LINEAR OPTICS CORRECTION AND …physics.indiana.edu/~shylee/info/LiuZ_thesis.pdfLINEAR OPTICS...

LINEAR OPTICS CORRECTION AND

OBSERVATION OF ELECTRON PROTON

INSTABILITY IN THE SNS ACCUMULATOR

RING

Zhengzheng Liu

Submitted to the faculty of the University Graduate School

in partial fulfillment of the requirement

for the degree

Doctor of Philosophy

in the Department of Physics,

Indiana University

Aug 2011

ii

Accepted by the Graduate Faculty, Indiana Univeristy, in partial fulfillment of the

requirements for the degree of Doctor of Philosophy.

Shyh-Yuan Lee, Ph.D.

David V. Baxter, Ph.D.

Doctoral

Committee

William M. Snow, Ph.D.

Rex Tayloe, Ph.D.

May 2011

iii

Copyright c©2011 by

Zhengzheng Liu

ALL RIGHTS RESERVED

iv

To my parents.

v

Acknowledgments

This dissertation could not have been accomplished without the guidance and

support of many people. I would like to take this opportunity to express my thanks

to many individuals that provided assistance towards this effort.

First of all I would like to thank to my thesis advisor, Professor Shyh-Yuan Lee, for

his guidance during my research and study at Indiana University. He is a supportive

and energetic advisor. His insight and enthusiasm in research has motivated many of

his students. I have benefited from both his direct supervision at Indiana University

and remote assistance in my research at ORNL.

After two years of course work and research at Indiana University, I joined the

Accelerator Physics Group in the Spallation Neutron Source of ORNL, where I was

offered the opportunity to transform my knowledge in accelerator physics into a real

machine. I was fortunate during my stay at ORNL to work with a group of highly

talented scientists.

I want to express my gratitude to my supervisor Dr. Jeffery A. Holmes at ORNL.

Dr. Holmes has proofread my thesis and made a lot of corrections from the grammar

to the physics. I am also indebted to many of my colleagues from the Accelerator

Physics Group: Dr. S. Danilov, Dr. J. Galambos, Dr. S. Cousineau, Dr. M. Plum,

Dr. A. Shishlo, Dr. T. Pelaia and Dr. C.K. Allen. They have helped me a lot on my

studies. It is my pleasure to be part of the AP team and to work with such a high

quality group of professionals.

I would like to give a special thanks to Dr. Xiaobiao Huang. He guided me in the

utilization of LOCO code and provided me many valuable suggestions. He deserves

special thanks for his generous help on my first project in ORNL. I would also like

to thank Professor D. V. Baxter, W. M. Snow and R. Tayloe as my dissertation

vi

committee members. They provided valuable suggestions and made my dissertation

active.

I owe my deepest gratitude to my family. I am indebted to my father and my

mother for their care and love throughout my life. They spared no effort to create

the best possible environment for me to grow up, which has made all of my accom-

plishments possible. Finally, I sincerely dedicate this dissertation to my family.

vii

Zhengzheng Liu

LINEAR OPTICS CORRECTION AND OBSERVATION

OF ELECTRON PROTON INSTABILITY IN THE SNS

ACCUMULATOR RING

The accumulator ring of the Spallation Neutron Source is a high intensity proton

storage ring. The choice of its operating tunes is critical. There was a relatively

large tune discrepancy ∼ 0.2 between model prediction and real measurement. As

a consequence, it was not possible to set the lattice using the model calculation.

The orbit response matrix (ORM) method, as programmed in the application code

LOCO, was employed to solve the optics discrepancy and calibrate the linear model.

Offline study shows that we can attribute most of the tune discrepancy to the errors of

quadrupole magnet power supplies, which is up to 2.9%. The results and discussions

of proved and potential optics improvement are presented in detail in the thesis.

Due to the high intensity of proton beam and the similarity of SNS and PSR,

collective instabilities, especially the electron-proton (e-p) instability, pose potential

limitations on the peak intensity and therefore become major concerns in the SNS

power-up plan. Therefore, although the e-p instability has not emerged in the normal

neutron productions yet, we have manipulated the machine setting to observe it in a

series of experiments. It shows that, the buncher voltage has little effect on instability

threshold and that the instability has a strong dependence on proton bunch shape.

Moreover, a potential mitigation of the e-p instability involves the use of a flat top

current profile with a short tail. Detailed observation and discussion can be found in

the thesis.

CONTENTS ix

Contents

Acceptance ii

Acknowledgments v

Abstract vii

1 Introduction 1

1.1 Overview of the SNS Accumulator Ring . . . . . . . . . . . . . . . . . 1

1.2 Beam Loss Mechanisms for High Intensity proton storage ring . . . . 10

1.2.1 ”First-turn” losses . . . . . . . . . . . . . . . . . . . . . . . . 10

1.2.2 Imperfection resonance . . . . . . . . . . . . . . . . . . . . . . 11

1.2.3 Space-charge effects . . . . . . . . . . . . . . . . . . . . . . . . 11

1.2.4 Coherent instabilities . . . . . . . . . . . . . . . . . . . . . . . 14

1.2.5 Contents of chapters . . . . . . . . . . . . . . . . . . . . . . . 15

2 Application of Orbit Response Matrix Method to the SNS ring 17

2.1 Orbit Response Matrix Method . . . . . . . . . . . . . . . . . . . . . 19

2.1.1 The Perturbed Orbit and Green’s Function . . . . . . . . . . . 19

2.1.2 Algorithm of ORM Method . . . . . . . . . . . . . . . . . . . 20

2.1.3 Dispersion Effect on Closed Orbit . . . . . . . . . . . . . . . . 22

2.1.4 LOCO Code and Its Algorithm . . . . . . . . . . . . . . . . . 23

x CONTENTS

2.1.5 Constraints in LOCO Fitting . . . . . . . . . . . . . . . . . . 29

2.2 Application of ORM to SNS Accumulator Ring . . . . . . . . . . . . 30

2.2.1 Measurement of Response Matrix . . . . . . . . . . . . . . . . 31

2.2.2 Uncover Quadrupole Gradient Errors . . . . . . . . . . . . . . 34

2.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3 Electron Cloud 47

3.1 Introduction of Physics of Electron Cloud Effect . . . . . . . . . . . . 47

3.2 Electron Cloud . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.2.1 Physics of Secondary Electron Emission . . . . . . . . . . . . 51

3.2.2 Two stream instability model for coasting beam . . . . . . . . 60

3.3 Observation of Electron-Proton instability at the SNS ring . . . . . . 63

3.3.1 Observation of multi turn and single turn electron accumulation 64

3.3.2 A particular observation of e-p instability with buncher voltages 69

3.3.3 Observation of e-p instability with intensity scan . . . . . . . 71

3.3.4 Effect of proton bunch shape on e-p instability . . . . . . . . . 76

3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4 Conclusion 91

Bibliography 97

LIST OF TABLES xi

List of Tables

1.1 Spallation neutron source primary parameters . . . . . . . . . . . . . 9

2.1 Response matrix calculation methods in LOCO [1] . . . . . . . . . . . 24

2.2 Comparison of measurement conditions . . . . . . . . . . . . . . . . 33

2.3 Fitting parameters and methods for ”June 2008” data set . . . . . . . 36

2.4 The weights of changes of fit parameters . . . . . . . . . . . . . . . . 38

2.5 Gradient errors based on data of June 2008 . . . . . . . . . . . . . . . 38

3.1 Main parameters of the model, used for SNS TiN coated and uncoated

stainless steel chamber. . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.2 Primary simulation parameters . . . . . . . . . . . . . . . . . . . . . 83

xii LIST OF TABLES

LIST OF FIGURES xiii

List of Figures

1.1 Layout of the spallation neutron source . . . . . . . . . . . . . . . . . 2

1.2 Schematic layout of the beam injection region: The solid black line

represents the closed orbits with and without kicks. The transverse

painting is controlled by the eight programmable, time-dependent kickers. 4

1.3 Longitudinal phase space distribution at the end of injection . . . . . 5

1.4 X-Y correlated and anti-correlated painting injection schemes: The

bumps move the closed orbit monotonically in time. For the x-y corre-

lated painting, phase spaces in both directions are painted from small

to large emittance. For the x-y anti-correlated painting, the total trans-

verse emittance is approximately constant during injection but requires

50% more vertical aperture clearance than does correlated painting. . 6

1.5 Linac beam pulsed structure . . . . . . . . . . . . . . . . . . . . . . . 7

1.6 Beam halo at the end of injection for (νx, νy) = (6.4, 6.3): Blue color

shows the halo due to space charge only; Red color shows the halo due

to systematic and random magnet field errors of magnitude 10−4, sex-

tupoles and quadrupole fringe fields; Yellow color shows an additional

effect of x,y misalignment of 0.5 mm and magnet tilt of 1 mrad. . . . 12

xiv LIST OF FIGURES

1.7 Beam halo at the end of injection for (νx, νy) = (6.23, 6.20): Blue color

shows the halo due to space charge only; Yellow color shows halo when

magnet errors of expected magnitude are included. . . . . . . . . . . 13

2.1 Correlation coefficient r and phase advance between neighboring quadrupoles

of SPEAR3 [1]: Stronger correlation is a result of smaller horizontal

phase advance (mod π), which implies that, two quadrupoles can be

physically set apart but can have strong correlation if the horizontal

phase advance between them is a multiple of π. . . . . . . . . . . . . 28

2.2 Converging path with or without constraints. Solid: no constraints;

Dash: with constraints. . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.3 Example of a good fit: the x axis is the error in the unit of standard

deviation, and the y axis shows the error distribution of the thousands

of points in the matrix in a histogram format. Therefore, the more

the points close to zero, the better the agreement between model and

measurement. After the fitting (black color), the error is reduced sig-

nificantly from before (red color). . . . . . . . . . . . . . . . . . . . . 35

2.4 Comparison of fitted cases (follow-up to Table 2.3): Plot on the left

shows percentage errors of grouped quadrupole gradients uncovered

from fit. The graph on the right is the comparison of chi2 fitting for

the three cases. The LOCO ”Constant Path Length” corresponds to

the green color. The χ2 fluctuates a lot before the 10th iteration. . . . 37

2.5 Energy shift with ”Constant Momentum” method: The magnitude of

energy shift is only 10−4. Therefore it does not have much effect on

the fitting result as shown in Figure 2.4. . . . . . . . . . . . . . . . . 39

LIST OF FIGURES xv

2.6 BPM gain factors and coupling: Default BPM gain is 1. Default BPM

coupling is 0. So the points close to 1 represent BPM gain and points

close to 0 stand for the BPM coupling. The left and right plot show

fitted horizontal and vertical BPM calibration values respectively. Blue

points correspond to ”Dec 2009” data and red points correspond to

”June 2008” data set. . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.7 Kicker strengths and coupling: Kicker strengths are normalized to 1.

Default coupling is 0. So the points close to 1 represent kicker normal-

ized strength and points close to 0 stand for the kicker coupling. The

left and right plot show fitted horizontal and vertical kicker calibration

values respectively. Blue points correspond to ”Dec 2009” data and

red points correspond to ”June 2008” data set . . . . . . . . . . . . . 42

2.8 Fittings with two different fitting parameter sets: One set (red color)

only includes 52 quad strengths, while the other (blue color) also in-

cludes BPM and corrector parameters as additions. The previous cor-

rections of quad group fitting were implemented initially. The two

subfigures represent the additional quad errors and iterative χ2, re-

spectively. As shown on the left, both cases uncover that the quad

group QV03a05a07 (point 33 to 44) has an additional error ∼ 0.6%

that was not discovered. Group QH04a06 (point 45 to 52) also has an

undiscovered error. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

xvi LIST OF FIGURES

3.1 Mechanism of electron cloud development for a long proton bunch: Red

arrow stands for electrons in the beam gap. They are captured and

oscillate inside the beam potential well as the beam intensity increases,

released as the intensity decreases and produce secondary electrons

(purple arrows) when they hit the wall. Another source of secondary

electrons is due to the lost protons(dark blue arrow). Those protons

hit the chamber and generate electrons(green arrow). The electrons

finally gain energy in the duration of the second half of bunch and

strike the wall to produce secondary electrons and this process can be

repeated. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.2 The SEY for stainless steel for SLAC standard 304 rolled sheet, chem-

ically etched and passivated but not conditioned. The parameters of

the fit are listed in Table 3.1 . . . . . . . . . . . . . . . . . . . . . . 54

3.3 The Monte Carlo scheme implemented in the C++ class ”controlled

emission surface”. The two M represent the macro-sizes of incident

and emitted electrons. If G < fdeath(E0), then Mout = 0; else if G >

fdeath(E0), Mout = Min ·δ/(nborn ·(1−fdeath(E0))), where G is a random

number between 0 and 1, δ is the SEY, fdeath(E0) is a user defined

function for the incident energy E0 and is usually equal to 0 if E0 >

1eV , and nborn is the number of emission procedure per impact event

and is usually equal to 1. . . . . . . . . . . . . . . . . . . . . . . . . . 56

LIST OF FIGURES xvii

3.4 Horizontal oscillations on the head of proton bunch at SNS. Measure-

ment was taken in 2008 with 2nd harmonic RF phase set to 5 deg.

The red line represents the BPM sum signal. The blue line is the BPM

difference signal with closed orbit offset substracted. Development of

the unstable oscillation can be seen in the progress from the upper to

the lower plots. It occurs at the head of proton bunch, which is an

evidence in favor of the multi-turn electron accumulation. . . . . . . . 66

3.5 Horizontal oscillations on the head and tail of proton bunch at SNS.

Measurement was taken in 2008 with 2nd harmonic RF phase set to 15

deg. The unstable oscillation first occurs at the head of proton bunch

and also emerges in the tail at a later time. This is evidence in favor

of the multipactor effect at the trailing edge, in addition to the multi

turn accumulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.6 Experimental proton beam profile near the end of injection with simu-

lated electron cloud. The proton longitudinal peak density, simulated

electron peak density and e-p growth rate have a linear relationship in

the lower plot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.7 SNS: Effect of buncher voltage on the instability threshold intensity.

The change of threshold intensity is very little and there is no clear

dependence on the buncher voltage. . . . . . . . . . . . . . . . . . . . 70

3.8 The instability frequency spectrum for 11 µC proton bunch. . . . . . 72

3.9 Frequency spectrum for different beam intensity of 15 µC, 17 µC and

21 µC. The instability occurs before the end of injection (15 µC ' 700

turn ). Therefore the development of instability shows the same trend.

The instability is developed stronger for higher intensity according to

the color bar. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

xviii LIST OF FIGURES

3.10 Strip filter for frequency spectrum: The upper plot is the general fre-

quency spectrum. Strip filter is applied to cut the three strip from the

spectrum with an example in the lower plot. The lower spectrum is

later inverted to time domain oscillation. . . . . . . . . . . . . . . . . 75

3.11 BPM difference signal for strip 1 on frequency spectrum. The red

represents the BPM sum signal after scaling and blue is the difference

signal after filtering. . . . . . . . . . . . . . . . . . . . . . . . . . . . 77







3.14 Filtered signals at turn 605 for the three strips. The center frequency

for the three strips has an interval of ∼ 10 MHz, which is not obvious in

the time domain. The locations of the three oscillations almost overlap

in the time domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

3.15 Parallel strip pattern was also observed for oscillation at proton bunch

head. Measurement was taken in 2008. . . . . . . . . . . . . . . . . . 81

3.16 Effect of trailing edge’s length (time duration). The sub-figure on the

left plots some example trapezoid distributions and the corresponding

electron clouds with the same color. The sub-figure on the right plots

the length of trailing edge versus the peak height of electron cloud.

And we can use a cubic polynomial fit to perfectly fit those points:

f(l) = 5.7× 10−6l3 − 1.3× 10−3l2 + 9.7× 10−2l − 2.239. . . . . . . . . 84

LIST OF FIGURES xix

3.17 Steepness factor versus peak height of electron cloud. Steepness factor

is defined as s = Ttail/Thead, where T is the time duration. The triangle

shape is changed from head-only (steepness=0), to tail-only (steepness

= ∞). When s > 10, the difference between triangles is very tiny and

thus the electron cloud looks saturated. . . . . . . . . . . . . . . . . . 85

3.18 Beam longitudinal profile evolution for different RF phases. ”ph1”,

”ph3” and ”ph5” denote 2nd harmonic RF phase −35 deg, −5 deg and

15 deg, respectively. The e-p instability develops from no (ph1) to

stronger with larger growth rate. Instability occurs near turn 700 ∼

800 for ph3 and ph5. . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

3.19 Horizontal spectrum for ph1, ph3 and ph5. . . . . . . . . . . . . . . . 87

xx LIST OF FIGURES

Introduction 1

Chapter 1

Introduction

1.1 Overview of the SNS Accumulator Ring

The Spallation neutron source (SNS) is an accelerator complex designed to deliver

1.5 MW of pulsed beam power at a repetition rate of 60 Hz. The accelerator complex

consists of the ion source and the front end, a 1 GeV full energy linac, an accumulator

ring and its transport lines, and a mercury target. Figure 1.1(a) shows the conceptual

drawing of the whole accelerator system [2].


storage ring. It accumulates protons for delivery onto a mercury target to produce

pulsed neutrons. The purpose of the accumulator ring is to serve the needs of neutron-

scattering research which requires short, extremely intense bunches of neutrons. To

achieve this requirement, the beam from linac must be compressed more than 1000

times. An H- pulse from linac is wrapped into the ring through stripper foils that

strip electrons to produce protons, and approximately 1016 turns of protons are ac-

cumulated. Then all the proton beam is kicked out at once, producing a pulse less

than 1 micro-second in duration that is delivered to target. A schematic layout of

2 1. Introduction

(a) Conceptual drawing of SNS

injection septum& bumps

ext. kickers

ext. septum

movablescatterer collimators

fixed

beam

beam gap kicker

instrumentationRF

(b) Schematic layout of the ring

Figure 1.1: Layout of the spallation neutron source

1.1 Overview of the SNS Accumulator Ring 3

the accumulator ring is shown in Figure 1.1(b) [3].

The ring lattice is four-fold symmetric with each super-period containing one

FODO arc section and one doublet straight section. The arc section consists of

four 8 meter long FODO cells, each with a horizontal betatron phase advance of 90

degrees. The dispersion-free straight section consists of one 12.5 meters and two 6.85

meters drift spaces, designed mainly for beam injection, collimation, extraction and

RF bunching. The total ring circumference is 248 meters.

The accumulator adopts a multi-turn H− stripping injection scheme. The inser-

tion consists of four dipole magnets to bump the circulating beam near to the vicinity

of a carbon stripper foil, through which the injected beam is stripped from H- to H+.

The foil lies inside the second chicane magnet, which was designed with a compli-

cated pole tip in order to direct stripped electrons to a collector. In addition, the

injection system also contains eight (four horizontal, four vertical) programmable,

time-dependent kickers to paint the desired transverse distribution into the ring, as

shown in Figure 1.2 [3]. The desired transverse beam distribution is achieved by in-

jection painting. With the long straight section provided by doublets that is shown

in Figure 1.2, beam injection is essentially decoupled from lattice tuning [3]. Two

injection schemes were proposed to paint the high intensity beam: correlated painting

in which both horizontal and vertical displacements from closed orbit are increased

in time, and anti-correlated painting in which one plane displacement is increased

which the other is decreased as shown in Figure 1.4 [3]. For the anti-correlated paint-

ing scheme, an extra vertical aperture which is about 50% of beam size is reserved

in the injection section to accommodate the orbit bump, as shown in Figure 1.4(b).

Therefore, although simulation using ORBIT proves that both schemes are capable

of producing satisfactory results for ring loss and target distribution, the correlated

painting scheme, which produces rectangular beam profiles, was selected over the

anti-correlated one, which produces round profiles [4]. Longitudinal painting was

4 1. Introduction

1814

290

-H

3000

110.8 mr

802.0 kG

12500

4080 8040

80

80

550700

400

150

10080

914 900

23813230

170

140

42.0 mr

3596

863

42.0 mr

100

44.3 mr

47.3 mr

890

1501

0.25 mr

2029

19030

foil @ 2.1 kG

2.5 kG45.0 mr

1220

3.0 mr

3.0 mr

3.60 kG

0.04 kG

3.3 kG

383

2.5 kG

1.3 GeV Horizontal

8390.80 kG9.39 mr

428

1800

100.0 mr3.7 kG

center line

(a) Horizontal layout : Blues are quadrupole doublets. Greens are horizontal dipole

kickers which control horizontal painting, and reds are chicane magnets. The green

dot at the second chicane magnet represents the primary stripper foil, and the long

green slice at the forth chicane is the secondary stripper foil which is used to collect

the escaped H− and partially stripped H0.

80

28

12500

17

46

13690

-8.17 mr 3.68 mr

80

0.53 kG (0.63 kG)839 4280.58 kG (0.70 kG)

center line

(b) Vertical layout: Yellows are the vertical dipole kickers. The painting in vertical

plane is much simpler, which uses only four vertical kickers.

Figure 1.2: Schematic layout of the beam injection region: The

solid black line represents the closed orbits with and

without kicks. The transverse painting is controlled by

the eight programmable, time-dependent kickers.


-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0 100 200 300 400 500 600 700 800 900

gam

ma

- gam

m0

time (ns)

Figure 1.3: Longitudinal phase space distribution at the end of

injection

omitted after simulations suggested that design constraints could be met without it.

The resulting longitudinal phase space beam distribution is shown in Figure 1.3 [3].

The multi-stage collimation section consists of movable primary scatterers and

three self-shielded collimators located in three consecutive drift spaces. At the design

acceptance of 480 πmm·mr, the expected collimation efficiency is about 95% [3]. The

main uncontrolled beam losses are expected to be at the injection region caused by

nuclear scattering of the foil, and from inefficiency of the collimation system. Beam

residual in the gap between subsequent linac bunches is cleaned by the beam-in-gap

(BIG) kicker together with the multi-stage collimation system [4].

Since the longitudinal painting is omitted, the bunch after accumulation is 645 ns

long, the same as the linac pulse, with full intensity of 1.5× 1014 protons. Extraction

of the accumulated beam happens in 1 milli-second after the injection process is com-

pleted. The beam is extracted from the accumulator ring in a single beam revolution

6 1. Introduction

x

injection end

injection begin

foil

y

(a) Correlated painting scheme

x

foil

injection begin

injection end

y

(b) Anti-correlated painting scheme

Figure 1.4: X-Y correlated and anti-correlated painting injection

schemes: The bumps move the closed orbit monoton-

ically in time. For the x-y correlated painting, phase

spaces in both directions are painted from small to

large emittance. For the x-y anti-correlated painting,

the total transverse emittance is approximately con-

stant during injection but requires 50% more vertical

aperture clearance than does correlated painting.


Macro-pulsestructure(for target)

2.4845 ns (1/402.5 MHz)

260 micro-pulses

645 ns 300 ns

945 ns (1/1.059 MHz)

1ms

16.7ms (1/60 Hz)

15.7ms

Mini-pulsestructure(for ring)

Micro-pulsestructure(for RF)

Figure 1.5: Linac beam pulsed structure

(945 ns). The maximum extraction rate is 60 Hz. The extraction system consists of

14 fast kickers (τ ∼ 200ns), 7 upstream of the straight section doublet and 7 down-

stream of the doublet, followed by a single Lambertson type septum magnet. During

the gap of the beam (τ ∼ 300ns), the kickers rise to their full strength and remain on

for ∼ 645 ns. To accommodate the injection and extraction scheme, the linac beam

pulsed structure is complex as shown in Figure 1.5. The extraction kickers deflect

the beam vertically, and the Lambertson septum will deflect the beam horizontally

in order to clear the quadrupole following the septum. The vertical deflection of the

closed orbit at the entrance of the septum is 168 mm. It corresponds to an acceptance

of 400 πmm ·mr at the entrance of the septum [3].

The main purpose of the ring RF system is to maintain the 300 ns gap for the

8 1. Introduction

rise time of the extraction kickers together with low peak beam current and large

momentum spread. It is a dual harmonic system with three f1 = 1.05MHz cavities

and one f2 = 2.11MHz cavity. For small beam current, the RF bucket area is

εbucket = 19eV ·s and the bunch emittance is εbunch = 14eV ·s with a full energy spread

of ±11.1MeV and a synchrotron period of 1400 turns at the edge of the bunch. For

high beam current, the longitudinal space charge impedance reduces the bucket area

and maximum energy spread. Beam loading effects are also a consideration. A full

energy spread of ±9.8MeV is typical with good beam loading compensation.

There are 32 arc dipole magnets and 52 quadrupole magnets. The inscribed di-

ameters of the vacuum chamber are 26 cm for 24 quadrupoles and 21 cm for the other

28 quadrupoles. Of the 26 cm quadrupoles, 16 are located in the straight sections

and 8 are located in high dispersion areas of the arcs. The nominal working point

in the transverse tune space, (Qx, Qy) = (6.23, 6.20), is decided by the quadrupole

gradients. However, a real machine is never perfect due to inevitable manufacturing

defects, installation errors and operation uncertainties. Therefore, some additional

magnets are needed to tune the machine. Every dipole, quadrupole and sextupole

magnet has its own corrector, and there is also a group of octupole correctors. How-

ever, the power supplies are limited and errors in some primary magnets have been

neglected at this moment. The currently utilized corrector package contains 24 hor-

izontal/28 vertical dipole correctors, trim quadrupoles for each quadrupole, and 28

skew quadrupoles. The independently controlled dipole correctors are used to cor-

rect the beam orbit and they are usually used in combination to produce ”bumps”

instead of global ”distortion”. The trim quadrupoles are used to control the betatron

tunes and they are powered by 16 power supplies. The skew quadrupoles are used to

compensate linear coupling and they are independently powered.

The diagnostics of the accumulator ring provide strong support to monitor the

beam. There are 44 bi-directional beam position monitors (BPMs), 82 beam loss


Ring Parameters Design Value Operation Unit

08/22/2011

Kinetic Energy Ek 1000 925 [MeV ]

Uncertainty δEk95% ±15 [MeV ]

Beam power on target 1.4 0.83 [MW ]

Pulse Length on target 645 [ns]

Average macro pulse H− 26 [mA]

Linac average beam current 1.6 [mA]

Ring circumference 248.0 [m]

Average radius 39.47 [m]

Repetition rate 60 [Hz]

Normalized emittance 347 π ·mm ·mr

Unnormalized emittance (99%) 160-240 π ·mm ·mr

Horizontal tune 6.23 6.19

Vertical tune 6.20 6.17

Transition energy γT 5.25

Slip factor η -0.198

Horizontal natural chromaticity -7.7

Vertical natural chromaticity −6.4

Electron bounce frequency 100-175 [MHz]

Ring rf frequency 1.058 [MHz]

Ring injection time 1.0 [ms]

Beam bunch intensity 1.6 [1014]perpulse

Ring space-charge tune spread 0.15

Table 1.1: Spallation neutron source primary parameters

10 1. Introduction

monitors (BLMs), 1 beam current monitor (BCM), 1 wall current monitor (WCM)

and 2 wire scanners in the ring. The primary parameter list is attached as Table 1.1

[3].

1.2 Beam Loss Mechanisms for High Intensity pro-

ton storage ring

For a high intensity proton ring such as the SNS accumulator ring, there are strict

requirements for the uncontrolled beam loss (0.01% of Ibeam for SNS). High beam loss

does not only affect machine avaliability, but may also damage devices in hot regions.

In this section, we will briefly describe some of the typical beam loss mechanisms that

apply to high intensity proton rings [5].

1.2.1 ”First-turn” losses

”First-turn” losses are associated with the injection. There are two major causes:

Scattering at the foil and excited H0 states.

The first mechanism is nuclear and large angle Coulomb scattering of the circu-

lating beam at the injection stripping foil. This led beam losses of 0.3− 0.5% in PSR

prior to the upgrade. By choosing a direct H− injection scheme [6] and minimizing

beam foil transversal, this loss can be significantly reduced.

In the second mechanism, a fraction of injected beam interacts in the stripping

foil and is converted to excited states of H0. When those H0 pass through the

magnetic field, they can be stripped by the Lorentz force. Depending on the time of

stripping, their subsequent trajectories can be outside the beam core. The H0s that

exit the foil will populate the various hydrogen states n, where n denotes the principal

quantum number. A widely used approach to describe the behavior of excited states

1.2 Beam Loss Mechanisms for High Intensity proton storage ring 11

of H0 is the fifth-order perturbation theory of Damburg and Kolosov. By choosing

an appropriate foil thickness, the production of H0 states can be strongly reduced.

In the SNS design, to prevent stripping of H0 in n = 4 and higher excited states, the

injection stripping foil is located at the downstream end of the injection dipole with

the field of subsequent dipole magnet 2.4 kG. The fringe field of the injection dipole

is shaped so that stripped electrons spiral down to where they can be easily collected.

With this design, reduction of this type of loss below 10−5 is expected.

1.2.2 Imperfection resonance

Machine resonances are a fundamental source of beam halo in circular accelerators.

Therefore we need to select carefully the working point and appropriate correction

schemes to reduce the beam loss to a 10−3 level or lower.

As an example of calculation by SNS design group, Figure 1.6 and 1.7 show

resulting beam halos for two different working points of SNS [3]. Figure 1.6 shows

the case of crossing several imperfection resonances, while Figure 1.7 shows no major

imperfection resonances for working point (6.23, 6.20), and this working point was

finally chosen to be the operation point of SNS.

1.2.3 Space-charge effects

After correcting imperfection resonances, most of the beam losses are associated with

space-charge effects. Space-charge driven resonances could be an important source

of halo formation as shown in Figure 1.6. The choice of working point should be

done by taking such resonances into account. In general, space-charge forces can be

alleviated by longitudinal manipulation (double RF, barrier cavity, etc) to enhance

bunching factor. Increasing injection energy and injection painting can also help to

alleviate space charge forces.

12 1. Introduction

0.00022 0.00024 0.00026 0.00028Total emittance pi m rad

5

10

15

20

25

30

%ofpartcilesoutside

Figure 1.6: Beam halo at the end of injection for (νx, νy) =

(6.4, 6.3): Blue color shows the halo due to space

charge only; Red color shows the halo due to system-

atic and random magnet field errors of magnitude 10−4,

sextupoles and quadrupole fringe fields; Yellow color

shows an additional effect of x,y misalignment of 0.5

mm and magnet tilt of 1 mrad.


0.00022 0.00023 0.00024 0.00025 0.00026 0.00027 0.00028Total emittance pi m rad

0.25

0.5

0.75

1

1.25

1.5

1.75

2

%ofpartcilesoutside

Figure 1.7: Beam halo at the end of injection for (νx, νy) =

(6.23, 6.20): Blue color shows the halo due to space

charge only; Yellow color shows halo when magnet er-

rors of expected magnitude are included.

14 1. Introduction

1.2.4 Coherent instabilities

All high intensity accelerators encounter coherent instabilities, which can strongly

limit the operation of a machine. The instability comes from the complex interac-

tion between the beam and its surrounding. Different from the usual studies of wake

fields, the electron-proton instability is generally a two-stream instability. In the elec-

tron proton instability, the typical electron bunch length can be of the same order

or even shorter than the wavelength of the instability driving force, therefore requir-

ing consideration of a bunch structure. However, the length of proton bunches are

usually much larger than the wavelength of instability driving force, which allows us

to consider them locally as a coasting beam. In another word, bunched beam struc-

ture could support oscillation modes within the bunch similar to those of a coasting

beam. However, the bunch length sets the resonance condition for the standing wave,

and moreover, there can be coupled bunch modes which are characterized by phase

between the oscillations from bunch to bunch.

The natural stabilizing mechanism against collective instabilities is the synchrotron

or betatron frequency spread of particles in the beam. This stabilization is known

as Landau damping. The spread in frequencies of the beam can come from several

sources, such as the dependence of the betatron frequencies on the energy of the

particles and the nonlinearities in the focusing system. Longitudinally, the source

of frequency spread depends on whether the beam is bunched or unbunched. For

bunched beams, a spread of synchrotron frequency can be induced by nonlineari-

ties in the rf focusing voltage. For unbunched beams, the spread comes from the

dependence of the revolution frequency on the particle energy. If a large spread of

frequencies provides a fast decay of center of mass, the instability is suppressed.


1.2.5 Contents of chapters

To understand the beam loss and enhance the machine performance, one needs to

build an accurate model to model the machine and beam dynamics. The online model

and Objective Ring Beam Injection and Tracking (ORBIT) code serve these two pur-

poses respectively. There are two corresponding subjects in this thesis: Chapter 2

presents the calibration of the linear optics of the linear model, which is motivated

by the betatron tune discrepancy between online model and BPM turn-by-turn mea-

surement. By using the Orbit Response Matrix Method (ORM), the major causes of

the discrepancy has been explored, and the resulted correction is implemented in the

online model. Detail analysis is discussed in this chapter.

Chapter 3 discussed the observations of electron-proton instability at the SNS

accumulator ring, which is produced mainly by manipulating the RF cavities and

accumulated beam intensity. The electron-proton instability is a potential obstacle for

the future SNS power upgrade and it may emerge in the normal neutron production.

A feedback system was designed to kill this coherent instability and is in progress.

Data analysis of the experiments and some electron-cloud simulation are presented.

A brief conclusion is drawn in Chapter4.

16 1. Introduction

Application of Orbit Response Matrix Method to the SNS ring 17

Chapter 2

Application of Orbit Response

Matrix Method to the SNS ring

The accumulator ring has a circumference of only 248 m, but it will hold 1.5 ×

1014 protons after accumulation of more than 1000 turns. Although all magnets

were designed to accommodate expected beam size of 4 MW beam power, beam

loss could prevent SNS from delivering its full production rate. The particle loss

can cause radiation damage to accelerator components, while consequent radiation

activation causes difficulties with machine maintenance. The loss requirement on the

accumulator ring is: an uncontrolled loss faction within 2 × 10−4 and a controlled

loss fraction within 1 × 10−3. Therefore, particular attention to the beam dynamics

issues in the ring was made during the design stage and is further needed for the real

machine operation.

It is important to understand the causes of the beam loss in order to reduce it.

The choice of operating point of tunes is critical for a high intensity ring. The hor-

izontal and vertical tune must be selected away from strong, low order resonances

to avoid emittance growth and beam loss. The tunes are determined by the lattice,

18 2. Application of Orbit Response Matrix Method to the SNS ring

especially the ring quadrupoles. However, the measured operating point in the SNS

ring displayed a large discrepancy of 0.2 from the predicted tune. Therefore we could

not get the desired working point by using the lattice model to set the machine. Al-

though the tune point (6.22, 6.20) was reached by manually tuning quadrupoles and

was subsequently used for the other production, it does not assure the desired optics

such as the betatron function and limits the flexibility to be adjusted. Therefore,

understanding of the real machine is necessary: Is the lattice model reliable by com-

paring with the measured optics? Are there any large lattice distortion sources such

as major magnet imperfection?

In this chapter, we will introduce a powerful optics calibration tool: the orbit re-

sponse matrix (ORM) method, along with its implementation in the program LOCO.

We use this method to solve the optics discrepancy and successfully calibrate the

linear model. Several ORM measurements were carried out and we will pick up the

highest quality data set to discuss the details. Using the ORM method, we deter-

mined that the current/field errors in the six main quadrupole power supplies were

the major source of the tune discrepancy between measurement and model. By im-

plementing these quadrupole gradient errors into the ring model, we can bring the

machine to a desired working point within a tolerable discrepancy range 0.003 ∼ 0.01

[7]. Also, the confirmed asymmetrical beta function suggests an optimization for the

base-tune lattice. Section 2.1 introduces orbit response matrix method and discusses

its implementation in the ”LOCO” code [8]. Section 2.2 shows the result of the ap-

plication of ORM method on the SNS ring and discusses the practical and potential

optics improvement.

2.1 Orbit Response Matrix Method 19

2.1 Orbit Response Matrix Method

2.1.1 The Perturbed Orbit and Green’s Function

An ideal reference closed orbit with perfect magnets passes through the center of most

magnets. The closed orbit is perturbed by dipole field errors, which may arise from

errors in dipole length, power supply, or quadrupole misalignment.

Consider a single thin dipole field error at a location s = s0 with a kick-angle

θ = δBdt/Bρ, where δBdt is the integrated dipole field error and Bρ = p0/e is the

momentum rigidity of the beam. The closed orbit condition is

M

y0

y′0

=

y0

y′0 − θ

(2.1)

where M is the one-turn transfer matrix of Equation (2.1) for an ideal accelerator.

The resulting new closed orbit at s0 is

y0 =β0θ

2 sinπνcosπν y′0 =

θ

2 sinπν(sinπν − α0 cos πν) (2.2)

where ν is betatron tune and α0, β0 are the values of betatron amplitude at kick dipole

location s0.

With the transfer matrix

M(s|s0) =

√β 0

− α√β

1√β

cosψ sinψ

− sinψ cosψ

1√β0

0

α0√β0

√β0

(2.3)

the new closed orbit in the accelerator becomes y(s)

y′(s)

co

= M(s|s0)

y0

y′0

(2.4)

or

yco(s) = G(s, s0)θs0 (2.5)


where

G(s, s0) =

√β(s)β(s0)

2 sinπνcos(πν − |ψ(s)− ψ(s0)|) (2.6)

is the Green’s function of Hill’s equation.

The Green’s function depends on the betatron function and phase advance ψ(s)−

ψ(s0) , which is actually determined by the quadrupole fields. Equation 2.6 shows

that, if we use a thin dipole kicker to artificially produce a dipole field kick θ(s0), and

measure the closed orbit with BPM at s, then the Green’s function can be calculated

numerically using yco/θ(s0). This reverse procedure can be implemented for n dipole

kickers and m BPMs, to generate an m× n matrix of Green’s functions: This is the

so-called Orbit Response Matrix (ORM) [10]. Measurement of the ORM can be then

used to model the accelerator.

2.1.2 Algorithm of ORM Method

As stated in Sec. 2.1.1, the orbit response to a small kick is the product of the kicker

strength and Green functions between the two points. The orbit response matrix

(ORM), as indicated by the name, is a matrix map composed of Green functions

between any two pairs of dipole corrector and BPM, which is Mm×n in the following

matrix expression:

Ym×n = Mm×nΘn×n (2.7)

where an element yi,j of Y matrix is the change of closed orbit at ith BPM due to a

small increase of kick at jth dipole corrector 4θj.

Due to the inevitable rolls of magnets and BPMs, a one-plane kick often triggers

cross-plane orbit deviation. Therefore the horizontal and vertical planes of the ORM

are often coupled as a result of this transverse coupling, i.e., Mxz and Mzx are nonzero

matrices:


M =

Mxx Mxz

Mzx Mzz

(2.8)

where the two diagonal blocks represent orbit responses due to in-plane kick and the

two off-diagonal blocks represent orbit responses due to cross-plane kick. The coupled

ORM can be derived from the 4D transfer matrix with closed orbit condition:

T

4x0

4x′04z0

4z′0

=

4x0

4x′0 − θ

4z0

4z′0

(2.9)

where T is the 4D one-turn transfer matrix at the location of kick θ. Solution of

Equation (2.9) gives the change of phase space coordinates per kick angle. Once

it is propagated to the other locations and the other plane, the ORM elements are

determined by

Mi,j =4yi4θj

(2.10)

One can also use a computing program, e.g. MAD or AT toolbox (which underlies

the tracking code for LOCO) [1], to calculate the closed orbit before and after a kick.

This includes the quadrupole effect of the sextupoles and the dipole effect of the

quadrupoles due to off-center beam. Since the calculation requires evaluating the

model once per kicker, it requires significantly longer time than the transfer matrix

method, especially in Jacobian matrix calculation.

Once the model and measured ORMs are obtained, we can minimize the chi-square

difference between these two matrices:

χ2 =∑i,j

(Gmodel,ij −Gmeas,ij)2

σ2i

≡∑k=i,j

E2k (2.11)


where σi is the measured noise level of the ith BPM, and ~Ek is the error vector, whose

length minimization is equivalent to minimizing χ2.

Assume Kl are model parameters varied to fit the response matrix. Then the

purpose of adjusting the Kl is to generate a 4 ~E such that [8]

~E +4 ~E = ~E +∂ ~E

∂Kl

4Kl = 0 (2.12)

Depending on the fit parameter, one can use an analytical calculation of ∂ ~E/∂Kl or

a lattice model to numerically calculate the derivative.

The model fitting commits to finding the 4Kl that best cancel the difference

between the measured and model response matrices.

4Kl = −

(∂ ~E

∂Kl

)−1

~E (2.13)

where (∂ ~E/∂Kl)−1 is the inverted matrix of ∂ ~E/∂Kl. Since there are far more data

points in the response matrix than fitting parameters, the equation is over-constrained

and Single Value Decomposition (SVD) [8] is used to invert the matrix ∂ ~E/∂Kl.

2.1.3 Dispersion Effect on Closed Orbit

As demonstrated by Equations (2.5) and (2.6), a dipole-kick θj at position sj will

change the closed orbit by G(si, sj)θj at location si. The circumference will therefore

be changed by ∆C = D(sj)θj. For ORM measurement, all dipole correctors are

fired and we need to evaluate the effect of the circumference change on the ORM

calculation:

• Constant momentum: Due to synchrotron radiation, we need to consider elec-

tron accelerator and proton accelerator separately. For an electron accelerator,

the RF cavities need to be on to compensate the energy loss. The change of


revolution period is ∆T = ∆C/βc = D(sj)θj/βc at a constant velocity βc.

Therefore to keep a constant momentum, the RF frequency must be synchro-

nized according to ∆f/f = −∆T/T . With this adjustment, the beam motion

will be on-momentum and the response matrix is G(si, sj). For proton ac-

celerator, the RF cavity is normally off during the ORM measurement. The

synchrotron radiation is negligible and therefore the beam is ”on-momentum”.

Hence the response matrix is also G(si, sj).

• Constant path length: For an electron accelerator, if the RF frequency is con-

stant during the ORM measurement, then the path length is constant. Since

the path length was changed by the dipole kick as mentioned previously, an

equivalent off-momentum variable with respect to the new closed orbit becomes

”δ” = 1/αc ×∆C/C0. The corresponding closed orbit is

xi = G(si, sj)θj +D(si)δ = G(si, sj) + (D(si)D(sj)

2πRαc)θj (2.14)

where αc is the momentum compaction factor, D(s) is the dispersion function,

and R is the mean radius of the accelerator. Therefore the response matrix

become G(si, sj) +D(si)D(sj)

2πRαc.

2.1.4 LOCO Code and Its Algorithm

The primary tool that we use to fit the optics is the Linear Optics from Closed Orbit

(LOCO) code[1, 8, 11]. The original FORTRAN code was written to correct the

optics of the NSLS X-Ray ring, and was applied soon after with the ALS optics. The

code was rewritten in MATLAB, including a user-friendly interface and many fitting

options. The MATLAB version of LOCO uses Accelerator Toolbox (AT) [12] as its

underlying tracking code, which was also developed in MATLAB.


Table 2.1: Response matrix calculation methods in LOCO [1]

Linear option ”Constant Momentum” ”Constant Path Length”

(default in LOCO)

Computing Base T T + Dispersion

Fit Dispersion? No Include as a column

Horizontal ORM Equation (2.16) Equation (2.15)

I Calculation of Model Response Matrix

The basic function of the LOCO code is to calculate a parameterized model response

matrix and fit it to a measured matrix. Due to calculation time considerations, a linear

approximation method is adopted to compute the model response matrix. It is based

on the numerically obtained 4D transfer matrices at each BPM and corrector (see

Equation (2.1)), the model dispersion function, and the model momentum compaction

factor. LOCO also has an option of nonlinear computing that iteratively searches for

a closed orbit in the presence of a corrector magnet kick. It is slower but includes

the nonlinear effects due to sextupoles and other nonlinear elements. However, the

time spent on a fully nonlinear calculation is not necessary for many accelerators on

the first couple of iterations. The following discussion will focus on the computing

options based on linear method.

There are two linear options to choose in LOCO [1]:

Mmodij = Mp0

ij +DiDj

αcL0

(2.15)

Mmodij = Mp0

ij +4pjpDmeasi (2.16)


In the LOCO code, the two options are called ”Constant Path Length” and ”Con-

stant Momentum”. However, these names worth to be questioned due to the explana-

tion in Section 2.1.3. For our convenience to keep the consistency with other papers

of LOCO, I will refer to these options using these names with quotation marks. In

Table 2.1, T is the Green’s function for an on-momentum particle. As explained in

Section 2.1.3, the additional term Dj/αcL0 is used for electron rings when the RF

cavity frequency is maintained. But for proton rings such as SNS, the RF cavities are

turned off during the ORM measurement. Therefore the LOCO default ”Constant

Path Length” method is not appropriate in this case.

The LOCO ”Constant Momentum” method introduces fitting of energy changes

at the correctors. By adding an additional term4pjpDmeasi to the orbit calculation, the

distortion of quadrupole gradient fits from unmodeled dispersion due to orbit errors

is hopefully to be eliminated. By introducing the measured dispersion into model,

although it sound a little odd, the method was chosen since changing the measured

response matrix causes other logical problems. Using this option, even with a perfect

fit of all LOCO parameters, the model and measured dispersion will not be forced to

match, i.e, it is designed to zero the weighted dispersion in the least squares algorithm.

In the followed data analysis, I will evaluate the magnitude of this additional term

and compare the outcomes of the calculation with and without the additional term.

Therefore in a practical way, for electron rings with the ”Constant Path Length”

method, LOCO also has an option to explicitly include the dispersion as a column in

the response matrix. Two benefits are brought by this method: additional weight is

added to the dispersion correction, and it constrains the BPM gain and corrector’s

calibration factors since the measurement of dispersion does not use steering magnets.

For our proton ring case, physically, as stated in Section 2.1.3, the ORM of SNS

uses Green’s function only, neither Equation (2.16) nor Equation (2.15). It can be

practically achieved in LOCO by setting the measured dispersion to be zero in the


”Constant Momentum” method.

II Minimization Algorithm

In general, to solve a nonlinear least squares problem, the common method is to

minimize the merit function:

f(~p) = χ2 =∑

[yi − y(xi; ~p)]2 (2.17)

where ~p is a vector of fitting parameters, (xi, yi) are measured data and y(x; ~p)is a

nonlinear model function. The Jacobian matrix J is defined as:

Jij =∂ri∂pj

(2.18)

where ri = yi− y(xi; p) is the component of the residual vector ~r with i = 1, 2, · · · , N

and N is the number of data points. In the ORM chi-square fitting, the change in the

response matrix due to the quadrupole gradient is not always linear. Therefore the

fitting must be iterated several times to converge to the best solution. The Gauss-

Newton method is the first minimization algorithm adopted by the original LOCO.

In this method, the solution propagates toward the minimum at each iteration by

4~p, which is determined by

JTJ4~p = −JT ~r0 (2.19)

where ~r0 is the residual vector of the previous iteration. This is essentially the method

adopted in LOCO, which has a similar specific Equation 2.13. The use of the JTJ

format instead of J is much faster using SVD on the former matrix, while the latter

has tens of times more rows.

In the real world, two parameters can be deeply coupled such that their contri-

butions to the merit function are very difficult to separate. As an extreme case,

it is impossible to determine the two fitting parameters p1 and p2 in χ2 =∑

[yi −


y(xi, p1 − p2)]2 because the merit function has no dependence on p1 + p2. The cor-

responding columns of the Jacobian matrix for the two parameters differ by only a

scaling constant, which means that the Jacobian matrix is rank deficient. In a less

severe case, the merit function may have weak dependence on p1 +p2 so that it can be

determined in principle. However, in such a case, noise in the experimental data con-

sequently have large error bar. In LOCO, coupling of adjacent quadrupole gradient

parameters leads to the similar behavior as the above example. In the real machine,

if two quadrupoles are placed next to each other with little space in between, their

perturbation to the linear optics behaves this way [13].

The integrated gradient of two magnets can be fitted accurately, but the individual

contributions are hard to distinguish. Detailed analysis shows that the betatron phase

advances between two quadrupoles can be used to determine their coupling strength.

The correlation coefficient that reflects the coupling between fitting parameters

can be written as

r12 =vT1 v2

‖ v1 ‖‖ v2 ‖(2.20)

where v1,2 are corresponding columns of the parameters p1,2 and ‖ • ‖ represents the

2-norm of its argument.

Due to the coupling between quadrupoles, some patterns of change of quadrupole

gradient are less restricted in LOCO. If these patterns form a null space in the param-

eter space, i.e., correspond to singular values considerably smaller than others, then

they can be removed by the proper selection of singular values. However, the less

restrictive patterns are rarely orthogonal but the patterns through SVD have to be

orthogonal. Therefore, the singular value spectrum is a smooth curve without a clear

cut. Completely removing any mode is a loss of information that could reduce the

accuracy of fitting by some level. Consequently, it takes a lot work to find an optimal

threshold. Unfortunately, some less restrictive patterns still leak into the solution


0 50 100 150 200 2500

0.2

0.4

0.6

0.8

1

s (m)

,

corr. coef. x /

y/

Figure 2.1: Correlation coefficient r and phase advance between

neighboring quadrupoles of SPEAR3 [1]: Stronger cor-

relation is a result of smaller horizontal phase advance

(mod π), which implies that, two quadrupoles can be

physically set apart but can have strong correlation if

the horizontal phase advance between them is a mul-

tiple of π.

even when a seemingly optimal threshold is applied. And SOLEIL’s experience [14]

shows the difficulty of obtaining good accuracy while keeping a quadrupole’s change

reasonably low by merely selecting the singular values.

The intrinsic degeneracy due to coupling between fitting parameters causes large

excursions in less restrictive directions. There is no ”unique” solution from the fitting.

Any solution with χ2 less than a certain amount from the global minimum is valid and

equivalent, and in principle should give the same lattice. In reality, there are some

”reasonable” solutions better than others. For example, with a smaller change of


quadrupole, the result may be more reliable. As a result, a constraint fitting method

is introduced in the next section to conquer this problem.

2.1.5 Constraints in LOCO Fitting

If unrestrictive excursions occur due to the coupling of fitting parameters K, it is

natural to put a penalty on such excursions. A penalty term can be added to the

merit function. A penalty can be devised by grouping the coupled fitting parameters,

figuring out if they are correlated positively or negatively, and then trying to minimize

the group. As an example, if 4K(i) and 4K(i+ 1) are coupled positively, then the

penalty term (4K(i) − 4K(i + 1))/σK can be added to the merit function with a

weight factor of 2. However, the configuration of these penalty terms is not easy. A

simpler approach is to put a penalty on each 4K directly:

χ2 =∑i,j

(Mmodel,ij −Mmeas,ij)2

σ2i

+1

σ24K

∑k

w2k4K2

k (2.21)

where σ4K is an overall normalization constant and w2k are individual weighting fac-

tors to constrain the corresponding quadrupoles, which should be adjusted according

to a trial performance. Once the appropriate set of weighting factors is found for a

lattice, there is usually no need to change it for other measurements. As a compar-

ison, removing singular values is equivalent to putting an infinite penalty weight on

the corresponding pattern. Therefore, the more ”gentle” constraint method is more

effective than the extreme approach of singular value selection.

Though the minimization algorithm has been modified, the input and output

remain the same as the original LOCO. The additional constraint terms change the

solution to the linear problem of each iteration. The global minimum of the original

problem does not change since only the gradient between successive iterations are

constrained with a cost. Therefore, the only change on the minimization algorithm


Figure 2.2: Converging path with or without constraints. Solid:

no constraints; Dash: with constraints.

is the path of converge. Figure 2.2 shows an illustration of the converging path [1],

point 0 is the initially guessed solution, M is the global minimum. Within the ellipse

that represents noise level, there is a sea of equivalent solutions. The unconstrained

path (solid arrows) takes large excursions but move quickly, while the constrained

path (dash arrows) goes more straightforward but slows down when enters the sea of

noise.

2.2 Application of ORM to SNS Accumulator Ring

The standard set of LOCO input data is composed of: a measured orbit response

matrix, a measured dispersion function and measured BPM resolution. Since SNS

uses a Java based hierarchy, XAL [15], as its application programming infrastructure,

instead of the Matlab based MML (Matlab Middle Layer) [16] that is used by SLAC,

the MML based LOCO code can not be used to directly measure and export the

lattice. Therefore, a Jython script was written to carry out the ORM measurement

2.2 Application of ORM to SNS Accumulator Ring 31

graphically. A real-time lattice can be exported in MAD8 format and converted by

Accelerator Toolbox [12] to an acceptable format for LOCO. The analysis of single

BPM data is shown in subsection 2.2.1.

2.2.1 Measurement of Response Matrix

The ORM measurement depends highly on the beam and machine status: Is the

single mini pulse cleanly chopped? Does the beam energy jitter more than 1% from

linac? Is the pulse to pulse variation large or not? Are the diagnostic instruments

and magnets performing well? Does the orbit response from kicks exceed the BPM

linear response range of ±20cm?

Based on these concerns, the general procedure of ORM measurement includes:

• Set up single mini pulse injection, adjust the gate and pulse width to clean the

satellite residue particles;

• Store beam for 500 turns, turn off ring RF cavities;

• Make sure that the BPM timing is right. Manually calibrate all ring BPM gains

and check the average start and end turn;

• Flatten the ring orbit, adjust the injection kickers if the betatron oscillation is

greater than 20cm or less than 5cm on one side.

• Activate sextupoles to eliminate chromaticity. But measurements without sex-

upoles are also needed for comparison.

• Activate skew quadrupoles to eliminate linear coupling. But measurement with-

out skew quadrupoles should also be carried out.

• Document all the settings as well as the live lattice.


• Launch applications ”beam-trigger.py” to continuously trigger beam and ”corrector-

probe-gui.py” to record BPM Turn-By-Turn data when every corrector’s field

is set to change by 0.004, 0.002, 0, -0.002, -0.004 Tesla.

• Use a pair of BPM after the last SCL cavity to measure the dispersion function

D = dx0dp/p0

= β2E× dx0dE

with visible change of energy obtained by varying phase

settings of the last SCL cavity.

As a preparation for LOCO input files, preliminary data analysis is necessary:

• Sinusoidal fitting of BPM turn-by-turn data. The fractional betatron tune and

closed orbit at that BPM are extracted from the fitting. Malfunctioning BPMs

and correctors are excluded.

• Linear Fitting of corrector kick angles versus closed orbit. The slopes obtained

are the elements of the response matrix.

• Linear Fitting of beam energy versus closed orbit. Dispersion is calculated by

β2E × slope.

• Prepare the lattice for the LOCO model. Integrate divided quadrupoles, sub-

stitute their names according to power supplies.

To confirm the data quality and LOCO fitting results, three response matrix mea-

surements have been implemented at different periods. Table 2.2 lists three ORM

measurements that have been done at three different times. To our surprise, although

the third data set (Dec 2009) implemented quadrupole power supply corrections and

thus significantly decreased initial tune discrepancy, it has a larger initial χ2/D.O.F

than the uncorrected cases (June 2008). If we look into the fitting parameter sensi-

tivities on χ2 after one iteration, for the uncorrected quadrupole case ”June 2008”:


Table 2.2: Comparison of measurement conditions

Experiment Jan 2008 June 2008 Dec 2009∗

Beam Energy 845 MeV 875 MeV 928 MeV

Number of bi-direction BPM 38 42 38

Horizontal Correctors 21 24 24

Vertical Correctors 26 28 26

Sextupole for chromaticity on on on

Skew Quadrupole for linear coupling on on on

Dispersion Measurement No Yes Yes

Initial χ2total/D.O.F 297.16 110.76 309.93

Main Quadrupole corrected? No No Yes

Measured Betatron Tune (6.222, 6.194) (6.240, 6.198) (6.227, 6.198)

Initial Model Tune (6.389, 6.343) (6.450, 6.348) (6.228, 6.218)∗

Initial Tune Discrepancy (0.167, 0.149) (0.210, 0.150) (0.001, 0.020)

Initial(βmaxx , βmaxy ) [m] (27, 14) (27, 14) (29, 13.5)


• ∆χ2(By changes of BPMs) = 37.6

• ∆χ2(By changes of correctors) = 96.9

• ∆χ2(By changes of quadrupoles) = 67.9

for the corrected quadrupole case ”Dec 2009”:

• ∆χ2(By changes of BPMs) = 56.0

• ∆χ2(By changes of correctors) = 131.1

• ∆χ2(By changes of quadrupoles) = 0.6

The comparison of the fitting parameter sensitivities gives a hint to the mystery of

larger χ2 for the corrected lattice, namely that the mystery is related to BPMs and

correctors. We pick the data set of ”June 2008” for our main discussion.

2.2.2 Uncover Quadrupole Gradient Errors

The 52 main quadrupoles are grouped to six power supplies in the machine. The

large discrepancies between model and measured tunes suggest errors in the main

quadrupole gradients, or, the quadrupole power supply problems. Therefore, it is

logical to group quadrupole gradients in the fitting as six independent fit parameters.

Quadrupoles strung together on a single power supply are varied together. In this

way, the results of fitting quadrupole gradients are more realistic.

As we picked ”June 2008” data set for discussion at this stage, Table 2.3 gives a

summary of the cases for comparison. The ”Green function only” method is physically

correct as explained in Section 2.1.3; the ”Constant Momentum” method used in

LOCO includes measured dispersion in the calculation to eliminate the systematic

error of dispersion; the ”Constant Path Length” method default in LOCO is actually

for electron accelerators, but we also include it into our discussion.


−40 −30 −20 −10 0 10 20 30 400

10

20

30

40

50

60

Error in Units of Standard Deviations

Num

ber

of P

oint

s (4

368

tota

l poi

nts)

Histogram: (Mmeas − Mmodel) / σbpm

Error before fittingError after fitting

Figure 2.3: Example of a good fit: the x axis is the error in the unit

of standard deviation, and the y axis shows the error

distribution of the thousands of points in the matrix

in a histogram format. Therefore, the more the points

close to zero, the better the agreement between model

and measurement. After the fitting (black color), the

error is reduced significantly from before (red color).


Table 2.3: Fitting parameters and methods for ”June 2008” data

set

Fitting Parameters Name of method Calculation formula

6 grouped gradients of

52 quadrupoles

”Green function only” Mmodij = Gij

84 BPM Gains and

couplings

”Constant Momen-

tum”(fit δp/p)

Mmodij = Gij +

4pjpDmeasi

52 corrector kick

strengths

”Constant Path

Length”

Mmodij = Gij +

DiDj

αcL0

Although we knew that the large tune discrepancy implies quadrupole gradient

error, to our surprise, the BPM and dipole correctors also played an important role.

This can be seen in the logarithmic magnitude of change in χ2 w.r.t the type of fitting

parameters, i.e. log10∂χ2

∂p, shown in Table 2.4.

The sensitivity of χ2 to dipole kick strengths is the second largest and nearly

comparable to the quadrupole gradients, which indicates a large error in dipole kickers.

This was confirmed by the other beam studies. The fitting results will be discussed

in order of sensitivity from large to small.

Figure 2.4 shows that the fit χ2 of the ”Green function only” method (Red line)

converges at ∼ 2.8, while the ”Constant Momentum” method (blue line) converges a

little lower at ∼ 2.3 due to the additional fitting parameters. The large fluctuations

in the ”Constant Path Length” method confirmed the inappropriateness because of

the measurement method. Quantitatively, Gi,j has a magnitude of ∼ 10m, while

the additional termDiDj

αcL0for ”Constant Path Length” max out at ∼ 1.9m, which is

comparable to the Green’s function. The magnitude of the ”fitted” term4pjpDmeasi


1 2 3 4 5 60.5

1

1.5

2

2.5

3

3.5

Main quadrupole power supplies

Per

cent

age

(%)

Before fitting, ν is (6.424, 6.322)

ν(6.243,6.20)ν(6.244, 6.20)ν(6.25, 6.202)

0 5 10 15 20 2510

0

101

102

103

Iteration

χ2 /D.O

.F

Green Function onlyFit δp/p at kickersConstant Path Length

Figure 2.4: Comparison of fitted cases (follow-up to Table 2.3):

Plot on the left shows percentage errors of grouped

quadrupole gradients uncovered from fit. The graph

on the right is the comparison of chi2 fitting for the

three cases. The LOCO ”Constant Path Length” cor-

responds to the green color. The χ2 fluctuates a lot

before the 10th iteration.


Table 2.4: The weights of changes of fit parameters

Type of fit parameter (p) magnitude of log10∂χ2

∂p

BPM gain factor 1

BPM coupling 1

Dipole kick strength 4

Dipole kick coupling −2

quadrupole gradients 5.5

Table 2.5: Gradient errors based on data of June 2008

main power supplies QH10a13 QH02a08 QH04a06

δB 3.13% 1.60% 0.82%

main power supplies QV11a12 QV01a09 QV03a05a07

δB 3.08% 1.58% 1.24%

for ”Constant Momentum” is only ∼ 10−4m and therefore has insignificant effect on

the ORM calculation.

The percentage error of the quadrupole gradient was calculated asKoriginal/Kfitted−

1. We dialed this error from the ”Green’s function only” fitting into the model by

changing the quadrupole field Bset to Bnew = Bset · (1 + δ) with δ from Table 2.5.

The quadrupole percentage errors extracted from the ”Green’s function only” and

”Constant Momentum” methods overall agree well, as shown in Figure 2.4, except

for quadrupole power supplies QH02a08 and QH04a06: the ”Constant Momentum”

method gives 1.54% and 0.84%, respectively.

After the fitting, νmodel becomes (6.244, 6.200), compared with the original model


0 5 10 15 20 25−1

−0.5

0

0.5

1

1.5

2x 10

−4 δ p/p at horizontal kickers

number of hor. kickers

fitte

d δ

p/p

afte

r 4

itera

tions

0 10 20 30 40 50−1

0

1

2

3

4Measured horizontal dispersion

number of hor. BPMsm

easu

red

hor.

dis

pers

ion

(m)

0 5 10 15 20 25 30−2

−1

0

1

2

3

4x 10

−4 δ p/p at vertical kickers

number of ver. kickers

fitte

d δ

p/p

afte

r 4

itera

tions

0 10 20 30 40 50−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15Measured vertical dispersion

number of ver. BPMs

mea

sure

d ve

r. d

ispe

rsio

n (m

)

Figure 2.5: Energy shift with ”Constant Momentum” method:

The magnitude of energy shift is only 10−4. There-

fore it does not have much effect on the fitting result

as shown in Figure 2.4.


value (6.424, 6.322), while νmeasure is (6.240, 6.198). This is a significant improvement

for the calibration of the accelerator model and was confirmed after implementation

in the ring.

In the model, the corrected BPM and steering magnet parameters are applied

to the model response matrix to best match the measurement using the following

equations:

xmeas

ymeas

=

gx,loco cx,loco

cy,loco gy,loco

xmodel

ymodel

(2.22)

δx,actual

δy,actual

=

cos θ

sin θ

∗ gain ∗ δx,meas (2.23)

Figures 2.6 and 2.7 present the fitting results for the BPMs and correctors. For

confirmation purpose, two independent data sets are plotted in the same graph. One

may notice that for the BPM gain, the two data sets overlap oppositely from horizontal

plane to vertical plane, i.e, if red points sit above blue points in the horizontal plane,

then they will be below in the vertical plane. This is caused by different linear

coupling values for these two independent measurements. Actually, the ”June 2008”

measurement may have had a larger linear coupling, so that the vertical kickers show

a coupling error up to 0.2 (sin(11.46) ' 0.2). There is a large discount ∼ 30% on

kicker strengths. This explains the surprising sensitivity of χ2 w.r.t kickers. The

BPMs have a gain of ∼ 80%, while the pattern should be related to dispersion,

either caused by energy mismatch in the measurement or by longitudinal position

misalignment in the model.

Due to the residual tune discrepancy, we also tried to fit the individual quadrupoles

on the basis of previous correction of quadrupole families. Due to the possibility of

degeneracy of fitting parameters, we have an option to fit BPM/corrector gain and

coupling factors together with individual quadrupoles. The comparison of the two fits


0 50 100 150 200 250−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Distance (m)

defa

ult g

ain=

1, c

oupl

ing=

0

Calibration for horizontal BPM

0 50 100 150 200 250−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Distance (m)

defa

ult g

ain=

1, c

oupl

ing=

0

Calibration for vertical BPM

Analysis from Dec 2009 dataAnalysis from June 2008 Data

Figure 2.6: BPM gain factors and coupling: Default BPM gain is

1. Default BPM coupling is 0. So the points close

to 1 represent BPM gain and points close to 0 stand

for the BPM coupling. The left and right plot show

fitted horizontal and vertical BPM calibration values

respectively. Blue points correspond to ”Dec 2009”

data and red points correspond to ”June 2008” data

set.


0 50 100 150 200 250−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Distance (m)

defa

ult g

ain=

1, c

oupl

ing=

0

Calibration for horizontal kicker

0 50 100 150 200 2500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Distance (m)

defa

ult g

ain=

1, c

oupl

ing=

0

Calibration for vertical kicker

Analysis from Dec 2009 dataAnalysis from June 2008 Data

Figure 2.7: Kicker strengths and coupling: Kicker strengths are

normalized to 1. Default coupling is 0. So the points

close to 1 represent kicker normalized strength and

points close to 0 stand for the kicker coupling. The

left and right plot show fitted horizontal and vertical

kicker calibration values respectively. Blue points cor-

respond to ”Dec 2009” data and red points correspond

to ”June 2008” data set


0 10 20 30 40 50 60−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

Main quadrupole power supplies

Per

cent

age

(%)

Fit 52 Quads OnlyFit Quad+BPM+CM)

1 2 3 4 51

2

3

4

5

6

7

8

9

Iteration

χ2 /D.O

.F

Fit 52 Quads OnlyFit Quad+BPM+CM)

Figure 2.8: Fittings with two different fitting parameter sets: One

set (red color) only includes 52 quad strengths, while

the other (blue color) also includes BPM and correc-

tor parameters as additions. The previous corrections

of quad group fitting were implemented initially. The

two subfigures represent the additional quad errors and

iterative χ2, respectively. As shown on the left, both

cases uncover that the quad group QV03a05a07 (point

33 to 44) has an additional error ∼ 0.6% that was not

discovered. Group QH04a06 (point 45 to 52) also has

an undiscovered error.


with and without BPM/correctors is presented in Figure 2.8. The points in the sub-

plot on the left is aligned in order of quadrupole power supplies: QV11a12, QH10a13,

QV01a09, QH02a08, QV03a05a07 and QH04a06. Each power supply provides current

to eight quadrupoles except for QV03a05a07, which supplies twelve quadrupoles. The

finally obtained χ2s for the two fits differ by 20% due to approximately 2% BPM gain

changes, which suggests that the fit with BPM/corrector (blue color) is a more reliable

one. The percentage error plot on the left suggests that the error of QV03A05A07

and QH04a06 has not been completely found in the pervious power supply fit, which

should be a result of the limitation of searching algorithm.

2.3 Conclusion

The uncalibrated lattice model of SNS accumulator ring contained significant optics

discrepancies between prediction and measurement. As a consequence, it was very

difficult to set the desired lattice using the model calculation. In the real operation,

due to that difficulty, we use an empirical base-tune method to set the working point

(6.23,6.20); the magnet tunes were all scaled on the basis of this empirical setting.

Therefore, although the working point is inherited from the base-tune lattice, unop-

timized optics were also inherited, such as the beta beating. Although this was not

a big problem at the current stage of operation, the asymmetry could be an obstacle

for the future power ramping due to the shrinkage of dynamic aperture.

We use the orbit response matrix (ORM) method to solve the optics discrepancies

and to calibrate linear models. Several ORM measurements and analysis were carried

out in the past few years. Since the ORM application code LOCO was generated for

the electron rings, we noticed the difference between proton and electron ring and

hence modified its ORM calculation method to better adopt the proton ring. The

quadrupole gradient errors in quadrupole power supplies were determined and con-

2.3 Conclusion 45

firmed by measurement, bringing the tune discrepancy between BPM measurement

and online model from 0.2 to 0.008. Hence, the desired working point can now be set

using online model.

We also tried to fit the individual quadrupole errors. However, the first result

shows degeneracy problem, which is needed to be further investigated. If allowed,

the quadrupole power supplies and dipole correctors should be directly measured and

corrected in the hardware level. And the ORM measurement should be repeated after

each long maintenance period or instrument calibrations.

Electron Cloud 47

Chapter 3

Electron Cloud

3.1 Introduction of Physics of Electron Cloud Ef-

fect

As a high intensity accelerator-based neutron source, SNS was designed to accommo-

date 1.5×1014 protons per pulse at 1 GeV kinetic energy. As the protons accumulate

in the storage ring to such a high intensity, collective beam effects emerge and cause

beam instabilities. These instabilities pose a limitation on the peak intensity and

therefore become major concerns in the SNS power-up plan. Among those observed

instabilities, the electron-proton instability (called e-p instability) is the strongest.

As its name indicates, this instability depends on the accumulation of electrons in

the vacuum chamber, which is associated with the ”electron cloud effect” (ECE).

Figure 3.1 explains the most probable mechanism of ECE development, which

involves multipactor effect that secondary electron emission in resonance with an

alternating electric field leads to exponential electron multiplication. It that can

drastically increase the electron density, and hence increase the instability rates of

coupling between trapped oscillating electrons and circulating proton beam. However,

48 3. Electron Cloud

Figure 3.1: Mechanism of electron cloud development for a long

proton bunch: Red arrow stands for electrons in the

beam gap. They are captured and oscillate inside the

beam potential well as the beam intensity increases,

released as the intensity decreases and produce sec-

ondary electrons (purple arrows) when they hit the

wall. Another source of secondary electrons is due to

the lost protons(dark blue arrow). Those protons hit

the chamber and generate electrons(green arrow). The

electrons finally gain energy in the duration of the sec-

ond half of bunch and strike the wall to produce sec-

ondary electrons and this process can be repeated.

3.1 Introduction of Physics of Electron Cloud Effect 49

this phenomenon, multipacting, plays different roles for different cases, for example,

coasting beam and bunched beam [17]. The detailed explanation is listed as follows.

For coasting beams, multipacting occurs due to proton beam instability. Electrons

accumulate in the proton potential well during beam injection. After reaching some

threshold density, unstable coupled oscillations are generated between electrons and

the proton beam.

For the bunched beam, there are two types of electron accumulation: single pass

and multi-pass. Single pass accumulation is related to the multipacting on the trailing

edge of proton beam, i.e, the second half of the bunch as mentioned in the Figure 3.1.

Consider stray electrons with zero initial kinetic energy near the wall. They oscillate

across the vacuum chamber and through the circulating proton beam due to coulomb

force. Therefore, for constant longitudinal beam density, those electrons gain zero

net energy when they reach the opposite wall. However, if the longitudinal density

is decreasing, the electrons gain energy after traveling across the chamber and thus

can hit the wall and produce secondary electrons. It is speculated that multipacting

can significantly increase the number of electrons on the beam trailing edge, if the

energy gain of electrons is above 50 eV for an aluminum wall. Moreover, if there is

a significant number of electrons present when the proton beam center passes, the

number of electrons will increase by a tremendous factor, depending on the material of

the wall. This process continues up to a point that the electron density is comparable

to the proton density. For the case of the SNS ring, the most probable places to apply

this mechanism are at the stripper foil which has a large density of electrons from the

very beginning. Finally, almost all electrons accumulated in a single pass disappear

in the beam gap due to their own space charge.

Compared with the single pass electron accumulation, multi-pass accumulation

has a more complicated origin. If the SEM coefficient, or the number of initial elec-

trons is too low to produce a significant electron density during a single pass, these


electrons can accumulate in a multi-turn process. This accumulation mechanism de-

pends strongly on the proton beam distribution, since the electron energy at the end

of the bunch depends on the steepness of the trailing edge, which is a typical feature

of the e-p instability.

However, the above mechanisms are too simple to explain the repeatability of

electron accumulation and e-p instability. Generally speaking, for bunched proton

beam, the electrons become unstable when the proton intensity is high enough. All

electrons existing in the gap are attracted to the center of the proton beam when

it passes through. Electron amplitude increases on the trailing edge of the proton

beam and they hit the wall near the very beginning of the proton beam gap. With

high proton intensity, the electron energy is large enough to produce more than one

electron on average, and those secondary electrons have low velocities and can survive

the gap without hitting the wall again. Slow accumulation of electrons is caused by

repeating this event, and the accumulation stops when the electron density is high

enough to repulse the secondary particles back to the wall, which corresponds to the

saturation of electron density.

Electron-proton (e-p) instability is induced when those background low-energy

electrons are trapped within the space-charge potential of the circulating proton beam.

Coupled transverse oscillations of the proton beam and trapped electrons can develop

and become larger and larger, leading to beam loss. Therefore, the e-p instability

requires a source of enough low energy electrons, stable trapping of the electrons, and

unstable coupled oscillations. This fast instability with beam loss was first observed

and studied in Los Alamos Proton Storage Ring (PSR). It has also been considered

during the design stage of the SNS accumulator ring due to the similarity of PSR

and SNS, leading to the Titanium Nitrate (TiN) coating for most of the SNS vacuum

chamber. However, the instability is still observable in SNS for some cases above the

beam intensity threshold. In this chapter, we will discuss the experimental observation

3.2 Electron Cloud 51

and qualitative analysis of the e-p instability in the SNS ring, provide comparison of

simulation and experimental results, and try to benchmark the effect of the proton

bunch distribution on the e-p instability.

3.2 Electron Cloud

3.2.1 Physics of Secondary Electron Emission

The simulation of the interaction of proton bunch and electron cloud is implemented

with the ORBIT code and ECE module. The proton bunch is propagated in the

ORBIT code through a series of ”Nodes”, which perform operations on a set of macro-

particles that form the bunch. The nodes include TEAPOT-like elements for drift and

magnets, rf buncher node, an injection foil node, longitudinal space charge node and

an auxiliary electron cloud node (ECN). Proton bunch is divided into longitudinal

slices by the longitudinal space charge node for realistic longitudinal space charge

simulation. However, this longitudinal slicing is not important for ECE because the

most interesting dynamics is on the transverse plane and therefore the ECN usually

has only a few or even one slice. Propagation of the proton bunch uses the location

s as independent variable until it encounters the ECN. During the passage through

the ECN, the bunch is frozen and calculated with the e-cloud module using time t

as independent variable. Macro electrons move in the electromagnetic field created

by themselves, the proton bunch, the perfectly conducting walls of beam pipe, and

any other external magnetic fields. New macro electrons are created as a result

of lost protons and by macro-electrons impacting the beam pipe. The collision of

macro-electrons is described with a probabilistic model of secondary electron emission

developed by M. Furman and M. Pivi [18], which will be discussed later in detail.

There can be one or more ECNs in the lattice. Each ECN has its own electron


cloud. One can define multiple ECNs in the ring to cover the most important sources

of secondary electrons. The length of each e-cloud region should be short enough to

guarantee small changes in physics parameters inside, while each region has its own

bunch of electrons with its own history and dynamics. The instability occurs due to

the interaction of electron clouds and proton bunch. The ECN performs action for

each proton slice by applying a momentum kick to every proton in the slice:

∆p = (Leff/Lec) · e · Eec(t) ·∆t (3.1)

where Lec and Leff are the defined length and effective length (induced to reduce the

calculation time) of ECN, Eec is the electric field created by the electron cloud, and

∆t is the passage time of a proton through the ECN.

Now consider the model of secondary electron emission that developed by Furman

and Pivi. As implied in Section. 3.1, there are four possible sources for the electron:

1) electrons produces at the injection region stripping foil; 2) electrons produced by

lost proton grazing the vacuum chamber; 3) secondary electron emission process; 4)

electrons produced by residual gas ionization. The two main sources considered for

proton storage rings of SNS and PSR are lost protons hitting the vacuum chamber

walls and secondary emission from electrons hitting the wall. The input ingredients

of the secondary emission model are the secondary-emission yield (SEY) δ(E0) and

the emitted-energy spectrum of the secondary electrons dδ/dE, where E0 is the in-

cident electron energy and E is the emitted secondary energy. The main result is

the set of probabilities for the generation of electrons, which is embodied in a Monte

Carlo procedure that generates simulated secondary-emission events given the pri-

mary electron energy and angle. The parameters related to the secondary emission

process were obtained from detailed fits to the measured SEY of stainless steel. The

main SEY parameters are the energy Emax at which δ(E0) is maximum and the peak


value itself.

In the Furman and Pivi model, the conventional picture of secondary emission is

summarized as follows: when a steady current I0 of electrons impinges on a surface, a

certain portion of the current Ie is backscattered elastically while the rest penetrates

into the material. Some of these electrons scatter from one or more atoms inside the

material and are reflected back out, which are called ”rediffused” electrons, with the

corresponding current Ir. The rest of the electrons interact in a more complicated

way with the material and yield the so-called ”true secondary electrons”, with corre-

sponding current Its. The yields for each type of electron are defined by δe = Ie/I0,

δr = Ir/I0, and δts = Its/I0. The total SEY is

δ = (Ie + Ir + Its)/I0 = δe + δr + δts (3.2)

For normal incidence (θ0 = 0), the θe(E0, θ0) and θr(E0, θ0) are well characterized by

empirical formulas from experiment data [18] such as

θe(E0, 0) = P1,e(∞) + [P1,e − P1,e(∞)]e−(|E0−Ee|/W )p/p (3.3)

and

θr(E0, 0) = P1,r(∞)[1− e−(E0/Er)r ] (3.4)

and the energy dependence of θts(E0, 0) is well fit experimentally by an approximately

universal function of the form

θts(E0, 0) =s(E0/Ets)θts

s− 1 + (E0/Ets)s(3.5)


2.0

1.5

1.0

0.5

0.0

δ

8006004002000

Incident electron energy (eV)

data δe (fit) δr (fit) δts (fit) δe+δr+δts (fit)

SEY for stainless steelnormal incidence

Figure 3.2: The SEY for stainless steel for SLAC standard 304

rolled sheet, chemically etched and passivated but not

conditioned. The parameters of the fit are listed in

Table 3.1


Table 3.1: Main parameters of the model, used for SNS TiN coated

and uncoated stainless steel chamber.

Backscattered electrons Coated Uncoated

P1,e(∞) 0.02 0.07

P1,e 0.5 0.5

Ee(eV ) 0 0

W (eV) 60 100

p 1 0.9

Rediffused electrons Coated Uncoated

P1,r(∞) 0.19 0.74

Er(eV ) 0.04 40

r 0.1 1.0

True secondary electrons Coated Uncoated

Ets(eV ) 246 310

δts 1.8 1.22

s 1.54 1.813

Total SEY Coated Uncoated

Et(eV ) 250 292

δt 2 2.05


Figure 3.3: The Monte Carlo scheme implemented in the C++

class ”controlled emission surface”. The two M repre-

sent the macro-sizes of incident and emitted electrons.

IfG < fdeath(E0), thenMout = 0; else ifG > fdeath(E0),

Mout = Min·δ/(nborn·(1−fdeath(E0))), whereG is a ran-

dom number between 0 and 1, δ is the SEY, fdeath(E0)

is a user defined function for the incident energy E0

and is usually equal to 0 if E0 > 1eV , and nborn is the

number of emission procedure per impact event and is

usually equal to 1.


A fit to stainless steel data is shown in Figure 3.2 and Table 3.1 shows the ex-

perimental parameters of different materials in the SNS vacuum chamber, such as

the TiN coated stainless steel for most parts of vacuum chamber and the uncoated

stainless steel parts due to instrument devices. In SNS, we also have a few places that

may have larger secondary emission coefficient and may be more responsible for the

electron cloud generation, such as places near the stripper foil (coated by aluminum)

where aluminum might be evaporated, and some ceramic breaks of the vacuum cham-

ber, bellows, etc. Unfortunately, we do not have an exact measured parameter list

and effective length for those regions. In order to uncover those suspicious hot regions

for electron cloud generation, electron detectors are often installed in the vicinity. In

SNS ring, there are five electron detectors, one in the injection region where there is a

coating of aluminum, three in straight sections with TiN coated walls, and the other

one in an arc. However, the detectors have not been fully commissioned and only

partially functional. We have not seen any substantial signals from those detectors

so far. In this case, the simulation tool, ORBIT, implemented with its electron cloud

module must play the primary role to help us understand the physics phenomena.

The basic simulation algorithm for secondary electron emission is described as follows:

Figure 3.3 describes the Monte Carlo scheme that is implemented in the C++ class

”Controlled emission surface”. The fdeath(E0) is a user defined probability function

to cut off the emitted macro-electron to save calculation time for low energy macro-

electrons. The process to calculate the macro-size and momentum of the new macro-

electron is as follows [19]:

• Remove the incident macro-electron from electron bunch;

• Calculate the SEY δ(E0, θ0) for the incident macro-electron (E0, θ0);

• Add a new macro-electron;


• Set the macro-size of the new macro-electron as δ×size of incident electron;

• Determine the type of the emission by calculating the following probabilities:

Pelastic backscattered = δel/δ (3.6)

Prediffused = δrd/δ (3.7)

Pn, true secondary =δtsδ· Pn,ts∑Memiss

i=1 Pi,ts(3.8)

where

Pelastic backscattered + Prediffused +

Memiss∑i=1

Pn,true secondary = 1 (3.9)

• Calculate the new electron’s energy by random sampling from the model spec-

trum;

• Calculate the new electron’s angle under the probability density cos θ where θ

is the angle normal to the surface;

• Calculate the new momentum of the new macro-electron at the impact point.

The procedure above demonstrates how the physical system simulates the accu-

mulation of electron cloud. The instability comes from the interaction between proton

bunch and electron cloud. When the proton bunch encounters the electron cloud re-

gion, it is frozen and passes through the region using time t as independent variable,

while it contributes to the electron dynamics in that region. The changes in proton

momentum due to the electron cloud are accumulated as kicks in an auxiliary grid

covering the proton bunch and are applied to the protons at the end of propagation

through the electron cloud region. Generally speaking, the simulation algorithm can

be summarized as three stages [20]:


• Preparation of calculation: This stage deals with the proton beam only. The

macro-particles of the proton bunch are distributed among CPUs, which is

carried out by the ParticleDistributor ORBIT class. According to the assigned

parameters in the class, the necessary 3D arrays are resized on each CPU. It

is necessary to have 5 distributed 3D grid objects: two for space charge proton

density and potential, three for accumulated kicks in x,y,z direction. The macro-

protons are binned into the space charge density grid and space charge potential

is calculated.

• EC buildup simulation: This stage simulates the primary electron generation,

motion of macro-electrons and multipactor, which is done by three nested loops:

The outside loop prepares EC potential as a field source for EC dynamics sim-

ulation. The number of steps (usually a few thousands) is determined by the

requirement of adiabatic change in electron cloud potential. Primary electrons

are generated in the beginning of this iteration by routines simulating proton

grazing the vacuum chamber or ionizing residual gas. They are randomly dis-

tributed between CPUs and reside at the same CPU during calculation. Then

the space charge potential of electrons is calculated, which is a sum of all po-

tentials over all CPUs. Before the end of this iteration, the momentum kicks to

proton bunch are accumulated.

The Intermediate loop is used to update the proton beam field source for the

Tracker, a class to move the macro-electron by using the combined force from

all registered electro-magnetic field sources. Usually it is sufficient to update

the fields simultaneously, which means the loop needs only one step.

The third loop implements the electron motion and Pivi-Furman model as de-

scribed in the previous paragraphs of this section.


• Proton coordinate updates: this stage applies the accumulated kicks from elec-

trons to protons.

3.2.2 Two stream instability model for coasting beam

As denoted in Section 3.1, a coupled oscillation between the proton beam and electron

cloud could happen in the transverse plane when the accumulation of electrons reaches

a threshold. If we assume both the proton and electron beam are coasting beams, with

the same transverse sizes and uniform distribution longitudinally and transversely, we

can simply describe the oscillation with coupled oscillator equations for the vertical

direction [23]:

(∂

∂t+ w0

∂

∂θ)2yp +Q2

βw20yp = −Q2

pw20(yp − ye) +Q2

psw20(yp − yp)) (3.10)

d2yedt2

= −Q2ew

20(ye − yp) +Q2

esw20(ye − ye) (3.11)

where yp and ye are the vertical displacements of the centroids of proton and electron

beams from the axis of the vacuum chamber, w0 is the angular revolution frequency,

θ is the azimuthal angle around the ring, Qβ is the betatron tune, and Qe and Qp are,

respectively, the bounce tune of electrons inside the proton beam and the oscillation

tune of protons inside the electron beam. Thus we have

Ω2e = (Qew0)2 =

4Nprec2

b(a+ b)C(3.12)

Ω2p = (Qpw0)2 =

4Nprpc2χe

b(a+ b)γC(3.13)

where χe is the neutralization factor, or the ratio of electrons to protons. rp is the

classical proton radius, re is the classical electron radius, and C is the circumference


of the ring. The negative signs on the right hand side of Equations (3.10) and (3.11)

indicate that the protons are focused by the electron beam and electrons are focused

by the proton beam. The Lorentz factor γ in Equation (3.13) is introduced because

the protons are circulating while the electrons are not. One may notice that there

is no magnetic field contribution. That is because the electron has essentially no

velocity although it sees a magnetic field from the proton, and the proton does not

see a magnetic field in a stationary electron beam. Here we consider uniformly and

cylindrically symmetric proton and electron beams of b(a + b) → 2a2. Image effects

in the walls of vacuum chamber are neglected.

The last term in the Equation (3.10) denotes the oscillations of the proton under

the self-field of the proton beam.

(Qpsw0)2 =4Nprpc

2

b(a+ b)γ3C(3.14)

which is proportional to the linear space charge tune shift of the proton beam. Sim-

ilarly, the last term in Equation (3.11) with Q2es = Q2

eχe is also proportional to the

space charge tune shift of the electron beam.

Averaging over yp and ye, the space charge terms, Q2ps and Q2

es are dropped, and

thus we obtain the equations for the beam centroid yp and ye. If there is a coherent

instability that occurs at Ω = Qw0, we can get

yp ∼ ei(nθ−Ωt) (3.15)

ye ∼ ei(nθ−Ωt) (3.16)

where n is the longitudinal harmonic number. The coupled equation can then be

solved to give


(Q2 −Q2e)[(n−Q)2 −Q2

β −Q2p]−Q2

eQ2p = 0 (3.17)

For a solution when Q is near Qe, we can expand Q around Qe. When Qp or

χe is large enough, the solution becomes complex and instability occurs. From the

stability condition

Qp ≤| (n−Qe)

2 −Q2β −Q2

p |2√Qe | n−Qe |

(3.18)

the limiting neutralization factor χe can be obtained. The growth rate above threshold

is given by

1

τ≈ Qpw0

2

√Qe

| n−Qe |(3.19)

The equation of motion of the electron, Equation (3.11), describes an undamped

oscillation driven by yp. However, there is another consideration of stability, namely,

Landau damping. To damp the electron oscillation, there must be a spread in the

electron bounce tune Qe, while spread in betatron tune Qβ is necessary to damp

proton oscillation. Therefore, to provide Landau damping to the coupled-centroid

oscillation, there must exist large enough spreads in both Qβ and Qe. A stability

condition based on Landau damping has been developed by Schnell and Zotter [21],

assuming parabolic distributions for the betatron tune and electron bounce tune, but

without consideration of space charge self-forces. The stability condition is given as

4Qβ

Qβ

4Qe

Qe

≥ 9π2

64

Q2p

Q2β

(3.20)

where the factor 9π2/64 is a form factor of the parabolic distributions. To apply the

Laslett-Sessler-Mohl criterion [22], 4Qβ can be interpreted as the half tune spread

of the betatron tune in excess of what is necessary to cope with the instabilities of

3.3 Observation of Electron-Proton instability at the SNS ring 63

the single proton beam. 4Qe can be interpreted as half tune spread of the electron

bounce tune in excess of what is necessary to cope with the instabilities of the single

electron beam.

The spread in the electron bounce frequency is difficult to measure. However, we

can infer the electron bounce frequency by measuring the coherent frequency of the

proton beam when instability occurs. PSR measured 4Qe/Qe ∼ 0.25. The limiting

Qp and neutralization χe can be computed as 0.18 and 3.4%, respectively. SNS has a

much higher threshold, limiting Qp and χe are 0.50 and 16.2% respectively. Further

increase in threshold requires larger spreads in Qe and Qβ and threshold χe can be

very sensitive to the distribution of betatron tune and electron bounce tune. Actually,

anti-damping can even happen unless there is a large enough overlap between 4Qβ

and 4Qe [23].

3.3 Observation of Electron-Proton instability at

the SNS ring

Due to the similarity of SNS accumulator ring and PSR, the electron-proton insta-

bility has been considered as a potential obstacle since the design stage. Inspired by

the research at PSR, most of the vacuum chamber at SNS was coated with a low

secondary electron emission Titanium Nitrate (TiN) material. In addition, the ring

was equipped with an electron collector near the stripping foil and space was reserved

space for solenoid magnets to reduce electron buildup in likely regions. As a result,

the neutralization threshold of the SNS ring is much higher than in PSR. We have not

observed e-p instability during the normal neutron production so far. However, we

can activate the e-p instability by increasing the proton intensity, varying the trans-

verse tune and chromaticity, or modifying the proton bunch distribution. We have


carried out a series of high intensity beam measurements between 2008 and 2010.

Besides the beam intensity, the shape of the proton bunch is presented as another

key factor of the e-p instability. Compared with PSR, the SNS ring has a second

harmonic RF cavity to add more freedom in bunch shape control, which can be used

to perform some interesting experiments as we will show in this section.

The e-p instability experiments were divided into several groups according to the

variables we manipulated. We will discuss these observations in orders in this section.

Due to the complexity of the e-p instability, quantitative comparison under different

machine conditions is difficult. Since we do not have functioning electron detectors

in the ring and hence can not identify the exact source and location of the electron

cloud, it is also very difficult to calibrate the electron cloud model of ORBIT code

to the experiment data. With these limits, the comparison will be split into two

parallel processes: experimental data analysis and simulation of the corresponding

experiments.

3.3.1 Observation of multi turn and single turn electron ac-

cumulation

The e-p instability is actually a two stream instability. An intense proton beam

forms a potential well to trap the electrons, which have opposite charge. The trapped

electrons can often accumulate to such an extent that they provide a potential well

for protons. Thus, the electrons can oscillate transversely in the potential well of

protons, while protons can oscillate transversely in the potential well of electrons.

This can give rise to the two stream instability. As we have mentioned in Section 3.1,

the accumulation of electrons is the key of this instability and this process can be

classified into two types for bunched beam: single pass accumulation and multi turn

accumulation. The multi turn accumulation is a result of surviving electrons in the


beam gap from previous turns. A clean beam gap will help eliminate the accumulation

of electrons. However, sometimes the trailing edge multipacting is so strong that

the accumulation produced by one single passage derives instability even before the

clearing gap is reached.

One way to separate the contribution of the two mechanisms is to measure elec-

trons from the passage of a single beam pulse. However, since we do not have func-

tioning electron detectors, we can not tell the weights of the two mechanisms by

experiment. In SNS, these two mechanisms are related and contribute to each other.

While the trailing edge multipacting leads to more ”cold electrons” in the beam gap,

the source electrons for trailing edge multipacting also originate in the bunch head.

Figure 3.4 and 3.5 present two examples of unstable oscillations obtained by in-

creasing RF 2.1’s phase from 5 deg to 15 deg. Such a small change resulted in a small

modification on the bunch shape as shown in Figure 3.6(a); induced a small increase

of the electron cloud; and brought larger multi-pactor effect on the trailing edge.

This suggests that we can control the multi-pacting on the trailing edge by varying

the bunch shape with 2nd harmonic RF. However, the multi-turn accumulation and

trailing edge accumulation tangle easily, i.e., electrons involved in the oscillation at

the bunch head may contribute to the trailing edge multi-pacting, while the electrons

from multipacting may survive the beam gap. When increasing the RF 2.1’s phase

from -5 deg to 15 deg, the bunch shape becomes more triangular. The tail can be

represented by a trial Bolzman function y = A2−A1

1+e(x−x0)/dx+ A2, and the slope at the

middle point is A2−A1

4dx. Based on the trial fitting, the slope of the three case ranges

from −0.112 to −0.128, which is small. The main difference is the peak longitudinal

density of the bunch. The growth rate of the e-p instability, the electron cloud peak

density, and the longitudinal proton peak density display a linear relationship, which

is shown in Figure 3.6(b).


turn number 550 turn number 600




Figure 3.4: Horizontal oscillations on the head of proton bunch at

SNS. Measurement was taken in 2008 with 2nd har-

monic RF phase set to 5 deg. The red line represents

the BPM sum signal. The blue line is the BPM differ-

ence signal with closed orbit offset substracted. Devel-

opment of the unstable oscillation can be seen in the

progress from the upper to the lower plots. It occurs

at the head of proton bunch, which is an evidence in

favor of the multi-turn electron accumulation.






Figure 3.5: Horizontal oscillations on the head and tail of proton

bunch at SNS. Measurement was taken in 2008 with

2nd harmonic RF phase set to 15 deg. The unstable

oscillation first occurs at the head of proton bunch and

also emerges in the tail at a later time. This is evidence

in favor of the multipactor effect at the trailing edge,

in addition to the multi turn accumulation.


0

20

40

60

80

100

120

140

160

180

200

0 200 400 600 800 1000 1200

line

dens

ity (

nC/m

)

time (ns)

RF 2.1 -5 degRF 2.1 5 deg

RF 2.1 15 deg

(a) proton bunch profile

174176

178180

182184

186188

190

55

60

65

70

7516

17

18

19

20

21

22

23

proton longitudinal peak density(nC/m)simulated electron peak density (nC/m)

grow

th r

ate

(1/m

s)

RF −5 deg

RF 5 deg

RF 15 deg

(b) comparison of the 3 cases

Figure 3.6: Experimental proton beam profile near the end of in-

jection with simulated electron cloud. The proton lon-

gitudinal peak density, simulated electron peak density

and e-p growth rate have a linear relationship in the

lower plot.


3.3.2 A particular observation of e-p instability with buncher

voltages

As we mentioned in Section 3.2.2, the stability condition of a coasting beam with

Landau damping can be expressed as [24]:

∆Q

Q

∆Qe

Qe

≥ 9π2

64

Q2p

Q2

Since increased momentum spread which gives larger tune spread, i.e.,

∆Q = |(n−Q)η − εQ|+N.L.

(n−Q) ' Qe

and εQ can be neglected compared with Qeη, we can expand the threshold condition

as

Np ≤π

2

R

re(

64

9π2

mp

me

γpβp)2 b(a+ b)

R2Qβ

1− fef 2e

(∆Qe

Qe

)2(η∆p

p)2F

where F is the filling factor for bunched beam. If we make the following assumptions:

1. The beam size is roughly constant within a small range of beam intensity; 2.

Constant fe. This is an assumption that the electrons surviving the gap saturate at

a lower intensity. 3. ∆Qe

Qeis roughly constant for cold electrons when the bunch shape

changes little with intensity. Then the threshold intensity Np will linearly increase

with (∆pp

)2, or, in another word, with buncher’s voltage V since (∆pp

)2 ∝ V .

The measurement of threshold intensity is easy. While we keep the machine

settings except the voltage of first harmonic RF stations 1.1 and 1.3, we change

the number of accumulation turns until 10% current loss is monitored from beam

current monitor 25I. This pair of values is then presented as a point on the threshold

intensity curve. The electron cloud conditon in PSR meets the above requirements

and therefore the measured threshold intensity shows a strong linear dependence

on buncher’s voltage [24, 25]. However, the same experiment done at SNS gives a

different result.


0 2 4 6 8 10 120

5

10

15

20

25

RF Buncher Voltage (kV)

Thr

esho

ld In

tens

ity (

uC)

measured pointslinear fit

measured on July, 2009

Figure 3.7: SNS: Effect of buncher voltage on the instability

threshold intensity. The change of threshold intensity

is very little and there is no clear dependence on the

buncher voltage.


As shown in Figure 3.7, different from the strong linear dependence at PSR,

buncher voltage has little effect on the instability threshold at SNS. A guess is that the

electron cloud has not saturated at SNS; therefore the previous calculation dominated

by Landau damping is not proper anymore.

3.3.3 Observation of e-p instability with intensity scan

In section 3.2.2, we demonstrated the scheme of coupled oscillation between coasting

proton beam and electron. The electron bounce frequency can be deduced as

Ωe = Qew0 =

√4λrec2

b(a+ b)(3.21)

where λ is the linear particle density of the proton beam. For coasting beam, it

is simple because the linear beam density does not change. However, the situation

becomes much more complicated for bunched beam, because the bounce frequency

of the electrons depends on their location inside the proton bunch and the wide

spread frequency may change due to evolution of the electrons. Nevertheless, the

formula is still applicable if we know the localized information, or, if the proton

bunch distribution is a FWHM profile, we can at least predict that frequency spread

will be 1/√

2 its mean value.

The intensity scan has been carried out several times from 2008 for the bunched

beam. The measurement on April 2008 [26] shows that the frequency of instability

does not change much when the intensity is increased from 5 µC to 10 µC. This

observation was later confirmed in 2009 with the intensity ramping up from 6 µC

to 21 µC. Here we take the data of 2009 as the example to further understand the

observation.

In the intensity scan of 2009, the machine was set to have natural chromaticity

and 1000 stored turns and all four RF stations were on to keep the beam bunched.


(a) Horizontal spectrum

(b) Vertical spectrum

Figure 3.8: The instability frequency spectrum for 11 µC proton

bunch.


When the intensity was increased to 11 µC, the instability emerged in both horizontal

and vertical plane as shown in Figure 3.8. The horizontal and vertical spectra show

similar pattern, also for higher intensities. Therefore, we take the horizontal data

from intensity scanning as examples to explore the physics.

Figure 3.9 shows the horizontal frequency spectrum for intensity 15 µC, 17 µC and

21 µC. The full intensity 21µC corresponds to 1000 accumulation turn, and therefore

15 µC corresponds to nearly 700 accumulation turn. After the accumulation, the

bunch was stored for extra 1000 turns. Instability occurs before the end of injection

for all three cases and therefore they share the repeat trend of instability development.

The thin black line marks the trend in Figure 3.9. For 15 µC, one may notice the

small distortion from the black line. It might be caused by pulse to pulse jittering

which is a common observation in high intensity experiments. One may notice that

for each spectrum, there are several parallel strips shifting from lower frequency to

higher frequency. To further look into the pattern, a strip filter is designed to bypass

the defined strip region only and zero out the rest spectrum.


Figure 3.9: Frequency spectrum for different beam intensity of 15

µC, 17 µC and 21 µC. The instability occurs before

the end of injection (15 µC ' 700 turn ). Therefore the

development of instability shows the same trend. The

instability is developed stronger for higher intensity

according to the color bar.


(a) General spectrum for 17 µC

(b) Spectrum strip after filtering

Figure 3.10: Strip filter for frequency spectrum: The upper plot

is the general frequency spectrum. Strip filter is ap-

plied to cut the three strip from the spectrum with

an example in the lower plot. The lower spectrum is

later inverted to time domain oscillation.


Figure 3.10(a) shows 17 µC vertical instability spectrum with three strips picked

for the application of strip filter. Figure 3.10(b) shows an example of strip 1 pass filter.

By inverting the filtered frequency into time domain, Figure 3.11, 3.12 and 3.13 show

the evolution of BPM difference signal of the three strips respectively. The shifting

of each strip from lower to higher frequency might be induced by the synchrotron

motion (synchrotron period ∼ 2000 turns) that ”pushed” oscillation slowly to the

head of the proton bunch. From eyeball view, the three high frequency oscillations

have almost the same development pattern of trailing edge multipacting. And they

have almost the same location as shown in Figure 3.14. The only obvious difference in

the time domain is their maximum amplitude. This parallel frequency strip pattern

is also observed in the case of multi-turn accumulation as shown in Figure 3.15, which

is induced by electrons survival in the beam gap and the coupled oscillation occurs

at the bunch head. One guess is that it is induced by the multipacting of electrons

thrown out of the proton bunch. After the initial accumulation of electrons, coherent

oscillation is developed and some electrons are thrown out by large oscillation. These

electrons repeat the multipacting process and accumulate again in proton bunch.

This scheme is nevertheless too simple to explain the pattern thoroughly for bunched

beam. Simulation is the only way to thoroughly model the electron accumulation

and instability in the bunched beam. Benchmark of simulation can be achieved once

we have functional electron detectors and thus the location and origin of electron

accumulation can be measured.

3.3.4 Effect of proton bunch shape on e-p instability

The proton bunch shape is very important for e-p instability. The bunch head attracts

electrons in beam gap [20] while the bunch tail produces strong multipactor effect.

The effect of bunch head on primary electron accumulation is small compared to


turn number 365 turn number 415 turn number 465



Figure 3.11: BPM difference signal for strip 1 on frequency spec-

trum. The red represents the BPM sum signal after

scaling and blue is the difference signal after filtering.


strip3strip2strip1sum signal

Figure 3.14: Filtered signals at turn 605 for the three strips. The

center frequency for the three strips has an interval of

∼ 10 MHz, which is not obvious in the time domain.

The locations of the three oscillations almost overlap

in the time domain.


(a) Horizontal Spectrum for case ”rf1”

(b) BPM difference signal at turn 800

Figure 3.15: Parallel strip pattern was also observed for oscillation

at proton bunch head. Measurement was taken in

2008.


the main E-P source of proton loss, and the subsequent effect on electron cloud line

density is much more tiny compared to the result from different tail slope [20]. At

SNS, by using the combination of first harmonic and second harmonic RF stations,

we produce a proton bunch of trapezoid shape with large flat top.

To study this effect more quantitatively and systematically, some simulation re-

sults of electron cloud will be introduced first. If simulating the electron cloud only,

we do not need to inject tens of millions macro-protons into the ring (To implicitly

describe the high frequency oscillation of e-p instability of 100MHz for example, it

means 100 slices comparing with the revolution frequency of 1MHz. But, considering

that a sinusoidal period needs 40 ∼ 50 points, there should be at least 4000 slices

longitudinally. Since every 2-D slice has 32× 32 grids at least and 10 macro-particles

per grid is a necessity, more than 32× 32× 10× 4000 = 40960000 macro-protons are

needed to satisfy the e-p instability simulation.). The physics parameters in these

simulations, inspired by the experiment done in October 2008 at SNS, is given

Figures 3.16 and 3.17 show the simulation results for different bunch shapes. In

general, the proton population on the trailing edge, and the distribution of this pop-

ulation, is very important for the electrons’ generation. The proton bunch for normal

productions has trapezoid shape with flat top. However, it can also turn to tri-

angle shape by RF modulation or specific injection painting. Due to the generally

longer trailing edge of triangle compared with trapezoid distribution, as shown in

Figure 3.16, the electron cloud generation is much stronger for the triangle shape and

thus induces e-p instability. This was also revealed in the experiment.

As an example to change the bunch shape with 2nd harmonic RF phase modula-

tion, Figures 3.18 and 3.19 show the evolution of beam longitudinal profile and their

frequency spectrum. Although the coupled oscillation happens mostly at the head

of proton bunches as we mentioned in Sec. 3.3.1, it implies important contribution

from trailing edge multipactor with changes of bunch shape. The flat top trapezoid


Parameter Value (Unit)

RING PARAMETER

Proton beam energy 0.890 (GeV)

Particle population 1.0612× 1014 ppp

Revolution period 963.2 (ns)

RF harmonics h1.1, h1.2, h1.3, h1.4 1, 1, 1, 2

RF V1.1, V1.2, V1.3, V2.1 9.0, 0.0, 10.5, 10.5 (Volt)

SIMULATION PHYSICS PARAMETERS

Averaged proton loss rate 10−6 per turn

Electron yield material (length) Stainless steel (100 m)

Injected proton distribution type Guassian transversely

Uniform energy and random phase longitudinally

Injected proton bunch emittance 26.81, 24 (mm ·mrad)

Injection β function 10.947, 12.795 m

Injection α 0.059, 0.055

NUMERICAL PARAMETERS

Number of macro-particles 500000

Number of beam slices 400

Table 3.2: Primary simulation parameters


0 200 400 600 800 1000 12000

20

40

60

80

100

120

140

160

180

200

Time duration (ns)

Line

den

sity

(nC

/m)

proton bunches with the same tail density but different tail lengths

proton bunch

electron bunch

(a) proton and electron distribution

100 150 200 250 3000

10

20

30

40

50

60

x: bunch tail length (ns)

y: e

lect

ron

clou

d lin

e de

nsity

(nC

/m)

max e density for every trapezoid distributioncubic polynomial fit

y = 5.7e−6 * x3 − 1.3e−3 * x2 + 9.7e−2 * x − 2.239

(b) bunch tail length vs. electron density

Figure 3.16: Effect of trailing edge’s length (time duration). The

sub-figure on the left plots some example trapezoid

distributions and the corresponding electron clouds

with the same color. The sub-figure on the right plots

the length of trailing edge versus the peak height of

electron cloud. And we can use a cubic polynomial

fit to perfectly fit those points: f(l) = 5.7× 10−6l3−

1.3× 10−3l2 + 9.7× 10−2l − 2.239.


0

50

100

150

200

250

0 200 400 600 800 1000 1200

Line

den

sity

(nC

/m)

time (ns)

(a) proton and electron distribution

0 5 10 15 20 25 30

−20

0

20

40

60

80

100

steepness factor

peak

hei

ght o

f ele

ctro

n cl

oud

(nC

/m)

peak height of electron cloudexpoenential fit

y = a*exp(b*x) + c* exp(d*x)a= 106.6b= 0.002907c= −108.7d= −0.5867

(b) steepness factor vs. electron density

Figure 3.17: Steepness factor versus peak height of electron cloud.

Steepness factor is defined as s = Ttail/Thead, where T

is the time duration. The triangle shape is changed

from head-only (steepness=0), to tail-only (steepness

= ∞). When s > 10, the difference between trian-

gles is very tiny and thus the electron cloud looks

saturated.


0 0.5 10

0.05

0.1

0.15

0.2

0.25

WCM waterfall signal for ph1

time(us)

Cur

rent

(A

)

0 0.5 10

0.05

0.1

0.15

0.2

0.25

ph3

time(us)0 0.5 1

0

0.05

0.1

0.15

0.2

0.25

ph5

time(us)

turn 200

turn 280

turn 360

turn 440

turn 520

turn 600

turn 680

turn 760

turn 840

turn 920

turn 1000

turn 1080

Figure 3.18: Beam longitudinal profile evolution for different RF

phases. ”ph1”, ”ph3” and ”ph5” denote 2nd har-

monic RF phase −35 deg, −5 deg and 15 deg, respec-

tively. The e-p instability develops from no (ph1) to

stronger with larger growth rate. Instability occurs

near turn 700 ∼ 800 for ph3 and ph5.


(a) ph1 (b) ph3

(c) ph5

Figure 3.19: Horizontal spectrum for ph1, ph3 and ph5.


shape is preferred to eliminate e-p instability and it can be achieved by setting the

voltage and phase of 2nd harmonic RF properly. The real proton bunch shape is often

complicated, i.e., the ”flat top” is not flat but declining, or the trailing edge is not

smoothly drooping but often concave-convex. Therefore it is difficult to fit all of the

shapes with a slope evaluation function such as the Boltzmann function. And hence

it is difficult to build a direct quantitative relationship between RF parameters and

e-p instability. We suggest to monitor the bunch shape using beam current monitor

while adjust the RF station to change the bunch shape. A general guideline is to

increase the flat top region and increase the slope of the trailing edge to eliminate the

instability.

3.4 Conclusion

Although SNS has coated most of the vacuum chamber with TiN inspired by the

electron cloud study at PSR, the e-p instability has still been observed at ∼ 5× 1013

ppp and higher intensity. Currently, the e-p instability does not emerge during the

normal productions. However, small modulation on the bunch shape, or unclean

beam gap, can trigger the instability easily. Therefore, it is a potential big problem

for the SNS power upgrade.

Due to the absence of effective electron detector, we cannot observe the electron

cloud directly. Only e-p instability can be directly monitored on the BPM oscillo-

scope. Although the lack of electron data limits the benchmark and simulation with

the electron cloud model, the observation itself has already generated a lot of use-

ful information and provided some mitigation guidelines. From a particular threshold

measurement, we found that the buncher voltage has little effect on instability thresh-

old, which behaves differently from PSR’s strong linear dependence. We also found

that the instability has a strong dependence on proton bunch shape. To mitigate

3.4 Conclusion 89

the e-p instability, flat top and short tail is highly preferred. A proposed solution,

other than the existing mitigation tools such as electron collector at injection area

and solenoid, is to use RF barrier cavity [27], which leads to flat longitudinal cur-

rent density. It needs further simulation study after the instability model is well

benchmarked.

A feedback system, whose purpose is to damp the high frequency oscillation, is

also developed at SNS, similar as PSR design [28]. Currently, this system has not

provided effective damping to the oscillations. One potential explanation is that

the power of the damping system is not big enough to damp the beam significantly.

Moreover, coherent tune shift in the beam is observed and it adds difficulties to the

efficiency of the feedback system.

Conclusion 91

Chapter 4

Conclusion


storage ring. It accumulates protons in approximately 1016 turns for delivery of 1.5

MW pulsed beam power onto a mercury target to produce pulsed neutrons. For such

a high intensity accumulator, high beam loss does not only affect machine availability,

but may also damage devices in hot regions. Therefore, there are strict requirements

for the uncontrolled beam loss, which requires good hardwares as desired, an accurate

model to set up the machine, and the understanding of the beam loss mechanisms. In

this dissertation, there are two subjects that were discussed as main parts: Chapter 2

presents the calibration of the linear optics of the linear model, which is motivated

by the betatron tune discrepancy between online model and BPM turn-by-turn mea-

surement. Chapter 3 discussed the observations of electron-proton instability at the

SNS accumulator ring, which is produced mainly by manipulating the RF cavities

and accumulated beam intensity.

We used the Orbit Response Matrix (ORM) method to solve the discrepancy

between the model predicted and the BPM measured tune. In the past few years,

several ORM experiments has been carried out with different machine setup. We

92 4. Conclusion

used a MATLAB code LOCO to calculate a parameterized model response matrix

and fit it to a measured matrix. We notice that the model ORM calculation in LOCO

was mainly invented and applied for electron rings. The different characteristics of

proton ring led to different experiment design, and therefore the ORM calculation

method should be modified to better adopt the proton ring case. The errors of

six quadrupole power supplies were determined by fitting the grouped quadrupoles

and BPM/corrector gains and couplings. Those errors range from 0.82% to 3.13%,

reducing the tune discrepancy from 0.2 to 0.008. This error set was confirmed by the

other independent experiments and has been implemented in the model. This study

is described in detail in Chapter 2.

Although SNS has coated most of the vacuum chamber with TiN, which was

inspired by the electron cloud study at PSR, the e-p instability has still been observed

at ∼ 5 × 1013 ppp and higher intensity. The e-p instability does not emerge in the

normal productions, but small modulation on the bunch shape, or unclean beam gap,

can trigger the instability easily. Therefore, it is a potential obstacle for the SNS power

upgrade. We have carried out a series of systematic experiment to examine the effect

of the beam intensity and bunch shape on the e-p instability development. Similar

as PSR observations, the e-p instability usually show strong dependence on the RF

buncher voltage. However, in a particular experiment of threshold measurement, we

observed little dependence. Our potential explanation for this mystery is that the

electron cloud saturated in the case of strong dependence, where Laudau damping

dominated the mechanism. In the particular experiment with a certain machine

condition, the electron cloud did not saturate and it balanced the Laudau damping.

However, since the electron detectors are not functioning, we are not able to directly

measure the electron cloud in SNS ring to prove the above explanation. The instability

experiments and simulations also show a strong dependence on the proton bunch

shape, which suggests the features of a better longitudinal beam profile with flat top

Conclusion 93

and short tail that can significantly reduce the possibility of e-p instability. These

observations can be found in Chapter 3.

94 4. Conclusion

BIBLIOGRAPHY 95

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