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FIFTH EDITION V Raghavan Materials Science and Engineering A First Course 1 H 1008 1s1 2 He 4003 1s2 3 Li 6939 2s1 4 Be 9012 2s2 5 B 1081 2s22p1 6 C 1201 2s22p2 7 N 1401 2s22p3 8 O 1600 2s22p4 9 F 1900 2s22p5 10 Ne 2018 2s22p6 11 Na 2299 3s1 12 Mg 2431 3s2 13 Al 2698 3s23p1 14 Si 2809 3s23p2 15 P 3097 3s23p3 16 S 3206 3s23p4 17 Cl 3545 3s23p5 18 Ar 3995 3s23p6 19 K 3910 4s1 20 Ca 4008 4s2 21 Sc 4496 3d14s2 22 Ti 4790 3d24s2 23 V 5094 3d34s2 24 Cr 5200 3d54s1 25 Mn 5494 3d54s2 26 Fe 5585 3d64s2 27 Co 5893 3d74s2 28 Ni 5871 3d84s2 29 Cu 6354 3d104s1 30 Zn 6537 3d104s2 31 Ga 6972 4s24p1 32 Ge 7259 4s24p2 33 As 7492 4s24p3 34 Se 7896 4s24p4 35 Br 7991 4s24p5 36 Kr 8380 4s24p6 37 Rb 8547 5s1 38 Sr 8762 5s2 39 Y 8891 4d15s2 40 Zr 9122 4d25s2 41 Nb 9291 4d45s1 42 Mo 9594 4d55s1 43 Tc 99 4d65s1 44 Ru 1011 4d75s1 45 Rh 1029 4d85s1 46 Pd 1064 4d10 47 Ag 1079 4d105s1 48 Cd 1124 4d105s2 49 In 1148 5s25p1 50 Sn 1187 5s25p2 51 Sb 1217 5s25p3 52 Te 1276 5s25p4 53 I 1269 5s25p5 54 Xe 1313 5s25p6 55 Cs 1329 6s1 56 Ba 1373 6s2 57 La 1389 5d16s2 72 Hf 1785 5d26s2 73 Ta 1809 5d36s2 74 W 1839 5d46s2 75 Re 1862 5d56s2 76 Os 1902 5d66s2 77 Ir 1922 5d9 78 Pt 1951 5d96s1 79 Au 1970 5d106s1 80 Hg 2006 5d106s2 81 Tl 2044 6s26p1 82 Pb 2072 6s26p2 83 Bi 2090 6s26p3 84 Po 210 6s26p4 85 At 211 6s26p5 86 Rn 222 6s26p6 87 Fr 223 7s1 88 Ra 226 7s2 89 Ac 227 6d17s2 THE PERIODIC TABLE 1 The atomic number is shown at the top 2 The middle line indicates the atomic mass in amu 3 The outer electron configuration is shown at the bottom 58 Ce 1401 4f26s2 59 Pr 1409 4f36s2 60 Nd 1443 4f46s2 61 Pm 147 4f56s2 62 Sm 1503 4f66s2 63 Eu 1520 4f76s2 64 Gd 1573 4f75d16s2 65 Tb 1589 4f85d16s2 66 Dy 1625 4f106s2 67 Ho 1649 4f116s2 68 Er 1673 4f126s2 69 Tm 1689 4f136s2 70 Yb 1730 4f146s2 71 Lu 1750 4f145d16s2 90 Th 232 6d27s2 91 Pa 231 5f26d17s2 92 U 238 5f36d17s2 93 Np 237 5f57s2 94 Pu 244 5f67s2 95 Am 243 5f77s2 96 Cm 247 5f76d17s2 97 Bk 247 5f86d17s2 98 Cf 251 5f107s2 99 Es 254 5f117s2 100 Fm 257 5f127s2 101 Md 256 5f137s2 102 No 254 5f147s2 103 Lw 257 5f146d17s2 Materials Science and Engineering 2004 by PHI Learning Private Limited New Delhi All rights reserved No part of this book may be reproduced in any form by mimeograph or any other means without permission in writing from the publisher Contents iv Contents Summary 45 Problems 46 Multiple Choice Questions 49 Sources for Experimental Data 52 Suggestions for Further Reading 52 4 Atomic Structure and Chemical Bonding 5380 STRUCTURE OF THE ATOM 54 41 The Quantum States 54 42 The Periodic Table 55 43 Ionization Potential Electron Affinity and Electronegativity 60 CHEMICAL BONDING 63 44 Bond Energy Bond Type and Bond Length 63 45 Ionic Bonding 65 46 Covalent Bonding 68 47 Metallic Bonding 71 48 Secondary Bonding 72 49 Variation in Bonding Character and Properties 74 Summary 76 Problems 77 Multiple Choice Questions 78 Suggestions for Further Reading 80 5 Structure of Solids 81119 51 The Crystalline and the Noncrystalline States 82 INORGANIC SOLIDS 83 52 Covalent Solids 83 53 Metals and Alloys 88 54 Ionic Solids 97 55 The Structure of Silica and the Silicates 103 POLYMERS 106 56 Classification of Polymers 106 57 Structure of Long Chain Polymers 107 58 Crystallinity of Long Chain Polymers 111 Summary 112 Problems 113 Multiple Choice Questions 116 Sources for Experimental Data 119 Suggestions for Further Reading 119 6 Crystal Imperfections 120147 61 Point Imperfections 121 62 The Geometry of Dislocations 126 63 Other Properties of Dislocations 131 64 Surface Imperfections 137 Contents v Summary 142 Problems 143 Multiple Choice Questions 146 Sources for Experimental Data 147 Suggestions for Further Reading 147 7 Phase Diagrams 148177 71 The Phase Rule 148 72 Singlecomponent Systems 150 73 Binary Phase Diagrams 151 74 Microstructural Changes during Cooling 157 75 The Lever Rule 159 76 Summary of Phase Diagram Rules 161 77 Some Typical Phase Diagrams 163 78 Other Applications of Phase Diagrams 170 Summary 171 Problems 171 Multiple Choice Questions 175 Sources for Experimental Data 177 Suggestions for Further Reading 177 8 Diffusion in Solids 178200 81 Ficks Laws of Diffusion 179 82 Solution to Ficks Second Law 181 83 Applications Based on the Second Law Solution 182 84 The Kirkendall Effect 189 85 The Atomic Model of Diffusion 190 86 Other Diffusion Processes 193 Summary 196 Problems 196 Multiple Choice Questions 199 Sources for Experimental Data 200 Suggestions for Further Reading 200 9 Phase Transformations 201237 91 Time Scale for Phase Changes 202 NUCLEATION AND GROWTH 204 92 The Nucleation Kinetics 205 93 The Growth and the Overall Transformation Kinetics 211 APPLICATIONS 213 94 Transformations in Steel 214 95 Precipitation Processes 220 96 Solidification and Crystallization 224 97 The Glass Transition 226 98 Recovery Recrystallization and Grain Growth 228 vi Contents Summary 232 Problems 232 Multiple Choice Questions 235 Sources for Experimental Data 237 Suggestions for Further Reading 237 10 Elastic Anelastic and Viscoelastic Behaviour 238259 ELASTIC BEHAVIOUR 239 101 Atomic Model of Elastic Behaviour 239 102 The Modulus as a Parameter in Design 243 103 Rubberlike Elasticity 246 ANELASTIC BEHAVIOUR 249 104 Relaxation Processes 249 VISCOELASTIC BEHAVIOUR 252 105 SpringDashpot Models 252 Summary 255 Problems 256 Multiple Choice Questions 257 Sources for Experimental Data 258 Suggestions for Further Reading 259 11 Plastic Deformation and Creep in Crystalline 260297 Materials PLASTIC DEFORMATION 261 111 The Tensile StressStrain Curve 261 112 Plastic Deformation by Slip 263 113 The Shear Strength of Perfect and Real Crystals 266 114 The Stress to Move a Dislocation 269 115 The Effect of Temperature on the Stress to Move a Dislocation 271 116 Multiplication of Dislocations during Deformation 274 117 Work Hardening and Dynamic Recovery 275 118 The Effect of Grain Size on Dislocation Motion 278 119 The Effect of Solute Atoms on Dislocation Motion 281 1110 The Effect of Precipitate Particles on Dislocation Motion 283 1111 Review of Strengthening Methods 285 CREEP 287 1112 Mechanisms of Creep 287 1113 Creep Resistant Materials 289 Summary 290 Problems 291 Multiple Choice Questions 294 Sources for Experimental Data 297 Suggestions for Further Reading 297 Contents vii 12 Fracture 298314 121 Ductile Fracture 298 122 Brittle Fracture 300 123 Fracture Toughness 304 124 The DuctileBrittle Transition 305 125 Fracture Mechanism Maps 307 126 Methods of Protection against Fracture 308 127 Fatigue Fracture 310 Summary 311 Problems 312 Multiple Choice Questions 313 Suggestions for Further Reading 314 13 Oxidation and Corrosion 315331 OXIDATION 316 131 Mechanisms of Oxidation 316 132 Oxidation Resistant Materials 318 CORROSION 319 133 The Principles of Corrosion 320 134 Protection against Corrosion 325 Summary 329 Problems 329 Multiple Choice Questions 330 Sources for Experimental Data 331 Suggestions for Further Reading 331 14 Conductors and Resistors 332354 141 The Resistivity Range 333 142 The Free Electron Theory 334 143 Conduction by Free Electrons 338 144 Conductor and Resistor Materials 341 145 Superconducting Materials 344 Summary 349 Problems 349 Multiple Choice Questions 351 Sources for Experimental Data 354 Suggestions for Further Reading 354 15 Semiconductors 355392 151 The Energy Gap in Solids 356 152 Intrinsic Semiconductors 361 153 Extrinsic Semiconductors 364 154 Semiconductor Materials 368 155 Fabrication of Integrated Circuits 369 156 Some Semiconductor Devices 379 viii Contents Summary 384 Problems 385 Multiple Choice Questions 387 Sources for Experimental Data 392 Suggestions for Further Reading 392 16 Magnetic Materials 393411 161 Terminology and Classification 394 162 Magnetic Moments due to Electron Spin 395 163 Ferromagnetism and Related Phenomena 397 164 The Domain Structure 399 165 The Hysteresis Loop 400 166 Soft Magnetic Materials 402 167 Hard Magnetic Materials 405 Summary 407 Problems 407 Multiple Choice Questions 409 Sources for Experimental Data 411 Suggestions for Further Reading 411 17 Dielectric Materials 412426 171 Polarization 413 172 Temperature and Frequency Effects 416 173 Electric Breakdown 419 174 Ferroelectric Materials 420 Summary 423 Problems 424 Multiple Choice Questions 424 Sources for Experimental Data 426 Suggestions for Further Reading 426 Appendix IProperties of Elements 427429 Appendix IIProperties of Engineering Materials 430 Index 431439 Since the first edition was published many new features have been added to this book They include new chapters and new sections on recent topics such as the oxide superconductors fabrication techniques used in manufacturing integrated circuits fullerenes and fracture mechanism maps In this edition the text has been updated and rewritten for greater clarity The diagrams have been improved and drawn using a computer software The author is thankful to Mr Narendra Babu for his assistance in preparing the diagrams Thanks are also due to the editorial and production team of the Publishers PHI Learning in particular to Mr KC Devasia for their assistance The author is grateful to his colleagues at IIT DelhiDr RK Pandey Dr SK Gupta Dr AN Kumar and Dr R Prasadfor their valuable suggestions V RAGHAVAN In keeping with modern trends the courses in engineering materials or engineering metals and alloys have been replaced by a course in Materials Science in many institutions in India and abroad Most of the curricula in metallurgy ceramics and other materialsoriented disciplines have also incorporated a first general course in materials science This book is intended for use in such courses as well as by the students of applied sciences Postgraduates who have had no previous exposure to the subject should also find this book useful In deciding the level at which this material is to be covered it has been assumed that the reader has a background in college level physics chemistry and mathematics Though not essential an elementary knowledge of physical chemistry and thermodynamics would be an added advantage A reasonably wide coverage in sufficient depth has been attempted giving the necessary importance to the physical mechanical chemical electrical and magnetic properties Consistent with the moderate size of the book the author has tried to emphasize the properties that are more structuresensitive Keeping in view the engineering applications numerous examples of real materials of technological importance have been discussed A number of colleagues of the author and over 1500 students who studied this first course during the last 10 years have contributed significantly in class testing and greatly improving this work In particular mention must be made of Drs EC Subbarao D Chakravorty LA Shepard MF Merriam CV Seshadri RK Mittal and Shri VM Kumar and ML Gandhi The author is grateful to Professor NM Swani for his encouragement and interest Special mention must be made of Professor Morris Cohen for providing the author an opportunity to teach materials science courses at the Massachusetts Institute of Technology January 1974 V RAGHAVAN SI UNITS Base Units Quantity Unit Symbol Length l metre m Mass m kilogram kg Time t second s Electric current I ampere A Temperature T kelvin K Amount of substance n mole mol Luminous intensity candela cd The mole is the amount of substance that contains as many elementary entities as there are atoms in 0012 kg of carbon12 The elementary entities may be atoms ions electrons other particles or groups of particles Supplementary Units Plane angle radian rad Solid angle steradian sr Derived Units with Special Names Quantity Special name Symbol Equivalence in other derived units base units Frequency hertz Hz s¹ Force weight newton N kg m s² Stress strength pascal Pa N m² kg m¹ s² pressure Energy work joule J N m kg m² s² quantity of heat Power watt W J s¹ kg m² s³ Electric charge coulomb C A s Electric potential volt V kg m² s³ A¹ Resistance ohm Ω V A¹ kg m² s³ A² Capacitance farad F C V¹ kg¹ m² s⁴ A² Magnetic flux weber Wb V s kg m² s² A¹ Magnetic flux tesla T Wb m² kg s² A¹ density Inductance henry H Wb A¹ kg m² s² A² 1012 tera T 109 giga G 106 mega M 103 kilo k 102 hecto h 101 deka da 101 deci d 102 centi c 103 milli m 106 micro µ 109 nano n 1012 pico p 1015 femto f 1018 atto a Note All multiple prefix symbols except kilo hecto and deka are written in capitals and all submultiple symbols are written in lower case letters Have a feel for SI units Sir Issac Newton the Apple and SI units Length unit 1 metre distance the apple travelled Time unit 1 second time of fall of the apple Force unit 1 newton weight of the apple When writing unit symbols do not use full stops plurals dots or dashes No degree symbol for kelvin write as K and not as K No kgm kilogram mass or kgf kilogram force The mass unit is kilogram kg and the force unit is newton N No space between the prefix symbol and the unit symbol eg meganewton should be written as MN and not as M N One space between two symbols for clarity eg metre second should be written as m s and not as ms which means millisecond All symbols associated with proper names are written with a capital eg A K N etc When they are written as a word the lower case is used throughout ampere kelvin and newton Prefix symbols for multiples and submultiples are preferred in steps of 103 Thus tera T giga G mega M kilo k milli m micro µ nano n and pico p are preferred prefix symbols Attach prefix to numerator and not to denominator eg use MN m2 instead of N mm2 even though both are identical Three digits are grouped together on either side of the decimal point eg 1256 637 83 60 023 Four digit number need not be so grouped eg 7386 06921 Some nonSI units generally accepted 1 degree celsius C 2 minutes hours days months and years 3 Angstrom Å for 1010 m 4 Electronvolt eV for energy equal to 1602 1019 J 1 electronvolt is the kinetic energy acquired by an electron when falling through a potential of 1 volt PHYSICAL CONSTANTS Avogadros number N 6023 1023 mol1 Boltzmanns constant k 1380 1023 J K1 8614 105 eV K1 Gas constant R 8314 J mol1 K1 Plancks constant h 6626 1034 J s Electronic charge e 1602 1019 C Electron rest mass m0 9109 1031 kg Velocity of light c 2998 108 m s1 Bohr magneton magnetic moment µB 9273 1024 J A m2 Permittivity of free space ε0 8854 1012 F m1 Permeability of free space µ0 4π 107 H m1 1257 106 H m1 Faradays constant F 9649 kC mol1 of electrons Atomic mass unit amu 1103N 1660 1027 kg Acceleration due to gravity g 981 m s2 1 inch 254 mm 1 nm 109 m 1 Å 1010 m 01 nm 1 1573 rad T C T 27315 K T F 59T 45967 K 1 per F 95 K1 1 kgf 981 N 1 lb 445 N 1 dyne 105 N 1 dynecm 103 N m1 1 atmosphere 0101 325 MN m2 1 bar 101 MPa 1 psi 689 kN m2 1 ksi 103 psi 689 MN m2 1 tonsqin 1546 MN m2 1 kgfcm2 981 kN m2 1 kgfmm2 981 MN m2 1 dynecm2 01 N m2 1 torr mm of Hg 1333 N m2 1 kgfmm3 0310 MN m3 1 ksi vin 110 MN m3 1 eV 1602 x 1019 J 1 erg 107 J 1 calorie 418 J 1 eVentity 9649 kJ mol1 1 ergcm 105 J m1 1 ergcm2 103 J m2 1 ergcm3 01 J m3 1 lbinin2 175 J m2 1 poise 01 Pa s 1 debye 333 x 1030 C m 1 mAcm2 10 A m2 1 A hr 36 kC 1 ohm cm 102 ohm m 1 mhocm 102 ohm1 m1 1 voltmil 39 370 V m1 1 cm2volt sec 104 m2 V1 s1 1 gauss 104 Wb m2 1 oersted 796 A m1 Name Forms Sound Name Forms Sound alpha α a nu ν n beta β b xi ξ x gamma γ Γ g omicron o o delta δ Δ d pi π Π p epsilon ε e rho ρ r zeta ζ z sigma σ Σ s eta η e tau τ t theta θ Θ th upsilon υ u iota ι i phi φ Φ ph kappa κ k khi χ kh lambda λ Λ l psi ψ Ψ ps mu μ m omega ω Ω o In this introductory chapter we briefly discuss the nature of Materials Science and Engineering After defining or explaining what Materials Science and Engineering is we classify engineering materials according to their nature and the various categories of applications Then we discuss the different levels of the internal structure of materials Finally we emphasize the importance of the structureproperty relationships in materials outlining the general approach of the ensuing chapters 2 Introduction empirical rules and extrapolates available information to unknown situations In this respect materials science and engineering draws heavily from the engineering sciences such as metallurgy ceramics and polymer science These in their own time have grown out of their interaction with the basic sciences of chemistry and physics 12 Classification of Engineering Materials Having defined the limits of materials that come under our purview we can classify them in three broad groups according to their nature i Metals and alloys ii Ceramics and glasses iii Organic polymers Metals are familiar objects with a characteristic appearance they are capable of changing their shape permanently and have good thermal and electrical conductivity An alloy is a combination of more than one metal Ceramics and glasses are nonmetallic inorganic substances which are brittle and have good thermal and electrical insulating properties Organic polymers are relatively inert and light and generally have a high degree of plasticity Figure 11 lists typical examples from each of these three groups of materials In addition examples of materials which lie between two groups are also shown Metals and Alloys Steels aluminium copper silver gold Brasses bronzes manganin invar Superalloys Boron rare earth magnetic alloys Ceramics and Glasses MgO CdS Al2O3 SiC BaTiO3 Silica sodalimeglass concrete cement Ferrites and garnets Ceramic superconductors Organic Polymers Plastics PVC PTFE polyethylene Fibres terylene nylon cotton Natural and synthetic rubbers Leather Si Ge GaAs Boridereinforced steel Metalreinforced plastics Fig 11 The three major groups of engineering materials Glass fibrereinforced plastics An alternative way of classifying materials is according to the three major areas in which they are used i Structures ii Machines iii Devices Structures not to be confused with the internal structure of a material refer to the objects without moving parts erected by engineers such as a concrete dam a steel melting furnace a suspension bridge and an oil refinery tower Machines include lathes steam and gas turbines engines electric motors and generators Devices are the most recent addition to engineering materials and refer to such innovations as a transistor a photoelectric cell piezoelectric pressure gauges ceramic magnets and lasers Invariably in each category of applications we find materials from all the three groups described above To give some examples an aircraft structure is built of aluminium alloys and plastics a steel melting furnace is built of refractory oxides and structural steel safety helmets are made of glassreinforced plastics Similarly we have metaloxide semiconductors The block diagram in Fig 12 depicts this interplay between material groups and categories of applications Metals and Alloys Ceramics and Glasses Polymers Engineering Materials Applications Structures Machines Devices Fig 12 Each category of engineering application requires materials from any or all of the three groups of materials 13 Levels of Structure The internal structure of a material simply called the structure can be studied at various levels of observation The magnification and resolution of the physical aid used are a measure of the level of observation The higher the magnification the finer is the level The details that are disclosed at a certain level of observation are generally different from the details disclosed at some other level Henry Sorby was one of the first to realize this when he wrote in 1886 Levels of Structure 3 4 Introduction Though I had studied the microscopical structure of iron and steel for many years it was not until last autumn that I employed what may be called high powers This was partly because I did not see how this could be satisfactorily done and partly because it seemed to me unnecessary I had found that in almost every case a power magnification of 50 linear showed on a smaller scale as much as one of 200 and this led me to conclude that I had seen the ultimate structure Now that the results are known it is easy to see that my reasoning was false since a power of 650 linear enables us to see a structure of an almost entirely new order We have now come a long way since Sorbys time Magnifications with matching resolutions of a million times linear are now common Depending on the level we can classify the structure of materials as Macrostructure Microstructure Substructure Crystal structure Electronic structure Nuclear structure Macrostructure of a material is examined with naked eye or under a low magnification The internal symmetry of the atomic arrangements in a crystalline material may reflect in the external form of a crystal such as quartz Large individual crystals of a crystalline material may be visible to the naked eye as in a brass doorknob by the constant polishing and etching action of the human hand and sweat Microstructure generally refers to the structure as observed under the optical microscope see Fig 13 This microscope can magnify a structure up to about 1500 times linear without loss of resolution of details of the structure The limit Fig 13 Crystal boundaries in nickel ferrite Fe2NiO4 magnified 900 times linear WD Kingery Introduction to Ceramics with permission from John Wiley New York Substructure refers to the structure obtained by using a microscope with a much higher magnification and resolution than the optical microscope In an electron microscope a magnification of 1 000 000 times linear is possible By virtue of the smaller wavelength of electrons as compared to visible light the resolving power also increases correspondingly so that much finer details show up in the electron microscope Another modern microscope is the field ion microscope It produces images of individual atoms and imperfections in atomic arrangements Crystal structure tells us the details of the atomic arrangement within a crystal It is usually sufficient to describe the arrangement of a few atoms within what is called a unit cell The crystal consists of a very large number of unit cells forming regularly repeating patterns in space The main technique employed for determining the crystal structure is the Xray diffraction The electronic structure of a solid usually refers to the electrons in the outermost orbitals of individual atoms that constitute the solid Spectroscopic techniques are very useful in determining the electronic structure Nuclear structure is studied by nuclear spectroscopic techniques such as nuclear magnetic resonance NMR and Mossbauer studies 14 StructureProperty Relationships in Materials Until recently it has been the practice in a course on engineering materials to list the composition treatment properties and uses of as many materials as possible The number and variety of engineering materials and applications have increased tremendously in recent years Now we have more than a thousand types of steel alone each with a specific composition thermal and mechanical history Therefore it is impossible to describe an adequate number of engineering materials in one course Moreover our knowledge of the internal structure of materials and how this structure correlates with the properties has rapidly advanced in recent decades So it is more interesting and appropriate to study some of the key factors that determine the structureproperty relationships rather than go for a fully descriptive account of a large number of materials This approach is adopted in this book The discussion of a structuredependent property is usually followed by typical applications The levels of structure which are of the greatest interest in materials science and engineering are the microstructure the substructure and the crystal structure The chemical mechanical electrical and magnetic properties are among the most important engineering properties We first develop the basic concepts pertaining to the levels of structure These include concepts in equilibrium and kinetics the geometry of crystals the arrangement of atoms in the unit cell of crystalline materials the substructural imperfections in crystals and the microstructure of single phase and multiphase materials We then discuss how changes in the structure are brought about and how they can be controlled to the best possible advantage Solid state diffusion and control of phase transformations by heat treatment are the main topics here In the latter half of the book corrosion among chemical properties elastic and plastic deformation among mechanical properties and several electrical and magnetic properties are discussed with numerous examples of typical engineering materials The gross composition of a material is important in determining its structure Yet for a given gross composition radical changes in the structure and properties can be brought about by subtle changes in the concentration and StructureProperty Relationships in Materials 7 distribution of minute quantities of impurities The same may also be possible by a thermal or a mechanical treatment that involves no change in the overall composition of the material Materials Science and Engineering deals more with this kind of changes rather than with the effect of gross composition on the properties Suggestions for Further Reading A Street and W Alexander Metals in the Service of Man Penguin Books 1976 DL Weaire and CG Windsor Eds Solid State SciencePast present and predicted Adam Hilger Bristol UK 1987 8 Suggestions for Further Reading CHAPTER 2 Equilibrium and Kinetics In this chapter we introduce the concept of stability and metastability using the mechanical analog of a tilting rectangular block We discuss the importance of the indefinite existence of materials in the metastable state After defining basic thermodynamic functions we also discuss the statistical nature of entropy and the role of the highenergy fraction of atoms in the statistical distribution in surmounting activation barriers for reactions in materials Units Quantity SI units Other units Temperature T kelvin K C F Pressure P megapascal MPa atmosphere psi kgcm² megnewton MN m2 dynecm² mm of Hg Internal energy E External energy PV Enthalpy H Gibbs free energy G thermal energy RT Activation energy Q Entropy S Specific heat Cv Cp joule per mole J mol1 calmole calgm per kelvin J mol1 K1 calmoleC calgmC Constants Avogadros number N 6023 10²³ mol1 Boltzmanns constant k 1380 1023 J K1 Gas constant R 8314 J mol1 K1 10 Equilibrium and Kinetics 21 Stability and Metastability The concept of stability is easily understood by considering a mechanical analog In Fig 21 a rectangular block of square crosssection is shown in 1 1 2 3 4 5 5 4 3 2 Fig 21 Various positions of a tilting rectangular block illustrate the concept of stability and metastability various tilted positions In position 1 the block is resting on the square base the arrow from the centre of mass indicates the line along which the weight acts In position 2 the block is tilted slightly to the right about one of its edges such that the line along which the weight acts is still within the square base The centre of mass has moved up due to the tilt In position 3 the tilt is increased to such an extent that the line of force just falls on the periphery of the base The centre of mass is now at the maximum possible height from the base Further tilting lowers it The line of force now falls outside the square base but within the rectangular base position 4 On coming to rest on the rectangular face position 5 the centre of mass is at the lowest possible position for all configurations of the block The centres of mass for the various positions of the block are joined by a curve The potential energy of the block is measured by the height of the centre of the mass from the base Position 5 corresponds to the lowest potential energy for all configurations and is correspondingly described as the most stable state or simply the stable state A system always tends to go towards the most stable state Position 3 has the maximum potential energy and is called an unstable state Positions 2 and 4 are also unstable states but do not have the maximum energy Position 1 is called a metastable state We can use this mechanical analogy to illustrate various equilibrium configurations of a system Figure 22 shows the potential energy of a system as a function of configuration The potential energy curve has two valleys and a peak At these positions the curve has zero slope that is the energy does not vary as a function of configuration for infinitesimally small perturbations Such configurations are called equilibrium configurations Corresponding to the terminology used for the tilting block we have stable equilibrium unstable equilibrium and metastable equilibrium see Fig 22 Even though the potential energy is a minimum in the metastable state it is not the lowest for all configurations of the system Due to the valley position after small perturbations the original configuration is restored in both stable and metastable equilibrium Such restoration does not occur in the case of unstable equilibrium Unstable Potential energy Metastable Activation barrier Stable Fig 22 Potential energy as a function of the configuration coordinate Configuration A metastable state of existence is very common in materials For example most metals at room temperature are stable only in the form of an oxide Oxygen is easily available in the surrounding air Yet a metal may not combine with oxygen at room temperature except for a very thin film on the surface It may exist in the metastable metallic state for an indefinite period This period could be centuries as is borne out by the unchanging state of some ancient metallic statues and pillars This fortunate set of circumstances enables us to use metals in many engineering applications In the mechanical analogy we used the block can be tilted and brought to the most stable configuration starting from a metastable state by an external supply of energy in the form of a hand push or a jiggling base In materials the most common source of such energy is the thermal energy As the temperature of a solid is increased from 0 K the atoms in the solid vibrate about their mean positions with increasing amplitude When the temperature is sufficiently high rotation of atoms or small groups of atoms also becomes possible in some solids The translational motion of atoms past one another is more characteristic of liquids and gases The energy associated with the vibrations rotations and translations aids in taking a material from a metastable state to a stable state In the above example of the oxidation of metals at sufficiently high temperatures thermal energy will aid the chemical combination of the metal with the surrounding oxygen taking it from the metastable to the stable state This of course poses the problem of protecting metals against oxidation when used at high temperatures Stability and Metastability 11 22 Basic Thermodynamic Functions The concepts of equilibrium and kinetics are intimately associated with the basic thermodynamic parameters Pressure P and temperature T are familiar intensive parameters As opposed to these we have extensive or capacity parameters which depend on the extent or quantity of the material that comprises the system The extensive thermodynamic functions are now described Internal energy E also denoted by U at temperature T is given by E E0 T0 Cv dT 21 where E0 is the internal energy of the material at 0 K and Cv is the specific heat at constant volume The enthalpy or the heat content of a material H is defined in a similar manner H H0 T0 Cp dT 22 where H0 is the enthalpy at 0 K and Cp is the specific heat at constant pressure E and H are related through P and V where V is the volume of the material H E PV 23 PV represents the external energy as opposed to the internal energy and is equal to the work done by the material at constant pressure in creating a volume V for itself For condensed systems like the liquid and the solid state at atmospheric pressure the PV term is negligible so that E H This approximation can be used in most of the problems concerning solid materials H0 represents the energy released when the individual atoms of the material are brought together from the gaseous state to form a solid at 0 K The gaseous state where there is no interaction between the atoms is taken as the reference zero energy state To indicate that the system has lost energy H0 is written with a negative sign As the temperature increases from 0 K the material absorbs heat from the surroundings and H increases The solid melts on reaching the melting point and a further quantity of heat ΔH called the enthalpy of fusion is added at the melting temperature When all the solid has melted the temperature of the liquid may further increase with the absorption of more energy All the energy that a system possesses is not available as work during a chemical change That part of the energy which can become available as work is called the Gibbs free energy or simply the Gibbs energy The part which cannot be released as work is called bound energy Another thermodynamic function called entropy defines the relationship between the total energy and the Gibbs energy At constant pressure the entropy S of a system is given by S T0 Cp dTT 24 The entropy of a material is zero at 0 K in contrast to the enthalpy and the internal energy terms which have nonzero negative values at 0 K The entropy increases with increasing temperature It is a measure of the thermal disorder introduced in the solid as it is heated above 0 K The solid state is characterized by the random vibrations of atoms about their mean positions In the liquid state the atoms have more freedom and can also move past one another resulting in greater disorder Correspondingly the entropy increases when a solid melts at constant temperature Example 21 Calculate the entropy increase when one mole of ice melts into water at 0ºC Solution The latent heat of fusion of ice ΔH 602 kJ mol1 80 calgm This heat is absorbed at the constant temperature Tm 0ºC The entropy increase in the process ΔS heat added temperature ΔH Tm 602 10³27315 2204 J mol1 K1 In addition to thermal entropy a system may also possess configurational entropy which is dependent on the configurations of the system Following Boltzmanns definition the configurational entropy can be written as S k ln w where k is Boltzmanns constant and w is the number of different configurations of equal potential energy in which the system can exist The next section gives a more detailed discussion of configurational and thermal entropies The Gibbs energy G is defined in terms of the enthalpy H and the entropy S G H TS As already pointed out the free energy represents the available part of the energy which can be converted to work As the temperature increases H increases but TS increases more rapidly than H and so G decreases with increasing temperature The Gibbs energy is used as a criterion of stability The most stable state of a material is that which has the minimum Gibbs energy For a process to occur spontaneously the Gibbs energy must decrease during the process Changes in thermodynamic quantities are always defined as the final value minus the initial value ΔG Gfinal Ginitial Then for a spontaneous process the free energy change ΔG during the process must be negative At a constant temperature and pressure we can write this condition for a spontaneous change as ΔG ΔH T ΔS 0 Only if there is no change in the entropy of a system during a process the enthalpy change ΔH can be used in place of ΔG as a criterion of stability In the example of the tilting block where no entropy change occurs during the tilt we were justified in defining the stable state as a state of lowest potential energy or enthalpy In general however the entropy change may not be negligible Several chemical reactions are known to be endothermic that is they absorb heat during the reaction making ΔH positive however they may occur spontaneously indicating that ΔG is negative In such cases T ΔS ΔH 23 The Statistical Nature of Entropy The entropy of a system has been defined by Eqs 24 and 25 The meaning of these equations can be understood with reference to physical processes Taking the configurational entropy first consider the arrangement of equal numbers of white and black spheres on 16 sites shown in Fig 23 Four possible configurations of these spheres are shown In Figs 23a and 23b the arrangement is disordered and random In Fig 23c every white sphere is surrounded by black spheres and every black sphere is surrounded by white spheres In Fig 23d all white spheres are separated from the black spheres It is easy to calculate the total number w of such distinguishable configurations that we can have with these spheres For generality if we call the total number of sites as N and the number of white spheres as n the number of black spheres is N n and it is easily seen that w N N n n For the above example substituting N 16 and n 8 we obtain w 12 870 Among these only two arrangements shown in Figs 23c and 23d are perfectly ordered and thus very unlikely to occur in a random choice The probability of the system existing in a disordered configuration is almost unity Equation 25 states that the configurational entropy increases as the logarithm of w One mole of a solid contains more than 1023 atoms If we mix two different kinds of atoms randomly in a solid we end up with an extremely large number of distinguishable configurations and an appreciable amount of configurational entropy Since w can never be less than one the configurational entropy is either zero or positive It is zero for an absolutely pure solid consisting of the same kind of atoms on all its sites or for a perfectly ordered solid like a compound Consider a mole of atomic sites N in a solid Let n atoms of B and N n atoms of A be mixed randomly on these sites The configurational entropy is zero before mixing The increase in entropy due to mixing is given by ΔS Sfinal Sinitial k lnN N n n The following Stirlings approximation is valid for n 1 ln n n ln n n Combining Eqs 210 and 211 we obtain ΔS k N ln N N n ln N n n ln n Equation 24 gives the thermal part of entropy as a function of temperature At 0 K we can visualize the atoms of a solid to be at rest so that there is no disorder due to temperature and consequently the entropy is zero As the temperature increases the atoms begin to vibrate about their mean positions in the solid with increasing frequency and amplitude The atoms can be considered to oscillate in three modes corresponding to the three orthogonal directions For many solids above room temperature the frequency of these oscillations reaches a constant value about 1013 s1 In contrast the amplitude of the oscillations continues to increase with increasing temperature The average energy per atom per mode of oscillation is called the thermal energy and is equal to kT where T is the temperature in kelvin and k is Boltzmanns constant equal to 1380 1023 J K1 For one mole of atoms the thermal energy becomes NkT RT where N is Avogadros number and R is the gas constant given by R Nk 6023 1023 1380 1023 8314 J mol1 K1 All the atoms of a solid do not vibrate with the same amplitude and energy at any instant The vibrating atoms interact with their neighbours and as a very large number of atoms are involved the interaction effects are very complex Consequently the vibrational energy of any particular atom fluctuates about the average value in a random and irregular way In other words there is a statistical distribution of vibrational energies between the atoms This distribution is the source of the thermal disorder The entropy given by Eq 24 should be associated with this distribution The thermal entropy arises from the vibrational energy distribution in a solid while the configurational entropy arises from the distribution in the configurational arrangements The Arrhenius plot for the reaction I2 H2 2HI SUMMARY 1 Calculate the increase in the enthalpy and the entropy of copper as it is heated from room temperature 300 K to 1000 K The specific heat in this temperature range is given by Cp 2261 627 103 T J mol1 K1 starting from room temperature required to double the reaction rate for a value of Q equal to i 100 kJ mol¹ and ii 200 kJ mol¹ Answer i 5C ii 25C Find the activation energy from an Arrhenius plot for a reaction that requires the following times for completion at the indicated temperatures Temperature K Time s 600 9360 700 275 800 073 900 001 Answer 200 kJ mol¹ Sources for Experimental Data I Barin O Knacke and O Kubaschewski Thermochemical Properties of Inorganic Substances SpringerVerlag Berlin 1973 and Supplement 1977 R Hultgren et al Eds Selected Values of the Thermodynamic Properties of the Elements American Society for Metals Metals Park Ohio 1973 Suggestions for Further Reading EA Guggenheim Thermodynamics NorthHolland Amsterdam 1967 CHP Lupis Chemical Thermodynamics of Materials NorthHolland Amsterdam 1983 22 Sources For Experimental DataSuggestions For Further Reading A reaction takes 500 min at 10C for completion It takes 1 min at 80C Find the time it would take at 40C Answer 25 min 24 Crystal Geometry and Structure Determination GEOMETRY OF CRYSTALS 31 The Space Lattices Before discussing the periodic patterns of atomic arrangements in crystals we need to look into arrangements of points in space in periodically repeating patterns This leads us to the concept of a space lattice A space lattice provides the framework with reference to which a crystal structure can be described A space lattice is defined as an infinite array of points in three dimensions in which every point has surroundings identical to that of every other point in the array As an example for ease of representation on paper consider a two dimensional square array of points shown in Fig 31 By repeated translation of Fig 31 A twodimensional square array of points gives a square lattice Two ways of choosing a unit cell are illustrated 1 1 1 1 b a 5 2 5 2 the two vectors a and b on the plane of the paper we can generate the square array The magnitudes of a and b are equal and can be taken to be unity The angle between them is 90 a and b are called the fundamental translation vectors that generate the square array To ignore end effects near the boundary we will assume that the array can be extended infinitely If we locate ourselves at any point in the array and look out in a particular direction that lies on the plane of the paper the scenery is the same irrespective of where we are Consider the immediate surroundings of a point in the array If we look due north or due east from this point we see another point at a distance of 1 unit Along northeast we see the nearest point at a distance of 2 units and along northnortheast the nearest point is at a distance of 5 units As this is true of every point in the array the array satisfies the definition given above and can be called a twodimensional square lattice Example 31 Draw a twodimensional pentagonal lattice Solution A regular pentagon has an interior angle of 108 As 360 is not an integral multiple of 108 pentagons cannot be made to meet at a point bearing a constant angle to one another Hence a pentagonal lattice is not possible On the other hand a square or a hexagonal twodimensional lattice is possible A space lattice can be defined by referring to a unit cell The unit cell is the smallest unit which when repeated in space indefinitely will generate the space lattice In the above example of the square lattice the unit cell is the square obtained by joining four neighbouring lattice points as shown in Fig 31 Since every corner of this square is common to four unit cells meeting at that corner the effective number of lattice points in the unit cell is only one Alternatively the unit cell can be visualized with one lattice point at the centre of the square and with none at the corners see Fig 31 A threedimensional space lattice is generated by repeated translation of three noncoplanar vectors a b and c It so turns out that there are only 14 distinguishable ways of arranging points in threedimensional space such that each arrangement conforms to the definition of a space lattice These 14 space lattices are known as Bravais lattices named after their originator They belong to seven crystal systems and are listed in Table 31 according to the crystal system The cubic system is defined by three mutually perpendicular translation vectors a b and c which are equal in magnitude The angle between b and c is α the angle between c and a is β and that between a and b is γ This notation about angles is general and should be consistently followed As shown in Table 31 there are three space lattices in the cubic crystal system the simple cubic the body centred cubic and the face centred cubic space lattices Example 32 Derive the effective number of lattice points in the unit cell of the three cubic space lattices Solution The unit cell in all these three cases is the cube The corners of a cube are common to eight adjacent cubes The faces are common to two adjacent cubes The body centre is not shared by any other cube So the effectiveness of a corner lattice point is 18 that of a face centred lattice point is 12 and that of the body centre is 1 Referring to Table 31 we can write Space lattice Abbreviation Effective number of lattice points in unit cell Simple cubic SC 1 Body centred cubic BCC 2 Face centred cubic FCC 4 TABLE 31 cont The Bravais Lattices IV Rhombohedral a b c α β γ 90 10 Simple Points at the eight corners of the unit cell V Hexagonal a b c α β 90 γ 120 11 Simple i Points at the eight corners of the unit cell outlined by thick lines or ii Points at the twelve corners of the hexagonal prism and at the centres of the two hexagonal faces VI Monoclinic a b c α γ 90 β 12 Simple Points at the eight corners of the unit cell 13 End centred Points at the eight corners and at two face centres opposite to each other VII Triclinic a b c α β γ 90 14 Simple Points at the eight corners of the unit cell Crystals have inherent symmetry A cubic crystal is said to have a fourfold rotation symmetry about an axis passing through the centres of two opposite faces of the unit cube During each complete rotation about this axis the crystal passes through identical positions in space four times The rotational translational and reflection symmetry operations constitute the symmetry elements of a crystal The crystal systems in Table 31 are arranged in order of decreasing symmetry the cube being the most symmetric and the triclinic system being the least symmetric The details of symmetry elements of a crystal are given in books on crystallography and will not be covered here After the cubic system the next less symmetric crystal system is the tetragonal system It is defined by three mutually perpendicular vectors only two of which are equal in magnitude There are two space lattices here Space lattice Abbreviation Effective number of lattice points in unit cell Simple tetragonal ST 1 Body centred tetragonal BCT 2 It is interesting to note that there is no face centred tetragonal space lattice Any array of lattice points that can be represented by an FCT cell can equally well be described by a BCT cell as illustrated in Fig When there are two such alternatives of the same crystal system available to describe the same array of lattice points the unit cell which has the smaller number of lattice points is chosen for the Bravais list As a final example let us examine the unit cell of the hexagonal system In order that the hexagonal symmetry becomes evident we can take the unit cell to be a regular hexagonal prism see Table 31 The effective number of lattice points in this unit cell is 3 For generating the entire space lattice by translation of the unit cell a smaller cell with only one lattice point is used refer Table 31 Example 33 List the lattice unit cell parameters required to specify fully the unit of each crystal system Solution Crystal system To be specified Total number of parameters axes angles Cubic a 1 Tetragonal a c 2 Orthorhombic a b c 3 Rhombohedral a α 2 Hexagonal a c 2 Monoclinic a b c β 4 Triclinic a b c α β γ 6 30 Crystal Geometry and Structure Determination Crystal Number of nearest Closest distance of approach monoatomic neighbours or atomic diameter SC 6 a BCC 8 32 a FCC 12 2 a The student should check the number of nearest neighbours listed above for each case In principle an infinite number of crystal structures can be generated by combining different bases and different lattice parameters with the same space lattice In Fig 34 three different bases are combined with the simple cubic lattice In Fig 34a the crystal is monoatomic with just one atom at each lattice point For clarity neighbouring atoms are shown separately Figure 34b illustrates a molecular crystal with a diatomic molecule at each lattice point The centre of the larger atom of the molecule coincides with a lattice point while the smaller atom is not at a lattice point In molecular crystals the basis is fully defined by giving the number and types of atoms the internuclear distance of separation between neighbours in the molecule and the orientation of the molecule in relation to the unit cell In Fig 34c the corner atoms of the cube are of one type but the atom at the body centre is of a different type The basis is two atoms the larger one in this case at a lattice point and the smaller one positioned halfway along the body diagonal at the body centre which is not a a c b Fig 33 Unit cells of monoatomic crystals of a simple cubic SC b body centred cubic BCC and c face centred cubic FCC structure lattice point In the crystal of course the unit cell can be shifted such that the body centre becomes a lattice point and the body corners are no longer lattice points This crystal should not be confused with the monoatomic BCC crystal where the body corner and the body centre atoms are of the same type The number of crystal structures known to exist runs into thousands This indicates that there can be more complex bases than those we have considered above For example one crystal form of manganese has the structure referred to the BCC space lattice with 29 atoms grouped together at each lattice point In protein structures the number of atoms in the basis may be as high as 10 000 Obviously the description of such bases would include a number of complex details 33 Crystal Directions and Planes It is necessary to use some convention to specify directions and planes in a crystal For this purpose the system devised by Miller known as Miller indices is widely used In Fig 35 the vector r passing through the origin o to a lattice point can be expressed in terms of the fundamental translation vectors a b and c which form the crystal axes as r rla r2b r3c 31 a c b Fig 34 Three different crystal structures referred to the same simple cubic lattice For clarity neighbouring atoms or molecules are shown separated Crystal Directions and Planes 31 where r1 r2 and r3 are integers The caxis is not shown in the figure as r is assumed to lie on the ab plane The components of r along the three axes are r1 2 r2 3 and r3 0 Then the crystal direction denoted by r is written as 230 in the Miller notation with square brackets enclosing the indices If there is a negative component along a crystal axis such as 2 it is written as 2 and read as bar 2 A family of directions is obtained by all possible combinations of the indices both positive and negative The family 230 203 203 302 320 etc is represented by 230 where the angular brackets denote the entire family Such a convention of representing a family is very convenient for cubic crystals Example 34 Find the family of crystal directions represented by cube edges face diagonals and body diagonals of the unit cube Give the number of members in each family Solution Direction Miller indices Number in the family Cube edge 100 6 Face diagonal 110 12 Body diagonal 111 8 It is left as an exercise to the student to write down the Miller indices of each member of the three families The magnitude of the vector r gives the magnitude of that crystal direction The crystal directions 230 460 and 1 ½ 0 all have the same direction but different magnitudes Since Miller indices for directions are usually specified as the smallest possible integers the differences in magnitude for the above three directions are indicated using the following convention 230 2230 and 12230 The Miller indices of a crystal plane are determined as follows Referring to the plane shown in Fig 36 Step 1 Find the intercepts of the plane along the axes a b and c the intercepts are measured as multiples of the fundamental vectors Step 2 Take reciprocals of the intercepts Step 3 Convert into smallest integers in the same ratio Step 4 Enclose in parentheses Example 35 Find the Miller indices of the direction r and the plane indicated by unit normal s in Fig 37 Solution The direction r does not pass through the origin in the figure As the choice of origin is arbitrary shift the origin so that r passes through it or alternatively draw a vector parallel to r and passing through the origin Then the Miller indices are found to be 110 The plane s passes through the origin Draw a parallel plane that makes the smallest integral intercepts on the coordinate axes The intercepts for the parallel plane drawn in Fig 37 are 3 2 and so that the Miller indices are given by 230 Some useful conventions and results of the Miller notation are as follows i Unknown Miller indices are denoted by symbols h k and l For example for an unknown family of directions we write hkl ii When the integers used in the Miller indices contain more than one digit the indices must be separated by commas for clarity eg 3 10 15 iii The crystal directions of a family are not necessarily parallel to one another Similarly not all members of a family of planes are parallel to one another The interplanar spacing between adjacent planes of Miller indices hkl dhkl is defined as the spacing between the first such plane and a parallel plane passing through the origin The spectrum of xrays emitted from a molybdenum target at 35 kV includes both types of radiation as illustrated in Fig 310 The wavelength of radiation used for crystal diffraction should be in the same range Xrays have wavelengths in this range and are therefore diffracted by crystals This property is widely used for the study of crystal structures semitransparent that is they allow a part of the xrays to pass through and reflect the other part the incident angle θ called the Bragg angle being equal to the reflected angle Referring to Fig 311 there is a path difference between rays reflected from plane 1 and the adjacent plane 2 in the crystal The two reflected rays will reinforce each other only when this path difference is equal to an integral multiple of the wavelength If d is the interplanar spacing the path difference is twice the distance d sin θ as indicated in Fig 311 The Bragg condition for reflection can therefore be written as nλ 2d sin θ where n is an integer and λ is the wavelength of the xrays used A first order reflection is obtained if n 1 a second order reflection occurs if n 2 and so on As sin θ has a maximum value of 1 for a typical value of interplanar spacing of 2 Å Eq 33 gives the upper limit of λ for obtaining a first order reflection as 4 Å There will be no reflection if λ is greater than 4 Å λ can be reduced indefinitely obtaining reflections from other sets of planes that have spacing less than 2 Å as well as an increasing number of higher order reflections A very small wavelength of the order of 01 Å is not necessarily an advantage as it tends to produce other effects such as knocking off electrons from the atoms of the crystal and getting absorbed in the process The wavelengths of the Kα radiation given in Table 32 for typical target metals lie in the right range The Bragg equation can be used for determining the lattice parameters of cubic crystals Let us first consider the value that n should be assigned A second order reflection from 100 planes should satisfy the following Bragg condition 2λ 2d100 sin θ or λ d100 sin θ Similarly a first order reflection from 200 planes should satisfy the following condition λ 2d200 sin θ We have earlier noted that the interplanar spacing of 100 planes is twice that for 200 planes So Eqs 34 and 35 are identical For any incident beam of xrays the Bragg angle θ would be the same as the two sets of planes in question are parallel As Eqs 34 and 35 are identical the two reflections will superimpose on each other and cannot be distinguished By a similar argument it can be shown that the third order reflection from 100 planes will superimpose on the first order reflection from 300 planes In view of such superimposition there is no need to consider the variations in n separately instead we take n to be unity for all reflections from parallel sets of planes such as 100 200 300 400 etc In a crystal it may turn out for example that there is no 200 plane with atoms on it Then what is designated as a 200 reflection actually refers to the second order reflection from 100 planes Example 37 A diffraction pattern of a cubic crystal of lattice parameter a 316 Å is obtained with a monochromatic xray beam of wavelength 154 Å The first four lines on this pattern were observed to have the following values Line θ in degrees 1 203 2 292 3 367 4 436 Determine the interplanar spacing and the Miller indices of the reflecting planes Solution Using the Bragg equation we can write the interplanar spacing d λ2 sin θ n is assumed to be 1 as higher order reflections superpose on the lower order ones for parallel sets of planes The d values can now be determined Since d ah² k² l² for cubic crystals h² k² l² can also be determined The results are tabulated below Line dhkl Å h² k² l² a²d² hkl 1 2220 2 110 2 1579 4 200 3 1288 6 211 4 1116 8 220 Starting from the lowest index plane we notice that there are no reflections corresponding to h² k² l² 1 3 and 5 There is no plane which has h² k² l² 7 Corresponding to h² k² l² 2 4 6 and 8 there are reflections from 110 200 211 and 220 planes respectively The powder method is a widely used experimental technique for the routine determination of crystal structures It is highly suitable for identification and for determination of the structures of crystals of high symmetry Here a monochromatic xray beam usually of Kα radiation is incident on thousands of randomly oriented crystals in powder form The powder camera called the DebyeScherrer camera consists of a cylindrical cassette with a strip of photographic film positioned around the circular periphery of the cassette The powder specimen is placed at the centre of the cassette in a capillary tube or pasted on a thin wire The tube the wire and the paste material must be of some nondiffracting substance such as glass or glue The xray beam enters through a small hole passes through the powder specimen and the unused part of the beam leaves through a hole at the opposite end The geometry of the powder method is illustrated in Fig 312 Consider a set of parallel crystal planes making an angle θ with the incident direction When this angle satisfies the Bragg equation there is reflection By virtue of the large number of randomly oriented crystals in the powder there are a number of possible orientations of this set of planes in space for the same angle θ with the incident direction So the reflected radiation is not just a pencil beam like the incident one instead it lies on the surface of a cone whose apex is at the point of contact of the incident radiation with the specimen Also the interplanar spacing d being the same for all members of a family of crystal planes they all reflect at the same Bragg angle θ all reflections from a family lying on the same cone After taking n 1 in the Bragg equation there are still a number of combinations of d and θ that would satisfy the Bragg law For each combination of d and θ one cone of reflection must result and therefore many cones of reflection are emitted by the powder specimen If the reflected cones were recorded on a flat film placed normal to the exit beam they will be in the form of concentric circles In the powder camera however only a part of each reflected cone is recorded by the film strip positioned at the periphery of the cylindrical cassette The recorded lines from any cone are a pair of arcs that form part of the circle of intersection When the film strip is taken out of the cassette and spread out it looks like Fig 312b Note that the angle between a reflected line lying on the surface of the cone and the exit beam is 2θ Therefore the angle included at the apex of the cone is twice this value 4θ Fig 312a When the Bragg angle is 45 the cone opens out into a circle and reflection at this angle will make a straight line intersection with the film strip at the midpoint between the incident and the exit points in Fig 312b When the Bragg angle is greater than 45 back reflection is obtained that is the reflected cones are directed towards the incident beam Bragg angles up to the maximum value of 90 can be recorded by the film of the powder camera which is not possible on a flat film placed in front of the exit beam The exposure in a powder camera must be sufficiently long to give reflected lines of good intensity The exposure time is usually a few hours After the film is exposed and developed it is indexed to determine the crystal structure It is easily seen that the first arc on either side of the exit point corresponds to the smallest angle of reflection The pairs of arcs beyond this pair have larger Bragg angles and are from planes of smaller spacings recall that d λ2 sin θ The distance between any two corresponding arcs on the spread out film is termed S Fig 312b S is related to the radius of the powder camera R S 4Rθ 36 where θ is the Bragg angle expressed in radians For easy conversion of the distance S measured in mm to Bragg angle in degrees the camera radius is often chosen to be 573 mm as 1 rad 573 In the powder method the intensity of the reflected beam can also be recorded in a diffractometer which uses a counter in place of the film to measure intensities The counter moves along the periphery of the cylinder and records the reflected intensities against 2θ Peaks in the diffractometer recording Fig 313 correspond to positions where the Bragg condition is satisfied 36 Structure Determination The determination of a complex crystal structure is often time consuming requiring a lot of patience and ingenuity A stepbystep procedure is followed in such cases first determining the macroscopic symmetry of the crystal then the space lattice and its dimensions and finally the atomic arrangement within the unit cell Measurement of the density of the crystal and the chemical composition also assist the process of structure determination In simple crystals of high symmetry the space lattice and its dimensions can be determined relatively easily If the crystal is monoatomic the space lattice together with the lattice parameters is a complete description of the crystal structure If on the other hand the basis is two or more atoms per lattice point the number and distribution of atoms within the unit cell can be determined only from quantitative measurements of the reflected intensities For such measurements the recording from a diffractometer is more useful than the pattern obtained from a powder camera The procedure for determining the structure of monoatomic cubic crystals is outlined below Combining Eq 32 for the interplanar spacing d with the Bragg equation we obtain sin²θ λ²4a²h² k² l² 37 n is assumed to be 1 for reasons already outlined θvalues can be determined from a powder pattern using Eq 36 Since monochromatic radiation is used in the powder technique the value of λ is known Then the unknowns in Eq 37 are the Miller indices of the reflecting planes that correspond to the measured angles of reflection For a given cubic lattice it is possible to list all combinations of h k and l and arrange h² k² l² in increasing order which will also be the increasing order of θ values as seen from Eq 37 The sin²θ values will be in the same ratio as h² k² l² if the assumed and actual lattices coincide Xrays with a wavelength of 154 Å are used to calculate the spacing of 200 planes in aluminum The Bragg angle for this reflection is 224 What is the size of the unit cell of the aluminium crystal Check your answer with table on back inside cover Determine the structure of the crystal in Example 37 Answer BCC Find the Miller indices of a plane that makes intercepts on a b and c axes equal to 3 Å 4 Å and 3 Å in a tetragonal crystal with the ca ratio of 15 Answer 436 Using a diffractometer and a radiation of wavelength 154 Å only one reflection from an FCC material is observed when 2θ is 121 What are the indices of this reflection What is the interplanar spacing Show that the next higher index reflection cannot occur Answer 111 0885 Å Explain why there is no endcentred cubic space lattice The acute angle between 101 and 101 directions in a tetragonal crystal with ca 15 is A 90 B 6738 C 5630 D 3369 The angle between 111 and 112 directions in a cubic crystal is A 0 B 45 C 90 D 180 If the first reflection from a FCC crystal has a Bragg angle θ of 215 the second reflection will have an angle θ of A 185 B 25 C 312 D 368 The rigidity of a solid arises from the fact that the atoms in the solid are held together by interatomic bonds The spatial arrangement of atoms in a solid is strongly influenced by the nature of these bonds which in turn is influenced by the electronic structure of the atoms Plancks constant h 6626 10³⁴ J s Electronic charge e 1602 10¹⁹ C Electron rest mass me 9109 10³¹ kg Velocity of light c 2998 10⁸ m s¹ STRUCTURE OF THE ATOM As the reader is aware the atom consists of a nucleus and surrounding electrons The nucleus is composed of protons and neutrons As the mass of the electron is negligible compared to that of protons and neutrons the mass of the atom depends mostly on the number of protons and neutrons in the nucleus The neutrons carry no charge the protons are positively charged and the electrons carry a negative charge The charge on a proton or an electron is equal to 1602 10¹⁹ C iii mℓ is the magnetic quantum number and defines the spatial orientation of the electron probability density cloud mℓ can take values ℓ ℓ 1 1 0 1 ℓ 1 and ℓ so that there are 2ℓ 1 values of mℓ iv ms is called the electron spin quantum number It defines the spin of the electron There are two possible spins spin up and spin down corresponding to ms 12 and 12 The Pauli exclusion principle states that no two electrons can have the same set of quantum numbers In other words only one electron can occupy a given quantum state When there are a number of electrons surrounding the nucleus of an atom they have to occupy different quantum states In the lowest energy state of the atom known as the ground state the electrons occupy the lowest energy levels without at the same time violating the Pauli exclusion principle The order of increasing energy of the orbitals is as follows 1s 2s 2p 3s 3p 4s 3d 4p 5s 4d 5p 6s 4f 5d 6p 7s The electron probability cloud of an sorbital is spherically symmetric as shown in Fig 41a The electron probability density ρ decreases with increasing distance from the nucleus The radial density 4πr²ρ measures the probability of finding the electron in a thin spherical shell of radius r The radial density is zero at the nucleus It increases to a maximum value at a radial distance r₀ from the nucleus where the electron is most likely to be found see Fig 41b Thereafter the radial density falls with further increase in the radial distance The energy of the 1s electron is equal to 218 10¹⁸ J 136 eV The reference state of zero energy is that in which the electron is at an infinite distance from the nucleus The negative value of the energy indicates that the system of the proton and the electron has lost energy to the surroundings as the electron was attracted to the 1s orbital from infinity The 2s orbital of the hydrogen atom which is normally vacant has a higher energy equal to 544 10¹⁹ J 34 eV From the Einstein relation Eq 42 Frequency of radiation emitted v 164 10¹⁸6626 10³⁴ 248 10¹⁵ Hz From Eq 43 Wavelength of radiation emitted λ 2998 10⁸248 10¹⁵ 121 10⁷ m 1210 Å The element next to hydrogen is the inert gas helium with an atomic number Z 2 It has two protons and two neutrons in its nucleus The charge of the protons is balanced by two electrons in the 1s orbital When n 1 l 0 ml 0 ms 12 Thus only two quantum states are possible for n 1 The two electrons of the 1s orbital are of opposite spins and have spin quantum numbers ms 12 and 12 When the helium atom is in the ground state the 2s orbital is vacant as in the case of hydrogen When n 2 we can have l 0 ml 0 ms 12 2s l 1 ml 1 ms 12 l 0 ms 12 ml 1 ms 12 Therefore the maximum number of electrons in the second principal orbital is eight Correspondingly there are eight elements in the second row of the periodic table The first element is the alkali metal lithium with Z 3 It has two electrons of opposite spin in the 1s orbital and a third electron in the 2s orbital This filling order is in accord with the Pauli exclusion principle and the minimum energy criterion This electronic configuration is denoted as 1s²2s¹ As can be seen from the first column of the periodic table a lone 1s¹ electron in the outermost principal orbital is the characteristic of all alkali metals Beryllium with Z 4 has an electronic configuration of 1s²2s² After beryllium the electrons start to fill the states of the second principal orbital Two linear combinations addition and subtraction of the electron density clouds corresponding to ml 1 and 1 yield the px and py orbitals The pz orbital results when ml 0 In contrast to the s orbital the probability cloud of a p orbital is not spherically symmetric Each of px py and pz orbitals has the maximum electron density along one of the three coordinate directions as shown in Fig 42 When bonds are formed by electrons of the p orbitals this directional nature of the orbital plays an important role in determining the bond angles The electron configuration of the elements from lithium to neon is summarized in Table 41 Fig 42 The electron density clouds of px py and pz orbitals are concentrated each along one of the three coordinate axes Hunds rule states that in order to reduce the electronelectron repulsive energy the number of electrons of the same spin in p d or f states should be maximum The filling order in the p orbital shown in Table 41 is in accord with this rule Nitrogen for example has three electrons in the p orbital They are distributed one each in the px py and pz orbitals so that all three can have the same spin The halogen fluorine is only one electron short for the full complement of p electronsa characteristic common to all the halogens When all the electrons are present in the p orbital the sum of the electron probability clouds along the three coordinate axes results in spherical symmetry Neon has all the six p electrons and has a spherically symmetric cloud around it This configuration is called the inert gas configuration All the elements at the end of a row in the periodic table are inert gases The third row of the periodic table starts as in the second row with an alkali metal sodium having a 1s¹ configuration in the outermost orbital complete electron configuration of sodium is 1s²2s²2p⁶³s¹ The row ends with the inert gas argon with configuration 1s²2s²2p⁶³p⁶ The number of quantum states in the third principal orbital is 18 since for n 3 we can have l 0 ml 0 ms 12 3s l 1 ml 1 ms 12 l 0 ms 12 3p ml 1 ms 12 l 2 ml 2 ms 12 ml 1 ms 12 ml 0 ms 12 3d ml 1 ms 12 ml 2 ms 12 However the 3d states corresponding to l 2 remain vacant till the fourth row is reached in accordance with the order of increasing energy of the orbitals Therefore the third row of the periodic table has only eight elements The fourth row starts with potassium of configuration 1s²2s²2p⁶³s² Next comes calcium After calcium the first transition series begins where the 3d orbitals begin to get filled The d orbitals are directional in nature like p orbitals and can hold a maximum of 10 electrons The order of filling of 3d orbitals is summarized in Table 42 60 Atomic Structure and Chemical Bonding When all the ten electrons are filled the d orbital acquires spherical symmetry and we move out of the transition series The copper atom with the outer configuration 3d104s1 has a spherical electron probability cloud around the nucleus Next to copper zinc has the outer configuration 3d104s2 The 4p orbitals begin to get filled from gallium onwards up to the inert gas krypton see the Periodic Table The fifth row contains the second series of transition elements and is similar to the fourth row in the order of filling The 4f and the 5f orbitals are filled in the sixth and the seventh rows of the periodic table There are 14 forbitals corresponding to the value of l 3 and ml 3 2 1 0 1 2 and 3 Elements with partially filled f orbitals are called the rare earth elements The sixth row has 32 elements in it while the seventh row goes up to lawrencium Z 103 In recent years some elements with Z 103 have been discovered 43 Ionization Potential Electron Affinity and Electronegativity We have already noted that electrons in the orbitals around the nucleus can absorb energy and get promoted to higher energy levels If sufficient energy is supplied an electron in the outer orbital can break away completely from the atom and become free The energy required to remove an electron in this manner is known as the ionization potential The energy to remove the outermost electron which is weakly bound to the atom is called the first ionization potential Figure 43 gives the first ionization potentials of the elements The potential is the least for the first atom in a row of the periodic table and increases as we Ionization potential eV 25 20 15 10 5 He Ne N O B Li Be Na Al S P Ar Kr Xe ZnAs Se Ga Rb In Cs Cd Sb Te Tl Hg Mg 10 20 30 40 50 60 70 80 Z Fig 43 First ionization potentials of the elements go to the right along the row The alkali metals have a lone s electron in their outermost orbital which can be removed with relative ease The inert gases on the other hand have the full complement of s and p electrons in their outermost principal orbital Removing an electron from the stable inert gas configuration requires a relatively large expenditure of energy Further as we go down a column of the periodic table the outermost electrons are less and less tightly bound to the nucleus Correspondingly among the alkali metals the first ionization potential is highest for lithium at the top of the column and lowest for cesium at the bottom Among the inert gases the potential is highest for helium and lowest for xenon When an electron is removed from the neutral atom there is a decrease in the mutual repulsion between the orbital electrons They can approach one another more closely and are therefore attracted more strongly to the nucleus resulting in the shrinking of all the orbitals As a result the energy required to remove the second and successive electrons becomes increasingly greater Example 42 Give the electronic configuration of a neutral iron atom a ferrous ion and a ferric ion Compare their sizes Solution Species Electronic configuration Radius Å Neutral iron Fe 1s22s22p63s23p63d64s2 124 Ferrous ion Fe2 1s22s22p63s23p63d6 083 Ferric ion Fe3 1s22s22p63s23p63d5 067 The size of the atom decreases considerably as more and more electrons are removed from the outer orbital Consider a system of a neutral atom and an extra electron The work done by this system when the extra electron is attracted from infinity to the outer orbital of the neutral atom is known as the electron affinity of the atom The electron affinities of some elements are shown in Table 43 The stable configuration of the inert gases has no affinity for an extra electron The halogens which are just one electron short to achieve the stable inert gas configuration have the largest electron affinities When an extra electron is added to a neutral atom there is a weakening of the attraction of the electrons to the nucleus resulting in an expansion of the electron orbitals and an increase in the size of the atom Ionization Potential Electron Affinity and Electronegativity 61 62 Atomic Structure and Chemical Bonding TABLE 43 Electron Affinities of Some Elements Values are given in eV and kJ mol1 For one electron H He 07 0 68 0 Li Be B C N O F Ne 054 0 054 113 02 148 362 0 52 0 52 109 19 143 349 0 Na Mg Al Si P S Cl Ar 074 0 04 190 080 207 382 0 71 0 39 183 77 200 369 0 Br Kr 354 0 342 0 I Xe 324 0 313 0 The tendency of an atom to attract electrons to itself during the formation of bonds with other atoms is measured by the electronegativity of that atom This is not the same as the electron affinity where the tendency of an atom to attract an isolated electron is measured Consider for example the bond formation between hydrogen and fluorine by the sharing of their outer electrons Fluorine has a greater tendency to attract the bonding electrons to itself than hydrogen Fluorine has thus a larger electronegativity than hydrogen Pauling has worked out empirically the electronegativities of elements Table 44 As seen from the table the halogens have the largest electronegativities and the alkali metals have the smallest Qualitatively the nonmetals such as the halogens are said to be electronegative A value of 2 on the electronegativity scale can be taken as the approximate dividing line between metals and nonmetals TABLE 44 Paulings Electronegativities of Elements H 21 Li Be B C N O F 10 15 20 25 30 35 40 Na Mg Al Si P S Cl 09 12 15 18 21 25 30 K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br 08 10 13 15 16 16 15 18 18 18 19 16 16 18 20 24 28 Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I 08 10 12 14 16 18 19 22 22 22 19 17 17 18 19 21 25 Cs Ba La Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At 07 09 11 13 15 17 19 22 22 22 24 19 18 18 19 20 22 CHEMICAL BONDING 44 Bond Energy Bond Type and Bond Length The forces between two atoms or ions as a function of their distance of separation r are schematically shown in Fig 44 When the distance of Bond Energy Bond Type and Bond Length 63 Fr F 0 Force r0 Fa r Fig 44 Interatomic forces and potential energy in a system of two atoms as a function of their distance of separation r 0 Potential energy W r0 W0 r separation is large the significant force is the attractive negative force Fa At closer distances of approach a repulsive positive force Fr also becomes significant and increases rapidly with decreasing distance of separation When the distance of separation is r0 the attractive and the repulsive forces exactly balance each other and the net force is zero This distance corresponds to stable equilibrium with a minimum in potential energy W The magnitude of the minimum energy W0 is called the bond energy 64 Atomic Structure and Chemical Bonding Bond energy should be expressed in kilojoules per mole of bonds kJ mol1 Other units such as electron volt per bond and kilocalories per mole of bonds are also used It is useful to remember that 1 eVbond is approximately equal to 100 kJ mol1 and that 1 kilocalorie is equal to 418 kilojoules The energy per mole of bonds is not necessarily equal to the enthalpy of atomization which is the energy required to convert one mole of atoms in a solid into its component atoms in the gaseous state This is so because each atom in the solid may form bonds with a number of neighbours and all these bonds are broken during vapourization If the bonds are discrete the enthalpy of atomization equals z2 times the energy per mole of bonds where z is the number of nearest bonding neighbours of an atom The factor 2 arises as each bond between a pair of neighbours is counted twice According to strength chemical bonds can be grouped into primary and secondary bonds Primary bonds have bond energies in the range 100 1000 kJ mol1 110 eVbond Covalent metallic and ionic bonds are all primary bonds Among these covalent and ionic bonds are generally stronger than metallic bonds Secondary bonds have energies in the range 150 kJ mol1 00105 eVbond one or two orders of magnitude smaller than those of primary bonds Examples of secondary bonds are van der Waals bonds and the hydrogen bond Generally van der Waals bonds are very weak Few materials have pure bonds of one type or the other Occurrence of mixed bonds in materials is the rule rather than the exception In fact empirical rules exist to apportion a fraction of a bond to a particular type Notwithstanding this it is useful to classify materials according to the bond type that is dominant in a given material This classification helps in predicting the approximate properties and behaviour of a material under a given set of conditions even if the actual behaviour may sometimes not agree with this prediction The length of a bond r0 in Fig 44 is defined as the centretocentre distance of the bonding atoms Strong bonds pull the bonding atoms closer together and so have smaller bond lengths as compared to weak bonds Primary bonds have lengths in the range 12 Å 0102 nm Secondary bond lengths are larger in the range 25 Å 0205 nm The length of a bond can be used to define atomic or ionic diameters When bonding is between two neighbouring atoms of the same kind the atomic diameter is simply equal to the bond length see Fig 45a For example the equilibrium distance between the atomic centres in the diatomic chlorine molecule is 181 Å which is also the diameter of the chlorine atom To indicate the character of the bond this can be called the covalent diameter of chlorine The equilibrium distance between two nearest bonding copper atoms in a copper crystal is 256 Å which is also the metallic diameter of copper Some ambiguity in this definition arises if the element in question exhibits different crystal forms For example the diameter of the iron atom is 248 Å when it is surrounded by eight neighbours in the BCC crystal and 254 Å when it has 12 nearest neighbours in the FCC crystal When two bonding atoms are of different types as in the ionic bonding the bond length is equal to the sum of their radii rc ra as illustrated in Fig 45b The equilibrium distance of separation shown in Fig 44 applies to 0 K where there is no thermal energy At higher temperatures under the influence of thermal energy atoms vibrate about their mean positions the amplitude of vibrations increasing with increasing temperature As can be seen from Fig 46 at temperature T1 the amplitude is a1b1 and at T2 T2 T1 the amplitude is a2b2 The corresponding mean spacings between the atoms are given by r0 and r0 As the repulsive force is short range in nature as compared to the attractive force the potential energy curve is steeper on the left side of r0 than on the right side Therefore r0 r0 r0 as shown in Fig 46 That is the mean bond length increases on heating In other words the material exhibits thermal expansion Ionic bonding forms between two oppositelycharged ions which are produced by the transfer of electrons from one atom to another The steps in the formation of an ionic bond between sodium and chlorine can be visualized as follows The minimum in the energy function can be obtained by setting dWdr 0 to yield the bond energy W0 and the equilibrium bond length r0 The strength of an ionic bond will depend on the relative magnitudes of the three energy terms on the right side of Eq 44 Considering ΔE first this must be as small as possible Electropostive elements such as the alkali metals have small ionization potentials Electronegative elements such as halogens have large electron affinities Hence ionic bonds form most readily between electropositive and electronegative elements ΔE is almost zero for electron transfer from a cesium atom to a chlorine atom Consequently these two elements form a bond that is dominantly ionic in character The Coulomb term is dependent on the charges of the ions forming the bond Ionic crystals with multivalent ions have generally stronger bonds and hence higher melting points than crystals with univalent ions Thus MgO BeO and Al2O3 are refractory oxides with melting points above 2000C MgO magnesia has important applications as a refractory in the steel making industry Al2O3 alumina crucibles can hold almost all common metals in the molten condition 68 Atomic Structure and Chemical Bonding 46 Covalent Bonding Covalent bonding occurs by the sharing of electrons between neighbouring atoms This is in contrast to the transfer of electrons from one atom to another in the ionic bonding For sharing with a net decrease in potential energy good overlap of the orbitals which will bring the shared electrons close to both the nuclei is necessary This occurs readily when there are vacant electron states in the outermost orbital of the bonding atoms When the overlapping orbitals are directionally oriented and not spherically symmetric good overlapping and substantial decrease in the potential energy can occur This also gives directionality to the covalent bond Consider first the formation of a hydrogen molecule When two hydrogen atoms are very far apart they do not interact and the lone electrons of the atoms stay in their respective 1s ground states When the atoms come closer the electron probability clouds of the 1s states overlap As the 1s orbitals can have two electrons of opposite spin the sharing of electrons between the two atoms takes place without having to promote the electrons to higher energy levels Both the electrons are close to both the nuclei and in fact spend much of the time in between the two nuclei The hydrogen molecule has a bond energy 436 kJ mol1 and a bond length of 074 Å The next element helium cannot form a covalent bond as this requires promotion of the 1s electrons to the second principal orbital Sharing of electrons and the formation of covalent bonds readily occurs between atoms which have unfilled p orbitals The p orbitals are directional in nature and hence permit efficient overlapping of the orbitals in the direction of the maximum electron probability density Fluorine with Z 9 and an electronic configuration 1s22s22p5 has a vacant state in the pz orbital Two fluorine atoms can come together such that the halffilled pz orbitals overlap As there are no other halffilled orbitals each fluorine atom forms only one bond In the elemental state fluorine forms the diatomic molecule F2 with a bond energy of 154 kJ mol1 and a bond length of 142 Å In oxygen to the left of fluorine the py and pz orbitals have one vacant state each Each of the unpaired electrons in these orbitals can share an electron with another atom In the water molecule the unpaired 1s electrons of two hydrogen atoms pair up with the two unpaired 2p electrons of oxygen If the overlap occurred without any distortion of the p orbitals the angle between the bonds would be the same 90 as between the unbonded p orbitals However the observed HOH bond angle in the water molecule is 104 The formation of the bond here is more appropriately visualized to occur between hybridized orbitals produced by the interaction between the 2s orbitals and the 2p orbitals of the oxygen atom The hybridized orbitals known as the sp3 orbitals are four in number and can hold a total of eight electrons Their mutual orientation is ideally the one between the four lines joining the centre to the four corners of a regular tetrahedron A regular tetrahedron is a solid figure made up of four faces all of which are equal and equilateral triangles The interbond angle corresponding to this orientation is 1095 In the water molecule two of the hybridized orbitals of oxygen are occupied by four electrons of oxygen and hence do not take part in bonding In the other two orbitals electrons are shared between two hydrogen atoms By a similar bonding mechanism the other elements of the sixth column form two bonds That is each atom is bonded to two neighbours The SSS bond in sulphur has a bond angle of 107 The SeSeSe bond in selenium and the TeTeTe bond in tellurium have a bond angle of 104 The above are examples of an endtoend overlap of p orbitals giving rise to what is known as a σ sigma bond When there is lateral overlap of p orbitals π pi bonds are said to form Double and triple bonds are examples of π bonds Figure 47 illustrates the difference between the endtoend overlap and the lateral overlap of p orbitals In the diatomic oxygen molecule one double bond holds together the two oxygen atoms with a bond energy of 494 kJ mol¹ and a bond length of 121 Å A fifth column element has all three p electrons unpaired Hence an atom here forms three covalent bonds The bond angles reflect the fact that the three p orbitals have a mutually perpendicular orientation Some examples are given in Table 46 Atomic number Bond energy kJ mol¹ Bond length Å Melting point C Boiling point C Phosphorus 15 PPP 99 110 214 Arsenic 33 AsAsAs 97 125 134 Antimony 51 SbSbSb 96 145 126 Bismuth 83 BiBiBi 94 156 105 These bond angles are closer to the 90 angle between unbonded p orbitals rather than to the tetrahedral angle of 1095 which is characteristic of hybridized orbitals With increasing atomic number the bond length increases and the bond strength decreases As we go down the column the increasing shielding effect of the electrons of the inner orbitals binds the outer electrons less and less tightly to the nucleus This explains the decrease in the bond strength as the atomic number increases Nitrogen in this column forms a diatomic molecule with a triple bond between two nitrogen atoms which has a bond energy of 942 kJ mol¹ and a bond length of 110 Å The diamond form of carbon and other fourth column elements share all the four hybridized orbitals with four neighbours The four bonds formed are of equal strength and bear the ideal tetrahedral angle of 1095 to one another They form crystals with a threedimensional network of covalent bonds The bonding characteristics are given in Table 47 47 Metallic Bonding The elements to the left of the fourth column in the periodic table exhibit metallic characteristics The sharing of electrons between neighbouring atoms now becomes delocalised as there are not enough electrons to produce the inert gas configuration around each atom The metallic sharing changes with time and the bonding electrons resonate between different atoms The metallic state can be visualized as an array of positive ions with a common pool of electrons to which all the metal atoms have contributed their outer electrons This common pool is called the free electron cloud or the free electron gas These electrons have freedom to move anywhere within the crystal and act like an allpervasive mobile glue holding the ion cores together This is in sharp contrast to the electrons in covalent bonding which are localised bind just two neighbouring atoms and stay with them This freedom makes the metallic bonds nondirectional Note that the ion cores have spherical electron density clouds around them The bond energies of common metals are listed in Table 49 TABLE 49 Bond Energies of Metals Bond energy No of bonding Melting Boiling Metal kJ moll neighbours of point C point C of bonds an atom Copper Cu 564 12 1083 2595 Silver Ag 475 12 961 2210 Gold Au 600 12 1063 2970 Aluminium Al 540 12 660 2450 Nickel Ni 716 12 1453 2730 Platinum Pt 943 12 1769 4530 Lead Pb 325 12 327 1725 Cobalt Co 708 12 1495 2900 Magnesium Mg 246 12 650 1107 Cadmium Cd 186 12 321 765 Zinc Zn 219 12 420 906 Sodium Na 270 8 98 892 Tungsten W 2123 8 3410 5930 Iron Fe 1040 8 1535 3000 Chromium Cr 993 8 1875 2665 Molybdenum Mo 1646 8 2610 5560 Vanadium V 1285 8 1900 3400 Niobium Nb 1808 8 2468 4927 Bond lengths can be derived from the data in the table on the back inside cover Metallic Bonding 71 Molecule Dipole moment C m Debye Bond energy kJ mol¹ Melting point C Boiling point C Water H₂O 62 10³⁰ 185 205 0 100 Ammonia NH₃ 49 10³⁰ 148 78 78 33 HF 67 10³⁰ 200 315 83 20 Example 44 From the enthalpy of fusion of ice estimate the fraction of hydrogen bonds that are broken when ice melts Solution The enthalpy of fusion of ice 602 kJ mol1 There are two moles of hydrogen bonds per mole of H2O in ice From Table 410 the hydrogen bond energy 205 kJ mol1 Assuming that all the heat absorbed during melting goes into breaking the bonds the fraction of bonds broken is given by 602205 2 015 As bonds are continuously broken and remade in a liquid this should be considered as a timeaveraged value Inert gas atoms have spherically symmetric electron probability clouds around them and therefore have no permanent dipole moments Yet inert gases form solid crystals at sufficiently low temperatures The bonding in such solids is called the van der Waals bonding It is the result of momentary fluctuations in the charge distribution around an atom Even though the time averaged electron probability distribution is spherically symmetric the electronic charge at any instant of time is concentrated locally resulting in a weak fluctuating dipole within the atom The electric field of this imbalance can induce a dipole moment in a neighbouring atom in such a way as to attract it The dipole in the second atom can in turn induce a dipole in a third atom in order to attract it and so on This dipoleinduceddipole attraction is nondirectional in nature When the atoms come closer a repulsion arises which can be explained as in other bonds on the basis of the Pauli exclusion principle The bond energies and the bond lengths of some inertgas crystals formed by van der Waals attraction are given in Table 411 The very low bond energies are associated with correspondingly large bond lengths 74 Atomic Structure and Chemical Bonding 49 Variation in Bonding Character and Properties We will examine some generalizations between properties and the bonding character of a material The solid state can be visualized as atoms vibrating about their mean positions on fixed atomic sites In the liquid state the atoms have also translational freedom and can slide past one another The bonds between atoms in the liquid are continuously broken and remade In the gaseous state the bonds are totally broken The thermal energy of atoms must be sufficient to achieve these disruptions in bonding The higher the bond strength the more will be the thermal energy required to break the bonds Correspondingly strongly bonded materials tend to have high melting and boiling temperatures Among the primary bonds covalent and ionic bonds are generally stronger than metallic bonds Hence covalent and ionic solids have high melting and boiling points The general trend of correlation between bond strength and the melting and boiling temperatures is already shown in several tables of this chapter When a solid consists of molecules held together by secondary bonds the melting and boiling points of the solid reflect only the strength of the secondary bonds between the molecules and not the strength of the primary bonds within the molecule Here the molten state and the gaseous state are to be visualized as consisting of units of molecules and not individual atoms of the molecule The silica SiO2 crystal which has a threedimensional network of SiO bonds without any secondary bonding has a melting point of 1723C During melting the SiO bonds are broken In sharp contrast to this methane CH4 has CH bonds in the molecule The molecules are held together by van der Waals bonds in the solid which melts at 182C Here the CH bond strength of 413 kJ mol1 which is even larger than the SiO bond strength of 375 kJ mol1 has no correlation with the melting point of methane Rather the van der Waals bond strength of 136 kJ mol1 is to be associated with the low melting temperature The thermal and electrical conductivities of a solid are to a large extent dependent on the presence of free electrons in the solid In ionic solids the electron transfer produces the stable inert gas configuration around both the cations and the anions Hence there are no free electrons in ionic solids Likewise covalent bonding produces the inert gas configuration around atoms sharing the electrons with the result that there are no free electrons here either Hence typical ionic and covalent materials are good thermal and electrical insulators Solids which have secondary bonds such as van der Waals bonds wholly or in addition to ionic or covalent bonds are also good insulators In contrast metals have free electrons and are therefore good conductors of heat and electricity The best known conductors of heat and electricity are copper silver and gold We noted that the thermal expansion of materials arises from the asymmetry of the potential energy versus distance curve refer Fig 46 Deep potential wells are more symmetrical about the equilibrium position r0 than shallow potential wells So the thermal expansion at a given temperature tends to be less for strongly bonded materials than for weakly bonded materials The thermal expansion coefficients of a number of materials at room temperature are given in Appendix I The mechanical properties of solids are dependent on the strength of the bonds as well as the directional nature of bonding Solids with strong and directional bonds tend to be brittle For example covalently bonded diamond is very hard and brittle As metallic bonds are relatively weak and nondirectional metals are soft ductile and malleable They can change their shape permanently without breaking Ionic solids fall in between covalent and metallic solids in that they may exhibit a very limited amount of ductility In any one row of the periodic table as we go from left to the right the metallic character of the bond decreases and the covalent character increases Metallic bonds being weaker than covalent bonds this transition is reflected in the increasing bond energy and the decreasing bond length from left to the right as shown in Table 412 for the third row of the periodic table Similarly the transition from covalent to metallic character as we go from top to the bottom of a column is seen in the decreasing bond energy in Table 47 TABLE 412 Properties of Elements of the Third Row Element Na Mg Al Si P S Cl Bond energy kJ mol1 27 25 54 176 214 243 242 Bond length Å 372 318 286 236 220 208 181 Melting point C 98 650 660 1410 44 119 101 Boiling point C 892 1107 2450 2680 280 445 35 Many important engineering metals and alloys belong to the three transition series Here in addition to one or two electrons in the outermost s orbital there are partially filled d orbitals When bonding occurs there is overlap of s orbitals as well as some overlap of d orbitals As d orbitals have directional characteristics the electronic structure of transition elements lends a partial covalent character to the bonding This reflects in their properties which fall between those of covalent crystals and typical metals The thermal and electrical conductivities of transition metals are lower than those of typical metals such as copper and aluminium The transition metals are also hard and not so ductile as copper or silver The melting points of transition metals are higher than those of typical metals Figure 49 shows the variation in bond energy melting point thermal expansion and density in the first transition series of elements Variation in Bonding Character and Properties 75 Fig 49 The variation in the bond strength melting point thermal expansion and density of the metals of the first transition series SUMMARY 1 The atom consists of a nucleus and surrounding electrons Each electron occupies a quantum state with a unique set of quantum numbers 2 With increasing atomic number the elements in the periodic table have increasing number of protons and electrons The order of occupation of quantum states by electrons is determined by the Pauli exclusion principle the Hunds rule and the minimum energy criterion 3 Ionization potential is the energy required to remove an electron from the outer orbital of an atom Electron affinity is the energy released when a free electron is added to the outer orbital The electronegativity of an atom is a measure of its tendency to attract bonding electrons to itself 4 The magnitude of the energy released when two atoms come together from a large distance of separation to the equilibrium distance is called the bond energy It is related to the enthalpy of atomization of the solid The centre to centre distance of the atoms at equilibrium is the bond length 5 Primary bond energies are in the range 1001000 kJ mol1 Secondary bond energies are in the range 150 kJ mol1 6 Electrons can be transferred from an electropositive atom to an electronegative atom producing ions of opposite sign and giving rise to the nondirectional ionic bond 7 Sharing of electrons between neighbouring atoms results in a covalent bond which is directional 8 A metal is an array of positive ions which are held together in a cloud of free electrons The metallic bond is nondirectional and generally weaker than ionic and covalent bonds 9 Melting and boiling points of materials increase with increasing bond strength Strong and directional bonds result in hard and brittle solids Free electrons are responsible for the high thermal and electrical conductivities of metals PROBLEMS 41 Find the minimum uncertainty in determining the position of a particle if the uncertainty in its momentum Dp is not to exceed 1030 kg m s1 Answer 1055 104 m 42 Using the rest mass of the electron and the mass of a cricket ball calculate the uncertainty in the velocity of each for Dp 1030 kg m s1 Answer 1 m s1 and 1029 m s1 43 Calculate the limits within which the energy difference of an electronic transition should be in order that the emitted or absorbed radiation is in the visible range 3900 to 7800 Å of the electromagnetic spectrum Answer 509 1019 J 318 eV to 255 1019 J 159 eV 44 Prepare a table similar to Table 42 showing the outer electron configuration of the second transition series of elements Answer 32 45 On the basis of values permissible for the four quantum numbers derive the number of quantum states corresponding to the fourth principal shell n 4 Answer 32 46 Give the electronic configuration of the fluorine atom and the F ion Compare their sizes 47 The heat of dissociation of the chlorine molecule is 1213 kJ mol1 The ClCl bond energy is 2424 kJ mol1 Reconcile this difference 48 Derive the units of the constant A in Eq 44 Compare it with the units of the dielectric constant in Chap 17 49 The potential energy W for the formation of a bond between two univalent ions is given by Eq 44 A 113 1011 in appropriate SI units The equilibrium spacing r0 250 Å Find the value of the constant B given m 9 Answer 492 10105 J m² Answers 1 C 2 A 3 C 4 B 5 C 6 D 7 B 8 D 9 D 10 C 11 D 12 A 13 A Suggestions for Further Reading MFC Ladd Structure and Bonding in Solid State Chemistry Ellis Horwood Chichester UK 1979 L Pauling The Nature of the Chemical Bond Cornell University Press Ithaca 1960 HH Sisler Electronic Structure Properties and the Periodic Law Reinhold Publishing Corporation New York 1963 80 Suggestions for Further Reading 82 Structure of Solids 51 The Crystalline and the Noncrystalline States The number and kind of nearest neighbours that an atom or an ion has in a solid is nearly the same for both the crystalline and the noncrystalline forms of the solid However the noncrystalline structure does not have the long range periodicity characteristic of the crystalline state In a crystal any number of integral lattice translations would take us from an atom located at a lattice point to another identical atom located at a different lattice point This is not true for the noncrystalline state As long range periodicity is the basis of diffraction effects the noncrystalline solids do not give rise to sharp diffraction patterns like crystals Several factors promote the formation of noncrystalline structures When primary bonds do not extend in all directions onedimensional chain molecules or twodimensional sheet molecules are formed Such units have to be aided by secondary bonding forces to form a threedimensional crystal Consider a structure consisting of long chain molecules In the molten state the chains persist and are like a bowl of wriggling earthworms see Fig 5la The wriggling refers to the translational motion of the chains past one another On cooling if the secondary bonding forces are not strong enough to exert themselves the chains cannot get straightened out of the tangle to become the orderly parallel arrangement of a crystal as in Fig 5lb The translational freedom is gradually lost during cooling till the noncrystalline glassy state is reached where only the vibrational degree of freedom remains The earthworms are frozen in place in the solid in some random configuration similar to that in Fig 51a a Fig 51 a The tangled up configuration of long chain molecules b the parallel array of the chain molecules characteristic of a crystal b The free energy of the crystalline state is always lower than that of the noncrystalline state Only when the free energy difference between the two states is large in magnitude the tendency to crystallize will be strong Some materials form a relatively open network structure of atoms where there is little free energy difference between an orderly array and a disorderly array of the units In such cases the tendency to crystallize will be weak A third factor that promotes the formation of noncrystalline structures is the rate of cooling from the liquid state Kinetic barriers exist along the path of transition from the liquid to the crystalline state Slow cooling rates allow enough time for crystallization while fast cooling rates may prevent crystallization altogether The rate of cooling which is considered as slow or fast will vary widely for different materials depending on the magnitude of the kinetic barrier For a metal the cooling rate required to prevent crystallization may be as high as 106 K s1 For a silicate cooling at the rate of a fraction of K per hour may be sufficient to prevent crystallization The crystal exhibits a sharp melting point in contrast to the noncrystalline material which gradually softens over a range of temperature As a result of the regularity of the arrangement the atoms or molecules in a crystal are more closely packed Hence the crystalline form has a higher density than the noncrystalline form The closer packing in the crystal tends to increase the average strength of the secondary bonds present INORGANIC SOLIDS 52 Covalent Solids Starting from the seventh column of the periodic table we note that the halogens are only one electron short to fill their outermost p orbitals and therefore each atom forms one covalent bond This results in diatomic molecules of F2 Cl2 Br2 and I2 Only weak secondary forces can hold these molecules together in a crystal Fluorine and chlorine are in the gaseous state at ambient temperatures bromine is a liquid and iodine forms an orthorhombic crystal The sixth column elements form two covalent bonds by the sharing of the two halffilled p orbitals of each atom A number of these elements form long zigzag chains which are held together by secondary bonds in a solid The zigzagging reflects the angular relationship between the p orbitals taking part in bonding It is difficult to produce a stable threedimensional structure by holding together long onedimensional chains with the aid of weak bonds Hence these elements are frequently in the noncrystalline form The two bonding electrons can also produce ring molecules such as S8 small molecules like H2O and H2S or doubly bonded diatomic molecules such as O2 These molecules are bonded together by secondary forces when they form a crystal For example H2O forms the hexagonal ice crystal with hydrogen bonds between the water molecules P As Sb and Bi of the fifth column with three halffilled p orbitals form three bonds Their structure consists of puckered sheets in which each atom has three nearest neighbours The angular relationships between the p orbitals are more or less preserved The sheets are held together by van der Waals forces Covalent Solids 83 84 Structure of Solids The intersheet bonding even though stronger than the interchain bonding is still weak so that these materials are also found in the noncrystalline form The covalent bond length within a sheet is smaller than the van der Waals bond length across two neighbouring sheets This difference however decreases as we go down the column from phosphorus to bismuth with the changing character of the primary bond Antimony and bismuth form in the molten state nearly close packed structures characteristic of typical molten metals with as many as 10 or 11 nearest neighbours on an average However they crystallize with three covalently bonded neighbours to an atom As a result of this they expand on solidification filling minute cavities and reproducing the mould details accurately a property that is exploited in type casting The three bonding electrons of the fifth column elements also produce small molecules such as NH3 and triply bonded N2 When each carbon atom forms three covalent bonds due to sp2 hybridization sheets of graphite are produced Unlike the puckered sheets of the fifth column elements the graphite sheets are planar as the three sp2 bonds are coplanar with an interbond angle of 120 The sheets are held together in a crystal by van der Waals bonds as depicted in Fig 52 The fourth bonding sp2 bonds Fig 52 Sheets of graphite are held together by secondary bonds in the crystal van der Waals bonds electron of carbon is delocalized and resonates between the three sp2 bonds Its mobility accounts for a 100fold increase in electrical and thermal conductivity in a direction parallel to the sheets as compared to the perpendicular direction The weak intersheet bonding explains the softness of graphite in sharp contrast to diamond the other crystalline form of carbon which is the hardest known mineral Consequently graphite is used as a lubricant It is also the lead in pencils where the softness is controlled by varying the proportion of clay to graphite giving different grades of pencils from 4B to 4H Its high sublimation temperature and appreciable electrical conductivity are utilized in resistance heating applications Also if the sheets are aligned in a fibre such that the sheets are parallel to the fibre axis the mechanical strength and the elastic modulus can be increased by orders of magnitude In the other crystal form of carbon diamond each atom forms four bonds as a result of sp3 hybridization These four bonds produce a threedimensional network of primary bonds Diamond exists in two crystal forms the cubic and the hexagonal In both the forms the directions of the bonds from any atom in the network are given by the lines joining the centre to the corners of a regular tetrahedron with an interbond angle of 1095 We will discuss only the better known diamond cubic DC structure The DC unit cell is shown in Fig 53a A plan view of the positions of atoms in the unit cell is shown in Fig 53b The numbers indicated at atom positions represent the height of those positions from the base of the cube the height of the unit cell being taken as unity Two numbers at the same position denote two atoms one above the other a b 01 12 01 14 34 34 12 12 14 12 01 01 Fig 53 a The diamond cubic DC unit cell b plan view of atom positions in the unit cell The numbers indicate the height from the base of the cell the total height of the cell being taken as unity 01 The space lattice of the DC crystal is FCC with two atoms per lattice point The basis has one atom at a lattice point say at one corner of the unit cell and the other atom at a point quarter way along the body diagonal which is not a lattice point The distance of separation between the two atoms of the basis which are nearest neighbours is a 34 where a is the lattice parameter This distance 154 Å defines the diameter of the carbon atom in the DC crystal The number of nearest neighbours of a carbon atom known as the coordination number is four This low coordination dictated by the covalent bonding results in a relatively inefficient packing of the carbon atoms in the crystal Covalent Solids 85 to 60 Å The tubes can be up to 1 mm in length As very small fibres of high strength they have the potential to be a very effective reinforcing material for a softer matrix eg a polymer The other elements of the fourth column Si Ge and gray tin also have the diamond cubic structure of carbon The lattice parameter of the DC type crystal increases with increasing atomic number as tabulated as follows Element C dia Si Ge Gray tin Atomic number 6 14 32 50 Lattice parameter Å 357 543 565 646 A large number of compounds with equal atomic fractions of two elements crystallize in forms closely related to the cubic and hexagonal forms of diamond They are made up of two elements of IV column or one element each of III and V columns or II and VI columns or I and VII columns of the periodic table Equal atomic fractions of these combinations give on an average four electrons per atom needed for the tetrahedral covalent bonding Some examples are IVIV Compound SiC IIIV Compounds AlP AlAs AlSb GaP GaAs GaSb InP InAs InSb IIIV Compounds ZnO ZnS CdS CdSe CdTe IVII Compounds CuCl AgI For example in cubic ZnS the sulphur atoms are at the body corners and the face centres of the unit cell The zinc atoms are at the 14 14 34 34 positions within the unit cell Si Ge and the compounds listed above form the vast majority of semiconductor crystals They are used in a number of solid state devices such as diodes transistors radiation detectors photoelectric devices solar batteries thermistors and lasers a b Fig 54 Structure of C60 molecule a the icosahedron b the truncated icosahedron Covalent Solids 87 88 Structure of Solids Diamond being very hard is used in wire drawing dies and as an abrasive in polishing and grinding operations Silicon carbide is cubic and is used as an abrasive and as heating element in furnaces 53 Metals and Alloys As metallic bonds are nondirectional each metal atom in a crystal tends to surround itself with as many neighbours as possible for minimizing the potential energy As a first approximation we can take the metal atoms to be hard incompressible spheres and look at the geometry of close packing of such equal sized spheres Close packing is used here to mean the closest possible packing Close packing along a row is obtained by arranging spheres in contact with one another along a row as shown in Fig 55a Another row of close packed spheres can be placed against the first row such that each sphere of the second row fits in the space between two adjacent spheres of the first row touching both of them as illustrated in Fig 55b Repetition of this step produces a close packed plane see Fig 55c a b c Fig 55 Close packing of spheres a along a row b on two adjacent rows c on a plane d threedimensional ABCABC stacking and e ABABA stacking d e A B C A A B B A A valleys of the A planes The A plane atoms are at lattice points while the B plane atoms are not refer to Table 31 for lattice point distribution in the hexagonal lattice The effective number of lattice points is 3 whereas the effective number of atoms in the unit cell is 6 three of which are from the B plane The basis therefore consists of two atoms per lattice point There are other ways of stacking close packed planes of spheres the only restriction being that no two adjacent close packed planes can have the same symbol A stacking such as ABBA is not permissible The packing efficiency of all close packed stackings is 074 that is 26 of the space in the close packing of equal sized spheres is empty The coordination number for all close packings is 12 that is each sphere has 12 nearest neighbours 6 in the same plane and 3 each in the two adjacent planes above and below Some crystals have arrangements of atoms which do not correspond to close packing The most important of these is the arrangement in which each sphere has 8 nearest neighbours This gives rise to the body centred cubic crystal the unit cell of which is shown in Fig 33b The effective number of atoms or lattice points in the unit cell is 2 The atoms touch each other along the body diagonal It can easily be shown that the packing efficiency of this arrangement is 068 only 6 lower than that for close packing The packing efficiency depends on the coordination number as shown in Table 51 TABLE 51 Coordination Number and Packing Efficiency Crystal Coordination number Packing efficiency Diamond cubic DC 4 034 Simple cubic SC 6 052 Body centred cubic BCC 8 068 Face centred cubic FCC 12 074 A B A Fig 58 The unit cell of the HCP crystal For clarity the close packed planes are shown separated Metals and Alloys 91 As a review the space lattices the unit cells and the sharing of atoms by the unit cells are illustrated below for BCC FCC and HCP crystals The space lattices BCC unit cell FCC unit cell HCP unit cell Metals and Alloys 95 96 Structure of Solids A number of metals dissolve in each other forming solid solutions Solid solutions are analogous to liquid solutions The mixing of the elements in the solid is on the atomic scale When a solute atom is much smaller than the solvent atom it may dissolve interstitially occupying a void space in the parent structure For example carbon is an interstitial solute in FCC iron and occupies the octahedral voids in the FCC structure When the solute and the solvent atoms are of comparable sizes the solute substitutes for the solvent atom on a regular atomic site For example a 70 Cu30 Zn alloy alpha brass has an FCC structure with copper and zinc atoms occupying randomly the atomic sites of the FCC crystal Hume Rothery has framed empirical rules that govern the formation of substitutional solid solutions Extensive solid solubility by substitution occurs when i the solute and the solvent atoms do not differ by more than 15 in diameter ii the electronegativity difference between the elements is small and iii the valency and the crystal structure of the elements are the same AgAu CuNi and GeSi systems satisfy the Hume Rothery conditions very well as shown in Table 52 These systems therefore form complete solid solutions that is the two elements mix in each other in all proportions Starting from pure silver for example the silver atoms can be continuously replaced by gold atoms in the FCC structure till pure gold is obtained TABLE 52 Parameters Relevant to the Hume Rothery Rules Crystal Radius of Electro System structure atoms Å Valency negativity AgAu Ag FCC 144 1 19 Au FCC 144 1 24 CuNi Cu FCC 128 1 19 Ni FCC 125 2 18 GeSi Ge DC 122 4 18 Si DC 118 4 18 This obviously is not possible in the case of Cu and Zn which have FCC and HCP structures respectively They therefore form solid solutions up to a certain extent only At room temperature up to 35 of zinc can dissolve in the FCC crystal of copper as the Hume Rothery conditions are partially satisfied However only about 1 of copper dissolves in the HCP structure of zinc The reason for this difference in behaviour seems to lie in the fact that an excess of bonding electrons for example when zinc is dissolved in copper is more easily accommodated than a deficiency of bonding electrons which is seen when copper is dissolved in zinc Substitutional solid solutions have usually a random arrangement of the constituent atoms on the atomic sites especially at elevated temperatures This is so as configurational entropy makes a greater contribution in lowering the free energy with increasing temperature recall that G H TS This random arrangement of the constituent atoms in a solid solution may change over to an ordered arrangement on cooling to lower temperatures if ordering lowers the enthalpy of the crystal sufficiently For example the solid solution of copper and zinc mixed in equal atomic proportions forms a disordered BCC structure at temperatures above 470C The atomic sites in the BCC crystal are fixed but the probability of finding a given atomic site occupied by a copper atom or a zinc atom is 05 that is in the same proportion as the concentration of the constituent atom Below 470C the alloy becomes ordered with all the copper atoms occupying the cube corners and the zinc atoms occupying the body centres or vice versa Such a structure is identical to the one shown in Fig 34c This ordering occurs as there is some preference for CuZn bonds which have a larger bond energy than CuCu or ZnZn bonds Note that in the ordered state all the zinc atoms have copper nearest neighbours and all the copper atoms have zinc neighbours that is all the bonds are of the CuZn type The above example of the 50 Cu50 Zn alloy is an intermediate structure that forms in a system of limited solid solubility Its crystal structure BCC is different from that of either copper FCC or zinc HCP If an intermediate structure exists only at a fixed composition it is called an intermetallic compound For example iron carbide Fe3C a common constituent of steels is an intermetallic compound It has a complex crystal structure referred to an orthorhombic lattice and is hard and brittle Intermediate structures which have appreciable difference in electronegativity of the constituent atoms obey the normal rules of valency Examples where magnesium is one of the constituent atoms are Mg2Si Mg3P2 MgS Mg2Ge and MgSe Some intermediate compounds which do not obey the normal rules of valency are called electron compounds Hume Rothery has shown that they occur at certain definite values of free electron to atom ratio in the alloy such as 3 2 21 13 and 7 4 Typical examples are CuZn 3 2 Cu5Zn8 21 13 and CuZn3 7 4 54 Ionic Solids In Chapter 4 we noted that the size of an atom increases on adding extra electrons and shrinks on removing electrons Consequently the cation is usually smaller than the anion Exceptions to this exist in a few cases such as RbF with Rb larger than F Ionic bonds are nondirectional Therefore each ion tends to surround itself with as many ions of opposite sign as possible to reduce the potential energy This tendency promotes the formation of close packed structures Unlike the metallic structures here the differences in size of the two or more ions forming the crystal should also be taken into account in the geometry of close packing Consider first the local packing geometry of one type of cation and one type of anion The cation is assumed to be the smaller ion The number of anions surrounding a central cation is called the coordination number or ligancy The Ionic Solids 97 98 Structure of Solids ligancy is a function of the ion sizes and can be worked out from space filling geometry when the following conditions corresponding to a stable configuration are satisfied simultaneously i An anion and a cation assumed to be hard spheres always touch each other ii Anions generally will not touch but may be close enough to be in contact with one another in a limiting situation iii As many anions as possible surround a central cation for the maximum reduction in electrostatic energy When the cation is very small compared to the anion it is easily seen that only two anions can be neighbours to the cation in order to satisfy all the above three conditions Consider next the configuration shown in Fig 511a Here the three surrounding anions are touching one another and also the central cation rcra 0155 0 rcra 0155 0155 rcra 0225 a b c Fig 511 Triangular coordination of anions around a central cation a the critical configuration b the unstable configuration and c stable but not critical configuration The ratio of the cation to anion radius rcra for this configuration is 0155 which can be worked out from the simple geometry Example 55 The triangular arrangement in Fig 511a is one of the limiting situations The radius ratio is said to be a critical value because for values of rcra smaller than 0155 the central cation will rattle in the hole and not touch all the three anions at the same time as illustrated in Fig 511b This violates condition i above and leads to instability When the radius ratio is less than 0155 the only way to satisfy all three conditions is to reduce the number of anions to 2 For values of rcra slightly greater than 0155 all the anions touch the central cation but do not touch one another as shown in Fig 511c All three conditions of stability are still satisfied This situation will prevail till the radius ratio increases to 0225 the next higher critical value corresponding to a tetrahedral four coordination At rcra 0225 the four surrouding anions touch one another and also the central cation This configuration is the same as that obtained by fitting the largest possible sphere in the tetrahedral void of a close packed structure see Example 54 A ligancy of five does not satisfy all the three conditions for stable configuration because it is always possible to have six anions as an alternative to any arrangement that contains five anions without a change in the radius ratio The critical condition for octahedral six coordination occurs at rcra 0414 which is the same as the size of the octahedral void in a close packed structure Ligancies of 7 9 10 and 11 are again not permissible The radius ratio ranges in which different values of ligancy are obtained are summarized in Table 53 At the end of the table the limiting case of rcra 1 is identified with configurations of close packing of equal sized spheres TABLE 53 Ligancy as a Function of Radius Ratio Ligancy Range of radius ratio Configuration 2 00 0155 linear 3 0155 0225 triangular 4 0225 0414 tetrahedral 6 0414 0732 octahedral 8 0732 10 cubic 12 10 FCC or HCP Example 55 Find the critical radius ratio for triangular coordination Solution The critical condition for triangular coordination is shown in Fig 511a The three anions touch one another as well as the central cation From the simple geometry we can write rc ra 2 3 2ra sin 60 1155ra rcra 0155 The ligancy rules outlined above are obeyed in a number of cases For example in the NaCl crystal the radius ratio rNarCl 054 which lies between 0414 and 0732 As listed in Table 53 the predicted ligancy is six The octahedral geometry of six chlorine ions surrounding a central cation is experimentally observed In MgO rMg2rO2 059 and again the octahedral coordination is observed In CsCl rCsrCl 091 This value lies in the range of 073210 The predicted coordination of eight anions surrounding a cation is observed In a marginal case such as CaF2 where rCa2rF 073 it is difficult to predict whether a sixfold or an eightfold coordination will occur It so turns out that the eight coordination is observed in this case that is every calcium cation is surrounded by eight fluorine anions Ionic Solids 99 100 Structure of Solids The SiO bond in silica as well as in silicates is about 50 ionic and 50 covalent Here the central silicon cation is surrounded by four oxygen anions located at the corners of a regular tetrahedron This arrangement satisfies both the ligancy rules as rcra 029 the tetrahedral coordination is predicted from Table 53 as well as the orientation relationships of sp3 bonds The stability criteria listed above for predicting the ligancy may not always be valid If directional characteristics of bonding persist to any significant degree the considerations based on the radius ratio alone will not lead to the correct prediction of ligancy In the abovediscussed example of SiO coordination the radius ratio criterion and the bond angle requirements happen to coincide In ZnS where the bond is more covalent than ionic the ligancy predicted from rZn2rS2 048 is octahedral Yet the fourfold coordination characteristic of sp3 bonding is what is observed In the formation of ionic crystals the ligancy rules described above determine the local packing around a cation The longrange arrangement of ions in the crystal is dependent on the following factors i In the crystal the overall electrical neutrality should be maintained whatever be the net charge on a local group of a cation and surrounding anions For example in NaCl where a cation is surrounded by six anions the net charge on NaCl6 is five Evidently this has to be neutralised in the long range arrangement ii The ionic bond being nondirectional the ions are packed as closely as possible in the crystal consistent with the local coordination iii If small cations with a large net charge are present in the crystal the cationcation repulsion will be high Then it may be necessary to have a long range arrangement that maximizes the cationcation distance even if close packing is not possible Such a situation usually arises when the charge on the cation increases to three or four When the cation charge is not more than two or at best three and when the radius ratio is in the range 04140732 the crystal structure can be described as a FCC or HCP packing of anions with the cations occupying all or part of the octahedral voids in the structure The fraction of octahedral voids that are filled depends on the number of cations to anions in the chemical formula Thus for the rock salt NaCl structure adopted by hundreds of binary ionic compounds rNarCl 054 and the anion packing is FCC with all octahedral voids filled with sodium cations Recall that there is one octahedral void per sphere in a close packed array A unit cell of NaCl crystal is shown in Fig 512 wtih the larger chlorine ions occupying the face centred cubic positions and the sodium ions in the octahedral voids The octahedral positions are at the body centre and at the midpoints of the cube edges Note that unlike in the monoatomic FCC crystal refer Fig 33c the chlorine ions do not touch one another along the face diagonal This is so because the radius ratio of 054 is greater than the size of the octahedral void in a close packed structure which is 0414 The FCC close packing is opened up here to the extent necessary to accommodate the sodium cations in the octahedral voids 102 Structure of Solids Several ionic crystals which have the radius ratio in the range of octahedral coordination are listed in Table 54 In alumina Al2O3 the cation positions in neighbouring planes of octahedral voids are staggered such that the mutual repulsion of the trivalent cations is minimized Due to multivalent ions the bond strength in Al2O3 is high producing a hard crystal with a high melting point TABLE 54 Structure of Some Ionic Crystals Fraction of octahedral Crystal rcra Anion packing voids with cations NaCl 054 FCC All MgO 059 FCC All CdCl2 057 FCC Half Al2O3 043 HCP Twothirds The electrical insulating properties of Al2O3 are excellent Al2O3 is used as a substrate for building integrated circuits and in spark plugs of automobiles In the Al2O3 structure it is possible to replace part or all of the Al3 ions by other ions provided the size difference between them is small The structure thus produced is a substitutional solid solution already discussed in connection with alloys Replacing a small fraction of the Al3 ions by other ions such as Cr3 and Fe3 results in the gemstones ruby and blue sapphire Ruby Cr3 ions added to Al2O3 is a LASER Light Amplification through Stimulated Emission of Radiation which is a crystal device Sapphire is very hard and is used in jewelled bearings and cutting tools When rcra is in the range 07321 the eightfold coordination is observed CsCl with rCsrCl 091 is a typical example of this structure Referring to Fig 34c the cesium ions are at the body centre and the chlorine ions are at the body corners The space lattice is simple cubic with a basis of one cesium ion plus one chlorine ion per lattice point In CaF2 also the coordination around the cation is eight with the difference that only one body centre for every two unit cells is occupied by Ca2 As an example of an ionic crystal with more than two types of ions consider the crystal structure of spinels Spinels are compounds with two different cations A2 and B3 and oxygen as the anion with the general formula AB2O4 Here the oxygen anions form the FCC packing For every four oxygen anions there are four octahedral sites and eight tetrahedral sites Out of these twelve only three are needed to fill the cations of the above formula In the normal spinel structure the A cations are in the tetrahedral voids and the B cations are in the octahedral voids Alternatively half of the B cations can occupy the tetrahedral voids while the remaining half of the B cations and all the A cations are randomly distributed in octahedral voids resulting in the inverse spinel structure In both normal and inverse spinels only half of the octahedral sites and 18 of the tetrahedral sites are filled When B2O3 Fe2O3 we have a series of compounds called ferrites where different A cations can be present in varying proportions Ferrites have the inverse spinel structure They are important soft magnetic materials refer Chap 16 104 Structure of Solids by two tetrahedra This arrangement maintains the electrical neutrality of the network as a whole In the crystalline form such as quartz the tetrahedra are arranged in a periodically repeating pattern see Fig 5l4a In the noncrystalline form silica glass the tetrahedra are randomly bonded to other tetrahedra as shown in Fig 5l4b Note that there is no difference in the siliconoxygen coordination between the crystalline and the noncrystalline forms Quartz is used in optical components It is also piezoelectric piezo press A mechanical stress applied to the crystal displaces the ions in the crystal and induces electric polarization Similarly an electric field will cause the crystal to be elastically strained Quartz is used in watches and clocks and for accurate frequency control in electronic circuits Fused silica glass is used in applications requiring low thermal expansion It is highly viscous even in the molten state because of the SiO bond between the tetrahedra In the threedimensional network of silica other oxides can be dissolved to yield a number of both crystalline and noncrystalline silicates Soda lime glass is a noncrystalline silicate with Na2O and CaO added to silica The alkali cations break up the network of the silicate tetrahedra as shown in Fig 515 For each Na2O introduced one SiO bridge is disrupted and the extra oxygen atom from Na2O splits up one common corner into separate corners The two sodium ions Na Na Na Na Na Fig 515 Addition of Na2O to silica introduces weaker bonds in the network Na stay close to the disrupted corner due to the electrostatic attraction The network at the corner is bonded through the NaO bonds The NaO bond being weaker than the SiO bond the viscosity of the glass is drastically reduced as a result of the alkali addition Along similar lines we can explain the much lower softening temperature of pyrex 80 SiO2 14 B2O3 4 Na2O as compared to fused silica 998 SiO2 When the tetrahedra share all corners the only cation in the structure is the silicon at the centre of the tetrahedron By introducing other cations a number of different structures can be produced These can be described by reference to the number of corners the tetrahedra share amongst themselves without an intervening link provided by a different cation According to the corners shared the structures of many minerals can be classified as island chain sheet and threedimensional network of tetrahedra as shown in Table 55 106 Structure of Solids produced when each tetrahedron is joined at two corners to two other tetrahedra directly that is by having a common oxygen with the remaining two corners joined through other cations This results in some interesting cleavage properties of the minerals A cleavage direction refers to the direction along which the bonds are broken during fracture The fibrous quality of asbestos which has a double chain structure is attributable to the cleavage along certain crystallographic directions which go through the weaker bonds of the structure Similarly mica with tetrahedra arranged in sheet form breaks parallel to the sheets The bond between sheets of clay is van der Waals which explains the softness of clay The excess charge on one side of the sheet compared to the other side provides ideal sites for absorption of the polar water molecules So clay has the characteristic plasticity when mixed with water Talc is soft like clay due to van der Waals bonding between sheets but does not absorb water like clay Cement and concrete are common building construction materials used in huge quantities Portland cement consists of a number of silicate minerals with the approximate composition given in Table 56 Cement mixed with water sets as a function of time due to several hydration reactions Water binds the sheet like silicate molecules together thereby hardening the cement Concrete is a sized aggregate of rocks embedded in a cement matrix which binds the rock pieces together TABLE 56 Approximate Composition of Portland Cement Constituent Symbol Percentage Tricalcium silicate C3S 45 Dicalcium silicate C2S 30 Tricalcium aluminate C3A 10 Tetracalcium aluminoferrite C4AF 8 Other bonding agents 7 POLYMERS Most organic polymers are based on the covalent bonds formed by carbon The electrons are bonded strongly by the localized sharing characteristic of covalent bonding so that polymers are good thermal and electrical insulators The relative inertness of polymeric materials they do not corrode in the sense that metals do can be attributed to covalent bonding 56 Classification of Polymers The word mer in Greek means a unit and so monomer stands for a single unit and polymer for many units joined together A polymer usually has thousands of The remaining double bond is necessary for producing an elastomer Natural rubber as tapped from a tropical tree is a viscous liquid in which the long polyisoprene molecules are able to flow past one another at room temperature An elastomer is produced by heating raw rubber with sulphur Sulphur forms covalent bonds with the carbon by using the remaining double bond in the monomer This reaction known as vulcanization produces additional links between chains called cross links The degree of vulcanization determines the stiffness of rubber With increasing cross links the rubber becomes more rigid for example a cycle tube is less vulcanized than a rubber pocket comb In the limit when all the double bonds are used up by sulphur bridges a threedimensional network of primary bonds results The material thus produced ebonite is hard and brittle The elastic properties of rubber are discussed in Chap 10 58 Crystallinity of Long Chain Polymers Long chain polymers are usually in the noncrystalline form or in the semicrystalline form In the noncrystalline form the long chains are randomly tangled with one another In the semicrystalline form parts of the polymer volume have the parallel chain arrangement while other parts are randomly oriented as shown in Fig 517 Sometimes it is possible to grow single crystals of a polymer These crystals have a folded chain structure where the same chain folds back and forth many times into a parallel arrangement For example the single crystal of polyethylene has an orthorhombic unit cell Fig 517 Many polymers are semicrystalline with random and parallel arrangements of chains in different regions In all polymers the long chains can be aligned to some extent by mechanical working Such an alignment promotes crystallinity As the chains are more closely packed in the crystalline form the density increases with increasing alignment of Crystallinity of Long Chain Polymers 111 510 Calculate the critical radius ratio for tetrahedral octahedral and eightfold coordination around a central cation in an ionic crystal 511 Distinguish between atom sites and lattice points in a monoatomic FCC crystal and a NaCl crystal 512 State the differences and common points between i CsCl ii a monoatomic BCC crystal and iii a monoatomic SC crystal 513 What are the factors that determine the density of a crystal Which of these factors is dominant in determining the decreasing order of density with increasing atomic number in the following elements Cu Z 29 8960 Ge Z 32 5320 and Se Z 34 4790 kg m3 514 Aluminium has an FCC structure Its density is 2700 kg m3 spgr 27 Calculate the unit cell dimension and the atomic diameter Answer 405 Å 286 Å 515 Iron changes from BCC to FCC form at 910C At this temperature the atomic radii of the iron atoms in the two structures are 1258 Å and 1292 Å respectively What is the per cent volume change during this structural change Answer 045 516 Xray analysis of a MnSi alloy with 75 atomic per cent of Mn and 25 atomic per cent of Si showed that the unit cell is cubic and the lattice parameter a 286 Å The density of the alloy is 6850 kg m3 Find the number of atoms in the unit cell Answer 2 517 Calculate the density of the CsCl crystal from the radii of the ions Cs 165 Å and Cl 181 Å Answer 4380 kg m3 518 Find the local packing arrangement in the ionic crystal CaO Assuming the crystal structure to be cubic calculate the density of the crystal Given the radii Ca2 094 Å and O2 132 Å Answer Octahedral 4030 kg m3 519 The bonds in diamond are predominantly covalent in character and so are the bonds along the chains of a long chain polymer Why does the polymer melt at a much lower temperature 520 The melting point of a polymer increases with increasing molecular weight of the chain Explain why this is so 521 Calculate the endtoend distance of an uncoiled chain molecule of polyethylene that has 500 mers in it The CC bond length is 154 Å Answer 1258 Å 522 The degree of polymerization of a polystyrene chain is 10 000 Calculate the molecular weight of the chain Answer 1 040 000 114 Problems 35 The minimum number of double bonds required in the monomer for cross linking is A 0 B 1 C 2 D 3 36 The bulkiest side group in the monomer is in A teflon B PVC C PTFE D polystyrene 37 The chemical formula of the isoprene molecule is A C4H5CH3 B C3H4CH3Cl C C4H3Cl D C2H3CH3 Answers 1 B 2 C 3 A 4 D 5 B 6 B 7 B 8 A D 9 B 10 A B C 11 C 12 B 13 D 14 C 15 C 16 C 17 B 18 B D 19 D 20 B 21 A 22 C 23 A 24 B 25 A 26 B 27 D 28 B 29 B 30 A 31 A B C D 32 C 33 A 34 A 35 C 36 D 37 A Sources for Experimental Data CA Harper Modern Plastics Handbook McGrawHill New York 2000 P Villars and LD Calvert Pearsons Handbook of Crystallographic Data for Intermetallic Phases Vols 13 American Society for Metals Metals Park Ohio 1985 Suggestions for Further Reading RC Evans An Introduction to Crystal Chemistry Cambridge University Press Cambridge 1964 AF Wells Structural Inorganic Chemistry Clarendon Press Oxford 1975 Sources for Experimental DataSuggestions for Further Reading 119 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 Zn O Zn O Zn O Zn O Zn O Zn O Zn O Zn O Zn O Zn O Zn O Zn O Zn 2 2 2 2 2 2 3 2 2 2 3 2 2 2 2 3 2 2 2 2 3 2 2 Fe O Fe O Fe O Fe O Fe O Fe O O Fe O Fe O Fe O O Fe O Fe We noted in Chap 5 that trivalent cations such as Fe3 and Cr3 can substitute for trivalent parent Al3 cations in the Al2O3 crystal If however the valency of the substitutional impurity is not equal to the parent cation additional point defects may be created due to such substitution For example a divalent cation such as Cd2 substituting for a univalent parent ion such as Na will at the same time create a vacant cation site in the crystal so that electrical neutrality is maintained Defect structures are produced when the composition of an ionic crystal does not correspond to the exact stoichiometric formula Such defect structures have an appreciable concentration of point imperfections Consider deviations from the stoichiometric formula in compounds of ZnO and FeO In ZnyO where y 1 the excess cations occupy interstitial voids Such a compound can be produced by heating a stoichiometric compound in zinc vapour The two electrons released from each zinc atom that enters the crystal stays around an interstitial cation as shown in Fig 63a In FexO where x 1 vacant cation sites are present Such a compound can be produced by heating a stoichiometric FeO in an oxygen atmosphere The two electrons required by each excess oxygen atom is donated by two Fe2 ions which become Fe3 ferric ions see Fig 63b a b Fig 62 Point imperfections in an ionic crystal a Frenkel defect and b Schottky defect a b 2 Zn 2 Zn Fig 63 Defects present in ZnO and FeO owing to deviations from stoichiometry Vacant cation sites are indicated by Point Imperfections 123 126 Crystal Imperfections 62 The Geometry of Dislocations Line imperfections are called dislocations Note that the word dislocations is used by convention to denote only line imperfections even if the word means any general discontinuity in the crystal Line imperfections are onedimensional imperfections in the geometrical sense The concept of a dislocation was first put forward in 1930s Abundant experimental evidence for the presence of dislocations in crystals has since accumulated The static and dynamic properties of dislocations have been studied in great detail because of the vital role they play in determining the structuresensitive properties of crystalline materials Dislocations are best understood by referring to two limiting straightline types i the edge dislocation and ii the screw dislocation Discussing the edge dislocation first Fig 65a shows a perfect crystal the top sketch depicting a threedimensional view and the bottom one showing the a Fig 65 An incomplete plane in a crystal results in an edge dislocation b atoms on the front face We can consider the perfect crystal to be made up of vertical planes parallel to one another and to the side faces If one of these vertical planes does not extend from the top to the bottom of the crystal but ends part way within the crystal as in Fig 65b a dislocation is present In the lower sketch notice the atomic arrangements on the front face In the perfect crystal the atoms are in equilibrium positions and all the bond lengths are of the equilibrium value In the imperfect crystal on the right just above the edge of the incomplete plane the atoms are squeezed together and are in a state of compression The bond lengths have been compressed to smaller than the equilibrium value Just below the edge the atoms are pulled apart and are in a 128 Crystal Imperfections defined as the boundary between the slipped and the unslipped parts of the slip plane This definition of a dislocation is in fact general and applies to any dislocation line In the above example we considered a particular case in which the displacement was one interatomic distance The magnitude and the direction of the displacement are defined by a vector called the Burgers vector BV which characterizes a dislocation line The Burgers vector of a dislocation is determined as follows Consider the perfect crystal shown in Fig 67a Starting from the point P if we go up by x number of steps x 4 in Fig 67a then take y steps to the right y 5 in the figure then x steps down and finally y steps to the left we end up at the starting point We have now traced a Burgers circuit in taking these steps We returned to the starting point because the region enclosed by the Burgers circuit is perfect with no line imperfection cutting across it If we now do the same operation on a crystal which has a dislocation in it as shown in Fig 67b starting from point P we end up at Q We need an extra step to return to point P or to close the Burgers circuit The magnitude and the direction of this step define the Burgers vector BV QP b 64 The Burgers vector is perpendicular to the edge dislocation line We have taken the Burgers circuit to be clockwise in the above example The direction of the Burgers vector depends on the direction of the circuit which can be clockwise or anticlockwise To avoid this ambiguity a unit vector t is first assigned to denote the direction of the dislocation line The direction vector is drawn tangential to the dislocation line at the point of interest Then a righthand screw RHS convention is followed in tracing the circuit If we place the end of an ordinary right handed screw on the paper and turn the head clockwise the screw tends to move into the plane of the paper If the vector t has a direction vertically into the plane of the paper as in Fig 67b using the RHS convention the Burgers circuit should be drawn clockwise t P b Q P a b Fig 67 Burgers circuits a in a perfect crystal and b in an imperfect crystal with an edge dislocation The other limiting type of dislocation is a screw dislocation Consider the hatched area AEFD on the plane ABCD in Fig 68a As before let the top part of the crystal over the hatched area be displaced by one interatomic distance to the left with respect to the bottom part as shown in Fig 68b As a different area is hatched here as compared to the edge dislocation case in Fig 66 for the same sense of displacement we now produce a screw dislocation at the boundary EF between the displaced and the undisplaced parts of the slip plane Let the t vector be defined such that it points from right to left as shown in Fig 68b The Burgers circuit is then drawn using the RHS convention on the right side face in a clockwise sense The Burgers vector is determined by the step needed to close the circuit In this case of a screw dislocation the Burgers vector b has the same direction as the t vector The Burgers vector is parallel to the screw dislocation line We note that there is no extra plane at the screw dislocation in contrast to what we saw in the edge dislocation Examine the atomic arrangement around the dislocation line If we go round the dislocation line once as illustrated in Fig 69 we move along the line by a step equal in magnitude to the Burgers vector If we turn a screw through one full rotation we move the screw backward or forward by a distance equal to its pitch This feature has led to the name screw dislocation The atomic bonds in the region immediately surrounding the dislocation line have undergone a shear distortion as seen clearly in Fig 69 The vertical atomic planes parallel to the side faces in the perfect crystal of Fig 68a have become continuous due to the displacement in Fig 68b They can be compared to a spiral staircase whose central pillar coincides with the screw dislocation line Screw dislocations are symbolically represented by or depending on whether the Burgers vector and the t vector are parallel or antiparallel These two cases are referred to as positive and negative screw dislocations The Geometry of Dislocations 129 D F C B A E a F t E b b Fig 68 A screw dislocation EF is created by displacement of the top part of the crystal with respect to the bottom across the hatched area 130 Crystal Imperfections Any general dislocation line is a combination of the edge and the screw types In Fig 610 the hatched area on the slip plane is bounded by a curved line If we do the same displacement over the hatched area as before the 16 15 1211 8 7 4 3 14 13 10 9 6 5 2 1 Fig 69 The atoms around a screw dislocation line are arranged in a spiral screwlike fashion b b t Fig 610 Displacement across the hatched area with a curved boundary produces a mixed dislocation line at the boundary boundary of the hatched area that lies within the crystal becomes a curved dislocation line The dislocation has pure screw character at the right face and pure edge character on the front face It has mixed character in between Note that the t vector goes into the plane of the paper on the front face It changes continuously its direction as the dislocation is curved and finally it emerges out on the right side face Burgers circuits are drawn on both the front face and the right side face following the RHS convention The circuit is clockwise on the front face as the t vector is facing inwards It is anticlockwise on the right side face as the t vector is emerging out of that face The Burgers vector b has the same magnitude and direction in both cases Thus we obtain the important result Other Properties of Dislocations 135 towards the right of the positive edge dislocation lying on it The successive positions of the dislocation during this motion are illustrated in Fig 612 The atoms have been labelled to show their positions before and after the dislocation motion It is important to note that there is no bodily shift of the incomplete plane during the motion and that the configuration of the dislocation is what is moving The configuration is shifted from one position to the next by small variations of bond lengths in the dislocation region The dislocation can eventually reach the surface in Fig 612 As soon as this happens the compressive and the tensile strains will be relieved at the free surface and we would have a step on the right side face This step at the surface does not have the compressive and tensile strains characteristic of the edge dislocation It should not be called a dislocation any more We say that the dislocation disappears on reaching the surface leaving a step behind The magnitude of the step is equal to the Burgers vector of the dislocation The motion of a dislocation on a plane that contains the direction vector t and the Burgers vector b is called the glide motion The motion of the edge dislocation that we considered with reference to Fig 612 is glide motion The dislocation loop in Fig 611 can glide on the slip plane as b and t lie on this plane at all points around the loop In fact it cannot glide on any other plane As there is only one plane which can contain both b and t when they are not parallel the glide plane is uniquely defined for a pure edge dislocation as well as for a mixed dislocation The unit normal defining the glide plane is given by the cross product of b and t For a screw dislocation b and t are parallel or antiparallel so that a glide plane is not uniquely defined The cross product of b and t is zero for a screw dislocation Example 63 A circular dislocation loop has edge character all round the loop What is the surface on which this dislocation can glide Solution By definition the Burgers vector is perpendicular to an edge dislocation line Also the Burgers vector is invariant The edge dislocation can glide only on a surface that contains both the Burgers vector and the t vector These considerations are satisfied only when the given dislocation moves on a cylindrical surface containing the loop The motion of an edge dislocation on a plane perpendicular to the glide plane is called climb motion As the edge dislocation moves above or below the slip plane in a perpendicular direction the incomplete plane either shrinks or increases in extent This kind of shifting of the edge of the incomplete plane is possible only by subtracting or adding rows of atoms to the extra plane Climb motion is said to be nonconservative This is in contrast to the glide motion which is conservative and does not require either addition or subtraction of atoms from the incomplete plane During climbing up of an edge dislocation the incomplete plane shrinks Atoms move away from the incomplete plane to other parts of the crystal During Surface imperfections are not stable in a thermodynamic sense They are present as metastable imperfections If the thermal energy is increased by heating a crystal close to its melting point many of the surface imperfections can be removed The grain boundary area decreases as a polycrystalline material is heated above 05Tm where Tm is the melting point in K Larger crystals grow at the expense of smaller crystals Even though the average size of a crystal increases during this grain growth the number of crystals decreases resulting in a net decrease in the grain boundary area per unit volume of the material Example 66 A metal heated to elevated temperatures exhibits grooves on the surface at positions where the grain boundaries meet the surface see Fig 618 Assume that the ratio of the surface energy of the free surface to that of the grain boundary is 3 to 1 Compute the angle at the bottom of the groove of a boundary which makes an angle of 90 with the external surface Fig 617 Annealing twins and grain boundaries in brass J Nutting and RG Baker The Microstructure of Metals with permission from The Institute of Metals London Surface Imperfections 141 in pairs such that the orientation change introduced by one boundary is restored by the other as shown in Fig 616 The region between the pair of boundaries is called the twinned region Twin boundaries are easily identified under an optical microscope Figure 617 shows the microstructure of brass with a number of twins in the grains Twins which form during the process of recrystallization are called annealing twins and those which form during plastic deformation of the material are called deformation twins 12 The stacking which has a stacking fault in the following is A ABCABABCABABCAB B ABCABABCABCABC C ABABABCABABABCABABABC D ABCABCABABCABCABABCABCAB 13 The stackings with the closest packing of equalsized spheres are A ABBA B ABCCAB C ACBACB D ABABCABABC 14 The local stacking arrangement at a stacking fault in a HCP crystal A ABCABABC B ABABCABABAB C ABABCABABC D ABCABABCABABC 15 The tilt angle of a tilt boundary in BCC iron a 287 Å with edge dislocations 7500 Å apart is A 004 B 02 C 002 D 033 rad 16 The maximum possible decrease in energy during grain growth in Cu grain boundary energy 05 J m2 of initial grain diameter of 03 mm is A 05 kJ m3 B 25 kJ m3 C 5 kJ m3 D 10 kJ m3 Answers 1 B 2 B 3 A 4 A D 5 A 6 A 7 A 8 A 9 D 10 C 11 A 12 B 13 C D 14 B 15 C 16 C Source for Experimental Data Institute of Metals London Dislocations and Properties of Real Materials 1985 Suggestions for Further Reading JP Hirth and J Lothe Theory of Dislocations McGrawHill New York 1984 A Kelly GW Groves and P Kidd Crystallography and Crystal Defects John Wiley Chichester UK 2000 Sources for Experimental DataSuggestions for Further Reading 147 152 Phase Diagrams 2200 T 2100 2000 Al2O3 10 30 50 cl 70 co cs 90 Cr2O3 Liquid tieline Liquidus Solidus Solid Cr2O3 Temperature C Fig 72 The Al2O3Cr2O3 phase diagram Microstructural changes in an overall composition c0 are sketched on the right Example 72 At atmospheric pressure pressure arbitrarily chosen a material of unknown composition shows four phases in equilibrium at 987 K What is the minimum number of components in the system Solution As pressure is arbitrarily chosen we can use the modified form of the phase rule as given in Eq 73 The minimum number of components corresponds to the minimum degree of freedom which is zero Taking F 0 we get 0 C 4 1 C 3 The simplest binary phase diagram is obtained for a system exhibiting complete liquid solubility as well as solid solubility The two components dissolve in each other in all proportions both in the liquid and the solid states Clearly the two components must have the same crystal structure besides satisfying the other Hume Rotherys conditions for extensive solid solubility see Sec 53 CuNi AgAu GeSi and Al2O3Cr2O3 are examples of such systems Figure 72 shows the phase diagram of Al2O3Cr2O3 Pure Al2O3 and pure Cr2O3 form the left and the right end of the composition axis They are arranged in alphabetical order from left to right The composition is read as per cent of Cr2O3 starting from 0 at left and going to 100 at the right end Alternatively the composition can be read as per cent of Al2O3 from right to left Temperature is shown along the yaxis There are only two phases on the phase diagram the liquid and the solid phases The singlephase regions are separated by a twophase region L S where both liquid and solid coexist In all binary phase diagrams a twophase region separates singlephase regions as given by the 121 rule As we move from a singlephase region 1 we cross into a twophase region 2 and then L S again into a singlephase region 1 The phase boundary between the liquid and the twophase region is called the liquidus The boundary between the solid and the twophase region is called the solidus When only one phase is present the composition axis gives the composition of that phase directly When two phases are present the compositions of the phases are not the same They should be read according to the following convention At the temperature of interest T a horizontal line called the tieline is drawn as shown in Fig 72 The points of intersection of the tieline with the liquidus and the solidus give respectively the liquid and the solid compositions cl and cs which are in equilibrium with each other Thus in Fig 72 at 2180C for an overall composition co 73 Cr2O3 27 Al2O3 we have The liquid composition cl 57 Cr2O3 43 Al2O3 and The solid composition cs 82 Cr2O3 18 Al2O3 The phase rule can be applied to this phase diagram using the modified form given in Eq 73 For the singlephase region liquid or solid from Eq 73 F 2 1 1 2 So both temperature and the composition of the phase can be independently varied within limits In the twophase region F 2 2 1 1 Here we have three variables i Temperature ii Composition of the liquid phase iii Composition of the solid phase As F 1 only one of these three is independent If we arbitrarily choose the temperature the compositions of the two phases are automatically fixed and are given by the ends of the tieline drawn at that temperature If we specify the composition of one of the phases arbitrarily the temperature and the composition of the other phase are automatically fixed There is no threephase equilibrium in systems exhibiting complete solid solubility Many pairs of elements and compounds are unlikely to satisfy the conditions for complete solid solubility For instance the size difference between two atoms or ions can be appreciably more than 15 as the table of atomic and ionic radii indicates see back inside cover of the book Similarly the other conditions for extensive solubility may not be satisfied The solid solubility is therefore limited in a number of binary systems But it is never zero However unfavourable the conditions for solid solubility are a very small quantity of any component will always dissolve in another component as this increases the configurational entropy and lowers the free energy of the crystal recall Problem 210 The solubility may be so small that for all practical purposes only the pure component may be shown on the phase diagram When solid solubility is limited and the melting points of the components are not vastly different a eutectic phase diagram usually results As an example Binary Phase Diagrams 153 This is true except when the phase boundary is a horizontal line corresponding to an invariant temperature Depending on the cooling rate and the other alloying elements present in cast iron the carbon may be present as graphite or cementite Gray cast iron contains graphite in the form of flakes Slow cooling rates and the presence of silicon promote the formation of graphite The microstructure of gray cast iron is shown in Fig 714 and consists of graphite flakes in a matrix of ferrite The Fig 714 Microstructure of gray cast iron magnified 140 times Graphite flakes are embedded in a matrix of ferrite AL Ruoff Introduction to Materials Science with permission from Prentice Hall Inc Englewood Cliffs New Jersey graphite flakes are sharp at their tips and act as stress raisers Due to this gray cast iron is brittle under tensile loads in spite of the softness of graphite as compared to the very hard cementite present in white cast iron The brittleness can be avoided by producing the graphite in the form of spherical nodules which do not have stressraising sharp ends as is done in malleable cast iron and SG spheroidal graphite iron Malleable cast iron is produced by heat treating white cast iron for prolonged periods at about 900C and then cooling it very slowly The silicon content in the alloy must be 1 or less to ensure that cementite and not flaky graphite forms during solidification The cementite decomposes to the more stable graphite during the subsequent heat treatment Graphite appears in the heattreated microstructure as approximately spherical particles of temper carbon SG iron also known as nodular iron is produced by making certain alloy additions such as Mg or Ce to molten iron Here the silicon content must be about 25 to promote graphitization The alloy additions have the effect of modifying the growth rate of graphite from the melt to be more or less equal in all directions so that nodules and not flakes of graphite are produced No subsequent heat treatment is required for SG iron Some Typical Phase Diagrams 169 3 The degree of freedom when ice water and water vapour coexist in equilibrium is A 1 B triple pt C 0 D 1 4 The degrees of freedom when FCC iron and BCC iron coexist in equilibrium are A 2 B 1 C 0 D 1 5 The phase boundary between alpha and alpha beta regions is called A liquidus B solidus C solvus D none of these 6 The reaction that yields two solid phases on cooling a single solid phase is called A eutectoid B peritectoid C eutectic D congruent 7 If one solid phase splits into two solid phases on heating the reaction is A eutectic B peritectic C eutectoid D peritectoid 8 The reaction that on heating one solid phase yields another solid phase plus one liquid phase is called A eutectic B eutectoid C peritectic D peritectoid 9 If alpha of 82 B and liquid of 57 B are in equilibrium in an alloy of 73 B the fraction of liquid is A 036 B 064 C 36 B D 0 10 If the fraction of liquid with 57 B which is in equilibrium with solid of 82 B is 07 the overall composition is A 03 B 745 B C 645 B D 25 B 11 In the eutectic phase diagram of AgCu system the solubility limit at 500C of copper is 3 in the Agrich phase and of Ag is 2 in the Curich phase In sterling silver 925 Ag 75 Cu the per cent of copper in the Agrich phase at 500C is A 9526 B 474 C 3 D 98 12 The eutectic mixture in a PbSn solder alloy should be 90 At the eutectic temperature alpha of 19 Sn liquid of 62 Sn and beta of 97 Sn are in equilibrium The possible compositions of the solder alloy are A 577 Sn B 61 Sn C 655 Sn D 663 Sn 13 The fraction of pearlite in a 055 C steel is A 055 B 031 C 069 D 0 14 At 30C hot chocolate liquid with 35 chocolate and 65 vanilla transforms to chocolate ripple eutectic mixture of vanilla containing 10 chocolate and chocolate containing 5 vanilla Just below 30C the fraction of chocolate ripple in a composition with 45 chocolate is A 017 B 083 C 041 D 059 176 Multiple Choice Questions 15 Zone refining will be more efficient if the ratio of impurity in the solid to that in the liquid is A 001 B 01 C 04 D 10 Answers 1 D 2 B 3 C 4 B 5 C 6 A 7 D 8 C 9 A 10 C 11 C 12 A C 13 C 14 B 15 A Sources for Experimental Data American Ceramic Society Phase Equilibria Diagrams Phase Diagrams for Ceramists Columbus Ohio Vol I 1964 to Vol XI 1993 TB Massalski Ed Binary Alloy Phase Diagrams Vols 13 ASM International Materials Park Ohio 1990 Suggestions for Further Reading WD Kingery HK Bowen and DR Uhlmann Introduction to Ceramics Wiley New York 1976 Chap 7 A Prince Alloy Phase Equilibria Elsevier Amsterdam 1966 FN Rhines Phase Diagrams in Metallurgy McGrawHill New York 1956 Sources for Experimental DataSuggestions for Further Reading 177 gradient Surface hardening improves the wear resistance of components such as gears without impairing the bulk mechanical properties such as toughness If the carbon content of the carburizing atmosphere remains constant it would give rise to a constant carbon concentration cs at the surface of the steel Fig 84a Carburizing atmosphere Steel cs c1 0 x a Steel c2 cs 0 x b Decarburizing atmosphere The initial carbon content of the steel is c1 We can then write c x 0 c1 x 0 c 0 t cs So from Eq 88 A cs B cs c1 With D cs and c1 known the amount and depth of carbon penetration as a function of time can be computed 834 Decarburization of Steel The opposite of carburization is decarburization Here the carbon is lost from the surface layers of the steel due to an oxidizing atmosphere that reacts with carbon to produce CO or CO2 The fatigue resistance of steels is lowered due to decarburization and therefore it should be avoided by using a protective atmosphere during the heat treatment of the steel When the heat treatment is carried out in a nonprotective atmosphere eg air the extent of decarburization Fig 84 Concentrationdistance profiles for a carburization and b decarburization Applications Based on the Second Law Solution 187 190 Diffusion in Solids Inert markers thin rods of a high melting point substance which is insoluble in the diffusion matrix are placed at the weld joint of the couple prior to the diffusion anneal These markers are found to shift during the anneal in the same direction as the slower moving species The extent of this shift is found to be proportional to the square root of the diffusion time This kind of movement indicates that the net mass flow due to the difference in diffusivities is being compensated by a bulk flow of matter in the opposite direction within the diffusion zone That is lattice planes are created on one side of the diffusion zone while they are destroyed on the other side and the resulting bulk flow carries the markers along Notice that the bulk flow occurs relative to the ends of the diffusion couple It is quite a different phenomenon from the diffusion process itself In many cases porosity is observed on the lowermelting component side indicating that the bulk flow does not fully compensate for the difference in diffusivities of the two species The following analogy of gaseous interdiffusion aids in the understanding of the Kirkendall effect Let hydrogen and argon at the same pressure be kept in two chambers interconnected through a tube A frictionless piston in the tube separates the gases On opening an orifice in the piston the gases interdiffuse The lighter hydrogen will diffuse faster resulting in a pressure difference that will tend to shift the piston in the same direction as the slower diffusing argon is moving 85 The Atomic Model of Diffusion Diffusion occurs as a result of repeated jumps of atoms from their sites to other neighbouring sites Even when atoms jump randomly a net mass flow can occur down a concentration gradient when a large number of such jumps take place The unit step in the diffusion process is a single jump by the diffusing species In interstitial diffusion solute atoms which are small enough to occupy interstitial sites diffuse by jumping from one interstitial site to another as illustrated in Fig 85a In vacancy diffusion atoms diffuse by interchanging a b c d Fig 85 Mechanisms of diffusion 218 Phase Transformations contraction of the caxis is obstructed by the carbon in between Note that there are no carbon atoms at the middle of the a1 and a2 axes This obstruction along the caxis results in a tetragonal product with ca ratio slightly greater than unity The ca ratio of the BCT martensite is a function of the carbon content and varies from 10 at 00 carbon BCC martensite to 108 at 12 carbon When the carbon atoms are not present ie in pure iron on rapid quenching from the FCC region the shear process converts the FCC to BCC iron Here the caxis contracts and the a1 and a2 axes expand to a sufficient degree to make them all equal The hardness of martensite is a function of its carbon content It increases rapidly with increasing carbon content reaching a more or less constant value of 65 on the Rc scale at about 06 carbon Fig 913 The approximate hardness 60 40 20 0 02 04 06 08 Hardness Rc Carbon Fig 913 The hardness of martensite on the Rc scale as a function of the carbon content of a steel values of the various transformation products in an eutectoid steel are given in Table 92 The corresponding tensile strengths are also indicated The tensile strength increases with increasing hardness However when the hardness is a very high value such as Rc 65 the steel is brittle resulting in poor tensile strength TABLE 92 Properties of the Transformation Products in a 08 Carbon Steel Hardness Tensile strength Constituent Rc scale MN m2 Coarse pearlite 15 710 Fine pearlite 30 990 Bainite 45 1470 Martensite 65 Martensite tempered at 250C 55 1990 222 Phase Transformations time axis in Fig 915 is on log scale so that the optimum peak is not obtained at room temperature even after several years The increase in the hardness in the initial stages of the ageing curves can be attributed to the precipitation process taking place progressively After reaching a peak value the hardness starts to decrease This phenomenon is called overageing As the precipitate particles are very fine in size they have a high surface to volume ratio Therefore they have a tendency to coalesce or coarsen that is a number of small particles merge to form a large particle For a given volume fraction of precipitate particles the coarsening process decreases the total number of precipitate particles and increases the interparticle spacing The hindrance to the dislocation motion is thereby reduced accounting for the decrease in hardness beyond the peak in the curve A temperature between 100 and 180C see Fig 915 would be the optimum ageing temperature where the ageing time is not unduly large An alloy aged to the optimum peak at such a temperature would not overage during service at room temperature Duralumin alloy corresponding to US specification 2024T6 can have a tensile strength of 500 MN m2 50 kgfmm2 as compared to the strength of aluminium which is about 100 MN m2 Another agehardening alloy of AlZn of specification 7075T6 has a tensile strength of 550 MN m2 These two alloys are technologically important as they are used as aircraft structural material An increase in strength by a factor of five as in the above cases makes a vital difference to the economics of air transportation where the ratio of the dead load nonpaying load to the pay load is an important criterion in the choice of a material of construction The density of these alloys is not very much higher than that of aluminium as the alloy content is less than 10 In recent years lithium the density of which is only 20 of that of aluminium has been added as an alloying element to precipitationhardened aluminium alloys In addition to lowering the density lithium increases the Hardness 180C 100C 20C 0 01 10 10 100 days Ageing time Fig 915 Ageing curves for a duralumin alloy crystallize the viscosity would change abruptly at the freezing temperature from a low value about 100 Pa s in the liquid state to a very high value about 1020 Pa s in the crystalline state a change of 18 orders of magnitude at the freezing temperature However crystallization does not occur due to kinetic barriers The viscosity gradually increases with decreasing temperature and attains high values only at low temperatures The highest rate of change of viscosity with temperature occurs around 1012 Pa s at the glass transition temperature Tg see Fig 917 This provides a convenient point for making a distinction between a glass noncrystalline solid and a supercooled liquid In fact a solid can be defined as a material with viscosity greater than 1012 Pa s irrespective of whether it is crystalline or not The supercooled liquid even though it has a higher free energy than the crystals is in internal equilibrium within itself That is the atomic configurations in the supercooled liquid are such that its free energy is a minimum for the liquidlike structure at that temperature At the glass transition temperature the thermal energy becomes insufficient for any further configurational adjustments to take place within a reasonable amount of time The freezing of the first few configurations on approaching Tg favours the freezing of other neighbouring configurations in a cooperative fashion The range of temperature over which the entire atomic configuration is rendered immobile except for atomic vibrations about their mean positions characteristic of a solid is some 10C for organic polymeric glasses and somewhat larger for inorganic glasses such as silicates and borates In glass transitions the cooling rate around Tg is important in the control of properties For a slower cooling rate the transition temperature is lower and the specific volume volume per unit mass at any temperature below the transition region is also smaller as illustrated in Fig 918 Obtaining a specific volume as close as possible to the equilibrium volume which of course is the specific Supercooled liquid liquid Fast cool Crystal Tg Specific volume Slow cool Tm Glass Fig 918 Specific volume of a material as a function of cooling rate Temperature The Glass Transition 227 228 Phase Transformations volume of the crystalline state is important in all glass applications Even small amounts of shrinkage over a period of time at service temperatures which are below Tg in measuring apparatus such as glass thermometers can lead to important errors Such shrinkage in optical glasses can lead to nonuniformity in the refractive index and consequent errors in observations made with their aid For minimum shrinkage in service the glass should be cooled as slowly as possible through the glass transition temperature A rapidly cooled glass can be reheated and annealed at a temperature just below Tg and cooled slowly to protect against future shrinkage In organic polymers the glass transition temperature is dependent on the molecular weight of the polymer Smaller molecular weights correspond to shorter chain lengths and a lower glass transition temperature Plasticizers increase the distance of separation between chains and reduce the interfering effect on each other and this lowers Tg PVC cable sheaths rain coats etc are manufactured at room temperature after lowering the glass transition temperature of polyvinyl chloride by the addition of a plasticizer Typical glass transition temperatures Tg of some common long chain polymers are compared with their melting points Tm as illustrated in Table 94 The ratio of TgTm lies in the range 04075 TABLE 94 Glass Transition Temperatures Tg and Melting Points Tm for Some Common Long Chain Polymers Polymer Tm K Tg K TgTm LD polyethylene 388 153 039 HD polyethylene 410 153 037 Polyvinylchloride 465 360 037 Polypropylene 445 257 058 Polystyrene 513 378 074 Polyacrylonitrile 593 380 064 66 Nylon 538 323 060 Polyester 528 348 066 Polyisoprene 303 220 073 98 Recovery Recrystallization and Grain Growth Recovery recrystallization and grain growth are phenomena intimately associated with the annealing of a plastically deformed crystalline material In crystalline materials the density of point imperfections and dislocations increases with increasing amount of plastic deformation carried out at temperatures below the range 0305Tm where Tm is the melting point in kelvin Plastic working below 0305Tm is called cold work Recall from Chap 6 that point imperfections and dislocations have strain energy associated with them Between 1 and 10 of the energy of plastic deformation is stored in 230 Phase Transformations Some wellknown empirical laws of recrystallization are 1 The higher is the degree of deformation the lower is the recrystallization temperature 2 The finer is the initial grain size the lower is the recrystallization temperature 3 Increasing the amount of cold work and decreasing the initial grain size produce finer recrystallized grains 4 The higher is the temperature of cold working the less is the strain energy stored in the material The recrystallization temperature is correspondingly higher 5 The recrystallization rate increases exponentially with temperature The recrystallization temperature is strongly dependent on the purity of a material Very pure materials may recrystallize around 03Tm while impure materials may recrystallize around 0506Tm For example aluminium of 99999 purity recrystallizes at 75C 348 K 037Tm Commercial aluminium recrystallizes at 275C 548 K 059Tm The recrystallization temperature Tr of some pure metals are compared with the melting point Tm as shown in Table 95 The ratio of TrTm lies in the range 03505 TABLE 95 Recrystallization Temperatures Tr and Melting Points Tm Metal Tm K Tr K TrTm Mg 923 473 051 Al 933 423 045 Ag 1235 473 038 Au 1337 473 035 Cu 1358 473 035 Ni 1726 873 051 Fe 1811 723 040 Pt 2042 723 035 Mo 2883 1173 041 Ta 3269 1273 039 W 3683 1473 040 During recrystallization the impurity atoms segregated at the grain boundaries retard their motion and obstruct the processes of nucleation and growth This solute drag effect can be exploited in raising the recrystallization temperature in applications where the increased strength of a cold worked material is to be maintained at the service temperature without letting it to recrystallize Recrystallization is also slowed down in the presence of second phase particles When the particle lies in the migrating boundary during recrystallization the grain boundary area is less by an amount equal to the crosssectional area of the particle When the boundary moves out it has to pull away from the particle and thereby create new boundary area equal to the crosssection of the particle This increase in energy manifests itself as a pinning action of the particle on the boundary Consequently the rate of recrystallization decreases Grain growth refers to the increase in the average grain size on further annealing after all the cold worked material has recrystallized As a reduction in the grain boundary area per unit volume of the material occurs during grain growth there is a decrease in the free energy of the material Consider a curved grain boundary The atoms on one side of the boundary have on an average more nearest neighbours than on the other side Therefore the atoms tend to jump across the boundary to increase their overall bond energy It is easy to see that the boundary must move towards its centre of curvature for the atoms to go into a position of greater binding This results in a tendency for larger grains to grow at the expense of smaller grains As the grains grow larger the curvature of the boundaries becomes less The rate of grain growth decreases correspondingly The state of binding on either side of a planar boundary is the same and therefore a planar boundary tends to remain stationary In practical applications grain growth is usually not desirable Incorporation of impurity atoms which give rise to the solute drag effect and insoluble second phase particles which produce the pinning action on migrating boundaries are effective in retarding grain growth as well The effect on mechanical and some physical properties of the phenomena discussed in this section are summarized in Fig 919 With increasing cold work the tensile strength increases but the electrical conductivity and the ductility decrease On recovery the electrical conductivity is mostly restored as it depends mainly on the presence of point imperfections On recrystallization the tensile strength decreases and the ductility increases to the values prior to cold working The microstructural changes are also sketched in Fig 919 During cold work the grains become elongated in the direction of working During recrystallization new equiaxed grains form During grain growth these new grains increase in size but decrease in number Tensile strength Electrical conductivity Internal stress Ductility Property Microstructure Recovery Recrystallization Grain growth Cold work Annealing temperature Fig 919 Effect of cold work recovery recrystallization and grain growth on some properties of crystalline materials Recovery Recrystallization and Grain Growth 231 10 The hardness of martensite in a steel is a function of A C content B cooling rate C Ni content D nose location 11 Martensitic transformations A are diffusioncontrolled B are shear processes C yield two products of different compositions D yield a hard product in steels 12 The ca ratio of martensite depends on the concentration of A Ni B Mn C C D N 13 Bainite has A the same morphology as austenite B a nonlamellar morphology of ferrite and cementite C the coarsest morphology among all the products from austenite D none of these 14 During overageing hardness A decreases B increases C is constant D increases abruptly 15 Overageing refers to A ageing above room temperature B ultrafine precipitate size C long ageing times D coarsening of precipitate particles 16 The maximum temperature up to which tungsten mp 3410C can be cold worked is approximately A 0C B 27C C 1200C D 1940C 17 Lead melts at 327C It is hot rolled at A 273C B 200C C room temperature D none of these 18 The free energy decrease during recrystallization comes mainly from A excess point defects B excess dislocations C grain boundaries D lower energy of the new crystal structure 19 The recrystallization rate increases with A increasing amount of cold work B higher working temperature C higher annealing temperature D decreasing initial grain size 20 Grain growth occurs in the temperature range A 0203 Tm B 04 Tm C 0410 Tm D Tm 236 Multiple Choice Questions Answers 1 B 2 C 3 C 4 D 5 D 6 D 7 A 8 B C D 9 A B C 10 A 11 B D 12 C 13 B 14 A 15 D 16 C 17 C 18 B 19 A C D 20 C Sources for Experimental Data ASM International Metals Handbook 10th ed Vol 1Properties and Selection Iron Steels and High Performance Alloys Materials Park Ohio 1990 WD Kingery HK Bowen and DR Uhlmann Introduction to Ceramics Wiley New York 1976 Chaps 810 Suggestions for Further Reading WD Kingery HK Bowen and DR Uhlmann Introduction to Ceramics Wiley New York 1976 Chaps 810 V Raghavan Solid State Phase Transformations PrenticeHall of India New Delhi 1987 Sources for Experimental DataSuggestions for Further Readin 237 102 The Modulus as a Parameter in Design The stiffness of a material is its ability to resist elastic deformation or deflection on loading The stiffness is dependent on the shape of the structural component For identical shapes it is proportional to the elastic modulus Therefore the elastic modulus is an important material parameter in mechanical design Materials with high stiffness and hence high modulus are often required Covalently bonded elements such as diamond have a very high modulus 1140 GN m2 However they are not suitable for use in engineering practice due to high cost nonavailability and brittleness Brittle materials cannot withstand accidental overloading during service and may fail in a catastrophic manner Hence they are not suitable as structural members even though they may have a high modulus Ductile elements such as metals withstand accidental overloading without catastrophic failure and as such are suitable for structural components Among the metals the elements of the first transition series offer a good compromise of adequate ductility and a moderately high modulus in the range 200 GN m2 The metals of the second and the third transition series have an even higher modulus but have the disadvantage of high density By suitable alloying the Youngs modulus of metals can be increased However the modulus being a The equilibrium distance of separation r0 between atoms shown in Figs 44 and 101 is applicable at 0 K where there is no thermal energy At higher temperatures under the influence of thermal energy the atoms vibrate about their mean positions the amplitude of the vibrations increasing with increasing temperature With more thermal energy we can visualize the bonds to be somewhat loosened up This reflects in a decrease in the elastic modulus with increasing temperature see Fig 102 In a majority of cases on heating from 0 K to the melting point the decrease in elastic modulus is in the range 1020 W Cu Al Mg Sn Ca In 100 10 0 02 04 06 08 10 TTm Youngs modulus GN m2 Fig 102 Youngs modulus for a few materials plotted against TTm The Modulus as a Parameter in Design 243 244 Elastic Anelastic and Viscoelastic Behaviour structureinsensitive property it can be increased only in proportion to the concentration of the higher modulus additive For producing a high modulus Fe based material reinforcement with TiB2 is a promising route With 50 vol of TiB2 particles in the Fe matrix there is an increase of more than 50 in the modulus The TiB2 particles are in stable equilibrium with Fe As the particles are approximately spherical the modulus is not dependent on direction as in fibrereinforced materials The Youngs moduli of some ionic solids are given below Material NaCl MgO Al2O3 TiC Silica glass Youngs modulus Y GN m2 37 310 402 308 70 Even though the modulus values of some of them are quite high they also suffer from the lack of ductility like covalent solids In spite of their plasticity polymers are not suitable for applications where high stiffness is required They have a low modulus as the chains are bonded together by secondary bonds The value of the modulus is dependent on the nature of the secondary bonding van der Waals or hydrogen bonding the presence of bulky side groups branching in the chains and crosslinking For example unbranched polyethylene has a Youngs modulus of 02 GN m2 whereas polystyrene with a large phenyl side group in the monomer has a modulus of 3 GN m2 Refer to Table 57 for comparison of the monomer structure Threedimensionally bonded network polymers such as phenol formaldehyde and fully crosslinked rubber ebonite have a modulus in the range 35 GN m2 It is evident that polymers as a whole have much lower moduli as compared to other primarily bonded materials This places a severe restriction on the use of polymers as structural components In composite materials an attempt is made to increase the stiffness without the disadvantages of brittleness Boron has a low density and is suitable for light weight applications and for air borne structures Its elastic modulus is one of the highest for elements Y 440 GN m2 but it is brittle It can be used as a reinforcing fibre for a ductile matrix such as aluminium In the AlB composite the elastic modulus is increased due to the presence of the boron fibres At the same time the disadvantages of the brittleness of boron are countered by the cushioning effect of the ductile matrix The ductile matrix stops a propagating crack if a fibre embedded in it breaks accidentally If the entire material were to consist of boron only a propagating crack would culminate in the fracture of the entire crosssection The Youngs modulus Yc of a composite in a direction parallel to the fibres can be expressed as a linear function of the moduli of the fibre and the matrix Yf and Ym Yc Vf Yf VmYm 108 where Vf and Vm are the volume fractions of the fibre and the matrix Thus a 40 vol of boron in an aluminium matrix can raise the Youngs modulus from 71 GN m2 for pure aluminium to 219 GN m2 for the composite This composite would then be as stiff as steel but less than onethird its density The 246 Elastic Anelastic and Viscoelastic Behaviour rather than the van der Waals bonds between chains This increases the modulus of the aligned polymer Polymeric fibres can have a modulus at least one order of magnitude higher than that of a nonaligned polymeric structure By extending this argument one would have expected that single crystals of a polymer should have a very high stiffness when stressed in a direction parallel to the chains as there is perfect alignment in a crystalline arrangement Unfortunately this has not been realized in practice as the single crystals tend to have a folded chain structure The fold tends to be a weak point in the parallel arrangement Some applications require a nearzero variation in elastic modulus with changes in ambient temperature Alloys of iron with 36 Ni and 5 Cr have this property and are called elinvars These alloys are used in tuning forks and radio synchronization where an invariant modulus is required Here the usual decrease in modulus with increasing temperature is compensated by a slight decrease in the interatomic distance due to a magnetic effect As may be expected the magnetic effect also affects the thermal expansion The related alloy of iron with 36 nickel called invar has zero coefficient of thermal expansion around room temperature 103 Rubberlike Elasticity Materials which undergo recoverable deformation of a few hundred per cent are called elastomers and exhibit rubberlike elasticity The stress is not proportional to strain in these materials in contrast to ordinary elastic materials Elastomers Fig 103 Aluminium coated fibres of silica are pressed together to produce a strong and light composite material Courtesy Rolls Royce Limited Derby UK are long chain molecules with some crosslinking between the chains This crosslinking is important because this feature is what keeps the molecules from slipping past one another permanently during stretching After crosslinking the translational motion of chains gets restricted to segmental mobility between crosslinking points When a stress is applied to an elastomer equilibrium in the molecular configuration is established fairly quickly so that we can ignore the time dependent aspects of stretching as a first approximation In the unstretched state the chain molecules are randomly coiled A large number of configurations of equal potential energy are then possible This large number of distinguishable arrangements means an appreciable configurational entropy and a low free energy On application of an external stress the coiled molecules respond by stretching out The stretching reduces the number of possible configurations and hence lowers the configurational entropy In the limit when the molecules are all fully stretched out the possible configuration is only one and the configurational entropy is zero recall Eq 25 When a stretched rubber is heated the increase in thermal energy tends to coil back the uncoiled molecules against the stretching force The coiling and uncoiling of a long chain molecule in an elastomer is schematically illustrated in Fig 104 Note that full stretching out does not mean Increase tensile stress Increase temperature High entropy Low entropy Fig 104 The coiling and uncoiling of an elastomer chain molecule as a function of tensile stress and temperature a change in the CC bond angle of 1095 along the backbone of the chain If this were to happen the enthalpy bond energy of the elastomer would also change in addition to the configurational entropy Experimental results indicate that the change in the enthalpy on stretching a rubber is zero The stretching process merely uncoils the coiled molecules but does not change the bond lengths or bond angles This behaviour is in contrast to what happens in an ordinary elastic material where bond lengths are clearly changed see Fig 101 Using this experimental result with the first and second laws of thermodynamics it can be shown that the stretching force F at temperature T is related to the entropy S and length L of the material as follows Rubberlike Elasticity 247 RC Progelhof and JL Throne Polymer Engineering Principles Carl Hanser Verlag Munich 1993 Suggestions for Further Reading NG McCrum CP Buckley and CB Bucknall Principles of Polymer Enginee ring Oxford University Press Oxford 1988 D Rosenthal Resistance and Deformation of Solid Media Pergamon Press New York 1974 Chaps 1 and 3 Suggestions for Further Reading 259 starts in some region the strain rate increases locally resulting in a rapid increase of the stress required to cause further deformation in that region The deformation then shifts to another region of the material where there is no necking Here the strain rate and hence the stress to cause deformation are smaller Some stainless steels and aluminium alloys with a very fine grain size exhibit superplastic behaviour The glass blower is able to pull his working material to very long rods without necking because the exponent m for glass approaches one If m 1 the material behaves like a viscous liquid and exhibits Newtonian flow see Eq 1017 112 Plastic Deformation by Slip Xray diffraction studies show that the crystalline order in the solid is not lost during plastic deformation even though more imperfections are introduced The atom movements are such that the crystal structure remains the same before and after plastic deformation There are two basic modes of plastic deformation called slip and twinning Slip is a shear deformation that moves atoms by many interatomic distances relative to their initial positions as illustrated in Fig 112b Steps are created at the surface of the crystal during slip but the orientation of all parts of the crystal remains the same before and after slip Twinning on the other hand changes the orientation of the twinned parts see Fig 112c Here the movement of an atom a Before slip or twinning b After slip c After twinning Fig 112 The slip mode and the twinning mode of plastic deformation Plastic Deformation by Slip 263 Normally dislocations are always present in crystals Whiskers are special crystals which are very thin and almost free of dislocations Such crystals can withstand stresses much higher than ordinary crystals without undergoing plastic deformation If however a dislocation is introduced accidentally for example at the surface the crystal abruptly loses all its strength and there is a big drop in the stress required to cause further strain which is permanent This is illustrated for a copper whisker in Fig 116 The maximum stress the whisker withstands is 600 400 200 Stress MN m2 0 2 4 6 Fig 116 A copper whisker deforms plastically as soon as a dislocation is created with a big drop in the stress required to cause further strain Elongation 700 MN m2 but on the introduction of a dislocation this stress falls precipitously to a much lower value and considerable plastic deformation ensues Compare the maximum stress that the copper whisker can carry with the CRSS of an ordinary copper crystal listed in Table 112 114 The Stress to Move a Dislocation The stress required to move a dislocation in a crystal in the absence of other imperfections and impurities has been computed by R Peierls and FRN Nabarro These calculations from first principles are not accurate enough to predict the CRSS of different crystals and to correlate it with the observed value However we are in a position to understand the differences in the plastic deformation behaviour of crystals in a qualitative way Consider the two edge dislocations sketched in Fig 117 In Fig 117a the dislocation is stiff that is no relaxing displacements have taken place in the adjacent planes surrounding the dislocation region In Fig 117b such displacements have occurred around the incomplete plane In the first case the The Stress to Move a Dislocation 269 282 Plastic Deformation and Creep in Crystalline Materials of the straight lines in the figure depends on the size difference For example zinc and nickel have a size difference of 003 Å with respect to copper rZn rCu 003 Å rNi rCu 003 Å This size difference is small and accordingly the strengthening effect of zinc and nickel is the least among the solutes shown in Fig 1113 The size difference between copper and tin is large and the slope of the line for tin is also large For the same size difference the smaller atom viz nickel produces a greater strengthening effect than zinc In addition to the size difference the intensity of the stress field around a solute atom is also dependent on the elastic modulus of the solute Nickel having a higher elastic modulus than zinc produces a more intense stress field and a greater strengthening effect In order to improve the strength of a crystal by solute strengthening it would appear that the maximum size difference coupled with the maximum concentration of the solute would give the best results However these two factors are mutually exclusive Recall from HumeRutherys rules the more the size difference between the solute and the solvent the smaller is the equilibrium solubility The solubility can be increased by producing a supersaturated metastable solid solution by quenching from an elevated temperature The effect of the size difference and the concentration of the solute described above for substitutional solutes is also valid for interstitial solutes The interstitial atoms are usually larger than the interstitial voids they occupy Here the strengthening effect can be very strong except that the equilibrium solubility tends to be small Nature has provided mankind with a unique reaction in steel which permits an unusually large solubility of interstitial carbon in iron in spite of a very unfavourable size effect Martensite in steels is a supersaturated solution of carbon in iron obtained by quenching the steel and not allowing the carbon to diffuse out of the iron lattice For example an eutectoid steel contains 08 carbon in martensite which is some 40 times more than the equilibrium solubility in ferrite 002 at the eutectoid temperature On quenching the carbon gets trapped in the interstitial positions as the austenite shears over to form the martensitic structure The stress field produced by the oversized carbon atoms is so intense that the dislocation motion is very effectively hindered Indeed it is so effective that it becomes necessary to temper the martensite to restore some ductility at the expense of some hardness In addition to solute strengthening the martensitic plates may contain either a high dislocation density or very fine transformation twins depending on the type of martensite obtained These features also aid in increasing the strength of martensite The carbon atoms in the ferrite phase are responsible for the occurrence of a sharp yield point in mild steel They segregate around the dislocation cores and reduce the total distortional energy The segregation is known as a Cottrell atmosphere and the pinning effect of the atmosphere on the dislocations raises the yield stress of the crystal The dislocations are strongly locked by the atmosphere and do not get freed for motion during plastic deformation Very few free dislocations are available at the start of the plastic 284 Plastic Deformation and Creep in Crystalline Materials a b Fig 1114 Cont b t b t t t b t t t b t t t b t t t t d c Fig 1114 A moving dislocation either a cuts through the precipitate particles or b bypasses them Electron micrographs depicting these two processes are shown in c and d c and d Courtesy FJ Humphreys and V Ramaswamy 25 If the melting point of polyethylene is 140C it will not creep at A room temperature B 0C C 60C D 196C 26 With melting points given in brackets tick those materials which will creep significantly at 180C A Pb 327C B Cu 1084C C Al 660C D W 3410C 27 TD thoria dispersed nickel has adequate creep resistance up to 07 Tm because A thoria has a high melting point B thoria does not dissolve in nickel C nickel has a high melting point D nickel gets work hardened during service Answers 1 A 2 B C 3 C 4 C 5 D 6 D 7 C 8 D 9 B 10 A 11 A 12 C 13 B 14 B 15 B 16 C 17 C 18 A 19 A 20 B 21 A 22 C 23 A C D 24 A 25 D 26 A C 27 B Sources for Experimental Data ASM International Metals Handbook 10th ed Vol 1 Irons and Steels and High Performance Alloys Specialty Steels and Heat Resistant Alloys pp 7551003 Materials Park Ohio 1990 A Kelly and RB Nicholson Strengthening Methods in Crystals Elsevier Amsterdam 1971 Suggestions for Further Reading RWK Honeycombe Plastic Deformation of Metals Edward Arnold London 1984 WD Kingery HK Bowen and DR Uhlmann Introduction to Ceramics John Wiley New York 1976 Chap 14 Sources for Experimental DataSuggestions for Further Reading 297 true stress in this region is increasing in spite of the fall in the load and the engineering stress Fully ductile materials will continue to neck down to an infinitesimally thin edge or a point and thus fail as the crosssection at the neck becomes so small that it cannot bear the load any longer The more common type of ductile fracture occurs when the reduced crosssection has still an appreciable area Here the cracks are found to nucleate at brittle particles either the natural kind found in multiphase materials eg cementite in steel or foreign inclusions eg oxide inclusions in copper When a brittle particle is present it is difficult to maintain compatibility in the neck region between the continuously deforming matrix and the nondeforming particle This results in the formation of very tiny voids near the matrixparticle interface If fracture initiates at pores in the neck region then the voids are already present The voids grow with increasing deformation and ultimately reach sizes of the order of a mm At this stage the material may tear apart see Fig 121 The effective a b c d Fig 121 ad Successive stages in the ductile fracture of a tensile test specimen e Scanning electron micrograph of the fractured surface of an aluminium alloy showing dimples suggestive of void growth e AS Argon Ed Physics of Strength and Plasticity by permission from the MIT Press Cambridge Mass e Ductile Fracture 299 310 Fracture Example 124 Iron tested at 196C is brittle It however becomes ductile at this temperature if a thin layer of silver is diffused along the grain boundaries of iron Explain this change in behaviour Solution Silver being an FCC metal has a low PeierlsNabarro stress and hence is ductile at 196C The ductile layer of silver at the grain boundaries of iron effectively prevents crack propagation from one grain of iron into a neighbouring grain With the cracks not propagating the stress can be increased sufficiently to initiate plastic deformation in iron 127 Fatigue Fracture Rotating shafts connecting rods aircraft wings and leaf springs are some examples of structural and machine components that are subjected to millions of cycles of alternating stresses during service The majority of failures of such components in service are due to fatigue A fatigue failure can occur even below the yield stress of a material For example the yield strength of mild steel is 220 MN m2 but it will fail at a stress of 140 MN m2 if it is subjected to a very large number of stress reversals The fatigue behaviour of a material is understood from the results of a fatigue test which are presented in the form of SN curves Samples of the material are subjected to alternating stresses of different levels The number of cycles of stress reversals N required to cause fracture is plotted against the applied stress level S as shown in Fig 127 500 300 100 103 Stress level S MN m2 Nylon 6 Polymethylmethacrylate Aluminium alloy 045 C steel 105 107 109 Number of cycles N Fig 127 The SN curves for different materials 4 A silicate glass has a relatively low fracture strength because A the Youngs modulus of glass is low B the cracks propagate before Griffith criterion is satisfied C plastic deformation during crack propagation causes fracture D the cracks are sharp and propagate as soon as Griffith condition is met 5 In brittle materials with atomically sharp cracks the stress concentration at the tip of the crack is a factor of A 2 B 200 C 5000 D 106 6 Liberty ships in World War II failed by brittle fracture due to A going above the ductilebrittle transition temperature B going below the ductilebrittle transition temperature C glass superstructure D defective riveting 7 The residual stresses in the interior of a tempered glass are A nil B tensile C compressive D highly compressive 8 An ionexchange method of strengthening will be effective for a sodium silicate glass if it is dipped in A LiNO3 B NaNO3 C KNO3 D none of these 9 Tick the methods that improve fatigue resistance of materials A fine grain size B shot peening C polishing the surface D decarburizing a steel 10 The fatigue strength of mild steel is A equal to its tensile strength B more than its tensile strength C equal to its yield strength D lower than its yield strength Answers 1 B 2 C 3 A 4 D 5 B 6 B 7 B 8 C 9 A B C 10 D Suggestions for Further Reading FA McClintock and AS Argon Eds Mechanical Behavior of Materials AddisonWesley Reading Mass 1966 Chaps 1517 AS Tetelman and AJ McEvily Jr Fracture of Structural Materials John Wiley New York 1967 314 Multiple Choice QuestionsaSuggestions for Further Readin 1000C while 17 chromium is used above 1000C 188 stainless steel which contains 18 Cr and 8 Ni is among the best commerciallyavailable oxidation resistant alloys 24 Cr 55 Al and 2 Co alloyed with iron known as kanthal is used for furnace windings up to 1300C 80 Ni20 Cr nichrome and 76 Ni 16 Cr and 7 Fe inconel have excellent oxidation resistance and good mechanical properties 10 Cr alloyed with nickel chromel and 2 Al 2 Mn and 1 Si alloyed with nickel alumel are used up to 1100C as heat resistant thermocouple wires Molybdenum in a protective atmosphere of hydrogen can be used for furnace windings up to 1500C Aluminium is normally covered with a highly protective oxide film Therefore there is usually no need to add alloying elements to aluminium to improve its oxidation resistance Similarly titanium forms a protective oxide layer Addition of aluminium beryllium or magnesium to copper improves its oxidation resistance CORROSION The direct losses due to corrosion of structural and machine components is estimated to be 30 billion dollars annually It is therefore essential for an engineer to understand the basic principles of corrosion and the methods of protection against corrosion Chromium steels Stainless steels Chrome iron Oxidatin rate arbitrary units 8 6 4 2 4 12 20 28 Chromium Fig 132 The oxidation rate of iron decreases with increasing chromium content Oxidation Resistant Materials 319 where V0 is the standard potential M is the metal ion concentration in the electrolyte M is the concentration of the metal in the electrode n is the valence of the metal ion and F is Faraday constant equal to 9649 kCmole of electrons Under standard conditions the second term on the right side of Eq 131 is zero If M 1 that is if the electrolyte is deficient in metal ions the potential decreases and goes more towards the active end of Table 133 If the electrode is in the alloyed condition M 1 and then the potential increases and goes towards the noble end 1332 The Galvanic Series The structure and composition of the alloys used in service are complex The environmental conditions which provide the electrolyte are also difficult to define in terms of M The role of environment in determining the corrosion rate of mild steel is seen from the data in Table 134 TABLE 134 Role of Environment on the Corrosion Rate of Mild Steel Environment Corrosion rate mm per year Dry and nonidustrial 0001 Marine humid and nonindustrial 002 Humid and industrial 02 To describe the tendency to corrode in a given environment common metals and alloys are arranged on a qualitative scale called the galvanic series For example the galvanic series for the sea water environment is given in Table 135 In sea water 188 stainless steel in the passive condition at the top of the table is the least active and magnesium at the bottom of the table is the most active TABLE 135 Galvanic Series in Sea Water Noble end 188 stainless steel passive Nickel passive Copper Brass Tin Lead 188 stainless steel active Mild steel Alclad Aluminium Zinc Active end Magnesium The Principles of Corrosion 321 324 Oxidation and Corrosion A concentration cell can also arise due to differences in oxygen concentration The cathodic reaction 135 takes place more readily where oxygen is available so that an oxygenrich region is cathodic with respect to an oxygendepleted region This results in crevice corrosion which occurs at inaccessible locations crevices deficient in oxygen Examples of such locations are the interfaces of two coupled pipes threaded connections and areas covered with rust or dirt Corrosion occurs just below the water line in a tank This location being deficient in oxygen is anodic to the metal just above the water line An underground pipeline that goes through impervious clays in some regions and through porous sands in some other regions may corrode in the clay region A galvanic cell can form due to different residual stresses in the same metal The stressed region is more active and is anodic with respect to a stressfree region Such stress cells can form between regions of different dislocation density in a coldworked metal or in a polycrystalline metal where the grain boundaries are anodic to the interior of the grains A bent wire is likely to corrode at the bend where it has been plastically deformed 1334 Polarization When a galvanic cell is short circuited a corrosion current flows through the cell This current would set up differences in concentration of the metal ions near the electrodes In the zinccopper cell due to continued dissolution of zinc at the anode the zinc ion concentration near the anode builds up Unless the diffusion rate of these ions is fast enough to keep the electrolyte composition uniform the potential of the zinc electrode will shift towards the noble end that is towards that of copper Similarly at the cathode the cupric ions are being used up in the cathode reaction resulting in a deficiency of these ions near the cathode This shifts the cathode potential towards the active side that is towards that of zinc In addition the two electrode potentials tend to move towards each other due to the limited rates of the anodic and the cathodic reactions which proceed by the reacting species crossing an activation barrier Figure 135 shows schematically Vcathode Vanode Potential V Cathodic polarization Anodic polarization IR drop Steady current Current I Fig 135 The anode and the cathode polarization as a function of current I the variations in the cathode and the anode potentials with increasing current density The electrodes are said to be polarized as the current flows A steady state current is established when the potential difference between the cathode and the anode becomes equal to the IR drop through the electrolyte 1335 Passivation A piece of iron or steel readily dissolves in dilute nitric acid but may become resistant or passive in concentrated nitric acid The concentrated acid oxidizes the iron effectively and produces a thin protective layer on the surface Dilute acid is not strong enough to oxidize and hence continues to attack The formation of an oxide film on a corroding metal can passivate it in a similar fashion On increasing the potential of a metal electrode the current density increases at first When the current density reaches a critical value it may abruptly fall to a much lower value and remain more or less constant for some further increase in potential This phenomenon is called passivation For chromium the critical current density just before passivation is about 200 A m2 The current after passivation is less than 01 A m2 This big drop in current density is associated with the simultaneous formation of a thin oxide layer on the metal surface The phenomenon of passivation affords an important means of corrosion prevention 134 Protection against Corrosion The methods of corrosion prevention are based on the above principles of corrosion From the table of electrode potentials one can deduce that use of noble metals will prevent corrosion However it is clear that the choice of a material is dependent on many other factors We can use noble metals only in very limited applications such as ornaments and delicate scientific instruments Alternatively we can design to avoid physical contact between dissimilar metals so that a galvanic couple does not form Yet we often come across designs such as a steel screw in brass marine equipment a PbSn solder on copper wire and a steel shaft in a bronze bearing When contact between dissimilar metals is unavoidable it is necessary to see that the metal forming the anode does not have a small surface area as compared to the cathode As the same corrosion current passes through the anode and the cathode a small anode area would mean a high current density at the anode and a consequent high rate of corrosion A copper nut and bolt is permissible on a large steel plate or pipe but not a steel bolt and nut on a copper article of large surface area In materials with twophase structures where the phases form galvanic cells at the microstructural level the contact between the two phases is inevitable We have already referred to the example of untempered tempered and overtempered martensites which corrode at different rates see Fig 134 The twophase structure of duralumin corrodes faster than pure aluminium However the optimum aged twophase structure is often required for obtaining the desired mechanical properties In the case of duralumin the corrosion prevention is Protection against Corrosion 325 326 Oxidation and Corrosion achieved by making Alclad which has two thin pure aluminium sheets covering either side of a duralumin sheet Removing a cathodic reactant such as dissolved oxygen from water by means of a chemical reagent may prevent corrosion For example sulphites are used for this purpose in boiler feed water and cooling water systems Inhibitors form a protective layer on the metal surface and prevent corrosion Anodic inhibitors are oxidizing anions such as nitrites and chromates For example nitrites promote the formation of a thin passivating oxide film on iron These must be used in sufficient quantity to cover the whole surface Otherwise small uncovered areas may lead to severe localized corrosion known as pitting Cathodic inhibitors generally form a thick protective film on the surface Vapour phase inhibitors consist of nitrite or bicarbonate anions attached to a heavy organic cation The inhibitor compound is placed alongside the metal part to be protected in the storage room The compound has a vapour pressure of about 103 atm at ambient temperature so that it evaporates rapidly to ensure its adequate availability in the vicinity of the metal surface Sewing needles are first wrapped in thin paper saturated with a vapour phase inhibitor ethanolamine acetate and then in thick black paper which retains the vapour Metallic coatings are used for corrosion prevention If a metal coating is noble with respect to the underlying metal it is necessary to avoid flaws in the coating such as cracks and pores Such flaws could initiate corrosion with the coating acting as the cathode and the underlying metal as the anode As the exposed part of the anode at a crack or pore is very small corrosion takes the form of a localized attack Tin on steel is an example exhibiting localized pinhole attack The tin coating on a steel article is produced either by dipping it in molten tin or by electroplating The most common use of tinplate is for making food containers The pinhole corrosion referred to above can take place on the outside of the container Inside many organic acids that are present in foodstuffs and fruit juices form complexes with tin The concentration of the stannous ions is thereby lowered The potential of tin decreases enough to make it anodic with respect to iron In the absence of a suitable cathodic reaction the corrosion rate is low If the metal coating is baser than the substrate eg zinc or aluminium on steel galvanic protection is offered to the underlying metal The coating is anodic and corrodes first Zn and Al however become passive after the initial attack Galvanized iron GI is produced by dipping a low carbon steel sheet in molten zinc bath at about 450C On cooling in air the zinc coating crystallizes forming zinc flowers Articles such as buckets and drums made of galvanized iron are very suitable for aqueous environments In the presence of oxygen zinc hydroxide is precipitated as a protective layer Aluminium coatings are deposited by the process of calorizing mainly to improve the oxidation resistance of steel Decorative chromium plating is done over a first coating of nickel on automobile exterior components Hard chromium plating produces a thicker wear resistant surface Nonmetallic coatings such as enamel oil paint and tar act by simply excluding water and oxygen and by providing a layer of high electrical resistance 328 Oxidation and Corrosion an elevated temperature as shown in Fig 136 The region immediately next to the chromium carbide particles is denuded of chromium and therefore becomes Chromium depleted region 12 Cr Carbide precipitate Normal composition 18 Cr Fig 136 Intergranular corrosion of 188 stainless steel anodic to the interior of the grain A galvanic cell is thus set up and corrosion occurs close to the grain boundaries Intergranular corrosion can be prevented in three ways i by reducing the carbon level in the stainless steel to less than 005 ii by quenching the stainless steel from a high enough temperature to prevent chromium carbide precipitation and iii by adding strong carbide forming elements to the steel such as Nb or Ti so that carbon precipitates as niobium carbide or titanium carbide and not as chromium carbide Some special types of corrosion are also known One such type is the dezincification During corrosion of alpha brass the zinc is preferentially dissolved from the brass leaving a spongy mass of copper of little strength Dezincification is prevented by adding 004 arsenic to brass The phenomenon of stress corrosion occurs under the combined action of a corrosive environment and a mechanical stress In season cracking a coldworked brass with high residual stresses cracks along the grain boundaries in environments containing ammonia The grain boundaries have piledup dislocations in their neighbourhood due to the cold working The high energy due to this causes the boundaries to become anodic with respect to the grain interior and selective dissolution of the boundary region occurs If the coldworked brass is stress relieved at 300C season cracking can be avoided Stress corrosion of austenitic stainless steels occurs in chloride environments The crack here is transgranular that is it propagates across the grains Addition of molybdenum or increasing the nickel content of the stainless steel reduces the tendency for this type of corrosion Caustic embrittlement of boilers occurs under the combined action of stress and a high concentration of hydroxyl ions in the environment Addition of tannins and phosphates to the environment prevents caustic cracking 6 The standard potential of a zinc electrode is 076 V its potential at 300 K at a zinc ion concentration of 0001 in the electrolyte is F 96490 Cmol A 085 V B 067 V C 072 V D 0 V 7 The potential of a galvanic cell of copper potential of 034 V and aluminium potential of 166 V is A 200 V B 132 V C 132 V D 0 V 8 The peak in corrosion rate of martensite occurs when A in untempered condition B tempered at 800C C tempered at about 400C D in spherodized state 9 Porefree coating is required when A coating is noble with respect to the protected metal B coating is base with respect to the protected metal C coating has the same potential as the protected metal D none of these 10 For cathodic protection at a current density of 10 mA m2 the quantity of zinc atomic weight 654 required per m2 of ship hull per year is F 96490 Cmol A 01 kg B 02 kg C 107 kg D 213 kg Answers 1 C 2 A C 3 A 4 B D 5 B 6 A 7 A 8 C 9 A 10 A Source for Experimental Data S Lamb Ed Practical Handbook of Stainless Steels and Nickel Alloys ASM International Materials Park Ohio 1999 Suggestions for Further Reading DA Jones Principles and Prevention of Corrosion PrenticeHall Englewood Cliffs New Jersey 1996 JC Scully The Fundamentals of Corrosion Pergamon Oxford 1975 Sources for Experimental DataSuggestions for Further Reading 331 141 The Resistivity Range Electrical resistivity or conductivity is probably the most remarkable of all physical properties in that it varies over 25 orders of magnitude To get a feel for this wide range Table 141 lists the electrical resistivity at room temperature of a number of materials which are important from the engineering point of view The materials fall into three broad categories TABLE 141 The Resistivity of Materials ohm m 109 107 105 103 101 101 103 Ag Cu Al Ni Sb Bi Ge Ge Si Au Fe Graphite doped pure pure Metals Semiconductors 105 107 109 1011 1013 1015 1017 Window Bakelite Porcelain Lucite PVC SiO2 glass Diamond Mica pure Rubber Nylon Polyethylene Insulators Conductors are metals and alloys Gold silver and copper are among the best conductors of electricity Therefore their electrical resistivities are the lowest as shown in Table 141 They are followed by aluminium whose resistivity is 60 higher than that of copper Transition metals such as iron and nickel are not as good conductors as the above Still poorer conductors are the semimetals of the fifth column eg antimony and bismuth Graphite with one of its bonding electrons resonating between the sp2 bonds also fall in this category of semimetals The electrical resistivity of conductors ranges from 109 to 103 ohm m The electrical conductivity being the reciprocal of resistivity ranges from 109 ohml ml to 103 ohm1 m1 When the resistivity is in the range 103103 ohm m we have the second category of materials known as semiconductors They form the base for a number of solid state devices Here the resistivity is a strong function of small concentrations of impurities Doped germanium with an impurity content of a few tens per million can have a resistivity about two orders of magnitude lower than that of pure germanium see Table 141 Pure silicon has a higher resistivity than pure germanium The third category of materials are insulators Common electrical insulating materials such as polyethylene bakelite lucite mica PVC rubber and porcelain fall in this category The resistivity range for this category extends from 104 to beyond 1017 ohm m Here a difference in resistivity of some twelve orders of The Resistivity Range 333 338 Conductors and Resistors 143 Conduction by Free Electrons We have already noted that the wave number k takes both positive and negative values For every electron moving with a certain speed in a direction there is another electron moving with the same speed in the opposite direction This equal and opposite velocity distribution in a neutral solid can be biased by an externally applied electric field to yield a net velocity in one direction With this biasing the solid conducts electricity Under an applied field the Ek relationship of Fig 141 gets modified to the distribution shown in Fig 144 The negatively charged T1 T0 T2 T1 T0 0 K 1 PE 0 EF E Fig 143 The FermiDirac distribution of free electrons at different temperatures Field E 0 EF k Fig 144 Electrons moving towards the positive end of the applied field acquire extra velocity while those moving in the opposite direction lose some velocity electrons are accelerated towards the positive end of the field The velocity of the fastest electron moving in the direction of the positive end has a larger magnitude than that of the fastest electron moving towards the negative end of the field Such redistribution is possible only when empty electron states are available immediately above the Fermi level This availability is a basic characteristic of conductors as opposed to semiconductors and insulators 346 Conductors and Resistors 1452 Type I and Type II Superconductors Type I or the ideal superconductor when placed in a magnetic field repels the flux lines totally till the magnetic field attains the critical value Hc The magnetization M is equal to H up to Tc where it drops to zero as shown in Fig 1410a Type II or hard superconductors are those in which the ideal behaviour is seen up to a lower critical field Hc1 beyond which the magnetiz ation gradually changes and attains zero at an upper critical field designated Hc2 see Fig 1410b The Meissner effect is incomplete in the region between Hc1 and Hc2 this region is known as the vortex region The normal behaviour is observed only beyond Hc2 The magnetic flux lines gradually penetrate the solid as the field is increased beyond Hc1 and the penetration is complete only at Hc2 M Type I M Super conducting Normal Hc Super conducting Type II Normal Vortex Hc1 Hc Hc2 H H b Fig 1410 The magnetization M versus the critical magnetic field Hc for a Type I and b Type II superconductors a Type II superconductors are of great practical interest because of the high current densities that they can carry The type II state is determined by the microstructural condition of the material Heavily coldworked and recovery annealed material have cell walls of high dislocation density and this microstructure effectively pins the magnetic flux lines and makes their penetration difficult Grain boundaries also exert a pinning action an extremely fine grain size is effective in increasing Hc2 Similarly the dispersion of very fine precipitates in the matrix with the interparticle spacing of about 300 Å results in optimum flux pinning The critical current density Jc also increases as Hc2 increases The very high current densities obtainable in a Nb40 Ti alloy at 42 K at a magnetic field strength of 09Hc2 are listed below as a function of the microstructural condition Microstructure Jc A m2 Recrystallized 105 Cold worked and recovery annealed 107 Cold worked and precipitation hardened 108 1453 Potential Applications Superconducting materials are already in use for producing very strong magnetic fields of about 50 Tesla which is much larger than the field obtainable from an electromagnet Such high magnetic fields are required in MHD power generators At high magnetic field strengths a conventional copper solenoid consumes about 3 MW whereas a superconducting magnet consumes about 10 kW In addition the copper solenoid will require about 2000 lmin of water circulation to avoid burning down due to Joule heating Magnetic energy can be stored in large superconductors and drawn as required to counter voltage fluctuations during peak loading Superconductors can be used to perform logic and storage functions in computers A Josephson junction consists of a thin layer of insulating material between two superconducting solids The unique currentvoltage characteristics associated with the Josephson junction are suitable for memory elements Switching times of the order of 10 ps l011 s have been measured Arising from the Meissner effect a superconducting material can be suspended in air against the repulsive force from a permanent magnet This magnetic levitation effect can be used in transportation As there is no I2R loss in a superconductor power can be transmitted through superconducting cables without loss With the liquid N2 environment the economics of such transmission has become more favourable for adoption 1454 New Developments Till the year 1986 the highest known transition temperature Tc was 23 K in the Nb3Ge alloy In 1986 Bednorz and Mueller reported a significant increase in Tc to 34 K in a LaBaCuO ceramic material This Nobel prizewinning discovery was soon followed by further big increases in the transition temperature The oxide with the nominal formula YBa2Cu3O7x has a transition temperature of 90 K This transition temperature is 13 K above the boiling point of liquid N2 77 K Compare this with the boiling point of liquid He 4 K and that of liquid H2 23 K which is a safety hazard That a superconductor can function in liquid nitrogen is itself a remarkable achievement A liquid nitrogen environment is far easier and cheaper to obtain than a liquid helium medium Further it takes about 25 times more energy to cool from 77 K to 4 K than from room temperature to 77 K The oxide YBa2Cu3O7x is prepared by heating compacted powder mixtures of Y2O3 BaCO3 and CuO in the right proportion to temperatures between 900 and 1100C BaCO3 decomposes at this temperature to BaO and CO2 This is often followed by another annealing treatment at 800C in an atmosphere of oxygen The heat treatment conditions such as the partial pressure of oxygen in the atmosphere are critical for obtaining a high Tc The crystal structure of the powder product obtained is related to the cubic perovskite structure as illustrated in Fig 1411 Three bodycentered cubic unit cells are stacked one above another The atom distributions in the unit cells are as follows Superconducting Materials 347 New compounds such as TlBiBaSrCaCuO are now known with a reproducible Tc of about 125 K Answers 1 A 2 A 3 C 4 C 5 B 6 C 7 B 8 C 9 A 10 C 11 D 12 A B 13 C 14 C 15 B 16 A 17 C 18 C 19 C 20 C 21 D 22 D 23 B C D 24 C 25 A B C 26 D 27 A B 28 C 29 A B C 30 A C 31 A 32 A 33 B 34 A B C Sources for Experimental Data GWA Dummer Materials for Conductive and Resistive Functions Hayden Book Co New York 1971 Metals Handbook 10th ed Vol 3 Special Purpose Materials ASM International Materials Park Ohio 1990 pp 8041024 Suggestions for Further Reading C Kittel Introduction to Solid State Physics Wiley New York 1976 Chaps 6 and 12 LE Murr AW Hare and NG Eror Introducing the Metal Matrix High Temperature Superconductor in Advanced Materials and Processes ASM International Materials Park Ohio 1987 Vol 14510 pp 3644 354 Sources for Experimental DataSuggestions for Further Reading 360 Semiconductors Brillouin zone contains all the energy levels up to the first Ek discontinuity the second zone contains all levels between the first and the second discontinuities and so on The first zone for an FCC crystal is a polyhedra bounded by planar surfaces of the 111 and 200 types these two corresponding to the first and the second allowed reflections for the FCC lattice The simple energy band representation is adequate for our further discussion The outermost energy band that is full or partially filled is called the valence band in solids The band that is above the valence band and that is empty at 0 K is called the conduction band Solids are classified on the basis of their band structure as metals semiconductors and insulators Metals are those solids which have vacant electron energy states immediately above the highest filled level of the valence band This can happen in two ways In the first case the valence band is only halffilled as shown in Fig 155a As already discussed under metallic conductivity the electrons here can respond to an externally applied field by acquiring extra velocity and moving into the empty states in the top half of the valence band In the second alternative a full valence band overlaps the conduction band as shown in Fig 155b so that there is no forbidden gap Valence band Conduction band Valence band a b Fig 155 Metals have partially filled or overlapping bands Monovalent metals such as Cu Ag and Au have one electron in the outermost orbital Correspondingly the outermost energy band is only half full in these metals Divalent metals such as Mg and Be have overlapping conduction and valence bands Therefore they also conduct even if the valence band is full The band structure of trivalent metals such as Al is similar to that of monovalent metals in that the outermost band is half full In elements of the fourth column the electrons in the outermost orbital are even in number as in the divalent metals The valence band is full but there is no overlap of the valence band with the conduction band here Taking the case of covalent diamond the potential energy of electrons is a strong function of their location in the crystal Correspondingly a forbidden gap exists and it is relatively large in magnitude 54 eV see Fig 156a As we move down the fourth column to the elements below diamond the electrons of the outermost orbital are farther away from the nucleus The effect of this increasing separation is dominant over the effect of the increasing charge on the nucleus so that the electrons are less tightly bound to the nucleus as we go down the column The potential energy of the electron is no longer a strong function of its location This reduces the energy gap from 54 eV in diamond to 11 eV in silicon Fig 156b 07 eV in germanium and a mere 008 eV in gray tin In lead the forbidden gap is zero Those materials which have an energy gap of about 23 eV or less are called semiconductors and those with a gap of more than 3 eV are known as insulators In contrast to metals both insulators and semiconductors have a finite forbidden energy gap the only difference between them arising from the magnitude of the energy gap 152 Intrinsic Semiconductors In Chapter 14 we discussed the mechanism of conduction in metals When an external electric field is applied the free electrons accelerate by moving into the vacant quantum states immediately above the Fermi level In semiconductors and insulators this is not possible as there is a forbidden gap In order to conduct the electrons from the top of the full valence band have to move into the conduction band by crossing the forbidden gap The field that needs to be applied to do this works out to be extremely large Take the example of silicon where the forbidden gap is about 1 eV This is approximately the energy difference between a location close to an ion core and another location away from the ion core The distance between these two locations is about 1 Å 1010 m Therefore a field gradient of 1 V1010 m 1010 V m1 is necessary to move an electron from the top of the valence band to the bottom of the conduction band Such a high field gradient is not realizable in practice Conduction band Conduction band Eg 54 eV Eg 11 eV Valence band Diamond insulator Silicon semiconductor a b Fig 156 The difference between a an insulator and b a semiconductor is in the magnitude of the energy gap Intrinsic Semiconductors 361 substitutes for a silicon atom in the diamond cubic structure Four of the five electrons in the outermost orbital of the phosphorus atom take part in the tetrahedral bonding with the four silicon neighbours The fifth electron cannot take part in the discrete covalent bonding It is loosely bound to the parent atom It is possible to calculate an orbit for the fifth electron assuming that it revolves around the positively charged phosphorus ion in the same way as for the 1s electron around the hydrogen nucleus There is however one important difference The electron of the phosphorus atom is moving in the electric field of the silicon crystal and not in free space as is the case in the hydrogen atom This brings in the dielectric constant of the crystal into the orbital calculations and the radius of the electron orbit here turns out to be very large about 80 Å as against 05 Å for the hydrogen orbit Such a large orbit evidently means that the fifth electron is almost free and is at an energy level close to the conduction band as shown in Fig 158b Excitation of the fifth electron into the conduction band takes place much more readily than excitation from the valence band of the silicon crystal The phosphorus atom is said to donate its electron to the semiconductor The energy level of the fifth electron is called the donor level see Fig 158b As the elements to the right of the fourth column donate negative charges electrons the semiconductors doped with them are called ntype semiconductors The energy required to excite the fifth electron into the conduction band is known as the ionization energy see Table 152 TABLE 152 Ionization Energies for Some Elements in Silicon and Germanium eV Impurity Silicon Germanium ntype Phosphorus 0044 0012 Arsenic 0049 0013 Antimony 0039 0010 ptype Boron 0045 0010 Aluminium 0057 0010 Gallium 0065 0011 Indium 016 0011 As compared to the energy gap the ionization energy of an impurity atom is very small So at room temperature a large fraction of the donor level electrons are excited into the conduction band This fraction is much larger than the fraction of electrons excited due to the intrinsic process that is from the valence band According to the law of mass action the product of the number of electrons in the conduction band and the number of holes in the valence band must be constant This condition drastically reduces the number of holes in the ntype semicond uctor The electrons in the conduction band become the majority charge carriers Consider the alternative process of doping a silicon crystal with a third column element such as Al Ga or In Aluminium has three electrons in the outer orbital While substituting for silicon in the crystal it needs an extra electron to complete the tetrahedral arrangement of bonds around it The extra electron can Extrinsic Semiconductors 365 368 Semiconductors excitation to the acceptor level become available in increasing numbers with increasing temperature So this region is called the extrinsic region From the slope of the straight line in this region the ionization energy of the impurity can be calculated At higher temperatures there is the second region called the exhaustion region Here the excitation of charge carriers due to impurities is nearing completion due to the exhaustion of unexcited impurity electrons or holes This is indicated by the flat region of almost zero slope In this region the slope can even become positive as in Fig 1510 due to the dominance of the temperature dependent mobility term in the conductivity equation Increasing temperature results in decreasing mean free path of the conduction electrons or holes So the mobility decreases with increasing temperature Above room temperature there is another linear region with a negative slope of large magnitude which corresponds to intrinsic conduction The slope difference between the extrinsic and intrinsic regions reflects the difference in magnitude between the impurity ionization energy and the energy gap Eg For stability of operation of a semiconductor device a relatively flat region where the conductivity does not vary appreciably with changes in ambient temperature is often desirable On the other hand if the device is to be used as a thermistor for measuring temperature maximum sensitivity in measurement is desirable 154 Semiconductor Materials Silicon Eg 11 eV is the most widely used semiconductor crystal It is available in abundance in the earths crust in the form of silica and silicates It has a moderately high melting point 1410C which is easily achieved in modern zone refining and crystal growing apparatus Germanium Eg 07 eV is the other elemental semiconductor crystal with a lower melting point 937C Apart from these there are a number of compound semiconductors formed by combinations of equal atomic fractions of fifth and thirdcolumn or sixth and secondcolumn elements The crystal structures of these compounds are related to the diamond structure as discussed in Chap 5 Table 153 lists the properties of some IIIV compound semiconductors TABLE 153 Properties of Some Semiconductor Compounds Compound Energy gap eV Mobility m2 V1 s1 Melting point C Electrons Holes GaP 226 0002 1350 AlSb 152 002 002 1050 GaAs 143 085 004 1240 InP 129 046 0015 1070 GaSb 078 04 007 705 InAs 035 23 0024 940 InSb 018 65 01 525 374 Semiconductors 1556 Photolithography Lithography as used in the manufacture of ICs is the process of transferring geometrical shapes on a mask to the surface of a silicon wafer The mask is to be prepared first An electron beam machine generates the design pattern reduced in size and transfers it to a photosensitive glass called the photomask on a small area of a few mm2 The photographic setup is then moved to the adjacent area and the glass is exposed again and so on in a stepandrepeat fashion As many identical areas are put on the mask as the number of chips that will ultimately fit on the wafer The photomask is positioned above the oxidized surface of the wafer the wafer surface is coated with a photoresist solution and is exposed to an intense source of radiation as shown in Fig 1512a Till recently UV light from a mercury lamp was used Now excimer lasers with wavelength in the range of 24801930 Å provide radiation of adequate intensity Excimer or excited dimer means a molecule consisting of an excited atom and an atom in its unexcited ground state After the exposure when the photoresist is developed it dissolves away from those areas where the light has fallen as illustrated in Fig 1512b In the next step of etching see Fig 1512c the etchant dissolves the oxide layer in the exposed areas only The photoresist is etchresistant After etching the left over layer of photoresist is removed and the geometrical pattern is left behind in the silica see Fig 1512d All the steps in the image transfer are thus complete Photomask Photoresist SiO2 Si b a Photoresist SiO2 Si d c Fig 1512 The photolithographic process Photoresists can be positive or negative Positive resists behave as described above ie they dissolve away from areas where the light has fallen Negative resists act in the opposite way For example polyisoprene combined with a photosensitive compound is a negative resist Once activated by the light source the compound transfers the absorbed energy to the polyisoprene chain molecules and crosslinks them The crosslinked polymer is insoluble in the developer The etching or window cutting can be through an acid medium wet chemical etching or by sputtering dry etching In reactive sputtering a mixture of CF4 and O2 is used in the discharge tube to generate a plasma which is weakly ionized gas mixture When the plasma is directed onto the wafer surface the ions collide with atoms in the exposed areas of the wafer surface transfer their kinetic energy and eject the atoms out of the surface The ions arrive mostly at normal incidence and the degree of etch anisotropy is high as compared to acid etching ie the difference in the lateral dimension between the etch image and the mask image is small The etching is also required to be selective ie it should not remove the mask material or the underlying substrate in which the previously processed device elements may be present 1557 Doping Impurity doping can be carried out by solid state diffusion under two conditions i a constant surface concentration of the dopant is maintained at a high enough temperature so that the dopant diffuses in and creates a characteristic concentration profile and ii a constant total quantity of the dopant is first deposited on the surface predeposition either by shorttime annealing or by other techniques such as ion implantation at a low temperature The concentration profile is then altered through a high temperature anneal in a non doping atmosphere drivein A pn junction is created by diffusing into the bulk semiconductor through a window in the oxide layer an impurity say a ptype where the bulk crystal has already been doped to be the ntype In practice the concentrationdistance profile of the dopant is approximated to two limiting cases the abrupt junction profile This is the case in a shallowlydiffused junction or ionimplanted junction In the linearlygraded junction the concentration profile is assumed to change linearly with distance When the impurity is diffused in through a window it moves downwards as well as sideways The final shape of the diffused region instead of being a flat box becomes a box with bulging sides and spherical corners This shape has an important effect on the junction breakdown characteristics Avalanche multiplication is the most important mechanism of junction breakdown The avalanche breakdown voltage imposes an upper limit on the reverse bias of diodes the collector voltage of bipolar transistors and on the drain voltage of MOSFETs If the electric field is high the carriers may carry enough energy such that their collisions with atoms generate electronhole pairs The newlycreated pairs in turn generate more pairs by collisions and the process multiplies like an avalanche eventually causing breakdown The avalanche breakdown voltage decreases with increasing impurity concentration It is also less for cylindrical shape of the sides of the diffused region and when the corners are spherical The diffusion parameters for common dopants are B P As Sb D0 cm2sec 076 385 24 0214 Q eV 346 366 408 365 Fabrication of Integrated Circuits 375 378 Semiconductors be fabricated In circuit simulation the complete drawing of the circuit is broken into different levels of IC processing eg gate electrodes on one level contact windows on another level and so on These levels are called masking levels The final IC is manufactured by sequentially transferring the features from each mask level by level to the wafer surface Between two successive image transfers by photolithography an ionimplant doping oxidation andor metallization may take place After circuit simulation it is also necessary to carry out a process simulation Suppose a CMOS process consists of nine lithographic steps six ion implantation and several diffusion annealing and oxidation steps The critical steps in the process are simulated on a computer noting that all steps are strongly interrelated Each of the thermal cycles in a process such as oxidation epitaxial growth and annealing can affect the vertical and lateral diffusion of the dopants interdiffusion between layers and so on Developing a new process especially with a new material requires careful evaluation of the sequence of thermal cycles to be adopted Figure 1513 illustrates a sevenmask process in building a circuit The entire circuit is built in each of the squares of the wafer which are separated Fig 1513 The building of an integrated circuit with seven masking steps 7th Masking step 6th Masking step 5th Masking step 4th Masking step 3rd Masking step 2nd Masking step 1st Masking step WAFER 380 Semiconductors be constant throughout the crystal in thermal equilibrium This results in the electron energy levels at the bottom of the conduction band in the npart to be lower than those in the ppart by an amount equal to the contact potential eV0 as shown in Fig 1514 At equilibrium there is no net current flowing across the pn junction The concentration of electrons in the conduction band on the pside is small These electrons can accelerate down the potential hill across the junction to the nside resulting in a current I0 which is proportional to their number The concentration of electrons in the conduction band on the nside is large in comparison due to the donor contribution However only a small number of these electrons can flow to the pside across the junction as they face a potential barrier This small number can be computed using the Boltzmann probability equation At equilibrium the current from the nside to the pside and the current in the opposite direction from the pside to the nside are the same equal to I0 There is an equal additional contribution to I0 from the flow of holes across the junction The concentration of holes is large in the pregion as compared to the nregion If an external voltage Vi is now applied to the crystal such that the pside becomes positive with respect to the nside the electron energy levels will change as shown in Fig 1515a Note that the potential for electrons is opposite in sense to the conventional method of representing the sign of an electric potential The barrier at the junction is now lowered by an amount eVi resulting in a greatly enhanced current flow in the forward direction that is from the n side to the pside This change in barrier does not affect the flow of electrons in the reverse direction from the pside to the nside as the flow here is still down the potential hill So the applied voltage causes a large net current flow in the forward direction nside pside Conduction band eV0 Conduction band n p Eg Valence band EF Valence band Fig 1514 The Fermi level is the same on both sides of a pn junction The contact potential at the junction gives rise to an energy barrier eV0 V 0 input signal The nregion here is called the emitter and the pregion is the base The other junction is reverse biased and connected to the output The nregion at the output end is called the collector Any input voltage Vi would reduce the contact potential at the emitterbase junction by an amount equal to eVi This would result in a large current flow from the emitter to the base region as given by Eq 1512 The width of the base region is kept as small as possible so that the electrons flowing into the base region are not lost by recombination with the holes that are dominant in the base region In an ideal transistor almost all the electrons emitted reach the collector by flowing through the base and down the potential hill at the basecollector junction A small change in the input voltage results in a large current from the emitter to the base and then on to the collector The current flowing through different regions being the same this large current can appear as a large voltage across the load resistor if its resistance is high Thus amplification results The above is a description of a simple npn transistor A number of more complex transistors are now known Two promising types are MOSFET metal oxidesemiconductor fieldeffect transistor and CCD chargecoupled device Details of the working of these devices are beyond the scope of this book This is however not to minimize their practical importance For example a single computer now contains more than a million transistors or one of its close relations MOSFET or CCD 1563 Junction Lasers Laser stands for Light Amplification by Stimulated Emission of Radiation When a pn junction is forward biased a large number of electrons flow across the junction from the nside to the pside This greatly increases the concentration of electrons in the conduction band of the pregion much above the equilibrium value These electrons eventually recombine with the holes in the pregion emitting in the process a coherent monochromatic beam of light the laser beam The wavelength of this radiation for a GaAs junction laser is 8700 Å This light emerges strongly where the junction meets the sidewalls of the crystal Mirrors are placed on opposite sides of the crystal to reflect the beam back and forth until it builds up in intensity Alternatively the flat boundary between the crystal and the air acts as a reflector As compared to silicon the IIIV semiconductor compound GaAs has the advantages of high signal speed low power consumption and large operating temperature range This combined with its laser properties makes it an ideal material for use in optical fibre and satellite communications and sophisticated supercomputers The more recent development is the sandwich laser in which a laseractive GaAs layer is sandwiched between two laserinactive GaAlAs layers one being ptype and the other ntype The GaAlAs layers have larger energy gaps than GaAs and confine electrons to the active region The excellent matching at the interface of the crystal structures of GaAs and GaAlAs with less than 1 difference in the lattice parameter ensures minimum strain and absence of plastic deformation during growth of the sandwich laser The above Some Semiconductor Devices 383 43 The functions of an oxide layer during IC fabrication can be to A mask against diffusion or ionimplant B insulate the surface electrically C produce a chemically stable surface D increase the melting point of silicon 44 A negative resist A becomes more soluble after UV exposure B becomes less soluble after UV exposure C gets crosslinked D dissolves during acid etching 45 The main difficulty in monolithic integration of Si and GaAs is the difference in the lattice parameters which is about A 01 B 4 C 25 D 100 46 Electromigration in metallization refers to the diffusion under the influence of current of A Al B Cu in AlCu alloy C Si D Na 47 Electromigration in an IC chip refers to A grain boundary diffusion of Al in Al interconnections B grain boundary diffusion of Cu in Al interconnections C diffusion of Si in Al interconnections D diffusion of oxygen in Si 48 A transistor has a collector current of 5 mA when the emitter voltage is 20 mV At 30 mV the current is 30 mA At 50 mV it is A 80 mA B 180 mA C 480 mA D 1080 mA Answers 1 D 2 D 3 D 4 A 5 A 6 D 7 A B C D 8 C 9 B 10 D 11 D 12 C 13 A 14 A 15 A B 16 C 17 D 18 A 19 C 20 B C 21 C 22 B 23 A 24 A 25 C 26 A 27 C D 28 B C 29 D 30 C 31 A 32 A 33 C 34 A 35 A 36 C 37 B 38 C 39 A C D 40 A 41 B 42 D 43 A B C 44 B C 45 B 46 A 47 A 48 D Multiple Choice QuestionsAnswers 391 Sources for Experimental Data P Gise and R Blanchard Modern Semiconductor Fabrication Technology PrenticeHall Englewood Cliffs NJ 1986 SP Keller Ed Handbook on Semiconductors Vol 3Materials Preparation and Properties North Holland Amsterdam 1980 Suggestions for Further Reading C Kittel Introduction to Solid State Physics John Wiley New York 1976 Chaps 7 and 8 KK Ng Complete Guide to Semiconductor Devices IEEE Press and Wiley Interscience New York 2002 SM Sze Physics of Semiconductor Devices John Wiley New York 1981 392 Sources for Experimental DataSuggestions for Further Reading exception however is a superconductor which is perfectly diamagnetic with a susceptibility value of 1 All the lines of force are repelled by the super conductor thus making them useful for the purpose of shielding out magnetic fields Some atoms and molecules possess intrinsic permanent magnetic moments In the absence of an externally applied field the moments of the atoms in a solid are randomly oriented with respect to one another and the solid as a whole has no net magnetic moment If an external field is applied the magnetic moments tend to align themselves parallel to the applied field so as to lower their potential energy It is well known that a suspended magnetic needle aligns itself spontaneously with the earths magnetic field Similarly there is a spontaneous tendency for the permanent moments of the atoms of the solid to align themselves in the direction of the field thereby intensifying the lines of force in the field direction This phenomenon is called paramagnetism and is illustrated in Fig 161b The aligning force on the permanent moments of the atoms with ordinary magnetic fields is rather small so that the paramagnetic effect is weak The paramagnetic susceptibility is small and positive of the order of 103 It decreases with increasing temperature as thermal energy tends to randomize the alignment Ferromagnetic solids are those in which the permanent magnetic moments due to electron spins are already aligned due to bonding forces The susceptibility is very large and positive for ferromagnetic materials in the range 102105 They strongly attract the magnetic lines of force as schematically shown in Fig 161c 162 Magnetic Moments due to Electron Spin Permanent magnetic moments can arise from three sources the orbital magnetic moment of the electrons corresponding to the quantum number ml the spin magnetic moment of electrons corresponding to the spin quantum number ms and the spin magnetic moment of the nucleus Of these the spin magnetic moments of the electrons are the only ones that are important from our point of view We will discuss only this source of magnetism a b c Fig 161 a A diamagnetic solid repels slightly the magnetic lines of force b A paramagnetic solid weakly attracts the lines of force c A ferromagnetic solid attracts them very strongly Magnetic Moments due to Electron Spin 395 398 Magnetic Materials It turns out that only when this ratio lies between 15 and 20 the exchange energy is positive and parallel spins are energetically favoured Among the common metals only Fe Co and Ni have positive exchange energy and are in the spontaneously magnetized state These metals or their ions form the basis of the majority of magnetic materials No overlap of the d orbitals occurs in the higher transition series as the d electrons are strongly attracted to their respective nuclei which have larger positive charges The only other elemental crystals where the exchange energy is appreciable belong to the first rare earth series such as gadolinium terbium and dysprosium Here the 4f electrons align themselves in a parallel fashion Thermal energy tends to randomize the aligned spins so that all ferromagnetic materials become paramagnetic at a sufficiently high temperature The transition temperature at which all the spin alignment is lost is called the Curie temperature Just below the Curie temperature the susceptibility can be 102 to 103 increasing with decrease in temperature as the alignment becomes more complete The Curie temperature is a function of the magnitude of the exchange energy Cobalt that has the highest exchange energy has the highest Curie temperature 1400 K Gadolinium with a small exchange energy has a Curie temperature below room temperature Example 162 Which of the two solids cobalt and gadolinium has the higher saturation magnetization at i 0 K and ii 300 K Solution i At 0 K the thermal energy kT which tends to randomize the spins is zero Therefore all the spins remain aligned in both Co and Gd As the net magnetic moment of Gd is 7 per atom as compared to 17 per atom in the Co crystal Gd will have the higher saturation magnetization The actual values will depend on the number of atoms per unit volume in the two cases ii At 300 K Gd is above its Curie temperature of 289 K Hence Gd will be paramagnetic at 300 K and will have negligible magnetization as compared to Co which has a much higher Curie temperature The interatomic distance can be changed within limits by alloying For example manganese in the elemental form is not ferromagnetic On alloying the exchange energy can become positive and the spins on neighbouring atoms can become aligned if the MnMn distance is increased by the right amount so that the ratio of the atomic diameter to the 3d orbital diameter falls in the range 1520 This happens in Heusler alloys Cu2MnSn and Cu2MnAl which are ferromagnetic In some compounds the constituent atoms may be antiferromagnetically coupled but can have different magnetic moments This would give rise to a net magnetic moment in each coupling and the sum of the moments of all the couplings can result in magnetization that is comparable in order of magnitude to ferromagnetism This phenomenon is called ferrimagnetism and is compared with ferromagnetism and antiferromagnetism in Fig 162 Ferrites of the general formula Me2Fe3 2 O2 4 can be ferrimagnetic The divalent metal cation Me2 in the formula is usually Fe2 Ni2 Zn2 Mg2 Co2 Ba2 Mn2 or some combination of these ions These compounds crystallize mostly as the inverse spinel structure described in Chap 5 As an example of the incomplete cancellation of the magnetic moments consider the crystal of magnetite Fe3O4 Magnetite is the lodestone of the ancient mariners The ferric ion with two 4s electrons and one 3d electron removed from the neutral iron atom has a magnetic moment of 5 units and the ferrous ion has 4 units All the ferric ions in magnetite are antiferromagnetically coupled so that the net magnetic moment of the compound arises from the ferrous ions only The magnetization M calculated on the basis of 4 net magnetic moments per Fe3O4 is in good agreement with the experimental value for magnetite Example 163 In nickel ferrite NiFe2O4 the ferric ions are antiferro magnetically coupled The magnetization is due to the nickel ions When zinc is added to nickel ferrite the magnetization of the crystal increases even though the zinc ions are not ferromagnetic Explain how this could happen Solution Referring to Chap 5 the inverse spinel structure of nickel ferrite has all the Ni2 and half of the Fe3 ions in the octahedral sites The other half of the Fe3 ions are in the tetrahedral sites The ferric ions in the two sites are antiferromagnetically coupled When zinc substitutes for the ferric ions in the tetrahedral sites the antiferromagnetic coupling will not exactly cancel as the number of ferric ions in the octahedral sites will be more This would increase the magnetization as there is an extra contribution from the uncoupled ferric ions of the octahedral sites 164 The Domain Structure Iron has a high Curie temperature of 1041 K 768C and consequently almost all the spins remain aligned at room temperature Yet at room temperature an ordinary piece of iron is not magnetic in the absence of an applied field To explain this discrepancy Weiss conceived the idea of magnetic domains The a b c Fig 162 a Ferromagnetic b antiferromagnetic and c ferrimagnetic coupling of electron spins in atoms denoted by arrows The length of an arrow is a measure of the magnetic moment of the atom The Domain Structure 399 400 Magnetic Materials unmagnetized iron crystal consists of a number of domains and within each domain the spins of all the atoms are aligned However the spins of adjacent domains are not parallel Due to the random orientation of the domains with respect to one another the magnetic moments cancel out and the crystal as a whole is not magnetic In the presence of an external field the domains tend to align themselves with the field resulting in a large net magnetization of the solid The process of magnetization of the iron crystal is not the aligning of the electron spins of the different atoms but it is the alignment of the various domains each of which is already ferromagnetic The domain concept neatly explains the magnetization behaviour of iron Ample experimental evidence is now available for the presence of magnetic domains The magnetostatic energy of a ferromagnetic solid can be reduced if a number of domains are arranged such that no poles exist at the surface and no lines of force go out of the material as illustrated in Fig 163 Hence a single domain tends to break up into several domains even if this means an increase in the domain boundary energy The domain size does not decrease indefinitely as at some stage the increasing domain boundary energy would oppose this N S S N S N N N N S S S N S N S NS N S N S a One domain b Ten domains 165 The Hysteresis Loop The BH curve for a typical ferromagnetic material is shown in Fig 164 As the applied field H is increased the magnetic induction B increases slowly at first and then more rapidly The rate of magnetic induction slows down again eventually attaining a saturation value Bs With further increase in the magnetic field there is no increase in the induction If the field is reversed the induction decreases slowly at first and reaches a residual value Br at zero field Br represents the amount of residual induction left in the specimen after the removal of the field If the application of the field is continued in the opposite direction the domains tend to reverse their alignment so that the remaining Fig 163 a With one domain the magnetic lines of force go out of the material b The magnetic lines do not go out if a number of domains are present induction is lost at a certain value of the reverse field called the coercive field Hc The process of reversal of domains continues to give a net magnetization in the opposite direction After saturation occurs in this direction restoring the original field direction completes the hysteresis loop There are two possible ways to align a random domain structure by applying an external field One is to rotate a domain in the direction of the field and the other is to allow the growth of the more favourably oriented domains at the expense of the less favourably oriented ones If we compare the domain structure with the grain structure of a polycrystalline material the boundaries separating the domains called domain walls are the analogue of grain boundaries The domain boundary energy is about 0002 J m2 The domain walls however are some two orders of magnitude thicker than the grain boundaries because there is a gradual transition from one domain orientation to the next across the wall Also the domain boundaries can exist within the grain Analogous to grain growth Chap 6 the domain walls can move such that the more favourably oriented domains grow at the expense of others In the earlier stages of magnetization below the saturation region of the hysteresis curve domain growth is dominant The growth is more or less complete as the saturation region is approached Thereafter the most favourably oriented fully grown domain tends to rotate so as to be in complete alignment with the field direction The energy required to rotate an entire domain is more than that required to move the domain walls during growth Consequently the slope of the BH curve decreases on approaching saturation Each time the hysteresis loop is traversed energy equal to the area of the loop is dissipated as heat The power loss due to hysteresis in a transformer core is dependent on the number of times the full loop is traversed per second Fig 164 Magnetic induction B as a function of the applied field H for a ferromagnetic material tracing a hysteresis loop Br Bs Hc B H The Hysteresis Loop 401 404 Magnetic Materials In recent years metallic glasses produced from ironbase alloys containing 1525 of Si B C offer substantial reduction in core losses Such an alloy cooled at a rate of 104 C s1 from the molten state does not crystallize but solidifies into a ribbonshaped metallic glass Owing to the larger concentrations of the impurity atoms the electrical resistivity is higher than that for the Fe4 Si alloy thereby reducing eddy current losses The absence of grain boundaries in the glassy matrix reduces hysteresis losses The total iron losses can be reduced to 3010 of that for the conventional FeSi alloy Such a reduction can save nearly a billion dollars in distribution transformers alone in a developed country like the US Soft magnets made of metallic glass are also used in phonograph cartridges and audio and computer tape heads FeSi alloys are suitable for operation at power frequencies of 5060 Hz They are not suitable in communications equipment where high sensitivity and Eddy current losses can be minimized by increasing the resistivity of the magnetic medium Iron which used to be the material for transformer cores is now almost entirely replaced by an FeSi solid solution with about 4 silicon which has a substantially higher resistivity than pure iron as illustrated in Fig 166 The reason for stopping at a concentration of 4 is due to fabrication problems as higher silicon content tends to make the iron brittle see the variation in the ductilebrittle transition temperature with silicon content in Fig 166 In addition the transformer core is laminated such that the resistance of the laminations is much more than that of a onepiece core Bs 22 21 20 19 18 80 60 40 20 0 2 4 6 Silicon Resistivity 108 ohm m 300 200 100 0 100 Ductilebrittle transition temperature C Fig 166 Variation in magnetization resistivity and ductilebrittle transition temperature as a function of silicon content in FeSi alloys Sources for Experimental Data FN Bradley Materials for Magnetic Functions Hayden New York 1971 ASM International Metals Handbook 10th ed Vol 2 SpecialPurpose Materials Materials Park Ohio 1990 pp 761803 Suggestions for Further Reading F Brailsford An Introduction to the Magnetic Properties of Materials Longmans London 1968 L Solymar and D Walsh Lectures on the Electrical Properties of Materials Oxford University Press Oxford 1984 Sources for Experimental DataSuggestions for Further Reading 411 No field Field a b c d Fig 171 Various polarization processes a electronic polarization b ionic polarization c orientation polarization and d space charge polarization Fig 171b The ionic polarizability is due to this shift of the ions relative to other oppositelycharged neighbours It should be distinguished from electronic polarization where the electron cloud of an atom shifts with reference to its own nucleus Ionic polarization is also independent of temperature In methane molecule CH4 the centre of the negative and the positive charges coincide so that there is no permanent dipole moment On the other hand in CH3Cl the positive and the negative charges do not coincide The electronegativity of chlorine being more than that of hydrogen recall Table 44 the chlorine atom pulls the bonding electrons to itself more strongly than hydrogen So this molecule carries a dipole moment even in the absence of an electric field When an electric field is applied on such molecules they tend to align themselves in the applied field see Fig 171c Recall that atoms with permanent magnetic moments tend to align themselves with the applied magnetic field giving rise to paramagnetism The polarization due to this alignment is called orientation polarization and is dependent on temperature With increasing temperature the thermal energy tends to randomize the alignment Polarization 415 420 Dielectric Materials 174 Ferroelectric Materials In materials known as ferroelectrics the dielectric constants are some two orders of magnitude larger than those in ordinary dielectrics Barium titanate is a ferroelectric with a relative dielectric constant of over 2000 compared to less than 10 for ordinary dielectrics listed in Table 172 The difference in the magnetic susceptibility between ferromagnetic and paramagnetic materials bears a direct analogy to this difference in the values of the dielectric constants Following the nomenclature in magnetism materials of very large dielectric constants are called ferroelectrics As in the ferromagnetic phenomenon the electric dipoles in a ferroelectric solid are all aligned in the same direction even in the absence of an electric field Defect breakdown is due to cracks and pores at the surface To decrease the possibility of surface shorting insulators are designed with lengthened surface paths Moisture from the atmosphere can collect on the surface discontinuities and result in breakdown Glazing is done on ceramic insulators to make the surface nonabsorbent Gases can collect at pores and cracks and the breakdown can occur due to a gas discharge A graphic case of surface breakdown of an insulator is shown in Fig 174 Fig 174 Surface breakdown of an insulator LH Van Vlack Physical Ceramics for Engineers by permission from AddisonWesley Reading Mass oxygen anions are at the face centres and the titanium ion is in the octahedral void at the body centre Only one out of four octahedral voids in the unit cell is occupied and this corresponds to the chemical formula with one titanium for every four species of the other kinds one barium plus three oxygen Above 120C barium titanate is a cubic crystal with the ion locations as described above In this state the centres of the negative and the positive charges coincide and there is no spontaneous dipole moment If the crystal is cooled to below the ferroelectric Curie temperature of 120C the titanium ion shifts to one side of the body centre as shown dotted in the front view of Fig 175b There is also a displacement of the neighbouring oxygen anions The crystal transforms from a cubic to a tetragonal structure on cooling through 120C The ca ratio of the tetragonal cell is 403 Å398 Å 1012 The centres of the positive and the negative charges do not coincide any longer and local dipoles are created throughout the crystal The dipoles of neighbouring unit cells are all aligned resulting in a large polarization in the solid Example 172 Calculate the polarization of a BaTiO3 crystal The shift of the titanium ion from the body centre is 006 Å The oxygen anions of the side faces shift by 006 Å while the oxygen anions of the top and bottom faces shift by 008 Å all in a direction opposite to that of the titanium ion The ferroelectric phenomenon is discussed here with reference to the classical example of barium titanate BaTiO3 The cubic unit cell of barium titanate crystal is shown in Fig 175a Barium ions are at the body corners the 008 Å 006 Å 006 Å Ti4 O2 Ba2 Fig 175 a Cubic unit cell of BaTiO3 crystal b The dashed circle in the middle of the front view shows the shifting of the titanium ion on cooling through the Curie temperature The shift of the oxygen anions is also shown Ferroelectric Materials 421 a b Sources for Experimental Data HL Saums and WW Pendleton Materials for Electrical Insulating and Dielectric Functions Hayden New York 1971 RW Sillars Electrical Insulating Materials and Their Application P Peregrinus Ltd Stevenage UK 1973 Suggestions for Further Reading AJ Dekker Electrical Engineering Materials PrenticeHall of India New Delhi 1977 Chaps 23 WD Kingery HK Bowen and DR Uhlmann Introduction to Ceramics Wiley New York 1976 Chap 18 426 Sources for Experimental DataSuggestions for Further Reading 428 Appendix IProperties of Elements Fermium Fluorine Francium Fm F Fr 100 9 87 102 17 220 27 Gadolinium Gallium Germanium Gold Gd Ga Ge Au 64 31 32 79 786 591 532 1932 2001 118 1364 1020 828 181 575 141 56 925 99 78 1312 30 937 1063 Hafnium Helium Holmium Hydrogen Hf He Ho H 72 2 67 1 1309 679 1364 243 601 107 137 67 2222 270 1461 259 Krypton Kr 36 157 Lanthanum Lawrencium Lead Lithium Lutetium La Lw Pb Li Lu 57 103 82 3 71 619 1136 053 985 2244 1827 1299 1776 104 290 45 812 38 157 115 84 920 327 181 1652 Indium Iodine Iridium Iron In I Ir Fe 49 53 77 26 731 494 225 787 1571 257 854 71 314 663 117 105 528 210 156 114 2454 1535 Magnesium Manganese Mendelevium Mercury Molybdenum Mg Mn Md Hg Mo 12 25 101 80 42 174 743 1355 1022 140 739 1481 939 257 226 61 498 44 198 328 650 1245 38 2610 Neodymium Neon Neptunium Nickel Niobium Nitrogen Nobelium Nd Ne Np Ni Nb N No 60 10 93 28 41 7 102 700 890 857 2061 659 108 998 275 127 707 38 100 193 105 1019 249 637 1453 2468 210 Atomic Density Molar Thermal Youngs Melting Element Symbol number 103 kg m3 volume expansion modulus point C 108 m3 106 K1 GN m2 Osmium Oxygen Os O 76 8 2257 843 47 540 2700 219 Palladium Phosphorus Pd P 46 15 1202 183 888 1692 115 124 124 46 1552 44 APPENDIX I cont Platinum Plutonium Polonium Potassium Praseodymium Promethium Protactinium Pt Pu Po K Pr Pm Pa 78 94 84 19 59 61 91 2145 195 086 677 154 909 1227 4547 2082 1500 895 55 23 83 679 90 73 170 965 255 35 33 42 100 1769 640 254 64 919 1027 1230 Radium Radon Rhenium Rhodium Rubidium Ruthenium Ra Rn Re Rh Rb Ru 88 86 75 45 37 44 50 2104 1244 153 122 45 885 827 5587 829 202 663 840 881 936 16 460 372 27 410 700 71 3180 1966 39 2500 Samarium Scandium Selenium Silicon Silver Sodium Strontium Sulphur Sm Sc Se Si Ag Na Sr S 62 21 34 14 47 11 38 16 749 299 479 233 1049 097 260 207 2007 1489 1648 1206 1028 2367 34 155 104 100 369 307 192 706 20 64 34 79 58 103 805 89 135 195 1072 1539 217 1410 961 98 768 119 Tantalum Technetium Tellurium Terbium Thallium Thorium Thulium Tin gray Titanium Tungsten Ta Tc Te Tb Tl Th Tm Sn Ti W 73 43 52 65 81 90 69 50 22 74 166 624 825 1185 1166 931 730 451 193 109 2045 1926 1725 1990 1815 1626 1063 953 655 806 1677 103 294 112 133 53 835 459 181 370 41 575 8 74 75 52 106 396 2996 2130 450 1356 303 1750 1545 232 1668 3410 Uranium U 92 1907 1248 126 186 1132 Vanadium V 23 61 835 83 132 1900 Xenon Xe 54 112 Ytterbium Yb 70 696 2486 2496 18 824 Yttrium Y 39 447 1989 120 65 1509 Zinc Zn 30 713 917 297 92 420 Zirconium Zr 40 649 1406 578 92 1852 Atomic Density Molar Thermal Youngs Melting Element Symbol number 103 kg m3 volume expansion modulus point C 108 m3 106K1 GN m2 Appendix IProperties of Elements 429 APPENDIX I cont 432 Index Carburization 186187 311 Carnot cycle 289 Cast irons 168169 251 430 Cathodic protection 327 Caustic embrittlement 328 Ccurves 213 215216 219 Cellulose 110 Cement 106 Cementite 166169 214215 219 Ceramics and glasses 3 Characteristic radiation 37 Charge coupled device 383 Chemical bonding see Bonding Chemical vapour deposition 376377 Chromel 319 Chromium 29 92 219220 318319 327328 Clay 106 Cleavage direction 106 Cleavage planes 372 379 Climb of dislocations 135136 288 Close packing geometry 8891 Coalescence 222 290 Coarsening 222 290 Coatings 326327 Coercive field 401 406 Coiled molecules 247 255 Cold work 228229 277278 285 Collector 383 Collision time 339 Columnar crystals 226 Components 149 Composite materials 244245 Concentration cell 323324 Concrete 106 Conduction band 360366 419 Conductivity electrical 74 165 194 231 339340 363 366368 thermal 74 430 Conductors 333 342 360 Configurational entropy 1315 124 247248 Constantan 343 Contact angle 210 Contact potential 380 Coordination number 85 91 97 Copolymers 112 Copper 71 74 76 164165 220221 223 269 271 273 281282 342 Coring 158 Corrosion 185186 319328 Cotton 110 Cottrell atmospheres 133 282 Coulomb attraction 66 Covalent bond 64 6870 75 Covalent crystals 8388 Cracks 219 300309 311 Creep 253 287290 307308 Creepresistant materials 289290 Critical nucleus 206 Critical resolved shear stress 265 267 Cross links 111 196 247248 Cross slip 136137 278 287 Crystal directions 3136 Crystallinity 111112 Crystallization 224226 Crystal planes 3236 Crystals geometry of 2436 orientation 372 structure determination of 4245 structure of 7 2931 83106 Crystal systems 2528 Cubic crystals 2931 85 90 100101 Cubic space lattices 2526 Curie temperature 298 421423 Czochralski method 370372 Damping capacity 251252 Dashpot model 252255 de Broglie wavelength 334335 Debye Scherrer camera 41 Decarburization 187188 311 Deformation 261286 Deformation twins 141 Degradation 107 Degree of polymerization 108 Degrees of freedom 149150 Dendritic structure 158159 Density 7576 83 86 112 120 427430 Devices 3 Dezincification 328 Diamagnetism 394395 Diamond 70 8586 271 274 360361 Diamond cubic structure 4445 8586 Dielectric constant 365 413 417418 Dielectric strength 418419 Diffractometer 42 43 Diffusion 178196 activation energy 184 193195 coefficient 179 182185 192193 interstitial 190192 250251 mechanisms of 190193 substitutional 192193 Index 433 Diffusional creep 288 Diodes 379382 Dipole moment 72 415 Dislocations 126137 267286 371 climb of 135136 288 cross slip of 136137 278 287 density of 133 277278 285 edge 126128 elastic energy of 131132 in cubic crystals 132 movement of 134137 269286 multiplication of 274275 pile up of 277 278 279 screw 129130 sources 274275 width of 270 Dispersion hardening 290 Domain structure 399401 Donor level 364365 Doping 189 364 375 Double bond 69 70 dstates 54 5960 75 397399 Ductilebrittle transition 305307 404 Ductile fracture 298300 307308 Ductility 167 Duralumin 185186 220222 326 Dynamic recovery 278 Ebonite 111 244 Eddy current loss 403405 Edge dislocation 126128 Einstein relationship 55 Elastic anisotropy 242 Elastic behaviour 239252 Elasticity 239252 255 Elastic moduli 239246 427430 Elastomers 110111 246249 Electrical conductivity 74 165 194 231 339340 363 366368 Electric breakdown 419420 Electrode potential 320321 Electromigration 377 Electron 54 Electron affinity 6162 Electron compounds 97 Electron diffraction 5 Electronegativity 62 415 Electronic polarization 414 416417 Electronic structure 7 334338 356360 Electron micrographs 5 279 284 299 Electron microscope 56 Electron probability density 5658 356357 Electron probe microanalyzer 6 Elements 427429 Elinvars 246 Elongation 164 261262 277278 285 430 Embryos 206 Emitter 383 Endurance limit 311 Energy bands 359367 Energy gap 359367 direct 369 in compounds 368 Energy product 406 Engineering materials 23 430 Engineering strain 261 Engineering stress 261 299 Enthalpy 1213 of atomization 64 67 of formation 124 125 of fusion 12 of motion 191 192 Entropy 1215 configurational 1415 124 247248 thermal 12 15 Epitaxial growth 376377 Epoxy 245 Equilibrium configurations 11 Equilibrium diagrams see Phase diagrams Error function 181182 ESD magnets 406 Etch pits 133 Ethylene 107109 Eutectic mixture 157158 160161 Eutectic phase diagram 153154 Eutectic reaction 154 156 165 Eutectic temperature 154155 Eutectoid reaction 155156 Eutectoid steel 166 214215 Exchange energy 397398 Excimer lasers 374 Exhaustion region 367368 Explosive forming 348 Extinction rules 4445 Extrinsic semiconductors 364368 Face centred cubic stacking 8990 Fatigue fracture 310311 Fatigue resistance 187 310311 FCC crystals 2930 90 276 FCC space lattice 2526 434 Index FCC stacking 8990 Fe 64 71 94 121122 268 271 273274 402404 FeFe3C phase diagram 165 FeO 123 194 Fe3O4 399 Fe phase diagram 150151 FermiDirac statistics 337 362 Fermi energy level 337338 362 379380 Ferrimagnetism 398399 Ferrite phase 166167 214215 282 Ferrites 102 399 405 406 Ferroelectric domains 422 Ferroelectric materials 420423 Ferromagnetism 395 397398 Fibre reinforcement 244245 252 Fibres 2 109110 Ficks first law 179180 Ficks second law 180182 applications based on 182189 solution to 181182 Fieldion micrograph 6122 Fieldion microscope 6 Float zone method 370 372 Forbidden gap 359361 Forward bias 380382 Fouriers law 179 Four parameter model 255 Fracture 261262 298311 brittle intergranular 307308 mechanism maps 307308 Fracture stress 302 Fracture toughness 304305 FrankRead source 274275 276 277 Free electrons 71 74 conduction by 338340 Free electron theory 334338 Free energy 1213 Frenkel defect 122123 125 193 Fullerenes 86 Gallium arsenide 368 369 383384 Galvanic cell 322324 Galvanic protection 326 Galvanic series 321 Galvanized iron 326 Garnets 402 405 Germanium 70 87 361 368 369 Gibbs free energy 13 124 Gibbs phase rule 148 161 Glass transition 226228 temperature 227228 Glide of dislocation 135 GP zones 223 Grain boundary 138 278280 sliding 288289 Grain growth 141 231 Grain size 225226 278280 286 ASTM number 279281 286 Graphite 8485 86 flakes 169 251 Gray cast iron 169 251 430 Griffith criterion 300304 Ground state 55 Growth kinetics 211212 Growth of crystals 158 204 224 370372 HallPetch equation 279 286 306 Halogens 83 Hardenability 219 Hard magnets 405406 Hardness of steel phases 218 HCP crystal 9091 92 94 275276 HCP stacking 9092 140 Heating elements 88 343 Heat of fusion 1213 224 Heat treatment 214223 225 309 Heisenberg principle 54 Heterogeneous nucleation 209211 224 Heusler alloys 398 Hexagonal close packing 9092 Hexagonal unit cell 27 29 36 High angle boundaries 139 High speed steel 219 High temperature materials 289290 Holes 363 366 Homogeneous nucleation 205209 Hookes law 239 252 HumeRotherys rules 96 152 282 286 Hunds rule 58 397 Hybridized orbitals 68 Hydrogen atom 5557 Hydrogen bond 64 72 110 244 Hydrogen electrode 320 Hydrogen molecule 68 Hysteresis loop 251 400402 Ice 73 83 Icosahedron 8687 Ideal perfect crystals 120 266268 Immobile dislocation 277 Impact test 305306 Imperfections in crystals 120142 436 Index Molecular beam epitaxy 376377 Molecular crystal 30 Monoatomic crystals 2930 Monomer 106 109 110 MOSFET 383 NaCl structure 100101 133 Natural rubber 110 Nd2Fe14B 406 Necking 261262 287 298299 NernstEinstein relation 194 Neutrons 54 Newtons law of viscous flow 252 Nichrome 319 342 243 Nickel 71 94 215 219 268 273 282 290 Nitriding 186 Nodular iron 169 Noncrystalline state 8283 226 Nonsteady state flow 180 Normalized steel 215 npn junction 382383 ntype semiconductors 365 Nuclear structure 7 Nucleation and growth 204 Nucleation kinetics 205211 Nucleus 54 in nucleation 206 Nylon 109110 228 418 430 Octahedral coordination 99 Octahedral voids 9394 Octahedron 94 Ohms law 179 339 Opaque glasses 223 Orbitals 5460 Ordered state 97 Order of reflection 39 Orlon 109 Orthorhombic cell 26 28 Overageing 222 283 286 Oxidation 316319 373 Oxygen molecule 69 Packing efficiency 86 9192 Paramagnetism 395 Partial dislocation 132 Passivation 325 327 Pauli exclusion principle 55 66 73 335 347 Paulings electronegatives 62 PbSn phase diagram 154 Pearlite 166167 214215 218 PeierlsNabarro PN stress 270271 272 273 Perfect crystal 266267 Periodic table 5560 Peritectic reaction 156 164 165 Peritectic system 155156 Peritectoid reaction 156 Permalloy 405 Permanent magnets 405406 Permeability 394 Permittivity 413 Perovskite 347 Petch equation 279 Phase 149 Phase rule 148149 Phase transformations 201231 applications 213231 free energy change in 202203 overall kinetics 211213 time scale 203 Phenolformaldehyde 244 Phonons 340 Phosphates 225 328 Photoconductor 384 Photolithography 374375 Photon detectors 384 Photoresists 374 Pibonds 69 Piezoelectric device 104 251 423 PillingBedworth ratio 316 Pinning action 230 Pipe diffusion 194 Pitting 326 Plancks constant 55 334 Plastic deformation 260286 Plasticizers 112 228 Plastics 2 107 Platinum 6 71 343 pn junction 379382 Point imperfections 120125 Poissons ratio 242 Polarization 324325 415417 422 Polyacrylonitrile 109 228 Polycrystalline 23 138 Polyester fibres 110 228 Polyethylene 108109 228 244 333 418 430 Polyisoprene 110111 228 Polymers 2 106111 228 244 245 251 307 Index 437 Polymethylmethacrylate 109 430 Polypropylene 109 228 Polystyrene 109 228 430 Polytetrafluroethylene 109 430 Polyvinylchloride 109 228 418 Porcelain 333 418 Portland cement 106 Potential energy 1011 63 239 240 266267 356 Powder method 4142 Precipitates 221223 Precipitation 220223 283284 Precipitation hardening 283284 286 Primary bonds 64 74 244 Prismatic plane 36 264 276 Properties of elements 427429 of engineering materials 430 structure correlation 7 Protons 54 pstates 5455 5760 396 ptype semiconductors 366 Pyramidal planes 264 276 Pyrex 104 Pyroceram 225226 309 Quantum numbers 5455 336 Quantum states 5455 336 Quartz 104 105 251 Radius ratio 98101 Rapid solidification processing 224 Rare earth elements 60 398 406 Rate of a reaction 1618 Rayon 110 Recovery 229 231 278 Recrystallization 138 141 229230 temperature 230 Rectifier 382 Refractive index 223 Refractory 67 170 289 308 Relaxation processes 249252 Relaxation time 249252 Residual induction 400 405406 Residual stresses 219 309 Resistivity range 333334 Resistors 343 Resolution 35 Resolved shear stress 265 Resonance 418 Resonating bond 84 Reverse bias 381383 Ring mechanism 191 Rockwell hardness 218 219 Rotating crystal method 40 Rubber 110111 333 418 430 elasticity 246249 Ruby 102 Scanning electron microscope 5 Schottky defect 122 125 193 Screw dislocations 129131 278 287 cross slip of 136137 278 287 Season cracking 328 Secondary bonds 64 7273 74 244 Seeding 210 Semiconductors 87 333 355369 379384 devices 379384 doping of 189 364365 375376 extrinsic 364368 intrinsic 361364 materials 368369 Sessile dislocation 277 SG iron 169 Shear modulus 131 242 267 Shear strain 252254 267 Shear strength 267 Shear stress 265268 Shot peening 311 Sigma bond 69 Silica 103104 105 Silicate glasses 225 226227 structural units 105 structures 104106 tetrahedron 103 Silicon 70 87 271 273 308 363366 369370 doping of 189 364 365 375 Silicon carbide 70 8788 430 Silicon nitride 289 Simple cubic crystal 30 136 Simple cubic space lattice 2526 Single crystals 23 138 224 370372 creep of 290 growth of 224 370372 semiconductor 370372 Slip 134135 263265 Slip directions 264 Slip lines 264 Slip plane 127 264 Slip systems 264 SN curves 312 438 Index Sodalime glass 104 226 334 Sodium chloride 100101 132133 244 264 271 Soft magnetic materials 402405 Solder alloy 171 325 Solidification 157158 224 Solid solutions 9697 102 281282 286 Solid state devices 87 224 355 379384 Solidus 153 154 Solute atoms 121122 281282 286 340 Solute drag effect 230 231 Solvus 154 Space lattices 2431 definition 24 sp2 bonds 84 sp3 bonds 68 85 100 Spheroidal graphite iron 169 Spinels 102 405 Spin of electrons 55 395396 Splat cooling 203 224 Springdashpot models 252255 Sputtering 375 Stability 1011 Stable equilibrium 11 Stacking faults 140 Stainless steels 263 273 318319 327328 430 Standard potential 320 Steady state flow 179180 Steels phase diagram 165167 properties 430 transformations in 214220 uses of 168 Stiffness 243 Strain energy 131 208 216 Strain hardening see Work hardening Strain rate 262 252254 271272 Strain rate sensitivity 262 Strength coefficient 262 Strength of materials data 218 285 430 Stress cells 324 Stress concentration 279 303 311 Stress corrosion 328 Stress intensity factor 305 Stressstrain curve 249 250 261262 277 Structure determination of 4245 levels of 37 property relationships 78 Structures 3 Structuresensitive property 120 Styrofoam 109 Substitutional impurity 121 Substructure 56 Superalloy 290 Supercomputers 383 Superconductors 346348 Supercooling 203206 Supermalloy 405 Superplastic 262263 Surface hardening 186187 Surface energies 137138 imperfections 137141 Surfaces 137138 Susceptibility 394395 Symmetry 28 Talc 106 Tannins 328 Teflon 109 Tempering 219 309 323 Tensile strength 261 430 Tensile stressstrain curve 261263 Terylene 110 Tetragonal martensite 217218 Tetragonal unit cell 26 28 Tetrahedral angle 68 85 Tetrahedral coordination 85 103 Tetrahedral void 93 Tetrahedron 68 Texture 402 Thermal conductivity 430 Thermal energy 11 18 179 208 212 272274 287 316 362 398 419 Thermal expansion 65 7576 246 427430 Thermal stresses 203 219 Thermodynamic functions 1214 Thermoelectric potential 343 Thermometers 228 343 Thermoplasts 107 Thermosets 107 Thoria 290 343 Tie line 152153 159 161162 Tilt boundary 139140 229 Titanium 76 94 290 TiB2 244 Tool steels 219 Transistor 379 382383 Transition metals 59 7576 271 274 397398 Index 439 Transition temperature ductilebrittle 305 404 glass 227 superconducting 344 Translucent glass 223 Tremolite 105 Trichlorosilane 370 Triple bond 70 Triple points 151 True strain 261 True stress 261 299 TTT diagrams 213214 216 Tungsten 71 289 343 Tungsten carbide 219 Turbine blades 289 290 t vector 128131 Twin boundary 140141 278279 Twinning 263264 Twins 141 Twist boundary 140 229 Ultimate tensile strength 261 Unit cells 7 2429 85 8991 100101 Unstable equilibrium 11 267 Vacancy 121 enthalpy of formation of 124 Vacancy diffusion 190 Valence band 360366 van der Waals bond 64 73 244 Vibrational frequency 15 18 191193 207 Viscoelastic behaviour 252255 Viscosity 226227 252 289 Viscous flow 252255 289 VLSI 369 384 Voids in close packing 9394 VoigtKelvin element 254255 Von Mises criterion 264 Vortex region 346 Vulcanization 111 Wafer 372373 376 378379 Water molecule 68 72 83 Wave form 335337 356357 Wave number 334335 356360 Whiskers 269 White cast iron 168 430 White radiation 37 Wood 110 Work hardening 261 275277 285 Xrays diffraction by 3741 263 wavelengths of 3738 YBa2Cu3O7x 347348 Yield point 261 282283 Yield strength 261 285 306 Youngs modulus 240243 300302 427430 Yttriumirongarnet 405 Zinc 71 94 164 268 281282 ZnO 87 123 194 ZnS 87 100 Zone refining 170 372