Elements of Structures and Defects of Crystalline Materials

By Tsang-Tse Fang

Elements of Structures and Defects of Crystalline Materials has been written to cover not only the fundamental principles behind structures and defects, but also to provide deep insights into understanding the relationships of properties, defect chemistry and processing of the concerned materials. Part One deals with structures, while Part Two covers defects. Since the knowledge of the electron configuration of elements is necessary for understanding the nature of chemical bonding, it is discussed in the opening chapter. Chapter Two then describes the bonding formation within the crystal structures of varied materials, with Chapter Three delving into how a material’s structure is formed.

In view of the importance of the effects of the structure distortion on the material properties due to the fields, the related topics have been included in section 3.4. Moreover, several materials still under intensive investigation have been illustrated to provide deep insights into understanding the effects of the relationships of processing, structures and defects on the material properties.

The defects of materials are explored in Part II. Chapter 4 deals with the point defects of metal and ceramics. Chapter 5 covers the fundamentals of the characteristics of dislocations, wherein physics and the atomic mechanics of several issues have been described in detail. In view of the significant influence of the morphologies including size, shape and distribution of grains, phases on the microstructure evolution, and, in turn, the properties of materials, the final chapter focuses on the fundamentals of interface energies, including single phase (grain) boundary and interphase boundary.

• Discusses the relationship between properties, defect chemistry and the processing of materials
• Presents coverage of the fundamental principles behind structures and defects
• Includes information on two-dimensional and three-dimensional imperfections in solids

• Language: English
• Publisher: Elsevier Science
• Release date: Jan 25, 2018
• ISBN: 9780128142691

Tsang-Tse Fang

Tsang-Tse Fang received the B.E. degree in Dept. of Metallurgy and Materials Engineering, National Cheng Kung University (NCKU), Taiwan, the M. E degree in Institute of Mineral and Materials Science, NCKU, Taiwan, and Ph.D. degrees in Dept. of Materials Engineering, North Carolina State University, U.S.A. in 1985. Since February 1986, he has been with Dept. of Materials Science and Engineering, NCKU, Taiwan, where he was an Associate Professor, and became a Professor in 1991. In 2006, he was honored as a distinguished professor. He was a visiting associate professor, Materials Research Lab., Penn. State University, U. S. A. from 1989 to 1990), joining the research concerning smart materials and a visiting scientist, Dept. of Chemical Engineering, Massachusetts Institute of Technology, U. S. A. from 1994 to 1995. His major research areas include: (i) ceramic processing: powder synthesis, forming, and sintering involving defect chemistry and microstructural evolution, and (ii) physical properties of electronic ceramics: dielectrics, ferroelectricity, electrical conduction (involving lithium-ion battery), magnetism, superconductivity, magnetoelectronics, thermoelectricity and multiferroics. In 1993, he received outstanding research award from National Science Council Taiwan. He was an Editor of the International Scholarly Research Network (ISRN Ceramics) until 2012. Presently he is an adjunct professor of Dept. of Materials Science and Engineering, NCKU, and an associate editor of journal of electronic materials, U.S.A.

Book preview

Elements of Structures and Defects of Crystalline Materials

Tsang-Tse Fang

National Cheng Kung University

Table of Contents

Cover image

Title page

Copyright

Preface

Acknowledgements

Part I: Structures of the Crystalline Materials

Part I. Structures of the Crystalline Materials

Chapter 1. The Electron Configuration of Atoms

Abstract

1.1 Atoms With a Single Electron

1.2 Atoms With More Than One Electron

Chapter 2. Bonding Within Crystal Structures

Abstract

2.1 Bonding in Ionic Crystals

2.2 Covalent Bonds

2.3 Metallic Bonds

2.4 Effect of Bonding on the Material Properties by Means of the Potential-Well Concept

Chapter 3. The Structures of Crystalline Crystals

Abstract

3.1 Arrangements of Atoms and Ions in Crystalline Solids: Space Lattice

3.2 Metallic Structures

3.3 Ionic Structures

3.4 Structural Distortion in Ionic Structures

3.5 Structure of Material Technological Interest

Part II: Defects of Crystalline Materials

Part II. Defects of Crystalline Materials

Chapter 4. Point Defects in Crystalline Materials

Abstract

4.1 Point Defects in Metals

4.2 Point Defects in Ionic Solids

Chapter 5. Line Defects in Crystalline Solids

Abstract

5.1 The Discrepancy Between the Theoretical and Observed Yield Stresses of Crystals

5.2 Observations of Dislocations

5.3 Crystallographic Slip

5.4 Elementary Geometric Characteristics of Dislocations

5.5 Critical Resolved Shear Stress

5.6 Plastic Flow Associated With the Slip of the Dislocation Motion

5.7 Stress Fields of Dislocations

5.8 Energy of a Dislocation

5.9 Line Tension of a Dislocation

5.10 Forces Upon Dislocations

5.11 The Bowing of a Dislocation

5.12 Force Between Dislocations

5.13 Peach–Koehler Equation

5.14 Reactions Between Dislocations

5.15 Extended Dislocations

Chapter 6. Two-Dimensional (Interfaces) and Three-Dimensional (Second Phases) Imperfections in Solids

Abstract

6.1 Grain Boundaries

6.2 Interphase Boundaries in Solids

6.3 Surface Tension, Surface Stress, and Surface Free Energy of Interfaces

6.4 Free Surface

6.5 Interfaces of Phases

6.6 Effect of Interface Curvature on the Equilibrium Pressure Between Two Phases

6.7 Effect of the Interface Curvature on the Equilibrium Solubility Between Two Phases

6.8 Equilibrium Vacancy Concentration Changes at Curved Surfaces: Driving Force for Sintering

Appendix I. Spectroscopic Notation

Appendix II. Band Structures of Mott–Hubbard Materials and the Transition Metal Ionic Compounds, and Polaron

Bibliography

Index

Copyright

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Preface

As sparked by the discovery of the high Tc superconductor, growth and characterization of the materials mainly concerned with structures and defects are at the forefront of the research in materials science. The significance is further boosted on account of the subsequent discovery of the materials with so-called colossal properties, including magnetoresistance and dielectric constant. These materials are of perovskite or perovskite-like structures with transition metal ions on the B site and show an enormous variety of intriguing electronic or magnetic properties. This variety is not only related to their chemical flexibility concerning the defects, but also to a larger extent related to the complex character concerning the interplay between transition metal ions in certain coordinations and oxygen. Since these new emerging materials with multiple functions provide the most interesting and challenging topics in fundamental physics and the cutting edge of new technologies, they have drawn considerable worldwide attention of scientists with highly interdisciplinary backgrounds. However, research on these subjects is experiencing a superfast development and facing more completion. In response to these challenges, the scientists have to gain and grasp the desired knowledge in a short time. For most existing textbooks, they either are inadequate, focusing on one of the structures and defects, or are voluminous, including a wide variety of subjects. By contrast, this book is designed not only to be concise, yet comprehensive to thoroughly cover the fundamental principles of the structures and defects of crystalline materials, but also to provide deep insights into understanding the relationships of properties, defect chemistry, and processing of concerned materials. What is more, the issue concerning the significant effects on the properties of the functional oxides due to the crystal fields is missing from most of the existing related textbooks. This issue has been included in Section 3.4 of this book. In the fast-paced world, it is more important than ever to understand the fundamentals. Thus this book has been focused on the physics and the atomic mechanics of the problems to cover sufficient fundamentals, deep enough to allow the readers to delve into the most recent literature without too much difficulty. In addition, this book appeal to those other than materials science and engineering backgrounds, thus the author has made an effort to write this book so that it does not assume much prior knowledge of the subject from the readers. Furthermore, there are several issues in most existing textbooks that come from an old viewpoint. This book has updated these issues including extensions or corrections, e.g., Section 1.2.2 Energetic d-Orbital Collapse of Free Neutral Atoms at the Beginning of the Transition Rows, Section 2.3.1 Formation Energy and Mechanism of Covalent Bonding, Section 3.4.5 Structure Distortions Arising From Asymmetric Electron Density or Inert Pair Effect, Section 5.6 Plastic Flow Associated With the Slip of the Dislocation Motion, and Section 6.3 Surface Tension, Surface Stress, and Surface Free Energy of Interfaces. Finally, it has to be stressed that for most functional materials, to fully understand the interactions between properties and defect chemistry and processing, the knowledge concerning both structures and defects is necessary since they are correlated. This book has included the essence of the knowledge concerning the author’s more than 30 years of research to provide deep insights into understanding the relationships of properties, defect chemistry, and processing.

This book is divided into two parts with six chapters. The first three chapters deal with the structures (Part I) and the remaining three chapters are concerned about the defects (Part II). Since the knowledge of the electron configuration of elements is necessary for understanding the nature of chemical bonding, it is discussed in the first chapter. This chapter is by no means intended to be comprehensive but to lend support to the aspects that are fundamental to the Section 3.4 of d electron concerned crystal fields. The determination of the material’s structure is mainly from knowledge of its constituents’ chemical bonding, which also has a great influence on the physical and chemical properties of materials. Chapter 2, Bonding Within Crystal Structures, thoroughly describes the bonding formation within the crystal structures of different materials, wherein the formation energy and mechanism of covalent bonding have been updated and the potential well concept has been established to correlate with the material properties. Chapter 3, The Structures of Crystalline Crystals, introduces the basic principles for constructing the material’s structure. In view of the importance of the effects of the structure distortion on the material properties due to the crystal fields, the related topics have been included in Section 3.4. Moreover, several materials being still under intensive investigation have been illustrated to provide deep insights into understanding the effects of the relationships of processing, structures, and defects on the material properties.

The defects of materials are explored in Part II. Chapter 4, Point Defects in Crystalline Materials, deals with the point defects of metal and ceramics. While it starts from a very basic level, it proceeds to fairly advanced interpretations and covers the major fundamentals of defect structures and chemistry. While it is impossible to include all topics related to line imperfections, i.e., dislocations, Chapter 5, Line Defects in Crystalline Solids, essentially covers in ample detail the fundamentals of the characteristics of dislocations, wherein physics and the atomic mechanics of several issues have been described in detail. In view of the significant influence of the morphologies including size, shape, and distribution of grains and phases on the microstructure evolution and in turn the properties of materials, the final chapter focuses on the fundamentals of interface energies including single-phase (grain) boundary and interphase boundary, wettability determining the morphologies of phases, and the change of surface stress at a curved surface leading to the kinetic processes like Ostwald ripening and sintering. The distinction between surface energy, surface tension, and surface stress of liquids and solids in terms of atomic viewpoint is updated.

Acknowledgements

The accomplishment of this incredible task is by the grace of the Lord who lavishes wisdom, knowledge, and understanding upon me for the fear of the Lord is the beginning of wisdom and knowledge of the Holy One is understanding. Many thanks are to Prof. L.H. Chen for the helpful discussions and valuable suggestions concerning dislocations. It is also wished to acknowledge the generous assistance from Profs. J.C. Kuo and W.D. Hsu who provided me the computer-aided graphs of structures. Especially, I am most grateful to my wife for not only her support and patience during the preparation of this book but also her drawing most of the graphs in this book.

Part I

Structures of the Crystalline Materials

Outline

Part I. Structures of the Crystalline Materials

Chapter 1 The Electron Configuration of Atoms

Chapter 2 Bonding Within Crystal Structures

Chapter 3 The Structures of Crystalline Crystals

Part I. Structures of the Crystalline Materials

Concerning the structures of perfect crystalline solids, they possess a perfectly ordered, periodic arrangement of atoms or ions. The understanding of the structure of atoms would be a priori to describe the arrangement of atoms. This is necessary since the arrangement of atoms is dependent on the atom bonding whose origins are found to be in the electronic structure of atoms. The following chapter will deal with the electron configuration of atoms.

Chapter 1

The Electron Configuration of Atoms

Abstract

Since the knowledge of the electron configuration of elements is necessary for understanding the nature of chemical bonding, and in turn the structure of a perfectly ordered, periodic arrangement of atoms or ions, it is discussed in the first chapter. This chapter starts with the basic knowledge concerning the construction of the electron configuration of elements, and it proceeds to fairly advanced interpretations to understand the related issues. Moreover, although this chapter does not give detailed derivation of the Schrödinger’s wave equation, the spirit of this equation is presented, wherein the quantum numbers can be clearly defined and the electron density contours are well represented. Furthermore, on the basis of the comparison of the radial distribution functions, the significant influence of the penetration and shielding of electrons on constructing the electron configuration, especially, the energetic d-orbital collapse of 3d transition elements, can be perceived. The interpretation of the electron configuration of the half-filled and fully-filled 3d transition elements has been updated. This chapter is by no means intended to be comprehensive but to lend support to the fundamentals that are referred to in Section 3.4 of chapter 3 concerning d electron effects on crystal fields.

Keywords

Electron configuration; penetration; shielding; d-orbital collapse; half-filled orbital; fully-filled orbital; spin–orbital coupling; transition metals; radial distribution function

1.1 Atoms With a Single Electron

To appreciate the energetics and shapes of an atomic orbital is to solve the so-called Schrödinger’s wave equation. The simplest case is the hydrogen atom whose electron behavior is described by the following usual form of the Schrödinger’s wave equation:

(1.1)

is the Laplacian operator, expressed in Cartesian coordinates as

E and V are the total energy and the potential energy of electron in a hydrogen atom, respectively; me , is the solution of Eq. (1.1). Since the orbital of the hydrogen atom is spherically symmetrical, the description of the orbital would be easier to be tackled in spherical coordinates, as shown in Fig. 1.1. Eq. (1.1) can further be transformed in spherical coordinates as

(1.2)

Figure 1.1 Setup of relationship between Cartesian coordinates, x, y, z, and the spherical polar coordinates r, θ, and cubes based on the Cartesian axes are used to identify the axes of contour maps of the electron density distribution for s, p, and d orbitals.

The wave function now is a function of the three variables of rand can be represented by three separated functions R(ras follows:

(1.3)

Substitute Eq. (1.3) into Eq. (1.2) and multiply throughout by r²sin²/R , yielding

(1.4)

Underlying Eq. (1.4), because both sides of this equation with different variables, the term of each side must be equal to the same constant m² for the equation to be valid.

For the term on the right, we obtain

(1.5)

. Let each side of this equation be equal to the same constant α, giving the following two equations:

(1.6)

(1.7)

The acceptable (physically appropriate) solutions are further selected for Eq. (1.5), (1.6), and (1.7). The solution of Eq. (1.5) is

(1.8)

, the values of the constant m 2, …, etc., actually indicating the magnetic quantum number. For the possible solution of Eq. (1.7), we find out that α=l(l+1), where l +2, …, etc., and this l refers to the orbital quantum number. Another quantum number is further introduced, i.e., n, the so-called principal quantum number, for the solution of Eq. (1.6). The values of n can be taken as l+1, l+2, l+3, …, etc. The characteristics of these three quantum numbers are briefly reviewed as follows:

1. n: principal quantum number

It specifies the major axis of an elliptical orbit and denotes the probable size of the orbit and the energy shell. The letters K, L, M, N, etc., corresponding to n=1, 2, 3, 4, etc., respectively, are used in X-ray and spectroscopic work. The total possible number of electrons with a given value of n is 2n².

2. l: orbital quantum number

It is a measure of the angular momentum of the orbit motion of the electron and determines the shape of the orbital for any given value of n. It can only assume the values 0, 1, 2, …, n–1. The maximum number of states is given by 2(2l+1). All orbitals with l=0 are called s orbitals (s: sharp), l=1 are called p orbitals (p: principle), l=2 are called d orbitals (d: diffused), and l=3 are called f orbitals (f: fundamental).

3. m: magnetic quantum number

In the presence of a magnetic field, m determines the spatial orientation of the plane of the orbit. It varies in integral steps from –l to l, including zero.

To specify a particular energy state or quantum state for an electron, a fourth quantum number is required in addition to the aforementioned three quantum numbers. The fourth quantum number arises from the fact that the electron itself has an angular momentum referred to as the spin. The spin quantum number, s, (s=±1/2), describes the angular momentum components of the spin and denotes the direction of electron spin.

The wave function of the hydrogen electron (Eq. (1.3)) can further be specified in terms of the quantum numbers as:

(1.9)

. The conventional contour maps of the electron density distribution for s, p, and d orbitals are represented in , where r varying with r of orbitals 1s, 2s, 3s, 3p, and 3d.

Figure 1.2 Radial distribution functions for orbitals of 1s, 2s, 3s, 3p, and 3d of the hydrogen atom, indicating the probability of finding the electron at a distance r from the nucleus for each atomic orbital and the comparison of the degree of penetration of the electron in the orbitals of different principal quantum numbers.

For each orbital, the r value of each maximum probability corresponds to the radius of the corresponding Bohr orbit. Here, it should be pointed out that there is a finite probability at an infinite distance from the nucleus different than the classical concept of the Bohr orbit.

Concerning the electron structure of single electrons like hydrogen atoms, the energy states can basically be solved from Eq. (1.6) by substituting V with –Ze²/r (where Z is atomic number; e is electron charge; and r is the distance of the electron from the nucleus) and expressed as:

(1.10)

Thus for the single electron case, the energy of each subshell is determined uniquely by the principal quantum number n, i.e., the 2s and 2p subshells are of identical energy and the 3s,