Physics and Technology for Engineers: Understanding Materials and Sustainability 3031320832, 9783031320835

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Physics and Technology for Engineers

R. Prasad

Physics and Technology for Engineers Understanding Materials and Sustainability

R. Prasad Aligarh Muslim University Aligarh, U.P, India

ISBN 978-3-031-32083-5 ISBN 978-3-031-32084-2 (eBook) https://doi.org/10.1007/978-3-031-32084-2 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Dedicated to my wife Sushma Mathur (12 Nov, 1950–24 May, 2023)

(Beautiful lady fought bravely with cancer for three years)

Preface

The idea of writing this manuscript originated from my interaction with B.Tech. students who often complaint of not having a book that covers both the fundamentals of physics and modern technologies. The need of such a book was also felt by the group of eminent faculty with whom I was involved in setting papers for competitive examinations of various technical boards/institutes. It was realised that the available books/materials on these topics are incomplete, lopsided or too detailed in some aspects but lacking in others. Moreover, in most of the available books, modern topics like sustainability and sustainable energy sources are not even touched. With the view of providing a balanced description of physics of engineering materials and modern technologies and with the aim of making readers aware of their moral responsibility towards sustainable development, the present text is developed in textbook format. The book contains around 220 illustrative figures and some 35 tables. The book is divided into ten chapters. Chapter 1 starts with the classification of engineering materials and their important properties. In order to link specific material properties and their dependence on atomic structure of constituent atoms/ molecules, details of atomic structure and of atomic/ionic/molecular bonding are provided in this chapter. Chapter 2 discusses electrical behaviour of solids including superconductors and associated physics. Chapter 3 details the origin and behaviour of magnetic materials and types of magnetism along with the fabrication of materials with desired magnetic properties. Important topics of modern physics, like discovery and properties of X-rays, dual nature of matter and instances of the failure of classical physics based on Newtonian Mechanics and Maxwell’s theory of Electromagnetic radiations, and their successful resolution in the frame work of quantum approach are discussed in Chap. 4. Basics of quantum mechanics, particularly of Schrodinger approach is provided in Chap. 5. Some simple one-dimensional problems of potential wells and of barrier transmission are discussed in this chapter. Since most microand macrosystems consist assemblies of large number of identical particles, their behaviour is generally predicted by the laws of statistics. Since microsystems obey quantum mechanics and have discrete energy levels, the appropriate statistics that may be applied to these systems is quantum statistics. Laws of quantum statistics, macro- and microstates, a prior equal probability of all microstates associated with a vii

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given macrostate, etc. are discussed in Chap. 6. Technique of optical fiber communication is discussed in Chap. 7, while details of laser technology and its applications are detailed in Chap. 8. Chapter 9 gives detailed description of nanomaterials, their advantages, reasons behind their special properties, fabrication techniques for nanostructures, membranes, sheets, tubes, etc. Chapter 10 is special as it discusses details of the concept of sustainability, as applied to different fields like the social, economic, environmental and sustainable energy sources. Methods conducive to a sustainable development that may be adopted by individuals, by socio-economic groups, cluster of groups, etc. are discussed in this chapter. It is expected that after going through this chapter, a reader will become aware of his/her social responsibility towards sustainable development and engineers in particular will participate in generating sustainable energy sources so that the benefits of natural resources are left largely undiminished for the future generations. Chapters of the book have the following special features: 1. The objective of the chapter is spelled out at the very beginning. 2. Sufficient number of self-assessment questions (SAQ), probing the understanding of the reader, is uniformly distributed over the text of each chapter. A serious reader is expected to satisfy himself/herself by answering these questions before proceeding further. 3. Solved examples are included, wherever required, to illustrate the technique of problem solving. 4. Problems with answers are provided at the end of each chapter. 5. Large number of short answer questions are included at the end of each chapter. 6. Since most of competitive examinations are based on multiple choice questions, sufficient number of multiple choice questions (MCQ) with answers is provided at the end of each chapter. A special feature of these MCQ is that in some cases more than one alternative may be correct, and therefore all correct alternatives must be marked for complete answer of the question. 7. Some long answer questions are also provided at the end of each chapter. 8. Each topic of the text is started from the very basics and is developed to the desired level; therefore, no other book or material is required for reading this text. It is expected that the book will prove useful for readers. I shall very much appreciate receiving feedback from readers on the following e-mail address: [email protected] Kellyville, NSW, Australia

R. Prasad

Acknowledgements

Present book, like my earlier publications, is the result of encouragement and support extended by my students, members of my research group and colleagues. I would like to express my gratitude to all my students, members of my research group and, in particular, Prof. B. P. Singh (Former Chairman, Department of Physics) who showed great enthusiasm for the book project. While he provided some initial material, his involvement in developing the text was limited due to other academic commitments. Nevertheless, I would like to record my sincere appreciation to Prof. Singh’s unwavering support, encouragement and the valuable discussions we frequently had. Support from Prof. Manoj K. Sharma, Prof. Sunita Gupta, Prof. M. M. Musthafa (Exchairman Department of Physics, Calicut University), Dr. Pushpendra P. Singh, Dr. D. P. Singh, Dr. Abhishek Yadav, Dr. Unnati, Dr. Mohd. Shuaib and Dr. M. Shariq Asnain is thankfully acknowledged. I wish to thank the Aligarh Muslim University, Aligarh, India, and my colleagues at the Physics Department and at the Department of Applied Physics, Z. H. College of Engineering and Technology, AMU, Aligarh, with whom I passed more than forty years of my active life. Last but not the least, I wish to thank all the members of my family, in particular my wife Sushma who in spite of being seriously ill, extended all possible support and encouragement for the completion of the project. Kellyville, Australia

R. Prasad

ix

Contents

1

Engineering Materials, Atomic Structure and Bounding . . . . . . . . . . 1.1 Classification of Condensed Matter . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Ceramics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.4 Composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Atomic Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Elements of Atomic Structure . . . . . . . . . . . . . . . . . . . . 1.2.2 Arrangement of Electrons in Atom . . . . . . . . . . . . . . . . 1.2.3 Shape and Orientation of Orbitals . . . . . . . . . . . . . . . . . 1.2.4 Electron Energy Level Diagram . . . . . . . . . . . . . . . . . . . 1.2.5 Electron Configuration of Elements . . . . . . . . . . . . . . . 1.2.6 Aufbau or Building-Up Principle . . . . . . . . . . . . . . . . . 1.2.7 Representing Electron Configuration . . . . . . . . . . . . . . 1.2.8 Valence Shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.9 Some Anomalous Electron Configurations . . . . . . . . . 1.3 Bonds Between Atoms and Ions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Electronegativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 The Octet Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Classification of Bonding . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 17 19 24 29 29 30 34 36 37 37 39 40 42 43 43 46 46

2

Electrical Behaviour of Condensed Matter . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Electron Energy Band Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Insulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Intrinsic Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Covalent Band Picture of Intrinsic Semiconductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Doped or Extrinsic Semiconductors . . . . . . . . . . . . . . . 2.4.4 Doping Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61 61 64 66 68 68 76 79 81

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2.4.5 2.4.6 2.4.7

2.5 2.6

3

4

n and p Type Semiconductors . . . . . . . . . . . . . . . . . . . . Compensated Semiconductor . . . . . . . . . . . . . . . . . . . . . Degenerate and Non-degenerate Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.8 Direct and Indirect Semiconductor . . . . . . . . . . . . . . . . 2.4.9 Compound Semiconductors . . . . . . . . . . . . . . . . . . . . . . 2.4.10 Current Flow in Semiconductor . . . . . . . . . . . . . . . . . . . 2.4.11 Temperature Dependence of Semiconductor Resistivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.12 Theoretical Calculation of Carrier Concentration in a Semiconductor . . . . . . . . . . . . . . . . 2.4.13 Hall Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.14 p–n Junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.15 Some Formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Semimetals and Half Metals . . . . . . . . . . . . . . . . . . . . . . Superconductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.2 BCS Theory of Superconductivity . . . . . . . . . . . . . . . . .

Magnetic Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Electric Current and Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . 3.3 Magnetic Dipole Moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Magnetic Moment of a Charged Particle Moving in a Circular Orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Classical to Quantum Mechanics . . . . . . . . . . . . . . . . . . 3.5 Magnetic (Dipole) Moment of Electron . . . . . . . . . . . . . . . . . . . . . 3.6 Magnetic Behaviour of Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1 Magnetic Induction B and Magnetic Field H . . . . . . . 3.7 Classification of Magnetic Materials . . . . . . . . . . . . . . . . . . . . . . . 3.7.1 Diamagnetic Materials . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.2 Paramagnetic Materials . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.3 Ferromagnetic Materials . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.4 Antiferromagnetic and Ferrimagnetic Materials . . . . . 3.8 Permanent Magnetic Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . X-rays, Dual Nature of Matter, Failure of Classical Physics and Success of Quantum Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Discovery, Production and Properties of X-rays . . . . . . . . . . . . . . 4.2.1 Production of X-rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Continuous X-rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Characteristic X-rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4 Mosley’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.5 X-ray Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85 90 90 90 91 92 95 96 99 101 108 113 114 116 117 128 137 137 139 141 143 145 146 150 151 154 154 158 164 176 182 191 191 192 192 194 198 201 205

Contents

4.2.6 Some Application of X-rays . . . . . . . . . . . . . . . . . . . . . . Dual Nature of Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Davisson and Germer Experiment . . . . . . . . . . . . . . . . . Some Examples of the Failures of Classical Approach and Success of Quantum Approach . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Stability of the Atom and the Nature of Atomic Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Photoelectric Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Quantum Theory of Photoelectric Effect . . . . . . . . . . . 4.4.4 Work Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.5 Residual Atom after the Emission of Photoelectron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Blackbody Radiations and Their Energy Distribution . . . . . . . . . 4.5.1 Wien’s Displacement Law . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Failure of Wien’s Distribution Law . . . . . . . . . . . . . . . . 4.5.3 Rayleigh and Jean’s Distribution Law . . . . . . . . . . . . . 4.5.4 Failure of Rayleigh–Jeans Distribution . . . . . . . . . . . . . Quantum Theory of Blackbody Radiations . . . . . . . . . . . . . . . . . . Compton Scattering of Gamma Rays . . . . . . . . . . . . . . . . . . . . . . . 4.7.1 Compton Wavelength . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.2 Compton Scattering by the Whole Atom . . . . . . . . . . . 4.7.3 Photon Interactions with Matter . . . . . . . . . . . . . . . . . . . 4.7.4 Some Applications of Compton Scattering . . . . . . . . . Specific Heat of Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.1 Dulong–Petit Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.2 Obtaining Dulong–Petit Law on the Basis of Classical Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.3 Problems with Dulong–Petit Law . . . . . . . . . . . . . . . . . Quantum Approach to Atomic Specific Heat of Solids . . . . . . . . 4.9.1 Einstein’s Theory for Specific Heat of Solids . . . . . . . 4.9.2 Investigating the Temperature Dependence of Einstein’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9.3 Drawbacks of Einstein’s Model . . . . . . . . . . . . . . . . . . . 4.9.4 Debye Theory of Atomic Specific Heat . . . . . . . . . . . . 4.9.5 Debye Temperature θD . . . . . . . . . . . . . . . . . . . . . . . . . .

208 210 212

Introduction to Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Postulates of Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 What Does Wavefunction Represent? . . . . . . . . . . . . . . 5.2.2 Properties of the Acceptable Wavefunction . . . . . . . . . 5.3 Observables and Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Time Evolution of a Quantum Mechanical System . . . . . . . . . . . 5.4.1 Schrodinger Time-Dependent Equation . . . . . . . . . . . . 5.4.2 Some Properties of Schrodinger Equation . . . . . . . . . .

267 267 268 269 270 270 271 271 272

4.3 4.4

4.5

4.6 4.7

4.8

4.9

5

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222 222 224 231 234 234 235 237 238 239 240 240 242 245 245 246 247 247 247 248 249 249 250 251 252 253 257

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5.5 5.6

Time-Independent Schrodinger Equation . . . . . . . . . . . . . . . . . . . About Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.1 Null Operator (O) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.2 Unity or Identity Operator ( Iˆ) . . . . . . . . . . . . . . . . . . . . 5.6.3 Linear Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.4 Hermitian Conjugate and Hermitian Operator . . . . . . . 5.6.5 Anti-hermitian Operator . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.6 Inverse Operator ( Aˆ −1 ) . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.7 Unitary Operator (Uˆ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.8 Some Properties of Hermitian Operators . . . . . . . . . . . 5.6.9 Algebra of Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.10 Operators for Some Dynamical Variables . . . . . . . . . . Measurement of a Dynamical Variable in Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.1 Expectation Value of a Dynamic Variable . . . . . . . . . . Some One-Dimensional Problems . . . . . . . . . . . . . . . . . . . . . . . . . 5.8.1 Energy States: Bound and Scattering States . . . . . . . . . 5.8.2 Quantum Mechanical Description of a Free Particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8.3 Particle in a One-Dimensional Asymmetric Infinite Potential Well . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8.4 Potential Barrier and Tunnelling . . . . . . . . . . . . . . . . . . Heisenberg Uncertainty Principle . . . . . . . . . . . . . . . . . . . . . . . . . . Correspondence Principle and Ehrenfest’s Theorem . . . . . . . . . .

273 274 275 275 275 275 276 276 276 276 277 279

Quantum Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Application of Quantum Statistics (Statistical Mechanics) to an Assembly of Non-interacting Particles . . . . . . . . . . . . . . . . . 6.3 Energy Levels, Energy States, Degeneracy and Occupation Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Distinguishable and Indistinguishable Particles . . . . . 6.3.2 Macrostate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Microstates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.4 Time Evolution of an Assembly . . . . . . . . . . . . . . . . . . 6.3.5 Postulate of Equal a Prior Probability of All Microstates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Quantum Statistical Probability of a Macrostate . . . . . . . . . . . . . 6.4.1 System Properties and Average Occupation Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 The Bose–Einstein Statistical Distribution . . . . . . . . . . . . . . . . . . 6.6 The Fermi–Dirac Statistical Distribution . . . . . . . . . . . . . . . . . . . . 6.7 The Maxwell–Boltzmann Statistical Distribution . . . . . . . . . . . . . 6.8 Relation Between Entropy and Thermodynamic Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

317 317

5.7

5.8

5.9 5.10 6

283 286 289 289 291 294 300 308 309

318 319 322 323 324 325 327 327 328 328 332 334 336

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The Distribution Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338

7

Optical Fiber Communication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Advantages of Optical Fiber Communication . . . . . . . . . . . . . . . . 7.3 Basics of Optical Fiber Communication . . . . . . . . . . . . . . . . . . . . 7.3.1 Optical Fiber Materials . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Frequently Used Wavelengths in Optical Transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.3 Principle of Total Internal Reflection . . . . . . . . . . . . . . 7.3.4 Types of Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.5 Rays Guided Through Fiber . . . . . . . . . . . . . . . . . . . . . . 7.3.6 Meridional and Skewed Rays . . . . . . . . . . . . . . . . . . . . . 7.3.7 Acceptance Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.8 Numerical Aperture (NA) . . . . . . . . . . . . . . . . . . . . . . . . 7.3.9 The V Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.10 Attenuation and Dispersion of Optical Signal . . . . . . . 7.4 Components of Optical Fiber Network Link . . . . . . . . . . . . . . . . . 7.5 Applications of Optical Fiber Transmission . . . . . . . . . . . . . . . . .

347 347 348 349 350

8

Laser Technology and Its Applications . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Electromagnetic Radiations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Interaction of Electromagnetic Radiation with Matter . . . . . . . . . 8.4 Einstein Prediction of Stimulated Emission . . . . . . . . . . . . . . . . . 8.5 Stimulated (or Induced) Emission of Photons . . . . . . . . . . . . . . . . 8.5.1 Population Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.2 Essential Requirements for Laser Action . . . . . . . . . . . 8.5.3 Pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.4 Three and Four Level Lasing Schemes . . . . . . . . . . . . . 8.5.5 Optical Resonator or Laser Cavity . . . . . . . . . . . . . . . . 8.6 Special Characteristics of Laser Light . . . . . . . . . . . . . . . . . . . . . . 8.7 Classification of Laser Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7.1 Solid State Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7.2 Dye (Liquid) Laser Source . . . . . . . . . . . . . . . . . . . . . . . 8.7.3 Gas Laser Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7.4 Excimer Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7.5 Mode Locking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7.6 Q-Switching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8 Some Applications of Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

377 377 377 379 383 386 388 389 390 391 393 399 405 406 415 417 422 422 423 425

9

Nanomaterials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Special Features of Nanomaterials . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Technology Used for the Study of Nanostructures . . . . . . . . . . . . 9.4 Techniques of Producing Nanostructures . . . . . . . . . . . . . . . . . . .

435 435 437 446 452

351 351 353 356 356 357 360 362 363 369 373

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Contents

9.4.1 9.4.2 9.4.3

Bottom-Up Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . 453 Top-Down Techniques of Fabricating Nanostructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456 Carbon Nanotubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462

10 Sustainability and Sustainable Energy Options . . . . . . . . . . . . . . . . . . . 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Social Sustainability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Economical Sustainability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Environmental Sustainability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.1 Atmosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.2 Mechanism of Trapping Heat by Greenhouse Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.3 Global Greenhouse Gas Emission by Human Activities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Global Warming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5.1 The Carbon Footprint . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5.2 Reducing and Offsetting Carbon Footprints . . . . . . . . . 10.6 Projections on Average Temperature Rise of 1.5 °C Above Pre-industrial Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.7 United Nation’s Efforts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.7.1 Outlook Scenarios: Computer Model-Based Scenarios Prepared by IEA . . . . . . . . . . . . . . . . . . . . . . . 10.8 Sustainability of Land Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.9 Sustainability of Water Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.9.1 Sustainability of River and Other Water Systems . . . . 10.10 Some Efforts for Improving the Sustainability of Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.10.1 A Unique Fight Against Climate Change; the Ice Stupa or Artificial Glacier . . . . . . . . . . . . . . . . . 10.11 Sustainable Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.11.1 Units of Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.11.2 Primary Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.11.3 Global Energy Production, an Overview . . . . . . . . . . . 10.11.4 Electricity: The Most Convenient Form of Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.11.5 Cost of Electricity by Source: Cost Metrics . . . . . . . . . 10.11.6 Energy Densities Associated with Prevalent Energy Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.11.7 Problem with Present Energy Mix . . . . . . . . . . . . . . . . . 10.12 Some Clean and Sustainable Sources . . . . . . . . . . . . . . . . . . . . . . . 10.12.1 Hydrogen as an Alternative Source of Energy . . . . . . . 10.13 Hydrogen Fuel Cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.14 Nuclear Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.14.1 Drawbacks of Fission Reactor . . . . . . . . . . . . . . . . . . . .

473 473 474 475 476 477 480 481 483 485 485 486 487 488 489 490 491 492 494 495 495 495 496 497 499 499 501 501 502 503 506 510

Contents

10.15

10.16 10.17

10.18

10.19

xvii

10.14.2 Plus Points of Fission Reactor . . . . . . . . . . . . . . . . . . . . 10.14.3 Accelerator-Driven Energy Amplifier . . . . . . . . . . . . . . Terrain Dependent Renewable Energy Sources . . . . . . . . . . . . . . 10.15.1 Geothermal Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.15.2 Hydroelectric Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . Wind Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solar Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.17.1 Solar Thermal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.17.2 Solar Photovoltaic (PV) Technology . . . . . . . . . . . . . . . Energy from Ocean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.18.1 Tidal Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.18.2 Ocean Thermal Energy . . . . . . . . . . . . . . . . . . . . . . . . . . Portable Sources of Sustainable Energy . . . . . . . . . . . . . . . . . . . . 10.19.1 Lithium-Ion Battery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.19.2 Super Capacitor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

511 512 514 515 516 518 519 519 521 523 523 524 525 526 528

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533

Chapter 1

Engineering Materials, Atomic Structure and Bounding

Objective Classification of condensed matter and its correlation with atomic structure and chemical bonding are important from the view point of engineers. These topics are discussed in this chapter in sufficient details. It is expected that after going through the chapter, the reader will be able to identify the special properties of different engineering materials and will also be able to correlate these special characteristics of different materials with their atomic structure, electron configuration and atomic/ molecular bonding. This will go a long way in selecting a proper material for specific engineering requirements as well as in fabricating materials with desired properties.

1.1 Classification of Condensed Matter It is known that matter, on the basis of their physical state, may be classified as solids, liquids, gases and plasma. The first three states are quite well known; however, the fourth state, plasma, is rather peculiar. At very high temperature atoms of the matter get ionised forming plasma that contains ionised atoms and electrons in a state of rapid motion. The flame of a burning candle is a typical example of plasma. In material science solids are defined as the matter having the property of crystallinity. Crystallinity means molecules, atoms or ions of the matter spaced at regular, repeating distances and angles from each other in three dimensions. In condensed matter atoms or molecules or ions almost touch each other, like that in solids; liquids also show properties of condense matter; however, most of the time liquids do not have crystallinity: they are amorphous. Some liquids called liquid crystal, as exception, do exhibit some regularity of structure over comparatively large distances, but they do not possess this regularity in three dimensions. In supercritical states of matter, at very high temperature and pressure, matter is essentially in gaseous state but relative separation between constituent atoms, etc. is of the order of that in solids © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 R. Prasad, Physics and Technology for Engineers, https://doi.org/10.1007/978-3-031-32084-2_1

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1 Engineering Materials, Atomic Structure and Bounding

and hence fall in the category of condensed matter. Gases are characterised by atoms/ molecules separated from each other by large distances. There are some interesting consequences of the above-mentioned classification: glass and several types of plastics like polyvinylchloride (PVC), for example, are defined as rigid, supercooled liquids. Most materials of engineering interest are either solids or rigid supercooled liquids. Material science also classifies matter in four broad classes: metals, ceramics, polymers and composites. Metals are characterised by their lustre, good conductors of heat and electricity and to some extent by their property of ductility. A ceramic is a material that is neither metal nor organic. It may be crystalline or glassy (rigid supercooled liquid) or both. Ceramic pottery is quite well known but clay, bricks, tiles, glass, concrete and cement are some other examples of ceramics. Ceramics, depending on their composition may be semiconducting, superconducting, ferroelectric or insulator; hence they are finding ever-increasing applications in solid state electronics, fiber optics, artificial joints, space shuttle tiles, micropositioners, chemical sensors, body armours, self-lubricating bearings, etc. Polymers are mostly organic substances made of long chains of molecules. Skin, hair and wood are examples of polymers. Another class of materials is called ‘composites’ that are combinations of two or more of the above-mentioned metals, ceramics and polymers. Composites are materials designed for specific goals to achieve a combination of properties not found in any single material. Then there are advanced materials that are finding applications in highly sophisticated technical fields like, electronics, space technology, computers, etc. Advanced materials include semiconductors, nanoengineered materials and biomaterials, etc.

1.1.1 Metals All materials are made up of atoms, either of the same or of different elements. Atoms of different elements are characterised by their Atomic Number Z and Atomic Mass Number A. Russian scientist Dmitri Mendeleev developed a table, called periodic table, where elements were arranged in order of increasing atomic number from left to right and from top to bottom. Elements in periodic table are arranged in groups and rows such that elements falling in a group exhibit similar chemical behaviour. Based on the observed similarity in their chemical properties, elements have been grouped together as (i) alkali metals, (ii) alkaline earth metals, (iii) transition metals, (iv) other metals, (v) halogens, (vi) noble gases, (vii) rare earth and lanthanoid elements, (viii) non-metals and (ix) actinoid elements. These different groups of elements are shown with different colours in Fig. 1.1 that shows periodic table. Materials of metal group are composed of one or more metallic elements, like gold, titanium, nickel, copper, iron and aluminium and often also contain very few atoms of non-metals like carbon, oxygen and nitrogen. These non-metallic atoms

1.1 Classification of Condensed Matter

3

Fig. 1.1 Periodic table of elements

are either deliberately mixed in controlled amount or are present as impurity in a metallic crystal. Impurity atoms play a crucial role in altering the properties of the metallic crystal. Atoms in metals and their alloys are arranged in a very regular fashion. Metals, in general and in comparison to ceramic and polymers are dense, have higher density. Table 1.1 shows the density of some typical metals, polymers, ceramics and composite materials. Composite shown in the last column of the table is made by mixing E-glass fiber of density 2.56 × 103 kg/m3 with cast polyamide of density 1.1 × 103 kg/m3 in different ratios, and the product density varied from (1.15 to 1.73) × 103 kg/m3 . Some important characteristic properties of metals are (a) ductility, (b) malleability, (c) lustre (d) large values of electrical and thermal conductivities and (e) high melting and boiling points. (a) Ductility is a mechanical property that may be described as material’s amenability to drawing into wire. In material science, ductility is defined by the degree to which a material can sustain plastic deformation under tensile stress before fracture or failure. Ductility is an important quality from the point of view of manufacturing, defining the suitability of the material for manufacturing operations such as cold working. It also tells how far the material can absorb mechanical overload. Ductility is often measured through the percent elongation and reduction in area at fracture in a tensile test. The fracture strain

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1 Engineering Materials, Atomic Structure and Bounding

Table 1.1 Density of different materials Metal

Polymer

Name of the Density Name of the material in kg/ material m3 × 103 Osmium

22.59

Platinum

21.50

Aluminium

2.80

Brass

8.50

Copper

8.96

Germanium Indium

Ceramics Density Name of in kg/ the m3 × material 103

Composite Density Name of in kg/ the m3 × material 103

Low-density 0.92 polyethylene

Boron carbide

2.50

Cellulose diacetate

1.36

Sintered silicon nitride

3.00

Transparent acrylonitrile

1.08

Zirconias 5.50

Bark, wood

0.24

Density in kg/m3 × 103

Cast 1.15–1.73 polyamide Plus E-glass fiber

5.30 22.50

Iron

7.87

Lead

11.30

is defined as the strain at which a test specimen fractures during a uni-axial tensile test. In uni-axial tensile test a bar of the specimen is pulled axially along the length. As a result the length of the specimen increases while its area of cross section decreases. At some value of the tensile deforming force, the specimen undergoes fracture. Percentage elongation or engineering strain may be defined as; Percentage elongation =

final length − initial length × 100 initial length

(1.1)

The percentage reduction in area is given as; Percentage reduction in area Initial area of cross section − final area of cross section = × 100 Initial area of cross section

(1.2)

Materials can undergo two types of fractures under tensile stress: brittle fracture and ductile fracture. In brittle fracture the fractured ends have irregular shape. The two types of fractures are shown in Fig. 1.2. Metals under tensile stress undergo ductile fracture, as they have the property to withstand plastic deformation. Metals have the ability to absorb more energy prior to fracture.

1.1 Classification of Condensed Matter

5

It is important to note that the property of ductility depends on the temperature at which tensile stress is applied. At some temperature, it is possible that a ductile material may change from ductile to brittle, or vice versa; it is therefore important to know the value of this critical temperature. In most cases of metals it has been observed that at lower temperature they are less ductile or more brittle, while their power of absorbing more tensile stress energy increases with temperature and their ductile strength, therefore, increases with temperature. The minimum temperature at which the metal changes from brittle to ductile is known as ductile to brittle transition temperature or (DBTT). (b) Malleability is a physical property of metals that defines their ability to be hammered, pressed or rolled into thin sheets without breaking. This property may also be defined as property of metals to deform and take new shapes under compression. The malleability of a metal can be measured by how much compressive stress or pressure the metal specimen can withstand without breaking. Both the properties of malleability and ductility originate from the crystal structure of metals, which in turn depends on the type of bonding between atoms and molecules of the materials. Most of metals and their alloys have one of the three types of crystalline structures (i) body-centred cubic (bcc), (ii) facecentred cubic (fcc) or (iii) hexagonal close packed (hcp). Unit cells of these structures are shown in Fig. 1.3. On application of some stress the layer of atoms in metal slides over the layer below without damaging the crystal structure, and the specimen assumes a new shape. As shown in Fig. 1.4i, sliding is relatively easy in case of (hcp) and (bcc) crystal structures as compared to the (fcc) structure (Fig. 1.4ii). Fig. 1.2 Ductile and brittle fractures

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1 Engineering Materials, Atomic Structure and Bounding

Fig. 1.3 Unit cells of a hexagonal close packed (hcp), b face-centred cubic (FCC) and c bodycentred cubic (bcc) crystal structures Fig. 1.4 In case of metal layer of atoms contained in a plane may slide over the layer below it

Sliding of atomic layers is possible only in metals because in metals and their alloys atoms are bound through metallic bonding. In metallic bonding all valence electrons of each atom get detached from its parent individual atom forming an electron cloud around all positively charged metallic ions. Since electrons are no more attached or associated with a particular atom, these electrons are called delocalised electrons or free electrons. Since electrons in this electron cloud are not associated with a particular positive atomic ion, the binding of individual atomic ion with electron cloud is quite weak. Therefore, on application of stress, layers of atomic positive ions may move relative to the layers below or above it, without damaging crystal structure. With the change in the shape of the metal specimen, the delocalised electron cloud also assumes a new shape

1.1 Classification of Condensed Matter

7

and orientation such that the overall structure, i.e. the crystalline structure of specimen does not break. Bonding between atoms of a material depends essentially on two factors; the size of atoms and their relative separation. There are several types of bonding that are found in different materials. We shall study more details about bonding in the next section; however, for present it may suffice that in metals and their alloys a special type of bonding called metallic bonding is found. To understand the nature of metallic bonding let us start with two atoms of the same metal with atomic number Z. When these two atoms are far apart, each atom has the positive nucleus that is surrounded by a number of electrons such that the total positive charge of the nucleus is counterbalanced by the total negative charge of electrons; therefore, there must be Z electrons in each of the two atoms. These atomic electrons are distributed around the nucleus at different distances in different discrete energy states. The group of electrons farthest from the nucleus is termed as valence electrons. Since they are farthest from the nucleus, valence electrons experience a very weak force of attraction by the positively charged nucleus, in scientific language one says that valence electrons in an atom are least bound. If N is the number of valence electrons, then (Z − N) electrons will be tightly bound with the nucleus and one defines the core of the atom as the nucleus plus (Z − N) tightly bound electrons. Obviously, the net charge on atomic core will be (+ Ne), where e is the unit of charge. Atomic core, therefore, behaves as a positive ion as shown in Fig. 1.5a. For example, the number of valence electrons in aluminium atom is 3, hence each positive core or ion of Aluminium has a charge + 3e (3 × 1.6 × 10–19 C).

Fig. 1.5 a Structure of the core and valence electrons in an isolated atom b arrangement of positive ions (cores) and delocalised electrons in a metallic specimen

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1 Engineering Materials, Atomic Structure and Bounding

Next let us consider that another similar atom is brought near to the first, so close that the two atoms start feeling the presence of the coulomb fields of each other. Since the cores of two atoms are tightly bound, they will not be much affected by the presence of the other atom, but loosely bound valence electrons of both atoms will feel almost equal force of attraction by the core of its parent atom as well as the core of the other atom. Thus valence electrons will now get associated to both atomic cores. This results in delocalisation of valence electrons, which means that valence electrons (numbering 2N) are now not confined to the field of one atom but are relatively free to move from the field of one atom to the field of the other atom. As a result a bond is formed between the cores of the two atoms making the system a diatomic molecule. Valence electrons now move around the diatomic molecule in fixed orbitals. If a third atom is now brought very near to the diatomic molecule, a tri-atomic molecule is formed having 3N number of delocalised electrons circulating around three atomic cores in specified orbitals. In this way a piece of a metal may be considered as a multiatom molecule with delocalised electron cloud around it, see Fig. 1.5b. Delocalised electrons may freely move from one core to the other and then to the other, free to move within the electron cloud but they cannot leave the electron cloud. It is because if any electron tries to leave the multiatomic molecule electron cloud, the molecule develops a net positive charge and pulls the electron back. Often one uses the term ‘free electrons’ for delocalised electrons as they are not attached firmly to any atom of the specimen and are free to move from the electric field of one atom to the electric field of any other atom of the multiatom molecule, yet they are bound to the molecule. The delocalised electrons are very large in number, but they move in some well-defined orbitals. Each electron has a fixed discrete value of energy. Not more than two electrons can have same energy as per Pauli’s exclusion principle. The energy differences are very small and, therefore, electron energy levels are closed spaced as shown in Fig. 1.6. As shown in this figure, energy levels up to some energy are filled with electrons, but there are large number of empty levels. When delocalised electrons absorb energy from some external source, for example if light is made to fall on metallic surface delocalised electrons may absorb incident light photons and may shift to their next higher excited state. Similarly, if the metal specimen is heated delocalised electrons absorb energy and shift to higher excited states. Availability of large number of empty levels for electrons plays important role in metals. The crystal structure of metals consists of regular arrangement of atomic cores (or positive ions left after losing valence electrons) in two-dimensional arrays, called crystal lattice, stacked one over the other in three dimensions. A rough and enlarged version of metallic crystal structure is shown in Fig. 1.7. Two lattices marked A and B are shown in the figure with large number of delocalised electrons moving all around constituting electron cloud. Though

1.1 Classification of Condensed Matter

9

Fig. 1.6 Closely spaced energy levels of delocalised electrons

Fig. 1.7 A rough and enlarged view of metallic crystal

the lattices A and B are shown quite well apart in the figure, but in actual case successive lattices almost touch each other. In metallic crystals positive ions in a lattice are very strongly bound with each other; therefore, it requires large amount of energy to break lattice structure. The strong binding of positive ion cores in metallic crystals is provided by the cloud of delocalised electrons which serves as a glue. However, the binding is much weak between two adjacent lattices. That is why application of small stress, capable of overcoming bonding between lattices, may slide lattices one over the other, without damaging lattices. Sliding of lattices with respect to each other results in giving a new shape to the specimen. Since cloud of delocalised electrons is firmly bound with lattices, it readjusts its orientation and other parameter to give stability to the new shape. It may thus be observed that properties of ductility and malleability in metals originate from metallic bonding that is characterised by delocalised electron cloud, large number of empty electronic states and very strong binding between positive ions in crystal lattices. (c) Different types of metal strengths

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1 Engineering Materials, Atomic Structure and Bounding

Strength of a material is judged by the ability of the material to resist deformation and failure under the action of external forces. Some important and frequently used strength indexes for metals are: (i) Tensile strength It is defined as the maximum load that a metal part can support without fracture when being stretched, divided by original cross sectional area of the material. Mostly it is expressed in units of pounds per square inch (PSI) or Pascal denoted as Pa. The unit, named after Blaise Pascal, is defined as one newton per square metre. (ii) Yield strength It is defined as the maximum stress a material component can withstand without permanent deformation or the stress at the yield point at which the specimen starts plastic deformation. (iii) Compressive strength In contrast to the tensile strength, the compressive strength of a material is defined as the maximum compressive stress a solid material can sustain without fracture under gradually applied load. It is a measure of the capacity of the material to withstand loads tending to reduce its size. (iv) Impact strength It is defined as the maximum impact or suddenly applied force a specimen can take before its failure. As a matter of fact it is a measure of how much energy the specimen may absorb at the limited state. (v) Shear strength It is defined as the maximum shear load a material can withstand before failing divided by its cross sectional area. (vi) Ultimate strength It is measured as the amount of utmost tensile, compressive or shearing stress that a given unit area of the given specimen sample can bear without deformation or breaking. Table 1.2 shows the values of tensile and yield strengths for some metals. (d) Lustre Freshly cut surfaces of all metals shine when light falls on them. This property of metals is called lustre. This happens because of the delocalised electron cloud and the availability of large number of empty electron energy levels. Although the assembly of delocalised electrons is called a cloud but electrons in this cloud move in systematic way. In electron cloud there are large number of electrons, each having a discrete energy and moving in a defined orbital. When light from some external source falls on these electrons, they absorb incident light of all frequencies and get shifted to their next excited level which is empty. However, electron excited states are very short-lived; Table 1.2 Tensile and yield strength for some metals

Type of metal

Yield strength in PSI Tensile strength in PSI

Aluminium-3003

21,000

22,000

Copper

28,000

–

Stainless steel-304

40,000

90,000

Titanium

37,000

63,000

1.1 Classification of Condensed Matter

11

excited delocalised electrons almost immediately revert back to their ground states emitting the photons of nearly the same energy or frequency that they have absorbed. As a result of the emission of light photons from the de-excitation of electrons, metal surface shines. Surfaces of some metals like silver, gold, etc. remain shining all the time as these metals do not chemically react with environmental chemicals and gases. But in case of some other metals surfaces shine only when they are freshly cut, after some time the metal chemically reacts with atmospheric chemicals and a thin layer of metal oxide, etc. get deposited on the surface making it dull. (e) Electrical and thermal conductivities Metals are good conductors of electricity and heat; they have large values for electric and thermal conductivities. Let us consider a specimen of some material of length L and area of cross section A, if now a voltage V is applied across the two opposite faces of the specimen an electric current I may flow through the specimen. According to Ohm’s law, current I will be proportional to V, i.e. V ∝ I or V = R I

(1.3)

In Eq. (1.3), the proportionality constant R is called the resistance of the given piece of specimen and is a measure of the opposition that the specimen has offered to the flow of current through it. A large value of R is an indication that the specimen has offered large opposition to the flow of current, i.e. it is not a good conductor of electricity. Resistance R is measured in units of Ohm (Ω). The magnitude of R depends on (i) the material and on two physical parameters, (ii) area of cross section A of the specimen and its (iii) length L. One may therefore write, R∝

L L or R = ρ A A

(1.4)

The constant of proportionality ρ in above equation is called the resistivity or specific resistance of the material and is measured in units of (Resistancemetre) or (Ohm meter) also represented as (Ω-m). In Eq. (1.4) if one puts L = 1 m and A = 1 m2 , then R = ρ, that means that resistivity of the material is equal in magnitude to the resistance offered between the two opposite faces of a cube of side 1 m. Resistivity is a property of the material of the specimen, and it does not depend on the physical size of the specimen. The reciprocal of resistance R is called conductance and is measured in units of Siemen denoted by S. 1 S = 1 ohm1 (Ω) . In older literature ‘moh’ has been used in place of Siemen as unit of conductance. The reciprocal of resistivity, ρ1 , is called the specific conductivity or simply conductivity of the material and is denoted by σ . Conductivity is measured in units of (Ω-m)−1 which may be written as Siemen per metre represented by (S/m).

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Figure 1.8 shows the bar graph for the range of values of conductivity for different materials. The conductivity of metals and alloys has high value but varies in a narrow range. Conductivity has largest range for composites, as expected. Conductivity value at room temperature (≈ 20 ◦ C) for some important metals and alloys is given in Table 1.3. It may be observed in this table that silver that has the maximum value of conductivity is the best conductor of electricity and, therefore, it is frequently used in making electrical connections in sophisticated electronic circuits. Large number of delocalised or free electrons in metals is responsible for their high electrical and thermal conductivities. Electron, being negatively charged, experiences a force F = −eE when subjected to an electric field of strength E. The negative sign in this expression tells that the force F is in a direction opposite to the direction of electric field E. Figure 1.9 shows a rectangular metallic rod of length ‘d’ which is connected to a battery of voltage V with a switch ‘sw’. When the switch is made on, face F 1 of the rod is connected to the positive terminal of the battery and face F 2 to the negative. A current of magnitude I flows through the metallic rod such that V = R I . As already mentioned, R is a measure of the opposition that the metallic rod offers to the flow of current. In the following we shall discuss the mechanism of current flow and the origin of resistance R. Fig. 1.8 Bar graph showing range of conductivity for different materials

Table 1.3 Conductivity for some metals and alloys at 20 °C

Material

Conductivity (S/ Material m)

Conductivity (S/ m)

Silver

6.30 × 107

Manganin

2.07 × 106

Copper

5.96 ×

Constantan 2.04 × 106

Gold

4.1 × 107

Aluminium 3.5 ×

107

107

Nichrome

9.09 × 105

GaAs

5 × 10–8 to 103

1.1 Classification of Condensed Matter

13

Fig. 1.9 Current flow through a metallic specimen

The interior of the rod has a crystalline structure consisting of positively charged core of metallic atoms (indicated by + symbol) arranged in a regular fashion in three dimensions. These sheets of positive ions are called crystal lattices and they are held at their positions by the balance of attractive forces of delocalised electron cloud and repulsive forces between nearby positive ions. Almost free (delocalised) electrons, in motion, each with its inherent velocity (indicated by brown coloured arrows) surround the crystal lattices. With the application of voltage V across the two opposite faces of the rod electric field E of magnitude E = Vd , directed from face F 1 to face F 2 , gets established between the two faces F 1 and F 2 inside the rod. The electric field applies a force F = |eE| directed from F 2 to F 1 , on each of the delocalised electron and imparts an additional velocity, say vad directed opposite to the field direction, to each electron. This additional velocity component on each electron is represented by a small black arrow in Fig. 1.9. Under the action of two velocities, each delocalised electron moves in the direction of the resultant velocity. However, the additional velocity component tries to make every electron in the road to rush towards face F 1 . At first it appears that all electrons in the metal rod will reach face F 1 in no time and will accumulate there. But that is not true. These free electrons mostly moving towards face F 1 collide with crystal lattice and their direction of motion and the magnitude of velocity both get changed at such collisions. Free electron– lattice collisions are very frequent, as a result though there is a net flow of negative charge in the rod from face F 2 towards face F 1 but on average number of free electrons per unit volume of the rod remains almost constant; there is no accumulation of electrons in any part of the rod. The net flow of negative charge (from F 2 to F 1 ) establishes the current I, and conventional current is assumed to flow from F 1 to F 2 . One may ask a question why collisions between electrons are not taken into account. The answer is that collisions between electrons are quite unlikely because of their negligibly small size and very small time that electrons take in crossing each other. Such electron collisions with crystal lattice are the major cause for randomisation of electron velocities. Larger the frequency of

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1 Engineering Materials, Atomic Structure and Bounding

electron–lattice collisions, lesser will be the amount of net charge flow towards face F 1 , resulting in lower current. It may, therefore, be realised that in metals the opposition R to the flow of current originates from electron–lattice collisions. Since inherent speed of electrons increases with temperature, the electron–lattice collision frequency strongly depends on the temperature of the metallic rod; at higher temperature the resistance R of the same specimen rod will be larger. This is confirmed from the experimental observed fact that the same specimen shows a larger value of resistance R or of resistivity ρ and a lower value for conductivity σ at higher temperatures. It is, therefore, required to specify the temperature while mentioning the resistance, resistivity or conductivity of a specimen. Large number of electrons colliding with lattice impart energy and momentum to it that sets the crystal lattice in vibratory motion. This vibratory lattice motion is quantised, i.e. the lattice in vibratory motion either absorb or emit energy in packets. Energy packets corresponding to the vibratory lattice motion are called ‘phonon’. There should be no confusion between phonons and photons; photon is the energy quanta of electromagnetic field, while phonon is the quanta of lattice vibration. Heat is a form of energy, and it may transmit from one place to another by three distinct methods; (i) radiation (ii) convection and (iii) conduction. Heat transfer through radiation does not require any medium; it is directly transferred as energy quanta, photons, from the source to the receiver. Sun light reaches Earth via radiation crossing a vast region of vacuum. However, both convection and conduction require some material medium to transfer heat. In convection, medium particles take heat energy and move away to transport heat energy. This happens in boiling of water when water molecules absorb heat from the hot source at the bottom of the container and move out to transfer heat energy to the top layer. In case of conduction on the other hand, medium particles do not move, instead they transfer heat energy to the next particle, and then to the next and so on. Though heat transfer from a hot body by radiations cannot be avoided, but in solid materials heat transfer essentially takes place via conduction. Thermal conductivity of a material, generally denoted by κ, tells about the ability of the material to let heat energy pass through it via the process of conduction. A large value of κ for a given material means that the material is a good conductor of heat. Metals and their alloys are good conductors both of heat and electricity. Let us consider a rectangular sheet of a material of thickness ∆x (m) and area of cross section A (m2 ) as shown in Fig. 1.10. Let the front face of the sheet be at temperature (T + ∆T ) (Kelvin K) and the back face at temperature T (K). Front surface being at a higher temperature will conduct some heat energy ∆Q (Joule J) in time ∆t towards the back surface. Experimentally it has been found that heat transfer through conduction ∆Q from front face to the back face is proportional to the area of cross section A of the surface, time ∆t, the

1.1 Classification of Condensed Matter

15

Fig. 1.10 Heat flow across the thickness of a rectangular sheet

temperature difference ∆T and is inversely proportional to the thickness ∆x. One may, therefore write, −∆Q ∝

A . (∆t) . (∆T ) Or ∆x

− ∆Q = κ

A.(∆t).(∆T ) . ∆x

Or ∆Q A . (∆t)(∆T /∆x) × [Joule per unit area per unit time per unit temperature gradient] × [whatt/metre Kelvin = W/m K] (1.5)

κ =−

The left side of Eq. (1.5) gives the amount of heat energy lost by the front surface per unit time per unit area, and, therefore there is a negative sign, the negative sign simply shows that energy is lost by the front surface. The units of thermal conductivity κ may be given as Joule per second per unit area per unit temperature gradient or watt per meter per Kelvin, i.e. (W/m K). In metals heat conduction also takes place through the delocalised electrons which are relatively free to move and transport heat. Since the number of free electrons in metals is large, the conductivity of metals and alloys is also large. Values of thermal conductivity for some metals and other materials are shown in Table 1.4. (f) High melting point of metals Metals in general have high melting and boiling points. When a solid melts, its crystalline structure gets destroyed and it changes its phase from solid to liquid. On further heating the liquid, at some temperature called boiling point, changes into gaseous phase. A high melting point means that the bonding between constituent atoms is very strong. In case of metals crystalline structure is protected by very strong bonding of multiatomic molecules, i.e. by the large

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1 Engineering Materials, Atomic Structure and Bounding

Table 1.4 Thermal conductivities of some materials Material

Thermal conductivity (W/mK)

Material

Thermal conductivity (W/mK) silver

Silver

403

Iron

94

Copper

401

Nickel

106

Gold

327

Diamond

1000

Aluminium

237

Fiber glass

0.04

binding energy of crystal lattices. Tightly bound lattices in metallic crystals require large amount of heat energy to break them that is why the melting and boiling points of metals are high. Further, the binding energy will be larger for the lattice that contains larger number of ions per mol. Hence, heavier metals have higher melting and boiling points. Melting and boiling points of some metals are shown in Table 1.5, which clearly shows the increase in both melting and boiling points with the atomic weight of the metal. When a piece of metal is looked through a high resolution microscope it is generally observed that it does not contain a single large crystal. Instead, there is large number of small crystals of different sizes packed together with different orientations. These small crystals are called grains and the two-dimensional surfaces between adjacent grains as grain boundaries. Grains have different crystallographic orientations. Process of grain formation in a metallic specimen starts when molten metal cools and crystallisation takes place. Small crystals having different orientations start growing simultaneously at different locations in the specimen and grow till they fuse. Grain size is an important parameter for given metal. Small grain size increases tensile strength and tends to increase ductility. However, grains of small size reduce the electric and thermal conductivities. Grain boundaries may be looked as 2-D crystal defects and tend to reduce thermal and electrical conductivities. In general, grain orientations are random, but they may be aligned to some extent by repeated rolling along one direction. SAQ: On heating metals mostly become more malleable, why? SAQ: Ductility of metals that has grains of small size is more, how can this be explained? Table 1.5 Melting and boiling points of some metals Material

Melting point (°C)

Boiling point (°C)

Material

Melting point (°C)

Boiling point (°C)

Aluminium

660

2515

Iron

1150–1593

2861

Silver

961

2162

Platinum

1770

3825

Copper

1084

2562

Tungsten

3400

5550

1.1 Classification of Condensed Matter

17

SAQ: Crystal lattice consists of positively charged ions but they are very tightly bound. Which forces provide this strong binding? SAQ: Current in metals is constituted by the flow of electrons; what will happen to these electrons when a current carrying wire is cut, will electrons go out of the wire? SAQ: How can one explain the reduction of conductivity in metals that have small grains? SAQ: Why do delocalised electrons in metals remain inside the specimen when they are not attached to any specific atom?

1.1.2 Ceramics It is difficult to define ceramics in the present context as ceramics cover a very wide range of inorganic materials that may contain metallic or non-metallic chemical elements and are produced by many different physical and chemical processes. There was a time when ceramics were defined as non-organic, non-metallic materials, having very low electric conductivity and produced essentially by high-temperature treatment. Some old literature also tried to define ceramics as ‘refractory’ materials, which in technical material science language means, materials capable of withstanding every day abuses like extreme temperatures, attacks from acid and alkalis and general wear and tear. However, the old definition is no more valid, today ceramic materials having metallic ions and novel properties including semiconductor and superconductivity produced by many different techniques are being used in industrial applications. Some broad features of ceramics, as compared to polymers and metals, are shown in Table 1.6. Sometime it becomes easier to define materials in terms of their properties; their behaviour on heating, on passing current through them, or putting them in water, etc. Such a classification becomes confusing, for example graphite, an allotrope of carbon, is considered a ceramic because it is non-metallic and inorganic. However, unlike most ceramics it is soft, wears easily and is a good conductor of electricity. Diamond which is another form of carbon on the same grounds is treated as ceramic, but properties of diamond are totally opposite of graphite; it is hard, very stable and does not wear out easily. With regard to mechanical behaviour, ceramics are relatively stiff and strong; stiffness and strength are comparable to metals. In addition, ceramics Table 1.6 Broad features of Ceramics Material

Property Chemical stability

Density

Melting point (°C)

Plasticity

Ceramic

Non-reactive

Low to medium

Medium to high

Brittle

Polymer

Very reactive

Very low

Low

Ductile to brittle

Metal

Reactive

Medium to high

Low to high

Ductile

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1 Engineering Materials, Atomic Structure and Bounding

are very hard, extremely brittle and susceptible to fracture. Figure 1.11 shows the spread of tensile strengths for different materials. In spite of being very hard, ceramics may be optically transparent, translucent or opaque. Type of bonding between constituent atoms is often used to classify a material; for example, metals are characterised by metallic bonding where delocalised electrons of constituent atoms provide the ‘glue’ for strong binding. Similarly, polymers have strong covalent bonds in long molecular chains, while relatively weak van der Waals bond binds one long molecular chain with the other. On the other hand, ionic bonds are found in non-metals. In ceramics all types of bonds exist. As an example, Al2 O3 , MgO, SiO2 , etc. have ionic bonds, while SiC, BiC, BN, Si3 N, Si2 N2 O, etc. show covalent bonding. It is interesting to observe how relative content of different types of bonds changes the melting point of ceramics (see Table 1.7). Solids generally have crystalline structure with grains, i.e. they are poly crystalline; microscopically there are several crystals having different orientations, fused together. Fine-grained pure alumina and glass are polycrystalline ceramics. Ruby, diamond, etc. on the other hand are large single crystal ceramics. Ceramics have many different chemical compositions: There are simple oxides that have high melting point, like ThO2 (melting point Tm = 3300 °C), MgO (Tm: 2825 °C), UO2 (Tm: 2810 °C), etc. Ferrites that are complex oxides like Fe3 O4 , SrF12 O19 ; Titanates: BaTiO3 (Tm: 1625 °C); SrTiO3 (Tm: 2080 °C); Nitrides: Si3 N4 (Tm: 1900–2600 °C); TaN (Tm: 3080 °C); Brides: Fig. 1.11 Bar graph showing the spread of tensile strengths of different materials

Table 1.7 Effect of bond type on melting point of ceramics

Ceramic compound

Ionic bond (%)

Covalent bond (%)

Melting point (°C)

SiC

11

89

2830

Si3 N4

30

70

1900

SiO2

51

49

1715

1.1 Classification of Condensed Matter

19

HfB2 (Tm: 3350 °C); ZrB2 (Tm: 3245 °C); Silicides: Hf5 Si3 (Tm: 2600 °C); WSi2 (Tm: 2160 °C); Halides: NaCl (Tm: 800 °C); Intermetallides: HfRe2 (Tm: 3160 °C), Nb3 Sn; Metal ceramics: WC-TiC-Co; Polymer ceramic: Synthetic resins. Material scientists occasionally divide ceramics into Traditional and Advanced. Bricks, pottery, glass, porcelain, etc. are time tested daily use general purpose ceramics. Out of these pottery is generally made from traditional clay while bricks, tiles, etc. are heavy clay products. However, coarse-grained refractors fired bricks; silica bricks find special use in making high-temperature oven, etc. Cement and concrete are other traditional ceramics used in building construction. Advanced ceramics are those that have been specially engineered, mostly since the early twentieth century, for highly technical and specific applications. For example, aluminium nitride (AlN) and beryllium oxide (BeO) ceramics have been developed to serve as heat-sink for electronic elements. These ceramics are also used as substrate in electronic packaging. Similarly, SiO2 and polymer-ceramic compounds are used as thermal protection shields. High-resistivity conductors-ceramics silicon carbide (SiC), zirconium oxide (ZrO2 ), molybdenum silicate (MoSi2 ), etc. are used for making heating elements and electrodes. Ceramics having magnetic and superconducting properties have also been developed. Complex ferrite and oxides of heavy metals like, (Ba, Sr) Fe12 O19 ; Y Co5 Sm2 (Co, Fe, Cu, Zr)17 and Nd2 Fe14 B, etc. are used for making hard magnets. Magnet-doped silicon dioxide (SiO2 ) and chromium-doped complex Be3 Al2 (Si3 )6 have been used to make artificial gem stones, former as topaz and the later as emerald. Artificial diamonds are made from ceramic ZiSiO4 . Advanced ceramics are finding extensive use in nuclear technology. UO2 , UC and PuO2 are used as nuclear fuels. Ceramics BeO, BeC2 and ZrO2 have been used to moderate fast neutrons to thermal energies in nuclear reactors. Similarly, ceramics B4 C, HfO2 and Sm2 O5 are used for neutron shielding. Some ceramics are biocompatible that means they are not harmful or toxic for living tissues. Such ceramics are used for making artificial joints, prostheses, cardiac valves and other implants. One may conclude by saying that ceramics are versatile materials that have applications in almost all walks of human life.

1.1.3 Polymers Polymers are long-chain molecules of very high, running into hundreds and thousands, molecular weight. It is for this reason that they are also called ‘macromolecules’. In old literature, when polymer science was not so developed, term ‘resins’ was used for polymers. As has been mentioned, polymers are substances made up of recurring structural units, each of which is regarded as derived from a specific compound. These building blocks or units are called monomer. The physical and the chemical properties of a polymer depend very strongly on the number of

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1 Engineering Materials, Atomic Structure and Bounding

monomer units in the chain. As an example take a simple case of normal alkane hydrocarbon series; The bond structures of first three members of the series are shown in Fig. 1.12a, the monomer of the series is given in figure (b), while the series formula is shown in part (c) of the figure. In series formula ‘n’ gives the number of monomers in that particular member. It is interesting to note that the physical and chemical properties of different members of the series change with the number ‘n’ of monomers in the chain. First four members of the series are gases. The fifth member ‘n-Pentane’ is a low viscosity fluid with boiling point of ≈ 36 °C. With the increase of the number of monomers in the chain, the viscosity and boiling points of the member increase. Some characteristics of series members and the value of ‘n’ for them are tabulated in Table 1.8. The boiling point of successive members of the alkane chain also increases with the number of monomers, but the rate of increase slows down such that the boiling point for the massive members of the series saturates at about 145 °C. Long-chain alkanes having 103 to 3 × 103 carbon atoms are known as polyethylene. A big difference between wax and polyethylene lies in their mechanical behaviour. While polyethylene is a tough plastic, wax is a brittle crystalline solid. Fig. 1.12 a Bond structure of first three members of alkane series, b monomer of the series, c general representation for the series

Table 1.8 Variation in some properties of alkane series with number of monomers

Number of monomers in the chain

Physical properties

1–4

Gases

5–8

Low-viscosity liquids

9–16

Medium-viscosity liquids

17–25

High-viscosity fluid

26–50

Crystalline solid, wax

50–1000

Semicrystalline solids

1000–5000

Tough plastic solid, polyethylene

~ 105

Fibers

1.1 Classification of Condensed Matter

21

The difference in mechanical properties arises of their structures. Figure 1.13 shows that in case of wax linear carbon chains of up to 50 carbons are linked with each other by van der Waals forces that are quite weak. Hence, wax is brittle. On the other hand in polyethylene monomer chains are quite big and are folded, as shown in Fig. 1.14. Further, several chains are linked together through entanglements which are difficult to break. The process of formation of a polymer from its monomer is called polymerisation. The number of monomers in a given polymer chain ‘n’ is called the degree of polymerisation (DP). The functionality of a monomer is the number of sites it has for bonding to other monomers under the given conditions of polymerisation reaction. While the exact molecular weight required for a substance to be called a polymer is a subject of continued debate, often polymer scientists put the number at about 25,000 g/mol. Polymers may be classified in many different ways; for example, Fig. 1.13 Structure of wax

Fig. 1.14 Structure of polyethylene

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1 Engineering Materials, Atomic Structure and Bounding

classification based on (i) molecular forces, (ii) heat treatment, (iii) source, (iv) structure and (v) mode of polymerisation, etc. (i) Classification based on molecular forces Two types of bonds are frequent in polymers; the hydrogen bond and (b) van der Waals bond. These bonds bind chains and monomers in a chain with each other. Accordingly, polymers may be classified as: (a) Elastomers Rubber—like solids fall in this category. In these polymers chains are coupled with each other by the weakest intermolecular force which permits the polymer to be stretched. However, there are some crosslinks between the chains that bring back the polymer to its original shape when deforming force is withdrawn. Examples are neoprene, buna-N, etc. (b) Fibers Fibers are those polymers which may be drawn into long filaments with lengths at least 100 times of their radii. This happens because of the strong bonds between chains, usually hydrogen bonds. As a result of strong intermolecular force, these materials are closely packed and have crystalline structure. Examples are polyesters (terylene), polyamides (nylon), etc. (c) Resins They are liquid polymers that are used as adhesives, sealants, etc. Examples are epoxy adhesives and polysulphide sealants. (ii) Classification based on heat treatment A polymer that may be given different shapes to make tough and hard utility articles by heating and/or by applying pressure is called plastic. Plastics may be further classified as: (d) thermoplastic (e) thermosetting plastic. (d) Thermoplastic polymers Some polymers become soft on heating and can be given any desired shape. However, on cooling they again become hard and tough. The process of heating, reshaping and becoming tough and hard on cooling can be repeated several times. The intermolecular forces in these plastics are stronger than that in elastomers and weaker than those in fibers. Sealing wax, nylon, PVC, etc. are some examples. (e) Thermosetting polymers Those plastic polymers that undergo some chemical changes on heating and become infusible mass which cannot be given any shape are called thermosetting plastic polymers. In such polymers, heating creates large number of new cross-linking bonds that convert it into an infusible mass. Bakelite is an example of such thermosetting polymer. (iii) Classification based on source Based on the source of the polymer there may be three classes: (f) Natural Polymers Polymers that are found in nature, in plants and animals are called natural polymers. Proteins, cellulose, barks, starch, and rubber, etc. are some examples.

1.1 Classification of Condensed Matter

23

(g) Semisynthetic polymers Derivative cellulose that is obtained by modifying it is called semisynthetic polymer. Cellulose acetate also called rayon and cellulose nitrate are examples of semisynthetic polymers. (h) Synthetic polymers Man-made polymers, like bakelite, polythene, synthetic rubber, etc. are examples of synthetic polymers. (iv) Classification based on structure On the basis of their structure polymers may be called: (i) Linear polymers These polymers contain long and straight chains of monomers with very little or no cross-linking of chains. Examples are high-density polythene, PVC, etc. (ii) Branched polymers These polymers have linear chains with few branches; examples are low-density polythene. (iii) Cross-linked polymers These polymers are made up of bi-functional and tri-functional monomers and, therefore, have strong covalent bonding between linear monomer chains. Cross-linked polymers are usually hard, do not melt or soften on heating and mostly do not dissolve. Examples are vulcanised rubber, formaldehyde resins, etc. (v) Classification based on mode of polymerisation Polymers may also be classified on their mode of polymerisation into two classes; (l) Addition polymers This type of polymers is produced by repeated addition of monomer molecules that have double or triple bonds. Addition polymers formed by the polymerisation of a single monomeric species are called homopolymer; polythene is an example of the same. nCH2 = CH2 →→ Polymerization →→ −(CH2 − CH2 )n − Ethene

Polythene (Homopolymer)

Polymers made by addition polymerisation of two different monomers are called copolymers; examples are Buna-S, Buna-N, etc. Figure 1.15 shows the addition polymerisation of two different monomers butadiene and styrene into Buna-s copolymer. (m) Condensation polymer When two different bi-functional or trifunctional monomers undergo repeated condensation reactions, they form a condensation polymer. In this type of polymerisation small molecules like, water, HCl, alcohol, etc. are usually eliminated. Examples are nylon 6.6, etc. If one of the two monomers is tri-functional or there are three different bi-functional monomers and they undergo condensation polymerisation, the resultant polymer has linkage sequences in two or three dimensions; they are called cross-linked polymer.

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1 Engineering Materials, Atomic Structure and Bounding

Fig. 1.15 An example of copolymerisation

SAQ: Name three types of bonds that are found in polymers. SAQ: What is the class of the polymer that converts into an infusible mass on heating? What is the reason for this change? SAQ: What is meant by the functionality of a monomer? SAQ: What do you understand by cross-linking in polymers? How does it happen? SAQ: What must be the properties of monomers that on polymerisation produce linked polymers?

1.1.4 Composites Modern technologies often require materials with very special properties which are not available in metals, metal alloys, ceramics and polymers. For example, aerospace scientists are always in lookout of materials which have very low density, very strong, highly resistant to abrasion and impact yet quite stiff. This amounts to asking for two apparently opposite characteristics in the same material, because strong and stiff materials are generally dense. Further, increasing the strength or stiffness, in general, decreases the impact strength. An answer to such problems comes from composites; these are materials produced by the combination of two or more of the three materials, namely metals, ceramics and polymers. A composite may be defined as a combination of two or more materials (often called phases) at a microscopic scale and have chemically distinct phases that results in better properties than those of the individual components used alone. Though heterogeneous at a microscopic scale, a composite is statistically homogeneous at macroscopic scale. In general, out of different phases in a composite, one particular material has volume wise larger concentration than others. This component with largest concentration (or bulk material) is called the ‘matrix’. The other material which is in relatively smaller amount is termed as the ‘reinforcement’. Reinforcements are primarily added to increase the mechanical strength, toughness and stiffness of the material.

1.1 Classification of Condensed Matter

25

The manmade composites may be divided into three categories, (a) polymer matrix composites (PMC), (b) metal matrix composites (MMC) and (c) ceramic matrix composites (CMC). (a) Polymer matrix composites Some polymers, in particular epoxies and polyesters, have a notable property that they may be easily moulded into desired complex shapes. But their drawback is that they do not possess high mechanical strength as metals. On the other hand materials like glass, boron and aramid have extremely high tensile and compressive strengths which are, however, not readily apparent in their solid forms. This happens because when stressed, randomly distributed surface ‘flaws’ (abnormalities due to impurity, etc.) in these materials make the solid crack or break much below its theoretical ‘breaking point’ stress. To overcome this problem, fibers of these materials (boron, glass and aramid) are drawn. The advantage in fibers is that though the random distribution of faults will still be same but only few fibers will be affected by these faults and a very large number of fibers will have no fault in them and will break at their theoretical break point. Therefore, a bundle of fibers will reflect more accurately the optimum performance of the material. It may, however, be realised that a bundle of fibers will show its tensile strength only in the direction of its length, just like in a rope. When these fibers are mixed as reinforcement with a polymer matrix, like that of polyester, the resulting composite shows exceptional mechanical strength comparable or even more than that of metals. When a stress is applied to the composite, the matrix material, polyester in this case, spreads the stress to fibers. Further the bulk matrix protects fibers from atmospheric wear and tear, abrasion and impact. High strengths and stiffness, ease of moulding into complex shapes makes the composite superior to metals in many ways. The overall strength of the composite depends on following factors: (i) (ii) (iii) (iv)

Properties of the matrix polymer Properties of the reinforcing fiber The ratio of the fiber to the polymer, called fiber volume fraction (FVF) Geometry and orientation of fibers in the matrix.

It is obvious that the mechanical strength of the composite will increase with the increase of FVF, but there are limits to which this ratio may be increased. Firstly, it is essential that all fibers must be fully rapped with polyester matrix from all sides so that they are not exposed. Further, the manufacturing process that involves mixing of reinforcement with matrix often produces faults and air-inclusion, which may become cause of breakdown. In case of ordinary applications, like boat-building industry FVF of 30–40% is quite enough. However, in more sophisticate applications like aero-industry FVF of around 70% have been obtained by advanced manufacturing methods. The orientation of the fiber in the composite is also important because the maximum tensile strength of the fiber is along its length and the tensile strength in direction normal to the length is negligible. The composite is, therefore, anisotropic which is in contrast to metals and alloys which are largely isotropic.

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1 Engineering Materials, Atomic Structure and Bounding

It is, therefore, very important when considering the use of the composite at the design stage to know the magnitude and direction of the load in the finished structure. If properly taken into account, the property of anisotropy of composites may be used to advantage, as composite material may be used only where there are locked stresses. There are four main types of direct loads that a composite may have to bear in a structure. They are; (i) Tension Load Fig. 1.16a shows the situation when tensile force is applied to a composite. Response of the composite to tensile load very much depends on the tensile strength of fiber reinforcement mixed with the polymer, since it is much higher than that of the matrix material. (ii) Compressive Load Application of compressive load to a composite is shown in Fig. 1.16b. In this case the adhesive and stiffness properties of the matrix polymer are very important as they have to maintain the fiber straight and to prevent them from buckling. (iii) Shear Load As shown in Fig. 1.16c a shear load attempts to slide adjacent layers of the reinforcement fiber over each other. In this case also the properties of the matrix polymer plays a crucial role, it should not only have good mechanical strength but should also have very good adhesive force with fiber so that it remains firmly attached with it. (iv) Flexure Load As shown in Fig. 1.16d, flexure load is a combination of tensile, compression and shear loads. Therefore, both the matrix and the reinforcement fiber must possess good adhesive and mechanical strengths. (b) Metal matrix composites Metal matrix composites have found usage in our lives from olden times. Metals like cast iron with graphite, steel with high carbide contents are all examples of metal matrix composites. Artefacts made of metal matrix composites as swords, body armours, chains, etc. are all found in excavation of old habitation sites.

Fig. 1.16 Four types of loads that may be applied to a composite

1.1 Classification of Condensed Matter

27

There are many ways to classify metal matrix composites. One very often used classification is based on the nature of reinforcement component; particles, layer, fiber. Fiber composites may further be classified as, continuous fiber composite and whisker composite materials. The continuous fiber metal composites may either be monofilament or multifilament types. The reinforcement material in metal matrix composites may have different objectives. The reinforcement by light metals opens up the possibility of the application of these light metal reinforced metallic composites in areas where weight reduction is the first requirement. Frequently used light metals as reinforcement are Al2 O3 and SiC. The development objectives of light metal reinforced composites are; (i) Increase in yield strength and tensile strength at room temperature and at higher temperatures, maintaining the minimum toughness or ductility (ii) Increase fatigue strength particularly at higher temperatures (iii) Increase in Young’s modulus (iv) Reduction of thermal elongation (v) Improvement in corrosion and thermal shock resistances (vi) Low density (vii) Mechanical compatibility with the matrix metal (thermal expansion coefficient that matches with matrix metal) (viii) Chemical compatibility (ix) Good process ability (x) Economic efficiency. Some of the above-mentioned objectives may be achieved by using nonmetallic inorganic reinforcements. Ceramic particles, fibers and carbon fibers are frequently used as reinforcement materials in metal matrix composites (MMC). (c) Ceramic matrix composites (CMC) Ceramic matrix composites mostly consist of ceramic fibers embedded in a ceramic matrix forming a ceramic fiber reinforced ceramic composite (CFRC). Carbon and carbon fibers that are also considered ceramic materials, along with fibers of other ceramic materials, have been used as reinforcement elements. Typical reinforcing fiber materials are; Carbon C, Silicon carbide SiC, Alumina Al2 O3 , Mullite or Alumina Silica Al2 O3 –SiO2 . Normally a ceramic matrix gets fractured by a tensile stress that produces an elongation of about 0.05% in the length. However, if normal ceramic matrix is reinforced by ceramic fibers, the fracture or cracks produced by excessive tensile stress get covered up by the extension of fibers. An essential requirement for complete recovery of the fracture site is that the matrix ceramic should also be able to slid and fill the fracture gap. This requires that the adhesive force between matrix and fiber is not very strong. A strong bond between the matrix and the fiber will require a very high elongation capability of the fiber bridging the fracture gap and would result in a brittle fracture. The adhesive force between the matrix ceramic and the fiber is reduced by coating the fiber with a thin

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1 Engineering Materials, Atomic Structure and Bounding

Fig. 1.17 Ceramic matrix composite with fiber reinforcement

layer of pyrolytic carbon or boron nitride. These coatings weaken the bound at the fiber–matrix interface. Figure 1.17 shows how a ceramic matrix fiber reinforced composite repairs fracture or crack sites caused by tensile stress by the elongation of fiber and sliding of the matrix ceramic. Ceramic matrix reinforced by fibers composites may display both; the high insulating or high conductivity properties. As a matter of fact the thermal and electrical properties of ceramic matrix composites strongly depend on the properties of its constituents, namely fibers, matrix, pores in matrix, etc. Further, fibers bring in anisotropy in behaviour of composites. Oxide ceramic matrix composites are very good insulators. Because of their high porosity their thermal insulation is much better. The use of carbon fibers increases the electrical conductivity, provided the fibers remain in contact with each other and with the voltage source. Silicon carbide (which is a ceramic and a semiconductor) matrix is a good thermal conductor. Electrical conductivity of SiC matrix decreases with the rise of temperature as it is semiconductor. Some important properties of ceramic matrix composites are; (1) (2) (3) (4)

High thermal shock and creep resistance High temperature resistance Excellent resistance to corrosion, wear and aggressive chemicals High tensile and compressive strengths, thus no sudden failure as compared to conventional ceramics.

Applications CMC have a wide range of applications, some of which are given below; • High-performance breaking systems

1.2 Atomic Structure

• • • • • • • • • • • • •

29

Heat exchangers Bullet proof armour Turbine blades Heating elements Gas-fired burner parts Hot pressed dies Stator vanes Thrust control flaps for jet engines Refractory components Filters for hot liquids Heat shield systems for space vehicles Rocket propulsion components Turbo jet engine components.

SAQ: Fibers used as reinforcement in ceramic matrix composites are coated or painted with some material. What is the need of this coating or painting? SAQ: What is ‘meant by temperature shock’? Why CMC used in break lining should have high resistance for temperature shock? SAQ: What are ‘light metal MMC’? Which light metals are often used? SAQ: Which polymers are frequently used as matrix material in PMC and why?

1.2 Atomic Structure Though from engineer’s point of view it is only important to know different properties of materials so that an appropriate specimen may be selected for the required use, but if it is required to modify some property or to develop a new material having desired properties, it is essential to know how and why different materials have different properties. Key to this lies in the atomic and the molecular bonding of different materials, i.e. how atoms and molecules are held together in different materials. Some details of atomic structure along with different types of primary and secondary bonds are discussed in the following.

1.2.1 Elements of Atomic Structure All materials are made up of molecules, each molecule in turn, is made up of atoms. Each atom when looked from a distance appears electrically neutral. However, on a closer look, each atom has at its centre a nucleus with positive charge Ze. Here, ‘e’ stands for a unit of charge e = 1.6 × 10–19 C. Nucleus contains certain number N of neutrons, each neutron being neutral, and Z number of proton each with + 1e charge. Total number of nucleons (neutron and protons together are called nucleons) in a nucleus is denoted by A, called atomic mass number and A = (N + Z). Number of

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1 Engineering Materials, Atomic Structure and Bounding

protons Z in a nucleus decides the amount of positive charge on the nucleus and (Z) is called the atomic number of the nucleus/atom. The nucleus of the atom is surrounded by a spherical or nearly spherical distribution of negatively charged cloud made of electrons, each electron denoted by the symbol ‘e’, has − 1e unit of negative charge. Symbol e is used to denote both the unit of charge as well as an electron, but this does not create any confusion as the context of its use immediately tells whether it is used for denoting electron or for charge. The total number of electrons in this cloud is Z, so that the total negative charge surrounding the nucleus of atomic number Z is – Ze. When looked from a distance (much larger than the size of the atom), both the total positive charge in the nucleus (+ Ze) and the total negative charge (− Ze) contained in electron cloud appear as if they are held at a point at the centre of the atom (centre of the nucleus). Since total negative charge is equal in magnitude to the total positive charge, the net charge at atom’s centre becomes zero. Thus atom, looked from a distance, appears electrically neutral.

1.2.2 Arrangement of Electrons in Atom Quantum mechanical model of atom was proposed by an Austrian scientist named Erwin Schrodinger in 1926. The model is based on formalism or mathematical recipe that has some axioms or postulates. These postulates have no foundation in classical physics. Correctness of these postulates is derived from the correctness of the predictions of the model. According to the quantum or wave model of the atom, the presence/motion of electrons in the atom is completely described in terms of a mathematical function called wavefunction of electron, denoted by Greek letter ψ. The wavefunction is supposed to contain all information about the electron which may be obtained by solving a differential equation called Schrodinger wave equation. of the) distance ‘r’ of the electron from the centre Wavefunction ψ(r ) is a function ( of the atom. The quantity ψ(r )∗ ψ(r ) gives the probability of finding an electron at a distance ‘r’ from the centre of the atom. Here ψ ∗ is the complex conjugate of ψ. Though in principle it is possible to write down and solve Schrodinger equation for any atom, however, it becomes complicated to do it for an atom that has many electrons. Therefore, for most of the time one solves Schrodinger equation for the simplest atom; the Hydrogen atom, the atom that has one proton in the nucleus and one electron moving around it. Results obtained for Hydrogen atom are then extended, with suitable modifications, for other atoms. On solving Schrodinger equation for hydrogen atom one gets a number of wavefunctions that are characterised by three quantum numbers, namely the principal quantum number ‘n’, azimuthal quantum number ‘l’ and the magnetic quantum number ‘ml ’. Each electron in the atom has a unique set of values for these quantum numbers (n, l and m l ). A fourth quantum number called magnetic spin quantum number (or simply spin quantum number) and denoted by m s is added to the list of three quantum numbers obtained by solving Schrodinger equation. This

1.2 Atomic Structure

31

additional quantum number does not appear in the solution of Schrodinger equation but is added to account for the two possible spin orientations of the electron. Electron has an inherent spin of value 21 ℏ, here ℏ is quantum mechanical unit of measuring spins. Magnetic spin quantum number (or simply spin quantum number) m s in case of electron can have only two possible values; + 21 ℏ or − 21 ℏ. Although in principal it is not possible to understand any quantum mechanical processes in classical terms, however, for the sake of understanding, the two inherent spin motions of electron may be associated with clockwise and anticlockwise directions of spin. Microscopic systems or entities that follow quantum mechanics also obey a law or principle called Pauli’s exclusion principle, according to which two electrons in a given system (or atom) cannot have the same values for all the four quantum numbers. [ ∗ The region] of three dimensional space around the nucleus where the funcψn,l,m l has maximum value is called the atomic orbital or simply tion ψn,l,m l orbital. Orbital is a region of space around the nucleus of the atom where probability of finding an electron with specified quantum numbers is a maximum. Classically, orbital may be associated with classic orbit or Bohr orbit of the electron. But with the difference that classic electron orbit is a well-defined sharp circular/elliptical path in which electron travels around nucleus, while orbital is a volume of space around the nucleus where the probability of finding electron with given set of quantum numbers is maximum. Since there may be many different combinations of electron quantum numbers, there are several orbitals for an atom. Let us understand the physical significance of these quantum numbers. (a) Principal quantum number ‘n’ Principal quantum number ‘n’ defines the energy level of the electron or principle shell in atom. In quantum mechanics particles can have only discrete values of energy. Principal quantum number ‘n’ can have only positive non-zero integer values, i.e. n may have values: 1, 2, 3, 4 and so on. Principal quantum number ‘n’ also determines the mean distance of the electron from the centre of the atom that is from the nucleus. An energy level with principle quantum number ‘n’ may accommodate a maximum of 2n2 electrons. Thus, Energy level for which n = 1, may have at the most 2 electrons. Energy level for which n = 2, may have at the most 8 electrons. … … Energy level for which n = 5, may have at the most 50 electrons. All electrons in a level of given principal quantum number ‘n’ have very nearly same energy, but their other quantum numbers (l, m l , and m s ) are different. As a matter of fact the maximum number of electrons 2n2 in energy level ‘n’ is nothing but the number of different valid combinations of the remaining three quantum numbers (l, m l , and m s ).

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1 Engineering Materials, Atomic Structure and Bounding

Let us now understand what is meant by the energy of an electron in an atom. In an atom electrons are held because of the attractive force between the positively charged nucleus and negatively charged electrons. This attractive force binds the electron with the nucleus; it means that some amount of work will have to be done (or some energy has to be spent) to take a given electron out of the grip of the nucleus. An electron that is not bound with the nucleus is a free electron, its binding energy with atom is zero. Electrons in an atom that are bound with the nucleus have negative binding energies. The electron that is nearest to the nucleus has the maximum negative (binding) energy or minimum absolute energy. The electron (in an atom) which is farthest from the nucleus will have minimum negative (binding) energy or maximum absolute energy. So when one talks about the energy of an electron in the atom it means the binding energy of the electron which is always negative. An electron near to the nucleus has less absolute energy while the one far away has more absolute energy. Therefore, absolute (binding) energy of electron in an atom increases from inner most to the outer most electrons. (b) Azimuthal quantum number l Azimuthal quantum number is related to the shape of the orbital. It may have only positive integer values including zero up to (n − 1). That is for a given value of ‘n’, l may have values; 0, 1, 2, 3, …, (n − 1). For example, if n = 1, then l may have only one possible values 0. If n = 2, then l may have two values 0 and 1, for n = 4, l = 0, 1, 2 and 3 and so on. Each value of l defines a sub-shell or sub-state within the principal energy level or shell or orbital defined by n. Different numerical values of l are assigned different lower case alphabets; l = 0; is called s-orbital; l = 1 is called p-orbital; l = 2, is d-orbital; l = 3, f-orbital etc. It is desirable at this stage to introduce the concept of multiplicity. As already mentioned, the azimuthal quantum number l defines the shape of the orbital in three-dimensional space around the nucleus. An orbital of a given shape may have several different orientations with respect to the nucleus. In quantum mechanics space is also quantised, and therefore, orientations are also quantised. It is found that an orbital defined by azimuthal quantum number l can have (2l + 1) different orientations. Which means that s-orbital can have only one orientation as for s state l = 0 and, therefore, (2l+1) is 1. The p-orbital (l = 1) may have (2 × 1 + 1 =) 3 different orientations and orbital f for which l = 3, (2 × 3 + 1 =) 7 different orientations. The factor (2l + 1) which gives the number of different orientations is also called the Multiplicity of the orbital. (c) Magnetic quantum number ml This quantum number specifies the abovementioned different orientations of orbitals. For a given value of l the magnetic quantum number m l may have (2l + 1) different values, starting from −l to + l in steps of unity. For example if l = 3 (f-orbital), then quantum number m l may have (seven different) values −3, −2, −1, 0, +1, +2, and + 3. Similarly, if l = 1, which means p-orbital, the three different values of m l will be: − 1, 0, + 1.

1.2 Atomic Structure

33

(d) The Magnetic spin quantum number m s As already mentioned, m s does not arise from Schrodinger equation, it is included to specify direction of the inherent spin of the electron. In classical term, if the electron is spinning in clockwise direction than ms = + 1/2 and if it is spinning in anticlockwise direction then ms = − 1/2. These assignments of + 1/2 and − 1/2 are totally arbitrary. Let us now consider a typical case, suppose there is an electron in principal orbital or shell defined by n = 2. We shall now workout what possible combinations of quantum numbers this electron may have. Possible values of azimuthal quantum number l that this electron may have are: l = 0 and l = 1. Since there are two possible values of l, there will be two sets of values for magnetic quantum number m l . The set corresponding to l = 0 will have only one value m l = 0. The set corresponding to l = 1, m l may have three values: m l = − 1, 0 and + 1. Now corresponding to each set of values of n, l and m l , spin quantum number ms may have two values: + 1/2 and − 1/2. Table 1.9 lists the sets of quantum numbers n, l, m l and m s for principle orbital of n = 2, such that at least one of these quantum numbers is different. If orbital − 2 has a single electron then it may have one of the eight different sets of quantum numbers. Each set defines a sub-orbital or sub-shell within principle shell-2. Table 1.9 tells that second principle shell (n = 2) has eight sub-shells. Since quantum numbers associated with each sub-shell are different, a maximum of eight electrons may be accommodated in second principle shell (Pauli’s exclusion principle). Table 1.9 Possible sets of different quantum numbers in orbital (shell) of principal quantum number n=2 Serial number

Principal quantum number n

Azimuthal quantum number l

Magnetic quantum number ml

Spin quantum number ms

1

2

0 (s)

0

+ 1/2

2

2

0 (s)

0

− 1/2

3

2

1 (p)

−1

+ 1/2

4

2

1 (p)

−1

− 1/2

5

2

1 (p)

0

+ 1/2

6

2

1 (p)

0

− 1/2

7

2

1 (p)

+1

+ 1/2

8

2

1 (p)

+1

− 1/2

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1 Engineering Materials, Atomic Structure and Bounding

It is easy to show that the first principal orbital (shell) has only two sub-shells with set of quantum numbers (n = 1, l = 0, m l = 0, m s = +1/2) and (n = 1, l = 0, m l = 0, m s = −1/2). It is left as an exercise to show that the third principal shell (n = 3) will have 18 sub-shells and that fourth principal shell 32 sub-shells. It follows from above that a maximum of two electrons can be accommodated in I-principal shell, a maximum of 8 electrons in II-principal shell, a maximum of 18 electrons in III-principle shell, a maximum of 32 electrons in IV-principal shell and so on. The principle shells or orbitals are also called electron energy levels and subshells as electron energy states. In n = 1 energy level there are two possible energy states. Similarly in n = 3, energy level there will be 18 energy states.

1.2.3 Shape and Orientation of Orbitals Orbital is a three-dimensional space round the nucleus where there is large probability of finding an electron. There may be several orbitals like, 1s (n = 1, l = 0), 2s (n = 2, l = 0), 1p (n = 1, l = 1), 3d (n = 3, l = 2) etc.. All these orbitals have different shapes. Radial probability distribution of electron in hydrogen atom in 1s orbital is shown in Fig. 1.18a. As may be observed in this figure, probability of finding the electron sharply increase with radial distance, reaches a maximum at around 0.1 nm from the nucleus and then starts dropping sharply, touching a very low value at around 0.2 nm and then becomes almost zero with in a small distance. The probability distribution for 1s orbital is symmetrical in all directions and, therefore, it appears spherical in 3-D space; the surface boundary diagram of 1s orbital is shown in Fig. 1.18b where it may be observed that almost 95% chance of finding the electron is in a spherical volume lying from 0.08 to 0.17 nm from the nucleus. In Fig. 1.18b the darkness of the colour shade indicates the probability, darker the colour higher the probability. Radial probability distribution of electron for 2s orbital (n = 2, l = 1) is shown in Fig. 1.19a. Note that in this case probability increases from r = 0 and attains a small maximum value at around r = 0.05 nm and then falls sharply to zero at about r = 0.1 nm. After touching zero value probability again raises and attains a maximum value at around r = 0.28 nm and then falls off sharply. In contrast to the case of 1s, radial probability distribution for 2s orbital shows two maximums, one smaller and the other larger. In the region between these two maximums probability of finding electron is zero. This region with probability zero is called the node. Like 1s orbital, radial probability distribution for 2s is also same in every direction. Boundary surface diagram of 2s orbital is given in Fig. 1.19b. Radial probability distribution for 2p orbital, shown in Fig. 1.20, is not symmetrical; it has different shapes along the X-, Y- and Z-directions. Though probability distribution has two bob structure in each direction, the orientation of these bobs

1.2 Atomic Structure

35

Fig. 1.18 a Electron probability distribution as a function of distance from the nucleus for orbital 1s. b Boundary surface diagram for 1s orbital

Fig. 1.19 a Radial probability distribution for 2s orbital. b Boundary surface diagram for 2s orbital

is different. These different orientations result from different values of magnetic quantum number m l . It can be shown that probability distributions for orbital-d will have five different orientations, orbital-f, 7 different orientations and so on.

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1 Engineering Materials, Atomic Structure and Bounding

Fig. 1.20 Three different orientations of electron probability distributions for 2p orbital

1.2.4 Electron Energy Level Diagram While introducing quantum numbers associated with electron, it was stated that principal quantum number ‘n’ essentially defines the energy of the electron. Electrons with principal quantum number n = 1 mean electrons that are nearest to the nucleus; most tightly bound to the nucleus, having largest negative binding energy and minimum absolute energy. Electrons with n = 2 are not as close to the nucleus as n = 1 electrons, have negative binding energy but less than that of n = 1 electrons; have absolute energy more than that of electrons of n = 1 orbital. It may thus be observed that actual (negative) binding energy of electrons decreases, absolute energy and distance from the nucleus increases as the value of principle quantum number ‘n’ increases. One may, therefore, draw an energy level diagram for electrons on a vertical scale as shown in Fig. 1.21. It may, however, be said that this energy diagram is based on quantum mechanical solution of Schrodinger equation for hydrogen atom. Fig. 1.21 Energy level diagram for electron in hydrogen atom

1.2 Atomic Structure

37

The fact that a given principal orbital or major shell contains sub-orbitals or subshells brings out the fact that an electron in different sub-shells of a given major shell will have different energies. It means that the energy of an electron is decided not only by the principle quantum number ‘n’, but it also depends on the value of the azimuthal quantum number ‘l’. Actually, an electron put in different sub-shells of a given major shell will have slightly different energies in different sub-shells.

1.2.5 Electron Configuration of Elements Though exact solution of Schrodinger equation is possible only for hydrogen atom, electron energy level scheme obtained for hydrogen atom may be extended, with some modifications, to obtain the electron configuration for atoms of other elements. By electronic configuration one means how electrons are distributed in different orbitals in an atom. From the analysis of hydrogen atom, it is known that principal orbital with n = 1 may accommodate a maximum of two electrons in sub-state 1s; principal orbital n = 2, a maximum of 8 electrons (two electrons in sub-state 2s and 6 electrons in sub-state 2p) and so on. The problem in case of atoms other than that of hydrogen is to find out the sequence of sub-orbitals with increasing energy.

1.2.6 Aufbau or Building-Up Principle The underlying principal of electronic configuration is that electrons in different orbitals of an atom are distributed in such a way that the atom has minimum energy. This means that electrons in an atom are filled in different orbitals in order of their increasing (absolute) energy. This means that first electrons are filled in orbitals that have minimum absolute (or maximum negative binding energy) energy. Once orbital with minimum absolute energy is filled, additional electrons will go to the orbital or state that has next higher value of absolute energy and so on. As already mentioned, the energy of orbitals increases with the value of principal quantum number ‘n’ in case of Hydrogen atom, but in case of other atoms that have more than one electron energy sequencing of orbitals is found to depend not on ‘n’ but on the sum (n + l). Following rules may be used to determine the energies of different orbitals and sub-orbitals. RULE-1: An orbital with a lower value of (n + l) has lower energy. For example let us consider two orbitals 4s and 3d . The value of (n + l) for 4 s is (4 + 0 = ) 4 and for 3d (n + l) = (3 + 2) = 5. Thus orbital 4s will have lower energy and will be filled before orbital 3d . RULE-2: If the value of (n + l) for two orbitals is same then orbital with lower value of n will have lower energy and will be filled before the other. For example consider the orbitals 3d and 4p the value of (n + l) for orbitals 3d = (3 + 2) = 5;

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1 Engineering Materials, Atomic Structure and Bounding

and for orbital 4p (n + l) = (4 + 1) = 5. So both these orbitals have same value of (n + l), hence according to RULE-2, orbital with lower value of n, that is 3d will have lower energy than that of orbital 4p. RULE-3: Hund’s rule: This rule concerns the distribution of electrons in a set of orbitals of same energy, i.e. in sub-shells or sub-orbitals. According to this rule if a number of orbitals of the same sub-shell are available then electrons distribute in such a way that each orbital is first singly occupied with same spin. For example consider the distribution of electrons in carbon atom. A carbon atom has six electrons, now first two electrons will be accommodated in the lowest energy orbital 1s; one electron each in sub-orbitals 1s+1/2 and 1s−1/2 . The remaining four electrons will go to orbitals 2s and 2p. Out of these four electrons two will be accommodated in states 2s+1/2 and 2s−1/2 . The remaining two electrons will now go to 2p orbital. Now 2p orbital may have six sub-orbitals 2px+1/2 , 2px−1/2 , 2py+1/2 , 2py−1/2 , 2pz+1/2 and 2pz−1/2 . Hund’s Rule says that the two electrons that will be accommodated in 2p orbital will not go to 2px , but will distribute such that one electron will go in state 2px+1/2 and the other in 2py+1/2 . Since there were only two electrons, state 2px−1/2, 2py−1/2 , 2pz+1/2 , 2pz−1/2 will remain unfilled. According to the rules mentioned above the sequence of orbitals with increasing energy comes out to be: 1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d < 5p < 6s. Energy sequence mentioned above is shown in Fig. 1.22, where energy of orbitals in increasing order is shown. An easy way of remembering the energy sequence of orbitals is shown in Fig. 1.23. So, the energy sequence of orbitals may be listed as: 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f …. Fig. 1.22 Energy level scheme of orbitals

1.2 Atomic Structure

39

Fig. 1.23 Sequencing of orbital energies

1.2.7 Representing Electron Configuration With the help of the rules discussed above it is possible to write the electron configuration for atom of any element. There are three ways of representing electron configurations. (a) Orbital notation method In this method the orbitals that have electrons are written in order of increasing energy and the number of electrons in each orbital are given as a superscript to the orbital. For example, nitrogen atom has seven electrons and its electronic configuration may be written as: 1s2 2s2 2p3 . Figure 1.24 explains the meanings of each character of the notation. (b) Orbital diagram method In this method orbitals having electrons are represented by boxes and are written in the order of increasing energies. Electrons in each orbital are represented by arrows, direction of arrows indicating the direction of electron spins. For example, the electron configuration of some elements atoms are given in the last column of Table 1.10.

Fig. 1.24 Meaning of each character of the notation

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1 Engineering Materials, Atomic Structure and Bounding

Table 1.10 Electronic configurations for some elements in different notations

(c) Short-hand form In this method the last completely filled orbital or shell is represented in terms of a noble gas. For example, the electron configuration of lithium in this notation may be written as [He] 2s1 . Electron configurations for some elements in different notations are given in Table 1.10.

1.2.8 Valence Shell Shell or orbital of highest energy (largest value of n) that has some electrons is called the valence shell or valence orbital, and the electrons it contains are called valence electrons. For example the valence shell for Calcium is 4s2 with two valence electrons; the valence shell for Argon is 3s2 3p6 orbital with (2 + 6 =) 8 valence electrons; Aluminium has 3s2 3p1 shell as valence shell and it has (2 + 1 =) 3 valence electrons. It is important to remember that all electrons in different sub-shells of the highest ‘n’ value shell (that has some electrons) are counted as valence electrons. The number of valence electrons in case of noble gases is eight. Therefore, it is concluded that eight electrons in valence shell of any atom make it very stable and chemically inert. Valence shell and valence electrons are important because most of the chemical and some physical properties of the atom are decided by the valence shell and valence electrons. It is valence electrons that take part in chemical reactions and decides the type of bonding with other atoms to make molecules.

1.2 Atomic Structure

41

Each orbital or shell/sub-shell can accommodate a fixed maximum number of electrons; for example, s-shell can accommodate a maximum of 2 electrons, p-shell a maximum of 6 electrons, d-shell a maximum of 10 electrons and f-shell a maximum of 14 electrons. When the valence shell of an atom is filled with maximum number of electrons it can hold, the valence shell is said to be completely filled and the atoms with completely filled valence shell are found to be chemically very stable and do not show chemical activity. Valence shells of inert gases like argon, neon, and helium, etc. are completely filled with eight electrons. In periodic table (see Fig. 1.1) elements are arranged such that all elements falling in one group (vertical column) have similar electron configurations of their valance shells. They have same number of valence electrons; for example configurations of members of group-II are given as: Be = [He] 2s2 , Mg = [Ne] 3s2 , Ca = [Ar] 4s2 , Sr = [Kr] 5s2 , Ba = [Xe] 6s2 , and Ra = [Rn] 7s2 . Each member of group-II has only two valence electrons in s-orbital. This is the reason that all members of a group of periodic table show similar chemical properties. The periodic table may also be taken as a guide to the order in which orbitals are filled. Figure 1.25 shows the classification of groups of elements in the periodic table according to the type of outer sub-shell being filled with electrons. In Fig. 1.25 of the periodic table different elements may be identified through their horizontal group number given in red colour and the vertical row number written in blue. For example, the element specified as 75 is 43Tc; 46 (coloured dark brown) is 72Hf and element specified as 125 (shown by light brown colour) is 48Cd. The important characteristic of this figure of periodic table is that all elements falling in a group (vertical column) have same general formula for the valence orbital. For example all members of group 1A the general formula for valence orbital is: ns1 where n is the principle quantum number. All members of group 2A have the common representation for the valence orbital as: ns2 ; for all members of group 3A, the general representation for valence orbital is: ns2 np1 ; … for members of the group 6A: ns2 np4 and so on. The outer electron configuration for element 72Hf is: 6s2 5d2 ; Fig. 1.25 Groups of elements in periodic table according to the type of sub-shells being filled with electrons

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1 Engineering Materials, Atomic Structure and Bounding

similarly, the valence electron configuration for element 48Cd is: 5s2 4d10 and for the element 43Tc the outer electron configuration is: 5s2 4d5 .

1.2.9 Some Anomalous Electron Configurations The electron configuration rules stated above holds good in most cases but there are four outstanding exceptions where these rules fail to give the correct electron configuration. These four cases are of: (a) Chromium Cr; According to the rules the electron configuration of Cr should be [Ar] 4s2 3d4 but actually it is [Ar] 4s1 3d5 (b) Similarly, for copper Cu, according to rules the electronic configuration should be[Ar] 4s2 3d9 but its actual configuration is: [Ar] 4s1 3d10 (c) Silver (Ag) according to rules should have electron configuration of [Kr] 5s2 4d9 but actual electron configuration for silver is: [Kr] 5s1 4d10 (d) Also in case of Gold (Au) according to rules the electronic configuration should be [Xe] 6s2 5d9 but the actual configuration is: [Xe] 6s1 5d10 As may be observed in all the above cases, an enhanced stability is acquired by half or fully filled sub-shells. Only valence electrons take part in chemical reactions and in forming molecules, etc. The inner electrons are generally well protected and mostly do not take part in combination processes. An American chemist, G. N. Lewis introduced a simple notation to represent valence electrons in an atom. These notations are called Lewis symbols. Lewis symbols for elements of second period of periodic table may be given as (Fig. 1.26). The dots around the chemical symbol of an element give the number of valence electrons in the atom of the element. The valency of the element is equal to the number of dots around it or 8 minus the number of dots around. Electron configurations discussed above apply to the ground states of atoms. However, when an atom is excited by giving some energy by an external source, say by heating, etc., few electrons from its valence orbital shift to the next higher orbital. Therefore, the electron configuration of an excited atom is different from the ground state configuration. Similarly, electron configuration of an atomic ion is different from that of the parent atom. SAQ: What is the difference between a classical electron orbit and quantum mechanical orbital?

Fig. 1.26 Lewis symbols for second group elements

1.3 Bonds Between Atoms and Ions

43

SAQ: Does the energy of all electrons in a given principal orbital exactly same? SAQ: Which quantum number includes Pauli’s exclusion principle in quantum description of electron’s motion in an atom? SAQ: Write electron configurations of a singly ionised Sodium ion and a doubly ionised lithium ion.

1.3 Bonds Between Atoms and Ions When two atoms are brought near to each other two types of Coulomb forces come into play; the repulsive forces between the positively charge nuclei of the two atoms and between their electron clouds and attractive forces between electron cloud of one atom and the nucleus of the other atom. The magnitude of both types of forces increases with the decrease of relative separation r. Figure 1.27a shows the variation of the attractive, repulsive and net forces between two atoms as a function of their mutual separation r. The intra-atomic separation r 0 corresponds to the mutual separation where attractive and repulsive forces cancel each other and the two atoms are in a state of equilibrium. It is well known that any force F may be converted into potential energy V (or force F may be derived from potential V ) using the mathematical operation given by expression; ) ( ∂F ∂F ∂F + + V = −gradF = − ∂x ∂y ∂z Net potential energy between the two atoms as a function of intra-atomic separation r, obtained by using the above expression, is shown in Fig. 1.27b. It may be observed in this figure that net attractive force gives rise to the negative potential energy V 0 which is responsible for the binding of the two atoms. The negative potential energy has its maximum (negative) value at a separation r0 , the equilibrium distance. Two atoms develop a bond only when they are at a relative separation of r 0 , at larger separation they do not bind with each other as shown in the figure. The negative binding energy decreases on both sides of r 0 , and, therefore, the two atoms are held fixed at a separation of r 0 . Further, larger the magnitude of − V 0 , more tightly the two atoms are bound with each other. Atoms bound with each other makes a molecule. As a matter of fact the magnitude of V 0 decides many properties, physical and chemical, of the pair of two atoms or the molecule. For example, a larger value of V 0 corresponds to a higher melting point.

1.3.1 Electronegativity Chemical reactivity of an atom is decided by the valence electrons in the outer most orbital of the atom. If the valence shell is completely filled, like that of inert gases,

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Fig. 1.27 a Attractive, repulsive and net force between two atoms as function of intra-atomic distance r. b Net potential energy between two atoms as a function of r

the atom has almost no chemical reactivity. However, in case the valence shell is only partially filled, the atom shows chemical reactivity. Chemical reactivity of atoms may be measured in terms of parameters called electronegativity or electropositivity. Figure 1.28 shows the periodic table of elements. The electronegativity of elements in periodic table starts from almost the middle of the periodic table and increases towards right. Electronegativity decreases towards left, and atoms with lower value of electronegativity are said to have electropositivity. In periodic table highly electronegative halogens and highly electropositive alkali metals are separated by the noble gases. Atoms of elements of group-1 and group-2 of the periodic table (enclosed in box A) have partially filled valence shell; they have only one or two electrons in their valence shell. These atoms having almost empty valence shell, with lower electronegativity, have the tendency to give up their electrons to other atoms of higher electronegativity when they come in contact with them. Since they have the tendency to give their electrons and by doing so they acquire net positive charge, these atoms are said to have electropositivity. For example, if we consider element potassium (K) its electron configuration is: 1s2 2s2 2p6 3s2 3p6 4s1 or [Ar]4s1 . The valence shell of potassium 4s1 contains only on electron. Potassium has the tendency to give up a electron and becomes positive ion; K − 1 electron = K+ positive ion (Cation).

1.3 Bonds Between Atoms and Ions

45

Fig. 1.28 Periodic table showing electronegative and electropositive elements

On the other hand atoms of elements on right side of the periodic table like, chlorine (Cl) has many electrons in its valence shell but is not completely filled. Electron configuration of chlorine is: [Ne] 3s2 3p5 . The valence shell of chlorine (3s2 3p5 ) has seven electrons, one electron short of the maximum number that (sp)orbital can accommodate. Chlorine is highly electronegative; it has the tendency of taking an electron and becoming a negative ion (anion); Cl + 1 electron = Cl− Negative ion (anion). Elements contained in box A in Fig. 1.28 are typical of metallic character (electropositive), and those contained in box B have characteristics which are intermediate between metals and non-metals, possessing different degrees of electronegativity.

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1.3.2 The Octet Rule Though several attempts were made to explain the bond formation between atoms on the basis of electron structure, it were Kossel and Lewis, who independently gave a satisfactory explanation in 1916. They studied the electron structure of noble gases and found that all of them have eight electrons in their valence shell. Based on this observation, Kossel and Lewis developed a theory for combination between atoms called ‘electronic theory of chemical bonding’. According to this theory atoms can combine either by transfer of valence electron from one atom to the other or by shearing of valence electrons in order to have an octet (eight) of electrons in their valence shells. This is called octet rule. The octet rule though useful but is not universal. There are some limitations of the octet rule. Octet rule essentially applies to atoms of the second group of periodic table.

1.3.3 Classification of Bonding It is clear from above that characteristics of valence electrons and the net force of attraction between two atoms create attractive bonds between atoms. Depending on their strength and other characteristics bonds between atoms may be divided into two classes: (a) primary bonds between atoms and (b) secondary bonds between atoms and molecules. Primary bonds may further be divided into three types: (i) ionic bond, (ii) covalent bond and (iii) metallic bond while secondary bonds are into two types: (i) van der Waals bond and (ii) hydrogen bond. (A) Primary atomic bonds Primary atomic bonds are characterised by large interatomic forces. Primary bonds involve valence electrons of interacting atoms and arise from the tendency of atoms to acquire stable electron structure of completed valence shell. They may be nondirectional or localised (directional) and may be produced by electron transfer, electron shearing or delocalisation of electrons (i) Ionic or electrovalent bond These bonds are formed when two atoms of very different values of electronegativity combine together to form a molecule. Ionic bonds are formed typically between highly electropositive (metallic) and electronegative (non-metallic) elements. For example when an atom of sodium with very low value of electronegativity (or high value of electropositivity) combines with an atom of chlorine which has very high value of electronegativity, an ionic bond gets established between the two atoms.

1.3 Bonds Between Atoms and Ions

47

Fig. 1.29 Formation of ionic bond in NaCl molecule

As shown in Fig. 1.29, sodium (Na) has only one electron in its valence shell 3s. Since sodium is electropositive it has the tendency of giving away this electron, chlorine on the other hand is highly electronegative, has 7 electrons in valence shell 3s2 3p5 and has the tendency of acquiring electrons. As a result, when an atom of sodium comes sufficiently close to the chlorine atom to be with in its Coulomb field, it gives its only valence electron to the chlorine atom. With this transfer of electron sodium atom becomes a positive ion and on receiving an extra electron from sodium, chlorine atom becomes a negative ion. A bond gets established between positive sodium ion and negative chlorine ion due to coulomb attraction between them. Thus ionic bonds are formed by the transfer of valence electron(s) from the lower electron negativity atom to the higher electronegative atom. Only valence electrons take part in bonding while the inner shell electrons and the nucleus of the atom do not take part in chemical bonding, being well shielded by valence electrons and large force of attraction between the nucleus and inner shell electrons. Therefore, in pictorial representations of bonds, the nucleus and inner shell electrons are often represented by the core. Core of the atom has a positive charge equal in magnitude of the number of electrons in the valence shell. Figure 1.29 explains the ionic bond formation in case of NaCl. As shown in this figure, on forming the bond the size of Na+ ion shrinks (as compared to the size of Na atom) while the size of negative Cl− ion also shrinks but not so much as that of sodium ion. The reason for this reduction of size in case of sodium ion is the fact that on losing the only valence electron the valence shell disappears. The size of sodium ion then reduces to the size of its core. The size of negative chlorine ion also decreases because with the increase of

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Fig. 1.30 a Ionic bonds are generally non directional. b Structure of an ionic solid

negative charge in the valence shell, attractive force by its core increases which results in a valence orbital of reduced size. Ionic bonds do not have any preferred direction; it is because of the fact that both the positive ion and the negative ion attract each other by forces of equal magnitudes, as shown in Fig. 1.30a. Ionic solids have large lattice energies ranging from 600 to 3000 kJ/mol and have high melting temperatures. Melting point for NaCl, for example, is 801 °C. The binding energy for NaCl is ≈ −7.42 × 10−19 J = 4.63 eV. Ionic solids have a regular arrangement of alternate positive and negative ions in three dimensions, as shown in Fig. 1.30b. Ionic solids are mostly ceramics; they are bad conductors of heat and electricity and often brittle. (ii) Covalent bonds The term covalent bond was coined by the American Chemist Irving Langmuir in 1919. Covalent bonds are formed when two atoms of either same or nearly same electronegativity join together. This typically happens in non-metals. In case of covalent bonding, transfer of electrons from one atom to the other does not take place. Instead, the two interacting atoms shear electrons to complete the octet (eight electrons each) or duplet in their valence shells. Large number of elements in periodic table have either s-shell or the combination of sp orbitals as their valence shell. The maximum number of electrons that may be accommodated in s-shell is 2, and for sp shell 8 (2 + 6), therefore, atoms try to complete electron duplet, if s-shell is valence shell or octet if sp shell is valence shell.

1.3 Bonds Between Atoms and Ions

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Fig. 1.31 a Covalent bonding in Cl2 molecule. b Lewis dot structure for Cl2 molecule

Shearing essentially means that some electrons of individual atoms get attached to both the atoms. Generally, s and sp shell electrons shear to attain noble gas electron configuration. Covalent bonds are formed when atoms of F, O, N, Cl, H, C, Si, Ge, etc. form molecules. Covalent bonds are also found in compounds like GaAs, CH4 (methane), C2 H6 (ethane), etc. Figure 1.31a shows how two atoms of chlorine with seven electrons each in their valence shells share one electron to complete octets resulting in stable Cl2 molecule. Lewis dot structure of Cl2 molecule is shown in Fig. 1.31b. In the above-mentioned example of Cl2 molecule, two atoms of chlorine share only one electron creating a single covalent bond. However an atom may have more than one covalent bonds. For example, in one molecule of methane (Ch4 ) carbon atom that has only four valence electrons share an electron each with four hydrogen atoms, creating four covalent bonds. This is shown in Fig. 1.32. The number of covalent bonds that an atom may form is given by (8 − N) where N is the number of valence electron in the atom. That is why a carbon atom with 4-valence electrons forms (8 – 4 =) 4 covalent bonds and Chlorine with seven valence electrons (8 – 7 =)1 covalent bond. In C2 H4 molecule the two carbon atoms share two electrons each to complete electron octet for both carbon atoms. Thus the two carbon atoms get coupled through a covalent bond doublet. The remaining two electrons of each carbon atom are sheared by single electrons of two hydrogen atoms per carbon atom, completing the duplet of hydrogen.

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Fig. 1.32 a Pictorial representation of four covalent bonds in methane (Ch4 ) molecule. b Lewis dot structure for methane molecule

Lewis dot structure of C2 H4 molecule is given in Fig. 1.33a. Nitrogen atom with 5 electrons in s2 p5 valence shell is unstable, but it makes a stable N2 molecule with three covalent bonds as shown in Fig. 1.33b. Bond Parameters (a) Bond length Bond length is defined as the equilibrium distance between the nuclei of two bonded atoms in a molecule. Each atom of the molecule contributes to the bond length. Figure 1.34a shows two atoms A and B that are bonded by covalent bound. Here R is the covalent bond length and the contributions of the two atoms to the bond length, r a and r b are the bound radii respectively of the two atoms. The covalent bond lengths for O–H, C–H, C=C molecules are respectively 96, 107, 120 (× 10–12 m).

Fig. 1.33 a Lewis dot structure of C2 H4 molecule with double covalent bonds in carbon atoms. b Lewis dot structure for N2 molecule with triple covalent bonds in nitrogen atoms

1.3 Bonds Between Atoms and Ions

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Fig. 1.34 a Covalent radii of the two atoms are r a and r b , while R the separation between the two nuclei is the bond length. b Bond angle in H2 O molecule

(b) Bond angle It is defined as the angle between the orbitals containing bounding electron pair around the central atom in a molecule or complex ion. Bond angle is expressed in degrees, and it gives some idea about the distribution of orbitals around the central molecule, which means the shape of the molecule. For example, in case of water molecule the bond angle is 104.50, as shown in Fig. 1.34b. (c) Band order In Lewis description of covalent bond, bond order is the number of covalent bond in the molecule, for example in H2 bond order is 1, in O2 bond order is 2 and in N2 the bond order is 3. (d) Polarity of bond When covalent bond is formed in two similar atoms like H2 , O2 , Cl2 , etc., the shared electrons are equally attracted by the two atoms and, therefore, the electron pair is situated exactly between the two identical atoms or nuclei. Such a bond is called a non-polar covalent bond. However, when two dissimilar atoms are coupled through a covalent bond, like HF molecule, the sheared pair of electrons shifts more towards the fluorine because of the larger force of attraction by it than that by hydrogen nucleus. The bond in this case is a polar covalent bond. Shifting of the paired electrons from the centre gives rise to the formation of an electric dipole. The molecule that has polar covalent bond behaves as a tiny electric dipole, often represented by the symbol μ. Dipole strength of such atoms is measured in a unit called Debye denoted by D. Further 1 D = 3.33564 × 10–30 C m. Dipole moment is a vector quantity, and the direction of the vector is indicated by direction

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of shift of the shared electron pair from the central position, i.e. in case of HF molecule from H atom towards F atom. Because of the associated dipole moment polar covalent bonds are said to have specific directions. (e) Bonding energy The bonding energies of covalent bonds may be very different; it may be very high, for example in case of diamond which is the hardest material having a melting point of > 3550 °C. On the other hand bonding energy may be very low as in the case of bismuth which has a low melting point of around 270 °C. No material has 100% ionic bonds or 100% covalent bonds; materials that have predominantly ionic bonds also have a small percentage of covalent bonds and vice versa. (iii) Metallic bond Metallic bonds are found in metals and their alloys. Such bonding occurs when atoms of low electronegativity join together. Since low electronegativity atoms have the tendency to lose their valence electrons; the interacting atoms lose all their valence electrons which form a cloud of delocalised electrons. No electron of the cloud is essentially attached with any particular atom rather all electrons are attached with positive cores of all atoms. Metallic bonding may be looked as an extreme case of covalent bonding; in covalent bonding nearby atoms shear their valence electrons but in metallic bonding all atoms shear their valence electrons. Electrons that are not bound to any particular atom are called delocalised electrons. Cloud of delocalised electrons works as glue to bind positively charged cores of atoms, which in absence of delocalised electron cloud will repel each other and break the crystalline structure of the metal. Alternately, a metallic crystal that has N number of atoms may be looked as an N-atomic molecule and the cloud of delocalised electrons as electrons moving in large number of different molecular orbitals. Since the number of molecular orbitals of a molecule having N atoms will be very large, criss-crossing each other, the delocalised electrons in these molecular orbitals appears as an electron cloud. Looking from the point of quantum mechanics, each electron of the electron cloud has a discrete set of closely spaced energy levels. Since the number of delocalised electrons in the cloud is very large, the total number of electron energy levels becomes very large, almost a continuum of levels. Only the low lying energy levels of the continuum are filled with electrons, but large number of electron levels is empty. If energy is supplied to these delocalised electrons by some external source, say by shining light on the metallic crystal, electrons absorb the incident light photons of all frequencies and go to their respective higher energy levels which were empty. Since the mean life of these excited states is very short (< 10−9 s), the excited electrons revert back to their lower energy states

1.3 Bonds Between Atoms and Ions

53

re-emitting photons of almost the same energy as they have absorbed. This is why freshly cut metallic surfaces shine when put in light. Metallic bond energies may vary by large factor, for example, the bond energy in case of Tungsten is 850 kJ/mol and its melting point is 3410 °C. While bond energy in case of mercury is only 68 kJ/mol, with melting point of − 39 °C. Since the force responsible for binding atomic cores in a metallic crystal is provided by the cloud of delocalised electrons that does not have a fixed shape, it is relatively easy to bend metals. Further, since ionic cores of metallic atoms may slide over each other, metals are ductile. Some details of these properties of metals have already been discussed earlier. Because of close packing of atoms, most of the metallic crystals are solid (except mercury), have high density, high melting point, etc. The electron cloud of nearly free delocalised electrons makes metals a good conductor of both electricity and heat. (B) Secondary bonds Secondary bonds, as their name suggests, are relatively weaker than primary bonds, may have strength of the order of 10 kJ/mol, but they are very important. Secondary bonds, also called van der Waals bonds, are present in most of the systems but are overlooked because of their low strength. Secondary bonds originate because of electric dipole nature of some molecules/atoms. Electric dipole Fig. 1.35a shows an electric dipole; two equal and opposite charges separated by a small distance, and the vector dipole moment represented by Greek letter μ = Q . x. Here Q is the magnitude of charge (in Coulomb) and x (in metre) is the distance between the two charges. Dipole moment is a vector along the direction from negative charge to the positive charge. (i) Fluctuating dipole bond Electric dipoles are often formed in microscopic systems like atoms and molecules if centres of positive and negative charge distributions in them do not coincide. A charge distribution if looked from a distance appears as if the total charge contained in the distribution is concentrated at a particular point, this point is called the charge centre. This is likely to happen in a molecule/big atom, the electron clouds of which may extend up to large distances from their nuclei and may have non-spheroid shape. As a result, the centre of positive charge gets slightly displaced with respect to the centre of negatively charged electron cloud. Such molecules/atoms are called polar molecules/atoms. Polar molecules or atoms have the tendency to align themselves such that their electric dipole moment vectors point in opposite directions. Oppositely directed dipoles attract each other creating a secondary (or van der Waals) bond, as shown in Fig. 1.35b. Molecular dipoles formed as a result of displaced charge centres are often referred as fluctuating dipoles. Typical examples are atoms of noble gas elements. Noble gas

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Fig. 1.35 a An electric dipole of dipole moment μ. b Attraction between dipole molecules/atoms give rise to secondary bonding

atoms have completely filled valence shell (s2 p6 ) and therefore, cannot form primary bonds. However, when two atoms of a noble gas come close to each other, they induce electric dipoles in each and these dipoles align to form a fluctuating dipole secondary bond, as shown in Fig. 1.36a. (ii) Permanent dipole bond However, there are molecules that are permanent dipoles, like that of NaCl, which have ionic bonding between Na+ and Cl− ions. In such molecules close packing and alignment results in formation of permanent dipole secondary bonds (Fig. 1.36b). (iii) Hydrogen (secondary) bond Hydrogen bond is also a secondary dipole bond but it is much stronger compared to other secondary bonds. Hydrogen bond energy may be as large as 50 kJ/mol. This bond is found

Fig. 1.36 a Fluctuating dipole secondary bond b permanent dipole secondary bond

1.3 Bonds Between Atoms and Ions

55

in molecules where hydrogen is covalently bound to elements of high electronegativity like, fluorine, oxygen or nitrogen. When hydrogen is bonded with covalent bond to strongly electronegative element ‘x’, the electron pair sheared between hydrogen and atom ‘x’, moves faraway from hydrogen atom. Since there is displacement of electrons towards ‘x’, the hydrogen atom acquires a small positive charge (q+) and the other atom ‘x’ a negative charge (q−). This results in molecule becoming polar and behaving like a dipole. Two polar molecules then make a secondary (hydrogen) bond. Covalent and hydrogen bonds between molecules of HF are shown in Fig. 1.37. It is because of the relatively strong hydrogen bonds between water molecules that the boiling point of water is so high as compared to other materials of same molecular weight. Figure 1.38 shows how hydrogen bonds bind large number of water molecules together. Each water molecule (H2 O) has two hydrogen atoms that are coupled to oxygen atom with covalent bonds. Each water molecule behaves as a dipole since the pairs of electrons in covalent bonds are slightly shifted towards the oxygen nucleus so that hydrogen atoms develop positive charge and oxygen atom negative charge. The dipole water molecules join together by hydrogen bonds. Often in pictorial representation, covalent bonds are shown by solid line and hydrogen bond by dashed line, as shown in part (b) of the figure. Properties of six types of bonds are summarised in Table 1.11. SAQ: SAQ: SAQ: SAQ: SAQ:

Primary cause of bound formation between atoms is? What works as a glue between positive cores in crystalline metals? Give one similarity between covalent and metallic bonding. What is the difference between fluctuating and permanent dipole bonds? Hydrogen bond is a fluctuating dipole or permanent dipole bond?

Short Answer Questions SA 1.1

Two metals A and B respectively, have face centred (fcc) and body centred (bcc) crystal structures. Which of the two metals will be more ductile and why? (II).

Fig. 1.37 Hydrogen bond between two molecules of HF

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Fig. 1.38 Hydrogen bonds between water molecules

Table 1.11 Summary of bond properties Bond class

Special characteristics

Typical bond energies

Examples

Ionic

Formed by electron transfer between two atoms of very different electronegativity, non-directional, very strong

300–6000 kJ/ mol or 5–10 eV/atom

NaCl, CaCl, LiF

Covalent

Formed between atoms of either same or 300–800 kJ/ Ge, Si, CH4 , nearly equal electronegativity, shearing of mol or 3–8 eV/ Diamond electrons, may be directional if two atoms are atom different

Metallic

Delocalisation of valence electrons, delocalised electron cloud works as glue to hold positive cores, good conductors of heat and electricity, non-directional

100–1000 kJ/ mol or 0.5–1.6 eV/ atom

All metals and their alloys

Dipole bond

Formed by alignment of either fluctuating or permanent dipoles of molecules/atoms, non-directional

Around 10 kJ/ mol or 0.05–0.2 eV/ atom

Noble gas molecules

Hydrogen bond

Relatively stronger secondary bonds

Around 50 J/ mol or 0.25–0.6 eV/ atom

H2 O, HF, HN

SA 1.2 SA 1.3 SA 1.4

What are delocalised electrons? Discuss their role in determining the resistivity of a metal. Why the resistivity of a metal increases with temperature? Why does a piece of metal become more malleable on heating and tough, brittle on cooling and hammering?

1.3 Bonds Between Atoms and Ions

SA 1.5 SA 1.6

SA 1.7 SA 1.8 SA 1.9 SA 1.10 SA 1.11 SA1.12 SA 1.13 SA 1.14

SA 1.15 SA 1.16 SA 1.17 SA 1.18 SA 1.19 SA 1.20

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How one may define a Ceramic? Are both Diamond and Graphite Ceramic, if yes, why? How can one differentiate between Metals, Ceramics and Polymers on the basis of bonding in them? Which property of Ceramics is very much affected by the relative strength of bond types in it? What are thermosetting and thermoplastic polymers? Give one example of each. Discuss the process of crack/fracture repair in ceramic fiber reinforced CMCs. What are light metal reinforced (MMC) and where are these used? Which Composite you will use for fabricating light but strong car body. Give reasons for your answer. State and explain the rules that govern the distribution of electrons in an atom. Magnesium nucleus has 12 protons, write the electronic configuration of magnesium atom in three notations. What will be the multiplicity of f-orbital if electron spin is neglected. What is the special feature of elements in a group of periodic table and how does it effects the chemical behaviour of elements, explain with an example. Electron configuration of 21 Sc is [Ar] 3d1 4s2 . What will be the configuration of 22 Ti? How electro negativities of interacting atoms decide the type of bond between them? How does the melting point of a material related to the strength of the bond in its molecules? The HF molecule has a permanent dipole moment or not? Explain your answer. What are delocalised electrons? How do they play role in creating atomic bonding? What type of bonds do you expect between long molecular chains in polymers?

Multiple Choice Questions Note: Some of the multiple choice questions may have more than one correct alternative; all correct alternatives must be ticked for complete answer in such cases. MC 1.1

Silicon carbide (SiC) and Molybdenum silicate (MoSi2 ) are; (a) Metals (b) Ceramics (c) used as heat shields (d) used for making electrodes ANS: (b, d)

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MC 1.2

1 Engineering Materials, Atomic Structure and Bounding

Which of the following is Elastomers? (a) Polysulphide (b) Terylene (c) Neoprene (d) Sealing wax

MC 1.3

ANS: (c) Pickup the thermoplastic polymer (s) from the following

MC 1.4

(a) Nylon (b) Terylene (c) Neoprene (d) Sealing wax Ceramic fibers used as reinforcement in (CMC) are coated with; (a) Polyethylene (b) Silicon carbide (c) Boron nitride (d) pyrolytic carbon

MC 1.5

ANS: (c), (d) Composite used for making brake linings is; (a) Light metal reinforced (MMC) (b) Carbon fiber reinforced (PMC) (c) Carbon fiber reinforced (CMC) (d) Carbide reinforced (MMC)

MC 1.6

ANS: (c) Which of the following is (are) used for making electric heating elements and electrodes. (a) Ceramic ZrO2 (b) Ceramic MoSi2 (c) Polymer neoprene (d) Polymer Buna-N

MC 1.7

ANS: (a), (b) Bakelite is (a) Metal matrix composite (b) Thermosetting polymer (c) Thermoplastic Polymer (d) Polymer matrix composite

MC 1.8

ANS: (b) Which of the following may be used for moderation of fast neutrons in reactors? (a) Polymer Bakelite (b) Ceramic BeO (c) polymer neoprene (d) Ceramic ZrO2

MC 1.9

ANS: (b), (d) Which of the following is/are not ceramic? (a) SiC (b) ZrO2 (c) MoSi2 (d) Sealing wax

MC 1.10

ANS: (d) Electron configuration of of 14 6 C will be;

12 6 C

is 1s2 2s2 2p2 , the electron configuration

(a) [He]2s2 2p2 (b) [He]2s2 2p3 (c) 1s2 2s2 2p2 (d) 1s2 2s2 2p1 ANS: (a), (c)

1.3 Bonds Between Atoms and Ions

MC 1.11

59

Azimuthal quantum number and maximum number of electrons that may be accommodated in f-orbital are respectively; (a) 2 and 7 (b) 3 and 7 (c) 2 and 14 (d) 3 and 14

MC 1.12

ANS: (d) Which of the following represent same electron configurations; (a) [He]2s2 2p2 (b) [He]2s2 2p3 (c) 1s2 2s2 2p2 (d) 1s2 2s2 2p1

MC 1.13

ANS: (a), (c) The electron configuration of deuteron atom is; (a) 1s2 (b) 2s2 (c) 1s1 (d) 1s2

MC 1.14

ANS: (c) Noble gas atoms have completed valence shells; they are bound by. (a) Ionic bonds (b) covalent bonds (c) fluctuating dipole bonds (d) permanent dipole bonds

MC 1.15

ANS: (c) Diamond, the strongest material has. (a) Ionic bonds (b) covalent bond (c) hydrogen bonds (d) fluctuating dipole bonds

ANS: (b) MC (1.16) Covalent bonds are formed when atoms of (a) very different electronegativity combine (b) nearly same electronegativity combine (c) carbon combine (d) hydrogen and fluorine combine MC 1.17

ANS: (b), (c) If bond energy of a material is 4.0 eV/atom, the type of bond and its melting point are respectively; (a) ionic, high (b) fluctuating dipole, high (c) hydrogen, high (d) covalent, low ANS: (a)

Long Answer Questions LA 1.1 Summarise important properties of metals that differentiate them from other materials. Explain how delocalised electrons are formed in metals and the role they play in deciding thermal and electrical conductivities of metals. LA 1.2 What are composites and how are they classified? Discuss important properties and applications of fiber reinforced Ceramic matrix composites.

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LA 1.3 How can one define a ceramic? Name two ceramics that have electrical properties opposite to each other and two that have similar electrical properties. Bring out differences between metals, polymers, ceramics and composites. LA 1.4 What are polymers? What types of bond are usually found in polymers? What is meant by the functionality of a monomer and how does it affect the structure of polymer? LA 1.5 Discuss quantum mechanical model of electron configuration in atoms and explain physical significance of different quantum numbers giving their possible values. What are the rules for filling electrons in different orbitals? Explain by giving some example. LA 1.6 What is meant by electronegativity? How does electronegativity decide nature of bonds between two atoms? Describe various types of primary bonds giving examples for each type. LA 1.7 What are secondary bonds and how do they differ from primary bonds? Give details of fluctuating and permanent dipole secondary bonds bringing out the points of difference between them. What kind of secondary bonds are found in water molecules? What are their special characteristics?

Chapter 2

Electrical Behaviour of Condensed Matter

Objective Electrical behaviour of solids will be discussed in this chapter. Basis of classifying solids as insulator, semiconductor, conductor and superconductors will be discussed in details. After the study of this chapter it is expected that the reader will be able to understand the behaviour of different crystalline solids when they are subjected to electric field.

2.1 Introduction A material may possess several intensive properties that do not depend on the amount of the material. These quantitative properties are often used as a metric by which the advantages of one material over the other can be compared for material selection for a specific purpose. The first and the most important electrical property of a material is its ‘resistivity’, (or specific resistance) generally denoted by Greek letter ‘ρ’ (rho). Resistivity is defined as the resistance offered by a unit cube (a block of 1 m × 1 m × 1 m) of the material between its opposite faces (see Fig. 2.1). The MKS unit of resistivity is ‘ohm-metre’ written as ‘Ω-m’ in short. The resistance R of a block of a material of length L and uniform area of cross section A may be written as L(m) R(Ω) = ρ(Ω-m)  2  A m Or   A m2 ρ(Ω-m) = R(Ω) L(m)

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 R. Prasad, Physics and Technology for Engineers, https://doi.org/10.1007/978-3-031-32084-2_2

(2.1)

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Fig. 2.1 a Resistance between two opposite faces of a 1 m × 1 m × 1 m cube is equal in magnitude to the resistivity of the material. b Resistance of a bar of length L and uniform area of cross section A is given by R = ρ LA

Both resistivity ρ and resistance R are the measure of the opposition that an electric current faces while passing through the given specimen, however, ρ is an intensive property of the material (it does not depend on the amount of the material) while R is an extensive property that depends both on the size and shape of the material. Resistivity ρ has a fixed value for a given material at a given temperature; however, for same material it has different values at different temperatures. For most substances the temperature dependence of resistivity is given as ρT = ρ0 (1 + K T )

(2.2)

In expression (2.2) ρT is the value of resistivity at absolute temperature T, ρ0 is the value at absolute zero and K is a constant, called the temperature coefficient of resistivity. The SI unit of K is Kelvin−1 . Reciprocal of resistivity is called conductivity and is denoted by Greek alphabet σ . Conductivity or specific conductance is a measure of the ease with which a current may pass through the specimen. The SI units of conductivity are reciprocal of the unit of resistivity and are ‘Siemens per metre’ which in short may be written as S/m or Sm−1 .   1 S = (2.3) σ m ρ(Ω-m) Magnitudes of the resistivity, conductivity and the coefficient of temperature (K) for some important materials are given Table 2.1. It may be observed in this table that the coefficient temperature K has both positive and negative values. For material where K is positive, the resistivity of the material increases with temperature. The resistivity of those materials that have negative value of K decreases with the increase of temperature.

2.1 Introduction

63

Table 2.1 Resistivity, conductivity and temperature coefficient for some materials Element/material

Resistivity (Ω-m) at 20 °C

Conductivity (S/m) at 20 °C

Temperature coefficient K (K)−1

Gold

2.44 × 10–8

4.10 × 107

3.40 × 10–3

Silver

1.59 ×

6.30 ×

107

3.80 × 10–3

Copper

1.68 × 10–8

5.96 × 107

4.00 × 10–3

Iron

9.70 ×

10–8

1.00 ×

107

5.01 × 10–3

Platinum

1.06 ×

10–7

9.43 ×

106

3.90 × 10–3

Gallium

1.40 × 10–7

7.10 × 106

4.00 × 10–3

Carbon (amorphous)

5.0 × 10–4

1.25 × 103

Carbon (graphite) Parallel to basal plane Perpendicular to basal plane

2.5 × 10–6 –5.0 × 10–6 3.0 × 10–3

Gallium arsenide (GaAs)

1.0 × 10–3 –1.0 × 108 1.0 × 10–8 –1.0 × 103

Germanium

4.60 × 10–1

2.17

Silicon

6.41 × 102

1.56 × 10–3

Diamond

1.00 ×

1.00 ×

Teflon

1.0 × 1023 to 1.0 × 1025

10–8

10–4

to 8.0 ×

1012

103

to 2.0 ×

− 0.5 × 10–3

2.0 × 105 –3.0 × 105 3.30 × 102

− 48.0 × 10–3 − 75.0 × 10–3

10–13

1.0 × 10–25 to 1.0 × 10–23

Elements/materials listed in Table 2.1 may be divided into three main classes; (a) Have very high or large value of resistivity of the order of ≈ 1025 –1012 (Ω-m). Such materials are called insulators. Insulators offer very high opposition to the flow of current. An ideal insulator will have infinite value of resistivity and will not allow current to pass through it. (b) Materials that have resistivity value in the range of 102 –10–2 (Ω-m) with negative value of coefficient K. These materials are called semiconductors and are extensively used for fabricating electronic devices. (c) Materials for which the resistivity has small (but non-zero) value ≈ 10–5 –10–10 (Ω-m) are called conductors, and these materials offer small opposition to the flow of current. It may also be observed that coefficient of temperature K for conductors has a positive value, meaning thereby that the resistance of a given piece of the specimen conductor will increase on increasing the temperature. (d) A fourth category of materials that is not included in the table is called superconductors. Superconductors show zero resistivity under some special conditions of temperatures, etc. Having zero resistivity or zero resistance, superconductors are very important materials as no energy is consumed/wasted in passing current through them.

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In the backdrop of atomic theory of matter, according to which all matter is made of atoms, the question arises as what is the reason for this difference in electrical behaviour of different materials? One way of explaining the difference between insulators, semiconductors, conductors and superconductors is in terms of the electron band theory of condensed matter.

2.2 Electron Energy Band Theory Most solids are crystalline; they have a regular arrangement of atoms in a pattern which is repeated in three dimensions. Electric current in solids may flow only by a net movement of charge carriers that is of electrons, under an electric field. Therefore, for the flow of current in case of solids it is essential that there are relatively free electrons that may move when an electric field is applied across it. In contrast, electric current in liquids may also flow due to the motion of ions under applied electric field, as it happens in case of electrolytes, etc. In order to understand the physics of current flow and resistivity in solids, it is required to know: (a) which electrons in the solid and (b) under what conditions these electrons may become carrier of current. For that one has to understand the electron configuration in different atoms and how electron configurations of crystals play a role in current flow. According to the quantum mechanical model of the atom, electrons of an isolated atom are distributed in discrete energy levels (or orbitals). Electron energy levels are discrete but closely spaced. Let us take the example of element Aluminium each atom of which has a total of 13 electrons. Electron configuration of Al-atom is 1s2 2s2 2p6 3s2 3p1 , and energy distribution of electrons in different energy levels for an isolated atom is shown in Fig. 2.2a. Electrons in highest energy shell (3s2 3p1 shell in this case) are called valence electrons. Valence electrons are least tightly bound with the nucleus of the atom and take part in chemical reactions, etc. If two atoms of Aluminium come very close to each other and form a diatomic molecule, the electron energy levels of the di –atomic molecule will be like the one shown in Fig. 2.2b. There will be two levels close to each other corresponding to the two atoms. If three atoms come close enough to form a tri-atomic molecule, each level will split into three closely spaced levels, and so on. In a very small crystal of Aluminium there is very large number of atoms (≈ 1025 atoms) packed very close to each other, and therefore, electron energy levels group up in bands of very closely spaced levels separated by band gaps, as shown in Fig. 2.2c. The energy band that contains valence electrons is called the valence band and the band next to it in energy the conduction band. The band energy gap between the valence and the conduction band is called forbidden energy gap. Energy band gaps are regions of energy where no energy level of the material exists, and no electron of the atoms of material may have that energy. It is obvious that in solids only valence electrons may take part in current flow, since other inner electrons are tightly bound with the nucleus. On application of an electric field to the crystal, valence electrons in the crystal are subjected to a force and

2.2 Electron Energy Band Theory

65

Fig. 2.2 a Electron configuration of an isolated atom of Aluminium. b Electron energy levels for a diatomic molecule of Aluminium. c Electron energy bands in an Aluminium crystal

try to gain energy. Valence electrons will be able to absorb energy only if there are vacant levels available at higher energies. As such valence electrons will be able to gain energy and move only if (i) there are vacant levels in the valence band and/or (ii) when valence band is completely filled then the forbidden energy gap is non-existent or small enough so that valence electrons may shift to the conduction band, which is completely empty. Thus current flow in crystalline solids is decided by the nature of the valence band and the forbidden energy gap. It may, however, be noted that a partially empty valence band may allow the flow of current but only up to an extent because of the limitation of available unoccupied levels in the valence band. On the other hand, if the forbidden energy gap is small or does not exist, then considerable number of electrons from valence band may take part in current flow as there will be large number of unoccupied levels available to the electrons in conduction band. SAQ: No electron level of the parent crystalline solid may exist in forbidden energy gap, however, if some other atoms are imbedded /mixed in the crystal structure, electron level corresponding to the other atoms may exist in forbidden energy gap or not? So far we made only a qualitative discussion of electron energy bands in a crystal. Though exact quantum mechanical calculations for many body systems are impossible, however, approximate calculations show that energy gap between two consecutive bands depends strongly on the relative separation of atoms. Figure 2.3 shows the variation of band gap with relative separation of atoms. It is clear from the figure,

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2 Electrical Behaviour of Condensed Matter

Fig. 2.3 Variation of band gap with relative separation of atoms

depending on the packing of atom in the crystal the two consecutive bands may be far apart (as at d1 ), may be separated by a small band gap (d2 ) or may overlap as at distance d3 . In the light of the above, three different situations may arise (i) forbidden energy gap is quite large, and valence shell is completely filled; (ii) valence shell is completely or partially filled but forbidden energy gap is small and (iii) forbidden energy gap between the valence and conduction bands is either very small or does not exist. Corresponding to these conditions materials may be divided into insulators, semiconductors and conductors. SAQ: What characteristics of atoms decide the size of forbidden energy gap? SAQ: Valence and conduction bands in a crystalline solid overlap, the solid will be a (insulator/conductor/semiconductor)? Choose the correct alternative.

2.3 Insulators A crystalline solid in which valence shell is completely filled and forbidden energy gap is quite large becomes an insulator. It is because in such materials valence electrons do not find vacant levels to move when an electric field is applied to the specimen. Since only the valence and the conduction bands play role in deciding the electrical nature of materials, it is customary to show only these bands in pictorial representation of band structures of solids. The band structure of a typical insulator is shown in Fig. 2.4a, where the valence band is completely filled with the maximum number of electrons it can hold and the conduction band is empty, however, the forbidden energy gap E g between the valence and conduction bands is quite large, larger than 5 eV. If the band gap is large, electrons do not acquire sufficient energy from the applied electric field to overcome the band gap and shift to conduction band

2.3 Insulators

67

Fig. 2.4 a Energy band diagram for an insulator b a specimen of length ‘d’ subjected to a voltage V develops an electric field E = V /d. c Table showing forbidden energy gap for some materials at specified temperature

where there is large number of vacant levels. Energy acquired by an electron in an electric field of strength E is eE (eV) See Fig. 2.4b. In case eE < E g , electrons will not be able to cross the forbidden energy gap, and no current will flow through the specimen. It may be noted that in an insulator there are no electrons in conduction band even at room temperature, and no electrons may jump from valence band into conduction band by the application of an electric field. Forbidden energy gap is temperature dependent, and therefore its value is quoted at a given temperature. The strength of the electric field inside any specimen may be increased by increasing the voltage V. However, when applied voltage V is increased beyond a given value, called breakdown voltage, the electric field becomes so large that it tears the binding of valence electrons with atomic core and electrons so released may constitute a current through the specimen. Hence, insulators may allow some current if the applied voltage is beyond its breakdown limit. An interesting every day example is the occurrence of lightning and thunder during raining season. Though air is insulator but under high electric field produced by oppositely charged clouds, insulation break down of air takes place producing lightning. The thickness of the insulating material plays a role in breakdown. Specific dielectric strength is often listed in terms of kilo-volt per inch (kV/in.) and for some materials listed in Table 2.2. There is no perfect insulator even the best insulator like ceramic or Teflon may allow very minute currents even below their breakdown voltage because of impurities present in their crystals and also because of leakage through surface. Plastic and rubber insulators that are flexible because of their fiber nature are used to cover

68 Table 2.2 Dielectric strength of some insulators

2 Electrical Behaviour of Condensed Matter

Material Air Porcelain Rubber

Dielectric strength (kV/in.) 50 150 600

Paper

1250

Teflon

1500

Glass

2500

Mica

5000

electric wires and cables. Ceramic insulators are generally used for high voltage applications. SAQ: What is the physical significance of specific dielectric strength of a material?

2.4 Semiconductors Those materials for which forbidden energy gap lies in the range of 0.2–3.0 eV and electron density at Fermi level of around 1020 m−3 are usually classified as common type semiconductors. These limits are not very rigid some synthetic materials that have almost 5 eV forbidden energy gap or even larger also behave as semiconductors. In principle four factors decide the electrical nature of materials, they are (i) the magnitude of the forbidden energy gap E g , (ii) the magnitudes of the crystal wave number or crystal momentum vector (k) at the bottom of the conduction band and the top of the valence band, (iii) number of available electron energy states around Fermi energy and (iv) the mobility of charge carriers. It may, however, be mentioned that all these properties are inter-related and are not totally independent.

2.4.1 Intrinsic Semiconductors Two elements in their very pure form (> 99.999… %) are natural semiconductors; they are Germanium (Ge) and Silicon (Si). These two naturally occurring semiconductors in their purest form, purity better than 1 part in billion, are called intrinsic semiconductors. Semiconductors in which some impurity atoms are deliberately mixed are termed as extrinsic or more frequently doped semiconductors. It is interesting to observe the difference in electrical properties of elements that are members of the 14th group of periodic table: 6 C, 14 Si, 32 Ge, 50 Sn and 82 Pb. These elements have many common properties, for example, all of them have four electrons in their valence shell, and all have diamond like crystal structure but their electrical properties are quite different; carbon in diamond form is one of the best insulator, Silicon is a semiconductor, Germanium is both a semiconductor and half metal, Tin

2.4 Semiconductors

69

(Sn) is a metalloid and lead (Pb) a metal. These differences in electrical behaviour originate from the difference in the size of their atoms and relative separation between successive atoms in their crystals. Relative atomic separation in crystalline structure and the number of valence electrons decides the magnitude of the forbidden energy gap. In case of diamond the forbidden energy gap is as large as 7 eV, and so it is insulator, while for Silicon and Germanium the forbidden energy gaps are, respectively, 1.02 eV and 0.66 eV, and therefore, the two elements Si and Ge are semiconductors. Tin has E g of the order of 0.01 eV, and so it is half metal. In the case of lead (Pb) the valence and conduction bands overlap, and so it is a metal. SAQ: Which two parameters of atoms of a crystalline solid decide the magnitude of the forbidden energy gap? (i) Purification of natural Silicon The starting material used for the fabrication of semiconductor devices is monocrystalline Silicon or Germanium. However, most importantly to technology, Silicon is the principle platform for semiconductor devices. Silicon is one of the most abundant elements in the crust of earth. The process to transform raw Silicon into a use-able single crystal substrate for modern semiconductor processes begins by mining for relatively pure SiO2 . The relatively pure Silicon dioxide is reduced with carbon in an electric furnace at temperatures ranging from 1500 to 2000 °C. The reduction process yields metallurgical grade (MG) Silicon of purity around 97%. However, this Silicon must be further purified to bring down impurities below the parts-per-billion level. Though several different methods may be used for further purification of MG-grade Silicon, however, the two frequently used methods are discussed here. (a) The trichorosliane method For further purification MG-Silicon is treated with HCl to form trichlorosilane (TCS) in a fluidized-bed reactor at 300 °C according to the following chemical reaction Si + 3HCl → H2 + SiHCl3 (Trichlorosilane TCS)

(2.4)

In the process of converting MG-Silicon to TCS, many impurities like Fe, Al and B get removed. The ultrapure TCS is subsequently vaporised, diluted with of H2 and flowed into a deposition reactor where it is transformed into elemental Silicon. The contamination level in this Silicon is typically of the order of 0.001 parts per billion (ppb). It may be further refined using the zone refining technique. (b) Zone refining technique William Pfann, Chemical Engineer who pioneered this technique, repeatedly passed a long tube filled with Germanium ingot horizontally through a series of electric heating coils. This melted portions of Germanium and allowed them to re-crystallise. The newly crystalline material was found to be purer than what came before. It was found that impurities become

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2 Electrical Behaviour of Condensed Matter

steadily concentrated in the molten portions, which were swept away. The technique has undergone several improvements since then. The present day zone refining technique purifies solids by passing a number of molten zones through the solid in one direction. Each zone carries a fraction of the impurities to the end of the solid charge, thereby purifying the remainder of the solid. The basic lay out of the process is demonstrated in Fig. 2.5a where the impure crystal of the element (Si or Ge) in the form of a rod is taken and a heating element that may be moved along the length of the rod from one end to the other is placed, say, at the extreme left position indicated by 1. When current is passed through the heating element the part of the semiconductor rod immediately below the heater melts while the remaining section of the rod remains in solid state. The heater is then moved slowly towards right to position 2 and then to 3 and so on. As result of the motion of heater, successive sections of the semiconductor rod go to molten state then turn back to solid crystalline states one after the other. Figure 2.5b shows a typical part of the road, where initially the heater was above section AB, and the section ABBA of the rod was in molten state. With time the heater has moved to the location above section BC of the rod, and now the section of the rod BCCB is slowly turning to molten state. In the mean time section ABBA, which was in molten state earlier, starts solidifying, and process of solidification or the process of re-crystalline starts from the face AA and slowly spreads towards the face BB. Two important points to note are as follows: (i) the mobility of impurity atoms/ ions is more in molten phase of the semiconductor compared to its solid crystalline phase, and (ii) the melting point of pure crystalline substance is always higher than the impure substance. As a result impurities diffuse from the part of the rod that is undergoing solidification to the part still in molten phase and pure material crystallises at a higher temperature than the molten material with impurities. Migration of impurities is shown by red arrows in Fig. 2.5b. In this way, impurities accumulate at the right side edge BB of section ABBA. When section BCCB of the rod changes to

Fig. 2.5 a Zone refining technique for improving purity of semiconductors. b Impurities shift from solid phase to the molten phase

2.4 Semiconductors

71

molten state, impurities sitting at face BB migrate to the molten section BCCB. With the motion of the heating element to the next section of the road, re-crystallisation in molten section BCCB starts from face BB and spreads towards face CC. Impurities in section BCCB collect at face CC, increasing the purity of the re-crystallised solid. Several cycle of motion of the heating element only in the direction from left to right leaves the semiconductor ultrapure. SAQ: Two identical pieces of Silicon semiconductor are given, one is ultrapure, and the other has high amount of impurities. If both pieces are heated which one will melt first and why? (c) Poly crystal to monocrystal The ultrapure elemental Silicon is, however, poly crystalline and required to be converted into monocrystalline form. To achieve that, the polycrystalline Silicon is mechanically broken into 1–3 in. chunks and these chunks then undergo surface etching and cleaning in a clean room environment. These chunks are then packed into a quartz crucible for melting in a Czochralski furnace. A mono crystalline Silicon seed is loaded on seed shaft in the upper chamber of Czochralski furnace. Slowly the seed crystal is lowered and dipped up about 2 mm depth in the molten Silicon. The seed is then withdrawn back up to the surface of the melt, and the melt is allowed to solidify at the surface. While pulling back the seed, the crucible and the seed are rotated in opposite directions to grow an almost round crystal. For proper growth of the monocrystalline crystal, the furnace must be very stable and free from vibrations. Speed of pulling up the seed and furnace temperature is two critical parameters. Once the growth process is complete, the monocrystal is left for cooling inside the furnace for a considerable period of time up to 7–8 h. The monocrystalline Silicon ingot so developed has the same orientation as the seed crystal (Fig. 2.6). In float zone crystal growth technique the end of a long polysilicon rod is locally melted and brought in contact with a monocrystalline Silicon seed. The molten zone is slowly moved by moving the heating element (like in zone refining) leaving behind a ultrapure Silicon monocrystal. SAQ: Why only ultrapure monocrystal semiconductors are used for fabricating electronic devices? SAQ: What is the principle on which the zone refining technique is bases? (d) Monocrystal to wafers The monocrystal (also called ingot) is then undergo several precision mechanical and chemical processes to cut it into wafer of desired size, shape and other requirement. First of these steps is multiwire slicing (MWS). As the name suggests, a multiwire slicing machine consists of two cylindrical rollers with very fine grooves numbering from several hundred to thousand itched on them. Ultrafine steel wires of diameter around 2 mm are stretched

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2 Electrical Behaviour of Condensed Matter

Fig. 2.6 Sketch of a Czochralaski furnace

in these roller groves. Both rollers are connected to the same motor and may rotate at high speed in same direction. Steel wires are painted with liquid abrasive paint and work as sharp cutting blades. When a monocrystal is pressed through the wire blades, wafers of Silicon are cut through the crystal. A rough sketch of multiwire slicing machine is shown in Fig. 2.7. The rough drawing is just to understand the principle of working, an actual multiwire saw is much complicated which has several cylindrical derives to guide the motion of wire blades and wire spools to maintain a continuous supply of new wire. Silicon and Germanium wafers are used for fabricating semiconductor devices. (ii) Fermi energy and Fermi level Electron being a particle having spin 1/2 ℏ is a Fermion and obeys quantum mechanical statistics called Fermi–Dirac statistics. According to this statistics at absolute zero temperature (0 K) electrons start filling lowest energy states of the system (like an atom) obeying exclusion principle, filling in to higher states after exhausting all lower energy states. The resulting structure of electrons is termed as ‘Electron Sea’ or ‘Fermi Sea’. The surface of this sea is called Fermi surface or level. This means at absolute zero temperature no electron can have energy larger than the Fermi energy Ef of the Fermi level. According to

2.4 Semiconductors

73

Fig. 2.7 Sketch of a multiple wire slicing machine

Fermi–Dirac statistics the probability ‘p(E)’ of finding an electron with energy E, at absolute temperature T (Kelvin K) is given as p(E) =

1 E−E f )/kT ( 1+e

(2.5)

Here, E f is the Fermi energy for the system and k is Boltzmann constant. Equation (2.5) gives the theoretical probability of finding an electron with energy E at temperature T (K). In an actual system there will be an electron with energy E only if an electron level with energy E actually exists. Often it so happens that the given system does not have an allowed energy level at some energy, for example, in case of crystalline solids no energy levels for electrons of their atoms exist in forbidden energy gap, in that case Eq. (2.5) will still give some probability of finding an electron for a level in forbidden energy gap. When one calculates probability of finding an electron with energy equal to Fermi energy, i.e. if E = E f then Eq. (2.5) gives   p Ef =

1 1 1 = = E f −E f )/kT ( 1 + 1 2 1+e

(2.6)

Expression (2.6) tells that probability of finding an electron with Fermi energy E f is 0.5, and that this probability does not depend on temperature. So if temperature is 0 K or 100 K, the value of p(E f ) will remain 0.5. Energy band picture for an intrinsic semiconductor at absolute zero and at some higher temperature T > 0 K is shown in Fig. 2.8. At T = 0 K, valence band contains all valence electrons, and the conduction band is empty, which has no electrons. Therefore, the probability of finding an electron at the bottom of the conduction band is 0, while that of finding an electron at the top of the valence band is 1. The probability of finding an electron

74

2 Electrical Behaviour of Condensed Matter

Fig. 2.8 Band structure of a semiconductor at a Zero Kelvin (0 K) b at temperature T (K) > 0 (K)

with probability p(E f ) = 0.5 will be at a point midway between the bottom of the conduction band and the top of the valence band, i.e., in the middle of the forbidden energy gap. Therefore Fermi energy level for an intrinsic semiconductor lies in the middle of the forbidden energy gap. Further, since p(E f ) does not depend on temperature, the Fermi level for intrinsic semiconductor will remain in the middle of the forbidden energy gap at all temperatures. Figure 2.8a shows the conduction band, valence band and Fermi level for intrinsic semiconductor at absolute zero temperature. As may be observed in this figure, all valence electrons of the intrinsic material at absolute zero temperature are in valence band, and the conduction band is totally empty. When temperature of an intrinsic semiconductor is raised above 0 K, electrons in valence band absorb energy from the surrounding environment and if the energy gained by a valence electron becomes equal or larger than the forbidden energy gap E g , the electron jumps to the conduction band leaving an electron vacancy in the valence band. This vacancy of electron which behaves as a positive charge is called ‘hole’. Hole is just a fictitious entity but it is very useful in understanding the physics of semiconductors. The concept of hole originates from the quantum mechanical treatment of current flow in semiconductors. It so happens that Schrödinger’s equation when applied to a semiconductor, under some approximations, separates out into two independent components one describing the motion of electrons and the other the motion of a positively charged particle. The absence of electron in the valence band is thus assumed to be the positive particle, hole, the motion of which is described by the second component of Schrodinger’s equation. For all practical purposes hole is treated as a positively charged particle with charge + 1 e and a mass very nearly (slightly more) equal to the mass of an electron me .

2.4 Semiconductors

75

Electrons that have shifted to the conduction band at T > 0 K behave as delocalised electrons or free electrons that are not attached to any particular atom of the semiconductor crystal. Conduction band electrons may be compared to the delocalised electron cloud in case of metals which is associated with the crystal lattice but not to any individual atom. When an intrinsic semiconductor specimen at T > 0 K is subjected to an electric field by applying a voltage across it, a current consisting of conduction band electrons and valence band holes may flow through the specimen. Thus at absolute zero a pure or intrinsic semiconductor behaves as an insulator while at a temperature T > 0 K, the same specimen behaves as a conductor. Further in a pure semiconductor specimen at any temperature T > 0 K the number of electrons in conduction band is always equal to the number of holes in valence band, also the number of electrons in conduction band (and holes in valence band) increases with the increase in the temperature T as more valence electrons may jump to the conduction band at higher temperature. Fermi level assumes added importance in case of semiconductors, it may be treated as a reference of energy; energy of electrons in conduction band increases as one moves upwards from the Fermi level, while the energy of holes increases as one goes downwards from the Fermi level. That means that an electron at the top of the conduction band is most energetic while a hole at the bottom of valence band has largest energy. Further, the probability of finding an electron say X units of energy above the Fermi level is same as the probability of finding a hole same X units below the Fermi level. Figure 2.9 shows the conduction and valence bands for intrinsic Germanium and intrinsic Silicon crystals at absolute zero and at temperature T > 0 K. Figure 2.9 is self-explanatory, telling that both intrinsic semiconductors have empty conduction bands at T = 0 K, and hence behave as insulator. However, the two points of interest are (i) at T > 0 K, the number of free electrons in conduction band in Ge is larger than the number of free electrons in Si, because of the smaller value of its forbidden energy gap. (ii) The energy of free electrons in conduction band increases vertically upward from the Fermi level, while energy of holes in the valence band increases vertically downwards from the Fermi level. Therefore, an electron at the top of the coduction band has highest energy amongs free electrons, while a hole at the bottom of the valence band is the one with highest energy amongst holes. Further in a specimen of an intrinsic material the number of holes is always equal to the number of free electrons in the conduction band. At any temperature T > 0 K, in an intrinsic material, new holes and free electrons keep generating on one hand, and on the other hand they also get annihiliated when a free electron falls back in valence band and recombines with a hole. The process of electron + hole generation and annihilation goes on simultaneously in such a way that the average number of holes and free celectrons remain constant over a period of time. SAQ: What is Fermi level and what is its physical significance?

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2 Electrical Behaviour of Condensed Matter

Fig. 2.9 a Intrensic Ge crystal at 0 K. b Intrensic Si crystal at 0 K. c Intrensic Ge crystal at T > 0 K. d Intrensic Si crystal at T > 0 K

SAQ: Do holes may move in intra atomic space like free electrons? Justify your answer.

2.4.2 Covalent Band Picture of Intrinsic Semiconductor In the previous section we studied the electron band theory and its application in distingushing different types of crystalline materials according to their electrical properties. An other equivalent way of describing electrical properties of crystalline solids is by using the covalent bond picture of these materials. Atoms of Silicon and Germanium in their intrinsic crystals are held together by covalent bonds. Both these atoms (Ge and Si) have four valence electrons in their valence shells. Electronic configuration and arrangnment of electrons in a 14 Si atom are shown in Fig. 2.10. Since only the valence electron of an atom take part in bonding and inner electrons plus nucleus does not play any role, it is convinient to represent an atom by a core (that has nucleus and inner electrons) with positive charge equal to the number of valence electrons, (+ 4e) in case of Silicon, and four valence electrons. As a matter of fact any atom with four valence electrons (like Si, Ge, Sn, Pb) may be represented by a core of + 4e charge and four electrons. An atom for example of Aluminium which has three valence electrons may be represented by a positively charged core having

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77

Fig. 2.10 Representing an atom of 14 Si with four valence electrons by a core of + 4e charge surrounded by four valence electrons

charge + 3e and three valence electrons and a penta valent atom (like phosphorus) by a core of + 5e charge and five electrons. In covalent bonding atoms share their electrons; in case of Si and Ge four neighbouring atoms share their one electron each forming the crystal lattice. Figure 2.11 shows the covalent bonding in Silicon or Germanium intrinsic crystal at absolute 0 K temperature. As may be seen in the figure, at T = 0 K all covalent bonds are intact and all valence electrons of each atom are shared by neighbouring four atoms. Since electrons are held in covalent bonds because of the force of attraction of nearby positively charged

Fig. 2.11 Pictorial representation of covalent bonding in intrinsic Silicon or Germanium crystal at absolute zero temperature

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cores, they cannot move even when an electric field of moderate strength is applied to the crystal, hence the crystal behaves as an insulator at T = 0 K. Covalent bonds are characterised by bond energy; the energy by which electrons are held within the bond. The covalent bond energy of Silicon is 1.08 eV and that of Germanium 0.66 eV. It means that if an electron in the covalent bond of Silicon crystal somehow gets energy either equal to 1.08 eV or large; it may break the covalent bond and will become a delocalized or free electron which may hop from one atom to the other in the intra atomic space of the crystal. It may be observed that the covalent bond energy is simply equal to the forbidden energy gap of band theory. Electrons may get energy in several ways, like by heating or by putting them in an electric field etc. If an intrinsic Silicon or Germanium crystal is heated to say some temperature T > 0 K, its electrons in covalent bonds will acquire temperature T > 0 K and will get thermal energy ≈ kT, where k is Boltzmann constant (k = 8.6 × 10–5 eV per Kelvin). The most likely form of this thermal energy is kinetic energy associated with vibrations, electrons which were stationary in covalent bonds at T = 0 K, starts vibrating when the temperature of the specimen is increased. When thermal energy of the electron increases beyond the bond energy (1.08 eV for Si and 0.66 eV for Ge), it may break the bond and come out of the bond becoming a delocalised electron, leaving a hole in the covalent bond. The structure of an intrinsic semiconductor crystal (Si or Ge) at T > 0 K is shown in Fig. 2.12 The process of bond breaking resulting in creation of electron–hole pairs and the opposite process of recombination of electron–hole pairs to remake some of the broken bonds simultaneously keep going in the crystal at temperature T > 0 K. Ultimately, equilibrium is reached when the rate of new electron–hole pair creation becomes equal to the rate of recombination of electron–hole pairs. Covalent bond breaking →→ free electron + hole

(2.7)

Free electron + hole →→ reformation of covalent bond

(2.8)

At equilibrium, rate of reaction given by Eq. (2.7) becomes equal to the rate of process represented by Eq. (2.8). As a result at any temperature T > 0 K; the average number of free electrons and holes per unit volume of the semiconductor becomes constant (Fig. 2.12). Further, at equilibrium the average number per unit volume (called the number density or carrier concentration) of free electrons and holes in the crystal is equal. The average number density of electrons and holes in an intrinsic semiconductor is equal and constant at a fixed temperature; however the average number density increases with the rise of temperature. Further, it is a common practice to call free or delocalised electrons simply as electrons and instead of saying average number density, simply to say number density. Though obvious, but one must remember that free electrons (these are the electrons which in electron band theory shifts to the conduction band on acquiring of energy) are free to move within crystal while holes

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Fig. 2.12 Breaking of covalent bonds in intrinsic semiconductor at T > 0 K

are always bound within the covalent bond and can move from one covalent bond to the next. If the (average) number densities of (delocalised) electrons and holes in an intrinsic semiconductor at temperature T are, respectively, denoted by n ie and n ih , then n ie = n ih

(2.9)

The value of free electron number density n ie for Silicon at 300 K (nearly room temperature) is of the order of ≈ 1.08 × 1010 cm−3 . When a specimen of intrinsic semiconductor at T > 0 K is subjected to an electric field by applying a voltage across its two opposite faces, the free electrons in the specimen moves within the enter atomic space of the crystal in a direction opposite to the electric field, as shown in Fig. 2.13. Holes that are bound in covalent bonds also shift from one covalent bond to the next in the direction of the imposed electric field (see Fig. 2.13). Resultant current I in the circuit is the sum of the electron and hole currents. Therefore, both the electrons and holes participate in current flow in a semiconductor. SAQ: Consider Fig. 2.13 and draw three successive steps showing the motion of a hole under applied voltage V.

2.4.3 Doped or Extrinsic Semiconductors Addition of a very small amount (of the order of 1 part in 107 parts) of impurities in an intrinsic semiconductor crystal may drastically change its electrical conductivity,

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Fig. 2.13 Motion of free electrons and holes under electric field

Fig. 2.14 Block diagram of hybrid medium current ion implanter

optical and structural properties. The crystal with controlled impurity added to it is called a doped or extrinsic semiconductor crystal. The process using which small amount of impurities is added in a controlled way is called doping. The reason why deliberately added impurities are mixed only in small amount is (i) to avoid any breakdown in crystal structure and (ii) addition of very small amount of impurity is sufficient to change the conductivity of the intrinsic semiconductor to the desired value. Addition of excessive impurity may turn a semiconductor into a conductor. Further, only two types of impurities are added; impurity atoms with either

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5 valence electrons (pentavalent) or with 3 valence electrons (trivalent). Pentavalent atoms that are often used as dopant are arsenic (As), phosphors (P) and Antimony (Sb). Similarly, trivalent impurities that are often used for doping are boron (B), Aluminium (Al), gallium (Ga) and indium (In). Pentavalent impurities are called donor impurities while atoms of trivalent impurities are termed as acceptor impurity.

2.4.4 Doping Technology Doping refers to the process of introducing impurity atoms in a semiconductor in a controllable manner in order to define the electrical properties of the semiconductor. Doping of an intrinsic semiconductor by donor (pentavalent impurity atoms) and acceptor (trivalent atoms) atoms may change the free electron and hole concentrations (number density) in the doped semiconductor from 1013 cm−3 to 1021 cm−3 . Controlled doping may also change the spatial distribution of carriers (free electrons and holes) in the specimen semiconductor quite accurately. Spatial variation in carrier concentration is required in fabricating devices like pn junction diode and transistors. (i) Ion implantation technology Ion implantation is the front line technology of the day for semiconductor doping. In this technology an accelerated beam of the dopant ion (like that of boron ion or phosphorous ion etc.) of controlled energy and flux is made to hit the semiconductor wafer/monocrystal. Depending on the energy of the beam, the incident dopant ions penetrate up to a certain depth in the target wafer/monocrystal. The number density of dopant atoms in the target wafer may be controlled by controlling the flux and time of irradiation by dopant. The incident dopant ion beam may be masked externally by placing sufficiently thick shields in the path of the ion beam; internally within the semiconductor channelling may take place by crystal structure that is not transparent to the ion beam. Internal channelling is called self-aligned implant. Main reason for the popularity of ion implantation technique is the accuracy with which dopants may be distributed in the target semiconductor. Ion implant technology got a big boost with the development of high current ion accelerators. Most of commercial ion implanters have a linear accelerator which may accelerate ions up to several MeV of energies. Ion source, which produces the desired ions and accelerating column, where ions are accelerated to higher energies, is the two main components of an accelerator. Earlier accelerators used cold cathode ion source that could deliver ion currents of only few hundred micro appears (10–6 A). However, in the year 1970 a new type of ion source called hot cathode ion source was developed that may deliver ion currents of few hundred milli appears (10–3 A) at around 80 keV ion energy. Today, different ion implanter machines are available that cover the entire range of both energies and beam

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current requirement for semiconductor fabrication industry. These machines may be grouped as medium current, high current and special implanters. Most of the medium current implanters that deliver ion beam currents of few mA use the concept of hybrid scanning. These systems have a hot cathode ion source where the ion beam is generated. The ion beam then passes through a system called mass analyser. Mass analyser separates out ions of different masses and focuses them at different points in its focal plane. Any undesired ion that may be present in the ion source may be rejected at this stage. The beam of selected desired ions is then accelerated/decelerated in an accelerating column. The beam coming out of the accelerating column is made to go through an energy filter that ensures that ions of only the desired energy may travel further. The energy filtered ion beam then passes through a scanning magnet that scans and collimates the ion beam. In one-dimensional scan, the scan magnet scans the collimated monoenergetic and highly pure ion beam in one direction say in horizontal direction. That means that the ion beam is slowly swept in horizontal direction at a certain sweep rate. The target holder which holds the semiconductor wafer is scanned in vertical direction at the same rate. As a result of this hybrid scan, the sample wafer is uniformly irradiated by the ion beam. Block diagram of a hybrid scan medium current ion implanter is shown in Fig. 2.14. In normal case ion beams of energies ranging from about 100 eV to 3 MeV and irradiation dose (number of incident ion per unit area) ranging from 1011 to 1016 ions cm−2 are used. After ion implantation the crystal structure is damaged by incident ions. Further, the dopant lions are electrically inactive as majority of them do not occupy positions in crystal lattice. The ion implanted wafer/crystal is then subjected to thermal annealing process that restores the damaged crystal structure and imbed implanted ions in crystal lattice turning then electrically active. Figure 2.15 shows the profile of implanted dopant ions inside the semiconductor wafer. Sides of the wafer are covered by mask material that does not allow incident ions to enter the semiconductor and define the volume of the wafer exposed to implantation. Since charged ions have a certain range in semiconductor material, the maximum deposition of dopant ions takes place at a depth equal to the range of the incident ion. Range depends on the energy of the ion and the material of semiconductor. Therefore, by selecting the proper energy of incident ion beam the depth of maximum deposition of dopant may be adjusted. SAQ: Explain why the impurity ion concentration shows a peak at certain depth in the semiconductor when impurity is introduced by ionimplanting method? (ii) Diffusion technology

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Fig. 2.15 a Process of ion implantation. b Dopant profile in the ion implanted semiconductor wafer. Highest dopant concentration at range equivalent depth

Diffusion is another important method of introducing dopant atoms in semiconductor material in a controlled way. In diffusion process dopant atoms are introduced in the wafer from gas phase. The diffusion process may be considered as a series of atomic movement of the dopant atom in crystal lattice. Figure 2.16 shows the two mechanisms of diffusion mechanism. When temperature of a semiconductor wafer/monocrystal is raised, the host atoms (atoms of semiconductor material) start vibrating. When kinetic energy gained by a particular host atom accede its binding energy with crystal lattice, the atom leaves the lattice and moves to the interstitial space. Thus a vacancy is created in crystal lattice that may be occupied by the dopant atom (see Fig. 2.16a) which are in gaseous phase. The other method of doping via diffusion is that initially the dopant atom diffuses in interstitial space, as shown in Fig. 2.16b and later on occupies a site in crystal lattice when on further heating a vacancy is created. Diffusion is the process in which atoms/molecules moves from region of higher concentration to the region of lower concentration. Therefore, concentration gradient is the driving force for diffusion. When a certain area of the surface

Fig. 2.16 a Vacancy mechanism of diffusion. b Interstitial mechanism of diffusion

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Fig. 2.17 a Process of diffusion. b Profile of diffused atoms

of a semiconductor wafer is exposed to dopant atoms in gaseous phase, dopant atoms diffuse from the region of higher concentration (container of gaseous dopant atoms) into wafer, where initially there were no dopant atoms (see Fig. 2.17a). The flux F (number of atoms per unit area per unit time). Is proportional to the gradient of concentration of dopant atoms, i.e. for a one-dimensional case F ∝−

dC dC or F = −D dx dx

(2.10)

is the concentration gradient of dopant atoms in direction x and Here, dC dx the negative sign signifies that motion of diffusing atoms is in the direction of higher to lower concentration. The constant of proportionality D is called diffusion coefficient. With diffusion of dopant from gaseous phase container into semiconductor wafer, the concentration difference on the two sides will decrease and assuming D to be constant, and flux of diffusing atoms will also decrease. Ultimately, diffusion of dopant atoms will stop when concentration of atoms in container and in wafer will become equal. The depth of diffusion in wafer essentially depends on the temperature of dopant atoms, higher the temperature larger the diffusion depth. Depth profile of diffused atoms in the wafer at a given temperature is shown in Fig. 2.17b. According to the law of conservation of matter, the change of dopant concentration with time must be equivalent to the local decrease of the diffusion flux, in the absence of a source or sink, therefore,   ∂F ∂ ∂C ∂C ∂ 2C ∂C =− = D or =D 2 ∂t ∂x ∂x ∂x ∂t ∂x Equation (2.11) is often called Fick’s Second law of diffusion.

(2.11)

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Boron is the most common trivalent impurity dopant in Silicon; whereas arsenic and phosphorous are used extensively as pentavalent impurities. These three elements are highly soluble in Silicon with solubilises exceeding 5 × 1020 atoms per cc in the diffusion temperature range between 800 and 1200 °C. These dopants can be introduced via several means, including solid sources (BN for B, As2 O3 for As and P2 O5 for P), liquid sources (BBr for B, AsCl for As and POCl3 for P) and gaseous sources as B2 H6 , AsH3 and PH3, respectively, for boron, arsenic and phosphorous. SAQ: Two exactly identical wafers of Silicon are doped using diffusion technology. The temperatures of diffusing gas are different for the two wafers. What will be the difference in the profile of doped impurity in two cases? (iii) Doping at monocrystal growth stage Impurities in controlled amount may also be introduced in a semiconductor monocrystal while it is being grown from poly crystalline melt. If calculated amount of either tri or pentavalent dopants is mixed with the molten polycrystalline a part of these impurities enter the monocrystal.

2.4.5 n and p Type Semiconductors (i) n-type semiconductor Doping an intrinsic semiconductor wafer with pentavalent impurity, like phosphorus, arsenic or Antimony, results in n-type semiconductor. As a result of controlled doping the pentavalent impurity atoms occupy regular places in crystal lattice. Since the number of doping atoms is much smaller than the number of intrinsic atoms (Ge or Si), the impurity pentavalent atoms are surrounded by atoms of intrinsic material on four sides. The impurity atom is surrounded by four covalent bonds of neighbouring atoms and four out of five valence electrons of impurity atom are held in these four covalent bonds. However, the fifth electron of impurity atom remains loosely attached with the parent atom. As a result at absolute zero temperature the n-type material behaves like an insulator, just like intrinsic material. At temperature T > 0 K, the loosely bound fifth electron of pentavalent impurity atom becomes free, leaves the parent atom and like other conduction band electrons may take part in current flow. With the loss of fifth electron the impurity atom gets ionised and becomes a positive ion. The covalent bond picture of ntype semiconductor is shown in Fig. 2.18. At any temperature T > 0 K, each impurity atom gives or donates one additional free electron to the conduction band, therefore, the pentavalent impurity is called donor impurity. Since the concentration of donor impurity is of the order of 107 impurity atoms per cubic centimetre, the number of additional free electrons provided by donor impurity

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Fig. 2.18 Covalent bond structure of an n-type semiconductor

is of the order of 107 free electrons per cm3 . Apart from free electrons given by impurity atoms, there are also some electron and holes that are created by the breaking of covalent bonds. But the number of electrons given by donor impurity per unit volume is much larger than the number of free electrons produced by covalent bond breaking. Thus, at T > 0 K, an n-type semiconductor has (a) large number of free electrons due to the ionisation of donor atoms + (b) large number of positive ions of donor atoms that are fixed in crystal lattice and cannot move + (c) few electrons due to bond breaking and + (d) few holes due to covalent bond breaking. Both free electrons and holes take part in current flow if an n-type semiconductor is subjected to electric field by applying voltage across it. Since the number of free electrons in n-type semiconductor is much larger than the number of holes, free electrons are called majority carriers and holes the minority carriers. Electron band diagram of an n-type semiconductor is shown in Fig. 2.19ii where the band picture for an intrinsic semiconductor is also given for comparison. Three major differences between the two diagrams may be observed in Fig. 2.19; (a) the conduction band of n-type semiconductor has large number of free electrons, most of which are due to the ionisation of donor impurity atoms. (b) There is a donor level, an energy state for impurity atoms, just below the bottom of conduction band in forbidden energy gap. Energy difference between the bottom of conduction band and donor level is equal to the binding energy of the fifth electron of donor atom which could not be accommodated in covalent bonds. It sometimes appear confusing that how there could be any level or energy state in forbidden energy gap? At this point it is important to realise that forbidden energy gap region is restricted only for energy states or levels of parent intrinsic atoms, atoms of Si or Ge cannot have energy levels in forbidden energy gap; however, other impurity atoms may have their energy states in the region of forbidden energy. Further, Fig. 2.19 refers to a temperature T > 0 K, the temperature at which most of the impurity donor atoms have ionised and thus one

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electron from each donor atom has jumped to the conduction band leaving a positive donor ion at donor level. Positive donor ions in Fig. 2.19 are represented by the + sign. (c) An other very crucial observation from the figure is that Fermi level of n-type material has shifted up words, towards the conduction band, from the middle of forbidden energy gap. This has happened because of the large number of electrons in conduction band as compared to the holes in valence band. In intrinsic material at any temperature T > 0 K, the number of electrons in conduction band is exactly equal to number of holes in valence band as both are generated by the process thermal braking of covalent bonds. An n-type semiconductor contains large number of free electrons as majority carriers, few holes from covalent breaking, as minority carriers (see Fig. 2.20) and large number of immobile positive ions of donor atoms fixed in crystal lattice.

Fig. 2.19 Electron band diagram of i an intrinsic semiconductor at T > 0 K ii an n-type semiconductor at T > 0 K Fig. 2.20 Pictorial representation of n-type semiconductor

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(ii) p-type semiconductor Doping of an intrinsic semiconductor by some trivalent impurity, like boron, Aluminium, or indium results in a p-type semiconductor. A trivalent impurity atom when trapped by tetravalent intrinsic atoms on all four sides develops four covalent bonds with four neighbouring atoms for the continuity of lattice structure. However, one of these covalent bonds has an electron vacancy as trivalent atom has only three electrons to shear in four covalent bonds. This electron vacancy is nothing but hole. Therefore each trivalent impurity atom has one hole in one of the covalent bonds associated with it. At absolute zero temperature all electrons in covalent bonds are fixed at their locations as they have no energy. When temperature of the p-type material is raised above 0 K, electrons may move within covalent bonds and by chance an electron from some other nearby covalent bond may drop in the vacant place, the hole associated with trivalent impurity atom. This impurity atom will then have all the four covalent bonds around it filled with electrons, and therefore the trivalent impurity atom will become a negative ion, while hole will be shifted in some other covalent bond. Thus at T > 0 K holes associated with trivalent impurity atoms become mobile and each impurity atom becomes a negative immobile ion fixed in crystal lattice. Since the trivalent impurity atoms accept an electron and add one additional hole per impurity atom, they are called accepter impurity. Covalent bond picture of a p-type semiconductor is given in Fig. 2.21. A p-type semiconductor at T > 0 K has large number of mobile holes which play the role of majority carriers while few free electrons produced by bond breaking work as minority carrier. Energy band diagram of an intrinsic and a p-type semiconductor at T > 0 K are shown in Fig. 1.22. It may be observed in this figure that for p-type material (i) conduction band has few free electrons essentially from covalent bond breaking

Fig. 2.21 Covalent bond picture of a p-type semiconductor

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Fig. 2.22 Energy band picture at T > 0 K for a intrinsic semiconductor. b p-type semiconductor

that are minority carriers; (ii) the Fermi level is below the middle of forbidden energy gap towards the valence band; (c) there is an acceptor level just above the top of valence band holding negative ions of impurity acceptor atoms; (d) there are large number of holes in valence band most of which are created by the ionisation of trivalent impurity acceptor atoms (Fig. 2.22). As in case of n-type semiconductor, the semiconductor is over all neutral, similarly, p-type material is also over all neutral. As shown in Fig. 2.23, a p-type semiconductor has holes as majority carriers, electrons produced by bond breaking as minority carriers and negative ions of acceptor impurity atoms fixed in crystal lattice as immobile charges. The majority charge carriers, electrons in case of n-type semiconductor and holes in p-type semiconductor are released by the impurity atoms only when Fig. 2.23 Pictorial representation of p-type semiconductor

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these atoms get ionised. Those impurity atoms that get ionised at room temperature and release majority carriers are called shallow impurities. The ionisation energy for shallow impurities is of the order of ≈ kT where k is Boltzmann constant and T ≈ 300 K.

2.4.6 Compensated Semiconductor When an intrinsic semiconductor is simultaneously doped by both the shallow donor and shallow acceptor impurities it is termed as compensated semiconductor. If N D and N A , respectively, denote the concentration of donor and acceptor impurities, when all impurity atoms have ionised, and N D = N A , then the compensated semiconductor behaves as an intrinsic semiconductor. However if N D > N A , compensated material behaves as n-type and if N A > N D then as p-type semiconductor. SAQ: In order to fabricate n-type and p-type semiconductors the intrinsic material is doped using compensated method; can you give a reason why simultaneous doping with donor and acceptor impurities is better than doping by a single type of impurity?

2.4.7 Degenerate and Non-degenerate Semiconductors When an intrinsic material is doped with small amount of impurity, the impurity atoms in crystal lattice are far apart and do not interact with each other. Such doped semiconductors are termed as non-degenerate semiconductors. The impurity level (donor level in case of n-type and acceptor level in p-type) in non-degenerate semiconductor is discrete and sharp. However, if impurity concentration is relatively high and impurity atoms in crystal lattice are near to each other, they interact and the impurity level does not remain a single discrete level but it becomes a band of many energy levels and sometimes this impurity band may overlap with the nearby (conduction band in case of n-type and valence band in case of p-type semiconductor) band of the semiconductor. Such semiconductors are called degenerate semiconductors. Most of the doped semiconductors that are frequently used are non-degenerate type. SAQ: Draw a rough sketch of energy band picture for a degenerate p-type material.

2.4.8 Direct and Indirect Semiconductor The free electrons in the conduction band of a semiconductor are not really free; their motion is constrained by the periodic potential of ions in crystal lattice. The effect of periodic potential is included by assigning an effective mass to electron, denoted by

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m ∗n . Similarly, the motion of holes in covalent bonds is also restricted, and therefore, holes are also assigned an effective mass m ∗h . The energy momentum relation for a conduction band electron may be written as E=

pc2 2m ∗n

Here pc is crystal momentum along a given crystal direction defined by the Miller index. In some semiconductor materials, like Silicon (Si) and gallium arsenide (GaAs), the maximum of valence band energy and the minimum of the conduction band energy lie along the direction defined by pc = 0. As a result in such semiconductors a transition across forbidden energy gap requires just the absorption or emission of energy. Those semiconductors for which the maximum of valence band energy and the minimum of conduction band energy lie in the same crystal direction or have same value of pc are called direct semiconductors. The time response of direct semiconductors to photon absorption/emission (Si and GaAs) is fast, and therefore they are frequently used in optoelectronic devices. On the other hand, those semiconductors in which the maximum of valence band energy and the minimum of conduction band energy have different values of crystal momentum pc are called indirect semiconductors. Transition across forbidden energy gap in indirect semiconductors is slow because of the different values crystal momentum. SAQ: What may happen when a semiconductor absorbs a photon? SAQ: The photon absorption and emission processes in direct semiconductors are fast; why?

2.4.9 Compound Semiconductors Several compounds of trivalent and pentavalent elements, like gallium nitrate (GaN), gallium arsenide (GaAs), indium phosphate (InP), etc., behave as semiconductor. Most of these compound semiconductors are direct semiconductors and have forbidden energy gaps in the range of 1.4–1.5 eV. These compound semiconductors are, therefore, very suitable for fabricating optoelectronic devices. A large band gap reduces background noise and makes devices stable. Covalent band structure of a compound semiconductor (InP) is shown in Fig. 2.24. The total number of valence electrons in the two atoms (3 + 5) is eight which distribute themselves in four covalent bands as shown in the figure. SAQ: Give a reason why GaAs is often used for fabricating optoelectronic devices.

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Fig. 2.24 Covalent band structure of a compound semiconductor

2.4.10 Current Flow in Semiconductor Both intrinsic and semiconductors doped with shallow impurities have free electrons and holes at room temperature. Paul Drude, a German physicist, argued that free electrons in semiconductors and also in conductors (metals), under some assumptions may be treated as molecules of an ideal gas. He argued that kinetic theory of gases which in principle is applicable to an ideal gas may also be applied to free electrons. The law of equipartition of energy, which follows from kinetic theory, says that 1/2 kT of energy is associated with each degree of freedom, a molecule of an ideal gas that has three degrees of freedom, at temperature T > 0 K possesses 3/2 kT of kinetic energy. As such a free electron in semiconductor at room temperature ≈ 300 K will have roughly 6.21 × 10–21 J as kinetic energy. If Vthe is the speed of electron at room temperature and 9.1 × 10–31 kg the mass of electron, then 1 2 × 9.1 × 10−31 × Vthe = 6.21 × 10−21 2

(2.12)

Solution of the above equation gives V the , the thermal velocity of a free electron at room temperature, to be of the order of 1.17 × 105 m/s. This means that at room temperature free electrons in a semiconductor are moving with high speed of the order of 105 m/s. If no external potential is applied to a semiconductor and if there is no charge gradient within the semiconductor, then free electrons in it will be moving in random directions with velocities of 105 m/s. Fast moving electrons frequently collide with vibrating crystal lattice (crystal lattice also vibrates because of temperature), and at each collision its direction of motion gets changed. Hence in absence of any external voltage and any static charge gradient, the net effect of

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lattice-electron collisions is that on average equal number of electrons move in all directions at a given instant, and hence there is no net current in any direction. Current may, however, be made to flow in a semiconductor piece by many processes. Some of the important processes that may cause current to flow in a semiconductor are (a) application of external voltage that generates an electric field in semiconductor, (b) accumulation of charge at some location that produces static charge gradient within semiconductor, (c) injecting additional excess charges, (d) high field operation. We shall, however, discuss in brief only first two mechanisms of current flow in a semiconductor. (i) Drift current Fig. 2.25 shows apiece of semiconductor of length D to which  an external voltage V is applied to establish an electric field E = VD within the semiconductor in X direction. Each free electron in semiconductor experiences a force F = −(eE) in direction opposite to the direction of electric field. Force F produces acceleration a = F/m ∗n in the motion of electron. Here m ∗n is the effective mass of the electron. If τ is the time between two successive collisions of the electron with vibrating crystal lattice and if it is assumed that electron comes to a momentary rest after each collision, then the velocity gained by the electron in direction opposite to the electric field v nD , called drift velocity, is given by v nD = a.τ = −

e.τ e.τ ε = −μe ε, where μe = ∗ ∗ mn mn

(2.13)

Drift velocity, velocity acquired by an electron in an external electric field, is proportional to the strength of the electric field, and the proportionality constant μe is called electron mobility. Electron mobility depends on collision time τ , which in turn depends on temperature. Similarly one may define hole mobility μh = me.τ∗ and hole drift velocity v hD = μh ε. p

Drift velocity v nD gets superimposed on each electron along with the thermal velocity Vthe under the influence of the electric field. It is easy to show that Fig. 2.25 Force experienced by an electron in an electric field

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electron current density due to drift velocity j De is given by j De = −e.n e .v nD = e.n e μe .ε

(2.14)

Similarly, the hole current density due to drift is given as j Dh = e.n h .v hD = e.n h μh .ε

(2.15)

And the total current density due to drift j D is given as j D = j De + j Dh = eε[n e μe + n h μh ]

(2.16)

ne , nh , μe , and μh in above expressions are, respectively, the concentration of electrons, concentration of holes, mobility of electrons and mobility of holes. But, J D = σ ε, hence, σ = e[n e μe + n h μh ]

(2.17)

(ii) Diffusion current Diffusion is a universal phenomenon in which particles or in general objects, irrespective of their charge, move out from a region of higher concentration to the region of lower concentration so as to equalise the concentrations on the two sides. If in a semiconductor there is concentration gradient of charge carriers, electrons and holes, the carriers will move from higher concentration side to the lower concentration side establishing diffusion currents. Charge carrier concentration gradient in a semiconductor may be produced by non-uniform doping or by injecting charge carriers of a given type by implantation. Let us consider the case where in a semiconductor piece there is electron concene in x direction. In absence of any electric field the free electration gradient dn dx trons will be moving in random directions with thermal velocity Vthe . Electrons will undergo collisions with vibrating crystal lattice every now and then and if τ is the average time between two successive collisions, then one may assign a mean free path λ = V the . τ . Under these conditions it can be shown that electron diffusion current density jeDiff and hole diffusion current density jhDiff are given by jeDiff = e.Vthe .λ.

dn e dn h and jhDiff = −e.Vthe .λ. dx dx

(2.18)

Putting Vthe .λ ≡ Dn for electron and Vthe .λ ≡ D p for holes, Eq. (2.18) reduces to

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jeDiff = e.Dn .

dn e dx

(2.19)

And jhDiff = −e.D p .

dn h dx

(2.20a)

Dn and Dp in above expressions are called diffusion coefficient, respectively, for electron and hole. Though it may appear surprising at first sight but diffusion coefficient is related to the corresponding mobility. It can be shown that Dn = μe

kT e

and D p = μ p

kT e

(2.20b)

2.4.11 Temperature Dependence of Semiconductor Resistivity As already discussed, resistivity of a material is a measure of the opposition offered to the flow of current by a unit cube of the material. If a voltage V is applied across the two opposite faces of a cube of a given material and a current I passes through the circuit, then resistivity ρ = V /I. For a fixed value of V, resistivity is inversely proportional to current I. At a given temperature, the value of current I depends on three factors: (i) the number density or concentration of charge carriers (both free electrons and holes are charge carriers in case of semiconductors while only free electrons are charge carriers in conductors), (ii) number of collisions per unit time between charge carriers and crystal lattice and (iii) trapping of charge carriers per unit time at trap locations. Higher concentration of charge carrier results in increased current and in turn decreases the resistivity. On the other hand motion of charge carriers under the influence of the applied electric field gets randomised due to charge carrier-lattice collisions which reduces current and increases resistivity. It may be said that collisions between charge carriers and crystal lattice are the most frequent and important as collisions between charge carriers themselves are very less probable on account of their extremely small size and relatively high speed of their motion. Even in a very pure crystal there are always some sites where charge carriers are trapped. Trapping of charge carriers reduces current in turn increasing the resistivity. However, the probability of charge carrier trapping does not very much depend on temperature. At room temperature (≈ 300 K), in a semiconductor doped with shallow impurity of the order of 1013 atoms per m3 (107 atom/cm3 ) there are around 1013 majority charge carriers per metre cube and some 104 –105 minority carriers per cubic metre due to covalent bond breaking. There is also a certain rate of lattice-carrier collision, say N c collisions per second. On raising the temperature above room temperature, more covalent bonds got broken and more charge carriers are produced. Generation

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of additional charge carriers increases almost exponentially with temperature. On the other hand, collision rate N c also increases but linearly with temperature. Thus, rate of generation of additional charge carriers wins over the increase in collision rate N c . The net result of temperature increase is that current I through the semiconductor increases with temperature, reducing the resistivity of the semiconductor. In conductors (metals) at room temperature all atoms of the conductor are ionised and contribute at least one free electron, therefore, the free electron concentration in conductors is of the order of 1028 free electrons per m3 . As such the resistivity of conductors is very small of the order of 10–8 (Ω-m) (see Table 2.1). When temperature is raised, in case of conductors there is no further increase in the number of charge carrier but the rate of electron–lattice collision increases. As a result current I decreases with temperature, increasing the resistivity of conductors. SAQ: The mobility (μ) and the coefficient of diffusion (D) are parameters of two totally independent causes of current flow in a semiconductor but these constants are related with each other. What is inter-connecting link of the two?

2.4.12 Theoretical Calculation of Carrier Concentration in a Semiconductor There are two types of charge carriers, such as free electrons in conduction band and holes in valence band in a semiconductor. The concentration or number of charge carriers per unit volume of the semiconductor which is also called the carrier number density depends on the concentration of dopant impurity and temperature. Dopant concentration essentially decides the concentration of majority carriers while temperature, that determines the rate of covalent bond breaking, controls the minority carrier concentration. Both conduction band and the valence band have large number of discrete energy states where carriers may reside. These energy states, though discrete, yet they are so closely packed in energy that one cannot talk of any individual state, instead one talks of the state or level density, i.e. the number of states per unit volume within energy E and (E + dE). The level density of allowed energy states for free electrons in conduction band may be denoted by N(E) and for holes in valence band by P(E). Carrier concentration for electrons and holes in a semiconductor may be theoretically calculated using the tools of quantum statistics. Let us first calculate the number density (concentration) of electrons in conduction band of a semiconductor. Let us denote by n e (E) the density of electrons (number density or number of electron per unit volume) in conduction band at energy E and (E + dE). This number density of electrons may be written as the product of the density of energy state at energy E and (E + dE) in conduction band, and the probability F(E) that the energy range E and (E + dE) is occupied by electrons. Therefore, n e (E) = N (E).F(E)

(2.21a)

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To calculate the number density in conduction band of electrons of all energies n e , one must integrate Eq. (2.21a) from the lower energy limit of E C , energy at the bottom the conduction band and E top , the energy at the top of the conduction band. Hence, ∫Etop ∫Etop ne = n e (E)dE = N (E)F(E)dE EC

(2.21b)

EC

Exact calculation of factor N(E) for electrons is impossible as in an actual semiconductor crystal electrons face a periodic potential due to lattice ions. Approximate value for N(E) assuming that electron behaves as a free particle in a box is given by  N (E) = 4π

2m ∗n h2

3/2 E 1/2

(2.21c)

Here m ∗n and h are, respectively, the effective mass of the electron and Planck’s constant. The probability F(E) that electron occupies the state of energy E is given by the Fermi–Dirac distribution function of quantum statistics as F(E) =

1  ( E−E f )



1+e

(2.21d)

kT

In Eq. (2.21d) E f stands for Fermi energy and k for Boltzmann constant while T is temperature in Kelvin. It is easy to verify that; ⎧ ⎪ ⎪ E(F) = 1 for E < E f All states with energy less than ⎨ Fermi energy are filled with electrons At T = 0 K ⎪ All states with energy larger than E(F) = 0 for E < E f ⎪ ⎩ Fermi energy are empty 1 1 It follows from Eq. (2.21d) that for E = E f ; F(E) = 1+e 0 = 2 = 0.5; that means the probability is 0.5 that the state with energy E f is filled with electron. Figure 2.26i shows the variation of the function N(E) (density of states with energy E and E + dE) with energy E. This graph basically represents Eq. (2.21c). It may be observed in this figure that N(E) increases as the square root of energy E. Variation of occupation probability function F(E) with energy E is shown in Fig. 2.26ii. It may be observed that function F(E) behaves differently, for two situations T = 0 K and T > 0 K. As shown in the figure, when temperature T = 0 K, F(E) = 1 for E < E f and F(E) = 0 for E > E f , (red curve); there is a sharp cut in the probability at Fermi energy. For T > 0 K; there is not a sharp cut in F(E), it has a value 1 up to some energy, then starts decreasing, becomes 0.5 at E = E f and decreases almost exponentially after that.

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Fig. 2.26 Graphical representation of N(E), F(E) and electron concentration ne at T = 0 K and T >0K

Figure 2.26iii, iv show the number per unit volume of electrons of all energies ne at T = 0 K and at T > 0 K, respectively. As shown in these figures, the value of ne is given by the shaded area enclosed between the energy axis (X-axis) and curves for functions N(E) and F(E).  For the case where (E − E f ) > 3 kT, 1 may be neglected in comparison to e in Eq. (2.21d) and F(E) may be written as  E−E − kT f

F(E) = e

E−E f kT

(2.21e)

Putting this value in Eq. (2.21b) one gets; ∫Etop ∫∞  ∗ 3/2  E−E f 2m n 1/2 − kT dE E e ne = n e (E)dE = 4π h2 EC

(2.21f)

EC

The upper limit of integration in above expression is changed from E top , (energy at the top of conduction band) to ∞ as F(E) approaches to zero exponentially for large energies. Equation (2.21f) on integration gives

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99



2m ∗n kT ne = 2 h2

3/2  E −E 

 E −E  − CkT f − CkT f e = NC e

(2.21g)

where  NC = 2

2m ∗n kT h2

3/2 (2.21h)

Similarly, number density of holes n p may be given as  np = 2

2m ∗p kT h2

3/2  E −E 

 E −E  − CkT f − CkT f e = Nv e

(2.21i)

Here,  NV = 2

2m ∗p kT

3/2 (2.21j)

h2

(i) Calculation of Fermi energy at T ≈ 0 K At T = 0 K, electron concentration ne may be given as ∫E f ne =



2m ∗n 4π h2

3/2



E

1/2

2m ∗n .1dE = 4π h2

3/2

3/2

2E f 3

0

Or

Ef =

3 √ 16 2π

2/3

h 2 2/3 0.121h 2 2/3 ne = ne ∗ m m∗

(2.21k)

2.4.13 Hall Effect American physicist Edwin Herbert Hall in 1879 observed that when an electric current is passed through a conductor that is placed in a magnetic field, a potential proportional to the current and the magnetic field develops across the conductor in a direction perpendicular to both the current and the magnetic field. This effect is called the Hall Effect and the developed potential difference as Hall voltage. From the measurements he made, Hall for the first time was able to determine the charge of the current carriers. Even today Hall Effect is used to steady the charge transport characteristics of metals and semiconductors.

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2 Electrical Behaviour of Condensed Matter

Layout of experiment for the study of Hall Effect is shown in Fig. 2.27. A slab of the conductor/semiconductor of length L in X direction, width w in Y-direction and thickness t in Z-direction is taken, and a voltage source V is connected to the two opposite faces so that an electric field ε (= V /L) in direction X is produced within the slab. Electric field ε establishes a current I x in positive X-direction through the slab. Current I x may be constituted by the flow of charges of only one polarity (in case of metallic block by electrons) or it may be produced by charges of opposite polarities (in case of semiconductor both electron and holes). However for simplicity we assume that the current I X is due to charge carriers of only one polarity. Let q be the charge of the carrier. The electric field ε exerts a force qε on each charge carrier in positive X-direction and imparts an additional average velocity vX , called the drift velocity, to each charge carrier. If n represents the concentration of charge carriers, then the current density jx may be written as jx = nqvx and current Ix = nqvx .(area of cross section of the slab) = nqvx w.t Or vx =

Fig. 2.27 Layout of Hall Effect experiment

Ix nq.w.t

(2.22a)

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101

If a magnetic field Bz is now applied in Z-direction, the charge carriers that constitute current I x will experience a force in Y-direction. The direction of force is given by Fleming’s left hand rule (see inset in the figure) and the magnitude by the expression FY = q B Z vx Force F Y will deflect charge carriers towards the top of the slab resulting in accumulation of charge carriers on the inside of the top surface of the slab. Accumulation of charges on the inner top face generates an electric field E Y in negative Y-direction. Electric field E Y will repel charge carriers and will oppose further accumulation of charges. Thus charge carriers will experience two opposite forces, one in positive Y-direction F Y due to magnetic field and the other in negative Y-direction due to electric field E Y generated by the accumulation of charges. Ultimately a state of equilibrium will reach when two opposite forces will become equal, and no further deflection of charge carriers will take place. In the state of equilibrium, nq Bz vx = nq E Y or B Z vx = E y

(2.22b)

Electric field E Y that is produced by the accumulation of deflected charge carriers, results in the development of a voltage V H , called Hall voltage, across Y-direction and may be measured experimentally. Further, ∫0 VH =

∫w E Y dy = −

w

E Y dy = −E Y w 0

Substituting the value of E Y from Eq. (2.22b) and of vx from Eq. (2.22a) in the above expression one gets

    B Z .Ix B Z .Ix 1 = −R H VH = − nq t t

(2.22c)

In expression (2.22c) V H , BZ and I X are all measurable quantities, and therefore, in an experiment one can determine the value of RH , called Hall coefficient. RH will have positive value if the charge of the carrier is positive and will be negative for charges of negative polarity. For n-type semiconductor slab RH will have negative value and for p-type semiconductor slab RH will be positive.

2.4.14 p–n Junction When two ends of the same intrinsic wafer or intrinsic semiconductor monocrystal are doped, one with n-type impurity of pentavalent atoms and the other by p-type trivalent impurity, a p–n junction is formed at the boundary of the two sides. It may

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be emphasised that if a p-doped crystal and another n-doped crystal are put together touching each other, p–n junction will not be formed. For p–n junction to form it is essential that same crystal or wafer be doped on one side by p-type impurity and on the other side by n-type impurity then only a p–n junction is formed at the boundary of the p- and n-type materials within the given wafer or crystal. As is shown in Fig. 2.28, the Fermi level for isolated n-type material is shifted upwards from the middle of the forbidden gap and it is shifted downwards for the isolated p-type material. However, when p- and n-type materials are developed on the same crystal, the Fermi level cannot be different on two sides because of the continuity of crystal structure. As a result the band structure of n-side is pulled down with respect to the band structure of the p-side to equalise the Fermi levels of the two sides, as shown in Fig. 2.29. The band structures of the p- and n-sides in a single crystal are shown separated from each other in Fig. 2.29 just to indicate how the band structure of n-side is pulled down with respect to the p-side to equalise the Fermi level on two sides. However in reality the two band structures touch each other at the physical boundary of the pand n-sides. Since the energy of free electrons in conduction band is measured up wards from the Fermi level, electrons in conduction band on p-side are more energetic as compared to the electrons in the conduction band on n-side. Similarly, holes on n-side have more energy than holes on p-side. Consider the instant when p–n junction got established on doping the two sides. Initially both the n-side and the p-side were electrically neutral, however, at the establishment of junction, concentration of electrons on n-side is larger than the concentration of electrons on p-side and similarly, the concentration of holes on pside is larger than of holes on the n-side. Because of the concentration difference, some electrons diffuse from n-side to p-side, and some holes diffuse from p-side to n-side. As a result of diffusion of electrons from n-side, the n-type semiconductor develops a positive charge; the amount of positive charge developed on n-side is proportional to the number of electrons lost by it due to diffusion. The positive

Fig. 2.28 Band structures of isolated intrinsic, n-type and p-type semiconductors

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103

Fig. 2.29 Energy band structures of the p- and n-sides at pn junction

charge acquired by n-side try to pull back negatively charged electrons and tries to stop further diffusion of electrons. Thus two opposite forces; force of diffusion that tries to transfer electrons from n-side to p-side and the force of attraction between electrons and positively charged p-side got balanced after the diffusion of some electrons from n-side to the p-side. This is called the state of equilibrium, in state of equilibrium that occurs after the diffusion of some electrons from n-side to p-side, there is no further diffusion of electrons from n-side to p-side. As already mentioned, initially some holes, which are majority carrier on p-side, diffuse to n-side, making p-side negatively charged. The amount of negative charge developed on p-side is proportional to the number of holes that have diffused to n-side. Again, at the state of equilibrium, that occurs after some holes have already diffused to n-side, there is no further diffusion of holes. The state of equilibrium is reached within a fraction of a second as soon as the p–n junction is formed. After the system attains equilibrium, there is no further diffusion of electrons from n-side and of holes from the p-side. Further, after the establishment of equilibrium, the p-side develops a negative potential and n-side a positive potential. The potential difference between the n-side and the p-side is called internal potential barrier and is denoted by V B (see Fig. 2.30). (i) Depletion layer Diffusion of electrons from the n-side leaves a sheath of uncovered positive immobile donor ions on the n-side of the junction and diffusion of holes a layer of immobile uncovered negative acceptor impurity ions on the p-side of the junction. Thus around the junction there is a layer of positive uncovered ion on the n-side and a layer of uncovered negative ions on the pside, this region which contains uncovered ions is called depletion layer. As is

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2 Electrical Behaviour of Condensed Matter

Fig. 2.30 p–n junction diode with bulk p- and bulk n-sides along with depletion layer. Internal potential barrier V B is also shown in the figure

obvious, no mobile charge carrier, electron or hole, may stay in this depletion region, as it will be swept by the positive or negative uncovered ions. Since no mobile charge can stay in depletion region, i.e. it is depleted of mobile charges, hence the name depletion layer. Depletion layer has some special properties: (i) no free mobile charge may stay in this region, (ii) since it has no charge carriers it is like an insulator or has very high resistance, (iii) there are equal amounts of positive and negative charges at the two ends of the depletion layer which in itself behaves like an insulator; therefore, depletion layer works like a parallel plate capacitor. The capacitance of depletion layer may be changed by applying potential drop across p–n junction, and thus it provides a capacitor whose capacitance may be varied by varying voltage across the junction. p–n junction is also called junction diode. It is because of the fact that p–n junction behaves like an electron tube diode. Since the total uncovered negative charge on p-side of the depletion layer must be equal to the total uncovered positive charge on the n-side; n p .x p = n e X e Here, n p , n e , X p , and X e are, respectively, the concentration of acceptor impurity, concentration of donor impurity, thickness or width of depletion layer on p-side and width of depletion layer on n-side. When the p- and n-sides are not doped to the same concentration (n p /= n e ) the depletion layer will extend more towards the side of lower doping concentration as shown in Fig. 2.31.

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105

Fig. 2.31 Depletion layer extends more on the side of lower doping concentration

SAQ: Which part of a p–n junction has maximum resistance and why? SAQ: There are positive ions of donor impurity atoms on the bulk n-side and negative ions of acceptor impurity on the bulk p-side but these ions are covered with respective charge carriers. Why do ions become uncovered in the depletion layer of a p–n junction? SAQ: Can you estimate the thickness of depletion layer for normal doping. (ii) Biasing of p–n junction diode Biasing of a device means providing required voltages to different terminals of the device. Figure 2.32a shows the symbol used for a p–n junction diode in electronic circuits. A junction diode has two terminals; a terminal connected to the p-side and the other terminal connected to the n-side. A source of voltage, a battery, may be connected between these two terminals in two different ways. When diode terminal attached to the p-side is connected to the positive terminal of the battery and the n-side to the negative terminal, the arrangement is called forward bias. However, if the p-side is connected to the negative terminal of the battery and the n-side to the positive, the arrangement is reverse bias. (a) Forward bias Figure 1.32b shows the forward bias arrangement. It may be recalled that in an unbiased p–n junction at equilibrium diffusion of charge carriers does not take place because of the internal potential barrier V B , which restricts any transfer of charges from one side to the other. In forward bias arrangement the battery potential V opposes or reduces the internal potential barrier V B . Reduction of internal potential barrier results in two events: (i) reduction in the width of the depletion layer because on pside holes get repelled by the external battery potential + V and covers some of the uncovered acceptor ions in depletion layer and similarly, on n-side electrons get pushed into depletion region by the negative external battery potential and cover some uncovered positive donor ions. (ii) As a result of the reduction of internal potential barrier, some of majority

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2 Electrical Behaviour of Condensed Matter

Fig. 2.32 a Symbol for p–n junction diode used in electronic circuit b forward bias junction c reverse biased p–n junction

carrier holes from p-side and some majority carrier electrons from the n-side start moving to the other side. This movement of majority charge carriers constitutes a forward current I f through the circuit as shown in Fig. 2.33. Initially, the forward current I f increases slowly till the depletion layer vanishes completely at battery potential V d when forward current suddenly rises almost exponentially. Potential V d is called knee potential (or on potential) and in a way equal to the internal potential barrier V B . For Silicon Fig. 2.33 Characteristics of a p–n junction diode

2.4 Semiconductors

107

p–n junction diode the knee potential has a value of 0.7 V and for Ge based p-n diode it is 0.3 V. On further increasing the forward bias voltage beyond V d , the voltage across the junction does not increase but forward current of larger value flows through the forward biased circuit. A p–n junction diode in forward bias above on-voltage V d behaves as a battery of 0.7 V in case of Silicon-based diode and a battery of 0.3 V in case of Germanium-based diode. When forward bias voltage is increased beyond V d , the depletion layer disappears and large number of majority carriers diffuse from both sides to the opposite side. Therefore, forward current if is essentially due to the diffusion of majority carriers and since the concentration of majority carriers is quite high (≈ 107 charge carrier cm−3 ) forward current I f of few milli amperes flows through the circuit. (b) Reverse bias Circuit diagram for reverse bias arrangement is shown in Fig. 2.32c. In reverse bias arrangement the external battery potential add up with the internal potential barrier V B . This results in the increase of the width of depletion layer. With enhanced barrier at junction (V + V B ) the majority carrier on the two sides do not cross the depletion layer of enhanced width. However, minority carriers, electron on the p-side and holes on n-side are pushed by the total barrier potential (V B + V ) across the depletion layer constituting reverse current I r . This flow of minority carriers from one side to the other is not due to diffusion instead it is due to the large potential difference across the depletion layer. Minority carrier current in reverse bias is drift current and is only of the order of few microamps. When reverse bias voltage is increased beyond the breakdown voltage V b (see Fig. 2.33), suddenly a large reverse current starts flowing in the circuit. This large current flows because of the breakdown of the crystal structure. Because of the large electric field inside the semiconductor crystal (established by large reverse bias voltage V b ), the atoms in crystal structure break down releasing large number of electrons. Graphs showing the variation of forward and reverse currents as a function of applied voltage are called p–n junction diode characteristics and are shown in Fig. 2.33 both for the forward and the reverse bias arrangements. SAQ: Forward current across a p–n junction is generally in mA while the reverse current is in μA. What is the reason for this difference in the magnitudes of the two currents? SAQ: It is known that reverse saturation current I r is very sensitive to the ambient temperature; for every 100 C rise of temperature it gets doubled. However the forward current I f is not so sensitive to temperature. Can you assign a reason for this difference? Semiconductor materials are the backbone of electronic industry. These materials are used in fabricating solid state electronic devices that are extensively used in modern analogue and digital electronics. p–n junctions developed in

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special conditions of doping at more than one place in a monocrystal give rise to bipolar junction transistors and field effect transistors.

2.4.15 Some Formulations Some important formulae that are applicable in case of semiconductors are given here without their derivations, which are beyond the scope of the present text. These formulae may be used to solve numerical problems. (1) If ne and np, respectively, denote the concentrations of free electrons and holes in doped semiconductor at temperature T and ni the concentration of free electrons or holes in the intrinsic semiconductor at same temperature T, then, n e .n p = n i2

(2.23)

(2) For a non-degenerate semiconductor doped with shallow dopant at room temperature, ne ∼ = N D and n p ∼ = NA

(2.24)

Here, ne and np are, respectively, electron and hole concentrations, while N D and N A are concentrations of donor and acceptor atoms, respectively. (3) Theoretical values of carrier densities: theoretical value for the concentration of free electrons ne and holes np in a semiconductor at temperature T, calculated using quantum statistics, is given as

 E −E  − CkT F ; n e = NC e

 NC ≡ 2

2π m ∗n kT h2

 23 (2.25)

And

 E −E  − FkT V ; n p = NV e

 NV ≡ 2

2π m ∗p kT

 23

h2

(2.26)

Here, E C is the energy at the bottom of the conduction band, E V the energy at the top of valence band, k Boltzmann constant, m ∗n , m ∗p , respectively, the effective masses of electron and hole and T temperature in Kelvin. (4) Positioning of Fermi level Fermi energy at T ≈ 0 K is given by Eq. (2.21k) as

Ef =

3 √ 16 2π

2/3

h 2 2/3 0.121h 2 2/3 ne = ne ∗ m m∗

2.4 Semiconductors

109

Exact calculations show that Fermi level for an intrinsic semiconductor is positioned according to the following relation; E Fint

 ∗ mp (E C − E V ) 3kT = + ln 2 4 m ∗n

(2.27)

Here, E C and E V are, respectively, the energies at the bottom of conduction band and the top of valence band. m ∗p and m ∗n are, respectively, the effective mass of hole and electron. Second term on right that depends on temperature is negligible at room temperature and varies slowly with temperature, therefore, it is often neglected. Hence Fermi level for intrinsic semiconductor is taken at the middle of the forbidden energy gap at all temperatures. n-type However, for n-type semiconductor, it may be shown that Fermi level E F is given by n-type

EF

= E C − kT ln

NC ND

(2.28)

Here E C is the energy at the bottom of conduction band, k Boltzmann constant, T is 3  2π m ∗n kT 2 and N D is concentration of donor impurity temperature in Kelvin, NC ≡ 2 2 h atoms. Equation (2.28) tells that Fermi level in n-type semiconductor shifts towards the bottom of conduction band with the increase in donor impurity concentration. p-type Similarly, for p-type material the position of Fermi level E F is given by n-type

EF

= E V + kT ln

NV NA

(2.29)

Here E V is the energy at the top of valence band, N A concentration of acceptor  2πm ∗ kT 23 p . Equation (2.29) tells that the Fermi level for impurity atoms and N V ≡ 2 h2 p-tpye material shifts towards the top of valence band with the increase in acceptor impurity concentration. Values of Some Important Constants Energy 1.6 × 10–19 J = 1.0 eV. Thermal energy corresponding to room temperature (300 K) = kT = 0.025 eV. Unit of charge e = 1.6 × 10–19 C. Mass of electron me = 9.1091 × 10–31 kg. Planck’s constant h = 6.62608 × 10–34 J s = 4.1357 × 10–15 eV s. Velocity of light c = 2.9979 × 108 m s−1 = 3.0 × 108 m/s. Boltzmann constant k = 1.3807 × 10–23 J K−1 = 8.6173 × 10–5 eV K−1 .

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2 Electrical Behaviour of Condensed Matter

Solved Example SE2.1 A semiconductor absorbs light photons of wavelength shorter than 1 μm and is transparent to photons of wavelengths larger than it. What is the magnitude of forbidden energy gap of the material? Solution: Light photons that are absorbed by the semiconductor will transfer valence band electrons to conduction band. It is obvious that the forbidden energy gap will be equal to the energy of the photon. Now photon energy E is given by   4.1357 × 10−15 (eV s) × 3.0 × 10−8 m s−1 hc = = 1.24 eV. E = hν = λ 1 × 10−6 (m) Therefore the forbidden energy gap E g = 1.24 eV. Solved Example SE2.2 The forbidden energy gap for a semiconductor crystal at temperature 300 K is 1.50 eV. Take the effective mass of electron and hole, respectively, as 0.1 me and 0.5 me and calculate the energy shift of Fermi level from the middle of the forbidden energy gap. Solution: We use Eq. (2.27) E Fint

 ∗ mp (E C − E V ) 3kT + ln = 2 4 m ∗n

Energy shift of Fermi level from middle of forbidden energy gap is E Fint

 ∗ mp 3kT (E C − E V ) = ln − 2 4 m ∗n 3 × 8.61 × 10−5 × 300 0.5m e ln = 19.37 × 10−3 × 1.609 4 0.1m e = 0.031 eV

=

Solved Example SE2.3 The intrinsic concentration of charge carriers in a semiconductor is 1 × 1018 m−3 . Calculate the conductivity of the semiconductor given that electron and hole mobilities are, respectively, 0.40 and 0.15 m2 V−1 s−1 . Solution: Conductivity σ = n i .e.[μe + μh ] = 1 × 1018 × 1.6 × 10−19 [0.40 + 0.15] = 0.088 Ω−1 m−1 . Solved Example SE2.4 Forbidden energy gap for a compound semiconductor is 1.4 eV at 300 K. The Fermi level for the doped semiconductor is shifted towards the valence band by 0.20 eV. What is the type of doping and what is the majority carrier concentration? Given that effective mass of electron is 0.06 me and of hole 0.5 me . Also calculate the concentration of minority carrier in doped semiconductor and the concentration of charge carriers in intrinsic semiconductor.

2.4 Semiconductors

111

Solution: It is given that after doping the Fermi level shifts towards the valence band. It means that the doping is done with acceptor impurity and that the material has become p-type after doping. The band structures of the semiconductor before doping (intrinsic material) and after doping (p-type material) are shown in Fig. 2.34. As indicated in the figure after doping (E F − E V ) = 0.5 eV and (E C − E F ) = 0.9 eV. To calculate concentration of majority carrier holes we use Eq. (2.26) given below;

 E −E  − FkT V ; n p = NV e

 NV ≡ 2

2π m ∗p kT

 23

h2

Let us first calculate the value of N V  NV ≡ 2

2π m ∗p kT

 23

h2



2π × 0.5 × 9.1 × 10−31 × 1.38 × 10−23 × 300 =2  2 6.63 × 10−34

 23

= 8.836 × 1024 . Next we calculate



 E −E   −19 − FkT V − 0.5×1.6×10 = 8.836 × 1024 e 1.38×10−23 ×300 . n p = NV e Or n p = 8.836 × 1024 × e−19.32 = 8.836 × 1024 × 4.068 × 10−9 = 3.59 × 1016 m−3 .

Fig. 2.34 Band structure before and after doping

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2 Electrical Behaviour of Condensed Matter

Next we calculate the concentration of minority carrier electrons using Eq. (2.25) given below,

 E −E  − CkT F ; n e = NC e

 NC ≡ 2

2π m ∗n kT h2

 23

.

Now, 

2π m ∗n kT NC ≡ 2 h2

 23



2π × 0.06 × 9.1 × 10−31 × 1.38 × 10−23 × 300 =2  2 6.63 × 10−34

3/2

= 2.70 × 1021 . Or 

 E −E 

−19 − CkT F − 0.9×1.6×10 = 2.70 × 1021 × e 1.38×10−23 ×300 = 2.70 × 1021 × e−34.78 n e = NC e Or n e = 2.70 × 1021 × 7.8568 × 10−16 = 2.12 × 106 m−3 . Concentration of charge carriers in intrinsic material ni is given as 2 √  n i = n e n p = 2.12 × 106 × 3.59 × 1016 = 2.76 × 1011 m−3 . Therefore, (i) majority carrier concentration n p = 3.59 × 1016 m−3 (ii) Minority carrier concentration n e = 2.12 × 106 m−3 and (iii) Charge carrier concentration n i = n ip = n ie = 2.76 × 1011 m−3 . Solved Example SE2.5 Given that number density of free electrons in gold at very low temperature ≈ 0 K is 6.0 × 1022 cm−3 , calculate the Fermi energy for gold. Take the effective mass of electron to be equal to its mass 9.1 × 10–31 kg. Solution: In the given problem number density of electrons is given in CGS units while the electron mass is in MKS units. Let us convert electron number density also in MKS units; given quantities are; n e = 6.0 × 1022 cm−3 = 6.0 × 1028 m−3 , the effective electron mass m ∗e = 9.1 × 10−31 kg. We use the formula given by Eq. (2.21k) for Fermi energy E F at T = 0 K. 2 2/3 n e which on substituting the values gives, E F = 0.121×h m∗ e

2  2/3 0.121 × 6.60 × 10−34  J EF = 6.0 × 1028 −31 9.1 × 10

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113

= 8.87 × 10−19 J =

8.87 × 10−19 eV = 5.548 eV. 1.6 × 10−19

2.5 Conductors Conductors are solids that are characterised by metallic bonding, having either overlapping conduction and valence bands or with negligible forbidden energy gap. Metals and their alloys are mostly conductors. Their specific resistivity lies in the range of (1–100) × 10–8 for metals and (1–100) × 10–6 Ω m for most of the alloys. Overlapping of conduction and valence bands is the outcome of the high degree of overlap in outer electron orbital’s of individual atoms in some crystals. As a result the conduction and valence bands become so broad that they overlap. In such materials the valence electrons are far away from the corresponding nucleus of the atom and are very loosely bound with its parent nucleus. Also in their crystalline structure the relative separation of atoms is large so that the forbidden energy gap is either zero or very small. Figure 2.35 shows the band structure of a conductor (a) at 0 K and (b) at T > 0 K. At absolute zero all valence electrons are bound and are not available for conduction of current. However, with the rise of ambient temperature more valence electrons become delocalised and at room temperature in most of conductors all valence electrons become delocalised or free and are available for conduction of current. It is reasonable to assume that at room temperature all valence electrons of all atoms in the given specimen of conductor are delocalised and are available as free charge carriers. Obviously, current may flow only when some voltage is applied to the conductor that establishes an electric field. In absence of any electric field a piece of conductor at room temperature has large number of free electrons that move in random directions with thermal velocity which is of the order of 105 m/s. The number density of free electrons in Silver at temperature 300 K is of the order of 5.8 × 1028 m−3 . These randomly moving electrons undergo frequent collisions with crystal lattice and are also trapped at sites of unionised impurity atoms. At each collision the velocity and direction of motion of electron get changed. When randomly moving free electrons are subjected to an electric field by applying an external voltage, a drift velocity gets superimposed on each electron in a direction opposite to the direction of the electric field. This results in the flow of current through the conductor. Opposition to the smooth flow of current is generated by frequent lattice-electron collisions. Larger the rate of collision more will be the opposition to the flow of current. Therefore, resistance or resistivity of conductors is essentially the result of electron–lattice collisions. With the rise of temperature, the electron density in a conductor does not increase because all valence electrons are already delocalised at room temperature, however, the electron–lattice collision rate increases with temperature and hence the resistivity of conductors increases with temperature. That is why the temperature coefficient of conductors has a positive value. On the other hand, in a semiconductor, all valence

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Fig. 2.35 Band structure of a conductor a at absolute zero temperature and b at a temperature higher than absolute zero

electrons are not free at room temperature, and the free electron density rapidly rises with temperature due to covalent bond breaking. Though electron–lattice collisions in semiconductors also rise with temperature, but the rate of increase of free charge carriers (electrons and holes) with temperature is much larger than the increase of collisions; therefore, the resistivity of semiconductors decreases with temperature. That is why semiconductors have negative value of temperature coefficient.

2.5.1 Semimetals and Half Metals (i) Semimetal or metalloid Semimetals are materials for which the bottom of the conduction band either just touches the top of the filled valence band or there is very little overlap between the two bands. As a result, there is a very small value for the density of electron states near Fermi level in semimetals, as compared to metals where there is considerable overlap of bands (see Fig. 2.36). With few energy states available around Fermi level, semimetals behave as semiconductors with negligible forbidden energy gap. Some semimetals, also called metalloids, are arsenic (As), Antimony (Sb) and tellurium (Tl). Some elements like carbon are found in two allotropic forms, one of which behaves like a metal (graphite) while the other (diamond) like a non-metal, such elements are also included in the list of metalloids (ii) Half metal It is often said that a given electron level can accommodate at the most two electrons, one with spin up and the other with spin down. However, on close examination it is found that each electron level is made up of two very closely placed electron states, one for spin up electron and the other for spin down

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Fig. 2.36 Energy band structure of a metal b semimetal

electron. As such one can build two separate state energy diagrams one for spin up electrons and the other for spin down electrons with their own valence and conduction bands. In some crystals where atoms are bound by metallic bonding, it so happen that valence band for electrons of one specific spin orientation is partially filled while there is a forbidden energy gap for electrons of other spin orientation. As a result when external voltage is applied, electrons with that spin orientation for which there are vacant states in valence band contribute to the flow of current. Electrons with opposite spin orientation do not contribute to current flow because of forbidden energy gap. Since only about half of the total electrons contribute to current flow, the material is termed as half metal. Examples are chromium oxide and lanthanum-strontium-magnetite that are half metals and are also ferromagnetic. Though all half metals are ferromagnetic but all ferromagnetic materials are not half metals. Energy band structure of a typical half metal is shown in Fig. 2.37. Solved Example SE2.6 Density of trivalent Aluminium metal is 2.7 g cm−3 , and its molecular mass is 27 g/mol; assuming that at room temperature all valence electrons are non-localised (or free), calculate the number density of free electrons in the metal. Solution: It is known that a gramme mole of an element contains 6.022 × 1023 atoms (Avogadro’s number) of the element. Therefore, 27 g of Al will contain 6.022 × 1023 atoms of Aluminium.

(2.30)

Also, the density D of Al is given as D = 2.78 g cm−3 . But density is equal to mass/ volume.

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Fig. 2.37 Energy band structures for a half metal a for spin down electrons b for spin up electrons

We calculate the volume V of 27 g of Aluminium using its density as V =

27 M = cm3 = 9.71 cm3 D 2.78

(2.31)

It follows from Eqs. (2.30) and (2.31) that 27 g of Aluminium has 6.022 × 1023 atoms that occupy a volume of 9.71 cm3 . Therefore, the number of atoms in 1 cm3 =

6.022×1023 9.71

= 0.62 × 1023 .

Since the valency of Aluminium is 3, and all valence electrons are free at room temperature, therefore each atom will contribute three free electrons at room temperature. Hence the number density of free electrons in Aluminium at room temperature n per cc is n = 3 × 0.62 × 1023 per cm3 = 1.86 × 1023 per cm3 = 1.86 × 1029 per m3 .

2.6 Superconductor Resistivity of conductors, particularly of metals is very low of the order of 10–8 Ω-m, and it originates essentially from the free electron–lattice or electron–phonon interactions. It is because of the resistivity that energy is lost when current is passed through a conductor. As expected, resistivity of conductors decreases further with the decrease of temperature. In some materials, generally at very low temperature near about absolute zero, it is found that their resistivity simply vanishes. The state of the

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material in which the resistivity of the material becomes zero is called the superconducting state and the property as superconductivity. The characteristic temperature below which resistivity becomes zero is called the critical or transition temperature and is denoted by TC . It is obvious that no energy loss occurs when current is established through a superconductor, no matter for how long current flows through it.

2.6.1 Background Research group of Dutch physicist Heike Kamerlingh Onnes, in 1911 found that the resistivity of a mercury column becomes zero when the temperature of the specimen was reduced below 4.15 K (see Fig. 2.38). Complete disappearance of electric resistance in some other metals and solids below a certain characteristic very low temperature was observed in some other materials also. Onnes and his students were studying the electrical behaviour of wires of different materials and found that the resistance of a mercury wire took a precipitous drop when temperature reached to about 4.15 K. The drop in resistance was enormous, the resistance of the wire dropped at least by a factor of one thousand, so much so that exact measurement of the resistance became impossible (see Fig. 2.38). In order to further investigate the phenomenon, Onnes’ group setup a current through the mercury wire in the form of a ring by connecting at two points of the wire a voltage source for an instant and then removing the voltage source. To their surprise, they observed that current kept flowing through the mercury wire ring without any reduction in its magnitude, so long as the temperature of the wire was kept below 4.15 K. The observed perpetual flow of current was only possible if flow of current does not encounter any opposition or resistance. As is known opposition to the flow of current in normal situation arises essentially from electron–lattice collisions and electron trapping at impurity sites. Disappearance of resistance in case of mercury Fig. 2.38 Superconducting transition at critical temperature T c = 4.15 K

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Table 2.3 Superconducting transition temperatures for some metals

Metals

Transition temperature (K)

Lead

7.19

Mercury

4.15

Tin

3.72

Indium

3.41

Aluminium

1.20

Zinc

0.88

wire at temperature below 4.15 K means that electron–lattice collisions have either suddenly cease to happen below the critical temperature, the temperature below which mercury wire exhibits superconductivity or at least lattice vibrations are not opposing the flow of current. Transition or critical temperature for superconductivity transition for some metals is listed in Table 2.3. Many well-known scientists including Nobel Lauriat John Bardeen tried to explain and give a theoretical background for superconductivity but they did not succeed. The reason why no theoretical explanation of the process could be given at that time was that the process of superconductivity is a typical quantum phenomenon, and quantum physics was not in place till 1920 or so. (i) Meissner effect In the meantime experimental studies on superconductivity continued and in 1933, two scientists Walter Meissner and Robert Ochsenfeld discovered another very interesting property of superconductivity; they found that any material in superconducting state repels the lines of external magnetic field (Bex ) so long as the applied magnetic field is below the critical value denoted by BC ex . It means that for external magnetic fields Bex < Bc ex a superconductor behaves as a perfect diamagnetic material. If a magnet is brought near to a superconducting material, the superconductor does not allow magnetic lines of force to penetrate through it, rather it repels them. The effect is called Meissner effect. Meissner effect is a typical example that also shows that superconductors are not just perfect conductors. A perfect conductor may be defined as a conductor which has a pure crystalline structure without any impurity or missing atom sites and has small value of resistivity. However, a superconductor is different from a perfect conductor as it behaves differently than a perfect conductor when a magnetic field is first applied and then switched off. Figure 2.39 shows a perfect or ideal conductor and a superconductor, initially the temperature of both the specimen is above critical temperature T C and both of them allows the passage of magnetic lines through them. With magnetic field (Bex < BC ex ) on, if the temperature of both specimens is reduced below critical temperature T C , the ideal conductor will allow magnetic lines to pass through it, as they were before the reduction of temperature. It is because magnetic properties, like susceptibility, of a perfect conductor do not change

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with temperature. However, in the case of superconducting specimen, reduction of temperature below T C transforms it into superconducting state. The magnetic susceptibility of superconducting state is different than that of the non-superconducting state. As a result, at the instant when temperature goes below T c , magnetic flux linked with the superconducting volume changes. This change in the magnetic flux induces surface currents at the outer skin of the superconducting volume such that a magnetic field exactly equal in magnitude but opposite in direction to the previously existing magnetic field is generated. The previously existing magnetic field gets cancelled by the induced magnetic field in the interior volume of the superconducting specimen. Hence no magnetic field stays in the superconducting volume. The induced magnetic field cancels the existing magnetic field so that the interior of the superconducting volume becomes free of all magnetic fields; no magnetic field and flux remain linked with the superconducting volume. Let us now discuss what happens when external magnetic field is switched off keeping the temperature T < T c . At the instant when external magnetic field is switched off, the flux linked with the ideal conductor goes to zero from the initial value Bex . The changing

Fig. 2.39 Behaviour of perfect or ideal conductor and superconductor with respect to magnetization and demagnetization

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magnetic flux induces surface current at the outer skin of the ideal conductor, which in turn establishes a magnetic field in the interior volume of the perfect conductor specimen. It may, however, be mentioned that the induced surface currents will be short lived as the resistivity of the conductor will dissipate energy, and currents will die out. In case of the superconducting specimen, no magnetic flux is lined with the specimen volume (as there is no magnetic field inside superconducting volume) hence at the instant when Bex is switched off no change in magnetic flux will take place. As such no induced currents will be generated. The interior and exterior of the superconducting volume will contain no magnetic fields after the external magnetic field is switched off. SAQ: How can one explain the total absence of magnetic field in the interior of a superconducting volume when some external magnetic field is applied to the superconductor? SAQ: When an external magnetic field is switched off from a normal conductor, the conductor retains magnetic field in its interior and around. How one can explain this retention of magnetic field? (ii) Magnetic field trapped in a superconducting ring Figure 2.40i shows a ring shaped superconductor specimen placed in an external magnetic field Bex at temperature T > T c . Science temperature is above T C , the specimen ring behaves as a normal material and magnetic lines penetrate through the opening of the ring. The magnetic flux ϕ linked with the opening of the ring is given by ϕ = Bex .(Area of the ring opening) In the next step temperature T is reduced below the critical temperature T C and the magnetic field Bex is switched off. With the decrease of temperature T below T c , the specimen ring becomes superconducting while switching off of magnetic field (Bex = 0) will induce an electric field E in the ring according to Faraday’s law. ∮ , here E is the electric field along the closed loop of superconEdl = − dϕ dt ducting ring and ϕ is the magnetic flux through the opening of the ring. But no electric field can exist in a superconductor, hence, ∮ = 0, that means the flux linked with the ring before it becomes Edl = − dϕ dt superconducting will continue to remain linked with the opening of the ring even after switching off of the magnetic field as shown in Fig. 2.40ii. SAQ: Why electric field cannot sustain in a superconductor? (iii) Superconductor type-I and type-II

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Fig. 2.40 Magnetic field trapped in the opening of a superconducting ring

It is known that not only metals but some other materials below their transition temperature Tc become superconductor. Further, if an external magnetic field Bex is applied across a superconductor specimen, when it is below its transition temperature, magnetic field Bin inside the superconductor stays zero. This is, however, true only when the magnitude of the externally applied magnetic field is below a certain value Bc ex . If the magnitude of externally applied magnetic field Bex is increased beyond the critical value Bc ex then the superconductor may respond in two different ways, depending on its type. In the case of type-I superconductor on increasing the strength of external magnetic field Bex beyond Bc ex , the superconductivity of the specimen just vanish, though its temperature is still below Tc . It behaves as an ordinary conducting material and magnetic field penetrates in the interior of the specimen. This is shown in Fig. 2.41a where a dotted vertical line at Bc ex divides the figure in two parts; where the specimen remains a superconductor and the part where superconductivity is totally lost in spite of its temperature being below critical temperature T c . Figure 2.41a shows the typical behaviour of a type-I superconductor. Most metals show type-I superconductivity. In case of type-II superconductors, there are two values of external critical magnetic fields Bc1 ex and Bc2 ex such that between these two magnetic field values the specimen remains partially superconducting as shown in Fig. 2.41b. For external magnetic fields greater than Bc2 ex , the type-II specimen also becomes non-superconductor, though its temperature is still below its critical temperature.

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Fig. 2.41 a Type-I semiconductor. b Type-II semiconductor

During the phase of partial superconductivity (between Bc1 ex and Bc2 ex ), cylindrical tubular regains in type-II specimen become ordinary or nonsuperconductor as shown in Fig. 2.42. External magnetic field penetrates only in these cylindrical volumes which are almost uniformly distributed over the specimen volume. Initially, just beyond Bc1 ex , the non-superconductor (normal) cylindrical volumes are very thin; however, their radius increases as external magnetic field is increased from Bc1 ex, and finally the nonsuperconducting material fills the total volume of the specimen at Bc2 ex and beyond. The origin of these normal-filament like structures is beyond the scope of the present text. Table 2.4 give a list of some compounds that are type-II superconductors. In most of type-II superconductors the value of the external field Bc2 ex is quite high therefore, they may withstand high magnetic fields without losing superconductivity altogether. Wires made of type-II superconductor are frequently used to build powerful electromagnets. For example, the upper critical external magnetic field Bc2 ex for wires of niobium-tin (Nb3 Sn) is as high as 24 T. These wires are used in making strong electro magnets for use in MRI and other imaging machines. The advantage of superconducting electromagnets is that current only has to be applied once to the wire, which are then formed into a closed loop and allow the current and the magnetic field to persist indefinitely as long as the temperature is kept below the critical temperature. Once current is established in superconducting magnet, the external power supply may be switched off. On the other hand the strongest magnetic field that may be produced using a permanent magnet may be only of the order of few Tesla. SAQ: What are type-II superconductors?

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Fig. 2.42 Normal or non-superconducting cylindrical volumes in type-II superconductor. External magnetic field penetrates through these filaments like normal volumes. The radius of these filaments increases with the increase of external magnetic field

Table 2.4 Compounds that are superconductor Compound

T C (K)

Compound

T C (K)

Compound

T C (K)

Nb3 Sn

18.1

PbMo6 S8

15.0

YPd2 B2 C

23.0

Nb3 Ge

23.2

HoN12 B2 C

Cs3 C60

19.0

UPd2 Al3

MgB2

39.0

(ET)2 Cu [Ni(CN)2 ]Br

7.5

UPt3

0.5

(TMTSF)2 ClO4

1.2

2.0 11.5

(iv) Stable levitation In order to demonstrate the Meissner effect, a high-temperature superconductor like YBa2 Cu3 O7 is taken and cooled below its critical temperature (93 K). A small and strong permanent magnet is then placed on top of the superconductor to show the repulsion of the small magnet by the superconductor. Repulsion results in the levitation, or hanging/floating of the small magnet above the superconductor. Repulsion originates from the mirror reflection of permanent magnets lines of force by the diamagnetic superconductor that forms an inverted mirror image of the permanent magnet. The situation is comparable to placing one magnet over another identical magnet to achieve the floating of the one over the other. In principle levitation of a small permanent magnet over the superconductor should be possible only if the size of the superconductor is much larger than the size of the permanent magnet. In case the two are of the same size, distortion of magnetic lines at the rim of the magnet and in its mirror image at superconductor will make levitation unstable and the magnet will topple down, instead of levitating. This is exactly what happens in case

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of two identical magnets, stable levitation is never achieved. However, stable levitation in case of a small permanent magnet and a superconductor of only slightly larger in size than the permanent magnet does take place. How can this be explained? Not only the levitation is stable but a small nudge causes the magnet to spring back to its original position as if some unseen springs are holding the magnet at its position of levitation. The answer is hidden in the properties of type-II superconductor and the trick done by the one who carries out the levitation experiment. A keen observation of the levitation experiment will reveal that levitation of small permanent magnet is not stable if a freshly cooled superconductor is taken and the magnet is made to float over it; the magnet falls off in a short time. However, to make levitation stable, the experimenter pushes the permanent magnet towards the superconductor and then the levitation over a superconductor of small dimensions becomes stable. So the trick to achieve stable levitation is to make the small magnet thrust towards the superconductor. The superconductor taken to demonstrate levitation is type-II superconductor, pushing or thrusting the permanent magnet towards the superconductor increases the strength of the external magnetic field applied to the superconductor beyond the first critical value Bc1 ex . The type-II superconductor goes into the region of partial superconductivity. Large number of filament like normal (non-superconducting) regions develops in the superconductor. Magnetic lines of force originating from the permanent magnet penetrate in these filament shaped non-superconducting regions as these regions are not diamagnetic, they are normal regions. Penetrating magnetic force lines provide a cluster of strings that holds and brings back the floating magnet if displaced from its position of levitation. Levitation by small type-II semiconductors is robust and very stable. Crucial role in it is played by the filament like normal regions that develop in type-II superconductors in the region of partial superconductivity. These filaments like structures are called by many different names; flux lines, fluxoids, vortices, fluxons, etc. In a pure monocrystal of a type-II superconductor it can be shown that these vertices are flexible, i.e. they may bend and may not be straight. However, in a superconducting crystal that has impurities and internal structural boundaries, vortices become straight and rigid. This is called flux or vortices pinning. Flux pinning also play important role in stable levitation. SAQ: In your opinion which type of superconductor is more useful and why? SAQ: What is pinning? Discuss the role played by pinning in stable Lavitation. (v) High Tc superconductors Since the discovery of superconductivity it has been the desire of all scientists to develop materials that may show superconductive behaviour at room temperature. Room temperature superconductors would have revolutionise almost

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every aspect of modern technology, particularly, power generation, its transportation, fabrication of high magnetic field electromagnets, etc. However, till date room temperature superconductors have not been fabricated or discovered. Two scientists, Georg Bendnorz and K. Alex Muller, working at IBM lab, discovered in 1986 a class of materials that showed superconductivity at liquid nitrogen (LN2 ) temperature. These materials called high-temperature or high T c superconductors are frequently used to demonstrate superconducting behaviour on bench-top using cooling by liquid nitrogen, which is easily available. The LN2 temperature is around 77 K, much higher than the critical temperature required for older superconducting materials. Table 2.5 lists some high Tc superconductors. Bendnorz and Muller got Nobel Prize for their discovery in 1987, within one year of their discovery. It was the fasted award of Nobel Prize ever. In most bench-top demonstration experiments one uses Yttrium-Barium-copper oxide (YBa2 Cu3 O7 ) with liquid nitrogen cooling. (vi) Isotope effect The single most important experimental observation that paved way for developing an explanation of superconductivity was the ‘isotope effect’ discovered in 1950, simultaneously at two laboratories; by Reynolds, Serin, Wright and Nesbitt at Rutgers University and by Maxwell working at the National Bureau of Standards. These experimentalists accurately measured the critical temperatures for superconductive transition in mercury samples of different isotopic mass distributions. It is known that mercury has seven stable isotopes with mass numbers; 196, 198, 199, 200, 201, 202 and 204. Out of these seven, the two most abundant isotopes are 200 Hg (23,1%) and 202 Hg (29.7%). It is possible to change the relative percentage of different isotopes in different samples, and these experimentalists measured the critical temperature for these different mercury samples that have different values of average isotopic mass. They found that the critical temperature was inversely proportional to the square root of the average isotopic mass of mercury sample, see Fig. 2.43. Similar experiments were later carried out with other materials, and the isotope effect was observed in each case. This showed that superconductivity has something to do with the nuclei of the atoms of the material. Inverse dependence of critical temperature on the square root of average mass was something that has been observed elsewhere also. For example, when a mass attached with a spring is given a push, the spring starts vibrating, and the frequency of vibration is found to be inversely proportional to the square root of the mass. In a crystalline solid it may be assumed that different atoms are connected with each other by some sort of springs and giving a push to any atom will set the complete atomic layer, the lattice in vibratory motion. Isotope Table 2.5 List of some high T c superconductors

Compound

T C (K) Compound

T C (K)

HgBa2 Ca2 Cu3 O8+x

135

Ti2 Ba2 Ca2 Cu3 O10+x

125

Bi2 Sr2 Ca2 Cu3 O10+x

107

YBa2 Cu3 O6+x

93

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Fig. 2.43 Isotope effect for Mercury

effect indicates that current flow in superconductor is not only an electronic phenomenon but it very much depends on lattice vibrations. The isotopic effect showed that although the electrical conductivity was known to arise because of the motion of free electrons and the resistivity essentially due to collisions between electron and lattice vibrations, but lattice vibrations below critical temperature somehow help in smooth flow of electrons setting perpetual current in superconductors. SAQ: What is the significance of the isotope effect of superconductor? (vii) Cooper pair American physicist Leon Cooper in 1956 described a process of binding of two electrons (or any other Fermions) in crystalline solids at very low temperature. This special type of binding between two electrons that are quite far apart develops on account of the interaction between the lattice vibration quanta phonon and electrons. It may sound strange as to how an attractive force may develop between two negatively charged electrons that should repel each other, but this happens because of the distortion produced in electric field of crystal lattice on account of its vibratory motion. It is known that in metals there is large number of free electrons since almost all atoms of the solid are ionised. The positive ions of constituent atoms are arranged in a regular pattern in 3-dimensions. A two dimension plane containing positive ions is termed as crystal lattice. Ions of the lattice are held at their place by electrostatic forces of mutual repulsion and ion–electron cloud attraction. Each lattice ion is in a sort of dynamic equilibrium under the two opposite forces. These dynamic forces between lattice ions may be compared to small springs which hold ions in the lattice. A

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little jerk or push to an ion of the lattice may make the whole lattice to undergo vibratory motion. This vibratory lattice motion is quantized, and the quanta of vibratory motion of the lattice are called phonon. Figure 2.44 shows two electrons numbered 1 and 2 moving in opposite directions. Negatively charged electrons attract positive ions of the lattice towards them, distorting the crystal lattice and setting it in vibratory motion. The density of positive charge in regions around the lattice distortion increases beyond its normal value and becomes centres of attraction between electrons and the distortion. Nearby electrons feel force of attraction by the region of increased charge density but this force of attraction is over powered by the force of mutual repulsion between nearby electrons. However, electrons far away get bound with each other as the force of mutual repulsion between distant electrons is very small and force of attraction due to charge distortion over rides the repulsion. In this way Cooper electron pairs are formed. This is the reason why the distance between the two electrons of Cooper pair may range from 50 to 100 nm or more. This is in comparison with the lattice separation; distance between two neighbouring ions of the lattice, of 0.1–0.4 nm. It is important to note that only those electrons that are far apart may form Cooper pairs. Figure 2.44 shows two electrons numbered 1 and 2 moving in opposite directions. Negatively charged electrons attract positive ions of the lattice towards them, distorting the crystal lattice and setting it in vibratory motion. The density of positive charge in regions around the lattice distortion increases beyond its normal value and becomes centres of attraction between electrons and the

Fig. 2.44 Moving electrons produce distortion in ion lattice and set it in vibratory motion. Distortion in lattice increases density of positive charge in small regions which attract electrons that are far away and create Cooper pairs

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Fig. 2.45 Phonon interaction between electrons of Cooper pair

distortion. Nearby electrons feel force of attraction by the region of increased charge density but this force of attraction is over powered by the force of mutual repulsion between nearby electrons. However, electrons far away get bound with each other as the force of mutual repulsion between distant electrons is very small and force of attraction due to charge distortion over rides the repulsion. In this way Cooper electron pairs are formed. This is the reason why the distance between the two electrons of Cooper pair may range from 50 to 100 nm or more. This is in comparison to the lattice separation; distance between two neighbouring ions of the lattice, of 0.1–0.4 nm. It is important to note that only those electrons that are far apart may form Cooper pairs. Figure 2.45 shows the quantum mechanical interaction between the two electrons of the Cooper pair. SAQ: What are phonons? What role do they play in the formation of Cooper pairs. SAQ: Cooper pairs are formed with far away electrons, explain.

2.6.2 BCS Theory of Superconductivity John Barden, Leon Cooper and J. Robert Schrieffer, in 1957 developed a quantum mechanical microscopic theory for superconductivity, which in short is termed as BCS theory. The theory explains the resistance less flow of current by paired electrons in some materials below critical temperature T C . The theory is based on the concept of Cooper pairs which are formed in superconducting specimen below the critical temperature. The salient features of the theory may be summarised as follows: • Phonon-(free) electron interactions in some materials, below critical temperature, give rise to the formation of Cooper pairs. • The binding energy of a cooper pair is very small; of the order of few milli-electron volts (≈ 10−3 eV). Therefore, to keep Cooper pairs intact, the superconducting specimen must be kept below the critical temperature T C . • Electrons 100 nm or more far apart from each other join to form Cooper pair. It is because, the force of repulsion between distant electrons is small and may be overcome by the force of attraction between the phonon and electrons. In case of

2.6 Superconductor

• • •

•

•

•

•

129

nearby electrons the force of repulsion is high and dominates over attraction by phonon. BCS theory requires that the linear momentum of the Cooper pair must be zero; therefore, the two electrons forming a Cooper pair must be moving in opposite direction. The bound state of two electrons as Cooper pair is lower in energy than the energy state of unbound free electron, the state corresponding to Cooper pair lies below the Fermi level. Formation of Cooper pair is a transient phenomenon. Suppose at a given instant a Cooper pair has two electrons marked-1 and 2. It is possible that at the next instant electron-1 of the Cooper pair may change its partner and another electron out of the large number of free electron may join with electron-1 to form the Cooper pair, at the next instant electron marked-1 may be replaced by some other electron in the Cooper pair and so on. In this way, almost all free electrons get coupled with each other in forming Cooper pairs. This results in inter-linking or coupling of all free electrons in the volume of the superconducting specimen and they move coherently. Superconductivity is a quantum mechanical phenomenon; each free electron in quantum mechanics is represented by a wavefunction that extends over the volume of the specimen. Wavefunctions of all free electrons therefore overlap generating a resultant wavefunction representing all free electrons together. The resultant wavefunction gives rise to the coherent behaviour of all free electrons. A Cooper pair has two electrons each with spin 1/2 ℏ and, therefore the spin of a Cooper pair may either be 0 or 1 ℏ. Particles with integer spin (0, 1 ℏ, 2ℏ…) are called Bosons and obey Bose–Einstein quantum statistics. Unlike electrons which follow Fermi- Dirac statistics and not more than two electrons with their spins in opposite directions can stay in an energy state, large number of Bosons may stay in a given energy state. As such all Cooper pairs in the superconducting specimen stay in the lowest energy state, the ground state. Simultaneous stay of all Cooper pairs in the ground state is referred as Condensation. In energy band diagram of a superconductor, the ground band of Cooper pairs is separated by a forbidden energy gap Eg from the energy band for free electrons, as shown in Fig. 2.46. The forbidden energy E g is related to the transition or critical temperature T c by the relation. E g = 3.53kTC , here k stands for Boltzmann constant. Energy band diagram for a metal is also shown in Fig. 2.46 for comparison. In the superconductive state the current flow is constituted by the motion of coherent Cooper pairs. In ordinary metal resistance essentially develops out of the inelastic collisions between the lattice and free electrons, however, in case of superconductors, where current is constituted by the coherent motion of Cooper pairs, inelastic scattering between lattice and coherent Cooper pairs is not possible. It is because for inelastic collisions the Cooper pairs must change into free electrons for which energy equivalent to forbidden energy is required. Below critical

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Fig. 2.46 Band diagram for a normal metal b superconductor

temperature this energy is not available. Hence current constituting motion of coherent Cooper pairs is free of scattering, without ant resistance. SAQ: Stable levitation is more easily achieved with type-II superconductors. Why? SAQ: In a bench-top levitation demonstration the permanent magnet is first pushed towards the superconductor, why? SAQ: What is meant by the transient nature of Cooper pair formation? SAQ: Why Cooper pair-lattice collisions do not take place in the current flow through a superconductor? Problems P2.1 Photons of frequency larger than 1 × 1015 Hz get absorbed by a semiconductor. What is the Forbidden energy gap of the material? ANS: 4.1357 eV P2.2 A semiconductor has a forbidden energy gap of 1.46 eV. Calculate the maximum wavelength of photons that will be absorbed by the material. ANS: 0.85 × 10−6 m P2.3 Copper has 8.5 × 1028 free electrons per m3 , calculate the valency of copper given that its molecular mass is 63.54 g/mol and density 8.96 g cm−3 . ANS: 1 P2.4 A metal at very low temperature near absolute zero has Fermi energy of 7.0 eV. Calculate the number density of free electrons in the metal assuming that the effective mass of electron is equal to its actual mass 9.1 × 10–3 1 kg.

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ANS: 8.28 × 1028 m−3 Short Answer Questions SA2.1 SA2.2 SA2.3 SA2.4 SA2.5 SA2.6 SA2.7 SA2.8

SA2.9 SA2.10 SA2.11 SA2.12 SA2.13 SA2.14

Define a semiconductor and differentiate it with insulator What is meant by the breakdown of an insulator? Define the dielectric strength of an insulator. Explain why the resistivity of a semiconductor decreases with temperature while the resistivity of a conductor increases with temperature. What is the underlying principle of zone refining technique? Give an outline of steps to obtain ultrapure Silicon from its ore. What is doping? Describe ion implantation technique of doping. Draw energy band diagram and covalent bond picture of a p-type Silicon semiconductor at T > 0 K, indicating Fermi level, acceptor level, etc. What is depletion layer and how does it form in a p–n junction? What are the special properties of depletion region? In depletion layer the total uncovered positive charge on n-side is equal to the total uncovered negative charge on p-side. Suppose in a p–n junction pside is heavily doped as compared to the n-side, explain why the depletion layer will extend more on the n-side. What is a Cooper pair? How is it formed? Discuss the meaning of condensation with reference to superconductivity. State salient features of BCS theory for superconductors A superconducting body acts as a perfect diamagnetic material, explain. How can one explain the absence of resistivity in superconductors? What is Hall Effect? What is its significance? Define Hall coefficient and write its units. Drive an expression for Hall voltage.

Multiple Choice Questions Note: Some of the following multiple choice questions may have more than one correct alternative. All correct alternatives must be marked for complete answer. MC2.1

Conductivity has units (a) Ω-m (b) Ω m−1 (c) Ω−1 m−1 (d) Siemen per metre

MC2.2

ANS: (c), (d) Dielectric strength is (a) The maximum electric fields that an insulator may withstand before electric breakdown (b) Maximum voltage that may be applied to an insulator before electric breakdown

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2 Electrical Behaviour of Condensed Matter

(c) Maximum current that may pass through a conductor before electric breakdown (d) Maximum voltage that may be applied to a conductor before electric break down MC2.3

ANS: (a) If ni , np and ne , respectively, denote the charge carrier concentration in intrinsic material, concentrations of holes in p-type material and the concentration of electrons in n-type material, then (a) n i > n p (b) n i > n e (c) n p > n i (d) n p n e = n i2

MC2.4

ANS: (c), (d) Resistance of a piece of a semiconductor decreases with the increase of temperature because; (a) The mean free path of electron–lattice collisions increases with the rise in temperature (b) Rate of lattice-electron collisions decreases with the rise of temperature (c) With the increase of temperature, the rate of increase of carrier concentration overtakes the rate of rise of lattice–electron collisions (d) With the increase of temperature both the rate of increase in carrier concentration and rate of increase of lattice-electron collisions are nearly equal

MC2.5

ANS: (c) When dopant concentration in a given piece of a semiconductor is increased, the resistance of the piece. (a) Increases (b) Decreases (c) Remains unaltered (d) May increase or decrease depending the material

MC2.6

ANS: (b) At room temperature, the thermal velocity of free electrons in a semiconductor is of the order of (a) 10−5

MC2.7

m s

(b) 100

m s

(c) 105

m s

(d) 1015

m s

Ans: (c) Zone refining technique is based on the principle that; (a) Mobility of impurity atoms is less in molten state, and the melting point of pure Silicon is lower than the impure Silicon

2.6 Superconductor

133

(b) Mobility of impurity atoms is more in molten state, and the melting point of pure Silicon is lower than the impure Silicon (c) Mobility of impurity atoms is less in molten state, and the melting point of pure Silicon is higher than the impure Silicon (d) Mobility of impurity atoms is more in molten state, and the melting point of pure Silicon is higher than the impure Silicon MC2.8

ANS: (d) Depth profile of implanted impurity diopant ions is (a) Uniform (b) Shows a single peak at the range of implanted ion (c) Uniformly decreases (d) Uniformly increases

MC2.9

ANS: (b) Given that d p and d e , respectively, represent the width of depletion layer on the p-side, and on the n-side while N p and N e, respectively, the concentration of dopants on the n- and p-sides and if d p = 1.5 d n , then; (a) N p = N e (b) N p = 3 N e (c) N p = 1.5 N e (d) N e = 1.5 Np

ANS: (d) MC2.10 Reverse current in a pn junction gets doubled for every 100 C rise of temperature because; (a) Reverse current is constituted by minority carriers that are produced by bond breaking which increases with temperature (b) Reverse current is constituted by majority carriers that are produced by bond breaking which increases with temperature (c) Reverse current is constituted by minority carriers that are produced by bond breaking which decreases with temperature (d) Reverse current is constituted by majority carriers that are produced by bond breaking which decreases with temperature ANS: (a) MC2.11 Which of the following always has zero magnitude in a superconducting material? (a) Electric field E (b) Potential difference V (c) Current I (d) Magnetic susceptibility χ ANS: (a), (b) MC2.12 In an experiment a ring shaped specimen of superconducting material is placed in a magnetic field of strength B at temperature T > T C . The temperature T is then reduced and maintained below T C . The magnetic field is switched off. Which of the following statement(s) will correctly describe the result of the experiment?

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2 Electrical Behaviour of Condensed Matter

(a) Magnetic field will remain trapped outside the ring opening but it will be zero inside the ring opening (b) Magnetic field will remain trapped inside the ring opening and will be zero outside the ring opening (c) Magnetic field will be zero both inside and the outside the ring opening (d) Magnetic field will remain trapped both inside and outside the ring opening ANS: (b) MC2.13 A current I x in positive X-direction passes through a slab of n-type semiconductor. If a magnetic field BZ in positive Z-direction is applied across the semiconductor slab, the deflection force due to magnetic field on free electrons constituting the current will be in the direction of (a) X-axis (b) Y-axis (c) Z-axis (d) 45° between X- and Z-axis ANS: (b) MC2.14 Units for Hall coefficient are (a)

m3 C

(b)

C m3

(c)

m3 A.s

(d)

A.s m3

ANS: (a), (c) Long Answer Questions LA2.1 LA2.2

LA2.3

LA2.4

LA2.5

LA2.6 LA2.7

Discuss the electron energy band theory of crystals and hence explain the classification of solids according to their electrical properties. What are the characteristics of Insulators? Why do they have very large value for resistivity? Draw sketches for the energy band pictures of an insulator and a conductor. Explain the phenomena of breakdown in insulators and define the dielectric strength. Draw labelled diagrams for the band structures of a p-type and an n-type semiconductor. What are acceptor and donor levels and how do these levels affect the Fermi level? With the help of a labelled energy band diagram discuss the formation of a pn junction. What is meant by thermal equilibrium and how it is achieved in case of a pn junction? Discuss the formation of depletion layer at junction boundary and list some of its properties. What is a pn junction diode? Discuss the current flow through a junction diode under forward and reverse bias and draw its current voltage characteristics. With necessary details describe ion implantation method of doping of semiconductor wafer. What is Meissner effect? How can it be explained? Discuss stable levitation and essential conditions to achieve it.

2.6 Superconductor

LA2.8

135

What are Cooper pairs and their condensation? Outline BCS theory of superconductivity and explain why lattice-Cooper pair collisions do not take place in superconductors. LA2.9 Give distinguishing features of type-I and type-II superconductors. Wires of type-II superconductors are often used for making strong electromagnets, explain. LA2.10 What is isotope effect and why it was so significant in developing a theory for superconductivity? Explain the process of Cooper pair formation via electron–phonon interaction. With the help of energy band diagrams explain the difference between a conductor and a superconductor. LA2.11 What is Hall Effect? Describe with the help of a diagram the setup for measuring Hall voltage and derive an expression for Hall voltage in terms of Hall coefficient.

Chapter 3

Magnetic Materials

Objective Origin of magnetism in matter, types of magnetisms, their properties, applications, etc. will be discussed in this chapter. It is expected that after reading this chapter the reader will be able to understand how magnetic properties develop in materials and how materials may be classified in terms of their magnetic behaviour. He will also learn how materials with desired magnetic properties may be developed.

3.1 Introduction Magnetic materials include a variety of materials that are used in diverse applications. It is interesting to note that magnetic materials are utilised in generation and distribution of electricity and in most cases, they are also used in the appliances that use that electricity. Magnetic materials are used for the storage of data on audio and video tapes and computer disks. Magnetic materials also have applications in the field of medicine, they are used in body scanners as well as a range of applications where they are attached to or implanted into the human body. Non-polluting electrical vehicles have very efficient motors that utilise advanced magnetic materials. The fact that Earth behaves like a bar magnet was known to Indian saints and seers many centuries ago, and they developed and devised codes called ‘Vastushastra’ for build temples and buildings based on Earth’s magnetic meridian. In the modern era, however, in 1600 William Gilbert published the first systematic experiments on magnetism in the pamphlet ‘De Magnet’. By the end of the eighteenth century, scientists have noticed many electrical and magnetic phenomenon but they all believed that these two branches of science, the electricity and the magnetism are quite independent/separate from each other and that perhaps there is no direct relationship between the two. Lightning, the electric phenomena, was well known to ancient people but first magnetic material that showed the power of attracting small iron pieces was the loadstone, mineral magnetite (Fe3 O4 ), found in form of small © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 R. Prasad, Physics and Technology for Engineers, https://doi.org/10.1007/978-3-031-32084-2_3

137

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3 Magnetic Materials

pieces of rock, perhaps in Greece. Then in July 1820, Danish natural philosopher Hans Christian Orsted published a pamphlet that showed clearly that electric current and magnetism are very closely related to each other. It is said that Orsted, born in August 1777 after completing his Ph.D. in philosophy in 1801, travelled through Europe, as was customary at that time, and met many scientists/philosophers of Germany, France, etc. One person he met was Johann Ritter, a scientist who believed that electricity and magnetism are related. Orsted might have been influenced by him. Orsted returned back to Copenhagen in 1803 and tried for a faculty position at University there, but initially did not succeed. However, he started private lecture courses, charging admission fee, which attracted many young people. Later he got a position in Copenhagen University, perhaps on account of his popularity through private lectures. Though most of the scientists at that time thought electricity and magnetism to be two totally independent attributes of matter, but there were some indications, for example it was observed that a magnetic compass if struck by lightning reverses its poles; that pointed to some connection between the two. From his writings/lectures it appears that Orsted believed that electricity and magnetic behaviour are two independent properties of all matter and that these properties might interfere with each other. During one of his lecture demonstration on 21 April 1820 while setting up his apparatus, Orsted noticed that whenever he switched on current in his circuit, the north pole of the magnetic needle placed nearby deflected a bit. He also noticed that the direction of deflection of the North Pole of the compass needle changed when the direction of current in his circuit was reversed and that the effect of current on the movement of magnetic needle may be shielded by interposing an insulator/dielectric material between the electric circuit and the magnetic needle. Orested published his findings which were mainly qualitative, in a pamphlet, circulated to other scientists, on 21 July 1820; but the effect was clear: An electric current generates a magnetic field. French scientist Andre-Marie Ampere opined that the production of magnetic field by electric current is the fundamental feature of magnetism. In fact, he went to the extent that all magnetism, permanent magnets including, is the result of currents in them. Later, some 160 years ago, in 1864, James Clerk Maxwell carried out the first profound unification of Nature’s two forces, the electric force and the magnetic force in form of his famous Maxwell’s equations. The present understanding is that stationary electric charges produce an electric field around them; charges moving with uniform speed (current elements) give rise to both the electric and the magnetic fields while accelerated charges radiate electromagnetic fields.

3.2 Electric Current and Magnetic Field

139

3.2 Electric Current and Magnetic Field Electric current produces a magnetic field which may be visualised as a pattern of circular magnetic field lines surrounding the current carrying wire. The direction of the magnetic field may be determined using the compass needle while a Hall probe may be used to determine the magnitude of the field. Careful experimental study of the direction and magnitude of the magnetic field produced by a current in an infinite (very long) straight wire reviled the right-hand rule, according to which if one aligns the thumb of the right hand with the direction of current flow in the straight wire, the curled fingers of the right-hand point in the direction of the magnetic field. The magnitude of the magnetic field strength B due to current I in an infinite (or a sufficiently long) straight conductor at a perpendicular distance r from the conductor is given as; B=

μ0 I 2πr

(3.1)

Here constant μ0 is the permeability of the free space, is a basic constant of nature related to the velocity of light c, having the value μ0 = 4π × 10−7 T m/A. Since the current carrying conductor is very long, the magnetic field strength B at the point of observation O depends only on the perpendicular (shortest) distance r from the conductor. Biot–Savart argued that each little segment dl of the current produces a magnetic field at the point of observation and that the total magnetic field due to the complete current carrying conductor (of any shape) is given by, μ I B→ = 0 4π

→ → ∫ − ∫ − μ0 I dl X rˆ dl × →r = 2 r 4π |r|3

(3.2)

− → In Eq. (3.2) dl is the vector element of length in the direction of the current flow and rˆ is the unit vector in the direction of vector distance r from dl to the point of observation O. The line integration is to be carried on the length of the current carrying conductor. Figure 3.1a shows the direction of magnetic field lines due to an infinite straight conductor carrying current I, Fig. 3.1b depicts the right-hand rule to specify the direction of magnetic field lines, Fig. 3.1c gives the magnitude of magnetic field strength at point O situated at a perpendicular distance r from the straight infinite conductor carrying current I and Fig. 3.1d shows the strength of magnetic field due to a current element dl at distance r from it as given by Biot–Savart law. A bar magnet suspended in Earth’s magnetic field orients itself in North–South direction. The North seeking end of the bar magnet is called the North end and the geographic South seeking end as the South end. When lines of magnetic field of a bar magnet are drawn using a compass needle, they appear to originate from a point on the North end and appear to terminate at a point on the South end. Since all magnetic

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3 Magnetic Materials

Fig. 3.1 a Lines of magnetic field of an infinite conductor carrying current. b Right-hand rule giving the direction of magnetic field. c Magnitude of magnetic field strength at a point O at distance r from an infinite conductor carrying current I. d strength of magnetic field due to current element dl at point O

field lines appear to originate from one point on the North end, this particular point is called the North Pole of the magnet, and similarly point on the south end where all magnetic field lines appear to terminate, the South Pole. Number of magnetic field lines per unit area around a point is a measure of the intensity of magnetic field at that point. It is obvious that the magnitude of magnetic field intensity at poles is a maximum. Figure 3.2a shows the magnetic field lines of a bar magnet drawn using a compass needle. Though when looked from outside it appears as if the lines of magnetic field of a bar magnet originate from North pole and terminate at South pole, but the fact is that magnetic field lines are continuous within the bar magnet as shown in Fig. 3.2b. Parallel magnetic field lines at some place indicate a uniform magnetic field in that region. North poles as well as the south poles of two bar magnets repel each other while the opposite poles attract. One special property of a magnet is that North and South poles occur in pairs, for example if one break a bar magnet into pieces, each piece develops North and South poles. It is to say that free North or South magnetic poles (mono poles) never occur in nature. Magnetic poles to some extant may be compared with electric charges, magnetic north pole like positive electric charge and magnetic south pole like negative electric charge.

3.3 Magnetic Dipole Moment

141

Fig. 3.2 Magnetic field lines of a bar magnet

However, there is one fundamental difference; it is possible to have an isolated single free positive or negative electric charge but it is not possible to have a free single magnetic north pole or a single free magnetic south pole. Magnetic North and South poles always occur in pair. A major point of difference between the electric and magnetic field lines is that electric field lines actually originate from positive charge and terminate at negative charge but magnetic field lines always form a closed loop. Though electric and magnetic fields are two quite different fields, however, it is observed that the electric field due to an electric dipole, (two equal and opposite charges separated by a small distance) is similar to the magnetic field of a bar magnet as may be seen in Figs. 3.2 and 3.3a. Lines of magnetic fields due to circular current loop are shown in Fig. 3.3b, c. It may be observed in these figures that the magnetic field due to a bar magnet (or a magnetic dipole) is similar to the magnetic field produced by a circular current loop. Therefore, a circular current loop is also called a magnetic dipole. SAQ: What information does the permeability of a material give about the magnetic behaviour of the material?

3.3 Magnetic Dipole Moment Magnetic dipole moment or simply magnetic moment of an object is a vector quantity used to measure the tendency of the object to interact with an external magnetic field and is represented by the symbol µ. The object’s intrinsic magnetic properties play an important role in deciding the tendency of its interaction with external magnetic field. The intrinsic magnetic properties of the object are often visualised as emanating from a tiny bar magnet with north and south poles, hence the name dipole moment. Figure 3.4a shows the magnetic dipole moment for a small bar magnet of length 2a and pole strength M when looked from a distance. A current carrying circular loop of area A (= πr 2 ) when looked from a distance produces a magnetic field that resembles the magnetic field of a tiny bar magnet and the magnetic (dipole) moment of the current loop is equal to IA, where I is the current in the loop and A is the area

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3 Magnetic Materials

Fig. 3.3 a Electric field lines (blue) and equipotential lines (red) due to an electric dipole b magnetic field lines due to a current loop with axis along Z-axis; the yellow line shows the segment of the current loop c current loop and associated magnetic field lines

enclosed by the current loop. The direction of the magnetic moment of the loop may be easily determined using the right-hand rule. → is placed in an external When an object having a magnetic dipole moment μ −−→ magnetic field B ex t , the object or the magnetic moment associated with the object experiences a torque that tries to align the dipole moment of the object in the direction of the external magnetic field. The magnitude and direction of the torque τ are given −−→ −−→ → B ex t as shown in Fig. 3.4c. In case when B ex t has the by the vector equation τ→ = μ× magnitude of 1, and the angle between magnetic moment μ and external magnetic field Bext is 90°, then τ = μ. One may, therefore, define the magnetic moment of an object as the maximum torque experienced by the object in unit magnetic field.

3.4 Magnetic Moment of a Charged Particle Moving in a Circular Orbit

143

Fig. 3.4 a Magnetic dipole moment of a small bar magnet. b Magnetic dipole moment of a circular current loop. c Torque experienced by a magnetic dipole moment in an external magnetic field. d Potential energy of a magnetic dipole in an external magnetic field

The potential energy U of the interaction between the magnetic moment and the −−→ → B ex t . The negative sign in expression for external magnetic field is given as U = −μ. U is included to indicate that the potential energy of interaction will be a minimum when both the magnetic moment and the external magnetic fields are parallel and will be a maximum when they point in opposite directions. It is arbitrarily chosen to assign a value zero to the potential energy U when magnetic moment points in a direction perpendicular to the external magnetic field.

3.4 Magnetic Moment of a Charged Particle Moving in a Circular Orbit Figure 3.5 shows a particle of mass m, charge + q moving with linear velocity v in a circular path of radius r. We will calculate the magnetic dipole moment µ and angular momentum L of the particle under classical approach. → of the particle. A particle Let us first calculate the magnetic (dipole) moment μ having an electric charge + q, moving in a circular closed path with linear velocity v (angular velocity ω = v/r ) crosses a fixed point on its circular path after each

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3 Magnetic Materials

Fig. 3.5 Magnetic dipole moment and angular momentum of a particle of mass m and charge q moving with linear velocity v in a circular path of radius r

time interval t = 2πr/v. The frequency of crossing the fixed point f = 1t = v/2πr and the current I constituted by the circulating charge + q and defined as the rate of flowing of charge, may be written as, I = +q f =

+qv 2πr

(3.3)

Current I flows in the same direction as the direction of motion of the electric charge + q, in case the charge q is negative, the current I will flow in the direction opposite to the direction of motion of the charge. The circular area A enclosed by current I is A = π r 2 and, therefore, the magnetic (dipole) moment of the current loop is given by, +qv ( 2 ) +qvr − → πr = μ| = IA = 2πr 2

(3.4)

Next, let us calculate the angular momentum L of the charged particle of mass m which is moving in the circular path of radius r with linear velocity v. By definition, the angular momentum is given as; | | | | L→ = r→ × p→ = r→ × (m . v→) or | L→ | = mr v sin θ

(3.5)

In Eq. (3.5) θ is the angle between the linear velocity v and the radius vector r, which in the present case of circular motion is always 90° and hence → = mrv | L|

(3.6)

The direction of L is given by the vector cross-product rule and in the present case angular momentum of the particle points in the upward direction, normal to the − → → μ and L plane containing r and v. It may be observed that if q is positive, both − → μ will point downwards while points are in the same direction but if q is negative, − angular momentum L will point upwards.

3.4 Magnetic Moment of a Charged Particle Moving in a Circular Orbit

145

Having obtained expressions for the magnetic moment of the moving charge q and its angular momentum we calculate the ratio, | | +qvr | |μ |→|=γ = 2 = q | L→ | mr v 2m

(3.7)

The above ratio of magnetic moment to angular momentum of a particle is called the Gyromagnetic ratio, represented by the symbol γ , and is an important parameter for microscopic particles. Equation (3.7) tells that units for γ are Coulomb per kilogram in SI system. Also, one may write, q → μ → = γ L→ = L 2m

(3.8)

3.4.1 Classical to Quantum Mechanics So far we considered the motion of a charged particle in classical limits and derived expressions for magnetic dipole moment and angular momentum. However, it is known that the behaviour of microscopic particles like neutron, proton and electron, etc., which are constituents of atom, is better understood in terms of quantum mechanical treatment. It is interesting to note that all classical expressions derived here holds good in quantum mechanical treatment also. However, there are two points of difference between the classical and the quantum mechanical treatments: (i) In quantum treatment it is not important wether the current flows in a circular path or not, current must flow in a closed loop enclosing some area A (which may not necessarily be a circular area) and that value of area may be used in above expressions. (ii) In case of classical treatment L may have different continuous values from zero to mv, depending on the value of angle θ, but in quantum physics L may assume only discrete values. In switching from classical to quantum treatment, the expression for L becomes, √ L = l(l + 1)ℏ (3.9) where l is either 0 or a positive integer, i.e. l may have values 0, 1, 2, 3, … and ℏ is the unit in which angular momentum is measured in quantum mechanics ℏ = 1.05457 × 10−34 J s. ‘l’ is called the orbital angular momentum quantum number and the state of motion of the particle is characterised according to the value of l; when l = 0 the state of relative motion is called the s-state, for l = 1, the p-state; for l = 2, the d-state; for l = 3, the f-state and so on. Thus in quantum mechanical treatment the magnetic dipole moment of a charged particle for orbital motion may be written as;

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3 Magnetic Materials

− →l q → L μ = 2m

(3.10)

and, L=

√ l(l + 1)ℏ

(3.11)

h where l may have values 0, 1, 2, 3, …; and ℏ = 2π , h being Planck’s constant. Further in quantum mechanics it is convenient to measure angular momentum in units of ℏ.

3.5 Magnetic (Dipole) Moment of Electron Electrons are constituents of all atoms and molecules. It is known that two kinds of motions, namely orbital motion around the atomic nucleus and inherent spin motion, are associated with electrons. Both these motions obey laws of quantum mechanics and have corresponding magnetic moments associated with them. Let us first consider the orbital motion of an electron. (i) Orbital motion of electron The magnetic moment μorb e of electron due to its orbital motion may be obtained from Eq. (3.10) by replacing (i) q by (− e) where minus sign indicates that the charge of electron is negative, e = 1.6 × 10−19 C is the charge of the electron and (ii) m by m e , the mass of electron. With these substitutions, we get −−→ e → μorb L e =− 2m e

(3.12)

Equation (3.12) tells that in case of electron, the direction of magnetic moment μorb e is opposite to the direction of angular momentum L, because of the negative charge on electron. The simplest way of measuring the magnetic (dipole) moment of a particle is to put the particle in some external field in a given direction and measure the interaction energy U, from which the magnitude of magnetic moment may be calculated. Suppose the external magnetic field is applied along the z-axis. Then the z-component of the magnetic moment will come into play. It is reasonable to assume that Eq. (3.12) will also hold for the z-components of μle and L, therefore, (

μorb e

) z

=−

e Lz 2m e

(3.13)

3.5 Magnetic (Dipole) Moment of Electron

147

Now quantum mechanics tells that the z-component of angular momentum is also quantised, i.e. L z = ml ℏ

(3.14)

Here, ml , called the magnetic quantum number, may have (2l + 1) different values, starting from For −l, (−l + 1), (−l + 2), (−l + 3), . . . , (l − 3), (l − 2), (l − 1), l. example if orbital angular momentum l = 0 (s-state), then magnetic quantum number m l may have only one value that is 0. If l = 1(p-state), magnetic quantum number m l may have three different values; − 1, 0, and + 1; if l = 2(d-state) then m l may have 5 different values, − 2, − 1, 0, + 1 and + 2 and so on. Substituting the value of L z from Eq. (3.13) in Eq. (3.14), one gets; (

)

μorb e z

[ ] e e eℏ =− Lz = − ml ℏ = − ml 2m e 2m e 2m e

(3.15)

The negative sign on the RHS of Eq. (3.15) simply indicates that the direction of μorb e is opposite to L z . [ ] eℏ Quantity 2m is defined as 1 Bohr Magneton represented either by Bm or e μB . Magnetic moment of atomic particle like electron is measured in units of Bohr magneton. Magnitude of 1 Bm or 1μB depends on the system of units. In eℏ while in Gaussian CGS system 1 Bm = 2meℏe c SI system of units 1 Bm = 2m e Metric (SI) Equivalent 9.274 × 10−24 J/T or m2 A CGS 9.274 × 10−21 erg/G eV 5.7883 × 10−5 eV/T Bohr magneton may be expressed by different dimension formulas depending on systems of units, some of them are; M L 3 T −1 Q −1 ; N m( A/m)−1 ; N M T −1 etc. As has been mentioned, in experimental measurement of the magnetic moment of a particle, an external magnetic field is applied in a certain direction (denoted by direction z) and the z-component of the magnetic moment is measured. Thus, any measurement of magnetic moment yields only the component of magnetic moment in a specified direction. Further, the measurement may not give a unique value of the component; it is because, from Eq. (3.15) the measurement may give one of the several possible values of magnetic quantum number

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3 Magnetic Materials

m l . Let us understand it by taking an example. Suppose one measures the component of magnetic moment of an electron due to its orbital motion in state − d where l = 2. The possible values of m l for the case (l = 2) are: − 2, − 1, 0, + 1 and + 2. The measurement may give any one of these values. That means when same measurement is carried out by different experimentalists or repeated, they may get different values for the same component of magnetic moment. To remove this ambiguity, it has become a convention to (i) treat the measured component of magnetic moment as the magnetic moment of the particle (not to call it a component) and (ii) out of the several possible values of m l , the maximum positive value is taken as the measured value. Suppose in a given case J is the maximum positive value of magnetic quantum number, then one writes; ] [ eℏ J (3.16) μorb = − J 2m e As already mentioned the negative sign in Eq. (3.16) may be dropped, taking in account the fact that the magnetic moment of electron due to its orbital motion is opposite in direction ( to the ) angular momentum. It follows from Eqs. (3.15) is measured in units of Bohr magneton (Bm ) and and (3.16) that when μorb e z proper sign of μorb is taken into account, then the ratio J ( orb ) μe z (in unit Bm ) ml

= 1 or

μorb J =1 J

(3.17)

Expression (3.17) may be generalised for any quantised magnetic dipole momentum (measured in Bohr magneton) and the corresponding angular momentum quantum number in the following form; μ(in Bm ) =g J

(3.18)

Equation (3.18) implies that if μ is the value of the magnetic moment of some particle measured in units of Bohr magneton (Bm ) and J is the corresponding quantum number associated with motion of the particle, then the ratio of the two may be represented by a constant g. The constant g is called the g-factor of the particle and depends on the nature of the particle as well as on the type and state of motion of the particle. For example, the value of g-factor for electron for its orbital motion georbit may be given as; μorb J = georbit = 1 J

(3.19)

3.5 Magnetic (Dipole) Moment of Electron

149

Thus, the g-factor (a dimension less constant) of electron for its orbital motion georbit = 1. Difference between Gyromagnetic ratio γ and the g-factor: Both represent the ratio of magnetic moment to angular momentum, but units for measuring the magnetic moment and the angular momentum are different. The ratio of magnetic moment to angular momentum quantum number when magnetic moment is measured in units of Bohr magneton, gives the g-factor. (ii) Spin motion of electron Motion of electron around the nucleus of the atom is termed as its orbital motion. Apart from that an electron also possesses an inherent motion called the spin motion. Spin or spin motion is purely a quantum mechanical concept that cannot be explained in terms of classical physics. It is reasonable to assume that the inherent spin motion of electron follows the same laws as are followed by its orbital motion. There is an angular momentum quantum number L associated with the orbital motion similarly there is a quantum number S associated with spin motion. One can write down an expression for spin motion corresponding to Eq. (3.18) of orbital motion as given below; spin

μs

(in units of Bm ) = g spin s

(3.20)

Or μspin (in units of Bohr magneton (Bm )) = gespin s s

(3.21)

spin

Precise measurement of spin magnetic moment μs of electron gives the value spin of ge for electron as 2.002318. Therefore, in case of electron; georbit = 1 and gespin = 2.002318 spin

The fact that ge ≈ 2, while georbit = 1 implies that spin motion of electron is twice as effective in producing magnetic field as the orbital motion. There are many electrons in an average atom and the total magnetic dipole moment due to all electrons in the atom may be calculated in two different ways: (a) In this method, called the j − j coupling method, the first step is to find the total angular momentum ji of the ith electron by quantum mechanically adding its orbital and spin angular momentums, ji = li + si . In the second step the total angular momentum J of all the electrons is determined by quantum √ ji . mechanical adding of total angular momentums ji ’s, of electrons; J = Once total angular momentum J is known one may calculate the magnetic moment of the atom due to all its electrons using the expression

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3 Magnetic Materials ( j− j)

( j− j )

μatom = gatom J

(3.22)

( j− j)

Here gatom is the g-factor for j − j coupling. It may be mentioned that quantum mechanical addition of angular momentums is quite different from the classical vector addition. (b) In second method, called (l − s) coupling, the total orbital angular momentum L of all electrons in the atom is determined √ by quantum mechanical adding of their orbital angular momentums, L = li and similarly total spin S of the atom is√ obtained by quantum mechanical adding of spins of individual electrons, S = si . The resultant angular momentum J of the atom is then obtained by quantum mechanical addition of L and S; J = L + S. (L+S) Expression μ(L+S) atom = gatom J may be used to obtain the magnetic moment of ( j− j ) (L+S) the atom. g-factors; gatom and gatom may have different values for the same atom. (iii) Magnetic moments of nuclear particles Nucleus of every atom contains protons and neutrons which also perform orbital and spin motions. Nucleons (both neutron and proton), therefore, like electrons, have magnetic moments. Magnetic moment of nucleons is measured in a unit eℏ , called nuclear magneton, denoted by Nm . 1 Nuclear Magneton (Nm ) = 2m p where m p is the mass of proton. One nuclear magneton is about 1836 times smaller than one Bohr magneton because a proto (1.6 × 10–27 kg) is roughly 1836 times heavier than an electron (9.1 × 10–31 kg). As a result, the contribution of magnetic moments of nucleons is neglected while discussing the magnetic moment of the atom. As a concluding remark it may be said that magnetic fields originate from electric currents.

3.6 Magnetic Behaviour of Solids A common man may categorise solid matter in two classes: either magnetic or nonmagnetic. Magnetic materials are those which attract some other materials like iron filing, lodestone, pins, etc., whereas non-magnetic solids, like wood, common salt, chalk, etc. do not show any effect of magnetic field. However, the fact is that the so-called non-magnetic materials are also affected by magnetic fields but the effect is so weak that very sensitive detecting instruments and high magnetic fields are required to demonstrate the effect. Magnetic behaviour of matter originates essentially from the spin and orbital motions of electrons in atoms/molecules or ions. Fortunately, all electrons in the atom or ions, etc. do not contribute to magnetic properties. According to quantum mechanical laws, an even number of electrons in a given energy level of the atom/ion, etc. orient their orbital and spin motions in such a way that their magnetic moments cancel out, resulting in no net magnetic field. Therefore, atoms, ions and molecules the outer valence shell of which have even number of electrons do not show magnetic

3.6 Magnetic Behaviour of Solids

151

properties. However, if there are odd number of electrons in the outer energy levels of an atom, ion or molecule, it has a magnetic dipole moment (and hence show magnetic properties) that may be attributed to the unpaired electrons. Large number of atoms, like that of bismuth, mercury, silver, copper, inert gases, etc. have closed shell electron structure with even number of electrons, and hence, do not show any magnetic properties of their own. However, when put in an external magnetic field, these atoms develop an induced magnetism opposite to the applied magnetic field. Such materials are called diamagnetic materials. Copper atom in ground state has electron configuration: 1s2 2s2 2p6 3s2 3p6 4s1 3d10 ; and is diamagnetic. Copper ion Cu1+ on loosing 4s1 electron also becomes a diamagnetic ion. Some other examples are; Copper (Cu) ground state: [Ar] 4s1 3d10 ; Copper ion (Cu1+ ):[Ar] 3d10 ; Lead (Pb) ground state: [Xe]6s2 4f14 5d10 6p2 ; Silver (Ag) ground state: [Kr] 4p6 5s1 4d10 ; Mercury (Hg) ground state: [Xe] 4f14 5d10 6s2 Atoms/ions with odd number of electrons in outer shell show magnetic properties that may be attributed to the orbital and spin motions of the unpaired electron; such atoms behave like small dipoles. Since atomic dipoles, in general, are randomly oriented, the bulk material may show inherent magnetism but quite weak in strength. These materials are classified as paramagnetic materials. In some special cases, tiny atomic (or ionic or molecular) dipoles may align themselves in different patterns giving rise to ferromagnetism, ferrimagnetism and antiferromagnetism. In order to study different types of magnetism in details we define the following important magnetic parameters that are frequently used to specify types of magnetism. SAQ: It is customary to neglect magnetic effects produced by the motion of protons present in the nucleus of the atom, while calculating magnetic dipole moment of the atom. Give justification. SAQ: Neutron possesses a magnetic dipole moment due to its spin motion which has negative value. What inference do you draw about the charge distribution within a neutron?

3.6.1 Magnetic Induction B and Magnetic Field H Magnetic induction may be defined in different ways; however, we describe here an experimental method of determining magnetic induction B at a point in a magnetic field. Suppose there is a magnetic field in free medium (vacuum or air) produced, say, by a permanent magnet (see Fig. 3.6). It is known that conductor of length dl carrying a current i and placed in this magnetic field at some point P will experience a force F, given by; F = B i dl sin θ ,

(3.23)

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3 Magnetic Materials

Fig. 3.6 Defining magnetic induction at a point P in a magnetic field

Here θ is the angle between the direction of current and the force. In case of unit current (i = 1), force perpendicular to the direction of current per unit length (dl = 1) is equal to the magnetic induction B at point P. Direction of B will be perpendicular to the direction of current flow. The SI units for magnetic induction B is tesla, represented by letter T. One tesla 1 T = one weber per square metre that corresponds to 104 gauss. 1 T may also be represented as one kilogram per second squared per ampere (kg s−2 A−1 ). Having defined B at point P in terms of the force F, one may now define the strength of magnetic field H at point P as, H=

B μ0

(3.24)

Here μ0 is the permeability of the medium that is the or air in the present case. ( Hvacuum ) or Newton per Ampere square The value of μ0 for air or vacuum is 4π 10−7 Henery meter m (N/A2 ). It may be noted that symbol μ is frequently used in science to denote different quantities, for example in optics μ is often used to indicate the refractive index, it also denotes the magnetic dipole moment, but here it is being used to represent another magnetic parameter of a medium called permeability. Permeability of a medium is a measure of the ease with which a magnetic field may be established in the medium. The SI units of magnetic field strength H is amperes per metre (A/m), while the CGS units are Oersted (Oe). Let us now consider the case when the medium surrounding the point of observation P is not vacuum or air but is some other material, say, an iron or brass sheet or any other material. The magnetic field H which is due to the permanent magnet at point P will induce some magnetic field at point P because of the magnetisation of the medium. If M denotes the induced magnetic dipole moment per unit volume

3.6 Magnetic Behaviour of Solids Table 3.1 Permeability and relative permeability of some materials

153

Material/medium

Permeability μ (H/ m)

Relative permeability μr

Vacuum/air

1.256637 × 10–6

1.000000

Copper

1.256629 × 10–6

0.999994

Iron (99.8% pure)

6.3 × 10–3

5000

Nickel

1.25 × × 10–4

Superconductors

0

10–4

– 7.54

100–600 0

of the surrounding medium, then the magnitude of magnetic induction at point P (in presence of the new medium) is given as’ B = μ0 (H + M) = μH

(3.25)

Here, μ is the permeability of the medium (iron or brass, etc.). The ratio μμ0 = μr is called the relative permeability of the medium is just a number/fraction without units. Permeability and relative permeability of some materials are given in Table 3.1. In an isotropic medium B and H are parallel and the permeability is a scalar quantity. However in an anisotropic medium B and H may not necessarily be parallel and the permeability is a tensor. The induced dipole moment per unit volume M, also called the intensity of magnetisation or simply magnetisation is the effect of the magnetising field H on the medium. Thus H is the ‘cause’ and M is the ‘result’. It is obvious that the ‘result’ M will be proportional to its ‘cause’ H, i.e. M ∝ H or M = χ H

(3.26)

The constant of proportionality denoted by Greek letter χ is called the magnetic susceptibility of the medium. Susceptibility χ is a parameter that demonstrates the type of the magnetic material and the strength of the induced magnetic field in the medium. Sometimes one uses mass susceptibility, denoted by χm , that may be obtained by dividing χ with the density ρ of the medium. Further, it follows from Eqs. (3.25) and (3.26) that; B = μH = μ0 (H + M) = μ0 (H + χ H) = μ0 H(1 + χ ) Or μH = μ0 H(1 + χ ) That gives;

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3 Magnetic Materials

μ r = (1 + χ )

(3.27)

Magnetic properties of different materials may be classified in terms of the sign and the magnitude of its susceptibility χ . Since units of M and H are same, bulk susceptibility χ is just a number/fraction and has no units and dimensions. However, the mass susceptibility χm has units of metre3 /kilogram in SI system.

3.7 Classification of Magnetic Materials Magnetic behaviour of all materials, depending on the sign and magnitude of their magnetic susceptibilities, may be classified into five categories: (a) diamagnetic, (b) paramagnetic, (c) ferromagnetic, (d) antiferromagnetic and (e) ferrimagnetic. Out of some 90 stable elements of the periodic table around 38 are diamagnetic, 48 paramagnetic, 03 ferromagnetic and only 01 antiferromagnetic at room temperature. No element in natural form is found to exhibit ferrimagnetisms; only some compounds such as mixed oxides show this type of magnetic behaviour.

3.7.1 Diamagnetic Materials Materials with bulk susceptibility χ having small negative value in the range of −10−6 to − 10−5 are classified as diamagnetic. Negative value of susceptibility means that in an applied magnetic field diamagnetic materials acquire a magnetisation which points opposite to the magnetising field H. Atoms of diamagnetic materials do not show any magnetism in absence of any external magnetic field (H); that essentially means that the outer shells of these atoms have even number of electrons that chancel out their magnetic moments (due to their spin and orbital motions) in pairs. However, when some external magnetic field H is applied to a diamagnetic atom, it alters the speeds of rotation of the two electrons of a pair in such a way that a net magnetic moment is generated that is opposite in direction to the applied field H. The magnitude of bulk susceptibility for most of the diamagnetic materials does not change with temperature neither it depends on the strength of the external magnetising field H. Since application of an external magnetic field H effects the spin and orbital motions of electrons of any atom, diamagnetic effect is produced in all types of atoms irrespective of their class of magnetism; atoms of paramagnetic, ferromagnetic, antiferromagnetic and ferrimagnetic materials also show diamagnetic behaviour, but the magnitude of diamagnetic affect is much too small in these atoms as compared to other more dominant effects. As such it may not be wrong to say that diamagnetism is universal and more fundamental than any other type of magnetism.

3.7 Classification of Magnetic Materials

3.7.1.1

155

Langevin’s Theory of Diamagnetism

Though diamagnetism is a quantum phenomenon, a classical theory for diamagnetism was proposed by Langevin in 1905. The classical theory was able to explain most of the observed characteristics of diamagnetic materials. Brief outline of the theory is presented here. Langevin’s theory is based on the assumption that electrons circulating (around the nucleus) in closed circular paths in an atom produce magnetic dipole pole moments, the sum of dipole moments in a diamagnetic atom which is not subjected to any external magnetic field is zero because of the equal number of electrons circulating in clockwise and anticlockwise directions. However, if atoms of a diamagnetic material are put in an external magnetic field H, they develop magnetic induction B = μH = μ0 (H + M). Induction B applies a torque τ on each electron of the atom. Torque τ makes each electron of the atom to undergo a rotational motion about the direction of B. This rotational motion around B is called Larmor precession, and the angular frequency ω L of Larmor precession is given by; ωL =

eB 2m e

(3.28)

Here e and me are respectively the charge and mass of the electron. The angular momentum L p associated with precessional motion is given as; ⟨ ⟩ L p = m e ωL r 2

(3.29)

⟨ ⟩ where r 2 is the mean square distance from an axis through the nucleus parallel to B. Larmor precession motion of electron will associate an additional magnetic dipole moment μep with each electron of the atom, given by; μep = −

e Lp 2m e

(3.30)

If each atom has Z electrons and number of atoms per unit volume (number density of atoms) in the material is N, then the total additional dipole moment per unit volume or magnetisation M that will develop in the material due to Larmor precession will be given as; ( M = Z N μep = −N Z

e Lp 2m e

) = −N Z

⟨ ⟩) e ( m e ωL r 2 2m e

(3.31)

Substituting the value of ω L from Eq. (3.28) in Eq. (3.31) one gets, ) ( eB (⟨ 2 ⟩) e r (m e ) M = −N Z 2m e 2m e

(3.32)

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3 Magnetic Materials

Fig. 3.7 a Electron undergoing Larmor precession. b H–M graph for a diamagnetic material

However, B = μ0 H putting this in Eq. (3.31) gives M e2 (⟨ 2 ⟩) = χ = −μ0 N Z r H 4m e

(3.33)

(⟨ ⟩) The value of r 2 may be calculated with reference to Fig. 3.7a in terms of the mean square radius ρ of the orbit as, ρ2 = x 2 + y2 + z2 But x 2 = y2 = z2 =

ρ2 3

r 2 = x 2 + y2 =

2 2 ρ 3

and

(⟨ ⟩) ⟨ ⟩ Therefore, r 2 = 23 ρ 2 substituting this in Eq. (3.33) gives, ( ) M e2 2 ⟨ 2 ⟩ e2 ⟨ 2 ⟩ ρ = −μ0 N Z ρ = χ = −μ0 N Z H 4m e 3 6m e

(3.34)

Expression (3.34) for the bulk magnetic susceptibility of diamagnetic materials tells: 1. Atoms of diamagnetic materials as a whole does not possess any permanent dipole moment

3.7 Classification of Magnetic Materials

157

2. Magnetisation in diamagnetic materials is produced by the influence of the external magnetic field which makes the electrons of the atom to undergo Larmor precessional motion. 3. Diamagnetic ⟨ ⟩ materials have a small negative value of susceptibility, small because ρ 2 is small. 4. Negative value of susceptibility means that the magnetisation produced by the magnetising field is opposite in direction to the magnetising field. B < μ0 H . 5. Bulk susceptibility of diamagnetic materials depends on the number of electrons per atom (Z), number density (N) of atoms (atoms per unit volume) and is independent of temperature of the specimen and the intensity of the magnetising field H. 6. Figure 3.7b shows the variation of M with H for a diamagnetic material. Same results regarding diamagnetism are obtained from a quantum mechanical approach using perturbation theory. Some important diamagnetic elements are: H, Be, B, C, N, F, Ne, Si, P, S, Cl, Ar, Cu, Zn, Ga, As, Se, Br, Kr, Ag, Cd, Sb, Hg, Pb, Bi, etc. Some diamagnetic ions: A given ion is diamagnetic or not depends on its electronic configuration. If the highest electron energy states have even number of electrons then the ion will be diamagnetic; for example Ca2+ ion has electron structure 1s2 2s2 2p6 3s2 3p6 has 6-electrons in the highest energy level 3p, hence it is diamagnetic. Thus electronic configurations of atoms and ions may reveal their magnetic properties. Diamagnetic materials, because of the negative sign of their susceptibility, get repelled by external magnetic fields and try to move towards the weaker part of the external field. Diamagnetism is observed in water, wood, and in most of organic molecules including most of living organism. The fact that water is diamagnetic may be demonstrated by rapping a very strong permanent magnet (super magnet made from rare earth compounds) with a very thin layer of water. The strong magnetic field repels water which produces a small dimple (bulge) in water layer. The bulge is small and may be observed only by a reflection microscope. Superconductors might be considered as perfect diamagnetic material (χ = −1) since they repel (or got repelled by) magnetic fields. Strong repulsion of strong diamagnetic materials by super magnets may give rise to stable levitation. Pyrolytic graphite an unusually strong diamagnetic material (χ = −40.9) can be levitated in a stable equilibrium in the field of a permanent super magnet made from compounds of rare earths. Levitation of living organisms, like frog, mice, etc. in the fields of superconducting magnets has also been demonstrated. SAQ: What causes the diamagnetic behaviour of some atoms?

158

3 Magnetic Materials

3.7.2 Paramagnetic Materials Except the atoms/ions or molecules of diamagnetic materials, atoms (ions/molecules) of all other materials that may exhibit paramagnetism, ferromagnetism, ferrimagnetisms or antiferromagnetism possess an inherent magnetic dipole moment; each atom/ion is like a tiny bar magnet. The inherent magnetic dipole moment comes essentially from the spin motion of electrons (as spin motion is twice as effective as orbital motion in producing magnetic effects) of unpaired electrons in outer shells of the atom/ion or molecule. Normally at temperature T > 0 K the inherent magnetic dipoles have random orientations because of thermal agitation. The resultant magnetisation, which is the vector sum of atomic dipole moments, is zero because of the random distribution of atomic dipoles and also for minimising the magnetic energy of the system. However, when some external magnetic field H is applied, randomly oriented inherent dipoles try to align in the direction of the applied magnetic field. At a given temperature T, each atomic dipole experience two torques: a torque τ H due to field H that tries to align the atomic dipole in the direction of H and a torque due to thermal motion τT that opposes the torque τ H of alignment. Since torque τ H has different magnitudes for different atomic dipoles all atoms of the specimen do not align completely, resultant magnetisation M generated from the partial alignment of inherent dipoles has a small value but it is in the same direction as that of H. This results in a small positive value for the bulk magnetic susceptibility χ . Materials with positive small values of bulk susceptibilities in the range of + 10–5 to + 10–3 are termed as paramagnetic. Large number of elements like O, Na, Rb, Sc, Al, Sn, etc. is paramagnetic. When atoms of a paramagnetic material are put in an external magnetic field H, two happenings take place; (i) the inherent magnetic dipole moments of atoms try to align along the direction of the field H giving rise to a magnetisation, say M 1 in the direction of H and (ii) the electrons of the atom starts Larmor precession around the direction of induction B which generates a magnetisation − M 2 in a direction opposite to H. Since M 1 is generally much larger than M 2 , the net result is a magnetisation of magnitude + (M 1 − M 2 ) pointing in the direction of H. Obviously, M 2 is a reflection of the diamagnetic behaviour the paramagnetic material.

3.7.2.1

Langevin’s Theory for Paramagnetism

Different theories have been proposed to explain paramagnetism of different types of materials. Langevin’s theory is applicable to those materials in which the inherent magnetic dipoles of neighbouring atoms do not interact with each other. This is often referred as theory for non-interacting localised electrons and holds for materials like hydrated salts of transition metals like CuSO4 .5H2 O and gases. The diamagnetic water molecules in such hydrated salts shield one magnetic dipole from the other while in dilute gases atomic magnetic dipoles are far apart to be in the interaction range.

3.7 Classification of Magnetic Materials

159

Fig. 3.8 Partially aligned inherent atomic magnetic dipole

Langevin assumed that each atom/ion of the material has a permanent or inherent dipole moment μ and that number density of atoms/ions in the specimen is N per unit volume. Further, it is assumed that there is no interaction between these inherent dipoles. On application of an external magnetic field H, each inherent magnetic dipole experiences a torque τ which tries to align the magnetic dipole in the direction of external magnetic field. However, on account of the thermal motion (due to temperature T of the specimen), all atomic dipoles could not align completely along the direction of H. Different atomic dipoles will align to different degrees with respect to the field H. Let us assume that a typical inherent dipole align itself at an angle θ with respect to the direction of the external magnetic field H (see Fig. 3.8). The potential energy E θ of this magnetic dipole in magnetic field H is given as, E θ = μ.H = −μ.H cos θ

(3.35)

The number n of atomic magnetic dipoles with energy E θ at temperature T of the specimen may be calculated using Maxwell Boltzmann statistics which gives, ) ( E − k θT

n = n0e

β

= n0e

) ( cos θ − −μH k T β

(

= n0e

μH cos θ kβ T

)

(3.36)

Here n0 is a constant and k β is Boltzmann constant. The number dn of magnetic dipoles that have their inclinations between θ and (θ + dθ ) with respect to the direction of H may be obtained by differentiating expression (3.36) with respect to θ. Therefore, ( dn = n 0

) ( μH cos θ ) μH e kβ T sin θ dθ kβ T

(3.37)

Each of the partially aligned magnetic dipole contributes an additional magnetic moment m = μ cos θ to the specimen in the direction of the applied field H (components of μ in direction perpendicular to the field H from different dipoles will cancel each other). The average value of the total additional magnetic moment, denoted by ⟨m⟩ may be obtained by dividing the sum of the components in field direction contributed by each atomic dipole by the total number of dipoles,

160

3 Magnetic Materials

{π ⟨m⟩ =

0

μ cos θ dn {π = 0 dn

{π 0

[ ( ) ( μ cos θ ) ] kβ T e μ cos θ n 0 kμH sin θ dθ βT {π 0

(

n0

μH kβ T

) ( μ cos θ ) e kβ T sin θ dθ

Or {π ⟨m⟩ =

0

[ ( μH cos θ ) ] μH cos θ e kβ T sin θ dθ {π 0

(

e

μH cos θ kβ T

)

(3.38) sin θ dθ

Let us make following substitutions in Eq. (3.38); y=

μH kβ T

and x = cos θ ; then dx = − sin θ dθ

Now, when θ = 0, x = 1 and when θ = π, x = −1. With above substitutions and change of limits, expression (3.38) becomes,

⟨m⟩ =

μ

]1 [ μ xy e yx − y12 e yx −1 = [ ]1 1 yx e y

{1

yx −1 xe dx

{1

−1

e yx dx

−1

Or [ y −y y μ ey + e y − ey 2 + [ y ] ⟨m⟩ = −y e − ey y

e−y y2

]

⎫ ] [⎧ y 1 e + e−y − =μ e y − e−y y

Or ) ( 1 ⟨m⟩ = μ coth y − = μL(y) y

(3.39)

Here L(y) is called Langevin function. The average magnetic moment ⟨m⟩ multiplied by the number of atoms per unit volume N (number density of atoms in the specimen) gives the additional magnetisation M generated due to partial alignment of inherent magnetic dipoles under the influence of applied external magnetic field H. Hence; M = ⟨m⟩N = N μL(y)

(3.40)

Depending on the value of parameter y there may be two possible situations; (i) For y ≫ 1, i.e. for large values of y

3.7 Classification of Magnetic Materials

161

Y ≫ 1 means that (y =) kμH ≫ 1 or μH ≫ kβ T βT The above condition corresponds to the case of high value of applied magnetic field H and very low temperature T of the specimen, the function L(y) approaches 1 and M becomes. M = N μ and susceptibility χ =

M Nμ = H H

(3.41)

It may be noted that in the case μH ≫ kβ T susceptibility of a paramagnetic substance is independent of temperature T of the specimen and is given by Nμ . The value of magnetisation (M = Nμ) of a paramagnetic substance at H low temperature and high magnetising field H is called the saturation value of magnetisation, often denoted by M s and corresponds to the complete alignment of all the inherent dipole moments of the specimen along the direction of the applied field H. When y ≪ 1, i.e. μH ≪ kβ T For y ≪ 1, e y and e−y may be expanded in the form of converging series as fellows, ey = 1 + y +

y2 Y3 + + ··· 2! 3!

e−y = 1 − y +

y2 Y3 − + ··· 2! 3!

and

With these substitutions, Langevin function L(y) becomes [⎧ L(y) =

−y ⎫

e +e e y − e−y y

⎡ ( 2 1+ 1 =⎣ ( − y 2 y+ ]

y2 2!

+

y4 4!

y3 3!

+

y5 5!

)

⎤ 1⎦ )− y ...

···

Since successive terms of the two series appearing in the numerator and the denominator of the above expression decrease very fast, it is enough to retain first two terms of the series to obtain the approximate value of function L(y). Or [ L(y) ∼ =

2 1 + y2 y3 y + 3.2.1

−

1 y

]

) ⎤ 2 1 + y2 1 )− ⎦ =⎣ ( 2 y y 1 + y6 ⎡(

162

3 Magnetic Materials

[( )( ] )( )−1 1 y2 y2 1 1+ 1+ = − y 2 6 y Or )(( )) ] [( )( 1 y2 y2 1 1+ 1− − y 2 6 y ] ( )[ 2 2 4 y y y 1 1+ − − −1 = y 2 6 12

L(y) ∼ =

Dropping the y4 term one gets, L(y) ∼ =

( )( 2 ) y y 1 = y 3 3

Thus magnetisation per unit volume M = N μL(y) =

μH N μy = Nμ 3 3kβ T

and magnetic susceptibility χ= where C =

N μ2 3kβ

M N μ2 C = = H 3kβ T T

(3.42)

is called Curie constant.

Expression (3.42) is called Curie law and tells that for the condition μH ≪ kβ T the magnetic susceptibility of paramagnetic materials is inversely proportional to the temperature. Variation of the ratio of magnetisation M to the saturation magnetisation M s with the value of the parameter y (= μH/k β T ) is shown in Fig. 3.9. It may be noted that for low temperature and high magnetising field, the ratio M/M s increases with H as the tangent to the curve for low H and high temperature. As already mentioned, the above derivation of susceptibility for paramagnetic materials is applicable only to those atoms/ions or molecules where there is no interaction between nearby inherent magnetic moments of the material. However, this assumption does not hold, particularly, for paramagnetic metals which contain large number of free electrons in conduction band. Application of an external magnetic field affects both the magnetic moments due to the orbital and spin motions in such cases. Since spin of an electron is totally a quantum mechanical concept, Pauli model of paramagnetism is a quantum mechanical model that is beyond the scope of this discussion. In Pauli model conduction electrons are considered essentially free and

3.7 Classification of Magnetic Materials

163

Fig. 3.9 Variation of the ratio M/M s with parameter y

under the applied external magnetic field H an imbalance between electrons with opposite spin is setup that leads to a low value of magnetisation in the direction of field H. This imbalance may be understood in terms of Fig. 3.10. Figure 3.10a shows the Fermi–Dirac distribution of electrons with opposite spins before the application of magnetising field H. As may be observed in this figure the two halves of the distribution for opposite spins are equal. However, on application of external magnetic field H, the energy of component with spin parallel to H decreases by the amount μb .H , while those with spin opposite to H increases by the same amount. Here μb represents the inherent magnetic dipole moment of the atom/ion. As a result of pulling down (in energy) of parallel spin distribution, some electrons from antiparallel side in the neighbourhood of Fermi level, fall into parallel spin side, increasing the number of electrons with parallel spin over the number of electrons with antiparallel spins. This slight imbalance in the number of electrons on the two sides gives rise to a small value of magnetisation in the direction of the applied field. The susceptibility, though essentially independent of the temperature but in case of paramagnetic metals, may have some temperature dependence because of the change in electronic band structure due to field H. Some metals like Al, Mg, Ti, V, etc.; some diatomic gases like O2 , and NO, ions of transition metals and rear earth metals and their salts along with rear earth oxides show paramagnetism. Many minerals and other materials found in nature are paramagnetic, for example pyrrhotite (Fe3 S8 ), ilmenite (FeTiO3 ), siderite (FeCO3 ), quartz (SiO2 ), etc. show paramagnetic behaviour. SAQ: Use expression (3.36) to calculate the number of dipoles that will have zero energy in the applied field H.

164

3 Magnetic Materials

Fig. 3.10 Fermi–Dirac distribution of electrons with spin up and down a before the application of magnetic field H, b after the application of magnetic field H

3.7.3 Ferromagnetic Materials Ferromagnetic materials are characterised by a large positive value of susceptibility (0.1–5 × 103 ) below a certain temperature, called Curie temperature or transition temperature or critical temperature. Ferromagnetic materials are made up of atoms/ ions that have inherent permanent magnetic dipole moments (like paramagnetic atoms essentially because of the spin of unpaired electron in valence orbital) and are arranged in a lattice such that dipole moments of atoms/ions in a group align parallel to each other. Ferromagnetism is the outcome of some sort of ordering of atomic/ionic magnetic dipoles in different parts of the material.

3.7.3.1

Weiss’s Theory of Ferromagnetism

French physicist Pierre Ernest Weiss in 1907 proposed a phenomenological theory to explain ferromagnetism. The theory is based on the assumption that neighbouring atomic magnetic dipoles, due to certain mutual exchange interactions, align themselves in groups, forming several very small regions of volume, called domains. Atomic (ionic or molecular) dipoles in a domain are all aligned parallel to each other generating a magnetisation Ms for each domain. Since the number of atomic dipoles and their direction of alignment are different for different domains, the magnitude and direction of magnetisation Ms has a different value for each domain. The resultant magnetisation M R of a given specimen of a ferromagnetic material may be obtained by taking vector sum of magnetisation of individual domains; MR =

Σ all domains

Ms

3.7 Classification of Magnetic Materials

165

At a temperature below Curie temperature T c , and in absence of any external magnetic field, magnetisations of different domains are randomly oriented so that the overall magnetisation M R is zero and the material does not show any magnetism. This is required for minimisation of magnetic energy of the specimen in absence of any external magnetic field. Although there may be large number of domains in a piece of a ferromagnetic material but for simplicity, the total volume of the specimen may be divided into four equal volumes having dipole moments aligned along the four basic directions as shown in Fig. 3.11c. The directions of magnetisation in different domains form a closed loop to minimise magnetic energy of the system and to avoid the leakage of magnetic flux, as shown in this figure. On application of an external magnetising field H, domains (and individual atomic dipoles) rotate in an effort to align their magnetisation in the direction of applied field H. Out of the large number of domains some domains already have their magnetisations in the direction of applied field H, such domains are called favourable domains. When magnetising field H is switched on the size (volume) of favourable domains in the specimen increases at the cost of the volume of unfavourable domains as shown in Fig. 3.11d. Increase in the size of favourable domains is achieved by the motion of domain walls. This re-alignment of domains results in generating a net magnetisation in the direction of field H. When temperature is increased, the thermal motions of atoms destroy domain structure and randomise the direction of atomic dipoles, resulting in zero magnetisation. The characteristic feature of the ferromagnetic order in each domain is a spontaneous magnetisation M s due to spontaneous alignment of atomic magnetic moments. The spontaneous magnetisation M s tends to lie in a specific direction determined by the shape and/or crystal structure. Further, the spontaneous magnetisation disappears beyond a certain temperature called Curie temperature. Weiss theory assumes that the spontaneous magnetisation M s of each domain is due to an internal ‘molecular’ magnetic field H i . Which is proportional to the spontaneous magnetisation of the domain, i.e. H i = nw M s

(3.43)

The constant of proportionality n w , called Weiss molecular field constant, is a measure of the extent of alignment of atomic dipoles in a domain. In order to completely align all atoms of a domain in a particular direction (n w = 100%), the internal molecular field H i must be quite large. The origin of such a large internal molecular field remained a mystery until Heisenberg introduced the idea of the exchange interactions in 1928. If H is the external magnetising field then the effective magnetic field acting on each atom or ion may be written as H e f f = H + H i = H + nw M s

(3.44)

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3 Magnetic Materials

Fig. 3.11 Magnetic domains in a ferromagnetic material (a) in absence of any external field (b) in presence of external magnetic field H. c In absence of any magnetising field area/volume occupied by domains in four basic directions is same. d On applying the magnetising field H, the area/volume of the domain having magnetisation parallel to the applied field H increases

Let N be the number of atoms per unit volume, J the total angular momentum quantum number of each atom, then the possible components of magnetic moment is, M j gμB where M j may have values, M j = J, ( J − 1), (J − 2) . . .−(J − 2), −(J − 1), −J and g is Lande’s splitting factor while μB is Bohr magneton. Hence, the potential energy of a dipole with component M j gμ B along H is P E = −M j gμ B .H

(3.45)

3.7 Classification of Magnetic Materials

167

Now from statistical mechanics it follows that the total magnetic moment per unit volume or magnetisation M along H is given by

M=

N

√+J −J

(M

M j gμB e

j gμB kβ T

)

)

(3.46)

M = N g J μB B J (x)

(3.47)

√+J −J

(M

e

j gμB kβ T

where kβ is Boltzmann constant. Equation (3.46) may also be written as;

where B J (x) =

( ) ( x ) 2J + 1 1 2J + 1 coth x− coth 2J 2J 2J 2J

(3.48)

and x=

g J μB (H + n w M) g J μB Heff = . kβ T kβ T

(3.49)

In case when there is no external magnetic field (H = 0), H eff = nw Ms and Eq. (3.49) reduces to x=

g J μB (n w Ms ) kβ T

(3.50)

kβ T x g J μB n w

(3.51)

Or Ms (T ) =

When temperature T goes to zero, T → 0, x → ∞ and B J (x) → 1 all atomic dipole magnetic moments of a domain completely align themselves parallel to the direction of magnetisation M s . It follows from Eq. (3.47) [M = N g J μB B J (x)], that in case when T → 0, B J (x) → 1, therefore, Ms (0) = N g J μ B

(3.52)

Dividing Eq. (3.51) by Eq. (3.52) one gets, Ms (T ) kβ T x = ( )2 Ms (0) N n w g J μβ

(3.53)

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3 Magnetic Materials

Also, when one divides Eq. (3.47) by Eq. (3.52), one gets, Ms (T ) = B J (x) Ms (0)

(3.54)

The value of M s (T ), spontaneous magnetisation at temperature T, may be found by solving Eqs. (3.53) and (3.54) simultaneously. Simultaneous solution to these equations may be obtained in two different ways; either by graphical method or by numerical analysis. Let us find the solution using graphical method. Figure 3.12 shows the plots of Eq. (3.53) for four values of temperature T; T > T C , T = T C , T 1 < T C and T 2 < T C . The function BJ (x) has also been plotted for J = 1/2 value in the figure. As may be observed in the figure, a solution of the two equations that gives a real value greater than zero for the ratio M s (T )/M s (0) occurs only for T < T C , when the two curves cut each other. For T > T c and T = T c the two curves cut each other only at t = 0. It means that a positive value of the ratio M s (T )/ M s (0) may be obtained only when the temperature is below the transition temperature or Curie temperature T C . Moreover, for temperatures less than T c , the ratio M s (T )/ M s (0) decreases as temperature increases, as one goes from T 2 to T 1 . It may be noted that in Fig. 3.12 temperature increases as one moves from right to left. Hence, for any temperature T < T c , the ratio M s (T )/M s (0) is inversely proportional of temperature, i.e. the spontaneous magnetisation decreases with increasing temperature and vanishes beyond temperature T C called ferromagnetic Curie temperature. Variation of the ratio M S (T )/M S (0) with temperature ratio (T /T C ) is shown in Fig. 3.13. The maximum value of spontaneous magnetisation occurs at absolute zero temperature where there is no thermal motion of atoms and all atomic dipoles in a given domain line-up in one direction under the influence of internal exchange interaction field Hi . Spontaneous magnetisation decreases with the increase of temperature becoming zero at Curie temperature T C . Since M S (T )/M S (0) depends on function Fig. 3.12 Graphical solution of Eqs. (3.53) and (3.54) for J = 1/2 to find the spontaneous magnetisation M s when T < T C . Equation (3.53) is also plotted for T > T c and T = TC

3.7 Classification of Magnetic Materials

169

BJ (x) from Eq. (3.54), the ratio depends on the total momentum J associated with the atom and hence rate of fall for the ratio is J dependent as shown in Fig. 3.13. In case when T > T C , when there is no spontaneous magnetisation, the material behaves as a paramagnetic substance and a small external field may be required to align some of the atomic dipoles to produce some magnetisation. The external field must be small to avoid the state of saturation. Now from Eq. (3.49) x=

g J μB (H + n w M) g J μB Heff = ≪ 1; since T is large and H is small kβ T kβ T

Further, from Eq. (3.47) M = N g J μB B J (x) But for small x ≪ 1, B j (x) ∼ = one gets; M = N g J μB B J (x) ∼ = N g J μB

( J +1 )

(

3J

x; putting this value in the expression above,

( ) ) J +1 J + 1 g J μB (H + n w M) x = N g J μB 3J 3J kβ T

Or M=

N g 2 μ2B ( J + 1) [H + n w M] 3kβ T

But magnetic susceptibility χ =

M , H

therefore dividing Eq. (3.55) by H gives;

[ ] N g 2 μ2B (J + 1) N g 2 μ2B (J + 1) M M = 1 + nw = χ= [1 + n w χ] H 3kβ T H 3kβ T Or Fig. 3.13 Variation of ratio M S (T )/M S (0) with temperature ratio T /T c

(3.55)

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3 Magnetic Materials

) N g 2 μ2B (J + 1) N g 2 μ2B (J + 1)n w χ= 1− 3kβ T 3kβ T

(

(3.56)

Defining Tc ≡

N g 2 μ2B (J + 1)n w 3kβ

(3.57)

Substituting the above value of T c in Eq. (3.56) gives; ( 1−

) Tc TC χ= T nw T

( Or χ =

TC nw

)

C 1 = T − Tc T − Tc

Or χ=

C T − Tc

(3.58)

( ) Here C = nTCw is a constant for a give material and is called Curie constant. Equation (3.58) defines Curie–Weiss law and gives the magnetic susceptibility of a ferromagnetic material above Curie temperature for low external fields. The variation of magnetic susceptibility of materials at temperatures higher than Curie temperature, where a ferromagnetic material behaves as a paramagnetic substance, is well explained by Curie–Weiss law. The Curie temperature Tc for different material may be determined experimentally by measuring susceptibility at different temperatures. Once T C is known other parameters of the material may also be determined; for example in case of ferromagnetic element Gd, the experimentally determined value of T C is 292 K; J = S = 7/2, g = 2, N = 3.0 × 1028 m−3 . This data gives M s (0) = Ng μB J = 1.95 M A m−1 and Bi = μ0 H i = 144 T. Figure 3.14 shows the temperature dependence of susceptibility for a paramagnetic material and for a ferromagnetic material. A ferromagnetic material undergoes phase transition at Curie temperature T C . This results in singularities in the behaviour of physical properties like susceptibility, magnetisation, specific heat, etc. (i) Exchange interactions It can be shown that interactions between atomic magnetic dipoles cannot generate a magnetic field strong enough to align all atoms of a domain in a particular direction, i.e. dipole interactions are not strong enough to generate the internal magnetic field H i ≈ 100T which is required to align atoms in a domain. Weiss in 1907, while proposing his theory simply assumed that a sufficiently strong internal field H i responsible for ferromagnetism is present in each domain, without giving any explanation for its generation. The riddle was solved in 1928 when Heisenberg introduced the concept of exchange interactions. Exchange interactions originate from electrostatic Coulomb repulsion between electrons of the neighbouring atoms and Pauli’s exclusion principle.

3.7 Classification of Magnetic Materials

171

Fig. 3.14 Temperature dependence of a, c paramagnetic substance b, d Curie–Weiss susceptibility of a ferromagnetic material

Pauli’s exclusion principle forbids two electrons in a given energy state to have same values of all their quantum numbers. Since production of exchange interactions are strictly quantum mechanical phenomenon, which is beyond the scope of this discussion, we will not go into its details. Heisenberg derived the following Hamiltonian, H for exchange interaction H = −2 j S1 S2 Here S1 and S2 are the spins of neighbouring atoms, and j is the exchange integral (do not confuse this J with total spin J). J > 0 indicates a ferromagnetic interaction favouring parallel spin alignment (↑↑) while J < 0 indicates an antiferromagnetic interaction favouring antiparallel spin alignment (↑↓). SAQ: What happens to the domain walls when the magnitude of the external field H is changed? (ii) Spin wave The lowest energy state of a ferromagnetic system occurs when all spins (spins of all atoms) are parallel to each other along the direction of magnetisation. However, when one of the spins tilts or get disturbed, it begins to precess around the direction of magnetisation. The disturbance so produced propagates as a wave through the system because of exchange interaction between neighbouring atoms as shown in Fig. 3.15a.

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3 Magnetic Materials

Fig. 3.15 A spin wave on the line of spin

Spin waves are analogues to lattice waves created by the oscillation of atoms about their equilibrium position. In spin waves spins precess around equilibrium magnetisation and precession of atoms are correlated through exchange interactions. Spin waves may be quantised, like quantisation of lattice waves with quanta called phonon. The quantised spin wave is called magnon. Elements like iron (Fe), cobalt (Co), nickel (Ni), gadolinium (Gd) and dysprosium (Dy) are ferromagnetic at room temperature. The Curie temperature for Fe, Co, Ni and Gd is respectively, 770 °C, 1131 °C, 358 °C and 565 °C; however, EuO has Curie temperature of 70 K (343 °C) and EuS even lower. (iii) Saturation magnetisation Msat Three parameters, namely the Curie temperature, saturation magnetisation and magnetocrystalline anisotropy, are called intrinsic properties of a magnetic material as they do not depend on the microstructure, i.e. on the grain size and grain orientation in the crystal. Saturation magnetisation (M sat ) tells about the maximum magnetic field that may be produced by a material. Msat depends on three factors: (a) strength of each atomic/ionic magnetic dipole (m), (b) packing density of atomic/ionic dipoles and (c) the degree of alignment of dipoles at a given temperature. Factor (a) depends on the nature of the atom and its electronic configuration while factor (b) is determined by crystal structure and the presence of any nonmagnetic elements within the structure. Factor (c), the degree of alignment of atomic/ionic dipoles depends on temperature; higher the temperature less will be alignment, and on magnetic anisotropy of the crystal. (iv) Magnetic anisotropy In crystalline magnetic materials it is often observed that there is one particular crystallographic direction magnetisation along which is easier as compared to the other directions. For example, in case of the hexagonal crystal structure of cobalt (Co) it is easy to magnetise in the crystal direction (001) as compared to any other direction. It is hard to align the magnetic dipoles along the directions ⟨1010⟩ which lie in the plane normal to the crystal axis 001). See Fig. 3.16b. This anisotropy is created by the coupling of electron orbitals to the crystal lattice. In

3.7 Classification of Magnetic Materials

173

Fig. 3.16 Easy and hard directions of magnetisation in hexagonal cobalt crystal

the easy direction of magnetisation this coupling is such that electron orbitals are in their lowest energy state. Magnetic anisotropy of a crystal is measured in terms of the anisotropy field Ha defined as the magnetic field needed to saturate the magnetisation in a hard direction. (v) Magnetic hysteresis in ferromagnetic substances A ferromagnetic material in absence of any external magnetic field behaves as a diamagnetic material because of random orientation of magnetisations of different domains. However, application of a small external magnetic field H a large magnetic induction B or magnetisation M in the direction of the applied field H is produced. When a ferromagnetic material is magnetised in a particular direction, it does not revert back to the state of zero magnetisation when the external magnetising field H is withdrawn. In order to bring a magnetised ferromagnetic specimen back to the state of zero magnetism, a magnetic field opposite in direction to the field H has to be applied. Figure 3.17 shows the behaviour of a ferromagnetic specimen below its Curie temperature, subjected to an alternating magnetic field. The strength of the magnetising field H is plotted on the X-axis while the magnitude of the magnetic flux density B or the strength of magnetisation M (B = μ(H + M)) is shown on the Y-axis. The starting point is the origin O when H = 0 and B = 0. As H is increased in a given direction the flux density B in the direction of H also increases reaching the point a (H 0 , B0 or M 0 ). Any further increase of H beyond H 0 does not increase B or M above B0 (or M 0 ). This is called the state of saturation. In the state of saturation, magnetisations of all domains in the

174

3 Magnetic Materials

specimen get maximum aligned in the direction of field H. When magnetising field H is reduced from the saturation value H 0 , the flux density B also reduces but it does not follow the path traced while increasing H. The variation of B with reducing H is shown by the part of the curve shown by abcd in the figure. The important aspect of the curve is that when H = 0 at point b, there remains a residual magnetisation Br or M r in the specimen. This indicates that the ferromagnetic specimen retains some sort of a memory about its previous magnetic history which persists even when the magnetising field is withdrawn. This happens because thermal motion of atoms does not fully randomise the magnetisation of magnetic domains (which were aligned in the direction of H) in the specimen and they remain partially aligned in the direction of field H even when H becomes zero. Now if the external magnetising field is applied in opposite direction (− H), the magnetic flux density B or magnetisation M of the specimen decreases from the value Br (or M r ) to a zero value at point C on the graph where the value of magnetising field is (− H c ). This value (− H c ) is called the coercive field value that ultimately removes any residual magnetisation of the specimen. Further increase in the magnitude of magnetising field in opposite direction (− H) generates magnetisation in the specimen in the direction along (− H) which reaches the saturation value (− B0 or− M 0 ) in opposite direction at point d on the graph. When the magnitude of magnetising field (− H) is reduced, B (or M) in the specimen follow the path indicated by the part defa of the curve, where point e corresponds to residual magnetisation (− Br or − M r ) in opposite direction and point f to coercive field (H c ) that is required to nullify any residual magnetisation of the specimen. On increasing H beyond H c , the magnetisation of the specimen increases and ultimately attains the saturation value at point a. The closed curve abcdefa shown in the figure is often called the B–H curve or hysteresis loop of the ferromagnetic specimen. The area enclosed by the hysteresis loop gives the amount of energy consumed in one cycle of magnetisation from zero to saturation value in one direction to the saturation vale in the other direction and back. The energy consumed in magnetisation per cycle given by the area of the B–H curve generally appears in the form of heat. This may be seen in case of a solenoid that is fed by alternating (ac) current generating magnetic field with alternating polarities at its axis, a ferromagnetic material like iron rod if placed at the core of the solenoid gets heated. Similarly, magnetic cores of transformers that undergo repeated cycles of magnetisation get heated. The shape, particularly, the area enclosed by the hysteresis loop has different values for different types of ferromagnetic materials. Ferromagnetic materials with large area of B–H loop are called Hard magnetic materials; they have large value of residual magnetism, large value of coercive field and consume large amount of energy in each cycle of magnetisation. Natural iron is such a material; hard magnetic materials are used to make permanent magnets on account of the large value of residual magnetisation and large value of coercive field. On the other hand materials with narrow hysteresis loop, that have

3.7 Classification of Magnetic Materials

175

Fig. 3.17 Magnetic hysteresis loop for a ferromagnetic material at T < T C

low value of residual magnetisation and coercive field are called soft magnetic materials. Soft magnetic materials may be easily magnetised and demagnetised without loss of much energy. Soft materials have high value of relative permeability (μr ) and low value of coercive field. Silicon steel is a very good example of soft magnetic materials and is often used as core material in solenoids, transformers, and relays that operate on alternating current generating alternating magnetic fields. Common soft magnetic materials are iron, iron–silicon alloys (with 1–5% silicon) and nickel–iron alloys, also called permeability alloys, with preferred nickel contents of 42–79%. Addition of molybdenum gives extra electrical resistivity and addition of copper results in higher initial permeability. Soft magnetic ceramics, also called ceramic magnets, have been originally made from iron oxide (Fe2 O3 ) with one or more divalent oxides like that of ZnO, MgO or NO. The mixture of these oxides is first calcined and grinded to powder, pressed to the desired shape and sintered. Vectolite is typical light weight and very high resistivity (like that of an insulator) magnet made by moulding ferric and ferrous oxides and cobalt oxide. Magnadur (BaO.Fe2 O3 ), made from BaCO3 (barium carbonate) and ferric oxide is also a soft magnet material. Table 3.2 lists the properties of some important soft magnetic materials. Magnets made from hard magnetic materials have strong resistance against demagnetisation (large coercive field) and large area of hysteresis loop. Details of some important hard magnetic materials are tabulated in Table 3.3.

176

3 Magnetic Materials

Table 3.2 Properties of some soft magnetic materials Material/trade name

% Composition by weight, remainder iron

Relative permeability initial maximum

Coercive field (Oe)

10,000

200,000

1.1

250

9000

0.9

High-purity iron

Impurity < 0.05%

Commercial iron

Impurity 0.2%

Transformer, M-15

2.2% Si

1500

7000

0.35

Armature M-43

0.95% Si

100

4100

0.94

Table 3.3 Hard magnetic materials Material

Composition Curie (% by weight) temperature (°C)

Alnico-5

50Fe, 24CO, 15Ni, 8Al, 3Cu

900

Alnico-8

34Fe, 35Co, 15 Ni, 7Al, 5Ti, 4Cu

Mn–Al–C

Ba ferrite

Coercive field H c (Oe)

Residual induction Br (T)

Preparation

Mechanical properties

620

1.25

Casting and annealing

Hard and brittle

860

1600

0.83

70Mn, 29Al, 0.5Ni, 0.5C

300

2700

0.61

Casting, extruding, annealing

Hard

Bao.6Fe2 O3

450

2100

0.43

Pressing, sintering

Brittle

Hard

3.7.4 Antiferromagnetic and Ferrimagnetic Materials Fifteen elements of the periodic table, namely O, Cr, Mn, Fe, Co, Ni, Nd, Sm, Eu, Gd, Tb, Dy, Ho, Er and Tm in their solid states show some sort of magnetic order. As expected, atoms of all these elements have unpaired electrons and associated magnetic dipole moments essentially from the spins of the unpaired electrons. Heisenberg has shown that the magnetic order in these fifteen elements originates from the exchange interactions between the electron clouds of atoms/ions/molecules subject to Pauli’s exclusion principle (i.e. from electrostatic interactions) and not from the mutual interaction between the magnetic dipoles of atoms or from the spin– orbit interactions in atoms, etc. The electrostatic exchange interactions may align the spin magnetic dipole moments of individual atom/ion/molecule either parallel or antiparallel to each other. For example, in case of H2 molecule the energy of the parallel (spin) alignment of two atoms (↑↑), called triplet state, and antiparallel alignment (↑↓), called singlet state, have different value that depends on the relative separation of the two atoms as shown in Fig. 3.18. Therefore, the singlet or the triplet alignment in the ground state of a martial depends on their spins and relative separation of atoms/ions, etc. in the crystalline structure.

3.7 Classification of Magnetic Materials

177

Fig. 3.18 Singlet and triplet state energies for H 2 molecule

According to Heisenberg theory of electrostatic exchange interactions the effective interaction between a pair of atoms/ions of spins S 1 and S 2 is given by ⇀

⇀

S1 . S2 =

1 for triplet; and 4

−

3 for singlet, 4

And the interaction energy U is given as ( ) )⇀ ⇀ 1( U = − E singlet − E triplet S1 . S2 + E singlet + 3E triplet 4 Or ⇀

⇀

U = −J S1 . S2 + Constant

(3.59)

Here E singlet and E triplet respectively, represents the energies of the singlet and triplet states. Factor J in Eq. (3.59), called ‘exchange coupling constant’ may have positive (J > 0) or negative (J < 0) values. J > 0 refers to the case when spin will orient in the same direction (triplet case) while J < 0 refers to the antiparallel alignment of spins (singlet case). In ferromagnetic materials spins and magnetic dipoles of adjacent atoms/ions in a given domain align parallel to each other, and it refers to the case when J > 0. In antiferromagnetic and ferrimagnetic materials, J < 0 and spinful atoms/ ions of such materials align their spins/magnetic moment in opposite directions in a domain. It may therefore be said that both ferromagnetic and antiferromagnetic/ ferrimagnetic materials have domains, which are created because of the electrostatic

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3 Magnetic Materials

exchange interactions but in ferromagnetic materials the spin or the magnetic dipole moments of neighbouring atoms/ions are aligned parallel to each other while in antiferromagnetic and ferrimagnetic materials the dipole moments of adjacent atoms/ ions are aligned opposite to each other in each domain. Antiferromagnetism was predicted by French scientist Louis Neel in 1936. It is experimentally difficult to detect antiferromagnetic material because these materials above a certain temperature (called Neel temperature) behave like a paramagnetic material above Curie temperature; and show no magnetism. The final direct confirmation of Neel’s theory was done by Harry Shull using neutron diffraction in 1938. Figure 3.19 shows the spin (and magnetic moment) alignments of different types of magnetic materials in their solid states. Figure 3.19a shows that atoms/ions/molecules of the material have no net spin and magnetic moment, since they have no unpaired electrons and, therefore, in absence of an external magnetising field they show no magnetism. However, when an external magnetising field H is applied, the material develops weak magnetisation opposite to the direction of the applied field H. They are diamagnetic. Susceptibility of diamagnetic substance has a negative small value and generally independent of the temperature. Figure 3.19b shows a paramagnetic material; atoms/ions/molecules of a paramagnetic material have unpaired electrons that give rise to an inherent magnetic dipole moment to each of them. Individual dipoles do not interact with each other. At room temperature (T > 0 K) and in absence of magnetising field H, the atomic/ionic/molecular dipoles are randomly distributed (to minimise the magnetic energy of the system) and a paramagnetic material does not show any magnetisation. However, application of magnetising field H results in partial alignment of dipoles in the direction of field H generating a magnetisation in the material. The magnitude of magnetisation increases with the increase of H reaching a saturation value when all dipoles in the specimen get aligned to field H. Susceptibility of paramagnetic materials is positive but small; variations of susceptibility χ and 1/χ of paramagnetic materials with temperature are shown in Fig. 3.20. Atoms/ions/molecules of ferromagnetic, antiferromagnetic and ferrimagnetic materials have unpaired electrons and, therefore, each atom/ion/molecule possesses a dipole moment (like that of paramagnetic material), but these dipoles interact with each other (unlike paramagnetism). Mutual interaction between dipoles arises from electrostatic exchange interaction. The exchange interaction does two things: (i) it creates domains in the specimen and (ii) spins or magnetic moments of all atoms in a domain are either aligned parallel or anti-parallel. Materials for which spins are aligned parallel are called ferromagnetic and those where spins are aligned anti-parallel are either antiferromagnetic or ferrimagnetic. In ferromagnetic materials all atoms/ions, etc. have their magnetic moments aligned parallel to each other in a given domain that gives each domain a finite value of magnetisation. In absence of magnetising field H and below Curie temperature T c , the orientations and magnitudes of magnetisations of different domains are random, such that the net magnetisation of the material is zero. When external magnetising field H is switched on, the domain walls in the sample move so as to increase the size

3.7 Classification of Magnetic Materials

Fig. 3.19 Alignment of spin/magnetic moments in different types of magnetic materials

Fig. 3.20 Variation of magnetic susceptibility with temperature

179

180

3 Magnetic Materials

of the favourable domain that results in the generation of a magnetisation in direction of H in the ferromagnetic material. Susceptibility of ferromagnetic materials is positive and large and it decreases with temperature. Temperature dependence of susceptibility and its reciprocal for ferromagnetic materials are shown in Fig. 3.20. It may be mentioned that above Curie temperature a ferromagnetic material behaves like a paramagnetic material. Both in antiferromagnetic and ferrimagnetic materials the spin (or dipole moment) alignment of neighbouring atoms/ions are in opposite directions as shown in Fig. 3.19d, e. The difference between the two classes of magnetism lies in the relative magnitudes of the magnetic moments of the adjacent atoms/ions. In case of antiferromagnetic materials the magnetic moments of adjacent atoms are almost equal and, therefore, in absence any magnetising field H and below Neel temperature T N a specimen of antiferromagnetic material does not show any magnetisation. However, in presence of the magnetising field H below T N , domain walls move to increase the area of the favoured domain generating a net magnetisation in the specimen along the direction of field H. The net magnetisation in case of antiferromagnetic materials in presence of the magnetising field H is smaller than that of ferromagnetic material. Susceptibility of an antiferromagnetic material is positive but small and varies with temperature as shown in Fig. 3.20. An antiferromagnetic material behaves like a paramagnetic material above Neel temperature T N . The only element that shows antiferromagnetism at room temperature is chromium (Cr) for which Neel temperature is 37 °C (310 K). Most antiferromagnetic materials are transition metal oxides (oxides of elements whose atom has a partially filled d-sub shell, like Cr, Mn, Fe, Co, Ni, etc.). The structure of magnetic unit cell of MnO, a typical antiferromagnetic material is shown in Fig. 3.21a. An antiferromagnetic material behaves like a paramagnetic material above Neel temperature T N . However, the variation of susceptibility of an antiferromagnetic material below Neel temperature depends on the angle between the magnetising field H and the plane defined by the magnetic moments of the neighbouring atoms/ions. When the magnetising field H is parallel to the two spins S 1 and S2 of neighbouring atoms, the susceptibility is denoted by χ P . In case H is perpendicular to S 1 S 2 , the susceptibility is denoted by χT . Temperature dependence of χ P and χT is shown in Fig. 3.21b.

3.7.4.1

Ferrimagnetisms

Some ceramics exhibit permanent magnetisation, called ferrimagnetism. Microscopic arrangement of magnetic moments in antiferromagnetic and ferrimagnetic materials is similar; in both the magnetic moments of adjacent atoms/ions is aligned antiparallel to each other. In case of antiferromagnetic materials the antiparallel magnetic moments completely cancel each other and, therefore, an antiferromagnetic material has no magnetisation in absence of any external magnetic field. However, in ferrimagnetic materials, the magnetic moments of the two adjacent atoms/ions are not exactly equal: moment of one is larger than the moment of the other and,

3.7 Classification of Magnetic Materials

181

Fig. 3.21 a Unit magnetic cell of antiferromagnetic MnO compound. b Temperature dependence of the susceptibility of antiferromagnetic material below Neel temperature

hence, ferrimagnetic materials show a magnetisation even in absence of an external magnetising field. Synthetic materials with ferrimagnetisms are called ferrites. Depending on the crystal structure, ferrites may be divided into three classes: (i) cubic (ii) hexagonal and (iii) garnet. Magnetite, Fe3 O4 is a well-known cubic ferrimagnetic substance. The chemical composition of the material is FeO.Fe2 O3 . It has two types of iron ions: ferrous, doubly charged Fe2+ ions and triply charged Ferric Fe3+ ions. The compound magnetite in crystalline state has spinel structure. A unit cell of crystal contains 56 ions out of which 24 are iron ions and 32 are oxygen ions. In 24 Fe- ions 16 are ferric and 8 ferrous. Only iron ions have magnetic moments. In each unit cell the Fe- ions are located in two different coordinate environments: (a) A tetrahedral one, where the Fe-ion is surrounded by four oxygen ions and the other (b) octahedral structure in which each Fe- ion is surrounded by six oxygen ions. Out of 16 ferric ions 8 are in coordinate environment (a) and the remaining 8 in environment (b). The spin or magnetic dipoles of ferric Fe- ions (Fe3+ ) in coordinate environment (a) and (b) are directed opposite to each other and, therefore, 8-ferric ions in both configurations cancel their magnetic moments. The resultant magnetic moment of the molecule, therefore, arises entirely from the 8-ferrous Fe-ions which are placed at octahedral sites. Each ferrous Fe-ion has six 3d electrons whose spin orientations are ↑↑↑↑↑↓ Therefore, each ferrous Fe-ion carries a magnetic moment of 4 Bohr magneton (Bm or μB ) while each ferric ion carries a magnetic moment of 5μB . Schematic representation of Fe3 O4 is shown in Fig. 3.22. The general chemical formula for cubic ferrites is: MO.Fe2 O3 ; where M is a divalent cation, often of Zn, Cd, Fe, Ni, Cu, Co or Mg. The general ionic formula for such ferrite may be written as M2+ O2− (Fe3+ )2 (O2− ), where M2+ may be ions like Mn2+ , Co2+ , Cu2+ etc. each of which may have a net magnetic moment different than 4μB . Magnetic moments in unit of Bohr magneton for some frequently used ions are: Cu2+ (1 μB ); Ni2+ (2 μB ); Co2+ (3 μB ); Fe2+ (4 μB ); Mn2+ (5 μB ); F3+ (5 μB ).

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Fig. 3.22 Spin arrangement of Ferric and ferrous ions in magnetite

Using different divalent ions and mixtures of different divalent ions it is possible to make ferrites with desired magnetic properties. Ferrites made by mixing two or more divalent ions are called mixed ferrite, an example is (Mn, Mg)Fe2 O4 . Hexagonal ferrites have the general formula AB12 O19 , where A is a divalent metal, like barium (Ba), lead (Pb) or strontium (Sr) and B is a trivalent metal like aluminium (Al), gallium (Ga) or iron (Fe). Hexagonal ferrites have an inverse spinel like crystal structure with hexagonal symmetry. A common example of hexagonal ferrite is BaFe12 O19 . A class of ferrites is called garnets that have a complicated structure which may be represented as M3 Fe5 O12 . M in this formula stands for some rear earth ion like yttrium (Y), gadolinium (Gd), samarium (Sm) or europium (Eu). Yttrium iron garnet (Y3 Fe5 O12 ) is one of the frequently used garnets often denoted as YIG. Saturation magnetisation of ferrites is not as large as that of ferromagnetic materials but their biggest advantage is that some ferrites are ceramics and excellent electric insulators. An insulator magnetic core of ceramic ferrite in high-frequency transformers eliminates eddy current losses. SAQ: What are ferrites? And why they are very important? How a ferrite of desired magnetic properties may be synthesised?

3.8 Permanent Magnetic Materials Permanent magnets are required in all walks of life, be it big particle accelerators used in research or tiny computer memories or fridge-magnetic stickers. Permanent magnets are characterised in terms of the maximum energy product, i.e. the area

3.8 Permanent Magnetic Materials

183

of the largest rectangle starting from the origin that may be drawn in the second quadrant of the B–H curve, as shown in Fig. 3.23. (BH)max tells about the magnetic energy stored in the material per unit volume and is treated as the magnetic figureof-merit of the material. Maximum energy product is often measured in units of kilo-Joule per cubic metre, (kJ m−3 ) in SI system or MGOe (Mega-gauss-Oersted) in electromagnetic system. Further, 1 MG Oe = 7.958 kJ m−3 . Research in the field of magnetic materials has led to almost an exponential rise in the magnitude of (BH)max in the twentieth century starting from 20 kJ m−3 in 1900 to around 450 kJ m−3 in 2000. Increase in the (BH)max value resulted in considerable reduction in the size of permanent magnets; for example a NdFeB magnet of 102 cc volume will contain roughly the same magnetic energy as a brass bond lodestone of 105 cc volume; a reduction of almost 103 in the size of the magnet. Oldest permanent magnetic material is Lodestone, naturally occurring iron oxide Fe2 O3 , though loadstone magnets produce low fields but they offer high resistance to demagnetisation. Magnetic carbon steels, developed in eighteenth century, are generally alloyed with chromium or tungsten to restrict domain wall movement and increase coercive field. Magnets made from carbon steel have large saturation field, order of magnitude larger than loadstone magnets, but have lower value of coercive field. Synthetic magnets made from alloys of aluminium, cobalt and nickel, called Alnico magnets, were first developed around 1930 and show considerably larger values for magnetic hardness as compared to carbon steel. They also have high Curie temperature of the order of 900 °C. Alnico is cast in a foundry. Magnets of desired pattern may be made by using sand moulds and pouring molten magnetic material in the mould. Alnico magnets may also be made by sintering process to form small and accurate magnets. Alnico magnets have high operating temperature, good corrosion resistance and long-term magnetic stability. However, their drawbacks are the high cost on account of Cobalt and low resistance to demagnetisation. Pushing two Alnico magnets in repulsion may demagnetise both of them. Ferrite (Fe3 O4 ) is manufactured using powder sintering technology and exact size tooling into range of industry standard sized disks, rings or blocks of 150 mm × 100 mm × 25 mm size. These blocks can then be sliced into smaller magnets. Ferrites are used extensively in loudspeakers and other security systems. The biggest advantage of ferrite magnets is their very low cost, high resistance to corrosion and good magnetic stability. Low level of magnetism is their only weakness. Fig. 3.23 Maximum energy product (BH)max defining rectangle

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3 Magnetic Materials

A new variety of magnets, called Cobalt–Platinum magnets, developed in 1950, their greatest advantage over the other magnetic materials is that they have excellent resistance to corrosion and therefore they are almost exclusively used in biomedical applications. Hard Ferrite Magnets, like BaFe12 O19 and SrFe12 O19 , are presently the most used commercial magnetic materials for decay or two. High cost of these magnets is their biggest weakness. Neodymium-iron-boron compound, Nd2 Fe14 B, a magnetic material, was first fabricated in 1982 by General Motors. Since then strong grade neodymium magnets are commercially available. It is claimed that a 50 mm × 50 mm × 25 mm N52 Neodymium magnet can support vertically a steel weight of about 110 kg. The mean value of (BH)max for Nd-magnets of different grades is around 326 kJ m−3 . These magnets have an elaborate manufacturing process consisting of vacuum melting, milling, pressing and sintering. These magnets have high magnetic strength, but their weakness is their low operating temperature. Magnetic rubber is produced by heavy loading of ferrite powder of strontium or barium base into synthetic elastic matrix like PVC. The rubbery magnetic material is then extruded to a desired shape or is produced in thin sheets by calendaring. Magnetic rubber may be cut into any shape using foam cutters. The strong points of magnetic rubber are: flexibility, ease to cut and good resistance to corrosion and the weak points are: low magnetic flux density and low value of operating temperature. SAQ: (BH)max , though defined in terms of the area of a specific rectangle has SI units of kJ per cubic metre. How can one reconcile the units of area in definition with units of volume in SI units? Some characteristic properties of materials used in fabricating permanent magnets are given in Table 3.4. Some important conversion factors are: 1 Gauss = 10–4 T (Tesla); 12.54 Oe = 1kA m−1 (kilo ampere per metre; 1 MGOe = 7.958 kJ m−3 (kilo Joule per cubic metre.

Table 3.4 Properties of some magnetic materials Material

Residual magnetisation Br (Tesla) T

Coercive field Hc (kA m−1 )

Maximum power product (BH)max (kJ m−3 )

Maximum temperature of operation (Kelvin) K

Neodymium

13

9170

335

353

Alnico

1.25

51.00

44

773

Ceramic + Fe2 O3

0.40

2.35

28

450

Samarium + Cobalt

1.10

774.00

225

620

Magnetic rubber

0.20

128.00

7

323

3.8 Permanent Magnetic Materials

185

Solved Example SE(3.1) Calculate the value of 1 Bohr magneton (MB or μB ) in SI units. −19

−34

qe h ×6.626×10 Solution: By definition 1 μB = 4πm = 1.6×10 = 9.274 × 10−24 A m2 . 4×π×9.1×10−31 e −19 Here, qe = charge on an electron = 1.6 × 10 Coulomb h = Planck’s constant = 6.626 × 10−34 J per Hz And m e = mass of an electron = 9.1 × 10−31 kg

Solved Example SE(3.2) A particle of mass 1 kg having charge of + 5.0 μC moving with a constant velocity of 1.5 × 104 m/s making an angle of θ with the direction of magnetic induction of strength 0.25 T, experiences a force of 1.7 × 10–2 N. What is the magnitude of angle θ ? Solution: The force F experienced by a particle of charge q moving with velocity v in a magnetic field of induction B making an angle θ with B is given as; F = q(v X B) = qv B sin θ Substituting the value of q = 5.0 × 10–6 C; v = 1.5 × 104 m/s; B = 0.25 T and F = 1.7 × 10–2 N, in the above expression, one gets 1.7 × 10−2 = 5 × 10−6 × 1.5 × 104 × 0.25 × sin θ Or sin θ =

1.7 × 10−2 = 0.906 5 × 10−6 × 1.5 × 104 × 0.25

Therefore, θ = sin−1 0.906 = 64.96◦ . Solved Example SE(3.3) Ferrite Fe3 O4 forms a cubic crystal with unit cell edge length (a) of 0.840 × 10–9 m. Each unit cell of the material contains 8 Fe2+ ions with each ion having a magnetic moment of 4 μB , and 16 Fe3+ ions that are non-magnetic. Calculate the saturation magnetisation Bs per unit cell in units of A/m. Solution: Total magnetic moment of a unit cell due to 8 Fe2+ ions M cell = 8 × 4 μB = 32 μB . )3 ( Volume of the unit cell Vcell = a 3 = 0.84 × 10−9 = 0.59 × 10−27 m3 . 32×(9.27×10−24 A m2 ) 32 μB = Saturation magnetisation per cell B = Mcell = s

Vcell

(on substitution of μB = 9.27 × 10−24 A m2 ). Or Bs = 5.02 × 105 A/m.

0.59×10−27 m3

0.59×10−27 m3

Problems P3.1 Two identical squares of sides 6 cm, made from conducting wire are placed side by side on a horizontal table as shown in the figure (P3.1) and a current

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3 Magnetic Materials

of 45.0 mA is made to flow through the squares. Determine the nature and magnitude of the force between the two squares.

ANS: Repulsive force of 78.0 × 10–10 N P3.2 Ferrite Mn3 O4 , where Mn is the diatomic metal magnetic, forms a cubic crystal with unit cell edge length (a) of 0.840 × 10–9 m. Each unit cell of the material contains 8 Mn2+ ions with each ion having a magnetic moment of 5 μB , and 16 Mn3+ ions that are non-magnetic. Calculate the saturation magnetisation Bs per unit cell. ANS: 6.25 × 105 A/m Short Answer Questions SA3.1 SA3.2

SA3-3

SA3.4

SA3.5

SA3.6

Write a note on diamagnetism giving Langvine’s theory for it. Why diamagnetism is called the fundamental magnetism? Give the major points of similarities and differences between ferromagnetic and ferrimagnetic materials. What causes domains in these magnetic materials? What are the required magnetic properties that a material used for making permanent magnets must have?. List some important materials used for making permanent magnets. Write the expression relating magnetic dipole moment of a charged particle with its angular momentum in classical limits and discuss how the expression may be modified for quantum limits. Define 1 Bohr magneton and give its value in SI units. Explain why in absence of any external magnetic field H and at a temperature T (a little above 0 K), diamagnetic, paramagnetic and ferromagnetic materials all show no magnetism. Also write the order of magnitude of bulk susceptibilities for these materials. What is exchange interaction and what is its origin? Discuss the role played by exchange interaction in case of ferro, antiferro and ferric magnetic materials.

3.8 Permanent Magnetic Materials

SA3.7 SA3.8

SA3.9

SA3.10 SA3.11

SA3.12 SA3.13

SA3.14

187

What are ferrites? Explain how magnetic material with desired value of saturation magnetisation may be fabricated using mixtures of ferrites. Draw a typical B–H curve for a ferromagnetic material indicating important characteristics of the curve. What does the area of the B–H curve represents? Define (BH)max and discuss the significance of this parameter. Explain how the application of an external magnetic field H in case of metals that have free electrons, causes an imbalance in the number of electrons with opposite spins that leads to a lower value of magnetisation in the direction of field H. Explain what is magnetic anisotropy and what causes it. What are the distinguishing features of magnetically hard and soft materials? Briefly outline applications of the two types of these materials. What are ceramic soft magnetic materials and where are they used. What is meant by singlet and triplet states of spin alignment? Discuss their role in magnetism. Draw rough sketches for the variation of bulk susceptibility χ and (1/χ ) with temperature for different types of magnetic materials and define the Curie and the Neel temperatures. Write a note on ferrimagnetism giving some examples.

Multiple Choice Questions Note: Some of the multiple choice questions may have more than one correct alternative. All correct alternatives must be marked for the complete answer of such questions. MC3.1

Biot–Savart law in mathematical form may be given as; → → { − { − { r (a) B→ = μ4π0 I dlrX2 rˆ (b) B→ = μ4π0 I dl|r|×→ (c) B→ = 2πI 3 → { − I dl ×→r 4μ0 π

MC3.2

− → dl X rˆ r2

(d) B→ =

r2

ANS: (a), (b) Magnetic dipole moment of an object is equal to the (a) Minimum torque experienced by the object in an external magnetic field (b) Minimum torque experienced by the object in an external magnetic field of strength unity (c) Maximum torque experienced by the object in an external magnetic field (d) Maximum torque experienced by the object in an external magnetic field of strength unity ANS: (d)

188

MC3.3

3 Magnetic Materials

A circular loop of wire carries a current I in clock-wise direction. The direction of magnetic field at a point P inside the loop is (a) Left to point P (b) right to point P (c) points out of the page (d) points into the page

MC3.4

ANS: (d) 1 Bohr magneton is (a) 9.274 × 10−24 J/T (b) 9.274 × 10−21 erg/gauss (c) 9.274 × 10−21 eV/T (d) 5.7883 × 10−5 eV/T

MC3.5

ANS: (a), (b) and (d) M L 3 T −1 Q −1 ; and N M T −1 are dimensional formulas for (a) Magnetic moment (b) Magnetisation (c) Bohr magneton (d) Magnetic induction

MC3.6

ANS: (c) ‘g’-factor for the spin motion of electron is around (a) 1 (b) 2 (c) 3 (d) 4

MC3.7

ANS: (b) Stick out the incorrect alternative from the followings: Bulk susceptibility of diamagnetic materials depends on (a) the number of electrons per atom (Z), (b) number density (N) of atoms (atoms per unit volume) (c) temperature of the specimen T (d) Intensity of the magnetising field H

MC3.8

ANS: (c), (d) Langevin’s theory for paramagnetism is not applicable to. (a) metal ions (b) hydrated salts of transition metals (c) dilute gases (d) diamagnetic substances

MC3.9

ANS: (a), (d) Exchange interaction in ferromagnetic materials is the result of (a) atomic dipole–dipole interaction and Pauli’s exclusion principle (b) lattice-dipole interaction and Pauli’s exclusion principle (c) spin-dipole interaction and Pauli’s exclusion principle (d) interaction between electron clouds and Pauli’s exclusion principle ANS: (d)

3.8 Permanent Magnetic Materials

189

MC3.10 Best material for the core of a transformer is (a) diamagnetic (b) paramagnetic (c) antiferromagnetic (d) ceramic ferrites ANS: (d) MC3.11 Which of the following are intrinsic magnetic properties of a crystalline magnetic solid? (a) Curie temperature (b) saturation magnetisation (c) crystalline magneto anisotropy (d) bulk susceptibility χ ANS: (a), (b), (c) Long Answer Questions LA3.1 What are the characteristics of ferromagnetic materials? Which interactions produce domains in magnetic materials? Discuss in details Weiss theory of ferromagnetism. LA3.2 Define magnetic permeability and susceptibility. What are paramagnetic materials? Discuss with necessary detail Langevin’s theory for paramagnetism and hence bring out the importance of Curie temperature. LA3.3 Explain what causes diamagnetism in materials the atoms/ions of which have no magnetic moment. Give a detailed account of Langevin’s theory for diamagnetism. LA3.4 What are the distinguishing features of diamagnetic, paramagnetic, ferromagnetic, antiferromagnetic and ferrimagnetic materials?. What forces/ interactions are responsible for these differences? Discuss the origin of exchange interactions and the role they play in magnetism.

Chapter 4

X-rays, Dual Nature of Matter, Failure of Classical Physics and Success of Quantum Approach

Objective After reading this chapter the reader is expected to: (i) understand the method of producing X-rays, their properties and applications, (ii) be able to grasp the concepts of the dual nature of matter, energy and matter waves, (iii) appreciate the pitfalls of classical theories of physics in explaining some important experimental observations and how the quantum approach is able to explain them.

4.1 Introduction The foundation of classical physics, considered to be propounded in pre-1900 era, rests with the laws of motion given by Newton, theory of electromagnetism given by Maxwell and the laws of thermodynamics. During this period, the X-rays were discovered and were found to have many properties that initially indicated their quite abnormal behaviour. The X-ray photons were found to be like light photons except they contain so much energy, that they penetrate thin opaque sheets. Though the Xrays have sufficient energy to penetrate thin films of material, but objects of higher densities were found to block X-rays, casting shadows on screen placed in front of these. It was also realised that X-rays, like the visible light, are electromagnetic in nature. During the same period a major and revolutionary idea of matter waves was proposed by de Broglie. The idea that moving material particles have an associate wave, i.e. the dual nature of matter, was experimentally confirmed by Davisson and Germer. However, there was some phenomenon, for example, the energy distribution in blackbody radiation spectrum, interaction of light with matter, the heat capacity of gases, stability of atom, its line spectra, etc., that could not be explained by the available classical physics. In this chapter, it is proposed to briefly discuss the production and properties of X-rays and the experiments to establish the dual nature of matter. Failure of classical physics in explaining some important phenomenon is © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 R. Prasad, Physics and Technology for Engineers, https://doi.org/10.1007/978-3-031-32084-2_4

191

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4 X-rays, Dual Nature of Matter, Failure of Classical Physics and Success …

discussed in the later part of the chapter. Quantum mechanical explanations of these anomalies are also included, towards the end of the chapter.

4.2 Discovery, Production and Properties of X-rays The discovery of X-rays, that was accidental, was announced by Wilhelm Conrad Roentgen, in December 1895. W. C. Roentgen, Physics professor at Wurzburg University, Bavaria, Germany, was studying the properties of cathode rays that are emitted when an electric discharge is made to pass through the two electrodes of a cathode tube filled with some gas at low pressure. Roentgen was specifically interested in whether cathode rays could pass through glass body of the cathode-ray tube. His cathode-ray tube was covered by thick black paper on all sides, but he was surprised to see that an incandescent green light nevertheless escaped and projected onto a fluorescent screen placed nearby. Roentgen found that these mysterious rays could penetrate through most substances and cast their shadows on the screen. Since the exact nature and properties of these rays were not known at that time Roentgen called them X-rays, the term ‘X’ usually used to describe the unknown quantity. Roentgen also found that the X-ray was capable of passing through human’s tissues leaving the shadow of bones. Almost immediately X-ray’s uses as a diagnostic tool to detect bone fractures become a common medical practice.

4.2.1 Production of X-rays The X-rays are generally produced in a specially designed vacuum tube invented by William Crookes which is often called the discharge tube or a cathode-ray tube. A typical sketch of an X-ray tube is given in Fig. 4.1. As shown in the figure, the X-ray tube consists of an evacuated glass tube which is fitted with two electrodes. The electrode which is kept at a negative potential is called cathode and the other electrode kept at a higher positive potential, the anode. A potential difference of the order of few tens of kV is maintained between the two electrodes. Often a heater element is also attached to the cathode which may be connected to a low-voltage source. When a current is passed through the heating element, the temperature of cathode increases and thermionic emission of electrons from cathode takes place. The emitted thermo-electrons get repelled by the negative potential at cathode and are attracted by the positive anode. Thus electrons get accelerated and impinge with high speed on to the anode when they get decelerated. Since accelerated/decelerated charged particles (electrons in this case) emit electromagnetic radiations, decelerated electrons in the X-ray tube emit electromagnetic radiations in the form of X-rays. The X-rays generated due to deceleration of electrons are termed as continuous, soft or Bremsstrahlung X-rays. Bremsstrahlung X-rays consist of X-ray radiations of all energies starting from a minimum energy

4.2 Discovery, Production and Properties of X-rays

193

Fig. 4.1 Schematic diagram of an X-ray tube. X-rays are produced when energetic electrons impinge on the anode material

E min to a maximum energy E max , hence the name continuous X-rays. X-rays are also produced when high-energy electrons hit the atoms of the anode material and shift the atomic electrons to their higher energy states, thus exciting the anode atoms. Since atoms cannot remain in excited states for long, excited atoms of the anode revert back to their ground states emitting X-rays. X-rays produced by the de-excitation of excited atoms are called characteristic X-rays and their energy depends on the atoms of anode material. Characteristic X-rays have X-rays of some very definite wavelengths that depend on the nature of the target atom and are found superimposed as sharp lines on the continuous background of Bremsstrahlung X-rays. Since their inception, the X-ray tubes have evolved and undergone several changes/improvements. A modern cold cathode X-ray tube is shown in Fig. 4.2. As shown in the figure, the modern X-ray tube contains an anticathode or target opposite the concave-shaped cathode, further the cathode is not heated and electrons are not emitted by cathode. The X-ray tube is filled by some inert gas like argon at low pressure. A spark plug is used to ignite the inert gas which on ionisation produces electrons. Electrons produced by the inert gas are accelerated between the concave cathode and the target or the anticathode. Any desired material may be attached to the anticathode in order to produce characteristic X-rays of that material. Concave cathode focuses the electron beam to a point on the target which helps in generating focused X-rays that produce sharp images of dens materials like bones, etc. These X-ray tubes do not have any heater at its cathode; therefore, they are referred as cold cathode tubes. Soon after the discovery of X-rays, efforts were made to study the properties of these rays. Experimental observations revel that X-rays have a high penetrating power, travel in straight line and cannot be deflected by electric field or the magnetic field. When the X-rays are incident on the photographic plates, they are found to

194

4 X-rays, Dual Nature of Matter, Failure of Classical Physics and Success …

Fig. 4.2 Modern X-ray tube

blacken the film. When X-rays pass through a gaseous medium, they ionise it and can also cause photoelectric emission similar to when light is incident on metallic cathode. The X-rays have broadly been categorised into two categories as continuous X-rays and discrete/characteristic X-rays.

4.2.2 Continuous X-rays Continuous, Bremsstrahlung, white or soft X-rays are produced when energetic electrons, accelerated by the potential difference between the anode and the cathode of an X-ray tube, get retarded on hitting the target (anode or anticathode) which is generally of some metal. Electrons approaching a metal target interact with atoms of the target in two ways: (i) they are repelled by the negative electron cloud of target atom and (ii) they get inelastically scattered by the positively charged nucleus of the target atom. In both these interactions, incident electrons lose energy. The amount of energy lost by each incident electron depends on several factors including its direction of motion, distances from the electron cloud and nuclei of target atoms and the angle of scattering. Different electrons of the incident electron beam lose different amounts of energies on interacting with target atoms. The energy lost by each electron is converted into an X-ray. In this way, a bunch of X-rays having a random energy distribution (or wave length) is produced when a beam of accelerated electrons hit the target electrode in an X-ray tube. Bremsstrahlung is a word of German language, which may be broken into two German words: Brems, which means break (stopping), and strahlung, that means

4.2 Discovery, Production and Properties of X-rays

195

Fig. 4.3 Deceleration of incident electron by the coulomb field of the electron cloud of the target atom produce Bremsstrahlung X-ray

radiations. Bremsstrahlung, therefore, stands for the radiations that are emitted by the stopping of electrons. The process of X-ray emission as a result of deceleration of electrons is called Bremsstrahlung. Figure 4.3 shows how an incident electron gets retarded in the Coulomb field of the electron cloud of the target atom, and the difference of kinetic energy ΔE at points A and B is converted into an X-ray. A figure depicting the emission of X-ray as a result of change of velocity of electron due to scattering by the positively charged nucleus of the target atom is shown in Fig. 4.4.

Fig. 4.4 Deceleration and deflection of an energetic electron by a positively charged nucleus, resulting into emission of X-ray photon

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4 X-rays, Dual Nature of Matter, Failure of Classical Physics and Success …

X-rays, like other components of electromagnetic radiation spectrum, exhibit both the wave and particle nature. The X-ray energy quanta are called photon. X-rays are characterised either by energy E x , frequency ν or the wavelength λ, which are related to each other through following expressions: E x = hν =

hc c ; ν= ; λ λ

p=

E hν = c c

(4.1)

Symbols h, c and p in above expressions stand respectively for Planck’s constant, speed of light and the linear momentum of X-ray photon. Intensity distributions of continuous X-rays as the function of wavelength and as the function of photon energy E x or frequency γ are shown in Fig. 4.5. It may be noted in Fig. 4.5a that X-ray distribution curve has a cut-off wavelength, denoted by λmin while on the higher wavelength side the curve extends almost up to infinity. Minimum wavelength λmin corresponds to the X-ray of the highest energy, similarly X-rays of longer wavelengths have smaller energies extending to almost zero energy. In Fig. 4.5b, the X-ray of highest energy has the largest frequency γ max and on the lower frequency side the distribution curve extends almost to the origin. An important property of continuous X-rays is that the cut-off point (λmin or γ max ) depends only on the potential difference V between the anode and the cathode of the X-ray tube and it does not depend on the target material. The minimum wavelength λmin and maximum energy or maximum frequency νmax of emitted X-ray corresponds to an incident electron losing all of its energy in a single collision and radiating it away in the form of a single X-ray photon. If we assume that total kinetic energy (K.E.) of the electron is converted into energy (hν) of the X-ray photon, then

Fig. 4.5 Intensity distribution of continuous X-rays as a function of a X-ray wavelength, b X-ray energy or frequency

4.2 Discovery, Production and Properties of X-rays

197

Kinetic energy lost by the electron ΔE = Energy hν of emitted X-ray Or  ΔE = hνmax = h hνmax =

c



λmin

hc λmin

(4.2) (4.3)

The λmin may be called as the cut-off wavelength, which will mainly depend on the value of accelerating voltage V applied across the anode and cathode. Thus, hνmax =

hc = eV λmin

(4.4)

And λmin =

hc eV

(4.5)

When one substitutes the values of h, c and e in Expression (4.5), one gets the simplified relation between λmin (in units of Å = 10−10 m) and the voltage V between the anode and cathode of the X-ray tube as,   12.398 × 103 λmin in units of Å = V (volts)

(4.6)

The total X-ray energy emitted per second (intensity) I depends on the atomic number Z of the target atom, the electric current i passing through the two electrodes of the X-ray tube and the potential difference V between the electrodes. Intensity I may be written as: X-ray intensity I = Ai Z V m

(4.7)

Here A is a constant and m is also a constant that has the value of ≈ 2 for most of the metals. The dumbbell-shaped spectrum of continuous X-rays has a broad peak which corresponds to the wavelength of the X-rays of maximum emission, the X-rays which have the highest number in intensity distribution. With the increase of the voltage V between the electrodes of the X-ray tube, the peak of maximum emission and the cut-off wavelength λmin both shifts towards the shorter wavelength and the total area of the intensity curve also increases as shown in Fig. 4.6. This happens because of the increase in the kinetic energy of electrons with voltage V, resulting in larger number of electron interactions in which more energy is lost in the form of X-rays, and because current of the X-ray tube i also increases with V.

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4 X-rays, Dual Nature of Matter, Failure of Classical Physics and Success …

Fig. 4.6 Variation of the continuous X-ray spectrum with the voltage between electrodes

4.2.3 Characteristic X-rays Continuous component of X-rays is produced from the deceleration of high-energy electrons in the X-ray tube by Coulomb fields of negative electron cloud and the positively charged nucleus of the target atom. The electrons in the X-ray tube are generally accelerated by high voltages of the order of few tens of kilovolts, have sufficient energy to initiate two other processes. (i) They may excite the atoms of the target material by shifting an electron from the lower energy state to a state of higher energy. (ii) High-energy electrons may ionise the atoms of the target material by pushing out some electrons from the atom. Both the excited and the ionised target atoms are unstable, they cannot remain in excited or ionised state for long and in most cases the unstable target atoms revert back to their ground states within 10−9 –10−7 s. Characteristic X-rays are emitted by the atoms of the target material when they de-excite from the ionised or excited states. The wavelength and intensity of these X-rays depend on the type of the atom of the target material, and therefore, they are called characteristic X-rays. The energy (or wavelength/frequency) and intensity of characteristic X-rays change with change of the target material attached with anode or anticathode of the X-ray tube (Fig. 4.7). Characteristic X-rays are produced when an incident electron of high energy interacts with a bound electron (say K-shell electron) and eject the K-shell electron out of the atom, creating a vacancy of electron in K-shell. Obviously, this is possible only if the energy of the incident electron is more than the binding energy of electron in K-shell. This is ionisation of the target atom. The ionised atom is unstable and

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Fig. 4.7 Representation of X-ray emission during the transition of an electron from L- to K-shell

may revert back to a state of lower excitation by the transfer of an electron from the next higher shell, the L-shell. The difference in the energy of the ionised atom with electron vacancy in K-shell and when electron vacancy is in L-shell, is emitted in the form of an X-ray. This X-ray may be denoted as X L→K and is called the K α X-ray. Similarly, the vacancy in the K-shell may be filled by an electron from the M-shell (instead of from L-shell) producing a X M→L -ray which is designated as K β X-ray. In this way X-rays of different energies may be produced by the de-excitation of an ionised target atom. In case the high-energy electron in the X-ray tube creates an electron vacancy in L-shell of the target atom (instead of the K-shell), then in that case electrons from M, N and other higher shells may fill the electron vacancy in L-shell producing L α , L β , L γ . . . X-rays. Figure 4.8 shows the electron energy level diagram of any general target atom and the transitions corresponding to various Xrays. In the energy level diagram it may be noted that the energy difference between successive energy levels decreases as one goes higher in energy. The maximum energy difference occurs between the K- and L-shells, and therefore, the energy of the K α X-ray is equal to (E L − E K ), where E L and E K are respectively the energy of the L-shell and of the K-shell. In an actual case, there are large number of high-energy electrons in the X-ray tube that hit the target almost simultaneously and ionise many target atoms producing vacancies in different shells of different atoms, as a result many characteristic X-rays like K α , K β , L α …, etc. are emitted by target atoms. Since atoms of different materials have different energy level diagrams, the energies of characteristic X-rays are different for different materials. The characteristic X-ray spectrum may therefore be treated as a finger print of the atom and is often used to identify different atoms. The characteristic X-ray spectrum is always found to be superimposed as sharp peaks on the background of continuous X-ray spectrum as shown in Fig. 4.9. Since the width of characteristic X-ray peaks is very small as

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Fig. 4.8 Schematic energy level diagram of a target atom and transitions corresponding to different characteristic X-rays

compared to the broad peak of continuous spectra, characteristic peaks are called characteristic lines. Most X-ray tubes operate at voltages of the order of 50 kV or less. Therefore, the maximum energy of continuous X-rays may be at the most 50 keV or a little less, when an electron accelerated to 50 keV energy loses all its energy in a single event producing an X-ray of 50 keV. As such the continuous X-ray spectra from such an X-ray tube will terminate at 50 keV energy. The maximum energy X-ray in characteristic spectra will correspond to K α line of the target atom. For mediumweight target elements the energy of K α line lies in the range of 50–80 keV energy. Evidently, in such cases, the characteristic X-ray lines are found to be superimposed on the high-energy tail of the continuous spectra, as shown in Fig. 4.9. As already mentioned, the characteristics (wavelength of maximum emission, intensity and cut-off or minimum wavelength) of the continuous X-ray spectrum depends essentially on the magnitude of the voltage between the anode and the cathode of the X-ray tube; however, the wavelengths or frequencies of characteristic Fig. 4.9 Peaks of characteristic X-ray spectrum superimposed on the background of continuous X-ray spectrum

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X-ray lines depend only on the atoms of the target material and do not change with the voltage between the two electrodes of the X-ray tube. SAQ: What is the basic origin of X-rays and in what respect their origin is different from that of gamma rays? SAQ: Which characteristic X-ray of a given atom will have highest energy and why?

4.2.4 Mosley’s Law With the discovery of large number of elements, attempts were made to arrange elements in some order. First such attempt was made by John Dalton in 1803 when he arranged elements according to the increasing atomic weight. Later, it was observed that groups of elements exhibit similar chemical properties suggesting the presence of recurring patterns of chemical behaviour. Around 1870 Dimitri Mendeleev developed what is called the periodic table of elements, where elements were largely placed according to their atomic weights and numbered consecutively. In this periodic table no physical meaning or significance was attached to the sequence number of the element. However, some anomalies were found in this arrangement of elements in the table; for example, the atomic weight of Cobalt was higher (58.93) than that of Nickel (58.69), but its chemical properties suggested that it should be placed before Nickel in the periodic table. Similarly, the atomic weight of argon was larger than that of potassium, and the atomic weight of tellurium was greater than that of iodine, but the chemical properties of both these elements suggested that they in spite of their heavier weights should precede the corresponding lower weight partner. These anomalies indicated that atomic weight is not the correct criteria for numbering elements in the periodic table. Soon it was realised that numbering of elements in periodic table should be done according to the number of electrons in the atom of the element which is equal to the number of units of positive charge on the nucleus of the atom and is denoted by Z, the atomic number. A final and definitive resolution of this anomaly was achieved by H. G. Moseley, an English physicist, who in 1913 published a research paper based on his analysis of characteristic X-ray spectra from several elements showing that the frequencies of characteristic X-ray lines are proportional to the squares of whole numbers that are equal to the atomic number plus a constant. Moseley used Bohr’s atomic model for the analysis of experimentally observed characteristic X-ray spectra from many elements. In order to appreciate Moseley’s analysis it is required to re-visit Bohr’s model of the atom. In Bohr’s model of the atom, it is assumed that the electron moves in a circular orbit around the positively charged point nucleus, balancing the centrifugal force by the attractive Coulomb force between the oppositely charged electron and the nucleus. The breakthrough in Bohr’s model was the quantization of electrons’ angular momentum. Using this model Bohr derived the following formula for the wavelength of electromagnetic radiations emitted during transitions of electron between quantized electron energy states in hydrogen-like atom.

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  1 1 1 = R 2 − 2 Z2 λ nf ni

(4.8)

Here, n i , n f respectively, denote the angular momentum of the electron in the initial and final states (n = 1 for K-shell, 2 for L-shell, 3 for M-shell and so on), Z the units of positive charge on the nucleus, i.e. the atomic number of the element. R is the Rydberg constant. Two different values for this constant may be used (a) if it is assumed that the nucleus is of infinite mass as compared to the mass of the electron, the value R∞ is used. This assumption is true for heavy elements (b) in case it is assumed that the mass of nucleus is finite of value M, then the value given by R is used which is more appropriate for light elements. m e · e4 4π cℏ3

(4.9)

 ·e4 Mm e M + m e 4π cℏ3

(4.10)

R∞ = and  R=

Symbols used in above expressions have following meanings: M = mass of the nucleus; m e = mass of electron; e = unit of charge = charge of electron; c = velocity of light and ℏ = rationalised Planck’s constant. Though Bohr’s model was developed only for the hydrogen atom with Z = 1, but it worked rather well in case of singly ionised helium atom for which Z = 2. It was stipulated that if the model is applicable for the cases of Z = 1 and Z = 2, then it may also be valid for heavier elements with Z > 2. And if so, it will be possible to order transitions between electron shells as a function of Z alone, providing an unambiguous meaning to the atomic number Z and ordering of elements according to Z in periodic table. In case of K α transition the frequency να of the transition between nf = 1 and n = 2 shells may be obtained from Eq. (4.8) of Bohr model as:   1 c 1 νk = = Rc 2 − 2 Z 2 λ 1 2 Or νk =

3Rc 2 Z 4

And √ νk =

/

3Rc Z 4

(4.11)

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With this background, we now discuss Moseley’s work and his observations. A systematic examination of the characteristic X-ray radiations was carried out by Moseley for large number of elements from Aluminium to gold. He recorded the Xray spectra for these elements on a photographic √ plate. The analysis of the spectra was done, and the square root of the frequency ( ν) of the particular characteristic X-ray radiation versus the ordinal of the element’s position in the periodic table, which we for the present denote by number N, was plotted. Figure 4.10 shows a representative graph where square root of the frequency of K α lines of different elements is plotted against the serial number N of the element in periodic table. It was observed that with the increase in the position of the element in the periodic table, the root of the frequency of the emitted radiation increases monotonically and the graph may be fitted with a straight line. Similar graphs for K β lines and for other lines of K and the L-series were plotted, and it was observed that for each characteristic X-ray the data points fall on a straight line. These straight lines may be characterised in terms of their slope which may be denoted by ‘a’ and their intercept ‘b’ on the X-axis. Moseley observed that the intercept ‘b’ for all members of a series is same, while each straight line has a different value of slope ‘a’ as shown in Fig. 4.10. √ On the basis of his findings Moseley reached to the conclusion that the ν for different series may be written as:

Fig. 4.10 Plot between the square root of the frequencies of K and L lines and the serial number N of the element in the periodic table

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√ νkα = akα (N − bk ) √ νkβ = akβ (N − bk ) ..................... ..................... √ ν Lα = a Lα (N − b L ) √ ν Lβ = a Lβ (N − b L ) ..................... √ ν Mα = a Mα (N − b M )

(4.12)

(4.13)

(4.14)

That in general may be written as: √ ν = a(N − b)

(4.15)

The constants a and b in above equations are respectively called the proportionality constant and the screening constant. Moseley also obtained the values of the proportionality and screening constants for different series from his experimental plots and found that they may be given in terms of the Rydberg constant R, as given below, / akα = / a Lα =

3Rc and bk = 1; akβ = 4

5Rc and bk = 7.4; a Lβ 36

/

8Rc and bk = 1, 9 / 3Rc and bk = 7.4. = 16

Substituting the above values, the root of frequency of K α -series transitions may be written as: / 3Rc √ νk = (4.16) (N − 1) 4 Expression (4.16) obtained by Moseley from analysis of K α -lines in characteristic X-ray spectrum of many elements is an empirical relation that may be compared with the theoretical expression (4.11) based on Bohr model for hydrogen-like atoms. Except of the screening constant (bk = 1) in Moseley’s empirical relation, the two expressions are identical if one takes N = Z. Moseley argued that the screening constant originates from the fact that the effective charge Z of the nucleus is reduced by the factor b due to the screening or shadowing of the nuclear charge by the electrons left in a given orbital after ionisation. For example, the K-shell of an atom may have at the most two electrons, and if one electron is removed for creating a vacancy in Kth shell, the remaining one electron in the K-shell will screen the nucleus and the effective charge of the nucleus will appear to be (Z − 1) units.

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Moseley assumed that there must be some physical attribute of atoms of the periodic table that increases in a regular fashion by some fixed amount, from one element to the next one. He postulated that this can be the charge (Ze) of the nucleus of the atom which is screened by the negatively charged electrons remaining in orbitals after the creation of vacancy in a shell. According to Mosley, the ordinal or serial number N of the element’s position in the periodic table is equal to the number Z , related to the positive charge (Z e) carried by the nuclei of the element. The number Z is referred to as the atomic number of the element and is exactly equal to the number of protons in the nucleus of the atoms of the materials emitting X-rays. It may be mentioned here that before the investigations carried out by Mosley, the arrangement of elements in the periodic table was in the ascending order of their atomic weights and on the basis of their chemical properties. The Mosley’s results could provide a direct method of determining the atomic number of the elements and helped in removing the discrepancies in the periodic table arrangements. As already mentioned, initially the positions of the transition metals Cobalt (Z = 27) and Nickel (Z = 28) were determined on the basis of the ascending order of their atomic weights as Ni = 58.71 and Co = 58.93 were changed. In the same way some empty positions for the still undiscovered elements were filled. For example, new elements Hafnium (Z = 72), Technetium (Z = 43) and Rhenium (Z = 75) were discovered as there were missing gaps at these values of atomic numbers. It may be noted that the difference in the magnitudes of proportionally constant ‘a’ for different members of a given series, for example, between aKα , aKβ , aKU , etc., is very small, and therefore, curves for different members of a series shown in Fig. 4.10 often merge together, if the resolution of the graph is not good. It might be of interest to know that Henry Gwyn Jeffreys Moseley was born in Weymouth, Dorset, England, on November 23, 1887. After is education at Trinity College, Oxford, he was appointed lecturer in Physics at Rutherford Laboratory, University of Manchester in 1910. His initial research work was on radioactivity, but later he carried out detailed studies on X-ray spectra and in 1913–14 published his famous Moseley law, which paved way to uniquely determine the atomic number of elements and their positioning in the periodic table. In 1914 he was drafted in army during the First World War and was shot in the head by a Turkish sniper at the battle of Suvla Bay. He died at the young age of 27 years. SAQ: What is the physical significance of Mosley’s law? SAQ: Calculate the magnitude of proportionality constant aMβ .

4.2.5 X-ray Diffraction X-rays, like other components of electromagnetic radiations, undergo diffraction. Diffraction occurs when a wave disturbance encounters an obstacle, object or aperture and bends or spreads round the edges of the obstacle. In case the incident radiation is monoenergetic of single wavelength, secondary waves originating from

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different points of the incident wave front interfere with each other giving rise to an interference pattern. The interference pattern produced by the diffraction of electromagnetic waves is characteristic of the obstacle, that is the interference pattern may be treated as the finger print of the diffracting obstacle. The resolution of the diffraction–interference pattern depends on the wavelength of the electromagnetic radiations; radiations of shorter wavelength have better resolution and may resolve structural details of objects/obstacles comparable in dimensions with the wavelength of the radiation. X-rays are electromagnetic radiations with wavelengths in the range of nanometer (10−9 m) and are therefore capable of deciphering structural details of crystals where atoms or group of atoms are arranged at regular and repetitive distance in nanometer dimensions. X-ray diffraction (XRD) is now a well-establish and routine procedure to determine the lattice parameters, arrangement of individual atoms in a single crystal and the phase analysis in case of polycrystalline materials and compounds. It may be recalled that crystals are made up of unit cells which are the simplest repeating structure of a crystalline solid. Arrangement of unit cells results in a crystal lattice, which is a specific three-dimensional arrangement of unit cells. Incident Xrays on diffraction from atoms of the crystal lattice produce a distinct interference pattern characteristic of the atomic arrangement in the crystal lattice (Fig. 4.11).

Fig. 4.11 Two-dimensional lattice with groups of crystal planes

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In order to understand X-ray diffraction by crystalline materials, it is required to understand crystal planes. Crystal planes are imaginary planes inside a crystal which have large number of atoms. In the same crystal lattice there may be several groups of crystal planes with different orientations. The important property of a group of crystal planes is that all crystal plains in a group are parallel to each other and are separated from each other by a fixed distance generally denoted by ‘d’. The separation between successive members of a group of crystal plane, the orientations of possible groups of planes, density of atoms in a group of particular orientation, etc. are all depend on the crystal structure. As you might be knowing that there are seven basic crystal systems (Triclinic, Monoclinic, Orthorhombic, Trigonal, Hexagonal, Tetragonal and Cubic systems) and that for each system there are specific values of crystal plane parameters. Therefore, it is possible to determine various parameters of crystal planes using X-ray diffraction and to identify the crystal system. Since the density of atoms in a crystal plane is large, it is reasonable to assume that each point on this plane acts as if an atom is placed there. This assumption is rigorously valid when the crystal plane is scanned by X-rays of wavelength of the order of nanometer (10−9 m), while inter-atomic distance in crystal planes is much smaller of the order of 10−10 m. With this assumption it is possible to replace atoms in the plane by the plane itself and consider the interaction of incident X-rays by the crystal plane instead of with individual atoms. Let us now consider a parallel beam of monoenergetic X-rays incident on a crystal. Individual X-ray on interaction with individual atom may be scattered or diffracted in any direction; similarly, X-rays falling on a group of crystal planes are diffracted in all possible directions. However, X-rays diffracted by two successive crystal planes of a group after diffraction in a particular direction may undergo constructive or destructive interference creating a patterns that is specific to the crystal lattice. Conditions for constructive interference between X-rays diffracted (or scattered) by two successive crystal planes of a group are derived in the following. Figure 4.12 shows two monoenergetic (wavelength λ) X-rays, denoted as 1 and 2, parallel to each other impinging on a family of crystal planes making an angle θ with planes. Let us consider two consecutive planes, plane-1 and plane-2 which scatter (or diffract) these rays as ray-3 and ray-4, respectively. The scattered rays 3 and 4 will undergo constructive interference if the phase difference between the rays (1 + 3) and rays (2 + 4) is either zero or is an integer multiple of 2π, i.e. if the phase difference between rays (1 + 3) and rays (2 + 4) = n(2π ), where n is an integer including 0, the rays will give a maximum of intensity. Since a phase difference of 2π is equivalent to a path difference of λ, where λ is the wavelength of X-rays, the condition for constructive interference in rays-3 and 4, becomes: Path difference between rays (1 + 3) and rays (2 + 4) = nλ where n = 1, 2, 3, . . .

(4.17)

To calculate the path difference between rays (1 + 3) and rays (2 + 4), we drop perpendiculars AC and AD from point A on ray-2 and ray-4, respectively. As is clear

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Fig. 4.12 Interference of monoenergetic X-rays scattered by successive crystal planes

from the figure, ray-2 travels a distance CB more than ray-1 and ray-4 travels the distance BD more than ray-3. Therefore, the total path difference (Δ-path) between the incident and scattered rays is: Δ-path = C B + B D = d sin θ + d sin θ = 2d sin θ

(4.18)

Hence, constructive interference between X-rays scattered from consecutive crystal planes will take place when: 2d sin θ = nλ where n = 1, 2, 3, . . .

(4.19)

The father and son W. H. Bragg and Lawrence Bragg for the first time derived the above condition of constructive interference between electromagnetic radiations scattered by consecutive members of a family of crystal planes in 1913; the condition is called Bragg’s law. It is easy to verify in Fig. 4.12 that the diffracted rays (2 and 4) are rotated from their original direction (incident rays 1 and 2) by angle 2θ. SAQ: The phenomenon of diffraction of X-rays indicates their particle nature or wave nature?

4.2.6 Some Application of X-rays X-rays are extensively used in medical world for detecting fractures in bones, detecting breaks/tearing of ligaments, sterilising of medical instruments, cloths, bandages, etc. High-energy X-rays are now used for exploring underground structures

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and are providing valuable information of archaeological interest without excavation and digging. Another very important field of application of X-rays is the characterisation of crystalline solids. X-ray diffraction patterns may give information about the size of unit cell, phases involved in the structure and many other characteristics. Powder XRD, the X-ray diffraction pattern from powdered crystals has become a powerful tool for crystal structure studies. Some details of this technique are provided here. In powder X-ray diffraction, a diffraction pattern is obtained from the powder of the material, rather than an individual crystal. Powder X-ray diffraction is easier and simpler than the crystal diffraction as no single individual crystal is required. PXRD is characterised by high sensitivity, reliability, depth profiling, easy sample preparation, convenient procedure, fast speed and effective resolution. Further, the data obtained from PXRD may be used both for qualitative as well as for quantitative analyses. However, there are some disadvantages in this method. Firstly, one uses X-rays that are harmful for human being, and there is always some chance of leakage and unwanted exposure of the person carrying out measurements. Secondly, and more importantly, analysis of data requires standard references to match the experimental pattern. Some of these difficulties have now been taken care by turn-key PXRD systems commercially available with automatic data handling, comparing with inbuilt standard references and finally providing the required information in the output. In commercially available PXRD systems, samples are generally in the form of finely divided powders and the pattern is generated by the diffraction of monoenergetic X-ray from surfaces coplanar to the sample holder. Figure 4.13 shows the simplified sketch of a power XRD system. In this configuration, the X-ray source and the detector are both rotated by the same angle to look for the intensity of diffracted X-rays. X-ray detector in old version of the system use to be a photographic plate; however, in modern XRD systems scintillation spectrometers are used to record the intensity of diffracted X-rays. Scintillation detectors are much more efficient, have better energy resolution which helps in deciphering the crystal structure. A typical spectra of diffracted X-rays as a function of the angle θ are shown in Fig. 4.14.

Fig. 4.13 Simplified sketch of a possible configuration of X-ray source, powder sample holder and X-ray detector for PXRD scanning. In this configuration the sample remains stationary while both the X-ray source and the detector move by the same angle

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Fig. 4.14 Typical spectra of diffracted X-rays

X-ray peaks at different angles of rotation in Fig. 4.14 originate from the X-rays diffracted by different families of crystallographic planes in the crystal. The intensity of the peak is proportional to the density of planes of a given family. As the crystal is rotated, the angle of incidence θ changes for different families of crystal planes and that family of planes for which Bragg’s law 2d sin θ = nλ gets satisfied diffracts X-rays producing constructive interference and a maximum in intensity.

4.3 Dual Nature of Matter The description of the motion of particles in classical physics got revolutionised with the hypothesis of Louis Victor de Broglie (pronounced as de Broy) that every radiation has a particle-like nature and vice versa. This was referred to as the dual nature of matter. It meant that a moving particle of matter should exhibit wavelike properties under suitable conditions. It was mentioned that since the Nature is symmetrical the matter and energy should also be symmetrical. Thus, both radiation as well as matter should not be different. It is often questioned as to what led de Broglie to postulate that matter should have wave-like properties and that waves must possess matter like aspects. Though no clear answer may be given, but it may be said that by 1924 when de Broglie proposed the dual nature of matter, Einstein’s mass–energy equivalence (1905), Bohr model of hydrogen atom (1913) and the phenomena of Compton scattering of gamma rays (1923) were already known. Einstein’s mass energy equivalence clearly indicated that mass is a form of energy; similarly, it was postulated in Bohr’s model of hydrogen atom that energy difference between two energy states of electron may be emitted in the form of electromagnetic waves during the process of de-excitation of the atom, showing that waves are also a form of energy. In other words, these two postulates clearly established the equivalence of mass, energy and waves. The only remaining question that wave may also behave like a particle was established by the phenomena of Compton scattering, where a high-energy gamma ray behaving like a particle strikes a stationary atomic electron, scatters the stationary electron

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in some direction, loses some energy and is itself scattered with reduced energy in the complimentary direction. All these facts might have led de Broglie to postulate matter waves. According to de Broglie, the wavelength associated with a material particle is given by, λ = h/ p, where h is Planck’s constant, while p = mv is the linear momentum of the particle of mass m moving with velocity v. The validity of the de Broglie relation can best be tested by the results of the experiment. It may be remarked that the above equation is satisfied by a photon as well because the linear momentum of a photon p = hν/c. Thus, hp = νc = λ. De Broglie’s postulate essentially says that a particle of mass m, moving with velocity v, has an associated wavelength λ = h/mv. In order to appreciate de Broglie wavelength λ, it is required to understand what is meant by the term wavelength. Wavelength is essentially the uncertainty in the position of a moving particle. The location of a moving particle at any instant in its path of motion is uncertain by the amount of its wavelength. It is obvious that if the physical dimension (the size) of the moving object is smaller than the associated de Broglie wavelength, then only it will be possible to carry out experiments to see the effect of wavelength. In case the size of moving object is larger than the associated wavelength, it may not be possible to experimentally detect the wavelength associated with the particle. It can be seen from the expression for de Broglie wavelength that a particle of large mass will exhibit a smaller wavelength and vice versa. This is the reason, why in our daily life the wave-like character of even a cricket ball thrown with 80 miles/h is not observed. In general de Broglie wavelength of only microscopic atomic and nuclear particles moving with high velocities may be measured experimentally. According to Einstein’s theory of relativity, the mass of a moving body depends on its speed of motion. For particles moving with high velocities relativistic express for mass given below must be used. m=/

m0 1−

(4.20) v2 c2

Here, m0 is the rest mass of the body. √ Since the momentum p and kinetic energy E are related by the expression, p = 2m E, as such the de Broglie wavelength may also be written as: λ= √

h 2m E

.

(4.21)

SAQ: What is the rest mass energy of an electron in MeV? SAQ: What is the value of de Broglie wavelength for a stationary body and what does it mean? Solved Example SE4.1 Calculate the de Broglie wavelength of a ball of mass 100 g moving with a speed of 30 m/s. Solution: Given, m = 0.10 kg; v = 30 m/s

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λ=

h 6.64 × 10−34 J s h  = 4.21 × 10−34 m  = = p mv (0.10 kg) × 30 ms

As can be seen from the value of the wavelength obtained above that it is so small that it is not even measurable with present day instruments. This shows the reason why the wave nature of macroscopic objects is not observable in our daily life. On the other hand, for microscopic particles the wave-like nature is significant and observable. As an example, let us consider that an electron of m and charge e is accelerated across a potential of V volts, then the electron gains a kinetic energy K .E. = eV . √ p2 Since K .E. = 2m , therefore, p = 2meV . h The de Broglie wavelength λ = √2meV . If in the above expression we substitute the numerical values of Planck’s constant h, mass of electron and charge of electron, then the above expression reduces to λ = 1.227 √ nm, where V is the accelerating potential in volts. For a value of V = 150 V, V

1.227 1.227 nm = 14.247 nm = 0.100 nm. This the value of wavelength comes out to be λ = √ 150 value is comparable to the order of spacing between the atomic planes in crystals. It indicates that the particle nature of electrons could be verified by crystal diffraction experiments similar to the X-ray diffraction. The experimental verification of the de Broglie hypothesis has been described in detail in the next section. Louise Victor de Broglie was awarded the 1929 Nobel Prize in Physics for his discovery of the wave nature of electrons.

4.3.1 Davisson and Germer Experiment Clinton Davisson and Lester Germer were involved in the study of the surface properties of Nickel samples using a beam of low-energy electrons at Bell laboratories, USA, since 1923. However, the remark by Walter M. Elsasser (scientist at Gottingen, Germany) that electron scattering by crystalline solids may be used to test the wave nature of electrons, led Davisson and Germer in 1927 to repeat their experiment, now with the view to look for the wave nature of electrons. Experiment carried out by Davisson and Germer provides direct verification of De Broglie hypothesis of the wave nature of moving bodies and demonstrated that moving electron has an associated wave. The typical experimental setup used by Davisson and Germer is given in Fig. 4.15. The thermionic emission of electrons from a hot tungsten filament was used to provide a beam of electrons. These electrons were accelerated by applying suitable potential difference with the help of a battery as shown in Fig. 4.15. The electron beam was collimated by allowing them to pass through a cylindrical arrangement with a fine slit. The electron beam is incident on the Nickel crystal having ordered arrangement of atoms, at normal incidence. Nickel atoms diffracted/

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Fig. 4.15 Schematic diagram of Davisson–Germer experimental arrangement

scattered the incident electrons. In the experimental setup Nickel target crystal had an arrangement of rotation at different angles in a plane. The intensity of the electrons scattered by the crystal in a given direction was measured with the help of movable detector. The whole experimental arrangement was placed in a highly evacuated chamber. Several experiments were carried out and intensity of scattered beam at different angles was recorded for different accelerating voltages. The observed curves are plotted in polar coordinates in Fig. 4.16. Surprisingly, instead of a continuous variation of scattered electron intensity with angle distinct maxima and minima were observed whose position depended on the electron energy. It was observed that there is a pronounced maximum that appear at ϑ = 50◦ angle of scattering (with respect to the direction of the incident beam), when the accelerating potential is 54 V. Further, increasing the accelerating potential indicated that the bump like maximum decreases and becomes almost insignificant as the accelerating potential reaches to 68 V. Figure 4.17(i) shows the variation of the intensity of 54 eV energy electrons with angle of scattering/diffraction by the Ni-crystal. As may be observed in this figure a prominent maximum in the observed intensity occurs at angle 50°. Figure 4.17(ii) shows the direction of the incident beam of electrons which is normal to the top face of the Ni-crystal. However, the incident beam makes an angle of 65° with family of dominant crystal planes that are responsible for the diffraction of incident electrons. Further, from X-ray diffraction experiments it was known that the separation between successive crystal planes d is 0.91 Å. The reason for the prominent bump like state at ϑ = 50◦ at accelerating potential of 54 V may be understood as due to the diffraction of electron waves by the crystal planes in the target Ni-sample. Figure 4.17(i) shows the electron diffraction pattern for 54 eV electrons as a function of diffraction angle.

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Fig. 4.16 Polar plots of the diffraction pattern of electrons by Nickel crystal for different accelerating voltages

Fig. 4.17 (i) Diffraction pattern of 54 eV electrons by Ni-crystal. (ii) Crystal plane family and diffraction geometry

The de Broglie wavelength associated with electrons accelerated through 54 V may be calculated as, 1.227 1.227 1.227 = 0.168 nm = 1.68 Å. nm = λde Brg = √ nm = √ 7.348 V 54 One may also calculate the wavelength of the waves that have produced the observed maximum using Bragg’s law. While applying Bragg’s law one must remember that the angle of scattering that the incident electron beam makes with the crystal plane is 65° and that the separation between planes is 0.91 Å, λBragg = n2d sin θ

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Substituting order of diffraction n = 1, θ = 65° and d = 0.91 Å in the above equation, the wavelength obtained from Bragg’s law λBragg comes out to be λBragg = 1 × 2 × 0.91 × sin 65 Å = 2 × 0.91 × 0.906 Å = 1.66 Å. It may be observed that there is very good agreement between the de Broglie wavelength λde Brg (= associated with 54 eV electrons = 1.68 Å) and the wavelength λBragg (= 1.66 Å) obtained using Bragg’s law. It clearly proves that a beam of electrons behaves both as a beam of material particles as well as a beam of waves of wavelength given by de Broglie formula. Davisson and Germer experiments do provide a direct verification of de Broglie hypothesis of the wave nature of moving bodies. Soon after the publication of the results from Davisson–Germer experiment many more and detailed experiments were performed all of which confirmed the existence of matter waves. The electron diffraction was studied by G. P. Thomson using X-ray powder diffraction method, where an accelerated fine beam of electrons was made to hit normally on to the thin metallic foil. The other side of the foil was exposed to a photographic plate. As the electron beam passes through the foil, electrons of the incident beam get diffracted by the grating like crystal structure in the foil forming a diffraction pattern on the photographic plate. The diffraction pattern of bright co-centric circular rings around a central spot got reviled when the photographic plated was developed. In order to test that the diffraction pattern formed on the photographic plate is due to the diffraction of electrons of the incident beam, a magnetic field was applied between the source of electrons and the metallic foil. As expected, the diffraction pattern got disturbed by the magnetic field confirming that the diffraction pattern is truly due to the diffraction of de Broglie waves associated with accelerated electrons. Since then many instruments particularly electron microscopes that use de Broglie waves associated with accelerated electrons have been constructed and are in wide use. Since the resolving power of a microscope depends on the wavelength of the waves used in the instrument, de Broglie waves of very small wavelength may be produced using high-energy electrons. De Broglie waves of very small wavelengths associated with high-energy electrons are used in electron microscopes. It might interest you to know that the postulate of matter waves proposed by de Broglie was a part of his Ph.D. thesis. Confirmation of waves associated with particles through several experiments involving diffraction and interference of particle waves put de Broglie’s theory on firm footing. However, two big questions regarding the matter waves also spring up in the background of dual nature of matter. One big question was: what is the velocity of the matter waves? And the second question was: what (material/field/ function) makes de Broglie waves? For example, in case of waves in a pond or sea, it is water that moves up and down making the wave, in a wave on a string, different segments of the string moves to generate wave and in electromagnetic waves (light) it is the intensity of electric and magnetic fields that varies with time and generate the

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electromagnetic wave; so the question is: time variation of which quantity constitutes matter waves? (a) Velocity of de Broglie wave Let us address the first question: what is the speed of de Broglie (or matter) waves. It may be recalled that in order to explain phenomenon like photoelectric effect and Compton scattering it was necessary to assume that electromagnetic (EM) waves (light and gamma rays) have an associated particle, called photon. The energy E phot of photon, the momentum pphot of photon and the wavelength/frequency νphot of electromagnetic wave are related according to the following relations: E phot = hνphot ; λphot =

hνphot E phot c h = ; νphot = ; pphot = , pphot c c λphot

(4.22)

and νphot λphot = c Here, ‘c’ is the velocity of light in vacuum and ‘h’ is Planck’s constant. De Broglie argued that if waves have a particle associated with them, then on the basis of symmetry, moving material objects must also have associated waves, which he called matter waves. De Broglie assumed that relations corresponding to the set of relations given by Eq. (4.22) may also be written for matter waves. Energy of the particle that carry matter waves E matt = hνmatt

(4.23)

But from Einstein’s mass energy relation E matt = mc2 = /

m0

c2

1−

(4.24)

v2 c2

Equating Eqs. (4.23) and (4.24) one gets, νmatt =

mc2 h

(4.25)

Also the wavelength λmatt of matter waves may be given as λmatt =

h =⎛ mv ⎝ /

h 1 2

1− vc2

⎞

(4.26)

 ⎠m 0 v

In analogy to the expression for the velocity of EM wave c = λphot νphot , one may write the velocity of matter waves as Vmatt = λmatt · νmatt =

c2 h mc2 · = mv h v

(4.27)

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In Eq. (4.27), Vmatt represents the velocity of the matter wave, while ν is the velocity of the particle of rest mass m0 . Now there is a contradiction. According to the theory of relativity, no material particle can move with the velocity of light (c), and therefore, the velocity v of the particle must be small than velocity of light,  the c2 i.e. ν < c, and hence the velocity of the matter wave Vmatt = v may exceed the velocity of light. This puts a big question mark; what does this mean? The answer is that we have to re-visit the dynamics of wave propagation. In general, two velocities may be associated with a wave; they are: (i) the phase velocity and (ii) the group velocity. (i) Phase velocity To understand the concept of phase velocity, let us consider a string or a wire held fixed at two points in the x-direction. Suppose the wire is plugged in the y-direction so that it vibrates in y-direction. The displacement ‘y’ of any point on wire at time ‘t’ may be given by,   y = A cos 2π ν t −

x Vphase

 (4.28)

Here ν is the frequency and Vphase the wave speed, i.e. the speed with which the wave travels down the wire. As a matter of fact Vphase is the speed with which the displacement y travels along the wire in x-direction. Pick up any particular displacement (in vertical direction) y1 at point X 1 of the wire at instant ‘t 1 ’. At the next instant t 2 , the same displacement y1 will move to another location x 2 of the wire, and next at instant t 3 , displacement y1 will reach at point x 3 on the wire. In this way the location of displacement y1 is moving along the length of the wire in the x-direction. No part of wire is moving in x-direction. Wire is moving (vibrating) in Y-direction. It may thus be observed that no material particle moves along the x-direction, while the location of a given displacement moves along the x-direction with speed Vphase , as shown in Fig. 4.18. This speed, with which the phase of the wave travels, is called the phase velocity of the wave. Since no material particle travels with the phase velocity, the phase velocity may have a value equal or even greater than the velocity of light. 2 V matt = cv given by Eq. (4.27) represents the phase velocity of the matter wave. Further, in matter waves no matter/particles are vibrating in y or in any other direction. (b) Group velocity Equation (4.28) represents a progressive wave,i.e. a wave  which is moving in xx direction with velocity V phase . A negative sign in t − Vphase tells that the disturbance   x is moving in +ve X-direction, while a positive sign in t + Vphase means that the wave is moving in negative X-direction. It is often more useful to write Eq. (4.28) in a slightly different form as given below,

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Fig. 4.18 In case of a wire stretched in x-direction and vibrating in y-direction, the location of a particular vertical displacement moves in x-direction with phase velocity

  y = A cos 2π ν t −



x Vphase

Or  y = A cos

2π νt −

2π νx Vphase

 (4.29)

But 2π ν = ω (the cyclic frequency of the wave) and Vphase = ν · λ. Substituting these values in Eq. (4.29) gives  y = A cos

ωt −

2π νx νλ



 = A cos

ωt −

2π x λ

 (4.30)

One defines the wave number (or wave vector) k = 2π . With this substitution and λ using the fact that cos(−θ ) = cos(θ ), Eq. (4.30) reduces to: y = A cos[(kx − ωt)] where

(4.31)

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ω = 2π ν =

2π m 0 c2 2π mc2 = √ h h 1 − v2 /c2

(4.32)

And k=

2π mν 2π m 0 v 2π = = √ λ h h 1 − v2 /c2

(4.33)

It follows from Eqs. (4.32) and (4.33) that: 2π ν ω = 2π = νλ = the velocity of the wave k λ

(4.34)

In three dimensions the wave equation becomes   y = A cos k→ · r→ − ωt ,

(4.35)

here k→ · r→ represents the dot product of vector r and vector k. Equations (4.31) and (4.35) represent in one dimension and in three dimensions travelling transverse waves which are polarised in y-direction. The question now arises that if these ways are matter waves associated with some material particle then where does the motion of the particle is hidden in these equations? Clearly, these waves represent the motion of the phase of the wave which travels with speed V phase . These waves do not represent the motion of any material body. Motion of a material body may be introduced in the wave equation only and only by superimposing one or more than one waves. When there is more than one wave with slightly different values of ω and k, travelling through the same region of space, they undergo interference and generate a wave packet. It may be shown that the wave packet produced by the superimposition of waves (or as a result of modulation of one wave by other waves) may represent a material body and the speed with which this wave packet travels is called the group velocity of the packet. The group velocity Vgroup may be the velocity of the material particle associated with de Broglie waves. Production of beats in acoustics, when two sound waves of slightly different frequencies modulate each other, is well known. The pitch of the resultant sound increases and decreases periodically. This phenomenon of the increase and decrease of sound intensity is called the beat formation. When beads are formed by two sound waves of slightly different frequencies, the number of beats or the repetition rate of beads depends on the frequency difference of the two sounds. It may be noted that though both sound waves individually travel with the same velocity (velocity of sound in air), but the rate of beat formation depends on the frequency difference of the two waves (Fig. 4.19). To keep our discussion simple, let us consider two one-dimensional waves y1 and y2 having same amplitudes A but slightly different cyclic frequencies ω and the wave numbers k, moving in the same part of the space. The waves may be represented as:

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Fig. 4.19 Superposition of two waves of slightly different frequencies produces wave packets that move with the group velocity

y1 = A cos(ωt − kx) and y2 = A cos[(ω + Δω)t − (k + Δk)x]. The two waves will interfere and the resultant wave may be represented as: y1 + y2 = A cos(ωt − kx) + A cos[(ω + Δω)t − (k + Δk)x]

(4.36)

Expression (4.36) may be simplified using the following relations:  cos α + cos β = 2 cos

α+β 2



 cos

α−β 2

 and cos(−θ ) = cos θ 

1 1 y1 + y2 = 2 A cos[(2ω + Δω)t − (2k + Δk)x)] cos (Δω)t − (Δk)x 2 2

 (4.37)

Since Δω and Δk are much small compared to 2ω and 2k, respectively, one may take 2ω + Δω ≈ 2ω and 2k + Δk ≈ 2k. With these approximations, Eq. (4.37) reduces to:  y1 + y2 = 2 A cos(ωt − kx) cos

Δk Δω t− x 2 2

 (4.38)

Equation (4.38) represents a wave of angular frequency ω and wave number k and wave number Δk . that has been modulated by a wave of angular frequency Δω 2 2 The phase velocity of this modulated wave (see Eq. 4.34) is given by Vphase = ωk .   Term cos Δω t − Δk x in Eq. (4.38) represents a wave packet which travels with the 2 2 Δω velocity Δki . The velocity with which the wave packet travels is called the group Δω . velocity, and therefore, the group velocity Vgroup = Δki

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The magnitude of the group velocity may be obtained using Eqs. (4.32) and (4.33). From Eq. (4.32) 2π m 0 c2 ω= √ h 1 − v2 /c2 Therefore, 2π mv dω =  3/2 2 dv h 1 − vc2

(4.39)

Also from Eq. (4.33) 2π m 0 v k= √ h 1 − v2 /c2 Differentiating k with respect to ν gives, 2π m dk =  3/2 2 dv h 1 − vc2

(4.40)

It follows from Eqs. (4.39) and (4.40) that the group velocity Vgroup =

dω dk 2π mv dω 2π m = / =  /   3/2 = v 3/2 2 2 dk dv dv h 1 − vc2 h 1 − vc2

(4.41)

Thus, the group velocity (velocity with which the wave packet or the envelope of the modulated wave travels) comes out to be equal to the velocity of the particle 2 associated with the matter waves, while the phase velocity of matter waves ωk = cv . SAQ: Show that in a non-dispersive medium group velocity of an EM wave is equal to the phase velocity. (b) What makes matter waves Waves are generated in space by the time variation/fluctuation of some material/ fields or some other quantity. For example, sound consists of pressure difference in the medium, water waves are produced by the fluctuation in heights of water column, and electromagnetic waves consists of fluctuating electric and magnetic fields. Matter waves or de Broglie waves are produced by the time variation/fluctuation of the wavefunction ψ of the associated material particle. Schrodinger while developing the wave theory of particles (also called the quantum mechanics) introduced the concept of wavefunction of a particle, denoted by Greek letter ψ, which is a function that contains all properties of the particle. Once

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the wavefunction for a system/particle is known, all properties of the system/particle are known, and the system or the particle is completely defined. Wavefunction ψ is generally complex and it cannot be directly measured. In general ψ is a function of time and position (x, y, z).The probability of finding the object, for which ψ is the wavefunction, at some point P (x, y, z) at time ‘t’ is proportional to the value of the quantity ψ ∗ ψ at point P. When the value of ψ ∗ ψ at point P is 1, it means that the object is there, and when the value is zero, it means that the object in not at point P. It may be noted that the wavefunction of an object tells about the probability of finding the object at some particular point in space and time, but it certainly does not mean that the object has spread out in a wave. Further properties of wavefunctions will be discussed while dealing with Schrodinger’s equation in quantum mechanics.

4.4 Some Examples of the Failures of Classical Approach and Success of Quantum Approach Classical physics is built on Newtonian mechanics, thermodynamics and Maxwell’s theory of electromagnetism. Many theories of classical physics break down when applied to microscopic systems or to objects moving with high velocities comparable to the speed of light. An essential feature of classical approach is the assumption that physical variables like energy, angular momentum, etc. vary continuously. As a matter of fact the classical approach could not even explain the existence of an atom. Many other processes observed experimentally could not be explained by the classical theories, for example the photoelectric effect and Compton scattering. However, the biggest problem was encountered in explaining the energy distribution of blackbody radiations and the specific heat of solids. Failure of classical approach in explaining the above mentioned processes will be discussed in the following.

4.4.1 Stability of the Atom and the Nature of Atomic Spectra Rutherford in 1911 carried out some ingenuous and revealing experiments in which thin gold foils were bombarded by energetic alpha particles. Scattering of incident alpha particles by scattering angles as large as 180° established that there is a body at the centre of each atom where total positive charge and more than 99% mass of the atom are contained. The central body was named ‘Nucleus of the atom’, term nucleus being borrowed from biology. Earlier, J. J. Thomson has already discovered electron in 1897 and experiments with cathode-ray tube have proved that electron is an essential constituent of all atoms. Soon after the discovery of atomic nucleus, several theories for the structure of nuclear atom were proposed. The most convincing model for atomic structure was the planetary model where it is assumed that electrons in an atom revolve round the nucleus in circular orbits at different distances from the

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nucleus, like planets revolve round the sun in solar system. The planetary model of the atom was readily accepted because of its simplicity and compelling similarities with the planetary system. For example, Coulomb force of attraction between the electron and the nucleus may well be compared to the gravitational force of attraction between the sun and the planet and that both these forces have almost similar dependence on distance. Planetary model of the atom was also attractive as philosophically it was in confirmation to the adage that big solar system is just an upscale of atomic system. At the first sight planetary model of the atom appeared to be more stable than the solar system, because in case of the solar system planets move in outer space where there is matter of very low density, planets therefore experience a force of drag which reduces the orbit of the planet and in long run every planet is expected to fall down into the sun. No such drag was expected in case of the atomic electrons as the planetary model assumes perfect vacuum around the nucleus where electrons revolve. The stability of the planetary model of the atom which was based on classical laws of physics (Newton’s laws of motion and Coulomb’s law) was questioned by the classical theory of electromagnetism put forward by Maxwell (1862) in the form of four equations. Maxwell’s theory says that a charge at rest has an electric field around it which is strongly coupled to the charge; a charge moving with uniform speed carry both the electric and the magnetic fields strongly attached to the charge in uniform motion. However, if the charge is accelerated, a part of the electric and magnetic fields which were strongly attached to the charge get detached and move out in space with the velocity of light. Thus an accelerated electric charge according to the classical theory of electromagnetism will radiate electromagnetic field and loses energy. Although no force of drag is faced by the electrons in the atom, but because electrons are assumed to have been moving in circular orbits, their direction of motion changing at each instant, they are in accelerated motion and must radiate energy. If so, the planetary atom which is based on classical physics is unstable from the same classical approach. It is estimated that all electrons of an average atom will spiral back into the nucleus within 10−8 s. Further, the dying atom should emit electromagnetic (EM) waves of all frequencies as electrons in different orbits will lose energy at different rates. In summery it may be said that planetary atom which was the only possible model of the atom is (i) unstable from the view point of classical physics and that (ii) while dyeing atom must emit EM radiations of all frequencies. Experimental facts, however, contradict both the above-mentioned predictions of the classical approach. Atoms are in general stable, and when they emit EM radiations, the radiations are not of all frequencies, and atoms on de-excitation emit discrete EM radiations of fixed energies. As a matter of fact that the emission spectra of atoms of each element of the periodic table is characteristic of the atom or the element, it is like the signature or the thumb impression of the atom/element. Several attempts, within the classical framework, have been made to explain discrete atomic spectra by making different assumptions about the motion of electrons in the atom, but none has been successful. Figure 4.20 shows a representative line spectrum of sodium atom. As may be observed in this figure, the spectrum has several lines in the ultraviolet region which

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Fig. 4.20 Emission spectra of sodium atom

lie in the invisible part of the EM spectrum; however, the two prominent lines called D1 and D2 lines of yellow colour (wave lengths 589.0 and 589.6 nm) are characteristic of sodium. It is the light from these lines which is used in sodium vapour lamps. There are few other lines in the sodium spectrum that lie in the region of infrared and are not shown in the figure. Sodium lamp Sodium vapour lamps are frequently used for street lighting and can be easily identified by their signature yellow light. Excited sodium atoms in visible region emit yellow light of two wavelengths 589.0 and 589.6 nm. Sodium lamp has the advantage that they are very efficient; almost 80% of the electrical energy given to the lamp is converted into visible yellow light. Further, the lumen output of the lamp does not drop with age and the light of yellow colour emitted by the lamp is the colour to which human eye is most sensitive. Problems associated with the classical planetary model of atom were satisfactorily addressed by the quantum mechanical model of Schrodinger which is discussed in Chap. 5 of the book. In quantum mechanical model it is shown that electrons in an atom are placed in different energy states defined by principal quantum number n, orbital quantum number l and magnetic quantum number m.

4.4.2 Photoelectric Effect The discovery of photoelectric effect is a story in itself. It is difficult to give the credit of discovering photoelectric effect to one person alone. First signatures of photoelectric effect appeared in an experiment carried out by German scientist Heinrich Rudolf Hertz in 1887 to identify electromagnetic waves. Electromagnetic waves, predicted by Maxwell in 1865, were a hot topic at that time. Hertz in an experiment tried to generate EM waves by producing a spark between two electrodes which were kept at a small distance from each other and were maintained at high potential difference. He observed that production of discharge becomes easier when the cathode was illuminated with ultraviolet light. He concluded that ultraviolet light falling on metallic cathode emits some radiations from the cathode that ionises the gas between the electrodes making it easier to conduct the spark. The next year, in 1888 another German scientist Wilhelm Hallwachs repeated Hertz experiment with a simple geometry. He

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took a clean circular plate of Zinc and mounted it on an insulating stand. The Zinc plate was attached by a conducting wire to a gold leaf electroscope. The electroscope was then given negative charger. In normal conditions the electroscope lost its negative charge very slowly. However, when the Zinc plate was illuminated with ultraviolet light, the electroscope lost its charge very fast. On the other hand if the electroscope was charged positively, there was no quick leakage of positive charge even when the Zinc plate was illuminated with ultraviolet light. The picture remained unclear till 1899 when Thomson conclusively proved that ultraviolet light falling on metallic cathode/Zinc plate causes electrons to be emitted from the target. The process of emission of electrons from metallic surfaces when illuminated with light is called photoelectric effect and the electrons the photoelectrons. Philipp Eduard Anton Lenard, who earlier worked as assistant to Hertz, studied in details the properties of electrons emitted from metallic bodies when illuminated by ultraviolet and other lights. A typical experimental arrangement to study the photoelectric effect is shown in Fig. 4.21. Here, a photosensitive plate C is placed opposite to the metallic plate A inside an evacuated glass tube. A potentiometer setup is used to apply desired value of potential difference across plates C and A. The evacuated glass tube has a quartz window through which EM radiations of desired frequency and intensity from source S may be made to fall on the photosensitive plate C. Quartz crystal is transparent to most parts of the EM spectrum, and therefore, quartz windows are used to allow electromagnetic radiations of wide wavelength range to enter the tube without any substantial absorption, and ordinary glass on the other hand absorbs most of the ultraviolet part of spectrum. Often, plate C is also called the cathode and plate A as anode as they are generally kept, respectively, at negative and positive potentials.

Fig. 4.21 Experimental setup to study the photoelectric effect

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The working of the experimental setup may be understood as follows. When light from source S, of a given frequency and intensity, is made to fall on the cathode plate C, photoelectrons with some kinetic energy (or speed) are emitted from the plate. Now if anode plate A is given a positive potential +V with respect to the cathode C, emitted electrons are attracted by plate A and are collected on it. Photoelectrons picked up by the anode plate A flow through the external circuit constituting the photoelectric current. The current, which may be in the range of few milli- to few microamperes depending on the frequency and the intensity of the incident EM radiations, may be recorded by the ammeter in the circuit. On the other hand if the polarity of the applied potential is reversed, i.e. plate C is given a positive potential with respect to plate A, photoelectrons emitted from plate C will be repelled by the potential at plate A, as a result the photoelectric current in the circuit will get reduced. Magnitude of potential V and its polarity may be easily controlled by the potentiometer and commutator combination shown in the figure. Under the reverse voltage condition when plate C is at positive potential and plate A is at negative potential, the magnitude of the current in external circuit will get reduced because of the repulsion of photoelectrons. Experiments using electromagnetic (light) radiations of different frequencies and intensities were carried out in which current through the external circuit was recorded for different magnitudes and polarities of the potential difference V between plates C and A. The main observations of these experiments were that the magnitude of the measured photoelectric current depends on the (i) material of surface emitting electrons, (ii) intensity of the radiations incident on plate C, (iii) potential difference V between the plates, only when plate A is at a negative potential with respect to C. These observations are point wise further elaborated in the following. (a) Dependence of photoelectric current on frequency of incident radiation In experiments where monochromatic EM radiations of different intensities and frequencies ν1 , ν2 , ν3 , . . ., etc. were made to illuminate the cathode C and photoelectric currents in the external circuit were recorded, it was found that (i) When plate A was at a positive potential +V with respect to plate C, the photoelectric current changed only with the intensity of the incident radiations and remained constant when EM radiations of different frequencies but of same intensities were incident on plate C, for all values of positive potential +V. Moreover, the photoelectric current was recorded immediately without any time lag at the instant the radiations hit the cathode. (ii) When EM radiations of same intensity but of different frequencies were incident on the cathode, as mentioned earlier, it was found that photoelectric current remained constant for all values of positive potential (+V ); however, the current becomes zero when the frequency of the incident radiation was reduced to some value ν 0 or below this value. The maximum frequency ν 0 at which no photoelectric current passes through the circuit is called the threshold or cutoff frequency. This indicates that no photoelectrons are emitted when EM radiations of cut-off frequency ν 0 or of frequency lower than this are made to

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shine the cathode, no matter what is the intensity or for how long the radiations are made to hit the cathode C. The magnitude of the threshold frequency ν 0 has been found to be different for different metallic cathode surfaces. (iii) In the case of reverse voltage setting, when plate A was kept at negative potential (−V ) with respect to plate C, it was observed that the photoelectric current for incident radiations of all intensities and frequencies decreased sharply with the increase in the magnitude of the negative potential of plate A, becoming zero for a the negative potential (−V s ). Negative potential (−V s ) where the photoelectric current (for all intensities and frequencies of incident radiations) becomes zero is called retarding potential or cut-off potential and has been found to have different values for different metallic surfaces used as cathode plate C. Further the cut-off potential does not depend on the intensity of the incident radiations. Let us now try to understand, within the framework of classical physics, the process of photoelectric effect. Classically, EM radiations like all other waves carry energy and are a mode of energy transfer. When EM radiations fall on a metallic plate, they deposit energy in the plate at a certain rate, rate being proportional to the intensity and the frequency of the radiations. Amount of energy deposited in the cathode plate will be proportional to the time for which the plate is exposed to radiations and also to the intensity and the frequency of the radiations. The cathode plate contains atoms of the metal which have electrons that are bound to the bulk material of the plate with some binding energy, say w. This w is called the work function of the metal and is equal to the amount of energy required to take an electron out of the metal surface. Obviously, the value of w depends on the metal used for cathode. According to classical physics, emission of photoelectrons from the cathode plate will happen only if the energy deposited by the incident radiations is at least equal or more than the work function w of the material. If classical picture of photoelectric effect is true, then there should be some time lag between the irradiation and recording of photoelectric current, particularly for very low-intensity and low-frequency incident EM radiations. Further, incident radiations of any frequency must be able to eject photoelectrons, and low-frequency radiations, which deposit energy at a lower rate, should be able to deposit the required energy w in a longer time of irradiation. Therefore, radiations of all frequencies must be able to produce photoelectrons and photoelectric current; the classical approach could not explain why radiations of frequency less than the threshold frequency could not produce photoelectric current. Photoelectric current is constituted by the number of photoelectrons collected per unit time at anode A. On the other hand, the rate at which photoelectrons are emitted from cathode plate will depend on the rate at which energy is deposited in cathode plate by incident radiations. If the intensity of incident radiations is high, more photoelectrons will be emitted per unit time, and hence there will be large photoelectric current (as observed experimentally). However, according to classical picture, the amount of energy deposited in cathode will increase with time, and therefore, the photoelectric current should not remain constant but should increase with time. This contradicts the experimental observations.

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Summing up it may be said that instantaneous emission of photoelectrons, occurrence of threshold frequency, dependence of photoelectric current only on the intensity of incident radiations are some major issues that could not be addressed by the classical approach. (b) Dependence of photoelectric current on the intensity of incident radiation In experiments where the intensity (I) of incident radiations of fixed frequency was made to hit cathode C and the voltage between plates C and A was kept constant, it was found that the current in the external circuit was directly proportional to the intensity of the incident radiations as shown by graph of Fig. 4.22. Let us analyse if linear relationship between the intensity of incident EM radiations and the photoelectric current may be explained by the classical approach. Classically, the only requirement for the emission of a photoelectron is that an amount of energy w (or larger than that) is deposited on the electron so that it may overcome its binding with the cathode. Since EM waves of higher frequencies deposit energy in the material of plate C at a faster rate, more photoelectrons should have been emitted by incident EM waves of higher frequencies. Since photoelectric current is proportional to the number of photoelectrons emitted per unit time from the plate; the photoelectric current should have been proportional to the frequency of the incident radiation as well as the intensity of the incident wave. In actual experiment no linear relationship between the frequency of the incident radiations and the magnitude of the photoelectric current has been observed. In the light of the above discussion, results of these experiments could not be explained on the basis of classical physics. (c) Dependence of photoelectric current on the potential difference across the two plates In some experiments, cathode plate C was illuminated with monoenergetic EM radiations of constant intensity and frequency ν (> ν0 ). The photoelectric current (I) in the external circuit was recorded for different settings of voltages between plates C and A. Graphs for two different values of intensities I 1 and I 2 (I 1 > I 2 ) of incident radiations of a given frequency are shown in Fig. 4.23. The potential difference is taken positive when plate A is at a higher potential than plate C. It may be observed in the figure that when potential difference is positive, the magnitude of current remains constant. In the case of reverse potential when A was negative with respect Fig. 4.22 Variation of photoelectric current with the intensity of the monoenergetic (fixed frequency) incident radiations

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to C, current decreases (for both intensities) and ultimately becomes zero at the same value of reverse (or retarding) potential −V s for all values of intensities of incident radiations. The above experimental observations may be explained by assuming that the incident EM radiations of frequency ν (> ν0 ) falling on the surface of plate C emit photoelectrons of all kinetic energies from zero up to a maximum value E max , where E max depends on the frequency of the incident radiations. Further, the number of photoelectrons emitted per unit time is proportional to the intensity of the incident radiations. In case when plate A is at a positive potential +V with respect to C, all photoelectrons (of all energies from 0 to E max ) emitted per unit time are collected by plate A, irrespective of the magnitude of potential V, and the photocurrent remains constant for all positive values of V. However in case of the reverse potential when plate A is at a negative potential −V n with respect to plate C, photoelectrons emitted from plate C get repelled by plate A; at small negative values of V n only some lowenergy photoelectrons are not able to reach A reducing the magnitude of photocurrent, but when negative potential increases, more energetic photoelectrons are also not able to reach plate A. Ultimately when V n attains the value −V s , even the most energetic electrons of energy E max are also cut off and photocurrent becomes zero. For V n = −V s , one may write: E max =

1 2 = eVs . m e vmax 2

(4.42)

Here m e , e and vmax are respectively the mass, charge and maximum velocity of emitted photoelectrons. The above experimental observation that photoelectrons emitted by a certain EM radiation of frequency ν will have a maximum kinetic energy is not supported by classical approach. According to classical approach, EM radiations will keep depositing energy in the target material, and therefore, energy available to photoelectrons in Fig. 4.23 Dependence of photoelectric current on the potential difference between plates C and A

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excess to their work function should also go on increasing with the length of time the EM wave is kept shining on the plate C. (d) Dependence of photoelectric current on the frequency of incident light and on the stopping potential Figure 4.24 shows the results of an experiment in which EM waves of three different frequencies ν1 , ν2 and ν3 with (v3 > ν2 > ν1 > v0 the threshold frequency) having same intensities were incident on plate C, one at a time, and the photoelectric current for both positive and negative voltage settings between plates C and A was recorded. Since the magnitude of the photocurrent for any positive voltage on plate A depends only on the intensity of the incident EM radiations, the straight-line curves for the three frequencies overlap on each other and are shown by a single straight line marked as saturation current. However, in case of retarding potential when plate A is at negative potential with respect to plate C, curves for EM radiations of three different frequencies cut the X-axis at different cut-off (or retarding) potentials v3 −V s1 , −V s2 and −V s3 . Since |V s3 | > |V s2 | > |V s1 |, the maximum kinetic energy E max of photoelectrons emitted by the EM wave of frequency ν 3 is highest. This shows that the maximum kinetic energy of photoelectrons depends on the frequency of the incident EM wave and increases with its frequency. It may be noted that retarding or cut-off potential, (−V s3 ), for example, is a v3 of photoelectrons emitted by EM measure of the maximum kinetic energy E max radiations of a given frequency ν 3 . If one draws a graph between the maximum kinetic energy of photoelectrons and the frequency of the incident EM waves for a given material (metal) on plate C, a straight-line graph marked M 1 in Fig. 4.25 is obtained. Similar graphs for other metals M 2 and M 3 are also parallel straight lines, but the straight lines for different target metals cut the x-axis at different points marked as ν0M1 , ν0M2 , ν0M3 , etc. It means that the threshold frequencies, the minimum

Fig. 4.24 Effect of variation of frequency of incident light on stopping potential

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Fig. 4.25 Graph showing the dependence of maximum kinetic energy of photoelectrons on the frequency of the incident EM wave for different target metals

frequency of the incident radiations below which no emission of photoelectrons takes place, have different values ν0M1 , ν0M2 , ν0M3 , etc. for different target metals. As already mentioned, classical theory of EM waves could not explain why waves with frequencies below the observed threshold frequency (ν 0 ) could not emit photoelectrons from the target metal. Classical approach also fails to explain why different metallic surfaces have different values of threshold frequencies. In the conclusion it may be said that classical electromagnetic theory put forward by Maxwell could not explain the following experimental observations on photoelectric effect: (a) no time lag between the emission of photoelectron and the striking of EM radiations on target metallic plate. (b) Independence of the maximum kinetic energy of photoelectrons from the intensity of impinging EM radiations. (c) The presence of a cut-off or threshold frequency, EM radiations of frequencies below the threshold frequency cannot emit photoelectrons no matter for how long the target material is irradiated by EM radiations. (e) Dependence of cut-off (threshold) frequency on the type of cathode surface Einstein’s quantum mechanical approach that explained and removed all the abovementioned drawbacks of classical theory will be discussed in the following.

4.4.3 Quantum Theory of Photoelectric Effect According to the classical picture of EM radiations put forward by Maxwell, radiations are electromagnetic waves, the energy contents of which are proportional to the square of the amplitude of the wave. However, this description of EM waves could not explain various experimental observations on the process of photoelectric effect as mentioned in the last section. Similar discrepancies were also encountered by Planck and others in explaining the energy distribution of EM radiations emitted by a blackbody. It was Planck who in 1900 made the bold proposition that energy is

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absorbed and emitted not continuously but in small energy packets which he called energy quanta. Einstein in 1905 proposed the quantum theory for photoelectric effect, borrowing the idea of Planck that energy is emitted or absorbed in energy packets or energy quanta. Einstein assumed that EM radiations are made up of tiny energy packets, called photons which move with the velocity of light in vacuum. The energy ν E pho of a photon of light of frequency ν is given by, ν E pho = hν

(4.43)

where h is Planck’s constant. Therefore, a beam of monochromatic light (EM radiations) of frequency ν may be considered to be a bundle of photons, each of energy hν, moving with the velocity c, the velocity of light in the medium. The intensity of the EM wave is proportional to the number of photons in the bundle. It may be mentioned that photon is not a material body or particle, in other words the rest mass of photon is zero,  and it existsEonly ν  pho ν ν = hν = . in motion and possesses energy (E pho = hν) and momentum ppho c c A photon when interacts with a material particle, like an electron, obeys the laws of conservation of energy and momentum. Photon aspect of EM wave is the counterpart of its wave aspect and is in confirmation with principle of duality. It may also be mentioned that the wave and the particle aspects of EM radiations do not show up simultaneously; when radiations show wave aspect (in interference and diffraction), it does not show photon or particle aspect, and similarly when radiations exhibit particle aspect (photoelectric effect, Compton scattering, etc.), they do not show wave aspect. Einstein explained the photoelectric effect assuming that when a metallic surface is illuminated with EM radiations of frequency ν, an incoming photon of EM radiation of energy hν hits an electron in the atom of the target metal. If the energy (hν) of the photon is more than the binding energy (w) of the electron with the bulk metal, the incident photon may be absorbed and the struck electron gets ejected from the atom as a photoelectron. The maximum kinetic energy E max of the emitted photoelectron will be very nearly equal to the difference (hν − w) of the photon energy and the electron binding energy in bulk metal. The electron binding energy with the bulk metal w is called the work function of the metal, which has different values for different metal surfaces. Figure 4.26 shows that a photon of an incident EM  radiations of frequency ν,  interacts with an electron of energy E pho (= hν) and linear momentum ppho = hν c the K-shell of the target atom. The photon imparts its total energy and the momentum to the target atom and vanishes. In case the energy deposited by the photon is more than the work function w of the electron, the electron may be ejected from the target metallic surface with some kinetic energy denoted by E elec and linear momentum pelec . To apply the laws of conservation of energy and momentum to the interaction, one may look to the interacting system before the interaction and after the interaction. There were two entities, the photon and the target atom, before interaction, and there are two entities, the photoelectron and the residual atom, after interaction. Energy

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and momentum need to be conserved between these entities. If it is assumed that the target atom is at rest, the linear momentum pumped by the photon must be shared between the photoelectron and the residual atom. As such, the residual atom must recoil in a particular direction to conserve the input linear momentum. Some energy, say E reco , is consumed in this recoil. It follows from the conservation of energy that:   E elec = E pho − w − E reco

(4.44)

Since E reco may have different values for different photoelectrons, the energy of emitted photoelectrons may differ from each other by the amount oi the recoil energy, which is very small. Another reason for the difference in the kinetic energies of photoelectrons is the depth of the atom (which has lost the photoelectron) from the front surface. When photoelectron is generated deep inside the metallic target, it may lose some of its kinetic energy in reaching the surface. Thus, differences in the value of recoil energies and in the energy loss while coming out of the metallic surface produce distribution in the kinetic energy of emitted photoelectrons. Maximum kinetic energy is possessed by the photoelectron which is produced just at the front surface of the target and for which recoil energy is a minimum. In Fig. 4.26 it is shown that the photoelectric effect is taking place with an electron of the K-shell. It is because the probability of photoelectric effect with electrons of inner shells, like the K- or the L-shells of the atom, is a maximum. The reason for this is the fact that in photoelectric effect conservation of linear momentum demands that the residual atom (left after the emission of photoelectron) must recoil. This may happen only when the emitted photoelectron was tightly bound with the atom and may easily transfer the excess linear momentum to the residual atom. Since K- and L-shell electrons are most tightly bound with the atom, photoelectric effect is more likely to take place with these electrons. Einstein’s quantum mechanical model of photoelectric effect may explain simultaneous emission of photoelectrons, without any time lag, with the irradiation of the target metal surface by EM radiations, if the energy of the incident photon is more than the work function w of the target metal. A rough estimate of the threshold frequency ν 0 may be made from the work function w of the target metal as:

Fig. 4.26 Emission of photoelectrons by the absorption of incident photon

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v0 ≈

w h

(4.45a)

Since different metallic surfaces have different values of the work function, the cut-off or threshold frequency has different value for different materials. The intensity of incident EM radiations, according to the quantum approach, is proportional to the number density (number per unit volume) of photons in the beam. The number of photoelectrons emitted per unit time is also proportional to the number density of photons in the incident bam, which in turn is proportional to the intensity of the beam. Therefore, the photoelectric current which is constituted by the emitted photoelectrons is also proportional to the incident beam intensity as shown in graph of Fig. 4.23. It may be remarked that the quantum approach to photoelectric effect given by Einstein has been able to explain all experimentally observed facts about photoelectric effect and could address the anomalies posed by the classical approach.

4.4.4 Work Function In general one talks about the ionisation energy of an atom, which is the energy required to take out an electron from the atom when it is in gaseous state. Situation changes when one considers atoms in a solid, particularly in case of metallic solids. The structure of a metallic solid may be described in terms of a positive ion lattice surrounded by a cloud of de-localised electrons. Since in metals, an electron of the cloud is not bound with an individual atom, the concept of ionisation energy is not applicable. Instead the concept of work function is used; work function (w) may be defined as the energy required in taking out one electron (of the electron cloud) to the surface of the bulk material. In general for metals the work function (w) is smaller than the ionisation energy (E ioniz ) of the corresponding atom. For example, in case of copper w = 3.76 eV and E ionz = 7.52 eV and for Silver w = 4.34 eV and E ionz = 8.68 eV.

4.4.5 Residual Atom after the Emission of Photoelectron The residual atom left after the emission of photoelectron is still excited and has an electron vacancy in one of the inner most shells, like in K- or L-shells. Electrons from higher shell, like M, N, …, etc., may fill the vacancy of the inner shell. This transfer of electron from the higher shell to the lower shell is accompanied with the emission of characteristic lines of the emission spectrum of the atom. For example, if photoelectron is ejected from K-shell, then an electron say from the M-shell may move to K-shell and quench the vacancy there, emitting K β X-rays of the atom. Sometimes it may happen that electron from the M-shell goes to K-shell, but no

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235

X-ray is emitted, instate excess energy (that might have gone out in the form of K β X-ray) is given directly to some outer shell electron (which is loosely bound) and that electron goes out of the atom. Electrons emitted in this way by the direct transfer of excess energy are called Auger electrons and the process Auger effect. Dependence of photoelectric effect on atomic number and energy of photon The probability of photoelectric effect depends on the energy of incident radiation and also on the atomic number Z of the target atom and may be represented by the empirical relation; pPhotoelectric ∝

Z 4.5 hν 7/2

(4.45b)

It follows from the above expression that the probability of photoelectric effect is more for atoms of higher atomic number Z (heavier materials) and decreases with the increasing energy of the incident photon. SAQ: Why the photoelectric effect is said to be a bound state phenomenon?

4.5 Blackbody Radiations and Their Energy Distribution Since in this section we will be dealing with thermal radiations, it will be appropriate to define thermal radiations. It is a common observation that metallic objects when heated emit electromagnetic radiations in the visible region and that the colour of the emitted radiations changes with the temperature of the body. In general, not only metals but all bodies emit EM radiations when they are at a temperature above the absolute zero. The emitted EM radiations contain waves of many frequencies (or wave lengths) the distribution of which depends both on the temperature and the material of the body. The electromagnetic radiations emitted on account of the temperature of any object are called thermal radiations. Thermal radiations emitted by a perfect blackbody are termed as blackbody radiations. Thermal radiations emitted by an object because of its temperature are quite different from the emission line spectra of atoms or band spectra emitted by excited molecules. The concept of blackbody and blackbody radiations in thermodynamics has originated from Kirchhoff’s law of thermal emission, given by German scientist Gustav Robert Kirchhoff in 1862. There are several ways to express Kirchhoff’s law of thermal emission. The original law which was in German may be translated in simple English as: ‘if there is a region of space surrounded on all sides by perfectly insulating boundaries so that no part of thermal radiations may leak through them and if each part of the boundary is at the same constant temperature T, then the space surrounded by the boundary is filled with thermal radiations which are characteristic of temperature T alone’. He named these radiations as blackbody radiations at temperature T and called the space surrounded by the boundary as the blackbody. The characteristics of blackbody radiations at temperature T will be same as that of

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thermal radiations emitted by a lump of lamp black held at constant temperature T, hence the name blackbody radiations. Needless to say that blackbody radiations are electromagnetic waves having a spectrum of frequencies or wavelengths distributed in definite proportions. The condition that each part of the boundary should be at the same constant temperature T ensures thermal equilibrium in the space bounded by the boundary or in the blackbody. Is it possible to make a real blackbody and to take out blackbody radiations from it for studying their properties? The answer is big NO, because once a part of the blackbody radiations are taken out of the cavity, thermal equilibrium will get disturbed. Wilhelm Wien (German scientist), however, suggested that thermal radiations very close (in character) to blackbody radiations may be obtained from a very small hole made in the wall of a large container of any material and of any shape, provided that the interior of the container (called thermal cavity) is kept at a constant temperature T (i.e. in thermal equilibrium). Since it is easy to make a source which may deliver thermal radiations very close in character to blackbody radiations, many and very precise studies of blackbody radiations at different temperatures have been made. A typical spectral distribution of energy density of blackbody radiations for three different absolute temperatures (4000, 5770 and 7000 K) are shown in Fig. 4.27. A close look of the observed spectral energy density distribution reveals that (i) at a given temperature, the energy is not distributed uniformly in the blackbody spectrum. (ii) As the temperature of the blackbody increases, the energy density of all wavelengths in the spectrum increases. (iii) The high frequency or short wavelength emission cutoff shifts towards left side, towards shorter wavelength. (iv) Also, the peak of the distribution plot, corresponding to the wavelength (or frequency) of maximum emission (λmax ), shifts towards the short wavelength side and gets a bit narrower. (v) The intensity of the radiation is found to change with the wavelength,

Fig. 4.27 Energy distribution of blackbody radiations at three different temperatures

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237

and it increases exponentially at a faster rate and decreases exponentially with a slower rate with the increase in wavelength.

4.5.1 Wien’s Displacement Law After careful analysis of blackbody thermal radiation spectra at many temperatures, Wien (1894–1896) put forward a relation between the wavelengths λmax and the absolute temperature T. Since the law was derived purely on the basis of experimental data of blackbody spectra without any theoretical backing, the law is empirical and is called Wien’s displacement law. Mathematically, the empirical Wien’s displacement law may be written in the following two forms: λmax · T = Constant = 2.899 × 10−3 m K

(4.46a)

νmax = Constant = 5.879 × 1010 Hz/K T

(4.46b)

Or

It may appear surprising to note that the second form of the law given by c in Eq. (4.46a). The Eq. (4.46b) cannot be obtained by substituting νmax = λmax reason is that the wavelength of maximum emission λmax is not a single wavelength, but wavelengths lying in the range λmax and (λmax + dλmax ) are all wavelengths of maximum emission. Similarly, frequency of maximum emission νmax is also not a single frequency, but all frequencies in the range νmax and (νmax + dνmax ) are all frequencies of maximum emission. Now, dλmax /= dνmax that means that the wavelength and the frequency do not change at the same rate; hence, the two constants are different. Wien also proposed a law that may give the energy distribution in blackbody spectrum. Unlike the displacement law, Wien derived his distribution law using the laws of thermodynamics and Maxwell Boltzmann distribution law for the speed of gas molecules. Essentially Wien used the concept of adiabatic compression of blackbody radiations contained in an enclosure to reach a higher temperature. In his derivation Wien made many approximations and his original derivation is quite involved and lengthy. Also, in the light of the quantum theory of radiations it is not of much relevance now. Skipping the derivation, Wien’s distribution law may be given as, E λ dλ =

C −D/λT e dλ. λ5

(4.47)

Here, E λ denotes the energy density contained in spectral range λ and (λ + dλ) of blackbody spectra. C and D are two constants their values for a given temperature T may be obtained by fitting the experimental spectrum at temperature T as such these

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constants are temperature dependent. Though Expression (4.47) was derived using the laws of classical thermodynamics and Maxwell Boltzmann distribution, but the values of constants C and D need to be determined from experimental data, and the expression is, therefore, semi-empirical. The total emissive power E of the blackbody at temperature T, which may be defined as the total energy radiated per unit time, may be obtained by integrating Eq. (4.47) between the limits λ = 0 to λ = ∞, i.e. ∫∞ E=

∫∞ E λ dλ =

0

0

C −D/λT e dλ = σ λ4 λ5

(4.48)

where σ is a constant at a given temperature T and depends on the values of constants C and D. Equation (4.48) is nothing but mathematical representation of well-known Stephan–Boltzmann law of thermodynamics. It may be observed that Stephan’s law that was quite well-established law could be derived from Wien’s distribution law. Similarly, it is possible to derive displacement law of Wien from his distribution law. Successful derivation of these two laws gave good support to Wien’s distribution law.

4.5.2 Failure of Wien’s Distribution Law Wien’s distribution law may explain the shape of observed blackbody spectrum at a given temperature, only qualitatively. The term λC5 e−D/λT of the distribution formula may be considered to have two parts (a) λC5 and (b) e−D/λT . For small values of λ exponential part (b) becomes large and over rides the effect of part (a); as a result for short wavelengths, the energy density rises almost exponentially. On longer wavelength side the exponential part (b) becomes very small and the fall in the energy goes almost as λ−5 . However several attempts to reproduce quantitatively the experimental energy distribution curve of blackbody radiations at a given temperature T for the whole range of wavelengths using Wien’s distribution formula failed; Wien’s distribution law reproduced the lower wavelength part of the experimental energy distribution but failed to reproduce the longer wavelength part of experimental distribution curve. Moreover, assuming a nonzero value for the wavelength λ, if temperature T is set to formula, it is observed that the total the value of infinity (∞) in Wien’s distribution ∫∞ energy emitted by the blackbody E = 0 E λ dλ at an infinite temperature remains finite. This is physically unjustified. In short, Wien’s distribution formula failed as it has two drawbacks: (i) could not reproduce the longer wavelength part of the experimental blackbody radiation distribution and (ii) it predicts a finite value of energy being radiated by a blackbody

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239

Fig. 4.28 Comparison of experimental blackbody spectra with predictions of Wien and Rayleigh–Jeans distributions

even at infinite temperature. Comparison of blackbody energy distributions obtained experimentally and predicted by Wien’s formula is shown in Fig. 4.28.

4.5.3 Rayleigh and Jean’s Distribution Law Strutt John William, Third Baron Rayleigh, a British mathematician better known as Lord Rayleigh and Sir James Jeans in 1905 proposed another energy distribution law known as Rayleigh–Jeans law to describe the energy distribution of blackbody radiations. They argued that a blackbody cavity in thermal equilibrium at temperature T (K) may be considered as if it is a cubical enclosure filled with standing EM waves of different frequencies. They assumed that the walls of the blackbody cavity contain some hypothetical oscillators that emit and absorb EM radiations of different frequencies; the waves emitted by a particular oscillator interfere with waves impinging on the oscillator producing standing waves. Putting the condition that standing waves must have nodes at container walls, calculated the number of modes of vibrations per unit volume dN in frequency range ν to (ν + dν) as: dN =

8π ν 2 dν. c3

(4.49)

In their derivation Rayleigh and Jeans assumed that the frequency ν of oscillators varies continuously. Next they calculated the average energy per mode of vibration E ave using the law of equipartition of energy of thermodynamics. The law says that each degree of freedom has energy 21 kB T (kB being Boltzmann constant), and since there could be two degrees of freedom: one corresponding to the kinetic energy and the other of potential energy, the average energy for each vibrating oscillator becomes E ave = kB T

(4.50)

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As such the average energy density of each oscillator of frequency in the range ν to (ν + dν) may be written as: E(ν) =

8π ν 2 8π ν 2 kB T E = ave c3 c3

(4.51)

Writing expression (4.51) in terms of wavelength, Rayleigh and Jeans obtained the following formula for energy density of blackbody radiations as: E λ dλ =

8π kB T dλ λ4

(4.52)

In Eq. (4.49) kB is Boltzmann constant and T the absolute temperature. It is worth noting that Rayleigh–Jeans law in comparison to Wien’s distribution law does not involve any new/unknown constants.

4.5.4 Failure of Rayleigh–Jeans Distribution According to Rayleigh–Jean’s formula, the energy density in blackbody radiations should always increase with the decrease in the value of the wavelength, which is contrary to the observed experimental spectra where the energy density decreases both for very short and for very large wavelengths. The big flaw of Rayleigh–Jean distribution formula is that for shorter wavelengths (high frequencies) the energy tends to become infinite. This is called ultraviolet catastrophe. It has been observed that Rayleigh–Jean distribution function can reproduce the long wavelength component (after the peak) of the experimental blackbody spectrum, while the Wien’s distribution formula may correctly reproduce the shorter wavelength component, below the peak, of the experimental spectra. Both of the distribution formulae derived on the basis of classical physics (thermodynamics and Maxwell Boltzmann distribution) fail to explain the blackbody spectrum in full. Figure 4.28 shows the experimental energy distribution of blackbody radiations along with predictions of Wien and Rayleigh–Jean.

4.6 Quantum Theory of Blackbody Radiations Karl Ernest Ludwig Marx Planck better known as Marx Planck in 1900 put forward the quantum theory for the energy density distribution of blackbody radiations. Like Rayleigh–Jean, he also assumed that the blackbody cavity is filled with electromagnetic waves which are continuously emitted and absorbed by some sort of oscillators giving rise to the formation of standing waves. Using the condition that these standing

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241

electromagnetic waves must have nodes at the boundaries of the cubical enclosure, like Rayleigh and Jeans, Planck also obtained the same expression for the average energy density of each oscillator of frequency in the range ν to (ν + dν) which may be written as: E(ν) =

8π ν 2 E ave c3

(4.53)

At this stage Planck made a drastic assumption that oscillator cannot have continuously variable energies; he said that oscillators may have only energies in integer multiples of the quantity hν, where h is Planck’s constant. This assumption means that there may be oscillators of energies, hν, 2hν, 3hν . . . nhν, where n is a positive integer. Oscillators with energies 21 hν or 34 hν etc. were not possible. Next he calculated the probability p(n) of the mode with energy E n = nhν in thermal equilibrium, using Boltzmann distribution law, p(n) =

  − kEnT

e

∑∞

B

n=0 e

  − kEnT

(4.54)

B

The average energy density for mode of frequency ν is therefore, E ave =

∞ 

E n p(n) =

n=0

∑∞ n=0

nhνe

∑∞ n=0

  − kEnT B

  − kEnT

e

(4.55)

B

In order to solve the above expression, we may substitute x = exp(−E/kT ). Thus, the denominator of the above expression becomes, n=∞ 

exp(−E n )/kT = 1 + x + x 2 + x 3 + · · ·

n=0

This is geometrical progression, and its addition will be given by: (1/(1 − x)) =

1 (1 − exp(−E/kT ))

(4.56)

Also, n=∞ 

  n E exp(−n E/kT ) = E x + 2x 2 + 3x 3 + 4x 4 + · · ·

n=0

xE

 d 1 + x + x2 + x3 + · · · dx

(4.57)

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d xE dx



1 1−x



E x exp kT Ex = =  E 2 (1 − x)2 1 − exp kT

(4.58)

So the average energy of oscillators: E ave =

E E exp(−E/kT ) = {1 − exp(−E/kT )} {exp(E/kT ) − 1}

(4.59)

This may be simplified into, hν . E ave =  hν e kB T − 1

(4.60)

Substituting this value of average energy in Eq. (4.53), one gets, E(ν) =

1 8π ν 2 8π hv3  hν  E = ave c3 c3 e kB T − 1

(4.61)

This is the Planck distribution function which reproduces the energy density distribution of blackbody spectrum for all frequency or wavelength regions. At the time Planck proposed his radical hypothesis, many scientists could not believe mainly because Planck could not explain why the energies should be quantized. Initially, his hypothesis explained only the experimental data on blackbody radiation. It was mentioned that if quantization was observed for a large number of different phenomena, then quantization would become a law. It was also remarked that one needs to develop a theory that might explain that law. As things worked out, Planck’s hypothesis was the starting point from which the modern physics grew and developed. Planck’s theory of blackbody radiations assumes that electromagnetic radiations in the blackbody enclosure may have only discrete energies and the oscillators could only lose or gain energy in the form of packets, referred to as quanta, of size hν, for a given oscillator of frequency ν. The energy quanta of electromagnetic radiations are called photons.

4.7 Compton Scattering of Gamma Rays Electromagnetic radiations are produced when charged bodies are either accelerated or decelerated (for example, the emission of continuous X-rays) and also when electrons shift from one shell of the atom to another shell (example, characteristic line spectra). In the case of the atomic line spectra, the energy of emitted photons

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depends on the difference of energy of atomic levels or shells. Photons emitted in atomic transitions may have energies of few ten of keV only. Atomic nucleus contains neutrons and protons which are also distributed in energy levels, as electrons are in an atom. Transitions of neutrons or of protons from a level of higher energy to a level of lower energy in the nucleus of an atom give rise to electromagnetic radiations of high energies, in the range of MeV. The electromagnetic radiations that have their origin in the nucleus are called gamma rays and are often denoted/represented by symbol γ . Gamma ray energies may lie in the range of few keV to few MeV. Gamma rays when incident on an atom are very unlikely to produce photoelectric effect, as gamma ray photons have considerable energy and probability of photoelectric effect decreases sharply with the energy of incident photon. Moreover, the photoelectric effect could be explained only assuming a particle nature (photon) of light. American physicist Arthur Holly Compton at Washington University in 1923 discovered a phenomena in which high-energy EM radiations when incident on an atom ejected an electron from the atom (like photoelectric effect) and a lower energy EM radiation (unlike photoelectric effect) also appeared in the output. This process was different from photoelectric effect as in photoelectric effect the incident photon is lost and no photon appears in the output; there are photoelectron and recoiling residual atom. Further, EM radiations’ characteristic of the incident atom is also emitted following photoelectric effect. The process, in which high-energy incident photon ejects an electron from the target atom and itself is scattered with reduced energy, is called Compton scattering. Compton scattering is initiated only by high-energy photons, like that of high-energy X-rays and gamma rays. It was not possible to explain Compton scattering on the basis of classical picture of electromagnetic radiations which assumes EM radiations as waves. Only the particle nature or quantum aspect of EM radiations can explain the Compton scattering. In quantum mechanical framework, a gamma ray is treated as quanta of energy which behaves as a particle. Compton scattering, in quantum approach, may be treated as the inelastic scattering of incident gamma ray photon with the free and stationary electron in the target atom. Since the energy E (= hν) of the incident gamma ray photon is much larger (≈ few hundred keV) than the binding energy Be (≈ few eV) of the outer electrons of the target atom, it is reasonable to assume that the outer shell electron of an atom is free (unbound) and stationary. Pictorial representation of inelastic scattering of high-energy photon by the stationary and free electron is shown in Fig. 4.29. As shown in the figure, a photon of wavelength λi impinges on the stationary and free electron and kicks it out in a direction making e . an angle φ with the incident direction with kinetic energy E kin The incident photon gets scattered in a direction making angle θ with direction of incidence, with reduced energy having a longer wavelength λf . One may apply the laws of conservation of energy and linear momentum to the inelastic collision between the photon and the stationary electron. However, one point must be kept in mind while applying the law of conservation of energy it is that photo always move with the velocity of light and that the ejected electron may be given high velocity

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Fig. 4.29 Pictorial representation of Compton scattering

where relativistic variation of mass may become significant, therefore, one must use m e c2 for the rest mass / energy of electron and the energy of electron after collision   e m 2e c4 + pe2 c2 ; here m e is the rest mass of the electron, pe E kin may be given as the final linear momentum of electron and c the velocity of light. Energy conservation hνi + m e c2 = hνf +

/  m 2e c4 + pe2 c2

(4.62)

Conservation of X-component of linear momentum hνi hνf = pe cos φ + cos θ c c

(4.63)

Conservation of the Y-component of linear momentum 0 = pe sin φ +

hνf sin θ c

(4.64)

Equations (4.62), (4.63) and (4.64) may be solved to get Compton equation. λf − λi = Δλ =

h (1 − cos θ ) mec

(4.65)

Compton equation (4.65) tells that Compton shift in the wavelength Δλ can have a minimum value of zero, when incident photon passes on along the incident direction without getting scattered by electron, and the magnitude of wavelength shift increases with the angle of scattering θ, attaining a maximum value (2h/me c) for backscattering (θ = 180°) of photon. In his original experiments, Compton bombarded carbon target with high-energy X-ray photons and recorded the scattered photon of lower energy. Compton could explain the experimental data assuming the particle nature of photon and inelastic

4.7 Compton Scattering of Gamma Rays

245

scattering of incident photon by stationary electron. It was the time when the particle aspect of photon suggested by photoelectric effect was still being debated, Compton’s analysis of his experiments gave a clear and independent evidence of particle-like behaviour of electromagnetic radiations.

4.7.1 Compton Wavelength The quantity mhe c is called the Compton wavelength of electron. In general the Compton wavelength of a particle of rest mass m0 is given as mh0 c . Physical significance of Compton wavelength may be derived from the de Broglie wavelength associated with a particle, according to which a particle of rest mass m0 moving with velocity v has an associated de Broglie wavelength λde-bro = mh0 v . Since the velocity of a moving particle cannot exceed the velocity of light c, the minimum value of the de Broglie wavelength may occur when the velocity of the particle is taken as c. Putting v = c, gives the de Broglie wavelength as mh0 c , which is the Compton wavelength of the particle. Since wavelength associated with a particle is a measure of the uncertainty in the position of the particle, a particle cannot be confined in a space smaller or equal to its Compton wavelength. For example, Compton wavelength of 6.626×10−34 J s −12 m is around 2.4 × 10−12 m, electron = mhe c = 9.1×10 −31 kg×3.0×108 m = 2.427 × 10 and therefore, an electron cannot be confined in a space equal or shorter than this. Since the size (radius) of an average nucleus is of the order of 10−14 m, two order of magnitudes is smaller than the Compton wavelength of electron: electron cannot be confined within the nucleus and cannot be a constituent of nucleus.

4.7.2 Compton Scattering by the Whole Atom In some experiments Compton recorded incident photons scattered by large angle without appreciable change in wavelength. Also in such events no electron was detected. Compton explained such events by assuming that the incident photon is scattered not by the electron of the atom, rather it is scattered by the atom as a whole. The change in wavelength Δλ in such scattering by atom may be written as: λf − λi = Δλ =

h (1 − cos θ ) Mato c

M ato in the above expression is the mass of the atom as a whole, and Mhato c is Compton wavelength of the atom. Since mass of the atom is very large, its Compton wavelength is very small; hence, change in wavelength is undetectable. Further, the target atom remains intact; no electron is ejected by the incident photon.

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4.7.3 Photon Interactions with Matter Photons, depending on their energy and the atomic number of the target atom, may interact with the atom, with the bound and the free electrons of the atom and with the nucleus of the target atom. Low-energy photons mostly interact with the bound electrons (K-shell or L-shell electrons) producing photoelectric effect. With the increase of energy the probability of photon interaction with loosely bound electrons of the atom increases, resulting in Compton scattering. In case the energy of the photon is larger than 1.02 MeV, it may annihilate producing an electron and positron pair: the process called pair production. The minimum energy of photon required to produce an electron positron pair is 1.02 MeV which is the sum of the rest mass energies of the electron and positron pair (0.51 + 0.51 = 1.02). Pair production takes place in the field of a nucleus, when a high-energy photon (E phot > 1.02 MeV) passes through the nuclear field. Nuclear field facilitates the recoil of the nucleus that is required for the conservation of momentum in pair production process. High-energy photons may also excite or disintegrate atomic nucleus. Very high-energy photons, with energies > 150 MeV, may create mesons. Figure 4.30 shows the variation with photon energy of the probability for photoelectric effect, Compton scattering and pair production in lead (Pb). Kink (towards the top) in the curve for photoelectric effect, called k-edge, shows that probability for photoelectric effect suddenly increases for photon energy corresponding to the binding energy of the K-shell electrons. Similar but less pronounced edges (not shown in the figure) also appear for L and M-shells. SAQ: What may be the order of magnitude for Compton wavelength of a neutron? Fig. 4.30 Probability of photoelectric effect, Compton scattering and pair production as a function of photon energy in lead (pb)

4.8 Specific Heat of Solids

247

4.7.4 Some Applications of Compton Scattering For the explanation and recording of the Compton effect, Compton was awarded a share of the Nobel Prize in physics in the year 1927. Not only the Compton effect represented the particle nature of light, but it is also important from the application point of view. The Compton scattering is of importance in material science where it is being used to get information regarding wavefunction of electrons in matter. It is also of importance in the field of radiobiology and radiation therapy. Compton scattering also has applications in X-ray astronomy and in getting signature of black hole. SAQ: In nature there are no free electrons, then why the Compton scattering is said to take place with free electrons?

4.8 Specific Heat of Solids Specific heat is defined as the amount of heat required to change the temperature of unit mass of a substance by unit degree temperature. If ‘m’ kg of a substance is given a heat energy of amount ΔQ which rises the temperature of the substance by Δθ, then the specific heat of the substance is Specific heat =

ΔQ 1 ΔQ and the heat capacity = m Δθ Δθ

Molar or atomic specific heat It is defined as the quantity of heat energy required to raise the temperature of 1 kg mol or 1 kg atom of any substance by unit degree. It is obvious that molecular or atomic specific heats are the products of the molecular weight or the atomic weight with the specific heat of the substance. Molecular or atomic specific heat of solids is generally denoted by C v . [In case of gases there may be two types of molecular/atomic specific heats: at constant volume C v and at constant pressure C p .] In the following discussion the term atomic specific heat will be used which will also mean molar specific heat in case the solid is a compound and not an element.

4.8.1 Dulong–Petit Law French chemist Pierre Louis Dulong and French physicist Alexis Therese Petit in 1819 on the basis of their observation of atomic specific heat for large number of solids gave an empirical law which states that ‘gram-atomic heat capacity (atomic specific heat) of an element is a constant: that is, it is same for all solid elements, about 6 cal per g atom per °C and it is independent of temperature’. More than 60 elements

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in solid state are found to have their atomic specific heat in the range of 5.38–6.93 cal/ g atom/°C with an average of 6.15. However, the gram-atomic specific heat for some light elements like Silicon (gram-atomic specific heat 4.95) and diamond with gramatomic specific heat of 1.46 cal/g atom/°C does not follow Dulong–Petit law. Further, it is found that the gram-atomic specific heat of solids depends on temperature and approaches zero at absolute zero of temperature. The MKS unit for atomic specific heat is J/kg atom/K and 1 J/kg atom/K = 4.2 × 103 cal/kg atom/K. Therefore, the average value of atomic specific heat of 6.16 cal/ g atom/°C observed by Dulong–Petit is equal to 25.67 × 103 J/kg atom/K.

4.8.2 Obtaining Dulong–Petit Law on the Basis of Classical Physics Dulong–Petit law may be derived assuming that the classical law of equipartition of energy of thermodynamics holds good. At absolute zero solids have a crystalline structure in which atoms or molecules of the solid are at rest being held at their place by mutual interaction. When energy in the form of heat is supplied to the solid, the atoms or the molecules start vibrating around their mean position. If the temperature is not very high, the vibratory motion has six degrees of freedom: three of translatory motion (associated with kinetic energy) and three of vibrational motion (associated with potential energy). Now, according to the law of equipartition of energy, 21 kB T of energy is associated with each degree of freedom, and hence, the total energy u associated with each atom (or molecule) at temperature T (K) is u = 6 × 21 kB T = 3kB T (J). If AV denotes the Avogadro number, that is the number of atoms/molecules in one kilo atom (or mole) of the solid, then the energy possessed by 1 kilo atom of the substance is U = AV u = 3kB AV T

(4.66)

Here kB is Boltzmann constant (= 1.380 × 10−23 J/K) and Avogadro’s number AV = 6.03 × 1026 atoms per kg But kB AV = R, where R is the gas constant having the value R = 8.4 × 103 J/kg atom/K. Substituting kB AV = R in Eq. (4.66), one gets U = 3RT

(4.67)

And the atomic (or molar) specific heat CV is given as, CV =

dU = 3R dT

(4.68)

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249

Equation (4.68) says that the atomic or molecular specific heat for all solids has a fixed value of 3R (≈ 6.8 cal/g atom/°C) and is independent of temperature. This confirms Dulong–Petit law.

4.8.3 Problems with Dulong–Petit Law According to Dulong–Petit law (supported by classical physics) atomic specific heat should be same for all solids and be independent of temperature. Experimental measurements do not support both the above predictions. Experiments indicate that atomic specific heat for metallic solids has a value near to 3R and also changes slowly with temperature, but for non-metallic solids the magnitude of atomic specific heat was quite away from 3R and that it also increases with the increase in temperature approaching the Dulong–Petit value. However, the biggest challenge to Dulong– Petit law comes from the variation of experimentally measured atomic specific heats at lower temperatures (see Fig. 4.31); it was observed that atomic specific heat of solids decreases rapidly with the decrease in temperature approaching a value zero at absolute zero of temperature. As such Dulong–Petit law, backed by classical theory, fails to explain the dependence of atomic specific heat of solids on temperature and its sharp fall approaching to zero at absolute zero.

4.9 Quantum Approach to Atomic Specific Heat of Solids Initially Einstein in 1905 used the concept that a solid contains quantum harmonic oscillators all having same energy to derive atomic specific heat of solids. Einstein’s formulation correctly predicted the temperature dependence of atomic specific heat, Fig. 4.31 Temperature dependence of atomic specific heat for some elements in solid state

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but quantitative agreement with experimental values was poor. Later, in 1912 Peter Debye modified the concept of atomic oscillators all of same energy and included oscillators with different values of quantized energies. Debye’s theory correctly predicted both the temperature dependence and the magnitudes of atomic specific heats of solids. Essentials of both Einstein and Debye theories are discussed in the following.

4.9.1 Einstein’s Theory for Specific Heat of Solids Einstein made following assumptions for absorption of heat energy by solids: (a) Atoms in a solid at absolute zero are stationary at their equilibrium position under the mutual interaction between different atoms. As such each atom at absolute zero has zero or no energy. (b) When heat energy is given to a solid, its atoms start vibrating with a characteristic frequency ν, and characteristic frequency has a different value for each solid. Each atom of a given solid vibrates with the same frequency ν. (c) Each atom of a solid has three degrees of freedom, like the molecule of a perfect gas. (d) Each vibrating atom of the solid behaves like a Planck’s oscillator, its energy is quantized, i.e. the energy of each vibrating atom is an integral multiple of hν. Each atom has same value of energy which is an integer multiple of hν. Under these assumptions, the mean energy per degree of freedom is not 21 kB T as in classical law of equipartition of energy, instead the mean energy per degree of freedom ∈ is given by: hν  ∈ =  hν e kB T − 1

(4.69)

Above expression for average energy of an oscillator has been derived earlier (see Eq. 4.60). Since each atom has three degrees of freedom, energy associated with each atom u becomes 3hν  u =  hν e kB T − 1

(4.70)

Also, one kg atom of a solid contains AV number (Avogadro’s number) of atoms, and energy U of 1 kg atom of solid is 3AV hν  U = AV u =  hν e kB T − 1

(4.71)

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251

And the atomic specific heat Cv is given by: 



hν   hν 2 e kB T dU = 3AV kB Cv =  hν 2 dT kB T e kB T − 1



hν = 3R kB T



2

e

hν kB T



 hν 2 e kB T − 1

(4.72)

  has the dimensions of temperature and is called Einstein’s The quantity hν kB temperature which is denoted by θE . Einstein’s temperature θE has a different value for each solid. (4.72) may be written in terms of Einstein’s temperature by  Equation  hν substituting kB = θE to get, 

θE Cv = 3R T

2



e

θE T



 θ 2 E eT −1

(4.73)

Expression given by Eq. (4.73) is called Einstein’s specific heat equation or relation. Experimental values of atomic specific heat for a given solid at different values of absolute temperature T are fitted in Eq. (4.73) to obtain the best fit value of Einstein’s temperature θE from which the characteristic frequency ν for the solid may be obtained; this value of the frequency is denoted by νE and is called Einstein’s frequency of the solid.

4.9.2 Investigating the Temperature Dependence of Einstein’s Equation The temperature dependence of Einstein’s equation may be investigated in two limits: when (a) θTE « 1 or (b) θTE ≫ 1. θE

(a) At high temperatures T ≫ θE , θTE « 1; and term e T ≈ 1.    2 θE And e T ≈ 1 + θTE + 2!1 θTE + · · ·  2   ≈ 1 + θTE neglecting higher order terms θTE , and  θ    E therefore, e T − 1 ≈ θTE .  θ    θE E Substituting e T ≈ 1 and e T − 1 ≈ θTE in Eq. (4.73), one gets,

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θE Cv = 3R T



2

e

θE T





 θ E eT −1

θE 2 = 3R T

2

1  θE 2 = 3R T

It may be observed that for temperatures T much higher than Einstein temperature θE , atomic specific heat of solids approaches the Dulong–Petit value 3R.   (b) For the case of low temperatures when θTE ≫ 1  e

θE T

   θE θE θE T T ≫1 −1 ≈e as e ≫ 1 if T

Equation (4.73) in this case reduces to, 

θE Cv = 3R T 

θE ≈ 3R T



2

2

e

θE T





 θ E eT −1 1



e

θE T

θE 2 ≈ 3R T

2



θE



e T  θ 2 E eT



Or  Cv = 3R

θE T

2

1



e

θE T



(4.74)

 2 In Eq. (4.74) on the right-hand side, there are two factors θTE and  1θE  . e T  θE 2 Factor T increases with the decrease of temperature T; however, the other factor  1θE  decreases exponentially with the decrease of temperature T. Rate of e

T

decrease with temperature of the second factor is much faster as compared to the rate of increase (with the decrease of temperature) of the first factor. As a result the second factor becomes zero for very low temperatures earlier than the first factor becomes infinite, hence, the atomic specific heat of solids approaches zero at absolute temperature T approaches zero.

4.9.3 Drawbacks of Einstein’s Model Though Einstein’s theory for specific heat of solids predicts that the atomic specific heat for all solids should approach Dulong–Petit value of 3R at high temperatures and it should approach zero at 0 K temperature, but it could not reproduce the

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253

experimental values of specific heats for most of the solids. This theory suffers from the following drawbacks: (i) Does not reproduce the experimental values of specific heats for most solids. (ii) Einstein temperature ϑE and Einstein frequency νE have no physical justification; they could not be associated with any property of the solid, like its elastic constants or melting point, etc.

4.9.4 Debye Theory of Atomic Specific Heat Einstein in his theory for specific heat of solids assumed that on receiving heat energy, each atom of the solid vibrates with the same frequency which is quantized. Debye, on the other hand, assumed that on heating, the solid as a whole, i.e. the crystal lattices in the solid, undergoes collective vibrations. Lattice vibrations are assumed to be quantized. The quanta that represent lattice vibration are called PHONON. Phonon is the counter part of photon which is the quanta of EM waves. In case of the blackbody radiations it was assumed that the blackbody cavity is filled with photons of different quantized frequencies; similarly, Debye assumed that on heating a solid it gets filled with phonons of different frequencies that have quantized energies. Often, it is said that a solid at some temperature above absolute zero is filled with a phonon gas. In case of blackbody radiations it was assumed that photons of all frequencies are present in the blackbody cavity. However, in case of vibrations in a solids phonons of all frequencies are not present, and phonon frequency is bound by the medium of its propagation which is the atomic lattice of the solid. So there is an upper limit on the frequency of phonon in the solid that depends on the elastic constants and the crystal structure of the material. Debye also assumed that phonon waves, that are elastic waves, travel with some finite speed in the solid medium like sound waves. On account of their interference, standing phonon waves are formed in the solid. There may be three types of standing waves in the solid, longitudinal waves of velocity C L and two types of transverse waves with two different states of polarisations with speed C T . Before proceeding further, let us point out the basic difference in Classical theory, Einstein’s theory and Debye theory of specific heat of solids. (i) Both the classical (Dulong–Petit) and Einstein’s models treat each atom of the solid independently. (ii) Debye model is more realistic, since it considers collective vibrations of many atoms. As a matter of fact if one atom of the solid vibrates, the neighbouring atoms are also set in vibratory motion. (iii) Einstein’s model assumes that all atoms vibrate with the same frequency. Debye model, on the other hand, considers the collective vibrations of different groups of atoms. A group with large number of atoms can vibrate with lower frequency while the group with fewer atoms may vibrate with higher frequency. Therefore, the solid will contain vibrating groups of atoms with different frequencies, and

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the frequencies are being quantized. Vibrating groups of atoms fill the solid medium with elastic waves (sound waves) which is being confined within the boundaries of the solid get reflected at boundaries and form standing waves. The wavelength of an elastic wave depends on the number of atoms in the vibrating group. The minimum wavelength λmin will correspond to the vibration of only few atoms and will be related to the lattice constant of the crystalline structure. It is obvious that the group of atoms that vibrates with minimum wavelength will have largest value of the frequency of vibrations νmax . Further, if υ is the speed of the elastic wave in the medium, then υ = λmin νmax . Here νmax is the frequency of the wave that has the minimum wavelength. It follows from here that for a given material there will be standing waves of several quantised frequencies up to the maximum frequency νmax . Also, the νmax will depend both on the crystal structure (lattice constants) and the speed of the elastic waves in the medium. It is known that that elastic waves may be of two types: longitudinal that may travel with some speed say C 1 and transverse that may travel with a different speed, say C 2 in the medium. Further, transverse waves may have two different states of polarisations; therefore, there may be three different types of elastic waves of each frequency ν. (iv) In Einstein model the energy E of the system is given by E(Eins) = Average energy associated with each mode × number of atoms. In Debye model system energy is given as E(Debye) =



Average energy associated with each mode of frequency ν

ν

× number of modes of frequency ν

(4.75)

In Einstein model the number of modes is taken equal to the number of atom in a kilomole (= Avogadro number AV ). Modes essentially mean the number of standing waves in the volume of the solid. Since the number of standing waves is quite large, one calculates the number of standing waves g(ν)dν in a small frequency interval ν and (ν + dε) and integrates it from zero to νmax to get the total numbers of modes. It can be shown that the number of modes g(ν)dν for frequency range ν and (ν + dν) for kg atom of the solid is given as: g(ν)dν =

9AV ν 2 3 νmax

(4.76)

Further, there are large number of phonons with different energies, and the average energy Eave associated with each mode of vibration may be calculated using quantum mechanical Maxwell Boltzmann statistics and is given as,

4.9 Quantum Approach to Atomic Specific Heat of Solids

hν

Eave =  e

hν kB T

−1

255



(4.77)

The system energy E ν corresponding to phonon of frequency ν may be calculated by putting the values of different factors from Eqs. (4.76) and (4.77) in Eq. (4.75) to get ⎛

⎞ hν

⎠ E ν = (Eν )(g(ν)dν) = ⎝  hν e kB T − 1



9AV ν 2 3 νmax

 (4.78)

The total energy of the system may be obtained by integrating the above expression over frequency ν ∫νmax E= 0

⎛ ⎞  ∫νmax ∫νmax 2 hν 9A 9A hν 3 ν V V ⎝  hν  hν ⎠  dν dν = 3 E ν dν = 3 νmax νmax kB T kB T e e − 1 − 1 0 0 (4.79) ⎤

⎡ ⎢ 9A ∫ νmax V d⎢ 3 0 ⎣ νmax

But CV = Or CV =

dE dT 9AV 3 νmax

= ∫ νmax 0

dT



hν 3

∂

⎥ 3 ⎞ dν ⎥ ⎛ hν hν ⎦ k T ⎝e B −1⎠

hν e kB T

−1

−1

=

∂T

=

9AV 3 νmax

9AV 3 νmax

∫ νmax

∫ νmax 0

0

hν 3

−1  hν ∂ e kB T −1 ∂T

dν

hν

hν 3

hνe kB T  hν 2 dν. 2 kB T e kB T −1

Or ⎤ ⎡ hν ∫νmax 4 kB T 9AV h ⎥ ⎢ ν e CV = 3 ⎣  hν 2 ⎦dν νmax kB T 2 e kB T − 1 0 2

(4.80)

Let us make the following substitutions in Eq. (4.80): h hνmax hνmax hν = θD (Debye temperature) and x = ; dx = dν, xmax = kB T kB T kB T kB T Therefore, 9AV h 2 CV = 3 νmax kB T 2

 ∫xmax  kB T 4 x  x e kB T h dx h (ex − 1)2 0

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9AV T 3 kB4 = 3 νmax h 3

∫xmax 0

  ∫xmax 9AV kB T 3 x 4 ex x 4 ex dx =  dx 3 (ex − 1)2 (ex − 1)2 hνmax kB

0

Or  CV = 9AV kB

T θD

3 ∫xmax 0

 x 4 ex dx (ex − 1)2

(4.81)

Temperature dependence of the atomic (or molar) specific heat of solids in case of Debye theory may be discussed through expression (4.81). The value of C V for high temperatures T >> θ D and for the case of low temperatures T