Special Topics in Accelerator Physics

725 Pages·2022·32.724 MB·English

by  Alexander Wu Chao

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World Scientifi c
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Published by
World Scientific Publishing Co. Pte. Ltd.
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Library of Congress Cataloging-in-Publication Data
Names: Chao, Alexander Wu, author.
Title: Special topics in accelerator physics / Alexander Wu Chao,
SLAC National Accelerator Laboratory, USA.
Description: New Jersey : World Scientific Publishing Co., [2022] |
Includes bibliographical references and index.
Identifiers: LCCN 2021056666 | ISBN 9789811253492 (hardcover) |
ISBN 9789811253508 (ebook for institutions) | ISBN 9789811253515 (ebook for individuals)
Subjects: LCSH: Particle accelerators.
Classification: LCC QC787.P3 C476 2022 | DDC 539.7/3--dc23/eng/20220120
LC record available at https://lccn.loc.gov/2021056666
British Library Cataloguing-in-Publication Data
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v
Preface
Accelerators as research and industrial tools are increasingly becoming a key
driver of the advances of a modern society. As accelerator science evolves to
meet the ever-demanding needs of the society, the field of accelerator physics
has grown and deepened over the past few decades, and many of its branches
havedevelopedintospecialtopicsofresearchbytheirownrights. Asresearches
in these topics mature, it is appropriate to start documenting this hard-earned
expertise by the accelerator community for readers inspired to venture into this
rich field of science. With this view, a selection of special topics is presented in
this volume, Special Topics in Accelerator Physics.
Thisvolumeisdedicatedtoarelativelysmallselectionofthesespecialtopics.
The choice of these topics is somewhat arbitrary, no doubt influenced by the
interest and experience of the author, but they are believed to present the
accelerator physics as a diversified and exciting field. Eight special topics are
included in this volume with no intention to pretend the list to be exhaustive.
Manymorespecialtopicscanreadilybeaddedtotheoneshappentobechosen.
The topics are not arranged in any particular order. The material on each
topic has been intended to be self-contained. Individual chapters have varying
lengths and assume different levels of basic knowledge of the readers. However,
thereaderisassumedtobeknowledgeableofbasicacceleratorphysicstoputthe
materialinsomecontext. Thebookispresentedasatextbook, withhomework
at the end of each section. The contents were derived from lectures delivered
at various schools and universities. The style will be emphasizing the physical
principles of the various subjects.
Iwouldliketothankthestudentsfortheirinterestandtheirmanyinquisitive
questions, from which I learned. Warm and heart-felt thanks go to many of
my colleagues, with whom the learning has been such a joy over the years.
Special thanks also go to Karl Bane, Gennady Stupakov, Xiujie Deng, Linda
Spentzouris, Eberhard Keil, Kohji Hirata, Yukihide Kamiya, Miguel Furman,
and Rudiger Schmidt, who generously allowed me to use their illustrations.
Alexander Chao
Stanford, January 2022
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Contents
Preface v
1 THE FOKKER–PLANCK EQUATION 1
1.1 Vlasov equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Potential well distortion . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 Fokker–Planck equation . . . . . . . . . . . . . . . . . . . . . . . 24
1.4 Linear coupled system . . . . . . . . . . . . . . . . . . . . . . . . 39
1.5 Transient beam distribution . . . . . . . . . . . . . . . . . . . . . 54
1.6 Quantum lifetime . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
1.7 Fokker–Planck normal modes . . . . . . . . . . . . . . . . . . . . 79
2 SYMPLECTIFICATION OF MAPS 84
2.1 Phase space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
2.2 Symplecticity condition . . . . . . . . . . . . . . . . . . . . . . . 85
2.3 Symplectification of a linear map . . . . . . . . . . . . . . . . . . 91
2.4 Higher order integrator. . . . . . . . . . . . . . . . . . . . . . . . 98
2.5 Canonical integrator . . . . . . . . . . . . . . . . . . . . . . . . . 105
3 TRUNCATED POWER SERIES ALGEBRA 115
3.1 Introducing TPSA . . . . . . . . . . . . . . . . . . . . . . . . . . 115
3.2 TPSA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
3.3 Higher order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
3.4 Special functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
3.5 Multiple inputs and outputs . . . . . . . . . . . . . . . . . . . . . 130
3.6 TPSA tool. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
4 LIE ALGEBRA 134
4.1 Symplecticity and Poisson bracket . . . . . . . . . . . . . . . . . 134
4.2 Taylor and Lie map representations . . . . . . . . . . . . . . . . . 142
4.3 Algebra of operator . . . . . . . . . . . . . . . . . . . . . . . . . . 151
4.4 Baker–Campbell–Hausdorff formula. . . . . . . . . . . . . . . . . 187
4.5 Localized radio-frequency cavity . . . . . . . . . . . . . . . . . . 214
4.6 Single sextupole. . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
4.7 Distribution of multipole . . . . . . . . . . . . . . . . . . . . . . . 240
4.8 Normal form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
4.9 Application away from resonance . . . . . . . . . . . . . . . . . . 278
4.10 Isolated resonance . . . . . . . . . . . . . . . . . . . . . . . . . . 294
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viii Contents
5 SPIN DYNAMICS OF PROTON 328
5.1 Thomas–BMT equation . . . . . . . . . . . . . . . . . . . . . . . 329
5.2 Spin motion in an accelerator . . . . . . . . . . . . . . . . . . . . 333
5.3 Spinor algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339
5.4 Depolarization resonance. . . . . . . . . . . . . . . . . . . . . . . 345
5.5 Spinor matrix formalism . . . . . . . . . . . . . . . . . . . . . . . 359
5.6 Siberian snake . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380
6 SPIN DYNAMICS OF ELECTRON 395
6.1 Some quantum aspects of synchrotron radiation . . . . . . . . . . 396
6.2 Spin precession — A recap . . . . . . . . . . . . . . . . . . . . . 403
6.3 Semiclassical description of spin effect on synchrotron radiation . 406
6.4 Radiative polarization . . . . . . . . . . . . . . . . . . . . . . . . 418
6.5 Polarization in a storage ring . . . . . . . . . . . . . . . . . . . . 425
6.6 Derbenev–Kondratenko formula . . . . . . . . . . . . . . . . . . . 438
6.7 SLIM formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . 445
6.8 Spin transparency . . . . . . . . . . . . . . . . . . . . . . . . . . 471
7 ECHO 479
7.1 Echoes are everywhere . . . . . . . . . . . . . . . . . . . . . . . . 479
7.2 Transverse echo . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482
7.3 Longitudinal echo. . . . . . . . . . . . . . . . . . . . . . . . . . . 506
7.4.3Echo-enabled harmonic generation . . . . . . . . . . . . . . . . . 518
7.5 Spin echo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531
8 BEAM-BEAM INTERACTION 541
8.1 The luminosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542
8.2 Beam-beam perturbation . . . . . . . . . . . . . . . . . . . . . . 548
8.3 Linear thin-lens model . . . . . . . . . . . . . . . . . . . . . . . . 555
8.4 Weak-strong nonlinear beam-beam effect . . . . . . . . . . . . . . 584
8.5 Coherent beam-beam effect . . . . . . . . . . . . . . . . . . . . . 643
8.6 Quadrupole mode. . . . . . . . . . . . . . . . . . . . . . . . . . . 671
8.7 Synchrobetatron mode . . . . . . . . . . . . . . . . . . . . . . . . 687
8.8 Higher order mode . . . . . . . . . . . . . . . . . . . . . . . . . . 691
Subject Index 709
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1
Chapter 1
The Fokker–Planck
Equation
The description of the motion of charged-particle beams in an accelerator pro-
ceeds in steps of increasing complexity. The first step is to consider a single-
particlepictureinwhichthebeamisrepresentedasacollectionofnoninteracting
single-particlesmovinginanenvironmentofprescribedexternalelectromagnetic
fields. The electromagnetic fields generated and carried by the particles are ig-
nored. Knowing the external fields, it is then possible to calculate the beam
motion to a high accuracy by repeating the calculation for each and every one
of the single-particles in the beam.
The actual beam consists of a large number, perhaps 1011, of particles.
Although possible in principle, it is clearly not practical to treat the actual
beam behavior using this single-particle approach. This is even more the case
if one includes the effects of the electromagnetic fields the beam generates, as
would be necessary if one wants to deal with the collective instability effects of
an intense beam.
One way to approach this problem is to supplement the single-particle pic-
ture by another qualitatively different picture. The commonly used tools in
accelerator physics for this purpose are the Vlasov equation1 and the Fokker–
Planck equation.2 These equations assume a continuum of beam distribution
and are strictly valid in the limit of infinite number of microparticles, each car-
rying an infinitesimal charge. The hope is that by studying the two extremes,
the single-particle picture and the continuum picture, we gain a handle of the
behavior of our 1011-particle beam, at least approximately.
Asmentioned,themostnotableuseofthecontinuumpictureisthestudyof
collective beam effects. However, the purpose of this chapter is not to address
1A.A.Vlasov,J.Phys.USSR9,25(1945).
2A.D. Fokker, Ann. Phys. 348, 810 (1914); M. Planck, Sitzungsberichte der Preussischen
AkademiederWissenschaftenzuBerlin,24,324(1917);S.Chandrasekhar,Rev.Mod.Phys.
15,1(1943).