Physics of FunctionalMaterialsHasse Fredriksson

KTH Stockholm, Sweden

and

Ulla Åkerlind

University of Stockholm, Sweden

Physics of Functional Materials

Physics of FunctionalMaterialsHasse Fredriksson

KTH Stockholm, Sweden

and

Ulla Åkerlind

University of Stockholm, Sweden

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Library of Congress Cataloging in Publication Data

Fredriksson, Hasse.Physics of Functional Materials / Hasse Fredriksson and Ulla Åkerlind.

p. cm.Includes bibliographical references and index.ISBN 978-0-470-51757-4 (cloth) — ISBN 978-0-470-51758-1 (pbk. : alk. paper)1. Physics. 2. Materials. I. Akerlind, Ulla. II. Title.QC21.3.F74 2008530.4—dc22

2007050191

British Library Cataloguing in Publication Data

A catalogue record for this book is available from the British Library

ISBN 978-0-470-51757-4 (HB)ISBN 978-0-470-51758-1 (PB)

Typeset in 10/12pt Times by Integra Software Services Pvt. Ltd, Pondicherry, IndiaPrinted and bound in Spain by Grafos SA, Barcelona

Frontpage figures Reproduced with permission from

1. Posten Frimärken AB, Sweden, Nobel Prize stamps celebrating Schrödinger and de Broglie.

2. Parmanand Sharma, TMR Tohuku University, Sendai Magnetic Domains observed by MFM technology.

Contents

Preface vii

1 Structures of Melts and Solids 11.1 Introduction 11.2 X-ray Analysis 21.3 The Hard Sphere Model of Atoms 101.4 Crystal Structure 121.5 Crystal Structures of Solid Metals 191.6 Crystal Defects in Pure Metals 241.7 Structures of Alloy Melts and Solids 30

Summary 39Exercises 42

2 Theory of Atoms and Molecules 452.1 Introduction 462.2 The Bohr Model of Atomic Structure 462.3 The Quantum Mechanical Model of Atomic Structure 482.4 Solution of the Schrödinger Equation for Atoms 572.5 Quantum Mechanics and Probability: Selection Rules 652.6 The Quantum Mechanical Model of Molecular Structure 712.7 Diatomic Molecules 752.8 Polyatomic Molecules 88

Summary 89Exercises 94

3 Theory of Solids 973.1 Introduction 973.2 Bonds in Molecules and Solids: Some Definitions 983.3 Bonds in Molecules and Nonmetallic Solids 1003.4 Metallic Bonds 1123.5 Band Theory of Solids 1253.6 Elastic Vibrations in Solids 1463.7 Influence of Lattice Defects on Electronic Structures in Crystals 151

Summary 157Exercises 163

4 Properties of Gases 1694.1 Introduction 1704.2 Kinetic Theory of Gases 1704.3 Energy Distribution in Particle Systems: Maxwell–Boltzmann Distribution Law 1744.4 Gas Laws 1784.5 Heat Capacity 1854.6 Mean Free Path 1924.7 Viscosity 1964.8 Thermal Conduction 1984.9 Diffusion 199

vi Contents

4.10 Molecular Sizes 2044.11 Properties of Gas Mixtures 2054.12 Plasma – The Fourth State of Matter 208

Summary 213Exercises 216

5 Transformation Kinetics: Diffusion in Solids 2195.1 Introduction 2205.2 Thermodynamics 2205.3 Transformation Kinetics 2365.4 Reaction Rates 2425.5 Kinetics of Homogeneous Reactions in Gases 2455.6 Diffusion in Solids 253

Summary 276Exercises 283

6 Mechanical, Thermal and Magnetic Properties of Solids 2876.1 Introduction 2876.2 Total Energy of Metallic Crystals 2886.3 Elasticity and Compressibility 2906.4 Expansion 2966.5 Heat Capacity 3036.6 Magnetism 317

Summary 333Exercises 337

7 Transport Properties of Solids. Optical Properties of Solids 3417.1 Introduction 3427.2 Thermal Conduction 3427.3 Electrical Conduction 3477.4 Metallic Conductors 3507.5 Insulators 3577.6 Semiconductors 3627.7 Optical Properties of Solids 375

Summary 388Exercises 394

8 Properties of Liquids and Melts 3998.1 Introduction 4008.2 X-ray Spectra of Liquids and Melts 4008.3 Models of Pure Liquids and Melts 4028.4 Melting Points of Solid Metals 4068.5 Density and Volume 4078.6 Thermal Expansion 4098.7 Heat Capacity 4128.8 Transport Properties of Liquids 4158.9 Diffusion 4168.10 Viscosity 4258.11 Thermal Conduction 4388.12 Electrical Conduction 439

Summary 443Exercises 449

Answers to Exercises 455

Index 475

Preface

The basic idea and the initial aim of this book, Physics of Functional Materials, was to provide the necessary knowledgein physics for a deeper interpretation of many of the solidification and crystallization processes that are treated in the bookMaterials Processing during Casting, written for the university undergraduate level and published in March 2006. The presentbook fulfils this requirement and is at such a mathematical level that a basic knowledge of mathematics at university level issufficient.

However, the book Physics of Functional Materials has a very wide and general character. It is by no means designedonly for the purpose described above. On the contrary, this book may be useful and suitable for students in various sciencesat the Master’s and PhD level, who have not taken mathematics and physics as major subjects. Examples of such sciencesare materials science, chemistry, metallurgy and many other scientific and technical fields where a basic knowledge of thefoundations of modern physics and/or properties of materials is necessary or desirable as a basis and a background for higherstudies.

Fundamental properties of different materials such as diffusion, viscosity, heat capacity and thermal and electrical conductionare examined more extensively in the present book than in available physics books of today. This book will fill a gap betweenthe demand for and supply of such knowledge.

The atomistic view of matter requires a genuine background of modern physics from atoms via molecules to solid-statephysics, primarily the modern band theory of solids, and the nature of bonds and crystal structure in solids. These topics aretreated in Chapters 1–3 and are applied in later chapters, particularly in Chapter 7.

The three first chapters on modern physics are followed by applications of classical physics of material properties. Basicthermodynamics, properties of gases, including the kinetic theory of gases and the Boltzmann distributions of velocity andenergy of molecules, transformation kinetics including chemical reactions and diffusion in solids, mechanical, thermal andmagnetic properties of matter including ferromagnetism, are treated in Chapters 4–6.

Chapter 7 deals with thermal and electrical conduction in solids and their optical properties, particularly electrical conductionin metals and semiconductors. Polarization phenomena in crystals and optical activity in solids and liquids are discussed. Inthe last chapter, a short survey of the material properties of liquids is given.

Physics of Functional Materials contains solved examples in the text and exercises for students at the end of each chapter.Answers to all the exercises are given at the end of the book.

Acknowledgements

We want to express our sincere thanks to Dr Gunnar Benediktsson (KTH, Stockholm) for many long and fruitful discussionsand support on solid-state physics. Dr Ulf Ringström (KTH, Stockholm) gave many valuable points of view on atomic andmolecular physics and optics. We also owe our gratitude to Dr Göran Grimvall and Dr Ragnar Lundell (KTH, Stockholm) forvaluable help concerning aspects of solid-state theory and solid mechanics, respectively.

Dr Jonas Åberg, Thomas Bergström and Hani Nassar (Department of Casting of Metals, KTH, Stockholm) gave us practicalsupport concerning the ever-lasting computer problems throughout the years. We thank them gratefully for their patience andunfailing help. We also owe our gratitude to Dr Gunnar Edvinsson (University of Stockholm) and Dr Thomas Antonsson(Department of Casting of Metals, KTH, Stockholm) for valuable computer help. Colleges at the Institutes of Physics at theUniversities of Uppsala and Lund and KTH, Stockholm, kindly allowed us to use freely their problem collections in physics.We thank them for this generous offer. Some of their exercises have been included in this book. We also thank Dr OlofBeckman, University of Uppsala, for permission to use some polarization figures.

viii Preface

We are most grateful for financial support from The Iron Masters Association in Sweden. Finally and in particular we wantto express our sincere gratitude to Karin Fredriksson and Lars Åkerlind. Without their constant support and great patiencethrough the years this book would never have been written.

Hasse FredrikssonUlla Åkerlind

Stockholm, SwedenMarch 2008

1Structures of Melts and Solids

1.1 Introduction 11.2 X-ray Analysis 2

1.2.1 Methods of X-ray Examination of Solid Materials 21.2.2 X-ray Examination and Structures of Metal Melts 4

1.3 The Hard Sphere Model of Atoms 101.3.1 Atomic Sizes 101.3.2 The Hard Sphere Model of Liquids 11

1.4 Crystal Structure 121.4.1 Crystal Structure of Solids 121.4.2 Types of Crystal Structures 131.4.3 Lattice Directions and Planes 151.4.4 Intensities of X-ray Diffractions in Crystal Planes 18

1.5 Crystal Structures of Solid Metals 191.5.1 Coordination Numbers 191.5.2 Nearest Neighbour Distances of Atoms 201.5.3 BCC, FCC and HCP Structures in Metals 21

1.6 Crystal Defects in Pure Metals 241.6.1 Vacancies and Other Point Defects 251.6.2 Line Defects 261.6.3 Interfacial Defects 28

1.7 Structures of Alloy Melts and Solids 301.7.1 Basic Concepts 301.7.2 Structures of Liquid Alloys 311.7.3 Structures of Solid Alloys 33

Summary 39Exercises 42

1.1 Introduction

Crystallization is the process of transferring a material from a liquid or a gas phase into a solid state of regular order. The threeaggregation states have widely different atomic structures and properties. In order to understand crystallization processes, it isessential to have a thorough knowledge of the atomic structure of both the melt or gas and of the new solid phase.

Physics of Functional Materials Hasse Fredriksson and Ulla Åkerlind© 2008 John Wiley & Sons, Ltd

2 Physics of Functional Materials

In this chapter, the atomic structures of melts and various solid phases will be examined. The structures of pure elementsand of alloys and chemical compounds will be discussed. The chapter starts with a short introduction to X-ray analysis, whichis a very important tool for investigating atomic structures.

1.2 X-ray Analysis

The energetic X-radiation was discovered in 1901 by the German physicist W. K. Roentgen. The source was an evacuatedtube where electrons, accelerated by an electric field, hit a metal target (Figure 1.1). When the electrons suddenly lost theirkinetic energy, continous X-ray radiation was emitted together with a few discrete X-ray wavelengths characteristic of thetarget atoms (Figure 1.2).

Cathode Anode

X-rays

Figure 1.1 The principle of an X-ray tube. Electrons are accelerated in a strong electric field between an anode and an indirectly heatedcathode, hit the anode and lose their kinetic energy successively during numerous collisions. Their kinetic energies are transformed intocontinuous X-radiation. The whole equipment is included in a highly evacuated tube.

KL

M

Kα-radiation

KLM

Kβ-radiation

(a) (b)

Figure 1.2 (a) The origin of the characteristic X-radiation: When a high-speed electron hits an electron in an inner shell of a target atom,both electrons leave the atom and a vacancy is left in its inner shell. The rest of the process is described in (b).(b) The vacancy is filled by an electron from an outer shell of the atom and an X-ray photon with a wavelength characteristic for the anodematerial is emitted. There are several alternative wavelengths, depending on the shell from which the jumping electron emanates. The X-raylines K� and K� shown in (a) and (b) appear simultaneously as numerous such processes occur at the same time.

X-ray analysis of materials has been used since the beginning of the 20th century to investigate the construction andstructure of materials. Measurements, calculations and interpretations of the results were made by hand for many decades.Today, completely automatic computer-based X-ray spectrometers, which perform both the experiments and the analysis ofthe results, are commercially available. However, it is necessary to understand the principles of their function, which aredescribed briefly below.

1.2.1 Methods of X-ray Examination of Solid Materials

A very important method to provide information on the structure of materials is X-ray diffraction measurements. In solidcrystalline materials, the diffraction pattern is caused by the atoms in the crystal lattice (Figure 1.3). If a crystal surface is

Structures of Melts and Solids 3

exposed to parallel monoenergetic X-radiation with a known angle of incidence, Bragg’s law gives the condition for coherence(in phase) of the diffracted waves:

θ θ

θθ

2θd

Figure 1.3 X-ray diffraction in a crystal.

2d sin � = p� (1.1)

whered = distance between the atomic planes� = angle between the incident ray and the atomic plane� = wavelength of the monoenergetic X-rayp = an integer �= 0.

If � is known and � can be measured, Bragg’s law can be used to determine the distance d between the atomic planes in thecrystal lattice.

An X-ray crystallographic examination of a single crystal will in principle be performed as described in Figure 1.4.

C

S

D

θ2θ

Figure 1.4 The principle of an X-ray spectrometer.

The radiation from an X-ray tube S falls on a turnable single crystal C. The angle between the incident and diffracted raysis 2�. The crystal is tilted stepwise. For every angle of incidence � the detector D is placed at the corresponding angle ofdiffraction 2� and the intensity of the diffracted radiation is measured. In this way, the intensity as a function of � is obtainedand the whole X-ray spectrum of the solid material can be obtained.

The method described above is time consuming. If the single crystal is replaced with a powder compound, consisting ofsmall crystals of the solid material, it is no longer necessary to tilt the specimen to find high-intensity angles. All possiblecrystallographic directions are present in the powder compound.

In practice, the Debye–Scherrer method of X-ray crystallographic investigations on metallic or other crystalline materials isthe most common one. The principle is the same as that described in Figure 1.4. The apparatus used is shown in Figure 1.5.The detector consists of a cylinder-shaped photographic film. The monoenergetic radiation from the X-ray tube passes througha narrow vertical slit and then the specimen, a thin powder compound, placed at the centre of the cylinder.

The diffracted radiation with the highest intensity has the angle of scattering 2� (Figure 1.7). The diffracted radiationhas a cone-shaped form. The radiation is registered on the photographic film as four slightly curved lines. These linesare characteristic of the solid material and will appear symmetrically around the direct beam when the film is developed.Figure 1.6b shows the appearence of the diffraction pattern of zinc.

The positions of the lines are measured. The pattern on the film is analysed and the various distances between the atomicplanes in the crystals are calculated. In this way, the structure of the solid material can be derived.

4 Physics of Functional Materials

2θ

Specimen

Film

Figure 1.5 X-ray examination according to the Debye–Sherrer method. Reproduced with permission from B. D. Cullity, Elements ofX-Ray Diffraction, © Addison-Wesley Publishing Company, Inc. (now under Pearson Education).

2θ = 180° 2θ = 0

(a)

(b)

Figure 1.6 (a) The appearence of a Debye–Sherrer pattern. (b) Debye–Sherrer pattern of zinc. Reproduced with permission fromB. D. Cullity, Elements of X-Ray Diffraction, © Addison-Wesley Publishing Company, Inc. (now under Pearson Education).

2θθ

Incident beam

Scattered beam

Figure 1.7 The angle 2� of scattering is the angle between the incident and scattered beams.

1.2.2 X-ray Examination and Structures of Metal Melts

By replacing the powder compound by a metal melt at constant temperature, the corresponding Debye–Sherrer measurementscan be made on a liquid metal.

The results deviate very much from those obtained for crystalline powders. Instead of a few sharp, well-defined lines oneobtains several wide, unsharp maxima and minima. The resulting patterns can be analysed and fairly detailed information onthe structures of the melts can be obtained. This is illustrated below by two examples.

Structures of Pure Metal Melts

The X-ray pattern of liquid gold at 1100 �C is shown in Figure 1.8. The intensity of the diffracted radiation is plotted versussin �/�, where � is half the scattering angle and � is the X-ray wavelength. In Figure 1.8, the main peak is found to the leftand a series of subsidary peaks to the right.

Structures of Melts and Solids 5

The short vertical lines in Figure 1.8 show the corresponding X-ray pattern of crystalline gold. The intensities of thediffracted waves are indicated by the heights of the lines. There is a clear correspondence between the positions along thehorizontal axis of the main peaks of liquid gold and the main lines of crystalline gold.

sin θλ

12500

Intensity

10000

7500

5000

2500

00 0.2 0.4 0.6 0.8

Figure 1.8 The intensity of scattered X-radiation as a function of sin �/�. The curve originates from scattering in liquid gold at 1100 °C.The vertical lines originate from scattering in crystalline gold. Reproduced with permission from N. S. Gingrich, The diffraction of X-raysby liquid elements, Rev. Mod. Phys. 15, 90–100. © 1943 The American Physical Society.

Intensity

0 0.30.20.1 0.60.50.4

sin θλ

Figure 1.9 The intensity of scattered X-radiation as a function of sin �/�. The curve originates from scattering in liquid zinc at 460 �C.Reproduced with permission from N. S. Gingrich, The diffraction of X-rays by liquid elements, Rev. Mod. Phys. 15, 90–100. © 1943 TheAmerican Physical Society.

An X-ray intensity spectrum such as those in Figures 1.8 and 1.9 supply the necessary basis to obtain information concerningthe structure of a liquid or solid metal. However, to interpret an intensity diagram and obtain information from it, such astype of structure and atomic distances, it is necessary to transform the X-ray spectrum into another type of diagram. Thebackground of this is given in next section.