 of Thermodynamic and Thermophysical Properties of the Actinide Carbides Figure 25 of Thermodynamic and Thermophysical Properties of the Actinide Carbides Figure 26 of Thermodynamic and Thermophysical Properties of the Actinide Carbides Figure 27 of Thermodynamic and Thermophysical Properties of the Actinide Carbides Figure 28a of Thermodynamic and Thermophysical Properties of the Actinide Carbides Figure 28b of Thermodynamic and Thermophysical Properties of the Actinide Carbides Figure 2 of Physical and Mechanical Properties of Copper and Copper Alloys Figure 5 of Physical and Mechanical Properties of Copper and Copper Alloys Figure 6 of The Actinides Elements: Properties and Characteristics Figure 10 of The Actinides Elements: Properties and Characteristics Figure 11 of The Actinides Elements: Properties and Characteristics Figure 12 of The Actinides Elements: Properties and Characteristics Figure 15 of The Actinides Elements: Properties and Characteristics Table 1 of The Actinides Elements: Properties and Characteristics Table 6 of The Actinides Elements: Properties and Characteristics Figure 25 of Fundamental Properties of Defects in Metals Table 1 of Fundamental Properties of Defects in Metals Table 7 of Fundamental Properties of Defects in Metals Table 8 of Fundamental Properties of Defects in Metals www.springer.com The following material is reproduced with kind permission of Taylor & Francis Figure 9 of Radiation-Induced Segregation Figure 6 of Radiation Effects in Zirconium Alloys Figure 1 of Dislocation Dynamics Figure 25 of Radiation Damage Using Ion Beams Figure 26 of Radiation Damage Using Ion Beams Figure 27 of Radiation Damage Using Ion Beams Figure 4 of Radiation-Induced Effects on Material Properties of Ceramics (Mechanical and Dimensional) Figure 7 of The Actinides Elements: Properties and Characteristics Figure 20 of The Actinides Elements: Properties and Characteristics Figure 18a of Primary Radiation Damage Formation Figure 18b of Primary Radiation Damage Formation Figure 18c of Primary Radiation Damage Formation Figure 18d of Primary Radiation Damage Formation Figure 18e of Primary Radiation Damage Formation Figure 18f of Primary Radiation Damage Formation Figure 1 of Radiation-Induced Effects on Microstructure Figure 27 of Radiation-Induced Effects on Microstructure Figure 5 of Performance of Aluminum in Research Reactors Figure 2 of Atomic-Level Dislocation Dynamics in Irradiated Metals Figure 3 of Atomic-Level Dislocation Dynamics in Irradiated Metals Figure 5 of Atomic-Level Dislocation Dynamics in Irradiated Metals Figure 10a of Atomic-Level Dislocation Dynamics in Irradiated Metals Figure 10b of Atomic-Level Dislocation Dynamics in Irradiated Metals Figure 10c of Atomic-Level Dislocation Dynamics in Irradiated Metals
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Figure 10d of Atomic-Level Dislocation Dynamics in Irradiated Metals Figure 12a of Atomic-Level Dislocation Dynamics in Irradiated Metals Figure 12b of Atomic-Level Dislocation Dynamics in Irradiated Metals Figure 12c of Atomic-Level Dislocation Dynamics in Irradiated Metals Figure 12d of Atomic-Level Dislocation Dynamics in Irradiated Metals Figure 16a of Atomic-Level Dislocation Dynamics in Irradiated Metals Figure 16b of Atomic-Level Dislocation Dynamics in Irradiated Metals Figure 16c of Atomic-Level Dislocation Dynamics in Irradiated Metals Figure 16d of Atomic-Level Dislocation Dynamics in Irradiated Metals Figure 16e of Atomic-Level Dislocation Dynamics in Irradiated Metals Figure 17a of Atomic-Level Dislocation Dynamics in Irradiated Metals Figure 17b of Atomic-Level Dislocation Dynamics in Irradiated Metals Figure 17c of Atomic-Level Dislocation Dynamics in Irradiated Metals Figure 17d of Atomic-Level Dislocation Dynamics in Irradiated Metals www.taylorandfrancisgroup.com
1.01
Fundamental Properties of Defects in Metals
W. G. Wolfer Ktech Corporation, Albuquerque, NM, USA; Sandia National Laboratories, Livermore, CA, USA
Published by Elsevier Ltd.
1.01.1 1.01.2 1.01.3 1.01.3.1 1.01.3.2 1.01.3.3 1.01.4 1.01.4.1 1.01.4.2 1.01.4.3 1.01.4.4 1.01.5 1.01.5.1 1.01.5.2 1.01.5.3 1.01.6 1.01.6.1 1.01.6.2 1.01.6.3 1.01.7 1.01.7.1 1.01.7.2 1.01.7.3 1.01.7.4 1.01.7.4.1 1.01.7.4.2 1.01.7.4.3 1.01.7.4.4 1.01.8 1.01.8.1 1.01.8.2 1.01.8.2.1 1.01.8.2.2 1.01.8.3 1.01.9 Appendix A A1 A2 Appendix B B1 B2 B3 B4 B5 References
Introduction The Displacement Energy Properties of Vacancies Vacancy Formation Vacancy Migration Activation Volume for Self-Diffusion Properties of Self-Interstitials Atomic Structure of Self-Interstitials Formation Energy of Self-Interstitials Relaxation Volume of Self-Interstitials Self-Interstitial Migration Interaction of Point Defects with Other Strain Fields The Misfit or Size Interaction The Diaelastic or Modulus Interaction The Image Interaction Anisotropic Diffusion in Strained Crystals of Cubic Symmetry Transition from Atomic to Continuum Diffusion Stress-Induced Anisotropic Diffusion in fcc Metals Diffusion in Nonuniform Stress Fields Local Thermodynamic Equilibrium at Sinks Introduction Edge Dislocations Dislocation Loops Voids and Bubbles Capillary approximation The mechanical concept of surface stress Surface stresses and bulk stresses for spherical cavities Chemical potential of vacancies at cavities Sink Strengths and Biases Effective Medium Approach Dislocation Sink Strength and Bias The solution of Ham Dislocation bias with size and modulus interactions Bias of Voids and Bubbles Conclusions and Outlook Elasticity Models: Defects at the Center of a Spherical Body An Effective Medium Approximation The Isotropic, Elastic Sphere with a Defect at Its Center Representation of Defects by Atomic Forces and by Multipole Tensors Kanzaki Forces Volume Change from Kanzaki Forces Connection of Kanzaki Forces with Transformation Strains Multipole Tensors for a Spherical Inclusion Multipole Tensors for a Plate-Like Inclusion
2 3 5 5 8 10 12 12 13 15 15 16 16 17 20 21 22 23 25 26 26 26 27 29 29 29 30 31 32 32 33 33 35 35 37 38 38 38 41 41 42 43 43 43 44
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Fundamental Properties of Defects in Metals
Abbreviations bcc CA CD dpa EAM fcc hcp INC IHG SIA Vac
Body-centered cubic Cavity model Center of dilatation model Displacements per atom Embedded atom method Face-centered cubic Hexagonal closed packed Inclusion model Inhomogeneity model Self-interstitial atom Vacancy
1.01.1 Introduction Several fundamental attributes and properties of crystal defects in metals play a crucial role in radiation effects and lead to continuous macroscopic changes of metals with radiation exposure. These attributes and properties will be the focus of this chapter. However, there are other fundamental properties of defects that are useful for diagnostic purposes to quantify their concentrations, characteristics, and interactions with each other. For example, crystal defects contribute to the electrical resistivity of metals, but electrical resistivity and its changes are of little interest in the design and operation of conventional nuclear reactors. What determines the selection of relevant properties can best be explained by following the fate of the two most important crystal defects created during the primary event of radiation damage, namely vacancies and self-interstitials. The primary event begins with an energetic particle, a neutron, a high-energy photon, or an energetic ion, colliding with a nucleus of a metal atom. When sufficient kinetic energy is transferred to this nucleus or metal atom, it is displaced from its crystal lattice site, leaving behind a vacant site or a vacancy. The recoiling metal atom may have acquired sufficient energy to displace other metal atoms, and they in turn can repeat such events, leading to a collision cascade. Every displaced metal atom leaves behind a vacancy, and every displaced atom will eventually dissipate its kinetic energy and come to rest within the crystal lattice as a self-interstitial defect. It is immediately obvious that the number of self-interstitials is exactly equal to the number of vacancies produced, and they form Frenkel pairs. The number of Frenkel pairs created is also referred to as the number of displacements, and their accumulated density is
expressed as the number of displacements per atom (dpa). When this number becomes one, then on average, each atom has been displaced once. At the elevated temperatures that exist in nuclear reactors, vacancies and self-interstitials diffuse through the crystal. As a result, they will encounter each other, either annihilating each other or forming vacancy and interstitial clusters. These events occur already in their nascent collision cascade, but if defects escape their collision cascade, they may encounter the defects created in other cascades. In addition, migrating vacancy defects and interstitial defects may also be captured at other extended defects, such as dislocations, cavities, grain boundaries and interface boundaries of precipitates and nonmetallic inclusions, such as oxide and carbide particles. The capture events at these defect sinks may be permanent, and the migrating defects are incorporated into the extended defects, or they may also be released again. However, regardless of the complex fate of each individual defect, one would expect that eventually the numbers of interstitials and vacancies that arrive at each sink would become equal, as they are produced in equal numbers as Frenkel pairs. Therefore, apart from statistical fluctuations of the sizes and positions of the extended defects, or the sinks, the microstructure of sinks should approach a steady state, and continuous irradiation should change the properties of metals no further. It came as a big surprise when radiation-induced void swelling was discovered with no indication of a saturation. In the meantime, it has become clear that the microstructure evolution of extended defects and the associated changes in macroscopic properties of metals in general is a continuing process with displacement damage. The fundamental reason is that the migration of defects, in particular that of self-interstitials and their clusters, is not entirely a random walk but is in subtle ways guided by the internal stress fields of extended defects, leading to a partial segregation of self-interstitials and vacancies to different types of sinks. Guided then by this fate of radiation-produced atomic defects in metals, the following topics are presented in this chapter: 1. The displacement energy required to create a Frenkel pair. 2. The energy stored within a Frenkel pair that consists of the formation enthalpies of the selfinterstitial and the vacancy. 3. The dimensional changes that a solid suffers when self-interstitial and vacancy defects are created,
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Fundamental Properties of Defects in Metals
4.
5.
6.
7.
and how these changes manifest themselves either externally or internally as changes in lattice parameter. These changes then define the formation and relaxation volumes of these defects and their dipole tensors. The regions occupied by the atomic defects within the crystal lattice possess a distorted, if not totally different, arrangement of atoms. As a result, these regions are endowed with different elastic properties, thereby changing the overall elastic constants of the defect-containing solid. This leads to the concept of elastic polarizability parameters for the atomic defects. Both the dipole tensors and elastic polarizabilities determine the strengths of interactions with both internal and external stress fields as well as their mutual interactions. When the stress fields vary, the gradients of the interactions impose drift forces on the diffusion migration of the atomic defects that influences their reaction rates with each other and with the sinks. At these sinks, vacancies can also be generated by thermal fluctuations and be released via diffusion to the crystal lattice. Each sink therefore possesses a vacancy chemical potential, and this potential determines both the nucleation of vacancy defect clusters and their subsequent growth to become another defect sink and part of the changing microstructure of extended defects.
The last two topics, 6 and 7, as well as topic 1, will be further elaborated in other chapters.
where mc2 is the rest energy of an electron and L ¼ 4 mM/(m þ M)2. The approximation on the right is adequate because the electron mass, m, is much smaller than the mass, M, of the recoiling atom. Changing the direction of the electron beam in relation to the orientation of single crystal film specimens, one finds that the threshold energy varies significantly. However, for polycrystalline samples, values averaged over all orientations are obtained, and these values are shown in Figure 1 for different metals as a function of their melting temperatures.1 First, we notice a trend that Td increases with the melting temperature, reflecting the fact that larger energies of cohesion or of bond strengths between atoms also lead to higher melting temperatures. We also display values of the formation energy of a Frenkel pair. Each value is the sum of the corresponding formation energies of a self-interstitial and a vacancy for a given metal. These energies are presented and further discussed below. The important point to be made here is that the displacement energy required to create a Frenkel pair is invariably larger than its formation energy. Clearly, an energy barrier exists for the recoil process, indicating that atoms adjacent to the one that is being displaced also receive some additional kinetic energy that is, however, below the displacement energy Td and is subsequently dissipated as heat. The displacement energies listed in Table 1 and shown in Figure 1 are averaged not only over crystal orientation but also over temperature for those metals
50 Displacement energy Td(eV) fcc Frenkel pair (eV) bcc Frenkel pair (eV)
Scattering of energetic particles from external sources, be they neutrons, electrons, ions, or photons, or emission of such particles from an atomic nucleus, imparts a recoil energy. When this recoil energy exceeds a critical value, called the threshold displacement energy, Td, Frenkel pairs can be formed. To measure this displacement energy, an electron beam is employed to produce the radiation damage in a thin film of the material, and its rise in electrical resistivity due to the Frenkel pairs is monitored. By reducing the energy of the electron beam, the resistivity rise is also reduced, and a threshold electron energy, Emin, can be found below which no Frenkel pairs are produced. The corresponding recoil energy is given by relativistic kinematics as 2mc 2 þ Emin m Emin ½1 E 4 1 þ Td ¼ LEmin min 2mc 2 þ LEmin 2mc 2 M
Displacement and Frenkel pair energies (eV)
1.01.2 The Displacement Energy 40
30
20
10
0
0
500
1000 1500 2000 2500 3000 3500 Melting temperature (K)
4000
Figure 1 Energies of displacement and energies of Frenkel pairs for elemental metals as a function of their melting temperatures.
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Fundamental Properties of Defects in Metals
Table 1 Element
Displacement and Frenkel pair energies of elemental metals Symbol
Z
Melt temp. ( K)
M
Td (eV)
Frenkel pair energy (eV) fcc
Silver Aluminum Gold Cadmium Cobalt Chromium Copper Iron Indium Iridium Magnesium Molybdinum Niobium Neodymium Nickel Lead Palladium Platinum Rhenium Tantalum Titanium Vanadium Tungsten Zinc Zirconium
Ag Al Au Cd Co Cr Cu Fe In Ir Mg Mo Nb Nd Ni Pb Pd Pt Re Ta Ti V W Zn Zr
47 13 79 48 27 24 29 26 49 77 12 42 41 60 28 82 46 78 75 73 22 23 74 30 40
107.9 26.98 197.0 112.4 58.94 52.01 63.54 55.85 114.8 192.2 24.32 95.95 92.91 144.3 58.71 207.2 106.4 195.1 186.2 181.0 47.90 50.95 183.9 65.38 91.22
fcc fcc fcc hcp hcp bcc fcc bcc tetragonal fcc hcp bcc bcc hcp fcc fcc fcc fcc hcp bcc hcp bcc bcc hcp hcp
1235 933.5 1337 594.2 1768 2180 1358 1811 429.8 2719 923.2 2896 2750 1289 1728 600.6 1828 2041 3458 3290 1941 2183 3695 692.7 2128
26.0 15.3 34.0 19.0 23.0 28.0 18.3 17.4 10.5 46.0 10.0 32.4 28.2 9.30 22.0 11.8 34.0 34.0 44.0 26.7 20.8 28.0 44.0 12.0 22.5
bcc
6.52 4.96 5.98 4.40
6.66
4.03
6.69 4.01 10.5 3.24 6.51 12.7 5.77 3.26 8.90
Source: Displacement energies from Jung, P. In Landolt-Bo¨rnstein; Springer-Verlag: Berlin, 1991; Vol. III/25, pp 8–11.
50 Td fcc Td bcc Td hcp and others Td, eV
40 Displacement energy (eV)
where the displacement energy has been measured as a function of irradiation temperature. For some materials, such as Cu, a significant decrease of the displacement energy with temperature has been found. However, a definitive explanation is still lacking. Close to the minimum electron energy for Frenkel pair production, the separation distance between the self-interstitial and its vacancy is small. Therefore, their mutual interaction will lead to their recombination. With increasing irradiation temperature, however, the self-interstitial may escape, and this would manifest itself as an apparent reduction in the displacement energy with increasing temperature. On the other hand, Jung2 has argued that the energy barrier involved in the creation of Frenkel pairs is directly dependent on the temperature in the following way. This energy barrier increases with the stiffness of the repulsive part of the interatomic potential; a measure for this stiffness is the bulk modulus. Indeed, as Figure 2 demonstrates, the displacement energy increases with the bulk modulus. Since the bulk modulus decreases with temperature, so will the displacement energy. The correlation of the displacement energy with the bulk modulus appears to be a somewhat better
30
20
10
0 0
50
100
150
200
250
300
350
400
Bulk modulus (GPa) Figure 2 Displacement energies for elemental metals as a function of their bulk modulus.
empirical relationship than the correlation with the melt temperature. However, one should not read too much into this, as the bulk modulus B, atomic
Fundamental Properties of Defects in Metals
volume O, and melt temperature of elemental metals approximately satisfy the rule BO 100kB Tm discovered by Leibfried3 and shown in Figure 3.
1.01.3 Properties of Vacancies 1.01.3.1
Vacancy Formation
The thermal vibration of atoms next to free surfaces, to grain boundaries, to the cores of dislocations, etc., make it possible for vacancies to be created and then diffuse into the crystal interior and establish an equilibrium thermal vacancy concentration of f EV TSVf eq ½2 CV ðT Þ ¼ exp kB T given in atomic fractions. Here, EVf is the vacancy formation enthalpy, and SVf is the vacancy formation entropy. The thermal vacancy concentration can be measured by several techniques as discussed in Damask and Dienes,4 Seeger and Mehrer,5 and Siegel,6 and values for EVf have been reviewed and tabulated by Ehrhart and Schultz;7 they are listed in Table 2. When these values for the metallic elements are plotted versus the melt temperature in Figure 4, an approximate correlation is obtained, namely EVf Tm =1067
½3
4000
Bulk mod.*atom. vol./(100 k)
3500 3000 2500
1500 1000 500 0
Using the Leibfried rule, a new approximate correlation emerges for the vacancy formation enthalpy that has become known as the cBO model8; the constant c is assumed to be independent of temperature and pressure. As seen from Figure 5, however, the experimental values for EVf correlate no better with BO than with the melting temperature. It is tempting to assume that a vacancy is just a void and its energy is simply equal to the surface area 4pR2 times the specific surface energy g0. Taking the atomic volume as the vacancy volume, that is, O ¼ 4pR3/3, we show in Figure 6 the measured vacancy formation enthalpies as a function of 4pR2g0, using for g0 the values9 at half the melting temperatures. It is seen that EVf is significantly less, by about a factor of two, compared to the surface energy of the vacancy void so obtained. Evidently, this simple approach does not take into account the fact that the atoms surrounding the vacancy void relax into new positions so as to reduce the vacancy volume VVf to something less than O. The difference VVrel ¼ VVf O
0
500
1000 1500 2000 2500 3000 3500 4000 Melt temperature (K)
Figure 3 Leibfried’s empirical rule between melting temperature and the product of bulk modulus and atomic volume.
½4
is referred to as the vacancy relaxation volume. The experimental value7 for the vacancy relaxation of Cu is 0.25O, which reduces the surface area of the vacancy void by a factor of only 0.825, but not by a factor of two. The difference between the observed vacancy formation enthalpy and the value from the simplistic surface model has recently been resolved. It will be shown in Section 1.01.7 that the specific surface energy is in fact a function of the elastic strain tangential to the surface, and when this surface strain relaxes, the surface energy is thereby reduced. At the same time, however, the surface relaxation creates a stress field in the surrounding crystal, and hence a strain energy. As a result, the energy of a void after relaxation is given by FC ½eðRÞ; e ¼ 4pR2 g½eðRÞ; e þ 8pR3 me2 ðRÞ
2000
5
½5
The first term is the surface free energy of a void with radius R, and it depends now on a specific surface energy that itself is a function of the surface strain e(R) and the intrinsic residual surface strain e* for a surface that is not relaxed. The second term is the strain energy of the surrounding crystal that depends on its shear modulus m. The strain dependence of the specific surface energy is given by g½e; e ¼ g0 þ 2ðmS þ lS Þð2e þ eÞe
½6
Here, g0 is the specific surface energy on a surface with no strains in the underlying bulk material.
6
Fundamental Properties of Defects in Metals
Table 2
Crystal and vacancy properties
Metal
Crystal
Melt temp. (K)
KO (eV)
Debye temp. (K)
EfV (eV)
Em v (eV)
g0 (J m2)
Ag Al Au Be Co Cr Cs Cu Fe Hf Ir K Li Mg Mn Mo Na Nb Nd Ni Os Pb Pd Pt Re Rh Ru Sb Sr Ta Ti Tl U V W Zn Zr
fcc fcc fcc hcp hcp bcc bcc fcc bcc hcp fcc bcc bcc hcp bcc bcc bcc bcc hcp fcc hcp fcc fcc fcc hcp fcc hcp rbh* fcc bcc hcp hcp bco** bcc bcc hcp hcp
1235 933.5 1337 1560 1768 2180 301.6 1358 1811 2506 2719 336.7 453.7 923 1519 2896 371 2750 1289 1728 3306 600.6 1828 2041 3458 2237 2607 904 1050 3290 1941 577.2 1408 2183 3695 693 2128
10.9 7.89 18.1 6.57 13.1 12.1 41.2 10.1 12.3 15.3 31.3 1.55 1.63 5.13 9.15 25.4 1.70 19.3 6.8 12.5 36.7 8.46 17.7 26.7 34 32.6 18.6 7.9
229.2 430.6 162.7
1.11 0.67 0.93 0.8
588.4
2.1
1.09 1.02 1.33 1.30 2.22 2.01
349.6 483.3
1.28 1.90
0.66 0.61 0.71 0.87 0.72 0.95 0.084 0.70 0.55
25.3 11.8 7.7 13 13.5 30.8 6.49 13.8
92.7 369.5
0.48 0.80
473.4 157.1 254.6
3.10 0.34 2.70
481.4
1.79
0.038 0.50 1.30 1.35 0.03 0.55 0.81 1.04
106.6 277.9
0.58 1.85 1.35 3.10 2.50
0.43 1.03 1.43 2.20 1.50
264.7
3.10
0.70
399.4 384.3
2.10 3.60 0.54
0.50 1.70 0.42 0.58
1.57 2.12 1.92 2.65 0.129 0.472 0.688 2.51 0.234 2.31 2.38 2.95 0.54 1.74 2.20 3.13 2.33 2.65 0.461 0.358 2.49 1.75 0.55 1.78 2.30 2.77 0.896 1.69
*
rbh: rhombohedral bco: body-centered orthorhombic
**
However, such a surface possesses an intrinsic, residual surface strain e*, because the interatomic bonding between surface atoms differs from that in the bulk, and for metals, the surface bond length would be shorter if the underlying bulk material would allow the surface to relax. Partial relaxation is possible for small voids as well as for nanosized objects. In addition to the different bond length at the surface, the elastic constants, mS and lS, are also different from the corresponding bulk elastic constants. However, they can be related by a surface layer thickness, h, to bulk elastic constants such that mS þ lS ¼ ðm þ lÞh ¼ mh=ð1 2nÞ
½7
where l is the Lame’s constant and n is Poisson’s ratio for the bulk solid. Computer simulations on
freestanding thin films have shown10 that the surface layer is effectively a monolayer, and h can be approximated by the Burgers vector b. For planar crystal surfaces, the residual surface strain parameter e* is found to be between 3 and 5%, depending on the surface orientation relative to the crystal lattice. On surfaces with high curvature, however, e* is expected to be larger. The relaxation of the void surface can now be obtained as follows. We seek the minimum of the void energy as defined by eqn [5] by solving @FC =@e ¼ 0. The result is eðRÞ ¼
ðmS þ lS Þe h e ¼ mR þ ðmS þ lS Þ ð1 2nÞR þ h
½8
and this relaxation strain changes the initially unrelaxed void volume
7
Fundamental Properties of Defects in Metals
4
4 bcc Hf/v, eV fcc Hf/v, eV hcp Hf/v, eV
3 2.5 2 1.5 1
0
2.5 2 1.5 1
0
1 2 3 4 5 Surface energy of a vacancy (eV)
6
Figure 6 Correlation between the surface energy of a vacancy void and the vacancy formation energy. 3.5 Vacancy formation energy and its bulk contribution (eV )
4 fcc Hf/v, eV bcc Hf/v, eV hcp Hf/v, eV
3.5 3 2.5 2 1.5 1 0.5 0
0
500 1000 1500 2000 2500 3000 3500 4000 Melting temperature (K)
Figure 4 Vacancy formation energies as a function of melting temperature.
Vacancy formation enthalpy (eV)
3
0.5
0.5 0
fcc Hf/v, eV bcc Hf/v, eV hcp Hf/v, eV
3.5 Vacancy formation enthalpy (eV)
Vacancy formation energy (eV)
3.5
0
5
10
15
20
25
30
35
40
Exp. value Computed values Bulk strain energy
3
2.5
Ni
2
1.5
1
0.5
0 –0.35
–0.3
Bulk modulus * atomic volume (eV)
4p 3 ½9 R 3 consisting of n aggregated vacancies, by the amount nO ¼
½10
Applying these equations to a vacancy, for which n ¼ 1 and R b, we obtain VVrel O
¼
3e 2ð1 nÞ
and for the vacancy formation energy
–0.2
–0.15
–0.1
–0.05
0
Vacancy relaxation volume
Figure 5 Vacancy formation energy versus the product of bulk modulus and atomic volume.
V rel ðRÞ ¼ 3nOeðRÞ
–0.25
½11
Figure 7 Vacancy formation energy and its dependence on the relaxation volume.
( EVf ¼ 4pR2
2 ) 2ð3 4nÞ VVrel g0 mb 9ð1 2nÞ O
rel 2 2 V þ mO V O 3
½12
This equation is evaluated for Ni and the results are shown in Figure 7 as a function of the vacancy relaxa tion volume VVrel =O. It is seen that relaxation volumes of 0.2 to 0.3 predict a vacancy formation energy comparable to the experimental value of 1.8 eV.
8
Fundamental Properties of Defects in Metals
Few experimentally determined values are available for the vacancy relaxation volume, and their accuracy is often in doubt. In contrast, vacancy formation energies are better known. Therefore, we use eqn [12] to determine the vacancy relaxation volumes from experimentally determined vacancy formation energies. The values so obtained are listed in Table 3, and for the few cases7 where this is possible, they are compared with the values reported from experiments. Computed values for the vacancy relaxation volumes are between 0.2O and 0.3O for both fcc and bcc metals. The low experimental values for Al, Fe, and Mo then appear suspect. The surface energy model employed here to derive eqn [12] is based on several approximations: isotropic, linear elasticity, a surface energy parameter, g0, that represents an average over different crystal orientations, and extrapolation of the energy of large voids to the energy of a vacancy. Nevertheless, this approximate model provides satisfactory results and captures an important connection between the vacancy relaxation volume and the vacancy formation energy that has also been noted in atomistic calculations. Finally, a few remarks about the vacancy formation entropy, SVf , are in order. It originates from the change in the vibrational frequencies of atoms surrounding the vacancy. Theoretical estimates based on empirical potentials provide values that range from 0.4k to about 3.0k, where k is the Boltzmann constant. As a result, the effect of the vacancy formation entropy on the magnitude of the thermal equilibrium vacancy coneq centration, CV , is of the same magnitude as the statistical uncertainty in the vacancy formation enthalpy. Table 3
1.01.3.2
Vacancy Migration
The atomistic process of vacancy migration consists of one atom next to the vacant site jumping into this site and leaving behind another vacant site. The jump is thermally activated, and transition state theory predicts a diffusion coefficient for vacancy migration in cubic crystals of the form DV ¼ nLV d02 expðSVm ÞexpðEVm =kB T Þ ¼ DV0 expðEVm =kB T Þ
½13
Here, nLV is an average frequency for lattice vibrations, d0 is the nearest neighbor distance between atoms, SVm is the vacancy migration entropy, and EVm is the energy for vacancy migration. It is in fact the energy of an activation barrier that the jumping atom must overcome, and when it temporarily occupies a position at the height of this barrier, the atomic configuration is referred to as the saddle point of the vacancy. It will be considered in greater detail momentarily. Values obtained for EVm from experimental measurements are shown in Figure 8 as a function of the melting point. While we notice again a trend similar to that for the vacancy formation energy, we find that EVm for fcc and bcc metals apparently follow different correlations. However, the correlation for bcc metals is rather poor, and it indicates that EVm may be related to fundamental properties of the metals other than the melting point. The saddle point configuration of the vacancy involves not just the displacement of the jumping atom but also the coordinated motion of other atoms that are nearest neighbors of the vacancy and of the jumping atom. These nearest neighbor atoms
Vacancy relaxation volumes for metals
Metal
g 0 (J m2)
m (GPa)
n
HfV (eV)
V rel V =V (model)
Ag Al Au Cu Ni Pb Pd Pt Cr a-Fe Mo Nb Ta V W
1.19 1.1 1.45 1.71 2.28 0.57 1.91 2.40 2.23 2.31 2.77 2.54 2.76 2.51 3.09
33.38 26.18 31.18 54.7 94.6 10.38 53.02 65.1 117.0 90.4 125.8 39.6 89.9 47.9 160.2
0.354 0.347 0.412 0.324 0.276 0.387 0.374 0.393 0.209 0.278 0.293 0.397 0.324 0.361 0.280
1.11 0.05 0.67 0.03 0.93 0.04 1.28 0.05 1.79 0.05 0.58 0.04 1.7, 1.85 1.35 0.05 2.0 0.3 1.4, 1.89 3.2 0.09 2.6, 3.07 2.2, 3.1 2.2 0.4 3.1, 4.1
0.247 0.005 0.311 0.003 0.262 0.003 0.259 0.005 0.236 0.004 0.282 0.005 0.239, 0.225 0.260 0.003 0.218 0.02 0.278, 0.245 0.191 0.004 0.284, 0.258 0.264, 0.228 0.298 0.028 0.201, 0.161
V rel V =V (experiment) 0.05, 0.38 0.15 to 0.5 0.25 0.2 0.24, 0.42 0.05 0.1
Fundamental Properties of Defects in Metals
lie at the corners of a rectangular plane as shown in Figure 9. As the jumping atom crosses this plane, they are displaced such as to open the channel. This coordinated motion can be viewed as a particular strain fluctuation and described in terms of phonon excitations. In this manner, Flynn11 has derived the following formula for the energy of vacancy migration in cubic crystals. EVm ¼
15C11 C44 ðC11 C12 Þa3 w 2½C11 ðC11 C12 Þ þ C44 ð5C11 3C12 Þ
½14
Here, a is the lattice parameter, C11, C12, and C44 are elastic moduli, and w is an empirical parameter that characterizes the shape of the activation barrier and can be determined by comparing experimental vacancy migration energies with values predicted by eqn [14]. Ehrhart et al.7,12 recommend that w ¼ 0.022 for fcc metals and w ¼ 0.020 for bcc metals.
Vacancy migration energy (eV)
2.5
In the derivation of Flynn,11 only the four nearest neighbor atoms are supposed to move, while all other atoms are assumed to remain in their normal lattice positions. On the other hand, Kornblit et al.13 treat the expansion of the diffusion channel as a quasistatic elastic deformation of the entire surrounding material. The extent of the expansion is such that the opened channel is equal to the cross-section of the jumping atom, and a linear anisotropic elasticity calculation is carried out by a variational method to determine the energy involved in the channel expansion. A vacancy migration energy is obtained for fcc metals of EVm ¼ 0:01727a3 C11
p02
p0 p1 p22 þ 29 p0 p2 þ 19 p22
½15
and the parameters pi will be defined momentarily. For bcc metals,14 the activation barrier consist of two peaks of equal height EVmax with a shallow valley in between with an elevation of EVmin , where
fcc Hm/v, eV hcp Hm/v, eV bcc Hm/v, eV
2
9
q0 q1 q22 q02 112 q0 p2 þ 883 q22
½16
s0 s1 s22 2 s0 0:29232s0 s2 þ 0:0413s22
½17
EVmax ¼ 0:003905a 3 C11 and
1.5
EVmin ¼ 0:002403a 3 C11 1
The parameters pi, qi, and si are linear functions of the elastic moduli with coefficients listed in Table 4. For example,
0.5
q1 ¼ 3:45C11 0:75C12 þ 4:35C44 0
0
500 1000 1500 2000 2500 3000 3500 4000 Melt temperature (K)
Figure 8 Vacancy migration energy as a function of melting temperature.
If the depth of the valley is greater than the thermal energy of the jumping atom, that is, greater than 32 kT , then it will be trapped and requires an additional activation to overcome the remaining barrier of
Table 4 expressions
Figure 9 Second nearest neighbor atom (blue) jumping through the ring of four next-nearest atoms (green) into adjacent vacancy in a fcc structure.
Coefficients
for
the
Kornblit
energy
Function
C11
C12
C44
p0 p1 p2 q0 q1 q2 s0 s1 s2
5.29833 0.86667 1.41903 6.36429 3.45 3.32143 3.62621 1.57190 1.46564
4.76499 0.3333 0.88570 3.66429 0.75 0.62143 2.88241 0.82810 0.72184
9.35238 1.9111 1.64444 12.92142 4.35 3.70714 11.30366 4.21984 3.68855
10
Fundamental Properties of Defects in Metals
max EV EVmin . As a result, Kornblit14 assumes that the
vacancy migration energy for bcc metals is given by
EVm ¼
EVmax ; max 2EV EVmin ;
if EVmax EVmin 32 kT if EVmax EVmin > 32 kT
½18
Using the formulae of Flynn and Kornblit, we compute the vacancy migration energies and compare them with experimental values in Figure 10. With a few exceptions, both the Flynn and the Kornblit values are in good agreement with the experimental results. The self-diffusion coefficient determines the transport of atoms through the crystal under conditions near the thermodynamic equilibrium, and it is defined as
m eq DSD ¼ DV CV ¼ nLV a 2 exp SVf þ SVm =k expðQSD =kT Þ ½19
0 expðQSD =kT Þ ¼ DSD
where the activation energy for self-diffusion is Q SD ¼ EVf þ EVm
½20
The most accurate measurements of diffusion coefficients are done with a radioactive tracer isotope of the metal under investigation, and in this case one obtains values for the tracer self-diffusion coeffiT ¼ fD cient DSD SD that involves the correlation factor f. For pure elemental metals of cubic structure, f is a constant and can be determined exactly by computation.15 For fcc crystals, f ¼ 0.78145, and for bcc crystals, f ¼ 0.72149.
Theoretical vacancy migration energy (eV)
2 Evm, Flynn, fcc Evm, Kornblit, fcc Evm, Flynn, bcc Evm, Kornblit, bcc
1.5
1
To determine the preexponential factor for selfdiffusion 0 ¼ nLV a 2 exp SVf þ SVm =k ½21 DSD requires the values for the entropy SVf þ SVm and for the attempt frequency nLV. Based on theoretical estimates, Seeger and Mehrer5 recommend a value of 2.5 k for the former. The atomic vibration of nearest neighbor atoms to the vacancy is treated within a sinusoidal potential energy profile that has a maximum height of EVm . For small-amplitude vibrations, the attempt frequency is then given by rffiffiffiffiffiffi 1 EVm for fcc and by nLV ¼ a M rffiffiffiffiffiffiffiffi 1 2EVm ½22 nLV ¼ for bcc a 3M crystals where M is the atomic mass. In contrast, Flynn11 assumes that the atomic vibrations can be derived from the Debye model for which the average vibration frequency is rffiffiffi 3 kYD ½23 nLV ¼ 5 h where YD is the Debye temperature and h is Planck’s constant. The calculated preexponential factors for some fcc metals according to the models by Seeger and Mehrer5 and by Flynn are listed in Table 5 together with experimental values. They are also shown in Figure 11. While the computed values are of the right order of magnitude, there exists no clear correlation between the experimental and theoretical values. In fact, the computed values change little from one metal to another, and the Flynn model predicts values about twice as large as the model by Seeger and Mehrer. Either model can therefore be used to provide a reasonable estimate of the preexponential factor where no experimental value is available. 1.01.3.3 Activation Volume for Self-Diffusion
0.5
When the crystal lattice is under pressure p, the selfdiffusion coefficient changes and is then given by 0
0
0.5
1
1.5
Experimental vacancy migration energy (eV) Figure 10 Comparison of computed vacancy migration energies according to models by Flynn and Kornblit with measured values.
2
0 DSD ðT ; pÞ ¼ DSD expðQSD =kT ÞexpðpVSD =kT Þ
½24
The activation volume VSD can be obtained experimentally by measuring the self-diffusion coefficient as a function of an externally applied pressure. Such measurements have been carried out only for a few
Fundamental Properties of Defects in Metals
11
Preexponentials for tracer self-diffusion
Metal
M
a (nm)
QD (K)
Em V (eV)
Experimental value
S&M
Flynn (m2 s1)
Ag Al Au Cu Ni Pb Pd Pt
107.9 26.98 197 63.54 58.71 207.2 106.4 195.1
0.409 0.405 0.408 0.361 0.352 0.495 0.389 0.392
229 430.6 162.7 349.6 481.4 106.6 278 240
0.66 0.61 0.71 0.70 1.04 0.43 1.0 1.4
4.5e6 4.7e6 3.5e6 1.6e5 9.2e5 6.65e5 2.1e5 6.0e6
3.00e6 5.69e6 2.29e6 3.55e6 4.39e6 2.11e6 3.53e6 3.11e6
5.90e6 1.08e6 4.16e6 7.02e6 9.18e6 4.01e6 6.46e6 5.66e6
bcc Fe
Seeger Mehrer Flynn
Experiment Brown and Ashby Wallace Wang et al.
Rb Na
10–5
Li K Cs Zn Tl Element
Theoretical preexponential (m2 s–1)
Table 5
10–6 10–6
Mg Cd Pt
10–5
0.0001
Experimental preexponential Do (m2 s–1)
Pb Ni
Figure 11 Comparison of preexponential factors for tracer self-diffusion as computed with two models and as measured.
fcc Fe Cu Al Ag
metals, and it has been found that the activation volumes have positive values. Therefore, self-diffusion decreases with applied pressure. However, it has been noticed that the self-diffusion coefficient at melting appears to be constant, and this can be explained by the fact that the melting temperature increases in general with pressure. It follows then from the condition d½ln DSD ðp; Tm ðpÞÞ=dpjp¼0 ¼ 0
that
Q dTm VSD ¼ 0 Tm dp p¼0
½25
where Tm0 is the melting temperature under ambient conditions. Brown and Ashby16 have used this relationship to evaluate the activation volumes for self-diffusion
0
0.2 0.4 0.6 0.8 1 Activation volume/atomic volume
1.2
Figure 12 Activation volumes of elements divided by their atomic volumes from experiments and from relationship [25], using the change of the melting temperature with pressure from different sources.
for a variety of metals. Using more recent values for the pressure derivative of the melting temperature by Wallace17 and Wang et al.,18 one obtains activation volumes as shown in Figure 12. They are in reasonably good agreement with the experimental values where they exist. With the exception of Pt, the predicted values are also similar, giving an activation volume of about 0.85O for fcc metals, 0.65O for hcp metals, and around 0.4O for bcc metals.
12
Fundamental Properties of Defects in Metals
Table 6
Activation volume for vacancy migration
Metal
VSD =O
VfV =V
Vm V =V
Ag Al Cu Ni Pb bcc Fe
0.872 0.835 0.895 0.841 0.791 0.655
0.753 0.689 0.741 0.764 0.718 0.722
0.119 0.146 0.154 0.077 0.073 0.067
The equilibrium vacancy concentration in a solid under pressure p is given by f E TSVf pV f eq exp V ½26 CV ðT Þ ¼ exp V kB T kB T where VVf is the vacancy formation volume. Since the self-diffusion coefficient is the product of the thermal vacancy concentration and the vacancy migration coefficient, the activation volume for self-diffusion is the sum of two contributions, namely VSD ¼ VVf þ VVm ¼ O þ VVrel þ VVm
½27
VVm
with being the activation volume for vacancy migration. If one takes the average of the predicted activation volumes shown in Figure 12, and the vacancy relaxation volumes from Table 3, one obtains values for VVm listed in Table 6 and also shown in Figure 12.
1.01.4 Properties of Self-Interstitials 1.01.4.1 Atomic Structure of Self-Interstitials The accommodation of an additional atom within a perfect crystal lattice remained a topic of lively debates at international conferences on radiation effects for many decades. The leading question was the configuration of this interstitial atom and its surrounding atoms. This scientific question has now been resolved, and there is general agreement that this additional atom, a self-interstitial, forms a pair with one atom from the perfect lattice in the form of a dumbbell. The configuration of these dumbbells can be illustrated well with hard spheres, that is, atoms that repel each other like marbles. Let us first consider the case of an fcc metal. In the perfect crystal, each atom is surrounded by 12 nearest neighbors that form a cage around it as shown on the left of Figure 13. When an extra atom is inserted in this cage, the two atoms in the center form a pair
Figure 13 An atom with its 12 nearest neighbors in the perfect fcc lattice, on the left, and a [001] self-interstitial dumbbell with the same nearest neighbors, on the right.
whose axis is aligned in a [001] direction. This [001] dumbbell constitutes the self-interstitial in the fcc lattice. The centers of the 12 nearest neighbor atoms are the apexes of a cubo-octahedron that encloses the single central atom in the perfect lattice, and it can be shown19 that the cubo-octahedron encloses a volume of VO ¼ 10O/3. However, around a self-interstitial dumbbell, this cubo-octahedron expands and distorts, and now it encloses a larger volume of V001 ¼ 4.435O. The volume expansion is the difference DV ¼ V001 V0 ¼ 1:10164 O
½28
which happens to be larger than one atomic volume. We shall see shortly that the volume expansion of the entire crystal is even larger due to the elastic strain field created by the self-interstitial that extends through the entire solid. We consider next the self-interstitial defect in a bcc metal. Here, each atom is surrounded in the perfect crystal by eight nearest neighbors as shown on the left of Figure 14. When an extra atom is inserted, it again forms a dumbbell configuration with another atom, and the dumbbell axis is now aligned in the [011] direction, as shown on the right of Figure 14. The cage formed by the eight nearest neighbor atoms becomes severely distorted. It is surprising, however, that the volume change of the cage is only DV ¼ 0:6418 O
½29
less than the volume of the inserted atom to create the self-interstitial in the bcc structure. The reason for this is that the bcc structure does not produce the most densely packed arrangement of atoms, and some of the empty space can accommodate the self-interstitial. In contrast, the fcc structure has in fact the densest arrangement of atoms, and disturbing it by inserting an extra atom only creates disorder and lower packing density. As already mentioned, the large inclusion volume DV of self-interstitials leads to a strain field
Fundamental Properties of Defects in Metals
Figure 14 On the left is the unit cell of the bcc crystal structure. The central atom shown darker is surrounded by eight nearest neighbors. On the right is the arrangement when a self-interstitial occupies the center of the cell.
throughout the surrounding crystal that causes changes in lattice parameter and that is the major source of the formation energy for self-interstitials. In order to determine this strain field, we treat in Appendix A the case of spherical defects in the center of a spherical solid with isotropic elastic properties. Although this represents a rather simplified model for self-interstitials, for vacancies, and for complex clusters of such defects, it is a very instructive model that captures many essential features. 1.01.4.2 Formation Energy of Self-Interstitials In contrast to the formation energy of vacancies, there exists no direct measurement for the formation energy of self-interstitials. We have mentioned in Section 1.01.2 that the displacement energy required to create a Frenkel pair is much larger than the combined formation energies of the vacancy and the self-interstitial. As pointed out, there exist a large energy barrier to create the Frenkel pair, namely the displacement energy Td, and this barrier is mainly associated with the insertion of the interstitial into the crystal lattice. However, although this barrier should be part of the energy to form a selfinterstitial, it is by convention not included. Rather, the formation energy of a self-interstitial is considered to be the increase of the internal energy of a crystal with this defect in comparison to the energy of the perfect crystal. In contrast, since vacancies can be created by thermal fluctuations at surfaces, grain boundaries, and dislocation cores by accepting an atom from an adjacent lattice site and leaving it vacant, no similar barrier exists. The activation energy for this process is simply the sum of the actual formation energy EVf and the migration energy EVm, that is, the energy for self-diffusion, QSD.
13
When Frenkel pairs are created by irradiation at cryogenic temperatures, self-interstitials and vacancies can be retained in the irradiated sample. Subsequent annealing of the sample and measuring the heat released as the defects migrate and then disappear provide an indirect method to measure the energies of Frenkel pairs. Subtracting from these calorimetric values, the vacancy formation energy should give the formation energy of self-interstitials. The values so obtained for Cu7 vary from 2.8 to 4.2 eV, demonstrating just how inaccurate calorimetric measurements are. Besides, measurements have only been attempted on two other metals, Al and Pt, with similar doubtful results. As a result, theoretical calculations or atomistic simulations provide perhaps more trustworthy results. For a theoretical evaluation of the formation energy, we can consider the self-interstitial as an inclusion (INC) as described in Appendix A. Accordingly, a volume O of one atom is enlarged by the amount DV, or in other words, is subject to the transformation strain eij ¼ dij
where 3 ¼ DV =O
½30
The energy associated with the formation of this inclusion is given in Table A2, and it can be written as 2K mO DV 2 ½31 U0 ¼ 3K þ 4m O This expression for the so-called dilatational strain energy provides a rough approximation to the formation energy of a self-interstitial in fcc metals when the above volume expansion results are used. However, as the nearest neighbor cells depicted in Figures 13 and 14 show, their distorted shapes cannot be adequately described with a radial expansion of the original cell in the ideal crystal as implied by eqn [31]. As the detailed analysis by Wolfer19 indicates, the [001] dumbbell interstitial in the fcc lattice does not change the cell dimension in the [001] direction. In fact, it shortens it slightly, implying that e33 ¼ 0.01005. The volume change is therefore due to the nearest neighbor atoms moving on average away from the dumbbell axis. This can be represented by the transformation strain components e11 ¼ e22 being determined by 2e11 þ e33 ¼
DV ¼ 1:10164 O
The transformation strain tensor for the [001] selfinterstitials in fcc crystals is then
14
Fundamental Properties of Defects in Metals 0
1 0:556 0 0 B C eij ¼ B 0:556 0 C @ 0 A 0 0 0:010 0 1 0 0:189 0 0 0:367 0 0 B C B B 0:189 0 ¼B 0:367 0 C @ 0 Aþ@ 0 0 0 0:377 0 0 0:367 ¼ dij þ~eij
The transformation strain tensor can again be separated into a dilatational and a shear part as 1 C C A ½32
and it can be divided into an dilatational part, dij, and a shear part, ~e ij , as shown. To find the transformation strain tensor for the [011] self-interstitial in bcc crystal, it is convenient to use a new coordinate system with the x3-axis as the dumbbell axis and the x1- and x2-axes emanating from the midpoint of the dumbbell axis and pointing toward the corner atoms. While the distance between these corner atoms and the central atom in the original bcc unit cell is the interatomic distance r0, in the distorted cell containing pffiffiffi the self-interstitial, their distance is reduced to 3r0 =2 as shown by Wolfer.19 As a result, pffiffiffi 3 1 0:134 e11 ¼ e22 ¼ 2 the remaining strain component is then determined by 2e11 þ e33 ¼
DV ¼ 0:6418 O
0
B eij ¼ B @ 0 0
U1 ¼
0:214 0
1
0
C B B 0 C Aþ@ 0:214
0:348
0
0
0:348
0
0
30.7 26.2 27.3 49.8 161.4 91.3 8.08 48.5 64.7 116.9 87.6 0.914 3.95 125.6 1.99 38.1 0.434 89.9 47.9 160.2
1.89 1.51 1.93 2.00 7.17 3.03 1.00 2.70 3.78 1.23 1.04 0.086 0.102 2.05 0.097 0.91 0.055 1.81 0.81 2.58
0.38 0.31 0.35 0.42 1.60 0.69 0.18 0.52 0.72 3.19 2.42 0.170 0.209 4.63 0.194 1.72 0.102 3.89 1.63 5.97
C 0 C A 0:696
2ð9K þ 8mÞmO I2 5ð3K þ 4mÞ
EIf U ¼ U0 þ U1
½34
½35
The dilatational and the shear strain energies for some elements are listed in Table 7 in the fourth and fifth column, respectively. For the fcc elements, the ratio of U1/U0 is about 0.2. In contrast, for the bcc elements (in italics) this ratio is about two.
dK/dP
dG/dP
Vrel/O Theory
102.3 76.1 170.7 137.7 354.7 183.7 44.7 192.7 283.0 161.9 167.7 3.30 12.1 261.7 6.90 172.3 2.20 225.0 155.7 311.0
1
and is equal to I2fcc ¼ 0:1068 and I2bcc ¼ 0:3633 for selfinterstitials in fcc and bcc metals, respectively. We consider now the total strain energy as a reasonable approximation for the formation energy of self-interstitials, namely
K (GPa)
Ag Al Au Cu Ir Ni Pb Pd Pt Cr Fe K Li Mo Na Nb Rb Ta V W
0
½33
where I2 ¼ ~e11~e22 ~e22~e33 ~e33~e11
Metal
U1 (eV)
0
The strain energy associated with the shear part can be shown (see Mura20) to be
Strain energies and relaxation volumes of self-interstitials U0 (eV)
0
¼ dij þ~eij
Table 7
G (GPa)
0:214
6.12 4.42 6.29 5.48 4.83 6.20 5.53 5.35 5.18 4.89 5.29 3.96 3.53 4.40 4.69 6.91 3.63 3.15 3.50 3.95
Elements in italics are bcc, all others are fcc. Experimental values from Ehrhart and Schultz.7
1.40 1.80 1.05 1.35 3.40 1.40 1.10 0.54 1.60 1.40 1.80 0.79 0.42 1.50 0.80 0.53 0.72 1.10 0.94 2.30
1.94 2.03 1.80 1.87 2.67 1.98 1.73 1.44 2.05 1.21 1.43 0.97 0.76 1.26 0.99 0.91 0.97 1.05 1.03 1.58
Experiment 1.9 0.4 1.55 0.3 1.8 1.86 0.3 1.1
1.1 0.2 1.1
Fundamental Properties of Defects in Metals
1.01.4.3 Relaxation Volume of Self-Interstitials If the elastic distortions associated with selfinterstitials could be adequately treated with linear elasticity theory, and if the repulsive interactions between the dumbbell atoms with their nearest neighbors were like that between hard spheres, then the volume change of a solid upon insertion of an interstitial atom would be equal to the volume change DV as derived above. This follows from the analysis of the inclusion in the center of a sphere given in Appendix A. From the results listed in Table A2, under column INC, we see that the volume change of the solid with a concentration S of inclusions is simply given by DV ¼ 3S V where 3 is the volume dilatation per inclusion as if it were not confined by the surrounding matrix. This remarkable result has been proven by Eshelby21 to be valid for any shape of the solid and any location of the inclusion within it, provided the inclusion and the solid can be treated as one linear elastic material. In other words, the elastic strains within the inclusion and within the matrix must be small. However, this is not the case for the elastic strains produced by self-interstitials. Here, the elastic strains are quite large. For example, the volume of the confined inclusion, also listed in Table A2 under column INC, is given by Du 3 3 1þn ¼ ¼ ¼ 3 u 1 þ o gE 3ð1 nÞ and so it is reduced to about 62% of the unconstrained volume for a Poisson’s ratio of n ¼ 0.3. This amounts to an elastic compression of 42% of the ‘volume’ of the self-interstitial in fcc materials, and 25% for the self-interstitial in bcc materials. Clearly, nonlinear elastic effects must be taken into account. Zener22 has found an elegant way to include the effects of nonlinear elasticity on volume changes produced by crystal defects such as self-interstitials and dislocations. If U represents the elastic strain energy of such defects evaluated within linear elasticity theory, and if one then considers the elastic constants in the formula for U to be in fact dependent on the pressure, then the additional volume change dV produced by the defects can be derived from the simple expression dV ¼
@U U @p K
½36
15
found by Schoeck.23 Its application to the strain energy of self-interstitials leads to the following result. 3K m0 =m þ 4mK 0 =K 1 U0 dV ¼ 3K þ 4m K 12ðK 0 m K m0 Þ m0 1 þ U1 ½37 þ ð3K þ 4mÞð9K þ 8mÞ m K Here, m0 and K 0 are the pressure derivatives of the shear and bulk modulus, respectively. The first term arising from the dilatational part of the strain energy was derived and evaluated earlier by Wolfer.19 It is the dominant term for the additional volume change for self-interstitials in fcc metals. Here, we evaluate both terms using the compilation of Guinan and Steinberg24 for the pressure derivatives of the elastic constants, and as listed in Table 7. The calculated relaxation volumes for selfinterstitials, VIrel ¼ DV þ dV
½38
are given in the eighth column of Table 7, and they can be compared with the available experimental values also listed. We shall see that the relaxation volume of selfinterstitials is of fundamental importance to explain and quantify void swelling in metals exposed to fast neutron and charged particle irradiations. 1.01.4.4
Self-Interstitial Migration
The dumbbell configuration of a self-interstitial gives it a certain orientation, namely the dumbbell axis, and upon migration this axis orientation may change. This is indeed the case for self-interstitials in fcc metals, as illustrated in Figure 15. Suppose that the initial location of the selfinterstitial is as shown on the left, and its axis is along [001]. A migration jump occurs by one atom of the dumbbell (here the purple one) pairing up with one nearest neighbor, while its former partner
Figure 15 Migration step of the self-interstitial in fcc metals.
16
Fundamental Properties of Defects in Metals
(the blue atom) occupies the available lattice site. Computer simulations of this migration process have shown25 that the orientation of the self-interstitial has rotated to a [010] orientation, and that this combined migration and rotation requires the least amount of thermal activation. Similar analysis for the migration of selfinterstitials in bcc metals has revealed that a rotation may or may not accompany the migration, and these two diffusion mechanisms are depicted in Figure 16. Which of these two possesses the lower activation energy depends on the metal, or on the interatomic potential employed for determining it. In general, however, the activation energies for self-interstitial migration are very low compared to the vacancy migration energy, and they can rarely be measured with any accuracy. Instead, in most cases only the Stage I annealing temperatures have been measured. In the associated experiments, specimens for a given metal are irradiated at such low temperatures that the Frenkel pairs are retained. Their concentration is correlated with the increase of the electrical resistivity. Subsequent annealing in stages then reveals when the resistivity declines again upon reaching a certain annealing temperature. The first annealing, Stage I, occurs when self-interstitials become mobile and in the process recombine with
vacancies, form clusters of self-interstitials, or are trapped at impurities. Table 8 lists the Stage I temperature,7 TIm , for pure metals as well as two alloys that represent ferritic and austenitic steels. For a few cases, an associated activation energy EIm is known, and in even fewer cases, a preexponential factor, D0I , has been estimated.
1.01.5 Interaction of Point Defects with Other Strain Fields 1.01.5.1
The Misfit or Size Interaction
Many different sources of strain fields may exist in real solids, and they can be superimposed linearly if they satisfy linear elasticity theory. If this is the case, we need to consider here only the interaction between one particular defect located at rd and an extraneous displacement field u0(r) that originates from some other source than the defect itself. In particular, it may be the field associated with external forces or deformations applied to the solid, or it may be the field generated by another defect in the solid. To find the interaction energy, we assume that the defect under consideration is modeled by applying a set of Kanzaki26 forces f(a)(R(a)), a ¼ 1, 2, . . ., z, at z atomic positions R(a) as described in greater detail
Figure 16 Two migration steps are favored by self-interstitials in bcc metals; the left is accompanied by a rotation, while the right maintains the dumbbell orientation.
Fundamental Properties of Defects in Metals Table 8 Annealing temperatures for Stage I and migration properties estimated from them for self-interstitials Metal
Stage I Tm I (K)
Cr Fe K Li Mo Na Nb Ta V W CrxFe1x Ag Al Au Cu Ir Ni Pb Pd Pt Rh Th Cr Fe Ni Be Cd Co Mg Re Sc Ti Zn Zr
36 23–144
