Spin Dynamics and Snakes in Synchrotrons 9810228058, 9789810228057

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Synchrotrons: ”

s

e S. Y. Le = —

ie

World Scientific

a

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:

Spin Dynamics and

SNAKes in

Synchrotrons

Digitized by the Internet Archive in 2023 with funding from Kahle/Austin Foundation

https://archive.org/details/spindynamicssnak0000lees

5ekDynamics US oy; nakes in" ee Gienicins

S. Y. Lee Indiana University, Bloomington, USA

Weve one Scientific e «New Jersey *London Hong Kong

Published by

World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite 1B, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

Library of Congress Cataloging-in-Publication Data Lee, S. Y. (Shyh- Yuan)

Spin dynamics and snakes in synchrotrons / S. Y. Lee.

cin: Includes bibliographical references and index.

ISBN 9810228058 1. Synchrotrons.

QC787.S9L44 539.7'35--de21

2. Nuclearspin.

3. Nuclear magnetic resonance.

1997 97-10519 CIP

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

Copyright © 1997 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any formor by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

This book is printed on acid-free paper.

Printed in Singapore by Uto-Print

To the loving memory of Yu Lin Lee

“Sewaa)© Qe ———

remag-® ©;

ee

~

:

Preface Since its introduction by Uhlenbeck and Goudsmit in 1925, spin has played an important role in understanding many phenomena within atomic, particle, nuclear, and condensed matter physics. The existence of spin-} particles reflects the beauty of the Dirac equation in quantum mechanics. In recent years, interest in spin physics among high energy and nuclear physics has grown remarkably, due mainly to successes in the standard model, and to the continuing research for a better understanding of the quantum chromodynamics. Advances in accelerator technology have also spurred a renewed interest in accelerating and storing highly aligned spin particles in synchrotrons and storage rings. Remarkable progress in developing polarized ion sources has been made in recent years, and a 1 mA H™ source with about 80% polarization may soon be a reality. Tremendous progress has also been made in developing polarized electron sources; particularly in using the strained GaAs to attain 80-90% polarization. With these advances in ion sources, efforts to accelerate and maintain this polarization have become a topic of interest, with several accelerator physicists devoting their lives to polarized beam research. Although there are a number of lecture notes and conference reports available for accelerator physics students, a coherent textbook is not available at present. This book is intended to fill the void. It evolved from lectures on spin physics, given during my past few years here at Indiana University. Although special topics in accelerator physics include spin dynamics, collective instabilities, nonlinear dynamics, and high brightness beams, each of these topics can be developed into a one-semester course. Furthermore, the number of students taking accelerator physics is relatively small at present, so each topic can be shortened to about 1/3 of its intended length. For students who are interested mainly in the phenomena of spin dynamics in hadron accelerators, Chaps. 3 and 4 are relevant. Chapter 5 deals with a special type of resonance for high energy storage rings with local spin rotators called Siberian Snakes, invented by Derbenev and Kondratenko. Students interested in electron polarization in storage rings, should concentrate on Chaps. 3 and 6. Chapter 7 examines various ideas on spin manipulation in storage rings, as well as different types of spin rotatvil

viii

PREFACE

ors. This chapter may serve as the driving force to promote further innovation in the design of spin manipulation tools. Each chapter in the textbook is followed by exercises. These exercises are intended to re-enforce physics concept discussed in the text, to derive some useful formula for future applications, and to provide an introduction to some published literatures related to the polarized beam dynamics. S.Y. Lee Bloomington, Indiana, U.S.A. January, 1997

Contents

Preface

02508

et

ke

Acknowled ginientst S205 1

Introduction I Splozsehamtoles LMS Curie Vetere EUXCLCTS CR

2

OR

Spin I Il. III

hs FA

Colliders oo we oe ee

ee

LOG. AO, Tee

I

Ge

oie as ia

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etn

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ME patedD «OAS

ee es ee.

ita PARA

SR

Se hs

es al

kA cc ORE

ANS I

Ne

I eI

ARIE i Ne

PAR

a

IPRS tM

7 Se

Depolarization Resonances The Imperfection Resonance Strength ....... et eng; CA. : J[nteinsic- Resonances. ...... . - sn) ame heduitet eit. . 6.01... Higher-Order Spin Depolarization Resonances ............. III.1 Spin Depolarization due to Linear Betatron Coupling ..... Iln2 Synchrotron Sideband Resonances yo. Ocha) say: «Geb> = 111.3. Nonlinear, Coupling Resonances, ..04 2m Genet 4 + Re:

EE EL-CISC DASE

4

OE

The Thomas—BMT equation I Derivation of the Thomas-BMT Equation. ............... II Spin Motion in Terms of the Particle Coordinates ........... Ill” -Spinor Equation forcthe Polarized: Beam piscpetunbee@evleg® s « otads = IV-"Spin TravisteruMateix- Snake. oaleu ration witha e Bete. 4 ds | DGS a ESEDE hes, SO Seth cg

3

eee he

EE. See

en

cones

aa sete

has

Effects of Spin Resonances I Spinoratya Constant. Acceleration Rate... > 2 2 11.5 Spin Dispersion Function, Chromaticity, and Depolarization Time Spin Resonance Correction... 6 6. 2 ee ee ee JIT Harmonic Closed Orbit Correction “".") 7°" 7 eee Rares IJI.2 Fast/Slow Acceleration Through Intrinsic Resonance .... . IIL.3. .Nonadiabatie Betatron ‘Tune.Jmmp: 4...2G G0. « 2 ge se HiA» .Etfect.ot synchrotron! Motion... aii toe 2 ee. cee Effect of Spim Rotator on, Spin. Motion, 4,4», «265 2.205 2. SBRTS [V.1 Properties.of Spin Closed Orbit. Vector... a3. si os oles LVi2:- +Spine Punesand-Spim Closed Orbit® :8is3 u-i4.07 =) eee ee ee Bffect of Overlapping Resonances, ou, 4 xe dees 4a eee VulOverlapping Resonances: «ce. < G54 anc ie Gut, a ger Boe V.2 Nearly Overlappmg Resonances . . S15. segue Sonam = V.3 Examples of Data Analysis for Overlapping Resonances ....

BEEF CISC Sigs

ae nie eran

oc

ek eGo

co ya nine cote

ee ar

eI

Spin Dynamics with Snakes I Spin Motion with Oneisnake7Nie)) Ste einrel me MeO ae. I.1 Snake. Experiments P48 PSRteig ) PE? 10) DOSE PS IOMIGE. 1:2 Partial Snaker : £24 22.23 2.2 ft 5 ee eee Il; Spin. Motion with Many onakes> occ. 5.52 2 s.0 aes es ere II.1 A Model with Two Snakes and a Local Spin Kick ....... II.2 Basic Requirements of Snake Configurations .......... ll.3°. .Spin. Tracking HierarchypMquatiousanossn aeutoseaet Sal. . IL.4 . .The Perturbed Spin Tune. . .... . .Beoienteell senivial — Ill. .Odd-Order Snake Resonances?) ppianeeqet, tiie wha sang IV. “Overlapping Resonances SP) OF 9B) ROMSSTIBIpuee aie ta IWsl* ‘Even Order’ Snake Nesonances =o oe eee ae eee IV.2 A Model for Even Order Snake Resonances. .......... Vi; sShakealrmpertections oc seca ue abs co) Go once aie ee VL Performance of Polarized Colliders... 4. 5 «2 co eee VI.1 Polarized Ion Source and Acceleration in Medium Energy Synchrotrons. ... sis aetawieaoA Jena) & pan. ”. VI.2 Acceleration in High Energy Accelerators. ........... V1.3 Constraints on Polarized Proton Colliders ...........

Exercise

49 51 52 54 54 57 60 64 66 67 68 70 70 is 76

80 87 88 91 92 93 94 95 97 98 100 104 105 107 108

109 110 Lid 113

CONTENTS 6

Electron Polarization I Effect of Electron Spin on Synchrotron Radiation ........... ik) Lhejspinck-quation.of Motion 29.,........ OS Se Oe ee 1 ee Ee Omen CH IARe PANY et ek ss rere tee es ae ee eS een In2? wlodiiied Snake"Conieuration , 60s. ee toe eee YT, II.8 The Compact Snake Configuration withm>2......... IAPs copliteSnake Configuration. i... 46 kes se nd Sup Gy 2 Gs PO atbialoNnakes er. ak eee ree 5) a) a cecteee se eote ol© Os PL eH elcaloiekes a wpe. etch oe ie te a eke ee Ae see IV Helicity State Spin Rotators for Lepton Colliders. ........... Richter=Schwitters Spin Rotator... w... 2. 1 or ss IV.1 Ene NN OpINeObalLOLe ain Geach wo aif teat ek Xp ge rys2 Exercise

Appendices A

Particle Motion in Synchrotron I Petatrati MotiGhemeie Met marern tsi See = pera he eRe se a Ses Action-Angle and the Courant-Snyder Invariant ........ it The Emittance and the Gaussian Distribution Function... . We Example of FODO Cell in the Thin Lens Approximation .. . 1.3 Eiect 01 D pole rrOrs 9 ai ine ei se 8s eusl SS wes 1.4

xl

CONTENTS

IL III

es Effect, of Quadrupole Field Error. .. . amitesiieiet aaaiost 1.6 Dispersion Hanetion 2002p Wane a hide. dol eht tp Joa. .Synchrotron.Motion:.“).0-..)....-.. MORON te-pellsupd nine sdk... Properties of Electron Beams in Storage Rings 4. 020°. .0.1)..

B

Spinor Algebra

C

The Enhancement

170 171 172 174

178 Function

BIDHOgraphy ss. Prater

179 eee oot

ae oe

ee tas oe te te oer

ioe

180 185

Acknowledgments My knowledge of spin physics has been gleaned from many masters — E.D. Courant, A.D. Krisch, L. Ratner, etc — from whom, | benefitted greatly through many years of collaboration and many enlightening discussions. I thank Steve Tepikian, who shared an office with me for about nine months at Brookhaven National Laboratory, and who brought my attention to an enduring hobby of more than 10 years. I would like to thank many colleagues who allowed me to present their data in this book. If there are any errors in interpretating their data or numerical calculations, the blame rests entirely with me. I like to thank Dr. Desmond Barber and Georg Hoffsteater for clarifying discussions on the spin closed orbit associated with electron polarization in storage rings. I would like to thank students and colleagues, particularly D. Li, H. Huang, M. Berglund, and M. Bai at Indiana University who helped me polish the lecture notes into a book form. During the course of this work, I appreciate encouragements from L. Ahrens, B. Brabson, D. Caussyn, A. Chao, R. Phelps, and R. Ruth. Finally, I thank Dennis Stoller, Allen Lee, and Virginia Harper for their help in editing this book.

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Chapter 1 Introduction Spin and the associated magnetic moment are fundamental properties of elementary particles. The spin quantum number has been employed to understand many phenomena in atomic, nuclear, elementary particle, solid state, and statistical physics.! The spin orbit interaction has been important in understanding atomic and nuclear physics. The magnetic moments of baryons also offer new insights into the study of the constituent quark model. High energy polarized beam collisions of elementary particles may also provide information necessary for a better understanding of the quantum chromodynamics. Following classical electrodynamics, the magnetic moment of a charged particle moving in a circular orbit is given by =

OL

Horbital = 5—-L, 2m

2

Spay

L=rxp,

where q and m are the charge and the mass of the particle respectively, and L is the orbital angular momentum. However, the magnetic moment of elementary particles and nuclei may not be related to their spin in such a simple minded magneto-mechanical relation. In fact, two fundamental difficulties in atomic spectra during 1920-1925,

i.e., the anomalous Zeeman effect and the multiplets,’ led Uhlenbeck and Goudsmit 1see Ref. [1] for a historical account of the spin quantum number; see Refs. [2, 3, 4, 5] for notes on its importance in high energy physics; see Refs. [6, 7] for studies in achieving high energy polarized beam collisions; and see Refs. [8, 9, 10, 11, 12] for a review of spin dynamics in synchrotrons.

?The mechanical energy of an electron is given by $(p+ £ A)? ~ ip +

2m.’

where an empirical value of g = 2 greatly simplified the interpretation of the atomic spectra. Nevertheless, the magneto-mechanical ratio g = 2 caused problems in the

spin-orbit interaction term.

In 1926, L.H. Thomas [13] showed that the spin orbit

coupling was consistent with g = 2 provided that the “Thomas precession” correction is included, where the Lorentz transformation in a circular orbit can introduce a correction term to the spin precessing frequency. In 1927, P.A.M. Dirac showed that g = 2 arose solely from charged particles satisfying the relativistic Dirac equation, which led to new concepts in physics such as anti-particles and the vacuum state. Since then, the spin quantum number has been conclusively established. The magnetic moments of elementary particles and nuclei are commonly expressed as é€

He

ASS

=

NEG

ae

é

bu

—-

Jom,

e€

—~

= Dae 2ithadron

E, S nuclei =

q (————

ey

iG rackel

where e and q are, respectively, the charges of elementary particles and nuclei; m; are the masses of these particles; S is the spin of an elementary particle; 4 represents the spin of the nuclei; and the g-factors are determined from experimental measurements. In general, the magnetic moment of a composite system is a superposition of magnetic moments of its constituents. Since the magnetic moment of an electron is 2000 times larger than that of a proton, the magnetic moments of atoms are mainly determined by the magnetic moment due to valence electrons in atomic shell orbits.° Similarly, the magnetic moments of nuclei are determined by nucleons. Although nuclei, in a first order approximation, can be represented by an independent particle model with neutrons and protons occupying shell model orbits, the magnetic moments of nuclei differ substantially from the prediction of the single particle shell model due to strong interaction. On the other hand, it is a pleasant surprise that the constituent quark model “accurately” predicts magnetic moments of nucleons and hyperons to

within +0.2 nuclear magnetons. °The magnetic moment of an atom with one valence electron is given by ff = ~(L

am

+ gS) ~~)

(J+), where J is the total angular momentum. Since J?, J,, L?, S? are good quantum numbers, the magnetic moment measured experimentally is the projection of the magnetic moment onto J, 1.e., [Ei = ic J) = 9, V/i(j + Ie, where g, is called the Landé g-factor.

4see e.g. K. Heller, p. 177 in Reef. [4].

Table 1.1 lists some fundamental spin properties of elementary particles and nuclei, where ~ eh eo he ae

= 5.788382 x 107!! MeV/T,

eh i ale 2m,

= 2.799454 x 107!3 MeV/T,

eh

= 3.152452 x 107!4 MeV/T,

Ne

2m»

are the Bohr, muon, and the nuclear magnetons respectively. The anomalous g-factor given by CeCe

ead 2

is listed in the fifth column of Table 1.1. Here the symbol a is usually used for leptons and G is normally used for hadrons.

Table 1.1: Magnetic properties of elementary particles.

M(MEV) +

ple wl JR

1 2 1 2 1 2 1 2 1 2 1 2 1 2 1+

g/2 1.001159652 1.001165923

0.001159652 0.001165923

2.7928474 -1.915680 -0.729 3.07

1.7928474

0.510999 105.65839 1784.1 938.27231 939.5656 1115.63 1189.37 1192.55 1197.43 1314.8 1321.32

1.48 -1.752 0.917

1875.613 2808.38 6533.83 21409.21

0.85699 -3.184 2.533 1.533

H(HesMu OT Hy) 1.0011596522 1.001165923 2.792847386 -1.91304275 —0.613 + 0.004 2.42 + 0.05 —1.160 + 0.025 —1.250 + 0.014 —0.6507 + 0.0025

-0.14301 -4.184 1.533 0.533

To facilitate spin physics experiments, an intense polarized beam must be produced and accelerated to high energies. Since the spin interacts with the magnetic field in the accelerator through magnetic dipole interaction, spin dynamics in a synchrotron

4

CHAPTER 1. INTRODUCTION

is governed by the Thomas-BMT equation (13, 14], which describes the evolution of the spin vector in the external magnetic field. For a beam of particles, the polarization vector is defined as the ensemble average of spin vectors. Parameters needed in specifying the alignment of spin vectors of particles in the beam bunch are illustrated in the following examples for the spin-+ and spin-1l cases.

I

Spin-+ Particles

We first consider a system of spin-4 particles, where the degree of polarization is defined as [Pre

N, — N_

Ni +N_

Here Ny are the numbers of particles in two-spin states 3, +%) along a quantization axis. For a 100% polarized beam, the spin states of all particles are quantized along a polarization axis. By varying the relative numbers of two spin states, a beam of arbitrary polarization may be obtained. In general, three parameters are needed to specify the polarization of spin-+ particles, i.e., two parameters for the direction of the quantization axis and one parameter for the N,/N_ ratio. Thus the polarization of the spin-} system is a vector characterized by a direction and a magnitude. It is usually called vector polarization, which is the ensemble average of the spin vector,

p= (¢), where & are Pauli spin matrices

01 0 -i a=(r og) @=Goo)

psi%p m= lo 1):

Any coherent linear combination of pure states of spin-} particles results in a 100% polarization with the polarization vector defined by a proper quantization axis. Any spin-4 system is cylindrically symmetric about the direction of polarization.

II

Spin-1 System

For a spin-1 system [15], there are three m-states along the quantization axis. If all particles iin the bunch occupy a pure m = 0 state, the vector polarization is zero, i.e., (S) == 0, yet the beam is polarized. This is called an aligned state. States of this type are characterized by the magnitude of the spin component S? along the alignment axis. Thus the polarization of spin-1 particles is characterized by the vector polarization and the alignment. Let us consider a beam composed of N,, No, and N_ particles along the quantization axis with m = +1, 0, and —1 respectively. The vector polarization is given by

P=(N,—N_)/(Ny + No + N_). An unpolarized non-aligned beam corresponds to

II]. SPIN-1 SYSTEM

5

equal mixing in all three states, i.e., Ny = No = N_. Since (5?) = S(S +1) = 2, the alignment can be defined as

Meas See ere

aN

ay 6NaN aro

Nz + No + N_

Ny +No+N_

When defined as such, an unpolarized beam has P = 0 and A = 0 as expected. The following examples show that a zero vector polarization does not imply a zero alignment, e.g.

Ny = No = N_ P=0, A=0 {Nea =0 Ra 0 Anan? Ne=Ney No= 0. -P:=0;, A=.

Since the alignment of the spin-1 system depends on the ensemble average of the square of the spin vector, we need five of the following quadratic spin functions (S?), (S?), ($?), (S,55), (S.S,), and (5,5,) with (S2 + S? + S?) = 2 to define the tensor polarization. Including the vector polarization, eight parameters are needed to specify a polarized spin-1 system. There are two conventions being used:

1. P; = (S;) 3 Pig =

25

es

5 (S55

GUS

+ 555i) —

2635,

Oo

Hy

age

Pi,

3

1

+ P33

= 0.

heads k

(Too) =f

Lo

+ Poo

3

——Tho= desu,

2

9 (38: — ay.

Ty,£1 = rs. Se ey

3 T2,41 +

55452 ae z04)5

Sues T2,42 =

5 (5s):

A beam with a nonzero P; or (Tj,) is called vector polarized. A beam with nonvanishing P;; or (Tq) is either called aligned, tensor polarized or having a rank-2 polarization. An analogy to help visualize these polarization parameters is the multipole moment expansion of static charge or mass distribution. A charged distribution can be completely specified by multipole moments, i.e., the monopole (or intensity), dipole, quadrupole, etc. A spin system can be specified completely with moments up to 275-poles, e.g. up to vector polarization for the spin-4 particle and up to rank 2 tensor polarization for the spin-1 particle. A special class of mixed states corresponds to a system exhibiting symmetry about an axis, e.g. deuterons produced in a polarized ion source, where certain substates in an external magnetic field are selected. Because of the axial symmetry about the quantization axis, such a system can be specified by four parameters; two for the direction of quantization axis, and two for the relative populations in three states.

Once the axis is specified, two parameters, P3 (or (Tyo)) and P33 (or (T20)), are sufficient to describe the system.

6

CHAPTER 1. INTRODUCTION

The plan of the book This book is intended to be used as a textbook for teaching the spin dynamics in synchrotrons. Chapter 2 derives the Thomas-BMT equation and casts the equation into the Frenet-Serret coordinates system [16]. Due to the periodic nature of synchrotrons, spin motion will encounter spin depolarization resonances. Chapter 3 evaluates the spin resonance strength of synchrotrons. Chapter 4 discusses spin motion in the presence of a single isolated resonance, the spin resonance correction schemes, and the effects of overlapping spin resonances. Chapter 5 discusses the spin motion in the presence of local spin rotators (called snakes) invented by Derbenev and Kondratenko [17]. Higher order snake resonances will also be discussed [18, 19, 20]. Chapter 6 examines the radiative polarization of electron beams discovered by Sokolov and Ternov [21]. Chapter 7 provides basic concepts in the design of snake and spin rotators.

Exercise

1

1. Verify the following Lamour frequencies for electrons, muons, protons, and deuterons.

proton

deuteron

28 GHz/Tesla | 135 MHz/Tesla | 42.6 MHz/Tesla | 6.5 MHz/Tesla 2. The Hamiltonian for the magnetic interaction of a charged particle with the magnetic field is given by =; ee = Aint t = —E-B, f= Iam g——S. ( lise Show that the equation of motion is given by

dS"

.\

&4

Odp

se

a TEXBS, 0, the state parallel to the magnetic field is lower than the state anti-parallel to the magnetic field in energy by 2B where py is the magnetic dipole moment of the nuclei. The spin of these nuclei will precess about the magnetic field direction at the Lamour precessing frequency. e According to the Boltzmann law, the equilibrium population is given by

Neo

—— Se

8 Ae FF

21 — 2B/kf,

where T is the temperature. Show that the fractional difference in the population for protons in room temperature is about 6.8 x 107°/Tesla.

ah SPIN-1 SYSTEM

7

e Show that the energy difference between these two states is equal to hf,, where f,, is the Lamour precessing frequency, and h is the Planck constant. e When arf field perpendicular to the sample at the Lamour frequency (resonance) is applied, the magnetic moments of these nuclei will flip between two states resulting a net absorption of energy. This can be detected electronically. Find the frequency for the rf field at a main magnetic field of 2T.°

4. In 1921, O. Stern and W. Gerlach proved the existence of the magnetic moment in atoms through a classic experiment in which they passed a beam of slowly moving neutral silver atoms through a region of non-uniform magnetic field. They observed that the atoms were separated into two bands. Since the Hamiltonian for a particle with magnetic moment ji in the magnetic field is Hing = —j/2- B, where pp = 95F with angular momentum J, the equations of motion are given by =

nlp

~

= SAXB, = V(G-B). Let the coordinate system be represented by the horizontal transverse axis @, the horizontal longitudinal axis $, and the transverse vertical axis Z. If the vertical field B, is symmetric with respect to z, and independent of the longitudinal coordinate s, show that the force on the atom is in the vertical direction. Given a beam of silver

atoms (A=109, m = 108.9 amu, » = 1 Bohr magneton) emitted from an oven with temperature 1300°K with velocity of (3kT/m)!/? = 545 m/s, show that the magnetic force for a 10 Tesla/meter magnetic field gradient is about 9.3 x 10-73 N. Compare this force with the gravitational force. Given a 10 Tesla/meter magnetic gradient, show that the beam will split into two lines at a spacing of

Az = 0.0034(Ag) - L, where A¢ is the length of the magnet and L is the distance between the magnet and the beam measuring instrument. . The Schrodinger equation is given by ine = H;,,V. The equation for the spin wave function for a spin-} particle with magnetic moment #f = 93,8 in the magnetic field

B is given by

e Let the coordinate system be represented by the horizontal transverse axis @, the horizontal longitudinal axis §, and the transverse vertical axis Z. Choosing the quantization axis along the magnetic field B = B,3, solve the spinor equation and show that the upper (V;) and lower (W_) components of the spinor wave function correspond respectively to the 5 and -} eigenstates. 5 This principle is called Nuclear Magnetic Resonance (NMR), which has been used extensively to measure the magnetic dipole moments of nuclei. Using the same principle, the Magnetic Resonance

Imaging (MRI) has been used extensively in medical diagnosis, where spatially varying magnetic fields are used to encode each point of a sample.

CHAPTER 1. INTRODUCTION e Now we add a sinusoidal perturbing field

AB = B,

(écoswt — Ssinwt),

the equation for the spin wave function becomes

oY=i FO

AONB

( Be

weit

(eet

SB;

Transform the Schrodinger equation into the resonance precessing frame and show that the solution of the Schroedinger equation is given by W(t) ete e223

7 Mt-to)F-Ae—Zutooa yy(ty)

where W(to) is the initial spinor at time to, and the unit vector n along which the spin vector is stationary is given by

N=

é

—Z-+ |

a -|

with A = \/6? + |e|?. Here w, = g>— B, is the Lamour spin precessing frequency, €=0, on is the perturbation strength, and 6 = w, —w is the frequency difference between the Lamour frequency and the driving frequency. The vector 7 is called the stable spin orbit. e Assuming an initial spin up state at to = 0, i.e.,

V4(to)=1,

W-(to) =0,

show that the probability for finding the particle in the spin up and down states are given by

aE ee a: At |W. (t)|°Dear? = cos 5 + y2 sin’ >

es en Se At

Plot the spin direction as a function of time.

e We define the polarization as P = |W4|? — |W_|?. As one sweeps through the frequency w of the perturbing field, plot the polarization as a function of w for the duration of time t = 1/2¢, 7/e, 27/e for an initial state P = 1 at to = 0. e Discuss your result if the perturbing field is given by AB = B, coswt @.

e Discuss your result if the perturbing field is given by AB = B, (Zcoswt+8$sin wt).

Chapter 2

The Thomas—BMT

equation

The equation of motion for the spin vector defined in the rest frame of the particle in

a synchrotron is given by the Thomas—BMT equation [{13, 14]

Coa

es

—eon =—S

|

:

~

1+Gy7)B 14+G)B a heeS

Gy

y

+ —— a)

\Exf

6

ke

: (2.1)

where S is the spin vector of a particle in the particle rest frame, B, and Bi are the transverse and longitudinal components of the magnetic fields in the laboratory frame with respect to the velocity Be of the particle. The vector E stands for the electric field, G is the anomalous gyromagnetic g-factor, and ymc? is the energy of the moving particle. This peculiar expression arises from the transformation of the electric and magnetic fields of the curvilinear laboratory reference frame to the particle rest frame. This is done in order to obtain a proper description of the spin—magnetic interaction. Section I derives the Thomas-BMT equation.’ Section II transforms the Thomas-—BMT equation into a spin equation that depends on the betatron coordinates of the particle. Section III casts the Thomas-BMT equation into the spinor formalism. Section IV discusses the solution of the spinor equation in terms of the spin transfer matrix.

I

Derivation of the Thomas—BMT

Equation

In the rest frame of a particle, the spin equation of motion satisfies the magnetic interaction:

7

ds

es

eee GE = Aix B= 95S, x B=AS, x Gq,

(2.2)

1For those who do not care about the details of the derivation of the Thomas—BMT equation, they can skip this section.

10

CHAPTER 2.

THE THOMAS-BMT EQUATION

where ji is the magnetic dipole moment, g is the gyromagnetic factor, i is the spin vector in the particle rest frame, and the vector Qa, can be considered as the angular velocity. For spin particles moving in a circular orbit with transverse magnetic fields in the laboratory frame, Lorentz transformation is needed to obtain a proper spin equation of motion. In this section we will derive the spin equation following the

covariant formulation of Thomas [13] and Bargmann, Michel, and Telegdi [14]. Let the 4-velocity of the charged spin particle be V‘ = (yc, y#), which is the proper time derivative of the position 4-vector, X* = (ct, 7), i.e., dX

dt

The corresponding covariant vectors are V; = (yc, —yv) and X; = (ct, —£), where the metric tensor is given by

Set

eT

TL -0 Peet

0 DO

0 0

0

0

-l

0

In the presence of the electromagnetic fields E and B , the equation of motion for the charged particle is given by dy

Ass me

—

=

dyv

“UO eE-v,

m —_— 7

=

=

~

e[E + 0v x B]

|

(2.5)

or equivalently dV' PRY: ean.

pase

—preetoe:BAV reztn (0s Bes=BB)pg

(2.6)

where the subscript R stands for the rest frame of the particle and F is the skew symmetric tensor of the electromagnetic fields, i.e.,

1 GC

0

-E,

-E,

—-E,

E, lyiy

0 ~iCe,

— Ob 0

« CB —cB,,

E,

-—cB,

cB,

Let us define the spin 4-vector ess = (S25),

(2.7)

0 In the rest frame of the particle,

the spin 4-vector reduces to S*, = (0,5,,). Now we decompose $, into components parallel and perpendicular to the velocity J, i.e.,

s

Bn =

s

%

Heh Pope

yea

_

=

paPlB -S,)+ 5,_ + 58

3

5.):

(2.8)

I. DERIVATION OF THE THOMAS-BMT EQUATION

11

In the Lorentz boosted frame, the spin 4-vector becomes (see Exercise 5.1.1),

Si

(Sox, St)

(18ao eroe are he

'5 ai

(2.9)

This equation, which relates the spin vector 5 in the laboratory frame and the spin vector S: in the rest frame will be used later to derive the spin equation of motion. In the presence of a magnetic field, #50 will not be zero, even though So = 0 in the rest frame. First of all, we consider the fact that the scalar product of the 4-vectors S and V is invariant under Lorentz transformation. In particular, SV is zero in the rest frame, i.e.,

SV = ye(So — 5-8) = 0.

(2.10)

Thus the time evolution of S' satisfies the following equation:

dS

dV

eh a V = SG rey Te

sal (2.11)

which reduces to

ds

dS

|

oy

Dovihig

hidean 2

Oe -($)| --2(s$)| ‘eer aa R

R

R

in the rest frame. From this we obtain

|e dt F

(= eS mc

ft, g5—S x B),

(2.12)

where, for simplicity, we neglect the subscript R on the right hand side of the equation. Now our task is to find a Lorentz covariant spin equation of motion, which reduces to

Eq. (2.12) in the rest frame. Lorentz covariant forms can be constructed from the 4-vectors S and V and the skew-symmetric tensor F’. The covariant form, which reduces to Eq. (2.12), must be

linear in both S and F’. Two possible covariant vectors are SF and V(SFV), where we have

way

gee (=5. EB. ~SoBi + 8 & B) =

(-s. E, Sx B) c

(2.13) in the rest frame

and

VASRV)

o= V7 (18: B —15,8.3- (8x B) 3) +

(2.14)

(75+ #,0 in the rest frame.

CHAPTER 2.

2

THE THOMAS-BMT

EQUATION

Thus we can express the spin equation of motion as

=: = -aSF —bV(SFV),

(2.15)

where parameters a and b are determined from the rest frame reduction. frame, Eq. (2.15) reduces to ds

A

= (¢ at be)Se E, aS x 5]. c

ae eS

In the rest

(2.16)

(£-1)

(2.17)

V(SFV)] ,

(2.18)

ot(ae(8xzB B-B) + L908 g —dGle =\e x vex He | 2S x B+ (9-278

—(g —2)7° (S- E- So (E-8))], G56 5 69s 2) aes ee dt

2m

1($xB

b) +55

(2.19) s

CNG m=2):|Pate eh = oy See

So =

Sox;

S =

Syie

(2.20)

(2.21)

For simplicity, we now only consider the magnetic field for applications in particle accelerators. Relating Sz; to S, through Eq. (2.9), we obtain

B-§,)

==—(g-2)77 (Sx B-B).

(2.22)

Bx B,

(2.23)

I. DERIVATION OF THE THOMAS-BMT EQUATION

13

we also obtain

gogas at at GS. nN: Go ge get at FxrerA)(0-5)2 +2= ALS, x8). (2.24) Similarly, using Eq. (2.20), we get d§

To

e

am E(R

7

~

sels (2

=~

2) Bg

=

2) 0 (5, x B) a) (2.25)

Equating Eqs. (2.24) and (2.25), we obtain the following equation for S,

oe He = Fy

\95u* B+ee (9-27

(8-3,) +B (Sxx B)B)-A)]} -B - 1) [(6i xxB) B)(8-55) +8 (5,

ea IR ii(1- ee TANS = gat

—-ly

»

ey

-

B xE

= = PBs)

:

The spin equation of motion becomes

dor =e leg xWe(8, arena = +9,),

(2.31)

which is the same as that of Eq. (2.1). Particle Position

Reference

Orbit

wy

Figure 2.1: The curvilinear coordinate system for particle motion in a synchrotron. Here £, 8, and Z are respectively the transverse radial, the longitudinal, and the transverse vertical

unit base vectors, and 7(s) is the reference orbit.

II

Spin Motion in Terms of the Particle Coordinates

The phase space coordinates of a particle in a synchrotron are usually expanded around the reference orbit, called the Frenet-Serret coordinate system, shown in Fig. 2.1. Following Courant and Ruth [16], we express the spin equation of motion in terms of the particle phase space coordinates in this section. Let 2,8, and 2 be see J.D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975).

I]. SPIN MOTION IN TERMS OF THE PARTICLE COORDINATES

15

the unit base vectors along the radially outward, the longitudinal, and the transverse vertical axes, respectively. For a planar circular accelerator, we have

dy” s —=-, ds’ .p

ds £ —=--, ds p

and

dz —=0, ds

(2332)

where p is the local radius of curvature of the reference orbit. Particle motion in the accelerator is then characterized by the betatron coordinate (z,s,z) as 0(s)+ 2&4 22,

(2.33)

where 79 (s) is the reference orbit with § = dro/ds. The velocity of a particle is given by

a=

dt ou FGF [ees (142) s4e9] woe

rare),

(2.34)

where the prime denotes differentiation with respect to the coordinate s, and the magnitude of the particle velocity ds 0)

is constant.

~ dt

One can also obtain 1

/

Slaw (2"- tes Serers), p p

(2.35)

Thus the transverse magnetic field is given by >

>

Bi= es(vx B) x

v

1

4

=D p (- “)(2 - *)Z+ a5

p

p

23]’

(2.36)

where Bp = ymv/e is the magnetic rigidity of the particle. Since the dipole guide field is given by B, = —[Bp] /p, the corresponding longitudinal field is then obtained from the Maxwell equation,

pe = =

OB, _ OB, aa

d1 \’ ~(8p)(2)

or

9. (2.37)

; 1

B, = —Bpz (=) Thus the magnetic field parallel to the particle orbital direction is given by

(2.38)

CHAPTER 2. THE THOMAS-BMT EQUATION

16

Using the relation o = er L Eq. (2.1) becomes

1S ciate sie s —adxF,

(2.40)

where @ is the bending angle in a dipole with d? = ds/p, s is the longitudinal path length and p is the radius of curvature. It is worth pointing out that the @ variable is constant in a region without dipoles. The vector

F = Fia+ F)3+ Ff,

(2.41)

can be expressed in terms of particle coordinates as, Fo

==

p2"(l 4 Go,

'

Fee

(1 Gy) 24 — p(ll+ @) (=)

Fz; =

—(1+Gy)+(1+ Gy) pe”.

(2.42)

Expressing the spin vector in terms of its components, i.e.,

S = S\8 + $28 + S32,

(2.43)

and using the fact that a = §, & = —3, we arrive at Ci

dS

(PES; FSS — (1+ Fs) 51 + iS,

“do

F,S;

=

=

(2.44)

F, So.

The unitarity condition of the spin equation is clearly satisfied by observing the fact that S? + S? + S? is time independent. Let us define Sy = S;+752, and Fy = Fi +7F>. Then the equations of spin motion become

d ave== +1GyS4

dSa>

42

aos

5 (FS

=f tF4S3,

= oe),

where we have used the average —((1 + F3)) = Gy.

Note that, when Fy = 0 and

F, = 0, the spin vector is given by S4 =

exe

See

S3 =

S30.

(2.45)

Here, the spin components S, precess about the vertical axis at Gy precession turns per orbital revolution. Thus G’y is called the spin tune, which is defined to be the number of spin precession around a spin closed orbit per orbital revolution.

III. SPINOR EQUATION FOR THE POLARIZED BEAM

III

17

Spinor Equation for the Polarized Beam

Let us define three 6-independent unit vectors (€1, €2,é3) to coincide with (2, 8, 2) at any azimuth in the ring. Equation (2.44) can then be expressed as

dS do

=aixS,

t= Gyés— Frey — Fre.

(2.46)

Defining a two-component spinor W such that the 7-th component of the spin vector is given by

S; = (Wlo,|V) = Vto,¥,

(2.47)

Eq. (2.46) becomes*

dU

a pet

aig = royle se Gy) —£

where H is the spin precessing kernel and the o,’s are the Pauli spin matrices (see Appendix B). The off-diagonal matrix element of the spin precession kernel is given

by E(0) = F, — th»,

(2.49)

which characterizes the spin depolarization kick by coupling the up and down components of the spinor wave function. Since the betatron coordinate of the particle is a periodic or quasi-periodic function of the dipole bend angle 0, we can expand the depolarization driving term € in Fourier series

eisyarare hh —

(2.50)

The Fourier amplitude ¢, is called the resonance strength, and the corresponding frequency is called the spin resonance tune. The spin equation of motion of Eq. (2.48) is equivalent to Eq. (2.1) or Eq. (2.46). These equations describe the classical spin motion in synchrotrons. Letting the two component spinor be

v= ey

(2.51)

where u and d are complex numbers representing the up and down components, we obtain

S,=u"dtud*,

S,= -i(u"d—ud"),

S3 = |u|? —|d|*.

(2:52)

Since the spin precessing kernel H is hermitian, we get |S| = |u|? + |d|? = (W|W) and d|S|/d@ =0. Thus we choose the normalization of the spinor wave function as

(O|Y) = ful? + fd]? = 1.

(2.53)

3 Although Eq. (2.48) for the spinor wave function is similar in form to the Schrodinger equation for the spin wave function (see Exercise 1.5), the polarization P or S is a strictly classical quantity, i.e., § is not a quantum mechanical operator.

18

IV

CHAPTER 2. THE THOMAS-BMT EQUATION

Spin Transfer Matrix

If the spin precessing kernel is a constant, the propagation of the spinor wave function is given by ;

W (02) = e277 (%2-%)

(6,) = t (82, 01) UV (01).

(2.54)

Here t (2, 0;) is called the spin transfer matrix (STM). Using the property that the polarization vector is real, we find that the matrix elements of the STM satisfy

bs Be ia

ey

(2.55)

The STM can generally parametrized in terms of the Pauli matrices t = tol — itjo, — it202 — it303.

(2.56)

The normalization of the spinor wave function implies that

ET

&

(2.57)

where / is the unit matrix. For accelerators, magnetic fields are piecewise constant, the spinor wave function is given by the product of spin transfer matrices, 1.e., N

WU (9) = [Tt (;41, 05) U (01) = (8,8;)V (8), jal

(2.58)

where 6 = Oy and 6; = 6,. The spin transfer matrix in one revolution around the accelerator is called a one turn map (OTM), which can be expressed as

CG.¢ ahem where v, is called the spin tune and 7, is called the spin closed orbit. If neo is independent of the turn number, then the repeated operation of the spin transfer matrix only increases the precession phase of the spin vector. Similarly, the propagation of the spin vector is given by Si Si Sy | =f | 24 S3 53

(2.59) 0

Here the real and orthogonal spin transfer matrix T is related to the t matrix elements

by

tot+t{—t}—t2 T= | 2tyte+tots) (tits — tote)

2(tyte — tots) t§—t}+t2-t2 A(tatz +toti)

2(tits + tote) 2(totz — tot) t2 -t? 12422

(2.60)

with >°; Ti; Tix = 6jx. In Chap. 3, the spin transfer matrix t for the spinor will be used to study spin resonances. Now we consider the following two examples:

IV. SPIN TRANSFER MATRIX

19

1. Spin motion in a perfect accelerator: spinor wave function is given by

For a perfect accelerator with € = 0, the

WV (0) = et 7

(0).

(2.61)

Thus the spinor wave function is precessing at a rate of Gy turns per revolution. Thus the spin tune is equal to Gy. 2. Spin Motion in an accelerator with local field error: The spin transfer matrix in one turn around the ring is given by M

iz ei Sytnn6

a3

Bees eniay’2}

(2.62)

where @ is the particle orbital angle from the observation point to the error field azimuth. The general error field can be expressed as that given by a spin precessing angle ~ around an axis fe, i.e., SE

HS

Whee = Cr BOS

SS

—

eA

ce

Ser OTN

5)

(2.63)

i)

with the precessing axis represented by the directional cosine Ne = COS X1€; + COS Y¥2E2 + COS X3€3.

(2.64)

The OTM then becomes M = e~*@1""8 (cos v—isin ycos x303] — ve ith Seed aks[cos x10, + cos x209] sin —. 5 We identify the OTM with M = e7'™s*e% where v, is the spin tune and fi, is the spin closed orbit given by Ne = cos 0, é; + cos O2é, + cos O3é3

(2.65)

to obtain

ae sin

Gyr sin 9 ©O8X3»

~

_ ~

cos TV, = cos GyT cos 5

2

sin TV, cos ®3 = sin Gym cos 5 + cos Gym sin 5 C08 Xa,

sin Ty, cos ®; = [cos Gy (a — 0) cos x; — sin Gy (m — 8) cos x9] sin a sin mv, cos ®2 = [cos Gy (m — 8) cos x2 + sin Gy (7 — @) cos x1] sin = Thus the spin tune and spin closed orbit are modified by the error field. When the error field rotates the spin ~ = 7 with respect to an axis on the horizontal

20

CHAPTER 2.

THE THOMAS-BMT EQUATION

plane, i.e., cosy3 = 0, then the spin tune is equal to ; and is independent of energy. This type of special spin rotator is called a snake, which has been proposed by Derbenev and Kondratenko.[17] in order to avoid spin resonances.* In accelerators with one snake, we obtain cos ®3°

=—

0;

cos®,

=

cos [Gy(7—-9)+ x1],

cos®,

=

sin [Gy(7-—8@)+ xi],

i.e., the spin closed orbit vector lies on the horizontal plane. Its orientation depends on the Gy value and its location in the ring, except at the symmetry point 9 = 7 where the spin closed orbit vector is independent of G’y and lies on the axis of the error field.

Exercise

2

1. Verify Eq. (2.9) by the following steps: e Decompose GR into components parallel and perpendicular to V, es,

ey

em OMAP

oP

ga B(8

S»)]-

show that

2. Verify Eq. (2.18) and Eq. (2.20) 3. Verify Eq. (2.27), in particular, verify

(6 x B)(6-S,) + B(S, x B-B)

=S, x By

4. Verify Eq. (2.44). 5. Verify Eq. (2.46) PVE,

:

i ; eve ‘ Spin resonances are coherent spin perturbing kicks arising from non-vertical error fields.

IV. SPIN TRANSFER MATRIX

21

6. A particle travels in a circular path with velocity 0 at time t, and velocity {+67 at time t + dt at an infinitesimally nearby position. The acceleration @ is given by @ = 6v/6t.

Let O’ and O” be the particle rest frames at time t and t + dt respectively, and let O be the laboratory frame.

The Thomas precession term can appear in the Lorentz

transformation from the reference frames O’ to O”, which can be accomplished by the following Lorentz transformations. OEM OYE EMO

The Lorentz transformation connecting coordinates (77, t’) of the reference frame O' moving with velocity v relative to the reference frame O with coordinates (fF, t) is given

by

where y = 1/\/1 — 6? with 6B = v/c. Tosimplify your calculation, we assume 60-0= 0. Please show that the infinitesimal Lorentz transformation from coordinate O’ to O” of the particle’s reference frames is given by t

"

—t

r=

!

rAd

oes 2

7 —

Avt! +7 x AQ,

where

Av= 760,

6Q= (y- 1)

bs . vx Ov i

are the effective velocity and the angular rotation frequency respectively. Thus the Lorentz transformation in the moving particle reference frame can be decomposed into a linear velocity component Av, and a rotation with angular frequency vector AQ), i.e., the coordinate axes in the electron’s rest frame precess with an angular velocity®

7. Please show that the cyclotron frequency rotation vector of a charged particle qg is given

by

jx B

oF A! B B e

=

segment aie

Eh

ym

ab aa

Bee

Ey,

as

where 3 is the velocity vector of the charged particle and E and B are the electromagnetic fields in the laboratory frame. 8. Combining

the previous exercise with

Eq. (2.1), show that the net spin precession

vector is given by

Ge 678. + 1 P@)B, F3(G— ym ms

e

As

-,

1

\E x B,

afta 1

ame

Ling

5For electrons orbiting in an atomic orbit with v < c, the angular frequency is approximately given by WT = pat x d, which cancels half of the spin orbit interaction in the magnetic energy

—ji-(B- 4,0 x E) of orbiting electrons in atomic systems. This term is called the Thomas correction term.

22

CHAPTER 2.

THE THOMAS-BMT

EQUATION

Find the magic value of y such that the radial electric field does not contribute to spin precession for electrons and muons. This cancellation has been employed in the muon g — 2 experiments, where radial electric fields have been used for beam focusing.® 9. Assuming an uncompensated solenoid field error with spin precessing angle x in an otherwise ideal storage ring, find the spin closed orbit for the polarized proton [22]. e If a beam of vertically polarized protons is injected into the storage ring, find the final polarization of the beam in the storage ring. e Let us inject the vertically polarized protons at 120 MeV into the ring. If you measure the vertical and radial polarizations at a 60° orbital bending angle location before the solenoid, what is your expected polarization as a function of the solenoidal precessing angle y? 10. Let the spinor wave function be expressed as

LEAGieiels where ¢o, $1, $2, and $3 are real quantities with

$+ Oi+ 42+ 93 =1. e Show that the components of the spinor wave function after passing through a dipole with bending angle @ are given by 4

Gy

.

x

ae

eG

dir demerger ¢1 = $1 cos

—

Gy

-

Gy

~

Gyé

$o = bo cos

2 sin on ,

2 = ¢2c0s

— ss eesbopltad a

+ ¢; sin ; |.S;|? = 1. 15. Verify Eq. (2.60) for the spin transfer matrix.

Chapter 3 Spin Depolarization Resonances In the Frenet-Serret curvilinear coordinate system, the Thomas-BMT given by

a5

Safele;

—=SxF 7 x

F,

equation is

2: (2.40)

where @ is the dipole bending angle with d@ = ds/p, s is the longitudinal path length and p is the radius of curvature. In a region without dipoles, @ is constant. The angular precessing vector F = F\z + F)8 + F3Z is given by

F, = —pz"(1+Gy), Fy = (1 + Gy)! — (1 + G)(EY’, Fey =

(LEG)

FUE Gy) pe.

Here Z, 8, and Z are respectively the basis-vectors in the radially outward, the longitudinal, and the vertical directions. Letting the three 9-independent unit vectors €),é€2, and é3 that coincide with 2, §, and 2 at any azimuth in the ring, Eq. (2.40) can be expressed as

—~=nxS,

t= Gyé3— Fei — Foéo.

(2.46)

Defining a two-component spinor W such that S; = (W|o;|W), the Thomas-BMT equation becomes

ON cuit i =peadelnhGy =p pb= esiok 9 (9-H) = SHY (ts,

mei’ he Pond

(2.48)

The off-diagonal matrix element, (0) = F; — 7F2, of the spin precessing kernel H characterizes the spin depolarization kick by coupling the up and down components of the spinor wave function. 20

CHAPTER 3. SPIN DEPOLARIZATION RESONANCES

26

Given the repetitive nature of particle motion in synchrotrons, &(@) can be expanded in Fourier series

(0) = F, — 1 =e, ™,

(2.50)

K

where @ is the orbital bending angle, €,, is the Fourier amplitude or the resonance strength, and K is the resonance tune. Table 3.1 lists essential characteristics of these spin resonances where k,j,m,mz, and m, are integers, v, and v, are the horizontal and the vertical betatron tunes, sy, is the synchrotron tune, and P is the superperiod of the accelerator. Here, intrinsic resonances arise from vertical betatron oscillations, imperfection resonances arise from vertical closed orbit distortion, synchrotron sideband resonances are due essentially to the spin phase modulation caused by the synchrotron motion and higher-order spin resonances originate from spin perturbation due to linear betatron coupling and nonlinear magnetic multipoles.

Table 3.1: Classification of spin resonances. Classification of resonances [3/P ae We RP V7 Ea integer

intrinsic resonances intrinsic synchrotron sideband resonances imperfection resonances

es

integer EMVyn

imperfection synchrotron sideband resonances

g+kP+m,zv,+mzvz

+ MVsy, | higher-order spin resonances

Using the Thomas-BMT equation, the spin resonance amplitude of the spin perturbing fields, is given by

ex f [ater AB,:

AB

strength, or the Fourier

iK6

where AB, is the radial perturbing field, and A By is the solenoidal perturbing field. In many synchrotrons, there is little or no solenoidal field. The transverse radial field arises mainly from the dipole roll, and/or the vertical displacement with respect to the center of quadrupoles. Neglecting the effect of dipole roll, the radial perturbing field is given by

OB, a 3 Pa 7

+ higher-order multipoles.

(3.2)

More specifically, the spin resonance strength in terms of particle coordinates is given

by

aes = fll + Gy)(p2" + iz’) — ip(1 + Gy =yea,

(3.3)

27

The dominant term in Eq. (3.3) is given by the pz” term in the integrand, i.e., é,

1+ Gy

-

iK

14+Gy

pute Keds = +

ae

; 2%

iK

{Be Keds.

(3.4)

Equation (3.4) reduces to Eq. (3.1) by using Maxwell’s equation:

OB, Ox

OB, gisd ABE. Oz

z=

(3.5)

The spin resonance strength is given by the Fourier component of the non-vertical magnetic flux density seen by the orbiting spin particles. The vertical displacement of the beam from the center of a quadrupole can be decomposed into two parts a —

(Bes =

Zoffset ) F ZB,

where (Zco — Zoftset) is the closed orbit displacement from the center of a quadrupole resulting from dipole rolls and/or quadrupole misalignments, Zofset is the offset of

the quadrupole alignment from the beam closed orbit, and zg describes the betatron motion of the orbiting particle. Since the Zogset is usually random, we will not discuss

it here. Following the basic beam dynamics (see Appendix A), we have

Zeo(8) = B;/"(s) iB aaah tag? k=-—oo

(3.6)

2

(3.7)

2 cos(vede + x) za(s) = (FX)! wy

where v, is the vertical betatron tune, €, is the normalized emittance, ¢,(s) is the

betatron phase, and f; is the Fourier amplitude of the error harmonic k given by

es sao f Oto

Z

ds,

$:(s) = - i

s

(3.8)

Spin resonances due to the closed orbit errors z,, are called the imperfection spin resonances. The imperfection resonance strength depends on quadrupole misalignment errors and/or dipole rolls. For a perfect machine with a zero closed orbit error, the imperfection resonance strength is zero. For most hadron accelerators with a random misalignment error of 0.1 mm in quadrupoles, the imperfection resonance strength without orbit correction is approximately given by [16] €imperfection ~ 2x

10°°Gy,

(3.9)

i.e., the imperfection resonance strength is proportional to y for a given closed orbit error and the uncorrected imperfection resonance strength at a 20 TeV accelerator

28

CHAPTER 3. SPIN DEPOLARIZATION RESONANCES

can be as large as 100. The imperfection resonance strength can, however, be greatly reduced by a proper closed orbit correction scheme.

The resonances arising from the betatron motion zg are called the intrinsic spin resonances. The intrinsic resonance strength is nearly independent of the machine alignment. Due to the adiabatic damping of the betatron motion for proton synchrotrons, the intrinsic resonance strength is proportional to \/yé,. Figure 3.1 shows the scaling property of the intrinsic depolarization resonances for some proton synchrotrons [16], where the maximum intrinsic resonance strength for most high energy hadron accelerators is approximately given by

intrinsic

oe

~

2

10

‘)

x=

=e

|oe Eel

:

107mm — mrad

—

In this chapter, we examine characteristic features of spin resonances. Section I discusses properties of imperfection spin resonances. Section I] studies intrinsic spin resonances. Section III examines higher-order spin resonances due to the linear betatron coupling, synchrotron motion, and higher order nonlinear magnetic multipoles.

RHIC AGS AGS

Booster

Tevatron

SPS 70 TeV

Figure 3.1: The family of maximum

GeV

booster

Booster

intrinsic resonance strengths for some accelerators

with normalized emittance €,, = 107 mm-mrad is plotted as a function of y. Note that the maximum strength of the intrinsic resonance is proportional to , [VES

I. THE IMPERFECTION RESONANCE STRENGTH

I

29

The Imperfection Resonance Strength

Substituting z.. into Eq. (3.3), we obtain éx =

L

a

Ge“ih ae, SN te fe all 82

g1/2() eikbz(s)

K

KO 7.

(3.11)

k=-00

To study essential characteristics of imperfection resonances, we will evaluate the integral in the thin lens approximation. We assume that the accelerator lattice is composed of P superperiods with M FODO cells in each superperiod.! These superperiods are connected by insertion sections where there is no dipole. Let the phase advance be 27y for each FODO cell, and the dipole bending angle in each half cell be m/PM. The integral of Eq. (3.11) becomes

ios

Gy

—

V2 fy

at

see Tika

a

a

k+kK

1s aoa

x [9p82/?(D) — 9, B22 F)e thea/MP) + x7},

(3.12)

where the enhancement function ¢,,(x) is given by sin Nrzx

hookt is Tiara ter

(3.13)

Here 2nv, = 27M Py is the total phase advance accumulated through all dipole cells that precess the spin vector, and X,; represents the contribution to the integral from the insertion region where there is no spin precession. Since the spin does not precess in the insertion straight section, there is no enhancement due to spin phase coherence. The quadrupole strengths g, and g, are the inverse of the focal lengths

of the focusing and the defocusing quadrupoles,’ and 3,(F’) and @,(D) are the values of the vertical betatron amplitude function evaluated at the horizontal focusing and defocusing quadrupole locations, respectively. It is known that the most important harmonic for the closed orbit is located at

k = +[v,], the integer nearest +v,. At K = mP +[v,], where m is an integer, the enhancement factor becomes ¢, (5H) ~ P. This enhancement arises from the fact that each superperiod contributes coherently in phase to the spin kick. 1The lattices of high energy storage rings are normally composed of straight sections and arcs. The straight insertion sections are used for high energy physics experiments, beam injection and extraction, and rf systems, etc. Arcs, composed of many FODO cells with dipoles, are used to transport beams for a complete revolution. Each FODO cell consists of regularly spaced quadrupoles and dipoles (see Appendix A). The superperiod is the number of identical repetitive building blocks

for the accelerator. 2TIn accelerator physics convention, a focusing quadrupole provides horizontal focusing and vertical defocusing. The quadrupole strengths g, and g,, are given respectively by g, ,= { B’dl/Bp at the focusing and defocusing quadrupoles.

30

CHAPTER 3. SPIN DEPOLARIZATION RESONANCES

Similarly, at the condition K = mPM +[v,], where m is an integer and [v,] is the nearest integer of 7) iat the enhancement factor becomes

K+k[=]

———

:

3.14

The contribution of each FODO cell adds up coherently to give an enhancement factor of M. When m is an odd integer the effect is particularly enhanced due to the addition between the focusing and defocusing quadrupoles, i.e.,

BBE) —0 OE et) SMT) moon, C18 (atk

at K

1/2( H

1/2( BP

=odd,

=mPM +{v,]. If m is an odd integer, the focusing and the defocusing quadru-

poles contribute additively to the spin kick. In the thin lens approximation, the ratio of resonance strengths for m = odd to m = even is approximately equal to ee Ip B.(D = BAF

ot

2

9p BAD) — gr Be(F

yr

This factor is about 3 for a FODO cell with 90° phase advance (u = 1/4). To obtain an adequate dynamical aperture in a high energy accelerator, a proper closed orbit correction is essential. Generally, there are harmonic and/or global orbit correction schemes. A closed orbit correction usually eliminates only integer error harmonics f;, of Eq. (3.7) near {v,]. The final closed orbit error would behave as

loo fey

ee

(3.16)

k=—0o

where the resonance structure of Eq. (3.7) has disappeared. The amplitude |Z;,| is expected to be nearly constant at |k — v,| © 0 and |Z;| = hene for |k — v,| > 0. The remnant closed orbit harmonics are normally limited by |k — v,| < v,/2. The resulting statistical orbit deviation can be approximated by

(22)=BulVRB =e's Line

(3.17)

where the factor of Zi,max = V6(Zj)'/? is used. The resonance strength generated from the closed orbit deviation of Eq. (3.16) becomes bx

=

1+G 7

{ou

ee

=2

k+kK (>)

[oBz!"(D) — gpB:!"(F)eW

pusoss y )r

|4 xi}. (3.18)

II. INTRINSIC RESONANCES

31

Using the estimation of Eq. (3.17), the maximum imperfection resonance strength is estimated to be

a myCUPMV3(22.)"7 ap |, (B2(F))

Ri.

(3.19)

B.(D)

A summary of important imperfection resonances is listed below in Table 3.2, where the synchrotron sideband resonances will be discussed in Sec.III.2. Table 3.2: Important imperfection spin resonances.

nPM +[v,]

| super strong imperfection resonances

RET V,| integer

very strong imperfection resonances regular imperfection resonances

Kimp + MVsyn | imperfection synchrotron sideband resonances

II

Intrinsic Resonances

Substituting the betatron amplitude zg of Eq. (3.7) into Eq. (3.3), we obtain =

ES

fea f-£2_,/8 se ,(s) cos (vz¢z(s)

+ €) ends:

(3.20)

Due to the periodic structure of the integrand -2= G.(s), the integral is not zero only for the harmonics

1 esi 7H ea eat

(a2)

where P is the superperiodicity of the synchrotron, i.e., the number of identical periodic building blocks in an accelerator. To study essential features of intrinsic resonances, we will make the following assumptions for the accelerator lattice: (1) We use thin lens approximation for quadrupole and dipole fields, (2) The accelerator is composed of P superperiods with M FODO cells per superperiod, (3) There is no dipole in the insertion region, and (4) The total accumulated phase advance in dipoles cells is vy, = MPyu, where 27 is the phase advance in each FODO cell. With these assumptions, Eq. (3.20) can be evaluated to give

G €K

s

—

M2 (~*)

Tv

oF) —9,\/B. De SP {Et [ES (ae

B.(F

—

9p

B.(D Je

se

om [Es (ae VBP) ~ 9 5 V6.(D)e mF") +Xz,]},

)+Xz,]

(3.22)

32

CHAPTER 3. SPIN DEPOLARIZATION RESONANCES

where g, and g, are quadrupole strengths of the FODO cell, 6,(F') and 6,(D) are the values of the vertical betatron function evaluated at the horizontal focusing and defocusing quadrupole locations, X*, are contributions from the insertion, E* and ee are enhancement factors due to the P superperiods and the M FODO cells respectively, 1.€., gt

i

Pib

es

ei2t(K4tuz)

be

1

cian Kee

ae

in(P—1)

a elt SS

M

cian

?

P

3.23

ay Be =

ee

‘=

(i =f ay

Cp

papers

a

K4%

LS

ae

Sea

.

”

The factor E= enhances the resonance strength P times at

PM

K = mP+v,.

Similarly,

the factor EX enhances the resonance strength M times at K = mPM+v,. Normally, we have M > P in high energy accelerators. Thus the enhancement due to E* is much more important. However, K can only be located at mP+v,. SincemPM+v, may not necessarily coincide with the resonance condition of Eq. (3.21), important resonances occur at those K = nP + v, such that they are closest to mPM + v,. Furthermore, dominant resonances are located at m = odd integers, where spin kicks due to focusing and defocusing quadrupoles add up coherently. Figure 3.2 shows

intrinsic resonance strengths for a RHIC lattice at ¢,, = 107 mm-mrad. In general, realistic synchrotrons do not have a perfect superperiodicity due to errors in the magnetic field. Thus weak intrinsic resonances can also occur at i

ipaeve

for all integer n. These weak resonances can be important to polarized beams in storage rings. Table 3.3 lists all intrinsic resonances by classification. The gradient error spin resonances arise from quadrupole field errors so that the periodicity of the machine is broken (see Appendix A).

Table 3.3: Intrinsic spin resonances.

DIS aaive De sey 1p 35 Wy Ene ats MV syn

Classification super strong intrinsic resonances near nPM +v very strong intrinsic resonances gradient error intrinsic resonances intrinsic synchrotron sideband resonances

B

III. HIGHER-ORDER SPIN DEPOLARIZATION RESONANCES

xis

T

T

33

T

RHIC Ve= vy 6

vy = 28.824 Ey = 10%

mm-mrad

Figure 3.2: The intrinsic resonance strengths at K = mP+v, for a RHIC lattice are shown as open circles and dots. The enhancement functions for the mPM + v, are shown as the dash and solid lines respectively. In this RHIC lattice, there are six arcs with 12 FODO cells in each arc. Because of the anti-symmetric arrangement, the superperiod is P = 3. Including the dispersion suppressor, the effective number of dipole cells per superperiod is M = 27.

III

Higher-Order Spin Depolarization Resonances

In realistic accelerators, higher-order depolarization resonances may become important due to the linear betatron coupling, large amplitude synchrotron oscillations, and higher-order magnetic multipoles. Higher order resonances can also be important for polarized beams in storage rings, where particles stay at a resonance condition for a long time. In this section, we will evaluate the resonance strength of higher order resonances.

III.1

Spin Depolarization due to Linear Betatron Coupling

In the presence of skew quadrupoles, solenoids, and vertical closed orbit deviations from the center of sextupoles, the horizontal and vertical betatron oscillations will be coupled. The linear betatron coupling is particularly important if the betatron tunes are near a linear coupling resonance line at v, — v, = ¢, where @ is an integer.

For electron storage rings, because the horizontal emittance is much larger than the vertical emittance, the linear coupling spin resonance can be enhanced. The vertical betatron motion in the presence of linear betatron coupling can be decomposed into oscillations with normal modes in which the amplitudes of each mode depends on the coupling strength. Using a single resonance approximation, the vertical

CHAPTER 3. SPIN DEPOLARIZATION RESONANCES

34

betatron coordinate near a linear coupling resonance can be described by ee

2&z

ae cos (v4¢. + x4) + Cz

see cos (vz¢z + Xz),

(3.24)

where 3, and (3, are the unperturbed betatron amplitude functions, €, and €, are the

horizontal and vertical emittances (actions) of the particle, and v4 are normal mode tunes given by ve = 5(et

1 ££) + 5A,

A=

(ve — vz — )? + |C_P.

Here v, and v, are unperturbed horizontal and vertical betatron tunes. The coupling coefficient is given by

|C_|

G

with

‘

C= = ts

3.25 )

|v. — Vz —L| +A

aoe oa

(

ee

ane

(3.26)

T

where ¢,,, are the horizontal and the vertical betatron phases respectively, and R is the average radius of the accelerator. The coupling strength function A,, is given by

|\B. B.) “let BT B/S {or abe (3) aDl}

Oat Bp *2B

oe

where oes is skew quadrupole gradient, Bj is the solenoid field strength, and a, are related to the derivatives of the horizontal and the vertical betatron amplitude functions. Substituting Eq. (3.24) into Eq. (3.4), the first term gives rise to the intrinsic spin depolarization resonance discussed in Sec. II, and the second term gives rise to the intrinsic coupling resonance. The linear coupling spin depolarization resonance is located at i conluk =n

+.mP

+ Vz,

(3.28)

where the resonance strength is proportional to Ccacoapinns

on (1 +

GY)C,/ec-

(3.29)

Depending on the coupling coefficient C, and the emittance, the coupling resonance strength may be as large as the intrinsic resonance strength discussed in Sec. II. For electron storage rings where the horizontal rms emittance of the beam is usually much larger than the rms vertical emittance the coupling resonance may be even more important than the intrinsic resonances. Figure 3.3, taken from Ref. [23], shows the electron polarization measured at 3.7 GeV in SPEAR where beam depolarization has

III. HIGHER-ORDER SPIN DEPOLARIZATION RESONANCES

35

° co)

3.52

360

368

E (GeV)

Figure 3.3: The electron polarization at SPEAR around 3.7 GeV [23]. Beam depolarization was observed at the imperfection resonance K = 8, the linear betatron coupling resonance and its synchrotron sidebands at 3+ Qz + MQsyn, the intrinsic resonance 3 + Q, and its synchrotron sidebands, and higher-order resonances at

been observed at the imperfection resonance

8+ Q, — Q, and -2+Q,4+Q,.

K = 8, the linear betatron coupling

resonance and its synchrotron sidebands at 3+ Qz + MQsyn, the intrinsic resonance

3 + Q, and its synchrotron sidebands, and higher-order resonances at 8 + Q, — Q, and -2+Q,+Q,.

Using an assumption identical to that of Sec. II, the dominant depolarization resonance strength is given by

éxomping © Oz S1(5)74 74(4(4,(BUF)—95Be(D)ewP*) + Xi, 1+

Gy

Ex

sr

+ E> [Ex (9, VBe(F) —9p/Ba(D)e*

7") + Xz}

(3.30)

where g, and g, are quadrupole strengths in the FODO cells, 6,(F') and @,(D) are, respectively, the values of the horizontal betatron function evaluated at the horizontal focusing and defocusing quadrupoles, and X%, are contributions from insertions. E+ and E= are enhancement factors due to P superperiods and M FODO cells respectively, i.e., oa

les =

1 — €2"(K+v2) 1

-

ein neee i

K+v

co

Ee 7

ihe ei27(—p*) 1

—e

eos

‘

oe

The horizontal coupling intrinsic resonances has similar characteristics as those of intrinsic resonances. When strong linear betatron coupling is introduced, the lattice superperiodicity becomes 1. The spin depolarization resonance strength will be distributed at Keane yy (3.32)

36

CHAPTER 3. SPIN DEPOLARIZATION RESONANCES

for all integers n, which is not necessarily an integral multiple of the superperiod P. However, the resonance at n = mP may still be more important. Table 3.4 lists spin resonances due to linear betatron coupling. Table 3.4: Spin resonances due to linear betatron coupling. Classification

very strong coupling resonances near nPM tv, strong coupling resonances gradient error coupling resonances coupling synchrotron sideband resonances

III.2.

Synchrotron Sideband Resonances

Including the vertical dispersion function, the vertical particle coordinate can be written as

$23.47, 8C

A ee 8

(3.33)

Pp

where D, is the vertical dispersion function (which can arise from a non-planar accelerator geometry and/or the linear betatron coupling). Because of the synchrotron motion, the momentum deviation a for an off-momentum particle is given by

A —? = cos (Vsyn9 + x),

(3.34)

P

where @ is the amplitude of the synchrotron motion and vy, is the synchrotron tune.

Substituting Eq. (3.33) into Eq. (3.4), zo contributes to the imperfection resonance, zg gives rise to the intrinsic resonance, C,2g generates the linear coupling resonance, and the term D,42 can produce synchrotron sidebands around imperfection resonances located at K = n+Vsyn (see Exercise 3.5). Since it is proportional to the product of the synchrotron amplitude 4@ and the vertical dispersion D,, the resonance strength is small. Note particularly that the dispersion function cannot give rise to synchrotron sidebands around intrinsic depolarization resonances. More generally, the intrinsic resonance strength is obtained from the Fourier amplitude of (1+ Gy) B,/Bp for the transverse error field. Since the factor (1+ Gy)/Bp is nearly independent of the particle momentum at high energies, the transverse perturbing fields cannot generate spin synchrotron sideband resonances. The spin resonances induced by a solenoidal field can indeed produce synchrotron sidebands. However, the solenoidal field is usually weak in synchrotrons. Thus we would expect a very small resonance strength for the synchrotron depolarization resonance. However, experimental results indicate that synchrotron sidebands are very important (see Fig. 3.3). We will now discuss a mechanism that enhances the synchrotron sideband resonances.

III. HIGHER-ORDER SPIN DEPOLARIZATION RESONANCES

37

Synchrotron Sideband Resonances Arising from Kinematic Effects We have seen that the resonance strengths of synchrotron sidebands induced by the vertical dispersion function are relatively small. However, the SPEAR data showed that synchrotron sidebands are very important. What is the mechanism for synchrotron sideband resonances? In the following discussion, we will show that the enhancement of synchrotron sideband resonances can arise from the kinematic effect. We consider a single primary spin resonance with € = €,e~'*®, where the spinor equation of motion is given by

a A)

=

Gy 2

;

apd

aye

“Oy

1K 0

(3.35)

Now, transforming the spinor wave function into the spin precessing frame with

U = erie Crier, we obtain

'

eli aaa nasil f Grd8) a dé 2 \ ex ei(KO-

(3.36)

(espe) +(Ke-f Gy 0

Uy.

(3.37)

Le

The spin precessing phase for an off-momentum particle becomes [ Gyd0 = Gyo8 + ——— B°G0 4 SiN syn9,

(3.38)

Vsyn

where @ is the amplitude of synchrotron oscillations, yomc? is the energy of the synchronous particle, and Vy, is the synchrotron tune. In particular, we note that the spin precessing phase has been greatly enhanced by the smallness of the synchrotron tune. The effective synchrotron amplitude becomes B’G 0 a

(3.39)

= ——a. Vsyn

Expanding the spin precessing phase in Fourier harmonics, the effective resonance driving term in the spinor equation becomes co

Patel dalllGydd) _

SF Se Juagiea

a Sule

(3.40)

m=—Co

Near a single dominant resonance at Gyo K + mvsyn, the effective spin resonance harmonic of the off diagonal matrix element becomes —i(K@— |] Gyd0) Epes fe ) = €,

—i(K Im(g)e

—Gy —mv

o—mvsyn)n)@

Thus, if the condition |e,Jm(g)| < Vsyn is satisfied, the effective resonance strength for the sideband resonance becomes €,.Jm(g) where the amplitude g is enhanced by

CHAPTER 3. SPIN DEPOLARIZATION RESONANCES

38

Pure

Pr/Po

0.6

©

1.494

1.495

1.496

1.497

rf frequency

1.498

1.499

1.5

(MHz)

Figure 3.4: The vertical (upper plot) and the radial (lower plot) polarizations measured at the IUCF Cooler Ring are shown as a function of the rf frequency of the rf solenoid. The solid lines are a one parameter fit to be discussed in Sec. V of Chap. 4.

ee : ‘ : ' a factor oe in comparison with the resonance strength discussed in last section. Thus the synchrotron sidebands are particularly enhanced in synchrotrons with large Gy and small vy,. Using the small argument expansion for Bessel functions, Table 3.5 lists effective resonance strengths for synchrotron sidebands.

Table 3.5: Effective strengths for synchrotron sidebands.

Resonances

Ee

tn A

Effective Strength

The physics of synchrotron sideband resonances can be understood as follows. The synchrotron motion gives rise to a spin phase modulation, which can generate sidebands. If one of the sidebands falls on the primary resonance, the spin is strongly perturbed. Thus the resonance strengths of synchrotron sidebands are proportional to the strength of the primary resonance. Because the synchrotron tune is relatively small so that the particle stays at the spin resonance condition for a long time, the resonance strength is particularly enhanced. For electron storage rings, the synchrotron tune is of the order 0.1 so that synchrotron sidebands are well-separated. The single resonance dominant model used above is applicable. There is an abundance of experimental data which has confirmed

III. HIGHER-ORDER SPIN DEPOLARIZATION RESONANCES the enhanced synchrotron sideband resonances.

39

For example, resonance sidebands at

3 + Vez £ MV syn, and 3+ Vz, + Vsy, are clearly observed in Fig. 3.3.

For proton machines where synchrotron tunes are normally about 107°, the: enhancement factor can be large. This means that all synchrotron sidebands overlap with the primary resonance, and the single resonance dominant analysis used above is not valid anymore. Section V in Chap. 4 will show that the effective resonance strength can be summed to obtain a value that is equal to the primary resonance strength with a phase shift. Experimental measurements of synchrotron sidebands in proton synchrotrons have not yet been systematically studied. Synchrotron sidebands have been observed in a low energy storage ring [24, 25]. Figure 3.4 shows the vertical and radial polarizations as a function of the rf solenoidal frequency at the IUCF Cooler Ring [26]. The proton beam energy was 104.14 MeV with a revolution frequency of 1.506 MHz. The spin tune was Gy = 1.9942 with a type-3 snake tune shift of about Av; = 0.00233 [25]. The rf solenoidal field strength corresponded to a resonance strength of €, = (4.1 + 0. 2) x 10-4. The synchrotron tune was 0.001271, which gave rise to the parameter g = eala = 0.30 with the momentum spread about

4 ~ 0.001. Thus the effective resonance “strengths were about Jo(g)e,

4.0 x 1074

for the primary resonance, and J,(g)e, = 6 x 107°. The solid lines, to be discussed in Sec. V.3 of Chap. 4, fit the data reasonably well with a single parameter a [27].

III.3.

Nonlinear Coupling Resonances

Since the equation for betatron motion is given by

z+EE

Kls)eRE

=,

3.41 (3.41)

AB

aBz : ; where K, = —, and the prime corresponds to the derivative with respect to the Bp? azimuthal distance s, the nonlinear magnetic field can also contribute to spin depolarization resonances. Generally, the nonlinear magnetic field can be expressed as

AB, + tA Bz = Bo ¥>(bn + tan)(x + iz)”,

(3.42)

n>2

where b,,,@, are normal and skew multipoles. We then obtain

ps

Bp

ae 2h

p

2Ges

p

ede

Ee (POR

p

PU

p

eA) 4.

(3.43)

The strengths of nonlinear depolarizing resonances can be obtained by substituting the nonlinear field multipoles into Eq. (3.3). For example, the spin depolarizing resonances for sextupoles are located at K =n+mP+v, +1,. Possible spin resonances from nonlinear magnetic multipoles are listed in Table 3.6.

CHAPTER 3. SPIN DEPOLARIZATION RESONANCES

40

Table 3.6: Spin resonances induced by nonlinear multiples.

01 Sodhinermslrou strwabd igeiotih lene aenllistin patond sextupole octupole

|ntvu,+y, |n+2v,+v,,n£3y,,nty,

+ 2p eht.e 2, |n+3yz,,ntVz,nt

Vz + 2,

In high energy accelerators, sextupoles are important for the chromatic correction in achieving beam stability. Thus nonlinear spin resonances arising from the sextupole may be important shown evidently in Fig. 3.3 for the SPEAR, where depolarization were observed at K = 8+ v7, —v, and —-2+1,4 1.

Exercise 3 1. Substituting zg of Eq. (3.7) into Eq. (3.4), show that the contribution of a FODO cell to the intrinsic resonance strength is

“K,FODO

v

:

aie:

Vue

To

me Ip VV B,(D

e (K8xop0 +a)

wy e5% (s-/8.(F) =Ip /B2(D)e"(K*rovo “ny Qn

:

(=)

% @ VJ B.(F) le

where 0,55, and p are respectively the bending angle and the phase advance of the FODO cell, g, and g, are strength of the focusing and defocusing quadrupoles. Verify

the intrinsic and imperfection resonance strengths shown in Eqs. (3.12) and (3.22). 2. The AGS can be considered as a machine made of 60 FODO cells. The circumference is 807.12 m and the horizontal and the vertical tunes are both 8.8. Calculate the intrinsic spin depolarizing resonance strength for K = v, and K = 60 — v, for a particle with a normalized emittance of 107 mm-mrad in the thin lens approximation.?

3. The Fermilab booster is composed of combined function magnets with 24 superperiods. The lattice is given by BeamLine= BD:RBEND

BF

S120 BF

S050BD

S600 BD

S050

;

L = 2.889612

ANGLE = 0.070743

K1=-—0.05771

BF: RBEND L = 2.889612 DRI EL xyz

ANGLE = 0.060158

K1= 0.05422

SXy77

Here all lengths are in meters, and the quadrupole strength K1 is in m~?. Using thin lens approximation, calculate the intrinsic resonance strength for a beam with 107 mm-mrad normalized emittance. 3In reality, the AGS is made of 12 superperiods (A,B,C,..,L), with five nearly periodic FODO cells in each superperiod., Thus intrinsic resonances are located at 12k + v, (k = integer). Out of these resonances,

vz and 60 +,

are most important.

II.

HIGHER-ORDER SPIN DEPOLARIZATION RESONANCES

41

4. We consider an accelerator with 48 FODO cells, which is divided into 36 dipole FODO cells and 12 straight FODO cells for injection, ejection, and rf stations. Let the betatron tune be 8.8 and the circumference be 600 meters. The accelerator can be arranged into machines a,b,c,d, and e, according to the dipole cell arrangement:

dipole cell18per period

2

12

Discuss the spin resonance location and strength for a beam with 107 mm-mrad normalized emittance with the above lattice arrangements.* 5. The linear synchrotron motion for an off-momentum particle is given by

Be Vy

{ prewar f= = ¢VBE Doe Meds.

(3.47)

(3.48)

CHAPTER 3. SPIN DEPOLARIZATION RESONANCES

42

Because of the simple pole structure, the vertical dispersion function can be approximated by D,(s) © aa

cos (ko, + x).

(3.49)

Thus the most important synchrotron sideband resonance due to the vertical dispersion function is located at K = n+ Vsy, closest to the number mPM + [v,]. Discuss the magnitude of the synchrotron sideband depolarization resonance arising from the anomalous vertical dispersion. Compare your result with the synchrotron sideband resonance arising from the kinematic effect discussed in Sec. III.2. . The method by which a set of quadrupoles powered to cancel a specific intrinsic spin resonance harmonic is called intrinsic resonance spin matching. Show that the induced resonance strength by the set of quadrupoles with focusing strength Ag(s) is given by =Bie rot [= fags Ag(

Bz, cos (vz¢z(s) + x)e* ds.

Here we assumed that Ag(s) is small so that the betatron motion is not strongly perturbed.

. A set of orbit correctors can be powered to excite a harmonic orbit oscillations in an accelerator in order to cancel the imperfection spin resonance. This method is usually called the harmonic spin resonance correction or harmonic spin matching. There are two contributions from the vertical closed orbit correctors with horizontal fields. The

first contribution arises from the integral of Eq. (3.1) and the second contribution arises from the closed orbit perturbation given by ihe LING

om y Bx (s)

3 k=—00

v2 Le —

a, b G2 =

finds

is

B.(s)

with f=

Qnv, — f

/B.(s Des\e) .-ikbz(s)

Discuss the relative importance of these two =

ds.

eat

. Near an isolated synchrotron sideband with Gyo + K + mlsyn, Eq. (3.37) becomes

ee dé

2

(€* Jin(gt— k

-Gro—mveyn) 6

EJ (Gyr

0

ememrised?

Son-

where g = BG70/Vsyn, and Jm(g) is the Bessel function. Using the inverse transformation vy) =e

Foy,

show that the spinor equation becomes

dv

i

Gyo

doe nD batenne

oe

~€ Im (g)e7(K-mveun)4 —GY0

Chapter 4 Effects of Spin Resonances The Thomas-BMT spinor VU

equation is equivalent to the equation for the two-component

UVink n5Se-4nOy sonicté Gn (2 Sree

(2.48)

where the spin vector is given by 5 = (U|G|V), 6 is the orbital bending angle, and the depolarization driving term € arises from the spin perturbing electric and magnetic fields. Since, as a function of 0, the betatron and synchrotron oscillations are quasiperiodic and the electric and magnetic fields are periodic, € can be expanded in Fourier harmonics as é =

Ss eed:

K

where the Fourier amplitude ¢€, is the resonance strength, and K is the resonance tune.

For an ideal particle in a perfect circular accelerator, where € = 0, the spinor is given by

W (Os) = e721 82 (G;) = t(8,, 0;)U(8,).

(4.1)

Here t is called the spin transfer matrix (STM). Using the STM, the evolution of the spinor wave function can be tracked by the spin mapping equation. The STM in one complete orbital revolution is called the one turn map (OTM). For example, the OTM in an ideal synchrotron is given by

£(0; + 2, 0;) =e 7°", where Gy is the number of spin precessing turns per revolution, called the spin tune. The spin vector that is invariant under the OTM is called the stable spin direction or the spin closed orbit. In this chapter, we-examine the effect of spin resonances on the polarization vector with some exact solvable models in Secs. I and II, where the spinor equation for a single spin resonance with €(0) = ee~'*® can be solved analytically. Using these 43

44

CHAPTER 4. EFFECTS OF SPIN RESONANCES

solvable models, we examine methods of spin resonance correction as well as some experimental results in Sec. II. The spin motion resulting from a local spin rotator is studied in Sec. IV. This section illustrates a possible usage of the local spin rotator for the spin resonance correction. Finally, we analyze the effect of overlapping resonances on polarization in Sec. V.

I

Spinor at a Constant Acceleration Rate

Since the energy gain per passage at the rf cavity is small, we assume a uniform particle acceleration rate in synchrotrons with Gy = ko + a@, where

_ dGy

(4.2)

Ode

is the acceleration rate for the spin tune. Now we transform Eq. (2.48) onto the spin

precession frame (interaction picture) with [28] ;

78

U(d) = cP So 1479 (8).

(4.3)

The equation of motion becomes dv

r/

0

3

~

wimecyte pels!

.

EN

rele

72

Pee velleanll ea

(4.4)

The effect of the spin resonance is greatest when the exponent in Eq. (4.4) reaches the stationary phase condition Gy = K = ko + ad.

(4.5)

Choosing the initial condition to be ko = K, and letting WF be the upper and lower components of the spinor wave function respectively, we obtain

dut

i

dv;

i

apis Diniaineae sagged Bese Gl nS cE

du

+

2

a Figg + : wt = 0.

(4.6) (4.7)

In the limit of |e] = 0, we have

d+ dé one i.e., if there is no resonance, W* are stationary in the rotating frame. This condition is useful for us to find the asymptotic expansion of confluent hypergeometric functions.

Letting z = 4a6?, Eq. (4.7) becomes

CisGaaectan sl

avy’

dee eae

=

Ah

(4.8)

I. SPINOR AT A CONSTANT ACCELERATION RATE

45

where the dimensionless resonance strength parameter q is given by

rte =

(4.9)

The solutions of Eq. (4.8) are confluent hypergeometric functions.!. independent solutions are

NT se) 2(0) = M(agq, =, ~ a6?),

Two linearly

3s. at w(0) = “bei M(1 —14q,= 5° —526"),

(4.10)

where M(z, a, b) is the confluent hypergeometric function. Since z(@) and w(@) should be stationary in the limit of € = 0, we assign a = a+20* to obtain a proper asymptotic expansion for the confluent hypergeometric functions. The solutions of Eq. (4.8) are

Wt (0) = Az(6) + Bw(0),

W7(6) = —Aw*(6) + Bz*(6),

(4.11)

where A and B are constants to be determined from the initial conditions and the corresponding solution for the V~ is obtained from Eq. (4.6), where z* and w* are complex conjugates of z and w, respectively. We can then write the spinor wave function as

Wi= (ilo) st) (a) =TO(B) =POT UAW).

(4.12)

Here the matrix T(0)T~'(6;) is the spin transfer matrix. It transforms the spinor wave function from an initial state at 6; to a final state 0. The essential idea is similar to the

wave propagation matrix method in the semi-classical theory of potential scattering [30] or the transfer matrix method in the betatron motion. Using the asymptotic relation of the confluent hypergeometric functions for large

angle |@|, we obtain 1

z(+6)

peti

—7q/2,-iqIn par?

alia) copeeft ——\2

ri ad

w(+0) > +2

V8a

athe

14/2. g-gn 58?

4.13 eee

(1—7q)

Thus the asymptotic condition with 0 >> 0 and 0; < 0 for VW, is given by

(0) = e-t2 (F231)(q) 2M) W(6,)

(4.14)

with

en 274 © weed uid=(“omy Se) C=: ~ at EDht

see Chap. 13 of Ref. [29].

+e ey, Pewates

CHAPTER 4. EFFECTS OF SPIN RESONANCES

46

10-3

1072

10-1 4nmq

10°

Figure 4.1: The polarization of the Froissart-Stora formula is plotted as a function of the spin resonance strength 47q = m|€|?/2a, where a is the acceleration rate.

It is easy to check det (U(q)) = 1. The final polarization is given by

a4 ieee (S3) = (W;|e5|V7)-=2e"*" — 1 = 2e°" 4a — 1.

(4.16)

This is the celebrated Froissart-Stora formula [31] for obtaining a final polarization after passing through an isolated spin resonance with resonance strength € at a constant acceleration rate a = a Figure 4.1 shows the final polarization as a function of the resonance strength at a constant acceleration rate for the Froissart—Stora formula, where 47q < 0.005 is required to maintain 99% polarization, and 47q > 5.3 in

order to attain —99% spin flip. These conditions are equivalent to |e| < 0.056,/a and |e] > 1.8,/a, respectively.

II

Spin Motion at Constant Gy or Slow Acceleration

Another interesting analytically solvable model corresponds to a slow or zero acceleration rate. At a constant spin tune G’y, we can transform the spinor equation into the resonance precessing frame by defining

Wi(9) = 22 Wi).

(4.17)

The spinor equation for Vx becomes

dV x H(i fay do 2

: 5)Wi = £[b05 : + eqor — €,02]VK,

(4.18)

I]. SPIN MOTION AT CONSTANT Gy OR SLOW ACCELERATION

47

Figure 4.2: Schematic drawing of the spin closed orbit vector passing through an isolated spin resonance. Note that the vertical component of the spin closed orbit changes sign after passing through the resonance.

where 6 = K — Gy. The solution of Eq. (4.18) is given by, VK (0;) = erlbosten e167) 84-8)

,.(8,) £ er (81-8 Reo FY, (9,),

(4.19)

where €, and ¢€, are the real and the imaginary parts of the resonance strength,

A= 1/6? + lel?

(4.20)

is the spin tune and Dies =

6

€

~—€é3 +

6,

A

A

€ ys é,

(4.21)

r

is the spin closed orbit in the resonance precessing frame.” The spin vector, which follows adiabatically the spin closed orbit, will have the polarization value $3 = 6/2.

At 6 = +|e|, the spin closed orbit vector tilts 45° away from the vertical axis. The system has three eigenvalues, 0 and +2, which correspond to three eigensolutions describing the spin vector along the spin closed orbit and the spin vectors precessing right/left with respect to ro. Figure 4.2 shows schematically the evolution of the spin closed orbit in passing through a spin resonance.

II.1

Effect of Tune Jump on Polarization

Note that the é3 component of the spin closed orbit vector, to of Eq. (4.21), changes sign in passing through a resonance. A sudden tune jump will cause a nonadiabatic ?The spin closed orbit depends on the spin resonance strength ¢,, and the proximity parameter 6. Thus the spin closed orbit is a function of the betatron

orbiting particle.

amplitude,

and the momentum

of the

48

CHAPTER 4. EFFECTS OF SPIN RESONANCES

transition of the spin closed orbit vector. The polarization that survives in the tune jump is the projection of the initial spin closed orbit vector onto the final spin closed orbit vector, 1.e.,

P, =

fda +lelp AyA2

(4.22)

where Py and P, are the final and initial polarizations, 6, = K,—Gy and 62 = Kz—Gy with K, and K, as the resonance tunes before and after the jump, and the 4,’s are

given by Eq. (4.20). The negative sign in Eq. (4.22) arises from the fact that, after the tune jump, the n3 component of the spin closed orbit in Eq. (4.21) changes sign while the particle spin vector is not affected by the tune jump process. For an optimal tune jump with 6, = —é, = 6, the polarization becomes 5? — |e|? P;t= = ———_-P,. Hy lef

4.23 (4.23)

To achieve a proper tune jump with 95% polarization survival, Av, = 2d95% ~ 12|e| is needed in the tune jump for the single particle model. Since the betatron tune jump is limited to Av, < 0.3 by the stopbands of the betatron resonances,*? the maximum intrinsic spin depolarization resonance strength that can be effectively overcome by the tune jump scheme is € < 0.025. However the actual polarization survival is the ensemble average of Eq. (4.23) over the bunch distribution. The applicability of the tune jump method may depend on the actual beam distribution (see Sec. III.3 of

Chap. 4).

II.2.

Eigenmodes

There are two orthonormal eigenmodes

ay Ur

bs

pte

1

€

i2d6

1

=(5

1 —tir6

reeseyRAce ei

—_

aE

re onto

a

with A = ,/6?+ |e/?. At Gy < K,A x6 > 1, we have

Vit

(4) et

Wy)

ee Chios

(4.25)

and at Gy > K, \ » —6 > 1, we have

vr

pene (i),

War

ee (Tt) ee

(4.26)

3Important betatron resonances are the half-integer stopbands. Near a half-integer stopband, the betatron amplitude functions become large.

II. SPIN MOTION AT CONSTANT Gy OR SLOW ACCELERATION

49

At a spin tune Gy far away from the resonance tune K’, Vx is basically the spin up component of the spinor and Vx the spin down component. The general solution of Eq. (4.18) can be expressed as a linear combination of these two eigenmodes: Wi

=

CyVK4

+

CoVKy

(4.27)

with C? + C? = 1. The polarization is obtained from the expectation value of o3 in the spinor wave function

(S3) = (Wlo3|V) = (1a? — |C,|?) + elle, cos(\6 + ¢),

(4.28)

where the phase angle ¢ is given by ¢ = arg(C C3). At |6| > |e|, the initial polarization is given by

(S3)o = |Ci|? — |C2]?.

(4.29)

Since the oscillatory term is averaged to zero, the average polarization around a resonance region is given by

(Op sQ =js aeteesa~ yk = Gy)? + le

‘ic

(4.30)

From Eq. (4.30) we find that |e| plays the role of the resonance width adequately. At a distance Gy — kK =

+e] away from resonance,

polarization is reduced by a

factor of 1/\/2. A minimum of +3|e| from a resonance is needed to maintain a 95% polarization.

II.3.

Spin Transfer Matrix in the Particle Rest Frame

Transforming the spinor wave function of Eq. (4.19) back to the particle rest frame, one finds U(6;)

ia

e~ 2K 8703 @ glboa tens

—

t(0,/, 0;)U(0;).

—€192\(94—8i) eg KHi03Y(Q,) (4.31)

The components of the spin transfer matrix t(0;,6;) are $11(03, 03).= pe leah 0,8) il.

toi(Oy,0;) = —th (Of, 6:),

t12(07,9:) = ibeg ems Gita).

(4.32)

tr2(Or,01) = t3, (47, 9),

with

sin [X(0y — 65)/2] =(

ee 2)1/2 b = —sin[A(0; — 0;)/2] = (1—a*)"!*,

6=K-—-—Gy,

2 2\1/2 A=(d°4+ el"),

c=arctan [5tam (X(0, — 0;)/2)],

d=arg (ce).

ne

CHAPTER 4. EFFECTS OF SPIN RESONANCES

50

The off-diagonal matrix elements t;2 and tg; are the depolarization driving terms,

where the effective strength parameter b oscillates with an amplitude |e|/A. The effect of the depolarization kicks gives rise to an average polarization given by Eq. (4.30). In the laboratory frame the spin closed orbit n.. of Eq. (4.21) is also precessing at a frequency K about the vertical axis. If the polarized protons were injected initially along the vertical polarization direction, the final spin vector would precess around the spin closed orbit n., at a precession frequency A and the net vertical spin vector would become lel? sin . 9 [5s A (S3) = [tal” 2 — [fire 2 = 1 — 26° : = 1 — 22 — 6;)}.

(4.34)

Here the time average becomes ((53)) = 67/A? (see Exercise 4.4). The spin vector precesses about the n,. at a rate of 1/|e| per revolution on resonance with A = |e|. Here the inverse of the resonance strength 1/|e| plays the role of spin precessing tune about the spin closed orbit. At the same time, the spin closed orbit precesses at a spin tune K about the vertical axis. If all particles in the bunch had identical spin tunes Gy, then the polarization vector would coherently precess around the spin closed orbit Nco Without depolarization.

Figure 4.3 shows the free spin precession data at the IUCF Cooler Ring [26]. The experimental procedure is given as follows. First, vertically polarized protons at 104.14 MeV were injected into the Cooler Ring, then, the rf solenoid was turned on for a time duration of 10.316 ms, and finally, the vertical polarization was measured as a function of the solenoidal rf frequency. The measured vertical polarization is plotted as a function of the rf solenoid frequency K fo, where the revolution frequency was fo = 1.505 MHz. The solid line corresponds to the theoretical prediction of Eq. (4.34) with the resonance strength determined from magnet calibration with « = 0.000354 and a resonance on-time of 10.316 ms, or 0; — 6; = 27 x 15525.58 (4 spin precession turns around the spin closed orbit). When the rf solenoid frequency is turned off after 10.316 ms, the vertical spin vector can be described by Eq. (4.34) with

d= (Kup +e, where v, is the spin tune, i.e., vy, = Gy + Av3. Note here that the spin tune of the polarized proton in the IUCF Cooler Ring is shifted by an amount Av; due to the electron cooling confinement elements (see Exercises 4.9 and 4.10 and Ref. [25]). This method has been applied to flip the beam polarization vector in synchrotrons [32]. On the other hand, if there is a sizable spin tune spread in the bunch, the spin vectors of particles in the bunch will decohere and the remaining polarization will be the projection of spin vectors onto the spin closed orbit. The measurable polarization in the accelerator will be the projection of the spin closed orbit onto the vertical axis.

II. SPIN MOTION AT CONSTANT Gy OR SLOW ACCELERATION

51

Vite £

1.0

If 0.5

IM F

Sle 0.0

—0.5

On frey f,., €

a .

1.4965

ve

1.497 Feel

time = 10.316 ms = 1.505000 MHz = 1.497125 MHz = 0.000353

1.4975

wee

Ss

4

1.498

[MHz]

Figure 4.3: The vertical polarization vs the rf solenoid frequency (free spin experiment). The solid line is obtained with parameters f,-,5 = 1.497125 MHz, fe, = 1.505 MHz, e = 0.000353

and the solenoid on-time of 10.316 ms.

II.4.

Spin Tune and Spin Closed Orbit

The OTM can generally be expressed in e~'™”**°'7, where v, is the spin tune and fi,, =

(cos $1, cos $2, cos $3) is the directional cosine of the spin closed orbit. Identifying the matrix elements of Eq. (4.32) with the OTM for a single spin resonance, the spin tune is given by

cos mv, = acos(c— Kn),

(4.35)

where

b= IAin md v—h—Gy,

a=(1—67)?

\ = 4/6? + |el?,

¢= arctan . tan Ar).

The corresponding spin closed orbit fico = (cos 1, cos $2, cos f3) is given by

cos $3 = —

asin(c— Kr) sin Tl;

,

cosg, =

_ bcos ® sin TY,

»

COs g2 =

bsin® .

sin TY,

?

(4.36)

where ® = (d+ Kn + K6;) is the spin precessing phase, d = arg (e*), and 6; is the initial reference orbiting angle. From one turn to the next, 6; will advance 27. The effect of the spin resonance on spin motion can be summarized as follows. The spin tune in the resonance rotating frame is A. This is the spin precession frequency around the spin closed orbit. At an imperfection resonance where K’ is

CHAPTER 4. EFFECTS OF SPIN RESONANCES

52

an integer, the spin tune is shifted away from the resonance by a magnitude equal to |e,.|. The fi.o vector is stationary at a location in the accelerator, where for every orbital revolution 6; advances 27. Thus the spin closed orbit exists for imperfection resonance. When the spin tune is exactly on resonance, where 6 = K — Gy = 0, the projection of % on the 2 axis vanishes. The vertical projection of the polarization vector changes sign as the spin tune passes through the resonance. At an intrinsic resonance where K is not an integer, the spin precessing phase ®

of Eq. (4.36) is not a constant (modulo 27) as 6; advances 27. Only the spin tune and the vertical component 7. - 63 of the ?,. do not change with 0;, i.e., the spin closed orbit vector A. precesses about the vertical axis with a precessing tune K.* Since K is not an integer, the spin closed orbit will not return to the same direction in successive revolutions. Thus the spin closed orbit does not exist in the laboratory frame.

Any polarization vector which deviates from fi,. will precess about the neo

with a precessing tune A. At the same time the 7,. vector also precesses about the vertical axis at a tune K. The condition that the n,, lies on the horizontal plane is given by

c— Kr =0 (modulo 27), which depends on the K value and the resonance strength.

The reason that the n,,

precesses around the vertical axis can be understood as follows. Since the betatron displacements of succeeding revolutions differ from each other, the resulting spin kicks are different in every revolution and thus the n,, vector is not stationary.

II.5

Spin Dispersion Function, Chromaticity, and Depolarization Time

How long will the polarization vector remain coherent for a fully polarized beam at time t = 0? If there is no spin tune spread, the fully polarized spin vector i lying along ‘> 0, the vertical closed orbit is not sensitive to k = Gy harmonic while the spin depends sensitively on k = Gy harmonic. In other words, an excellent global closed orbit correction does not guarantee an excellent spin resonance correction at |Gy —v,| >> 0. Since the polarization depends on error harmonics of the machine, one may be able to use polarization data to obtain a better measurement of quadrupole misalignment errors. Such procedures have been used to analyze the vertical machine misalignment from the AGS polarized proton experimental data. Unfortunately, the machine misalignment error may change with machine survey, ground motion, tidal action, etc. These error analyses have only temporary value in machine operation.

III.2.

Fast/Slow Acceleration Through Intrinsic Resonance

When the resonance strength is weak, a fast acceleration rate through a resonance can be used to overcome the spin depolarization resonance. Using the Froissart—Stora formula of Eq. (4.16), the required acceleration rate per revolution is given by AE > 100|e|? GeV/turn

for 90% polarization preservation,

AE < 1.724|e|?

for —90% spin flip.

Using AGS as an example, where we have « = 0.015 at K = vy, for a particle at a normalized emittance of 10 7 mm-mrad, the acceleration rate should be 22 MeV/turn or greater to achieve a 90% polarization. In reality, the acceleration rate of the beam in AGS is about 0.15 MeV/turn, which corresponds to a = 4.56 x 10~°. Therefore it is difficult to maintain beam polarization by fast acceleration for the intrinsic resonance at Gy = v,. Similarly, if one employs the method of fast passage through the resonance by ramping the betatron tune, one needs

Av, = 190|e|?/turn.

58

CHAPTER 4. EFFECTS OF SPIN RESONANCES

Alternately, if the acceleration rate were less than 0.39 MeV/turn, it seemed to be able to achieve —90% spin flip based on the Froissart-Stora formula. However, the feasibility of achieving spin flip by using a slow acceleration rate is burdened by the fact that the beam polarization is an ensemble average of spin vectors of particles in the bunch. A beam bunch is composed of particles with different betatron amplitudes and phases. Let J be the emittance® or twice of the action of a particle, and let the distribution function be p(/). The polarization of the beam after passing through a resonance is given by

Ps ‘a [aes i 1p(t)at, fpar =1. (B=

(4.39)

Using a Gaussian bunch distribution (see Appendix A.1) and using the fact that the intrinsic resonance strength is proportional to the square root of the particle emittance, nee

lO = le(0)F°=£

0

we obtain

le

Din

bee

\p

_ theo

in) RE Rg eDaneman

(40)

Here J is the rms emittance of the beam (see Exercise 4.11 for the Kapchinskij— Vladimirskij beam distribution function). To achieve a +90% polarization, the required acceleration rate should be

AE > 196|e(Jo)|? GeV/turn, AE < 0.544]e(Io)|? GeV/turn

for 90% polarization, for —90% spin flip.

For a beam with rms normalized emittance of [7 = 2.57 mm-mrad’ the acceleration

rate should be 11 MeV/turn or higher to achieve a 90% polarization. Similarly, the acceleration rate should be 30 keV/turn or less in order to achieve —90% spin flip for the bunch beam in the AGS. In this case, the bucket area becomes too small to contain the beam bunch. Experiments with a slow acceleration rate at KEK have achieved —85% spin flip [38, 39]. At a slow acceleration rate, synchrotron oscillations may cause multiple crossings of the spin resonance. Thus it is not an easy task to overcome the intrinsic resonance through fast or slow acceleration. Using a tune jump to-overcome the intrinsic resonance we find that

Av, > 375\e(Io)|? /turn ® Although it is inappropriate to use the emittance of a “particle,” we loosely define the term “the emittance of a particle” as two times the betatron action of the particle. A proper definition of the emittance of a “beam” is the phase space area occupied by beam particles. "Here, the 95% of the beam is given by 6Jp = 157 mm-mrad.

III. SPIN RESONANCE

CORRECTION

59

is needed to achieve 90% polarization. For the AGS, e(Jo) = 0.0075 for Jp = 2.57 mm mrad rms normalized emittance. We obtain Av, > 0.02 per turn.

Table 4.2 summarizes experimental results for the AGS polarized proton run [35] where the first column shows the resonance tune, the second column provides the resonance strength calculated for a particle with 107 mm-mrad normalized emittance, and the third column shows the expected polarization for the 10 mm-mrad particle after passing through the resonance. The fourth column displays the measured polarization after passing through the resonance. Using the ensemble average of a Gaussian distribution with Jo = 2.57 mm-mrad

as the initial beam rms emittance the polar-

ization of Eq. (4.40) is listed in the fifth column, which corresponds well with the measured polarization for the 0 + v, and 24 — v, resonances. Using the measured polarization shown in the fourth column, the equivalent normalized beam emittance is shown on the sixth column. The measured data can be understood as a larger emittance beam at higher energy. The increase of beam emittance is due essentially to the vertical betatron tune jump [36]. Table 4.2: Intrinsic resonance strengths of the AGS.

0.015 0.0006 0.0058 0.0124 0.0010 0.00155 0.0268

;

0.0075 < 0.0003 0.0036 0.0076 0.00092 0.0016

;

The polarized proton data at Saturne for v, and 8 — v, resonances [37] are particularly interesting. A full spin flip was achieved by using the following resonance crossing procedures:

te =-—4.1s"!, v, = 2.69s"!, Without

increasing resonance

Gy=13.2s"! Gy=13.9s"!

for the

0+, resonance,

for the 8 — v, resonance.

crossing rates by a tune ramp, these two resonances

fully depolarized the beam, contradictory to theoretical calculations which show that the spin should be fully flipped at the nominal acceleration rate. A higher ramping rate through the resonance should result in less spin flip; yet, experimental results indicate the contrary. A possible yet improbable explanation is that these tune quadrupoles excite coherent betatron oscillations. Thus the effective spin resonance strength is increased for the bunch. Another possible explanation is that the number of synchrotron oscillations is reduced so that the effect of multiple crossings is reduced.

CHAPTER 4. EFFECTS OF SPIN RESONANCES

60

Adiabatic Spin Flip with Coherent Betatron Oscillations One interesting idea is to increase the intrinsic resonance width for all particles in the bunch by exciting a coherent betatron oscillation of the bunch during a resonance crossing.

For example, we consider the 0 + v, resonance of the AGS. The resonance

strength needed to achieve —99% spin flip is « = 0.012, which corresponds to a normalized emittance kick of 7t mm mrad. Therefore the kick amplitude of the beam should be (7 + €9)t mm

mrad, where é is the normalized emittance of the beam.

In reality, the beam is composed of particles with different actions. The polarization is the ensemble average of the spin of all particles. When the kicked beam with Gaussian distribution is accelerated through an intrinsic spin resonance, the final polarization, given by the ensemble average of the Froissart-Stora formula over the

beam distribution, becomes (see Exercise 4.13)

py eae) oe

(8x04)? ee

14+ wo

}-1,

|€o|? wo=— a

(4.41)

where 3,4 and a, are the vertical betatron function and the rms beam size at the kicker

location, 0; is the kicker angle, and € is the resonance strength of the rms particle with action Io = €yms/2. The condition for achieving a polarization of —99% is

BexOx > 3.3(1 +

a

T |€o|?

ye

Oz.

(4.42)

Clearly, the method is not applicable to weak spin resonances where tleé9|?/a < 1. The problem associated with this scheme is that the kicked beam can decohere and results in beam emittance blowup. To solve the difficulty of the beam emittance dilution, one can use an rf dipole to create 1:1 parametric resonance islands and use octupole to adjust the amplitude of the 1:1 parametric resonance islands in the betatron phase space. Slow tune ramp quadrupoles can be used to transport the beam bunch adiabatically into the stable fixed point of the outer parametric resonance island. The resulting beam bunch in the potential well of the outer parametric resonance island can execute coherent betatron oscillations without emittance dilution. After passing through the spin resonance, the betatron tune can be adiabatically brought back to the normal operational condition. Experimental test of such bunched beam manipulation is needed for polarized proton acceleration.

III.3.

Nonadiabatic Betatron Tune Jump

The nonadiabatic betatron tune jump method has been successfully applied in the AGS, the ZGS, and the KEK PS and PS Booster in overcoming the intrinsic resonances. The tune jump is accomplished through changing the betatron tune within a

III. SPIN RESONANCE

CORRECTION

61

few orbital revolutions so that the spin motion can avoid the effect of spin resonance. For example, using 10 ferrite quadrupoles (0.5 m each) located at high G, locations in the AGS, the vertical betatron tune can be changed by an amount of Av, = 0.3 in about 2.5 ys (or about one revolution). The change in the betatron tune is given by A

1 eu ze € Sie v= z= =} 7 |B:AKds & 21 VeAk);,b

AK

—

o32 On Bot

-=&B! BB

2, have not been observed due to their small widths.

1.2

Partial Snake

For low or medium energy synchrotrons, where imperfection resonance strengths are of the order of 0.01 or less, partial snakes may be used to overcome imperfection resonances. The spin vector experiences a periodic spin kick each time passing through a partial snake. Thus a partial snake can generate an equivalent spin resonance strength given by €

_¢

(5.10)

oe Oe at all integer ’, where ¢ is the snake precession angle. If the partial snake strength dominates the spin dynamics, i.e.,

b > 2 Empl,

(5.11)

where €imp 1s the spin imperfection resonance strength due to the machine closed orbit errors, the spin closed orbit is mainly determined by the spin precession angle ¢ and the unperturbed spin tune Gy.

II. SPIN MOTION WITH MANY SNAKES

93

At the location 6 = 180° from the partial snake, the spin closed orbit fi¢, lies on the

plane spanned by the vertical axis é3 and the snake axis fi, (see Sec. IV.1 in Chap. 4). The angle between the spin closed orbit and the vertical axis depends on Gy, €, and ¢. Particularly, Eq. (4.54) gives an example of closed orbit for the type I solenoidal partial snake (see Fig. 4.10), where the spin closed orbit coincides with the snake axis at Gy = integer. When the beam is accelerated through Gy = integers, the resulting polarization can be described by the Froissart—Stora formula: Py = De7Mie ttn P/20 9

(5.12)

ye

where a = ion is the acceleration rate. In order to achieve better than 99% spin flip in each passage through integer resonances, the snake strength must satisfy

?

ae > |e,.| + 2Va.

(5.13)

The measured polarization at a 5% snake for the AGS is shown in Fig. 4.11, where the spin was observed to flip at every passage through integer Gy values. A weak partial snake cannot overcome intrinsic resonances. Thus a tune jump or other possible intrinsic resonance correction method is needed in the low energy polarized beam acceleration. A question then arises: what is the minimum strength of a partial snake in order to overcome both the intrinsic and the imperfection resonances? A stronger partial snake with which the spin tune deviates substantially from an integer, i.e., |v; — $| < 0.3 so that the intrinsic resonance can be avoided if the

betatron tunes are located in the range of (0.8, 1) or (0, 0.2). Figure 5.4 shows the results of numerical spin simulation for a 50° partial snake with an intrinsic resonance strength 0.02 and an imperfection resonance strength 0.002 respectively. In this case, the second order snake resonance® located at v, =

1 — a> — ().93 is also visible in

Fig. 5.4.

II

Spin Motion with Many Snakes

To achieve an effective tune jump, the resonance strength of the rms particle should be less than 0.02, i.e., e([o) < 0.02. Similarly, the harmonic correction is limited to low energy due to difficulties in orbit corrections for many high harmonics. For high energy colliders, such as the relativistic heavy ion collider (RHIC) at Brookhaven, the Tevatron at Fermilab, and the LHC in CERN, the resonance strength is of the order of 0.1 to 5 (see Fig. 3.1), and the conventional spin resonance correction method will not work. With a single snake, the problem is alleviated slightly. However, since the spin 3The second order snake resonance is determined by the condition v, + 2v, = integer.

CHAPTER 5. SPIN DYNAMICS WITH SNAKES

94 1.0

Tat

7

SHAne

[Io

Sel

Noa

pol

TIT IREa

Emt= 0-02 Gunn 002

™

0.6

} 0.4

Lf

tT

|

1

|

i

|

|

i

!

a

0.2

| !

"| l

0.0

4

4

at

0.5

alae

4

1

0.6

4

|

ve

4

cs

0.7

Fall

4

4.

4

0.8

4

dh

—A____A

4

0.9

1

Vy,

Figure 5.4: The vertical polarization after passing through resonances with strengths Gnt = 0.02 and éimp = 0.002 as a function of the vertical tune with a 50° snake.

tune shift increases linearly with the resonance strength [Eq. (5.6)], the capability of a single snake is limited to a resonance strength about ¢ ~ 0.2. At a larger resonance strength, one needs two or more spin rotators.

In accelerators with two or more snakes, the spin closed orbit is lying along the vertical axis and is energy independent. Because of the smaller orbital angle difference 6; —0; between snakes, the effective spin perturbing strength parameter 6 in Eq. (4.33) is also smaller. This section discusses essential characteristics of spin motion in accelerators with many snakes.

II.1_

A Model with Two Snakes and a Local Spin Kick

We consider a perfect circular accelerator with two snakes —10, and —io2 separated by an orbital angle 7. The OTM is given by

[éaehent az?

[—to,] e

AM

ae

at

"2 7% =103.

The trace of the spin transfer matrix is zero, thus the spin tune is

spin closed orbit is vertical.

(5.14)

and the stable

Now we introduce a small constant local spin angular

precessing kick y about an axis n, in the horizontal plane, the spin transfer matrix becomes T; =

IXAL-aG-

ena2

hos:

(5.15)

II. SPIN MOTION WITH MANY SNAKES

95

Because ni, is in the horizontal plane, the evolution of the spin transfer matrix at the nth revolution becomes

Gi 2 Bor

aa)

al

leesla

if nm = even,

Tae PRRs

if n = odd.

(5.16)

This means that the spin-perturbing kicks cancel each other every two revolutions. Thus the snake is very effective in correcting the imperfection resonances. Extending the model a step further by assuming that the kick strength is different in each revolution, the spin transfer matrix becomes =

II EE) =

PL 59

el le ho ™xm Aix: *lios]” :

(5.17)

m=1

The vertical spin vector at the nth turn becomes

n sf )=1—2sin?

l< 2 Ey>(-1)" "xm :

(5.18)

pra |

Now if these spin perturbation kicks were due to a betatron motion, they would be correlated with Xm = Xo cos 2mnv,. (5.19) where

v, is the vertical betatron

tune.

When

the vertical betatron

tune is vy, =

; + integer, each kick adds BP coherently. The spin vector will precess around the ni, ae ata precessing tune of 3°,or it takes a revolutions to complete one turn about the nz axis. The snake is inettectivelsin dealing with this type of spin perturbation called the first order snake resonance, where the betatron tune is equal to the spin tune. On the other hand, this spin perturbing kicker can be used to precess the spin direction and invert the polarization direction.

II.2

Basic Requirements of Snake Configurations

Let (¢1,¢2,...,¢n,) be the snake axes of N, snakes in an accelerator and let 4;:41 be the azimuthal orbit rotation angle between the ith, and (2 + 1)th snakes shown schematically in Fig. 5.5. The one turn spin transfer matrix for a perfect planar synchrotron is given by

_No=1 e7# Flo, —9)o3, —iFin, o II [ena ee, k4193 p13ZA, Fei gta

=

—

(5.20)

k=1 where the spin tune v, and the spin closed orbit vector n, me mel 3

—

“6

3

BLES sie

The spin

(5.56)

VI. PERFORMANCE

OF POLARIZED

COLLIDERS

115

The resulting spin tune spread is about 0.005 for a beam with 107 mm-mrad normalized emittance at 250 GeV. Combining all the possible sources, we expect the total spin tune spread to be about 0.006 by taking quadrature of all sources. Betatron Tune Spreads and Modulations For an accelerator with two snakes, we find, from many numerical simulations, that the 9th order snake resonances are not important if the intrinsic resonance strength is kept below Gn < iN, = 0.4. The available tune space is about 0.03.

The control

of power supply ripples needed to avoid nonlinear betatron resonances for the orbit stability is enough to limit the snake resonance modulations. At the injection energy, the space charge tune spread can be as large as 0.02 for RHIC. However the corresponding spin resonance strength at low energy is also about a factor of 3 smaller. Thus the spin is less susceptible to field errors at injection. Beam-—Beam Interaction, Nonlinear Depolarizing Resonances The linear component of the beam—beam interaction gives rise to tune spread in the beam. The spin perturbation due to the beam—beam interaction is more important in the ete~ colliders, where the linear beam—beam tune shift is about 0.05. For a hadron collider, a typical tolerable linear beam—beam tune shift is 0.02, which is small relative to the available tune space of 0.03. Since the betatron tunes of high energy colliders should be chosen to be free from high order betatron resonances, the betatron tunes are also free from snake resonances. Higher order snake resonances should not be important provided that the spin tune spread is small. The most important spin resonances, arising from a two-family sextupole chromatic correction scheme, are located at Kan

Bik vetvara

M bw pF v,,,.

(5.57)

The corresponding maximum resonance strength is given by

cx

1+Gy

NF[6G BeBe PM(Sel + |Sol)

(5.58)

where S,, and S,, are respectively the integrated sextupoles strengths B’0/Bp at the focusing and defocusing quadrupole locations. Because the emittance decreases with energy, the sextupole spin resonance strength is energy independent in hadron storage rings. For RHIC, the resonance strength is about 1.5 x 1074 at a normalized emittance of 10

mm mrad.

At the lowest order snake resonance v,

tv, + vz = integer beam

depolarization will occur, which has been observed in the IUCF Cooler Ring at a 100% snake [25]. Higher order snake resonances are not important.

CHAPTER 5. SPIN DYNAMICS WITH SNAKES

116 Uncorrected

Solenoid

Field at IP

High energy particle detectors usually use solenoid magnets for particle tracking. The solenoid can contribute to the imperfection resonance strength €imp,sol =

(1+ G) f Bde an

Bp

5.59

(

)

For a 5 Tesla-meter integrated solenoid field strength, the resonance strength is about 0.02 at the injection energy and 0.003 at 250 GeV for RHIC. To achieve a helicity state collision, two spin rotators are needed. The spin transfer matrix at the IP can be expressed as eit 71 6829892 e-4G 1 — p15 9803

(5.60)

Thus the combination of 90° spin rotation along the é, direction and the solenoidal field in the €2 direction gives rise to a spin rotation about the vertical axis, i.e., the so called “type-3 snake.” The amount of spin precession is proportional to the solenoidal field precessing angle 9,, which is about 0.017 radian at 250 GeV. The corresponding spin tune shift is about 0.003, which is tolerable at 250 GeV in RHIC. At a lower energy, the solenoid field should be decreased accordingly. Since this is a systematic effect, cancelation of the type-3 snake effect can be arranged, e.g., the use of a local solenoid compensation scheme. One can also correct this tune shift with a small adjustment in snake axes.

Linear Coupling Linear coupling, which arises from skew quadrupoles and solenoids, is important in high energy colliders. Linear coupling limits the tune space available and can cause coupling snake resonances at

Q,+Q.

= integer,

¢ = integer.

Linear coupling correction in RHIC can minimize the coupling snake resonances. can also choose the betatron tune to avoid these snake resonances. Polarization

(5.61) One

Lifetime

The constraints listed in this section are important in preserving the beam depolarization. With careful manipulation of the operation conditions, the polarization lifetime should be at least as long as the beam lifetime. Adiabatic modulation within the tolerable limit does not affect the beam polarization. The spin vector will adiabatically follow the spin closed orbit. Nonadiabatic process, arising from rf noise at the spin precession frequency, can indeed cause beam depolarization. We consider a single dipole with strength 6, at

VI. PERFORMANCE

OF POLARIZED

COLLIDERS

117

the modulation frequency v, fo, which is about 39 kHz for RHIC. The corresponding induced spin precessing kick is Gy0,. The number of turns in which the spin is perturbed to 80% of the original polarization is given by,

__ arccos [0.8]

>=

Ga,

(5.62)

Now we consider a similar angular kick to the orbital motion. If there were a rf source at vs;fo, one would expect to have a similar angular kick at the frequency qfo, where q is the fractional part of the betatron tune. The orbital survival turn number is given by -

ING

(3),

(5.63)

where (3) is the average betatron amplitude, and A is the dynamical aperture. Using A = 0.01 m, (8) = 20 m for RHIC, we find that the orbital lifetime is not longer than the polarization lifetime. Indeed, any rf source at high frequencies around the synchrotron and betatron tunes, are dangerous to the stability of orbital motion. Similarly, any rf source at the spin tune can cause beam depolarization. These high frequency rf sources should be addressed carefully in hardware design. Synchrotron radiation is known also to limit the polarization lifetime of electron beam. However, synchrotron radiation is not important in RHIC energy.

Exercise 5 1. Verify the relation |7,;|? + |712|? = 1 for the r matrix given by Eqs. (5.6) and (5.24). 2. Let the spin precession angle for a single snake be y, where a partial snake corresponds to y < m and a 100% snake corresponds to vy = 7. e A partial snake is equivalent to imperfection spin resonance at all integers. One can use the Froissart-Stora formula to express the polarization in a polarized beam acceleration through integer Gy values, i.e., Py = Qe~ TF t Kimp|?/20 =) P.

’

where €imp is the spin resonance strength resulting from machine imperfections,

a = d(Gy)/d@ is the acceleration rate. Show that the condition for 99.9% spin flip is given by xX —> —

|leimp| +2.1/7a@. va

e Show that the second order snake resonance occurs at the betatron tune

y; ee =e

Xx

118

CHAPTER 5. SPIN DYNAMICS WITH SNAKES e Using linear response spin tracking equation of Eq. (5.8), estimate the proximity parameter 6 = Kk — j that the second order snake resonance can be measured

experimentally for a given resonance strength |e]. What is the minimum |e| such that the second order snake resonance can be measured?

3. Verify the snake requirements of Eqs. (5.21) and (5.22). 4. Verify Eq. (5.35) by using the linear response method in the spin tracking equation of Eq. (5.28). Using the property of the enhancement function ¢,(z) of Appendix C, show that the envelope of the vertical polarization around an intrinsic spin resonance

is given by Eq. (5.37). 5. Show that the equivalent imperfection resonance strength due to the error in the snake precession angle is given by [Eq. (5.47)] egmilt

Ag

€imp “|

’

where Ad is the deviation of the snake precession angle from 180°. 6. In astorage ring with many snakes, many snake configurations are possible. The snake superperiod P, is defined as the number of repetitive periods in one revolution for the spin motion. The spin superperiod requires both repetitive snake arrangement and the orbital motion which affect spin precession. For example, six snakes can be arranged as follows:

os

4

obs

4/2. -0 n/G. 0) -ce/6°>>6 m (Ge) af3, ce[2 20/2.

6 m/2 m/6 5SaJG

Show that the condition for the first order snake resonance becomes Kereta

= integer.

Ss

Use this result to predict snake resonance condition for the snake configuration listed in the above table (See Ref. [45]). Study the effect of snake superperiod on higher order snake resonances.

7. Verify Eqs. (5.57) and (5.58) for the important resonance tune and the corresponding resonance strength that arise from the sextupole magnets located in the arc FODO cells.®

SChromatic correction sextupoles are usually distributed in two or four families located near the focusing and the defocusing quadrupoles. This exercise assumes a two-family sextupole correction scheme.

Chapter 6 Electron Polarization Electrons differ from protons in two aspects. First of all, the electron g-factor is nearly equal to 2 and the anomalous g-factor is small with

= yang ag = 0.00115965. Secondly, electrons emit synchrotron radiation in dipoles and replenish their energy in rf cavities. The classical radiation power spectrum is continuous, extending up to a maximum at the critical frequency w, given by!

3y%c

wee

Do

where c is the speed of light, y is the relativistic factor of electrons, and p is the bending radius of dipoles. The critical photon energy is given by

ago aCe Oh,

Ue = hw, ray

where h is the Planck’s constant. For example, the critical photon energy in the LEP electron storage ring with p = 3096.175 m is u. = 0.09 MeV at EF = 50 GeV and u, = 0.7 MeV at F = 100 GeV. The radiation spectrum falls off exponentially beyond w, as e“/“e with the total integrated radiation power given by E4 Poleastcal —

where

uni

4 Tr Re

eucttna

eG

yo

27p

m

SL cl ecaakaieeNne

:

1In accelerator physics, the cyclotron frequency is usually denoted by wo = Bc/R, where R is the mean radius of a synchrotron, fc is the speed of the orbiting particle. Do not confuse the critical frequency w, with the cyclotron frequency.

119

120

CHAPTER 6. ELECTRON POLARIZATION

The corresponding average radiation power in an isomagnetic circular accelerator is given by

1 ign =

UE tegatcalt — ea) fTE easical ds =

F4 CCE ans on Rp = 4. 2 «10 3—_(GeV oie fs),

where E is in GeV and the average radius R and the bending radius p are in meters. The synchrotron radiation is basically a quantum mechanical process, where the electromagnetic radiation is emitted in quanta of energy u = hw. The total number of photons NV emitted per second and the moments of energy distribution are given

respectively by [47]

ee 15V3.P ep

8

wike

pe

7

a

eet ae

Gavigg, to Eh agetgtremil

—) =

kmagtons a hagahiq

BOW 3-0 Ba

mpg

ae

(6.1)

When photons are emitted, the energy of the electron will decrease by the same discrete amount. Thus the corresponding instantaneous radiation power will be lessened. To the first order in h, the quantum correction to the radiation power is given by [48] Dome q an

] i(=8/3 = LF classical

E

y

(6.2)

where F is the electron energy. The quantum mechanical correction factor is of the order of 10-° and cannot easily be measured. However, the quantum effect is evidently observable in the equilibrium phase space distribution due to radiation damping (a classical phenomena) and radiation excitation

(a quantum mechanical effect). The equilibrium momentum width for an isomagnetic ring is given by (22)? —

ER

55

hw, _ C

“SUE

=

?

dep

where C, = 3.84 x 107'* m and J, is the damping partition number. For example, we obtain %E ~ 8 x 107‘ at 50 GeV in LEP. Similarly, the transverse beam emittance for an isomagnetic storage ring is given by (see Appendix A.III) gs

Ce

2

4

(H) dipole

J xp

where H is the dispersion action and J, is the horizontal damping partition number. In this chapter, electron polarization due to the emission of photons will be discussed. Section I discusses the the Sokolov-Ternov radiative polarization. Section II examines the Thomas-BMT equation in the presence of Sokolov-Ternov effect. Section III studies the effect of spin diffusion. Section IV reviews methods of numerical simulations. Section V reviews polarized beam experiments in some electron storage rings. Section VI examines the essential requirements of polarization wigglers. Section VII studies the problems associated with the ete~ polarized colliders.

I. EFFECT OF ELECTRON SPIN ON SYNCHROTRON

I

RADIATION

121

Effect of Electron Spin on Synchrotron Radiation

In addition to the discreteness of photon emission, an electron has an intrinsic spin quantum number. The angular momentum carried by the spin is S= she, where ¢ are Pauli matrices. Including the spin correction, the radiation power is given by [21] 55 Ft

Pascsical

pe

neal era

BOO

eg

(6.3)

ae

where 7 be the spin orientation in the electron’s rest frame before photon emission, and Z is the direction of the magnetic field that bends electrons. Averaging overall spin orientations n, Eq. (6.3) reduces to Eq. (6.2). The spin correction is also very small and the spin dependent component of synchrotron radiation power is given by

N (u) = —Pay

Ore

(6.4)

Ucn

The important effect is the disparity in the transition rate. spin flip transition rate is given by [21]

The instantaneous

1 Wa Wo-| Le 9 (* 8)? + —(n- 4)],

(6.5)

where 8 is the direction of particle motion, Z is the quantization axis, and

Rie

5/3 reyrh

a 5/3 ue

g saiaee ER

has

(6.6)

is the average spin flip transition rate. Expressing the spin dependent fractional radiation power in terms of Wo, we obtain U

The instantaneous

8

spin flip transition

power

~

|

is the product of the transition rate

and the energy carried by each photon, i.e., Ptransition =

Wohw.

Thus the spin flip

transition power spectrum is much smaller than the classical power by a factor of (hw,/E)* ~ 107". Even if the power of radiation is small, the disparity in the transition rate may eventually populate more electrons (positrons) with spin antiparallel (parallel) to the magnetic field because electrons (positrons) with spin parallel (anti-parallel) to the magnetic field have a larger spin flip transition rate. Let 2 be the vertical direction along the guide field. Let the quantum states of electrons be either parallel (¢) or anti-parallel (|) to the guide field z. The spin flip transition rates become 1 Whe

8

5 Woll +. —),

1 WwW, lez je?

The equilibrium polarization becomes

ks (S3) = paleSes

(6.23)

The left plot of Fig. 6.1 taken from Ref. [23] shows the measured polarization time vs the beam energy. The solid line shows the polarization time T,,, calculated from Eq. (6.10). The resulting polarization level should be of the order of 85-88%. The right plot of Fig. 6.1 taken from Ref. [50] shows the measured polarization (P) vs the theoretical P,.7/Te_-

III. QUANTUM SPIN DIFFUSION AND ENHANCEMENT

III

127

Quantum Spin Diffusion and Enhancement

Since the emission of photons in synchrotron radiation is a nonadiabatic statistical process, the spin vector can be excited in each photon emission if the corresponding spin closed orbit? is altered as well. Furthermore, if there is a preferred direction for the photon emission, the effect can be reflected in the actual degree of spin alignment [525 53, 54,)55, 56; 57, 585,59]. Since an electron in a storage ring emits hundreds to thousands of photons per revolution (see Table 6.1), the average energy of each photon emitted is small and the effect. of photon emission can be treated perturbatively. Let 2(E) be the spin closed orbit of a particle. As the particle emits a photon with energy 5£F, the spin closed orbit is changed to n(E — dE). We now define the spin dispersion function or the Derbenev—Kondratenko vector as

(6.24) where fies = is usually used to denote the derivation of the spin closed orbit with respect to the Bee chit of the particle. Sincen is the spin closed orbit, we Both the spin closed orbit and the spin dispersion function depend and synchrotron motion. We consider a particle with spin lying along its spin closed orbit 1.€., S =n. Whena photon is emitted at t = 0+ with energy wu, its orbit becomes n;. Thus the spin vector Sis given by

have D-rA=0. on the betatron

n at time t = 0, new spin closed

S=fyt es

(6.25)

The spin vector 5 precesses about the new spin closed orbit. Since the spin precession frequency is much faster than the synchrotron phase space damping rate, the final polarization is given by i!

u2

Cordes iheAone w1~ 510.po, 1+ |B.)

E

The loss of polarization is equal to ie

s3 u

Multiplying the rate of photon emission per second W and averaging over the circumference, the loss of polarization is given is

© =e

encks

u?

-*

FBP ¥ yds}5 = 4B, N))S.

(6.26)

?The spin closed orbit defined along the closed orbit is also a function of the betatron amplitude. Due to betatron motion, the spin closed orbit changes direction each revolution. A complete betatron motion along the betatron ellipse gives rise to a closed curve for the spin closed orbit.

128

CHAPTER 6. ELECTRON POLARIZATION

This process contributes to the spin diffusion. If the spin closed orbit were independent of the momentum of the particle, i.e., D, = 0, there would be no spin diffusion in the photon emission. Now we consider another mechanism in which the emission of a photon depends on

the spin orientation [see Eq. (6.7)]. Let the initial spin of the electron be S= 1+ Ww, where || < |7| so that the initial polarization is given by P; © 1— 4|w|?. When a photon of energy u is emitted at t = 0*, the new spin closed orbit becomes 71, i.e., >

be

Ss

oy

pe

~

U

fatioh Die

The final polarization can be expressed as Py % 1 — $|t + D,#|?. The change of polarization is given by

(6.27)

AP = P; — P, = -w- D,

where the second term contributes to the spin diffusion discussed earlier, and the first term depends on the initial spin orientation. The spin perturbation rate can be obtained by multiplying the rate of photon emission and averaging over all phase space coordinates, i.e., dS3

=

—

U

>

—

U

Zips =-w-D(N—=)=S-D,(N—). Wp ale gala Tgp ais ThwRt

(6.28)

Here N is the number of photons emitted a second and we have used the fact that n is perpendicular to D, to arrive at @- D, = §- D,. Since we have to average over all orientations of the spin vector Ss the spin-independent component of (N=) will be averaged to zero. The component of (N’¢) that depends on the spin orientation is

given by [see Eq. (6.7)] u iNice) =>

=

Re

Bes,

z E2

(6.29)

Thus the orientation-dependent polarization driving term becomes

dS;

hw,

ie

=

F2

a Pit

2 aS

: D.),

(6.30)

where § should be replaced by the &"spin matrix, and the quantum mechanical equi-

valent relation is {(5-2),(S-D,)}= -Ds. vere over all phase space coordinates along the storage ring, we obtain

dS3

a

hw,

(Fay Pav? +Ds).

(6.31)

Combining all driving terms in the spin equation of motion, we get

8 Que gg! sill Comers e(1+ay)B, Fem 2 XS+ Wall — 515-8)? + BP) —(Wasa — 2 - By).

dS

(6.32)

III. QUANTUM SPIN DIFFUSION AND ENHANCEMENT

129

The equilibrium polarization is given by the Derbenev-Kondratenko (DK) formula:

(6.33) where § is replaced by 6 and Z is replaced by 6 = eT for a more general condition. UXVU The corresponding damping time is given by

7 = (Wo(1 — =(A- 6)? + =|D,)))

(6.34)

Thus the diffusion processes also shorten the polarization time and reduce the attainable polarization similarly to that of Eq. (6.23). Some observations for the DK formula are listed as follows:

e In a storage ring with a uniform field, i.e., 2 = 6 with D, = 0 and a-d = 0, (S) = Pop.

e Spin depolarizing resonances due to inhomogeneous magnetic fields tilt the spin closed orbit m away from the quantization axis b. In the absence of spin dispersion, i.e., D, = 0, the polarization is given by

(5) = (i byPt=

id

(6.35)

ceria

which is equal to Eq. (6.20). Thus the 7 vector carries the information about the spin resonances.

The term (4s |D.I?) in the denominator reflects the quantum fluctuation of synchrotron radiation. It reduces the degree of polarization.

e The term (x6 . D,) in the numerator of Eq. (6.33) arises from the fact that the power of synchrotron radiation depends on the initial spin orientation. This mechanism can either decrease or increase the degree of polarization. Derbenev and Kondratenko [52] pointed out that if m and D, were given by

A

4

iA= —-\[—b+,/—6, n+ Vin”

EW

7

‘

7

D, = ——(\/—b fia Vidas+1/—5), Valhee

6.3 ee)

the polarization would have been

me 95%.

(6.37)

130

CHAPTER 6. ELECTRON POLARIZATION

This theory was extended further by Bell [55], Hand, and Skuja [56], and Barber and Mane [57] to include the effect of synchrotron radiation on the transverse betatron oscillations, where the spin closed orbit depends also on the transverse phase space coordinates. The resulting equilibrium polarization is bE (S) SY Po

(6-2 —6- D, Pe

aa) DA

7

Mex

(pp(l — 9(%- 8)?+3331D.)

;

(6.38)

— aya (* X aa.) + sorlap,l°))

where ymc — OB, js the derivative of the spin closed orbit on the vertical momentum due to synchrotron radiation. Here the spin closed orbit dependence on the vertical betatron momentum is explicitly expressed. At high energy, sl ae < |Ds |, the equilibrium polarization reduces to the DK formula. Since (Dy precesses about the vertical axis at the spin tune, the time average of a (po . Ds) in i the DK formula is small. In this case, we obtain

(8) = Py,

(ph Bois . (= 2(n- 6)? + BIB,))

(6.39)

where the numerator solely consists of the information of spin resonances while the denominator includes information about the spin diffusion, etc. Since the spin closed orbit revolves about the vertical axis at a spin tune per revolution, the vertical polarization is the projection of the spin closed orbit onto the vertical axis. Letting P be the maximum attainable polarization, we arrive at the final vertical polarization

(6.40)

(Sa) = (a6)? P, which can be expressed as Eq. (6.21) for many nearly-overlapping resonances. For phenomenological data analysis, the polarization is usually expressed as

(5)

Teh

Pn

1

ee

1

TOF

ee

1

Ta

(6.41)

where 7,,. 1s the polarization time without radiation diffusion and spin resonance

depolarization, and 74 is the depolarization time. The relation 7,,,P,,= 7(S) has been verified experimentally (See Fig. 6.1). The dependence of the depolarization rate 1/Tg on spin quantum diffusion, beam—beam resonances, etc., is usually expressed as

1 —= Td

1

1 + —+

T diff

Tb—b

1

interactions, spin depolarization

free,

(6.42)

Tres

This implies that all depolarization mechanisms are independent random processes.

On the other hand, a combination of Eq. (6.23) and Eq. (6.39) seems to indicate that the polarization should be expressed as P

O) Pome Tec aal

ES

IV. NUMERICAL

IV

SIMULATION OF ELECTRON POLARIZATION

Numerical

Simulation of Electron

131

Polarization

Numerical simulations have been exceedingly useful in providing guidance in the design and attainment of electron radiative polarization. There are a number of use-

ful spin program SODOM tracking The

tracking programs such as the SLIM program by A. Chao [53], the SMILE by S. Mane [54], the SITROS program by J. Kewisch et al. [60], and the program by K. Yokoya [61]. In this section we will only discuss the linear program by A. Chao. first spin tracking program (SLIM) was written by A. Chao [53], who ex-

pressed the polarization vector of a nearly polarized electron beam with

S=Atarn +t pb, where fi is the unit vector along the spin closed orbit, rm and @ are two orthonormal unit vectors perpendicular to the spin close orbit, and a,@ < 1. The corresponding polarization is given by a

1

1,plated (98

Ny

ts

1

i

=.

2 labor x

The evolution of the a and ( can be obtained from the Thomas-BMT equation which depends on the particle phase space coordinates. Chao also introduced the 8-dimensional state vector:

D =

(6.44)

in order to describe the extended phase space. The propagation of the state vector of the electron can be performed by the transport matrices:

PiltuPi=( yee pt) Po 2x6

(6.45)

2x2

where the Mgx6 is the transfer matrix for the particle phase space coordinates, 0gx2 is a matrix with 0’s (see Exercise 6.5), [2x2 is the 2 by 2 unit matrix, and N2x¢ is the 2

by 6 matrix describing the evolution of the spin vector in the Thomas—BMT equation. Spin transfer matrices for accelerator elements are listed in Exercise 6.6 [53]. The transformation of the base vector requires an extra phase factor which describes the spin-phase precession per revolution. Thus the spin transfer matrix at the end of one revolution is transformed with a phase rotation: a ¢ tera

=

cos2mv, (sin2mv,

—sin Say a Cie cos2mv,

A (6.46)

132

CHAPTER 6. ELECTRON

POLARIZATION

where v, is the spin tune obtained from the spin-phase precession through a complete revolution in the storage ring. Tracking through accelerator elements in one complete

revolution gives us the one turn map (OTM) T, from which the spin closed orbit n and the spin dispersion function D, for the DK formula can be obtained. The closed orbit can be obtained by solving for the eigenvector of the OTM (see Exercises 2.10 and 2.11 for an alternative method of finding the spin closed orbit). Let E, and 2»; be the eigenvectors and the eigenvalues of the OTM at a location s = so in the storage ring, i.e.,

T(s0)Ex =AxEx,

k= +1,+2,+3, +4.

(6.47)

The eigenvalues can be expressed as \4, = e*'”*, where 14,23 are respectively the

betatron and synchrotron tunes and 14 is the spin tune. matrix is real, we have NES ON PSE

Because the spin transfer

Starting from the closed orbit at s = so the closed orbit along the ring may be obtained by tracking through accelerator elements. Because the effect of spin on the orbital motion is neglected in this model, the eigenvector E:4 has no orbital components. By definition, the electron that follows the spin closed orbit has a = 6 = 0. To find the DK vector, we consider an electron that follows the spin closed orbit along n. When a photon with energy u is emitted at s = so, the electron is left at a state

P(so) =

0 0 0 0 0

(6.48)

—u/E 0 0 After decomposition of this state in terms of eigenstates, i.e.,

P(so)=

>)

k=+1,£2,43

AxEx(so) +

(6.49)

Ias) (°K) So

one can find A; and a and G from eight linear equations. This task can be easily accomplished by employing the symplectic property of the orbital motion, 1.€.,

Mox6J Moxe = J

(6.50)

IV. NUMERICAL SIMULATION OF ELECTRON POLARIZATION with

0 Tak () a0 Natasa toh lilitasl oo,lage Plank Outen OCA DAHON

ou

0.0 dn leni Gaad.0 = den Oman) Ueshithie © “Oneapallincc= | (ON stf8c0

133

(6.51)

Let e, and A; be eigenvectors and eigenvalues of the transport matrix Mgxe, i-€.,

Me Sel

= tAer.)

hi Sat

lect

2h

(6.52)

The symplectic condition then gives rise to the normalization ee; ='0,

Ant vigsesk,

(6.53)

Since the scalar product is purely imaginary, the normalization is given by

ie =

a

ee

(6.54)

Using the orthonormal relation, we obtain

Ags

uIP (64) ~iBis( 80)

0

(6.55)

where Exe is the 2th component of the eigenvector E;. The remaining two equation can be solved to obtain

a

slaps Slu

(3)

Eo —

(Im

(Ej;ksEx7)

)

Im ( Exs Exs)

6.56

(

)

The DK vector is then given by

p--s| J Im(Ej,Eur)n + SD Im(EjExs)A} R=36253

(6.57)

k=1,2,3

This method has been used extensively in the simulation of electron polarization in storage rings. The tracking scheme has been extended to higher orders by including higher order spin kicks perturbatively in the program SMILE by S. Mane [54]. He has been able to reproduce a rich spectrum of higher order spin resonances shown clearly in the SPEAR data of Fig. 3.3 [23], where synchrotron sidebands also play a very important role in the spin depolarization. Further refinements in the photon emission have been accomplished by J. Kewisch et al. [60] in the tracking program SITROS. Extension to include all synchrotron sidebands has been achieved in the program SODOM by K.

Yokoya [61]. Both the SITROS and SODOM programs have been applied successfully in the tracking simulations of HERA and LEP electron polarization. Lie algebra has also recently been used in the spin tracking [62].

134

CHAPTER 6. ELECTRON POLARIZATION

V_

Experimental Measurements

Polarized electron beams serve two important functions in particle physics. Polarized beam collisions at high energy may provide important information on physics relevant to the standard model, and the single beam polarization can be used to calibrate the beam energy. Employing the resonance depolarization technique, the uncertainty of

beam energies in LEP [43], VEPPs [44], TRISTAN [49], and PETRA

[51] has been

determined to within 1 MeV. Note that the rms natural beam energy spread is of the order of a few MeV to about 40 MeV in these machines.* Because of the sensitivity of the spin depolarization mechanism, energy variations of electron beams in LEP due to the change of water level in Lake Geneva have been observed. Furthermore, the measured beam energy variation with respect to the time of day has been used to verify geological simulations of gravitational pull during a lunar cycle. Electron polarization in storage rings is measured with polarimeters which can be constructed through techniques based on Mott scattering, Moller scattering, Compton scattering, etc. [63]. Mott polarimeters utilize the spin-dependent Mott scattering between polarized electrons and the unpolarized high Z nuclei. Moller polarimeters employ the spin dependent Moller scattering between polarized electrons and polar-

ized electrons in a magnetized permendur foil (49% iron, 49% cobalt, 2% vanadium). Compton polarimeters use spin dependent Compton scattering between circularly polarized photons and the high energy polarized e* beams. Spin resonances of electron storage rings are classified in Table 6.2. The relative importance of these resonances has been discussed in Chap. 3. Table 6.2: Classification of spin resonances in electron storage rings. ay RP =p, t Vien RP 2, Svar

RR + [ve] & 1X cs keane, kP

+

Maly

a5 MzVz

=e MV

syn

Classification intrinsic resonances and sidebands intrinsic coupling resonances and sidebands dominant imperfection resonances and sidebands imperfection resonances and sidebands nonlinear spin resonances

Here [vz,z] are integers near the betatron tunes, k,m,m,, and m, are integers, and Vsyn is the synchrotron tune. Since the stored electrons attain polarization through spin dependent synchrotron radiation, one can avoid strong intrinsic and imperfection resonances by changing the betatron tunes, the lattice superperiodicity, and the ay value of electrons. The general strategy for attaining a good beam polarization is

given as follows: Since regular imperfection resonances are separated by about 440 MeV and the natural energy spread is of the order of 10 MeV or more, the electron beam polarization °The rms energy spread can be obtained from Table 6.1 with o, = E (o,/E), where E is the beam energy.

V. EXPERIMENTAL MEASUREMENTS

135

can encounter a constant depolarization driving force arising from the betatron and synchrotron motion and the closed orbit errors. Since the closed orbit correction mainly minimizes the error harmonics closest to the betatron tunes while the spin is sensitive only to error harmonics near the spin tune (see Table 6.1 for the spin tune), a global closed orbit correction may not be effective in overcoming spin resonances. This fact has been discussed in Sec. III.1 of Chap. 4 which deals with polarized protons. Similar results apply to electron storage rings. The synchrotron tune for electron storage rings is usually of the order of 0.1 and so all synchrotron sidebands can be considered as isolated resonances. The main conclusion to be drawn from Sec. III.1 in Chap. 3 is that the only way to correct synchrotron sidebands is to minimize the resonance strength of the primary resonance. This can be achieved by either harmonic spin matching, changing the betatron tune, and/or betatron spin matching. These methods are similar to those of proton synchrotrons. In this section, results of existing polarized beam experiments of some storage rings will be reviewed. T

|i

ao; Lets

T

|

T

T

aE

SPEAR

md

siei! DO

a

T

| a

|

T

T

J© is @

i!

|

Deep

dy b

ly | fe

BeO

>

d

T

fea

oh

1.00

T

(Dp

dD

B/ Rares ® S 0.00

S 1

1

aa)

1

S |

3.55

--

e 1

‘.

i

|

3.6

@ 4

a

|

we

Pesal,

4

ore, ly re ed

3.65

E (GeV) Figure 6.2: The polarization of e+ single beam in SPEAR is shown as a function of the beam energy [23]. The solid line is a fit to the eye by using single resonance model with the following parameters: v,; = 5.279662, v, = 5.182604, v, = 0.04276, o, = 8.7 x 10~*, and a = 6o0,. The resonance strengths listed in Table 6.2 should be considered as the effective resonance strengths.

V.1

SPEAR

Data

We begin with the detailed data by Johnson et al. [23]. The first systematic study of spin resonances in a storage ring was carried out in the polarization beam experiments in SPEAR. Figure 3.3, taken from Ref. [23], shows the measured beam polarization vs

136

CHAPTER 6. ELECTRON POLARIZATION

the beam energy. there were

Many depolarization spin resonances were observed.

K =8, 3+%,

Particularly,

3+%

primary resonances with synchrotron sidebands and nonlinear resonances at 8 + vz — Vz, Vz tv, —2. These polarization data have been extensively studied by Chao [53],

using the program SLIM, by Mane, using the program SMILE [58], and by Buon [59], using the statistical analytic approach. Instead of repeating the spin tracking, we will use Eq. (6.21) for the nearly overlapping resonance approximation to fit the data and extract the spin resonance information. As we have discussed in Chap. 3, synchrotron sideband resonances can be particularly enhanced through a large spin chromaticity and a small synchrotron tune. The physics of this phenomena may be explained as follows: Synchrotron oscillations give rise to a spin phase modulation. It is well-known that phase modulation generates sidebands, which can be thought of as a nonlinear beating between two frequencies. If one of the synchrotron sidebands falls within the width of the primary spin resonance, the spin is depolarized. This situation is particularly important for electron storage rings where the synchrotron tune is about 0.03 to 0.1 so that these synchrotron sidebands are well separated from their primary resonance line. Following Sec. III.2 of Chap. 3, we note that the resonance strength of the synchrotron sidebands at K + mv, is given by €xJm(g) with g = B?Gy0a/v,, where @ is

the synchrotron amplitude and v, is the synchrotron tune. For electron storage ring, the synchrotron amplitude is not a free parameter. The rms beam momentum spread is given by C SAp/p =

1/2

($+)

y~

Se

iex 10-4

for SPEAR at 3.6 GeV with the damping partition number J, = 2. If we choose CB V6oAp/p> we obtain g = 0.42. The solid line in Fig. 6.2 is a theoretical fit

with Eq. (6.21) [27]. The free parameters used in the theoretical fit are effective spin resonance strengths €g, €341,, €3+v,, and €g4,,-,, listed in Table 6.3.

Table 6.3: SPEAR machine parameters that fit the polarization data.

ve

__{[es | eatve |totus |eetrenve

0.04276 || 0.03 |0.008 | 0.001 | 0.001

The resonance strength derived from the data fitting can be tested by Eq. (3.1) with the SPEAR lattice. Unfortunately, this process involves the closed orbit error and the linear and nonlinear betatron coupling which may not be known with certainty. Furthermore, these effective resonance strengths should include the effect of spin diffusion due to quantum fluctuation.

V. EXPERIMENTAL MEASUREMENTS

137

HERA WITH SPIN ROTATORS

3

4-May-94

Ss

cS Ss Ss we wv (%) TION LARIZA iS)i PO.

Rise ofelectron polarization at HERA Longitudinal polarization at HERMES

= S&S Ss

3”

1532

134,

J36)

138

94

+342

44 TIME (h)

Figure 6.3: The measured longitudinal polarization attained at HERA with the use of mini-spin rotator [50]. The later experiments achieved 70% polarization after some energy adjustment.

V.2

VEPP Polarized Beam Experiments

Using the resonance depolarization method, VEPP2M and VEPP4 have been able to determine, very precisely, the energies of many mesons (see the review article of

Ref. [44] and reference therein). In particular, Fig. 6 of Ref. [44] showed many spin resonances from VEPP4, which contained no synchrotron sidebands. The absence of synchrotron sidebands may have resulted from the smallness of the synchrotron tune in VEPP4, causing all sidebands to fall within the width of the primary resonance

(see Sec. V.1 in Chap. 4).

V.3.

TRISTAN

Polarization Experiments

Electron polarization has also been observed in TRISTAN [49] at an energy of 14.76 GeV and 28.863-29.083 GeV. Harmonic matching has been applied to the nearby integer harmonic, K = 66, resulting in a 75% polarization. The polarization data has

also been used for beam energy calibration [49].

V.4.

HERA

Polarized Beam Experiments

Polarized beam experiments have been successfully carried out in HERA at DESY [50]. Using harmonic correction with eight vertical closed orbit bumps similar to those of PETRA [51, 64], a polarization of about 65-70% was obtained. Figure 6.1 shows the measured polarization vs the equilibrium polarization derived from the polarization

time measurements [50]. The longitudinal polarization at the interaction region is obtained by a system of

138

CHAPTER 6. ELECTRON

POLARIZATION

mini-spin rotators designed by J. Buon and K. Steffen [65] for the energy range of 29

GeV to 35 GeV (see Sec. V.2 of Chap. 7). Figure 6.3 taken from Ref. [50] shows the longitudinal polarization achieved with the mini-spin rotator system. An impressive effort to achieve better beam alignment called the beam base alignment has been attempted [66]. By measuring the closed orbit variation due to the variation of an individual (or a small set of) quadrupole strength, the relative alignment between the BPM and the center of the quadrupole can be calibrated. If this procedure is successful and the storage ring does not alter its alignment, i.e., the ground level does not changed with time the required harmonic orbit correction strength can be greatly reduced.

V.5

LEP Polarization Experiments

The polarized electron program in LEP has been contemplated since its initial design phase (see Ref. [67] and references therein). Because the polarization time is long (see Table 6.1), an asymmetric polarization wiggler [68, 69] has been considered. With the existing damping wiggler, the polarization time is about 90 minutes at 45.6 GeV beam energy. Two interesting experimental phenomena observed in LEP polarized beam experiments will be discussed below.

The TidExperiment In order to precisely calibrate the beam energy, a transverse radial dipole field with a maximum field integral B,D ~ 1.6 Gauss-m was used to induce spin depolarization [43]. The apparent variation of LEP beam energy with time led to a very successful polarized beam experiment: the TidExperiment, where the electron beam energy was measured by the resonant depolarization method as a function of time over a complete Earth revolution in a full Moon condition. Because of tidal action the shape of the Earth’s crust is deformed and the gravitational acceleration constant and the horizontal strain of the Earth’s surface are altered. A simple measure that relates the variation of the gravitational constant to

the horizontal strain is given by [43] AC

C

Ag =

Se

aaa

(6.58)

where C’ is the nominal circumference of the LEP ring, and AC is the variation in the circumference arising from the tidal action. Through geological simulations, the strain constant has been determined to be ag, = —0.16, i.e., 16% of the gravitational

variation is translated into the horizontal strain.

The frictional energy variation is

related to the circumference by

cllanentsA sae:

(6.59)

V. EXPERIMENTAL MEASUREMENTS

139

LEP TidExperiment

11 Nov. 1992 . Relative energy chonge measured by resonent depolorizction

— Tide prediction : -strain/a, (trom C.£. Fiecher)

E (tide=0) -100

-200

= -120

-80

40

0

40

80

120

0

4

AQeor (42g!)

812

Tee.

20

24

Time (hours)

Figure 6.4: The left plot shows the variation of the beam energy measured through spin tune vs the variation of the gravity constant. The right plot shows the fractional variation of the LEP beam energy vs the 24-hr period during the full Moon. (Data courtesy of M. Placidi

et al. [43]) where a, =

1.85 x 1074 is the momentum

compaction factor for the 90°/60° optics

used in the experiment. With g ~ 975 cm/s’, the fractional energy variation vs the gravity variation was predicted to be AE

Fa = Mite (Aglem/s?]),

miige"” = —0.887 /ugal,

(6.60)

where 1 pgal=1 x 10~® cm/s”. The left plot of Fig. 6.4, taken from Ref. [43], shows the measured fractional energy variation vs the measured gravity variation in gal. The measured result was KP" = —0.86 + 0.08/ygal. The right plot of Fig. 6.4 shows a 24-hr cycle of the tidal action [43]. Observation of Spin Flip In the course of energy calibration measurements, spin flip has also been observed. This method and its implications will be discussed as follows. The induced spin resonance strength which corresponds to a transverse radial rf dipole of 1.6 G-m in the spin tune measurement is « = 1.7 x 10~° (see Exercise 6.6). If the spin tune is changed by 0.01 in 100 ms, the equivalent acceleration rate is

dK ee ay

1.42 x 107° /rad.

Using the Froissart-Stora formula, the final polarization is 0.9998 of the original polarization, i.e., the spin will not be depolarized in a single sweep. Continuous forward

CHAPTER 6. ELECTRON POLARIZATION

140

(%) P

PPS IDY

22:30

22:35

22:40

22:45

Daytime

Figure 6.5: When the rfdipole field was applied to electron beams in LEP, spin was observed to flip. This result may indicate that the spin diffusion in LEP is not as important as expected [43].

and backward sweeps for five min has produced no depolarization either. This result was used to argue that this polarization diffusion rate was small. On the other hand, if the sweep of Av = 0.05 in At = 5 min was applied, which

corresponded to a = 2.36 x 107'° /rad., a complete depolarization was observed. The expected final polarization in this case is P; = 0.06P;, i.e., a complete depolarization. Some interesting observations attained in these experiments were the spin flips, which indicated that the spin of individual electrons followed the spin closed orbit during the frequency sweep. During the sweep, all particles were forced to precess about the spin closed orbit in the resonance rotating frame. The spin tune spread and the effect of quantum fluctuation seemed to be small. There is much to learn from these experiments. At a lower spin tune sweep rate, Fig. 6.5 taken from Ref. [43] shows the spin flip induced by the rf radial dipole.

VI

Effects of Polarization Wigglers

In LEP, the polarization time at 46.5 GeV was about 300 minutes and the luminosity lifetime was only about 200 minutes. To circumvent this difficulty, Montague [68],

Blondel, and Jowett [69] proposed to use asymmetric polarization wigglers to shorten the polarization time. This section discusses effects of wiggler on the polarization and other beam dynamics properties.

VI. EFFECTS OF POLARIZATION

WIGGLERS

141

The wiggler, which is able to enhance polarization, is a set of three dipoles:

Ee

B-(—)

By(L+)

ve

B(+>),

where By are, respectively, the magnetic fields in the same direction and opposite to the guide field, and L4 are the corresponding lengths of these two types of dipoles. In order to avoid closed orbit perturbation, the necessary condition is BLL_+ B,L, = 0. Now we define the wiggler asymmetry parameter as

The asymmetry arises from the condition that |B,| >> |B_|. With the asymmetric wiggler, the polarization rate is given by [See Eq. (6.6)] a 5V3hrey? Apw

8m

Cys rel f = NwL+ Bi (1 + 1/r?) —)°?——_ 1 + (A) st PIB dsPas ==0,[1 Mee

:

: (6.61)

where a, = Wo of Eq. (6.23) is the nominal polarization rate, N,, is the number of wigglers, B is the regular dipole field and R is the average radius. To reduce the polarization time from 300 minutes to less than 36 minutes, the factor in the square bracket must be larger than 8. The wiggler also introduces reverse bends into the storage ring. The resulting Sokolov—Ternov polarization in an otherwise perfect machine with the wiggler is given

by

‘a

1/r?)/27pB? pe W, — WL EY panier 1+ NwyL+|By|3(1 Sate elect scet—tel eee Beds |e Wi +W. 114+ N,L4)Bs/2(1 + 1/r?)/2mpB3

6.62 )

In a wiggler dominated machine, i.e., N,,L4|B,|* >> 27pB°, the equilibrium polarization becomes

(3).

roa

Teh, Psr sa

(6.63)

To attain a sizable equilibrium polarization, e.g., (S) > 0.8, the requirement for the asymmetry parameter is r > 6.

The conditions that a,y > a, and r > 6 essentially determine the total number and characteristics of polarization wigglers. Some examples of wiggler design have been suggested in Ref. [69]. Now we evaluate the impact of the polarization wiggler on the beam dynamics. Because of the wiggler, the horizontal emittance is given by xz

(ile?)

*" Je(1/p?)

(H) [1+ Nol |Bsl(H)u(1 + 1/r?)/2mpB°(H)s sats 1+ NwL4|B,PQ+1/r)/2npB? |’

(6.64)

142

CHAPTER 6. ELECTRON POLARIZATION

where J, is the damping partition number, B is the regular dipole field, and (H), and (H),, are respectively the average of the H-function in regular dipoles and in wigglers. To minimize emittance growth, the polarization wigglers should be located at. dispersion free straight sections so that (H),, * 0. The resulting natural emittance becomes smaller, where the emittance decreasing factor is equal to the phase space damping time decreasing factor:

ey

a=

:

=

Peete

C8u

hee).

67 R

p? ~

3Rp

pop

NL+|By\?(1 + 1/r)

ee 27 pB?

However, since the damping wiggler increases the quantum particle beam energy, the momentum spread will increase by

ae 2

E

1

hl

(6.65)

fluctuation in the

3

J,(1/P%)

VY "Spe

[1+ NvL+|BsPCL + 1/r?)/20pB [1+ NvL+|By|?(1 + 1/r)/2mpB?

|’

(6.66)

where J, is the damping partition number for the longitudinal phase space. Using the prescribed wiggler requirement given earlier, we easily find that the momentum spread can increase by about a factor of 2.5. This can cause dynamical aperture problems in storage ring operation [69].

VII

Polarized ete” Colliders

In storage rings, electrons and positrons are polarized through the radiative polarization of the Sokolov—Ternov effect. The resulting polarization has be employed to determine the beam energy to a high degree of precision. Besides the beam energy calibration, scattering of high energy polarized electrons off polarized protons and polarized nucleus, e.g., at SLAC, CEBAF and HERA, can be used to determine the spin structure of nucleons. High energy e*e™ collisions can also be used to test the standard electro-weak model, where helicity effects are expected to be important (see

Vol. 1 of Ref. [67] for detailed discussions). In electron storage rings, beam polarization can be obtained by the Sokolov—-Ternov effect. On the other hand, polarized electron source is needed in linear collider and in CEBAF facility. At SLC, polarized electrons are produced by the strained Ga-

As photocathode [70] attaining a polarization of 80-90%. achieve very accurate measurement

This has enabled SLC to

of the left-right asymmetry parameter for the

Z production (see M. Woods in Ref. [5]). Electron polarized sources with a high quantum efficiency will continue to play important roles in future linear colliders. For many high energy polarized beam experiments, the polarization figure of merit is given by

ag [Opa

c(edt.

(6.67)

VII.

POLARIZED

e+ e~ COLLIDERS

143

Since the polarization time is usually short for lower energy storage rings, the figure of merit can be optimized by maximizing the asymptotic polarization value. On the other hand, the polarization time of a high energy storage ring (such as LEP) is long, and asymmetric wigglers are needed to optimize the polarization time and the equilibrium polarization value. The luminosity of an ete™ collider is given by

L=f oie

(6.68)

d

410702

where f is the frequency of the bunch encounter, Ny are-the number of e* particles per bunch, and o,,, are the horizontal and vertical beam sizes at the interaction point (IP). Helicity state collisions of polarized electrons can be achieved by a pair of spin rotators at each IP. The Richter-Schwitters spin rotator [72], to be discussed in Sec. IV of Chap. 7, provides a very simple scheme for high energy ete™ colliders. A more compact Buon-Steffen spin rotator has been successfully commissioned in HERA [65]. Compton polarimeters are usually used to measure high energy electron polarization. The polarization of individual bunches can also be adjusted to enhance beam polarization measurements. For example, the Blondel scheme of polarized beam configuration Ci,

>

*

*

=

*

=

*

wm

etlet

allows a measurement of Apr, P; and P_ from a set of experimental data in one experiment. External polarimeters can also be used to monitor the evolution of polarization with time and possible differences between one bunch and another. There are many spin depolarization resonances in high energy storage rings. All resonances near the ay value in high energy experiments should be compensated. If the ay value of an experiment is near an intrinsic spin resonance, the betatron tune and the machine superperiodicity can be re-matched to avoid the important resonance. On the other hand, if the ay value is near an imperfection resonance, harmonic closed orbit correction may be the only solution. Finally, the effects of beam polarization due to beam—beam interaction are not well-known. Experimental data from PETRA [51] LEP [43] and SLC [71] seem to indicate that there was little depolarization due to beam—beam interaction until the beam lifetime itself began to be affected by the beam—beam interaction. Theoretical studies on this topic remain wide open.

Exercise

6

1. Verify the polarization damping time for storage rings listed in Table 6.1.

2. Show that the equilibrium solution of Eq. (6.17) is given by Eq. (6.19).

144

CHAPTER 6. ELECTRON

POLARIZATION

3. Let us write the solution of Eq. (6.17) as

SPE SES

BeeSesy:

where $§2, 557 are equilibrium solutions of Eq. (6.19), and S} and S§ are transient solutions. e Show that the transient spin component satisfies dst Ti

=

dS3 os

Age

(#216 —

=)54

t

ile

t

t

-lelfS4 Sree te 2, the free space can be minimized with d = 0. For the split snake configuration with 1 < m < 2, the snake can be decomposed into two parts, where each part is a local spin rotator and the combined result of these two parts functions as a snake. The spin rotation angle ¢ and the snake axis angle ¢, are given by

cos g= cos’, + cos my, sin? wz, cos ¢, =

2

(7.12)

ae

y

:

es aul: n( HS ve) sin? Be— simC a

sin$

2sin

“Y=

sin

sind, = lp sin $

w,

an 2

itp cn

cos” Ey(7.13) .

2

a =e any cos? . — .(7.14)

Note here that w, and mz, (instead of ~,) are the relevant variables to determine g. Once ¢ and 7~, are chosen, my, is determined by Eq. (7.12). The symmetric

properties of Eq. (7.6) remains valid.

II.3.

The Compact

Snake Configuration with m > 2

The total length of a snake can be minimized by a properly chosen parameter m. Assuming that the distance between adjacent magnets is £,, the closed orbit condition with d = 0 is given by 1

(m — 1) (2. + fy +5 (m=1) 4.) =£€, +0,

Palys

(715)

where ¢, and @, are lengths of magnets H and V respectively. The total length of the snake configuration becomes L = 6é, + 4@, + 6£,. The snake strength and the snake axis of this configuration are determined by Eqs. (7.12) and (7.14). The orbit displacements are given by

Dy = (le + £2 +20)Yo ==, Dz = (mb, +l, + 2b) pz Gy’

Gy

(7.16)

Table 7.1 compares snake configurations for m = 2 and m = 2.2 snakes by assum-

ing 2 Tesla magnetic fields for both H and V magnets and £, = 0.15 m. Note that the compact snake configuration, with m obtained from Eq. (7.15), has a total length that is 1 to 2 meters shorter than that of the Steffen snake configuration with m = 2. The horizontal orbit displacement is reduced slightly, while the vertical orbit displacement

remains the same in both cases [see Eqs. (7.12), (7.14), and (7.16)].

The total in-

tegrated { Bdé is also slightly smaller for the compact snake configuration. Thus the

Il. SNAKE WITH TRANSVERSE FIELD DIPOLES

+E,

ja

155

ania

Figure 7.4: A schematic drawing of split Steffen snake. Here two (V’,—V’") pairs are used to correct the vertical closed orbit and a (—H,#H) pair at the center region corrects the horizontal at the center section.

compact snake configuration offers the advantage of producing a smaller horizontal orbit displacement in a snake and requiring a smaller total integrated field strength. Table 7.1: Length and Orbit Displacements for Snakes.

II.4

Split Snake Configuration

When a snake is separated into two parts at the symmetry point, the combined effect on spin motion is still governed by Eqs. (7.12) and (7.14) provided that there is no net spin precession in the region between two halves of the snake. In the case of 1 < m < 2, the snake can be separated into two parts. The distance between these two half-snakes depends on the parameter m. The nuisance of a nonzero closed orbit value D, at the mid-section of the snake can be corrected by two pairs of vertical compensation dipoles, V' and —V’ located at both ends of the snake shown schematically in Fig. 7.4. The half snake can be used either as a spin rotator or as a space saver, where a half-snake

(VV, VV GHe SV all, V)

156

CHAPTER

7. DESIGN OF SPIN ROTATORS

occupies a straight section smaller than the full snake. The accelerator lattice that employs the above half-snake scheme must possess two adjacent straight sections separated by a single quadrupole. The resulting horizontal orbital displacement is

slightly larger [see Eq. (7.15)]. Due to a large horizontal orbit displacement in the mid-section with a long drift space, this split snake configuration is not useful for an interaction region of a collider, where the distance between two halves of a snake is large. A practical design of a split snake with the proper orbit correction is schematically shown in Fig. 7.4. Using orbit correctors, the snake can be split into two; each part can be used as a spin rotator. The split snake can also function as a spin rotator for the helicity states. Using the half snake as a spin rotator (along with its dual function as a snake), the helicity states of the spin particle can be attained. For a spin up particle passing through the half snake, spin components at the mid-section of the snake become

S, = —sin my, siny,,

S, = —sin? ne aire.

See 0.

eas

Letting ¢y be the angle of the spin relative to the radial ¢ axis, we arrive at tan om

=

Ss

oe

(7.18)

Since mip, and wp, are related through Eq. (7.12), dy depends solely on w, and is independent of m.

II.5

Partial Snakes

Partial snakes [80, 81] may be used to overcome weak imperfection resonances. The required spin rotation angle ¢ (the snake strength) is given by (See Exercise 5.2)

p

57 = |¢mpl + 2V/a,

(7.19)

where

es

dGy

with

_ Gy

Tog Sohal

®:

is the resonance crossing rate, w is the angular revolution frequency of the circulating particle, and mp is the imperfection resonance strength. Typically, a < 1074 for most low energy accelerators with energies less than 30 GeV, e.g., Qag, © 5- 107°. Fora low energy synchrotron such as the AGS, the imperfection resonance strength varies as see Chap. 3) €imp

x

0.2(22,)¥/?¥,

III. HELICAL SNAKES

157

where (z2,)!/? is the rms closed orbit displacement in meters. With a harmonic correction scheme reaching o, < 2.5 mm, we expect, based on Eq. (7.19), the required snake strength to be

¢ > 10°*yn. Therefore, a 5% snake may be enough to correct imperfection resonances for y up to 25: The snake configuration discussed in Sect. II can also be applied to design partial snakes. The total length of the partial snake can be minimized by a properly chosen m. Once the number m is chosen, the snake can be adiabatically turned on and off by a single rampable power supply. Using the Taylor series expansion of Eq. (7.12), we obtain = 2m|vzpzI,

(7.20)

where the snake strength ¢ depends bilinearly on 7, and 7. For a given snake strength ¢ we find that ~, and ~, satisfy the hyperbolic equation. If a snake is powered by a single power supply for H and V magnets, the snake strength will depend quadratically on the current of power supply. The corresponding snake axis angle ¢, is given by 7 m

be —2-(E=I be te,

(7.21)

Therefore for all partial snakes with a small ~, and w, that minimizes

the orbit

displacement and the total snake length, the resulting snake axis lies close to the longitudinal direction. In contrast, the type II snake, with ¢, = 0 or 180°, requires a large w,. This characteristic appears in all kinds of partial snake configurations. Because of the smaller snake strength requirement at lower energies [see Eq. (7.17)], the actual transverse closed orbit displacement can be lessened. Thus generalized snake structure can also be applied to a partial snake, which minimizes the orbit displacement and the total length of a snake. Such an optimization is usually needed

for small accelerators where the length of the straight section is small.

III

Helical Snakes

Helical snakes with a wiggler magnetic field can often minimize the orbit displacements for particle motion inside the snake [77, 78, 79]. The magnetic field of a helical snake is given by B= B,, cosks €; + By,sinks é3,