Interatomic Bonding in Solids E-Book

Valim Levitin

• Herausgeber: Wiley-VCH
• Kategorie: Fachliteratur
• Sprache: Englisch

Seitenzahl: 453

Veröffentlichungsjahr: 2013

Beschreibung

The connection between the quantum behavior of the structure elements of a substance and the parameters that determine the macroscopic behavior of materials has a major influence on the properties exhibited by different solids. Although quantum engineering and theory should complement each other, this is not always the case.

This book aims to demonstrate how the properties of materials can be derived and predicted from the features of their structural elements, generally electrons. In a sense, electronic structure forms the glue holding solids together and it is central to determining structural, mechanical, chemical, electrical, magnetic, and vibrational properties. The main part of the book is devoted to an overview of the fundamentals of density functional theory and its applications to computational solid-state physics and chemistry.

The author shows the technique for construction of models and the computer simulation methods in detail. He considers fundamentals of physical and chemical interatomic bonding in solids and analyzes the predicted theoretical outcome in comparison with experimental data. He applies first-principle simulation methods to predict the properties of transition metals, semiconductors, oxides, solid solutions, and molecular and ionic crystals. Uniquely, he presents novel theories of creep and fatigue that help to anticipate, and prevent, possibly fatal material failures.

As a result, readers gain the knowledge and tools to simulate material properties and design materials with desired characteristics. Due to the interdisciplinary nature of the book, it is suitable for a variety of markets from students to engineers and researchers.

Leseprobe

Contents

Preface

1 Introduction

2 From Classical Bodies to Microscopic Particles

2.1 Concepts of Quantum Physics

2.2 Wave Motion

2.3 Wave Function

2.4 The Schrödinger Wave Equation

2.5 An Electron in a Square Well: One-Dimensional Case

2.6 Electron in a Potential Rectangular Box: k-Space

3 Electrons in Atoms

3.1 Atomic Units

3.2 One-Electron Atom: Quantum Numbers

3.3 Multi-Electron Atoms

3.4 The Hartree Theory

3.5 Results of the Hartree Theory

3.6 The Hartree–Fock Approximation

3.7 Multi-Electron Atoms in the Mendeleev Periodic Table

3.8 Diatomic Molecules

4 The Crystal Lattice

4.1 Close-Packed Structures

4.2 Some Examples of Crystal Structures

4.3 The Wigner–Seitz Cell

4.4 Reciprocal Lattice

4.5 The Brillouin Zone

5 Homogeneous Electron Gas and Simple Metals

5.1 Gas of Free Electrons

5.2 Parameters of the Free-Electron Gas

5.3 Notions Related to the Electron Gas

5.4 Bulk Modulus

5.5 Energy of Electrons

5.6 Exchange Energy and Correlation Energy

5.7 Low-Density Electron Gas: Wigner Lattice

5.8 Near-Free Electron Approximation: Pseudopotentials

5.9 Cohesive Energy of Simple Metals

6 Electrons in Crystals and the Bloch Waves in Crystals

6.1 The Bloch Waves

6.2 The One-Dimensional Kronig–Penney Model

6.3 Band Theory

6.4 General Band Structure: Energy Gaps

6.5 Conductors, Semiconductors, and Insulators

6.6 Classes of Solids

7 Criteria of Strength of Interatomic Bonding

7.1 Elastic Constants

7.2 Volume and Pressure as Fundamental Variables: Bulk Modulus

7.3 Amplitude ofLattice Vibration

7.4 The Debye Temperature

7.5 Melting Temperature

7.6 Cohesive Energy

7.7 Energy of Vacancy Formation and Surface Energy

7.8 The Stress–Strain Properties in Engineering

8 Simulation of Solids Starting from the First Principles (“ab initio” Models)

8.1 Many-Body Problem: Fundamentals

8.2 Milestones in Solution of the Many-Body Problem

8.3 More of the Hartree and Hartree–Fock Approximations

8.4 Density Functional Theory

8.5 The Kohn–Sham Auxiliary System of Equations

8.6 Exchange-Correlation Functional

8.7 Plane Wave Pseudopotential Method

8.8 Iterative Minimization Technique for Total Energy Calculations

8.9 Linearized Augmented Plane Wave Method

9 First-Principle Simulation in Materials Science

9.1 Strength Characteristics of Solids

9.2 Energy of Vacancy Formation

9.3 Density of States

9.4 Properties of Intermetallic Compounds

9.5 Structure, Electron Bands, and Superconductivity of MgB2

9.6 Embrittlement of Metals by Trace Impurities

10 Ab initio Simulation of the Ni3Al-based Solid Solutions

10.1 Phases in Superalloys

10.2 Mean-Square Amplitudes of Atomic Vibrations in γ′-based Phases

10.3 Simulation of the Intermetallic Phases

10.4 Electron Density

11 The Tight-Binding Model and Embedded-Atom Potentials

11.1 The Tight-Binding Approximation

11.2 The Procedure of Calculations

11.3 Applications of the Tight-Binding Method

11.4 Environment-Dependent Tight-Binding Potential Models

11.5 Embedded-Atom Potentials

11.6 The Embedding Function

11.7 Interatomic Pair Potentials

12 Lattice Vibration: The Force Coefficients

12.1 Dispersion Curves and the Born–von Karman Constants

12.2 Fourier Transformation of Dispersion Curves: Interplanar Force Constants

12.3 Group Velocity of the Lattice Waves

12.4 Vibration Frequencies and the Total Energy

13 Transition Metals

13.1 Cohesive Energy

13.2 The Rectangular d Band Model of Cohesion

13.3 Electronic Structure

13.4 Crystal Structures

13.5 Binary Intermetallic Phases

13.6 Vibrational Contribution to Structure

14 Semiconductors

14.1 Strength and Fracture

14.2 Fracture Processes in Silicon

14.3 Graphene

14.4 Nanomaterials

15 Molecular and Ionic Crystals

15.1 Interaction of Dipoles: The van der Waals Bond

15.2 The Hydrogen Bond

15.3 Structure and Strength of Ice

15.4 Solid Noble Gases

15.5 Cohesive Energy Calculation for Noble Gas Solids

15.6 Organic Molecular Crystals

15.7 Molecule-Based Networks

15.8 Ionic Compounds

16 High-Temperature Creep

16.1 Experimental Data: Evolution of Structural Parameters

16.2 Physical Model

16.3 Equations to the Model

16.4 Comparison with the Experimental Data

17 Fatigue of Metals

17.1 Crack Initiation

17.2 Periods of Fatigue-Crack Propagation

17.3 Fatigue Failure at Atomic Level

17.4 Rupture of Interatomic Bonding at the Crack Tip

18 Modeling of Kinetic Processes

18.1 System of Differential Equations

18.2 Crack Propagation

18.3 Parameters to Be Studied

18.4 Results

Appendix A Table of Symbols

Appendix B Wave Packet and the Group and Phase Velocity

Appendix C Solution of Equations of the Kronig-Penney Model

Appendix D Calculation of the Elastic Moduli

Appendix E Vibrations of One-Dimensional Atomic Chain

References

Index

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Author

Prof. Dr. Valim Levitin

Friedrich-Ebert-Str. 6635039 MarburgGermany

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Cover Design Adam Design, Weinheim

This book is dedicated to my wife Lydia and my son Viktor with sincere gratitude for understanding, encouragement and help.

Preface

The progress in many areas of industry and technology depends increasingly on the development of new materials and processing techniques. Materials science explores fundamental properties of solids. The knowledge acquired in this area is of great importance for production and everyday life.

The aim of materials science is to explain fundamentals of processes in solids and to predict the material behavior in real conditions of operation. Achievements of condensed matter physics and chemistry combined with various experimental data make it possible to solve these problems.

A variety of solids is related to the distinction of a type and strength of the interatomic bonding.

The minute particles, which a solid consists of, have the extraordinary quantum features. However, there is a gap between quantum theory on the one hand and engineering on the other hand. Even the principal notions and terms are different. The quantum physics operates with such notions as electron, nucleus, atom, energy, the electronic band structure, wave vector, wave function, Fermi surface, phonon, and so on. The objects in the engineering material science are: crystal lattice, microstructure, grain size, alloy, strength, strain, wear properties, robustness, creep, fatigue, and so on.

It is important to understand a connection between a quantum behavior of the structure elements of a substance and parameters that determine the macroscopic properties of materials.

Mechanical, electrical, magnetic properties had, for many years, a basic meaning in order to study and predict the behavior of a material in designs, components, circuits, and engines. The technique of determining the properties by the trial and error method is still widely used.

Many properties of materials can now be determined directly from the fundamental equations for systems nuclei–electrons providing new insights into critical problems in materials science, physics, and chemistry. “Increasingly, electronic structure calculations are becoming tools used by both experimentalists and theorists to understand characteristic properties of matter and to make predictions for real materials and experimentally observable phenomena [1].”

The trend of the knowledge development is as follows: one goes from the known electronic structure of atoms to features of molecules and solids. In other words, one tries to derive the peculiarities of interatomic bonding in solids from the peculiarities of electrons in free atoms. Such research in condensed matter physics and chemistry has been performed a lot over the last 20 years mainly caused by advances in calculation techniques owing to the design of numerical algorithms.

There are also applications of quantum theory for instance in the onset of a failure in a material. The failure starts on the atomic scale when an interatomic bonding is stressed beyond its yield-stress threshold and breaks. The initiation and diffusion of point defects in crystal lattice turn out to be a starting point of many failures. These events occur in a stress field at certain temperatures. The phenomena of strain, fatigue crack initiation and propagation, wear, and high-temperature creep are of particular interest. The processes of nucleation and diffusion of vacancies in the crystal lattice determines the material behavior at many operation conditions.

The subject of this book is of an interdisciplinary nature. The theme is at a junction of physics, chemistry, materials science, computer simulation, physical metallurgy, and crystallography. The goal of the book is particularly to demonstrate how the properties of materials can be derived and predicted proceeding from the quantum properties of their structural elements. We would like to show the methods of the model construction and simulation.

Thanks to the research of Walter Kohn and his collaborators the first principle simulation of structure and properties of solids is available. This simulation using contemporary density functional theory has proved to be a reliable and computationally tractable tool in condensed matter physics and chemistry.

One feels, however, that we are now on the threshold with regard to the applications in materials science. This is because, as a result of the progress in the area, we now deal with much more complex structures of real materials. Also, the practitioners (engineers and technologists) are not sufficiently acquaint enough with up-to-date methods of structural investigations, simulation, and computations.

This book covers the important topics of quantum physics, chemistry, simulation, and modeling in solid state theory, an application of electronic and atomic properties to service performance of materials. The book is not an exhaustive survey of the applications. Other authors would have chosen different topics. Nevertheless, I hope that the book will introduce the reader into this vast area of solid-state physics and chemistry and its applications.

Science as well as engineering is very differentiated now. We need an interdisciplinary approach to processes in science, technology, and engineering. In addition, another view of events, phenomena and laws can be very useful in our own specialty. On the other hand, a presentation of theories and laws has to be done at an accessible level.

I hope that this book can be used for courses such as:

solid-state physics and chemistry as engineering fundamentals;

the computer simulation of solids;

condensed matter fundamentals in modern technology;

quantum physics as basis of materials behavior.

This book is appropriate for final-year undergraduates and first-year graduate students. First of all I mean students in the area of materials science, solid-state physics, chemistry, engineering, materials technology, and machine building. The book may also be used as a preparatory material for students starting a doctorate in condensed matter physics or for recent graduates starting research in these fields of industry. It should be used as a textbook in an upper-level graduate engineering course. I believe the book will be useful for researches and practitioners from industry sectors including metallurgical, mechanical, chemical, and structural engineers.

I would like to suggest that the reader would take up the simulation of a crystal structure, which is of interest for him or her. The study of solids by means of the first-principle simulation based on the density functional theory allows one to obtain new and useful data. The definite advantage of the method is a possibility in a variation of a composition, a type of the crystal structure and the unit cell. The computer simulation is a promising addition to experimental results. Besides, it is a creative and fascinating occupation for a researcher.

1

Introduction

The experiment is the supreme judge of any physical theory.

Lev Landau

The behavior of a solid in the force, electric, or magnetic field depends on the type and energy of interatomic bonding.

As a first approximation, we can consider processes, which determine properties and structures of solids, at five levels, namely: the electronic, the atomic, the microscopic, the mesoscopic, and the macroscopic levels. Table 1.1 presents these conventional levels. At each level, the processes take place in a space dimension given by a characteristic length. The characteristic length is a linear dimension, where a corresponding process occurs. It goes without saying that there are no clear boundaries between characteristics defining these levels. Some physical phenomena have significant manifestations on more than one level of length. The corresponding experimental techniques and some methods used for the theoretical study and the simulation of the phenomena are also shown in Table 1.1.

The smallest length scale of interest is about tenths and hundredths of a nanometer. On this scale one deals in a system directly with the electrons, which are governed by the Schrödinger equation of quantum physics. The techniques that have been developed for solving this equation are extremely computationally intensive. These calculations are theoretically the most rigorous; their data are also used for developing and validating more approximate but computationally more facilitated descriptions.

The electronic level of properties of solids is of primary importance. This is no mere chance that an important subfield of condensed matter physics and chemistry is focused on the electrons in solids. Basic sciences are fundamentally concerned with understanding and exploiting the properties of interacting electrons and atomic nuclei. With this comes the recognition that, at least in principal, almost all problems of materials can be and should be addressed within quantum theory. An understanding of the behavior of electrons in solids is essential for explanation and prediction of solid state properties.

In a sense, electrons form the glue holding solids as whole, and are central in determining structural, mechanical, electrical, and vibrational properties. The understanding of strength, plasticity, electric properties, magnetism, superconductivity, and most properties of solids requires a detailed knowledge of the “electronic structure,” which is the term associated with the study of the electronic energy levels. The concept of the energy is central in physics. Nearly all physical properties are related to total energies or to differences between total energies.

Table 1.1 The levels of properties in solids.

The atomic level spans from nanometers to micrometers. Here, theoretical and experimental techniques are well developed, requiring the specification with parameters fitted to electronic-structure calculations. The most important feature of atomic simulation is that one can study a system of a relatively large number of atoms.

Above the atomic level the relevant length scale is 1 μm.

The terms simulation, modeling, calculating, and computing all refer to formulating and solving various equations which describe, explain, and predict properties of materials. If we also want to study formation and breaking of bonds, optical properties, and chemical reactions, we have to use the principles of the quantum theory as the basis for our simulation.

Most simulations utilize idealized crystalline symmetry, thus diverging from accurate description of technologically important “real materials.” This is to make models more tractable or solvable at all.

Ab initio quantum chemistry has now achieved such a level of maturity that it can satisfactorily predict most properties of isolated, relatively small molecules from a theoretical point of view. One now attempts to apply the theory to more complicated and expensive experimental observations. However, there is an even greater need for the computer simulation of solids to be equally predictive. Good correspondence with experiments is the criterion, and it is the accuracy of this correspondence which measures the worth of simulations rather than pure numerical precision of results.

Major advances in prediction of the structural and electronic properties of solids come from two sources: improved performance of hardware and development of new algorithms, and their software. Improved hardware follows technical advances in computer design and electronic components. Such advances are frequently characterized by the Moore law, which states that computer power doubles every 2 years or so. This law has held true for the past 20 or 30 years and one expects that it will hold for the next decade, suggesting that such technical advances can be predicted. In clear contrast to hardware, the development of new high performance algorithms did not show such rapid growth. Nonetheless, over the past half century, most advances in the theory of the electronic structure of matter have been made with new algorithms as opposed to better hardware. One may reasonably expect these advantage to continue. Physical concepts such as density functional theory and pseudopotentials coupled with numerical methods such as iterative methods have permitted one to examine much larger systems than one could handle solely by more and more power hardware.

This book consists of 18 chapters. This introduction is the first one.

A succinct description of the quantum physics fundamentals is presented in Chapter 2. I recall particle–wave dualism, uncertainty principle, concepts of wave motion. The wave function and the Schrödinger equation and an abstract notion of k-space are discussed.

Chapter 3 is devoted to atoms. One-electron atom and multi-electron atoms of chemical elements are considered. The probability density functions (orbitals) for electrons are illustrated. The Hartree theory is presented as a first method of approximation that has been proposed in order to calculate wave functions and energies of electrons in atoms. The covalently bonded diatomic molecules are subject of the consequent consideration.

Chapter 4 deals with the crystal lattice. Here, we discuss a basic concept of reciprocal lattice in detail and present the Wigner–Seitz cell and the Brillouin zone. These notions are commonly used in any description of the energy of electrons in solids.

We consider a homogenous gas that consists of free electrons in Chapter 5. Notions of exchange energy and correlation energy of electrons are introduced. The theory enables one to calculate some macroscopic properties of simple metals, which have the ns1 external electronic shell. The calculated cohesive energy of simple metals turns out to fit the experimental values satisfactorily.

Chapter 6 is devoted to behavior of electrons in a crystal lattice. The obvious Kronig–Penny model demonstrates the influence of periodic potential of crystal lattice on the electronic structure. The Bloch waves in the crystal lattice are described. Description of conductors, insulators, and semiconductors follows consideration of a general structure of energetic bands.

Some criteria of strength of the interatomic bonding in solids are treated in Chapter 7. Especially, we consider elastic constants, amplitudes of atomic vibrations, melting temperature, and the energy of the vacancy formation.

A technique of the solid simulation starting from the first principles (ab initio theory) is the subject of Chapter 8. We study milestones in solution of the many-body problem. We describe the density functional theory as an essence of the technique. The Kohn–Sham approach, pseudopotential method, iterative technique of calculations are described here. These methods enable one to determine and calculate the equilibrium structure of a solid quantitatively and self-consistently.

The content of Chapter 9 sheds light on validity and application for different solids of the theory, which has been considered in previous chapters. The calculated values of cohesive energy and bulk modulus are compared with the experimental results. We present data on superconductivity, the embrittlement of metals, the electronic density of states, properties of intermetallic compounds, and the energy of vacancy formation.

The Ni3Al-based solid solutions are the subject of Chapter 10. Experimental methods and also the computer simulating technique are used for these technologically important intermetallic compounds. An increase in elastic constants as a result of the replacement of aluminum atoms by the atoms of 3d and 4d transition elements is described and discussed. We demonstrate the electron density distributions that evidence delocalization of electrons in alloyed intermetallic compounds.

Chapter 11 deals with the tight-binding and the embedded-atom models of solid state. The method of the local combination of atomic orbitals is described. We present examples of the technique application. Description of atom systems in the embedded-atom method, embedding functions and applications are considered. In conclusion the reader will find the review of interatomic pair potentials.

The crystal lattice vibration and the force coefficients are the subject of Chapter 12. We describe the experimental dispersion curves and conclusions that follow from their examination. The interplanar force constants are introduced. Group velocity of lattice waves is computed and discussed. It allows one to make conclusions about the interatomic bonding strength. Energy of atomic displacements during lattice vibration (that is propagation of phonons) is related to electron structure of metals.

The transition metals are presented in Chapter 13. We describe their structure, physical models, cohesive energy, density of states for these metals.

Chapter 14 contains data on band structure, the covalent bond strength and properties of semiconductors. Here we describe the graphene, a material that “should not exist.” Here, we also dwell on the nanomaterials.

Chapter 15 is devoted to the bonding nature in molecular and ionic crystals. We recall the dipole–dipole, dipole-induced and dispersion intermolecular forces. The van der Waals and hydrogen bonds are considered. We discuss intermolecular structure and strength of ice and the solid noble gases. The description of organic molecular crystals is presented. In conclusion we consider ionic crystals and calculate their interatomic bonding.

Fundamentals of the high-temperature creep in metals is described in Chapter 16.

A physical mechanism of fatigue is reported in Chapter 17.

Finally, in Chapter 18 we present instances of modeling of kinetic processes in solids by a system of differential equations.

2

From Classical Bodies to Microscopic Particles

Let us first of all recall some peculiarities of the microscopic world which are studied by quantum physics. The laws of this world are different from those of our “usual” macroscopic world.

The interatomic bonding in solids is ensured by substances that have unusual properties compared with our daily experiences. The laws of the quantum physics deal with energy and motion of microscopic objects.

2.1 Concepts of Quantum Physics

The fundamentals of quantum physics are as follows:

the electromagnetic radiation that reveals as a particle flux;

the particle–wave duality of microscopic objects;

the discreteness of their energy;

the uncertainty principle;

the probabilistic nature of the space position of the microscopic particles.

We now briefly discuss these phenomena.

Thermal radiation is electromagnetic radiation emitting from the surface of a body as a result of its temperature. The radiation intensity (that is, the energy divided by wavelength) emitted by a heated body as a function of temperature and wavelength is shown in Figure 2.1. The curves have a maximum. Note that the maximum of the intensity shifts to shorter wavelengths as the temperature increases.

Laws of classical physics can be used to derive an equation which describes the intensity of blackbody radiation as a function of frequency for a fixed temperature – the result is known as the Rayleigh–Jeans law.1) Although the Rayleigh–Jeans law agrees with experimental data for low frequencies (long wavelengths), it diverges as the frequency increases; physicists of the beginning of nineteenth century even named this discrepancy as an “ultraviolet catastrophe.”

Figure 2.1 Dependence of a radiation of the blackbody on the wavelength and temperature. Take note of three areas of the blackbody spectrum (infrared, visible, and ultraviolet).

The electromagnetic radiation can diffract and interfere, that is it possesses certain wave properties. However, the electromagnetic radiation behaves also as a flow of particles in other phenomena (the Compton effect, photoelectric emission).

It turns out that the waves, which heated bodies radiate, are identical to the flow of particles that are called photons. According to the Planck equation the energy of a photon can be expressed as

(2.1)

(2.2)

There is an experimental evidence that microscopic particles move according to the laws of wave motion. On the one hand, the electron is a particle with a certain rest mass and a certain charge. On the other hand, flow of electrons can diffract, that is to say the electrons behave as waves when they interact with atoms of a solid. This experimental evidence forms the basis for considering a microscopic object as a dual particle–wave one. According to the de Broglie ideas any moving particle associates with a wave. The wavelength λ of the moving particle is given by

(2.3)

It can be seen from (2.3) that for macroscopic bodies having a relatively large mass the wavelength λ is too little to be detected.

Figure 2.2 presents a scheme of a moving quantum particle and a wave packet that is associated with the particle. The motion of a quantum particle can be determined by the wave vector k. The direction of the wave vector coincides with the wave propagation. The magnitude of this vector is expressed as

(2.4)

The classical expression of total energy E of a body of a mass m and velocity v is the sum of the body’s kinetic and potential energy

(2.5)

where p is the momentum. Equation (2.5) does not restrict the energy of a classical body, it may vary continuously. We shall see further that energy of a quantum object varies discretely if its motion is limited by a certain finite part of the space.

Combining (2.2)–(2.5)we obtain the total energy of a quantum particle

(2.6)

Let us compare how the motion parameters can be measured in the classic physics and in the quantum physics. For the sake of simplicity we consider onedimensional motion of a body.

The basic equation of the classical mechanics asserts that the acceleration a of the body is directly proportional to the sum of the applied forces ∑iFi and inversely proportional to the mass m of the body. An external resultant force is the cause of a change in the velocity. The Newton Second Law is given by2)

(2.7)

(2.8)

and

(2.9)

where v0 and x0 are an initial velocity and an initial path, respectively.

It is important that there are not any restrictions on decreasing the errors in measurement Δa and Δt. One can decrease the errors by improving methods of measurements.

However, the situation becomes completely different when we deal with microscopic particles. The point is that a measurement of the particle position changes its state.

The Heisenberg uncertainty principle states that an experiment cannot simultaneously determine the exact values of the position and the momentum of a particle. The precision of measurement is inherently limited by the measurement process itself such that

(2.10)

where the momentum px is known to within an uncertainty (that is, an error) of Δpx and the position x at the same time to within an uncertainty Δx. The Heisenberg principle has nothing to do with improvements in instrumentation leading to better simultaneous determinations of px and x. The principle rather says that even with ideal instruments we can never measure better than ΔpxΔx ≥ ħ/2.

Let us illustrate the Heisenberg principle by considering motion of an electron in a hydrogen atom. The position of the electron and the momentum is assumed to be measured with an uncertainty equal to 0.01% – a commonly acceptable accuracy in engineering. The centripetal force equals to the force of electrostatic attraction between the electron and the nuclei. The velocity of the electron can be found from the equation

(2.11)

Thus, the coordinate of the electron becomes undefinable even if we determine its momentum with an error of 0.01%. If we determine the coordinate of the electron then its momentum becomes undefinable.

2.2 Wave Motion

A wave motion can be characterized by several important parameters: amplitude, wavelength, and frequency. The equation of the wave propagation in a medium represents a dependence of the deviation u of a chosen point relative to its equilibrium position on its coordinate x at the time t. The general expression of this dependence for a plane harmonic wave has the following form:

(2.12)

where u(x, t) is the displacement from the equilibrium position, A is the amplitude, k is the magnitude of the wave vector that is determined by (2.4), ω is the angular frequency.

Note that the function in (2.12) is complex. Generally speaking, one should equate only the real part of the expression as only it has the physical sense. One could also use functions sine or cosine for the wave equation as follows,

(2.13)

However, the exponential form (2.12) is more convenient, especially when summing waves with different amplitudes and phases. It is just convenient to sum waves with the same ω on the complex plane. The transition from one form of the equation to the other one can be done using the Euler formula

(2.14)

The graph of the transverse traveling wave is shown in Figure 2.3.

When the traveling wave reflects from the interface of two media, a standing wave is formed as a result of interactions of the direct and inverse waves. The specific features of the standing wave are appearance of both nodes and antinodes and absence of one-way transfer of the energy in the medium. The equation of the standing wave can be obtained by summing (2.13) with the opposite signs of the second term:

Figure 2.3 The graph of the traveling wave moving from left to right. The two curves correspond to two successive retention intervals; t2 > t1.

(2.15)

The graph of the standing wave (Figure 2.4) shows the evolution of displacements over time.

Let us return to Figure 2.2. In Appendix B we expand the concepts of wave packet and group and phase velocities.

There are two velocities associated with the moving wave packet, the phase velocity and the group velocity. The phase velocity of a wave vp is the rate at which the phase of the wave propagates in space. This is the speed at which the phase of any one frequency component of the wave travels. It is expressed as

Figure 2.4 The graph of the standing wave. 1, 2, 3,…, 6 are the positions in successive time intervals: 0; 0.1T; 0.2T; …; 0.5T.

Figure 2.5 Three-dimensional image of a plane wave.

(2.16)

The group velocity of a wave is the velocity with which the overall shape of the amplitudes, which is known as the envelope of the wave, propagates through the space. The group velocity is given by

(2.17)

Figures 2.5 and 2.6 show a plane wave. Equation of the plane wave is given by

2.3 Wave Function

We are forced to replace the deterministic notion of a state of a microscopic particle by a probabilistic determination of a state. Consequently, the deterministic equation of the state has to become a probabilistic one. By “state of a particle” we mean its location and its energy.

As the quantum particle possesses wave properties it is logical to describe its state by a corresponding function.

It is appropriate to introduce a wave function Ψ for description of the state of the particle. In the one-dimensional case, the wave function depends upon coordinate x and time t. The function Ψ(x, t) is related to the probability that the particle is in a point x at a moment t. The certain quantum equation that describes behavior of a quantum particle ought to be a differential one since it tells us about a force acting on the particle by specifying potential energy corresponding to the force. We are particularly interested in how particle state changes.

A mathematical expression for a wave function can be taken in a form of a traveling wave, such as

(2.19)

The wave function is a complex-valued one. One should consider the wave function (2.19) as a part of the Schrödinger theory. The connection between the behavior of a quantum particle and properties of the wave function Ψ(x, t) is expressed in terms of the probability density p(x, t). This value determines the probability, per unit length of the x axis of finding the particle at the coordinate x at the time t. The probability density is given by

(2.20)

where the symbol Ψ*(x, t) represents the complex conjugate of Ψ(x, t). The probability p(x, t)dx that the particle is situated at a coordinate between x and x + dx equals to Ψ(x, t) Ψ*(x, t)dx. Thus, only square of the wave function has certain physical meaning.

The following property says that the particle is always somewhere in the space as the probability of such an event has to be one,

(2.21)

2.4 The Schrödinger Wave Equation

An equation, which describes states of a microscopic particle, must satisfy all assumptions concerning quantum properties of the particle.

The time-dependent differential equation for the wave function Ψ(x, t) bears the name of Schrödinger. For a one-dimensional case it is given by

(2.22)

Equation (2.22) is a general equation that describes peculiarities of the objects in the microscopic world. The differential equation has particular solutions,

(2.23)

(2.24)

(2.25)

(2.26)

Substituting (2.25) and (2.26) into (2.22) and canceling we arrive at

(2.27)

which coincides with (2.6) for total energy of the quantum particle.

It might seem that the Schrödinger equation is unfounded. As a matter of fact, it cannot be derived either from laws of classic physic or from any known dependences. However, the equation is substantiated by correctly predicting of a number of natural phenomena.

The wave function is linear in Ψ(x, t). That is, if Ψ1(x, t) and Ψ2(x, t) are two different solutions to the Schrödinger wave equation then any of their linear combinations,

(2.28)

where c1 and c2 are some constants, is also a solution.

For three-dimensional cases, the Schrödinger equation is expressed as

(2.29)

where ∇ is the Laplacian operator in rectangular coordinates,

(2.30)

and r is the radius-vector of a point where the particle is found. This equation contains a space variable r and the time variable t. The solution of the time-dependent equation (2.29) may be separated into time-dependent and time-independent parts.

(2.31)

Let us separate variables in (2.29). The time-dependent part can be expressed as

(2.32)

where E is the total energy of the particle.

Substituting

(2.33)

into (2.29) we arrive at the time-independent Schrödinger equation

(2.34)

The wave functions ψ(r) are called the eigenfunctions of the Schrödinger equation. The term originates from the German adjective “eigen” (own). The time-independent Schrödinger equation (2.34) does not contain the imaginary unit i. In one-dimensional cases it involves only one independent variable x and is an ordinary differential equation. Three-dimensional version involves more independent variables and is therefore a partial differential equation.

Differential equations may have a wide variety of possible solutions. Acceptable solutions to the time-independent Schrödinger equation must satisfy certain requirements. An eigenfunction ψ and its derivative dψ/dx must be finite, single valued, and continuous.

Certain values of energy E that correspond to solutions ψ(r) are called the eigenvalues of the potential V(r).

2.5 An Electron in a Square Well: One-Dimensional Case

A free electron can have any value of energy. However, the state of an electron changes if its motion is limited by external interactions.

We consider first the one-dimensional case. The square well potential is used in quantum physics to represent a situation when a particle is confined to some region of space by forces holding it in the region. This model represents an infinite potential barrier placed at the ends of an interval 0 ≤ x ≤ L. The infinite square well potential is written as

(2.35)

where L is the length of the well. In other words, the electron cannot overstep the limits of the segment. This simple model is useful because the motion of the electron in the isolated atom is also confined to some area.

(2.36)

The states of the electron are described by the Schrödinger equation

(2.37)

A possible solution of (2.37) is

(2.38)

where A is a proportionality constant. It follows from the boundary conditions (2.35) that

(2.39)

Substituting (2.38) and (2.39) into (2.36) we arrive at

(2.40)

and

(2.41)

A probability to find the electron within the segment dx equals to ψ2dx,

(2.42)

Two basic results are as follows.

Figure 2.7 The squared wave functions of an electron for a one-dimensional model with an interval of 0–L. The segment L is taken to be 0.3 nm in length.

Substituting solution (2.38) to (2.37) one obtains

(2.43)

2.6 Electron in a Potential Rectangular Box: k-Space

We are looking for a solution of the three-dimensional time-independent Schrödinger equation,

(2.44)

(2.45)

Or

(2.46)

where kx, ky, kz are the components of the wave vector k:

(2.47)

(2.48)

The energy of the electron can take only discrete values. They are given by

(2.49)

The n values can again be interpreted as quantum numbers which define energetic states of the electron and therefore determine the allowed values of energy. It can be seen from (2.48) that the values of k and n are closely related.

Quantum numbers of the electron in the ground state are equal to 1, 1, 1; corresponding energy of the electron equals to 3π2ħ2/2mL2. The first excited state of the electron is determined by one of three combinations of quantum numbers: 2, 1, 1; 1, 2, 1 and 1, 1, 2. Any of three sets corresponds to the same value of energy, namely 6π2ħ2/(2mL2). A state, to which several combinations of the quantum numbers correspond to, is called a degenerated state. In this case it is the triply degenerated state.

The three-dimensional time independent Schrödinger equation (2.44) has a family solutions of the (2.46) type. Every solution has corresponding wave vector k with components kx, ky, kz given by (2.48). The k vectors constitute a vector space called k-space (reciprocal space and phase space are other terms used in the literature). The concept of k-space is extremely useful by considering the interaction of a radiation with a matter.

For the real-space with a coordinate system x, y, z that defines a three-dimensional crystal there is a k-space. The k-space is a mathematical image direct bounded with the real-space. The real-space and the k-space are presented in Figure 2.8.

1) A blackbody is an ideal body that completely absorbs all radiant energy falling upon it with no reflection and that radiates at all wavelengths with a spectral energy distribution dependent on its absolute temperature.

3

Electrons in Atoms

This chapter is devoted to the electronic structure of atoms.

We examine at first an one-electron atom. A state of the electron is described by quantum numbers. We consider a more complex problem of a multi-electron atom further and turn our attention to theories that describe the many-body problem.

D.R. Hartree and V.A. Fock made an important contribution to the science of systems consisting of a large number of electrons and nuclei. We review the Hartree theory, which gives an approximation method for the determination of the ground-state eigenfunction and the corresponding eigenvalue of a many-body quantum system.

Finally, we will reintroduce the atomic structures of chemical elements.

3.1 Atomic Units

Atomic units are used in quantum physics.

The Bohr radius a0 0.529 177 × 10–10 m is a unit of length. This unit is called Bohr (atom unit, a.u.).

The rest mass of the electron, which is equal to me 9.109 534 × 10–31 kg, is accepted as the unit of the mass.

3.2 One-Electron Atom: Quantum Numbers

The theory of one-electron atom seems to be the simplest one but it is also the most important one.

An one-electron atom contains two particles, that is, a positively charged nucleus and the negatively charged electron. The two particles are bound together by the Coulomb attraction force. In fact, both particles move around a common center of masses. The ratio of the proton mass to the electron mass mp/me is 1835.55. Thus, the massive nucleus may by considered as almost completely stationary.

A coordinate of the electron is determined by the vector r(x, y, z). In the rectangular coordinate system one has . The electron moves under the influence of the Coulomb potential given by

(3.1)

where 1/(4π ε0) is the Coulomb law constant, –e is the electron charge, Z is the number of the element in the periodic table, +Ze is the nucleus charge.

The time-independent Schrödinger equation for one-electron atom is expressed as

(3.2)

or, as a functional equation,

(3.3)

where H is called the Hamiltonian. Hamiltonian is the operator associated with energy of the electron. Solutions of the Schrödinger equation (3.2) exist only for certain values of energy; these values are called eigenvalues of energy. To each eigenvalue of energy Ei corresponds an eigenfunction ψi, so that

Figure 3.1 Eigenvalue and eigenfunction in quantum physics.

As before, we require that an eigenfunction ψ(x, y, z) and its derivatives ψ′(x, y, z) must be finite, single valued and continuous. Such functions are called “well-behaved.”

There are three unknown independent variables in (3.2), namely x, y, z. One should separate the variables in order to split the partial differential equation into a set of three ordinary differential equations, each involving only one coordinate. However, the separation of variables cannot be carried out when rectangular coordinates are employed because the Coulomb potential energy (3.1) could not be represented as a product of functions each depending on one variable only.

However, a solution can be found in spherical polar coordinates. This solution can be represented as a product of functions each dependent on one coordinate. These are the coordinates r, θ, φ, where r is the length of the segment connecting the electron with the nucleus (the origin), θ and φ are the polar and azimuthal coordinates, respectively. Because of this simplification of the potential, it is possible to carry out the separation of variables in the time-independent Schrödinger equation. This now can be written instead of (3.2) as

(3.4)

The change of coordinates allows one to find solutions of the time-independent Schrödinger equation of the form

(3.5)

The reader may refer to the book [2] for mathematical aspects of this approach. Here, we state the results.

The solutions of the ordinary differential equations in the assumed product form of (3.5) depend on three integer values that have certain physical meaning. These values are denoted by n, l and ml. They are called quantum numbers.

The quantum numbers characterize the following properties of the electron in atom, respectively:

the energy of the electron;

its angular momentum

1)

;

the projection of the orbital angular momentum onto direction of the external magnetic field.

The total energy of the electron En is found to be finite only if it has one of the values

(3.6)

Figure 3.2 The dependence of the Coulomb potential V on distance from nucleus r and the values of energy for an one-electron atom. En is negative and increases with principal quantum number n as 1/n2. The negative energy means that it takes the energy in order to remove the electron from the atom. En → 0 if n → ∞.

Figure 3.2 illustrates the dependence of energy of the electron in one-electron atom on its distance from the nucleus.

Thus, the principal quantum number n determines the energy of the electron.

The atom is a three-dimensional system. Consequently, the system has an angular momentum. The function Θ(θ) turned out to be dependent on the quantum number l. l