This introduction to the concepts and methods of quantum mechanics employs the analysis of one-dimensional problems to offer students a quantitative understanding of atomic, molecular, solid-state, and nuclear physics. Applications of these concepts and methods help answer the most intriguing questions of modern physics: What holds matter together? Holds it apart? How does the variety of chemical properties of different elements arise? How do electrons move through solids? Why do nuclei that occur in nature possess only certain combinations of protons and neutrons?
The text presents meaningful problems by topic — supplemented by ample illustrations, applications, and exercises — that address the most intriguing questions of modern physics. Answers to selected problems appear in the appendix. Geared toward science and engineering majors, this volume is also appropriate for independent study by those who have completed a general physics course.
About the Author
Robert L. Sproull is Professor Emeritus and former President of the University of Rochester. He directed the Advanced Research Projects Agency of the U.S. Department of Defense, served as Chairman of the Defense Science Board, and is a member of the Roundtable Council of the National Academy of Sciences.
W. Andrew Phillips is a University Demonstrator and Lecturer in Physics at the University of Cambridge, where he is a Life Fellow of Queens' College.
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The Quantum Physics of Atoms, Solids, and Nuclei
By Robert L. Sproull, W. Andrew Phillips
Dover Publications, Inc.Copyright © 1980 Robert L. Sproull and W. Andrew Phillips
All rights reserved.
ATOMS, NUCLEI, AND PARTICLES
An analytical introduction to much of the basic physics developed in the twentieth century is presented in this book. The new physics has been of great intrinsic interest, almost a new science in itself, and in addition has provided applications in engineering that are already considerable and are expanding rapidly. The study of modern physics has led to new devices and energy sources, to more convenient and accurate instruments, to new materials of construction, and to a clearer understanding of existing materials.
This book is primarily concerned with physical laws and processes, but applications in other sciences and in engineering will be described frequently. Television camera tubes, transistors, nuclear reactors, and other devices will be analyzed as part of the application of the basic physics; but most of the applications will be found in other science and engineering courses and in engineering practice.
The atom can be said to mark the boundary between nineteenth and twentieth century physics. Although the idea that matter is composed of atoms was popular in the nineteenth century, it was only at the end of the century that consistent measurements of atomic size became available, and the atomic theory placed on a sound quantitative footing. Since 1900 physics has been increasingly concerned with the internal structure of atoms, and such studies have provided an important element in the development of twentieth century physics. In this first chapter we parallel the historical development by describing a number of experiments that have been used to investigate atomic structure in more and more detail, and in so doing we introduce many important concepts and theories, useful later in the book. At various places in this chapter, as indeed in the first few chapters, it will frequently be necessary to assert some properties of particles and to assume the existence of sources and detectors. These assertions and assumptions will be justified only much later in the book. However, in the later chapters we attempt a uniformly analytical approach; that is, we make assertions or conclusions only by logical arguments based on experiments or on theories well tested by experiment.
This chapter opens with an account of a modern experiment that effectively allows one to see atoms and gives estimates of atomic sizes and binding energies. The properties of two subatomic particles, the electron and the proton, are described in Sec. 1-3, and this is followed by an account of the first investigation of the structure of the atom. The conclusion of this investigation is that the atom consists of a tiny, positively charged core, the nucleus, surrounded by one or more electrons. Although most of our discussion of the nucleus is left to Chapters 11 and 12, Sec. 1-6 gives some idea of the sizes and binding energies of nuclei. However, to provide a background to nuclear physics, the variation of mass with velocity and the famous Einstein E = Mc2 relation are presented in Sec. 1-5. Finally, a section is devoted to particle physics, where we try to give an idea of the properties of highly energetic particles.
As the chapter progresses, we look at the atom (or nucleus) on a finer and finer spatial scale and in general at effects that occur at higher and higher energies. These spatial and energy scales bring us to the most important conclusion of this chapter, that the problem of the interactions among particles can be divided into two separate problems: (1) The binding together of protons and neutrons into the nucleus at the center of the atom; (2) the motions of the electrons around the nucleus. This division can be achieved because (1) the energies of interaction between protons and neutrons in a nucleus are very much greater, by a factor of between 102 and 106, than the energies of interaction between electrons and nuclei; (2) a nucleus is much smaller than an atom. Therefore the size and energy scales of atomic experiments are such that the nuclei remain unchanged; we can consider the nucleus as a heavy particle and can ignore its size and internal structure. Nuclear experiments, on the other hand, involve such large energies and small distances that the presence of electrons surrounding the nucleus are of little consequence. Therefore we can deal separately with atomic physics (Chapters 3 to 6) and nuclear physics (Chapters 11 and 12). Furthermore, we shall see in Sec. 1-7 that the whole field of particle physics can be considered almost independently of the question of the structure of the nucleus.
1-2 THE ATOM
The idea that matter is composed of atoms is so basic to modern physics that it is difficult to realize that it was only at the turn of the century that scientists were finally convinced of the truth of the atomic theory. Philosophically, the idea is of long standing, but scientists were persuaded of the truth of the theory only when consistent measurements of atomic size became available. Perhaps the most important of these measurements was provided by X-ray diffraction experiments; these will be described in Sec. 3-5. A more recent technique, known as field-ion microscopy, is described in this section and comes as close as we can get to an actual picture of an atom.
In the field ion microscope a very fine needle of tungsten, niobium, or some other refractory metal is placed at the center of an evacuated chamber (as shown in Fig. 1-1a). A large potential difference is established between the needle, which is positive, and a negative or ground electrode close to the outer wall. The glass surface of the chamber is coated with a phosphor so that a flash of light is produced when it is struck by energetic particles (just as a television tube produces a flash of light when the screen is struck by electrons). Typically, the radius of curvature of the needle is approximately 100 nm, so that a voltage of 5000 V applied between needle and screen gives an electric field at the tip of the needle of 5 × 1010 Vm-1 (Problem 1-1). When a small amount of helium gas is let into the chamber, a pattern of spots appears on the phosphor, as shown in Fig. 12.
Even without any detailed explanation, this picture gives strong evidence that matter is not continuous but discrete on a scale of approximately 10-10 m. This scale can be evaluated by combining a measurement of the spot separation on the screen with the magnification of about 107, the ratio of the effective radii of the tip and the chamber.
The explanation of the effect makes use of the well-known concentration of electric field on the more pointed parts of a conductor (Problem 1-1). The average field at the tip of the needle is large, because of the small radius of curvature, but on an atomic scale (Fig. 1-1b) there are field variations between one atom and the next, and there is field enhancement at the bumps. A helium atom striking the tips may lose an electron through ionization (described in Sec. 5-4), which occurs preferentially at points where the field is largest. The resulting positive ion is strongly repelled from the surface and travels almost exactly along a radius, starting from the point at which it lost an electron. The pattern of spots on the phosphor produced by the helium ions therefore provides an enormously magnified image of the surface irregularities. As can be seen in Fig. 1-2, individual atoms can be resolved.
X-ray diffraction experiments are visually less dramatic but give more precise measurements of the distances between atoms in solids. If it is assumed that the atoms are packed so closely together that they touch, these experiments give a measurement of the size of atoms. Other estimates of atomic size are provided by studies of collisions between atoms in a gas (Sec. 2-5). All these experiments give the same result: atoms have average diameters of just over 10-10 m.
1-3 THE ELECTRON AND THE PROTON
It is a little ironic that just as precise measurements of atomic size were becoming available, evidence was also accumulating that atoms had an internal structure. The electron was the first subatomic particle to be identified from studies of the internal glow observed when an electrical discharge is struck in a gas at low pressure. We now know that such a discharge consists of a mixture of negatively charged electrons, positively charged ions, and neutral gas atoms. The charged particles are accelerated by the electric field and by colliding with neutral atoms can produce more electrons and ions so that the discharge is self-sustaining. Electrons are attracted to the anode, and by making a hole in the anode we can obtain a beam of electrons. The term cathode ray was first applied to the electrons, as it was thought that they originated at the cathode.
The identification of the cathode rays as electrons was made by J.J. Thomson in 1897 through a measurement of the ratio e/m, where e is the charge on the electron and m is the mass. This ratio, as we shall see, labels a particle almost unambiguously, although, as described in Sec. 1-5, at velocities close to the velocity c of light the mass of a particle depends on its velocity. The value of e/m that is quoted for a particular particle is, strictly speaking, the ratio e/m0 appropriate to zero velocity. Since throughout this book we deal with velocities very much less than c, we usually omit the subscript. Where both m and m0 appear in an equation m will be the actual mass and m0 will be the mass of the electron at rest. A similar convention will be followed where necessary for other particles.
The original apparatus used by Thomson is shown in Fig. 1-3. Although his result for e/m is now known to be rather inaccurate, we describe his method, since the principles involved are essentially the same as those used in more recent experiments. Electrons from the discharge (on the left) pass through the pair of slits which define their direction, and then between a pair of aluminum deflection plates. The end of the tube is coated with a phosphor, a material that emits light when struck by electrons and so allows them to be seen. The physics of this process is described in Sec. 9-6. A potential difference can be applied between these plates to give an electric field, and a magnetic field can also be generated in the same region of the tube by means of external coils (not shown in Fig. 1-3). Notice that the design of the system resembles very closely that of a modern cathode-ray tube, although in a modern tube electrons are generated by heating a metallic filament. This process is known as thermionic emission, and is described in Sec. 9-8.
A deflection of the electrons can be produced by applying a potential Vd between the two plates. The force on the electron of charge -e (e > 0) is -eVd/d, so when they leave the region of electric field the electrons have a transverse velocity equal to
-[eVd/md] [l/v] (1-1)
where l is the length of the plates and v is the initial velocity along the tube. The angular deflection, if small, is given by
θ = -[eVd/md] [l/v2] (1-2)
The sign of the deflection, of course, gives the sign of the charge on the particle.
If instead of an electric field the electrons move through a magnetic field the force is
F = -ev x B (1-3)
where F, v, and B are all vectors. The direction of this force is at right angles to both v and B. Since v is perpendicular to B the magnitude of F is just
F = -evB
This force produces a deflection φ given by
φ = -[eB/m] [l/v] (1-4)
In the actual experiment B was adjusted so that φ = θ which, on equating eqs. 1-2 and 1-4 gives
v = Vd/Bd (1-5)
Combining eq. 1-5 with eq. 1-2 gives
e/m = Vdθ/B2ld (1-6)
All the quantities on the right of eq. 1-6 can be measured, so that e/m can be found.
The best modern value, obtained with much greater precision than was possible in Thomson's experiment, is
e/m = 1.759 x 1011 cKg-1
Even though Thomson obtained a value roughly half this, his result was highly significant. Experiments on electrolysis had previously shown that the ratio of charge to mass for hydrogen, the smallest atom, was approximately 1000 times smaller than Thomson's value for the electron. The implication was clear: particles smaller than the atom could exist.
The e/m ratio of the electron is larger than the similar ratio for any other particle or aggregate of particles. This fact and the relative ease with which electrons can be emitted from solids are responsible for the great usefulness of electrons in vacuum-tube devices. Particles with smaller charge-to-mass ratios are more sluggish in electric and magnetic fields. If such particles were used in electron tube devices, the tubes could be used only at low frequencies.
Thomson's experiment gave a value only for the ratio e/m. The charge on the electron was measured by another classic experiment, the Millikan oil-drop experiment. The apparatus is shown in Fig. 1-4. A pair of horizontal, parallel condenser plates is mounted inside an enclosure, which prevents drafts and allows the pressure to be varied. Except in very precise work (where the pressure must be varied to determine small corrections), the chamber is filled with ordinary air at atmospheric pressure. An atomizer is used to spray fine drops of nonevaporating oil between the plates. A telescope with horizontal hairs permits a single drop to be observed and the vertical velocity measured. The velocity is determined by measuring the time required for the drop to rise or fall the fixed distance d between the images of the hairs. This distance is usually considerably less than D, the spacing between the plates.
A source of ionizing radiation that can be turned on or off is provided. This source can be an X-ray tube or an ultraviolet arc. The process of ionization will be considered in detail in later chapters; at this point, all we need to know is that it is possible to remove an electron from an atom by X rays or by ultraviolet light. If this atom is a gas atom, a positive ion and an electron are formed, either one of which may be captured by the oil drop. If this atom is one of the atoms making up the oil drop, the oil drop will attain a positive charge. Thus, while the ionizing radiation is turned on, the oil drop can have its charge either increased or decreased but always by an integral number of electron charges.
A falling drop of the size used in this experiment reaches its terminal velocity very quickly ([much less than] 1 s). When the drop moves at this constant velocity, the drag caused by the viscosity of air is equal in magnitude and opposite in sign to the other forces (gravitational or gravitational plus electrical) acting on the drop. For spherical drops, this viscous force has been found by ordinary hydrodynamics experiments to be
F = 6πηav (1-7)
where η is the viscosity of the medium, a is the radius of the sphere, and v is its terminal velocity. Equation 1-7 is called Stokes law (it requires a correction of a few percent for the very small drops used to measure e, but this correction can be found by varying the pressure of the air in the chamber). Drops of the size encountered in this experiment are spherical because of the surface tension of the oil.
The procedure of this experiment is illustrated in Fig. 1-5. A drop is selected, its time t0 of fall through a distance d is measured with no electric field, and its terminal velocity is then determined from the relation v0 = d/t0 (velocities downward are considered positive). Since the drop is not accelerating, the sum of the forces on it must be zero. Therefore the sum of the gravitational force Mg (downward) and the viscous force F (upward) must be zero, and
Mg = -6πηav0 = 6πηa d/t0 (1-8)
where g is the acceleration of gravity (9.80 ms2) and M is the mass of the drop. Since the drop is spherical,
M = 4/3πa3ρ (1-9)
where ρ is the density of the oil. Equations 1-8 and 19 have two unknowns (M and a); therefore M and a can be determined from them. Because the drop is so small, neither M nor a can be measured directly; the order of magnitude of a is 10-6 m.
Excerpted from Modern Physics by Robert L. Sproull, W. Andrew Phillips. Copyright © 1980 Robert L. Sproull and W. Andrew Phillips. Excerpted by permission of Dover Publications, Inc..
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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Table of Contents
1 Atoms, Nuclei and Particles
1-1 Introduction 1
1-2 The Atom 3
1-3 The Electron and the Proton 6
1-4 The Nuclear Binding Energies 15
1-5 Relativistic Effects 19
1-6 Nuclear Binding Energies 25
1-7 Particles and Particle Accelerators 37
2 Assemblies of Particles
2-1 Introduction 57
2-2 Energy of Random Motion 58
2-3 Maxwell Distribution of Kinetic Energies and Velocities 63
2-4 Boltzmann Distribution 69
2-5 Collisions, Mean Free Paths, and Atomic Sizes 72
3 Quantum Phenomena
3-1 Introduction 81
3-2 Black-body Radiation 82
3-3 The Photoelectric Effect 88
3-4 Line Spectra 94
3-5 X-Ray Line Spectra 99
3-6 The Continuous X-Ray Spectrum 108
3-7 Excitation Potentials 110
3-8 The Compton Effect 113
3-9 Electron Diffraction 118
3-10 Neutron Diffraction 121
3-11 The Stern-Gerlach Experiment 124
3-12 Summary 126
4 Introductory Quantum Mechanics
4-1 Introduction 135
4-2 The Wavefunction and the Schrödinger Equation 136
4-3 The Square Well Potential 141
4-4 The Free Particle, Wave Packets, and the Uncertainty Principle 145
4-5 Potential Wells and Potential Barriers 156
4-6 Operators and Observables 169
4-7 Measurement, Expectation Values, and Superposition States 178
4-8 Eigenvalues and Eigenfunctions of the Simple Harmonic Oscillator 186
4-9 Approximate Methods 196
4-10 Radiation Theory 204
4-11 Summary 211
5 Atomic Structure and Spectra
5-1 Introduction 221
5-2 The Hydrogen Atom 222
5-3 The Exclusion Principle 234
5-4 Electronic Structure of Atoms 236
5-5 X-Ray Spectra 245
5-6 Optical Spectra 252
5-7 Lasers 258
6-1 Introduction 273
6-2 The KCI Molecule and Ionic Binding 274
6-3 The Hydrogen Molecule Ion 279
6-4 Molecular Orbitals 283
6-5 Covalen Bonding in Molecules 291
6-6 Molecular Spectra and Dissociation 297
7 Binding and Energy Bands in Solids
7-1 Introduction 309
7-2 Inoic and Covalent Crytals 310
7-3 Metallic Crystals 315
7-4 Energy Bands, Atomic Energy Level Approach 320
7-5 Enery Bands, Nearly Free Electron Approach 329
8 Electrical, Thermal, and Magnetic Properties of Solids
8-1 Introduction 341
8-2 Conductors and Non-Conductors of Electricity 342
8-3 Fermi Distribution of Electron Energies 348
8-4 Conduction Band Electrons in Metals and Alloys 351
8-5 Thermal Properties of Solids 360
8-6 Electrical Conductivity of Metals and Alloys 369
8-7 Superconductivity 377
8-8 Magnetic Properties of Solids 387
9 Imperfections in Solids
9-1 Introduction 407
9-2 Types of Imperfections 408
9-3 Diffusion and Ionic Conductivity 418
9-4 Optical Absorption 421
9-5 Photoconductivity 426
9-6 Luminescence 432
9-7 Slip and Strength of Metals and Alloys 438
9-8 Surface Physics 444
10-1 Introduction 467
10-2 Intrinsic Semiconductors 468
10-3 n- and p-Type Semiconductors 476
10-4 Semiconductor-Insulator Boundaries 484
10-5 Applications of MOS Devices 495
10-6 p-n Junctions 500
10-7 Junction Transistors 516
11 Nuclear Physics
11-1 Introduction 533
11-2 Radioactivily; α Emission 534
11-3 Size and Constituents of Nuclei 540
11-4 β Emission and Electron Capture 548
11-5 γ Emission and Internal Conversion 552
11-6 Nuclear Stability 556
11-7 Nuclear Reactions 569
11-8 Nuclear Forces 578
12 Experimental and Applied Nuclear Physics
12-1 Introduction 591
12-2 Nuclear Fission 592
12-3 Nuclear Reactors 598
12-4 Nuclear Fusion 607
12-5 Interaction between Charged Particles and Matter 611
12-6 Detectors for Nuclear Particles 616
12-7 Application of Radioactive Nuclides 623
A Physical Constants 633
B Conversion of S.I. to cgs Units 635
C Periodic System of the Elements 639
D Atomic Masses and Atomic Weights 641
E Rutherford Scattering 647
F The Density of Normal Modes 653
G Pulse Spectra and the Uncertainty Principle 657
H Answers to Problems 663