Graduate Texts in Physics

Helmut Wiedemann

Particle
Accelerator
Physics
Fourth Edition

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Graduate Texts in Physics

Series Editors
Sadri Hassani
Illinois, USA
W.J. Munro
Kanagawa, Japan
Richard Needs
Cambridge, UK
William T. Rhodes
Florida, USA
Martin Stutzmann
Garching, Germany
Andreas Wipf
Jena, Germany

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Graduate Texts in Physics
Graduate Texts in Physics publishes core learning/teaching material for graduateand advanced-level undergraduate courses on topics of current and emerging fields
within physics, both pure and applied. These textbooks serve students at the
MS- or PhD-level and their instructors as comprehensive sources of principles,
definitions, derivations, experiments and applications (as relevant) for their mastery
and teaching, respectively. International in scope and relevance, the textbooks
correspond to course syllabi sufficiently to serve as required reading. Their didactic
style, comprehensiveness and coverage of fundamental material also make them
suitable as introductions or references for scientists entering, or requiring timely
knowledge of, a research field.
More information about this series at
www.springer.com/series/8431

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Helmut Wiedemann

Particle Accelerator Physics
Fourth Edition

123


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Helmut Wiedemann
Emeritus Professor of Applied Physics
and of the Stanford Synchrotron
Radiation Laboratory
Stanford University
Stanford
California, USA

ISSN 1868-4513
Graduate Texts in Physics
ISBN 978-3-319-18316-9
DOI 10.1007/978-3-319-18317-6

ISSN 1868-4521

(electronic)

ISBN 978-3-319-18317-6

(eBook)

Library of Congress Control Number: 2015945573
Open Access This book was originally published with exclusive rights reserved by the Publisher in 2015 and was
licensed as an open access publication in November 2019 under the terms of the Creative Commons Attribution
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To my sons and students


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Preface to Fourth Edition

Just 20 years have passed since the first edition. During those years, the book
has gone through several phases starting with the two volume edition 1 and 2.
Finally in 2007, both volumes and the book on Synchrotron Radiation have been
combined into the Third-Edition as one volume to serve as a textbook for students
and beginners as well as a reference book for the practitioners. Now it has become
necessary to review the text and upgrade to include new developments. It also has
become apparent that the decision for the Third-Edition to eliminate introductory
accelerator physics was not correct. Use of this text for beginners is quite broad,
and the introduction to accelerator physics is desired. Therefore, three chapters
have been added at the beginning to introduce a variety of accelerators and their
functioning. In support to teaching, many problems with solutions have been added
for those chapters. The author also tried to distinguish between introductory chapters
and chapters which lead to more detailed subjects and show proofs. Chapters which
can be skipped on a first reading have been labeled with a star :
As mentioned, the text includes many problems with and without solutions. The
idea was to give solutions for the beginners while more advanced problems are
not suitable for solutions in a textbook. Accelerator physics is not a collection
of homework problems. Many questions and problems are rather complex and
need to be treated in context with their impact on other systems. In most cases,
there is no one optimum solution. Individual parameter choices must be made and
modified according to their impact on other systems. Choices in beam dynamics,
for example, have an impact on magnet design or RF-system parameters, etc.
affecting the design of power supplies or financial budget. Straightforward design
choices permeate through almost all other components requiring careful evaluation.
Often the consequence of one parameter choice on other systems will become

apparent only after considerable further design optimization. Unfortunately, often
compromises must be made because of financial considerations. Work in accelerator
physics includes often several approximations, and the designer should not hesitate
to start over again with new insight. All this cannot be included in problem solutions
in a textbook. However, it seemed to the author interesting to throw up such design
problems which the interested reader can use to make his/her choices.
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Finally, in the last chapter on Free Electron Lasers, a short introduction into the
components of a SASE-FEL facility is given. This introduction must be short and
limited to the discussion of issues and function of main components in this text.
Much more detail is required to design such a facility and a dedicated textbook is
desirable.
I would like to thank all staff at Springer Publishing, especially the Editor Dr.
Christian Caron, Production Coordinator Mrs. Birgit Muench, the Production Editor
and Manager Ms. Shanthi Ramamoorthy, and Ms. Fathima Rizwana for their careful
editing, support, and help before and during the production process.
Chiang Mai, Thailand
February, 2015

Helmut Wiedemann

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Preface to Third Edition

This issue of Particle Accelerator Physics is intended to combine the content of two
earlier volumes and the volume on synchrotron radiation into one reference book.
This book is designed for the serious scientist and student to acquire the underlying
physics of electron accelerator physics. Introductory discussions on various types
of accelerators have been eliminated, being well documented in the literature. Beam
optics has been formulated in a general way as to be applicable also to proton and ion
beams. Following the requests of many readers many solutions to exercises are given
in the Appendix. Breaking with the author’s preference, Standard International units
are used in this edition. In Appendix B, transformation rules are given to convert
formulae between SI and cgs systems. In the process of rewriting the texts, known
typographical and real errors have been corrected. The author wishes to express his
sincere appreciation to all readers pointing out such errors.
I would like to thank all staff at Springer who have contributed to the publication
of this text. Foremost, I thank Dr. Christian Caron for his suggestion and encouragement to combine several textbooks into one reference volume. For the expert editing
and cover design I thank Mrs. Birgit Muench and her staff. Finally, it is a pleasure to
thank Ms. Bhawna Narang from Techbooks for her patient and thorough preparation
of the proofs and final printing.
Nakhon Ratchasima, Thailand
March 2007

Helmut Wiedemann

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Preface to First Edition, Volume I

The purpose of this book is to provide a comprehensive introduction into the
physics of particle accelerators and particle beam dynamics.Particle accelerators
have become important research tools in high energy physics as well as sources of
incoherent and coherent radiation from the far infra red to hard X-rays for basic and
applied research. During years of teaching accelerator physics, it became clear that
the single most annoying obstacle to get introduced into the field is the absence of
a suitable textbook. Indeed most information about modern accelerator physics is
contained in numerous internal notes from authors working mostly in high energy
physics laboratories all over the world.
This text intends to provide a broad introduction and reference book into the
field of accelerators for graduate students, engineers, and scientists summarizing
many ideas and findings expressed in such internal notes and elsewhere. In doing
so, theories are formulated in a general way to become applicable for any kind
of charged particles. Writing such a text, however, poses the problem of correct
referencing of original ideas. I have tried to find the earliest references among
more or less accessible notes and publications and have listed those although the
reader may have difficulty to obtain the original paper. In spite of great effort to be
historically correct, I apologize for possible omissions and misquotes. This situation
made it necessary to rederive again some of such ideas rather than quote the results
and refer the interested reader to the original publication. I hope this approach
will not offend the original authors, but rather provides a broader distribution

of their original ideas, which have become important to the field of accelerator
physics.
This text is split into two volumes. The first volume is designed to be selfcontained and is aimed at newcomers into the field of accelerator physics, but also
to those who work in related fields and desire some background on basic principles
of raccelerator physics. The first volume therefore gives an introductory survey of
fundamental principles of particle acceleration followed by the theory of linear beam
dynamics in the transverse as well as longitudinal phase space including a detailed
discussion of basic magnetic focusing units. Concepts of single and multi-particle
beam dynamics are introduced.
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Preface to First Edition, Volume I

Synchrotron radiation, its properties and effect on beam dynamics and electron
beam parameters, is described in considerable detail followed by a discussion of
beam instabilities on an introductory level, beam lifetime and basic lattice design
concepts. The second volume is aimed specifically to those students, engineers,
and scientists who desire to immerse themselves deeper into the physics of particle
accelerators. It introduces the reader to higher order beam dynamics, Hamiltonian
particle dynamics, general perturbation theory, nonlinear beam optics, chromatic
and geometric aberrations, and resonance theory. The interaction of particle beams
with rf fields of the accelerating system and beam loading effects are described
in some detail relevant to accelerator physics. Following a detailed derivation of
the theory of synchrotron radiation particle beam phenomena are discussed while
utilizing the Vlasov and Fokker Planck equations leading to the discussion of
beam parameters and their manipulation and collective beam instabilities. Finally,

design concepts and new developments of particle accelerators as synchrotron
radiation sources or research tools in high energy physics are discussed in some
detail.
This text grew out of a number of lecture notes for accelerator physics courses at
Stanford University, the Synchrotron Radiation Research Laboratory in Taiwan, the
University of Sao Paulo in Brazil, the International Center for Theoretical Physics
in Trieste and the US Particle Accelerator School as well as from interaction with
students attending those classes and my own graduate students.
During almost 30 years in this field, I had the opportunity to work with numerous
individuals and accelerators in laboratories around the world. Having learned
greatly from these interactions, I would like to take this opportunity to thank all
those who interacted with me and have had the patience to explain their ideas,
share their results, or collaborate with me. The design and construction of new
particle accelerators provides a specifically interesting period to develop and test
theoretically new ideas, to work with engineers and designers, to see theoretical
concepts become hardware and to participate in the excitement of commissioning
and optimization. I have had a number of opportunities for such participation at
the Deutsches Elektronen Synchrotron, DESY, in Hamburg, Germany and at the
Stanford University at Stanford, California and am grateful to all colleagues who
hosted and collaborated with me. I wished I could mention them individually and
apologize for not doing so.
A special thanks goes to the operators of the electron storage rings SPEAR and
PEP at the Stanford Linear Accelerator Center, specifically to T. Taylor, W. Graham,
E. Guerra, and M. Maddox, for their dedicated and able efforts to provide me during
numerous shifts over many years with a working storage ring ready for machine
physics experimentation.

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Preface to First Edition, Volume I

xiii

I thank Mrs. Joanne Kwong, who typed the initial draft of this text and introduced
me into the intricacies of TEX typesetting. The partial support by the Department
of Energy through the Stanford Synchrotron Radiation Laboratory in preparing
this text is gratefully acknowledged. Special thanks to Dr. C. Maldonado for
painstakingly reading the manuscript. Last but not least, I would like to thank my
family for their patience in dealing with an “absent” husband and father.
Palo Alto, CA, USA
December 1992

Helmut Wiedemann


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Preface to First Edition, Volume II

This text is a continuation of the first volume on “Basic Principles and Linear Beam
Dynamics”. While the first volume has been written as an introductory overview into
beam dynamics it does not include more detailed discussion of nonlinear and higher
order beam dynamics or the full theory of synchrotron radiation from relativistic

electron beams. Both issues are, however, of fundamental importance for the design
of modern particle accelerators. In this volume beam dynamics is formulated within
the realm of Hamiltonian dynamics leading to the description of multiparticle beam
dynamics with the Vlasov equation and including statistical processes with the
Fokker Planck equation. Higher order perturbations and aberrations are discussed
in detail including Hamiltonian resonance theory and higher order beam dynamics.
The discussion of linear beam dynamics in Vol. I is completed here with the
derivation of the general equation of motion including kinematic terms and coupled
motion. Building on the theory of longitudinal motion in Vol. I the interaction of
a particle beam with the rf system including beam loading, higher order phase
focusing and combination of acceleration and transverse focusing is discussed. The
emission of synchrotron radiation greatly affects the beam quality of electron or
positron beams and we therefore derive the detailed theory of synchrotron radiation
including spatial and spectral distribution as well as properties of polarization. The
results of this derivation are then applied to insertion devices like undulator and
wiggler magnets. Beam stability in linear and circular accelerators is compromised
by the interaction of the electrical charge in the beam with its environment leading
to instabilities. Theoretical models of such instabilities are discussed and scaling
laws for the onset and rise time of instabilities derived. Although this text builds up
on Vol. I, it relates to it only as a reference for basic issues of accelerator physics
which could be obtained as well elsewhere. This volume is aimed specifically to
those students, engineers and scientists who desire to acquire a deeper knowledge
of particle beam dynamics in accelerators. To facilitate the use of this text as a
reference many of the more important results are emphasized by a frame for quick
detection. Consistent with Vol. I we use the cgs system of units. However, for the
convenience of the reader who is used to the system of international units conversion
factors have been added whenever such conversion is necessary, e.g. whenever
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Preface to First Edition, Volume II

electrical or magnetic
p units are used. These conversion factors are enclosed in
square brackets like 4 0 and should be ignored by those who use formulas in
the cgs system. The conversion factors are easy to identify since they include only
the constants c; ; 0 ; 0 and should therefore not mixed up with other factors in
square brackets. For the convenience of the reader the source of these conversion
factors is compiled in the appendix together with other useful tools.
I would like to thank Joanne Kwong, who typed the initial draft of this text and
introduced me into the intricacies of TEX typesetting and to my students who guided
me by numerous inquisitive questions. Partial support by the Division of Basic
Energy Sciences in the Department of Energy through the Stanford Synchrotron
Radiation Laboratory in preparing this text is gratefully acknowledged. Special
thanks to Dr. C. Maldonado for painstakingly reading the manuscript and to the
editorial staff of Springer Verlag for the support during the preparation of this text.
Palo Alto, CA, USA
March 1994

Helmut Wiedemann

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Contents


Part I

Introduction

1

Introduction to Accelerator Physics . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
1.1 Short Historical Overview . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
1.2 Particle Accelerator Systems . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
1.2.1
Main Components of Accelerator Facilities . . . . . . . . . . . .
1.2.2
Applications of Particle Accelerators .. . . . . . . . . . . . . . . . . .
1.3 Definitions and Formulas . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
1.3.1
Units and Dimensions . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
1.3.2
Maxwell’s Equations.. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
1.4 Primer in Special Relativity. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
1.4.1
Lorentz Transformation .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
1.4.2
Lorentz Invariance . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
1.4.3
Spatial and Spectral Distribution of Radiation . . . . . . . . .
1.4.4
Particle Collisions at High Energies . .. . . . . . . . . . . . . . . . . .
1.5 Principles of Particle-Beam Dynamics .. . . . . . . . .. . . . . . . . . . . . . . . . . .
1.5.1

Electromagnetic Fields of Charged Particles . . . . . . . . . . .
1.5.2
Vector and Scalar Potential . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
1.5.3
Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
1.5.4
Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
1.5.5
Lorentz Force . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
1.5.6
Equation of Motion . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
1.5.7
Charged Particles in an Electromagnetic Field . . . . . . . . .
1.5.8
Linear Equation of Motion . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
1.5.9
Energy Conservation .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
1.5.10 Stability of a Charged-Particle Beam .. . . . . . . . . . . . . . . . . .
References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .

3
3
7
7
10
11
11
13
14
15

18
22
24
26
26
27
28
30
30
31
33
34
35
37
41

2

Linear Accelerators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
2.1 Principles of Linear Accelerators .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
2.1.1
Charged Particles in Electric Fields . . .. . . . . . . . . . . . . . . . . .
2.1.2
Electrostatic Accelerators.. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .

43
43
44
45
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3

Contents

2.2

Electric Field Components .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
2.2.1
Electrostatic Deflectors . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
2.2.2
Electrostatic Focusing Devices . . . . . . . .. . . . . . . . . . . . . . . . . .
2.2.3
Iris Doublet .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
2.2.4
Einzellens.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
2.3 Acceleration by rf Fields . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
2.3.1
Basic Principle of Microwave Linear Accelerators . . . .
References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .

48
48
49
51
52

54
54
57

Circular Accelerators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
3.1 Betatron .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
3.2 Weak Focusing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
3.3 Adiabatic Damping.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
3.4 Acceleration by rf Fields . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
3.4.1
Microtron .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
3.4.2
Cyclotron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
3.4.3
Synchro-Cyclotron .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
3.4.4
Isochron Cyclotron.. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
3.4.5
Synchrotron . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
3.4.6
Storage Ring.. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
3.4.7
Summary of Characteristic Parameters . . . . . . . . . . . . . . . . .
References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .

59
60
63
66
68

68
70
73
74
75
77
77
79

Part II
4

5

Tools We Need

Elements of Classical Mechanics. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
4.1 How to Formulate a Lagrangian? .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
4.1.1
The Lagrangian for a Charged Particle
in an EM-Field . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
4.2 Lorentz Force .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
4.3 Frenet-Serret Coordinates .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
4.4 Hamiltonian Formulation . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
4.4.1
Cyclic Variables .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
4.4.2
Canonical Transformations .. . . . . . . . . . .. . . . . . . . . . . . . . . . . .
4.4.3
Curvilinear Coordinates . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .

4.4.4
Extended Hamiltonian . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
4.4.5
Change of Independent Variable . . . . . .. . . . . . . . . . . . . . . . . .
References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .

83
85
85
86
87
88
90
90
93
95
96
98

Particle Dynamics in Electro-Magnetic Fields . . . . . .. . . . . . . . . . . . . . . . . .
5.1 The Lorentz Force .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
5.2 Fundamentals of Charged Particle Beam Optics.. . . . . . . . . . . . . . . . .
5.2.1
Particle Beam Guidance . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
5.2.2
Particle Beam Focusing .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
5.3 Equation of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .

99
99

100
100
102
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Contents

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5.4

Equations of Motion from the Lagrangian and Hamiltonian .. . . .
5.4.1
Equations of Motion from Lagrangian .. . . . . . . . . . . . . . . . .
5.4.2
Canonical Momenta.. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
5.4.3
Equation of Motion from Hamiltonian.. . . . . . . . . . . . . . . . .
5.4.4
Harmonic Oscillator . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
5.4.5
Action-Angle Variables . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
5.5 Solutions of the Linear Equations of Motion . . .. . . . . . . . . . . . . . . . . .
5.5.1
Linear Unperturbed Equation of Motion .. . . . . . . . . . . . . . .
5.5.2

Matrix Formulation . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
5.5.3
Wronskian .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
5.5.4
Perturbation Terms . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .

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112
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114
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118
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Electromagnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
6.1 Pure Multipole Field Expansion .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
6.1.1
Electromagnetic Potentials and Fields
for Beam Dynamics .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
6.1.2
Fields, Gradients and Multipole Strength Parameter .. .
6.1.3
Main Magnets for Beam Dynamics . . .. . . . . . . . . . . . . . . . . .
6.1.4

Multipole Misalignment and “Spill-down” .. . . . . . . . . . . .
6.2 Main Magnet Design Criteria .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
6.2.1
Design Characteristics of Dipole Magnets.. . . . . . . . . . . . .
6.2.2
Quadrupole Design Concepts . . . . . . . . .. . . . . . . . . . . . . . . . . .
6.3 Magnetic Field Measurement .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
6.3.1
Hall Probe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
6.3.2
Rotating Coil . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
6.4 General Transverse Magnetic-Field Expansion .. . . . . . . . . . . . . . . . . .
6.4.1
Pure Multipole Magnets . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
6.4.2
Kinematic Terms .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
6.5 Third-Order Differential Equation of Motion .. .. . . . . . . . . . . . . . . . . .
6.6 Longitudinal Field Devices . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
6.7 Periodic Wiggler Magnets . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
6.7.1
Wiggler Field Configuration .. . . . . . . . . .. . . . . . . . . . . . . . . . . .
6.8 Electrostatic Quadrupole .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .

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Part III

7

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128
131
137
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138
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148
152
153
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167
168
172
174

Beam Dynamics

Single Particle Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
7.1 Linear Beam Transport Systems . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
7.1.1
Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
7.2 Matrix Formalism in Linear Beam Dynamics . .. . . . . . . . . . . . . . . . . .
7.2.1

Driftspace.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
7.2.2
Quadrupole Magnet .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
7.2.3
Thin Lens Approximation . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
7.2.4
Quadrupole End Field Effects .. . . . . . . .. . . . . . . . . . . . . . . . . .

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7.3

Focusing in Bending Magnets . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
7.3.1
Sector Magnets . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
7.3.2
Fringe Field Effects .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .

7.3.3
Finite Pole Gap .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
7.3.4
Wedge Magnets . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
7.3.5
Rectangular Magnet.. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
7.3.6
Focusing in a Wiggler Magnet . . . . . . . .. . . . . . . . . . . . . . . . . .
7.3.7
Hard-Edge Model of Wiggler Magnets . . . . . . . . . . . . . . . . .
7.4 Elements of Beam Dynamics . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
7.4.1
Building Blocks for Beam Transport Lines .. . . . . . . . . . . .
7.4.2
Isochronous Systems. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .

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8

Particle Beams and Phase Space . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
8.1 Beam Emittance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
8.1.1
Liouville’s Theorem . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
8.1.2
Transformation in Phase Space. . . . . . . .. . . . . . . . . . . . . . . . . .
8.1.3
Beam Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
8.2 Betatron Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
8.2.1
Beam Envelope.. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
8.3 Beam Dynamics in Terms of Betatron Functions .. . . . . . . . . . . . . . . .
8.3.1
Beam Dynamics in Normalized Coordinates . . . . . . . . . . .
8.4 Dispersive Systems .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
8.4.1
Analytical Solution . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
8.4.2
3 3-Transformation Matrices . . . . . . . .. . . . . . . . . . . . . . . . . .
8.4.3
Linear Achromat . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
8.4.4
Spectrometer . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
8.4.5
Measurement of Beam Energy Spectrum .. . . . . . . . . . . . . .
8.4.6
Path Length and Momentum Compaction . . . . . . . . . . . . . .
References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .


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9

Longitudinal Beam Dynamics .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
9.1 Longitudinal Particle Motion . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
9.1.1
Longitudinal Phase Space Dynamics .. . . . . . . . . . . . . . . . . .
9.2 Equation of Motion in Phase Space . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
9.2.1
Small Oscillation Amplitudes . . . . . . . . .. . . . . . . . . . . . . . . . . .
9.2.2
Phase Stability . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .

9.2.3
Acceleration of Charged Particles . . . . .. . . . . . . . . . . . . . . . . .
9.3 Longitudinal Phase Space Parameters .. . . . . . . . . .. . . . . . . . . . . . . . . . . .
9.3.1
Separatrix Parameters .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
9.3.2
Momentum Acceptance.. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
9.3.3
Bunch Length . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
9.3.4
Longitudinal Beam Emittance .. . . . . . . .. . . . . . . . . . . . . . . . . .
9.3.5
Phase Space Matching . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .

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9.4

Higher-Order Phase Focusing . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
9.4.1
Dispersion Function in Higher Order .. . . . . . . . . . . . . . . . . .
9.4.2
Path Length in Higher Order . . . . . . . . . .. . . . . . . . . . . . . . . . . .
9.4.3
Higher Order Momentum Compaction Factor .. . . . . . . . .
9.4.4
Higher-Order Phase Space Motion .. . .. . . . . . . . . . . . . . . . . .
9.4.5
Stability Criteria . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .

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289
291
292
296
302

10 Periodic Focusing Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .

10.1 FODO Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
10.1.1 Scaling of FODO Parameters.. . . . . . . . .. . . . . . . . . . . . . . . . . .
10.1.2 Betatron Motion in Periodic Structures . . . . . . . . . . . . . . . . .
10.1.3 General FODO Lattice . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
10.2 Beam Dynamics in Periodic Closed Lattices . . .. . . . . . . . . . . . . . . . . .
10.2.1 Hill’s Equation . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
10.2.2 Periodic Betatron Functions . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
10.2.3 Periodic Dispersion Function.. . . . . . . . .. . . . . . . . . . . . . . . . . .
10.2.4 Periodic Lattices in Circular Accelerators . . . . . . . . . . . . . .
10.3 FODO Lattice and Acceleration .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
10.3.1 Lattice Structure.. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
10.3.2 Transverse Beam Dynamics and Acceleration . . . . . . . . .
References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .

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Part IV


Beam Parameters

11 Particle Beam Parameters .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
11.1 Definition of Beam Parameters . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
11.1.1 Beam Energy .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
11.1.2 Time Structure . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
11.1.3 Beam Current . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
11.1.4 Beam Dimensions .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
11.2 Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
11.2.1 Robinson Criterion .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
11.3 Particle Distribution in Longitudinal Phase Space.. . . . . . . . . . . . . . .
11.3.1 Energy Spread .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
11.3.2 Bunch Length . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
11.4 Transverse Beam Emittance . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
11.4.1 Equilibrium Beam Emittance.. . . . . . . . .. . . . . . . . . . . . . . . . . .
11.4.2 Emittance Increase in a Beam Transport Line .. . . . . . . . .
11.4.3 Vertical Beam Emittance . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
11.4.4 Beam Sizes . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
11.4.5 Beam Divergence . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
11.5 Variation of the Damping Distribution . . . . . . . . . .. . . . . . . . . . . . . . . . . .
11.5.1 Damping Partition and Rf-Frequency .. . . . . . . . . . . . . . . . . .

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11.6 Variation of the Equilibrium Beam Emittance . .. . . . . . . . . . . . . . . . . .
11.6.1 Beam Emittance and Wiggler Magnets . . . . . . . . . . . . . . . . .
11.6.2 Damping Wigglers . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
11.7 Robinson Wiggler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
11.7.1 Damping Partition and Synchrotron Oscillation .. . . . . . .
11.7.2 Can We Eliminate the Beam Energy Spread? . . . . . . . . . .
11.8 Beam Life Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
11.8.1 Beam Lifetime and Vacuum .. . . . . . . . . .. . . . . . . . . . . . . . . . . .
11.8.2 Ultra High Vacuum System. . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .


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380
382
382
384
385
386
395
399

12 Vlasov and Fokker–Planck Equations . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
12.1 The Vlasov Equation .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
12.1.1 Betatron Oscillations and Perturbations . . . . . . . . . . . . . . . .
12.1.2 Damping .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
12.2 Damping of Oscillations in Electron Accelerators.. . . . . . . . . . . . . . .
12.2.1 Damping of Synchrotron Oscillations . . . . . . . . . . . . . . . . . .
12.2.2 Damping of Vertical Betatron Oscillations . . . . . . . . . . . . .
12.2.3 Robinson’s Damping Criterion . . . . . . . .. . . . . . . . . . . . . . . . . .
12.2.4 Damping of Horizontal Betatron Oscillations . . . . . . . . . .
12.3 The Fokker–Planck Equation . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
12.3.1 Stationary Solution of the Fokker–Planck Equation . . .
12.3.2 Particle Distribution within a Finite Aperture . . . . . . . . . .
12.3.3 Particle Distribution in the Absence of Damping . . . . . .
References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .

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408
410

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412
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422
422
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13 Equilibrium Particle Distribution . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
13.1 Particle Distribution in Phase Space .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
13.1.1 Diffusion Coefficient and Synchrotron Radiation . . . . . .
13.1.2 Quantum Excitation of Beam Emittance.. . . . . . . . . . . . . . .
13.2 Equilibrium Beam Emittance .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
13.2.1 Horizontal Equilibrium Beam Emittance . . . . . . . . . . . . . . .
13.2.2 Vertical Equilibrium Beam Emittance . . . . . . . . . . . . . . . . . .
13.3 Equilibrium Energy Spread and Bunch Length .. . . . . . . . . . . . . . . . . .
13.3.1 Equilibrium Beam Energy Spread . . . .. . . . . . . . . . . . . . . . . .
13.3.2 Equilibrium Bunch Length . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
13.4 Phase-Space Manipulation .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
13.4.1 Exchange of Transverse Phase-Space Parameters .. . . . .
13.4.2 Bunch Compression . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
13.4.3 Alpha Magnet . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
13.5 Polarization of a Particle Beam . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .

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14 Beam Emittance and Lattice Design. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
14.1 Equilibrium Beam Emittance in Storage Rings .. . . . . . . . . . . . . . . . . .
14.1.1 FODO Lattice . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
14.1.2 Minimum Beam Emittance .. . . . . . . . . . .. . . . . . . . . . . . . . . . . .
14.2 Absolute Minimum Emittance .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
14.3 Beam Emittance in Periodic Lattices . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
14.3.1 The Double Bend Achromat Lattice (DBA) . . . . . . . . . . . .

14.3.2 The FODO Lattice . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
14.3.3 Optimum Emittance for Colliding Beam
Storage Rings . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .

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461
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Part V

472
472

Perturbations

15 Perturbations in Beam Dynamics . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
15.1 Magnet Field and Alignment Errors .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
15.1.1 Self Compensation of Perturbations . .. . . . . . . . . . . . . . . . . .
15.2 Dipole Field Perturbations .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
15.2.1 Dipole Field Errors and Dispersion Function.. . . . . . . . . .
15.2.2 Perturbations in Open Transport Lines.. . . . . . . . . . . . . . . . .
15.2.3 Existence of Equilibrium Orbits. . . . . . .. . . . . . . . . . . . . . . . . .
15.2.4 Closed Orbit Distortion .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
15.2.5 Statistical Distribution of Dipole Field

and Alignment Errors .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
15.2.6 Dipole Field Errors in Insertion Devices.. . . . . . . . . . . . . . .
15.2.7 Closed Orbit Correction . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
15.2.8 Response Matrix . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
15.2.9 Orbit Correction with Single Value
Decomposition .SVD/ . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
15.3 Quadrupole Field Perturbations . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
15.3.1 Betatron Tune Shift . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
15.3.2 Optics Perturbation Due to Insertion Devices . . . . . . . . . .
15.3.3 Resonances and Stop Band Width . . . .. . . . . . . . . . . . . . . . . .
15.3.4 Perturbation of Betatron Function.. . . .. . . . . . . . . . . . . . . . . .
15.4 Chromatic Effects in a Circular Accelerator . . . .. . . . . . . . . . . . . . . . . .
15.4.1 Chromaticity . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
15.4.2 Chromaticity Correction .. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
15.4.3 Chromaticity in Higher Approximation.. . . . . . . . . . . . . . . .
15.4.4 Non-linear Chromaticity .. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
15.5 Kinematic Perturbation Terms .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
15.6 Perturbation Methods in Beam Dynamics . . . . . .. . . . . . . . . . . . . . . . . .
15.6.1 Periodic Distribution of Statistical Perturbations .. . . . . .
15.6.2 Periodic Perturbations in Circular Accelerators . . . . . . . .
15.6.3 Statistical Methods to Evaluate Perturbations . . . . . . . . . .

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xxiv

Contents

15.7 Control of Beam Size in Transport Lines . . . . . . .. . . . . . . . . . . . . . . . . .
References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .


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16 Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
16.1 Lattice Resonances .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
16.1.1 Resonance Conditions . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
16.1.2 Coupling Resonances . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
16.1.3 Resonance Diagram .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
16.2 Hamiltonian Resonance Theory . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
16.2.1 Non-linear Hamiltonian.. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
16.2.2 Resonant Terms . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
16.2.3 Resonance Patterns and Stop-Band Width .. . . . . . . . . . . . .
16.2.4 Half-Integer Stop-Band .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
16.2.5 Separatrices.. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
16.2.6 General Stop-Band Width . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
16.3 Third-Order Resonance . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
16.3.1 Particle Motion in Phase Space . . . . . . .. . . . . . . . . . . . . . . . . .
References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .

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17 Hamiltonian Nonlinear Beam Dynamics .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
17.1 Higher-Order Beam Dynamics . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
17.1.1 Multipole Errors.. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
17.1.2 Non-linear Matrix Formalism . . . . . . . . .. . . . . . . . . . . . . . . . . .
17.2 Aberrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
17.2.1 Geometric Aberrations.. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
17.2.2 Filamentation of Phase Space . . . . . . . . .. . . . . . . . . . . . . . . . . .
17.2.3 Chromatic Aberrations.. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
17.2.4 Particle Tracking . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
17.3 Hamiltonian Perturbation Theory . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
17.3.1 Tune Shift in Higher Order . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .

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Part VI

Acceleration

18 Charged Particle Acceleration . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
18.1 Rf-Waveguides and Cavities . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
18.1.1 Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
18.1.2 Rectangular Waveguide Modes . . . . . . .. . . . . . . . . . . . . . . . . .
18.1.3 Cylindrical Waveguide Modes . . . . . . . .. . . . . . . . . . . . . . . . . .
18.2 Rf-Cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
18.2.1 Square Cavities . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
18.2.2 Cylindrical Cavity.. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
18.2.3 Energy Gain . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
18.2.4 Rf-Cavity as an Oscillator . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .
18.2.5 Cavity Losses and Shunt Impedance ... . . . . . . . . . . . . . . . . .

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