Intense Electron and Ion Beams S.I

Intense Electron and Ion Beams S.I. Molokovsky A.D. Sushkov Intense Electron and Ion Beams

With 140 Figures

123 Professor Dr. Sc. Sergey Ivanovich Molokovsky Professor Dr. Sc. Aleksandr Danilovich Sushkov State Electrotechnical University, Electronics Department Prof. Popov street 5, 197376 Saint Petersburg, Russia [email protected], [email protected]

Library of Congress Control Number: 2004117893

ISBN-10 3-540-24220-1 Springer Berlin Heidelberg NewYork ISBN-13 978-3-540-24220-1 Springer Berlin Heidelberg NewYork This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broad- casting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media. springeronline.com © Springer-Verlag Berlin Heidelberg 2005 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant pro- tective laws and regulations and therefore free for general use.

Typesetting and prodcution: PTP-Berlin, Protago-TEX-Production GmbH, Berlin Cover design: design & production GmbH, Heidelberg Printed on acid-free paper SPIN 10967867 57/3141/YU 543210 Preface

Intense charged-particle beams currently have many diﬀerent applications. High-current electron beams are used in various microwave tubes: O-type tubes (klystrons, travelling wave tubes, etc.), M-type tubes, gyrotrons, and free-electron lasers. High-current electron and ion beams are applied as tools in the installations of charged-particle beam technology. High-current electron and ion accelerators have been developed for industrial applications and physical experiments. This book is devoted to intense charged-particle beams used in vacuum tubes, particle beam technology, and installations for physical experiments (free-electron lasers, accelerators, etc.). The following scope of topics is con- sideredinthisbook:

• Physics and basic theory of intense charged-particle beams • Methods of electric and magnetic field computation as applied to the development of the systems used for the charged-particle beam formation and focusing • Methods of analytic and numerical analysis of charged-particle beam motion in self-consistent fields • Computation and design of the charged-particle guns for initial formation of pencil, hollow and strip beams Computation and design of the systems for focusing (transport) beams by different types of electric and magnetic fields • Computation and design of the systems for formation of charged-particle probes used in charged-particle beam technology • Multiple beam charged-particle systems, peculiarities of their design, and computation • Peculiarities of relativistic beam motion and focusing • Methods of experimental investigation of intense particle beams

Therefore, this book contains both the physics and theory of intense charged- particle beams and the design of their optical systems for beam formation and focusing. The methods of successive approximation, as well as the method of synthesis, are considered. The corresponding algorithms, formulas, and graphics are presented. The book reﬂects the authors’ experience in the development and experimental investigation of charged-particle optical systems VI Preface with an emphasis made on the systems for microwave tubes and particle-beam technology. The book is recommended to people who are studying or work with vacuum electronics and charged-particle beam technology: students, postgraduate students, engineers, research workers. The material presented in the book grew out of lectures given by one of the authors over many years to students and postgraduate students of the electronics faculty of St. Petersburg State Electrotechnical University. Two editions of this book have been published in Russian. At preparation of the English version of this book a chapter devoted to multiple-beam systems was added and supplementary references were in- cluded. Chapters 2–5, 7–11 and the Appendix were written by Professor S.I. Molokovsky, Chap. 12 by Professor A.D. Sushkov, and Chaps. 1 and 6 were prepared in collaboration. The authors express great appreciation to the head of the Radioelectronic Department Dr. V.B. Yankevich and our colleague Dr. V.A. Ivanov for their help preparing the English version of this book and for their continued interest in it. The authors would also like to express deep gratitude to the staﬀ and managers of Ingredient Ltd. (St. Petersburg) for their support of this work.

S.I. Molokovsky, A.D. Sushkov St. Petersburg, May 2005 Contents

1 Introduction to Particle-Beam Formation ...... 1 1.1 Application and Parameters ofIntenseCharged-ParticleBeams ...... 1 1.2 OpticalSystemsofElectronDevices ...... 4 1.2.1 ElectronGuns ...... 4 1.2.2 ElectronGunswithBeamCurrentControl ...... 6 1.3 FocusingSystemofElectronDevices...... 9 1.3.1 MagneticFocusingwithUniformField ...... 9 1.3.2 MagneticPeriodicFocusing...... 11 1.3.3 Periodic Electrostatic Focusing ...... 12 1.3.4 Hollow-BeamFocusing ...... 13 1.4 OpticalSystemsofParticle-BeamTechnology ...... 14 1.4.1 Installations of Electron Beam Welding andMelting ...... 14 1.5 IonOpticalSystems ...... 16

2 Methods of Fields Calculation ...... 19 2.1 General Equations of Electrostatic Fields ...... 19 2.2 Electrostatic Field Calculation. Dirichlet Problem ...... 20 2.2.1 MethodoftheSeparationofVariables ...... 20 2.2.2 MethodofGreenFunction ...... 23 2.2.3 MethodoftheIntegralEquations ...... 26 2.2.4 MethodofFiniteDiﬀerence ...... 27 2.3 Calculation of Electrostatic Field: Cauchy Problem ...... 30 2.3.1 MethodofAnalyticalContinuation ...... 31 2.4 Computer Calculation of Electrostatic Fields ...... 36 2.4.1 MethodofFiniteDiﬀerence ...... 36 2.4.2 MethodofIntegralEquations ...... 38 2.4.3 MethodofGreenFunction ...... 41 2.5 GeneralEquationsofMagneticField ...... 43 2.5.1 MagneticVectorPotential ...... 44 2.5.2 ScalarMagneticPotential ...... 46 2.6 NumericalCalculationofMagneticField ...... 47 2.6.1 FieldofaSolenoid ...... 47 2.6.2 FieldofaRing-ShapedMagnet ...... 48 VIII Contents

2.6.3 Technique of Solution ofNonlinearMagneticProblems ...... 50

3 Fundamentals of Charged-Particle Motion ...... 55 3.1 EquationsofMotioninNewtonForm ...... 55 3.2 LawofEnergyConservation ...... 56 3.3 Lagrange Equations of Motion ...... 58 3.4 HamiltonEquations ...... 59 3.5 Hamilton–Jacobi Equation ...... 60 3.6 EquationsofMotioninAxiallySymmetricFields...... 62 3.6.1 EquationsofMotioninNewtonForm ...... 62 3.6.2 Lagrange and Hamilton Equations of Motion ...... 66 3.6.3 Hamilton–Jacobi Equation of Motion ...... 67 3.6.4 MotionofElectronsinCylindricalMagnetron...... 67 3.7 MotioninPlanarTwo-DimensionalFields ...... 73 3.7.1 EquationsofMotioninNewtonForm ...... 73 3.7.2 Lagrange and Hamilton Equations ...... 73 3.8 NumericalCalculationofCharged-ParticleTrajectories .... 74 3.8.1 MethodofTaylorSeries ...... 74 3.8.2 Runge–Kutta Method of Fourth Order of Accuracy . 75 3.8.3 SpatialMethodofTrajectoryTracing ...... 76 3.9 Electrostatic Charged-Particle Lenses ...... 77 3.9.1 Types of Axially Symmetric Electrostatic Lenses . . . 77 3.9.2 Focusing Properties of Electrostatic Lenses ...... 77 3.10 MagneticLenses ...... 86 3.10.1 ThinMagneticLensApproximation ...... 86 3.10.2 ThickMagneticLens ...... 87 3.10.3 ParametersofSomeMagneticLenses ...... 88 3.10.4 MagneticLenseswithPermanentMagnets...... 91

4 Motion of Intense Charged-Particle Beams ...... 95 4.1 PeculiaritiesofIntenseParticle-BeamMotion ...... 95 4.1.1 EﬀectsofSelf-FieldsandVelocitySpreading ...... 95 4.1.2 Mathematical Description ofMultiple-VelocityBeams ...... 96 4.2 SimpliﬁedModelsofCharged-ParticleBeams ...... 98 4.2.1 MonovelocityModels ...... 98 4.2.2 Hydrodynamic (Laminar) Model of Flow ...... 100 4.2.3 QuasihydrodynamicFlowModel ...... 102 4.3 MethodsofSolutionofMotionEquations ...... 102 4.3.1 Method of Curvilinear Coordinates ...... 102 4.3.2 Method of Expansion in Series. ParaxialTrajectoryEquation ...... 104 4.4 ApproximateMethodofSpace-ChargeAccount ...... 105 Contents IX

4.5 Motion of Charge-Beams inChannelsFreefromExternalFields ...... 107 4.6 Influence of Residual Gases on Electron-Beam Motion ..... 110 4.6.1 Effect of Space-Charge Neutralization byPositiveIons ...... 110 4.6.2 Process of Neutralization of the Electron Beam inaLongChannel ...... 110 4.6.3 Effect of Ion Background onElectron-BeamMotion ...... 112 4.6.4 IonFocusing...... 112 4.6.5 Scattering of Electron Beam byMoleculesofResidualGas ...... 114 4.7 EstimationofEffectofThermalElectronVelocities ...... 117 4.7.1 CalculationofTransverseParticleDisplacement.... 117 4.7.2 CalculationofCurrent-DensityRedistribution ..... 119 4.8 MethodsofSolutionofSelf-ConsistentProblems ...... 121 4.8.1 Method of Successive Approximation ...... 121 4.8.2 Method of “Step by Step” (or Algorithm Modeling of Transient Process) ...... 123

5 Electron Guns ...... 125 5.1 TheProblemofElectron-BeamFormation...... 125 5.2 GunsforFormationofStripBeams ...... 126 5.2.1 ParallelStripBeam ...... 126 5.2.2 Wedge-ShapedBeam ...... 128 5.2.3 Strip-Beam Forming in Crossed Fields ...... 129 5.3 GunsforSolidAxiallySymmetricBeams ...... 133 5.3.1 ParallelCylindricalBeam ...... 133 5.3.2 PierceGunforConvergentBeam ...... 134 5.3.3 SynthesisofGunsbyOvcharov’sMethod...... 138 5.4 GunsforFormationofHollowAxisymmetricBeams ...... 141 5.4.1 Parallel Tubular Beam ...... 141 5.4.2 ConvergentHollowBeam...... 141 5.4.3 Magnetron Guns for Formation of Hollow Beams . . . 142 5.5 ElectronGunswithBeam-CurrentControl ...... 145 5.5.1 GunswithGridControlElectrodes ...... 146 5.5.2 Electron Gun with Diaphragm Control Electrode . . . 151 5.6 PrincipleofComputer-AidedDesignofElectronGuns ..... 151 5.6.1 SynthesisofElectronGun ...... 152 5.6.2 Electron-GunAnalysis ...... 152

6 Electron and Ion Sources with Field and Plasma Emitters ...... 157 6.1 SomePeculiaritiesofFieldandPlasmaEmitters ...... 157 6.2 Low-CurrentField-EmissionGuns ...... 157 X Contents

6.3 Multiple-TipField-EmissionCathodes ...... 159 6.4 High-CurrentElectronSourceswithExplosiveEmission .... 160 6.5 Liquid-MetalEmitters ...... 160 6.6 PlasmaSourcesofChargedParticles...... 161 6.6.1 Duoplasmatrons...... 161 6.6.2 Cold-CathodeSources ...... 162 6.6.3 High-FrequencyIonSources ...... 163 6.6.4 MicrowaveIonSources ...... 163 6.7 ExtractingofChargeParticlesandBeamForming...... 165 6.7.1 Extraction of Ions from Planar Plasma Boundary . . 165 6.7.2 SystemofExtractionwithExpansionCup...... 166 6.7.3 ProbeSystemofExtraction ...... 167

7 Magnetic Focusing Systems ...... 169 7.1 FocusingSystemswithUniformField ...... 169 7.1.1 PrincipleofFocusing ...... 169 7.1.2 Balanced (Brillouin) Axially Symmetric Beam ..... 170 7.1.3 UnbalancedBeaminUniformField...... 172 7.1.4 BeamFormedwithPartiallyShieldedGun ...... 174 7.1.5 BalancedNonlaminarBeam ...... 176 7.1.6 MagneticSystemsforUniformField ...... 180 7.2 SystemsofReversed-andPeriodic-FieldFocusing ...... 181 7.2.1 PrincipleofReversed-FieldFocusing ...... 181 7.2.2 EstimationofBeamDisturbances ...... 183 7.2.3 PrincipleofPeriodic-FieldFocusing ...... 184 7.2.4 AnalysisofPeriodicFocusing ...... 185 7.2.5 Stability of Periodic Focusing ...... 187 7.3 FocusingofHollowBeamsinUniformField ...... 188 7.3.1 Equilibrium Hollow Beam ...... 189 7.3.2 InjectionofHollowBeamsinUniformField ...... 191 7.3.3 Boundary Trajectory Pulsation ...... 192 7.4 FocusingofStripBeams ...... 193 7.4.1 Uniform-FieldFocusing ...... 193 7.4.2 PeriodicMagneticFocusing...... 195 7.5 Computer-AidedDesignofMagneticSystems ...... 196

8 Electrostatic Focusing Systems ...... 199 8.1 FocusingSystemforSolidAxiallySymmetricalBeams ..... 199 8.1.1 Equilibrium Cylindrical Beam ...... 199 8.1.2 Focusing System withPeriodicPotentialDistribution ...... 201 8.1.3 PeriodicSystemofAperturedDisks ...... 203 8.1.4 System of Unipotential Electrostatic Lenses ...... 203 8.2 SystemsforFocusingofStripBeams...... 210 8.2.1 Equilibrium Strip Beam ...... 210 Contents XI

8.2.2 Periodic Electrostatic System ...... 211 8.3 FocusingSystemsforHollowBeams ...... 211 8.3.1 Periodic Electrostatic Focusing ...... 211 8.3.2 Centrifugal Electrostatic Focusing ...... 213

9 Optical Systems of Technological Installations ...... 215 9.1 ElectronProbeofWeldingInstallation ...... 215 9.1.1 EﬀectofThermalVelocities ...... 216 9.1.2 AberrationsofOpticalSystem ...... 216 9.1.3 Eﬀects of Self-Fields and Scattering of Electrons . . . 219 9.2 OpticalSystemforElectron-BeamLithography ...... 219 9.2.1 SystemwithThermalCathode ...... 219 9.2.2 SystemwithColdFieldEmitter...... 221 9.3 IonProbes ...... 221

10 Intense Relativistic Charged-Particle Beams ...... 223 10.1 Relativistic Equations of Motion ...... 223 10.1.1 LawofEnergyConservation ...... 224 10.1.2 Equations of Motion in Axially Symmetry Fields . . . 224 10.2 Intense Relativistic Beams in Vacuum Channels ...... 225 10.2.1 Electron Beam Spread under Self-Fields Action .... 225 10.2.2 Beam Current Limitation ...... 227 10.2.3 Brillouin Relativistic Electron Beam ...... 229 10.3 Neutralized Relativistic Beams ...... 232 10.3.1 BeamsinSpaceFreefromExternalFields ...... 232 10.3.2 Neutralized Magnetically Conﬁned Beams ...... 232 10.4 Beam-Motion Computation ...... 232 10.4.1 AxiallySymmetricLensSystems ...... 233 10.4.2 QuadrupleLensSystems ...... 234

11 Multiple-Beam Electron-Optical Systems ...... 237 11.1 PeculiaritiesandApplicationofMultipleBeams...... 237 11.2 Multiple-BeamGunsandMagneticSystems ...... 239 11.2.1 ElectronGuns ...... 239 11.2.2 MagneticFocusingSystems...... 241 11.3 PeculiaritiesofMultipleSystems ...... 243 11.3.1 AnalysisofStray-FieldsAction ...... 243 11.3.2 TransverseFields...... 244 11.4 EstimationofEﬀectsofTransverseFields ...... 248 11.5 AnalysisofBeamInteraction ...... 250

12 Methods of Experimental Investigation of Beams ...... 255 12.1 ClassiﬁcationofMethods ...... 255 12.2 ThePin-HoleChamberMethod ...... 258 12.3 ApplicationofModiﬁedPin-HoleChamber ...... 261 XII Contents

Appendix ...... 265

References ...... 269

Index ...... 279