Charged particle beams - s humphries

Charged Particle Beams
I-1
Charged Particle Beams
Stanley Humphries, Jr.
Department of Electrical and Computer
Engineering
University of New Mexico
Albuquerque, New Mexico
Originally published in1990 by John Wiley and Sons (QC786.H86 1990, ISBN 0-471-
60014-8). Copyright ©2002 by Stanley Humphries, Jr. All rights reserved. Reproduction
of translation of any part of this work beyond that permitted by Section 107 or 108 of the
1976 United States Copyright Act without the permission of the copyright owner is
unlawful. Requests for permission or further information should be addressed to Field
Precision, Attn: Stanley Humphries, PO Box 13595, Albuquerque, NM 87192.
Charged Particle Beams
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Preface
________
Charged Particle Beams is the product of a two-term course sequence that I taught on
accelerator technology and beam physics at the University of New Mexico and at Los Alamos
National Laboratory. The material for the two terms was divided into the dynamics of single
charged particles and the description of large groups of particles (the collective behavior of
beams). A previous book, Principles of Charged Particle Acceleration (available on the
Internet at covered single particle topics such as linear
transfer matrices and the operation of accelerators. The new book is an introduction to charged-
particle-beam physics.
In writing Charged Particle Beams my goal was to create a unified description that would be
useful to a broad audience: accelerator designers, accelerator users, industrial engineers, and
physics researchers. I organized the material to provide beginning students with the background
to understand advanced literature and to use accelerators effectively. This book can serve as an
independent reference. Combining Charged Particle Beams with Principles of Charged

Particle Acceleration gives a programmed introduction to the field of particle acceleration. I
began my research on particle beams with a background in plasma physics. This change in
direction involved a difficult process of searching for material, learning from experts, and
seeking past insights. Although I found excellent advanced references on specialized areas, no
single work covered the topics necessary to understand high-power accelerators and
high-brightness beams. The difficulties I faced encouraged me to write Charged Particle
Beams. The book describes the basic ideas behind modern beam applications such as stochastic
cooling, high-brightness injectors and the free-electron laser.
Charged Particle Beams
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I was fortunate to have abundant help creating this book. Richard Cooper of Los Alamos
National Laboratory applied his proofreading ability to the entire manuscript. In additional to
mechanical corrections, his suggestions on technical points and emphasis were invaluable. The
creation of this book was supported in part by a sabbatical leave from the Department of
Electrical and Computer Engineering at the University of New Mexico. David Woodall. former
Chairman of the Department of Chemical and Nuclear Engineering at the University of New
Mexico, suggested the idea of the accelerator course sequence. I am grateful for his support
during the development of the courses.
Several people contributed advice on specific sections of the book. Commentators included
Kevin O'Brien of Sandia National Laboratories, John Creedon of Physics International
Company, Brendan Godfrey of the Air Force Weapons Laboratory, Edward Lee of Lawrence
Berkeley Laboratory, William Herrmannsfeldt of the Stanford Linear Accelerator Center, and
Carl Ekdahl of Los Alamos National Laboratory. I would also like to thank A. V. Tollestrup of
Fermi National Accelerator Laboratory for permission to paraphrase his article (coauthored by
G. Dugan) on Elementary Stochastic Cooling.
I want to express appreciation to the students in my beam physics course at the University of
New Mexico and at the Los Alamos Graduate Center. Through their contributions, I clarified and
expanded the material over several years. Los Alamos National Laboratory supported the
courses since their inception. I want to thank Robert Jameson and Alan Wadlinger of the
Accelerator Technology Division for their encouragement. The efforts of the Instructional

Television Center of UNM made it feasible to present classes at Los Alamos. I have also taught
the material in short course format. I am grateful to Thomas Roberts and Stanley Pruett for
organizing a course at the Strategic Defense Command.
Several accelerator science groups helped in the development of material for the book. I have
worked closely with the Heavy Ion Fusion Accelerator Research Group at Lawrence Berkeley
Laboratory for several years. I want to thank Henry Rutkowski, Thomas Fessenden, Denis Keefe
and Edward Lee for their suggestions on the book and for providing the opportunity to work in
the field of accelerator inertial fusion. The long-term support of Charles Roberson of the Office
of Naval Research has been critical for accelerator research at the University of New Mexico.
The University has also received generous research support from Groups CLS-7 and P-14 of the
Los Alamos National Laboratory. I am grateful to Roger Bangerter and the late Kenneth Riepe
who initiated the UNM program on vacuum arc plasma sources. I would also like to thank Carl
Ekdahl – much of the material in this book evolved from spirited discussions on high-current
beam physics.
Charged Particle Beams
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During the composition of this book, I had the opportunity to participate in several research
programs on high-power accelerators. I would like to thank Ralph Genuario and George
Fraser of Physics International Company, Sidney Putnam of Pulse Sciences Incorporated, Robert
Meger of the Naval Research Laboratory, Martin Nahemow of the Westinghouse Research and
Development Center, Richard Adler of North Star Research Corporation, Daniel Sloan of
CH2M-Hill, Kenneth Moses of Jaycor, and R. Bruce Miller of Titan Technologies. I would like
to acknowledge two meetings that I attended during the creation of the book. The first is the
NATO Workshop on High Brightness Beams in Pitlochry, Scotland. I express my appreciation to
Anthony Hyder for organizing this workshop. I have particularly enjoyed participating in the
U.S. Particle Accelerator Schools organized by Melvin Month.
Finally, I would to thank John Wiley and Sons Incorporated for graciously reverting the
copyright on this book so I could prepare this Internet version.
STANLEY HUMPHRIES, JR.
Albuquerque, New Mexico

November 2002
Charged Particle Beams
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Contents
________
1. Introduction 1
1.1. Charged particle beams 1
1.2. Methods and organization 6
1.3. Single-particle dynamics 9
2. Phase space description of charged particle beams 20
2.1. Particle trajectories in phase space 22
2.2. Distribution functions 28
2.3. Numerical calculation of particle orbits with beam-generated forces 32
2.4. Conservation of phase space volume 36
2.5. Density and average velocity 46
2.6. Maxwell distribution 49
2.7. Collisionless Boltzmann equation 52
2.8. Charge and current density 56
2.9. Computer simulations 60
2.10. Moment equations 65
2.11. Pressure force in collisionless distributions 71
2.12. Relativistic particle distributions 76
Charged Particle Beams
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3. Introduction to beam emittance 79
3.1. Laminar and non-laminar beams 80
3.2. Emittance 87
3.3. Measurement of emittance 93
3.4. Coupled beam distributions, longitudinal emittance, normalized
emittance, and brightness 101

3.5 Emittance force 107
3.6. Non-laminar beams in drift regions 109
3.7. Non-laminar beams in linear focusing systems 113
3.8. Compression and expansion of non-laminar beams 128
4. Beam emittance - advanced topics 133
4.1. Linear transformations of elliptical distributions 134
4.2. Transport parameters from particle orbit theory 145
4.3. Beam matching 150
4.4. Non-linear focusing systems 157
4.5. Emittance in storage rings 167
4.6. Beam cooling 174
5. Introduction to beam-generated forces 187
5.1. Electric and magnetic fields of beams 188
5.2. One-dimensional Child law for non-relativistic particles 195
5.3. Longitudinal transport limits for a magnetically-confined electron beams 204
5.4. Space-charge expansion of a drifting beam 211
5.5. Transverse forces in relativistic beams 216
Charged Particle Beams
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6. Beam-generated forces - advanced topics 224
6.1. Space-charge-limited flow with an initial injection energy 225
6.2. Space-charge-limited flow from a thermionic cathode 227
6.3. Space-charge-limited flow in spherical geometry 232
6.4. Bipolar flow 239
6.5. Space-charge-limited flow of relativistic electrons 242
6.6. One-dimensional self-consistent equilibrium 246
6.7. KV distribution 256
7. Electron and ion guns 262
7.1. Pierce method for gun design 263
7.2. Medium perveance guns 271

7.3. High perveance guns and ray tracing codes 277
7.4. High current electron sources 283
7.5. Extraction of ions at a free plasma boundary 289
7.6. Plasma ion sources 300
7.7. Charged-particle extraction from grid-controlled plasmas 315
7.8. Ion extractors 322
8. High power pulsed electron and ion diodes 328
8.1. Motion of electrons in crossed electric and magnetic fields 329
8.2. Pinched electron beam diodes 337
8.3. Electron diodes with strong applied magnetic fields 346
8.4. Magnetic insulation of high power transmission lines 351
8.5. Plasma erosion 356
8.6. Reflex triode 364
8.7. Low-impedance reflex triode 370
8.8. Magnetically-insulated ion diode 377
Charged Particle Beams
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8.9. Ion flow enhancement in magnetically-insulated diodes 388
9. Paraxial beam transport with space-charge 395
9.1. Envelope equation for sheet beams 396
9.2. Paraxial ray equation 400
9.3. Envelope equation in a quadrupole lens array 407
9.4 Limiting current for paraxial beams 412
9.5. Multi-beam ion transport 419
9.6. Longitudinal space-charge limits in RF accelerators and induction linacs 423
10. High current electron beam transport under vacuum 432
10.1. Motion of electrons through a magnetic cusp 433
10.2. Propagation of beams from an immersed cathode 439
10.3. Brillouin equilibrium of a cylindrical electron beam 445
10.4. Interaction of electrons with matter 451

10.5. Foil focusing of relativistic electron beams 457
10.6. Walle-charge and return-current for a beam in a pipe 470
10.7. Drifts of electron beams in a solenoidal field 477
10.8. Guiding electron beams with solenoidal fields 482
10.9. Electron beam transport in magnetic cusps 490
11. Ion beam neutralization 501
11.1. Neutralization by comoving electrons 502
11.2. Transverse neutralization 511
11.3. Current neutralization in vacuum 517
11.4. Focal limits for neutralized ion beams 522
11.5. Acceleration and transport of neutralized ion beams 528
Charged Particle Beams
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12. Electron beams in plasmas 535
12.1. Space-charge neutralization in equilibrium plasmas 536
12.2. Oscillations of an un-magnetized plasma 540
12.3. Oscillations of a neutralized electron beam 546
12.4 Injection of a pulsed electron beam into a plasma 552
12.5. Magnetic skin depth 563
12.6. Return current in a resistive plasma 569
12.7. Limiting current for neutralized electron beams 577
12.8. Bennett equilibrium 583
12.9. Propagation in low-density plasmas and weakly-ionized gases 587
13. Transverse instabilities 592
13.1. Instabilities of space-charge-dominated beams in periodic
focusing systems 594
13.2. Betatron waves on a filamentary beam 610
13.3. Frictional forces and phase mixing 615
13.4. Transverse resonant modes 622
13.5. Beam breakup instability 631

13.6. Transverse resistive wall instability 640
13.7. Hose instability of an electron beam in an ion channel 645
13.8. Resistive hose instability 655
13.9. Filamentation instability of neutralized electron beams 664
14. Longitudinal instabilities 674
14.1. Two-stream instability 675
14.2. Beam-generated axial electric fields 687
14.3. Negative mass instability 697
14.4. Longitudinal resistive wall instability 704
Charged Particle Beams
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15. Generation of radiation with electron beams 720
15.1. Inverse diode 722
15.2. Driving resonant cavities with electron beams 736
15.3. Longitudinal beam bunching 749
15.4. Klystron 762
15.5. Traveling wave tube 772
15.6. Magnetron 781
15.7. Mechanism of the free-electron laser 796
15.8. Phase dynamics in the free-electron laser 803
Bibliography
Index
Introduction Charged Particle Beams
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1
Introduction
__________
1.1 CHARGED PARTICLE BEAMS
A charged particle beam is a group of particles that have about the same kinetic energy and
move in about the same direction. Usually, the kinetic energies are much higher than the thermal

energies of particles at ordinary temperatures. The high kinetic energy and good directionality of
charged particles in beams make them useful for applications. Although we often associate
accelerators with the large machines of high-energy physics, charged particle beams have
continually expanding applications in many branches of research and technology. Recent active
areas include flat-screen cathode-ray tubes, synchrotron light sources, beam lithography for
microcircuits, thin-film technology, production of short-lived medical isotopes, radiation
processing of food, and free-electron lasers.
The importance of accelerators for applications in research and industry sometimes
overshadows beam physics as an intellectual discipline in its own right. The theory of charged
particle beams is much more than a tool to design machines - it is one of the richest and most
active areas of classical physics. In our study of charged particle beams, we shall gain a
comprehensive understanding of applied electromagnetism and collective physics.
Despite the practical importance and underlying unity of beam physics, the field has not yet
achieved a strong identity like plasma physics. Although there are many specialized review
papers and texts, few general works cover the full range of beam processes. There are several
reasons for fragmentation in the field. Accelerator scientists are largely goal-oriented,
concentrating on the theory and technology to solve the problem at hand. Each large accelerator
has its own mission and its own group of scientists. Because of the broad range of required
beam parameters, different accelerators use a diversity of technologies that often have little in
Introduction Charged Particle Beams
2
(1.1)
common. Although there are large differences in technology, we shall see that a few basic
principles enter into the the design of all accelerators and beam transport devices. As the
problems of accelerators become more challenging and beam applications become more
sophisticated, it is increasingly important for accelerator scientists to share their insights and
expertise. In recent years, there have been
several efforts to emphasize the unity of the field and to promote communication between
researchers. In the United States, examples include the Particle Accelerator Conference with its
steadily increasing attendance from all areas of accelerator research, the U.S. Particle

Accelerator School and its educational publications, and the recently-formed American Physical
Society Division of Accelerator Physics.
This book was written to guide students entering accelerator science and to provide
researchers with a comprehensive reference. It contains a unified treatment of beam physics at an
introductory level. This book and a previous one, Principles of Charged Particle
Acceleration, provide a bridge to carry students to advanced work in specialized fields of
accelerator science and beam theory. Principles of Charged Particle Acceleration reviews the
fundamentals of single particle dynamics. The book describes how accelerators work, from
small low-current devices to the largest machines of high-energy and high-power research. The
present book concentrates on problems of beam physics, the acceleration and control of large
numbers of charged particles. The range of topics is extensive, with reference material for
designers and users of all types of accelerators. Colored equation numbers indicate important
relationships.
In this section, we begin by reviewing properties of charged particles and particle beams. Sect.
1.2 discusses some of the problems of collective physics and outlines the organization of the
book. Sect. 1.3 summarizes some results from Principles of Charged Particle Acceleration that
will be useful for many of the derivations. The goal of beam theory is to describe how the
multitude of particles in a beam interact with one another. For this purpose, we need not consider
the internal structure of charged particles. Usually, it is sufficient to represent a particle as point
entity with two properties: charge, q, and rest mass, m
o
. We assume that the particle
characteristics are constant during acceleration and transport. In this book, we will not examine
the effects of finite particle dimensions and quantum properties such as spin. Except for
specialized applications, these properties have little effect on the formation and acceleration of
beams.
Much of the material in this book applies to any charged particle, from the elementary particles
of high-energy physics research to hyper-velocity charged dust projectiles. Familiar applications
usually involve one of two types of particles: electrons or ions. The electron is an elementary
particle with the following characteristics:

Introduction Charged Particle Beams
3
(1.2)
(1.3)
(1.4)
(1.5)
(1.6)
We shall apply SI units exclusively throughout the book with the exception of the electron volt,
a useful unit for the energy of individual particles.
Ions are composite particles. An ion is an atom missing one or more electrons. The following
quantities characterize an ion:
A: the atomic mass number, equal to the total number of protons and neutrons in the nucleus.
Z: the atomic number, equal to the number of electrons in the neutral atom.
Z
*
: the charge state of the ion, equal to the number of electrons removed from the atom.
The proton is the simplest ion - it is a hydrogen atom with its single electron removed. The
proton is an elementary particle with charge and mass:
We denote the charge and mass of other ions as:
The rest energy of a particle equals the rest mass multiplied by the square of the speed of light.
The rest energy of an electron is: If the kinetic energy of a particle
approaches or exceeds its rest energy, we must use relativistic equations of motion. The SI energy
unit of joules is not convenient for individual charged particles. The standard energy unit in beam
physics is the electron volt (eV). One electron volt equals the change in kinetic energy of an
electron or proton that crosses a potential difference of 1 V, or:
The electron rest energy in electron volts is:
When an electron accelerates through a potential difference of 5.11 × 10
5
V, the kinetic energy
equals the rest energy. Electrons are relativistic when they have kinetic energy above about 100

Introduction Charged Particle Beams
4
(1.7)
Figure 1.1. Particle orbits in a plasma compared with a beam.
Arrows represent velocity vectors
keV (10
5
eV). The Newtonian equations of motion are approximately correct for electron beams
with kinetic energy below this level. The proton rest mass exceeds the electron mass by a factor
of 1843 - the proton rest energy is correspondingly higher:
Because of the high rest energy, we can use Newtonian dynamics to predict the motion of ions in
many applications.
Although single charged particles may be useful for some physics experiments, we need large
numbers of energetic particles for most applications. A flux of particles is a beam when the
following two conditions hold:
1. The particles travel in almost the same direction.
2. The particles have a small spread in kinetic energy.
A beam is an ordered flow of charged particles. A disordered set f particles, such as a thermal
plasma, is not a beam. Figure 1.1 illustrates the difference between a beam and a plasma. The
relationship between a charged-particle beam and a plasma is analogous to the relationship
between a laser and a light bulb. The photons from a laser are directed and monochromatic. The
degree of order in a flow of particles is called coherence. A high level of coherence is essential
for most applications. For example, the minimum spot size of a scanning electron microscopic
depends on the parallelism of the electrons in the beam.
Several quantities are useful to characterize charged particle beams, including 1) type of
particle, 2) average kinetic energy, 3) current, 4) power, 5) pulse length, 6) transverse dimension,
7) parallelism, and 8) energy spread. The parameters of charged particle beams for applications
extends over a remarkably large range. Table 1.1 gives estimates of high and low values for
beam
properties. No other field of engineering or applied physics extends over such a broad parameter

space.
Introduction Charged Particle Beams
5
Table 1.1. Charged particle beam parameters for applications
Property Lower limit Upper limit Range
Mass m
e
, 9.1×10
-31
kg 238m
p
, 4.0×10
-25
kg 10
6
Charge e, 1.6×10
-19
C -100e, 1.6×10
-17
C10
2
Kinetic energy #1 eV 10
12
eV 10
12
Current 10
-9
A10
6
A10

15
Power <1 W 10
12
W >10
12
Pulse length <10
-10
s Continuous >10
10
Dimension 10
-6
m >1 m >10
6
Angular spread 10
-6
radians 1 radian 10
6
In conventional accelerators, particle mass spans the range from electrons to the heavy ions
used for nuclear physics and accelerator inertial fusion. The mass of a uranium ion is 238 times
that of the proton, or 440,000 times the electron mass. The charge state of particles in most
accelerators is q = ±e. Heavy ion accelerators are an exception. In these machines,
highly-stripped ions (Z
*
> 50) result when a medium energy beam passes through a thin foil. The
multiply-charged ions then accelerate to high kinetic energy in a linear accelerator.
The kinetic energy of beams for applications spans about twelve orders of magnitude. At the
low end, we shall encounter energies less than 1 eV when we study electron emission from a
thermionic cathode. The current achievements of high-energy physics accelerators define the high
end of the energy spectrum, about 1 TeV (10
12

eV).
The beam current in present devices spans an even broader range, -10
15
. Ion and electron
microprobes have a current of about 1 nA = 10
-9
A. Despite the low flux of such beams, we must
apply collective beam theory to predict the minimum spot size. At the other extreme, pulsed ion
or electron diodes generate beams with current exceeding 1 MA = 10
6
A.
To characterize beam power, we must distinguish between average power and peak power.
Many accelerators have a pulsed duty cycle. The highest peak power, over 10
12
W for -50 ns,
occurs in experiments on inertial fusion. At the low end, commercial devices such as CRT tubes
operate continuously at power levels below 1 W. Continuous machines define the upper limit on
beam pulse length. Resonant accelerators generate trains of very short pulses. Pulse durations
may be less than 100 ps = 10
-10
s.
The maximum transverse dimension of charged particle beams is immense if we include
astrophysical jets. In conventional applications, industrial sheet beam irradiators create the largest
beams, about 2 m in length. Scanning electron microscopes generate small beam spots less than 1
Introduction Charged Particle Beams
6
:m in diameter. The parallelism of orbits in beams also has a wide range. Accelerators under
development for defense applications have a requirement on angular divergence of about 1
:rad.
At the other extreme, intense pinched electron beams may have a divergence angle approaching a

radian, with a spread in longitudinal kinetic energy comparable to the directed energy.
1.2. Methods and organization
The central issue in beam physics is the solution of collective problems involving large numbers
of particles. The orbits of the particles depend on electric and magnetic fields. The fields, in turn,
result partly from contributions of the beam particles. Therefore, the field values depend on the
positions and velocities of all particles. An exact prediction of beam behavior demands the
simultaneous calculation of every particle orbit. The challenge is formidable - a low current beam
may contain more than 10
10
particles. Clearly, exact solutions are impossible, even with the most
powerful computers.
Collective physics is a science of approximation. Predictions involve insight and experience -
to solve problems, we must eliminate unnecessary material but preserve the essential processes.
Beam physics can be difficult for beginning students because there are no cut-and-dried methods.
Each calculation demands a careful analysis and a reduction with simplifying assumptions. One
goal of this book is give some insight into collective problems. The material of the book was
organized with this goal in mind:
1. The order of topics is from the simplest to the most complex. Ideally, the reader should
follow the text from beginning to the end. The early chapters give background material necessary
to understand advanced subjects like beam instabilities.
2. In collective problems, the initial analysis and reduction is as important as the correct
mathematical solution of the equations. The best mathematical methods are useless if the
statement of the problem is not physically correct. Therefore, we shall concentrate on setting up
problems, carefully listing all limiting conditions. After defining the governing equations, we
shall apply
straightforward mathematical methods to find a solution.
3. A frustrating problem in many advanced works on beam physics is that the derivations often
have missing steps. These leaps may be obvious to the author but are obscure to non-experts. To
avoid this difficulty, we shall follow all stages of derivations at the expense of some repetition.
This book is an introductory text. It does not address sophisticated methods of mathematical

analysis, the history of beam physics, or the vast range of advanced literature. The
references in Appendix1 are a good starting point for further reading on advanced topics.
The material in Chaps. 1 through 6 is the foundation for later chapters. Chapter 2 is a capsule
summary of collective physics with an emphasis on charged particle beam theory. Collective
physics organizes information about the motion of large numbers of particles. Rather than
calculate the orbits of individual particles, we try to identify general trends in behavior. The
Introduction Charged Particle Beams
7
best way to organize information about particles is to plot orbit vectors in phase-space. The
theorem of conserved particle density in phase space leads to the fundamental equation of
collective physics, the Boltzmann equation. From this relationship, we derive moment equations
that describe the conservation of particles, momentum and energy in large groups.
The introductory accelerator theory of Principles of Charged Particle Acceleration
concentrates on the orbits of single particles or on laminar beams where all orbits are similar. In
Chapter 3, we remove this limitation and study beams where the particles have random spreads in
direction and energy. Real beams always have such a diversity of orbits - to design accelerators,
we must understand the limitations set by beam imperfections. Chapter 3 defines emittance, a
quantity that characterizes the parallelism of beams. The principle of emittance conservation has
extensive applications to accelerators and beam optics systems.
Chapter 4 discusses consequences of beam emittance in low current beams with small
space-charge forces. The first three sections define the transport parameters of a beam and review
transport theory. This theory is useful for the the design of beam transport systems. Section 4.4
reviews imperfections in charged particle lenses and how they contribute to the growth of beam
emittance. The final two sections discuss the importance of beam emittance in storage rings and
beam colliders. We shall study methods that circumvent the principle of phase-volume
conservation to produce beams with low spreads in direction and energy.
Chapter 5 discusses equilibrium effects of beam-generated electric and magnetic fields. The
chapter introduces the idea of self-consistent calculations. Here, we follow the motion of beam
particles in fields that depend on the instantaneous position of all other particles. The Child
derivation is the prototype calculation of a self-consistent beam equilibrium. It leads to the Child

limit, a constraint on the current density from a beam extractor. Chapter 6 uses the expressions
for the fields generated by equilibrium beams to calculate one-dimensional current flow in several
practical cases. The chapter also introduces the KV distribution, a starting point for self-consistent
models of two-dimensional equilibria.
Chapters 7 and 8 introduce methods to create beams, while Chapters 9 through 12 discuss beam
transport and acceleration. Chapter 7 deals with electron and ion guns at low to medium current.
Sections 7.1 through 7.3 review design techniques for guns. Section 7.4 discusses electron
sources, while Sections. 7.5 and 7.6 review ion sources and ion extraction from plasmas. The
final two sections describe methods to generate large-area, high-current ion beams.
Chapter 8 is devoted to high-power pulsed electron and ion diodes. These devices use pulsed
power technology to generate beams with very high current. Sect. 8.1 discusses the motion of
electrons in crossed electric and magnetic fields - the resulting equations are also useful for
conventional devices like the magnetron. The next two sections review the generation of pulsed
electron beams. Sections 8.4 and 8.5 discuss two important processes for diode technology,
magnetic insulation and plasma erosion. The final four sections cover methods to create pulsed
ion beams with current density far beyond the Child limit.
In Chapter 9, we begin the study of beam transport. This chapter discusses the effect of
space-charge and emittance on beams in conventional accelerators. The beams in these devices
are
paraxial - particle orbits make small angles with respect to the axis. Sections 9.1 through 9.3
Introduction Charged Particle Beams
8
derive envelope equations for beams in several focusing systems. These equations, based on
transverse force balance, are important first-order design tools for beam optics systems. Section
9.4 applies the equations to define the maximum beam current in accelerators. Sect. 9.5 describes
multiple-beam transport, a method to circumvent current limitations. The final section reviews
limitations on beam power set by axial space-charge forces.
Chapters 10 through 12 describe methods to control high-power electron and ion beams.
Chapter 10 concentrates on high-current electron beams in vacuum - the material is useful for
applications such as microwave tubes. Solenoid lenses are effective for containment of

low-energy electron beams - the first three sections describe electron motion and linear beam
propagation in axial magnetic fields. Section 10.5 describes methods to focus relativistic beams
with thin foils or meshes. As background, Section 10.4 summarizes the scattering and energy loss
of electrons passing through matter. Section 10.6 derives the charge and current distributions
induced by beams in surrounding metal structures - the relationships are important for latter
calculations of beam stability. Sections 10.7 through 10.9 treat the steering and focusing of
high-current electron beams in curved transport systems.
In Chapter 11, we turn our attention to high-current ion beams. Moderate energy ions have low
velocity - for the same current and energy, an ion beam has higher space-charge than an electron
beam. Therefore, it is difficult to transport high flux ion beams through vacuum. High current ion
beams must be neutralized - the addition of low-energy electrons reduces the beam-generated
electric fields. Sections 11.1 through 11.3 describe methods to add electrons to ion beams. Section
11.4 reviews focal limits on neutralized ion beams, while Section 11.5 describes methods to
control and to accelerate high-flux ion beams.
Chapter 12 discusses the propagation of electron beams through plasmas. We shall review the
properties of plasmas that affect their response to all types of pulsed beams. Sections 12.1 and
12.2 introduce two basic plasma quantities, the Debye length and the plasma frequency. Sections
12.3 applies the theory of plasma oscillations to describe the transverse motion of an electron
beam in an ion column. Sections 12.4 through 12.6 concentrate on plasma responses to pulsed
electron beams. Sections. 12.7 though 12.9 review the properties of beam equilibria in plasmas
and processes that limit beam current and propagation length.
Chapters 13 and 14 cover instabilities, spontaneous departures from equilibrium driven by the
free energy of beams. The theoretical description of instabilities is a challenge - we must handle
time-dependent effects of beam-generated electric and magnetic fields with self-consistent
methods. The field of beam instabilities is broad. It would take an entire chapter just to list the
processes covered in the literature. Instead, we shall concentrate on a few important examples.
The detailed discussions illustrate methods and insights that have application to the full range of
instability calculations.
Most beam instabilities involve the transfer of axial kinetic energy to undesired random
motions. We use the term transverse instability when the energy contributes to transverse particle

motion. These disturbances can lead to increased emittance or to sweeping motions of the beam.
Sections 13.2 through 13.4 review background material. Section 13.2 classifies transverse beam
oscillations in focusing systems. Section 13.3 summarizes the effects of wall resistance and a
spread of particle momentum on coherent oscillations. Section 13.4 reviews the theory of
Introduction Charged Particle Beams
9
(1.8)
transverse resonant modes in accelerator cavities. These modes can effectively couple the beam
kinetic energy to transverse oscillations. The other sections in the chapter describe collective
instabilities for a broad range of accelerators.
Chapter 14 discusses longitudinal instabilities. Here, the kinetic energy of a mono-energetic
beam couples to an axial velocity spread. The resulting momentum dispersion can interfere with
beam containment and focusing. Section 14.2 derives reference expressions for axial electric
fields in perturbed beams. The other sections contain descriptions of specific instabilities.
In Chapter 15 we shall study an important application of charged particle beams, the generation
of electromagnetic radiation. The description of radiation sources intimately involves beam
theory. Besides reviewing some practical microwave devices, the chapter introduces material of
general interest. Section 15.2 covers the use of resonant cavities as impedance transformers.
Cavities convert beam energy at high voltage and low current to microwave energy at low
voltage and high current. Section 15.3 describes axial beam bunching, a process critical to the
operation of the klystron and RF accelerators. The other sections cover several ways to convert
the energy of a beam to electromagnetic radiation: the inverse diode, the klystron, the traveling
wave tube, and the magnetron. To conclude the book, Sections 15.7 and 15.8. introduce the
theory of the free-electron laser.
1.3. Single-particle dynamics
In the following chapters, we shall study the behavior of ordered groups of charged particles.
Although much of the material is introduced as needed, the reader must have a good preliminary
knowledge of single-particle dynamics. The companion book Principles of Charged Particle
Acceleration provides the necessary prerequisites. In this section, we shall summarize important
background equations for later reference. The symbol [CPA] appears throughout this book to

reference sections of Principles of Charged Particle Acceleration with relevant supplementary
material.
A. Particle dynamics
We shall construct theories of collective beam behavior by summing over the orbits of many
individual charged particles. We use equations of motion to predict single particle orbits. Four
quantities specify the status of a charged particle: the rest energy, m
o
, the charge, q, the vector
position, x, and the vector velocity, v. In most of the derivations in following chapters, q and m
o
are constant while x and v change. The velocity causes a change in position:
Although the rest mass of a particle is constant, the special theory of relativity states that the
inertia of a particle observed in a frame of reference depends on the magnitude of its speed in that
Introduction Charged Particle Beams
10
(1.9)
(1.10)
(1.11)
(1.12)
(1.13)
(1.14)
(1.15)
(1.16)
(1.17)
frame. In Cartesian coordinates, the particle speed is
Relativistic dynamics uses the special function of v:
where c is the speed of light:
The quantity ( is always greater than unity because the observed speed of a particle can never
equal or exceed c. Another useful parameter is the ratio of the particle speed to c:
Substituting Eq. (1.12) in Eq. (1.10) gives:

The inertia of a particle is proportional to (. The apparent mass is:
The particle momentum, a vector quantity, equals:
In response to a force, F, the momentum changes as:
Introduction Charged Particle Beams
11
(1.18)
(1.19)
(1.20)
(1.21)
(1.22)
(1.23)
(1.24)
(1.25)
(1.26)
If the force is a known function of x and v, we can use numerical methods to calculate the orbit
of a particle by a simultaneous solution of Eqs. (1.8) and (1.17). In relativistic dynamics, the total
energy of a particle is mc
2
= (m
o
c
2
. The kinetic energy equals the total energy minus the rest
energy:
Newtonian dynamics describes the motion of low-energy particles when:
In the non-relativistic limit, ( • 1. Here, the equations of particle dynamics are:
Equations (1.20)-(1.24) have a simpler form than the relativistic equivalents - it is usually easier
to find analytic solutions to non-relativistic problems.
Sometimes, we can use modified Newtonian equations to describe the transverse motion of
relativistic particles in beams. In most beams, the transverse velocities of particles are much

smaller than the axial velocities. We shall consistently use the coordinate z as the average
direction of beam motion; therefore, v
x
, v
y
n v
z
. For transverse particle motion with no
acceleration in z, the total particle energy is almost constant. If we take
( as a constant, Eqs. (1.8)
and (1.17) become:
Similar equations hold for the y-direction. Equations (1.25) and (1.26 ) have the form of
Newtonian equations of motion with a modified mass, (m
o
.
Introduction Charged Particle Beams
12
(1.27)
(1.28)
(1.29)
(1.30)
(1.31)
(1.32)
(1.33)
B. Electromagnetic forces
The motion of charged particles in accelerators depends almost entirely on electromagnetic
forces. The Lorentz force expression for a particle with charge q and velocity v is:
The values of electric and magnetic field in Eq. (1.27) are evaluated at the instantaneous position
of the particle.
We calculate electric and magnetic fields from the Maxwell equations using known

distributions of charge and current density. The charge density, D, is a scalar quantity with units
coulombs per cubic meter. The current density, j, is a vector quantity with units of amperes per
square meter. Several sources can contribute to the net charge density, including charges
deposited on electrodes by external power supplies (D
applied
), displaced charges in dielectric
materials (D
dielectric
), and the charge of beam particles moving freely through vacuum (D
space
). The
total current density may have contributions from applied currents in magnet coils (j
applied
), from
atomic currents in ferromagnetic materials (j
atomic
), and from beam particles (j
space
). The Maxwell
equations are:
The summation symbols denote the sum of all contributions to the charge and current density.
The constants in Eqs. (1.28) and (1.31) are:
and
Introduction Charged Particle Beams
13
(1.34)
(1.35)
(1.36)
(1.37)
(1.38)

(1.39)
(1.40)
(1.41)
Often, the portion of the electric field created by charges in dielectric materials is linearly
proportional to the total electric field in the material. In this case Eq. (1.28) becomes,
where
, is a constant that depends on the material, , $ ,
o
. The symbol D
free
represents all
contributions to the charge density except the charges bound in the dielectric. We can make
similar definitions for the effect of atomic currents in ferromagnetic materials. In the special case
that field components arising from dielectric and ferromagnetic materials are linearly proportional
to the total fields in the materials, we can write the Maxwell equations in an alternative form:
The constant : depends on the magnetic properties of the material; here, j
free
is the total current
density from all sources except the ferromagnetic material. We can characterize dielectrics in
terms of the relative dielectric constant:
and magnetic materials in terms of the relative magnetic permeability:
C. Coordinate transformations
In derivations of the following chapters, it is often useful to change between different frames of
Introduction Charged Particle Beams
14
(1.42)
(1.43)
(1.44)
(1.45)
(1.46)

(1.47)
(1.48)
reference. We often define two special frames. The stationary frame is the rest frame of the
physical devices that accelerate and confine a beam. In this frame, the beam moves at average
velocity v
z
= $
z
c. The beam rest frame moves at velocity $
z
c relative to the stationary frame. In
this frame, particles are at rest if the beam has no axial velocity spread.
The Lorentz transformations relate position and velocity between two frames of reference in
relative motion. Suppose that we determine the position and velocity of a particle in a frame that
we consider stationary - the measured quantities are (x,y,z,v
x
,v
y
,v
z
). Consider the viewpoint of an
observer who moves relative to our frame at speed $c in the +z-direction. The observer measures
the position and velocity of the particle as (x’,y’,z’, v
x
', v
y
', v
z
'). The Lorentz transformations give
relationships between the quantities measured in the two frames of reference:

where . Equations. (1.42)-(1.48) hold if we define the origin of time so that z = z'
at t = t' = 0.
Some derivations in later chapters are simplified by transforming electric and magnetic fields
between frames in relative motion. Suppose we measure the quantities (E
x
, E
y
, E
z
, B
x
, B
y
, B
z
) in a
frame that we consider stationary. In a frame in relative motion, the measured field components
are (E
x
’, E
y
’, E
z
’, B
x
’, B
y
’, B
z
’) . If the frame moves at relative velocity v = v

z
z = $cz, the electric
and magnetic fields are related by:
Introduction Charged Particle Beams
15
(1.49)
(1.50)
(1.51)
(1.52)
(1.53)
(1.54)
(1.55)
(1.56)
(1.57)
D. Transfer matrices
Most beam transport devices, such as charged particle lenses and bending magnets, apply
transverse forces that are linearly proportional to the distance of a particle from a preferred axis.
Suppose a device produces a linear transverse force in the x-direction over an axial length. We
want to compare the particle orbit at the exit of the device to the orbit at the entrance. To specify
the orbit of the particle in the x-direction, we must give its position, x, and velocity, v
x
. The
convention in charged-particle optics is to represent particle orbits in terms of their angle relative
to the main axis, rather than the transverse velocity. In the limit that v
x
n v
z
, the angle is
We can symbolize the entrance orbit as a vector, [x
o

, x
o
']. The exit vector is [x
1
, x
1
']. If the
x-directed forces in the device are linear, then we can express the exit vector as a linear
combination of the entrance vector components:
In matrix notation, the relationship is:
The quantities a
mn
depend on the distribution of forces. The focusing effect of any
one-dimensional linear device is specified by the four numbers, a
mn
. The matrix of Eq. (1.57) is