Smithells Metals Reference Book [8 ed.] 0750675098, 9780750675093, 9780080480961

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Smithells Metals Reference Book

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Smithells Metals Reference Book Eighth Edition Edited by

W. F. Gale

PhD Professor, Auburn University, Materials Research and Education Center, Auburn, AL, USA

T. C. Totemeier

PhD Staff Scientist, Idaho National Engineering and Environmental Laboratory, Idaho Falls, ID, USA

A Co-Publication with

amsterdam • boston • heidelberg • london • new york • oxford paris • san diego • san francisco • singapore • sydney • tokyo

Elsevier Butterworth-Heinemann The Boulevard, Langford Lane, Kidlington, Oxford, OX5 1GB 200 Wheeler Road, Burlington, MA 01803, USA First published 1949 Second edition 1955 Third edition 1962 Fourth edition 1967 Fifth edition 1976 Reprinted 1978 Sixth edition 1983 Seventh edition 1992 Paperback edition (with corrections) 1998 Eighth edition 2004 Copyright © 2004 Elsevier Inc. All rights reserved All rights reserved. No part of this publication may be reproduced in any material form (including photocopying or storing in any medium by electronic means and whether or not transiently or incidentally to some other use of this publication) without the written permission of the copyright holder except in accordance with the provisions of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London, England W1T 4LP. Applications for the copyright holder’s written permission to reproduce any part of this publication should be addressed to the publisher. Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone: (+44) 1865 843830, fax: (+44) 1865 853333, e-mail: [email protected]. You may also complete your request on-line via the Elsevier homepage (http://www.elsevier.com), by selecting ‘Customer Support’ and then ‘Obtaining Permissions’

British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library

Library of Congress Cataloguing in Publication Data A catalogue record for this book is available from the Library of Congress ISBN 0 7506 7509 8 For information on all Elsevier Butterworth-Heinemann publications visit our website at http://books.elsevier.com

Typeset by Charon Tec Pvt. Ltd, Chennai, India Printed and bound in The Netherlands

Contents

Preface Acknowledgements

xv xvii

Disclaimer List of contributors

xviii

1 Related designations

1–1

xix

Related designations for steels – Related designations for aluminium alloys – Related designations for copper alloys – Related designations for magnesium alloys

2 Introductory tables and mathematical information 2.1 Conversion factors SI units – Conversion to and from SI units – Temperature conversions, IPTS-48 to IPTS-68 – Corrosion conversion factors – Sieve Nos to aperture size – Temperature scale conversions 2.2 Mathematical formulae and statistical principles Algebra – Series and progressions – Trigonometry – Mensuration – Co-ordinate geometry – Calculus – Introduction to statistics

3 General physical and chemical constants Atomic weight and atomic numbers – General physical constants – Moments of inertia – Periodic system 3.1 Radioactive isotopes and radiation sources Positron emitters – Beta energies and half-lives – Gamma energies and half-lives – Nuclides for alpha, beta, gamma and neutron sources

4 X-ray analysis of metallic materials 4.1 Introduction and cross references 4.2 Excitation of X-rays X-ray wavelengths 4.3 X-ray techniques X-ray diffraction – Specific applications – Crystal geometry 4.4 X-ray results Metal working – Crystal structure – Atomic and ionic radii 4.5 X-ray fluorescence

v

2–1 2–1

2–13

3–1

3–5

4–1 4–1 4–1 4–11 4–39 4–49

vi

Contents

5 Crystallography 5.1 The structure of crystals Translation groups – Symmetry elements – The point group – The space group 5.2 The Schoenflies system of point- and space-group notation 5.3 The Hermann–Mauguin system of point- and space-group notation Notes on the space-group tables

6 Crystal chemistry 6.1 Structures of metals, metalloids and their compounds Structural details – Comparison of Strukturbericht and Pearson nomenclature

7 Metallurgically important minerals

5–1 5–1 5–3 5–3

6–1 6–1

7–1

Minerals, sources, and uses

8 Thermochemical data 8.1 Symbols 8.2 Changes of phase Elements – Intermetallic compounds – Metallurgically important compounds 8.3 Heat, entropy and free energy of formation Elements – Intermetallic compounds – Selenides and tellurides – Intermetallic phases 8.4 Metallic systems of unlimited mutual solubility Liquid binary metallic systems 8.5 Metallurgically important compounds Borides – Carbides – Nitrides – Silicides – Oxides – Sulphides – Halides – Silicates and carbonates – Compound (double) oxides – Phosphides – Phosphides dissociation pressures – Sulphides dissociation pressures 8.6 Molar heat capacities and specific heats Elements – Alloy phases and intermetallic compounds – Borides – Carbides – Nitrides – Silicides – Oxides – Sulphides, selenides and tellurides – Halides 8.7 Vapour pressures Elements – Halides and oxides

9 Physical properties of molten salts

8–1 8–1 8–2 8–8 8–15 8–20

8–39

8–51

9–1

Density of pure molten salts – Densities of molten salt systems – Density of some solid inorganic compounds at room temperature – Electrical conductivity of pure molten salts – Electrical conductivity of molten salt systems – Surface tension of pure molten salts – Surface tension of binary molten salt systems – Viscosity of pure molten salts – Viscosity of molten binary salt systems

10 Metallography 10.1 10.2

Macroscopic examination Etching reagents for macroscopic examination Microscopic examination Plastics for mounting – Attack polishing – Electrolytic polishing solutions – Reagents for chemical polishing – Etching – Color etching – Etching for dislocations

10–1 10–2 10–7

Contents 10.3

10.4 10.5 10.6 10.7

Metallographic methods for specific metals 10–29 Aluminium – Antimony and bismuth – Beryllium – Cadmium – Chromium – Cobalt – Copper – Gold – Indium – Iron and steel – Lead – Magnesium – Molybdenum – Nickel – Niobium – Platinum group metals – Silicon – Silver – Tantalum – Tin – Titanium – Tungsten – Uranium – Zinc – Zirconium – Bearing metals – Cemented carbides and other hard alloys – Powdered and sintered metals Electron metallography and surface analysis techniques 10–74 Transmission electron microscopy – Scanning electron microscopy – Electron spectroscopy and surface analytical techniques Quantitative image analysis 10–81 Scanning acoustic microscopy 10–82 Applications in failure analysis 10–83

11 Equilibrium diagrams 11.1 11.2 11.3 11.4

Index of binary diagrams Equilibrium diagrams Acknowledgements Ternary systems and higher systems

12 Gas–metal systems 12.1

The solution of gases in metals Dilute solutions of diatomic gases – Complex gas–metal systems – Solutions of hydrogen – Solutions of nitrogen – Solutions of oxygen – Solutions of the noble gases – Theoretical and practical aspects of gas–metal equilibria

13 Diffusion in metals 13.1 13.2 13.3

Introduction Methods of measuring D Steady-state methods – Non-steady-state methods – Indirect methods, not based on Fick’s laws Mechanisms of diffusion Self-diffusion in solid elements – Tracer impurity diffusion coefficients – Diffusion in homogeneous alloys – Chemical diffusion coefficient measurements – Self-diffusion in liquid metals

14 General physical properties 14.1

vii

The physical properties of pure metals Physical properties of pure metals at normal temperatures – Physical properties of pure metals at elevated temperatures 14.2 The thermophysical properties of liquid metals Density and thermal expansion coefficient – Surface tension – Viscosity – Heat capacity – Electrical resistivity – Thermal conductivity 14.3 The physical properties of aluminium and aluminium alloys 14.4 The physical properties of copper and copper alloys 14.5 The physical properties of magnesium and magnesium alloys 14.6 The physical properties of nickel and nickel alloys 14.7 The physical properties of titanium and titanium alloys 14.8 The physical properties of zinc and zinc alloys 14.9 The physical properties of zirconium alloys 14.10 The physical properties of pure tin 14.11 The physical properties of steels

11–1 11–1 11–7 11–524 11–533

12–1 12–1

13–1 13–1 13–4 13–8

14–1 14–1 14–9 14–16 14–19 14–22 14–25 14–28 14–29 14–29 14–29 14–30

viii

Contents

15 Elastic properties, damping capacity and shape memory alloys 15.1

15.2 15.3

Elastic properties Elastic constants of polycrystalline metals – Elastic compliances and elastic stiffnesses of single crystals – Principal elastic compliances and elastic stiffnesses at room temperature – Cubic systems (3 constants) – Hexagonal systems (5 constants) – Trigonal systems (6 constants) – Tetragonal systems (6 constants) – Orthorhombic systems (9 constants) Damping capacity Specific damping capacity of commercial alloys – Anelastic damping Shape memory alloys Mechanical properties of shape memory alloys – Compositions and transformation temperatures – Titanium–nickel shape memory alloy properties

16 Temperature measurement and thermoelectric properties 16.1 16.2 16.3

Temperature measurement Fixed points of ITS-90 Thermocouple reference tables Thermoelectric materials Introduction – Survey of materials – Preparation methods

17 Radiative properties of metals

15–1 15–1

15–8 15–37

16–1 16–1 16–4 16–10

17–1

Spectral normal emittance of metals – Spectral emittance in the infra-red metals – Spectral emittance of oxidised metals

18 Electron emission 18.1 18.2 18.3 18.4 18.5 18.6

Thermionic emission Element-adsorbed layers – Refractory metal compounds – Practical cathodes Photoelectric emission Photoelectric work function – Emitting surface Secondary emission Emission coefficients Auger emission Oxidised alloys – Insulating metal compounds Electron emission under positive ion bombardment Field emission Second Townsend coefficient γ electrons released per positive arriving

19 Electrical properties 19.1 19.2

Resistivity Pure metals – Alloys – Copper alloys – EC Aluminium Superconductivity Transition temperatures and critical fields of elements – Critical temperatures of superconducting compounds

20 Magnetic materials and their properties 20.1 20.2

Magnetic materials Permanent magnet materials Alnico alloys – Ferrite – Rare earth cobalt alloys – Neodymium iron boron – Bonded materials – Other materials – Properties, names and applications

18–1 18–1 18–4 18–5 18–6 18–8 18–8

19–1 19–1 19–7

20–1 20–1 20–2

Contents 20.3

20.4 20.5 20.6 20.7

Magnetically soft materials Silicon–iron alloys – Ferrites and garnets – Typical properties of silicon steels – Typical properties of some magnetically soft ferrites – Garnet material – Nickel–iron alloys – Amorphous alloy material High-saturation and constant-permeability alloys Magnetic powder core materials Magnetic temperature-compensating materials Non-magnetic steels and cast irons

21 Mechanical testing 21.1 21.2 21.3 21.4 21.5 21.6 21.7

Hardness testing Brinell hardness – Rockwell hardness – Rockwell superficial hardness – Vickers – Micro-hardness – Hardness conversion tables Tensile testing Standard test pieces Impact testing of notched bars Izod test – Charpy test Fracture toughness testing Linear-elastic (KIc ) – K–R curve – Elastic-Plastic (JIc , CTOD) Fatigue testing Load-controlled smooth specimen tests – Strain-controlled smooth specimen tests – Fatigue crack growth testing Creep testing Non-destructive testing and evaluation Ultrasonic – Radiography – Electrical and magnetic methods – Acoustic emission testing – Thermal wave imaging

22 Mechanical properties of metals and alloys 22.1

Mechanical properties of aluminium and aluminium alloys Alloy designation system for wrought aluminium – Temper designation system for aluminium alloys 22.2 Mechanical properties of copper and copper alloys 22.3 Mechanical properties of lead and lead alloys 22.4 Mechanical properties of magnesium and magnesium alloys 22.5 Mechanical properties of nickel and nickel alloys Directionally solidified and single crystal cast superalloys 22.6 Mechanical properties of titanium and titanium alloys 22.7 Mechanical properties of zinc and zinc alloys 22.8 Mechanical properties of zirconium and zirconium alloys 22.9 Tin and its alloys 22.10 Steels 22.11 Other metals of industrial importance 22.12 Bearing alloys

23 Sintered materials 23.1 23.2 23.3 23.4 23.5 23.6 23.7

The PM process The products Manufacture and properties of powders Powder manufacture – Properties of metal powders and how they are measured Properties of powder compacts Comparison tables of standard sieves – Properties of PM grade sponge iron powder Sintering Ferrous components Copper-based components

ix

20–9

20–16 20–17 20–17 20–18 21–1 21–1 21–8 21–9 21–12 21–16 21–19 21–20

22–1 22–1 22–26 22–46 22–49 22–61 22–82 22–93 22–94 22–95 22–98 22–157 22–160 23–1 23–1 23–1 23–2 23–4 23–8 23–8 23–9

x

Contents 23.8 23.9 23.10 23.11 23.12 23.13 23.14 23.15 23.16 23.17

Aluminium components Determination of the mechanical properties of sintered components Heat treatment and hardenability of sintered steels Case hardening of sintered steels Steam treatment Wrought PM materials Refractory metals – Superalloys – Copper – Lead – Aluminium – Ferrous alloys – Aluminium matrix composites Spray forming Injection moulding Hardmetals and related hard metals ISO classification of carbides according to use Novel and emerging PM materials

24 Lubricants 24.1

Introduction Main regimes of lubrication 24.2 Lubrication condition, friction and wear 24.3 Characteristics of lubricating oils Viscosity – Boundary lubrication properties – Chemical stability – Physical properties 24.4 Mineral oils 24.5 Emulsions 24.6 Water-based lubricants 24.7 Synthetic oils Diesters – Neopentyl polyol esters – Triaryl phosphate esters – Fluorocarbons – Polyglycols 24.8 Greases Composition – Properties 24.9 Oil additives Machinery lubricants – Cutting oils – Lubricants for chipless-forming – Rolling oils 24.10 Solid lubricants

25 Friction and wear 25.1

25.2

Friction Friction of unlubricated surfaces – Friction of unlubricated materials – Friction of lubricated surfaces – Boundary lubrication – Extreme pressure (EP) lubricants Wear Abrasive wear – Adhesive wear – Erosive wear – Fretting wear – Corrosive wear

26 Casting alloys and foundry data 26.1 26.2 26.3 26.4 26.5 26.6 26.7 26.8

Casting techniques Patterns—crucibles—fluxing Pattern materials – Crucibles and melting vessels – Iron and steel crucibles—fluxing Aluminium casting alloys Copper base casting alloys Nickel-base casting alloys Magnesium alloys Zinc base casting alloys Steel castings Casting characteristics – Heat treatment

23–13 23–13 23–15 23–15 23–15 23–15 23–28 23–29 23–31 23–36

24–1 24–1 24–1 24–2 24–3 24–6 24–7 24–7 24–8 24–11 24–13

25–1 25–1

25–12

26–1 26–1 26–10 26–20 26–39 26–52 26–56 26–68 26–70

Contents 26.9

Cast irons Classification of cast irons – General purpose cast irons – Compacted graphite irons – Applications of special purpose cast irons 26.10 Acknowledgements

27 Engineering ceramics and refractory materials 27.1 27.2 27.3 27.4 27.5 27.6 27.7 27.8

Physical and mechanical properties of engineering ceramics Prepared but unshaped refractory materials Aluminous cements Castable materials Mouldable materials Ramming material Gunning material Design of refractory linings

28 Fuels 28.1 28.2 28.3

Coal Analysis and testing of coal – Classification – Physical properties of coal Metallurgical cokes Analysis and testing of coke – Properties of metallurgical coke Gaseous fuels, liquid fuels and energy requirements Liquid fuels – Gaseous fuels – Energy use data for various metallurgical processes

29 Heat treatment 29.1 29.2

29.3

30.3

30.4 30.5 30.6 30.7

26–84 26–100 27–1 27–1 27–7 27–7 27–8 27–14 27–14 27–14 27–14 28–1 28–1 28–8 28–13

29–1

General introduction and cross references 29–1 Heat treatment of steel 29–1 Introduction – Transformations in steels – Hardenability – Hardenability measurement – Austenitisation – Annealing – Quenching – Tempering – Austempering – Martempering – Carburising – Carbonitriding Heat treatment of aluminium alloys 29–66 Introduction to aluminium heat treating – A brief description of aluminium physical metallurgy – Defects associated with heat treatment – Solution treatment – Quenching – Ageing (natural and artificial) – Appendix: Quenchants

30 Metal cutting and forming 30.1 30.2

xi

Introduction and cross-references Metal cutting operations Turning – Boring – Drilling – Reaming – Milling Abrasive machining processes Surface grinding – Cylindrical grinding – Centreless grinding – Plunge grinding – Creep feed grinding – Honing – Microsising – Belt grinding – Disc grinding Deburring Metal forming operations Introduction – Massive forming operations – Sheet-metal forming operations – Superplasticity Machinability and formability of materials Definitions – Formability Coolants and lubricants Liquid metal working fluids – Gaseous fluids and gaseous-liquid mixtures

30–1 30–1 30–1 30–4

30–5 30–5 30–8 30–9

xii

Contents 30.8

30.9

Non-traditional machining techniques Electrical Discharge Machining (EDM) – Fast Hole EDM drilling – Waterjet and abrasive waterjet machining – Plasma cutting – Photochemical machining – Electrochemical machining – Ultrasonic machining – Lasers Occupational safety issues Machine guarding – Hazardous materials – Noise exposure – Ergonomics

31 Corrosion 31.1 31.2 31.3 31.4

31.5

Introduction Types of corrosion Uniform corrosion Galvanic corrosion – Erosion, cavitation, and fretting corrosion Localised forms of corrosion Crevice corrosion – Dealloying corrosion – Environmental cracking—stress corrosion cracking and corrosion fatigue cracking – Hydrogen damage – Intergranular corrosion – Pitting corrosion Biocorrosion

32 Electroplating and metal finishing 32.1 32.2 32.3 32.4 32.5 32.6 32.7 32.8 32.9 32.10 32.11

Polishing compositions Cleaning and pickling processes Anodising and plating processes Electroplating process Plating processes for magnesium alloys Electroplating process parameters Miscellaneous coating processes Plating formulae for non-conducting surfaces Methods of stripping electroplated coatings Conversion coating processes Glossary of trade names for coating processes Wet processes – Dry processes

33 Welding 33.1 33.2 33.3 33.4 33.5

33.6

Introduction and cross-reference Glossary of welding terms Resistance welding The influence of metallurgical properties on resistance weldability – The resistance welding of various metals and alloys Solid-state welding Friction welding – Ultrasonic welding – Diffusion welding Fusion welding The fusion welding of metals and alloys—ferrous metals – Non-ferrous metals – Copper and copper alloys – Lead and lead alloys – Magnesium alloys – Nickel and nickel alloys – Noble metals – Refractory metals – Zinc and zinc alloys – Dissimilar metals Major standards relating to welding, brazing and soldering

34 Soldering and brazing 34.1 34.2 34.3

Introduction and cross-reference Quality assurance Soldering General considerations – Choice of flux – Control of corrosion – Solder formulations – Cleaning – Product assurance

30–10

30–12

31–1 31–1 31–1 31–1 31–4

31–10 32–1 32–1 32–2 32–6 32–8 32–16 32–17 32–18 32–19 32–20 32–21 32–23

33–1 33–1 33–1 33–6 33–10 33–15

33–41 34–1 34–1 34–1 34–2

Contents 34.4

34.5

Brazing General design consideration – Joint design – Precleaning and surface preparation – Positioning of filler metal – Heating methods – Brazeability of materials and braze alloy compositions Diffusion soldering or brazing

35 Vapour deposited coatings and thermal spraying 35.1 35.2 35.3

Physical vapour deposition Evaporation – Sputtering – Ion cleaning Chemical vapour deposition Thermal spraying Combustion wire spraying – Combustion powder spraying – Electric wire arc spraying – High velocity oxy-fuel spraying (HVOF) – Plasma spraying

36 Superplasticity

xiii 34–9

34–13 35–1 35–1 35–2 35–13

36–1

Investigations on superplasticity of metal alloys – Summary of research on internal stress superplasticity

37 Metal-matrix composites

37–1

Properties of reinforcing fibres at room temperature – Typical interactions in some fibre-matrix systems – Properties of aluminium alloy composites – Properties of magnesium alloy composites – Properties of titanium alloy composites – Properties of zinc alloy composites – Properties of co-deformed copper composites

38 Non-conventional and emerging metallic materials 38.1 38.2 38.3

38.4 38.5 38.6

Introduction Cross references Structural intermetallic compounds Sources of information on structural intermetallic compounds – Focus of the section – The nature of ordered intermetallics – The effect of ordering on the properties of intermetallics – Overview of aluminide intermetallics – Nickel aluminides – Titanium aluminides – Dispersion strengthened intermetallics and intermetallic matrix composites – Processing and fabrication of structural intermetallics – Current and potential applications of intermetallics Metallic foams Metallic glasses Metallic glasses requiring rapid quenching – Bulk metallic glasses Mechanical behaviour of micro and nanoscale materials Mechanics of scale – Thin films – Nanomaterials – Nanostructures

39 Modelling and simulation 39.1 39.2 39.3 39.4 39.5 39.6 39.7 39.8 39.9 39.10

Introduction Electron theory Thermodynamics and equilibrium phase diagrams Thermodynamics of irreversible processes Kinetics Monte Carlo simulations Phase field method Finite difference method Finite element method Empirical modelling: neural networks

38–1 38–1 38–1 38–1

38–18 38–20 38–23

39–1 39–1 39–1 39–3 39–4 39–5 39–6 39–7 39–9 39–9 39–10

xiv

Contents

40 Supporting technologies for the processing of metals and alloys 40.1 40.2 40.3

40.4

Introduction and cross-references Furnace design Introduction – Types of furnaces – Heat calculations – Refractory design – Vacuum furnaces – Cooling Vacuum technology Introduction – Pressure units and vacuum regions – Pressure measurement – Pumping technologies – Vacuum systems – Residual gas analysis – Safety – Selective bibliography Metallurgical process control Metals production and processing – Modelling and control of metallurgical processes – Process control techniques – Instrumentation for process control – Process control examples in metals production – Keywords in process control

41 Bibliography of some sources of metallurgical information Index

40–1 40–1 40–1 40–8

40–16

41–1 I–1

Preface

Over ten years have elapsed since the last edition of Smithells was published. Hence, a key objective of the editors for the present edition was to update the existing content. Thus, as can be seen from the table below, extensive changes have been made. In addition, the editors wished to expand the coverage of Smithells, both with respect to topics that have been overlooked in previous editions (such as metal working and machining) and to include aspects of metallurgy that have grown in importance, since the last edition was published (for example modelling).

Chapter

Title

Changes Rewritten completely, to reflect current designations. Major new section added on the statistics needed for process and quality control, etc. No major changes.

4

Related designations Introductory tables and mathematical information General physical and chemical constants X-ray analysis of metallic materials

5 6

Crystallography Crystal chemistry

1 2 3

7 8 9 10

Metallurgically important minerals Thermochemical data Physical properties of molten salts Metallography

11

Equilibrium diagrams

12

Gas–metal systems

13 14

Diffusion in metals General physical properties

15

Elastic properties, damping capacity and shape memory alloys Temperature measurement and thermoelectric properties

16

Numerous edits to text, references updated and obsolete information removed. Numerous edits to text. Updated comprehensively to bring all crystal structure and lattice parameter data into accord with commonly accepted values. Data table updated; introductory paragraphs rewritten. No major changes. No changes. Rewritten completely to reflect modern specimen preparation techniques for light microscopy and contemporary practice in both electron microscopy and analytical techniques. New section added on applications in failure analysis. Numerous new references. Updated comprehensively to bring all phase diagrams into accord with commonly accepted data. Section on solutions of hydrogen updated completely and re-written. Numerous corrections, plus some new data. All data on solid pure metals has been updated to match commonly accepted values. The section on liquid metals has been rewritten and now includes a more detailed narrative introduction. Minor corrections and updating of references. New section on thermoelectric materials, with narrative and data. Additional thermocouple data. (continued)

xv

xvi

Preface

Chapter

Title

Changes

17 18 19 20

Radiating properties of metals Electron emission Electrical properties Magnetic materials and their properties Mechanical testing

Text rewritten completely. Minor corrections. New references added on superconducting materials. Text updated. Includes new references on contemporary magnetic materials. Existing text edited extensively. Sections on tensile testing and fracture toughness testing re-written completely. New sections on fatigue testing, creep testing and non-destructive evaluation added. Minor editing of existing material; notes added concerning the applicability of the standards cited. New section added to provide data for cast Ni-base alloys. Edited extensively and updated. Extensive revisions; new data tables. Extensive revisions; new data tables. Extensive revisions, in particular extension and updating of the compositional and property data for foundry alloys. Extensive revisions, including new data on the properties of ceramics and standards for these materials. Extensive revisions, in particular to energy use data and standards. Rewritten entirely to reflect contemporary practice in steel and aluminium alloy heat-treatment. Provides a much greater level of detail than the previous edition. New chapter providing a comprehensive overview of metal cutting and forming by both traditional and non-traditional techniques. Replaces a chapter that was focused narrowly on laser metal working. Rewritten completely. Includes a new section on biocorrosion. No changes. Extensive revisions, in particular with respect to EN and ISO standards. Extensive revisions, principally to expand coverage with respect to processes and to include lead-free solders. Coverage expanded to include a new section on thermal spraying. Some revision of content on vapour deposition. Rewritten completely to provide narrative and extensive tabular information on both microstructural and internal-stress superplastic systems. Property data expanded and updated. New chapter providing narrative and tabular information on structural intermetallic compounds, metal foams, metallic glasses and micro/nanoscale materials. Includes numerous references to both overviews and original research. New chapter providing an overview of the methods used to model metallic materials. New chapter containing information on furnace design, vacuum technology and metallurgical process control. New chapter comprised of a guide to major works of reference, texts, journals, conferences, databases and specialist search tools.

21

22

Mechanical properties of metals and alloys

23 24 25 26

Sintered materials Lubricants Friction and wear Casting alloys and foundary data

27 28

Engineering ceramics and refractory materials Fuels

29

Heat treatment

30

Metal cutting and forming

31

Guide to corrosion control

32 33

Electroplating and metal finishing Welding

34

Soldering and brazing

35

Vapour deposited coatings and thermal spraying Superplasticity

36

37 38

Metal-matrix composites Non-conventional and emerging metallic materials

39

Modelling and simulation

40

Supporting technologies for the processing of metals and alloys Bibliography of some sources of metallurgical information

41

Acknowledgements ‘Data! Data! Data!’ he cried impatiently. ‘I can’t make bricks without clay.’

Sherlock Holmes, in The Adventure of the Copper Beeches by Sir Arthur Conan Doyle The editors wish to thank Mr Daniel Butts and Miss Dina Taarea of the Materials Research and Education Center (MREC) at Auburn University (AU), for their extensive, tireless and immensely valuable assistance with many aspects of the preparation of this work. Additional help from Mr. Venu Gopal Krishnardula and other members of the Physical Metallurgy and Materials Joining research group at the AU–MREC is acknowledged with thanks. WFG is also very grateful to his wife, Dr Hyacinth Gale of the NSF Center for Advanced Vehicle Electronics (CAVE) at AU, for her unstinting help with this work. The editors would like to thank the many contributors to this edition, without whom this work would not of course have been possible. A debt to both the contributors and editors of previous editions of this work should also be recorded. The editors would also like to remark that it is a little over half a century since the first edition of Smithells was published. Although many of the topics in this book (bulk metallic glasses, nanomaterials, etc.) lay far in the future in 1949, none of these would have been recorded here if it were not for the initial efforts of C. J. Smithells. The following bodies kindly provided permission to reproduce copyrighted material for this edition, as acknowledged at the relevant location(s) in the text: The Aluminum Association The American Foundry Society (AFS) ASM International Association of Iron and Steel Engineers (AISE) ASTM International Cambridge University Press Copper Development Association CRC Press Marcel Dekker Metallurgical and Materials Transactions National Center for Manufacturing Sciences (NCMS) North American Die Casting Association (NADCA) The Timken Company In addition to the above, the following allowed use of copyrighted material in the previous edition: British Standards Institute Genium Publication Corporation Institute of Gas Engineers International Atomic Energy Agency McGraw-Hill Book Company W. F. Gale, Auburn, AL T. C. Totemeier, Idaho Falls, ID

xvii

Disclaimer

Although great care was exercised in the preparation of this volume, the contributors, editors and publisher provide no warranty of any kind (explicit or implied) as to the accuracy, reliability or applicability of the data and information contained herein. All data in this volume are provided for general guidance only and cannot be used as evidence of: merchantability; suitability for engineering design; fitness for a particular purpose; safety in storage, transportation or use; or other similar or related purposes. Information provided in this volume is derived from the relevant professional literature, or other sources believed to be reputable by the contributors and editors, as specified in the detailed references provided. Where copyrighted material has been employed, this is used with permission of the copyright owner. Copyrighted material, used with permission, is indicated at the relevant location in the text. Mention of trademarked product or corporate names is purely for the purposes of identification and these remain the property of their owners and there is no intent to infringe upon ownership rights therein. Neither the presence, nor the absence of discussions of materials, processes, products or other items provides any indication as to the extent to which such items actually exist, are available commercially, or constitute original inventions. All information within this volume is used entirely at the reader’s own risk. Since the circumstances in which the contents of this volume might be employed will differ widely, the contributors, editors and publisher can not guarantee favourable results and expressly deny any and all liability connected with the use (and consequences of the use) of the information and data contained herein. This volume is intended for readers with suitable professional qualifications, supplemented by sufficient experience in the field, to be capable of making their own professional judgement as to the reliability and appropriateness of the information and data contained within. Before using any of this information or data, a suitably qualified professional/chartered engineer must be consulted, possible risks analysed/managed and rigorous testing conducted under the actual conditions to be experienced. Under no circumstances, whether covered in the disclaimer above or not, shall the liability of the contributors, editors or publisher of this work exceed the original purchase price of this volume and this shall constitute the sole remedy available to the purchaser (and subsequent readers) of this book. No liability is accepted for consequential or implied damages of any kind resulting from use of this volume.

xviii

Contributors Contributors to this edition*

Chapters/Sections

J. R. Alcock Advanced Materials Department, Cranfield University, Cranfield, Bedfordshire, UK P. N. Anyalebechi Padnos School of Engineering, Grand Valley State University, Grand Rapids, MI, USA S. I. Bakhtiyarov Solidification Design Center, Auburn University, Auburn, AL, USA H. K. D. H. Bhadeshia Department of Materials Science and Metallurgy, University of Cambridge, Cambridge, UK JT. Black Department of Industrial and Systems Engineering, Auburn University, Auburn, AL, USA D. A. Butts Materials Research and Education Center, Auburn University, Auburn, AL, USA V. Dayal Department of Aerospace Engineering and Engineering Mechanics, Iowa State University, Ames, IA, USA J. C. Earthman Department of Chemical and Biochemical Engineering, University of California at Irvine, Irvine, CA, USA J. W. Fergus Materials Research and Education Center, Auburn University, Auburn, AL, USA H. S. Gale NSF Center for Advanced Vehicle Electronics, Department of Electrical and Computer Engineering, Auburn University, Auburn, AL, USA W. F. Gale Materials Research and Education Center, Auburn University, Auburn, AL, USA R. M. German Department of Engineering Science and Mechanics, Pennsylvania State University, University Park, PA, USA W. Gestwa Poznan University of Technology, Poznan, Poland C. Hammond School of Materials, University of Leeds, Leeds, UK

20 12 14.2 39 30 6, 11, 19.2, 20-Appendix 21.7 31.5 41 41 6, 10.5, 11, 28.3.3, 38.1–38.5, 41 23 29.2 4, 5

*Where material was retained from the Seventh Edition, this was not the work of the individuals shown above.

xix

xx

Contributors

Contributors to this edition*

Chapters

T. Hornig Materials Science Institute, RWTH (Aachen University of Technology), Aachen, Germany W. E. Lee Department of Engineering Materials, The University of Sheffield, Sheffield, South Yorkshire, UK D. Y. Li Department of Chemical and Materials Engineering, University of Alberta, Edmonton, Alberta, Canada E. Lugscheider Materials Science Institute, RWTH (Aachen University of Technology), Aachen, Germany D. S. MacKenzie Houghton International, Valley Forge, PA, USA S. Maghsoodloo Department of Industrial and Systems Engineering, Auburn University, Auburn, AL, USA M. F. Modest Department of Mechanical Engineering, Pennsylvania State University, University Park, PA, USA S. B. Newcomb Sonsam Ltd, Newport, Co. Tipperary, Ireland T. G. Nieh Lawrence Livermore National Laboratory, Livermore, CA, USA R. A. Overfelt Solidification Design Center, Auburn University, Auburn, AL, USA G. Ozdemir Department of Industrial and Systems Engineering, Auburn University, Auburn, AL, USA L. N. Payton Department of Industrial and Systems Engineering (now with Materials Research and Education Center), Auburn University, Auburn, AL, USA P. J. Pinhero Idaho National Engineering and Environmental Laboratory, Idaho Falls, ID, USA D. L. Porter Argonne National Laboratory - West, Idaho Falls, ID, USA B. C. Prorok Materials Research and Education Center, Auburn University, Auburn, AL, USA M. Przylecka Poznan University of Technology, Poznan, Poland D. Pye Pye Metallurgical Consulting Inc., Meadville, PA, USA R. J. Reid CCLRC Daresbury Laboratory, Daresbury, Warrington, Cheshire, UK M. A. Reuter Resource Engineering Section, Department of Applied Earth Sciences, TU Delft, Delft, The Netherlands

35.1, 35.2

*Where material was retained from the Seventh Edition, this was not the work of the individuals shown above.

27 24, 25 35 29.3 2.2.7 17 10.4 36 26 2.2.7 30 33 22 38.6 29.2 29.2, 40.2 40.3 40.4

Contributors Contributors to this edition*

Chapters

K. Seemann Materials Science Institute, RWHT (Aachen University of Technology), Aachen Germany M. R. Scheinfein Department of Physics and Astronomy, Arizona State University, Tempe, AZ, USA (Now at Department of Physics, Simon Fraser University, Burnaby, British Columbia, Canada) C. A. Schuh Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA C. Shannon Department of Chemistry, Auburn University, Auburn, AL, USA Y. Sohn Advanced Materials Processing and Analysis Center, University of Central Florida, Orlando, FL, USA D. Taarea Materials Research and Education Center, Auburn University, Auburn, AL, USA

35.3

T. C. Totemeier Idaho National Engineering and Environmental Laboratory, Idaho Falls, ID, USA G. E. Totten G.E. Totten & Associates, LLC, Seattle, WA, USA H. Tsai Department of Mechanical Engineering, Michigan State University, East Lansing, MI, USA G. F. Vander Voort Buehler Ltd., Lake Bluff, IL, USA A. Williams Department of Fuel and Energy, University of Leeds, Leeds, UK P. J. Withers Manchester Materials Science Centre, University of Manchester/UMIST, Manchester, UK Y. Yang Resource Engineering Section, Department of Applied Earth Sciences, TU Delft, The Netherlands B. Yaryar Department of Mining Engineering, Colorado School of Mines, Golden, CO, USA Y. Zhou Department of Mechanical Engineering, University of Waterloo, Waterloo, ON, Canada

xxi

18

36 16.3 13 14.1, 19.2, 20-Appendix, Tables 26.28, 26.34, 26.46, 26.47 1, 10.7, 21, 22 29.2 15 10.1–10.3 28 37 40.4 7 33, 34

*Where material was retained from the Seventh Edition, this was not the work of the individuals shown above.

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1

Related designations

The following tables of related designations are intended as a guide to alloy correspondence on the basis of chemical composition. The equivalents should not be taken as exact, and in all cases of doubt the relevant national specification or standard should be consulted. The tables do not represent an exhaustive list of all alloys available; the references listed at the end of the chapter are more complete sources. In the case of the United Kingdom, France, and Germany, the tables refer to designations and standards that have recently been superseded by European (EN) and international (ISO) standards. The older standards have been referenced because the alloy designations are still in common use, and because the new standards use in some cases several different designations for chemically identical alloys, depending on the product form. A list of all designations is beyond the scope of this book; the references should be consulted for further details. Table 1.1 lists designations for steels, with subsections for steels of different types. Designations for wrought aluminium alloys are listed in Table 1.2; cast aluminium alloys are shown in Table 22.2. Table 1.3 gives related copper alloy designations, subdivided into coppers, brasses, bronzes, and nickel silvers. Magnesium alloys cast and wrought are in Table 1.4. Related designations for nickel and titanium alloys are listed in Sections 22.5 and 22.6, respectively. Unified number designations—UNS designations are five-digit numbers prefixed by a letter that characterises the alloy system as shown below. UNS Letter Designation1 A Aluminium and aluminium alloys B Copper and copper alloys D Specified mechanical properties steels E Rare earth and rare earth like metals and alloys F Cast irons and cast steels G AISI and SAE carbon and alloy steels H AISI H-steels J Cast steels (except tool steels) K Miscellaneous steels and ferrous alloys L Low melting metals and alloys M Miscellaneous nonferrous metals and alloys N Nickel and nickel alloys P Precious metals and alloys R Reactive and refractory metals and alloys S Heat and corrosion resistant (stainless) steels T Tool steels W Welding filler metals

1–1

1–2

Related designations

Table 1.1

RELATED DESIGNATIONS FOR STEELS

Nominal composition

USA AISI/SAE (UNS)

UK BS 970 (En)

1.1.1 Carbon steels C 0.06 Mn 0.35 C 0.08/0.13 Mn 0.3/0.6 C 0.13/0.18 Mn 0.3/0.6 C 0.17/0.23 Mn 0.3/0.6 C 0.22/0.28 Mn 0.3/0.6 C 0.27/0.34 Mn 0.6/0.9 C 0.31/0.38 Mn 0.6/0.9 C 0.36/0.44 Mn 0.6/0.9 C 0.42/0.50 Mn 0.6/0.9 C 0.47/0.55 Mn 0.6/0.9 C 0.55/0.65 Mn 0.6/0.9 C 0.65/0.76 Mn 0.6/0.9 C 0.74/0.88 Mn 0.6/0.9 C 0.90/1.04 Mn 0.3/0.5

1006 (G10060) 1010 (G10100) 1015 (G10150) 1020 (G10200) 1025 (G10250) 1030 (G10300) 1035 (G10350) 1040 (G10400) 1045 (G10450) 1050 (G10500) 1060 (G10600) 1070 (G10700) 1080 (G10800) 1095 (G10950)

030A04 045M10 (32A) 050A15 050A20 (2C, 2D) 080A25 080A30 (5B) 080A35 (8A) 080A40 (8C) 080M46 080M50 080A62 (43D) 080A72 080A83 060A99

Germany DIN (Wk. No.)

Japan JIS

Russia GOST

Sweden SIS

France AFNOR

China GB

Ck7 (1.1009) Ck10 (1.1121) Ck15 (1.1141) Ck22 (1.1151) Ck25 (1.1158) Ck30 (1.1178) Ck34 (1.1173) Ck40 (1.1186) Ck45 (1.1191) Ck50 (1.1206) Ck60 (1.1221) Ck67 (1.1231) 80Mn4 (1.1259) Ck101 (1.1274)

—

08Fkp

14 1147

XC6FF

05F

S9Ck

10kp

14 1311

CC10, AF34

10F

S15Ck

15kp

14 1370

XC15

15F

S20C

20kp

14 1450

XC18, C20

20F

S25C

M26

—

XC25

1025

S30C

30G

—

XC32

30

S35C

35

14 1572

XC35, C35

35

S40C

40

—

AF60, C40

40

S45C

45

14 1672

XC45

45

S50C

50

14 1674

XC50

50

S58C

60, 60G

14 1678

XC60

60

—

70, 70G

14 1770

XC70

70

—

80

14 1774

XC80

80

SUP 4

—

14 1870

XC100

—

Related designations

1.1.2 Carbon-manganese and free-cutting steels C 0.10/0.16 1513 130M15 Mn 1.1/1.4 (G15130) (201) C 0.15/0.21 1518 120M19 Mn 1.1/1.4 (G15180) C 0.18/0.24 1522 150M19 Mn 1.1/1.4 (G15220) (14A, 14B) C 0.30/0.38 1536 120M36 Mn 1.2/1.55 (G15360) (15B) C 0.36/0.45 1541 150M40 Mn 1.35/1.65 (G15410) C 0.35/0.43 1139 216M36 Mn 1.35/1.65 (G11390) (15AM) S 0.13/0.20 C 0.40/0.48 1144 226M44 Mn 1.35/1.65 (G11440) S 0.24/0.33 C 0.12 1212 — Mn 0.7/1.0 (G12120) S 0.16/0.23 C 0.13 1213 230M07 Mn 0.7/1.0 (G12130) S 0.24/0.33 1.1.3 Alloy steels C 0.1/0.2 Mn 0.3/0.6 Cr 0.6/0.95 Ni 2.75/3.25 C 0.23/0.28 Mn 0.7/0.9 Cr 0.4/0.6 Mo 0.2/0.3

12Mn6 (1.0496) 21Mn4 (1.0469) 20Mn6 (1.1169) 36Mn5 (1.1167) 36Mn6 (1.1127) 35S20 (1.0726)

1–3

—

—

—

12M5

—

SMnC420

20GLS

14 2135

20M5

18MnSi

SMn21

20G2

14 2165

20M5

18MnVB

SMn1

35GL

14 3562

32M5

35Mn2

SMn21

40G2

14 2120

40M5

40Mn2

SUM41

A40G

14 1957

35MF4

Y40Mn

45S20 (1.0727)

SUM43

A40

14 1973

45MF6

—

9S20 (1.0711)

SUM21

A11

—

12MF4

—

9SMn28 (1.0715)

SUM22

—

14 1912

S250

—

3415

655M13 (36A)

14NiCr14 (1.5752)

SNC22H

12ChHN3A

—

12NC15

12CrNi3

4125

708A25

25CrMo4 (1.7218)

SCCrM1

25ChGM

14 2225

25CD45

—

(continued)

1–4

Related designations

Table 1.1

RELATED DESIGNATIONS FOR STEELS—continued

Nominal composition C 0.35/0.40 Mn 0.7/0.9 Mo 0.2/0.3 C 0.28/0.33 Mn 0.4/0.6 Cr 0.8/1.0 Mo 0.15/0.25 C 0.33/0.35 Mn 0.7/0.9 Cr 0.6/1.10 Mo 0.15/0.25 C 0.38/0.43 Mn 0.75/1.00 Cr 0.8/1.10 Mo 0.15/0.25 C 0.38/0.43 Mn 0.6/0.8 Cr 0.7/0.9 Ni 1.65/2.00 Mo 0.2/0.3 C 0.18/0.23 Mn 0.45/0.65 Mo 0.45/0.60 C 0.13/0.18 Mn 0.7/0.9 Cr 0.7/0.9 C 0.17/0.22 Mn 0.7/0.9 Cr 0.7/0.9 C 0.28/0.33 Mn 0.7/0.9 Cr 0.8/1.1

USA AISI/SAE (UNS)

UK BS 970 (En)

Germany DIN (Wk. No.)

Japan JIS

Russia GOST

Sweden SIS

France AFNOR

China GB

4037

605H37

GS-40MnMo43

SCCrM1

25ChGM

14 2225

25CD45

—

4130 (G41300)

708A30

—

SCM2 SCM430

30ChMA

14 2233

30CD4

30CrMo

4135 (G41350)

708A37 (19B)

CS-34CrMo4 (1.7220)

SCM3H

35ChM

14 2234

35CD4

35CrMo

4140 (G41400)

708A40 (19A)

42CrMo4 (1.7225)

SCM4

38ChM

14 2244

42CD4

42CrMo

4340 (G43400)

816M40 (110)

40NiCrMo6 (1.6565)

SNCM8

40ChMA

14 2541

35NCD6

—

4419 (G44190)

—

GS-22Mo4 (1.5419)

SCPH11

—

—

18MD4.05

—

5115 (G51150)

527A17

16MnCr5 (1.7131)

SCr21

15Ch

14 2127

16MC5

15Cr

5120 (G51200)

—

20CrMnS3 (1.7121)

SCr22

20Ch

—

20MC5

20Cr

5130 (G51300)

530A30 (18A)

30MnCrTi4 (1.8401)

SCr2H

27ChGR

—

28C4

30Cr

Related designations

C 0.33/0.38 Mn 0.6/0.8 Cr 0.8/1.05 C 0.38/0.43 Mn 0.7/0.9 Cr 0.7/0.9 C 0.98/1.10 Mn 0.25/0.45 Cr 1.3/1.6 C 0.48/0.53 Mn 0.7/0.9 Cr 0.8/1.1 V 0.15 C 0.18/0.23 Mn 0.7/0.9 Cr 0.4/0.6 Ni 0.4/0.7 Mo 0.15/0.25 C 0.38/0.43 Mn 0.75/1.0 Cr 0.4/0.6 Ni 0.4/0.7 Mo 0.15/0.25

1–5

5135 (G51350)

530A36 (18C)

38Cr4 (1.7043)

SCr3H

35Ch

—

38C4

35Cr

5140 (G51400)

530A40 (18D)

42Cr4 (1.7045)

SCr4H

40Ch

14 2245

42C4

40Cr

E52100 (G52986)

535A99 (31)

105Cr5 (1.2060)

SCr5

SchCh15

14 2258

100C6

—

6150 (G61500)

735A50 (47)

GS-50CrV4 (1.8159)

SUP10

50ChF

14 2230

50CV4

50CrVA

8620 (G86200)

805A20

21NiCrMo2 (1.6523)

SNCM21

AS20ChGNM

14 2506-03

20NCD2

20CrNiMo

8640 (G86400)

945A40 (10C)

40NiCrMo2 (1.6546)

SNCM6

38ChGNM

—

40NCD2

—

301S81

X7CrNiAl17 7 (1.4564)

SUS631

—

—

177F00

1Cr17Ni7Al

1.1.4 Stainless steels C 0.9 17-7 PH Cr 16/18 (S17700) Ni 6.5/7.75 Al 0.75/1.5

(continued)

1–6

Related designations

Table 1.1

RELATED DESIGNATIONS FOR STEELS—continued

Nominal composition C 0.15 Mn 7.5/10.0 Cr 17/19 Ni 4/6 N 0.25 C 0.15 Mn 2.0 Cr 17/18 Ni 8/10 C 0.8 Mn 2.0 Cr 18/20 Ni 8/10.5 C 0.03 max Mn 2.0 Cr 18/20 Ni 8/12 C 0.25 Mn 2.0 Cr 24/26 Ni 19/22 C 0.8 Mn 2.0 Cr 16/18 Ni 10/14 Mo 2/3 C 0.03 max Mn 2.0 Cr 16/18 Ni 10/14 Mo 2/3

USA AISI/SAE (UNS)

UK BS 970 (En)

202 (S20200)

Germany DIN (Wk. No.)

Japan JIS

Russia GOST

Sweden SIS

France AFNOR

China GB

284S16

X8CrMnNi18 8 (1.3965)

SUS202

1Ch17N3G8AE

14 2357

—

1Cr18Mn8Ni5N

302 (S30200)

302S25 (58A)

X12CrNi18 8 (1.4300)

SUS302

Ch18N9

14 2331

302F00 Z10CN18.9

1Cr18Ni9

304 (S30400)

304S15 (58E)

X5CrNi18 10 (1.4301)

SUS304

0Ch18N10

14 2332

304F01 Z5CN18.09

0Cr18Ni9

304L (S30403)

304S11 (58E)

X2CrNi19 11 (1.4306)

SUS304L

03Ch18N11

14 2352

304F11 Z2CN18.10

00Cr18Ni10

310 (S31000)

310S24

X12CrNi25 21 (1.4845)

SUSY310

Ch25N20

—

310F00 Z12CN25.20

2Cr25Ni20

316 (S31600)

316S16 (58J)

X15CrNiMo17 13 3 (1.4436)

SUS316

SW-04Ch19Ni1M3

14 2343

316F00

0Cr17Ni20Mo2

316L (S31603)

316

X2CrNiMo18 14 3 (1.4435)

SUSY316L

03Ch16N15M3

14 2348

22CND17.12

00Cr17Ni14Mo2

Related designations

C 0.08 Mn 2.0 Cr 16/18 Ni 10/14 Mo 2/3 Ti 5x(C+N)/0.7 C 0.08 Mn 2.0 Cr 17/19 Ni 10/14 Mo 2/3 Nb 10xC/1.10 C 0.08 Mn 2.0 Cr 17/19 Ni 9/12 Ti 5xC min C 0.08 Mn 2.0 Cr 17/19 Ni 9/13 Nb+Ta 10xC min C 0.15 Mn 1.0 Cr 11.5/13.0 C 0.15 Mn 1.0 Cr 11.5/13.5 Si 1.0 C 0.15 min Mn 1.0 Cr 12/14 Si 1.0

1–7

316Ti (S31635)

320S17 (58J)

X6CrNiMoTi17 12 2 (1.4571)

—

08Ch17N13M2T

14 2350

Z6CNDT17.12

0Cr18Ni12Mo2Ti

316Cb (S31640)

318C17

X6CrNiMoNb17 12 2 (1.4580)

SCS22

0Ch18N9MB

—

Z4CNDNb18.12M

—

321 (S32100)

321S20 (58B/C)

X6CrNiTi18 10 (1.4541)

SUS321

08Ch18N10T

14 2337

Z6CNT18.10

0Cr18Ni11Ti

347 (S34700)

347S31

X5CrNiNb18 10 (1.4546)

SUS347

Ch18N11B

14 2338

Z4CNNb19.10M

0Cr18Ni11Nb

403 (S40300)

410S21 (56A)

X10Cr12 (1.4006)

SUS403

15Ch13L

14 2301

Z15C13

1Cr12

410 (S41000)

—

X15Cr13 (1.4024)

SUS416

12Ch13

14 2302

Z12C13M

1Cr13

420 (S42000)

420S37 (56C)

X20Cr13 (1.4021)

SUS420J2

20Ch13

14 2304

Z20C13N

2Cr13

(continued)

1–8

Related designations

Table 1.1

RELATED DESIGNATIONS FOR STEELS—continued

Nominal composition C 0.12 Mn 1.0 Cr 16/18 C 0.48/0.58 Mn 8/10 Cr 20/22 Ni 3.25/4.5 N 0.28/0.5 1.1.5 Tool steels C 0.8/1.0 Cr 3.75/4.5 Mo 4.5/5.5 W 5.5/6.75 V 1.75/2.2 C 0.65/0.8 Cr 3.75/4.5 W 17.25/18.75 V 0.9/1.3 C 1.5/1.6 Cr 3.75/5.0 Mo 1.0 max W 11.75/13.0 V 4.5/5.25 Co 4.75/5.25 C 0.35/0.45 Si 0.8/1.2 Cr 3.0/3.75 Mo 2.0/3.0 V 0.25/0.75

USA AISI/SAE (UNS)

UK BS 970 (En)

Germany DIN (Wk. No.)

Japan JIS

Russia GOST

Sweden SIS

France AFNOR

China GB

430 (S43000)

430S15 (60)

X8Cr17 (1.4016)

SUS430

12Ch17

14 2320

Z8C17

1Cr17

21-4N (S63008)

349S52

X5CrMnNiN21 9 (1.4871)

SUH36

5Ch20N4AGN

—

Z52CMN21.09

Y5Cr21Mn9Ni4N

M2 (T11302rc)

BM2

S6-5-2 (1.3343)

SKH51

R6AM5

14 2722

6-5-2

W6Mo5Cr4V2

T1 (T12001)

BT1

S18-0-1 (1.3355)

SKH2

R18

—

18-0-1

W18Cr4V

T15 (T12015)

BT15

S12-1-4-5 (1.3202)

SKH10

R10K5F5

—

12-1-5-5

—

H10 (T20810)

VH10

X32CrMoV3 3 (1.2365)

SKD7

3Ch3M3F

—

32DCV28

—

Related designations

C 0.33/0.43 Si 0.8/1.2 Cr 4.75/5.0 Mo 1.1/1.6 V 0.3/0.6 C 0.32/0.45 Si 0.8/1.2 Cr 4.75/5.5 Mo 1.1/1.75 V 0.8/1.2 C 0.26/0.36 Cr 3.0/3.75 W 8.5/10.0 V 0.3/0.6 C 1.4/1.6 Cr 11.0/13.0 Mo 0.7/1.2 V 1.1 max C 2.0/2.25 Cr 11.0/13.5 W 1.0 max V 1.0 max C 0.4/0.55 Si 0.15/1.2 Cr 1.0/1.8 W 1.5/3.0 V 0.15/0.30 C 0.7/1.5 C 0.8/1.5 V 0.15/0.35

1–9

H11 (T20811)

BH11

X41CrMoV5 1 (1.7783)

SKD6

4Ch5MF5

—

FZ38CDV5

—

H13 (T20813)

BH13

X40CrMoV5 1 (1.2344)

SKD61

4Ch5MF15

14 2242

Z40CDV5

—

H21 (T20821)

BH21

X30WCrV9 3 (1.2581)

SKD5

3Ch2W8F

14 2730

Z30WCV9

—

D2 (T30402)

BD2

X155CrMo12 1 (1.2379)

SKD11

—

14 2310

Z200C12

3-2 Cr12MoV

D3 (T30403)

BD3

X210CrW12 (1.2436)

SKD1

Ch12

—

Z200C12

3-1 Cr12

S1 (T41901)

BS1

45WCrV7 (1.2436)

SKS41

4ChW2S

14 2710

55WC20

2-2 5CrW2Si

W1 (T72301) W2 (T72302)

BW1B

C125W1 (1.1663) 100V1 (1.2833)

SK2

U12

14 1880

Y(2) 120

3 85

SKS43

—

—

Y105V

1-8V

BW2

1–10

Related designations

Table 1.2

RELATED DESIGNATIONS FOR WROUGHT ALUMINIUM ALLOYS

UNS

ISO—Nominal composition

A91050 A91070 A91080 A91100 A91145 A91200 A91350 A92011 A92014 A92017 A92024 A92031 A92117 A92219 A92618 A93003 A93004 A93103 A93105 A94032 A94043 A94047 A95005

Al99.5 Al99.7 Al99.8 Al99.0Cu — Al99.0 — Al-Cu6BiPb Al-Cu4SiMg Al-Cu4MgSi Al-Cu4Mg1 Al-Cu2NiMgFeSi Al-Cu2Mg Al-Cu2Mg1.5Fe1Ni1 Al-Mn1Cu — Al-Mn1 Al-MnMg — Al-Si12 Al-Mg1

USA/Japan 1050 1070 1080 1100 1145 1200 1350 2011 2014 2017 2024 2031 2117 2219 2618 3003 3004 3103 3105 4032 4043 4047 5005

UK Former BS

Germany DIN (Wk. No.)

Russia GOST

Sweden SIS

France Former NF

China GB

1B

Al99.5 (3.0255) Al99.7 (3.0275) Al99.8 (3.0285) — — Al99.0 (3.0205) EAl99.5 (3.0257) AlCuBiPb (3.1655) AlCuSiMn (3.1255) AlCuMg1 (3.1325) AlCuMg2 (3.1355) — AlCu2.5Mg0.5 (3.1305) — — AlMnCu (3.0517) AlMn1Mg1 (3.0528) AlCu2.5Mg0.5 (3.1305) AlMn0.5Mg0.5 (3.0505) — S-AlSi5 (3.2245) — AlMg1 (3.3315)

Al A7 — A2 AE, AT A2 A00 — 1380, AK8 AK10, D17 1160, D16 — 1180, D18 1201 1141, AK4-1 1400, AMts AMts2 D18 — — — — 1510, AMg1

4007 4005 4004 — 4010 4008 4355 4338 — — — —

A5 A7 A8 A45 — A4 — A-U5PbBi A-U4SG A-U4G A-U4G1 A-U2N A-U2G

L4 L2 L1 — — — — — — LD9, LD10 LY6, LY9, LY12 — —

— 4054 — — — — 4225 — 4106

— A-M1 A-M1G — — — — A-S12 A-G0.6

— — — — — — LT1 — —

1A — — 1C 1E, E57S FC1 H15 — 2L97, 2L98, L109, L110 H12 3L86 H16 N3 — N3 N31, E4S 38S N21 N21 N41

Related designations

A95050 A95052 A95056 A95083 A95086 A95154 A95251 A95454 A95456 A95754 A96060 A96061 A96063 A96082 A96101 A96463 A97005 A97020 A97072 A97075

Al-Mg1.5 Al-Mg2.5 Al-Mg5 Al-Mg4.5Mn Al-Mg4 Al-Mg3.5 Al-Mg2 Al-Mg3Mn — Al-Mg3 — Al-Mg1SiCu Al-MgSi Al-Si1MgMn — — — Al-Zn4.5Mg Al-Zn1 Al-Zn6MgCu

5050 5052 5056 5083 5086 5154 5251 5454 5456 5754 6060 6061 6063 6082 6101 6463 7005 7020 7072 7075

3L44 2L55 N6, A56S N8 N5 N4 N51 — — — H20, E91E H9 H30 91E BTR6 H17 — 2L95, L160, L161, L162, C77S

AlMg1.5 (3.3318) AlMg2.5 (3.3523) AlMg5 (3.3555) AlMg4.5Mn (3.3547) AlMg4.5 (3.3345) — AlMg2Mn0.3 (3.3525) AlMg2.7Mn (3.3537) — AlMg3 (3.3525) AlMgSi0.5 (3.3206) AlMgSiCu (3.3211) — AlMgSi1 (3.2315) E-AlMgSi0.5 (3.3207) Al99.85MgSi (3.2307) AlZnMgCu0.5 (3.4345) (3.4335) AlZn1 (3.4415) AlZnMgCu1.5 (3.4365)

— 1520, AMg2 AMg5 — 1540, AMg4 1530, AMg3 — — 45Mg2, AMg6 1530, AMg3 — 1330, AD33 1310, AD31 — 1310, AD31

— 4120 4146 4140 — — — — — 4133 4103

1910,m ATsM

4212 4425 — —

AtsP1 1950, V95

4104 4212 4102

A-G1.5 A-G2.5C — A-G.5MC A-G4MC AG3 A-G2M A-G2.5MC A-G5MC A-G3M A-GS A-GSUC — A-SGM0.7 A-GS/L A85-GS — A-Z5G — A-Z5GU

1–11

— LF2 LF10, LF5-1, LF51 LF4 — — — — — — — — — — — — — LB1 LC4, LC9

1–12

Table 1.3 UNS

Related designations

RELATED DESIGNATIONS FOR COPPER ALLOYS

ISO—Nominal composition

USA

UK Former BS

Germany DIN (Wk. No.)

Japan JIS

Russia GOST

Sweden SIS

France Former NF

China GB

C103 C101

C1020 C1100 C1220 — — — — C1700 C1720 — —

V3 — — — — — — MTsBB BHT 1.7 BHT 1.9, M2, M3 — Br.Kh, Br.Kh.5, Br.Kh.7, Br.Kh.8

5011 5010 5030 5105 5013 — — 5055 — — — —

Cu/c1, Cu/c2 Cu/a1, Cu/a2 — — Cu/a3 — — — CuBe1.7 — — —

Cu-1, Cu-2 T2 TAg0.1 — — — — QCd1 QBe1.7 QBe2 — —

C2100 C2200 C2300 C2400 C2600 C2680 C2740 C2800 — C3603 C3561 — — — C4641

L96, LT96 — L85, L87 L80 L69, L70, LMSH68-.06 L66, L68 L63 L58, L60 LS64-2 LS63-3 LS59-1, LS59-1L, LS59-1V — — LO70-1, LOAs70-1-005 LO60-1, LO62-1

— — 5112 5114 5122 — 5150 5163 5165 — 5168 5168 5170 5220 —

U-Z5 U-Z10 U-Z15 U-Z20 U-Z30 U-Z33 U-Z37 U-Z40 — U-Z36Pb3 — U-Z39Pb2 — U-Z29E1 —

H96 H90 H85 H80 H70 — H62, H63 H59 — HPb63-3 — — — HSn70-1 HSn62-1

1.3.1 Coppers C10200 Cu-OF C11000 Cu-ETP, Cu-FRHC C11600 CuAg0.1 C12200 Cu-DHP C12500 Cu-FRTP C14200 CuAs(P) C14500 Cu-Te, Cu-Te(P) C16200 CuCd1 C17000 CuBe1.7 C17200 CuBe2 C17500 CuCo2Be C18200 CuCr1

OF ETP STP DHP FRTP DPA DPTE CDA 162 CDA 170 CDA 172 CDA 175 CDA 182

A 3/1 A 2/1

SE-Cu (2.0070) E-Cu57 (2.0060) CuAg0.1 (2.1203) SF-Cu (2.0090) (2.008) CuAs(P) (2.0150) CuTeP (2.1646) CuCd1 (2.1266) CuBe1.7 (2.1245) CuBe2 (2.1247) CuCo2Be (2.1285) CuCr (2.1291)

1.3.2 Brasses C21000 CuZn5 C22000 CuZn10 C23000 CuZn15 C24000 CuZn20 C26000 CuZn30 C26800 CuZn33 C27400 CuZn37 C28000 CuZn40 C35300 CuZn35Pb2 C36000 CuZn36Pb3 C37700 CuZn38Pb2 C38000 CuZn43Pb2 C38500 CuZn39Pb3 C44300 CuZn28Sn C46400 CuZn38Sn1

CDA 210 CDA 220 CDA 230 CDA 240 CDA 260 CDA 268 CDA 274 CDA 280 CDA 353 CDA 360 CDA 377 CDA 380 CDA 385 CDA 443 CDA 464

CZ125 CZ101 CZ102 CZ103 CZ106 CZ107 CZ108 CZ109 CZ119 CZ124 CZ120 CZ122 CZ121 CZ111 CZ112

CuZn5 (2.0220) CuZn10 (2.0230) CuZn15 (2.0240) CuZn20 (2.0250) CuZn30 (2.0285) CuZn33 (2.0280) CuZn37 (2.0320) CuZn40 (2.0360) CuZn36Pb1.5 (2.0331) CuZn36Pb3 (2.0375) CuZn38Pb2 (2.0380) CuZn43Pb2 (2.0402) CuZn39Pb3 (2.0401) CuZn28Sn (2.0470) CuZn38Sn (2.0530)

C106 C104 C107 C109 C108 CB101

Related designations

1–13

1.3.3 Phosphor bronzes C51000 CuSn4 C51100 CuSn4 C51900 CuSn6

CDA 510 CDA 511 CDA 519

PB102 PB101 PB103

— — CuSn6 (2.1020)

C5101 C5101 C5191

Br.OF5.5-.3 Br.OF4-.25 Br.OF6.5-.15

— — 5428

— — U-E5P

C52100

CuSn8

CDA 521

PB104

CuSn8 (2.1030)

—

—

U-E9P

C52400

CuSn10

CDA 524

—

—

—

—

U-E9P

—

C54400

CuSn4Pb4Zn3

CDA 544

—

—

C5441

Br.OF7-.2 Br.OF7-.4 Br.O10-1 Br.OF10-1 Br.OSTs4-4-.25 Br.OSTs4-4-2.5

— QSn4-0.3 QSn6.5-0.1 QSn6.5-0.4 QSn7-0.2

—

—

QSn4-3 QSn4-4-2.5 QSn4-4-4

1.3.4 Aluminium-silicon bronzes C60800 CuAl5 CDA 608 C62300 CuAl10Fe3 CDA 623 C63000 CuAl10Fe5Ni5 CDA 630

CA101 CA103 CA105

CuAl5As (2.0918) CuAl10Fe (2.0938) CuAl11Fe6Ni5 (2.0978)

— C6161 C6031

C65500 C68700

CS101 CZ110

CuSi3Mn (2.1525) CuZn20Al (2.0460)

CN102 CN104, NS108, NS109 CN106, CN107 NS106 NS104 NS107

CuNi10Fe1 (2.0872) CuNi20Fe (2.0878) CuNi30Fe (2.0882) CuNi18Zn (2.0740) CuNi12Zn24 (2.0730) —

CuSi3Mn1 CuZn20Al2

CDA 655 CDA 687

1.3.5 Copper-nickel and nickel silvers C70600 CuNi10Fe1Mn CDA 706 C71000 CuNi20Mn1Fe CDA 710 C71500 CuNi30Mn1Fe CDA 715 C75200 CuNi18Zn20 CDA 752 C75700 CuNi12Zn24 CDA 757 C77000 CuNi18Zn27 CDA 770

— — —

U-A6 — —

— —

BRA5 Br.VAZh Br.AZhN10-4-4L Br.AZhN9-4-4 Br.KMTs3-1 LA77-2

— —

— U-Z22A2

— QAl9-4 QAl/10-4-4 QA/11-6-6 QSi3-1 —

C7060 C7100 C7150 C7521 — C7701

MNZhMTs10-1.5-1 — HM70, MH33 MNTs — —

5667 — 5682 5246 5243 —

U-N10 — U-N30 U-Z22N18 — U-Z27N18

BFe10-1-1 — BFe30-1-1 — — —

1–14 Table 1.4

UNS

Related designations RELATED DESIGNATIONS FOR MAGNESIUM ALLOYS ISO— Nominal composition

USA ASTM

UK BS 2970

Germany DIN (Wk. No.)

Japan JIS

Russia GOST

Europe AECMA

France AFNOR

1.4.1 Cast alloys M11101

Mg-Al9Zn

AZ101A

MAG 3

—

—

MGS6, ML6

—

—

M11810

Mg-Al8Zn

AZ81A

MAG 1

G-MgAl8Zn1 (3.5812)

MB2, MS2

ML5

MG-C-61

G-A9

M11910

Mg-Al9Zn

AZ91A

—

GD-MgAl9Zn (3.5912)

MDC1A

—

—

—

M11914

—

AZ91C

MAG 3

G-MgAl9Zn (3.5912)

MC2

—

—

G-A9Z1

M12330

Mg-RE3Zn2Zr

EZ33A

MAG 6

G-MgSE3Zn2Zr1 (3.5103)

MC8

—

MG-C-91

G-TR3Z2

M13320

Mg-Th3Zn2Zr

HZ32A

MAG 8

MgTh3Zn2Zr1 (3.5105)

—

ML14

MG-C-81

G-Th3Z2

M16410

Mg-Zn4REZr

ZE41A

MAG 5

MgZn4SE1Zr1 (3.5101)

—

—

MG-C-43

G-Z4TR

M16510

Mg-Zn5Zr

ZK51A

—

—

MC6

ML12, VM65

—

—

M16610

Mg-Zn6Zr

ZK61A

—

—

MC7

VM65-3

—

—

M19995

Mg99.95

9995A

—

H-Mg99.95 (3.5002)

—

—

—

—

1.4.2 Wrought alloys M11311

Mg-Al3Zn1

AZ31B

MAG S-1110

MgAl3Zn (3.5312)

MT1, MB1, MS1

—

Mg-P-62

G-A3Z1

M11600

Mg-Al6Zn1

AZ61A

—

MgAl6Zn (3.5612)

MT2, MB2, MS2

MA3

Mg-P-63

G-A6Z1

M11800

Mg-Al8Zn

AZ80A

MAG 7

MgAl8Zn (3.5812)

MB3, MS3

MA5

Mg-P-61

G-A8Z1

REFERENCES 1. ‘Metals and Alloys in the Unified Numbering System’, 8th edition, Society of Automotive Engineers, Warrendale, PA, 1999. 2. W. C. Mack, ed., ‘Worldwide Guide to Equivalent Irons and Steels’, 4th edition, ASM International, Materials Park, OH, 2000. 3. W. C. Mack, ed., ‘Worldwide Guide to Equivalent Nonferrous Metals and Alloys’, 3rd edition, ASM International, Materials Park, OH, 1996. 4. D. L. Potts and J. G. Gensure, ‘International Metallic Materials Cross-Reference’, 3rd edition, Glenium Publishing, Schenectady, NY, 1988. 5. R. B. Ross, ‘Metallic Materials Specification Handbook’, 4th edition, Chapman and Hall, London, 1992.

2

Introductory tables and mathematical information

2.1 Conversion factors Conversion factors into and from SI units are given in Table 2.5. The table can also be used to convert from one traditional unit to another. Convenient multiples or sub-multiples of SI units can be derived by the application of the prefix multipliers given in Table 2.4. Table 2.6 gives commonly required conversions. The majority of the conversion factors are based upon equivalents given in BS 350:Part 1:1983 ‘Conversion Factors and Tables’. Throughout the conversions the acceleration due to gravity (g) has been taken as the standard acceleration 9.806 65 m s−1 . Units containing the word force like ‘pounds force’ are converted to SI units using this value of g. The B.t.u. conversions are based on the definition accepted by the 5th International Conference on Properties of Steam, London, 1956, that 1 B.t.u. lb−1 = 2.326 J g−1 exactly. Conversions to joules are given for three calories; calories (IC) is the ‘international table calorie’ redefined by the 1956 conference referred to above as 4.186 8 J. Calories (15◦ C) refers to the calorie defined by raising the temperature of water at 15◦ C by 1◦ C and calories (US thermochemical) is the ‘defined’ calorie used in some USA work and is defined at 4.184 J exactly. The conversions are grouped in alphabetical order of the physical property to which they relate but are not alphabetical within the groups.

2.1.1

SI units

In this edition quantities are expressed in SI (Système International) units. Where c.g.s. units have been used previously only SI units are given. However, familiar units in general technical use have been retained where they bear a simple power of ten relation to the strict SI unit. For instance density is given as g cm−3 and not as kg m−3 . Where Imperial units have been used (e.g. in Mechanical Properties, etc.) data are given in both SI units and Imperial units. The basic units of the SI system are given in Table 2.1, derived units with special names and symbols in Table 2.2 and derived units without special names in Table 2.3. Multiples and sub-multiples of SI units are formed by prefixes to the name of the unit. The prefixes are shown in Table 2.4. The prefixed unit is written without a hyphen—for instance a thousand million newtons is written giganewton—symbol GN. The name of the unit is written with a small letter even when the symbol has a capital letter, e.g. ampere, symbol A. In the case of the kilogram, the multiple or sub-multiple is applied to the gram—for instance a thousand kilograms is written Mg. In this edition stress is expressed in Pascals (Pa). A pascal (Pa) is identical to a newton per square metre (N m−2 ) and a megapascal (MPa) is identical to a newton per square millimetre (N mm−2 ). PRINTED FORM OF UNITS AND NUMBERS

The symbol for a unit is in upright type and unaltered by the plural. It is not followed by a full stop unless it is at the end of a sentence. Only symbols of units derived from proper names are in the upper case.

2–1

2–2

Introductory tables and mathematical information

When units are multiplied they will be printed with a space between them. Negative indices are used for units expressed as a quotient. Thus newtons per square metre will be N m−2 and metres per second will be m s−1 . The prefix to a unit symbol is written before the unit symbol without a space between and a power index applies to both the symbols. Thus square centimetres is cm2 and not (cm)2 . Numbers are printed with the decimal point as a full stop. For long numbers, a space and not a comma is given between every three digits. For example π = 3.141 592 653. When a number is entirely decimal it will begin with a zero, e.g. 0.5461. If two numbers are multiplied, a × sign is used as the operator. HEADING OF COLUMNS IN TABLES AND LABELLING OF GROUPS

The rule adopted in this edition is that the quantity is obtained by multiplying the unit and its multiple given at the column head by the number in the table. For example when tabulating a stress of 2 × 105 Pa the heading is stress, below which appears 105 Pa, with 2.0 appearing in the table. If no units are given in the column heading, the values given are numbers only. In graphs the power of ten and units by which the point on the graph must be multiplied are given on the axis label. TEMPERATURES

The temperature scale IPTS–68 has been replaced by the International Temperature Scale of 1990 (ITS–90). For details of this see chapter 16, where Table 16.1 gives the differences between ITS–90 and EPT–76 and between ITS–76 and between ITS–90 and IPTS–68. Figure 16.1 gives differences (t90 −t68 ) between ITS–90 and IPTS–68 in the range −260◦ C to 1064◦ C. Table 2.7 gives conversions between the old IPTS–68 and the old IPTS–48. Table 2.1

BASIC SI UNITS

Quantity

Name of unit

Unit symbol

Length Mass Time Electric current Thermodynamic temperature Luminous intensity Amount of substance Plane angle Solid angle

metre kilogram second ampere kelvin candela mole radian steradian

m kg s A K cd mol rad sr

From ‘Quantities, Units and Symbols’, Royal Society, 1981.

Table 2.2

DERIVED SI UNITS WITH SPECIAL NAMES

Quantity

Name of unit

Symbol

Equivalent

Definition

Activity (radioactivity) Absorbed dose (of radiation) Dose equivalent (of radiation) Energy Force Stress or pressure Power Electric charge Electric potential

becquerel gray sievert joule newton pascal watt coulomb volt

Bq Gy Sv J N Pa W C V

s−1 J kg−1 J kg−1 Nm J m−1 N m−2 J s−1 As W A−1

m2 kg s−2 m kg s−2 m−1 kg s−2 m2 kg s−3 sA m2 kg s−3 A−1 (continued)

Conversion factors

2–3

Table 2.2

DERIVED SI UNITS WITH SPECIAL NAMES—continued

Quantity

Name of unit

Symbol

Equivalent

Definition

Electric resistance Electric capacitance Electric conductance Magnetic flux Inductance Magnetic flux density Luminous flux Illumination Frequency

ohm farad siemens weber henry tesla lumen lux hertz

 F S Wb H T lm lx Hz

V A−1 C V−1 A V−1 Vs Vs A−1 Wb m−2 cd sr cd sr m−2 s−1

m2 kg s−3 A−2 m−2 kg−1 s4 A2 m−2 kg−1 s3 A2 m2 kg s−2 A−1 m2 kg s−2 A−2 kg s−2 A−1 cd sr m−2 cd sr s−1

From ‘Quantities, Units and Symbols’, Royal Society, 1981. Note: Symbols derived from proper names begin with a capital letter. In the definition the steradian (sr) is treated as a base unit.

Table 2.3

SOME DERIVED SI UNITS WITHOUT SPECIAL NAMES

Quantity

SI unit

Symbol

Area Acceleration Angular velocity Calorific value Concentration Current density Density Diffusion coefficient Electrical conductivity Electric field strength Electrical resistivity Entropy Exposure (to radiation) Heat capacity Heat flux density Latent heat Luminance Magnetic field strength Magnetic moment Molar volume Moment of inertia Moment of force Molar heat capacity Permittivity Permeability Radioactivity Speed (velocity) Specific volume Specific heat-mass Specific heat-volume Surface tension Thermal conductivity

square metre metre/second squared radian/second joule/kilogram mole/cubic metre ampere/square metre kilogram/cubic metre square metre/second siemens/metre volt/metre ohm metre joule/kelvin coulomb/kilogram joule/kelvin watt/square metre joule/kilogram candela/square metre ampere/metre joule/tesla cubic metre/mole kilogram/square metre newton metre joule/kelvin mole farad/metre henry/metre 1/second metre/second cubic metre/kilogram joule/kilogram kelvin joule/cubic metre kelvin newton/metre watt/metre kelvin

m2 m s−2 rad s−1 J kg−1 mol m−3 A m−2 kg m−3 m2 s−1 S m−1 V m−1 m J K−1 C kg−1 J K−1 W m−2 J kg−1 cd m−2 A m−1 J T−1 m3 mol−1 kg m−2 Nm J K−1 mol F m−1 H m−1 s−1 m s−1 m3 kg−1 J kg−1 K−1 J m−3 K−1 N m−1 W m−1 K−1 (continued)

2–4

Introductory tables and mathematical information

Table 2.3

SOME DERIVED SI UNITS WITHOUT SPECIAL NAMES—continued

Quantity

SI unit

Symbol

Thermoelectric power Viscosity–kinematic Viscosity–dynamic Volume Wave number

volt/kelvin square metre/second pascal second cubic metre 1/metre

V K−1 m2 s−1 Pa s m3 m−1

Table 2.4

PREFIXES FOR MULTIPLES AND SUB-MULTIPLES USED IN THE SI SYSTEM OF UNITS

Sub-multiple

Prefix

Symbol

Multiple

Prefix

Symbol

10−1

deci centi milli micro nano pico femto atto

d c m µ n p f a

10 102 103 106 109 1012 1015 1018

deca hecto kilo mega giga tera peta exa

da h k M G T P E

10−2 10−3 10−6 10−9 10−12 10−15 10−18

From ‘Quantities, Units and Symbols’, Royal Society, 1981.

Table 2.5

CONVERSION FACTORS

To convert B to A multiply by

A

B

To convert A to B multiply by

102 3.937 008 × 10 3.280 84 1.019 716 × 10−1

Acceleration centimetres/second squared inches/second squared feet/second squared standard acceleration due to gravity

metres/second squared metres/second squared metres/second squared metres/second squared

10−2 2.54 × 10−2 3.048 × 10−1 9.806 65

2.062 65 × 105 3.437 75 × 103 5.729 58 × 10 1.591 55 × 10−1 6.366 20 × 10

seconds minutes degrees revolutions grades

radians radians radians radians radians

4.848 14 × 10−6 2.908 88 × 10−4 1.745 33 × 10−2 6.283 19 1.570 80 × 10−2

5.729 58 × 10 1.591 55 × 10−1 3.437 75 × 103 9.549 27

degrees/second revolutions/second degrees/minute revolutions/minute

1028 1.550 003 × 103

barn square inches

Angle—plane

Angular velocity radians/second radians/second radians/second radians/second

1.745 33 × 10−2 6.283 19 2.908 88 × 10−4 1.047 20 × 10−1

Area square metres square metres

10−28 6.451 6 × 10−4 (continued)

Conversion factors Table 2.5

2–5

CONVERSION FACTORS—continued

To convert B to A multiply by 1.076 391 × 10 1.195 990 3.861 02 × 10−7 2.471 052 × 10−4 10−4 2.471 052 2.5 × 10−1 1.562 5 × 10−3 2.683 92 × 10−2 4.308 86 × 10−11 2.388 46 ×10−4

A

B

square feet square yards square miles acres hectares acres acres square miles

square metres square metres square metres square metres square metres hectares roods acres

Calorific value—volume basis British thermal units/cubic foot joules/cubic metre therms/UK gallon joules/cubic metre kilocalories/cubic metre joules/cubic metre

2.390 06 × 10−4

Calorific value—mass basis British thermal units/pound joules/kilogram International joules/kilogram kilocalories/kilogram thermochemical joules/kilogram kilocalories/kilogram

10−1

Compressibility square centimetres/dyne metres/newton

4.299 × 10−4 2.388 46 × 10−4

Density 10−3 1.603 59 × 10−1 6.242 80 × 10−2 3.612 73 × 10−5 1.002 241 × 10−2 8.345 434 × 10−3 7.015 673 × 10

grams/cubic centimetre ounces/gallon (UK) pounds/cubic foot pounds/cubic inch pounds/gallon (UK) pounds/gallon (US) grains/gallon (UK)

kilograms/cubic metre kilograms/cubic metre kilograms/cubic metre kilograms/cubic metre kilograms/cubic metre kilograms/cubic metre kilograms/cubic metre

104

Diffusion coefficient square centimetres/second square metres/second

2.997 93 × 109 10−1

electrostatic units electromagnetic units

2.997 93 × 109 10−1

electrostatic units electromagnetic units

2.997 93 × 105 10−5 10−4 6.452 × 10−4 9.290 2 × 10−2 2.777 778 × 10−7 1.019 72 × 10−1 2.373 04 × 10 7.375 62 × 10−1 3.725 06 × 10−7 9.869 23 × 10−3

To convert A to B multiply by 9.290 3 × 10−2 8.361 27 × 10−1 2.589 99 × 106 4.046 86 × 103 104 4.046 86 × 10−1 4 6.40 × 102 3.725 89 × 10 2.320 80 × 1010 4.186 8 × 103 2.326 × 103 4.186 8 × 103 4.184 × 103 10 103 6.236 03 1.601 85 × 10 2.767 99 × 104 9.977 64 × 10 1.198 26 × 102 1.425 38 × 10−2 10−4

Electric charge coulombs coulombs

3.335 64 × 10−10 10

amperes amperes

3.335 64 × 10−10 10

Electric current

Electric current density electrostatic units amperes/square metre electromagnetic units amperes/square metre amperes/square centimetre amperes/square metre amperes/square inch amperes/square metre amperes/square foot amperes/square metre Energy—work—heat kilowatt hours joules kilogram force metres joules foot poundals joules foot pounds force joules horsepower hours joules 3 litre (dm ) atmospheres joules

3.335 64 × 10−6 105 104 1.55 × 103 1.076 4 × 10 3.6 × 106 9.806 65 4.214 01 × 10−2 1.355 82 2.684 52 × 106 1.013 25 × 102 (continued)

2–6

Introductory tables and mathematical information

Table 2.5

CONVERSION FACTORS—continued

To convert B to A multiply by

A

B

To convert A to B multiply by

2.388 46 × 10−4 8.850 34 9.478 17 × 10−4 107 6.241 808 × 1018 9.478 13 × 10−9 2.388 46 × 10−1 2.389 201 × 10−1 2.390 057 × 10−1

kilocalories (IC) inch pounds force British thermal units ergs electron volts therms (Btu) calories (IC) calories (15◦ C) calories (US thermochemical)

joules joules joules joules joules joules joules joules joules

4.186 8 × 103 1.129 9 × 10−1 1.055 06 × 103 10−7 1.602 1 × 10−19 1.055 06 × 108 4.186 8 4.185 5 4.184

2.388 46 × 10−1 5.265 62 × 10−4

Entropy calories (IC)/degree centigrade British thermal unit/degree Fahrenheit

joules/kelvin joules/kelvin

4.186 8 1.899 11 × 103

105 3.596 94 1.019 72 × 102 2.248 09 × 10−1 7.233 01 1.003 61 × 10−4 1.124 047 × 10−4 1.019 72 × 10−1

dynes ounces force grams force pounds force poundals UK tons force US tons force kilograms force

newtons newtons newtons newtons newtons newtons newtons newtons

10−5 0.278 014 9.806 65 × 10−3 4.448 22 0.138 255 9.964 02 × 103 8.896 422 × 103 9.806 65

√ newton/ (metre3 )

9.806 55 × 103

1.019 72 × 10−4 9.100 42 × 10−7 4.062 73 × 10−7 3.162 26 × 10−6

4.299 23 × 10−4 2.388 46 × 10−4 3.345 52 × 10−1 1.019 72 × 10−1 7.500 64 × 103 1010 106 9.979 84 × 109 3.937 01 × 10 3.280 84 1.093 61 6.213 71 × 10−4 5.396 118 × 10−4 5.399 568 × 10−4 1.81 × 10−1 2.5 × 10−1 10−1 1.25 × 10−1

Force

Fracture toughness (kilograms √ force/square centimetre) (centimetre) (kilopounds force/square inch) √ (inch) √ (tons force/square inch) (inch) √ hectobars (millimetre)

√ newton/ (metre3 ) √ newtons/√(metre3 ) newtons/ (metre3 )

Heat—see Energy Heat flow rate—see Power Latent heat British thermal units/pound joules/kilogram calories (IC)/gram joules/kilogram foot pounds force/pound joules/kilogram kilogram force metres/kilogram joules/kilogram Leak rate lusec (micron Hg litre/second)

1.098 85 × 106 2.461 4 × 106 3.162 3 × 105

2.326 × 103 4.186 8 × 103 2.989 07 9.806 65

joules/second

1.333 22 × 10−4

metres metres metres metres metres metres metres metres metres yards rods, poles, etc. chains furlongs

10−10 10−6 1.002 02 × 10−10 2.54 × 10−2 3.048 × 10−1 9.144 × 10−1 1.609 344 × 103 1.853 184 × 103 1.852 × 103 5.5 4.0 10.0 8.0

Length angstroms (Å) microns (µ) kx units inches feet yards miles miles (naut UK) miles (naut Int) rods, poles or perches chains furlongs miles (UK)

(continued)

Conversion factors Table 2.5

CONVERSION FACTORS—continued

To convert B to A multiply by 1.66 × 10−1 8.33 × 10−3 1.644 7 × 10−4 1.8939 × 10−4

107 3.417 17 × 103 2.373 04 × 10 5.467 47 × 104

A

B

fathoms cable lengths nautical miles miles (UK)

feet fathoms feet feet

To convert A to B multiply by 6.0 1.2 × 102 6.080 × 103 5.280 × 103

Magnetic conversions—see Magnetic units and conversion factors, Chapter 20 Moment of force—see Energy Moment of inertia grams centimetre squared kilograms metre squared 10−7 pounds inch squared kilograms metre squared 2.926 40 × 10−4 pounds foot squared kilograms metre squared 4.214 01 × 10−2 ounces inch squared kilograms metre squared 1.829 00 × 10−5

7.233 01 105

Momentum foot pounds/second kilogram metres/second gram centimetres/second kilogram metres/second

drams (Av) ounces (Av) pounds (Av) stones (Av) quarters (Av) hundredweights (Av) tons (Av) grains or minims (Apoth) scruples (Apoth) drams (Apoth) ounces (Apoth or Troy) grains (Troy) tonnes (metric) tons (short 2000 lb) metric carats (CM)

8.921 80 × 103 1.843 348 2.048 16 × 10−1 2.949 357 × 10 3.227 055

pounds/acre pounds/square yard pounds/square foot ounces/square yard ounces/square foot

5.599 73 × 10−2 6.719 71 × 10−1 2.015 91

pounds/inch pounds/foot pounds/yard

107 3.412 14 8.598 45 × 10−1 7.375 61 × 10−1 2.388 46 × 10−1 1.341 022 × 10−3 1.359 62 × 10−3

1.382 55 × 10−1 10−5

Mass

5.643 819 × 102 3.527 399 × 10 2.204 622 6 1.574 731 × 10−1 7.873 650 × 10−2 1.968 415 × 10−2 9.842 035 × 10−4 1.543 236 × 104 7.716 180 × 102 2.572 063 × 102 3.215 072 × 10 1.543 237 × 104 10−3 1.102 311 × 10−3 5.0 × 103

2.7 × 10−11

2–7

kilograms kilograms kilograms kilograms kilograms kilograms kilograms kilograms kilograms kilograms kilograms kilograms kilograms kilograms kilograms

Mass per unit area kilograms/square metre kilograms/square metre kilograms/square metre kilograms/square metre kilograms/square metre Mass per unit length kilograms/metre kilograms/metre kilograms/metre

Power—Heat flow rate ergs/second watts British thermal units/hour watts kilocalories (IC)/hour watts foot pounds force/second watts calories (IC)/second watts horsepower watts metric horsepower (CV) (PS) watts

curie

Pressure—see Stress Radioactivity becquerel

1.771 85 × 10−3 2.834 95 × 10−2 4.535 923 7 × 10−1 6.350 293 1.270 059 × 10 5.080 23 × 10 1.016 05 × 103 6.479 89 × 10−5 1.295 978 × 10−3 3.887 93 × 10−3 3.110 35 × 10−2 6.479 885 × 10−5 103 9.071 85 × 102 2.0 × 10−4 1.120 85 × 10−4 5.424 912 × 10−1 4.882 432 3.390 57 × 10−2 3.051 52 × 10−1 1.785 80 × 10 1.488 16 4.960 55 × 10−1 10−7 2.930 71 × 10−1 1.163 1.355 82 4.186 8 7.457 × 102 7.354 99 × 102 3.7 × 1010 (continued)

2–8

Introductory tables and mathematical information

Table 2.5

CONVERSION FACTORS—continued

To convert B to A multiply by

A

102

rem

3.876 × 103

roentgen

10−3 2.388 46 × 10−4 2.388 46 × 10−4 1.858 63 × 10−1 1.019 72 × 10−1 10−6 2.388 459 × 10−4 1.491 066 × 10−5

B Radiation–absorbed dose sievert Radiation exposure coulomb/kilogram

Specific heat capacity—mass basis joules/gram degree centigrade joules/kilogram kelvin calories*/gram degree centigrade joules/kilogram kelvin British thermal units/ joules/kilogram kelvin pound degree Fahrenheit foot pounds force/ joules/kilogram kelvin pound degree Fahrenheit kilogram force metres/ joules/kilogram kelvin kilogram degree centigrade Specific heat—volume basis joules/cubic centimetre degree joules/cubic metre kelvin centigrade kilocalories*/cubic metre degree joules/cubic metre kelvin centigrade British thermal units/cubic foot joules/cubic metre kelvin degree Fahrenheit Stress

1.450 377 × 10−4 6.474 881 × 10−8 10 10−5 10−7 1.019 716 × 10−7 7.500 638 × 10−3 7.500 638 × 10−3 7.500 638 9.869 233 × 10−6 1 6.474 8807 × 10−2 103 6.852 178 × 10−2 5.710 148 × 10−3 1 1.8 1.8 10−2 2.388 46 × 10−3 8.598 45 × 10−1 5.777 91 × 10−1 6.933 47

pounds force/square inch UK tons force/square inch dynes/square centimetre bars hectobars kilograms force/square millimetre torrs millimetres of mercury micron of mercury atmospheres pascals UK tons force/square inch dynes/centimetre pounds force/foot pounds force/inch

To convert A to B multiply by 10−2 2.58 × 10−4 103 4.186 8 × 103 4.186 8 × 103 5.380 32 9.806 65

106 4.186 8 × 103 6.706 61 × 104

newtons/square metre newtons/square metre newtons/square metre newtons/square metre newtons/square metre newtons/square metre

6.894 76 × 103 1.544 43 × 107 10−1 105 107 9.806 65 × 106

newtons/square metre newtons/square metre newtons/square metre newtons/square metre newtons/square metre megapascals

1.333 22 × 102 1.333 22 × 102 1.333 22 × 10−1 1.013 250 × 105 1 1.544 43 × 10

Surface tension newtons/metre newtons/metre newtons/metre

Temperature interval degrees Celsius (centigrade) kelvins degrees Fahrenheit kelvins degrees Rankine kelvins Thermal conductivity watts/centimetre degree watts/metre kelvin centigrade calories/centimetre second watts/metre kelvin degree centigrade kilocalories/metre hour degree watts/metre kelvin centigrade British thermal unit/foot hour watts/metre kelvin degree Fahrenheit British thermal unit inch/square watts/metre kelvin foot hour degree Fahrenheit

10−3 1.459 39 × 10 1.751 268 × 102 1 5.55 × 10−1 5.55 × 10−1 102 4.186 8 × 102 1.163 1.730 73 1.442 28 × 10−1 (continued)

Conversion factors Table 2.5

2–9

CONVERSION FACTORS—continued

To convert B to A multiply by

A

B

To convert A to B multiply by

Time 1.66 × 10−2 2.77 × 10−4 1.157 41 × 10−5 1.653 44 × 10−6 3.170 98 × 10−8 1.141 552 5 × 10−4 3.280 84 1.968 504 × 102 3.6 3.728 227 × 10−2 2.236 94 1.942 60 1.943 85 1.136 × 10−2 10 1.019 72 × 10−1 6.719 71 × 10−1 2.088 542 × 10−2

minutes hours days weeks years years

feet/second feet/minute kilometres/hour miles/minute miles/hour UK knots International knots UK miles/minute

seconds seconds seconds seconds seconds hours Torque—see Energy Velocity metres/second metres/second metres/second metres/second metres/second metres/second metres/second feet/second

Viscosity—dynamic* poise newton seconds/square metre kilogram force seconds/square newton seconds/square metre metre poundal seconds/square foot newton seconds/square metre pound force seconds/square foot newton seconds/square metre

104 1.550 03 × 103 1.076 392 × 10 5.580 011 × 106 3.875 009 × 104 3.6 × 103

stokes square inches/second square feet/second square inches/hour square feet/hour square metres/hour

6.102 37 × 104 3.531 473 × 10 1.307 95 103 2.199 69 × 102 2.641 72 × 102 1.759 755 × 10−3 2.199 69 × 10−1 3.519 508 × 10

cubic inches cubic feet cubic yards litres gallons (UK) gallons (US) pints (UK) gallons (UK) fluid ounces (UK)

Viscosity—kinematic square metres/second square metres/second square metres/second square metres/second square metres/second square metres/second

6.0 × 10 3.600 × 103 8.64 × 104 6.048 × 105 3.153 6 × 107 8.760 × 103 0.304 8 5.08 × 10−3 0.277 778 2.682 40 × 10 0.447 04 0.514 773 0.514 444 8.8 × 10 10−1 9.806 65 1.488 16 4.788 03 × 10 10−4 6.4516˙ × 10−4 9.290 3 × 10−2 1.792 111 × 10−7 2.580 640 × 10−5 2.77 × 10−4

Volume cubic metres cubic metres cubic metres cubic metres cubic metres cubic metres cubic centimetres litres litres Work—see Energy

* newton seconds/square metre (Ns/m2 ) ≡ pascal seconds (Pa s).

1.638 71 × 10−5 2.831 68 × 10−2 7.645 55 × 10−1 10−3 4.546 09 × 10−3 3.785 41 × 10−3 5.682 61 × 102 4.549 09 2.841 306 × 10−2

2–10 Table 2.6

Introductory tables and mathematical information COMMONLY REQUIRED CONVERSIONS

Acceleration Angle

g = 32 feet/second squared 1 radian

Area

1 acre 1 hectare

Density Energy

1 gram/cubic centimetre 1 calorie (IC) 1 kilowatt hour 1 British thermal unit 1 erg 1 therm 1 horsepower hour

Force

1 dyne 1 pound force 1 UK ton force 1 kilogram force

Length

1 angstrom unit (Å) 1 micron (µm) 1 micron (µm) 1 thousandth of an inch 1 inch 1 foot 1 yard 1 mile 1 ounce (Av) 1 ounce (Troy) 1 pound (Av) 1 hundredweight 1 UK ton (Av) 1 short ton (2 000 lbs) 1 carat (metric) 1 horsepower 1 pound force/square inch (p.s.i.) 1 UK ton force/square inch 1 bar 1 hectobar 1 kilogarm force/square centimetre 1 kilogram force/square millimetre 1 torr = 1 millimetre of mercury 1 atmosphere 1 pascal

Mass

Power Stress

Surface tension

1 dyne/centimetre

Velocity

1 foot/second 1 mile/hour

Volume

1 cubic inch 1 cubic foot 1 cubic yard 1 litre 1 litre l UK gallon 1 UK gallon

m s−2

= =

9.806 65 metres/second squared 57.295 8 degrees

◦

4 046.86 square metres 10 000 square metres

m2 m2

= = = = = = =

1 000 kilograms/cubic metre 4.186 8 joules 3.6 megajoules 1 055.06 joules 10−7 joules 105.506 megajoules 2.684 52 megajoules

kg m−3 J MJ J J MJ MJ

= = = =

10−5 newtons 4.448 22 newtons 9 964.02 newtons 9.806 65 newtons

N N N N

= = = = = = = = = = = = = = = = = = = = = = = = =

10−10 metres 10−6 metres 0.039 37 × 10−3 inches 25.4 micrometres 2.54 centimetres 30.48 centimetres 91.44 centimetres 1.609 344 kilometres 28.349 5 grams 31.103 5 grams 453.592 grams 50.802 3 kilograms 1 016.05 kilograms 907.185 kilograms 0.2 grams 745.7 watts 6.894 76 kilopascals 15.444 3 megapascals 100 kilopascals 10 megapascals 98.006 5 kilopascals 9.806 65 megapascals 133.322 pascals 101.325 kilopascals 1 newton/square metre

m m in µm cm cm cm km g g g kg kg kg g W kPa MPa kPa Mpa kPa MPa Pa kPa N m−2

=

1 millinewton/metre

mN m−1

= =

1.097 28 kilometres/hour 1.609 344 kilometres/hour

km h−1 km h−1

= = = = = = =

16.387 1 cubic centimetres 28.316 8 cubic decimetres 0.764 555 cubic metres 1 cubic decimetre 1.759 75 UK pints 4.546 09 cubic decimetres 0.160 544 cubic feet

cm3 dm3 m3 dm3 pint dm3 ft3

= =

Conversion factors

2–11

Table 2.7 CORRECTIONS TO TEMPERATURE VALUES IPTS–48 TO IMPLEMENT IPTS–68 (IPTS–68)–(IPTS–48) IN ◦ C t68

0

−10

−20

−30

−40

−50

−60

−70

−80

−90

−100

t68 ◦ C

−100 −0

0.022 0.000

0.013 0.006

0.003 0.012

−0.006 0.018

−0.013 0.024

0.013 0.029

−0.005 0.032

0.007 0.034

0.012 0.033

(0.008 at O2 point) 0.022 0.029

−100 −0

t68 ◦ C

0

10

20

30

40

50

60

70

80

90

100

t68 ◦ C

0 100 200 300 400 500 600 700 800 900 1 000

0.000 0.000 0.043 0.073 0.076 0.079 0.150 0.39 0.67 0.95 1.24

−0.004 0.004 0.047 0.074 0.075 0.082 0.165 0.42 0.70 0.98 1.27

−0.007 0.007 0.051 0.075 0.075 0.085 0.182 0.45 0.72 1.01 1.30

−0.009 0.012 0.054 0.076 0.075 0.089 0.200 0.47 0.75 1.04 1.33

−0.010 0.016 0.058 0.077 0.074 0.094 0.23 0.50 0.78 1.07 1.36

−0.010 0.020 0.061 0.077 0.074 0.100 0.25 0.53 0.81 1.10 1.39

−0.010 0.025 0.064 0.077 0.074 0.108 0.28 0.56 0.84 1.12 1.42

−0.008 0.029 0.067 0.077 0.075 0.116 0.31 0.58 0.87 1.15 1.44

−0.006 0.034 0.069 0.077 0.076 0.126 0.34 0.61 0.89 1.18 —

−0.003 0.038 0.071 0.076 0.077 0.137 0.36 0.64 0.92 1.21 —

0.000 0.043 0.073 0.076 0.079 0.150 0.39 0.67 0.95 1.24 —

0 100 200 300 400 500 600 700 800 900 1 000

t68 ◦ C

0

100

200

300

400

500

600

700

800

900

1 000

t68 ◦ C

1 000 2 000 3 000

— 3.2 5.9

1.5 3.5 6.2

1.7 3.7 6.5

1.8 4.0 6.9

2.0 4.2 7.2

2.2 4.5 7.5

2.4 4.8 7.9

2.6 5.0 8.2

2.8 5.3 8.6

3.0 5.6 9.0

3.2 5.9 9.3

1 000 2 000 3 000

From BS 1826:1952 Amendment No. 1, 2 February 1970. Example: 1 000◦ C according to IPTS–48 would be corrected to 1001.24◦ C to conform to IPTS–68. For conversions from IPTS–68 to ITS–90 see Table 16.1.

2–12

Introductory tables and mathematical information

Table 2.8

CORROSION CONVERSION FACTORS

The following conversion factors relating loss in weight and depth of penetration are useful in the assessment of corrosion. density of metal in grams/cubic centimetre = d density of metal in kilograms/cubic metre = 103 d To convert B to A multiply by 10−3 d −1 3.65 × 10−1 d −1 8.76 d −1 3.937 × 10−2 d −1 1.44 × 10 d −1 3.45 × 102 d −1 1.201 × 10 d −1 Table 2.9

To convert A to B multiply by

A

B

millimetres millimetres/year millimetres/year thousandths of an inch (mils) mils/year mils/year mils

grams/square metre grams/square metre per day grams/square metre per hour grams/square metre

103 d 2.74 d 1.14 × 10−1 d 2.54 × 10 d

grams/square metre per day grams/square metre per hour ounces/square foot

6.96 × 10−2 d 2.90 × 10−3 d 8.326 × 10−2 d

TEST SIEVE MESH NUMBERS CONVERTED TO NOMINAL APERTURE SIZE FROM BS 410:1969

Wire cloth test sieves were designated by the mesh count or number. This method, widely used until 1962, was laid down in previous British Standards—BS 410. Sieves are now designated by aperature size: see BS 410:1969, for full details. The table gives the previously used mesh numbers with the corresponding nominal aperture sizes, the preferred average wire diameters in the test sieves and the tolerances. Aperture tolerances

Mesh No. 3 3 21 4 5 6 7 8 10 12 14 16

18 22 25 30 36 44 52 60 72 85 100 120

Nominal aperture size mm

Preferred average wire diameter in test sieve mm

Max. tolerance for size of an individual aperture mm +

Tolerance for average aperature size mm ±

Intermediate tolerance mm +

5.60 4.75 4.00 3.35 2.80 2.36 2.00 1.70 1.40 1.18 1.00

1.60 1.60 1.40 1.25 1.12 1.00 0.90 0.80 0.71 0.63 0.56

0.50 0.43 0.40 0.34 0.31 0.26 0.24 0.20 0.18 0.17 0.15

0.17 0.14 0.12 0.10 0.084 0.071 0.060 0.051 0.042 0.035 0.030

0.34 0.29 0.28 0.23 0.20 0.17 0.16 0.14 0.11 0.11 0.09

µm

µm

µm +

µm ±

µm +

850 710 600 500 425 355 300 250 212 180 150 125

500 450 400 315 280 224 200 160 140 125 100 90

128 114 102 90 81 71 64 58 53 51 48 46

30 28 24 20 17 14 15 13 12 11 9.4 8.1

79 71 66 55 51 43 40 36 33 31 29 27 (continued)

Mathematical formulae and statistical principles Table 2.9

2–13

TEST SIEVE MESH NUMBERS—continued

Aperture tolerances

Mesh No.

Nominal aperture size mm

Preferred average wire diameter in test sieve µm

Max. tolerance for size of an individual aperture µm +

Tolerance for average aperture size µm ±

Intermediate tolerance µm +

150 170 200 240 300 350 400

106 90 75 63 53 45 38

71 63 50 45 36 32 30

43 43 41 41 38 38 36

7.4 6.6 6.1 5.3 4.8 4.8 4.0

25 25 24 23 21 21 20

Notes: (1) No aperture size shall exceed the nominal by more than the maximum tolerance. (2) The average aperture size shall not be greater or smaller than the nominal by more than the average tolerance size. (3) Not more than 6% of the apertures shall be above the nominal size by more than the intermediate tolerance.

For perforated plate sieve sizes with square or round holes—see BS 410:1969. Other national standards for test sieves may be found for France in NF X11–501, for Germany in DIN 4188 and for USA in ASTM E11–61. 2.1.2 Temperature scale conversions The absolute unit of temperature, symbol K, is the kelvin which is 1/273.16 of the thermodynamic temperature of the triple point of water—see Section 16. Practical temperature scales are Celsius (previously Centigrade) symbol ◦ C, and Fahrenheit, symbol ◦ F. An absolute scale based on Fahrenheit is the Rankine, symbol ◦ R. Where K, C, F and R, represent the same temperature on the Kelvin, Celsius, Fahrenheit and Rankine scales, conversion formulae are: 5 K = C + 273.15 C = (F − 32) 9 9 F = C + 32 R = F + 459.67 5 Rapid approximate conversions between Celsius and Fahrenheit scales can be obtained from Figure 2.1.

2.2 Mathematical formulae and statistical principles 2.2.1 Algebra IDENTITIES a2 − b2 = (a − b)(a + b)

a2 + b2 = (a − ib)(a + ib)

where i =

a3 − b3 = (a − b)(a2 + ab + b2 )

√

−1

a3 + b3 = (a + b)(a2 − ab + b2 )

an − bn = (a − b)(an−1 − an−2 b + · · · + bn−1 )

an + bn = (a + b)(an−1 − an−2 b + · · · + bn−1 )

(a ± b)2 = a2 ± 2ab + b2

(a ± b)3 = a3 ± 3a2 b + 3ab2 ± b3

when n is odd

2–14 °C 500

Introductory tables and mathematical information °F

°C 1000

900

450

850

950

800 400

750

900

700 350

650

850

550

400 350

150

300

650

250 100

1750 1450

2650 1950

3550

1700

2600

3500

1650 1400

2550 1900

3450

1600

2500

3400

1550

200

600

100

Figure 2.1

3300

1300 1200

2200 1700

3100

1250

2150

3050

1200 1150

2100 1650

3000

1150

2050

2950 1600

1550

1050

1850

2400

2350

2950

4250

2900

2850

2800 5050 5000 2750 4950

4050

4000

2700

3900

4900 4850

3950 2650

4800 4750

3850 2600

2100

4700

3800

4650

3750 2550

2050

4600

3700

4550

3650 2000

5150 5100

4100

2150

5250 5200

4150

2200

5350 5300

4200

2750 1500

4350

°F 5400

4300

2800

1900

1000

2900

°C 3000

4400

2850

1950

4450

2250 3150

2000

2450

3200 1750

2250

1100

°F 4500

3250

1350

1100

°C 2500

2300

1800

2300

950 500

2400 2350

1000

50 0

3350

1250

550

50

2450

1300

1050

150

1850

1350

1400

700

°F 3600

750 450

°C 2000

2700

1450

500 250

°F

1800

800

300

°C 1500

1500

600

200

°F

2500

Nomogram for approximate interconversion between Celsius and Fahrenheit temperature scales

Mathematical formulae and statistical principles (a + b)n = an + nan−1 b + +··· +

n(n − 1) n−2 2 a b 2!

n  
n n−x x n(n − 1)(n − 2) · · · (n − r + 1) n−r r a b a b + · · · + bn = x r! x=0

RATIO AND PROPORTION

If a c = b d then c+d a+b = b d and a−b c−d = b d In general, c e a = = = ··· = b d f



pan + qcn + ren + · · · pbn + qd n + rf n + · · ·

1/n

where p, q, r and n are any quantities whatever. LOGARITHMS

If ax = N ,

then x = loga N

loga MN = loga M + loga N

M = loga M − loga N N loga (M p ) = p loga M loga

1 log M r a 1 logb N = × loga N loga b

loga (M 1/r ) =

In particular, loge N = 2.302 58509 × log10 N

log10 N = 0.434 294 48 × loge N loge N ≡ ln N

2–15

e = 2.718 28

log10 N ≡ lg N

THE QUADRATIC EQUATION

The general quadratic equation may be written ax2 + bx + c = 0 √ −b ± b2 − 4ac Solution x = 2a

Introductory tables and mathematical information

2–16 If

 = (b2 − 4ac) the roots are real and equal if  = 0 the roots are imaginary and unequal if  < 0 the roots are real and unequal if  > 0 Also, if the roots are α and β, α+β = − αβ =

c a

b a

THE CUBIC EQUATION

The general cubic equation may be written y3 + a1 y2 + a2 y + a3 = 0 If we put y = x − 31 a1 the equation reduces to x3 + ax + b = 0, where a = a2 − Solution

1 2 a 3 1

x =z+v

and or

2 3 1 a − a1 a2 + a3 27 1 3   √ z−v z+v − ±i 3 , 2 2 b=

where z=



v=



3

1 − b+ 2



b2 a3 + 4 27



a3 b2 + 4 27

and 3

1 − b− 2

Alternatively,∗  x=2

 −a θ + 2kπ cos 3 3

(k = 0, 1, 2)

where cos θ = −

b 2



−a3 27

−1/2

If =

a3 b2 + 4 27

there are two equal and one unequal root if  = 0 three real roots if  < 0 one real and two complex roots if  > 0 ∗ The

second form of the solution is particularly useful when  < 0 (i.e. in the case of three real roots).

Mathematical formulae and statistical principles 2.2.2

2–17

Series and progressions

NUMERICAL SERIES

n (n + 1) 2

1 + 2 + 3 + ··· + n =

12 + 22 + 32 + · · · + n2 =

n (n + 1)(2n + 1) 6

13 + 23 + 33 + · · · + n3 =

n2 (n + 1)2 4

ARITHMETIC PROGRESSION

a, a + d, a + 2d, . . . , a + (n − 1)d Sn =

n [2a + (n − 1)d] 2

where Sn denotes the sum to n terms. GEOMETRIC PROGRESSION

a, ar, ar 2 , . . . , ar n−1 Sn =

a(1 − r n ) a(r n − 1) = (r − 1) (1 − r)

where Sn denotes the sum to n terms. If

r2 < 1 S∞ =

and

n → ∞,

a (1 − r)

TAYLOR’S SERIES

f (x) = f (a) + (x − a)f ′ (a) +

(x − a)2 ′′ (x − a)3 ′′′ f (a) + f (a) + · · · 2! 3!

MACLAURIN’S SERIES

f (x) = f (0) + xf ′ (0) +

x2 ′′ x3 f (0) + f ′′′ (0) + · · · 2! 3!

In the following series, the region of convergence is indicated in parentheses. If no region is shown, the series is convergent for all values of x. BINOMIAL SERIES

(1 ± x)n = 1 ± nx +

n(n − 1) 2 n(n − 1)(n − 2) 3 x ± x + ··· 2! 3!

(x2 < 1)

(1 ± x)−n = 1 ∓ nx +

n(n + 1) 2 n(n + 1)(n + 2) 3 x ∓ x + ··· 2! 3!

(x2 < 1)

2–18

Introductory tables and mathematical information

LOGARITHMIC SERIES

x2 x3 x4 ± − ± · · · (x2 < 1) 2 3 4   x3 x5 (1 + x) = 2 x+ + + ··· (x2 < 1) loge (1 − x) 3 5       1 x−1 3 1 x−1 5 x−1 loge x = 2 + + ··· (x > 0) + x+1 3 x+1 5 x+1    3  5 1 1 x x x loge (a + x) = loge a + 2 + + ··· + 2a + x 3 2a + x 5 2a + x loge (1 ± x) = ±x −

(a > 0, x + a > 0) EXPONENTIAL SERIES

e = 1+1+

1 1 1 + + + ··· 2! 3! 4!

ex = 1 + x +

x2 x3 x4 + + + ··· 2! 3! 4!

ax = 1 + x loge a +

(x loge a)2 (x loge a)3 + + ··· 2! 3!

TRIGONOMETRIC SERIES

sin x = x −

x5 x7 x3 + − + ··· 3! 5! 7!

cos x = 1 −

x2 x4 x6 + − + ··· 2! 4! 6!

tan x = x +

2x5 17x7 62x9 x3 + + + + ··· 3 15 315 2835

sin−1 x = x + cos−1 x =

1 x3 1 3 x5 1 3 5 x7 · + · · + · · · + ··· 2 3 2 4 5 2 4 6 7

π − sin−1 x 2

tan−1 x = x −

x5 x7 x3 + − + ··· 3 5 7

(−1 ≤ x ≤ 1)

If x = 1, cot−1 x =

 π π − 0 for all i, and in general xg ≤ x. For the data of  1/19 = 130.564 812 < 134.894 7 = x. Geometric mean has applications in Example 1, xg = ( 19 i=1 xi ) DOE (design of experiments) where at least 2 responses from each experimental unit are measured. n −1 = (n/ (1/xi )), xi = 0 for all i, i.e. xh The Harmonic mean is defined as xh = i=1 (1/xi )/n is the inverse of the average inverses of xi ’s. For the data of Example 1: 19
i=1

(1/xi ) = (1/87) + (1/103) + (1/130) + · · · + (1/129) = 0.150 325 6 −→ xh =

19 = 126.392 3 < xg . 0.150 325 588 522 53

The Harmonic mean has applications in ANOVA (analysis of variance) when the design is unbalanced and gives the average sample size over all levels of a factor, and in general xh ≤ xg ≤ x. The mode is the observation with the highest frequency. For the data of Example 1, MO1 = 87 and MO2 = 145 because both observations 87 and 145 have a frequency f = 2 (i.e. the data is bimodal). Most populations that should not be stratified for the purpose of sampling generally have a single mode. If a manufacturing product dimension has more than one mode, then there are quality problems with the process that manufactures the product. This is due to the fact that the ideal situation occurs when there is a single mode with a very high frequency at the ideal target of the product dimension.

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Introductory tables and mathematical information

MEASURES OF VARIABILITY

We present two measures of variability: the 1st measure, defined by R/d2 , is used in statistical process control when the size of the sample n ≤ 12. The random variable R represents the sample range defined as R = x(n) − x(1) , where x(n) is the nth order statistic (i.e. the sample maximum) and x(1) is the sample minimum (i.e. the 1st order statistic). The parameter d2 = E(R/σ); the quantity R/σ, where σ is the process standard deviation, is called the relative range in the field of statistics and E represents the expected value operator. For a normal universe, the values of d2 are given in Table 2.12 below. For n > 15, the sampling distribution of sample range, R, becomes unstable and hence it is inadvisable to use R as a measure of variation regardless of the underlying parent population. The 2nd measure of variability, and the most common, is the standard deviation (stdev) = S. In order to compute the stdev, S, we must always compute the variance first. The sample variance,  SV, is the average of deviations of xi (i = 1, 2, . . ., n) from their-own-mean squared, i.e. SV = ni=1 (xi − x)2 /n, but SV generally underestimates the population variance σ 2 because E(SV) = ((n − 1)/n)σ 2 , where E stands for the expected-value operator. This implies that SV  is a biased estimator of σ 2 . In order to get rid of this bias, the quantity S2 =(1/(n − 1)) ni=1 (xi − x)2 is used  as a measure of variability. Another reason for dividing the corrected sum of squares 2 by (n − 1), instead of n, in order to obtain a measure of variation, is the CSS = ni=1 (xi − x) n 2 fact that the CSS = i=1 (xi − x) has only (n − 1) degrees of freedom because of the constraint n 2 to as the sample varii=1 (xi − x) ≡ 0. For the above two stated reasons, S is loosely referred √ ance. The square root of S2 provides the stdev, S, and dividing S by n gives the (estimated) √ standard error of the mean σˆ x = S/ n. A simple binomial expansion of (xi − x)2 shows that the n 2  n 2  CSS = ni=1 (xi − x)2 = ni=1 xi2 − i=1 xi is called i=1 xi /(n) = USS − CF, where USS = n 2 2 the uncorrected sum of squares and CF = i=1 xi /n = n(x) is called the correction factor. For the data of Example 1, the value of the correction factor CF = 2 5632 /19 = 345 735.210 526 316, the USS = 872 + 1032 + 1302 + · · · + 1292 = 368 501, and thus the CSS = 368 501 − 345 735.210 53 = 227 65.789 473 7. Since there are 19 sample values, then there are 18 degrees of freedom and as a result S2 = 22 765.789 473 7/18 = 1 264.766 1 minutes.2 Clearly, the units of variance is that of the data’s squared and hence the true measure of variation is the positive square √ root of variance, namely S = S2 = (1 264.766 1)1/2 = 35.56 355 minutes. Although in Example 1 the sample size n = 19 is a bit too large to use σˆ = R/d2 as a measure of variation, we will compute this statistic for comparative purposes. The values of d2 for n = 16, 17, 18, 19, and 20 are 3.532, 3.588, 3.640, 3.689, and 3.735, respectively. The sample range for the data of the example is R = 211 − 87 = 124, and hence σˆ = R/d2 = 124/3.689 = 33.613 445 4. For majority of samples encountered in practice, the value of σˆ ≤ S. Note √ that σˆ without a subscript represents the stdev of individual elements in the sample while σˆ x = S/ n gives the estimate of the standard error of x given √ by se(x) = σ/ n. As yet another example, for the n = 10 time to failures of incandescent lamps (2 120, 2 250, 2 690, 4 450, 4 800, 4 950, 5 700, 5 840, 5 888, √ 6 990 hours), S = 1 687.018 264 while σˆ x = R/d2 = 1 582.196 231 32, and σˆ x = 1 687.018 264/ 10 = 533.482 02. HISTOGRAM

Grouping a data set in the form of a histogram is appropriate only for large samples, say n > 40 units. In order to illustrate the construction of a histogram for a large data set, we will go through an example in a stepwise fashion.

Example 2 The data set given below consists of n = 58 observations on shear strength of spot welds made on a certain type of sheet (arbitrary units).

Table 2.12 n

2

VALUES OF d2 FOR A NORMAL UNIVERSE

3

4

5

6

7

8

9

10

11

12

13

14

15

d2 1.128 1.693 2.059 2.326 2.534 2.704 2.847 2.970 3.078 3.173 3.258 3.336 3.407 3.472

Mathematical formulae and statistical principles

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5 434, 5 112, 4 820, 5 378, 5 027, 4 848, 4 755, 5 207, 5 049, 4 740, 5 248, 5 227, 4 931, 5 364, 5 189, 4 948, 5 015, 5 043, 5 260, 5 008, 5 089, 4 925, 5 621, 4 974, 5 173, 5 245, 5 555, 4 493, 5 640, 4 986, 4 521, 4 659, 4 886, 5 055, 4 609, 5 518, 5 001, 4 918, 4 592, 4 568, 4 723, 5 388, 5 309, 5 069, 4 570, 4 806, 4 599, 5 828, 4 772, 5 333, 4 803, 5 138, 4 173, 5 653, 5 275, 5 498, 5 582, 5 188. The 1st task in developing a histogram is to decide on the number of subgroups, denoted by C. ∼ √n and Sturges’ practical guideline C = ∼ 1 + 3.3 log10 (n). It can The two guidelines for C are√C = n ≥ 1 + 3.3 log (n), and thus we generally select 1 + 3.3 log10 (n) ≤ be verified that for n ≥ 40, 10 √ √ C ≤ n with the option that both 1 + 3.3 log10 (n) and n can be either rounded down √ or up to the next positive integer in order to obtain the value of C. For our shear strength data, 58 = 7.62 and 1 + 3.3 log10 (58) = 6.82, and hence 6 ≤ C ≤ 8. We settle on the value of C = 6 subgroups (or classes) for our histogram. The 2nd task is to determine the length, , of a subgroup, where it is mandatory to have equal class interval length. This is given by  = R/C, where it is necessary to always round up to the same number of decimals as the data. For our shear strength data, R = 5 828 − 4 173 = 1 655 and hence the length of each subgroup is equal to  = 1 655/6 = 276. Note that if the value of R/C is not rounded up in order to obtain , then in most cases the C subgroups will not span over the entire data range. The 3rd step in developing a histogram is to determine the class limits and boundaries for each subgroup, bearing in mind that class limits must have the same number of decimals as the original data while boundaries must have one more decimal than the original data. Table 2.13 gives the frequency distribution for the shear strength data, which clearly shows that the upper limit of the 1st subgroup is 4 448 and the lower class limit of the 2nd subgroup is 4 449. However, the upper boundary of the 1st class is UB1 = 4 448.5 which is equal to the lower boundary of the 2nd subgroup LB2 , i.e. UB1 = LB2 = 4 448.5. In a similar manner, UB2 = LB3 = 4 724.5, UB3 = LB4 = 5 000.5, UB4 = LB5 = 5 276.5, and UB5 = LB6 = 5 552.5. Further, j = UBj − LBj =  = 276 for all j, and because  there cannot be any overlapping of successive subgroups, then the Cj=1 fj must always add to n (=58 in this case). The midpoint of the jth subgroup is given by mj = (UBj + LBj )/2, or the average of upper and lower class limits of the same subgroup. Table 2.13 shows that m1 = (4 448 + 4 173)/2 = 4 310.5, m2 = (4 724.5 + 4 448.5)/2 = 4 586.5, . . ., m6 = 5 690.5; note that  must equal to mj − mj−1 for j = 2, 3, 4, . . . , C, i.e.  = mj − mj−1 . The histogram from Minitab software is provided in Figure 2.2.

Table 2.13

FREQUENCY DISTRIBUTION FOR DATA IN EXAMPLE 2

Subgroup Limits fj Class Limits fj

4 173–4 448 1 5 001–5 276 20

4 449–4 724 9 5 277–5 552 8

4 725–5 000 14 5 553–5 828 6

Frequency

20

20

10 14 9

8

6

0 4310.5

Figure 2.2

4586.5

4862.5 5138.5 SHST

Histogram of Shear Strength (SHST) Data

5414.5

5690.5

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Introductory tables and mathematical information

If an experimenter does not possess the raw data but only the histogram is available, then the computing formulas for the mean and standard deviation for grouped data may be used as given below: xg =

C 1
(mj × fj ), n j=1

where the subscript g stands for grouped, and S2g

   C C ( Cj=1 mj × fj )2 1 
2 1
2 . (mj − g) × fj = mj × fj − = n−1 n−1 n j=1

j=1

The raw mean and standard deviation for the shear strength data are x = 5 057.552 and S = 344.680 9, while those of the histogram in Figure 2.2 are xg = 5 067.1207 and Sg = 341.218 3. THE BOXPLOT

A boxplot is a graphical measure of variability, and it can also assess if the data contain mild or extreme outliers. For the sake of illustration, we will obtain the boxplot for the data of Example 2 in a stepwise manner. Step 1. Draw a vertical line through the median x˜ = x0.50 = 5 046. Step 2. Draw vertical lines through the 25th percentile x0.25 = Q1 and the 75th percentile x0.75 = Q3; then connect at the bottom and the top to make a (rectangular) box. For the data of Example 2, the box is shown below. x0.25

x0.50

x0.75

4806

5046

5275

Step 3. Compute the values of interquartile range, IQR = Q3 − Q1, 1.5 × IQR, and 3 × IQR. For the Example 2, IQR = 5 275 − 4 806 = 469, 1.5 × IQR = 703.50 and 3 × IQR = 1 407.00. Then all data points less than Q1 − 3 × IQR = 4 806 − 1 407 = 3 399 or larger than Q3 + 3 × IQR = 5 275 + 1 407 = 6 682 are extreme outliers. Example 2 data has no extreme outliers because no data point lies outside the interval [3 399, 6 682]. Next determine if there are any mild outliers by obtaining the interval [x0.25 − 1.5 × IQR, x0.75 + 1.5 × IQR] = [4 102.5, 5 978.5]. Since no data point falls outside the interval [4 102.5, 5 978.5], then Example 2 has no mild outliers. Step 4. Draw whiskers from Q1 = x0.25 and Q3 = x0.75 to the smallest and largest order statistics that are not outliers. Note that only outliers for which assignable causes are found, and the corresponding corrective actions taken, should be removed from the data for further analysis, or else the outlier must remain as a part of the data. LAWS OF PROBABILITIES

Definition.

The occurrence probability (pr) of an event A is defined as the function

P(A) =

N(A) N(U)

Mathematical formulae and statistical principles

2–33

Top tank Port 2

Body

Bottom tank

Port 1 Figure 2.3

Heat exchanger geometry for use with example 3

iff (if and only if) all the N(U) elementary outcomes in the universe U are equally likely, and N(A) are those elementary outcomes that are favourable to the occurrence of the event A.

Example 3 An aluminium-alloy heat exchanger is furnace brazed so that the tubes, fins, and side rails, plus the headers used to attach the top and bottom tanks come out of the furnace as a single unit that makes up the body of the heat exchanger. After brazing, the top and bottom tanks are crimped into place. The orientation of the top and bottom tanks will determine the relative position of the ports through which coolant enters and leaves the heat exchanger. Thus, for the heat exchanger shown is in Figure 2.3, there are 4 possible outcomes because of the body symmetry. The two tanks can be assembled correctly as shown or vice versa on the other side of the body. The two unacceptable assemblies occur when port 1 (at the left of bottom tank) and port 2 (at the right of top tank) are on different sides. Letting F represent a port on the front and B a port on the back, the set of all possible outcomes for the assembly is U = {(F1 , F2 ), (F1 , B2 ), (B1 , B2 ), (B1 , F2 )}. The conforming units are those with either (F1 , F2 ) or (B1 , B2 ) outcomes. The function P(A) must satisfy the following axioms: (1) 0 ≤ P(A) ≤ 1 because 0 ≤ N(A) ≤ N(U). (2) P(U) ≡ 1, and P(φ) ≡ 0, i.e. the occurrence pr of the null set φ is identically zero. (3) IfA and B are mutually exclusive (MUEX), then P(A ∩ B) = P(φ) = 0, and P(A ∪ B) = P(A) + P(B); further, if A1 , A2 , A3 , . . . is a countably infinite jointly MUEX events, then ∞  ∞ 
P Ui=1 Ai = P(Ai ) i=1

The event A ∩ B occurs iff both A and B occur simultaneously, and A ∪ B occurs iff at least one of the two events occur. The event A′ occurs iff A does not occur, i.e. the event A′ is the complement of the event A. Since A and A′ are MUEX events, then A ∪ A′ = U and as a result P(A ∪ A′ ) = 1 = P(A) + P(A′ ) → P(A′ ) = 1 − P(A). Let events A and B belong to the same sample space U. Then in general P(A ∪ B) = P(A) + P(B) − P(A ∩ B) for all events A and B in the same universe U.

Example 4 A repairman claims that the pr that an air conditioning compressor is ‘all right’ is P(A) = 0.85, and P(B) = 0.64 that its fan motor is all right, and 0.45 that both the compressor and fan belt are all right. Can his claim be true?

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Introductory tables and mathematical information

P(A ∪ B) = P(A) + P(B) − P(A ∩ B) = 0.85 + 0.64 − 0.45 = 1.04. Clearly, his claim is false! If he revises his claim to P(B) = 0.55, then his claim would be correct because P(A ∪ B) = 0.95 in which case the probability that neither A nor B occur is P[(A ∪ B)′ ] = P(A′ ∩B′ ) = 0.05. PERMUTATIONS

The total number of different ways that n distinct objects can be permuted is given by n Pn

= n!

As an example, consider the 3 distinct objects A, B, C. The permutations of these objects are ABC, ACB, BAC, BCA, CAB, CBA, of which there are six. Therefore, 3 P3

= 3! = 6

The total number of ways that r(≤n) objects can be selected from n objects and then permuted is given by n Pr

=

n! (n − r)!

For example, consider 4 objects A, B, C and D. Then the elements of 4 P2 are: AB, BA, AC, CA, AD, DA, BC, CB, BD, DB, CD, DC, of which there are 12 = 4!/(4 − 2)!, i.e. the permutations of 4 objects taken two at a time is equal to 4 P2 = 12. COMBINATIONS

The total number of different ways that r distinct objects can be selected from n (r ≤ n) is given by   n! n C = n r r = nPr /r! = (n − r)!r! . For example, consider the combinations of the 4 objects A, B, C and D, taken 2 at a time: AB, AC, DA, BC, DB, CD, of which there are 6 = 4 C2 = 4!/(2! × 2!) of them. Note that AB and BA are the same element of while they are two distinct elements of 4 P2 .

4 C2 ,

Example 5 A production lot contains 80 units of which only 6 are nonconforming (NC) to customer specifications. A random sample of size n = 10 is drawn from the lot without replacement and the number of NC units in the sample is counted. What is the pr that the sample contains exactly 2 NC units? What is the pr that the sample contains at least one NC unit? Let the variable X represent the number of NC units in the random sample of size n = 10. Then,    6 74 8 2 (6 C2 )(74 C8 ) =   = 0.137 305 316 27, P(X = 2) = 80 80 C10 10 and P(X ≥ 1) = 1 − P(X = 0) = 1 −

74 C10 80 C10

= 0.563 674 217 2.

Note that the procedure outlined in the above example is generally referred to as sampling inspection in the field of quality control (QC). Its objective often is to rectify an outgoing lot by first deciding whether to accept or reject a lot, and in case the lot is rejected, then it is subjected to 100% screening in order to remove all defective units before shipment to customer.

Mathematical formulae and statistical principles

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RANDOM VARIABLES AND THEIR FREQUENCY FUNCTIONS

A random variable (rv) is a real-valued function defined on the universe U. If the range space of a rv is discrete, then its frequency function is generally referred to as a pr mass function (pmf), and if its range space is continuous, then its frequency function is called a pr density function (pdf). For example, the rv of the Example 5 is discrete because its range space Rx = {0, 1, 2, 3, 4, 5, 6} and its pmf is given by    6 74 10 − x x   , p(x) = 80 10

x = 0, 1, 2, 3, 4, 5, 6

In fact the above pmf is called the hypergeometric in the field of statistics. Clearly, if p(x) is a pmf,  then it is absolutely necessary that Rx p(x) ≡ 1, and similarly, if the rv, X, is continuous with pdf  f(x), then it is paramount that Rx f (x)dx ≡ 1 (or 100%). The population mean, µ, and population variance, σ 2 , of a rv are given by (the symbol E used below represents the expected-value operator). 
 xp(x), if X is discrete   R x µ = E(X) =     xf (x) dx, if X is continuous  Rx

and


 (x − µ)2 p(x), if X is discrete   R x σ 2 = V(X) = E[(X − µ)2 ] =     (x − µ)2 f (x) dx, if X is continuous  Rx

The most two important pmfs in the field of statistics are the binomial and Poisson. THE BINOMIAL PMF

Consider n > 1 independent (or Bernoulli) trials in successions, where the experimental result can be classified as either ‘success’ or ‘failure’. The binomial rv, X, is defined as X = ‘The number of generic successes observed in n trials’, where Rx = {0, 1, 2, 3, . . . , n −1, n}, and p = pr of a success at each single trial. Then, the binomial pmf is given by b(x; n, p) =nCx px (1 − p)n−x The PDF (pr distribution function) in this equation is called the binomial with parameters n and p and is denoted cumulative distribution function (cdf) is given by F(x) = P(X ≤ x) = by Bin(n, p). Its B(x; n, p) = xi=0 b(i; n, p) = xi=0 n Ci × pi × q n−i , where q = 1 − p represents the failure pr at each trial. It can be verified that the binomial pmf, b(x; n, p), has a mean of E(X) = np and a variance σ 2 = npq.

Example 6 A manufacturing process produces parts which are, on the average, 1% NC to customer specifications. A random sample of n = 30 items is drawn from a conveyor belt. Compute the pr that the sample contains exactly x = 2 NC units. b(2; 30, 0.01) = 30 C2 (0.01)2 (0.99)28 = 0.032 83. The pr that the sample contains at most two NC units is given by P(X ≤ 2) = B(2; 30, 0.01) = 0.996 682 3, and the pr that the sample contains at least two NC units is given by

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Introductory tables and mathematical information

P(X ≥ 2) = 1 − P(X ≤ 1) = 1 − B(1; 30, 0.01) = 0.036 148. The expected number of NC units in the sample is given by µ = n × p = 0.30 and the variance of X is given by V(X) = σ 2 = npq = 0.297. Note that in the context of the Example 6 the statistic pˆ = X/n is called the sample fraction NC and is used as a point estimate of process fraction NC, p. The binomial expansion of (a + b)n is given by n
n−i i (a + b)n = b nCn−i a i=0

which shows that

n

x=0

b(x; n, p) =

THE POISSON DISTRIBUTION

n

x x=0 n Cx p (1

− p)n−x = (p + q)n = 1n = 1.

Consider many Bernoulli trials (i.e. n → ∞) such that occurrence pr of success at each trial is small (say p < 0.15) and average number of successes per time unit, µ = n × p = λ, is a constant. Let the rv, X, denote the number of Poisson events (or generic successes) that occur during one time interval of unit length. Then the PDF (or pr distribution function) of X is given by p(x; λ) =

λx −λ e , x!

x = 0, 1, 2, 3, 4, . . .

(2.1)

It can be shown that the Poisson pmf in (2.1) is simply the limiting distribution for a Bin(n, p) as n → ∞ and simultaneously as p → 0 but the product n × p stays fixed at a rate equal to λ, and is generally required that n × p ≤ 20. For n × p > 20, the process becomes practically Gaussian (a pdf that will be defined later).

Example 7 The number of accidents per day in a certain city is Poisson distributed at an average rate of 5 accidents/day. The pr of no accidents occurring in the city during the next day is given by p(0; 5) = (50 /0!) e−5 = 0.006 738. The pr of at most 4 accidents occurring during the next day is  P(X ≤ 4) = Fx (4; 5) = 4x=0 (5x /x!) e−5 = 0.440 493 3. The Poisson distribution can be used to compute probabilities over intervals of length t (t = 1 unit of time). Let Y = number of Poisson events occurring during an interval of length t (t = 1) where the average number of Poisson events per unit of time (t = 1) is λ = µ = np. Then the average number of Poisson events per interval of length t is E(Y) = λt. As a result the pmf for the rv Y is given by p(y; λt) =

(λt)y −λt e , y!

y = 0, 1, 2, 3, 4, . . .

Consider the Poisson process of Example 7. We wish to compute the pr of exactly 6 accidents occurring during the next 2 days. Then, λ = np = 5 accidents/day, t = 2 days → λ × t = 10 accidents/two days → P(Y = 6) = (106 /6!) e−10 = 0.144 458 2, i.e. Y is p(y; 10) read as Poisson distributed at an average rate of 10 accidents per two days. The pr that there will be exactly 11 accidents in the city during the next 3 days is P(Y = 11) = (1511 /11!) e−15 = 0.066 287 4. The pr that there will be at least 11 accidents in the next 3 days is given by P(Y ≥ 11) = 1 − P(Y ≤ 10) = 1 −  x −15 = 0.881 536. FY (10; 15) = 1 − 10 x=0 (15 /x!)e It can be verified that the long-term (or weighted) average of X, as expected, is given by µ = E(X) = λ and V(X) = λ =√ µ. As a result, all Poisson processes have a CV (coefficient of variation = σ/µ) equal to (100/ λ)%. CONTINUOUS FREQUENCY FUNCTIONS

If the range space of a rv is continuous, then the occurrence pr of a single point in the range space b is zero, but the pr over a real interval [a, b] is given by a f (x) dx, where f(x) is the probability density function (pdf) of the rv X. There are numerous pdfs in the field of statistics, each of which has specific applications. The three most commonly encountered in practice are the uniform, normal (or Gaussian), and the exponential. The normal distribution is used to approximately model the distribution of almost any manufactured dimension, and the exponential has extensive applications in life testing and in Markov processes (specifically in queuing theory). The uniform distribution has extensive applications in statistical simulations because of the fact that all continuous cumulative distribution functions in the universe are uniformly distributed over the interval [0, 1].

Mathematical formulae and statistical principles

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THE UNIFORM pdf

A continuous rv, X, is said to be uniformly distributed over the real interval [a, b] iff its pdf is given by   1 , f (x) = b − a  0,

a≤x≤b elsewhere.

This is expressed as X ∼ U(a, b). The population mean of a rv X ∼ U(a, b) is given by µ = E(X) = (a + b)/2 and its population variance is given by V(X) = σ 2 = (b − a)2 /12. THE NORMAL (OR GAUSSIAN) DISTRIBUTION N(µ, σ 2 )

A continuous rv, X, is said to be normally distributed with mean µ and variance σ 2 iff its pdf is given by 2

f (x; µ, σ) = C e−((1/2)((x−µ)/σ) ) = C Exp[ − (x − µ)2 /(2σ 2 )],

−∞ < x < ∞

where C is a normalising constant and must be evaluated in such a manner that the total area under the pdf is exactly equal to 1 (or 100% pr), i.e. we must require that 

∞

−∞

2

C e−((1/2)((x−µ)/σ) ) dx = 1,

which leads to C2 = 1/(2πσ 2 ). Therefore, the only √ unique value of C that makes the value of the above improper integral equal to 100% is C = 1/(σ 2π). These developments show that a continuous rv, X, is N(µ, σ 2 ) [read as normally distributed or Gaussian with mean µ and variance σ 2 ] only if its pdf is given by f (x; µ, σ) =

1 2 √ e−((1/2)((x−µ)/σ) ) , σ 2π

−∞ < x < ∞.

The graph of the normal pdf in the equation above is symmetrical and is provided in Figure 2.4. It can easily be verified that the modal point of a N(µ, σ 2 ) distribution, as shown in Figure 2.4, occurs at x = µ. Secondly, its points of inflection occur at x = µ ± σ.Therefore, one invariant property of all normal distributions is the fact that the distance between the line x = µ and the point of inflection is exactly one standard deviation σ. Because of the enormous importance of this distribution, we list its first 4 moments, where the kth moment about a real constant C is given by E[(X − C)k ].

s

m Figure 2.4

The modal point of a N(µ, σ 2 ) distribution

0.05 x0.95

x

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Introductory tables and mathematical information

It can be verified that the 1st four moments of any Gaussian distribution are µ′1 = µ, V(X) = µ2 = σ 2 , α3 ≡ µ3 /σ 3 = 0, and α4 = µ4 /σ 4 = 3.00, where µ4 = E[(X − µ)4 ] is called the 4th central moment of X, and µ′k represents the kth origin moment. It must be emphasised that the normal distribution has unlimited applications in all facets of human life (for example, in all sciences, in all fields of engineering, in agriculture, pharmacy and medicine, etc.). In fact the distribution of every dimension that is manufactured can be modelled closely by a normal curve! This implies that over 75% of all parent populations in the universe are assumed normally distributed with some mean µ and some variance σ 2 . Application examples will now follow.

Example 8 The design tolerances (or consumers’ specifications) on length of steel pipes is 30 ± 0.25 cm. The dimension or length of a pipe, X, is N(30, 0.006 4 cm2 ). We wish to compute the pr that a randomly selected pipe has length below 30.10 cm? P(X < 30.10 cm) =



30.10 −∞

1 2 √ e−((1/2)((x−µ)/σ) ) dx, σ 2π

where µ = 30.00 cm and σ = 0.08 cm. To simplify the integrand, let (x − µ)/σ = z; then, dz/dx = 1/σ or dz = dx/σ. Substituting the transformation z = (x − µ)/σ into the above integral results in: P(X < 30.10) =



1.25

−∞

1 2 √ e−z /2 dz = 2π



1.25 −∞

φ(z)dz = (1.25).

√ 2 The pdf φ(z) = (1/ 2π)e−z /2 is called the standard normal density function because its mean is zero and its variance is equal to 1. Unfortunately, φ(z) has no closed-form simple antiderivative, and therefore its cdf has been tabulated for different values of z (using numerical integration) in tables in almost all statistical texts. For example, see pages 722–723 of J. L. Devore ‘Probability and Statistics for Scientists and Engineers’, Duxbury Press, Belmont, CA, 2000. The cdf of the standard normal density N(0, 1) is universally denoted by (z). For example, from Table A.3, p. 723 of Devore,

(1.5) = 0.933 19, (−1.5) = 0.066 81, (1.96) = 0.975, (−1.96) = 0.025, (3) = 0.998 65,

(−3) = 0.001 35 [Note that (−z) = 1 − (z)]. Thus, P(X < 30.10) = FX (30.10) = P(Z ≤ 1.25) =

(1.25) = 0.894 35. The graph of φ(z) is shown in Figure 2.5. Microsoft Excel will provide both normal prs and the normal inverse. The reader can open Excel, click on Insert, Function, Statistical, scroll down to NORMSDIST, insert the desired value of z, then Excel will provide the corresponding value of cdf of z, that is denoted by (z). Figure 2.5 graphs the standard normal density (z) versus z.

1

⌽(1.25)

0 Figure 2.5

Standard normal density (z) versus z

0.106 1.25

z

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Suppose a pipe is selected at random. We wish to compute the pr that its length falls within 2σ of the mean µ = 30 cm. P(|X − µ| ≤ 2σ) = P(29.84 ≤ X ≤ 30.16)   X−µ 30.16 − 30 29.84 − 30 =P ≤ ≤ 0.08 σ 0.08 = P(−2 ≤ Z ≤ 2) = (2) − (−2) = 0.977 25 − 0.022 75 = 0.954 50. Further, the pr that the length of a pipe is within 3 sigma of µ is given by P(|X − µ| ≤ 3σ) = P(−3 ≤ Z ≤ 3) = (3) − (−3) = 0.998 65 − 0.001 35 = 0.997 30. Finally, the P(µ − 4σ ≤ X ≤ µ + 4σ) = (4) − (−4) = 0.999 968 33 − 0.000 031 671 = 0.999 936 66. Next, the fraction of the pipes that are not conforming to the lower specification limit LSL = 29.75 cm and upper specification limit USL = 30.25 cm is called the fraction NC (FNC) of the process, denoted by p, and is computed below. p = FNC = P(X < 29.75) + P(X > 30.25) = P(Z ≤ −3.125) + P(Z ≥ 3.125) = (−3.125) + [1 − (3.125)] = 2 × (−3.125) = 2 × 0.000 889 025 = 0.001 778 051. Suppose that the process in the Example 8 is not centred at µ = 30.00 cm. Then, the process fraction NC under the condition that µ = 30.00 cm will always exceed p = 0.001 778 (at the same value of σ). For example, if µ = 30.05 cm, then ZL = −3.75, pL = 0.000 088 42, ZU = 2.50, pU = 0.006 21, and p = 0.006 298 1 at σ = 0.08 cm. THE PERCENTILES OF A NORMAL DISTRIBUTION

For the standard normal density, N(0, 1), the 95th percentile is denoted by z0.05 , and is defined such that the P(Z ≤ z0.05 ) = 0.95. Similarly, the 90th percentile of Z is defined as P(Z ≤ z0.10 ) = 0.90. This definition implies that for the standard normal density function (STNDF), (z0.05 ) = 0.95 and (z0.10 ) = 0.90. Table A.3 on page 723 of Devore shows that z0.05 = 1.645 while z0.10 = 1.282. By symmetry of the STNDF, it follows that the 5th percentile of Z is equal to z0.95 = −1.645 and the 10th percentile z0.90 = −1.282, i.e. the P(Z ≤ −1.645) = (−1.645) = 0.05 and the P(Z ≤ −1.282) = (−1.282) = 0.10. It should be noted that z0.05 = 1.645 is also called the 5 percentage point of the STNDF and z0.10 = 1.282 is called the 10 percentage point of a STNDF. It turns out that in the field of statistics, a percentage point, such as zα , is the distance from the origin beyond which the right tail pr is exactly equal to α. Further, for the STNDF, it should be clear that zα = −z1−α for all 0 < α < 1. Figure 2.5 shows that z0.105 65 = 1.250 and hence z0.894 35 = −1.250. If the rv, X, is normally distributed but either µ = 0, or σ = 1 (or both), the 95th percentile of X is defined as x0.95 such that the P(X ≤ x0.95 ) = 0.95, and similarly the 90th percentile of X is defined such that P(X ≤ x0.90 ) = 0.90. Furthermore, in general x0.95 = − x0.05 or xp = x1−p . THE REPRODUCTIVE PROPERTY OF THE NORMAL DISTRIBUTION

Suppose X1 , X2 , . . ., Xn are n normally distributed rvs with means µi and variances σi2 and correlation coefficients ρij = σij /σi σj , where σij = E[(Xi − µi ) × (Xj − µj )] is called the covariance between the two random variables Xi and Xj (i = j). Further, c1 , c2 , . . ., cn are known or given constants. Then it has been proven in the theory of statistics that the linear combination (LC). Y = c1 X1 + c2 X2 + · · · + cn Xn = is also normally distributed with mean E(Y) =

n
i=1

ci µi

n
i=1

ci Xi

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and variance V(Y) =

n
i=1

c2i σi2 + 2

n n−1



ci cj σij ,

i=1 j>i

  n n−1
n n


ci cj σij . i.e., Y ∼ N  ci µi , c2i σi2 + 2 i=1

i=1

i=1 j>i

If the dimensions Xi and Xj are mutually independent for all i = j, then σij = 0, and the V(Y) reduces  to σy2 = ni=1 c2i σi2 . Note that zero covariance does not always imply independence.

Example 8 (continued)

Three steel pipes are selected at random from the N(30, 0.006 4) process and placed endto-end. Compute the pr that the total length of the assembly exceeds 90.24 cm. [n = 3, each ci = 1, and each Xi ∼ N(30.00, 0.006 4)]. →Y = X1 + X2 + X3 → E(Y) = 30 + 30 + 30 = 90.00, V(Y) = 0.006 4 + 0.006 4 + 0.006 4 = 0.004 8 →Y ∼ N(90, 0.019 2), and hence   y − µy 90.24 − 90.00 P(Y > 90.24 cm) = P = P(Z > 1.732 051) > σy 0.138 564 = 1 − (1.732 051) = 1 − 0.958 368 = 0.041 632 3. THE CENTRAL LIMIT THEOREM (CLT)

If X1 , X2 , . . ., Xn are independent, but not necessarily normal, rvs with E(Xi ) = µi and V(Xi ) = σi2 , then the distribution of Y = X1 + X2 + · · · + Xn , as n → ∞, approaches normality with  n 2 E(Y) = ni=1 µi and V(Y) n = i=1 σi . Note that the more skewed the frequency functions of the individual Xi ’s in the sum i=1 Xi are, the more slowly (in terms of n) the distribution of Y approaches normality. For example, if the Xi ’s are uniformly distributed, then for n > 8, the distribution of the sum, Y = ni=1 Xi , is approximately normal, while if Xi ’s are exponentially distributed, then a value  of n > 200 is needed. The uniform skewness is α3 = E (x − µ/σ)3 = 0, while that of exponential distribution is α3 = 2.00. In case Xi ’s have the same mean and variance, then the distribution of Y approaches normality with E(Y) = n µ and V(Y) = n σ 2 . A general conservative rule of thumb is that a value of n > 25α23 and simultaneously n > 5α4 (and in most applications at least n > 30) is needed in order for the distribution of the sumY = ni=1 Xi to become approximately Gaussian.Again, the quantity α3 = E[(X − µ)3 /σ 3 ] is called the coefficient of skewness, or simply the skewness of X. The closer α3 (X) is tozero, the smaller n is needed for an adequate normal approximation to the distribution of the sum ni=1 Xi .

Example 9

A large freight elevator can transport a maximum of 10 000 kg of load. A load of cargo containing 50 boxes of a certain product is transported on the elevator every hour. The weights of the boxes are independent (not normal but identically distributed) each with E(Xi ) = 190 kg and V(Xi ) = 900 kg2 . Estimate (using the CLT) the pr that the maximum limit is exceeded on a given hour and disrupts the transporting process?  Solution: Y = a Cargo Load = 50 i=1 Xi → E(Y) = 50 × 190 = 9 500 and V(Y) = 50 × 900 = 45 000.00. P(Y > 10 000 kgs) = P(Z > (10 000 − 9 500/212.132)) = P(Z > 2.357 02) = 1−

(2.357 023) = 1 − 0.990 789 = 0.009 211 1. THE NORMAL APPROXIMATION TO THE BINOMIAL

Since a  binomial rv, XBin , is the sum of n independent Bernoulli rvs, i.e. then the binomial rv XBin = ni=1 Xi , where each Xi is Bernoulli (i.e. Xi has a binomial distribution with the parameter n ≡ 1) with E(Xi ) = p and V(Xi ) = pq. As a result of the CLT, as n → ∞, the binomial pmf can be approximated by a normal pdf with E(X) = np and V(X) = npq. However, because a binomial rv has a discrete range space, a correction for continuity has to be applied as will be illustrated in the following example. The approximation is adequate when n is large enough such that the product np > 15 and 0.10 < p < 0.90. If np < 15, we would recommend the Poisson approximation to the Binomial. The general rule of thumb was that n must be large enough such that n > 25α23 and simultaneously n > 5α4 , where for a single Bernoulli trial α23 = (q − p)2 /(p × q), and α4 = (1 − 3p + 3p2 )/(p × q) = (1 − 3 pq)/

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√ (p × q). It can be shown that for a Bin(n, p) distribution the value of skewness is α3 = (q − p)/ npq 4 4 and the binomial α4 = E[((X − µ)/σ) ] = E(Z ) is given by α4 = (1 + 3(n − 2)pq/npq).

Example 10 Consider a binomial distribution with n = 50 trials and p = 0.30 (so that µ = np = 15). Note that we would like to have np > 15, but 25α23 = 19.05 implies that the requirement on skewness n = 50 > 25α23 is easily satisfied and similarly, 5α4 = 14.876 2 shows that the α4 requirement is also satisfied because n > 14.876 2. Then the range space of the discrete rv, XD , is Rx = {0, 1, 2, 3, . . . , 50}. The exact pr of attaining exactly 12 successes in 50 trials is given by b(12; 50, 0.30) = P(XD = 12) = 50 C12 (0.30)12 (0.70)38 = 0.083 83. To apply the normal distribution (with µ = 15 and σ 2 = npq = 10.50) in order to approximate the P(XD = 12), we 1st have to select an interval on a continuous scale, XC , to represent XD = 12. Clearly this continuous interval on XC has to be (11.5, 12.5), i.e. we will have a correction of 0.50 for continuity. In short, XD = 12 ∼ = (11.5, 12.5)C . Thus,   12.5 − 15 11.5 − 15 P(XD = 12) ∼ ≤Z≤ = P(11.5 ≤ XC ≤ 12.5) = P 3.240 4 3.240 4 = P(−1.080 1 ≤ Z ≤ −0.771 52) = (−0.771 52) − (−1.080 1) = 0.220 200 − 0.140 044 = 0.080 157. Since the exact pr was 0.083 83, the percent relative error in the above approximation is   0.083 83 − 0.080 157 % Relative Error = × 100 = 4.382%. 0.083 83 Next we compute the exact pr that X exceeds 12, i.e. P(XD > 12) = 1 − B(12; 50, 0.30) = 1 − 0.222 865 8 = 0.777 134. The normal approximation to this binomial pr is P(XC ≥ 12.5) = P(Z ≥ −0.77 152) = 0.779 80 ∼ = P(XD > 12). The % error in this approximation is 0.343%. As stated earlier, the normal approximation to the binomial should improve as n increases and as p → 0.50. To illustrate this fact, consider a binomial distribution with parameters n = 65 and p = 0.40. Then, µ = np = 26, σ = (npq)1/2 = 3.949 7, and P(XD = 22) =65 C22 (0.40)22 (0.60)43 = 0.061 7. The normal approximation to this b(22; 65, 0.40) is P(21.5 ≤ XC ≤ 22.5) = P(−1.139 33 ≤ Z ≤ − 0.886 15) = 0.187 77 − 0.127 28 = 0.060 5 with relative % error of 1.935%.  x 65−x = 0.188 327. The normal Further, P(XD < 23) = B(22; 65, 0.40) = 22 x=0 65 Cx (0.40) (0.60) approximation to this binomial cdf corrected for continuity (cfc) is P(XC ≤ 22.5) = P(Z ≤ −0.886 15) = (−0.886 15) = 0.187 769 2 with an error of 0.296 2%. THE EXPONENTIAL pdf

A continuous rv, X, with pdf f(x; λ) = λe − λx is said to be exponentially distributed at the average rate of λ > 0. The cdf of the exponential density is given by  x F(x) = λe−λt dt = 1 − e−λx . 0

The exponential pdf has widespread and enormous applications in reliability engineering (or life testing) and in stochastic processes (specifically queuing theory). The majority of its applications occur because of its relation to the Poisson distribution as shown below. Consider a Poisson event that occurs at a rate of λ per unit of time. The average number of occurrences during an interval of length t is λt. Let X(t) represent the number of Poisson events occurring during an interval of length t, i.e. RX(t) = {0, 1, 2, 3, . . .}. Then from the Poisson pmf P[X(t) = 0] =

(λt)0 −λt e = e−λt . 0!

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Next, define a continuous rv, T, as follows: T = the time between the occurrences of 2 successive Poisson events (or T = intercurrence or intervening time of 2 successive Poisson events). Clearly, the following two events are equivalent. [X(t) = 0] = [T > t]. Thus, P(T > t) = P[X(t) = 0] = e−λt = 1 − P(T ≤ t) = 1 − F(t) −→ F(t) = 1 − e−λt −→ f (t) = dF(t)/dt = λe−λt

The above developments show that if the number of occurrences of an event is Poisson distributed, then the occurrence time, T, to the next Poisson event measured from the last occurrence is exponentially distributed.

Example 11 The number of downtimes of a computer network is Poisson distributed with an average of 0.20 downtime/week. Compute the pr of exactly 2 failures next week. p(2; 0.20) =

(0.20)2 × e−0.20 = 0.016 4. 2!

Next, the pr that the time between 2 future successive downtimes exceeds one week is given by P(T > 1) = P[X(1) = 0] = e−0.20 = 0.818 73. The pr that exactly 2 failures occur in the next 3 weeks is computed as follows. Y = number of failures/(3 weeks) −→ µ = λt = (0.20) × 3 = 0.60 P(Y = 2) =

(0.60)2 −0.60 e = 0.098 79. 2!

The pr that the time between 2 successive downtimes (or failures) exceeds 3 weeks is P(T > 3) = P[X(3) =Y = 0] = e−0.60 = 0.548 812. The pr that the time between next 2 failures will be shorter than 2 weeks is given by P(T < 2) = P[X(2) ≥ 1] = 1 − P[X(2) = 0] = 1 − e−0.40 = 0.329 7. This pr can also be computed by the direct integration of the exponential pdf as P(T < 2) = 2 −0.20t dt = 1 − e−0.20×2 = 0.329 7. 0 0.20 e Finally, the pr that the time (since the last system failure) to the next downtime will be less than 4 weeks is P(T < 4) = 1 − e−0.80 = 0.550 7. In this last example, the P(T < 4) also is referred to as the unreliability at time t = 4 weeks, and therefore, the reliability at time t = 4 weeks is defined as R(4) = P(T > 4) = 1 − P(T ≤ 4) = 0.449 3. Note that by reliability at 4 units of time we mean  ∞the pr that a system survives beyond t = 4, i.e. in general R(t) = Pr(survival > t) = P(T > t) = t f(x) dx, where f(t) represents the mortality (or failure) density function of the rv T. 2.2.7.3

Statistical inference

By statistical inference (SI) we mean estimation and/or test of hypothesis. There are two types of estimates: point and (confidence) interval estimate. A point estimate, or estimator,∗ is a random variable (or statistic) before the sample is drawn, but it becomes a numerical value after the sample is drawn estimating a population parameter (such as µ and σ). For example, any one of the statistics, the sample mean x, the median x˜ , the sample mode MO, etc., can be considered as point estimators of the population mean µ. If the parameter under consideration is the population variance σ 2 , then we may use any one of the statistics S2 =

n 1
(xi − x)2 , n−1 i=1

σˆ 2 =

n 1
(xi − x)2 , n i=1

∗ The term estimator, θˆ , is applied to the random variable, and the word estimate is reserved for the numerical value of θˆ taken on by the random variable after the sample is drawn and the experimental data have been inserted.

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2

or (R/d2 ) as estimators of σ , where R is the sample range, d2 = E (R/σ), and the rv w = R/σ is called the relative range in the field of QC. Note that any symbol with a hat atop it implies an estimator. The question is then how one chooses among several available estimators of a given process parameter? The obvious answer is to select the estimator that is the most accurate, i.e. whose value comes closest to the true value of the parameter being estimated in the long run. Since a point estimator, θˆ , is a random variable (i.e. it changes from sample to sample), it has a pr density function, and for θˆ to be an ‘accurate’ estimator, the pdf of θˆ should be closely concentrated about the population parameter θ. The criterion used to decide which one of the two estimators (θˆ 1 or θˆ 2 ) of the parameter θ is better now follows. The statistic θˆ 1 is said to be a more accurate estimator of θ than θˆ 2 iff E[(θˆ 1 − θ)2 ] < E [(θˆ 2 − θ)2 ]. Since E[(θˆ − θ)2 ] is defined to be the mean square error of θˆ , i.e. MSE(θˆ ) = E[(θˆ − θ)2 ], then θˆ 1 is a more accurate estimator than θˆ 2 iff MSE (θˆ 1 ) < MSE (θˆ 2 ). PROPERTIES OF POINT ESTIMATORS

An estimator is said to be consistent iff the limit (n → ∞) of θˆ = θ. If the population is finite of size N, then θˆ is consistent iff limit of θˆ = θ as n → N. For example, x is a consistent estimator of µ for both finite and infinite populations because limit of x = µ as n → N. However, S2 is not a consistent estimator of σ 2 for finite populations but S2 becomes consistent as N →∞. One of the most important property of a point estimator is the amount of bias in the estimator. The amount of bias in a point estimator is defined as B(θˆ ) = E(θˆ ) − θ = E(θˆ − θ) and, therefore, θˆ is an unbiased estimator iff E(θˆ ) = θ (i.e. B = 0). Users make the common mistake that an unbiased estimator is one whose value is equal to the parameter it is estimating. This is completely false! We now give the relationship between the accuracy of an estimator θˆ , measured by its MSE, and the amount of bias in θˆ . It can be shown that MSE(θˆ ) = V(θˆ ) + B2 = [E(θˆ 2 ) − E2 (θˆ )] + B2 This equation clearly shows that MSE (θˆ ) = V(θˆ ) iff θˆ is an unbiased estimator of θ.

Example 12 Consider an infinite population (not necessarily a normal population) with parameters µ and σ 2 , and a simple random sample of size of n > 1 is drawn. Then E(x) = µ, i.e. x is an unbiased estimator of the population mean µ. The identity n
i=1

(xi − x)2 ≡

n
i=1

[(xi − µ) − (x − µ)]2 ≡ · · · ≡




(xi − µ)2 − n(x − µ)2 ,

can be used to further prove that E (S2 ) = σ 2 only for infinite populations. That is, S2 is an unbiased estimator of process variance σ 2 for any infinite population. The relative efficiency (REL-EFF) of θˆ 1 to θˆ 2 is defined as MSE(θˆ 2 )/MSE(θˆ 1 ). As an example, consider a random sample of size n = 6 from a population with unknown mean µ. Let θˆ 1 = (x1 + x2 + x3 + x4 + x5 + x6 )/6 = x, and θˆ 2 = (2x1 + x4 − x6 )/2. Then, E (θˆ 1 ) = E (θˆ 2 ) = µ, i.e. both θˆ 1 and θˆ 2 are unbiased estimates of µ but the MSE (θˆ 1 ) = σ 2 /6, MSE(θˆ 2 ) = 1.5σ 2 so that the REL-EFF of θˆ 1 to θˆ 2 = (1.5σ 2 )/(σ 2 /6) = 900%. SAMPLING DISTRIBUTIONS OF STATISTICS WITH UNDERLYING NORMAL PARENT POPULATIONS

In carrying out statistical inference, we will assume that the underlying distribution (or the parent population) is Gaussian (or approximately so) in which case the sampling distribution of sample

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elements x1 , x2 , . . ., xn is normal and mutually independent of each other. A summary of the most commonly occurring normally-related statistics that are used to conduct SI on different population parameters will now follow. (i) Suppose X ∼ N(µ, σ 2 ) and a random  sample of  size n has been drawn with sample values n x1 , x2 , . . ., xn . Since the statistic x = i=1 xi /n is a linear combination of normally and 2 independently distributed (NID) rvs, then√x is also N(µ, σ√ /n). This implies that the sampling distribution (SMD) of Z = (x − µ)/(σ/ n) = ((x − µ) n)/σ is that of the standardised normal distribution N(0, 1) so that Z can be used for SI on µ if the value of σ is known. (ii) If σ is unknown and has to be estimated from the sample statistic S, then the SMD of √ ((x − µ) n)/S = (x − µ)/σˆ x is that of a Student’s t with (n − 1) degrees of freedom. (iii) To compare the means of two normal populations, we make use of the fact that the SMD of x1 − x2 is N(µ1 − µ2 , (σ12 /n1 ) + (σ22 /n2 )). In case the two population  variances are unknown but equal, i.e. σ12 = σ22 = σ 2 , then rv [(x1 − x2 ) − (µ1 − µ2 )]/[Sp (1/n1 ) + (1/n2 )] has a t sampling distribution with ν = n1 + n2 − 2 df, where S2p =

(n1 − 1)S21 + (n2 − 1)S22 CSS1 + CSS2 = . n1 + n2 − 2 n1 + n2 − 2

(iv) It can be shown that if Z ∼ N(0, 1), then the rv Z2 has a chi-square (χ2 ) distribution with 1 degree of freedom (df). Using this fact, then it can be shown that the SMD of S2 (n − 1)/σ 2 = ni=1 (xi − x)2 /σ 2 follows a χ2 with (n −1) df (not n degrees of freedom). (v) If S2x represents the sample variance of machine X from a random sample of size nx and S2y represents that of machine Y, then from statistical theory the sampling distribution of S2x /σx2 )/(S2y /σy2 follows (Sir R.A.) Fisher’s F with numerator degrees of freedom ν1 = nx − 1, and denominator degrees of freedom ν2 = ny − 1, where ny is the number of items sampled at random from machine Y. CONFIDENCE INTERVALS FOR ONE PARAMETER OF A NORMAL UNIVERSE

Let (1 − α) be the confidence interval (CI) coefficient (or confidence level) for the population mean µ. In the case of known σ for the normal underlying distribution, the two-sided confidence interval for µ is given by √ √ x − Zα/2 σ/ n ≤ µ ≤ x + Zα/2 σ/ n, (2.2a) where Zα/2 is the (α/2) × 100 percentage point of a standardised normal distribution. For example, Z0.025 = 1.96 is the same as the 97.5 percentile (or the 2&1/2 percentage point) of a N(0, 1) distribution. If the quality characteristic of interest, X, is of smaller-the-better (STB) type (such as loudness of a compressor, or radial force harmonic of a tire), then from an engineering standpoint it is best to obtain an upper one-sided confidence interval for µ given by √ (2.2b) 0 < µ ≤ x + Zα σ/ n. Conversely, if the quality characteristic, X, is of larger-the-better (LTB) type (such as tensile strength, efficiency, etc) then it is best to provide a lower one-sided confidence interval for µ, which is given by √ x − Zα σ/ n ≤ µ < ∞. (2.2c) If the population standard σ, of the normal universe is unknown and has to be estimated  deviation,  by sample statistic S = (1/(n − 1)) ni=1 (xi − x)2 , then in equations (2.2) for the CI on µ, replace σ by its estimator S and replace Zα with tα;n−1 where tα;n−1 is the α × 100 percentage point of a Student’s t distribution with (n − 1) degrees of freedom.

Example 13 A normal process manufactures wires with a strength lower specification limit LSL = 8.28 units and variance σ 2 = 0.029 93 units2 (‘units’ is used to indicate some measure of strength). A random  sample of size n = 25 items gave 25 i=1 Xi = 217.250 units. Our objective is to obtain the point and 95% interval estimators (1 − α = 0.95) for the process mean µ.

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The point estimator of the process mean strength, µ, is µ ˆ =x=

25 1
xi = 8.690 units. 25 i=1

Since µ is an LTB type parameter, then we develop a 95% lower CI for µ (note that there are no concerns about strength being too high from the consumers’ standpoint). The lower √ 95% confidence limit for µ, from (2.2c), is given by µ = X − Z × σ/ n = 8.690 − 1.645 × L 0.05 √ 0.173/ 25 = 8.690 − 0.0569 = 8.633 1 units. Thus, we are 95% confident that µ lies in the interval 8.633 1 ≤ µ < ∞. The number interval [8.633 1 units, ∞] no longer has a pr of 0.95 to √ contain the true value of µ because that pr is either 0 or 1. However, the random interval x − 1.645σ/ n ≤ µ < ∞ has a pr of 95% to contain µ before the sample is drawn because x is a rv before the random drawing of the n sample values. This implies that if we wish to test the null hypothesis H0 : µ = 8.621 versus the one-sided alternative H1 : µ > 8.621 at the level of significance of α = 0.05, then our 95% CI: 8.6331 ≤ µ < ∞ dictates that we must reject H0 in favour of H1 at the 5% level of significance because the hypothesised value of process mean µ0 ≡ 8.621 is outside the 95% CI, or µ0 ≡ 8.621 < µL = 8.633 1 units. On the other hand, our 95% CI does not allow us to conclude that µ > 8.650 units at the 5% level because 8.650 is inside the 95% CI 8.633 1 ≤ µ < ∞. With the above example we have demonstrated that all confidence intervals in the universe are tests of hypotheses in disguise because a lower (1 − α)100% one-sided CI for µ provides all possible right-tailed tests at the level of significance α for the parameter µ. In the above example if the process variance σ 2 were unknown and had to  be estimated from the sample, then the sample values must be used to compute the USS = ni=1 xi2 , say that for the above example n 2 2 the uncorrected sum of squares were computed to be i=1 xi = 1 888.790 units . Then, the 2 2 CSS = 1 888.790 − 217.25 /25 = 0.887 50, S = 0.887 5/24 = 0.037 0, and S = 0.192 3 units would the point estimate of the unknown parameter σ. The corresponding√95% lower confidence limit for √ µ is given by µL = x − t0.05;24 × S/ n = 8.690 − 1.711 × 0.192 3/ 25 = 8.624 units, which results in the confidence interval 8.624 ≤ µ < ∞. CONFIDENCE INTERVALS FOR PARAMETERS OF TWO NORMAL UNIVERSES

The Student’s t distribution must be used (except when the two process variances are known) in order to compare two population means while the Fisher F distribution is used to compare two process variances. We provide an example of how to compare two normal population means.

Example 14 For the sake of illustration consider the Experiment reported in the Journal of Waste and Hazardous Materials (Vol. 6, 1989), where X1 = weight percent calcium in standard cement, while X2 = weight percent calcium in cement doped with lead. Reduced levels of calcium would cause the hydration mechanism to become blocked and allow water to attack various locations of the cement structure. Ten samples of standard cement gave x1 = 90.0% with S1 = 5.0 while n2 = 15 samples of lead-doped cement resulted in x2 = 87.0 with S2 =4.0. Assuming that X1 ∼ N(µ1 , σ 2 ) and X2 ∼ N(µ2 , σ 2 ), then the rv [(x1 − x2 ) − (µ1 − µ2 )]/[Sp (1/n1 ) + (1/n2 )] has a t sampling distribution with ν = n1 + n2 − 2df. The expression for the two-sided 95% CI forµ1 − µ2 is given by x1 − x2 − t0.025;23 × Sp (1/n1 ) + (1/n2 ) ≤ µ1 − µ2 ≤ x1 − x2 + t0.025;23 × Sp (1/n1 ) + (1/n2 ), where t0.025;23 = 2.069 and S2p = [9(25) + 14(16)]/(23) = 19.521 74. Substitution of sample results into the above expression yields (µ1 − µ2 )L = 3 − 2.069 × 1.8038 = −0.732 and (µ1 − µ2 )U = 3 + 3.732 = 6.732. Since this 95% CI, −0.732 ≤ µ1 − µ2 ≤ 6.732, includes zero, then there does not exist a significant difference between the two population parameters µ1 and µ2 at the 5% level, i.e. the null hypothesis H0 : µ1 − µ2 = 0 cannot be rejected at the 5% level of significance. This implies that doping cement with lead does not significantly (at the 5% level) alter water hydration mechanism. Note that the null hypothesis H0 : µ1 − µ2 = 7 must be rejected at the 5% level of significance because 7 is outside the 95% CI. When the variances of the two independent normal populations, σ12 and σ22 , are unequal and unknown, then the CI for µ1 − µ2 must be obtained from the fact that the SMD of the rv (x1 − x2 ) − (µ1 − µ2 )  S21 S22 + n1 n2

2–46

Introductory tables and mathematical information

follows a Student’s t distribution with df ν given by the following equation.

ν=

(S21 /n1 + S22 /n2 )2 (S2 /n2 )2 (S21 /n1 )2 + 2 n1 − 1 n2 − 1

(2.3)

where Max(n1 − 1, n2 − 1) < ν < n1 + n2 − 2. As a general rule of thumb, we would recommend against using the pooled t procedure outlined in Example 14 if S21 > 2S22 , or vice a versa. From the above discussions we conclude that in case S21 > 2S22 , then the t statistic for testing the null hypothesis H0 : µ1 − µ2 = δ is given by (x1 − x2 ) − δ t0 =  . S21 S22 + n1 n2 If the test is 2-sided, then the acceptance interval for testing H0 : µ1 − µ2 = δ at a level of significance 0.05 is given by (−t0.025;ν , t0.025;ν ), where ν is given in equation (2.3).

2.2.7.4 Statistical process control (SPC) The objective of SPC is to test the null hypothesis that the value of a process parameter is either at a desired specified value, or at a value that has been established from a long-term data. This objective is generally carried out through constructing a Shewhart control chart from m > 1 subgroups of data. Further, it is assumed that the underlying distribution is approximately Gaussian, and for moderately large sample sizes, it is also assumed that the SMD of the statistic used to construct the Shewhart chart is also Gaussian. When a sample point goes out of control limits, the process must be stopped in order to look for assignable causes, and if one is found, then corrective action must be taken and the corresponding point should be removed from the chart. In case no assignable (or special) causes are found, then the control chart has led to a false alarm (or a type I error). Since false alarms are very expensive and disruptive to a manufacturing process, all Shewhart charts are designed in such a manner that the pr of committing a type I error, α, is very small. The standard level of significance, α, of all charts are set at α = 0.002 7. When departures from the underlying assumptions are not grossly violated, then a Shewhart control chart will generally lead an experimenter to 27 false alarms in 10 000 random samples. Moreover, setting the value of α at 0.002 7, will correspond to three-sigma control limits for a control chart as long as the normality assumption is tenable. We will discuss only two types of charts: (1) Charts for continuous variables, and (2) Charts for attributes, where the measurement system merely classifies a unit either as conforming to customer specifications or nonconforming to specifications (i.e. success/failure, 0/1, defective/effective, etc.). SHEWHART CONTROL CHARTS FOR VARIABLES

As an example, suppose we wish to control the dimension of piston ring inside diameters, X, with design specifications X: 74.00 ± 0.05 mm. The rings are manufactured through a forging process. Since the random variable X is continuous, then we need two charts; one to control variability (or σ), and a second chart for controlling the process mean µ. If subgroup sample sizes, ni , are identical and lie within 2 ≤ n ≤ 15, then an R-chart (i.e. range-chart) should be used to monitor variability, but for n > 15, an S-chart should be used for control of variation. This is due to the fact that the SMD of sample range, R, becomes unstable for moderate to large sample sizes. Samples of sizes ni (i = 1, 2, . . . , m) are taken from the process, generally at equal intervals of time, (where hourly or daily samples, or samples taken at different shifts, are the most common), and the number of subgroups m should generally lie within 20 ≤ m ≤ 50. Samples should be taken in such a manner as to minimise the variability within samples and maximise the variability among samples, a concept that is consistent with design of experiments (DOE). Such samples are generally referred to as rational subgroups.

Mathematical formulae and statistical principles

2–47

R- AND X-CHARTS (FOR 2 ≤ n ≤ 15)

In practice the R-chart should be constructed first in order to bring variability in a state of statistical control, followed by developing the x-chart for the purpose of monitoring the process mean. In order to use the R-chart for monitoring process variation, the subgroup sample sizes ni (i = 1, 2, . . ., m) must be the same, i.e. ni = n for all i, or else an R-chart cannot be constructed. All univariate control charts consist of a central line, CNTL, a lower control limit LCL, and an upper control limit UCL. Further, nearly in all cases, LCL = CNTL − 3 × se (sample statistic) and UCL = CNTL + 3 × se (sample statistic), where in the case of the R-chart the sample statistic will be the sample range R, while for the x-chart the sample statistic will be the sample mean x. The pertinent formulas are provided below. CNTLR = R =

m 1
Ri m i=1

Note that we are taking the liberty to use the terminology standard error, se, as the estimate of the Stdev of the sample statistic. The se(R) = σˆ R = d3 R/d2 , the values of d2 are given inTable 2.12 for n = 2, 3, . . . , 15, d32 = V(R/σ), and the values of d3 for a normal universe are given in Table 2.14. Since the most common of all sample sizes for constructing an R- and x-Chart is n = 5, we compute σˆ R only for n = 5. From Tables 2.12 & 2.14, se(R) = 0.864 × R/2.326 = 0.371 45 × R. Then for n = 5, the LCLR = R − 3 × 0.371 45 × R = R(1 − 1.114 4) → LCLR = 0, and UCLR = 2.114 36 × R. In fact, it can be shown that the LCLR = 0 for all sample sizes in the range 2 ≤ n ≤ 6, but LCLR > 0 for all n > 6.  √ the se(x) = σ/ n. The central line for an x-Chart is given by CNTLx = 1/m m i=1 xi = X and √ Since a point estimate of σ is given by σˆ x = R/d2 , then σˆ x =R/d2 n, as a result the √ √ LCLx = X − 3 × se(x) = X − 3R/d2 n, and UCLx = X + 3R/d2 n. For samples of size n = 5, these last two control limits reduce to LCLx = X − 0.5768R,

and

UCLx = X + 0.5768R.

S- AND x-CHARTS

If subgroup sample sizes differ and/or n > 12, then process variation must be monitored by an S-chart. The most common occurrence of an S-chart is when the sampling design is not balanced, i.e. ni ’s (i = 1, 2, . . ., m) are not the same, then the experimenter has no option but to use an S-chart for the control of variation. The central line is given by CNTLS = Sp =



m 2 i=1 (ni − 1)Si i=m i=1 (ni − 1)

1/2

=

 m
i=1

1/2

(ni − 1)S2i /(N − m)

j=n where N = i=1 ni , and the quantity (ni − 1)S2i = j=1i (xij − xi )2 is called the corrected sum of squares within the ith subgroup. It can be shown (using the properties of χ2 ) that for a  normal universe the E(S) = c4 σ, where the constant c4 = 2/(n − 1) × Ŵ(n/2)/Ŵ[(n − 1)/2] lies in the interval (0.797 884 5, 1) for all n ≥ 2 and its limit as n → ∞ is equal to 1. Further, the authors have shown that for n ≥ 20 the value of c4 can be approximated to 5 decimals by c4 ∼ = 4n2 − 8n + 3.875/(4n − 3)(n − 1). These discussions imply that, in the long-run, the statistic S underestimates the population standard deviation σ, and hence an unbiased estimator of σ for a normal universe is given by σˆ x = S/c4 (this is due to the fact that E(S) = c4 σ). i=m

Table 2.14 n

2

THE VALUES OF d3 FOR A NORMAL UNIVERSE

3

4

5

6

7

8

9

10

11

12

13

14

15

d3 0.853 0.888 0.880 0.864 0.848 0.833 0.820 0.808 0.797 0.787 0.778 0.770 0.763 0.756

2–48

Introductory tables and mathematical information

To compute the se(S), we make use of the fact that V(S) = E(S2 ) − [E(S)]2 = σ 2− (c4 σ)2 = (1−c24 )σ 2 . This development implies that se(S) = (1 − c24 )1/2 σˆ x = (1 − c24 )1/2 (S/c4 ) = S (c4 )−2 − 1. Thus the control limits are given by   LCLi (S) = Sp − 3 Sp (c4 )−2 − 1 = [1 − 3 (c4 )−2 − 1]Sp   UCLi (S) = Sp + 3 Sp (c4 )−2 − 1 = [1 + 3 (c4 )−2 − 1]Sp .

Note that the LCLS = 0 when 2 ≤ ni ≤ 5, but LCLS > 0 when n > 5. If the sampling design is balanced, i.e. ni = n > 12 for all i, then replace Sp in the previous two equations with  S = (1/m) m i=1 Si . Once variability is in a state of statistical control (i.e. all sample Si ’s lie within their own control limits), then an x-chart is developed to monitor the process mean. The central line for an x-chart is given by CNTLx = x = where N =

i=m i=1

m


ni xi /N

i=1

√ √ ni . Since the V(xi ) = σ 2 /ni , then se(xi ) = σˆ x / ni = (Sp /c4 ni ), and as a result

Sp LCLi (x) = x − 3 √ , c4 ni

and

Sp UCLi (x) = x + 3 √ . c4 ni

(2.4)

Note that if the sampling design is unbalanced, then all points on the S- and x-charts have the same CTLNs but every point on both charts has its own control limits due to differing sample sizes. When the sampling scheme is balanced (i.e. ni = n for all i), then in equation (2.4) replace Sp with  S = (1/m) m i=1 Si . SHEWHART CONTROL CHART FOR FRACTION NONCONFORMING (THE P-CHART)

As an example, consider an injection moulding process that produces instrument panels for an automobile. The occurrence of splay, voids, or short shots will make the panel defective. Thus we have a binomial process where each panel is classified as defective (or 1) or as conforming (or 0). The binomial rv, X, represents the number of nonconforming panels in a random sample of size ni , where it is best to have at least m > 20 subgroups in order to construct the p-chart, where p is the FNC of the process. The sample FNC is given by pˆ = X/n, and if √ n > 30 and np and nq > 10, then the SMD of pˆ is approximately normal with mean p and se(ˆp) = pq/n, where q = (1 − p) is the process fraction conforming (or process yield). The central line is given by m m Xi ni pˆ i =p = i=1 CNTLp = i=1 m N n i i=1

 where N = i=m i=1 ni is the total number of units inspected by attributes in all m samples, Xi represents the number of NC units in the ith  subgroup, and pˆ i = Xi /ni is the sample FNC of the ith subgroup. Since the estimate of the se(ˆpi ) = p(1 − p)/ni , then the control limits for the ith subgroup is given   by LCLi (p) = p − 3 p(1 − p)/ni , and UCLi (p) = p + 3 p(1 − p)/ni . Note that, when subgroup sizes differ on a Shewhart p-chart, then every sample FNC, pˆ i , has its own control limit. If the difference between maximum and minimum sample sizes do not exceed 10 units, then a p-chart based on average sample size should be constructed for monitoring process FNC. In all cases the central line stays the same, but the average control limits simplify to   p(1 − p) p(1 − p) , and UCL(p) = p + 3 , LCL(p) = p − 3 n n  where n = (1/m) m i=1 ni . The reader is cautioned to the fact that if a p-chart based on an average sample size is used to monitor process FNC, then all points (i.e. all sample FNCs) that are close to their average control limits (whether in or out of control) must be checked against their own limits to determine their control nature.

Mathematical formulae and statistical principles Table 2.15

2–49

DATA FOR TEXTILE PROCESS EXAMPLE

Sample number (i) Square metres ci ni ui LCLi UCLi

1 180 9 1.8 5.00 1.123 13.012

2 150 15 1.5 10.00 0.555 13.579

3 120 6 1.2 5.000 0.00 14.348

4 90 5 0.90 5.556 0.00 15.474

5 150 16 1.5 10.667 0.555 13.579

6 160 10 1.6 6.25 0.762 13.372

7 120 4 1.2 3.333 0.00 14.348

8 140 12 1.4 8.571 0.327 13.807

9 130 14 1.3 10.769 0.072 4 14.062

10 175 9 1.75 5.143 1.038 13.096

SHEWHART CONTROL CHART FOR NUMBER OF NONCONFORMITIES PER UNIT (THE u-CHART)

Since the construction of the u-chart is not as straight forward as the others discussed thus far, we will describe the methodology through an example. In practice, it is best to have at least m = 20 subgroups, but herein for simplicity we will use m = 10 samples of differing sizes. Consider a textile process that produces oilcloth in lots of differing sizes, measured in square metres. An inspector selects m = 10 lots at random and counts the number of defects, ci , in each lot (or sample). The data is displayed in Table 2.15. Note that in Table 2.15, because of different square metres, we have arbitrarily let 100 square metres equal to one unit, although 50, or, 10, or any other convenient square metres would work just as well. Further, ui = ci /ni represents the number of defects per unit. Note that because of differing sample  sizes, it would be erroneous to compute the average number of defects per unit from (1/m) m i=1 ui = 7.028 9, where this last formula would work only if all ni ’s were identical. The correct formula for the central line is given by m m 100 ni ui i=1 ci = 7.067 1 = i=1 = CNTLu = u = m i n 14.15 i=1 i i=1 ni

It is well known that the number of defects per unit, c, follows a Poisson process and hence its variance is also given by E(c), i.e. V(c) can be approximated by u. Unfortunately, the terminology and notation for a u-chart has been somewhat confusing in statistical and QC literature and we anticipate no change. Therefore, herein we attempt to remedy the notational problem to some extent. First of all, the fifth row of Table 2.15 actually provides the average number of defects per unit for the ith sample, and hence the proper notation for the fifth row should be ui (not ui as is used in QC literature) because a bar is generally placed on averages in the field of statistics. This implies that a u-chart is actually a u-chart because it is the average number of defects per unit that is plotted on this chart. Secondly, the central line should be called u because the CNTL gives the grand average of all average number of defects per unit. These discussions lead to the fact that firstly V(u) = V(c)/n, and secondly V(u) can be estimated by u/n. Since we do not wish to deviate from QC literature terminology, we will stay number of defects per√ unit with CNTL as u with the existing√notation and let ui represent the average √ and the se(u) = u/ni . Thus, the LCLi (u) = u − 3 u/ni , and UCLi (u) = u + 3 u/ni . The values of control limits for all the m = 10 samples are provided in the last two rows of Table 2.15. Table 2.15 clearly shows that each ui is well within its own control limits, implying that the process is in a state of excellent statistical control. Note that in all cases when the value of LCL became negative, a zero LCL was assigned in row 6 of Table 2.15. This example provides a good illustration of a process that is in an excellent state of statistical control, but one that is in all pr not capable of meeting customer specifications due to the fact that u = 7.067 1 is too large and customers in today’s global market will generally demand lower average number defects per unit. If this manufacturer does not improve its process capability through quality improvement (QI) methods, it may not survive very long in global competition. Further, SPC is not a QI method but simply an on-line procedure to monitor process quality and to identify where the quality problems lie. After problems are identified, then off-line methods (DOE or Taguchi methods) can be applied to fine-tune a process.

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3

General physical and chemical constants

Table 3.1

ATOMIC WEIGHTS AND ATOMIC NUMBERS OF THE ELEMENTS

Name

Symbol

Atomic number %

Actinium Aluminium Americium Antimony Argon Arsenic Astatine Barium Berkelium Beryllium Bismuth Boron Bromine Cadmium Caesium Calcium Californium Carbon Cerium Chlorine Chromium Cobalt Copper Curium Dysprosium Einstenium Erbium Europium Fermium Fluorine Francium Gadolinium Gallium Germanium Gold Hafnium Helium Holmium Hydrogen Indium Iodine

Ac Al Am Sb Ar As At Ba Bk Be Bi B Br Cd Cs Ca Cf C Ce Cl Cr Co Cu Cm Dy Es Er Eu Fm F Fr Gd Ga Ge Au Hf He Ho H In I

89 13 95 51 18 33 85 56 97 4 83 5 35 48 55 20 98 6 58 17 24 27 29 96 66 99 68 63 100 9 87 64 31 32 79 72 2 67 1 49 53

International atomic weights* 1971 12 C = 12 (227) 26.981 54 (243) 121.75 39.948 74.921 6 (210) 137.34 (247) 9.012 18 208.980 4 10.81 79.904 112.40 132.905 4 40.08 (251) 12.011 140.12 35.453 51.996 58.933 2 63.546 (247) 162.50 (254) 167.26 151.96 (257) 18.998 40 (223) 157.25 69.72 72.59 196.966 5 178.49 4.002 60 164.930 4 1.007 9 114.82 126.904 5

International atomic weights* 1971 12 C = 12

Name

Symbol

Atomic number %

Iridium Iron Krypton Lanthanum Lawrencium Lead Lithium Lutetium Magnesium Manganese Mendelevium Mercury Molybdenum Neodymium Neon Neptunium Nickel Niobium (Columbium) Nitrogen Nobelium Osmium Oxygen Palladium Phosphorus Platinum Plutonium Polonium Potassium Praseodymium Promethium Protactinium Radium Radon Rhenium Rhodium Rubidium Ruthenium Samarium Scandium Selenium

Ir Fe Kr La Lr Pb Li Lu Mg Mn Md Hg Mo Nd Ne Np Ni Nb

77 26 36 57 103 82 3 71 12 25 101 80 42 60 10 93 28 41

192.22 55.847 83.80 138.9055 (260) 207.2 6.941 174.97 24.305 54.938 0 (258) 200.59 95.94 144.24 20.179 237.048 2 58.69 92.906 4

N No Os O Pd P Pt Pu Po K Pr Pm Pa Ra Rn Re Rh Rb Ru Sm Sc Se

7 102 76 8 46 15 78 94 84 19 59 61 91 88 86 75 45 37 44 62 21 34

14.006 7 (259) 190.2 15.9994 106.42 30.973 8 195.09 (244) (209) 39.098 140.908 (145) 231.035 9 226.025 (222) 186.2 102.905 5 85.4678 101.07 150.36 44.955 9 78.96 (continued)

3–1

3–2

General physical and chemical constants

Table 3.1

ATOMIC WEIGHTS AND ATOMIC NUMBERS OF THE ELEMENTS—continued

Name

Symbol

Atomic number %

Silicon Silver Sodium Strontium Sulphur Tantalum Technetium Tellurium Terbium Thallium Thorium

Si Ag Na Sr S Ta Tc Te Tb Tl Th

14 47 11 38 16 73 43 52 65 81 90

International atomic weights* 1971 12 C = 12 28.085 5 107.868 22.989 8 87.62 32.06 180.9479 (98) 127.60 158.925 204.383 232.038

Name

Symbol

Atomic number %

Thulium Tin Titanium Tungsten Uranium Vanadium Xenon Ytterbium Yttrium Zinc Zirconium

Tm Sn Ti W U V Xe Yb Y Zn Zr

69 50 22 74 92 23 54 70 39 30 40

International atomic weights* 1971 12 C = 12 168.934 118.71 47.88 183.85 238.029 50.9414 131.30 173.04 88.905 9 65.39 91.224

*Atomic Weights of the Elements 1981, Pure and Applied Chemistry 1983, 55 (7), 1101–1136. A value given in brackets denotes the mass number of the isotope of longest known half-life. Because of natural variation in the relative abundance of the isotopes of some elements, their atomic weights may vary. Apart from this they are considered reliable to ±1 in the last digit, or ±3 if that digit is subscript.

Table 3.2

GENERAL PHYSICAL CONSTANTS

The probable errors of the various quantities may be obtained from the reference given at the end of the table. Quantity

Symbol

Value

Units

Acceleration due to gravity (standard) Atmospheric pressure (standard) Atomic mass unit Atomic weight of electron Avogadro number Boltzmann’s constant Bohr radius Bohr magneton Charge in electrolysis of 1 g hydrogen Compton wavelength of electron Classical electron radius Compton wavelength of proton Compton wavelength of neutron Density of the earth (average) (core) Density of mercury (0◦ C, A0 ) Density of water (max) Electronic charge Electron rest mass Electron volt energy Electron magnetic moment Faraday constant Fine structure constant Gas constant Gravitational constant Ice point (absolute value) Litre (12th CGPM 1964) (1963 weights and measures) Neutron rest mass Planck’s constant Proton rest mass

gn A0 mu N A mc NA k a0 µB F/H λC re λc,p λc,n δ

9.806 65 1.013 25 × 105 1.660 565 5 × 10−27 5.486 802 6 × 10−4 6.022 045 × 1023 1.380 662 × 10−23 5.291 770 6 × 10−11 9.274 078 × 10−24 9.572 378 × 104 2.426 308 9 × 10−12 2.817 938 0 × 10−15 1.321 409 9 × 10−15 1.319 590 9 × 10−15 5.518 × 103 1.072 × 104 1.359 508 × 104 9.999 72 × 102 1.602 189 2 × 10−19 9.109 534 × 10−31 1.602 192 × 10−19 9.284 832 × 10−24 9.648 456 × 104 7.297 350 6 × 10−3 8.314 41 6.672 0 × 10−11 2.731 5 × 102 1.0 exactly × 10−3 1.000 028 × 10−3 1.674 954 3 × 10−27 6.626 176 × 10−34 1.672 648 5 × 10−27

m s−2 Pa kg u* mol−1 J K−1 m J T−1 C m m m m kg m−3 kg m−3 kg m−3 kg m−3 C kg J J T−1 C mol−1 — J K−1 mol−1 Nm2 kg−2 K m3 m3 kg Js kg

D0 δm (H2 O) e me Eρ µe F α R0 G T0 1 mn h mp

(continued)

General physical and chemical constants Table 3.2

3–3

GENERAL PHYSICAL CONSTANTS—continued

Quantity

Symbol

Value

Units

Radiation constant—first Radiation constant—second Rydberg’s constant Stefan–Boltzmann constant Velocity of light Volume of ideal gas (0◦ C, A0 ) Wien’s constant Zeeman displacement

8πhc c2 = hc/k R∞ σ c V0 λmT

4.992 563 × 10−24 1.438 786 × 10−2 1.097 373 177 × 107 5.670 32 × 10−8 2.997 924 580 × 108 2.241 36 × 10−2 2.897 8 × 10−3 4.668 58 × 10

Jm mK m−1 Wm−2 K−4 m s−1 m3 mol−1 mK m Wb−1

∗u

= unified atomic mass unit (amu) based on u =

1 12

of the mass of 12 C.

REFERENCES ‘Quantities, Units and Symbols’, The Royal Society, 1981.

Table 3.3 MOMENTS OF INERTIA Moment of inertia = Mk 2 where M is mass and k radius of gyration. Body

Dimensions

Axis*

k2

Uniform thin rod

length l

Rectangular lamina

sides a and b

l 2 /12 l 2 /3 (a2 + b2 )/12

Circular lamina

radius r

Annular lamina

radii r1 and r2

Through centre, perpendicular to rod Through end, perpendicular to rod Through centre, perpendicular to plane of lamina Through centre, parallel to side b Through centre, perpendicular to plane of lamina Through any diameter Through centre perpendicular to plane of lamina Through any diameter Through centre, perpendicular to face ab Through any diameter Through any diameter Through any diameter Longitudinal axis through centre Through centre perpendicular to longitudinal axis Longitudinal axis through centre Through centre perpendicular to longitudinal axis Longitudinal axis through centre Through centre perpendicular to longitudinal axis Longitudinal axis through apex Through centre of gravity perpendicular to longitudinal axis Through centre along axis a

Rectangular solid

sides a, b and c

Sphere Spherical shell Thin spherical shell Right circular cylinder

radius r radii r1 and r2 radius r radius r length l

Hollow circular cylinder

radii r1 and r2 length l

Thin hollow circular cylinder

radius r length l

Right circular cone

height h base radius r

Ellipsoid

semi-axis a, n and c

a2 /12 r 2 /2 r 2 /4 (r12 + r22 )/2 (r12 + r22 )/4 (a2 + b2 )/12 2r 2 /5 2(r15 − r25 )/5(r13 − r23 ) 2r 2 /3 r 2 /2 r 2 /4 + l 2 /12 (r12 + r22 )/2 (r12 + r22 )/4 + l 2 /12 r2 r 2 /2 + l 2 /12 3r 2 /10 3(r 2 + h2 /4)/20 (b2 + c2 )/5

*If the moment of inertia Iq about an axis through the centre of gravity is known, then the moment, I , about any other parallel axis may be obtained from I = Iq + Mh2

where h is the distance from the centre of gravity to the parallel axis.

General physical and chemical constants

3–4

2

He 2 4·0026

1 H 1 1·007 94

3

Li 2-1 6·941

11

Na

Be 2-2 9·012 18

5

12

13

B 2-3 10·81

4

Mg

Al

6

C 2-4 12·011

14

Si

N 2-5 14·0067

8

O 2-6 15·999

9

15

16

17

7

P

S

F 2-7 18·998

10

Ne 2-8 20·183

Cl

18

Ar

2-8-1

2-8-2

2-8-3

2-8-4

2-8-5

2-8-6

2-8-7

2-8-8

22·9898

24·312

26·9815

28·0855

30·9738

32·064

35·453

39·948

19 K -8-8-1 39·102

20 Ca -8-8-2 40·08

21 Sc -8-9-2 44·956

31 Ga -8-18-3 69·72

32 Ge -8-18-4 72·59

33 As -8-18-5 74·9216

34 Se -8-18-6 78·96

35 Br -8-18-7 79·909

36 Kr -8-18-8 83·8

37

38

39

49

50

52

53

Rb

Sr

-18-8-2

-18-9-2

85·47

87·62

88·905

Cs

56

Ba

57

40

Y

-18-8-1

55

22 Ti 23 V 24 Cr 25 Mn 26 Fe 27 Co 28 Ni -8-10-2 -8-11-2 -8-13-1 -8-13-2 -8-14-2 -8-15-2 -8-16-2 47·88 50·942 51·996 54·938 55·847 58·9332 58·69

La

Zr 41

Nb 42

Mo 43

Tc 44

Ru 45

Rh 46

Pd

-18-10-2 -18-12-1 -18-13-1 -18-13-2 -18-15-1 -18-16-1 -18-18-0 91·224

71

Lu 72

92·906

Hf 73

95·94

Ta 74

(98)

W 75

101·07

Re 76

102·905 106·42

Os 77

Ir 78

Pt

-18-8-1

-18-8-2

-18-9-2

-32-9-2 -32-10-2 -32-11-2 -32-12-2 -32-13-2 -32-14-2 -32-15-2 -32-16-2

132·905

137·34

138·91

174·47

178·49

Rare earths

87

Fr

-18-8-1 (223)

Figure 3.1

88

Ra

-18-8-2 (226)

89 Ac -18-9-2 (227)

The Periodic Table

180·948

183·85

186·2

190·2

192·2

195·09

29 Cu -8-18-1 63·54

47

Ag

30 Zn -8-18-2 65·39

48

Cd

In

Sn

51

Sb

Te

I

54

Xe

-18-18-1

-18-18-2

-18-18-3

-18-18-4

-18-18-5

-18-18-6

-18-18-7

-18-18-8

107·870

112·40

114·82

118·71

121·75

127·6

126·904

131·30

79

Au

80

Hg

81

Tl

82

Pb

83

Bi

84

Po

85

At

86

Rn

-32-18-1

-32-18-2

-32-18-3

-32-18-4

-32-18-5

-32-18-6

-32-18-7

-32-18-8

196·967

200·59

204·83

207·19

208·98

(209)

(210)

(222)

58 Ce 59 Pr 60 Nd 61 Pm 62 Sm 63 Eu 19-9-2 20-9-2 -22-8-2 -23-8-2 -24-8-2 -25-8-2 150·35 151·96 140·12 140·908 144·24 (145) 64 Gd 65 Tb 66 Dy 67 Ho 68 Er 69 Tm 70 Yb -25-9-2 -26-9-2 -28-8-2 -29-8-2 -30-8-2 -31-8-2 -32-8-2 157·25 158·925 162·50 164·93 167·26 168·934 173·04

Cf 99 Es 100 Fm 101 Md 102 No 103 Lr 104 Rf U 93 Np 94 Pu 95 Am 96 Cm 97 Bk 98 90 Th 91 Pa 92 -19-9-2 -20-9-2 -21-9-2 -22-9-2 -23-9-2 -24-9-2 -25-9-2 -26-9-2 -28-9-2 -29-8-2 -30-8-2 -31-8-2 -32-8-2 (237) (244) (243) (247) (247) (251) (257) (258) (259) (262) (261) (254) 232·038 (231) 238·03

KEY. Atomic number Outer electron shells

x

x –x–x (x)

Chemical symbol

Atomic weight (x) Mass Number of longest known isotope

Radioactive isotopes and radiation sources

3–5

3.1 Radioactive isotopes and radiation sources Tables 3.4, 3.5 and 3.6 are so arranged as to assist in selecting an isotope with a given half-life and decay radiation energy, for positron, beta and gamma emitters. Tables 3.7 to 3.12 list the most commonly used commercially available alpha, beta and neutron sources. Table 3.4

POSITRON EMITTERS (USEFUL NUCLIDES)

Isotope Oxygen-15 Nitrogen-13 Bromine-80 Carbon-11 Manganese-52 m Gallium-68 Fluorine-18 Titanium-45 Scandium-44 Iron-52 (Daughter: Mn-52 m) Zinc-62 Gallium-66

Half-life 2.0 min 10.0 min 17.4 min 20.4 min 21.1 min 68.0 min 109.8 min 3.1 h 3.9 h 8.3 h 9.3 h 9.4 h

Copper-64 Niobium-90 Cobalt-55

12.7 h 14.6 h 17.5 h

Arsenic-72

26.0 h

Nickel-57

36.1 h

Gold-194

39.5 h

Bromine-77 Yttrium-87 Iodine-124

56 h 80.3 h 4.2 d

Manganese-52 Caesium-132 Iodine-126

5.6 d 6.5 d 13.0 d

Vanadium-48

16.2 d

Arsenic-74 Rubidium-84

17.8 d 32.8 d

Cobalt-58 Cobalt-56

70.8 d 78.8 d

Yttrium-88 Zinc-65 Sodium-22

106.6 d 243.9 d 2.60 yr

Aluminium-26

7.2 × 105 yr

β+ energies MeV % 1.73–99.9 1.20–99.8 0.85–2.6 0.96–99.8 2.63–96.4 0.82–1.1 1.90–87.9 0.63–100 1.04–84.8 1.48–94.4 0.80–56.0 0.61–7.6 0.36–1.0 0.77–0.7 0.92–4.1 1.78–0.4 4.15–49.6 0.65–17.9 1.50–53 0.44–0.27 0.48–0.03 0.65–0.46 0.81–0.5 1.87–6.0 2.50–64.0 2.64–0.1 3.33–16.5 0.30–0.41 0.46–0.87 0.72–5.7 0.84–33.1 1.16–0.64 1.49–1.0 0.34–0.73 0.45–0.16 0.81–0.29 1.53–11.2 2.14–11.2 0.58–29.6 0.6–0.3 0.47–0.2 1.13–0.8 0.56–0.1 0.70–50.0 1.99–0.09 0.94–25.7 1.54–3.4 0.78–14.4 1.66–11.9 0.48–15.0 0.42–1.1 1.46–18.8 0.76–0.22 0.33–1.5 0.55–89.8 1.89–0.06 1.17–81.8

3–6

General physical and chemical constants

Table 3.5

BETA ENERGIES AND HALF-LIVES

Half-life β− -energy MeV

3.0

69 Zn,

49 Ca,

80 Br,

128 I,

2.0–3.0

233Th

38 Cl,

155 In,

1.5–2.0

80 Br,

101Tc,

38 Cl,

81 Se,

123m Sn,

49 Ca,

70 Ga,

101 Mo,

155 Sm,

88 Rb,

116m In,

61 Co,

105 Ru, 149 Nd,

75 Ge,

117 Cd,

152m Eu,

171 Er,

65 Ni,

117 Cd, 171 Er,

97 Nb,

127Te, 171 Er

117 Cd,

52V,

31 Si,

41Ar,

97 Nb,

105 Ru,

117 Cd,

171 Er,

176m Lu,

177Yb

105 Ru,

117 Cd,

132 I,

199 Pt,

101 Mo,

116m In, 128 I,

233Th

51Ti,

80 Br,

108Ag,

85m Kr,

152m Eu,

61 Co,

132 I,

149 Nd,

56 Mn,

117 Cd,

117 In,

97 Nb,

132 I,

165 Dy,

65 Ni,

149 Nd,

152m Eu

75 Ge,

105 Ru, 139 Ba, 171 Er

56 Mn, 75 Ge,

165 Dy,

101 Mo,

52V,

41Ar,

117 Cd,

56 Mn,

132 I,

28 Mg,

77 Ge,

28 Mg,

43 K,

188 Re

77 Ge,

187 W, 197 Pt

64 Cu, 97 Zr,

159 Gd, 194 Ir,

28 Mg, 77 Ge,

65 Ni,

139 Ba,

97 Zr,

188 Re,

72 Ga,

130 I,

187 W, 197 Pt

43 K,

97 Zr,

159 Gd,

187 W,

24 Na,

43 K,

197 Pt

77 Ge,

130 I,

194 Ir

42 K,

77 Ge,

188 Re,

199 Pt

88 Rb,

88 Rb

165 Dy,

41Ar,

49 Ca,

144 Pr

132 I,

105 Ru,

80 Br,

101 Mo,

38 Cl, 51Ti,

65 Ni,

105 Ru,

10 hours–1 day

28 Mg/28Al, 77 Ge,

194 Ir

42 K,

97 Zr,

187 W,

43 K,

97 Zr,

194 Ir

142 Pr,

72 Ga

130 I,

72 Ga,

130 I,

194 Ir,

77 Ge,

142 Pr,

188 Re,

72 Ga,

130 I,

194 Ir,

72 Ga,

109 Pd,

188 Re,

72 Ga,

130 I,

72 Ga,

188 Re,

Radioactive isotopes and radiation sources

1–10 days 67 Cu,

10–100 days

76As,

77As,

33 P,

131 I,

132Te,

133 Xe,

124 Sb,

166 Ho,

169 Er,

175Yb,

192 Ir,

82 Br,

143 Ce,

177 Lu,

99 Mo, 105 Rh,

47 Sc,

121 Sn

169 Er,

47 Ca, 76As,

111Ag,

143 Ce,

161Tb,

115m Cd, 182Ta

149 Pm, 151 Pm, 147 Nd,

160Tb,

191 Os,

198Au,

67 Cu,

82 Br,

131 I,

175Yb,

199Au,

76As,

99 Mo,

133 Xe,

166 Ho,

177 Lu,

222 Rn + D.P. 47 Sc,

77As,

115 Cd,

67 Cu,

105 Rh, 131 I,

193 Os,

99 Mo, 111Ag, 131 I,

186 Re,

193 Os,

203 Hg,

46 Sc,

95 Zr,

124 Sb,

126 I,

129m/129Te,

141 Ce,

181 Hf,

86 Rb,

147 Nd,

185 W

47 Ca,

76As,

115 Cd,

129m/129Te,

74As,

160Tb,

166 Ho,

90Y,

154 Eu,

228 Ra,

228Th + D.P.,

84 Rb,

95 Zr,

95 Zr,

74As,

126 I,

210 Pb,

134 Cs,

227Ac,

226 Ra + D.P.,

182Ta

85 Kr,

125 Sb,

152 Eu,

226 Ra + D.P.,

90 Sr,

137 Cs,

152 Eu,

154 Eu,

137 Cs,

152 Eu,

154 Eu,

228 Ra/228Ac,

129m/129Te,

126 I,

140 Ba,

127m Te, 170Tm

144 Ce/144 Pr,

36 Cl,

204Tl,

227Ac + D.P.,

226 Ra + D.P.,

228 Ra/228Ac,

228Th + D.P. 123 Sn,

182Ta

40 K,

154 Eu,

227Ac + D.P.,

226 Ra + D.P.,

228 Ra/228Ac, 238 U/234m Pa

91Y,

59 Fe,

114m/114 In,

86 Rb,

182Ta

115m Cd, 124 Sb, 129m/129Te,

124 Sb,

94 Nb,

134 Cs,

228Th + D.P.,

32 P,

99Tc,

152 Eu,

228Th + D.P.,

193 Os,

140 La,

87 Rb,

134 Cs,

228 Ra + D.P.,

149 Pm, 140 Ba/140 La

122 Sb,

60 Co,

125 Sb,

140 Ba/140 La

60 Co,

154 Eu,

228 Ra/228Ac,

228Th + D.P.

160Tb 110m/110Ag,

144 Ce/144 Pr

90 Sr/90Y,

106 Ru/106 Rh, 226 Ra + D.P., 228Th + D.P.,

228 Ra/228Ac,

238 U/234m Pa 222 Rn + D.P.

129 I,

228Th + D.P.

140 Ba/140 La, 76As,

152 Eu,

154 Eu,

140 Ba,

89 Sr,

99 Mo,

144 Ce,

192 Ir,

160Tb,

222 Rn + D.P.

222 Rn + D.P.

147 Pm,

227Ac + D.P.,

110m Ag,

147 Nd,

222 Rn + D.P.

140 La,

63 Ni,

106 Ru,

235 U + D.P.

143 Pr,

122 Sb,

123 Sn,

160Tb,

198Au,

192 Ir

110m Ag,

140 Ba,

124 Sb,

210 Bi,

76As,

91Y,

103 Ru,

143 Ce,

47 Ca,

14 C,

93 Zr,

155 Eu,

115m Cd, 182Ta

143 Ce,

140 La,

151 Pm, 186 Re,

3 H,

85 Kr,

125 Sb,

233 Pa

114m/114 In,

115m Cd, 124 Sb,

233 Pa

115 Cd,

59 Fe,

103 Ru,

149 Pm, 151 Pm, 153 Sm, 129m/129Te,

222 Rn + D.P.

110m Ag, 144 Ce,

235 U + D.P.,

151 Pm, 153 Sm, 141 Ce,

122 Sb,

129m/129Te,

>1 year

238 U/234Th

222 Rn + D.P.

111Ag,

45 Ca,

103 Ru,

151 Pm, 161Tb,

199Au,

59 Fe,

95 Nb,

222 Rn + D.P.

77As,

35 S,

100 days–1 year

3–7

106 Ru/106 Rh, 226 Ra + D.P.

106 Ru/106 Rh, 226 Ra + D.P.,

3–8

General physical and chemical constants

Table 3.6

GAMMA ENERGIES AND HALF-LIVES

Half-life γ-energy MeV

3.0

66 Ga,

49 Ca,

66 Cu,

155 Sm,

2.0–3.0

56 Mn,

27 Mg,

116m In,

1.5–2.0

51Ti,

38 Cl,

49 Ca,

101 Mo,

116m In,

49 Ca,

88 Rb

88 Rb,

144 Pr

44 Sc,

117 Cd, 56 Mn,

65 Ni,

132 I,

56 Mn, 68 Ga,

117 Cd,

56 Mn,

132 I

66 Ga

66 Ga,

139 Ba,

65 Ni,

97 Nb,

132 I,

66 Ga,

55 Co,

194 Ir

28 Mg/28Al, 72 Ga,

142 Pr,

24 Na,

72 Ga,

64 Cu,

77 Ge,

194 Ir

42 K,

77 Ge,

43 K,

77 Ge,

109 Pd, 187 W,

43 K,

72 Ga,

130 I,

42 K,

97 Zr,

55 Co,

194 Ir

Radioactive isotopes and radiation sources

1–10 days 47 Sc,

77As,

97 Ru,

111Ag, 131 I,

133m Xe,

10–100 days 67 Cu,

77 Br,

99 Mo,

111 In,

132Te,

133 Xe,

67 Ga,

82 Br,

105 Rh,

115 Co,

133m Ba,

140 La,

59 Fe,

73As,

103 Pd/103m Rh,

103 Ru/103m Rh,

114m In,

125 I,

131m Xe,

140 Ba,

129m/129Te,

100 days–1 year 83 Rb, 105Ag, 131 Ba,

141 Ce,

57 Co,

113 Sn,

139 Ce,

170Tm,

195Au

75 Se,

119m Sn,

144 Ce,

181 W,

>1 year 110m Ag,

127m Te,

153 Gd,

182Ta,

44Ti,

129 I,

154 Eu,

210 Pb,

227Ac + D.P.,

149 Pm,

151 Pm,

147 Nd,

160Tb,

169Yb,

228Th + D.P.,

174Yb,

177 Lu,

186 Re,

185 Os,

191 Os,

192 Ir,

235 U + D.P.,

193 Os,

200 Bi,

224 Ra, 47 Ca,

77 Br,

99 Mo,

115 Cd,

161Tb,

197 Hg,

67 Cu,

87Y,

105 Rh, 131 I,

177 Lu,

193 Os,

224 Ra,

225Ac

52 Mn,

77As,

87Y,

111Ag,

124 I,

143 Ce,

153 Sm,

186 Re,

206 Bi,

224 Ra, 47 Ca,

72As, 82 Br,

111Ag,

140 La,

151 Pm,

193 Os,

161Tb,

67 Ga,

97 Ru,

111Ag, 132 Cs,

76As,

223 Ra,

72As,

97 Ru,

115 Cd,

131 I,

115m Cd,

124 Sb,

129m Te/129Te,

181 Hf,

76As,

82 Br,

99 Mo,

122 Sb,

132 Cs,

160Tb,

223 Ra, 48V,

74As,

85 Sr,

233 Pa

56 Co,

83 Rb,

103 Ru,

147 Nd,

129m Te/129Te,

140 Ba,

193 Os,

198Au,

223 Ra

222 Rn + D.P.,

183 Re,

124 Sb,

151 Pm, 166 Ha,

169Yb,

114m In,

149 Pm, 161Tb,

51 Cr,

103 Ru,

198Au,

222 Rn + D.P.,

77 Br,

7 Be,

103 Pd,

140 Ba/140 La,

183 Re,

231 Pa,

233 Pa

181 Hf,

185 Os,

85 Sr,

105Ag,

126 I,

75 Se,

110m Ag,

106 Ru/106 Rh,

113 Sn/113m In

147 Nd,

105Ag,

52 Mn,

76As,

97 Ru,

124 I,

143 Ce,

166 Ho,

206 Bi,

67 Ga,

77 Br,

99 Mo,

131 I,

149 Pm,

186 Re,

52 Mn, 77 Br,

72As, 82 Br,

124 I,

206 Bi,

222 Rn + D.P.

166 Ho,

132 Cs,

198Au,

46 Sc,

232 U,

235 U + D.P.,

57 Co,

65 Zn,

106 Ru/106 Rh,

127m Te,

88Y,

110m Ag,

144 Ce/144 Pr

134 Cs,

207 Bi,

233 U,

26Al,

231 Pa, 239 Pu 85 Kr,

154 Eu,

207 Bi,

208 Po,

134 Cs,

152 Eu

137 Cs/137m Ba,

228 Ra/228Ac,

134 Cs,

152 Eu,

227Ac + D.P.,

228Th + D.P., 238 U/234m Pa

48V,

56 Co,

114m In,

115m Cd,

95 Nb,

126 I,

129m Te/129Te,

140 Ba/140 La,

95 Zr,

46 Sc,

48V,

56 Co,

91Y,

105Ag,

124 Sb,

129m Te/129Te, 160Tb

88Y,

182Ta,

210 Po

110m Ag,

84 Rb,

126 I,

94 Nb,

154 Eu,

226 Ra + D.P.,

207 Bi,

227Ac + D.P.,

228 Ra/228Ac,

228Th + D.P.,

160Tb,

192 Ir

74As,

54 Mn,

106 Ru/106 Rh,

131 Ba,

185 Os,

86 Rb,

133 Ba,

154 Eu,

125 Sb,

192 Ir,

105Ag,

115m Cd,

233 U,

237 Np,

94 Nb,

226 Ra + D.P.,

84 Rb,

59 Fe,

22 Na,

131 Ba,

83 Rb,

124 Sb,

230Th,

239 Pu,

228 Ra/228Ac,

160Tb,

58 Co, 89 Sr,

226 Ra + D.P.,

228Th + D.P.,

192 Ir,

126 I,

125 Sb,

152 Eu,

208 Po,

227Ac + D.P.,

175 Hf,

84 Rb,

232 U,

241Am

131 Ba,

58 Co,

155 Eu,

125 Sb,

152 Eu,

226 Ra + D.P.,

238 U/234Th,

225Ac

122 Sb,

143 Ce,

203 Hg,

181 Hf,

151 Pm,

175Yb,

222 Rb + D.P. 47 Ca,

175 Hf,

225Ac

143 Ce,

206 Bi,

199Au,

222 Rn + D.P.,

140 La,

153 Sm,

166 Ho,

109 Cd,

133 Ba,

228 Ra/228Ac,

143 Ce,

153 Sm,

3–9

238 U/234m Pa

65 Zn,

110m Ag,

106 Ru/106 Rh, 123 Sn,

144 Ce/144 Pr,

182Ta

131 Ba,

22 Na,

60 Co,

154 Eu,

226 Ra + D.P.,

26Al,

40 K,

134 Cs,

152 Eu,

152 Eu,

154 Eu,

207 Bi,

227Ac + D.P., 228 Ra/228Ac,

228Th + D.P., 238 U/234m Pa

72As,

124 I,

166 Ho,

76As,

132 Cs,

206 Bi,

82 Br,

140 La,

56 Co,

124 Sb,

58 Co,

84 Rb,

140 Ba/140 La

88Y,

110m Ag

106 Ru/106 Rh,

228 Ra/228Ac,

222 Rn + D.P. 72As,

140 La,

76As,

206 Bi,

222 Rn + D.P. 72As

26Al,

207 Bi,

226 Ra + D.P.,

228Th + D.P.,

238 U/234m Pa 124 I,

48V,

56 Co,

140 Ba/140 La 56 Co

124 Sb,

88Y,

106 Ru/106 Rh,

144 Ce/144 Pr

26Al,

228Th + D.P.

226 Ra + D.P.,

3–10 Table 3.7

General physical and chemical constants NUCLIDES FOR ALPHA SOURCES

Nuclide

Half-life

α-energies MeV

Associated β and γ radiation MeV

Americium-241 Lead-210 (+daughters)

432.2 yr 22.3 yr

5.44, 5.48 5.305

Plutonium-238 Plutonium-239 Polonium-210 Radium-226 (+daughters)

87.74 yr 2.41 × 104 yr 138.4 d 1 600 yr

5.352, 5.452, 5.495 5.096, 5.134, 5.147 5.305 4.589–7.68

γmax 0.060 βmax 1.17 γmax 0.80 (very weak) γ 0.096 (very weak) γmax 0.451 (weak) γ 0.80 (very weak) βmax 3.26 γmax 2.43

Table 3.8

NUCLIDES FOR BETA SOURCES

Nuclide

Half-life

βmax MeV

Associated α and γ radiation MeV

Carbon-14 Cerium-144 + praseodymium-144 Iron-55 Krypton-85 Lead-210 + bismuth-210

5 730 yr 284.3 d 2.7 yr 10.72 yr 22.3 yr

0.159 2.98 0.0052 0.67 1.17

Nickel-63 Promethium-147 Ruthenium-106 + rhodium-106 Strontium-90 + yttrium-90 Thallium-204 Tritium Yttrium-90

96 yr 2.62 yr 368.2 d 29.12 yr 3.78 yr 12.35 yr 64.0 h

0.066 0.225 3.6 2.27 0.77 0.018 2.27

— γ γ γ α γmax — — γ — — — —

0.034–2.19 0.0059–0.0065 0.51 5.30 0.80 (very weak) 0.43–2.41

NUCLIDES FOR NEUTRON SOURCES—POLONIUM-210 (ALPHA, N) SOURCES WITH VARIOUS TARGETS

Table 3.9

Target Aluminium Beryllium Boron Fluorine-19 Lithium Magnesium Oxygen-18 Sodium Mock fission

Neutrons s−1 TBq−1 5.4 × 105 6.8 × 107 5.4 × 106 2.7 × 106 1.4 × 106 8.1 × 105 2.7 × 107 1.1 × 106 1.1 × 106

Neutron energy (MeV) Mean

Maximum

— 4.3 — 1.4 0.48 — — — 1.6

2.7 10.8 5.0 2.8 1.32 — 4.3 — 10.8

Radioactive isotopes and radiation sources Table 3.10

NUCLIDES FOR NEUTRON SOURCES—(GAMMA, N) SOURCES

Nuclide

Half-life

Target

Antimony-124 Radium-226 (+daughters) Radium-226 (+daughters)

60.20 d 1 600 yr 1 600 yr

Thorium-228 (+daughters) Thorium-228 (+daughters)

1.91 yr 1.91 yr

Beryllium Beryllium Deuterium (heavy water) Beryllium Deuterium (heavy water)

Table 3.11

Observed neutron s−1 TBq−1

Observed neutron energy keV

4.3 × 107 3.5 × 107 —

24.8 700 (max) 120

— 3.2 × 107

827 197

NUCLIDES FOR NEUTRON SOURCES—(ALPHA, N) SOURCES WITH BERYLLIUM TARGETS

Nuclide

Half-life

Actinium-227 (+daughters) Americium-241 Lead-210 (+daughters) Plutonium-239 Polonium-210 Radium-226 (+daughters) Thorium-228 (+daughters)

21.77 yr 432.2 yr 22.3 yr 2.41 × 104 yr 138.4 d 1 600 yr 1.91 yr

Table 3.12

3–11

Neutrons s−1 TBq−1 4.9 × 108 5.9 × 107 6.2 × 107 3.8 × 107 6.8 × 107 4.2 × 108 6.8 × 108

Gamma emission µGy h−1 at 1 m from 106 neutrons s−1 80 11.4 88 17 0.4 600 300

SPONTANEOUS FISSION NEUTRON SOURCE

Nuclide

Half-life

Neutrons

Californium-252

2.64 yr (effective)

2.3 × 109

s−1

mg−1

Gamma emission µGy h−1 at 1 m from 106 neutrons s−1 0.7

REFERENCES ‘Radionuclide Transformations, Energy and Intensity of Emissions’, ICRP Publication 38, Pergamon, Oxford, 1983. ‘Radioactive Decay Data Tables’, DOE/TIC-11026, Technical Information Center, U.S. Department of Energy, 1981.

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4

X-ray analysis of metallic materials

4.1 Introduction and cross references X-rays are very short wavelength electromagnetic waves. Their range encompasses the interatomic distances in crystalline materials, typically 0.5 to 2.5 Å, which permit the analysis of crystalline and to a lesser degree amorphous materials by X-ray diffraction (XRD). When atoms are irradiated with radiation of sufficient energy, they emit characteristic X-ray spectra, fluorescence, which are analysed in X-ray fluorescent analysis (XRF) to give element analysis. Individual electrons are emitted from an irradiated surface with energies related to the electron levels in the atom. The energies of these electrons are analysed in X-ray photoelectron spectroscopy (XPS) to provide an elemental analysis of surface atoms together with information on the chemical bonding between them. This chapter reviews the applications of XRD to the investigation of metallic materials. The subject is well covered in detail in several standard textbooks including ‘X-ray Metallography’ by A. Taylor;1 ‘Structure of Metals’ by C. S. Barrett and T. B. Massalski;2 ‘X-ray Diffraction Procedures’ by H. P. Klug and L. E. Alexander.3 ‘Elements of X-ray Crystallography’ by L. V. Azaroff,4 ‘Structure Determination by X-ray Crystallography’ by M. F. C. Ladd and R. A. Palmer,5 ‘An Introduction to X-ray Crystallography’ by M. M. Woolfson,6 ‘Elements of X-ray Diffraction’ by B. D. Cullity and S. R. Stock,7 and ‘The Basics of Crystallography and Diffraction’ by C. Hammond.8 Chapter 5 discusses crystallography and crystal structure data can be found in Chapter 6.

4.2 Excitation of X-rays X-rays are produced (excited) when an electron experiences a sudden acceleration or (more commonly in practice) deceleration as, for example, when a beam of fast-moving electrons strikes an atom. The frequency ν or wavelength λ of the radiation produced as a result of energy loss E is given by the equation E = hν =

hc λ

The maximum value of E, Emax , and corresponding smallest value of λ occurs when all the electron energy is lost on impact, i.e. Emax = Ve =

hc λswl

where V = accelerating voltage and λswl = short wavelength limit or cut-off wavelength. Substitution of numerical values gives λswl =

12.34 V

where λswl is expressed in Ångstrom units and V in kilovolts. However, most electrons undergo repeated collisions of varying energy losses until finally stopped giving rise to a whole range of λ values – the so-called ‘white’ or ‘Brehmstrallung’ radiation. If the intensity of the radiation is plotted against the wavelength, for relatively low applied voltages a curve is obtained rising from zero to a more or less pronounced maximum at a wavelength of about 4/3 λswl , falling again to a low value with increasing wavelength. It has been found that for this

4–1

4–2

X-ray analysis of metallic materials

curve of so-called ‘white radiation’ both the total radiation and the intensity of the peak are closely proportional to the atomic number of the target material. Thus if white radiation is required a target of one of the heavy metals should be selected, and since the total intensity (the area under the curve) is proportional to the square of the applied voltage the latter should be as high as possible. On increasing the voltage beyond a certain limit, which is different for different targets, a line spectrum begins to appear, superimposed upon the continuous or white spectrum, the wavelengths of the lines being constant and characteristic of the target. The lowest critical voltage at which these lines appear is that which endows the bombarding electrons with just sufficient energy to eject electrons from one of the inner shells in the target atoms. Each vacancy as it occurs is filled by an electron jumping in from one of the outer shells, the jump being accompanied by an emission of energy of frequency ν given by the Bohr relationship hv = W1 − W2

where W1 − W2 represents the change in potential energy of the system and h is Planck’s constant. The K series of lines is excited when electrons are ejected from the K shell and their places taken by electrons from the L, M or N shells. It consists principally of five lines Kα1 , Kα2 , Kβ1 , Kβ3 and Kβ2 , the α lines being associated with jumps from the L shell, β1 and β3 with those from the M shell, and β2 with those for the N shell. Other series of lines are emitted when the bombarding electrons eject electrons from the L, M or N shells. The K emissions which are the most intense are generally used in XRD while all the spectra are employed in XRF. If V0 is the critical voltage at which the K spectra first appear and e is the electron charge, then the energy needed to remove an electron from the K shell is V0 e which is equivalent to radiation of frequency vk and wavelength λk given by V0 e = hvk =

hc λk

where c is the velocity of light. This critical wavelength λk , termed the K absorption edge of the material, together with slightly shorter wavelengths are heavily absorbed in the material. In XRD, to avoid fluorescence (high background) and low penetration into the sample a radiation is chosen which is on the long wavelength side of any absorption edge of the sample. The empirical expression I = K(V − V0 )1.7 where K is a constant, relates the intensity, I , of the characteristic radiation to the difference between the tube voltage V and the critical voltage V0 of the target material. It is found that the relative intensity of the characteristic radiation to that of the continuous wavelength (white) radiation is greatest for tube voltages of about four times the critical voltage. A method of achieving partial monochromatisation of the X-ray beam is to reduce the intensity of the Kβ radiation with respect to that of the Kα by using a filter of an element with K absorption edge between the Kβ and Kα wavelengths. The thickness of the filter is usually chosen to reduce the intensity of Kβ to about 1/600 of the transmitted Kα . Generally, the element preceding the target element in the periodic table will act as a Kβ filter. Filters cannot separate the α1 and α2 components and where they are not resolved in a diffraction pattern, d values are calculated using a weighted average wavelength where α1 , the stronger of the two, is given twice the weight of α2 . Full monochromatisation can be achieved with a single crystal monochromator oriented such that only Kα1 component is diffracted. Monochromators in the form of a single crystal lamina bent to produce focusing are widely used in diffractometry to remove all unwanted radiation. Construction of focusing monochromators is described in ‘International Tables for X-ray Crystallography’.9 Table 4.1 lists wavelengths, minimum excitation potentials and Kβ filters for targets frequently used in XRD. Table 4.2 gives minimum excitation potentials for characteristic K, L, M, and N spectra while Tables 4.3 and 4.4 give K and L emission spectra and L111 absorption edges for all elements. Crystals suitable for monochromators are listed with comments on their performance in Table 4.5. 4.2.1

X-ray wavelengths

The X-unit defined by Siegbahn and very nearly equal to 10−3 Å was based on the atomic spacing in calcite as determined from the expression:   4M 1/3 2d = ρNV

Excitation of X-rays

4–3

where M = molecular weight of calcite, ρ = density of calcite, N = Avogadro’s number, V = volume of calcite rhombohedron with unit distance between the faces considered. kX units were widely used up to about 1945 but were then gradually replaced by Ångström units as more precise methods for determining X-ray wavelengths were introduced. The nanometer (10−9 m) is now declared to be the standard unit for X-ray studies. However, in view of the much greater familiarity with Å, and the very simple conversion factor, in this edition the Ångström unit is used unless the contrary is stated. 1 kX unit = 1.002 02 Ångström units 1 Ångström unit, Å = 10−10 m or 10−1 nanometers

Emission lines are frequently quoted in keV 1 keV =

12.3975 Wavelength in Å

WAVELENGTHS, EXCITATION POTENTIALS, AND β FILTERS FREQUENTLY USED IN CRYSTALLOGRAPHIC WORK

Table 4.1

Target

Element

Atomic No.

Cr

24

Mn

25

Fe

26

Co

27

Ni

28

Cu

29

W

74

Zn

30

Au

79

Line

Wavelength* Å

Kα1 Kα2 Kα Kβ1 Kα1 Kα2 Kα Kβ1 Kα1 Kα2 Kα Kβ1 Kα1 Kα2 Kα Kβ1 Kα1 Kα2 Kα Kβ1 Kα1 Kα2 Kα Kβ1 Lα1 Lα2 Kα1 Kα2 Kα Kβ1 Lα1 Lα2

2.289 62 2.293 51 2.290 9 2.084 80 2.101 75 2.105 69 2.103 1 1.910 15 1.935 97 1.939 91 1.937 3 1.756 53 1.788 92 1.792 78 1.790 2 1.620 75 1.657 84 1.661 69 1.659 1 1.500 10 1.540 51 1.544 33 1.541 8 1.392 17 1.476 35 1.487 42 1.435 11 1.438 94 1.436 4 1.295 22 1.276 39 1.287 77

Kα2 Kα2 Kα

0.785 88 0.790 10 0.787 29

β-filter

Element

Absorption edge

Mass absorption coefficient µ/ρ

5.98

V

2.267 5

77.3

0.009

0.016

6.54

Cr

2.070 1

71

0.011

0.016

7.1

Mn

1.895 4

61.9

0.012

0.016

7.71

Fe

1.742 9

58.6

0.014

0.018

8.29

Co

1.607 2

51.6

0.015

0.018

8.86

Ni

1.486 9

48.0

0.019

0.021

Cu

1.380 2

42.0

0.019

0.021

Cu

1.380 2

42.0

0.019

0.021

14.4

Ga

1.195 7

37.0

0.028

0.047

80.5

Sr

0.769 69

18.1

0.053

0.210

Excitation potential† kV

12.1 9.65

Material content g cm−2

Thickness mm

(continued)

4–4

X-ray analysis of metallic materials

WAVELENGTHS, EXCITATION POTENTIALS, AND β FILTERS FREQUENTLY USED IN CRYSTALLOGRAPHIC WORK—continued

Table 4.1

Target

Element

Atomic No.

Zr Mo

40 42

Rh

45

Pd

46

Ag

∗

47

β-filter

Line

Wavelength* Å

Kβ Kα1 Kα2 Kα Kβ1 Kα1 Kα2 Kα Kβ1 Kα1 Kα2 Kα Kβ1 Kα1 Kα2 Kα Kβ1

0.701 70 0.709 26 0.713 54 0.710 7 0.632 25 0.613 25 0.617 61 0.614 7 0.545 59 0.585 42 0.589 80 0.586 9 0.520 52 0.559 36 0.563 78 0.560 9 0.497 01

Material content

Absorption edge

Mass absorption coefficient µ/ρ

g cm−2

Thickness mm

Zr

0.688 8

17.2

0.069

0.108

23.2

Ru

0.559 5

15.4

0.077

0.064

24.4

Rh or Ru

0.534 1

14.6

0.091

0.073

13.1

0.096

0.079

Excitation potential† kV

Element

20

25.5

Pd or Rh

0.559 5 0.509 0 0.534 1

λKα is here defined as (2λKα1 + λKα2 )/3. optimum voltage for operating a tube with raw alternating currents is approximately 5 times the excitation potential, and 4 times the excitation potential with fully smoothed direct current, but normally 80 kV cannot be exceeded owing to the danger of electrical breakdown.

† The

Table 4.2

EXCITATION POTENTIALS IN kV FOR CHARACTERISTIC X-RAY SPECTRA

Atomic No. and element

K

L

M

N

Atomic No. and element

K

L

M

N

11 Na 12 Mg 13 Al 14 Si 15 P 16 S 17 Cl 19 K 20 Ca 21 Sc 22 Ti 23 V 24 Cr 25 Mn 26 Fe 27 Co 28 Ni 29 Cu 30 Zn 31 Ga 32 Ge 33 As 34 Se 35 Br 37 Rb

1.07 1.30 1.55 1.83 2.13 2.46 2.81 3.69 4.02 4.48 4.94 5.44 5.96 6.51 7.08 7.67 8.29 8.94 9.62 10.32 11.05 11.81 12.59 13.41 15.13

— — — — — — 0.24 0.34 0.40 0.46 0.53 0.60 0.68 0.76 0.85 0.92 1.01 1.10 1.19 1.29 1.41 1.52 1.65 1.78 2.05

— — — — — — — — — — — — — — — — — — — — — — — — —

— — — — — — — — — — — — — — — — — — — — — — — — —

49 In 50 Sn 51 Sb 52 Te 53 I 55 Cs 56 Ba 57 La 58 Ce 59 Pr 60 Nd 62 Sm 63 Eu 64 Gd 65 Tb 66 Dy 67 Ho 68 Er 69 Tm 70 Yb 71 Lu 72 Hf 73 Ta 74 W 75 Re

27.80 29.06 30.35 31.66 33.01 35.80 37.24 38.75 40.27 41.81 43.37 46.63 48.29 50.00 51.76 53.55 55.36 57.22 59.07 61.02 63.01 65.01 67.09 69.18 71.28

4.21 4.42 4.69 4.93 5.16 5.68 5.92 6.24 6.53 6.81 7.10 7.71 8.02 8.35 8.67 9.01 9.35 9.71 10.06 10.45 10.82 11.23 11.63 12.04 12.46

0.83 0.88 0.94 1.01 1.08 1.21 1.29 1.36 1.43 1.51 1.58 1.72 1.80 1.88 1.96 2.04 2.13 2.22 2.31 2.41 2.50 2.60 2.69 2.80 2.91

0.12 0.13 0.15 0.17 0.19 0.23 0.25 0.27 0.29 0.30 0.32 0.35 0.36 0.38 0.40 0.42 0.43 0.45 0.47 0.50 0.51 0.54 0.57 0.59 —

(continued)

Excitation of X-rays Table 4.2

4–5

EXCITATION POTENTIALS IN kV FOR CHARACTERISTIC X-RAY SPECTRA—continued

Atomic No. and element

K

L

M

N

Atomic No. and element

K

L

M

N

38 Sr 39 Y 40 Zr 41 Nb 42 Mo 44 Ru 45 Rh 46 Pd 47 Ag 48 Cd

16.03 16.96 17.92 18.90 19.91 22.02 23.12 24.23 25.40 26.59

2.20 2.38 2.52 2.69 2.86 3.21 3.39 3.60 3.81 4.00

— — 0.43 0.48 0.51 0.59 0.62 0.67 0.72 0.77

— — 0.05 0.05 0.06 0.06 0.07 0.08 0.10 0.11

76 Os 77 Ir 78 Pt 79 Au 80 Hg 81 Tl 82 Pb 83 Bi 90 Th 92 U

73.54 75.77 78.02 80.42 82.69 85.28 87.66 90.03 109.27 115.54

12.91 13.35 13.80 14.29 14.77 15.27 15.79 16.31 20.36 21.66

3.03 3.15 3.28 3.41 3.55 3.69 3.84 3.99 5.17 5.54

0.64 0.67 0.71 0.79 0.82 0.86 0.89 0.96 1.33 1.44

Table 4.3

K EMISSION LINES AND K ABSORPTION EDGES IN Å

Line transition intensity rel. to Kα1

α2 KL11 50

3 Li 4 Be 5B 6C 7N 8O 9F 10 Ne 11 Na 12 Mg 13 Al 14 Si 15 P 16 S 17 Cl 18 A 19 K 20 Ca 21 Sc 22 Ti 23 V 24 Cr 25 Mn 26 Fe 27 Co 28 Ni 29 Cu 30 Zn 31 Ga 32 Ge 33 As 34 Se 35 Br 36 Kr 37 Rb 38 Sr 39 Y 40 Zr 41 Nb 42 Mo 43 Tc 44 Ru

228.0 114.0 67.6 44.7 31.6 23.62 18.32 14.61 11.910 1 9.890 0 8.341 73 7.127 91 6.160 5.374 96 4.730 7 4.194 74 3.744 5 3.361 66 3.034 2 2.752 16 2.507 38 2.293 61 2.105 78 1.939 98 1.792 85 1.661 75 1.544 39 1.439 00 1.343 99 1.258 01 1.179 87 1.108 82 1.043 82 0.984 1 0.929 69 0.879 43 0.833 05 0.790 15 0.750 44 0.713 59 0.679 32+ 0.647 41

α1 KL111 100

β3 KM11 15

β1 KM111 15

β2 KN11,111 5

K Absorption edge

8.339 34 7.125 42 6.157 5.372 16 4.727 8 4.191 80 3.741 4 3.358 39 3.030 9 2.748 51 2.503 56 2.289 70 2.101 82 1.936 04 1.788 97 1.657 91 1.540 56 1.435 16 1.340 08 1.254 05 1.175 88 1.104 77 1.039 74 0.980 1 0.925 553 0.875 26 0.828 84 0.785 93 0.746 20 0.709 30 0.675 02+ 0.643 08

— — — — — — — — — — — — — — — — — — — — — — — — — — 1.392 6 — 1.208 35 1.129 36 1.057 83 0.992 68 0.933 27 0.879 0 0.829 21 0.783 45 0.741 26 0.702 28 0.666 34 0.632 87 0.601 88+ 0.573 07

— — — — — — — 14.452 11.575 9.521 7.960 6.753 5.796 5.032 4.403 4 3.886 0 3.453 9 3.089 7 2.779 6 2.513 91 2.284 40 2.084 87 1.910 21 1.756 61 1.620 79 1.500 14 1.392 22 1.295 25 1.207 89 1.128 94 1.057 30 0.992 18 0.932 79 0.878 5 0.828 68 0.782 92 0.740 72 0.701 73 0.665 76 0.632 29 0.601 30+ 0.572 48

— — — — — — — — — — — — — — — — — — — — — — — 1.744 2 1.608 9 1.488 6 1.381 09 1.283 72 1.196 00 1.116 86 1.045 00 0.979 92 0.920 46 0.866 1 0.816 45 0.770 81 0.728 64 0.689 93 0.654 16 0.620 99 0.590 24+ 0.561 66

226.5 — — 43.7 31.0 23.3 — — — 9.511 7 7.951 1 6.744 6 5.786 6 5.018 2 4.396 9 3.870 7 3.436 45 3.070 16 2.757 3 2.497 3 2.269 0 2.070 1 1.896 4 1.743 3 1.608 1 1.488 0 1.380 4 1.283 3 1.195 7 1.116 5 1.045 0 0.979 78 0.919 95 0.865 47 0.815 49 0.769 69 0.727 62 0.688 77 0.652 91 0.619 77 (0.589 1) 0.560 05 (continued)

4–6

X-ray analysis of metallic materials

Table 4.3

K EMISSION LINES AND K ABSORPTION EDGES IN Å—continued

Line transition intensity rel. to Kα1

α2 KL11 50

α1 KL111 100

β3 KM11 15

45 Rh 46 Pd 47 Ag 48 Cd 49 In 50 Sn 51 Sb 52 Te 53 I 54 Xe 55 Cs 56 Ba 57 La 58 Ce 59 Pr 60 Nd 61 Pm 62 Sm 63 Eu 64 Gd 65 Tb 66 Dy 67 Ho 68 Er 69 Tm 70 Yb 71 Lu 72 Hf 73 Ta 74 W 75 Re 76 Os 77 Ir 78 Pt 79 Au 80 Hg 81 Tl 82 Pb 83 Bi 84 Po 85 At 86 Rn 87 Fr 88 Ra 89 Ac 90 Th 91 Pa 92 U

0.617 63 0.589 82 0.563 80 0.539 42 0.516 54 0.435 24 0.474 83 0.455 78 0.437 83 0.420 87+ 0.404 84 0.389 67 0.375 31 0.361 68 0.348 75 0.336 47 0.324 80 0.313 70 0.303 12 0.293 04 0.283 42 0.274 25 0.265 486 0.257 11 0.249 10 0.241 42 0.234 08 0.227 02 0.220 31 0.213 83 0.207 61 0.201 64 0.195 90 0.190 38 0.185 08 0.179 96 0.175 04 0.170 29 0.165 72 0.161 30+ 0.157 05+ 0.152 94+ 0.148 96+ 0.145 12+ 0.141 41+ 0.137 83 0.134 34+ 0.130 97

0.613 28 0.585 45 0.559 41 0.535 01 0.512 11 0.425 92 0.470 35 0.451 30 0.433 32 0.416 34+ 0.400 29 0.385 11 0.370 74 0.357 09 0.344 14 0.331 85 0.320 16 0.309 04 0.298 45 0.288 35 0.278 72 0.269 53 0.260 76 0.252 37 0.244 34 0.236 66 0.229 30 0.222 22 0.215 50 0.209 01 0.202 78 0.196 79 0.191 05 0.185 51 0.180 20 0.175 07 0.170 14 0.165 38 0.160 79 0.156 36+ 0.152 10+ 0.147 98+ 0.143 99+ 0.140 14+ 0.136 42+ 0.132 81 0.129 33+ 0.125 95

0.546 20 0.521 12 0.497 69 0.475 73 0.455 18 0.424 67 0.417 74 0.400 66 0.384 56 0.369 41+ 0.355 05 0.341 51 0.328 69 0.316 52 0.304 98 0.294 03 0.283 63+ 0.273 76 0.264 33 0.255 34 0.246 83 0.238 62 0.230 83 0.223 41 0.216 36 0.209 6+ 0.203 09+ 0.196 86+ 0.190 89 0.185 18 0.179 70 0.174 43 0.169 37 0.164 50 0.159 81 0.155 32 0.150 98 0.146 81 0.142 78 0.138 92+ 0.135 17+ 0.131 55+ 0.128 07+ 0.124 69+ 0.121 43+ 0.118 27 0.115 23+ 0.112 30

+ Interpolated

values.

β1 KM111 15 0.545 61 0.520 52 0.487 07 0.475 10 0.454 54 0.431 84 0.417 09 0.399 99 0.383 91 0.368 72+ 0.354 36 0.340 81 0.327 98 0.315 82 0.304 26 0.293 30 0.282 90+ 0.273 01 0.263 58 0.254 60 0.246 08 0.237 88 0.230 12 0.222 66 0.215 56 0.208 84 0.202 31+ 0.196 07 0.190 09 0.184 37 0.178 88 0.173 61 0.168 54 0.163 68 0.158 98 0.154 49 0.150 14 0.145 97 0.141 95 0.138 07+ 0.134 32+ 0.130 69+ 0.127 19+ 0.123 82+ 0.120 55+ 0.117 40 0.114 35+ 0.111 39

β2 KN11,111 5 0.535 03 0.510 23 0.487 03 0.465 33 0.445 00 0.431 75 0.407 97 0.391 10 0.375 23+ 0.360 26+ 0.346 11 0.332 77 0.320 12 0.308 16 0.296 79 0.286 1+ 0.275 9+ 0.266 2 0.256 92 0.248 16 0.239 7+ 0.231 7+ 0.224 1+ 0.216 7+ 0.209 8+ 0.203 3+ 0.196 9+ 0.190 8+ 0.185 10 0.179 51 0.174 15 0.169 90 0.164 05 0.159 29 0.154 72 0.150 30 0.146 04 0.142 01 0.138 07 0.134 28+ 0.130 67+ 0.127 08+ 0.123 68 0.120 39 0.117 21+ 0.114 15+ 0.111 18+ 0.108 27+

K Absorption edge 0.533 8 0.509 2 0.485 8 0.464 09 0.443 88 0.424 68 0.406 63 0.389 72 0.373 79 0.358 49 0.344 74 0.331 37 0.318 42 0.306 47 0.295 16 0.284 51 0.274 3 0.264 62 0.255 52 0.246 80 0.238 40 0.230 46 0.222 90 0.215 66 0.208 9 0.202 23 0.195 84 0.189 81 0.183 93 0.178 37 0.173 11 0.167 80 0.162 86 0.158 16 0.153 44 0.149 23 0.144 70 0.140 77 0.137 06 — — — — — — 0.112 93 — 0.106 80

Excitation of X-rays

Table 4.4

4–7

L EMISSION SPECTRA AND ABSORPTION EDGES IN Å

Line transition intensity rel. to α1

l Liii Mi 30

η Lii Mi 10

α2 Liii Miv 10

17 Cl 18 A 19 K 20 Ca 21 Sc 22 Ti 23 V 24 Cr 25 Mn 26 Fe 27 Co 28 Ni 29 Cu 30 Zn 31 Ga 32 Ge 33 As 34 Se 35 Br 36 Kr 37 Rb 38 Sr 39 Y 40 Zr 41 Nb 42 Mo 43 Tc 44 Ru 45 Rh 46 Pd

67.90 56.30+ 47.74 40.96 35.59 31.36 27.77 24.78 22.29 20.15 18.292 16.693 15.286 14.02 12.953 11.965 11.072 10.294 9.585 — 8.363 6 7.836 2 7.356 3 6.918 5 6.517 6 6.150 8 — 5.503 5 5.216 9 4.952 5

67.33 55.9+ 47.24 40.46 35.13 30.89 27.34 24.30 21.85 19.75 17.87 16.27 14.90 13.68 12.597 11.609 10.734 9.962 9.255 — 8.041 5 7.517 1 7.040 6 6.606 9 6.210 9 5.847 5 — 5.205 0 4.921 7 4.660 5

— — —

α1 Liii Mv 100

— — — 36.33 31.35 27.42 24.25 21.64 19.45 17.59 15.972 14.561 13.336 12.254 11.292 10.436 1 9.670 9 8.990 0 8.374 6 7.817+ 7.325 1 7.318 3 6.869 7 6.862 8 6.455 8 6.448 8 6.077 8 6.070 5 5.731 9 5.724 3 5.414 4 5.406 6 — 5.114 8+ 4.853 8 4.845 8 4.605 5 4.597 4 4.375 9 4.367 7

L Absorption edges

β1 Lii Miv 80

β4 Li Mii 20

β3 Li Miii 30

β2 Liii Nv 60

β5 Lii Oiv, v 60

γ1 Lii Niv 40

Li

Lii

Liii

— — — — 31.02 27.05 23.88 21.27 19.11 17.26 15.666 14.271 13.053 11.983 11.023 10.175 9.414 1 8.735 8 8.125 1 7.576+ 7.075 9 6.623 9 6.212 0 5.836 0 5.492 3 5.177 1 4.887 3+ 4.620 6 4.374 1 3.990 2

— — — — — — — — — — — — — — — — — — — — 6.820 7 6.402 6 6.018 6 5.668 1 5.345 5 5.048 8 — 4.523 0 4.288 8 4.071 1

— — — — — — — — — — — — — — — — — — — — 6.787 6 6.367 2 5.983 2 5.633 0 5.310 2 5.013 3 — 4.486 6 4.252 2 4.034 6

— — — — — — — — — — — — — — — — — — — — — — — 5.586 3 5.237 9 4.923 2 — 4.371 8 4.131 0 3.908 9

— — — — — — — — — — — — — — — — — — — — — — — — — — — — — —

— — — — — — — — — — — — — — — — — — — — — — — 5.384 3 5.036 1 4.725 8 — 4.182 2 3.943 7 3.724 6

52.084 43.192 36.352 31.068 26.831 23.389 20.523 18.256 16.268 14.601 13.343 12.267 11.269 10.330 9.535 8.729 8.107 7.467 6.925 6.456 6.006 5.604 5.193 1 4.893 8 4.5911 4.320 7 4.064 3 3.841 3 3.641 6 3.430 0

61.366 50.390 42.020 35.417 30.161 26.831 23.702 21.226 18.896 17.169 15.534 14.135 12.994 11.840 10.613 9.965 9.128 1 8.421 2 7.752 3 7.165 3 6.653 8 6.185 6 5.709 8 5.370 9 5.024 7 4.713 3 4.427 1 4.176 5 3.949 0 3.713 6

61.672 50.803 42.452 35.827 30.457 27.184 24.070 21.596 19.248 17.484 15.831 14.448 13.258 12.106 10.855 10.228 9.376 7 8.662 4 7.987 1 7.422 7 6.875 2 6.399 6 5.914 1 5.573 7 5.226 0 4.909 3 4.625 4 4.366 3 4.138 9 3.896 9 (continued)

4–8

X-ray analysis of metallic materials

Table 4.4

L EMISSION SPECTRA AND ABSORPTION EDGES IN Å—continued

Line transition intensity rel. to α1 47 Ag 48 Cd 49 In 50 Sn 51 Sb 52 Te 53 I 54 Xe 55 Cs 56 Ba 57 La 58 Ce 59 Pr 60 Nd 61 Pm 62 Sm 63 Eu 64 Gd 65 Tb 66 Dy 67 Ho 68 Er 69 Tm 70 Yb 71 Lu 72 Hf 73 Ta 74 W 75 Re 76 Os

l Liii Mi 30 4.707 6 4.480 1 4.268 7 4.071 7 3.888 3 3.717 0 3.557 5 — 3.267 0 3.135 5 3.006 2.891 7 2.784 1 2.676 0 — 2.482 3 2.394 8 2.312 2 2.235 2 2.158 9 2.086 0 2.015 1.955 0 1.894 2 1.836 0 1.781 5 1.728 4 1.678 2 1.630 6 1.585 0

η Lii Mi 10 4.418 3 4.193 2 3.983 3 3.788 8 3.607 7 3.438 3 3.279 8 — 2.993 2 2.862 7 2.740 2.620 3 2.512 2.409 4 — 2.218 2 2.131 5 2.049 4 1.973 0 1.897 4 1.826 4 1.756 6 1.696 3 1.635 6 1.577 9 1.523 3 1.471 1 1.421 1 1.373 4 1.327 9

α2 Liii Miv 10

α1 Liii Mv 100

4.162 9 3.965 0 3.780 7 3.608 9 3.448 4 3.298 5 3.157 9 — 2.902 0 2.785 5 2.675 3 2.570 6 2.472 9 2.380 7 2.292 6 2.210 6 2.131 5 2.057 8 1.987 5 1.919 9 1.856 1 1.795 5 1.738 1 1.682 9 1.630 3 1.580 5 1.532 9 1.487 4 1.444 0 1.402 3

4.154 4 3.956 4 3.771 9 3.599 9 3.439 4 3.289 2 3.148 6 3.016 6+ 2.892 4 2.776 0 2.665 7 2.561 5 2.463 0 2.370 4 2.282 2 2.199 8 2.120 9 2.046 8 1.976 5 1.908 8 1.845 0 1.784 3 1.726 8+ 1.671 9 1.619 5 1.569 6 1.522 0 1.476 4 1.432 9 1.391 2

β1 Lii Miv 80 3.934 7 3.738 2 3.555 3 3.384 9 3.225 7 3.076 8 2.937 4 — 2.683 7 2.568 2 2.458 9 2.356 1 2.258 8 2.166 9 2.079 7 1.998 1 1.920 3 1.846 8 1.776 8 1.710 6 1.647 5 1.587 3 1.530 4 1.475 7 1.423 6 1.374 1 1.327 0 1.281 8 1.238 6 1.197 3

β4 Li Mii 20

β3 Li Miii 30

β2 Liii Nv 60

β5 Lii Oiv, v 60

γ1 Lii Niv 40

3.870 2 3.682 0 3.507 0 3.343 4 3.190 1 3.046 6 2.912 1 — 2.666 6 2.555 3 2.449 3 2.349 7 2.255 0 2.166 9 — 2.001 0 1.925 5 1.854 0 1.786 4 1.721 0 1.659 5 1.600 7 1.544 8 1.491 4 1.440 6 1.392 2 1.345 8 1.301 6 1.259 2 1.218 4

3.833 1 3.645 0 3.469 8 3.305 9 3.152 6 3.008 9 2.874 3 — 2.628 5 2.516 4 2.410 5 2.310 9 2.217 2 2.126 8 2.042 1 1.962 4 1.886 7 1.815 0 1.747 2 1.682 2 1.620 3 1.561 6 1.506 3 1.452 3 1.401 4 1.353 0 1.306 8 1.262 7 1.220 3 1.179 6

3.703 4 3.514 1 3.338 4 3.175 1 3.023 4 2.882 2 2.750 5 — 2.511 8 2.404 4 2.303 0 2.208 7 2.119 4 2.036 0 1.955 9 1.882 2 1.811 8 1.745 5 1.683 0 1.623 7 1.567 1 1.514 0 1.464 0 1.415 5 1.370 1 1.326 4 1.284 5 1.244 6 1.206 6 1.169 8

— — — — — — — — — — — — — — — 1.847 0 1.777 2 1.713 0 1.651 0 1.588 4 1.537 8 1.484 8 1.434 9 1.387 0 1.341 8 1.297 6 1.255 5 1.215 5 1.177 2 1.140 5

3.522 6 3.335 6 3.162 1 3.001 2 2.851 6 2.712 4 2.582 4 — 2.348 0 2.241 5 2.141 8 2.048 7 1.961 1 1.877 9 1.798 9 1.727 2 1.657 4 1.592 4 1.530 3 1.472 7 1.417 4 1.364 1 1.315 3 1.267 7 1.222 3 1.179 0 1.137 9 1.098 6 1.061 0 1.025 0

L Absorption edges Li 3.238 2 3.084 3 2.933 3 2.788 8 2.633 0 2.502 7 2.389 8 2.274 5 2.172 5 2.083 4 1.978 9 1.890 8 1.813 1 1.737 6 1.668 4 1.601 1 1.538 2 1.478 7 1.422 7 1.369 3 1.319 4 1.270 9 1.227 1 1.181 4 1.140 1 1.098 6 1.060 8 1.025 0 0.990 1 0.955 7

Lii 3.494 8 3.322 4 3.155 0 2.994 9 2.823 0 2.682 5 2.553 2 2.430 6 2.312 7 2.202 2 2.100 3 2.009 4 1.923 1 1.842 4 1.765 8 1.694 4 1.625 9 1.560 8 1.501 1 1.443 6 1.390 0 1.337 2 1.288 6 1.241 5 1.197 5 1.153 1 1.112 6 1.074 4 1.036 5 1.001 3

Liii 3.672 9 3.500 7 3.321 5 3.169 5 2.996 4 2.851 6 2.719 0 2.593 3 2.472 3 2.361 1 2.257 9 2.164 1 2.077 1 1.994 5 1.918 9 1.844 6 1.774 9 1.709 6 1.648 4 1.590 3 1.535 3 1.482 4 1.431 4 1.385 2 1.340 4 1.295 7 1.254 3 1.215 3 1.177 2 1.141 4

Excitation of X-rays

77 Ir 78 Pt 79 Au 80 Hg 81 Tl 82 Pb 83 Bi 84 Po 85 At 86 Rn 87 Fr 88 Ra 89 Ac 90 Th 91 Pa 92 U 93 Np 94 Pu +

Interpolated values.

1.540 9 1.499 5 1.459 6 1.421 6 1.384 8 1.349 9 1.316 1 1.282 9 — — — 1.167 2 — 1.115 1 1.090 8 1.067 1 1.042 8 1.022 6

1.284 5 1.242 9 1.202 7 1.164 1.127 7 1.092 4 1.058 6 — — — — 0.907 4 — 0.854 5 0.829 5 0.805 1 0.780 9 0.759 1

1.362 5 1.324 3 1.287 7 1.252 6 1.218 8 1.186 5 1.155 4 1.125 5+ 1.096 7+ 1.069 0+ 1.042 3 1.016 6 0.991 8+ 0.967 9 0.944 8+ 0.922 6 0.901 1 0.880 3

1.351 3 1.313 0 1.276 4 1.241 2 1.207 4 1.175 0 1.143 9 1.113 9 1.085 0+ 1.057 2+ 1.030 5 1.004 7 0.979 9+ 0.956 0 0.932 8 0.910 6 0.889 1 0.868 3

1.157 8 1.119 9 1.083 5 1.048 7 1.015 1 0.982 9 0.952 0 0.922 0 0.893 5+ 0.866 1+ 0.839 4 0.813 8 0.789 0+ 0.765 2 0.742 3 0.720 0 0.698 5 0.677 7

1.179 6 1.142 2 1.106 5 1.072 2 1.039 2 1.007 5 0.976 9 0.947 5 — — — 0.840 7 — 0.792 6 0.769 9 0.748 0 0.726 7 0.706 2

1.140 9 1.103 9 1.067 9 1.033 6 1.000 6 0.969 1 0.938 6 0.909 1 0.881 4+ 0.854 4+ 0.827 9+ 0.802 7 0.778 2+ 0.754 8 0.732 3 0.710 3 0.689 2 0.668 7

1.135 3 1.102 0 1.070 2 1.039 8 1.010 3 0.982 2 0.955 2 0.929 4 — — 0.858 0.835 4 — 0.793 5 0.773 7 0.754 7 0.736 2 0.718 5

1.105 9 1.072 4 1.040 4 1.009 9 0.980 6 0.952 6 0.925 6 0.899 6 — — — 0.806 3 — 0.764 7 0.745 2 0.726 3 0.708 1 0.690 7

0.990 9 0.958 0 0.926 5 0.896 5 0.867 5 0.839 7 0.813 1 0.787 5 0.762 9 0.739 3+ 0.716 5+ 0.694 6 0.673 5+ 0.653 1 0.633 6+ 0.614 8 0.596 5 0.578 9

0.924 25 0.894 05 0.863 78 0.835 31 0.807 87 0.781 53 0.756 49 0.732 19 0.709 15 0.686 75 0.665 37 0.644 61 0.624 8 0.606 1 0.587 5 0.569 7 0.553 1 0.536 6

0.967 00 0.934 84 0.902 77 0.877 90 0.843 55 0.815 52 0.789 10 0.763 77 0.738 73 0.715 29 0.692 90 0.671 14 0.650 0 0.630 1 0.610 6 0.591 9 0.574 2 0.557 1

4–9

1.106 0 1.073 1 1.040 3 1.009 4 0.979 68 0.951 12 0.924 59 0.897 61 0.872 34 0.848 45 0.825 29 0.802 84 0.781 6 0.761 5 0.741 4 0.722 3 0.703 9 0.686 4

4–10

X-ray analysis of metallic materials

Table 4.5

CRYSTALS FOR PRODUCING MONOCHROMATIC X-RAYS

Properties of crystal

Properties of reflection Crystal

Reflection

Spacing Å

Peak intensity

Breadth

Crystal imperfection

Stability

Mechanical properties

Special uses

β alumina

0 002 0 004

11.24 5.62

Weak Weak–medium

Moderate

Great

Perfect

Hard, brittle

Mica

001 004

10.1 2.53

Weak

Small

Fair

Flexible, easily cleaved

Gypsum

020

7.60

Medium–strong

Very small

Poor

Soft, flexible

Pentaerythritol

002

4.40

Very strong

Moderate

Negligible for selected specimens Good specimens hard to find Great

For long wavelengths, but usable crystals hard to obtain For point-focusing devices; exhibits irradiation effects

Poor

Soft, easily deformed

1 011

3.35

Weak–medium

Very small

Negligible

Perfect

Can be elastically bent

Potassium bromide

200

3.29

Medium–strong

Moderate

Negligible

—

Fluorite

111 220

3.16 1.94

Medium–strong Very strong

Moderate

Small

Slightly deliquescent Perfect

Moderately hard

Urea nitrate

002

3.14

Strong

Very large

Very great

Very poor

Very easily deformed

Calcite

200

3.04

Medium

Small

Negligible

Perfect

Moderately soft

Rock salt

200

2.82

Medium–strong

Large

Great

Aluminium

111

2.33

Very strong

Moderate to large

—

Slightly deliquescent Good

Diamond Lithium fluoride

111 200

2.05 2.01

Weak Very strong

Very small Small-moderate

Negligible Negligible

Perfect Perfect

Graphite

002

3.35

Very strong

Small

Negligible

Perfect

Can be plastically bent in warm supersaturated saline Soft, can be seeded and grown to shape, then plastically shaped at room temperature Very hard Hard, can be plastically bent at high temperature Easily shaped

Quartz

For small-angle scattering; focusing long wavelengths General purposes; exhibits irradiation effects For small-angle scattering; focusing — For eliminating harmonics; general purposes; short wavelengths For large specimens; soon decays For small-angle scattering; isolation of α1 or α2 For focusing For focusing; diffuse scattering

For eliminating harmonics For focusing; diffuse scattering; general purposes Widely used for monochromators on diffractometers

X-ray techniques

4–11

4.3 X-ray techniques The techniques, involving X-rays, which are commonly employed in the investigation of metallic materials are X-radiography, X-ray diffraction analysis, X-ray small angle scattering, X-ray fluorescence, and X-ray photoelectron spectroscopy (XPS). The basis of these techniques and their metallurgical applications are given in Table 4.6.

4.3.1

X-ray diffraction

Diffraction effects are produced when a beam of X-rays passes through the three-dimensional array of atoms which constitutes a crystal. Each atom scatters a fraction of the incident beams, and if certain conditions are fulfilled then the scattered waves reinforce to give a diffracted beam. These conditions are expressed by the Bragg equation: nλ = 2dhkl sin θ where n is an integer, λ the wavelength of the incident beam, dhkl the interplanar spacing of the hkl planes, and θ the angle of incidence on the hkl planes. A randomly oriented single crystal irradiated with monochromatic radiation is unlikely to have any planes at the correct orientation to satisfy the Bragg equation. Hence when single crystals or more usually individual grains are examined they are either rotated in the X-ray beam or continuous wavelength (white) radiation is used as in the Laue technique. In the case of powder or fine grained material sufficient crystals are irradiated for some to have planes oriented to satisfy the Bragg equation. The number is increased by rotating or translating the specimen. A series of concentric diffraction cones with semiapex angle 2θ are produced which intersect a narrow film placed around the specimen in a series of arcs, which is the diffraction pattern. This is either recorded on film or scanned with an electronic radiation detector. 4.3.1.1

Experimental methods

The experimental methods of X-ray diffraction and their common metallurgical applications are reviewed in Table 4.7. Film methods are included (only to illustrate particular applications) as they have virtually been replaced by computer automated diffractometer methods. Fully automated diffractometer systems are now commercially available for all XRD applications, ranging from routine rapid qualitative and quantitative analysis to detailed single crystal structure analysis. With these systems, diffraction data are collected and processed automatically, peaks are identified and 2θ, d, peak intensity and width and integrated intensities are measured. The data are stored in preparation for further evaluation specific to the applications, for example phase analysis and quantification, crystallite size and strain determination from peak breadths and crystallographic evaluations. These are obtained from the determination of unit cell, indexing of peaks and lattice parameter measurements. The speed of data collection can be increased some 50 to 100 fold by replacing the scintillation counter previously supplied as the standard detector on conventional diffractometers with a Position Sensitive Detector (PSD). With this conversion a complete diffraction pattern can be obtained in a few minutes. The increase in speed is achieved by simultaneously recording diffractions over a 2θ range of about 12◦ as compared with only, typically, 0.05◦ with a conventional detector. There are two different modes of operation, stationary and scanning. In the stationary mode diffractions over a 2θ range of about 12◦ are simultaneously recorded, similar to, but much faster than recording on film. In the scanning mode, the detector is moved round the 2θ circle like a conventional detector. The scanning mode is often called Continuous Position Sensitive Proportional Counter (CPSC).10 Particular applications of each mode are: Stationary mode—small angle scattering phase transition residual stress determination Scanning mode—to produce a rapid throughput of specimens phase transition where a complete diffraction pattern is needed use with a high temperature chamber

4–12

X-ray analysis of metallic materials

Table 4.6

METALLURGICAL INVESTIGATION TECHNIQUES INVOLVING X-RAYS

Technique

Principle of technique

Metallurgical applications

X-radiography

Sample placed between divergent X-ray beam and detector (usually a photographic film). Dark areas on negative show regions of low X-ray absorption (i.e. low density in specimen such as cracks) and vice versa Beam of X-rays of specific wavelength (λ) diffracted at certain angles (θ) by crystal planes of appropriate spacing (d) which satisfies the Bragg equation, i.e. nλ = 2d sin θ Angular position of diffraction peaks, shape of peaks, and intensity peaks give information on crystal structure and physical state

Crack and other defect detection in castings, welded fabrications, etc. Very widely used inspection and quality assurance technique

X-ray diffraction

Small angle X-ray scattering

X-ray fluorescence

X-ray photo-electron spectroscopy (XPS)

Large regions (e.g. 10 → 10 000 Å) of inhomogeneous electron density distributions (e.g. vacancy clusters or Guinier–Preston zones) cause scattered radiation pattern near main beam Electrons ejected from inner shells of atoms cause emission of X-rays (fluorescence) characteristic of atomic species— exciting radiation may be photons (γ and X-rays), positive ions or electrons. Fluorescent X-rays analysed by wavelength or energy dispersive spectrometers. Electrons from inner shells of atoms are ejected by X-radiation of specific wavelength. Energy of ejected electron measured by spectrometer gives information on binding energy of electron shells

Identification and quantitative analysis of crystalline chemical compounds (phase analysis and quantification, e.g. retained austenite determination) Crystal structure determination Residual macro- and micro-stress analysis and crystallite size Texture (preferred grain orientation) measurement Detection of cold work and imperfections (by stacking faults, etc.) N.B. All results obtained from thin (typically 50 µm) surface layer Determination of size, shape and composition (in terms of electron density) of Guinier–Preston zones, vacancy clusters, etc. in metallic transmission technique therefore limited to thin samples—typically 10–50 µm Elemental analyses Na to U routine. C, N, O, F require specifically designed equipment. Hence used for routine analysis of composition of ores, semi-finished and finished metals and their alloys. Quantitative analysis normally employs calibrated standards. N.B. Results are obtained from surface layer of approx. 2 µm thickness Qualitative and quantitative analysis of surface layers, XPS signals typically come from depths of less than 50 Å. Elemental analysis He to U including, C, N, O, F. Information on the state of bonding of analysed elements

X-ray techniques

Table 4.7

4–13

X-RAY DIFFRACTION TECHNIQUES

Experimental technique

Description

Metallurgical applications

Single crystal Single-crystal camera Weissenberg camera Precession camera Automated single crystal diffractometer

Small single crystal or metallic grains oriented on a two-axis goniometer with a prominent crystallographic axis along the goniometer axis. Crystal rotated in a monochromatic or filtered X-ray beam. Produces a spot diffraction pattern related to the symmetry of the crystal.

Determination of unit cell dimensions

Back reflection or transmission Laue camera

Crystal structure analysis (i.e. determination of atom positions and thermal vibrations) etc.

Stationary sample, single crystal or polycrystalline irradiated with white, continuous wavelength radiation. The back scattered, (back reflection) and forward scattered transmission diffraction patterns are recorded on flat films perpendicular to the incident beam. The symmetry of the diffraction pattern is related to the symmetry of the crystal in the beam direction. The transmission pattern can be electronically recorded on large-area detectors.

Texture studies on deformed (worked) metallic materials

Cylindrical compact of powder or wire sample (approx. 5 mm long and 0.5–1 mm in diameter) rotated (to avoid texture effects) in monochromatic X-ray beam. Diffraction cones intersect narrow cylindrical film (coaxial with sample) to give line spectrum.

Phase identification Quantitative analysis of mixtures of chemical compounds usually with aid of calibrated standards Composition of alloy phases by correlation of lattice parameters with varying constituent elements

Glancing angle camera

Diffraction spectra obtained by irradiating surface or edge sample at glancing angle and recording one half of total diffraction spectra (other half absorbed by sample) on photographic film.

Examination of bulk samples, simultaneous detection and analysis of surface film (e.g. oxides) and parent metal substrate

Guinier camera

Flat sample positioned on the circumference of the camera is irradiated by a monochromatic or filtered beam of X-rays diverging from a film around the circumference.

Improved resolution and inherently low background aids identification and comparison of specimens with small differences of structure. Diffractions at low Bragg angles (high d values) can be studied

Diffractometry

Flat sample irradiated by a diverging beam of monochromatic or filtered X-rays. Detector rotated at twice the angular speed of the specimen to maintain the Bragg-Brentano focusing conditions.

Phase identification and quantification (usually with calibrated standards) Studies of crystal imperfections, e.g. stacking faults, microstrain, etc. by detailed measurement of spectra profiles With appropriate attachments the following are possible: Residual stress measurement Texture (preferred orientation) determination Thermal expansion parameter measurement Phase transition monitoring

Powder or fine-grained sample Debye-Scherrer camera

Confirmation of crystal symmetry Determination of crystal orientation

4–14

X-ray analysis of metallic materials

4.3.1.2 Accessory attachments for diffractometers The range of applications of diffractometers is increased by accessory attachments for particular applications. These include: • Specimen holder with rotation and translation scanning for analysing coarse grained solid and

powder samples.

• Specimen changers for the sequential analysis of up to 40 specimens. • Chambers for high and low temperature measurement. Temperature control and data collection

can be automated and with a CPSC, rapid data collection system, a 3D display of 2θ and intensity against temperature can be obtained in a few hours. • Preferred orientation is measured with a Texture Attachment. Intensity data are automatically collected and results displayed as either a conventional pole figure or as orientation distribution functions (ODF).11 • Attachments for quantitatively measuring residual stress. • Diffraction patterns can be obtained from thin surface films by irradiating the surface at a shallow angle to increase the effective thickness of the film.

4.3.2

Specific applications

4.3.2.1 Phase identification and quantitative measurements Phase identification and quantification depend on the accurate measurement of the interplanar (d) spacings and relative intensities of the diffractions in a diffraction pattern. For routine phase identification the observed d spacings and relative intensities are compared with standard X-ray data listed in the X-ray Powder Diffraction File (PDF) published by the International Center for Diffraction Data (ICDD), formerly the Joint Committee on Powder Diffraction Standards (JCPDS).12 The file contains data for over 90 000 materials and is regularly updated. It can be obtained on computer files for computer search and match procedures which have virtually replaced the previously used manual procedures. Interactive graphic search programmes are commonly employed. After entering any chemical information, the PDF files are automatically searched and possible matches listed. These are subsequently subtracted from the unknown pattern and the residual displayed for further analysis. With a mixture, the intensities of the file patterns can be varied to produce the best fit. In practice, the success in analysing a mixture often depends on the availability of additional information. For example chemical analysis of individual phases or particles carried out by EDAX in a SEM, or a manual separation of phases for separate XRD analysis. The patterns can then be subtracted from the unknown pattern. Quantitive determinations are carried out by comparing integrated intensities of selected diffractions. In order to allow for absorption, powders are usually analysed by adding a known fraction of standard calibrating powder and comparing the intensities of a diffraction from the component to be analysed with the intensity of a diffraction from the internal standard. Methods are thoroughly discussed in references 3, 9, 10 and 11. X-ray diffraction analysis is used for the identification of atmospheric pollutants. Examples include silica13 and asbestos particles collected on multipore filters. The method requires the taking of an X-ray scan directly from the filter and determining the amount of pollutant by comparison with calibration curves prepared from similar filters containing known amounts of the pollutants. As little as 2 µg cm−2 of silica can be detected by this method. Most metallurgical samples of interest, however, are solid and cannot be analysed by the above methods. In this case either a calibration curve is constructed from well characterised standards, showing, for precisely defined diffraction conditions, the variation in intensity of a particular diffraction with percentage of the phase to be analysed. Alternatively, theoretical intensities are calculated for diffractions from each phase to be analysed and the ratios of their amounts determined from the observed intensity ratios. The method is illustrated below for the determination of retained austenite in steels but can be extended to other systems. X-ray data for calculating diffraction intensities are listed in Tables 4.8, 4.9, 4.10 and 4.11.

X-ray techniques Table 4.8

ANGLES BETWEEN CRYSTALLOGRAPHIC PLANES IN CRYSTALS OF THE CUBIC SYSTEM

(HKL)

(hkl)

Values of α, the angle between (HKL) and (hkl)

100

100 110 111 210 211 221 310 311 320 321 110 111 210 211 221 310 311 320 321 111 210 211 221 310 311 320 321 210 211 221 310 311 320 321 211 221 310 311 320 321

0◦ 45◦ 54◦ 44′ 26◦ 34′ 35◦ 16′ 48◦ 11′ 18◦ 26′ 25◦ 14′ 33◦ 41′ 36◦ 43′ 0◦ 35◦ 16′ 18◦ 26′ 30◦ 19◦ 28′ 26◦ 34′ 31◦ 29′ 11◦ 19′ 19◦ 6′ 0◦ 39◦ 14′ 19◦ 28′ 15◦ 48′ 43◦ 5′ 29◦ 30′ 61◦ 17′ 22◦ 12′ 0◦ 24◦ 6′ 26◦ 34′ 8◦ 8′ 19◦ 17′ 7◦ 7′ 17◦ 1′ 0◦ 17◦ 43′ 25◦ 21′ 19◦ 8′ 25◦ 9′ 10◦ 54′ 70◦ 54′ 0◦ 32◦ 31′ 25◦ 14′ 22◦ 24′ 11◦ 29′ 79◦ 44′ 0◦ 17◦ 33′ 15◦ 15′ 21◦ 37′ 65◦ 0◦ 23◦ 6′ 14◦ 46′ 0◦ 15◦ 30′ 72◦ 45′ 0◦ 64◦ 37′

110

111

210

211

221

310

311

320 321

221 310 311 320 321 310 311 320 321 311 320 321 320 321 321

4–15

90◦ 90◦ 63◦ 65◦ 70◦ 71◦ 72◦ 56◦ 57◦ 60◦ 90◦ 50◦ 54◦ 45◦ 47◦ 64◦ 53◦ 40◦ 70◦ 75◦ 61◦ 54◦ 68◦ 58◦ 71◦ 51◦ 36◦ 43◦ 41◦ 58◦ 47◦ 29◦ 33◦ 33◦ 35◦ 49◦ 42◦ 37◦ 29◦ 77◦ 27◦ 42◦ 45◦ 42◦ 27◦ 84◦ 25◦ 40◦ 37◦ 32◦ 75◦ 35◦ 41◦ 36◦ 22◦ 27◦ 77◦ 21◦ 69◦

26′ 54′ 32′ 34′ 27′ 19′ 42′

90◦

46′ 44′

71◦ 73◦ 76◦ 63◦ 90◦ 66◦ 55◦

52′ 46′ 58′ 54′ 32′ 2′ 52′ 44′ 35′ 31′ 19′ 53′ 52′ 5′ 49′ 3′ 36′ 45′ 13′ 33′ 16′ 48′ 24′ 37′ 12′ 24′ 16′ 27′ 17′ 18′ 1′ 53′ 51′ 17′ 52′ 19′ 19′ 6′ 11′ 19′ 37′ 11′ 9′ 47′ 4′

90◦ 90◦ 74◦ 30′ 90◦ 34′ 13′ 22′ 26′

90◦ 90◦ 77◦ 5′

54′ 28′

78◦ 41′ 67◦ 48′

79◦ 6′

90◦ 78◦ 54′ 79◦ 58′ 72◦ 53◦ 56◦ 53◦ 45◦ 66◦ 41◦ 53◦ 48◦ 47◦ 58◦ 60◦ 55◦ 40◦ 83◦ 38◦ 58◦ 59◦ 49◦ 36◦

1′ 8′ 47′ 24′

36◦ 55◦ 52◦ 40◦ 85◦ 50◦ 54◦ 49◦ 46◦ 35◦ 85◦ 31◦ 73◦

52′ 6′ 8′ 29′ 9′ 29′ 10′ 52′ 11′ 23′ 45′

8′ 55′ 18′ 11′ 7′ 55′ 30′ 33′ 12′ 44′ 57′ 12′ 50′ 40′ 42′

24′

90◦ 66◦ 79◦ 63◦ 64◦ 82◦ 60◦ 61◦ 60◦ 65◦ 75◦ 75◦ 63◦ 49◦ 90◦ 63◦ 65◦ 72◦ 68◦ 57◦ 53◦ 67◦ 74◦ 47◦ 90◦ 62◦ 65◦ 61◦ 62◦ 48◦ 90◦ 38◦ 81◦

25′ 29′ 26′ 54′ 15′ 15′ 26′

78◦ 28′ 90◦ 72◦ 39′ 73◦ 34′

90◦

54′ 2′ 45′ 5′ 6′

68◦ 70◦ 70◦ 74◦ 82◦ 90◦ 83◦ 56◦

9′ 13′ 32′ 12′ 35′

75◦ 83◦ 80◦ 82◦

37′ 4′ 27′ 18′ 41′

83◦ 83◦ 84◦ 79◦ 63◦

37′ 57′ 14′ 21′ 33′

90◦

8′ 35′ 45′ 28′

72◦ 79◦ 84◦ 53◦

33′ 1′ 58′ 44′

84◦ 16′ 90◦

58′ 17′ 5′ 31′ 9′

84◦ 75◦ 71◦ 67◦ 53◦

47′ 28′ 12′ 23′ 37′

13′ 47′

44◦ 25′ 85◦ 54′

90◦ 38′ 8′ 24′ 12′

82◦ 53′ 90◦

30′ 56′

84◦ 42′ 74◦ 30′

59◦ 32′ 85◦ 82◦ 72◦ 58◦ 50◦

12′ 44′ 5′ 45′

90◦ 63◦ 36′ 60◦

4–16

X-ray analysis of metallic materials

Table 4.9 Class of reflection hkl

SYMMETRY INTERPRETATIONS OF EXTINCTIONS*

Condition for non-extinction (n = an integer) h + k + 1 = 2n h + k = 2n h + l = 2n k + l = 2n  % h + k = 2n h + l = 2n k + l = 2n

≎ h, k, l, all even or all odd −h + k + l = 3n h + k + l = 3n

∗

Interpretation of extinction

Symbol of symmetry element

Body centred lattice C-centred lattice B-centred lattice A-centred lattice

I C B A

Face centred lattice

F

Rhombohedral lattice indexed on hexagonal reference system Hexagonal lattice indexed on rhombohedral reference system

R H

From M. J. Buerger, ‘X-ray Crystallography’, John Wiley & Sons, New York, 1942.

Table 4.10

MULTIPLICITY FACTORS FOR POWDER PHOTOGRAPHS

Laue or point-group symmetry ¯ 43m, 43, m3m 23, m3

hkl

hhl

48 2 × 24

24 24

Cubic system 0kl 0kk 24 2 × 12

12 12

hhh

00l

8 8

6 6

Hexagonal and rhombohedral systems

Laue or point-group symmetry

hkil

¯ hh2hl

¯ 0k kl

hki0

¯ hh2h0

¯ 0k k0

000l

¯ 62m, 6mm, 62, 6/mmm ¯ 6/m 6, 6, ¯ 3m, 32, 3m 3,3¯

24 2 × 12 2 × 12 4×6

12 12 12 2×6

12 12 2×6 2×6

12 2×6 12 2×6

6 6 6 6

6 6 6 6

2 2 2 2

Tetragonal system

Laue or point-group symmetry

hkl

hhl

0kl

hk0

hh0

0k0

00l

¯ 42m, 4mm, 42, 4/mmm ¯ 4/m 4, 4,

16 2×8

8 8

8 8

8 2×4

4 4

4 4

2 2

Orthorhombic system

Laue or point-group symmetry

hkl

0kl

h0l

hk0

h00

0k0

00l

mm, 222, mmm

8

4

4

4

2

2

2

Monoclinic system

Laue or point-group symmetry

hkl

h0l

0k0

m, 2, 2/m

4

2

2

Laue or point-group symmetry 1, 1¯

Triclinic system hkl 2

Where the multiplicity is given, for example, as 2 × 6, this indicates two sets of reflections at the same angle but having different intensities.

X-ray techniques Table 4.11

ANGULAR FACTORS

sin2 θ

1 + cos2 2θ sin 2θ

1 + cos2 2θ

2 2 21 3 3 12 4 4 12 5 6 7 8 9 10 12 14 16 18 20

0.0000 0.0003 0.0006 0.0011 0.0019 0.0027 0.0037 0.0049 0.0061 0.0076 0.0109 0.0149 0.0193 0.0243 0.0302 0.0432 0.0581 0.0762 0.0955 0.1170

∞ 57.272 38.162 28.601 22.860 19.029 16.289 14.231 12.628 11.344 9.411 8.025 6.980 6.163 5.506 4.510 3.791 3.244 2.815 2.469

22 12 25 27 12

0.1465 0.1786 0.2133

2.121 1.845 1.622

11.086 8.730 7.027

30

0.2500

1.443

5.774

32 12 35 37 12

0.2887 0.3290 0.3706

1.300 1.189 1.105

4.841 4.123 3.629

40

0.4131

1.046

3.255

42 12 45

0.4564 0.5000

1.011 1.000

2.994 2.828

4.3.2.2

Determination of retained austenite in steel

θ◦ 0 1

1 2

4–17

1 + cos2 2θ

sin θ cos θ

θ◦

sin2 θ

1 + cos2 2θ sin 2θ

∞ 6563 2916 1639.1 1048 727.2 533.6 408.0 321.9 260.3 180.06 131.70 100.31 78.80 63.41 43.39 31.34 23.54 18.22 14.44

45 47 12

0.500 0.5436

1.000 1.011

2.828 2.744

50

0.5868

1.046

2.731

52 21 55 57 21

0.6294 0.6710 0.7113

1.105 1.189 1.300

2.785 2.902 3.084

60

0.7500

1.443

3.333

62 21

65 67 12 70 72 74 76 78 80

0.7868 0.8214 0.8536 0.8830 0.9045 0.9240 0.9415 0.9568 0.9698

1.622 1.845 2.121 2.469 2.815 3.244 3.791 4.510 5.506

3.658 4.071 4.592 5.255 5.920 6.749 7.814 9.221 11.182

81 82 83 84 85 85 21 86 86 12 87 87 21 88 88 12 89 90

0.9755 0.9806 0.9851 0.9891 0.9924 0.9938 0.9951 0.9963 0.9973 0.9981 0.9988 0.9993 0.9997 1.000

6.163 6.980 8.025 9.411 11.344 12.628 14.231 16.289 19.029 22.860 28.601 38.162 57.272 ∞

12.480 14.097 16.17 18.93 22.78 25.34 28.53 32.64 38.11 45.76 57.24 76.35 114.56 ∞

2

sin2 θ cos θ

An important example of quantitative phase analysis by X-ray diffraction is the determination of retained austenite in steels. The method is based on the comparison of the integrated diffracted X-ray intensities of selected (hkl) reflections of the martensite and austenite phases. The necessary formulae and reference data are given below; for more details of the experimental methods the definitive paper by Durnin and Ridal14 should be consulted. The integrated intensity of a diffraction line is given by the equation: I(hkl) = n2 Vm (LP) e−2m (F)2

(4.1)

in which I(hkl) = integrated intensity for a special (hkl) reflection; n = number of cells in cm3 ; V = volume exposed to the X-ray beam; (LP) = Lorentz-Polarisation factor; m = multiplicity of (hkl); e−2m = Debye–Waller temperature factor; F = structure factor (which includes f , the atomic scattering factor.) For n2 V we may substitute V /v2 , in which v is the volume of the unit cell. If the ratio between the integrated intensities of martensite and austenite is denoted by P: P=

Vα vγ2 mα (LP)α eα−2m (Fα )2 I martensite (α) = I austenite (γ) Vγ vα2 mγ (LP)γ eγ−2m (Fγ )2

(4.2)

4–18

X-ray analysis of metallic materials

Each factor is determined from the International Tables9 and depends on the reflection used. A factor G is then determined for each combination of α and γ peaks used; hence: P=

Vα 1 × Vγ G

(4.3)

If α and γ are the only phases present: Vγ =

1 1 + GP

(4.4)

Hence, measurement of the ratio (P) of two diffraction peaks and calculation of the factor G will give the volume fraction of austenite Vγ . The factors involved in the calculation of G for two steels—16.8%Ni–Fe(0.35%C) and NCMV (a Ni–Cr–Mo–V steel with composition wt % 0.43C; 0.31Si; 0.57Mn; 0.009S; 0.005P; 1.69Ni; 1.36Cr; 1.08Mo; 0.24V; 0.11Cu), Mo, Co and Cr radiation and a selection of hkl peaks have been extracted from the ‘International Tables for X-Ray Crystallography’9 by Durnin and Ridal14 and are presented in Table 4.12. These factors may be used to calculate G for different radiations and peaks. The results are presented in Table 4.13. When the alloy compositions are being investigated the factors which make up G must be determined from the International Tables.9 Accuracy obtainable using a diffractometer is in the region of 0.5% for the range 1.5–38 volume percentage of austenite. X-ray diffraction determination accuracy thus compares favourably with other techniques such as metallography, dilatometry and saturation magnetisation intensity methods which are all inaccurate below 10% austenite content. The main source of error in X-ray determination of retained austenite comes from overlapping carbide peaks. The carbides and their diffraction peaks most likely to cause problems are summarised in Table 4.14. 4.3.2.3

X-ray residual stress measurements

Residual stresses can be divided into two general categories, macrostresses where the strain is uniform over relatively large distances and microstresses produced by non-uniform strain over short distances, typically a few hundred Å. Both types of stress can be measured by X-ray diffraction techniques. The basis of stress measurement by X-ray diffraction is the accurate measurement of changes in interplanar d spacing caused by the residual stress. When macrostresses are present the lattice plane spacing in the crystals (grains) change from their stress-free values to new values corresponding with the residual stress and the elastic constants of the material. This produces a shift in the position of the corresponding diffraction, i.e. a change in Bragg angle θ. Microstresses however give rise to non-uniform variations in interplanar spacing which broaden the diffractions rather than cause a shift in their position. Small crystal size also give rise to broadening.

Measurement of Macro-residual stress1,2,3,7,15 The working equation used in most X-ray stress analysis is σφ =

dψ d⊥ 1 E × × d⊥ 1+v sin2 ψ

(1)

where σψ is the surface stress lying in a direction common to the surface and the plane defined by the surface normal and the incident X-ray beam. E and ν are Young’s modulus and Poisson’s ratio respectively, dψ and d⊥ are the interplanar spacings of planes with normals parallel to the surface normal and at an angle ψ to the surface normal. These angles are related to the direction of the incident X-ray beam, as shown in Fig. 4.1. Thus σ can be determined from two exposures, one with the incident beam inclined at θ to the surface to measure d⊥ and the other at an angle ψ the first to measure dψ . This technique is known as the two-exposure technique. Alternatively equation (1) can be rewritten as   1+ν (2) dψ = d⊥ σφ sin2 ψ + d⊥ E

X-ray techniques

Table 4.12

4–19

INTENSITY FACTORS FOR DIFFERENT RADIATIONS AND PEAKS14

Peak Material

Radiation

Factor

16.8%Ni–Fe NCMV 16.8%Ni–Fe 16.8%Ni–Fe Both compositions Both

Mo Mo Co Cr All Mo Co Cr Mo Mo Co Cr Mo Mo Co Cr All All

Bragg angle, θ

16.8%Ni–Fe NCMV 16.8%Ni–Fe 16.8%Ni–Fe 16.8%Ni–Fe NCMV 16.8%Ni–Fe 16.8%Ni–Fe Both Both

Multiplicity, m Lorentz and polarisation (LP) Debye–Waller temp. e−2m Atomic scattering, fo Structure factor, F 1/V 2 (V is volume of unit cell)

α200

α211

14.41 17.73 14.40 17.70 38.67 49.89 53.15 78.05 6 24 29.46 18.84 3.44 2.73 2.81 9.26 0.910 0.869 0.910 0.869 0.908 0.869 0.912 0.869 15.1 13.4 14.7 13.1 10.78 9.14 13.19 11.54 2 2 1.79 × 10−3 kx units

γ200

γ220

11.49 16.32 11.49 16.32 29.99 44.92 39.74 64.65 6 12 47.56 22.55 5.79 2.83 3.29 4.01 0.943 0.889 0.943 0.889 0.941 0.889 0.943 0.889 17.0 14.0 16.6 13.7 12.69 9.84 15.09 12.24 4 4 4.68 × 10−4 kx units

γ311 19.21 19.21 55.85 — 24 15.78 2.96 — 0.847 0.848 0.852 — 12.8 12.5 8.60 — 4

4–20

X-ray analysis of metallic materials

Table 4.13

AUSTENITE DETERMINATION FACTOR G FOR DIFFERENT RADIATIONS AND PEAK COMBINATIONS14

Peak combination Material

Radiation

α200–γ200

α200–γ220

α200–γ311

α211–γ200

α211–γ220

α211–γ311

16.8%Ni–Fe NCMV 16.8%Ni–Fe 16.8%Ni–Fe

Mo Mo Co Cr

2.22 2.23 2.52 1.66

1.36 1.38 1.40 2.50

1.50 1.51 2.15 —

1.16 1.15 1.09 0.17

0.71 0.72 0.61 0.26

0.78 0.78 0.93 —

Table 4.14

INTERFERENCE OF ALLOY CARBIDE LINES WITH AUSTENITE AND MARTENSITE LINES14

Austenite and martensite ‘d’ spacings

Fe3 C

M6 C

V 4 C3

Mo2 C or W2 C

WC

Cr23 C6

Cr7 C3

(200)γ 1.80 Å (200)α 1.43 Å (220)γ 1.27 Å (211)α 1.17 Å (311)γ 1.08 Å

Clear Clear Weak overlap Strong overlap Weak overlap

Clear Strong overlap Strong overlap Clear Medium overlap

Clear Clear Weak overlap Clear Clear

Clear Clear Strong overlap Weak overlap Weak overlap

Clear Weak overlap Weak overlap Weak overlap Clear

Strong overlap Weak overlap Medium overlap Medium overlap Strong overlap

Strong overlap Weak overlap Clear Medium overlap Clear

4–21

Diffr ac

Normal to surface

ted X-ra ys No inc rm a lin l t ed o to hkl su pla rfa n e ce s at an gle c

X-ray techniques

ys

ra

90 ⫺

␧wc

n

de

ci

In

90 ⫺ u

w

u

s⫽0

tX

w

sw

Figure 4.1

This shows that dψ is a linear function of sin2 ψ. The intercept on the d axis gives d⊥ and the slope d⊥ ((1 + v)E)σφ. A positive slope corresponds to a tensile stress and a negative slope to compression. This technique takes advantage of measurements involving a number of dψ values. The determination of interplanar spacings depends on the accurate measurement of the corresponding Bragg angle θ where dhkl =

λ 2 sin θhkl

Back reflection diffractions are used as the highest sensitivity to changes in interplanar spacing are obtained as θ tends to 90◦ . In practice the specimen is rotated through ψ between exposures (or the tube and detector together through the same angle on portable systems for measuring large samples). Due to the limited penetration of commonly used X-ray wavelengths, typically 4 µm for chromium radiation to 11 µm for molybdenum in steel, only surface stresses are measured. Stresses at lower depths in the sample are determined by repeating measurements after removing (electropolishing) layers of known thickness. The measured values are subsequently corrected for changes in stress resulting from the removal of upper layers. Residual stress in small components can be measured on a converted diffractometer by adding a specimen support table with three orthogonal adjustments to permit the surface of the component to be brought into the beam. Fully automated portable systems are available for determining residual stress in large components. Typical accuracy for steel is ±1.5–3.0 × 107 Pa (±1–2 ton in−2 ). MEASUREMENT OF MICROSTRESSES2,16,17

A worked surface gives rise to broadened diffractions due to a combination of microstresses and small crystallite size. An approximate method to separate the two effects is to assume that the breadth is the sum of the separate broadening from each effect.

4–22

X-ray analysis of metallic materials

The relationships between diffraction breadth β and average crystallite size ε and mean stress σ¯ are: 1. Small crystallite alone βs.c. ≃

kλ εhkl cos 

where k is a constant ≃1 and εhkl is the linear dimension perpendicular to the measured hkl plane. 2. Microstresses alone 4σ¯ tan  βM.S. ≃ Ehkl where Ehkl is the elastic constant perpendicular to hkl. When both effects are present, then β = βs.c. + βM.S. λ 4σ¯ tan  = + εhkl cos  Ehkl which can be rewritten in the form: 1 4σ¯ sin  β cos  = + λ εhkl λEhkl If only these defects are present, then a plot of β cos /λ against sin  should be a straight line, with intercept on the β cos /λ axis giving 1/εhkl and the slope 4σ/λE ¯ hkl . For non-cubic materials, for example tetragonal and hexagonal, separate plots should be made for the hk0 and 001 results. In the above formulae, diffraction breadth is measured as either the breadth in radians at half the peak height (HPHW) or as the integral breadth which is the integrated intensity of the diffraction divided by the peak height. It corresponds to the width of a rectangle having the same area and height as the diffraction. The latter is particularly useful in analysing peaks with partially resolved α1 α2 doublets. The measured breadths include the instrumental breadth which is independent of any crystal defects. This can be measured for subsequent subtraction from the measured breadths by running a scan from a defect-free specimen. 4.3.2.4 Preferred orientation Preferred orientation can be represented in two ways, either as a convential pole figure or as an inverse pole figure. A conventional pole figure shows the distribution of a low-index pole—normal to a crystallographic plane, over the whole specimen. With a cubic metal, these are generally constructed for {100}, {110} and {111} planes while for hexagonal metals usually the basal (0001) planes ¯ or {1011} ¯ planes are selected. A high density of poles shows the preferred direction of the {1010} pole with respect to the sample. An inverse pole figure on the other hand shows how the grains are distributed with respect to a particular direction in the sample. Inverse pole figures are usually constructed for the principal directions of the sample, for example the extrusion, radial, and tangential directions in extruded material. The method is rapid, and data for a single direction can be determined from a conventional diffractometer scan taken from a surface which is perpendicular to the required direction. The method is based on the fact that all diffractions recorded on a conventional scan come from planes which are parallel to the surface and their intensities are related to the number of grains (volume of material) which has this orientation. The diffraction intensities are compared to those from a sample having random orientation. These can be either theoretical calculated values or values measured directly from a corresponding scan taken from a sample with random orientation. These relative intensities are called texture coefficients (TC) and are expressed mathematically as TC(hkl) =

1 n

0 I(hkl) /I(hkl) &0 n 0 I(hkl) i(hkl)

where I = measured integrated intensity of a given hkl diffraction I 0 = corresponding intensity for the same hkl diffraction from a random sample n = total number of diffractions measured.

X-ray techniques

4–23

The TChkl values are proportional to the number of grains (volume of sample) which are oriented with an hkl plane parallel to the sample surface. The values can be plotted on a partial sterographic projection and contour lines drawn through the plotted points to produce an ‘Inverse Pole Figure’ for that particular specimen direction. High values show the preferred grain orientation in the specimen direction.

4.3.2.5 Specimen preparation Methods for preparing standard diffractometer specimens are discussed in references 1–3. The most common method is to pack loose powder into a flat cavity in an aluminium specimen holder, taking care not to introduce preferred orientation. Another method is to mix the powder into a slurry and smear some over a glass cover slip. Coarse powders which tend to slide out of the cavity during a scan can be mixed with petroleum jelly or gum tragacanth which themselves give negligible diffraction. A problem when analysing a small amount of material is diffraction and scatter from the specimen holder. These effects are reduced by using a single crystal holder of a low element material which has been polished so the surface is just off a Bragg plane. Most laboratories have developed procedures and specimen holders for non-standard applications. These have been surveyed by D. K. Smith and C. S. Barrett and the results published in ‘Advances in X-ray Analysis’.18 One application is the analysis of air-moisture reactive powders. A simple solution is to fill and seal the powder in thin walled glass quills inside a dry box. After removal from the glove box the quills are mounted in raft fashion across a recessed specimen holder. Sealed cells are also used to avoid handling delicate quills inside a glove box. One such method for lithium compounds was to load the powder inside a glove box into a recessed specimen holder and cover the powder with a flanged 0.001 in. thick aluminium foil dome. Petroleum jelly was lightly smeared round the flange to hold the dome on to the specimen holder and make a moisture seal. Metal samples submitted for XRD analysis are generally metallurgically mounted and polished samples. Due to the limited penetration of the X-ray beam into the sample it is always advisable to first electropolish the surface. Useful electropolishes are listed in Chapter 10, Table 10.4. An electropolish frequently used in preparing specimens for retained austenite determinations is 7 vol% perchloric acid 69 vol% ethanol 10 vol% glycerol 14 vol% water At a voltage of 23 V. Electrolyte maintained at temperature below 12◦ C. EXTRACTION TECHNIQUES

Precipitates that are present at very low concentrations can be concentrated by dissolving away the matrix and collecting the residue. Typical solutions used for extraction are as follows.19 (i) Carbides and certain intermetallic compounds (a) Immersion overnight in 5 or 10% bromine in methanol, or (b) Electrolytically in 10% hydrochloric acid in methanol at a current density of 0.07 A cm−2 . Extraction for 4 h provides sufficient powder residue. The intermetallic phases which can be extracted using these solutions are sigma phase, certain of the Laves phases, Fe3 Mo2 etc. (ii) Gamma prime, eta phase, M(CN) (a) Electrolytically in 10% phosphoric acid in water at a current density of 10 A dm−2 . Time of extraction 4 h. It is sometimes necessary to add a small amount of tartaric acid in order to prevent the formation of tantalum and tungsten hydroxides, or (b) Electrolytically in 1% citric acid plus 1% ammonium sulphate in water. Current density 2 A dm−2 , duration 4 h. For quantitative work this solution is preferred to the phosphoric acid electrolyte. Certain of the Laves phases may also be extracted using either of these solutions. (iii) Precipitates and intermetallic compounds in chromium Immersion overnight in 10% hydrochloric acid. This has been used to extract carbides and borides. Precipitates can also be concentrated by heavily etching a surface to leave them proud of the surface as used to produce replicas in electron microscopy.20

4–24

X-ray analysis of metallic materials

Table 4.15

CRYSTAL GEOMETRY

System

dhkl = Interplanar spacing

V =Vol. of unit cell

Cubic

h2 + k 2 + l 2 1 = d2 a2

V = a3

Tetragonal

1 h2 k2 l2 = 2 + 2 + 2 d2 a a c

V = a2 c

Orthorhombic

k2 h2 l2 1 = 2 + 2 + 2 d2 a b c

V = abc

Rhombohedral* Hexagonal†

(h2 + k 2 + l 2 ) sin2 α + 2(hk + kl + hl)( cos2 α − cos α) 1 = d2 a2 (1 − 3 cos2 α + 2 cos3 α)  2  l2 4 h + hk + k 2 1 + 2 = d2 3 a2 c

√ V = a3 (1 − 3 cos2 α + 2 cos3 α) V =

√ 3 2 a c = 0.866 a2 c 2

Monoclinic

k2 2hl cos β h2 l2 1 + 2 + − = 2 2 2 d2 b a sin β c sin2 β ac sin2 β

V = abc sin β

Triclinic

1 1 = 2 (s11 h2 + s22 k 2 + s33 l 2 + 2s12 hk + 2s23 kl + 2s13 hl) d2 V

√ V = abc (1 − cos2 α − cos2 β − cos2 γ + 2 cos α cos β cos γ)

X-ray techniques

4–25

System

φ = angle between planes h1 k1 l1 and h2 k2 l2

Cubic

h1 h2 + k1 k2 + l1 l2 cos φ = √ 2 [(h1 + k12 + l12 )(h22 + k22 + l22 )]

Tetragonal

(h1 h2 /a2 ) + (k1 k2 /a2 ) + (l1 l2 /c2 ) cos φ = √ {[(h21 /a2 ) + (k12 /a2 ) + (l12 /c2 )][(h22 /a2 ) + (k22 /b2 ) + (l22 /c2 )]}

Orthorhombic

(h1 h2 /a2 ) + (k1 k2 /b2 ) + (l1 l2 /c2 ) cos φ = √ 2 2 {[h1 /a ) + (k12 /b2 ) + (l12 /c2 )][(h22 /a2 ) + (k22 /b2 ) + (l22 /c2 )]}

Rhombohedral∗

(h1 h2 + k1 k2 + l1 l2 ) sin2 α + (k1 l2 + k2 l1 + l1 h2 + l2 h1 + h1 k2 + h2 k1 )( cos2 α − cos α) cos φ = √ {[(h21 + k12 + l12 ) sin2 α + 2(h1 k1 + k1 l1 + h1 l1 )( cos2 α − cos α)][(h22 + k22 + l22 ) sin2 α + 2(h2 k2 + k2 l2 + h2 l2 )( cos2 α − cos α)]}

Hexagonal†

h1 h2 + k1 k2 + 21 (h1 k2 + h2 k1 ) + 43 (a2 /c2 ) · l1 l2 cos φ = √ 2 {[h1 + k12 + h1 k1 + 34 (a2 /c2 )l12 ][h22 + k22 + h2 k2 + 34 (a2 /c2 ) · l22 ]}

Monoclinic

(h1 h2 /a2 ) + k1 k2 sin2 β/b2 ) + (l1 l2 /c2 ) − [(l1 h2 + l2 h1 ) cos β/ac)] cos φ = √ 2 2 2 2 {[h1 /a ) + (k1 sin β/b2 ) + (l12 /c2 ) − (2h1 l1 cos β/ac)][(h22 /a2 ) + (k22 sin2 β/b2 ) + (l22 /c2 ) − (2h2 l2 cos β/ac)]}

Triclinic

cos φ =

dhl k1 l1 · dh2 k2 l2 [s11 h1 h2 + s22 k1 k2 + s33 l1 l2 + s23 (k1 l2 + k2 l1 ) + s13 (l1 h2 + l2 h1 ) + s12 (h1 k2 + h2 k1 )] V2 where s11 = b2 c2 sin2 α s22 = a2 c2 sin2 β s33 = a2 b2 sin2 γ

∗ Rhombohedral axes. † Hexagonal axes, co-ordinates

hkil where i = −(h + k).

s12 = abc2 (cos α cos β − cos γ) s23 = a2 bc (cos β cos γ − cos α) s13 = ab2 c (cos γ cos α − cos β)

4–26

X-ray analysis of metallic materials

4.3.2.6

Formulae and crystallographic data

Formulae for calculating interplanar dhkl spacings from lattice parameters and data for calculating intensities together with other useful information on crystal symmetry are given in Tables 4.8, 4.9, 4.10, 4.11 and 4.15. INTENSITIES

The relative intensities of diffractions recorded on a diffractometer scan are given by the formula 1∝ where



  1 + cos2 2θ sin2 cos θ

1 + cos2 θ

sin2 θ cos θ

|F|2 · T · p · A

 |F|2 · T · p · A = Combined polarisation Lorentz factor (Table 4.11)

F = structure factor involving the summation of scattering from all atoms in the unit cell 2 2 T = temperature factor, e−(B sin θ/λ ) p = multiplicity factor, Table 4.10 A = an absorption factor. For a diffractometer specimen of effectively infinite thickness, A = K/µ where K is a constant and µ the linear absorption coefficient of the sample. A is therefore independent of θ. A criterion for this condition is that the thickness of the specimens is >3.2/µ. Tables 4.16 and 4.17 give values of Mean Atomic Scattering Factors and Mass Absorption Coefficients, respectively.

X-ray techniques

Table 4.16

4–27

MEAN ATOMIC SCATTERING FACTORS*

Sin θ −1 λ Å

0.0

H H−1 He Li Li+1

1.000 2.000 2.000 3.000 2.000

Li−1 Be Be+1 Be+2 B

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

0.811 1.064 1.832 2.215 1.935

0.481 0.519 1.452 1.741 1.760

0.251 0.255 1.058 1.512 1.521

0.130 0.130 0.742 1.269 1.265

0.071 0.070 0.515 1.032 1.025

0.040 0.040 0.358 0.823 0.818

0.024 0.024 0.251 0.650 0.647

0.015 0.015 0.179 0.513 0.510

0.010 0.010 0.129 0.404 0.403

0.007 0.007 0.095 0.320 0.319

0.005 0.005 0.071 0.255 0.254

0.0035 0.0035 0.054 0.205 0.203

4.000 4.000 3.000 2.000 5.000

2.176 3.067 2.583 1.966 4.066

1.743 2.067 2.017 1.869 2.711

1.514 1.705 1.721 1.724 1.993

1.269 1.531 1.535 1.550 1.692

1.033 1.367 1.362 1.363 1.534

0.826 1.201 1.188 1.180 1.406

0.654 1.031 1.022 1.009 1.276

0.516 0.878 0.870 0.855 1.147

0.408 0.738 0.735 0.721 1.016

0.323 0.620 0.618 0.606 0.895

0.257 0.519 0.520 0.508 0.783

0.205 0.432 0.436 0.427 0.682

B+1 B+2 B+3 C C+2

4.000 3.000 2.000 6.000 4.000

3.471 2.757 1.979 5.126 3.686

2.551 2.290 1.919 3.581 2.992

1.962 1.928 1.824 2.502 2.338

1.688 1.707 1.703 1.950 1.910

1.536 1.552 1.566 1.685 1.672

1.410 1.414 1.420 1.536 1.533

1.283 1.278 1.274 1.426 1.429

1.154 1.144 1.132 1.322 1.332

1.028 1.016 0.999 1.218 1.233

0.908 0.896 0.877 1.114 1.131

0.798 0.786 0.767 1.012 1.030

0.698 0.687 0.669

C+3 C+4 N N+3 N+4

3.000 2.000 7.000 4.000 3.000

2.842 1.986 6.203 3.772 2.890

2.487 1.945 4.600 3.227 2.619

2.133 1.880 3.241 2.635 2.306

1.874 1.794 2.397 2.172 2.038

1.697 1.692 1.944 1.869 1.837

1.564 1.579 1.698 1.682 1.690

1.447 1.459 1.550 1.558 1.573

1.335 1.338 1.444 1.461 1.472

1.225 1.219 1.350 1.373 1.375

1.116 1.104 1.263 1.287 1.281

1.012 0.994 1.175 1.199 1.188

0.913 0.893 1.083 1.112 1.097

N−1 O O+1 O+2 O+3

8.000 8.000 7.000 6.000 5.000

6.688 7.250 6.493 5.647 4.760

4.631 5.634 5.298 4.776 4.151

3.186 4.094 4.017 3.771 3.410

2.364 3.010 3.016 2.924 2.745

1.929 2.338 2.356 2.327 2.246

1.694 1.944 1.956 1.948 1.913

1.551 1.714 1.717 1.716 1.701

1.446 1.566 1.567 1.568 1.562

1.352 1.462 1.461 1.463 1.463

1.263 1.374 1.374 1.378 1.382

1.170 1.296 1.296 1.301 1.308

O−1 F F−1 Ne Na

9.000 9.000 10.00 10.00 11.00

7.836 8.293 9.108 9.363 9.76

5.756 6.691 7.126 7.824 8.34

4.068 5.044 5.188 6.987 6.89

2.968 3.760 3.786 4.617 5.47

2.313 2.878 2.885 3.536 4.29

1.934 2.312 2.323 2.794 3.40

1.710 1.958 1.972 2.300 2.76

1.566 1.735 1.747 1.976 2.31

1.46 1.587 1.596 1.760 2.00

1.373 1.481 1.486 1.612 1.78

1.294 1.496 1.399 1.504 1.63

Na+ Mg Mg+2 Al Al+1

10.00 12.00 10.00 13.00 12.00

9.551 10.50 9.66 11.23 10.94

8.390 8.75 8.75 9.16 9.22

6.925 7.46 7.51 7.88 7.90

5.510 6.20 6.20 6.77 6.77

4.328 5.01 4.99 5.69 5.70

3.424 4.06 4.03 4.71 4.71

2.771 3.30 3.28 3.88 3.88

2.314 2.72 2.71 3.21 3.22

2.001 2.30 2.30 2.71 2.70

1.785 2.01 2.01 2.32 2.32

1.634 1.81 1.81 2.05 2.04

1.220

1.322 1.419 1.52 1.524 1.65 1.65 1.83 (continued)

4–28

X-ray analysis of metallic materials

Table 4.16

MEAN ATOMIC SCATTERING FACTORS*—continued

Sin θ −1 λ Å

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

Al+2 Al+3 Si Si−3 Si−4 P S S−1 S−2 Cl

11.00 10.00 14.00 11.00 10.00 15.00 16.00 17.00 18.00 17.00

10.40 9.74 12.16 10.53 9.79 13.17 14.33 15.00 (15.16) 15.33

9.17 9.01 9.67 9.48 9.20 10.34 11.21 11.36 (10.74) 12.00

7.65 7.98 8.22 8.34 8.33 8.59 8.99 8.95 (8.66) 9.44

6.79 6.82 7.20 7.27 7.31 7.54 7.83 7.79 (7.89) 8.07

5.70 5.69 6.24 6.25 6.26 6.67 7.05 7.05 (7.22) 7.29

4.71 4.69 5.31 5.30 5.28 5.83 6.31 6.32 (6.47) 6.64

3.88 3.86 4.47 4.44 4.42 5.02 5.56 5.57 (5.69) 5.96

3.22 3.20 3.75 3.73 3.71 4.28 4.82 4.83 (4.93) 5.27

2.71 2.70 3.16 3.14 3.13 3.64 4.15 4.16 (4.23) 4.60

2.33 2.32 2.69 2.67 2.68 3.11 3.56 3.57 (3.62) 4.00

2.05 2.04 2.35 2.34 2.33 2.69 3.07 3.08 (3.13) 3.47

1.84 1.84 2.07 2.06 2.35 2.66 2.67 (2.71) 3.02

Cl−1 A K K+1 Ca

18.00 18.00 19.00 18.00 20.00

16.02 16.30 16.73 16.68 17.33

12.22 12.93 13.73 13.76 14.32

9.40 10.20 10.97 10.96 11.71

8.03 8.54 9.05 9.04 9.64

7.28 7.56 7.87 7.86 8.26

6.64 6.86 7.11 7.11 7.38

5.97 6.23 6.51 6.51 6.75

5.27 5.61 ' 5.95 5.94 6.21

4.61 5.01

4.00 4.43

3.47 3.90

3.03 3.43

Ca+1 Ca+2 Sc Sc+1 Sc+2 Sc+3 Ti Ti+1 Ti+2 Ti+3 V V+1 V+2 V+3 Cr

19.00 18.00 21.00 20.00 19.00 18.00 22.00 21.00 20.00 19.00 23.00 22.00 21.00 20.00 24.00

17.21 16.93 18.72 18.50 17.88 17.11 19.96 19.61 18.99 18.24 20.90 20.56 19.94 19.19 21.84

14.35 14.40 15.38 15.43 15.27 14.92 16.41 16.55 16.52 16.27 17.23 17.37 17.35 17.11 18.05

11.70 11.70 12.39 12.43 12.44 12.38 13.68 13.64 13.75 13.82 14.39 14.36 14.46 14.54 15.11

9.63 9.61 10.12 10.13 10.18 10.22 11.53 11.53 11.50 11.58 12.15 12.15 12.12 12.19 12.78

8.26 8.25 8.60 8.61 8.64 8.68 9.88 9.98 9.86 9.84 10.43 10.44 10.41 10.38 10.98

7.38 7.38 7.64 7.64 7.65 7.76 8.57 8.56 8.58 8.55 9.05 9.05 9.07 9.04 9.55

6.75 6.75 6.98 6.98 6.98 6.98 7.52 7.52 7.52 7.53 7.95 7.95 7.96 7.97 8.39

Cr+1 Cr+2 Cr+3 Mn Mn+1

23.00 22.00 21.00 25.00 24.00

21.50 20.89 20.15 22.77 22.44

18.20 18.18 17.96 18.88 19.02

15.07 15.18 15.26 15.84 15.79

12.78 12.75 12.82 13.41 13.42

10.99 10.97 10.94 11.54 11.55

9.54 9.56 9.53 10.04 10.04

8.40 8.40 8.41 8.84 8.84

5.39

4.84

4.32

3.83

5.70

5.19

4.69

4.21

6.21 6.22 6.45 6.45 6.45 6.44 6.65

5.70 5.70 5.96 5.96 5.96 5.96 5.95

5.19 5.18 5.48 5.48 5.48 5.49 5.36

4.68 4.68 5.00 5.00 5.01 5.02 4.86

4.53 5.53 4.54 4.56 4.43

7.05

6.31

5.69

5.15

4.70

7.44

6.67

6.01

5.45

4.97

7.85

7.03

6.34

5.75

5.25

X-ray techniques

Mn+2 Mn+3 Mn+4 Fe Fe+1

23.00 22.00 21.00 26.00 25.00

21.84 21.10 20.30 23.71 23.39

19.01 18.80 18.42 19.71 19.85

15.90 15.99 15.97 16.56 16.52

13.38 13.45 13.54 14.05 14.05

11.53 11.50 11.53 12.11 12.12

10.06 10.03 10.00 10.54 10.54

8.84 8.85 8.82 9.29 9.29

Fe+2 Fe+3 Fe+4 Co Co+1

24.00 23.00 22.00 27.00 26.00

22.79 22.06 21.26 24.65 24.33

19.85 19.65 19.28 20.54 20.68

16.62 16.71 16.71 17.29 17.25

14.02 14.08 14.18 14.69 14.70

12.09 12.06 12.09 12.67 12.68

10.56 10.54 10.50 11.05 11.04

9.29 9.30 9.28 9.74 9.74

Co+2 Co+3 Ni Ni+1 Ni+2

25.00 24.00 28.00 27.00 26.00

23.74 23.01 25.60 25.28 24.69

20.69 20.50 21.37 21.52 21.53

17.35 17.44 18.03 17.98 18.08

14.66 14.72 15.34 15.34 15.30

12.66 12.63 13.25 13.25 13.24

11.07 11.04 11.56 11.55 11.58

9.74 9.76 10.20 10.20 10.19

Ni+3 Cu Cu+1 Cu+2 Cu+3 Zn Zn+2 Ga Ga+1 Ga+2 Ge Ge+2 Ge+4 As As+1

25.00 29.00 28.00 27.00 26.00 30.00 28.00 31.00 30.00 28.00 32.00 30.00 28.00 33.00 32.00

23.97 26.54 26.22 25.64 24.93 27.48 26.59 28.43 28.12 26.84 29.37 28.50 27.02 30.32 30.02

21.35 22.21 22.35 22.37 22.20 23.05 23.32 23.89 24.03 23.90 24.73 24.91 24.45 25.58 25.72

18.18 18.76 18.71 18.81 18.91 19.50 19.55 20.35 20.19 20.39 20.99 21.03 21.18 21.74 21.68

15.36 15.98 15.99 15.95 16.00 16.64 16.60 17.29 17.30 17.30 17.95 17.92 18.05 18.61 18.63

13.20 13.82 13.83 13.81 13.77 14.40 14.40 14.98 14.99 14.93 15.57 15.57 15.53 16.16 16.16

11.56 12.07 12.07 12.09 12.07 12.59 12.61 13.11 13.10 13.11 13.63 13.65 13.60 14.16 14.16

10.21 10.66 10.66 10.65 10.68 11.12 11.12 11.59 11.60 11.61 12.06 12.06 12.07 12.54 12.54

As+2 As+3 Se Br Kr

31.00 30.00 34.00 35.00 36.00

29.45 28.75 31.26 32.21 33.16

25.76 25.62 26.42 27.27 28.12

21.77 21.88 22.49 23.24 24.00

18.58 18.61 19.28 19.95 20.62

16.16 16.11 16.75 17.35 17.95

14.18 14.17 14.69 15.22 15.76

12.53 12.53 13.02 13.50 13.98

4–29

8.25

7.39

6.67

6.06

5.53

8.66

7.77

7.01

6.37

5.82

9.08

8.14

7.35

6.68

6.11

9.46

8.52

7.70

7.00

6.40

9.91

8.90

8.05

7.32

6.70

10.33

9.29

8.40

7.64

6.99

10.76

9.68

8.76

7.97

7.29

11.19

10.07

9.11

8.30

7.60

11.62 12.06 12.50

10.46 10.86 11.26

9.47 9.84 10.21

8.63 8.97 9.31

7.91 8.21 8.53 (continued)

4–30

X-ray analysis of metallic materials

Table 4.16 Sin θ −1 λ Å

MEAN ATOMIC SCATTERING FACTORS*—continued

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

Rb Rb+1 Sr Y Zr

37.00 36.00 38.00 39.00 40.00

34.11 33.82 35.06 36.01 36.96

28.97 29.11 29.83 30.68 31.54

24.75 24.70 25.51 26.28 27.04

21.29 21.31 21.96 22.64 23.32

18.55 18.55 19.15 19.76 20.37

16.30 16.30 16.84 17.39 17.94

14.47 14.48 14.96 15.46 15.95

12.94

11.66

10.58

9.65

8.84

13.39 13.84 14.29

12.07 12.48 12.89

10.95 11.32 11.70

9.99 10.34 10.68

9.16 9.48 9.80

Zr+4 Nb Mo Mo+1 Tc Ru Rh Pd Ag Ag+1 Cd In Sn Sb Te I Xe Cs Ba La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho

36.00 41.00 42.00 41.00 43.00 44.00 45.00 46.00 47.00 46.00 48.00 49.00 50.00 51.00 52.00 53.00 54.00 55.00 56.00 57.00 58.00 59.00 60.00 61.00 62.00 63.00 64.00 65.00 66.00 67.00

34.72 37.91 38.86 38.59 39.81 40.76 41.72 42.67 43.63 43.37 44.58 45.53 46.49 47.45 48.40 49.36 50.32 51.27 52.23 53.19 54.15 55.11 56.07 57.02 57.98 58.94 59.91 60.87 61.83 62.79

31.39 32.40 33.25 33.39 34.12 34.98 35.84 36.70 37.57 37.71 38.44 39.31 40.17 41.05 41.92 42.79 43.66 44.54 45.41 46.29 47.16 48.04 48.92 49.80 50.68 51.56 52.45 53.33 54.21 55.10

27.25 27.81 28.57 28.51 29.34 30.12 30.89 31.67 32.44 32.38 33.22 34.00 34.78 35.57 36.35 37.14 37.93 38.72 39.51 40.30 41.09 41.89 42.69 43.48 44.28 45.08 45.88 46.68 47.49 48.29

23.39 24.01 24.69 24.72 25.38 26.07 26.76 27.46 28.16 28.18 28.85 29.56 30.26 30.96 31.67 32.38 33.09 33.80 34.51 35.23 35.94 36.66 37.38 38.10 38.82 39.55 40.27 41.00 41.73 42.46

20.31 20.98 21.60 21.59 22.21 22.83 23.46 24.08 24.71 24.70 25.34 25.97 26.60 22.74 27.87 28.51 29.16 29.80 30.44 31.09 31.74 32.39 33.04 33.69 34.35 35.01 35.66 36.33 36.99 37.65

17.92 18.49 19.04 19.04 19.60 20.16 20.72 21.28 21.85 21.85 22.42 22.99 23.56 24.14 24.71 25.29 25.87 26.46 27.04 27.63 28.22 28.81 29.40 29.99 30.59 31.19 31.79 32.39 32.99 33.59

15.97 16.45 16.95 16.96 17.46 17.96 18.47 18.98 19.50 19.50 20.02 20.53 21.05 21.58 22.10 22.63 23.16 23.69 24.22 24.76 25.30 25.84 26.38 26.92 27.46 28.01 28.56 29.11 29.66 30.21

14.74 15.20

13.31 13.73

12.08 12.46

11.04 11.39

10.13 10.45

15.65 16.12 16.58 17.05 17.52

14.15 14.57 14.99 15.42 15.85

12.85 13.24 13.63 14.02 14.42

11.74 12.10 12.46 12.82 13.19

10.78 11.11 11.45 11.78 12.12

17.99 18.46 18.93 19.41 19.89 20.37 20.86 21.34 21.83 22.32 22.81 23.31 23.80 24.30 24.80 25.30 25.80 26.31 26.81 27.32

16.28 16.71 17.15 17.59 18.03 18.47 18.92 19.36 19.81 20.26 20.71 21.17 21.62 22.08 22.54 23.00 23.46 23.93 24.39 24.86

14.81 15.21 15.61 16.02 16.42 16.83 17.24 17.65 18.07 18.48 18.90 19.32 19.74 20.16 20.58 21.01 21.44 21.87 22.30 22.73

13.56 13.93 14.30 14.67 15.05 15.42 15.80 16.18 16.57 16.95 17.34 17.72 18.11 18.51 18.90 19.29 19.69 20.09 20.49 20.89

12.46 12.80 13.15 13.49 13.84 14.19 14.54 14.90 15.25 15.61 15.97 16.33 16.69 17.05 17.42 17.79 18.16 18.53 18.90 19.27

X-ray techniques

Er Tm Yb Lu Hf Ta W Re Os Ir Pt Au Au+1 Hg Hg+2 Tl Tl+1 Tl+3 Pb Pb+3 Bi Po At Rn Fe Ra Ac Th Pa U

68.00 69.00 70.00 71.00 72.00 73.00 74.00 75.00 76.00 77.00 78.00 79.00 78.00 80.00 78.00 81.00 80.00 78.00 82.00 79.00 83.00 84.00 85.00 86.00 87.00 88.00 89.00 90.00 91.00 92.00

63.75 64.71 65.67 66.64 67.60 68.56 69.52 70.49 71.45 72.42 73.38 74.35 74.14 75.31 74.65 76.27 76.07 75.03 77.24 76.00 78.20 79.17 80.13 81.10 82.07 83.03 84.00 84.97 85.93 86.90

55.98 56.87 57.75 58.64 59.53 60.42 61.31 62.20 63.09 63.98 64.87 65.77 65.88 66.66 66.90 67.55 67.67 67.82 68.45 68.71 69.34 70.24 71.13 72.03 72.93 73.82 74.72 75.62 76.52 77.42

49.10 49.90 50.71 51.52 52.33 53.14 53.95 54.76 55.58 56.39 57.21 58.02 57.96 58.84 58.79 59.66 59.59 59.71 60.48 60.53 61.30 62.12 62.94 63.76 64.58 65.41 66.23 67.06 67.88 68.71

43.19 43.92 44.66 45.39 46.13 46.86 47.60 48.34 49.08 49.83 50.57 51.31 51.35 52.06 52.05 52.81 52.84 52.76 53.56 53.50 54.30 55.05 55.80 56.56 57.31 58.06 58.82 59.57 60.33 61.09

38.31 38.98 39.65 40.32 40.99 41.66 42.33 43.01 43.68 44.36 45.04 45.72 45.70 46.40 46.41 47.08 47.06 47.08 47.77 47.46 48.45 49.14 49.82 50.51 51.20 51.89 52.58 53.27 53.97 54.66

34.20 34.81 35.42 36.03 36.64 37.25 37.87 38.48 39.10 39.72 40.34 40.96 40.97 41.59 41.59 42.22 42.23 42.24 42.85 42.87 43.47 44.10 44.73 45.36 45.99 46.63 47.26 47.90 48.53 49.17

* Condensed from ‘International Tables of X-ray Crystallography’, Kynoch Press, Birmingham, England, 1962. Note: For elements of atomic number 22 or more factors are from Thomas–Fermi–Dirac Statistical Model.

30.76 31.32 31.88 32.44 33.00 33.56 34.12 34.69 35.26 35.82 36.39 36.96 36.97 37.54 37.53 38.12 38.12 38.10 36.69 38.68 39.27 39.85 40.43 41.01 41.59 42.17 42.75 43.34 43.93 44.51

4–31

27.83 28.34 28.85 29.37 29.88 30.40 30.92 31.44 31.96 32.48 33.01 33.53

25.33 25.80 26.28 26.75 27.23 27.70 28.18 28.66 29.14 29.63 30.11 30.60

23.17 23.60 24.04 24.48 24.92 25.36 25.80 26.25 26.70 27.14 27.59 28.04

21.29 21.70 22.11 22.51 22.92 23.33 23.74 24.16 24.57 24.99 25.41 25.83

19.65 20.03 20.40 20.78 21.17 21.55 21.93 22.32 22.70 23.09 23.48 23.87

34.06

31.08

28.50

26.25

24.27

34.60

31.59

28.68

26.68

24.67

35.13

32.08

29.42

27.11

25.07

35.66 36.19 36.73 37.26 37.80 38.34 38.88 39.42 39.96 40.50

32.57 33.06 33.55 34.05 34.55 35.04 35.54 36.05 36.55 37.05

29.87 30.33 30.79 31.25 31.71 32.17 32.64 33.10 33.57 34.04

27.53 27.96 28.38 28.81 29.24 29.67 30.10 30.54 30.97 31.41

25.46 25.86 26.26 26.66 27.06 27.46 27.87 28.27 26.68 29.09

X-ray analysis of metallic materials

4–32

Table 4.17

MASS ABSORPTION COEFFICIENT µ/ρ, CORRECTED FOR SCATTERING*

Radiation

Absorber H

1

He

2

Li

3

Be

4

B

5

C

6

N

7

O

8

F

9

Ne

10

Na

11

Ag Kα = 0.5609 Kβ = 0.4970

Pd 0.5869 0.5205

Rh 0.6147 0.5456

Mo 0.7107 0.6323

Zn 1.4364 1.2952

Cu 1.5418 1.3922

Ni 1.6591 1.5001

Co 1.7902 1.6207

0.371 0.366 0.195 0.190 0.187 0.178 0.229 0.208 0.279 0.245 0.400 0.333 0.544 0.433 0.740 0.570 0.943 0.710 1.308 0.965 1.644 1.198

0.373 0.368 0.197 0.192 0.191 0.181 0.239 0.215 0.295 0.257 0.432 0.356 0.597 0.471 0.821 0.628 1.056 0.790 1.471 1.082 1.857 1.350

0.375 0.370 0.199 0.194 0.196 0.185 0.251 0.224 0.314 0.270 0.464 0.383 0.658 0.515 0.916 0.696 1.187 0.884 1.662 1.218 2.106 1.528

0.381 0.376 0.207 0.200 0.217 0.200 0.307 0.258 0.393 0.327 0.625 0.495 0.916 0.700 1.313 0.980 1.737 1.276 2.463 1.792 3.147 2.275

0.425 0.414 0.348 0.306 0.611 0.493 1.245 0.958 1.965 1.490 3.764 2.815 6.132 4.557 9.337 6.917 12.79 9.459 18.53 13.96 23.96 18.05

0.434 0.421 0.384 0.334 0.716 0.572 1.498 1.149 2.386 1.806 4.603 3.446 7.524 5.603 11.48 8.53 15.72 11.67 22.79 16.91 29.45 21.87

0.446 0.430 0.430 0.369 0.850 0.673 1.823 1.393 2.927 2.213 5.680 4.257 9.311 6.950 14.22 10.60 19.48 14.51 28.23 21.03 36.46 27.19

0.462 0.442 0.490 0.414 1.025 0.804 2.245 1.711 3.628 2.742 7.075 5.310 11.62 8.698 17.76 13.28 24.34 18.19 35.26 26.37 45.50 34.05

Fe 1.9373 1.7565 0.482 0.458 0.569 0.474 1.254 0.978 2.796 2.130 4.545 3.438 8.899 6.697 14.64 11.00 22.39 16.80 30.67 23.02 44.41 33.36 57.24 43.05

Mn 2.1031 1.9102 0.509 0.479 0.673 0.554 1.556 1.209 3.526 2.688 5.757 4.365 11.31 8.542 18.63 14.05 28.48 21.48 39.01 29.43 56.43 42.62 72.65 54.95

Cr 2.2909 2.0848

Ti 2.7496 2.5138

0.545 0.506 0.813 0.661 1.960 1.520 4.474 3.439 7.378 5.614 134.53 11.02 23.95 18.16 36.61 27.76 50.10 38.02 72.39 55.01 93.07 70.83

0.658 0.595 1.261 1.010 3.260 2.533 7.636 5.884 12.58 9.674 24.83 19.08 40.96 31.47 62.52 48.07 85.38 65.72 123.0 94.82 157.5 121.7

X-ray techniques

Mg

12

Al

13

Si

14

P

15

S

16

Cl

17

A

18

K

19

Ca

20

Sc

21

Ti

22

V

23

Cr

24

Mn

25

2.171 1.568 2.671 1.917 3.330 2.379 3.992 2.842 5.014 3.561 5.796 4.109 6.486 4.592 8.238 5.830 9.872 6.984 10.66 7.543 12.08 8.548 13.54 9.585 15.67 11.11 17.39 12.34

2.458 1.774 3.030 2.174 3.783 2.703 4.539 3.234 5.705 4.056 6.599 4.684 7.385 5.238 9.382 6.651 11.24 7.970 12.14 8.608 13.75 9.754 15.41 10.94 17.83 12.67 19.78 14.08

2.794 2.014 3.450 2.474 4.312 3.082 5.179 3.692 6.513 4.824 7.536 5.356 8.435 5.992 10.72 7.611 12.83 9.120 13.86 9.851 15.70 11.16 17.58 12.51 20.34 14.49 22.54 16.08

4.202 3.024 5.208 3.736 6.525 4.672 7.852 5.614 10.43 7.403 11.44 8.172 12.81 9.148 16.26 11.62 19.48 13.93 21.00 15.03 23.65 17.02 26.56 19.06 30.67 22.03 33.94 24.42

32.15 24.26 39.90 30.14 50.06 37.84 60.04 45.47 84.24 57.03 86.38 65.68 95.83 73.06 120.4 92.09 142.5 109.3 151.7 116.7 169.0 130.6 185.8 144.2 210.7 164.3 228.3 179.1

39.49 29.35 48.98 36.55 61.42 45.73 73.58 54.88 92.00 68.74 105.6 79.03 116.9 87.75 146.6 110.3 173.1 130.7 183.8 139.2 204.3 155.3 224.0 171.0 253.1 194.2 273.3 210.7

48.86 36.47 60.53 45.24 75.87 56.75 90.78 68.02 113.3 85.08 129.8 97.70 143.5 108.3 179.6 135.9 211.5 160.6 223.9 170.7 248.0 189.9 271.0 208.4 305.0 235.9 327.8 255.0

69.92 45.65 75.37 56.57 94.41 70.92 112.8 84.90 140.5 106.0 160.6 121.5 177.2 134.4 221.3 168.3 259.7 198.4 274.1 210.3 302.5 233.2 329.2 255.1 368.7 287.5 394.1 309.5

76.55 57.65 94.57 71.35 118.4 89.40 141.1 106.8 175.5 133.2 200.1 152.3 220.2 168.1 274.0 210.0 320.6 246.7 337.1 260.6 370.5 288.0 401.0 313.6 446.7 351.8 57.20 376.6(k)

97.01 73.50 119.6 90.82 149.4 113.7 178.0 135.6 220.8 168.7 251.1 192.5 275.3 211.8 341.5 263.8 398.0 308.9 416.5 325.0 455.3 357.4 489.9 387.3 55.67 431.9 72.59 54.90

124.1 94.60 152.7 116.7 190.8 146.0 226.4 173.7 280.0 215.5 317.4 245.2 346.7 268.9 428.1 333.6 496.6 389.0 516.8 407.3 561.2 445.5 68.38 479.7(k) 70.08 53.30 93.00 70.78

4–33

209.0 161.9 255.9 198.7 318.8 247.9 375.9 293.3 461.5 361.5 518.5 408.1 560.9 443.7 684.6 545.0 783.4 628.2 802.4 649.0 98.45 75.84 116.2 89.57 118.8 91.70 157.4 121.6 (continued)

X-ray analysis of metallic materials

4–34

Table 4.17

MASS ABSORPTION COEFFICIENT µ/ρ, CORRECTED FOR SCATTERING*—continued

Radiation

Absorber Fe

26

Co

27

Ni

28

Cu

29

Zn

30

Ga

31

Ge

32

As

33

Se

34

Br

35

Kr

36

Rb

37

Sr

38

Y

39

Zr

40

Ag Kα = 0.5609 Kβ = 0.4970

Pd 0.5869 0.5205

Rh 0.6147 0.5456

Mo 0.7107 0.6323

Zn 1.4364 1.2952

Cu 1.5418 1.3922

Ni 1.6591 1.5001

Co 1.7902 1.6207

Fe 1.9373 1.7565

Mn 2.1031 1.9102

Cr 2.2909 2.0848

Ti 2.7496 2.5138

19.91 14.14 21.83 15.52 25.18 17.93 26.55 18.93 29.30 20.92 31.04 22.20 33.49 24.00 36.29 26.07 38.31 27.59 41.95 30.29 44.18 31.98 47.51 34.49 50.74 36.95 54.50 39.83 57.66 42.29

22.63 16.11 24.79 17.68 28.59 20.42 30.12 21.54 33.22 23.80 35.16 25.24 37.90 27.27 41.04 29.60 43.28 31.29 47.33 34.32 49.79 36.20 53.47 39.01 57.03 41.74 61.16 44.93 64.61 47.66

25.79 18.41 28.23 20.19 32.53 23.30 34.24 24.58 37.74 27.13 39.90 28.76 42.98 31.04 46.49 33.66 48.97 35.56 53.50 38.96 56.19 41.05 60.26 44.18 64.18 47.22 68.71 50.76 72.45 53.76

38.74 27.92 42.31 30.55 48.62 35.19 51.05 37.03 56.09 40.78 59.12 43.11 63.43 46.39 68.36 50.14 71.69 52.78 77.97 57.61 81.49 60.46 83.32 64.78 92.04 68.92 97.92 73.72 14.94 77.63(k)

254.9 200.2 271.5 215.7 303.5 243.0 40.41 30.53 45.73 34.57 49.49 37.44 54.70 41.34 60.44 45.81 65.14 49.52 72.75 55.26 78.20 59.47 85.88 65.39 93.51 71.30 102.5 78.24 110.7 84.64

303.7 235.6 322.0 251.5 45.99 281.7(k) 49.61 36.91 56.10 41.77 60.69 45.22 66.91 49.90 74.03 55.26 79.72 49.57 88.97 66.55 95.57 71.56 104.9 78.61 114.1 85.63 124.9 93.87 134.8 101.5

362.5 283.9 48.18 301.6(k) 56.15 42.49 61.32 45.82 69.30 51.83 74.92 56.08 82.55 61.85 91.25 68.44 98.19 73.73 109.5 82.31 117.5 88.44 128.8 97.1 140.0 105.6 153.1 115.7 165.1 124.9

53.48 342.8(k) 60.17 45.02 69.88 53.17 76.35 57.31 86.23 64.78 93.15 70.05 102.6 77.20 113.3 85.36 121.8 91.88 135.7 102.5 145.4 110.0 159.2 120.6 172.9 131.2 188.8 143.5 203.3 154.8

67.25 50.61 75.48 56.85 89.03 67.17 95.81 72.28 108.1 81.65 116.7 88.22 128.3 97.14 141.6 107.3 152.1 115.4 169.2 128.6 181.2 137.9 198.2 151.0 214.8 164.0 234.3 179.2 251.9 193.0

85.29 64.56 95.66 72.47 112.7 85.49 121.2 92.01 136.7 103.9 147.3 112.1 161.9 123.3 178.4 136.1 191.3 146.2 212.7 162.7 227.4 174.3 248.2 190.6 268.7 206.7 292.5 225.5 313.9 242.5

109.2 83.16 122.4 93.29 144.1 110.0 154.7 118.2 174.2 133.3 187.6 143.8 205.9 158.0 226.6 174.1 242.6 186.8 269.2 207.6 287.4 222.0 313.2 242.4 338.3 262.4 367.6 285.8 393.6 306.7

184.4 142.6 206.1 159.6 242.1 187.7 259.3 201.3 291.1 226.4 312.5 243.5 341.8 266.7 374.8 293.0 399.8 313.3 441.8 346.9 469.4 369.5 509.2 401.8 547.3 433.0 591.3 469.2 629.3 501.0

X-ray techniques

Nb

41

Mo

42

Tc

43

Ru

44

Rh

45

Pd

46

Ag

47

Cd

48

In

49

Sn

50

Sb

51

Te

52

I

53

Xe

54

61.23 45.09 63.87 47.23 66.43 49.36 69.16 51.65 72.82 54.68 13.32 56.28(k) 14.42 10.20 15.11 10.70 16.14 11.42 16.96 12.00 17.94 12.71 18.54 13.14 20.15 14.29 21.03 14.93

68.49 50.73 71.30 53.07 84.01 55.37 76.88 57.83 14.34 61.10(k) 15.17 10.75 16.41 11.64 17.20 12.21 18.36 13.03 19.29 13.69 20.41 14.50 21.09 14.99 22.91 16.29 23.91 17.01

76.65 57.14 79.63 59.67 14.22 62.14(k) 15.07 64.78(k) 16.37 11.63 17.32 12.31 18.74 13.32 19.63 13.97 20.95 14.91 22.01 15.66 23.29 26.58 24.04 17.14 26.12 18.63 27.24 19.44

16.23 82.02(k) 19.90 14.22 21.57 15.42 22.85 16.35 24.81 17.75 26.22 18.78 28.35 20.31 29.68 21.28 31.65 22.71 33.21 23.85 35.10 25.23 36.21 26.04 39.30 28.29 40.94 29.50

119.5 97.71 145.9 111.9 157.0 120.7 165.2 127.2 178.0 137.3 186.7 144.3 200.2 155.0 207.8 161.3 219.6 170.9 228.2 178.0 238.9 186.8 243.8 191.2 261.7 205.9 269.5 212.7

145.4 109.6 177.2 133.8 190.6 144.1 200.2 151.7 215.5 163.5 225.7 171.6 241.6 184.1 250.5 191.2 264.2 202.2 274.0 210.2 286.4 220.3 291.8 225.0 312.5 241.7 321.2 249.1

177.8 134.8 216.5 164.4 232.5 176.9 243.9 185.9 262.1 200.2 284.0 209.8 292.9 224.7 303.1 233.1 319.1 246.1 330.2 255.3 344.5 267.1 350.1 272.3 374.0 291.9 383.3 300.2

218.7 166.9 265.8 203.1 285.1 218.2 298.6 229.1 320.3 246.3 334.2 257.6 356.5 275.5 368.1 285.2 386.7 300.6 399.2 311.2 415.3 324.9 420.9 330.5 448.3 353.4 458.0 362.5

270.5 207.7 328.3 252.6 351.3 271.0 367.2 283.9 393.1 304.7 409.2 318.1 435.5 339.5 448.4 350.7 469.9 368.7 483.5 380.9 501.5 396.5 506.5 401.2 537.4 428.7 546.8 438.3

336.5 260.5 407.5 316.2 435.1 338.5 453.7 354.0 484.5 379.1 502.9 394.8 533.7 420.4 547.8 433.1 572.0 454.0 586.3 467.5 605.9 485.1 609.2 490.3 643.4 520.6 651.2 530.2

420.9 328.8 508.4 398.3 541.5 425.4 563.0 443.7 599.2 473.9 619.9 492.1 655.4 522.4 670.1 536.4 696.8 560.4 710.6 574.6 730.8 594.0 730.6 597.6 766.8 631.5 642.2(lI ) 639.6(L)

4–35

668.6 534.1 802.1 643.2 847.8 682.6 874.3 707.0 922.2 749.4 944.7 777.1 988.1 812.1 998.4 825.9 102.5 853.9 103.0 864.9 975.4 883.5(lI ) 513.1(lII ) 727.3(lI ) 223.0(lIII ) 447.8(lII ) 243.9 470.1(lII ) (continued)

X-ray analysis of metallic materials

4–36

Table 4.17

MASS ABSORPTION COEFFICIENT µ/ρ, CORRECTED FOR SCATTERING*—continued

Radiation

Absorber Cs

55

Ba

56

La

57

Ce

58

Pr

59

Nd

60

Pm

61

Sm

62

Eu

63

Gd

64

Tb

65

Dy

66

Ho

67

Er

68

Ag Kα = 0.5609 Kβ = 0.4970

Pd 0.5869 0.5205

Rh 0.6147 0.5456

Mo 0.7107 0.6323

Zn 1.4364 1.2952

Cu 1.5418 1.3922

Ni 1.6591 1.5001

Co 1.7902 1.6207

Fe 1.9373 1.7565

Mn 2.1031 1.9102

Cr 2.2909 2.0848

Ti 2.7496 2.5138

22.40 15.90 23.31 16.56 24.76 17.60 26.34 18.74 28.04 19.96 29.29 20.87 30.91 22.04 31.99 22.84 33.72 24.09 34.66 24.79 36.38 26.04 37.83 27.11 39.83 28.57 41.73 30.03

25.46 18.13 26.48 18.87 28.12 20.06 29.91 21.35 31.82 22.73 33.24 23.76 35.05 25.09 36.28 25.98 38.22 27.41 39.26 28.18 41.20 29.60 42.82 30.80 45.06 32.45 47.19 34.02

29.00 20.71 30.16 21.56 32.01 22.93 34.03 24.37 36.21 25.95 37.80 27.11 39.85 28.61 41.22 29.63 43.41 31.24 44.57 32.11 46.75 33.72 48.56 35.06 51.07 36.93 53.46 38.70

43.54 31.39 45.21 33.64 47.94 34.64 50.89 36.82 54.07 39.16 56.37 40.87 59.34 43.08 61.28 44.54 64.43 46.90 66.04 48.14 69.14 50.48 71.69 52.41 75.26 55.11 78.62 57.66

283.1 224.2 290.2 230.7 303.7 242.2 317.9 254.5 332.7 267.5 341.4 275.7 353.4 286.8 358.6 292.5 370.1 303.6 371.8 306.9 324.1(lI ) 316.6(L) 217.7(lII ) 323.2(L) 230.1(lII ) 333.8(L) 241.4(lII ) 292.2(lI )

336.6 261.9 344.3 268.9 359.2 281.6 375.0 295.1 391.1 309.1 400.1 317.7 412.6 329.3 417.2 334.6 326.5(iI ) 345.9(L) 367.1(iI ) 348.1(L) 249.3(lII ) 357.6(L) 259.4(iII ) 309.8(lI ) 135.7 212.9(lII ) 139.0 223.6(lII )

400.6 314.9 408.5 322.5 424.8 336.8 442.0 352.0 459.3 367.6 467.9 376.6 480.4 388.9 411.6(lI ) 393.8(L) 276.0(lII ) 340.7(lI ) 284.1(iII ) 345.1(lII ) 142.9 356.7(lI ) 152.7 242.5(iII ) 165.0 256.0(lII ) 168.9 129.1

476.9 379.1 484.6 387.1 501.8 402.9 519.8 419.7 537.5 436.4 461.4(lI ) 445.5(L) 304.1(iII ) 458.0(L) 314.9(lII ) 390.9(lI ) 153.1 404.7(lI ) 162.3 268.1(lII ) 175.1 281.9(lII ) 186.9 143.4 201.7 155.1 206.1 158.7

567.0 456.9 573.3 464.7 590.7 481.8 512.8(lII ) 499.6(L) 333.4(lII ) 517.3(L) 349.4(lII ) 442.4(lI ) 169.4 458.5(lI ) 174.2 300.5(lII ) 189.4 317.7(iII ) 200.4 154.2 215.8 166.5 229.9 177.8 247.5 191.9 252.7 196.1

671.3 550.1 565.9(lI ) 556.8(L) 359.0(lII ) 574.2(L) 383.1(lII ) 497.1(lI ) 190.6 518.3(lI ) 205.7 337.4(iII ) 211.3 357.0(lII ) 217.1 167.7 235.6 182.4 248.7 193.0 267.3 208.0 284.1 221.7 305.0 238.7 311.0 243.7

669.7(lI ) 659.8(L) 417.0(lII ) 555.1(lI ) 197.7 576.6(lI ) 218.0 374.9(lII ) 240.1 186.2 258.4 200.9 265.2 206.4 272.2 212.1 294.7 230.2 310.3 243.1 332.6 261.3 352.5 277.8 377.3 298.4 383.9 304.2

271.1 212.4 293.3 230.0 322.9 254.1 354.5 279.7 388.3 307.2 415.7 329.9 425.3 338.1 435.2 346.4 468.1 374.0 489.6 392.6 520.7 518.1 547.4 442.6 580.7 471.9 587.9 479.1

X-ray techniques

Tm

69

Yb

70

Lu

71

Hf

72

Ta

73

W

74

Re

75

Os

76

Ir

77

Pt

78

Au

79

Hg

80

43.04 30.93 44.49 32.02 46.41 33.45 53.58 38.66 55.53 40.13 57.57 41.66 59.37 43.03 60.97 44.25 63.40 46.09 65.68 47.82 67.84 49.48 69.90 51.07

48.63 35.11 50.25 36.33 52.40 37.93 60.48 43.82 62.63 45.46 64.88 47.16 66.87 48.69 68.63 50.05 71.32 52.10 73.93 54.02 76.19 55.87 78.46 57.62

55.06 39.91 56.87 41.28 59.26 43.08 68.34 49.75 70.74 51.58 73.24 53.49 75.42 55.19 77.36 56.69 80.33 58.98 83.09 61.12 85.67 63.16 88.15 65.11

80.81 59.37 83.28 61.30 86.60 63.85 99.63 73.60 102.9 76.16 106.3 78.81 109.1 81.12 111.6 83.17 115.6 86.32 119.2 89.24 122.5 91.96 125.7 94.57

117.6 198.6(iII ) 120.6 206.2(lII ) 123.5 215.4(lII ) 124.9 302.4(lII ) 133.3 103.3 142.1 110.3 151.4 117.7 158.5 123.6 168.4 131.7 178.5 140.0 188.3 148.1 198.8 156.8

142.2 230.9(lII ) 145.6 110.8 149.0 113.5 151.0 114.8 160.9 122.6 171.3 130.8 182.0 139.3 190.5 146.1 201.9 155.3 213.7 164.8 224.9 173.9 237.1 183.8

172.7 132.2 176.6 135.4 180.6 138.6 183.3 140.3 195.1 149.7 207.3 159.4 219.8 169.5 229.6 177.5 242.9 188.3 256.3 199.4 269.2 210.0 283.2 221.6

210.6 162.3 215.2 166.1 219.8 169.9 223.6 172.4 237.4 183.5 251.7 195.1 266.3 207.0 277.7 216.4 292.8 229.1 308.2 242.0 322.7 254.3 338.6 267.7

257.8 200.4 263.2 204.9 268.5 209.4 273.8 212.8 290.0 226.1 306.7 239.9 323.6 253.9 336.5 264.9 353.7 279.6 370.9 294.5 386.9 308.6 404.8 324.0

316.9 248.7 323.0 254.0 329.1 259.2 336.5 264.1 355.4 279.9 374.6 296.2 394.0 312.7 408.4 325.3 427.3 342.1 451.9 359.0 463.2 374.8 482.7 392.3

390.6 310.1 397.3 316.1 404.1 322.1 414.7 329.2 436.4 347.9 458.3 366.9 479.9 385.7 495.4 400.2 515.6 419.0 535.0 437.6 552.3 454.7 572.6 474.0

4–37

595.2 486.3 602.2 493.5 609.2 500.7 632.0 516.3 658.0 540.7 682.9 564.8 706.0 588.1 719.6 603.7 736.1 623.5 749.6 641.7 758.7 656.8 773.1 675.9 (continued)

X-ray analysis of metallic materials

4–38

Table 4.17

MASS ABSORPTION COEFFICIENT µ/ρ, CORRECTED FOR SCATTERING*—continued

Radiation

Absorber Tl

81

Pb

82

Bi

83

Po

84

At

85

Rn

86

Fr

87

Ra

88

Ac

89

Th

90

Pa

91

U

92

∗

Ag Kα = 0.5609 Kβ = 0.4970

Pd 0.5869 0.5205

Rh 0.6147 0.5456

Mo 0.7107 0.6323

Zn 1.4364 1.2952

Cu 1.5418 1.3922

Ni 1.6591 1.5001

Co 1.7902 1.6207

Fe 1.9373 1.7565

Mn 2.1031 1.9102

Cr 2.2909 2.0848

71.77 52.52 73.74 54.07 76.60 56.26 79.42 58.44 81.40 60.05 82.38 60.91 83.36 61.77 85.43 63.47 86.25 64.30 88.04 65.81 90.63 67.91 93.18 69.94

80.49 59.23 82.63 60.93 85.78 63.37 88.86 65.77 90.97 67.53 91.97 68.44 92.98 69.35 95.17 91.20 95.95 72.04 97.82 73.66 88.60(iI ) 75.95(L) 89.92(lI ) 78.17(L)

90.37 66.87 92.67 68.75 96.13 71.44 99.50 74.11 101.7 76.00 102.7 76.96 103.7 77.92 106.0 79.91 106.7 80.75 97.75(iI ) 82.48(L) 72.19(iII ) 84.95(L) 74.20(lII ) 87.38(L)

128.4 96.90 131.2 99.31 135.7 103.0 139.9 106.5 122.1(lI ) 108.8(L) 122.5(lI ) 109.8(L) 91.18(lII ) 110.8(L) 95.03(lII ) 113.1(L) 97.50(lII ) 102.1(lI ) 100.3(lII ) 75.07(lII ) 103.2(lII ) 77.4(lII ) 105.9(iII ) 79.70(lII )

206.7 163.6 215.4 171.1 226.8 180.0 237.7 190.3 249.2 200.2 249.0 201.1 258.8 210.1 266.6 217.8 273.6 224.7 278.6 230.6 290.1 240.2 294.9 246.7

245.7 191.2 255.6 199.6 268.3 210.3 280.3 220.6 293.1 231.6 291.9 231.7 302.2 241.1 309.9 248.9 314.6 255.8 320.5 261.1 332.8 272.2 336.7 275.2

292.5 229.9 303.1 239.2 317.4 251.5 330.3 263.1 344.4 275.5 341.5 274.7 351.8 284.8 358.7 292.7 364.3 299.5 366.1 304.0 378.6 316.0 380.7 320.3

348.2 276.8 359.5 287.2 375.0 301.0 388.5 313.7 403.4 327.4 397.8 325.1 407.2 335.5 412.0 342.7 415.4 348.8

414.0 333.6 425.6 344.7 441.5 359.9 454.6 373.4 469.6 388.1 459.8 383.4 466.7 393.1 467.3 398.6

490.3 401.7 500.5 413.0 516.2 429.2 527.3 442.5 540.7 457.5 524.3 448.6 526.0 456.2

576.1 481.9 583.1 492.2 596.0 508.1 602.0 519.5 611.0 533.2

Reproduced by permission from the International Tables for X-ray Crystallography.

Ti 2.7496 2.5138

X-ray results

4–39

4.4 X-ray results 4.4.1

Metal working

Table 4.18

GLIDE ELEMENTS OF METAL CRYSTALS

Low temperatures Structure

Metal

Cubic, face centred

Al Cu Ag Au Pb Ni Cu-Au α-Cu-Zn α-Cu-Al Al-Cu Al-Zn Au-Ag

Cubic, body centred

α-Fe

Glide plane

Tetragonal

Rhombohedral

4.4.2

     ¯ [101]

(111)                 

β-Cu-Zn

Hexagonal, close packed

Glide direction

                 

W Mo K Na

α-Fe-Si, 5% Si  Mg   Zn  Cd   Be  ' Ti Zi

Elevated temperatures

¯ [111] ¯ [111] ¯ [111] ¯ [111] ¯ [111] ¯ [111] ¯ [111] — ¯ [111] — —

(101) (112) (123) (112) (112) (123) (112) — (110) (112)(?) (110)

(0001) or ¯ (1010) ¯ (1010) or (0001)

   

              

¯ [1120]

[001] [001] — ¯ [101]

              

β-Sn (white)

(110) (100) (101) (121)

As Sb Bi Hg

— — ¯ and (111) [101] ¯ (111) [101] (100) — and complex

Glide plane

Glide direction

(100)

[101]

—

—

— — — — (110) (123) (110) (123) — — —

— — — — ¯ [111] ¯ [111] ¯ [111] ¯ [111] — — —

¯ (1011) or ¯ (1012)

(110) — — — — — —

              

¯ [1120]

Most closely packed Lattice plane

Lattice direction

  1(111)  2(100)  3(110) 

        

¯ 1[101] 2[100] 3[112]

—

—

1(101) 2(100) 3(111) — — — — — — —

1[111] 2[100] 3[110] — — — — — — —

¯ (1010) (0001) (0001) (0001) ¯ (1010) ¯ (1010)

¯ [111] — —

1(100) 2(110) 3(101) —

— — — —

¯ 1(110) ¯ 2(111) — —

            

¯ [1120]

1[001] 2[111] 3[100] 4[101] ¯ 1[101] — — —

Crystal structure

Crystal structural data for free elements are given in Table 4.25. The coordination number, that is the number of nearest neighbours in contact with an atom, is listed in column 4 and the distances in column 5. In complex structures, such as α Mn where the coordination is not exact, no symbol is used and the range of distances between near neighbours is given. A co-ordination symbol x in column 4 indicates that each atom has x equidistant nearest neighbours, at a distance from it (in kX-units) specified in column 5. The symbol x, y indicates that a

4–40

X-ray analysis of metallic materials

Table 4.19

PRINCIPAL TWINNING ELEMENTS FOR METALS

Crystal structure B.c.c. F.c.c. Rhombohedral (As, Sb, Bi) All c.p.h. Some c.p.h.

%

Hg Diamond cubic (Ge) Tetragonal (In, β-Sn)

Orthorhombic α-U

        

Twinning plane, K1

Twinning direction, η1

Second undistored plane, K2

Direction, η2

{112} {111} {110} ¯ {1012} ¯ {1121} ¯ {1122} ¯ {1123} {135}

¯

¯

¯

¯

¯

¯

— ¯

¯ {112} ¯ {111} {001} ¯ {1012} {0001} ¯ {112¯ 4} — ¯ {111}



¯

¯

¯

— ¯

0.707 0.707

{111} {301} {101} {130} {172} {112} {121} {176}

¯

¯

¯

¯

¯ {111} ¯ {101} ¯ {101} ¯ {110} {112}



0.707

¯ {111}



Shear

0.299 0.228 0.228 0.329 0.214

From C. S. Barrett and T. B. Massalski, ‘Structure of Metals’, 1980.2

given atom has x equidistant nearest neighbours, and y equidistant neighbours lying a small distance further away. These distances are given in column 5. In complex structures, such as α-Mn, where the co-ordination is not exact, no symbol is used, and the range of distances between near neighbours is given in column 5. The Goldschmidt atomic radii given in column 6 are the radii appropriate to 12-fold co-ordination. In the case of the f.c.c. and c.p.h. metals the radius given is one-half of the measured interatomic distance, or of the mean of the two distances for the hexagonal packing. In the case of the b.c.c. metals, where the measured interatomic distances are for 8-fold co-ordination, a numerical correction has been applied. In some cases, where the pure element crystallises in a structure having a low degree of co-ordination, or where the co-ordination is not exact, it is possible to find some compound or solid solution in which the element exists in 12-fold co-ordination, and hence to calculate its appropriate radius. In a few cases no correction for co-ordination has been attempted, and here the figures, given in parentheses, are one-half of the smallest interatomic distances. It should be emphasised that the Goldschmidt radii must not be regarded as constants subject only to correction for co-ordination and applicable to all alloy systems: they may vary with the solvent or with the degree of ionisation, and they depend to some extent on the filling of the Brillouin zones. Ionic radii vary largely with the valency, and to a smaller extent with co-ordination. The values given in column 8 are appropriate to 6-fold co-ordination, and have been derived either by direct measurement or by methods similar to those outlined for the atomic radii. All are based, ultimately, on the value of 1.32 Å obtained for O2+ ions by Wasastjerna,21 using refractivity measurements. Ionic radii are also affected by the charge on neighbouring ions: thus in CaF2 the fluorine ion is 3% smaller than in KF, where the metal ion carries a smaller charge. It is not possible to give a simple correction factor, applicable to all ions: the effect is specific and is especially marked in structures of low co-ordination. Figures in arbitrary units indicating the power of one ion to bring about distortion in a neighbour (its ‘polarising power’), and indicating the susceptibility of an ion to such distortion (its ‘polarisability’) are given in columns 9 and 10, respectively. The crystal structures of alloys and compounds are listed in Chapter 6, Table 6.1. Other sources of data are references 12 and Pearson22 which is particularly valuable as the variation of lattice parameters with composition as well as structure is given. Structures are generally referred to standard types which are listed in Pearson and in Table 6.2 in Chapter 6. Further information on crystallography can be obtained from International Tables for X-ray Crystallography.9

X-ray results Table 4.20

ROLLING TEXTURES IN METALS AND ALLOYS

Texture

Texture Metal or alloy

1

Face-centred cubic Cu Cu Cu* Cu 70%-Zn 30% Cu 70%-Zn 30%* Cu + 12 at. % Al Cu + 1.5 at. % Al

¯ (110)/[112] ¯ (123)/[121] ¯ (123)/[121] ¯ (110)/[112] ¯ (110)/[112] ¯ (123)/[121] ¯ (110)/[112]

Cu + 3 at. % Au Cu + 29.6 at. % Ni Cu + 49 at. % Ni Ni Au Au + 10 at. % Cu Al Al Al + 2 at. % Cu Al + 1.25 at. % Si Al + 0.7 at. % Mg

Ag Pb + 2 wt. % Sb

Body-centred cubic α-Fe α-Fe Mo W V Fe + 4.16 wt. % Si

∗

4–41

¯ (123)/[121] ¯ (123)/[121] ¯ (123)/[121] ¯ (123)/[121] ¯ (123)/[121] ¯ (110)/[112] ¯ (123)/[121] ¯ (110)/[112] ¯ (110)/[112] ¯ (123)/[121] ¯ (110)/[112]

¯ (110)/[112] (110)/[011]

2

3

1

Body-centred cubic (continued) Fe+35 wt % Co (100)/[011] Fe+35 wt % Ni (100)/[011] β-Brass (100)/[011]

c = 1.5847 a

Be,

¯ (110)/[112]

Ti,

c = 1.5873 a

Zr,

c = 1.5893 a

¯ (110)/[112]

Mg, ¯ (123)/[121]

3

¯ (112)/[110]

¯ (111)/[112] ¯ (111)/[110]

¯ (111)/[112]

¯ (112)/[110] Scatter increases with Si Content

¯ (111)/[112]

¯ (0001)/[1010] (0001) tilted approx. 25–30◦ round RD out of rolling plane; ¯ parallel RD [1010] ¯ (0001)/[1120] (0001) tilted 30–40◦ round RD out of rolling plane; ¯ parallel Rd [1010]

c = 1.6235 a

(0001) parallel rolling plane

c =1.623 a

(0001) parallel rolling plane

β-Co,

¯ (112)/[110]

Straight-reverse rolling treatment. From A. Taylor, ‘X-ray Metallography’, John Wiley and Sons Inc.

2

Hexagonal close-packed

¯ (110)/[112]

(100)/[011] (100)/[011] (100)/[011] (100)/[011] (100)/[011] (100)/[011]

Metal or alloy

Zn,

c = 1.8563 a

Cd,

c = 1.8859 a

Mg + Al ( DNb

 1 773–1 923         1 673–1 873      1 573–1 773     1 423–1 673    1 423–1 623      1 348–1 573 IIIa(i)    1 223–1 448      1 165–1 398        966–1 298           966–1 298

127

(a) This is a representative selection from a larger table of values in reference 127. Nb

V 0 20 40 60 80 100

Nb

W (wt %) 10 20 30 40 50 60 70 80 90

1.6 × 10−2 1.95 × 10−2 2.3 × 10−2 2.8 × 10−2 3.3 × 10−2 3.8 × 10−2

 910.3   343.3    293.1 268.0     263.8  263.8

81.45 22.2 1.97 1.4 × 10−2 7.4 × 10−3 3 × 10−3 1.8 × 10−3 1.0 × 10−3 6.0 × 10−4

 439.6   418.7     376.4     280.1   272.1  252.0     289.3     236.1    228.2

DV ∼ 3–5 × DNb

1 678–2 023

IIa(i)

150

— — — — — — — — —

2 273–2 673

IIa(i)

119

(continued)

Mechanisms of diffusion Table 13.4

CHEMICAL DIFFUSION COEFFICIENT MEASUREMENTS—continued

Element 1 At.%

Element 2 At.%

Nb 5 30 95 0 20 40 60 80 100 —

Zr

(a)

0–5

A cm2 s−1

Q kJ mol−1

D cm2 s−1

— — — ∼10−2 ∼4 × 10−2 ∼10−1 ∼3 × 10−1 ∼2 ∼10 1.2

 217.7 247.0 332.9  ∼196.8    ∼209.3    ∼255.4 301.4    ∼349.5    ∼389.4 349.26

— — — — — — — — — —

13–95

Temp. range K

Method

Ref.

1 173–1 873

IIa

179

1 718–1 963

IIa(i)

115

2 070–2 370

IIIa(ii), p.c.

269

(a) Values taken from smoothed plots of A and Q against composition. Ni

O (a) (b)

(a) Ni + 0.58%Si.

s.c. (b) Ni + 0.48%Si. p.c. Ni Pb 0–3 Ni Pb

10 20 30 40 50 60 70 80 90

7.9×104 9.5×104

309.4 311.5

(s.c.) (p.c.)

12.1

241

—

) Internal oxidation method 623–1 273 IIIb(ii), p.c.

∼0.66

105.9

—

558–593

859◦ C 0.68 1.03 1.52 1.72 1.97 1.40 0.84 0.56 0.50

˜ × 1011 D 950◦ C 1.30 2.01 3.54 5.37 6.61 5.42 3.06 1.90 1.25

1 019◦ C 1.79 3.37 7.39 11.7 16.1 15.6 11.3 5.58 2.30

0.86 1.55 0.55 0.38 0.42 0.18 0.12 0.11

233.8 234.2 217.4 209.3 209.9 202.7 204.0 212.1

— — — — — — — —

1 073–1 473 1 173–1 573

II

1 150◦ C 18.6    30.0    70.4     98.2  1.58  131    104     58.7   35.0

201 202

46

IIa(i)

144

II (EPMA)

320

Ni 15.4 23.1 32.2 42.2 52.1 62.1 73.5 86.6

Pd — — — — — — — —

Ni 0–14.9

Pt 0–100

7.9 × 10−4

180.5 — Figure 13.32

1 316–1 674 1 223–1 573

IIa(ii) IIa(i)

28 218

Ni wt. %

Si 0– < 1 2.3

1.5 0.3(a)

258.3 240

— —

1 393–1 573 1 373–1 523

IIa(ii) IIa(i), p.c.

112 311

Ni — — — — — — — —

Sn 0 1 2 3 4 5 6 7

260   259     257    255 251    253    251   250 14 10 DSn 3.4   3.7 4.2   6.0

—

1 223–1 473         

IIa(i), p.c.

261

— — — —

0.8 2.2 2.2 4.5

2.5 2.5 2.3 2.3 1.9 2.5 2.4 2.3 1014 DNi 1.9 2.2 1.8 3.0

Ni

Ti 0–0.9

0.86

257.1

900–1 200

—

1373

—

1 373–1 573

        IIa(ii)

23 (continued)

13–96 Table 13.4

Diffusion in metals CHEMICAL DIFFUSION COEFFICIENT MEASUREMENTS—continued

Element 1 At.%

Element 2 At.%

Ni Ta2 Ni TaNi TaNi2 TaNi3 TaNi8

Ta

Ni 0–sol. limit

U (γ)

Ni

V 0–16.5

Ni

W 0–1.5 0–5 1 2 3 4 5 6 7 8 9 10 11 12

Ni

Zn  5 95

δ γ′ γ′′ — — — — — —

5 10 15 20 25 30

A cm2 s−1

Q kJ mol−1

D cm2 s−1

2.6 × 10−3 2.1 0.1 1.7 × 10−5 0.9 × 10−2

230.7 306.9 250.4 133.6 334.1

2 500

192.6

 —   — —  —   —

0.287 11.1 0.86 2.24 2.16 2.11 2.07 2.04 2.01 1.98 1.95 1.94 1.92 1.90 1.89 1.05×103 × exp (−0.142CNi ) ANi 8.3 × 10−2 AZn 0.176 7.1 × 10−2 1.2 × 10−1 3.0 × 10−2 ln A = ˜ 2.4 × 10−2 Q −5.6

Temp. range K

Method

Ref.

1 423–1 573

II

236

—

1 123–1 273

IIc

54

247.9

—

1 373–1 573

IIa(ii)

47

321.5 294.8 303.0   303.5     303.8     304.2    304.5    304.8 305.2    305.6    805.9     306.3     306.7  367.2

— — — — — — — — — — — — — —

1 423–1 563 1 373–1 573

IIa(ii) IIa(ii)

23 47

1 273–1 589

IIa(i)

180

180.0

—

873–1 273

IIa(i)

181

— — —

483–873

II

237

—

1 073–1 323

IIIa(i)

286

Ic

204

Combined data, several sources

121

QNi 182.1 QZn 203.1 85.0 90.9 96.7(a) ∼265 ∼252 ∼242 ∼236 ∼219 ∼208

(a) Values estimated from a graph. O Sol. soln range O 0–1.13

Pt

9.3

326.6

—

1 708–1 777

1.05 × 10−2

110.43

—

298–1 673

1.3 × 10−3 1.3 × 102

46.1 209.2

—

1 713–1 973 1 273–1 473

IIa(ii) IIa(ii)

159 248

0.45 0.14

201 138.2

— —

573–1 223 1 023–1 623

IIIa(i), p.c. IIIa(i) and IIa(ii)

130 123 and 287

Ta

O 25–220 p.p.m.

Th (β) (α)

O α-sol. soln β-sol. soln

Ti

(continued)

Mechanisms of diffusion Table 13.4

CHEMICAL DIFFUSION COEFFICIENT MEASUREMENTS—continued

Element 1 At.%

Element 2 At.%

O

V

O Sol. soln range Sol. soln range Sol. soln range Sol. soln range

Zr (α) (α) (β)

A cm2 s−1

Q kJ mol−1

D cm2 s−1

2.46 × 10−2

123.5

—

% 1.32 0.0661 16.5 0.977

13–97

Temp. range K

Method

Ref.

333–2 098

Various

247

Various

131

III

183 306 132 132

201.8 184.2 229.0 171.7

— — — —

563–1 773 563–923 923–1 773 1 322–1 473



(β)

2.63 × 10−2

118.1

—

1 273–1 573

(α) (Zircalloy) (β) (Zircalloy)

0.196

171.7

—

1 273–1 773

IIa(ii) and IIc IIIc(i)

0.045 3

118.1

—

1 273–1 773

IIIa(i)

(a) Reference 131 reviews all published data. Quoted A’s and Q’s are ‘best mean values’. Data slightly better represented by two Arrhenius expressions, above and below 650◦ C. Pb

— — —

Sn 0–2 Sol. soln range 0 5 10

Pb

T1 0–2 0.53

Pd 0 10 20 30 40 50 60 70 80 90 100

Pt — — — — — — (a) — — — — —

4.0

99.6 100.5

— — —

— — —

0.025 1.03

81.2 103.0

∼4 × 10−3 ∼4 × 10−3 ∼5 × 10−3 ∼5 × 10−3 ∼6 × 10−3 ∼8 × 10−3 ∼1.0 × 10−2 ∼1.3 × 10−2 ∼2.5 × 10−2 ∼5.0 × 10−2 ∼9.5 × 10−2

∼54    ∼54     ∼54.5    ∼55     ∼56  ∼57   ∼58     ∼59     ∼61     ∼64  ∼67

— ˜ increases with D conc. of Sn  3 × 10−11 9.5 × 10−11 2 × 10−10

518–573 443 and 454

IIa(ii) IIa(i)

10 135

523

IIa(i), p.c.

284

— Almost independent of conc.

493–558 533–588

IIa(ii) IIa(i)

29 113

IIa(i), p.c.

290

IIa(i)

238

IIa(i)

140

—

1 335–1 676

(a) Rough values estimated from graphs. Pd

Pt 2 ) 50 55 65  77 80 85

Ti (β) (γ) (δ) (ε) (η)

W (β) (γ)(a) (ε)(a) (α)

1.26 × 10−3 1.6 × 10−8 3.6 × 10−4 1.6 × 10−6 6.4 × 10−4

3.1 × 102 4.7×10−3 3.3 × 10−3 4.4 × 10−2 % 1.8×10−2 1.2 × 10−2 1.3 × 10−2 *

131.9 44.8 129.4 84.2 745.3

 582.0 350.0 343.7 385.2 ) 326.6 315.7 310.7

— —  — —  — DTi = 3.54 × 10−9 DPd = 1.32 × 10−9 DTi /DPd — — —

 973–1 273          973–1 273 

1 173 1 173 1 073

         

 1 573–2 016       1 746–2 016  1 573–2 016       1 573–1 973 

(a) The γ and ε are two new phases observed during the diffusion experiments and not previously reported. (b) Arrhenius plots not always linear. Q and A derived from measurements at higher temperatures. (continued)

13–98

Diffusion in metals

Table 13.4

CHEMICAL DIFFUSION COEFFICIENT MEASUREMENTS—continued

Element 1 At.%

Element 2 At.%

Pu 2 15

Ti (β) (β)

Pu 1.75 3.50 5.25 7.0 8.75 10.50 12.25 14.0 15.75 Pu 20 30 40 50 60 (δ) 0.115

                        

1.15 20 30 40 50

U

(α)

Zr

A cm2 s−1

Q kJ mol−1

9.4 × 10−4 2.3 × 10−3

123.9 127.7

'

 0.14 × 10−7    0.15 × 10−7      0.18 × 10−7    0.28 × 10−7  0.44 × 10−7    0.88 × 10−7     1.18 × 10−7   −7   2.0 × 10  2.57 × 10−7

56.1 57.4 59.0 63.6 68.2 74.9 78.7 83.7 86.2

            

7 × 10−1 1 × 10−2 1.5 × 10−3 2.5 × 10−4 9 × 10−5 5.89 × 10−6 0.1

(εβ) 4.2–11.4 (α)

11.1 AZr 8 × 10−1 7.5 2 × 10−1 1.5 × 10−4

(εβ)

Rh 3 ) 60 70 90

W (α)

Ru 5 39 ) 70 90

W (α) (σ)

(ε) (β)

(β)

*

1.3 × 10−6 1.5 × 10−6 3.1 × 10−6 2.5 × 10−6

5.5 × 10−3 1.2 × 10−5 * 1.8 × 10−5 1.0 × 10−5

184.2 144.4 119.3 98.4 77.5 83.7  226.1

           

272.1 QZr 188.4 205.2 167.5 121.4

 242.8   174.6 181.7   174.2

D cm2 s−1

Temp. range K

Method

Ref.

—

1 173–1 373

IIa(ii)

217

683–813

IIa(i)

134

Probably a significant contribution to ˜ from g.b. D diffusion

— — — — — — — —

1 023–1 173   973–1 123 IIa(i)   973–1 143 973–1 143 624–748

189 IIa

188

973–1 073  QPu    184.2  175.8   113.0   117.2

IIa(ii)

192

Same T range as ˜ for D

189

—

1 573–2 073

IIa(i)

140

— —

1 573–2 298 2 058–2 298

—

1 573–2 298

142

APu 6 × 10−1 4 × 10−1 1 × 10−3 3 × 10−2

(a)

391.5 255.4 ) 207.2 239.5

(a) Arrhenius plots not always linear. Q and A derived from measurements at highest temperatures. S ∼0.01 Si Sol. soln range

Ni

Sn 1.0 8.0

Ti (β)

2.0 Sn Sol. soln range 0–3.9

2.3 × 106

376.8

—

1 273–1 473

IIIa(ii)

20

188.4

—

1 123–1 323

IIc

54

8.4 × 10−7 2.7 × 10−4

64.1 124.8

IIa(i)

26

(β)

—

—

Zr (α)

3.10−4

U (γ)

(β)

0–5

—

Sr Sol. soln

U (γ)

92.1

 Increases 1 273–1 523    linearly 1 363–1 523  with C  * −9 )  DSn = 9.18 × 10   1 523 DTi = 2.65 × 10−9 —

873–1 123

IIa(ii)



136

6.9 × 10−4

150.7

—

1 373–1 573

IIa(ii)

0.07

212

—

1 605–1 970

IIIa(ii), p.c.

271

2.38 × 10−3

196.8

—

1 073–1 273

IIc

109

(continued)

Mechanisms of diffusion Table 13.4

CHEMICAL DIFFUSION COEFFICIENT MEASUREMENTS—continued

Element 1 At.%

Element 2 At.%

TA

W ‘Ta rich’ ‘W rich’ 20–80 70

— 30 Ti  10.0   20.0    30.0    40.0   50.0 60.0    70.0    80.0    90.0  95.0

U

(γ)

165.5

Q kJ mol−1

1.78 4.16 × 10−2 1.0 ATa = 1.8 AW = 0.17

498.2 — 418.7 — 546.4 QTa = 553.9 — QW = 510.8

 11.10−3    1.4 × 10−3     1.6 × 10−3    −3  4.0 × 10  9.5 × 10−3  2.6 × 10−3     2.6 × 10−3   −3   2.2 × 10  −3   1.1 × 10 0.46 × 10−3 —

18.0

 153.2   138.2    145.7    160.8   175.8 165.0    165.0    157.0    141.5  126.4

—

16.5–18 Ti (α)

A cm2 s−1

QU = 161.2

QTi = 167.5

—

—

1.25 × 10−2 6.0 × 10−3

173.3 165.8

(β) (β)

V Sol. soln range 0–10 2.0

(β)

3.5

—

—

20 40 60 80

8.3 × 10−4 1.5 × 10−3 4.4 × 10−3 1.3 × 10−2 2.4 × 10−2 1.1 × 10−2 8.1 × 10−4 4.1 × 10−4 1.6 × 10−4 1.6 × 102 4.8 × 10−2 0.8 8.2

197.6 199.3 203.9 206.8 203.5 186.2 153.2 139.8 123.9  191.3   212.7 261.7   295.2

2.69 1.7 × 10−12 1.8 × 10−2 1.4 × 10−2 3.3 × 10−3 5 × 10−7 2.7 × 10−3 1.2 × 10−6 2.4 × 10−3 1.7 × 10−6 1.6 × 10−3 2.0 × 10−6 1.5 × 10−3 2.2 × 10−6 1.3 × 10−3

261 49.4 167.9 164.5 146.5 65.7 142.8 71.6 140.7 74.1 136.5 18.0 134.8 75.4 131.9

10 20 30 40 50 60 70 80 90 — — — — Ti 0–4 (α) (β) 10 25 40 50 65 80 90 50.5

Zr — 0–10 0–10

D cm2 s−1

—

DTi DU 5.8×10−9 2.2×10−8 % 1.2 × 10−9 4.7 × 10−8 2.9 × 10−9 9.5 × 10−9 4.1 × 10−9 1.6 × 10−8 3.91 × 10−15 4.7 × 10−15 — Dep. on c in range 2–12% v. slight ) * DTi = 1.31 × 10−9 −9 DV = 14.9 × 10  —   —    —    — —  —    —    —   —

*

Temp. range K

Method

Ref.

2 373–2 773

IIa

190

1 573–2 373

IIa(ii), p.c.

259

                     1 223–1 348      

IIa(i)                          

1 348 1 223 1 273 1 323

) 873 973 1 173–1 573 1 173–1 523  1 523

137

) IIa(ii) IIa(ii)

114

IIa(i)

26

IIa(i)

240

 

923–1 323

—

1 173–1 573

— — — — 1 103–1 323 — — — — — — 75.4

1 700–1 953 873–1 073 1 173–1 573  1 103–1 323       923–1 103     1 103–1 323     923–1 103     1 103–1 323    923–1 103 1 103–1 323      —     —    923–1 103     1 103–1 323      1 073

— — DZr = 5.1 × 10−10 DTi = 3.2 × 10−10

13–99

289

IIIa(ii), p.c. IIa(ii) IIa(ii)

271 114 114

IIa(i)

239

923–1 103 1 103–1 323

(continued)

13–100

Diffusion in metals

Table 13.4

CHEMICAL DIFFUSION COEFFICIENT MEASUREMENTS—continued

Element 1 At.%

Element 2 At.%

U ‘low’

W

U (β) (γ) U

Xe 0–10−6 0–10−6 Zr 10 20 30 40 50 60 70 80 90 95 10–95

(γ)

                            

A cm2 s−1

Q kJ mol−1

D cm2 s−1

Temp. range K

Method

Ref.

1.8 × 10−2

389.4

—

2 243–3 003

IIIb

241

9 × 10−7 108

96.3 410.3

— —

973–1 323 1 083–1 333

IIIb(ii)

149

9.5×10−4 1.3 × 10−4 0.35 × 10−4 0.4 × 10−4 0.8 × 10−4 0.63 × 10−4 0.55 × 10−4 3.2 × 10−4 78 × 10−4 870 × 10−4

 134.0   119.7     110.1     114.7    124.3  124.3    124.3     143.6     171.7    196.8

—

1 223–1 348

IIa(i)

138

DU and DZr Figures 13.33 and 13.34

1 123–1 313

IIa(i)

139

REFERENCES TO TABLE 13.4 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. . 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40.

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Mechanisms of diffusion 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79. 80. 81. 82. 83. 84. 85. 86. 87. 88. 89. 90. 91. 92. 93. 94. 95. 96. 97. 98. 99. 100. 101. 102. 103. 104. 105. 106. 107. 108. 109. 110. 111. 112.

13–101

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13–102

Diffusion in metals

113. W. Seith and J. Herrmann, Z. Electrochem., 1940, 46, 213. 114. R. P. Elliot, US Rep. AD. 290336, March 1962. 115. C. S. Hartley, J. E. Steedly and L. D. Parsons, US Rep. ML-TDR-64-316, December 1964, and ‘Diffusion in B.C.C. Metals’, Am. Soc. Met, 1965, p. 35. 116. Y. Adda and J. Philibert, C. R. Acad. Sci. Paris, 1958, 246, 113; Rep C.E.A.-880, March 1958. 117. A. M. Rodin and V. V. Surenyants, Phys. Met. Metallogr., 1960, 10, (2), 58. 118. D. Calais, M. Beyeler, M. Mouchnino, A. van Craeynest and Y. Adda, C. R. Hebd. Séanc. Acad. Sci. Paris, 1963, 257, 1285. 119. E. P. Nechiporenko et al., Fiz. Met. Metalloved., 1971, 32, 89. 120. P. Gröbner, Hutnické listy, 1955, 10, 200. 121. F. J. M. Baratto and R. E. Reed-Hill, Mat. Sci. Eng., 1980, 43, 97. 122. A. F. Gerds and M. W. Mallett, J. Electrochem. Soc., 1954, 101, 175. 123. I. F. Sokirianskii, D. V. Ignatov and A. Y. Shinyaev, Fiz. Met. Metalloved., 1969, 28, 287. 124. M. W. Mallett. J. Belle and B. B. Cleland, J. Electrochem. 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Mechanisms of diffusion 184. 185. 186. 187. 188. 189. 190. 191. 192. 193. 194. 195. 196. 197. 198. 199. 200. 201. 202. 203. 204. 205. 206. 207. 208. 209. 210. 211. 212. 213. 214. 215. 216. 217. 218. 219. 220. 221. 222. 223. 224. 225. 226. 227. 228. 229. 230. 231. 232. 233. 234. 235. 236. 237. 238. 239. 240. 241. 242. 243. 244. 245. 246. 247. 248. 249. 250. 251. 252. 253. 254. 255. 256.

13–103

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Diffusion in metals

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Mechanisms of diffusion

~ Q 150

~ Q (kJ/mol)

175

* Q Cd

125

⫺3 ~ ~ Log Do(Do: m2/s)

13–105

⫺4 Do*Cd

~ Do

⫺5

⫺6

0

5

10

15

20

Figure 13.9 The concentration dependences of ˜ and D ˜ in Ag-Cd alloys291 Q

25

At. % Cd

⫺11 Present work Butrymowicz et al.

~ D

Iorio et al. Tomizuka et al.

⫺12

D *Cd(O)

~ Log (D/m2 s⫺1)

1073 K 1023 ⫺13

973 923

⫺14

⫺15

873

0

5

10 15 At. % Cd

20

25

Figure 13.10 Concentration dependence of the interdiffusion ˜ at five coefficients, D, temperatures between 873 and 1073 K determined by use of Ag-25Cd diffusion couples291

Diffusion in metals

13–106 260 Diffusion coefficient (⫻10⫺8 cm2 s⫺1)

240

10⫻10⫺8cm2 s⫺1 9 8 400°C 7 6 5 4 3 2 1

220 200 180 160 140 120

610°C

100 80 60 40

470°C

20 0

40 42 44 46 48 50 52 54 At. % Zn

Figure 13.11 Chemical diffusion coefficients in AgZn alloys11 Table 13.4

175

~ Q /kJ mol⫺1)

Q*Zn Rothman & Peterson ~ Q

Present work

150

(a)

125 ⫺4

~ log (Do / m2 s⫺1)

log D *oZn Rothman & Peterson ~ log Do

Present work

⫺5

(b) ⫺6

0

Figure 13.12

5

10 At. % Zn

15

20

˜ (a) and log D ˜ o (b) in Ag-Zn α phase312 Concentration dependence of Q

Mechanisms of diffusion

13–107

⫺12

Ag-Zn a phase

~ log (D/m2 s⫺1)

⫺13

At. % Zn

⫺14

18 15 12 9

⫺15

6 3 0 ⫺16 0.9

1.0

1.1

1.2

1.3

T⫺1/10⫺3 K⫺1

0.05

D ⫻ D10 (cm2 s⫺1)

15

At. % Mn 0.1

0.15

Figure 13.13 Temperature dependence of the interdiffusion coefficients in Ag-Zn α phase312

0.2

10

5 650°C 0

0.1

625°C 600°C 0.2 0.3 Wt. % Mn

0.4

Figure 13.14 alloys21

Chemical diffusion in AlMn

10⫺11

~ D (m2 s⫺1)

A alloy/Al couple Present B alloy/Al couple work24

873 K 10⫺12 823 K(Bückle)21

823 K

774 K(Mehl)25

773 K 10⫺13 0

0.1

0.2

0.3 0.4 At. % Si

0.5

0.6

Figure 13.15 alloys21,25

Chemical diffusion in AlSi

13–108

Diffusion in metals

955°C

10⫺10

895°C 868°C 850°C

800°C

10⫺11

~ D (cm2 /sec)

755°C

720°C

10⫺12

10⫺13

10⫺14

0

20

40

60 At. % Co

Figure 13.16a

Interdiffusion in b.c.c. FeCo alloys318

80

100

Mechanisms of diffusion 10⫺9 1300°C

1250°C

1200°C

10⫺10

D (cm2 /sec)

1150°C

1100°C

10⫺11 1050°C 1000°C

10⫺12 0

10

20

30

40

50

60

70

80

90

100

At. % Co Figure 13.16b

Interdiffusion in f.c.c. FeCo alloys318

⫺21 1300°C

1250°C

⫺22

~ In D

1200°C

⫺23 1125°C

⫺24

⫺25

20

40

60

At. % Pt

80 Figure 13.17

Interdiffusion in Co-Pt218

13–109

765

806

866

940

906

Diffusion in metals 1066 1030 1000 983

13–110

u°C Levasseur and Philibert

109

63Ni

in Cu

%

90

%

95

~ D (cm2 s⫺1)

1010

1011

1013 0.7

%

%

% % 50 40 30%

20

60

1012 64Cu

80

%

70

Masson

%

in Ni

0.8

0.9 103 T (K)

(a)

Levasseur and Philibert Method of Matano Method of Hall

109

66

10 1010

°C

°C 00

10 ~ D (cm2 s⫺1)

0°C

94

3°C

98

1011

6°

86

6°C

C

6°C

90

80

5°C

76

1012

0°C

71 1013

Cu 0

(b) Figure 13.18

50

100

At. % Cu Interdiffusion in Cu-Ni55 (a) as function of I/T; (b) As a function of composition

Mechanisms of diffusion

10⫺4 e1⫺Zn g ⫺Zn

10⫺5

120

10⫺6

DO(m2 s⫺1)

~ Q (kJ mol⫺1)

~ DO

100

~ Q 80

75

Figure 13.19

⫺6.0

Log DCu, DSn (cm2 s⫺1)

⫺6.5

80 At. % Zn

85

Chemical diffusion AgZn alloys221

550

Temperature (°C) 500 450

DCu

⫺7.0 ⫺7.5

DSn

⫺8.0 ⫺8.5 ⫺9.0 ⫺9.5

⫺10.0 11.5 12.0 12.5 13.0 13.5 14.0 14.5 4 Temperature 10 T (K) Figure 13.20

Partial diffusion coefficients for Cu and Sn in δ-phase CuSn alloys60

13–111

Diffusion in metals

~ D g (cm2 s⫺1)

13–112

10⫺6 5

2

10⫺7 25

30 35 Wt. % Sn

40

Figure 13.21 Chemical diffusion coefficients for γ-phase of Cu-Sn systems at 706.5◦ C60

5

2

DCu,DSn (cm2 s⫺1)

DCu 10⫺6 DSn 5

2

10⫺7 25

30

35

40 Figure 13.22 Partial diffusion coefficients for Cu and Sn in γ CuSn alloys at 706.5◦ C

Wt. % Sn

Dx (cm2 s⫺1 ⫻ 10⫺9)

6

DZn

D

4

2

DCu

780°C

0

10

20

Concentration (At. % Zn)

30

Figure 13.23 Chemical and partial diffusion coefficients in α-Cu-Zn system at 780◦ C61

Mechanisms of diffusion

13–113

20

Dx (cm2 s⫺1 ⫻ 10⫺9)

DZn

15 D

10

DCu

5

855°C 20

10

30

20

Concentration (At. % Zn)

Dx (cm2 s⫺1 ⫻ 10⫺9)

40

Figure 13.24 Chemical and partial diffusion coefficients in α-Cu-Zn systems at 855◦ C61

DZn

30 D

20

10

DCu

915°C 0

10 20 Concentration (At. % Zn)

30

Figure 13.25 Chemical and partial diffusion coefficients in α-Cu-Zn system at 915◦ C61

Diffusion in metals

13–114 54

1.25%C

D⫻ 1011 (cm2 s⫺1)

46 38 30 22

0.02%C

14 6

0

8

16

24

32

40

48

56

64

γ

Dc ⫻10⫺7 (cm2 s⫺1)

At. % Mn 3.5

Figure 13.26 Chemical diffusion coefficients in γ-Fe-Mn alloys with 0.02 and 1.25 wt%C. Temperature 1200◦ C67

950°C

3.0

920°C 880°C

2.0 1.0

0

0.2

0.4

0.6

0.8

1.0

1.2

Wt. % Si

Figure 13.27 Effect of Si content on chemical diffusion of C in γFe145

1061°C 1053°C 1000°C

10⫺9

968°C

D (cm2 s⫺1)

931°C 10⫺10

10⫺11

10⫺12

Pd

10

20

30

40

50 At. % Cu

Figure 13.28

Interdiffusion in the Cu-Pd system160

60

70

80

90

Cu

Mechanisms of diffusion 10⫺9

10⫺7

13–115

1 atm 20 kbar

~ D (cm2 s⫺1)

~ D (cm2 s⫺1)

10⫺8

10⫺9

1 atm

10⫺10

40 kbar

10⫺11

40 kbar 20 kbar 10⫺10 6.0

6.4

6.8

7.2

7.6

8.0

10⫺12 6.0

104 T (K)

Figure 13.29 Chemical diffusion in α-Fe V 10% alloy at 1, 20 and 24 kbar pressure

10⫺9

6.4

6.8

7.2 104 T (K)

1356ºC 1315ºC

D (cm2 s⫺1)

1200ºC

10

⫺10

1130ºC

⫺11

Fe Figure 13.31

8.0

Figure 13.30 Chemical diffusion in γ Fe V 0.7% alloy at 1, 20 and 40 kbar pressure78

1258ºC

10

7.6

At. % Ni Chemical diffusion coefficients in the Fe-Ni system160

Ni

13–116

Diffusion in metals 1300ºC

1250ºC ⫺22

1200ºC

1160ºC ⫺23 1125ºC

~ In D

1100ºC ⫺24 1010ºC

⫺25

950ºC

⫺26

⫺27

D (cm2 s⫺1)

10⫺8

20

40

60 80 At. % Pt

{ 1000°C { 950°C

Figure 13.32 Chemical diffusion coefficients in the Ni-Pt system218

DU D Zr DU D Zr

10⫺9

10⫺10

Figure 13.33

20

40

60 At. % U

80

100

Partial diffusion coefficients DU and DZr in γ U-Zr alloys139

Mechanisms of diffusion

DU /DZr

10

950ºC 1000ºC 1040ºC

5

1 20

Figure 13.34

40

60 At. % U

80

100

Ratio of partial diffusion coefficients in γ U-Zr alloys139

10⫺7

1500°C

10⫺8

D (cm2 s⫺1)

10⫺9

1200°C

10⫺10

1000°C

10⫺12

850°C 10⫺13

20

40

60 At. % Hf

Figure 13.35a

Chemical diffusion in Hf-Zr alloys

80

100

13–117

13–118

Diffusion in metals DO

10 DO ⫻ 10⫺3 (cm2 s⫺1)

Q (k cal mol⫺1)

50

Q

40

20

40

60

80

9 100

At. % Hf A and Q for chemical diffusion in Hf-Zr alloys235

Figure 13.35b

10⫺9

10⫺10

~ D (cm2 s⫺1)

173°C

149°C

129°C

10⫺11

115°C

10⫺12

0

10

20

30 At. % In

Figure 13.36

Chemical diffusion in In-Pd alloys245

40

50

60

Mechanisms of diffusion

13–119

400°C

380°C 360°C 10⫺8 340°C 320°C 300°C

10⫺9

~ D (cm2 s⫺1)

250°C

10⫺10

10⫺11

78

80

82

84

86

88

At. % Zn Figure 13.37 Table 13.5 Element

Chemical diffusion in ε Cu-Zn alloys251 SELF-DIFFUSION IN LIQUID METALS

A

Q

Li 1.44 × 10−3 12.0 9.29 Na 8.6 × 10−4 K 7.6 × 10−4 8.46 D = 5.344 × 10−10 T 2 − 2.443 × 10−5 8.29 Rb 6.6 × 10−4 D = 3.824 × 10−10 T 2 − 1.479 × 10−5 Cs 4.8 × 10−4 7.79 40.7 Cu 1.46 × 10−3

Temp. range ◦C

Method

Ref.

465–723 375–563 355–563 354–868 330–503 337–856 323–473 1 413–1 543

IIIb(ii), Capillary IVa(i), Capillary IVa(i), Capillary IIa(ii), Capillary Electrotransport method IIa(ii), Capillary Calculated IIIb(ii), Capillary

1 and 2 2 2 22 2 and 3 22 2 and 4 5 (continued)

13–120

Diffusion in metals

Table 13.5

SELF-DIFFUSION IN LIQUID METALS—continued

Element

A

Q

Temp. range ◦C

Ag

5.8 × 10−4

32.1

1 243–1 573

Zn

8.2 × 10−4

21.3 23.4 13.9 18.5

723–893 703–893 603–783 603–773 273–567 248–525 292–556 304–674 443–1 023 623–1 073 540–956 628–1 925 543–1 048 623–783 540–980 ' 1 610–1 680 1 510–1 630 710–873

1.2 × 10−3 Cd 3.62 × 10−4 7.54 × 10−4 Hg D = 4.48 × 10−10 T 1.854 D = 4.34 × 10−9 T 3/2 − 4.81 × 10−6 Ga D = 2.2 × 10−10 T 2 D = 6.01 × 10−9 T 3/2 − 1.60 × 10−5 In 2.89 × 10−4 Tl 3.17 × 10−4 Sn 3.24 × 10−4 D = [1.8 + 0.012(T − TM )]10−5 D = 6.85 × 10−11 T 2 Pb 2.37 × 10−4 Bi 8.3 × 10−5 Fe 1 × 10−2 4.3 × 10−3 Te 1.36 × 10−3

10.2 15.2 11.6 24.7 10.5 65.7 51.1 23.7

(a) (b)

Method

Ref.

IIIb(ii), Capillary IIIb(ii), Capillary IIIb(ii), Capillary IIIb(ii), Capillary IIIa(i), Capillary IIIb(ii), Capillary IIa(ii), shear cell IIa(ii), Capillary IIa(ii), Shear cell IIIb(ii), Capillary IIIb(ii), Capillary IIa(ii), Capillary IIa(ii), Shear cell IIa(ii), Capillary IIIb(ii), Capillary IIIb(ii), Capillary

6 7 8 9 23 11 10 12 13 14 15 21 16 17 9 18

IIb(ii), Capillary

19

IVb(ii), Capillary

20

(a) Average values. A and Q both increase with temperature. (b) Supercooled.

REFERENCES TO TABLE 13.5 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24.

A. Ott and A. Lodding, Z. Naturforsch., 1965, 20a, 1578. S. J. Larsson, C. Roxbergh and A. Lodding, Phys. Chem. Liquids, 1972, 3, 137. A. Nordén and A. Lodding, Z. Naturforsch., 1968, 22a, 215. A. Lodding, Z. Naturforsch., 1972, 27a, 873. J. Henderson and L. Young, Trans. Met. Soc. AIME, 1961, 221, 72. V. G. Leak and R. A. Swalin, Trans. Met. Soc. AIME, 1964, 230, 426. N. H. Nachtrieb, E. Fraga and C. Wahl, J. Phys. Chem., 1963, 67, 2353. W. Lange, W. Pippel and F. Bendel, Z. Phys. Chem., 1959, 212, 238. M. Mirshamshi, A. Cosgarea and W. Upthegrove, Trans. Metall. Soc., AIME, 1966, 236, 122. E. F. Broome and H. A. Walls, Trans. Met. Soc. AIME, 1968, 242, 2177. R. E. Meyer, J. Phys. Chem., 1961, 65, 567. S. Larsson et al., Z. Naturforsch., 1970, 25a, 1472. E. F. Broome and H. A. Walls, Trans. Met. Soc. AIME, 1969, 245, 739. A. Lodding, Z. Naturforsch., 1956, 11a, 200. N. Petrescu and L. Ganovici, Rev. Roum. de Chim., 1976, 21, 1293. A. Bruson and M. Gerl, Phys. Rev. B, 1980, 21, 5447 and J. de Phys., 1980, 41, 533. U. S¯odervall, H. Odelius, A. Lodding, G. Frohberg, K.-H. Kratz and H. Wever, Proc. ‘SIMS V’ Conference, Washington D.C., 1985. N. Petrescu and M. Petrescu, Revue Roum. Chim., 1970, 15, 189. L. Yang, M. T. Simnad and G. Derge, Trans. Met. Soc. AIME, 1956, 206, 1577. L. Nicoloiu, L. Ganovici and I. Ganovici, Rev. Roum. Chim., 1970, 15, 1713. G. Careri, A. Paoletti and M. Vincentini, Nuovo Cimen., 1958, 10, 1088. M. Hseih and R. A. Swalin, Acta Met., 1974, 22, 219. I. Ganovici and L. Ganovici, Rev. Roum. de Chim., 1970, 15, 213. M. Shimoji and T. Itami, Atomic Transport in Liquid Metals. Diffusion and Defect Data, 1986, 43, 1.

14

General physical properties

14.1 The physical properties of pure metals Many physical properties depend on the purity and physical state (annealed, hard drawn, cast, etc.) of the metal. The data in Tables 14.1 and 14.2 refer to metals in the highest state of purity available, and are sufficiently accurate for most purposes. The reader should, however, consult the references before accepting the values quoted as applying to a particular sample.

Table 14.1

THE PHYSICAL PROPERTIES OF PURE METALS AT NORMAL TEMPERATURES

Density (g/cm3 ) at 25◦ C [i]∗

Thermal conductivity (W/m K) at 25◦ C [ii]

Mean specific Resistivity (10−8 m) Temp. coeff. of resistivity heat at T (◦ C) [iv] (J/kg K) at (10−3 K−1 ) 25◦ C [iii] T (◦ C) Resistivity 0–100◦ C

Coeff. of expansion at 25◦ C [i]

Metal

Melting point (◦ C) [i]

Boiling point (◦ C) [i]

Aluminium Antimony Arsenic Barium Beryllium

660.323(∗ ) 630.63(∗ ) {817}∗ 727 1 287

2 519 1 587 616∗ 1 897 2 471

2.70 6.68 5.727∗ 3.62 1.85

247 25.9 — 18.4 210

897 207 329 204 1 825

20 0 20 0 20

2.654 8 39.0 33.3 6 000 4 (aa)

4.5 5.1 — — 9.0

23.1 11.0 5.6 20.6 11.3

Bismuth Cadmium Caesium Calcium Cerium

271.40 321.069 (∗ ) 28.44 842 798

1 564 767 671 1 484 3 433

9.79 8.69 1.93 1.54 6.77

8.2 97.5 18.42 196 11.3

122 232 242 647 192

0 0 20 0 25

106.8 6.83 20 3.91 75

4.6 4.3 4.8 4.57 8.7

13.4 30.8 97 22.3 6.3

Chromium Cobalt Copper Dysprosium Erbium

1 907 1 495 (∗ ) 1 084.62 (∗ ) 1 412 1 529

2 671 2 927 2 562 2 567 2 868

7.15 8.86 8.96 8.55 9.07

67 69.04 398 10.7 14.5

449 421 385 170 168

0 20 20 25 25

12.9 6.24 1.673 57 107

2.14 6.6 4.3 1.19 2.01

4.9 13.0 16.5 9.9 12.2

Gadolinium Gallium Germanium Gold Hafnium

1 313 29.764 6 (∗ ) 937 1 064.18 (∗ ) 2 233

3 273 2 204 2 830 2 856 4 603

7.90 5.91 5.32 19.3 13.3

10.5 33.49 58.6 317.9 23

236 371 320 129 144

25 20 25

140.5 174 (bb) 46 (cc) 2.35 35.1

0.9/1.76 — — 4.0 4.4

9.4 (100◦ C) 18 5.75 14.2 5.9

Holmium Indium Iridium Iron Lanthanum

1 474 156.598 5 (∗ ) 2 446 (∗ ) 1 538 918

2 700 2 072 4 428 2 861 3 464

8.80 7.31 22.5 7.87 6.15

16.2 83.7 147 80.4 13.4

165 233 131 449 195

25 20 20 20 0 to 26

87 8.37 5.3 9.71 57

1.71 5.2 4.5 6.5 2.18

11.2 32.1 6.4 11.8 12.1

Lead Lithium Lutetium Magnesium Manganese

327.462 180.5 1 663 650 1 246

1 749 1 342 3 402 1 090 2 061

11.3 0.534 9.84 1.74 7.3

33.6 44.0 16.4 155 7.79

129 3 582 154 1 023 479

20 0 25 20 20

20.648 8.55 79 4.45 185 (α)

4.2 4.35 — 4.25 —

28.9 46 9.9 24.8 21.7

(continued)

14–1

General physical properties

14–2 Table 14.1

THE PHYSICAL PROPERTIES OF PURE METALS AT NORMAL TEMPERATURES—continued

Density (g/cm3 ) at 25◦ C [i]∗

Thermal conductivity (W/m K) at 25◦ C [ii]

Mean specific Resistivity (10−8 m) heat at T (◦ C) [iv] (J/kg K) at 25◦ C [iii] T (◦ C) Resistivity

Temp. coeff. of resistivity Coeff. of (10−3 K−1 ) expansion 0–100◦ C at 25◦ C [i]

Metal

Melting point (◦ C) [i]

Boiling point (◦ C) [i]

Mercury Molybdenum Neodymium Nickel Niobium

−38.83 2 623 1 021 1 455 (∗ ) 2 477

356.73 13.533 6 4 639 10.2 3 074 7.01 2 913 8.90 4 744 8.57

8.21 142 16.5 82.9 52.3

140 251 190 444 265

50 0 25 20 0

98.4 5.3 64 6.84 12.5

1.0 4.35 1.64 6.8 2.6

60.4 4.8 9.6 13.4 7.3

Osmium Palladium Platinum Plutonium Polonium

3 033 1 554.9 1 768.4 640 254

5 012 2 963 3 825 3 228 962

22.59 12.0 21.5 19.7 9.20

— 70 71.1 6.5 —

130 246 133 142∗ —

20 20 20 107 —

9.5 10.8 10.6 141.4 —

4.1 4.2 3.92 — —

5.1 11.8 8.8 46.7 23.5

Potassium Praeseodymium Radium Rhenium Rhodium

63.38 931 700 3 186 1 964

759 3 520 — 5 596 3 695

0.89 6.77 5 20.8 12.4

108.3 12.5 — 71.2 150

757 193 — 137 243

0 25 — 20 20

6.15 68 — 19.3 4.51

5.7 1.71 — 4.5 4.4

83.3 6.7 — 6.2 8.2

Rubidium Ruthenium Samarium Scandium Selenium

39.30 2 334 1 074 1 541 220.5

688 4 150 1 794 2 836 685

1.53 12.1 7.52 2.99 4.79

58.3 — 13.3 15.8 2.48

363 238 197 568 321

20 0 25 22 0

12.5 7.6 88 61 (ave) 12

4.8 4.1 1.48 — —

9 6.4 12.7 10.2 37

Silicon Silver Sodium Strontium Tantalum

1 412 961.78 (*) 97.72 777 3 017

3 270 2 162 883 1 382 5 458

2.34 10.5 0.97 2.64 16.4

156 428 131.4 — 54.4

705 235 1 228 301 140

0 20 0 20 25

10 1.59 4.2 23 12.45

— 4.1 5.5 — 3.5

7.6 18.9 71 22.5 6.3

Terbium Tellurium

1 356 450

3 230 988

8.23 6.24

11.1 5.98–6.02

182 202

20 23

11 600 — 4.36 × 107 —

Thallium Thorium Thulium

304 1 750 1 545

1 473 4 788 1 950

11.8 11.7 9.32

47 77 16.9

129 113 160

0 0 25

18 13 79

5.2 4.0 1.95

Tin Titanium Tungsten Uranium Vanadium

231.928 (∗ ) 1 668 3 422 1 135 1 910

2 602 3 287 5 555 4 131 3 407

7.26 4.51 19.3 19.1 6.0

62.8 11.4 160 27.6 31.0

228 523 132 116 489

0 20 27 — 20

11 (dd) 42 5.65 30 (ee) 24.8–26

4.6 3.8 4.8 3.4 3.9

22.0 8.6 4.5 13.9 8.4

Ytterbium Yttrium

819 1 522

1 196 3 345

6.90 4.47

38.5 17.2

155 298

29 57 (ff)

1.30 2.71

26.3 10.6

Zinc Zirconium

907 419.527 (∗ ) 1 855 4 409

7.14 6.52

113 21.1

388 278

25 from 20–250 20 —

5.916 40

4.2 4.4

30.2 5.7

(∗ ) Defined fixed point of ITS-90—see Ch. 16 {} Rare earths and rare metals ∗ Densities of higher allotropes not at 20◦ C

(aa) annealed, commercial purity (bb) for a-axis; 8.1 for b-axis and 54.3 for c-axis (cc) Ohm cm for intrinsic germanium at 300 K

(10.3 1.7 || c axis 27.5 ⊥ c axis 29.9 11.0 13.3

(dd) for white tin (ee) Crystallographic average (ff) for polycrystalline material

[i] Reprinted with permission from Handbook of Chemistry and Physics 82nd Edition (12–219). Copyright CRC Press, Boca Raton, Florida [ii] Ref. 47 [iii] Reprinted with permission from Handbook of Chemistry and Physics 82nd Edition (4–133). Copyright CRC Press, Boca Raton, Florida [iv] Ref. 49

The physical properties of pure metals Table 14.2

Metal

THE PHYSICAL PROPERTIES OF PURE METALS AT

Temperature t◦ C

Coefficient of expansion 20−t ◦ C 10−6 K−1

Resistivity at t ◦ C µ cm

Thermal conductivity at t ◦ C Wm−1 K−1

Specific heat at t ◦ C J kg−1 K−1

— 237 — 240 — 236 — 231 — 218

900 — 938 — 984 — 1 030 — 1 076 —

Aluminium

20 27 100 127 200 227 300 327 400 427

— — 23.9 — 24.3 — 25.3 — 26.49 —

Antimony

20 100 500

— 8.4–11.0 9.7–11.6

40.1 59 154

18.0 16.7 19.7

Beryllium

20 27 100 127 200 227 300 327 427 500 527 627 700

— — 12 — 13 — 14.5 — — 16 — — 17

3.3 3.76 — 6.76 — 9.9 — 13.2 16.5 — 20.0 23.7 —

180 — 152 — 130.2 — 117.7 — — 103.0 — — 85.8

Bismuth

20 100 250

— 13.4 —

117 156 260

Cadmium

0 20 27 100 127 227 300

— — — 31.8 — — (38)

Chromium

20 27 100 127 327 400 527 700 727

— — 6.6 — — 8.4 — 9.4 —

20 100 200 300 400 600 800 1 000 1 200

— 12.3 13.1 13.6 14.0 — — — —

Cobalt

2.67 2.733 3.55 3.87 4.78 4.99 6.99 6.13 7.30 8.70

14–3

ELEVATED TEMPERATURES†

205 214 239 1 976 — 2 081 — (2 215) — (2 353) — — (2 621) — — (2 889)

References* 7, 8, 9, 50

7, 10, 6

11, 50

8.0 7.5 7.5

121 130 147

6.8 — — — — — —

97.5 84 96.8 87.9 94.7 92 104.7

— 230 — 239 — — 260

7, 13, 14, 50, 51

13.2 12.7 18(152◦ C) 15.8 24.7 31(407◦ C) 34.6 47(652◦ C) —

— 93.7 — 90.9 80.7 — 71.3 — 65.4

444 — 490 — — 582 — 649 —

7, 15, 16, 50

5.86 9.30 13.88 19.78 26.56 40.2 58.6 77.4 91.9

— — — — — — — — —

434 453 478 502 527 575 716 800 883

7, 12

42, 45

(continued)

14–4

General physical properties

Table 14.2

THE PHYSICAL PROPERTIES OF PURE METALS AT ELEVATED TEMPERATURES† —continued

Temperature t◦ C

Coefficient of expansion 20−t ◦ C 10−6 K−1

Resistivity at t ◦ C µ cm

Thermal conductivity at t ◦ C Wm−1 K−1

Specific heat at t ◦ C J kg−1 K−1

20 27 100 127 200 227 427 500 827 1 000

— — 17.1 — 17.2 — — 18.3 — 20.3

— 1.725 — 2.402 — 3.090 5.262 — — —

394 401 394 393 389 386 366 341(538◦ C) 339 244(1037◦ C)

385 — 389 — 402 — — (427) — (473)

7, 17, 16, 18, 50

20 27 100 127 500 527 900 927

— — 14.2 — 15.2 — 16.7 —

2.2 2.271 2.8 3.107 6.8 6.81 11.8 —

293 317 293 311 — 284 — 255

126 — 130 — 142 — 151 —

7, 50

Hafnium

20 27 100 127 200 227 327 400 1 000 1 400 1 800

— — — — — — — 6.3 6.1 6.0 5.9

35.5 34.0 46.5 48.1 60.3 63.1 78.5 84.4 — — —

(22.2) — 22.0 — 21.5 — — 20.7 — — —

144 — 148 — 152 — — 160 185 — —

43, 44, 48, 50

Iridium

20 100 500 1 000

— 6.8 7.2 7.8

5.1 6.8 15.1 —

148(0◦ C) 143 — —

Iron

20 27 100 127 200 227 327 400 527 600 727 800

— — 12.2 — 12.9 — — 13.8 — 14.5 — 14.6

10.1 9.98 14.7 16.1 22.6 23.7 32.9 43.1 57.1 69.8 — 105.5

73.3 80.2 68.2 69.5 61.5 61.3 54.7 48.6 43.3 38.9 32.3 29.7

444 — 477 — 523 — — 611 — 699 — 791

7, 20, 50

Lead

20 27 100 127 200 227 300 327

— — 29.1 — 30.0 — 31.3 —

20.6 21.3 27.0 29.6 36.0 38.3 50 —

34.8 35.3 33.5 34.0 31.4 32.8 29.7 31.4

130 — 134 — 134 — 138 —

7, 6, 11, 50

Metal Copper

Gold

130 134 142 159

References*

19

(continued)

The physical properties of pure metals Table 14.2

THE PHYSICAL PROPERTIES OF PURE METALS AT

Metal Magnesium

Molybdenum

Nickel

Niobium

Palladium

14–5

ELEVATED TEMPERATURES† —continued

Coefficient of expansion 20−t ◦ C 10−6 K−1

Resistivity at t ◦ C µ cm

Thermal conductivity at t ◦ C Wm−1 K−1

Specific heat at t ◦ C J kg−1 K−1

— — 26.1 — 27.0 — — 28.9 — —

4.2 4.51 5.6 6.19 7.2 7.86 9.52 12.1 11.2 12.8

167 156 167 153 163 151 149 130 — 146 (extp)

1 022 — 1 063 — 1 110 — — 1 197 — —

7, 50

— —

5.7 5.52 7.6 8.02 17.6 18.4 — 31 — 46 — 77

142 138 138 134 121 118 105 105 94.6 84 88 —

247 — 260 — 285 — — 310 — 339 (mean) — —

7, 21, 22, 23, 50, 51

— — 13.3 — 13.9 — 14.4 — 14.8 — 15.2 — — 16.3 —

6.9 7.20 10.3 11.8 15.8 17.7 22.5 25.5 30.6 32.1 34.2 35.5 — 45.5 —

88 90.7 82.9 80.2 73.3 72.2 63.6 65.6 59.5 — 62.0 67.6 71.8 — 76.2

435 — 477 — 528 — 578 — 519 — 535 — — 595 —

24, 50

0 20 27 200 227 400 527 600 727 800 927 1 000

— — —

15.2 14.6 — 25.0 — 36.6 — 48.1 — 59.7 — 71.3

53.3 — 53.7 — 56.7 — 61.3 — 64.4 — 67.5 —

— 268 — 271 — 284 — 292 — 301 — 310

20 100 500 1 000

— 11.1 12.4 13.6

10.8 13.8 27.5 40

75 74 — —

Temperature t◦ C 20 27 100 127 200 227 327 400 427 527 20 27 100 127 500 527 927 1 000 1 327 1 500 1 727 2 500 20 27 100 127 200 227 300 327 400 427 500 527 727 900 927

— — —

5.2 5.7

— — —

— — — —

5.75 6.51

7.19 7.39 7.56 7.72 7.88

243 247 268 297

References*

42, 45, 50, 51

19

(continued)

14–6

General physical properties

Table 14.2

Metal Platinum

Plutonium

THE PHYSICAL PROPERTIES OF PURE METALS AT ELEVATED TEMPERATURES† —continued

Temperature t◦ C

Coefficient of expansion 20−t ◦ C 10−6 K−1

Resistivity at t ◦ C µ cm

Thermal conductivity at t ◦ C Wm−1 K−1

Specific heat at t ◦ C J kg−1 K−1

20 27 100 127 500 527 927 1 000 1 127 1 500 1 527

— — 9.1 — 9.6 — — 10.2 — 11.31 —

10.58 10.8 13.6 14.6 27.9 28.7 — 43.1 — 55.4 —

72 71.6 72 71.8 — 75.6 82.6 67 87.1 63 96.1

134 — 134 — 147 — — 159 — 176 —

7, 19, 23, 25, 50

145.8 141.6 107.8 107.4 100.7 110.6

(8.4) — — — — —

131 138 145 153 154 144

42, 45

26, 16, 27

20 α → α 100 α → α 200 α → β 300 α → γ 400 α → δ 500 α → ε

47 203 173 181 109 101 ( 12.4 ||-axis 4.7 ⊥-axis 7.29(2 000◦ C)

References*

' 18.7 25 132

—

134 138 209(2 527◦ C)

—

4.7 6.2 14.6 —

149 147 — —

243 255 289 331

19

20 27 100 127 500 527 900 927

— — 19.6 — 20.6 — 22.4 —

16.3 1.629 2.1 2.241 4.7 4.91 7.6 —

419 429 419 425 (377) 396 — 361 (extp)

234 — 222 — (230) — (243) —

7, 28

Tantalum

20 27 100 127 500 527 627 1 500 1 527 2 327 2 500 2 727

— — 6.5 — 6.6 — — — — — — —

13.5 13.5 17.2 18.2 35 35.9 40.1 71 — — 102 —

57 57.5 54 57.8 — 59.4 59.8 — 63.4 65.8 — 66.5

138 — 142 — 151 — — 167 — — 234(2 727◦ C) —

7, 29, 30, 31, 32

Thallium

20 100 200

— 30 —

16.6 — —

46 45 45

134 138 142

33, 6

Tin

0 20 27 100 127 200 227

— — — 23.8 — 24.2 —

11.5 12.6 — 15.8 — 23.0 —

— 65 66.6 63 62.2 60 59.6

— 222 — 239 — 260 —

Rhenium

Rhodium

Silver

20 100 2 500 20 100 500 1 000

8.5 9.8 10.8

48

7, 34

(continued)

The physical properties of pure metals Table 14.2

Metal

14–7

THE PHYSICAL PROPERTIES OF PURE METALS AT ELEVATED TEMPERATURES† —continued

Temperature t◦ C

Coefficient of expansion 20−t ◦ C 10−6 K−1

Resistivity at t ◦ C µ cm

Thermal conductivity at t ◦ C Wm−1 K−1

Specific heat at t ◦ C J kg−1 K−1

References*

Titanium

0 20 27 100 127 200 227 327 400 527 600 800 927

— — — 8.8 — 9.1 — — 9.4 — 9.7 9.9 —

39 54 — 70 — 88 — — 119 — 152 165 —

— 16 21.9 15 20.4 15 19.7 19.4 14 19.7 13 (13) 22.2

— 519 — 540 — 569 — — 619 — 636 682 —

Tungsten

20 27 100 127 500 527 927 1 000 1 727 2 000 3 000

— — 4.5 — 4.6 — — 4.6 — 5.4 6.6

5.4 5.44 7.3 7.83 18 18.6 — 33 — 65 100

167 174 159 159 121 125 112 111 98 93 —

134 — 138 — 142 — — 151 — — —

37, 33, 38

— — — —

30 59 55.5 54

27 38 40 42.3

116 186 176 160

Expansion anisotropic 42, 45

— — 8.3 — 9.6 — — — 10.4

24.8 20.2 31.5 28.0 — 53.1 58.7 — —

— — 31 — 36.8 — — 35.2 —

492 — 505 — 570 — — 603 636

42, 45

113 — 109 — 105 — 101 — 96

389 — 402 — 414 — 431 — 444

39, 13, 40, 41, 6

Uranium

20 α 600 α 700 β 800 γ

Vanadium

20 27 100 127 500 527 627 700 900

Zinc

∗ †

20 27 100 127 200 227 300 327 400

— — 31 — 33 — 34 — —

5.96 6.06 7.8 8.37 11.0 10.82 13.0 13.49 16.5

35, 36

Items from the CRC Materials Science and Engineering Handbook, 3rd Edition, are reprinted with permission. Copyright CRC Press, Boca Raton, Florida. Data in this table are from multiple sources and may not be fully consistent.

14–8

General physical properties

REFERENCES TO TABLES 14.1 AND 14.2 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52.

W. Slough, Private Communication, Chemical Standards Division, NPL, 1972. M. J. Swan, Private Communication, Electrical Science Division, NPL, 1972. G. W. C. Kaye and T. H. Laby, ‘Tables of Physical and Chemical Constants’, Longmans, London, 1966. ‘Thermophysical Properties of Matter’, TPRC Data Series, Volume 4. R. J. Corrnecini and J. Gniewek, Nat. Bureau of Stds. Monograph; 1960. ‘Thermophysical Properties of Matter’, TPRC Series Volume 1; 1970. US Bur. Stds. Circular C447, ‘Mechanical Properties of Metals and Alloys’, Washington, 1952. R. Hase, R. Heierberg and W. Walkenborst, Aluminium, 1940, 22, 631. T. G. Peason and H. W. L. Phillips, Met. Rev., 1957, 2, 305. H. Tsutsumi, Sci. Rep. Tôhoku Univ. (1), 1918, 7, 100. R. W. Powell, Phil. Mag., 1953, 44, 657. E. F. Northrup and V. A. Suydam, J. Franklin Inst., 1913, 175, 160. Saldau, Z. Metallogr., 1915, 7, 5. S. Grabe and E. J. Evans, Phil. Mag., 1935, 19, 773. R. W. Powell and R. P. Tye, J. Inst. Metals, 1957, 85, 185. C. F. Lucks and H. W. Deem, ASTM Special Tech. Pubn., 1958, No. 227. C. S. Smith and E. W. Palmer, Trans. AIMME, 1935, 117, 225. C. J. Meechan and R. R. Eggleston, Acta Met., 1954, 2, 680. R. F. Vines, ‘The Platinum Metals and their Alloys’, New York, 1941. BISRA, ‘Physical Constants of Some Commercial Steels at Elevated Temperatures’, London, 1953. C. Zwikker, Physica, 1927, 7, 73. E. P. Mikol, US Atomic Energy Comm. Publ. ORNL–1131, 1952. O. H. Kirkorian, UCRL–6132, TID–4500, Sept. 1960, USA. Mond Nichel Co. Ltd., Nickel Bull., 1951, 24, 1. K. S. Krishnan and S. C. Jain, Br. J. Appl. Phys., 1954, 5, 426. C. Agte, H. Alterthum et al., Z. Anorg. Chem., 1931, 196, 129. R. E. Taylor and R. A. Finch, US Atomic Energy Com. Rep., 1961 NAA–SR–6034. US Bur. Min. Circular C412, ‘Silver, its Properties and Industrial Uses’, Washington, 1936. L. Malter and D. B. Langmuir, Phys. Rev., 1939, 55, 743. M. Cox, Phys. Rev., 1943, 64, 241. I. B. Fieldhouse et al., WADC Tech. Rep. 55–495, 1956. N. S. Rasor and J. D. McClelland, WADC Tech. Rep. 56–400, 1957. A. E. van Arkel, ‘Reine Metalle’, Berlin, 1939. Int. Tin R. and D. Council. Tech. Publ. B1, 1937. E. S. Greiner and W. C. Ellis, Metals Tech., Sept., 1948. L. Silverman, J. Metals, N.Y., 1953, May, p. 631. C. J. Smithells, ‘Tungsten’, London, 1952. V. S. Gurnenyuk and V. V. Lebeden, Fizica Metall., 1961, 11, 29. F. L. Uffelmann, Phil. Mag. (7), 1930, 10, 633. Lees, Phil. Trans. R. Soc., 1908, A208, 432. Dewar and Fleming, Phil. Mag., 1893, 36, 271. C. A. Hampel, ‘Rare Metals Handbook’, Chapman & Hall, London, 1961. H. K. Adenstedt, Trans. A.S.M., 1952, 44, 949. R. P. Cox et al., Ind. Eng. Chem., 1958, 50, 141. Thermochemical Data Section. Met. Ref. Book. R. W. Powell, R. P. Tye and M. J. Woodman, J. Less Common Metals, 1967, 12, 1. CRC Handbook of Chemistry and Physics, 82nd edition, p. 12-219. ASM Metals Handbook, p. 115. CRC Handbook of Chemistry and Physics, 82nd edition, p. 4-133. ASM Metals Reference Book, p. 143. CRC Handbook of Chemistry and Physics, 82nd edition, p. 12-45, 12-221. CRC Materials Science and Engineering Handbook, 3rd edition, p. 384–389.

The thermophysical properties of liquid metals

14.2 The thermophysical properties of liquid

14–9

metals∗

Accurate and reliable information on the thermophysical properties of molten metals becomes increasingly significant as new casting techniques are developed and advancement is made in numerical modelling of these processes. To obtain precise thermophysical data for molten metals and alloys, it is necessary to know the special features of different techniques and to utilise the most suitable one for the specimen of interest. 14.2.1 Density and thermal expansion coefficient Knowledge of the density of liquid metals is crucial in most theories related to the liquid state and for the simulation of the contraction that occurs during solidification. There are several methods for measuring the density of high-melting liquid metals: balanced columns, pycnometer, immersedsinker, maximum-bubble, etc. However, the application of all these techniques is limited due to reaction between liquid metal sample and the apparatus. Therefore, an electromagnetic levitation technique can be considered as a good alternative for density measurements in molten metals. The density (ρ) of the specimen can, of course, be determined as ρ=

m , V

where m and V are the mass and the volume of the specimen, respectively. Assuming a spherical shape for the specimen, the volume can be determined from the radius of the droplet. Repeating the density measurements at different temperatures, the thermal expansion coefficient (β) can be simulated as β=

1 ∂V . V ∂T

The determination of the thermal expansion coefficient of the materials requires the measurement of the linear size of the specimen as a function of the temperature. It is experimentally found that the variation of the density of most liquid metals and alloys with temperature (T ) is well represented by a linear equation ρ = ρm +

∂ρ (T − Tm ), ∂T

where ρm is the density of the liquid metal or alloy at its melting point Tm . However, for certain metals (aluminium, gallium, antimony) this relationship is not linear. Table 14.3a presents density data available for liquid metals at their melting point and their temperature dependence (∂ρ/∂t) from [1, 2]. 14.2.2 Surface tension The surface tension is determined by the microscopic structure of the liquid near the surface. At a liquid–vapour interface the density changes severely from a high value in the liquid state to a very low value in the gas phase. Therefore, surface atoms experience an attraction toward the liquid phase, which is the cause of the surface tension. Due to its energetic and entropic origin, it is necessary to calculate the free energy of the system in order to determine the surface tension. Thus the surface tension is determined, as the additional free energy required to generate a unit surface area separating the liquid from its vapour phase. There are many techniques for surface tension measurements: sessile drop, pendant drop, maximum bubble pressure, maximum pressure in a drop, detachment or maximum pull, capillary-rise, drop weight, and oscillating drop methods. The sessile-drop technique has been widely used due to its many advantages. The sessile drop method utilises a molten drop resting on a horizontal ceramic substrate, and allows measurements over a wide range of temperatures. However, the surface tension data is affected by contaminants. To avoid the contamination effects the surface tension of a liquid droplet can be measured by exciting surface oscillations. The frequency of the oscillations is related to the surface tension. ∗

For the physical properties of molten salts, see Chapter 9.

14–10

General physical properties

Table 14.3a

THE THERMOPHYSICAL PROPERTIES OF LIQUID METALS

Density, surface tension and viscosity Density

Surface tension Viscosity

Metal Ag Al As Au B Ba Be Bi Ca Cd Ce Co Cr Cs Cu Fe Fr Ga Gd Ge Hf Hg

In Ir K La Li Mg Mn Mo Na Nb Nd Ni Os P Pb Pd Pr Pt Pu Rb Re Rh Ru S Sb Se Si Sn Sr Ta Te Th Ti

Temp. Tm K

ρm 103 kg m−3

−(∂ρ/∂T ) 10−1 kg m−3 K−1

1 233.7 933 1 090 1 336 2 350 1 000 1 556 544 1 138 594 1 077 1 766 2 148 301.6 1 356 1 809 291 302.8 1 585 1 207 2 216 234.13 273 293 373 429.6 2 716 336.5 1 203 453.5 924 1 514 2 880 369.5 2 741 1 297 1 727 3 000 317 600 1 825 1 208 2 042 913 311.9 3 431 2 239 2 700 392 903.5 490 1 683 505 1 043 3 250 724 1 964 1 958

9.33 2.385 5.22 17.36 2.08 3.321 1.690 10.05 1.365 8.01 6.685 7.76 6.29 1.84 8.000 7.03 2.35 6.10 7.14 5.49 11.1 13.691 13.595 13.546 13.352 7.03 20.0 0.827 5.955 0.518 1.590 5.76 9.34 0.927 7.83 6.688 7.905 20.1 — 10.678 10.49 6.611 18.91 16.65 1.48 18.8 10.8 10.9 1.819 6.483 4.00 2.53 6.98 2.37 15.0 5.80 10.5 4.13

9.1 3.5 5.4 15 — 2.7 1.2 11.8 2.2 12.2 2.3 10.9 7.2 5.7 8.0 8.8 7.92 5.6 — 4.9 — 2.436 — — — 6.8 — 2.4 2.4 1.0 2.6 9.2 — 2.35 — 5.3 11.9 — — 13.2 12.3 2.5 28.8 14.1 4.5 — — — 8.00 8.2 11.7 3.5 6.1 2.6 — 7.3 — 2.3

γm mN m−1

−(dγ/dT ) mN m−1 K−1

ηmp mN s m−2

η0 mN s m−2

E kJ mol−1

966 914 — 1 169 1 060 277 1 390 378 361 570 740 1 873 1 700 70 1 303 1 872 62 718 810 621 1 630 498 — — — 556 2 250 115 720 398 559 1 090 2 250 191 1 900 689 1 778 2 500 52 458 1 500 — 1 800 550 86 2 700 2 000 2 250 61 367 106 865 560 303 2 150 180 978 1 650

0.19 0.35 — 0.25 — 0.08 0.29 0.07 0.10 0.26 0.33 0.49 0.32 0.06 0.23 0.49 0.044 0.10 0.16 0.26 0.21 0.20 — — — 0.09 0.31 0.08 0.32 0.14 0.35 0.2 0.31 0.10 0.24 0.09 0.38 0.33 — 0.13 0.22 — 0.17 0.10 0.06 0.34 0.30 0.31 0.07 0.05 0.1 0.13 0.09 0.10 0.25 0.06 0.14 0.26

3.88 1.30 — 5.0 — — — 1.8 1.22 2.28 2.88 4.18 — 0.68 4.0 5.5 0.765 2.04 — 0.73 — 2.10 — — — 1.89 — 0.51 2.45 0.57 1.25 — — 0.68 — — 4.90 — 1.71 2.65 — 2.80 — 6.0 0.67 — — — 12 1.22 24.8 0.94 1.85 — — 2.14 — 5.2

0.453 2 0.149 2 — 1.132 — — — 0.445 8 0.065 1 0.300 1 — 0.255 0 — 0.102 2 0.300 9 0.369 9 — 0.435 9 — — — 0.556 5 — — — 0.302 0 — 0.134 0 — 0.145 6 0.024 5 — — 0.152 5 — — 0.166 3 — — 0.463 6 — — — 1.089 0.094 0 — — — — 0.081 2 — — 0.538 2 — — — — —

22.2 16.5 — 15.9 — — — 6.45 27.2 10.9 — 44.4 — 4.81 30.5 41.4 — 4.00 — — — 2.51 — — — 6.65 — 5.02 — 5.56 30.5 — — 5.24 — — 50.2 — — 8.61 — — — 5.59 5.15 — — — — 22.0 — — — — — — — — (continued)

The thermophysical properties of liquid metals Table 14.3a

14–11

THE THERMOPHYSICAL PROPERTIES OF LIQUID METALS—continued

Density

Surface tension Viscosity

Metal

Temp. Tm K

ρm 103 kg m−3

−(∂ρ/∂T ) 10−1 kg m−3 K−1

Tl U V W Yb Zn Zr

575 1 406 2 185 3 650 1 097 692 2 123

11.35 17.27 5.36 17.6 — 6.575 5.8

13.0 10.3 3.2 — — 9.8 —

γm mN m−1

−(dγ/dT ) mN m−1 K−1

ηmp mN s m−2

η0 mN s m−2

E kJ mol−1

464 1 550 1 950 2 500 — 782 1 480

0.08 0.14 0.31 0.29 — 0.17 0.20

2.64 6.5 — — 1.07 3.85 8.0

0.298 3 0.484 8 — — — 0.413 1 —

10.5 30.4 — — — 12.7 —

The temperature dependence of surface tension is related to the surface entropy and the surface excesses. Therefore, the changes in the structure of the liquid specimen with temperature are reflected in the temperature coefficients of the surface tension. The variation of the surface tension with the temperature for most liquid metals can be expressed by a linear relationship γT = γm −

dγ (T − Tm ), dT

where γT and γm are the surface tensions at temperature T and at the melting point Tm , respectively. Experimental values of surface tension for liquid metals at their melting points, and their temperature coefficients of surface tension are presented in Table 14.3b. The data are taken from Allen.3,2 14.2.3 Viscosity Viscosity is one of the most important transport properties of molten metals. It is related to the internal friction within the liquid and provides some information about the structure of the material. The most crucial hydrodynamic criteria such as the Reynolds number, the Rayleigh number, the Hartmann number, and the Marangoni number contain viscosity. The existing methods for measuring viscosities of liquids are restricted for liquid metals and alloys due to their low viscosities, high melting points, and chemical reactivity. The capillary, oscillating-vessel, rotational, and oscillating plate methods are most suitable techniques for determination of the viscosity of liquid metals and alloys. The viscosity η relates the shear stress τ to the shear rate γ˙ τ = ηγ. ˙ By analogy to the Wiedemann–Franz–Lorenz law and the Stokes–Einstein relation, there is the following relationship between viscosity and surface tension  γ 15 κB T = , η 16 m where κB is the Boltzmann’s constant and m is the atomic mass. Since the viscosity measurements are not so responsive to convection as diffusion measurements, the diffusivity D may be estimated from the viscosity data using Stokes–Einstein theory D=

κB T . 6πRη

For most liquid metals and alloys the variation of viscosity with temperature may be determined as   E η = η0 exp , RT where η0 and E are constants, and are given in the Table 14.3a for liquid metals,4,5 and R is the gas constant, 8.314 4 J K−1 mol−1 .

14–12

General physical properties

Table 14.3b

THE THERMOPHYSICAL PROPERTIES OF LIQUID METALS

Heat capacity, thermal conductivity and electrical resistivity

Metal Ag

Al

As Au

B Ba Be Bi

Ca Cd

Ce

Co Cr Cs

Cu

Fe Fr Ga

Gd Ge Hf

Temperature K

Heat capacity J g−1 K−1

Thermal conductivity Wm−1 K−1

Electrical resistivity µ m

1 233.7 1 273 1 373 1 473 1 573 1 673 933 973 1 073 1 173 1 273 1 090 1 336 1 373 1 473 1 573 1 673 2 350 1 000 1 556 544 573 673 773 873 973 1 138 594 673 773 873 1 077 1 273 1 473 1 766 2 176 301.6 373 473 673 873 1 073 1 873 1 356 1 373 1 473 1 673 1 873 1 809 291 973 302.8 373 473 573 1 623 1 207 1 273 2 500

0.283 0.283 0.283 0.283 0.283 — 1.08 1.08 1.08 1.08 — — 0.149 0.149 0.149 0.149 0.149 2.91 0.228 3.48 0.146 0.143 0.147 5 0.137 5 0.133 6 — 0.775 0.264 0.264 0.264 0.264 0.25 0.25 0.25 0.59 0.78 0.28 0.265 0.240 0.21 0.22 0.25 — 0.495 0.495 0.495 0.495 0.495 0.795 0.142 0.134 0.398 0.398 0.398 0.398 0.213 0.404 0.404 —

174.8 176.5 180.8 185.1 189.3 193.5 94.05 95.37 98.71 102.05 105.35 — 104.44 105.44 108.15 110.84 113.53 — — — 17.1 15.5 15.5 15.5 15.5 15.5 — 42 47 54 61 — — — — — 19.7 20.2 20.8 20.2 18.3 16.1 4.0 165.6 166.1 170.1 176.3 180.4 — — — 25.5 30.0 35.0 39.2 — — — —

0.172 5 0.176 0 0.184 5 0.193 5 0.202 3 0.211 1 0.242 5 0.248 3 0.263 0 0.277 7 0.292 4 2.10 0.312 5 0.318 0 0.331 5 0.348 1 0.363 1 2.10 1.33 0.45 1.290 — — — — — 0.250 0.337 0.343 0 0.351 0 0.360 7 1.268 1.294 1.310 1.02 0.316 0.370 0.450 0.565 0.810 1.125 1.570 — 0.200 0.202 0.212 0.233 0.253 1.386 0.87 — 0.26 0.27 0.28 0.30 0.278 0.672 0.727 2.18 (continued)

The thermophysical properties of liquid metals Table 14.3b

Metal Hg

Ho In

K

La

Li

Mg

Mn Mo Na

Nb Nd Ni P Pb

Po Pr Pt Pu Ra Rb

14–13

THE THERMOPHYSICAL PROPERTIES OF LIQUID METALS—continued

Temperature K

Heat capacity J g−1 K−1

Thermal conductivity Wm−1 K−1

Electrical resistivity µ m

234.13 273 293 373 773 1 273 1 733 1 773 429.6 473 673 873 336.5 373 473 773 1 273 1 773 1 203 1 273 1 373 1 473 453.5 473 673 873 1 073 1 273 1 873 923 973 1 073 1 273 397 2 880 370 373 473 673 873 1 073 1 273 1 473 2 741 1 297 1 727 317 600 673 773 873 1 073 1 273 527 1 208 2 043 913 1 233 311.8 373 473

0.142 0.142 0.139 0.137 0.137 — — 0.203 0.259 0.259 0.259 0.259 0.820 0.810 0.790 0.761 0.838 — 0.057 5 0.057 5 0.057 5 0.057 5 4.370 4.357 4.215 4.165 4.148 4.147 4.36 1.36 1.36 1.36 1.36 0.838 0.57 1.386 1.385 1.340 1.278 1.255 1.270 1.316 1.405 — 0.232 0.620 — 0.152 0.144 0.137 0.135 — — — 0.238 0.178 — 0.136 0.398 0.383 0.364

6.78 7.61 8.03 9.47 12.67 8.86 ∼0.000 4 — 42 — — — 53.0 51.7 47.7 37.8 24.4 15.5 21.0 — — — 46.4 47.2 53.8 57.5 58.6 58.4 52.0 78 81 88 100 — — 89.7 89.6 82.5 71.6 62.4 53.7 45.8 38.8 — — — — 15.4 16.6 18.2 19.9 — — — — — — — 33.4 33.4 31.6

0.905 0.940 0.957 1.033 1.600 3.77 ∼1 000 1.93 0.323 0 0.333 9 0.436 1 0.513 1 0.136 5 0.154 0.215 0.444 0.110 1.38 1.43 1.50 1.56 0.240 — — — — — — 0.274 0.277 0.282 — 0.40 0.605 0.096 4 0.099 0.134 0.224 0.326 0.469 — — 1.05 1.26 0.850 2.70 0.948 5 0.986 3 1.034 4 1.082 5 1.169 1.263 3.98 1.38 0.73 1.33 1.71 0.228 3 0.273 0 0.366 5 (continued)

14–14

General physical properties

Table 14.3b

THE THERMOPHYSICAL PROPERTIES OF LIQUID METALS—continued

Metal

Temperature K

Heat capacity J g−1 K−1

Thermal conductivity Wm−1 K−1

Electrical resistivity µ m

Re Ru

773 1 273 1 773 3 431 2 700

0.348 0.378 — — —

26.1 17.0 8.0 — —

0.689 0 1.71 5.32 1.45 0.84

392 903.5 973 1 073 1 273 1 812

0.984 0.258 0.258 0.258 0.258 0.745

— 21.8 21.3 20.9 — —

>1010 1.135 1.154 1.181 1.235 1.31

490 1 683 1 773 1 873 1 345 505 573 673 773 1 273 1 043 3 269 1 638 723 773 873 1 073 1 273 1 958 576 673 773 1 873 1 406 1 473 1 573 2 185 3 650 1 803 1 097 692.5 773 873 1 073 2 123

0.445 1.04 1.04 1.04 0.223 0.250 0.242 0.241 0.24 0.26 0.354 — — 0.295 0.295 0.295 0.295 0.295 0.700 0.149 0.149 0.149 — 0.161 0.161 0.161 0.780 — 0.377 — 0.481 0.481 0.481 0.481 0.367

0.3 — — — — 30.0 31.4 33.4 35.4 — — — — 2.5 3.0 4.1 6.2 — — 24.6 — — — — — — — — — — 49.5 54.1 59.9 60.7 —

∼106 0.75 0.82 0.86 1.90 0.472 0 0.490 6 0.517 1 0.543 5 0.670 0.58 1.18 2.44 5.50 4.80 4.30 3.9 3.8 1.72 0.731 0.759 0.788 1.88 0.636 0.653 0.678 0.71 1.27 1.04 1.64 0.374 0.368 0.363 0.367 1.53

S Sb

Sc Se Si

Sm Sn

Sr Ta Tb Te

Ti Tl

Tm U

V W Y Yb Zn

Zr

The thermophysical properties of liquid metals

14–15

14.2.4 Heat capacity Heat capacity (C) is one of the most essential thermodynamic properties of metals. A knowledge of the heat capacity and its temperature dependence allows prediction of the enthalpy and entropy of a material. It is determined as C=

δq , dT

where δq is an infinitesimal heat quantity added to (or withdrawn from) the matter and dT is the resulting infinitesimal temperature change. The specific heat is traditionally measured by adiabatic calorimetry techniques. The calorimeter is isolated from the environment, and requires long relaxation times. Heat capacities determined at constant volume or at constant pressure are of most general importance. Table 14.3b lists values of constant-pressure heat capacity for liquid elements at different temperatures.6–8 14.2.5 Electrical resistivity Information on the electrical resistivity of molten metals and alloys is especially important in many metallurgical processes such as electroslug remelting, electromagnetic stirring in continuous casting, electrolysis, and induction melting in foundries. Due to the disordered arrangement of ions in the liquid state, molten metals and alloys exhibit higher (∼1.5–2.3 times) electrical resistivity than those in the solid state. However, relatively few studies have been reported on the electrical resistivity of molten metals and alloys, particularly at elevated temperatures, since the measurements are extremely difficult. The methods of electrical resistivity measurements can be categorised into three groups: direct resistance measurements using contact probes, contactless inductive measurements, and noncontact containerless measurement techniques. The technique of choice for solid materials is the direct resistance four-probe method, which is based upon application of Ohm’s law. Although this direct method can be applied to low melting point, non-reactive liquid materials, reactions between the probes and the molten sample preclude using the four-probe method with high-melting point materials. Also, this technique is limited to materials with very narrow freezing ranges. Inductive techniques for measuring electrical resistivity are contactless and thus prevent chemical reactions between molten samples and contacting probes as in the direct method. There are two different types of contactless methods. The rotating field method is usually based on the phenomenon that when a metal sample rotates in a magnetic field (or the magnetic field rotates around a stationary sample), circulating eddy currents are induced in the sample, which generate an opposing torque proportional to the electrical conductivity of the sample.9,10 In liquid metals and alloys the applied magnetic field also causes significant rotation of the liquid in the crucible, which decreases the angular velocity between the field and the sample. In both direct resistance and contactless inductive techniques chemically reactive components contained in the crucible may easily contaminate the molten sample, altering its electrical properties. The container may provide heterogeneous nucleants that generate early solid-phase nucleation. Electromagnetic levitation is another technique for containerless measurements of electrical conductivity in liquid metals.11 It combines the containerless positioning method of electromagnetic levitation with the contactless technique of inductive electrical conductivity measurement. The electrical resistivity ρe of liquid metals (except Cd, Zn, Hg, Te and Se) increases linearly with increasing temperature ρe = aT + b, where a and b are temperature coefficients of the resistivity, and are determined experimentally. The electrical resistivities of most liquid metallic elements at different temperatures are shown in Table 14.3b.6–8 14.2.6 Thermal conductivity Accurate measurement of thermal conductivity of liquid metals and alloys is usually more difficult than the measurement of electrical conductivity and thermal diffusivity. The source of difficulty is mainly related to problems with making accurate heat flow measurements. Also there is possibility

General physical properties

14–16

of some flows in the liquid sample. Thermal conductivity is directly related to the change in the atomic vibrational frequency. For a number of non-metallic substances it is found that λ √ = 2.4 × 10−3 , M where λ is the thermal conductivity, M is the molecular weight. Since free electrons are responsible for the electrical and thermal conductivities of conductors in both solid and liquid states, many researchers use the Wiedemann-Franz-Lorenz law to relate the thermal conductivity to the electrical resistivity: πκ2 λρe = 2 ≡ L0 , T 3e where κ is the Boltzmann constant, e is the electron charge. The constant L0 =

π 2 κ2 = 2.45 × 10−8 W  K −2 , 3e2

is the Lorenz number. The validity of this relationship was confirmed experimentally with high accuracy by many researchers. The thermal conductivity values of various liquid metals at different temperatures are given in Table 14.3b.6–8 REFERENCES FOR SECTION 14.2 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

D. J. Steinberg, Metall. Trans., 1974, 5, 1341. T. Iida and R. I. L. Guthrie, ‘The Physical Properties of Liquid Metals’, Clarendon Press, Oxford, UK, 1988. B. C. Allen, ‘Liquid Metals’ (ed. S. Z. Beer), Marcel Dekker, New York, NY, 1972, 162–212. R. T. Beyer and E. M. Ring, ‘Liquid Metals’ (ed. S. Z. Beer), Marcel Dekker, New York, NY, 1972, p. 450. L. J. Wittenberg and D. Ofte, Techniques of Metals Research, 1970, 4, 193. A. V. Grosse, Revue Hautes Temp. & Refrac., 1966, 3, 115. D. R. Stull and G. C. Sinke, ‘Thermodynamic Properties of the Elements’, Amer. Chem. Soc., 1956. R. Hultgren et al., ‘Selected Values of Thermodynamic Properties’, Wiley, 1963. S. I. Bakhtiyarov and R. A. Overfelt, J. Materials Science, 1999, 34, 945–949. S. I. Bakhtiyarov and R. A. Overfelt, Acta Materialia, 1999, 47, 4311–4319. S. I. Bakhtiyarov and R. A. Overfelt, Annals of New York Academy of Sciences, New York, NY, 2002.

14.3 The physical properties of aluminium and aluminium alloys THE PHYSICAL PROPERTIES OF ALUMINIUM AND ALUMINIUM ALLOYS AT NORMAL TEMPERATURES

Table 14.4a

Sand cast

Material Al Al–Cu Al–Mg Al–Si Al–Si–Cu

Nominal composition % Al Al Cu Cu Cu Mg Mg Mg Si Si Si Cu Si Cu

99.5 99.0 4.5 8 12 3.75 5 10 5 11.5 10 1.5 4.5 3

Density g cm−3

Coefficient of expansion 20–100◦ C 10−6 K−1

Thermal conductivity 100◦ C Wm−1 K−1

Resistivity µ m

Modulus of elasticity MPa × 103

2.70 2.70 2.75 2.83 2.93 2.66 2.65 2.57 2.67 2.65 2.74

24.0 24.0 22.5 22.5 22.5 22.0 23.0 25.0 21.0 20.0 20.0

218 209 180 138 130 134 130 88 159 142 100

3.0 3.1 3.6 4.7 4.9 5.1 5.6 8.6 4.1 4.6 6.6

69 — 71 — — — — 71 71 — 71

2.76

21.0

134

4.9

71 (continued)

The physical properties of aluminium and aluminium alloys

14–17

THE PHYSICAL PROPERTIES OF ALUMINIUM AND ALUMINIUM ALLOYS AT NORMAL TEMPERATURES—continued

Table 14.4a

Sand cast Nominal composition %

Material Al–Si–Cu–Mg∗ Al–Cu–Mg–Ni (Y alloy) Al–Cu–Fe–Mg Al–Si–Cu–Mg–Ni (Lo-Ex)

∗

Si Cu Mg Cu Mg Ni Cu Fe Mg Si Cu Mg Ni Si Cu Mg Ni

17 4.5 0.5 4 1.5 2 10 1.25 0.25 12 1 1 2 23 1 1 1

Density g cm−3

Coefficient of expansion 20–100◦ C 10−6 K−1

Thermal conductivity 100◦ C Wm−1 K−1

Resistivity µ m

Modulus of elasticity MPa × 103

2.73

18.0

134

8.6

88

2.78

22.5

126

5.2

71

2.88

22.0

138

4.7

71

2.71

19.0

121

5.3

71

2.65

16.5

107

—

88

Die cast.

Table 14.4b

THE PHYSICAL PROPERTIES OF ALUMINIUM AND ALUMINIUM ALLOYS AT NORMAL TEMPERATURES

Wrought

Specification

Nominal composition %

Condition

1199

Al

Sheet

1080A 1050A 1200

Al Al Al

99.992 99.8

Extruded Sheet

99.5

Extruded Sheet

99

Extruded Sheet

Density g cm−3

Coefficient of expansion 20–100◦ C 10−6 K−1

H111 H18

2.70

23.5

H111 H18

2.70

23.5

H111 H18

2.71

23.5

H111 H18

2.71

23.5

T4 T6

2.8 2.8

22 22

239 234 239 234 230 230 230 230 226 226 226 226 142 159

T3 T6

2.77 2.77

23 23

151

5.7 5.7

73 73

T8

2.59

23.6

88.2

9.59

76

T8

2.58

23.9

84

9.59

75

H111 H12 H14 H16 H18

2.74

23.0

180

3.9

0.003 0

69

151

4.8

0.002 4

—

∗

Extruded 2014A

2024 2090 2091

3103

Cu Mg Si Mn Cu Mg Mn Cu Li Zr Cu Li Mg Zr Mn

4.4 0.7 0.8 0.75 4.5 1.5 0.6 2.7 2.3 0.12 2.1 2.0 1.50 0.1 1.25

Sheet

Extruded

Thermal conductivity 100◦ C Wm−1 K−1

Resistivity µ cm 2.68 2.70 2.68 2.74 2.76 2.79 2.80 2.82 2.85 2.87 2.89 2.86 5.3 4.5

Temp. coeff. of resistance 20–100◦ C 0.004 2 0.004 2 0.004 2 0.004 2 0.004 2 0.004 1 0.004 1 0.004 1 0.004 1 0.004 0 0.004 0 0.004 0

Modulus of elasticity MPa × 103 69 69 69 69 69 69 69 69 69 69 69 69 74

(continued)

14–18

General physical properties

THE PHYSICAL PROPERTIES OF ALUMINIUM AND ALUMINIUM ALLOYS AT NORMAL TEMPERATURES—continued

Table 14.4b

Wrought

Specification 5083 5251

5154A 5454

Mg Mn Cr Mg Mn

4.5 0.7 0.15 2.0 0.3

Mg 3.5 Mg Mn Cr Li Mg Li Li Mg Mg Si Cu Cr Mg Si Mg Si

2.7 0.75 0.12 2.0 3.0 2.0 3.0 2.0 1.0 0.6 0.2 0.25 0.5 0.5 0.5 0.5

Mg Si Mn 6082 Mg Si 6463 Mg Si Al–Cu–Mg–Si Cu (Duralumin) Mg Si Mn Cu Mg Si Mn Al–Cu–Mg–Ni Cu (Y alloy) Mg Ni Al–Si–Cu–Mg Si (Lo–Ex) Cu Mg Ni Al–Zn–Mg Zn Cu Mn Mg 7075 Zn Mg Cu Cr 8090 Li Cu Mg Zr

1.0 1.0 0.7 1.0 1.0 0.65 0.4 4.0 0.6 0.4 0.6 4.5 0.5 0.75 0.75 4.0 1.5 2.0 12.0 1.0 1.0 1.0 10.0 1.0 0.7 0.4 5.7 2.6 1.6 0.25 2.5 1.3 0.95 0.1

Al–Li Al–Mg–Li Al–Li–Mg 6061

6063 6063A 6082

Coefficient of expansion Density 20–100◦ C g cm−3 10−6 K−1

Nominal composition % Condition∗ Sheet Sheet Extruded Sheet

H111 2.67 H12 H14 H111 2.69 H13 H16

Thermal Temp. conductivity coeff. of Modulus of ◦ 100 C Resistivity resistance elasticity −1 −1 ◦ Wm K µ cm 20–100 C MPa × 103

24.5

109

6.1

0.001 9

71

24

155

4.7

0.002 5

70

23.5

4.9 5.3 5.4 5.7 5.1

0.002 3 0.002 1 0.002 1 0.001 9

— 70 — — 70

24

Sheet Sheet

H111 2.68 H22 H24 T6 2.56 T6 2.52

147 142 138 134 147

— —

— —

— —

— —

77 79

Sheet

T6

—

—

—

—

84

Bar

H111 2.7 T4 2.7 T6 2.7

23.6 23.6 23.6

180 154 167

2.70

23.0

2.7 2.7 2.7

24 24 24 23 23

193 201 197 209 201 172 184

3.5 3.3 3.5 3.2 3.3 4.1 3.7

2.69 2.71 2.71 2.80

23.0 23.4 23.4 22.5

188 193 209 201 147

3.6 3.4 3.1 3.3 5.0

0.003 3 0.003 5 0.002 3

69 — 69 69 73

Extruded Sheet

H111 H14 2.67

Extruded

T4 T6 Bar T4 T5 T6 Bar/Extruded T4 T6 Sheet Bar Sheet

T4 T6 T5 T6 T6

2.46

68.9 68.9 68.9 0.003 3 0.003 5

0.003 1 0.003 1

71 — 69 69 69 69 69

Sheet

T4 T6

2.81

22.5

147 159

5.2 4.5

0.002 2 0.002 6

73 —

Forgings

T6

2.78

22.5

151

4.9

0.002 3

72

Forgings

T6

2.66

19.5

151

4.9

0.002 3

79

2.91

23.5

151

4.9

0.002 3

—

2.80

23.5

130

5.7

0.002 0

72

2.55

21.4

Forgings

Extrusion

Plate

T6

93.5

9.59

77

(continued)

The physical properties of copper and copper alloys

14–19

THE PHYSICAL PROPERTIES OF ALUMINIUM AND ALUMINIUM ALLOYS AT NORMAL TEMPERATURES—continued

Table 14.4b

Wrought

Specification Al–Cu–Mg–Si (Duralumin)

Al–Cu–Mg–Ni (Y alloy) Al–Si–Cu–Mg (Lo-Ex) Al–Zn–Mg

7075

8090

Nominal composition % Cu Mg Si Mn Cu Mg Si Mn Cu Mg Ni Si Cu Mg Ni Zn Cu Mn Mg Zn Mg Cu Cr Li Cu Mg Zr

4.0 0.6 0.4 0.6 4.5 0.5 0.75 0.75 4.0 1.5 2.0 12.0 1.0 1.0 1.0 10.0 1.0 0.7 0.4 5.7 2.6 1.6 0.25 2.5 1.3 0.95 0.1

∗ O = Annealed. H111 = Annealed. H12,22 = Quarter hard.

∗

Condition

Density g cm−3

Coefficient of expansion 20–100◦ C 10−6 K−1

Thermal conductivity 100◦ C Wm−1 K−1

Resistivity µ cm

Temp. coeff. of resistance 20–100◦ C

Modulus of elasticity MPa × 103

Sheet

TF

2.80

22.5

147

5.0

0.002 3

73

Sheet

TB TF

2.81

22.5

147 159

5.2 4.5

0.002 2 0.002 6

73 —

Forgings

TF

2.78

22.5

151

4.9

0.002 3

72

Forgings

TF

2.66

19.5

151

4.9

0.002 3

79

2.91

23.5

151

4.9

0.002 3

—

2.80

23.5

130

5.7

0.002 0

72

2.55

21.4

Forgings

Extrusion

TF

Plate

93.5

H14,24 = Half hard. H16,26 = Three-quarters hard. H18,28 = Hard.

9.59

77

T4 = Solution treated and naturally aged. T6 = Solution treated and artificially aged. See also pp. 22–1 and 22–2.

14.4 The physical properties of copper and copper alloys Table 14.5

THE PHYSICAL PROPERTIES OF COPPER AND COPPER ALLOYS AT NORMAL TEMPERATURES

Composition %

Melting point of Density liquidus −3 ◦C g cm

Coefficient of expansion 25–300◦ C 10−6 K−1

Electrical conductivity Thermal 20◦ C conductivity %IACS∗ Wm−1 K−1 Refs.

Oxygen-free high conductivity copper

Cu 99.99+

8.94

1 083

17.7

101.5

399

1

Tough pitch HC copper Phosphorus-deoxidised non-arsenical copper Deoxidised arsenical copper Silver bearing copper

O2 0.03 P 0.005–0.012 P 0.013–0.050 P 0.03 As 0.35 O2 0.02 Ag 0.05

8.92 8.94 8.94

1 083 1 083 1 083

17.7 17.7 17.7

101.5 85–96 70–90

397 341–395 298–372

2,3,4 5 5

45

177

2

101

397

2,3

Material

8.94 8.92

10.82 17.4 1 079

17.7

(continued)

14–20

General physical properties

THE PHYSICAL PROPERTIES OF COPPER AND COPPER ALLOYS AT NORMAL TEMPERATURES—continued

Table 14.5

Material Tellurium copper Chromium copper Beryllium copper

Composition %

Density g cm−3

Melting point of liquidus ◦C

Cu Te

8.94

1 082

Cu Cr Be Co Be Co

99.5 0.5 99.4 0.6 1.85 0.25 0.5 2.5

Coefficient of expansion 25–300◦ C 10−6 K−1 17.7

8.89

1 081

17

8.25

1 000

17

8.75

1 060

17

Electrical conductivity 20◦ C %IACS∗

Thermal conductivity Wm−1 K−1

Refs.

98

382

2

45(1)

82(2)

167 188

6

17(1) 23(2) 23(1) 47(2)

84 105 126 210

7 7

Cadmium copper

Cu Cd

99.2 0.8

8.94

1 080

17

85

376

6

Sulphur copper

Cu S

99.65 0.35

8.92

10.75

17

95

373

5

Cap copper

Cu Zn

95 5

8.85

1 065

18.1

56

234

5

Cu Zn Cu Zn Cu Zn

90 10 85 15 80 20

8.80

1 040

18.2

44

188

5

8.75

1 020

18.7

37

159

5

8.65

1 000

19.1

32

138

5

Cu Zn Cu Zn Cu Zn Cu Zn

70 30 67 33 63 37 60 40

8.55

965

19.9

28

121

5

8.50

940

20.2

27

121

5

8.45

920

20.5

26

125

5

8.40

900

20.8

28

126

5

Aluminium brass CuZn22Al2

Cu Zn Al

76 22 2

8.35

1 010

18.5

23

101

5

Naval brass CuZn36Sn

Cu Zn Sn

62 37 1

8.40

915

21.2

26

117

5

Free cutting brass CuZn39Pb3

Cu Zn Pb

58 39 3

850

900

20.9

26

109

3, 5

Cu Zn Pb

58 40 2

8.45

910

20.9

26

109

3, 5

20–25

Gilding metals CuZn10 CuZn15 CuZn20 Brass CuZn30 CuZn33 CuZn37 CuZn40

Hot stamping brass CuZn40Pb2

High tensile brass Nickel silver 10% 12% 15%

Cu 54–62 Others 7 max. Zinc—balance Cu Ni Zn Cu Ni Zn Cu Ni Zn

62 10 28 62 12 26 62 15 23

8.3–8.4

990 approx.

21 approx.

88–109

5

8.60

1 010

16.4

8.31

37

8, 9

8.64

1 025

16.2

7.71

30

8, 9

8.69

1 060

16.2

7.01

27

8, 9 (continued)

The physical properties of copper and copper alloys

14–21

THE PHYSICAL PROPERTIES OF COPPER AND COPPER ALLOYS AT NORMAL TEMPERATURES—continued

Table 14.5

Composition %

Material 18% 25%

Phosphor bronze CuSn3P CuSn5P CuSn7P CuSn8P Copper-nickel CuNi5Fe CuNi10FeMn CuNi30FeMn Silicon bronze Aluminium bronze CuAl5 CuAl8Fe CuAl10Fe5Ni5

Cu Ni Zn Cu Ni Zn

62 18 20 62 25 13

Sn P Sn P Sn P Sn P

3.5 0.12 5 0.09 7 0.12 8 0.05

Ni Fe Mn Ni Fe Mn Ni Fe Mn

5.5 1.2 0.5 10.5 1.5 0.75 31.0 1.0 1.0

Si Mn

3 1

Cu Al Cu Al Fe Al Fe Mn Ni

95 5 9 8 2 9.5 4.0 1.0 5.0

Density g cm−3

Melting point of liquidus ◦C

Coefficient of expansion 25–300◦ C 10−6 K−1

8.72

1 100

16.0

8.82

1 160

8.85

Electrical conductivity 20◦ C %IACS∗

Thermal conductivity Wm−1 K−1

Refs.

6.3

28

8, 9

17.0

5.1

21

8, 9

1 070

18.8

18.8

85

5, 10

8.85

1 060

18.0

16.8

75

5,10

8.80

1 050

18.5

14.0

67

5, 10

8.80

1 040

18.0

14.0

63

10

8.94

1 121

17.5

12.5

67

11

8.94

1 150

17.1

8.0

42

11

8.90

1 238

16.6

4.5

21

11

8.52

1 028

18.0

8.1

50

5

8.15

1 065

18.0

17.7

85

2

7.8

1 045

17.0

14.0

0

5

7.57

1 060

17.0

13

62

5

∗ The

International Annealed Copper Standard is material of which the resistance of a wire 1 metre in length and weighing 1 gram is 0.153 28 ohm at 20◦ C. 100% IACS at 20◦ C = 58.00 MS m−1 . (1) Solution heat treated. (2) Fully heat treated (to maximum hardness).

REFERENCES TO TABLE 14.5 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

OFHC® Copper—Technical Information, Americal Metal Climax Inc., 1969. R. A. Wilkins and E. S. Bunn, ‘Copper and Copper Base Alloys’, New York, 1943. C. S. Smith, Trans. AIMME, 1930, 89, 84. C. S. Smith, Trans. AIMME, 1931, 93, 176. Copper Development Association, Copper and Copper Alloy Data Sheets, 1968. Copper Development Association, High Conductivity Copper Alloys, 1968. Copper Development Association, Beryllium Copper, 1962. M. Cook, J. Inst. Metals, 1936, 58, 151. International Nickel Limited, Nickel Silver Engineering Properties, 1970. M. Cook and W. G. Tallis, J. Inst. Metals, 1941, 67, 49. International Nickel Limited, Cupronickel Engineering Properties, 1970.

14–22

General physical properties

14.5 The physical properties of magnesium and magnesium alloys Table 14.6

Material Pure Mag Mg–Mn Mg–Al Mg–Al–Zn

Mg–Zn–Mn Mg–Zn–Zr

THE PHYSICAL PROPERTIES OF SOME MAGNESIUM AND MAGNESIUM ALLOYS AT NORMAL TEMPERATURE

Nominal composition† % Mg (MN70)Mn (AM503)Mn AL80Al Be (AZ31)Al Zn (A8)Al Zn (AZ91)Al Zn (AZM)Al Zn (AZ855)Al Zn (ZM21)Zn Mn (ZW1)Zn Zr (ZW3)Zn Zr (Z5Z)Zn Zr (ZW6)Zn Zr

99.97 0.75 approx. 1.5 0.75 approx. 0.005 3 1 8 0.5 9.5 0.5 6 1 8 0.5 2 1 1.3 0.6 3 0.6 4.5 0.7 5.5 0.6

Condition

Density at 20◦ C g cm−3

Sol.

Liq.

Coeff. of thermal expansion 20–200◦ C 10−6 K−1

T1 T1 T1 T1

1.74 1.75 1.76 1.75

650 650 650 630

651 651 640

27.0 26.9 26.9 26.5

167 146 142 117

3.9 5 5.0 6

1 050 1 050 1 050 1 050

A A A A

T1

1.78

575

630

26.0

(84)

10.0

1 050

A

1.81 1.81 1.83 1.83 1.83 1.80

475∗

600

1 000 1 000 1 000 1 000 1 000 14 000

C

610

13.4 — 14.1 — — 14.3

A

510

84 84 84 84 84 79

C

595

27.2 27.2 27.0 27.0 27.0 27.3

A

470∗

475∗

600

27.2

79

14.3

1 000

A

AC AC T4 AC AC T4 AC T6 T1

Melting point ◦C

Thermal conductivity Wm−1 K−1

Electrical resistivity µ cm

Specific heat 20–200◦ C J kg−1 K−1

Weldability by argon arc process‡

1.80

T1

1.78

T1

1.80

625

645

27.0

134

5.3

1 000

A

T1

1.80

600

635

27.0

125

5.5

960

C

AC T6

1.81

560

640

27.3

113

6.6

960

C

T5

1.83

530

630

26.0

117

6.0

1 050

C

—

C

A

T1

27.0

Relative damping capacity§

A A

The physical properties of magnesium and magnesium alloys

Mg–Y–RE–Zr

Mg–RE–Zn–Zr

Mg–Th–Zn–Zr∗∗

Mg–Ag–RE–Zr

(WE43)Y RE() Zr (WE54)Y RE() Zr (ZRE1)RE ZN Zr (RZ5)Zn RE Zr (ZE63)Zn RE Zr (ZTY)Th Zn Zr (ZT1)Th Zn Zr (TZ6)Zn Th Zr (QE22)Ag RE(D) Zr (EQ21)RE(D) Ag Cu Zr

4.0 3.4 0.6 5.1 3.0 0.6 2.7 2.2 0.7 4.0 1.2 0.7 6 2.5 0.7 0.8 0.5 0.6 3.0 2.2 0.7 5.5 1.8 0.7 2.5 2.0 0.6 2.2 1.5 0.07 0.7

AC T6

1.84

550

640

26.7

51

14.8

966

A

AC T6

1.85

550

640

24.6

52

17.3

960

A

AC T5

1.80

545

640

26.8

100

7.3

1 050

A

AC T5

1.84

510

640

27.1

113

6.8

960

B

AC T6

1.87

515

630

27.0

109

5.6

960

A

T1

1.76

600

645

26.4

121

6.3

960

A

AC T5

1.83

550

647

26.7

105

7.2

960

A

AC T5

1.87

500

630

27.6

113

6.6

960

B

AC T6

1.82

550

640

26.7

113

6.85

1 000

A

AC T6

1.81

540

640

26.6

113

6.85

1 000

A

14–23

B

(B)

(continued)

14–24

Table 14.6

General physical properties

THE PHYSICAL PROPERTIES OF SOME MAGNESIUM AND MAGNESIUM ALLOYS AT NORMAL TEMPERATURE—continued

Material Mg–Zn–Cu–Mn

MG–Ag–RE–∗∗ Th–Zr

Mg–Zr

Nominal composition† % (ZC63)Zn Cu Mn (ZC71)Zn Cu Mn

6.0 2.7 0.5 6.5 1.3 0.8

(QH21)Ag RE(D) Th Zr (ZA)Zr

2.5 1.0 1.0 0.7 0.6

AC Sand cast. T4 Solution heat treated. T5 Precipitation heat treated. T6 Fully heat treated. † Mg–Al type alloys normally contain 0.2–0.4% Mn to improve corrosion resistance. ∗∗ Thorium containing alloys are being replaced by alternative Mg alloys.

Sol.

Liq.

Coeff. of thermal expansion 20–200◦ C 10−6 K−1

1.87

465

600

26.0

122

5.4

962

B

T6

1.87

465

600

26.0

122

5.4

62

B

AC T6

1.82

540

640

26.7

113

6.85

1 005

A

—

AC

1.75

650

651

27.0

(146)

(4.5)

1 050

A

A

Condition

Density at 20◦ C g cm−3

AC T6

Melting point ◦C

T1 Extruded, rolled or forged. RE Cerium mischmetal containing approx. 50% Ce. ∗ Non-equilibrium solidus 420◦ C. () Estimated value. RE(D) Mischmetal enriched in neodynium. RE() Neodynium + Heavy Rare Earths.

Thermal conductivity Wm−1 K−1

Electrical resistivity µ cm

‡ Weldability rating: A Fully weldable. B Weldable. C Not recommended where fusion welding is involved.

Specific heat 20–200◦ C J kg−1 K−1

Weldability by argon arc process‡

Relative damping capacity§

§ Damping capacity rating: A Outstanding. B Equivalent to cast iron. C Inferior to cast iron but better than Al-base cast alloys.

The physical properties of nickel and nickel alloys

14–25

14.6 The physical properties of nickel and nickel alloys THE PHYSICAL PROPERTIES OF WROUGHT NICKEL AND SOME HIGH NICKEL ALLOYS AT ROOM TEMPERATURE

Table 14.7

Alloy∗ Nickel Nickel 205

Coefficient of expansion Density 20–100◦ C g cm−3 10−6 K−1

Specific Thermal Electrical heat conductivity resistivity J kg−1 K−1 Wm−1 K−1 µ cm

8.89 8.89

13.3 13.3

456 456

74.9 75.0

9.5 9.5

8.83

13.9

423

21.7

51.0

8.91

15.5

380

29.4

41.2

8.46

13.7

419

17.4

61.4

8.88 8.42

14.9 13.3

421 460

19.5 14.8

52.0 103

8.11

13.75

448

11.2

119

8.36

11.6

419

13.6

122

8.44

12.8

410

9.8

129

8.19

13

435

11.4

125

Al Nb 825

12.6

425

12.0

122

8.3

12.2

14.26

107.5

8.08

14.9

460

12.4

101.7

8.05

14.7

500

12.3

108

Nominal composition %

99.4 99.6 30 Monel† alloy 400 1.5 1.0 31.0 Monel 450 0.7 Rem. 29 Monel alloy K-500 2.8 0.5 Cupro-nickel 55 Inconel† alloy 600 16 6 60.5 Inconel 601 23.0 1.4 15.1 55.7 21.5 Inconel 617 12.5 9.0 1.2 0.1 22 Inconel alloy 625 4 9 52.5 19.0 18.8 Inconel 718 5.2 3.1 0.9 0.5 15 Inconel alloy X-750 7 2.5 78.0 Inconel MA 754 20.0 1.0 0.6 35.5 INCO 330 44.8 18.5 1.2 35.0 38.4 INCO 020 20.0 3.5 2.5 0.6

Ni Ni Cu Fe Mn Ni Fe Cu Cu Al Ti Cu Cr Fe Ni Cr Al Fe Ni Cr Co Mo Al C Cr 0.3 Nb 0.3 Mo Ni Cr Fe Nb Mo Ti Al Cr 0.6 Fe 0.8 Ti Ni Cr Fe Y203 Ni Fe Cr Si Ni Fe Cr Cu Mo Nb

Ti Al

(continued)

14–26

General physical properties

THE PHYSICAL PROPERTIES OF WROUGHT NICKEL AND SOME HIGH NICKEL ALLOYS AT ROOM TEMPERATURE—continued

Table 14.7

Alloy∗

Nominal composition % 49.0 22.5 19.5 7.0 2.0 59.0 16.0 15.5 5.5 4.0 48.3 22.0 18.5 9.0 1.5 0.6 0.1 45 21 0.4 32 21 3 40 18 2

Ni Cr Fe Mo Cu Ni Mo Cr Fe W Ni Cr Fe Mo Co W C Fe Cr Ti Fe Cr Mo Fe Cr Si

Ni Span† alloy C-902 47 5.5 2.5 Hastelloy B 2 28 Hastelloy C 4 16 16 9 Hastelloy alloy X 21 18

Fe Cr Ti Mo Mo Cr Mo Cr Fe

Nimonic† alloy 75

Cr Ti Cr Ti Al Cr Ti Al Cr Co Ti Cr Co Mo Cr Co Mo

INCO G-3

INCO C-276

INCO-HX

Incoloy† alloy 800 Incoloy alloy 800 H‡ Incoloy alloy 825

Incoloy alloy DS

Nimonic alloy 80A

Nimonic alloy 81

Nimonic alloy 90

Nimonic alloy 105

Nimonic alloy 115

Coefficient of expansion Specific Thermal Electrical conductivity resistivity Density 20–100◦ C heat −3 −6 −1 −1 −1 −1 −1 10 K J kg K Wm K µ cm g cm

20 0.4 20 2.0 1.5 30 1.8 1.0 20 17 2.4 15 20 5 14 13 3

0.4

2 1.0

8.3

12.2

14.26

107.5

8.89

12.2

427

9.8

122.9

8.23

13.3

461

11.6

116

7.95

142.0

460

11.5

93

Cu Ti 814

14.0

441

11.1

113

7.91

14.2

450

12.0

108

8.10

7.6

502

12.1

101

9.22

10.3

373

11.1

137

8.64

10.8

406

10.1

125

8.23

13.8

485

9.1

118

8.37

11.0

461

11.7

102

8.19

12.7

460

11.2

117

8.06

11.1

461

10.9

127

8.18

12.7

445

11.5

114

Al

0.5 Al

1.4

Al

5 1.2

Al Ti

8.01

12.2

419

10.9

131

5 4

Al Ti

7.85

12.0

444

10.6

139 (continued)

The physical properties of nickel and nickel alloys

14–27

THE PHYSICAL PROPERTIES OF WROUGHT NICKEL AND SOME HIGH NICKEL ALLOYS AT ROOM TEMPERATURE—continued

Table 14.7

Alloy∗ Nimonic alloy 263

Nimonic alloy 901

Nimonic alloy PE16

Nimonic PK33

Astroloy

Rene 41

Rene 95

Udimet 500

Udimet 700

Waspaloy

∗ Where

Nominal composition % 20 20 6 13 35 6 16 32 3 18.0 14.0 7.0 2.25 2.1 54.8 15.0 17.0 5.3 4.0 3.5 55.4 19.0 11.0 11.0 1.5 3.1 61.5 14.0 8.0 3.5 3.5 3.5 2.5 53.7 18.0 18.5 4.0 2.9 2.9 55.5 15.0 17.0 5.0 4.0 3.5 58.7 19.5 13.5 4.3 1.3 3.0

Cr Co Mo Cr Fe Mo Cr Fe Mo Cr Co Mo Ti Al Ni Cr Co Mo Al Ti Ni Cr Co Mo Al Ti Ni Cr Co Mo Nb Al Ti Ni Cr Co Mo Al Ti Ni Cr Co Mo Al Ti Ni Cr Co Mo Al Ti

2 0.5

Ti Al

3

Ti

1.0 1.0

Ti Al

Density g cm−3

Coefficient of expansion 20–100◦ C 10−6 K−1

Specific heat J kg−1 K−1

Thermal conductivity Wm−1 K−1

Electrical resistivity µ cm

8.36

11.1

461

11.7

115

8.16

13.5

419

—

—

8.02

11.3

544

11.7

110

8.21

12.1

419

11.3

126

9

130.8

7.91

8.25

8.7

8.02

11.1

7.91

19.6

8.19

10.7

120.3

124

trade marks apply to the name of an alloy there may be materials of similar composition available from other producers who or may not use the same suffix along with their own trade names. The suffix alone e.g. Alloy 800 is sometimes used as a descriptive term for the type of alloy but trade marks can be used only by the registered user of the mark. † Registered Trade Mark. ‡ A variant on alloy 800 having controlled carbon and heat treatment to give significantly improved creep-rupture strength.

14–28

General physical properties

14.7 The physical properties of titanium and titanium alloys Table 14.8

PHYSICAL PROPERTIES OF TITANIUM AND TITANIUM ALLOYS AT NORMAL TEMPERATURES

Material IMI designation

Nominal composition %

CP Titanium

Commercially pure Cu 2.5 Pd 0.2 Al 2.0 Mn 2.0 Al 5.0 Sn 2.5 Al 6.0 V 4.0 Al 4.0 Mo 4.0 Sn 2.0 Si 0.5 Al 4.0 Mo 4.0 Sn 4.0 Si 0.5 Sn 11.0 Zr 5.0 Al 2.25 Mo 1.0 Si 0.2 Sn 11.0 Mo 4.0 Al 2.25 Si 0.2 Al 6.0 Zr 5.0 Mo 0.5 Si 0.25 Al 5.5 Sn 3.5 Zr 3.0 Nb 1.0 Mo 0.3 Si 0.3 Al 5.8 Sn 4.0 Zr 3.5 Nb 0.7 Mo 0.5 Si 0.35 C 0.06

IMI 230 IMI 260/261 IMI 315 IMI 317 IMI 318 IMI 550

IMI 551

IMI 679

IMI 680

IMI 685

IMI 829

IMI 834

Temp. coefficient of resistivity 20–100◦ C µ cm K−1

Specific heat 50◦ C J kg−1 K−1

48.2

0.002 2

528

+3.4

70 48.2 101.5

0.002 6 0.002 2 0.000 3

— 528 460

— — +4.1

6.3

163

0.000 6

470

+3.2

8.0

5.8

168

0.000 4

610

+3.3

4.60

8.8

7.9

159

0.000 4

—

—

4.62

8.4

5.7

170

0.000 3

400

+3.1

4.84

8.0

7.1

163

0.000 4

—

—

4.86

8.9

7.5

165

0.000 3

—

—

4.45

9.8

4.8

167

0.000 4

—

—

4.53

9.45

7.8

—

—

530

—

4.55

10.6

—

—

—

—

Density g cm−3

Coefficient of expansion 20–100◦ C 10−6 K−1

Thermal conductivity 20–100◦ C Wm−1 K−1

4.51

7.6

16

4.56 4.52 4.51

9.0 7.6 6.7

13 16 8.4

4.46

7.9

4.42

—

Resistivity 20◦ C µ cm

Magnetic suscept. 10−6 cgs units g−1

The physical properties of pure tin

14–29

14.8 The physical properties of zinc and zinc alloys Table 14.9

PHYSICAL PROPERTIES OF ZINC AND ZINC ALLOYS

Nominal composition

Material Zn Polycrystalline ZnAlMg BS1004A ZnAlCuMg BS1004B ZnAlCuMg ILZRO 12 (ZA12) ZA27

Coefficient of Density expansion g cm−3 10−6 K−1

99.993% Zn

7.13 (25◦ C) 4% Al 0.04% Mg 6.7 4% Al 1% Cu 0.04% Mg 11% Al 1% Cu 0.02% Mg

6.7

27% Al 2.3% Cu 0.015% Mg

5.0

6.0

Electrical Melting Thermal conductivity point conductivity % IACS (liquids) Wm−1 K−1 20◦ C Condition ◦ C

39.7 (20–250◦ C) 27 (20–100◦ C) 27 (20–100◦ C) 28 (20–100◦ C)

113

28.27

Cast

419.46

113

27

387

109

26

115

28.3

Pressure die cast Pressure die cast Chill cast

26 (20–100◦ C)

123

29.7

Chill cast

487

388 432

14.9 The physical properties of zirconium alloys Table 14.10

Alloy Zirconium 10 Zirconium 30 Zircalloy II

Zr 702

Zr 704 Zr 705 Zr 706

PHYSICAL PROPERTIES OF ZIRCONIUM ALLOY

Coefficient of expansion 20–100◦ C 10−6

Electrical resistivity µ cm

Composition %

Density g cm−3

Thermal cond. at 25◦ C Wm−1 K−1

Commercially pure Cu 0.55 Mo 0.55 Sn 1.5 Fe 0.12 Cr 0.10 Ni 0.05 Commercially pure with up to 4.5 Hf Cr+Fe 0.2–0.4 Sn 1–2 Nb 2.5 O2 0.18 Nb 2.5 O2 0.16

6.50

21.1

5.04

—

6.55

25.3

5.93

—

6.55

12.3

5.67

—

6.51

22

5.89

39.7

6.57

—

—

—

6.64

17.1

6.3

55

6.64

17.1

6.3

55

See also Table 26.36 page 26–52.

14.10 The physical properties of pure tin Melting point Boiling point Vapour pressure at

727◦ C 1 127◦ C 1 527◦ C Volume change of freezing Expansion on melting Phase transformation

α⇄β

231.9◦ C 2 270◦ C 7.4× 10−6 mmHg 4.4× 10−2 mmHg 5.6 mmHg 2.7% 2.3% 13.2◦ C

General physical properties

14–30

Density at 20◦ C Specific heat at 20◦ C Latent heat of fusion Latent heat of evaporation Linear expansion coefficient at 0–100◦ C Thermal conductivity at 0–100◦ C Electrical conductivity at 20◦ C Electrical resistivity at 20◦ C Temperature coefficient of electrical resistivity at 0–100◦ C Thermal EMF against platinum cold junction at 0◦ C hot junction at 100◦ C Superconductivity, critical temperature (Tc ) Viscosity Surface tension Gas solubility in liquid tin: Oxygen at 536◦ C Oxygen at 750◦ C Hydrogen at 1 000◦ C Hydrogen at 1 300◦ C Nitrogen

7.28 g/cm3 222 J kg−1 K−1 59.6 kJ kg−1 2 497 J kg−1 23.5×10−6 K−1 66.8 Wm−1 K−1 15.6 IACS 12.6 µ cm 0.004 6 K−1 +0.42 mV 3.722 K 0.013 82 poise at 351◦ C 0.011 48 poise at 493◦ C 548 mN m−1 at 260◦ C 529 nM m−1 at 500◦ C 0.000 18% 0.004 9% 0.04% 0.36% Very low

14.11 The physical properties of steels PHYSICAL PROPERTIES OF STEELS

Table 14.11

Thermal properties (see Notes) Material and condition Composition %

Temperature ◦C

Specific gravity g cm−3

Specific heat J kg−1 K−1

Coefficient of thermal expansion 10−6 K−1

Thermal conductivity Wm−1 K−1

Electrical resistivity µ cm

Carbon steels C Mn

0.06 0.4

'

Annealed

C Mn

0.08 0.31

'

Annealed

C Mn

' 0.23 En 3 0.6 060A22

Annealed

RT 100 200 400 600 800 1 000

7.87

— 48.2 520 595 754 875 —

— 12.62 13.08 13.83 14.65 14.72 13.79

65.3 60.3 54.9 45.2 36.4 28.5 27.6

12.0 17.8 25.2 44.8 72.5 107.3 116.0

RT 100 200 400 600 800 1 000

7.86

— 482 523 595 741 960 —

— 12.19 12.99 13.91 14.68 14.79 13.49

59.5 57.8 53.2 45.6 36.8 28.5 27.6

13.2 19.0 26.3 45.8 73.4 108.1 116.5

RT 100 200 400 600 800 1 000

7.86

— 486 520 599 749 950 —

— 12.18 12.66 13.47 14.41 12.64 13.37

51.9 51.1 49.0 42.7 35.6 26.0 27.2

15.9 21.9 29.2 48.7 75.8 109.4 116.7 (continued)

The physical properties of steels Table 14.11

14–31

PHYSICAL PROPERTIES OF STEELS—continued

Thermal properties (see Notes) Material and condition Composition % C Mn

' 0.42 En 8 0.64 060A42

Annealed

C Mn

0.80 0.32

'

Annealed

C Mn

1.22 0.35

'

Annealed

C Mn

' 0.23 En 14 1.51 150M19

Annealed

C Mn Ni

0.13 0.61 0.12



Annealed

 C 0.40 1% Ni Mn 0.67 En 12 Ni 0.80 Hardened 850◦ C OQ Tempered 600◦ C (1 h) OQ  C 0.37 Mn–Mo Mn 1.56 En 16 Mo 0.26 605A37 Hardened 845◦ C OQ Tempered 600◦ C (1 h)  C 0.37 Mn–Mo Mn 1.48 En 17 Mo 0.43 608M38 Hardened 850◦ C OQ Tempered 620◦ C (1 h) OQ

Temperature ◦C

Specific gravity g cm−3

Specific heat J kg−1 K−1

Coefficient of thermal expansion 10−6 K−1

Thermal conductivity Wm−1 K−1

Electrical resistivity µ cm

RT 100 200 400 600 800 1 000

7.85

— 486 515 586 708 624 —

— 11.21 12.14 13.58 14.58 11.84 13.59

51.9 50.7 48.2 41.9 33.9 24.7 26.8

16.0 22.1 29.6 49.3 76.6 111.1 122.6

RT 100 200 400 600 800 1 000

7.85

— 490 532 607 712 616 —

— 11.11 11.72 13.15 14.16 13.83 15.72

47.8 48.2 45.2 38.1 32.7 24.3 26.8

17.0 23.2 30.8 50.5 77.2 112.9 119.1

RT 100 200 400 600 800 1 000

7.83

— 486 540 599 699 649 —

— 10.6 11.25 12.88 14.16 14.33 16.84

45.2 44.8 43.5 38.5 33.5 23.9 26.0

18.4 25.2 33.3 54.0 80.2 115.2 122.6

RT 100 200 400 600 800 1 000 0 100 200 400 600 800 1 000

7.85

— 477 511 590 741 821 — 435 494 528 599 754 833 657

— 11.89 12.68 13.87 14.72 12.11 13.67

46.1 46.1 44.8 39.8 34.3 26.4 27.2

19.7 25.9 33.3 52.3 78.6 110.3 117.4 16.3 22.6 29.6 48.2 74.2 110.0 119.4

— 11.90 12.55 13.75 14.45

— 49.4 46.9 40.6 34.8

21.9 26.4 33.4 52.0 77.5

RT 100 200 400 600

7.84

Low alloy steels ∗— 7.85 486 507 544 586

RT 100 200 400 600

7.85

∗— 456 477 532 599

— 12.45 13.20 14.15 14.80

— 48.2 45.6 39.4 33.9

25.4 30.6 39.1 60.0 88.5

RT 100 200 400 600

7.85

∗— 482 494 519 595

— 12.45 13.00 13.90 14.75

— 45.6 44.0 39.4 33.9

22.5 27.2 34.3 52.5 77.5 (continued)

14–32

General physical properties

Table 14.11

PHYSICAL PROPERTIES OF STEELS—continued

Thermal properties (see Notes) Material and condition Composition %  C 0.32 1% Cr Mn 0.69 En 18B Cr 1.09 530A32 Annealed

 C 0.39 1% Cr Mn 0.79 En 18D Cr 1.03 530A40 Hardened 850◦ C OQ Tempered 640◦ C (1 h) OQ  C 0.28/0.33   Mn 0.4/0.6  Si 0.2/0.35 1% Cr–Mo  Cr 0.8/1.1    Mo 0.15/0.25 Hardened and tempered

Temperature ◦C

Specific gravity g cm−3

Specific heat J kg−1 K−1

Coefficient of thermal expansion 10−6 K−1

Thermal conductivity Wm−1 K−1

Electrical resistivity µ cm

RT 100 200 400 600 800 1 000

7.84

— 494 523 595 741 934 —

— 12.16 12.83 13.72 14.46 12.13 13.66

48.6 46.5 44.4 38.5 31.8 26.0 28.1

20.0 25.9 33.0 51.7 77.8 110.6 117.7

RT 100 200 400 600

7.85

∗— 452 473 519 561

— 12.35 13.05 14.40 15.70

— 44.8 43.5 37.7 31.4

22.8 28.1 35.2 53.0 78.5

42.7 — 42.7 — 40.6 — 37.3 — 31.0 — 28.1 30.1

— 22.3 27.1 34.2 — 52.9 — 78.6 — 110.3 117.1 122.2

— 12.25 12.70 13.70 14.45

— 42.7 42.3 37.7 33.1

22.2 26.3 32.6 47.5 64.6 —

0 RT 100 200 300 400 500 600 700 800 1 000 1 200

7.85

— — 477 515 544 595 657 737 825 883 — —

 C 0.41  1% Cr–Mo Mn 0.67 En 19 Cr 1.01  708A42 Mo 0.23 Hardened 850◦ C OQ Tempered 600◦ C (1 h) OQ  C 0.4  1% Cr–Mo Mn 0.4 En 20B Cr 1.1  Mo 0.7 Hardened and tempered

RT 100 200 400 600

7.83

RT 100 200 400 600 800

7.85

— 12.3 12.6 13.7 14.4 —

— 41.9 41.9 38.9 32.7 26.0

 C 0.4  3% Cr–Mo–V  Mn 0.6 En 40C Cr 3.0 897M39  Mo 0.8   V 0.2 Hardened and tempered  C 0.35   Mn 0.59 Low Ni–Cr–Mo Ni 0.20  En 19 Cr 0.88   Mo 0.20 Annealed

RT 100 200 400 600

7.83

— 12.5 12.9 13.5 14.0

— 37.7 37.7 34.8 31.0

RT 100 200 400 600 800 1 000

7.84

— 12.67 13.11 13.82 14.55 11.92 13.86

42.7 42.7 41.9 38.9 33.9 26.4 28.1

∗— — 473 519 561

— 477 515 595 737 883 —

—

21.1 27.1 34.2 52.9 78.6 110.3 117.1 (continued)

The physical properties of steels Table 14.11

14–33

PHYSICAL PROPERTIES OF STEELS—continued

Thermal properties (see Notes) Material and condition Composition %  C 0.23  3% Cr–W–Mo–V Mn 0.45    Si 0.45 Cr 2.87  W 0.59   Mo 0.51   V 0.77 Hardened and tempered  C 0.32 3% Ni Mn 0.55 En 21 Ni 3.47 Annealed

 C 0.33  3% Ni–Cr Mn 0.50 En 23 Ni 3.4   Cr 0.8 Hardened and tempered  C 0.41  1 Ni 1.43 1 2 % Ni–Cr–Mo En 24 Cr 1.07  Mo 0.26 817M40 Hardened 830◦ C OQ Tempered 630◦ C (1 h) OQ  C 0.32  1 Ni 2.60 2 2 % Ni–Cr–Mo En 25 Cr 0.67  Mo 0.51 826M31 Hardened 830◦ C OQ Tempered 650◦ C OQ  C 0.34   Mn 0.54 3% Ni–Cr–Mo Ni 3.53  En 27 Cr 0.76   Mo 0.39 Hardened and tempered

 C 0.29 1 Ni 4.23 4 4 % Ni–Cr En 30A Cr 1.26 Hardened 820◦ C AC Tempered 250◦ C (1 h)  C 0.18 2% Ni–Mo Ni 1.76 En 34 Mo 0.20 665A17 Blank carburised 920◦ C Hardened 800◦ C OQ

Temperature ◦C

Specific gravity g cm−3

Specific heat J kg−1 K−1

Coefficient of thermal expansion 10−6 K−1

Thermal conductivity Wm−1 K−1

— 11.9 12.4 13.1 13.6 14.1

38.5 33.6 33.1 30.6 29.3 28.9

35.5 39.0 46.2 63.0 85.4 —

Electrical resistivity µ cm

RT 100 200 400 600 800

7.83

RT 100 200 400 600 800 1 000

7.85

— 482 523 590 749 604 —

— 11.20 11.80 12.90 13.87 11.10 13.29

36.4 37.7 38.9 36.8 32.7 25.1 27.6

25.9 32.0 39.0 56.7 81.4 112.2 118.0

RT 100 200 400 600 800 1 000

7.85

— 494 523 599 775 557 —

— 11.36 12.29 13.18 13.72 10.69 13.11

34.3 36.0 36.8 36.4 31.8 26.0 27.6

25.6 31.7 38.7 56.7 81.7 111.5 117.8

RT 100 200 400 600

7.84

— — 12.40 13.60 14.30

24.8 29.8 36.7 55.2 79.7

RT 100 200 400 600

7.85

— — 11.55 13.10 13.85

27.7 32.1 38.7 57.3 82.5

RT 100 200 400 600 800 1 000

7.86

RT 100 200

RT 100 200

7.83

7.85

— 486 523 607 770 636 —

— 11.63 12.12 13.12 13.79 10.67 12.96

33.1 33.9 35.2 35.6 30.6 26.8 28.5

27.7 33.7 40.6 58.2 82.5 111.4 117.6

— 10.55 12.00

— 27.6 29.7

37.0 41.6 49.3

— 12.50 13.10

24.9 29.6 37.1

(continued)

14–34

General physical properties

Table 14.11

PHYSICAL PROPERTIES OF STEELS—continued

Thermal properties (see Notes) Material and condition Composition %  C 0.15  1 Ni 4.25 4 4 % Ni–Cr–Mo En 39B Cr 1.18  Mo 0.20 835A15 Blank carburised 920◦ C Hardened 800◦ C OQ  C 0.39  Low alloy steel  En 100 Mn 1.35 Ni 0.65 945M38  Cr 0.48   Mo 0.17 Hardened 850◦ C OQ Tempered 620◦ C (1 h) OQ  C 0.39  Low Ni–Cr–Mo Ni 1.39 En 110 Cr 1.02  816M40 Mo 0.14 Hardened 840◦ C OQ Tempered 650◦ C (1 h) OQ  C 0.17 43 % Ni–Cr Ni 0.86 En 351 Cr 0.71 635A14 Blank carburised 910◦ C Hardened 820◦ C OQ  C 0.17  1 Ni 1.25 1 4 % Ni–Cr–Mo En 353 Cr 1.02  Mo 0.15 815A16 Blank carburised 910◦ C Hardened 810◦ C OQ  C 0.16  2% Ni–Cr–Mo Ni 2.00 En 355 Cr 1.50  822A17 Mo 0.20 Blank carburised 910◦ C Hardened 810◦ C OQ  C 0.48  2% Si–Cu Mn 0.90 Si 1.98  Cu 0.64 Annealed

 C 0.05   2% B Mn 0.3  Si 0.7  B 1.96   Al 0.03 Hot worked

Temperature ◦C

Specific gravity g cm−3

Specific heat J kg−1 K−1

Coefficient of thermal expansion 10−6 K−1

Thermal conductivity Wm−1 K−1

Electrical resistivity µ cm

RT 100 200

7.85

— 11.30 12.55

36.3 40.1 46.7

RT 100 200 400 600

7.86

— 12.00 12.75 14.00 14.75

24.7 28.2 34.0 52.0 74.7

RT 100 200 400 600

7.84

— 12.00 12.65 13.65 14.30

24.8 29.2 35.6 54.0 78.0

RT 100 200

7.85

— 12.80 13.10

29.1 34.2 41.1

RT 100 200

7.87

— 11.30 12.45

31.8 36.6 43.2

RT 100 200

7.84

— 11.80 12.30

RT 100 200 400 600 800 1 000

7.73

— 498 523 603 749 528 —

7.72

461

0 RT 100 200 400 600 800

— 11.19 12.21 13.35 14.09 13.59 14.54 — — 10.0 11.0 11.9 11.8 13.3

34.5 39.2 45.7

25.1 28.5 30.1

41.9 47.0 52.9 68.5 91.1 117.3 122.3 24.9 — 30.9 38.7 57.4 81.9 — (continued)

The physical properties of steels Table 14.11

14–35

PHYSICAL PROPERTIES OF STEELS—continued

Thermal properties (see Notes) Material and condition Composition %  C 0.10   4% B Mn 0.14 Si 0.43  B 4.2    Al 0.53 As cast

Typically 1 C 0.10 2 % Mo–B Mo 0.5 ‘Fortiweld’ B 0.004 Normalised and stress-relieved 600◦ C  C 0.10 5% Cr Si 1.0 max AISI 502 Cr 4.0/6.0 Annealed

Temperature ◦C 0 RT 100 200 300 400 500 600 700 800 1 000 RT 100 200 400 600 700 30 100 200 400 600 800 1 200

Specific gravity g cm−3

Specific heat J kg−1 K−1

7.40

523

7.86

∗ 440

465 494 557 632 674

7.7

High alloy steels 7.6

Coefficient of thermal expansion 10−6 K−1

Thermal conductivity Wm−1 K−1

— — 9.5 — 10.4 — 11.2 — 11.8 — 13.0

Electrical resistivity µ cm 39.9 — 50.6 61.5 72.3 83.3 — 106.5 — 129.4 —

12.00 12.55 13.25 14.30 15.10 15.40

46.1 45.2 44.4 41.5 36.9 35.2

— 11.0 11.6 12.6 13.3 — —

36.0 — 35.2 — — 26.8 26.8

— 13.0 13.0 13.0 14.0 —

22.2 — — — — 31.4

80

20.0 24.5 31.0 48.5 74.5 88.0

 C 0.45  Mn 0.5 3% Cr–3% Si Si 3.5   Cr 3.5 Hardened and tempered

RT 100 300 500 700 900

 C 0.45  8% Cr–3% Si Mn 0.5 En 52 Cr 8.0   401S45 Si 3.4 Hardened and tempered

RT 100 300 500 700 900

7.6

— 13.0 13.0 13.0 14.0 —

22.2 — — — — 31.4

80.0 — 110.0

 C 0.40 Mn 0.3 11% Cr Cr 11.5 Hardened and tempered

RT 100 300 500 700 750

7.75

— 10.0 11.0 12.0 12.0 —

23.5 — — — — 24.3

60.0 — — — — 119.0

 C 0.12 9% Cr–Mo Cr 9.0 Mo 1.0 Normalised and tempered

RT 100 200 400 600 700

7.78

11.15 11.30 11.60 12.10 12.65 12.85

26.0 26.4 26.8 27.6 26.8 26.8

49.9 55.5 63.0 79.5 97.5 106.5

∗ 402

427 461 528 595 624

(continued)

14–36

General physical properties

Table 14.11

PHYSICAL PROPERTIES OF STEELS—continued

Thermal properties (see Notes) Material and condition Composition %  C 0.20  Mn 0.4    11% Cr–Mo–V–Nb Cr 11.0 Mo 0.5    V 0.7   Nb 0.15 Hardened and tempered  C 0.13  13% Cr Mn 0.25 En 56B Cr 12.95  420S29 Ni 0.14 Annealed  C 0.07 Mn 0.8 17% Cr Cr 17.0 Annealed  C 0.06 Mn 0.8 21% Cr Cr 21.0 Annealed

 C 0.22 Cr 30.4 30% Cr–Ni Ni 0.26 Hardened and tempered ' C 1.22 Mn 1.30 13% Mn 1 050◦ C Air-cooled

 C 0.28 Mn 0.89 28% Ni Ni 28.4 ◦ 950 C, WQ

 C 0.10  12% Cr–4% Al Mn 0.60 AISI 406 Cr 12.0   Al 4.5 Softened

Temperature ◦C

Specific gravity g cm−3

RT 100 200 400 600 800

7.75

RT 100 200 400 600 800 1 000

7.74

Specific heat J kg−1 K−1

Coefficient of thermal Thermal expansion conductivity 10−6 K−1 W m−1 K−1

Electrical resistivity µ cm

— 9.3 10.9 11.5 12.1 12.2 — 473 515 607 779 691 —

— 10.13 10.66 11.54 12.15 12.56 11.70

26.8 27.6 27.6 27.6 26.4 25.1 27.6

48.6 58.4 67.9 85.4 102.1 116.0 117.0

— 10.0 11.0 12.0

21.8

62.0

— 10.0 11.0 11.0 12.0 13.0

21.8

62.0

— 10.0

12.6

80.0

RT 100 200 300

7.7

RT 100 300 500 700 900

7.76

RT 100

7.90

RT 100 200 400 600 800 1 000

7.87

— 519 565 607 704 649 673

— 18.01 19.37 21.71 19.86 21.86 23.13

13.0 14.6 16.3 19.3 21.8 23.5 25.5

66.5 75.7 84.7 100.4 110.0 120.4 127.5

RT 100 200 400 600 800 1 000

8.16

— 502 519 540 586 586 599

— 13.73 15.28 17.02 17.82 18.28 18.83

12.6 14.7 16.3 18.9 22.2 25.1 27.6

82.9 89.1 94.7 103.9 111.2 116.5 120.6

RT 100 300 500 600 700 850

7.42

502

— 11.0 12.0 12.0 — 13.0 —

— 25.1 — 28.5 —

122 125 129 — 136 — 141

482

(continued)

The physical properties of steels Table 14.11

14–37

PHYSICAL PROPERTIES OF STEELS—continued

Thermal properties (see Notes) Material and condition Composition %

Temperature ◦C

 C 0.72   Mn 0.25 Ni 0.07 4% Cr–18% W  Cr 4.26   W 18.5 Annealed 830◦ C

RT 100 200 400 600 800 1 000

 C 0.16  16% Cr–Ni Mn 0.2 En 57 Ni 2.5   431S29 Cr 16.5 Softened  C 0.08   18% Cr–8% Ni Mn 0.3/0.5 En 58A Ni 8   302S25 Cr 18/20 1 100◦ C WQ  C 0.12  18% Cr–11% Ni  Mn 1.5  (Nb stabilised) Ni 11.0 En 58G  Cr 17.5    347S17 Nb 1.2 Softened  C 0.22   20% Cr–8% Ni Mn 0.6  Cr 20.0 (Ti stabilised)  Ni 8.5    Ti 1.2 Softened  C 0.15   19% Cr–14% Ni Mn 0.8  Ni 14 (Nb stabilised)  Cr 19    Nb 1.7 Softened  C 0.30 18% Cr–8% Ni–W  En55 Mn 0.6    Si 1.5 Ni 8    Cr 20   W 4 Softened  C 0.12 18% Cr–8% Ni–Al  Mn 0.3    (Ti stabilised) Ni 8.5 Cr 18.5    Ti 0.8   Al 1.4 Normalised and tempered

RT 100 300 500 RT 100 200 400 600 800 1 000

Specific gravity g cm−3 8.69

7.7

7.92

Specific heat J kg−1 K−1

Coefficient of thermal expansion 10−6 K−1

Thermal conductivity Wm−1 K−1

Electrical resistivity µ cm

— 410 435 502 599 716 —

— 11.23 11.71 12.20 12.62 12.97 12.44

24.3 26.0 27.2 28.5 27.2 26.0 27.6

40.6 47.2 54.4 71.8 92.2 115.2 120.9

— 482

— 10 11 12

18.8 — — 24.3

72.0 — — 103.0

— 511 532 569 649 641 —

— 14.82 16.47 17.61 18.43 19.03 —

15.9 16.3 17.2 20.1 23.9 26.8 28.1

69.4 77.6 85.0 97.6 107.2 114.1 119.6

15.9 — 17.2 18.8 20.1

72

RT 100 300 500 700

7.9

— 16.0 18.0 18.0 19.0

RT 100 300 500 700 900

7.72

— 15.0 15.0 16.0 17.0 18.0

RT 100 200 400 600

7.92

— 17.0 17.2 17.6 18.6

RT 100 300 500 700 900 1 050 RT 100 300 500 700 900

7.8

7.67

16.0 17.0 17.0 18.0 18.0 — — 15 15 15 16 17

82

— 15.1 16.8 20.1 24.3 13

85

29

125

18.0

85

26.0

125 (continued)

14–38

General physical properties

Table 14.11

PHYSICAL PROPERTIES OF STEELS—continued

Thermal properties (see Notes) Material and condition Composition %  C 0.10  Mn 0.3 12% Cr–12% Ni Ni 12.5  En 58D Cr 12.5 Softened  C 0.10   15/10/6/1 Mn 6.0  Cr 15.0 Cr–Ni–Mn–Mo  Ni 10.0    Mo 1.0 Solution treated 1 100◦ C  C 0.27  Mn 1.25 11% Cr—36% Ni Ni 36   Cr 11 Softened  C 0.1    Mn 1.3  Si 1.2 30% Cr–Ni  Ni 1.8    Cr 29.0 Softened  C 0.1  Mn 1.0 14% Cr–63% Ni Ni 63   Cr 14 Softened  C 0.30  Mn 3.0    17% Cr–17% Ni–Mo  –Co–Nb Ni 17.5  Cr 16.5  Mo 3.0    Nb 2.5    Co 7.0 Softened  C 0.4  Mn 0.9   13% Cr–13% Ni  –W–Nb Si 1.4 Ni 13.0  Cr 13.0   W 2.3   Nb 0.9 Normalised C 0.4   Mn 0.8   Si 1.0   13% Cr–13%  Ni–W–Mo–Co–Nb Ni 13  Cr 13  W 2.5   Mo 2.0    Nb 3.0  Co 10.0 Solution treated

Specific gravity g cm−3

Specific heat J kg−1 K−1

Coefficient of thermal Thermal expansion conductivity 10−6 K−1 Wm−1 K−1

RT 100

8.01

490

— 18

15.5 16.8

70 77

RT 100 200 400 600 700

7.94

∗ 477

14.80 15.70 16.75 18.25 18.95 19.30

12.6 13.8 15.4 18.8 21.8 23.0

74.1 80.0 86.7 99.4 108.4 114.4

RT 100 300 500

8.08

— 14 15 16

12.1 — — 18.4

97 — — 117

RT 200 800 1 000 1 100

7.5

10 11 13 13

15.9

88

26.4

126

RT 100 200 400 600 800 1 000

8.1

12.6

105

28.9

110

Temperature ◦C

RT 100 300 500 700

RT 100 200 400 600 800

RT 100 200 400 600 800

494 511 536 557 565

12.0 12.5 13.5 14.5 15.5 16.5

8.0

— 15 16 16 17

8.03

— 16.8 17.3 18.3 18.9 19.3

8.13

— 15.6 15.8 16.9 17.3 18.0

12.6

Electrical resistivity µ cm

93.8

— 13.4 17.2 18.8 22.2 25.5

(continued)

The physical properties of steels Table 14.11

14–39

PHYSICAL PROPERTIES OF STEELS—continued

Thermal properties (see Notes) Material and condition Composition %  C 0.27  Mn 0.77   20% Cr–10%  Ni 10.5    Ni–46% Co Cr 19.1 Mo 2.2    Nb 1.4    V 3.0    Co 46.6 Solution treated and aged '

Plain carbon B.S. 1 617A A 950◦ C, N 950◦ C

C Mn

0.11 0.35

C Mn

' 0.4 Plain carbon 0.5 BS 1 760 A 900◦ C, OQ 830◦ C T 650◦ C

C Mn Mo

 0.17 Carbon Mo 0.74 BS 1 398 0.50 A 920◦ C, SR 650◦ C

C Mn

0.25 1.55

'

1 21 % Mn BS 1 456A A 950◦ C, WQ 910◦ C, T 660◦ C

Specific Specific heat Temperature gravity J kg−1 ◦C g cm−3 K−1 RT 100 200 400 600 800

8.26

Coefficient of thermal Thermal Electrical expansion conductivity resistivity 10−6 K−1 Wm−1 K−1 µ cm — 14.8 15.0 15.2 15.9 16.8

— 14.7 16.3 19.7 23.0 26.0

100 200 300 400 500 600 100 200 300 400 500 600

12.2 12.6 13.2 13.6 13.9 14.2 11.8 12.4 12.8 13.3 13.7 14.2

48.6

19.5

42.3

23.5

100 200 300 400 500 600

12.4 12.8 13.1 13.4 13.8 14.2

100 200 300 400 500 600

13.2 13.3 13.7 14.1 14.7 15.2

Cast steels

C Cr Ni Mo

 1 0.29  1 2 % Cr Ni Mo 1.80 BS 1 458 ◦ ◦ 0.46  OQ 900 C, T 660 C 0.52

100 200 300 400 500 600

12.5 12.7 13.0 13.4 13.9 14.4

C Ni Cr Mo

 0.34  1 2.82 2 2 % Ni Cr Mo BS 1 459 0.74  ◦ ◦ 0.42 OQ 850 C, T 640 C

100 200 300 400 500 600

12.0 12.3 12.6 13.0 13.5 13.9

C Cr Mo

 0.24 3% Cr Mo 3.23 BS 1 461 0.51 OQ 900◦ C, T 690◦ C

100 200 300 400 500 600

12.2 12.4 12.7 12.9 13.3 13.6

24.2

27.6

39.4

27.3

(continued)

14–40

General physical properties

Table 14.11

PHYSICAL PROPERTIES OF STEELS—continued

Thermal properties (see Notes) Material and condition Composition %  C 0.1 5% Cr Mo Cr 4.06 BS 1 462 Mo 0.57 N 950◦ C, T 680◦ C

C Cr Mo Si

 0.13  9% Cr Mo 8.29 BS 1 463 1.1   OQ 900◦ C, T 690◦ C 1.1

Temperature ◦C

Specific gravity g cm−3

Specific heat J kg−1 K−1

Coefficient of thermal expansion 10−6 K−1

100 200 300 400 500 600

11.8 12.0 12.3 12.5 12.7 13.0

100 200 300 400 500 600

11.9 11.6 11.7 11.7 11.8 11.9

C Cr Ni Si

 0.27  13% Cr 12.1 BS 1 630 1.07  OQ 930◦ C, T 730◦ C 1.16

100 200 300 400 500 600

11.5 11.8 12.4 12.6 12.7 12.9

C Cr

' 0.47 Carbon Cr 0.85 BS 1 956 A N 870◦ C, T 635◦ C

100 200 300 400 500 600

12.5 12.9 13.2 13.4 13.5 13.6

C Ni

' 1 0.1 3 2 % Ni 3.35 BS 1 504–503 WQ 880◦ C, T 650◦ C

100 200 300 400 500 600

11.3 11.9 12.2 12.7 13.5 13.6

100 200 300 400 500 600

11.8 12.4 12.6 13.3 13.7 13.9

C 0.19 Cr 1.13 Mo 0.5



1 41 % Cr Mo BS 1 504–621 N 920◦ C, T 625◦ C

 C 0.13 13% Cr Mn 0.80 BS 1 630 A Cr 12.5 Hardened and tempered  C 0.25 13% Cr Mn 0.70 BS 1 630 C Cr 12.5 Hardened and tempered  C 0.07   Si 0.70 18% Cr–8% Ni Mn 0.80 BS 1 631 A  Ni 8.5    Cr 18.0 Normalised

Thermal conductivity Wm−1 K−1

Electrical resistivity µ cm 37.1

25.1

28.7

Cast corrosion-resisting steels 100 7.73 482 11.0 300 11.0 12.0 500 600

24.7 — — 27.6

56

100 300 500 600

7.75

482

11.0 11.0 12.0 —

24.3 — — 26.0

57

100

7.93

502

17.0

16.3

72

(continued)

The physical properties of steels Table 14.11

14–41

PHYSICAL PROPERTIES OF STEELS—continued

Thermal properties (see Notes) Material and condition Composition %  C 0.08  Si 1.00   Mn 0.50 18% Cr, 8% Ni Nb Ni 9.00  BS 1 631 B Nb  Cr 18.0   Nb 0.9 Normalised  C 0.12  Si 1.50   Mn 0.80 18% Cr, 8% Ni Ti Ni 9.00  BS 1 631 B Ti  Cr 19.00  Ti 0.6 Normalised  C 0.06  Si 0.70   Mn 0.70 19% Cr, 12% Ni, 1 Ni 12.0   3 2 % Mo BS 1 632 A  Cr 19.0   Mo 3.6 Water quenched  C 0.07  Si 1.00   18% Cr, 10% Ni Mn 1.00 2 21 % Mo Ni 10.5   BS 1 632 B  Cr 18.0   Mo 2.75 Normalised  C 0.08  Si 1.00    18% Cr, 10% Ni, Mn 0.50 Ni 10.5 2 12 % Mo Nb  Cr 18.0   BS 1 632 C Nb  Mo 2.75   Nb 0.90 Normalised  C 0.10  Si 1.50    18% Cr, 10% Ni, Mn 0.80 Ni 10.0 2 21 % Mo Ti  Cr 18.0    BS 1 632 C Ti Mo 2.75   Ti 0.60 Normalised  C 0.06  Si 0.70   18% Cr, 8% Ni, Mn 0.60 2 12 % Mo Ni 8.5   BS 1632 D  Cr 18.0   Mo 2.5 Normalised

Specific Specific heat Temperature gravity J kg−1 ◦C g cm−3 K−1

Coefficient of thermal Thermal Electrical expansion conductivity resistivity 10−6 K−1 Wm−1 K−1 µ cm

100 300 500 700

7.93

502

17.0 18.0 18.0 19.0

15.9 — — 20.1

100

7.78

444

17.0

15.5

RT 100

7.96

502

— 16.0

16.3

RT 100 200 400 500 600 800

7.96

502

— 16.5 16.9 17.2 17.4 17.9 19.0

16.3

73

RT 100

7.96

502

16.3 17.6

73

RT 100

7.78

448

— 17.0

15.5

78

RT 100

7.93

502

— 16.0

16.3

— 16.0

70

(continued)

14–42

General physical properties

Table 14.11

PHYSICAL PROPERTIES OF STEELS—continued

Thermal properties (see Notes) Material and condition Composition %

 C 0.25  Si 0.70 13% Cr Mn 0.70  BS 1 648 A Cr 12.5 Hardened and tempered  C 0.40  Si 0.80 27% Cr Mn 0.90  BS 1 648 B Cr 29.0 Tempered

Temperature ◦C

RT 100 300 500 600

Specific gravity g cm−3

Specific heat J kg−1 K−1

Cast heat-resisting steels 7.75 482

Coefficient of thermal expansion 10−6 K−1

Thermal conductivity Wm−1 K−1

Electrical resistivity µ cm

— 11.0 11.0 12.0 —

24.3 — — — 26.0

57

RT 100 200 400 600 800 1 000

7.63

482

— 10.2 10.8 11.0 11.5 12.4 13.3

20.9

70

RT 100 200 400 600 800 1 000

7.63

482

— 10.2 10.8 11.0 11.5 12.4 13.3

20.9

70

RT 100 500 800 1 100

7.74

502

— — 17.8 18.5 19.6

— 15.5 — 26.8 —

80

RT 100 300 500 700 900 1 000

7.92

435

— 13.6 14.5 15.4 16.5 17.7 18.3

10.9

86

 C 0.20   25% Cr 12% Ni Si 1.20 Mn 1.30 BS 1 648 E  Ni 13.0    Cr 25.0 Normalised

RT 100 200 400 600 800 1 000

7.92

544

— 16.5 16.6 16.9 17.6 18.2 18.7

13.8

85

 C 0.20  Si 1.00   25% Cr 12% Ni Mn 0.80 3% W Ni 12.0   BS 1 648 E  Cr 23.0   W 3.0 Normalised  C 0.20   Si 1.50 25% Cr 20% Ni Mn 1.00  BS 1 648 F Ni 20.0    Cr 25.0 Normalised

RT 100 300 500 700 900 1 000

7.90

502

— 15.0 16.0 16.0 17.0 19.0 —

12.6

87

RT 100 200 400 600 800 1 000

7.90

544

— 16.5 16.9 17.5 18.3 19.2 20.0

 C 1.70  Si 0.70 27% Cr Mn 0.70  BS 1 648 C Cr 27.0 Tempered C Si Mn Ni Cr C Si Mn Ni Cr W

 0.30   1.50 20% Cr 10% Ni 1.50 BS 1 648 D  10.0    20.0  0.35  1.50   21% Cr 8% Ni 0.80 4% W 7.0   BS 1 648 D  21.0   4.0

26.8 —

29.3 15.9

90

(continued)

The physical properties of steels Table 14.11

14–43

PHYSICAL PROPERTIES OF STEELS—continued

Thermal properties (see Notes) Material and condition Composition %

Temperature ◦C

Specific gravity g cm−3

Specific heat J kg−1 K−1

Coefficient of thermal expansion 10−6 K−1

Thermal conductivity Wm−1 K−1

Electrical resistivity µ cm

12.6

88

 0.35   25% Ni 15% Cr 0.90 0.75 BS 1 648 G  25.0    15.0

RT 100 300 500 700 900 1 000

7.90

502

— 15.0 16.0 17.0 17.0 18.0

 C 0.50   Si 2.00 Mn 1.50 35% Ni 15% Cr  Ni 35.0    Cr 15.0 Cast  C 0.50   Si 2.0  Mn 1.50 40% Ni 20% Cr  BS 1 648 H Ni 40.0    Cr 20.0 Cast  C 0.50   Si 2.00 60% Ni 15% Cr Mn 1.50 BS 1 648 K  Ni 60.0    Cr 15.0

RT 100 500 800 1 100

7.93

460

— — 16.0 16.5 17.6

— 13.4

100

RT 100 500 800 1 100

8.02

460

— — 16.0 16.4 17.4

— 13.4 — 23.9 —

105

RT 100 500 800 1 100

8.12

460

— — 14.2 15.3 16.5

— 13.4 — 23.0 —

108

C Si Mn Ni Cr

29.3

Cast

Notes: ∗ The values are a mean from RT up to the temperature quoted. 1. Where specific heats are quoted at temperatures above RT the values have been determined over a range of 50◦ C up to the temperature quoted. 2. Coefficients of expansion are mean values from RT up to the temperature quoted. 3. Electrical resistivity values are uncorrected for dimensional changes of the specimen with temperature. Original dimensions as at RT.

REFERENCES TO TABLE 14.11 1. ‘Metals Handbook’, 4th edn. 2. J. Woolman and R. A. Mottram, ‘Mechanical and Physical Properties of BS En Steels (BS 970, 1950), Pergamon Press. 3. Sundry technical information issued by industrial organizations e.g. British Steel Corporation, Mond Nickel Co. Ltd.

14–44

General physical properties

Table 14.12

SOME LOW TEMPERATURE THERMAL PROPERTIES OF A SELECTION OF STEELS

There is particular interest in the thermal properties (especially the thermal expansion) of steels used under conditions well below normal atmospheric temperature, and available information is set out below in respect of some such steels.

Material and condition Analyses %

Temperature ◦C

Coefficient of thermal expansion 10−6 K−1

Typically' C 0.09 9 Ni Ni 9 Double normalised and tempered

−200 −150 −100 −50 RT 100 200 300

−9.5 −9.7 −9.9 −10.2 10.5 11.0 11.7 12.3

 C 0.42   Ni 1.58 1 12 Ni–Cr–Mo Cr 1.19   Mo 0.24 Hardened 840◦ C OQ Tempered 650◦ C (1 h)/AC  C 0.12 1 Mo 0.54 2 Mo–B B 0.003 Hardened 960◦ C OQ Tempered 700◦ C ( 21 h) AC  C 0.27 Cr 3.14 3 Cr–Mo Mo 0.49 Hardened 900◦ C OQ Tempered 650◦ C (1 h) AC  C 0.09    Mn 6.23    Ni 9.88    Cr 14.88 15 Cr–10 Ni–6 Mn–Mo–V–B–Nb Mo 1.01    V 0.28    B 0.003   Nb 0.94 ◦ 1 150 C AC  C 0.13  Ni 4.16 4 21 Ni–Cr–Mo Cr 1.23  Mo 0.19 Blank carburised 890◦ C AC 820◦ C ( 41 h), transferred to 580◦ C ( 12 h) OQ  C 0.41  Cr 3.14 3 Cr–1 Mo–V Mo 0.97  V 0.20 Hardened 930◦ C OQ Tempered 700 ( 21 h) AC

−150 −100 −50 RT

−10.4 −11.2 −11.8 12.1

−150 −100 −50 RT

−10.5 −11.2 −11.8 12.2

−150 −100 −50 RT

−9.8 −10.3 −10.8 11.5

−150 −100 −50 RT

−14.7 −15.3 −15.7 16.2

−150 −100 −50 RT

−9.4 −9.8 −10.2 10.8

−150 −100 −50 RT

Thermal conductivity Wm−1 K−1 16.0 19.5 23.0 26.5 29.5 32.0 34.0 34.5

−9.7 −10.1 −10.5 11.2

(continued)

The physical properties of steels Table 14.12

14–45

SOME LOW TEMPERATURE THERMAL PROPERTIES OF A SELECTION OF STEELS—continued

Material and condition Analyses %  C 0.12  Ni 3.10 3 21 Ni–Cr–Mo Cr 0.91  Mo 0.16 Blank carburised 910◦ C AC Hardened 840◦ C ( 14 h) OQ Tempered 760◦ C OQ ' C 0.99 1 Cr 1.47 1 C–1 2 Cr

Hardened 850◦ C AC Tempered 650◦ C ( 21 h) AC  C 0.17  Ni 1.74 2 Ni–Mo Cr 0.2   En 34 Mo 0.22 Blank carburised 910◦ C AC Hardened 870◦ C OQ Tempered 770◦ C OQ ' C 0.11 Ni 3.04 3 Ni

910◦ C AC

Blank carburised Hardened 870◦ C OQ Tempered 770◦ C OQ

Temperature ◦C −150 −100 −50 RT

Coefficient of thermal expansion 10−6 K−1

Thermal conductivity Wm−1 K−1

−10.1 −10.6 −11.0 11.6

−150 −100 −50 RT

−9.6 −10.6 −11.6 12.3

−150 −100 −50 RT

−8.5 −9.5 −10.4 11.3

−150 −100 −50 RT

−9.9 −10.5 −11.0 11.5

∗ Thermal expansion values shown for temperatures other than RT are the mean values from RT to that temperature. For RT the instantaneous value is given.

REFERENCE TO TABLE 14.12 1. Sundry technical information issued by British Steel Corporation and Mond Nickel Co. Ltd.

This page intentionally left blank

15

Elastic properties, damping capacity and shape memory alloys

15.1 Elastic properties The elastic properties of a metal reflect the response of the interatomic forces between the atoms concerned to an applied stress. Since the bonding forces vary with crystallographic orientation the elastic properties of metal single crystals may be highly anisotropic. However, polycrystalline metals and alloys with a randomly oriented grain structure behave isotropically. Table 15.1 lists elastic constants for polycrystalline metals and alloys in an isotropic condition. Any preferred orientation or texture resulting from rolling, drawing or extrusion, for example, will result in departures from the listed values to a degree that depends upon the elastic anisotropy of the individual crystals (which may be deduced from the single crystal elastic constants of Tables 15.2 to 15.6 that follow) and the nature and extent of the preferred orientation. Since the elastic properties are determined by the aggregate response of the interatomic forces between all the atoms in the metal, the presence of small quantities of solute atoms in dilute alloys or their rearrangement by heat treatment will have relatively little effect on the magnitudes of their elastic constants. Consequently, the elastic constants of all the plain carbon and low alloy steels will be approximately the same unless some preferred orientation is present. Similarly with Cu-, Al- and Ni- base dilute alloys, etc. In the case of concentrated alloys there may be larger variations in elastic moduli, especially where there is a drastic change in the relative proportions of different phases in a multiphase alloy. In the case of ideal solid solutions the elastic moduli vary linearly with atom fraction. The elastic moduli of non-ideal solid solutions may show positive or negative deviations from linearity. Ordering produces an increase in elastic moduli. Increase in temperature causes a gradual decrease in elastic moduli. The decrease is fairly linear over wider ranges of temperature but sharply increases in magnitude as the melting point is approached. Discontinuities are observed when structural transformations occur. Ferromagnetic materials having a high degree of domain mobility may exhibit considerably higher elastic moduli below the Curie point in the presence of a high magnetic field. The lower elastic moduli in the absence of a magnetic field are due to magnetostrictive dimensional changes caused by stress-induced domain movement.

15–1

15–2

Elastic properties, damping capacity and shape memory alloys

Table 15.1

ELASTIC CONSTANTS OF POLYCRYSTALLINE METALS AT ROOM TEMPERATURE

Metal Aluminium Antimony Barium Beryllium Bismuth Brass 70Cu 30Zn Cadmium Caesium Calcium Cast Iron—Grey, BS 1452:1977 Grade 150 Grade 180 Grade 220 Grade 260 Grade 300 Grade 350 Grade 400 —Blackheart malleable BS 310:1972 Grades B340/12 to B290/6 Pearlitic malleable BS 3333:1972 Grades P4440/7 to P540/5 Whiteheart malleable BS 309:1972 Grades W340/3, W410/4 Nodular BS 2789:1973 Grades 370/17, 420/12 Grades 500/7, 600/3 Grades 700/2, 800/2 (pearlitic, normalised) pearlite 700/2, 800/2 (hardened, tempered) Cerium Chromium Cobalt Constantan 45Ni 55Cu Copper Cupro-nickel 70Cu 30Ni Duralumin Gallium Germanium Gold Hafnium Incoloy 800 20Cr, 32Ni bal Fe Indium Invar 64Fe 36Ni Iridium Iron (pure) Lanthanum Lead Lithium Magnesium Manganese

Young’s modulus GPa

Rigidity modulus GPa

Bulk modulus GPa

Poisson’s ratio

Ref.

70.6 54.7 77.9 12.8 318 34.0 100.6 62.6 1.7 19.6

26.2 20.7 19.3 4.86 156 12.8 37.3 24.0 0.65 7.9

75.2 — — — 110 — 111.8 51.0 — 17.2

0.345 0.25–0.33 — 0.28 0.02 0.33 0.35 0.30 0.295 0.31

1 2, 3 4 2 5 2 1 5 2 2, 6

100 109 120 128 135 140 145

40 44 48 51 54 56 58

— — — — — — —

0.26 0.26 0.26 0.26 0.26 0.26 0.26

7, 8 7, 8 7, 8 7, 8 7, 8 7, 8 7, 8

169

67.6

—

0.26

7, 9

172

68.8

—

0.26

7, 9

176

70.4

—

0.26

7, 9

169 169–174 176

66 65.9 68.6

— — —

0.275 0.275 0.275

7, 10 7, 10 7, 10

172

67.1

—

0.275

7, 10 11, 12 1 13, 16 1 1 14 1 2 2 15, 16 17, 18 38 2 1 16, 19, 20, 21 1 2, 12 1 23, 28 1 2, 24

33.5 279 211 162.4 129.8 144 70.8 9.81 79.9 78.5 141 196 10.6 144 528

13.5 115.3 82 61.2 48.3 53.8 26.3 6.67 29.6 26.0 56 73 3.68 57.2 209

— 160.2 181.5 156.4 137.8 — 75.4 — — 171 109 — — 99.4 371

0.248 0.21 0.32 0.327 0.343 0.34 0.345 0.47 0.32 0.42 0.26 0.334 0.45 0.259 0.26

211.4 37.9 16.1 4.91 44.7 191

81.6 14.9 5.59 4.24 17.3 79.5

169.8 — 45.8 — 35.6 —

0.293 0.28 0.44 0.36 0.291 0.24

(continued)

Elastic properties Table 15.1

15–3

ELASTIC CONSTANTS OF POLYCRYSTALLINE METALS AT ROOM TEMPERATURE—continued

Metal Manganese–copper 70Mn 30Cu (high damping alloy) Molybdenum Monel 400 63–70Ni, 2Mn, 2.5Fe, bal Cu Nickel Nickel silver 55Cu, 18Ni, 27Zn Nimonic 80A 20Cr, 2.3Ti, 1.8Al, bal Ni (fully heat-treated) Niobium Ni–span C902 (constant modulus alloy) Osmium

Young’s modulus GPa

Rigidity modulus GPa

Bulk modulus GPa

Poisson’s ratio

Ref.

—

—

25

93

22.4

324.8 185

125.6 66

261.2 —

0.293 0.32

1 26

199.5 132.5 222

76.0 49.7 85

177.3 132 —

0.312 0.333 0.31

1 1 27

104.9 186 559

37.5 66 223

170.3 — 373

0.397 0.41 0.25

1 28 16, 21, 29 18, 19, 21, 29 16, 19, 29, 30 31 15

Palladium

121

43.6

187

0.39

Platinum

170

60.9

276

0.39

— —

0.18 0.35

334

0.26

276 — 286 — — 103.6 — 160–169 168.7 165 165.3 165.2

0.26 0.30 0.25 0.447 0.42 0.367 0.34 0.27–0.3 0.293 0.296 0.287 0.295

2, 16, 32 16, 19, 29 2 18, 21, 29 15 2, 33 1 2, 23 34 1 1 1 1

— — — — 166

0.30 0.25–0.29 0.27–0.3 0.27–0.3 0.283

35 36 36 1, 36 1

12.0 196.3 — 28.5 54.0 58.2 108.4 311 319 97.9 158 — 69.4 89.8

0.28 0.342 0.16–0.3 0.45 0.26 0.357 0.361 0.28 0.22 0.20 0.365 0.265 0.249 0.38

2, 6 1 15 2, 6 2, 6 1 1 1 1 37 1 12 1 17, 18

Plutonium Potassium (−190◦ C)

87.5 3.53

Rhenium

466

34.5 1.30 (room temp.) 181

Rhodium Rubidium Ruthenium Selenium Silicon Silver Sodium Steel—Mild 0.75C 0.75C (hardened) Tool 0.98C, 1.03 Mn, 0.65 Cr, 1.01 W Tool 0.98C, 1.03 Mn, 0.65 Cr, 1.01 W (hardened) Maraging Fe–18Ni 8Co 5Mo Stainless austenitic (Fe–18Cr, 8–10 Ni) Stainless, ferritic (Fe–13Cr) Stainless, martensitic (Fe–13Cr, 0.1–0.3C) Stainless, martensitic (Fe–18Cr, 2Ni, 0.2C) Strontium Tantalum Tellurium Thallium Thorium Tin Titanium Tungsten Tungsten carbide Uranium Vanadium Yttrium Zinc Zirconium

379 2.35 432 58 113 82.7 6.80 208–209 210 201.4 211.6 203.2

147 0.91 173 — 39.7 30.3 2.53 81–82 81.1 77.8 82.2 78.5

186 190–201 200–206 200–215 215.3

72 74–86 78–79 80–83 83.9

15.7 185.7 47.1 7.90 78.3 49.9 120.2 411 534.4 175.8 127.6 66.3 104.5 98

6.03 69.2 16.7 2.71 30.8 18.4 45.6 160.6 219 73.1 46.7 25.5 41.9 35

15–4

Elastic properties, damping capacity and shape memory alloys

REFERENCES TO TABLE 15.1 1. G. Bradfield, ‘Use in Industry of Elasticity Measurements in Metals with the help of Mechanical Vibrations’, National Physical laboratory. Notes on Applied Science No. 30, HMSO, 1964. 2. W. Köster, Z. Electrochem. Phys. Chem., 1943, 49, 233. 3. W. Köster, Z. Metall., 1948, 39, 2. 4. ‘Metals Handbook’, Amer. Soc. Metals, Vol. 1, 1961. 5. D. J. Silversmith and B. L. Averbach, Phys. Rev., 1970, B1, 567. 6. S. F. Pugh, Phil. Mag. Ser. 7, 1974, 45, 823. 7. H. T. Angus, ‘Cast Irons, Physical and Engineering Properties’, Butterworths, London, 1976. 8. ‘Engineering Data on Grey Cast Irons’, Brit. Cast Iron Res. Assoc., 1977. 9. ‘Engineering Data on Malleable Cast Irons’, Brit. Cast Iron Res. Assoc., 1974. 10. ‘Engineering Data on Nodular Cast Irons’, Brit. Cast Iron Res. Assoc., 1974. 11. M. Rosen, Phys. Rev., 1969, 181, 932. 12. J. F. Smith, C. D. Carlson and F. H. Spedding, J. Metals, 9; Trans. AIME, 1957, 209, 1212. 13. ‘Physical and Mechanical Properties of Cobalt’, Cobalt Information Centre, Brussels, 1960. 14. ‘Cupro-Nickel Alloys, Engineering Properties’, Publ. 2969, Inco Europe, London, 1966. 15. Landolt-Börnstein, ‘Zahlenwerte und Funktionen’, Vol. 2, Part 1, Springer-Verlag, Berlin, 1971. 16. A. S. Darling, Int. Met. Rev., 1973, 91. 17. Private communication, Imperial Metal Industries, Witton, Birmingham. 18. A. S. Darling, Proc. Inst. Mech. Eng., 1965, Pt 3D, 180, 104. 19. W. Köster, Z. Metall., 1948, 39, 1. 20. A. Roll and H. Motz, Z. Metall., 1957, 48, 272. 21. K. H. Schramm, Z. Metall., 1962, 53, 729. 22. P. W. Bridgman, Proc. Amer. Acad. Arts Sci., 1922, 57, 41. 23. O. Bender, Ann. Phys., 1939, 34, 359. 24. M. Rosen, Phys. Rev., 1968, 165, 357. 25. D. Birchon, Engineering Mater and Design, 1964, 7, 606. 26. ‘Wrought Nickel-Copper Alloys, Engineering Properties’, Publ. 7011, Inco Europe, London, 1970. 27. ‘Nimonic Alloy 80A’, Publ. 3663, Henry Wiggin Ltd., Hereford, 1975. 28. ‘Controlled Expansion and Constant Modulus Nickel–Iron Alloys’, Publ. 6710, Inco Europe, London, 1967. 29. W. Köster, Z. Metall., 1948, 39, 111. 30. E. Grüneisen, Ann. Phys., 1908, 25, 825. 31. ‘Plutonium Handbook’, Ed. O. J. Wick, Gordon and Breach, New York, 1967, p. 39. 32. T. E. Tietz, B. A. Wilcox and J. W. Wilson, Standford Res. Instit. Calif., Report SU-2436, 1959. 33. R. L. Templin, Metals and Alloys, 1932, 3, 136. 34. J. Woolman and R. A. Mottram, ‘The Mechanical and Physical Properties of the British Standard En Steels’, Vol. 1, Pergamon, Oxford, 1964. 35. ‘18% Nickel Maraging Steels’, Publ. 4419, Inco Europe, London, 1976. 36. J. Woolman and R. A. Mottram, ‘The Mechanical and Physical Properties of the British Standard En Steels’, Vol. 3, Pergamon, Oxford, 1969. 37. ‘Commercial Uranium’, Brit. Nuclear Fuels Ltd., Warrington. 38. ‘Incoloy 800’, Publ. 3664, Henry Wiggin Ltd., Hereford, 1977.

15.1.1 Elastic compliances and elastic stiffnesses of single crystals Single crystals are generally anisotropic and therefore require many more constants of proportionality than isotropic materials. The relations between stress and strain are defined by the generalised Hooke’s law, which states that the strain components are linear functions of the stress components and vice versa. That is, εxx = S11 σxx + S12 σyy + S13 σzz + S14 σyz + S15 σzx + S16 σxy εyy = S21 σxx + S22 σyy + S23 σzz + S24 σyz + S25 σzx + S26 σxy ..................................................................................... εxy = S61 σxx + S62 σyy + S63 σzz + S64 σyz + S65 σzx + S66 σxy and correspondingly σxx = C11 εxx + C12 εyy + C13 εzz + C14 εyz + C15 εzx + C16 εxy ....................................................................................... σxy = C61 εxx + C62 εyy + C63 εzz + C64 εyz + C65 εzx + C66 εxy

Elastic properties

15–5

where σxx , σyy , σzz and σyz , σzx , σxy represent normal and shear stresses, respectively; εxx , εyy , εzz and εyz , εzx , εxy represent normal and shear strains, respectively. The elastic constants Sij and Cij are called the elastic compliances and elastic stiffnesses, respectively. Many of the constants are equal, the number of independent constants decreasing with increasing crystal symmetry. For example, in the hexagonal system there are five independent constants, while in the cubic system there are only three independent elastic compliances S11 , S12 , S44 with corresponding elastic stiffnesses C11 , C12 , C44 . The tensile and shear moduli will vary with orientation in a single crystal of a cubic metal according to 1 = S11 − 2[(S11 − S12 ) − 12 S44 ] (l 2 m2 + m2 n2 + l 2 n2 ) E 1 = S44 − 2[(S11 − S12 ) − 12 S44 ] (l 2 m2 + m2 n2 + l 2 n2 ) G where l, m, n are the direction cosines of the specimen axis with respect to the crystallographic axes. For an isotropic crystal S44 = 2(S11 − S12 ) and C44 = 21 (C11 − C12 ) hence E=

1 S11

and

G=

1 S44

Therefore, the degree of anisotropy is conveniently specified by 2(S11 − S12 ) S44

or

(C11 − C12 ) 2C44

15.1.2 Principal elastic compliances and elastic stiffnesses at room temperature The units are TPa−1 for Sij (elastic compliances) and GPa for Cij (elastic stiffnesses). Table 15.2

CUBIC SYSTEMS (3 CONSTANTS)

Metal

S 11

S 44

S 12

C 11

C 44

C 12

Ref.

Ag Al Au Ca Cr Cs (78 K) Cu Fe Ge Ir

22.9 16.0 23.4 94.0 3.08 1 676 15.0 7.67 9.73 2.24 2.28 1 215 1 339 315 2.71 549 6.56

22.1 35.3 23.8 83.0 9.98 676 13.3 8.57 14.9 3.72 3.90 531 526 104 9.0 233 35.2

−9.8 −5.8 −10.7 −31.0 −0.49 −762 −6.3 −2.83 −2.64 −0.67 −0.67 −558 −620 −144 −0.74 −250 −2.29

123 108 190 16.0 346 2.49 169 230 129 600 580 3.71 3.69 13.4 459 7.59 245

45.3 28.3 42.3 12.0 100 2.06 75.3 117 67.1 270 256 1.88 1.90 9.60 111 4.30 28.4

92.0 62.0 161 8.0 66.0 1.48 122 135 48.0 260 242 3.15 3.18 11.3 168 6.33 132

1 1, 2 1 3 1 4 1 1 1 5 6 7 8 9 1 1 1, 10, 11

K Li Mo Na Nb

(continued)

15–6

Elastic properties, damping capacity and shape memory alloys

Table 15.2

CUBIC SYSTEMS (3 CONSTANTS)—continued

Metal

S 11

S 44

S 12

C 11

C 44

C 12

Ref.

Ni (zerofield) Ni (saturation field) Pb Pd Pt Rb Si Sr Ta Th

7.67 7.45 93.7 13.7 7.35 1 330 7.74 148 6.89 27.2 27.4 101 6.76 2.49

8.23 8.08 68.0 14.0 13.1 625 12.6 174 12.1 20.9 22.0 91 23.2 6.35

−2.93 −2.82 −43.0 −6.0 −3.08 −600 −2.16 −60 −2.57 −10.7 −10.9 −46 −2.32 −0.70

247 249 48.8 224 347 2.96 165 14.7 262 75.3 77.0 40.8 230 517

122 124 14.8 71.6 76.5 1.60 79.2 5.74 82.6 47.8 45.5 11.0 43.2 157

153 152 41.4 173 251 2.44 64 9.9 156 48.9 50.9 34.0 120 203

1, 12 1 1 1 13 14 1 15 1, 16 17 18 19 1 1

Tl V W

Table 15.3

HEXAGONAL SYSTEMS (5 CONSTANTS)

Metal Be Cd Co Dy Er Gd

Hf Ho Mg Nd Pr Re Ru Sc Tb Tl Ti Y Zn Zr

S C S C S C S C S C S S C C S C S C S C S C S C S C S C S C S C S C S C S C S C S C

11

33

44

3.45 292 12.2 116 5.11 295 16.0 74.0 14.1 84.1 18.3 18.0 66.7 67.8 7.16 181 15.3 76.5 22.0 59.3 23.7 54.8 26.6 49.4 2.11 616 2.09 563 12.5 99.3 17.4 69.2 104 41.9 9.69 160 15.4 77.9 8.22 165 10.1 144

2.87 349 33.8 50.9 3.69 335 14.5 78.6 13.2 84.7 16.1 16.1 71.9 71.2 6.13 197 14.0 79.6 19.7 61.5 18.5 60.9 19.3 57.4 1.70 683 1.82 624 10.6 107 15.6 74.4 31.1 54.9 6.86 181 14.4 76.9 27.7 61.8 8.0 166

6.16 163 51.1 19.6 14.1 71.0 41.2 24.3 36.4 27.4 48.3 48.1 20.7 20.8 18.0 55.7 38.6 25.9 60.9 16.4 66.5 15.0 73.6 13.6 6.21 161 5.53 181 36.1 27.7 46.0 21.8 139 7.20 21.5 46.5 41.1 24.3 25.3 39.6 30.1 33.4

12 −0.28 24 −1.2 42 −2.37 159 −4.6 25.5 −4.2 29.4 −5.7 −5.7 25.0 25.6 −2.48 77 −4.3 25.6 −7.8 25.7 −9.50 24.6 −11.3 23.0 −0.80 273 −0.58 188 −4.30 39.7 −5.2 25.0 −83.0 36.6 −4.71 90.0 −5.10 29.2 −0.60 31.1 −4.0 74

13 −0.05 6 −8.9 41 −0.94 111 −3.2 21.8 −2.6 22.6 −3.8 −3.6 21.3 20.7 −1.57 66 −2.9 21.0 −5.0 21.4 −3.90 16.6 −3.80 14.3 −0.40 206 −0.41 168 −2.20 29.4 −3.60 21.8 −11.6 29.9 −1.82 66.0 −2.70 20.0 −7.0 50.0 −2.4 67

Ref. 1 1 1 1 1 1 1 1 1 1 20 21 20 21 22 22 1 1 1 1 23, 24 23, 24 25, 26 25, 26 1 1 20 20 27 27 21, 28 21, 28 29, 30 29, 30 1 1 31 31 1, 32 1 1 1

Elastic properties Table 15.4

TRIGONAL SYSTEMS (6 CONSTANTS)

Metal As

S C S C S C S C S C S C S C

Bi B Hg (83 K) Sb Se Te

Table 15.5

S C S C

Sn

Table 15.6 Metal Ga U

S C S C

11

33

44

12

13

14

Ref.

30.6 130 25.7 62.3 — 467 154 36.0 16.0 101 131 18.6 53.4 34.4

140 58.7 41.1 37.0 — 473 45.0 50.5 29.6 44.8 41 76.1 24.3 70.8

45.0 22.5 113 11.5 — 198 151 12.9 39.1 39.6 112 14.8 52.1 32.7

20.5 30.3 −7.8 23.1 — 241 −119 28.9 −6.1 31.4 −13 7.3 −16.1 9.0

−56.0 64.3 −11.2 23.4 — — −21 30.3 −6.0 27.0 −40 25.2 −13.6 24.9

1.7 −3.7 −21.4 7.3 — 15.1 −100 4.7 −12.4 22.1 56 5.6 26.7 13.1

33 33 1, 34 1, 34 — 35 36 36 1 1 1 1 1 1

TETRAGONAL SYSTEMS (6 CONSTANTS)

Metal In

15–7

11

33

44

66

12

13

Ref.

149 45.2 42.4 73.2

199 44.9 14.8 90.6

154 6.52 45.6 21.9

83 12.0 42.1 23.8

−44 40 −32.4 59.8

−96 41.2 −4.3 39.1

1 1 1 1

ORTHORHOMBIC SYSTEMS (9 CONSTANTS)

11

12

33

44

55

66

12.2 100 4.91 215

14.0 90.2 6.73 199

8.49 135 4.79 267

28.6 35.0 8.04 124

23.9 41.8 13.6 73.4

24.8 40.3 13.4 74.3

12 −4.4 37.0 −1.19 46.5

13 −1.7 33.0 0.08 21.8

23 −2.4 31.0 −2.61 108

Ref. 1 1 37 37

REFERENCES TO TABLES 15.2–15.6 1. Landolt-Börnstein, ‘Numerical Data and Functional Relationships in Science and Technology’, New Series, Group III, Vol. 2, Berlin, Springer-Verlag, 1979. 2. C. Gault, P. Boch, A. Dauger, Phys. Stat. Solidi, 1977, a43, 625. 3. M. Taut and H. Eschrig, Phys. Stat. Solidi, 1976, b73, 151. 4. F. J. Kollarits and T. Trivisonno, J. Phys. Chem. Solids, 1968, 29, 2133. 5. H. G. Purwins, H. Hieber and J. Labusch, Phys. Stat. Solidi, 1965, 11, k63. 6. R. E. Macfarlane, J. A. Rayne and C. K. Jones, Phys. Letters, 1966, 20, 234. 7. P. A. Smith and C. S. Smith, J. Phys. Chem. Solids, 1965, 26, 279. 8. G. Fritsch and H. Bube, Phys. Stat. Solidi, 1975, a30, 571. 9. H. C. Nash and C. S. Smith, J. Phys. Chem. Solids, 1959, 9, 113. 10. E. Walker and M. Peter, J. Appl. Phys., 1977, 48, 2820. 11. D. M. Schlader and J. F. Smith, J. Appl. Phys., 1977, 48, 5062. 12. K. Salama and J. A. Alers, Phys. Stat. Solidi, 1977, a41, 241. 13. R. E. Macfarlane, J. A. Rayne and C. K. Jones, Phys. Letters, 1965, 18, 91. 14. C. A. Roberts and R. Meister, J. Phys. Chem. Solids, 1966, 27, 1401. 15. S. S. Mathur and P. N. Gupta, Acustica, 1974, 31, 114. 16. W. L. Stewart et al., J. Appl. Phys., 1977, 48, 75. 17. P. E. Armstrong, O. N. Carlson and J. F. Smith, J. Appl. Phys., 1959, 30, 36. 18. J. D. Greiner, D. T. Peterson and J. F. Smith, J. Appl. Phys., 1977, 48, 3357. 19. M. S. Shepard and J. F. Smith, Acta Met., 1967, 15, 357. 20. E. S. Fisher and D. Dever, Trans. Met. Soc. AIME, 1967, 239, 48. 21. S. B. Palmer, E. W. Lee and M. N. Islam, Proc. Roy. Soc., 1974, A338, 341.

15–8

Elastic properties, damping capacity and shape memory alloys

E. S. Fisher and C. J. Renken, Phys. Rev., 1964, 135A, 482. J. D. Greiner et al., J. Appl. Phys., 1976, 47, 3427. J. T. Lenkkeri and S. B. Palmer, J. Phys., 1977, F7, 15. J. D. Greiner et al., J. Appl. Phys., 1973, 44, 3862. S. B. Palmer and C. Isci, Physica, 1977, 86–88, 45. E. S. Fisher and D. Dever, Proc. Rare Earth Res. Conf., Coronado Calif., 1968, Vol. 7, p. 237. K. Salama, F. R. Brotzen and P. L. Donoho, J. Appl. Phys., 1972, 43, 3254. R. Weil and A. W. Lawson, Phys. Rev., 1966, 141, 452. R. W. Ferris, M. L. Shepherd and J. F. Smith, J. Appl. Phys., 1963, 34, 768. J. F. Smith and J. A. Gjevre, J. Appl. Phys., 1960, 31, 645. D. P. Singh, S. Singh and S. Chendra, Ind. J. Phys., 1977, A51, 97. N. G. Pace and G. A. Saunders, J. Phys. Chem. Solids, 1971, 32, 1585. A. M. Lichnowski and G. A. Saunders, J. Phys., 1977, C10, 3243. I. M. Silvestrova et al., Mater. Res. Bull., 1974, 9, 1101. H. B. Huntington, ‘Solid State Physics’, (ed. F. Seitz and D. Turnbull), New York, Academic Press, 1958, Vol. 7, p. 213. 37. E. S. Fisher and H. J. McSkimin, J. Appl. Phys., 1958, 29, 1473. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36.

15.2 Damping capacity The damping capacity of a metal measures its ability to dissipate elastic strain energy. The existence of this property implies that Hooke’s law is not obeyed even at stresses well below the conventional elastic limit. In a perfectly elastic solid in vibration, stress and strain are always in phase and no energy is dissipated. There are two important types of damping: anelastic and hysteretic. In an anelastic solid there is a lag between the application of stress and the attainment of the resulting equilibrium strain; unless the stress changes exceedingly slowly. Processes with this characteristic give rise to an energy loss that reaches a peak at a critical frequency of vibration. An hysteretic solid has a stress–strain curve on loading that does not coincide with that on unloading. The area between the two curves is proportional to the energy loss and does not vary with the frequency with which the load cycle is traversed but changes in a complex fashion with peak stress. Damping from this class of mechanism is often high and since it does not vary with frequency is of particular interest to the engineer since it can contribute to vibration and noise reduction and can limit the intensity of vibrational stress under resonant conditions and thus minimise fatigue failure. Table 15.7 lists the specific damping capacity of a number of commercial alloys including some of very high damping that might be of interest to vibration engineers. In all cases the damping is predominantly of the hysteretic type. Table 15.7

THE SPECIFIC DAMPING CAPACITY OF COMMERCIAL ALLOYS AT ROOM TEMPERATURE

The specific damping capacity which is normally measured on solid cylinders stressed in torsion is defined as the ratio of the vibrational strain energy dissipated during one cycle of vibration to the vibrational strain energy at the beginning of the cycle.

Alloy Cast irons High carbon inoculated flake iron Spun cast iron Non-inoculated flake iron Inoculated flake iron Austenitic flake graphite Alloyed flake graphite Nickel-copper austenitic flake Undercooled flake graphite titanium/CO2 treated

Composition % 2.5% C, 1.9% Si, 1.0% Mn, 20.7% Ni, 1.9% Cr, 0.13% P 3.54% C, 3.39% G.C., 1.9% Si, 0.4% Mn, 0.38% P 3.3% C, 2.2% Si, 0.5% Mn, 0.14% P, 0.03% S 3.3% C, 2.2% Si, 0.5% Mn, 0.14% P, 0.03% S 2.5% C, 1.9% Si, 1% Mn, 20.7% Ni, 1.9% Cr, 0.03% P, 0.03% S 3.14% C, 2% Si, 0.6% Mn, 0.7% Ni, 0.4% Mo, 0.14% P, 0.03% S 2.55% C, 1.9% Si, 1.25% Mn, 15.2% Ni, 7.3% Cu, 2% Cr, 0.03% P, 0.04% S 3.27% C, 2.2% Si, 0.6% Mn, 0.35% Ti, 0.14% P, 0.03% S

Specific damping capacity %

Surface shear stress MPa

19.3

34.5

10.8

34.5

8.5

34.5

7.3

34.5

7.1

34.5

5.3

34.5

3.9

34.5

3.9

34.5 (continued)

Damping capacity

15–9

THE SPECIFIC DAMPING CAPACITY OF COMMERCIAL ALLOYS AT ROOM TEMPERATURE—continued

Table 15.7

Alloy

Composition %

Annealed ferritic nodular

3.7% C, 1.8% Si, 0.4% Mn, 0.76% Ni, 0.06% Mg, 0.03% P, 0.01% S, 6%), all wt %

Reference: D. Birchon, Engineering Materials and Design, Sept., Oct., 1964; also 307, 312.

0.5

15–10

Elastic properties, damping capacity and shape memory alloys

15.2.1 Anelastic damping Of interest to the physical metallurgist is the fact that a phase lag between stress and strain can give rise to a peak in energy dissipation or damping as a function of temperature or frequency. Several quite distinct atomic processes have been identified with damping peaks and measurements on these peaks in a wide variety of metals and alloys have been used to give diffusion data and to study precipitation, ordering phenomena and the properties of dislocations, point defects and grain boundaries. Table 15.8 identifies the damping peaks found in a number of pure metals and alloys with the relaxation process thought to be involved and also give an indication of the magnitude of the damping peak height. Detailed information on the specific mechanisms involved can be obtained from the reviews below and the references given for the respective damping peaks. The main types of peak that are observed are as follows. In cold worked pure metals movement of dislocation lines results in a number of low temperature peaks known as Bordoni peaks. Interaction of dislocations with point defects give rise to a further series of unstable peaks at higher temperatures; these are now called Hasiguti peaks. In alloys the stress-induced redistribution of solute atoms results in two types of peak, the Zener-type peak in substitutional solid solutions and the Snoek-type peak in interstitial solid solutions. The interaction of interstitial solute atoms with substitutional solute atoms give rise to a modified Snoek peak in ternary alloys. In cold worked alloys the interaction of interstitial solute atoms with dislocations results in the Köstler-type peaks at higher temperatures than the Snoek-type peaks. In pure metals and alloys the stress-induced migration of grain boundaries and/or polygonised (sub-grain) boundaries give rise to a further series of high temperature damping peaks. In the ideal case, for a relaxation process having a single relaxation time (τ) the logarithmic decrement (δ) will be given by δ = π

ωτ 1 + ω2 τ 2

where ω is the angular frequency and  is the modulus defect. The relaxation time, being diffusion controlled, varies with temperature according to an Arrhenius equation of the form τ = τ0 exp (H/RT) where τ0 is a constant, H is the activation energy controlling the relaxation process and T is the absolute temperature. The condition for maximum damping (δp ) is that ωτ = 1 and hence δp =

π (= πQ−1 max) 2

The modulus defect which is a measure of the strength of the relaxation can be highly orientation dependent and therefore the values given in the table below must be interpreted with caution. It must also be noted that for many measured damping peaks a distribution of relaxation times is found to be present. This leads to a broader peak being observed than would be present if a single relaxation time were operative. The decrement will be given by δ=π


i

i

ωτi 1 + ω2 τi2

where each i refers to a component of the total peak that has the same form as that of a peak arising from a single relaxation time. This theoretical aspect of the analysis of broad peaks in terms of a spectrum of τ’s has been comprehensively dealt with by A. S. Nowick and B. S. Berry in IBM Journal of Research and Development, 1961, 5(4), 297–311, 312–20. VIEWS 1. C. Zener, ‘Elasticity and Anelasticity of Metals’, Chicago: Chicago University Press, 1948. 2. K. M. Entwistle, ‘Progress in Non-Destructive Testing’ (edited by E. G. Stanford, J. H. Fearnon), vol. 2, p. 191, London: Heywood, 1960. 3. D. H. Niblett and J. Wilks, Adv. Phys., 1960, 9, 1. 4. K. M. Entwistle, Metall. Rev., 1962, 7, 175. 5. A. S. Nowick and B. S. Berry, ‘Anelastic Relaxation in Crystalline Solids’, Academic Press, 1972.

Damping capacity

15–11

In Table 15.8, metals and alloys are listed in alphabetical order with the highest concentration constituent first. The values of the modulus defect are deduced using the equation =

2δp −1 = 2Qmax π

−1 where δp is the peak decrement and Qmax is the peak value of Q−1 . These relationships are valid only if the damping arises from a process having a single relaxation time. In most cases a distribution of relaxation times exists and the peaks are broader, but there is only rarely sufficient published data to permit this distribution to be deducted. As many authors do not make it clear which damping units they use, the quoted values of Modulus Defect in the table must not be interpreted too precisely. The aim is to record the existence of peaks and list the suggested mechanism giving rise to them, and the values of  serve to indicate the approximate strength of the relaxation. If detailed quantitative data are sought peaks should be analysed in particular cases using the method of Berry and Nowick.

Table 15.8

Alloy Ag Ag

Ag

Ag

Ag Ag–Au Ag–Cd Ag–Cd Ag–In Ag–In Ag–In Ag–In Ag–In Ag–Sb Ag–Sn Ag–Sn Ag–Sn

ANELASTIC DAMPING

Composition and physical condition

Peak temp.

Frequency Hz

Modulus defect

Activation energy kJ mol−1

Mechanism or type

Ref.

99.999% Ag (CW∗ 16% at 4.2 K) 99.999% Ag single crystal deformed at RT 43.7% [121] 5.4% [111] Single crystals deformed at (5.4% [111]) RT 99.99% Ag

37 K 50 K ∼50 K

0.7 0.7 ∼600

2.6 × 10−2 1.2 × 10−2 13 × 10−4

0.84 0.84 —

Bordoni type Bordoni type Bordoni type

1, 2 1, 2 3

∼50 K

∼600

6 × 10−3

—

Bordoni type

3

173 K

103

—

21.3

4

(CW∗ at RT)

200 K

103

—

35.6 etc.

99.998% Ag (84% area reduction, then annealed at 500◦ C for 1 h) 99.999% Ag 25–80 At. % Au 42 At. % Au

163◦ C 356◦ C

1.04 0.98

38 × 10−4 39 × 10−4

92.0 177

Point defect/ dislocation interactions Point defect/ dislocation interaction Grain boundary Grain boundary

150 K 364– 398◦ C 460◦ C ∼230◦ C — 220◦ C 367– 452◦ C 580– 536 K — — ∼500 K

1.5 1.0

1.3 × 10−2 2 × 10−3 – 2 × 10−2 0.104 max — — 4 × 10−2 9.2 × 10−2 – 1.38 × 10−1 < 3.5 × 10−3 – 7 × 10−3 6.5 × 10−4 – 6 × 10−3 ∼20 × 10−3

92 175.7–

Grain boundary Zener

6 7

165.3 152.3 123.0 146.9 159–188

Grain boundary Zener Zener Zener Grain boundary

8 9 9 10 6, 11

152.8– 130.5 159 133.1 146.9– 139.7 — 172–188

Zener

12

Zener

9

Zener

13

Zener Grain boundary

14 6, 11

136 —

Zener Zener

15 6, 11

131.8 171–184

Zener Grain boundary

15 6, 11

29 At. % Cd 39 At. % Cd 32 At. % Cd 0.9–32 At. % Cd 9.6 At. % In– 17.9 At. % In 7.5 At. % In– 15.6 At. % In 10.8 At. % In– 18.1 At. % In 16 At. % In 1 At. % In– 16 At. % In 6.3 At. % Sb 0.93 At. % Sn 8.1 At. % Sn 0.9 At. % Sn– 8.0 At. % Sn

270◦ C 355– 450◦ C ∼510 K 200◦ C ∼550 K 390– 440◦ C

0.36 ∼1 ∼1 0.6 1.5 ∼1 ∼1 10−3 –

10−1 1.6 1.5

10−3 –10−1 1.5 10−3 –10−1 1.5

2.9 × 10−2 9.5 × 10−2 – 1.27 × 10−1 ∼5 × 10−3 1.3 × 10−3

∼5 × 10−3 1.08 × 10−1 – 1.27 × 10−1

4 5 5

(continued)

Elastic properties, damping capacity and shape memory alloys

15–12 Table 15.8

ANELASTIC DAMPING—continued

Activation energy kJ mol−1

Mechanism or type

Ref.

142–136

Zener

16

2.3

Bordoni type

17, 18

— — — —

11.6 18.3 —

Bordoni type Bordoni type ? ?

19

1.08 × 103 1.06 × 103 4 × 10−2 2 × 103

1.6 × 10−3 2.4 × 10−3 ∼10−2 ∼2 × 10−3

16.3 28.9 24 —

Bordoni type Bordoni type Hasiguti type Hasiguti type

20

130 K

∼1

∼30 × 10−3

—

22

99.994% [111] [100] [110]

139 K 153 K 196 K

107 – 5 × 107

— — —

4.1 6.0 19.5

Bordoni type with contribution from impurity–dislocation interactions Bordoni type Bordoni type Bordoni type

Al

99.999% Al (CW∗ 18% at RT)

213 K

103

—

38.5

Point defect/ dislocation interaction

4

Al

99.6% Al reduced by 69%

270◦ C

5 × 10−1

∼1.2 × 10−1

—

24

Al

99.6% Al reduced by 99.3%

270◦ C

5 × 10−1

∼2 × 10−1

—

400◦ C

5 × 10−1

∼10−1

—

∼300◦ C

∼1

48.2

340◦ C

2.32

10−3 –10−2 depending on CW 7.85 × 10−2

Relaxation of stresses by shear deformation and recrystallisation Relaxation of stresses by shear deformation and recrystallisation Associated with grain boundaries Associated with dislocations

Composition and physical condition

Peak temp.

Frequency Hz

Modulus defect

15 At. % Zn– 30 At. % Zn 99.99% Al (CW∗ 3% at RT) 99.999% Al deformed at 20 K then at 80 K

280– 232◦ C 24 K

0.68 1.2 × 104

1.2 × 10−2 – 1.4 × 10−2 1.5 × 10−4

70 K 100 K 115 K 155 K

3 3 3 3

99.99% Al (CW∗ 6% at RT) 99.999% Al 2 h at 470 K, then CW∗ by 0.5% at 77 K 99.999% Al deformed at 85 K at 10−4 and cycled 102 times

83 K 119 K 110 K 155 K

Al

Alloy Ag–Zn Al Al

Al Al Al

Al Al

99.999% Al deformed by 65%, annealed and deformed again 99.96% Al area reduced by 75% annealed at 325◦ C for 2 h

Al

99.991% Al

275◦ C

0.69

Al

99.999% Al CW∗ 4% then irradiated by neutrons at 80 K, annealed 360 K 99.9999wt% Al (bamboo boundaries) 99.999wt% (3 h at 402◦ C) 99.999wt% (single crystal sheet) 99.9999wt% Al (polycrystalline wires) (grain size ≪ wire dia.) (grain size ≫ wire dia.) 99.999wt% (single crystal)

110 K

2.5

∼300◦ C

∼1

1.44 eV

396◦ C

8 × 10−3

0.52

400◦ C

1.3

Al Al Al Al

Al Al–Ag

20% Ag (quenched from 520◦ C, aged at 155◦ C)

1.4 × 10−1 1.6 × 10−2

144.3

21

23

24

24 25

Grain boundary

5

134

Grain boundary

26

—

Point defect/ dislocation interaction

241

Grain boundary

251

Polygonisation

256

Dislocation network

258

1.5 eV

272 210◦ C 170◦ C 365◦ C

∼1 ∼1 1

140◦ C

0.25

184 eV 1.2 × 10−2

105–113

Grain boundary Not known Point defects/ dislocation interaction Stress induced change of local degree of precipitation

265 27

(continued)

Damping capacity Table 15.8

ANELASTIC DAMPING—continued

Alloy

Composition and physical condition

Peak temp.

Frequency Hz

Modulus defect

Al–Ag

2.5% Ag–30% Ag annealed

∼140– ∼170◦ C

∼1

∼5 × 10−4 ∼8 × 10−3

155

Al–Ag

15% Ag (quenched from 200◦ C)

160– 210◦ C

0.45 0.45

∼4 × 10−3 ∼4 × 10−3

— —

Al–Ag

Al + 30% Ag (1) quenched (2) aged at 520 K Al–2, 5, 10, 30wt% Ag (1 mm dia. wires of g.s. 0.75 mm)

410 K

—

8 × 10−3

92

420 K 400 K; 450 K; variable

— ∼1

14 × 10−3

110

175◦ C

0.9

2 × 10−3

128(117)

120◦ C

0.5

2.4 × 10−3

92

580◦ C

0.3

0.44

20◦ C

2 × 103

6 × 10−4 max

56.5

Al–Fe

Al–0.25% Fe (annealed at 600◦ C for 6 h) Al–0.06% Fe

∼280◦ C ∼310◦ C ∼280◦ C

20– 2 × 102 ∼1

117 167 ∼140

Al–Fe

Al–0.16% Fe–0.5% Fe

310–360◦ C 440–480◦ C

∼1 ∼1

∼5 × 10−4 ∼2 × 10−3 Depends on grain size Depends on grain size

Al–Fe–Ce

Rapidly quenched A–8.6wt% Fe–3.8wt% Ce (extrusion compacted, rapidly solidified) Al–0.6% Mg– 0.6% Si (quenched from 480◦ C and aged at 230◦ C for 26 21 h) Al–0.03at% Mg (0.5% tensile strain at 208 K)

475 K

0.8

96 × 10−4

150

215◦ C

2.25

5.5 × 10−2

243 K; 333 K

∼1

RT 165◦ C

2.5 × 104 1.5

1.2 × 10−5 7 × 10−4

— 116.3

∼40◦ C

∼2

—

66.9

∼80◦ C ∼120◦ C ∼227◦ C ∼203◦ C

∼2 ∼2 ∼2 ∼2

— — — —

— 102.5 125.5 125.5

∼100− ∼200◦ C 440 K

0.4–98

10−4 –2 × 10−3

135.0

Al–Ag

Al–Cu Al–Cu Al–Cu Al–Cu– Mg–Si Al–Fe

Al–Mg– Si Al–Mg Al–Mg Al–Mg

Al–Mg

Al–Mg Al–Mg Al–Mn– Fe–Si

4% Cu (quenched from 520◦ C reverted at 200◦ C) 4% Cu (quenched, reverted, aged at 200◦ C for 144 h) Al–0.015wt% Cu (3 h at 402◦ C) Quenched from 500◦ C, aged at 50◦ C

2% Mg 5.45% Mg (quenched from 500◦ C, aged at 250◦ C) 7.5% Mg (annealed at 400◦ C then quenched)

7.5% Mg (annealed at 400◦ C, cooled slowly) 0.93% Mg–12.1% Mg Al–1.07wt% Mn– 0.52wt% Fe–0.11wt% Si (cold worked and precipitated)

1

Activation energy kJ mol−1

15–13

Mechanism or type

Ref.

Diffusion controlled relaxation of partial dislocations around precipitates Associated with γ Associated with clustering Zener in solid solution Zener in γ phase Zener; partial reversion of G.P. zones; gamma precipitation Zener Stress induced change of shape of precipitate Polygonisation

28

29 247 247 301

30, 31 30 256

Stress induced ordering of complex atom group ? ? Grain boundary

32

Grain boundary Related to relaxation of stresses and precipitation of Fe on grain boundaries Zener

34 34

126

Stress induced diffusion of solutes (?)

35

0.32 eV; 0.22 eV

Dragging of solute atoms and atom/vacancy pairs Thermoelastic Zener

299

Relaxation of solute clusters Zener Zener ?

38

Zener

39

Ke, 3 times wider than single relaxation

305

∼140 200

33 33 34

263

36 37

(continued)

15–14 Table 15.8

Elastic properties, damping capacity and shape memory alloys

ANELASTIC DAMPING—continued

Composition and physical condition

Peak temp.

Frequency Hz

Insoluble Si particles in Al (solution-treated and aged)

420 K

∼1

110–180◦ C

1

Al–Zn

3.7% Zn (quenched from 450◦ C to RT)

21◦ C

0.26

A1050/SiC Au

Composite 99.999% Au. (CW∗ 16% at 4.2 K)

300 K 43 K 65 K 77 K

0.07 0.5 0.5 0.5

Au

99.999% Au (annealed at 1 170 K for 4 h, then CW∗ 3% at 77 K) 99.999% Au (CW∗ 16% at 70 K)

120 K 180 K 210 K

4 × 10−2 – 2 × 103

∼10−4 ∼5 × 10−3 ∼10−4

130 K 190 K 210 K 160 K

4.0 4.0 4.0 ∼1

4 × 10−3 2.2 × 10−2 6 × 10−3 ∼2 × 10−3

Alloy Al–Si Al–Si

Au Au

Au Au

99.999% Au (quenched) from 700◦ C) 99.999% Au (quenched from 800◦ C) 99.999% (quenched from 1 000◦ C)

Modulus defect

Activation energy kJ mol−1 0.92 eV

∼2 × 10−3

53

1.4 × 10−1 1.6 × 10−1 1.6 × 10−1

9.62 18.4 18.4 — 35.7 58 177 (two stages) 21.3 32.84 34.7 68(?)

230 K

∼2 × 10−3

—

∼290 K

∼1

∼4 × 10−3

—

210 K 220 K

∼1 ∼1

∼4 × 10−4

57.7 —

—

Au

99.999 9% Au (quenched from 1 000◦ C)

0◦ C

∼10

7 × 10−4 – 8 × 10−3

62.7

Au

99.99% Au (CW∗ , annealed at 600◦ C) 99.999 8% Au (CW∗ 36% annealed, 650–870◦ C) 99.999 95% Au (annealed at 900◦ C) Au–42 At. % Ag– 15 At. % Zn 10 At. % Cu– 90 At. % Cu 10 At. % Cu– 90 At. % Cu

330◦ C

0.7

404◦ C

1.0 1.0

Depends on grain size 4.4 × 10−2 3.2 × 10−2

141.4

238◦ C

∼400◦ C

∼1

∼10−1

435

260◦ C

0.7

0.11

146

326– 392◦ C 552– 753◦ C

1.0

3.8 × 10−3 – 0.57 0.14 0.76

114.6– 165.3 201.7– 342.3

175– 250◦ C 490 K (quenched only) 635 K (all)

1.0

5 × 10−2 – 2 × 10−2

—

Au Au Au–Ag–Zn Au–Cu Au–Cu Au–Cu Au–Cu

Au–Cu– Zn Au–Cu– Zn Au–Cu– Zn

10 At. % Cu– 90 At. % Cu Au–25at% Cu (quenched) (quenched and annealed) (quenched and annealed) 42 At. % Cu– 15 At. % Zn 21 At. % Cu– 17 At. % Zn 63 At. % Cu– 17 At. % Zn

1.0

∼1

144.3 242.7

Ref.

Relaxation at Al–Si interface Migrates and falls to zero during ageing; vacancy/Si clusters Stress induced ordering of zinc atom–vacancy complexes Interface Bordoni type Bordoni type Bordoni type

298 311 312 40

252 1, 4 1, 4 1, 4

Hasiguti type Hasiguti type Hasiguti type

21 21 21

Hasiguti type Hasiguti type Hasiguti type Associated with dislocations Associated with dislocations Associated with dislocations Stress induced reorientation of divacancies Hasiguti type Stress induced reorientation of divacancies Grain boundary

42 42 42 43

Grain boundary Associated with grain boundaries Sliding at grain boundaries Zener

47 47

Zener

50

Adsorption of solute atoms on grain boundaries Grain boundaries

48

Point defects/order

267

43 43 44 44 45 46

48 49

48

Zener

760 K (all)

Grain boundary

380◦ C

0.5

0.14

—

300◦ C

0.5

0.50

—

0.5

2.6 × 10−2

—

340◦ C

Mechanism or type

Order–disorder peak Order–disorder peak Zener

49 49 49 (continued)

Damping capacity Table 15.8

ANELASTIC DAMPING—continued

Alloy

Composition and physical condition

Peak temp.

Frequency Hz

Modulus defect

Au–Fe

5% Fe

∼1

Au–Fe

7% Fe

Au–Fe

10% Fe–27% Fe

365◦ C– 535◦ C 380◦ C– 564◦ C 390◦ C

15–15

Activation energy kJ mol−1

Mechanism or type

Ref.

159 177.8 159 177.8 151–151.9

Zener Precipitation of Fe Zener Precipitation of Fe Zener

51 51 51 51 51

Au–Ni

30 At. % Ni

397◦ C

1.0

2.4 × 10−3 1.5×10−2 1.9 × 10−2 – 1.2×10−2 3.5 × 10−2 – 1.5 × 10−2 —

182.0

Zener

52

Au–Ni

Au–30 At. % Ni (quenched)

∼380◦ C

∼0.5

∼10−1

88.3

53

Au–Ni

7.7 At. % Ni– 90.8 At. % Ni 15 At. % Zn Stoichiometric AuZn annealed

397◦ C 652◦ C 250◦ C 260◦ C– 290◦ C 210 K 135◦ C

1.0

—

0.7 0.2–1

4.2 × 10−2

182.0– 251.0 218 140

Zener, modified by quenched–in vacancies Zener

49 54

1.0 1.0

— —

51.9 101.7

Zener Concerned with short-range order (?) Bordoni type Solute atom/defect interaction

98.6% Be (annealed)

213 K

∼1

∼3 × 10−4

50.2

56

135◦ C

∼1

∼5 × 10−4

100

56

0.5◦ C

1.22

10−2

63.6

Cold work induced line defect interaction (?) Solute interaction with lattice (?) Snoek type

20◦ C 20◦ C 215 K (two peaks) 263 K (two peaks) 297 K 410 K 600 K

0.75 1.0 103

0.26 0.29 max —

80 79.5 40.6

Zener Zener Bordoni type (?)

58 59 4

103

—

51.0

103

4 4

— ∼2 × 10−3

36.8 72.3

280◦ C

1–16 × 105

4 × 10−4

159

Point defect/ dislocation interaction Twin boundaries (?) Movement of divacancies and dislocations Motion of C atom pairs in lattice

111◦ C

103

3 × 10−3

—

340◦ C 430◦ C 150◦ C 310◦ C ∼370◦ C

1.5– 1.8 × 104 1.5– 1.8 × 104

∼2.5 × 10−2 ∼2.8 × 10−2 ∼2.9 × 10−2 ∼3.1 × 10−2

— — — —

38◦ C

—

2 × 10−3

—

160◦ C ∼36◦ C

1.0 3

1.5 × 10−3 ∼4 × 10−4

101.7 115.1

155◦ C

1

up to 2.2 × 10−3

85.8

Au–Zn Au–Zn Be

Be

Be–Fe–O Cd–Mg Cd–Mg Co Co

Be–0.4% Fe + interstitial impurities including oxygen Cd–29.3% Mg 5% Mg–30% Mg 99.23% Co (0.69% N) 99.23% Co (0.69% N) CW∗ at RT

Co

99.999% Co (quenched)

Co–C

Co–Ni

Heated in C atmosphere at 1 050◦ C and quenched Co–37.8% Fe– 8.7% Cr. In magnetic field of 0.6 × 103 A/M Co–2% Ni

Co–Ni

Co–23% Ni

Co–Ni– Cr–W

Co–22wt% Ni– 22wt% Cr–14wt% W (wrought alloy) 99.8% Cr

Co–Fe–Cr

Cr Cr–N Cr–N Cr–N

Cr–0.004 5% N 35 ppm N (quenched from 83◦ C) (Annealed in NH3 at 1 150◦ C for 48 h)

∼1 ∼1

10– 5 × 102

∼10−2

52

55 55

Electron spin redistribution at Curie point α–β phase transformation α–β phase transformation Snoek effect (C–C pairs)

57

60 61 62 63 63 63 63 259

Electron spin redistribution at Neel temp. (40◦ C) Snoek type Magnetomechanical damping Snoek type

64 65 66 67

(continued)

15–16 Table 15.8

Alloy Cr–Re–N Cu

Elastic properties, damping capacity and shape memory alloys

ANELASTIC DAMPING—continued

Composition and physical condition

Peak temp.

Frequency Hz

Modulus defect

Activation energy kJ mol−1

Mechanism or type

Cr–35% Re (quenched from 1 000◦ C in NH3 atmosphere) 99.999% Cu (CW∗ at 77 K 99.999% Cu (single crystal) CW∗ 5% at 77 K

130◦ C 190◦ C

1 1

∼10−3 ∼10−3

89.1 126.4

Snoek type Snoek type

68

— — 38 K

∼1 ∼1 1.09 × 104

— — 2 × 10−3

4.34 11.6 4.2

Niblett–Wilks type Bordoni type Bordoni type

79 K

1.09 × 104

4 × 10−3

11.7

Bordoni type

Ref.

Cu

99.999% Cu (single crystal 100 orientation)

80◦ C

13

34 × 10−4

—

Bordoni type

69 69 19, 41, 70, 71, 19, 41, 70, 71 238

Cu

99.999% Cu (single crystal 110 orientation) Electrolytic Cu (Fatigued, 4 × 105 cycles)

70◦ C

13

11.4 × 10−4

—

Bordoni type

238

140 K

0.3

∼4 × 10−4

—

72

225 K

0.3

∼4 × 10−4

—

240 K

0.3

∼2 × 10−4

—

165 K

0.3

∼1 × 10−4

—

30 K 70 K 190 K 148 K

6 × 102 6 × 102 6 × 102 1.0

∼10−3 ∼3 × 10−3 ∼10−4 4 × 10−3

— — — 31.0

170 K

1.0

1.2 × 10−2

33.9

238 K

1.0

1 × 10−2

41.4

5

Cu

Cu

Cu Cu

Cu Cu

Cu

99.999% Cu (deformed 2.5% quenched in liquid He) 99.999% Cu (CW∗ 5% at 77 K)

99.999% Cu (annealed at 500◦ C for 4 h) 99.99% Cu (area reduced 47%, annealed at 600◦ C for 2 h) 99.999 9% Cu

1.0

2.11 × 10−2

157

Dislocationdivacancies interaction Dislocationvacancies interaction Dislocationinterstitials interaction Dislocationinterstitials interaction Niblett–Wilks type Bordoni type Hasiguti type Point defect, dislocation interaction Point defect, dislocation interaction Rotation of split interstitials (?) Grain boundary

216◦ C

1.17

1.65 × 10−2

132

Grain boundary

416◦ C

∼1

∼0.2

435

735◦ C

Sliding at grain boundaries Sliding at grain boundaries Grain boundary Grain boundary  Dislocation/ silver atom/  point defect interaction Bordoni type

215◦ C

∼1

∼0.2

169.5

5.0 1

4 × 10−2

—

156.9 154.7

Cu Cu–Ag

99.999% Cu Cu–0.71% Ag

300◦ C 550◦ C

Cu–Ag

Cu–0.1 At. % Ag CW∗ 5%

223 K 283 K

5 × 103 5 × 103

2.6 × 10−4 2.5 × 10−4

— —

2% Al–10% Al (Deformed 3% at RT) Cu–15wt% Al (cold worked)

145 K

1.5 × 107

—

∼24

−60◦ C

∼2

0.04

Cu–Al Cu–Al

72 72 72 73 73 73 74 74 74 75

48, 76 76 75 237 250 77 308 (continued)

Damping capacity Table 15.8

ANELASTIC DAMPING—continued

Alloy

Composition and physical condition

Peak temp.

Frequency Hz

Modulus defect

Activation energy kJ mol−1

Cu–16.8 At. % Al Cu–13wt% Al −7.9wt% Ni (solutiontreated at 1 223 K and cooled at various rates)

360◦ C 323 K at all rates

0.66 ∼520–680

7 × 10−3

174.9

Cu–Al Cu–Al–Ni

Cu–Au

Cu–25at% Au (annealed)

Cu–Au

Cu–1.5at% Au (cold worked and annealed at 100◦ C) (Aged at 575◦ C for 3 min)

Cu–Co Cu–Ga Cu–Fe

Cu–Fe Cu–Ni

Cu–16 At. % Ga 0.5% Fe–10% Fe

Up to 1.5% Fe (Quenched from 820◦ C) 25% Ni–75% Ni (Quenched from 720◦ C to 240◦ C)

Others at high rates 680 K

∼1

800 K −50◦ C

∼1

230◦ C

1

330◦ C 320◦ C– 350◦ C 480◦ C– 580◦ C

1.0 ∼1

800◦ C– 850◦ C 34 K 150 K

Zener

266

Zener Grain boundary

33 79

∼1

209

Connected with precipitation of Fe on Cu grains

79

∼1

∼10−1

∼125

80

∼1

—

79.9

∼1

—

111.3 151

Connected with ageing of alloy Associated with precipitation Associated with precipitation Zener

∼1

—

∼800◦ C

∼1

∼5 × 10−2

208

590– 726◦ C 381◦ C

∼1

Cu–Ni– Zn

5.6 At. % Ni– 94.9 At. % Ni 20 At. % Zn– 10 At. % Ni

0.129– 0.112 5.3 × 10−3

Cu–Ni– Zn

Cu2 Ni1.15 Zn0.92 single crystal

76 K

6.9

1

310 75

81 81 82 83

368–264

Associated with recrystallisation (?) Associated with dislocations Grain boundary

197

Zener

85

3 × 10−2

—

Zener

244

1.2 × 10−2

244 86

1.74 eV 151–205

Ordering Overdamped resonance of dislocations Overdamped resonance of dislocations Precipitation of K1 Relaxation at ppt. interfaces in stacking faults Grain boundary Grain boundary

— 159–178

Thermoelastic Zener

88 12

182.0– 161.1 172(?)

Zener

89

Grain boundary

90

Cu–Pd

0.01% Pd–0.3% Pd

71 K 30–150 K

6.9 5×106

10−4 –10−3

— —

Cu–Pt

0.01% Pt–0.3% Pd

30–150 K

5×106

10−4 –10−3

—

Cu–Si Cu–Si

Cu + 5.09 wt. % Si Cu − 5.09wt. % Si (precipitation treated)

200◦ C 200◦ C

3 3

7.5 × 10−3

118 28 kcal mol−1

Cu–Sn

3% Sn–9% Sn

366◦ C 490–500◦ C

3 1.8

0.14–0.152

Cu–Zn Cu–Zn

(α) Cu–31% Zn 10% Zn–30% Zn

R.T. 290–350◦ C

6 × 103 0.7

Cu–Zn

17.6 At. % Zn– 29.4 At. % Zn Cu–30% Zn

657–614 K

1.3

425◦ C

0.5

Cu–Zn

78 281

— 159

∼2 × 10−2

Cu–Ni

Zener Twin boundary relaxation of gamma-mart.

184.9

∼1

Cu–Ni

Ref.

7 × 10−4 (depends on ageing) 2 × 10−2 Depends on grain size Depends on grain size

∼600◦ C

Cu–3% Ni (Annealed at 1 000◦ C) Cu–45% Ni (Reduced by 90% at RT)

Mechanism or type

Grain boundary Interaction between dislocations and Au clusters Grain boundary

2.1 × 10−3

Cu–Ni

∼580◦ C

15–17

× 10−4 1.8 At. conc Zn 4

3.5 × 10−3 – 9.2 × 10−3 0.12

83 84

86 248 306

87

(continued)

15–18

Elastic properties, damping capacity and shape memory alloys

Table 15.8

ANELASTIC DAMPING—continued

Alloy

Composition and physical condition

Peak temp.

Frequency Hz

Modulus defect

Activation energy kJ mol−1

Cu–Zn

(β) Cu–45 At. % Zn (Quenched from 400◦ C)

70◦ C

0.9

2.6 × 10−3

69.5

Diffusion of Zn accelerated by vacancies

91

Cu–Zn

(β) Cu–45 At. % Zn

177◦ C

0.9

1.4 × 10−3

130

91 91

Cu–Zn

(α − β) Cu–43 At. % Zn

285◦ C

0.9

8 × 10−3

159

Cu–Zn

(β − γ) Cu–50 At. % Zn

190◦ C

0.9

2.2 × 10−3

130

Cu–Zn–Al

77% Cu–89% Cu 5% Zn–20% Zn 2% Al–8% Al Cu–29.5% Zn–2.4% Al

623–672 K

∼1

2 × 10−3 – 8.5 × 10−3

—

Stress induced reorientation of Cu atom pairs Stress relaxation at β–α interfaces Stress relaxation at β–γ interfaces Zener

593 K

1

150.6

Zener

93

615 K ∼50 K

1 105

3.27 × 10−2 ∼1.5 × 10−3

163.2 —

198 K

103

—

44.4

Fe

Armco (CW∗ at RT)

230 K

103

—

54.8

Fe Fe

CW∗

275◦ C 526◦ C

2.9 1.03

1.55 × 10−3

174 192

Zener Associated with motion of kinks in dislocations Point defect/dislocation interaction Point defect/dislocation interaction Köster type Grain boundary

93 94

Fe

Cu–17.0% Zn–9.0% Al (Re-electrolytic) (CW∗ 5% at RT, in magnetic field of 7.5 × 104A/m) Armco (CW∗ at RT)

3.5 × 10−2

110 K

1.4

Magnetic relaxation of point defects Magnetic relaxation of point defects Relaxation of self -interstitials Movement of Al within tetrahedral lattice lattice Movement of Al within tetrahedral lattice Movement of Al within tetrahedral lattice Zener Zener

246

99

Cu–Zn–Al Cu–Zn–Al Fe

Fe

Fe–Al

40% at RT 99.98% Fe (0.029% O) (CW∗ ) Nb–0.014% C

152◦ C 283◦ C

1 1

∼9 × 10−3 ∼1 × 10−3

— —

410◦ C

1

∼1 × 10−3

—

Dislocation/solute interaction

169

259◦ C

0.57

4 × 10−3

139.3

Snoek type

High purity single crystal + 180 at ppm H. CW at 320 K

270 K

1.3 × 103

3.0 × 10−2

—

Snoek–Köster type

170, 171 239

Nb

Nb–Al Nb–Al Nb–C Nb

Snoek type due to N Snoek type due to O Relaxation of ordered cluster of interstitials near dislocations Snoek type due to N Snoek type due to O

167 167 167 168

169 169

(continued)

Damping capacity Table 15.8

ANELASTIC DAMPING—continued

Alloy

Composition and physical condition

Peak temp.

Frequency Hz

Modulus defect

Activation energy kJ mol−1

Nb single crystal R. ratio 2 500

α1 143 K

0.5

10 × 10−3

29.7

α2 121 K

0.5

6 × 10−3

19.8

180–220 K

2–2 × 107

5 × 10−4 –

24.1

Nb

Nb–Mo Nb

Nb Nb

1.3% Mo–16% Mo (CW∗ 30% at RT) 99.97% Nb (CW∗ ) (Superconducting type) (Normal type) (CW∗

Zone refined 5% at RT) 99.9% Nb (CW∗ )

3.24 K

8 × 104

1 × 10−4 ∼2 × 10−6

2.08 K

8 × 104

∼4 × 10−6

0.15

8.8

10−4

—

2 × 104 – 1 × 105 2×104 1 × 105

∼2 × 10−4

1.6

∼1 × 10−4 –

24.1

11–19 K (Broad) ∼30 K

Nb–Mo–O

Nb–N Nb–N Nb–N–T Nb (–O)

High purity single crystal CW at 320 K Nb–5.3 At. % Mo– 0.16 At. %O (Homogenised at 1 200◦ C for 15 min) Nb-0.018% N Nb–0.066% N (CW∗ 10%) Nb–0.50 at% N– 0.3at% T (Tritium) ‘Pure’ nobium (wire)

153, 166 160 160 161

167◦ C

0.6

∼10−2

—

274◦ C

0.55

1.1 × 10−2

145.6

Snoek type

170, 171

500◦ C

0.31

2.4 × 10−2

201

Köster type

173

57 K; 85 K

∼1

N/H pairs; N/T pairs

297

300–350 K

∼1

Extrinisic double-kink/O atoms Intrinsic doublekink on screw disloc. Thermal unpinning of screw disloc. from O Snoek type Snoek type

287

0.6 eV 0.8 eV

152◦ C 168◦ C

0.6 2.13

Nb–O

Nb–50 to 1 000 ppm O (worked polycrystals)

543–575 K 692–726 K 618 K (low O) 621 K (high O) 420 K

1 1 1

Nb (–Ta)

249

—

Nb–0.18 At. % O Nb–0.026% O

Nb–0.56at% O– 0.3at% T (tritium) Nb–0.03wt% Ta (70 ppm O, 30 ppm N) (neutron irradiated and annealed 1.25 mm wires)

Motion of dislocations Bordoni type (?)

249

1.2 × 10−2

Nb–O Nb–O

Nb–O–T

Formation of kinkpairs in non-screw dislocation Kink diffusion in screw dislocation Type of Bordoni mechanism Motion of dislocations

1.3 × 103

∼500 K

Nb–1.2 At. % O– 0.11 At. % N (CW∗ 34%)

Ref.

190 K

350–360 K

Nb–O–N

Mechanism or type

Motion of dislocations Interaction of dislocations and impurity atoms Motion of dislocation Snoek type

∼180 K Nb

0.18

15–25

∼10−2 1.6 × 10−2

— 114.2 2.00–2.06 eV 1.68–1.81 eV 1.49 eV

1

162 162 239 172

172 174, 171 270

1.50 eV

∼1

∼5 × 10−3

—

500 K

∼1

∼1 × 10−3

—

50 K; 80 K

∼1

Segregation of O atoms to dislocations Segregation of N atoms to dislocations O/H pairs; O/T pairs

470 K

O plus irradiation defects

670 K

N plus irradiation defects

175 175 297 283

(continued)

15–26 Table 15.8

Elastic properties, damping capacity and shape memory alloys

ANELASTIC DAMPING—continued

Alloy

Composition and physical condition

Peak temp.

Frequency Hz

Modulus defect

Activation energy kJ mol−1

Nb–Ti

Nb–48% Ti

100◦ C

0.6

∼7 × 10−3

100.0

Snoek due to O impurities

176

Nb–Ti

Nb–48% Ti (N atmosphere at 1 200◦ C for 1 h) Nb–5at% Ti (plus D)

340◦ C

0.6

∼10−2

—

Snoek due to N

176

50 K 100 K 170 K

20 × 103 20 × 103 20 × 103

∼200 K 645 and 685 K

2–2 × 107 ∼1

Nb–2, 6 and 12 at% W–N

645, 685 and 745 K

∼1

Nb–1.0% Zr Nb–1% Zr–O + traces of N

∼200 K ∼500◦ C

2–2 × 107 5 × 104

1 × 10−3 —

— 143 and 153 respectively 143, 153 and 167 respectively — 110.9

∼500◦ C

5 × 104

—

111.3

∼500◦ C

5 × 104

—

123.4

∼500◦ C

5 × 104

—

146.4

∼500◦ C

5 × 104

—

147.3

138 K

3 × 10−4

155 K

103

2 × 10−3 – — 35 × 10−3 −2 1 × 10 –0.10 — Up to — 1.6 × 10−3 Up to — 3.0 × 10 — 29.7

350 K

103

—

51.0

397 K

103

—

69.4

Nb–Ti

Nb–V Nb–0.3% V Nb–W–N Nb–N (wires)

Nb–Zr Nb–Zr

Nb–Zr

Ni Ni

Ni

—

Zone refined (CW∗ ) Zone refined single crystal (CW∗ ) 99.99% Ni at RT)

(CW∗

3 × 104

248 K 145–123 K ∼2 × 104 223–263 K ∼2 × 104

Mechanism or type

Ref.

273

2.5 × 10−3

Ti-D complexes reorientation Dislocation damping Snoek

153 285

Dislocation damping Snoek due to O in Nb Substitutional– interstitial process involving O Substitutional– interstitial process involving O pairs Snoek due to N in Nb Substitutional– interstitial process involving N

153 177

Niblett and Wilks type Bordoni type Associated with dislocation reactions Bordoni type

178

Point defect/dislocation interaction Point defect/dislocation interaction Point defect/dislocation interaction Stress induced reorientation of interstitial Ni atom pairs Magneto-mechanical damping

177 177 177 177

178 179 179 4 4 4

Ni

99.9% Ni

70◦ C

0.7

—

77.0

Ni

99.999 9% Ni

150◦ C

1

∼0.1

—

Ni

99.99% Ni. (Area reduced 90%, annealed at 905◦ C for 1 h) 99.98% Ni

432◦ C

1.41

3.4 × 10−2

308

Grain boundary

5

440– 460◦ C 630–

0.5

0.10

—

183

0.5

0.12

—

720◦ C 280◦ C

Grain boundary % Stress relaxation at polygonised boundaries

0.5

—

—

Ni

Ni–Al–C

Ni–2% Al–0.5% C (quenched)

Diffusion of C in Ni–Al

180, 181

182

183 184 (continued)

Damping capacity Table 15.8

ANELASTIC DAMPING—continued

Alloy

Composition and physical condition

Peak temp.

Frequency Hz

Ni–B

Ni (pure)

∼1

Ni–C Ni–C Ni–Cr

Ni–0.0035wt% B (quenched) (cold-worked) 0.5% Cr

Ni–Cr

Ni–33.3at% Cr

Ni–Cr Ni–Cr–Ce

Ni–20wt% Cr Ni–20wt% Cr– 180 at ppm Ce Ni–20% Cr–1.87% C (quenched) Ni–20% Cu

470◦ C; 670◦ C 550◦ C 230◦ C ∼430◦ C 530– 800◦ C 390◦ C; 570◦ C ∼700◦ C ∼700◦ C

∼1 0.5 ∼1 2 0.7–1.5

Modulus defect

Activation energy kJ mol−1

Mechanism or type

Ref.

All grain boundary

300

Diffusion of C in Ni Dislocation–C Grain boundary

184 276 185

Long range order

282

370 200

Grain boundary Grain boundary

257 257 186

—

Diffusion of C in Ni–Cr Magnetomechanical damping Magnetic ordering Magnetomechanical damping Grain boundary

—

‘Blocking’ peak

187

275 eV

Atomic diffusion

262

— Depends on grain size

∼1 ∼1

15–27

—

250◦ C

0.9

—

98.3

∼140◦ C

1

∼10−2

—

(Varies) ∼200◦ C

1 ∼1

∼10−2

— —

∼450◦ C

∼1

600– 700◦ C 147◦ C

∼1

70–80 K

2.7

∼10−3

12.34– 16.19

Stress induced ordering of H pairs in β phase

188

120 K ∼150 K

∼103 ∼103

∼2 × 10−3 ∼5 × 10−3

— —

189 189

99.999% Pd (Annealed, electrolytically loaded with D) 40 At. % D– 73 At. % D 80at% Pd–20at% Si with 1–2at% H (rapidly solidified amorphous) (Deformed 2.9%, annealed at 1 080 K for 1 h)

78–86 K

2.7

—

15.94– 20.71

Snoek type due to H Pinning of dislocations by interstitial impurities Stress induced ordering of D pairs in β phase

180 K

100

∼70 K

5 × 103 – 6.5 × 104

125 K

Pt

99.999% Pt (CW∗ 16% at 4.2 K) ‘Pure’

Pu–Al

Pu–5 At. % Al

Pu–Ce Re Se

Ni–Cr–C Ni–Cu Ni–Zr

Pd–Cu–Si Pd–H

Pd–H

Pd–D

Pd–Si Pt

Pt

0.1% Zr–0.5% Zr

Amorphous Pd– 6at% Cu–16.5at% Si (1 mm wire) 99.999% Pd (Annealed, electrolytically loaded with H) 40 At. % H– 75 At. % H PdH (β phase) (Strained)

2 × 10−3 – 8 × 10−4 1 × 10−3 – 3 × 10−4 < 1 × 10−4 – 2 × 10−4

182 182 187 187

188

0.31 eV

Short range diffusion of H

293

5 × 10−6

11.6

190

0.8

8 × 10−3

28.0

Associated with dislocations grouped into subgrain boundaries Bordoni type

940– 1 090 K ∼65 K

10−2 –1

—

275

191

107

—

—

Recrystallisation peak Co-operative electron transition

Pu–6 At. % Ce

∼65 K

107

—

—

— (amorphous)

1 400 K 30 K

1.1 160 × 106

4 × 10−2

586

Co-operative electron transition Gain boundary Attenuation of ultrasonic shear waves

1, 2

192 192 193 302

(continued)

15–28 Table 15.8

Alloy Si

Si Si

Elastic properties, damping capacity and shape memory alloys ANELASTIC DAMPING—continued

Composition and physical condition

Peak temp.

Frequency Hz

Modulus defect

Activation energy kJ mol−1

Surface damaged [100] by polishing [111]

160 K

∼2 × 10−5

∼2 × 10−5

29

200 K

∼2 × 103

∼4 × 10−5

29

689 K

1.2 × 103

∼2 × 10−4

130

655◦ C

105

1.4 × 10−5

134.7

∼900 K ∼1 300 K (max.) widened 398 K

∼1 ∼1

Electronic relaxation Relaxation in surface layers due to dislocation formation

195, 196 277

1.2 × 103

∼3 × 10−5

68

194

1.2 × 103

∼2 × 10−4

96 80.0

Pure n-type [111] (Quenched from 1 000◦ C to 77 K) Single crystal

Si

Si whiskers (annealed) (thermally cycled)

Si–Cu

Si (Cu doped)

Mechanism or type

Ref.

Associated with dislocations Associated with dislocations Migration of Ovacancy complex

132 132 194

Si–Li–B

Si–Li–0.01% B

626 K 210◦ C

Si–O

Single crystal

1 030◦ C

105

6 × 10−4

246.0

Sn Ta

99.99% Sn 99.99% Ta (CW∗ 7%)

80◦ C 24.6 K

Zone refined (CW∗ 12% at RT)

124 K (α peaks) 202 K (β peaks) 170 K

3 × 10−2 3.1 × 10−4

79.5 3.7

Ta

3 × 102 2.26 × 104

Migration of interstitial Cu Precipitation of Cu Reorientation of Li+ B− pairs Stress-induced diffusion of interstitial oxygen Grain boundary Bordoni type

9 × 10−4

24.3

Bordoni type

151

0.8

1.3 × 10−3

41.4

Bordoni type

151

17

19 × 10−3

—

Snoek–Köster type

239

150 K

0.7

—

4 × 10−3

240

1 100◦ C 1 230◦ C

0.65 1.0

7 × 10−2 5.6 × 10−2

418 406

Dislocation kink formation Grain boundary Grain boundary

338◦ C

0.55

2.6 × 10−2

161.1

Snoek type

27 K

2 × 104 –

∼5 × 10−5

1.6

2.4 × 10−3 – 2.7 × 10−3 —

—

Associated with dislocations Bordoni type (?)

Ta Ta Ta Ta Ta–C Ta–H Ta–H

Ta–H

Ta–N

Ta–O

(Single crystal < 111 > orientation) CW∗ 3.1% (Single crystal) — 99.89% (Annealed at 1 700◦ C) Ta–0.1% C (CW∗

99.9% Ta H charged) —

1.7%

8.5 At. % H–42.4 At. %H

Ta–0.11% N

Ta–0.081% O

1.74 × 104

0.8

6.5 × 10−5

100 K

105 1.75 × 102

190 K

1.75 × 102

1◦ C

1.75 × 102

6 × 10−3 – 1.6 × 10−2

52.3

36◦ C

1.75 × 102

3 × 10−3 – 1.9 × 10−2

—

54◦ C

1.75 × 102

2.6 × 10−3 – 8.5 × 10−3

—

334◦ C

0.6

9 × 10−2

156.9

362◦ C

0.6

9 × 10−2

167.4

137◦ C

0.6

8 × 10−2

104.6

0.6

8 × 10−2

104.6

162◦ C

34.7

194 197 195, 196 198 199

193 142

Stress induced ordering in the Ta2 H phase Stress induced ordering in Ta2 H phase Long range ordering of H in Ta2 H phase Short range ordering of H in Ta2 H phase Diffusion of interstitial N Diffusion of interstitial N atom pairs Diffusion of interstitial O Diffusion of interstitial O atom pairs

200, 171 162 201, 162 201, 162 201 201 201 202 202 203, 204 203, 204

(continued)

Damping capacity Table 15.8

15–29

ANELASTIC DAMPING—continued

Composition and physical condition

Peak temp.

Frequency Hz

Modulus defect

Activation energy kJ mol −1

Mechanism or type

Ref.

340◦ C

0.5

4.4 × 10−3

151

Köster type

205

∼340◦ C

0.8 0.8

∼6 × 10−3

156.9 168.2

Snoek type due to N Snoek type due to N

206

Te

Ta–0.01% O (CW∗ 30%) Ta–(1.3 At. % Re– 3.8 At. % Re)– 600 ppm N —

∼400 K

1.44 × 104

∼10−4

—

207

Ti

99.7% Ti (CW∗ at RT)

220 K 305 K

103 103

— —

33.9 42.3

336 K

103

—

51.9

133 K; 173 K 198 K; 223 K 775◦ C 675– 725◦ C ∼−83 to −113◦ C ∼−53 to 7◦ C 116 K

1

Recombination of election-hole pairs Bordoni type (?) Point defect/dislocation interaction Point defect/dislocation interaction Bordoni

1

Hasiguti

Alloy Ta–O Ta–Re–N

Ti

High purity (cold worked)

Ti Ti–Al

99.6% Ti 0.04% Al–0.12% Al

Ti–Al–V

Ti–6wt% Al–4wt% V (alpha/beta quenched)

Ti–Al–V

Ti–6wt% Al–4wt% V (commercially heat-treated)

∼380◦ C

1.0 1.0 1

∼10−3

0.38 2.5 × 10−2 −0.4

201 218–293

1

2 × 103

11.1 kcal mol−1 259.4 29

Snoek H peak in alpha phase Grain boundary Associated with vacancies in 2 coexisting electronic environments Diffusion of interstitial H H in beta phase Alpha phase Beta/omega shear trans; ditto diffusion

Ti–H

Ti–0.15% H

273 K

1.2

1.6 × 10−2

Ti–5wt% Mo–5wt% V–8wt% Cr–3wt% Al Ti–11.5wt% Mo– 6wt% Zr–4.5wt% Sn (Beta III) (beta solid solution) Ti–25 At. % Nb (Quenched from 1 000◦ C)

130◦ C 45◦ C −170◦ C; 250◦ C 177 K

∼105

6.15 × 10−3

29

Ti–Nb

0.04% Nb–0.12% Nb

1.0

Ti–10 At. % Ni (Quenched from 1 000◦ C)

∼105

2.7 × 10−2 – 3.7 × 10−2 2.15 × 10−3

213–264

Ti–Ni

625– 675◦ C 152 K

Ti–Ni

TiNi Intermetallic compound. (Ni–49% Ti). Annealed at 800◦ C

203 K

∼1

∼10−2

36.6

∼1

∼10−2

—

—

350◦ C

600◦ C

∼1 ∼1

∼10−3

— —

450◦ C

1.0

1.2 × 10−2

200.8

Ti–2 At. % O

261

289 K

4 × 10−2 1.75 × 10−2

Ti–O

Hydride precipitation

Snoek H peak in beta phase

1.0 ∼105

Ti–Ni

208 208

5.6 kcal mol−1

715◦ C 152 K

Ti–Nb

223–313 K

309

Grain boundary Grain boundary

2 × 103

Ti–0.05% Au Ti–10 At. % Cr (Quenched from 1 000◦ C)

Ti–Mo– Zr–Sn

4

Dislocation/hydrogen

Ti–Au Ti–Cr

Ti

4 4

62.8 0.29 eV 0.49 eV

∼1

∼2 × 10−2

29

292

Associated with vacancies in 2 coexisting electronic environments Grain boundary Associated with vacancies in 2 coexisting electronic environments Dislocation motion (?) Fine structure of 203 K peak Impurity effect (?) Transition from TiNi (II) to TiNi (I) Diffusion of interstitial O in presence of impurities

208 209

210 253 284

209

208 209

211 211 211 211 212, 213,

(continued)

Elastic properties, damping capacity and shape memory alloys

15–30 Table 15.8

ANELASTIC DAMPING—continued

Alloy

Composition and physical condition

Peak temp.

Frequency Hz

Modulus defect

Activation energy kJ mol −1

Mechanism or type

Ref.

Ti–O

Ti–0.6 At. % O

660– 650◦ C 140 K

1.0

2.5 × 10−2

188.3

Grain boundary

214

∼105

2.2 × 10−3

29

209

209

Ti–V

Ti–20 At. % V (Quenched from 1 000◦ C)

Ti–V

Ti–50 At. % V (Quenched from 1 000◦ C)

161 K

∼105

2 × 10−3

29

Ti–V

0.02% V–0.12% V

1.0

3 × 10−2 – 0.42

230–335

Ti–V–H Ti–Zr

25at% V 0.02% Zr–0.12% Zr

16 251–502

Snoek type Grain boundary

254 208

U

99.9% U (CW∗ at RT)

600– 700◦ C 84 K/123 K 650– 700◦ C 155 K 202 K

Associated with vacancies in 2 coexisting electronic environments Associated with vacancies in 2 coexisting electronic environments Grain boundary

103 103

23.0 42.3

4 4

V–C

—

162◦ C

0.55

6.4 × 10−3

114.2

V–H

99.99% V + 600 ppm H

170 K

500

13×10−4

—

V–H

—

18.5 K

2 × 102 – 105 ∼

∼6 × 10−5

1.1

∼10−5

—

285 K

∼

∼10−5

—

Bordoni type (?) Point defect/dislocation interaction Diffusion of interstitial C Point defect/dislocation interaction Dislocation damping Point defect/dislocation interaction Point defect/dislocation interaction

∼7 × 10−4

50

8 × 10−3 – 7.2 × 10−2 2.2 × 10−2

37.87

250 K

2/1.5 1.0

V–H

99.99% + H in soln

203 K

V–H

195 K

V–N

1.2 At. % H– 14.5 At. % H —

2 × 102 – 105 75

272◦ C

1.0

V–O

—

174◦ C

0.55

W

99.999 8% W

W W

High purity (CW∗ 3% at 400◦ C) 99.99% W single crystal

3.1 × 10−2 – 3.6 × 10−2 — —

142.7

1.2 × 10−2

122.6

— —

— —

469 46

150 K

—

33

Stress induced ordering in β phase Diffusion of interstital H Diffusion of interstitial N Diffusion of interstitial O Grain boundary Movement of vacancies Movement of divacancies

165 K

1.5 × 104

10−3

24.3

Bordoni type

∼300◦ C

∼1

∼2 × 10−4

146

∼400◦ C

∼1

∼2 × 10−4

188

Snoek type of uncertain origin Snoek type associated with C impurities Grain boundary

W

Commercial purity

1 250◦ C

0.94

0.28

481–523

W

(CW∗

1 535◦ C

∼70

∼0.2

—

10−2 –0.25

∼0.6

477

0.35

0.56

619

Primary recrystallisation peak Recrystallisation peak Grain boundary

∼70

∼0.1

—

Grain boundary

at RT)

W

Zone refined

W

Commercial purity

1 600– 1 650 K 1 900◦ C

W

(CW∗ at RT, annealed at 3 000◦ C)

∼2 000◦ C

208

215 243 216 216 216 216 217 218 215 219 219 219 151, 200 221 221 193, 222 223 224 222, 225 223

(continued)

Damping capacity Table 15.8

Alloy W–Re Zn

Zn Zn Zn–Al

Zr

15–31

ANELASTIC DAMPING—continued

Composition and physical condition

Peak temp.

Frequency Hz

Modulus defect

Activation energy kJ mol −1

W–20% Re 99.999% Zn (Compressed 2.4% at RT)

1 950◦ C ∼100 K

1.08 ∼2 × 107

∼5 × 10−2 —

510 5.8

∼170 K

∼2 × 107

—

15.4

∼230 K

∼2 × 107

—

19.2

140–220 K

1 × 103

383 K

∼1

∼0.3

99.999wt% Zn (worked single crystals) 99.999% Zn (Annealed at 100◦ C for 12 h) 22wt% Al (2 h at 360◦ C, w.q.) 22wt% Al–0.3wt% Cu (2 h at 360◦ C, w.q.) As above + 0.021wt% Mg (2 h at 360◦ C, w.q.) 99.999% Zr (7 k bar pressure) High purity (deformed and annealed alpha Zr)

∼25◦ C

95

1–3

9 × 10−2

∼25◦ C

1–3

3 × 10−2

∼25◦ C

1–3

1.44 × 10−2

80 K 250 K 480◦ C; 530◦ C; 600◦ C

1.5 × 105 1.5 × 105 4

∼5 × 10−4 ∼2 × 10−3

28 13

Mechanism or type

Ref.

Grain boundary Dislocation movements in basal plane Dislocation movements in prismatic plane Dislocation movements in pyramidal plane Bordoni

223 226

271

Grain boundary

227

226 226

255

Zr

99.9% Zr (CW∗ at RT)

200–220 K 305 K

103 103

9 × 10−4 —

33.9(?) 42.3

Zr

—

336 K

103

—

51.0

Zr

99.9% Zr

600◦ C

1.0

4.6 × 10−2

218–243

Bordoni type (?) Dislocation/O interaction; longitudinal and transverse motion in dislocation core Bordoni type(?) Point defect/dislocation interaction Point defect/dislocation interaction Grain boundary

860◦ C

1.0

0.28

—

α–β transformation

∼220 K

1.3 × 104

Up to 1.5 × 10−3

33.8

Bordoni type

230, 231 230, 232 233

Zr(–O)

Zr–Cu

228 228 295

41, 229 4 4

Zr–Cu– Mo Zr–H

Zr–up to 2.5% Cu (Annealed at 900◦ C for 1 h, CW∗ 10%) Zr–0.5% Cu–0.5% Mo (Deformed 10%) Zr–0.89% H

∼230 K

1.3 × 104

2 × 10−3

33.8

Bordoni type

233

228 K

1.0

4 × 10−3

48.5

231

Zr–H

Zr–1.28% H

5◦ C

1.0

2.4 × 10−2

71.1

Zr–H

Zr–1.15% H

∼50◦ C

∼2 × 104

∼1 × 10−3

—

234

∼130◦ C

∼2 × 104

∼1 × 10−2

—

1.0

5 × 10−3

—

Diffusion of interstitial H Diffusion of interstitial H atom pairs Associated with δ and γ phases Associated with δ and γ phases Associated with ZrH precipitate

∼1

∼7 × 10−3 ∼10−2 2 × 10−4 – 4 × 10−4 1.36 × 10−2

—

Grain boundary

235

201

235

∼2 × 10−4

1.9

Diffusion of O in lattice Associated with vacancies in 2 coexisting electronic environments ?

Zr–H

Zr–0.26% H

230◦ C

Zr–Hf–O

Zr–(0.005% Hf–1% Hf)–O

530◦ C 540◦ C 422◦ C

∼1

Zr–Mo

Zr–6 At. % Mo (Quenched from 1 000◦ C)

213 K

∼105

Zr–Nb

Zr–5% Nb

12 K

105

29

231

234 231

209

236 (continued)

Elastic properties, damping capacity and shape memory alloys

15–32 Table 15.8

ANELASTIC DAMPING—continued

Composition and physical condition

Peak temp.

Frequency Hz

Zr–Nb

5% Nb–25% Nb (CW∗ )

40 K 160 K

105

Zr–Nb

12 At. % Nb– 75 At. % Nb (Quenched from 1 000◦ C) Zr–1.95% O

163–207 K

∼105

420◦ C

∼1

Zr–10 At. % V (Quenched from 1 000◦ C)

193 K

∼105

Alloy

Zr–O Zr–V

∗

Modulus defect

Activation energy kJ mol −1

∼6 × 10−4 Depends on CW 1.5 × 10−4 – 5.9 × 10−3

4.8 29

Depends on O concn 1.5 × 10−4

201

29

29

Mechanism or type Jahn-Teller type Stress induced reorientation of atoms Associated with vacancies in 2 coexisting electronic environments Diffusion of O in lattice Associated with vacancies in 2 coexisting electronic environments

Cold worked.

REFERENCES TO TABLES 15.7 and 15.8 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41.

S. Okuda, J. Phys. Soc. Japan, 1963, 18 (Suppl. 1), 187. S. Okuda, Appl. Phys. Letters, 1963, 2, 163. B. M. Mecs and A. S. Nowick, Phil. Mag., 1968, 17, (147), 509. R. R. Hasiguti, N. Igata and G. Kamoshita, Acta Met., 1962, 10, 442. J. N. Cordea and J. W. Spretnak, Trans. Met. Soc. AIME, 1966, 236 (12), 1685. L. Rotherham and S. Pearson, J. Metals, 1956, 8, 881, 894. T. J. Turner and G. P. Williams, Jr., J. Phys. Soc., Japan, 1963. 18 (Suppl. II), 218. T. J. Turner and G. P. Williams, Jr., Acta Met., 1960, 8, 891. B. Mills, Phys. Status, Solidii, 1971, 6a (1), 55. T. J. Turner and G. P. Williams, Jr., Acta Met., 1962, 10, 305. S. Pearson, RAE Rep No Met., 1953, 71. B. N. Finkel’shteyn and K. M. Shtrakhman, Phys. Met. Metallogr., 1964, 18 (4), 132. G. P. Williams, Jr. and T. J. Turner, Phys. Status Solidii., 1968, 26 (2), 645. B. G. Childs and A. D. Le Claire, Acta Met., 1954, 2, 718. G. P. Williams, Jr. and T. J. Turner, Phys. Status Solidii, 1968, 26 (2), 645. A. S. Nowick, Phys. Rev., 1952, 88, 925. E. Lax and D. H. Filson, Phys. Rev., 1959, 114, 1273. L. J. Bruner, Phys. Rev., 1960, 118, 399. J. Völkl, W. Weinländer and J. Carsten, Phys. Status Solidii, 1965, 10 (2), 739. W. J. Baxter and J. Wilks, Acta Met., 1963, 11, 978. W. Benoit, B. Bays, P. A. Grandchamp, B. Vittoz, G. Fantozzi and P. Gobin, J. Phys. Chem. Solids, 1970, 31 (8), 1907. J. L. Chevalier, P. Peguin, J. Perez and P. Gobin, J. Phys.(D), 1972, 5 (4), 777. M. Mongy, K. Salama and O. Beckman, Solid State Commun., 1963, 1 (7), 234. A. A. Galkin, O. I. Datsko, V. I. Zaytsev and G. A. Matinin, Phys. Met. Metallogr., 1969, 28 (1), 207. J. Perez and P. Gobin, Phys. Status Solidi, 1967, 24 (2), K167. T. S.-Kê Phys. Rev., 1947, 72, 41. A. C. Damask and A. S. Nowick, J. Appl. Phys., 1955, 26, 1165. G. Schoeck and E. Bisogni, Phys. Status Solidii, 1969, 32 (1), 3. R. E. Miner, T. L. Wilson and J. K. Jackson, Trans. Met. Soc., AIME, 1969, 245 (6), 1375. B. S. Berry and A. S. Nowick, NACA Tech. Note, 1958, 4225. I. N. Fitzpatrick, Ph.D. Thesis, University of Manchester, 1965. K. M. Entwistle, J. Inst. Met., 1953–1954, 82, 249. E. A. Attia, Brit. J. Appl. Phys., 1967, 18 (9), 1343. B. Ya Pines and A. A. Karmazin, Phys. Met. Metallogr., 1970, 29 (1), 206. K. J. Williams, Acta Met., 1967, 15 (2), 393. R. H. Randall and C. Zener, Phys. Rev., 1940, 58, 473. W. G. Nilson, Canad. J. Phys., 1961, 39, 119. B. N. Dey and M. A. Quader, Canad. J. Phys., 1965, 43 (7), 1347. J. Belson, D. Lemercier, P. Moser and P. Vigier, Phys. Status Solidii, 1970, 40, 647. H. Haefner and W. Schneider, Phys. Status Solidii, 1971, 4a, K221. S. Okuda, J. Appl. Phys., 1963, 34, 3107.

Ref. 236 236 209

235 209

Damping capacity 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79. 80. 81. 82. 83. 84. 85. 86. 87. 88. 89. 90. 91. 92. 93. 94. 95. 96. 97. 98. 99. 100. 101. 102. 103. 104. 105. 106. 107. 108. 109.

15–33

S. Okuda and R. R. Hasiguti, Acta Met., 1963, 11, 257. C. H. Neuman, J. Phys. Chem. Solids, 1966, 27 (2), 427. S. Okuda and R. R. Hasiguti, J. Phys. Soc. Japan, 1964, 19 (2), 242. D. G. Franklin and H. K. Birnbaum, Acta Met., 1971, 19 (9), 965. W. Köster, L. Bangert and J. Hafner, Z. Metall, 1956, 47, 224. D. R. Mash and L. D. Hall, Trans. AIMME, 1953, 197, 937. M. E. De Morton and G. M. Leak, Metal Sci. J., 1967, 1, 166. A. Pirson and C. Wert, Acta Met., 1962, 10, 299. G. K. Mal’tseva, V. S. Postnikov and V. V. Usanov, Phys. Met. Metallogr., 1963, 16 (2), 120. B. A. Mynard and G. M. Leak, Phys. Status Solidii, 1970, 40 (i), 113. C. Ang, J. Sivertson and C. Wert, Acta Met., 1955, 3, 558. J. R. Cost, Acta Met., 1965, 13 (12), 1263. K. Mukherjee, J. Appl. Phys., 1966, 37 (4), 1941. C. Ang and K. T. Kamber, J. Appl. Phys., 1963, 34, 3405. Choh-Yi Ang and K. T. Kamber, J. Appl. Phys., 1963, 34 (11), 3405. M. J. Elias and R. Rawlings, J. Less-common Metals, 1965, 9 (4), 305. J. Lulay and C. Wert, Acta Met., 1956, 4, 627. J. Enrietto and C. Wert, Acta Met., 1958, 6, 130. R. Kamel and K. Z. Botros, Phys. Status Solidii, 1965, 12 (1), 399. G. Mah and C. A. Wert, Trans. Met. Soc. AIME, 1968, 242 (7), 1211. K. P. Belov, G. I. Katayev and R. Z. Levitin, J. Appl. Phys., 1960, 31 (Suppl. 1), 1535. V. N. Belko, B. M. Darinskiy, V. S. Postnikov and I. M. Sharshakov, Phys. Met. Metallogr., 1969, 27 (1), 140. M. E. Fine, E. S. Greiner and W. C. Ellis, J. Metals, N. Y., 1951, 191, 56. M. E. De Morton, J. Appl. Phys., 1962, 33, 2768. M. J. Klein, J. Appl. Phys., 1967, 38 (2), 819. M. J. Kelin and A. H. Claver, Trans. Met. Soc., AIME, 1965, 233 (11), 1771. M. J. Klein, Trans. Met. Soc., AIME, 1965, 233 (1), 1943. S. Okuda, J. Appl. Phys., 1963, 34 (10), 3107. D. H. Niblett and J. Wilks, Phil. Mag., 1956, 1, 415. H. S. Sack, Acta Met., 1962, 10, 455. P. Bajons and B. Weiss, Scripta Metall., 1971, 5, 511. B. M. Mecs and A. S. Nowick, Acta Met., 1965, 13 (7), 771. M. Koiwa and R. R. Hasiguti, Acta Met., 1963, 11, 1215. D. T. Peters, J. C. Bisseliches and J. W. Spretnak, Trans. met. Soc. AIME, 1964, 230 (3), 530. M. E. De Morton and G. M. Leak, Acta Met., 1966, 14 (9), 1140. H. Kayano, K. Kamigaki and S. Koda, J. Phys. Soc. Japan, 1967, 23 (3), 649. C. Y. Li and A. S. Nowick, Phys. Rev., 1956, 103, 294. A. A. Karmazin and V. I. Startsev, Phys. Met. Metallogr., 1970, 29 (6), 191. V. S. Postnikov, S. A. Ammer, A. T. Kosilov and A. M. Belikov, Phys. Met. Metallogr., 1966, 21 (5), 121. B. N. Dey, Scripta Metall., 1968, 2 (9), 501. J. T. A. Roberts and P. Barrand, Scripta Metall, 1969, 3 (1), 29. V. S. Postnikov, I. V. Zolotukhin and I. S. Pushkin, Phys. Met. Metallogr., 1968, 26 (4), 147. J. T. A. Roberts, Metall. Trans., 1970, 1 (9), 2487. M. G. Coleman and C. A. Wert, Trans. Met. Soc. AIME, 1966, 236 (4), 501. A. Ikushima and T. Kaneda, Trans. Japan Inst. Metals, 1968, 9 (Suppl). K. J. Marsh, Acta Met., 1954, 2, 530. R. H. Randall, F. C. Rose and C. Zener, Phys. Rev., 1939, 56, 343. K. M. Shtrakhman, Phys. Met. Metallogr., 1967, 24 (3), 116. T. S. Kê, J. Appl. Phys., 1948, 19, 285. L. M. Clareborough, Acta Met., 1957, 5, 413. K. M. Shtrakhman, YU. S. Logvinenko, V. F. Grishchenko and Yu. V. Piguzov, Soviet Phys. solid St., 1971, 13 (5), 1238. K. M. Shtrakhman, Soviet Phys. Solid St., 1967, 9 (6), 1360. K. Takita and K. Sakamoto, Scripta Metall., 1970, 4 (5), 403. A. I. Surin and M. S. Blanter, Phys. Met. Metallogr., 1970, 29 (1), 199. G. M. Leak, Proc. Phys. Soc., Lond., 1961, 78, 1520. J. Delaplace, J. Hillairet and A. Silvent, C.r. hebd. Séanc. Acad. Sci., Paris, 1966 (c), 262 (4), 319. D. B. Fishbach, Acta Met., 1962, 10, 319. K. Tanaka, J. Phys. Soc. Japan, 1971, 30 (2), 404. W. R. Thomas and G. M. Leak, Nature, Lond., 1955, 176, 29. Y. Hayashi and T. Sugeno, Acta Met., 1970, 18 (6), 693. C. A. Wert, Phys. Rev., 1950, 79, 601. R. Blackwell, Nature, Lond., 1966, 211 (5050), 733. P. Barrand and G. M. Leak, Acta Met., 1964, 12 (10), 1147. K. Kamber, D. Keefer and C. Wert, Acta Met., 1961, 9, 403. A. E. Lord and D. N. Beshers, Acta Met., 1966, 14 (12), 1659. G. W. Miles and G. M. Leak, Proc. Phys. Soc., Lond., 1961, 78, 1529. T. Maeda, Japan J. Appl. Phys., 1971, 10 (10), 1299. P. Barrand, Acta Met., 1966, 14 (10), 1247.

15–34 110. 111. 112. 113. 114. 115. 116. 117. 118. 119. 120. 121. 122. 123. 124. 125. 126. 127. 128. 129. 130. 131. 132. 133. 134. 135. 136. 137. 138. 139. 140. 141. 142. 143. 144. 145. 146. 147. 148. 149. 150. 151. 152. 153. 154. 155. 156. 157. 158. 159. 160. 161. 162. 163. 164. 165. 166. 167. 168. 169. 170. 171. 172. 173. 174. 175. 176.

Elastic properties, damping capacity and shape memory alloys

P. Barrand, Metal Sci. J., 1967, 1, 127. P. Barrand, Metal Sci. J., 1967, 1, 54. C. R. Ward and G. M. Leak, Metallurgical, ital., 1970, 62 (8), 302. I. G. Ritchie and R. Rawlings, Acta Met., 1967, 15 (3), 491. W. R. Heller, Acta Met., 1961, 9, 600. R. Gibala, Acta Met., 1967, 15 (2), 428. T. S. Kê and C. T. Tsien, Phys. Met. Metallogr., 1957, 4 (2), 78. V. Kandarpa and J. W. Spretnak, Trans. Met. Soc. AIME, 1969, 245 (7), 1439. G. J. Couper and R. Kennedy, J. Iron Steel Inst., 1967, 205 (6), 642. E. T. Stephenson, Metall. Trans., 1971, 2 (6), 1613. J. D. Fast and M. B. Verrijip, J. Iron Steel Inst., 1955, 180, 337. R. S. Lebyedev and V. S. Postnikov, Phys. Met. Metallogr., 1959, 8 (2), 134. D. Siddell and Z. C. Szkopiak, Metall. Trans., 1972, 3 (7), 1907. Yu. V. Grdina, Ye. E. Glikman and Yu. V. Piguzov, Phys. Met. Metallogr., 1966, 21 (4), 90. D. A. Leak, W. R. Thomas and G. M. Leak, Acta Met., 1955, 3, 501. G. Szabó-Miszenti, Acta Met., 1970, 18 (5), 477. J. Stanley and C. Wert, J. Appl. Phys., 1961, 32, 267. R. M. Jamieson and R. Kennedy, J. Iron Steel Inst., 1966, 204 (2), 1208. H. Sekine, T. Inoue and M. Ogasawara, Japan. J. Appl. Phys., 1967, 6 (21), 272. J. D. Fast and J. L. Meijering, Philips Res. Rep., 1953, 8, 1. W. Hermann, Solid State Commun., 1968, 6 (9), 641. C. F. Burdett, Phil Mag., 1968, 18 (154), 745. B. M. Mecs and A. S. Nowick, Appl. Phys. Letters, 1966, 8 (4), 75. A. Zuckerwar and W. Pechhold, Z. Angew. Phys., 1968, 24 (3), 134. K. Ohori and K. Sumino, Phys. Status Solidii, 1972(a), 9 (1), 151. P. D. Southgate, Proc. Phys. Soc. Lond., 1960, 76, 385, 398. L. N. Aleksandrov, Yu. N. Golobokov, V. N. Orlov and F. L. ‘Edel’ man, Soviet Phys. Solid St., 1969, 10 (9), 2269. F. Calzecchi, P. Gondi and S. Mantovani, J. Appl. Phys., 1969, 40 (12), 4798. E. Bisogni and C. Wert, US Air Force, Sci. Res. Rep., 1961, Contract AF49(638)672. T. Alper and G. A. Saunders, Phil. Mag., 1969, 20 (164), 225. M. E. De Morton, Phys. Status Solidii, 1968, 126, K73. V. S. Postnikov, I. V. Zolotukhin, V. N. Burmistrov and I. M. Sharshakov, Phys. Met. Metallogr., 1969, 28 (4), 210. M. J. Murray, J. Less-common Metals, 1968, 15 (4), 425. T. D. Gulden and J. C. Shyne, J. Inst. Metals, 1968, 96 (5), 139. T. D. Gulden and J. C. Shyne, J. Inst. Metals, 1968, 96 (5), 143. D. P. Seraphim and A. S. Nowick, Acta Met., 1961, 9, 85. J. M. Roberts, Trans. Japan Inst. Metals, 1968, 9 (Suppl.), 69. R. T. C. Tsui and H. S. Sack, Acta Met., 1967, 15 (11), 1715. H. L. Caswell, J. Appl. Phys., 1958, 29, 1210. S. Koda, K. Kamigaki and H. Kayano, J. Phys. Soc. Japan, 1963, 18 (Suppl. 1), 195. A. V. Siefert and F. T. Worrel, J. Appl. Phys., 1951, 22, 1257. R. H. Chalmers and J. Schultz, Acta Met., 1962, 10, 466. H. Mühlbach, Phys. Status Solidii, 1969, 36 (1), K33. R. Gibala, M. K. Korenko, M. F. Amateau and T. E. Mitchell, J. Phys. Chem. Solids, 1970, 3 (8), 1889. G. Rieu, J. De Fouquet and A. Nadeau, C.r. hebd. Séanc. Acad. Sci., Paris, 1970 (c), 270 (3), 287. S. Z. Bokshtein, M. B. Bronfin, et al., Soviet Phys. solid St., 1964, 5 (11), 2253. Yu. V. Piguzov, W. D. Werner and I. Ya. Rzhevskaya, Phys. Met. Metallogr., 1967, 24 (3), 179. R. H. Schnitzel, Trans. Met. Soc. AIME, 1964, 230 (3), 609. M. J. Murray, Phil. Mag., 1969, 20 (165), 561. A. A. Belyakov, V. P. Yelyutin and Ye. I. Mozzhukhin, Phys. Met. Metallogr., 1967, 23 (2), 115. E. J. Kramer and C. L. Bauer, Phys. Rev., 1967, 163 (2), 407. J. Schultz, Bull. Am. Phys. Soc., 1964, 9, 214. F. M. Mazzolai and M. Nuovo, Solid State Commun., 1969, 7 (1), 103. J. Filloux, H. Harper and R. H. Chalmers, Bull. Am. Phys. Soc., 1964, 9, 230. M. W. Stanley and Z. C. Szkopiak, J. Inst. Metals, 1966, 94 (2), 79. M. F. Amateau, R. Gibala and T. E. Mitchell, Scripta Metall., 1968, 2 (2), 123. M. F. Amateau, T. E. Mitchell and R. Gibala, Phys. Status Solidii, 1969, 36 (1), 407. R. A. Hoffman and C. A. Wert, J. Appl. Phys., 1966, 37 (1), 237. F. Schlät, Trans. Japan Inst. Metals, 1968, 9 (Suppl.), 64. E. Davenport and G. Mah, Metall. Trans., 1970, 1 (5), 1452. R. W. Powers and M. V. Doyle, J. Metals, N.Y., 1957, 9, 1285. R. W. Powers and M. V. Doyle, J. Appl. Phys., 1959, 30, 514. C. Vercaemer and A. Clauss, C.r. hebd. Séanc. Acad. Sci., Paris 1969 (c), 269 (15), 803. D. H. Boone and C. Wert, J. Phys. Soc. Japan, 1963, 18 (Suppl. 1), 141. C. Y. Ang, Acta Met., 1953, 1, 123. D. J. Van Ooijen and A. S. Van Der Goot, Acta Met., 1966, 14 (8), 1008. G. Vidal and H. Bibring, C.r. hebd. Séanc. Acad. Sci., Paris, 1965, 260 (3), 857.

Damping capacity 177. 178. 179. 180. 181. 182. 183. 184. 185. 186. 187. 188. 189. 190. 191. 192. 193. 194. 195. 196. 197. 198. 199. 200. 201. 202. 203. 204. 205. 206. 207. 208. 209. 210. 211. 212. 213. 214. 215. 216. 217. 218. 219. 220. 221. 222. 223. 224. 225. 226. 227. 228. 229. 230. 231. 232. 233. 234. 235. 236. 237. 238. 239. 240.

15–35

R. E. Miner, D. F. Gibbons and R. Gibala, Acta Met., 1970, 18 (4), 419. A. W. Sommers and D. N. Beshers, J. Appl. Phys., 1966, 37 (13), 4603. P. S. Venkatesan and D. N. Beshers, J. Appl. Phys., 1970, 41 (1), 42. A. Seeger, P. Schiller and H. Kronmüller, Phil. Mag., 1960, 5, 853. P. Schiller, H. Kronmüller and A. Seeger, Acta Met., 1962, 10, 333. J. T. A. Roberts and P. Barrand, Acta Met., 1967, 15 (11), 1685. O. I. Datsko and V. A. Pavlov, Phys. Met. Metallogr., 1958, 6 (5), 122. T. S. Kê, Acta Phys. Sin., 1955, 11 (5), 405. V. N. Gridnev, A. I. Yefimov and N. P. Kushnareva, Phys. Met. Metallogr., 1967, 23 (4), 142. Y. S. Avraamov, L. N. Belyakov and B. G. Livshits, Phys. Met. Metallogr., 1958, 6 (1), 104. V. M. Azhazha, N. P. Bondarenko, M. P. Zeydlits and B. I. Shapoval, Phys. Met. Metallogr., 1970, 29 (2), 101. R. R. Arons, J. Bouman, M. Witzenbeek, P. T. A. Klaase, C. Tuyn, G. Leferink and G. De Vries, Acta Met., 1967, 15 (1), 144. R. R. Arons, C. Tuyn and G. De Vries, Acta Met., 1967, 15 (10), 1673. J. Coremberg and F. M. Mazzolai, Solid State Commun., 1967, 6 (1), 1. V. O. Shestopal, Phys. Met. Metallogr., 1968, 26 (6), 176. M. Rosen, G. Erez and S. Shtrikman, J. Phys. Chem. Solids, 1969, 30 (5), 1063. R. Schnitzel, J. Appl. Phys., 1959, 30, 2011. L. N. Aleksandrov, M. I. Zotov, R. Sh. Ibragimov and F. L. ‘Edel’ man, Soviet Phys. Solid St., 1970. 11 (7), 1494. P. D. Southgate, Proc. Phys. Soc. Lond., 1957, 70 (B), 804. P. D. Southgate, Proc. Phys. Soc. Lond., 1960, 76, 385, 398. B. S. Berry, J. Phys. Chem. Solids, 1970, 13 (8), 1827. L. Rotherham, A. D. N. Smith and G. B. Greenough, J. Inst. Metals, 1951, 79, 439. L. Verdini and L. A. Vienneau, Canad. J. Phys., 1968, 46 (23), 2715. R. W. Powers and M. V. Doyle, J. Appl. Phys., 1957, 28, 255. P. Kofstad and R. A. Butera, J. Appl. Phys., 1963, 34, 1517. R. W. Powers and M. V. Doyle, Acta Met., 1956, 4, 233. R. W. Powers and M. V. Doyle, Acta Met., 1955, 3, 135. R. W. Powers and M. V. Doyle, Trans. AIMME, 1959, 215, 655. G. Schoek and M. Mondino, J. Phys. Soc. Japan, 1963, 18 (Suppl. 1), 149. A. A. Sagues and R. Gibala, Scripta Metall., 1971, 5 (8), 689. G. Arlt and W. Hermann, Solid State Commun., 1969, 7 (1), 75. J. Winter and S. Weinig, Trans. AIMME, 1959, 215, 74. J. E. Doherty and D. F. Gibbons, Acta Met., 1971, 119 (4), 275. W. Köster, L. Bangert and M. Evers, Z. Metall., 1956, 47, 564. R. R. Hasiguti and K. Iwasaki, J. Appl. Phys., 1968, 39 (5), 2182. W. J. Bratina, Acta Met., 1962, 10, 332. J. N. Pratt, W. J. Bratina and B. Chalmers, Acta Met., 1954, 2, 203. D. Gupta and S. Weinig, Acta Met., 1962, 10, 292. R. W. Powers and M. V. Doyle, Acta Met., 1958, 6, 643. G. Cannelli and F. M. Mazzolai. J. Phys. Chem. Solids, 1970, 31 (8), 1913. R. A. Butera and P. Kofstad, J. Appl. Phys., 1963, 34, 2172. R. W. Powers, Acta Met., 1954, 2, 604. L. N. Aleksandrov and V. S. Mordyuk, Phys. Met. Metallogr., 1966, 21 (1), 101. R. H. Chalmers and J. Schultz, Phys. Rev. Letters, 1961, 6, 273. R. H. Schnitzel, Trans. Met. Soc. AIME, 1965, 233 (1), 186. L. H. Aleksandrov, Phys. Met. Metallogr., 1962, 13 (4), 143. I. Berlec, Metall. Trans, 1970, 1 (10), 2677. V. O. Shestopal, Phys. Met. Metallogr., 1968, 25 (6), 148. V. P. Yelyutin and A. K. Natanson, Phys. Met. Metallogr., 1963, 15 (5), 89. H. Kayano, J. Phys. Soc. Japan, 1969, 26 (3), 733. G. Roberts, P. Barrand and G. M. Leak, Scripta Metall., 1969, 3 (6), 409. J. E. Doherty and D. F. Gibbons, J. Appl. Phys., 1971, 42 (11), 4502. P. L. Gruzin and A. N. Semenikhin, Phys. Met. Metallogr., 1963, 15 (5), 128. W. J. Bratina and W. C. Winegard, J. Metals, N.Y., 1956, 8, 186. K. Bungardt and H. Preisendanz, Z. Metall., 1960, 51, 280. V. Y. Ivanov, B. I. Shapoval and V. M. Amonenko, Phys. Met. Metallogr., 1961, 11 (1), 55. P. Boch, J. Petit, C. Gasc and J. De Fouquet, C.r. hebd. Séanc. Acad. Sci., Paris, 1968 (c), 266 (9), 605. H. L. Brown, P. E. Armstrong and C. P. Kempter, J. Less-common Metals, 1967, 13 (4), 373. J. L. Gacougnolle, S. Sarrazin and J. De Fouquet, C.r. hebd. Séanc. Acad. Sci., Paris 1970 (c), 270 (2), 158. C. W. Nelson, D. F. Gibbons and R. F. Hehemann, J. Appl. Phys., 1966, 37 (13), 4677. S. Karashima and K. Saito, J. Jap. Inst. Metals, 1973, 37(3), 326. H. Farman and D. H. Niblett, ‘Proc. 3rd Euro. Conf. Int. Frict.’, Manchester, 1980, Pergamon Press, p. 7. H. Schulz, U. Rodrian and M. Maul, ‘Proc. 3rd Euro. Conf. Int. Frict’., Manchester, 1980, Pergamon Press, p. 19. H. E. Schaeffer, H. Schulz and H. P. Stark, ‘Proc. 3rd Euro. Conf. Int. Frict’., Manchester, 1980, Pergamon Press, p. 25.

15–36 241. 242. 243. 244. 245. 246. 247. 248. 249. 250. 251. 252. 253. 254. 255. 256. 257. 258. 259. 260. 261. 262. 263. 264. 265. 266. 267. 268. 269. 270. 271. 272. 273. 274. 275. 276. 277. 278. 279. 280. 281. 282. 283. 284. 285. 286. 287. 288. 289. 290. 291. 292. 293. 294. 295. 296. 297.

Elastic properties, damping capacity and shape memory alloys

F. Baudraz and R. Gotthardt, ‘Proc. 3rd. Euro. Conf. Int. Frict.’, Manchester, 1980, Pergamon Press, p. 67. S. M. Seyed Reihani, G. Fantozzi, C. Esnouf and G. Revel, Scripta Met., 1979, 13(8), 1011. H. Mizubayashi, S. Okuda and M. Daikubara, Scripta Met., 1979 13(12), 1131. A De Rooy, P. M. Bronsveld and J. Th M. De Hosson. ‘Proc. 3rd. Euro. Conf. Int. Frict.’, Manchester, 1980, Pergamon Press, p. 149. J. N. Lomer and C. R. A. Sutton, ‘Proc. 3rd Euro. Conf. Int. Frict.’, Manchester, 1980, Pergamon Press, p. 199. M. Weller and J. Diehl, ‘Proc. 3rd Euro. Conf. Int. Frict.’, Manchester, 1980, Pergamon Press, p. 223. R. Schaller and W. Benoit, ‘Proc. 3rd Euro. Conf. Int. Frict.’, Manchester, 1980, Pergamon Press, p. 311. M. Mondino and R. Gugelmeier, ‘Proc. 3rd Euro. Conf. Int. Frict.’, Manchester, 1980, Pergamon Press, p. 317. R. Klam, H. Schulz and H. E. Schaeffer, Acta Met., 1979, 278, 205. K. Iwasaki, K. Lücke and G. Sokolowski, Acta. Met., 1980, 28, 855. B. L. Cheng and T. S. Ke, Phys. Status Solidii, 1988, 107, 177. H. Tezuka et al., J. Nucl. Mater., 1988, 155/7A, 340. S. J. Ding, W. B. Li and G. P. Yang, Rare Met. (China), 7, 99. K. Kato, O. Yoshinari and K. Tanaka, Jpn Inst. Met., 1988, 29, 251. M. Tagami, T. Othani and T. Usami, J. Jap. Inst. Light Met., 1988, 38 (2), 107. S. C. Yan and T. S. Ke, Phys. Status Solidii, 1987, 104, 715. F. Cosandey et al., Scripta Metall., 1988, 22, 395. L. D. Zhang, J. Shi and T. S. Ke, Phys. Status Solidii, 1986, 98, 151. S. Wang, T. Dai and C. Shi, Acta Metall. Sin., 22, A441. S. Chen, J., S. Zhang and Z. Xu, Acta Met. Sin., 22, 379. T. Enjo and T. Kuroda, Trans. JWRA, 1986, 15, 41. S. Sinnema et al., Rapidly Quenched Metals, 1985, 1, 719. K. E. Vidal, W. N. Weins and R. A. Winholz, High Strength Powder Metallurgy Aluminium Alloys, II, 1986, 255. S. Asano and S. Tamaoka, Scr. Metall, 1986, 20, 1151. T. S. Ke and C. M. Su, Phys. Status Solidii, 1986, 94, 191. K. Iwasaki, J. Phys. Soc. Jpn, 1986, 55, 546. K. Iwasaki, J. Phys. Soc. Jpn, 1986, 55, 845. S. Asano and M. Kasaoka, J. Jpn. Inst. Met., 1986, 50, 391. S. Asano and M. Usui, J. Jpn. Inst. Met., 1985, 49, 945. G. Li, Z. Pan and J. Zhang, Acta Metall. Sin., 1985, 21, 21. T. Yokoyama, Scr. Metall., 1985, 19, 747. P. Cui, Q. Huang, T. S. Ke and S. C. Yan, Phys. Status Solidii, 1984, 86, 593. G. Canelli et al., J. Phys. F: Met. Phys., 1984, 14, 2507. C. Esnouf et al., Acta Metall., 1984, 32, 2175. S. Asano and H. Seki, J. Jpn. Inst. Met., 1984, 48, 694. B. Purniah and R. Ranganathan, Phil. Mag. A, 1983, 47, L23. S. A. Antipov, A. I. Drozhzhin and A. M. Roshchupkin, Fiz. Tverd. Tela, 1983, 25, 1392. C. M. Mo and P. Moser, Phys. Status Solidii (a), 1983, 78, 201. J. N. Daou, P. Moser and P. Vajda, J. Phys. (Orsay), 1983, 44, 543. S. Asano and K. Oshima, Trans. Jpn. Inst. Met., 1982, 23, 530. H. Mitani, N. Nakanishi and K. Suzuki, J. Jpn. Inst. Met., 1980, 44, 43. V. F. Belostotskii, T. V. Golub and I. G. Polotskii, Metalofizika, 1982, 4, 106. K. Hakomori, N. Igata and K. Miyahara, J. Jpn. Inst. Met., 1980, 44, 474. P. F. Gobin, J. Merlin and G. Vigier, Ti and Ti Alloys, 1982, 3, 1691. V. E. Bakhrushin, A. V. Novikov and Y. A. Pavlov, Izv. V.U.Z. Chernaya Metall., 1982, 7, 113. H. B. Chen, N. Igata and K. Miyahara, Scr. Metall., 1982, 16, 1039. N. Igata, M. Masamura and H. Murakami, Mech. Props of B.C.C. Metals, 1982, 75. H. Muhlbach, Phys. Status Solidii (a), 1982, 69, 615. H. B. Chen, N. Igata, K. Miyahara and T. Uba, 7th Int. Conf. on Internal Friction and Ultrasonic Attenuation in Solids, 1981, Inst. de Genie Atomique, Lausanne, Paper 1.B.11. B. Dubois and M. Lebienvenu, 7th Int. Conf. on Internal Friction and Ultrasonic Attenuation in Solids, 1981, Inst. de Genie Atomique, Lausanne, Paper 7.A.6. G. Haneczok, J. Moron and T. Poloczek, 7th Int. Conf. on Internal Friction and Ultrasonic Attenuation in Solids, 1981, Inst. de Genie Atomique, Lausanne, Paper 6.C.2. J. Du, 7th Int. Conf. on Internal Friction and Ultrasonic Attenuation in Solids, 1981, Inst. de Genie Atomique, Lausanne, Paper 6.B.4. K. Agyeman, E. Armbruster, H. Guntherodt and H. U. Kunzi, 7th Int. Conf. on Internal Friction and Ultrasonic Attenuation in Solids, 1981, Inst. de Genie Atomique, Lausanne, Paper 4.A.6. C. Li, W. Li, Z. Lui and G. Yang, 7th Int. Conf. on Internal Friction and Ultrasonic Attenuation in Solids, 1981, Inst. de Genie Atomique, Lausanne, Paper 3.C.4. I. G. Ritchie and K. W. Sprungmann, 7th Int. Conf. on Internal Friction and Ultrasonic Attenuation in Solids, 1981, Inst. de Genie Atomique, Lausanne, Paper 3.A.1. K. Sakamoto and M. Shimada, 7th Int. Conf. on Internal Friction and Ultrasonic Attenuation in Solids, 1981, Inst. de Genie Atomique, Lausanne, Paper 1.A.14. R. Hanada, Scr. Metall., 1981, 15, 1121.

Shape memory alloys 298. 299. 300. 301. 302. 303. 304. 305. 306. 307. 308. 309. 310. 311. 312.

15–37

T. Mori, T. Mura and Mokabe, Phil. Mag. A, 1981, 44, 1. K. Qing-hu, G. Ting-sui, P. Zheng-liang and W. Zhong-guang, Acta Phys. Sinica, 1980, 25, 1180. S. Sato and H. Suto, Trans. Jpn. Inst. Met., 1980, 21, 83. W. Bernoit and R. Schaller, Mem. Sci. Rev. Metall., 1979, 76, 521. G. Bellessa and J. Y. Duquesne, J. Phys. C: Solid State Phys., 1980, 13, 215. S. Asano, M. Shibata and Tsunoda, Scr. Metall, 1980, 14, 377. C. Esnouf, G. Fantozzi, G. Revel and S. M. Seyed-Reihani, 3rd European Conf. Internal Friction and Ultrasonic Attenuation in Solids, Manchester, 1979, 1979. C. Diallo and M. Mondini, 3rd European Conf. Internal Friction and Ultrasonic Attenuation in Solids, Manchester, 1979, 1979. R. Gugelmeier and M. Mondini, 3rd European Conf. Internal Friction and Ultrasonic Attenuation in Solids, Manchester, 1979, 1979. R. L. Crosby, J. L. Holman and L. A. Neumeier, U.S. Dept of Interior, Bureau of Mines Rep. Invest. No. 8383, 32pp. M. Hirabayashi, M. Iseki and M. Koiwa, 6th Int. Conf. Internal Friction and Ultrasonic Attenuation in Solids, Tokyo, 1977, 659. A. Isore, L. Miyada, K. Tanaka and S. Watanabe, 6th Int. Conf. Internal Friction and Ultrasonic Attenuation in Solids, Tokyo, 1977, 605. K. Iwasaki, J. Phys. Soc. Jpn, 1978, 45, 1583. I. Brough, K. M. Entwistle and P. Fuller, Acta Met., 1978, 26, 1055. N. Nagai, 1979 (1976), U.S. Patent No. 4, 134, 758.

15.3 Shape memory alloys* 15.3.1 Mechanical properties of shape memory alloys Most shape memory alloys have compositions at which the crystallographic structure can change reversibly and reproducibly from a higher temperature phase with higher symmetry to a lower temperature phase with lower symmetry by a small change in temperature or by a change in mechanical stress at temperatures just above the transformation temperature at zero stress. In most shape memory alloys (and in all the industrially useful alloys), the change of structure usually occurs over a narrow range of temperature by means of a self-accommodating martensitic transformation, during which a small amount of heat is evolved or absorbed depending on the direction of the temperature change. This usually gives rise to a thermal hysteresis of about 10 to 40◦ C over which the parent phase (usually referred to as ‘austenite’) and the martensite can co-exist. If a stressed shape memory alloy is thermally cycled through its martensitic transformation temperature, the strain–temperature relationship will take the form of a closed hysteresis loop similar in shape to the B–H curve of ferromagnetic alloys. If a shape memory alloy is cooled to below its Mf temperature, it undergoes little change in shape or volume. If it is then deformed plastically to a new shape, it will recover its original undeformed shape on re-heating to a temperature above its As temperature. The amount of strain which can be recovered in this way is not unlimited but depends on the nature of the alloy. For example, the maximum recoverable strain is about 8% in Ti–Ni alloys and 10–12% in Cu–Zn alloys (although the latter cannot be achieved in industrially useful alloys). On cooling through the transformation to the martensitic state, the temperatures at which the transformation starts and finishes at zero applied stress are denoted by Ms and Mf respectively. On re-heating, the temperatures at which the reverse transformation to the high temperature phase takes place are As and Af respectively. These temperatures can be determined experimentally by thermal or dilatometric analysis or by changes in electrical resistivity. The Ms temperature is raised progressively by applied stress; the Md temperature is the highest temperature at which the transformation can be induced by stress. Figure 15.1 illustrates these points.11 If an alloy is deformed above the Ms but below the Md temperature, a stress-induced martensitic strain can be obtained. This is completely recovered on unloading (see Figures 15.4 and 15.7). Shape memory does not always appear to depend on the martensitic transformation. Small amounts of shape memory can be obtained in primary solid solutions of alloys of low stacking fault energy.16 Examples are shown in Table 15.9, e.g. Cu–Al and Cu–Si primary solid solutions and some stainless steels. Although the latter undergo martensitic transformations to the alpha prime martensite, this is too brittle to deform. Shape memory is only found when the stainless steel is deformed at very low temperatures but above the Ms to form both delta and some alpha martensite. 20% deformation ∗ Additional

references on this topic may be found in Chapter 38.

Elastic properties, damping capacity and shape memory alloys

15–38

may be needed to obtain 1.0–1.5% shape memory strain and it has proved to be of little industrial relevance so far. Though most work has concentrated on Ti–Ni and related alloys because of their industrial importance, the shape memory phenomenon has been demonstrated in a wide range of alloys, some of which are listed in Table 15.9. Ms temperatures can be varied continuously by changing the composition. Examples include: (i) changing Ni content in Ti–Ni alloys; (ii) partially replacing Ni by Fe, Co or Pd in Ti–Ni alloys; and (iii) changing Al and Zn contents in Cu–Al–Zn alloys or by partially replacing either element by others such as Sn, Mn, etc. Such variations also change the character of the alloys. Table 15.9 shows typical compositions for which data are published but it is possible to derive additional alloys within the ternary and more complex systems. Note that if shape memory alloys are cooled under stress, the Ms temperature is raised in direct proportion to stress below Mo (see Figure 15.7). Note also that the Ms , etc temperatures are not exact in that for a given composition they can be changed by heat-treatment and by cold-working. Figure 15.8 illustrates an example of the extent to which the hysteresis can be widened in a Cu-based alloy.37

I)

i (II

Uniaxial expansion

Ni T

Ni Ti

(II)

Mf

Ms

As

Af Md

Temperature Figure 15.1 Typical uniaxial dimensional change behaviour for drawn wire. Ti–55.0% Ni–0.07%C. After W. B. Cross et al.11

Table 15.9

COMPOSITIONS AND TRANSFORMATION TEMPERATURES OF SHAPE MEMORY ALLOYS

Alloy composition wt%(1) Au–28at% Cu–46 At.% Zn Au–47.5% Cd Au–12.9% Cu–25.5% Zn Au–15.2% Cu–28.0% Zn

Ms temperature at zero stress at ◦C −15 60 −100 −50

As temperature at zero stress ◦C 70

Maximum shape memory strain %

1.55 ∼1.0; brittle g.b. phase

Reference 6 13 32 32 (continued)

Shape memory alloys

15–39

Table 15.9 COMPOSITIONS AND TRANSFORMATION TEMPERATURES OF SHAPE MEMORY ALLOYS—continued

Alloy composition wt%(1)

Ms temperature at zero stress at ◦C

As temperature at zero stress ◦C

Au–16.0% Cu–32.3% Zn

−118

Au–22.3% Cu–31.4% Zn Au–28.7% Cu–31.1% Zn Cu–2.50% Al–31.75% Zn Cu–3.94% Al–25.60% Zn Cu–4.00% Al–26.10% Zn

−64 < −196 −105 54 24

Cu–6.00% Al–22.00% Zn Cu–7.50% Al–17.00% Zn Cu–11.75% Al–6.00% Zn Cu–10.50% Al–7.25% Zn Cu–11.25% Al–4.75% Zn Cu–11.75% Al–2.50% Zn Cu–4.90% Sn–31.25% Zn Cu–1.75% Si–34.50% Zn Cu–2.25% Si–31.25% Zn Cu–3.25% Si–27.50% Zn Cu–12.00% Al–2.00% Mn Cu–11.25% Al–4.25% Mn Cu–10.75% Al–6.00% Mn Cu–10.40% Al–7.00% Mn Cu–10.60% Al–7.00% Mn Cu–11.00% Al–7.00% Mn Cu–11.10% Al–7.00% Mn Cu–12.50% Al–1.00% Fe Cu–12.50% Al–8.00% Fe Cu–13.25% Al–2.75% Ni Cu–8.0% Al Cu–4.0% Si Cu–14.2% Al–4.3% Ni Cu–40% Zn Cu–34.7% Zn–3.0% Sn

−50 −10 50 140 170 250 −70 −140 −50 75 240 160 100 83 52 5 −10 300 250 82 N.A. N.A. −20 −70 −52

N.A. N.A. −15 −120 −50

Fe–15% Cr–15% Ni–15% Co Fe–20% Cr–10% Ni–1% Al Fe–20% Mn–3.75% Ti Fe–20% Cr–15% Ni Fe–17% Cr–19% Ni Fe–30% Mn–6.5% Si Fe–24% Mn–3% Si Fe–26% Mn–4% Si Fe–27% Mn–3% Si Fe–28% Mn–3% Si Fe–28% Mn–4% Si Fe–29% Mn–4% Si Fe–32% Mn–4% Si Fe–33% Mn–4% Si Fe–34% Mn–4% Si Fe–35% Mn–4% Si Ti–50.2at% Ni Ti–50.2at% Ni Ti–50.2at% Ni

< −196 < −196 < −196 < −196 > −196 < 20 123 106 103 102 62 88 27 22 8 −10 35 10 ∼25

−196 to 40 −196 to 40 −196 to 40 −196 to 40 −196 to 40 20 to 400 188 182 176 164 162 164 134 131 127 123 50 57 ∼55

Ti–56.4% Ni Ti–55.5% Ni Ti–55.0% Ni

5 10 35

48 23

Maximum shape memory strain % ditto; Zn too high 2.15 ?; Zn too low 10.25 > 2.0 14% (5–6% reversible) 4.8 4 2.8 3.9 2 1.9 1 6.3 6 2.95 0.95 0.45 0.45

1.75 1.1 2.9 1.05 1.4 4.5 (polycrystal) 8.5 (single crystal) 1.45 0.45 1.4 1.05 0.9 2.1 1.5–1.75 1.5–1.75 1.5–1.75 1.5–1.75 1.5–1.75 1.5–1.75 1.5–1.75 1.5–1.75 1.5–1.75 1.5–1.75 1.4 (reversible) 5.5% (∼4% reversible)

Reference 32 32 32 15 23 20 15 15 15 15 15 15 15 15 15 15 15 15 15 21 21 21 21 15 15 15 16 16 8 8 9

16 16 16 16 16 22 24 24 24 24 24 24 24 24 24 24 17 18 19 26 26 26 (continued)

15–40

Elastic properties, damping capacity and shape memory alloys

Table 15.9 COMPOSITIONS AND TRANSFORMATION TEMPERATURES OF SHAPE MEMORY ALLOYS—continued

Alloy composition wt%(1) Ti–53.7% Ni Ti–54.8% Ni Ti–52–56% Ni Ti–53.5% Ni Ti–54.0% Ni Ti–54.5% Ni Ti–55.0% Ni Ti–55.5% Ni Ti–56.0% Ni Ti–56.5% Ni Ti–51at% Ni TiNi(2) TiCo (3) TiNix Co1−x

TiFe Ti–35% Nb Ti–51.4% Ni–3.57% Co 0.3 in. rod 0.003 in. foil 0.01 in. wire Ti–55.0% Ni-0.07% C 0.2 in. rod 0.003 in. foil 0.01 in. wire Ti–54.6% Ni–0.06% C 0.625 in. rod 0.003 in. foil 0.01 in. wire Ti–47.0% Ni–7% Cu Ti–44.5% Ni–10% Cu Ti–32.0% Ni–22% Cu Ti–27.0% Ni–29% Cu Ti–47.0at% Ni–3at% Fe Ti–52.85% Ni–0.28% Fe

Ms temperature at zero stress at ◦C 75 62 ∼RT 98 140 170 140 30 −25 −50 28 166 −238 166 where x = 1 −238 where x = 0 ∼ − 269 −175

As temperature at zero stress ◦C

175

Maximum shape memory strain %

8.0

62

Reference 26 27 1 2 2 2, 4 2 2 2 2 7 3 3 3

184

∼2.5

10

−51 −65 −73

−40 −45 −51

6 to 10 6 to 10 6 to 10

11 11 11

21 27 18

60 43 43

6 to 10 6 to 10 6 to 10

11 11 11

43 38 32 63 52 74 20 −180 34

71 66 54

6 to 10 6 to 10 6 to 10

11 11 11 27 27 27 27 30 14, 33, 34

Ti–47.0at% Ni–9at% Nb

−90

Ti–50at% Pd Ti–50at% Pd Ti–50at% Pd Ti–40.0at% Pd–10.0at% Ni Ti–30.0at% Pd–20.0at% Ni Ti–20.0at% Pd–30.0at% Ni Ti–10.0at% Pd–40.0at% Ni Ti–50.0at% Ni Ti–5.0at% Pt–45.0at% Ni Ti–10.0at% Pt–40.0at% Ni Ti–20.0at% Pt–30.0at% Ni Ti–30.0at% Pt–20.0at% Ni Ti–50.0at% Pt Ti–44at% Pd–6at% Fe Ti–42at% Pd–8at% Fe Ti–40at% Pd–10at% Fe Ti–38at% Pd–12at% Fe Ti–36at% Pd–14at% Fe Ti–34at% Pd–16at% Fe

533 510 563 403 241 95 −18 55 29 18 300 619 1 070 321 293 173 96 25 −49

−88 35 −56 up to 55 573 520 580 419 230 90 −26 80 36 −27 263 626 1 040 335 250 178 99 25 −45

8 (0.094 rad in torsion) ∼2.0 2.39 2.66 4.38 1.84

2.17 5.0

30 28, 29 36 35 35 35 35 35 35 35 35 35 35 36 28, 29 28 28 28 28 28 (continued)

Shape memory alloys

15–41

Table 15.9 COMPOSITIONS AND TRANSFORMATION TEMPERATURES OF SHAPE MEMORY ALLOYS—continued

Alloy composition wt%(1) Ti–42at% Pd–4at% Fe Ti–40at% Pd–6at% Fe Ti–38at% Pd–8at% Fe ZrRu(4) ZrRh(4) ZrPd(4) U–4.0% Mo U–4.5% Mo U–5.0% Mo U–6.0% Mo U–4.0% No U–5.0% Nb U–7.0% Nb U–14at% Nb U–9at% Nb(5) U–12at% Nb(5) U–15at% Nb(5) U–18at% Nb(5)

Ms temperature at zero stress at ◦C

As temperature at zero stress ◦C

228 119 18 −233 380 727

216 124 3

160 0 0 0 −196

200 80 50 −50 ∼250 ∼200 ∼100 90 150 150 60 0

Maximum shape memory strain %

3.2

4.8

(1) (2)

Reference 28 28 28 3 3 3 31 31 31 31 31 31 31 25 5 5 5 5

Unless otherwise stated. See also Figure 15.2.3 See Figures 15.3. (4) Non-linear interpolation between their compounds is possible. Relationship is of form log (M K) ∞ x where x is in the range e s 0–1 in Zr–Rux –Rh1−x etc. (5) M and A not accurately determined but are within these temperature ranges. s s (3)

PHYSICAL AND MECHANICAL PROPERTIES OF A TITANIUM–55% NICKEL SHAPE MEMORY ALLOY

Table 15.10 Property

Value

Density M.p. Magnetic permeability Electrical resistivity

6.45 g cm−3 1 310◦ C < 1.002 80 µ cm at 20◦ C 132 µ cm at 900◦ C 10.4 × 10−6 ◦ C −1

Coefficient of thermal expansion (24–900◦ C) Mechanical properties at 20◦ C (i.e. below Ms ) 0.2% yield stress UTS Elongation % Reduction in area % Impact (unnotched) Fatigue (rotating beam)

207 MPa 861 MPa 22 20 159 J at 20◦ C 95 J at 80◦ C >25 × 106 cycles at 483 MPa

Note If recovery is prevented mechanicaly, the TiNi will exert a stress and is capable of doing work when heated to above the As temperature. Samples of Ti–55.0% Ni–0.07%C were capable of exerting a stress of up to 758 MPa at 171◦ C (see Figure 15.5). The amount of mechanical work which this alloy was capable of doing was 17–20 J cm−3 on heating from 24◦ C to 171◦ C.

15–42

Elastic properties, damping capacity and shape memory alloys 200

150

Temperature (°C)

100

50

0

⫺50

Harrison et al. Hanlon et al. Purdy et al.

⫺100

⫺150

47

48

49 50 51 Nickel atomic (%)

52

53

Logarithm*martensitic transition temperature, K

Figure 15.2 The dependence of the transformation temperature Ms on composition of Ti–Ni alloys, after K. N. Melton39

6.0

5.0

4.0

3.0 6.0

6.5 7.0 Free electron concentration

TiFe Figure 15.3

See Table 15.9 after F. E. Wang and W. J.

TiCo Buehler3

TiNi *Natural logarithm

Shape memory alloys

15–43

Cu⫺39.8% Zn Ms ⫽ ⫺125°C 125

⫺60°C

Stress (MPa)

100

⫺75°C ⫺88°C

75

⫺99°C 50 25 0

0

2

4 6 Strain (%)

8

10

Figure 15.4 Stress–strain curves for a Cu–Zn single crystal loaded in tension above Ms . As the Ms temperature is approached, the stress required to induce martensite is lowered, after C. M. Wayman and T. W. Duerig38

120

Maximum recovery stress, (Ib ⫻ 103) in⫺2

100

80

60

40

2.54 mm dia rod 0.51 mm dia wire 0.38 mm dia wire 0.25 mm dia wire 0.15 mm thick foil 0.08 mm thick foil

20

Note: Strain temperature ⫽ 25°C(75°F)

0

2

4

6

8 Initial strain, %

10

12

14

Figure 15.5 Maximum recovery stress versus initial strain curves for Ti–55.0% Ni–0.0% C. See Table 15.9, after W. B. Cross et al.11

Elastic properties, damping capacity and shape memory alloys Work done/vol.,10⫺3 J mm⫺3

15–44

Pre strain 39° (0.097 radius shear strain)

14 12 10 8 6 4 2 0

Figure 15.6 torque.14

2

4

6 8 10 12 14 16 18 20 Restraining torque, Nm

The relationship between the capacity of TiNi alloy to do work in torsion and the restraining

Cu⫺39.8% Zn

Stress (MPa)

150

s e

100

50 Ms 0 ⫺130 ⫺120

As ⫺100

⫺70

T (°C) Figure 15.7 Plotting the plateau stresses such as shown in Figure 15.4 as a function of temperature gives a linear plot which obeys the Clausius–Clapeyron relationship. The alloy’s zero stress As and Ms are marked on the ordinate.38

Mf

AS

A′S

Strain

First heating after preconditioning

Second heating after preconditioning MS

Af

A′f

Temperature Figure 15.8 The preconditioning process is a one-time displacement of As and Af . Once recovery is complete, martensite can be reformed, after which As and Af are restored to their original values, after Duerig et al.37

Shape memory alloys

15–45

REFERENCES FOR SECTION 15.3 For data on designs and applications using shape memory alloys, see ‘Engineering Aspects of Shape Memory Alloys’, edited by T. W. Duerig, K. N. Melton, D. Stöckel and C. M. Wayman, published by Butterworth–Heineman Ltd, London, 1990. For a general review of shape memory materials, their properties and applications, please refer to ‘Shape Memory Materials’, edited by K. Otsuka and C. M. Wayman, published by Cambridge University Press, 1998. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39.

W. J. Buehler and R. C. Wiley, US Patent 3 174 851. A. G. Rozner and W. J. Buehler, US Patent 3 351 463. F. E. Wang and W. J. Buehler, US Patent 3 558 369. G. B. Brook, unpublished data. R. J. Jackson, J. F. Boland and J. L. Frankeng, US Patent 3 567 523. N. Nakanishi, et al., Phys. Letters, 1971, 37A, 61. F. E. Wang, et al., J. Appl. Phys., 1968, 39, 2166. K. Otsuka, et al., Scripta Metall., 1972, 6, 377. J. D. Eisenwasser and L. C. Brown, Met. Trans. AIME., 1972, 3, 1359. C. Baker, Met. Sci. J., 1971, 5, 92. W. B. Cross, et al., NASA Report CR—1433, Sept. 1969. H. U. Schuerch, NASA Report CR—1232, Nov. 1968. D. S. Lieberman, T. A. Read and M. S. Wechsler, J. Appl. Phys., 1957, 28, 532. G. B. Brook, et al., Fulmer Research Inst. Rep. No R662/4A, Feb., 1977. G. B. Brook and R. F. Iles, British Patent Specification No. 1346047. G. B. Brook, R. F. Iles and P. L. Brooks, in ‘Shape Memory Effects in Alloys’ (ed. J. Perkins), 1975, Plenum Press, New York, p. 477. S. Edo, J. Mat. Sci. Letters, 1989, 24, 3991. H. Tamura, Y. Susuki and T. Todoroki, Proc. Int. Conf. Martensitic Transform., 1986, p. 736 (Jap. Inst. Met.). Y. Liu and P. G. McCormick, Acta Met. Mat., 1990, 38, 1321. L. Contardo and G. Guenin, Acta Met. Mat., 1990, 38, 1267. M. L. Blasquez, C. Lopez and C. Gomez, Metallography, 1989, 23, 119. J. S. Robinson and P. G. McCormick, Scripta Met., 1989, 23, 1975. S. S. Leu and C. T. Hu, Mat. Sci. and Eng., 1989, A117, 247. M. Sade, K. Halter and E. Hornbogen, J. Mat. Sci. Letters, 1990, 9, 112. R. A. Vandermeer, J. C. Ogle and W. B. Snyder, Scripta Met., 1978, 12, 243. V. N. Ermakov, V. I. Kolomytsev, V. A. Lobodyuk and L. G. Khandros, Metall. Term. Obr. Metallov, 1981, 5, 57. R. H. Bricknell, K. N. Melton and O. Mercier, Met. Trans., AIME, 1979, 10A, 693 (See also US Patent 4 144057, 1979 for additional compositions in Ti–Ni–Cu– (Co, Fe, Al, Cr) alloys). K. Enami, T. Yoshida and S. Nenno, Proc. Int. Conf. Martensitic Transform., 1986, p. 103 (Jap. Inst. Met.). K. Enami, Y. Miyasaka and H. Takakura, MRS Int. Conf. on Adv. Mat., 1989, 9, 135. J. A. Simpson, T. Duerig and K. M. Melton, European Patent, 1985, No. 0187452. G. B. Brook and R. F. Iles, British Patent Specification, 1969, No. 1315653. G. B. Brook and R. F. Iles, Gold Bulletin, 1975, 8, 16. D. Powley and G. B. Brook, 12th Aerospace Mechanisms Symp., NASA Conf. Publ. 2080, 1978, 119. G. B. Brook, Inst. Metallurgists Conf. on Phase Transf., 1979, Ser. 3, 2 (11), VI-1–3. P. G. Lindquist and C. M. Wayman in ‘Engineering Aspects of Shape Memory Alloys’ (ed. T. W. Duerig et al.), 1990, Butterworth–Heinemann, London, p. 58. H. C. Donkersloot and J. H. N. Van Vucht, J. Less Common Metals, 1970, 20, 83. T. W. Duerig, K. N. Melton and J. L. Proft in ‘Engineering Aspects of Shape Memory Alloys’ (ed. T. W. Duerig et al.), 1990, Butterworth–Heinemann, London, p. 130. C. M. Wayman and T. W. Duerig in ‘Engineering Aspects of Shape Memory Alloys’ (ed. T. W. Duerig et al.), 1990, Butterworth–Heinemann, London, p. 3. K. N. Melton in ‘Engineering Aspects of Shape Memory Alloys’ (ed. T. W. Duerig et al.), 1990, Butterworth– Heinemann, London, p. 21.

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16 Temperature measurement and thermoelectric properties 16.1 Temperature measurement* The unit of the fundamental physical quantity known as thermodynamic temperature, symbol T , is the Kelvin, symbol K, defined as the fraction 1/273.16 of the thermodynamic temperature of the triple point of water.† Because of the way earlier temperature scales were defined, it remains common practice to express a temperature in terms of its difference from 273.15 K, the ice point. A thermodynamic temperature, T , expressed in this way is known as a Celsius temperature, symbol t, defined by: t/◦ C = T /K − 273.15 The unit of Celsius temperature is the degree Celsius, symbol ◦ C, which is by definition equal in magnitude to the Kelvin. A difference of temperature may be expressed in Kelvin or degrees Celsius. The International Temperature Scale of 1990 (ITS-90) defines both International Kelvin Temperatures, symbol T90 , and International Celsius Temperatures, symbol t90 . The relation between T90 and t90 is the same as that between T and t, i.e.: t90 /◦ C = T90 /K − 273.15 The unit of the physical quantity T90 is the Kelvin, symbol K, and the unit of the physical quantity t90 is the degree Celsius, symbol ◦ C, as is the case for the thermodynamic temperature T and the Celsius temperature t. The International Temperature Scale of 19901 was adopted by the International Committee of Weights and Measures at its meeting in 1989, in accordance with the request embodied in Resolution 7 of the 18th General Conference of Weights and Measures of 1987. This Scale supersedes the International PracticalTemperature Scale of 1968 (amended edition of 1975) and the 1976 Provisional 0.5 K to 30 K Temperature Scale. The ITS-90 extends upwards from 0.65 K to the highest temperature practicably measurable in terms of the Planck radiation law using monochromatic radiation. The ITS-90 comprises a number of ranges and subranges throughout each of which temperatures T90 are defined. Several of these ranges or subranges overlap, and where such overlapping occurs differing definitions of T90 exist: these differing definitions have equal status. For measurements of the very highest precision there may be detectable numerical differences between measurements made at the same temperature but in accordance with differing definitions. Similarly, even using one definition, at a temperature between defining fixed points two acceptable interpolating instruments (e.g. resistance thermometers) may give detectably differing numerical values of T90 . In virtually all cases these differences are of negligible practical importance and are at the minimum level consistent with a scale of no more than reasonable complexity: for further information on this point, see ‘Supplementary Information for the ITS-90’.2 The ITS-90 has been constructed in such a way that, throughout its range, for any given temperature the numerical value of T90 is a close approximation to the numerical value of T according ∗ †

For details of emissivity, as needed for pyrometry—see Chapter 17. ‘Comptes Rendus des Séances de la Treizième Conférence Générale des Poids et Mesures (1967–1968)’, Resolutions 3 and 4, p. 104.

16–1

16–2

Temperature measurement and thermoelectric properties

to best estimates at the time the scale was adopted. By comparison with direct measurements of thermodynamic temperatures, measurements of T90 are more easily made, are more precise and are highly reproducible.3 There are significant numerical differences between the values of T90 and the corresponding values of T68 measured on the International Practical Temperature Scale of 1968 (IPTS-68), see Figure 16.1 and Table 16.1. Similarly there were differences between the IPTS-68 and the International Practical Temperature Scale of 1948 (IPTS-68), and between the International Temperature Scale of 1948 (ITS-48) and the International Temperature Scale of 1927 (ITS-27).3 Between 0.65 K and 5.0 K T90 is defined in terms of the vapour-pressure temperature relations of 3 He and 4 He. Between 3.0 K and the triple point of neon (24.556 1 K) T90 is defined by means of a helium gas thermometer calibrated at three experimentally realisable temperatures having assigned numerical values (defining fixed points) and using specified interpolation procedures. Between the triple point of equilibrium hydrogen (13.803 3 K) and the freezing point of silver (961.78◦ C) T90 is defined by means of platinum resistance thermometers calibrated at specified sets of defining fixed points and using specified interpolation procedures. Table 16.1

THE DIFFERENCES BETWEEN ITS-90 AND EPT-76, AND BETWEEN ITS-90 AND IPTS-68

(T90 − T76 )/mK T90 /K 0 0 10 −0.6 20 −2.2

1

2

3

4 −1.1 −3.2

5 −0.1 −1.3 −3.5

6 −0.2 −1.4 −3.8

7 −0.3 −1.6 −4.1

8 −0.4 −1.8

9 −0.5 −2.0

−0.7 −2.5

−0.8 −2.7

−1.0 −3.0

4 −0.006 −0.006 −0.008 −0.006 −0.003 0.005 0.007 0.008 0.008

5 −0.003 −0.005 −0.007 −0.007 −0.002 0.005 0.008 0.008 0.008

6 −0.004 −0.004 −0.007 −0.007 −0.001 0.006 0.008 0.008 0.008

7 −0.006 −0.004 −0.007 −0.007 0.000 0.006 0.008 0.008 0.009

8 −0.008 −0.005 −0.006 −0.006 0.001 0.007 0.008 0.008 0.009

9 −0.009 −0.006 −0.006 −0.006 0.002 0.007 0.008 0.008 0.009

(T90 − T68 )/K T90 /K 0 10 20 −0.009 30 −0.006 40 −0.006 50 −0.006 60 0.003 70 0.007 80 0.008 90 0.008

1

2

3

−0.008 −0.007 −0.006 −0.005 0.003 0.007 0.008 0.008

−0.007 −0.008 −0.006 −0.005 0.004 0.007 0.008 0.008

−0.007 −0.008 −0.006 −0.004 −0.004 −0.007 −0.008 −0.008

T90 /K 100 200

0.009 0.011

10 0.011 0.010

20 0.013 0.009

30 0.014 0.008

40 0.014 0.007

50 0.014 0.005

60 0.014 0.003

70 0.013 0.001

80 0.012

90 0.012

(t90 − t68 )/◦ C t90 /◦ C 0 −100 0.013 0 0.000

−10 0.013 0.002

−20 0.014 0.004

−30 0.014 0.006

−40 0.014 0.008

−50 0.013 0.009

−60 0.012 0.010

−70 0.010 0.011

−80 0.008 0.012

−90 0.008 0.012

10 −0.002 −0.028 −0.040 −0.039 −0.051 −0.083 −0.118 0.24 0.32 −0.03 −0.20

20 −0.005 −0.030 −0.040 −0.039 −0.053 −0.087 −0.122 0.28 0.29 −0.06 −0.21

30 −0.007 −0.032 −0.040 −0.040 −0.056 −0.090 −0.125* 0.31 0.25 −0.08 −0.22

40 −0.010 −0.034 −0.040 −0.040 −0.059 −0.094 −0.08 0.33 0.22 −0.10 −0.23

50 −0.013 −0.036 −0.040 −0.041 −0.062 −0.098 −0.03 0.35 0.18 −0.12 −0.24

60 −0.016 −0.037 −0.040 −0.042 −0.065 −0.101 0.02 0.36 0.14 −0.14 −0.25

70 −0.018 −0.038 −0.039 −0.043 −0.068 −0.105 0.06 0.36 0.10 −0.16 −0.25

80 −0.021 −0.039 −0.039 −0.045 −0.072 −0.108 0.11 0.36 0.06 −0.17 −0.26

90 −0.024 −0.039 −0.039 −0.046 −0.075 −0.112 0.16 0.35 0.03 −0.18 −0.26

100 −0.26 −0.79 −0.59

200 −0.30 −0.85 −1.69

300 −0.35 −0.93 −1.78

400 −0.39 −1.00 −1.89

500 −0.44 −1.07 −1.99

600 −0.49 −1.15 −2.10

700 −0.54 −1.24 −2.21

800 −0.60 −1.32 −2.32

900 −0.66 −1.41 −2.43

t90 /◦ C 0 100 200 300 400 500 600 700 800 900 1 000 t90 /◦ C 1 000 2 000 3 000

0

0 0.000 −0.026 −0.040 −0.039 −0.048 −0.079 −0.115 0.20 0.34 −0.01 −0.19 0 −0.72 −1.50

* A discontinuity in the first derivative of (t90 − t68 ) occurs at a temperature of t90 = 630.6◦ C when (t90 − t68 ) = −0.125◦ C.

Temperature measurement

16–3

Temperature difference (t 90 ⫺ t 68) °C

0.02 0

0.4

⫺0.02 0.2

⫺0.04 ⫺200

0

⫺0.2

0

200

400

0

0

100

⫺0.01

⫺0.2

⫺0.02

⫺200

Figure 16.1

0

200

400 t 90°C

600

800

1000

The differences, (t90 − t68 ), between ITS-90 and IPTS-68 in the range from –260◦ C to 1 064◦ C

Above the freezing point of silver (961.78◦ C) T90 is defined in terms of a defining fixed point and the Planck radiation law. The defining fixed points of the ITS-90 together with some selected secondary reference points are listed in Table 16.2. Full details of the practical realisation of this scale can be found in refs. 2 and 3. THE DEFINING FIXED POINTS OF ITS-90 AND SOME SELECTED SECONDARY REFERENCE POINTS

Table 16.2

Equilibrium state Cd superconducting transition point Zn superconducting transition point Al superconducting transition point 4 He lambda point In superconducting transition point 4 He boiling point Pb superconducting transition point *Triple point of equilibrium hydrogen *Boiling point of equilibrium hydrogen at a pressure of 33 330.6 pascals (25/76 standard atmosphere) *Boiling point of equilibrium hydrogen *Ne triple point Ne boiling point *O2 triple point N2 triple point N2 boiling point *Ar triple point O2 condensation point Kr triple point CO2 sublimation point *Hg triple point H2 O freezing point

T90 (K)

t90 (◦ C)

0.519 0.851 1.179 6 2.176 8 3.414 5 4.222 1 7.199 6 13.803 3 17.035 7 20.271 1 24.556 1 27.098 54.358 4 63.150 77.352 83.805 8 90.196 115.776 194.685 234.315 6 273.15

−218.791 6 −210.000 −195.798 −189.344 2 −182.954 −157.374 −78.465 −38.834 4 0 (continued)

16–4

Temperature measurement and thermoelectric properties

THE DEFINING FIXED POINTS OF ITS-90 AND SOME SELECTED SECONDARY REFERENCE POINTS—continued

Table 16.2

Equilibrium state

T90 (K)

*H2 O triple point *Ga melting point H2 O boiling point *In freezing point *Sn freezing point Bi freezing point Cd freezing point Pb freezing point *Zn freezing point S boiling point Cu–Al eutectic melting point Sb freezing point *Al freezing point Ag–Cu eutectic melting point *Ag freezing point *Au freezing point *Cu freezing point Ni freezing point Co freezing point Pd freezing point Pt freezing point Rh freezing point Ir freezing point W freezing point

273.16 302.914 6 373.124 429.748 5 505.078 544.553 594.219 600.612 692.677 717.764 821.313 903.78 933.473 1 053.09 1 234.93 1 337.33 1 357.77 1 728 1 768 1 827 2 041 2 235 2 719 3 693

t90 (◦ C) 0.01 29.764 6 99.974 156.598 5 231.928 271.403 321.069 327.462 419.527 444.614 548.163 630.63 660.323 779.94 961.78 1 064.18 1 084.62 1 455 1 495 1 554 1 768 1 962 2 446 3 420

* Defining point of ITS-90. For details of experimental techniques used in the realisation of these fixed points see references 2 and 3. All except the triple points and the hydrogen boiling point at 33 330.6 Pa are at a pressure of 101 325 Pa (1 standard atmosphere).

16.2 Thermocouple reference tables The introduction of ITS-90 on 1st January 1990 has led to an international programme, coordinated by the Comité Consultatif de Thermométrie, for the revision of the internationally agreed reference tables for thermocouples. It will take a few years, however, for the new tables to be adopted by national and international standards organizations. Meanwhile the following old tables (BS 4937 and IEC 584-1 1977) based upon IPTS-68 should be used together with the differences t90 − t68 given in Table 16.1 and Figure 16.1.

Table 16.3

THERMAL ELECTROMOTIVE FORCE (millivolts) OF ELEMENTS RELATIVE TO PLATINUM*

A positive sign means that in a simple thermoelectric circuit the element is positive to the platinum at the reference junction (0◦ C). The e.m.f. generated by any two elements, A and B, can also be found from this table. It is the algebraic difference, Ae − Be , between the values (Ae and Be ) for the e.m.f. generated by each relative to platinum; a positive sign indicates that A is positive to B at the reference junction, and a negative sign that A is negative to B. Cold junction at 0◦ C Temperature of hot junction ◦ C

Aluminium Antimony Bismuth Cadmium Caesium Calcium Carbon

−200

−100

+100

+200

+0.45 — +12.39 −0.04 +0.22 — —

−0.06 — +7.54 −0.31 −0.13 — —

+0.42 +4.89 −7.34 +0.91 — −0.51 +0.70

+1.06 +10.14 −13.57 +2.32 — −1.13 +1.54 (continued)

Thermocouple reference tables

16–5

THERMAL ELECTROMOTIVE FORCE (millivolts) OF ELEMENTS RELATIVE TO PLATINUM*—continued

Table 16.3

Cold junction at 0◦ C Temperature of hot junction ◦ C

Cerium Cobalt Copper Germanium Gold Indium Iridium Iron Lead Lithium Magnesium Mercury Molybdenum Nickel Palladium Potassium Rhodium Rubidium Silicon Silver Sodium Tantalum Thallium Thorium Tin Tungsten Zinc

−200

−100

+100

+200

— — − 0.19 −46.00 − 0.21 — − 0.25 − 3.10 + 0.24 − 1.12 + 0.31 — — + 2.28 + 0.81 + 1.61 − 0.20 + 1.09 +63.13 − 0.21 + 1.00 + 0.21 — — + 0.26 + 0.43 − 0.07

— — − 0.37 −26.62 − 0.39 — − 0.35 − 1.94 − 0.13 − 1.00 − 0.09 — — + 1.22 + 0.48 + 0.78 − 0.34 + 0.46 +37.17 − 0.39 + 0.29 − 0.10 — — − 0.12 − 0.15 − 0.33

+ 1.14 − 1.33 + 0.76 +33.9 + 0.74 + 0.69 + 0.65 + 1.98 + 0.44 + 1.82 + 0.44 + 0.06 + 1.45 − 1.48 − 0.57 — + 0.70 — −41.56 + 0.74 — + 0.33 + 0.58 − 0.13 + 0.42 + 1.12 + 0.76

+ 2.46 − 3.08 + 1.83 +72.4 + 1.77 — + 1.49 + 3.69 + 1.09 — + 1.10 + 0.13 + 3.19 − 3.10 − 1.23 — + 1.61 — −80.58 + 1.77 — + 0.93 + 1.30 − 0.26 + 1.07 + 2.62 + 1.89

*The numerical values given in this table should be taken only as a guide since thermoelectric properties are very sensitive to impurities and state of anneal.

THERMAL ELECTROMOTIVE FORCE (millivolts) OF SOME BINARY ALLOYS RELATIVE TO PLATINUM WITH JUNCTIONS AT 0◦ and 100◦ C*

Table 16.4 Metal A: Metal B: %A

Lead Tin

Tin Copper

Zinc Copper

Gold Silver

Gold Palladium

Nickel Copper

Tin Bismuth

Antimony Cadmium

Antimony Bismuth

0 10 20 30 40 50 60 70 80 90 100

+0.44 +0.44 +0.44 +0.44 +0.45 +0.45 +0.44 +0.44 +0.43 +0.42 +0.42

+0.76 +0.53 +0.56 +0.65 +0.65 +0.69 +0.72 +0.62 +0.54 +0.48 +0.42

+0.76 +0.54 +0.53 +0.54 +0.51 +0.54 +0.47 +0.87 +0.66 +0.98 +0.76

+0.74 +0.55 +0.48 +0.47 +0.47 +0.48 +0.49 +0.49 +0.50 +0.59 +0.78

−0.57 −0.85 −1.25 −1.42 −1.69 −2.44 −2.97 −2.63 −0.46 −0.05 +0.78

+0.76 −2.63 −3.08 −3.54 −4.03 −3.64 −3.06 −2.54 −2.49 −1.93 −1.48

−7.34 +4.00 +3.52 +2.56 +2.10 +1.77 +1.14 +0.95 +0.78 +0.60 +0.42

+0.90 +1.52 +2.88 +6.4 +12.2 +23.1 +44.4 +21.5 +12.8 +8.1 +4.89

−7.34 −8.82 −7.31 −5.66 −4.05 −2.51 −1.06 +0.32 +1.79 +3.31 +4.89

*The numerical values given in these tables should be taken only as a guide since thermoelectric properties are very sensitive to impurities and state of anneal.

Table 16.5

ABSOLUTE THERMOELECTRIC POWER OF PLATINUM

Temperature (K) Thermoelectric power (µVK−1 )

300 −5.05

400 −7.66

500 600 700 800 900 −9.69 −11.33 −12.87 −14.38 −15.97

1 000 −17.58

1 100 −19.03

1 200 −20.56

16–6

Temperature measurement and thermoelectric properties

Table 16.6

PLATINUM–10% RHODIUM/PLATINUM THERMOCOUPLE TABLES—TYPE S

Reference junction at 0◦ C t68 /◦ C

0

10

20

30

0 0 100 200 300 400 500 600 700 800 900 1 000 1 100 1 200 1 300 1 400 1 500 1 600 1 700

0 0 645 1 440 2 323 3 260 4 234 5 237 6 274 7 345 8 448 9 585 10 754 11 947 13 155 14 368 15 576 16 771 17 942

−53 55 719 1 525 2 414 3 356 4 333 5 339 6 380 7 454 8 560 9 700 10 872 12 067 13 276 14 489 15 697 16 890 18 056

−103 113 795 1 611 2 506 3 452 4 432 5 442 6 486 7 563 8 673 9 816 10 991 12 188 13 397 14 610 15 817 17 008 18 170

−150 173 872 1 698 2 599 3 549 4 532 5 544 6 592 7 672 8 786 9 932 11 110 12 308 13 519 14 731 15 937 17 125 18 282

Table 16.7

60

70

80

90

t68 /◦ C

365 1 109 1 962 2 880 3 840 4 832 5 855 6 913 8 003 9 126 10 282 11 467 12 671 13 883 15 094 16 296 17 477 18 612

432 1 190 2 051 2 974 3 938 4 933 5 960 7 020 8 114 9 240 10 400 11 587 12 792 14 004 15 215 16 415 17 594

502 1 273 2 141 3 069 4 036 5 034 6 064 7 128 8 225 9 355 10 517 11 707 12 913 14 125 15 336 16 534 17 711

573 1 356 2 232 3 164 4 135 5 136 6 169 7 236 8 336 9 470 10 635 11 827 13 034 14 247 15 456 16 653 17 826

0 0 100 200 300 400 500 600 700 800 900 1 000 1 100 1 200 1 300 1 400 1 500 1 600

40 50 e.m.f ./µV −194 235 950 1 785 2 692 3 645 4 632 5 648 6 699 7 782 8 899 10 048 11 229 12 429 13 640 14 852 16 057 17 243 18 394

−236 299 1 029 1 873 2 786 3 743 4 732 5 751 6 805 7 892 9 012 10 165 11 348 12 550 13 761 14 973 16 176 17 360 18 504

PLATINUM–13% RHODIUM/PLATINUM THERMOCOUPLE TABLES—TYPE R

Reference junction at 0◦ C t68 /◦ C

0

10

20

30

0 0 100 200 300 400 500 600 700 800 900 1 000 1 100 1 200 1 300 1 400 1 500 1 600 1 700

0 0 647 1 468 2 400 3 407 4 471 5 582 6 741 7 949 9 203 10 503 11 846 13 224 14 624 16 035 17 445 18 842 20 215

−51 54 723 1 557 2 498 3 511 4 580 5 696 6 860 8 072 9 331 10 636 11 983 13 363 14 765 16 176 17 585 18 981 20 350

−100 111 800 1 647 2 596 3 616 4 689 5 810 6 979 8 196 9 460 10 768 12 119 13 502 14 906 16 317 17 726 19 119 20 483

−145 171 879 1 738 2 695 3 721 4 799 5 925 7 098 8 320 9 589 10 902 12 257 13 642 15 047 16 458 17 866 19 257 20 616

Table 16.8

60

70

80

90

t68 /◦ C

363 1 124 2 017 2 997 4 039 5 132 6 272 7 460 8 696 9 978 11 304 12 669 14 062 15 470 16 882 18 286 19 670 21 006

431 1 208 2 111 3 099 4 146 5 244 6 388 7 582 8 822 10 109 11 439 12 808 14 202 15 611 17 022 18 425 19 807

501 1 294 2 207 3 201 4 254 5 356 6 505 7 703 8 949 10 240 11 574 12 946 14 343 15 752 17 163 18 564 19 944

573 1 380 2 303 3 304 4 362 5 469 6 623 7 826 9 076 10 371 11 710 13 085 14 483 15 893 17 304 18 703 20 080

0 100 200 300 400 500 600 700 800 900 1 000 1 100 1 200 1 300 1 400 1 500 1 600

40 50 e.m.f ./µV −188 232 959 1 830 2 795 3 826 4 910 6 040 7 218 8 445 9 718 11 035 12 394 13 782 15 188 16 599 18 006 19 395 20 748

−226 296 1 041 1 923 2 896 3 933 5 021 6 155 7 339 8 570 9 848 11 170 12 532 13 922 15 329 16 741 18 146 19 533 20 878

PLATINUM–30% RHODIUM/PLATINUM–6% RHODIUM THERMOCOUPLE TABLES—TYPE B

Reference junction at 0◦ C t68 /◦ C

0

10

20

30

40

0 100 200 300 400

0 33 178 431 786

−2 43 199 462 827

−3 53 220 494 870

−2 65 243 527 913

−0 78 266 561 957

50 e.m.f ./µV 2 92 291 596 1 002

60

70

80

90

t68 /◦ C

6 107 317 632 1 048

11 123 344 669 1 095

17 140 372 707 1 143

25 159 401 746 1 192

0 100 200 300 400

(continued)

Thermocouple reference tables

16–7

PLATINUM–30% RHODIUM/PLATINUM–6% RHODIUM THERMOCOUPLE TABLES—TYPE B—continued

Table 16.8

Reference junction at 0◦ C t68 /◦ C

0

10

20

30

500 600 700 800 900 1 000 1 100 1 200 1 300 1 400 1 500 1 600 1 700 1 800

1 241 1 791 2 430 3 154 3 957 4 833 5 777 6 783 7 845 8 952 10 094 11 257 12 426 13 585

1 292 1 851 2 499 3 231 4 041 4 924 5 875 6 887 7 953 9 065 10 210 11 374 12 543 13 699

1 344 1 912 2 569 3 308 4 126 5 016 5 973 6 991 8 063 9 178 10 325 11 491 12 659 13 814

1 397 1 974 2 639 3 387 4 212 5 109 6 073 7 096 8 172 9 291 10 441 11 608 12 776

60

70

80

90

t68 /◦ C

1 560 2 164 2 855 3 626 4 474 5 391 6 374 7 414 8 504 9 634 10 790 11 959 13 124

1 617 2 230 2 928 3 708 4 562 5 487 6 475 7 521 8 616 9 748 10 907 12 076 13 239

1 674 2 296 3 003 3 790 4 652 5 583 6 577 7 628 8 727 9 863 11 024 12 193 13 354

1 732 2 363 3 078 3 873 4 742 4 680 6 680 7 736 8 839 9 979 11 141 12 310 13 470

500 600 700 800 900 1 000 1 100 1 200 1 300 1 400 1 500 1 600 1 700

40 50 e.m.f ./µV 1 450 2 036 2 710 3 466 4 298 5 202 6 172 7 202 8 283 9 405 10 558 11 725 12 892

1 505 2 100 2 782 3 546 4 386 5 297 6 273 7 308 8 393 9 519 10 674 11 842 13 008

Table 16.9 NICKEL–CHROMIUM/COPPER–NICKEL THERMOCOUPLE—TYPE E (Chrome–Constantan) Reference junction at 0◦ C t68 /◦ C

0

10

20

30

40 50 e.m.f ./µV

60

70

80

90 t68 /◦ C

−200 −8 824 −9 063 −9 274 −9.455 −9 604 −9 719 −9 797 −9 835 −100 −5 237 −5 680 −6 107 −6 516 −6 907 −7 279 −7 631 −7 963 −8 273 −8 561 0 0 −581 −1 151 −1 709 −2 254 −2 787 −3 306 −3 811 −4 301 −4 777 0 0 591 1 192 1 801 2 419 3 047 3 683 4 329 4 983 5 646 100 6 317 6 996 7 683 8 377 9 078 9 787 10 501 11 222 11 949 12 681 200 13 419 14 161 14 909 15 661 16 417 17 178 17 942 18 710 19 481 20 256 300 21 033 21 814 22 597 23 383 24 171 24 961 25 754 26 549 27 345 28 143 400 28 943 29 744 30 546 31 350 32 155 32 960 33 767 34 574 35 382 36 190 500 36 999 37 808 38 617 39 426 40 236 41 045 41 853 42 662 43 470 44 278 600 45 085 45 891 46 697 47 502 48 306 49 109 49 911 50 713 51 513 52 312 700 53 110 53 907 54 703 55 498 56 291 57 083 57 873 58 663 59 451 60 237 800 61 022 61 806 62 588 63 368 64 147 64 924 65 700 66 473 67 245 68 015 900 68 783 69 549 70 313 71 075 71 835 72 593 73 350 74 104 74 857 75 608 1 000 76 358

−200 −100 0 0 100 200 300 400 500 600 700 800 900

Table 16.10 IRON COPPER–NICKEL THERMOCOUPLE TABLES—TYPE J (Iron–Constantan) Reference junction at 0◦ C t68 /◦ C

0

10

20

30

40 50 e.m.f ./µV

60

70

80

90 t68 /◦ C

−200 −7 890 −8 096 −100 −4 632 −5 036 −5 426 −5 801 −6 159 −6 499 −6 821 −7 122 −7 402 −7 659 0 0 −501 −995 −1 481 −1 960 −2 431 −2 892 −3 344 −3 785 −4 215 0 0 507 1 019 1 536 2 058 2 585 3 115 3 649 4 186 4 725 100 5 268 5 812 6 359 6 907 7 457 8 008 8 560 9 113 9 667 10 222 200 10 777 11 332 11 887 12 442 12 998 13 353 14 108 14 663 15 217 15 771 300 16 325 16 879 17 432 17 984 18 537 19 089 19 640 20 192 20 743 21 295 400 21 846 22 397 22 949 23 501 24 054 24 607 25 161 25 716 26 272 26 829

−200 −100 0 0 100 200 300 400

(continued)

16–8

Temperature measurement and thermoelectric properties

Table 16.10

IRON COPPER–NICKEL THERMOCOUPLE TABLES—TYPE J—continued

Reference junction at 0◦ C t68 /◦ C

0

10

20

30

500 600 700 800 900 1 000 1 100 1 200

27 388 33 096 39 130 45 498 51 875 57 942 63 777 69 536

27 949 33 683 39 754 46 144 52 496 58 533 64 355

28 511 34 273 40 382 46 790 53 115 59 121 64 933

29 075 34 867 41 013 47 434 53 729 59 708 65 510

60

70

80

90

t68 /◦ C

30 782 36 671 42 922 49 354 55 553 61 459 67 240

31 356 37 280 43 563 49 989 56 155 62 039 67 815

31 933 37 893 44 207 50 621 56 753 62 619 68 390

32 513 38 510 44 852 51 249 57 349 63 199 68 964

500 600 700 800 900 1 000 1 100

70

80

40 50 e.m.f ./µV 29 642 35 464 41 647 48 076 54 341 60 293 66 087

30 210 36 066 42 283 48 716 54 948 60 876 66 664

Table 16.11 COPPER/COPPER–NICKEL THERMOCOUPLE TABLES—TYPE T (Copper–Constantan) Reference junction at 0◦ C t68 /◦ C

0

10

20

30

40 50 e.m.f ./µV

60

90 t68 /◦ C

−200 −5 603 −5 753 −5 889 −6 007 −6 105 −6 181 −6 232 −6 258 −100 −3 378 −3 656 −3 923 −4 177 −4 419 −4 648 −4 865 −5 069 −5 261 −5 439 0 0 −383 −757 −1 121 −1 475 −1 819 −2 152 −2 475 −2 788 −3 089 0 0 391 789 1 196 1 611 2 035 2 467 2 908 3 357 3 813 100 4 277 4 749 5 227 5 712 6 204 6 702 7 207 7 718 8 235 8 757 200 9 286 9 820 10 360 10 905 11 456 12 011 12 572 13 137 13 707 14 281 300 14 860 15 443 16 030 16 621 17 217 17 816 18 420 19 027 19 638 20 252 400 20 869

−200 −100 0 0 100 200 300

Table 16.12 NICKEL–CHROMIUM/NICKEL–ALUMINIUM THERMOCOUPLE TABLES—TYPE K (Chromel–Alumel) Reference junction at 0◦ C t68 /◦ C

0

10

20

30

40 50 e.m.f ./µV

60

70

80

90 t68 /◦ C

−200 −5 891 −6 035 −6 158 −6 262 −6 344 −6 404 −6 441 −6 458 −100 −3 553 −3 852 −4 138 −4 410 −4 669 −4 912 −5 141 −5 354 −5 550 −5 730 0 0 −392 −777 −1 156 −1 527 −1 889 −2 243 2 586 −2 920 −3 242 0 0 397 798 1 203 1 611 2 022 2 436 2 850 3 266 3 681 100 4 095 4 508 4 919 5 327 5 733 6 137 6 539 6 939 7 338 7 737 200 8 137 8 537 8 938 9 341 9 745 10 151 10 560 10 969 11 381 11 793 300 12 207 12 623 13 039 13 456 13 874 14 292 14 712 15 132 15 552 15 974 400 16 395 16 818 17 241 17 664 18 088 18 513 18 938 19 363 19 788 20 214 500 20 640 21 066 21 493 21 919 22 346 22 772 23 198 23 624 24 050 24 476 600 24 902 25 327 25 751 26 176 26 599 27 022 27 445 27 867 28 288 28 709 700 29 128 29 547 29 965 30 383 30 799 31 214 31 629 32 042 32 455 32 866 800 32 277 33 686 34 095 34 502 34 909 35 314 35 718 36 121 36 524 36 925 900 37 325 37 724 38 122 38 519 38 915 39 310 39 703 40 096 40 488 40 879 1 000 41 269 41 657 42 045 42 432 42 817 43 202 43 585 43 968 44 349 44 729 1 100 45 108 45 486 45 863 46 238 46 612 46 985 47 356 47 726 48 095 48 462 1 200 48 828 49 192 49 555 49 916 50 276 50 633 50 990 51 344 51 697 52 049 1 300 52 398 52 747 53 093 53 439 53 782 54 125 54 466 54 807

−200 −100 0 0 100 200 300 400 500 600 700 800 900 1 000 1 100 1 200

Thermocouple reference tables

16–9

NICKEL–CHROMIUM–SILICON/NICKEL–SILICON (NICROSYL/NISIL) THERMOCOUPLE TABLES—TYPE N

Table 16.13

Reference junction at 0◦ C t68 /◦ C

0

10

20

30

40 50 e.m.f ./µV

60

70

80

90 t68 /◦ C

−200 −3 990 −4 083 −4 162 −4 227 −4 277 −4 313 −4 336 −4 345 −100 −2 407 −2 612 −2 808 −2 994 −3 170 −3 336 −3 491 −3 634 −3 766 −3 884 0 0 −260 −518 −772 −1 023 −1 268 −1 509 −1 744 −1 972 −2 193 0 0 261 525 793 1 064 1 340 1 619 1 902 2 188 2 479 100 2 774 3 072 3 374 3 679 3 988 4 301 4 617 4 936 5 258 5 584 200 5 912 6 243 6 577 6 914 7 254 7 596 7 940 8 287 8 636 8 987 300 9 340 9 695 10 053 10 412 10 773 11 135 11 499 11 865 12 233 12 602 400 12 972 13 344 13 717 14 092 14 467 14 844 15 222 15 601 15 981 16 362 500 16 744 17 127 17 511 17 896 18 282 18 668 19 055 19 443 19 831 20 220 600 20 609 20 999 21 390 21 781 22 172 22 564 22 956 23 348 23 740 24 133 700 24 526 24 919 25 312 25 705 26 098 26 491 26 885 27 278 27 671 28 063 800 28 456 28 849 29 241 29 633 30 025 30 417 30 808 31 199 31 590 31 980 900 32 370 32 760 33 149 33 538 33 927 34 315 34 702 35 089 35 476 35 862 1 000 36 248 36 633 37 018 37 403 37 786 38 169 38 552 38 934 39 316 39 696 1 100 40 076 40 456 40 835 41 213 41 590 41 966 42 342 42 717 43 091 43 464 1 200 43 836 44 207 44 578 44 947 45 315 45 682 46 048 46 413 46 777 47 140 1 300 47 502 Table 16.14

−200 −100 0 0 100 200 300 400 500 600 700 800 900 1 000 1 100 1 200

TUNGSTEN–5% RHENIUM/TUNGSTEN–26% RHENIUM—TYPE C4

Reference junction at 0◦ C t68 /◦ C

0

10

20

30

0 100 200 300 400 500 600 700 800 900 1 000 1 100 1 200 1 300 1 400 1 500 1 600 1 700 1 800 1 900 2 000 2 100 2 200 2 300

0 1 451 3 089 4 863 6 731 8 655 10 606 12 558 14 494 16 397 18 257 20 066 21 819 23 514 25 148 26 722 28 236 29 688 31 078 32 404 33 660 34 839 35 932 36 922

135 1 607 3 261 5 047 6 921 8 849 10 801 12 753 14 686 16 585 18 440 20 244 21 991 23 680 25 308 26 876 28 384 29 830 31 214 32 533 33 782 34 953 36 036 37 015

272 1 765 3 434 5 231 7 112 9 044 10 997 12 947 14 877 16 773 18 623 20 421 22 163 23 846 25 468 27 030 28 531 29 971 31 349 32 661 33 902 35 065 36 138 37 107

412 1 925 3 609 5 416 7 304 9 239 11 192 13 142 15 069 16 960 18 805 20 598 22 334 24 010 25 627 27 183 28 678 30 112 31 483 32 788 34 022 35 177 36 240 37 197

40 50 e.m.f ./µV 554 2 087 3 785 5 601 7 496 9 434 11 388 13 336 15 260 17 147 18 987 20 774 22 504 24 175 25 785 27 335 28 824 30 252 31 617 32 915 34 142 35 288 36 341 37 286

698 2 250 3 962 5 788 7 688 9 629 11 583 13 529 15 450 17 333 19 168 20 950 22 674 24 339 25 943 27 486 28 969 30 391 31 749 33 041 34 260 35 397 36 441 37 374

60

70

80

90

t68 /◦ C

845 2 415 4 140 5 975 7 881 9 824 11 778 13 723 15 640 17 519 19 349 21 125 22 843 24 502 26 100 27 637 29 114 30 530 31 882 33 166 34 378 35 506 36 539 37 460

993 2 581 4 319 6 163 8 074 10 019 11 974 13 916 15 830 17 704 19 529 21 299 23 012 24 664 26 256 27 788 29 259 30 668 32 013 33 291 34 494 35 614 36 637 37 545

1 144 2 749 4 500 6 352 8 267 10 215 12 169 14 109 16 020 17 889 19 709 21 473 23 180 24 826 26 412 27 938 29 402 30 805 32 144 33 415 34 610 35 721 36 733 37 629

1 296 2 918 4 681 6 541 8 461 10 410 12 364 14 302 16 208 18 073 19 888 21 647 23 347 24 988 26 568 28 087 29 546 30 942 32 274 33 538 34 725 35 827 36 828 37 711

0 100 200 300 400 500 600 700 800 900 1 000 1 100 1 200 1 300 1 400 1 500 1 600 1 700 1 800 1 900 2 000 2 100 2 200 2 300

REFERENCES 1. The International Temperature Scale of 1990, Metrologia, 1990, 27, 3–10 and 167. 2. ‘Supplementary Information for the ITS-90 and Techniques for Approximating the ITS-90’, BIPM, Pavillion de Breteuil, F-92312 Sèvres Celex, France, 1990. 3. T. J. Quinn, ‘Temperature’, 2nd edn, Academic Press, London, 1990. 4. Values calculated using coefficients from: P. C. Fazio et al., Annual Book of ASTM Standards, Volume 14.03, Temperature Measurement, ASTM, Philadelphia, PA, 1993.

16–10

Temperature measurement and thermoelectric properties

16.3 Thermoelectric materials 16.3.1 Introduction Thermoelectric devices convert thermal energy from a temperature gradient into electrical energy (Seebeck effect), or electrical energy into a temperature gradient (Peltier effect).1,2 Thermoelectric materials are evaluated on the basis of their thermoelectric figure-of-merit, Z, which is defined by: Z=

σS 2 κ

In this equation, σ is the electrical conductivity (−1 m−1 ), S is the Seebeck coefficient (VK−1 ), and κ is the thermal conductivity (Wm−1 K−1 ). Z has units of reciprocal temperature, and the product ZT is referred to as the dimensionless figure-of-merit. Good thermoelectric materials have large values of Z (or ZT ). From a materials perspective, one seeks to maximise σ and S while minimising κ. The Seebeck coefficient and the electrical conductivity are strong functions of the Fermi energy, which, in turn, depends on the free carrier concentration, the carrier effective mass and temperature. The Seebeck coefficient of most metals is typically too small for them to have any practical application beyond a simple thermocouple. On the other hand, many semiconductors have Seebeck coefficients on the order of hundreds of µV/K. Good thermoelectric materials must also have a reasonable electrical conductivity since charge transport is the basis by which heat is transferred, and resistive heating will lower the overall efficiency. In addition, the material must be a good thermal insulator to maintain the thermal gradient established by the motion of charge carriers. The relationships among the transport properties that influence the figure-of-merit are shown in Figure 16.2. The dependence of Z on n is the basis of the general rule of thumb suggested by Ioffe that the best thermoelectric materials will be relatively highly doped semiconductors (n ∼ 1020 cm−3 ).3 The effect of doping is limited in practice by the fact that although the lattice part of the thermal conductivity is not very sensitive to the free carrier concentration, n, the electronic component clearly is. It is also well known that there is a correlation between the band gap and Z for many materials. The band gap rule states that EG ∼ 10 kBTM , where TM is the temperature where ZT is a maximum and kB is Boltzmann’s constant.4 The choice of majority carrier is important in determining the sign of the Peltier and Seebeck coefficients. In a recent review, Mahan offered several criteria for good thermoelectric materials.5 In terms of their band structures, good thermoelectric materials will have multiple degenerate bands, high carrier mobilities, and large carrier effective masses. In addition, good thermoelectric materials must have a low thermal conductivity. There are three strategies to minimise the thermal conductivity: the use of heavy atoms, complex crystal structures and alloys. In general, heavy elements have lower thermal conductivities than lighter elements, heat is transferred less efficiently in crystals with a higher density of optical phonons, and alloying reduces the lattice thermal conductivity. Alloying also lowers the charge mobility, but the effect is greater for the lattice thermal conductivity. Materials used today in thermoelectric devices can be grouped into three categories based on the device operation temperature: the Group V chalcogenides based on Bi2Te3 -Sb2Te3 alloys for refrigeration (ca. 300 K); the Group IV chalgogenides based on PbTe for power generation (700 K); and Si-Ge solid solutions for power generation at high temperature (1 100–1 200 K). Examples of the best thermoelectric materials in these temperature ranges are given in Table 16.15. Generally, thermoelectric devices suffer from poor efficiency. For example, current thermoelectric refrigerators are about one third as efficient as their conventional (Freon based) analogues. As a result, thermoelectric devices tend to be important in niche applications where the stability and long life afforded by an all solid-state device outweighs the efficiency issue. 16.3.2 Survey of materials Bi2Te3 is composed of hexagonal, closest-packed Te layers with Bi atoms occupying two-thirds of the octahedral holes. The crystal consists of a sequence of atomic layers Te-Bi-Te-Bi-Te. Three such stacks sandwiched together along the c-axis of the crystal form the hexagonal unit cell, which repeats every nine Te layers. The band gap of Bi2Te3 is 0.15 eV, and increases upon doping. Both p- and n-type materials can be prepared by alloying with compounds such as Sb2Te3 or Bi2 Se3 , which have the same crystal structure as Bi2Te3 . Bi2Te3 is the most important thermoelectric material for cooling applications at temperatures near 300 K. All devices that operate near room temperature are based on Bi2Te3 and its alloys. Alloys are used in order to lower the thermal conductivity and thereby increase ZT.

Seebeck coefficient, S, V K⫺1 Electrical conductivity, s, Ω⫺1 m⫺1

Thermoelectric materials

16–11

s

Thermal conductivity κ, W m⫺1 K⫺1

S

κel

κlattice

Figure-of-merit, Z, K⫺1

Z ⫽ sS2/κ

Semiconductors

Insulators

Metals

Free carrier concentration, n Figure 16.2 Schematic depiction of the dependence of the Seebeck coefficient, electrical conductivity, thermal conductivity, and thermoelectric figure-of-merit on the free carrier concentration THERMOELECTRIC MATERIALS FOR REFRIGERATION AND POWER GENERATION

Table 16.15 Material

Type

T (K)

ZT

Bi24 Sb68Te142 Se6 Bi0.5 Sb1.5Te3.13 Bi1.75 Sb0.25Te3.13 PbTe SiGe SiGe

p p n n p n

300 300 300 700 1 100 1 200

0.96 0.90 0.96 1.10 0.505 0.938

PbTe has the NaCl crystal structure and a band gap of 0.31 eV at room temperature. PbTe and its alloys are good thermoelectrics near 700 K and are used in power generation applications. PbTe frequently is alloyed with Sn to form Pb1−x SnxTe, which is a good n-type thermoelectric with a ZT

16–12

Temperature measurement and thermoelectric properties

of about 1.1 at 800 K. The p-type material is not thermally stable and has poor mechanical properties and therefore is not used in any devices. TAGS (tellurium-antimony-germanium-silver) alloys make good p-type thermoelectric materials in a temperature range similar to that of PbTe (which is n-type), and they are often used together. In fact, TAGS were developed as a direct result of the search for a replacement for p-PbTe. TAGS is an alloy of GeTe with AgSbTe2 (10–20%). The 20% alloy has a ZT > 2, which is among the highest ever reported. The measured value of Z was 3 × 10−3 K−1 at T = 700 K.6 The transport properties of Si make it an excellent choice for high temperature power generation, although its thermal conductivity is too high. Typically, Si is alloyed with Ge to lower the thermal conductivity. Overall, Si-Ge solid solutions are the best materials for high temperature applications. The composition that gives a maximum ZT is approximately Si0.7 Ge0.3 . These materials are used in the thermonuclear thermoelectric generators that supply power for deep-space vehicles such as the Voyager spacecraft. Bi-Sb is the only material other than Bi2Te3 that displays good thermoelectric performance near room temperature. The ZT is fairly low, however, only about 0.6 near 300 K. Interestingly, ZT can be increased by almost a factor of 2 by placing the material in a magnetic field. This is known as the thermomagnetic effect. The skutterudites have the formula MX3 where M = Co, Rh, Ir and X = P, As, Sb. The parent compound, CoSb3 , was first discovered in the town of Skutterude in Norway. One of the most interesting properties of the skutterudites is the extremely high hole mobility in the p-type materials (ca. 2 000 cm2 /Vs). Unfortunately, the thermal conductivities of these compounds are too high for thermoelectric applications. Due to their open crystal structures, however, these compounds may be doped with ‘rattler’ atoms that function to lower the overall thermal conductivity of the material without significantly altering the carrier mobility. The family of compounds called the ‘filled’ skutterudites has the general formula RM4 X12 . These compounds are cubic with a 34-atom unit cell. The rare earth atom (R) acts as the ‘rattler’ atom. These materials are promising candidates for high temperature applications. For example, Sales et al. report a value of ZT of about 1 for LaFe3 CoSb12 at T ∼ 800 K.7 For comparison, the ordinary skutterudites have ZT’s of about 0.3. The 3-fold increase in ZT in the filled skutterudites is attributed to the lowering of the thermal conductivity due to the lanthanum ‘rattler’ atom. Quantum wells are interesting composite materials that are under development for potential thermoelectric applications. It has long been known that modulation doping can increase carrier mobilities, and it is hoped that the Seebeck coefficient can be enhanced in quantum well structures in an analogous fashion. In multiple quantum well structures, the thermal conductivity is lowered due to the presence of multiple interfaces that increase phonon scattering. This is an example of an ‘electron transmitting/phonon blocking’ structure. Recently Venkatasubramanian et al. have reported a ZT of 2.4 for Bi2Te3 /Sb2Te3 multiple quantum wells prepared by molecular beam epitaxy.8 The thickness of the Bi2Te3 layers in these materials was only 1.0 nm. 16.3.3 Preparation methods Most thermoelectric materials are either compounds or solid solutions, and are easily prepared using standard solid-state synthesis procedures.9 A variety of melt techniques, including both stoichiometric and non-stoichiometric melts, are commonly used, typical examples being the Bridgman method and liquid phase eptitaxy, respectively. Powder metallurgy, in which a compact powder is sintered at elevated temperature, is also often used. Other synthetic approaches, including mechanical alloying and PIES (pulverised and intermixed sintering method) have also been developed. For the fabrication of thin film structures, molecular beam epitaxy, liquid phase epitaxy as well as electrochemical deposition all can be used. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9.

T. J. Seebeck, Abh. K. Akad. Wiss. Berlin, 1821, 289. J. C. A. Peltier, Ann. Chem. Phys., 1834, 56, 371. A. F. Ioffe, Semiconductor Thermoelements and Thermoelectric Cooling, Infosearch, Ltd, London, 1957. G. D. Mahan, J. Appl. Phys., 1989, 65, 1578. G. D. Mahan, Solid State Physics, 1998, 51, 81. S. K. Plachkova, Phys. Stat. Solid. A, 1984, 83, 349. B. C. Sales, D. Mandrus and R. K. Williams, Science, 1996, 272, 1325. R. Venkatasubramanian, E. Siivola, T. Colpitts and B. O’Quinn, Nature, 2001, 413, 597. D. M. Rowe, Ed., CRC Handbook of Thermoelectrics, CRC Press, New York, 1995.

17

Radiative properties of metals

The ability of a surface to radiate energy is governed by the material of which the surface is composed and its physical condition. Any attempt, therefore, to place a numerical value on its radiating ability must be related to a definition of the surface condition. It is usual to choose smooth polished surfaces for this purpose and thus arrive at values which are comparable from one metal to another. A perfect radiator (blackbody) provides a standard of comparison for defining the radiating ability of any other body or surface by determining the ratio of the emission of the surface to that of a blackbody when they are at the same temperature. An examination of the ratios thus obtained shows that the radiating ability of a metal surface varies with wavelength, temperature and angle of emission. The definition of the emittance, as this ratio is called, must therefore take into account these variations. DEFINITIONS OF EMITTANCE

Spectral, directional emittance, ελ , of a surface is the ratio of the energy emitted over an infinitesimally small wavelength range at wavelength λ into a specified direction, per unit area of the surface, to the energy emitted by a unit area of a black surface at the same temperature. The emittance in a direction normal to the surface, called the normal, spectral emittance, εnλ , is most commonly employed. The spectral, hemispherical emittance, ελ , is a directional average of the spectral, directional emittance, and gives the ratio of emitted radiative flux of the given surface to that of a blackbody at the same temperature. While normal emittances are the ones that are generally measured and tabulated, it is the hemispherical emittance that governs heat transfer rates between surfaces. For metals, the hemispherical emittance is always a little greater than its normal value, usually by about 10 to 20 percent. Total emittance is a weighted spectral average of the spectral emittance, and is the ratio of radiative emission of a surface over the entire spectrum, compared that of a blackbody at the same temperature. Again, we distinguish between directional and hemispherical values; and again, the more important hemispherical values exceed the total, normal emittance by roughly 10 to 20 percent for metals. The absorptance (α) and reflectance (ρ), for an opaque polished surface, are defined as the ratio of the rate of absorption or reflection of energy to the rate of incidence of radiative energy. Since the incident energy must be either reflected or absorbed, the sum of reflectance and absorptance must be unity. For spectral, directional values the absorptance is equal to the emittance and, with reasonable accuracy, this is often also true for hemispherical and/or total values.1 Thus, ε∼ =α=1−ρ Hence the emittance may be derived from the reflectance, which is sometimes more convenient than a direct determination. The following equation relates the normal reflectance of a well-polished surface to the complex index of refraction, given by m = n − ik, where n is the real part (‘refractive index’) and k is the complex part (‘absorptive index’ or ‘extinction coefficient’): ρnλ =

n2 + k 2 + 1 − 2n n2 + k 2 + 1 + 2n

17–1

17–2

Radiative properties of metals

where the absorptive index is related to the fraction of light transmitted perpendicularly through a layer of thickness d by exp (−4πdk/λ), λ being the wavelength in air. Normal emittance follows as εnλ = 1 − ρnλ =

4n n2 + k 2 + 1 + 2n

For a homogeneous material the complex index of refraction can be related to material properties through Maxwell’s electromagnetic wave theory,1    2  2 ε 1 ε σλ 2 n = + + 2 ε0 2πc0 ε0 ε0    2  2 σλ ε 1 ε + −  k2 =  2 ε0 2πc0 ε0 ε0

where σ is the electrical conductivity, ε and ε0 are the electrical permittivity of the material and of vacuum, respectively, c0 is the speed of light and λ the wavelength, both in vacuum or air. For metals with their large electrical conductivities, and for reasonably large wavelengths, σλ ≫ 2πc0 ε, leading to  σλ n≈k≈ ≫1 2πc0 ε0 and 0.0667 0.365 , εnλ ∼ − = √ σλ σλ

σλ in −1 ,

(17.1)

which is commonly known as the Hagen-Rubens relation.1 The equations show that the spectral emittance of metals should increase as the wavelength decreases and this is in general agreement with experiment. However, these relations are valid only for sufficiently large wavelengths and, indeed, there generally is a lack of agreement between experimental and theoretical values in the visible and ultraviolet (see Fig. 17.1 for tungsten). EFFECTS OF SURFACE TEMPERATURE

The Hagen-Rubens √ relation predicts that the spectral, normal emittance of a metal should be proportional to 1/ σ. Since the electrical conductivity is approximately inversely proportional to temperature, the spectral emittance should, therefore, be proportional to the square root of absolute temperature for long enough wavelengths. This trend should also hold for the spectral, hemispherical emittance. Experiments have shown that this is indeed true for many metals. A typical example is given in Fig. 17.1, showing the spectral dependence of the hemispherical emittance for tungsten for a number of temperatures.2,3 Note that the emittance for tungsten tends to increase with temperature only beyond a crossover wavelength of approximately 1.3 µm, while the temperature dependence is reversed for shorter wavelengths. Similar trends of a single crossover wavelength have been observed for many metals. DIRECTIONAL DEPENDENCE OF EMITTANCE

The spectral, directional reflectance of a perfectly smooth surface is governed by the so-called Fresnel’s relations.1 For metals in the infrared this implies relatively small and constant values of emittance for most angles of emission, accompanied by a sharp increase in emittance at near-grazing angles, followed by a sharp drop back to zero at 90◦ off-normal. As an example, experimental results for platinum at 2 µm are compared with Fresnel predictions in Fig. 17.2.1 For shorter wavelengths the directional behaviour follows the behaviour of nonmetals, i.e. relatively large and constant values of emittance for most angles of emission, accompanied by a gradual decrease beyond about 60◦ off-normal until zero emittance is reached at 90◦ .

Radiative properties of metals

17–3

0.5

Spectral emissivity

0.4

0.3

0.2

2100 K 1700 K 1300 K

0.1 300 K

0

0.5

1.0

1.5 2.0 Wavelength, ␮m

2.5

3.0

Relative emissivity

Figure 17.1 Spectral emittance of tungsten as a function of wavelength for different temperatures. Dotted lines calculated from equation 17.1. References 2 and 3

1.16 1.12

a

1.08 b

1.04 1.00

0 10 20 30 40 50 60 70 80 90 Angle of emission, degrees

Figure 17.2 Spectral, directional emittance of platinum at λ = 2 µm

TOTAL EMITTANCES

The total, normal or hemispherical emittances are calculated by integrating spectral values over all wavelengths, with the blackbody emissive power as a weight function. Since the peak of the blackbody emissive power shifts toward shorter wavelengths with increasing temperature, it follows that hotter surfaces emit a higher fraction of energy at shorter wavelengths, where the spectral emittance is greater, resulting in an increase in total emittance. The total, normal reflectance and emittance may be evaluated from the simple Hagen-Rubens relation. While this relation is not accurate across the entire spectrum, it does predict the emittance trends correctly in the infrared, and it does allow an explicit evaluation of total, normal emittance. Using Equation (17.1) in the weighted spectral averaging process and retaining the first two terms of the series expansion, leads to εn = 0.578(T /σ)1/2 − 0.178(T /σ),

T in K,

σ in −1 cm−1

(17.2)

The total, hemispherical emittance of a metal may be evaluated in a similar fashion, using Fresnel’s relations for directional dependence, leading to slightly larger values, or1 ε(T ) = 0.766(T /σ)1/2 − [0.309 − 0.089 9 ln(T /σ)](T /σ), T in K,

σ in −1 cm−1 (17.3)

Radiative properties of metals

Relative emissivity

17–4 2.0 1.8 1.6

a

1.4 1.2

b

1.0 0

10 20 30 40 50 60 70 80 90 Angle of emission, degrees

Figure 17.3 Total, hemispherical emittance of various polished metals as function of temperature. Reference 4

Because the Hagen-Rubens relation is only valid for large values of (σλ), this relation only holds for small values of (T /σ), i.e. the temperature of the surface must be such that only a small fraction of the blackbody emissive power comes from short wavelengths (where the Hagen-Rubens relation is not applicable). For pure metals, with its electrical conductivity inversely proportional to absolute temperature, the total emittance should be approximately linearly proportional to temperature. Comparison with experiment (Fig. 17.3)4 shows that this nearly linear relationship holds for many metals up to surprisingly high temperatures; for example, for platinum (T /σ)1/2 = 0.5 corresponds to a temperature of 2700 K. TEMPERATURE MEASUREMENT AND EMITTANCE

Radiation pyrometers, both spectral and total, are usually calibrated in terms of blackbody radiation and, thus, measure what is known as radiance temperature. The radiance temperature, Tr , measured by a total radiation pyrometer is related to the true temperature, T , by the formula T = Tr /ε1/4 where ε is the total emittance of the surface. True temperature always exceeds radiance temperature. For an optical pyrometer which measures irradiance from a narrow spectral interval only, the radiance temperature Tr is related to the true temperature by the equation 1 λ 1 1 = − ln T Tr C2 ελ where ελ is the spectral emittance and C2 = 1.438 8 cm K is the constant in Wien’s approximation for the blackbody emissive power. For the same emittances the correction is considerably greater for total radiation than for spectral radiation. For metallic surfaces the difference in correction is even greater, since the spectral emittance in the visible region for a given temperature is always greater than the total emittance. EMITTANCE VALUES

The values of emittance given in Tables 17.1 to 17.5 pertain, as far as is known, to emission from plane polished or plane unoxidised liquid metal surfaces. Emittance tends to be considerably higher if the surface is oxidised and/or rough. In any practical application, therefore, the values in the tables must be used with discretion and, where precise measurement is of importance, a determination of the emittance should be made for the prevalent surface conditions. As a rough guide the emittances for various oxidised surfaces are given in Tables 17.4 and 17.5.

Radiative properties of metals

17–5

␧t ⫽ 0.05

2000

0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75

1000 900 800 700 600 500

Correction, °C

400 300 200

0.80

100 90 80 70 60 50 40

0.85 0.90

30

0.95

20

10 600 700

800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800

Observed temperature, °C 1000 900 800 700 600 500 400

5

␧ ␭⫽ 0.0

0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80

300 200

Correction, °C

Figure 17.4 Correction to radiation pyrometer readings for total emittance

100 90 80 70 60 50 40

0.85

30

0.90

20

10 9 8 7 6 5

0.95

4 3 2

1 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900

Observed temperature, °C

Figure 17.5 Correction to optical pyrometer readings for spectral emissivity λ = 0.65 µm, C2 = 1.438 cm K

17–6

Radiative properties of metals

Table 17.1

SPECTRAL NORMAL EMITTANCE OF METALS FOR WAVELENGTH OF 0.65 µm

Temperature ◦ C Metal

600

800

1 000

1 200

1 400

1 600

1 800

2 000

2 500

3 000

References

Beryllium Chromium Cobalt Copper Erbium Germanium Gold Hafnium Iridium Iron Manganese Molybdenum Nickel Niobium Osmium Palladium Platinum Rhenium Rhodium Ruthenium Silicon Silver Tantalum Thorium Titanium Tungsten Uranium Yttrium Zirconium Alloys Cast iron Nichrome (in hydrogen) Steel

— — — — — — 0.16–0.18 — — — — — 0.36 — — — — — — — — — 0.47 — — — — — —

— — — 0.11 — 0.55 0.16–0.19 — — 0.37 — 0.37–0.43 0.35 — — 0.40 0.29–0.31 — 0.25 — 0.63 0.055 0.46 — 0.48 — 0.19–0.36 — —

0.37 — 0.33–0.38 0.10 0.55 — 0.16–0.21 — 0.36 0.36 — 0.36–0.42 0.34 0.37 0.52 0.37 0.29–0.31 — 0.22 0.42 0.57 0.055m 0.45 0.38 0.48 0.46–0.48 0.19–0.36 — 0.48

0.37 — 0.34–0.37 0.10m 0.55 — 0.13m — 0.34 0.35 0.59 0.35–0.42 — 0.37 0.44 0.34 0.29–0.31 — 0.19 0.35 0.52 — 0.44 0.38 0.48 0.43–0.48 0.34m — 0.45

— 0.39 0.35–0.37 0.11m 0.55 — — 0.45 0.32 0.35 0.59m 0.34–0.41 — 0.37 0.40 0.30 0.29–0.31 0.42 0.18 0.32 0.46 — 0.42 0.38 0.47 0.42–0.47 0.34m 0.35 0.42

— — 0.37m 0.12m 0.38m — — — — 0.37m — 0.34–0.41 — 0.37 0.38 0.37m 0.29–0.31 0.42 0.16 0.31 0.48m — 0.41 — — 0.42–0.47 — — 0.39

— — — 0.14m — — — — — — — 0.33–0.40 — 0.37 0.38 — — 0.41 — 0.31 — — 0.40 — — 0.41–0.47 — — 0.36

— 0.39m — — — — — — 0.30 — — 0.32–0.39 — 0.37 0.38 — — 0.41 — 0.31 — — 0.39 — — 0.40–0.47 — — —

— — — — — — — — — — — 0.31–0.37 — 0.40m — — — 0.40 — — — — 0.38 — — 0.38–0.46 — — —

— — — — — — — — — — — — — — — — — — — — — — 0.36 — — 0.36–0.45 — — —

5 6 6,7 8 6 9 5 10 6, 11 5 6 5 5 12 13 14 5 15 14 13 9 8 5 12 16, 17 5 5 6 18

— — —

0.37 0.35 0.35–0.40

0.37 0.35 0.32–0.40

0.37 0.35 0.30–0.40

0.37 0.35 —

0.40m — 0.37m

— — —

— — —

— — —

— — —

— 5 5

m Value

for molten state.

Radiative properties of metals

Table 17.2

SPECTRAL EMITTANCE IN THE INFRA-RED OF METALS AT HIGH TEMPERATURES

Metal Cobalt

Copper

Iridium

Iron

Molybdenum

Nickel

Niobium

17–7

Wavelength µm

Temperature ◦C

1.0

1.2

1.4

1.5

1.6

1.8

2.0

2.5

3.0

3.5

4.0

4.5

References

800 1 000 1 200 762 901 985 827 1 227 1 727 2 127 800 1 000 1 200 1 245 1 327 1 727 2 527 800 1 000 1 200 1 110 827 1 227 1 727 2 127

— — — — — 0.049 0.229 0.233 0.243 0.247 — — — 0.340 0.335 0.300 0.260 — — — — 0.345 0.335 0.320 0.315

0.26 0.26 0.26 — — — 0.203 0.213 0.228 0.233 0.294 0.294 0.291 0.316 — — — 0.295 0.293 0.290 0.292 — — — —

— — — — — — 0.185 0.194 0.210 0.219 — — — 0.298 — — — 0.267 0.269 0.271 0.270 — — — —

— — — 0.031 0.079 0.037 — — — — — — — 0.290 0.185 0.195 0.210 — — — 0.250 0.23 0.25 0.26 0.27

— — — — — — 0.167 0.180 0.199 0.207 0.264 0.267 0.300 0.282 — — — 0.250 0.252 0.253 — — — — —

— — — — — 0.034 0.152 0.169 0.188 0.199 — — — 0.268 — — — 0.230 0.232 0.235 — — — — —

0.21 0.21 0.22 0.029 0.065 — 0.140 0.160 0.180 0.192 0.237 0.245 0.252 0.260 0.140 0.170 0.193 0.215 0.219 0.223 0.290 0.19 0.21 0.23 0.25

— — — — 0.052 0.032 — — — — 0.217 0.227 0.235 0.248 — — — — — — 0.205 — — — —

— 0.18 0.19 — 0.043 0.031 — — — — — — — 0.240 0.115 0.155 0.185 — — — 0.187 — — — —

— — — — 0.038 — — — — — — — — 0.235 — — — — — — 0.174 — — — —

— — — 0.025 0.032 0.030 — — — — — — — 0.225 0.114 0.145 0.185 — — — 0.162 — — — —

— — — — — — — — — — — — — 0.218 — — — — — — — — — — —

5 5 5 19 20 21 22 22 22 22 23 23 23 23 5 5 5 23 23 23 23 5 5 5 5 (continued)

17–8

Radiative properties of metals

Table 17.2

Metal Platinum Rhenium

Tantalum

Titanium Tungsten

Zirconium

SPECTRAL EMITTANCE IN THE INFRA-RED OF METALS AT HIGH TEMPERATURES—continued

Wavelength µm

Temperature ◦C

1.0

1.2

1.4

1.5

1.6

1.8

2.0

2.5

3.0

3.5

4.0

4.5

References

1 127 1 537 2 118 2 772 1 427 1 927 2 527 750 1 327 2 127 2 527 1 127 1 327 1 727

— 0.36 0.36 0.36 0.295 0.310 0.330 0.490 0.385 0.37 0.36 0.46 0.444 0.442

0.257 — — — — — — — — — — — — —

— — — — — — — 0.510 — — — — — —

0.227 0.29 0.30 0.32 0.220 0.245 0.290 0.500 0.28 0.292 0.30 0.422 — 0.375

— — — — — — — — — — — — — —

— — — — — — — — — — — — — —

0.193 0.25 0.27 0.29 0.190 0.215 0.270 0.455 0.21 0.245 0.26 0.386 0.368 0.357

— 0.23 0.24 0.26 — — — — — — — 0.360 — 0.351

0.151 — — — 0.170 0.192 0.240 0.525 0.13 0.18 — 0.348 0.343 0.342

— — — — — — — 0.575 — — — — — 0.330

0.130 — — — 0.150 0.180 0.230 0.600 0.095 0.15 — — 0.325 —

— — — — — — — — — — — — — —

5 15 15 15 5 5 5 24 5 5 5 25 25 25

Radiative properties of metals Table 17.3

17–9

SPECTRAL NORMAL EMITTANCE OF METALS AT ROOM TEMPERATURE

Derived from reflectance data by formula ελ = 1 − ρλ Wavelength µm Metal

10.0

9.0

5.0

3.0

1.0

0.6

0.5

References

Aluminium Antimony∗ Bismuth Cadmium Chromium Cobalt Copper Gold Iridium Iron Lead Magnesium* Molybdenum Nickel Niobium Palladium Platinum Rhodium Silver Tantalum Tellurium Tin Titanium Tungsten Vanadium Zinc

0.02–0.04 — 0.08 — — — 0.021 0.015 — — — — 0.15 — 0.04 — 0.05 — 0.02 — — — 0.05–0.12 0.03 0.06–0.09 0.03

— 0.28 — 0.02 0.08 0.04 — 0.015 0.04 — 0.06 0.07 — 0.04 — 0.03 — 0.05 — 0.06 0.22 0.14 — — — —

0.03–0.08 0.31 0.12 0.04 0.19 0.15 0.024 0.015 0.06 — 0.08 0.14 0.16 0.06 0.06 0.10 0.06 0.07 0.02 0.07 0.43 0.24 0.10–0.18 0.05 0.07–0.11 0.05

0.03–0.12 0.35 0.26 0.07 0.30 0.23 0.026 0.015 0.09 — — 0.20 0.19 0.12 0.14 0.12 0.11 0.08 0.02 0.08 0.47 0.32 0.25–0.33 0.07 0.10–0.17 0.08

0.08–0.27 0.45 0.72 0.30 0.43 0.32 0.030 0.020 0.22 0.41 — 0.26 0.42 0.27 0.29 0.28 0.24 0.16 0.03 0.22 0.50 0.46 0.37–0.49 0.40 0.36–0.50 0.50–0.61

— 0.47 0.76 — 0.44 — 0.080 0.080 — 0.48 — 0.27 — — 0.55 0.37 0.36 0.21 0.03 0.55 0.51 — — 0.44–0.49 0.42–0.57 0.42–0.58

— — 0.75 — 0.45 — 0.36 0.45 — 0.49 — 0.28 — — — 0.42 0.40 0.24 0.03 0.62 — — — — 0.43–0.59 —

5 26 27 28 26 28 5 5 28 14 29 26 30 31 5 18,14,32 33 26 5 28 26 28 5 5 5 5

∗ Values

for spectral, directional emittance at 15◦ only.

SPECTRAL EMITTANCE OF OXIDISED METALS FOR WAVELENGTH OF 0.65 µm

Table 17.4

Oxide formed on smooth surfaces. For oxides in the form of refractory materials, values of emittance widely different from those below may be given and will be dependent on the grain size. Metal

ε0.65

Metal

ε0.65

Aluminium Beryllium Chromium Cobalt Copper Iron Magnesium Nickel Niobium Tantalum Thorium Titanium

0.30 0.61 0.60 0.77 0.70 0.63 0.20 0.85 0.71 0.42 0.57 0.50

Uranium Vanadium Yttrium Zirconium

0.30 0.70 0.60 0.80

Alloys Cast iron Nichrome Constantan Carbon steel Stainless steel

0.70 0.90 0.84 0.80 0.85

References 5 and 6.

17–10

Table 17.5

Radiative properties of metals

TOTAL NORMAL EMITTANCE OF METALS

Temperature ◦ C Metal

20

100

500

1 000

1 200

1 400

1 600

2 000

2 500

3 000

References

Aluminium Beryllium Bismuth Chromium Cobalt Copper Germanium Gold Hafnium Iron Lead Magnesium Mercury Molybdenum Nickel Niobium Palladium Platinum Rhenium Rhodium Silver Tantalum Tin Titanium Tungsten Uranium (α-phase) Uranium (γ-phase) Zinc Zinc (galvanised iron) Zirconium Alloys Brass Cast iron (cleaned) Nichrome Steel (polished) Steel (cleaned)

— — — — — — — — — 0.05 — — — 0.065 — — — — — — — 0.03 — — — — — — — —

0.038 — 0.06 0.08 0.15–0.24 — — 0.02 — 0.07 0.63 0.12* 0.12 0.08 — — — — — — 0.02–0.03 0.04 0.07 0.11 — — — 0.07 0.21 —

0.064 — — 0.11–0.14 0.34–0.46 0.02 0.54 0.02 — 0.14 — — — 0.13 0.09–0.15 — 0.06 0.086 — 0.035 0.02–0.03 0.06 — — 0.05 0.33* — — — —

— 0.55 — — — — — — — 0.24 — — — 0.19 0.14–0.22 0.12 0.12 0.14 0.22 0.07 — 0.11 — — 0.11 — 0.29–0.40* — — 0.22

— 0.87 — — — 0.12m — — 0.30 — — — — 0.22 — 0.14 0.15 0.16 0.25 0.08 — 0.13 — — 0.14 — — — — 0.25

— — — — — — — — 0.31 — — — — 0.24 — 0.16 — — 0.27 0.09 — 0.15 — — 0.17 — — — — 0.27

— — — — — — — — 0.32 — — — — 0.27 — 0.18 — — 0.29 — — 0.18 — — 0.19 — — — — —

— — — — — — — — — — — — — — — 0.21 — — — — — 0.23 — — 0.23 — — — — —

— — — — — — — — — — — — — — — — — — — — — 0.28 — — 0.27 — — — — —

— — — — — — — — — — — — — — — — — — — — — — — — 0.30 — — — — —

34 35 — 5 5 5 36 — 37 5 38 19 — 5 5 39 14 5 40 14 5 5 41 42 5 5 5 43 41 44

— — — — —

0.059 0.21 — 0.13–0.21 0.21–0.38

— — 0.95 0.18–0.26 0.25–0.42

— — 0.98 0.55–0.80 0.50–0.77

— — — — —

— — — — —

— 0.29m — ——

— — — — —

— — — — —

— — — — —

41 5 5 5 5

∗ Value for total hemispherical m Value for molten state.

emissivity.

Radiative properties of metals Table 17.6

17–11

TOTAL NORMAL EMITTANCE OF OXIDISED METALS

The values depend on the degree of oxidation and the grain size. Unless stated otherwise, the following are results obtained for metals oxidised in general above 600◦ C. Metal

200◦ C

400◦ C

600◦ C

800◦ C

1 000◦ C

References

Alumium Brass Chromium Copper (red heat for 30 min) Copper (stably oxidised at 760◦ C) Copper (extreme oxidation) Cast iron Cast iron (strongly oxidised) Iron (red heat for 30 min) Lead Molybdenum (oxide volatile in vaccum above 540◦ C) Monel Nickel (stably oxidised at 900◦ C) Nimonic (buffed, oxidised at 900◦ C) Nimonic (buffed, oxidised at 1 200◦ C) Niobium (oxidised and annealed)∗ Palladium Stainless steel (stably oxidised at high temperature) Stainless steel (red heat in air for 30 min) Stainless steel (buffed, stably oxidised at 600◦ C) Stainless steel (polished, oxidised at high temperature) Stainless steel (shot blasted stably oxidised at 600◦ C) Tantalum (red heat for 30 min) Zinc

0.11 0.61 — 0.15

0.15 0.60 0.09 0.18

0.19 0.59 0.14–0.34 0.23

— — — 0.24

— — — —

38 38 14 5

—

0.40–0.50

0.60–0.66

—

—

5

—

0.88

0.92

—

—

5

0.64 0.95

0.71 —

0.78 —

— —

— —

38 —

0.45

0.52

0.57

—

—

5

0.63 —

— 0.84

— —

— —

— —

— 5

0.41 0.15–0.50

0.44 0.33–0.51

0.47 0.44–0.57

— 0.49–0.71

— —

38 5

—

0.46

—

—

—

45

—

0.72

—

—

—

45

—

—

—

0.74

—

5

0.03 —

0.05 0.80–0.87

0.076 0.84–0.91

— 0.89–0.95

0.124 —

0.12–0.25

0.17–0.30

0.23–0.37

0.30–0.44

∼

5

—

0.41

0.44

0.54

—

5

—

—

0.65–0.70

—

0.73–0.83

5

—

0.65

0.67

—

—

5

0.42

0.42

0.42

—

—

5

—

0.11

—

—

—

38

∗ Value

14 5

for total hemispherical emissivity.

REFERENCES 1. 2. 3. 4.

M. F. Modest, ‘Radiative Heat Transfer’, McGraw-Hill Publishing Company, New York, NY, 1993. W. W. Coblentz, Bull. US Bur. Stand., 1918, 14, 312. W. Weniger and A. H. Pfund, Phys. Rev., 1919, 14, 427. W. J. Parker and G. L. Abbott, ‘Symposium on Thermal Radiation of Solids’, (ed. S. Katzoff), NASA SP-55, 1965, 11–28. 5. Y. S. Touloukian and D. P. DeWitt (editors), ‘Thermophysical Properties of Matter’, Vol. 7, ‘Thermal Radiative Properties—Metallic Elements and Alloys’, IFI/Plenum, New York, 1970.

17–12 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45.

Radiative properties of metals

G. K. Burgess and R. G. Wallenberg, Bull. US Bur. Stand., 1915, 11, 591. H. B. Wahlin and H. W. Knop, Jr., Phys. Rev., 1948, 74, 687. C. C. Bidwell, Phys. Rev., 1914, 3, 439. F. G. Allen, J. Appl. Phys., 1957, 28, 1510. M. L. Shaw, J. Appl. Phys., 1966, 37, 919. O. K. Husmann, J. Appl. Phys., 1966, 37, 4662. L. V. Whitney, Phys. Rev., 1935, 48, 458. R. W. Douglass and E. F. Adkins, Trans. Met. Soc. AIME, 1961, 221, 248. H. T. Betz, O. H. Olsen, B. D. Schurin and J. C. Morris, WADC-TR-56-222 (Part 2), 1957, 1–184 (AD 202 493). D. T. F. Marple, J. Opt. Soc. Amer., 1956, 46, 490. F. J. Bradshaw, Proc. Phys. Soc., 1950, B63, 573. H. Seemuller and D. Stark, Z. Phys., 1967, 198, 201. S. C. Furman and P. A. McManus, USAEC, GEAP-3338, 1960, 1–46. W. G. D. Carpenter and J. H. Sewell, RAE Rpt: CHem-538, 1962, 1–6 (AD 295 648). D. J. Price, Proc. Phys. Soc., 1947, 59, 118. A. E. Anderson, Univ. of Calif., M.S. Thesis, 1962, 1–57. T. B. Barnes, J. Opt. Soc. Amer., 1966, 56, 1546. L. Ward, Proc. Phys. Soc., 1956, 69, 339. J. G. Adams, Northrup Corp., Novair Div., 1962, 1–259 (AD 274 555). J. A. Coffman, G. M. Kibler, T. F. Lyon and B. D. Acchione, WADD-TR-60-646 (Part 2), 1963, 1–183, (AD 297 946). W. W. Coblentz, Bull. US Bur. Stand., 1911, 7, 197. J. G. Adams, Northrup Space Labs., Hawthorne, Calif., NSL-62-198, 1962, 1–101. W. W. Coblentz, Publ. Carneg. Instn. Wash. No. 65, 1906, p. 91; Bull. US Bur. Stand., 1906, 2, 457. H. Schmidt and E. Furthmann, Mitt. K. Wilhelm. Inst. Eisenforsch. Dusseld., 1928, 10, 225. R. V. Dunkle and J. T. Gier, Inst. of Eng. Res., Univ. of Calif., Berkeley, Progress Rpt., 1953, 1–73 (AD 16 830). E. Hagen and H. Rubens. Ann. Phys. Lpz., 1900 (4), 1, 352. E. O. Hulburt, Astrophys. J., 1915, 42, 203. R. A. Seban, WADD-TR-60-370 (Part 2), 1962, 1–72 (AD 286 863). E. Schmidt, V. A. W. Hauzeitschr and A. G. Erftwerk, Aluminium, 1930, 3, 91. S. Konopken and R. Klemm, NASA-SP-31, 1963, 505–513. V. F. Brekhovskikh, Inz.-fiz. Zh., 1964, 7 (5), 66. D. L. Timrot, V. Yu. Voskresenskii and V. E. Peletskii, High Temp., 1966, 4, 808. C. P. Randolf and M. J. Overholzer, Phys. Rev., 1913, 2, 144. G. L. Abbott, WADD-TR-61-94 (Part 3), 1963, 1–30 (AD 435 825) (AD 436 887). G. B. Gaines and C. T. Sims, J. Appl. Phys., 1963, 34, 2922. T. T. Barnes, W. E. Forsythe and E. Q. Adams, J. Opt. Soc. Amer., 1947, 37, 804. J. T. Bevans, J. T. Gier and R. V. Dunkle, Trans. ASME, 1958, 80, 1405. P. F. McDermott, Rev. Sci. Instrum., 1937, 8, 185. D. L. Timrot and V. E. Peletskii, High Temp., 1965, 3, 199. A. H. Sully, E. A. Brandes and R. B. Waterhouse, Br. J. Appl. Phys., 1952, 3, 97.

18

Electron emission*

Under normal conditions electrons are prevented from leaving a metal by a potential step at the surface. In the absence of an electric field, the height of this potential step is called the work function φ. Electrons can, however, escape if they are given enough energy. This energy can be supplied in a number of different ways, giving rise to the various types of electron emission.

18.1 Thermionic emission When a metal is heated, some electrons with energies near the Fermi level are enabled to escape by acquiring extra thermal energy. An adjacent anode carrying a sufficiently positive potential will collect all the electrons emitted, and the saturated emission current will flow. A further increase of anode potential causes a positive field at the metal surface; this lowers the potential barrier slightly and increases the current. The ‘zero field’ saturated emission current per unit area of the cathode J , is related to the temperature according to the Richardson-Dushman equation. J = AT 2 exp (−eφ/kT ) where A is a constant, e the magnitude of the electronic charge, k Boltzmann’s constant and T the absolute temperature. For a metal, the theoretical value of A is 1.2 MA m−2 . In practice φ usually has a temperature dependence and this results in a different value of A. The work function φ cannot be calculated reliably, but tends to increase with the density of the metal. The observed values of A and φ are shown in Table 18.1 for polycrystalline surfaces of a number of metals. To calculate the emission from the values of A and φ in the table, the emission formula may be written as log10 J = log10 A + 2 log10 T − 504 0 φT −1 where J is in kA m−2 , T is in K. The work function of a metal is lowered when a layer of a more electropositive material is adsorbed on its surface. This increases the thermionic emission, which has a maximum value when the adsorbed layer is approximately monatomic. Table 18.2 shows typical values of thermionic constants for various such surfaces. The emission may also be increased by a coating of a refractory metallic compound, usually about 100 µm thick. The thermionic emission follows the usual law but in the case of a semiconductor layer the quantities A and φ have a different significance. Table 18.3 shows the emission constants for various carbides, borides and oxides, and the emission available at a particular operating temperature. For a thermionic cathode to be technically useful it must have an adequate emission at a temperature where the rate of evaporation is not excessive. This limits the practical cathodes to a relatively small number. Table 18.4 gives the most important of these with their normal maximum operating temperatures and maximum operating emission densities for a generally acceptable life. An ‘L’ cathode consists of a block of porous tungsten, the front emitting surface of which is activated with barium. A reaction between the tungsten and barium oxide at the rear surface produces free barium which diffuses through the porous tungsten. ∗

For applications in electron microscopy, see Section 10.4.

18–1

18–2

Electron emission

Table 18.1

THERMIONIC PROPERTIES OF THE ELEMENTS

Element

A kAm−2 K−2

φ V

Barium Beryllium Caesium Calcium Carbon Chromium Cobalt Copper Hafnium Iridium Iron α Iron γ Molybdenum Nickel

600 3 000 1 600 600 150 1 200 410 1 200 220 1 200 260 15 550 300

2.11 3.75 1.81 2.24 4.5 3.90 4.41 4.41 3.60 5.27 4.5 4.21 4.15 4.61

Element

A kAm−2 K−2

φ V

Niobium Osmium Palladium Platinum Rhenium Rhodium Silicon Tantalum Thorium Titanium Tungsten Uranium Zirconium

1 200 1 100 000 600 320 1 200 330 80 1 200 700 — 600 60 3 300

4.19 5.93 4.9 5.32 4.96 4.8 3.6 4.25 3.38 3.9 4.54 3.27 4.12

References: 1, 2, 3, 4 (General): 5 (Hf). 6 (Nb, Ta, Re, Os, Ir); 7 (Be, Cr, Cu). THERMIONIC PROPERTIES OF REFRACTORY METALS WITH ADSORBED ELECTROPOSITIVE LAYERS

Table 18.2

Surface

A kAm−2 K−2

φ V

Tungsten–barium Tungsten–caesium Tungsten–cerium Tungsten–lanthanum Tungsten–strontium Tungsten–thorium Tungsten–uranium Tungsten–yttrium Tungsten–zirconium Molybdenum–thorium Tantalum–thorium

15 32 80 80 — 30 32 70 50 15 15

1.56 1.36 2.71 2.71 2.2 2.63 2.84 2.70 3.14 2.59 2.52

Reference 8.

Table 18.3

THERMIONIC PROPERTIES OF REFRACTORY METAL COMPOUNDS

Compound

A kAm−2 K−2

φ V

Emission kAm−2

TaC TiC ZrC ThC2 SiC UC CaB6 SrB6 BaB6 LaB6 CeB6 ThB6 PrB6 NdB6 ThO2 CeO2 La2 O3 Y 2 O3 BaO/SrO (oxide cathode)

3 250 3 5 500 640 330 26 1.4 160 290 36 5 — — 50 10 9 10 1–10

3.14 3.35 2.18 3.5 3.5 2.9 2.9 2.7 3.5 2.7 2.6 2.9 3.12 4.6 2.6 2.3 2.5 2.4 1.0

3 at 2 000 K 63 at 2 000 K 40 at 2 000 K 40 at 2 000 K 4 at 2 000 K 50 at 2 000 K 0.12 at 1 670 K 0.036 at 1 670 K 0.018 at 1 670 K 7 at 1 670 K 1.7 at 1 670 K 0.022 at 1 670 K

References: 9 (Carbides), 10 (Borides), 11 (Oxides), 12 (UC).

20 at 1 900 K 26 at 1 900 K 8 at 1 900 K 13 at 1 900 K See comments in text

Thermionic emission

18–3

EMISSION AT THE NORMAL MAXIMUM OPERATING TEMPERATURE OF PRACTICAL CATHODES

Table 18.4

Cathode

Operating temperature K

Emission kA m−2

Tungsten Tantalum Rhenium BaO/SrO on nickel d.c. ‘Oxide cathode’ pulse BaO/SrO Ni. Matrix type ‘L’ cathode Impregnated tungsten ThO2 on W or Ir LaB6 on Re LaB6 bulk Thoriated tungsten

2 500 2 400 2 400 1 100 1 100 1 150 1 360 1 350 1 900 1 450 1 900 1 900

3 8 0.5 10 100 20 30 50 10 0.5 50 10

6

B La lk Bu

103

Max on substrate

W

102

O Th

Th

6

B

R

W

10⫺1

Ta

e

100

or

2

ed

10

iat

104

La

Emission density, A m⫺2

105

‘O xid e’ Ba pu lse riu d Im m ‘O pr ni x id eg ck e ‘L’ ’c na el te ma on ca d tr th W ix tinu od ou e

s

106

10⫺2 800 1000 1200 1400 1600 1800 2000 2200 2400 2600 Temperature, K Figure 18.1

Thermionic emission of practical cathodes as a function of temperature

Impregnated tungsten is a porous tungsten block whose pores are filled with barium calcium aluminate by infiltration in the molten phase at 1 700◦ C. The emitting surface is partly the compound and partly barium activated tungsten. The thoriated tungsten is a high density rod or wire containing about 1% thorium oxide. The surface is carburised to form a layer of W2 C. In operation a reaction between the oxide and the carbide produces free thorium, which activates the surface. Figure 18.1 (reference 13) shows the saturated emissions available from some of these cathodes as a function of temperature. In practice it is usual to operate cathodes at rather less than the saturated emission. The full lines show the region where a useful life is obtainable. The cathode life is usually limited by the evaporation rate and lowering the temperature increases the life considerably. Typically,

18–4

Electron emission

lowering the temperature of an ‘L’ or impregnated tungsten cathode by 80 K increases its life by an order of magnitude. For an oxide cathode, the continuously drawn emission must be limited to about 10 kA m−2 , but if the current is drawn in microsecond pulses the pulse current may be increased to 100 kA m−2 . The emissions shown on Figure 18.1 are obtainable only in an environment free from oxidising gases. In general the partial pressure of gases such as O2 , CO2 , and H2 O should not exceed approximately 10−5 Pa. The emission is unaffected by rare gases except when the cathode is bombarded excessively with positive ions; the presence of H2 may counteract to some extent the effect of oxidising gases.

18.2 Photoelectric emission When light of sufficiently high frequency is incident on the surface of a metal, electrons are emitted. In order that electrons shall be emitted with zero velocity from a metal at absolute zero of temperature, the energy of the light photons must equal the energy corresponding to the work function. Thus there is a threshold frequency ν0 at which hν0 = eφ where h is Planck’s constant. If the frequency of the light is ν, where ν is greater than ν0 , the maximum energy of the emitted electrons is hν-hν0 . At temperatures above absolute zero the threshold is not sharp since light of a frequency less than ν0 can liberate a small number of electrons. An experiment measuring the energy of photoelectrons emitted for monochromatic light of known energy enables φ to be determined fairly accurately, the accuracy improving as the temperature is lowered. Table 18.5 shows the values of φ for a number of metals determined using the photoelectric effect. The values obtained in this way should correspond to the thermionic values in Table 18.1; discrepancies are probably due to contamination of the surface. For practical uses it is required to obtain the maximum photoelectric current for a given light flux. For extracting photocurrent, the efficiency of the conversion process (yield) must be separated from effects due to space charge in shielding the surface. The photoelectric efficiency of a surface may be defined in various ways, the most fundamental being the quantum efficiency Y. This is the ratio of the number of electrons released to the number of incident photons. For clean metals this is very low (10−4 approximately) so they are not often used. The efficient photoemitters as used in photocells and photomultipliers are semiconductors with a low effective photon threshold. They are usually formed by combination of one or more alkali metals with an evaporated thin film of

Table 18.5

PHOTOELECTRIC WORK FUNCTIONS

Surface

φ

Surface

φ

Aluminium Antimony Arsenic Barium Beryllium Bismuth Boron Cadmium Caesium Calcium Carbon Chromium Cobalt Copper Gallium Germanium Gold Iron Iridium Lead Lithium Manganese Mercury

4.2 4.1 5.1 2.5 3.4 4.4 4.5 4.0 1.9 2.9 4.8 4.4 4.0 4.5 3.9 4.8 4.8 4.4 4.6 4.0 2.4 3.8 4.5

Molybdenum Nickel Palladium Platinum Potassium Rhenium Rhodium Rubidium Silicon Silver Sodium Strontium Tantalum Tellurium Thallium Thorium Tin Titanium Tungsten Uranium Zinc Zirconium

4.2 4.9 5.0 5.3 2.2 5.0 4.6 2.1 4.2 4.7 2.2 2.7 4.1 4.8 3.8 3.5 4.3 4.1 4.5 3.6 4.3 3.8

References: 1, 2, 3 (General); 14 (Cu, Ag, Al); 15 (Re).

Secondary emission

18–5

PROPERTIES OF EFFICIENT PHOTOELECTRIC EMITTING SURFACE

Table 18.6

Surface

Photoelectric quantum efficiency Y

Photon threshold energy eV

Na3 Sb K3 Sb Rb3 Sb Cs3 Sb NaK3 Sb CsNaK3 Sb

0.02 0.07 0.10 0.25 0.30 0.40

3.1 2.6 2.2 2.05 2.0 1.55

Reference 16.

antimony (apart from the type consisting of caesium on oxidised silver). Table 18.6 gives the value of maximum photoelectric yields for various such surfaces. This maximum is reached at photon energies 1–1.5 eV above the threshold value, which is also shown. The corresponding wavelength of light λ in nm is related to the photon energy eφ in electron volts by the (non-relativistic) relationship λ = 1.24 × 103 φ−1

18.3 Secondary emission When electrons (primaries) are incident upon a surface of a solid, electrons (secondaries) are produced which leave the surface in the direction from which the primaries arrive. The total flow of secondaries consists of: 1. Primaries elastically scattered. 2. Primaries reflected inelastically with an energy loss of some tens of volts. 3. True secondaries with an energy independent of the primary energy and a mean value of about 10 eV. Electrons with an energy up to about 50 eV are usually considered to be true secondaries. The ratio of the total flow of secondaries to that of the primaries is called the secondary emission coefficient δ. As the primary electron energy is increased from zero, δ rises to reach a maximum value δmax for a primary energy Vmax in the range of 200–2 000 eV for metals, and it falls off more slowly at energies above Vmax . The shape of the curve relating δ to V is approximately similar for most metals, and a (universal) curve normalised to δmax and Vmax is shown in Figure 18.2. The values of δmax and Vmax are shown for most metals in Table 18.7. These values are for clean smooth polycrystalline surfaces. It is impossible to remove oxide films from many metals, such as aluminium or magnesium, by heating. These metals are usually deposited as clean layers either by evaporating in high vaccum or by sputtering. Alternatively the surface of the bulk metal may be cleaned by sputtering in an electric discharge in argon. It should be noted that the total (integrated) secondary emission is reduced by roughening the surface, as some of the secondaries released in the valleys in the surface may be intercepted by the adjacent high spots. An example of this is carbon; the value of δmax for polished graphite is approximately 1, while that for soot is only about 0.5. The secondary emission of metal oxides is usually higher than that of metals. Surfaces with high values of δ are used in secondary electron multipliers and are prepared by oxidising metals containing small quantities, usually about 2%, of magnesium, beryllium or aluminium. Oxidised metal surfaces with caesium evaporated on to them also have high values of δ and are used in photomultipliers. Table 18.8 shows maximum (at V /Vmax = 1) values of δ obtained from various such surfaces. Table 18.9 shows the secondary emission from a number of insulating metal compounds either as evaporated films (e) or surface layers on the parent metal (s). These layers must be very thin to avoid accumulating a charge. The secondary electrons originate normally within 10 nm of the surface, so provided films are thicker than this a true value of δ will be obtained.

Electron emission

18–6

1.0

d/ dmax

0.8

0.6

0.4

0.2

0

0

Figure 18.2

Table 18.7

0.5

1.0

1.5

2.0 V/ Vmax

2.5

3.0

3.5

4.0

Normalised curve of secondary emission as a function of primary voltage

MAXIMUM SECONDARY EMISSION COEFFICIENTS

Element

δmax

Vmax

Element

δmax

Vmax

Aluminium Antimony Barium Beryllium Bismuth Boron Cadmium Calcium Carbon (Graphite) Caesium Chromium Cobalt Copper Dysprosium Erbium Gadolinium Gallium Germanium Gold Holmium Indium Irdium Iron Lead Lithium Magnesium

0.97 1.30 0.85 0.5 1.15 1.2 1.59 0.60 1.02 0.72 1.10 1.35 1.28 0.99 1.05 1.04 1.08 1.08 1.79 1.02 1.40 1.55 1.30 1.10 0.52 0.97

300 600 300 200 550 150 800 200 300 400 400 500 600 900 1 100 600 600 400 1 000 900 500 700 200 500 100 275

Manganese Mercury Molybdenum Nickel Niobium Palladium Platinum Potassium Rhenium Rubidium Ruthenium Silicon Silver Sodium Strontium Tantalum Terbium Thallium Thulium Thorium Tin Titanium Tungsten Ytterbium Zinc Zirconium

1.35 1.30 1.20 1.35 1.20 1.65 1.60 0.53 1.30 0.90 1.40 1.10 1.56 0.82 0.72 1.25 1.02 1.40 1.05 1.10 1.35 0.90 1.35 1.04 1.40 1.10

200 600 350 450 350 550 720 175 800 300 570 250 800 300 400 600 900 800 1 100 800 500 280 650 800 800 350

References: 17; 18 (Pt, Ir, Ru); 19 (Rare earth metals).

18.4 Auger emission When the energy distribution of secondaries is examined closely, small peaks can be seen superimposed on the basically smooth background. These peaks can be made much more visible by electronic differentiation, and have been shown to originate in Auger transitions, as follows (reference 22). A

Auger emission

18–7

SECONDARY EMISSION FROM OXIDISED ALLOYS AND PHOTOCELL SURFACES

Table 18.8

Oxidised alloy Ag–Mg Ag–Be Ni–Be Cu–Be Cu–Mg Cu–Mg–Al Photocell surfaces Ag–O–Cs Ni–O–Cs Ag–O–Rb Ag–Sb–Cs Sb–Cs

δmax

Vmax

10–16 6 5–10 5 12.5 10

600 500 600 500 900 700

6–10 5.7 5.5 8.0 12

500 500 800 500 450

Reference 20.

SECONDARY EMISSION FROM INSULATING METAL COMPOUNDS

Table 18.9 Material

δmax

Vmax

Li F (e) Na F (e) NaCl (e) KCl (e) RbCl (e) Cs Cl (e) Na Br (e) KI (e) NAI (e) Ca F2 (e) Ba F2 (e) Mg F2 (e) BeO (s) MgO (s) Al2 O3 (s) Cu2 O (s) PbS (s) MoS2 (s) WS2 (s) ZnS (s) MoO2 (s) Ag2 O (s) SiO2 (e) Cs2 O (s)

5.6 5.7 6–6.8 7.5–8.0 5.8 6.5 6.25 5.6 5.5 3.2 4.5 4.1 3.4–8 2.4–17.5 1.5–3.2 1.19–1.25 1.2 1.10 0.96–1.04 1.8 1.09–1.33 0.90–1.18 2.2 2.3–11

— — 600 1 500 — — — — — — — 410 200–400 400–1 600 350–1 300 440 500 — — 350 — — 300 800

Reference 21.

primary electron removes an electron from an electron shell in an atom. Subsequently another electron in a higher energy shell transfers to the vacancy. The energy thus released is given to a third electron which is emitted. It may be seen that the energy of the third electron is independent of the energy of the primary and is characteristic of three energy levels in the excited surface atom. A large number of these Auger energies have been measured for many elements and they provide identification of the element concerned. Auger electrons have a very short mean free path and hence, in order to escape the surface, must originate within the top 10 nm of the surface. Fractions of a monolayer of an element can be detected using Auger electron spectroscopy. This relatively new technique of analysis appears to have a considerable number of possible uses.

Electron emission

18–8 90 85 80 75 70

MNN

65

Atomic number

60 55 50 45 40

LMM

35 30 25 20 15 10

KLL

5 0

0

200

400

600

800

1000

1200 1400

1600 1800

2000

2200

Electron energy, eV Figure 18.3

Strongest Auger emission peaks as a function of the atomic number

The energy of the primary bombarding electrons is usually greater than 2.5 keV, and the Auger peaks are detected from about 50 eV up to 2 kV. Figure 18.3 shows the Auger energies of the stronger peaks plotted against the atomic number Z. The letters are the electron levels involved in the Auger transition (reference 23).

18.5 Electron emission under positive ion bombardment When an electrical discharge takes place between two electrodes in a low gas pressure the cathode is bombarded with positive ions and emits electrons. These electrons are essential for maintaining the discharge. The number of electrons released for each arriving ion is usually called γ, the second Townsend coefficient. The coefficient is generally approximately constant for positive ion energies from zero up to about 1 kV. The energy required to release the electron is supplied by neutralisation of the positive ion as follows. When the ion is very close to the metal surface the electrostatic force is sufficient to extract an electron which neutralises the positive ion. This releases a photon of energy (I − φ)e, where I is the ionisation potential of the gas atom. This photon can then release a photoelectron from the metal provided (I − φ)e > φe or I > 2φ. The value of γ thus tends to increase with increasing I and decreasing φ. Values of γ for various inert gas ions and metals are shown in Table 18.10. At energies above a few keV the value of γ usually increases approximately linearly with energy, the extra electrons being released as a result of kinetic energy transfer.

18.6 Field emission When a very high positive electric field is applied to the surface of a metal, the potential just outside the metal becomes more positive than the Fermi level in the metal. The work function barrier, instead of being a step becomes very thin, and electrons can ‘tunnel’ through the barrier and be emitted. This emission is usually called field emission (sometimes tunnel emission), and the emission density

Field emission

18–9

SECOND TOWNSEND COEFFICIENT γ ELECTRONS RELEASED PER POSITIVE ION ARRIVING

Table 18.10

Metal

Ion

γ

Ion energy eV

Tungsten (outgassed) Tungsten (outgassed) Tungsten (outgassed) Tungsten (outgassed) Tungsten (outgassed)

Ne+ He+ A+ Kr+ Xe+

0.25 0.24 0.10 0.05 0.02

0–1 000 0–1 000 0–1 000 0–1 000 0–1 000

Tantalum (outgassed) Tantalum (gas covered) Tantalum (gas covered)

A+ He+ He++

0.02 0.2 0.7

100 500 500

Molybdenum (outgassed)

He++ He+ Ne++ Ne+ A++ A Kr

0.8 0.22 0.7 0.2 0.35 0.08 0.05

0–1 000 0–1 000 0–1 000 0–1 000 0–1 000 0–1 000 0–1 000

Nickel

He+ Ne+ A+

0.7 0.4 0.1

800 800 800

Reference 24.

Table 18.11

FIELD EMISSION FROM TUNGSTEN

φ = 2.0 eV Field 109 V m−1

log10 J A m−2

1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6

6.98 8.45 9.49 10.23 10.89 11.40 11.82 12.16 12.45

φ = 4.5 eV

Current from 10−14 m2 Field A 109 V m−1

log10 J A m−2

1 × 10−7 2.8 × 10−6 3.1 × 10−5 1.9 × 10−4 7.8 × 10−4 2.5 × 10−3 6.6 × 10−3 1.5 × 10−2 2.8 × 10−2

0.67 5.57 8.06 9.59 10.62 11.36 11.94 12.39 12.76 13.32

2 3 4 5 6 7 8 9 10 12

Current from 10−14 m2 A 4.7 × 10−14 3.4 × 10−9 1.1 × 10−6 3.4 × 10−5 4.2 ×10−4 2.3 × 10−3 8.8 × 10−3 2.4 × 10−2 5.8 × 10−2 2.1 × 10−2

φ = 6.3 eV Field 109 V m−1

log10 J A m−2

2 4 6 8 10 12 14 16 18 20

−8.0 3.12 7.25 9.34 10.66 11.52 12.16 12.65 13.04 13.36

Current from 10−14 m2 A 10−22 1.3 × 10−11 1.8 × 10−7 2.2 × 10−5 4.6 × 10−4 3.3 × 10−3 1.5 × 10−2 4.5 × 10−2 1.1 × 10−1 2.3 × 10−1

Reference 25.

is related to the field by the Fowler-Nordheim law. Table 18.11 shows values of log10 (emission density) for various fields for clean tungsten (φ = 4.5 V) and also for a barium contaminated surface (φ = 2.0 V) and an oxygen contaminated surface (φ = 6.3 V). High fields which produce appreciable emissions usually result from field concentration at the tips of small projections, spikes or whiskers on metal surfaces. As these usually have submicron tip diameters the emission current in A for an area of 10−14 m2 is also shown in Table 18.11. The field at the tips of emitting projections is greater than the ‘macroscopic’ field at an electrode by a factor which varies between about 1 000 for a rough surface, down to less than 100 for a highly polished and voltage conditioned hard metal surface. When the field emission from a projection reaches an appreciable fraction of an ampere, the projection will melt and vaporise and this may initiate electrical breakdown between the electrodes in vacuum. Alternatively the field emission may heat the anode electrode and release gas or metal vapour which can also initiate a breakdown. From

18–10

Electron emission

Table 18.11 the necessary tip field is likely to be 2 × 109 − 1010 V m−1 , depending on work function while the macroscopic field will usually be of the order of 1% of this. Single field emitting sources can be made, consisting of a point 0.1–1.0 µm diameter etched on the end of a refractory metal wire. An anode electrode near the point and carrying a positive potential of a few kV is sufficient to cause field emission. The point usually emits over most of its approximately hemispherical tip. The intensity of the emission varies with direction, as different crystal planes on its surface have different work functions (reference 25). The close packed planes, e.g. the (110) plane of a body centred crystal such as tungsten, have the highest work function and lowest field emission. This effect is shown visually in a field electron microscope where the emission from the tip is viewed on a screen. This arrangement has been used extensively to study surface migration and adsorption phenomena. Measurements can however only be carried out in ultra high vacuum (pressure 0.15% but 10−3 /450–80 3·10−2 /— 1·10−1 /570

3 3 3 3 3 26 27 3 28 28 22 29

670

8·10−4 /450

30

180

2·101 /575

31

2·10−2 /538 5·10−2 /300 4·100 /— 10−2 /— 10−2 –10−1 /— 2·10−2 /550 —/250 —/400, d = 10 µm 10−2 /— 2·10−3 /— 3·10−4 /— 10−2 –10−1 /— 10−1 /— 10−2 /—

8 32 33 33 33 34 35 25

1 130

1·10−2 /475

37

>1 020 700 600 1 100 420 1 100 640 600 580 >200

2·10−2 /538 6·10−3 /300 2·10−3 /— 10−3 /300 10−3 /— 7·10−3 /300 10−3 /— 10−2 /— 5·10−3 /300 5·10−3 /—

8 38 39 40 9 38 9 9 41 42

22 23 24 3 25

33 33 33 33 33 36

(continued)

36–4

Superplasticity

Table 36.1 Material [wt %]

INVESTIGATIONS ON SUPERPLASTICITY OF METAL ALLOYS—continued

Grain size [µm]

Temp. range [◦ C]

Strain rate range [s−1 ]

m

ef [%]

Optimum condition [s−1 ]/[◦ C]

Ref.

10−2 /300

36

— —

43 44

Engineering Aluminium Alloys Al-2004 (Supral 100): Al–6Cu–0.4Zr–0.25Mg–0.1Fe–0.05Zn–1 300

6·10−3 /—

2–4

750–850

2·10−5 –6·10−1

0.68

>5 500

6·10−3 /750

3–6

680

2·10−3

—

620

X = 0.5

680

2·10−3

—

1 000

X = 0.5

90

550 550–750

2·10−3

— —

1 300 377

— 3·10−2 /700

91 92

—

3–6 — 70

10−3 –4·10−2

0.56

(continued)

36–6

Superplasticity

Table 36.1

INVESTIGATIONS ON SUPERPLASTICITY OF METAL ALLOYS—continued

Material [wt %] Cu–28.1Zn–15.0Ni– 13.3Mn Cu–38.4Zn–15.7Ni– 0.13Mn Cu–38.5Zn–3Fe Cu–40Zn Cu–40Zn–1.5Pb Iron Alloys Fe–0.03C–4Ni–3Mo– 1.6Ti Fe–0.03C–4Ni–3Mo– 1.6Ti + 600 ppm B Fe–(0.13–0.34)C Fe–0.42C–1.9Mn Fe–0.88C–1.15Mn– 0.22V–0.5W– 0.48Cr–0.31Si Fe–0.91C–0.45Mn– 0.12Si Fe–1.11C–0.48Mn– 1.45Cr–9.30Al Fe–1.17C–5.12Al Fe–1.25C–0.65Mn– 0.1Si Fe–1.25C–10Al– 1.5Cr–0.54Mn Fe–1.32C–0.6Mn– 1.68Cr–1.9Al Fe–1.34C–3.05Al Fe–1.51C–3.38Al Fe–1.55C–5.22Al Fe–1.6C–0.73Mn– 0.28Si Fe–1.92C–0.82Mn– 0.3Si Fe–2.4C Fe–3C Fe–3C–1.5Cr Fe–3.26C–4.1N– 2.1Cr–0.38Si– 0.45Mn Fe–4.3C–0.18Si Stainless Steels Fe–20Cr–10Ni–0.7Ni Fe–21.3Cr–4.2Mn– 2.1Ni–0.1Si– 0.74Cu Fe–23Cr–4Ni–0.1N Fe–24.66Cr–6.82Ni– 2.79Mo–0.85Mn– 0.48Si–0.28W– 0.14N Fe–25Cr–7Ni–3Mo

Grain size [µm]

Temp. range [◦ C]

Strain rate range [s−1 ]

m

ef [%]

Optimum condition [s−1 ]/[◦ C]

Ref.

1–3

570

10−5 –2·10−3

0.5

620

2·10−4 /—

1–3

462–652

10−5 –2·10−3

0.5

750

6·10−4 /580

10–80 3 2 000– 3 000

500–800 560–640 670–720

2·10−5 –10−2 3·10−5 –6·10−3 4·10−4 –2·10−3

0.5 0.63 0.56

350 410 192

2·10−4 /700 — 6·10−4 /700

93, 94 93, 94 95 96 97

6

900–960

4·10−5 –4·10−3

0.53

>500

8·10−4 /800

98

900–960

4·10−5 –4·10−3

634

2·10−4 /900

98

2–12 1.4–2.6 700

5·100 /920

106

1–2

700–780

0.43

—

—

103

7·10−6 –10−2 10−4 –8·10−2

1–2 1–2 1–2 1.5–3

750–850 750–850 750–850 600–800

10−4 –8·10−2 10−4 –8·10−2 7·10−5 –3·10−2

0.5 0.49 0.63 0.5

583 783 450 760

8·10−3 /750

8·10−3 /750 2·10−3 /850 10−4 /630

104 104 104 105

1.5–3

600–800

2·10−4 –2·10−3

0.5

380

2·10−4 /650

105

2 2 1 1–2

630–725 650–700 630–725 650–750

2·10−5 –5·10−3

2·10−5 –5·10−4 2·10−5 –5·10−3 7·10−5 –2·10−3

0.5 0.5 0.5 0.46

500 940 1 410 210

2·10−4 /700

2·10−4 /700 2·10−4 /700 2·10−4 /675

107 107 107 108

1–2

650–750

9·10−5 –3·10−3

0.41

150

2·10−4 /675

108

— 10–20

700–1 000 900–1 050

3·10−4 –2·10−2 10−4 –6·10−2

0.65 0.56

527 —

—/800 —

109 110

7.5 1 600

— 10−2 /—

110 113

(continued)

Superplasticity Table 36.1

36–7

INVESTIGATIONS ON SUPERPLASTICITY OF METAL ALLOYS—continued

Grain size [µm]

Temp. range [◦ C]

2–3

1 050–1 350

10−3 –100

0.6

3–4

700–1 020

2·10−6 –2·10−2

0.54

73.1

300–500

8·10−5 –8·10−2

—

43.9

300–500

8·10−5 –8·10−2

—

66.8 0.2 1–20

300–500 250–350 250–300

8·10−5 –8·10−2 10−4 –100 4·10−4 –10−2

— 0.5 0.62

1 0.5–3 300

2·10−3 /725 10−2 /700 6·10−5 /—

162 161 163

4·10−4 –4·10−3

0.57

910

10−3 /700

164

590–735 600–750

3·10−4 –5·10−2 3·10−4 –5·10−3

0.44 0.85

692 658

3·10−3 /695 3·10−4 /685

165 166

1.1 0.6–2.2 0.5–1.5 1 0.4–2 0.6–2.2 2 2.1 1.3–3.7 2.4 2.5 2.6 1–5 2.3–4.6 2.5–4.2

20–220 250–350 20–250 60–200 200–250 200–250 220–300 200 177–252 200 200 230 175–250 136–230 150–230

10−5 –4·10−2 2·10−6 –2·10−2 7·10−5 –2·10−1 3·10−7 –3·10−1 3·10−5 –2·10−2 2·10−6 –2·10−1 8·10−4 –8·10−1 10−5 –100 c 10−7 –10−1 6·10−6 –4·10−1 10−5 –100 10−5 –10−1 c 3·10−9 –2·10−2 10−7 –2·10−2 10−5 –100

0.5 0.53 0.55 0.5 0.5 0.6 0.44 0.41 0.45 — — 0.5 0.5 0.5 0.5

700 — >1 000 — 500 900 1 200 — — 1 800 2 900 2 550 — — 2 900

27 167 168 169 170 167 171 172 173 174 175 176 177 178 179

4–8 — 0.5–1.8 1 0.6–2.2 2.5 2 2–4 2–4 2–4 2–4 2–4

25–250 25–350 250 180 22–325 150–190 20 275 275 275 275 275

3·10−4 –3·10−1 2·10−5 –10−2 b 2·10−6 –2 10−6 –10−1 2·10−6 –2·10−2 8·10−4 2·10−4 –4·10−2 10−5 –1 10−5 –1 10−5 –1 10−5 –1 10−5 –1

— 0.7 0.5 0.4 0.5 0.54 0.34 0.17 0.39 0.36 0.27 0.22

770 900 — — — 290 180 102 620 500 132 150

10−2 /102 — a 2·10−1 /250 — 10−1 /250 2·10−2 /250 — — — — 10−2 /— 7·10−2 /— a 2·10−5 –2·10−3 a 10−4 –10−2 10−2 /230, d = 2.5 µm 3·10−2 /250 2·10−2 /250 — — — —/190 — — — — — —

Optimum condition based on strain rate sensitivity, m, instead of elongation.

b Testing performed in compression. c Testing performed in double-shear configuration. d Dispersion-strengthened alloy. e Tested under a 2 kV/cm applied

electric field.

158 154

180 181 182 183 167 184 185 186 186 186 186 186

36–10 Table 36.2

Superplasticity INVESTIGATIONS ON INTERMETALLIC ALLOYS EXHIBITING SUPERPLASTICITY

Note that compositions are given in atomic %. Material [at %]

Grain size [µm]

Iron Aluminide Alloys Fe3 Al Alloys Fe–27Al 100–800 Fe–28Al–2Ti 100 100 FeAl Alloys Fe–36.5Al Fe–36.5Al–1Ti Fe–36.5Al–2Ti

>100 >100 350 >100

Iron Silicide (Fe3 Si) Alloys Fe–14Si–0.25B 72 Fe–18Si–0.25B 72 Nickel Aluminide Alloys NiAl Alloys NiAl 200 Ni–28.5Al–20.4Fe 30–50 Ni3 Al Alloys Ni–15.5Al–7.4Cr– 0.4Zr–0.09B Ni–24Al–0.24B

0.05 6 13–20 1.6

Nickel Silicide (Ni3 Si) Alloys Ni–17.2Si–3.3V– 15 1.1Mo Ni–17.3Si–3.3V– 8–20 2.3Mo 14 Ni3 (Si,Ti) 4–14 Titanium Aluminide Alloys TiAl Alloys Ti–40Al 0.5 Ti–43Al 5 Ti–43.8Al–12.1V 454 231

264 265 260 266 266

— — — — 510 500 580 695 >490 —

266 267 264 260 268 268 268 268 102 269

—

270

>100 —

271 272

>30

273

—

274

—

275, 276 277 260 278 279

— — — —

T

650–925 120 832–932 — 830–1030 4.2 Cycles of chemical composition through cyclic hydrogen introduction 830–1 010 4.2

Ti–6Al–4V Ti–6Al–4V

T T

760–981 840–1 030

120 7.5

Ti–6Al–4V

T

7.5

Ti–6Al–4V/5 vol% TiBW Ti–6Al–4V/10 vol% TiBW Ti3Al, Super α2

T T T

Various, 840–990 840–1 030 840–1 030 950–1 150

— 260– 398 —

278, 280 277 281, 282 283

7.5 7.5 4–15

390 260 610

284 282 285

135

(continued)

36–13

36–14

Superplasticity

SUMMARY OF RESEARCH ON INTERNAL STRESS SUPERPLASTICITY—continued

Table 36.4

Material

Mismatch

U

T

U

T

Zn

A

Zn

A

Zn

A

Zn–30 vol% Al2 O3P

A,C

Zr Zr Zr

T T T

Zr/2 vol% ZrH2

T

Temp. range [◦ C] Various, 200–450 613–713 720–820 60–150 110–330 50–250 100–300 150–350 200–300 230–330 250–350 270–370 290–390 100–300 150–350 750–950 813–913 810–910 810–940 23–425

Frequency [hr−1 ]

ef [%]

Ref .

50–170

—

286

—

—

260

14.4 8.5 12–180

—

251

>200

287

90

—

288

120

—

289

— — 6–15

— — >270

263 260 290

0.036

—

291

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Trans., 1996, 27A, 2837. 255. M. Y. Wu and O. D. Sherby, Scr. Metall., 1984, 18, 773. 256. Y. C. Chen, G. S. Daehn and R. H. Wagoner, Scr. Metall. Mater., 1990, 24, 2157. 257. C. Y. Huang and G. S. Daehn, p. 135 in ‘Superplasticity and Superplastic Forming 1995’ (eds A. K. Ghosh and T. R. Bieler), TMS, Warrendale PA. 1996. 258. G. Gonzales-Doncel, S. D. Karmarkar, A. P. Divecha and O. D. Sherby, Comp. Sci. Technol., 1989, 35, 105. 259. J. C. LeFlour and R. Locicero, Scripta Metall., 1987, 21, 1071. 260. G. W. Greenwood and R. H. Johnson, Proc. Roy. Soc.-London, 1965, 283A, 403. 261. M. Zamora and J. P. Poirier, Mech. Mater., 1983, 2, 193.

36–20 262. 263. 264. 265. 266. 267. 268. 269. 270. 271. 272. 273. 274. 275. 276. 277. 278. 279. 280. 281. 282. 283. 284. 285. 286. 287. 288. 289. 290. 291.

Superplasticity

V. E. Hornbogen and G. Wassermann, Z. Metallk., 1956, 47, 427. M. G. Lozinsky and I. S. Simeonova, Acta Metall., 1961, 9, 689. M. de Jong and G. W. Rathenau, Acta Metall., 1959, 7, 246. F. Clinard and O. D. Sherby, Acta Metall., 1964, 12, 911. P. Zwigl and D. C. Dunand, Metall. Mater. Trans., 1998, 29A, 565. M. de Jong and G. W. Rathenau, Acta Metall., 1961, 9, 714. D. Oelschlagel and V. Weiss, Trans. Am. Soc. Metals, 1966, 59, 143. T. B. Massalski, S. K. Bhattacharyya and J. H. Perepezko, Metall. Trans., 1978, 9A, 53. K. Kitazono, E. Sato and K. Kuribayashi, Scripta Mater., 2001, 44, 2695. K. Kitazono, E. Sato and K. Kuribayashi, Scr. Mater., 1999, 41, 263. R. S. Sundar, K. Kitazono, E. Sato and K. Kuribayashi, Intermetallics, 2001, 9, 279. P. Zwigl and D. C. Dunand, Materials Science and Engineering, 2001, 298, 63. C. Chaix and A. Lasalmonie, Res Mech., 1981, 2, 241. N. Furushiro, H. Kuramoto, Y. Takayama and S. Hori, Trans. ISIJ, 1987, 27, 725. Y. Takayama, N. Furushiro and S. Hori, p. 753 in ‘Titanium Science and Technology’ (eds G. Lutjering, U. Zwicker and W. Bunk), Deutsche Gesellschaft fur Metallkunde, Munich. 1985. R. Kot, G. Krause and V. Weiss, p. 597 in ‘The Science, Technology and Applications of Titanium’ (eds R. I. Jaffe and N.E. Promisel), Pergamon, Oxford. 1970. C. Schuh and D. C. Dunand, Scripta Mater., 1999, 40, 1305. D. C. Dunand and P. Zwigl, Metall. Mater. Trans., 2001, 32A, 841. D. C. Dunand and C. M. Bedell, Acta Metall. Mater., 1996, 44, 1063. C. Schuh and D. C. Dunand, J. Mater. Res., 2001, 16, 865. C. Schuh and D. C. Dunand, Int. J. Plastic., 2001, 17, 317. C. Schuh and D. C. Dunand, Acta Mater., 2001, 49, 199. C. Schuh and D. C. Dunand, Scripta Mater., 2001, 45, 631. C. Schuh and D. C. Dunand, Acta Mater., 1998, 46, 5663. A. G. Young, K. M. Gardner and W. B. Rotsey, Journal of Nuclear Materials, 1960, 2, 234. M. Y. Wu, J. Wadsworth and O. D. Sherby, Metall. Trans., 1987, 18A, 451. K. Kitazono, R. Hirasaka, E. Sato, K. Kuribayashi and T. Motegi, Acta Mater., 2001, 49, 473. M. Y. Wu, J. Wadsworth and O. D. Sherby, Scr. Metall., 1987, 21, 1159. P. Zwigl and D. C. Dunand, Metall. Mater. Trans., 1998, 29A, 2571. K. Nuttall and D. P. McCooeye, p. 129 in ‘Mechanical Behavior of Materials’, The Society of Materials Science, Japan. 1974.

37

Metal-matrix composites

Metal-matrix composites are engineered materials typically comprising reinforceants of high elastic modulus and high strength in a matrix of a more ductile and tougher metal of lower elastic modulus and strength. The metal-matrix composite has a better combination of properties than can be achieved by either component material by itself. Normally, the objective of adding the reinforceant is to transfer the load from the matrix to the reinforceant so that the strength and elastic modulus of the composite are increased in proportion to the strength, modulus and volume fraction of the added material. Improvements in non-mechanical properties can also be achieved by adding a second phase, for example electrical conductivity, as compared with conventional alloying for strength. The reinforcement can take one of several forms. The least expensive and most readily available on the market are the particulates. These can be round but are usually irregular particles of ceramics, of which SiC and Al2 O3 are most frequently used. Composites reinforced by particulates are isotropic in properties but do not make best use of the reinforceant. Fine fibres are much more effective though usually more costly to use. Most effective in load transfer are long parallel continuous fibres. Somewhat less effective are short parallel fibres. Long fibres give high axial strength and stiffness, low coefficients of thermal expansion and, in appropriate matrices, high creep strength. These properties are very anisotropic and the composites can be weak and brittle in directions normal to that of the fibres. Where high two-dimensional properties are needed, cross-ply or interwoven fibres can be used. Short or long randomly oriented fibres provide lower efficiencies in strengthening (but are still more effective than particulates). These are most frequently available as SiC whiskers or as short random alumina (‘Saffil’) fibres or alumino-silicate matts. Long continuous fibres include drawn metallic wires, mono-filaments deposited by CVD or multifilaments made by pyrolysis of polymers. The properties of some typical fibres are compared in Table 37.1. The relative prices are given as a very approximate guide. Because most composites are engineered materials, the matrix and the reinforceant are not in thermodynamic equilibrium and so at a high enough temperature, reaction will occur between them which can degrade the properties of the fibre in particular and reduce strength and more especially fatigue resistance. As many composites are manufactured by infiltration of the liquid metal matrix into the pack of fibres, reaction may occur at this stage. Some typical examples of interaction are listed in Table 37.2. In order to obtain load transfer in service, it is essential to ensure that the reinforceant is fully wetted by the matrix during manufacture. In many cases, this requires that the fibre is coated with a thin interlayer which is compatible with both fibre and matrix. Commonly, this also has the advantage of preventing deleterious inter-diffusion between the two component materials. The data on most coatings are proprietary knowledge. However, it is well known that silicon carbide is used as an interlayer on boron and on carbon fibres to aid wetting by aluminium alloys. The routes for manufacturing composites are still being developed but the most successful and lowest cost so far is by mixing particulates in molten metal and casting to either foundry ingot or as billets for extrusion or rolling. This is applied commercially to aluminium alloy composites. Another practicable route is co-spraying in which SiC particles are injected into an atomised stream of aluminium alloy and both are collected on a substrate as a co-deposited billet which can then be processed conventionally. This is a development of the Osprey process and can be applied more widely to aluminium and other alloys. Other routes involve the infiltration of molten metal into fibre pre-forms of the required shape often contained within a mould to ensure the correct final shape. This can be done by squeeze casting or by infiltrating semi-solid alloys to minimise interaction between the fibre and metal. Fibres can also be drawn through a melt to coat them and then be consolidated by hot-pressing. To reduce interaction, solid state methods can be used, e.g. cold isostatic pressing (CIP) and sintering or hot isostatic pressing (HIP) of metal powders mixed with short fibres. Diffusion-bonding

37–1

37–2

Metal-matrix composites

Table 37.1

PROPERTIES OF REINFORCING FIBRES AT ROOM TEMPERATURE (BASED ON REFERENCES 9 and 16)

Density g cm−3 19.2 7.8 2.6 3.4 2.6 3.2 3.9 3.5 2.0 1.9 3.0

Weibull modulus

Fracture stress MPa

Elastic modulus GPa

Coefficient of thermal expansion K−1 × 106

Price relative to glass fibre

2 500 2 500 3 500 3 800 2 500 10 000 1 500 2 000 3 000 4 200 850

400 210 400 450 200 700 380 300 600 300 150

5.0 15.0 8.0 4.5 4.5 4.5 7.0 7.0 0 0 —

1 000–50 30–1 250 500 100 150 100 25 1 000 100 20

—

Fibre

Form

Preparation route

Diameter µm

Tungsten Steel Boron SiC SiC SiC ∼Al2 O3 ∼Al2 O3 C (high modulus) C (med. strength) Alumino-silicate Glass (27% SiO2 ) Alumino-silicate Glass (47% SiO2 ) S-Glass

Wire Wire Cont. mono-filament Cont. mono-filament Cont. multi-filament Whisker (random, short) Multi-filament Random short fibres Cont. multi-filament Cont. multi-filament Random short fibres

Drawn Drawn Chemical vapour depos. Chemical vapour depos. Polymer fibre pyrolysis Polymer fibre pyrolysis Oxide/salt fibre pyrolysis Polymer fibre pyrolysis Polymer fibre pyrolysis Polymer fibre pyrolysis

10–500 10–250 150 150 10–15 0.1–2.0 15–25 2–4 10 8 3

Random short fibres

Drawn from melt

3

2.7

1 750

105

Cont. multi-filament

Drawn from melt

3–20

2.5

4 000

90

7–19 8 1.8 6 3–8 5–8

3.0

1 2.5

Metal-matrix composites Table 37.2

TYPICAL INTERACTIONS IN SOME FIBRE-MATRIX SYSTEMS

Temperature of significant interaction ◦ C

References

System

Potential interaction

Al-C

Formation of Al4 C3 at interface. Degradation of C fibre properties. No significant reaction at normal fabrication temperatures B2 O3 reacts with Al to form borides.

550 ∼495

9

770

10

Boride formation; interlayer of SiC needed. Interfacial layer of LiAl5 O8 on liquid infiltration. No significant reaction below melting point. Al3 C3 and Si can form in liquid Al. Formation of iron aluminides. Directionally solidified eutectic; dissolution and re-precipitation. No interaction up to melting point. Formation of Fe7 W6 : dissolution of fibre. No significant reaction at m.p. of alloy provided O and N avoided during infiltration. Slight reaction to pit fibre. NiAl2 O4 spinel forms in air. Ni activates recrystallisation of fibre. Ni activates recrystallisation of fibre. Formation of nickel silicides. Recrystallisation of fibre. Degradation of creep properties. Formation of TiB2 . Formation of TiC, TiSi2 and Ti5 Si3 .

500 ∼650 m.p. 660 >700 500 1 200

9 13 9 12 9 9

1 083 1 000

9 9 11, 14

1 100 1 100 1 150–1 300 800 800 1 000 900 750 700

9

Al-Al2 O3 Al-oxide (Al2 O3 -SiO3 -B2 O3 ) Al-B Al/Li-Al2 O3 Al-SiC Al-steel Co-TaC Cu-W Fe-W Mg(AZ91)-C Ni-Al2 O3 Ni-C(I) Ni-(II) Ni-SiC Ni-W Ti-B Ti-SiC

37–3

9

9 9 9 9 9 9

laminates of layers of fibres and metal foil is also effective. Another successful method of fabricating continuous fibre composites is to pre-coat the individual fibres with metal-matrix by PVD prior to consolidation by hot pressing. Very fine composite microstructures can be made by co-deforming a dispersed metal in a metal matrix, for example, Cu-Cr, Cu-Nb, in these cases very high strengths can be achieved for large drawing strains. The properties which can be achieved in composites include higher strength, higher stiffness, improved high-temperature properties, lower or matched coefficients of thermal expansion and improved wear resistance. This is usually at the expense of conventional ductility but in most cases improved fracture toughness can be obtained. The following tables illustrate what has been achieved in aluminium, magnesium, titanium, zinc and copper alloys. The improvement in strength is often larger in the weaker alloys with less benefit being realised in the stronger alloys, e.g. see Table 37.3. All the alloys show an improvement in elastic modulus when reinforced.

37–4

Metal-matrix composites

Table 37.3

MECHANICAL PROPERTIES OF ALUMINIUM ALLOY COMPOSITES AT ROOM TEMPERATURE

Fracture toughness MPa m−1/2 29.7 24.1 22 21.5 17.9 — — 25.3 18.0 18.8 — 17.7 — — — —

Form

1.0 0.6 0.2 0.25

Extrusion

T6

Cu Mg Si Mn

4.4 0.7 0.8 0.75

Extrusion

T6

Cu Mn V Cu Mg Si Fe Ni Zn Mg Cu Cr Li Cu Mg Zr

6.0 0.3 0.1 2.0 1.5 0.9 0.9 1.0 5.6 2.2 1.5 0.2 2.5 1.3 0.95 0.1

Sheet Extrusion

T6 T6

Sheet Extrusion Extrusion

T6 T6 T6

Nil 10% Al2 O3 15% Al2 O3 20% Al2 O3 13% SiC 20% SiC 30% SiC Nil 10% Al2 O3 15% Al2 O3 20% Al2 O3 10% SiC 8.2% SiC Nil 15% Al2 O3 20% Al2 O3 ∼ 10% SiC Nil 13% SiC

276 297 317 359 317 440 570 414 483 476 483 457 448 290 359 359 396 320 333

310 338 359 379 356 585 795 483 517 503 503 508 516 414 428 421 468 400 450

20.0 7.6 5.4 2.1 4.9 4.0 2.0 13.0 3.3 2.3 0.9 1.8 4.5 10.0 3.8 3.1 3.3 — 6.0

69.9 81.4 87.6 98.6 89.5 120.0 140.0 73.1 84.1 91.7 101.4 91.2 82.5 73.1 88.3 91.7 93.6 75.0 89.0

Extrusion

T6

Nil 12% SiC

617 597

659 646

11.3 2.6

Extrusion (18 mm)

T6 T6

Nil 12% SiC

480 486

550 529

5.0 2.6

6061

Mg Si Cu Cr

2014A

2219

8090

Elastic modulus GPa

% particulate

Nominal composition

7075

Elongation %

Heat treatment

Base alloy

2618

Ultimate tensile stress MPa

0.2% proof stress MPa

Density g cm−3

References

— 28.9

2.71 2.81 2.86 2.94 — — — 2.80 2.92 2.97 2.98 — — — — — — — —

1, 2 1, 2 1, 2 1, 2 1 3 3 1, 2 1, 2 1, 2 1, 2 1 1 1, 2 1, 2 1, 2 1, 6 1, 6 1, 6

71.1 92.2

— —

— —

6 6

79.5 100.1

— —

— —

1, 6 1, 6

Metal-matrix composites Table 37.4

37–5

MECHANICAL PROPERTIES OF ALUMINIUM ALLOY COMPOSITES AT ELEVATED TEMPERATURES

Base alloy

Nominal composition Form

Heat % treatment particulate

6061

Mg Si Cu Cr

1.0 0.6 0.2 0.25

Extrusion T6

2014A Cu Mg Si Mn

4.4 0.7 0.8 0.75

Extrusion T6

15% Al2 O3 15% Al2 O3 15% Al2 O3 15% Al2 O3 15% Al2 O3 15% Al2 O3 15% Al2 O3 15% Al2 O3 15% Al2 O3 15% Al2 O3 15% Al2 O3 15% Al2 O3 15% Al2 O3 15% Al2 O3

0.2% proof Temperature stress ◦C MPa

Tensile strength MPa

References

22 93 150 204 260 316 371 22 93 150 204 260 316 371

359 331 303 262 179 117 69 503 (483) 490 (434) 434 (379) 338 (310) 214 (172) 110 (76) 55 (41)

1 1 1 1 1 1 1 1 1 1 1 1 1 1

317 290 269 241 172 110 62 476 (413) 455 (393) 407 (352) 317 (283) 200 (159) 103 (62) 55 (35)

Figures in parentheses are for basic alloy without particulate.

Table 37.5

Base alloy

MECHANICAL PROPERTIES OF MAGNESIUM ALLOY COMPOSITES AT ROOM TEMPERATURE

Nominal composition Form

ZK60A Mg 5.5 Zn 0.5Zr

MG 12Li

Table 37.6

Base alloy Ti-6Al-4V

Extruded rod

Squeeze infiltration Squeeze infiltration Squeeze infiltration Extruded rod Extruded rod

% reinforcement

0.2% proof Tensile Elastic stress strength Elongation modulus MPa MPa % GPa References

Nil 15% SiC (partic.) 20% SiC (partic.) 15% SiC (whisker) 20% B4 C (partic.) Nil

260 330 370 450 405 —

325 420 455 570 490 80

15.0 4.7 3.9 2.0 2.0 8.0

44 68 74 83 83 —

5 5 5 5 5 7

12% Al2 O3 (fibre)

—

200

3.5

—

7

24% Al2 O3 (fibre)

—

280

2.0

—

7

Nil

—

75

10.0

45

7

20% SiC(whisker)

—

338

0.8

112

7

MECHANICAL PROPERTIES OF TITANIUM ALLOY COMPOSITES

Form

Particulate Particulate Particulate Particulate Particulate Ti-6Al-4V SCS6 fibre Ti-15Al-3V SCS6 fibre

0.2% proof % Temperature stress ◦ MPa reinforceant C 10% TiC 10% TiC 10% TiC 10% TiC 10% B4 C 35% SiC 35% SiC

21 427 538 649 21 20 20

Figures in parentheses are for basic alloy without reinforcement.

Tensile strength MPa

800 806 476 (393) 524 (510) 414 (359) 455 (441) 369 (221) 317 (310) — 1 055 (890) — 1 634 (1 024) — 1 572 (772)

Elastic Elongation modulus % GPa 1.13 1.70 (11.6) 2.40 (8.5) 2.90 (4.2) — 1.1 0.7

References

106–120 4 — 4 — 4 — 4 205 5 213 15 210 15

37–6

Metal-matrix composites

Table 37.7

MECHANICAL PROPERTIES OF ZINC ALLOY COMPOSITES AT ROOM TEMPERATURE

Base alloy ZA-12

Nominal composition Zn-12% Al

As cast As rolled Extruded

ZA-27

Zn-27% Al

As cast

As rolled Extruded

Form

Tensile strength MPa

Elongation %

Elastic modulus GPa

References

Nil Nil 10% SiC 20% SiC Nil 20% Al2 O3 Nil 10% SiC 20% SiC 50% SiC Nil 10% SiC 20% SiC Nil 12% SiC 16% SiC Nil 20% Al2 O3

300 350 323 373 313 532 410 396 330 310 393 370 349 340 382 466 314 382

5 22 0 0 — — 2 0 0 0 16 3 0 — — — — —

87 115 119 129 102 120 73 92 110 220 85 89 102 83 105 130 78 100

8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8

MECHANICAL PROPERTIES OF CO-DEFORMED COPPER COMPOSITES AT ROOM TEMPERATURE

Table 37.8 System

Ultimate strength MPa

Drawing strain

References

Cu-15wt% Cr Cu-18vol% V Cu-30vol% Fe Cu-12vol% Nb Cu-12vol% Nb Cu-12vol% Nb Cu-12vol% Nb Cu-20vol% Nb Cu-20vol% Nb Cu-20vol% Nb Cu-20vol% Nb

906 1 850 1 500 400 810 1 180 1 520 430 970 1 410 1 840

7.0 11.0 8.5 3.1 8.2 10.3 11.9 3.1 8.2 10.3 11.9

17 18 18 19 19 19 19 19 19 19 19

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.

Private communication, Alcan Aerospace, 1989. C. Baker, Alcan International, 1990. F. H. Froes, Materials Edge, May/June, 1989, 17. S. Abkowitz and P. Weihrauch, Adv. Mat. and Proc., 1989 (7), 31. F. H. Froes and J. Wadsworth, BNF 7th Int. Conf., 1990, Paper 1. J. White et al., in ‘Adv. Mat. Tech. Int.’, ed. G. B. Brook, publ. Sterling Publ. Ltd, 1990, 96. J. F. Mason et al., J. Mat. Sci., 1989, 24, 3934. J. A. Cornie et al., ASM Conf. on Cast Metal Matrix Composites, Chicago, Sept. 1988, 155. R. Warren, in ‘Adv. Mat. Tech. Int.’, ed. G. B. Brook, Publ. Sterling Publ. Ltd, 1990, 100. J. Yang and D. D. L. Chung, J. Mat. Sci., 1989, 24, 3605. T. Iseki, T. Kameda and T. Maruyama, J. Mat. Sci., 1984, 19, 1692. S. P. Rawal, L. F. Allard and M. S. Misra, in ‘Interfaces in Metal-Matrix Composites’, Proc. Symp. AIME, New Orleans, March, 1986 (ed. A. K. Dhinga and S. G. Fishman), 211. A. R. Champion et al., Proc. Int. Conf. on Composite Mat., AIME, ICCM, 1978, 883. M. H. Richman, A. P. Levitt and E. S. DiCesare, Metallography, 1973, 6, 497. S. M. Jeng, J. M. Yang and C. J. Yang, Mat. Sci. Eng. A, 1991, A138, 169. A. Parvizi-Majidi, in Mat. Sci. & Tech.—A Comprehensive Treatment, (Ed. R. W. Cahn, Phaasen, E. J. Kramer), VCH, Weinheim, 13, 1994, pp 27–88. K. Adachi, T. Sumiyuki, T. Takeuchi and H. G. Suzuki, J. Jap. Inst. Metals, 1997, 61, 397. A. M. Russell, S. Chumbley and Y. Tian, Adv. Eng. Mat., 2000, 2, 11. W. A. Spitzig and P. D. Krotz, Scripta Metal, 1987, 21, 1143.

38

Non-conventional and emerging metallic materials

38.1 Introduction Metallic materials that are outside the mainstream of conventional metals and alloys, with respect to composition, structure and/or processing, are discussed in this chapter. The materials considered are as follows: a. b. c. d.

structural intermetallic compounds; metallic foams; metallic glasses; micro- and nanoscale materials;

These materials are in different stages of development. Metallic glasses are technologically mature materials, with long-established commercial applications, although there continue to be exciting developments in the area of bulk metallic glasses. Metallic foams are the subject of a quite intense research effort at the time of writing, but have already found commercial success in a number of markets. Structural intermetallic compounds have been the subject of major world-wide research activities dating back over many years. However, significant barriers remain to the widespread commercial implementation of these materials. Metallic nanomaterials feature in much contemporary research and the details of these are still very much subject to change. However, work on the fundamentals of these materials has reached a point where this can usefully be summarised.

38.2 Cross references The following contain information that is relevant to the contents of this Chapter: a. b. c. d. e. f.

crystallographic data on intermetallics—Chapter 6; shape memory alloys—Chapter 15; thermoelectric materials—Chapter 16; superconducting materials—Chapter 19; magnetic materials—Chapter 20; metal matrix composites—Chapter 37.

38.3 Structural intermetallic compounds Intermetallic compounds are employed commonly to strengthen conventional alloys. Two examples are: • Age-hardenable aluminium alloys, strengthened by a variety of intermetallic phases, as discussed

in Section 29.3.

• Nickel-base superalloys (for which an overview can be found in reference 1), with a γ/γ ′

microstructure. In this case γ represents a nickel-base solid solution and γ ′ is the intermetallic phase Ni3 (Al,Ti).

More recently, intermetallic compounds have attracted great interest as monolithic (stand-alone) structural materials, for high-temperature applications. Table 38.1 summarises some of the

38–1

38–2

Non-conventional and emerging metallic materials

SOME EXAMPLES OF INTERMETALLIC COMPOUNDS OF INTEREST FOR HIGH-TEMPERATURE APPLICATIONS

Table 38.1

Phases listed in brackets, within a given system, are relatively technologically immature. This table does not represent an exhaustive list, especially with respect to the ‘exotic intermetallics’. For example, numerous ternary compounds are not included. System Aluminides Ni Ti Fe Refractory metal Silicides

Compounds of interest Ni3Al, NiAl Ti3Al, TiAl, (TiAl3 ) Fe3Al, FeAl NbAl3 etc. MoSi2 , (NbSi2 ), (TiSi2 ), (CrSi2 ), (Co2 Si), (Ti5 Si3 ) etc.

intermetallics that have been considered as structural materials (see [2] for discussion of the screening criteria needed for selecting structural intermetallic compounds). Much of the interest in structural intermetallics has centred on the aluminides of nickel (Ni3Al and NiAl) and titanium (Ti3Al and TiAl). However, other aluminides feature prominently in the literature on intermetallics. In particular, interest has focused on the iron aluminides Fe3Al and FeAl (see some of the general references in the nickel aluminide section). However, other aluminide systems, for example refractory metal aluminides (e.g. [3–7]), have also been investigated. There is also a body of work on non-aluminide phases, such as silicides, in particular MoSi2 (see e.g. [8, 9] for overviews). Nonetheless, in the interests of brevity, the discussion in this section is confined to the nickel and titanium aluminides. 38.3.1 Sources of information on structural intermetallic compounds General overviews of intermetallic compounds, may be found in [10–14], including a historical perspective in the case of the first of these references. Other useful sources of collected work on structural intermetallics are the proceedings of the following: • The Materials Research Society (MRS) has sponsored a long-running symposium series,

‘High-Temperature Ordered Intermetallic Alloys’.15

• The Minerals, Metals and Materials Society (TMS), in some cases in conjunction with ASM

International, has organised a number of conferences and symposia on structural intermetallics. An example of these is the ‘International Symposium on Structural Intermetallics’ series.16

A useful compilation of mechanical and other property data, collected up to the early to mid 1990s, for aluminides and other intermetallics, may be found in [17]. 38.3.2 Focus of the section At the time of writing, there has been nearly 20 years of intense research effort in the US, Europe, Japan and elsewhere on structural intermetallics and sporadic research dating back far longer. Following this work, the general features of the leading structural intermetallic systems are now relatively mature, for good or ill. In contrast, the detailed compositions and processing paths for these materials are still evolving. Hence, this section concentrates on the general features of these materials, and a large body of alloy-specific numerical data are not included (since the latter are still subject to change). 38.3.3 The nature of ordered intermetallics In a solid-solution (A–B), the solute atoms (B) sit on random sites in the solvent (A) lattice. Small solute atoms (e.g. C in Fe) become interstitials and larger solute atoms (e.g. Cr in Fe) substitute for the solvent atoms. In contrast to a solid-solution, in an intermetallic compound (Ax By ) the constituent (A and B) atoms occupy well defined sites within the structure of this phase. Hence, intermetallic

Structural intermetallic compounds

B2

38–3

L12

A

B

L10 Figure 38.1 The crystal structures of NiAl (B2), Ni3Al (L12 ) and TiAl (L10 ). Crystallographic data for these phases (and many other intermetallics) may be found in [18]. A significant body of crystal structure data is available in Chapter 6

compounds are referred to as ‘ordered’ and random solid-solutions as ‘disordered’.∗ For example, Al atoms occupy random substitutional sites in the face centred cubic (FCC) Ni–Al solid solution. In contrast, ordering of this solid solution, to form the intermetallic compound Ni3Al, involves placing the aluminium atoms at the cube corners and the nickel atoms at the face centre sites of the unit cell (Figure 38.1). Note, this ordering process results in a change in Bravais lattice and Ni3Al is primitive cubic, not FCC.18 Ordering is often described in terms of the production of a ‘superlattice’ that is superimposed on the disordered phase. Not all ordered intermetallic phases have crystal structures that can be related to those of their constituent metals. Consider an example.The Ni–Al phase diagram (which can be found in Chapter 11), contains several intermetallic compounds, including the phase NiAl. NiAl has the CsCl structure (Figure 38.1), whereas both nickel and aluminium are FCC. Given the free electron model of metallic bonding, it might seem at first sight that metals of the same crystal structure should be fully mutually soluble (and this can be the case, as in the Ni–Cu system, for example). However, the Hume-Rothery rules (see e.g. [19, 20]) describe the circumstances under which solid-solubility is limited. These circumstances and the resulting phases are as follows: a. Size misfit between the solvent and solute atoms. This leads to a ‘size factor’ compound. Examples include Laves phases, such as MgZn2 and MX type interstitial compounds, for example Ti(C,N). b. Large differences in electronegativity between the solvent and solute. In this case, there will be a tendency towards ionic bonding. Thus ionically bonded compounds, such as Mg2 Sn can form. ∗ The

term ‘disordered’ in the present context refers to a random solid solution. Confusingly, however, the same term is sometimes applied to amorphous metals (see Section 38.5), even though a disordered solid solution possesses a well defined Bravais lattice and an amorphous metal does not.

38–4

Non-conventional and emerging metallic materials

c. Unsuitable valence electron (e) to atom (a) ratios. The formation of ‘electron compounds’ occurs at specific e/a values. For example, there is a large class of 1.5 e/a electron compounds, sharing the B2 structure of ordered β brass. Note, only in case ‘b’ above will the normal rules of valency be followed in the chemical formula of the intermetallic. A detailed discussion of the nature of solid solutions and the breakdown in solubility may be found in [21]. The same volume also includes a useful discussion of the structure of intermetallics.22 Regardless of the circumstances leading to ordering, a key point is that the formation of an intermetallic will result in unlike nearest neighbour atoms (e.g. in NiAl, the nearest neighbour of each aluminium atom is a nickel atom and vice versa). Thus, intermetallic formation is favoured when the energy of unlike neighbour bonds is lower than that of like neighbour bonds (the opposite is also possible, in which case cluster formation will occur), so that: ordering is favourable if: clustering is favourable if: the solid-solution remains random if:

2EAB < EAA + EBB 2EAB > EAA + EBB 2EAB = EAA + EBB

Note, however, that bonding in intermetallics can be quite complex and the reader is referred to work on structural modelling of intermetallics (see [23] and specific references in the text; Chapter 39 of the present work contains an overview of the modelling of materials). Some intermetallics remain strongly ordered up to their melting point (for example NiAl), whereas other intermetallics (such as those in the Au–Cu system) undergo disordering at high temperatures, as can be seen by perusal of the phase diagrams contained in Chapter 11. The transition from ordered to disordered may be sudden (as in Cu3Au, for which a well defined disordering temperature is seen) or gradual (as in the case of disordering of β-brass), depending on the system involved. The degree of order is usually represented by the long-range order (LRO) parameter (S), which is defined24 as follows: S=

p−r 1−r

where p = fraction of species X on X sites r = atom fraction of species X in the alloy. Thus, for a fully ordered alloy, S equals unity and the value of S falls as the degree of order decreases. In cases where a solid-solution is not completely random, but does not possess a welldefined superlattice, the short-range order (SRO) parameter (σ) can be employed. The SRO parameter (see [25] for the original discussion), at a given temperature, is defined as: σ=

q − qr qm − qr

where q = fraction of unlike nearest neighbour atoms observed qr = fraction of unlike nearest neighbour atoms that would be present if the solid-solution were completely random qm = fraction of unlike nearest neighbour atoms that would be present under the condition of maximum possible order. Two important features of ordered intermetallic compounds are antisite defects and antiphase (domain) boundaries (APBs or APDBs). Consider a notional intermetallic AB. In this case, an antisite defect would be an A atom on a B site, or vice versa. An APB would be formed, as illustrated in Figure 38.2 (for early work on the observations of antiphase domains, see [26, 27]). The domains at the top and bottom of the figure are both perfectly ordered. However, there is an antiphase displacement between the ordering of these two domains. As a result, AA and BB bonds are formed at the boundary between the domains (i.e. the APB). In the event that the intermetallic is strongly ordered, the creation of an antisite defect will carry a large energy penalty and the APB energy (APBE) will be high, in some cases to the point where APBs can not be produced. Note, however, in intermetallics with a wide compositional range, the presence of structural antisite defects provides a possible mechanism for accommodating deviations

Structural intermetallic compounds

38–5

A B

Figure 38.2 Schematic representation of an APB (the plane of the APB corresponds to the dashed line at the centre of the figure)

from stoichiometry, as an alternative to the formation of structural vacancies. Such structural antisite defects can be a normal part of the structure of a non-stoichiometric intermetallic and can form even in strongly ordered alloys (this is discussed further, with respect to NiAl, in Section 38.3.6 below). In long period superlattice structures, such as those that form in the Au–Cu system, regularly spaced APBs form an integral part of the structure of the intermetallic (a concise discussion of these and other superlattice structures is contained in [28]). However, in the case of intermetallics of interest as structural materials, APBs are of the greatest importance with respect to their role in the behaviour of dislocations. Issues related to both antisite defects and APBE values are discussed further in the next section and in more detail (again with respect to NiAl) in Section 38.3.6. 38.3.4 The effect of ordering on the properties of intermetallics In cases where there is a strong affinity between the constituent species, the melting point of an intermetallic can be significantly higher than that of any of its constituent elements. For example, as can be seem from the Ni–Al phase diagram in Chapter 11, the melting point of stoichiometric NiAl is nearly 200◦ C higher than that of nickel and far exceeds the melting temperature of aluminium. In a very strongly ordered intermetallic: • Diffusion of the constituents of the intermetallic will be impeded, when vacancy—atom inter-

changes lead to the creation of antisite defects. The simplest example of this phenomenon∗ is B2 compounds (as shown in Figure 38.3), such as NiAl. In NiAl, strong ordering forces diffusion to occur via cyclic mechanisms requiring a series of co-ordinated vacancy—atom interchanges.29 Diffusion is a random walk process and so coordinated cyclic jumps have a much lower probability than a single vacancy—atom interchange event. Thus, ordering will significantly reduce

∗ The

effect of ordering on diffusion can be seen most clearly by looking at the effect of temperature on diffusion coefficients in reversibly ordered intermetallics.

38–6

Non-conventional and emerging metallic materials A B Vacancy B2 phase

Vacancy on A sublattice

Antisite defect Figure 38.3 The B2 structure, showing that a single vacancy—atom interchange event will result in the formation of an antisite defect THE EFFECT OF DEVIATIONS FROM STOICHIOMETRY ON THE SELF DIFFUSION COEFFICIENT OF NICKEL (DNi ) IN NiAl, AT A TEMPERATURE OF 1 000◦ C 33

Table 38.2

Ni (at. %)

DNi (m2 s−1 )

50 48.5 58

∼1 × 10−16 7 × 10−16 5 × 10−14

the observed diffusion coefficients∗ Non-stoichiometric intermetallics can have quite different diffusion coefficients than stoichiometric compounds (see Table 38.2). This follows because deviations from stoichiometry are accommodated by constitutional point defects, These constitutional point defects can be either antisite atoms (as in nickel-rich NiAl), or structural vacancies (as in aluminium-rich NiAl—see references in the discussion of NiAl). • The possible Burger’s vectors may be restricted, to those which do not involve the formation of an APB (see Figure 38.4). Hence the number of slip systems may be severely restricted. The operation of the remaining available slip systems may be changed significantly (in some cases such that these are difficult to activate). • Other properties may be affected, for example the rate of work hardening can increase on ordering and recrystallisation may become sluggish (for an overview of the latter, see [34]). The formation of a high melting temperature intermetallic, in which diffusion (and possibly dislocation motion) may be difficult, raises the possibility of employing such materials under conditions ∗

Diffusion of ternary additions in B2 compounds is influenced by the extent to which the alloying addition partitions to one of the two sublattices (i.e. Ni sublattice and Al sublattice in the case of NiAl). An example of recent work on diffusion in B2 compounds with ternary additions can be found in [30]. See [31] for recent work on site occupancy in NiAl and FeAl with ternary additions. The exact character of the intermetallic is very important in determining the extent to which ordering influences diffusion. For example, in the case of L12 type Ni3Al, diffusion on the Al sublattice might be expected to be much more difficult than diffusion on the Ni sublattices, since only the former would disrupt local order. However, in recent work on Ni3Al, the intrinsic diffusion coefficients of Ni and Al were found to be almost the same,32 which suggests that Al can diffuse readily by an antisite mechanism. ¯ In discussing diffusion in an intermetallic Ax By , it is especially important to distinguish between the interdiffusion coefficient (D) and the self diffusion coefficients (DA and DB ). These are related to the atom fractions of A and B (XA and XB ) by ¯ = (XB DA + XA DB ) F D

The value of the parameter F depends on the character of the intermetallic and in some cases can actually be larger than unity.33

Structural intermetallic compounds

B2

38–7

A B Creates APB No APB formed

L12 Figure 38.4 Some possible Burgers vectors in the B2 and L12 systems. Notice that 1/2 B2 and 1/2 L12 lead to the creation of an APB, whereas B2 does not

where creep resistance is required (see [35] for an overview of creep in structural intermetallic compounds). However, it must be cautioned that in cases where dislocation motion is difficult and/or an inadequate number of slip systems operate, problems are experienced with loss of lowtemperature ductility.As will be discussed below, the lack of room-temperature ductility is perhaps the largest single obstacle impeding the widespread commercial application of structural intermetallics. An overview of plastic deformation in intermetallic compounds is available in [36]. 38.3.5 Overview of aluminide intermetallics Much of the work on monolithic intermetallic compounds has aimed at enabling the use of these materials in high-temperature structural applications, typically as an alternative to nickel-base superalloys. The driving force behind this work has largely been the relatively low densities of nickel and titanium aluminide intermetallics, when compared with those of typical nickel-base superalloys (Table 38.3). Given the low density of these materials, some specific properties of aluminide intermetallics, such as specific stiffness, are attractive. However, it should be cautioned that the creep and environmental resistance of these materials do not necessarily rival those of single crystal γ/γ ′ superalloys, as can be seen in Table 38.4 (an overview of oxidation and corrosion of intermetallics can be found in [37]). Note also the wide variation in fracture toughness between these different alloys. These various points will be discussed further in subsequent sections. 38.3.6 Nickel aluminides The Ni–Al binary system contains several intermetallic phases. However engineering interest has focused on Ni3Al and NiAl. The aluminium-rich phases in this system have relatively low melting

38–8

Non-conventional and emerging metallic materials

THE DENSITY (ρ), ROOM-TEMPERATURE YOUNG’S MODULUS (E) AND SPECIFIC STIFFNESS (E/ρ) OF SOME COMMON INTERMETALLIC COMPOUNDS, IN COMPARISON WITH THOSE OF NICKEL-BASE SUPERALLOYS AND TITANIUM ALLOYS

Table 38.3

Material Ni3Al NiAl SC PC Ti3Al S A TiAl S A

ρ (Mg m−3 ) 7.5 ( 5.9

E/ρ (kPa g−1 m3 ) References

180 95–270∗ 170

24 ( 16–46

4.3 4.1–4.7

( 145

( 31–35

3.8 3.7–3.9

( 176

( 45–48

Ni-Base superalloys ( SC ∼8.0–9.0 PC

Ti alloys

E (GPa)

∼4.5–4.9

∼ 125# ∼ 200

105–125

(

38, 39           

38–41

         

14–25

For density data see Chapter 14 For modulus data see Chapter 22 and [39]

21–28

For density data see Chapter 14 For modulus data see Chapter 22

These data are for polycrystalline alloys unless denoted otherwise (SC = single crystal, PC = polycrystalline, S = stoichiometric binary phase, A = typical ternary alloys, ∗ = depending on orientation, # = oriented).

points (and are very brittle) and so are not candidates for structural applications. The structure and properties of Ni3Al and NiAl diverge considerably.

Ni3Al The primitive cubic L12 type structure of γ ′ –Ni3Al phase is the same as that of the similar phase in γ/γ ′ superalloys, although the latter has a composition that is usually closer to Ni3 (Al,Ti), as has already been remarked. In the binary Ni–Al system, Ni3Al has quite a narrow range of composition. However, ternary alloying of this phase is possible (for an overview of partitioning behaviour and phase stabilisation in the γ/γ ′ /β system see [42]). Dislocations in γ ′ travel in the form of pairs.13 Each dislocation in such a pair is of the 1/2 type (but see discussion below on further dissociation). This Burgers vector is the same as that observed in disordered FCC metals and alloys—see e.g. [43] for a concise overview of dislocations in FCC metals). A dislocation with a Burger’s vector of 1/2 is a perfect dislocation in the disordered γ phase. However, with respect to the superlattice of γ ′ , 1/2 is a partial dislocation, since this displacement places Al onto Ni sites (Figure 38.4). The terminology describing such dislocations is far from standardised, but the term ‘superlattice partial’ dislocation will be used here (superpartial is another common term). Given that 1/2 is a superlattice partial, the first dislocation in the pair creates an APB, which is then removed by the second dislocation. Hence, the two 1/2 dislocations in a pair are separated by a ribbon of APB, the formation of which is opposed by the APBE. The spacing between the superlattice partials is determined by equilibrium of the tension, induced by the APB, and the elastic repulsion of the like 1/2 dislocations. Such pairs of dislocations, bound together by APBs, are often referred to as ‘superdislocations’. The APBE in γ ′ is relatively low. Measured and calculated values of the APBE in γ ′ have been summarised in [46].These lie in the range of 90–196 mJ m−2 forAPBs on {100} and 110–283 mJ m−2 for APBs on {111} (compared with stacking fault energies in the range of 5–40 mJ m−2 in the same system). Given the relatively low APBE, the spacing between the 1/2 dislocations can be quite large. Just as in disordered FCC metals and alloys, the 1/2 dislocations dissociate into pairs of 1/6 partial dislocations (these are partials with respect to the fundamental FCC structure and not just to the primitive cubic superlattice). Each pair of 1/6 partials is separated by a ribbon of stacking fault (SF), so that the spacing of these partials is influenced by the SF energy (SFE) in the same fashion as in an FCC metal or alloy. Note, in the case of ordered γ ′ , the result

Structural intermetallic compounds

Table 38.4

38–9

STRUCTURES AND REPRESENTATIVE PROPERTIES OF NICKEL AND TITANIUM ALUMINIDES, IN COMPARISON WITH NICKEL-BASE SUPERALLOYS

See the footnotes to the table for the sources of these data. It should be noted that, within these general classes of material, there are very significant variations in properties depending on the alloy and processing route chosen. Also, some information on the current generation of alloys remains proprietary at the time of writing. Thus, these data should be treated only as a general guide. Likely maximum operational temperature of alloys based on this phase (◦ C)

Type of alloy

Usual notation for major phase

Structure of major phase(s)1

Ni3Al

γ′

L12

NiAl3

β

B2

Ti3Al4

α2

D019

TiAl4

γ

Ni-Base3

γ/γ ′

1 2 3

Due to creep

Due to oxidation2

Representative room temperature properties (polycrystals unless otherwise stated) Type of alloy (see text for details)

Yield stress (MPa)

1 000

>1 100

Microalloyed with B5 Single crystal5

∼240–∼700 1 100

≫1 100

Stoichiometric5,6 Stoichiometric single crystal5,6

180 1 200

760

650

L10

1 1007

α2 + β 9 γ/α10 2

Single crystal7

Data from [18]. For more detailed comments on oxidation see [44]. Data from [40] except where indicated otherwise. 4 Data from [41] except were indicated otherwise. 5 Data from [17]. 6 Data from [45]. 7 Additional data from [39]. 8 See references to multiphase NiAl alloys in text. 9 In this alloy, β indicates a B2 phase, typically stabilised by Nb additions. 10 The best ductility is produced by duplex alloys (equiaxed γ + lamellar γ/α ), whereas fully lamellar (γ/α ) 2 2 materials offer the maximum fracture toughness and creep resistance.

Ultimate tensile strength (MPa) ∼700–∼1 200 ∼200–∼900 ' 260 125–183

Tensile ductility (%) 20–30 10–50 % ∼0 (∼20 for β − (γ)/ (γ ′ ) ternary alloys)8

Fracture toughness √ (MPa m) (

≥60 (alloyed with Cr) % 5–6 (>15 for eutectics)8

700–990

800–1 140

2–10

13–30

400–650

450–800

1–4

10–20

∼ 10

30–35

>1 100

10 MPa m, compared to about 5–6 MPa m for binary NiAl). The poor room temperature ductility of NiAl can be addressed by the addition of a significant volume fraction of a ductile second phase, for example γ ′ or a number of disordered FCC solid solution γ phases (see e.g. [128–139] for a representative selection of the research that has been conducted on this topic). In this case, the second phase serves two functions. Firstly, the ductile phase serves as an ‘adhesive’ that bonds the NiAl together, so that the limited number of slip systems in the latter ceases to be a problem. Secondly, glissile dislocations are ‘injected’ into the β-phase, so that the available slip systems in the β-phase work more efficiently (additional slip systems are not necessarily activated). In these circumstances, it is very important to have a morphology that removes the high angle β–β grain boundaries, that would initiate cracks in the absence of non slip in the β-phase. Also the occurrence of a suitable orientation relationship (i.e. Kurdjumov–Sachs or Nishiyama–Wasserman between β and γ or γ ′ ) facilitates slip transfer to the β-phase. Two phase β–γ ′ alloys can of course be formed in the binary Ni–Al binary system. However, the formation of a β–γ alloy requires ternary alloying, for example with iron. Although useful toughening or ductilisation of NiAl can be achieved by the use of multiphase microstructures of the two types discussed above, there is a price to be paid. Increases in density result from the use of alloys that are nickel-rich and/or have extensive additions of heavy alloying elements. Thus, there is reason to question the extent to which these multiphase alloys would be useful in density critical applications (e.g. aerospace). Furthermore, creep resistance may be compromised as a result of: • the presence of significant volume fractions of low creep resistance second phases, in the form

of contiguous regions.

• moving the β-phase away from stoichiometry.

An additional problem (see e.g. [140–146]) is that the microstructures developed in complex multiphase NiAl alloys can be quite complex and in some cases show limited stability. Those alloys that attempt to combine ductilisation and/or toughening (through the addition of γ or γ ′ or an eutectic involving a body centred cubic metal), with creep resistance (via the formation of β′ ), can be especially problematical with regard to the stability of the desired microstructure. There have been a number of studies (e.g. [147–151]) on micro- and nanostructured NiAl alloys. The intent of these studies was to enhance the fracture toughness of this alloy. In some cases, this work avoided sacrificing the low density of stoichiometric NiAl, whereas a significant portion of this effort focused on nickel-rich materials. In the latter case, there are overlaps with work on γ ′ and martensite-containing NiAl-derived materials, as discussed elsewhere in this section. Although these efforts have met with at least some success, the microstructural stability of these materials at high-temperature and their likely creep resistance remain concerns. For a general overview of grain size effects in intermetallics see [152]. As with much of the work on structural intermetallics, the intended applications of NiAl as a structural material are mostly in aerospace (e.g. gas turbine stators and rotors). However, unless the low temperature ductility/toughness of this phase can be improved, without compromising its low density, it is difficult to envisage large scale structural application of this material. Nickel-rich NiAl will undergo a martensitic transformation,∗ by twinning, to an L10 type martensite. This transformation is thermoelastic and is capable of showing shape memory. The martensite start (Ms ) temperature of NiAl is strongly dependent on composition and ranges from 30 K (–243◦ C) for a 60 at. % Ni alloy to 1 146 K (873◦ C) for 69 at. % Ni material.161 Thus NiAl offers Ms temperatures that are far higher than for a commercial shape memory alloy (SMA), such as NiTi (Nitinol). Hence, there has been some interest in using NiAl as a high-temperature SMA.162–166 However, the maximum recoverable strain from the shape memory effect (SME) in NiAl is very low (≪1 %). Also both the diffusional and diffusionless phase transformations in nickel-rich NiAl are quite complex (see [167–200] for some of the literature on this topic) and many of the possible product phases do not show does have somewhat better fracture toughness √ the SME. The L10 martensite √ (KIC > 10 MPa m versus around 5–6 MPa m) than stoichiometric NiAl. Indeed, the formation of ∗

In common with many other phases based on the B2 type β CuZn structure (see e.g. [153, 154,]), even stoichiometric NiAl possesses a modulated ‘tweed’ microstructure. This microstructure involves small displacements from the nominal lattice of the β-phase (not to be confused with spinodal decomposition, which can occur in some NiAl-derived alloys). An important cause of this ¯ shear. In NiAl (and some other β phases), this modulation is that the β phase is intrinsically unstable with respect to {110} instability is manifested in the formation of shear displacement waves, with propagation vectors of and polarisation vectors ¯ of . Notwithstanding the shear displacements, there have been suggestions that the tweed microstructure is not necessarily a precursor of the martensitic transformation of β. Some of the literature on the tweed microstructure can be found in [155–160]. Information on the elastic properties of NiAl is available in [160].

38–14

Non-conventional and emerging metallic materials

martensite been considered as a possible toughening route for NiAl (see references in the discussion of micro- and nanostructured NiAl alloys). Nonetheless, the addition of a significant volume fraction of non-shape-memory ductilising second phase is still likely to be needed. This will further dilute the already low recoverable strain obtainable with NiAl. 38.3.7 Titanium aluminides The intermetallic phases in the Ti–Al system that have attracted major interest as structural materials are D019 type α2 –Ti3Al and L10 type γ–TiAl, both of which have a significant range of composition in the Ti–Al binary system (see Chapter 11). TiAl3 is D022 in the Ti–Al system, but an L12 structure can be stabilised by ternary alloying (see e.g. [201]). The use of TiAl3 would be attractive from the standpoint of both density (3.37 Mg m−3 ) and oxidation resistance (since this phase contains enough aluminium to be a reliable alumina former). However, alloys based on TiAl3 are very brittle (and seem likely to remain brittle for the foreseeable future) and so will not be discussed here. See for example [202] for work on dislocations in L12 stabilised TiAl3 . The interest in the titanium aluminides has primarily been driven by potential applications in aerospace, both as engine components and airframe materials. Unlike the nickel aluminides, the titanium aluminides do not offer a sufficiently high operating temperature to compete with nickelbase superalloys in applications such as aero gas turbine blades and guide vanes. Instead, the titanium aluminides are largely intended to serve in applications at intermediate temperatures (2 ∼0.8–2

E =Young’s modulus, σyc = compressive yield stress, σc = compressive strength, σt = tensile strength, εe = elastic elongation to failure, εpc = compressive plastic strain to failure.

Mechanical behaviour of micro and nanoscale materials

38–23

437

These materials can also have unusual magnetic properties. For some recent examples of work in the area of amorphous–nanocrsystalline composites, see.438–443 Research has also been conducted444 on composites with a BMG matrix, plus high volume fractions (up to ∼50 vol. %) of coarse crystalline metallic particles (with particle sizes up to 200 µm). There has also been interest in metal fibre reinforced BMG matrix composites.445 As with any metallic glass, BMGs are metastable and will crystallise if heated to a suitable temperature. Depending on the intended application, this can be a problem (if unwanted devitrification occurs during processing or service), or a benefit (as in controlled devitrification to produce the composite materials discussed above). Some recent research on crystallisation and the thermal stability of BMGs can be found in [446–452]. In addition to thermally-induced devitrification, it is also possible to precipitate metastable nanocrsystalline phases under electron irradiation.453 It should be cautioned that low-temperature annealing, below the glass transition temperature (Tg ) can lead to embrittlement in some cases.454 For work on the effect of heat treatment on the mechanical properties of BMGs, see for example [455].

38.6 Mechanical behaviour of micro and nanoscale materials Until recently, the field of mechanics of materials has dealt mainly with how solids behave when acted upon by static and dynamic forces. The resulting interplay of stress and strain in controlling how materials deform is organised into a continuum describing material behaviour. However, with the advent of microelectronics and micro-electro-mechanical systems (MEMS), traditional continuum mechanics has been unable to explain the deleterious effects that stresses cause in thin film devices. Consequently, considerable work has been carried out towards investigating and understanding the origin of distinctive thin film behaviour as well as determining means to minimise and control its effects. In addition to stresses in thin films, recent advances in nanotechnology and the arrival of carbon nanotubes have again challenged traditional theories describing material behaviour. In these systems, typically one or more of the structure’s dimensions is in the order of 1 to 100 nm in size, and so the mechanisms controlling deformation revolve around interatomic (or molecular) forces rather than defect generation, density and mobility. The study of these issues no longer lies only in the field of mechanics but has become cross-disciplinary and must borrow from other fields, such as physics and materials science. This section is focused on introducing the reader to recent advances in the micro and nanomechanics field and providing a basic understanding of the issues involved therein. 38.6.1 Mechanics of scale It has been known for quite some time that materials and structures with small-scale dimensions do not behave mechanically in the same manner as their bulk counterparts. This aspect was first observed in thin films where stresses were found to have deleterious effects on the film’s structural integrity and reliability. This occurs as one of a specimen’s physical dimensions begins to approach that of the microstructural features. The material’s mechanical properties then begin to exhibit a dependence on the specimen size. In metallic thin films, this translates to plastic yielding occurring at increased stresses, over their bulk counterparts. Although this phenomenon was observed as early as 1959,456 no consensus or common basic understanding of it has yet been arrived upon. Besides plastic behaviour, other mechanical properties, such as fracture and fatigue also exhibit size effects. Each of these properties operates on a characteristic length scale that can be compared with the characteristic dimensions of microelectronics, microdevices or nanodevices. This is shown schematically in Figure 38.8, which utilises a logarithmic length scale map beginning at the atomic scale and onwards up to the macro world. On the left of Figure 38.8 are four categories of devices and the regime where their dimensions fit on the length scale. On the right are regimes indicating where dimensional size-effects begin to affect the material’s mechanical properties and theories used to predict behaviour. Elastic properties are dependent on the nature of bonding of the material and only exhibit size effects at the atomic scale. In contrast, plastic, fatigue and fracture properties all exhibit size effects in the micrometre and submicrometre regime. These properties all depend on defect generation and evolution, the mechanisms of which operate on characteristic length scales.457 In terms of devices, these fundamental changes in mechanical behaviour occur in the size scale of the MEMS and microelectronics regimes, and thus, a better understanding of inelastic mechanisms is required to better predict their limits of strength. The right side of Figure 38.8 lists the theories used to predict behaviour and the regimes on the length scale where they are applicable. These include classical plasticity at the upper end of the scale and dislocation/interatomic mechanics at the lower end. Classical plasticity is described as traditional

38–24

Non-conventional and emerging metallic materials Macro world 10 1 Conventional components

m

Classical plasticity

10⫺1

• Continuum mechanics • No internal length-scale parameter

10⫺2 10⫺3 MEMS devices

10⫺4

Fracture

10⫺5 10⫺6

Microelectronics and thin films

mm

Fatigue ␮m Plasticity

10⫺7 Dislocation mechanics

10⫺8 10⫺9 Nano devices

Figure 38.8

10⫺10

nm o

Elasticity

A Atomic world

Molecular dynamics and quantum mechanics • Current computational power can only handle systems approximately 106 atoms in size

Illustration of length-scale effects on the mechanical properties of materials

continuum mechanics, which uses traditional relationships describing the interplay between stress and strain and is applicable for predicting behaviour from a range of approximately 100 µm and up. Dislocation mechanics (or molecular mechanics) is at the other end of the scale and involves the generation, mobility, and interaction between individual dislocations. The latter is applicable only at the lower end of the scale since it is based on numerical simulation and is consequently limited by current computational power, i.e. systems approximately one million atoms in size. Also, at the lower end is intermolecular mechanics that deals with the interatomic potentials between bonded atoms. In the region between classical and dislocation/interatomic mechanics is a grey area that continues to elude prediction of behaviour. This is a critical region where many size effects fit on the length scale. There has been solid work in the field by a number of researchers who have developed theories and models to explain thin film behaviour.458–463 Unfortunately, these tend to address the subject within a particular range of length scale and some with only particular geometrically induced effects. An additional issue also stems from a lack of experimentally resolved real-world phenomena in the literature. Moreover, no consensus has yet been reached on exactly how the length scale affects the mechanical properties of thin films, e.g. failure mode and fracture toughness, yield strength, etc. The higher yield point of metallic thin films mentioned above is likely the result of a combined interaction of strain hardening and constraint of dislocation motion due to specimen size 464 and/or an effect of grain boundary dislocation sources 465 and geometrically necessary dislocations.466 Clearly, a better understanding of inelastic mechanisms is required to better predict limits of strength in this region. Several pioneering studies have identified experimentally the existence of size effects on the plasticity of metals.467–471 These include: nanoindentation data showing a strong size effect in finding that hardness decreased as indentation depth increased;468,472–479 bending metal strips of varying thickness around a rigid rod of scaling diameter whereby when strained to the same degree the thinner strips required a larger bending moment471 and applying a torque load to copper rods of varying diameter whereby an identical applied twist resulted in an increase in strength by a factor of three for the smallest wire over the largest.470 In these pioneering studies, the size dependence of the mechanical properties has been considered to be a result of non-uniform straining.470,475,480 It was shown that classical continuum plasticity cannot predict the size dependence under these conditions. A new theory termed strain gradient plasticity has been proposed to describe this behaviour, but is only applicable under particular geometrically induced effects.461–463,470,480–483 However, a comprehensive theory that predicts behaviour in this intermediate region continues to elude researchers.

Mechanical behaviour of micro and nanoscale materials Nanoindenter wedge tip

Suspended membrane

Mirau objective

Figure 38.9

38–25

(100

) Si

waf

er

3-D schematic view of the membrane deflection experiment (MDE)

38.6.2 Thin films Although thin films have been employed widely in protective coatings, microelectronics and microelectro-mechanical systems, little is understood about their mechanical behaviour. This deficiency stems from difficulties in performing actual mechanical tests on these microscale structures. Since the physical dimensions of thin films are of a very small magnitude (from a few hundred µm down to as small as 1 nm), conventional mechanical testing methods have not been successful at measuring properties. Specimens in this size range are easily damaged through handling and they are difficult to position to ensure uniform loading along specimen axes. They are also difficult to attach to the testing instrument’s grips, in a consistent fashion. Testing has been shown to suffer from inadequate load resolution as well as having analytical solutions that are highly sensitive to precise dimensional measurements. Several small-scale testing techniques have been employed to investigate size effects on mechanical properties. A number of reviews detail the particulars of these methods.459,484–486 One of the more noteworthy micro-scale tensile tests, called the membrane deflection experiment (MDE), was developed and involves the stretching of freestanding, thin-film membranes in a fixed-fixed configuration with sub-micrometre thickness.487–490 In this technique, the membrane is attached at both ends and spans a micromachined window located beneath the membrane (see Figure 38.9). A nanoindenter applies a line-load at the centre of the span to achieve a deflection. Simultaneously, an interferometer focused on the lower side of the membrane records the deflection. The geometry of the membranes is such that they contain tapered regions to eliminate boundary-bending effects. The result is direct tension, in the absence of strain gradients, of the gauged regions of the membrane, with load and deflection being measured independently. The MDE test has certain advantages, for instance, the simplicity of sample microfabrication and ease of handling lend confidence to repeatability. The loading procedure is straightforward and accomplished in a highly sensitive manner, while preserving the independent measurement of stress and strain. It can also test specimens of widely varying geometry, with thicknesses from sub-micrometre to several micrometres and width from one micrometre to tens of micrometres. Recent work, employing the MDE technique, on tensile testing of thin gold films of submicrometre thickness has shown that size effects do indeed exist in the absence of strain gradients.487–490 In these studies, grain size was held constant at approximately 250 nm while specimen thickness and width were varied systematically. Figure 38.10(a) consists of diagrams that show the side view of the three studied membranes with different thicknesses. Each thickness has a characteristic number of grains composing the thickness. Stress–strain plots for these films are shown in Figure 38.10(b) and show that the yield stress more than doubled when film thickness was decreased, with the thinner specimens exhibiting brittle-like failure and the thicker samples a strain softening behaviour. It is believed that these size effects stem from the limited number of grains through the thickness, which limits the number of dislocation sources and active slip systems; hence, other deformation modes such as grain rotation and grain boundary shearing accompanied by diffusion become dominant. These data show great promise of an ability to test ever-smaller specimens, generating information that is expected to have a great impact on the development of micro and nanoscale devices. When

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Non-conventional and emerging metallic materials 400

0.3 ␮m Stress (MPa)

0.5 ␮m

t ⫽ 0.3 ␮m t ⫽ 0.5 ␮m t ⫽ 1.0 ␮m

350 300 250 200 150 100

1.0 ␮m

50 0 0

(a)

(b)

0.002 0.004

0.006 0.008 0.010

Strain

Figure 38.10 Thin film size effects (a) Schematic representation of the number of grains within the film thickness (b) Stress–strain plot for gold membranes, 0.3, 0.5 and 1.0 µm thick 488

coupled with finite element modelling, this information should be able to provide an accurate description of nanoscale structures and features, in order to predict behaviour. These sorts of data are important in exploiting micro and nanoscale properties in the design of more reliable devices with increased functionality. A compilation of mechanical behaviour for various thin film materials is given in [491]. 38.6.3 Nanomaterials Nanomaterials have been receiving increased attention in recent years due to the promising and unique properties that result from their distinctive structure. They are described physically as materials possessing a nominal grain size in the range of 1 to 100 nm, a condition where the interface to volume ratio, or in other words the grain boundary area, greatly increases over traditionally coarsegrained systems. These characteristics are expected to influence strongly a material’s chemical and physical properties. Thus, the major focus of study in the nanomaterials field is how properties change as grain size is reduced to the nanometre scale. Recent reviews on nanomaterials can be found in the literature.492–495 By controlling the grain size, properties such as the hardness, wear and fatigue resistance, ductility, electric resistivity, magnetic properties, etc. can be optimised.492–498 An example of these improvements is the yield strength increase observed with decreasing grain size in the Hall-Petch relationship.499,500 This relationship which predicts realistically increases in material strength as grain size is reduced, is as follows: k σy = σ0 + √ d

(1)

where σy is the yield strength, σ0 is a friction stress below which dislocations will not move, d is the grain diameter and k is an experimental constant. A similar relationship exists for material hardness. This expression predicts, that as the grain size is reduced, the yield strength will increase. This behaviour has indeed been demonstrated experimentally in many material systems. Extrapolation of the Hall-Petch relationship to the nanometre scale predicts an enormous increase in strength and hardness. However, experimental measurements of these properties actually fall well below the extrapolation. The understanding of relationships between nanostructure and mechanical properties of nanocrystalline materials is in its infancy however, and much research remains to be accomplished. Several models have been developed to account for Hall-Petch behaviour;501–504 graphical representations of three leading models are given in Figure 38.11. The first is a traditional approach and involves the pile-up of dislocations at the boundary of a grain.501 A grain boundary is considered the ultimate obstacle for a dislocation to overcome. Thus dislocations generated in a grain, in response to an applied stress, will travel along slip planes until a boundary is met, whereby, the dislocations then arrest and begin piling-up. After a critical number of dislocations pile-up, a number dependent on the misorientation of the neighbouring grain sharing the boundary, stress in the boundary causes dislocations to arise in the neighbouring grain. This allows yielding to propagate across the boundary. This theory is applicable only when the grain size is large enough to support the generation of at least two dislocations. Further grain refinement would then result in a plateau of strength or

Mechanical behaviour of micro and nanoscale materials s

s

s

(a)

s

s

(b)

38–27

(c)

s

Figure 38.11 Schematic representations of models proposed to explain Hall-Petch behaviour, (a) Cottrell,501 (b) Li 502,503 and (c) Meyers 504

1 000

100 nm

25 nm

10 nm Chokshi et al. Kumar et al. Valiev et al. Chokshi et al. Gertsman et al. Sanders et al. Suryanarayanan et al. Neimann et al. Hayashi et al. Feltham & Meakin Linear Hall-Petch fit for Present analysis Present analysis

t [MPa]

800 600 400 200 0 0.0

0.1

0.2 (d [nm])⫺1/2

0.3

0.4

Figure 38.12 Compilation of yield stress data on nanocrystalline copper on a Hall-Petch plot. Note that the references given in the plot are from [506]

even a reduction,505 the so-called negative Hall-Petch behaviour. Other theories include dislocation generation at grain boundary ledges 502,503 or as a consequence of stress concentrations at grain boundaries resulting from elastic anisotropy.504 It has not yet been agreed upon as to whether dislocation based deformation processes are active in the nanosized grain size regime or are merely suppressed because of the difficulty in activating sources. In moving from coarse-grained to nano-grained materials three types of deformation, based upon the Hall-Petch gradient dσ/d(d −1/2 ), have been documented experimentally. These include: coarse-grained materials exhibiting a constant positive Hall-Petch gradient, a reduction in the positive gradient when a critical grain size is reached and finally the so-called negative Hall-Petch behaviour with a negative gradient.492,494,495 Masumura et al.506 have collected data on the yield stress of nanocrystalline copper from various sources and summarised them on a Hall-Petch plot (see Figure 38.12). The plot shows a continued increase in strength, with decrease in grain size down to a critical size of approximately 15–25 nm. Here, the transition to negative Hall-Petch behaviour is distinct where the difference in strength between the grain sizes can readily be seen. Recent work in molecular dynamics (MD) simulations have also predicted that, at very small grain sizes, further grain refinement would lead to a decrease in strength.507–509 Beyond hardness and strength, other additional aspects of nanocrystalline materials are affected by their distinctive structure. These include elasticity, ductility and toughness, and superplastic behaviour. In terms of material elastic response, the first documented evidence showed that the elastic moduli of nanocrystalline materials can sometimes be only a fraction of their coarse-grained counterparts.510–514 The agreed upon explanation of this is rooted in extrinsic defects such as pores

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Non-conventional and emerging metallic materials

and cracks that result from the compaction of powders. Experimental results on pore-free Ni–P, produced by electroplating, have verified that modulus does return to coarse-grained levels with the removal of pores,515 also confirmed by other studies.513,516 Ductility and toughness of coarse-grained materials has been shown to depend on dislocations and cracks, both their interplay or competition, the effect of which are determined by grain size, whereby a reduction in grain size limits the propagation of each and results in a corresponding increase in fracture toughness.517,518 In nanocrystalline materials, one would then expect further grain refinement to be a simple extrapolation of coarse-grained behaviour. However, experimental results have been inconclusive in this respect, a consequence of specimen inconsistencies such as porosity, flaws, and surface conditions as well as the testing methodology.519 This may be the case for metallic materials, however, ductility has been observed in nanocrystalline ceramic materials such as CaF2 and TiO2 at temperatures as low as 80 to 180◦ C.520,521 Superplasticity is the ability of a material to withstand elongations in excess of 1000% without the occurrence of necking or fracture. Traditionally this occurs in materials at elevated temperatures, T > 0.5 Tm , containing a second phase that inhibits grain growth, and possessing a fine grain size, on the order of 10 µm.522,523 Microscopic examination of these materials has concluded that grain boundary sliding accompanied by considerable grain rotation is the incipient mechanism. As grain size is further refined, so is the temperature where superplasticity occurs. However, superplasticity in nanocrystalline materials has not been observed much below 0.5 Tm , most likely a result of nanocrystalline creep being comparable to coarse grained creep as exhibited by creep measurements in this regime.524 Nevertheless, several experimental investigations have found enhancements in superplastic behaviour of nanocrystalline materials above 0.5 Tm .525–527 38.6.4 Nanostructures As structures move beyond the sub-micrometre to the nanometre scale, description of mechanical behaviour will be focused on concerns other than the traditional ensemble of defects. For instance, the length scale of a typical dislocation and the volume of material required for it to have significant influence on deformation is large compared with the typical nanovolume. Therefore, it can be argued that beyond a certain point, other types of defects, surface forces, and intermolecular processes will control behaviour. Nanostructures can be described as either a grain structure with a nominal size in the range of 1 to 100 nm or structures with one or more dimensions on the same order. An example of a nanostructure is a carbon nanotube, which is a molecular scale fibrous structure made of carbon. Carbon nanotubes, discovered by Sumio Iijima in 1991, are a subset of the family of fullerene structures.528 Note that fullerenes were originally discovered by Curl, Kroto and Smalley, for which they were awarded the 1996 Nobel Prize in Chemistry. The simplest description of their structure would be to imagine a flat plane of carbon graphite rolled into a tube, much the same as a piece of paper (see Figure 38.13). (a)

(b)

Armchair face

(c)

(d)

(e)

Zigzag face Figure 38.13 Schematic drawings of a 2-D graphene sheet (a) rolled-up sheet (b) armchair (c) zigzag (d) chiral (e) nanotubes [(c), (d), and (e) from reference [536]]

Mechanical behaviour of micro and nanoscale materials

38–29

Like the paper, the graphene plane can be rolled in several directions to achieve varying structures. Since their discovery, many scientists have been fascinated by their ensemble of unique properties. A dominant characteristic of nanostructures is that they possess a rather large surface area to volume ratio. As this ratio increases, interfaces and interfacial energy as well as surface topography, are expected to acquire a commanding role in deformation and failure processes. The picture of nanoscale behaviour can be viewed as the following. At the larger end of the length scale, i.e. 50 to 100 nm, dislocation generation and motion will continue to dictate behaviour. As the grain size or structural dimensions proceed below this, behaviour control will transition to surface and intermolecular mechanisms. A few experimental studies detail the tensile testing of carbon nanotubes.529–535 The main difficulties that have to be overcome in the testing of these nanoscale structures are similar to those for thin films; specimen handling, position, gripping and alignment. Young’s modulus was found to be in the TPa regime and the tensile strength was found to be in the 11–63 GPa range, depending on the technique. These rather large numbers are reflective of the nanotubes’ very low defect density. Clearly there are many things still to be learned about materials at small scales. Understanding the mechanics of these materials and structures and the competition and interplay between their deformation mechanisms will be essential to predicting their behaviour. Such knowledge is essential to their application in nanoscale electronics and devices. These types of concerns compose the field of nanomechanics, the foundation of which is currently being laid out. REFERENCES 1. C. T. Sims, N. S. 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L. Wong, D. Ostrander, U. Erb, G. Palumbo and K. Aust, pp 85–93 in Nanophases and Nanostructured Materials (Editors: R. D. Shull and J. M. Sanchez), TMS, Warrendale, PA, 1993.

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Non-conventional and emerging metallic materials

516. M. N. Rittner, J. R. Weertman, J. A. Eastman, K. B. Yoder and D. S. Stone, pp 399–406 in Processing and Properties of Nanocrystalline Materials (Editors: C. Suryanarayana, J. Singh and F. H. Froes), TMS: Warrendale, PA, 1996. 517. R. Thomson, pp 2207–2291 in Physical Metallurgy (Editors: R. W. Cahn and P. Haasen), 4th Edition, Elsevier Science, Amsterdam: 1996. 518. P. Nagpal and I. Baker, Scripta Metall. Mater., 1990, 24, 2381–2384. 519. C. Koch, pp. 93–111 in Nanostructure Science and Technology, R & D Status and Trends in Nanoparticles, Nanostructured Materials and Nanodevices, (Editors: R. W. Siegel, E. Hu and M. C. Roco), KluwerAcademic Publishers, Dordrecht, The Netherlands, 1999. 520. J. Karch, R. Birringer and H. Gleiter, Nature, 1987, 330, 556–558. 521. R. Bohn, T. Haubold, R. Birringer and H. Gleiter, Scripta Metall. Mater., 1991, 25, 811–816. 522. J. W. Edington, K. N. Melton and C. P. Cutler, Prog. Mater. Sci., 1976, 21, 63–170. 523. M. M. I. Ahmed and T. G. Langdon, Metall. Trans. A, 1977, 8A, 1832–1833. 524. P. G. Sanders, M. Rittner, E. Kiedaisch, J. R. Weertman, H. Kung and Y. C. Lu, Nanostructured Materials, 1997, 9, 433–440. 525. R. S. Mishra and A. K. Mukherjee, Mater. Sci. Forum, 1997, 243, 315–320. 526. R. S. Mishra, R. Z. Valiev and A. K. Mukherjee, Nanostructured Materials, 1997, 9, 473–476. 527. V. A. Valitov, G. A. Salishchev and S. K. Mukhtarov, Russ. Metall., 1994, 109–112. 528. S. Iijima, Nature, 1991, 354, 56–58. 529. F. Li, H. M. Cheng, S. Bai, G. Su and M. S. Dresselhaus, Appl. Phys. Lett., 2000, 77, 3161–3163. 530. M. F. Yu, B. S. Files, S. Arepalli and R. S. Ruoff, Phys. Rev. Lett., 2000, 84, 5552–5555. 531. M. F. Yu, O. Lourie, M. J. Dyer, K. Moloni, T. F. Kelly and R. S. Ruoff, Science, 2000, 287, 637–640. 532. D. A. Walters, L. M. Ericson, M. J. Casavant, J. Liu, D. T. Colbert, K. A. Smith and R. E. Smalley, Appl. Phys. Lett., 1999, 74, 3803–3805. 533. Z. W. Pan, S. S. Xie, L. Lu, B. H. Chang, L. F. Sun, W. Y. Zhou, G. Wang and D. L. Zhang, ibid., 1999, 74, 3152–3154. 534. M. B. Nardelli, B. I. Yakobson and J. Bernholc, Phys. Rev. Lett., 1998, 81, 4656–4659. 535. H. D. Wagner, O. Lourie, Y. Feldman and R. Tenne, Appl. Phys. Lett., 1998, 72, 188–190. 536. M. S. Dresselhaus, G. Dresselhaus and R. Saito, Carbon, 1995, 33, 883–891.

39

Modelling and simulation

39.1 Introduction The design of useful materials generally involves the simultaneous optimisation of large numbers of parameters, often in circumstances where the interactions between the parameters are ill-defined. This chapter describes some of the methods by which the task can be enhanced using quantitative models which cover a range of disciplines and which include both rigour and empiricism. From the scientific point of view, modelling can lead to the creation of new theory capable of dealing with complexity; the practical goal is to accelerate the design process and to minimise the use of resources. Each of the following sections is intended to communicate the concepts associated with a particular method; appropriate references are provided where the details can be sought.

39.2 Electron theory A metal is created when the atoms are brought sufficiently close together so that the electrostatic repulsion in transferring a valency electron between adjacent atoms is offset by the gain due to delocalisation of the electron.1 This enables the valency electrons to move within the metal. The essential characteristics of the metallic state can be understood in terms of a single-electron wave function which in one-dimension (x) and for the description of stationary states takes the time-independent form: ψ{x} = C exp{ikx} where k is the wave number, C is a constant. All such wave functions must become zero at the boundaries of the metal so that allowed values of k are discrete, each of which defines a quantum state. The Pauli exclusion principle permits only two electrons with opposite spins to occupy each state. The energy of each state scales with k 2 and the highest occupied state at 0 K is said to have the Fermi energy. The delocalised electrons feel only a weak electrostatic field (‘pseudopotential’), from the positively charged atomic cores. This is because the attraction of the valence electrons to the positive cores is greatly reduced by the repulsion from the core electrons. The delocalised electrons are also partly screened from each other by small exclusion zones (positive holes). It is useful to understand why the electrons are able to move within the potential of the screened positive ion cores without being scattered. Bloch showed that the effect of the periodic potential u{x} is simply to modulate the free-electron wave function by a term u{x}: ψ{x} = u{x} exp{ikx} Difficulties only arise when the wave vector k is such that the wave satisfies the Bragg condition; the electron cannot then, on average, move forward or backward through the reflecting planes. In wave theory this is represented by using two functions corresponding to ±k, one of which places the maximum electron density where the positive potential is lowest, giving a solution with an energy that is lower than that of a completely free electron. Similarly the other solution has an energy higher than that of a free electron. This introduces band gaps in the distribution of electron energies. Electrons are able to move, and hence the metallic state is able to exist, if the valency bands are only partly filled.

39–1

39–2

Modelling and simulation

We have so far considered the metallic state in terms of a single-electron wave function; this fails to account for the vast numbers of electrons that are interacting within the metal. There will be Coulomb correlations between the electrons. Their spins may also correlate. The density functional method which is the basis of much of the modern theory of metals, provides an expression of these effects whilst at the same time exploiting the one-electron equation. The energy E of an electron gas is written as a function of a function (i.e. a functional) in terms of the electron density n and a radial coordinate r:    2 1 e E{n} = V {r}n{r} dr + n{r1 }n{r2 } dr1 dr2 + T {n} + Exc {n} 2 r12 where the first term is the pseudopotential due to the ionic cores, the second term is the classical Coulomb interaction energy (e is the electronic charge), the third term is the kinetic energy (i.e. Fermi energy) and the fourth term accounts for the exchange and correlation effects. The use of this equation, and the adaptations necessary to account for various complications (e.g. to take account of partly covalent characteristics), represents the skill in using electron theory. The most rigourous application of the theory requires the largest amount of computing power and hence is limited to circumstances where there is strict periodicity in the atomic structure so that only small numbers of atoms need to be considered with periodic boundary conditions. It is then possible to calculate, for example, the cohesive energy of arbitrary crystal structures, the elastic properties and surface energies. Some calculations for iron are illustrated in Figure 39.1, which shows the cohesive energy as a function of the density and crystal structure. Of all the test structures, hexagonal close-packed iron (h.c.p.) is found to show the highest cohesion and therefore should represent the most stable form. This contradicts experience, because body-centred cubic iron (b.c.c.) is the equilibrium form at low temperatures. This discrepancy arises because the model ignores magnetic effects—it is ferromagnetism which stabilises b.c.c. iron over the h.c.p. form. The example illustrates how all models begin with intentional or unintentional simplifying assumptions and yet can be useful. The electron theory not only highlights the importance of magnetic terms, but in addition makes it possible to study crystal structures of iron which have yet to be achieved in practice. The diamond form of iron would have a density of only 5 g cm−3 . Unfortunately, the calculations show that the difference in energy between the diamond and b.c.c. forms is so large that it is improbable for the b.c.c. → diamond transformation to be induced by alloying. The methodology has been used to see whether certain intermetallic compounds can be forced to occur into desirable crystal structures.3 ⫺35 Diamond cubic ⫺40

ev/atom

⫺45

Cubic–P

Hexagonal–P

⫺50

b.c.c.

⫺55

f.c.c. h.c.p.

0.8

1.0

1.2 Ω/Ω0

1.4

Figure 39.1 Plot of cohesive energy (0 K) versus the normalised volume per atom for a variety of crystal structures of iron. Hexagonal–P and Cubic–P are primitive structures2

Thermodynamics and equilibrium phase diagrams

39–3

One application of the theory is to derive empirical potentials representing pairwise or higher order interactions between neighbouring atoms. Such potentials are used in molecular dynamics simulations in which the motion of hundreds of thousands of atoms can be followed simultaneously.4

39.3 Thermodynamics and equilibrium phase diagrams Electron theory is not yet able to give sufficiently accurate calculations of phase diagrams; thermodynamic models which exploit experimental databases are much more suited for this purpose.5–7 The basic information to be gained from a binary phase diagram in which temperature is plotted against concentration, is the chemical composition and proportion of each phase at a specified temperature and average composition. Plotting such diagrams becomes tedious for ternary alloys, and ceases to be useful for higher order alloys. This is not a problem in practice because the information to be gained from a multicomponent, multiphase diagram is identical to that expressed for a binary system. The volume fractions and chemical compositions can readily be communicated numerically instead of using complicated diagrams, as in Table 39.1. Given thermodynamic data, calculations such as those presented in Table 39.1 are based on the simple requirement that the chemical potential must be uniform at equilibrium: β

µαi = µi = . . .

for i = 1, 2, 3, . . .

and

phase = α, β . . .

(1)

j µi

where represents the chemical potential of component i. There are many thermodynamic models for expressing the chemical potential as a function of the mixing the solutes. Each model will have a contribution from the entropy of mixing (SM ) and enthalpy of mixing (HM ) to derive a free energy of mixing (GM ). The equations defining these quantities depend on the assumptions: in ideal mixing the solutes are dispersed at random and HM = 0; the regular solution model has HM = 0 but it is still assumed that SM can be calculated assuming random mixing; quasichemical theory also has a finite enthalpy of mixing but does a better estimation of SM . The method adopted for generalised computation of phase diagrams recognises that in practice it is necessary to implement calculations over a wide range of concentrations, parameters and phases. The molar free energy of mixing in a binary solution is often written as: GM = e G + Na kT [(1 − x) ln{1 − x} + x ln{x}]

(2)

where Na is Avogadro’s number, k is the Boltzmann constant, x is the mole fraction of solute, T is the absolute temperature. The last term in equation 2 comes from the ideal entropy of mixing. The excess Gibbs free energy e G expresses the deviation from an ideal solution and is written empirically for a binary solution with components A and B as:
e GAB = xA xB LAB,i (xA − xB )i (3) i

The excess free energy of a ternary solution can as a first approximation be expressed in terms of purely binary interactions:
LAB,i (xA − xB )i e GABC = xA xB i

+ xB xC

+ xC xA Table 39.1


i


i

LBC,i (xB − xC )i

(4)

LCA,i (xC − xA )i

EQUILIBRIUM PHASE MIXTURE FOR Fe–0.1C–0.2Si–0.5Mn–9Cr–1W wt%, AT 873 K

Phase fractions and concentrations are as mole fractions Phase

Fraction

C

Si

Mn

Cr

Ferrite M23 C6 Laves

0.976 9 0.022 3 0.000 8

0.000 001 9 0.206 896 6 0.000 000 0

0.006 067 5 0.000 000 0 0.000 000 0

0.005 124 6 0.001 977 6 0.000 000 0

0.083 947 3 0.624 445 9 0.134 653 0

W 0.001 998 7 0.036 302 9 0.333 333 3

39–4

Modelling and simulation

The advantage of the representation embodied in equation 4 is that for the ternary case, the relation reduces to the binary problem when one of the components is set to be identical to another, e.g. B ≡ C. There might exist ternary interactions, in which case a term xA xB xC LABC,0 is added to the excess free energy. If this does not adequately represent the deviation from the binary summation, then it can be converted into a series which properly reduces to a binary formulation when there are only two components:  1 xA xB xC LABC,0 + (1 + 2xA − xB − xC )LABC,1 3

1 (1 + 2xB − xC − xA )LBCA,1 3  1 + (1 + 2xC − xA − xB )LCAB,1 3

+

The method can clearly be extended to deal with any number of components, with the great advantage that few coefficients have to be changed when the data due to one component are improved. The experimental thermodynamic data necessary to derive the coefficients may not be available for systems higher than ternary so high order interactions are often set to zero.

39.4 Thermodynamics of irreversible processes Equilibrium as described above represents a state in which there is no perceptible change no matter how long one observes the state. A process of change (i.e. an irreversible process) is by contrast associated with the dissipation of free energy and is best described using kinetic theory. Before dealing with full-blown kinetic theory, it is possible to study a compromise model in which thermodynamics is used to represent systems in which there is apparently no change but free energy is nevertheless being dissipated. This is the case of the steady state, for example, diffusion across a constant gradient; neither the flux nor the concentration at any point changes with time, and yet the free energy of the system is decreasing since diffusion occurs to minimise free energy. The rate at which energy is dissipated is the product of the temperature and the rate of entropy production (i.e. T σ) with:8, 9 T σ = JX

(5)

where J is a generalised flux of some kind, and X a generalised force. In the case of an electrical current, the heat dissipation is the product of the current (J ) and the electromotive force (X ). Provided the flux–force sets can be expressed as in equation 5, the flux must naturally depend in some way on the force. It may then be written as a function J {X } of the force X . It is found that for small deviations from equilibrium, the flux is proportional to force, Table 39.2. There are many circumstances where a number of irreversible processes occur together. In a ternary Fe–Mn–C alloy, the diffusion flux of carbon depends not only on the gradient of carbon, but also on that of manganese. Thus, a uniform distribution of carbon will tend to become inhomogeneous in the presence of a manganese concentration gradient. Similarly, the flux of heat may not depend on the temperature gradient alone; heat can be driven also by an electromotive force. Electromigration Table 39.2

EXAMPLES OF FORCES AND THEIR CONJUGATE FLUXES

z is distance, φ is the electrical potential in volts, and µ is a chemical potential Force Electromotive force (e.m.f.) =

Flux ∂φ ∂z

Electrical current

−

1 ∂T T ∂z

Heat flux

−

∂µi ∂z

Diffusion flux

Kinetics

39–5

involves diffusion driven by an electromotive force. When there is more than one dissipative process, the total energy dissipation rate can still be written.
Tσ = J i Xi . (6) i

For multiple irreversible processes, it is found experimentally that each flow Ji is related not only to its conjugate force Xi , but also is related linearly to all other forces present. Thus, Ji = Mij Xj

(7)

with i, j = 1, 2, 3, . . .. Therefore, a given flux depends on all the forces causing the dissipation of energy. Onsager has shown that to maintain dynamic equilibrium, and provided that the forces and fluxes are chosen from the dissipation equation and are independent, Mij = Mji . An exception occurs with magnetic fields in which case there is a sign difference Mij = −Mji .

39.5 Kinetics Many kinetic processes are not consistent with the steady-state. The vast majority of such processes in metals involve aspects of nucleation and growth; the theory for nucleation and growth is standard and is assumed here, in order to focus on the estimation of the volume fraction, which requires impingement between particles to be taken into account. This is done using the extended volume concept of Kolmogorov, Johnson, Mehl and Avrami.10 Suppose that two particles (Fig. 39.2) exist at time t; a small interval δt later, new regions marked a, b, c & d are formed assuming that they are able to grow unrestricted in extended space whether or not the region into which they grow is already transformed. However, only those components of a, b, c & d which lie in previously untransformed matrix can contribute to a change in the real volume of the product phase (α):   Vα dV α = 1 − (8) dVeα V where it is assumed that the microstructure develops at random. The subscript e refers to extended volume, V α is the volume of α and V is the total volume. Multiplying the change in extended volume by the probability of finding untransformed regions has the effect of excluding regions such as b, which clearly cannot contribute to the real change in volume of the product. For a random distribution of precipitated particles, this equation can easily be integrated to obtain the real volume fraction, ( ' Vα Vα = 1 − exp − e (9) V V

c b a d

Time ⫽ t

Time ⫽ t ⫹ ⌬t

Figure 39.2 An illustration of the concept of extended volume. Two precipitate particles have nucleated together and grown to a finite size in the time t. New regions c and d are formed as the original particles grow, but a & b are new particles, of which b has formed in a region which is already transformed

39–6

Modelling and simulation

Temperature

Cooling curve

j ⫽ 0.05

j ⫽ 0.10 Isothermal transformation curves

⌬t i ti

Time Figure 39.3

The Scheil method for converting between isothermal and anisothermal transformation data

The extended volume Veα is straightforward to calculate using nucleation and growth models and neglecting any impingement effects. Solutions typically take the form: ξ = 1 − exp{−kA t n }

(10)

where kA and n characterise the reaction as a function of mechanism of transformation, time, temperature and other variables. Reactions sometimes do not happen in isolation. For example, a steel designed to serve at 600◦ C over a period of 30 years may contain more than six different kinds of precipitates so that it can sustain a load without creeping. A simple modification for the simultaneous formation of two precipitates (α and β) is that the relation between extended and real space becomes a coupled set of two equations,11,12     Vα + Vβ Vα + Vβ (11) dV α = 1 − dVeα and dV β = 1 − dVeβ V V This can be done for any number of reactions happening together. The resulting set of equations must in general be solved numerically. A popular method of converting between isothermal and anisothermal transformation data is the additive reaction rule of Scheil.10 A cooling curve is treated as a combination of a sufficiently large number of isothermal reaction steps. Referring to Figure 39.3, a fraction ξ = 0.05 of transformation is achieved during continuous cooling when
ti =1 ti i

(12)

with the summation beginning as soon as the parent phase cools below the equilibrium temperature. The rule can be justified if the reaction rate depends solely on ξ and T . Although this is unlikely, there are many examples where the rule has been empirically applied with success. Reactions for which the additivity rule is justified are called isokinetic, implying that the fraction transformed at any temperature depends only on time and a single function of temperature.

39.6 Monte Carlo simulations The complete description of microstructure requires both field and feature parameters; the former describes averaged quantities such as volume fraction and the amount of surface per unit volume;

Phase field method 1

1

2

2

2

2

3

3

3

1

1

1

1

2

2

2

2

2

4

1

1

1

1

2

2

2

2

2

4

1

1

1

1

2

2

2

2

2

4

4

1

1

3

3

2

2

2

4

4

4

4

3

3

3

3

3

4

4

4

4

4

3

3

3

3

3

3

4

4

4

3

3

3

3

3

3

3

4

4

39–7

Figure 39.4 Mapping of variable defining crystallographic orientation; the orientation domains are separated by grain boundaries. In this example, the elements within each grain are arranged in a square pattern

the latter includes for example, size distributions. One disadvantage of the Avrami method is that it yields only the field parameters since information about individual particles is lost during the conversion from extended to real space. Methods such as Monte Carlo and phase field simulations permit the visualisation of microstructural development and can therefore reveal feature parameters. The simulation of microstructure using the Monte Carlo method begins with the definition of domains using an appropriate variable which can take on discrete values out of a set of possible values.4 A case where the variable is used to represent crystallographic orientation is illustrated in Figure 39.4, with grain boundaries located wherever the variable changes value. We shall consider isothermal grain coarsening to illustrate the method. Having defined the starting grain structure as in Figure 39.4, the elements located at the grain boundaries are identified and are the only ones considered in further calculations. Each of these grain boundary variables is then randomly perturbed. If the perturbation leads to an overall reduction in energy, it is accepted. If it leads to an increase E in energy, then it is accepted only if the term exp{−E/kT } ≤ p where p is a random, computer-generated number between 0 and 1. The accepted perturbations lead to a new grain structure once all the elements have been sampled. The process of sampling of all of the elements defined in the model corresponds to one Monte Carlo step, equivalent to the progress of time. Since each of the elements illustrated in Figure 39.4 is much coarser than an atom, care must be taken to investigate the effect of this ‘coarse-graining’; indeed, the symmetry of the pattern in which the elements are arranged within each grain is known to influence the outcome.4

39.7 Phase field method Consider the growth of a precipitate which is separated from the matrix by an interface. There are then three distinct quantities: the precipitate, matrix and interface. The interface is described as an evolving surface whose motion is controlled according to the boundary conditions imposed to describe the physical mechanisms which lead to the growth of one phase at the expense of the other. This mathematical description categorises the boundary as a two-dimensional surface with no width or structure, a sharp interface. By contrast, the phase-field method13 begins with a description of the entire microstructure, including the interface, in terms of a single variable known as the order parameter. The precipitate and matrix each have a particular value of the order parameter. Likewise, the interface is located by the position where the order parameter changes from its precipitate-value to its matrix-value. The order parameter is thus continuous as one traverses the precipitate and enters the matrix (Fig. 39.5).

Modelling and simulation

39–8

Interface

Order parameter

Order parameter

Interface

Distance

(a) Figure 39.5

(b)

Distance

(a) Sharp interface. (b) Diffuse interface

The range over which it changes is the width of the interface. The set of values of the order parameter over the whole microstructure is the phase field. The order parameter changes smoothly through the interfacial region from one limiting value to the other, in a phase-field model. A theory which tracks the dynamics of the order parameter across the entire phase field, would allow the evolution of the microstructure to be calculated without the need to track the interface. Such a theory would require the free energy as a function of the order parameter so that the field can follow a path which leads to a maximisation of entropy production. Two examples of phase-field modelling are as follows: the first is that of grain growth, where the order parameter is not conserved since the amount of grain surface per unit volume decreases with grain coarsening. The second conserves the order parameter. When composition fluctuations evolve into precipitates, it is the average chemical composition which is conserved. We now consider this second example in a little more detail. In solutions that tend to exhibit the clustering of atoms, it is possible for a homogeneous phase to become unstable to infinitesimal perturbations of chemical composition. The free energy of a solid solution which is chemically heterogeneous can be factorised into three components.14 First, there is the free energy of a small region of the solution in isolation, given by the usual plot of the free energy of a homogeneous solution as a function of chemical composition. The second term comes about because the small region is surrounded by others which have different chemical compositions. This gradient term is an additional free energy in a heterogeneous system, and is regarded as an interfacial energy describing a ‘soft interface’ of the type illustrated in Figure 39.5b. In this example, the soft-interface is due to chemical composition variations, but it could equally well represent a structural change. The third term arises because a variation in chemical composition also causes lattice strains in the solid-state. We shall neglect these coherency strains. The free energy per atom of an inhomogeneous solution is then given by: gih =



[g{c0 } + υ3 κ(∇c)2 ]dV

(13)

where g{c} is the free energy per atom in a homogeneous solution of concentration c0 , υ is the volume per atom and κ is called the gradient energy coefficient. gih is often referred to as a free energy functional since as in density functional theory, it is a function of a function. Equilibrium in a heterogeneous system is obtained by minimising the functional, subject to the requirement that the average concentration is maintained constant: 

(c − c0 )dV = 0

where c0 is the average concentration. Spinodal decomposition can therefore be simulated using the functional defined in equation 13. The system would initially be set to be homogeneous but with some compositional noise. It would then be perturbed, allowing those perturbations which reduce free energy to survive. In this way, the whole decomposition process can be modelled without explicitly introducing an interface. The interface is instead represented by the gradient energy coefficient.

Finite element method a

39–9

b

h

C0 C1

C2 Z

Figure 39.6

Finite difference representation of diffusion

39.8 Finite difference method The finite difference method is useful in the numerical solution of second order boundary value problems.15 Consider one-dimensional diffusion in a concentration gradient along a coordinate z (Fig. 39.6). The concentration profile is divided into slices, each of thickness h. The matter entering a unit area of the face at a in a time increment τ is given approximately by Ja = −Dτ(C1 − C0 )/h. That leaving the face at b is Jb = −Dτ(C2 − C1 )/h. If C1′ is the new concentration in slice 1, then the net gain in solute is (C1′ − C1 )h so that C1′ − C1 =

Dτ (C0 − 2C1 + C2 ) h2

(14)

This allows the concentration at a point to be calculated as a function of that at the two neighbouring points. By successively applying this relation at each slice, and advancing the time τ, the entire concentration profile can be estimated as a function of time. The approximation here is that we have applied Fick’s first law to each slice, i.e. assumed that the concentration gradient within each slice is constant. Since the profile is in fact a curve, the approximation will be better for smaller values of h, but the computation times will be correspondingly longer. Considerations of numerical accuracy are vital in such methods; the accuracy can be assessed by changing h and seeing whether it makes a significant difference to the calculated profile.

39.9 Finite element method In finite element analysis, continuous functions are replaced by piecewise approximations. Thus, a finite element representation of a circle would be a circumscribed polygon, with each edge being a finite element. We shall discuss this in terms of the forces involved in the deflection of springs.16 It is assumed that for any spring, the force varies linearly with displacement: F = kδ where k is the stiffness of the spring. Consider the forces at the nodes of a spring in a system of springs at equilibrium, as illustrated in Figure 39.7a. Since F1 = −F2 , / 0 / 0  k −k δ1 F1 = −k k F δ 2

2

The forces at the nodes of the springs illustrated in Figure 39.7b are therefore   8 9  9    8 0 0 0 0 δ1 k1 −k1 0 F1 δ1 F2 = 0 k2 −k2 δ2 F2 = −k1 k1 0 δ2 δ3 0 0 0 0 F3 δ3 0 −k2 k2 9  8   8 9 8 k1 δ1 −k1 k1 −k1 0 F1 0 0 0 δ2 ≡ −k1 k1 + k2 F2 = −k1 k1 0 + 0 k2 −k2 0 −k2 δ3 0 −k2 k2 0 0 0 F3 3 45 6 45 3 component stiffnesses

0 −k1 k2

overall stiffness

9  δ1 δ2 δ3 6

Modelling and simulation

39–10

d2

d1 1

2

(a)

F1 d1

F2

d2

1

d3

2

(b)

F1

Figure 39.7

Forces on springs

3 F2

F3

x2

x1 x2

x1 w1

w2

u

x3

Input nodes u(1)

w3

Σ

(1) w1

(1) w2

x3 (1) w3

(1)

)

(1)

tanh(Σwj xj ⫹ u

Hidden unit

j

w (2)h ⫹ u(2) y

Output node

(a)

(b)

Figure 39.8

y

(a) A neural network representation of linear regression. (b) A non-linear network representation

This simple case illustrates how the properties of the elements can be combined to yield an overall response function.

39.10

Empirical modelling: neural networks

There are problems in materials science where the concepts might be understood but which are not amenable to thorough scientific treatment. Empirical regression then becomes useful. By fitting data to a specified relationship, an equation is obtained which relates the inputs xj via weights wj and a constant θ to obtain an estimate of the output y = j wj xj + θ. Because the variables are assumed to be independent, this equation can be stated to apply over a certain range of the inputs. This linear regression method can be represented as a neural network (Fig. 39.8a). The inputs xi define the input nodes, and there is an output node. Each input is multiplied by arandom weight wi and the products are summed together with a constant θ to give the output y = i wi xi + θ. The summation is an operation which is hidden at the hidden node. Since the weights and the constant θ were chosen at random, the value of the output will not match with experimental data. The weights are changed systematically until a best-fit description of the output is obtained as a function of the inputs; this operation is known as training the network. The network can be made non-linear as shown in Figure 39.8b. As before, the input data xj are (1) multiplied by weights (wj ), but the sum of all these products forms the argument of a hyperbolic tangent:  
(1) wj xj + θ  with y = w(2) h + θ (2) (15) h = tanh  j

where w(2) is a weight and θ (2) another constant. The strength of the hyperbolic tangent transfer function is determined by the weight wj . The output y is therefore a non-linear function of xj , the

Empirical modelling: neural networks x1 (a)

x2

x3

39–11

Input nodes

(b) 2

y

1

f{xj }

Hidden units

y

Output node

(c)

Figure 39.9 (a) Three different hyperbolic tangent functions; the ‘strength’ of each depends on the weights. (b) A combination of two hyperbolic tangents to produce a more complex model. (c) A two hidden-unit network

function usually chosen being the hyperbolic tangent because of its flexibility. The exact shape of the hyperbolic tangent can be varied by altering the weights (Fig. 39.9a). A one hidden-unit model may not be sufficiently flexible. Further degrees of non-linearity can be introduced by combining several of the hyperbolic tangents (Fig. 39.9b), permitting the neural network method to capture almost arbitrarily non-linear relationships. The number of tanh functions per input is the number of hidden units; the structure of a two hidden unit network is shown in Figure 39.9c. The function for a network with i hidden units is given by y=


i

(2)

wi hi + θ (2)



where hi = tanh 


j



(1) (1) wij xj + θi 

(16)

Notice that the complexity of the function is related to the number of hidden units. A neural network like this can capture interactions between the inputs because the hidden units are nonlinear. Appropriate measures must be taken to avoid overfitting. With complex networks it is not possible to easily specify a range of applicability. MacKay has developed a particularly useful treatment of neural networks in a Bayesian framework;18 instead of calculating a unique set of weights, a probability distribution of sets of weights is used to define the fitting uncertainty. The error bars therefore become large when data are sparse or locally noisy. REFERENCES 1. A. H. Cottrell, Introduction to the Modern Electron Theory of Alloys, Institute of Materials, London, U.K., 1989. 2. A. T. Paxton, M. Methfessel and H. M. Polatoglou, Physical Review B, 1990, 41, 8127. 3. D. G. Pettifor, Materials Science and Technology, 1992, 8, 345. 4. D. Rabbe, Computational Materials Science, Wiley–VCH, Germany, 1998. 5. L. Kaufman, Prog. in Materials Science, 1969, 14, 57. 6. M. Hillert, Hardenability Concepts with Applications to Steels, eds D. V. Doane and J. S. Kirkaldy, TMS– AIME, Warrendale, Pennsylvania, U.S.A., 1977, 5. 7. K. Hack, SGTE Casebook: Thermodynamics at Work, Institute of Materials, London, U.K., 1996. 8. D. G. Miller, Chemical Reviews, 1960, 60, 15. 9. K. G. Denbigh, Thermodynamics of the Steady State, John Wiley and Sons, Inc., New York, U.S.A., 1955. 10. J. W. Christian, Theory of Transformations in Metals and Alloys, Part 1, 2nd edition, Pergamon Press, Oxford, U.K., 1975. 11. J. D. Robson and H. K. D. H. Bhadeshia, Materials Science and Technology, 1997, 13, 631. 12. S. J. Jones and H. K. D. H. Bhadeshia, Acta Materialia, 1997, 45, 2911. 13. J. A. Warren, IEEE Computational Science and Engineering, Summer 1995, 38. 14. J. W. Cahn and J. E. Hilliard, Journal of Chemical Physics, 1959, 31, 688. 15. J. Crank, The Mathematics of Diffusion, 2nd edition, Clarendon Press, Oxford, 1975. 16. K. M. Entwistle, Basic Principles of Finite Element Analysis, Institute of Materials, London, U.K., 1999. 17. D. J. C. MacKay, Neural Computation, 1992, 4, 415. 18. H. K. D. H. Bhadeshia, ISIJ International, 1999, 39, 966.

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40

Supporting technologies for the processing of metals and alloys

40.1 Introduction and cross-references This chapter brings together three topics that support the processing of metals and alloys, namely: the design of metallurgical furnaces, vacuum technology and the control of metallurgical processes. Material in this chapter relates to the following: • The statistical basis of process control is discussed in Chapter 2. • Thermocouple-based temperature measurement and information on emissivity relevant to

pyrometry are outlined respectively in Chapters 16 and 17.

• Foundry practice and heat-treatment are considered respectively in Chapters 26 and 29. • Refractory materials form the topic of Chapter 27. • For a discussion of fuels, see Chapter 28.

40.2 Furnace design 40.2.1 Introduction There are many considerations to be made when designing a metallurgical processing furnace. A furnace is not just simply a box of bricks; it is a unit which encompasses many engineering aspects including: – – – – – – – – – – – – – – – – – – – – –

Furnace safety Energy input Temperature measurement Temperature control Temperature uniformity Temperature recording Mechanical handling Door mechanisms Heat loss Batch or continuous operation Pre-cleaning Post-cleaning Method of quenching Quench media Quench agitation Heat transfer from quench medium Furnace atmosphere Generation of furnace atmosphere Control of furnace atmosphere Mass (load weight) of work pieces Heat transfer to load.

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Supporting technologies for the processing of metals and alloys

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The above are just a few of the considerations that are necessary in the design of the metallurgical processing furnace. Furnace design requires the engineer to have, or to have access to, multiple disciplines which range from: – – – – –

Mechanical engineering Electrical engineering Metallurgical engineering Physics Chemistry.

These disciplines will assist the engineer in designing a cost-effective, functioning, metallurgical processing furnace that will process the necessary work in an efficient and safe manner to the benefit of both the furnace operator and the resulting product’s end user. Metallurgical processing furnaces can be categorised into two main groups, which relate to the types of materials to be processed. – Ferrous materials – Non-ferrous materials Within the above two groups, further subdivisions can be made in relation to heating methods. These subdivisions are:

Electrical heating The electrical heating methods and techniques can be further subdivided into elemental type heating systems and other electrical methods of heating such as laser, radio frequency, electron beam, infrared. The latter four types of heating will not be considered in this section.

Gas heating Furnaces can be further categorised into three distinctive groupings which relate to the general types of application. These distinctive groupings are: Ovens: An oven is considered to be a low operating temperature system. This means that the unit will operate from room temperature up to approximately 300◦ C (600◦ F). Heat treatment furnaces: A heat treatment furnace will operate over a temperature range from 150◦ C (300◦ F) up to (as in the case of vacuum furnaces) 1 650◦ C (3 000◦ F), although specialist furnaces can reach far higher temperatures. Melting and holding furnaces: These furnaces will melt both ferrous and non-ferrous materials ranging from lead to aluminium, brasses, cast irons and steels. The focus of the discussion in this section will be on the design of heat treatment furnaces only, i.e. this means furnaces that are capable of heat treating aluminium alloys and various types of steels, whose only function is to accomplish phase changes within the selected metal.

40.2.2 Types of furnaces There are three simple furnace configurations, which are as follows: – Horizontal furnaces – Pit furnaces – Bottom load elevator hearth furnaces. The first group of furnaces to be considered will be the horizontal furnace configuration.

40.2.2.1

Box furnaces

The box furnace is perhaps the most simple of all of the heat treatment furnaces. This is usually of a rugged and simple construction. The furnace will literally be a box which will contain insulation material (refractory insulation or thermal blanket insulation). The furnace usually has a range of operating temperatures from 150◦ C (300◦ F) up to 1 200◦ C (2 200◦ F). Some high-temperature box furnaces that are heated by silicon carbide elements are capable of reaching temperatures above 1 300◦ C (2 400◦ F). Heating methods, such as the use of MoSi2 elements, for higher temperature

Furnace design

40–3

vertical furnaces are not discussed here. The furnace can have either a single access door or an entry and exit door at each end of the furnace. The furnace heating system can be:

Direct heating In this instance, the products of combustion (if gas-fired) can provide an atmosphere. This atmosphere can be either oxidising, carburising or decarburising. If the furnace is heated electrically, then heat transfer will be either by convection, conduction or both, depending on the furnace’s maximum operating temperature, with radiant heat transfer becoming important at high temperature.

Indirect heating A furnace heated by indirect methods can have its heat source located in an external heating chamber, and the heated air driven by an air blower which will distribute the heated air into the work processing chamber through a series of dampers, which are set to distribute the heated air and ensure good temperature uniformity. 40.2.2.2 Integral quench furnaces The integral quench furnace is perhaps the ‘workhorse’ of the heat treatment furnaces. The design of the integral quench furnace dates back to the late 1940s. The original aim of this type of furnace was to transform a box type (batch) furnace into a continuous/semi continuous type furnace. The design of the furnace integrates the use of two separate process chambers, the first chamber being the entry vestibule which contains the quench tank and is separated by an intermediate insulated door. Behind the intermediate door is the high heat chamber in which the heat treatment procedure could be carried out. The high heat chamber contains the furnace heating system and the furnace atmosphere system, as well as the mechanical handling equipment which enables the workload to be transferred into the furnace from an external load table, through the entry vestibule and on into the heating chamber. Upon completion of the heat treatment procedure, the workload is transferred back into the front entry vestibule and on to an elevator table which will lower the workload into the quench medium (if required). If the process is, for example, an annealing process, then the workload can sit on the elevator table without being immersed into the quench medium. This will allow a slow cool. The integral quench furnace is used for many different heat treatment processes, such as: – – – – – –

Annealing Carburising Carbonitriding Ferritic nitrocarburising Solutionising Neutral hardening.

The limiting factors of the integral quench furnaces can be both the volume of quench medium in the quench tank and size limitations of the heating chamber. Generally, these types of furnaces will have a temperature limitation of 950◦ C (1 750◦ F). Many advances have been made since the late 1940s in terms of heating efficiency, operating costs, work load handling and most importantly, process control in terms of atmosphere control, temperature control, temperature uniformity and minimal heat losses. 40.2.2.3 Pit furnaces As the name implies, this type of furnace can be installed into a pit or be mounted on the shop floor. It will be the size of the furnace, height clearance above the furnace, and foundation/floor conditions that will determine the positioning of the furnace. These furnaces are also known as vertical air circulation furnaces. This means that the heated air/atmosphere within the furnace is circulated throughout the heating chamber, which provides good temperature uniformity. The air circulation is accomplished by the use of a fan, located either in the base of the furnace or in the furnace lid. This type of furnace is used typically for the following heat treatment processes: – – – –

Tempering Stress relieving Normalising Annealing

40–4 – – – –

Supporting technologies for the processing of metals and alloys Nitriding Ferritic nitrocarburising Carbonitriding Carburising.

The furnace construction is very simple, once again, and offers excellent uniformity of both temperature and atmosphere (if used). Improved temperature uniformity is a result of the shape of the furnace chamber, which is circular. 40.2.2.4 Horizontal car bottom furnaces The car bottom furnace is designed usually to process large products such as fabrications, pressure vessels, ingots, forgings and castings. The heat treatment procedures usually carried out in these furnaces are: – – – – –

Annealing Normalising Stress relieving Hardening Tempering.

The furnace consists of a static steel framework into which is installed the heating system (gas or electric), insulation (refractory brick or low thermal mass fibre), entry door (which can be a single entry door, or two doors i.e. one front, one rear) and finally the car bottom. If the furnace is of the single door entry type, then it is supplied usually with one insulated refractory brick/castable transportable base on wheels. The drive mechanism of the hearth can be an external steel rope and pulley system, or a motorised drive system located on the underside of the hearth. If the furnace has a two-door system, then a single hearth can still be used on a straight through basis or two load transportation cars can be utilised i.e. one for processing and one for loading/unloading. If the furnace is to be used for hardening followed by quenching, then a lifting or tilting mechanism will be necessary to discharge the load into the quench tank. Another variation of the car bottom type of furnace is the lift off bell furnace. The bell can be circular and cylindrical, or a box type furnace. The circular/cylindrical bell furnace can be heated by electrical methods or by gas firing. The heating system is generally on the external side of the bell. The box type furnace is usually fired within the insulated box and can be fired by either electrical or gaseous heating methods. 40.2.2.5 Continuous furnaces The continuous type of furnace is used normally for high-volume production of manufactured parts. The mechanical handling of the work piece within this type of furnace can be designed for the particular product that the furnace is to handle, such as: Ingots, slabs bars: These products can be handled within the furnace grouping known as walking beam furnaces. This work handling method is managed by two sets of beams within the furnace heating chamber. The first group of beams are static beams, which are fixed to the hearth of the furnace. A second group of beams is able to move. This group of beams manipulates the products through the furnace. This type of furnace is generally fired internally by gas and air mixtures. Smaller components (e.g. small billets): These can be heated inside a rotary hearth furnace. This type of furnace is round in shape and is of a static type. The only moving parts of this furnace being the entry/exit doors and the rotary hearth upon which the work is placed. The furnace can be heated either by gaseous means or electrically. The furnace atmosphere can either be generated from the products of combustion, from direct gas firing, or by any generated atmosphere such as an endothermic atmosphere or a nitrogen/methanol atmosphere. If these types of atmospheres are used, then the firing method will be encapsulated in the burner combustion tubes. Mass production: The continuous type of furnace for this application can be a roller hearth furnace. In this type of furnace, the heating method can be either direct or indirect heating, using gas or electrical heating. The work is transported through the furnace on specially designed work load trays that are placed (with the work) on to driven rollers. Each of these

Furnace design

40–5

rollers are driven from a master drive system and each roller is supported on either side of the roller with water cooled bearings. The furnaces are usually long and have an entry and exit door at each end of the furnace. Where the integrity of the atmosphere is vital, the choice of furnace would be a pusher type furnace. This type of furnace is usually heated indirectly and the heating method (gas or electrical) is encapsulated in the heating tubes. Generally, the furnaces are designed for atmosphere conditions using either endothermic or nitrogen/methanol atmospheres. Because of these types of atmospheres, a front and rear vestibule is generally fitted to the furnace. Generally these furnaces are fitted with an atmosphere circulation system compromising of multiple fans located either in the roof or side walls of the furnace. This type of furnace can maintain a very accurate atmosphere for a variety of processes. When small components are required to be heat-treated, the furnace of choice would be a shaker hearth furnace or a continuous mesh belt furnace. A shaker hearth furnace (a.k.a. a shuffle hearth furnace) comprised of a long horizontal tetragonal furnace with a single hearth that moves rapidly forwards and backwards over a distance of approximately 150 mm (6 in). This causes the work load to move forward through the heating chamber. A variety of processes can be accomplished in this type of furnace, such as hardening, tempering, carbonitriding and carburising. Usually, at the end of the table, the work is allowed to free fall into a vertical chute which is immediately above the quench tank. The disadvantage of this type of furnace is that there is no control of the presentation of the work to the quench medium, therefore the problem of distortion becomes quite high. A mesh belt furnace is a long horizontal tetragonal type of furnace that is capable of holding the furnace atmosphere as it is with the shaker hearth furnace. The difference between the two furnaces is simply in the method of transportation of the work. In the mesh belt furnace, the mesh belt is a woven belt of heat resistant stainless-steel which transports the work through the heating chamber and discharges it either into a quench tank or into a controlled cooling area. The mesh belt can be returned to the driven end of the furnace either externally or internally. If the belt is returned externally, it means that the belt has to be reheated. These furnaces can be heated either electrically or by gaseous methods. The heat source is usually encapsulated, irrespective of which method is chosen. Rotary barrel furnaces contain a heat resisting process barrel which is fitted internally with a screw drive mechanism. The work is loaded, on a continuous basis, through an external sealed feeding system. The barrel is rotated by an external drive system which can be either a direct chain drive or a roller drive system. Sometimes, the whole furnace will tilt to discharge the workload into the quench medium, or the parts will be screw driven out of the furnace through a sealed discharge aperture and into the quench medium. The single most important problem with any of the above furnaces is that of the potential for distortion to occur, simply because the work is allowed to free fall into the quench medium, without any control of how the component is presented to the quench medium. 40.2.3 Heat calculations Irrespective of the type of furnace selected, it is necessary to calculate the amount of energy that is required to raise the temperature of the workload from ambient up to the metallurgical processing temperature. Such calculations can range from simple ‘back of the envelope’ methods (e.g. energy requirement = specific heat capacity × gross load mass × [target temperature – ambient temperature]) to sophisticated heat transfer models. The former type of calculation will be familiar to readers, whilst the latter forms an entire branch of the technical literature and hence cannot be discussed here. Once the heating requirement calculations have been accomplished, then it is necessary to calculate the hearth length. The furnace length will be determined by considering the following parameters and factors which are in relation to the operation process parameters. Note that load support fixtures as well as furnace support and carrier trays must be considered in the heating calculation. The process parameters to be considered are listed below: – – – – – –

Total work throughput (mass per hour) Number of trays per hour Trays and support mass per hour Number of trays per hour required for the desired rate of production Method of heating Process time at temperature.

It is necessary to consider the heating method to determine if there is space to equip the furnace with either heating elements or burners and burner tubes.

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Supporting technologies for the processing of metals and alloys

40.2.4 Refractory design The choice of insulation materials are wide and varied (see Chapter 27 for more detail). REFRACTORY BRICKS VERSUS CERAMIC FIBRE BLANKETS

The maximum operating temperature of the furnace will determine the primary choice of insulation material, in addition to which air velocity and atmosphere movement within the process chamber must be considered. On this basis, a choice must be made between refractory brick and a low mass ceramic fibre material. It cannot be stated that one type of material is universally better or worse than the other. Instead, the choice of insulating material is application specific and depends upon the heat up and cool down rate required, maximum furnace operating temperature and the amount of wind abrasion that will occur within the furnace process chamber. Another consideration in regard to temperature, will be the cold face temperature. That is the temperature that will occur on the external steel walls of the furnace. The accepted safe outside wall temperature is 65◦ C (150◦ F) maximum. However, if the furnace maximum operating temperatures is say, 1 000◦ C (1 800◦ F), then the external wall temperature will be determined by: The hot face insulation of the furnace wall is made up of refractory material (in the form of bricks or a fibre blanket as discussed above). Behind the hot face insulation material will be another course of lower temperature heat resistance material, followed by possibly a third or fourth layer, to reduce the transfer of the heat that is generated at the hot face, in order to maintain a safe external wall temperature. The disadvantage of using refractory brick is that the insulating wall thickness is generally quite high, which means additional weight within the furnace construction. A further disadvantage of all refractory bricks (which is probably the primary disadvantage) is the thermal energy necessary to heat the furnace chamber up to its operating temperature. The major advantage of refractory brick however, is that it has good heat retention and storage capacity. Generally (excluding fire brick) the brick is very porous. The degree of porosity will determine the bricks’ heat storage capacity. The high porosity bricks can work against the heat treatment practitioner, in so much as, if the work load requires a controlled carbon atmosphere, or a carburising atmosphere, then the brick will begin to soak up the carbon atmosphere (depending of course on the carbon potential of the furnace atmosphere). This will mean a frequent burnout of the process chamber. Another consideration when using refractory brick, is the iron content. Generally when firing the furnace electrically, one should choose a brick with a low iron content. Excess iron can lead to ‘tracking’ or small arcs occurring at the point of mounting the element onto the wall. If the furnace is gas fired, then the iron content is not as critical to the refractory selection process. Low mass ceramic fibre blanket is not very capable of heat storage. However, it has the ability to heat up to the operating temperature very quickly when compared with refractory brick. The blanket is very easy to install, and in order to reduce the effects of wind erosion, due to air movement within the process chamber, a surface hardener can be applied to the surface of the blanket. Periodic inspections must be made to assess the condition of the surface hardener. The surface can also be enhanced for longer life by the application of wire mesh with a hardener painted over the mesh. The mesh is generally made of stainless steel. The low mass ceramic fibre blanket is not very good for use with controlled carbon atmosphere potentials. The blanket has a far greater porosity than the brick. Thus the blanket will also take carbon away from the furnace atmosphere, making it difficult to control the atmosphere accurately. MODULES

When contemplating the use of low mass fibre, consideration can be given to the use of preformed square blanket modules. This will reduce the installation time dramatically as a result of increased ease of application. PREFORMS

Another insulation material to consider, particularly when designing and building repeat types of furnaces and sometimes with complex designs, are pre-formed/vacuum formed insulation modules. MORTAR

Careful consideration should be given to the selection of mortar used to install and cement the brick together. The mortar should be applied:

Furnace design

40–7

– Uniformly – Thinly – With a small joint clearance. The mortar should have similar insulating characteristics to those of the refractory brick. When mixing the mortar, the consistency of the mortar will be determined by the amount of water added. The mixed mortar should have a firm consistency. DRYOUT

Any new or re-bricked furnace must be dried out before commencing heat treatment operations. This is not as critical when the furnace has been relined with, for example, a low mass thermal insulation blanket, simply because the fibre is flexible and can accommodate any moisture that might be present within the blanket. A furnace that is insulated with refractory brick will absorb atmospheric moisture as well as moisture contained in the mortar. If the furnace is raised from room temperature to operating temperature immediately after bricking or re-bricking, the residual moisture present in the brick will expand very rapidly and will have the potential to crack the brick. Therefore the furnace should be raised to a temperature of say, 105◦ C (220◦ F), and held at this temperature for up to 12 hours. This should be done with the furnace doors open, to allow the moisture bearing air to disperse from the furnace process chamber. The next step in the dry out procedure will be to raise the furnace temperature up to say 260◦ C (500◦ F) and hold it for approximately 12 hours, again with the doors open. After this segment is completed, the furnace can be heated up to perhaps 535◦ C (1 000◦ F) and held for approximately six hours. At this point the furnace doors can be closed. The furnace temperature can then be ramped up to say 750◦ C (1 400◦ F) then the furnace can be heated up to its process temperature with a ramp rate of no more than typically 35◦ C per hour (100◦ F per hour). This is to reduce the risk of thermal shock to the refractory brick, thus minimising the potential to crack the refractory, which might occur if the temperature is raised too quickly. In the case of a new furnace, then it is most important to adhere to the manufacturer’s dry out procedure. The above description of dry out, gives both the furnace engineer and the service technician an idea of the potential for cracking the refractory, if the furnace is raised to temperature too quickly. 40.2.5 Vacuum furnaces (see also section 40.3 on vacuum technology) The dry out procedure of a vacuum furnace is somewhat different to that of a refractory lined furnace. Vacuum furnaces can be both heated and insulated in two different fashions: All metal This means refractory metals and stainless steels for both heat shield and elements. Graphite Graphite or carbon reinforced carbon can be used as the heating elements. The insulation can be manufactured from pure graphite or graphite fibres with a sealed reflective surface on the hot face of the insulation material. Both oxygen and moisture can be detrimental to each of the above insulating materials, insomuch as oxygen contamination of a metallic heat shield will disturb the reflective surface of the insulating metal by oxidising the metal surface. If graphite or fibrous graphitic material is used as an insulator or a heating element, and is left exposed to atmosphere, it will absorb moisture. The presence of moisture within the vacuum chamber will make it increasingly difficult to pump down to the appropriate pressure level, due to the evaporation of moisture. Moisture can be a technician’s enemy in terms of vacuum operation, due to its ability to migrate anywhere within the vacuum process chamber. Once moisture is in the chamber, and in particular within the fibrous insulation materials, the moisture will ‘stick’ to the insulating fibres and will be difficult to remove on pump down. Therefore, when a new vacuum furnace is shipped to its end-user, it is usually shipped with the door clamped down and the interior under vacuum conditions. Once the furnace is in position at the installation site, the vacuum will be broken, and the door opened. Once the installation is complete, the door is closed and the vacuum pump started to take the furnace down to its ultimate vacuum level. The temperature of the furnace is raised to its maximum operating temperature and allowed to hold at this temperature for a short period of time. After the outgassing period of time has expired, the furnace is allowed to cool down to ambient temperature. The vacuum pump is then turned off and the leak up rate of the process chamber is noted. The normal leak up rate of any type of commercial-scale vacuum furnace that is cold, clean and empty should be in the range of 0.7 Pa (5 mtorr) per hour maximum.

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Supporting technologies for the processing of metals and alloys

The act of pumping down the vessel and raising the furnace up to its maximum operating temperature is designed to ensure that not only the insulating material, but also all other materials of construction within the heating chamber, have completely outgassed. If the vessel then leaks up at a rate greater than 0.7 Pa (5 mtorr) per hour, it means that there is a potential leak within the process chamber. It is then necessary to begin to establish the nature of the problem. There are two types of problems that produce the appearance of leaks, namely: – Real leak (actual physical leak) – Apparent leak (internal or virtual leak or outgassing) REAL LEAK

A real leak is a leak through a hole in the furnace wall. This means that somewhere within the construction of the furnace vessel there is a passage which will allow air, or moisture to leak into the process chamber, thus steadily raising the furnace chamber vacuum pressure. The leak can be at any vacuum sealing point, such as the main door, power feedthroughs, the thermocouple feedthroughs, recirculation fan seal, or any other aperture into the furnace. APPARENT LEAK

An apparent leak is a situation where there is a rate of rise in pressure, but after leak detection procedures are completed, no leak is visible, yet the pressure continues to rise. This is the most difficult type of vacuum problem to handle, simply because it has no external cause. The apparent leak up rate is being caused by something within the process chamber that is outgassing. This can be caused, of course, by a pin hole leak within a weldment on the inner wall of the chamber. This allows water from the water jacket to migrate into and vaporise in the process chamber. Another cause of this condition can be oxides on furnace support fixtures that are outgassing, or the components being processed outgassing. This last type of apparent ‘leak’ is the most difficult to locate. Poorly designed vacuum systems can include constrictions that make pump down of some regions difficult, thus producing the appearance of a leak (i.e. a virtual leak). Assuming that the vacuum furnace is tight and that there are no apparent or real leaks in the furnace system, then the vacuum furnace should be kept closed and under partial vacuum when not in use. 40.2.6 Cooling Once the work has been processed, it needs to be cooled. The cooling rate of the work piece will be determined by the desired metallurgy necessary to impart the required mechanical properties to the material. Cooling can be accomplished (see Chapter 29 for more detail) as follows: Rapidly (by a variety of quenching processes) Slowly The word ‘slowly’ in this instance, is relative. The work piece can be cooled down in still air or forced air (fan cooling). It can also be cooled down in air with a specially designed cooling chamber which would be surrounded by a water jacket. The work piece could be cooled down in a specially designed cooling chamber but insulated with a low mass ceramic fibre blanket. The work piece could further be cooled down by quenching into a liquid salt bath at a predetermined temperature. The physical act of cooling can be accomplished in either the batch type operation or a continuous operation.

40.3 Vacuum technology 40.3.1 Introduction In recent years, the field of vacuum technology has expanded greatly in terms of volume production of equipment and in the diversity of equipment available. To a large extent, this has been driven by the semiconductor industry with its requirements for many large scale wafer fabs operating at high vacuum levels, although other industrial uses of vacuum have also burgeoned. These cover such diverse applications as coating window glass with reflective films, plasma processing of baby’s disposable diapers, freeze drying, etc. Figure 40.1 shows some typical industrial processes using vacuum and the range of pressures over which they tend to operate. Despite this growth, the underlying technologies of pumping and measurement of vacuum have changed little. Such advances as there have been have tended to be better implementations of existing technologies.

Vacuum technology

40–9

Freeze drying Electron beam welding Metal melting Degassing Electron and TV tubes Evaporative coating Semiconductor production Sputter coating 10⫺12

10⫺9

10⫺6

10⫺3

103

1

Pressure in mbar Figure 40.1 Table 40.1

Some industrial processes using vacuum UNITS OF PRESSURE

Unit

Common area of usage

Inches of water Torr mbar

Industry where a ‘negative’ pressure is required e.g. vacuum forming Very widely used in industry Widely used in laboratories and often in industry Largely confined to standards laboratories

Pascal

Table 40.2

Equivalent in mbar 2.49 0.752 1.0 0.01

Comment Widely used, but definitely ‘non-standard’ Non-standard but refuses to die! An alternative unit in the SI system The SI unit

VACUUM RANGES WITH APPROPRIATE PRESSURE MEASUREMENT TECHNIQUES

Pressure region

Pressure range (mbar)

Pressure measurement technique

Low (rough) vacuum Medium (fine) vacuum High vacuum Ultra high vacuum (UHV)

1 000–1 1–10−4 10−4 –10−9