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#### Principles of Charged Particle Acceleration

**By Stanley Humphries Jr.**

**Dover Publications, Inc.**

**Copyright © 2012 Stanley Humphries, Jr.**

All rights reserved.

ISBN: 978-0-486-32063-2

All rights reserved.

ISBN: 978-0-486-32063-2

CHAPTER 1

*Introduction*

This book is an introduction to the theory of charged particle acceleration. It has two primary roles:

1. A unified, programmed summary of the principles underlying all charged particle accelerators.

2. A reference collection of equations and material essential to accelerator development and beam applications.

The book contains straightforward expositions of basic principles rather than detailed theories of specialized areas.

Accelerator research is a vast and varied field. There is an amazingly broad range of beam parameters for different applications, and there is a correspondingly diverse set of technologies to achieve the parameters. Beam currents range from nanoamperes (10-9 A) to megaamperes (106 A). Accelerator pulselengths range from less than a nanosecond to steady state. The species of charged particles range from electrons to heavy ions, a mass difference factor approaching 106. The energy of useful charged particle beams ranges from a few electron volts (eV) to almost 1 TeV (1012 eV).

Organizing material from such a broad field is inevitably an imperfect process. Before beginning our study of beam physics, it is useful to review the order of topics and to define clearly the objectives and limitations of the book. The goal is to present the theory of accelerators on a level that facilitates the design of accelerator components and the operation of accelerators for applications. In order to accomplish this effectively, a considerable amount of potentially interesting material must be omitted:

1. Accelerator theory is interpreted as a mature field. There is no attempt to review the history of accelerators.

2. Although an effort has been made to include the most recent developments in accelerator science, there is insufficient space to include a detailed review of past and present literature.

3. Although the theoretical treatments are aimed toward an understanding of real devices, it is not possible to describe in detail specific accelerators and associated technology over the full range of the field.

These deficiencies are compensated by the books and papers tabulated in the bibliography.

We begin with some basic definitions. A *charged particle* is an elementary particle or a macroparticle which contains an excess of positive or negative charge. Its motion is determined mainly by interaction with electromagnetic forces. *Charged particle acceleration* is the transfer of kinetic energy to a particle by the application of an electric field. A *charged particle beam* is a collection of particles distinguished by three characteristics: (1) beam particles have high kinetic energy compared to thermal energies, (2) the particles have a small spread in kinetic energy, and (3) beam particles move approximately in one direction. In most circumstances, a beam has a limited extent in the direction transverse to the average motion. The antithesis of a beam is an assortment of particles in thermodynamic equilibrium.

Most applications of charged particle accelerators depend on the fact that beam particles have high energy and good directionality. Directionality is usually referred to as *coherence.* Beam coherence determines, among other things, (1) the applied force needed to maintain a certain beam radius, (2) the maximum beam propagation distance, (3) the minimum focal spot size, and (4) the properties of an electromagnetic wave required to trap particles and accelerate them to high energy.

The process for generating charged particle beams is outlined in **Table 1.1**. Electromagnetic forces result from mutual interactions between charged particles. In accelerator theory, particles are separated into two groups: (1) particles in the beam and (2) charged particles that are distributed on or in surrounding materials. The latter group is called the external charge. Energy is required to set up distributions of external charge; this energy is transferred to the beam particles via electromagnetic forces. For example, a power supply can generate a voltage difference between metal plates by subtracting negative charge from one plate and moving it to the other. A beam particle that moves between the plates is accelerated by attraction to the charge on one plate and repulsion from the charge on the other.

Electromagnetic forces are resolved into electric and magnetic components. Magnetic forces are present only when charges are in relative motion. The ability of a group of external charged particles to exert forces on beam particles is summarized in the *applied electric and magnetic fields.* Applied forces are usually resolved into those aligned along the average direction of the beam and those that act transversely. The axial forces are acceleration forces; they increase or decrease the beam energy. The transverse forces are confinement forces. They keep the beam contained to a specific cross-sectional area or bend the beam in a desired direction. Magnetic forces are always perpendicular to the velocity of a particle; therefore, magnetic fields cannot affect the particle's kinetic energy. Magnetic forces are confinement forces. Electric forces can serve both functions.

The distribution and motion of external charge determines the fields, and the fields determine the force on a particle via the Lorentz force law, discussed in **Chapter 3**. The expression for force is included in an appropriate equation of motion to find the position and velocity of particles in the beam as a function of time. A knowledge of representative particle orbits makes it possible to estimate average parameters of the beam, such as radius, direction, energy, and current. It is also possible to sum over the large number of particles in the beam to find charge density ρb and current density *j*b. These quantities act as source terms for beam-generated electric and magnetic fields.

This procedure is sufficient to describe low-current beams where the contribution to total electric and magnetic fields from the beam is small compared to those of the external charges. This is not the case at high currents. As shown in **Table 1.1**, calculation of beam parameters is no longer a simple linear procedure. The calculation must be *self-consistent.* Particle trajectories are determined by the total fields, which include contributions from other beam particles. In turn, the total fields are unknown until the trajectories are calculated. The problem must be solved either by successive iteration or application of the methods of collective physics.

Single-particle processes are covered in this book. Although theoretical treatments for some devices can be quite involved, the general form of all derivations follows the straight-line sequence of **Table 1.1**. Beam particles are treated as test particles responding to specified fields. A continuation of this book addressing collective phenomena in charged particle beams is in preparation. A wide variety of useful processes for both conventional and high-power pulsed accelerators are described by collective physics, including (1) beam cooling, (2) propagation of beams injected into vacuum, gas, or plasma, (3) neutralization of beams, (4) generation of microwaves, (5) limiting factors for efficiency and flux, (6) high-power electron and ion guns, and (7) collective beam instabilities.

An outline of the topics covered in this book is given in **Table 1.2**. Single-particle theory can be subdivided into two categories: *transport* and *acceleration.* Transport is concerned with beam confinement. The study centers on methods for generating components of electromagnetic force that localize beams in space. For steady-state beams extending a long axial distance, it is sufficient to consider only transverse forces. In contrast, particles in accelerators with time-varying fields must be localized in the axial direction. Force components must be added to the accelerating fields for longitudinal particle confinement (phase stability).

Acceleration of charged particles is conveniently divided into two categories: electrostatic and electromagnetic acceleration. The accelerating field in electrostatic accelerators is the gradient of an electrostatic potential. The peak energy of the beam is limited by the voltage that can be sustained without breakdown. Pulsed power accelerators are included in this category because pulse lengths are almost always long enough to guarantee simple electrostatic acceleration.

In order to generate beams with kinetic energy above a few million electron volts, it is necessary to utilize time-varying electromagnetic fields. Although particles in an electromagnetic accelerator experience continual acceleration by an electric field, the field does not require prohibitively large voltages in the laboratory. The accelerator geometry is such that inductively generated electric fields cancel electrostatic fields except at the position of the beam.

Electromagnetic accelerators are divided into two subcategories: nonresonant and resonant accelerators. Nonresonant accelerators are pulsed; the motion of particles need not be closely synchronized with the pulse waveform. Nonresonant electromagnetic accelerators are essentially step-up transformers, with the beam acting as a high-voltage secondary. The class is subdivided into linear and circular accelerators. A linear accelerator is a straight-through machine. Generally, injection into the accelerator and transport is not difficult; linear accelerators are useful for initial acceleration of low-energy beams or the generation of high-flux beams. In circular machines, the beam is recirculated many times through the acceleration region during the pulse. Circular accelerators are well suited to the production of beams with high kinetic energy.

The applied voltage in a resonant accelerator varies harmonically at a specific frequency. The word *resonant* characterizes two aspects of the accelerator: (1) electromagnetic oscillations in resonant cavities or waveguides are used to transform input microwave power from low to high voltage and (2) there is close coupling between properties of the particle orbits and time variations of the accelerating field. Regarding the second property, particles must always be at the proper place at the proper time to experience a field with accelerating polarity. Longitudinal confinement is a critical issue in resonant accelerators. Resonant accelerators can also be subdivided into linear and circular machines, each category with its relative virtues.

In the early period of accelerator development, the quest for high kinetic energy, spurred by nuclear and elementary particle research, was the overriding goal. Today, there is increased emphasis on a diversity of accelerator applications. Much effort in modern accelerator theory is devoted to questions of current limits, beam quality, and the evolution of more efficient and cost-effective machines. The best introduction to modern accelerators is to review some of the active areas of research, both at high and low kinetic energy. The list in **Table 1.3** suggests the diversity of applications and potential for future development.

*Particle Dynamics*

Understanding and utilizing the response of charged particles to electromagnetic forces is the basis of particle optics and accelerator theory. The goal is to find the time-dependent position and velocity of particles, given specified electric and magnetic fields. Quantities calculated from position and velocity, such as total energy, kinetic energy, and momentum, are also of interest.

The nature of electromagnetic forces is discussed in **Chapter 3**. In this chapter, the response of particles to general forces will be reviewed. These are summarized in laws of motion. The Newtonian laws, treated in the first sections, apply at low particle energy. At high energy, particle trajectories must be described by relativistic equations. Although Newton's laws and their implications can be understood intuitively, the laws of relativity cannot since they apply to regimes beyond ordinary experience. Nonetheless, they must be accepted to predict particle behavior in high-energy accelerators. In fact, accelerators have provided some of the most direct verifications of relativity.

This chapter reviews particle mechanics. **Section 2.1** summarizes the properties of electrons and ions. **Sections 2.2–2.4** are devoted to the equations of Newtonian mechanics. These are applicable to electrons from electrostatic accelerators of in the energy range below 20 kV. This range includes many useful devices such as cathode ray tubes, electron beam welders, and microwave tubes. Newtonian mechanics also describes ions in medium energy accelerators used for nuclear physics. The Newtonian equations are usually simpler to solve than relativistic formulations. Sometimes it is possible to describe transverse motions of relativistic particles using Newtonian equations with a relativistically corrected mass. This approximation is treated in **Section 2.10**. In the second part of the chapter, some of the principles of special relativity are derived from two basic postulates, leading to a number of useful formulas summarized in **Section 2.9.**

**2.1 CHARGED PARTICLE PROPERTIES**

In the theory of charged particle acceleration and transport, it is sufficient to treat particles as dimensionless points with no internal structure. Only the influence of the electromagnetic force, one of the four fundamental forces of nature, need be considered. Quantum theory is unnecessary except to describe the emission of radiation at high energy.

This book will deal only with ions and electrons. They are simple, stable particles. Their response to the fields applied in accelerators is characterized completely by two quantities: mass and charge. Nonetheless, it is possible to apply much of the material presented to other particles. For example, the motion of macroparticles with an electrostatic charge can be treated by the methods developed in **Chapters 6–9**. Applications include the suspension of small objects in oscillating electric quadrupole fields and the acceleration and guidance of inertial fusion targets. At the other extreme are unstable elementary particles produced by the interaction of high-energy ions or electrons with targets. Beamlines, acceleration gaps, and lenses are similar to those used for stable particles with adjustments for different mass. The limited lifetime may influence hardware design by setting a maximum length for a beamline or confinement time in a storage ring.

An electron is an elementary particle with relatively low mass and negative charge. An ion is an assemblage of protons, neutrons, and electrons. It is an atom with one or more electrons removed. Atoms of the isotopes of hydrogen have only one electron. Therefore, the associated ions (the proton, deuteron, and triton) have no electrons. These ions are bare nucleii consisting of a proton with 0, 1, or 2 neutrons. Generally, the symbol *Z* denotes the atomic number of an ion or the number of electrons in the neutral atom. The symbol *Z** is often used to represent the number of electrons removed from an atom to create an ion. Values of *Z** greater than 30 may occur when heavy ions traverse extremely hot material. If *Z** = *Z,* the atom is fully stripped. The atomic mass number *A* is the number of nucleons (protons or neutrons) in the nucleus. The mass of the atom is concentrated in the nucleus and is given approximately as *Am*p, where *m*p is the proton mass.

Properties of some common charged particles are summarized in **Table 2.1**. The meaning of the rest energy in **Table 2.1** will become clear after reviewing the theory of relativity. It is listed in energy units of million electron volts (MeV). An electron volt is defined as the energy gained by a particle having one fundamental unit of charge (*q* = ± *e* = ± 1.6 × 10-19 coulombs) passing through a potential difference of one volt. In mks units, the electron volt is

1 eV = (1.6 × 10-19 C)(1 V) = 1.6 × 10-19 J.

Other commonly used metric units are keV (103 eV) and GeV (109 eV). Relativistic mechanics must be used when the particle kinetic energy is comparable to or larger than the rest energy.

There is a factor of 1843 difference between the mass of the electron and the proton. Although methods for transporting and accelerating various ions are similar, techniques for electrons are quite different. Electrons are relativistic even at low energies. As a consequence, synchronization of electron motion in linear accelerators is not difficult. Electrons are strongly influenced by magnetic fields; thus they can be accelerated effectively in a circular induction accelerator (the betatron). High-current electron beams (? 10 kA) can be focused effectively by magnetic fields. In contrast, magnetic fields are ineffective for high-current ion beams. On the other hand, it is possible to neutralize the charge and current of a high-current ion beam easily with light electrons, while the inverse is usually impossible.

*(Continues...)*

Excerpted fromPrinciples of Charged Particle AccelerationbyStanley Humphries Jr.. Copyright © 2012 Stanley Humphries, Jr.. Excerpted by permission of Dover Publications, Inc..

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