THEORY AND MODELING OF INTENSE ION BEAMS AND DIAGNOSTIC

MEASUREMENTS IN ACCELERATOR FRONT ENDS

By

Chun Yan Jonathan Wong

A DISSERTATION

Michigan State University

in partial fulfillment of the requirements

Submitted to

for the degree of

Physics – Doctor of Philosophy

2019

ABSTRACT

THEORY AND MODELING OF INTENSE ION BEAMS AND DIAGNOSTIC

MEASUREMENTS IN ACCELERATOR FRONT ENDS

By

Chun Yan Jonathan Wong

Beam dynamics in front ends of ion accelerators is often subject to strong space charge and
significant transverse coupling due to magnetic fields in the source and the use of focusing solenoids
in the beam line. This thesis starts with an overview and a brief primer on beam dynamics to frame
two research themes: simulation and analytic studies on beam dynamics, and improvements on
transverse beam diagnostics. Each theme is first developed in a general treatment, followed by
applications to the front end of the Facility for Rare Isotope Beams (FRIB).

The first research theme includes: (1) numerical modelling of accelerator front ends, and (2)
analytic studies of coupled transverse beam dynamics. In (1), a set of readily adaptable simulation
tools based on the PIC code Warp is constructed to model the transport and manipulation of
intense multi-species beams with high levels of detail. The tools are applied to investigate beam
evolution in the FRIB front end including the species separation process. In (2), general analytic
results on beams with rotational symmetry and the formalism of eigen-emittances are derived to
elucidate non-axisymmetric initial conditions at the ion source and beam states advantageous for
solenoid transport. The results are employed to construct LEBT/MEBT (low/medium energy beam
transport) tuning schemes currently adopted at FRIB for transverse matching of strongly magnetized
beams into the RFQ and cryomodules respectively.

The second research theme presents improvements on transverse beam diagnostics including:
(1) corrections in the conventional data analysis model of high-resolution phase space measurements
with Allison scanners, and (2) error minimization in beam matrix measurements and enhanced
tomography capabilities using beam profile monitors. All techniques and algorithms under this
theme have been verified using measurements at FRIB.

To my family

iii

ACKNOWLEDGEMENTS

First and foremost, I would like to express my heartfelt gratitude to my advisor, Professor Steven
Lund, for his unfailing guidance and support. He granted me the freedom to pursue a variety of
different projects, and always made time with a single knock on his door to share his wisdom in
physics as well as in life. As a scientist, his excellence and integrity set the standards I aspire to
meet, and his enthusiasm is a constant source of encouragement. As a mentor and teacher, his
patience, his vision for education, and his strong commitment to preparing all students to succeed,
continue to inspire me to contribute to the accelerator physics community. I was very fortunate to
have the joy of learning from him.

It has been my great privilege to work with the Accelerator Physics (AP) Department at FRIB. I
thank Professor Peter Ostroumov for maintaining an exciting research environment in which I was
nurtured, and for giving me invaluable opportunities to participate in beam studies. His remarkable
intuition and astute judgment played an important role in shaping the course of my research. I am
grateful to Professor Yue Hao who always expressed genuine interest in my work and well-being.
I greatly benefited from his immense knowledge and career advice.

I am also indebted to many congenial collaborators in AP: Kei Fukushima for many fruitful
discussions on beam simulations; Tomofumi Maruta for his physical insight in beam tuning and
his generous supervision of my experimental studies; Alexander Plastun for his prowess in beam
optics and his perceptive observation of the devils lying in the details; Takashi Yoshimoto for
his intriguing work; Tong Zhang for the joint efforts in application development that brought my
understanding of accelerator control system to a whole new level; and Qiang Zhao for sharing his
practical wisdom and extensive experience. They taught me a lot and their work constantly fueled
my interest and curiosity. I would like to take this opportunity to wish AP and the FRIB project
resounding success in the years to come.

I owe many thanks to Professor Steven Lund (chair), Professor Philip Duxbury, Professor Yue
Hao, Professor Peter Ostroumov, Eduard Pozdeyev and Professor Vladimir Zelevinsky for serving

iv

on my thesis committee and providing invaluable feedback. Professor Yoshishige Yamazaki also
shared thoughtful comments on countless occasions. I gratefully acknowledge Professors Martin
Berz and Kyoko Makino for their courses and textbooks on beam physics. The unique techniques
and approaches I learned have been very beneficial to my understanding and research work.

I would like to thank Scott Cogan, Diego Omitto and Rebecca Shane for their technical support
on Allison scanner measurements at FRIB; Guillaume Machicoane and Jeff Stetson for their insight
on ECR ion sources and the associated beam transport; and Haitao Ren for his support on modeling
of the front end. Special thanks must be given to Steven Lidia for many enlightening conversations
on beam diagnostics that provided much-needed reality checks and refocused my work on the
important problems.

My research updates were regularly presented at a meeting that mainly consists of MSU graduate
students in accelerator physics. I thank Daniel Alt, Michael Balcewicz, Crispin Contreras, Robert
Hipple, Bryan Isherwood, Andrew LaJoie, Didi Luo, Derek Neben, Christopher Richard, and
Safwan Shanab for their questions and feedback that helped me understand my work. I am also
grateful to Harry He, Felix Marti and Alfonse Pham who were only too willing to provide their
astute comments. I appreciate the assistance from Kim Crosslan at the Department of Physics and
Astronomy that allowed me to meander through all the paperwork in graduate school.

I am deeply indebted to the US Particle Accelerator School for the invaluable learning oppor-
tunities and social interactions that have been immensely beneficial to my thesis research and my
growth in the field of accelerator science. I am grateful to the staff Susan Winchester and Irina
Novitski for making every session such a memorable and addicting experience.

Last but not least, I owe a special debt to Professor Simon Yu, my master’s thesis advisor at the
Chinese University of Hong Kong, for guiding me into the field of accelerator physics and laying
the foundation in both knowledge and research mentality.

v

TABLE OF CONTENTS

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.
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LIST OF TABLES .
LIST OF FIGURES .
CHAPTER 1

.
.
.
.
INTRODUCTION .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 High Intensity Ion Accelerators . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Facility for Rare Isotope Beams (FRIB)
. . . . . . . . . . . . . . . . . . . . . . .
1.3 Electron Cyclotron Resonance (ECR) Ion Sources . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . .
1.4 Beam Dynamics Challenges in Accelerator Front Ends

ix

x

1
1
2
4
5

CHAPTER 2 A PRIMER ON BEAM PHYSICS . . . . . . . . . . . . . . . . . . . . . .
2.1 Particle Motion and Phase Space . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Symplectic Dynamics, Beam Matrices and Emittance Conservation

7
7
. . . . . . . . 10
Symplectic Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2.1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2.2 Beam Matrices
2.2.3 Emittance Conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3
2D Linear Phase Space Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4 Axisymmetry and Conservation of Canonical Angular Momentum . . . . . . . . . 14

.

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.

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3.1 Warp Code .
3.2 Beam Line Element Models
3.3 Simulations of FRIB Front End

CHAPTER 3 SELF-CONSISTENT SIMULATIONS . . . . . . . . . . . . . . . . . . . . 16
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
. . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.3.1
Initial Conditions & Loading . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.3.2 Lattice Design & Matching Methodology . . . . . . . . . . . . . . . . . . 19
3.3.3 Envelope Model for Multi-Species Beam Transport
. . . . . . . . . . . . . 23
Simulation Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.3.4
3.3.5 Warp Simulation Results: Case Comparisons
. . . . . . . . . . . . . . . . 26
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.4 Further Work .

.

.

.

.

.

CHAPTER 4 THEORY OF TRANSVERSE COUPLED BEAM DYNAMICS, WITH

4.1 Beams with Rotational Symmetry . . . . . . . . . . . . . . . . . . . . . . . . .

APPLICATIONS TO THE FRIB FRONT END . . . . . . . . . . . . . . . 34
. 34
4.1.1 Effect of Symmetry on Beam Moments
. . . . . . . . . . . . . . . . . . . 34
4.1.2 Derivation of Moment Relations from Symmetry . . . . . . . . . . . . . . 37
4.1.3 Correspondence between Discrete and Continuous Rotational Symmetry
. 39
. . . . . . . . . . . . . . . . . . . . 42
4.1.4 Example: Beams with C3 Symmetry
. 42
. 44
4.2 Coupled Beam Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.2.1 Axisymmetric Beams and Eigen-emittances . . . . . . . . . . . . . . . . . 46
4.2.2 Magnetized Beam .
. 47

2nd Order Moments . . . . . . . . . . . . . . . . . . . . . . .
3rd Order Moments
. . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.1.4.1
4.1.4.2

vi

. . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . .

4.2.3 Mirror-Symmetric Beam Line Does Not Preserve Axial-symmetry . . . . . 49
. 51
4.2.4 Linear Solenoid Transport
4.2.5 Quasi-axisymmetric Conditions
. 52
4.3 Applications to FRIB Front End . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.3.1 LEBT: Before Species Selection . . . . . . . . . . . . . . . . . . . . . .
. 55
4.3.2 LEBT: After Species Selection . . . . . . . . . . . . . . . . . . . . . . . . 56
4.3.3 MEBT and LS1 .
. 61
4.3.3.1 Optics Design and Matching Recipe . . . . . . . . . . . . . . . . 62
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
Further Work .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.3.4

.

.

CHAPTER 5 TRANSVERSE PHASE-SPACE MEASUREMENTS WITH ALLISON-

.

.

.

.

.

. .

5.2.2

Introduction .

TYPE EMITTANCE SCANNERS . . . . . . . . . . . . . . . . . . . . . . 65
5.1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.2 Additional Geometric Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.2.1 Geometric Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
Longitudinal Asymmetry in E-dipole Placement (l1 (cid:44) l2) . . . . . 68
5.2.1.1
Finite Slit Thickness d (cid:38) Slit Width s . . . . . . . . . . . . . . . 68
5.2.1.2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
Four Cases .
Particle Simulations
. . . . . . . . . . . . . . . . . . . . . . . . 70
5.2.2.1
Particle Transmission Quantified . . . . . . . . . . . . . . . . . . 72
5.2.2.2
5.2.2.3 Analytic Results
. . . . . . . . . . . . . . . . . . . . . . . . . . 75
Two Corrections in Data Analysis . . . . . . . . . . . . . . . . . . . . . . 78
5.2.3.1 Voltage-to-Angle Relation . . . . . . . . . . . . . . . . . . . . . 78
5.2.3.2 Voltage-Dependent Weight Compensation . . . . . . . . . . . . . 78
5.2.4 Analysis of Synthetic Data . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.3 Corrections for Degenerate Phase Space Measurements . . . . . . . . . . . . . . . 83
Example: FRIB Allison Scanner . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.4
5.4.1
FRIB Front End . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.4.2 Quadrupole Scan Measurements . . . . . . . . . . . . . . . . . . . . . . . 90
Total Current Normalization Anomaly . . . . . . . . . . . . . . . . . . . . 93
5.4.3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.5 Conclusion . .

5.2.3

. .

.

.

.

.

.

.

CHAPTER 6 OPTIMIZED TRANSVERSE PHASE-SPACE MEASUREMENTS WITH

6.1 Review of Linear Algebra .

BEAM PROFILE MONITORS . . . . . . . . . . . . . . . . . . . . . . . . 97
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
. . . . . . . . . . . . . . . . . . . . . . . . 98
6.1.1 Least Squares Approximation
6.1.2
Singular Value Decomposition . . . . . . . . . . . . . . . . . . . . . . . . 99
6.1.3 Vector Norms and Matrix Norms . . . . . . . . . . . . . . . . . . . . . . . 99
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.1.4 Condition Number
. 104
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
1st Order Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
2nd Order Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
. . . . . . . . . . . . . . . . . . . . . . . 107
. 108

6.2.1 Notation .
6.2.2
6.2.3
6.2.4 A Limitation of Solenoid Scans

6.2 Quadrupole Scan and Solenoid Scan: Working Principles . . . . . . . . . . . . .

6.3 Quadrupole Scan and Solenoid Scan: Error Quantification and Minimization . .

. .

. .

.

vii

6.3.1 Error Quantification . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3.2 Error Minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. 108
. 108
6.4 Spatial and Phase Space Tomography . . . . . . . . . . . . . . . . . . . . . . . . . 111
. 111
. 114
6.5 Example: FRIB Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.6 Further Work .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

Spatial Tomography
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
Phase Space Tomography . . . . . . . . . . . . . . . . . . . . . . . . . .

6.4.1
6.4.2

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APPENDICES .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
APPENDIX A EXTENSIONS ON ALLISON SCANNER ANALYSIS . . . . . . . . 123
APPENDIX B ALTERNATIVE PROOF OF RELATIONS GOVERNING 2ND

ORDER MOMENTS OF CN BEAMS . . . . . . . . . . . . . . . . . 136
APPENDIX C GENERATING A 4D COUPLED DISTRIBUTION . . . . . . . . . . 144
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

BIBLIOGRAPHY .

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viii

LIST OF TABLES

Table 3.1: Beam Parameters of U34+ based on the VENUS-like source. For definitions,
see also Sec. 3.3.3 and ??. Temperatures and envelope parameters are the
same for all ion species (indices suppressed). Canonical angular momentum,
radial normalized emittance, and kinetic energy/rigidity vary with ion species. . . 21

Table 4.1: Beam states entailed by the requirement of 2nd-order axisymmetry in the lower

LEBT. (cid:88) and X denote true and false respectively.

. . . . . . . . . . . . . . . . 60

Table 4.2: First set of quadrupole scan parameters

. . . . . . . . . . . . . . . . . . . . . . 61

Table 5.1: Geometric parameters applied in examples corresponding to the four geometric

models in Fig. 5.2 .

. .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

Table 5.2: Analytic formulae corresponding to the geometric models in Fig. 5.2 for V0 ≥
0. Results for V0 < 0 obey the same formulae with V0 → |V0| and the x-axis
reversed.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

. .

.

.

. .

.

Table 5.3: Beam parameters of the three distributions in Fig. 5.8.

. . . . . . . . . . . . . . 84

Table 5.4: Beam parameters of the two distribution in Fig. 5.10 compared against those

of the waterbag distribution . . . . . . . . . . . . . . . . . . . . . . . . . . .

. 87

Table 5.5: Allison scanner geometries at FRIB front end, before and after modification

in Feb 2018. . .

. .

. .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

Table 5.6: Beam parameters of corresponding to the phase space distribution at VQ3 =
. . . . . . .

1000 V in Fig. 5.13, computed with and without correction terms.

. 92

Table 5.7: Solution of Eq. (5.11) for the quadrupole scan in Sec. 5.4.2, with measurement

results x f calculated using conventional and improved analysis method respectively. 93
Table 6.1: Quadrupole parameters for Scan 1 . . . . . . . . . . . . . . . . . . . . . . . . . 116

Table 6.2: Quadrupole parameters for Scan 2 . . . . . . . . . . . . . . . . . . . . . . . . . 117

Table 6.3: Beam moments and emittances corresponding to the phase space distributions

plotted in Fig. 6.3.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

Table 6.4: Comparison between beam moments of the reconstructed spatial distribution

in Fig. 6.11 and the values obtained from wire scanner measurements.

. . . . .

. 119

ix

LIST OF FIGURES

Figure 1.1: High intensity ion accelerators (Image courtesy of Prof. Jie Wei at FRIB) [1] .

.

Figure 1.2: Schematic of FRIB superconducting driver linac. Note that part of the front
end is located on ground level, a complete illustration of the front end is
shown in Fig. 1.3 .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

Figure 1.3: Schematic of the FRIB front end with emphasis on the location of diagnostics
devices. Abbreviations are listed below: BCM - beam current monitor; BPM
- beam position monitor; FC - Faraday cup; MHB - multi-harmonic buncher;
PM - profile monitor.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

Figure 1.4: Schematic of an electron cyclotron resonance (ECR) ion source (Image cour-

tesy of Dr. Daniele Leitner at LBNL).

. . . . . . . . . . . . . . . . . . . . . .

Figure 1.5: Measured charge state distribution of a uranium beam from the VENUS ion

source [9].

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 2.1: Frenet-Serret coordinate system. Image adapted from Ref. [11]
Figure 2.2: Ellipse in x-x(cid:48) phase space [17].
Figure 3.1: CST Studio model of S4 solenoids at FRIB and the calculated fields at different

. . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . .

. 13

2

3

4

5

6

8

radii within the clear bore aperture radius R = 164.2/2 mm.

. . . . . . . . . . 19

Figure 3.2: CST Studio model of a Q7 electrostatic quadrupole triplet at FRIB with

dimensions and potential plots.

. . . . . . . . . . . . . . . . . . . . . . . . . . 20

Figure 3.3: CST Studio model of a Q7 electrostatic quadrupole triplet at FRIB. Dimen-

sions and potential contours at various axial locations are shown.

. . . . . . . . 21

Figure 3.4: CSS lattice design used in Warp simulations. . . . . . . . . . . . . . . . . . .

. 22

Figure 3.5: Comparisons between numerical solutions of the envelope equation and Warp
simulation results in the transport line upstream of the CSS. These correspond
to simulation cases 3 and 4 described in Sec. 3.3.4.

. . . . . . . . . . . . . . . 24

Figure 3.6: Accessible lattice functions β and α at the entrance of the first dipole when the
Bpeak in both solenoids varied independently between 0 T and 1.5 T. Results
are shown for different space charge neutralization factors. Observe that there
are regions of initial Twiss parameters that are inaccessible, and the region of
accessible Twiss parameters decreases with decreasing neutralization factor

. . 25

x

Figure 3.7: Warp simulation results of Uranium species separation and collimation in the
beam evolution from the ECR source extraction through the midpoint of the
charge selection system showing collimation to two target species for high
intensity FRIB operation.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

Figure 3.8: Rms envelops σx,y a) and normalized rms-emittance εx,y b) of the reference
species U34+ in the CSS. c) shows the centroid of U33+ and x y-particle
projections of both target species at the CSS mid-point for two cases. Solid
curves are x-plane and dashed curves are y-plane. Colors flag Warp sim-
ulation cases. Black solid lines denote MADX design values in a) and c).
The axial midplane of the CSS is indicated in c) (z in FRIB coordinates with
origin outside machine). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Figure 3.9: Extraction geometry of the VENUS-like ECR ion source. The electrode in
red has negative voltage to prevent backflow of electrons. (Image courtesy of
Dr. Daniel Winklener.)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Figure 3.10: Beam sizes in the post-extraction beam line downstream of the VENUS-like
ECR at FRIB, obtained from Warp simulations with different homogeneous
neutralization factors.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Figure 4.1: Simulated spatial distribution of Ar9+ ions at the extraction plane of the
Artemis [5] ECR ion source at FRIB. The inner circle indicates the extraction
aperture. x-y and u-v denote two possible transverse coordinate systems.
Subsequent analysis will show that, despite severe triangulation, the rms
envelope and emittance of the beam is the same along any transverse direction.
Simulation results courtesy of the code developed by Dr. Vladirmir Mironov
at JINR-Dubna [25, 26]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

Figure 4.2: A schematic of transverse focusing and bending elements in the FRIB front
end. Note that Segment 2 and 5 are mirror-symmetric about the axial
mid-plane. Lengths are not drawn to scale and all quadrupoles are normal
quadrupoles (i.e. upright orientation).

. . . . . . . . . . . . . . . . . . . . . . 55

Figure 4.3: An example of FRIB LEBT tuning results where the focusing strengths of 17
elements (10 quadrupoles and 7 solenoids) were optimized to achieve desired
beam conditions at the RFQ entrance.
(Image courtesy of Dr. Tomofumi
Maruta at FRIB Accelerator Physics Department)

. . . . . . . . . . . . . . . . 58

Figure 4.4: An example of FRIB LEBT tuning results where the tuning scheme introduced
in Sec. 4.3.2 was applied to optimize elements strengths in three stages.
(Image courtesy of Dr. Tomofumi Maruta at FRIB Accelerator Physics
Department) .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

. .

. .

xi

Figure 5.1: Side-view of the FRIB Allison scanner. . . . . . . . . . . . . . . . . . . . . . . 69

Figure 5.2: Four geometric cases and their respective characteristic trajectories. Case I:
l1 = l2 = l and d = 0; Case II: l1 (cid:44) l2 and d = 0; Case III: l1 = l2 = l and
d (cid:44) 0; Case IV: l1 (cid:44) l2 and d (cid:44) 0. zi is the longitudinal position where the
beam particles enter the device and where the x-x(cid:48) phase space is measured.
Figure 5.3: Blue dots denote particles that are transmitted in the simulations described
in Sec. 5.2.2.1. They map x-x(cid:48) phase space regions at the device entrance
z = zi that contribute to the data point at x = 0 mm. Results are shown for
all four geometric cases in Fig. 5.2 for (a) V0 = 500V and (b) V0 = 1000V
respectively. Geometric parameters for each case are listed in Table 5.1.
Black lines define the phase space grid with increments determined by the
scanning step sizes. Dashed blue lines indicate the nominal position x = 0
and angle x(cid:48) = x(cid:48)
ref corresponding to the data point. The red rectangles denote
. .
the device resolution predicted by the conventional treatment (i.e. Case I).

. . 71

. 73

.

. .

. .

Figure 5.4:

Illustrates how the shrunk phase space area attains its shape in Case III of
Fig. 5.3b. .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
Figure 5.5: Angular transmission factor T(x(cid:48)) for Cases I to IV in Fig. 5.3b. . . . . . . . . . 75
Figure 5.6: Voltage-to-angle relation for the selected beam. The red curve (analytic)

overlays the blue curve (ballistic simulation). . . . . . . . . . . . . . . . . . . . 79

Figure 5.7: Plots of a) angular resolution ∆x(cid:48), and b) angular correction factor as a
ref. The red curve (analytic) overlays the blue curve (ballistic
. .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

function of x(cid:48)
simulation).

. .

Figure 5.8: Uniform phase space projection of the KV distribution, compared against
results of synthesized measurement data analyzed using conventional and
improved analysis methods. The left plot shows a sampling (1 out of 1000
particles) used to represent the distribution. The middle and right plots are
fine mesh density plots with the same color scale.

. . . . . . . . . . . . . . .

. 83

Figure 5.9: Phase space region contributing to the data point at x = 0 mm and V0 = 200
V (presentation format as detailed in Fig.5.3), where the Allison scanner
geometry corresponds to Case IV of Table 5.1 except for increased slit width
to s = 300 µm. Scanning step sizes in x and V0 are 0.5 mm and 20 V respectively. 84

xii

Figure 5.10: Waterbag distribution (left column), compared against synthesized measure-
ment data analyzed without (middle column) and with (right column) correc-
tions for degenerate measurements. The upper row corresponds to intensity,
whereas the lower row shows all non-zero intensity points in the same color
to highlight extent errors. Black and white ellipses denote the ideal boundary
of the waterbag distribution analyzed. In the left column sampled particles (1
out of 1000) are plotted. Measurement data are synthesized with s = 300 µm,
and scanning step sizes of 0.5 mm and 10 V.

. . . . . . . . . . . . . . . . . . . 86

Figure 5.11: Fraction of particles that fall outside the ellipse (orientation defined by Twiss
parameters) with area πε in each of the three distributions in Fig. 5.10. The
vertical dashed line corresponds to the sharp phase space edge of the waterbag
distribution at ε/εrms = 6. The blue curve approaches ε/εrms = 6 if the lower
range of the ordinate is extended.

. . . . . . . . . . . . . . . . . . . . . . . . . 87

Figure 5.12: Upstream end of the low energy beam transport (LEBT) line of the FRIB

front end. .

.

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

Figure 5.13: y-y(cid:48) phase-space projections measured at the FRIB front end during a quadrupole

scan. VQ3 denotes the voltage at the last quadrupole upstream of the Allison
scanner.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

.

.

.

.

.

.

.

Figure 5.14: Projected beam currents from Allison scanner measurements in a quadrupole
scan, compared against the beam current measured at the Faraday Cup slightly
upstream (see Fig. 5.12).

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

Figure 5.15: Screenshot of measured currents on the x-direction Allison scanner on the
Artemis beam line of FRIB, plotted against the bias voltage of the Faraday
cup on the scanner. The scanner position and voltage were fixed at a phase
space coordinate with high density throughout the measurements. . . . . . . . . 95

Figure 6.1: Schematic of the beam line section containing the first profile monitor at the

FRIB Front End. .

. .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

Figure 6.2: Schematic of a wire scanner at the FRIB front end.

Image courtesy of

Tomofumi Maruta at FRIB.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

Figure 6.3: A diagonal wire only gives the total current in each diagonal because and each
grid cell along a diagonal always contributes equally to a ray. The alternating
diagonals are plotted using chessboard coloring on the right.

. . . . . . . . . . 113

Figure 6.4: A wire intersecting a 2 : 1 grid at 45o is equivalent to the case of a wire

intersecting an even grid at 63o. . . . . . . . . . . . . . . . . . . . . . . . . . . 113

xiii

Figure 6.5: For a rotated grid, the wires are no longer vertical, horizontal or diagonal

with respect to it.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

Figure 6.6: Spatial tomography of several synthesized distributions.

. . . . . . . . . . . . . 115

Figure 6.7: Schematic of the beam line section containing the first profile monitor at the
FRIB Front End. The profile monitor is located at the position designated by
f and the beam was reconstructed at position i. Green blocks denote identical
(Q7 type) electrostatic quadrupoles. . . . . . . . . . . . . . . . . . . . . . . . . 116

Figure 6.8: Projection angles on the initial phase space corresponding to quadrupole scan

parameters in Table 6.1 and Table 6.2.

. . . . . . . . . . . . . . . . . . . . . . 117

Figure 6.9: Histograms of the normalized x-emittance with random errors applied to the
measurement results. The applied errors have a truncated Gaussian distribu-
tion where 3σ = 1% for Scan 1 and 3σ = 10% for Scan 2. . . . . . . . . . . . . 118

Figure 6.10: Comparison between tomographic reconstruction of the x-x(cid:48) phase space
distribution, and the measured distribution at the Allison scanner propagated
to the same location. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

Figure 6.11: Spatial tomography with a grid rotated by 20o clockwise with respect to the

standard x-y coordinate system. . . . . . . . . . . . . . . . . . . . . . . . . . . 119

Figure A.1: Potential contours of the FRIB Allison scanner in Fig. 5.1, including details

of the fringe structure in the vicinity of the entrance slit (right).

. . . . . . . . . 125

Figure A.2: How (a) shifting a trajectory for fixed angle and (b) changing the angle for

fixed initial position allow one to calculate quantities listed in Table 5.2.

. . . . 126

Figure A.3: Reversing particle velocity allows one to check results when the E-dipole

placement is asymmetric.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

Figure A.4: Comparison between transmission ratio in Case III and IV when weight

correction effect is large.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

Figure A.5: Beam region is defined by the red ellipse. . . . . . . . . . . . . . . . . . . . . . 130

Figure A.6: Histogram of current values at data points outside the defined beam region. . . . 130

Figure A.7: Non-zero data points after background subtraction. . . . . . . . . . . . . . . . . 131

Figure A.8: Non-zero data points of the beam distribution, before and after applying the

island filter.

.

. .

. .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

xiv

Figure A.9: Beam parameters setup screen for the Allison scanner GUI. (Image courtesy

of Zhang Tong from the Accelerator Physics Department at FRIB).

. . . . . . . 134

Figure A.10:Noise correction screen for the Allison scanner GUI. (Image courtesy of

Zhang Tong from the Accelerator Physics Department at FRIB). . . . . . . . . . 135

xv

CHAPTER 1

INTRODUCTION

This chapter overviews high intensity ion accelerators and introduces the Facility for Rare Isotope
Beams (FRIB). Electron cyclotron resonance (ECR) ion sources and beam dynamics challenges at
front ends of high intensity ion accelerators are discussed.

1.1 High Intensity Ion Accelerators

A summary of high intensity ion accelerators operating or being planned around the globe is

shown in Fig. 1.1. They serve a variety of purposes including:

1. The intensity frontier in high energy physics, which uses secondary or tertiary beams of

neutrinos, muons and kaons (e.g. PIP-II, USA; J-PARC, Japan).

2. Nuclear physics (e.g. FRIB, USA; RIKEN, Japan; RISP, Korea).

3. Spallation neutron production (e.g. SNS, USA; ISIS, UK; ESS, Sweden).

4. Accelerator driven system (ADS) for nuclear waste transmutation and subcritical power

generation (e.g. CADS, China; MYRRHA, Belgium).

5. Plasma physics with beam-driven warm dense matter or high-energy-density physics (e.g.

FAIR, Germany; HIAF, China).

A comprehensive overview as of 2014 can be found in Ref. [1]. A common metric of high intensity
ion accelerators is the average beam power on target defined by (maximum beam energy × average
beam current). Proton accelerators have already achieved beam power above 1 MW, and multiple
MW to >10 MW facilities are presently under construction. For heavy ion accelerators, FRIB is set
to advance the continuous-wave (CW) power frontier by two orders of magnitude when it reaches
full operating specifications at 400 kW. The FRIB accelerator is described in the next section.

1

Figure 1.1: High intensity ion accelerators (Image courtesy of Prof. Jie Wei at FRIB) [1]

1.2 Facility for Rare Isotope Beams (FRIB)

The Facility for Rare Isotope Beams (FRIB) [2] is currently being constructed at Michigan State
University. FRIB consists of a CW superconducting radiofrequency (SRF) linear accelerator that
aims to accelerate all stable isotopes to energies >200 MeV/u and attain up to 400 kW CW beam
power on target. Secondary rare isotope beams generated at the target will be separated in-flight
and transferred to experimental halls as fast, stopped or re-accelerated beams. Re-acceleration
capabilities are enabled by a SRF ReAccelerator (ReA).

A schematic of the FRIB driver linac is shown in Fig. 1.2.

It is approximately 470 m in
length and consists of over 300 superconducting quarter-wave and half-wave coaxial resonators
distinguishable into four types that are suited to different beam velocities [3]. The linac has a folded
“paper-clip” layout which facilitates beam collimation after stripping at the end of linac segment 1
to enhance acceleration efficiency. A novel liquid lithium stripper will be employed [4].

The warm front end section of FRIB is illustrated in Fig. 1.3. There will be two electron
cyclotron resonance ion sources (ECRIS, see section below) at the front end: ECRIS-1, ARTEMIS

2

Figure 1.2: Schematic of FRIB superconducting driver linac. Note that part of the front end is
located on ground level, a complete illustration of the front end is shown in Fig. 1.3

[5], is a normal conducting source that was commissioned in September 2016; ECRIS-2, VENUS-
like [6], is a superconducting source that can operate at higher frequency to produce higher currents
and higher charge states in heavy ions like uranium. ECRIS-2 will be commissioned in two years’
time. The beam undergoes three stages of acceleration in the front end (see Fig. 1.3):

1. Accelerated by a 20 kV to 30kV potential upon extraction from the ECR.

2. Accelerated to 12 keV/u by an electrostatic gap prior to charge selection.

3. Accelerated to 500 keV/u by the RFQ.

Species selection is achieved by a charge selection system that consists of two 90-degree bends for
generating sizable charge separation. The lattice includes six electrostatic quadrupoles and the two
dipoles include a linear field gradient to enhance dispersion for species separation. After charge
selection, the beam line has transverse focusing (magnetic solenoids and electrostatic quadrupoles)
and bending elements (electrostatic) for transporting the beam to the RFQ with design parameters,
as well as choppers and bunchers for longitudinal prebunching of the beam before it enters the
RFQ.

A variety of diagnostic devices are installed along the beam line for monitoring beam transport
[7]. Wire profile monitors are the most abundant, and play a key role in trajectory corrections
throughout the LEBT. There are also scintillator view screens and phase space measurement
devices in the form of Allison-type scanners (operational) and pepper-pot emittance meters (not

3

Figure 1.3: Schematic of the FRIB front end with emphasis on the location of diagnostics
devices. Abbreviations are listed below: BCM - beam current monitor; BPM - beam position
monitor; FC - Faraday cup; MHB - multi-harmonic buncher; PM - profile monitor.

yet operational). We will discuss improved diagnostic measurements with Allison scanners in
Chapter 5, and with profile monitors in Chapter 6.

1.3 Electron Cyclotron Resonance (ECR) Ion Sources

Electron cyclotron resonance (ECR) ion sources [8] are widely used in high intensity heavy
ion accelerators because they can stably produce high currents of multiply charged ions with high
phase space density from a large variety of stable elements. A schematic of an ECR ion source
is shown in Fig. 1.4. The plasma is confined axially by two solenoids that constitute a magnetic
bottle, and radially by the sextupole field. Microwaves heat the electrons in the plasma, which in
turn collide with ions to produce a high density of high charge state ions that are continuous (DC
beam) extracted.

The beam extracted from an ECR ion source contains ions of many different charge states. An

4

Figure 1.4: Schematic of an electron cyclotron resonance (ECR) ion source (Image courtesy of
Dr. Daniele Leitner at LBNL).

example beam from a VENUS-like ECR tuned for producing maximum current of U33+ has charge
states ranging from 22+ to 40+ as shown in Fig. 1.5.

The magnetic fields of the ECR ion source also induce complication for the dynamics of the
extracted beam. Firstly, there exists a strong axial magnetic field where the ions are extracted, which
gives ions a large canonical angular momentum. Secondly, due to the sextupole in the source, the
extracted density profile of the beam is markedly non-axisymmetric.

1.4 Beam Dynamics Challenges in Accelerator Front Ends

Accelerator front ends must preserve beam quality from the source while eliminating unwanted
ions. This is necessary to achieve maximum beam power while avoiding machine damage down-
stream due to loss of ions in the distribution tail. This section overviews challenges in understanding
beam dynamics in accelerator front ends.

Ion beams in accelerator front ends have low energies and high intensities. Both factors
enhance space-charge effects in the beam evolution. This is further complicated by electron
neutralization caused by ionization of residue gas in the beam pipe. Beams are typically extracted
at a high magnetic field, which imparts large canonical angular momenta onto beam ions when the

5

Figure 1.5: Measured charge state distribution of a uranium beam from the VENUS ion source
[9].

downstream transport is axisymmetric. Thus the transverse phase space of the beam is correlated.
There are also many types of lattice elements, which have nonlinear fields with overlapping fringes.
Transverse-longitudinal coupling also occurs as a result of beam acceleration and bunching.

For heavy ion accelerators which employ ECR-type ion sources, there are further complications
due to characteristics of ECR beams. The multi-species beam has strong space-charge forces with
intricate inter-species interactions that are partially electron neutralized. Next, beam manipulations
are required for species selection. Furthermore, the initial phase space distribution of the beam is
non-axisymmetric. Optimal tuning of the front end in the presence of these complexities is essential
to preserving beam brightness (current / emittance2) from the source and suppressing downstream
beam losses in delicate and expensive SRF cavities. This is critical in high power CW machines
like FRIB.

6

CHAPTER 2

A PRIMER ON BEAM PHYSICS

This chapter overviews key beam physics results relevant to subsequent chapters in this dissertation.

2.1 Particle Motion and Phase Space

The relativistic Hamiltonian of a charged particle in an electromagnetic field is given by [10]:

(cid:17)2
where m is the mass, q is the charge, φ is the scalar potential and −→
P is the canonical momentum
which is related to the kinetic momentum −→p and vector potential −→
A by:

−→
P − q
A

H = eφ + c

m2c2 +

(cid:114)

(cid:16)−→

(2.1)

−→
−→
P = −→p + q
A

(2.2)

In an accelerator, particles evolve about a reference orbit (also called the design trajectory)
that is often a plane curve. Therefore, it is convenient to employ the Frenet-Serret coordinate
system with vanishing torsion to describe particle motion. The notation commonly employed in
the accelerator community is illustrated in Fig. 2.1. s is the path length along the curve from some
initial point whereas the local radius of curvature is denoted by ρ(s).

After adopting the Frenet-Serret coordinate system, the description of particle motion can be
further simplified by noting that the location of beam line elements are defined in s, not t, so it is
desirable to use s as the independent variable. Moreover, particles typically only deviate slightly
from the reference orbit. Normalizing by the longitudinal momentum and expressing phase space
coordinates as differences from those of a reference particle that follows the reference orbit with the
design energy, the coordinates become small quantities that enable simplifying approximations.

The sequence of coordinate changes outlined above can be achieved via a series of canonical
transformations [12]. The resulting Hamiltonian in the absence of applied electric fields and self

7

Figure 2.1: Frenet-Serret coordinate system. Image adapted from Ref. [11]

(cid:18)

fields is:

H = −

1 +

x
ρ

(cid:19)(cid:34)(cid:16)1 + δ

2(cid:17)

(cid:18)

+

px − qAx
p0

(cid:19)2

(cid:18)

+

py − qAy
p0

(cid:19)2(cid:35)1/2

(cid:18)

−

1 +

(cid:19) qAs

p0

x
ρ

+ (1 + δ)

(2.3)

where phase space variables are (x, px, y, py, ∆z, δ). px and py are conjugate momenta to x and y,
which are coordinates in the Frenet-Serret coordinate system after the transformations. Longitudinal
phase space coordinates ∆z and δ express position and energy deviations from the reference particle
as follows:

∆z = −v0(t − t0)
p − p0
p0

δ =

(2.4)

(2.5)

where p0 and v0 are the momentum and velocity of the reference particle respectively, and t0 is
the time at which the reference particle arrives at position s. Transverse magnetic fields can be
expressed solely with As, so Ax and Ay are taken to be zero. Then px and py are related to the

8

mechanical momentum and thus the slope via:

−→p · ˆx
p0
−→p · ˆy
p0

px =

py =

≈ x(cid:48)(1 + δ) ≡ dx
ds
(cid:48)(1 + δ) ≡ dy
≈ y
ds

(1 + δ)

(1 + δ)

(2.6)

(2.7)

Given a beam line element, its vector potential −→

A = As ˆs can be inserted into the Eq. (2.3) to
derive Hamilton’s equations that govern the single-particle dynamics. It is a common approximation
to assume linear motion, whereupon all terms higher than 2nd order in the Hamiltonian are
discarded. Discarding terms in the Hamiltonian guarantees that the resulting linear dynamics is
symplectic, which implies nice properties including the preservation of phase space volume that
we exploit in Sec. 2.2.

With the assumption of small momenta in the linearization of the equations of motion, the
difference between x(cid:48), y(cid:48) and px, py is a higher order term that can be neglected. Therefore, it is
customary to describe linear dynamics using the following formalism:

ˆx(s = s f ) = M(s f |si) ˆx(s = si)

(2.8)

where the ˆx is the phase space vector (x, px, y, py, ∆z, δ) and the its initial (s = si) and final (s = s f )
values are related by a transfer matrix M.

This dissertation mainly studies transverse motion.

If the 2nd-order Hamiltonian does not
couple transverse and longitudinal phase space coordinates, it can be separated into a transverse
part and a longitudinal part. The transverse part will evolve independently as follows:

(2.9)

(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)

(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172)s=s f

x
x(cid:48)

y

y(cid:48)

(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)

(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172)s=si

x
x(cid:48)

y

y(cid:48)

= M⊥

where M⊥ is the 4 × 4 transfer matrix. Furthermore, if the x-x(cid:48) and y-y(cid:48) phase space are not

9

coupled, M⊥ becomes block diagonal, whereupon the dynamics can be further simplified into:

x(cid:48)

(cid:169)(cid:173)(cid:173)(cid:171) x
(cid:169)(cid:173)(cid:173)(cid:171) y

y(cid:48)

(cid:170)(cid:174)(cid:174)(cid:172)s=s f
(cid:170)(cid:174)(cid:174)(cid:172)s=s f

x(cid:48)

(cid:169)(cid:173)(cid:173)(cid:171) x
(cid:169)(cid:173)(cid:173)(cid:171) y

y(cid:48)

(cid:170)(cid:174)(cid:174)(cid:172)s=si
(cid:170)(cid:174)(cid:174)(cid:172)s=si

= Mx

= My

where Mx and My are 2 × 2 matrices.

Analytic forms of transfer maps for standard beam line elements such as dipoles, quadrupoles

and solenoids are readily available in the literature [13].

2.2 Symplectic Dynamics, Beam Matrices and Emittance Conservation

(2.10)

(2.11)

(2.12)

(2.13)

(2.14)

2.2.1 Symplectic Condition

Transfer matrices obey the symplectic condition [14]:

MTSM = S

where S is the block diagnonal symplectic matrix given by:

S =

(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)

s
0
0
s
...
0
0 · · ·

and

(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172)
(cid:170)(cid:174)(cid:174)(cid:172)

· · · 0
0
0
...
. . .
s
0

0 =(cid:169)(cid:173)(cid:173)(cid:171)0 0

0 0

with s =(cid:169)(cid:173)(cid:173)(cid:171) 0

1
−1 0

(cid:170)(cid:174)(cid:174)(cid:172)

2.2.2 Beam Matrices

The second order moments of the distribution of a beam can be naturally represented as the elements
of a matrix. Such a matrix is called the beam matrix or sigma matrix, and is commonly denoted

10

by σ. The dimension of σ depends on the dimension of the phase space under question. In this
dissertation, we often use the 4 × 4 sigma matrix which comprises the 2 × 2 sigma matrices as its
blocks. For a beam with n particles indexed with subscript i that runs from 1 to n, define

y(cid:48)(cid:17)

i

x(cid:48)

y

σ =

x

1
n

x
x(cid:48)

y

y(cid:48)

i=1

(cid:16)
(cid:170)(cid:174)(cid:174)(cid:172)

(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)
(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172)i
n

=(cid:169)(cid:173)(cid:173)(cid:171)σxx σx y
σyy =(cid:169)(cid:173)(cid:173)(cid:171) (cid:104)yy(cid:105)
n


σT
x y σyy

(cid:104)yy(cid:48)(cid:105)

(cid:104)xx(cid:105) =

1
n

i=1

where:

σxx =(cid:169)(cid:173)(cid:173)(cid:171) (cid:104)xx(cid:105)

(cid:104)xx(cid:48)(cid:105)

(cid:170)(cid:174)(cid:174)(cid:172)

(cid:104)xx(cid:48)(cid:105)
(cid:104)x(cid:48)x(cid:48)(cid:105)

and

2.2.3 Emittance Conservation

(cid:170)(cid:174)(cid:174)(cid:172)

σx y =(cid:169)(cid:173)(cid:173)(cid:171) (cid:104)x y(cid:105)

(cid:104)x(cid:48)y(cid:105)

(cid:170)(cid:174)(cid:174)(cid:172)

(cid:104)x y(cid:48)(cid:105)
(cid:104)x(cid:48)y(cid:48)(cid:105)

(cid:104)yy(cid:48)(cid:105)
(cid:104)y(cid:48)y(cid:48)(cid:105)

xi xi

etc.

(2.15)

(2.16)

(2.17)

(2.18)

Following the treatment of Ref. [15], invariants composed of 2nd-order moments of a beam can be
extracted by invoking the symplecticity of the linear transfer map. Consider 4D phase space vector:

(cid:16)

x =

x(cid:48)

x

y

y(cid:48)(cid:17)T

Denote x(s = s2) as x2, x(s = s1) as x1. For x2 = M⊥x1,

σ2S = x2xT2S

= M⊥x1xT1MT⊥S
= M⊥σ1SM−1⊥

11

which shows that σ1S and σ2S are similar matrices and have the same eigenvalues. Therefore, the
eigenvalues of σS are invariants of the motion.

When the transverse motion is decoupled, the proof above also applies individually to 2D phase
spaces x-x(cid:48) and y-y(cid:48) where M⊥ should be replaced by Mx or My, and σ by σxx or σyy. The
resulting conserved eigenvalues can be expressed as the familiar 2D emittances:

(cid:113)(cid:104)xx(cid:105) (cid:104)x(cid:48)x(cid:48)(cid:105) − (cid:104)xx(cid:48)(cid:105)2
(cid:113)(cid:104)yy(cid:105) (cid:104)y(cid:48)y(cid:48)(cid:105) − (cid:104)yy(cid:48)(cid:105)2

εx =

εy =

2.3 2D Linear Phase Space Motion

This section briefly discusses the special status of 2D phase space ellipse in linear motion and
its connection to the Twiss parameters. The treatment is inspired by Ref. [16] and omits discussion
of periodic motion because accelerator front ends are single-pass systems.

Consider an ellipse in x-x(cid:48) phase space (see Fig. 2.2) defined by:

γx2 + 2αxx(cid:48) + βx(cid:48)2 = ε

(2.21)

The equation has redundancy in the sense that the ellipse remains unchanged if one rescales both
sides by the same factor. To remove the redundancy, the normalization βγ−α
2 = 1 is conventionally
chosen because it renders the ellipse area πε. Hence, given ε, two out of the three parameters α, β
and γ completely determines the orientation of the ellipse including its extent and critical points.
In practice, α and β are normally the two parameters used because they naturally connects to an
envelope description of a beam.

The phase space ellipse at s = s1 can be written in matrix form as:

(2.19)

(2.20)

(2.22)

(2.23)

where

(cid:16)

x

α1 β1

x(cid:48)(cid:17)(cid:169)(cid:173)(cid:173)(cid:171)γ1 α1
det(cid:169)(cid:173)(cid:173)(cid:171)γ1 α1

α1 β1

x(cid:48)

(cid:169)(cid:173)(cid:173)(cid:171) x
(cid:170)(cid:174)(cid:174)(cid:172)
(cid:170)(cid:174)(cid:174)(cid:172) = ε
(cid:170)(cid:174)(cid:174)(cid:172) = 1

12

Figure 2.2: Ellipse in x-x(cid:48) phase space [17].

from the normalization constraint. It can be shown that, under uncoupled linear dynamics, an ellipse
in x-x(cid:48) phase space transforms into another ellipse with the same area but different orientation.
Take the transfer map from s1 to s2 to be Mx, the ellipse at s = s2 becomes:

(2.24)

x

α1 β1

x(cid:48)

(cid:16)

x

x(cid:48)(cid:17) MT
(cid:169)(cid:173)(cid:173)(cid:171) x
(cid:169)(cid:173)(cid:173)(cid:171)γ1 α1
(cid:170)(cid:174)(cid:174)(cid:172) = ε
(cid:170)(cid:174)(cid:174)(cid:172) Mx
(cid:169)(cid:173)(cid:173)(cid:171)γ2 α2
(cid:169)(cid:173)(cid:173)(cid:171)γ1 α1
(cid:170)(cid:174)(cid:174)(cid:172) Mx
(cid:170)(cid:174)(cid:174)(cid:172) = MT
MT

det(cid:169)(cid:173)(cid:173)(cid:171)γ2 α2
(cid:169)(cid:173)(cid:173)(cid:171)γ1 α1
(cid:170)(cid:174)(cid:174)(cid:172) = det
(cid:170)(cid:174)(cid:174)(cid:172) Mx
(cid:17) det(cid:169)(cid:173)(cid:173)(cid:171)γ1 α1
= det(cid:16)MT
(cid:170)(cid:174)(cid:174)(cid:172) det(Mx)

α1 β1

α1 β1

α1 β1

α2 β2

x

x

x

α2 β2

= 1

where:

is represents an ellipse with a different orientation but the same area because

Since a phase space ellipse only changes orientation under uncoupled linear optics, it is the
ideal geometric object in phase space for describing a beam. For a beam that is elliptical in phase

13

space, α and β are sufficient to describe the orientation and determine how it evolves under linear
optics. α, β and γ, are called the Twiss parameters of a beam. For a beam that is not elliptical in
phase space, there is no unique way to define its Twiss parameters. The commonly adopted strategy
is to calculate its second order moments and use them to define an ellipse by asking: if a uniformly
filled ellipse were to have these second order moments, what would its Twiss parameters be? The
correspondence to beam moments under this perspective is summarized by:

βx =

γx =

αx =

(cid:104)xx(cid:105)
εx
(cid:104)x(cid:48)x(cid:48)(cid:105)
εx
− (cid:104)xx(cid:48)(cid:105)

εx

(2.25)

(2.26)

(2.27)

where εx is the rms emittance defined in Eq. (2.19). The beam is then represented by such an rms
phase space ellipse, with area εx,or 4εx depending on the convention. The rms phase space ellipse
should behave just as another other ellipse in phase space, provided the optics is uncoupled and
linear.

2.4 Axisymmetry and Conservation of Canonical Angular Momentum

We know from Noether’s Theorem [14] that every continuous symmetry of the Lagrangian of
a system gives rise to a conserved quantity. For a charged particle in an axisymmetric beam line,
the conserved quantity is the canonical angular momentum which can be derived as follows.
The Lagrangian for a relativistic charged particle in an electromagnetic field is given by:

L = −mc2

γ

− eφ + e−→

v · −→

A

(2.28)

is the velocity and γ is the Lorentz factor. In cylindrical coordinates, the Lagrangian

v ≡ d−→x

dt

where −→
reads:

L(r, (cid:219)r, θ, (cid:219)θ, z, (cid:219)z) = −

(cid:112)1 − ((cid:219)r2 + r2 (cid:219)θ

mc2

2 + (cid:219)z2)/c2 − qφ + q(cid:0)(cid:219)r Ar + r (cid:219)θ Aθ + (cid:219)zAz(cid:1)

(2.29)

where r is the radial coordinate and θ is the azimuthal angle. For an axisymmetric system, neither
φ(r, z) nor −→
A(r, z) depends on θ. Therefore, θ becomes a cyclic coordinate of L, and the canonical

14

angular momentum:

∂L
∂ (cid:219)θ

≡ Pθ = γmr2 (cid:219)θ + qr Aθ

(2.30)

is a conserved quantity. Since the conservation of Pθ holds for every particle, for a single species
mononenergetic beam:

(cid:68)

(cid:69)

(cid:104)Pθ(cid:105) = γm

r2 (cid:219)θ

+ q (cid:104)r Aθ(cid:105) = const

(2.31)

This relation places a strong constraint on transverse single-particle dynamics in axisymmetric
focusing systems. In practice, asymmetries due to misalignments and field errors will induce some
degree of violation to this constraint.

15

CHAPTER 3

SELF-CONSISTENT SIMULATIONS

This chapter describes the development of a set of self-consistent simulation tools that are readily
adaptable to simulations of accelerator front ends with high levels of detail. Simulations are crucial
to understanding the beam dynamics because the effects of non-ideal applied fields, intricate lattice
geometries and intense non-uniform space-charge cannot be described purely analytically. We
applied these simulation tools to analysis the VENUS-like beam line at the FRIB front end at full
power. Many effects were discovered with causes including beam magnetization, space charge,
field nonlinearities and their interplay.

3.1 Warp Code

Warp [18, 19] is a open source particle-in-cell (PIC) code well-suited for self-consistent mod-
eling of high intensity ion beams. In additional to an extensive list of standard beam line elements
such as dipoles, quadrupoles and solenoids; the code enables time-dependent fields and detailed
specifications of conductor surfaces and applied fields from imported field maps. This capability,
discussed in more detail in the section below, is particularly useful for modeling accelerator front
ends given the variety of beam elements involved. Warp also allows simulations with an arbitrary
number of particle species including multiple ions with multiple charge states, which is necessary
in the modeling of beams extracted from an ECR ion source.

Warp simulations can be run in various modes including: 1) full 3D; 2) axisymmetric (r − z);
and 3) transverse slice (x − y). The code has a variety of electrostatic and electromagnetic field
solvers (including capabilities like adaptive and static mesh refinement) and is built around an
adaptable python interface. Assuming there is no axisymmetry, the user can choose between full
3D mode and slice mode depending on whether longtudinal self-field is important. The beam can be
initialized with pseudoequilibrium distributions adapted for space-charge[20], or with distributions
obtained from other simulations or experimental data.

16

We have written a set of wrapper modules in Python that allows one to conveniently:

1. Set up beam lines by specifying the locations of focusing elements, conductors and beam

collimators at several levels of detail.

2. Generate initial beam conditions given the current and phase space parameters of each ion

species.

3. Assign diagnostics and record simulation results.

4. Control global properties and parameters of the simulation.

Transverse x-y slice simulations are employed that adjust timesteps of individual particles advanced
to advance slice-to-slice for a specified lattice advance step. Self-fields are generated by solving
Poisson’s equation on a discretized spatial mesh with the correct transverse conducting aperture
of the accelerator lattice. Applied electric and magnetic fields (including longitudinal electric) are
imported via gridded potential and field elements. This model neglects relatively weak longitudinal
self-fields. However, it retains correct longitudinal applied forces from lattice elements. The field
solutions are consistent with local bend radii ρ of the tight dipoles used for charge selection in the
front end (see overview in Chapter 1).

These modules constitute a versatile code framework for simulations of ion accelerator front
ends using Warp. The framework is highly adaptable to different accelerator facilities and different
operating points, which helps expand its utility within the community. While we have not employed
this functionality in the studies below, Warp also supports parallelization to allow high-detail runs
and parametric studies.

3.2 Beam Line Element Models

The transverse spatial extent of low energy beams in accelerator front ends can be similar to
the extent of the beam pipe and apertures of focusing elements. In fact, it is not uncommon to have
∼ 1% beam loss in front ends, which indicate scraping during beam transport. Therefore, the outer
particles in the beam distribution can evolve beyond the good field radius of beam line elements and

17

experience substantial field nonlinearities. Furthermore, large transverse-to-longitudinal aspect
ratios in focusing elements increase nonlinearities and cause fringe fields to leak out far beyond the
longitudinal extent of the element, sometimes overlapping with those from adjacent elements. The
Warp simulations properly model these effects.

Fig. 3.1 shows the CST Studio[21] model of S4 solenoids employed in the front end at FRIB.
One can observe that field nonlinearties grow significantly as one goes to larger radii. Fringe fields
also extend up to 15 cm outside the solenoid on both sides, which is half as long as the coil length.
The CST Studio model of Q7 electrostatic qaudrupoles are shown Fig. 3.2 and Fig. 3.3. While the
field is relatively more uniform, the Q7 in the triplet assembly contains collimators that can intercept
beam particles that might be lost on electrode structures. The Warp simulations employ particle
scraping on the detailed aperture structures like the electrostatic quadrupoles. Both examples show
that it is necessary to implement detailed field models to perform realistic simulations of accelerator
front ends.

3.3 Simulations of FRIB Front End

3.3.1

Initial Conditions & Loading

x =(cid:10)x2(cid:11)). The 35 kV extraction potential sets the kinetic energy of each species, and

2

Table 3.1 lists beam parameters used in the Warp simulations. An initial axisymmetric waterbag
distribution adapted for space-charge [20] is injected with no centroid offset at the extraction point
of the ECR. Each species’ envelope extent fills the puller electrode aperture (i.e. Rpuller = 2σx
where σ
the longitudinal velocity spread and radial (thermal component) emittance arise from a 3 eV ion
temperature. Since the beam aperture radius at extraction is relatively small in the sextupole field
of the ECR ion source, we assume nearly constant beam canonical angular momentum (cid:104)Pθ(cid:105) in the
initial transport from the ECR. The average canonical angular momentum of each species is set
birth Bbirth/2, where
z(peak 1) Bz(r = 0)dz is the average axial B-field between the peaks of the solenoidal
field of the ECR. Since the B-field at launch differs from Bbirth, for (cid:104)Pθ(cid:105) = const as specified, an

by assuming all ions are born at the same magnetic field, i.e. (cid:104)Pθ(cid:105) = q(cid:10)r2(cid:11)
Bbirth =Þ z(peak 2)

18

Figure 3.1: CST Studio model of S4 solenoids at FRIB and the calculated fields at different radii
within the clear bore aperture radius R = 164.2/2 mm.

angular velocity is injected upon each ion of the species with:

Here, [Bρ] is the particle rigidity, and we assume(cid:10)r2(cid:11)

birth
launch

Bbirth − Blaunch

(cid:48)
0 =

θ

(cid:34) (cid:10)r2(cid:11)
(cid:10)r2(cid:11)

(cid:35)
birth =(cid:10)r2(cid:11)

/(2[Bρ]).

(3.1)

launch. Simulated beam species

and currents for U operation are given in Ref. [9].

3.3.2 Lattice Design & Matching Methodology

The initial beam line from the ECR ion source can be divided into two distinct sections:
the
axisymmetric transfer line from the ECR to the charge selection system (CSS), and the CSS which
produces large dispersion for species selection.

19

395.2 mm296 mm164.2 mm456.1 mmyokecoilhalf lengthRadial FieldAxial FieldCross Section (Side View)Cross Section (Perspective View)Figure 3.2: CST Studio model of a Q7 electrostatic quadrupole triplet at FRIB with dimensions
and potential plots.

For species selection with no loss in target species and preservation of beam quality, the CSS

should:

1) Be linearly achromatic.

2) Generate large dispersion with small βx (x-betatron lattice function) at collimation point to

maximize selection resolution with vertical slits.

3) Output the beam with well-controlled envelopes for downstream transport.

Requirements (1) and (3) can be met in an idealized linear optic limit with a symmetric beam
envelope in the CSS by exploiting mirror symmetry about the axial mid-point of the CSS (strength
of each element equals that of its mirror counterpart) and setting D(cid:48) = αx = αy = 0 (D = bend-
plane dispersion function and αj = −β(cid:48)
j/2, j = x, y) at the axial mid-point. The axisymmetric

20

70 mm 90 mm200 mm239 mm96 mm100 mm70 mm150 mm130 mmBetweeen electrode & endplateEnd of electrodeMid-quadFigure 3.3: CST Studio model of a Q7 electrostatic quadrupole triplet at FRIB. Dimensions and
potential contours at various axial locations are shown.

Table 3.1: Beam Parameters of U34+ based on the VENUS-like source. For definitions, see also
Sec. 3.3.3 and ??. Temperatures and envelope parameters are the same for all ion species (indices
suppressed). Canonical angular momentum, radial normalized emittance, and kinetic
energy/rigidity vary with ion species.

Quantity
Transverse Temperature
Canonical Angular Momentum
Radial Normalized Emittance
Radial Envelope
Radial Envelope Angle
Kinetic Energy
Rigidity
Longitudinal Temperature

T

Symbol
(cid:104)Pθ(cid:105) /(mc)
rms
√
ε
nr
2σx
σr =
σ(cid:48)
r = dσr/dz

T(cid:107)

Value at Launch

3 eV

0.305 mm-mrad
0.015 mm-mrad

2.82 mm

0

3 eV

Ek (before/after ES Gap)
[Bρ] (before/after ES Gap)

5.00 / 12.00 keV/u

7.13×10−2 / 1.10 ×10−1 Tm

initial beam envelope conditions are adjusted to satisfy requirement (2) and achieve an “attractive”
beam envelope devoid of large excursions.

The CSS design procedure was carried out with the code MADX [22] using a hard-edge lattice
and single-particle dynamics. Results are shown in Fig. 3.4. The lattice is achromatic with zero

21

in

= 0) and final dispersion (Dout = D(cid:48)

initial (Din = D(cid:48)
out = 0), and the envelopes are symmetric
about mid-plane with lattice functions satisfying βin = βout and αin = −αout to ideally recover
axisymmetry on exit. The 1 m-long 90-degree bend dipoles have slanted poles that generate a
focusing strength equivalent to a superimposed quadrupole with κ0 = 0.365 m−2. The three
electrostatic quadrupoles (ESQs) in the triplet have axial length 20.7 cm with κ1 = 6.62 m−2, κ2 =
-14.8 m−2 and κ3 = 7.80 m−2. The initial envelope has βin = 4.383 m and αin = 0.306, which (see
below) are consistent with upstream matching from the ECR.

Figure 3.4: CSS lattice design used in Warp simulations.

The transport line from the ECR must be tuned to deliver the required lattice functions entering
the CSS. We assume no initial centroid offset (Din = D(cid:48)
= 0). Applying εx = εeff/2 (see
x/εx to convert between lattice functions and envelope
Sec. 3.3.3), βx = σ
size, the excitation of the two upstream solenoids are tuned to match the required lattice functions.

x/εx and αx = −σxσ(cid:48)
2

in

22

s (m)0.01.2.3.4.5.6.7.8.0.00.51.01.52.02.53.0Dx(m)βxβyDx69.0970.1371.1772.2173.2574.2990o dipole90o dipoleESQ TripletESQ Triplet3.3.3 Envelope Model for Multi-Species Beam Transport

Beam evolution in the axisymmetric transport line from the ECR to the 1st dipole of the CSS can
be described by a multi-species envelope equation:

(cid:48)(cid:48)
r j =
σ

+ 


qjV(cid:48)
2Ek j

(cid:48)
r j +
σ

qjV(cid:48)(cid:48)
4Ek j

σr j −

Q js fs

σr j
+ σ

2
rs

2
r j

+

(cid:18) qj Bz0

(cid:19)2

j /(cid:16)

(cid:17)2

mj βjc

.

σr j
2mj βjc
2 + (cid:104)Pθ(cid:105)2
rms
ε
r j
3
r j

(3.2)

(3.3)

σ

s,species

√
2σx j =

σ
Details of this model can be found in Ref. [23]. In brief:
j indexes the jth species; qj and mj
are the charge and mass of the species; βj and Ek j are the relativistic factor and kinetic energy
√
2(cid:104)x2(cid:105)j is the radial sigma of the species charge profile; V and
of the species; σr j =
B0z are the on-axis applied potential (of accelerating gaps) and the axial magnetic field which both
can vary in s; (cid:104)Pθ(cid:105)j = const is the canonical angular momentum of the species; and fj is the
fractional electron neutralization factor ( fj = 0 corresponds to neutralized). The canonical angular
momentum measured by (cid:104)Pθ(cid:105) j /(mj βjc) contributes to the beam’s expansion in the same manner
as the thermal geometric emittance

(cid:118)(cid:116) Tj

rms
r j =
ε

2

j c2 R

mj β

rms
nr j

at injection with ε
= βj εr j = const (conserved normalized emittance) set by the initial value
emerging from the source. In Eq. (3.3), R = 4 mm is the ECR extraction aperture and the value of
rms
ε
r j

corresponds to a uniform density beam species filling the aperture. In this model

2
ε
eff,j

= ε

rms
r j

2

+

(cid:104)Pθ(cid:105)2
(mj βjc)2

j

(3.4)

constitutes an effective phase-space volume of the beam. The envelope model also assumes
Gaussian density profile species are maintained in the evolution. The species kinetic energy Ek is
set consistently with the applied on-axis potential V of electrostatic gaps. The envelope equations
and energy equations of all species constitute a system of ordinary differential equations (ODEs)
that can be numerically integrated along with an equation for the species energy using the linear

23

field components extracted from the 3D element models [B0z(s) = Bz(r = 0, s) for solenoids and
V(s) for electrostatic gaps]. The integration begins at the ECR extraction point and ends at the
entrance of the first dipole. Figure 3.5 shows that the envelope equation achieves good agreement
with self-consistent Warp simulations both with and without space charge.

A python based envelope package was constructed within the Warp interface to integrate the
coupled nonlinear envelope equations 3.2 and the energy gain equation. This allowed the same
lattice setup to be used as the full simulations. Even with little numerical optimization, this envelope
model is more than 100 times faster than relatively efficient Warp transverse slice simulations of the
same section. This efficiency allows parametric exploration of improved matching. Successively
finer scans of the two solenoid fields (minimum increments being 0.002 Tesla) are carried out
until target envelope conditions are matched by hitting target envelope parameters entering the
achromatic CSS.

Figure 3.5: Comparisons between numerical solutions of the envelope equation and Warp
simulation results in the transport line upstream of the CSS. These correspond to simulation cases
3 and 4 described in Sec. 3.3.4.

A caveat is appropriate for the matching procedure above: the lattice design is not solely dictated
by the CSS, because the design incoming lattice functions may be unattainable within the range of
possible solenoid ex citations. Fig 3.6 summarizes accessible lattice functions at the entrance of
the first dipole for different space charge neutralization values. Colored dots indicate points where

24

ECRCSSsolsolgrated gapOn-Axis Peak Field35kV50kVFigure 3.6: Accessible lattice functions β and α at the entrance of the first dipole when the Bpeak
in both solenoids varied independently between 0 T and 1.5 T. Results are shown for different
space charge neutralization factors. Observe that there are regions of initial Twiss parameters that
are inaccessible, and the region of accessible Twiss parameters decreases with decreasing
neutralization factor

accessible solenoid excitations can be found for the two solenoids connecting the ECR to the CSS
to achieve the indicated lattice function values βx = βy ≡ β and αx = αy ≡ α = β(cid:48)/2. If the CSS
operating point is incompatible with the matching capabilities of the solenoids, an alternative CSS
operating point with attainable initial lattice functions must be selected.

3.3.4 Simulation Cases

Fields from lattice elements in the axisymmetric transport line are imported from r-z electrostatic
and magnetostatic simulations with Poisson [24] and CST Studio [21]. Warp also allows hard edge
elements, and for present purposes, all elements in the CSS are modeled as hard-edge equivalent
here to simplify comparisons to the MADX design lattice. Simulations with space charge include a
uniform neutralization factor of 75% ( f = 0.75) except inside the grated gap (protected by a guard
electrode included in V) and the CSS lattice downstream of the 1st electrostatic quadrupole.

To illustrate effects of space charge and canonical angular momentum in the CSS, we launch

25

beams with the same effective phase-space area (εeff) as calculated from Eq. (3.4), but with different
relative compositions from thermal emittance and canonical angular momentum. A “magnetized”
beam has phase space contributions consistent with Table 3.1. A “thermal” beam has the same
phase-space area (emittance) but with zero canonical angular momentum ((cid:104)Pθ(cid:105)j = 0). That is, for
the thermal beam is unmagnetized with increased thermal temperature Tj for the same effective
phase-space area measured by εeff,j. The following cases were run:

1) Thermal beam, no space charge.

2) Thermal beam, with space charge.

3) Magnetized beam, no space charge.

4) Magnetized beam, with space charge.

Note that case 4 is close to what is expected in the laboratory since canonical angular momentum
dominates the effective phase space volume.

3.3.5 Warp Simulation Results: Case Comparisons

Before presenting how the case comparisons of Sec. 3.3.4 illuminate different effects in the beam
dynamics, it is helpful to observe the phenomena of species separation and collimation in the
charge selection section as illustrated in Fig. 3.7. The Warp simulation results correspond to the
realistic case 4 with charge neutralization factor f = 0.75. For the two-species transport (U33+
and U34+) planned for the full power operation at FRIB, the optics is tuned with a hypothetical
U33.5+ ion as the reference. As seen in the upper panel of Fig. 3.7 showing beam species extents
in the x-plane, only U33+ and U34+ (purple and dark green) species make it past the final 4-jaw
collimator aperture. Numerous species, including high space-charge intensity oxygen, are lost in the
dipole and on the slits incorporated within the following electrostatic quadrupole triplet structure.
Numerous species are also lost on the gate valve structure. Neutralization ends at the start of the first
electrostatic quadrupole which has strong electric sweeping fields. This is also the axial location

26

of a gate valve. Corresponding line-charge measures of space-charge intensity are given in the
middle panel of Fig. 3.7. This shows that most cumulative space-charge intensity emerging from
the ECR is scraped. Simple analytic estimates of the depression in local particle phase advance due
to beam space-charge are included in the lower panel of the figure. Only the region downstream
of the four jaw collimator is show since the estimates are based on single species beam models.
The depression measure shown is σ/σ0 the ratio of space-charge depressed phase advance to the
single particle phase advance (σ/σ0 = 1 for single particle dynamics with zero beam intensity and
σ/σ0 → 0 at the transport limit with full space-charge depression). Results are consistent with a
roughly 10% space-charge depression (σ/σ0 = 0.9). We note that this operating point would be
representative of a highest intensity facility operation. While space-charge is not negligible post
species separation, it is not large and will further drop post acceleration in the RFQ. Lower current
beams presently being used in facility commissioning have lower space-charge intensity and single
particle dynamics downstream of species separation will be a good approximation in these cases.
Warp PIC simulations of the 4 cases in the CSS are contrasted to understand deviations from
the MADX optical design due to space charge, non-constant emittance, beam canonical angular
momentum, and nonlinear optical effects. In this case the four simulations are run through the 2nd
dipole of the CSS. Identifying the source of deviations from the ideal design with reference to the
four test cases outlined in Sec. 3.3.4 will help guide future optimization of the CSS. Figure 3.8
shows the rms beam envelope (a) and emittance (b) evolution of the reference species U34+, and
the centroid offset of U33+ along with mid-plane xy phase space projections (c). Contrasting the
simulation cases, we observe the following behavior:

• Beam rotation due to canonical angular momentum.

• In cases with space charge, non-reference species have centroid offset upon exiting the CSS.

• RMS emittance growth in the CSS which is mostly reversed on exiting the CSS.

• RMS emittance exchange between the x- and y-planes for magnetized beams.

27

Figure 3.7: Warp simulation results of Uranium species separation and collimation in the beam
evolution from the ECR source extraction through the midpoint of the charge selection system
showing collimation to two target species for high intensity FRIB operation.

Distribution x y projections in Fig. 3.8c illustrate how a magnetized beam becomes tilted in
real space at the CSS midplane due to canonical angular momentum ((cid:104)Pθ(cid:105) (cid:44) 0; dropping species
subscripts henceforth) whereas a thermal beam ((cid:104)Pθ(cid:105)) does not. Large initial (cid:104)x y(cid:48)(cid:105) and (cid:104)x(cid:48)y(cid:105)
moments at the start of the dipole in the case of a magnetized beam causes (cid:104)x y(cid:105) to evolve in the
non-axisymmetric lattice which manifests as a significant distribution tilt in the middle of the CSS.
This tilt suggest that more elaborate shaped electrodes for final beam collimation may be beneficial.
However, effective tuning of such a system would require improved understanding of the initial
state of the beam emerging from the ECR including the degree of magnetization (value of (cid:104)Pθ(cid:105)).

28

Figure 3.8: Rms envelops σx,y a) and normalized rms-emittance εx,y b) of the reference species
U34+ in the CSS. c) shows the centroid of U33+ and x y-particle projections of both target species
at the CSS mid-point for two cases. Solid curves are x-plane and dashed curves are y-plane.
Colors flag Warp simulation cases. Black solid lines denote MADX design values in a) and c). The
axial midplane of the CSS is indicated in c) (z in FRIB coordinates with origin outside machine).

29

U33+U34+Therm w/ SCMag w/ SCa)b)c)(mid CSS)(mid CSS)xyxyWhile space charge of the multi-species beam has little impact on the envelope evolution in x
and y, insofar as the matching is carried out equivalently, the cases with and without space charge
in Fig. 3.8c show that space charge creates a ∼mm shift in the (cid:104)x(cid:105) centroid of U33+. However, the
canonical angular momentum has little impact on either the envelope extent or the dispersive shift of
the centroid. The space charge induced centroid shift at mid CSS suggests that optimal adjustment
of the four jaw collimator will vary with ECR current extracted. This further complicates the
impact of magnetization induced tilt on the beam x y profile.

Figure 3.8b shows significant emittance growth inside the 1st bending dipole that is largely
removed on exiting the 2nd dipole of the achromatic CSS beamline for simulation cases 1) – 3).
Slight asymmetries for case 2) (thermal beam with space-charge) are likely induced by nonlinear
space-charge fields that vary within the CSS evolution upstream and downstream of the midplane.
Effects from nonlinear terms in the dipole arising from large x/ρbend may also contribute to
asymmetries. The emittance increase is stronger for thermal beams than for magnetized beams –
regardless of space charge. For case 4) (magnetized beam with space-charge) there is significant
emittance exchange between the x- and y-planes that results in significantly different final emittance
values. For single particle dynamics beams in quadrupoles and dipoles, there is no coupling between
planes regardless of beam rotation. For beams aligned with the x-y symmetry axes with charge
density nearly constant on elliptical symmetry surfaces, space-charge only induces weak coupling.
However, for a rotated x y beam profile, space-charge coupling between the planes is enhanced.
More analysis is necessary to probe if this exchange comes with net growth due to nonlinearities
but the effect appears largely exhange. Understanding effects like these and predicting the x y tilt
angle of the beam will help guide more effective choices of apertures and slit positions for optimal
species collimation at higher intensities anticipated for high-power FRIB operating modes.

3.4 Further Work

Simulation studies of the FRIB front end can be improved by including the physics of ion
extraction at the ECR. We have completed the first step by including the extraction electrodes

30

into the lattice as conductors. A schematic of the electrodes of the VENUS-like ECR is shown
in Fig. 3.9. We have yet to study the beam dynamics with the extraction electrodes included,
and we expect to see interesting effects because experiments have shown substantial changes in
beam conditions downstream when the extraction electrode positions are adjusted. Nonlinearities
in extraction also have maximal effect near the emitting plasma meniscus of the source. Proper
resolution of these effects will likely require r-z or full 3D (to include sextupole distortions) of the
beam in the extraction region.

A further improvement will be the use of realistic initial beam conditions derived from ECR
simulations. A group at JINR, Dubna, Russia has developed sophisticated simulations of ECR ion
sources [25, 26]. We briefly interacted with Dr. Vladimir Mironov on a visit to FRIB. If such
steady-state simulations were carried out for the ECRs at FRIB, they could be run to accumulate
a population of extracted ions. Then, Warp simulations can be set up to birth randomly chosen
extracted ions and model their extraction in 3D mode until steady state is reached. Results of such
simulations (perhaps also needing proper plasma meniscus modeling) could reveal much of the ion
beam extraction physics that is presently omitted.

Another important piece of physics that is currently missing from the Warp simulations is self-
consistency in the electron neutralization model. Fig. 3.10 shows that the beam evolution heavily
depends on the degree of neutralization, so a clear understanding is essential to front end design and
tuning. Not only do we lack a detailed model electron neutralization effects, even in the primitive
model of a homogeneous neutralization factor, we are not certain what it should be. Tentative steps
towards developing a simulation neutralization model were presented in the Ph.D thesis of Daniel
Winklener [27]. Extensions of such complicated models may be necessary for more fully realistic
modeling. A critical detail may be transitions between neutralized and unneutralized (swept by
electrostatic fields) regions. Lack of uniformity in neutralization factors there may enhance the
nonlinear component of space-charge forces.

A final problem concerns the generation of 4D coupled phase space distributions. The ini-
tialization of uncoupled distributions with high space charge intensity is treated comprehensively

31

Figure 3.9: Extraction geometry of the VENUS-like ECR ion source. The electrode in red has
negative voltage to prevent backflow of electrons. (Image courtesy of Dr. Daniel Winklener.)

Figure 3.10: Beam sizes in the post-extraction beam line downstream of the VENUS-like ECR at
FRIB, obtained from Warp simulations with different homogeneous neutralization factors.

32

In Sec. 3.3.1, coupling is inserted into an uncoupled distribution by the addition
in Ref. [20].
of an angular velocity as described by Eq. (3.1). Alternatively, we have derived in Appendix C
a method of generating a 4D coupled distribution with any set of 2nd-order moments via linear
transformation. It should be beneficial to study how these treatments connect with each other and
whether they produce simulation artifacts.

33

CHAPTER 4

THEORY OF TRANSVERSE COUPLED BEAM DYNAMICS, WITH APPLICATIONS

TO THE FRIB FRONT END

At the time of this writing, the analysis in this chapter pertaining to beams with rotational symmetry
is under preparation for submission for publication [28].

4.1 Beams with Rotational Symmetry

This section develops analytic tools to describe moments of beam distributions with discrete or
continuous rotational symmetry. Their implications and possible applications are also discussed.

4.1.1 Effect of Symmetry on Beam Moments

Define transverse beam moments

(cid:68)

xb1 x(cid:48)b2 yb3 y

(cid:48)b4(cid:69) ≡

Þ ∞

−∞

Þ ∞
Þ ∞
Þ ∞
Þ ∞
Þ ∞
Þ ∞
−∞ F(x, x(cid:48), y, y(cid:48))xb1 x(cid:48)b2 yb3 y(cid:48)b4dxdx(cid:48)dydy(cid:48)
−∞

−∞ F(x, x(cid:48), y, y(cid:48))dxdx(cid:48)dydy(cid:48)

Þ ∞

−∞
−∞

−∞

−∞

where F(x, x(cid:48), y, y(cid:48)) is the distribution function in transverse phase space and b1, b2, b3, b4 ∈ Z≥0.
The moment is said to be of k-th order with k = b1 + b2 + b3 + b4. Note that the moments are
calculated assuming each variable has zero mean, i.e. all four first order moments vanish. This
corresponds physically to the beam centroid following the design (reference) orbit and must be true
for beams with rotational symmetry, as will be proved in subsequent arguments.

For a beam with n-fold rotational symmetry (i.e. Cn), the distribution function F(x, x(cid:48), y, y(cid:48))
is invariant under a rotation by θ = 2π/n. This implies all transverse beam moments remain
unchanged under the transformation:

(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)

(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172)

x
x(cid:48)

y

y(cid:48)

(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)

cos θ
0

− sin θ

0
cos θ
0

(cid:55)→

0

− sin θ

34

sin θ
0
cos θ
0

0
sin θ
0
cos θ

(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172)

(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)

(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172)

x
x(cid:48)

y

y(cid:48)

(4.1)

For an axisymmetric beam (i.e. beam with continuous rotational symmetry), the invariance of F
and beam moments holds for any rotation angle θ. This is the only piece of information that we
need to extract from the rotational symmetry of the beam. No assumption on space-charge intensity
is made or implied. The rest of this section explores its consequences and applications.

Given a beam distribution F with rotational symmetry, its beam moments must satisfy certain
constraints in order for them to be invariant under the corresponding rotational transformations.
Since rotations amount to linear transformations among phase space coordinates, one expects the
constraints to arise from systems of linear equations among beam moments of the same order.
Formally, denote (x, x(cid:48), y, y(cid:48)) as (x1, x2, x3, x4) and given the rotation matrix:

where θ is a rotation angle of the symmetry, the relation:

R =

0

− sin θ

− sin θ

cos θ
0

0
cos θ
0

sin θ
0
cos θ
0

0
sin θ
0
cos θ

(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172)
R3j xj(cid:170)(cid:174)(cid:172)b3(cid:169)(cid:173)(cid:171) 4


(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)
R1j xj(cid:170)(cid:174)(cid:172)b1(cid:169)(cid:173)(cid:171) 4

R2j xj(cid:170)(cid:174)(cid:172)b2(cid:169)(cid:173)(cid:171) 4

C(a1, a2, a3, a4)(cid:68)






=


j=1

a1

a2

a3

a4

(cid:68)

xb1
1 xb2

2 xb3

3 xb4
4

=

(cid:69)

(cid:42)(cid:169)(cid:173)(cid:171) 4

(cid:69)

j=1

(cid:68)

xb1
1 xb2

2 xb3

3 xb4
4

j=1
holds for every b1, b2, b3, b4 ∈ Z≥0. Eq. (4.3) can be rewritten in the form:

j=1

(4.2)

(4.3)

(4.4)

R4j xj(cid:170)(cid:174)(cid:172)b4(cid:43)
(cid:69)

xa1
1 xa2

2 xa3

3 xa4
4

where a1, a2, a3, a4 ∈ Z≥0 and a1 +a2 +a3 +a4 = b1 +b2 +b3 +b4. Each coefficient C(a1, a2, a3, a4)
is a polynomial of matrix elements of R, which means it is either zero or a polynomial of degree k
in sin θ and cos θ.

Thus, the set of all k-th order beam moments in Eq. (4.4) for non-negative integers bj form a

35

linear system:

(cid:68)

(cid:68)
(cid:68)
(cid:68)

(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)

3, x0
2, x0
1 x0
xk
4
xk−1
3, x0
2, x0
x1
1
4
xk−1
x0
3, x0
2, x1
1
4
...
3, xk−1
1 x0
x0
2, x1
4
1 x0
x0
2, x0
3, xk
4

(cid:69)

(cid:68)

(cid:69)

(cid:69)
(cid:69)
(cid:69)

(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172)

(cid:68)

(cid:68)
(cid:68)
(cid:68)

(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)

3, x0
2, x0
1 x0
xk
4
xk−1
3, x0
2, x0
x1
1
4
xk−1
x0
3, x0
2, x1
1
4
...
3, xk−1
1 x0
x0
2, x1
4
1 x0
x0
2, x0
3, xk
4

(cid:69)

(cid:68)

(cid:69)

(cid:69)
(cid:69)
(cid:69)

(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172)

= A(θ)

(4.5)

that describes relations among beam moments resulting from invariance under rotation by angle θ.
A(θ) is a function of θ because the coefficients C(a1, a2, a3, a4) in Eq. (4.4) depend on θ, and so the
relations change with the order of rotational symmetry.

While Eq. (4.5) is a general result that already contains the properties of beam moments under
rotational symmetry, the expressions are complicated because most elements in A(θ) are non-zero.
It is difficult to know whether or how the equations in Eq. (4.5) can be manipulated to obtain simple
equality relations that are practically useful. One trick to achieving some simplification uses the
with 0 ≤ m < n. Therefore, for a beam with Cn symmetry,
fact that Eq. (4.5) holds for all θ = 2mπ
n
a sum over the set of θ that preserves symmetry can be taken to obtain:
1 x0
2, x0
3, x0
xk
4
xk−1
3, x0
2, x0
x1
1
4
xk−1
x0
3, x0
2, x1
1
4
...
3, xk−1
x0
1 x0
2, x1
4
2, x0
x0
1 x0
3, xk
4

1 x0
2, x0
3, x0
xk
4
xk−1
3, x0
2, x0
x1
1
4
xk−1
x0
3, x0
2, x1
1
4
...
3, xk−1
x0
1 x0
2, x1
4
2, x0
x0
1 x0
3, xk
4

(cid:18)2mπ

n−1


(cid:68)
(cid:68)
(cid:68)

(cid:69)
(cid:69)
(cid:69)

(cid:69)
(cid:69)
(cid:69)

(4.6)

m=0

(cid:69)

(cid:69)

(cid:68)

(cid:68)

=

1
n

(cid:69)

A

n

(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)

(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172)

Each coefficient contains summations of polynomials P with two trigonometric variables:

(cid:68)

(cid:19)

(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)

(cid:68)
(cid:68)
(cid:68)
(cid:68)
(cid:18)2mπ

n

(cid:19)(cid:19)

(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172)

(cid:69)
n−1


m=0

(cid:18)

(cid:18)2mπ

(cid:19)

n

sin

P

,cos

which may be reducible. For k = 2, this method invokes the same identities as the alternative
proof presented in Appendix B, and enables the derivation of simple relations among 2nd order

36

(4.7)

beam moments. The k = 2 results suggest that simple equality relations exist for k > 2, but the
method of Eq. (4.6) is difficult to generalize to k > 2 because the elements of A [i.e. coefficients
C(a1, a2, a3, a4) of Eq. (4.4)] grow exceedingly complicated as the moment order k increases. We
employ a more efficient, yet less intuitive, method to derive moment relations from symmetry in
Sec. 4.1.2.

4.1.2 Derivation of Moment Relations from Symmetry

Having developed the intuition on the relationship between rotational symmetry and beam moments
in Sec. 4.1.1, we devised a more abstract and efficient method to derive clean expressions of the
equality relations. The method was inspired by Ref. [16], and is readily applicable to moments of
arbitrary order in beams with any rotational symmetry.

Define two complex conjugate pairs composed of transverse phase space coordinates:

(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172)

(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)

w

w

w(cid:48)
w(cid:48)

(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172)

0
0
0
e−iθ

Upon a coordinate rotation by θ,(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)

w

w

w(cid:48)
w(cid:48)

(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172)

(cid:55)→

eiθ
0
0
0

0
e−iθ
0
0

0
0

eiθ
0

In analogy to Sec. 4.1.1, we construct complex moments(cid:68)
Z≥0 and a1 +a2 +a3 +a4 = k. The real and imaginary parts of(cid:68)
(cid:48)a3 w(cid:48)a4(cid:69) (cid:55)→ ei(a1−a2+a3−a4)θ(cid:68)

wa1 wa2 w

(cid:68)

wa1 wa2 w(cid:48)a3 w(cid:48)a4(cid:69) where a1, a2, a3, a4 ∈
wa1 wa2 w(cid:48)a3 w(cid:48)a4(cid:69) each comprises

(cid:48)a3 w(cid:48)a4(cid:69)

wa1 wa2 w

(4.10)

sums of physical k-th order beam moments. Upon a rotation by θ, the complex moment undergoes
the following transformation in accordance with Eq. (4.9):

(4.8)

(4.9)

(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172)

w

w

w(cid:48)
w(cid:48)

≡

(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)

(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172)

x + iy
x − iy
x(cid:48) + iy(cid:48)
x(cid:48) − iy(cid:48)

(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)
(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)

37

If rotation by θ is a symmetry of the beam, (cid:68)

wa1 wa2 w(cid:48)a3 w(cid:48)a4(cid:69) remains unchanged upon the

(cid:68)

transformation which gives:

(cid:48)a3 w(cid:48)a4(cid:69)

(cid:48)a3 w(cid:48)a4(cid:69)
due to symmetry. For every 4-tuple(a1, a2, a3, a4) where ei(a1−a2+a3−a4)θ (cid:44) 1,(cid:68)

= ei(a1−a2+a3−a4)θ(cid:68)

wa1 wa2 w

wa1 wa2 w

Eq. (4.11) is the key equation that efficiently generates equality relations among beam moments

(4.11)

wa1 wa2 w(cid:48)a3 w(cid:48)a4(cid:69)

=

0 must follow; this gives:

Re(cid:16)(cid:68)
Im(cid:16)(cid:68)

(cid:48)a3 w(cid:48)a4(cid:69)(cid:17)
(cid:48)a3 w(cid:48)a4(cid:69)(cid:17)

wa1 wa2 w

wa1 wa2 w

= 0

= 0

(4.12)

(4.13)

For a beam with n-fold rotational symmetry, each 4-tuple (a1, a2, a3, a4) satisfying the following

conditions:

Condition 4.1.1. a1 + a2 + a3 + a4 = k.

Condition 4.1.2. (a1 + a3) − (a2 + a4) > 0.

Condition 4.1.3. (a1 + a3) − (a2 + a4) (cid:44) 0 (mod n).

gives two unique relations on the k-th order transverse moments of the beam, in the form of

Eq. (4.12) and Eq. (4.13).

Condition 4.1.1 is merely a restatement of the moment being k-th order, while condition 4.1.3
asserts ei(a1−a2+a3−a4)θ (cid:44) 1. Condition 4.1.2 removes redundant relations for the following reason.
Suppose a 4-tuple (a1, a2, a3, a4) fulfills conditions 4.1.1 and 4.1.3 but not 4.1.2:

(cid:68)
(cid:68)

(cid:48)a3 w(cid:48)a4(cid:69)
(cid:48)a4 w(cid:48)a3(cid:69)

wa1 wa2 w

wa2 wa1 w

Its complex conjugate:

(a1 + a3) − (a2 + a4) < 0

(4.14)

(a2 + a4) − (a1 + a3) > 0

(4.15)

= 0

= 0

38

is equivalent to and contains the same information as:

(cid:68)

(cid:48)b3 w(cid:48)b4(cid:69)

wb1 wb2 w

(4.16)
Therefore, the relations generated by (a1, a2, a3, a4), which violates condition 4.1.2, are already
covered by another 4-tuple (b1, b2, b3, b4) that satisfies condition 4.1.2.

(b1 + b3) − (b2 + b4) > 0

= 0

angle θ keeps(cid:68)

For an axisymmetric beam (i.e. beam with continuous rotational symmetry), rotation by any

wa1 wa2 w(cid:48)a3 w(cid:48)a4(cid:69) unchanged. Therefore, ei(a1−a2+a3−a4)θ (cid:44) 1 can always be

satisfied for some θ. In this case, conditions 4.1.1 and 4.1.2 suffice for a 4-tuple (a1, a2, a3, a4) to
impose unique relations upon k-th order transverse moments.

An immediate corollary of these analytic arguments is the fact that any beam with rotational
symmetry must have zero means (i.e. first order centroid moments) in all transverse phase
space coordinates. For moments of order k = 1, when n (cid:44) 1, (a1, a2, a3, a4) = (1,0,0,0) and
(a1, a2, a3, a4) = (0,0,1,0) always satisfy conditions 4.1.1, 4.1.2 and 4.1.3. Therefore, (cid:104)w(cid:105) = 0 and
(cid:104)w(cid:48)(cid:105) = 0 must hold, which imply (cid:104)x(cid:105) = (cid:104)y(cid:105) = 0 and (cid:104)x(cid:48)(cid:105) = (cid:104)y(cid:48)(cid:105) = 0 respectively.

4.1.3 Correspondence between Discrete and Continuous Rotational Symmetry

Armed with a general method for deriving relations imposed upon beam moments due to the
rotational symmetry possessed by the beam, it is illuminating to ask: how do continuous and
discrete rotational symmetries manifest themselves differently in beam moment relations? This
question was part of the motivation for developing the analytic tools in Sec. 4.1.2, and will lead to
arguments that enable simple proofs of certain useful results.

We begin by counting the number of equality relations governing k-th order transverse moments

of an axisymmetric beam. We define:

• ξ(k) as the number of 4-tuples (a1, a2, a3, a4) that satisfy conditions 4.1.1;
• χ(k) as the number of 4-tuples (a1, a2, a3, a4) that satisfy conditions 4.1.1 and 4.1.2; and

•

ˆχ(k) as the number of 4-tuples (a1, a2, a3, a4) that satisfy conditions 4.1.1 and (a1 + a3) −
(a2 + a4) = 0.

39

ξ(k) is the number of distinct k-th order moments, which applies equally to real moments

wa1 wa2 w(cid:48)a3 w(cid:48)a4(cid:69). It is equivalent to the total number

of ways to distribute k indistinguishable balls into 4 distinguishable boxes, which gives

(cid:104)xa1 x(cid:48)a2 ya3 y(cid:48)a4(cid:105) and complex moments(cid:68)
ξ(k) =(cid:169)(cid:173)(cid:173)(cid:171)k + 3
(cid:170)(cid:174)(cid:174)(cid:172) =

k

(k + 3)!
3!k! =

(k + 3)(k + 2)(k + 1)

3 × 2 × 1

(4.17)

To find χ(k), we use the fact that the number of 4-tuples satisfying (a1 + a3) − (a2 + a4) > 0

and (a1 + a3) − (a2 + a4) < 0 are equal. Therefore,

χ(k) =

1
2 [ξ(k) − ˆχ(k)]

(4.18)

For odd k, ˆχ(k) = 0 because(a1+a3)−(a2+a4) = 0 cannot occur. For even k,(a1+a3)−(a2+a4) = 0
occurs whenever a1 + a3 = a2 + a4 = k/2. The number of 2-tuples (a1, a3) such that a1 + a3 = k2
equals k2 + 1, likewise for (a2, a4). Since (a1, a3) and (a2, a4) are independent, ˆχ(k) = 0 equals

. Summarizing these results with Eq. (4.17) and Eq. (4.18), we obtain

(cid:16) k2 + 1(cid:17)2

χ(k) =

(k+3)(k+2)(k+1)

(cid:20)(k+3)(k+2)(k+1)

3×2×1

3×2×1

1
2
1
2

−(cid:16) k2 + 1(cid:17)2(cid:21)

= k(k+2)(2k+5)

24

if k is odd,

if k is even.

(4.19)



Since each vanishing complex moment generates two relations, the total number of equality relations
governing k-th order transverse moments of an axisymmetric beam is 2 χ(k).

2 χ(k) is the maximum number of moment relations that can be imposed upon k-th order
moments of a beam due to rotational symmetry. For beams with discrete rotational symmetry Cn,
the number of relations ≤ 2 χ(k) because some 4-tuples (a1, a2, a3, a4) which satisfy condition 4.1.1
and condition 4.1.2 may violate condition 4.1.3.

Let m(n, k) be the number of 4-tuples (a1, a2, a3, a4) that satisfy conditions 4.1.1 and 4.1.2,
but not condition 4.1.3, for the given n. Thus the number of moment relations governing a beam
with Cn symmetry is given by 2[ χ(k) − m(n, k)]. We discuss the implications of m(n, k) = 0 and
m(n, k) (cid:44) 0 below.

We introduce the term “k-th order axisymmetry” to denote the set of 2 χ(k) relations obeyed
by k-th order moments of an axisymmetric beam. A beam is said to be “k-th order axisymmetric”

40

if it obeys k-th order axisymmetry. Note that k-th order axisymmetry merely concerns properties
of k-th order moments, so a beam can be k-th order axisymmetric but not l-th order axisymmetric
for any l (cid:44) k. k-th order axisymmetry is a much weaker notion than axisymmetry, i.e. continuous
rotational symmetry, which implies k-th order axisymmetry for all k.
When m(n, k) = 0, a Cn beam is k-th order axisymmetric, i.e.

its k-th order moments obey
the same set of relations as those of an axisymmetric beam. This is true despite the fact that Cn
and axisymmetry are different rotational symmetries. This equivalence indicates that k-th order
axisymmetry renders the Cn beam indistinguishable from an axisymmetric beam in terms of k-th
order beam moments. As a result, the Cn beam will share all properties of an axisymmetric beam

that only depend on k-th order moments. For example,(cid:68)

xk(cid:69) of an axisymmetric beam remains the

same regardless of the orientation of the transverse coordinate system, so the same must hold for
the Cn beam as well.

m(n, k) = 0 is quite common and occurs in more than half of all (n, k) pairs. Referring again to
conditions 4.1.2 and 4.1.3, to ensure that both of them are always satisfied, (a1 + a3)−(a2 + a4) = 0
(mod n) cannot occur unless (a1 + a3)−(a2 + a4) = 0. It is easy to show that this condition always
holds if k < n, or if k is even and n is odd, or vice versa.

A corollary of the above statement is that, given n, k = n is the smallest k at which m(n, k) (cid:44) 0
occurs. This agrees with the heuristic picture where, the higher the degree of discrete rotational
symmetry, the closer the beam is to being axisymmetric and the higher the order of moments it takes
for deviations to start. Let S(Cn) and S(axisymmetry) be the sets of unsimplified relations derived
from Eq. (4.11) governing n-th order moments of a Cn beam and an axisymmetric beam respectively.
When k = n, m(n, n) = n + 1, so S(axisymmetry) contains 2(n + 1) more elements (i.e. moment
relations) than S(Cn). With n being the lowest moment order at which Cn and axisymmetry
are distinguishable, the difference between S(Cn) and S(axisymmetry) can be interpreted as a
manifestation of the difference between Cn and axisymmetry in terms of beam moments. Such
a perspective can allow one to employ beam moments to quantify how close a Cn beam is from
being axisymmetric. The quantification may be useful if one would like to gauge the impact of

41

elements with Cn symmetry on the beam distribution, or if one would like to generate a Cn beam
via maximum entropy methods with prior knowledge on beam moments.

The general arguments developed in this subsection will be applied to analyze the 2nd and 3rd

order moments of a beam with C3 symmetry in Sec. 4.1.4.

4.1.4 Example: Beams with C3 Symmetry

To demonstrate the utility of the analytic tools developed in Sec. 4.1.1 and 4.1.2, we derive equality
relations governing the moments of a beam with C3 symmetry, and compare them against those of
an axisymmetric beam. Beams with C3 symmetry are directly relevant to ECR ion sources which
contain a sextupole (C3) in additional to solenoids (axisymmetric). Simulation results of an ECR
beam at the extraction plane is shown in Fig. 4.1 where the spatial distribution of ions extracted is
visibly triangulated due to the sextupole field of the ECR. Note that the dominant triangulation is
shown over the plasma chamber extent but is still manifest within the extraction aperture.

4.1.4.1

2nd Order Moments

For k = 2, the complex moments satisfying conditions 4.1.1 to 4.1.3 can easily be found to be
(cid:104)ww(cid:105), (cid:104)ww(cid:48)(cid:105) and (cid:104)w(cid:48)w(cid:48)(cid:105). The equality relations they impose on transverse beam moments are
given as follows:

(cid:10)ww
(cid:10)w

w

(cid:48)

(cid:104)ww(cid:105) = 0

(cid:48)(cid:11) = 0
(cid:48)(cid:11) = 0

(cid:104)x y(cid:105) = 0
(cid:104)x y
(cid:104)x(cid:48)

(cid:48)(cid:105) = −(cid:104)x(cid:48)
(cid:48)(cid:105) = 0

y

y(cid:105)

⇒
⇒
⇒

(cid:104)xx(cid:105) = (cid:104)yy(cid:105)
(cid:104)xx(cid:48)(cid:105) = (cid:104)yy
(cid:48)(cid:105)
(cid:48)(cid:105)
(cid:104)x(cid:48)x(cid:48)(cid:105) = (cid:104)y
(cid:48)

y

42

Figure 4.1: Simulated spatial distribution of Ar9+ ions at the extraction plane of the Artemis [5]
ECR ion source at FRIB. The inner circle indicates the extraction aperture. x-y and u-v denote
two possible transverse coordinate systems. Subsequent analysis will show that, despite severe
triangulation, the rms envelope and emittance of the beam is the same along any transverse
direction. Simulation results courtesy of the code developed by Dr. Vladirmir Mironov at
JINR-Dubna [25, 26].

Together these constraints give:

(cid:104)xx(cid:105) = (cid:104)yy(cid:105)
(cid:48)(cid:105)
(cid:104)xx(cid:48)(cid:105) = (cid:104)yy
(cid:104)x(cid:48)x(cid:48)(cid:105) = (cid:104)y
(cid:48)
(cid:48)(cid:105)
(cid:104)x y(cid:105) = 0
(cid:104)x(cid:48)
(cid:48)(cid:105) = 0
y
(cid:48)(cid:105) = −(cid:104)x(cid:48)
(cid:104)x y

y

y(cid:105)

(4.20)

(4.21)

(4.22)

(4.23)

(4.24)

(4.25)

An alternative proof of the fact that Eq. (4.20) to (4.25) holds for a beam with 3-fold rotational
symmetry is provided in Appendix ??. That proof utilizes trigonometric identities and is applicable
to any n-fold rotational symmetry where n > 2.

Eq. (4.20) to (4.25) are the same set of relations that are obeyed by a beam with continuous

43

rotation symmetry. Therefore, following the argument in Sec. ??, we can immediately conclude
that, not only are the 2nd-order beam moments in x-x(cid:48) and y-y(cid:48) identical, they must be identical in
any transverse direction. This can also be proved by writing out moments in an arbitrary direction
with u = x cos θ + y sin θ, but the approach is much more tedious than the indistinguishability
argument and is not easily extendable to higher order moments. The results also imply that the
azimuthal orientation of the ECR sextupole (see Fig. 4.1) has no effect on 2nd-order moments.
Small errors (e.g. mechanical misalignment) may introduce a limited degree of violation to this
useful idealization.

3rd Order Moments

4.1.4.2
For k = 3, we use Eq. (4.19) to obtain χ(3) = 10, which means 10 complex moments satisfy
conditions 4.1.1 and 4.1.2. We also know that, since the moment order k = 3 is the same as n, 4
complex moments must violate condition 4.1.3, namely:

(cid:104)www(cid:105)

(cid:10)www
(cid:10)ww
(cid:10)w

w

w

w

(cid:48)

(cid:48)

(cid:48)

(cid:48)(cid:11)
(cid:48)(cid:11)
(cid:48)(cid:11)

They provide no information because e3iθ = 1 for θ = 2π/3.

The 6 complex moments which also satisfy condition 4.1.3 give:

(cid:104)www(cid:105) = 0 ⇒
= 0 ⇒

(cid:68)
www(cid:48)(cid:69)
(cid:48)(cid:11) = 0 ⇒
(cid:10)www
(cid:10)ww
(cid:48)(cid:11) = 0 ⇒
w(cid:48)(cid:69)
(cid:68)
w(cid:48)(cid:69)
(cid:68)

(cid:104)xxx(cid:105) = −(cid:104)x yy(cid:105)
(cid:48)(cid:105) = (cid:104)x(cid:48)
2(cid:104)x yy
(cid:104)xxx(cid:48)(cid:105) = −(cid:104)x(cid:48)
(cid:104)xx(cid:48)x(cid:48)(cid:105) = −(cid:104)x y
= 0 ⇒ 2(cid:104)x(cid:48)
(cid:48)(cid:105) = (cid:104)x y
(cid:48)
= 0 ⇒ (cid:104)x(cid:48)x(cid:48)x(cid:48)(cid:105) = −(cid:104)x(cid:48)

yy(cid:105) − (cid:104)xxx(cid:48)(cid:105)
yy(cid:105)
(cid:48)
(cid:48)(cid:105)
(cid:48)(cid:105) − (cid:104)xx(cid:48)x(cid:48)(cid:105)
(cid:48)(cid:105)
(cid:48)

ww
(cid:48)

yy

w

w

w

(cid:48)

(cid:48)

(cid:48)

y

y

y

y

(cid:48)(cid:105)

(cid:104)yyy(cid:105) = −(cid:104)xx y(cid:105)
2(cid:104)xx(cid:48)
(cid:104)xx y
(cid:104)x(cid:48)x(cid:48)
2(cid:104)x(cid:48)x y
(cid:48)
(cid:104)y

(cid:48)(cid:105) − (cid:104)yyy
y(cid:105) = (cid:104)xx y
(cid:48)(cid:105) = −(cid:104)yyy
(cid:48)(cid:105)
(cid:48)
(cid:48)(cid:105)
y(cid:105) = −(cid:104)yy
y
(cid:48)(cid:105) = (cid:104)x(cid:48)x(cid:48)
y(cid:105) − (cid:104)yy
(cid:48)(cid:105) = −(cid:104)x(cid:48)x(cid:48)

(cid:48)(cid:105)

(cid:48)

y

y

y

(cid:48)

(cid:48)(cid:105)

y

44

Further combination of these relations show that they impose the following independent conditions
on third order moments:

(cid:104)xxx(cid:105) = −(cid:104)x yy(cid:105)
(cid:104)yyy(cid:105) = −(cid:104)xx y(cid:105)
(cid:48)(cid:105) = (cid:104)x(cid:48)
yy(cid:105) = −(cid:104)xxx(cid:48)(cid:105)
(cid:104)x yy
(cid:104)xx(cid:48)
(cid:48)(cid:105) = −(cid:104)yyy
(cid:48)(cid:105)
y(cid:105) = (cid:104)xx y
(cid:104)x(cid:48)
(cid:48)(cid:105) = (cid:104)x y
(cid:48)
(cid:48)(cid:105) = −(cid:104)xx(cid:48)x(cid:48)(cid:105)
y
(cid:104)xx(cid:48)
(cid:48)(cid:105) = (cid:104)x(cid:48)x(cid:48)
(cid:48)(cid:105)
y(cid:105) = −(cid:104)yy
(cid:104)x(cid:48)x(cid:48)x(cid:48)(cid:105) = −(cid:104)x(cid:48)
(cid:48)
(cid:48)
(cid:48)(cid:105) = −(cid:104)x(cid:48)x(cid:48)
(cid:104)y

(cid:48)(cid:105)
(cid:48)(cid:105)

yy

y

y

y

y

(cid:48)

y

(cid:48)

y

y

Conversely, all third order moments of an axisymmetric beam must vanish. The same conclusion
applies to all odd-order moments of an axisymmetric beam as well. This can be seen immediately
if one applies a rotation by θ = π. This difference in moment relations between a C3 beam and an
axisymmetric beam yields interesting implications for beam transport.

Whereas the approximation of linear optics is often made, beam transport inevitably involve
nonlinear terms in the transfer map. Consider a beam with some specific rotational symmetry
entering a beam line with the nonlinear map:

4


i=1

4


4


i=1

j=1

xl =

Rli xi +

i )
Tli j xi xj + O(x3

(4.26)

where subscripts can all attain integer values from 1 to 4 with l, m denoting final phase space
coordinates and i, j, k denoting initial phase space coordinates. Second order moments of the
beam exiting the beam line are given by:

4


4


i=1

j=1

(cid:10)xi xj

(cid:11) +

4


4


4


(cid:16)

i=1

j=1

k=1

(cid:104)xl xm(cid:105) =

RliRm j

RliTm jk + RmiTl jk

(cid:17)(cid:10)xi xj xk

(cid:11) + O(cid:16)(cid:68)

(cid:69)(cid:17)

x4
i

(4.27)

45

which have a dependence on initial moments higher than second order due to nonlinear terms in
the map. For an axisymmetric beam, all of its 3rd order moments vanish, so the leading correction
to linear optics arises from 4th order moments. For a C3 beam, however, 3rd order moments are
probably nonzero and can affect the final second order moments. Hence, linear optics is a better
approximation in the sense that residual terms neglected are high order when the incoming beam
is axisymmetric than when it has C3 symmetry.

Last but not least, we note that all 3rd order moments of a C2 beam also vanish. Thus
the discussion in the paragraph above also holds if we substitute axisymmetry by C2, the most
commonly occurring rotational symmetry in beam lines due to the ubiquity of quadrupoles. This
fact may render linear optics a surprisingly poor assumption when it is applied to C3 beams. The
assumption may be poor because second order terms in the transfer map act more strongly on
the rms evolution of C3 beams than on that of axisymmetric or C2 beams, and the poorness may
be surprising because most of our experience is built upon the less error-prone cases of C2 and
axisymmetry.

4.2 Coupled Beam Dynamics

This section presents several results on linear coupled beam dynamics in single particle dynam-

ics (i.e. self fields are neglected).

4.2.1 Axisymmetric Beams and Eigen-emittances
This subsection derives the relation between εx, (cid:104)x y(cid:48)(cid:105) and the eigen-emittances for an axisymmetric
beam. An axisymmetric beam has sigma matrix with the following (generally non-zero) 2nd-order
moment elements:

(4.28)

(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)

σ =

(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172)

(cid:104)xx(cid:105)
(cid:104)xx(cid:48)(cid:105)
0
(cid:104)x y(cid:48)(cid:105)

0

(cid:104)xx(cid:48)(cid:105)
(cid:104)x(cid:48)x(cid:48)(cid:105) − (cid:104)x y(cid:48)(cid:105)
− (cid:104)x y(cid:48)(cid:105)
(cid:104)xx(cid:105)
(cid:104)xx(cid:48)(cid:105)

0

(cid:104)x y(cid:48)(cid:105)
0
(cid:104)xx(cid:48)(cid:105)
(cid:104)x(cid:48)x(cid:48)(cid:105)

46

From the eigenvalues of σS, one obtains the two eigen-emittances:

ε1 = εx +(cid:10)x y
ε2 = εx −(cid:10)x y

(cid:48)(cid:11)
(cid:48)(cid:11)

where εx is the usual rms emittance defined by Eq. (2.19). It follows that:

(cid:10)x y

εx =

(cid:48)(cid:11) =

ε1 + ε2

(cid:16) ε1 − ε2

2

2

(cid:17)

(4.29)

(4.30)

(4.31)

(4.32)

4.2.2 Magnetized Beam
Consider an axisymmetric beam in an axisymmetric beam line going from si to s f where Bz0(si) (cid:44) 0
and Bz0(s f ) = 0

At s = si, due to the presence of axial magnetic field, canonical coordinates ˜x and laboratory

coordinates x are related by

(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)

K =

0

˜x = Kx
0
0
1
1 −ks/2 0
0
0
0
0
ks/2 0
1

1
0

(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172)

where

ks =

qB
mv

The sigma matrix written in terms of canonical coordinates is given by

σi = ˜xi˜xT

i = KxixT

i KT

(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)

σi =

0

0
0
1
1 −ks/2 0
0
0
0
0
ks/2 0
1

1
0

(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172)

(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)

(cid:104)xx(cid:105)i
(cid:104)xx(cid:48)(cid:105)i
0
(cid:104)x y(cid:48)(cid:105)i

(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172)

(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)

(cid:104)x y(cid:48)(cid:105)i
0
(cid:104)xx(cid:48)(cid:105)i
(cid:104)x(cid:48)x(cid:48)(cid:105)i

0
1

0 ks/2
1
0
0
0
0 −ks/2 1
0
0
0
1

0

(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172)

0

(cid:104)xx(cid:48)(cid:105)i
(cid:104)x(cid:48)x(cid:48)(cid:105)i − (cid:104)x y(cid:48)(cid:105)i
− (cid:104)x y(cid:48)(cid:105)i
(cid:104)xx(cid:105)i
(cid:104)xx(cid:48)(cid:105)i

0

47

At s = s f ,

σ f =

(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)

(cid:104)xx(cid:105) f
(cid:104)xx(cid:48)(cid:105) f
0
(cid:104)x y(cid:48)(cid:105) f

(cid:104)xx(cid:48)(cid:105) f
(cid:104)x(cid:48)x(cid:48)(cid:105) f
− (cid:104)x y(cid:48)(cid:105) f

0

0

− (cid:104)x y(cid:48)(cid:105) f
(cid:104)xx(cid:105) f
(cid:104)xx(cid:48)(cid:105) f

(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172)

(cid:104)x y(cid:48)(cid:105) f
0
(cid:104)xx(cid:48)(cid:105) f
(cid:104)x(cid:48)x(cid:48)(cid:105) f

(cid:10)x y

(cid:48)(cid:11)

f

=(cid:10)x y

(cid:48)(cid:11)

ks
2 (cid:104)xx(cid:105)i

+

i

From conservation of canonical angular momentum:

(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)

σ f =

(cid:104)xx(cid:105) f
(cid:104)xx(cid:48)(cid:105) f
0

(cid:104)x y(cid:48)(cid:105)i + ks2 (cid:104)xx(cid:105)i

(cid:104)xx(cid:48)(cid:105) f
(cid:104)x(cid:48)x(cid:48)(cid:105) f

− (cid:104)x y(cid:48)(cid:105)i − ks2 (cid:104)xx(cid:105)i

0

0

− (cid:104)x y(cid:48)(cid:105)i − ks2 (cid:104)xx(cid:105)i

(cid:104)xx(cid:105) f
(cid:104)xx(cid:48)(cid:105) f

(cid:104)x y(cid:48)(cid:105)i + ks2 (cid:104)xx(cid:105)i

0
(cid:104)xx(cid:48)(cid:105) f
(cid:104)x(cid:48)x(cid:48)(cid:105) f

We can solve for eigenvalues of σiS and σ f S to get eigen-emittances:

(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172)

2
i; I,II =




2
i,x +

±(cid:16)

k2
2 (cid:104)xx(cid:105)2
s

ks (cid:104)xx(cid:105)i + 2(cid:10)x y

(cid:48)(cid:11)

i + 2ks (cid:104)xx(cid:105)i

(cid:10)x y

(cid:48)(cid:11)

+(cid:10)x y

(cid:48)(cid:11)2

i

i
+ k2

i

(cid:17)(cid:113)
(cid:10)x y

2


i,x

(cid:48)(cid:11)

i

s (cid:104)xx(cid:105)i /4 + ks (cid:104)xx(cid:105)i (cid:104)x y(cid:48)(cid:105)i

+(cid:10)x y

(cid:48)(cid:11)2

i ± 
 f ,x

(cid:16)

ks (cid:104)xx(cid:105)i + 2(cid:10)x y

(cid:48)(cid:11)

(cid:17)

i

2
f ; I,II = 


2
f ,x




+

k2
4 (cid:104)xx(cid:105)2
s

i + ks (cid:104)xx(cid:105)i

Eigen-emittances are invariants, i.e.

Therefore, the final x-emittance is given by:


i; I,II = 
 f ; I,II

2
f ,x




= 


2
i,x +

k2
4 (cid:104)xx(cid:105)2
s

i + ks (cid:104)xx(cid:105)i

48

(cid:10)x y

(cid:48)(cid:11)

i

4.2.3 Mirror-Symmetric Beam Line Does Not Preserve Axial-symmetry

In a beam line consisting of quads and dipoles only, equations of motion in x and y are decoupled.
Thus the transfer matrix is block diagonal, with the general form:

(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)

M =

m11 m12

0
0

0
0
0 m33 m34
0 m43 m44

m21 m22
0
0

(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172)

Suppose there is a symmetric beam line, i.e.

M(s f |si) = M1M2 . . . Mn−1MnMnMn−1 . . . M2M1

where

M(sm|si) = MnMn−1 . . . M2M1

M(s f |sm) = M1M2 . . . Mn−1Mn

(cid:170)(cid:174)(cid:174)(cid:172)

Proof:

m21 m11

m21 m22

Where si, sm and s f stand for the start, mid- and end point respectively.

Lemma: if Mx(sm|si) =(cid:169)(cid:173)(cid:173)(cid:171)m11 m12
(cid:170)(cid:174)(cid:174)(cid:172) then Mx(s f |sm) =(cid:169)(cid:173)(cid:173)(cid:171)m22 m12
(cid:169)(cid:173)(cid:173)(cid:171) x
= Mx(sm|si)(cid:169)(cid:173)(cid:173)(cid:171) x
(cid:170)(cid:174)(cid:174)(cid:172)m
(cid:170)(cid:174)(cid:174)(cid:172)i
=(cid:169)(cid:173)(cid:173)(cid:171) x
Mx(s f |sm)(cid:169)(cid:173)(cid:173)(cid:171) x
(cid:170)(cid:174)(cid:174)(cid:172)m

By time-reversal:

x(cid:48)

(cid:170)(cid:174)(cid:174)(cid:172)i

−x(cid:48)

x(cid:48)

−x(cid:48)

49

where

(cid:170)(cid:174)(cid:174)(cid:172)

0
0 −1

0
0 −1

m21 m22

m21 m11

0
0 −1

x(cid:48)

0
0 −1

0
0 −1

(cid:169)(cid:173)(cid:173)(cid:171) x

x(cid:48)

=(cid:169)(cid:173)(cid:173)(cid:171)1
(cid:169)(cid:173)(cid:173)(cid:171) x
Mx(s f |sm)(cid:169)(cid:173)(cid:173)(cid:171)1
(cid:170)(cid:174)(cid:174)(cid:172)
(cid:170)(cid:174)(cid:174)(cid:172)m
x (sm|si)(cid:169)(cid:173)(cid:173)(cid:171)1
Mx(s f |sm) =(cid:169)(cid:173)(cid:173)(cid:171)1
(cid:170)(cid:174)(cid:174)(cid:172) M−1
T =(cid:169)(cid:173)(cid:173)(cid:171)1
(cid:170)(cid:174)(cid:174)(cid:172)
(cid:170)(cid:174)(cid:174)(cid:172) =⇒(cid:101)Mx =(cid:169)(cid:173)(cid:173)(cid:171)m22 m12
Mx =(cid:169)(cid:173)(cid:173)(cid:171)m11 m12
My =(cid:169)(cid:173)(cid:173)(cid:171)m33 m34
(cid:170)(cid:174)(cid:174)(cid:172) =⇒(cid:101)My =(cid:169)(cid:173)(cid:173)(cid:171)m44 m34
(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)
(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)
(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172)m
(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172)i
(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)

0
0
1
0 −1 0
0
0
0
1
0 −1
0

(cid:104)x y(cid:105)
(cid:104)x y(cid:48)(cid:105)
(cid:104)x(cid:48)y(cid:105)
(cid:104)x(cid:48)y(cid:48)(cid:105)

(cid:104)x y(cid:105)
(cid:104)x y(cid:48)(cid:105)
(cid:104)x(cid:48)y(cid:105)
(cid:104)x(cid:48)y(cid:48)(cid:105)

0

0
0
1
0 −1 0
0
0
0
1
0 −1
0

0
0

= Mx ⊗ My

m43 m44

m43 m33

(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)

(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172)

M−1

0

0
0

(cid:101)M =

(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)

(cid:104)x y(cid:105)
(cid:104)x y(cid:48)(cid:105)
(cid:104)x(cid:48)y(cid:105)
(cid:104)x(cid:48)y(cid:48)(cid:105)

(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) f

(cid:170)(cid:174)(cid:174)(cid:172)i
(cid:170)(cid:174)(cid:174)(cid:172)

(cid:170)(cid:174)(cid:174)(cid:172)
(cid:170)(cid:174)(cid:174)(cid:172)

(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172)
(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172)i

(cid:16)(cid:101)MxMx

(cid:17) ⊗(cid:16)(cid:101)MyMy

=

(cid:17)(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)

(cid:104)x y(cid:105)
(cid:104)x y(cid:48)(cid:105)
(cid:104)x(cid:48)y(cid:105)
(cid:104)x(cid:48)y(cid:48)(cid:105)

50

For (cid:104)x y(cid:105)i = 0, (cid:104)x(cid:48)y(cid:48)(cid:105)i = 0 and (cid:104)x y(cid:48)(cid:105)i = −(cid:104)x(cid:48)y(cid:105)i:

(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)

(cid:104)x y(cid:105)
(cid:104)x y(cid:48)(cid:105)
(cid:104)x(cid:48)y(cid:105)
(cid:104)x(cid:48)y(cid:48)(cid:105)

(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) f

(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)

=

2(m11m22 + m12m21)m34m44 − 2(m33m44 + m34m43)m12m22
−4m12m22m33m43 + (m11m22 + m12m21)(m33m44 + m34m43)
4m11m21m34m44 − (m11m22 + m12m21)(m33m44 + m34m43)
−2(m11m22 + m12m21)m33m43 + 2(m33m44 − m34m43)m11m21

(cid:104)x y

(cid:48)(cid:105)i

which do not obey any of (cid:104)x y(cid:105) f = 0, (cid:104)x(cid:48)y(cid:48)(cid:105) f = 0 or (cid:104)x y(cid:48)(cid:105) f = −(cid:104)x(cid:48)y(cid:105) f in general. In fact,

(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172)

(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)

(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) f

(cid:104)x y(cid:105)
(cid:104)x y(cid:48)(cid:105)
(cid:104)x(cid:48)y(cid:105)
(cid:104)x(cid:48)y(cid:48)(cid:105)

(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)

0
L/2
−L/2
0

(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172)

=

only if Mx = My

4.2.4 Linear Solenoid Transport

Section 4.1 proves that the 2nd-order moments of an axisymmetric beam must obey the following
relations:

(cid:104)xx(cid:105) = (cid:104)yy(cid:105)
(cid:104)xx(cid:48)(cid:105) = (cid:104)yy
(cid:48)(cid:105)
(cid:104)x(cid:48)x(cid:48)(cid:105) = (cid:104)y
(cid:48)(cid:105)
(cid:48)
(cid:104)x y(cid:105) = 0
(cid:48)(cid:105) = 0
(cid:104)x(cid:48)
y
(cid:48)(cid:105) = −(cid:104)x(cid:48)
(cid:104)x y

y

y(cid:105)

We call a beam that fulfills these relations 2nd-order axisymmetric, which is a much weaker notion
than axisymmetric because it does not impose any requirements on higher order moments. This

51

if a
subsection shows that 2nd-order axisymmetry is preserved by linear solenoid transport, i.e.
2nd-order axisymmetric beam enters a linear solenoid, it will retain its 2nd-order axisymmetry
upon exit. Note that in this context we are neglecting space charge forces which, depending on the
distribution, may exert non-axixymmetric self-fields.

An axisymmetric beam will still be axisymmetric after passing through a solenoid, which
means that it obeys 2nd-order axisymmetry both upstream and downstream. Furthermore, when
the solenoid is linear, the final beam moments are only functions of initial beam moments of the same
order. Therefore, a linear solenoid has a transfer map that preserves 2nd-order axisymmetry. This is
a general property of the transfer maps of linear solenoids that is agnostic to beam information other
than 2nd-order moments. So as long as a beam has 2nd-order axisymmetry, it will be preserved by
a linear solenoid.

Lastly, we note that, for a linear solenoid, not only does 2nd-order axisymmetric upstream
imply 2nd-order axisymmetric downstream, considerations from time reversal also dictates 2nd-
order axisymmetric downstream implies 2nd-order axisymmetric upstream.

4.2.5 Quasi-axisymmetric Conditions

As shown in Sec. 4.2.4, linear solenoid optics preserves 2nd-order axisymmetry, so it would be ideal
for a beam to obey the corresponding relations when it enters solenoid transport. In this subsection,
we define a weaker set of conditions which we call quasi-axisymmetry. A non-axisymmetric
beam satisfying its conditions have properties similar to that of an axisymmetric beam in solenoid
transport. The conditions are weaker because it does not require εx = εy, which is a necessary
condition for an axisymmetric beam. This has practical importance when a beam with εx (cid:44) εy has
to be matched into a solenoid transport line using only normal quadrupoles.

52

The quasi-axisymmetry conditions read:

αx = αy

βx = βy
(cid:104)x y(cid:105) = 0

(cid:48)(cid:11) = 0
(cid:48)(cid:11) = −(cid:10)x(cid:48)

(cid:10)x(cid:48)
(cid:10)x y

y

(4.33)

(4.34)

(4.35)

(4.36)

(4.37)

y(cid:11)

In terms of 2D phase space projections, the first two conditions describe rms ellipses in x-x(cid:48) and
y-y(cid:48) phase spaces which have the same orientation and sizes of the ratio εx : εy. Therefore,

(cid:104)xx(cid:105) =

(cid:10)xx(cid:48)(cid:11) =
(cid:10)x(cid:48)x(cid:48)(cid:11) =

εy
εx
εy
εx
εy
εx

(cid:104)yy(cid:105)

(cid:10)yy
(cid:10)y

(cid:48)(cid:11)
(cid:48)(cid:11)

(cid:48)

y

(4.38)

(4.39)

(4.40)

If we express moments of a quasi-axisymmetric beam in an arbitrary transverse direction in

terms of the moments in x, x(cid:48) and y, y(cid:48), we obtain, for u = x cos θ + y sin θ:

(cid:104)uu(cid:105) = (cid:104)xx(cid:105) cos2
= (cid:104)xx(cid:105) cos2
= (cid:104)xx(cid:105) cos2
= (cid:104)xx(cid:105)

(cid:20)

1 +

θ + 2 (cid:104)x y(cid:105) sin θ cos θ + (cid:104)yy(cid:105) sin2
θ + (cid:104)yy(cid:105) sin2
θ + (cid:104)xx(cid:105) εy
εx
sin2

(cid:18) εy

θ
sin2

(cid:19)

(cid:21)

− 1

θ

θ

εx

θ

(4.41)

Applying Eq. (4.38), one can check that Eq. (4.41) correctly recovers (cid:104)uu(cid:105) = (cid:104)xx(cid:105) and (cid:104)uu(cid:105) = (cid:104)yy(cid:105)
at θ = 0 and θ = π/2 respectively. Similarly,

(cid:10)uu(cid:48)(cid:11) =(cid:10)xx(cid:48)(cid:11)(cid:20)
(cid:10)u(cid:48)u(cid:48)(cid:11) =(cid:10)x(cid:48)x(cid:48)(cid:11)(cid:20)

(cid:19)
(cid:19)

(cid:18) εy
(cid:18) εy

εx

εx

1 +

53

(cid:21)
(cid:21)

1 +

− 1

sin2

θ

− 1

sin2

θ

(4.42)

(4.43)

The Twiss parameters and emittance along u are given by:

(cid:113)(cid:104)uu(cid:105) (cid:104)u(cid:48)u(cid:48)(cid:105) − (cid:104)uu(cid:48)(cid:105)2

Applying Eq. (4.41) to Eq. (4.43), it can be shown that:

εu =

αu =

βu =

− (cid:104)uu(cid:48)(cid:105)
εu
(cid:104)uu(cid:105)
εu

εu = εx

(cid:18) εy

(cid:20)
(cid:19)
= εx +(cid:0)εy − εx(cid:1) sin2

1 +

εx

− 1

sin2

(cid:21)

θ

θ

= αx

− (cid:104)uu(cid:48)(cid:105)
εu
(cid:104)uu(cid:105)
εu

=

αu =

βu =

− (cid:104)xx(cid:48)(cid:105)

=

(cid:104)xx(cid:105)
εx

εx

= βx

(4.44)

(4.45)

(4.46)

(4.47)

(4.48)

(4.49)

Eq. (4.48) and Eq. (4.49) states that αu and βu has no dependence on θ and always equal their
counterparts in x. This means that, for a quasi-axisymmetric beam, despite εx (cid:44) εy, the Twiss
parameters are the same in all transverse directions. On the other hand, Eq. (4.47) shows that εu
changes with θ. εu attains its maximum and minimum value along x and y respectively if εx > εy,
and vice versa.

4.3 Applications to FRIB Front End

This section shows how the results developed in this chapter can be applied to the FRIB front
end. Fig. 4.2 shows a schematic of the FRIB front end where only elements relevant to transverse
dynamics are displayed. The beam line is also divided into segments (1) through (7) which will
play different roles in beam tuning. Section 4.3.1 discusses how results from Sec. 4.1 clarify the
beam dynamics in segments 1 and 2 and motivate investigation on an unresolved phenomenon.
Section 4.3.2 employs arguments in Sec. 4.2 to elucidate attainable beam states in segments 3 to
6 and presents a tuning scheme that can be readily implemented in the laboratory for achieving
optimal matching into the RFQ. Section 4.3.3 analyzes the intertwined problems of how the optics

54

Figure 4.2: A schematic of transverse focusing and bending elements in the FRIB front end.
Note that Segment 2 and 5 are mirror-symmetric about the axial mid-plane. Lengths are not
drawn to scale and all quadrupoles are normal quadrupoles (i.e. upright orientation).

in Linac Segment 1 (following segment 7) should be designed and how the MEBT (segment 7)
should match a non-axisymmetric beam into Linac Segment 1. Space charge is relatively weak
after the mid-plane of the CSS (segment 2), hence it is neglected in the discussion on tuning in
segments 3 to 7.

4.3.1 LEBT: Before Species Selection

Using the analytic tools developed Sec. 4.1, we proved in Sec. 4.1.4 that a beam with C3 symmetry
has the same emittance in any transverse direction. This result should hold in the presence of
multi-species space charge, chromatic aberrations and radial field nonlinearities because none of
these effects break the discrete rotational symmetry of the system.

This conclusion quickly revealed a conundrum at FRIB because diagnostic measurements often
show >10% difference between εx and εy at the mid-point and end-point of the charge selection
system (i.e. segment 2). Prior to this analysis, the difference used to be attributed to the sextupole

55

of the ECR ion source that possibly results in a triangulated beam, symmetry arguments from
Sec. 4.1 invalidated such an explanation and gave impetus to a more thorough inquiry.

We have identified two possible reasons for εx (cid:44) εy to occur in the charge selection system,
neither of which we understand fully: (A) the beam has εx (cid:44) εy entering segment 2; and (B)
emittance evolution in the dipole or prior to charge separation. Further investigation is underway
on both fronts to explain this phenomenon.

If the possibility A is true, the results in Sec. 4.1 compels us to argue by transposition that
axisymmetry is broken in segment 1. Thus we searched for plausible causes and concluded that
an old solenoid with poorly designed current leads may create strong multipole moments in the
downstream fringe fields.

Possibility B may also be true because εx (cid:44) εy was observed in Warp simulation results in
Sec. 3.3 due to an interplay between space charge and beam magnetization, albeit with a smaller
percentage difference. Furthermore, the discussion at the end of Sec. 4.1 shows that the evolution of
second order moments of a beam with C3 symmetry is likely more susceptible to higher order terms
in the transfer map than that of an axisymmetric beam. Since the transfer map of the 90-degree
combined function dipole may contain significant nonlinearities, the use of axisymmetric beams in
previous simulations could induce large errors in modeling the dipole transport. This suggests that
further simulation studies should include non-axisymmetric initial conditions with C3 symmetry
to probe how this changes the beam dynamics in segment 2.

4.3.2 LEBT: After Species Selection

After charge selection, the LEBT has to match the beam into the RFQ while avoiding beam loss.
The design of the optics can be done in two steps: (A) obtain complete information on transverse
coupled beam moments after segment 2 via diagnostic measurements; and (B) tune quadrupoles
and solenoids via optimization to accomplish design goals. There are a total of 17 knobs available
for tuning the beam, they comprises the focusing strengths of 10 quadrupoles and 7 solenoids. The
vertical drop only has six knobs rather than twelve because only two of the four quadrupole triplets

56

are independent in its design as a mirror-symmetric achromat.

Design goals include small beam sizes at locations with small apertures (e.g. within the buncher

and chopper), as well as:

αx = αy = αin

βx = βy = βin

εx = εy

(4.50)

(4.51)

(4.52)

at the RFQ entrance where αin and βin denote the design Twiss parameters entering the RFQ.

The most direct scheme is to employ all 17 knobs as optimization parameters and construct a
penalty function that incorporates all design goals as constraints. This approach was adopted by the
Accelerator Physics Department at FRIB. An example of optics design 1 based on the direct scheme
is plotted in Fig. 4.3. Note that there is strong emittance exchange in the lower LEBT solenoid
transport (segment 6) which causes the beam to enter the RFQ with large emittance differences in
x and y. Such emittance exchange is a common problem that occurs because the beam does not
fulfill 2nd order axisymmetry in the solenoid transport line.

We designed an easily implementable tuning scheme based on results in coupled beam dynamics
developed in Sec. 4.2. This treatment delineates attainable and preservable beam states in each
beam line segment, which allow us to split the optimization into separate stages with well-defined
goals.

In the ensuing analysis, it is important to distinguish between beam line segments that: (A) con-
tain solenoids, which couple the dynamics in x and y; and (B) contain dipoles and/or quadrupoles,
which does not couple the couple the dynamics in x and y. It is proved in Sec. 4.2.4 that linear

1provided by Dr. Tomofumi Maruta

57

Figure 4.3: An example of FRIB LEBT tuning results where the focusing strengths of 17
elements (10 quadrupoles and 7 solenoids) were optimized to achieve desired beam conditions at
the RFQ entrance. (Image courtesy of Dr. Tomofumi Maruta at FRIB Accelerator Physics
Department)

solenoids preserve the set of 2nd-order axisymmetric conditions:

(cid:104)xx(cid:105) = (cid:104)yy(cid:105)
(cid:104)xx(cid:48)(cid:105) = (cid:104)yy
(cid:48)(cid:105)
(cid:48)(cid:105)
(cid:48)
(cid:104)x(cid:48)x(cid:48)(cid:105) = (cid:104)y
(cid:104)x y(cid:105) = 0
(cid:104)x(cid:48)
(cid:48)(cid:105) = 0
y
(cid:48)(cid:105) = −(cid:104)x(cid:48)
(cid:104)x y

y

y(cid:105)

For dipoles and quadrupoles, εx and εy are conserved quantities under linear optics (see Chapter 2).
We extract from these results three conditional statements on beam states and beam transport:

Corollary 4.3.1. (2nd-order axisymmetry) → (εx = εy)

58

"Catchall" OptimizationCorollary 4.3.2. In linear optics, for a beam line consisting of solenoids and drifts, (incoming
beam is 2nd-order axisymmetric) ↔ (outgoing beam is 2nd-order axisymmetric)

Corollary 4.3.3. In linear optics, for a beam line consisting of normal dipoles, quadrupoles and
drifts, (incoming beam has εx = εy) ↔ (outgoing beam has εx = εy)

We start at the end (RFQ entrance) and work backwards. To achieve Eq. (4.50) to (4.52) at the
RFQ entrance, it is highly desirable for the beam to be 2nd-order axisymmetric in the lower LEBT.
2nd-order axisymmetry guarantees:

αx = αy

βx = βy

εx = εy

throughout the solenoid transport, which means the beam optics in identical in any transverse
direction and reduces to an effective 1D (radial) design. Hence, matching into the RFQ entrance
with Twiss parameters αin and βin becomes a problem of tuning 4 solenoids to meet 2 constraints.
Not only it the goal easily attainable, the problem also affords additional freedom to accommodate
other constraints, such as small beam size at the multi-harmonic buncher.

The next step concerns how 2nd-order axisymmetry can be achieved in the lower LEBT. A
summary of the problem can be found in Table 4.1. To have 2nd-order axisymmetry after segments
5 and 6 implies having εx = εy as well (see Corollary 4.3.1), and Corollary 4.3.3 in turn implies
εx = εy before the vertical drop. Concerning the conditions after the charge selection system,
the results in Sec. 4.2.3 show that it takes Mx = My for an uncoupled mirror symmetry beam
line to preserve 2nd-order axisymmetry. With only 3 quadrupoles on each half, segment 2 cannot
fulfill this condition even if the beam is 2nd-order axisymmetric before segment 2. Therefore,
the beam generally lacks 2nd-order axisymmetry after segment 2.
In practice, as we noted in
Sec. 4.3.1, measurements show that the beam almost always fail to even fulfill εx = εy after the
charge selection system. Remaining uncertainties in the design are denoted by question marks and
will be resolved below.

59

After segment 2 (charge selection system)
After segment 3 (upper LEBT quad.)
After segment 4 (upper LEBT sol.)
After segment 5 (vertical drop)
After segment 6 (lower LEBT)

εx = εy

X
?
(cid:88)
(cid:88)
(cid:88)

2nd-order axisymmetric

X
?
?
(cid:88)
(cid:88)

Table 4.1: Beam states entailed by the requirement of 2nd-order axisymmetry in the lower LEBT.
(cid:88) and X denote true and false respectively.

Two possible schemes to tune the LEBT are shown in Table 4.2. We proposed Scheme 2 which
tunes for εx = εy in the upper LEBT and uses the vertical drop to achieve 2nd-order axisymmetry.
Scheme 1, which tunes for 2nd-order axisymmetry in the upper LEBT, is also discussed because it
was an adaptation of the FRIB baseline design. Arguments below will illustrate why Scheme 2 is
preferable.

Scheme 1 is unlikely to work for the following reasons. Firstly, a prerequisite of the scheme
is εx = εy after the charge selection system, otherwise εx = εy cannot hold after segment 3
either (by corollary 4.3.3). However, measurement results have shown that this prerequisite almost
always fails to hold. Furthermore, even if εx = εy is true after segment 2, tuning for 2nd-order
axisymmetry after segment 3 is difficult to succeed. In general, after εx = εy removes one condition,
the beam must satisfy five more conditions to attain 2nd-order axisymmetry. However, segment
3 only contains four quadrupoles, so the task employs a four-parameter optimization to meet five
conditions, which is unlikely to produce favorable operating points even if we disregard aperture
limits and other secondary considerations.

Before explaining how Scheme 2 works, we can show by deduction that it is the only possible
arrangement. Since εx (cid:44) εy after segment 2, it entails (by Corollary 4.3.3) εx (cid:44) εy after segment 3
, which in turn implies that 2nd-order axisymmetry cannot hold after segment 3 (by Corollary 4.3.1)
and, consequently, after segment 4 (by Corollary 4.3.2).
The tuning in Scheme 2 is performed in three stages:

1. Stage 1 (segment 3 and 4): attain εx = εy at the entrance of segment 5. Seven knobs to tune

for one condition.

60

After segment 2 (charge selection system)
After segment 3 (upper LEBT quad.)
After segment 4 (upper LEBT sol.)
After segment 5 (vertical drop)
After segment 6 (lower LEBT)

εx = εy

(cid:88)
(cid:88)
(cid:88)
(cid:88)
(cid:88)

X
(cid:88)
(cid:88)
(cid:88)
(cid:88)

εx = εy

X
X
(cid:88)
(cid:88)
(cid:88)

Scheme 1

2nd-order

axisymmetric

Scheme 2

2nd-order

axisymmetric

X
X
X
(cid:88)
(cid:88)

Table 4.2: First set of quadrupole scan parameters

2. Stage 2 (segment 5): attain 2nd-order axisymmetry at the entrance of segment 6. Six knobs

to tune for five conditions.

3. Stage 3 (segment 6): match beam into RFQ design parameters. Four knobs to tune for two

conditions.

Each optimization contains more parameters than constraints, so it is likely that a favorable solution
will be found even with secondary constraints such as a small beam size at the multi-harmonic
buncher upstream of the RFQ.

Fig. 4.4 shows an example optics design based on Scheme 2 of Table 4.2. The optimization
was implemented by Dr. Tomofumi Maruta using a linear optics envelope code FLAME[29].
While emittance exchange occurs in the solenoids of segment 3, as a result of the optimization,
the solenoid strengths are arranged such that εx = εy when the exchange ends. There exists small
emittance exchange in the lower LEBT because the optimization does not attain perfect 2nd-order
axisymmetry after the vertical drop, but the exchange is much smaller than the direct optimization
design in Fig. 4.3, and εx is very close to εy at the RFQ entrance. This scheme is presently
employed for front end tuning in FRIB and has superseded previous methods.

4.3.3 MEBT and LS1

Despite tuning efforts in the LEBT, the beam often has different emittances in x and y (εx (cid:44) εy)
downstream of the RFQ. With only normal quadrupoles in the MEBT, it is impossible to achieve

61

Figure 4.4: An example of FRIB LEBT tuning results where the tuning scheme introduced in
Sec. 4.3.2 was applied to optimize elements strengths in three stages. (Image courtesy of Dr.
Tomofumi Maruta at FRIB Accelerator Physics Department)

εx = εy and hence axisymmetry at the LS1 entrance. This subsection addresses three intertwined
problems under these circumstances: 1) how to design the optics for the solenoid transport line
LS1; 2) how to define the acceptance condition; and 3) how to match a non-axisymmetric beam
into the design optics.

In the following treatment, we first propose a design and matching recipe, and then discuss the

underlying beam dynamics and why it will work.

4.3.3.1 Optics Design and Matching Recipe

Firstly, the transverse optics of the LS1 solenoid transport line should be designed assuming the
incoming beam is axisymmetric. This reduces the design to a 1D problem (optics and beam
parameters identical along any transverse direction due to axisymmetry) which is easy to handle
and is how it was done in the conceptual and engineering design of FRIB. The 1D design defines
αin and βin, the incoming beam parameters required for matching, and εA, the maximum acceptable
incoming emittance (i.e. acceptance).

62

Optimization:Stage 1Stage 2Stage 3Coupling Coeff.Next, we use the quasi-axisymmetric conditions defined in Sec. 4.2.5 to match a beam with

εx (cid:44) εy into an axisymmetric optics by requiring:

αx =αy = αin

βx =βy = βin
(cid:104)x y(cid:105) = 0

(cid:48)(cid:11) = 0
(cid:48)(cid:11) = −(cid:10)x(cid:48)

y(cid:11)

(cid:10)x(cid:48)
(cid:10)x y

y

This is achieved by tuning the MEBT quadrupoles. The acceptance condition is fulfilled if εx ≤ εA
and εy ≤ εA.

To explain why the recipe works, we first note the transfer matrix of a linear solenoid can be

decomposed as [30]:

Msol = MRMF

where MR is a rotation matrix and MF is focusing in both transverse directions:

(4.53)

(4.54)

(4.55)

(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)
(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)

MR =

MF =

cos ks(cid:96)

0

− sin ks(cid:96)

0

sin ks(cid:96)

0

cos ks(cid:96)

0

sin ks(cid:96)

0

cos ks(cid:96)

0

0

− sin ks(cid:96)

0

cos ks(cid:96)
−ks sin ks(cid:96)

1
sin ks(cid:96)
ks
cos ks(cid:96)

0
0

0
0

cos ks(cid:96)
−ks sin ks(cid:96)

(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172)

cos ks(cid:96)
0
0

(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172)

0
0
1
sin ks(cid:96)
ks
cos ks(cid:96)

ks = Bz0/2[Bρ], (cid:96) is the effective length of the solenoid and Bz0 is the effective field.

When a quasi-axisymmetric beam enters a linear solenoid, since the incoming beam has the
same Twiss parameters in every transverse direction, and the focusing force is axisymmetric, the
beam should retain quasi-axisymmetry upon exiting the solenoid. We arrive at this conclusion just

63

by invoking symmetry, but Eq. (4.53) tells us more. In addition to focusing, the linear solenoid
also azimuthally rotates the beam by the Larmor angle ks(cid:96) [13]. This is relevant to the dynamics
because Eq. (4.47) shows that εu, the emittance measured along ˆu where u = cos θ + sin θ, is
dependent on θ. This means the transverse directions along which the beam size is maximum and
minimum rotate by angle ks(cid:96) after passage through the solenoid.

The beam evolution can be described using the following picture. The axisymmetric optics
design and acceptance together defines a cylindrical envelope that varies in size along the solenoid
transport line. For a quasi-asymmetric beam that is matched to the optics design with εx = εA > εy,
the beam is spatially an ellipse as it enters the solenoid transport. The ellipse rotates in each solenoid
and its two points of maximum extent always touches the cylindrical envelope. Thus a matched
quasi-axisymmetric beam within acceptance guarantees that the beam is spatially bounded by the
design cylindrical envelope throughout the solenoid transport.

4.3.4 Further Work

In practice, the LS1 is not purely a solenoid transport because there exists quadrupole kicks from
the quarter-wave SRF cavities. While the quadrupole kicks can be treated as small perturbations, it
will be interesting to study how the solenoid optics can be designed such that the effect from such
perturbations can be minimized.

Additionally, construction and alignment errors can introduce further (hopefully small) symme-
try breaking terms. Likewise, space charge nonlinearities of a nonuniform and non-axisymmetric
beam can muddle the overall picture. Nevertheless, considerations presented in this chapter pro-
vide a clear ideal basis for optimal tuning whose robustness against reasonable error sources can
be evaluated with detailed simulations.

64

CHAPTER 5

TRANSVERSE PHASE-SPACE MEASUREMENTS WITH ALLISON-TYPE

EMITTANCE SCANNERS

The content of this chapter and Appendix A is an extended version of the work published in
Ref. [31].

5.1

Introduction

Allison-type emittance scanners [32], or Allison scanners for short, are widely used to efficiently
measure projected 2D phase space distributions of low-energy beams. An Allison scanner (see
Fig. 5.1) consists of an entrance slit-plate (slit width s), an aligned exit slit-plate (slit width s)
with an integrated Faraday cup, and a bipolar-biased electric dipole (voltage ±V0) placed between
the two slits. The scanner assembly is translated mechanically (typically in discrete steps with
a stepper motor) to change the slit position, and the dipole voltage V0 is swept at each step to
select transmittable angles by varying the bending strength. The scanner samples one grid point
in phase space for each coordinate and “E-dipole” voltage setting, with the beam density taken to
be proportional to the current collected by the Faraday cup. The entire projected distribution is
measured by recording currents collected over a range of position and voltage values that samples
the full phase-space projection.

Analysis of Allison scanner data requires a voltage-to-angle conversion and the angular resolu-
tion of the device. Idealized relations were derived in Ref. [32] and summarized in Ref. [33]. They
constitute the conventional analysis model which assumes the scanner has an idealized geometry
with uniform hard-edge dipole fields, thin slit-plates and longitudinally symmetric placement of
the E-dipole.

Extending upon the work in [34], Sec. 5.2 augments the conventional analysis model to account
for two commonly occurring geometric features that can lead to significant corrections due to:
1) the effective longitudinal asymmetry in E-dipole placement between the slits, and 2) the finite

65

slit-plate thickness. Their effects on particle transmission are studied using both simulations and
analytic expressions which show excellent agreement. Asymmetry in E-dipole placement alters the
slope of the linear relation between dipole voltage and particle angle. Finite slit thickness causes
the weight of data points to vary as a decreasing function of their voltage values. The reduction
becomes significant at large voltages and when the slits’ thickness-to-width ratio (cid:38) 1.

When the interval between data points is smaller than the device resolution, the current measured
at each data point is a weighted sum of the actual current densities at the data point and its neighbors.
Sec. 5.3 shows that a detailed accounting of the phase space region contributing to each data point
allows one to deconvolve the effective blur induced by overlapping data points and resolve the
beam distribution more accurately. Reconstruction procedures are illustrated and verified using
numerically generated phase-space distributions so that improvements are clearly characterized.

Section 5.4 applies the improved model to experimental data from the Allison scanner used in
the front-end of the Facility for Rare Isotope Beams (FRIB) at Michigan State University. Raw
measurement data first undergo a noise removal scheme, then the processed data are analyzed with
both the conventional model and the improved model. Comparisons reveal that the improved model
produces more consistent results among a series of scans with changing focusing upstream. Beam
moments are modified significantly relative to conventional treatments, with differences growing
with the angular divergence of the beam distribution.

The Python code tools used to perform the data analysis and noise removal are described and
made available [35]. The modular programs are readily adaptable to other Allison scanners via a
change of geometric input parameters.

Section 5.5 concludes this study with an outline of directions for future improvements.

5.2 Additional Geometric Features

This section first describes how asymmetric E-dipole placement and finite slit thickness can be
modeled in Allison scanners. Effects induced are investigated using simulations that probe particle
transmission given the device geometry and E-dipole voltage. The simulation results are accurately

66

modeled by analytic expressions for typical choices of scanner parameters. The implications for
data analysis are demonstrated through simulated measurements of ideal particle distributions
chosen to highlight those effects.

5.2.1 Geometric Model

With reference to a realistic Allison scanner (see Fig. 5.1), the geometric information relevant
to particle transmission and measurement data analysis can be captured using a model with six
parameters. A schematic of the model for measuring the x-x(cid:48) phase space is illustrated in Case IV
of Fig. 5.2 where

• L is the E-dipole plate length;

• l1 is the distance between the entrance-slit and the E-dipole plate;

• l2 is the distance between the exit-slit and the E-dipole plate;

• g is the gap between the E-dipole plates;

• s is the slit width; and

• d is the slit thickness.

The model assumes that particle transmission has no y-dependence, which is a good approximation
if no particle approaches the vertical extent of the device. This requires the vertical extent of the
scanner to be much larger than 2(|ycen| + yrms) where ycen and yrms are the beam centroid position
and rms width in the vertical direction.

The geometric model hinges upon a correct definition of l1, l2 and d, so it is important to
identify what performs the function of a slit. Figure 5.1 shows the design for the FRIB Allison
scanner [lidia2016overview]. On each side of the E-dipole there is a thick plate with an opening
(see zoom insert) that has large-angled relief cuts and a straight channel of minimum width. The
relief cuts allow using thicker plates that enable cooling and have higher mechanical stability. The

67

relief cuts do not intercept any ions that would have otherwise passed through the slits, since the
opening angle (30◦ in Fig. 5.1) far exceeds the transverse angles of beam ions. Hence the slit, i.e.
the structure that limits particle transmission, is only taken to be the minimum-width channel, with
width s = 60 µm and thickness d = 254 µm in Fig. 5.1.

The relief cut is immaterial to the slit-to-E-dipole distance l1 in Fig. 5.1, hence l1 = 2.01 mm.
In contrast, l2, the distance between the dipole plate and the exit slit, must take into account
the longitudinal extent of the relief cut because it opens towards the E-dipole. Therefore, l2 =
(2.06 + 3.18 − 0.254)mm = 4.986 mm. Note that the relief cuts as illustrated in a real system in
Fig. 5.1 induce significant asymmetry in E-dipole placement with l2 = 4.986 mm > l1 = 2.01 mm.
Conventional treatments model an Allison scanner with thin slits and symmetry in E-dipole
placement, which corresponds to d = 0 and l1 = l2. However, real devices often deviate from these
assumptions as discussed below.

5.2.1.1 Longitudinal Asymmetry in E-dipole Placement (l1 (cid:44) l2)

Effective longitudinal asymmetry in E-dipole placement commonly arises in Allison scanners,
probably unintentionally, because relief cuts on thick entrance- and exit-plates have been made in
the same direction[36, 37]. This asymmetry was also present in the early implementation of the
FRIB Allison scanner where relief cuts on both plates face the incoming beam as shown in Fig. 5.1.
Since the relief cuts have no effect on particle transmission, it effectively displaces the end slit but
not the entrance slit to make l2 > l1 in the FRIB design. We find that such asymmetries affect the
transmitted angles for a given V0.

5.2.1.2 Finite Slit Thickness d (cid:38) Slit Width s

The slit thickness d can be neglected when it is much smaller than the slit width s. This idealization
starts to break down as Allison scanner designs employ decreasing slit widths to improve resolution.
Recent examples include slit widths of s ≤ 100 µm in plans presented for GSI FAIR[38] and
s = 38 µm implemented at TRIUMF[39]. Fig. 5.1 details the slit plate presently employed in the

68

Figure 5.1: Side-view of the FRIB Allison scanner.

Table 5.1: Geometric parameters applied in examples corresponding to the four geometric
models in Fig. 5.2
Case L [mm]
71.85
I
71.85
II
III
71.85
71.85
IV

Symmetric
Asymmetric
Symmetric
Asymmetric

l2 [mm]
3.498
4.986
3.498
4.986

s [µm]

d [µm] E-dipole Placement

l1 [mm]
3.498
2.01
3.498
2.01

g [mm]
7.91
7.91
7.91
7.91

60
60
60
60

0
0
254
254

Slits
Thin
Thin
Thick
Thick

FRIB scanner where s = 60 µm and the slit thickness (not including irrelevant longitudinal extent
of the 30◦ relief cut) is 254 µm. This approximately 4 : 1 aspect ratio effectively produces a narrow
channel, which can scrape particles that would have otherwise passed through a slit-plate with no
thickness.

5.2.2

Four Cases

To investigate the effects of asymmetry in dipole placement and finite slit thickness, four Allison
scanner models (see Fig. 5.2) are specified using the parameters listed in Table 5.1. The parameters
in Case IV correspond to the geometric model of the FRIB Allison scanner in Fig. 5.1, which
has asymmetric E-dipole placement (l1 (cid:44) l2) and thick slits (d (cid:44) 0). To mimic how the analysis
would have been done if one or both geometric features are omitted, the FRIB design is modified
into examples of Cases I to III (see Table 5.1) with thin slits (d = 0) and/or symmetric E-dipole

69

(cid:1)(cid:2)(cid:3)(cid:2)(cid:4)(cid:5)(cid:6)(cid:7)(cid:2)(cid:8)(cid:8)(cid:9)(cid:2)(cid:3)(cid:2)(cid:4)(cid:7)(cid:5)(cid:10)(cid:11)(cid:2)(cid:8)(cid:8)(cid:12)(cid:5)(cid:13)(cid:7)(cid:2)(cid:8)(cid:8)(cid:12)(cid:5)(cid:13)(cid:14)(cid:2)(cid:8)(cid:8)(cid:1)(cid:2)(cid:15)(cid:5)(cid:7)(cid:10)(cid:2)(cid:8)(cid:8)(cid:16)(cid:17)(cid:1)(cid:18)(cid:2)(cid:19)(cid:20)(cid:21)(cid:22)(cid:23)(cid:1)(cid:24)(cid:2)(cid:25)(cid:21)(cid:23)(cid:22)(cid:24)(cid:2)(cid:2)(cid:2)(cid:19)(cid:13)(cid:26)(cid:2)(cid:3)(cid:2)(cid:14)(cid:13)(cid:2)(cid:8)(cid:16)(cid:17)(cid:1)(cid:18)(cid:2)(cid:19)(cid:20)(cid:21)(cid:22)(cid:23)(cid:1)(cid:24)(cid:2)(cid:25)(cid:21)(cid:23)(cid:22)(cid:24)(cid:2)(cid:2)(cid:2)(cid:27)(cid:19)(cid:13)(cid:28)(cid:23)(cid:29)(cid:23)(cid:30)(cid:23)(cid:31)(cid:2)(cid:32)(cid:33)(cid:34)(cid:14)(cid:13)(cid:8)(cid:12)(cid:11)(cid:35)(cid:2)(cid:8)(cid:15)(cid:5)(cid:7)(cid:10)(cid:2)(cid:8)(cid:8)(cid:15)(cid:13)(cid:20)(cid:36)(cid:20)(cid:20)(cid:8)(cid:2)(cid:20)(cid:37)(cid:2)(cid:26)(cid:21)(cid:17)(cid:22)(cid:38)(cid:39)(cid:21)(cid:17)(cid:22)(cid:151)(cid:151)(cid:151)placement (l1 = l2). Note that Case I corresponds to the conventional analysis model. All four cases
have identical E-dipole length L, gap g and total inter-slit distance L + l1 + l2; thus all differences
in analysis results arise from the geometric features.

Characteristic trajectories in Fig. 5.2 illustrate the range of angles that can be transmitted at

each voltage setting due to the finite slit width. We define:

ref: angle at which an ion would enter and exit the slits at the same x-position 1,

• x(cid:48)
• x(cid:48)
• ∆x(cid:48) = x(cid:48)max − x(cid:48)

max / min: maximum/minimum transmittable angle,

min: angular resolution,

where x(cid:48) ≡ dx/dz and x = 0 corresponds to the device center-line. The x(cid:48)max-trajectory touches
the lower edge of the entrance slit and the upper edge of the exit slit, and vice versa for the
x(cid:48)
min-trajectory. When the slits have finite thickness, one has to determine whether the extreme
trajectory should touch the corner on the upstream or downstream side of each slit by selecting the
combination that is not scraped. This complicates analytic modeling and results in a longitudinal
shift of the extreme trajectories as visualized in Case III and IV of Fig. 5.2.

5.2.2.1 Particle Simulations

Transmission properties of an Allison scanner design can be investigated using particle simulations.
Two types of simulations are employed, namely “ballistic” and “realistic”, depending on the model
of the dipole field. Realistic simulations employ a numerically calculated fringe field model for the
dipole and include non-paraxial effects due to energy change. Ballistic simulations take the Allison
scanner as an ideal dipole with uniform hard-edge fields over the axial extent L, with free drifts
and collimating apertures on either side. Energy change from crossing E-dipole potential lines is
neglected. Given V0, the particle trajectory consists of two straight lines outside the dipole and a
parabola inside, and is scraped if it hits either of the slit plates. Space charge and scattering effects

1see Appendix A.3.2 for a subtle distinction in Case IV

70

Figure 5.2: Four geometric cases and their respective characteristic trajectories. Case I:
l1 = l2 = l and d = 0; Case II: l1 (cid:44) l2 and d = 0; Case III: l1 = l2 = l and d (cid:44) 0; Case IV: l1 (cid:44) l2
and d (cid:44) 0. zi is the longitudinal position where the beam particles enter the device and where the
x-x(cid:48) phase space is measured.

are neglected in this study. A condition is derived in Appendix A.1 to show that non-paraxial
effects are typically tiny. Details of the realistic model are given in Appendix A.2.

Most simulation results shown are obtained using ballistic simulations because they are much
faster than realistic simulations and can be summarized using analytic formulae. Results in
Sec. 5.2.3 show that the differences between ballistic and realistic simulations are small for typical
scanner geometries.

The device center-line is chosen as the x-axis of the coordinate system for simplicity of de-
scription. To analyze particle transmission at a given V0 value, the incoming angle which sends a

71

particle to the same position at both slits, i.e. x(cid:48)
ref, is first obtained numerically via root-finding.
Then, assuming that the phase space distribution is uniform over the acceptance region, an ensem-
ble of incident particles are generated under the following conditions. Transversely, the particles
have a uniform distribution about x(cid:48) = x(cid:48)
ref and x = 0 with sufficient widths to fully populate the
acceptance. Longitudinally, all particles have the same initial position zi and lie on the plane where
the particles enter the device (see Fig. 5.2). For slits with finite thickness, the particles are aligned
with the outer plane of the entrance slit. Every particle is advanced with ballistic simulations and
all that can pass through the exit slit are recorded.

Using 40Ar9+ ions with kinetic energy E = 12 keV/u as test particles, the phase space region that
can be transmitted at V0 = 500V and V0 = 1000V for the four geometric cases in Table 5.1 are plotted
in Fig. 5.3 for ballistic simulations. The red rectangle represents the position resolution ∆xideal and
angle resolution ∆x(cid:48)
ideal predicted by the conventional treatment ([32, 33], also reproduced under
Case I in Table 5.2), where

(5.1)

∆xideal = s,
∆x(cid:48)

ideal =

2s
L + 2l .

A comparison of Cases I and III (l1 = l2) against Cases II and IV (l1 (cid:44) l2) reveals that x(cid:48)
ref
changes as a result of longitudinal asymmetry, even though the parameters are chosen such that all
cases have the same inter-slit distance.

The effects of finite slit thickness are manifested in Cases III and IV where the phase space area
transmitted shrinks in both dimensions because d ≈ 4s (cid:44) 0. Furthermore, the shrinkage is larger
in Fig. 5.3a than in Fig. 5.3b - this indicates that this effect increases with V0, which corresponds to
a larger selected reference angle x(cid:48)
ref. An illustration of the details of how the shrunk phase space
area attains its shape is given by Fig. 5.4.

5.2.2.2 Particle Transmission Quantified

To quantify the transmission properties at a specified value of V0, assuming the phase space
distribution is uniform about x = 0 and x(cid:48) = x(cid:48)
ref, the density of transmitted particles in phase space

72

(a) V0 = 500V

(b) V0 = 1000V

Figure 5.3: Blue dots denote particles that are transmitted in the simulations described in
Sec. 5.2.2.1. They map x-x(cid:48) phase space regions at the device entrance z = zi that contribute to
the data point at x = 0 mm. Results are shown for all four geometric cases in Fig. 5.2 for (a)
V0 = 500V and (b) V0 = 1000V respectively. Geometric parameters for each case are listed in
Table 5.1. Black lines define the phase space grid with increments determined by the scanning
step sizes. Dashed blue lines indicate the nominal position x = 0 and angle x(cid:48) = x(cid:48)
ref
corresponding to the data point. The red rectangles denote the device resolution predicted by the
conventional treatment (i.e. Case I).

can be defined as

f(x, x(cid:48)) =



1

0

if (x, x(cid:48)) ∈ A
if (x, x(cid:48)) (cid:60) A

,

(5.2)

where A = {(x, x(cid:48))| particle at (x, x(cid:48)) is transmittable}. Graphically, in Fig. 5.3, f(x, x(cid:48)) = 1 inside
the blue area flagging the simulated phase space particles that are transmittable and f(x, x(cid:48)) = 0
outside.

Two quantities that distill useful information concerning the transmitted phase space area are

the angular transmission factor:

T(x(cid:48)) =

1

f( ˜x, x(cid:48))d ˜x,

(5.3)

Þ ∆xideal/2

∆xideal

−∆xideal/2

73

41.041.542.042.543.043.544.0Angle x′ [mrad]Case Id=0l1=l2Case IIId≠0l1=l2−0.050.000.05Position x [mm]42.543.043.544.044.545.045.5Angle x′ [mrad]Case IId=0l1≠l2−0.050.000.05Position x [mm]Case IVd≠0l1≠l283.584.084.585.085.586.086.5Angle x′ [mrad]Case Id=0l1=l2Case IIId≠0l1=l2 0.050.000.05Position x [mm]87.087.588.088.589.089.590.0Angle x′ [mrad]Case IId=0l1≠l2 0.050.000.05Position x [mm]Case IVd≠0l1≠l2Figure 5.4:

Illustrates how the shrunk phase space area attains its shape in Case III of Fig. 5.3b.

and total transmission factor

W =

1
∆x(cid:48)

ideal

Þ x(cid:48)

ref+∆x(cid:48)
x(cid:48)
ref−∆x(cid:48)

ideal/2
ideal/2

T( ˜x(cid:48))d ˜x(cid:48)

.

(5.4)

Here, ∆x(cid:48) and ∆x(cid:48)
ideal are the ideal position and angular resolution of the device described by
Eq. (5.1), and ˜x and ˜x(cid:48) are dummy variables used to denote integration over position and angle
respectively.

The angular transmission factor T(x(cid:48)) represents the fraction of particles entering the slit with
angle x(cid:48) that is transmitted, assuming a uniform distribution in initial x over the slit width. T(x(cid:48))
can be interpreted as a normalized projection of the blue areas in Fig. 5.3 onto the coordinate x(cid:48)
by integrating over x. T(x(cid:48)) corresponding to the four cases in Fig. 5.3b is plotted in Fig. 5.5. For
Cases I and II with thin slits, T(x(cid:48)) = 1 when x(cid:48) = x(cid:48)
ref and decreases linearly until T(x(cid:48)) = 0 at
ideal. Thick slits in Cases III and IV diminish both the range of transmittable x(cid:48) and
x(cid:48) = x(cid:48)
the fraction of particles transmitted at a given x(cid:48).

ref ± ∆x(cid:48)

74

xzllddxzxzFigure 5.5: Angular transmission factor T(x(cid:48)) for Cases I to IV in Fig. 5.3b.

ideal defined by the device resolution.

The total transmission factor W represents the area ratio between the transmittable phase space
region and ∆xideal × ∆x(cid:48)
It can be thought of as the ratio
between the blue area and area of the red rectangle in any individual plot in Fig. 5.3. Alternatively,
it is the ratio between the area under T(x(cid:48)) and the area enclosed by the two vertical dotted lines in
Fig. 5.5. W always satisfies W ≤ 0.5 with W = 0.5 corresponding to the ideal geometry (Case I).
This result means that within the rectangular phase space region defined by the resolution obtained
conventionally, no more than half of the area is actually transmittable. The fact that W decreases
with V0 in Cases III and IV leads to significant corrections in the data analysis.

5.2.2.3 Analytic Results

Exploiting the simple geometry of particle trajectories in free drifts and ideal dipole fields, all
quantities relevant to particle transmission defined in this section can be derived analytically
following the steps outlined in Appendix A.3. Table 5.2 summarizes the formulae derived for all
four cases. The formulae presented have been numerically verified.
The linear coefficient in the voltage-to-angle relation giving x(cid:48)

ref as a function of V0 is shown
to change with asymmetry in dipole placement. Expressions for T(x(cid:48)) in Cases I to III agree with

75

−0.8−0.6−0.4−0.20.00.20.40.60.8Relative Angle x′−x′ref [mrad]0.00.20.40.60.81.0Angular Transmission T(x′−x′ref) Ɗx′ideal2−Ɗx′ideal2Δase IΔase IIΔase IIIΔase IV(cid:104)1 − (x(cid:48)

Fig. 5.5 where T(x) is maximum at x(cid:48) = x(cid:48)

refd/s)(cid:105) in Case III arise from effects resulting from the finite slit thickness. The factor is

ref and declines linearly to 0 at x(cid:48) = x(cid:48)

max/min. Factors of

≤ 1 and decreases linearly with increasing V0 since x(cid:48)
the blue areas in Fig. 5.3 when d (cid:44) 0.

ref ∝ V0. This effect causes the shrinkage of

T(x(cid:48)) and W in Case IV are more complicated due to the coupling between asymmetric dipole
placement and finite slit thickness. Details of the derivation can be found in Appendix A.3.2.
However, in most relevant cases, T(x(cid:48)) and W of Case III and Case IV are very close except at large
angles. This can be observed in Fig. 5.5 where the two plots almost overlap even at V0 = 1000 V
(Fig. A.4 in Appendix A.3.2 highlights the differences when they are discernible). Therefore, the
discussion concerning Case III typically applies to Case IV with only small modifications.

The agreement between analytic formulae and simulation results, both ballistic and realistic, is
demonstrated in the next section where we discuss how the additional geometric features lead to
two corrections.

76

Table 5.2: Analytic formulae corresponding to the geometric models in Fig. 5.2 for V0 ≥ 0. Results for V0 < 0 obey the same formulae
with V0 → |V0| and the x-axis reversed.
I

IV

III

II

symmetric (l1 = l2)

thin (d = 0)

asymmetric (l1 (cid:44) l2)

(cid:16)1 + l2−l1

thin (d = 0)
qV0L
gE

L+l1+l2

(cid:17)

1
2

Case
Dipole placement
Slit thickness
x(cid:48)
ref
x(cid:48)max − x(cid:48)
ref
x(cid:48)
ref − x(cid:48)
min
∆x(cid:48)
T(x(cid:48)) for x(cid:48) ≥ x(cid:48)
ref
T(x(cid:48)) for x(cid:48) < x(cid:48)
ref

W

1
2

qV0L
gE
s

L+2l

s

L+2l
2s
L+2l
x(cid:48)max−x(cid:48)
x(cid:48)max−x(cid:48)
ref
x(cid:48)−x(cid:48)
min
x(cid:48)
ref−x(cid:48)
min
1
2

symmetric (l1 = l2)

thick (d (cid:44) 0)

(cid:19)

(cid:18)

1
qV0L
gE
2
s−x(cid:48)
refd
L+2l+d
s−x(cid:48)
refd
L+2l+d
(cid:18)
1 − x(cid:48)
(cid:18)
1 − x(cid:48)
(cid:19)2
1 − x(cid:48)
1 − x(cid:48)

refd
s

refd
s
refd
s
refd
s

(cid:19)
(cid:19)

2s

L+2l+d
x(cid:48)max−x(cid:48)
x(cid:48)max−x(cid:48)
ref
x(cid:48)−x(cid:48)
min
x(cid:48)
ref−x(cid:48)
min
1
2

(cid:18)

1
2

(cid:17)

asymmetric (l1 (cid:44) l2)

thick (d (cid:44) 0)
qV0L
gE

L+l1+l2

(cid:16)1 + l2−l1
(cid:16)
(cid:18)

s−x(cid:48)
refd
L+l1+l2+d
s − L+2l1
L+2l2 x(cid:48)
refd
x(cid:48)
1 − L+l1+l2
refd
L+2l2
s

(cid:17)

1

2s

(cid:19)

L+l1+l2+d

L+l1+l2+d

see Appendix A.3.2

see Appendix A.3.2

see Appendix A.3.2

s

L+l1+l2

L+l1+l2

s

2s

L+l1+l2
x(cid:48)max−x(cid:48)
x(cid:48)max−x(cid:48)
ref
x(cid:48)−x(cid:48)
min
x(cid:48)
ref−x(cid:48)
min
1
2

77

5.2.3 Two Corrections in Data Analysis

Analytic and simulation results above reveal two significant corrections on the data analysis of
Allison scanners relative to the conventional treatment that assumes symmetric dipole placement
and thin slits [32, 33]. One correction arises from asymmetry in dipole placement and alters
the proportionality constant in the linear relation between dipole voltage (V0) and selected angle
(x(cid:48)
ref). The other correction adjusts the weights of data points as a function of x(cid:48)
ref to compensate
for scraping due to the finite slit thickness. Both corrections can be applied using the quantities
analytically expressed in Table 5.2. We illustrate these effects for the FRIB Allison scanner
geometry shown in Fig. 5.1 for an 40Ar9+ ion with kinetic energy E = 12 keV/u.

5.2.3.1 Voltage-to-Angle Relation

As shown in Cases II and IV of Table 5.2, when the dipole placement is asymmetric, the reference
angle x(cid:48)
ref has an additional factor [1 + (l2 − l1)/(L + l1 + l2)] relative to the symmetrical model
results (Case I and III). For the FRIB Allison scanner, l2−l1 = 2.976 mm, so the relief cuts generate
substantial effective asymmetry. In Fig. 5.6, x(cid:48)
ref versus V0 is plotted for the symmetrical treatment
(Table 5.2, Case I or III), improved analytic results (Table 5.2, Case II or IV), and both ballistic
and realistic numerical simulations. Both types of simulations agree almost perfectly with analytic
results to show a slope of 88.4 mrad/kV, which deviates from the symmetrical result 85.2 mrad/kV
by roughly 4% as a consequence of the effective asymmetry. The close correspondence between
realistic simulations and analytic results shows that fringe-fields and non-paraxial effects impose
minimal changes on particle trajectories for typical device geometries and beam parameters, thus
validating the analytic calculations as an accurate and efficient way to obtain x(cid:48)
ref.

5.2.3.2 Voltage-Dependent Weight Compensation
As mentioned in Sec. 5.2.2, thick slits cause W to decrease as a function of x(cid:48)
ref. To understand
implications of this effect, it is important to delineate what the blue area, red rectangle and black

78

Figure 5.6: Voltage-to-angle relation for the selected beam. The red curve (analytic) overlays the
blue curve (ballistic simulation).

rectangle represent in Fig. 5.3.

Phase-space scans are typically performed by measuring the current at regular discrete steps in
slit position (x) and voltage (V0). After voltage-to-angle conversion, the data are represented by a
grid in phase-space where each grid represents a rectangle of size (position step) × (angle step). In
general, grid dimensions will not coincide with the spatial and angular resolution of the device. In
Fig. 5.3, the black rectangle denotes the phase space grid while the conventional device resolution
defines the red rectangle. In this study, scanning steps exceed the respective device resolution in
both position and angle, as is typical of scans taken in FRIB (see Sec. 5.4).

Assuming a uniform distribution of particles within each grid, the actual beam current falling

within each phase space grid cell should be calculated by

Igrid = Ifc × Black Rectangle Area

Red Rectangle Area × 1
W ,

(5.5)

where Ifc is the Faraday cup current collected at the data point, and W is the ratio between blue
area and red rectangular area defined in Sec. 5.2.2.

With thin slits, Eq. (5.5) merely rescales all data points uniformly because W = 1/2 is a constant.
Thus one does not need to distinguish between Igrid and Ifc when one calculates moments of the

79

02004006008001000Dipole Voltage V0 [V]0102030405060708090Angle x′ref [mrad]Conventional AnalysisSimulation (Ballistic)Simulation (Realistic)Analyticbeam distribution in phase space with measured data, unless one also wants to compute the total
beam current accurately to verify normalizations.

With thick slits, W is not constant and decreases quadratically in (x(cid:48)
ref × d/s) (see Table 5.2).
Therefore, Ifc has to be rescaled differently depending on the angle x(cid:48)
ref. The larger the slit
thickness-to-width ratio d/s, the more important this effect becomes. This effect must be taken
into account for accurate moment calculations.

In Fig. 5.7, the angular resolution ∆x(cid:48) and 1/W are plotted as a function of x(cid:48)

ref for the FRIB
Allison scanner. Curves for conventional treatment (Table 5.2, Case I), improved analytic results
(Table 5.2, Case II or IV), and numerical simulations are shown. Note again the minimal difference
between ballistic and realistic simulation results.

ref = 70 mrad, W−1 ≈ 4 as opposed to W−1 = 2 when x(cid:48)

The correction introduced by the effect of finite slit thickness can be substantial. For example,
in Fig. 5.7b, when x(cid:48)
ref = 0 mrad. Therefore,
if we apply the conventional model where all data points have equal weights, 0-mrad data points
would wrongly weigh twice as much as ±70-mrad data points. Failure to correct the weights will
distort beam measurements and moment evaluations with increasing amplitude as beam angular
extent increases.

In the laboratory, increasing the angular extent of the beam phase space distribution at the
Allison scanner by adjusting the applied focusing lattice is often needed to obtain sufficient angular
information. An order-of-magnitude inspection based on ideal device resolution in Eq. (5.1) reveals
the reason. (L + 2l) typically ∼ 0.1 m, therefore

∆xideal
∆x(cid:48)
ideal

(cid:39) L + 2l
2

(cid:39) 1
20

mm
mrad .

(5.6)

Since measurements usually take scanning steps exceeding the device resolution, Eq. (5.6) says
that if a similar level of detail is desired in both dimensions, the distribution’s angular extent in
mrad should be around 20 times its position counterpart in mm. This often forces one to employ
relatively narrow scanner slit width s at the expense of reduced Faraday cup currents that lead to
greater noise issues, particularly for heavy-ion beams whose currents are typically lower than that
of proton beams.

80

(a)

(b)

Figure 5.7: Plots of a) angular resolution ∆x(cid:48), and b) angular correction factor as a function of
x(cid:48)
ref. The red curve (analytic) overlays the blue curve (ballistic simulation).

81

020406080100Angle x′ref [mrad]0.80.91.01.11.21.31.41.51.6Angular Re olution Ɗx′ [mrad]Conventional Analy i Simulation ΔBalli tic)Simulation ΔReali tic)Analytic020406080100Angle x′ ef [m ad]0123456T ansmission Facto  Inve sed W−1Conventional AnalysisSimulation (Ballistic)Simulation (Realistic)Analytic5.2.4 Analysis of Synthetic Data

The effects of the two corrections on data analysis can be tested using synthetic measurement data
as follows.

After generating an idealized distribution of particles in phase space with given Twiss parameters
and emittance, the transmission of each particle at each (x,V0) pair of the scan can be tested using
ballistic simulations. The particle number transmitted at each (x,V0) setting is recorded to emulate
the measurements that would have been taken by an Allison scanner. It is unnecessary to test every
(x,V0) pair for a given particle as many pairs are ruled out based on the device resolution. The
artificial data set thus synthesized is processed using both the conventional and improved analysis
to analyze errors in beam moments and phase space distortions in comparison with the idealized
distribution.

Beam moments can be defined in terms of a continuous particle distribution function represent-

ing the beam as:

(cid:10)g(u, u(cid:48))(cid:11) ≡

Þ ∞

−∞

Þ ∞
Þ ∞
Þ ∞
−∞ f(u, u(cid:48))g(u, u(cid:48))dudu(cid:48)
−∞

−∞ f(u, u(cid:48))dudu(cid:48)

(5.7)

(5.8)

(5.9)

where f(u, u(cid:48)) is the distribution function in u-u(cid:48) phase space, and g(u, u(cid:48)) = ua1u(cid:48)a2 with a1 + a2
being the order of the moment. Here, u represents either x or y. Beam moments can be equivalently
calculated from the particle perspective as:

for a distribution with N discrete particles, and within a gridded phase space as:

g(ui, u(cid:48)
i)

1
N

N

(cid:10)g(u, u(cid:48))(cid:11) =
i=1
n

m
(cid:10)g(u, u(cid:48))(cid:11) =

m
i=1
n

i=1

j=1 Ji j g(uj, u(cid:48)
i)

j=1 Ji j

where Ji j is the current density in u-u(cid:48) phase space in the i, j cell. Equation (5.9) is equivalent to
discretizing Eq. (5.7).

A data set for the Allison scanner geometry in Case IV of Table 5.1 is generated using a 12
keV/u 40Ar9+ beam with KV distribution [20, 40]. The KV distribution has uniform projected

82

Figure 5.8: Uniform phase space projection of the KV distribution, compared against results of
synthesized measurement data analyzed using conventional and improved analysis methods. The
left plot shows a sampling (1 out of 1000 particles) used to represent the distribution. The middle
and right plots are fine mesh density plots with the same color scale.

density with an x-x(cid:48) phase space ellipse. This simple uniform projection makes it easy to visualize
distribution distortions. The KV distribution and results from conventional and improved analysis
methods are plotted in Fig. 5.8. Although the KV distribution has a uniform x-x(cid:48) projection
within an elliptical boundary by construction, in the synthesized measurement, the beam density
artificially diminishes strongly at larger angles under the conventional analysis, whereas the correct
uniformity is accurately preserved under the improved analysis.

Beam moments and the associated Twiss parameters αx, βx with rms emittance εrms are listed
in Table 5.3. The results of the improved analysis match the simulated distribution very closely,
whereas there is > 10% deviation in the Twiss parameters if the conventional analysis is used. The
conventional analysis also underestimates the total beam current by 25%. These deviations in the
conventional analysis will amplify if the angular extent of the KV distribution increases.

5.3 Corrections for Degenerate Phase Space Measurements

In the previous section, all analysis was made assuming that the measurement step size of the
scan always exceeds the device resolution. This assumption is illustrated in Fig. 5.3, where the red
rectangle denoting the device resolution is completely encompassed by the black grid defined by
the interval between data points. In this section, we discuss how to correct measurement results

83

−15−10−5051015Position x [mm]−80−60−40−20020406080Angle x′ [mrad]KV Distrib tion−10−50510Position x [mm]Conventional Analysis−10−50510Position x [mm]Improved Analysis0.00.20.40.60.81.00.00.20.40.60.81.0Table 5.3: Beam parameters of the three distributions in Fig. 5.8.

Parameters
(cid:104)xx(cid:105) [mm2]
(cid:104)xx(cid:48)(cid:105) [mm mrad]
(cid:104)x(cid:48)x(cid:48)(cid:105) [mrad2]
αx
βx [m]
εrms [mm mrad]
Itotal [mA]

Distribution

KV

25.0
192.0
1600
-3.43
0.446
56.0
2.00

Conventional

Analysis

20.9
156.5
1304
-3.00
0.400
52.1
1.49

Improved
Analysis

25.0
192.3
1603
-3.43
0.447
56.0
2.00

Figure 5.9: Phase space region contributing to the data point at x = 0 mm and V0 = 200 V
(presentation format as detailed in Fig.5.3), where the Allison scanner geometry corresponds to
Case IV of Table 5.1 except for increased slit width to s = 300 µm. Scanning step sizes in x and
V0 are 0.5 mm and 20 V respectively.

when this assumption is no longer valid.

Intuitively, when the device resolution exceeds the grid size, the current measured has contri-
butions from multiple grid cells. Assuming this occurs only in the angle coordinate but not in the
position coordinate, Fig. 5.9 shows an example where the phase space region extends over five grid
cells vertically. The current measured at a data point is thus a weighted sum of the current densities
from contributing grid cells. Assuming uniform current density in each grid cell, the weighting
factor equals the overlapping area between the blue quadrilateral and the grid cell. Therefore, for
each column in the phase space plot, if the array of angular values are numbered from 1 to n, a

84

Angle x' [mrad]system of linear equations can be written down

(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)

A1,1
...
An,1

· · · A1,n
...
. . .
· · · An,n

(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172)

(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)

(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) =

(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)

j1
...
jn

(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) ,

I1
...
In

(5.10)

where In is the current measured at the n-th angle value, An,m is the area in m-th grid occupied by
the phase space region contributing to In, and jn is the current density at the n-th grid. Fig. 5.9
shows several elements of the matrix A, where In corresponds to the current measured at V0 = 200
V in this case.

Utilizing detailed information from Sec. 5.2 on the phase space region contributing to a data
point, the matrix A can be obtained via transmission simulations or numerical integration of
T(x(cid:48) − x(cid:48)
ref) (see Table 5.2). The current densities j can thus be solved for as unknowns to
deconvolve the information intermixed in current measurements.

Corrections for degenerate measurements are tested using a synthesized data set generated with
the methods described in Sec. 5.2.4 using a waterbag distribution. The waterbag distribution [20,
40] is applied corresponding to uniform transverse energy in phase space up to a sharp cutoff value,
with self-field energy neglected. The Allison scanner geometry is the same as Case IV of Table 5.1
except s = 300 µm (i.e. wider slits), while the scanning step size in V0 is 10V. The uncorrected
and corrected phase space distributions are plotted in Fig. 5.10. Beam parameters are listed in
Table 5.4.

Note that the uncorrected phase space distribution appears smoother than the corrected distribu-
tion because the uncorrected current measured at each grid point takes contributions from multiple
neighboring grid cells, thereby causing a blurring effect. In other words, while the grid size is
determined by the scanning step size and can be made arbitrarily small, details finer than the device
resolution are misleading unless corrections are applied to deconvolve overlapping measurements.
Figure 5.10 shows that the uncorrected distribution also has ghost tails that leak out of the 100%
beam ellipse on both sides of the angular extent of the distribution. The effect is illustrated clearly
when all non-zero points are colored equally (second row in Fig. 5.10). This arises because currents

85

Figure 5.10: Waterbag distribution (left column), compared against synthesized measurement
data analyzed without (middle column) and with (right column) corrections for degenerate
measurements. The upper row corresponds to intensity, whereas the lower row shows all non-zero
intensity points in the same color to highlight extent errors. Black and white ellipses denote the
ideal boundary of the waterbag distribution analyzed. In the left column sampled particles (1 out
of 1000) are plotted. Measurement data are synthesized with s = 300 µm, and scanning step sizes
of 0.5 mm and 10 V.

measured at points outside the ellipse also take contribution from interior grid points, so the extent
of the leakage is determined by the device resolution. Fig. 5.11 shows that such ghost tails can
cause significant overestimation of particles with large angles. Therefore, a correct characterization
of degenerate measurements and device resolution is potentially important for studies of beam halo
where particles outside the core can extend to large angles.

The treatment above assumes the device resolution only exceeds the step size in x(cid:48) but not in
x. This is quite common because the angular resolution limit is much more likely to be reached
by the angular step size, rather than the same occurring in the position dimension, due to the ease
of electronically sweeping and recording voltage steps relative to mechanical translation of the

86

Figure 5.11: Fraction of particles that fall outside the ellipse (orientation defined by Twiss
parameters) with area πε in each of the three distributions in Fig. 5.10. The vertical dashed line
corresponds to the sharp phase space edge of the waterbag distribution at ε/εrms = 6. The blue
curve approaches ε/εrms = 6 if the lower range of the ordinate is extended.

Table 5.4: Beam parameters of the two distribution in Fig. 5.10 compared against those of the
waterbag distribution

Conventional Phase Space
Correction

Analysis

9.03
11.91
27.48
-1.16
0.88
10.31
2.00

9.03
11.91
25.01
-1.30
0.99
9.17
2.00

Parameters
(cid:104)xx(cid:105) [mm2]
(cid:104)xx(cid:48)(cid:105) [mm mrad]
(cid:104)x(cid:48)x(cid:48)(cid:105) [mrad2]
αx
βx [m]
εrms [mm mrad]
Itotal [mA]

Waterbag
Distribution

9.00
12.00
25.00
-1.33
1.00
9.00
2.00

87

0246810Relative Emi  ance ε/εrms10−310−210−1100Frac ion of Par icles Ou side Area πεεmax in idealwa erbag dis ribu ionWa erbag Dis ribu ionw/o Degeneracy Correc ionsw/ Degeneracy Correc ionsscanner (see also last paragraph of Sec. 5.2.3). If the step size in x is smaller than s, the method
above can be extended to 2D whereby all points in the plot have to be solved at once. This can
result in a very large matrix inversion problem unless contributions are limited to some level of
nearest-neighbor zones.

5.4 Example: FRIB Allison Scanner

This section applies the improved analysis model on measurements at the front end of the
Facility for Rare Isotope Beams (FRIB). Sec. 5.4.1 describes the FRIB front end. Section 5.4.2
highlights the discrepancies between conventional and improved analysis results, and validates the
improved treatment by demonstrating that it delivers more consistent results among a series of
measurements performed in a quadrupole scan.

The underlying Python code tools we employ for data analysis of Allison scanner measurements
have been made publicly available in a git [41] software repository [35]. Tools include an adaptable
code to implement the ideal and improved analysis methods described in Sec. 5.2, and algorithms
to process measurement data for noise thresholding. These tools are self-contained, documented,
and readily adaptable to a whole variety of applications. They are made available as a community
resource. The version used in this study corresponds to November, 2018. Future algorithm advances
will also be posted and documented in the repository. The algorithms have also been incorporated
into a graphical user interface (GUI) developed by Tong Zhang that runs as an application in the
FRIB control system. Information on this GUI is given in Appendix A.5.

5.4.1

FRIB Front End

The front end of the Facility for Rare Isotope Beams (FRIB) [2] began commissioning in June
2017. A schematic of the upstream end of the low energy beam transport (LEBT) line is shown
in Fig. 5.12. Beam ions are produced in an Artemis-type ECR source [5] with a 15 kV extraction
voltage. The beam then traverses a short transport section with solenoid focusing and an electrostatic
gap biased to accelerate the target ion species to 12 keV/u. Species are separated horizontally (in

88

Figure 5.12: Upstream end of the low energy beam transport (LEBT) line of the FRIB front end.

x) via a large dispersion generated in a 90◦ magnetic dipole. This is followed by an electrostatic
quadrupole triplet and a four-jaw collimator to scrape unwanted ion species in a high dispersion
region. Two Allison scanners, one for each transverse direction, are located downstream of the
four-jaw collimator. The collimator jaws are adjusted such that only the target species and possibly
traces of contaminant ions whose mass-to-charge ratios M/Q differ from the target ion species by
< 1% remain.

The initial implementation of the Allison scanner had geometries corresponding to Case IV in
Table 5.1. Motivated by this work, the direction of the relief cuts on the entrance slit plate was
flipped in February 2018, thus greatly reducing the longitudinal asymmetry in E-dipole placement.
Table 5.5 compares the geometries before and after the modification. All measurements in this
section were obtained under the new device geometry, hence the corrections have negligible

89

ARTEMIS ECRQuadrupole Triplet4-Jaw CollimatorAllison ScannerFaraday CupSolenoids90o DipoleElectrostatic GapTable 5.5: Allison scanner geometries at FRIB front end, before and after modification in Feb
2018.

Design
Before Symmetrization
After Symmetrization

L [mm]
71.85
71.85

l1 [mm]
2.01
4.938

l2 [mm]
4.986
4.986

g [mm]
7.91
7.91

s [µm]

60
60

d [µm]
254
254

contribution from asymmetry and mainly arise from thick slit effects.

5.4.2 Quadrupole Scan Measurements

A set of six measurements were made on a 50 µA (measured separately by Faraday cup) Ar9+ ion
beam using the y-plane Allison scanner, where VQ3, the voltage at the last quadrupole in the triplet
upstream, varied from 0V to 5000V at 1000V intervals.

Each scan was performed with y-spatial steps of 1 mm and voltage steps of 20 V. Raw data
were processed using the noise removal procedures outlined in Appendix A.4. The resulting data
set was analyzed using both the conventional and improved methods as outlined in Sec. 5.2. Three
y-y(cid:48) phase-space projections (with corrections) at different VQ3 are shown in Fig. 5.13.

To illustrate the differences between the conventional and improved analysis methods, we
compute moments of the beam measured at VQ3 = 1000 V both with and without geometric
corrections. The calculations with geometric corrections use the post-Feb 2018 geometry in
Table 5.5, while the geometry’s symmetric (l = [l1 + l2]/2 = 4.962 mm on both ends) and thin-slit
(d = 0 µm) counterpart gives results with both geometric features ignored.

(cid:10)y

Results are listed in Table 5.6. All second-order moments have centroids subtracted, i.e.,

2(cid:11) ≡(cid:10)(y − (cid:104)y(cid:105))2(cid:11). We observe differences up to ∼10% in beam moments, thus confirming the

importance of corrections for accurate measurements.

While the example above shows the magnitude of the corrections, it does not provide evidence
on which result is more valid. If the corrections introduced in the analysis constitute an “improved”
treatment over the conventional one, a set of quadrupole scan measurements analyzed with the
former should exhibit better consistency than those analyzed with the conventional methods. This
check is performed below.

90

Figure 5.13: y-y(cid:48) phase-space projections measured at the FRIB front end during a quadrupole
scan. VQ3 denotes the voltage at the last quadrupole upstream of the Allison scanner.

91

−10−50510152025Position y [mm]−40−2002040Angle y′ [mrad]VQ3=1000 Vεrms=20.1 mm mrad−100102030Position y [mm]−20−100102030Angle y′ [mrad]VQ3=3000 Vεrms=20.7 mm mrad−100102030Position y [mm]−20−100102030Angle y′ [mrad]VQ3=5000 Vεrms=20.0 mm mradTable 5.6: Beam parameters of corresponding to the phase space distribution at VQ3 = 1000 V in
Fig. 5.13, computed with and without correction terms.

Conventional Phase Space Percentage
Correction Difference

Analysis

Parameters
(cid:104)y(cid:105) [mm]
(cid:104)y(cid:48)(cid:105) [mrad]
(cid:104)yy(cid:105) [mm2]
(cid:104)yy(cid:48)(cid:105) [mm mrad]
(cid:104)y(cid:48)y(cid:48)(cid:105) [mrad2]
εrms [mm mrad]
Itotal [µA]

7.02
4.36
8.84
17.8
76.9
19.0
36.8

7.09
4.64
9.76
19.6
80.8
20.1
39.6

1.0%
6.0%
9.4%
9.2%
4.8%
5.5%
7.1%

Conclusive evidence of improvement arises from first order moments of the beam. Measurement
results of a quadrupole scan and the linear transfer map at each setting can be used to form a system
of linear equations:

(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)

M(VQ3 = V1)

...

M(VQ3 = Vn)

(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172)

I2
...
I2

(cid:169)(cid:173)(cid:173)(cid:171)xi

x0

(cid:170)(cid:174)(cid:174)(cid:172) =

(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)

xf(VQ3 = V1)

...

xf(VQ3 = Vn)

(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) .

(5.11)

x(cid:48)(cid:17)T

(cid:16)

Here, VQ3 is the voltage of the last quad, I2 is the 2× 2 identity matrix, M(VQ3) is the 2× 2 transfer
matrix consisting of the quadrupole at VQ3 and the quadrupole-to-scanner drift, and x =
is the state vector with xi being the initial centroid position and angle entering the quad, x0 being
the position offset and tilting angle of the Allison scanner with respect to the centerline of the quad,
and xf(VQ3) being the measured centroid position and angle corresponding to the measurement at
VQ3.

x

Equation (5.11) solves for the initial centroid position and angle of the beam, as well as the
position offset and tilting angle of the Allison scanner. xf(VQ3) is computed differently depending
on whether the conventional or improved analysis is applied. Results listed in Table 5.7 show that
solutions from the improved analysis are much more realistic. Tilting angles of 33.6 mrad would not
go undetected in mechanical alignment, nor would a beam with centroid angles (cid:104)x(cid:48)(cid:105) = 37.7 mrad
transport through the downstream beam line without significant loss. Therefore, the experimental
data support the validity of the improved analysis over the conventional treatment.

92

Table 5.7: Solution of Eq. (5.11) for the quadrupole scan in Sec. 5.4.2, with measurement results
x f calculated using conventional and improved analysis method respectively.

Analysis
Conventional
Improved

Beam
x(cid:48)
i

xi [mm]
-5.9
-0.7

[mrad]
37.7
5.7

Scanner
x(cid:48)
0 [mrad]
-33.6
-1.2

x0 [mm]
-18.0
3.0

The large differences between the two sets of solutions in Table 5.7 can be explained as follows.
xf(VQ3) from the conventional analysis can be viewed as a perturbation to xf(VQ3) from the
improved analysis. The sensitivity of the solutions xi and x0 to the perturbation can be quantified
by the condition number [42] of the coefficient matrix in Eq. (5.11). Quadrupole scan parameters
corresponding to larger condition numbers will be more sensitive to the perturbation. For the case
presented in Table 5.7, the condition number is large. Choosing scan parameters corresponding to
a well-conditioned coefficient matrix with a smaller condition number will be conducive to error
minimization. Such issues have been explored in Sec. 6.3

5.4.3 Total Current Normalization Anomaly

Despite the successful experimental verification described in Sec. 5.4.2, the quadrupole scan
measurements also present a conundrum. Using Eq. (5.5) and detailed knowledge of the phase
space area contributing to each data point summarized in Table 5.1, it is possible to calculate
the current within each grid cell in an Allison scanner measurement. The summation of all grid
cell current contributions should be consistent with the total beam current that can be accurately
measured by the full-beam Faraday cup (see Fig. 5.12) located at approximately the same beamline
location as the Allison scanner.

Results for the quadrupole scan in Sec. 5.4.2 are shown in Fig. 5.14. The beam current is 50 µA
according to direct Faraday cup measurements. However, the projected total currents from Allison
scanner measurements are consistently ≈ 40 µA, which are 20% smaller than expected. Note that
the improved analysis does make a small small correction, and the agreement is even worse if the
conventional analysis is applied. But the discrepancies are still large.

93

Figure 5.14: Projected beam currents from Allison scanner measurements in a quadrupole scan,
compared against the beam current measured at the Faraday Cup slightly upstream (see Fig. 5.12).

Causes of the discrepancy are unknown and remain a subject of investigation. The bias voltage
on the Allison scanner Faraday Cup, which has a great impact on the current reading, has been
eliminated as a possible cause by the measurement results in Fig. 5.15. When ions hit the Faraday
cup, secondary electrons may escape and amplify the measured current. Therefore, a bias voltage
has to be applied to a Faraday cup to suppress such “ghost” currents. Fig. 5.15 shows that the
measured current on the Faraday cup converges at bias voltages below -100 V, and FRIB usually
employs a bias voltage of - 150 V. To amplify the current by 20%, the bias voltage has to be adjusted
to roughly -25 V, which is clearly wrong because the curve has a large slope at that point which
indicates that there exist strong ghost currents.

Upcoming studies will compare the current discrepancy at beam currents ranging from 50 µA
to 200 µA. If the discrepancy factor is not the same, the results will suggest there are effects at play
other than an unknown scaling factor in the data processing.

To the author’s knowledge, the literature contains no comparison between the projected total
current from Allison scanner measurements and direct measurements with a Faraday cup. Under-
standing reasons for this anomaly would help establish the underlying efficacy of precision Allison

94

010002000300040005000Quadrupole Voltage VQ3 [V]0102030405060Projected Beam Current [A]Current Measured at Faraday CupConventionalImprovedFigure 5.15: Screenshot of measured currents on the x-direction Allison scanner on the Artemis
beam line of FRIB, plotted against the bias voltage of the Faraday cup on the scanner. The scanner
position and voltage were fixed at a phase space coordinate with high density throughout the
measurements.

scanner measurements. Current is in some sense the lowest order beam moment - the fact that it
is not reproduced accurately by Allison scanner measurements is worrisome and unavoidably casts
doubt on results on higher-order phase space moments. The author invites the community to collect
data on this issue on other machines to potentially gain insight into the anomaly.

5.5 Conclusion

We incorporated two important geometric features for Allison scanners into an improved model
that extends the conventional analysis. Both effects were modeled with analytic formulae that have
been verified by particle simulations. Asymmetric E-dipole placement alters the voltage-to-angle

95

relation for the selected beam, whereas slit thickness (cid:38) slit width introduces additional scraping
and requires a data point weight correction that is quadratic in voltage. These effects increase with
the angular extent of the beam distribution, and can significantly change results of phase space
measurements relative to conventional analysis methods. Applying these corrections to Allison
scanner data at the FRIB front end led to (∼10%) changes in beam moments; these corrections were
crucial for obtaining consistent results in a quadrupole scan, which in turn provided experimental
verification for the improved analysis methods. Detailed knowledge of the phase space area
contributing to each data point also allows one to deconvolve overlapping measurements when the
device resolution exceeds the spacing between data points. This was demonstrated using synthetic
data sets.

Python programs incorporating the improved analysis and noise removal for measurement data
are made available [35]. They are readily applicable to any Allison scanner given its device geometry
to implement accurate analytic models. Tools will be updated as continued improvements are made.
An easy-to-use GUI was developed as an FRIB application and is described in Appendix A.5.

96

CHAPTER 6

OPTIMIZED TRANSVERSE PHASE-SPACE MEASUREMENTS WITH BEAM

PROFILE MONITORS

This chapter discusses how measurements from beam profile monitors can be utilized to obtain
information on the transverse phase space of the beam consistent with linear particle dynamics.
Linear dynamics over a limited transport length is a reasonable assumption in FRIB downstream
of the charge selection system. Examples shown here primarily relate to wire-scanner profile
monitors and tomography as employed in the FRIB front end (see Chapter 3 and 4). However,
methods presented can be broadly applicable.

Section 6.1 reviews results from linear algebra that are relevant to improving measurements
from phase space diagnostics. Section 6.2 describes how quadrupole and solenoid scans can be
used to measure the beam phase space moments. Statistical methods to quantify the errors in such
measurements and an efficient scheme to minimize the errors are given in Sec. 6.3. In addition to
obtaining the first and second order moments of the beam distribution, methods of tomography can
be employed to reconstruct the phase space in detail. Techniques for tomographic reconstruction
of the spatial density profile and 2D phase space are discussed in Sec. 6.4.

At the timing of this writing, the analysis in this chapter is under preparation for submission for

publication [43].

6.1 Review of Linear Algebra

This section reviews concepts and tools in linear algebra that are relevant to beam diagnostics
using profile monitors. The treatment is based upon Ref. [42, 44, 45] which emphasize numerical
aspects of the subject.

We restrict our discussion to systems of linear equations Ax = b with full column rank, i.e.
A ∈ Rm×n where m ≥ n and rank(A) = n. This corresponds to the case where there are at least
as many equations as unknowns. In our applications, b consists of measurement data, x consists

97

of the unknowns (e.g. initial conditions upstream or spatial density in a grid) and A describes the
linear relationship between measurement data and unknowns. In practice, a sufficient number of
measurements is often made in beam diagnostics to over-constrain the system (i.e. make m > n) to
allow better accounting for measurement uncertainties.

6.1.1 Least Squares Approximation

Generally, an over-determined system of linear equations Ax = b has no exact solution. In that
case, it is useful to find the least squares solution that minimizes a quadratic measure of deviation
r(cid:124)r for r = Ax − b.

The least squares solution x can be found as follows. Require r to be an extremum with

∇x r(cid:124)r = 0

Then:

∇x (cid:0)x(cid:124)

(cid:124)
A

(cid:124)b + b(cid:124)b(cid:1) = 0

Ax − 2x(cid:124)
A
(cid:124)
Ax − 2A
2A
(cid:124)
A

(cid:124)b = 0
Ax = A

(cid:124)b

Hence, the least squares solution can be solved by inverting A(cid:124)Ax = A(cid:124)b, giving

which is called the normal equation, and

is defined as the pseudo-inverse of A. Here, B−1 denotes the inverse of a square matrix B and A†
denotes the transpose of a (generally non-square) matrix A. For a square matrix B with full rank,
the pseudo-inverse becomes the normal inverse, i.e. B† = B−1.

98

(cid:124)

x =(cid:0)A
A(cid:1)−1
A(cid:1)−1
A† ≡(cid:0)A

(cid:124)

(cid:124)b,
A

(cid:124)
A

,

(6.1)

(6.2)

6.1.2 Singular Value Decomposition
Given a matrix A ∈ Rm×n where m ≥ n, it can always be factorized by the singular value
decomposition (SVD) which decomposes A as follows:

where U ∈ Rm×m and V ∈ Rn×n are both orthogonal matrices (i.e. UUT = UTU = I), and

(6.3)

(6.4)

A = UΣVT

(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172)

(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)

0
σ1
0 σ2
...
...
0
0
...
...
0
0

. . .

. . .
. . .

0
0
...

. . . σn
...
...
0
0

Σ =

∈ Rm×n

is a rectangular diagonal matrix. σi’s are called the singular values of A and they are typically
arranged such that σ1 ≥ σ2 ≥ · · · ≥ σn.

From AT AV = V ΣTΣ, we observe that V consists of the eigenvectors of AT A. Similarly, U
consists of the eigenvectors of AAT. Since rank(AAT) = n, the last m − n columns of U has to be
filled in to complete an orthonormal basis.

Note that for square matrices, the set of singular values σi(A) and eigenvalues λi(A) are different

in general and should not be confused.

6.1.3 Vector Norms and Matrix Norms
A vector norm is a function (cid:107).(cid:107) : Rn → R with the three properties:

1. (cid:107)x(cid:107) = 0 =⇒ x = 0 (positive definite)

2. (cid:107)x + y(cid:107) ≤ (cid:107)x(cid:107) + (cid:107)y(cid:107) (satisfies triangle inequality)

3. (cid:107)αx(cid:107) = α (cid:107)x(cid:107) (absolutely homogeneous)

99

In colloquial terms, a vector norm measures the size of a vector.

The family of vector p-norms of an n-dimensional vector x are defined as:

(cid:32) n


i=1

(cid:107)x(cid:107)p =

(cid:33)1/p

|xi|p

.

They include the most commonly used vector norms:

(cid:113)|x1|2 + |x2|2 + · · · + |xn|2

(cid:107)x(cid:107)1 = |x1| + |x2| + · · · + |xn|
(cid:107)x(cid:107)2 =
(cid:107)x(cid:107)∞ = max(|x1| ,|x2| , . . . ,|xn|)

L1-norm (Manhattan norm)
L2-norm (Euclidean norm)
L∞-norm (infinity norm)

(6.5)

(6.6)

(6.7)

(6.8)

Unless otherwise specified, the length or magnitude of a vector in physics is usually synonymous
with its Euclidean L2-norm.

A matrix norm is a function (cid:107).(cid:107) : Rm×n → R that measures the size of a matrix. It has similar

properties to a vector norm:

1. (cid:107) A(cid:107) = 0 =⇒ A = 0 (positive definite)
2. (cid:107) A + B(cid:107) ≤ (cid:107) A(cid:107) + (cid:107)B(cid:107) (satisfies triangle inequality)
3. (cid:107)αA(cid:107) = α (cid:107) A(cid:107) (absolutely homogeneous)

A common family of matrix norms are called induced norms, whereby the matrix norm is defined
in terms of vector norms as follows:

(cid:107) A(cid:107) ≡ max
(cid:107)x(cid:107)(cid:44)0

(cid:107) Ax(cid:107)
(cid:107)x(cid:107)

(6.9)

Note that both norms on the R.H.S. are vector norms. The induced norm essentially finds, for a
given vector norm, the maximum factor by which a matrix can increase such a vector norm. Two
corollaries of the definition are:

(cid:107) A(cid:107) = max
(cid:107)x(cid:107)=1

(cid:107) Ax(cid:107)

(cid:107) Ax(cid:107) ≤ (cid:107) A(cid:107) (cid:107)x(cid:107)

∀ (cid:107)x(cid:107) (cid:44) 0

100

(6.10)

(6.11)

We will use (cid:107) A(cid:107)2 to denote the induced norm of A corresponding to the L2 vector norm. This

is the only matrix norm we will use in the rest of the chapter.

As an example of the definition of induced norms, we prove two results below that will be useful

in the discussion on condition number in Sec. 6.1.4. Firstly we prove

(cid:107) A(cid:107)2 = max
(cid:107)x(cid:107)2=1

(cid:107) Ax(cid:107)2 = σ1(A),

(6.12)

where σ1(A) denotes the largest singular value of A.

Proof:

(cid:107) Ax(cid:107)2 =

=

√
√
√

x(cid:124)A(cid:124)Ax
x(cid:124)V Σ(cid:124)U(cid:124)UΣV(cid:124)x
x(cid:124)V Σ(cid:124)ΣV(cid:124)x

max
(cid:107)x(cid:107)2=1

=

√
(cid:107) Ax(cid:107)2 = max
(cid:107)x(cid:107)2=1
= max

(cid:107)V(cid:124)x(cid:107)2=1

x(cid:124)V Σ(cid:124)ΣV(cid:124)x
√
x(cid:124)V Σ(cid:124)ΣV(cid:124)x

(cid:112)y(cid:124)Σ(cid:124)Σy

(y = V

(cid:124)x)

(∵ VV

(cid:124) = I)

= max
(cid:107)y(cid:107)2=1
= max(σi)

(cid:13)(cid:13)(cid:13)A†(cid:13)(cid:13)(cid:13)2

Next, for the pseudo-inverse A† ≡ (A(cid:124)A)−1 A(cid:124) defined in Sec. 6.1.1,

=

1
σn(A)

(6.13)

where σn(A) denotes the smallest singular value of A.

101

Proof:

(cid:124)

(cid:124)

= V

(cid:124)
A

ΣV
(cid:124)

A = V Σ

A(cid:1)−1
(cid:0)A
Σ(cid:1)−1
(cid:124)(cid:0)Σ
Σ(cid:1)−1
(cid:124) = V(cid:0)Σ
A(cid:1)−1
(cid:13)(cid:13)(cid:13)A†(cid:13)(cid:13)(cid:13)2
(cid:13)(cid:13)(cid:13)V(cid:0)Σ
Σ(cid:1)−1
(cid:16)(cid:0)Σ
Σ(cid:1)−1

= σ1

A

=

V
(cid:124)

Σ

Σ

(cid:124)

(cid:124)

(cid:124)

(cid:124)

(cid:0)A

(cid:124)

U
(cid:124)

(cid:124)

(cid:124)(cid:13)(cid:13)(cid:13)2
(cid:124)(cid:17)

U

Σ

=

1
σn(A)

6.1.4 Condition Number

When we solve Ax = b, the solution x will be changed by perturbations both in A and in b, which
we denote as δA and δb respectively. The condition number of A, commonly denoted by κ(A),
is a number that quantifies the sensitivity of the solution to such perturbations. In our context, b
typically represents beam measurements with error δb, and A is constructed from linear transfer
maps with uncertainties δA. We first show the results for a special case and follow with the general
case.

To illustrate the framework, we first consider the special case where δA = 0, A ∈ Rn×n and

rank(A) = n. Let x and ˜x be the solutions to:

Using:

Ax = b
A˜x = b + δb

Ax = b
(cid:107)b(cid:107) = (cid:107) Ax(cid:107)
(cid:107)b(cid:107) ≤ (cid:107)x(cid:107) (cid:107) A(cid:107)
(cid:107)x(cid:107) ≤ (cid:107) A(cid:107)
1
(cid:107)b(cid:107)

102

(6.14)

(6.15)

(6.16)

δx =(cid:101)x − x
(cid:13)(cid:13)(cid:13)
(cid:13)(cid:13)(cid:13)A−1
(cid:107)δx(cid:107) ≤(cid:13)(cid:13)(cid:13)A−1(cid:13)(cid:13)(cid:13) (cid:107)δb(cid:107)

δx = A−1
δb

(cid:107)δx(cid:107) =

δb

we obtain

where κ(A) is the condition number of A defined as:

(cid:107)x(cid:107) ≤ κ(A)(cid:107)δb(cid:107)
(cid:107)δx(cid:107)
(cid:107)b(cid:107)
(cid:13)(cid:13)(cid:13)A†(cid:13)(cid:13)(cid:13)2

=

σ1(A)
σn(A) ≥ 1

κ(A) ≡ (cid:107) A(cid:107)2

(6.17)

(6.18)

(6.19)

The relative change in the solution measured by (cid:107)δx(cid:107) /(cid:107)x(cid:107) is bounded by the relative change
in the measurement data (cid:107)δb(cid:107) /(cid:107)b(cid:107) multiplied by κ(A). Therefore, in the context of diagnos-
tic measurements, the smaller κ(A) is, the less sensitive the results are to relative errors in the
measurements.

The treatment can be extended to the general case where A ∈ Rm×n and there are perturbations

to both A and b.

From Sec. 6.1.1, finding the least squares solution to Ax = b is equivalent to solving the normal

equation

(cid:124)
A

Ax = A

(cid:124)b.

(6.20)

Applying perturbations to both A and b and leaving only linear terms, the normal equation becomes

(cid:124)
δA

(cid:124)
Ax + A

(cid:124)
δAx + A

Aδx = δA

(cid:124)
(cid:124)b + A

δb.

(6.21)

It can be shown, analogously to Eq. (6.18), that the relative change in x is bounded by the

relative changes in A and b multiplied by powers of κ(A):

(cid:18) (cid:107)b(cid:107)2

(cid:107) Ax(cid:107)2

(cid:19)

(cid:107)δb(cid:107)2
(cid:107)b(cid:107)2

+

(cid:107)δA(cid:107)2
(cid:107) A(cid:107)2

(6.22)

(cid:107)δx(cid:107)2
(cid:107)x(cid:107)2

≤ κ(A)2 (cid:107)r(cid:107)2
(cid:107) Ax(cid:107)2

(cid:107)δA(cid:107)2
(cid:107) A(cid:107)2

+ κ(A)

103

Proof:

(cid:13)(cid:13)(cid:13)A†(cid:13)(cid:13)(cid:13)2 (cid:107)(δb − δAx)(cid:107)2
(cid:19)

= κ(A) are used
(cid:107)δAx(cid:107)2
(cid:107) A(cid:107)2 (cid:107)x(cid:107)2

+

(cid:107)δx(cid:107)2 =

=

(cid:107)δx(cid:107)2
(cid:107)x(cid:107)2

δA

δA

+

(cid:124)

(cid:124)

(cid:124)

(cid:124) (b − Ax) + A† (δb − δAx)

δx =(cid:0)A
A(cid:1)−1
(cid:13)(cid:13)(cid:13)(cid:0)A
(cid:124) (b − Ax)(cid:13)(cid:13)(cid:13)2
(cid:13)(cid:13)(cid:13)A† (δb − δAx)(cid:13)(cid:13)(cid:13)2
A(cid:1)−1
≤(cid:13)(cid:13)(cid:13)(cid:0)A
A(cid:1)−1(cid:13)(cid:13)(cid:13)2
(cid:13)(cid:13)(cid:13)A†(cid:13)(cid:13)(cid:13)2 (cid:107)(δb − δAx)(cid:107)2
(cid:13)(cid:13)δA
(cid:124)(cid:13)(cid:13)2 (cid:107)(b − Ax)(cid:107)2 +
(cid:13)(cid:13)(cid:13)(cid:0)A
A(cid:1)−1(cid:13)(cid:13)(cid:13)2
(cid:13)(cid:13)δA
(cid:124)(cid:13)(cid:13)2 (cid:107)(b − Ax)(cid:107)2 +
(cid:19)
(cid:18)(cid:107)δb(cid:107)2
(cid:107) A(cid:107)2 (cid:107) A(cid:107)2
(cid:107) A(cid:107)2 (cid:107) A(cid:107)2
≤ κ(A)2 (cid:107)δA(cid:124)(cid:107)2
A(cid:1)−1(cid:13)(cid:13)(cid:13)2 (cid:107) A(cid:107)2
where(cid:13)(cid:13)(cid:13)(cid:0)A
(cid:13)(cid:13)(cid:13)A†(cid:13)(cid:13)(cid:13)2
(cid:107) A(cid:107)2
(cid:18)
(cid:124)
(cid:18) (cid:107)δb(cid:107)2
(cid:18) (cid:107)b(cid:107)2

+ κ(A)
2 = κ(A)2 and (cid:107) A(cid:107)2
(cid:107)δb(cid:107)2
+ κ(A)
(cid:107) A(cid:107)2 (cid:107)x(cid:107)2

≤ κ(A)2 (cid:107)δA(cid:124)(cid:107)2
(cid:107) A(cid:107)2 (cid:107)x(cid:107)2
≤ κ(A)2 (cid:107)δA(cid:107)2
(cid:107)r(cid:107)2
(cid:107) A(cid:107)2
(cid:107) Ax(cid:107)2
= κ(A)2 (cid:107)r(cid:107)2
(cid:107)δA(cid:107)2
(cid:107) Ax(cid:107)2
(cid:107) A(cid:107)2

(cid:107)r(cid:107)2
(cid:107) A(cid:107)2
+ κ(A)
+ κ(A)

(cid:107) Ax(cid:107)2
(cid:107) Ax(cid:107)2

(cid:107)δA(cid:107)2 (cid:107)x(cid:107)2
(cid:107) A(cid:107)2 (cid:107)x(cid:107)2
(cid:107)δb(cid:107)2
(cid:107)b(cid:107)2

(cid:107)δA(cid:107)2
(cid:107) A(cid:107)2

+

(cid:124)
(cid:107)r(cid:107)2
(cid:107) A(cid:107)2

(cid:107)δAx(cid:107)2
(cid:107) A(cid:107)2

(cid:107) A(cid:107)2
(cid:107) A(cid:107)2

+

(cid:107) A(cid:107)2

(cid:19)

(cid:19)

+

6.2 Quadrupole Scan and Solenoid Scan: Working Principles

This section describes a formulation to extract 1st and 2nd order moments of a beam at a location
upstream of quadrupoles or solenoids via beam profile measurements with less information at a
common downstream location taken under varying focusing strengths. This method is typically
called a “quadrupole scan” for quadrupole optics, and “solenoid scan” for solenoid optics. One
or more quadrupoles or solenoids may be varied. The method is routinely used to support beam
trajectory correction and envelope matching under linear single particle dynamics. It is applicable to
any measurements that can measure beam profiles including view screens, wire scanners and beam
position monitors. Space charge forces, both linear and nonlinear, are neglected in this formulation
because it relies on transfer maps which cannot include dependence on the beam distribution.

104

6.2.1 Notation

The notation used in this section are defined as follows. Greek letters α, β, µ, ν denote 4D-phase
space indices 1, 2, 3, 4 with x1 = x, x2 = x(cid:48), x3 = y, x4 = y(cid:48). M is the 4 × 4 transverse
transfer matrix and (cid:104). . . (cid:105) denotes beam moments, i.e. statistical average over the beam distribution.
Subscripts i and f denote the initial and final values with respect to the reference orbit. The final
values are to be distinguished from the values measured on the profile monitor, which are denoted
by the subscript m.

6.2.2

1st Order Moments

The final position and initial phase space coordinates are related by:

(cid:11)
(cid:11)

(cid:10)xβ
(cid:10)xβ

M1β

M3β

4

4


β=1

β=1

(cid:104)x(cid:105) f =

(cid:104)y(cid:105) f =

i

i

(6.23)

(6.24)

If the coordinate system of the profile monitor is misaligned to that of the beam line, the measured
x and y centroid positions are different from those with respect to the beam’s reference orbit:

(cid:104)x(cid:105)m = (cid:104)x(cid:105) f + xshift =

(cid:104)y(cid:105)m = (cid:104)y(cid:105) f + yshift =

+ xshift

+ yshift

i

i

(6.25)

(6.26)

(cid:11)
(cid:11)

(cid:10)xβ
(cid:10)xβ

M1β

M3β

4

4


β=1

β=1

Therefore, the transverse shifts resulting from misalignments can be treated as unknowns along
with the initial first order moments. One obtains two equations from each measurement. With n

105

sets of measurement data, they can be solved from the system of equations

(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)

Row 1 of M1 1 0
Row 3 of M1 0 1
Row 1 of M2 1 0
Row 3 of M2 0 1
...
Row 1 of Mn 1 0
Row 3 of Mn 0 1

...

...

(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172)

(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)

(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172)

(cid:104)x(cid:105)i
(cid:104)x(cid:48)(cid:105)i
(cid:104)y(cid:105)i
(cid:104)y(cid:48)(cid:105)i
xshift

yshift

=

(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)

(cid:104)x(cid:105)1
(cid:104)y(cid:105)1
(cid:104)x(cid:105)2
(cid:104)y(cid:105)2
...
(cid:104)x(cid:105)n
(cid:104)y(cid:105)n

(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172)

(6.27)

In the symbolism of Ax = b, b consists of the measured data and A is the coefficient matrix. The
first 4 columns of A depend on the linear transfer map corresponding to each measurement, whereas
the last two columns are (1,0,1,0, . . . ,1,0)(cid:124) and (0,1,0,1, . . . ,0,1)(cid:124) respectively. This treatment
thus incorporates any shifts induced by misalignment.

6.2.3

2nd Order Moments

Denoting:

4


β=1

x f ,α =

Mαβxi,β

(6.28)

The second order moments at the measurement point are related to the initial second order moments
as follows:

(cid:10)(xα − (cid:104)xα(cid:105))(xµ −(cid:10)xµ
(cid:11))(cid:11)
(cid:42) 4

4

(cid:11))
Mαβ(xβ −(cid:10)xβ
4

4

(cid:10)(xβ −(cid:10)xβ
(cid:11) whose results must cancel. The related centroid subtraction
the shift affects both xµ and(cid:10)xµ

Note that the equation holds regardless of whether the profile monitor is displaced, because

f
Mµν(xν − (cid:104)xν(cid:105))

(cid:11))(xµ −(cid:10)xµ

(cid:43)
(cid:11))(cid:11)

i

MαβMµν

β=1

ν=1

=

=

β=1

ν=1

i

106

is omitted in the notation below for ease of presentation, in which case final values of 2nd-order
spatial moments can be written as:

β=1

4

4

4


β=1

ν=1

4

4

4


ν=1

β=1

ν=1

(cid:10)xβxν
(cid:10)xβxν
(cid:10)xβxν

(cid:11)
(cid:11)
(cid:11)

M1βM1ν

M1βM3ν

M3βM3ν

(cid:104)xx(cid:105) f =

(cid:104)x y(cid:105) f =

(cid:104)yy(cid:105) f =

i

i

i

(6.29)

(6.30)

(6.31)

Each profile measurement can give some or all of the above 2nd-order spatial moments. Then,

the 10 initial 2nd-order moments can be solved via a system of linear equations Ax = b where:

x =(cid:0)(cid:104)xx(cid:105)i,(cid:104)xx(cid:48)(cid:105)i,(cid:104)x y(cid:105)i,(cid:104)x y

(cid:48)(cid:105)i,(cid:104)x(cid:48)x(cid:48)(cid:105)i,(cid:104)x(cid:48)

y(cid:105)i,(cid:104)x(cid:48)

(cid:48)(cid:105)i,(cid:104)yy(cid:105)i,(cid:104)yy

(cid:48)(cid:105)i,(cid:104)y

(cid:48)

(cid:48)(cid:105)i

y

y

(cid:1)(cid:124)

and b with n measurements reads:

b =(cid:0)(cid:104)xx(cid:105)1 ,(cid:104)x y(cid:105)1 ,(cid:104)yy(cid:105)1 , . . . ,(cid:104)xx(cid:105)n ,(cid:104)x y(cid:105)n ,(cid:104)yy(cid:105)n

(cid:1)(cid:124)

A is the coefficient matrix that depends on the transfer map for each measurement.

6.2.4 A Limitation of Solenoid Scans

For an ideal, aligned solenoid, transport symmetries in linear optics mandate:

M11M14 = M12M13

M31M34 = M32M33

M11M34 + M31M14 = M12M33 + M32M13

(6.32)

(6.33)

(6.34)

The coefficient matrix A has 10 columns corresponding to the 10 initial 2nd order moments.
However, the columns corresponding to (cid:104)x y(cid:48)(cid:105)i and (cid:104)x(cid:48)y(cid:105)i are always identical. Therefore, only the
sum of two terms, (cid:104)x y(cid:48)(cid:105)i + (cid:104)x(cid:48)y(cid:105)i can be solved but not each term individually.

107

6.3 Quadrupole Scan and Solenoid Scan: Error Quantification and Mini-

mization

6.3.1 Error Quantification

In quadrupole and solenoid scans, the beam matrix is solved using a linear system of equations
Ax = b. The error in the solution x can be quantified via statistical methods.

There are many sources of errors in a quadrupole / solenoid scan including: misalignments, field
deviations and measurement errors. These errors constitute perturbations to A and b in Ax = b.
Given estimates of the magnitude of each error, one can apply random errors and solve for Ax = b
in each case.

Frequency count of the solutions give rise to a probability distribution of x. With the additional
condition that unphysical answers are discarded, this treatment is equivalent to a Bayesian analysis
with a uniform prior on the domain of physicality.

6.3.2 Error Minimization

From the discussion on condition number in Sec. 6.1.4, it is clear that the error of quadrupole and
solenoid scan results can be minimized by choosing a set of scan parameters that give a coefficient
matrix A with as small a condition number as possible. The condition number κ(A) is defined in
Eq. (6.19). This requirement also requires having transport without beam loss in each measurement.
The use of condition number in error minimization of beam measurements is inspired by the work at
the GSI Helmholtz Centre for Heavy Ion Research (GSI) [46, 47]. However, the papers by GSI did
not appear to analyze how to choose scan parameters such that the condition number is minimized.
Better choice of scan parameters can reduce the condition number and thereby decrease the relative
uncertainty of results.

A simple estimate can show that an exhaustive search over all possible sets of scan parameters
is impractical. Suppose there are two knobs (e.g. focusing strenghs in a quadrupole doublet) where
each knob can attain 10 values, thus giving 100 possible settings in total. To choose a set of scan

108

parameters for four measurements, the number of possible combinations equal:

(cid:169)(cid:173)(cid:173)(cid:171)100

4

(cid:170)(cid:174)(cid:174)(cid:172) =

100!
4! × 96! ≈ 4 × 106

.

For scans with more knobs and more measurements, the number can be orders of magnitude larger.
It would be very computationally inefficient to build all possible coefficient matrices, compute their
singular values, and select the one with the smallest condition number.

Therefore, one has to rely on other ideas to efficiently obtain a set of scan parameters. The Paul
Scherrer Institute (PSI) proposed choosing quadrupole parameters that correspond to discrete steps
in particle phase advance between the measurement and reconstruction point [48]. However, the
phase advance describes rotation in normal coordinates, whereas the actual rotation in phase space
depends on the orientation of the invariant ellipses, which is different for each focusing setting
and may not even exist (in a single-pass system, there is no need for a cell to satisfy the stability
condition).

Building upon PSI’s idea, we believe the viewpoint from tomography is the most direct and
beneficial. Quadrupole settings alter the linear map from zi to z f and in effect provide a different
projection of the initial phase space onto the 1D spatial measurements of the beam profile monitor.
If we choose a set of measurements which correspond to a diverse range of projection angles, the
corresponding matrix A should have a low condition number which reduces uncertainties in the
solution.

The projection angle corresponding to the linear map of a quadrupole transport line can be

found as follows. The initial and final phase space coordinates are related by:

Consider a line in x f -x(cid:48)
is given by:

f

phase space defined by x f = x0 where x0 in a constant. Its image in xi-x(cid:48)

i

x(cid:48)
f

m21 m22

(cid:169)(cid:173)(cid:173)(cid:171)x f
(cid:170)(cid:174)(cid:174)(cid:172) =(cid:169)(cid:173)(cid:173)(cid:171)m11 m12
(cid:169)(cid:173)(cid:173)(cid:171)xi
(cid:170)(cid:174)(cid:174)(cid:172)
(cid:170)(cid:174)(cid:174)(cid:172)
(cid:170)(cid:174)(cid:174)(cid:172) =(cid:169)(cid:173)(cid:173)(cid:171) m22 −m12
(cid:169)(cid:173)(cid:173)(cid:171)x0
(cid:169)(cid:173)(cid:173)(cid:171)xi
(cid:170)(cid:174)(cid:174)(cid:172)

−m21 m11

x(cid:48)
f

x(cid:48)
i

x(cid:48)
i

(cid:170)(cid:174)(cid:174)(cid:172)

109

(6.35)

(6.36)

Figure 6.1: Schematic of the beam line section containing the first profile monitor at the FRIB
Front End.

which are parametric equations of xi and x(cid:48)
in xi-x(cid:48)

is given by

i

with x(cid:48)
f

as the parameter. Hence the slope of the line

i

tan θ = −m11
m12

(6.37)

where θ corresponds to the projection angle on the initial phase space. The argument is illustrated
by Fig. 6.1. Here we employ normalized dimensionless coordinates in measuring x and x(cid:48) (e.g.
normalization by 1 mm and 1 mrad respectively).

We must add one caveat concerning the dimensionless coordinates in the aforementioned
treatment. The projection angle is actually dependant on the choice of units for the phase space
coordinates. For instance, in Fig. 6.1, if the unit of angle is changed from mrad to rad, the plot is
compressed 1000-fold in y and the angle θ will decrease drastically. Thus we must ask: what units
should we choose for the phase space coordinates? The most natural answer is a choice of units
such that the beam has roughly equal sizes in the two dimensions. Heuristically, it often suffices
to use the units adopted at a certain facility, e.g. mm for position and mrad for angles at FRIB,
because the distribution typically has similar extent in both coordinates in these units. This is likely
the reason why such units are chosen in the first place.

In a more sophisticated treatment, two measurements with projection angles close to 0 and π/2
respectively, should be made first. They give a good estimate of the beam size in the original phase
space and the associated projection angles will change minimally under a change of units. Then,

110

xix'ix'fxfMM-1θthe phase space units can be rescaled in accordance with the beam extent (measured by rms size or
maximum extent), and the projection angle corresponding to each setting can be recalculated under
the new units. Suppose mm and mrad are used as units of position and angle respectively. The new
projection angles will be given by:

α =

x[mm]
x(cid:48)[mrad]

tan θnew = α tan θ

(6.38)

(6.39)

The choice of settings for other measurements can now be selected as before, with a diverse spread
of projection angles (under the rescaled coordinates) being the primary criterion.

Techniques explored in this section and their connection to the condition number κ are presented

in FRIB examples in Sec. 6.5.

6.4 Spatial and Phase Space Tomography

6.4.1 Spatial Tomography

Spatial tomography harnesses information from beam profile monitors to reconstruct the beam’s
image in x y space. One approach is the algebraic reconstruction technique (ART) which solves
for densities on a grid as unknowns in a system of linear equations. A projection consists of many
steps where each individual step is called a ray. A ray can be visualized as a line of finite thickness
that samples the distribution and threads a path through the grid. For an m × n grid, the system of
equations reads:

where ρj is the discretized current density in the j-th grid cell, Ii is the measured current of the
i-th ray, and A is the projection matrix with ai j being the area of the j-th grid cell that falls under
the i-th ray.

111

(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)

a1,1
a2,1
...
ak,1

. . . a1,m×n
. . . a2,m×n
...
. . . ak,m×n

...

(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172)

(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)

(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172)

ρ1

ρ2
...

ρm×n

(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)

I1

I2
...
Ik

(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172)

=

(6.40)

Figure 6.2: Schematic of a wire scanner at the FRIB front end. Image courtesy of Tomofumi
Maruta at FRIB.

In FRIB and several other facilities, the wires in beam profile monitors are oriented vertically,
horizontally and diagonally. A schematic of a wire scanner device at FRIB is shown in Fig. 6.2;
it consists of two diagonal wires and one vertical wire. If we attempt to apply ART to conduct
spatial tomography with such a system, we quickly run into problems due to severe redundancy in
the system of equations.

For an m× n grid, it can be observed that horizontal and vertical wires only provide information
on the total current in each row and column, thereby giving at most m and n equations respectively.
This is true regardless of how dense the steps are in each projection.

While a diagonal wire can couple information from different rows and columns, it is not effective
and only provide no more than m + n − 1 equations. This can be seen by coloring the grid as one
would a chessboard. There are m + n − 1 diagonals of squares that alternate in color, and the wire
can only give information on the total current in each of the diagonals. The argument above is
illustrated in Fig. 6.3.

If a diagnostic station contains vertical, horizontal and two diagonal wires, for a m × n grid, the
total number of equations < 4m + 4n − 2. The number is not equal to 4m + 4n − 2 due to further
redundancies among the information provided by individual wires (e.g. the total current). In all
cases of interest, 4m + 4n − 2 is much smaller than mn which is the number of equations required
for solving the current densities in all grid cells.

112

xyFigure 6.3: A diagonal wire only gives the total current in each diagonal because and each grid
cell along a diagonal always contributes equally to a ray. The alternating diagonals are plotted
using chessboard coloring on the right.

Figure 6.4: A wire intersecting a 2 : 1 grid at 45o is equivalent to the case of a wire intersecting
an even grid at 63o.

The most direct solution would be to change the physical orientation of the wires. However, this
is costly to do, and, perhaps more importantly, the existing orientation is optimal for measuring beam
moments (cid:104)xx(cid:105), (cid:104)x y(cid:105) and (cid:104)yy(cid:105). Beam moments should take precedence over spatial tomography.
The other remedies are indirect ways of altering the wire angles. The first method is to use
grids with uneven horizontal and vertical sizes. This amounts to a rescaling of one of the axis, and
a diagonal that intersects an uneven grid at 45o is equivalent to the a wire intersecting a regular
grid at a different angle. An example is shown in Fig. 6.4. However, this method only removes the
redundant information from diagonal wires, not horizontal and vertical, and the tunability is low
because a grid with vastly uneven sizes in the two directions is unlikely to be desirable.

A more effective and implementable method is to rotate the grid on which the densities are
to be solved by making a judicious choice of transformed coordinaces. The coordinate system is

113

133355224446rescaleuneven grid (1:2)even grid (1:1)45o63oFigure 6.5: For a rotated grid, the wires are no longer vertical, horizontal or diagonal with
respect to it.

an abstract construction anyway and does not have to be aligned with the axis of the beam line
coordinates. This allows wires that are horizontal, vertical and diagonal with respect to the beam
line to intersect the grid at angles of one’s choice. Fig. 6.5 shows a grid that is rotated by 25o with
respect to the standard one.

The ART technique for spatial tomography was tested using synthesized measurements from
several numerically generated particle distributions. Both uniform and waterbag distributions are
standard, while hollow distributions have been observed at FRIB and dual peak will be relevant
when FRIB operates dual-species transport. The tests assumed the profile monitor has three
wires: one vertical, one horizontal and one diagonal. The reconstruction results for both standard
and rotated grids are plotted in Fig. 6.6. One can observe that the rotated grid produces more
truthful reconstructions for the uniform and waterbag distributions. For hollowed and dual peak
distributions, it may be beneficial to employ a variety of rotated coordinates and to combine their
results. Optimized choices for such treatment will require further study.

6.4.2 Phase Space Tomography

Unlike spatial tomography, phase space tomography can be conducted with as many projections as
one desires. Hence, the reconstruction can be conducted using a standard grid, and the choice of
focusing parameters between the profile monitor and reconstruction point can be chosen with the

114

Standard GridRotated GridFigure 6.6: Spatial tomography of several synthesized distributions.

115

UniformWaterbagHollowDual PeakParticle DistributionStandard GridRotated GridFigure 6.7: Schematic of the beam line section containing the first profile monitor at the FRIB
Front End. The profile monitor is located at the position designated by f and the beam was
reconstructed at position i. Green blocks denote identical (Q7 type) electrostatic quadrupoles.

V1
V2
V3
V4

-2657
4513
-4295
300

-2657
4513
-4295
1300

-2657
4513
-4295
2300

-2657
4513
-4295
3300

-2657
4513
-4295
4300

-2657
4513
-4295
5300

Table 6.1: Quadrupole parameters for Scan 1

same reasoning as Sec. 6.3.2. Examples of phase space tomography at FRIB are shown in Sec. 6.5.

6.5 Example: FRIB Measurements

To test the implementation of the methods developed in this chapter, two quadrupole scans were
conducted using the first profile monitor at the FRIB Front End on 2019/01/18. A schematic of
the relevant beam line section is shown in Fig. 6.7. There are two quadrupole doublets between
the measurement and reconstruction points. The voltages applied to the four quadrupoles are
listed in Table 6.1 for Scan 1 and Table 6.2 for Scan 2. The parameters in Table 6.1 is typical of
how quadrupole scans were usually performed where only the focusing strength of the quadrupole
immediately upstream of the profile monitor is varied. The parameters in Table 6.2 were chosen
using the techniques described in Sec. 6.3.2 to sample the initial distribution from a wide range of
projection angles. The projection angles on the initial phase space corresponding to each scan are
shown in Fig. 6.8. Note that Scan 2 has a much wider spread of angles than Scan 1.

The performance of Scan 1 and Scan 2 in both beam matrix measurements and phase space
tomography are compared. The phase space distribution measured at the upstream Allison scanner

116

-500
3000
-4500
4000

-4000
4500
-4500
4000

-4500
-4000
V1
5500
4500
V2
-3000 -4500
V3
500
3500
V4
Table 6.2: Quadrupole parameters for Scan 2

-1500
4000
-4500
4000

Figure 6.8: Projection angles on the initial phase space corresponding to quadrupole scan
parameters in Table 6.1 and Table 6.2.

is propagated to the reconstruction location to serve as a benchmark. Hard edge equivalent transfer
matrices are applied for the quadrupoles. As discussed in Sec. 6.2, beam moments are calculated
from a system of linear equations where the coefficient matrix A depends on the choice of scan
parameters. In the current example, the condition number κ(A) as defined in Sec. 6.1.4 equals 466
and 14.5 for Scan 1 and Scan 2 respectively. With this > 30-fold difference in the condition number,
the errors in Scan 1 are expected to be much larger than those in Scan 2. This has been verified
by applying random errors to measurement results and solving for the emittance in each case. The
histograms corresponding to 10000 perturbed solutions in each case are plotted in Fig. 6.9, where
the standard deviation of x-emittance εx is 0.029 mm-mrad and 0.005 mm-mrad for Scan 1 and
Scan 2 respectively. Despite the fact that applied errors for Scan 1 are chosen to be 10 times smaller
than those applied for Scan 2, the measurement errors in Scan 1 are still 6 times larger. This strongly
illustrates the viability of the scan methodology presented in Sec. 6.3.2 to reduce uncertainties in
quadrupole scans.

117

−4−2024xi [mm]−4−2024x′i [mrad]Measurement Set 1−4−2024xi [mm]−4−2024Measurement Set 2(a) Scan 1

(b) Scan 2

Figure 6.9: Histograms of the normalized x-emittance with random errors applied to the
measurement results. The applied errors have a truncated Gaussian distribution where 3σ = 1%
for Scan 1 and 3σ = 10% for Scan 2.

Figure 6.10: Comparison between tomographic reconstruction of the x-x(cid:48) phase space
distribution, and the measured distribution at the Allison scanner propagated to the same location.

The fact that the initial beam conditions are better reconstructed in Scan 2 than in Scan 1
also manifest itself in tomographic reconstruction. Fig. 6.10 compares the reconstructed phase
space distribution from Scan 1 and Scan 2 against the distribution propagated from Allison scanner
measurements. The tomography results from Scan 1 are very noisy, whereas the reconstructed
phase space of Scan 2 agrees much more closely with Allison scanner measurements. The beam
moments and emittances calculated from the reconstructed phase space distribution are listed in
Table 6.3.

The algorithms for spatial tomography with rotated grids presented in Sec. 6.4.1 was also tested
as part of the experiment. Fig. 6.11 shows the reconstructed x-y spatial density distribution of
the beam on a grid that is rotated by 20o. Beam moments of the reconstructed phase space are

118

0.000.020.040.060.080.100.120.140.16εx [mm-mrad]0100200300400500600700FrequencyAllison ScannerUnperturbed Solution0.060.070.080.090.10εx [mm-mrad]025050075010001250150017502000FrequencyAllison ScannerUnperturbed SolutionPhase Space Tomography from Measurem Set 1Phase Space Tomography from Measurem Set 2Propagated Distribution from Allison Scanner MeasurementsTomography Tomography

Scan 1

Scan 2

Projected from
Allison Scanner

(cid:104)xx(cid:105) [mm2]
(cid:104)xx(cid:48)(cid:105) [mm-mrad]
(cid:104)x(cid:48)x(cid:48)(cid:105) [mrad2]
εx normalized [mm-mrad]

21.4
56.3
220.0
0.199

6.49
22.9
132.7
0.107

7.09
21.4
105.6
0.087

Table 6.3: Beam moments and emittances corresponding to the phase space distributions plotted
in Fig. 6.3.

Figure 6.11: Spatial tomography with a grid rotated by 20o clockwise with respect to the
standard x-y coordinate system.

Wire Scanner Measurements Rotated Grid Tomography

(cid:104)xx(cid:105) [mm2]
(cid:104)x y(cid:105) [mm2]
(cid:104)yy(cid:105) [mm2]

44.2
24.7
44.6

46.9
24.6
51.3

Table 6.4: Comparison between beam moments of the reconstructed spatial distribution in
Fig. 6.11 and the values obtained from wire scanner measurements.

transformed back into x-y and compared against moments directly obtained from beam profile
measurements with the wire scanner. Table 6.4 shows the results where the agreement is quite
close, thus validating the algorithms as a potential expansion of the diagnostic capabilities of wire
scanners at FRIB.

119

6.6 Further Work

We have developed error minimization techniques and phase space tomography algorithms for
quadrupole scans with profile monitors and tested them using the wire scanner data at FRIB. Further
quadrupole scan experiments should be performed which can serve multiple purposes:

1. Test the method of making projects on rescaled phase space outlined at the end of Sec. 6.3.2.

2. Develop the above method into a general automated algorithm for online selection of optimal

quad scan parameters.

3. Further testing and development of automated 2D phase space tomography as an additional

output of quad scans.

4. Use the first profile monitor to benchmark beam moments and reconstructed phase space from

quadrupole scans against Allison scanner measurements propagated to the same location.

We would like to emphasize the significance of item 4, where the beam’s 2D phase space roughly
half way between the Allison scanner and 1st profile monitors is obtained using distinct methods
and compared. Good agreement in such a benchmark will lend confidence to the analysis methods
of both devices as well as the transfer maps employed.

Two extensions on the error minimization techniques will be highly beneficial. Firstly, the
choice of optimal quad scan parameters in Sec. 6.3.2 only applies to 2D phase spaces x-x(cid:48) and
y-y(cid:48). The four coupled moments are also measured by the quadrupole scan, and they are likely
even more susceptible to measurement errors because they have one more unknown than 2D phase
spaces. An understanding of error minimization for coupled moments should be the next step in
advancing the physical model behind quadrupole scans.

An even more difficult problem is how error minimization techniques can be applied to solenoid
scans, where nine coupled moments are solved simultaneously (not ten, see Sec. 6.2.4). One
plausible idea is to generalize the projection angle picture in 2D and apply it to 4D. For a given set
of solenoid strengths, each measured moment correspond to some projection on the initial 4D phase

120

space, which can be represented by a point on a unit 3-sphere. However, with three angles instead
of one, the idea of a diverse range of projection angles becomes much more fuzzy. Even if one
associates a “spread” in projection angles by inter-point geodesic distances, it is not obvious how
an optimal set can be chosen efficiently. It may be possible to reduce dimensionality by employing
rotated frame Larmor variables or other choices of canonical variables in pure solenoid scans. That
may enable the formulation of a procedure analogous to that developed for quadrupole scans. Much
investigation remains to be done, and we hope such a pursuit may also provide a physical picture
on why (cid:104)x y(cid:48)(cid:105) and (cid:104)x(cid:48)y(cid:105) cannot be measured separately in a solenoid scan.

121

APPENDICES

122

APPENDIX A

EXTENSIONS ON ALLISON SCANNER ANALYSIS

In this appendix, we include details associated with the Allison scanner analysis (Chapter 5)
including: estimates on the range of validity of the paraxial approximation employed (A.1); a
numerical model used to check analytic results (A.2); key steps in the derivation of analytic formulae
(A.3); noise removal schemes for experimental measurement data (A.4); and implementation of
the algorithms in the FRIB control system (A.5).

A.1 Non-paraxial Effects: When They are Negligible

This appendix estimates the magnitude of non-paraxial effects in an Allison scanner and provides

a simple condition to determine whether they are negligible.

Consider an idealized device geometry, i.e. symmetric E-dipole placement and thin slits, which
corresponds to Case I in Fig. 5.2. Given V0, a reference particle with the corresponding angle x(cid:48)
ref
reaches maximum x−displacement at the axial center (z = l + L/2). The ratio between the axial
kinetic energy decrement ∆E ≡ E − Emin at this point and the original axial kinetic energy E of
the ion measures the strength of non-paraxial effects.

Taking uniform hard-edge dipole fields and the paraxial approximation,



For x(z = 0) = 0 and x(cid:48)(z = 0) = x(cid:48)

ref = 1
2

qV0L
gE , the particle position at the center of the device is

x(cid:48)(cid:48)(z) =

− qV0
gE

if l ≤ z ≤ L + l

0 otherwise

.

(cid:19)

(cid:18)3L
qV0
gE L
qV0
gE L2

.

8 + l

1
x(z = l + L/2) =
2
≈ 1
5

The energy change ∆E is maximum at z = l + L/2 where the particle has its closest approach to

123

the E-dipole. The fractional energy change is

∆E
E =

1
E

x(z = l + L/2)

g/2

qV0 =

(cid:19)2

(cid:18) L

g

(cid:19)2(cid:18) qV0

E

.

2
5

For non-paraxial effects to be negligible, such energy gain should be much smaller than the

kinetic energy of the beam, i.e. ∆E (cid:28) E, or

(cid:19)2

2
5

(cid:19)2(cid:18) qV0
(cid:18) L
(cid:19)2(cid:18)
(cid:18)Q

E

g

(cid:28) 1.

(cid:19)2

40

A

eV0

E0 [eV/u]

(cid:28) 1.

Taking the approximation L/g ≈ 10 for a typical plate spacing, this condition reduces to

Here, for ions we take E0 = E/A to be the kinetic energy per atomic mass unit and q = Qe with Q
being the charge state and e the elementary charge. Non-paraxial effects are negligible at V0 values
that satisfy the above condition. For example, consider the condition applied to the E0 = 12 keV/u
40Ar9+ ion beam measured in the FRIB front end in Sec. 5.4. Even at V0 = 1000 V, the condition
reduces to ≈ 0.014 (cid:28) 1, which is well satisfied. Thus it is not surprising that the realistic and
ballistic simulations agree well for typical parameters.

A.2 Realistic Model

A Python [49] code is employed to numerically integrate particle equations of motion in a
realistic x-z 2D field map of the device geometry that is generated from the electrostatic field code
POISSON [24]. Ex = −∂φ/∂x and Ez = ∂φ/∂z electric field data are exported from POISSON
onto a high-resolution, uniform x-z mesh (dx = dz = 0.2 mm), and then imported into the Python
code. Fields at the particle position are calculated using bilinear interpolation from the gridded
field data [50].

In Fig. A.1, the applied field potential φ of the FRIB Allison scanner is contoured showing
enhanced detail near the entrance slit. Only half the geometry in the zoomed figure is contoured
since φ(−x, z) = −φ(x, z).

124

Figure A.1: Potential contours of the FRIB Allison scanner in Fig. 5.1, including details of the
fringe structure in the vicinity of the entrance slit (right).

Particles are advanced with non-relativistic equations of motion

d2x
dt2 =
d2z
dt2 =

q
m
q
m

Ex,

Ez.

The independent variable can be transformed exactly from time t to axial coordinate z giving

(cid:16) qEx

m x(cid:48)(cid:17)

x(cid:48)
t(cid:48)
m − qEz
m t(cid:48)3
− qEz

t(cid:48)2



d
dz

where (cid:48) ≡ d/dz, and t(cid:48) = 1/vz.





x

t
x(cid:48)
t(cid:48)



=

(A.1)

This numerical model includes full fringe field effects entering and exiting the dipole field
region, as well as non-paraxial effects due to energy change as the particle crosses potential lines.
Image charges, beam space charge and scattering effects are neglected.

The state vector in Eq. (A.1) describing the particle trajectory is advanced using the ODE
package within Scientific Python (SciPy)[51] for specified initial particle coordinate x, angle x(cid:48),
and dipole voltage V0 (field data scaled). The code takes into account scraping on all boundaries.
To solve for x(cid:48)
max,min, note that the corresponding trajectory must touch the slits at two points (see
Fig. 5.2). We employ a numerical root-finding procedure to solve for the initial x(cid:48) that connects
the upstream point to the downstream point. x(cid:48)
ref is solved analogously with the condition x = 0 at
both ends.

125

z [cm]x [mm]Electric Dipole Plate (V = V0)Electric Dipole Plate (V = -V0)x [mm]z [mm]0-5-10-15510150123456701234560123456(a)

(b)

Figure A.2: How (a) shifting a trajectory for fixed angle and (b) changing the angle for fixed
initial position allow one to calculate quantities listed in Table 5.2.

A.3 Analytic Results

Appendix A.3.1 sketches key steps in the derivation of the expressions in Table 5.2 using
the geometric models shown in Fig. 5.2. Complications involved in Case IV are outlined in
Appendix A.3.2.

A.3.1

Sketch of Derivation Using Geometric Models

Table 5.2 is mostly derived based on two simple rules: 1) trajectories can be shifted in x-position;
and 2) changing the initial x(cid:48) = x(cid:48)
at z = zi by δx(cid:48) results in a displacement δx = (z − zi)δx(cid:48)
i
downstream. Subscripts i and f denote initial (entrance) and final (exit) locations respectively.

In Fig. A.2, these principles are applied to Case III (symmetric E-dipole placement, thick slits)
to illustrate several calculations. We observe that among a uniform spatial distribution of particles
entering the slit with x(cid:48)
ref, a fraction s1/s is collimated by the slit plate due to its thickness, where
s1 = x(cid:48)
ref), which
manifests itself in Fig. 5.3 as a horizontal shrinkage of the blue area in Case III in comparison with
Case I.

in the transmission coefficient T(x(cid:48)

refd. This explains the factor

(cid:19)

(cid:18)

1 − x(cid:48)

refd
s

To calculate x(cid:48)max, we note that the corresponding trajectory touches the slit plates at the
positions shown in Fig. A.2. Consider a particle with the same xi entering the slits with angle x(cid:48)
ref.
Applying the second rule, we obtain s2 = (x(cid:48)max − x(cid:48)
refd as calculated

ref)(d + 2l + L), while s1 = x(cid:48)

126

Thick slitlldds1Thick slitlldd(a)

(b)

Figure A.3: Reversing particle velocity allows one to check results when the E-dipole placement
is asymmetric.

above. Therefore,

s2 = s − s1,
ref)(d + 2l + L) = s − x(cid:48)
refd,
s − x(cid:48)
refd
d + 2l + L ,

x(cid:48)max − x(cid:48)

(x(cid:48)max − x(cid:48)

ref =

which is one of the results in Table 5.2.

Note that the derivation above assumes bending within the E-dipole, which does not hold when
V0 = 0. In that case, instead of ∆x(cid:48) = 2s/(L + 2l + d), one can draw straight trajectories between
furthest corners of the slits to see that ∆x(cid:48) = 2s/(L + 2l + 2d). Since d (cid:28) L + 2l, the difference is
very small and will be neglected.

(cid:44) −x(cid:48)
f

The calculations for asymmetric E-dipole placement were carried out using the same principles,
but one must note that x(cid:48)
due to the asymmetry. This makes calculations in Case IV
i
(asymmetric E-dipole placement, thick plate) much more complicated due to different scraping
factors on the two ends. Rather than showing the tedious calculations, it is interesting to observe a
succinct way to check the validity of the expressions in Case IV of Table 5.2 via direction reversal.
Fig. A.3a shows Case IV trajectories and Fig. A.3b their direction-reversed counterparts, where
the trajectories corresponding to maximum and minimum angles are interchanged depending on the
direction of incoming particles. For the x(cid:48) = x(cid:48)max trajectory in the normal case, the final particle
angle equals x(cid:48)max − qV0L
min for the direction-reversed case. Therefore, we can

gE , which equals −x(cid:48)

127

check whether

(cid:19)
(cid:18) L + 2l2
(cid:32) L + 2l∗

ˆL

+

2

ˆL

qV0L
gE

(cid:34)1

1
2
= −

qV0L
gE

2

(cid:32)

(cid:33)

s − x(cid:48)
refd
ˆL + d
− 1
ˆL + d
1 + l∗

(cid:33)(cid:35)

− qV0L
gE
s − L + 2l∗
1
L + 2l∗
2
2 remains constant and l∗

x(cid:48)
refd

where inter-slit distance ˆL = L + l1 + l2 = L + l∗
2 = l1. All
results for which the E-dipole placement is asymmetric have been verified with these procedures.

1 = l2, l∗

A.3.2 Results for Case IV

Procedures sketched above are applied to Case IV to obtain, for l2 > l1:

ref ≤ x(cid:48)

for x(cid:48)
for ˜x(cid:48) ≤ x(cid:48) < x(cid:48)
ref
for x(cid:48) < ˜x(cid:48)

,

2(l2 − l1)d

(L + l1 + l2 + 2d)(L + 2l2)

(cid:21)

,



T =

W = D1

c1

x(cid:48)max−x(cid:48)
x(cid:48)max−x(cid:48)
ref
x(cid:48)
ref−x(cid:48)
ref− ˜x(cid:48) c2 + x(cid:48)− ˜x
x(cid:48)
x(cid:48)
ref− ˜x(cid:48) c1
˜x(cid:48)−x(cid:48)
˜x(cid:48)−x(cid:48)
min
L + l1 + l2

c2

2s2

,

(cid:33)

(cid:32)
(cid:18)

(cid:20)
1 −
1 − x(cid:48)
refd
s
1 − ˜x(cid:48)d
(c1s)2 + (c2s)2
2(L + l1 + l2 + d) +

(cid:19)

s

,

,

˜x(cid:48) = x(cid:48)
ref

c1 =

c2 =

where

D1 =

ref − ˜x(cid:48)).
ref only if l1 (cid:44) l2 and d (cid:44) 0, so the case ˜x(cid:48) ≤ x(cid:48) < x(cid:48)

2(c1 + c2)(x(cid:48)

s

Observe that ˜x(cid:48) (cid:44) x(cid:48)
the E-dipole placement is asymmetric and slits are thick.

ref is only relevant when

The reason there exists a third region in T(x(cid:48)) is associated with a subtlety in the definition of
x(cid:48)
ref. Previously, x(cid:48)
ref is defined as the angle at which particles enter and exit the slits at the same
x-position, with no distinction on which side of the slits we mean when the slits are thick.

128

Figure A.4: Comparison between transmission ratio in Case III and IV when weight correction
effect is large.

Taking the inner facing side of the entrance slit as z = 0, x(cid:48)

ref is defined as the angle at which
particles attain the same position at z = 0 and z = ˆL where ˆL = L + l1 + l2 is the inter-slit distance.
Then there is another angle ˜x(cid:48) with which the particles attain the same position at z = −d and
z = ˆL + d. x(cid:48)
ref = ˜x(cid:48) in Case III because the entering and exiting angles are the same. However, in
Case IV, since the entering and exiting angles are different due to the asymmetry, x(cid:48)
ref (cid:44) ˜x(cid:48). When
the d/s ratio is large, the effects can be significant as shown in Fig. A.4 for d/s (cid:39) 4.

A.4 Noise Removal Scheme

The noise removal scheme for raw data is illustrated using an example measurement from the
FRIB Front End (see Sec. 5.4). The scheme first defines the beam region and uses the data points
outside the beam region to characterize the background. Then the background is subtracted and
unrealistic islands in the distribution are filtered. Details of the procedure are discussed below.
Such noise thresholding is crucial for heavy ion beam measurements because the total beam current
is typically ∼50 µA; this leaves many data points with merely ∼ nA of collected currents which
may only be ∼10 times larger than noise fluctuations. Algorithms implementing the noise removal
scheme applied in this study are incorporated in the Python programs available in Ref. [35].

The noise removal scheme employed here has no capability to correct for ghost signals caused
by particles that scatter after impacting the E-dipole plates; such ghost signals have been mitigated

129

−0.20−0.15−0.10−0.050.000.050.100.150.20x′−x′ref [mrad]0.0000.0250.0500.0750.1000.1250.1500.1750.200T(x′−x′ref) x′ref̃x′̃ase IIĨase IVFigure A.5: Beam region is defined by the red ellipse.

Figure A.6: Histogram of current values at data points outside the defined beam region.

by using E-dipole plates whose surfaces have a staircase profile to reduce grazing incidences [33,
52].

A.4.1 Specifying the Beam Region

Figure A.5 shows an ellipse that surrounds the entire beam distribution, thereby defining a region
that contains the beam. Such a region can be designated by user input or generated automatically.
It should be self-consistent in the sense that, after background subtraction and island filter, no
non-zero data point touches the boundary specifying the beam region.

130

−10−505101520Position x [mm]−40−2002040Angle x′ [mrad]−1.0−0.50.00.51.01.52.02.53.0−1.0−0.9−0.8−0.7−0.6Current I [nA]0100200300400500600700800Count+2σ−2σFigure A.7: Non-zero data points after background subtraction.

A.4.2 Background subtraction

After designating the beam region, all exterior data points are used to characterize the background.
In this example, the background noise has an average µ = −0.778 nA with standard deviation
σ = 0.079 nA. Fig. A.6 shows a histogram of the measured currents in exterior data points, where
almost all values lie below µ + 2σ. Therefore, a µ + 2σ cutoff is typically applied for interior data
points, whereby any point with current < µ + 2σ is regarded as pure noise and assigned the value
0. Subsequently, all remaining non-zero data points have µ subtracted from them to correct for the
background. Note that µ < 0 in this case, which is likely due to amplifier characteristics. Fig. A.7
shows all non-zero data points after applying background subtraction to the raw data in Fig. A.5.

A.4.3

Island Filter

The beam distribution after background subtraction often contains spurious islands that probably
arise from noise that exceeds the cutoff value. Fig. A.8 zooms in on the upper right sector of the
beam distribution in Fig. A.7 to show a number of islands before and after they are filtered.

The island filter we employ is a modification of the median filter widely used in speech and
image processing [53]. For each non-zero data point, the island filter examines neighbors in an n
x n grid and counts the number of non-zero points. If the number < k, the data point in question
is assigned a zero value after the filter has processed all data points. The filter can be applied

131

−10−505101520Position x [mm]−40−2002040Angle x′ [mrad]Figure A.8: Non-zero data points of the beam distribution, before and after applying the island
filter.

until it no longer has any effect on the distribution, which may require several iterations to enable
layer-by-layer removal of clusters of islands.

In Fig. A.8, n = 5 and k = 3, which means that if fewer than 3 data points amongst 24
neighboring points are non-zero, the data point at center is deemed part of a noise island and
assigned the value zero. The filter takes two iterations to reach convergence in this case.

A.5

Implementation in FRIB Control System

The improved analysis of Allison scanner measurements described in this dissertation and
Refs. [31, 34] have been incorporated into applications within the FRIB Control System. The
initial application was made available to members of the Accelerator Physics Department in the
form of a Jupyter notebook [54] which can be used both online and offline. The online mode provides
the functionality of setting up and starting a scan using the notebook. Once the measurement data
is ready, it is read from the control system, saved and analyzed with the algorithm. The data is
saved in such a way that they can be readily analyzed by running the notebook offline.

Tong Zhang from the Acceleartor Physics Department at FRIB incorporated the algorithms
into a graphical user interface (GUI) that eases usage. The interface is set up to allow enhanced
configurability while being straightforward to use with defaults for operators not familiar with

132

5.07.510.012.515.017.520.0Position x [mm]2025303540Angle x′ [mrad]Unfiltered5.07.510.012.515.017.520.0Position x [mm]Filteredphysics issues. and prepares it for transition into a tool for the operators. The new software also
communicates better with the control system and allows real-time update of the data and scanner
status throughout a scan. Two screenshots of the software are shown in Fig. A.9 and Fig. A.10.

133

Figure A.9: Beam parameters setup screen for the Allison scanner GUI. (Image courtesy of Zhang Tong from the Accelerator Physics
Department at FRIB).

134

Figure A.10: Noise correction screen for the Allison scanner GUI. (Image courtesy of Zhang Tong from the Accelerator Physics
Department at FRIB).

135

APPENDIX B

ALTERNATIVE PROOF OF RELATIONS GOVERNING 2ND ORDER MOMENTS OF

Cn BEAMS

This appendix presents an alternative, more direct, proof of the fact that, for all n > 2, the 2nd order
moments of a beam with n-fold rotational symmetry (i.e. Cn) obey the same relations as those of
an axisymmetric beam, i.e.:

(cid:104)xx(cid:105) = (cid:104)yy(cid:105)
(cid:104)xx(cid:48)(cid:105) = (cid:104)yy
(cid:48)(cid:105)
(cid:104)x(cid:48)x(cid:48)(cid:105) = (cid:104)y
(cid:48)(cid:105)
(cid:48)
(cid:104)x y(cid:105) = 0
(cid:48)(cid:105) = 0
(cid:104)x(cid:48)
y
(cid:48)(cid:105) = −(cid:104)x(cid:48)
(cid:104)x y

y

y(cid:105)

(B.1)

This is true regardless of the orientation of the coordinate axis on the transverse plane. Methods
outlined here would become increasingly cumbersome when applied to higher-order moments.

B.1 Spatial Density and Local Velocity

Define F(r, θ, vr, vθ) as the beam distribution function in 4D transverse phase space where r
and θ are the usual cylinderical-polar spatial coordinates and vr and vθ denote the corresponding
angles in a cylindrical polar representation. The usual transverse particle angles (i.e.
transverse
velocity normalized by vz, x(cid:48) = vx/vz) are related to vr and vθ by:

x(cid:48) = vr cos θ − vθ sin θ
(cid:48) = vr sin θ + vθ cos θ

y

136

A conveniently normalized spatial distribution function f(r, θ) withÞ 2π

0

Þ ∞

0

f(r, θ)rdrdθ = 1 is

defined as:

f(r, θ) =

Þ 2π

0

Þ ∞
Þ ∞
Þ ∞
Þ ∞
Þ ∞

−∞
−∞

0

−∞ F(r, θ, vr, vθ)dvr dvθ
−∞ F(r, θ, vr, vθ)rdrdθdvr dvθ

The local average flow angle of particles within the distribution is denoted (cid:174)V(r, θ) = Vr(r, θ)ˆr +
Vθ(r, θ) ˆθ (components vary locally as a function of r and θ) and is obtained by averaging over
angular degrees of freedom:

Þ ∞
Þ ∞
Þ 2π
Þ ∞

−∞
0

(cid:174)V(r, θ) =

Þ ∞

Þ ∞

−∞ F(r, θ, vr, vθ)(vr ˆr(θ) + vθ
0

ˆθ(θ))dvr dvθ
−∞ F(r, θ, vr, vθ)rdrdθdvr dvθ

−∞

For a beam with n-fold rotational symmetry, its spatial density and local flow angle of the particle

distribution must remain unchanged upon a coordinate rotation by angle φ = 2jπ/n ∀j ∈ Z, i.e.

f(r, θ) = f(r, θ +
Vr(r, θ) = Vr(r, θ +
Vθ(r, θ) = Vθ(r, θ +

2jπ
)
n
2jπ
)
n
2jπ
)
n

(B.2)

(B.3)

(B.4)

A sufficient, but not necessary, condition for these relations to hold is the even stronger statement
that F(r, θ, vr, vθ) satisfies

F(r, θ, vr, vθ) = F(r, θ +

2jπ
n , vr, vθ) ∀j ∈ Z

In that case, Eqs. (B.2)–(B.3) can be verified from:

f(r, θ) =

=

Þ 2π
Þ 2π

0

0

Þ ∞
Þ ∞
Þ ∞
Þ ∞
Þ ∞
Þ ∞
Þ ∞
Þ ∞
Þ ∞
Þ ∞

−∞
−∞
−∞
−∞
2jπ
)
n

0

0
= f(r, θ +

−∞ F(r, θ, vr, vθ)dvr dvθ
−∞ F(r, θ, vr, vθ)rdrdθdvr dvθ
−∞ F(r, θ +
n , vr, vθ)dvr dvθ
2jπ
n , vr, vθ)rdrdθdvr dvθ
−∞ F(r, θ +
2jπ

137

and

Vr(r, θ) = ˆr(θ) · (cid:174)V(r, θ)
−∞

0

Þ ∞
Þ ∞
Þ ∞
Þ ∞
Þ ∞
−∞ F(r, θ, vr, vθ)vr dvr dvθ
Þ ∞
Þ ∞
−∞ F(r, θ, vr, vθ)rdrdθdvr dvθ
Þ ∞
Þ ∞
Þ ∞
−∞
−∞ F(r, θ + 2nπ3 , vr, vθ)vr dvr dvθ
−∞ F(r, θ + 2nπ3 , vr, vθ)rdrdθdvr dvθ
−∞
2nπ
3 )

−∞

Þ 2π
Þ 2π

0

=

=

0

0
= Vr(r, θ +

(B.5)

Similarly, Vθ(r, θ) = Vθ(r, θ + 2nπ3 ).
B.2 Useful Trigonometric Identities

In this section, several trigonometric identities are proved in anticipation of their utility in

subsequent arguments. The first identity is:

(cid:18)

n


j=1

(cid:18)

n


j=1

sin

(cid:19)

cos

θ +

4jπ
n

=

θ +

4jπ
n

= 0 ∀n > 2

(B.6)

(cid:19)

(cid:19)(cid:21)

Identity 1 can be efficiently proven in a complex number representation with Euler’s theorem

n


eix = cos x + i sin x with i ≡ √−1:
(cid:20)
(cid:18)
(cid:17) (cid:44) 1, this implies that:
For exp(cid:16) 4πi

exp

j=1

θ +

i

n

(cid:20)

(cid:18)

exp

i

θ +

4jπ
n

(cid:19)(cid:21)

(cid:18)4πi

n

(cid:19) n

(cid:19)(cid:21)

j=1

4jπ
n

= 0

4jπ
n

= exp

(cid:18)

(cid:20)

i

θ +

exp

n


j=1

The fact that both the real and imaginary parts of the L.H.S. must vanish individually gives us
Eq. (B.6). Note that when n = 2, the proof does not work because exp(4πi/2) = 1.

138

Next we define sums:

j=1
to prove two corollaries of Identity 1 as:

(cid:19)

(cid:18)

θ +

2mπ
n

θ +

sin

j=1

S2 ≡ n

S3 ≡ n

S4 ≡ n


j=1

cos

(cid:18)
cos2(cid:18)
sin2(cid:18)

θ +

θ +

(cid:19)

(cid:19)
(cid:19)

2jπ
n
2jπ
n
2jπ
n

S2 = 0 ∀n > 2
S3 = S4 ∀n > 2

(B.7)

(B.8)

(B.9)

(B.10)

(B.11)

Identity 2 can be proven using the trigonometric identity sin x cos x = 1

2 sin(2x) to rewrite S2

as

(cid:19)

(cid:18)

(cid:18)

S2 ≡ n

n


j=1
1
2

=

j=1

sin

2θ +

4jπ
n

(cid:19)

(cid:18)

sin

(cid:19)

cos

θ +

2jπ
n

θ +

2jπ
n

It then follows from Identity 1 with θ → 2θ that S2 = 0 ∀n > 2.

Similarly, Identity 3 can be proven using the trigonometric identities cos2 x = 1

2 + 1

2 cos(2x)

139

and sin2 x = 1

2 − 1

2 cos(2x) and adding in zero symmetrically:

S3 ≡ n


1
2

j=1
n
2 +

cos2(cid:18)
n

n

n
2 + 0
2 − 1
sin2(cid:18)
n
n


j=1

j=1

2

=

=

=

=

j=1
≡ S4

θ +

2jπ
n

(cid:19)

(cid:19)

(cid:18)
(cid:18)

θ +

2jπ
n

cos

2θ +

4jπ
n

by Identity 1
4jπ
n

2θ +

cos

(cid:19)
(cid:19)

by Identity 1

Showing that S3 = S4 ∀n > 2.

B.3 Proof of Beam Moment Relations

In this section, we prove the 2nd order axisymmetric beam moment relations in Eq. (B.1) hold
for a beam with Cn symmetry by the following trick. For such a beam, all moments remain the same
when the coordinate system rotates by 2jπ/n where j is an integer. By summing over j = 1 to j = n,
the trigonometric identities derived in Sec. B.2 can be applied to simplify resulting trigonometric
functions and show that the 2nd order axisymmetric moment constraints hold. All relations are
proven for completeness, but with decreasing levels of detail in each successive sub-proof as the
underlying manipulations are analogous.

Proof of (cid:104)x y(cid:105) = 0:
To prove (cid:104)x y(cid:105) = 0, we note that

(cid:104)x y(cid:105) =

Þ 2π

Þ ∞

0

0

r2 cos θ sin θ f(r, θ)rdrdθ

140

By symmetry, for all integer j,

Using Eq. (B.12), we can write:

(cid:19)
(cid:19)

)rdrdθ

f(r, θ +

2jπ
n
f (r, θ) rdrdθ

(B.12)

(cid:19)

f (r, θ) rdrdθ

(cid:19)
(cid:19)

sin

sin

(cid:18)
(cid:18)
(cid:19)

2jπ
n
2jπ
n

(cid:18)

θ +

θ +

2jπ
n
2jπ
n

(cid:18)

(cid:18)
(cid:18)

θ +

θ +

r2 cos

r2 cos

0

0

Þ 2π
Þ ∞
Þ 2π
Þ ∞
Þ 2π
Þ ∞
n

Þ ∞
Þ 2π

0

0

0

0

r2 cos

θ +

2jπ
n

sin

θ +

2jπ
n

r2S2 f(r, θ)rdrdθ

0

0

by Identity 2

(cid:104)x y(cid:105) =

=

n(cid:104)x y(cid:105) =

(cid:104)x y(cid:105) =

j=1
1
n
= 0

Proof of (cid:104)xx(cid:105) = (cid:104)yy(cid:105):

(cid:104)xx(cid:105) =

=

(cid:104)yy(cid:105) =

=

0

0
1
n

Þ 2π
Þ ∞
Þ ∞
Þ 2π
Þ 2π
Þ ∞
Þ ∞
Þ 2π

0

0

0

0
1
n

0

0

r2 cos2

θ f(r, θ)rdrdθ

r2S3 f(r, θ)rdrdθ

r2 sin2

θ f(r, θ)rdrdθ

r2S4 f(r, θ)rdrdθ

from Identity 3:

S3 = S4

∴ (cid:104)xx(cid:105) = (cid:104)yy(cid:105)

141

Proof of (cid:104)xx(cid:48)(cid:105) = (cid:104)yy(cid:48)(cid:105):

0

0
1
n

Þ ∞
Þ 2π
Þ ∞
Þ 2π
Þ ∞
Þ 2π
Þ ∞
Þ 2π

0

0

0

0
1
n

0

0

(cid:104)xx(cid:48)(cid:105) =

=

(cid:104)yy

(cid:48)(cid:105) =

=

r cos θ [Vr(r, θ) cos θ − Vθ(r, θ) sin θ] f(r, θ)rdrdθ
[S3Vr(r, θ) − S2Vθ(r, θ)] f(r, θ)r2drdθ

r sin θ [Vr(r, θ) sin θ + Vθ(r, θ) cos θ] f(r, θ)rdrdθ
[S4Vr(r, θ) + S2Vθ(r, θ)] f(r, θ)r2drdθ

from Identity 3:

from Identity 2:

S3 = S4
S2 = 0
∴ (cid:104)xx(cid:48)(cid:105) = (cid:104)yy

(cid:48)(cid:105)

Henceforth, we will suppress angular arguments of Vr(r, θ), Vθ(r, θ), and f(r, θ) to further

abbreviate.

Proof of (cid:104)x y(cid:48)(cid:105) = −(cid:104)x(cid:48)y(cid:105):

(cid:104)x y

(cid:48)(cid:105) =

=

(cid:104)x(cid:48)

y(cid:105) =

=

0

0
1
n

Þ 2π
Þ ∞
Þ 2π
Þ ∞
Þ 2π
Þ ∞
Þ ∞
Þ 2π

0

0

0

0
1
n

0

0

r cos θ [Vr sin θ + Vθ cos θ] f rdrdθ
r [S2Vr − S3Vθ] f rdrdθ

r sin θ [Vr cos θ − Vθ sin θ] f rdrdθ
r [S2Vr + S4Vθ] f rdrdθ

from Identity 3:

from Identity 2:

S3 = S4
S2 = 0
(cid:48)(cid:105) = −(cid:104)x(cid:48)

y(cid:105)

∴ (cid:104)x y

142

Proof of (cid:104)x(cid:48)x(cid:48)(cid:105) = (cid:104)y(cid:48)y(cid:48)(cid:105):

(cid:104)x(cid:48)x(cid:48)(cid:105) =

=

(cid:48)

(cid:104)y

(cid:48)(cid:105) =

y

=

0

0
1
n

Þ 2π
Þ ∞
Þ ∞
Þ 2π
Þ ∞
Þ 2π
Þ ∞
Þ 2π

0

0

0

0
1
n

0

0

(cid:104)

(cid:104)

(cid:105)

(cid:105)

f rdrdθ

f rdrdθ

[Vr cos θ − Vθ sin θ]2 f rdrdθ
r − 2S2VrVθ + S4V2

S3V2

θ

[Vr sin θ + Vθ cos θ]2 f rdrdθ
r + 2S2VrVθ + S3V2

S4V2

θ

from Identity 3:

from Identity 2:

S3 = S4
S2 = 0
∴ (cid:104)x(cid:48)x(cid:48)(cid:105) = (cid:104)y

(cid:48)

(cid:48)(cid:105)

y

Proof of (cid:104)x(cid:48)y(cid:48)(cid:105) = 0:

(cid:104)x(cid:48)

(cid:48)(cid:105) =

y

=

Þ ∞
Þ 2π
Þ 2π
Þ ∞

0

(cid:110)

0
1
n

0

0

(cid:104)

[Vr cos θ − Vθ sin θ][Vr sin θ + Vθ cos θ] f rdrdθ

(cid:105)

+ VrVθ [S3 − S4](cid:111)

f rdrdθ

S2

V2
r + V2

θ

from Identity 3:

from Identity 2:
∴ (cid:104)x(cid:48)

y

(cid:48)(cid:105) = 0

S3 = S4
S2 = 0

143

APPENDIX C

GENERATING A 4D COUPLED DISTRIBUTION

In this appendix, we present a method for generating an arbitrary 4D coupled distribution using a
similarity transformation.

Recall the 4D sigma matrix is given by:
(cid:104)xx(cid:48)(cid:105)
(cid:104)x(cid:48)x(cid:48)(cid:105)
(cid:104)x(cid:48)y(cid:105)
(cid:104)x(cid:48)y(cid:48)(cid:105)

(cid:104)xx(cid:105)
(cid:104)xx(cid:48)(cid:105)
(cid:104)x y(cid:105)
(cid:104)x y(cid:48)(cid:105)

σ =

(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)

(cid:104)x y(cid:105)
(cid:104)x(cid:48)y(cid:105)
(cid:104)yy(cid:105)
(cid:104)yy(cid:48)(cid:105)

(cid:104)x y(cid:48)(cid:105)
(cid:104)x(cid:48)y(cid:48)(cid:105)
(cid:104)yy(cid:48)(cid:105)
(cid:104)y(cid:48)y(cid:48)(cid:105)

(cid:16)

y(cid:48)(cid:17). Since σ is a real symmetric matrix, ∃ orthogonal matrix R (i.e.

=(cid:10)xx(cid:124)(cid:11)

(C.1)

where x(cid:124) =
x(cid:48)
R(cid:124) = R−1) such that:

x

y

where λj, j = 1,4 denote the four eigenvalues of σ.

The method can be described as follows. One first generates a 4D completely uncoupled

distribution in the transformed coordinates ˜x which satisties:

(cid:10)˜x˜x(cid:124)(cid:11) =

Then, transforming ˜x in accordance with:

λ1
0
0
0

0

λ2
0
0

0
0

λ3
0

0
0
0

λ4

(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172)
(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172)
(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172)

R(cid:124)

σR =

λ1
0
0
0

0

λ2
0
0

0
0

λ3
0

0
0
0

λ4

(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)
(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)

˜x (cid:55)→ x = R˜x

144

(C.2)

(C.3)

(C.4)

will result in a 4D coupled distribution with the specified moments.

Proof.

σ =(cid:10)xx(cid:124)(cid:11)

(cid:104)xx(cid:48)(cid:105)
(cid:104)x(cid:48)x(cid:48)(cid:105)
(cid:104)x(cid:48)y(cid:105)
(cid:104)x(cid:48)y(cid:48)(cid:105)
0
0
0

λ2
0
0

λ3
0

(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172)

(cid:104)x y(cid:48)(cid:105)
(cid:104)x(cid:48)y(cid:48)(cid:105)
(cid:104)yy(cid:48)(cid:105)
(cid:104)y(cid:48)y(cid:48)(cid:105)

(cid:104)x y(cid:105)
(cid:104)x(cid:48)y(cid:105)
(cid:104)yy(cid:105)
(cid:104)yy(cid:48)(cid:105)
0
0
0

(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172)

λ4

R(cid:124)

=

(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)

(cid:104)xx(cid:105)
(cid:104)xx(cid:48)(cid:105)
(cid:104)x y(cid:105)
(cid:104)x y(cid:48)(cid:105)

(cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171)

λ1
0
0
0

= R

= R(cid:10)˜x˜x(cid:124)(cid:11) R(cid:124)
=(cid:10)R˜x˜x(cid:124)R(cid:124)(cid:11)

Note that no assumption is made on the form of the decoupled distribution in ˜x, insofar as the
2nd-order moments satisfy Eq. (C.3). However, how distributions in ˜x should be interpreted after
the transformation may warrant careful considerations (see the discussion at the end of Sec. 3.4).
This formulation can be applied to generate a wide variety of coupled distributions with spec-
ified 2nd-order moments to simulate magnetized beams emerging from ECR sources. Additional
symmetry arguments (see Chapter 4) together with physics considerations may also be exploited to
reduce the number of free parameters.

145

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