Measurement and Control of Charged Particle Beams 3 Berlin Heidelberg New York Hong Kong London Milan Paris Tokyo Particle Acceleration and Detection http://www.springer.de/phys/books/pad/ The series Particle Acceleration and Detection is devoted to monograph texts dealing with all aspects of particle acceleration and detection research and advanced teaching. The scope also includes topics such as beam physics and instrumentation as well as applications. Presentations should strongly emphasise the underlying physical and engineering sciences. Of particular interest are • contributions which relate fundamental research to new applications beyond the immeadiate realm of the original field of research • contributions which connect fundamental research in the aforementionned fields to fundamental research in related physical or engineering sciences • concise accounts of newly emerging important topics that are embedded in a broader framework in order to provide quick but readable access of very new material to a larger audience The books forming this collection will be of importance for graduate students and active researchers alike. Series Editors: Professor Christian W. Fabjan CERN PPE Division 1211 Genève 23 Switzerland Professor Franco Bonaudi CERN PPE Division 1211 Genève 23 Switzerland Professor Alexander Chao SLAC 2575 Sand Hill Road Menlo Park, CA 94025 USA Professor Franceso Ruggiero CERN SL Division 1211 Genève 23 Switzerland Professor Rolf-Dieter Heuer DESY Gebäude 1d/25 22603 Hamburg Germany Professor Takahiko Kondo KEK Building No. 3, Room 319 1-1 Oho, 1-2 1-2 Tsukuba 1-3 1-3 Ibaraki 305 Japan M.G. Minty F. Zimmermann Measurement and Control of Charged Particle Beams With 172 Figures Dr. Michiko G. Minty DESY - MDE Notkestrasse 85 22607 Hamburg Germany E-mail: michiko.minty@desy.de Dr. Frank Zimmermann CERN, AB Division, ABP Group 1211 Geneva 23 Switzerland E-mail: frank.zimmermann@cern.ch Cover picture by courtesy of CERN. Cataloging-in-Publication Data applied for A catalog record for this book is available from the Library of Congress. Bibliographic information published by Die Deutsche Bibliothek. Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de. ISSN 1611-1052 ISBN 978-3-540-44187-8 Springer-Verlag Berlin Heidelberg New York Open Access This book was originally published with exclusive rights reserved by the Publisher in 2003 and was licensed as an open access publication in November 2019 under the terms of the Creative Commons Attribution distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license if changes were made. The images or other third party material in this book may be included in the book's Creative Commons license, unless indicated otherwise in a credit line to the material or in the Correction Note appended to the book. For details on rights and licenses please read the Correction https://doi.org/10.1007/978-3-662-08581-3_13. If material is not included in the book's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. ©TheEditor(s)(ifapplicable)andTheAuthor(s) 2003, correctedpublication 2019 The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. The original version of this book was revised. The correction to this book can be found at https://doi.org/10.1007/978-3-662-08581-3_13 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, Typesetting: Author and LE-TEX GbR, Leipzig using a Springer L A TEX macro package Production: LE-TEX Jelonek, Schmidt & Vöckler GbR, Leipzig Cover design: eStudio Calamar Steinen Cover production: design & production GmbH, Heidelberg Printed on acid-free paper 54/3141/YL - 5 4 3 2 1 0 We dedicate this book to the memory of Prof. Dr. Bjorn Wiik, whose charismatic and visionary leadership continues to guide us towards new directions in accelerator physics. Preface The intent of this book is to bridge the link between experimental obser- vations and theoretical principles in accelerator physics. The methods and concepts, taken primarily from high energy accelerators, have for the most part already been presented in internal reports and proceedings of accelera- tor conferences, a portion of which has appeared in refereed journals. In this book we have tried to coherently organize this material so as to be useful to designers and operators in the commissioning and operation of particle accelerators. A point of emphasis has been to provide, wherever possible, experimental data to illustrate the particular concept under discussion. Of the data pre- sented, most are collected from presently existing or past accelerators and we regret the problem of providing original data some of which appear in less accessible publications – for possible omissions we apologize. Regarding the uniformity of the text, particularly with respect to symbol definitions, we have taken the liberty to edit certain representations of the data while trying to maintain the essence of the presented observations. Throughout the text we have attempted to provide references which are readily available for the reader. In this monologue we describe practical methods for measuring and ma- nipulating various beam properties, and illustrate these concepts with many examples, which are taken from our working experience at CERN, DESY, SLAC, IUCF, KEK, LBNL, FNAL, and other laboratories. In Chaps. 2, 3, 4, 7 and 8 we discuss a present various techniques which can be employed to verify or correct the transverse and longitudinal optics, to optimize the beam orbit, and to measure or vary the beam emittances. Other chapters are devoted to special topics, such as transverse manipulations in photoinjectors (Chap. 5), beam collimation (Chap. 6), polarization (Chap. 9), injection and extraction (Chap. 10), and beam cooling (Chap. 11). Some basic knowledge of accelerator physics is a necessary prerequisite for following the material presented. This monologue results from many years of practice in accelerator physics and from teaching at various particle accelerator schools. We are grateful to our many students for their enthusiasm and especially for their interesting ideas and questions. We express our gratitude to Prof. S.Y. Lee, former or- VIII Preface ganizer of the United States Particle Accelerator Schools, for suggesting and encouraging this work. We thank most gratefully our mentors and colleagues with whom we had the pleasure to work or who have supported our professional car- reers, including Chris Adolphsen, Ron Akre, Gianluigi Arduini, Ralph Ass- mann, Karl Bane, Desmond Barber, Walter Barry, Martin Breidenbach, Reinhard Brinkmann, Karl Brown, David Burke, John Byrd, Yunhai Cai, John Cameron, Alex Chao, Ernest Courant, Martin Donald, Frank-Josef Decker, Martin Donald, Jonathan Dorfan, Don Edwards, Helen Edwards, Paul Emma, Alan Fischer, Etienne Forest, John Fox, Joseph Frisch, Alexan- der Gamp, Hitoshi Hayano, Sam Heifets, Linda Hendrickson, Thomas Himel, Georg Hoffst ̈ atter, Albert Hofmann, John Irwin, Keith Jobe, Witold Koza- necki, Wilhelm Kriens, Alan D. Krisch, Kiyoshi Kubo, S.Y. Lee, Gregory Loew, Douglas McCormick, Lia Merminga, Phil Morton, Steve Myers, Yuri Nosochkov, Katsunobu Oide, Toshiyuki Okugi, Ewan Paterson, Nan Phin- ney, Robert Pollock, Pantaleo Raimondi, Ina Reichel, Tor Raubenheimer, Burton Richter, Robert Rimmer, Thomas Roser, Marc Ross, Francesco Ruggiero, Giovanni Rumolo, Ron Ruth, Shogo Sakanaka, Matthew Sands, Frank Schmidt, Peter Schm ̈ user, John Seeman, Mike Seidel, Robert Sie- mann, William Spence, Christoph Steier, Gennady Stupakov, Mike Sulli- van, Nobu Terunuma, Dieter Trines, James Turner, Junji Urakawa, Albrecht Wagner, Nick Walker, David Whittum, Helmut Wiedemann, Uli Wienands, Bjorn Wiik, Ferdinand Willeke, Perry Wilson, Mark Woodley, Yiton Yan, and Michael Zisman. We would especially like to thank our colleagues who have gratuitously contributed to the examples and figures presented in this book. Last but not least, we also thank our editor Dr. Christian Caron and his team from Springer Verlag including Gabriele Hakuba, Sandra Thoms, and Peggy Glauch for their patience, continuous encouragement, and valuable help. Hamburg and Geneva, Michiko G. Minty April 2003 Frank Zimmermann Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Review of Transverse Linear Optics . . . . . . . . . . . . . . . . . . . . . . 6 1.2 Beam Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.3 Review of Longitudinal Dynamics . . . . . . . . . . . . . . . . . . . . . . . 12 1.4 Transverse and Longitudinal Equations of Motion . . . . . . . . . 14 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2 Transverse Optics Measurement and Correction . . . . . . . . . . 17 2.1 Betatron Tune . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.1.2 Fast Fourier Transform (FFT) and Related Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.1.3 Swept-Frequency Excitation . . . . . . . . . . . . . . . . . . . . . . 25 2.1.4 Phase Locked Loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.1.5 Schottky Monitor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.1.6 Multi-Bunch Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.2 Betatron Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.2.1 Harmonic Analysis of Orbit Oscillations . . . . . . . . . . . . 28 2.3 Beta Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.3.1 Tune Shift Induced by Quadrupole Excitation . . . . . . . 30 2.3.2 Betatron Phase Advance . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.3.3 Orbit Change at a Steering Corrector . . . . . . . . . . . . . . 34 2.3.4 Global Orbit Distortions . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.3.5 β ∗ at Interaction or Symmetry Point . . . . . . . . . . . . . . . 36 2.3.6 R Matrix from Trajectory Fit . . . . . . . . . . . . . . . . . . . . . 37 2.4 Detection of Quadrupole Gradient Errors . . . . . . . . . . . . . . . . . 40 2.4.1 First Turn Trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.4.2 Closed-Orbit Distortion . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.4.3 Phase Advance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.4.4 π Bump Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.5 Multiknobs, Optics Tuning, and Monitoring . . . . . . . . . . . . . . . 43 2.6 Model-Independent Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.7 Coherent Oscillations and Nonlinear Optics . . . . . . . . . . . . . . . 48 2.7.1 Beam Response to a Kick Excitation . . . . . . . . . . . . . . . 48 X Contents 2.7.2 Coherent Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.7.3 Detuning with Amplitude . . . . . . . . . . . . . . . . . . . . . . . . 50 2.7.4 Filamentation due to Nonlinear Detuning . . . . . . . . . . . 52 2.7.5 Decoherence due to Chromaticity and Momentum Spread . . . . . . . . . . . . . . . . . . . . . . . . . . 53 2.7.6 Resonance Driving Terms . . . . . . . . . . . . . . . . . . . . . . . . . 53 2.7.7 Tune Scans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 2.8 Betatron Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 2.8.1 Driving Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 2.8.2 First Turn Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 2.8.3 Beam Response after Kick . . . . . . . . . . . . . . . . . . . . . . . . 58 2.8.4 Closest Tune Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 2.8.5 Compensating the Sum Resonance . . . . . . . . . . . . . . . . . 60 2.8.6 Emittance near Difference Resonance . . . . . . . . . . . . . . 61 2.8.7 Emittance near Sum Resonance . . . . . . . . . . . . . . . . . . . 62 2.8.8 Coupling Transfer Function . . . . . . . . . . . . . . . . . . . . . . . 63 2.8.9 Excursion: Flat Versus Round Beams . . . . . . . . . . . . . . 63 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3 Orbit Measurement and Correction . . . . . . . . . . . . . . . . . . . . . . . 69 3.1 Beam-Based Alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.1.1 Quadrupole Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.1.2 Quadrupole Gradient Modulation . . . . . . . . . . . . . . . . . . 75 3.1.3 Sextupole Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 3.1.4 Sextupole Movement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 3.1.5 Structure Alignment Using Beam-Induced Signals . . . 79 3.2 One-to-One Steering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 3.3 Lattice Diagnostics and R Matrix Reconstruction . . . . . . . . . . 82 3.4 Global Beam-Based Steering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 3.5 Singular Value Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 3.6 ‘Wake Field Bumps’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 3.7 Dispersion-Free Steering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 3.8 Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 3.9 Orbit Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 3.10 Excursion – AC Dipole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 4 Transverse Beam Emittance Measurement and Control . . . 99 4.1 Beam Emittance Measurements . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.1.1 Single Wire Measurement . . . . . . . . . . . . . . . . . . . . . . . . 101 4.1.2 Multiple Wire Measurement . . . . . . . . . . . . . . . . . . . . . . 104 4.1.3 Graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 4.1.4 Emittance Mismatch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 4.2 Beta Matching in a Transport Line or Linac . . . . . . . . . . . . . . 116 4.3 Equilibrium Emittance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 Contents XI 4.3.1 Circumference Change . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 4.3.2 RF Frequency Change . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 4.3.3 Wigglers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 4.4 Linac Emittance Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 4.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 4.4.2 BNS Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 4.4.3 Trajectory Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 4.4.4 Dispersion-Free Steering . . . . . . . . . . . . . . . . . . . . . . . . . . 129 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 5 Beam Manipulations in Photoinjectors . . . . . . . . . . . . . . . . . . . . 133 5.1 RF Photoinjector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 5.2 Space-Charge Compensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 5.3 Flat-Beam Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 6 Collimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 6.1 Linear Collider . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 6.2 Storage Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 7 Longitudinal Optics Measurement and Correction . . . . . . . . 149 7.1 Synchronous Phase and Synchrotron Frequency . . . . . . . . . . . 150 7.2 Dispersion and Dispersion Matching . . . . . . . . . . . . . . . . . . . . . 152 7.2.1 RF Frequency Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 7.2.2 RF Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 7.2.3 RF Amplitude or Phase Jump . . . . . . . . . . . . . . . . . . . . 155 7.2.4 Resonant Correction of Residual Dispersion . . . . . . . . . 155 7.2.5 Higher-Order Dispersion in a Transport Line or Linac 156 7.3 Momentum Compaction Factor . . . . . . . . . . . . . . . . . . . . . . . . . . 159 7.3.1 Synchrotron Tune . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 7.3.2 Bunch Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 7.3.3 Lifetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 7.3.4 Path Length vs. Energy . . . . . . . . . . . . . . . . . . . . . . . . . . 162 7.3.5 Beam Energy via Resonant Depolarization . . . . . . . . . . 163 7.3.6 Change in Field Strength for Unbunched Proton Beam . . . . . . . . . . . . . . . . . . . . . . 164 7.4 Chromaticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 7.4.1 RF Frequency Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 7.4.2 Head-Tail Phase Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 7.4.3 Alternative Chromaticity Measurements . . . . . . . . . . . . 167 7.4.4 Natural Chromaticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 7.4.5 Local Chromaticity: d β/ d δ . . . . . . . . . . . . . . . . . . . . . . . 168 7.4.6 Chromaticity Control in Superconducting Proton Rings . . . . . . . . . . . . . . . . . . 168 XII Contents 7.4.7 Application: Measuring the Central Frequency . . . . . . 170 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 8 Longitudinal Phase Space Manipulation . . . . . . . . . . . . . . . . . . 175 8.1 Bunch Length Compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 8.2 Bunch Length Precompression . . . . . . . . . . . . . . . . . . . . . . . . . . 178 8.3 Bunch Coalescing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 8.4 Bunch Splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 8.5 Harmonic Cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 8.6 Energy Spread . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 8.7 Energy Compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 8.8 Beam Loading and Long-Range Wake Fields . . . . . . . . . . . . . . 197 8.9 Multi-Bunch Energy Compensation . . . . . . . . . . . . . . . . . . . . . . 202 8.10 Damping Partition Number Change via RF Frequency Shift 203 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 9 Injection and Extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 9.1 Transverse Single-Turn Injection . . . . . . . . . . . . . . . . . . . . . . . . . 211 9.2 Multi-Turn Injection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 9.2.1 Transverse Multi-Turn Injection . . . . . . . . . . . . . . . . . . . 214 9.2.2 Longitudinal and Transverse Multi-Turn Injection . . . 216 9.2.3 Longitudinal Multiturn Injection . . . . . . . . . . . . . . . . . . 217 9.2.4 Phase-Space Painting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 9.3 H − Charge Exchange Injection . . . . . . . . . . . . . . . . . . . . . . . . . . 219 9.4 Resonant Injection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 9.5 Continuous Injection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 9.6 Injection Envelope Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 9.7 Fast Extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 9.8 Kickers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 9.9 Septa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 9.10 Slow Extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 9.11 Extraction via Resonance Islands . . . . . . . . . . . . . . . . . . . . . . . . 232 9.12 Beam Separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 9.13 Crystal Extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 10 Polarization Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 10.1 Equation of Spin Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 10.2 Thomas-BMT Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 10.3 Beam Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 10.4 Spinor Algebra Using SU(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 10.5 Equation of Spin Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 10.6 Periodic Solution to the Equation of Spin Motion . . . . . . . . . . 243 10.7 Depolarizing Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 10.8 Polarization Preservation in Storage Rings . . . . . . . . . . . . . . . . 246 Contents XIII 10.8.1 Harmonic Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 10.8.2 Adiabatic Spin Flip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 10.8.3 Tune Jump . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 10.9 Siberian Snakes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 10.10 Partial Siberian Snakes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 10.11 RF Dipole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 10.12 Single Resonance Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 11 Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 11.1 Damping Rates and Fokker–Planck Equation . . . . . . . . . . . . . . 263 11.2 Electron Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 11.2.1 Basic Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 11.2.2 Estimate of the Cooling Rate . . . . . . . . . . . . . . . . . . . . . 268 11.2.3 Optical Functions at the Electron Cooler . . . . . . . . . . . 271 11.2.4 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 11.3 Stochastic Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 11.3.1 Basic Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 11.3.2 Application: Emittance Growth from a Transverse Damper . . . . . . . 276 11.4 Laser Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 11.4.1 Ion Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 11.4.2 Electron Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 11.5 Thermal Noise and Crystalline Beams . . . . . . . . . . . . . . . . . . . . 282 11.6 Beam Echoes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 11.6.1 Illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 11.6.2 Calculation of Transverse Echo . . . . . . . . . . . . . . . . . . . . 286 11.6.3 Measurements of Longitudinal Echoes . . . . . . . . . . . . . . 290 11.6.4 Measurements of Transverse Echoes . . . . . . . . . . . . . . . . 292 11.7 Ionization Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 11.8 Comparison of Cooling Techniques . . . . . . . . . . . . . . . . . . . . . . . 297 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 12 Solutions to Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 C 1 Correction to: Measurement and Control of Charged Particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Beams Symbols Constants a 0.0011596 anomalous part of the electron magnetic moment c 2.9979 × 10 8 m/s speed of light in vacuum C d 2.1 × 10 3 m 2 GeV − 3 s − 1 C q 3.84 × 10 − 1 C Q ≈ 2 × 10 − 11 m 2 GeV − 5 e 1.6 × 10 − 19 C electric charge G 1.79285 anomalous part of the proton magnetic moment h 6.626075 × 10 − 34 J s Planck’s constant m e 511 keV/c electron mass m p 928.28 MeV/c protron mass N A 6.0221 × 10 23 mol − 1 Avogadro’s number r e 2.817940 × 10 − 15 m classical electron radius μ B 5.78838 × 10 − 11 MeV T − 1 Bohr magneton Frequent Abbreviations BNS damping named after Balakin, Novokhatsky, and Smirnov BPM Beam Position Monitor CCS Chromatic Correction Section DF Dispersion-Free DFS Dispersion-Free Steering Drift Drift space (a field-free region) FEL Free Electron Laser FF Final Focus IP Interaction Point Linac Linear accelerator OTM One Turn Map Quad Quadrupole magnet (QF focusing, QD defocusing) SASE Self-Amplified Spontaneous Emission SVD Singular Value Decomposition XVI Symbols Acronyms of Accelerator Facilities and Projects AGS Alternating Gradient Synchrotron at BNL ALS Advanced Light Source at LBNL ANL Argonne National Laboratory in Chicago APS Advanced Photon Source at ANL ASSET Accelerator Structure Setup Facility at SLAC ATF (BNL) Accelerator Test Facility at BNL ATF (KEK) Accelerator Test Facility at KEK BEPC Beijing Positron Electron Collider BNL Brookhaven National Laboratory on Long Island CERN European Organization for Nuclear Research in Geneva CESR Cornell Electron Storage Ring CLIC Compact Linear Collider CTF CLIC Test Facility DESY Deutsches Elektronen-Synchrotron in Hamburg ESRF European Synchrotron Radiation Facility in Grenoble FFTB Final Focus Test Beam at SLAC FNAL Fermi National Accelerator Laboratory near Chicago HERA Hadron-Elektron Ring-Anlage at DESY IUCF Indiana University Cyclotron Facility ISR Intersecting Storage Rings at CERN JLC Japanese or Joint Linear Collider KEK High Energy Accelerator Research Organization in Tsukuba KEKB KEK B factory LBNL Lawrence Berkeley National Laboratory in Berkeley LEP Large Electron Positron Collider at CERN LHC Large Hadron Collider under construction at CERN NLC Next Linear Collider NLCTA NLC Test Accelerator PEP Proton-Electron-Positron Project at SLAC PEP-II SLAC B factory PETRA Positron-Elektron Tandem Ring-Anlage PS Proton Synchrotron at CERN Recycler permanent magnet antiproton ring at FNAL RHIC Relativistic Heave Ion Collider SLAC Stanford Linear Accelerator Center near San Francisco SLC SLAC Linear Collider SPEAR Stanford Positron Electron Accelerating Ring SPring-8 third generation synchrotron radiation facility in Japan SPS Super Proton Synchrotron at CERN TESLA Tera Electron Volt Energy Superconducting Linear Accelerator Tevatron TeV proton collider at FNAL TRISTAN former electron-positron collider at KEK TTF TESLA Test Facility ZGS Zero Gradient Synchrotron at ANL Symbols XVII Alphanumeric Symbols A atomic mass in units of the proton mass [ m p ] B x,y,z transverse and longitudinal magnetic fields [T] B r , B φ radial and angular components of magnetic field [T] B ⊥ , B ‖ components of magnetic fields perpendicular and parallel to the particle velocity [T] B mag mismatch parameter [1] C circumference of a circular accelerator [m] D(s) dispersion function [m] D x,y Sands’ number for total guide field configuration [1] E particle energy [GeV] E x,y,z transverse and longitudinal electric fields [V/m] E r , E φ radial and angular components of electric field [V/m] f quadrupole focal length [m] f coll average bunch collision frequency in a collider [Hz] f rev revolution frequency in a circular accelerator [kHz] f rf accelerating rf frequency [MHz] f x,y transverse betatron frequencies [kHz] f s synchrotron oscillation frequency [Hz] F x,y,z transverse and longitudinal Lorentz force [N] F r , F φ radial and angular components of Lorentz force [N] F ( q ) longitudinal aperture function [1] g Lande g-factor [1] G curvature function of the design orbit [1/m] h harmonic number, h = f rf f rev [1] H Hamiltonian [m] H x,y horizontal/vertical dispersion invariant [m] i ( t ) beam current in time domain [A] I ( ω ) beam current in frequency domain [As] i dc dc component of beam current [A] i b component of beam current at rf frequency (=2 i dc ) [A] I x,y action variables [m] J x,y transverse damping partition numbers [1] J longitudinal damping partition number [1] k normalized quadrupole strength [m − 2 ] k h ratio of voltages of harmonic cavities and accelerating rf [1] k l loss factor [V/C] k γ,b ratio of energies of emitted photons and beam energy [1] XVIII Symbols Alphanumeric Symbols, continued K integrated quadrupole strength [m − 1 ] L superperiod length in a periodic lattice [m] L luminosity of a collider [cm − 2 s − 1 ] m normalized sextupole strength [m − 3 ] M integrated sextupole strength [m − 2 ] m x mass of particle x [GeV/c 2 ] n s stable spin direction [1] N ppb number of particles per bunch [1] N t number of turns [1] p x , p y , p z components of the particle momentum vector [GeV/c] q overvoltage factor [1] Q quality factor [1] Q x , Q y transverse betatron tunes, also called ν x,y [1] Q s synchrotron tune [1] Q I,II eigenmodes of betatron oscillations (for coupled systems) [1] Q ′ x , Q ′ y horizontal and vertical chromaticity [1] R cavity impedance [ Ω ] R l loaded cavity impedance [ Ω ] R i,j point-to-point transfer matrix from i to j [m, 1, m − 1 ] s longitudinal coordinate along beamline [m] S x , S y , S z components of the beam polarization [1] t time measured in laboratory rest frame [s] T rev revolution period [s] T rf period of rf acceleration [s] U 0 energy loss per turn due to synchrotron radiation [eV] U hom energy loss per turn due to higher order modes [eV] V c cavity voltage [MV] W ⊥ , W ‖ transverse and longitudinal components of the wakefields, [m − 2 ] and [m − 1 ] x, y horizontal and vertical position coordinates [m] x ′ , y ′ horizontal and vertical angle coordinates [1] x co , y co transverse coordinates representing central trajectory offset [m] x β , y β tranverse coordinates representing offset due to betatron motion [m] x δ , y δ transverse coordinates representing offset due to energy deviation [m] x d , y d position offset due to quadrupole misalignment [m] x b , y b position offset due to BPM electronic offset [m] x m y m measured position offset seen by a BPM [m] X 0 radiation length, [m] or [m 4 /g] z longitudinal coordinate (relative to bunch center) [m] Z atomic number [1] Symbols XIX Greek and Latin Symbols α c momentum compaction factor [1] α p rate of spin precession [s − 1 ] α x , α y Twiss parameter, α = − 1 2 dβ ds [1] β relativistic velocity factor, β = v c [1] β c cavity coupling parameter [1] β x , β y Twiss parameter, beta function [m] β ∗ x,y beta function at a collider interaction point [m] χ 2 chi-squared parameter used in minimization algorithms [1] δ relative momentum deviation from ideal particle [1] strength of depolarizing resonances [1] x,y transverse beam emittance [m rad] x,y,N normalized transverse beam emittance [m rad] z longitudinal beam emittance, [m rad] or [eV s] γ Lorentz factor, γ = E mc 2 [1] γ x , γ y Twiss parameter ( β x,y γ x,y = (1 + α x,y 2 )) [1/m] γ t transition energy [1] κ ± coupling parameter [1] λ an eigenvalue λ rf rf wavelength, λ rf = 2 πf rf [m] μ nonlinear tune shift with amplitude parameter [1/m] μ particle magnetic moment [MeV/T] μ x,y phase advance argument, μ x,y = 2 πφ x,y [1] ν x,y transverse betatron tunes, also called Q x,y [1] ν s spin tune [1] ω r angular revolution frequency [s − 1 ] Ω solid angle [steradian] Ω x,y transverse angular betatron frequencies [s − 1 ] Ω s angular synchrotron frequency [s − 1 ] φ b phase of beam relative to rf crest [1] φ l loading angle [1] φ x,y horizontal and vertical betatron phase [1] φ x synchronous phase angle [1] φ z tuning angle [1] Ψ spin wave function [1] XX Symbols Greek and Latin Symbols, continued ρ local bending radius [m] ρ w bending radius in a wiggler magnet [m] σ cross section for scattering processes, [m 2 ] or [barn] σ ij ij -th element of the beam matrix, Σ beam , [m 2 , m, 1] σ δ rms relative momentum spread [1] σ x,y rms transverse beam sizes [m] σ z rms bunch length [m] Σ beam beam matrix θ kick angle induced by a corrector magnet [1] τ beam lifetime [s] τ f fill time of a structure or cavity [s] τ x,y transverse damping times [s] τ δ longitudinal damping time [s] τ q quantum lifetime [s] 1 Introduction Particle accelerators were originally developed for research in nuclear and high-energy physics for probing the structure of matter. Over the years ad- vances in technology have allowed higher and higher particle energies to be attained thus providing an ever more microscopic probe for understanding el- ementary particles and their interactions. To achieve maximum benefit from such accelerators, measuring and controlling the parameters of the acceler- ated particles is essential. This is the subject of this book. In these applications, an ensemble of charged particles (a ‘beam’) is ac- celerated to high energy, and is then either sent onto a fixed target, or col- lided with another particle beam, usually of opposite charge and moving in the opposite direction. In comparison with the fixed-target experiments, the center-of-mass energy is much higher when colliding two counter-propagating beams. This has motivated the construction of various ‘storage-ring’ colliders, where particle beams circulate in a ring and collide with each other at one or more dedicated interaction points repeatedly on successive turns. A large number of particles, or a high beam current, is desired in almost all applica- tions. The colliders often require a small spot size at the interaction point to maximize the number of interesting reactions or ‘events’. The charged particles being accelerated are typically electrons, positrons, protons, or antiprotons, but, depending on the application, they can also be ions in different states of charge, or even unstable isotopes. Often the beams consist of several longitudinally separated packages of particles, so- called ‘bunches’, with empty regions in between. These bunches are formed under the influence of a longitudinal focusing force, usually provided by the high-voltage rf field, which also serves for acceleration. If the trajectory of a high-energy electron or positron is bent by a magnetic field, it emits energy in the form of synchrotron radiation. The energy loss per turn due to synchrotron radiation increases with the fourth power of the beam energy and decreases only with the inverse of the bending radius. This limits the energy attainable in a ring collider. The maximum energy ever obtained in a circular electron-positron collider – more than 104 GeV per beam – was achieved in the Large Electron Positron Collider (LEP) at the European laboratory CERN in Geneva, Switzerland, with a ring circumference of almost 27 km. This chapter has been made Open Access under a CC BY 4.0 licens e. For details on rights and licenses please read the Correction https://doi.org/10.1007/978-3-662-08581-3_13 © The Author(s) 2003 M. G. Minty et al., Measurement and Control of Charged Particle Beams, https://doi.org/10.1007/978-3-662-08581-3_1