Callister 5th ed_english Ciencia de los materiales

Fundamentals of Materials Science and Engineering

An Interactive e Tex t

F I F T H E D I T I O N

Fundamentals of MaterialsScience and Engineering

An Interactive e Te x t

William D. Callister, Jr.Department of Metallurgical Engineering

The University of Utah

John Wiley & Sons, Inc.New York Chichester Weinheim Brisbane Singapore Toronto

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Preface

Fundamentals of Materials Science and Engineering is an alternate version ofmy text, Materials Science and Engineering: An Introduction, Fifth Edition. Thecontents of both are the same, but the order of presentation differs and Fundamen-tals utilizes newer technologies to enhance teaching and learning.

With regard to the order of presentation, there are two common approachesto teaching materials science and engineeringone that I call the traditionalapproach, the other which most refer to as the integrated approach. With thetraditional approach, structures/characteristics/properties of metals are presentedfirst, followed by an analogous discussion of ceramic materials and polymers. Intro-duction, Fifth Edition is organized in this manner, which is preferred by manymaterials science and engineering instructors. With the integrated approach, oneparticular structure, characteristic, or property for all three material types is pre-sented before moving on to the discussion of another structure/characteristic/prop-erty. This is the order of presentation in Fundamentals.

Probably the most common criticism of college textbooks is that they are toolong. With most popular texts, the number of pages often increases with each newedition. This leads instructors and students to complain that it is impossible to coverall the topics in the text in a single term. After struggling with this concern (tryingto decide what to delete without limiting the value of the text), we decided to dividethe text into two components. The first is a set of core topicssections of thetext that are most commonly covered in an introductory materials course, andsecond, supplementary topicssections of the text covered less frequently. Fur-thermore, we chose to provide only the core topics in print, but the entire text(both core and supplementary topics) is available on the CD-ROM that is includedwith the print component of Fundamentals. Decisions as to which topics to includein print and which to include only on the CD-ROM were based on the results ofa recent survey of instructors and confirmed in developmental reviews. The resultis a printed text of approximately 525 pages and an Interactive eText on the CD-ROM, which consists of, in addition to the complete text, a wealth of additionalresources including interactive software modules, as discussed below.

The text on the CD-ROM with all its various links is navigated using AdobeAcrobat. These links within the Interactive eText include the following: (1) fromthe Table of Contents to selected eText sections; (2) from the index to selectedtopics within the eText; (3) from reference to a figure, table, or equation in onesection to the actual figure/table/equation in another section (all figures can beenlarged and printed); (4) from end-of-chapter Important Terms and Conceptsto their definitions within the chapter; (5) from in-text boldfaced terms to theircorresponding glossary definitions/explanations; (6) from in-text references to thecorresponding appendices; (7) from some end-of-chapter problems to their answers;(8) from some answers to their solutions; (9) from software icons to the correspond-ing interactive modules; and (10) from the opening splash screen to the supportingweb site.

vii

The interactive software included on the CD-ROM and noted above is the samethat accompanies Introduction, Fifth Edition. This software, Interactive MaterialsScience and Engineering, Third Edition consists of interactive simulations and ani-mations that enhance the learning of key concepts in materials science and engi-neering, a materials selection database, and E-Z Solve: The Engineers EquationSolving and Analysis Tool. Software components are executed when the user clickson the icons in the margins of the Interactive eText; icons for these several compo-nents are as follows:

Crystallography and Unit Cells Tensile Tests

Ceramic Structures Diffusion and Design Problem

Polymer Structures Solid Solution Strengthening

Dislocations Phase Diagrams

E-Z Solve Database

My primary objective in Fundamentals as in Introduction, Fifth Edition is topresent the basic fundamentals of materials science and engineering on a levelappropriate for university/college students who are well grounded in the fundamen-tals of calculus, chemistry, and physics. In order to achieve this goal, I have endeav-ored to use terminology that is familiar to the student who is encountering thediscipline of materials science and engineering for the first time, and also to defineand explain all unfamiliar terms.

The second objective is to present the subject matter in a logical order, fromthe simple to the more complex. Each chapter builds on the content of previous ones.

The third objective, or philosophy, that I strive to maintain throughout the textis that if a topic or concept is worth treating, then it is worth treating in sufficientdetail and to the extent that students have the opportunity to fully understand itwithout having to consult other sources. In most cases, some practical relevance isprovided. Discussions are intended to be clear and concise and to begin at appro-priate levels of understanding.

The fourth objective is to include features in the book that will expedite thelearning process. These learning aids include numerous illustrations and photo-graphs to help visualize what is being presented, learning objectives, WhyStudy . . . items that provide relevance to topic discussions, end-of-chapter ques-tions and problems, answers to selected problems, and some problem solutions tohelp in self-assessment, a glossary, list of symbols, and references to facilitateunderstanding the subject matter.

The fifth objective, specific to Fundamentals, is to enhance the teaching andlearning process using the newer technologies that are available to most instructorsand students of engineering today.

Most of the problems in Fundamentals require computations leading to numeri-cal solutions; in some cases, the student is required to render a judgment on thebasis of the solution. Furthermore, many of the concepts within the discipline of

viii Preface

Preface ix

materials science and engineering are descriptive in nature. Thus, questions havealso been included that require written, descriptive answers; having to provide awritten answer helps the student to better comprehend the associated concept. Thequestions are of two types: with one type, the student needs only to restate in his/her own words an explanation provided in the text material; other questions requirethe student to reason through and/or synthesize before coming to a conclusionor solution.

The same engineering design instructional components found in Introduction,Fifth Edition are incorporated in Fundamentals. Many of these are in Chapter 20,Materials Selection and Design Considerations, that is on the CD-ROM. Thischapter includes five different case studies (a cantilever beam, an automobile valvespring, the artificial hip, the thermal protection system for the Space Shuttle, andpackaging for integrated circuits) relative to the materials employed and the ratio-nale behind their use. In addition, a number of design-type (i.e., open-ended)questions/problems are found at the end of this chapter.

Other important materials selection/design features are Appendix B, Proper-ties of Selected Engineering Materials, and Appendix C, Costs and RelativeCosts for Selected Engineering Materials. The former contains values of elevenproperties (e.g., density, strength, electrical resistivity, etc.) for a set of approxi-mately one hundred materials. Appendix C contains prices for this same set ofmaterials. The materials selection database on the CD-ROM is comprised ofthese data.

SUPPORTING WEB SITEThe web site that supports Fundamentals can be found at www.wiley.com/

college/callister. It contains student and instructors resources which consist of amore extensive set of learning objectives for all chapters, an index of learning styles(an electronic questionnaire that accesses preferences on ways to learn), a glossary(identical to the one in the text), and links to other web resources. Also includedwith the Instructors Resources are suggested classroom demonstrations and labexperiments. Visit the web site often for new resources that we will make availableto help teachers teach and students learn materials science and engineering.

INSTRUCTORS RESOURCESResources are available on another CD-ROM specifically for instructors who

have adopted Fundamentals. These include the following: 1) detailed solutions ofall end-of-chapter questions and problems; 2) a list (with brief descriptions) ofpossible classroom demonstrations and laboratory experiments that portray phe-nomena and/or illustrate principles that are discussed in the book (also found onthe web site); references are also provided that give more detailed accounts of thesedemonstrations; and 3) suggested course syllabi for several engineering disciplines.

Also available for instructors who have adopted Fundamentals as well as Intro-duction, Fifth Edition is an online assessment program entitled eGrade. It is abrowser-based program that contains a large bank of materials science/engineeringproblems/questions and their solutions. Each instructor has the ability to constructhomework assignments, quizzes, and tests that will be automatically scored, re-corded in a gradebook, and calculated into the class statistics. These self-scoringproblems/questions can also be made available to students for independent study orpre-class review. Students work online and receive immediate grading and feedback.

Tutorial and Mastery modes provide the student with hints integrated within eachproblem/question or a tailored study session that recognizes the students demon-strated learning needs. For more information, visit www.wiley.com/college/egrade.

ACKNOWLEDGMENTSAppreciation is expressed to those who have reviewed and/or made contribu-

tions to this alternate version of my text. I am especially indebted to the followingindividuals: Carl Wood of Utah State University, Rishikesh K. Bharadwaj of SystranFederal Corporation, Martin Searcy of the Agilent Technologies, John H. Weaverof The University of Minnesota, John B. Hudson of Rensselaer Polytechnic Institute,Alan Wolfenden of Texas A & M University, and T. W. Coyle of the Universityof Toronto.

I am also indebted to Wayne Anderson, Sponsoring Editor, to Monique Calello,Senior Production Editor, Justin Nisbet, Electronic Publishing Analyst at Wiley,and Lilian N. Brady, my proofreader, for their assistance and guidance in developingand producing this work. In addition, I thank Professor Saskia Duyvesteyn, Depart-ment of Metallurgical Engineering, University of Utah, for generating the e-Gradebank of questions/problems/solutions.

Since I undertook the task of writing my first text on this subject in the early1980s, instructors and students, too numerous to mention, have shared their inputand contributions on how to make this work more effective as a teaching andlearning tool. To all those who have helped, I express my sincere thanks!

Last, but certainly not least, the continual encouragement and support of myfamily and friends is deeply and sincerely appreciated.

WILLIAM D. CALLISTER, JR.Salt Lake City, Utah

August 2000

x Preface

Contents

xi

Chapters 1 through 13 discuss core topics (found in both print and onthe CD-ROM) and supplementary topics (in the eText only)

LIST OF SYMBOLS xix

1. Introduction 1

Learning Objectives 21.1 Historical Perspective 21.2 Materials Science and Engineering 21.3 Why Study Materials Science and Engineering? 41.4 Classification of Materials 51.5 Advanced Materials 61.6 Modern Materials Needs 6

References 7

2. Atomic Structure and Interatomic Bonding 9

Learning Objectives 102.1 Introduction 10

ATOMIC STRUCTURE 10

2.2 Fundamental Concepts 102.3 Electrons in Atoms 112.4 The Periodic Table 17

ATOMIC BONDING IN SOLIDS 18

2.5 Bonding Forces and Energies 182.6 Primary Interatomic Bonds 202.7 Secondary Bonding or Van der Waals Bonding 242.8 Molecules 26

Summary 27Important Terms and Concepts 27References 28Questions and Problems 28

3. Structures of Metals and Ceramics 30


CRYSTAL STRUCTURES 31

3.2 Fundamental Concepts 313.3 Unit Cells 323.4 Metallic Crystal Structures 33

xii Contents

3.5 Density ComputationsMetals 373.6 Ceramic Crystal Structures 383.7 Density ComputationsCeramics 453.8 Silicate Ceramics 46 The Silicates (CD-ROM) S-1

3.9 Carbon 47 Fullerenes (CD-ROM) S-3

3.10 Polymorphism and Allotropy 493.11 Crystal Systems 49

CRYSTALLOGRAPHIC DIRECTIONS ANDPLANES 51

3.12 Crystallographic Directions 513.13 Crystallographic Planes 543.14 Linear and Planar Atomic Densities

(CD-ROM) S-4

3.15 Close-Packed Crystal Structures 58

CRYSTALLINE AND NONCRYSTALLINEMATERIALS 62

3.16 Single Crystals 623.17 Polycrystalline Materials 623.18 Anisotropy 633.19 X-Ray Diffraction: Determination of

Crystal Structures (CD-ROM) S-6

3.20 Noncrystalline Solids 64Summary 66Important Terms and Concepts 67References 67Questions and Problems 68

4. Polymer Structures 76

Learning Objectives 774.1 Introduction 774.2 Hydrocarbon Molecules 774.3 Polymer Molecules 794.4 The Chemistry of Polymer Molecules 804.5 Molecular Weight 824.6 Molecular Shape 874.7 Molecular Structure 884.8 Molecular Configurations

(CD-ROM) S-11

4.9 Thermoplastic and ThermosettingPolymers 90

4.10 Copolymers 914.11 Polymer Crystallinity 924.12 Polymer Crystals 95


5. Imperfections in Solids 102


POINT DEFECTS 103

5.2 Point Defects in Metals 1035.3 Point Defects in Ceramics 1055.4 Impurities in Solids 1075.5 Point Defects in Polymers 1105.6 Specification of Composition 110 Composition Conversions

(CD-ROM) S-14MISCELLANEOUS IMPERFECTIONS 111

5.7 DislocationsLinear Defects 1115.8 Interfacial Defects 1155.9 Bulk or Volume Defects 1185.10 Atomic Vibrations 118

MICROSCOPIC EXAMINATION 118

5.11 General 1185.12 Microscopic Techniques

(CD-ROM) S-17

5.13 Grain Size Determination 119Summary 120Important Terms and Concepts 121References 121Questions and Problems 122

6. Diffusion 126

Learning Objectives 1276.1 Introduction 1276.2 Diffusion Mechanisms 1276.3 Steady-State Diffusion 1306.4 Nonsteady-State Diffusion 1326.5 Factors That Influence Diffusion 1366.6 Other Diffusion Paths 1416.7 Diffusion in Ionic and Polymeric

Materials 141Summary 142Important Terms and Concepts 142References 142Questions and Problems 143

7. Mechanical Properties 147

Learning Objectives 1487.1 Introduction 1487.2 Concepts of Stress and Strain 149

ELASTIC DEFORMATION 153

7.3 StressStrain Behavior 1537.4 Anelasticity 1577.5 Elastic Properties of Materials 157

Contents xiii

MECHANICAL BEHAVIORMETALS 160

7.6 Tensile Properties 1607.7 True Stress and Strain 1677.8 Elastic Recovery During Plastic

Deformation 1707.9 Compressive, Shear, and Torsional

Deformation 170

MECHANICAL BEHAVIORCERAMICS 171

7.10 Flexural Strength 1717.11 Elastic Behavior 1737.12 Influence of Porosity on the Mechanical

Properties of Ceramics (CD-ROM) S-22

MECHANICAL BEHAVIORPOLYMERS 173

7.13 StressStrain Behavior 1737.14 Macroscopic Deformation 1757.15 Viscoelasticity (CD-ROM) S-22

HARDNESS AND OTHER MECHANICAL PROPERTYCONSIDERATIONS 176

7.16 Hardness 1767.17 Hardness of Ceramic Materials 1817.18 Tear Strength and Hardness of

Polymers 181

PROPERTY VARIABILITY AND DESIGN/SAFETYFACTORS 183

7.19 Variability of Material Properties 183 Computation of Average and Standard

Deviation Values (CD-ROM) S-287.20 Design/Safety Factors 183


8. Deformation and StrengtheningMechanisms 197


DEFORMATION MECHANISMS FOR METALS 198

8.2 Historical 1988.3 Basic Concepts of Dislocations 1998.4 Characteristics of Dislocations 2018.5 Slip Systems 2038.6 Slip in Single Crystals (CD-ROM) S-318.7 Plastic Deformation of Polycrystalline

Metals 2048.8 Deformation by Twinning

(CD-ROM) S-34

MECHANISMS OF STRENGTHENING INMETALS 206

8.9 Strengthening by Grain SizeReduction 206

8.10 Solid-Solution Strengthening 2088.11 Strain Hardening 210

RECOVERY, RECRYSTALLIZATION, AND GRAINGROWTH 213

8.12 Recovery 2138.13 Recrystallization 2138.14 Grain Growth 218

DEFORMATION MECHANISMS FOR CERAMICMATERIALS 219

8.15 Crystalline Ceramics 2208.16 Noncrystalline Ceramics 220

MECHANISMS OF DEFORMATION AND FORSTRENGTHENING OF POLYMERS 221

8.17 Deformation of SemicrystallinePolymers 221

8.18a Factors That Influence the MechanicalProperties of Semicrystalline Polymers[Detailed Version (CD-ROM)] S-35

8.18b Factors That Influence the MechanicalProperties of Semicrystalline Polymers(Concise Version) 223

8.19 Deformation of Elastomers 224Summary 227Important Terms and Concepts 228References 228Questions and Problems 228

9. Failure 234


FRACTURE 235

9.2 Fundamentals of Fracture 2359.3 Ductile Fracture 236 Fractographic Studies (CD-ROM) S-38

9.4 Brittle Fracture 2389.5a Principles of Fracture Mechanics

[Detailed Version (CD-ROM)] S-38

9.5b Principles of Fracture Mechanics(Concise Version) 238

9.6 Brittle Fracture of Ceramics 248 Static Fatigue (CD-ROM) S-53

9.7 Fracture of Polymers 2499.8 Impact Fracture Testing 250

xiv Contents

FATIGUE 255

9.9 Cyclic Stresses 2559.10 The SN Curve 2579.11 Fatigue in Polymeric Materials 2609.12a Crack Initiation and Propagation


9.12b Crack Initiation and Propagation(Concise Version) 260

9.13 Crack Propagation Rate(CD-ROM) S-57

9.14 Factors That Affect Fatigue Life 2639.15 Environmental Effects (CD-ROM) S-62

CREEP 265

9.16 Generalized Creep Behavior 2669.17a Stress and Temperature Effects


9.17b Stress and Temperature Effects (ConciseVersion) 267

9.18 Data Extrapolation Methods(CD-ROM) S-65

9.19 Alloys for High-Temperature Use 2689.20 Creep in Ceramic and Polymeric

Materials 269Summary 269Important Terms and Concepts 272References 272Questions and Problems 273

10 Phase Diagrams 281


DEFINITIONS AND BASIC CONCEPTS 282

10.2 Solubility Limit 28310.3 Phases 28310.4 Microstructure 28410.5 Phase Equilibria 284

EQUILIBRIUM PHASE DIAGRAMS 285

10.6 Binary Isomorphous Systems 28610.7 Interpretation of Phase Diagrams 28810.8 Development of Microstructure in

Isomorphous Alloys (CD-ROM) S-67

10.9 Mechanical Properties of IsomorphousAlloys 292

10.10 Binary Eutectic Systems 29210.11 Development of Microstructure in

Eutectic Alloys (CD-ROM) S-70

10.12 Equilibrium Diagrams HavingIntermediate Phases or Compounds 297

10.13 Eutectoid and Peritectic Reactions 29810.14 Congruent Phase Transformations 301

10.15 Ceramic Phase Diagrams (CD-ROM)S-77

10.16 Ternary Phase Diagrams 30110.17 The Gibbs Phase Rule (CD-ROM) S-81

THE IRONCARBON SYSTEM 302

10.18 The IronIron Carbide (FeFe3C)Phase Diagram 302

10.19 Development of Microstructures inIronCarbon Alloys 305

10.20 The Influence of Other AlloyingElements (CD-ROM) S-83


11 Phase Transformations 323


PHASE TRANSFORMATIONS IN METALS 324

11.2 Basic Concepts 32511.3 The Kinetics of Solid-State

Reactions 32511.4 Multiphase Transformations 327

MICROSTRUCTURAL AND PROPERTY CHANGES INIRONCARBON ALLOYS 327

11.5 Isothermal TransformationDiagrams 328

11.6 Continuous Cooling TransformationDiagrams (CD-ROM) S-85

11.7 Mechanical Behavior of IronCarbonAlloys 339

11.8 Tempered Martensite 34411.9 Review of Phase Transformations for

IronCarbon Alloys 346

PRECIPITATION HARDENING 347

11.10 Heat Treatments 34711.11 Mechanism of Hardening 34911.12 Miscellaneous Considerations 351

CRYSTALLIZATION, MELTING, AND GLASSTRANSITION PHENOMENA IN POLYMERS 352

11.13 Crystallization 35311.14 Melting 35411.15 The Glass Transition 35411.16 Melting and Glass Transition

Temperatures 35411.17 Factors That Influence Melting and

Glass Transition Temperatures(CD-ROM) S-87

Contents xv


12. Electrical Properties 365


ELECTRICAL CONDUCTION 366

12.2 Ohms Law 36612.3 Electrical Conductivity 36712.4 Electronic and Ionic Conduction 36812.5 Energy Band Structures in Solids 36812.6 Conduction in Terms of Band and

Atomic Bonding Models 37112.7 Electron Mobility 37212.8 Electrical Resistivity of Metals 37312.9 Electrical Characteristics of Commercial

Alloys 376

SEMICONDUCTIVITY 376

12.10 Intrinsic Semiconduction 37712.11 Extrinsic Semiconduction 37912.12 The Temperature Variation of

Conductivity and CarrierConcentration 383

12.13 The Hall Effect (CD-ROM) S-9112.14 Semiconductor Devices (CD-ROM) S-93

ELECTRICAL CONDUCTION IN IONIC CERAMICSAND IN POLYMERS 389

12.15 Conduction in Ionic Materials 38912.16 Electrical Properties of Polymers 390

DIELECTRIC BEHAVIOR 391

12.17 Capacitance (CD-ROM) S-9912.18 Field Vectors and Polarization

(CD-ROM) S-101

12.19 Types of Polarization (CD-ROM) S-10512.20 Frequency Dependence of the Dielectric

Constant (CD-ROM) S-106

12.21 Dielectric Strength (CD-ROM) S-107

12.22 Dielectric Materials (CD-ROM) S-107

OTHER ELECTRICAL CHARACTERISTICS OFMATERIALS 391

12.23 Ferroelectricity (CD-ROM) S-10812.24 Piezoelectricity (CD-ROM) S-109


13. Types and Applicationsof Materials 401


TYPES OF METAL ALLOYS 402

13.2 Ferrous Alloys 40213.3 Nonferrous Alloys 414

TYPES OF CERAMICS 422

13.4 Glasses 42313.5 GlassCeramics 42313.6 Clay Products 42413.7 Refractories 424

Fireclay, Silica, Basic, and SpecialRefractories(CD-ROM) S-110

13.8 Abrasives 42513.9 Cements 425

13.10 Advanced Ceramics (CD-ROM) S-11113.11 Diamond and Graphite 427

TYPES OF POLYMERS 428

13.12 Plastics 42813.13 Elastomers 43113.14 Fibers 43213.15 Miscellaneous Applications 43313.16 Advanced Polymeric Materials

(CD-ROM) S-113


Chapters 14 through 21 discuss just supplementary topics, and arefound only on the CD-ROM (and not in print)

14. Synthesis, Fabrication, and Processingof Materials (CD-ROM) S-118

Learning Objectives S-11914.1 Introduction S-119

FABRICATION OF METALS S-119

14.2 Forming Operations S-11914.3 Casting S-12114.4 Miscellaneous Techniques S-122

xvi Contents

THERMAL PROCESSING OF METALS S-124

14.5 Annealing Processes S-12414.6 Heat Treatment of Steels S-126

FABRICATION OF CERAMIC MATERIALS S-136

14.7 Fabrication and Processing of GlassesS-137

14.8 Fabrication of Clay Products S-14214.9 Powder Pressing S-14514.10 Tape Casting S-149

SYNTHESIS AND FABRICATION OF POLYMERSS-149

14.11 Polymerization S-15014.12 Polymer Additives S-15114.13 Forming Techniques for Plastics S-15314.14 Fabrication of Elastomers S-15514.15 Fabrication of Fibers and Films S-155

Summary S-156Important Terms and Concepts S-157References S-158Questions and Problems S-158

15. Composites (CD-ROM) S-162


PARTICLE-REINFORCED COMPOSITES S-165

15.2 Large-Particle Composites S-16515.3 Dispersion-Strengthened Composites

S-169

FIBER-REINFORCED COMPOSITES S-170

15.4 Influence of Fiber Length S-17015.5 Influence of Fiber Orientation and

Concentration S-17115.6 The Fiber Phase S-18015.7 The Matrix Phase S-18015.8 PolymerMatrix Composites S-18215.9 MetalMatrix Composites S-18515.10 CeramicMatrix Composites S-18615.11 CarbonCarbon Composites S-18815.12 Hybrid Composites S-18915.13 Processing of Fiber-Reinforced

Composites S-189

STRUCTURAL COMPOSITES S-195

15.14 Laminar Composites S-19515.15 Sandwich Panels S-196


16. Corrosion and Degradation ofMaterials (CD-ROM) S-204


CORROSION OF METALS S-205

16.2 Electrochemical Considerations S-20616.3 Corrosion Rates S-21216.4 Prediction of Corrosion Rates S-21416.5 Passivity S-22116.6 Environmental Effects S-22216.7 Forms of Corrosion S-22316.8 Corrosion Environments S-23116.9 Corrosion Prevention S-23216.10 Oxidation S-234

CORROSION OF CERAMIC MATERIALS S-237

DEGRADATION OF POLYMERS S-237

16.11 Swelling and Dissolution S-23816.12 Bond Rupture S-23816.13 Weathering S-241


17. Thermal Properties (CD-ROM) S-247

Learning Objectives S-24817.1 Introduction S-24817.2 Heat Capacity S-24817.3 Thermal Expansion S-25017.4 Thermal Conductivity S-25317.5 Thermal Stresses S-256


18. Magnetic Properties (CD-ROM) S-263

Learning Objectives S-26418.1 Introduction S-26418.2 Basic Concepts S-26418.3 Diamagnetism and Paramagnetism S-26818.4 Ferromagnetism S-27018.5 Antiferromagnetism and

Ferrimagnetism S-27218.6 The Influence of Temperature on

Magnetic Behavior S-27618.7 Domains and Hysteresis S-27618.8 Soft Magnetic Materials S-28018.9 Hard Magnetic Materials S-282

Contents xvii

18.10 Magnetic Storage S-28418.11 Superconductivity S-287


19. Optical Properties (CD-ROM) S-297


BASIC CONCEPTS S-298

19.2 Electromagnetic Radiation S-29819.3 Light Interactions with Solids S-30019.4 Atomic and Electronic Interactions

S-301

OPTICAL PROPERTIES OF METALS S-302OPTICAL PROPERTIES OF NONMETALS S-303

19.5 Refraction S-30319.6 Reflection S-30419.7 Absorption S-30519.8 Transmission S-30819.9 Color S-30919.10 Opacity and Translucency in

Insulators S-310APPLICATIONS OF OPTICAL PHENOMENA S-311

19.11 Luminescence S-31119.12 Photoconductivity S-31219.13 Lasers S-31319.14 Optical Fibers in Communications S-315


20. Materials Selection and DesignConsiderations (CD-ROM) S-324


MATERIALS SELECTION FOR A TORSIONALLYSTRESSED CYLINDRICAL SHAFT S-325

20.2 Strength S-32620.3 Other Property Considerations and the

Final Decision S-331

AUTOMOBILE VALVE SPRING S-332

20.4 Introduction S-33220.5 Automobile Valve Spring S-334

ARTIFICIAL TOTAL HIP REPLACEMENT S-339

20.6 Anatomy of the Hip Joint S-33920.7 Material Requirements S-341

20.8 Materials Employed S-343

THERMAL PROTECTION SYSTEM ON THE SPACESHUTTLE ORBITER S-345

20.9 Introduction S-34520.10 Thermal Protection SystemDesign

Requirements S-34520.11 Thermal Protection

SystemComponents S-347

MATERIALS FOR INTEGRATED CIRCUITPACKAGES S-351

20.12 Introduction S-35120.13 Leadframe Design and Materials S-35320.14 Die Bonding S-35420.15 Wire Bonding S-35620.16 Package Encapsulation S-35820.17 Tape Automated Bonding S-360

Summary S-362References S-363Questions and Problems S-364

21. Economic, Environmental, andSocietal Issues in Materials Scienceand Engineering (CD-ROM) S-368


ECONOMIC CONSIDERATIONS S-369

21.2 Component Design S-37021.3 Materials S-37021.4 Manufacturing Techniques S-370

ENVIRONMENTAL AND SOCIETALCONSIDERATIONS S-371

21.5 Recycling Issues in Materials Scienceand Engineering S-373Summary S-376References S-376

Appendix A The International System ofUnits (SI) 439

Appendix B Properties of SelectedEngineering Materials 441

B.1 Density 441B.2 Modulus of Elasticity 444B.3 Poissons Ratio 448B.4 Strength and Ductility 449B.5 Plane Strain Fracture Toughness 454B.6 Linear Coefficient of Thermal

Expansion 455B.7 Thermal Conductivity 459

xviii Contents

B.8 Specific Heat 462B.9 Electrical Resistivity 464B.10 Metal Alloy Compositions 467

Appendix C Costs and Relative Costsfor Selected Engineering Materials 469

Appendix D Mer Structures forCommon Polymers 475

Appendix E Glass Transition and MeltingTemperatures for Common PolymericMaterials 479

Glossary 480Answers to Selected Problems 495Index 501

List of Symbols

The number of the section in which a symbol is introduced orexplained is given in parentheses.

xix

A areaA angstrom unitAi atomic weight of element i (2.2)

APF atomic packing factor (3.4)%RA ductility, in percent reduction in

area (7.6)a lattice parameter: unit cell

x-axial length (3.4)a crack length of a surface crack

(9.5a, 9.5b)at% atom percent (5.6)

B magnetic flux density (induction)(18.2)

Br magnetic remanence (18.7)BCC body-centered cubic crystal

structure (3.4)b lattice parameter: unit cell

y-axial length (3.11)b Burgers vector (5.7)C capacitance (12.17)Ci concentration (composition) of

component i in wt% (5.6)Ci concentration (composition) of

component i in at% (5.6)Cv , Cp heat capacity at constant

volume, pressure (17.2)CPR corrosion penetration rate (16.3)CVN Charpy V-notch (9.8)

%CW percent cold work (8.11)c lattice parameter: unit cell

z-axial length (3.11)c velocity of electromagnetic

radiation in a vacuum (19.2)D diffusion coefficient (6.3)

D dielectric displacement (12.18)d diameterd average grain diameter (8.9)

dhkl interplanar spacing for planes ofMiller indices h, k, and l (3.19)

E energy (2.5)E modulus of elasticity or Youngs

modulus (7.3)E electric field intensity (12.3)

Ef Fermi energy (12.5)Eg band gap energy (12.6)

Er(t) relaxation modulus (7.15)%EL ductility, in percent elongation

(7.6)e electric charge per electron

(12.7)e electron (16.2)

erf Gaussian error function (6.4)exp e, the base for natural

logarithmsF force, interatomic or mechanical

(2.5, 7.2)F Faraday constant (16.2)

FCC face-centered cubic crystalstructure (3.4)

G shear modulus (7.3)H magnetic field strength (18.2)

Hc magnetic coercivity (18.7)HB Brinell hardness (7.16)

HCP hexagonal close-packed crystalstructure (3.4)

HK Knoop hardness (7.16)HRB, HRF Rockwell hardness: B and F

scales (7.16)

nn number-average degree ofpolymerization (4.5)

nw weight-average degree ofpolymerization (4.5)

P dielectric polarization (12.18)PB ratio PillingBedworth ratio (16.10)

p number of holes per cubic meter(12.10)

Q activation energyQ magnitude of charge stored

(12.17)R atomic radius (3.4)R gas constantr interatomic distance (2.5)r reaction rate (11.3, 16.3)

rA , rC anion and cation ionic radii (3.6)S fatigue stress amplitude (9.10)

SEM scanning electron microscopy ormicroscope

T temperatureTc Curie temperature (18.6)TC superconducting critical

temperature (18.11)Tg glass transition temperature

(11.15)Tm melting temperature

TEM transmission electronmicroscopy or microscope

TS tensile strength (7.6)t time

tr rupture lifetime (9.16)Ur modulus of resilience (7.6)

[uvw] indices for a crystallographicdirection (3.12)

V electrical potential difference(voltage) (12.2)

VC unit cell volume (3.4)VC corrosion potential (16.4)VH Hall voltage (12.13)Vi volume fraction of phase i (10.7)v velocity

vol% volume percentWi mass fraction of phase i (10.7)

wt% weight percent (5.6)

xx List of Symbols

HR15N, HR45W superficial Rockwell hardness:15N and 45W scales (7.16)

HV Vickers hardness (7.16)h Plancks constant (19.2)

(hkl) Miller indices for acrystallographic plane (3.13)

I electric current (12.2)I intensity of electromagnetic

radiation (19.3)i current density (16.3)

iC corrosion current density (16.4)J diffusion flux (6.3)J electric current density (12.3)

K stress intensity factor (9.5a)Kc fracture toughness (9.5a, 9.5b)

KIc plane strain fracture toughnessfor mode I crack surfacedisplacement (9.5a, 9.5b)

k Boltzmanns constant (5.2)k thermal conductivity (17.4)l length

lc critical fiber length (15.4)ln natural logarithm

log logarithm taken to base 10M magnetization (18.2)

M n polymer number-averagemolecular weight (4.5)

M w polymer weight-averagemolecular weight (4.5)

mol% mole percentN number of fatigue cycles (9.10)

NA Avogadros number (3.5)Nf fatigue life (9.10)n principal quantum number (2.3)n number of atoms per unit cell

(3.5)n strain-hardening exponent (7.7)n number of electrons in an

electrochemical reaction (16.2)n number of conducting electrons

per cubic meter (12.7)n index of refraction (19.5)

n for ceramics, the number offormula units per unit cell (3.7)

List of Symbols xxi

x lengthx space coordinateY dimensionless parameter or

function in fracture toughnessexpression (9.5a, 9.5b)

y space coordinatez space coordinate lattice parameter: unit cell yz

interaxial angle (3.11), , phase designations

l linear coefficient of thermalexpansion (17.3)

lattice parameter: unit cell xzinteraxial angle (3.11)

lattice parameter: unit cell xyinteraxial angle (3.11)

shear strain (7.2) finite change in a parameter the

symbol of which it precedes engineering strain (7.2) dielectric permittivity (12.17)

r dielectric constant or relativepermittivity (12.17)

.

s steady-state creep rate (9.16)T true strain (7.7) viscosity (8.16) overvoltage (16.4) Bragg diffraction angle (3.19)

D Debye temperature (17.2) wavelength of electromagnetic

radiation (3.19) magnetic permeability (18.2)

B Bohr magneton (18.2)r relative magnetic permeability

(18.2)e electron mobility (12.7)h hole mobility (12.10)

Poissons ratio (7.5) frequency of electromagnetic

radiation (19.2)

density (3.5)

electrical resistivity (12.2)

t radius of curvature at the tip ofa crack (9.5a, 9.5b)

engineering stress, tensile orcompressive (7.2)

electrical conductivity (12.3)* longitudinal strength

(composite) (15.5)c critical stress for crack

propagation (9.5a, 9.5b)fs flexural strength (7.10)m maximum stress (9.5a, 9.5b)m mean stress (9.9)m stress in matrix at composite

failure (15.5)T true stress (7.7)w safe or working stress (7.20)y yield strength (7.6)

shear stress (7.2)c fibermatrix bond strength/

matrix shear yield strength(15.4)

crss critical resolved shear stress(8.6)

m magnetic susceptibility (18.2)

SUBSCRIPTSc composite

cd discontinuous fibrous compositecl longitudinal direction (aligned

fibrous composite)ct transverse direction (aligned

fibrous composite)f finalf at fracturef fiberi instantaneous

m matrixm, max maximum

min minimum0 original0 at equilibrium0 in a vacuum

C h a p t e r 1 / Introduction

A familiar item that is fabricated from three different material types is the beveragecontainer. Beverages are marketed in aluminum (metal) cans (top), glass (ceramic) bot-

tles (center), and plastic (polymer) bottles (bottom). (Permission to use these photo-

graphs was granted by the Coca-Cola Company.)

1

L e a r n i n g O b j e c t i v e sAfter careful study of this chapter you should be able to do the following:

1. List six different property classifications of mate-rials that determine their applicability.

2. Cite the four components that are involved in thedesign, production, and utilization of materials,and briefly describe the interrelationships be-tween these components.

3. Cite three criteria that are important in the mate-rials selection process.

1.1 HISTORICAL PERSPECTIVEMaterials are probably more deep-seated in our culture than most of us realize.Transportation, housing, clothing, communication, recreation, and food produc-tionvirtually every segment of our everyday lives is influenced to one degree oranother by materials. Historically, the development and advancement of societieshave been intimately tied to the members ability to produce and manipulate materi-als to fill their needs. In fact, early civilizations have been designated by the levelof their materials development (i.e., Stone Age, Bronze Age).

The earliest humans had access to only a very limited number of materials,those that occur naturally: stone, wood, clay, skins, and so on. With time theydiscovered techniques for producing materials that had properties superior to thoseof the natural ones; these new materials included pottery and various metals. Fur-thermore, it was discovered that the properties of a material could be altered byheat treatments and by the addition of other substances. At this point, materialsutilization was totally a selection process, that is, deciding from a given, ratherlimited set of materials the one that was best suited for an application by virtue ofits characteristics. It was not until relatively recent times that scientists came tounderstand the relationships between the structural elements of materials and theirproperties. This knowledge, acquired in the past 60 years or so, has empoweredthem to fashion, to a large degree, the characteristics of materials. Thus, tens ofthousands of different materials have evolved with rather specialized characteristicsthat meet the needs of our modern and complex society; these include metals,plastics, glasses, and fibers.

The development of many technologies that make our existence so comfortablehas been intimately associated with the accessibility of suitable materials. An ad-vancement in the understanding of a material type is often the forerunner to thestepwise progression of a technology. For example, automobiles would not havebeen possible without the availability of inexpensive steel or some other comparablesubstitute. In our contemporary era, sophisticated electronic devices rely on compo-nents that are made from what are called semiconducting materials.

1.2 MATERIALS SCIENCE AND ENGINEERINGThe discipline of materials science involves investigating the relationships that existbetween the structures and properties of materials. In contrast, materials engineeringis, on the basis of these structureproperty correlations, designing or engineeringthe structure of a material to produce a predetermined set of properties. Throughoutthis text we draw attention to the relationships between material properties andstructural elements.

2

4. (a) List the three primary classifications of solidmaterials, and then cite the distinctive chemi-cal feature of each.

(b) Note the other three types of materials and,for each, its distinctive feature(s).

1.2 Materials Science and Engineering 3

Structure is at this point a nebulous term that deserves some explanation.In brief, the structure of a material usually relates to the arrangement of its internalcomponents. Subatomic structure involves electrons within the individual atomsand interactions with their nuclei. On an atomic level, structure encompasses theorganization of atoms or molecules relative to one another. The next larger struc-tural realm, which contains large groups of atoms that are normally agglomeratedtogether, is termed microscopic, meaning that which is subject to direct observa-tion using some type of microscope. Finally, structural elements that may be viewedwith the naked eye are termed macroscopic.

The notion of property deserves elaboration. While in service use, all materi-als are exposed to external stimuli that evoke some type of response. For example,a specimen subjected to forces will experience deformation; or a polished metalsurface will reflect light. Property is a material trait in terms of the kind andmagnitude of response to a specific imposed stimulus. Generally, definitions ofproperties are made independent of material shape and size.

Virtually all important properties of solid materials may be grouped into sixdifferent categories: mechanical, electrical, thermal, magnetic, optical, and deterio-rative. For each there is a characteristic type of stimulus capable of provokingdifferent responses. Mechanical properties relate deformation to an applied loador force; examples include elastic modulus and strength. For electrical properties,such as electrical conductivity and dielectric constant, the stimulus is an electricfield. The thermal behavior of solids can be represented in terms of heat capacity andthermal conductivity. Magnetic properties demonstrate the response of a material tothe application of a magnetic field. For optical properties, the stimulus is electromag-netic or light radiation; index of refraction and reflectivity are representative opticalproperties. Finally, deteriorative characteristics indicate the chemical reactivity ofmaterials. The chapters that follow discuss properties that fall within each of thesesix classifications.

In addition to structure and properties, two other important components areinvolved in the science and engineering of materials, viz. processing and perfor-mance. With regard to the relationships of these four components, the structureof a material will depend on how it is processed. Furthermore, a materials perfor-mance will be a function of its properties. Thus, the interrelationship betweenprocessing, structure, properties, and performance is linear, as depicted in theschematic illustration shown in Figure 1.1. Throughout this text we draw attentionto the relationships among these four components in terms of the design, production,and utilization of materials.

We now present an example of these processing-structure-properties-perfor-mance principles with Figure 1.2, a photograph showing three thin disk specimensplaced over some printed matter. It is obvious that the optical properties (i.e., thelight transmittance) of each of the three materials are different; the one on the leftis transparent (i.e., virtually all of the reflected light passes through it), whereasthe disks in the center and on the right are, respectively, translucent and opaque.All of these specimens are of the same material, aluminum oxide, but the leftmostone is what we call a single crystalthat is, it is highly perfectwhich gives riseto its transparency. The center one is composed of numerous and very small single

Processing Structure Properties Performance

FIGURE 1.1 The four components of the discipline of materialsscience and engineering and their linear interrelationship.

crystals that are all connected; the boundaries between these small crystals scattera portion of the light reflected from the printed page, which makes this materialoptically translucent. And finally, the specimen on the right is composed not onlyof many small, interconnected crystals, but also of a large number of very smallpores or void spaces. These pores also effectively scatter the reflected light andrender this material opaque.

Thus, the structures of these three specimens are different in terms of crystalboundaries and pores, which affect the optical transmittance properties. Further-more, each material was produced using a different processing technique. And, ofcourse, if optical transmittance is an important parameter relative to the ultimatein-service application, the performance of each material will be different.

1.3 WHY STUDY MATERIALS SCIENCE AND ENGINEERING?Why do we study materials? Many an applied scientist or engineer, whether mechan-ical, civil, chemical, or electrical, will at one time or another be exposed to adesign problem involving materials. Examples might include a transmission gear,the superstructure for a building, an oil refinery component, or an integrated circuitchip. Of course, materials scientists and engineers are specialists who are totallyinvolved in the investigation and design of materials.

Many times, a materials problem is one of selecting the right material from themany thousands that are available. There are several criteria on which the finaldecision is normally based. First of all, the in-service conditions must be character-ized, for these will dictate the properties required of the material. On only rareoccasions does a material possess the maximum or ideal combination of properties.Thus, it may be necessary to trade off one characteristic for another. The classicexample involves strength and ductility; normally, a material having a high strengthwill have only a limited ductility. In such cases a reasonable compromise betweentwo or more properties may be necessary.

A second selection consideration is any deterioration of material propertiesthat may occur during service operation. For example, significant reductions inmechanical strength may result from exposure to elevated temperatures or corrosiveenvironments.

Finally, probably the overriding consideration is that of economics: What willthe finished product cost? A material may be found that has the ideal set of

4 Chapter 1 / Introduction

FIGURE 1.2Photograph showing the light

transmittance of three aluminum oxidespecimens. From left to right: single-crystal material (sapphire), which is

transparent; a polycrystalline and fullydense (nonporous) material, which is

translucent; and a polycrystallinematerial that contains approximately 5%

porosity, which is opaque. (Specimenpreparation, P. A. Lessing; photography

by J. Telford.)

1.4 Classification of Materials 5

properties but is prohibitively expensive. Here again, some compromise is inevitable.The cost of a finished piece also includes any expense incurred during fabricationto produce the desired shape.

The more familiar an engineer or scientist is with the various characteristicsand structureproperty relationships, as well as processing techniques of materials,the more proficient and confident he or she will be to make judicious materialschoices based on these criteria.

1.4 CLASSIFICATION OF MATERIALSSolid materials have been conveniently grouped into three basic classifications:metals, ceramics, and polymers. This scheme is based primarily on chemical makeupand atomic structure, and most materials fall into one distinct grouping or another,although there are some intermediates. In addition, there are three other groupsof important engineering materialscomposites, semiconductors, and biomaterials.Composites consist of combinations of two or more different materials, whereassemiconductors are utilized because of their unusual electrical characteristics; bio-materials are implanted into the human body. A brief explanation of the materialtypes and representative characteristics is offered next.

METALSMetallic materials are normally combinations of metallic elements. They have largenumbers of nonlocalized electrons; that is, these electrons are not bound to particularatoms. Many properties of metals are directly attributable to these electrons. Metalsare extremely good conductors of electricity and heat and are not transparent tovisible light; a polished metal surface has a lustrous appearance. Furthermore,metals are quite strong, yet deformable, which accounts for their extensive use instructural applications.

CERAMICSCeramics are compounds between metallic and nonmetallic elements; they are mostfrequently oxides, nitrides, and carbides. The wide range of materials that fallswithin this classification includes ceramics that are composed of clay minerals,cement, and glass. These materials are typically insulative to the passage of electricityand heat, and are more resistant to high temperatures and harsh environments thanmetals and polymers. With regard to mechanical behavior, ceramics are hard butvery brittle.

POLYMERSPolymers include the familiar plastic and rubber materials. Many of them are organiccompounds that are chemically based on carbon, hydrogen, and other nonmetallicelements; furthermore, they have very large molecular structures. These materialstypically have low densities and may be extremely flexible.

COMPOSITESA number of composite materials have been engineered that consist of more thanone material type. Fiberglass is a familiar example, in which glass fibers are embed-ded within a polymeric material. A composite is designed to display a combinationof the best characteristics of each of the component materials. Fiberglass acquiresstrength from the glass and flexibility from the polymer. Many of the recent materialdevelopments have involved composite materials.

SEMICONDUCTORSSemiconductors have electrical properties that are intermediate between the electri-cal conductors and insulators. Furthermore, the electrical characteristics of thesematerials are extremely sensitive to the presence of minute concentrations of impu-rity atoms, which concentrations may be controlled over very small spatial regions.The semiconductors have made possible the advent of integrated circuitry that hastotally revolutionized the electronics and computer industries (not to mention ourlives) over the past two decades.

BIOMATERIALSBiomaterials are employed in components implanted into the human body forreplacement of diseased or damaged body parts. These materials must not producetoxic substances and must be compatible with body tissues (i.e., must not causeadverse biological reactions). All of the above materialsmetals, ceramics, poly-mers, composites, and semiconductorsmay be used as biomaterials. For example,in Section 20.8 are discussed some of the biomaterials that are utilized in artificialhip replacements.

1.5 ADVANCED MATERIALSMaterials that are utilized in high-technology (or high-tech) applications are some-times termed advanced materials. By high technology we mean a device or productthat operates or functions using relatively intricate and sophisticated principles;examples include electronic equipment (VCRs, CD players, etc.), computers, fiber-optic systems, spacecraft, aircraft, and military rocketry. These advanced materialsare typically either traditional materials whose properties have been enhanced ornewly developed, high-performance materials. Furthermore, they may be of allmaterial types (e.g., metals, ceramics, polymers), and are normally relatively expen-sive. In subsequent chapters are discussed the properties and applications of anumber of advanced materialsfor example, materials that are used for lasers,integrated circuits, magnetic information storage, liquid crystal displays (LCDs),fiber optics, and the thermal protection system for the Space Shuttle Orbiter.

1.6 MODERN MATERIALS NEEDSIn spite of the tremendous progress that has been made in the discipline of materialsscience and engineering within the past few years, there still remain technologicalchallenges, including the development of even more sophisticated and specializedmaterials, as well as consideration of the environmental impact of materials produc-tion. Some comment is appropriate relative to these issues so as to round outthis perspective.

Nuclear energy holds some promise, but the solutions to the many problemsthat remain will necessarily involve materials, from fuels to containment structuresto facilities for the disposal of radioactive waste.

Significant quantities of energy are involved in transportation. Reducing theweight of transportation vehicles (automobiles, aircraft, trains, etc.), as well asincreasing engine operating temperatures, will enhance fuel efficiency. New high-strength, low-density structural materials remain to be developed, as well as materi-als that have higher-temperature capabilities, for use in engine components.


References 7

Furthermore, there is a recognized need to find new, economical sources ofenergy, and to use the present resources more efficiently. Materials will undoub-tedly play a significant role in these developments. For example, the direct con-version of solar into electrical energy has been demonstrated. Solar cells employsome rather complex and expensive materials. To ensure a viable technology,materials that are highly efficient in this conversion process yet less costly mustbe developed.

Furthermore, environmental quality depends on our ability to control air andwater pollution. Pollution control techniques employ various materials. In addition,materials processing and refinement methods need to be improved so that theyproduce less environmental degradation, that is, less pollution and less despoilageof the landscape from the mining of raw materials. Also, in some materials manufac-turing processes, toxic substances are produced, and the ecological impact of theirdisposal must be considered.

Many materials that we use are derived from resources that are nonrenewable,that is, not capable of being regenerated. These include polymers, for which theprime raw material is oil, and some metals. These nonrenewable resources aregradually becoming depleted, which necessitates: 1) the discovery of additionalreserves, 2) the development of new materials having comparable properties withless adverse environmental impact, and/or 3) increased recycling efforts and thedevelopment of new recycling technologies. As a consequence of the economics ofnot only production but also environmental impact and ecological factors, it isbecoming increasingly important to consider the cradle-to-grave life cycle ofmaterials relative to the overall manufacturing process.

The roles that materials scientists and engineers play relative to these, aswell as other environmental and societal issues, are discussed in more detail inChapter 21.

R E F E R E N C E S

The October 1986 issue of Scientific American, Vol.255, No. 4, is devoted entirely to various advancedmaterials and their uses. Other references forChapter 1 are textbooks that cover the basic funda-mentals of the field of materials science and engi-neering.Ashby, M. F. and D. R. H. Jones, Engineering Mate-

rials 1, An Introduction to Their Properties andApplications, 2nd edition, Pergamon Press, Ox-ford, 1996.

Ashby, M. F. and D. R. H. Jones, Engineering Mate-rials 2, An Introduction to Microstructures, Pro-cessing and Design, Pergamon Press, Oxford,1986.

Askeland, D. R., The Science and Engineering ofMaterials, 3rd edition, Brooks/Cole PublishingCo., Pacific Grove, CA, 1994.

Barrett, C. R., W. D. Nix, and A. S. Tetelman, ThePrinciples of Engineering Materials, PrenticeHall, Inc., Englewood Cliffs, NJ, 1973.

Flinn, R. A. and P. K. Trojan, Engineering Ma-terials and Their Applications, 4th edition, JohnWiley & Sons, New York, 1990.

Jacobs, J. A. and T. F. Kilduff, Engineering Materi-als Technology, 3rd edition, Prentice Hall, Up-per Saddle River, NJ, 1996.

McMahon, C. J., Jr. and C. D. Graham, Jr., Intro-duction to Engineering Materials: The Bicycleand the Walkman, Merion Books, Philadel-phia, 1992.

Murray, G. T., Introduction to Engineering Materi-alsBehavior, Properties, and Selection, Mar-cel Dekker, Inc., New York, 1993.

Ohring, M., Engineering Materials Science, Aca-demic Press, San Diego, CA, 1995.

Ralls, K. M., T. H. Courtney, and J. Wulff, Intro-duction to Materials Science and Engineering,John Wiley & Sons, New York, 1976.

Schaffer, J. P., A. Saxena, S. D. Antolovich, T. H.Sanders, Jr., and S. B. Warner, The Science and

Design of Engineering Materials, 2nd edition,WCB/McGraw-Hill, New York, 1999.

Shackelford, J. F., Introduction to Materials Sciencefor Engineers, 5th edition, Prentice Hall, Inc.,Upper Saddle River, NJ, 2000.

Smith, W. F., Principles of Materials Science and

Engineering, 3rd edition, McGraw-Hill BookCompany, New York, 1995.

Van Vlack, L. H., Elements of Materials Scienceand Engineering, 6th edition, Addison-WesleyPublishing Co., Reading, MA, 1989.


C h a p t e r 2 / Atomic Structure andInteratomic Bonding

This micrograph, whichrepresents the surface of a

gold specimen, was taken

with a sophisticated atomic

force microscope (AFM). In-

dividual atoms for this (111)

crystallographic surface

plane are resolved. Also

note the dimensional scale

(in the nanometer range) be-

low the micrograph. (Image

courtesy of Dr. Michael

Green, TopoMetrix Corpo-

ration.)

Why Study Atomic Structure and Interatomic Bonding?

An important reason to have an understandingof interatomic bonding in solids is that, in someinstances, the type of bond allows us to explain amaterials properties. For example, consider car-bon, which may exist as both graphite anddiamond. Whereas graphite is relatively soft and

9

has a greasy feel to it, diamond is the hardestknown material. This dramatic disparity in proper-ties is directly attributable to a type of interatomicbonding found in graphite that does not exist indiamond (see Section 3.9).

L e a r n i n g O b j e c t i v e sAfter careful study of this chapter you should be able to do the following:

1. Name the two atomic models cited, and note thedifferences between them.

2. Describe the important quantum-mechanicalprinciple that relates to electron energies.

3. (a) Schematically plot attractive, repulsive, andnet energies versus interatomic separationfor two atoms or ions.

(b) Note on this plot the equilibrium separationand the bonding energy.

4. (a) Briefly describe ionic, covalent, metallic, hy-drogen, and van der Waals bonds.

(b) Note what materials exhibit each of thesebonding types.

2.1 INTRODUCTIONSome of the important properties of solid materials depend on geometrical atomicarrangements, and also the interactions that exist among constituent atoms ormolecules. This chapter, by way of preparation for subsequent discussions, considersseveral fundamental and important concepts, namely: atomic structure, electronconfigurations in atoms and the periodic table, and the various types of primaryand secondary interatomic bonds that hold together the atoms comprising a solid.These topics are reviewed briefly, under the assumption that some of the materialis familiar to the reader.

A T O M I C S T R U C T U R E2.2 FUNDAMENTAL CONCEPTS

Each atom consists of a very small nucleus composed of protons and neutrons,which is encircled by moving electrons. Both electrons and protons are electricallycharged, the charge magnitude being 1.60 1019 C, which is negative in sign forelectrons and positive for protons; neutrons are electrically neutral. Masses forthese subatomic particles are infinitesimally small; protons and neutrons have ap-proximately the same mass, 1.67 1027 kg, which is significantly larger than thatof an electron, 9.11 1031 kg.

Each chemical element is characterized by the number of protons in the nucleus,or the atomic number (Z).1 For an electrically neutral or complete atom, the atomicnumber also equals the number of electrons. This atomic number ranges in integralunits from 1 for hydrogen to 92 for uranium, the highest of the naturally oc-curring elements.

The atomic mass (A) of a specific atom may be expressed as the sum of themasses of protons and neutrons within the nucleus. Although the number of protonsis the same for all atoms of a given element, the number of neutrons (N) may bevariable. Thus atoms of some elements have two or more different atomic masses,which are called isotopes. The atomic weight of an element corresponds to theweighted average of the atomic masses of the atoms naturally occurring isotopes.2

The atomic mass unit (amu) may be used for computations of atomic weight. Ascale has been established whereby 1 amu is defined as of the atomic mass of

10

1 Terms appearing in boldface type are defined in the Glossary, which follows Appendix E.2 The term atomic mass is really more accurate than atomic weight inasmuch as, in thiscontext, we are dealing with masses and not weights. However, atomic weight is, by conven-tion, the preferred terminology, and will be used throughout this book. The reader shouldnote that it is not necessary to divide molecular weight by the gravitational constant.

2.3 Electrons in Atoms 11

the most common isotope of carbon, carbon 12 (12C) (A 12.00000). Within thisscheme, the masses of protons and neutrons are slightly greater than unity, and

A Z N (2.1)

The atomic weight of an element or the molecular weight of a compound may bespecified on the basis of amu per atom (molecule) or mass per mole of material.In one mole of a substance there are 6.023 1023 (Avogadros number) atoms ormolecules. These two atomic weight schemes are related through the followingequation:

1 amu/atom (or molecule) 1 g/mol

For example, the atomic weight of iron is 55.85 amu/atom, or 55.85 g/mol. Sometimesuse of amu per atom or molecule is convenient; on other occasions g (or kg)/molis preferred; the latter is used in this book.

2.3 ELECTRONS IN ATOMSATOMIC MODELSDuring the latter part of the nineteenth century it was realized that many phenomenainvolving electrons in solids could not be explained in terms of classical mechanics.What followed was the establishment of a set of principles and laws that governsystems of atomic and subatomic entities that came to be known as quantummechanics. An understanding of the behavior of electrons in atoms and crystallinesolids necessarily involves the discussion of quantum-mechanical concepts. How-ever, a detailed exploration of these principles is beyond the scope of this book,and only a very superficial and simplified treatment is given.

One early outgrowth of quantum mechanics was the simplified Bohr atomicmodel, in which electrons are assumed to revolve around the atomic nucleus indiscrete orbitals, and the position of any particular electron is more or less welldefined in terms of its orbital. This model of the atom is represented in Figure 2.1.

Another important quantum-mechanical principle stipulates that the energiesof electrons are quantized; that is, electrons are permitted to have only specificvalues of energy. An electron may change energy, but in doing so it must make aquantum jump either to an allowed higher energy (with absorption of energy) orto a lower energy (with emission of energy). Often, it is convenient to think ofthese allowed electron energies as being associated with energy levels or states.

Orbital electron

Nucleus

FIGURE 2.1 Schematic representation ofthe Bohr atom.

12 Chapter 2 / Atomic Structure and Interatomic Bonding

These states do not vary continuously with energy; that is, adjacent states areseparated by finite energies. For example, allowed states for the Bohr hydrogenatom are represented in Figure 2.2a. These energies are taken to be negative,whereas the zero reference is the unbound or free electron. Of course, the singleelectron associated with the hydrogen atom will fill only one of these states.

Thus, the Bohr model represents an early attempt to describe electrons in atoms,in terms of both position (electron orbitals) and energy (quantized energy levels).

This Bohr model was eventually found to have some significant limitationsbecause of its inability to explain several phenomena involving electrons. A resolu-tion was reached with a wave-mechanical model, in which the electron is consideredto exhibit both wavelike and particle-like characteristics. With this model, an elec-tron is no longer treated as a particle moving in a discrete orbital; but rather,position is considered to be the probability of an electrons being at various locationsaround the nucleus. In other words, position is described by a probability distributionor electron cloud. Figure 2.3 compares Bohr and wave-mechanical models for thehydrogen atom. Both these models are used throughout the course of this book;the choice depends on which model allows the more simple explanation.

QUANTUM NUMBERSUsing wave mechanics, every electron in an atom is characterized by four parameterscalled quantum numbers. The size, shape, and spatial orientation of an electronsprobability density are specified by three of these quantum numbers. Furthermore,Bohr energy levels separate into electron subshells, and quantum numbers dictatethe number of states within each subshell. Shells are specified by a principal quantumnumber n, which may take on integral values beginning with unity; sometimes theseshells are designated by the letters K, L, M, N, O, and so on, which correspond,respectively, to n 1, 2, 3, 4, 5, . . . , as indicated in Table 2.1. It should also be

0 0

1 1018

2 1018

(a) (b)

5

10

15

n = 1 1s

Ene

rgy

(J)

Ene

rgy

(eV)

n = 2

n = 3

2s

3s

2p

3p3d

1.5

3.4

13.6

FIGURE 2.2 (a) Thefirst three electron

energy states for theBohr hydrogen atom.

(b) Electron energystates for the first three

shells of the wave-mechanical hydrogenatom. (Adapted fromW. G. Moffatt, G. W.Pearsall, and J. Wulff,

The Structure andProperties of Materials,Vol. I, Structure, p. 10.

Copyright 1964 byJohn Wiley & Sons,

New York. Reprintedby permission of John

Wiley & Sons, Inc.)


1.0

0

(a) (b)

Orbital electron Nucleus

Prob

abili

ty

Distance from nucleus

FIGURE 2.3 Comparison ofthe (a) Bohr and (b) wave-mechanical atom models interms of electrondistribution. (Adapted fromZ. D. Jastrzebski, TheNature and Properties ofEngineering Materials, 3rdedition, p. 4. Copyright 1987 by John Wiley & Sons,New York. Reprinted bypermission of John Wiley &Sons, Inc.)

Table 2.1 The Number of Available Electron States in Some of theElectron Shells and Subshells

PrincipalNumber of ElectronsShellQuantum Number

of StatesSubshellsNumber n Designation Per Subshell Per Shell1 K s 1 2 2

2 Ls 1 2

8p 3 6

s 1 23 M p 3 6 18

d 5 10

s 1 2

4 Np 3 6

32d 5 10f 7 14


noted that this quantum number, and it only, is also associated with the Bohr model.This quantum number is related to the distance of an electron from the nucleus,or its position.

The second quantum number, l, signifies the subshell, which is denoted by alowercase letteran s, p, d, or f ; it is related to the shape of the electron subshell.In addition, the number of these subshells is restricted by the magnitude of n.Allowable subshells for the several n values are also presented in Table 2.1. Thenumber of energy states for each subshell is determined by the third quantumnumber, ml . For an s subshell, there is a single energy state, whereas for p, d, andf subshells, three, five, and seven states exist, respectively (Table 2.1). In the absenceof an external magnetic field, the states within each subshell are identical. However,when a magnetic field is applied these subshell states split, each state assuming aslightly different energy.

Associated with each electron is a spin moment, which must be oriented eitherup or down. Related to this spin moment is the fourth quantum number, ms , forwhich two values are possible ( and ), one for each of the spin orientations.

Thus, the Bohr model was further refined by wave mechanics, in which theintroduction of three new quantum numbers gives rise to electron subshells withineach shell. A comparison of these two models on this basis is illustrated, for thehydrogen atom, in Figures 2.2a and 2.2b.

A complete energy level diagram for the various shells and subshells using thewave-mechanical model is shown in Figure 2.4. Several features of the diagram areworth noting. First, the smaller the principal quantum number, the lower the energylevel; for example, the energy of a 1s state is less than that of a 2s state, which inturn is lower than the 3s. Second, within each shell, the energy of a subshell levelincreases with the value of the l quantum number. For example, the energy of a3d state is greater than a 3p, which is larger than 3s. Finally, there may be overlapin energy of a state in one shell with states in an adjacent shell, which is especiallytrue of d and f states; for example, the energy of a 3d state is greater than that fora 4s.

Principal quantum number, n

Ene

rgy

1

s

sp

sp

sp

sp

df sp

sp

df

d

d

d

f

2 3 4 5 6 7

FIGURE 2.4 Schematicrepresentation of the relativeenergies of the electrons for thevarious shells and subshells. (FromK. M. Ralls, T. H. Courtney, and J.Wulff, Introduction to MaterialsScience and Engineering, p. 22.Copyright 1976 by John Wiley &Sons, New York. Reprinted bypermission of John Wiley & Sons,Inc.)


ELECTRON CONFIGURATIONSThe preceding discussion has dealt primarily with electron statesvalues of energythat are permitted for electrons. To determine the manner in which these statesare filled with electrons, we use the Pauli exclusion principle, another quantum-mechanical concept. This principle stipulates that each electron state can hold nomore than two electrons, which must have opposite spins. Thus, s, p, d, and fsubshells may each accommodate, respectively, a total of 2, 6, 10, and 14 electrons;Table 2.1 summarizes the maximum number of electrons that may occupy each ofthe first four shells.

Of course, not all possible states in an atom are filled with electrons. For mostatoms, the electrons fill up the lowest possible energy states in the electron shellsand subshells, two electrons (having opposite spins) per state. The energy structurefor a sodium atom is represented schematically in Figure 2.5. When all the electronsoccupy the lowest possible energies in accord with the foregoing restrictions, anatom is said to be in its ground state. However, electron transitions to higher energystates are possible, as discussed in Chapters 12 and 19. The electron configurationor structure of an atom represents the manner in which these states are occupied.In the conventional notation the number of electrons in each subshell is indicatedby a superscript after the shellsubshell designation. For example, the electronconfigurations for hydrogen, helium, and sodium are, respectively, 1s1, 1s2, and1s22s22p63s1. Electron configurations for some of the more common elements arelisted in Table 2.2.

At this point, comments regarding these electron configurations are necessary.First, the valence electrons are those that occupy the outermost filled shell. Theseelectrons are extremely important; as will be seen, they participate in the bondingbetween atoms to form atomic and molecular aggregates. Furthermore, many ofthe physical and chemical properties of solids are based on these valence electrons.

In addition, some atoms have what are termed stable electron configurations;that is, the states within the outermost or valence electron shell are completelyfilled. Normally this corresponds to the occupation of just the s and p states forthe outermost shell by a total of eight electrons, as in neon, argon, and krypton;one exception is helium, which contains only two 1s electrons. These elements (Ne,Ar, Kr, and He) are the inert, or noble, gases, which are virtually unreactivechemically. Some atoms of the elements that have unfilled valence shells assume

Incr

easi

ng e

nerg

y

3p3s

2s

1s

2p

FIGURE 2.5 Schematic representation of thefilled energy states for a sodium atom.


stable electron configurations by gaining or losing electrons to form charged ions,or by sharing electrons with other atoms. This is the basis for some chemicalreactions, and also for atomic bonding in solids, as explained in Section 2.6.

Under special circumstances, the s and p orbitals combine to form hybrid spn

orbitals, where n indicates the number of p orbitals involved, which may have avalue of 1, 2, or 3. The 3A, 4A, and 5A group elements of the periodic table (Figure2.6) are those which most often form these hybrids. The driving force for theformation of hybrid orbitals is a lower energy state for the valence electrons. Forcarbon the sp3 hybrid is of primary importance in organic and polymer chemistries.

Table 2.2 A Listing of the Expected Electron Configurationsfor Some of the Common Elementsa

AtomicElement Symbol Number Electron ConfigurationHydrogen H 1 1s1

Helium He 2 1s2

Lithium Li 3 1s22s1

Beryllium Be 4 1s22s2

Boron B 5 1s22s22p1

Carbon C 6 1s22s22p2

Nitrogen N 7 1s22s22p3

Oxygen O 8 1s22s22p4

Fluorine F 9 1s22s22p5

Neon Ne 10 1s22s22p6

Sodium Na 11 1s22s22p63s1

Magnesium Mg 12 1s22s22p63s2

Aluminum Al 13 1s22s22p63s23p1

Silicon Si 14 1s22s22p63s23p2

Phosphorus P 15 1s22s22p63s23p3

Sulfur S 16 1s22s22p63s23p4

Chlorine Cl 17 1s22s22p63s23p5

Argon Ar 18 1s22s22p63s23p6

Potassium K 19 1s22s22p63s23p64s1

Calcium Ca 20 1s22s22p63s23p64s2

Scandium Sc 21 1s22s22p63s23p63d 14s2

Titanium Ti 22 1s22s22p63s23p63d 24s2

Vanadium V 23 1s22s22p63s23p63d 34s2

Chromium Cr 24 1s22s22p63s23p63d 54s1

Manganese Mn 25 1s22s22p63s23p63d 54s2

Iron Fe 26 1s22s22p63s23p63d 64s2

Cobalt Co 27 1s22s22p63s23p63d 74s2

Nickel Ni 28 1s22s22p63s23p63d 84s2

Copper Cu 29 1s22s22p63s23p63d 104s1

Zinc Zn 30 1s22s22p63s23p63d 104s2

Gallium Ga 31 1s22s22p63s23p63d 104s24p1

Germanium Ge 32 1s22s22p63s23p63d 104s24p2

Arsenic As 33 1s22s22p63s23p63d 104s24p3

Selenium Se 34 1s22s22p63s23p63d 104s24p4

Bromine Br 35 1s22s22p63s23p63d 104s24p5

Krypton Kr 36 1s22s22p63s23p63d 104s24p6

a When some elements covalently bond, they form sp hybrid bonds. This is espe-cially true for C, Si, and Ge.

1

H1.0080

3

Li6.939

4

Be9.0122

11

Na22.990

12

Mg24.312

19

K39.102

20

Ca40.08

37

Rb85.47

38

Sr

21

Sc44.956

39

Y87.62

55

Cs132.91

56

Ba137.34

5

B10.811

13

Al26.982

31

Ga69.72

49

In114.82

81

Tl204.37

6

C12.011

14

Si28.086

32

Ge72.59

50

Sn118.69

82

Pb207.19

7

N14.007

15

P30.974

33

As74.922

51

Sb121.75

83

Bi208.98

8

O15.999

16

S32.064

34

Se78.96

52

Te127.60

84

Po(210)

9

F18.998

17

Cl35.453

35

Br79.91

53

I126.90

85

At(210)

10

Ne20.183

2

He4.0026

18

Ar39.948

36

Kr83.80

54

Xe131.30

86

Rn(222)

22

Ti47.90

40

Zr91.2288.91

72

Hf178.49

23

V50.942

41

Nb92.91

73

Ta180.95

24

Cr51.996

42

Mo95.94

74

W183.85

25

Mn54.938

43

Tc(99)

75

Re186.2

26

Fe55.847

44

Ru101.07

76

Os190.2

27

Co58.933

45

Rh102.91

77

Ir192.2

28

Ni58.71

46

Pd106.4

78

Pt195.09

29

Cu63.54

29

Cu63.54

47

Ag107.87

79

Au196.97

30

Zn65.37

48

Cd112.40

80

Hg200.59

66Dy

162.50

98

Cf(249)

67Ho

164.93

99

Es(254)

68Er

167.26

100

Fm(253)

69Tm

168.93

101

Md(256)

70Yb

173.04

102

No(254)

71Lu

174.97

103

Lw(257)

57La

138.91

89

Ac(227)

58Ce

140.12

90

Th232.04

59Pr

140.91

91

Pa(231)

60Nd

144.24

92

U238.03

61Pm

(145)

93

Np(237)

62Sm

150.35

94

Pu(242)

63Eu

151.96

95

Am(243)

64Gd

157.25

96

Cm(247)

65Tb

158.92

97

Bk(247)

87

Fr(223)

88

Ra(226)

Atomic number

Symbol

Metal

Nonmetal

Intermediate

Atomic weight

KeyIA

IIA

IIIB IVB VB VIB VIIBVIII

IB IIB

IIIA IVA VA VIA VIIA

0

Rare earth series

Actinide series

Acti-nideseries

Rareearthseries

2.4 The Periodic Table 17

The shape of the sp3 hybrid is what determines the 109 (or tetrahedral) anglefound in polymer chains (Chapter 4).

2.4 THE PERIODIC TABLEAll the elements have been classified according to electron configuration in theperiodic table (Figure 2.6). Here, the elements are situated, with increasing atomicnumber, in seven horizontal rows called periods. The arrangement is such that allelements that are arrayed in a given column or group have similar valence electronstructures, as well as chemical and physical properties. These properties changegradually and systematically, moving horizontally across each period.

The elements positioned in Group 0, the rightmost group, are the inert gases,which have filled electron shells and stable electron configurations. Group VIIAand VIA elements are one and two electrons deficient, respectively, from havingstable structures. The Group VIIA elements (F, Cl, Br, I, and At) are sometimestermed the halogens. The alkali and the alkaline earth metals (Li, Na, K, Be, Mg,Ca, etc.) are labeled as Groups IA and IIA, having, respectively, one and twoelectrons in excess of stable structures. The elements in the three long periods,Groups IIIB through IIB, are termed the transition metals, which have partiallyfilled d electron states and in some cases one or two electrons in the next higherenergy shell. Groups IIIA, IVA, and VA (B, Si, Ge, As, etc.) display characteristicsthat are intermediate between the metals and nonmetals by virtue of their valenceelectron structures.

FIGURE 2.6 The periodic table of the elements. The numbers in parentheses arethe atomic weights of the most stable or common isotopes.


As may be noted from the periodic table, most of the elements really comeunder the metal classification. These are sometimes termed electropositive elements,indicating that they are capable of giving up their few valence electrons to becomepositively charged ions. Furthermore, the elements situated on the right-hand sideof the table are electronegative; that is, they readily accept electrons to formnegatively charged ions, or sometimes they share electrons with other atoms. Figure2.7 displays electronegativity values that have been assigned to the various elementsarranged in the periodic table. As a general rule, electronegativity increases inmoving from left to right and from bottom to top. Atoms are more likely to acceptelectrons if their outer shells are almost full, and if they are less shielded from(i.e., closer to) the nucleus.

A T O M I C B O N D I N G I N S O L I D S2.5 BONDING FORCES AND ENERGIES

An understanding of many of the physical properties of materials is predicated ona knowledge of the interatomic forces that bind the atoms together. Perhaps theprinciples of atomic bonding are best illustrated by considering the interactionbetween two isolated atoms as they are brought into close proximity from an infiniteseparation. At large distances, the interactions are negligible; but as the atomsapproach, each exerts forces on the other. These forces are of two types, attractiveand repulsive, and the magnitude of each is a function of the separation or in-teratomic distance. The origin of an attractive force FA depends on the particulartype of bonding that exists between the two atoms. Its magnitude varies with thedistance, as represented schematically in Figure 2.8a. Ultimately, the outer electronshells of the two atoms begin to overlap, and a strong repulsive force FR comesinto play. The net force FN between the two atoms is just the sum of both attractiveand repulsive components; that is,

FN FA FR (2.2)

1

H

3

Li4

Be

11

Na12

Mg

19

K20

Ca

37

Rb38

Sr

21

Sc

39

Y

55

Cs56 57 71

Ba La Lu

5

B

13

Al

31

Ga

49

In

81

Tl

6

C

14

Si

32

Ge

50

Sn

82

Pb

7

N

15

P

33

As

51

Sb

83

Bi

8

O

16

S

34

Se

52

Te

84

Po

9

F

17

Cl

35

Br

53

I

85

At

10

Ne

2

He

18

Ar

36

Kr

54

Xe

86

Rn

22

Ti

40

Zr

72

Hf

23

V

41

Nb

73

Ta

24

Cr

42

Mo

74

W

25

Mn

43

Tc

75

Re

26

Fe

44

Ru

76

Os

27

Co

45

Rh

77

Ir

28

Ni

46

Pd

78

Pt

29

Cu

47

Ag

79

Au

30

Zn

48

Cd

80

Hg

87

Fr88

Ra89 102

Ac No

2.1

1.0 1.5

0.9 1.2

0.8 1.0

0.8

1.3

1.0

0.7 0.9 1.1 1.2

2.0

1.5

1.6

1.7

1.8

2.5

1.8

1.8

1.8

1.8

3.0

2.1

2.0

1.9

1.9

3.5

2.5

2.4

2.1

2.0

4.0

3.0

2.8

2.5

2.2

1.5

1.41.2

1.3

1.6

1.6

1.5

1.6

1.8

1.7

1.5

1.9

1.9

1.8

2.2

2.2

1.8

2.2

2.2

1.8

2.2

2.2

1.9

1.9

2.4

1.6

1.7

1.9

0.7 0.9 1.1 1.7

IA

IIA

IIIB IVB VB VIB VIIBVIII

IB IIB

IIIA IVA VA VIA VIIA

0

FIGURE 2.7 The electronegativity values for the elements. (Adapted from LinusPauling, The Nature of the Chemical Bond, 3rd edition. Copyright 1939 and 1940,3rd edition copyright 1960, by Cornell University. Used by permission of thepublisher, Cornell University Press.)

2.5 Bonding Forces and Energies 19

which is also a function of the interatomic separation, as also plotted in Figure2.8a. When FA and FR balance, or become equal, there is no net force; that is,

FA FR 0 (2.3)

Then a state of equilibrium exists. The centers of the two atoms will remain separatedby the equilibrium spacing r0 , as indicated in Figure 2.8a. For many atoms, r0 isapproximately 0.3 nm (3 A). Once in this position, the two atoms will counteractany attempt to separate them by an attractive force, or to push them together bya repulsive action.

Sometimes it is more convenient to work with the potential energies betweentwo atoms instead of forces. Mathematically, energy (E) and force (F) are related as

E F dr (2.4)Or, for atomic systems,

EN r

FN dr (2.5)

r

FA dr r

FR dr (2.6)

EA ER (2.7)

in which EN , EA , and ER are respectively the net, attractive, and repulsive energiesfor two isolated and adjacent atoms.

+

(a)

(b)

Interatomic separation r

Interatomic separation r

Repulsive force FR

Attractive force FA

Net force FN

Att

ract

ion

Rep

ulsi

on

Forc

e F

Repulsive energy ER

Attractive energy EA

Net energy EN

+

0

0

Att

ract

ion

Rep

ulsi

on

Pot

enti

al e

nerg

y E

r0

E0

FIGURE 2.8 (a) Thedependence of repulsive,attractive, and net forces oninteratomic separation fortwo isolated atoms. (b) Thedependence of repulsive,attractive, and net potential

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