Kevin Jensen, PhD is a research physicist in the Materials and Systems Branch, Materials Science and Technology Division, at the Naval Research Laboratory. Since 2001, he has been a visiting senior research scientist at the University of Maryland’s Institute for Research in Electronics and Applied Physics (IREAP). Dr. Jensen joined the theory section of the Vacuum Electronics Branch at NRL in 1990. He earned a doctorate in physics from New York University in 1987. He has been and is Principal Investigator for several research programs investigating the application of electron sources (particularly field and photoemission sources) to microwave devices and Free Electron Lasers. Over the years, he has authored or coauthored over 150 articles and conference proceedings. He became a Fellow of the American Physical Society in 2009 for his contributions to the theory and modeling of electron emission sources for particle accelerators and microwave tubes. He presently serves on the Editorial Board of Journal of Applied Physics.

• Acknowledgements xiiiPart I: Foundations1 Prelude 32 Units and evaluation 72.1 Numerical accuracy 72.2 Atomic-sized units 82.3 Units based on emission 113 Pre-quantum models 133.1 Discovery of electron emission 133.2 The Drude model and Maxwell–Boltzmann statistics 133.3 The challenge of photoemission 194 Statistics 254.1 Distinguishable particles 254.2 Probability and states 284.3 Probability and entropy 304.4 Combinatorics and products of probability 335 Maxwell–Boltzmann distribution 375.1 Classical phase space 375.2 Most probable distribution 395.3 Energy and entropy 415.4 The Gibbs paradox 425.5 Ideal Gas in a potential gradient 445.6 The grand partition function 455.7 A nascent model of electron emission 466 Quantum distributions 496.1 Bose–Einstein distribution 496.2 Fermi–Dirac distribution 506.3 The Riemann zeta function 506.4 Chemical potential 526.5 Classical to quantum statistics 566.6 Electrons and white dwarf stars 577 A box of electrons 617.1 Scattering 617.2 From classical to quantum mechanics 617.3 Moments and distributions 637.4 Boltzmann’s transport equation 648 Quantum mechanics methods 738.1 A simple model: the prisoner’s dilemma 738.2 Matrices and wave functions 789 Quintessential problems 919.1 The hydrogen atom 929.2 Transport past barriers 1029.3 The harmonic oscillator 110Part II: The canonical equations10 A brief history 12110.1 Thermal emission 12110.2 Field emission 12210.3 Photoemission 12310.4 Secondary emission 12410.5 Space-charge limited emission 12410.6 Resources and further reading 12411 Anatomy of current density 12711.1 Supply function 12811.2 Gamow factor 12811.3 Image charge potential 13112 Richardson–Laue–Dushman equation 13512.1 Approximations 13512.2 Analysis of thermal emission data 13613 Fowler–Nordheim equation 13913.1 Triangular barrier approximation 14013.2 Image charge approximation 14113.3 Analysis of field emission data 14513.4 The Millikan–Lauritsen hypothesis 14614 Fowler–Dubridge equation 14914.1 Approximations 14914.2 Analysis of photoemission data 15315 Baroody equation 15515.1 Approximations 15515.2 Analysis of secondary emission data 16015.3 Subsequent approximations 16116 Child–Langmuir law 16316.1 Constant density approximation 16416.2 Constant current approximation 16516.3 Transit time approximation 16817 A General thermal–field–photoemission equation 17317.1 Experimental thermal–field energy distributions 17517.2 Theoretical thermal–field energy distributions 17617.3 The N(n,s,u) function 18117.4 Brute force evaluation 18917.5 A computationally kind model 19317.6 General thermal–field emission code 198Part III: Exact tunneling and transmission evaluation18 Simple barriers 20918.1 Rectangular barrier 20918.2 Triangular barrier: general method 21318.3 Triangular barrier: numerical 22219 Transfer matrix approach 22719.1 Plane wave transfer matrix 22719.2 Airy function transfer matrix 23320 Ion enhanced emission and breakdown 24520.1 Paschen’s curve 24520.2 Modified Paschen’s curve 24720.3 Ions and the emission barrier 250Part IV: The complexity of materials21 Metals 25721.1 Density of states, again 25721.2 Spheres in d dimensions 25921.3 The Kronig Penny model 26121.4 Atomic orbitals 26421.5 Electronegativity 26621.6 Sinusoidal potential and band gap 26921.7 Ion potentials and screening 27222 Semiconductors 27722.1 Resistivity 27722.2 Electrons and holes 27922.3 Band gap and temperature 28122.4 Doping of semiconductors 28122.5 Semiconductor image charge potential 28622.6 Dielectric constant and screening 28723 Effective mass 29123.1 Dispersion relations 29123.2 The k ⋅ p method 29323.3 Hyperbolic relations 29623.4 The alpha semiconductor model 29923.5 Current and effective mass 30124 Interfaces 30324.1 Metal–insulator–metal current density 30324.2 Band bending 31024.3 Accumulation layers 31124.4 Depletion layers 31924.5 Modifications due to non-linear potential barriers 32425 Contacts, conduction, and current 32925.1 Zener breakdown 32925.2 Poole–Frenkel transport 32925.3 Tunneling conduction 33325.4 Resonant tunneling in field emission 33626 Electron density near barriers 34126.1 An infinite barrier 34126.2 Two infinite barriers 34426.3 A triangular well 34626.4 Density and dipole component 34827 Many-body effects and image charge 35327.1 Kinetic energy 35327.2 Exchange energy 35427.3 Correlation term 35627.4 Core term 35727.5 Exchange-correlation and a barrier model 36028 An analytic image charge potential 36328.1 Work function and temperature 36328.2 Work function and field 36328.3 Changes to current density 366Part V: Application physics29 Dispenser cathodes 37129.1 Miram curves and the longo equation 37129.2 Diffusion of coatings 37529.3 Evaporation of coatings 39129.4 Knudsen flow through pores 39329.5 Lifetime of a sintered wire controlled porosity dispenser cathode 39930 Field emitters 40330.1 Field enhancement 40330.2 Hemispheres and notional emission area 40630.3 Point charge model 40830.4 Schottky’s conjecture 41230.5 Assessment of the tip current models 41530.6 Line charge models 41730.7 Prolate spheroidal representation 42030.8 A hybrid analytic-numerical model 42530.9 Shielding 43330.10 Statistical variation 43831 Photoemitters 44331.1 Scattering consequences 44631.2 Basic theory 44831.3 Three-step model 44931.4 Moments model 45131.5 Reflectivity and penetration factors 45731.6 Lorentz–Drude model of the dielectric constant 45831.7 Scattering contributions 46631.8 Low work function coatings 47831.9 Quantum efficiency of a cesiated surface 48532 Secondary emission cathodes 48732.1 Diamond amplifier concept 48732.2 Monte Carlo methods 49432.3 Relaxation time 49932.4 Monte Carlo and diamond amplifier response time 51633 Electron beam physics 52533.1 Electron orbits and cathode area 52633.2 Beam envelope equation 52833.3 Emittance for flat and uniform surfaces 53333.4 Emittance for a bump 54533.5 Emittance and realistic surfaces 563Part VI: AppendicesAppendix 1 Summation, integration, and differentiation 569A1.1 Series 569A1.2 Integration 569A1.3 Differentiation 577A1.4 Numerical solution of an ordinary differential equation 582Appendix 2 Functions 585A2.1 Trigonometric functions 585A2.2 Gamma function 585A2.3 Riemann zeta function 585A2.4 Error function 587A2.5 Legendre polynomials 587A2.6 Airy functions 588A2.7 Lorentzian functions 590Appendix 3 Algorithms 591A3.1 Permutation algorithm 591A3.2 Birthday algorithm 592A3.3 Least squares fitting of data 593A3.4 Monty Hall algorithm 595A3.5 Wave function and density algorithm 596A3.6 Hydrogen atom algorithms 598A3.7 Root-finding Methods 601A3.8 Thermal–field algorithm 604A3.9 Gamow factor algorithm 606A3.10 Triangular barrier D(E) 607A3.11 Evaluation of Hc(u) 608A3.12 Transfer matrix algorithm 610A3.13 Semiconductors and doping density 616A3.14 Band bending: accumulation layer 618A3.15 Simple ODE solvers 619A3.16 Current through a metal–insulator–metal diode 622A3.17 Field emission from semiconductors 624A3.18 Roots of the quadratic image charge barrier 626A3.19 Zeros of the airy function 627A3.20 Atomic sphere radius rs 629A3.21 Sodium exchange-correlation potential 631A3.22 Field-dependent work function 632A3.23 Digitizing an image file 632A3.24 Lattice gas algorithm 633A3.25 Evaluation of the point charge model functions 636A3.26 Modeling of field emitter I(V) data 638A3.27 Modeling a log-normal distribution of field emitters 640A3.28 Simple shell and sphere algorithm 643A3.29 Gyftopoulos–Levine work function algorithm 645A3.30 Poisson distributions 648A3.31 Electron–electron relaxation time 650A3.32 Resistivity and the Debye temperature 651A3.33 Orbits in a magnetic field 655A3.34 Trajectory of a harmonic oscillator 657A3.35 Trajectories for emission from a hemisphere 658A3.36 Monte Carlo and integration 660References 663Index 683

Kevin Jensen, PhD is a research physicist in the Materials and Systems Branch, Materials Science and Technology Division, at the Naval Research Laboratory. Since 2001, he has been a visiting senior research scientist at the University of Maryland’s Institute for Research in Electronics and Applied Physics (IREAP). Dr. Jensen joined the theory section of the Vacuum Electronics Branch at NRL in 1990. He earned a doctorate in physics from New York University in 1987. He has been and is Principal Investigator for several research programs investigating the application of electron sources (particularly field and photoemission sources) to microwave devices and Free Electron Lasers. Over the years, he has authored or coauthored over 150 articles and conference proceedings. He became a Fellow of the American Physical Society in 2009 for his contributions to the theory and modeling of electron emission sources for particle accelerators and microwave tubes. He presently serves on the Editorial Board of Journal of Applied Physics.

• Acknowledgements xiiiPart I: Foundations1 Prelude 32 Units and evaluation 72.1 Numerical accuracy 72.2 Atomic-sized units 82.3 Units based on emission 113 Pre-quantum models 133.1 Discovery of electron emission 133.2 The Drude model and Maxwell–Boltzmann statistics 133.3 The challenge of photoemission 194 Statistics 254.1 Distinguishable particles 254.2 Probability and states 284.3 Probability and entropy 304.4 Combinatorics and products of probability 335 Maxwell–Boltzmann distribution 375.1 Classical phase space 375.2 Most probable distribution 395.3 Energy and entropy 415.4 The Gibbs paradox 425.5 Ideal Gas in a potential gradient 445.6 The grand partition function 455.7 A nascent model of electron emission 466 Quantum distributions 496.1 Bose–Einstein distribution 496.2 Fermi–Dirac distribution 506.3 The Riemann zeta function 506.4 Chemical potential 526.5 Classical to quantum statistics 566.6 Electrons and white dwarf stars 577 A box of electrons 617.1 Scattering 617.2 From classical to quantum mechanics 617.3 Moments and distributions 637.4 Boltzmann’s transport equation 648 Quantum mechanics methods 738.1 A simple model: the prisoner’s dilemma 738.2 Matrices and wave functions 789 Quintessential problems 919.1 The hydrogen atom 929.2 Transport past barriers 1029.3 The harmonic oscillator 110Part II: The canonical equations10 A brief history 12110.1 Thermal emission 12110.2 Field emission 12210.3 Photoemission 12310.4 Secondary emission 12410.5 Space-charge limited emission 12410.6 Resources and further reading 12411 Anatomy of current density 12711.1 Supply function 12811.2 Gamow factor 12811.3 Image charge potential 13112 Richardson–Laue–Dushman equation 13512.1 Approximations 13512.2 Analysis of thermal emission data 13613 Fowler–Nordheim equation 13913.1 Triangular barrier approximation 14013.2 Image charge approximation 14113.3 Analysis of field emission data 14513.4 The Millikan–Lauritsen hypothesis 14614 Fowler–Dubridge equation 14914.1 Approximations 14914.2 Analysis of photoemission data 15315 Baroody equation 15515.1 Approximations 15515.2 Analysis of secondary emission data 16015.3 Subsequent approximations 16116 Child–Langmuir law 16316.1 Constant density approximation 16416.2 Constant current approximation 16516.3 Transit time approximation 16817 A General thermal–field–photoemission equation 17317.1 Experimental thermal–field energy distributions 17517.2 Theoretical thermal–field energy distributions 17617.3 The N(n,s,u) function 18117.4 Brute force evaluation 18917.5 A computationally kind model 19317.6 General thermal–field emission code 198Part III: Exact tunneling and transmission evaluation18 Simple barriers 20918.1 Rectangular barrier 20918.2 Triangular barrier: general method 21318.3 Triangular barrier: numerical 22219 Transfer matrix approach 22719.1 Plane wave transfer matrix 22719.2 Airy function transfer matrix 23320 Ion enhanced emission and breakdown 24520.1 Paschen’s curve 24520.2 Modified Paschen’s curve 24720.3 Ions and the emission barrier 250Part IV: The complexity of materials21 Metals 25721.1 Density of states, again 25721.2 Spheres in d dimensions 25921.3 The Kronig Penny model 26121.4 Atomic orbitals 26421.5 Electronegativity 26621.6 Sinusoidal potential and band gap 26921.7 Ion potentials and screening 27222 Semiconductors 27722.1 Resistivity 27722.2 Electrons and holes 27922.3 Band gap and temperature 28122.4 Doping of semiconductors 28122.5 Semiconductor image charge potential 28622.6 Dielectric constant and screening 28723 Effective mass 29123.1 Dispersion relations 29123.2 The k ⋅ p method 29323.3 Hyperbolic relations 29623.4 The alpha semiconductor model 29923.5 Current and effective mass 30124 Interfaces 30324.1 Metal–insulator–metal current density 30324.2 Band bending 31024.3 Accumulation layers 31124.4 Depletion layers 31924.5 Modifications due to non-linear potential barriers 32425 Contacts, conduction, and current 32925.1 Zener breakdown 32925.2 Poole–Frenkel transport 32925.3 Tunneling conduction 33325.4 Resonant tunneling in field emission 33626 Electron density near barriers 34126.1 An infinite barrier 34126.2 Two infinite barriers 34426.3 A triangular well 34626.4 Density and dipole component 34827 Many-body effects and image charge 35327.1 Kinetic energy 35327.2 Exchange energy 35427.3 Correlation term 35627.4 Core term 35727.5 Exchange-correlation and a barrier model 36028 An analytic image charge potential 36328.1 Work function and temperature 36328.2 Work function and field 36328.3 Changes to current density 366Part V: Application physics29 Dispenser cathodes 37129.1 Miram curves and the longo equation 37129.2 Diffusion of coatings 37529.3 Evaporation of coatings 39129.4 Knudsen flow through pores 39329.5 Lifetime of a sintered wire controlled porosity dispenser cathode 39930 Field emitters 40330.1 Field enhancement 40330.2 Hemispheres and notional emission area 40630.3 Point charge model 40830.4 Schottky’s conjecture 41230.5 Assessment of the tip current models 41530.6 Line charge models 41730.7 Prolate spheroidal representation 42030.8 A hybrid analytic-numerical model 42530.9 Shielding 43330.10 Statistical variation 43831 Photoemitters 44331.1 Scattering consequences 44631.2 Basic theory 44831.3 Three-step model 44931.4 Moments model 45131.5 Reflectivity and penetration factors 45731.6 Lorentz–Drude model of the dielectric constant 45831.7 Scattering contributions 46631.8 Low work function coatings 47831.9 Quantum efficiency of a cesiated surface 48532 Secondary emission cathodes 48732.1 Diamond amplifier concept 48732.2 Monte Carlo methods 49432.3 Relaxation time 49932.4 Monte Carlo and diamond amplifier response time 51633 Electron beam physics 52533.1 Electron orbits and cathode area 52633.2 Beam envelope equation 52833.3 Emittance for flat and uniform surfaces 53333.4 Emittance for a bump 54533.5 Emittance and realistic surfaces 563Part VI: AppendicesAppendix 1 Summation, integration, and differentiation 569A1.1 Series 569A1.2 Integration 569A1.3 Differentiation 577A1.4 Numerical solution of an ordinary differential equation 582Appendix 2 Functions 585A2.1 Trigonometric functions 585A2.2 Gamma function 585A2.3 Riemann zeta function 585A2.4 Error function 587A2.5 Legendre polynomials 587A2.6 Airy functions 588A2.7 Lorentzian functions 590Appendix 3 Algorithms 591A3.1 Permutation algorithm 591A3.2 Birthday algorithm 592A3.3 Least squares fitting of data 593A3.4 Monty Hall algorithm 595A3.5 Wave function and density algorithm 596A3.6 Hydrogen atom algorithms 598A3.7 Root-finding Methods 601A3.8 Thermal–field algorithm 604A3.9 Gamow factor algorithm 606A3.10 Triangular barrier D(E) 607A3.11 Evaluation of Hc(u) 608A3.12 Transfer matrix algorithm 610A3.13 Semiconductors and doping density 616A3.14 Band bending: accumulation layer 618A3.15 Simple ODE solvers 619A3.16 Current through a metal–insulator–metal diode 622A3.17 Field emission from semiconductors 624A3.18 Roots of the quadratic image charge barrier 626A3.19 Zeros of the airy function 627A3.20 Atomic sphere radius rs 629A3.21 Sodium exchange-correlation potential 631A3.22 Field-dependent work function 632A3.23 Digitizing an image file 632A3.24 Lattice gas algorithm 633A3.25 Evaluation of the point charge model functions 636A3.26 Modeling of field emitter I(V) data 638A3.27 Modeling a log-normal distribution of field emitters 640A3.28 Simple shell and sphere algorithm 643A3.29 Gyftopoulos–Levine work function algorithm 645A3.30 Poisson distributions 648A3.31 Electron–electron relaxation time 650A3.32 Resistivity and the Debye temperature 651A3.33 Orbits in a magnetic field 655A3.34 Trajectory of a harmonic oscillator 657A3.35 Trajectories for emission from a hemisphere 658A3.36 Monte Carlo and integration 660References 663Index 683

Kevin Jensen, PhD is a research physicist in the Materials and Systems Branch, Materials Science and Technology Division, at the Naval Research Laboratory. Since 2001, he has been a visiting senior research scientist at the University of Maryland’s Institute for Research in Electronics and Applied Physics (IREAP). Dr. Jensen joined the theory section of the Vacuum Electronics Branch at NRL in 1990. He earned a doctorate in physics from New York University in 1987. He has been and is Principal Investigator for several research programs investigating the application of electron sources (particularly field and photoemission sources) to microwave devices and Free Electron Lasers. Over the years, he has authored or coauthored over 150 articles and conference proceedings. He became a Fellow of the American Physical Society in 2009 for his contributions to the theory and modeling of electron emission sources for particle accelerators and microwave tubes. He presently serves on the Editorial Board of Journal of Applied Physics.

• Acknowledgements xiiiPart I: Foundations1 Prelude 32 Units and evaluation 72.1 Numerical accuracy 72.2 Atomic-sized units 82.3 Units based on emission 113 Pre-quantum models 133.1 Discovery of electron emission 133.2 The Drude model and Maxwell–Boltzmann statistics 133.3 The challenge of photoemission 194 Statistics 254.1 Distinguishable particles 254.2 Probability and states 284.3 Probability and entropy 304.4 Combinatorics and products of probability 335 Maxwell–Boltzmann distribution 375.1 Classical phase space 375.2 Most probable distribution 395.3 Energy and entropy 415.4 The Gibbs paradox 425.5 Ideal Gas in a potential gradient 445.6 The grand partition function 455.7 A nascent model of electron emission 466 Quantum distributions 496.1 Bose–Einstein distribution 496.2 Fermi–Dirac distribution 506.3 The Riemann zeta function 506.4 Chemical potential 526.5 Classical to quantum statistics 566.6 Electrons and white dwarf stars 577 A box of electrons 617.1 Scattering 617.2 From classical to quantum mechanics 617.3 Moments and distributions 637.4 Boltzmann’s transport equation 648 Quantum mechanics methods 738.1 A simple model: the prisoner’s dilemma 738.2 Matrices and wave functions 789 Quintessential problems 919.1 The hydrogen atom 929.2 Transport past barriers 1029.3 The harmonic oscillator 110Part II: The canonical equations10 A brief history 12110.1 Thermal emission 12110.2 Field emission 12210.3 Photoemission 12310.4 Secondary emission 12410.5 Space-charge limited emission 12410.6 Resources and further reading 12411 Anatomy of current density 12711.1 Supply function 12811.2 Gamow factor 12811.3 Image charge potential 13112 Richardson–Laue–Dushman equation 13512.1 Approximations 13512.2 Analysis of thermal emission data 13613 Fowler–Nordheim equation 13913.1 Triangular barrier approximation 14013.2 Image charge approximation 14113.3 Analysis of field emission data 14513.4 The Millikan–Lauritsen hypothesis 14614 Fowler–Dubridge equation 14914.1 Approximations 14914.2 Analysis of photoemission data 15315 Baroody equation 15515.1 Approximations 15515.2 Analysis of secondary emission data 16015.3 Subsequent approximations 16116 Child–Langmuir law 16316.1 Constant density approximation 16416.2 Constant current approximation 16516.3 Transit time approximation 16817 A General thermal–field–photoemission equation 17317.1 Experimental thermal–field energy distributions 17517.2 Theoretical thermal–field energy distributions 17617.3 The N(n,s,u) function 18117.4 Brute force evaluation 18917.5 A computationally kind model 19317.6 General thermal–field emission code 198Part III: Exact tunneling and transmission evaluation18 Simple barriers 20918.1 Rectangular barrier 20918.2 Triangular barrier: general method 21318.3 Triangular barrier: numerical 22219 Transfer matrix approach 22719.1 Plane wave transfer matrix 22719.2 Airy function transfer matrix 23320 Ion enhanced emission and breakdown 24520.1 Paschen’s curve 24520.2 Modified Paschen’s curve 24720.3 Ions and the emission barrier 250Part IV: The complexity of materials21 Metals 25721.1 Density of states, again 25721.2 Spheres in d dimensions 25921.3 The Kronig Penny model 26121.4 Atomic orbitals 26421.5 Electronegativity 26621.6 Sinusoidal potential and band gap 26921.7 Ion potentials and screening 27222 Semiconductors 27722.1 Resistivity 27722.2 Electrons and holes 27922.3 Band gap and temperature 28122.4 Doping of semiconductors 28122.5 Semiconductor image charge potential 28622.6 Dielectric constant and screening 28723 Effective mass 29123.1 Dispersion relations 29123.2 The k ⋅ p method 29323.3 Hyperbolic relations 29623.4 The alpha semiconductor model 29923.5 Current and effective mass 30124 Interfaces 30324.1 Metal–insulator–metal current density 30324.2 Band bending 31024.3 Accumulation layers 31124.4 Depletion layers 31924.5 Modifications due to non-linear potential barriers 32425 Contacts, conduction, and current 32925.1 Zener breakdown 32925.2 Poole–Frenkel transport 32925.3 Tunneling conduction 33325.4 Resonant tunneling in field emission 33626 Electron density near barriers 34126.1 An infinite barrier 34126.2 Two infinite barriers 34426.3 A triangular well 34626.4 Density and dipole component 34827 Many-body effects and image charge 35327.1 Kinetic energy 35327.2 Exchange energy 35427.3 Correlation term 35627.4 Core term 35727.5 Exchange-correlation and a barrier model 36028 An analytic image charge potential 36328.1 Work function and temperature 36328.2 Work function and field 36328.3 Changes to current density 366Part V: Application physics29 Dispenser cathodes 37129.1 Miram curves and the longo equation 37129.2 Diffusion of coatings 37529.3 Evaporation of coatings 39129.4 Knudsen flow through pores 39329.5 Lifetime of a sintered wire controlled porosity dispenser cathode 39930 Field emitters 40330.1 Field enhancement 40330.2 Hemispheres and notional emission area 40630.3 Point charge model 40830.4 Schottky’s conjecture 41230.5 Assessment of the tip current models 41530.6 Line charge models 41730.7 Prolate spheroidal representation 42030.8 A hybrid analytic-numerical model 42530.9 Shielding 43330.10 Statistical variation 43831 Photoemitters 44331.1 Scattering consequences 44631.2 Basic theory 44831.3 Three-step model 44931.4 Moments model 45131.5 Reflectivity and penetration factors 45731.6 Lorentz–Drude model of the dielectric constant 45831.7 Scattering contributions 46631.8 Low work function coatings 47831.9 Quantum efficiency of a cesiated surface 48532 Secondary emission cathodes 48732.1 Diamond amplifier concept 48732.2 Monte Carlo methods 49432.3 Relaxation time 49932.4 Monte Carlo and diamond amplifier response time 51633 Electron beam physics 52533.1 Electron orbits and cathode area 52633.2 Beam envelope equation 52833.3 Emittance for flat and uniform surfaces 53333.4 Emittance for a bump 54533.5 Emittance and realistic surfaces 563Part VI: AppendicesAppendix 1 Summation, integration, and differentiation 569A1.1 Series 569A1.2 Integration 569A1.3 Differentiation 577A1.4 Numerical solution of an ordinary differential equation 582Appendix 2 Functions 585A2.1 Trigonometric functions 585A2.2 Gamma function 585A2.3 Riemann zeta function 585A2.4 Error function 587A2.5 Legendre polynomials 587A2.6 Airy functions 588A2.7 Lorentzian functions 590Appendix 3 Algorithms 591A3.1 Permutation algorithm 591A3.2 Birthday algorithm 592A3.3 Least squares fitting of data 593A3.4 Monty Hall algorithm 595A3.5 Wave function and density algorithm 596A3.6 Hydrogen atom algorithms 598A3.7 Root-finding Methods 601A3.8 Thermal–field algorithm 604A3.9 Gamow factor algorithm 606A3.10 Triangular barrier D(E) 607A3.11 Evaluation of Hc(u) 608A3.12 Transfer matrix algorithm 610A3.13 Semiconductors and doping density 616A3.14 Band bending: accumulation layer 618A3.15 Simple ODE solvers 619A3.16 Current through a metal–insulator–metal diode 622A3.17 Field emission from semiconductors 624A3.18 Roots of the quadratic image charge barrier 626A3.19 Zeros of the airy function 627A3.20 Atomic sphere radius rs 629A3.21 Sodium exchange-correlation potential 631A3.22 Field-dependent work function 632A3.23 Digitizing an image file 632A3.24 Lattice gas algorithm 633A3.25 Evaluation of the point charge model functions 636A3.26 Modeling of field emitter I(V) data 638A3.27 Modeling a log-normal distribution of field emitters 640A3.28 Simple shell and sphere algorithm 643A3.29 Gyftopoulos–Levine work function algorithm 645A3.30 Poisson distributions 648A3.31 Electron–electron relaxation time 650A3.32 Resistivity and the Debye temperature 651A3.33 Orbits in a magnetic field 655A3.34 Trajectory of a harmonic oscillator 657A3.35 Trajectories for emission from a hemisphere 658A3.36 Monte Carlo and integration 660References 663Index 683