Fundamentals of Inorganic Chemistry [3]


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Table of contents :
Cover Page
Half Title Page
Title Page
Copyright
Foreword
Dedication
Preface to the Second Edition
Preface to the First Edition
Contents
Volume 3
12. Bonding in Metals and Metal Clusters: Electrical Conductivities of Solids: Semiconductors and Superconductors
12.1 General Properties of Metals
12.2 Crystal Structure of the Metals and the Effects of Lattice Structure on their Properties
12.3 The Free Electron Theory of Metallic Bonding
12.4 The Bloch Theory (i.e. Zone Theory) of Metallic Bonding
12.5 Molecular Orbital Theory or Band The,ory of Metallic Bonding
12.6 Valence Bond Approach of Metallic Bonding
12.7 Solid Solutions : Alloys : Intermetallic Compounds
12.8 Metal-Metal Bonds in Metal Clusters
12.9 Semiconductors and their Properties in the Light of Band Theory
12.10 Semiconductivity and Metallic Conductivity of Metal Oxides: Hopping Mechanism of Conductivity
12.11 Electrical Conductivity in One Dimensional Solids: Peierls Distortion
12.12 Conducting Organic Substances: Organic Metals, Organic Semiconductors, Molecular Inorganic Superconductors
12.13 Superconductivity
12.14 Electrical Conductivity of Solid Electrolytes
Solved Numerical Problems
Exercise-XII
13. Hydrogen Bonding and Other Weaker Chemical Forces Including Supramolecular Systems
13.1 Hydrogen Bonding
13.2 Hydrates and Clathrates or Cage Compounds
13.3 The Nature of Intermolecular Forces: van der Waals Forces: Lennard-Jones Potential: The 6-12 Potential
13.4 Interaction Between Ionic and Covalent Compounds
13.5 Noncovalent Interactions in Supramolecular Systems and Molecular Recognition
13.6 Energetics of Dissolution
13.7 Energetics of Phase Transitions - Melting, Boiling, Sublimation
Exercise-XIII
14. Acids and Bases and Ionic EquiUbria in Aqueous Solutions
14.1 Definitions of Acids and Bases
14.2 Arrhenius Theory of Acids and Bases
14.3 Bronsted-Lowry Concept: The Proton Donor-Acceptor Concept: Protonic Concept: The Conjugate Acid-Base Theory
14.4 Mechanism of Buffer Action: Buffer Capacity
14.5 Selection of Acid-Base Indicators in Acid-Base Titrations
14.6 Hammett Acidity Function: Superacids
14.7 The Lewis Concept: The Electron Pair Donor-Acceptor System
14.8 Strength of Bronsted (i.e. Protonic) Acids and the Governing Factors Involved and the Properties of Bronsted Acids and Bases
14.9 Some Important Factors Governing the Acid-Base Strength (both Bronsted and Lewis Acid-Base Systems)
14.10 Lux-Flood Concept
14.11 Solvent System Definition
14.12 Usanovich Concept
14.13 A Generalised Concept of Acids and Bases
14.14 Concept of Ultimate Acids and Bases
14.15 Hard and Soft Acids and Bases (HSAB)
14.16 Surface Acids and Solid Acids: Heterogenous Acid-Base Reactions
14.17 Aqueous Solutions of Salts: Hydrolysis of Salts: Solubility Product of Sparingly Soluble Salts
Solved Numerical Problems
Exercise-XIV
15. Nonaqueous Solvents
15.1 Introduction
15.2 Classification of Solvents
15.3 Characteristics of Ionising Solvents
15.4 Liquid Ammonia as a Solvent
15.5 Liquid Hydrogen Fluoride as a Solvent
15.6 Liquid Hydrogen Cyanide as a Solvent
15.7 Acetic Acid (CH3CO2H) as a Solvent
15.8 Sulfuric Acid as a Solvent
15.9 Fluorosulfonic Acid (HSO3F) as a Solvent and Superacids
15.10 Liquid Sulfur Dioxide as a Solvent
15.11 Bromine Trifluoride as a Solvent
15.12 Oxyhalides as Solvents
15.13. Molten Salts (i.e. Ionic Liquids) as Solvents
Exercise-XV
16. Redox Systems and Electrode Potential: Application of Electrode Potentials: Electroanalytical Techniques
16.1 Some Preliminary Aspects of Redox Reactions
16.2 Electrode Potentials: Standard Potentials
16.3 Formal (Conditional) Potentials (cf. Conditional Stability Constant vs. Thermodynamic Stability Constant)
16.4 Effects of Different Factors on Electrode Potential: Application of Such Effects in Analytical Chemistry
16.5 Periodic Trends of Electrode Potential
16.6 Function of Zimmermann-Reinhardt Solution (in Titrating Fe(II) by KMnO4 in the Presence of Chloride)
16.7 Instability of Some Species in Aqueous Solution
16.8 EMF Diagrams
16.9 Equilibrium Constant from the Standard Electrode Potentials
16.10 Disproportionation and Comproportionation Reactions
16.11 Potential Profile in a Redox Titration
16.12 Selection of a Redox Indicator
16.13 Potentiometric Titrations
16.14 Practical Applications: Electrochemical Cells and Batteries - Primary Cells, Secondary Cells, Fuel Cells : Corrosion
16.15 Kinetic Factors in Electrode Process and Electron Transfer Reactions
16.16 Photochemical Reactions: Photoredox Reaction and Photochemical Splitting of Water
16.17 Ellingham Diagram: Reduction of Metal Oxides: Carbon - A Potential Reducing Agent
16.18 Hydrometallurgy
16.19 Electrode Potentials in Nonaqueous Systems
16.20 Examples of Some Common Catalysed Redox Reactions
16.21 Redox Activity in Relation to Explosive Action
16.22 Chemistry of Some Important Electroanalytical Techniques
Solved Numerical Problems
Exercise-XVI
Numerical Problems
Appendix 16A: Chemistry of Explosives
Appendix 16B: Chemical Clock Reactions: Oscillating Reactions
Appendices
Appendix I: Units and Conversion Factors
Appendix II: Some Physical and Chemical Constants
Appendix III: Wavelength and Colours
Appendix IV: Names, Symbols, Atomic Numbers and Atomic Weights of the Elements
Appendix V: Some Useful Mathematical Relationships
Appendix VI: Books Consulted
Index
Back Cover
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Inorganic

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Fundamental Concepts of

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Chemistry lc

SECOND EDITION

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Volume 3

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This Page is Intentionally Left Blank

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Revised Second Edition

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Asim K. Dos MSc (Gold Medalist, CU), PhD (CU), DSc (Visva Bharan)

Mahua Das MSC (CU), PhD (Visva Bharati)

Former Research Associate, Department of Chemistry Visva Bharati University, Santiniketan 731235 West Bengal (India)

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Professor of Chemistry Visva Bharati University, Santiniketan 731235 West Bengal (India)

I] CBS

CBS Publishers & Distributors Pvt Ltd New Delhi • Bengaluru • Chennai • Kochi • Kolkata • Mumbai Bhubaneswar • Hyderabad • Jharkhand • Nagpur • Patna • Puna • Uttarakhand

Disclaimer Disclaimer Science and technology are constantly changing fields. New research and experience broaden the scope of Science and technology ore· constantly information and knowledge. The authors have tried their best in giving information available to them while chongingftelds. New research and expetience preparing the material for this book. Although, all efforts have been made to ensure optimum accuracy of the broaden the scope of information and material, yet it is quite possible some errors might have been left uncorrected. The publisher, printer knowledge. The .• the authors hoveand tried their best In. gMng.lnformatlon available to them while the authors will not be held responsible for any inadvertent errors, omissions or inaccuracies.

preparing the material for this. book. Although, all efforts have been made to ensure optimum accuracy of the mote(fQl, yet it is quite possible some errors might hove been left uncorrected. The· publisher, printer and authors· will not be held responsible for any inadvertent errors or inaccuracies.

eISBN: 978-93-890-1752-6

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Copyright © Authors and Publisher

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Second eBook Edition: 2019

All rights reserved. No part of this eBook may be reproduced or transmitted in any form or by any means,

electronic or mechanical, including Fundamental Concepts of photocopying, recording, or any information storage and retrieval system

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without permission, in writing, from the authors and the publisher. Inorganic Chemistry Revised Second Published by SatishEdition Kumar Jain and produced by Varun Jain for CBS Publishers & Distributors Pvt. Ltd.

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ISBN:Corporate 978-81-239-1868-6 Office: 204 FIE, Industrial Area, Patparganj, New Delhi-110092 Ph: +91-11-49344934; Fax: +91-11-49344935; Website: www.cbspd.com; www.eduport-global.com; Copyright © Authors and Publisher E-mail: [email protected]; [email protected] Revised Second Edition: 2016 Head Office: CBS PLAZA, 4819/XI Prahlad Street, 24 Ansari Road, Daryaganj, New Delhi-110002, India. Reprint: 2011,2013,2015,2018 Ph: +91-11-23289259, 23266861, 23266867; Fax: 011-23243014; Website: www.cbspd.com; First Edition: 2000 E-mail: [email protected]; [email protected] Reprint: 2001,2003,2005,2008,2009

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Dr. A.V. Saha, DSc

Foreword

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Head of the Department of Chemistry Ramakrishna Mission Residential College P.O. Narendrapur 743508, 24-Parganas (8) West Bengal, India

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Present-day inorganic chemistry is no more a collection of unrelated facts. The scenario has undergone a.drastic change over the last fifteen to twenty years with the applicati~n' of kinetic, thermodynamic and structural studies to inorganic substances and with newer techniques providing newer information. Often the information gathered, has made theoreticians develop/modify valency theories and principles. In this situation every teacher and student realise the importance of a textbook that will help them develop the concepts and understanding ofthe subject. There are attempts by a few authors at achieving this goal but those seldom cover the whole curricula followed by most of the Indian Universities and Institutes. In many of these books the authors virtually neglect the evolutionary developments of the subject, creating undesirable lacunae in the readers' understanding. Here is an honest and sincere attempt at bridging these gaps and presenting a comprehensive textbook on concepts and understanding to the readers. The treatment of every topic is elaborate and is marked by remarkable clarity and the author has not compromised herewith the volume of his work. The book Fundamental Concepts ofInorganic Chemistry covers the inorganic chemistry curricula at the BSc (honours) and MSc preliminaries. A large number of exercises and problems essential for modern teaching, have been incorporated meticulously at the end of each chapter to bridge the gaps, if any, in the understanding of the subject. Hope this earnest effort of Dr. Das, a teacher of distinction, will receive well-deserved acclaim from the students and teachers of this subject.

A.V. Saha

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my teachers and well-wishers who paved the way to reach here

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One who h~s shraddha acquires knowledge

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Preface to the Second Edition

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I do endeavour. He gives the strength and patience.

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My beloved readers are requested first to accept my unconditional apology for this long time taken to shape the second edition of the book. Though the second edition incorporates some aspects of coordination chemistry, the readers desire the complete Coordination Chemistry which is expected to appear in the next volumes and the process is in progress. I feel delighted to record the warm response which the first edition of the book has received from the students and teachers throughout the country. I have tried my best to incorporate all the suggestions received from the readers. The present edition has been thoroughly revised and substantially enlarged and, in fact, no single page has been left untouched. However, I have taken all the measures to retain the features for which the first edition has been so popular. In this second edition of the book, scope of the book has been broadened by adding new topics and revisions of the earlier sections. The new topics are atomic units; NMR active nuclei and principles of NMR spectroscopy; nucleosynthesis of elements after big bang; nuclear waste management; BoseEinstein condensate (BEC): fifth state of matter; bonding properties of d- and }-orbitals: comparison; application of Woodward-Hoffmann orbital symmetry rules; structure and properties of tluoroalkyl radicals and ions; structure and properties of zeolites and clay materials; aluminosilicates and surface acids; heteropoly and isopoly acid anions; hypervalency; molecular orbital diagrams of polyatomic molecules in terms of group orbitals (i.e. TASOs); 3c-4e versus 3c-2e bonding systems; geometry of ·the molecules in terms of Walsh diagrams; isolobal fragments; reaction-mechanism of the main group elements; different aspects ofmaterial science; PSPET- Wade's rule, Jemmi's rule; chemistry ofboranes and carboranes; metal clusters and carbonyl clusters; Zintle ions; symmetry elements and point groups; fullerenes; intercalation compounds ofgraphite; band theory of solids - band splitting and band bending; electrical and magnetic properties of solids; piezoelectric solids; organic metals and organic semiconductors; superconductivity; supra~olecular chemistry; aqueous chemistry of amino acids; surface

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Fundamental Concepts of Inorganic Chemistry

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acids and solid acids; agostic interaction; ionic equilibria and related numerical problems; inorganic photochemistry; lithium battery; fuel cells; solar cells; EMF diagrams - Frost diagram, Pourbaix diagram; Tafel equation; stability field of water; chemistry of explosives; electroanalytical techniquespolarography, coulometry, ampereometry, LSV, CV, polarography; chemical clock reC\ctions and oscillating chemical reactions. In preparing the revised manuscript, I have freely consulted the books and reviews of different authors and I have borrowed their ideas whenever it has been required. I am extremely grateful and indebted to these authors. In fact, I have picked up the flowers from these gardens to prepare the garland to worship the goddess of learning. The golden period of my student life at Ramakrishna Mission Residential College, Narendrapur, gives the foundation of my career. There I got a gifted teacher, Prof. AV Saha whose teaching in the subject is still remaining vibrant in my memory and his teaching is reflected at many places. of this book. The author is thankful to Prof. KL Sebastian, IPC, IISc, Bangalore; Prof. S Basu, Burdwan University; Prof. GN Mukherjee, Sir Rashbehari Ghose, Professor of Chemistry; Dr A Sen, Ramakrishna Mission Centenary College, Rahara, 24 Parganas; Dr MN Bishnu, City College, Kolkata; Prof. RN Mukherjee, lIT, Kanpur; Dr SP Banerjee, Retired Professor, Vivekananda Mahavidyalaya, Burdwan; Dr S Roy Chowdhury, Dinabandhu Andrews College, Garia, Kolkata; Prof. K Dey, Kalyani University; Prof. A Sarkar, Kalyani University; Dr. G Mukherjee, Kalna College, Burdwan; Dr N' Koley, Retired Professor, Raghunathpur College, Purulia; Prof. B Pathak, Retired Professor, Presidency College, Kolkata; Dr B Saha, Burdwan University; Dr H Karak, St Paul's College, Kolkata, for their encouragement and suggestion. I am thankful to all my colleagues for extending their moral support. I am specially thankful to my colleague Dr M Seikh for discussion on many topics and procuring some reference materials. I ~m extremely grateful to Mr SK Jain, Managing Director, CBS Publishers and Distributors, New Delhi for his continued support. I am also thankful to Mr YN Arjuna, Senior Director (Publishing, Editorial and Publicity) and the staff involved in DTP for taking the troubles in processing the revised manuscript. I am thankful to my PhD students, Dr M Islam and Sri R Bayen for their cooperation. I am indebted to my family members Dr (Mrs) Mahua Das (wife), Ankita Das (elder daughter) and Udita Das (younger daughter) for their moral support without which it would have been impossible to do this task. Despite my best efforts, some mistakes and misconceptions might have crept in and for these inconveniences, I beg to be pardoned. Suggestions and criticisms are always welcome from the readers. Santiniketan

Asim K. Das

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Preface to the First Edition

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- "No great work can be achieved by humbug. It is through love, a passion for truth; and tremendous energy, that all undertakings are accomplished. "

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:- "I hold every man a traitor who, having been educated at their expense pay not the least heed to them. " Swami Vivekananda

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As a student and also as a teacher, the author has experienced that for a systematic and comprehensive coverage of the present subject, one is forced to consult various books on different disciplines to collect the reading materials. Consequently, the students are specially constrained. Keeping this idea in mind, it has been attempted here to present a complete textbook on the subject. In a logical sequence, the book deals extensively with different aspects such as atomic structure (both classical and wave mechanical) and atomic spectra, fundamentals of quantum mechanics and wave mechanics, nuclear chemistry and radiation chemistry, different theories of valence for~es and chemical forces including band and Bloch theory of solids, solid-state chemistry, acids and bases, nonaqueous solvents and redox potentials. It also covers the different aspects of materials science which is emerging with a great promise. Thus the present book covers the curricula followed by most of the Indian Universities and Institutes at the SSc and MSc levels. It also aims to help the students preparing for competitive examinations like NET, GATE, SLET, etc. Adequate stress on the basic theories and concepts has been given everywhere to rationalise the presentation. The ideas have been very often illustrated through solution of related numerical problems. Each chapter is ended with various types of questions and problems to afford an opportunity to the students for self-evaluation. In writing a book of this nature, one accumulates indebtedness to the previous authors of different books. The books which have been consulted are listed separately and gratefully acknowledged. The author expresses his deep sense of gratitude to Dr AV Saha, a gifted teacher, from whom the author had started to learn and understand the subject in his student life in Ramakrishna Mission Residential College, Narendrapur, for writing the foreword. The author's association for a fairly long period with the said Institute was highly fruitful in shaping the present ideas and the author must express his indebtedness to his teachers and the authorities of the Institute. The author is grateful to his wife, Dr M Das for various types of help, assistance and cooperation.

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Fundamental Concepts of Inorganic Chemistry

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The author is specially grateful to Mr SK Jain, Managing Director ofeSS Publishers and Distributors, New Delhi for his earnest interest in publishing the book. Thanks are also due to .his colleagues and students. In conclusion, the author's attempt will be amply rewarded, if it is found helpful to the students and teachers. In spite ofall precautions, some errors might have crept in. Constructive criticism and valuable suggestions from the readers will be most welcome.

Asim K. Das

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Contents

Foreword

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Preface to the Second Edition

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Preface to the First Edition

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VOLUME 3

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12. Bonding in Metals and Metal Clusters: Electrical Conductivities of Solids: Semiconductors and Superconductors

1110-1249 1110

12.2 Crystal Structure of the Metals and the Effects of Lattice Structure on their Properties

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12.1 General Properties of Metals

1114

12.4 The Bloch Theory (i.e. Zone Theory) of Metallic Bonding Basic Concepts of Bloch Theory; Brillouin Zone, Forbidden Zone and Conductivity

1131

12.5 Molecular Orbital Theory or Band The,ory of Metallic Bonding Formation of Energy Bands and Brillouin Zones; Band Structure ofNontransition Metals; Band Structure of Transition Metals; Valence and Conduction Bands; Explanation of Electrical Conductivities of Solids in the Light of Band Theory; Optical Properties of Solids in the Light of Band Theory; Magnetic Properties Including the Temperature Independent Paramagnetis (TIP) (Le. Pauli Paramagnetism) of the Metals in Terms of

1136

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12.3 The Free Electron Theory of Metallic Bonding Drude-Lorentz Classical Free Electron Theory: Electron Sea Model.; Sommerfeld's Quantum Mechanical Free Electron Theory and Fermi Dirac Distribution Function; Quantitative Aspects of Some Properties of Metals; Electrical Conductivity and Hall-Effect

xii

Fundamental Concepts of Inorganic Chemistry Band Theory; Ferromagnetism in Transition and Inner-transition Metals; Band Structure of Some Inorganic Solids and their Properties

1154

12.7 Solid Solutions : Alloys : Intermetallic Compounds Interstitial Alloys; Substitutional Alloys, Intermetallic Compounds and Hume-Rothery Rule

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12.6 Valence Bond Approach of Metallic Bonding Valence Bond Approach for the Short Period Metals; Valence Bond Approach for the Long Period Metals; Merits ofthe Valence Bond Approach; Demerits ofthe Valence Bond Theol)'

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12.9 Semiconductors and their Properties in the Light of Band Theory Characteristic Features of Intrinsic Semiconductor; Characteristic Features of Extrinsic or Doped Semiconductors; Temperature Dependence of Semiconductor Conductivity; Optical Properties of Solids; Uses of Semiconductors: Direct and Indirect Gap Semiconductors; Some Useful Semiconductor Materials and their Properties; Characteristics of p-n Junctions; Characteristic Features of p-n-p and n-p-n Junction Transistors; Some Applications of Semiconductor Devices

1181

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12.8 Metal-Metal Bonds in Metal Clusters Binuclear Compounds; Trinuclear Compounds; Octahedral Clusters; Chevrel Phase; Octahedral Metal Cluster; Carbonyl Clusters; Metal-only Clusters (Zintle Cluster Ions); Metal-Metal Bonds in Stacked Polymeric Structure; Conditions for Metal-Metal Bonding

12.10 Semiconductivity and Metallic Conductivity of Metal Oxides: Hopping Mechanism of Conductivity

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12.11 Electrical Conductivity in One Dimensional Solids: Peierls Distortion Electrical Conductivity in KCP [Ptossium tetracyanoplatinate(II)] and Peierls Distortion; Electrical Conductivity in Polythiazyl (SN)x (Le. ~ [SN] - a one dimensional metal); Electrical Conductivity in trans-Polyacetylene (C2H2 )x or (CH)x .- an Organic Semiconductor

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12.12 Conducting Organic Substances: Organic Metals, Organic Semiconductors, Molecular Inorganic Superconductors Introduction; Some Representative Examples of Organic Salts as Conducting Substances: Organic Superconductors

1218

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12.13 Superconductivity Concept of Superconductivity; Characteristic Features ofSuperconducting Materials; BCS Theory of Superconductivity; Superconducting Materials; Structural Features of Some Copper-containing Superconducting Ceramic Materials (Warm Superconductors); Application of Superconductivity

1214

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12.14 Electrical Conductivity of Solid Electrolytes

1232

Solved Numerical Problems

1236

Exercise-XII

1241

13. Hydrogen Bonding and Other Weaker Chemical Forces Including Supramolecular Systems

1250-1342

13.1 Hydrogen Bonding Types of Hydrogen Bonding Depending on the Nature of Electron Clouds on Donor (D); Types of Hydrogen Bonding Depending on the Position of Donor (D); Participation of

1250

Contents

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C-H Bonds in H-Bonding; Syinmetrical and Unsymmetrical Hydrogen Bonding; Hydrogen Bond Energy and Bond Length; Effects of Intermolecular H-Bonding; Effects of Intramolecular H-Bonding; Detection of H-Bonding; Theories of Hydrogen Bonding; Agostic Interaction vs. Hydrogen Bonding Interaction; Comparison among H-bridge Bond, H-Bond and Agostic Interaction

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13.2 Hydrates and Clathrates or Cage Compounds

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Hydrates; Clathrates or Cage Compounds; Configurational Entropy and Thermodynamic Aspects of Clathrate Compounds

13.3 The Nature of Intermolecular Forces: van der Waals Forces: Lennard-Jones Potential: The 6-12 Potential

1285

13.4 Interaction Between Ionic and Covalent Compounds

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The Attractive Intermolecular Forces; Total Intermolecular Attractive Forces (van der Waals Forces); The Repulsive Intermolecular Forces; Lennard-Jones Potential

1295

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,Ion-Dipole Interaction; Ion-Induced Dipole Interaction; Metal-Ligand (neutral) Interaction in Terms of Ion-Dipole Interaction and Polarisati-on of the Metal Ions and Ligands; HardHard Intera~tion (Electrostatic Interaction) and Soft-Soft Interaction (Mutual Polarisation)

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13.5 Noncovalent Interactions in Supramolecular Systems and Molecular Recognition

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Interactions in Supramolecules; The Lock and Key Principle in Supramolecular Assemblies; Representative Examples of Supramolecular System; Double Helical Structure of DNA; Macrocyclic Systems (Synthetic Molecular Receptors for Cations); Molecular Recognition of Specific Metal Ions by Macrocyclic Systems; Crown Ethers as Receptors; Spherical • Recognition of Metal Ions; Molecular Recognition of NH: and RNH; (Primary Alkyl Ammonium Cation) by [18]-Crown-6; Selective Perching or Nesting of NH: or RNH; on Crown-6; Cryptands; Synthetic Molecular Receptors for Different Types of Substrates - Cations, Anions, Neutral Molecules; Application of Macrocyclic Crown Ethers and Cryptands; Polyazamacrocycles and Polyazacryptands as Anion Receptors (Synthetic Receptors); Molecular Recognition by Macrocyclic Systems; Calixarenes as Receptors; Cyclodextrins (CDs) as Natural Receptors; Cyclophanes as Synthetic Receptors for Apolar Guests; H-Bond Directed Molecular Assembly; Metal Coordinated Self-assembly; Molecular Switch; Catenanes and Rotaxanes; Inverse Crown Ethers: Novel Anion Receptors

13.6 Energetics of Dissolution

1324

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Thermodynamic Aspects and Conditions of Dissolution; Born Equation and Ion Solvation; Energetics of Solubilities of Ionic Salts in Polar Solvents; Solubilities of Ionic Salts in Nonpolar Solvents; Solubilities of Nonpolar Solutes in Nonpolar Solvents; Solubilities of Nonpolar Solutes in Polar Solvents

13.7 Energetics of Phase Transitions - Melting, Boiling, Sublimation

1332

Phase Transition in Covalent Compounds; Ionic Compounds; Ionic-Covalent Compounds (i.e. Fajans Type Compounds); Ease of Phase Transition Depending on the Nature of Involved Chemical Forces

Exercise-XIII

1336

14. Acids and Bases and Ionic EquiUbria in Aqueous Solutions 1343-1554 14.1 Definitions of Acids and Bases

1343

xiv

Fundamental Concepts of Inorganic Chemistry

14.2 Arrhenius Theory of Acids and Bases

1343 1344

m yl

ib ra

ry

14.3 Bronsted-Lowry Concept: The Proton Donor-Acceptor Concept: Protonic Concept: The Conjugate Acid-Base Theory Basic Concepts of Bre>nsted-Lowry Theory; Relative Strength of Acids and Bases : Differentiating and Levelling Solvents in the Light of Brt>nsted-Lowry Concept; Merits of the Bre>nsted-Lowry Concept; Limitations ofthe Brt>nsted-Lowry Concept; Cosolvating Agents and Acid-Base Strength; Application of Bre>nsted-Lowry Concept in Calculating pH ofAqueous Solutions: A Generalised Approach to Calculate the pH Values of Different Types of Aqueous Solutions; Acid-Base Behaviour of Amino Acids in Aqueous Solution: Isoelectric Point; The Chemistry of Proton in Water: Proton Transport Process 14.4 Mechanism of Buffer Action: Buffer Capacity

1370 1372

14.6 Hammett Acidity Function: Superacids Concept of Hammett Acidity Function (H); Concept of Superacids and Examples of Superacids; Application of Superacids

1382

14.7 The Lewis Concept: The Electron Pair Donor-Acceptor System Basic Concepts and Characteristic Features ofthe Lewis Theory; D.ifferent Types of Lewis Acids; Classification ofthe Lewis Bases; Lewis Acid-Base Reactions and Redox Reactions; Some Typical Examples of Lewis Acids and Bases, Lewis Acid-Base Adducts; Covalent' and Ionic Contributions in the Stabilities of the Adducts : Drago-Wayland Equation; Demerits of Lewis Acid-Base Theory; Solvent Properties in the Light of Lewis AcidBase Concept; Change of Bond Length in Lewis Acid-Base Interaction

1386

t.m e/

th

ea

lc

he

14.5 Selection of Acid-Base Indicators in Acid-Base Titrations

1414

k

H

er e

14.8 Strength of Bronsted (i.e. Protonic) Acids and the Governing Factors Involved and the Properties of Bronsted Acids and Bases Strength of Different Protonic Acids and Conjugate Acids of Some Bases; Born-Haber Cycle and Thermodynamic Parameters to Determine the Acid Strength: Acid Strength of Hydracids and Carboxylic Acids: Gas Phase Acidity; Factors Governing the ~trength of Oxyacids; Hydrolysis and Aqua Acid Strength of Metal Ions in the Light of Bre>nstedLowry Concept; Basicity ofthe Anions; Periodic Variation of Acidic and Basic Character; Nucleophilicity and Basicity; Acid and Base Catalysis

1437

C

lic

14.9 Some Important Factors Governing the Acid-Base Strength (both Br6nsted and Lewis Acid-Base Systems) Inductive Effect: Entropic Effect, d-Orbital Participation; State of Hybridisation of the Central Atom; Steric Factors: F-Strain and B-Strain; Solvation Effects (cf. H-bonding); Resonance and 1t-Bonding and the Possibility of d-Orbital Participation in the Reacting Species; Effect of Resonance Stabilisation in the Product; Effect of H-Bonding on Acid Strength 14.10 Lux-Flood Concept

1466

14.11 Solvent System Definition

1468

14.12 Usanovich Concept 14.13 A Generalised Concept of Acids and Bases

1472 1472

Contents xv

14.14 Concept of Ultimate Acids and Bases 14.15 Hard and Soft Acids and Bases (HSAB)

1472 1473

ib ra

ry

Basic Concept and Principle of Hard and Soft Acids a,nd Bases (HSAB) .Theory; Basis of Classification of the Acids and Bases as Hard or Soft; Characteristics of Hard and Soft Species; Symbiosis in Hardening or Softening a Centre; Acid-Base Strength versus HSAB Principle; Application of HSAB Principle; Theoretical Background of HSAB Principle

m yl

14.16 Surface Acids and Solid Acids: Heterogenous Acid-Base Reactions 14.17 Aqueous Solutions of Salts: Hydrolysis of Salts: Solubility Product of Sparingly Soluble Salts

1492 1494

Hydrolysis of Salts; Solubility Product and Activity Product of Sparingly Soluble Salts; Application of the Principle of Solubility Product

lc

1555-1576

ea

15. Nonaqueous Solvents

1555 1555

th

15.1 Introduction 15.2' Classification of Solvents

1510 1541

he

Solved Numerical Problems Exercise-XIV

1556

t.m e/

15.3 Characteristics of Ionising Solvents

1558

15.5 Liquid Hydrogen Fluoride as a Solvent

1564

15.6 Liquid Hydrogen Cyanide as a Solvent

1565

15.7 Acetic Acid (CH3C02H) as a Solvent

1566

er e

15.4 Liquid Ammonia as a Solvent

15.8 Sulfuric Acid as a Solvent

1567

15.9 Fluorosulfonic Acid (HS03F) as a Solvent and Superacids

1568 1569

15.11 Bromine Trifluoride as a Solvent 15.12 Oxyhalides as Solvents

1570 1571

lic

k

H

15.10 Liquid Sulfur Dioxide as a Solvent

15.13. Molten Salts (i.e. Ionic Liquids) as Solvents

1572

Exercise-XV

C

1575

16. Redox Systems and Electrode Potential: Application of Electro.de Potentials: Electroanalytical Techniques

1577-1783

16.1 Some Preliminary Aspects of Redox Reactions Ion Electron Method of Balancing Redox Reactions; Oxidation Number and Rules for Calculating Oxidation Number; Oxidative Addition and Reductive Elimination Reaction; Equivalent Weights ofOxidants and Reductants; Complementary and Noncomplementary Redox Reactions; Redox and Acid-Base Reactions; Disproportionation and Comproportionation Reactions

1577

xvi

Fundamental Concepts of Inorganic Chemistry

16.2 Electrode Potentials: Standard Potentials

1596

ib ra

16.3 Formal (Conditional) Potentials (cf. Conditional Stability Constant vs. Thermodynamic Stability Constant)

ry

Electrical Double Layer and Genesis of Electrode Potential; Determination of Electrode Potential; Standard Electrode Potential: Nemst Equation: Stoichiometric Standard Potential and Formal Potential : Concept of pE; Sign Conventions of the Electrode Potentials; Standard Electrode Potentials of Metals and Electrochemical Series; Standard Reduction Potentials; Reversible Chemical Cells: Cells with and without Transference; Galvanic vs. Electrolytic Cell: 'Ohmic Potential and Observed Cell emf

m yl

16.4 Effects of Different Factors on Electrode Potential: Application of Such Effects in Analytical Chemistry

1607 1610

th

ea

lc

he

Effects ofpH on Electrode Potentials due to the Participation 9f fr or 011 in Electrode Process (cf. Pourbaix diagrams); Effects of pH on Electrode Potentials due to the Formation of Sparingly Soluble Hydroxo-species (cf. Pourbaix diagrams);, Effects of Precipitation (other than hydroxides) on Electrode Potential,S; Effects of Complexation' on Electrode Potentials; Chemistry of Aqua Regia and Dissolution of Noble Metals: E.nhancement ofOxidisability of Metals through Complexation; Attack of Noble Metals by HF and by a Mixture of HF and HN0 3 : Enhancement of Oxidisability of Metals through Complexation; Dissolution of Antimony by a Mixture ofNitric Acid and Tartaric A~id : Enhancement of Oxidisability of Metals through Complexation I

16.5 Periodic Trends of Electrode Potential

1640

t.m e/

Variation of Standard Oxidation Potential of the Metals; Variation of Oxidising Power of the Oxyanions in a Group

1643

16.7 Instability of Some Species in Aqueous Solution

1644

er e

16.6 Function of Zimmermann-Reinhardt Solution (in Titrating Fe(II) by KMn04 in the Presence of Chloride) 16.8 EMF Diagrams

1647

H

Latimer Diagram; Frost Diagrams (Ebsworth Diagram); Pourbaix Diagram or PotentialpH (E-pH) Diagram

1661

k

16.9 Equilibrium Constant from the Standard Electrode Potentials

1663

16.11 Potential Profile in a Redox Titration

1672

C

lic

16.10 Disproportionation and Comproportionation Reactions

16.12 Selection of a Redox Indicator

1677

16.13 Potentiometric Titrations

1680.

16.14 Practical Applications: Electrochemical Cells and Batteries Secondary Cells, Fuel Cells : Corrosion

Primary Cells,

Basic Requirements of an Electrochemical Cell to Act as a Power Source; Representative Examples ofPrimary and Secondary Cells; Some Representative Fuel Cells; Electrochemistry of Corrosion; Electrolytic Cell vs. Galvanic Cell: Electrolysis of Solutions and Electrodeposition; Decomposition Voltage of Electrolysis and Electrolytic Separations of Metals; Some Other Practical Applications of the Knowledge of Electrode Potential

1686

Contents

16.15 Kinetic Factors in Electrode Process and Electron Transfer Reactions

xvii

1704

ib ra

ry

Importance of Kinetic Factors; Importance of Overpotential (in Electrode Process and Cell Potential): Tafel Equation; Redox Reactions Through Electron Transfer; Redox Reactions Through Atom Transfer

16.16 Photochemical Reactions: Photoredox Reaction and Photochemical Splitting of Water 1708

m yl

Some Representative Photochemical Reactions Illustrating the Characteristic Features; Photochemical Splitting of Water and Photochemistry of Ru(bpy)}+; Ti02, an Important Photocatalyst; Direct Photochemical Reduction ofDintrogen; Charge Transfer Band and Redox Stability

1719

16.18 Hydrometallurgy

1722

ea

16.19 Electrode Potentials in Nonaqueous Systems

lc

he

16.17 Ellingham Diagram: Reduction of Metal Oxides: Carbon - A Potential Reducing Agent

1723 1724

16.21 Redox Activity in Relation to Explosive Action

1726

th

16.20 Examples of Some Common Catalysed Redox Reactions 16.22 Chemistry of Some Important Electroanalytical Techniques

1727

t.m e/

Polarographic Method of Analysis; Cyclic Voltammetry (CV); Amperometric Titration; Coulometric Analysis

1752 1763

Numerical Problems

1770

er e

Solved Numerical Problems Exercise-XVI

1773

Appendix 168: Chemical Clock Reactions: Oscillating Reactions

1777

H

Appendix 16A: Chemistry of Explosives

k

Appendices

C

lic

Appendix Appendix Appendix Appendix Appendix Appendix

Index

A-1-A-12

I: Units and Conversion Factors II: III: IV: V: VI:

A-I

Some Physical and Chem~cal Constants A-4 A-5 Wavelength and Colours Names, Symbols, Atomic N_~Plbers and Atomic "Yeights of the Elements A-6 Some Useful Mathematical Relationships A-8 A-II Books Consulted 1-1-1-18

xviii

Fundamental Concepts of Inorganic Chemistry

VOLUME 1 1. Classical and Vector Models of Atoms

1-110 1

1.2 Anode Rays Production of Anode Rays; Properties of Anode Rays; Analysis of Positive Rays by Thomson's Parabola Method; Aston's Mass Spectrograph; Dempster's Mass Spectrometer; Bainbridge Mass Spectrometer

8

m yl

ib ra

ry

1.1 Cathode Rays and Electron Electric Discharge through Gases and Production of Cathode Rays; Properties ofCathode Rays; Detennination of Charge to Mass Ratio (elm) ofan Electron by Thomson's Method; Detennination of Charge of an Electron by Millikan's Oil-Drop Method

he

1.3 The Neutron as a Fundamental Particle The Discovery of Neutron; Properties of Neutron

16

18

1.5 The Hydrogen Spectrum Balmer Formula; Rydberg Fonnula; Ritz Combination Principle

18

ea

lc

1.4 Some Important Properties of the three Fundamental Particles

20

1.7 Scattering of a-Particles and Rutherford's Nuclear Model of Atom Rutherford's Nuclear Model of Atom; Rutherford's Theory of a-Particle Scattering; Nuclear Dimension from the a-Particle Scattering Experiment

21

1.8 Bohr's Atomic Model for the Hydrogen-like Systems Bohr's Atomic Model; Numerical Values ofRadius and Energy in Bohr's Atom; Correction for the Finite Mass of the Nucleus; Bohr's Theory and the Spectral Lines in Hydrogenlike Systems; Ionisation and Resonance Potential for Hydrogen and Hydrogen-like Systems in the Light of Bohr's Theory; Bohr's Theory and Correspondence Principle; Moseley's Law in the Light of Bohr's Theory; Franck-Hertz Experiments (Nobel Prize in Physics, 1925) in the Light of Bohr's Atomic Model; Merits of the Bohr's Atomic Model; Drawbacks in the Bohr's Model

25

k

H

er e

t.m e/

th

1.6 Thomson's Plum-Pudding Model of Atom

Model Orbits and Quantum Defect in Theory; Achievements of the Model

45

1.10 Vector Model of the Atom Concept of Spatial Quantisation; Concept of the Spinning Electron; Quantum Numbers in the Vector Model; Four Quantum Number System and Pauli Exclusion Principle; Coupling Schemes and Atomic States in the Vector Model; L-S and Coupling Schemes; Spatial Quantisation of the Resultant Vectors (i.e. !, S, J) in Atoms or Ions; Determination of Microstates and Russel-Saunders Tenns; A Simple Working Procedure to Determine the Term Symbols; Hund's Rules to Detennine the Ground State Tenn;

52

C

lic

1.9 Fine Structure of Spectra and Sommerfeld's Atomic Sommerfeld's Atomic Model; Penetrating Power of the Polyelectronic Systems in the Light of Sommerfeld's Sommerfeld's Theory; Drawbacks in the Sommerfeld's

J-]

-:,.

Contents

xix

Derivation of the Ground State Tenns in the Light of Hund's Rules; Spectral Selection Rules in the Vector Model; Intensity Rules for Spectral Transitions in the Vector Model; Fine or Multiplet Structure of the Spectral Lines in the Light of Vector Model; Nuclear Spin and Hyperfine Structure of the Spectral Lines; Orbital and Spin Magnetic Moments

1.11 Electronic Configuration Scheme for Many Electron Systems Rules for Placing the Electrons in Different Energy Levels

m yl

Solved Numerical Problems

ib ra

ry

ofan Electron; Stem-Gerlach Experiment-An Evidence in Favour of Space Quantisation and Spin Quantum Number; Explanation ofthe Magneto-Optic Phenomenon-the Zeeman Effect; Electronic Structure of the Atoms; Merits of the Vector Model

2. Origin of the Quantum Theory

97 lOS

111-132 111

lc

2.1 Introduction

he

£xercise-I

86

111

2.3 Different Laws and Theories of Black-Body Radiations

113

ea

2.2 Nature of Black-Body Radiations

2.4 The Photoelectric Effect

th

The Stefan-Boltzmann Law; Wien's Law; Rayleigh-Jeans Law; Planck's Quantum Theory

118

2.5 Compton Effect

t.m e/

Characteristics of Photoelectric Effect; Einstein's Theory of Photoelectric Effect; Work Function and Ionisation Potential

121

er e

Theoretical Background of Compton Effect; Compton Shifts at Different Cases; Characteristic Features of Compton Effect

126 127 130

H

2.6 Pair Formation Solved Numerical Problems Exercise-II

k

3. Fundamentals of Wave Mechanics

133-184 133 133

lic

3.1 Introduction 3.2 Wave Particle Daality and de Broglie's Matter Wave

C

de Broglie Wavelength for Matter Waves; The de Broglie Wavelength for the Macroscopic and Microscopic Bodies; Difference between de Broglie Matter Wave and Electromagnetic Wave; The de Broglie Electron Wave; Relativistic Correction in the de Broglie Electron Wave; Verification of the de Broglie Electron Wave by Davisson and Germer's Experiment; Bohr's Quantum Restriction from the de Broglie Concept

3.3 The

Heisenberg~s Uncertainty

Principle and its Implications

U·ncertainty Principle; Illustration of the Uncertainty Principle in the Determination of the Position ofa Particle by a Microscope: Compton Effect and the Uncertainty Principle; Applicability of the Uncertainty Principle to Large and Small Particles; Some Important Applications of the Uncertainty Principle

140

xx

Fundamental Concepts of Inorganic Chemistry 145

. 3.5 Some Applications of the Schr6dinger's Wave Equation Free Particle; Particle in a One-Dimensional Box; Particle in a Three-Dimensional Box; Importance of the Model - Particle in a Box; Some Applications of the Model - Particle . in a Box; Quantum Mechanical Tunnelling Effect

150

ib ra

ry

3.4 Schr6dinger's Wave Equation Representation ofSchr6dinger's Wave Equation; Physical Significance ofSchr6dinger's Equation and Hamiltonian Operator; Eigen Values and Eigen Functions; Physical Significance of the Wave Function (V); Nonnalised and Orthogonal Wave Functions

m yl

3.6 Approximate Methods of Solving the Schr6dinger's Wave Equation Variation Method; The Linear Combination Method

he

Solved Numerical Problems Exercise-III

4. Wave Mechanical Model of Atom

lc

Appendix 3A: Application of Variation Method in Hydrogen Atom

165 167 177 180

185-235 185

4.2 Wave Mechanical Model for Polyelectronic Systems Self-Consistent Field (SCF) Method; Concept of Slater Orbitals; Concept of Shielding and Quantum Defect; Slater's Rules in Calculating the Shielding Constant (S) and Effective Nuclear Charge (Z*)

219

lic

k

H

er e

t.m e/

th

ea

4.1 Wave Mechanical Model for the Hydrogen-like Atoms or Ions Wave Functions for the Hydrogen-like Systems; Wave Mechanical Genesis of the Quantum Numbers; Important Features of a Dirac Atom Developed from the Relativistic Wave Mechanics; The Radial Wave Function and Radial Probability Distribution of Electron CJoud in Hydrogenic Systems; The Physical and Chemical Significance of the Nodal Points; Angular Wave Functions and Orbital Shapes; Electron Cloud Density Representation of the Orbitals in Hydrogen-like Systems; Symmetry of the Orbitals; Concept of Atomic Orbitals in Wave Mechanics in Relation to the Orbits in Classical Mechanics; Energy of the Electron in Hydrogen-like Systems; The Four Quantum Numbers; Pauli Exclusion Principle in the Light of Wave Mechanics

4.3 Bonding Properties of d- and I-Orbitals

225 226

Exercise-IV

234

C

Solved Problems

5. Atomic Nucleus and its Structure 5.1 The Atomic Nucleus Composition of the Nucleus; Size of the Nucleus; Mirror Nuclei and Nuclear Radius; Nuclear Density; Shape· of the Nucleus; Total Angular Momentum of the Nucleus: Magnetic Properties ofthe Nucleus and Nuclear Magnetic Resonance (NMR) Frequency and NMR Spectra of Some Representative Examples; Classification of the Nuclides;

236-308 236

Contents xxi Nuclear Spin Isomerism in Diatomic Molecules: Ortho- and Para-Hydrogen: Intensity of the Rotational Lines in the Band Spectra of Diatomic Molecules 253

5.3 Nuclear Stability Even-Odd Nature of the Nucleons (Harkins' Rule); The Neutron to Proton Ratio and Stability of the Nuclides; The Neutron to Proton Ratio and Different Modes of Decay; Packing Fraction and Nuclear Stability; Mass Defect and Nuclear Binding Energy (NBE)

268

m yl

ib ra

ry

5.2 The Isotopes Discovery and Characterisation of Nonradioactive Isotopes; Isotopic Shift in Optical Spectra; Characteristic Features of the Isotopes; Isotopic Composition of the Elements; Uses of Isotopes; Separation of Isotopes

he

5.4 The Nuclear Forces Nature ofthe Nuclear Forces; Exchange Force as the Nuclear Force; The Nuclear Potential and Nuclear Potential Barrier

lc

5.5 Nuclear Models Fenni Gas Model; The Liquid Drop Model; Nuclear Shell Model; The Optical Model

282 296

ea

5.6 The Fundamental Particles

278

305

6. Radioactivity and Radiation Chemistry

309-382

Solved Numerical Problems

t.m e/

th

Exercise-V

299

309

6.2 Types of Radioactive Emanations

310

6.3 Properties of Alpha, Beta and Gamma Radiations Alpha (a)-rays; Beta (~)-Rays; Gamma (y) Rays; Comparison among the Different Types of Radioactive Emanations

311

6.4 Detection and Measurement of Radioactivity

315

H

er e

6.1 Discovery of Natural Radioactivity

320

6.6 Different Modes of Radioactive Decay

322

6.7 The Fluorescence and Auger Effect

323

6.8 Disintegration Chain and Soddy-Fajan Group Displacement Law

324

C

lic

k

6.5 Units of Radioactivity

6.9 The Decay Kinetics and Parent-Daughter Growth Relationships Case I : The Daughter Nucleus is Stable; Case II : When the Daughter Element is Radioactive

325

6.10 Radioactive Equilibrium

332

6.11 Energetics of the Disintegration Processes

336

6.12 Some Important Aspects of Alpha Decay

338

6.13 Some Important Aspects of Beta Decay

340

6.14 Some Important Aspects of Gamma Emission

341

xxii

Fundamental Concepts of Inorganic Chemistry

6.15 The Disintegration Series

342

6.16 Artificial or Induced Radioactivity

345

Discovery of Artificial Radioactivity; Production of Artificial Radioactivity

6.17 Applications of Radioactive Isotopes

347

ib ra

ry

Application of Radioisotop~s in the Field of Chemistry; Application of Radioactivity in Biological Fields; Application of Radioactivity in Agriculture; Application ofRadioactivity in the Medical Field; Application ofRadioactivity in Industry; Application of Radioactivity . in Age Determination

6.18 Some Important Aspects of Radiation Chemistry

363

m yl

Radiation Chemistry; Characteristics of Interaction of Medium Energy Charged Particles with Matter; Interactions ofNeutrons with Matter; Interaction of Gamma-Radiation with Matter; Radiation Dosimetry; Radiolysis of Water

he

Solved Numerical Problems

lc

Exercise-VI

383 384 385 386 390

th

Artificial Transmutations and Nuclear Reactions Bethe's Notation of Nuclear Reactions Nuclear Reactions Versus Chemical Reactions Classification of Nuclear Reactions Conservation Laws in Nuclear Reactions

.

t.m e/

7.1 7.2 7.3 7.4 7.5

379 383-447

ea

7. Nuclear Reactions

371

Conservation of Energy; Conservation of Linear Momentum; Conservation of Angular Momentum; Conservation of Neutrons and Protons

392

7.7 Nuclear Reaction Cross-Section

392

7.8 Mechanisms of Nuclear Reactions

393

H

er e

7.6 Threshold Energy of a Projectile for a Nuclear Reaction

k

The Compound Nucleus Theory; Oppenheimer-Phillips Mechanism in Stripping Reactions; Direct Nuclear Reaction Mechanism

lic

7.9 Charged Particle Accelerators

397

The Linear Accelerator; The Cyclotron

C

7.10 Filling in the Gaps in the Periodic Table: The Man-Made

EI~ments

401

7.11 Transuranic and Translawrencium Elements: The Man-Made Elements

402

7.12 Nuclear Fission

406

, Historical Background; Characteristic Features of Nuclear Fission; Bohr-Wheeler Theory ofNuclear Fission: Liquid Drop Model; Charge Distribution in Fission Products: Principle of Equal Charge Displacement

7.13 Fission Chain Reaction: A Source of Atomic Energy Factors Controlling a Fission Chain Reaction; The Fertile and Fissile Nuclides as the Nuclear Fue!s; Atom Bomb; Fenni's Four Factor Formula and Nuclear Reactor or Atomic

415

Contents xxiii Pile for the Controlled Nuclear Fission; Breeder Reactor; A Natural Fission Reactor : Okla Phenomenon; Nuclear Reactors in India; Recovery of Uranium and Plutonium from Spent Fuel

7.14 Nuclear Fusion: Thermonuclear Reaction

. 424

ib ra

ry

Characteristic Features of Nuclear Fusion; Hydrogen and Cobalt Bomb; Stellar Energy; Controlled Fusion Reactions: An Innocent and a Never-Ending Source of Energy; Cold Fusion

7.15 The Origin and Evolution of Elements: Nucleosynthesis of Elements

429

m yl

Nucleosynthesis of Elements from the Primordial Element Hydrogen in Stars; Cosmic Abundance of Elements

Solved Numerical Problems

he

Exercise-VII

lc

8. Periodic Table and Periodic Trends of Different Properties of Elements

ea

8.1 Historical Background of the Development of Periodic Classification

435 444

448-541 448

th

Law of Triads; Pattenkofer's Rule of Integral Multiple; Chancourtois Law of Telluric Screw; Newlands' Law of Octaves; Mendeleev's and Lothar Meyer's Periodic Laws

t.m e/

8.2 Mendeleev's Periodic Law and Periodic Table

451

Characteristics of Mendeleev's Short Periodic Table

8.3 Characteristics and Usefulness of the Mendeleev's Periodic Table

455

er e

Systematic Classification of the Elements; Correction of Atomic Weights; Prediction of the Missing Elements

8.4 Defects in the Mendeleev's Periodic Table

456

8.5 Long Form of the Modern Periodic Table

458

k

H

Important Segments of the Long Form of the Modern Periodic Table; Advantages of the Long Form of Periodic Table; Defects of the Long Form Periodic Table

lic

8.6 Extended Periodic Table: Superheavy Elements: Postactinides and Superactinides

C

8.7 Classification of the Elements Based on the Electronic Configuration and Chemical Affinities

462 464

Classification Based on the Position ofthe Differentiating Electron; Bohr's Classification Based on Electronic Configuration; Goldschmidt's Geochemical Classification ofElements Based on Chemical Affinity

8.8 Characteristics and Position of the Border Line Elements Position of the Zn-Farnily; Positions of Lu, Th and Lr; Position of the Lighter Actinides;

467

4[Series (Lanthanides) vs. 5fSeries (Actinides)

8.9 Lanthanide and Actinide Contractions (i.e. I-Contractions) Nature ofthe!-Contraction; Explanation of.fContraction; Effects ofLanthanide Contraction

469

xxiv Fundamental Concepts of Inorganic Chemistry 8.1 0

Periodit~

Trends of Z* and Size of the Atoms and Ions

473

Two Important Factors (n and Z*) to Determine the Size of Atoms and Ions; Periodic Trends of Z* (Effective Nuclear Charge); Variation of Size along the Periods; Variation of Size in a Group

478

ry

8.11 Variation of Atomic Volumes in the Periodic Table

ib ra

Atomic Volume: Definition and Limitation of the Parameter; Nature of Variation Trend 2); Steps to Select a Point Group (excluding the special point groups with multiple high order axes); Symmetry Elements and Optical Activity; Symmetry Elements and Dipole Moment; Symmetry Number and Point Groups

m yl

Exercise-X

Appendix lOA: Fluoroalkyl Radicals and Carbocations and Fluorocarbenes

lc

11.1 Characteristic Properties of Ionic Compounds

he

11. Structure, Bonding and Properties of Ionic Solids and Solid-State Chemistry

962 972

977-1109 977 980

11.3 Electron Density (ED) Maps and Ionic Radii

982

th

ea

11.2 Different Types of Ions and Electronic Configurations of the Ions Involved in Ionic Bonding

t.m e/

Ionfc Radii from X-ray Electron Density Maps; Ionic Radii from Other Methods; Shannon's Crystal Radii; Comparison among Shannon's Crystal radii, Goldschmidt's Radii and Pauling's Radii

11.4 Factors Affecting the Ionic Radii

989

11.5 Energetics of Ionic Bond Formation: Born-Haber Cycle

994

er e

Born-Haber Cycle in Ionic Bond Formation; Drawbacks in the Concept of Born-Haber Cycle; Factors Favouring the Formation of Ionic Bonds

H

11.6 Applications of Born-Haber Cycle 11.7 Theoretical Aspects of Crystal Forces: Lattice Energy of Ionic Crystals

997 1003

lic

k

Born-Lande Equation of Lattice Energy; Verification of the Born-Lande Equation; Modification of the Born-Lande Equation; Kapustinskii Equation; Factors Affecting the Lattice Energy; Hardness of Ionic Compounds

C

11.8 Radius Ratio Rule and Preferred Structures of Ionic Compounds

1014

Limiting Radius Ratios for Some Common Geometries; Application of the Radius Ratio Rule; Limitations of the Radius Ratio Rule

11.9 Some Basic Crystal Geometries (i.e. Unit Cells) Involved in the Structure of Crystalline Solids Simple Cubic (se) Unit Cell; Body Centred Cubic (bee) Unit Cell; Face, Centred Cubic (fcc) Unit Cell; Diamond Cubic (de) Unit Cell; Graphite Layer Lattice; Close Packing Models of Spheres; The Body Centred (bee) Lattice in Terms ofClose Packing of Spheres; Characteristic Features of hep, fcc and bee Structure

1024

xxx

Fundamental Concepts of Inorganic Chemistry

11.10 Structures of Ionic Crystals

1035

Stru'ctures of Ionic Crystals of AB Type; Structures of Ionic Crystals of AB 2 Type; Structures of Ionic Crystals of AB 3 Type; Structure of Ionic Crystals of A2B3 Type: Corundum (6:4 Coordination Crystals); Structure of Ionic Crystals Consisting of Polyatomic Ions; Comparison of the Basic Ionic Crystals

ry

11.11 Structures of Mixed Metal Oxides

1042

ib ra

The Spinel and Inverse Spinel Structure; The Ilmenite Structure (FeTi03); The Perovskite Structure (CaTi03); Mixed Oxides - Coloured Minerals and Gem Quality Crystals

11.12 Crystal Structure in Relation to Superconductivity and Ferroelectric Property

1047

m yl

YBa 2Cu 30 7_x ; BaTi03

11.13 Deviations from Simple Ionic Structures

1049

he

Layer Lattices (Le. Two Dimensional Lattices); Chain Lattices (i.e. One Dimensional Lattices)

1051

lc

11.14 Structures of Silicates and Isopoly and Heteropoly Acids: Representative Examples of Complex Ionic Crystal 11.15 Crystal Defects in Ionic Solids

ea

Chemistry of Silicates and Cement; Isopoly and Heteropoly Acids

1063

th

Stoichiometric (i.e. Intrinsic) Defects; Nonstoichiometric (or Berthollide) (i.e. Extrinsic) Defects

11.16 Magnetic Materials: Magnetic Properties

1076

t.m e/

Tenninology in Describing the Magnetic Properties and Different Units; Classification of Magnetic Substances; Cooperative Magnetism - Aniferromagnetism, Canted Magnetism; Ferrimagnetism and Ferromagnetism; Magnetic Properties of Metals; Temperature Dependence of Magnetic Susceptibility of Different Types of Magnetic Materials

11.17 Properties of Ferroelectric Materials and Piezoelectric Materials

1088 1091

11.19 Glasses and Glass Tral\sition

1092

H

11.18 Carbides

er e

Ferroelectric Materials; Piezoelectricity and Piezoelectric Materials

1094

11.20 Structure of Liquids: General Aspects: Hole Theory of Liquids

1095

Exercise-XI

1102

lic

k

Solved Numerical Problems

C

Appendices Appendix

A·1-A·12

I: Units and Conversion Factors

Appendix II: Some Physical and Chemical Constants Appendix Appendix Appendix Appendix

Index

III: IV: V: VI:

A-I A-4

Wavelength and Colours A-5 Names, Symbols, Atomic Numbers and Atomic Weights of the Elements A-6 Some Useful Mathematical Relationships A-8 A-II Books Consulted

1·1-1·20

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VOLUME 3

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12. Bonding in Metals and Metal Clusters: Electrical Conductivities of Solids: Semiconductors and Superconductors

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13. Hydrogen Bonding and Other Weaker Chemical Forces Including Supramolecular Systems

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14. Acids and Bases and Ionic Equilibria in Aqueous Solutions 15. Nonaqueous Solvents

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16. Redox System and Electrode Potential: Application of Electrode Potentials: Electroanalytical Techniques

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Appendices Units and Conversion Factors Some Physical and Chemical Constants Wavelength and Colours Names, Symbols, Atomic Numbers and Atomic Weights of the Elements Appendix V: Some Useful Mathematical Relationships Appendix VI: Books Consulted

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Appendix I: Appendix II: Appendix III: Appendix IV:

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Bonding in Metals and Metal Clusters: Electrical Conductivities of Solids: Semiconductors and Superconductors 12.1 GENERAL PROPERTIES OF METALS

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Before to discuss any theory in explaining the bonding mechanism in metals, we should highlight the general properties of metals. Any model proposed regarding the bonding mechanism must be able to explain the general behaviour of metals. The general properties are discussed below. (i) Electrical (0') and thermal (S) conductivities: The metals are good conductors of electricity and heat. The electrical conductivity decreases with the increase oftemperature. Both the conductivities are attained due to the movement of the electrons. This represents the electronic conduction. In an aqueous solution of an ionic compound, the electrical conductivity gets developed by the movement ofthe ions (Le. ionic conduction). In solid ionic crystals, the weak electrical conductivity arises mainly due to the crystal defects. Because of the similarity in the mechanism of both electrical and thermal conductivities, a good electricity conducting metal is also a good conductor of heat. The ratio ofthermal (S) and electrical (cr) conductivity of different metals is more or less constant (irrespective of the nature of metals) at a particular temperature. This is known as Wiedemann-Franz's Law. (ii) Metallic lustre: The metal surfaces are very much bright lustrous in appearance. The electrons are excited to the higher energy levels by absorbing the visible light and the excitation energies are emitted back in radiation when they jump down from the excited levels. In deexcitation, all possible wavelengths in the visible spectrum are emitted. Thus the metal surfaces act as good reflectors of all wavelengths. This is why, the metal shows the shiny silver colour. Here, gold (yellow) and copper (red) are exceptions. Probably, these absorb at some specific wavelengths preferably. (iii) Photoelectric and thermionic emission: Some of the metals give an emission of electrons on being excited by photons, i.e. lights (photoelectric effect), or by heat (thermionic effect). The excitation energy is different for different metals. It indicates that the binding forces experienced by the electrons are different for different metals. 1110

Bonding in Metals and Metal Clusters: Electrical Conductivities of Solids 1111

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(iv) High cohesive forces: Because of the very high cohesive forces, they show high melting and boiling points. The cohesive force measured by the increase of melting points runs as : Or III (13) metals> Or II (2) metals> Or I (1) metals. lIn the first transition series, it increases up to vanadium with the increase of the number of unpaired d-electrons and then it falls with the pairing of d-electrons. Thus in general it appears that with the increase ofthe number ofelectrons available for bonding purpose, the cohesive force increases. The heat of sublimation of a metal is much greater than the heat of dissociation of the molecular species, M2 (say), if it exists in vapour. It indicates that in the solid assemblage, the cohesive energy is very large. . (v) High density: The metals generally show very high density because ofthe close-packed crystalline structure. However, the density varies with the variation of the lattice structure.

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(vi) Malleability and ductility: The metals'show very high melting points indicating the high resistance towards the rupture of the crystal, but they are malleable and ductile which indicate that they are not much resistant towards deformation. The malleability and ductility depend on the crystal structure and crystal imperfections (e.g. point defects, grain boundaries, dislocations, interstitial impurity, etcJ. Generally thefcc lattices are more malleable and ductile compared to the hcp

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lattices.

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(vii) Elasticity: Metals can tolerate high stress indicating high elasticity of the metals. It also depends on the crystal structure.

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(viii) Solid solutions - alloys and nonstoichiometric intermetallic compounds: Through substitution at the lattice sites or through accommodation in the interstitial spaces, such nonstoichiometric compounds are formed. To have such compounds or solutions some conditions such as radius ratio rule (for interstitial accommodation) and Hume-Rothery rules (for substitution) are to be followed. (ix) Efectropositive character: The metals are generally electropositive in nature.

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(x) Chemical reactivity and state of subdivision : The finely divided particles are more reactive than the crystalline lump. The increased reactivity for the finely divided particles arises due to the large surface area and no requirement of lattice energy to break down the lattice.

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12.2 CRYSTAL STRUCTURE OF-THE METALS AND THE EFFECTS OF LATTICE STRUCTURE ON THEIR PROPERTIES

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12.2.1 'Crystal Structure of the Metals Almost all the metals crystallise in fcc or bcc or hcp models. The alkali metals crystallise in the bcc structure while the coinage metals crystallise in the fcc structure. Be, Mg, Zn and Cd of Or II adopt the hcp structure but Ca and Sr are in the fcc structure. Ba being very large in size adopts the bcc structure. For the transition metals, because of the different allotropic forms, it is very difficult to predict the crystal structure ofthe transition metals. Different allotropic forms adopt different structures. For example, Fe has got three allotropic forms (1, y and 0 which exist at different temperature ranges. The a and 0 forms adopt the bcc structure while the y form attains the fcc structure. The crystal structure of some common metals are given below. bcc : alkali metals, Ba, a-Cr, a- and o-Fe, V, Nb, Ta, Mo, W, V, etc.

1112

Fundamental Concepts of Inorganic Chemistry

fcc: Cu, Ag, Au, Rh, Jr, Pd, Pt, AI, Ni, Pb, y-Fe (910°C to 1410° C), etc. hcp : Be, Mg, Zn, Cd, Ti, Zr, HI, Ru, as, etc.

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At a higher temperature, the lattice points vibrate with the higher vibrational amplitudes. The bcc structure being an open structure gives the better provision to accommodate such higher vibrational amplitudes. These increased vibrations give an increased entropy of the system to lower down the free energy. This is why, the allotropic forms which are stable at a higher temperature adopt the bc~ structure. For example, in the case of iron, the allotropic from which exists above 14100 C is in the bce s.tructure while the other allotropic form existing in the range 910°C to 141 O°C is in the fcc structure. In the same line of argument, the tendency of the alkali metals to adopt the bce structure can be explained. The alkali metals undergo melting at a lower temperature range (20° -200°C). Thus even at the room temperature, the lattice vibrations are sufficiently high and these can be better accommodated by the bcc structure to increase the thermal entropy.

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12.2.2 Effects of the Crystal Structure on the Properties of Metals

Metal

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Different properties are strongly dependent on the structure of the crystal lattice. These are discussed below. (i) Melting point, cohesive energy and density: The bcc metals are generally soft with the low densities and low melting points. Thus, the low cohesive energy of the bcc metals is evidenced in the properties of the alkali metals. Thus the alkali metals having the bcc structure remarkably differ in properties from the coinage metals (i.e. Cu, Ag, Au) which crystallise in the fcc structure. Here it is interesting to note that both the alkali and coinage metals are having the same nurgber of valence electrons. This aspect is illustrated by considering the following pair. Lattice structure

Rb Cu

er e

bcc fcc

At. Wt.

85.50 63.40

No. o/valence electrons

(kg m-3 )

Melting point (C)

1480 8920

18.8 1083

Density

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In the bcc structure, the packing efficiency is 680/0 while in the fcc and hcp structure it is 740/0. Thus the density is reduced in the bcc structure. The fewer interactions due to the lower number of nearest neighbours (i.e. coordination numbers) reduce the cohesive energy. Besides this, the

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higher random vibrational amplitudes in the bcc structure reduce the' cohesive energy and decrease the melting points.

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A comparison of the different properties of bcc, fcc and hcp structures is already given in Sec.

11.9.8.

(ii) Malleability and ductility: Malleability and ductility ofa metallic crystal depend on the number of slip planes. The slip planes are the planes of highest atomic density. Because ofthe highest close-packed array ofthe atoms in the slip plane, there is a very high cohesive force to prevent the breaking. In a particular crystal, the highest close-packed planes are most widely placed and consequently the resistance to slip for such planes is less compared to the other sets of planes. The slip direction is the direction along which the slip occurs within the slip planes. The slip forces (a shear deformation causing the movement of atoms by many interatomic distances with respect to their initial positions) is illustrated in Fig. 12.2.2.1.

(iI)

(iv)

(iii)

(b)

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Slip planes

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(a)

1113

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Bonding in Metals and Metal Clusters: Electrical Conductivities of Solids

(c)

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Fig. 12.2.2.1. (a) Schematic representation of slip process in a 2-dimensional lattice under a shearing force. (b) Schematic representation of slip process. (c) Slip planes and slip directions in a fcc lattice (see Fig. 11.9.3.3 for lattice planes).

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In afcc lattice (cg. Fig. 11.9.3.3 for lattice planes), the slip planes are {Ill} perpenqicular to the C 3 axis (Le. body diag0nal) and these correspond to four planes (Ill), (Ill), (Ill), (Ill) and the slip directions are the family (i.e. face diagonals). The slip plane together with the slip direction (close-packed directions along which the slip occurs) constitutes the slip system. It leads to 12 slip systems in a fcc metallic cl)'stal. In a hcp lattice the slip plane is the hexagonal basal plane perpendicular to the C6 axis of the unit cell and it leads to 3 slip systems. It is expected: larger the number ofslip systems, more the malleability and ductility. This is why, the metals (e.g. Cu, Ag, Au, Pt, etc.) which crystallise in afcc structure are more ductile and malleable than the metals (e.g. Zn, Cd, Mg, etc.) having the hcp structure. In a bee structure the more common slip planes are {IIO} and the slip directions are the family. In the bee metals, {I 12} and {I23} planes (having slip directions < 111» can act also as slip planes. Here it may be noted that though both the bcc andice lattices are characterised by 12 slip systems, the slip plane of the fcc lattice is more closel)" packed (Le. of higher ~!omic density). In the bee lattice, the slip planes are not closely packed and they cannot slip under the slip forces. Rather, the bcc metals are soft and brittle. The malleability and ductility get drastically reduced with the increase of crystal imperfections such as point defects, grain boundaries, dislocations, interstitial accommodation ofsome impurities, etc.

1114

Fundamental Concepts of Inorganic Chemistry

12.3 THE FREE ELECTRON THEORY OF METALLIC BONDING 12.3.1 Drude-Lorentz Classical Free Electron Theory: Electron Sea Model

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The foundation of classical free electron theory was first laid by Drude (1900) and developed by Lorentz (1923). In this theory, it is believed that in a metal crystal the valence electrons are free to move throughout the crystal consisting of positive ions. The free electrons follow the laws ofclassical mechanics and they are treated as electron gas. Thus a metal crystal can be considered as an array of positive atomic cores (called kernels) immersed in a sea of mobile electrons which are held through electrostatic forces. This is why, the model is referred to as electron sea model or electron pool model, or electron gas model. In this theory, the following assumptions are made.

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(i) The electrons do not experience any repulsion among themselves. (ii) The potential field due to the ions is considered to be uniform throughout the crystal, but there exists a very high potential at the surface to imprison the electrons within the crystal. (iii) The free electrons obey the laws of kinetic theory of gases and their energy distribution pattern is governed by the Maxwell-Boltzmann statistics.

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The model can explain some properties of metals at least qualitatively. These are discussed below. (a) Electrical conductivity: When an external field is applied, the free electrons move towards the positive field. Such a directed flow of electrons explains its electrical conductivity. It can establish the Ohm's law. But the temperature dependence of resistivity predicted from this theory is not quantitatively correct. These aspects have been discussed later. (b) Thermal conductivity: When a piece of metal is heated at one end, the free electrons at that region acquire a higher kinetic energy and move to the cooler end. In both the thermal and electrical conduction, the free electrons are involved in the same way. Thus it is expected that both the thermal (S) and electrical conductivities (0) should be correlated. It has been found that for a large number of metals at a particular temperature, Sia is more or less constant which is independent ofthe nature ofthe metals. This relation is popularly known as Wiedemann-Franz's law. But quantitatively, the ratio (Sla]) called Lorenz number (L) obtained from the classical free electron theory differs from the actual value. (c) Metallic lustre: The free electrons may be excited to higher energy states by .absorbing the incident light. During coming back from the excited states to the ground state, it may occur in multisteps giving rise to the emission of radiations of all probable wavelengths in the visible range. It explains the metallic lustre. This fact can be better rationalised in terms of Sommerfeld's quantum mechanical free electron theory which will be discussed later. (d) Thermionic emission and photoelectric effect : The energy distribution among the free electrons is governed by the Maxwell-Boltzmann law. At room temperature, the fraction of total free electrons having sufficient energy to surmount the potential barrier (Le. the potential at the surface) is negligibly small, but as the temperature is raised, the fraction will increase (n = no exp (-Elk B ]), n = number of electrons having energy at least E, no = total number of electrons, k B = Boltzmann constant and T= temperature in Kelvin scale) and consequently the thermionic current will be increased. In the same way, the photoelectric effect can be explained. (e) Malleability and ductility: The electrostatic force is not directional and it is uniform throughout the crystal. Thus a layer of kernels (Le. positive ions) along with the free electrons can glide on a plane by experiencing a shearing stress. The relative positions of the kernels suffering

Bonding in Metals and Metal Clusters: Electrical Conductivities of Solids

1115

the shifting remain the same, because the free electrons are equally probable everywhere throughout the system. This fact can explain the malleability and ductility:

Limitations of the Classical Free Electron Theory (ct. Sec. 12.3.3 tor quantitative analysis)

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Classicalfree electron gas theory using the Boltzmann distribution law can explain several properties including thermionic emission, photoelectric emission, etc. But the theory fails totally to explain the specific heat, quantitative treatment of thermal conductivity, temperature independent paramagnetism, etc. For example if all the free electrons are subjected to Boltzmann distribution of energies and they contribute to the specific heat then the electronic specific heat becomes sufficiently high. But in reality,

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the electronic specific heat ofmetal is negligible. It indicates that all thefree electrons do notparticipate in determining the electronic specific heat ofmetals. Similarly, the assumption in classical free electron

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theory that all the free electrons participate equally in determining the other properties like thermal conductivity, electrical conductivity, magnetic susceptibility is not correct. To remove the difficulties in classical free electron theory, some modifications were required and it was mainly done by Sommerfeld by using the Fermi-Dirac statistics instead of Maxwell-Boltzmann statistics. The refined quantum mechanical treatment indicates that not all the electrons but the

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electrons present close to a certain level (ctliled Fermi level) participate to these phenomena.

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12.3.2 Sommerfeld's Quantum Mechanical Free Electron Theory and Fermi Dirac Distribution Function

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(A) Essential Features of Sommerfeld Model

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Sommerfeld introduced the laws of quantum mechanics instead of classical mechanics to explain the behaviour of the free electrons within a metal crystal. In terms of wave particle duality, the electron is associated with a de Broglie wavelength A as follows:

A=!!-=~

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...( 12.3.2.1) mu p where h = Planck's constant, m = mass of an electron, u = velocity of the electron. It can be expressed in terms of wave number, k which is defined as : ...( 12.3.2.2) k = 2rc/A Therefore, 2rclk = hlmu

k

or,

k = (21t/ h) mu = (2rc/ h)p

..(12.3.2.3)

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Thus the wave number (k) is directly proportional to the momentum (P) of the electron. _ (Note: Sometimes, k is defined as k = l/A as it has been done in the chapters on atomic structure, v = 1/ A). The potential energy (U) within the crystal is uniform and constant which can be taken arbitrarily to be zero. This assumption does not hamper anything when we are interested with the energy difference between the different states. The total energy can be considered as the kinetic energy. The kinetic energy is given by :

...(12.3.2.4) or,

...( 12.3.2.5)

1116 Fundamental Concepts of Inorganic Chemistry

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[Note: In reality, in the above equations, m should be replaced by the effec.tive mass m*. The mass of an electron in a crystal is different from the free electron mass. The detail calculation in this aspect is beyond the scope ofth~ present book.] The above relation between E and k is parabolic (Fig. 12.3.2.1) in nature. k is actually a vector quantity which is required to represent the velocity ofthe electron both in magnitude and direction. This is why, both th~ positive and negative values of k are shown in Fig. 12.3.2.1. Now let us consider the permissible energy values. All the values given by Eqn. 12.3.2.5 are not acceptable. The present problem can be considered as a particle in a box of one dimension (see Sec. 3.5.2) and the electron is totally confined within the box (Le. one dimensional crystal lattice box). The probabilities of finding the particle at the two extreme ends are zero (Le. nodes of the electron wave exist at the ends). Hence, the longest wavelength associated with the electron which can exist within the box is 2L (where L = length of the box) (Fig. (A 12.3.2.2).

-k

0

+k

m yl

Fig. 12.3.2.1. The parabolic relationship between E (kinetic 27t energy) and. k(=T' wave

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number) of a free ~fectron.

=.?f)

ea

=

1,2,3,4, ...)

...( 12.3.2.6)

t.m e/

A == 2L/n, (n

th

In general we have :

\V

=

sin (27tX/A)

==

= 2L.

n'

H

A = 27t k and

or k

= n7t / L

...( 12.3.2.7)

h2k 2

n

87t m

8mL

k

E

2Eco E1

o'-------------' x----.

1

o

L

Fig. 12.3.2.2. The de Broglie wave lengths of the electrons in a one dimensional crystal lattice box of metals.

2 2

h

E=-2-=--2

lic

E

:::J C

:::J

sin (n7tX/L), (cf. Sec. 3.5.2)

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It leads to :

3~

(A = ~)

(A=2f)

Waves of this nature has the form :

c:

... (12.3.2.8)

By considering a three dimensional cubic box we have : 2

C

h 2 2 2 (n x +ny +n_) ...(12.3.2.9) 8mL where nx ' ny and n= are the quantum numbers in the specified three directions. Here we can recall the results (see Sec. 3.5.3) obtained by using the Schrodinger's general equation. In fact, the results obtained in both cases are identical. Now let us consider the energy difference between the successive levels. The permitted k-values are inversely proportional to L (cf. k = n7t/L). Hence, for any real crystal of an appreciable length, the energy difference between the successive levels is negligibly small. It makes the energy levels almost quasicontinuous. In other words, the relation between E and k is quasicontinuous. E=;--2

Bonding in Metals and Metal Clusters: Electrical Conductivities of Solids

1117

(B) Fermi-Dirae Energy Distribution Among the Electrons

h"l

(n1t)2 -L ,(cf. Eqn.

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22

k h

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Now let us consider the energy distribution among the free electrons. The electrons obey the' Pauli's exclusion principle and thus each level can accommodate maximum two electrons with opposite spins. At 0 K, the highest level which can be occupied is called Fermi level and the corresponding energy is called Ep Thus E F denotes the topmost energy leveljilled up at the ground state. For one-dimensional system, the energy of the n-th level is given by : =-2-=-2-

.-----------. 2

(N1t)2 2L

...( 12.3.2.1 0)

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h E F = 87t 2 m

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12.3.2.8) 81t m 81t m If there are N-electrons, then Fermi energy level (Le. topmost filled level) is nF = N/2 (assuming N to be an even number). This EF is given by : En

I 1 + exp [(E - E F )/ kBT]

...(12.3.2.11)

t.m e/

is Boltzmann constant. It is shown in Fig. 12.3.2.3.

er e

~here kB

=

th

P(E)

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The energy distribution is governed by the Fermi-Dirac statistics. At 0 K, all the levels up to the Fermi level are filled in and with the increase of temperature, the population at higher levels goes to increase and consequently, the vacancies below E F increase. At temperature above 0 K, the probability of occupation P(E) of an energy level E (>EF ) by the electrons is given by :

1.0 I-----_....!--f_ _-+-t

~TK

ooAo-

0""'---------------'-------EF E

_

k

rY.o""__

H

GJ

------+

(a)

(b)

C

lic

E--+

Fig. 12.3.2.3. The Fermi-Dirac distribution of energies among the free electrons at different temperatures.

Here it is important to mention that Boltzmann distribution function is a special case of Fermi-Dirac distributionfunction. For energies well above EF" 1 «exp[(E - EF)/kB1] and Eqn. 12.3.2.11 reduces to Boltzmann distribution law.

P(E)

~

exp{-(E - EF )/ kBT}

... (12.3.2.12)

Now let us consider the Fermi-Dirac statistics to calculate the number of energy states present in per unit volume having the energy values between E and E + dE. This gives the measure of density of energy states (DOS) and it is given by :

1118

Fundamental Concepts of Inorganic Chemistry

... (12.~2.13) Assuming two electrons to occupy a particular energy state (Pauli's exclusion principle), the number of electrons having energies between E and E + dE in volume V is given by : x P(E)

(2:2)[8~]3/2 VE II2

=

V x Fermi distribution function

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= 2 x Number of energy states in volume = 2VZ(E)dE

=(2:)[8~]3/2 2

h

VE I12 dEP(E) ...(12.3.2.14)

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h

Ik

dE l+exp[(E-EF ), B T ]

ib ra

N(E)dE

Characteristics of Fermi-Dirac energy distribution

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The relationship between N(E) and E is parabolic (cf. Fig. 12.3.2.4). The following conclusions are evident. (i) At 0 K, Emax = E F and N(E) is finite (:;t: 0); and N(E) = 0 for E > E F; in terms of P(E), P(E) = 0 for E> E F and P(E) = 1 for E S EF . (ii) At T K > 0 K, P(E) > 0 for E > E F and P(E) = 0.5 for E = E F ; only in the region of EF" P(E) begins to decrease slig~tly from unity. (iii) Thus the Fermi level can\be defined as the level which has got a 50% probability ofoccupation at any temperature great~r than 0 K [Le. P(E) = 0.5 for E = EF when T K > 0 K]. The highest energy leve~ occupied at O.K is called the Fermi level but at the working temperature above 0 K, orily a small fraction of the electrons from the level just below the Fermi level will be promoted to the levels just above the Fermi level. This number of excited electrons is quite negligible compared to the number of electrons present in a band. This is why, for practical purposes, E F is considered to denote the energy of the highest level occupied at ordinary temperature.

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k

H

At a temperature greater than 0 K, for Emax > EF" N(E) is finite. Fig. 12.3.2.4 indicates that all the electrons are not gaining the thermal energy with the increase of temperature. In other words, even

C

T2 K

GJ OK

CU ~

1 E----+ (a)

~

1

EF (b)

Fig. 12.3.2.4. The Fermi-Dirac energy distribution among the electrons at different temperatures.

Bonding in Metals and Metal Clusters: Electrical Conductivities of Solids

1119

at much higher temperature, some electrons possess the kinetic energy which is associated with some electrons at 0 K. In fact, with the rise of temperature, not all the electrons but only a fraction (in the order of TITF == kBTIEF ) of the electrons residing close to the Fermi level are gaining the thermal energy, i.e. all the electrons are not participating to contribute in the electronic specific heat ofmetal.

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The important conclusion is that with the increase of temperature, all the electrons do not gain the thermal energy kBT, but the electrons within the range of kBT from the Fermi level gain the thermal energy. Thus at a temperature greater than 0 K, only a small fraction of the empty energy levels above the Fermi level is occupied and consequently only a smallfraction ofthe energy levels lying below Fermi level remains unoccupied. Generally, TF (Fermi temperature) is of the order of 5 x 104 K, and thus even at very high temperature only a small fraction (= TIT~ of the electrons gain the thermal energy. Illustration

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• Calculate peE) for E== (EF + 0.10) eVand E== (EF - 0.10) eV at 800 K.

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E-EF =:;> O.IOeV =1.45 5 kJ3-T (8.62 x 10- eVIK) (800 K) (i) For E== (EF-+ 0.10) eV, i.e. 0.10 eVabovethe Fermi level:

=

1 =0.19 (i.e. 19%) exp (1.45) + 1

th

P(E)

t.m e/

(ii) For E== (EF - 0.10) eV, Le. 0.10 eV below the Fermi level. P(E) =

1

1 + exp (- 1.45)

= 0.81 (i.e. 81%)

In other words, the probability of vacance below EF

==

1 - P(£)

==

0.19 (i.e. 19%).

H

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• Estimate the fraction of the electrons that are excited above E F on heating from 0 K to 300 K of a piece of silver (EF == 5.48 eV). 21 kBT ~ 4 x 10- J = 0.025 eV

= 0.025 =0.0046

(Le.0.46%).

5.48

k

The fraction approximately given by: kaT EF

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(C) Some Important Results of Fermi-Dirac Distribution Function

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(a) At 0 K, peE) == 1 for E S EF and then the number of electrons in volume V (== a 3 ) is given by : N= jN(E)dE =

(~)[8:2a2r/2

7I/2 E

( ~)[8mh a2 ]3/2 3. 2

2

3

( ~3 ) [8hm ]3/2 VE F3/

2

2

dE

[E 3/2 ]EF 0

...( 12.3.2.15)

1120

Fundamental Concepts of Inorganic Chemistr/

The density of electrons (Le. number of electrons per unit volume) is given by : / _ (~ [8m b -- NIv - 3) f1

I

32

J

E 3/2

/

F

ry

It leads to :

ib ra

...( 12.3.2.16)

"21 muF2 =EF For Sodium, EF == 3.2 eV == 3.2

x

1.6

x

andkBTF =EF

10- 19 J; uF = 1.1

x

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(b) 'The terms Fermi temperature (TF ), Fermi velocity (u F ) are defined as follows:

... (12.3.2.17)

106 m S-I.

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(c) Now let us calculate the mean energy of an electron at 0 K. The total energy (Va) in volume Vat o K is given by :

o

=

f

2VZ(E)dE P(E)E, (where P(E)

ea

V

lc

EF

er e

t.m e/

th

o

= I for E :s; EF )

...( 12.3.2.18)

H

Thus the mean energy Eo at 0 K is : ~)

3

E" =}j=SEF

...( 12.3.2.19)

k

(Using Eqn. 12.3.2.13 for Z(E)dE and Eqn. 12.3.2.15 for N).

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(0) Electrical Conductivity in Terms of Sommerfeld's Free Electron Model

C

In the ,absence of any electrical field, there is no net motion of the electrons in a particular direction, because for every moving electron there is an electron (in terms of probability) which moves with the same speed in the opposite direction. Thus Fig. 12.3.2.5(a) obtained from Eqn. 12.3.2.5 remains symmetrically filled in up to E F (at ground state), but when an external field is applied, the electrons' produce a net momentum towards the positive pole. When the external field is applied, the electrons moving towards the positive end of the field are accelerated. Consequently, such electrons move faster to the positive end of the field compared to the electrons travelling in the opposite direction. In other words, the electrons travelling in the direction of the field (Le. towards the positive end) are of relatively lower energy and thus the degeneracy of the energy levels is lost in the presence ofthefield. The lower energy levels are more populated than

Bonding in Metals and Metal Clusters: Electrical Conductivities of Solids 1121 Vacant energy levels +

o

-k

+k

m yl

-k

ib ra

ry

Field

he

(a) (b) Fig. 12.3.2.5. (a) In the absence of field, energy levels are degenerate. (b) In the presence of field, one half band of lower energy is more populated giving rise to net flow of electrons towards the positive pole of the field (ct. Fig. 12.5.5.1).

er e

t.m e/

th

ea

lc

the higher energy levels giving rise to a netflow ofelectrons in the direction ofthefleld. The removal of degeneracy in the presence of the external field modifies the energy band (Fig. 12.3.2.5). For this modification to occur, some empty energy levels must be present just immediately above the Fermi level. In other words, it can be stated that electrical conductivity can occur only when the band is not co.mplctely filled. Such substances are describd as electronic conductors. Conclusion: • Incompletely filled bands called (conduction band) can allow the electrical conduction~ • For electrical conduction, there must be some vacant energy levels just above the Fermi level. • For good electrical conductivity, the band width and density of states (DOS) of the conduction band must be sufficiently large for an effective delocalisation of the conducting electrons through the lattice. These aspects \vill be discussed in detail later.

H

12.3.3 Quantitative Aspects of Some Properties of Metals (a) Ohm's law and electrical conductivity

C

lic

k

Let us first consider the classical free electron theory. The electrical force introduces an acceleration if) to the electrons towards the positive pole, but the velocity does not go to increase limitlessly. In classical free electron theory, the moving electron will collide with the positive ion cores and it will lead to the lattice-electron scattering. Here it should be mentioned that the collisions may arise from other sources. The obstacles are random lattice-vibration, impurities, different point imperfections, dislocations, grain boundaries, etc. These have been discussed in (c) of this section. Restricting our attention to the lattice-electron scattering only as in classicalfree electron theory, it is assumed that at each collision, the moving electrons will loose their velocity. After the collision, the velocity will again increase upto the next collision. Between the successive collisions, the increase in velocity is called drift velocity CUd). Thus the drift velocity measures the extra velocity (towards the positive end of the field) that the electrons acquire between the successive collisions over their normal velocity in the absence of electrical field (Fig. 12.3.3.1). The electrons will lose the drift velocity at the time of collision.

1122

Fundamental Concepts of Inorganic Chemistry Collisions

ib ra

ry

~.~

---+ TIme

+-- t---+

m yl

r;ig. 12.3.3.1. Attainment of drift velocity (u d) of an electron in the direction of the field (Le. towards to the positive pole) after each collision.

lc

he

The mean collision time t (Le. the average time difference between the successive collisions) and the acquired drift velocity (u d ) can be correlated with the electical potential gradient (E) as follows in the classical law. f= ud/t, and electrical force = Ee m (ujt) = Ee; or, ud = Eet/m

...(12.3.3.1)

ea

Thus,

th

Thus the average drift velocity CUd) acquired is given by : 1. 1 ud =-(o+ud)=-u d

t.m e/

2

...( 12.3.3.2)

2

The flux (J) of charge due to the moving electrons gives the current density,

1 2 J=neu d =-(ne tE/m)

(12.3.3.3)

2

er e

(Note: Some authors define, J = neud = ne 2tE/m.) Here n gives the number of electrons per unit volume moving towards the field. The conductivity

a = J/ E = ne 2t/2m

IcrE = ne tE12m = J I 2

k

Therefore,

H

(a) is defined as the flux per unit potential gradient, Le.

...(12.3.3.4)

lic

This is an expression of Ohm's law.

C

For copper, n = 8.5 conductivity ( cr

x

1028 m- 3 (assuming one electron per Cu-atom), t = 2

= ; : ) becomes 2.4

x

10- 14 s (at 300 K), the

x 10 7 Ohm- 1 m- 1, 23

n = 6.023 x lOx 8960 x 10 63.54

3

= 8.50 x 10 28

where density = 8960 kg m- 3, m ~ rest mass of an electron = 9. 1

x

m -3 ,

10-31 kg

The above expression (J = aE) can be transformed into the familiar form of Ohm's law. Let us consider a conductor of length 1 and crosssectional area A. If the potentials at the two ends are £1 and £2' then the potential gradient (E) is given by : ...( 12.3.3.5) £ == (E2 -:- E 1)/1 = I1E/I

Bonding in Metals and Metal Clusters: Electrical Conductivities of Solids 1123 The current (I) carried by the conductor is related with the current density (.I) as : 1 == JA. It leads to: cr EA

The conductance (G) is defined as : G == crAil

==

1

crt1EA II

== ~.

...( 12.3.3.6)

It leads to : t1E

R'

R (resistance) == IIG and 1== GM: ==

1M

or,

==

IR

IA commonform of Ohm's law.

Resistivity (p) is defined as : E == Jp. It leads to : _ t1E _ EI _ EI _ J pi _ pi R ----------

ry

==

ib ra

1 == JA

...(12.3.3.7)

H

er e

t.m e/

th

ea

lc

he

m yl

...( 12.3.3.8) 1 1 JA JA A This is another commonform of Ohm's law. The units are: Ohm (n) for resistance (R), siemens (S) for conductance (0), Le. IS == In-I; n m for resstivity (p); S m- I for conductivity (cr). In strict classical free electron theory (by Drude-Lorenz), it has been mentioned that the latticeelectron scattering (Le. collision between the moving electron and positive ion core at the lattice site) is the main obstacle for the moving electron. But in reality, many other obstacles are responsible (cf. (c) of this Section) for giving the electrical resistance. However, the concept of collision to destroy the drift velocity can lead to the Ohm's law as discussed above. This is an important success of DrudeLorentz classical free electron theory. In classical free electron theory, current is carried by the free electrons each moving with an average drift velocity (u d ) towards the field. But in quantum mechanical treatment, the current is carried by the electrons moving with the Fermi velocIty (u F ) in the direction of the field near the Fermi surface. Thus the inner electrons are not relevant so far as the electrical conduction is concerned. The quantum mechanical treatment leads to : 2

J = crE = (ne t m

2

JE = ( mU ne 1 JE F

...(12.3.3.9)

lic

k

where I == tu F' I == mean free path. This result is very much similar to that of classical free electron theory.

C

(b) Temperature Dependence of Electrical Conductivity

In the classical free electron theory, the electron-lattice scattering (Le. collision between the moving electron and positive ion-core at the lattice sites) is considered to be the main obstacle for the moving electron. This is why, the mean free path (I) distance travelled between the successive collisions is constant and it is of the order ofinteratomic spacing. Thus the temperature dependence of conductivity arises from the variation of the average velocity of the electron. With the increase o/temperature, the average velocity increases and consequently the number 0/ collisions increases to reduce. the conductivity. The quantitative relationship can be deduced as follows. From the concept of equipartition ofenergy, we get:

1124

Fundamental Concepts of Inorganic Chemistry

~mc2='ikT· 2

Of,

C

2

B

'

(root mean square velocity) == ~3k B T / m

( 12.3.3.10)

u =~Ud =~(:)

(12.3.3.11)

We have:

ry

d

ib ra

If I is the mean free path, then the time interval between the successive collisions is given by : ... (12.3.3.12)

t==l/c

ud _~(Ee)(~)_~(Eel m

- 2

c - 2

Ij;, _ Eel m) ~3kBT - 2~3kBT m

Therefore, the current density is given by : nEe 21

~

he

_ J=neu d =

m yl

Thus,

2,,3kB Tm

Thus the conductivity (0) is given by: .

...(12.3.3.13)

...( 12.3.3.14)

lc

..--------.--. 2

ce

Jr;

...(12.3.3.15)

2~3kBTm

th

cr

ne I

or p (resistivity)

ce .JT

...(12.3.3.16)

t.m e/

i.e.

J /E=

ea

0=

Thus the classical free electron theory indicates the temperature dependence of resistivity (p) as :

p oc .fi , but this is not experimentally verified. The experimental observation indicates that p is proportional to T. In fact, in arriving the relationship from the classical free theory of electron, it was assumed that the mean free path (l) is almost independent of temperature. In fact, in classical free

C

lic

k

H

er e

electron theory, collision between the moving electron and positive ion core (residing at the lattice sites) is considered and I is ofthe order ofinteratomic spacing. According to quantum mechanics, the Fermi velocity (u F) of the conduction electrons changes very slightly with the rise of temperature, but the mean free path (I) depends on the temperature. It leads to the temperature dependence of conductivity. Quantum mechani~ally, it can be shown that the mean free path (I) is inversely proportional to the absolute temperature (Le. I ex: liT). It explains the observation, p oc T by considering the result of quantum mechanical treatment (cf. Eqn. 12.3.3.9) Le. : 2

J

= [ne

/)

mU F

E ex:

~T (cr. Eqn.12.3.3.9, where I ce~) T

(c) Sources of resistivity (p) and temperature dependence (experimental observation) of resistivity or conductivity (0- = 1/p)

The resistivity of a conductor increases with the increase of temperature, impurities and crystal imperfections. The factors which can reduce the mean collision time (t) and mean free path (I) will increase the resistivity. The important factors are discussed below. (i) Phonons and temperature: Above 0 K, the lattice points vibrate randomly about their equilibrium positions. Such vibrations behaving as elastic waves in the crystal are called phonones. These

Bonding in Metals and Metal Clusters: Electrical Conductivities of Solids 1125

lc

he

m yl

1

ib ra

ry

random lattice vibrations destroy the ideal periodicity of the crystal and interfere with the moving electrons. Thus the electron-phonon interaction reduces the More impure mean free path (l) and conductivity decreases with temperature, because with the increase of temperature, Impure the random lattice vibrations (i.e. phonones) increase. c.. At 0 K, in a pure metallic crystal, there is no phonon Pure or lattice imperfection to interfere with the moving electron and theoretically the mean free path tends to be infinite. Consequently, conductivity should be also infinite. o - - -.... T(K) The resistivity due to phonon-electron interaction is described as phonon resistivity (Pr) which bears the Fig. 12.3.3.2. Variation' of resistivity (p) of a metal with temperature (Mathessen following relationship with temperature (T in Kelvin rule). scale) as follows: Pr oc T, (at higher temperature) (12.3.3.17)

H

er e

t.m e/

th

ea

3 and, PI' oc T , (at very low temperature) ( 12.3.3.18) The above relationships have been experimentally verified in many cases and Proriginated from the electron-phonon interaction is called thermal part of resistivity (cf. Mattheissen rule). (ii) Solute ato,,:,s or impurities: Presence offoreign atoms as impurities gives the effective scattering centres for the electrons. In fact, the local field around a foreign atom is different from that present in the remaining portion of the crystal. These local fields are responsible for electron scattering. It leads to increased collision numbers to reduce the mean free path and conductivity. (iii) Plastic deformation: The crystal inperfections like point defects, dislocations and grain boundaries can cause the scattering of electrons and it leads to increase the resistivity. At 0 K, there is no crystal imperfections to introduce the resistivity. The scattering by impurities is more important than the scattering by crystal defects. Mattheissen rule states that the total resistivity (p) is additive and it can be expressed in the following form P == Pr+ Pr ••• (12.3.3.19)

C

lic

k

where Pr (thermal part of resistivity) is caused by the thermal phonons; Pr (residual part of resistivity) is caused by the impurities and other crystal imperfections. Pr is dependent on temperature while Pr is independent of temperature (ignoring the effect of crystal defects). In many cases (e.g. pure ell), the plot Df P versus T leads to zero intercept, but the finite intercept indicates the presence of impurity and the magnitude of intercept depends on the concentration of impurity.

(d) Heat capacity of the electron gas

According to the classical free electron theory, it is suggested that all the valence electrons (Le. free electrons) will absorb the thermal energy. According to the law of equipartition of energy, every electron possesses an average kinetic energy given electrons per mole of a metal is given by :

by 'i kBT . Thus the total kinetic energy of the 2

1126 Fundamental Concepts of Inorganic Chemistry 3

U=-kBTxN A

••• (12.3.3.20)

2

= number of valency electron per metal atom, e.g. x = 1 for alkali metals, N A = Avogadro number). If the metal is heated, the free electrons will absorb the energy and the electronic specific heat per mole of the metal is : dU 3 3 . [CY]el =- =- kBN AX = - Rx, (R = universal gas constant) . ...(12.3.3.21) dT 2 2 . For an alkali metal (Le. x = 1), it becomes about 12.5 J mol- 1 K-l and this value is much higher than the experimentally observed one.

ib ra

ry

(x

m yl

In fact, the total molar specific heat capacity of a metal arises from the electronic specific heat and lattice vibration, Le. :

lic

C

and

k

H

er e

t.m e/

th

ea

lc

he

[Cy ltotaJ = [Cy ]el + [Cy llat ...( 12.3.3.22) It has been experimentally found that for alkali metals, [Cy]total is itself 3R and the contribution of [Cylel is negligibly small. It supports the fact that all the free electrons will not participate in sharing the thermal energy. Thus the law of equipartition of energy and classical Maxwell-Boltzmann distribution function are not applicable. From the Fermi-Dirac distribution function, it is evident that on heating from 0 K, all the electrons do not gain the thermal energy; only the electrons within the energy range of kBT from the Fermi level are excited (cf. Figs. 12.3.2.3 and 12.3.2.4). The excited electrons gain the thermal energy of the order of kBT and go to the higher unoccupied energy states. Thus, if we consider the total valence electrons (TVE) NA per mole (for alkali metals), then only a fraction of the order of TITF (= kBTIEF ) of the TVE (= NA ) can only be excited thermally at temperature TK. These electrons lie within the energy range of kBTfrom the Fermi level having energy EF . To excite all the electrons, the required energy is EF (= kBTF' TF ~ 5 x 104 K). Thus the total external energy due to thermal excitation of the electrons is :

U=N(~:) (% kBT) = [N;:~) (%) T

2

...(12.3.3.23)

A

(Taking x = 1 for alkali metals)

dU (kBT) =3R (kBT) [CY]el =-=3N AkB dT EF EF (0,'

=0,

,T

...(12.3.3.24)

= 3RkB / EF)

For sodium metal, E F ~ 3.2 eV, T= 300 K, kBT= 0.025 eV and it leads to [CY]el = 0.008R. Thus, only about 1% of the free electrons participate in electronic specific heat. The lattice vibration contributes (at higher temperature) to [Cyltotal and this contribution is 3R and thus the contribution of [Cvl el is negligible to [Cyltotal. This has been experimentally verified. More detailed calculation shows : '"'J

...(12.3.3.25)

Bonding in Metals and Metal Clusters: Electrical Conductivities of Solids

1127

Thus for alkali metals (Le. x == 1), we have:

,...-----------------. 2

[Cylel

= r.

2

(!-) TF

R, (quantum mechanically)

[Cylel = ~ R, (classically)

ry

2

[Cv llDlal = [Cv leI + [Cv 10/ = aT + pT

3

ib ra

.The total specific heat is the sum of electronic specific heat and lattice specific heat (= PT3).

[Cvl/otal AT 2 ---( 12.3.3.26) T This temperature dependence (a., p constants) is valid at low temperature range and it has been experimentally verified (cf. Fig. 12.3.3.3). At higher temperature, [Cvl /at is independent of temperature.

ea

lc

he

m yl

--~=a.+p

Le.

th

lev]

t.m e/

I 0

JI~

tanO

=~3

I

----+ T(K)

----+

r2

er e

1: [Cy]total = aT + J3 T3, 2: [CY]lat = PT3 , 3: [Cv]el = (l T

Fig. 12.3.3.3. Dependence of specific heat of metals on temperature.

k

H

(e) Relationship between electrical (a) and thermal conductivity (5) : Wiedemann-Franz law and Lorenz number

lic

The thermal conductivity due to the free electrons is given by :

1-

C

S=-[Cvle,lnc 3

...(12.3.3.27)

[C lei = ~ k B =

average electronic specific heat of an electron, n = density of the electrons, / = 2 mean free path, c == root mean square velocity. It leads to : where

y

B (~kB) = n/kB J3k B T

S = }-n/ J3k T 3 m and

0"=

ne 2 1

22m

f.i"i7C,(cf.Eqn.12.3.3.15) 2 3k BTm

v

...( 12.3.3.28)

1128

Fundamental Concepts of Inorganic Chemistry

It leads to :

~ = 3T (k:

J

= LT,

(classical free electron theory) ...(12.3.3.29)

ry

L= Lorenz number =3 (k: J

ea

lc

he

m yl

ib ra

The ratio and Sand cr at a particular temperature is constant and the ratio depends linearly on absolute temperature (irrespective of the nature of the metals). This is Wiedemann-Franz law. Thus the classical free electron theory can explain the law but the calculated Lorenz number is different from the experimental value. Now let us examine the situation in the light of quantum mechanics. For the heat transport phenomenon, not all the electrons but the electrons close to the Fermi level participate in thermal conduction. The velocity of such electrons can be calculaed. 1 2 -mu F = E F 2 This thermal conductivity in terms of quantum mechanics is given by : 2

th

- v lei I n UF' [C - v lei = average electronic specific heat of an electron = -1t S == -1 [C 3 2 2

(k T) k _B-

~

B

t.m e/

:;- - (kBT)] -kBnuFI ,(cf.Eqn.12.3.3.25 for [Cv]el) E 1 [1t -' 2

F

]

~[nkB2TUL,/] [~2 = [~] 3 nk~Tt r

mU

...(12.3.3.30)

m

F

er e

6

( using, E F =

~mu~, / = tU F )

lic

k

H

Quantum mechanical calculation for electrical conductivity (cr) yields:

C

Thus,

a

2

= he

t, (cf. Eqn. 12.3.3.9)

m

s/cr=( ~2)(k:

J

T=LT, (Quantum mechanical theory) 2

r]

...(12.3.3.31)

This is Wiedemann-Franz law and L [ = 1t ( k: nicely agrees with the experimental value. 3 However, this Lorenz number is close to 3(kB / e)2 obtained from classical theory with a Maxwellian distribution of velocity. Note: Here it has been assumed that the mean collision time (t) is the same for both the conductivities. This is a reasonable approximation when the electrons are the carriers for the both types of conductivities. But the phonons can also transport thermal energy to some significant extent.

Bonding in Metals and Metal Clusters: Electrical Conductivities of Solids 1129 (f) Paramagnetic susceptibility of conduction of electrons

= nJlB2 H / kBT

But experimentally the spin susceptibility ( X =

~)

...(12.3.3.32)

m yl

M

ib ra

ry

There are some ferromagnetic transition metals. The nonferromagnetic transition metals very often show the temperature independent paramagnetism (T1P) known as Pauli paramagnetism which is quite unexpected from the classical theory. An electron is associated with a magnetic moment of one Bohr magneton, JlB (1 BM). The Curie-type paramagnetic contribution leads to the magnetisation : M = nJl~H / kBT in the presence of field H. This result is achieved by considering the fact that the probability an electron will be lined up parallel to the field exceeds the probability of antiparallel alignment by roughly JlIII/kBT. It lea~s to net magnetisation for n-electrons per unit volume (in .terms ofclassical statistics).

is indendent of temperature and its value is

t.m e/

th

ea

lc

he

very small compared to the one obtained from the above relation. The experimental result can be explained by considering the Fermi-Dirac distribution function instead of Boltzmann function. In the absence of any ext~mal field, half of the electrons are with spin along the +z-direction while the other half is aligned with spins in the -z-direction. Thus, in absence of any field, no net magnetisation.results. When a magnetic field (H) is applied in the z-direction, the energy of the electrons with parallel spins is lowered by the amount JlIII while the energy of the electrons with the antiparallel spins is raised by JlIII. This produces an unstable situation. This is why, the electrons residing very close to the Fermi level start to change from their antiparallel arrangement to parallel arrangement and it causes a net magnetisation. In fact, the electrons lying within the range of kBTfrom the Fermi level canflip their spins to align themselves parallel to the field. Thus the fraction T / TF (= kBT / EF ~ 0.02 for EF ~ 1 eV at T ~ 300 K) ofthe total electrons can participate in the magnetisation process and thus the susceptibility (X) is given by :

C

lic

k

H

er e

E

2JJ s H Density .....---t-------of state

--~

(i) (No field)

(ii i) ~

(In thepr;;=sence of magnetic field)

Fig. 12.3.3.4. Explanation of Pauli paramagnetism (Le. TIP) in metals. (i) In the absence of field (Le. H = 0). (ii) Instantaneous and unstable arrangement of the parallel and antiparallel spins (with respect to the direction of field). (iii) Equilibrium and stable arrangement of the parallel and antiparallel spins to maintain the Fermi level in the presence of the field. Total energy (E) = kinetic and magnetic energy of the electrons.

1130

Fundamental Concepts of Inorganic Chemistry 2

M = n~BH

2

x.!- = n~BH

kBT M

...(12.3.3.33)

kBTF

2

n~B

= kBTp

,

~ ..(12.3.3.34)

(independent ojtemperature)

ry

X= H

TF

m yl

ib ra

It indicates indicates that for EF == 3 eV, at the room temperature range only 1% of the electrons contribute to the paramagnetism. This temperature independent paramagnetism is known as Pauli paramagnetism and it is mainly found in nonferromagnetic transition metals and other metals like Na, K, Li, Al etc. This Pauli paramagnetism is important in such transition metals where the density of energy states (DOS) in the d-band at the Fermi level is very high. In fact, because of this fact, their electronic specific heat is also very high.

ea

lc

he

The simple metals are often diamagnetic because in such cases, the diamagnetism due to the filled shells outweighs the Pauli paramagnetism, if any, due to the conduction electrons. Note: It may be noted that Van Vleck Paramagnetism (a 2nd order Zeeman effect) and DiamagJletism are also temperature independent. 12.3.4 Electrical Conductivity and Hall-Effect

t.m e/

th

In some metals (e.g. Zn, Cd) and p-type semiconductors, not the electrons but the positive holes are the predominant electricity carriers. To determine the nature of the electricity carrier, consideration of sign of.the Hall-potential is extremely important. The Hall-effect indicates the effect ofa magnetic field Magf'1etic field " N ..... , on electrical conductivity.

,. -- . . r

h

.

I

,

C

lic

k

H

er e

Let us consider a current of density Jx passing J «I through a metal strip (in the xy-plane) in the \ x-direction. Now if a magnetic field of strength H z is placed in the z-direction, Le. perpendicular to Fig. 12.3.4.1. Representation of hall effect. the metal strip, then the electricity carriers (say electrons) will be deflected and it iwill develop an electrical field (Ey ) along the width in the y-direction. This will create a potential difference (~H' called Hall-potential) between M and N (see Fig. 12.3.4.1). Let us illustrate the phe'oomena by considering the movement of the electrons before and after appl ication of the magnetic field. When the electricity flows in the +x direction, the electrons move along the -x direction. When the magnetic field is introduced, the moving electrons tend to bend towards downward (Fig. 12.3.4.2). Consequently, the electrons accumulate on the lower surface and the positive charges appear on the upper surface. This separation of positive and negative charges between the upper and lower surface of the strip produces the Hall potential acting along the y-direction. Therefore, to balance the effects of the magnetic and electric field, we have:

Heu x We have: Jx

==

==

neux ' (n

eEy ; ==

(ux

==

velocity of the electrons in the x-direction)

number iof electrons moving towards the field per unit volume).

Bonding in Metals and Metal Clusters: Electrical Conductivities of Solids

1131

---+. Direction of current flow Upper surface (accumulation of positive charge) +

+

+

+

+

+ Ux

--------

y

+

e

;.------.x

ry

+

z

m yl

Hall field . . . . - . - - - - - - - - - - - Lower surface (accumulation of negative charge)

ib ra

[email protected]

Fig. 12.3.4.2. Development of Hall potential and motion of electrons in a conductor in presence of a magnetic field In the z-direction.

Ey == HJx / ne

he

Combining the above two relations, we get:

lc

...( 12.3.4.1)

The Hall-potential ~H is defined as, ~H == lEy, (I == width of the strip in the y-direction).

ea

th

IHJx

~H = - - = RHIHJx

Therefore,

...( I2.3.4.2)

t.m e/

ne

H

er e

where RH == I/ne is called Hall-coefficient. From the measurements of ~H' I, Hand Jx ' RH can be determined from which n is determined. The Hall-coefficient (R H ) may have both positive and negative values. The negative value indicates that the electrons are the carriers while the positive value indicates the holes as carriers. The Hall-coefficients for some common metals are given below.

Cu

Ag

Au

Na

Zn

Cd

-8.4 -7.2 -25.0 +3.3 +6.0

Upper surface (accumulation of negative charge) Hall field

I

+

+

+

+

+

+

+

Lower surface (accumulation of positive cha rge)

Fig. 12.3.4.3. Motion of positive holes in a conductor or p-type semiconductor in the presence of a magnetic field in the z-direction.

k

R H / (10- 11 m 3 C-I) -5.5

- - - -.... Current

C

lic

Note: If the electrons are carriers then the Hall potential is developed in the negative y-direction. On the other hand, if the positive holes are the carriers, the Hall potential is developed in the opposite direction because the electrons and the positive holes move in the opposite directions (Fig. 12.3.4.3). Negative value of RH is quite expected from the free electron theory, but the positive value of RH is quite unexpected.

12.4 THE BLOCH THEORY (i.e. ZONE THEORY) OF METALLIC BONDING 12.4.1 Basic Concepts of Bloch Theory In the free electron theory, the electrons are supposed to travel in a uniform field throughout the metallic but such a over-simplification is not tenable in reality. In a metallic crystal, the positively

~rystal,

1132 Fundamental Concepts of Inorganic Chemistry

4>k

m yl

ib ra

ry

charged kernels are periodically oriented. The principal interaction between the electrons and kernels is electrostatic in nature. When an electron is at the closest position to a kernel, the potential energy is minimum while it is maximum when the electron is at the farthest distance from the kernel. As the kernels are periodically arranged, the potentialfield also varies periodically. Thus the pote!ltial energy is considered as zero, when the electron lies between the kernels and it is highly negative at the very close position to the kernel as shown in Fig. 12.4.1.1 which represents a one dimensional array of the kernels. By considering the periodic potential of the lattice points, behaviour of the electron can be expressed, by ~k which is a product of the wave function of a free electron, 'IIx and a function of the lattice periodicity, u(x), Le. :

= \jI(x) u(x) = exp(ikx)u(x), k = 27t A.

lc

he

~k is known as Bloch wave function named after F. Bloch (1928). Here we shall discuss the matter in a qualitative way.

~

e> ~

cg

Cl.

-Ve

th

0

t.m e/

~

Positively charged kernel

-·n·nen·ner+--d----+

Q)

o

ea

i +Ve

er e

Fig. 12.4.1.1. Periodic variation of potential energy in the field of kernel (one dimensional array) in a metallic crystal.

C

lic

k

H

Such a periodic potential predicts that the lattice points (Le. kernels) influence the behaviour of the electrons within the crystal. Thus it is reasonable that the electrons are not equally available everywhere within the crystal. In some regions, the probability is very high while in some regions, the probability is very low. This aspect can be rationalised by considering the Bragg's reflection. The electrons having wavelength comparable with the spacing (d) of the lattice planes may suffer Bragg's reflection under suitable conditions as given by :

nA. == 2d sin e, n == I, 2, 3,

... ( 12.4. 1. 1)

where e is the glancing angle of the electron wave at the lattice plane from which it is reflected. Thus it is evident that an electron cannot travel throughout the crystal due to the Bragg's reflection. The Bragg's relation (Eqn. 12.4.1.1) can be expressed in tenns of wave number (k) which is defined as k == 21t/A. Thus, or,

k=~

... (12.4.1.2)

k sin e = n1t d

... (12.4.1.3)

dsin

e

,:

.

:

!

Bonding in Metals and Metal Clusters: Electrical Conductivities of Solids 1133



From Fig. 12.4.1.2 showing the Bragg's reflection in a two dimensional square lattice, it is evident that k sin e is the component (kx ) of the wave number k in the x-direction (normal to the Bragg's vertical lattice plane).

'J'

I I

~



m yl



n1t

. • t· . kx

k{ = d' (lattice plane perpendicular to x-axis)



lc



n1t



• • •

= k sin 0 = n1t I

~

d



ea

(lattice plane perpendicular to y-axis)

ry



ib ra

...(12.4.1.4)

is given by,

= d'

,..-d---+e

Kernels

In general, it can be concluded that if the component of k in the direction normal to a lattice plane is n7fld, then the electron wave will experience a Bragg's reflection from the lattice plane. Thus the condition for Bragg's reflection

ky

1 d

I

he

kx =n1t/d

Thus,



t.m e/

th

Fig. 12.4.1.2. Bragg's reflection of the electron where n = 1 (first order reflection), n = 2 (second wave in a metallic crystal. order reflection), and so on. Here it is evident that the critical values of the components of k for the Bragg's reflection are inversely proportional to the interplanar distance d. Thus the planes (hkl, in terms of Miller indices)

having the largest value ofdhkl

[= J

2

1

2

2

) will be the first planes to influence the behaviour

h +k +1

er e

of the electrons through the Bragg's reflection.

H

Ifwe consider a simple cubic (sc) system, the planes of the family {100}, Le. 010 (horizontal) and 100 (vertical) planes, will act as the first reflecting planes. The second most widely apart planes in the family {110} will be the second set of the reflecting planes.

C

lic

k

The reflecting planes will confine the electrons in a particular zone within the three dimensional crystal. These zones are called Brillouin zones which are the polyhedra formed in k-space by the different sets of reflecting planes. When the vector k is considered to have the components kx' ky ' k= along the x, y and z axes of crystal respectively, it defines a k-space in which the Brillouin zones exist. The zone in which the 1st order reflections (i.e. n = 1) are confined is called the 1st Brillouin zone, similarly for the 2nd order reflection there is a 2nd Brillouin zone and so on. It is interesting to note that the area (in two dimensions) of each zQne in a particular crystal is constant. In the k-space, two dimensional Brillouin zones are shown in Fig. 12.4.1.3. The first zone is confined by kx = ± 1t/d and ky = ± 1t/d. When the component of k increases so that, Ikx . yl > 1t/d, the electrons do not fit in the first zone. The 2nd zone accommodates the electrons for which Ikx . yl = 1t/d to 21t/d. In a particular zone up to the limiting value of k, the energy varies with k quasicontinuously ~s predicted from the free electron theory but as k goes to attain the limiting value, it deviates and at the

limiting value, there is an energy gap.

1134 Fundamental Concepts of Inorganic Chemistry

2nd Brillouin zone



ry

d -21£

ib ra

d

m yl

1st Brillouin zone 1 -21£

he

Id Fig. 12.4.1.3. Two dimensional Brillouin zones in k-space. 12.4.2 Brillouin Zone, Forbidden Zone and Conductivity

ea

lc

Let us first consider the one dimensional case and for the three dimensional case we can proceed in the same way. From the Sommerfeld's free electron theory, the energy of an electron in a one dimensional metallic crystal is given by :

th

Ex = h 2 k; /(87t 2 m), (cf. Eqn. 12.3.2.4)

t.m e/

From this parabolic Ex - kx relation (see Fig. 12.3.2.1), it has been found that the energy levels are

quasicontinuous. But because ofthe existence ofthe Brillouin zones, for k = ± n7t (see Eqn. 12.4.1.4), x

d

lic

k

H

er e

there is a discontinuity of energy. For such limiting values, the electron waves are Bragg reflected back and forth and the electron waves appear as standing waves rather than the travelling waves. The Schrodinger's equation for such standing waves at the critical values of kx leads to the two possibilities having energies E1 and E2, and there is no energy level between E) and E2 • The lower value corresponds to the highest value of the n-th Brillouin zone while the higher value corresponds to the lowest value of the (n + 1)-th Brillouin zone. Thus there is an energy gap at the critical values of kx • This 3spect is represented in Fig. 12.4.2.1. I'his E-k plot determines the shape ofthe valence and conduction bands. It determines the properties of direct gap and indirect gap materials (cf. Sec. 12.9.8). The energy gap between the minimum of

C

the conduction band and maximum ofthe valence band is denoted by Eg (jo,bidden energy gap). The energy gap between the Brillouin zones explains the conductivity of the solid. Before the limiting values of k, the N(~E curves show the nonnal parabolic behaviour (shown by dotted line) but when the interaction starts at the limiting values of k due to the Brillouin zones, it deviates. The N(~E relation for two successive zones is represented in Fig. 12.4.2.2. The energy zone may be separated or overlapped depending on the nature ofthe matter. The seplrated zones (i.e. existence of a forbidden gap) are well understood, but for the overlapping zones it requires an explanation. The magnitude of aforbidden energy gap depends on the direction in the crystal lattice. If the highest energy level of a zone exceeds the lowest energy level of the next higher zone then the overlapping occurs. It occurs generally for the metals.

Bonding in Metals and Metal Clusters: Electrical Conductivities of Solids

1135

E

-2n

T

0

-n

7

n ~

lc

-3n d

he

m yl

ib ra

ry

t

21t

3n d

7

ea

Fig. 12.4.2.1. The relationship between E and k; development of forbidden energy gaps at some critical

t.m e/

th

values of k.

er e

"",.

---+

E

,

~

J\ I

I

I I

'"

\

~ Zone II I

---+E

H

Fig. 12.4.2.2. N(E)-E curve: separated Brillouin zones and overlapping zones.

C

lic

k

The basic reason for the difference in conducting power among the conductors, insulators, semiconductors and semimetals can be interpreted in terms of the zone theory. • Conductors: Where the topmost Brillouin zone is incompletely filled, or a vacant (or partially filled) Brillouin zone overlaps with a filled (or partially filled) Brillouin zone, the electrons are available for conducting electri'city. For example, in alkali metals, there is a half-filled Brillouin zone; for the alkaline earth metals, a filled Brillouin zone overlaps with a vacant Brillouin zone. • Insulat:)rs: When the topmost Brillouin zone is completely filled in and it is separated by a large energy gap from the next vacant Brillouin zone, the substance shows no conducting power. The forbidden energy gap is so high that it cannot be surmounted by the available thermal energy 3t ordinary temperature. Generally, if the forbidden energy gap exceeds the value 3 eV, the substance is regarded as an insulator. • Semiconductors: If the energy gap "between a filled zone and a vacant zone is not too high, then some of the electrons can move to the vacant zone by gaining the thermal energy available at room temperature. Such substances are called semiconductors. With the increase of temperature, the number

1136

Fundamental Concepts of Inorganic Chemistry

er e

t.m e/

th

ea

lc

he

m yl

ib ra

ry

of electrons capable of passing into the higher zone increases, and consequently, the conducting power increases. Thus at 0 K, a semiconductor will be an insulator. Generally for the semiconducto/,s, the energy difference lies in the range 2-3 eVe The properties of the semiconductors will be disc·ussed elaborately in Sec. 12.5.6. • Semimetals: In the band gap, there is no energy level (Le. zero density ofstates). In some cases, a filled band of lower energy may just coincides with an empty band of higher energy. In such cases, there is a zero density of states (DOS) at their junction (cf. Fig. 12.5.1.5). In fact, the conduction band edge (Sec. 12.5.8) slightly overlaps with the valence band edge and it leads to a small concentration of the electrons in the conduc.tion band and holes in the valency band. Such solids are described as semimetals which are characterised by their low metallic conductivities. In fact, only a few electrons can participate in carrying the electricity. Graphite shows this semimetallic conductivity in the directions parallel to the sheets of carbon layer (Sec. 10.11.2). Bismuth, antimony, arsenic also show the semimetallic properties. • Temperature dependence of electrical conductivity in superconductor, conductor (metals) and semiconductor: In conductors (e.g. metals), the conductivity decreases with the rise of temperature (cf. Sec. 12.3.3). In the case of superconductors (Sec. 12.13) below Tc ' the conductivity increases with the decrease of temperature. In semiconductors, the conductivity increases with the rise of temperature. A particular substance can behave in different ways in different temperature ranges. A superconductor above Tc shows the metallic conduction (Le. cr decreases wtih the rise of temperature). Bismuth in solid state shows the metallic conduction at lower temperature but on melting (544 K) there occurs a phase transition and in the liquid state, it shows the semiconductor property (Le. cr increases with the increase of 1). It is quite interesting to note that the monoxides of the early 3d-metals (e.g. TiO, va, etc.) show the metallic conduction while the monoxides of the late members (e.g. MnO, FeO, CoO, NiO) show the semiconductor property. This has been explained by considering their bonding characteristics (cf Sec. 12.10).

H

12.5 MOLECULAR ORBITAL THEORY OR BAND THEORY OF METALLIC BONDING

lic

k

12.5.1 Formation of Energy Bands and Brillouin Zones

C

According to LeAD-MOT, if two atoms having one orbital on each atom undergo ~ombination, two molecular orbitals, one bonding and another antibonding are formed. For such three combining atoms, three molecular orbitals, i.e. bonding, nonbonding, and antibonding, are formed. Thus for an array of such n atoms, there will be n molecular orbitals. These are multicentred MOs. For a real metallic

crystal, n is significantly large. In this situation the large number of multicentred molecular orbitals .formed remain closely packed and they form an energy band (Fig. 12.5.1.1) in which the energy levels

are quasicontinuous (see Sommerfeld's quantum mechanical free electron theory). In a particular energy band, the topmost levels arefully antibonding in nature and the lowest energy levels are fully bonding in nature, and the intermediate levels are ofintermediate character; It happens so for different possible combinations of a large number of atomic orbitals coming from a three dimensional array of the combining atoms.

Bonding in Metals and Metal Clusters: Electrical Conductivities of Solids

~ C')

c

(1)

,

i

~

C

W

(ASMO)

\ \

I

w

I

cr*

1, ---

,

I

\

I

\

I

\

,--3s

ib ra

\

\

3s (1 AO)

ry

~

~

(1)

1137

\

,

I

I

\ \

,

(1 AO)

I

\

\

(SMD)

J

m yl

\

cr

---+ Internuclear distance

(b)

he

Actually. the band consists of multicentred MOs

I,I ", "I

lc

~

ea

~ (1)

C

W

I ~

~

1

t.m e/

th

(1)

(5 AOs)

-'

~~,

,

\\ \ \\ \\

,

\

(d)

er e

~

Most anti-bonding

H

~

II ,

~~~

\

----+ Internuclear distance (c)

I,

Q)

c:

w

lic

k

I

Fermi level

C

1

-----+

Internuclear distance (e)

~(ADs) 3s

3s (ADs) 0)

C

~

.5 .0 t3

~

EF

Q)

c:

W

1

Nonbonding region

0) Q)

'(ii.S "0 as -c as ~ § m

>.

C')

Most bonding (f)

(g)

DOS

Fig. 12.5.1.1. Illustration of formation of 3s-band in Na by using the 351 atomic orbitals (AOs) of sodium. (a) and (b) : Interaction of 35 AOs of two Na-atoms giving rise to a a-bonding molecular orbital (SMO) and a a* antibonding molecular orbital (ASMO). (c) and (d) : Interaction among 10 Na-atoms. (e) and (f) : Interaction among a farge number of Na-atoms (say one mole) to produce the 3s-energy band which is half-filled. (g) Half-filled 3s-band of Na (Energy vs. DOS, cf. Fig. 12.5.1.4).

1138

Fundamental Concepts of Inorganic Chemistry

t.m e/

th

ea

lc

he

m yl

ib ra

ry

The different valence orbitals form such different energy bands. Due to the lo~ degree of overlap in the case of inner orbitals, the band width for the inner orbitals is negligibly small. Rather it is reasonable to consider that the inner electrons are basically associated with their own atomic orbitals. The different energy bands are called Brillouin Zones (cf. Bloch theory). The electrons are distributed in order of increasing the energy states ofthe energy bands (cf. Aufbau principle of electron filling in atoms). The two successive energy bands may be separated or overlapped (see Figs. 12.5.1.2, 12.5.1.3 and 12.5.2.1). The energy gap between the successive Brillouin zones is called forbidden energy gap. The overlapping occurs when the highest energy level of the lower energy band exceeds the energy of the lowest energy level of the next higher energy band. In the case of overlapping bands, before the completion of electron filling in the lower band, the filling of the next higher overlapping band starts. According to Pauli's exclusion principle, a band containing n levels can accommodate maximum 2n electrons. The band width depends on the degree of overlapping which is controlled by the energy difference of the involved atomic orbitals and internuclear separation. Thus the energy difference between the most ABMO and most BMO gives the measure of band width. This difference is fixed for a particular band. The magnitude of energy separation between the different energy bands depends on the above two factors as illustrated 'in Figs. 12.5.1.2-3.

p-band

1

1

~

2l Q,)

2> Q,)

c

~

-2l Q)

Band gap

w

1

c

c

w +-

Vacant levels

s-band

er e

w

Vacant p-band

~

Fermi level Filled levels

(b)

(c)

s-band

k

(a)

H

s-band

C

lic

Fig. 12.5.1.2. Possibility of overlapping between the 5- and p-bands. (a) The two bands do not overlap to produce the energy gap. It indicates that the interaction among the atoms is relatively weak Le. band widths are smaller. (b) Overlapping of two bands (as in Ca) where the interaction among the atoms is relatively stronger, Le. band widths are larger. (c) For the ns'n{iJ electronic configuration as in the case of K, the s-band remains just half filled and the p-band remains vacant. Here s-band acts the conduction band. • Band width: It is determined by the energy difference between the most bonding and antibonding

combination of the atomic orbitals. Whatever may be the number of combining atomic orbitals, all the energy levels will be confined within these limits. Efficient overlap of the orbitals gives a large band width while inefficient overlap gives a narrow band width. Obviouly, DOS is relatively higher in a narrow band.

Bonding in Metals and Metal Clusters: Electrical Conductivities of Solids 1139

~

0) ~

.Q)

c

~

3d

e>

ry

P 2 } Valence electron 25 levels

Q)

c: W

I

w

4s

m yl

1s (Inner electron levels)

ib ra

I

(Internuclear distance)

-----+ (Internuclear distance)

he

Fig. 12.5.1.3. Formation of different energy bands; band width, a function of degree of overlapping .

C

H

lic

k

1

er e

t.m e/

th

ea

lc

• Density of states (DOS) in an energy band: The number energy levels (Le. multicentred MOs) per unit energy increment gives the measure of density of states. Thus DOS denotes the number of energy states per unit volume in the interval E and E + dE. It may be noted that a particular energy band is not uniformly packed, Le. DOS is not the same throughout the band. In three dimensions, the centre ofa band,is most heavily populated (i.e. DOS is maximum) while the edges (both topmost and low~st) are poorly populated (i.e. DOS is less) (Fig. 12.5.1.4). The levels (Le. MOs) close to the ed§es . are either fully bonding (in the lowest edge) or fully antibonding (in the topmost edge) in char-acter. There is only one way for the formation of a fully bonding MO or a fully anti-bonding MO. On the other hand, the MOs at the centre of the band are having the intermediate character of bonding and antibonding. In a three dimensional array of the combining atoms, there are many possible ways

~

e> Q)

c

w

Fermi level s-band

I ---. Density of states (DOS) (a)

---+ Density of states (b)

Density of states (DOS) (c)

Fig. 12.5.1.4. (a) Representation of density of states (DOS) i.n the s- and p-bands in a typical metal (e.g. Na) where the half-filled s-band is the conduction band. (b) Overlapping of filled s-band and vacant p-band as in Ca. (c) Overlapping of (n - 1)d band and ns band in transition metals. Relative positions of the d- and s-bands depend on the cJ1 configuration.

1140

Fundamental Concepts of Inorganic Chemistry

of linear combinations to produce the multicentred MOs having the intermediate bonding and antibonding character.

Empty band (conduction band)

--------------------------------------------------------------------

1

t.m e/

th

rc-Band

ea

lc

he

rr*-Band

m yl

ib ra

ry

In the band gap there is no energy level. If a filled band just coincides with an energy band then at the junction, DOS is zero (Fig. 12.5.1.5) and such solids (Le. semimetals) are weakly conducting. Graphite is an example of semimetal. In graphite, the p-orbitals perpendicular to the layer undergo n-type combination to produce n-BMOs and 7t*-ABMOs. These 7t-BMOs of lower energy produce the valence band while the 7t*-ABMOs produce the conduction band. Actually, here the filled VB and empty CB are generated through the splitting of the half-filled tr-band in the region of nonbonding energy levels (cf. Figs. 12.5.2.3, 12.11.3.2). These two bands reside in a way to mak~ the band gap almost zero. The term semimetal is meaningful in terms of electrical conductivity but not in terms of chemical properties of metals and nonmetals.

~

Density of States (DOS)

Fig. 12.5.1.5. Position of filled and empty bands in a typical semimetal (graphite) (cf. Question No. 164).

C

lic

k

H

er e

The DOS in the d-band is much higher than that in the s- or p-band. The d-band is also narrower. This aspect has been discussed in explaining the properties of transition metals (cf. Sec. 12.5.3) • Merits of band theory of solids: It may nicely explain the electrical conductivities of solids and the properties of transition and nontransition metals. These aspects have been discussed in subsequent sections. Photoelectric properties of the materials can also be nicely interpreted. In fact, for the photoelectric materials, the energy gap between a filled band and an empty band is comparable to the visible/ultraviolet photon energies. The high reflectivity or luster of the metals can be explained in terms of band theory. In metals, the unoccupied energy levels lie close to the occupied levels and the energy leve!s are close-packed (Le. quasicontinuous). Consequently, metals can absorb and subsequently emit radiation of all wavelengths in the visible range. It gives the ",eta/lie luster and high reflectivity. Optical properties (cf. Sec. 12.5.8) of the materials having a finite gap between the filled and empty band can also be explained. 12-.5.2 Band Structure of Nontransition Metals

• Alkali m~tals (Gr. I): Let us consider the metallic crystal of lithium atoms, Le. Lin' From the electronic configuration of Li (1 s22s 1), the 2s-orbitals will form an energy band containing n-molecular orbitals closely packed. Energy of the levels is given by :

Bonding in Metals and Metal Clusters: Electrical Conductivities of Solids 1141

( 21tm)

E = J + 2K cos -n-

t.m e/

th

ea

lc

he

m yl

ib ra

ry

where m == 0, ± 1, ± 2, ....; J == Coulombic integral, K == exchange integral, n == number of atomic orbitals contributed, by n atoms. It indicates that whatever may be the value of n, the energy levels are confined within the energy limits, J + 2K and J - 2K. The difference between these two limiting values of energy gives the measure of the band width. Obvio~sly, for a large value of n (~ 1023 cm-3), the energy levels will form a continuum within the said energy limits. This continuum is described as an energy band. For Lin' the band will have the maximum capacity 2n, to accommodate the electrons but the available electrons are n. Hence the band will remain only half-filled (see Fig. 12.5.2.la). Thus for each alkali metal, the highest energy band remains always half-filled. For sodium, probably 3s and 3p bands overlap. It may also happen for other alkali metals. • Alkaline earth metals (G r. 2): Let us illustrate for beryllium. For the beryllium (1 s22s 2) crystal, we have two bands, i.e. 2s and 2p bands, having the electron accommodating capacities 2n and 6n respectively. If the bands were separated, the 2s band would remain full-filled while the 2p band would remain vacant. But in fact, these two bands are mutually overlapping and the 2s electrons can occupy both the bands. Thus no band is completely filled for the alkaline earth metals (see Fig. 12.5.2.1b).

C

2p

"0 C

.Q +J_

ro°"O

co:::JC "Oro

a.c.o

N'§'

Half-filled

zone

H

er e

2s

Mos

Aos

Mos

(nLi)

(Lin)

(nBe)

(Ben)

lic

k

Aos

(a)

(b)

C

Fig. 12.5.2.1. Formation of energy bands consisting of multicentred MOs in alkali metals (a) and alkaline earth metals (b).

• Gr-III (13) metals: For the Gr-III (13) elements (ns 2np l) like AI, the outermost np band remains partially filh.:d in while the ns band is completely filled in. Thus the Gr-III (13) elements are comparable with the Gr-I (1) elements, but in the Gr III (13) elements, probably the filled ns band and partially filled np band overlap.

• Band Splitting: Different Types of Band Structure for Gr IV (14) Elements - why? (see Question No. 164)

For the Gr I( 1), 11(2), 111(3) metals, the general observation is that the ns and np bands overlap. Each

1142 Fundamental Concepts of Inorganic Chemistry

'0

2sp3

'I'I 'I

'1// /1,/

ry

\\,

\\ \\ \\ \

Sp3_ cr band (VB)

ib ra

(C)n

\

ns-band (Filled)

ns-band (Partly filled) \

Filled

m yl

ns-band (Half-filled) '----y--J (a)

.

Vacant

(~)n\"

np-band (Partly filled)

np-band (Partly filled)

sp3- cr *-band (CB)

\

v

v

Overlapping of ns and np band

Overlapping of ns and np band

J

V'

(d ) Band splitting

he

(b) (c) Fig. 12.5.2.2. Schematic representation of the band structure of (a) alkali metals (Gr 1), (b) alkaline

ea

lc

earth metals(Gr 2), (c) Gr 13 (III) metals, (d) Gr 14 (IV) elements (e.g. diamond, Si, Ge), III-IV (Le. 13-14) compounds like GaAs, II-VI (12-16) compounds having 4 valence electrons (in average) per atom (cf. Fig. 12.9.1.2 for the band structure of GaAs). .

t.m e/

th

Note: • For (b) and (c), the ns and np bands are shown horizontaUy shifted just for the clarity of the representation.• For (a), (b) and (c), in reality the ns and np bands mix and it-is better to describe the band as n(s - p) band having 4x energy levels for the (A)x cluster. However, for the sake of simplicity, ns and. np bands are shown separately.

C

lic

k

H

er e

band is constituted by the multicentred SMOs, NSMOs and ABMOs. Thus the overlapped s-p band contains SMOs, NBMOs and ABMOs. For the Gr IV(14) elements like C (diamond), Si and Ge, the s-p band (say sp3 band) splits into two bands where the lower band consists of SMOs and the higher band consists of ABMOs. What is the cause of this band splitting? The exact quantum mechanical explanation lies beyond this book. The nonmathematical explanation indicates that for the Gr. I, II and III metals, the coordination number (eN) in their crystals (as in bee, fcc, hcp) is large with few valence electrons. On the other hand, for nonmetals (e.g. Gr IV or 14, V or 15), the CN is less (cf. 4 in diamond structure) with large number of valence electrons (e.g. 4 for Gr IV, 5 for Gr V). For the (A)x cluster, the number of total energy levels in the n(s-p) band consisting of all SMOs, NSMOs and AB~Os is 4x and full capacity of the band to accommodate the electrons is 8x. For the (A)x cluster of Gr I, II and III metals, having the valence electrons lx, 2x and 3x respectively, mainly the bonding energy levels of the n(s-p) band will"be filled in. They will also earn cohesive energy through th~ large CN. Thus the system is stabilised. For the (A)x cluster of Gr IV elements having 4x valence electrons, the n(s-p) band will be just half-filled and the highest occupied energy levels are mainly the nonbonding ones. These nonbonding electrons cannot stabilise the system. Under the circumstances, if the band splits at the region of nonbonding levels into two bands: lower band constituted by the bonding MOs and higher band constituted by the antibonding MOs, and the lower band having the bonding energy levels is filled in completely by the 4x electrons, then the system is stabilised more. Here, it is not required to place any electron in the nonbonding or antibonding energy levels. This stabilisation is specially important for systems having lower eN.

Bonding in Metals and Metal Clusters: Electrical Conductivities of Solids

1143

Region of nonbonding energy levels

For C (diamond), Si and Ge, they maintain the tetrahedral structure and each centre (ns 2np 2) is sp3 hybridised. These sp3 hybrid orbitals are expected to undergo -a a-type combination to produce a spJ-band consisting of multicentred a-BMOs, a-NBMOs and a-ABMOS as usual and the band will be just half-filled in. To stabilize the system (as discussed above), the half-filled spJ band

//~

j ///~ .,. L:.:.J EF

' ,',

'/

,,:

Eg

'

ry

,,

(5 - p)-a band (2x energy levels)

ib ra

"

,

5p3-band Le. (5 - p)

(s-p)-cr* band (2x energy levels)

(Diamond structure)

lc

he

m yl

undergoes splitting (cf. Figs. 12.5.2.2-3) in band (4x energy levels) the region ofNBMOs to give a lower energy Fig. 12.5.2.3. Schematic representation of splitting of band (i.e. filled VB) constituted mainly by n(s-p) or spLband for the (A)xof Gr IV(~4) elements. the a-HMOs and a higher energy band (i.e. Note : Absence of nonbonding energy levels and empty CB) constituted mainly by the presence of a high energy gap (E g) stabl1lse the a-AMBOs. The lower energy band (Le. VB) diamond structure. is called the spJ a-band while the higher energy band is called the spJ a*-band and the energy gap (Eg ) between these two bands mainly resides

t.m e/

th

ea

in the region of nonbonding energy levels. Here the lower band (Le. VB) isjust completely filled in and the higher band (Le. CB) is vacant and thus the system avoids the placement of electrons in the nonbonding levels. The energy gap (Eg ) decreases in the sequence: C > Si > Ge (cf. Table 12.5.5.1). For the heavier congeners like Sn and Ph, sp3-hybridisation (required for the diamond structure) is not energetically feasible (cf. Sec. 9.13.4, 10.3) and they very often adopt the cubic close packing structure with high coordination number. If, at all, they adopt the diamond structure (Le. splitting of the spJ-band into spJ--(J band spJ-a* band), the band gap becomes very small because of an inefficient sp3-sp3 overlap for the larger ·internuclear separation and more diffused character of the orbitals of the heavier congeners. This concept also explains the insulator property of diamond and

C

lic

k

H

er e

semiconductor property ofSi and Ge. Here, it is important to mention that a large band gap (due to splitting of the spJ-band) in the diamond structure stabilises the system because of the absence of nonbonding levels. But if the band gap decreases, probability ofthe appearance ofthe nonbonding levels increases and the diamond structure becomes lessfavourable. This is why,for Sn and Ph, the diamond structure is notfavoured. The small band gap for Sn and Pb destroys the advantage ofdiamond structure having no nonbonding en~rgy levels. In such cases (Le. Sn, Pb), the close packing system having high coordination number stabi I ises better than the diamond structure. For Pb, the 6s 2 electron pair is not available for bonding (cf. inert pair effect) and the 6p 2 electrons occupy the p-band. The p-band remains partially filled in and it shows the metallic conduction. In fact, it adopts the close packed structure. • Gr IV (14) elements: The electronic configuration ns 2np 2 indicates that the ns-band will remain completely filled in but the np-band should be partly filled in. It.should lead the metallic property to the elements like C, Si and Ge. Their electricaJ conductivity indicates that in their band structure there is no partly filled band and there is a band gap between the vacant conduction band (CB) and filled valence band (VB). In fact, quantum mechanically it can be shown that the nature of band structure of C, Si and Ge is different from that of metals like Na, AI, etc. Here we have discussed the nonmathematical

1144

Fundamental Concepts of Inorganic Chemistry

approach to explain the band structure of diamond and the similar band structure is also present for Si and Ge. The Eg values decrease from diamond (C) to Si to Ge (cf. Table 12.5.5.1). For the heavier congeners,

ea

lc

he

m yl

ib ra

ry

because of the more diffused character of the atomic orbitals and larger internuclear separation, the overlapping of the sp3 orbitals becomes less efficient. It makes energy difference between the a- and a*-MOs less. It explains the gradual decrease of Eg in diamond structure for the heavier congeners. For the metals like Sn and Pb, the normal band structure i.e. filled ns-band and partly filled np-band can explain their properties. However below 291 K, tin adopts the diamond structure. Graphite (cf. Sec. 10.11.2, Question No. 164): Band structure of graphite (C) is quite interesting. rrhe p-orbitals perpendicular to the layers undergo a 1t-type combination to produce a half-filled 1t-band which undergoes splitting (cf. Figs. 12.5.2.3, 12.11.3.2) in the region of NBMOs to give a lower energy band (Le. filled in VB) constituted mainly by the 1t-BMOs and a higher energy band (i.e. empty CB) constituted mainly by the 1t-ABMOs. Astonishingly, the energy gap (Eg ) between the n-band (VB) and 1t*-band (CB) is zero but at the boundary surface of these two bands, DOS (density of states) is almost zero (cf. Fig. 12.5.1.5). It gives the semimetallic property of graphite. The metallic conduction in the intercalates of graphite has been discussed in terms of its band structure in Sec. t 0.11.2.

th

12.5.3 Band Structure of Transition Metals (ct. Sec. 12.5.7 tor magnetic properties)

lic

k

H

er e

t.m e/

From the knowledge of electronic configuration of the transition metals, it has appeared that the (n - l)d levels are very much comparable in energy with the ns orbitals and even in some cases the (r l)d levels may fall below the ns level. This fact becomes predominant when the d electron population dCilsity is significantly high. Because of this fact, in the metallic crystals, the (n - l)d and ns bands undergo overlapping (cf. Figs. 12.5.1.3, 4). From the eltrctronic configuration of(n - 1)d1- 10 ns l - 2, it is evident that the capacity to accommodate the electrons of the(n - l)d band is five times greater than that of the ns band. This is why, the energy levels in the d-band are more compact i.e. DOS is very high. For the elements (e.g. Cu, Ag, Au) having electronic configurations (n -1 )d10ns 1, it is expected that the (n - l)d band is completely filled in while the ns band is just half-filled in. This half-filled ns band can explain the metallic properties. Now let us consider the following elements which are the true representatives of the transition elements and these metals show the ferromagnetism.

C

Fe : 3cJ6 4s 2 ; Co : 3d74s 2 ; Ni : 3d8 4s 2 For the above cases, if the (n - l)d and ns bands were mutually noninteracting, then we would have the full-filled ns band and partially filled (n -1)d band, but in reality, the two bands are mutually interacting and the electrons are redistributed between the bands. From the experimental observations, in contrast to the electronic configurations for the isolated gaseous atoms given above, the electrons are distributed in the solid state as follows: Fe : 3d7· 84so. 2 ; Co : 3d8· 3 4s o.7 ; Ni : 3cP· 4 4s0. 6 Thus in solid state both the (n - l)d and ns bands are partially filled in. Because of the existence of holes in both the d and s bands, the d-s electron transition between the d and s bands may occur. The possibilities of these d-s electron transitions can explain the high

Bonding in Metals and Metal Clusters: Electrical Conductivities of Solids 1145

ry

electronic specific heat and low electrical and low thermal conductivities of the transition metals compared to those of the nontransition metals. The magnetic moment arises from the presence of unpaired electron. The number of unpaired electrons in the 3d metals in solid state changes compared to those in the respective isolated atoms because of the overlapping interaction of the 3d and 4s bands. Thus, the 4s electrons spend some time in 3d orbitals in solid state and it reduces the number ofunpaired electrons

45 band

ib ra

3d band

Fig. 12.5.3.1. Qualitative representhe 3d-band. The number of unpaired electrons in the 3dtation of band structure of iron where band decreases due to the overlapping interaction of the 4s both 4s and 3d bands remain partly band, if the 3d band remains itself half-filled or more than filled. half-jilled as in the case ofFe, Co and Ni. Here, it is important to mention that for the late members of the 3d series, the energy of the 3d band decreases and its energy may be even less than that of the 4s band.

lc

he

m yl

i~

~

CD c:

I! w

.

{

-

electron:

0.3 hole

------------- --------------:-:~~:-:-:- :-:-:~~:-:-:-

:=:=:__=:=:=: =:=:=:__=:=:=:

band

er e

4.7

45

th

Fermi surface

t.m e/

0.6

ea

CJ)

'W'

H

3d band

5

4.4

4.7

electron (J..) electron (I) .......... --.-......- -.....'---

(a)

---------------45

band

-.~

electron (J..) electron (i) .....~ -____ 'W'

3d band

(b)

C

lic

k

Fig. 12.5.3.2. Qualitative band structure of nickel. (a) Above Curie temperature (Tc)' Equal number of holes in both 3d (~) and 3d (t) sub-bands, Le. net magnetisation from the 3d-band in zero. (b) At T < Tc' say T = a K. Filled 3d (i) sub-band is more stable than the partly filled 3d (J,) sub-band as the exchange interaction in the filled 3d (i) sub-band is more. The net magnetic moment 0.6 J-lB per Ni-atom arises from the 0.6 hole in the 3d (t) sub-band which is energetically less stable.

In fact, if the 3d- and 4s-bands were noninteracting in the solid state, the number of unpaired electrons would not change. For example, there are four unpaired electrons per Fe atom in the isolated condition and the corresponding magnetic moment is 4 MB (1 MB = magnetic moment of an electron spin = 9.27 x 10-24 A m2 ), but in solid state, Fe has the corres-ponding value of 2.2 MB only per atom. Similarly Co (with 3 unpaired electrons per isolated 'atom) shows the magnetic moment of only 1.7 8M per atom in solid state; Ni (with 2 unpaired electrons per isolated atom) shows only the value of 0.6 BM per atom in the solid state. Interestingly, in Gd (a member of 4.fseries), the magnetic moment

1146

Fundamental Concepts of Inorganic Chemistry

corresponding to 7 unpaired electrons (4f7) remains unchanged in solid state indicating non-overlapping

Half-filled 4s-band

interaction of the 4jorbitals (i.e. 4jband) with the other energy bands.

Fermi surface

1

overlapping interaction, more directional property giving rise to a large number of non bonding combinations and larger number of AOs) are responsible to produce the higher density of states (DOS) in the d-band compared to the s- and p-band.

Fig. 12.5.3.3. Qualitative band structure of copper. Both the 3d (t) and 3d (~) subbands are filled in i.e. net magnetisation from the 3d-band is zero.

~-

ry

The density ofstates (DOS) in the d-band is very high (cf. Fig. 12.5.1.4). The 3d-orbitals are buried to some extent in the 3s and 3p core. Consequently, dd overlapping is not quite efficient and this will narrow the d-band. The more directional property ofd-orbitals compared to that of s- and p-orbitals will generate a larger number of AD combinations having nonbonding overlap. Each atom can provide five dorbitals for combination. All these factors (i.e. poor

----- -------I ~t

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t~~~~~~~~~~~~~f~~~~~~~~~~~ 5

5

ea

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--electron (J,) electron (i) Half....... ~ filled Filled 3d band

C

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The observed magnetic moments in solid states of Fe, Co and Ni can be explained by considering the number of vacancies (== number of unpaired electrons in the 3dband) per atom in the 3d band of the respective elements. It is considered that the unpaired electrons in the 4s band having much less density o/states (DOS) (cf. Fig. 12.5.1.4) do not participate in determining the paramagnetism (cf Sec. 12.5.7). Thus the number of unpaired electrons in the 3d-band determines the saturation magnetisation and thus the magnetic moment per atom is obtained in the unit 0/8M. From the proposed electronic configurations of the solid metals, the number of vacancies per atom in the d band for Fe, Co, Ni and Cu are 2.2, 1.7, 0.6 and 0 respectively. These are supported experimentally. For example, in copper the 3d band is completely filled in and it is found diamagnetic as expected. This has been/urther verified/rom the alloying systems: Ni-Cu, Ni-Zn, etc. The doping of Cu in Ni means the addition of extra d electron to the solvent crystal lattice. The added extra electrons from the s-band of copper. preferably occupy the vacancies in the (n-l)d band of nickel due to energetic reasons. Thus the number of holes in the dband of the solvent metal (Le. Ni, At. No. 28) decreases with the increase of solute atoms (Le. Cu, At. No. 29) and consequently the magnetic moment decreases with the increase of copper content in the alloy. At about 60 % Cu and 400/0 Ni (in atomic percentage), the magnetic moment falls to almost zero. Similar observations have been found in many other alloy systems. Distribution offerromagnetic property among the transition metals can be explained by considering the band theory and this aspect has been discussed in Sec. 12.5.8. There it has been pointed out that formation o/3d band for the late members of the 3d series is not very much energetically favourable in the solid state (because of the contraction of the 3d orbitals). In fact, such d-bands possess some atomic character. Because of the poorer d-d overlap, the band width is very small and the energy levels in the d-band are not efficiently delocalised. However, the overlapping interaction of the 3d band (having some atomic character) with the 4s band in solid state is evidenced from their magnetic moments and high electronic specific heat.

Bonding in Metals and Metal Clusters: Electrical Conductivities of Solids

1147

m yl

ib ra

ry

• Variation of cohesive energy among the transition metals: For the (n - 1)J5ns 1 configuration, the d- and s-bands become just filled in i.e. the BMOs of the bands are just filled in (ignoring the effect of mixing of d- and s-bands). Similarly, for (n - 1)dJOns 2 configuration as in Zn, all the BMOs and ABMOs of the d- and s-bands are filled in. Thus, it is reasonable to expect the most stable situation for metallic bonding at (n - 1)d5ns l and the most unfavourable situation at (n - l)d I Ons 2 • Thus for the 3d series, the cohesive energy should be maximum at Cr(3d54s l ) and minimum at Zn(3d 104s 2 ). It is true for Zn but the maximum arises at V (Fig. 12.5.3.4) and there is a drop of cohesive energy at Cr and Mn. The drop of cohesive energy at Cr(3d54s 1) and Mn(3J54s 2) arises due to non-aufbau filling (Fig. 12.5.3.4) of the d-band. It causes the placement of electrons in some ABMOs to produce more unpaired electrons. It occurs due to two reasons: gain of exchange energy and reduction in electron-electron repulsion. The placement of electrons in some ABMOs of d-band (i.e. nonaufbau filling) weakens the metal-metal bond Le. reduces the cohesive energy. It explains the drop of cohesive energy at Cr and

(1)

>-

'(j) ~ (1)

"

'-,

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... ",,,,,,-- ... ,

:::' ,

'

-§ ~

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()

(1)

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Sc To V Cr Mn Fe Co Ni Cu

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1

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Mn. After Mn, the cohesive energy increases and it remains almost constant for Fe, Co and Ni for which both the d- and s-orbitals remain partly filled due to overlapping of the bands. For these metals, Fermi level represents the highest level occupied in both the bands (Fig. 12.5.3.1-3). After lVi, ~ohesive energy decreases in the sequence Ni > Cu > Zn. The d-bands of Cu and Zn are completely filled in and gradual addition of electrons are accommodated in the s-band and the Fermi level is detennined by the topmost level occupied in the s-band (Fig. 12.5.3.3). ThusFF increases as: Zn > Cu > Ni. It is due to the fact that the electrons in the narrow d-band are more stable than in the wide s-band (Fig. 12.5.1.4). 'rhus, it explains the cohesive energy sequence (a measure of metal-metal bond strength): Ni> Cu > Zn (Fig. 12.5.3.4).

k

(a) For 3d-series

~n 1

La Hf Ta W Re Os Ir Pt Au Hg

(b ) For 5d-series

••

(c) Aufbau filling of ad-band

(d) Nonaufbau filling of a d-band

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Fig. 12.5.3.4. (a) and (b): Qualitative representation of variation of cohesive energy for the 1st and 3rd t.ransition series (Le. 3d and 5d series). (c) and (d): Aufbau and nonaufbau filling of d-bands. Here, it is worth mentioning that in contrast to 3d series, in 4d and 5d series, there is no remarkable drop in cohesive energy at Mo and W (congeners of Cr) and at Tc and Re (congeners of Mn). It indicates thatfor the heavier congeners of4d and 5d series, the nonaufbau jilling ofa d-band is not occurring. In fact, for the members of 4d and 5d series, the electron-electron repulsion is relatively less because of their larger size. Consequently, the systems do not experience any urge to promote the electrons to the ABMOs through unpairing to avoid the electron-electron repulsion. The cohesive energy follows the sequence: 5d series> 4d series> 3d series. In fact, the larger d-orbitals can overlap better and the efficiency of d-d overlap runs as: 5d-5d> 4d-4d> 3d-3d.

1148

Fundamental Concepts of

Inorgani~

Chemistry

12.5.4 Valence and Conduction Bands

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The energy bands produced by the inner orbitals are narrower compared to the bands formed by the valence atomic orbitals (see Figs. 12.5.1.2-3). The energy level below which all the energy levels may be completely filled in at 0 K is referred to as Fermi level. For carrying electricity, there should be some vacant levels in the vicinity above the Fermi level, so that the electrons can move to the vacant levels in the presence ofan external electrical potential. The band which provides the accommodation for the conducting electrons is called the conduction band (CD). When the valence band is incompletely filled, the valence band is itself a conduction band as in the case of alkali metals. In some cases, the valence band may undergo overlapping withthe conduction band as in the case of alkaline earth metals. Sometimes, a forbidden energy gap exists between the conduction and valence bands. The magnitude of the forbidden energy gap is extremely important in determining the properties of solids, in connection with their electrical conductivities. .

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12.5.5 Explanation of Electrical Conductivities of Solids in the Light of Band Theory

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Depending on the degree of efficiency of electrical conductivity, the solids can be classified as conductors (e.g. metals), semimetals (e.g. graphite), insulators (e.g. diamond) and semiconductors (e.g. silicon). When there is no external potential, the net wave momentum of the electrons is zero, be~ause for each moving electron, there is an electron (in terms of probability) moving with the same speed in the opposite direction (see Figs. 12.3.2.5 and 12.5.5.1). But when an electrical potential is applied, the number of electrons moving towards the positive end increases giving rise to a net wave momentum (i.e.. net flow) to that direction. In fact, in presence of an external fields, the electrons moving towards the positive end are of lower energy. The electrons moving towards the negative end are of 'higher energy. To accommodate these two types of electrons, the energy band is modified where the two sets of energy levels are no longer equally populated (Fig. 12.5.5.1). Thus the electrical conductivity is attained (see Figs. 12.3.2.5 and 12.5.5.1). Thus the basic condition to satisfy the criterion is : there must be some empty levels just immediately above the Fermi level. In a filled band, no such redistribution ofthe electrons is possible. The incompletely filled bands are called conduction bands. For good electrical conductivity, the band width and density of states (DOS) of the conduction

lic

k

+

C



(a) No field

E /F

(b) Field applied

Fig. 12.5.5.1. Change of electron wave momentum direction in the half-filled energy band in the presence of an external field. (a) Two sets of energy levels equally pop~lated. (b) One set of orbitals more populated (cf. Fig. 12.3.2.5).

Bonding in Metals and Metal Clusters: Electrical Conductivities of Solids 1149

ry

band should be sufficiently large to give an effective delocalisation of the electron through the lattice. Thus for electrical conductivity we should consider the important factors: presence of a partially filled band (called conduction band), band width (good overlapping gives a larger band width) and DOS (density of states) of the conduction band.

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(A) Conductors (i.e. Metals) (cf. Sees. 12.5.2-3 for band structure)

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(8) Semimetals (e.g. graphite)

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m yl

The substances in which the highest valence band is partially filled in or the conduction band and valence band are overlapping are described as conductors, because the electrons can easily move above the Fermi level to carry out the electrical conduction. In fact, in such cases, the forbidden energy gap (Eg ) is almost zero. This condition is attained for the metals. This is why, they are good conductors. The Gr-I (J) metals are having the outermost electronic configuration ns 1 (i.e. alkali metals: Na, K, Li, Cs, etc; coinage metals: Cu, Ag, Au). Such metals having one valence electron on each atom provide a half-filled conduction band (see Fig. 12.5.2.1). Similarly, the Gr-III (13) metals having the outermost electronic configuration ns 2np 1 (e.g. At) also provide a partially filled conduction band. On the other hand, in Gr-II (2) metals (e.g. Be), the outermost ns valence band and np conduction band undergo overlapping (see Figs. 12.5.1.4 and 12.5.2.1). In the transition metals the ns and (n - l)d bands undergo interaction giving rise to the vacancies in both the bands.

t.m e/

In such cases (Fig. 12.5.1.5), Eg = 0, but at the junction of conduction band and valence band, DOS = 0. These are relatively weakly conducting but the conducting behaviour is similar to that of metals. Bismuth, antimony and arsenic show the semimetallic property. (C) Insulators

Empty conduction band

C

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k

H

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In the insulator substances, the highest filled valence band and the empty conduction band are so widely separated (see Fig. 12.5.5.2) that the electrons from the valence band c~nnot travel to the conduction band under the ordinary electrical potential. Diamond is a good example in this class. In diamond, the forbidden energy gap (Eg ) is '- 5.5 eVe Generally, the systems having Eg' > 3 eV are referred to as insulators.

Empty conduction band

f

--------E+9 -------·EF =Eilled:~alence:bar:ld=

(Insulator)

(Intrinsic semiconductor)

Fig. 12.5.5.2. Forbidden energy gaps (Eg ) in insulators and intrinsic semiconductors.

1150

Fundamental Concepts of Inorganic Chemistry

Definition of Eg

It should be remembered that Eg value gives the energy difference between the minimum ofthe conduction band (CD) and the maximum of the valency band (VB) (cf. Fig. 12.9.4.1).,

ry

Na-band

(overlapped 3s and 3p bands)

ib ra

(CB)

7 eV

(VB).

m yl

• Band structures of ionic compounds MX and MO solids (cf. Fig. 9.11.4.1 ): Generally, the ionic solids are insulators because .of the high band gap i.e. Eg value is high. Let us illustrate by considering the band structure of NaCI (I-VII Le. 1-17 compound). In NaCI crystal, the CI-centres are in contact and their 3s and 3p valence orbitals overlap to produce a narrow 3(s-p) band consisting of 4x energy levels for (NaCl)x crystal. The

CI-band

t.m e/

th

ea

lc

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(overlapped 3s valence orbitals of Na-centres will also overlap to produce a and 3p bands) 3 (s-p ) band but the overlap integral is very much small. The energy difference between the sodium band and chlorine band is about 7 s-p band (Filled) eV. The high band gap allows the movement of electrons from the Fig. 15.5.5.3. Band structure of higher energy Na-band to the lower energy CI-band. It makes NaCI NaGI. E g value for the I-VII ionic i.e. Na+CI-. For (NaCl)x the number of available valence compounds runs in the electron is 8x (7x from xCI-centres, and Ix from xNa-centres) which sequence: NaF > NaGI > NaBr; is just sufficient to completely fill in the chlorine band. Thus the KF > KGI > KBr > KI. Na-band remains vacant. The high band gap (,..., 7 eV) is due to the electronegativity difference between lva and CI. The higher electronegativity of chlorine places the

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chlorine band far below the sodium band (Fig. 12.5.5.3). In the same way, the band structure of MO(M= Ca, Mg, Ba, A.~r) (II-VI compounds) can be explained. T'he filled 2(s-p) band of oxygen acts as the valence band and the conduction band i.e. n(s-p) band of M, remains vacant. The high band gap makes MO insulator. The band structure of AgX(X= CI, Br) can be explained in the same way and Eg decreases for the heavier halides (Fig. 9.11.4.1). It explains the colour of silver halides and their nonconducting properties. In AgBr (Eg = 2.7 eV), visible light can

H

excite an electron to VB to CB. This property makes it useful in photography.

lic

k

• Band structure of Gr-IV (14) elements (Sec. 12.5.2): In the case of Gr-IV (14) elements with the increase of atomic number, the energy gap (Eg ) generally decreases (Sec. 12.5.2). Infact, Eg value gives the measure ofionisation energy (in the electrical field ofthe crystal) required to raise an electron from the valence band to the conduction band. It generally decreases with the increase of

C

dielectric constant ofthe crystal and the principal quantum number of the valence shell (see Sec. 12.5.2). The Eg values for some Gr-IV (14) elements are given in Table 12.5.5.1. From the variation of Eg

values, it is evident that diamond is an insulator but others are semiconductors (cf. Fig. 12.5.2.2). Table 12.5.5.1. Forbidden energy gaps and dielectric constants in some Gr-IV (14) elements

Element

c

Si

Ge

Sn (grey)

E?, (eV)

5.5

1.1

0.7

0.08

Dielectric constant

5.7

EO

12

EO

16

EO

Bonding in Metals and Metal Clusters: Electrical Conductivities of Solids

1151

In fact, between.the semiconductor (intrinsic) and insulator, there is no sharp line of demarcation. In the intrinsic semiconductors, the forbidden energy gap (Eg ) is relatively smaller (see Fig. 12.5.5.2) so that some electrons may pass into the conduction band even at ordinary temperature.

(0) Semiconductors

12.5.6 Optical Properties of Solids in the Light of Band Theory

m yl

This aspect has been discussed in detail in Sec. 12.9.4.

ib ra

ry

The substances which lie between the conductors and insulators in conducting power are called semiconductors. In these cases, the forbidden energy gap (Eg ) is not too high. Hence, some of the electrons can be thermally excited to the conduction band. The Eg values generally lie in the range 2-3 eVe The semiconductors can be classified as intrinsic semiconductors and extrinsic semiconductors.

he

12.5.7 Magnetic Properties Including the Temperature Independent Paramagnetism (TIP) (i.e. Pauli Paramagnetism) of the Metals in Terms of Band Theory (cf. Sees. 12.5.3, 12,3.3f, Fig. 12.3.3.4)

lic

k

H

er e

t.m e/

th

ea

lc

The paramagnetic properties of some transition metals have been explained in terms of their band structure in Sec. 12.5.3. The origin of temperaiure independent paramagnetism (Pauli paramagnetism) has been explained (Sec. 12.3.3f, Fig. 12.3.3.4) by considering the quantum mechanical free electron model of the metals. This Pauli paramagnetism is mainly important for the nonferromagnetic transition metals where the density ofenergy states (DOS) at the Fermi level in the d-band is very high. In fact, this high DOS at the Fermi level can also explain the high electronic specific heat of such metals. For other simple metals, the Pauli paramagnetism (TIP) from the conduction electrons is not so strong and they are generally diamagnetic or weakly paramagnetic (cf. Sec. 12.3.3 and Fig. 12.3.3.4). The .origin of TIP can als~ be qualitatively explained in terms of band theory. With the increase of temperature, some of the" paired electrons in the topmost levels of a Brillouin zone will be unpaired and excited to the higher energy levels. The free spin states of the unpaired electron contribute to the paramagnetism. The spin paramagnetism (i.e. Langevin paramagnetism) decreases with the increase of temperature (cf. Curie equation). Thus, though the increased temperature produces more unpaired electrons to increase the paramagnetism, the opposing contribution developed from their spin paramagnetism makes the resultant paramagnetism almost independent of temperature. Distribution of ferromagnetic properties among the transition metals has been discussed separately in Sec. 12.5.8.

12.5.8 Ferromagnetism in Transition and Inner-transition Metals (ct.' Sec. 11.16.3)

C

Parallel alignment of magnetic moments of individual magnetic centres leads to ferromagnetism (cf. Sec. 11.16.3). This interaction exists in the materials having unpaired d- and .felectrons which may align parallel. This alignment is not so strong to form the covalent bonds through overlapping ofthe orbitals bearing the unpaired electrons. In fact, such overlapping interactions will lead to the antiferromagnetic coupling through spin pairing. The substance having a higher Curie temperature (Tc ) is a stronger ferromagnetic one. It is interesting to compare the magnetic properties of the 3d metals (cf. electrical conductivity of the monoxides of 3d-metals, cf. Sec. 12.10). For the early members ofthe 3d-series, the radius ofthe 3d-orbitals is sufficiently large to allow the formation of 3d-band through overlapping. Pairing of electrons in the 3d-band occurs to bring about the antiferromagnetic interaction. For the late members

1152

Fundamental Concepts of Inorganic Chemistry

ib ra

ry

(e.g. Fe, Co, Ni) of the 3d-series, the radii of the 3d-orbitals shrink and consequently formation of the 3d-band becomes energetically less favoured. In fact, for such metals, the uppermost d-bands are of strong atomic character. In such cases, the unpaired electrons in the 3d-orbitals of the neighbouring atoms align their spins in aparallelfashion (no overlapping and consequently no requirement of Pauli's exclusion principle). The driving force for the parallel alignment of the unpaired electrons can be understood by considering the spin dependent electrostatic energy. It arises from an exchange interaction between the electrons in different quantum states provided the spin quantum numbers of both the states are the same (i.e. parallel alignment of the spins). The exchange energy for such interaction between the centres i and j is given by:

m yl

E ex = -2J,f~j

he

where S; and Sj are the spin angular moments associated with the i-th and j-th state resp~ctively, Jij is the exchange integral. In general, the exchange integral is negative giving the E ex positive (i.e. no stabilisation through the parallel alignment of the electrons), but the exchange integral value may become positive in some cases (e.g. Fe, Co, Ni etc.). The exchange integral depends on the ratio, rab/ro (where rab = interatomic distance in the crystal, ro = orbital radius). It has been found that Jij becomes positive when rab I > 3. For Gd (a 4fblock element, i.e. ro refers to the radius of the 4f orbital), the exchange integral is positive to allow the ferromagnetic interaction.

ea

lc

'0

r uh

/

2.24

rv =

Tc (K)

=

8M per atom

* ferromagnetic

Cr

Mn

t.m e/

Ti

th

Table 12.5.8.1. Magnetic properties of some metals

2.36

2.94

Fe*

Co*

Ni* "

Gd*

3.26 1043 2.2

3.64 1400 1.7

3.96 631 0.6

3.12 288 7.1

lic

k

H

er e

Here it is important to mention that in the ferromagnetic materials, the parallel alignment of the electrons earns the stabilisation through Eex but it is partly offset by the rise of Fermi level and the consequent is the increase of the average kinetic energy of the electrons. Thus it leads to increase the average Fermi velocity of the electrons and it leads to a greater physical separation between the electrons to disfavour the overlapping interaction.

r

Nature of interaction depending on 'ab value

'0

C

• ~ < 3: The interacting orbitals are sufficiently large and close to allow the overlapping leading ro to an antiferromagnetic interaction i.e. spin pairing. ab > 3: The interacting orbitals are not sufficiently large and close to allow the overlapping ro but they are at an optimum distance to experience an exchange interaction (Le. parallel alignment) leading to a ferromagnetic interaction.



r

• rab » 3 : The orbitals ofthe adjacent metal centres are widely separated and they remain mutually ro

noninteracting i.e. the electrons of the orbitals behave independently.

Bonding in Metals and Metal Clusters: Electrical Conductivities of Solids Fig. 12.5.8.1 (Betlle-Slater curve) indicates that Jij becomes positive for some optimum values of r o• Ifro is quite large (as in the case of early members of the 3d series) i.e. r ab / r o < 3.0, then overlapping ofthe concerned orbitals will occur to favour the antiferromagnetic interaction and Jij becomes negative. On the other hand, if r 0 is small (as in the case of late members of d- and .fseries elements where the d- and.forbitals are strongly attracted towards their respective nuclei) i.e. rab / r0 > 3.0, then the inter-acting electrons cannot come

1153

o~----+-----------

ib ra

ry

----+ rablro

Fig. 12.5.8.1. Dependence of exchange integral (Jij) on 'ab / ratio for ferromagnetic ,interaction (Bethe-Slater curve).

'0

m yl

!;'ufficiently close together to allow the overlapping. interaction but they can allow the exchange interaction.

C

lic

k

H

er e

t.m e/

th

ea

lc

he

This is why, besides the said late 3d-metals, the .ferromagnetic interaction is also favoured to some members ofthe first rare earth series (i.e. 4.fseries) where the f-orbitals are sufficiently contracted to make rab / r o > 3.0. It may be mentioned that if r o value is too small (Le. rab / r 0 » 3) then the distance between the interacting electrons is so long that they cannot experience exchange interaction Le. the electrons behave independently. Thermal energy tends to randomise the parallel alignment ofthe electrons requiredforferromagnetic interaction. In fact, above Tc a ferromagnetic material becomes a normal paramagnetic substance. The Curie temperature is a function of exchange energy. Fig. 12.5.8.1 indicates that for Co, the exchange energy is maximum (i.e. J lj is maximum positive) and it registers the highest T c (~ 1400 K), and Gd with a small exchange energy shows a very low Tc (~ 288 K). The importance of the rab / r 0 parameter to determine the exchange energy is illustrated in some alloy system involving Mn. In Mn, the 3d orbitals are sufficiently large to form a d-band (i.e. overlapping interaction leading to an antiferromagnetic interaction) and it is indicated by rab / r o (~ 2.94). But the ratio is not far away from the required value for ferromagnetic interaction. On alloying with some nonferromagnetic elements (e.g. As, Sb, Cu, Al etc.), the Mn-Mn distance is increased by the right amount so that overlapping of the d-orbitals does not occur but exchange interaction can arise. In fact, such alloys (e.g. "eusler alloys Cu 2MnSn, Cu 2MnAl) are found ferromagnetic. The magnetic properties (Le. observed magnetic moments) of Fe, Co and Ni have been explained in Sec. 12.5.3 in terms of band theory. For Fe, Ni and Co, it is believed that the 4s electrons spend some time in their respective 3d orbitals to reduce the number of unpaired electrons in the d-band in solid state. It occurs so due to the overlapping of 3d and 4s bands. But in Gd, no such interaction to reduce the number of unpaired (41') occurs.

12.5.9 Band Structure of Some Inorganic Solids and their Properties (a) Band structure ofGr-IV(14) elements: It has been already discussed in Sec. 12.5.2. The related isoelectronic III-V (i.e. 13-15) compounds (used as compound semiconductors) like GaP, GaAs, etc. can be treated in the same way. (b) Band structure of NaCI (i.e. I-VII or 1-17 compound): It has been already discussed in Sec. 12.5.5. In the same way II-VI (i.e. 2-16) compounds like MgO can be explaine~. (c) Band structure of transition metal oxides (MO): This has been .discussed in Sec. 12.10.

1154

Fundamental Concepts of Inorganic Chemistry

12.6 VALENCE BOND APPROACH OF METALLIC BONDING 12.6.1 Valence Bond Approach for the Short Period Metals

".'

Li- Li

I

Li

Li

Li

I

Li

m yl

Li -

Li ~

ib ra

ry

The idea of classical 2c-2e covalent bond can also be extrapolated in explaining the metallic bond. Let us first illustrate the lithium crystal. From the electronic configuration (Li =, I s22s 1), lithium should form only one covalent bond. To explain the possible interaction with all other neighbouring centres, the following types of canonical forms (in a two dimensional crystal) are possible.

he

By considering the pair Li+ (ls2), Li- (ls 22s 2 ) (within the metallic crystal, the increased dielectric constant reduces the ionisation energies), Li- can form two covalent bonds through the participation of 2p orbitals. Thus the following types of canonical forms are also possible. Li- - Li Li+ I ~ Li Li+ Li -

lc

Li-Li

I~

Li-

ea

Li-Li

Li

er e

t.m e/

th

In a three dimensional crystal, a large number of canonical forms can be considered. Thus the resonance energy is very high. This high resonance energy can explain the high cohesive energy in a metallic crystal. Thus with the increase of the number of valence electrons, an atom can form a larger number of bonds which in turn yields a higher cohesive energy. In the same way, for the two short period elements, (Le. He to Ne, Na to Ar) the concept of valence bond theory can be utilised to explain the bonding pattern. For the two short period elements, only four orbitals (one's' and three 'p') are available. Hen~e for the members having valence shell electrons four or more than four, there would be no vacant orbital described as metallic orbital. These vacant metallic orbitals are responsible for carrying electricity. This is why the elements of Gr-IV (14) and higher groups are nonmetals.

H

12.6.2 Valence Bond Approach for the Long Period Metals

C

lic

k

For the long period metals, the conclusion regarding the bonding is not so straightforward in contrast to the case of short period elements. The (n - l)d electrons come into picture and the systems get complicated. All the available valence orbitals may not participate in the bonding purpose. The problem can be compared with the concept ofd-orbital participation in forming the isolated molecules. Similarly, the relative extent of participation of the ns and np orbitals for the heavier ones, depends on many factors (see inert pair effect). Behaviour of tin : To illustrate the matter, let us consider the case of tin (Sn), a Gr-IVA (14) element. The following possibilities of electron distribution in forming the bonds may exist. Forms

Electronic Configuration

Sn(A)

4d105s 15p 3

Sn(B)

4dl 05s 25p 2

Covalency

No. ofmetallic orbitals per atom

o

Bonding in Metals and iV1etal Clusters: Electrical Conductivities of Solids 1155 It is evident that the Sn(A) is an insulator while Sn(B) is metallic in character. The grey tin having the diamond structure is probably Sn(A) which is stable below 18°C while the stable white tin (Temp. > 18°C) having the metallic properties is Sn(B). Probably both the forms contribute always but the

ry

distribution function depends on the temperature. Here Sn(B) shows the inert pair effect. Behaviour of zinc: Now let us consider the case of Zn which may have the following possible electronic configurations for bonding in the metallic crystal. Forms

Electronic Configuration

Covalency

Zn(A)

3d84s 14p 3

6(J2 + sl + p3)

Zn(B)

3JJ4s 14p 2

4(d 1 + sl + p2)

Zn(C)

3d lO4s 14p l

2 (sl + pI)

ib ra

No. ofmetallic orbitals

m yl

0

2

t.m e/

th

ea

lc

he

Here, the Zn(B) and Zn(C) show the metallic properties while the form Zn(A) should show the nonmetallic behaviour. Though the three different forms are not reported separately, it is believed that all the forms contribute to the Zn-structure and their relative proportions are different which depend on the conditions (e.g. pressure, temperature, etc.). Behaviour ofnickel : Lastly, we shall consider such an example in which all the unpaired electrons in the d-shell may not participate in the bonding purpose. Such orbitals containing unpaired electrons are called nonbonding orbitals. These orbitals remain nonbonding because of the steric hindrance. These nonbonding orbitals are important in explaining the paramagnetism of the metals. This is illustrated by considering the possible electronic configurations of Ni in the metallic crystal. Electronic configuration

No. of Covalency

Ni(A)

3d'4s 14p 3

Ni(B)

3cfl4s 14p 2

er e

Forms

No. of metallic orbitals

No. of nonbonding orbitals

6(J2 + sl + p3)

0

2

6(cf3 + sl + p2)

1

0

H

Thus Ni(A) gives the paramagnetic but nonmetallic covalent structure while Ni(B) yields the diamagnetic but metallic structure. The actual structure consists of both the forms and their relative

lic

k

contribution can be obtained from the observed paramagnetism. In this way, the valence bond concept can be extended to explain the properties of other transition ·metals.

C

12.6.3 Merits of the Valence Bond Approach

(i) Trends of cohesive energy: The number of bonds formed by an atom in a resonating structure depends on the number of unpaired valence electrons available for the bonding purpose, Thus the increase of cohesive energy in the sequence, Or III (13) > Gr II (2) > Gr I (1) is quite reasonable. Similarly, for the transition elements in which the d-electrons participate in bonding, the highest cohesive energy is reached for the metals (e.g. V, Cr) in which the highest number of unpaired valence electrons (from both (n - I)d and ns leve~s) is available (cf. Fig. 12.5.3.4). (ii) Electrical conductivity: To show the electrical conductivity, there must be some vacant energy levels called metallic orbitals just above the occupied level, so that the conducting electrons can travel into the vacant levels in the presence of an electrical field. For example, in the lithium

1156

Fundamental Concepts of Inorganic Chemistry

ea

lc

he

m yl

ib ra

ry

crystal, some of the 2p levels (Li- utilises 2s and one 2p orbital for the bonding purpose) are always available for conducting the electricity. These vacant 2p orbitals are lying just above the occupied levels. For the Gr III (13) elements, the !Vacant np level can perform the same task. For the Gr IV (14) elements (e.g. C, Si, Ge, Sn) the four valence electrons occupy the ns and np levels and no vacant level exists just above the occupied levels. I'his is why, these are nonconductors. However, the metallic property of some allotropic forms of the heavier congeners (e.g. Sn) can be explained by considering the redistribution of electrons in the valence shell (cf. Sec. 12.6.2). The electrical conductivity for the heavier elements and transition metals has been explain~d in Sec. 12.6.2. (iii) The distribution of metallic properties in the periodic table: It has already been discussed that for the two short periods elements, the members of Gr I (1), II (2) and III (13) are metals because of the presence of vacant metaillic orbitals. This has been verified. For the long period elements, the prediction is not so straight forwrard (discussed in Sec. 12.6.2). (iv) Paramagnetic properties of the metals: The presence of nonbondig orbitals with unpaired electrons can explain the paramagnetic properties of the metals (see Sec. 12.6.2). (v) Explanation ofa110 tropicforms : It can explain the existence of allotropic forms of a metal fr~m the difference in bonding patterns. The properties of the allotropes like grey tin and white tin have already been explained in terms of VBT in Sec. 12.6.2.

th

12.6.4 Demerits of the Valence Bond Theory

er e

t.m e/

Compared to the molecular orbital theory, the valence bond thoery appears more empirical. By progressing through the theoretical argument in MOT, we can better explain the different properties of metals, but in the VB theory to speculate the bonding pattern we ought to depend more on the observation rather tl}an the theory.

12.7 SOLID SOLUTIONS: ALLOYS: INTERMETALLIC COMPOUNDS

C

lic

k

H

When two solid metals or one metal and another nonmetal are mixed through heating, the following possibilities may arise. (i) Formation of an ionic compound. (ii) Formation of an interstitial alloy. (iii) Formation of a subtitutional alloy. (iv) Formation of a simple mixture. Formation of an ionic compound occurs when the electronegativities of the component elements are widely apart. These are better treated as ionic solids. Here we shall pay attention to the second and. third possibilities. 12.7.1 Interstitial Alloys

Metals adopt the close-packed array leaving some octahedral (Oh) and tetrahedral (Td ) holes. These holes may be occupied by some solute elements provided their sizes will fit in the holes properly. To occupy the tetrahedral and octah~dral holes, the following conditions must be satisfied. But if the

Bonding in Metals and Metal Clusters: Electrical Conductivities of Solids 1157 solute impurities are very small, then no efficient packing in the holes occurs. The required radius ratios are: rsolute/rsolvent == 0.225 to 0.414 (for Td holes); and rsolute/rsolvent

= 0.414 to 0.732 (for 0h holes).

he

m yl

ib ra

ry

Generally, the smallest element, hydrogen occupies the tetrahedral sites while the other elem'ents like carbon, nitrogen and boron occupy the octahedral sites. In all cases, the basic metallic structures of the solvent metal remain unchanged. The interstitial solutes remarkably influence the properties of the metal such as hardness, malleability and ductility. Because of the occupation of the interstitial sites by th~ solute atoms, the ease of slipping of one layer over the other is affected. Thus the hardness and melting points are significantly increased. In addition to these, sometimes, the chemical inertness is also significantly increased. In this connection, the various forms of steel depending o'n the amount of interstitial carbides are extremely important (cf. Sec. 11.18). 12.7.2 Substitutional Alloys, Intermetallic Compounds and Hume-Rothery Rule

0

0

lic

0

k

H

er e

t.m e/

th

ea

lc

If the solute atoms can substitute the solvent atoms at their lattice sites then the substitutional alloys are formed. The substitution can go on either in an prdered or disordered way. If the arrangement is random, it leads to a random substitutional alloy while the ordered arrangement produces the ordered substitutional alloy called superlattiee (see Fig. 12.7.2.1). The random substitution leads to an increase ofentropy which lowers the free energy (~G = ~H - T~. This is why, the substitutional alloy specially at a higher temperature is disordered and on lowering the temperature, it tends to have the ordered superlattice structure if the ordering of the lattice points lowers the enthalpy (~H). For example, the solid solution of Cu and Zn in equal atomic proportions (i.e. 1 : 1) adopts the disordered substitutional alloy in the bee lattice at above 470°C. But on lowering the temperature, it tends to adopt the ordered bee lattice in which the unit cells place the ('u atoms at the corners and the Zn atoms at the centres. At temperature above 470°C, in the above disorqered solid solution, the probability of finding a Cu or Zn atom at any lattice point is 0.5. In lowering the temperature, it attains the bee structure in such a way that each Cu atom gets surrounded by eight Zn atoms and each Zn atom is coordinated by eight eu atoms. Probably the driving force of this ordered arrangement arises due to the more favourable Cu-Zn interactions rather than the Cu-C'u or Zn-Zn interactions.

• •0

C

0

0

O. O. 0 0

0

0

0



•0 0

0 0

Interstitial alloy

0



0

0

0

• • • 0

0

0

0



0 0



0



0



0

Random substitutional alloy



0

0 0



0







0 0

0 0

Superlattice





Fig. 12.7.2.1. Interstitial and substitutional alloy.

In this connection it is worth mentioning that in the 1: 1 (atomic proportions) Cu-Zn alloy, the crystal structure (which is bee) is different from that of either pure Cu (fcc) or Zn (hep).

1158 Fundamental Concepts of Inorganic Chemistry Similarly, Cu-Au system gives a disordered structure at above 450°C, but on cooling the more ordered superlattice structure tends to be adopted. To have the continuous miscible solution through substitution, some conditions are to be satisfied. These are formulated as Hume-Rothery rules as follows.

ry

(i) Metallic radii: The solute and solvent metal atoms must be similar in size. Their metallic radii should not differ by more than 15%.

ib ra

(ii) Crystal structure: Both the solute and solvent metals must have the same crystal structure. (iii) Electronegativity: The electronegativities ofthe two metals should be very much comparable and the number of valence electrons should be the same.

m yl

(iv) Chemical properties: The chemical properties of the two metals should be similar.

th

ea

lc

he

(v) Solid solutions vselectron compounds: The systems, e.g. Cu-Au, Ag-Au, Cu-Ni, Ge-Si, etc. which satisfy (see Table 12.7.2.1) the Hume-Rothery rules fully can form solid solutions at any proportion. Such systems can form superlattice structures on cooling. On the other hand, the systems such as Cu-Zn, Na-K, etc. which only satisfy some of the conditions can form the solid solutions in a limited range through disordered substitution. Above the limited range of solubility, such systems tend to produce some special types of intermetallic compounds (described as electron compounds) which cannot be explained in terms of common concept of valence. This aspect has been discussed later (cf. Table 12.7.2.2).

System

Metallic radii (pm) *

% difference in radius

Cu (fcc)

128

12.5

Au (fcc)

144

Ag (fcc)

144

Au (fcc)

144

eu (fcc)

128

lic

k

H

(b) Ag-Au

(c) Cu-Ni

Crystal structure o/the components

er e

(a) eu-Au

C

(d) Ge-Si

(e) Cu-Zn

(f) Na-K

t.m e/

Table 12.7.2.1. Relevant parameters of the Hume-Rothery rules for some systems.

Ni (fcc)

124

Ge (de)

122

Si (de)

118

eu (fcc)

128

Zn (hcp)

138

Na (bee)

190

K (bee)

235

* Metallic radii for 12 coordination ** Pauling's value *** Miscible only at limited proportions

Miscibility No. o/valency Electroelectrons negativity ** 1.9

At any ratio

2.4 1.9

0.0

2.4 3.2

3.4

7.8

23.7

1

1.9

2

1.8

4

1.8

4

1.8

1

1.9

2

1.6 0.9 0.8

*** ***

Bonding in Metals and Metal Clusters: Electrical Conductivities of Solids

1159

Cu

+ Zn «35%)

h + Zn (45 - 50%) a-p ase • p-phase (fcc, random (bcc, intermediate substitution compound of approx. (solid solution) stoichiometry of CuZn) •

+ Zn

lc

he

(fcc)

m yl

ib ra

ry

In Table 12.7.2.1, the first four systems (i.e. a, b, c, d) satisfy all the conditions of Hume-Rothery rules and they form miscible solid solutions at any proportion. At room temperature, up to ~ 35% Zn (in hcp), it can get dissolved in the fcc lattice of Cu; and only'" 1% Cu can get dissolved in the hcp lattice structure of Zn. In the Na-K system, because of the large difference in their size, they form solutions in a limited range. The systems (e.g. Cu-Zn, Na-K, etc.) in which only one or two conditions of the Hume-Rothery rules are satisfied will form the random substitutional alloys in a limited range. Above this limited range, their compositions can be represented very often by the chemical formulae based on the relative number of the component atoms of each kind per unit cell of the alloy. Such compounds are referred to as intermetallic compounds which differ in the lattice structure with respect to those of the component metals.

0. ll-phase +• Zn (97-100%) £-phase +• Zn (82-880/0) 0 K, is given by : l+exp(E g /2k B T)

...(12.9.1.7)

I

lc

P(E) = exp [-E g /2k B T]

. .(12.9.1.8)

ea

I

he

For an intrinsic semiconductor Eg is close to about 1 eV (as in the case of silicon) and it is much larger than kBT(~ 0.026 eV at room temperature). Thus we can reasonably assume exp (E/2kBD» 1 and above relation reduces to .:

2

n j = nenh

t.m e/

th

Thus under the circumstances, the Fermi-Dirac distribution law is reduced to Maxwell-Boltzmann distribution law. • Intrinsic concentration ofthe carriers: In an intrinsic semiconductor, ne = nh = n j (called intrinsic concentration ofthe charge carrier, the relationship is called mass-action law), Le. nenh = n;.

=

N c exp[-(Ec -EF)/kBT]Nv exp[-(EF -Ev)/kBT] NcNv exp[-(Ec -Ev)/kBT]

er e

NC1V V exp [- E g / kBT]

Eg =

k B T In

NcNv 2

...(12.9.1.9b)

nj

H

Le.

...( 12.9.1.9a)

By using the above relation, n j can be calculated and it has been experimentally verified.

C

lic

k

• Conductivity of an intrinsic semiconductor: In terms of mobility (average drift velocity per unit potential gradient) of the electrons in the CB and holes in the VB, the conductivity (a j ) of an' intrinsic semiconductor can be expressed as follows: a j = neeue + nheu h = n j (u e + uh)e, (n e = nh = n j )

••• (

12.9.1.1 0)

where ue and uh are the mobilities of the electrons and holes respectively. By using Eqn. 12.9.1.9a for' n i , we have:

a j = ~NcNve(ue + uh ) exp (- Eg /2k BT)

a o exp (- Eg /2k BT), [a o = ~ e(ue + uh )] or,

E g /2k B T = A - Eg /2k B T (A = some constant = In a 0) Ina;

= Ina o

-

...(12.9.1.11) ...(12.9.1.12)

1184

Fundamental Concepts of Inorganic Chemistry

ry

The quantity A contains the term JNcNv which depends on temperature (cf. N c or N v ex: '(3/2) and the mobility term (u) which also depends on temperature. Increase of temperature leads to more collision between the charge carriers and phonons. Thus the mobility ofthe charge carriers decreases with the rise of temperature. The mobility (in a pure sample where impurity scattering is absent) has been found to vary with temperature as: u ex: 1 3/ 2 (considering only scattering by phonons). Thus the term A (= In CJo) overall becomes more or less temperature independent (compared to the exponential

ib ra

term), but the exponential term is highly temperature dependent.

k

H

er e

t.m e/

th

ea

lc

he

m yl

From the above argument, it is evident that for temperature dependence of Ina;, the second exponential term is more important. With the increase of temperature, the number of charge carriers increases exponentially and this is why the conductivity (CJi ) increases very rapidly. The constancy of the term A and linear relationship between Inaland liT has been experimentally verified. From the plot of Ina; vs. IIT(cf. Fig. 12.9.3.1), the forbidden energy gap (Eg ) in an intrinsic semiconductor can be estimated. • Intrinsic compound semiconductors: Besides the elemental semiconductors (e.g. Si, Ge), many compound semiconductors like III-V (Le. 13-15) compounds (e.g. GaAs, Ga.P, etc.), II-VI (Le. 12-16) compounds (e.g. CdS, CdTe, etc.), IV-IV (Le. 14-14) compounds (e.g. SiC) are quite important. These are discussed in Sec. 12.9.5. In the given compound semiconductors, total valence electron is four per atom. They probably undergo sp3 hybridisation to adopt the diamond structure and the sp3-band undergoes splitting in the region of nonbonding levels to produce the valence band constituted by the a-BMOs and the conduction band constituted by the a*-ABMOs (cf. Fig. 12.5.2.2). The a-band (VB) is completely filled in and the a*-band (CB) is vacant. There is a finite energy gap (Eg ) between these two bands. Fig. 12.9.1.2. Qualitative representation of the band A similar band structure occurs for diamond structure of GaAs (a III-V intrinsic compound where Eg is relatively high. Band structure of semiconductor (cf. band structure of Gr IV (14) elements; Sec. 12.5.2). GaAs is shown in Fig. 12.9.1.2.

lic

12.9.2 Characteristic Features of Extrinsic or Doped Semiconductors

C

In these semiconductors, a small amount of impurity called dopant is added deliberately to a pure sample of intrinsic semiconductor. This is why, this type of semiconductor is called extrinsic or doped or impurity semiconductor. The Inethod of addition of an impurity is called doping. Depending on the nature of impurity, the extrinsic semiconductors can be classified as n-type and p-type extrinsic semiconductors. It should be mentioned that by doping of suitable impurities, the conductivity of the semiconductor is increased appreciably.

(i) n-type extrinsic semiconductors Let us consider the crystal of silicon in which a small amount of a Gr-V (i.e. Or 15 in modern version) element such as P or As or Sb is doped. Each impurity atom substitutes it silicon atom in the diamond

Bonding in Metals and Metal Clusters: Electrical Conductivities of Solids

:~=!.:_=....

\ •



I .,

" •

\

\ •

' •

'

I'

'

I

~.



,\

2c-2e "'.""""---~II . '. I • \ " • \ I ...J. Hole covalent \• ' \ ()41 , • I bond ..... --... ....-- ...' . ' • ~.. •• •

~... \ .... - - ..., . \

, • \ .... - - ..., •

'~' Si nucleus ","'\.1 "

,I

\.' \.' '.' ' ... - - ......... - - .... , • • •• ••• ..... - _ ........ - _... •

I

\ .... - - ..., . ,

I

\.'

·

.~:~j.:.- !~.



,..--

. ' - - ... iaIl . . -- . . \.' I • \ .... - - ..., •

Surplus electron



,

I,

I • \

I

I

, •

"

\

Pentavalent \ . , \• ' \• , impurityatom ..... --.... ....-- .....' (donor) (e.g.· ~!.... ~.... P, As, etc. , . \ - " •

'.'

~

I.\--'. \

" •

"

: •



"

. . . Tnvalentlmpunty • •' . - -• • •--.~. • • • • • ~. atom (acceptor), ~ • .... - _ ..........- _.... e.g. B, AI, etc. • .... - - ' ..... - _ .....

ib ra

I • '.

I

· -- · -t

2c-1e covalent bond

ry

~

·A· ~~~~~n~~o~~-2e...:_j.:.~rJ!.. ·

.•·

Valence electrons

1185

~.',-- ......... - - .... ~ (b)

(c)

m yl

(a)

he

Fig. 12.9.2.1. Schematic representation of formation of n-type (c) and p-type (b) semiconductors through doping in the intrinsic silicon semiconductor (a).

C

lic

k

H

er e

t.m e/

th

ea

lc

cubic structure. The Gr-V (15) element utilises its four electrons out of its five valence electrons to bind tetrahedrally with the neighbouring Si-atoms. Thus, the fifth electron on each impurity atom remains unbound and these are loosely held (Figs. 12.9.2.1 and 12.9.2.2). Each Si-atom having four valence electrons bind tetrahedrally with four adjacent atoms. Thus the fifth electron on each impurity atom renzain$ free in the absence of covalent inleraction with the adjacent centres. This fifth electron on each impurity atom remains in the electric field of silicon and in such a field, the dielectric constant (cf. Table 12.5.5.1) is significantly high. Because of this fact, the fifth electron is loosely bound to the nucleus. In fact, this fifth electron is almost free and it can be easily ionised to give the electrons into the conduction bond (CB). T~is is why, such impurity atoms are called donor atoms. It may be mentioned here that the formal charge, + 1 of phosphorus (Fig. 12.9.2.2b) attracts the

==Si

+

Si - - - - Si - - - Si

I

II 2c-le bond

I

==

I

Positive hole developed due to the removal of one electron to form 4 tetrahedral bonds around the acceptor impurity atom (B)

== Si ......- - - Si - - - - Si - - - Si ==

II

II

II

II

Excess electron from the donor impurity atom (P)

Fig. 12.9.2.2. Schematic representation of p-type (a) and n-type (b) semiconductor.

Fundamental Concepts of Inorganic Chemistry

Empty

Empty

Conduction Band

Conduction Band

Ec; 1+--E1E ED • e- e e- e e- ee- eeg

ry

i

Ec EA

L Donor Impurity Level

ib ra

1186

E'E

Ev

Ev

Eilled:'\lalence:Bana

m yl

Eilloo:'\lalence:Bana

(a)

(b)

lc

he

Fig. 12.9.2.3. Conduction band, valence band and impurity level in extrinsic semiconductors. (a) type semiconductors, (b) p-type semiconductor.

n-

lic

k

H

er e

t.m e/

th

ea

electrons to some extent. Consequently these are relatively less free and they remain present not in the conduction band but in a band close to the conduction band. Thus fifth electrons coming from the impurities lie in the energy levels close to the conduction band and such energy levels form a new band or level called impuri!}' donor level which is placed in the vicinity of the CB (cf. Fig. 12.9.2.3a). The doped impurity atoms provide the electrons in the impurity donor level and these electrons are ultimately promoted to the CB for electrical conductivity. This is why, this type of semiconductor is referred to as n-type semiconductor. The required energy in promoting the electrons (Le. fifth electrons ofthe impurity centres) from the impurity donor level to the CB is called ionisation energy (E1£) which is much smaller compared to the forbidden energy gap Eg (Table 12.9.2.1). At room temperature, a large fraction of the electrons is excited from the impurity donor level to the CB while only a small number of the electrons is excited from the VB t() the CB. Thus the total number of the electrons (n e) in the CB is much larger than the number oj ~4.ules (nh ) in the VB, i.e. n e » n h . We can ignore the contribution of holes in the impurity donor level in electrical conductivity because in the impurity level, the energy levels are discrete (i.e. DOS in small) (when the concentration of the impurity atoms is small) and they cannotform a continuous band (cf. Fig. 12.9.2.4). Thus in the n-type semiconductor, the electrons

C

Table 12.9.2.1. Ionisation energies (E,E) of some doped impurities and forbidden energy gap (Eg ) in some intrinsic semiconductors

Type

Impurity .

Ionisation energy (E1FJ. e V

Forbidden energy gap (EgJ. eV

Ge

Si

Ge

Si

0.7

1.1

n-type

P As Sb

0.012 0.013 0.010

0.044 0.049 0.039

p-type

B Al Ga In

0.010 0.010 0.011

0.045 0.057 0.065 0.160

-6:Ol1

Bonding in Metals and Metal Clusters: Electrical Conductivities of Solids

1187

Empty CB Empty CB

Donor impurity levels (Discrete

Acceptor imputirty levels (Discrete energy levels)

Intrinsic semiconductor

---.

DOS

ib ra

---. DOS

m yl

---. DOS

ry

energy levels)

p-type semiconductor

n-type semiconductor

ea

lc

he

Fig. 12.9.2.4. Schematic representation of band structure of intrinsic semiconductor, n-type semiconductor and p-type semiconductor. Note: Energy levels in the impurity level are discrete and they cannot produce a continuous band. Obviously, in these energy levels, DOS is also very small. In such discrete energy levels, no effective delocalisation of electron or positive hole is possible. Consequently such levels cannot participate in electrical conduction.

t.m e/

th

are the majority carriers of electricity while the holes are the minority carriers. In naming this type of semiconductor, n is coined from the negative charge carrying property of the electrons. In such cases, concentration of the electrons equals to the number of donor atoms (nd)' Thus, the conductivity is given by: ... (12.9.2.1)

er e

(ii) p-type extrinsic semiconductors

C

lic

k

H

If an impurity (e.g. B, AI, Ga, In, etc.) from Gr-III (i.e. Gr-13 in modem version) is doped in a silicon crystal, then there will be three available valence electrons from each impurity atom. In Si-crystal, each Si-atom is tetrahedrally bound with the adjacent Si-atoms. If a B-atom substitutes one Si-atom, then the B-atom binds with three Si-atoms through the normal 2c-2e covalent bonds while it binds with the fourth Si-atom through an electron deficient 2c-le bond. In other words, it may be considered that to maintain the four normal covalent bonds (2c-2e) around the impurity atom, one electron must be withdrawn from an adjacent Si-Si bond (which is automatically converted into an electron deficient 2c-le bond). It makes the impurity atom to carry the extra negative charge and the positive charge resides nearby (Fig. 12.9.2.2a). This is why, the impurity atom like B is called acceptor and it accepts an electron from the VB of silicon. The electron deficiency in a bond (i.e. 2c-le bond) is equivalent to a positive hole signifying the missing of an electron. The number of such positive holes is the number of impurity atoms doped. The positive holes remain in a level called impurity acceptor level close to the VB of silicon. During ionisation, electrons from the VB move to the impurity acceptor level. Thus the positive holes in the VB are generated. The required energy to promote the electrons from the VB to the impurity acceptor level is very small compared to Eg • Thus, the thermal excitation promotes the electrons from the VB to the impurity acceptor level and thus the number of positive holes (nh) generated in the VB equals the number of acceptor atoms (n a ). The number of electrons promoted to the CB from the VB is negligibly small.

1188 Fundamental Concepts of Inorganic Chemistry

ry

The electrons promoted to the impurity acceptor level do not contribute significantly to the conductivity because the energy levels in the impurity acceptor level are not continuous and they cannotform a band (i.e. DOS is very small). Thus the positive holes in the VB are the majority carriers ofthe electricity. This is why, this type ofsemiconductor is called p-type semiconductor. The conductivity is given by : crp = enauh ••. (12.9.2.2)

he

m yl

ib ra

• Position of Fermi level in an extrinsic semiconductor: When the intrinsic semiconductor is doped to make it a n-type semiconductor, the Fermi level goes up from the middle position of the energy gap. It resides close to the impurity donor levelfrom whie,h the electrons can easily pass into the conduction band. On the other hand, for the p-type semiconductor, the Fermi level shifts towards the valency band (VB). In fact, the Fermi level resides close to the imJ'urity acceptor level. The exact position of the Fermi level can be calculated as discussed below. In a n-type semiconductor, the number of electrons (n e ) in the CB is approximately e~ual to the number of donor atoms (n d) doped. Thus we can write:

(Nnc )

ea

.

E F = Ec - kBT In

or,

lc

ne =nd =Nc exp[-(Ec -EF )lkB T](cf.Eqn.12.9.1.2) .•. (12.9.2.3)

th

d

t.m e/

Similarly, it can be shown that for a p-type semiconductor (by using the Eqn. 12.9.1.4). EF = Ev + kBT In (

Note: At T= 0 K, EF(n-type) =

1

2"

:~ )

(E c + ED) and EF(P-type) =

..(12.9.2.4) 1

2"

(EA + E v)' With the increase of

H

Conduction band

er e

temperature, EF (for n-type) decreases while EF (for p-type) increases gradually).

Ec

E E----------. F 0

Conduction band

CB

Ec

~

e> Q)

c

lic

k

Ec

Conduction band

C

,... ........

W

EF

-==--C

~,--

EF

Ev

A- .... ,, ---.-.--------/

---yJ--------B-' ~

1

2 E9

E ----------. A

E Valence band

(a)

Valence band

(b)

v

E

F

Valence band

(c)

VB Temperature (d)