Space-charge flow

목차 (Table of Contents)

• CONTENTS
• FOREWORD = ⅶ
• PREFACE = xi
• LIST OF SYMBOLS = xxv
• CHAPTER Ⅰ. THE GENERAL DYNAMICS OF SPACE-CHARGE FLOW = 1
• 1. INTRODUCTION = 1
• 1.1. The Space-charge-flow Problem = 1
• 1.2. The Contents of the Chapter = 2
• 2. THE BASIC EQUATIONS = 3
• 2.1. Introduction = 3
• 2.2. The Electromagnetic Field Equations = 4
• 2.3. The Lorentz Force Law = 5
• 2.4. The Hydrodynamic Equations = 8
• 3. SOME INVARIANTS OF THE FLOW = 12
• 3.1. Introduction = 12
• 3.2. The Lagrange Invariant, Poincare Invariant, and Busch's Theorem = 12
• 3.3. Liouville's Theorem = 17
• 4. CYCLOTRON FREQUENCY AND LARMOR FREQUENCY = 20
• 5. THE CLASSIFICATION OF ELECTRON BEAMS = 23
• 5.1. Introduction = 23
• 5.2. Congruent and Noncongruent Flow = 24
• 5.3. The Equations of the Space-charge Field for Congruent Flow = 28
• 5.4. The Equations of the Space-charge Field for Noncongruent Flow = 29
• 6. BOUNDARY CONDITIONS AT AN EMITTER = 30
• 7. SCALING LAWS = 32
• 7.1. Introduction = 32
• 7.2. General Scaling = 32
• 7.3. Transverse Scaling = 38
• CHAPTER Ⅱ. EXACT SPACE-CHARGE-FLOW SOLUTIONS = 45
• 1. INTRODUCTION = 45
• 2. ONE-DIMENSIONAL SPACE-CHARGE-FLOW SOLUTIONS = 46
• 2.1. Introduction = 46
• 2.2. Rectilinear Flow in a Planar Diode = 47
• 2.3. Space-charge Flow in Crossed Electric and Magnetic Fields = 51
• 2.4. Solutions with a Component of Magnetic Field Normal to the Cathode = 57
• 3. ORTHOGONAL CURVILINEAR COORDINATE SYSTEMS BASED ON THE STREAMLINES OF THE FLOW = 60
• 3.1. Introduction = 60
• 3.2. Normal Congruent Flow = 61
• 3.3. A Solution in Cylindrical Polar Coordinates = 67
• 4. THE USE OF THE SEPARATION OF VARIABLES WITH THE ACTION FUNCTION = 69
• 4.1. Introduction = 69
• 4.2. Normal Congruent Flow with the Separation in Additive Form = 70
• 4.3. Normal Congruent Flow with the Separation in Product Form = 72
• 4.4. Skew-congruent Flow = 81
• 5. THE SEPARATION OF VARIABLES AND THE LORENTZ FORCE LAW = 84
• 5.1. Introduction = 84
• 5.2. The Hydrodynamic Equation of Motion = 85
• 5.3. Numerical Methods and the Direct Application of the Lorentz Equation of Motion = 89
• CHAPTER Ⅲ. PARAXIAL-RAY EQUATIONS = 97
• 1. INTRODUCTION = 97
• 2. PARAXIAL EQUATION FOR THE BOUNDING SURFACE OF A ROUND SOLID BEAM WITH A RECTILINEAR AXIS = 99
• 2.1. Mathematical Preliminaries = 99
• 2.2. Detailed Derivation of the Paraxial Equation = 101
• 2.3. Discussion of Paraxial Forces = 104
• 2.4. Assumptions Inherent in Paraxial Theory = 105
• 3. PARAXIAL EQUATION FOR THE BOUNDING SURFACE OF SHEET BEAMS = 106
• 4. THE LAMINAR-BEAM MODEL = 107
• 5. THE PARAXIAL EQUATIONS INSIDE A SOLID AXIALLY SYMMETRIC BEAM = 109
• 5.1. Introduction = 109
• 5.2. Derivation of the Equations = 109
• 5.3. The Matrix Formulation = 111
• 5.4. An Illustrative Example = 113
• 6. CURVILINEAR SHEET BEAMS = 114
• 6.1. Introduction = 114
• 6.2. Establishment of the Metric = 114
• 6.3. Legrange's Equation for r = 116
• 6.4. Discussion of the Paraxial-ray Equation = 119
• 6.5. The Paraxial Equations Inside a Curvilinear Sheet Beam = 121
• 7. ANNULAR BEAMS = 122
• 7.1. Space-charge Fields in a Cylindrical System = 122
• 7.2. Paraxial-ray Equations for Annular Beams = 122
• 8. PHYSICAL ACTION OF A LENS = 123
• 8.1. Phenomenological Description = 123
• 8.2. Paraxial-ray Analysis of a Lens = 124
• 8.3. The Thin Lens = 129
• 8.4. The Aperture Lens = 130
• 9. THE USE OF PARAXIAL THEORY FOR ELECTRONGUN DESIGN = 131
• 10. SPREADING OF A DRIFTING BEAM = 138
• CHAPTER Ⅳ. UNIFORM FOCUSING = 145
• 1. INTRODUCTION = 145
• 2. GENERAL TREATMENT OF UNRIPPLED ROUND BEAMS = 146
• 2.1. Some Relations Pertaining to the Beam Remote from the Cathode = 146
• 2.2. Relation of the Beam to the Cathode = 148
• 3. UNRIPPLED ISOAXIAL-VELOCITY ROUND BEAMS = 151
• 3.1. Classification of Beams = 151
• 3.2. The Brillouin Solid Beam = 153
• 3.3. The Brillouin Hollow Beam = 157
• 3.4. Harris Flow = 160
• 4. PARAXIAL ANALYSIS OF ROUND BEAMS = 163
• 4.1. Uniform Magnetic Focusing ; Equilibrium Radii = 163
• 4.2. First Integral of the Paraxial Equation ; Beam Stiffness = 165
• 4.3. Second Integral of the Paraxial Equation ; Resonant Frequency and Wavelength = 167
• 4.4. Application to a Brillouin Hollow Beam = 169
• 4.5. Harris Flow = 170
• 5. ISOROTATIONAL BEAMS ; UNRIPPLED THEORY = 172
• 6. PERTURBATIONS IN THIN CROSSED-FIELD ANNULAR BEAMS = 175
• 6.1. Paraxial Theory = 175
• 6.2. Stability = 176
• 7. UNRIPPLED RECTILINEAR SHEET BEAMS = 180
• 7.1. General Development = 180
• 7.2. Brillouin Flow = 181
• 7.3. Rectilinear Crossed-field Flow = 183
• 8. RIPPLING IN SHEET BEAMS = 184
• 8.1. Introduction = 184
• 8.2. Rippling in a Brillouin Sheet Beam = 185
• 8.3. Rippling in a Crossed-field Sheet Beam = 186
• 9. MAGNETIC-FIELD PERTURBATIONS = 186
• 9.1. General Discussion = 186
• 9.2. Rippling Caused by a Field Reversal = 188
• 9.3. Beam Rotation Caused by a Field Reversal ; Magnetic Mirrors = 192
• 9.4. Adiabatic Tapering of the Magnetic Field = 194
• 9.5. Transverse Fields = 196
• CHAPTER Ⅴ. PERIODIC FOCUSING = 200
• 1. INTRODUCTION = 200
• 2. PERIODIC PERMANENT MAGNET(PPM) FOCUSING OF ROUND UNIPOTENTIAL BEAMS = 204
• 2.1. Introduction = 204
• 2.2. Stability of a Beam with Negligible Space Charge = 209
• 2.3. Focusing with Space Charge = 212
• 2.4. Hollow Beams and Beams with a Radial Variation of Density = 218
• 2.5. Tapering of a Periodic Magnetic Field = 223
• 3. PERIODIC ELECTROSTATIC FOCUSING OF BEAMS WITH A RECTILINEAR AXIS = 225
• 3.1. Introduction = 225
• 3.2. Space-charge Balance Conditions for a Sheet Beam = 229
• 3.3. Effect of Improper Entrance Conditions on the Beam Focusing = 231
• 3.4. An Example of Strip-beam Focusing = 234
• 3.5. Electrostatic Focusing of a Thick Beam = 237
• 3.6. Asymmetrical Sheet-beam Focusing = 240
• 3.7. Space-charge Balance Conditions for a Round Beam = 242
• 3.8. The Effect of Improper Entrance Conditions on the Focusing of a Round Beam = 243
• 3.9. The Consequences of Nonlaminarity = 244
• 3.10. An Example of the Electrostatic Focusing of a Cylindrical Beam = 245
• 3.11. Periodic Electrostatic Focusing of a Thick Round Beam or a Hollow Beam = 247
• 3.12. The Bifilar Helix = 249
• 4. PERIODIC DEFLECTION FOCUSING = 252
• 4.1. Introduction = 252
• 4.2. General Formulation for the Electrostatic Focusing of a Strip Beam Uniform in the z Direction = 254
• 4.3. Slalom Focusing = 256
• CHAPTER Ⅵ. THERMAL EFFECTS AND NONLAMINAR FLOW = 263
• 1. INTRODUCTION = 263
• 2. THE PLANAR DIODE = 265
• 2.1. Introduction = 265
• 2.2. The Boundary Conditions at the Cathode = 265
• 2.3. The Mathematical Formalism = 268
• 2.4. The Solutions of the Equations for the Potential in Normalized Forms = 271
• 2.5. The Calculation of the Position and Effect of the Potential Minimum = 275
• 3. PARAXIAL OPTICS FOR THERMAL BEAMS = 276
• 3.1. Introduction = 276
• 3.2. The Imaging Properties of Thermal Beams = 277
• 3.3. The Trajectories of Thermal Electrons = 280
• 4. THE CURRENT DENSITY AT THE AXIS OF AN ELECTRON BEAM = 282
• 4.1. Introduction = 282
• 4.2. Langmuir's Bounding Expression for the Current Density on the Axis of a Cylindrical System = 283
• 4.3. The Current Density of the Axis of a Perfect Cylindrical Focusing System = 285
• 5. THE CURRENT DENSITY AT ANY POINT IN A CYLINDRICAL BEAM = 289
• 5.1. Introduction = 289
• 5.2. The Basic Integration to Find Current Densities = 290
• 5.3. The Density Variation for $$r_e$$／σ ≫ 1 = 296
• 6. SOME APPLICATIONS OF THE CYLINDRICAL-BEAM FORMULAS = 298
• 6.1. Introduction = 298
• 6.2. The Properties of a Cylindrical Beam Emitted from a Shielded Gun = 299
• 6.3. The Properties of a Thermally Diffused Cylindrical Beam in a Planar Diode = 305
• 6.4. The Properties of a Thermally Diffused Cylindrical Beam in a Brillouin Focusing System = 308
• 6.5. A Linear Treatment of the Thermal Spread in a Round Drifting Beam = 310
• 7. THE CURRENT DENSITY AT ANY POINT IN A SHEET BEAM = 311
• 7.1. Introduction = 311
• 7.2. The Basic Integration to Compute Final Current Densities = 312
• 7.3. A Linear Treatment of the Current Density in a Drifting Sheet Beam = 316
• 8. THE MODIFICATION OF THE THERMAL-SPREAD FORMULAS DUE TO THE CHANGE IN SPACE-CHARGE FORCES = 318
• 8.1. Introduction = 318
• 8.2. The Modification in a Paraxial Beam = 318
• 8.3. Application of the Method of Sec. 8.2. to a Drifting Sheet Beam = 320
• 8.4. An Alternative Analysis of the Current Density in a Thermally Diffused Drifting Beam = 322
• 8.5. The Spherical Diode = 323
• CHAPTER Ⅶ. ANALYTICAL METHODS OF ELECTRODE DESIGN = 331
• 1. INTRODUCTION = 331
• 2. SOME SIMPLE SOLUTIONS OF TWO-DIMENSIONAL BOUNDARY-VALUE PROBLEMS = 334
• 2.1. Introduction = 334
• 2.2. Solution under Different Boundary Conditions = 336
• 2.3. The Instability of Solutions under Cauchy Conditions and the Stability under Dirichlet Conditions = 338
• 2.4. The Stability of Solutions of Hyperbolic Equations under Cauchy Conditions = 340
• 3. THE SOLUTION OF THE TWO-DIMENSIONAL EXTERIOR-BOUNDARY- VALUE PROBLEM = 341
• 3.1. Introduction = 341
• 3.2. Electrodes to Maintain a Parallel Beam = 342
• 3.3. Extension to an Arbitrary Potential and Field on a Line = 343
• 3.4. Extension to an Arbitrary Potential and Field Defined on an Arbitrary Curve = 345
• 3.5. Application to a Simple Problem of a Crossed-field Gun = 346
• 3.6. A More Complicated Application to Crossed-field Guns = 348
• 3.7. The Beam-forming Electrodes near a Curvilinear Strip Beam = 351
• 3.8. Summary of the Potentialities of the Method and Its Pitfalls = 355
• 4. EXTERIOR-BOUNDARY-VALUE PROBLEMS FOR AXIALLY SYMMETRIC AND PLANAR SYSTEMS = 356
• 4.1. Introduction = 356
• 4.2. Electrodes for Axially Symmetric Pencil Beams = 357
• 4.3. The Mathematical Formalism = 361
• 5. SOLUTION OF THE AXIALLY SYMMETRIC PROBLEM BY EMBEDDING IT IN A THIRD DIMENSION = 364
• 5.1. Introduction = 364
• 5.2. The Potentials Required to Maintain a Rectilinear Sheet Beam = 365
• 5.3. Extension of the Method to the Case of Arbitrary Potentials and Fields Prescribed, for Axially Symmetric Beams, on a Prescribed Curve = 368
• CHAPTER Ⅷ. APPROXIMATE THEORETICAL METHODS OF ELECTRODE DESIGN = 377
• 1. INTRODUCTION = 377
• 2. ANALYSIS OF A STEADY-ELECTRON-FLOW PROBLEM, NEGLECTING SPACE-CHARGE FORCES = 379
• 2.1. Introduction = 379
• 2.2. The Flow, Neglecting Space-charge Forces = 379
• 2.3. Boundary Conditions at the Emitting Surface = 381
• 3. THE ANALYSIS OF AN ELECTRODE SYSTEM, USING SPACE-CHARGE SIMULATION = 382
• 3.1. Introduction = 382
• 3.2. The Iteration Procedure = 383
• 3.3. Convergence and Rate of Convergence = 386
• 3.4. Uniqueness of the Iterative Solutions = 389
• 3.5. Accuracy of the Method = 393
• 3.6. Errors Due to the Discrete Nature of the Space-charge Simulation = 395
• 3.7. Errors Due to Mesh Size in the Difference Analog for the Potential = 395
• 4. ANALYSIS AND SYNTHESIS OF ELECTRODE SYSTEMS BY THE SUPERPOSITION PRINCIPLE = 399
• 4.1. Introduction = 399
• 4.2. The Determination of One Electrode System from Another to Produce the Same Beam = 401
• 4.3. The Analytic Determination of the Space-charge Potential = 403
• 5. THE ANODE-HOLE PROBLEM = 407
• 5.1. Introduction = 407
• 5.2. The Determination of the Effect of the Anode Hole with Space-charge-simulation Equipment = 409
• 5.3. The Effect of the Anode Hole in Very Low Perveance Guns = 409
• 5.4. The Use of a Dummy Cathode to Determine Effects in Medium-perveance Guns = 410
• 5.5. The Use of the Superposition Method in Medium-perveance Guns = 412
• 5.6. The Compensation for the Deteriorating Effect of the Anode Hole = 413
• 6. ELECTRODE ANALYSIS FOR PARAXIAL BEAMS = 415
• 6.1. General Simplifications in the Methods Due to the Paraxial Assumptions = 415
• 6.2. Simplifications Due to the Form of the Potential = 417
• 7. AN EVALUATION OF THE METHODS OF ANALYZING ELECTRODE CONFIGURATIONS = 419
• CHAPTER Ⅸ. ANALOGS USED FOR ELECTRODE DESIGN = 423
• 1. INTRODUCTION = 423
• 1.1. Background = 423
• 1.2. The Plan of the Chapter = 424
• 2. EQUATIONS OBEYED BY THE POTENTIAL FOR STEADY FLOW IN VACUUM AND ITS ANALOGS = 424
• 2.1. Introduction = 424
• 2.2. The Equations for Steady Planar or Axially Symmetric Flow = 425
• 2.3. The Equations for Steady Current Flow in an Electrolyte = 426
• 2.4. The Difference-equation Approximation = 426
• 2.5. Scaling = 430
• 3. RELAXATION TECHNIQUES = 430
• 3.1. Introduction = 430
• 3.2. The Relaxation Method for Two-dimensional Systems = 431
• 3.3. Overrelaxation = 433
• 3.4. The Effect of Mesh Size = 433
• 3.5. The Relaxation Technique in Axially Symmetric Systems = 436
• 4. THE RESISTANCE-NETWORK ANALOG = 436
• 4.1. Introduction = 436
• 4.2. The Equation for the Resistance Network = 437
• 4.3. The Analog of the Axially Symmetric Poisson's Equation = 437
• 4.4. The Simulation of Boundary Conditions = 439
• 4.5. Termination of the Network = 441
• 4.6. Description of the Network = 443
• 4.7. Accuracy of the Resistance Network = 444
• 5. RESISTANCE-PAPER ANALOG = 445
• 5.1. Introduction = 445
• 6. CHARACTERISTICS OF ELECTROLYTIC TANKS = 445
• 6.1. Introduction = 445
• 6.2. Physical Description = 446
• 6.3. Tanks with Solid Electrolytes = 450
• 6.4. The Simulation of Space Charge by Current Sources = 451
• 7. THE RUBBER-MEMBRANE ANALOG = 454
• 7.1. Description of the Analog = 454
• 8. AN EVALUATION OF THE METHODS OF SOLVING POISSON'S EQUATION = 456
• 9. FIELD AND TRAJECTORY PLOTTING = 457
• 9.1. Introduction = 457
• 9.2. Field Plots in an Electrolytic Tank or Resistance Network = 458
• 9.3. The Basic Equations for Ray Tracing = 461
• 9.4. Ray Tracing by Graphical Means = 463
• 9.5. Plotting Trajectories by Integrating the Lorentz Force Law = 465
• 9.6. Automatic Trajectory Tracing with a Computer = 467
• APPENDIX A. THE VECTOR OPERATORS IN ORTHOGONAL CURVILINEAR COORDINATES = 471
• 1. INTRODUCTION = 471
• 2. THE EXPRESSIONS FOR THE VECTOR OPERATORS IN A GENERAL ORTHOGONAL COORDINATE SYSTEM = 471
• 3. THE VECTOR OPERATORS IN SOME SPECIAL COORDINATE SYSTEMS = 475
• APPENDIX B. CONFORMAL TRANSFORMATIONS AND THE PRINCIPLE OF ANALYTIC CONTINUATION = 478
• 1. INTRODUCTION = 478
• 2. THE PROPERTIES OF CONFORMAL TRANSFORMATIONS = 478
• 3. THE PRINCIPLE OF ANALYTIC CONTINUATION = 480
• APPENDIX C. MATRIX THEORY FOR PERIODIC SYSTEMS = 482
• APPENDDC D. HILL'S AND MATHIEU'S EQUATIONS = 485
• APPENDIX E. THE SPHERICAL AND CYLINDRICAL DIODES = 490
• Index = 497
• Physical Constants in MKS Units = inside front cover