Handbook of Physics [1 ed.]

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H A N D B O O K OF P H Y S I C S

McGRAW-HILL HANDBOOKS

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Handbook: Handbook: Handbook: Handbook:

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HANDBOOK OF PHYSICS Prepared by a staff of specialists Edited by

E. U . C O N D O N , Ph.D. Way man Crow Professor of Physics Washington University, St. Louis Former Director, National Bureau of Standards Washington, D. C.

H U G H O D I S H A W , D.Sc. Executive Director, U. S. National Committee for the International Geophysical Year, National Academy of Sciences, Washington, D. C. Former Assistant to the Director, National Bureau of Standardsy Washington, Z>. C.

M c G R A W - H I L L BOOK CO M P AN Y , INC. New York Toronto London

1958

HANDBOOK

OF

PHYSICS

Copyright © 1958 by the McGraw-Hill Book Company, Inc. Printed in the United States of America. All rights reserved. This book, or parts thereof, may not be reproduced in any form without permission of the publishers. Library of Congress Catalog Card Number: 57-6387

T H E M APLE PRESS COMPANY, YO R K, PA.

Contributors

Milton Abramowitz

Clyde R. Burnett

Chief, Computation Laboratory National Bureau o f Standards {deceased)

Assistant Professor The Pennsylvania State University

Lawrence H. Aller

Herbert B. Callen

Professor o f Astronom y University o f M ichigan

Professor o f Physics University o f Pennsylvania

Franz L. Alt

G. M. Clemence

Assistant Chief A pplied Mathematics Division National Bureau o f Standards

Scientific Director U.S. Naval Observatory

R. D. Arnold Operations Analyst Operations Evaluation Group Massachusetts Institute o f Technology

E. U. Condon W ay man Crow Professor o f Physics Washington University

E. Richard Cohen

Professor o f Physics

Research Advisor Atom ics International A Division o f North Am erican Aviation, Inc.

University o f Pennsylvania

L. E. Copeland

John Bardeen

Senior Research Chemist Portland Cement Association Research and Development Laboratories

K . R. Atkins

Professor o f Physics and Professor o f Electrical Engineering U?iiversity o f Illinois

Richard A. Beth Physicist, Brookhaven National Laboratory

R. Byron Bird Professor o f Chemical Engineering University o f W isconsin

John P. Blewett Senior Physicist Brookhaven National Laboratory

Robert R. Brown Assistant Professor University o f C alifornia, Berkeley

C. F. Curtiss Associate Professor o f Chemistry University o f W isconsin

Jesse W. M. DuMond Professor o f Physics California Institute o f Technology

Leonard Eisenbud Physicist Bartol Research Foundation

Churchill Eisenhart C hief Statistical Engineering Laboratory National Bureau o f Standards

Sanborn C. Brown

William E. Forsythe

Associate Professor o f Physics Massachusetts Institute o f Technology

Lamp Division General Electric Company

William Fuller Brown, Jr.

M. M. Frocht

Professor o f Electrical Engineering University o f M innesota

Research Professor o f Mechanics and Director o f Experimental Stress A nalysis Illinois Institute o f Technology

Stephen Brunauer Principal Research Chemist Portland Cement Association Research and Development Laboratories

Lawrence S. Germain University o f California Radiation Laboratory Livermore

CONTRIBUTORS

Walter Gordv

Walter H. Johnson, Jr.

Professor of P hysics, Duke University

Research Associate School o f Physics University o f M innesota

R. W. Gurney Research Associate University o f Bristol {deceased)

Andrew Guthrie Heady Nucleonics Division U.S. Naval Radiological Defense Laboratory

David Halliday Professor o f Physics University of Pittsburgh

Walter J. Hamer Chiefy Electrochemistry Section National Bureau o f Standards

R. W. Hayward Radioactivity Section National Bureau o f Standards

Max J. Herzberger Senior Research Associate

Deane B. Judd Physicist National Bureau o f Standards

C. Lanczos Senior Professor Dublin Institute fo r Advanced Studies

Gerald L. Landsman Assistant Director fo r Research and Development M ilitary Electronics Division M otorola, Inc.

Howard R. Lillie Staff Research M anager Corning Glass W orks

Julian Ellis Mack Professor o f Physics University o f W isconsin

Kodak Research Laboratories

Robert Maurer

E. L. Hill

Professor o f Physics University o f Illinois

Professor o f Physics University of M innesota

John A. Hippie Director of Rcscarchy Philips Laboratories

J. G. Hirsehberg

J. Rand McNally, Jr. Physicist Oak Ridge National Laboratory

Walter C. Michels

University o f W isco/m a

M arion Reillcy Professor o f Physics B ryn M awr College

Joseph 0 . Hirschfelder

Elliott W. Mont roll

Professor o f Chemistry and Director o f The University o f ir im m sm Naval Research Laboratory University o f W isconsin

Research Professor Institute fo r Fluid Dynam ics and A pplied Mathematics University o f M aryland

Alan J. Hoilman

Philip M. Morse

Consultant Management Consultation Services General Electric Company

Professor o f Physics Massachusetts Institute o f Technology

Theodor Ilurliinann

C h ief o f the Reactor Group

Reaktor Ltd. W nerenlingen Switzerland

National Bureau o f Standards

Uno Ingarrl Associate Professor o f Physics Massachusetts Institute o f Technology Fritz John Professor o f Mathematics Institute o f Mathematical Sciences New York University

( 1. O. Muehlhause

Hamid H. Nielsen Professor o f Physics The Ohio State University

Alfred 0 . Nier Professor t Tniversity o f M innesota

Richard M. Noyes Professor o f Chemistry University o f Oregon

CONTRIBUTORS

Hugh Odishaw

Lloyd P. Smith

Executive Director U.S. National Committee fo r the International Geophysical Year National Academ y o f Sciences

President

Chester H. Page

Professor o f Physics Carnegie Institute o f Technology

Consultant to the Director National Bureau o f Standards

Ray Pepinsky Research Professor o f Physics and Directory The Groth Institute The Pennsylvania State University

B. Peters

Research (fc Advanced Development Division Avco M anufacturing Corporation

Roman Smoluchowski

Edward S. Steeb Physicist M iam i University

William E. Stephens Professor o f Physics University o f Pennsylvania

Professor o f Physics Tata Institute o f Fundamental Research Bombay

A. H. Taub

Louis A. Pipes

Olga Taussky

Professor o f Engineering University o f California Los Angeles

Research Associate (Mathematics) California Institute o f Technology

Karl S. Quisenberry Assistant Professor University o f Pittsburgh

Norman F. Ramsey Professor o f Physics Harvard University

Markus Reiner Professor o f A pplied M echanics Israel Institute o f Technology H aifa

M. E. Rose Chief Physicist Oak Ridge National Laboratory

Frederick D. Rossini Sillinian Professor and Heady Department o f Chemistry and Directory Chemical and Petroleum Research Laboratory Carnegie Institute o f Technology

Raymond J. Seeger National Science Foundation

Rescar eh Professor o f A pplied Mathematics University o f Illinois

Alan M. Thorndike Physicist Brookhaven National Laboratory

John Todd Professor o f Mathematics California Institute o f Technology

C. Tompkins Professor o f Mathematics and Directory Numerical A nalysis Research University o f California

V. Vand Associate Professor o f Physics The Pennsylvania State University

Arthur R. von Hippel Director o f the Laboratory fo r Insulation Research and Professor o f Electrophysics Massachusetts Institute o f Technology

John Archibald Wheeler Professor o f Physics Princeton University

Eugene P. Wigncr Palmer Physical Laboratory Princeton University

E. J. Seldin

R. E. Wilson

Physicist National Carbon Research Laboratories A Division o f Union Carbide Corporation

Assistant to Laboratory Manager and Heady Administration Tucson Engineering Laboratory Hughes A ircraft Company

Harold K . Skramstad Assistant Chief fo r Systems Data Processing Systems Division National Bureau o f Standards

vii

John Gibson Winans Associate Professor o f Physics University o f Jrtsamsin

viii

CONTRIBUTORS H ugh C. W olfe

W . J. Y ou d en

Heady Physics Department Cooper Union School of Engineering

Applied Mathematics Division National Bureau of Standards

R u eben E. W ood

M arvin Zelen

Professor of Chemistry The George Washington University

M athematician National Bureau of Standards

Preface

T h is book was first planned nearly ten years ago when we were closely associated a t the N ation al Bureau of Standards. W e set our­ selves the problem of m aking a judicious selection from the v a st literatu re of physics of m aterials which m ight reasonably be called “ W h at every ph ysicist should kn ow .” A s the planning w ent forw ard we becam e increasingly aw are of w h at a difficult task we had undertaken. T h e literature of physics has be­ com e so great, and is grow ing a t such a rate, th a t it is v e ry difficult for a p h ysicist to be really well-inform ed on more than a rela tively narrow sp ecialty w ithin the subject. N evertheless the u n ity of the science is such th a t m uch research progress depends considerably 011 u tilization of advan ces in one p art to provide the m eans for solving problem s in another. Therefore it is necessary for physicists to m ake strong efforts to resist tendencies tow ard over-specialization. One w ay in w hich the rapid ity of progress has com plicated our task is the tenden cy for parts of the book to becom e out-of-date while being set up in typ e. W e have m ade efforts to avoid this b y m aking more than the usual num ber of additions and corrections while the book was going through galley proof. Our than ks are due the contributing specialists for their willingness to go to the extra trouble of m aking their chapters as up-to-date as possible in spite of this difficulty. B y the v e ry nature of the preparation and publication process, a handbook cannot be com pletely current w ith journal literature, and there is variation even am ong the chapters, as revealed b y their refer­ ences. W ithin this restriction, we believe th a t the H andbook fulfills its function as a one-volum e com pendium . I t is our sincere hope th at physicists the world over will find this selection of m aterials to be a useful one. W e think th at there is con­ siderable econom y of effort to be gained in a one-volum e synthesis of the principal parts of the science in th at so m an y techniques find use again and again in different parts of the su bject and only need to be explained once in a w ork of this kind. W e will appreciate receiving suggestions from readers as to how the book's usefulness m ay be im proved in fu tu re editions. E. U. CONDON HUG1I ODISHAW

Contents

Contributors Preface ix

PART

v

1 • M ATHEM ATICS

Chapter 1

A rithm etic b y Franz L. A lt

1-4

1. Numbers and Arithmetic Operations. 2. Logical Foundation of Arithmetic. 8. Digital Computing Machines.

Chapter 2

Algebra b y Olga Taussky

1-10

1. P oly n om ia ls. 2. Algebraic Equations in One Unknown, Com­ plex Numbers. 3. Equations of Degree 2 (Quadratic Equa­ tions. 4> Equations of Degree 3 (Cubic Equations). 5. Equa­ tions of Degree 4 (Biquadratic Equations), 6. Equations of Degree n. 7. Discriminants and General Symmetric Functions. 8. Computational Methods for Obtaining Roots of Algebraic Equations. 9. Matrices. 10. Determinants. 11. Systems of Linear Equations. 12. Numerical Methods for Finding the In­ verse of a Matrix and for Solving Systems of Linear Equations. 13. Characteristic Roots of Matrices and Quadratic Forms. 14. Computation of Characteristic Roots of Matrices. 15. Func­ tions of Matrices and Infinite Sequences. 16. Hypercomplex Systems or Algebras. 17. Theory of Groups.

Chapter 3

Analysis by John Todd

1-22

1. Real Numbers, Limits. 2. Real Functions. 3. Finite Differ­ ences. 4 • Integration. 5. Integral Transforms. 6. Functions of Several Real Variables. 7. Complex Numbers. 8. Series of Func­ tions. 9. Functions o f a Complex Variable. 10. Conforinal Mapping. 11. Orthogonality. 12. Special Functions.

Chapter 4

Ordinary Differential Equations b y Olga Taussky

1-59

1. Introduction. 2. Simple Cases. 3. Existence Theorems. 4• Methods for Solution. 5. Examples of Well-known Equations. 6. Some General Theorems. 7. Nonhomogcneous Equations, Green’s Function. 8. Numerical Integration o f Differential Equations. 9. Systems of Simultaneous Differential Equations.

Chapter 5

Partial Differential Equations b y Fritz John

1-66

1. General Properties. 2. First-order Equations. 3. Ellip­ tic Equations. 4• Parabolic Equations of Second Order. 5. Hyperbolic Equations in Two Independent Variables. 6. Hyper­ bolic Equations with More than Two Independent Variables. 7. Numerical Solution of Partial Differential Equations.

xii

CONTENTS

Chapter 6

Integral Equations b y M . Abram ow itz

1-90

1. Integral Equat ions of the Second Kind. 2. Symmetric Kernels. 3. Nonsymnietrie Kernels. 4• Integral Equations of the First Kind, 5. Voltcrra’s Equation. 6. Nonlinear Integral Equation.

Chapter 7

Operators b y Olga Tausskv

1-95

1. Vector Spaces, Abstract Hilbert Spaces, llilbert Space. Definition of Operator or Transformation. 3. Spectrum Bounded Operators, Eigenvalues, and Eigenfunctions.

Chapter 8

G eom etry b y A, J* Hoffm an

2. of

1-97

1. Definition and Assumptions. 2. Projective Plane. 3. Projec­ tive Group. 4• Correlations, Polarities, and Conics. 5. Projec­ tive Line. 6. Subgroups of the Projective Group. 7. Affine Group and Plane. S. Euclidean Group and Plane. 9. Conics. 10. Angles. 11. Triangles. 12. Polygons. 13. Hyperbolic Group and Plane. 14. Elliptic Group and Plane.

Chapter 9

Vector Analysis b y E. U. Condon

1-103

1. Addition o f Vectors. 2. Scalar and Vector Products. 3. Vec­ tors and Tensors in Oblique Coordinates. 4• Gradient of Scalar and Vector Fields. 5. Divergence of a Vector Field. 6. Curl of a Vector Field. 7. Expansion Formulas. 8. Orthogonal Curvi­ linear Coordinates. 9. Transformation o f Curvilinear Coordi­ nates.

Chapter 10

Tensor Calculus b y C, Lanzcos

1-111

1. Scalars, Vectors, Tensors. 2. Analytic Operations with Vec­ tors. 3. Unit Vectors; Components. 4. Adjoint Set o f Axes. 5. Covariant and Contravariant Components o f a Vector. 6. Transformation of the Basic Vectors Vi. 7. Transformation of Vector Components. 8. Radius Vector R. 9. Abstract Defini­ tion of a Vector. 10. Invariants and Covariants. 11. Abstract Definition of a Tensor. 12. Tensors o f Second Order. 13. Ein­ stein Sum Convention. 14. Tensor Algebra. 15. Determinant Tensor. 16. Dual Tensor. 17. Tensor Fields. 18. Differentia­ tion o f a Tensor. 19. Covariant Derivative o f the Metrical Ten­ sor. 20. Principles o f Special and General Relativity. 21. Curvi­ linear Transformations. 22. Covariant Derivative of a Tensor. 23. Covariant Derivative of the Metrical Tensor. 24. Funda­ mental Differential Invariants and Covariants of Mathematical Physics. 25. Maxwell Electromagnetic Equations. 26. Curva­ ture Tensor of Ricmann. 27. Properties of Ilicmann Tensor. 28. Contracted Curvature Tensor. 29. The Matter Tensor of Einstein. 30. Einstein’s Theory of Gravity.

Chapter II

Calculus o f Variations by C. 8 . Tom pkins

1-123

1. Maxima and Minima of a Function o f a Single Variable. 2. Minima of a Function o f Several Variables. 3. Minima of a Definite Integral— the Euler Equations. 4• Examples. 5. Other First Variations: Weierstrass Condition, Corner Conditions, Oneside Variations. 6. Parametric Problems. 7. Problems with Variable End Points. 8. Isopcriinetric Problems— the Problem of Bolza. 9. Second Variations. 10. Multiple-integral Problems. 11. Methods of Computation. 12. Conclusion.

xiii

CONTENTS

Chapter 12

Elem ents o f Probability b y Churchill Eiscnhart and Marvin Zclcn 1-134 1. Probability. 2. Random Variables and Distribution Func­ tions. 3. Distributions in n Dimensions. 4- Expected Values, Moments, Correlation, Covariance, and Inequalities on Distribu­ tions. 5. Measures of Location, Dispersion, Skewness, and Kurtosis. 6*. Characteristic Functions and Generating Functions. 7. Limit Theorems. 8. The Normal Distribution. 9. Discrete Distributions. 10. Sampling Distributions.

Chapter 13

PART

Statistical Design o f Experiments b y W . J. Y ouden

1-165

2 • M E C H A N IC S OF P A R T IC L E S AND RIGID RODIES

Chapter I

K inem atics b y E. U. Condon

2-3

1. Velocity and Acceleration. 2. Kinematics of a Rigid Body. 3. Euler’s Angles. 4« Rclativistic Kinematics. 5. Vector Alge­ bra of Space-Time.

Chapter 2

Dynamical Principles b y E. U. Condon

2-11

1. Mass. 2. Momentum. 3. Force. 4• Impulse. 3. Work and Energy: Power. 6. Potential Energy. 7. Central Force: Colli­ sion Problems. 8. System of Particles. 9. Lagrange’s Equations. 10. Ignorable Coordinates. 11. Hamilton’s Equations. 12. Relativistic Particle Mechanics. 13. Variation Principles.

Chapter 3

Theory o f Vibrations b y E. U. Condon

2-21

1. Simple Harmonic Motion. 2. Damped Harmonic Motion. 3. Forced Harmonic M otion. 4• Mechanical Impedance. 3. Two Coupled Oscillators. 6. Small Oscillations about Equilib­ rium. 7. Oscillations with Dissipation. 8. Forced Oscillations of Coupled Systems. 9. General Driving Force. 10. Physical Pendulum. 11. Nonharmonic Vibrations.

Chapter 4

Orbital M otion by E. U. Condon

2-28

1. Motion under Constant Gravity. 2. Effect of Earth’s Rota­ tion. 3. General Integrals of Ccntral-forcc Problem. 4• Differ­ ential Equation for Orbit. 3. Motion under Invcrse-square-law Attraction. 6. Motion in Elliptic Orbit.

Chapter 5

Dynam ics o f Rigid Bodies b y E. U. Condon

2-33

1. Angular Momentum. 2. Kinetic Energy. 3. Equations of M otion. 4• Rotation about a Fixed Axis. 5. Rotation about a Fixed Point with No External Forces. 6. Asymmetrical Top.

Chapter 6

Q uantum Dynamics b y E. U. Condon

2-38

1. Particle Waves. 2. The Schroedingcr Wave Equation. 3. Matrix Representations. 4• The Harmonic Oscillator. 3. Angu­ lar Momentum. 6. Central-force Problems. 7. The Dynamical Equation. S. Perturbation Theory for Discrete States. 9. Variation Method. 10. Identical Particles. 11. Collision Prob­ lems.

xiv

CONTENTS

Chapter 7

Gravitation by Hugh C. W olfe I. Inversc-square Law. 2. Gravitational Constant, G. celeration of Gravity g and Geophysical Prospecting.

Chapter 8

2-55 8. Ac­

Dynam ics o f the Solar System b y G . M . Clémence 2-60 1. Introduction. 2. Equations of Motion. 8. Method of Solu­ tion. 4- Form of Solution. 5. Precession and Nutation. 6. Frames of Reference. 7. Determination of the Precession. 8. Perturbations of Planets and Satellites. D. Determination of Time. 10. Relativity. 11. National Ephcmcrides. 12. Celestial Navigation. 13. Astronomical Constants.

Chapter 9

Control M echanism s b y Harold K . Skram stad and Gerald L. Landsm an 2-69 1. Introduction. 2. Differential Equation Analysis. 8. Frequency-rcsponse Analysis. 4- System Improvement by Compen­ sation. 5. Steady-state Error. 0. Other Methods of Analysis.

PART

3 • M E C H A N IC S OF DEFORMADLE BODIES

Chapter 1

K inem atics and D ynam ics b y E. U. Condon

5-5

1. Kinematics of Continuous Media. 2. Stress. 3. Equations of Motion. 4* Molecular Standpoint. 5. Energy Relations for Fluid. 6. Strain. 7. Hooke’s Law. S. Viscosity.

Chapter 2

Fluid M echanics b y R. J. Scegcr

5-14

1. Statics of Fluids. 2. In viscid-fluid Dynamics. 3. Irrotational, Continuous Flows of Inviscid Fluids. 4- Discontinuous Flows of In viscid Fluids. 5. Vortex Flows of Inviscid Fluids. 6*. Flows of Compressible, Inviscid Fluids. 7. Flows of Viscous Fluids. S. Turbulence. !). Fluids with Heat. 10. Flows in Electric and Magnetic Fields.

Chapter 3

Rhcology by M . Reiner

5-49

1. Introduction. 2. Second-order Effects in Elasticity and Vis­ cosity. 8. Rheological Properties. 4* Complex Bodies. 5. Vol­ ume Changes. 6. Strength. 7. Microrheological Aspects. 8. Rheometry.

Chapter 4

Wave Propagation in Fluids b y A . II. Tauh

5-59

1. Conservation Laws. 2. Small Disturbances. 3. Interactions of Waves of Small Amplitude. 4. Small Disturbances in Shallow Water. Í). Plane Waves of Finite Amplitude. 0. Formation and Decay of Shocks in One Dimension. 7. Spherical Waves of Finite Amplitude. 8. Effect o f Viscosity and Heat Conduction.

Chapter 5

Statics of Elastic Bodies b y Richard A. Belli 1. Elastic Bodies and Structures. Beams. 4* Columns. 5. Torsion.

5-64

2. The Elastic Moduli.

3.

CONTENTS

Chapter 6

xv

Experimental Stress Analysis b y M . M . Frocht

3-78

1. Two-dimensional Stresses and Strains. 2. Bonded Wireresistance Strain Gauges. 3. Photoelasticity. 4* Two-dimen­ sional Photoelasticity. 5. Three-dimensional Photoelasticity. 6. Photoplasticity. 7. Dynamic Photoelasticity. 8. Brittle Coat­ ings. 9. X Rays.

Chapter 7

Vibrations o f Elastic Bodies; Wave Propagation in Elastic Solids by Philip M . Morse 3-97 1. Equation of M otion; Energy and Intensity. 2. Plane Waves in Homogeneous Media. 3. Spherical Waves, Green’s Tensor for Isotropic Media. 4- Reflection from a Plane Interface, Surface Waves. 5. Waves in a Plate. 6. Waves along a Cylindrical Rod. 7. Standing Waves. S. Transverse Oscillations of Rods and Plates. 9. Scattering of Elastic Waves.

Chapter 8

Acoustics b y Uno Ingard.

3-112

1. Limits of Frequency and Sound Pressure. 2. General Linear Equations of Sound Propagation. 3, Kirehhoff’s Formula in a M oving Medium. 4* Boundary Conditions. Impedance and Absorption Coefficients. 5. Second-order Quantities. 6. Elec­ tromechanical Analogues. 7. The “ Natural” Sources of Sound. 8. Generation of Sound by Turbulent Flow. 9. Radiation from a Simple Source in a Moving Medium. 10. Radiation from a Moving Sound Source. 11. The Doppler Effect. 12. Radiation and Scattering. 13. Technical Aspects of Sound Generation. 14. The Human Voice and Speech Mechanism. 15. Propagation of Sound in the Atmosphere. 16. Propagation in Tubes. 17. Propagation o f Large-ainplitude Waves. 18. Acoustic Streaming. 19. Absorption Materials. 20. Unavoidable Sound Absorption. 21. Microphones. 22. Microphone Calibration. 23. Other Meas­ urements. 24. The Ear and Hearing. 25. Room Acoustics. 26. Transmission of Sound in Building Structures. 27. Genera­ tion. 28. Measurements. 29. Applications.

PART

4 • ELECTRICITY

Chapter I

AND

M AGNETISM

Basic Electrom agnetic Phenom ena b y E. U. Condon

4-3

1. Electrostatic Charge and Coulom b’s Law. 2. Electric Field and Potential. 3. Conductors and Dielectrics. 4■ Forces and Energy in the Electric Field. 5. Ohm’s Law and Electromotive Force. 6. Magnetic Fields Due to Permanent Magnets. 7. Magnetic Fields Due to Electric Currents. 8. Magnetization and Molecular Currents. 9. Electromagnetic Induction. 10. Relativistic Formulation.

Chapter 2

Static Electric ami M agnetic Fields b y E. U. Condon

4-19

1. Field Due to Given Charge Distribution. 2. Force on a Rigid Charge Distribution. 3. Interaction of Two Rigid Charge Dis­ tributions. 4• Conductor in a Given Field. 5. System of Con­ ductors. 6. Magnetic Field Due to a Given Current Distribution. 7. Force on a Rigid Current Distribution. 8. Mutual Inductance and Self-inductance. 9. Magnetic Interaction of Conductors.

CONTENTS Chapter 3

Electric Circuits by Louis A. Pipes

4-28

1. General Considerations. 2. Fundamental Eleetric-eircuit Pa­ rameters. 3. Kirehhoff’s Laws. Jh Laws of Combination of Circuit Parameters. 5. Applications o f the Fundamental Laws. 6. Energy Relations. 7. The Mesh Equations of a General Net­ work. 8. Energy Relations in a Network. 9. General Solution of the Mesh Equations: Transient Phenomena. 10. Examples of Simple Transients. 11. Nodal Equations of the General Network: Duality. 12. Alternating Currents. 13. Power, Effective, or Root-mean-square Values; Series Resonance. 14. Impedances in Series and Parallel: Parallel Resonance. 15. Transmission of Power. 16. General A-C Network: Network Theorems. 17. Two-terminal Networks; Foster’s Reaction Theorem. IS. Fourterminal Networks in the A-C Steady State. 19. Wave Propaga­ tion along a Cascade of Symmetric Structures. 20. Filters. 21. Nonlinear Problems in Electric-eircuit Theory.

Chapter 4

Electronic Circuits b y Chester II. Page

4-47

1. General Considerations. 2. Nonlinear-positive-resistanee Ele­ ments. 3. Negative Resistance. 4. Nonlinear Reactance. 5. Active Circuits.

Chapter 5

Electrical M easurem ents b y W alter C. M ichels 4-55 1. Standards. 2. Defleetion Instruments; the D ’Arsonval Gal­ vanometer. 3. Direct-current Ammeters and Voltmeters. 4• Alternating-current Meters; Electrodynamic Instruments. 5. Null Detectors. 6. Potentiometers. 7. Bridges; the Four-arm Bridge. 8. Measurements Using Resonant Circuits. 9. Meas­ urements at Ultrahigh Frequencies; Distributed Parameters.

Chapter 6

C on duction : M etals and Sem iconductors b y John Bardeen

4-73

1. General Relations. 2. Semiconductors. 3. Thermoelectric and Transverse Effects. 4. Solutions of the Boltzmann Equation. 5. Scattering Mechanisms. 6. Temperature Variation.

Chapter 7

Dielectrics by A. von Ilippcl

4-103

1. Introduction. 2. Complex Permittivity and Permeability. 3. Polarization and Magnetization. 4. Macroscopic Description of Dielectrics by Various Sets of Parameters. 5. Molecular Mechanisms of Polarization. 6. Resonance Polarization. 7. Relaxation Polarization. 8. Piezoelectricity and Ferroeleetrieity. 9. Polarization by Migrating Charge Carriers. 10. Electric Breakdown.

Chapter 8

M agnetic Materials b y W illiam Fuller Brown, Jr.

4-126

1. Basic Concepts. 2. M acroscopic Theory. 3. Classical M icro­ scopic Theory. 4• Quantum-mechanical Concepts. 5. Diamag­ netism. 6. Paramagnetism. 7. Saturation in Paramagncties and Spontaneous Magnetization in Ferromagnetics. 8. Ferromagnetic Domains and the Magnetization Curve. 9. Magnetomeehanical Phenomena in Ferromagnetics. 10. Dynamic Phenomena.

CONTENTS

Chapter 9

xvii

Electrolytic Conductivity and Electrode Processes by Walter J. Ilanier and Reuben E. Wood 4-138 1. Electrolytic and Electronic Conduction. 2. Electrolytic Con­ ductors. 3. Ionization, 4> Degree o f Ionization. 5. Ionic Charge and the Faraday. 6. Electrolytic Conductivity. 7 . Equivalent and Molar Conductance. S. Measurements of Elec­ trolytic Conductivity. 9. Significance of Equivalent Conduct­ ance. 10. Ionic Conductances and Transference Numbers. 11. Ionic Mobilities. 12. Interionic Attraction and Electrolytic Conductivity. 13. High-field Effects in Conductance. 14. Con­ ductance at High Frequencies. 15. Electrochemical Thermody­ namics. 16. Galvanic Cells at Equilibrium. 17. Galvanic Cells Not at Equilibrium. 18. Batteries.

Chapter 10

Conduction o f Electricity in Gases b y Sanborn C. Brown

4-159

1. Probability of Collision. 2. Diffusion. 3. Electron Mobility. 4. Ionic M obility. 5. The Ratio D/n for Electrons. 6. Ambipolar Diffusion. 7. Electron Attachment. S. Ion Recombina­ tion. 9. Electron-Ion Recombination. 10. Neutral Atoms and M olecules. 11. Ion ization by C ollision. 12. H igh-frequ en cy Breakdow n. 13. Low -pressure D -C Breakdow n. I^ .A tm o s pheric-pressure Spark. 15. Low-pressure Glow Discharge. 16. Arc Discharges. 17. Plasma Oscillations.

PART

5 • HEAT

Chapter 1

AND

THERMODYNAMICS

Principles o f Therm odynam ics b y E. U. Condon 5-3 1. The Nature of Heat. 2. First Law of Thermodynamics. 3. Second Law of Thermodynamics. 4• Absolute Temperature Scale. 5. Third Law of Thermodynamics. 6. Equilibrium Con­ ditions. 7. Relations between Thermodynamic Functions. 8. Phase Equilibria of Single-component Systems. 9. Systems of Several Components. 10. Chemical Equilibrium.

Chapter 2

Principles o f Statistical M echanics and K inetic Theory o f Gases b y E. W . M on troll 5-11 1. Scope of Statistical Mechanics. 2. Identification of Tempera­ ture with Molecular Motion and the Maxwell Velocity Distribu­ tion. 3. Mean Free Path and Elementary Theory of Transport Processes. 4• The Boltzmann Equation and the Systematic Kinetic Theory of Gases. 5. The Boltzmann H Theorem. 6. Averages in Equilibrium Statistical Mechanics and the Liouvillo Equation. 7. The Microcanonical and Canonical Ensembles. 8. The Partition Function and the Statistical Basis of Thermody­ namics. 9. Some Simple Examples. 10. Molecular Distribution Functions. 11. Calculation o f Thermodynamic Quantities from Molecular Distribution Functions. 12. The Integrodifferential Equations for the Distribution Functions. 13. Theory of Fluctua­ tions and the Grand Canonical Ensemble.

Chapter 3

T herm om etry and Pyronietry b y R. E. Wilson and R. D. Arnold 5-30 1. Thermodynamic

Temperature

Scale.

2. The

International

xviii

CONTENTS Temperature Scale. 3. Calibration of Temperature Measuring Instruments. 4- Temperature Scales below the Oxygen Point.

Chapter 4

The Equation of State and Transport Properties o f Gases and Liquids b y R. B. Bird, J. O. Hirschfelder, and C. F. Curtiss 5-42 1. The Potential Energy of Interaction between Two Molecules. 2. The Equation of State of Dilute and Moderately Dense Gases. 3. The Equation of State of Dense Gases and Liquids. 4* The Transport Coefficients of Dilute Gases. 5. The Transport Coeffi­ cients of Dense Gases and Liquids. 6. Some Applications o f the Principle of Corresponding States.

Chapter 5

Heat Transfer b y E. U. Condon

5-66

1. Heat Conductivity. 2. Equation o f Heat Conduction. 3. Simple Boundary Value Problems. 4• Cooling of Simple Bodies. 5. Point Source Solutions. 6. Periodic Temperature Change. 7. Natural Heat Convection. 8. Forced Heat Convection. 9. Condensation and Evaporation. 10. Radiative Ileat Transfer.

Chapter 6

Vacuum Technique by Andrew Guthrie

5-78

1. The Vacuum Circuit— Conductance. 2. Flow o f Gases through Tubes. 3. Pumping Speed and Evacuation Rate. 4- Vacuum Pumps. 5. Vacuum Gauges. 6. Components and Materials. 7. Leak-detection Instruments and Techniques.

Chapter 7

Surface Tension, Adsorption b y Stephen Brunaucr and L . E. Copeland 5-94 1. The Thermodynamic Theory of Capillarity. 2. The Surface Tension and Total Surface Energy of Liquids and Solids. 3. Adsorption on Liquid Surfaces. 4- Adsorption on Solid Surfaces. Physical Adsorption of Gases and Vapors. 5. Chemical Adsorp­ tion of Gases on Solids. 6 . Adsorption on Solids from Solutions.

Chapter 8

Chem ical T herm odynam ics b y Frederick 1). Rossini

5-119

1. Introduction. 2. Useful Energy; Free Energy; Criteria of Equilibrium. 3. Equilibrium Constant and Change in Free Energy for Reactions of Ideal Gases. 4- Fugacity; Standard States. 5. Solutions: Apparent and Partial Molal Properties. 6. The Ideal Solution. 7. The Dilute Real Solution. 8. Equilib­ rium Constant and the Standard Change in Free Energy. 9. Thermodynamic Calculations.

Chapter 9

Chem ical Kinetics b y Richard M . Noyes

5-140

R e s u l t s o k K i n e t i c O iis e u v a t io n s . 1. Introduction. 2. Ex­ perimental Techniques. 3. Orders of Chemical Reactions. 4Consecutive Reactions. 5. Reversible Reactions. 0. Effect of Temperature. T h e o r e t i c a l In te r p r e ta tio n of C h e m ic a l K in e t ic s . 7. Introduction. S. Collision Theory of Biinolccular Gas Reactions. 9. Collision Theory o f Unimolecular Gas Reac­ tions. 10. Statistical-Thermodynamic Theory of Reaction Kinet­ ics. 11. Theoretical Estimation of Energies of Activation. 12.

CONTENTS Consecutive

xix

13. Reactions in Solution. E l u c i d a ­ 14. Criteria for a Satisfactory Mechanism. 15. Reactions Involving Nonrcpetitive Steps. 16. Chain Reactions. 17. Branching Chains. IS. Photochemistry. 19. Heterogeneous Reactions.

tio n

Chapter 10

Reactions.

o f C h e m i c a l M e c h a n is m .

Vibrations o f Crystal Lattices and T herm odynam ic Properties o f Solids b y E. W . M on troll

5-150

1. Introduction. 2. Debye Theory of Heat Capacities. 3. Theory of Born and von Ivdrmdn. 4• Equation of State of Crystals.

Chapter 11

Superfluids b y K . R. Atkins 1. Liquid Helium.

PART

5-159

2. Superconductivity.

6 • OPTICS

Chapter 1

Electrom agnetic Waves b y E. U. Condon

6-5

I. Nature of Light. 2. States of Polarization. 3. Maxwell Field Equations. 4• Poynting Theorem. 5. Plane Waves in Isotropic Media. 6. Reflection and Refraction at a Plane Boundary. 7. Plane Waves in Anisotropic Media. 8. Optical Activity. 9. Waveguides and Transmission Lines. 10. Black-body Radiation. II . Radiation from Oscillating Charge Distribution. 12. Quan­ tization of the Radiation Field.

Chapter 2

Geom etrical Optics b y Max Ilcrzbcrger

6-20

1. Introduction. I. G E N E R A L T H E O R Y . 2. Optical Form of the General Variation Problem. 3. General Problem of Geo­ metrical Optics. 4• Characteristic Function of Hamilton. Laws of Fermat and of Malus-Dupin. Descartes’ Law of Refraction. Lagrange Bracket. II. A N A T O M Y . R a y T r a c i n g . 5. The Refraction Law. 6. Tracing a Ray through a Surface of Rotation. 7. Special Surfaces. 8. Transfer Formulas. 9. General Formu­ las. Diapoint Computation. B a s i c T o o l s o f O p t ic s . 10. The Characteristic Functions. 11. The Direct Method. L a w s o f Im a g e F o r m a t io n . 12. Image of a Point. Caustic. 13. Image of the Points of a Plane. 14. The Image of the Points of Space. 15. The Characteristic Function W for a Single Surface. 16. The Direct Method and the Addition o f Systems. III. D IA G N O ­ SIS. G a u s s ia n O p t i c s . 17. Introduction. 18. General Laws. 19. Focal Points and Nodal Points. 20. Viewing through an Instrument. 21. Distance of Conjugated Points from the Origins and Their Magnification. 22. Gaussian Brackets. 23. Expres­ sion of Basic Data of Gaussian Optics with the Help of Gaussian Brackets. 24. Vignetting. A n a l y s i s o f a G i v e n O p t i c a l S y s te m . 25. Introduction. 26. Seidel Aberrations. 27. Exten­ sion of Seidel Theory to Finite Aperture and Field. 2S. The Spot-diagram Analysis and the Diapoint Plot. IV. T H E R A P Y . 29. Correction o f an Optical System. V. P R O PH Y LA X IS. 30. Introduction. 31. Dispersion of Glass. 32. Color-corrected System of Thin Lenses. A p p e n d ix . 33. Intensity Considera­ tions. 34. Some Historical Remarks.

CONTENTS

XX

Chapter 3

Photom etry and Illum ination b y E. S. Stccb, Jr., and W . E. Forsythe 6-47 1. Visual Photometry. 2. Physical Photometry: The Spherical Integrator. 3. Photometry Spectral Response vs. Luminosity Curve. 4- Production of Light, 6. Radiant Energy. 6. Light Sources.

Chapter 4

Color Vision ami Colorimetry b y Deane 11. Judd

6-64

1. Definition of Color. 2. Types of Color Vision. 3. Tristimulus Values. 4> Theories of Color Vision. 5. Chromaticity Diagrams. 6. Photoelectric Colorimeters. 7. Colorimetry by Difference.

Chapter 5

Diffraction and Interference b y C. II. Burnett, J. G . Hirschbcrg, and J. E. M ack 6-77 1. Geometrical Optics as an Approximation. 2. General Aspects of Diffraction and Interference. 3. Diffraction. 4- Resolution and Fringe Shape. 5. Two-beam Interference. 6. Equal-amplitude Multibeam Interference. 7. Geometrically Degraded Ampli­ tude Multibcam Interference.

Chapter 6

M olecular Optics b y E. U. Condon

6-109

1. Molecular Rcfractivity. 2. Dispersion. 3. Absorption and Selective Reflection. 4- Crystalline Double Refraction. 5. Fara­ day Effect; Cotton-M outon Effect. 6. Kerr Effect. 7. Optical R ota tory Power. 8. P h otocla sticity. 9 . Flow B irefringence: Maxwell Effect. 10. Pleochroism. 11. Light Scattering.

Chapter 7

Fluorescence and Phosphorescence by J. G . W inans and E. J. Scldin

6-128

1. Introduction. 2. Fluorescence o f Gases and Vapors. 3. Gen­ eral Theory of Quenching o f Fluorescence. 4• Polarization of Resonance Radiation. 9. Stepwise Excitation o f Fluorescence in Gases. 6*. Optical Orientation of Nuclei. 7. Sensitized Fluores­ cence. 8. Selective Reflection. 9. Rcemission. 10. Fluorescence in Liquids. 11. Therinoluminesceiice. 12. Phosphorescence.

Chapter 3

Optics and Relativity Theory b y E. L. Hill

6-150

I. Introduction. 2. The Special Theory of Relativity. 3. The Transformation Formulas o f Special Relativity. 4 . The Trans­ formation Equations for Plane Waves. 5. The Dynamical Proper­ ties of Photons. G. Aberration of Light. 7. Doppler Effect. 8. The Experiment of Ives and Stilwell. 9. The MiehelsonMorley Experiment. 10. The Kennedy-Tliorndikc Experiment. II. Generalizations of the Lorcntz Transformation Group. 12. Electromagnetic Phenomena in M oving Media. 18. The Special Theory of Relativity and Quantum Mechanics. 14. The General Theory of Relativity. 16. Cosmological Problems. 10. Recent Developments.

xxi

CONTENTS

PART

7 • ATOMIC

Chapter I

PHYSICS

Atom ic Structure b y E. U. Condon

7-3

1. Nuclear Atom Model. 2. Atomic Weights. 3. Periodic Table. 4. Atomic Units. 5. Theory of Atomic Energy Levels. 6. Series. Isoeleetronic Sequences. 7. Magnetic Spin-orbit Interaction. 8. Two-electron Spectra. 9. Ionization Potentials. 10. Zeeman Effect.

Chapter 2

A tom ic Spectra, Including Zeem an and Stark Effects b y J. Rand M cN ally, Jr.

7-25

1. Introduction. 2. Spectroscopic Nomenclature. 3. Space Quantization. 4• Classical Theory of Spectra. 5. Wave Mechan­ ics. G. Interaction Energy and Fine Structure. 7. Zeeman Effect. 8. Intensity of Zeeman Components. 9. The Stark Effect. 10. Intensity of Stark Lincs.

Chapter 3

A tom ic Line Strengths b y Lawrence Aller 1. Atomic Radiation Processes. 2. Strengths. 3. Continuous Atomic Forbidden Lines. 5. The Atomic 6. Experimental Determination of / cations of the Theory.

Chapter 4

7-48

Formulas and Tables for Line Absorption Coefficients. 4• Line Absorption Coefficient. Values. 7. Tests and Appli­

Hypcrfinc Structure and A tom ic Beam M ethods b y N orm an F. Ram sey 7-53 1. Introduction. 2. Multipole Interactions. 3. Magnetic Dipole Interactions. 4• Electric Quadrupole Interaction. 5. Magnetic Octupole Interaction. 6. Optical Studies of Hyperfine Structure. 7. Atomic Beam-deflection Experiments. 8. Atomic Beam Mag­ netic Resonance Experiments. 9. Hydrogen Fine Structure. The Lamb Shift.

Chapter 5

The Infrared Spectra o f M olecules b y Iiarald II. Nielsen

7-64

1. Introduction. 2. The Energies o f a Molecule. 3. The Vibra­ tion of a Molecule. 4* The Rotational Energies of Molecules. 5. The Energy of Interaction, E*. 6. The Selection Rules for the Rotator. 7. The Interpretation o f Band Spectra. 8. The Raman Spectroscopy o f Molecules. 9. Resonance Interactions of Levels.

Chapter 6

Microwave Spectroscopy b y W alter Gordy

7-82

I. Introduction. 2. The Microwave Spectroscope. S. Micro­ wave Spectra of Free Atoms. 4• Pure Rotational Spectra. 5. Inversion Spectra. 6. Electronic Effects in Molecular Spectra. 7. Nuclear Effects in Molecular Spectra. S. Stark and Zeeman Effects in Rotational Spectra. 9. Shapes and Intensities o f M icro­ wave Absorption Lincs. 10. Electronic Magnetic Resonance in Solids.

CONTENTS

xxii Chapter 7

Electronic Structure o f Molecules b y E. U. Condon 7-700 1. Energy Levels of Diatomic Molecules. 2. Electronic Band Spectra of Diatomic Molecules. 3. Franck-Condon Principle. 4* Dissociation Energy. o. Continuous and Diffuse Spectra. Prcdissociation. 6. Hydrogen Molecule. 7. Sketch of Chemical Bond Theory. 8. Bond Energies, Lengths, and Force Constants. 9. Ionic Bonds and Dipole Moments.

Chapter 8

X Rays b y E. U. Condon

7-770

1. Main Phenomena. 2. Emission: Continuous Spectrum. 3. Emission: Characteristic Line Spectrum. 4* Absorption. 5. Angular Distribution of Photoelectrons. 6. Intensity Measure­ ment. 7. Internal Conversion: Auger Effect. S. Pair Produc­ tion. .9. Coherent Scattering. 10. Incoherent Scattering: Comp­ ton Effect.

Chapter 9

Mass Spectroscopy and Ionization Processes by John A. Hippie

7-131

1. Introduction. 2. Study of Ionization Processes. 3. Ionization of Atoms by Electron Impact. 4* Diatomic Molecules. 5. Poly­ atomic Molecules. 6. Analysis.

Chapter 10

F undam ental Constants o f A tom ic Physics by Jesse W . M . DuM ond and E. Richard Colicn 7-143 1. The Group Known as the Atomic Constants. 2. The Pioneer Work and Methods of It. T. Birgc and Others Prior to 1049. 3. Data of Greatly Increased Accuracy Subsequent to 1949. 4• Con­ sistency Diagrams and Graphic M ethods: The Ellipsoid o f Error. 5. The Method of Least Squares. 6. Calculation of Standard Errors and Correlation Coefficients. 7. Rejection o f Certain Input Data in the Present Least-squarcs Adjustment. 8. Choice o f the Unknowns and the Primitive Observational Equations. 9. The Auxiliary Constants and Equations. 10. Formation o f the Linearized Equations o f Observation in Five Variables. 11. The Least-squares Solution. 12. Illustrative Example o f Computa­ tion of the Standard Deviation of a Function of Tabular Values Obtained in the Present Least-squares Analysis. 13. Discussion of the Results. 14. Variance Analysis. 1955 Adjustments. 15. Recent Developments (1958).

PART

8 • TI1E

Chapter 1

SOLID

STATE

Crystallography and X -r a y Diffraction b y R. Pcpinsky and V. Vand

8-3

1. Classical Crystallography. 2. X -ray Diffraction: Experimen­ tal. 3. Theory of X -ray Scattering. 4» Fourier Transforms. 5. The Phase Problem.

Chapter 2

The Energy-hand Theory o f Solids b y Herbert lb Cullen

8-24

1. The Born-O ppenheim er A pproxim ation . 2. Determ inantal Wave Functions and the Ilartree-Foek Equations. 3. The Fermi

CONTENTS

xxiii

Hole and the Exchange Term. 4* The Consequences of Sym­ metry. 5. Properties of Bloch Functions. 6. Some Qualitative Comments. 7. Momentum Eigenfunctions. S. The Wannier Function. 9. Perturbations of Periodicity. 10. Techniques of Calculation.

Chapter 3

Ionic Crystals b y R. W . Gurney

8-43

1. The Perfect Ionic Lattice. The Cohesive Energy. 2. The Born-IIaber Cycle. 3. Dielectric Constant. 4♦ Electronic En­ ergy Levels. 5. Positive Holes. 6. Excited Electronic States of a Crystal. 7. Lattice Imperfections. Schottky Defects. S. Frenkel Defects. 9. Ionic Conductivity. 10. M obility of Lat­ tice Defects. 11. Crystals with Nonstoichiometric Composition. 12. Trapped Electrons and Positive Holes. 13. The F band and the V band. 14. Photoconductivity. 15. Crystals Containing F Centers. 16. Dielectric Breakdown in Ionic Crystals. 17. Ionic Crystals in Photographic Emulsions.

Chapter 4

Flow o f Electrons and Holes in Sem iconductors b y John Bardeen 8-52 1. Introduction. 2. Basic Equations. S. Examples of Flow. Space-charge Layers and Metal-Semiconductor Contacts.

Chapter 5

Photoelectric Effect by R. J. M aurer

4-

8-66

1. General Considerations. 2. The Spectral Distribution Func­ tion. 3. The Energy Distribution Function. 4• Semiconductors and Insulators.

Chapter 6

Therm ionic Em ission by Lloyd P. Sm ith 1. Uniform Pure Metal Crystals. 2. Polycrystalline 3. Metals with Adsorbed Monolayers.

Chapter 7

Glass by II. R. Lillie

8-74 Metals.

8-83

1. Definition. 2. Glass Types. 3. Glass rium Phases. 5. Attainment of the Vitreous State. 6. Rates of Crystal Growth. 7. The Transformation. S. Viscositv-Temperature Relations. 9. Equations for Viscosity Variations. 10. Stress Release and Annealing. 11. Optical Properties. 12. Electrical Properties. 13. Thermal Properties. 14. Mechanical Properties. 15. Radiation Absorption. 16. Glass Sealing.

Chapter 8

Phase Transform ations in Solids b y R. Snioluchowski

8-108

1. Classical Phase Transformations. 2. Transformations of Higher Order. 3. Order-disorder Theory. 4* Orientational Transitions. 5. Nucléation and Growth. 6. Shear Transformations. 7. Rate of Ordering. S. Crystallographic Factors Affecting Transforma­ tion Rate.

PART

9 • NUCLEAR

Chapter 1

PHYSICS

General Principles o f Nuclear Structure b y Leonard Eiscnbud and Eugene P. W igncr

9-4

I. G E N E R A L FE A TU R E S OF NU CLEI. 1. Nuclear Composi­ tion. 2. Nuclear Masses: Binding Energies. 3. Types of Nuclear

xxiv

CONTENTS ^Instability. Spontaneous and Induced Transformations. II. SY S T E M A T IC S OF STABLE NUCLEI. D e t a i l s o f B in d in g en ergy S u r fa c e s . III. PR O P E R T IE S OF N U C LEA R ST A T E S : G r o u n d S t a t e s . 1. The Size of the Nuclei. IV. SUR­ V E Y OF N U C LEA R REACTIO N S. 1. Types of Reaction, Cross Sections, Excitation Functions. 2, Resonance Processes. 3. Direct Processes. 4- Table of Most Important Reactions. V. T W O B O D Y SY STEM S: I n t e r a c t i o n s b e t w e e n N u c l e o n s . 1. Internucleon Forces. 2. Saturation Properties and Internucleon Forces. 3. Charge Independence of Nuclear Forces: The Isotopic or Isobaric Spin. Quantum Number. VI. N U C LEA R M ODELS. I. T h e U n i f o r m M o d e l . I . General Remarks. 2. Powder and Shell Models. S. Supermultiplet Theory. V II. N U C LEA R M ODELS. II. I n d e p e n d e n t P a r t i c l e M o d e l s . I. General Features of the Independent Particle or Shell Models. 2. The L-S Coupling Shell Model. 3. Comparison of the L-S and j-j Shell Models. 4• The j - j Coupling Shell Model. 5. Coupling Rules for the j - j Model. 6. Normal States and Low Excited States. 7. Magnetic and Quadrupolc Moments. 8. Problems of the j - j Model. V III. N U C LEA R M ODELS. III. M a n y - p a r t i c l e M o d e l s . 1. The «-particle Model. 2. Collective M odel. 3. Comparison of the j - j and the Collective Models. IX . N U C LE A R R E A C TIO N S. 1. C l o s e C o l l i s i o n s . I. The Collision Matrix. 2. Qualitative Discussion of Resonance Phenomena. S. Derivation of the Resonance Formula. 4• Dependence of the Parameters on the Size of the Internal Region. 5. Radioactivity. 6. The Clouded Crystal-ball Model. 7. The Intermediate Coupling or Giant Resonance Model. X . N U C LEA R RE A C TIO N S. II. S u r f a c e R e a c tio n s . 1. Angular Distribution in Stripping Reactions. 2. Electric Excitation. X I. IN T E R A C T IO N W IT H ELEC­ T R O N -N E U T R IN O FIELDS. 1. Theory of 0 Decay. 2. Allowed and Forbidden Transitions. 3. Shape of the Spectrum. 4• Total Transition Probability. X II. E L E C T R O M A G N E T IC T R A N S I­ TIO N S IN C O M PLE X NUCLEI. /. Introduction. 2. Radi­ ative Transitions. 3. Single-particle Matrix Elements.

Chapter 2

M easurem ent o f Nuclear Masses b y W alter II, Johnson, Jr., Karl S. Qiiiscnberry, and A. O. Nier 9-55 1. Nuclear Transformations and Atomic Masses. 2. Atomic Masses from Mass Spectroscopy. 3. Calculations of Atomic Masses. 4• The Atomic Mass Table. 5. Nucleon Bindingencrgy Systematics.

Chapter 3

Nuclear M om ents b y N orm an F. Ramsey

9-63

1. Introduction. 2. Optical Spectroscopy. 3. Molecular Beam Experiments. 4• Nuclear Paramagnetic Resonance Experiments. 5. Microwave Spectroscopy and Paramagnetic; Resonance Experi­ ments. 3. Results of Nuclear Moment Measurements.

Chapter 4

Alpha Particles and Alpha Radioactivity by W illiam W . Stephens and Theodor H urlim ann

9-72

L Alpha Particles. 2. Passage of Alpha Particles through Matter. 3. Scattering of Alpha Particles. 4- Alpha-partiele Radioactivity.

xxv

CONTENTS

Chapter 5

Beta Radioactivity b y M . E. Rose

9-90

1. Decay Processes. 2. Formulation of the Beta Interaction (Classical). 3. Selection Rules and Transition Probabilities. 4» Energy Spectra and Angular Correlation. 5. Symmetry Opera­ tions in /3 Decay. 6. Breakdown o f the Conservation of Sym­ metry in /3 Decay. 7. Evaluation o f the Coupling Constants. 3. Recent Theoretical Developments. 9. Meson Decay.

Chapter 6

Nuclear Electrom agnetic Radiations b y R. W . Hayward

9-106

1. Introduction. 2. Direct Nuclear Transitions. 3. Other Phe­ nomena Involving the Nuclear Electromagnetic Field. 4• Inter­ action of Gamma Rays with Matter. 5. Experimental Detection of Nuclear Gamma Rays.

Chapter 7

Neutron Physics b y C. O. M uelilhause

9-125

1. Fundamental Properties. 2. Interactions with Individual Nuclei. 3. Interactions with Unordcred Matter. 4• Interactions with Ordered Matter. 5. Interactions with Fundamental Par­ ticles.

Chapter 8

Nuclear Reactions b y David Halliday

9-139

1. Introduction. 2. Energetics. 3. Experimental Determination of Q. 4• Centcr-of-mass Coordinates. 5. Cross Section. 6. Method of Partial Waves. 7. Elastic Scattering Cross Sections. 8. The Reaction Cross Section. 9. The Compound Nucleus. 10. Nuclear Resonances. 11. Nuclear Resonances— Theory. 12. The Statistical Model. 13. The Optical Model.

Chapter 9

Acceleration o f Charged Particles to High Energies b y John P. Blcwett 9-153 1. Introduction. 2. The Cockcroft-W alton Accelerator. 3. The Van De Graafif Electrostatic Generator. 4• The Betatron. 5. Principles of Synchronous Accelerators. 6. The Linear Accelera­ tor. 7. The Cyclotron and the Synchrocyclotron. 8. The Electron Synchrotron. 9. The Proton Synchrotron. 10. Strong Focusing Principle. 11. Application of Strong Focusing to Accel­ erators. 12. Conclusion.

Chapter 10

C loud-eham hcr and Em ulsion Technique b y Robert R. Brown and Lawrence S. G erm ain

9-167

A. C l o u d - c h a m b e r T e c h n i q u e . 1. Drop Formation. 2. Sensi­ tive Time. 3. Construction and Operation. 4• Illumination and Photographic Arrangements. 5. Measurements. B. E m u ls io n T e ch n iq u e . 6. Types of Emulsion. 7. Processing the Emul­ sions. 8. Protecting the Emulsion. 9. Examining the Emulsion. 10. Measurements Made in the Emulsion.

Chapter 11

Fission by John Archibald Wheeler

9-177

1. Survey of Fission. 2. The Compound Nucleus and Models of Nuclear Structure. 3. Fission and the Unified Nuclear Model. 4. The Fission Chain Reaction.

CONTENTS

xxvi Chapter 12

Cosm ic Rays b y B. Peters

9-201

I n tr o d u c tio n . 1. Brief History of Cosmic-ray Research. 2. Schematic Outline of the Principal Cosmic-ray Phenomena Occur­ ring in the Atmosphere. P r im a r y C o s m i c - r a y P a r t i c l e s . 3. The Relative Abundance of Various Primary Nuclei. 4• The Influence of the Earth’s Magnetic Field. Geomagnetic Theory. 5. Primary Intensity and Energy Spectrum. S e c o n d a r y C o s m ic r a y P a r tic le s . 6. The n Meson. 7. The Charged x Meson. 8. The Neutral x Meson. 9. Heavy Mesons and Hyperons. N u c le a r C o llis io n s . 10. The Process of Star Formation. 11. Identification of Secondary Particles and Their Production Spec­ trum. 12. Multiplicity and Angular Distribution of Mesons Pro­ duced in Nuclear Collisions. IS. The Interaction Mean Free Path for Nucleons. 14. The Interaction Cross Section of Heavy Primary Nuclei. D e v e l o p m e n t o f t h e N u c l e o n i c C a s c a d e in t h e A tm o sp h e re . 15. High-energy Protons and Neutrons in the Atmosphere. 16. Low-energy Nucleons. 17. The Slow-neutron Component. T h e E l e c t r o n i c C o m p o n e n t . IS. The Develop­ ment of Electronic Cascades. A l t i t u d e V a r i a t i o n . 19. Inten­ sity Variation of Various Cosmic-ray Components with At­ mospheric Depth. C o s m ic R a d i a t i o n b e l o w G r o u n d . 20. Composition of Underground Radiation. 21. The Energy Spec­ trum of n Mesons Below Ground. 22. Meson Showers Under­ ground. 23. Extensive Air Showers. V a r i a t i o n s o f C o s m ic r a y I n t e n s i t y in T im e . 24. Periodic Variations. 25. Non­ periodic Variations. 26. Problems Connected with the Origin of Cosmic Rays.

Chapter 13

M eson Physics b y Alan M . Thorndike 1. Introduction.

9-245

T y p es o f M eson s and H y p e ro n s an d T h e ir

2. n Mesons. 3. x Mesons. 4• K Mes­ ons. 5. Hyperons. 6. Antiprotons. P r o d u c t i o n o f M e s o n s . 7. Production of x Mesons. 8. Production of Heavy Mesons and Hyperons. N u c l e a r I n t e r a c t i o n s o f M e s o n s . 9. Nuclear Absorption of Stopped Mesons. 10. Nuclear Interactions in Flight. D ecay

S ch em es.

Units anilConversion Factors Index

1 15

Pari 1

Chapter 1 1. 2.

1. 2. 3. 4. 5. 6.

1. 2. 3. 4. 5. 6.

1. 2. 3. 4. 5.

1-7

1-10

1-13 1-14 1-15

1-12

9. M atrices..................................................... 10. Determinants.............................................. 1 1 . Systems of Linear Equations.................. 12. Numerical Methods for Finding the In­ verse of a Matrix and for Solving Sys­ tems of Linear Equations........................ 13. Characteristic Roots of Matrices and Quadratic Form s........................................ 14. Computation of Characteristic Roots of M atrices....................................................... 15. Functions of Matrices and Infinite Sequences..................................................... 16. Hypercomplex Systems or Algebras. . . 17. Theory of Groups......................................

1-22 1-25 1-29 1-31 1-36 1-37

7. 8. 9. 10. 11. 12.

1-39 1-40 1-42 1-46 1-49 1-52

1-10 1-11 1-11 1-11 1-11 1-12

1-15 1-16 1-17 1-18 1-18 1-19

Complex Num bers.......................... Series of Functions......................... Functions of a Complex Variable. Conformal M apping....................... Orthogonality................................... Special Functions............................

7. Nonhomogcncous Equations, Green’s Function...................................................... 1-63 8. Numerical Integration of Differential Equations.................................................... 1-64 9. Systems of Simultaneous Differential Equations.................................................... 1-64

1-59 1-59 1-60 1-61 1-61 1-62

Partial Differential Equations by Fritz John

General Properties..................................... First-order Equations............................... Elliptic Equations..................................... Parabolic Equations of Second O rd e r.. Hyperbolic Equations in Tw o Inde-

Chapter 6

Digital Computing Machines.

Ordinary Differential Equations by Olga Taussky

Introduction................................................ Simple Cases............................................... Existence Theorem s.................................. Methods for Solution................................ Examples of Well-known Equations. . . Some General Theorem s..........................

Chapter 5

3.

Analysis by John Todd

Real Numbers, Lim its............................. Real Functions......................................... Finite Differences..................................... Integration................................................. Integral Transform s................................. Functions of Several Real V ariables...

Chapter 4

1-4 1-6

Algebra b y Olga Taussky

1 . Polynom ials.. . . ........................................ 2. Algebraic Equations in One Unknown, Complex Numbers..................................... 3. Equations of Degree 2 (Quadratic Equa­ tions)............................................................. 4. Equations of Degree 3 (Cubic Equa­ tions)............................................................. 5. Equations of Degree 4 (Biquadratic Equations)................................................... 6. Equations of Degree n .......................... 7. Discriminants and General Symmetric Functions..................................................... 8. Computational Methods for Obtaining Roots of Algebraic E quations................

Chapter 3

Mathematics

A rithm etic b y Franz L. Alt

Numbers and Arithmetic Operations.. Logical Foundation of Arithm etic

Chapter 2

•

pendent Variables...................................... 1-79 6. Hyperbolic Equations with More than Tw o Independent Variables................... 1-84 7. Numerical Solution of Partial Differ­ ential Equations........................................ 1-86

1-66 1-69 1-72 1-76

Integral Equations b y M . Abraniowitz 4. Integral Equations of the First K in d . . 1-93 5. Volterra’s Equation................................... 1-93 6. Nonlinear Integral E quation.................. 1-94

1. Integral Equations of the Second K ind. 1-90 2. Symmetric Kernels.................................... 1-90 3. Nonsymmetric Kernels............................ 1-92 1-1

1 -2

MATHEMATICS

Chapter 7

Operators by Olga Taussky

1 . Vector Spaces, Abstract Hilbert Spaces, Hilbert Space.............................................. 2. Definition of Operator or Transforma-

Chapter 8 1. 2. 3. 4. 5. 6. 7.

1-97 1-97 1-97 1-98 1-98 1-98 1-99

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

Euclidean Group and Plane.................... C onics........................................................... Angles........................................................... Triangles...................................................... P olygons....................................................... Hyperbolic Group and Plane.................. Elliptic Group and Plane.........................

1-99 1-89 1-100 1-100 1-101 1-101 1-101

6. 7. 8. 9.

Curl of a Vector Field.............................. Expansion Formulas.................................. Orthogonal Curvilinear Coordinates. .. Transformation of Curvilinear Coordi­ nates..............................................................

1-107 1-108 1-108 1-109

1-111 1-111 1-111 1-112 1-112 1-113 1-113 1-113 1-113 1-115 1-115 1-115 1-115 1-116 1-117 1-118 1-118

18. Differentiation of a T ensor...................... 1-118 19. Co variant Derivative of the Metrical T en sor........................................................... 1-119 20. Principles of Special and General Rela­ tiv ity ............................................................. 1-119 2 1 . Curvilinear Transformations................. 1-119 22. Covariant Derivative of a T en sor 1-119 23. Covariant Derivative of the Metrical Ten sor........................................................... 1-120 24. Fundamental Differential Invariants and Covariants of MathematicalPhysics 1-120 25. Maxwell Electromagnetic E quation s... 1-121 26. Curvature Tensor of Riem ann.............. 1-121 27. Properties of Riemann T en sor.............. 1-121 28. Contracted Curvature T en sor............... 1-121 29. The Matter Tensor of Einstein............ 1-122 30. Einstein’s Theory of G ra vity ................ 1-122

Calculus o f Variations by C. B. T om p kins

1 . Maxima and Minima of a Function of a Single Variable........................................... 2. Minima of a Function of Several Variables...................................................... 3. Minima of a Definite Integral— the Euler Equations......................................... 4. Examples...................................................... 5. Other First Variations: Weierstrass Condition, Corner Conditions, One-side

Chapter 12

8. 9. 10. 11. 12. 13. 14.

Tensor Calculus b y C. Lanezos

Scalars, Vectors, Tensors......................... Analytic Operations with V ectors Unit Vectors; Com ponents...................... Adjoint Set of A xes................................... Covariant and Contravariant Com­ ponents of a V ector................................... Transformation of the Basic Vectors Vt* Transformation of Vector Components Radius Vector R ........................................ Abstract Definition of a V ector............. Invariants and Covariants...................... Abstract Definition of a Tensor............. Tensors of Second O rder.......................... Einstein Sum Convention........................ Tensor Algebra........................................... Determinant T ensor.................................. Dual Tensor................................................ Tensor Fields..............................................

Chapter II

tion ................................................................ 1-95 Spectrum of Bounded Operators, Eigen­ values, and Eigenfunctions...................... 1-96

Vector Analysis by E. U. Condon

1 . Addition of Vectors.................................... 1-103 2. Scalar and Vector Products.................... 1-104 3. Vectors and Tensors in Oblique Coordi­ nates.............................................................. 1-104 4. Gradient of Scalar and Vector Fields 1-106 5. Divergence of a Vector Field.................. 1-107

Chapter 10

3.

Geom etry by A. J. Hoffm an

Definition and Assumptions.................... Projective Plane......................................... Projective G roup....................................... Correlations, Polarities, and Conics. .. Projective L ine........................................... Subgroups of the Projective G ro u p .. . . Affine Group and Plane...........................

Chapter 9

1-95

1-123 1-123 1-124 I-125

Variations.................................................... 6. Parametric Problem s................................ 7. Problems with Variable End P oin ts. . . 8. Isopcrimctric Problems— the Problem of Bolza........................................................ 9. Second Variations...................................... 10. Multiple-integral Problems..................... 11. Methods of C om putation........................ 12. Conclusion

1-127 1-128 1-129 1-129 1-130 1-131 1-132 1-133

Elem ents o f Prohuhility b y Ghureliill Eisenhart ami Marvin Zelen

1 . Probability................................................... 1-134 2. Random Variables and Distribution

3.

Functions.................................... Distributions in n Dimensions

1-135 1-139

MATHEMATICS 4. Expected Values, Moments, Correla­ tion, Covariance, and Inequalities on D istributions............................................... 1-140 5. Measure of Location, Dispersion, Skew­ ness, and K urtosis..................................... 1-141 6. Characteristic Functions and Generat-

Chapter 13

ing Functions.............................................. 7. Limit Theorem s......................................... 8. The Normal D istribution........................ 9. Discrete Distributions.............................. 10. Sampling Distributions..........................

Statistical Design o f Experiments hy W . J. Y ouden

1 -3 1-143 1-147 1-149 1-151 1-161

1-165

Chapter 1 Arithmetic By F R A N Z L. ALT, National Bureau of Standards

Instead of the decimal system, in which powers of 10 play a fundamental role, systems based on other integers arc occasionally used. Thus a system using the base b contains b digits, whose values arc 0, 1,2, . . . ,5 — 1. In particular, the binary system, with base 2, has only 0 and 1 for digits. A number like 101.1001 is understood to mean

1. N u m b ers and A r ith m e tic O p era tion s N u m b ers. It is possible to define numbers and prove statements about them without specific recourse to experience or intuition, using only a few simple con­ cepts of logic. For this reason mathematical state­ ments are considered infallible except for demonstra­ ble errors in reasoning, while statements in the physical sciences are subject to empirical verification or revision. Such a definition of numbers, together with a list of their fundamental properties which can be proved logically on the basis of this definition, is given in Sec. 2. For convenience, numbers are written in the decimal system. In this system, ten integers arc represented by special symbols, called digits: 0, 1,2,3,4,5,6,7,8,9. Any positive integer greater than 9 is represented by a group of digits

1 X 2 H 0 X 2 1 + 1X2» + 1 X 2 -1 + o X 2-2 + 0 X 2- 3 + 1 X 2- 4 = 4 + 1 + H + y l 6 = 5.5625 The binary system is used in some types of computing machines, as arc the systems based on 4, 8, and 16. The duodecimal system (base 12) has a certain his­ torical importance. Other number systems are used only infrequently. A d d itio n a n d S u b tr a c tio n . The simplest arith­ metic operation is addition, a + b. The two numbers on which the operation is carried out are called terms; if they arc to be distinguished, the first is called augend, the second addend (or occasionally auctor). The result of the operation is called the sum. Because of the associative law of addition, according to which a + (b + c) = (a + b) + c (sec Sec. 2), it is per­ missible to write sums of more than two terms:

dndn i • • • d^dido where 0 < di < 9 for i = 0,1,2, . . . ,n. is understood to represent the number = 0

whose roots are the 2 m powers of those of the original equation. If there is one root a of the original equa­ tion whose modulus exceeds that of all the rest, say, M > |j3| ^ 1?! ^ • • • , then for comparatively small values of m (m )

■1

(m)

and a can be obtained by taking logarithms. 9. Matrices An n X m matrix A — [a**], with i = 1, • • • ,n and k — 1, • • • ,m, is an array of numbers arranged in n rows and m columns:

«21

«12 a 22

* -’ • • '* •

«

, a ni

««2

• 1’ *

«M W _

an

«1TO 2m

Certain operations arc defined for such arrays: if another matrix B — [6tft] has the same number of rows and columns, then the sum of the two matrices is A + B — [dik + &»*]

f(x ) = A$(x — a ) 3 + A\(x — a )2 + A 2(x — a) + A 3

1

1 -13

x 3 - 7a;2 + 11* - 3 = 0 has a root between 0 and 1

The product of a number r and a matrix A — [at-*] gives the matrix r A = [ra,-*]. If [&»*] is a matrix with m rows and p columns, then the product A B of [a**] and [bik] is the n X p matrix [c«], where Cik — dubiic +

+ dinbnk

In general the product A B differs from the product B A . If the rows and columns of a matrix are inter­ changed, the resulting matrix is called the transpose A ' of A. The transposed matrix of a product A B is B 'A '. Matrices are clearly connected with linear trans­ formations of variables or linear substitutions, and it is from them that the product definition originates. Of particular usefulness are the square matrices for which m = n. There the elements a«, 1 = 1, • • • tnf form the principal or main diagonal; their sum is called the trace of the matrix. If a square matrix coincides with its transpose, it is called symmetric; if it

MATHEMATICS

1 -1 4

coincides with the complex conjugate of its transpose, it is called Hcrmitian; if it coincides with its negative transpose, it is called skew or antisymmetric. The square matrix of n rows with ones in the principal diagonal and zeros everywhere else is called the unit matrix or A matrix all of whose elements are zeros is called a zero matrix. A square matrix A of n rows for which a matrix B exists such that A B = / » is called nonsingular; otherwise it is called singular. The matrix B is unique, is called the inverse matrix of A y and is usually denoted by A ~ l. It is also easily seen that A “ 1A — I n. In general, A B 7* BA for arbitrary matrices A and B; however, for any two matrices A,Z? we have trace AB = trace B A . Both this property and the fact that ( A “ 1)“ 1 — A do not hold in general for infinite matrices. Also the fact that the inverse matrix is unique is in general not true for infinite matrices. Explicit expressions for the inverse of a nonsingular matrix will be mentioned later (Sec. 12); these expres­ sions are, however, of little use for the computation of the inverse of numerical matrices. An orthogonal matrix is a real matrix whose transpose coincides with the inverse; a unitary matrix is one for which the complex conjugate transpose coincides with the inverse. Using the product definition for general matrices, a system of linear equations «nXi +

• • • + a\mx m = hi

a2\Xi +

• • • +

«nlXl +

• • * + QnmXm = bn

a2mx m = ¿>2

(2 .1 2 )

can be written in the abbreviated form Ax = b

(2.13)

where x stands for the column vector (xi, . . . ,x m) and b for (bi, . . . ybn). If n = m and the matrix A is nonsingular, the solution of the system can be written symbolically as x = A “ hi

(2.14)

If it is desirable to transform the unknowns x» in (2.13) to a set of unknowns y = i/» which arc con­ nected with the Xi by the transformation y — Bx, where B is again a nonsingular n X n matrix, then it appears that the system (2.13) is equivalent with the system B A B -h j = Bu The matrix A is said to have been transformed by B and B A B ~ l is also called similar to A. Similar matrices have the same traces. Another very impor­ tant concept, which is invariant under transforma­ tions, is the n characteristic roots or eigenvalues or proper values of the matrix (see Sec. 13).

numbers 1, . . . ,n and the sign ± is chosen accord­ ingly as the permutation is even or odd. The value of the determinant is rarely computed from this expression, but by using some of the properties of determinants: (1) A matrix and its transpose have the same determinants. (2) If all numbers in a fixed row (or column) are multiplied by the same number, the determinant is multiplied by that number too. (3) If a multiple (by the same number) of the elements of a row (or column) is added to another row (or column), the value of the determinant is unchanged. (4) Denote by Aik the value of the subdeterminant of the matrix obtained from A by omitting the ith row and the Ath column, multiplied by ( — l ) i+*. The sub­ determinant alone is called a minor; A»-* is called the eofactor of the element a»*. The determinant can be expressed in the form |«»*| = an A n + au A u +

0 = an A h -|- anAkz - { - • • •

This is, however, not generally of much use for the computation of the inverse of a numerical matrix (see Sec. 12). (5) The determinant of the product of two matrices is equal to the product of the deter­ minants of the two matrices. (6) From the definition of the determinant it is evident that a matrix which has a row or a column of zeros has a vanishing deter­ minant. Further, the determinant vanishes if there is a linear dependence between the rows or the columns of the matrix, that is, if numbers on, . . . fa n exist sueh that not all oa — 0 and ttlflit + Of2«2t + • • • + otnaHi = 0 or nrjaii + « 2« i2 * * * 4"

2/

i = 1, . . . fn 0

A square matrix is singular if and only if its deter­ minant vanishes. Although much work has been done on properties of special determinants, only three will be mentioned. Let xi, . . . ,x„ be n unknowns; then the correspond­ ing Vandermonde determinant is |«a|, where a»* = Xik~ x. The value of this determinant is |1

_ Xt)

i m, such a system has a solution only if there is a linear dependency between the equa­ tions so that certain of them are a consequence of the others. The case n — m has been discussed. If n < m and the system is homogeneous, it will always have solutions; if it is inhomogeneous, only if other conditions are fulfilled. Any two solutions of an inhomogeneous system differ by a solution of the corresponding homogeneous system. 12. Numerical Methods for Finding the Inverse of a Matrix and for Solving Systems of Linear Equations A practical method for the solution of a system of linear equations is the elimination process. Consider the system x + 2y + 32 + Aw 2x + 3y + 4z + 5w 3x + Ay + 4z + bw Ax + 5y + 5z + 8w Subtract multiples of the first equation from the other three so as to eliminate x in each of these, giving = = =

Now subtract multiples of (2.20) from (2.21) and (2.22) to eliminate y: -z - w = -7 -z + w = 1

(2.23) (2.24)

Now eliminate z by subtracting (2.23) from (2.24): 2w = 8

10 — 4

Substitute in (2.23) to get 2 = 3, then in(2.20) to obtain y = 2, and finally in (2.16) to obtain x = 1. This method applies whether the system of equa­ tions is symmetric or not. It is important, in prac­ tice, to reorder the equations and the variables, if necessary, in order to ensure that the “ pivotal coef­ ficients” are not too small. The pivotal coefficients are those of x in (2.16), of y in (2.20), of z in (2.23). Checks on the numerical work have been devised; they are discussed in books on numerical analysis. Essentially equivalent to this process is one due to Cholesky. Given a symmetric matrix A, it is pos­ sible to determine a lower triangular matrix L such that A - LV A lower triangular matrix is one for which a » = 0 for i < k. Observe that A ~ l = (L U )~1 = (L ')- 1L-1 = (L -O 'L " 1

Xi = • • • = x n = 0

- y - 2z - Zw —2y — hz — 7w - Z y - 7z - Sw

1 -1 5

- 2 0 (2.20) —47 (2.21) - 5 9 (2.22)

Thus the problem of inversion of A is essentially reduced to that of L, a very much simpler problem. L" 1 is itself another lower triangular matrix. It is not necessary to go through the whole process of inversion of A for the solution of Ax = b. Replace A x = 6 by LL'x = bf and replace this by Ly — b

and

L'x = y

which is solved first for the vector y and then for the vector x. As an example consider the case of the inversion of A and the solution of A x = 6, where

2 A = r 21 3 L3 4

3i 4 4J

14i b = r20 L23J

Assume 0 01 b c 0 Id e /J fa A —

fa

0 L0

b dl c e 0 /J

Equate elements in the first row and obtain a2 = 1 , ab — 2, ad — Z from which, assuming a = 1, we find b = 2, d — 3. On equating elements in the second row, 52 c 2 _ 3 anci bd Ar Ce — 4, which gives c — -H , say, and e = 2i. Finally, we obtain d2 + e2 + f 2 = 4, giving / = -K , say. - 30 (2.16) The important point is that there have been no = 40(2.17) simultaneous equations to solve. The inversion of L = 43be (2.18) can accomplished in a similar way. Assume = 61(2.19) 1 0 0*1 f a 0 O'] f l 0 01 2 i0 6 r 0 = 0 1 0 3 2i iJ I d e /J LO 0 lJ We find successively a = 1, 6 = 2i, d = —i , c — —i, e — 2i, f = —i. Hence

MATHEMATICS

1 -1 6 A -i =

ri 0

2i -¿ H f 1 - i 2i\\ 2 i

L0

0 — i J L —i

-n

-I

:

0 —i

It follows that the vector x is a solution system of the homogeneous system of equations

0*1 0

2i — iJ

(A — A /n)x = 0

a

W c can find j- from this as

or we can use the alternative method already out­ lined, which wc now follow in detail. Having obtained L, wc consider the equation Ly = b, that is,

1 2 3

î

2lii

“ 1

r - 1

U ly j

-

r - i

L23J

and obtain successively //1 = 14, y% — Si, yz = 3i. Wc then consider L'x = y, that is, r 1

0

2

Lü

0

i

2i

x2

iJ Ix zl

=

r 14i

8/

L 3/J

and obtain successively j*3 = 3, x 2 = 2, X\ 1 as before. Iterative processes can be applied to obtain ap­ proximately the inverse of a matrix or the solution of a system of linear equations. Of these wc mention the so-called relaxation process and the process indi­ cated by A'„+i = A'„(2 — A X n) which, when it con­ verges, converges to A -1. It is not possible to dis­ cuss these in detail here. In general, the methods already described are preferable unless a good approxi­ mation is already known or can be found easily. A good approximation can be easily obtained, for instance, when the matrix is nearly diagonal or nearly triangular. Gauss used a method for solving a system of linear equations with a symmetric matrix which is substan­ tially equivalent to the reduction of a quadratic form to a sum of squares (see also Sec. 13). His method is applied to the solving of normal equations which arc obtained from a system A x — b by going over to A 1A x = A'h] the matrix A here is assumed to be an n X m matrix with n < m. The normal equations arc derived when the least-sqnare method is applied to the given system. 13. Cliarneleristie lloots of Matrices and Quadratic Forms Like the trace of a matrix its determinant too is invariant, under transformation (see Sec. 9); both facts follow from the invariance of the characteristic roots. These numbers X|, . . . , A„ have the properly that A x — Xtx

(2.25)

for a suitable vector x (different from the null vector), called the modal vector or characteristic vector that belongs to A,.

Hence the determinant |A — A /w| = 0. This is an algebraic equation of degree n in X, the characteristic equation of A, and this implies that there are n (not necessarily different) values A, (as stated above). If the coefficients of the powers of X arc investigated, Xn_1 has as coefficient the trace of A and the constant term is |