Electronic Devicesand CircuitsMILLMAN & HALKIASINTERNATIONAL STUDENT EDI"

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ELECTRONIC DEVICES

AND CIRCUITS

Jacob Millman, Ph.D.

Professor of Electrical Engineering

Columbia University

Christos C. Halkias, Ph.D.

Associate Professor of Electrical Engineering

Columbia University

INTERNATIONAL STUDENT EDITION

McGRAW-HILL BOOK COMPANY*New York St. Louis San Francisco Diisseldorf

London Mexico Panama Sydney Toronto

KOGAKUSHA COMPANY, LTD.Tokyo

ELECTRONIC DEVICES AND CIRCUITS

INTERNATIONAL STUDENT EDITION

Exclusive rights by Kogokusha Co., Ltd., for manufactureand export from Japan. This book cannot be re-exportedfrom the country to which it it coniigned by KogakushaCo., Ltd., or by McGraw-Hill Book Company or any of itisubsidiaries.

XI

Copyright 1967 by McGraw-Hill, Inc. All Rights Re-served. No part of this publication may be reproduced,stored in a retrieval system, or transmitted, in any formor by any meant, electronic, mechanical, photocopying,recording, or otherwise, without the prior written permis-sion of the publisher.

Library of Congress Catalog Card Number o7-16934

TOSHO J'HINTINQ CO., LTD., TOKYO, JAPAN

PREFACE

This book, intended as a text for a first course in electronics for elec-trical engineering or physics students, has two primary objectives: topresent a clear, consistent picture of the internal physical behavior ofmany electronic devices, and to teach the reader how to analyze anddesign electronic circuits using these devices.

Only through a study of physical electronics, particularly solid-state science, can the usefulness of a device be appreciated and itslimitations be understood. From such a physical study, it is possibleto deduce the external characteristics of each device. This charac-terization allows us to exploit the device as a circuit element and todetermine its large-signal (nonlinear) behavior. A small-signal(linear) model is also obtained for each device, and analyses of manycircuits using these models are given. The approach is to consider acircuit first on a physical basis, in order to provide a clear under-standing and intuitive feeling for its behavior. Only after obtainingsuch a qualitative insight into the circuit is mathematics (throughsimple differential equations) used to express quantitative relationships.

Methods of analysis and features which are common to manydifferent devices and circuits are emphasized. For example, Kirch-hoff's, Thevenin's, Norton's, and Miller's theorems are utilized through-out the text. The concepts of the load line and the bias curve areused to establish the quiescent operating conditions in many differentcircuits. Calculations of input and output impedances, as well ascurrent and voltage gains, using small-signal models, are made for awide variety of amplifiers.

A great deal of attention is paid to the effects of feedback oninput and output resistance, nonlinear distortion, frequency response,and the stabilization of voltage or current gains of the various devicesand circuits studied. la order that the student appreciate the differentapplications of these circuits, the basic building blocks (such as untunedamplifiers, power amplifiers, feedback amplifiers, oscillators, and powersuppliers) are discussed in detail.

For the most part, real (commercially available) device charac-teristics are employed. In this way the reader may become familiarwith the order of magnitude of device parameters, the variability ofthese parameters within a given type and with a change of temperature,the effect of the inevitable shunt capacitances in circuits, and the effectof input and output resistances and loading on circuit operation. These

vii

viii / PREFACE

considerations are of utmost importance to the student or the practicing engi-neer since the circuits to be designed must function properly and reliably inthe physical world, rather than under hypothetical or ideal circumstances.

There are over 600 homework problems, which will test the student'sgrasp of the fundamental concepts enunciated in the book and will give himexperience in the analysis and design of electronic circuits. In almost allnumerical problems realistic parameter values and specifications have beenchosen. An answer book is available for students, and a solutions manualmay be obtained from the publisher by an instructor who has adopted the text.

This book was planned originally as a second edition of Millman's"Vacuum-tube and Semiconductor Electronics" (McGraw-Hill Book Com-pany, New York, 1958). However, so much new material has been addedand the revisions have been so extensive and thorough that a new title for thepresent text seems proper. The changes are major and have been madenecessary by the rapid developments in electronics, and particularly by thecontinued shift in emphasis from vacuum tubes to transistors and other semi-conductor devices. Less than 25 percent of the coverage relates to vacuumtubes; the remainder is on solid-state devices, particularly the bipolar tran-sistor. In recognition of the growing importance of integrated circuits andthe field-effect transistor, an entire chapter is devoted to each of these topics.But to avoid too unwieldy a book, it was decided not to consider gas tubes,silicon-controlled rectifiers, polyphase rectifiers, tuned amplifiers, modulation,or detection circuits. The companion volume to this book, Millman andTaub's "Pulse, Digital, and Switching Waveforms" (McGraw-Hill BookCompany, New York, 1965), gives an extensive treatment of the generationand processing of nonsinusoidal waveforms.

Considerable thought was given to the pedagogy of presentation, to theexplanation of circuit behavior, to the use of a consistent system of notation,to the care with which diagrams are drawn, and to the many illustrative exam-ples worked out in detail in the text. It is hoped that these will facilitate theuse of the book in self-study and that the practicing engineer will find the textuseful in updating himself in this fast-moving field.

The authors are very grateful to P. T. Mauzey, Professor H. Taub,and N. Voulgaris, who read portions of the manuscript and offered con-structive criticism. We thank Dr. Taub also because some of our materialon the steady-state characteristics of semiconductor devices and on tran-sistor amplifiers parallels that in Millman and Taub's "Pulse, Digital, andSwitching Waveforms." We acknowledge with gratitude the influence ofDr. V. Johannes and of the book "Integrated Circuits" by Motorola, Inc.(McGraw-Hill Book Company, New York, 1965) in connection with Chapter15. We express our particular appreciation to Miss S. Silverstein, adminis-trative assistant of the Electrical Engineering Department of The City College,for her most skillful service in the preparation of the manuscript. We alsothank J. T. Millman and S. Thanos for their assistance.

Jacob MillmanChristos C. Halkias

CONTENTS

Preface

Electron Ballistics and Applications 1

1-1 Charged Particles 11-2 The Force on Charged Particles in an Electric Field1-3 Constant Electric Field S1-4 Potential 61-5 The eV Unit of Energy 71-6 Relationship between Field Intensity and Potential1-7 Two-dimensional Motion 81-8 Electrostatic Deflection in a Cathode-ray Tube 101-9 The Cathode-ray Oscilloscope 121-10 Relativistic Variation of Mass with Velocity IS1-11 Force in a Magnetic Field 151-12 Current Density 161-13 Motion in a Magnetic Field 171-14 Magnetic Deflection in a Cathode-ray Tube 201-15 Magnetic Focusing 211-16 Parallel Electric and Magnetic Fields 241-17 Perpendicular Electric and Magnetic Fields 261-18 The Cyclotron SI

Energy Levels and Energy Bands 36

2-1 The Nature of the Atom 362-2 Atomic Energy Levels S82-3 The Photon Nature of Light 402-4 Ionization

x / CONTENTS

3

5

6

Conduction in Metals 52

3-1 Mobility and Conductivity 623-2 The Energy Method of Analyzing the Motion of a

Particle 543-3 The Potential-energy Field in a Metal 573-4 Bound and Free Electrons 693-5 Energy Distribution of Electrons 603-6 The Density of States 863-7 Work Function 683-8 Thermionic Emission 693-9 Contact Potential 703-10 Energies of Emitted Electrons 713-1

1

Accelerating Fields 743-12 High-field Emission 763-13 Secondary Emission 75

Vacuum-diode Characteristics 774-1 Cathode Materials 774-2 Commercial Cathodes 804-3 The Potential Variation between the Electrodes4-4 Space-charge Current 824-5 Factors Influencing Space-charge Current 864-6 Diode Characteristics 874-7 An Ideal Diode versus a Thermionic Diode4-8 Rating of Vacuum Diodes 894-9 The Diode as a Circuit Element 90

* 7

80

88 8

Conduction in Semiconductors 95

5-1 Electrons and Holes in an Intrinsic Semiconductor 965-2 Conductivity of a Semiconductor 975-3 Carrier Concentrations in an Intrinsic Semiconductor 995-4 Donor and Acceptor Impurities 1085-5 Charge Densities in a Semiconductor 1055-6 Fermi Level in a Semiconductor Having Impurities 1055-7 Diffusion 1075-8 Carrier Lifetime 1085-9 The Continuity Equation 1095-10 The Hall Effect 113

Semiconductor-diode Characteristics 115

6-1

6-26-3

6-4

6-5

6-6

6-7

6-8

115Qualitative Theory of the p-n JunctionThe p-n Junction as a Diode 117Band Structure of an Open-circuited p-n JunctionThe Current Components in a p-n Diode 12$Quantitative Theory of the p-n Diode CurrentsThe Volt-Ampere Characteristic 127The Temperature Dependence of p-n CharacteristicsDiode Resistance 1S2

9

120

124

ISO

CONTENTS / xi

6-9 Space-charge, or Transition, Capacitance CV6-10 Diffusion Capacitance 1386-11 p-n Diode Switching Times 1406-12 Breakdown Diodes 1486-13 The Tunnel Diode 1476-14 Characteristics of a Tunnel Diode 153

134

156

166

175

Vacuum-tube Characteristics 156

7-1 The Electrostatic Field of a Triode7-2 The Electrode Currents 1597-3 Commercial Triodes 1617-4 Triode Characteristics 1627-5 Triode Parameters 16$7-6 Screen-grid Tubes or Tetrodes7-7 Pentodes 1697-8 Beam Power Tubes 1 717-9 The Triode as a Circuit Element 1737-10 Graphical Analysis of the Grounded-cathode Circuit7-11 The Dynamic Transfer Characteristic 1787-12 Load Curve. Dynamic Load Line 1797-13 Graphical Analysis of a Circuit with a Cathode

Resistor 1817-14 Practical Cathode-follower Circuits 184

Vacuum-tube Small-signal Models and Applications 187

8-1 Variations from Quiescent Values 1878-2 Voltage-source Model of a Tube 1888-3 Linear Analysis of a Tube Circuit 1908-4 Taylor's Series Derivation of the Equivalent Circuit8-5 Current-source Model of a Tube 1968-6 A Generalized Tube Amplifier 1978-7 The Thevenin's Equivalent of Any Amplifier 1998-8 Looking into the Plate or Cathode of a Tube 2008-9 Circuits with a Cathode Resistor 2048-10 A Cascode Amplifier 2078-11 Interelectrode Capacitances in a Triode 2098-1

2

Input Admittance of a Triode 2118-13 Interelectrode Capacitances in a Multielectrode

Tube 2158-14 The Cathode Follower at High Frequencies 216

194

Transistor Characteristics 220

9-1 The Junction Transistor 2209-2 Transistor Current Components 2229-3 The Transistor as an Amplifier 2259-4 Transistor Construction 2269-5 Detailed Study of the Currents in a Transistor9-6 The Transistor Alpha 2309-7 The Common-base Configuration 23

1

227

xlf / CONTENTSCONTENTS / xitt

M. 10

11

12

9-8

9-9

9-109-11

9-129-13

9-149-159-169-179-18

The Common-emitter Configuration 234The CE Cutoff Region 237The CE Saturation Region 239Large-signal, DC, and Small-signal CE Values of CurrentGain 242The Common-collector Configuration 243Graphical Analysis of the CE Configuration 244Analytical Expressions for Transistor Characteristics 47Analysis of Cutoff and Saturation Regions 251Typical Transistor-junction Voltage Values 256Transistor Switching Times 267Maximum Voltage Rating 260

Transistor Biasing and Thermal Stabilization 263

10-1 The Operating Point 26310-2 Bias Stability 28510-3 Collector-to-Base Bias 26810-4 Self-bias, or Emitter Bias 27110-5 Stabilization against Variations in Vbe and for the

Self-bias Circuit 27610-6 General Remarks on Collector-current Stability 28010-7 Bias Compensation 28S10-8 Biasing Circuits for Linear Integrated Circuits 28510-9 Thermistor and Sensistor Compensation 28710-10 Thermal Runaway 28810-11 Thermal Stability 290

Small-signal Low-frequency Transistor Models 294

11-1 Two-port Devices and the Hybrid Model 29411-2 Transistor Hybrid Model 2961 1-3 Determination of the h Parameters from the

Characteristics 29811-4 Measurement of h Parameters 30211-5 Conversion Formulas for the Parameters of the Three

Transistor Configurations 30511-6 Analysis of a Transistor Amplifier Circuit Using h

Parameters S0711-7 Comparison of Transistor Amplifier Configurations 31211-8 Linear Analysis of a Transistor Circuit 31611-9 The Physical Model of a CB Transistor S1611-10 A Vacuum-tube-Transistor Analogy 319

Low-frequency Transistor Amplifier Circuits 323

12-1 Cascading Transistor Amplifiers 32312-2 n-stage Cascaded Amplifier 32712-3 The Decibel 33212-4 Simplified Common-emitter Hybrid Model 33312-5 Simplified Calculations for the Common-collector

Configuration 335

13

12-6 Simplified Calculations for the Common-baseConfiguration SS9

12-7 The Common-emitter Amplifier with an EmitterResistance 340

12-8 The Emitter Follower 34612-9 Miller's Theorem 34812-10 High-input-resistance Transistor Circuits 35012-11 The Cascode Transistor Configuration 36612-12 Difference Amplifiers 357

The High-frequency Transistor 363

13-1 The High-frequency T Model 36313-2 The Common-base Short-circu it-current Frequency

Response 36613-3 The Alpha Cutoff Frequency 36613-4 The Common-emitter Short-circuit-current Frequency

Response S6813-5 The Hybrid-pi (n) Common-emitter Transistor

Model 36913-6 Hybrid-pi Conductances in Terms of Low-frequency

h Parameters 37113-7 The CE Short-circuit Current Gain Obtained with the

Hybrid-pi Model 37613-8 Current Gain with Resistive Load S7813-9 Transistor Amplifier Response, Taking Source

Resistance into Account 380

14 Field-effect Tronsistors 384

390

15

14-1 The Junction Field-effect Transistor14-2 The Pinch-off Voltage VP 38814-3 The JFET Volt-Ampere Characteristics14-4 The FET Small-signal Model 39214-5 The Insulated-gate FET (MOSFET) 39614-6 The Common-source Amplifier 40014-7 The Common-drain Amplifier, or Source Follower14-8 A Generalized FET Amplifier 40314-9 Biasing the FET 40614-10 Unipolar-Bipolar Circuit Applications 4**

14-11 The FET as a Voltage-variable Resistor (WE) 4*$14-12 The Unijunction Transistor 415

Integrated Circuits 418

15-1 Basic Monolithic Integrated Circuits 418

15-2 Epitaxial Growth 42815-3 Masking and Etching 4%415-4 Diffusion of Impurities 4&5

15-5 Transistors for Monolithic Circuits 430

15-6 Monolithic Diodes 4$415-7 Integrated Resistors 436

402

xlv / CONTENTS

* 16

17

18

15-8 Integrated Capacitors and Inductors 48815-9 Monolithic Circuit Layout 44015-10 Integrated FieJd-effect Transistors 44415-11 Additional Isolation Methods 449

Untuned Amplifiers 450

16-1

16-216-316-4

16-5

16-616-716-816-9

16-10

16-11

16-12

Classification of Amplifiers 460Distortion in Amplifiers 46$Frequency Response of an Amplifier 462The AC-coupled Amplifier 455Low-frequency Response of an /eC-ooupled Stage 467High-frequency Response of a Vacuum-tube Stage 468Cascaded CE Transistor Stages 460Step Response of an Amplifier 466Bandpass of Cascaded Stages 467Effect of an Emitter (or a Cathode) Bypass Capacitoron U)w-frequency Response 468Spurious Input Voltages 472Noise 47S

Feedback Amplifiers and Oscillators 48017-1

17-2

17-3

17-4

17-5

17-6

17-7

17-817-9

17-1017-11

17-1217-1317-1417-1517-1617-1717-1817-1917-2017-21

Classification of Amplifiers 48OThe Feedback Concept 48SGeneral Characteristics of Negative-feedbackAmplifiers 488Effect of Negative Feedback upon Output and InputResistances 491Voltage-series Feedback 498A Voltage-series Feedback Pair 602Current-series Feedback 604Current-shunt Feedback 508Voltage-shunt Feedback 612The Operational Amplifier 614Basic Uses of Operational Amplifiers 517Electronic Analog Computation 620Feedback and Stability 522Gain and Phase Margins 624Sinusoidal Oscillators 625The Phase-shift Oscillator 628Resonant-circuit Oscillators 680A General Form of Oscillator Circuit 582Crystal Oscillators 686Frequency Stability 687Negative Resistance in Oscillators 588

Large-signal Amplifiers 542

18-1 Class A Large-aignal Amplifiers 54218-2 Second-harmonic Distortion 64418-3 Higher-order Harmonic Generation 546

CONTENTS / xv

18-4 The Transformer-coupled Audio Power Amplifier18-5 Power Amplifiers Using Tubes 55818-6 Shift of Dynamic Load Line 55618-7 Efficiency 55618-8 Push-Pull Amplifiers 66818-9 Class B Amplifiers 66018-10 Class AB Operation 564

549

**. 19 Photoelectric Devices 56619-1 Photocmissivity 66619-2 Photoelectric Theory 56819-3 Definitions of Some Radiation Terms 57119-4 Phototubes 57819-5 Applications of Photodevices 57519-6 Multiplier Phototubes 67819-7 Photoconductivity 58019-8 The Semiconductor Photodiode 58819-9 Multiple-junction Photodiodes 58619-10 The Photovoltaic Effect 687

20 Rectifiers and Power Supplies 59220-1 A Half-wave Rectifier 69220-2 Ripple Factor 59720-3 A Full-wave Rectifier 69820-4 Other Full-wave Circuits 60020-5 The Harmonic Components in Rectifier Circuits20-6 Inductor Filters 60320-7 Capacitor Filters 60620-8 Approximate Analysis of Capacitor Filters 60920-9 L-section Filter 61120-10 Multiple L-section Filter 61620-1

1

11-section Filter 61720-12 fl-section Filter with a Resistor Replacing the

Inductor 62020-13 Summary of Filters 62120-14 Regulated Power Supplies 62120-15 Series Voltage Regulator 62320-16 Vacuum-tube-regulated Power Supply 629

602

Appendix A Probable Values of General PhysicalConstants 633

Appendix B Conversion Factors and Prefixes 634Appendix C Periodic Table of the Elements 635Appendix D Tube Characteristics 636

Problems 641

Index 745

1 ELECTRON BALLISTICS

AND APPLICATIONS

In this chapter we present the fundamental physical and mathemati-cal theory of the motion of charged particles in electric and magneticfields of force. In addition, we discuss a number of the more impor-tant electronic devices that depend on this theory for their operation.

The motion of a charged particle in electric and magnetic fields ispresented, starting with simple paths and proceeding to more complexmotions. First a uniform electric field is considered, and then theanalysis is given for motions in a uniform magnetic field. This dis-cussion is followed, in turn, by the motion in parallel electric and mag-netic fields and in perpendicular electric and magnetic fields.

1-1 CHARGED PARTICLES

The charge, or quantity, of negative electricity of the electron hasbeen found by numerous experiments to be 1.602 X 10-" C (coulomb).The values of many important physical constants are given in Appen-dix A. Some idea of the number of electrons per second that repre-sents current of the usual order of magnitude is readily possible. Forexample, since the charge per electron is 1.602 X 10~ 19 C, the numberof electrons per coulomb is the reciprocal of this number, or approxi-mately, 6 X 10 18 . Further, since a current of 1 A (ampere) is the flowof 1 C/sec, then a current of only 1 pA (1 picoampere, or 10-12 A)represents the motion of approximately 6 million electrons per second.Yet a current of 1 pA is so small that considerable difficulty is experi-enced in attempting to measure it.

In addition to its charge, the electron possesses a definite mass.A direct measurement of the mass of an electron cannot be made, butthe ratio e/m of the charge to the mass has been determined by a1

2 / ELECTRONIC DEVICES AND CIRCUITS Sec. 7-2

number of experimenters using independent methods. The most probablevalue for this ratio is 1.759 X 10 11 C/kg. From this value of e/m and thevalue of e, the charge on the electron, the mass of the electron is calculatedto be 9.109 X lO"31 kg.

The charge of a positive ion is an integral multiple of the charge of theelectron, although it is of opposite sign. For the case of singly ionized parti-cles, the charge is equal to that of the electron. For the case of doubly ionizedparticles, the ionic charge is twice that of the electron.

The mass of an atom is expressed as a number that is based on the choiceof the atomic weight of oxygen equal to 16. The mass of a hypothetical atomof atomic weight unity is, by this definition, one-sixteenth that of the mass ofmonatomic oxygen. This has been calculated to be 1.660 X 10-27 kg. Hen^e,in order to calculate the mass in kilograms of any atom., it is necessary only tomultiply the atomic weight of the atom by 1.660 X 10~" kg. A table of atomicweights is given in Appendix C.

The radius of the electron has been estimated as 10-16 m, and that of anatom as 10~10 m. These are so small that all charges are considered as masspoints in the following sections.

Classical and Wave-mechanical Models of the Electron The foregoingdescription of the electron (or atom) as a tiny particle possessing a definitecharge and mass is referred to as the classical model. If this particle is sub-jected to electric, magnetic, or gravitational fields, it experiences a force, andhence is accelerated. The trajectory can be determined precisely using New-ton's laws, provided that the forces acting on the particle are known. In thischapter we make exclusive use of the classical model to study electron ballistics.The term electron ballistics is used because of the existing analogy between themotion of charged particles in a field of force and the motion of a falling bodyin the earth's gravitational field.

For large-scale phenomena, such as electronic trajectories in a vacuumtube, the classical model yields accurate results. For small-scale systems,however, such as an electron in an atom or in a crystal, the classical modeltreated by Newtonian mechanics gives results which do not agree with experi-ment. To describe such subatomic systems properly it is found necessary toattribute to the electron a wavelike property which imposes restrictions on theexactness with which the electronic motion can be predicted. This wave-mechanical model of the electron is considered in Chap. 2.

1-2 THE FORCE ON CHARGED PARTICLES IN AN ELECTRIC FIELDThe force on a unit positive charge at any point in an electric field is, by definition,the electric field intensity at that point. Consequently, the force on a positivecharge q in an electric field of intensity is given by q, the resulting force

Sec. 7-3 ELECTRON BALLISTICS AND APPLICATIONS / 3

being in the direction of the electric field. Thus,

(1-1)

where f is in newtons, q is in coulombs, and is in volts per meter. Boldfacetype is employed wherever vector quantities (those having both magnitudeand direction) are encountered.

The mks (meter-kilogram-second) rationalized system of units is foundmost convenient for the subsequent studies. Therefore, unless otherwisestated, this system of units is employed.

In order to calculate the path of a charged particle in an electric field,the force, given by Eq. (1-1), must be related to the mass and the accelerationof the particle by Newton's second law of motion. Hence

dt (1-2)

where m = mass, kga = acceleration, m/sec*v = velocity, m/sec

The solution of this equation, subject to appropriate initial conditions, givesthe path of the particle resulting from the action of the electric forces. If themagnitude of the charge on the electron is e, the force on an electron in thefield is

f-

-S (1-3)

The minus sign denotes that the force is in the direction opposite to the field.In investigating the motion of charged particles moving in externally

applied force fields of electric and magnetic origin, it is implicitly assumedthat the number of particles is so small that their presence does not alter thefield distribution.

1-3 CONSTANT ELECTRIC FIELDSuppose that an electron is situated between the two plates of a parallel-platecapacitor which are contained in an evacuated envelope, as illustrated in Fig.1-1- A difference of potential is applied between the two plates, the directionof the electric field in the region between the two plates being as shown. Ifthe distance between the plates is small compared with the dimensions of theplates, the electric field may be considered to be uniform, the lines of forcepointing along the negative X direction. That is, the only field that is presentis along the -X axis. It is desired to investigate the characteristics of themotion, subject to the initial conditions

* = va X = x when ( = (1-4)

4 / ELECTRONIC DEVICES AND CIRCUITS Sac. 7-3

i d-

e-*-

Fig. 1-1 The one-dimenstona) electricfield between the plates of a parallel-plate capacitor.

This means that the initial velocity vex is chosen along e, the lines of force,and that the initial position x of the electron is along the X axis.

Since there is no force along the Y or Z directions, Newton's law statesthat the acceleration along these axes must be zero. However, zero acceler-ation means constant velocity; and since the velocity is initially zero alongthese axes, the particle will not move along these directions. That is, the onlypossible motion is one-dimensional, and the electron moves along the X axis.

Newton's law applied to the X direction yields

or

e = 7ttax

68a, = = constm d-5)

where represents the magnitude of the electric field. This analysis indicatesthat the electron will move with a constant acceleration in a uniform electricfield. Consequently, the problem is analogous to that of a freely falling bodyin the uniform gravitational field of the earth. The solution of this problemis given by the well-known expressions for the velocity and displacement, viz.,

v. = tv, + aj. x = x + vOIt + lad* d-6)

provided that a- = const, independent of the time.It is to be emphasized that, if the acceleration of the particle is not a con-

stant but depends upon the time, Eqs. (1-6) are no longer valid. Under thesecircumstances the motion is determined by integrating the equations

dV;

dland

dxdl

= vx (1-7)

These are simply the definitions of the acceleration and the velocity, respec-tively. Equations (1-6) follow directly from Eqs. (1-7) by integrating thelatter equations subject to the condition of a constant acceleration.

Sec 1-4 ELECTRON BALLISTICS AND APPLICATIONS / 5

EXAMPLE An electron starts at rest on one plate of a plane-parallel capacitorwhose plates are 5 cm apart. The applied voltage is zero at the, instant the elec-tron is released, and it increases linearly from zero to 10 V in 0,1 Msec.f

a. If the opposite plate is positive, what speed will the electron attain in50 nsec?

b. Where will it be at the end of this time?c. With what speed will the electron strike the positive plate?

Solution Assume that the plates are oriented with respect to a cartesian systemof axes as illustrated in Fig. 1-1. The magnitude of the electric field intensity is

a. 6 =

whence

X = 2 X 109*5 X 10-* 10"7

V/m

a* = ^ - - = = (1.76 X 10)(2 X 10()at m M

= 3.52 X 10M( m/sec J

Upon integration, we obtain for the speed

v, = T ax dt = 1.76 X 10*V

At t = 5 X 10~ a sec, vx = 4.40 X 10* m/sec.

6. Integration of vx with respect to (, subject to the condition that x =when t = 0, yields

x m j* Vz dt = P 1.76 X \0*H*dt = 5.87 X 10,9f 3

At t m 5 X 10"" sec, x = 7.32 X 10~ 3 m = 0.732 cm,

c. To find the speed with which the electron strikes the positive plate, wefirst find the time t it takes to reach that plate, or

/ x Y / 0.05 Y[ 1 = f J - 9.46 X 10'\5.87 X 10'7 \5.87 X 10'V

Hence

1.76 X 10M/ S = 1.76 X 10(9.46 X 10" 8)* - 1.58 X 10 m/sec

1-4 POTENTIAL

The discussion to follow need not be restricted to uniform fields, but x maybe a function of distance. However, it is assumed that Ex is not a function

t 1 /^ec = 1 microsecond = 10~sec. 1 nsec = 1 nanosecond = 10-*sec. Conversionfactors and prefixes are given in Appendix B.

6 / ELECTRONIC DEVICES AND CIRCUITS Sec. 1-4

of time. Then, from Newton's second law,

e&a_

dvxm ~~ dt

Multiply this equation by dx = vx dt, and integrate. This leads to

/ &x dx = v, dv* (1-8)571 JXa JVoi

The definite integral

/* &x dx

is an expression for the work done by the field in carrying a unit positivecharge from the point x to the point x.

By definition, the potential V (in volts) of point x with respect to point x isthe work done against the field in taking a unit positive charge from xa to x. Thusf

V m - &x dx (1-9)By virtue of Eq. (1-9), Eq. (1-8) integrates to

eV = m(vx* - *,*) (1-10)

where the energy eV is expressed in joules. Equation (1-10) shows that anelectron that has "fallen" through a certain difference of potential V in goingfrom point xa to point x has acquired a specific value of kinetic energy andvelocity, independent of the form of the variation of the field distributionbetween these points and dependent only upon the magnitude of the potentialdifference V.

Although this derivation supposes that the field has only one component,namely, 8* along the X axis, the final result given by Eq. (1-10) is simply astatement of the law of conservation of energy. This law is known to bevalid even if the field is multidimensional. This result is extremely impor-tant in electronic devices. Consider any two points A and B in space, withpoint B at a higher potential than point A by VBA . Stated in its mostgenera] form, Eq. (1-10) becomes

qVzA = fymA* wu>s* (1-11)

where q is the charge in coulombs, qVBA is in joules, and v* and vB are thecorresponding initial and final speeds in meters per second at the points A andBy respectively. By definition, the potential energy between two points equals thepotential multiplied by the charge in question. Thus the left-hand side of Eq.(1-11) is the rise in potential energy from A to B. The right-hand side repre-sents the drop in kinetic energy from A to B. Thus Eq. (1-11) states that therise in potential energy equals the drop in kinetic energy, which is equivalentto the statement that the total energy remains unchanged.

t The symbol w used to designate "equal to by definition."

Sec. 1-5 ELECTRON BALLISTICS AND APPLICATIONS / 7

It must be emphasized that Eq. (1-11) is not valid if the field varies with time.If the particle is an electron, then e must be substituted for q. If the

electron starts at rest, its final speed v, as given by Eq. (1-11) with vA 0,vB = v, and VBa = V, is

or

-Mv = 5.93 X 10 6F*

d-12)

(1-13)

Thus, if an electron "falls" through a difference of only 1 V, its final speed

is 5-93 X 106 m/sec, or approximately 370 miles/sec. Despite this tremen-dous speed, the electron possesses very little kinetic energy, because of its

minute mass.It must be emphasized that Eq. (1-13) is valid only for an electron starting

at rest. If the electron does not have zero initial velocity or if the particleinvolved is not an electron, the more general formula [Eq. (1-11)] must be used.

1-5 THE eV UNIT OF ENERGY

The joule (J) is the unit of energy in the mks system. In some engineeringpower problems this unit is very small, and a factor of 10 3 or 10 8 is introducedto convert from watts (1 W = 1 J/sec) to kilowatts or megawatts, respectively.However, in other problems, the joule is too large a unit, and a factor of 10~7

is introduced to convert from joules to ergs. For a discussion of the energiesinvolved in electronic devices, even the erg is much too large a unit. Thisstatement is not to be construed to mean that only minute amounts of energycan be obtained from electron devices. It is true that each electron possessesa tiny amount of energy, but as previously pointed out (Sec. 1-1), an enor-mous number of electrons is involved even in a small current, so that con-siderable power may be represented.

A unit of work or energy, called the electron volt (eV), is defined as follows:

1 eV = 1.60 X 10- 19 J

Of course, any type of energy, whether it be electric, mechanical, thermal, etc.,may be expressed in electron volts.

The name electron volt arises from the fact that, if an electron falls througha potential of one volt, its kinetic energy will increase by the decrease inpotential energy, or by

eV - (1.60 X 10- 19 C)(l V) = 1.60 X 10" 19 J = 1 eV

However, as mentioned above, the electron-volt unit may be used for any typeof energy, and is not restricted to problems involving electrons.

The abbreviations MeV and BeV are used to designate 1 million and1 billion electron volts, respectively.

8 / ELECTRONIC DEVICES AND CIRCUITS Sec, 1-6

1-6 RELATIONSHIP BETWEEN FIELD INTENSITY AND POTENTIALThe definition of potential is expressed mathematically by Eq. (1-9). If theelectric field is uniform, the integral may be evaluated to the form

-

J* dx= -&x(x - Xo) = V

which shows that the electric field intensity resulting from an applied potentialdifference V between the two plates of the capacitor illustrated in Fig. 1-1 isgiven by

V V*

=

x^Ja = ~d (1-14)where 6, is in volts per meter, and d is the distance between plates, in meters.

In the general case, where the field may vary with the distance, thisequation is no longer true, and the correct result is obtained by differentiatingEq. (1-9). We obtain

dVax (1-15)

The minus sign shows that the electric field is directed from the region ofhigher potential to the region of lower potential.

1-7 TWO-DIMENSIONAL MOTIONSuppose that an electron enters the region between the two parallel plates of aparallel-plate capacitor which are oriented as shown in Fig. 1-2 with an initialvelocity in the

-fX direction. It will again be assumed that the electric fieldbetween the plates is uniform. Then, as chosen, the electric field is in thedirection of the Y axis, no other fields existing in this region.

The motion of the particle is to be investigated, subject to the initialconditions

fz = % x =

(1-16)vv = y = ) when t =

v, = z =

Since there is no force in the Z direction, the acceleration in that direction is

!4-

-

r-=5

Fig, 1-2 Two-dimensional electronic motionin a uniform electric field.

Sec. 1-7 ELECTRON BALLISTICS AND APPLICATION > / 9

zero. Hence the component of velocity in the Z direction remains constant.Since the initial velocity in this direction is assumed to be zero, the motionmust take place entirely in one plane, the plane of the paper.

For a similar reason, the velocity along the X axis remains constant andequal to vox. That is,

H = Mm

from which it follows that

x = vext (1-17)

On the other hand, a constant acceleration exists along the Y direction, andthe motion is given by Eqs. (1-6), with the variable x replaced by y;

where

vy = av t

eOy = =

m

V =Wmd

(1-18)

(1-19)

and where the potential across the plates is V = Vd . These equations indi-cate that in the region between the plates the electron is accelerated upward,the velocity component vv varying from point to point, whereas the velocitycomponent vx remains unchanged in the passage of the electron between theplates.

The path of the particle with respect to the point is readily determinedby combining Eqs. (1-17) and (1-18), the variable ( being eliminated. Thisleads to the expression

^2 y (1-20)

which shows that the particle moves in a parabolic path in the region betweenthe plates.

EXAMPLE Hundred-volt electrons are introduced at A into a uniform electricfield of 10* V/m, as shown in Fig. 1-3. The electrons are to emerge at thepoint B in time 4.77 nsec.

a. What is the distance AB?b. What angle does the electron beam make with the horizontal?

Fig. 1-3 Parabolic path of an electron ina uniform electric field.

J^

10 / ELECTRONIC DEVICES AND CIRCUITS Sec. 7-8

Solution The path of the electrons will be a parabola, as shown by the dashedcurve in Fig. 1-3, This problem is analogous to the firing of a gun in the earth'sgravitational field. The bullet will travel in a parabolic path, first rising becauseof the muzzle velocity of the gun and then falling because of the downward attrac-tive force of the earth. The source of the charged particles is called an electrongun, or an ion gun.

The initial electron velocity is found using Eq. (1-13).

R, = 5.93 X 10 s -s/lOO = 5.93 X 10 s m/secSince the speed along the X direction is constant, the distance AB = xte given by

x = (v cos 6)t = (5.93 X 10 fi cos 0)(4.77 X 10~) = 2.83 X 10~ 2 cos 8Hence we first must find 8 before we can solve for x. Since the acceleration a, inthe Y direction is constant, then

y = (v sin 8)t ^Oyt*

and y = at point B, or

v sin 9*i?"i- 1(1.76 X 10") (10*) (4.77 X 10"*) - 4.20 X 10* m/sec

and

, . 4.20 X 10 s AmM

a. x = 2.83 X 10-* X 0.707 = 2.00 X 10"* m = 2.00 cm

1-8 ELECTROSTATIC DEFLECTION IN A CATHODE-RAY TUBEThe essentials of a cathode-ray tube for electrostatic deflection are illustratedin Fig. 1-4. The hot cathode A' emits electrons whieh are accelerated towardthe anode by the potential Va . Those electrons which are not collected bythe anode pass through the tiny anode hole and strike the end of the glassenvelope. This has been coated with a material that fluoresces when bom-

Anode

Cathode*kS r

Vertical-deflectingplates

+ Vd + us

Fluorescent screen

Fig. 1-4 Electrostatic deflection in a cathode-ray tube.

See. 7-8 ELECTRON BALLISTICS AND APPLICATIONS / II

barded by electrons. Thus the positions where the electrons strike the screenare made visible to the eye. The displacement D of the electrons is deter-mined by the potential Vd (assumed constant) applied between the delectingplates, as shown. The velocity vox with which the electrons emerge from theanode hole is given by Eq. (1-12), viz.,

\ m (1-21)

on the assumption that the initial velocities of emission of the electrons fromthe cathode are negligible.

Since no field is supposed to exist in the region from the anode to thepoint 0, the electrons will move with a constant velocity %* in a straight-linepath. In the region between the plates the electrons will move in the para-bolic path given by y = ^{ajv^x2 according to Eq. (1-20). The path is astraight line from the point of emergence M at the edge of the plates to thepoint P' on the screen, since this region is field-free.

The straight-line path in the region from the deflecting plates to the screenis, of course, tangent to the parabola at the point M. The slope of the lineat this point, and so at every point between M and P', is [from Eq. (1-20)

J

tan* = ^l m *JdxJz~i vax2

From the geometry of the figure, the equation of the straight line MP' isfound to be

(1-22)

since x = I and y = ^aJ a/* at the point M

.

When y = 0, z = 1/2, which indicates that when the straight line MP' isextended backward, it will intersect the tube axis at the point O', the centerpoint of the plates. This result means that O' is, in effect, a virtual cathode,and regardless of the applied potentials Va and Vd, the electrons appear toemerge from this "cathode" and move in a straight line to the point P*.

At the point P't y = D, and x - L + $L Equation (1-22) reduces to

ttjjuTD =

By inserting the known values of ay ( = eVd/dm) and vox , this becomeslLVdD -2dVa (1-23)

This result shows that the deflection on the screen of a cathode-ray tube isdirectly proportional to the deflecting voltage Vd applied between the plates.Consequently, a cathode-ray tube may be used as a linear-voltage indicatingdevice.

The electrostatic-deflection sensitivity of a cathode-ray tube is defined as

12 / ELECTRONIC DEVICES AND CIRCUITS Sec. 1-9

the deflection (in meters) on the screen per volt of deflecting voltage. Thus

B D ILS = Vd = 2dVa (1-24)

An inspection of Eq. (1-24) shows that the sensitivity is independent of boththe deflecting voltage Vd and the ratio e/m. Furthermore, the sensitivityvaries inversely with the accelerating potential Va .

The idealization made in connection with the foregoing development, viz.,that the electric field between the deflecting plates is uniform and does notextend beyond the edges of the plates, is never met in practice. Consequently,the effect of fringing of the electric field may be enough to necessitate correc-tions amounting to as much as 40 percent in the results obtained from anapplication of Eq. (1-24). Typical measured values of sensitivity are 1.0 to0.1 mm/V, corresponding to a voltage requirement of 10 to 100 V to give adeflection of 1 cm.

U9 THE CATHODE-RAY OSCILLOSCOPEAn electrostatic tube has two sets of deflecting plates which are at right anglesto each other in space (as indicated in Fig. 1-6). These plates are referred toas the vertical-deflection and horizontal-deflection plates because the tube is ori-ented in space so that the potentials applied to these plates result in verticaland horizontal deflections, respectively. The reason for having two sets ofplates is now discussed.

Suppose that the sawtooth waveform of Fig. 1-6 is impressed across thehorizontal-deflection plates. Since this voltage is used to sweep the electronbeam across the screen, it is called a sweep voltage. The electrons are deflected

Vertical-deflectionplates

Horizontal- deflectionplates

Verticalsignal

voltage v.

Horizontalsawtoothvoltage

Electron beam

Ftg. 1-5 A waveform to be displayed on the screen of acathode-ray tube is applied to the vertical-deflection plates,and simultaneously a sawtooth voltage is applied to the hori-zontal-deflection plates.

Sec. 1-70 ELECTRON BALLISTICS AND APPLICATIONS / 13

Voltage

Fig. 1 -6 Sweep or sawtooth voltage

for a cathode-ray tube.Time

linearly with time in the horizontal direction for a time T. Then the beamreturns to its starting point on the screen very quickly as the sawtooth voltagerapidly falls to its initial value at the end of each period.

If a sinusoidal voltage is impressed across the vertical-deflection plateswhen, simultaneously, the sweep voltage is impressed across the horizontal-deflection plates, the sinusoidal voltage, which of itself would give rise to avertical line, will now be spread out and will appear as a sinusoidal trace onthe screen. The pattern will appear stationary only if the time T is equal to,or is some multiple of, the time for one cycle of the wave on the vertical plates.It is then necessary that the frequency of the sweep circuit be adjusted tosynchronize with the frequency of the applied signal.

Actually, of course, the voltage impressed on the vertical plates may haveany waveform. Consequently, a system of this type provides an almostinertialess oscilloscope for viewing arbitrary waveshapes. This is one of themost common uses for cathode-ray tubes. If a nonrepeating sweep voltage isapplied to the horizontal plates, it is possible to study transients on the screen.This requires a system for synchronizing the sweep with the start of thetransient.

'f

A commercial oscilloscope has many refinements not indicated in theschematic diagram of Fig. 1-5. The sensitivity is greatly increased by meansof a high-gain amplifier interposed between the input signal and the deflectionplates. The electron gun is a complicated structure which allows for acceler-ating the electrons through a large potential, for varying the intensity of thebeam, and for focusing the electrons into a tiny spot. Controls are also pro-vided for positioning the beam as desired on the screen.

1-10 RELATIVISTIC VARIATION OF MASS WITH VELOCITYThe theory of relativity postulates an equivalence of mass and energy accord-ing to the relationship

W = mc* (1-25)where W = total energy, J

m = mass, kgc = velocity of light in vacuum, m/sec

t Superscript numerals are keyed to the References at the end of the chapter.

T4 / ELECTRONIC DEVICES AND CIRCUITS Sec. 7-70

According to this theory, the mass of a particle will increase with its energy,and hence with its speed.

If an electron starts at the point A with zero velocity and reaches thepoint B with a velocity v, then the increase in energy of the particle must begiven by the expression eV, where V is the difference of potential betweenthe points A and B, Hence

eV = mc 2 rrioC* (1-26)

where mc 3 is the energy possessed at the point A. The quantity m is knownas the rest mass, or the electrostatic mass, of the particle, and is a constant,independent of the velocity. The total mass m of the particle is given by

m =VI - 7c' (1-27)

This result, which was originally derived by Lorentz and then by Einsteinas a consequence of the theory of special relativity, predicts an increasing masswith an increasing velocity, the mass approaching an infinite value as thevelocity of the particle approaches the velocity of light. From Eqs. (1-26)and (1-27), the decrease in potential energy, or equivalently, the increase inkinetic energy, is

eV = m** (X

- i\ (1-28)

This expression enables one to find the velocity of an electron after it hasfallen through any potential difference F. By defining the quantity vx as thevelocity that would result if the relativistic variation in mass were neglected,i.e.,

J2eV

(1-28) can be solved for v, the true velocity of the particle. The

Vn = (1-29)

then Eq.result is

v = c 1 -1 "li

(1-30)(1 + ArV2c)*_

This expression looks imposing at first glance. It should, of course,reduce to v = vN for small velocities. That it does so is seen by applying thebinomial expansion to Eq. (1-30). The result becomes

'*Mfr-W*-'") (1-31)From this expression it is seen that, if the speed of the particle is much less

than the speed of light, the second and all subsequent terms in the expansioncan be neglected, and then v = vN , as it should. This equation also servesas a criterion to determine whether the simple classical expression or the moreformidable relativistic one must be used in any particular case. For example,

Swc. I-W ELECTRON BALLISTICS AND APPLICATIONS / 15S

if the speed of the electron is one-tenth of the speed of light, Eq. (1-31) showsthat an error of only three-eighths of 1 percent will result if the speed is takenas Vft instead of v.

For an electron, the potential difference through which the particle mustfall in order to attain a velocity of 0.1c is readily found to be 2,560 V. Thus,if an electron falls through a potential in excess of about 3 kV, the relativisticcorrections should be applied. If the particle under question is not an elec-tron, the value of the nonrelativistic velocity is first calculated. If this isgreater than 0.1c, the calculated value of 0jy must be substituted in Eq. (1-30)and the true value of v then calculated. In cases where the speed is not toogreat, the simplified expression (1-31) may be used.

The accelerating potential in high-voltage cathode-ray tubes is sufficientlyhigh to require that relativistic corrections be made in order to calculate thevelocity and mass of the particle. Other devices employing potentials thatare high enough to require these corrections are x-ray tubes, the cyclotron,and other particle-accelerating machines. Unless specifically stated otherwise,nonrelativistic conditions are assumed in what follows.

1-11 FORCE IN A MAGNETIC FIELD

To investigate the force on a moving charge in a magnetic field, the well-known motor law is recalled. It has been verified by experiment that, if aconductor of length L, carrying a current of /, is situated in a magnetic field ofintensity B, the force / acting on this conductor is

/. - BIL (1-32)

where fm is in newtons, B is in webers per square meter (Wb/m2),t / is in am-peres, and L is in meters. Equation (1-32) assumes that the directions of /and B are perpendicular to each other. The direction of this force is perpen-dicular to the plane of I and B and has the direction of advance of a right-handed screw which is placed at O and is rotated from I to B through 90, asillustrated in Fig. 1-7. If I and B are not perpendicular to each other, only thecomponent of I perpendicular to B contributes to the force.

Some caution must be exercised with regard to the meaning of Fig. 1-7.If the particle under consideration is a positive ion, then I is to be taken alongthe direction of its motion. This is so because the conventional direction ofthe current is taken in the direction of flow of positive charge. If the current's due to the flow of electrons, the direction of I is to be taken as opposite tothe direction of the motion of the electrons. If, therefore, a negative charge

t One weber per square meter (also called a testa) equals 10* G. A unit of more prac-tical size in most applications is the milliweber per square meter (mWb/m 1), which equals10 G. Other conversion factors are given in Appendix B.

16 / ELECTRONIC DEVICES AND CIRCUITS Sec. 1-12 Sec. M3 ELECTRON BALLISTICS AND APPLICATIONS / 17

L"

o.^

-90'T

Fig. 1-7 Pertaining to the determination of the direc-tion of the force fm on a charged particle in amagnetic field.

lorv*

moving with a velocity v~ is under consideration, one must first draw I anti-parallel to v~ as shown and then apply the "direction rule."

If N electrons are contained in a length L of conductor (Fig. 1-8) and ifit takes an electron a time T sec to travel a distance of L m in the conductor,the total number of electrons passing through any cross section of wire inunit time is N/T. Thus the total charge per second passing any point, which,by definition, is the current in amperes, is

T = *!T (1-33)

The force in newtons on a length L m (or the force on the N conduction chargescontained therein) is

BIL = BNeL

Furthermore, since L/T is the average, or drift, speed v m/sec of the electrons,the force per electron is

f = eBv (1-34)

The subscript m indicates that the force is of magnetic origin. To sum-marize: The force on a negative charge e (coulombs) moving with a componentof velocity r (meters per second) normal to a field B (webers per square meter)is given by eBv~ (newtons) and is in a direction perpendicular to the plane of Band y~, as noted in Fig. 1-7. f

1-12 CURRENT DENSITY

Before proceeding with the discussion of possible motions of charged particlesin a magnetic field, it is convenient to introduce the concept of eurrent density.

t In the crosa-product notation of vector analysis, fm m eB x v~. For a positive ionmoving with a velocity v+

, the force Is fm = ev+ X B.

mN electrons D-i

Fig. T-8 Pertaining to the determination of themagnitude of the force fm on a charged particlein a magnetic field.

This concept is very useful in many later applications. By definition, thecurrent density, denoted by the symbol J, is the current per unit area of theconducting medium. That is, assuming a uniform current distribution,

"i (1-35)where J is in amperes per square meter, and A is the cross-sectional area (inmeters) of the conductor. This becomes, by Eq. (1-33),

r _N*

J TA

But it has already been pointed out that T L/v. Then

_

_Nev

J~LA (1-36)

From Fig. 1-8 it is evident that LA is simply the volume containing the Nelectrons, and so N/LA is the electron concentration n (in electrons per cubicmeter). Thus

(1-37)N

n = LA

and Eq. (1-36) reduces to

J = nev = pv (1-38)

where p = ne is the charge density, in coulombs per cubic meter, and v is inmeters per second.

This derivation is independent of the form of the conducting medium.Consequently, Fig. 1-8 does not necessarily represent a wire conductor. Itmay represent equally well a portion of a gaseous-discharge tube or a volumeelement in the space-charge cloud of a vacuum tube or a semiconductor.Furthermore, neither p nor v need be constant, but may vary from point topoint in space or may vary with time. Numerous occasions arise later inthe text when reference ia made to Eq. (1-38).

1-13 MOTION IN A MAGNETIC FIELDThe path of a charge particle that is moving in a magnetic field is now investi-gated. Consider an electron to be placed in the region of the magneticfield. If the particle is at rest, / = and the particle remains at rest. Ifthe initial velocity of the particle is along the lines of the magnetic flux,there is no force acting on the particle, in accordance with the rule associatedwith Eq. (1-34). Hence a particle whose initial velocity has no componentnormal to a uniform magnetic field will continue to move with constant speedalong the lines of flux.

18 / ELECTRONIC DEVICES AND CIRCUITS Sec. L13

Field-freeregion

x

X

K Magneticfield Into

* paper

Fig. 1-9 Circular motion of an electron in a

transverse magnetic field.

Now consider an electron moving with a speed v to enter a constantuniform magnetic field normally, aa shown in Fig. 1-9. Since the force fmis perpendicular to v and so to the motion at every instant, no work is doneon the electron. This means that its kinetic energy is not increased, andso its speed remains unchanged. Further, since v and B are each constantin magnitude, then fm is constant in magnitude and perpendicular to thedirection of motion of the particle. This type of force results in motion in acircular path with constant speed. It is analogous to the problem of a masstied to a rope and twirled around with constant speed. The force (whichis the tension in the rope) remains constant in magnitude and is always directedtoward the center of the circle, and so is normal to the motion.

To find the radius of the circle, it is recalled that a particle moving ina circular path with a constant speed v has an acceleration toward the centerof the circle of magnitude v 3/R, where R is the radius of the path in meters.Then

from which

The corresponding angular velocity in radians per second is given by

_

v__

eBR m

The time in seconds for one complete revolution, called the period, is

m _ 2t __ 2irnicd eB

For an electron, this reduces to

3.57 X 10-11T =B

(1-39)

(1-40)

(1-41)

(1-42)

In these equations, e/m is in coulombs per kilogram and B in webers per squaremeter.

S*c. 1-13 ELECTRON BALLISTICS AND APPLICATIONS / T9

It is noticed that the radius of the path is directly proportional to thespeed of the particle. Further, the period and the angular velocity are inde-pendent of speed or radius. This means, of course, that faster-moving particleswill traverse larger circles in the same time that a slower particle moves in itssmaller circle. This very important result is the basis of operation of numer-ous devices, for example, the cyclotron and magnetic-focusing apparatus.

EXAMPLE Calculate the deflection of a cathode-ray beam caused by the earth'smagnetic field. Assume that the tube axis is so oriented that it is normal to thefield, the strength of which is 0.6 G. The anode potential is 400 V; the anode-screen distance is 20 cm (Fig. 1-10).

Solution According to Eq. (1-13), the velocity of the electrons will be

pm = 5.93 X 10* Vibo = 1.19 X 107 m/sec

Since 1 Wb/m* = 10* G, then B = 6 X 10" B Wb/m a . From Eq. (1-39) the radiusof the circular path is

R = 1.19 X 107

= 1.12 m = 112 cm(e/m)B 2.76 X 10" X 6 X 10" 5

Furthermore, it is evident from the geometry of Fig. 1-10 that (in centimeters)

112s = (112 - D)* + 202

from which it follows that

D* - 2242) + 400 =

The evaluation of D from this expression yields the value D = 1.8 cm.This example indicates that the earth's magnetic field can have a large effect

on the position of the cathode-beam spot in a low-voltage cathode-ray tube. If

Fig. 1-10 The circular path of an elec-tron in a cathode-ray tube, resulting fromthe earth's transverse magnetic field(normal to the plane of the paper).This figure is not drawn to scale. (112-0)

20 / ELECTRONIC DEVICES AND CIRCUITS Sec. I -U

the anode voltage is higher than the value used in this example, or if the tube isnot oriented normal to the field, the deflection will be less than that calculated.In any event, this calculation indicates the advisability of carefully shielding acathode-ray tube from stray magnetic fields.

1-14 MAGNETIC DEFLECTION IN A CATHODE-RAY TUBE

The illustrative example in Sec. 1-13 immediately suggests that a cathode-ray tube may employ a magnetic as well as an electric field in order to accom-plish the deflection of the electron beam. However, since it is not feasibleto use a field extending over the entire length of the tube, a short coil furnishinga transverse field in a limited region is employed, as shown in Fig. 1-1 1. Themagnetic field is taken as pointing out of the paper, and the beam is deflectedupward. It is assumed that the magnetic field intensity B is uniform inthe restricted region shown and is zero outside of this area. Hence theelectron moves in a straight line from the cathode to the boundary of themagnetic field. In the region of the uniform magnetic field the electronexperiences a force of magnitude eBv, where v is the speed.

The path OM will be the arc of a circle whose center is at Q. The speedof the particles will remain constant and equal to

J2eVa(1-43)

The angle "5

where, by Eq. (1-39),

mvR =>eB

(1-44)

(1-45)

In most practical cases, L is very much larger than I, so that little error will

i^ Magnetic field

*?i*l out of paper

Fig. 1-1 T Magnetic deflection in acathode-ray tube.

S. M5 ELECTRON BALLISTICS AND APPLICATIONS / 21

be made in assuming that the straight line MP', if projected backward, willpass through the center 0' of the region of the magnetic field. Then

D L tan

22 / ELECTRONIC DEVICES AND CIRCUITS Sec. J -15

a constant longitudinal magnetic field, the axis of the tube coinciding withthe direction of the magnetic field. A magnetic field of the type here con-sidered is obtained through the use of a long solenoid, the tube being placedwithin the coil. Inspection of Fig. 1-12 reveals the motion. The Y axisrepresents the axis of the cathode-ray tube. The origin is the point at whichthe electrons emerge from the anode. The velocity of the origin is v

,the

initial transverse velocity due to the mutual repulsion of the electrons beingVoX. It is now shown that the resulting motion is a helix, as illustrated.

The electronic motion can most easily be analyzed by resolving thevelocity into two components, vv and v 9t along and transverse to the magneticfield, respectively. Since the force is perpendicular to B, there is no accelera-tion in the Y direction. Hence vv is constant and equal to vv . A force eBv tnormal to the path will exist, resulting from the transverse velocity. Thisforce gives rise to circular motion, the radius of the circle being mv9/eB t withv9 a constant, and equal to y- The resultant path is a helix whose axis isparallel to the Y axis and displaced from it by a distance R along the Z axis,as illustrated.

The pitch of the helix, defined as the distance traveled along the directionof the magnetic field in one revolution, is given by

V = v^T

where T is the period, or the time for one revolution.(1-41) that

27T771V =

~eB^

It follows from Eq.

(1-48)

If the electron beam is defocused, a smudge is seen on the screen whenthe applied magnetic field is zero. This means that the various electronsin the beam pass through the anode hole with different transverse velocitiesv, and so strike the screen at different points. This accounts for the appear-ance of a broad, faintly illuminated area instead of a bright point on the screen.As the magnetic field is increased from zero the electrons will move in helicesof different radii, since the velocity t> that controls the radius of the pathwill be different for different electrons. However, the period, or the time totrace out the path, is independent of vex, and so the period will be the samefor all electrons. If, then, the distance from the anode to the screen is madeequal to one pitch, all the electrons will be brought back to the Y axis (thepoint 0' in Fig. 1-12), since they all will have made just one revolution.Under these conditions an image of the anode hole will be observed on thescreen.

As the field is increased from zero, the smudge on the screen resultingfrom the defocused beam will contract and will become a tiny sharp spot(the image of the anode hole) when a critical value of the field is reached.This critical field is that which makes the pitch of the helical path just equalto the anode-screen distance, as discussed above. By continuing to increase

Sec. T-T5 ELECTRON BALLISTICS AND APPLICATIONS / 23

Y

Fig. 1-12 The helical path of an

electron introduced at an angle (not

90) with a constant magnetic field. Electronicpath

the strength of the field beyond this critical value, the pitch of the helixdecreases, and the electrons travel through more than one complete revolution.The electrons then strike the screen at various points, so that a defocusedspot is again visible. A magnetic field strength will ultimately be reachedat which the electrons make two complete revolutions in their path from theanode to the screen, and once again the spot will be focused on the screen.This process may be continued, numerous foci being obtainable. In fact, thecurrent rating of the solenoid is the factor that generally furnishes a practicallimitation to the order of the focus.

The foregoing considerations may be generalized in the following way:If the screen is perpendicular to the Y axis at a distance L from the point ofemergence of the electron beam from the anode, then, for an anode-cathodepotential equal to Va , the electron beam will come to a focus at the center of thescreen provided that L is an integral multiple of p. Under these conditions,Eq, (1-48) may be rearranged to read

e

m8ir2 K~ n2

L*B 2(1-49)

where n is an integer representing the order of the focus. It is assumed, inthis development, that eVa ~ fynvey 2 , or that the only effect of the anodepotential is to accelerate the electron along the tube axis. This implies thatthe transverse velocity xoz> which is variable and unknown, is negligible incomparison with voy . This is a justifiable assumption.

This arrangement was suggested by Busch, and has been used 2 to measurethe ratio e/m for electrons very accurately.

A Short Focusing Coil The method described above of employing alongitudinal magnetic field over the entire length of a commercial tube isnot too practical. Hence, in a commercial tube, a short coil is wound around

24 / ELECTRONIC DEVICES AND CIRCUITS Sec. 1-16

Rg. 1-13 Parallel electric and magnetic fields.

the neck of the tube. Because of the fringing of the magnetic lines of flux,a radial component of B exists in addition to the component along the tubeaxis. Hence there are now two components of force on the electron, onedue to the axial component of velocity and the radial component of the field,and the second due to the radial component of the velocity and the axialcomponent of the field. The analysis is complicated, 8 but it can be seenqualitatively that the motion will be a rotation about the axis of the tube and,if conditions are correct, the electron on leaving the region of the coil maybe turned sufficiently so as to move in a line toward the center of the screen.A rough adjustment of the focus is obtained by positioning the coil properlyalong the neck of the tube. The fine adjustment of focus is made by con-trolling the coil current.

1-16 PARALLEL ELECTRIC AND MAGNETIC FIELDSConsider the case where both electric and magnetic fields exist simultaneously,the fields being in the same or in opposite directions. If the initial velocityof the electron either is zero or is directed along the fields, the magnetic fieldexerts no force on the electron, and the resultant motion depends solely uponthe electric field intensity . In other words, the electron will move in adirection parallel to the fields with a constant acceleration. If the fields arcchosen as in Fig. 1-13, the complete motion is specified by

vv = Vey at y = v

2d / ELECTRONIC DEVICES AND CIRCUITS

-Z

Sec. 1-17

Fig. 1-15 The projection of the path in theXZ plane is a circle.

180-e

+ Z\ u sin \p = u.

By^use of either the relationship T = 2r/w or Eq. (1-42), there is obtainedT = 4,75 X 10~8 sec, and hence less than one revolution is made before thereversal.

The point P' in space at which the reversal takes place is obtained by con-sidering the projection of the path in the XZ plane (since the Y coordinate Ualready known). The angle 8 in Fig. 1-15 through which the electron has rotatedis

9 - rf - 1.32 X 10 X 2.06 X 10" 8 = 2.71 rad = 155The radius of the circle is

fi^,

=(5 X 10)(0.6)

to 1.32 X 108

From the figure it is clear that

X = R sin (180 - $) = 2.27 sin 25 = 0.957 cmZ = R + R cos (180 - $) = 2.27 + 2.05 = 4.32 cmc. The velocity is tangent to the circle, and its magnitude equals v a sin *> =

5 X 10' X 0.6 = 3 X 10" m/sec. At 9 = 155, the velocity components are9, - -#, cos (180 - 6) - -8 X 10 s cos 25 = -2.71 X 10 8 m/secv =

f. = v sin (180 - 6) = 3 X 10 sin 25 = 1.26 X 10 8 m/sec

1-17 PERPENDICULAR ELECTRIC AND MAGNETIC FIELDSThe directions of the fields are shown in Fig. 1-16. The magnetic field isdirected along the - 1' axis, and tho electric field is directed along the

-Xaxis. The force on an electron due to the electric field is directed along the+X axis. Any force due to the magnetic field is always normal to B, and

Sec. M7 ELECTRON BALLISTICS AND APPLICATIONS / 27

Fig. 1-16 Perpendicular electric and magnetic fields.

hence lies in a plane parallel to the XZ plane. Thus there is no componentof force along the Y direction, and the Y component of acceleration is zero.Hence the motion along Y is given by

L = U = 1*0 y = vovt (1-51)assuming that the electron starts at the origin.

// the initial velocity component parallel to B is zero, the path lies entirelyin a plane perpendicular to B.

It is desired to investigate the path of an electron starting at rest at theorigin. The initial magnetic force is zero, since the velocity is zero. Theelectric force is directed along the +X axis, and the electron will be acceler-ated in this direction. As soon as the electron is in motion, the magneticforce will no longer be zero. There will then be a component of this forcewhich will be proportional to the X component of velocity and will be directedalong the +Z axis. The path will thus bend away from the +X directiontoward the +Z direction. Clearly, the electric and magnetic forces interactwith one another. In fact, the analysis cannot be carried along further,profitably, in this qualitative fashion. The arguments given above do, how-ever, indicate the manner in which the electron starts on its path. This pathwill now be shown to be a cycloid.

To determine the path of the electron quantitatively, the force equationsmust be set up. The force due to the electric field is e& along the +X direc-tion. The force due to the magnetic field is found as follows: At any instant,the velocity is determined by the three components vx , vv> and v, along thethree coordinate axes. Since B is in the Y direction, no force will be exertedon the electron due to vy . Because of vx , the force is eBvx in the -\-Z direc-tion, as can be verified by the direction rule of Sec. 1-11. Similarly, the forcedue to v, is eBvt in the X direction. Hence Newton's law, when expressedin terms of the three components, yields

j- dvx i,fx = m -j- = e8 eBvz

at

dv. _f

'= m dl

= eBv'

By writing for convenience

eBm

= and U= B

(1-52)

(1-53)

28 / ELECTRONIC DEVICES AND CIRCUITS

the foregoing equations may be written in the form

~dlN oju biVz dvt

Tt= +m*

Sec, 7-17

(1-54)

A straightforward procedure is involved in the solution of these equations.If the first equation of (1-54) is differentiated and combined with the second,we obtain

d 2vx dvt (1-55)

This linear differential equation with constant coefficients is readily solvedfor vx . Substituting this expression for vx in Eq. (1-54), this equation can besolved for vt . Subject to the initial conditions , = ,= 0, we obtain

vx = u sin (d p = u u cos tat (1-56)

In order to find the coordinates x and z from these expressions, each equa-tion must be integrated. Thus, subject to the initial conditions x = z = 0,

4i tk

x = - (1 cos at) z = ut - sin o)t

If, for convenience,

8 s at and Q = -

then

x = 0(1 - cos 8) z = Q(8 - sin 8)

where u and a? are as defined in Eqs. (1-53).

(1-57)

(1-58)

(1-59)

Cycloid a! Path Equations (1-59) are the parametric equations of a com-mon cycloid, defined as the path generated by a point on the circumference of a circleof radius Q which rolls along a straight line, the Z axia. This is illustratedin Fig. 1-17. The point P, whose coordinates are x and z (y = 0), representsthe position of the electron at any time. The dark curve is the locus of thepoint P. The reference line CC is drawn through the center of the generatingcircle parallel to the X axis. Since the circle rolls on the Z axis, then OCrepresents the length of the circumference that has already come in contactwith the Z axis. This length is evidently equal to the arc PC (and equals Qd).The angle 8 gives the number of radians through which the circle has rotated.From the diagram, it readily follows that

x = Q - Qcos8 z = Q& - Q sin 8 (1-60)which are identical with Eqs. (1-59), thus proving that the path is cycloidalas predicted.

S. M7 ELECTRON BALLISTICS AND APPLICATIONS / 29

Fig. 1-17 The cydoidol path of an electron in perpen-

dicular electric and magnetic fields when the initial

velocity is zero.

The physical interpretation of the symbols introduced above merelyas abbreviations is as follows:

u represents the angular velocity of rotation of the Tolling circle.

8 represents the number of radians through which the circle has rotated.Q represents the radius of the rolling circle.Since u = wQ, then u represents the velocity of translation of the center of

the rolling circle.

From these interpretations and from Fig. 1-17 it is clear that the maximumdisplacement of the electron along the X axis is equal to the diameter of therolling circle, or 2Q. Also, the distance along the Z axis between cusps isequal to the circumference of the rolling circle, or 2vQ. At each cusp thespeed of the electron is zero, since at this point the velocity is reversing its

direction (Fig. 1-17). This is also seen from the fact that each cusp is alongthe Z axis, and hence at the same potential. Therefore the electron has gainedno energy from the electric field, and its speed must again be zero.

If an initial velocity exists that is directed parallel to the magnetic field,the projection of the path on the XZ plane will still be a cycloid but theparticle will now have a constant velocity normal to the plane. This path

30 / ELECTRONIC DEVICES AND CIRCUITS

might be called a "cycloidal helical motion."(1-59), with the addition of Eqs. (1-51).

Sec. 1-17

The path is described by Eqs.

Straight Line Path As a special case of importance, consider that the elec-tron is released perpendicular to both the electric and magnetic fields so thatvox = vay = and vot ^ 0. The electric force is eS along the -\-X direction(Fig. 1-16), and the magnetic force is eBv

32 / ELECTRONIC DEVICES AND CIRCUITSSee. 1. 18

0* Magnetic fleld(Into paper)

6-*

fig. 1-1? The trocholdal paths of electrons inperpendicular electric and magnetic fields.

two halves of a shallow, hollow, metallic "pillbox" which has been split alonga diameter as shown; a strong magnetic field which is parallel to the axis ofthe dees; and a high-frequency ac potential applied to the dees.

A moving positive ion released near the center of the dees will be acceler-ated in a semicircle by the action of the magnetic field and will reappear atpoint 1 at the edge of dee I. Assume that dee II is negative at this instantwith respect to dee I. Then the ion will be accelerated from point 1 to point 2across the gap, and will gain an amount of energy corresponding to the poten-tial difference between these two points. Once the ion passes inside the metaldee II, the electric field is zero, and the magnetic field causes it to move in thesemicircle from point 2 to point 3. If the frequency of the applied ac poten-tial is such that the potential has reversed in the time necessary for the ion to

Dees

Particle orbit

(schematic)

Fig. 1-20 The cyclotron principle.

South poleVacuumchamber

s*. its ELECTRON BALLISTICS AND APPLICATIONS / 33

go from point 2 to point 3, then dee I is now negative with respect to dee II,and the ion will be accelerated across the gap from point 3 to point 4. Withthe frequency of the accelerating voltage properly adjusted to this "resonance"value, the ion continues to receive pulses of energy corresponding to thisdifference of potential again and again.

Thus, after each half revolution, the ion gains energy from the electricfield, resulting, of course, in an increased velocity. The radius of each semi-circle is then larger than the preceding one, in accordance with Eq. (1-39),so that the path described by the whirling ion will approximate a planar spiral.

EXAMPLE Suppose that the oscillator that supplies the power to the dees of agiven cyclotron imparts 50,000 eV to heavy hydrogen atoms (deuterons), eachof atomic number 1 and atomic weight 2.0147, at each passage of the ions acrossthe accelerating gap. Calculate the magnetic field intensity, the frequency ofthe oscillator, and the time it will take for an ion introduced at the center of thechamber to emerge at the rim of the dee with an energy of 5 million electron volts(5 MeV). Assume that the radius of the last semicircle is 15 in.

Solution The mass of the deuteron is

at = 2.01 X 1.66 X 10~" = 3.34 X 10"" kg

The velocity of the 5-MeV ions is given by the energy equation

hrw* - (5 X 10 s) (1.60 X 10"") = 8.00 X lO"" J

/2 X 8.00 X 10-l>\* nnfl . ,\ 3.34 X 10-" /3.34 X 10-

The magnetic field, given by Eq. (1-39),

(3.34 X 10-") (2.20 X 10 T)B= =eR (1.60 X 1Q-)(15 X 2.54 X 0.0!)

= 1.20 Wb/m*

is needed in order to bring these ions to the edge of the dees.The frequency of the oscillator must be equal to the reciprocal of the time of

revolution of the ion. This is, from Eq. (1-41),

._

1_

eB

T 2-rm1.60 X 10-" X 1.202tt X 3.34 X 10""

= 9.15 X 10 B Hit = 9.15 MHiSince the ions receive 5 MeV energy from the oscillator in 50-keV steps, they

must pass across the accelerating gap 100 times. That, is, the ion must make50 complete revolutions in order to gain the full energy. Thus, from Eq. (1-41),the time of flight is

1= SOT - 50 X 19.15 X 10 8

* Hi hertz = cycles per second

m 5.47 X 10-* sec = 5.47 usee

MHz = megahertz (Appendix B).

34 / ELECTRONIC DEVICES AND CIRCUITS Sec. 1-78

In order to produce a uniform magnetic field of 1.2 Wb/m2 over a circulararea whose radius is at least 15 in., with an air gap approximately 6 in. wide, anenormous magnet is required, the weight of such a magnet being of the order of60 tons. Also, the design of a 50-kV oscillator for these high frequencies andthe method of coupling it to the dees present some difficulties, since the dees arein a vacuum-tight chamber. Further, means must be provided for introducingthe ions into the region at the center of the dees and also for removing the high-energy particles from the chamber, if desired, or for directing them against atarget.

Sc. M ELECTRON BALLISTICS AND APPLICATIONS / 35

hollow cylinder, since there is need for a magnetic field only transverse to thepath. This results in a great saving in weight and expense. The dees of thecyclotron are replaced by a single-cavity resonator. Electrons and protonshave been accelerated to the order of a billion electron volts (Bev) in synchro-trons. 8 The larger the number of revolutions the particles make, the higherwill be their energy. The defocusing of the beam limits the number of allow-able cycles. With the discovery of alternating-gradient magnetic field focusing,*higher-energy-particle accelerators (70 BeV) have been constructed. 10

The bombardment of the elements with the high-energy protons, deu-terons, or helium nuclei which are normally used in the cyclotrons rendersthe bombarded elements radioactive. These radioactive elements are of theutmost importance to physicists, since they permit a glimpse into the consti-tution of nuclei. They are likewise of extreme importance in medical research,since they offer a substitute for radium. Radioactive substances can be fol-lowed through any physical or chemical changes by observing their emittedradiations. This "tracer," or utagged-atom," technique is used in industry,medicine, physiology, and biology.

F-M Cyclotron and Synchrotron It is shown in Sec. 1-10 that if an elec-tron falls through a potential of more than 3 kV, a relativistic mass correctionmust be made, indicating that its mass increases with its energy. Thus, ifelectrons were used in a cyclotron, their angular velocity would decrease astheir energy increased, and they would soon fall out of step with the high-fre-quency field. For this reason electrons are not introduced into the cyclotron.

For positive ions whose mass is several thousand times that of the elec-tron, the relativistic correction becomes appreciable when energies of a fewtens of millions of electron volts are reached. For greater energies than these,the ions will start to make their trip through the dees at a slower rate and Blipbehind in phase with respect to the electric field. This difficulty is overcomein the synchrocyclotron, or f-m cyclotron, by decreasing the frequency of theoscillator (frequency modulation) in accordance with the decrease in the angu-lar velocity of the ion. With such an f-m cyclotron, deuterons, a particles,and protons have been accelerated to several hundred million electron volts. 7

It is possible to give particles energies in excess of those for which therelativistic correction is important even if the oscillator frequency is fixed,provided that the magnetic field is slowly increased in step with the increasein the mass of the ions so as to maintain a constant angular velocity. Suchan instrument is called a synchrotron. The particles are injected from a gun,which gives them a velocity approaching that of light. Since the radius ofthe orbit is given by R = mv/Be and since the ratio m/B is kept constant andv changes very little, there is not much of an increase in the orbit as the energyof the electron increases. The vacuum chamber is built in the form of adoughnut instead of the cyclotron pillbox. The magnet has the form of a

REFERENCES

1. Millman, J., and H. Taub: "Pulse, Digital, and Switching Waveforms," chaps. 14and 19, McGraw-Hill Book Company, New York, 1965.

2. Goedicke, E. : Eine Neubestimmung der spezifischen Ladung des Electrons nach derMethode von H. Busch, Physik. Z., vol. 36, no. 1, pp. 47-63, 1939.

3. Cosslett, V. E.: "Introduction to Electron Optics," Oxford University Press, FairLawn, N.J., 1946.

4. Millman, J., and S. Seely: "Electronics," 2d ed., p. 35, McGraw-Hill Book Com-pany, New York, 1951.

5. James, G., and R. C. James: "Mathematics Dictionary," D. Van Nostrand Com-pany, Inc., Princeton, N.J., 1949.

6. Livingston, M. S.: The Cyclotron, I, J. Appl. Phys., vol. 15, pp. 2-19, January,1944; The Cyclotron, II, ibid., pp. 128-147, February, 1944.Livingston, M. S.: Particle Accelerators, Advan. Electron., Electrochem. Eng., vol. 1,pp. 269-316, 1948.

7. Brobeck, W. M., E. 0. Lawrence, K. R. MaeKenzie, E. M. McMillan, R. Serber,D. C. Sewell, K. M. Simpson, and R. L. Thornton: Initial Performance of the 184-inch Cyclotron of the University of California, Phys. Rev., vol. 71, pp. 449-450,April, 1947.

8. Livingston, M. S., J. P. Blewett, G. K. Green, and L. J. Haworth: Design Study fora Three-Bev Proton Accelerator, Rev. Set. Tnstr., vol. 21, pp. 7-22, January, 1950.

9. Courant, E. D., M. S. Livingston, and H. 8. Snyder: The Strong-focusing Syn-chrotron: A New High Energy Accelerator, Phys. Rev., vol. 88, pp. 1190-1196,December, 1952.

10. Livingston, M. S., and J. P. Blewett: "Particle Accelerators," chap. 15, McGraw-Hill Book Company, New York, 1962.

2 /ENERGY LEVELS ANDENERGY BANDS

In this chapter we begin with a review of the basic atomic properties

of matter leading to discrete electronic energy levels in atoms. Wealso examine some selected topics in quantum physics, such as thewave properties of matter, the Schrodinger wave equation, and thePauli exclusion principle. We find that atomic energy levels arespread into energy bands in a crystal. This band structure allows usto distinguish between an insulator, a semiconductor, and a metal.

2-1 THE NATURE OF THE ATOMIn order to explain many phenomena associated with conduction ingases, metals, and semiconductors and the emission of electrons fromthe surface of a metal, it is necessary to assume that the atom has

loosely bound electrons which can be torn away from it.Rutherford, 1 in 1911, found that the atom consists of a nucleus of

positive charge that contains nearly all the mass of the atom. Sur-

rounding this central positive core are negatively charged electrons.

As a specific illustration of this atomic model, consid