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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.
viii
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
3.12 BPM difference signal for strip 2 on frequency spectrum. The red
represents the BPM sum signal after scaling and blue is the difference
signal after filtering. . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.13 BPM difference signal for strip 3 on frequency spectrum. The red
represents the BPM sum signal after scaling and blue is the difference
signal after filtering. . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
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].
The accumulator ring of the Spallation Neutron Source is a high intensity proton
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.
1.1 Overview of the SNS Accumulator Ring 5
-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.
1.1 Overview of the SNS Accumulator Ring 7
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
1.1 Overview of the SNS Accumulator Ring 9
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.
1.2 Beam Loss Mechanisms for High Intensity proton storage ring 13
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 Beam Loss Mechanisms for High Intensity proton storage ring 15
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)
20 2. Application of Orbit Response Matrix Method to the SNS ring
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:
2.1 Orbit Response Matrix Method 21
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)
22 2. Application of Orbit Response Matrix Method to the SNS ring
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
2.1 Orbit Response Matrix Method 23
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.
24 2. Application of Orbit Response Matrix Method to the SNS ring
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)
2.1 Orbit Response Matrix Method 25
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
26 2. Application of Orbit Response Matrix Method to the SNS ring
”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 −
2.1 Orbit Response Matrix Method 27
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
28 2. Application of Orbit Response Matrix Method to the SNS ring
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
2.1 Orbit Response Matrix Method 29
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
30 2. Application of Orbit Response Matrix Method to the SNS ring
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.
32 2. Application of Orbit Response Matrix Method to the SNS ring
• 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”:
2.2 Application of ORM to SNS Accumulator Ring 33
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)
34 2. Application of Orbit Response Matrix Method to the SNS ring
• ∆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.
2.2 Application of ORM to SNS Accumulator Ring 35
−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).
36 2. Application of Orbit Response Matrix Method to the SNS ring
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
2.2 Application of ORM to SNS Accumulator Ring 37
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.
38 2. Application of Orbit Response Matrix Method to the SNS ring
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
2.2 Application of ORM to SNS Accumulator Ring 39
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.
40 2. Application of Orbit Response Matrix Method to the SNS ring
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
2.2 Application of ORM to SNS Accumulator Ring 41
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.
42 2. Application of Orbit Response Matrix Method to the SNS ring
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
2.2 Application of ORM to SNS Accumulator Ring 43
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.
44 2. Application of Orbit Response Matrix Method to the SNS ring
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.
46 2. Application of Orbit Response Matrix Method to the SNS ring
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
50 3. Electron Cloud
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
52 3. Electron Cloud
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
3.2 Electron Cloud 53
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)
54 3. Electron Cloud
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
3.2 Electron Cloud 55
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
56 3. Electron Cloud
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.
3.2 Electron Cloud 57
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;
58 3. Electron Cloud
• 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]:
3.2 Electron Cloud 59
• 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.
60 3. Electron Cloud
• 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
3.2 Electron Cloud 61
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
62 3. Electron Cloud
(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
64 3. Electron Cloud
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
3.3 Observation of Electron-Proton instability at the SNS ring 65
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).
66 3. Electron Cloud
turn number 550 turn number 600
turn number 650 turn number 700
turn number 750 turn number 800
turn number 850 turn number 900
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.
3.3 Observation of Electron-Proton instability at the SNS ring 67
turn number 550 turn number 600
turn number 650 turn number 700
turn number 750 turn number 800
turn number 850 turn number 900
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.
68 3. Electron Cloud
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 Observation of Electron-Proton instability at the SNS ring 69
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.
70 3. Electron Cloud
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.
3.3 Observation of Electron-Proton instability at the SNS ring 71
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.
72 3. Electron Cloud
(a) Horizontal spectrum
(b) Vertical spectrum
Figure 3.8: The instability frequency spectrum for 11 µC proton
bunch.
3.3 Observation of Electron-Proton instability at the SNS ring 73
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.
74 3. Electron Cloud
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.
3.3 Observation of Electron-Proton instability at the SNS ring 75
(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.
76 3. Electron Cloud
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
3.3 Observation of Electron-Proton instability at the SNS ring 77
turn number 365 turn number 415 turn number 465
turn number 515 turn number 565 turn number 615
turn number 665 turn number 715 turn number 765
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.
78 3. Electron Cloud
turn number 365 turn number 415 turn number 465
turn number 515 turn number 565 turn number 615
turn number 665 turn number 715 turn number 765
Figure 3.12: BPM difference signal for strip 2 on frequency spec-
trum. The red represents the BPM sum signal after
scaling and blue is the difference signal after filtering.
3.3 Observation of Electron-Proton instability at the SNS ring 79
turn number 365 turn number 415 turn number 465
turn number 515 turn number 565 turn number 615
turn number 665 turn number 715 turn number 765
Figure 3.13: BPM difference signal for strip 3 on frequency spec-
trum. The red represents the BPM sum signal after
scaling and blue is the difference signal after filtering.
80 3. Electron Cloud
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.
3.3 Observation of Electron-Proton instability at the SNS ring 81
(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.
82 3. Electron Cloud
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
3.3 Observation of Electron-Proton instability at the SNS ring 83
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
84 3. Electron Cloud
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.
3.3 Observation of Electron-Proton instability at the SNS ring 85
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.
86 3. Electron Cloud
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.
3.3 Observation of Electron-Proton instability at the SNS ring 87
(a) ph1 (b) ph3
(c) ph5
Figure 3.19: Horizontal spectrum for ph1, ph3 and ph5.
88 3. Electron Cloud
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.
90 3. Electron Cloud
Conclusion 91
Chapter 4
Conclusion
The accumulator ring of the Spallation Neutron Source is a high intensity proton
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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Recommended