beam uniformization and low frequency rf cavities in

BEAM UNIFORMIZATION AND LOW FREQUENCY RF CAVITIES IN COMPACT ELECTRON STORAGE RINGS

Alfonse N. Pham

Submitted to the faculty of the University Graduate School in partial fulfillment of the requirement for the degree Doctor of Philosophy in the Department of Physics, Indiana University

June, 2014

ii Accepted by the Graduate Faculty, Indiana Univeristy, in partial fulfillment of the requirement for the degree of Doctor of Philosophy.

Doctoral Committee

Shyh-Yuan Lee, Ph.D.

John P. Carini, Ph.D.

Rick Van Kooten, Ph.D.

W. Michael Snow, Ph.D.

May 6, 2014

iii

c 2014 Copyright Alfonse N. Pham

iv

To my loving family.

v

Acknowledgments

As I reflect on my graduate experience, I have come to realize the difficulty of adequately acknowledging all those who have contributed greatly to my development as a person, scientist, and scholar. Over the course of the last five years, my path has crossed with many exceptional individuals. These encounters range from simple exchanges of pleasantries to lasting friendships that have forever altered my life. I would like to take the opportunity here to acknowledge these individuals, from those whose encouragement helped me find the resolve to carry forth, to those unwavering pillars of scholastic truth upon which I stand, and the vast ocean of love and support of those who dwell in between. I am very fortunate to have had some of the best physics instructors leading the graduate curriculum at Indiana University (IU). They included professors: Michael S. Berger, Hal Evans, Herbert A. Fertig, Steven A. Gottlieb, Chuck J. Horowitz, Rick J. Van Kooten, and W. Michael Snow. I have received much encouragement in the Department of Physics from professors and scientists: John M. Beggs, Dobrin Bossev, John P. Carini, Alex R. Dzierba, and Brian D. Serot. The staff in the Department of Physics who have supported me along the way, especially Erin Arthur and Anne Foley. A special thanks to Bob Noel, head of Swain Library, who was fearless while on the hunt for obscure articles and scholarly works. While working with the Emissaries for Graduate Student Diversity, I have received much support from Assistant Dean Yolanda Trevino, and Associate Dean Maxwell Watson who share my passion for promoting cultural diversity and heterogeneity of thought in academia. To the Indiana University Cyclotron Facility (IUCF) whose welcoming and knowl-

vi edgeable personnel has taken me in as their own, you will always have special place in my heart. To Bob Brown, Scott Coffey, Michael Landreth, Maggie M. Ochoada, Richard Seybert, Bruce Shei, and Moya Wright who have supported me at the facility. My deepest gratitude to Doug McCammon, Stephen Anderson, and Nick Venstra of the RF Electronics Group who went above and beyond to support me with their expertise and enriched my life with their humor. To members of the ALPHA Group at the IUCF: Larry Boot, Steve Clark, Jak Doskow, Gary East, Robert Ellis, Bob Pollock, Tom Rinckel, and Chandra Romel, it has been a pleasure working with you. I would like to especially thank Larry Boot whose artisanal knowledge in machining and metalwork has been instrumental in the construction of the ALPHA RF cavity. I would also like to thank Jak Doskow who has helped enriched many of the engineering drawings shown in this dissertation with his artistic flair. To my friend and colleague, K. Y. Ng, thank you for providing me with your experience and insight. I have had the pleasure of working alongside many of the staff, instructors, and participants of the United State Particle Accelerator School (USPAS). This program has greatly fostered my development as a beam physicist and provides a venue for scientists in the field to interact. The amazing staff that keeps the program running twice a year, snow or influenza, includes: William Barletta, Irina Novitski, and Susan Winchester with guest appearances by Josh O’Connell, Nino Strothman, and Steve Conlon. I would like to especially thank Susan Winchester for her kindness and unwavering support of me all these years. I will forever be indebted to my doctoral advisor, Professor S. Y. Lee, who has patiently guided my growth towards becoming an accelerator physicist. His knowledge, enthusiasm, and insight into the field has been an inspiration that has fueled my passion for the subject matter. In working within the Accelerator Physics Group at

vii the IUCF, I had the wonderful privilege of working and growing alongside colleagues: Hong-Chun Chao, Alper Duru, Jeff Eldred, Kun Fang, Zhenghao Gu, Kilean Hwang, Yichao Jing, Jeff Kolski, Ao Liu, Honghuan Liu, Tianhuan Luo, Michael Ng, Xiaoying Pang, Patrick McChesney, Xiaozhe Shen, Jack Yi-Chun Wang, and Haisheng Xu. Together we overcame whatever challenges that Professor S. Y. Lee has placed on our paths and shared countless memories. To the all the good friends who I have had the privilege of knowing, I thank you for all the richness you bring to my life. To the people dearest to my heart to whom I have dedicated my life of inquiry. My supportive girlfriend, Cara, whose graduate path fatefully intertwined with my own. I am grateful for your love, kindness, and support. There is nothing in this earthly world that we cannot overcome. My younger sister, Tien, who is currently embarking on her own journey, thank you for being a wonderful sibling. The legacy of our parents and that of our ethnic origin burdens heavily on our shoulders. We will persevere together. My loving and supportive parents, Hien and Lieu, who have endured and sacrificed far too much in their lifetimes so their children would have opportunities in the new world. No words or actions will ever convey enough gratitude for your devoted efforts. I owe my life, likeness, dreams, and everything to you. You are my guiding light in stormy seas. You are the rock, the foundation on which all things are possible. Thank you for your unconditional love.

viii

I found a loose string dangling from the hem of my existence. When I pulled it, the world around me began to unravel. --William Reschke

ix

Alfonse N. Pham

BEAM UNIFORMIZATION AND LOW FREQUENCY RF CAVITIES IN COMPACT ELECTRON STORAGE RINGS

An electron storage ring is currently under construction at Indiana University for extreme environment radiation effects experiments, x-ray production, and particle beam dynamics experiments. For an electron bunch to be successfully stored for long durations, a radio-frequency (RF) resonant structure will be used to provide an adequate RF bucket for longitudinal focusing and replenishment of energy electrons loses via synchrotron radiation. Due to beam line space limitation that are inherent to compact circular particle accelerators, a unique ferrite-loaded quarter-wave RF resonant cavity has been designed and constructed for use in the electron storage ring. The physics of particle accelerators and beams, ferrite-loaded RF resonant cavity theory, and results of the Poisson-SUPERFISH electromagnetic field simulations that were used to guide the specification and design of the RF cavity will be presented. Low-power resonant cavity characterization measurements were used to benchmark the performance of the RF cavity. High-power characterization and measurements with electron beams will be used to validate the performance of the cavity in the electron storage ring. To fulfill the requirements for radiation effect experiments, the storage ring manipulation of beams that utilizes a phase space beam dilution method have been developed for the broadening of the radiation damped electron bunch with longitudinal particle distribution uniformity. The method relies on phase modulation applied

x to a double RF system to generate large regions of bounded chaotic particle motion in phase space. These region are formed by a multitude of overlapping parametric resonances. Parameters of the double RF system and applied phase modulation can be adjusted to vary the degree of beam dilution. The optimal RF parameters have been found for maximal bunch broadening, uniform longitudinal particle distribution, and bounded particle diffusion. Implementation of the phase space dilution method in the electron storage ring will be presented. This novel method has applications in alleviating adverse space charge effects pertaining to high intensity beams, particle bunch distribution uniformization, and industrial radiation effects experiments.

Contents Acceptance

ii

Acknowledgments

v

Abstract

ix

1 Introduction

1

1.1

Accelerator Coordinate System . . . . . . . . . . . . . . . . . . . . .

2 Betatron Motion of Particles

4 7

2.1

Electromagnetic Fields in Accelerators . . . . . . . . . . . . . . . . .

7

2.2

Transverse Equations of Motion . . . . . . . . . . . . . . . . . . . . .

11

2.3

Transfer Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

2.3.1

Stability of Betatron Motion . . . . . . . . . . . . . . . . . . .

15

2.3.2

Courant-Snyder Parametrization

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

17

Floquet Transformation . . . . . . . . . . . . . . . . . . . . . . . . .

19

2.4.1

21

2.4

Betatron Phase and Tune . . . . . . . . . . . . . . . . . . . .

3 Synchrotron Motion of Particles

23

3.1

Longitudinal Equation of Motion . . . . . . . . . . . . . . . . . . . .

24

3.2

Hamiltonian of Synchrotron Motion . . . . . . . . . . . . . . . . . . .

30

3.2.1

30

Synchrotron Frequency and Tune . . . . . . . . . . . . . . . . xi

xii

CONTENTS 3.3

Phase-Stability of Synchrotron Motion . . . . . . . . . . . . . . . . .

4 Transmission Line Theory

31 37

4.1

Telegrapher’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . .

38

4.2

Lossless Transmission Line . . . . . . . . . . . . . . . . . . . . . . . .

41

4.2.1

Terminated Lossless Transmission Line . . . . . . . . . . . . .

42

4.2.2

Lossless Transmission Line Terminated in a Short . . . . . . .

43

4.2.3

Lossless Transmission Line Terminated in a Open . . . . . . .

44

5 Electron Storage Rings

47

5.1

Larmor’s Formula and Synchrotron Radiation . . . . . . . . . . . . .

49

5.2

Radiation Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51

5.2.1

Damping of Synchrotron Motion . . . . . . . . . . . . . . . . .

51

5.2.2

Damping Partition . . . . . . . . . . . . . . . . . . . . . . . .

52

5.2.3

Damping of Betatron Motion . . . . . . . . . . . . . . . . . .

54

5.2.4

Summary of Radiation Damping Parameters . . . . . . . . . .

58

Case Example: ALPHA Storage Ring . . . . . . . . . . . . . . . . . .

59

5.3

6 Ferrite-loaded Quarter-wave Coaxial Resonator Cavity

69

6.1

Ferrite-Loaded Transmission Line Theory . . . . . . . . . . . . . . . .

71

6.2

Design of Ferrite-loaded Cavity . . . . . . . . . . . . . . . . . . . . .

74

6.2.1

Physical Design Constraints . . . . . . . . . . . . . . . . . . .

75

6.2.2

Storage Ring Cavity Requirements . . . . . . . . . . . . . . .

78

6.2.3

Ferrite Material Characterization . . . . . . . . . . . . . . . .

82

6.2.4

Resonant Cavity Simulations . . . . . . . . . . . . . . . . . . .

91

6.2.5

Application of Transmission Line Theory . . . . . . . . . . . . 111

6.3

Construction of Ferrite-loaded Cavity . . . . . . . . . . . . . . . . . . 115 6.3.1

Resonant Frequency Tuning Capacitor . . . . . . . . . . . . . 120

CONTENTS 6.3.2 6.4

6.5

6.6

6.7

xiii RF Power Coupling . . . . . . . . . . . . . . . . . . . . . . . . 120

Cavity Characterization Measurements . . . . . . . . . . . . . . . . . 122 6.4.1

Cavity Impedance Matching Network . . . . . . . . . . . . . . 124

6.4.2

Scattering Parameter S11 and SWR . . . . . . . . . . . . . . . 129

6.4.3

RF Gap Voltage and Shunt Impedance . . . . . . . . . . . . . 131

6.4.4

Quality Factor and Rs /Q . . . . . . . . . . . . . . . . . . . . . 139

6.4.5

Summary of Low-Power Cavity Characterization . . . . . . . . 142

Field Perturbation Bead-pull Measurements . . . . . . . . . . . . . . 143 6.5.1

Slater’s Perturbation Theorem . . . . . . . . . . . . . . . . . . 143

6.5.2

Application of Slater’s Perturbation Theorem . . . . . . . . . 152

6.5.3

Bead-pull Measurement Results . . . . . . . . . . . . . . . . . 153

Calibrated Probe Measurements . . . . . . . . . . . . . . . . . . . . . 158 6.6.1

Description of Cavity Pickup . . . . . . . . . . . . . . . . . . . 158

6.6.2

Calibration of Cavity Probes . . . . . . . . . . . . . . . . . . . 159

Cavity Measurements with Beam . . . . . . . . . . . . . . . . . . . . 163 6.7.1

Coherence-Decoherence of Synchrotron Oscillation . . . . . . . 163

6.7.2

Measurement of Synchrotron Sidebands . . . . . . . . . . . . . 167

7 Phase Space Dilution of Beams 7.1

7.2

173

The Double RF Model . . . . . . . . . . . . . . . . . . . . . . . . . . 176 7.1.1

Choice of phase difference parameter ∆φo

. . . . . . . . . . . 179

7.1.2

Synchrotron Tune and Resonance Strengths . . . . . . . . . . 182

7.1.3

Modulation Tune and Amplitude . . . . . . . . . . . . . . . . 188

Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 7.2.1

Phase Difference Parameter ∆φo = 30◦ . . . . . . . . . . . . . 192

7.2.2

Choice of Modulation Tune and Amplitude . . . . . . . . . . . 202

7.2.3

Phase Difference Parameter ∆φo = 45◦ . . . . . . . . . . . . . 205

7.2.4 7.3

Modulation Phase ηm . . . . . . . . . . . . . . . . . . . . . . . 219

Implementation of the Dilution Method . . . . . . . . . . . . . . . . . 222 7.3.1

Alternative Implementation with an AC Dipole . . . . . . . . 223

8 Conclusion

229

Appendix A Synchrotron Radiation

235

A.1 Liénard–Wiechert Potentials . . . . . . . . . . . . . . . . . . . . . . . 235 A.2 Larmor’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 Appendix B Derivation of Ferrite-loaded Cavity Quantities

238

B.1 Effective Permittivity and Permeability . . . . . . . . . . . . . . . . . 238 B.1.1 Line Capacitance . . . . . . . . . . . . . . . . . . . . . . . . . 238 B.1.2 Line Inductance . . . . . . . . . . . . . . . . . . . . . . . . . . 240

Bibliography

241

Curriculum Vitae

xiv

List of Tables 2.1

Multipoles fields most used in particle accelerators with their respective functional descriptions. . . . . . . . . . . . . . . . . . . . . . . . . . .

3.1

10

Summary of the synchrotron phase requirements for phase stability in circular particle accelerators operating with beam energy γo below and above transition energy γT . . . . . . . . . . . . . . . . . . . . . . . . .

6.1

Summary of the geometric constraints in the initial design process of the 15 MHz quarter-wave resonant cavity. . . . . . . . . . . . . . . . .

6.2

78

Summary of the geometric parameters of the M4 C21A ferrite rings to be utilized in the 15 MHz quarter-wave resonant cavity. . . . . . . . .

6.3

35

84

Summary of the final geometric parameters of the ferrite-loaded quarterwave cavity optimized using Poisson-Superfish. . . . . . . . . . . . . . 113

6.4

Summary of transmission line parameters relevant to the resonant frequency tuning calculations for the 15 MHz ferrite-loaded quarter-wave RF cavity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

6.5

Summary of the low-power cavity characterization measurement for the 15 MHz ferrite-loaded quarter-wave RF cavity. . . . . . . . . . . . 142

xv

List of Figures 1.1

Frenet-Serret Curvilinear Coordinate System used in the field of accelerator physics. This coordinate system will be adopted for use throughout the entire dissertation. . . . . . . . . . . . . . . . . . . . . . . . .

3.1

5

Graphical illustration of the principle of phase stability for circular particle accelerators operating with beam energies (a.) below transition energy (γo < γT ) where η < 0, and (b.) above transition energy (γo > γT ) where η > 0. The initially higher energy particle is shown in blue, initially lower energy particle is shown in red, and synchronous particle is shown in black. . . . . . . . . . . . . . . . . . . . . . . . .

4.1

A schematic lumped-element circuit model of an ideal transmission line of length ∆z at an arbitrary time t. . . . . . . . . . . . . . . . . . . .

4.2

42

Transmission line terminated into a short circuit where the impedance ZL (z = 0) = 0 with the source displaced z = −` distance away. . . . .

4.4

39

Transmission line terminated into an arbitrary load of impedance ZL with the source displaced z = −` distance away. . . . . . . . . . . . .

4.3

34

44

Transmission line terminated into an open circuit where the impedance ZL (z = 0) = ∞ with the source displaced z = −` distance away. . . . xvi

45

LIST OF FIGURES 5.1

xvii

Diagram depicting the ALPHA project in the commissioning stages. (From left to right) An electron bunch is generated and accelerated to full energy. The injection line transports the electron bunch to the storage ring accounting for all lattice matching considerations. The bunch is injected into the storage ring through the Lambertson septum. Depending on mode of operation, extraction is accomplished with a magnetic bumper or a traveling-wave fast-extraction kicker. The manipulated bunch is transported through the extraction line to the device under test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.2

61

Detailed diagram of the ALPHA storage ring with important components labeled in their respective locations in the compact storage ring. The red dashed-line represent the direction of beam propagation where the injection line is to the right and the extraction line is to the left. .

5.3

62

The momentum compaction factor αc (black), horizontal damping partition number Jx (green), vertical betatron tune νz (red), and horizontal betatron tune νx (blue) as a function of the gradient damping wiggler bending radius ρw . The horizontal dotted-line represent the region near the isochronous condition where αc ∼ 0. When Jx < 0, uncontrolled growth of the horizontal betatron amplitude is encountered. 65

5.4

Damping time of the energy spread amplitude τE (black), damping time of the horizontal betatron amplitude τx (blue), and damping time of the vertical betatron amplitude τz (red) as a function of the gradient damping wiggler bending radius ρw . The damping times have been calculated for Eo = 25 MeV electron beam and the focusing gradient of the damping wiggler is B1 /B0 ≈ 1.9 m−1 . . . . . . . . . . . . . . .

66

xviii 5.5

LIST OF FIGURES The horizontal betatron function, vertical betatron function, and horizontal dispersion function as a function of the lattice displacement parameter s where the gradient damping wiggler bending radius was set at ρw = 0.75 m in the vicinity of the isochronous condition. The results were obtained from the MAD-X lattice simulation software suite. 67

6.1

Cross-sectional schematic diagram of a quarter-wave resonant structure loaded with ferrite rings with permittivity and permeability = e o and µ = µe µo . (a) Side-view diagram depicting longitudinal dimensions where d1 is the thickness of the ferrite rings (shown in gray), d2 is the spacing between ferrite rings that could be composed of metallic cooling plates or air, and ` is the overall length of the structure. The ceramic gap (shown black) will allow for the interaction of the particles and the fields generated by the structure. (b) Perspective-view diagram depicting the radial geometry of the ferrite-loaded quarterwave structure where the radial distance from the origin to the inner conductor is r1 , to inner radius of the ferrite ring is r2 , to the outer radius of the ferrite ring is r3 , and to the outer conductor is r4 . . . . .

6.2

73

Diagram of the ALPHA storage ring populated by magnetic elements, vacuum pumps, and vacuum diagnostics. The blue hatched region depicts the planned location for the RF resonant cavity, between Dipole #3 and the Damping Wiggler #2. . . . . . . . . . . . . . . . . . . . .

6.3

76

(top) Vacuum pipe portion of the RF cavity with dimensions labeled in inches. (bottom) Prospective view of the vacuum pipe showing the mounting brackets and ceramic (acceleration) gap. . . . . . . . . . . .

77

LIST OF FIGURES 6.4

xix

Touschek lifetime τT as a function of the damping wiggler bending radius ρw where the peak RF voltage is Vp = 0.1 kV (blue), Vp = 0.5 kV (red), Vp = 1 kV (black), and Vp = 2 kV (green). The lifetime calculations were carried out for electrons with beam energy of Eo = 50 MeV. The peak in the lifetimes are for ρw in the vicinity of the isochronous condition. . . . . . . . . . . . . . . . . . . . . . . . . . .

6.5

A simple two-element model of the ferrite ring where Lp is the ideal inductance and Rp is the ideal resistance in parallel. . . . . . . . . . .

6.6

83

85

The relative permeability µ0 of a single Toshiba M4 C21A ferrite ring enclosed in the heated aluminum test fixture as a function of frequency from 10–110 MHz at temperatures from 25◦ C up to 150◦ C. (Data courtesy of K. Y. Ng and D. Wildman). . . . . . . . . . . . . . . . . . . .

6.7

88

The ferrite core loss µ00 of a single Toshiba M4 C21A ferrite ring enclosed in the heated aluminum test fixture as a function of frequency from 10– 110 MHz at temperatures from 25◦ C up to 150◦ C. (Data courtesy of K. Y. Ng and D. Wildman). . . . . . . . . . . . . . . . . . . . . . . .

6.8

89

A similar test fixture constructed at Los Alamos National Laboratory designed to measure the impedance of a ferrite ring using a vector network analyzer. (Courtesy of C. Beltran) . . . . . . . . . . . . . . .

6.9

90

Characterization summary of the M4 C21A ferrite. The solid line corresponds to the real component of the magnetic permeability and the dot-dashed line corresponds to the imaginary component of the magnetic permeability as a function of frequency. The lines in blue and red represents the complex permeability (real and imaginary) with ferrite core at 25◦ C and 125◦ C, respectively. . . . . . . . . . . . . . . . . . .

92

xx

LIST OF FIGURES 6.10 An example output of a Poisson-Superfish simulation run where a slice of the a quarter-wave resonant structure is shown axially symmetric about the horizontal axis. The geometry of this structure had an inner conductor radius of r1 = 2.84 cm, outer radius of r4 = 25 cm, and overall cavity length of ` = 35 cm yielding a resonant frequency of 160.28 MHz. The electric field contours are depicted with magenta lines where the density of lines represent the regional electric fields strengths and the magnetic fields are depicted with circles where the size of the circles represents the regional magnetic field strengths. The scales in the horizontal and vertical axes are in units of cm. . . . . . .

94

6.11 Resonant frequency of the quarter-wave RF structure where no ferrite rings were loaded as a function of the overall cavity length ` and outer conductor radius r4 where (a) r4 = 10.79 cm, (b) 15 cm, (c) 20 cm, (d) 25 cm, (e) 30 cm, and (f) 35 cm. The inner conductor radius of the structure was held fixed at r1 = 2.84 cm in the Poisson-Superfish simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

96

6.12 An example output of a Poisson-Superfish simulation run where a slice of the a quarter-wave resonant structure is shown axially symmetric about the horizontal axis. The geometry of this structure had an inner conductor radius of r1 = 5.72 cm, outer radius of r4 = 25 cm, and overall cavity length of ` = 35 cm yielding a resonant frequency of 172.8 MHz. The electric field contours are depicted with magenta lines where the density of lines represent the regional electric fields strengths and the magnetic fields are depicted with circles where the size of the circles represents the regional magnetic field strengths. The scales in the horizontal and vertical axises are in units of cm. . . . . . . . . . .

98

LIST OF FIGURES

xxi

6.13 Resonant frequency of the quarter-wave RF structure where no ferrite rings were loaded as a function of the overall cavity length ` and outer conductor radius r4 where (a) r4 = 10.79 cm, (b) 15 cm, (c) 20 cm, (d) 25 cm, (e) 30 cm, and (f) 35 cm. The inner conductor radius of the structure was held fixed at r1 = 5.72 cm in the Poisson-Superfish simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

99

6.14 Cross-sectional diagram of a quarter-wave resonating structure with numbers corresponding to the relative position of a single ferrite ring introduced within the simulation. . . . . . . . . . . . . . . . . . . . . 100 6.15 Resonant frequency of the quarter-wave RF structure as a function of the location of a single ferrite ring with respect to the accelerating gap calculated using Poisson-Superfish. Geometric parameters such as the overall length and inner radius were held fixed at ` = 37.32 cm and r1 = 5.72 cm as the outer conductor radius r4 was allowed to vary. The material properties of the ferrite ring was set to µe = 43 and e = 13 for nominal room temperature operation. . . . . . . . . . . . . . . . . 102 6.16 Poisson-Superfish simulation result with a single M4 C21A ferrite ring placed at position #5 where the inner conductor radius is r1 = 5.72 cm, outer radius of r4 = 10.79 cm, and overall cavity length of ` = 37.29 cm yielding a resonant frequency of 85.43 MHz. The electric field contours are depicted with magenta lines where the density of lines represent the regional electric fields strengths and the magnetic fields are depicted with circles where the size of the circles represents the regional magnetic field strengths. The scales in the horizontal and vertical axes are in units of cm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

xxii

LIST OF FIGURES

6.17 Resonant frequency of the quarter-wave RF structure as a function of number of ferrite rings loaded into the cavity using Poisson-Superfish where the first ferrite ring was loaded into the position closest to the accelerating gap. Geometric parameters such as the overall length and inner radius were held fixed at ` = 37.32 cm and r1 = 5.72 cm as the outer conductor radius r4 was allowed to vary. The material properties of the ferrite ring was set to µe = 43 and e = 13 for nominal room temperature operation. . . . . . . . . . . . . . . . . . . . . . . . . . . 105

6.18 Poisson-Superfish simulation result where three M4 C21A ferrite rings were placed in positions furthest from the accelerating gap ∼1 cm apart. The inner conductor radius was r1 = 5.72 cm, outer radius r4 = 10.79 cm, and overall cavity length ` = 37.29 cm yielding a resonant frequency of 48.01 MHz. The electric field contours are depicted with magenta lines where the density of lines represent the regional electric fields strengths and the magnetic fields are depicted with circles where the size of the circles represents the regional magnetic field strengths. The scales in the horizontal and vertical axises are in units of cm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

6.19 Resonant frequency of the quarter-wave RF structure as a function of the magnetic permeability µ0 of the ferrite material. Geometric parameters such as the overall length, inner radius, and outer radius of the structure were held fixed at ` = 37.32 cm, r1 = 5.72 cm, and r4 = 10.79 cm, respectively and the ficed material properties were set to e = 13, in the Poisson-Superfish simulation. . . . . . . . . . . . . . 108

LIST OF FIGURES

xxiii

6.20 Poisson-Superfish simulation result where seven M4 C21A ferrite rings are loaded and thus filling the cavity. The inner conductor radius was r1 = 5.72 cm, outer conductor radius r4 = 30 cm, and overall cavity length ` = 37.29 cm yielding a resonant frequency of 44.96 MHz. The electric field contours are depicted with magenta lines where the density of lines represent the regional electric fields strengths and the magnetic fields are depicted with circles where the size of the circles represents the regional magnetic field strengths. The scales in the horizontal and vertical axises are in units of cm. . . . . . . . . . . . . . . . . . . . . 110

6.21 Poisson-Superfish simulation result where seven M4 C21A ferrite rings are loaded and thus filling the cavity. The inner conductor radius was r1 = 5.72 cm, outer conductor radius r4 = 10.79 cm, and overall cavity length ` = 37.29 cm yielding a resonant frequency of 30.29 MHz. The electric field contours are depicted with magenta lines where the density of lines represent the regional electric fields strengths and the magnetic fields are depicted with circles where the size of the circles represents the regional magnetic field strengths. The scales in the horizontal and vertical axises are in units of cm. . . . . . . . . . . . . . . . . . . . . 112

6.22 The fundamental resonant frequency, 3rd , and 5th harmonic solutions to the transcendental quarter-wave transmission line equation given in Eq. (6.26).

Four curves represents 1/ωo Cgap where (a) Cgap =

600 pF, (b) 102.74 pF, (c) 50 pF, and (d) 25 pF. The intersection points of the fundamental resonant frequency at 15 MHz, 3rd harmonic at 70.76 MHz, and 5th harmonic at 135.91 MHz was calculated for the gap capacitance of Cgap = 102.74 pF. . . . . . . . . . . . . . . . . . . 116

xxiv

LIST OF FIGURES

6.23 Engineering drawing showing the assembled vacuum ceramic break and specific welding requirements for each joint in the construction of the 15 MHz quarter-wave resonant cavity. . . . . . . . . . . . . . . . . . . 117 6.24 Engineering drawing showing an example of an end piece machined for use on the 15 MHz quarter-wave resonant cavity. This particular OFHC part fastens the rolled inner conductor to the vacuum ceramic break mounting flange. . . . . . . . . . . . . . . . . . . . . . . . . . . 119 6.25 Cut-away engineering drawing showing an example of an OFHC end piece machined and mounted for use on the 15 MHz quarter-wave resonant cavity. These end pieces fastens the rolled OFCHC inner and outer conductor to the mounting flange welded on the vacuum ceramic break pipe. The green G-10 guides ensure the correct placement of the ferrite rings. The 3-point stand supports the entire cavity so that the weight of the ferrite rings are not on the vacuum beam pipe. . . . . . 121 6.26 Cut-away engineering drawing of the 15 MHz quarter-wave ferrite-loaded RF cavity with important components denoted. (a) Inductively coupled RF drive points between ferrite #6 and #7, (b) seven M4 C21A ferrite rings in placement, (c) high-voltage variable capacitor, (d) accelerating gap made of a high-purity alumina ceramic welded in place with low-expansion Kovar fittings (e) capacitor tuning motor and fixture for remote cavity tuning. . . . . . . . . . . . . . . . . . . . . . . 123 6.27 Smith Chart depicting the complex reflection coefficient at the input of the ferrite loaded cavity without matching in the range from 6 MHz to 29 MHz with 1601 points. The data in the red circle was recorded at 14.999 MHz, and the red squares recorded at 15.013 MHz and 14.984 MHz, respectively. . . . . . . . . . . . . . . . . . . . . . . 125

LIST OF FIGURES

xxv

6.28 Smith Chart depicting the complex reflection coefficient at the input of the ferrite loaded cavity with tee-stub matching in the range from 6 MHz to 29 MHz with 1601 points. The data in the red circle was recorded at 14.999 MHz, and the red squares recorded at 15.013 MHz and 14.984 MHz, respectively. . . . . . . . . . . . . . . . . . . . . . . 128 6.29 S11 scattering parameter and SWR as a function of frequency from 1–75 MHz. When the fundamental resonant frequency of the cavity is tuned to be at 15 MHz with SWR = 1.074, the 3rd harmonic resonant frequency will be located at 57.891 MHz where SWR = 2.648. The sharp increases in the SWR around 25 MHz and 45 MHz is due to the response of the ferrite material. . . . . . . . . . . . . . . . . . . . . . 130 6.30 Photograph of the Agilent E2697A high-voltage probe placement across the accelerating gap inside the RF cavity for the gap voltage measurements. Data was acquired through the Agilent DSO81204A digital oscilloscope. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 6.31 Mean peak RF gap voltage in units of Vp as a function of the linearscaled transmitted RF power in units of watt over the output RF power range 0-20 dBm in increments of 1 dBm steps from a RF signal generator source at the frequency of 15 MHz. The error bars are magnified 100 times their actual value to be able represent them. . . . . . . . . 135 6.32 Mean peak RF gap voltage in units of Vp as a function of the log-scaled transmitted RF power in units of dBm over the output RF power range 0-20 dBm in increments of 1 dBm steps from a RF signal generator source at the frequency of 15 MHz. The error bars are magnified 100 times their actual value to be able represent them. . . . . . . . . . . . 136

xxvi

LIST OF FIGURES

6.33 Mean RF cavity shunt impedance Rs in units of Ω as a function of the log-scaled transmitted RF power in units of dBm over the output RF power range 0-20 dBm in increments of 1 dBm steps from a RF signal generator source at the frequency of 15 MHz. The error bars are magnified 10 times their actual value to be able represent them. A shunt impedance of Rs = (5.565 ± 0.003) kΩ was calculated. . . . . . 138 6.34 Measurement of the S21 scattering parameter as a function of frequency with a vector network analyzer. The half-power frequency bandwidth, with 3 dB points shown in blue, was found to be ∆f = 228.227 kHz about a resonant frequency of fo = 14.997, shown in red. The resulting loaded Q-factor was found to be Qloaded = 65.71 ± 0.06. . . . . . . . . 141 6.35 (a) The unperturbed resonant structure of arbitrary geometry with total volume Vo and total surface area So enclosing fields E and H within a lossless medium at the resonant frequency of ωo . (b) An exaggerated depiction of the perturbed resonant structure where the volume and surface area changed by the amount ∆V and ∆S with fields E and H at resonant frequency of ω. . . . . . . . . . . . . . . . 145 6.36 (a) The unperturbed resonant structure of arbitrary geometry with total volume Vo and total surface area So enclosing fields Eo and Ho within a lossless medium at the resonant frequency of ωo . (b) The perturbed resonant structure where the medium within been changed by the amount ∆ and ∆µ with fields E and H at resonant frequency of ω. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 6.37 Photograph of the ferrite-loaded quarter-wave cavity assembled on the bead-pull test stand. A vector network analyzer was used to measure the resonant frequency shift due to the introduction of a small metallic bead into the accelerating gap. . . . . . . . . . . . . . . . . . . . . . . 154

LIST OF FIGURES

xxvii

6.38 Raw frequency-shift data due to the introduction of a spherical metallic bead of radius rbead = 5 mm with step increments of ∆z = 5 mm measured with vector network analyzer. The overall negative linear slope is due the fact that the experiment was carried out in less than ideal conditions. Conditions when the environment monotonically increased in ambient temperature. . . . . . . . . . . . . . . . . . . . . . . . . . 156

6.39 Frequency-shift data due to the introduction of a spherical metallic bead of radius rbead = 5 mm with step increments of ∆z = 5 mm measured with vector network analyzer and corrected for monotonically increasing ambient temperature. The shunt impedance deduced from the corrected data was Rs = 5898.6 Ω. . . . . . . . . . . . . . . . . . . 157

6.40 Direct measurement of the RF gap voltage with a high-voltage probe and measurement using two different 1 pF cavity sample probes as function of DAC control parameter that is proportional to the input RF power. The errorbars have been magnified by 10 to allow for display. The data was used to establish calibration constants for the 1 pF cavity sample probes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

6.41 Direct measurement of the RF gap voltage with a high-voltage probe and measurement of E-Field cavity sample probes as function of DAC control parameter that is proportional to the input RF power. The errorbars have been magnified by 10 to allow for display. The data was used to establish calibration constants for the E-Field cavity sample probes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

xxviii

LIST OF FIGURES

6.42 Raw data from a single port of a beam position monitor in the ALPHA storage ring. The signal was acquired on a 10GS/s digital oscilloscope where the vertical scale was set as 90 mV/div, horizontal scale 100 µs/div, and each data point separated by 100 ps. About five complete synchrotron oscillations were observed over a time span of 100 µs in the data above. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 6.43 Peak RF gap voltage deduced from beam position monitor data of synchrotron oscillation as a function of pre-amplifier signal generator RF drive into a 1 kW linear RF power amplifier with 60 dB gain shown in blue. Measurement was compared to peak RF gap voltage predictions from the shunt impedance shown in red. The measurement versus theory deviation is shown in black. The measurements was taken when the ALPHA lattice had momentum compaction factor of αc ∼ 0.33 and electron beam energy of Eo ∼ 23 MeV. . . . . . . . . . . . . . . . 166 6.44 Spectrum analyzer sweeping one port of a beam position monitor in the storage ring. The center frequency was set to the 2nd harmonic of the revolution frequency at 29.886 MHz. The sidebands of the central peak manifest the synchrotron oscillation of the beam centroid within the RF bucket generated by the RF cavity. . . . . . . . . . . . . . . . 169 6.45 Peak RF gap voltage deduced from beam position monitor data of synchrotron side-bands as a function of DAC control parameter. Measurement deduced from left side-band is shown in blue and from the right side-band shown in red. The measurements were taken when the ALPHA lattice had momentum compaction factor of αc ∼ 0.2 and electron beam energy of Eo ∼ 23 MeV. . . . . . . . . . . . . . . . . . 170

LIST OF FIGURES

xxix

6.46 Mean peak RF gap voltage deduced from beam position monitor data of synchrotron side-bands as a function of DAC control parameter. Measurement was compared to the peak RF gap voltage obtained from a lightly coupled 1 pF capacitor are the accelerating gap. Deviation of gap voltage deduced from synchrotron side-bands from measurement of the 1 pF probe is shown in black. The measurements were taken when the ALPHA lattice had momentum compaction factor of αc ∼ 0.2 and electron beam energy of Eo ∼ 23 MeV. . . . . . . . . . . . . . . . . . 172

7.1

(left) Observed growth of the squared bunch length σs2 as a function of time for phase modulation driven diffusion when the phase difference ∆φo between the two RF subsystems were varied. Data was obtained by digitizing beam position monitor signals at 1 ns resolution where the phase modulation frequency of fm = 1400 Hz and modulation amplitude of am = 100◦ was applied to a double RF system configured with parameters r = 0.11 and h = 9. (right) The squared bunch length σt2 obtained from numerical simulations for two different phase difference parameters ∆φo = 180◦ and 245◦ , showing agreement of measurements with the simulation results [11]. . . . . . . . . . . . . . . . . . . . . . 174

7.2

(a) The potential V (φ)/νs as a function of φ where the double RF system parameters are r = 1/2, h = 2, and ∆φo = 30◦ . (b) The phase space torus of motion corresponding to total energies E1 = −0.1, E2 = 0, E3 = 0.1, E4 = 0.2, and E5 = 0.3. . . . . . . . . . . . . . . . 178

xxx 7.3

LIST OF FIGURES Potential V (φ)/νs as a function of phase φ where the double RF system parameters are r = 1/2 and h = 2. Labels are in the vicinity of the boxes denoting each respective potential minimum where the phase difference parameters are (a) ∆φo = −60◦ , (b) −30◦ , (c) 0◦ , (d) 30◦ , and (e) 60◦ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

7.4

Potential minimum offset φo as a function of the phase difference parameter ∆φo when r = 1/2 and h = 2. The boxes denotes location of the potential minimum with (a) offset φo = −27.122◦ for ∆φo = −60◦ , (b) φo = −30◦ for ∆φo = −30◦ , (c) φo = 0◦ for ∆φo = 0◦ , (d) φo = 30◦ for ∆φo = 30◦ , and (e) φo = 27.122◦ for ∆φo = 60◦ . Note the insensitivity of φo as ∆φo is varied in the vicinity of ∆φo = 30◦ . . . . . . . . 183

7.5

Normalized synchrotron tune Qs /νs and relevant harmonics as a function of torus extrema φ1 and φ2 . The νm /νs = 2 horizontal dotted-line intercepts depict the parametric resonances driven by phase modulation where the double RF parameters are r = 1/2, h = 2, and ∆φo = 30◦ .185

7.6

Normalized synchrotron tune Qs /νs and relevant harmonics as a function of the action J. The νm /νs = 2 horizontal dotted-line intercepts depict the parametric resonances driven by phase modulation where the double RF parameters are r = 1/2, h = 2, and ∆φo = 30◦ . The vertical dotted-line where J = 0.092 depict the initial location of the bunch at the origin of phase space. . . . . . . . . . . . . . . . . . . . 186

7.7

Magnitude of the resonance strength |gn (J)| as a function of the action J for n : 1 parametric resonances where n = 1, 2, 3, 4. The double RF parameters are chosen to be r = 1/2 and h = 2 where |gn (J)| associated with ∆φo = 30◦ are depicted with solid-lines and ∆φo = 45◦ with dashed-lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

LIST OF FIGURES 7.8

xxxi

Poincaré surface of section depicting the non-diffusive longitudinal phase space structure with parameters am = 8◦ and ∆φo = 30◦ . Strong resonance islands observed near the origins of phase space confine particles and inhibits diffusion. The existence of well-behaved tori bounding the entire phase space structure is observed. . . . . . . . . . . . . . . 193

7.9

The evolution of the small bunch of 1 × 104 particles with initial rms length σφ = 1 × 10−3 rad and momentum spread σδ = 1 × 10−3 as it undergoes the diffusion process where r = 1/2, h = 2, am = 58◦ , and ∆φo = 30◦ . Particles are observed to be smoothly streaming in the chaotic region about the resonance islands rather than confined to any particular chains of islands associated with strong resonances. . . . . 195

7.10 Poincaré surface of section depicting the diffusion of the Gaussian bunch of σφ = 1×10−3 rad and σδ = 1×10−3 initially at the phase space origin after 1.5 million revolutions where am = 58◦ and ∆φo = 30◦ . The red points depict the single-particle tracking of test particles placed initially near the edge to show that the structure is indeed bounded. . . 196 7.11 Square of the longitudinal bunch sizes σφ and σδ as a function of revolution number for am = 58◦ and ∆φo = 30◦ as the diffusion process evolves linearly with time and equilibrates after about 1.5 × 105 revolutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 7.12 The rms bunch spreads σφ and σδ as a function of modulation amplitude am where ∆φo = 30◦ . The hatched region depict the approximate regions where no diffusion was observed. Beyond am ∼ 115◦ results in particle loss. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

xxxii

LIST OF FIGURES

7.13 Poincaré surface of section depicting the non-diffusive longitudinal phase space structure on the verge of particle loss where am = 104◦ and ∆φo = 30◦ . Note that instead of the well-behaved tori, the structure is bounded by layers of overlapping resonances that will drive particle loss. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 7.14 (color) Parameter map of the rms bunch length as a function of the modulation amplitude and modulation fraction where ∆φo = 30◦ . The dark blue regions represent parameters where the particles are confined and no diffusion occurs. The dark red regions represent parameters where the diffusion is unbounded and particles were lost in the process. 204 7.15 (color) Parameter map of the rms normalized momentum spread as a function of the modulation amplitude and modulation fraction where ∆φo = 30◦ . The dark blue regions represent parameters where the particles are confined and no diffusion occurs. The dark red regions represent parameters where the diffusion is unbounded and particles were lost in the process. . . . . . . . . . . . . . . . . . . . . . . . . . 206 7.16 Normalized synchrotron tune Qs /νs and its multiples as a function of the action J. The νm /νs = 2 horizontal dotted-line interception illustrates the parametric resonances driven by phase modulation where the double RF parameters are r = 1/2, h = 2, and ∆φo = 45◦ . The vertical dotted-line where J = 0.106 depict the initial location of the bunch at the origin of phase space and the circular intercept points depict the two relevant 8 : 3 resonances that are required to be disrupted to enable a larger chaotic region to be filled. . . . . . . . . . . . . . . 208 7.17 Poincaré surface of section depicting the non-diffusive longitudinal phase space structure with parameters am = 7◦ , νm /νs = 2, and ∆φo = 45◦ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

LIST OF FIGURES

xxxiii

7.18 The rms bunch spreads σφ and σδ as a function of modulation amplitude am where ∆φo = 45◦ . The hatched region depict the approximate regions where no diffusion was observed. The solid band depict the approximate region where beam loss was encountered. Data points where the error is much less than 1% have negligible error bars. . . . 211 7.19 Poincaré surface of section depicting the diffusion of a small bunch where am = 23◦ and ∆φo = 45◦ just as the bunch sizes sharply increase for the second time. Note the islands of the 8 : 3 resonance at the edge of the diffused structure just before collapse and subsequent shrinking. 212 7.20 (Color) Black: Poincaré surface of section of a small Gaussian bunch of rms length σφ = 0.001 rad initially at the phase space center at the last modulation periods of 1.5 million revolutions where am = 28◦ and ∆φo = 45◦ . The red points depict single-particle tracking of test particles placed initially near the edge of the phase space structure. . 214 7.21 Histogram of 30 bins spanning the entire length comparing the linear particle distribution of the two different double RF settings after diffusion. The black line depict the turn-by-turn averaged of the individual bins 1 × 106 revolutions after diffusion, confirming the stability of the structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 7.22 Longitudinal bunch sizes as a function of revolution number for am = 28◦ and ∆φo = 45◦ as the diffusion process evolves and equilibrate after about 6.5 × 104 revolutions. . . . . . . . . . . . . . . . . . . . . 216 7.23 (color) Parameter map of the rms bunch length as a function of the modulation amplitude and modulation fraction where ∆φo = 45◦ . The dark blue regions represents parameters where the particles confined and no diffusion occurs. The dark red regions represents parameters where the diffusion is unbounded and particles are loss. . . . . . . . . 217

7.24 (color) Parameter map of the rms normalized momentum spread as a function of the modulation amplitude and modulation fraction where ∆φo = 45◦ . The dark blue regions represents parameters where the particles confined and no diffusion occurs. The dark red regions represents parameters where the diffusion is unbounded and particles are loss. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 7.25 Poincaré surface of sections depicting the diffusive longitudinal phase space structures for varying modulation phase ηm where am = 28◦ , ∆φo = 30◦ , and νm = 1.6νs . The phase space structure is seen to rotate about the potential well minimum. . . . . . . . . . . . . . . . . 220 7.26 Poincaré surface of sections depicting the phase space structures for varying modulation phase η where am = 58◦ , ∆φo = 30◦ , and νm = 2.0νs . When η = 60◦ , the initial particle bunch at the origin of phase space is confined within the large 1 : 1 resonance island and diffusion was inhibited. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 âc 7.27 The product the dispersion function and peak magnetic field Dx (so )B as a function of the gradient damping wiggler bending radius ρw and injection beam energy. Two different phase space beam dilution parameters, ∆φo = 30◦ am = 58◦ and ∆φo = 45◦ am = 28◦ , are depicted. 227

xxxiv

Chapter 1 Introduction

The application of particle accelerators has become ubiquitous in our modern age. As particle beam technology matures, advancements and new techniques have helped scale down the size and cost of these complex and prohibitively expensive scientific machines. The development of compact particle accelerators has provided new scientific tools with wide-ranging capabilities to smaller research institutions such as university laboratories and medical facilities. These compact accelerators have been used to produce and manipulate particle beams for material science and radiation effect experiments. They have also been used for the production of x-rays critical for x-ray crystallography and spectroscopy experiments. As a laboratory for beam and accelerator physics, it has been utilized to explore the physics of low energy particle beams including space charge effects, and higher-order particle beam dynamics. The research advances made while operating compact machines have been used to enhance beam performance at larger particle accelerator facilities and provide guidance for the design of the next generation of particle accelerators that are to come.

1

2

1. Introduction The design and operation of a compact storage ring requires special hardware con-

siderations to address the smaller circumference ring. Requirements of experiments will further drive the specification of hardware implemented. A compact 15 MHz ferrite-loaded quarter-wave RF cavity has been designed and built for the Advanced Electron-Photon Facility (ALPHA) at Indiana University. The ALPHA ring is a 50– 100 MeV, four-folded symmetric 20 m electron storage ring that has the capability of momentum compaction factor αc variability using a pair of gradient damping wiggler magnets. The compact RF resonant structure built will provide a single RF bucket for longitudinal focusing and replenishment of the energy loss via synchrotron radiation of an electron bunch stored in the ring. Due to large-angle coulomb scattering of electrons in the stored bunch, momentum exchanged from transverse motion to longitudinal motion occurs. If these single event momentum exchanges are large enough, the closed-orbit of the two interacting particles will be displaced enough such that the particle are loss in the aperture limits of the ring. Since the ALPHA storage ring is Touschek limited, beam dynamics of large-angle coulomb scattering will used to specify the performance specifications of the RF cavity. The radiation effects experiments will require an electron bunch, that have undergone radiation damping, to be broadened up to 40 ns before extraction where further transverse beam manipulation in the extraction beam line will be applied. The broadening of the electron bunch will enhance the duty factor for experiments that utilizes direct electron impingement. Broadening also provides for the reduction of the peak current and increase in the average current of the stored electron bunch. This will allow for the accumulation of large total bunch charge while circumventing the adverse effects of space-charge in high charge density bunches. Due to the size of the compact storage ring, conventional methods for bunch broadening cannot be implemented and/or cannot fulfill the longitudinal beam profile requirements of the

Introduction

3

radiation effects experiments1 . Some of these method include: RF barrier bucket manipulation, equilibration of longitudinal particle distribution in a harmonic cavity system, and applied band-limited phase noise for beam excitation. Motivated by the experiments at the Indiana University Cooler Storage Ring in 1997, a method of phase space beam dilution was developed. By applying the appropriate phase modulation to the RF system, parametric resonances are excited. When many of these parametric resonances are strongly driven, they will eventually overlap and are disrupted to form large regions of chaotic particle motion in phase space. These region of chaos will be useful for bunch broadening if a suitably large chaotic region is generated. Chaotic regions that are bounded by well-behaved tori will help prevent particle loss during the rapid diffusion. The initially small radiation damped particle bunch placed in an optimal location that will be allowed to diffuse into the large region of chaos. The generation of large bounded region of chaotic particle motion is optimized by changing the characteristics of the applied phase modulation. To ensure that the initial particle bunch is located in a region that promotes rapid diffusion, the relative phase difference between the two RF cavities in the double RF system can be varied. Optimal parameters for the phase modulated double RF system have been found to maximize the process of phase space beam dilution where bunch broadening and longitudinal particle distribution uniformity can be obtained. This dissertation will be structured as follows. Fundamental concepts in accelerator physics and beams such as the Frenet-Serret coordinate system, betatron motion, and synchrotron motion will be reviewed in Chapters 1–3. Concepts in transmission line and resonant circuit theory will be reviewed in Chapter 4. The physics of 1

See Chapter 5 for the ALPHA Project requirements.

4

1. Introduction

electron storage rings and details of the ALPHA project as a case example will be given in Chapter 5. The design, construction, and characterization measurements of the 15 MHz ferrite-loaded quarter-wave coaxial RF resonant cavity will be presented in Chapter 6. The method of phase space beam dilution and discussion of its implementation will be presented in Chapter 7. The dissertation will conclude in Chapter 8.

1.1

Accelerator Coordinate System

The motion of particles in a particle accelerator are governed largely by electromagnetic fields that are utilized to guide the particle trajectory in a well-defined orbit known as the reference trajectory (or design trajectory). Particles typically execute motions that slightly deviate from this reference trajectory that is continuous and differentiable in three-dimensional Euclidean space. Thus a coordinate system referenced to this design trajectory has been adopted to describe particle motion in a particle accelerator and is classified into a general class of coordinate systems known as the curvilinear coordinate system. The curvilinear coordinate system in three-dimensional Euclidean space is often referred to as the Frenet-Serret coordinate system. As depicted in Fig. 1.1, let ro (s) be the reference trajectory, where s represent the arc length in which particles move along the curve. Here the bold text will represent vectors and this convention will be adopted for the entire dissertation. The three unit vectors that form the orthonormal basis for the right-handed Frenet-Serret curvilinear

1.1 Accelerator Coordinate System

5

zˆ r xˆ sˆ ro

Reference Trajectory

Figure 1.1: Frenet-Serret Curvilinear Coordinate System used in the field of accelerator physics. This coordinate system will be adopted for use throughout the entire dissertation.

6

1. Introduction

coordinate system are sˆ(s) =

dro (s) , ds

xˆ(s) = −ρ(s)

dˆ s(s) , ds

zˆ(s) = xˆ(s) × sˆ(s),

(1.1)

where ρ(s) is the radius of curvature at every given arc length s. The Frenet-Serret formulas in the transverse direction are given as xˆ0 (s) =

1 sˆ(s) + τ (s)ˆ z (s), ρ(s)

zˆ0 (s) = −τ (s)ˆ x(s)

(1.2)

where the prime, 0, denotes differentiation with respect to s, and τ (s) is the torsion of the curve. We will restrict our discussion to particle motion in the planar geometry, i.e., where τ (s) = 0. The slightly deviated particle trajectory around the reference orbit can be expressed as r(s) = ro (s) + xˆ x(s) + z zˆ(s).

(1.3)

Chapter 2 Betatron Motion of Particles

The motion of particles in the transverse plane, that is orthogonal to the direction of beam propagation, is commonly referred to as betatron motion. The name is attributed to the observation of electrons oscillating about a fixed ideal reference orbit within a time-varying magnetic field induction accelerator called the betatron. The first successful operation of the betatron was demonstrated by Donald W. Kerst in 1940 at the University of Illinois where 2.2 MeV electrons were produced. The mathematical formalism of particle motion within particle accelerators have been well-established in the last few decades after the betatron [1, 2, 3, 4, 5, 6].

2.1

Electromagnetic Fields in Accelerators

In general, particle accelerators can be divided into two categories, linear accelerators and circular accelerators. Particles are guided through the design trajectories in both 7

8

2. Betatron Motion of Particles

types of machines by means of electromagnetic fields given by the Lorentz force law F = e [E + v × B]

(2.1)

where E and B are the electromagnetic fields, e is the charge of the particles, and v is the particle velocity. The basic task of guiding, focusing, and higher-order augmentation of beam trajectories is often accomplished by using magnetic fields as opposed to electric fields due to current technological limitations. This fact could be intuitively understood by considering highly relativistic particles, e.g. electrons, where v → c traversing a static magnetic element with a readily obtainable field of 1 T. To achieve the comparable effect using a static electric fields instead, one would need an electric field of about 3 × 108 V m−1 which well beyond the capabilities of current technology1 . Thus conventional beam optics used in modern accelerators comprises mostly of magnetic elements. This distinction will become less well-defined for non-relativistic particles as the required electric field scales directly with the particle velocity. In the accelerator coordinate system, the particle velocity is assumed to be parallel to sˆ so that only the transverse magnetic fields in xˆ–ˆ z plane will be considered. For particle trajectories confined to a circular path in the sˆ–ˆ x plane under the influence of the applied magnetic field, the relationship known as the magnetic rigidity for a charged particle is given as ρB [T m] =

p A = 3.336 × × p¯ [GeV c−1 u−1 ] , e Z

(2.2)

where ρ is the radius of curvature (bending radius), B is the magnetic field orthogonal to the sˆ–ˆ x plane, A is the atomic mass number, eZ in the total charge of the particle, and p¯ is the momentum of the particle in units of GeV c−1 per atomic mass unit. Since the transverse dimensions of the beam is assumed to be small compared to 1

Due to electrical breakdown (also known as dielectric breakdown).

2.1 Electromagnetic Fields in Accelerators

9

the bending radius, also known as the paraxial approximation where x, z ρ, R, the magnetic field may be expanded about the design trajectory to obtain Bz = Bz0 + Bx =

∂Bz 1 ∂ 2 Bz 2 1 ∂ 3 Bz 3 x+ x + x + ..., ∂x 2! ∂x2 3! ∂x3

1 ∂ 2 Bx 2 1 ∂ 3 Bx 3 ∂Bx z+ z + z + ..., ∂z 2! ∂z 2 3! ∂z 3

(2.3a) (2.3b)

where particles are assumed to be deflected only in the sˆ–ˆ x plane. Considering only non-skewed magnetic terms, when Eq. (2.3a) is combined with the magnetic rigidity in Eq.(2.2), the multipole expansion of the effective magnetic field can be found to be

N

1 X Kn n 1 1 1 Bz = + x = + K 1 x + K 2 x2 + K 3 x3 + . . . , ρB ρ n=1 n! ρ 2 6

(2.4)

where the nth order effective field coefficient is given as Kn =

1 ∂ n Bz . ρB ∂xn

(2.5)

Here the 1/ρ term corresponds to the dipole field, K1 corresponds to the quadrupole field, K2 corresponds to the sextupole field, K3 corresponds to the octupole field, etc. A similar line of thought can be use to expand the magnetic fields in the zˆ transverse direction. A more generalized approach with derivations directly from Maxwell’s equations can be used to find the magnetic vector potential (alternatively, the scalar potential) in the longitudinal direction to ascertain all relevant transverse magnetic fields utilized in particle accelerators. The longitudinal magnetic vector potential is summarized in a power series as " As = Bo Re where i =

√

∞ X bn + ian n=0

n+1

# (x + iz)n+1 ,

(2.6)

−1, and the operator Re[. . .] represents the real part of the complex ar-

gument. Since Bz =

∂As , ∂x

Bx =

∂As , ∂z

and b0 = 1 was chosen as the normalization such

10


that Bo is the dipole strength, the resulting effective multipole field that influences particles in an accelerator becomes ∞

1X 1 (Bz + iBx ) = ∓ (bn + ian )(x + iz)n , ρB ρ n=0 with the multipole coefficients given as 1 ∂ n Bz bn = , Bo n! ∂xn x=z=0

1 ∂ n Bx an = , Bo n! ∂z n x=z=0

(2.7)

(2.8)

where b0 corresponds to the dipole, a0 to the dipole roll, b1 to the quadrupole, a1 to the skew quadrupole, b2 to the sextupole, a2 to the skew sextupole, b3 to the octupole, a3 to the skew octupole, and etc. The ∓ sign convention in Eq. (2.7) is used to distinguish positive (with a − sign) and negative (with a + sign) particle charge given the accelerator coordinate system defined in the previous chapter. The general effects of each conventional optical element in the accelerator plane (ˆ s–ˆ x plane) is given in Table 2.1.

Optical Elements Dipole Quadrupole Sextupole Octupole

Multipole Coefficients e 1 = B0 ρ p e ∂Bz K1 = p ∂x e ∂ 2 Bz K2 = p ∂ 2x e ∂ 3 Bz K3 = p ∂ 3x

Description of Effects Bending and Steering Focusing and defocusing Chromaticity corrections Higher-order field corrections

Table 2.1: Multipoles fields most used in particle accelerators with their respective functional descriptions.

2.2 Transverse Equations of Motion

2.2

11

Transverse Equations of Motion

The Hamiltonian for particle motion in the Frenet-Serret coordinate system can be expressed as (

(ps − eAs )2 + (px − eAx )2 + (pz − eAz )2 H = eΦ + c m2 c2 (1 + x/ρ)

)1/2 ,

(2.9)

where the conjugate momentum in the Cartesian coordinate system, P, can be transformed into the Frenet-Serret coordinate system by making use of the generating function of the third kind, G = F3 (p, Q, t), that depends only on the old generalized momenta and the new generalized coordinates to give x ∂F3 ∂F3 = 1+ = P · xˆ, P · sˆ, px = − ps = − ∂s ρ ∂x

pz = −

∂F3 = P · zˆ. ∂z

(2.10)

The vector potential in the Cartesian coordinate system, A, can be transformed in a similar fashion to give x As = 1 + A · sˆ, ρ

Ax = A · xˆ,

Az = A · zˆ.

(2.11)

In the field of accelerator physics and particle beam dynamics, it is often more convenient to express the independent variable in terms of the particle displacement s rather than time t. Using the relation dH = (∂H/∂px ) dpx + (∂H/∂ps ) dps = 0, the evolution of the generalized coordinates with respect to an independent variable is −1 dx dx ds ∂ (−ps ) 0 = = , etc., (2.12) x = ds dt dt ∂px where the prime (0 ) denotes differentiation with respect to s. A new Hamiltonian ˜ = −ps can be expressed as where H #1/2 " 2 (H − eΦ) ˜ =− 1+ x H − m2 c2 − (px − eAx )2 − (pz − eAz )2 − eAs , (2.13) 2 ρ c

12


where the phase space coordinates are (x, px , z, pz , t, −H). From the Hamilton expressed in Eq. (2.13) and the magnetic fields given in the previous section, the linearized transverse equations of particle motion that give rise to betatron motion is given as 2 x ρ−x Bz po 1+ , x − =± ρ2 ρB p ρ 2 Bx po x 00 z =∓ 1+ , ρB p ρ 00

(2.14a) (2.14b)

where p = po + ∆p is the momentum of a given particle, po is the momentum of the reference particle, and e is the charge of the particle. In Eq.(2.14), higher-order terms are neglected and the upper signs and lower signs corresponds to positively and negatively charged particles, respectively. Consider the on-momentum particle where p = po and keeping the magnetic field expansion up to first order, Eq. (2.3) become ∂Bz x = ∓B0 + B1 x, ∂x ∂Bz z = B1 z, Bx = ∂x

Bz = ∓B0 +

(2.15a) (2.15b)

where the quasi-stationary Ampere’s Law in a charge free region (∇ × B = 0) gives B1 =

∂Bz ∂x

=

∂Bx ∂z

and B1 is the quadrupole field gradient function evaluated in the

closed orbit. The betatron equations of motion in Eq. (2.14) now becomes x00 + Kx (s)x = 0,

z 00 + Kz (s)z = 0,

(2.16)

with Kx (s) = where K1 (s) =

B1 (s) ρB

1 ∓ K1 (s), ρ2

Kz (s) = ±K1 (s)

(2.17)

is the effective focusing function and the upper and lower signs

corresponds to positively and negatively charged particles. If the functions Kx (s) and

2.3 Transfer Matrices

13

Kz (s) are periodic over a length L, i.e., Kx (s + L) = Kx (s) and Kz (s + L) = Kz (s), Eq. (2.16) can be referred to as Hill’s equations with solutions completely described by Floquet Theory that will be revisited later. When the field index (also known as the focusing index) is defined as n(s) = ±ρ2 K1 (s),

(2.18)

The equations of motion in Eq. (2.16) can now be written as x00 +

1 (1 − n) x = 0, ρ2

z 00 +

n z = 0, ρ2

(2.19)

and the condition for simultaneous focusing of both transverse motions is 0 < n < 1.

(2.20)

This is known as the weak focusing condition that allows for stable particle motion in both transverse dimensions. Letting y and y 0 represent either the horizontal, where y = x and y 0 = x0 , or the vertical, where y = z and y 0 = z 0 , phase space coordinates, the generalized Hill’s equation is given as y 00 + Ky (s)y = 0,

(2.21)

where Ky (s) is the focusing function for a given phase space coordinate.

2.3

Transfer Matrices

Since conventional magnet elements in an accelerators are often designed to have the desirable uniform or near-uniform magnetic fields, the focusing function Ky (s) is

14


assumed to be piece-wise constant. To briefly introduce the transfer matrix formalism, the solution to Eq. (2.21) is found for a finite Ky (s) to be   1/2   a cos K s + b , K > 0, y    y(s) = as + b, K = 0,      a cosh Ky1/2 s + b , K < 0,

(2.22)

where the integration constants a and b are determined by initial conditions. For a given initial state vector at position so 

y(so )



, y(so ) =  0 y (so )

(2.23)

the betatron transfer matrix M (s|so ) operates on the initial-state vector to generate the final-state vector at some later position s  y(s) = 

y(s) y 0 (s)

  = M(s|so )y(so ).

(2.24)

Straight sections or segments of accelerators where no magnetic field is applied is often referred to as drift space where Ky = 0, and the transfer matrix through these elements is given as  Mdrif t (s|so ) = 

1 ` 0 1

 ,

(2.25)

where ` = s − so is the length of the element. For segments of the accelerator subjected to quadrupole fields with constant fo-


15

cusing function Ky , the transfer matrix through these elements is given as    1/2 −1/2 1/2  Ky sin Ky `     cos Ky `  Ky > 0,  ,   1/2 1/2 1/2   −Ky sin Ky ` cos Ky `    Mquad (s|so ) =        1/2 −1/2 1/2  |Ky | sinh |Ky | `   cosh |Ky | `      , Ky < 0,   1/2  |Ky |1/2 sinh |Ky |1/2 ` cosh |Ky | ` (2.26) where Ky > 0 refers to the focusing quadrupole and Ky < 0 refers to the defocusing quadrupole. In the thin-lens approximation when ` → 0, the transfer matrix given in Eq. (2.26) reduces to 

 1 0  for focusing quadrupoles, Mfocus =  −1/f 1   1 0  for defocusing quadrupoles, Mdefocus =  1/f 1

(2.27a)

(2.27b)

where the focal length f is given by f = lim |K1y `| . `→0

The individual betatron transfer matrix can be concatenated to produce an overall transfer matrix through many elements such that M(s|so ) = M(s|sn ) . . . M(s3 |s2 )M(s2 |s1 )M(s1 |so ).

2.3.1

(2.28)

Stability of Betatron Motion

Suppose we have a 2 × 2 transfer matrix of a given unit cell expressed as   a b , M= c d

(2.29)

16


where a, b, c, and d are constants resulted in the multiplication of many smaller matrix elements that form the cell. From Eq. (2.29), the eigenvalue scan be obtained from the characteristic equation λ2 − (a + d)λ + 1 = 0,

(2.30)

where det(M) = 1 was used and is true in general for all transfer matrices in linear beam optics. It follows that the eigenvalues, the roots of the characteristic equation in Eq. (2.30), can be written as 1 λ1 = tr (M) + 2 and 1 λ2 = tr (M) − 2

s

s

2 1 tr(M) − 1 2 1 tr(M) 2

(2.31)

2 −1

(2.32)

where tr(M) is the trace of transport matrix M. Suppose we let tr(M) = 2 cos Φ, where Φ is the phase advance of a periodic cell. When tr(M) ≤ 2, Φ is a real number. When tr(M) > 2, Φ is a complex number. The eigenvalues can be expressed in terms of the phase advance to be λ1 = eiΦ ,

λ2 = e−iΦ .

(2.33)

The initial beam coordinates (y0 , y00 ) can be expressed as a linear superposition of eigenvectors given as   y  0  = Av1 + Bv2 , y00

(2.34)

where v1 and v2 are eigenvectors associated with eigenvalues λ1 and λ2 , respectively. If the transport matrix M is periodic for every revolution, after the mth revolution the particle coordinates becomes     ym y m   = Mm  0  = Aλm 1 v1 + Bλ2 v2 . 0 ym y00

(2.35)


17

The stability of particle motion through a periodic cell that makes up the accelerator m ring requires that the eigenvalues λm 1 and λ2 not grow as a function of revolution

number m. Thus a necessary condition for the stability of betatron motion requires that the betatron phase advance Φ be a real number, or alternatively, the requirement can be expressed as |tr(M)| ≤ 2.

(2.36)

When the condition in Eq. (2.36) is met, uncontrolled blowup of the transverse emittance due to the linear optics will not occur.

2.3.2

Courant-Snyder Parametrization

For a stable orbit where Eq. (2.36) is met, the matrix M in Eq. (2.29) can be parameterized. Let the diagonal element of matrix M be written as a = cos Φ + k1 ,

d = cos Φ − k1 ,

(2.37)

where k1 is a real number. Using the fact that det(M) = ad − bc = 1 and ad = cos2 Φ + k12 , the expression bc = −k12 − sin2 Φ

(2.38)

can be found. Since |cos Φ| < 1 implies that sin Φ 6= 0, the constant k1 can be expressed as k1 = α sin Φ so that bc = − 1 + α2 sin2 Φ.

(2.39)

By expressing the anti-diagonal elements as b = β sin Φ,

c = −γ sin Φ,

(2.40)

18


parameters α, β, and γ must be related by

γ=

1 + α2 , β

or βγ = α2 = 1.

(2.41)

Thus the most general form for the transport matrix M can be written as  M=

cos Φ + α sin Φ −γ sin Φ

β sin Φ



 = I cos Φ + J sin Φ = eJΦ , cos Φ − α sin Φ

(2.42)

where β, α, and γ are called Courant-Snyder parameters or Twiss parameters, Φ is the phase advance, I is the unit matrix, and 



α β  J= −γ −α

(2.43)

with tr(J) = 0 and J2 = −I. The sign ambiguity of the phase advance Φ can be resolved by requiring that β > 0 when |tr(M)| ≤ 2. This also implies that γ > 0. The Courant-Snyder parameters β, α, and γ should not be confused with parameters related to the Lorentz factor and the momentum compaction factor. The transformation of the Courant-Snyder parameters from an initial location s1 to a final location s2 can be expressed as      2 2 β M11 −2M11 M12 M12 β           α = −M11 M21 M11 M22 + M12 M21 −M12 M22  α ,      2 2 γ M12 −2M21 M22 M22 γ 2

1

where Mij are the matrix elements of the M(s2 |s1 ) transport matrix.

(2.44)

2.4 Floquet Transformation

2.4

19

Floquet Transformation

Due to the periodicity of the focusing functions Kx,z (s) = K(s), the solution to the second-order differential equation given in Eq. (2.16) can be obtained by performing the Floquet transformation. Floquet’s theorem states that Hill’s equation has quasiperiodic solutions. The transformation is applied by assuming the solution of the form y(s) = Aw(s)eiψ(s) ,

y ∗ (s) = Bw(s)e−iψ(s) ,

(2.45)

where a is a constant, w(s) is the amplitude function, and ψ(s) is the phase function. Since both the solutions in Eq. (2.45) separately satisfy Hill’s equation, substituting the first term of Eq. (2.45) into Eq. (2.21) yields A

w00 − wψ 02 + Kw − i (2w0 ψ 0 + wψ 00 ) eiψ = 0.

(2.46)

The real and imaginary terms on the left-hand-side of Eq. (2.46) must each be equal to zero for the solution to satisfy Hill’s Equation. A simple integration of the imaginary term in Eq. (2.46) yields ψ0 =

1 , w2

(2.47)

where the constant of integration was set equal to one since it may be absorbed into w when the constant A and B in Eq. (2.45) are rescaled. The real term in Eq. (2.46) now becomes w00 − wψ 02 + Kw = 0.

(2.48)

Eq. (2.48) is known at the betatron envelope equation and Eq. (2.47) the phase equation. Since any solution to Hill’s equation is a linear superposition of the linearly independent solution, shown in Eq. (2.45), the most general transport matrix M(s2 |s1 ) transforming the particle state at point s1 to point s2 along the beam line

20


is given as  M(s2 |s1 ) =

w2 cos ψ − w2 w10 sin ψ w1  0 (1+w1 w10 w2 w20 ) w1 w20 − sin ψ − w2 − w1 cos ψ w1 w2



w1 w2 sin ψ w1 w2

cos ψ +

w1 w20

sin ψ

, (2.49)

where the amplitudes are w1 = w(s1 ) and w2 = w(s2 ), the derivative of the amplitudes with respect to s are w10 = w0 (s1 ) and w20 = w0 (s2 ), and the phases are ψ1 = ψ(s1 ) and ψ2 = ψ(s2 ) with ψ = ψ(s2 ) − ψ(s1 ). Suppose the particle traverses a length L = s2 − s1 of a periodic beam line where the focusing function K(s) satisfies the periodic boundary condition K(s) = K(s+L), the amplitude and phase function will become w10 = w20 = w0 ,

w1 = w2 = w,

ψ(s1 + L) − ψ(s1 ) = Φ.

(2.50)

Comparing the general transport matrix expressed in periodic amplitude and phase function in Eq. (2.49) to the Courant-Snyder parameterized transport matrix in Eq. (2.42), the relationships β = w2 ,

α = −ww0 = −

β0 , 2

(2.51)

are obtained. The transport matrix in Eq. (2.49) can now be expressed in terms of Courant-Snyder parameters by  M(s2 |s1 ) = 

q

β1 β2

√

(cos ψ + α1 sin ψ)

√ 1 α2 sin ψ + − 1+α β1 β2

α √1 −α2 β1 β2

cos ψ

q

β1 β2

β1 β2 sin ψ



, (cos ψ − α2 sin ψ)

(2.52)

where β1 , α1 ,and γ1 are Courant-Snyder parameters at location s1 , β2 , α2 ,and γ2 are Courant-Snyder parameters at location s2 , and the phase ψ = ψ2 − ψ1 .

2.4 Floquet Transformation

2.4.1

21

Betatron Phase and Tune

Consider a circular particle accelerator with circumference C = P L where P is the number of identical superperiods and L is the length of a single superperiod. From Eq. (2.47) and Eq. (2.51), the transverse phase advance in a superperiod can be expressed as Z Φy = 0

L

ds , β(s)

(2.53)

where β(s) is the betatron function through a single superperiod. The betatron tune, denoted as Qy or νy , is defined as the number of betatron oscillations in one revolution given as P Φy 1 Qy = νy = = 2π 2π

Z s

s+C

ds . βy (s)

(2.54)

The betatron tune is related to the angular betatron frequency by ωy(β) = 2πfy(β) = 2πνy fo , (β)

where fy

(2.55)

is the betatron frequency, and fo is the revolution frequency of the circu-

lating particle. The general solution to Hill’s equation can be expressed as q y(s) = A βy (s) cos (Φy (s) + ξy ) , where the A and ξy are constants determined from initial conditions.

(2.56)

22


Chapter 3 Synchrotron Motion of Particles

The essence of a particle accelerator, as apparent in its name, is the production of electric fields to accelerate charged particles or replenish energy lost via synchrotron radiation. Since magnetic fields act only orthogonally to the direction of particle propagation, it is not hard to see why electric fields are preferential in the art of particle acceleration. There are two basic classes of electric fields produced in accelerating structures, time-stationary electrostatic fields and time-varying electrodynamic fields. Historically, electrostatic accelerators were the first particle accelerators to be built and studied. Due to the simplistic nature of static fields, these particle accelerators are still popular today out numbering all other types of accelerators. However, electrostatic accelerating fields are limited by voltage breakdown1 and thus are better-suited for studies in lower particle energy regime. The voltage breakdown limit was circumvented through the use of time-varying 1

The practical voltage limit of electrostatic fields is about 30 MV when submerged in gases with

high dielectric constant, such as sulfur hexafluoride, allowing for operation at higher voltages

23

24

3. Synchrotron Motion of Particles

fields in the radio frequency that were popular during the post-World War II era as military technology and advancements in radar and communication systems became increasingly available to particle accelerator researchers. Although more complex in its implementation, RF acceleration ushered in the era of high energy particle accelerators. An important milestone in RF acceleration was the discovery of synchrotron phase-stability that was independently discovered by E. M. McMillan and V. Veksler in the 1940s which we will revisit in more detail later in the chapter. The study of charged particle motion interacting with the accelerating fields is known as longitudinal beam dynamics or synchrotron motion of particles. For the sake of simplicity, we will forego the 6-dimensional Hamiltonian derivation for both synchrotron and betatron motion and adopt a treatment where the Hamiltonian is based on the revolution frequency and energy of the particles. Although this choice of formalism will lack the essential link between synchrotron and betatron motion (synchro-betatron coupling), it simplifies the synchrotron equations of motion so that the physics can easily be extracted and understood. Since the energy gain or lost by the particles passing through the RF structure depends strongly on both the synchronization with the RF field and the arrival time of off-momentum particles, it will be here that we begin our exploration into the synchrotron motion of particles.

3.1

Longitudinal Equation of Motion

Suppose there exists an ideal particle that is synchronized with the RF phase φ = φs at a revolution frequency of f = fo and momentum p = po . This ideal particle is known as the synchronous particle or the reference particle. And as the synchronous particle traverses a RF cavity, it will lose or gain energy of Eo = eV sin φs per revolution.

3.1 Longitudinal Equation of Motion

25

In reality though, a particle bunch consists of many particles with momenta and longitudinal displacement deviating from that of the synchronous particle. A particle with momentum p will have its own off-momentum orbit given by xco + δD(s), where xco is the closed-orbit of the synchronous particle, D(s) is the dispersion function, and δ = (p − po )/po is the fractional momentum deviation of the particle. For a given accelerating RF structure, let the longitudinal electric field that a circulating charged particle sees at the gap be E = Eo sin (φRF (t) + φs ) ,

(3.1)

where φRF (t) = hωo t is the RF phase, ωo = βo c/Ro is the angular revolution frequency, Eo is the amplitude of the electric field, βo c and Ro are the particle speed and the mean radius of the orbiting reference particle, h is an integer number known as the harmonic number, and φs is the phase of the synchronous particle with respect to the RF wave. Suppose a synchronous particle traverses an accelerating gap of length g, the energy gain by the reference particle in each successive passage is given as +g/2β Z oc

∆E = eEo βo c

sin (hωo t + φs ) dt = eEo gT sin φs ,

(3.2)

−g/2βo c

where the transit time factor is defined to be T =

sin (hg/2Ro ) , (hg/2Ro )

(3.3)

and e is the charge of the particle being accelerated. The effective voltage that the orbiting particle experiences is V = Eo gT . The transit time arises from the fact that the particle traverses a gap of finite length (or within a finite time interval) so that the energy gain is the time-average of the electric field in the gap for each successive passage. Note that as the gap length approaches zero, the effective voltage maximizes.

26


In applications however, the high fields associated with short accelerating gap lengths may cause electric field breakdown and damaging electrical arcing at the gap. Since the reference particle is synchronized with the RF wave, with angular frequency of ωRF = hωo and angular revolution frequency of ωo = βo c/Ro , it encounters the same RF voltage every revolution given the same synchronous phase φs . Thus the acceleration rate for the synchronous particle is E˙ o = efo V sin φs ,

(3.4)

where the dot denotes a time-derivative. Now lets consider a non-synchronous (off-momentum) particle with small deviation in parameters from the synchronous particle, ω = ωo + ∆ω,

φ = φs + ∆φ,

p = po + ∆p,

E = Eo + ∆E,

θ = θs + ∆θ,

where ωo , φs , θs , po , and Eo are the angular revolution frequency, RF phase, azimuthal orbital angle, momentum, and energy of the synchronous particle. Terms denoted with ∆ represents small deviations from their respective parameters. The relation between the phase coordinates and the orbital angle is given as ∆ω =

d 1 d 1 dφ ∆θ = − ∆φ = − , dt h dt h dt

(3.5)

where ∆φ = φ − φs = −h∆θ and the minus sign was chosen as a convention. The first equation of synchrotron motion as the time evolution of the phase-angle variable φ can be expressed as φ˙ = −h (ω − ωo ) = −h∆ω.

(3.6)


27

The scaling relationship of an off-momentum particle with respect to the synchronous particle can be shown to be βRo ∆ω = − 1, ωo βo R

(3.7)

where the mean radius of a circular accelerator is expanded in δ to be R = Ro 1 + α0 δ + α1 δ 2 + α2 δ 3 + α3 δ 4 + . . . .

(3.8)

The momentum compaction factor is defined to be αc =

1 dR 1 = α0 + 2α1 δ + 3α2 δ 2 + 3α3 δ 3 + . . . ≡ 2 , Ro dδ γT

(3.9)

where γT (or more precisely γT mc2 ) is the transition energy. Most accelerators are designed to have lattices with γT ∈ R where α0 > 0. In these machines, higher energy particles (δ > 0) have longer closed-orbit path lengths compared to that of the reference particle. There exist lattices that are designed to achieve the quasiisochronous condition where αc = 0, that is, when the closed-orbit path length to first-order is independent of the particle momentum. Recently, accelerators have been designed to have γT ∈ Z where α0 < 0. In these machines, higher energy particles have shorter closed-orbit path lengths. The fractional angular revolution frequency deviation can be obtained by combining Eq. (3.7) and Eq. (3.9) to give ∆ω = −η(δ)δ. = − η0 + η1 δ + η2 δ 2 + . . . δ, ωo where the ηn are the nth-order phase slip factors given as 1 η0 = α0 − 2 , γo η1 =

3βo2 + α1 − α0 η 0 , 2γo2

(3.10)

(3.11a) (3.11b)

28


η2 = −

βo2 (5βo2 − 1) 3βo2 α0 α1 2 η − + α − 2α α + + α , ... 2 0 1 0 0 2γo2 γo2 2γo2

(3.11c)

In the linear approximation, the angular revolution frequency deviation can be expressed as ∆ω = −η0 ωo δ =

1 1 − 2 2 γo γT

ωo δ.

(3.12)

In accelerators with γT ∈ R and operating below transition energy (γo < γT ), higher energy particles (δ > 0) have higher revolution frequencies and longer closed-orbit path lengths. This counter-intuitive statement can be understood by considering that the speed of the higher energy particles compensates fully for the longer path lengths. On the other hand, accelerators operating above transition energy (γo > γT ) will have higher energy particles with lower revolution frequencies and longer closed-orbit path lengths. The particles in these machines can be said to exhibit characteristics of having “negative mass.” When Eq. (3.6) and Eq. (3.12) are combined, the first equation of synchrotron motion can be expressed as, φ˙ = hωo η0 δ,

(3.13)

that is, the time evolution of the phase-angle variable as a function of the fractional momentum deviation. A second equation of motion describes the time evolution of the conjugate energy difference or fractional momentum deviation variable (depending the of choice of canonical coordinates) and is required to complete the description of particles executing synchrotron motion. The energy gained by the non-synchronous particle per revolution is eV sin φ and the rate at which the energy is gained is ω E˙ = eV sin φ. 2π

(3.14)


29

By using the relation, d ∆E 1 ˙ 1 ˙ 1 ˙ ∆ω 1 ˙ ∆(1/ωo ) ˙ ˙ + ... = , (3.15) E − Eo = ∆E − E 2 ≈ ∆E + E ω ωo ωo ωo ωo ∆E dt ωo the second equation of synchrotron motion in terms of the energy-difference between the synchronous and non-synchronous particles becomes d ∆E 1 = eV (sin φ − sin φs ) . dt ωo 2π

(3.16)

Using the definition of fractional momentum deviation variable written in terms of fractional energy deviation, δ=

∆p ωo ∆E = 2 , po β E ωo

(3.17)

taking its time-derivative, and substituting in Eq. (3.16), we obtain the second equation of synchrotron motion in terms of the fractional momentum variable, δ˙ =

ωo eV (sin φ − sin φs ) . 2πβ 2 E

(3.18)

In summary, the pair of expressions that form the synchrotron equations of motion were derived to be hωo2 η ∆E ˙ , φ= 2 β E ωo ˙ ∆E 1 = eV (sin φ − sin φs ) , ωo 2π

(3.19a) (3.19b)

for the phase-space coordinates (φ, ∆E/ωo ) and φ˙ = hωo ηδ,

(3.20a)

ωeV δ˙ = (sin φ − sin φs ) , 2πβ 2 E

(3.20b)

for the phase-space coordinates (φ, δ).

30


3.2

Hamiltonian of Synchrotron Motion

The synchrotron equations of motion seen in Eq. (3.19) and Eq. (3.20) can be derived from their respective Hamiltonian given by 1 hηωo2 H= 2 β 2E

∆E ωo

2 +

eV [cos φ − cos φs + (φ − φs ) sin φs ] , 2π

(3.21)

and 1 ωo eV H = hωo ηδ 2 + [cos φ − cos φs + (φ − φs ) sin φs ] , 2 2πβ 2 E

(3.22)

where the independent variable is time t for phase-space coordinates (φ, ∆E/ωo ) and (φ, δ), respectively. Although these Hamiltonians are valid and have been used as the basis of many tracking routines where the synchrotron motion is well-decoupled from the betatron motion, the equations are in a form that is incompatible with the Hamiltonian for betatron motion. In order to fully articulate the sychro-betatron coupling, a fully consistent treatment2 will be required but the theoretical framework is well beyond the scope of this dissertation.

3.2.1

Synchrotron Frequency and Tune

Utilizing either of the Hamiltonian given in Eq. (3.21) or Eq. (3.22), the linearized equation of motion expanded to first-order about φ = φs can be derived as d2 hωo2 eV η0 cos φs (φ − φ ) = (φ − φs ) . s dt2 2πβ 2 E 2

(3.23)

S. Y. Lee, et al., Effects of Tune Modulation on Particles Trapped in One-Dimensional Resonance

Islands, Phys. Rev. E 49, 5706 (1994).

3.3 Phase-Stability of Synchrotron Motion

31

Apparent from the second-order differential equation in Eq. (3.23), the angular synchrotron frequency and synchrotron tune can be written as s heV |η0 cos φs | , ωs = ωo 2πβ 2 E s heV |η0 cos φs | ωs Qs = = . ωo 2πβ 2 E

(3.24)

(3.25)

The absolute value of the η0 cos φs term ensures that the synchrotron frequency and tune are always real. The absolute value will be lifted in the next section to show that regions of phase stability will depend on the energy of the particle γo and the momentum compaction factor αc . In practice, the typical synchrotron tunes measured in proton synchrotrons are of the order of ∼ 10−3 and in electron storage rings are of the order of ∼ 10−1 .

3.3

Phase-Stability of Synchrotron Motion

The general solution to the linearized equation of motion with respect to φ shown in Eq. (3.23) can be written as ∆φ(t) = φ(t) − φs = Aeωs t ,

(3.26)

where the constant A depends on initial conditions. The synchrotron frequency is rewritten where the absolute value has been lifted s s heV cos φs 1 heV cos φs 1 1 ωs = ωo αc − 2 = ωo − 2 , 2πβ 2 E γo 2πβ 2 E γT2 γo

(3.27)

where the zero-th order phase slip factor η has been defined in Eq. (3.11a) and the momentum compaction factor is related to the transition energy γT by αc ≡

1 . γT2

(3.28)

32


For the purpose of demonstrating phase stability, the effects of damping are assumed to be negligibly small and can be ignored. When the argument of the square root in Eq. (3.27) is negative overall, the synchrotron frequency will be an imaginary number. From Eq. (3.26), an imaginary synchrotron frequency argument of the exponent will ensure stable oscillatory particle motion in the longitudinal direction. Closer inspection of Eq. (3.27) reveals that there are in fact two regions in which phase stability can be attained. For particle accelerators where the beam energy γo is less than the transition energy γT , i.e., the resulting phase slip factor η < 0, the cos φs must be positive to maintain phase stability. This would mean that the synchrotron phase φs must be configured in the range −π/2 ≤ φs ≤ π/2 for the argument of the square root in Eq. (3.27) to remain negative. For particle accelerators where the beam energy γo is greater than the transition energy γT , i.e., the resulting phase slip factor η > 0, the cos φs must be negative to maintain phase stability. This would mean that the synchrotron phase φs must be configured in the range π/2 ≤ φs ≤ 3π/2 for the argument of the square root in Eq. (3.27) to remain negative. This argument is further refined by taking into account the synchronization of particle with the sinusoidal varying field of the accelerating RF cavity. For circular particle accelerators operating with beam energy below transition energy (γo < γT where η < 0), Eq. (3.12) shows that more energetic particles (δ > 0) will revolve with larger revolution frequency (∆ω/ωo > 0) with respect to the synchronous particle and will arrive at the RF cavity before the synchronous particle. Eq. 3.12 can alternatively be expressed in the change in the revolution period ∆τ = τ − τo with respect to the synchronous period τo given as ∆τ = ηδ = τo

1 1 − 2 2 γT γo

∆p . po

(3.29)

Eq. 3.12 and Eq. 3.29 different only by a sign due to the inverse relationship of


33

revolution frequency to the revolution period. More energetic particles (δ > 0) will have a shorter revolution period (∆τ /τo < 0) compared to the synchronous particle when η < 0. Suppose the RF cavity have a time-varying field of the form V (t) = Vpeak sin (ωRF t + φs ) ,

(3.30)

where Vpeak is the amplitude, and ωRF is the angular frequency of the time-varying field. To provide a restoring force so that synchrotron oscillations are generated about the synchronous particle, the accelerating RF voltage must have a positive slope. This way, higher energy particles that arrive at the RF cavity early will receive a smaller longitudinal kick than lower energy particles that arrive after the synchronous particle. This is depicted in Fig. 3.1a where a higher energy particle that initially arrives first at the RF accelerating cavity (blue) receives a smaller longitudinal kick compared to the later lower energy particles (red). As a result, the blue particle will lag behind by ∆τ and arrive after the synchronous particle after one synchrotron period τo . The synchronous phase φs must then be in the range 0 ≤ φs ≤ π/2 to ensure phase stability is maintained while still replenishing energy that is loss via synchrotron radiation. For circular particle accelerators operating with beam energy above the transition energy (γo > γT where η > 0), Eq. (3.12) shows that more energetic particles (δ > 0) will revolve with smaller revolution frequency (∆ω/ωo < 0) and larger revolution period (∆τ /τo > 0) with respect to the synchronous particle and will arrive at the RF cavity after the synchronous particle. To provide a restoring force so that synchrotron oscillations are generated about the synchronous particle, the accelerating RF voltage must have a negative slope. This way, higher energy particles that arrive at the RF cavity later will receive a smaller longitudinal kick than lower energy particles that arrived before the synchronous particle. This is depicted in Fig. 3.1b where a lower

34


τo − ∆τ τo τo + ∆τ

η0 (b.) φ = ωRF t + φs

Figure 3.1: Graphical illustration of the principle of phase stability for circular particle accelerators operating with beam energies (a.) below transition energy (γo < γT ) where η < 0, and (b.) above transition energy (γo > γT ) where η > 0. The initially higher energy particle is shown in blue, initially lower energy particle is shown in red, and synchronous particle is shown in black.


Lattice Configuration

Phase Requirement

γo < γT , η < 0

0 ≤ φs ≤ π/2

γo > γT , η > 0

π/2 ≤ φs ≤ π

35

Table 3.1: Summary of the synchrotron phase requirements for phase stability in circular particle accelerators operating with beam energy γo below and above transition energy γT .

energy particle that initially arrives first at the RF accelerating cavity (red) receives a larger longitudinal kick compared to the later higher energy particles (blue). As a result, the red particle will lag behind by ∆τ and arrive after the synchronous particle after one synchrotron period τo . The requirement that the synchronous phase φs must be in the range π/2 ≤ φs ≤ π to ensure phase stability is maintained while still replenishing energy that is lost via synchrotron radiation. Table 3.1 summarizes the criteria for phase stability in circular particle accelerators operating below and above transition energy. Transition energy crossing is an important aspect of operating circular particle accelerators. For injected beams with energies below the transition energy, typical for proton synchrotrons, severe beam loss is encountered for beams accelerated near the transition energy. This could conceptually be seen in Eq. (3.27). When γo → γT , the synchrotron frequency vanishes and thus the phase focusing effect also vanishes. To circumvent beam loss when crossing the transition energy, careful management of the synchrotron phase φs must designed into the RF control electronics. When the beam energy approaches the transition energy, a phase jump must be performed from φs to π − φs in order to maintain stable oscillations about the synchronous particle.

36


Chapter 4 Transmission Line Theory

The transmission of electrical impulses over extended distances was first demonstrated in 1729 by the English scientist Stephan Gray using a single-wire transmission line made of dampened hemp twine suspended by silk threads. Although the discovery was quite significant, it did not lend itself to be of practical use until the 1830s where it was used in the application of telegraph communications. The mathematical understanding of transmission lines flourished in the years to follow building on the works of James C. Maxwell, Oliver Heaviside, and William Thomson or better known as Lord Kelvin. Transmission line theory can be seen as a link to bridge the connection between electromagnetic field analysis and electrical circuit theory. The main difference between transmission line theory and conventional circuit theory are ranges of electrical wavelengths considered as compared to the physical dimension of the network. Transmission line theory is often utilized when the electrical wavelength in radio frequencies are comparable to physical dimension of the networks being analyzed. Thus it is im37

38

4. Transmission Line Theory

perative that much care and consideration be taken in the fabrication of components operating in the radio frequencies regime to prevent interference and damage. Since the quarter-wave resonant RF structure to be built for the ALPHA storage ring can be modeled using transmission line theory, it will be important to introduce its key concepts so that it may be extended to include ferrite loading. This chapter will serve as an overview into the field transmission line theory starting with the derivation of the telegrapher’s equations from both circuit theory and Maxwellian formalism and will expand to the consequences of these equations with transmission line terminated into a short circuit, open circuit, and an arbitrary load.

4.1

Telegrapher’s Equation

A transmission line is often represented as a lumped-element circuit as shown in Fig. 4.1 where R, L, G, and C are the series resistance per unit length with units of Ω/m, series inductance per unit length with units of H/m, shunt conductance per unit length with units of S/m, and shunt capacitance per unit length with units of F/m. The series resistance R represents the resistance due to the finite conductivity of the conductor, the series inductance L represents the total self-inductance of the two conductors, the shunt conductance G is due to the dielectric loss in the material between the conductors, and the shunt capacitance C is due to the close proximity of the two conductors. R and G represents losses in the transmission line, this idea will be revisited in a later section. An actual transmission line of finite length is then modeled as a cascade of lumped-elements circuits. Starting from the circuit in Fig. 4.1, applying Kirchhoff’s voltage and current laws

4.1 Telegrapher’s Equation

39 I(z + ∆z, t)

I(z, t) R∆z

L∆z

+

+

V (z, t)

G∆z

C∆z

V (z + ∆z, t)

−

− ∆z

Figure 4.1: A schematic lumped-element circuit model of an ideal transmission line of length ∆z at an arbitrary time t.

will yield a pair of equation given as ∂I(z, t) + V (z + ∆z, t), ∂t

(4.1a)

∂V (z + ∆z) + I(z + ∆z, t). ∂t

(4.1b)

V (z, t) = R∆zI(z, t) + L∆z I(z, t) = G∆zV (z + ∆z, t) + C∆z

Taking the limit as ∆z → 0, we arrive at a pair of differential equations lim

V (z + ∆z, t) − V (z, t) ∂V (z, t) ∂I(z, t) = = −RI(z, t) − L , ∆z→0 ∆z ∂z ∂t

(4.2a)

I(z + ∆z, t) − I(z, t) ∂I(z, t) ∂V (z, t) = = −GV (z, t) − C . ∆z→0 ∆z ∂z ∂t

(4.2b)

lim

Assuming harmonic time dependency, where V (z, t) = V (z)eiωt and I(z, t) = I(z)eiωt , the decoupling the spatially-dependent term from the time-dependent term can be accomplished. The time-dependent term will be suppressed for notational simplicity. The differential equation becomes ∂V (z, t) = − (R + iωL) I(z, t) ∂z

⇒

dV (z) = − (R + iωL) I(z), dz

(4.3a)

∂I(z, t) = − (G + iωC) V (z, t) ∂z

⇒

dI(z) = − (G + iωC) V (z). dz

(4.3b)

40


The resulting pair of linear differential equations describing the voltage and current on an ideal transmission line as a function of displacement and time is known as the telegrapher’s equations, developed by Oliver Heaviside in the 1880s. Heaviside’s model for the transmission line included uniformly distributed inductance in a telegraph line that can be, in practice, added discretely to diminish attenuation and correct for distortions of the source signal. Unfortunately, the importance of his results were not immediately recognized until later. Now substituting Eq. (4.3a) and Eq. (4.3b) into one another, a pair of second-order differential equations can be obtained and simultaneously solved to arrive at the solution for V (z) and I(z). d2 V (z) − γ 2 V (z) = 0, dz 2

(4.4a)

d2 I(z) − γ 2 I(z) = 0, dz 2

(4.4b)

where γ = α + iβ =

p

(R + iωL) (G + iωC),

(4.5)

is the complex wave number as a function of frequency. The solution for Eq. (4.4a) and Eq. (4.4b) can be found as V (z) = Vo+ e−γz + Vo− e+γz ,

(4.6a)

I(z) = Io+ e−γz + Io− e+γz ,

(4.6b)

where e−γz represents wave propagation in the positive z-direction and e+γz represents wave propagation in the negative z-direction. Substituting Eq. (4.6a) into Eq. (4.3a) gives I(z) =

+ −γz γ − Vo− e+γz , Vo e R + iωL

(4.7)

4.2 Lossless Transmission Line

41

and by inspection, the characteristic impedance can be defined as r R + iωL R + iωL V+ −V − Zo = = = o+ = −o . γ G + iωC Io Io

(4.8)

V (z) and I(z) can now be written in terms of the characteristic impedance as V (z) = Vo+ e−γz + Vo− e+γz ,

(4.9a)

Vo+ −γz Vo− +γz e − e . Zo Zo

(4.9b)

I(z) =

4.2

Lossless Transmission Line

It is a good place to pause and introduce the assumption of the lossless transmission line. It can be seen from Eq. (4.5) and Eq. (4.8) that the wave number and the characteristic impedance are complex values that are frequency dependent. In the case where the losses in the line are negligible, the simplification R = G = 0 can be applied to give

√ γ = α + iβ = iω LC,

(4.10)

where α = 0, √ β = ω LC.

We can see that the term α that was responsible for the attenuation of the waves in a lossy transmission line has been set to zero to reflect the conditions of the lossless line. The characteristic impedance reduces to a real number given as r L Zo = . C

(4.11)

42

4. Transmission Line Theory IL = I(0) + V (z), I(z), Zo

VL = V (0)

ZL

− z = −`

z=0

Figure 4.2: Transmission line terminated into an arbitrary load of impedance ZL with the source displaced z = −` distance away.

4.2.1

Terminated Lossless Transmission Line

Suppose a transmission line is terminated into an arbitrary load ZL with the source at an arbitrary distance away as shown in FIG. 4.2. It is of great interest to find the expression for voltage and current at an arbitrary distance from a terminating load. Starting with Eq. (4.9a) and Eq. (4.9b), the voltage and current at the load (at z = 0) is related to the load impedance Zo , ZL =

VL V (0) V + + Vo− = = o+ Zo . IL I(0) Vo − Vo−

(4.12)

The voltage reflection coefficient Γ, can now be defined as Γ=

ZL − Zo Vo− = , + Vo ZL + Zo

(4.13)

which is the amplitude of the reflected voltage wave normalized to the amplitude of the incident voltage wave. Now our expression for the total voltage and current on a line can be written as V (z) = Vo+ e−iβz + Γe+iβz ,

(4.14a)


I(z) =

43 Vo+ −iβz e − Γe+iβz . Zo

(4.14b)

It is now convenient to define the voltage standing wave ratio as a measure of the mismatch of a line given as VSWR =

1 + |Γ| 1 − |Γ|

(4.15)

where the VSWR is a real number such that 1 ≤ VSWR ≤ ∞. A VSWR = 1 implies a matched load, whereas VSWR = ∞ implies a completely mismatched load. The impedance looking into a transmission lines are position-dependent as the both the voltage and current in the line are also position-dependent as seen in Eq. (4.9a) and Eq. (4.9b). At a distance z = −` from the load, the input impedance looking into a line toward an arbitrary load is V (−`) 1 + Γe−2iβ` Zin = = Zo . I(−`) 1 − Γe−2iβ`

(4.16)

Using Eq. (4.13), a more useful expression for the input impedance can be obtained, (ZL + Zo ) e+iβ` + (ZL − Zo ) e−iβ` = Zo Zin = Zo (ZL + Zo ) e+iβ` − (ZL − Zo ) e−iβ`

ZL + iZo tan β` Zo + iZL tan β`

.

(4.17)

This important result is known as the transmission line impedance equation and special cases of this general result will be considered in the next sections.

4.2.2

Lossless Transmission Line Terminated in a Short

In a special case of the transmission line impedance equation in Eq. (4.17), the line is terminated in a short circuit where ZL = 0 at the location of the load (z = 0). From

44

4. Transmission Line Theory IL = I(0) + V (z), I(z), Zo

VL = V (0)

ZL = 0

− z = −`

z=0

Figure 4.3: Transmission line terminated into a short circuit where the impedance ZL (z = 0) = 0 with the source displaced z = −` distance away.

Eq. (4.13), we see that a shorted transmission line gives Γ = −1, and Eq. (4.15), the VSWR is infinite. The total voltage and current on the line from Eq. (4.14a) and Eq. (4.14b) gives V (z) = Vo+ e−iβz − e+iβz = −2iVo+ sin βz,

(4.18a)

Vo+ −iβz 2V + e + e+iβz = + o cos βz. Zo Zo

(4.18b)

I(z) =

At the location of the load, the voltage is VL = V (0) = 0 and the current maximizes as expected for a short circuit. At a distance z = −` from the load, the input impedance looking into a line toward a short circuit is Zin =

4.2.3

V (−`) = iZo tan β`. I(−`)

(4.19)

Lossless Transmission Line Terminated in a Open

Now consider the transmission line impedance equation where the line is terminated in an open circuit where ZL = ∞ at the location of the load (z = 0). From Eq. (4.13),


45 IL = I(0)

+ V (z), I(z), Zo

VL = V (0)

ZL = ∞

− z = −`

z=0

Figure 4.4: Transmission line terminated into an open circuit where the impedance ZL (z = 0) = ∞ with the source displaced z = −` distance away.

we see that a shorted transmission line gives Γ = +1, and Eq. (4.15), the VSWR is infinite once again. The total voltage and current on the line from Eq. (4.14a) and Eq. (4.14b) gives V (z) = Vo+ e−iβz + e+iβz = 2Vo+ cos βz, I(z) =

−2iVo+ Vo+ −iβz sin βz. e − e+iβz = + Zo Zo

(4.20a) (4.20b)

At the location of the load, the current is IL = I(0) = 0 and the voltage maximizes as expected for a open circuit. At a distance z = −` from the load, the input impedance looking into a line toward a open circuit is Zin =

V (−`) = −iZo cot β`. I(−`)

(4.20c)

46


Chapter 5 Electron Storage Rings

Charge particles undergoing acceleration or deceleration emit electromagnetic radiation. In the late 19th century, scientists William Crookes and Johann Hittorf, working on what are now known as cathode ray tubes (CRT), discovered that photographic plates in the vicinity of the tubes became fogged or contained anomalous shadows. The efforts of Hermann Helmholtz, Philipp von Lenard, Heinrich Hertz, and many other scientists culminated in the first x-ray photograph published by Wilhelm Röntgen in 1896. This led to the discovery of Bremsstrahlung (braking) radiation as the mechanism for the x-ray production through the deceleration of a charged particle, i.e., energetic electrons from a cathode colliding with the anode in a CRT. Although naturally occurring synchrotron radiation from distant stars have been observed for some time, the confirmation of the mechanism for electromagnetic radiation produced by the acceleration of charges particles did not arrive until 1947. Observation of the Ivanenko-Pomeranchuk radiation1 were made at the 70 MeV General 1

D. D. Ivanenko, I. Ya. Pomeranchuk, Dokl. Akad. Nauk SSSR 44 (1944), p. 343

47

48

5. Electron Storage Rings

Electric Synchrotron by scientists Frank Elder, Anatole Gurewitsch, Robert Langmuir, and Herb Pollock. After the machine for which the phenomena was first confirmed, the moniker, synchrotron radiation, stuck and is used to this day. Mathematical expressions for electromagnetic radiation in circular orbits were derived by Alfred-Marie Liénard in 1898, and the modern synchrotron radiation theory used today was based on the work of Jullian Schwinger in 1946 [14]. The important results pertaining to electron storage rings are summarized below:

• The angular distribution of synchrotron radiation is a maximum in the direction of electron propagation with angular spread proportional to 1/γ, where γ is the Lorentz factor. • Synchrotron radiation is plane polarized in the orbit plane of the electron, and elliptically polarized outside this plane. • Synchrotron radiation spans a continuous spectrum with half of the total radiated power at frequencies below the critical frequency ωc = (3cγ 3 )/2ρ and the other half of radiated power at frequencies above ωc . • Quantum mechanical corrections will become important in the regime where the critical energy of the radiated photon, c = ~ωc = (3~cγ 3 )/2ρ, is comparable to the beam energy, E = γmc2 . For electron beams, this occurs around beam energies of 103 TeV, therefore quantum corrections is mostly negligible at the current operational beam energies.

The synchrotron radiation emitted from charges particles in a storage ring greatly affects particle motion. This is particularly true in the handling of electrons and positrons in circular accelerators. For protons and heavy ions, synchrotron radiation

5.1 Larmor’s Formula and Synchrotron Radiation

49

loss is small so are typically neglected. The study of radiative effects will be important in the design of an electron storage ring. In this chapter, basic radiation theory will be introduced to understand the properties of an electron storage ring. Topics such as radiation damping, beam lifetime, and equilibrium bunch sizes will be visited. A brief introduction of the design parameters of the ALPHA storage ring will be presented.

5.1

Larmor’s Formula and Synchrotron Radiation

According to Larmor’s theorem, the instantaneous power radiated by an accelerating electron of charge e is given as 1 2e2 v˙ 2 2re Pγ = = 4πo 3c3 3mc

dp dp · dt dt

,

(5.1)

where re = e2 /(4πo mc2 ) is the classical radius of the electron. More details on the derivation of Larmor’s formula is given in Appendix A. The relativistic generalization of Larmor’s formula, originally obtained by A. Liénard, is given as " 2 # 2 2re 1 dE dp Pγ = − 2 , 3mc dτ c dτ

(5.2)

where τ = γt is the proper-time. The first term in Eq. (5.2) represents acceleration perpendicular to the direction of particle propagation and the second term represents acceleration parallel to the direction of particle propagation. Since the synchrotron radiation power arising from circular motion is at least a factor of 2γ 2 larger that the power from linear motion, the assumption 2 2 1 dE dp 2 dτ c dτ

(5.3)

50


can be made. Synchrotron radiation from a linear accelerator is extremely small; e.g., a 1 TeV linear accelerator that is 10 km in length and equipped with accelerating structures of field gradient 100 MV/m will radiate total energy less than 0.04 eV. The synchrotron radiation power of electrons confined to circular motion by a constant magnetic field of B is 2re Pγ = 3mc

dp dτ

2 =

2re β 4c E 4 β 4 c3 e2 (γevB)2 = Cγ 2 = Cγ E 2 B 2 , 3mc 2π ρ 2π

(5.4)

where γevB is the confining force of the magnetic field, v = βc, ρB = p/e is the momentum rigidity, ρ is the radius of curvature, and the radiation constant [15] is defined as Cγ =

4π ro = 8.846 × 10−5 m/GeV3 , 3 (mc2 )3

(5.5)

for electrons. The total energy radiated for an electron with energy Eo in one revolution is I Urad =

β 3 Eo4 Pγ dt = Cγ 2π

Z

2πR

0

1 ds 3 4 = Cγ β Eo R , 2 ρ ρ

(5.6)

where R is the average radius of the storage ring. For a back-of-an-envelope calculation, the isomagnetic assumption can be evoked where the field is a constant in all dipole magnets. The energy loss via synchrotron radiation per revolution becomes Urad =

Cγ β 3 Eo4 , ρ

(5.7)

and the average synchrotron power radiated is

Urad cCγ β 3 Eo4 = , Pγ = To 2πRρ where the revolution period is To = (2πR)/βc.

(5.8)

5.2 Radiation Damping

5.2

51

Radiation Damping

Since the synchrotron radiation power loss is proportional to E 4 /ρ2 and an electric field provided by the RF cavity replenishes the energy in the longitudinal direction, a net longitudinal damping effect is attained. Furthermore, the electrons loses energy in a cone with angular spread of 1/γ from the direction of propagation. This provides the mechanism for transverse phase space damping.

5.2.1

Damping of Synchrotron Motion

To model the behavior of synchrotron radiation loss that is energy dependent, i.e., to incorporate Eq. (5.4), the energy requirement per revolution is expanded about the synchronous energy expressed as dU U (E) = Uo + ∆E + . . . , dE E=Eo

(5.9)

where Uo is the synchrotron enery loss from a synchronous particle, and Eo is the synchronous energy. The equation of motion can now be written as d d2 ∆E + 2αE ∆E + ωs2 ∆E = 0 2 dt dt where

s ωs =

heV |ηo cos φs | 2πβ 2 Eo

(5.10)

(5.11)

is the angular synchrotron frequency, 1 dU ωo dU αE = = 2To dE E=Eo 4π dE E=Eo

(5.12)

is the synchrotron damping coefficient, To is the revolution period, and ωo is the angular revolution frequency. Eq. (5.10) is the familiar equation of a damped oscillator.

52


Since the damping coefficient (damping rate) is typically small compared to the synchrotron frequency (synchrotron period), the assumption αE ωs can be made. The solution of the damped oscillator can now be expressed as ∆E(t) = Ao e−αE t cos (ωs t − θo )

(5.13)

where the terms Ao and θo are determined by initial conditions.

5.2.2

Damping Partition

To evaluate the damping coefficient αE , the ratio dU/dE must be determined. Consider a particle with energy deviating slightly from the synchronous energy and that this particle follows a dispersive trajectory. The path element ds0 traversing a dipole magnet of bending radius (radius of curvature) ρ along a dispersive trajectory with a displacement x can be expressed as x ds = 1 + ds ρ 0

(5.14)

where ds represents the path element along the ideal path of the synchronous particle. For an electron, the assumption ds0 /dt = c can be made. The synchrotron radiation energy loss per revolution can be expressed as I I I 1 1 x 0 Urad = Pγ dt = Pγ ds = Pγ 1 + ds. c c ρ

(5.15)

The horizontal displacement x(s) of an off-energy particle for a given lattice with dispersion function D(s) is x(s) = D(s)

∆E Eo

(5.16)

where the displacement due to betatron motion is hxβ i = 0. Eq. (5.15) becomes I 1 D ∆E Urad = Pγ 1 + ds. (5.17) c ρ Eo


53

Differentiating Eq. (5.17) with respect to energy yields the expression I 1 dUrad dPγ D dPγ ∆E Pγ = + + ds. dE c dE ρ dE Eo Eo

(5.18)

Since the energy deviation ∆E varies periodically about the synchronous energy, on average h∆E/Ei = 0. Eq. (5.18) then becomes I dUrad 1 dPγ D Pγ = + ds. dE c dE ρ Eo

(5.19)

The first term on the right hand side of Eq. (5.19), dPγ /dE, can be obtained by using Eq. (5.4) such that dPγ =2 dE

Pγ Pγ dB + Eo Bo dE

=2

Pγ Pγ D dB + Eo Bo Eo dx

(5.20)

where Bo is the dipole field strength. The change in synchrotron radiation energy loss with respect to an off-energy particle in Eq. (5.19) can be expressed as I 1 Pγ Pγ D dB D Pγ dUrad = + 2 + ds dE c Eo Bo Eo dx ρ Eo I Uo 2 dB 1 1 = DPγ + 2+ ds Eo cUo ρ Bo dx

(5.21)

The synchrotron damping coefficient in Eq. (5.12) can now be written as αE =

ωo Uo Uo (2 + D) = (2 + D) , 2To Eo 4πEo

where D is the damping repartition number defined as I 1 1 2 dB D= DPγ + ds. cUo ρ Bo dx

(5.22)

(5.23)

The damping repartition number can alternatively be expressed in terms of synchrotron radiation integrals [15, 16] given as I 1 1 D= D(s) + 2Kx (s) ds I2 ρ2 (s)

(5.24)

54


where the second synchrotron radiation integral is defined as I 1 ds, I2 = 2 ρ (s)

(5.25)

the quadrupole gradient function is defined as Kx (s) =

1 dB(s) . ρBo dx

(5.26)

The damping repartition number is dependent only on the lattice configuration of a given storage ring.

5.2.3

Damping of Betatron Motion

The emission of synchrotron radiation from particles in circular motion provides a damping mechanism for longitudinal motion. Since the radiation is emitted in a light cone with angular spread of 1/γ about the direction of propagation, betatron motion (transverse particle motion) is also dampened. Consider the general solution to Hill’s Equation in the vertical plane and its propagation derivative given as p z βz (s) cos (ψz (s) + ξz ) , r z z 0 (s) = − [αz (s) cos (ψz (s) + ξz ) + sin (ψz (s) + ξz )] . βz (z) z(s) =

(5.27) (5.28)

where βz (s) is the vertical beta function, αz (s) = −βz0 (s)/2 is the correlation function, z is the emittance, ψz (s) is the phase advance, and ξz is a constant phase factor. To simplify the derivation, the assumption that the betatron function is relatively constant is used, i.e., αz (s) = 0, and that there is no dispersion in the vertical plane. Eq. (5.27) and Eq. (5.28) can be written as z = A cos φ z0 = −

A sin φ βz

(5.29) (5.30)

5.2 Radiation Damping where A =

55

p z βz (s) is the betatron oscillation amplitude and φ = ψz (s) + ξz is the

betatron phase. It follows that 2

A2 = A2 cos2 φ + A2 sin2 φ = z 2 + (βz z 0 ) .

(5.31)

As the electron executes betatron motion about the design trajectory, it loses energy via synchrotron radiation and regains energy from the accelerating RF cavity. The photons that are emitted are in the direction of propagation, therefore there is a slight decrease δp in momentum due to the “recoil” of photon emission, but no changes to the angle z 0 = p⊥ /pk . As the electron traverses the RF cavity, it gains energy in the longitudinal direction but not in the transverse. The resulting angle after the RF cavity can be written as [15, 19] p⊥ p⊥ ≈ z + δz = pk + δp pk 0

0

δp 1− p

=z

0

δp 1− p

,

(5.32)

and the change in angle resulting from synchrotron radiation loss and RF cavity interaction immediately follows, δz 0 = −z 0

δE . Eo

(5.33)

The change in the betatron oscillation amplitude in Eq. (5.31) resulting from the change in angle is h i 2 δ A2 = 2AδA = δ z 2 + δ (βz z 0 ) = 2βz2 z 0 δz 0 ,

(5.34)

where the emission of photons do not change the position of the electrons, i.e., δ(z 02 ) = 0, and using Eq. (5.33), AδA = −βz2 z 02

δE . Eo

(5.35)

Since the motion of the electron is oscillatory, the time-averaged squared angle, hz 02 i using Eq. (5.30) is given as A2 hz i = 2πβz2 02

Z 0

2π

sin2 φ dφ =

A2 . 2βz2

(5.36)

56


Eq. (5.35) now becomes the vertical fractional betatron amplitude found to be Uo ∆A =− . A 2Eo

(5.37)

The amplitude A thus decreases, i.e., the vertical betatron oscillation are damped. The vertical damping constant can then be defined as αz =

Uo , 2To Eo

(5.38)

where the time-evolution of the vertical betatron amplitude is ∆A(t) = Ao e−αz t ,

(5.39)

and Ao is determined by initial conditions. Synchrotron radiation damping of the horizontal betatron motion is a bit more involved as it includes the effect of dispersion in the bending plane. Consider the horizontal displacement and its derivative from the design trajectory x = xβ + xD ,

(5.40)

x0 = x0β + x0D .

(5.41)

The displacement due to betatron motion is given by xβ = A cos φ, x0β = − where A =

A sin φ, βx

(5.42) (5.43)

p x βx (s) is the horizontal betatron oscillation amplitude and φ = ψx (s)+

ξx is the phase advance. The displacement and its derivative due to an off-momentum particle in a dispersive region is given as ∆E , Eo ∆E x0D = Dx0 (s) , Eo xD = Dx (s)

(5.44) (5.45)


57

where Dx (s) is the dispersion function in the horizontal plane and ∆E is the offenergy particle (off-momentum) particle. The energy of the electron is altered by an amount u due to photon emission. As with the case in the vertical plane, the emission of synchrotron radiation will not affect the displacement or angle of the propagating particle leading to u , Eo u δx0β = −δx0D = −Dx0 . Eo

(5.46)

δxβ = −δxD = −Dx

(5.47)

From Eq.5.42 and Eq. (5.43), the following relationship can be obtained A2 = x2β + βx x0β

2

(5.48)

.

The change in betatron amplitude due to the emission of synchrotron radiation becomes AδA = xβ δxβ + βx2 x0β δx0β = − Dx xβ βx2 Dx0 x0β

u . Eo

(5.49)

The radiation energy loss u of an electron in an element length ds0 of the dispersive path is given as Pγ (xβ ) Pγ (xβ ) 0 ds = − u=− c c

xβ 1+ ρ

ds

(5.50)

where Eq. (5.14) was used. To express the radiated power loss as a function of xβ , Eq. (5.4) is expanded as a function of the betatron displacement such that Pγ(xβ ) = Po +

2Po dB xβ Bo dx

(5.51)

where Po is the radiation power loss of the design trajectory. Combining Eq. (5.49), Eq. (5.50), and Eq. (5.51) 2 dB x β Po AδA = xβ Dx 1 + xβ + . Bo dx ρ cEo

(5.52)

58


Since the off-momentum closed orbit, hxβ i = hx0β i = 0 over the betatron phase, and hx2β i = A2 /2 over the betatron phase, the fractional betatron amplitude can be expressed as ∆A Uo = A 2Eo

I

Dx (s)

1 1 Uo + 2Kx (s) ds =D . 2 ρ (s) I2 2Eo

(5.53)

Eq. (5.53) implies that that there is a increase in horizontal betatron amplitude due to synchrotron radiation. The emission of photons excites the betatron motion in a beam. If the phase space damping effect of the RF cavity is included, where the derivation is similar to the vertical betatron motion damping, the horizontal fractional betatron amplitude can be expressed as AδA = − (1 − D)

Uo = −αx To 2Eo

(5.54)

where the horizontal damping coefficient is αx =

Uo , 2To Eo

(5.55)

and the time-evolution of the horizontal betatron amplitude is A(t) = Ao e−αx t

5.2.4

(5.56)

Summary of Radiation Damping Parameters

The radiation damping coefficients in all orthogonal coordinates for a beam in circular motion is given as αx = Jx

hPγ i , 2Eo

αz = Jz

hPγ i , 2Eo

αE = JE

hPγ i , 2Eo

(5.57)

where hPγ i = Uo /To is the average synchrotron radiation power, and the damping partition numbers are defined as Jx = 1 − D,

Jz = 1,

JE = 2 + D.

(5.58)

5.3 Case Example: ALPHA Storage Ring

59

An important theorem stating that the sum of three damping partition numbers is an invariant quantity, called the Robinson Theorem [17, 18], follows Jx + Jz + JE = 4.

(5.59)

In circular accelerators with planar geometry, Jx and JE can vary by augmenting D with the Robinson’s Theorem still strictly adhered. The damping time constant is defined as the time it takes for the initial amplitude to decay to 1/e of it’s original value. The corresponding damping time constants are as follow 2Eo 4πRρ 2Eo = = To , Jx hPγ i cCγ Jx Eo3 Jx Uo 4πRρ 2Eo 2Eo = = To , τz = 3 Jz hPγ i cCγ Jz Eo Jz Uo 2Eo 4πRρ 2Eo τE = = = To , 3 JE hPγ i cCγ JE Eo JE Uo τx =

(5.60) (5.61) (5.62)

where Eo is the beam energy, R is the average radius of the storage ring, ρ is the radius of curvature, Cγ is the radiation constant, and To is the revolution period.

5.3

Case Example: ALPHA Storage Ring

The Advanced Electron Photon Facility (ALPHA) is a joint collaboration between the Indiana University Center for Exploration of Energy and Matter (IU CEEM) and the Crane Naval Surface Warfare Center (NSWC Crane) [20, 22]. The facility was designed for the manipulation of an electron bunch for radiation effects experiments on sensitive avionics and electronics in radiation-rich environments. The ALPHA electron storage ring has a circumference of 20 m, designed with a four-fold symmetry [21]. Electrons from a 50-100 MeV linear accelerator (linac) are transported by the injection line into the Lambertson septum for full energy injection. The compact

60


ring was designed with betatron tunes νx = 1.75, νz = 0.75. A pair of gradient damping wigglers [24, 25, 26] enables variability of the momentum compaction factor αc and the damping repartition number D making the ring an ideal laboratory to study beam dynamics near the transition energy (i.e., isochronous condition). The ALPHA storage ring has two modes of operation: single-revolution debunching of the electron beam from the linac, and multi-revolution accumulation of charge with bunch broadening. Electron beams produced in the linac will inherently manifest characteristic features that are remnant of their production and acceleration. The linac for ALPHA operates in the common S-band (2.856 GHz) frequency from a klystron that provides RF power to the accelerating sections. Electron bunches from the linac will have the characteristic S-band micro-structure that will interfere with the device under test (DUT) upon beam delivery. To suppress this effect, the ALPHA storage ring was designed to rotate the micro-structures in phase space such that the resulting bunch upon exiting the storage will be a direct current (DC) beam with ripples within experimental tolerances. This mode will be used to supply beams to experiments requiring lower currents and higher repetition rates. The multi-revolution mode is used to accumulate charge for radiation experiments that require higher beam currents. An electron bunch in the storage that is initially broadened to about 40 ns will be injected with more electrons from the linac for charge accumulation for a total charge bunch of up to 600 nC. The stacking of multiple injection is accomplished by utilizing a pair of closed-orbit bumper magnets with decaying supply current applied at each subsequent stacking cycle. These bumper magnets provides a localized closed-orbit distortion at the insertion point for beam injection. Once the desired total bunch charge is attained, the long bunch is extracted from the storage ring using a 100 kV traveling-wave fast-extraction kicker with a rise


Figure 5.1: Diagram depicting the ALPHA project in the commissioning stages. (From left to right) An electron bunch is generated and accelerated to full energy. The injection line transports the electron bunch to the storage ring accounting for all lattice matching considerations. The bunch is injected into the storage ring through the Lambertson septum. Depending on mode of operation, extraction is accomplished with a magnetic bumper or a travelingwave fast-extraction kicker. The manipulated bunch is transported through the extraction line to the device under test.

61

62


Bumper Magnet2 Bumper Magnet1

Lambertson Septum Fast-Extraction Kicker

Damping Wiggler2 RF Cavity

Damping Wiggler1

Figure 5.2: Detailed diagram of the ALPHA storage ring with important components labeled in their respective locations in the compact storage ring. The red dashed-line represent the direction of beam propagation where the injection line is to the right and the extraction line is to the left.

time of 10 ns or less. If the bunch broadening is obtained by the phase space dilution method presented in Chapter 7, the resulting electron bunch will be broadened up to 40 ns with longitudinal particle distribution uniformity. Fig. 5.2 shows a more detailed diagram of the ALPHA storage rings with all the key accelerator component labeled at their respective location in the storage ring. Four dipole magnets [30] from the Cooler Injector Synchrotron [27, 28, 29], built in 1996, were re-purposed for use in the ALPHA storage ring. The dipole magnet has an effective length of 2 m and a vertical gap of 5.82 cm. A radius of curvature of ρ = 1.273 m is required to keep the electron beam is circulation. This ρ corresponds


63

to dipole magnetic fields of Bo = 0.131 T, 0.197 T, and 0.262 T for electron beam energies of Eo = 50 MeV, 75 MeV, and 100 MeV, respectively. The magnetic bumpers are used to introduce a localized close-orbit distortion around the region of the Lambertson septum. This will shift the closed-orbit of the circulation electrons into the field region of the Lambertson septum for injection and extraction in single-revolution mode. The fields in the magnetic bumper are allowed to exponentially decay for the four-turn injection scheme in the multi-revolution accumulation mode. The RF system in the electron storage ring serve two primary objectives: to replenish energy loss by the electrons due to synchrotron radiation, and provide a RF bucket for longitudinal particle focusing. The 15 MHz operating frequency of the RF cavity that is comparable to the revolution frequency will enable a single bucket for an electron bunch to be stored in the ALPHA storage ring. This RF operating frequency was chosen to be able to maximally broaden the bunch for the radiation effects experiments. A proposed second RF cavity is currently being designed to upgrade the current RF system to a double RF system. The double RF system used in conjunction with RF phase modulation will allow for the execution of the phase space dilution technique for particle bunch broadening. Each of the gradient damping wiggler magnet [24, 25, 26] sets (within the pair) was constructed from three combined function C-type magnets. The middle C-type magnet is 0.2 m in length with a magnet gap of 4 cm. The outer two C-type magnets are 0.1 m in length with a magnet gap of 3.587 cm. The ratio of the quadrupole field to the dipole field is B1 /Bo ≈ 1.9 m−1 , where B1 = dB/dx. By changing the supply current for the gradient damping wiggler magnet pair, the effective bending radius ρw associated with the damping wiggler magnet is altered. Fig. 5.3 shows the relationship between the relevant lattice parameters (e.g., betatron tunes, momentum compaction

64


factor, and horizontal damping partition number) as a function of gradient damping wiggler bending radius ρw . The horizontal damping partition number Jx , shown as the green line in Fig, 5.3, increases with decreasing ρw (increase in damping wiggler magnet supply current). In Section 5.2.2, Eq. (5.56), Eq. (5.57), Eq. (5.58) implies that in the condition where Jx < 0, uncontrolled growth of the horizontal betatron amplitude occurs. This would mean that the ALPHA storage ring must operate with gradient damping wiggler configured so that the condition ρw . 2 m is maintained to ensure proper radiation damping of the horizontal betatron amplitude. The momentum compaction factor αc decreases with decreasing ρw . The phase slip factor 1 ηo = αc − 2 γ

(5.63)

vanishes when αc approaches zero for highly relativistic beams of energy proportional to γ. For the ALPHA lattice, this occurs when ρw ∼ 0.75 m. This special lattice configuration is known as the isochronous condition. Throughout the entire tuning range of the ρw parameter, both the horizontal betatron tune νx and vertical betatron tune νz will remain relatively constant. The damping times of the oscillatory amplitudes in the three orthogonal coordinates are show in Fig 5.4 as function of the ρw . The damping times were calculated for beam energy Eo = 25 MeV and the focusing gradient of the damping wiggler is B1 /B0 ≈ 1.9 m−1 . The rate of radiation damping is enhanced for the horizontal and vertical amplitudes as ρw decreases as implied by the decreasing damping time. Operation with higher energy beam (Eo > 25 MeV) further enhances the radiation damping time. The lattice functions such as the horizontal and vertical betatron functions and


2.5

65

2.5

2

νx

1.5

2

1.5 Jx

1

1 νz

0.5

0.5 αc

0

0

-0.5

-1 0.5

-0.5

0.7

1.0 2.0 3.0 4.0 5.0 Damping Wiggler Bending Radius ρw (m)

-1

Figure 5.3: The momentum compaction factor αc (black), horizontal damping partition number Jx (green), vertical betatron tune νz (red), and horizontal betatron tune νx (blue) as a function of the gradient damping wiggler bending radius ρw . The horizontal dotted-line represent the region near the isochronous condition where αc ∼ 0. When Jx < 0, uncontrolled growth of the horizontal betatron amplitude is encountered.

66


1000

Damping Times τx , τz , and τE (s)

τx

τz 100

τE

10

0.5

1 Damping Wiggler Bending Radius ρw (m)

1.5

Figure 5.4: Damping time of the energy spread amplitude τE (black), damping time of the horizontal betatron amplitude τx (blue), and damping time of the vertical betatron amplitude τz (red) as a function of the gradient damping wiggler bending radius ρw . The damping times have been calculated for Eo = 25 MeV electron beam and the focusing gradient of the damping wiggler is B1 /B0 ≈ 1.9 m−1 .


67 β x,z (m), D x (m)

7.

6.

5.

4.

3.

2.

1.

0.0

-1. 0.0

βx

2.

βz

4.

Dx

6. 8. 10. 12. 14. 16. 18. 20. s (m)

Figure 5.5: The horizontal betatron function, vertical betatron function, and horizontal dispersion function as a function of the lattice displacement parameter s where the gradient damping wiggler bending radius was set at ρw = 0.75 m in the vicinity of the isochronous condition. The results were obtained from the MAD-X lattice simulation software suite.

68


the horizontal dispersion function are shown in Fig. 5.5 as a function of the lattice displacement parameter s where ρw = 0.75 m and the chicane that is located at 10 m was turned off. These lattice functions were obtained from the MAD-X (Methodical Accelerator Design) program developed by CERN [31, 32].

Chapter 6 Ferrite-loaded Quarter-wave Coaxial Resonator Cavity

Long before the development and wide spread use of application-tailored ferrite materials, there existed only accounts of a strange substance that has captured the human imagination. In his work, The Natural History, Pliny the Elder (23–79 AD) recounts the legend of a shepherd named Magnes, who upon taking his herd to pasture, discovered magnets when the nails of his shoe and iron ferrule of his staff adhered to the ground. For the shepherd’s namesake, the substance became known as magnets, or perhaps due to the fact that materials exhibiting ferromagnetism were commonly found in an ambiguous city named Magnesia. No one really knows. Though their origin is still debated, magnets and the nature of magnetism have evoked heated arguments among philosophers on the physical and metaphysical mechanisms of the natural world with influences that can been seen to this day. One early instance in written history that mentions naturally occurring ferromagnets came from the writ-

69

70

6. Ferrite-loaded Quarter-wave Coaxial Resonator Cavity

ings of the Greek philosopher Thales of Miletus [33], circa 600 BC, describing the peculiar attractive properties of lodestones. Another early observation of lodestones in Chinese literature came from the scholar Guan Zhong (725–654 BC) who describes the attractive nature of the stones as “loving stones” [34]. Ancient sailors are thought to have suspend lodestones in a bowl of water to be used for maritime navigation. Although ancient and enigmatic its history, ferrite materials has become ubiquitous in the modern technological age with applications in electronics, telecommunication, and computational technology. Ferrite materials are classified into two major categories: soft ferrites and hard ferrites. Soft ferrite materials have low coercivity (the ability of material to change magnetization without significant energy dissipation), low remanence after magnetization, and typically high resistivity to prevent eddy currents in the core. In contrast, hard ferrite materials have high coercivity and high remanence after magnetization. Common household permanent ferrite magnets and lodestones are typically under the classification of hard ferrites. In particle accelerators, the application of specialized ferrite materials is typically found implemented in high power RF accelerating structures. These RF structures are called ferrite-loaded or ferrite-dominated resonant cavities. Acceleration RF resonant cavities used in particle accelerators functions to provide RF buckets for particle bunches (longitudinal focusing) and serves to replenish energy loss due to synchrotron radiation. The implementation of ferrite-loaded cavities become very important in synchrotrons with low revolution frequencies (∼10–25 MHz) where lower harmonic numbers are desired, in beam transport line where space limitations demands for a compact structure, or when rapid frequency ramping are required during the acceleration of charged particles. The introduction of ferrite materials into the resonating medium of a RF cavity allows for more compact structure designs to operate at lower

6.1 Ferrite-Loaded Transmission Line Theory

71

resonant frequencies and the current biasing of the ferrite material will enable another mechanism to tune the resonant structure. In this chapter, transmission line theory will be extended to model the behavior of ferrite-loaded quarter-wave axial resonant cavities. A discussion of the electrical properties, such as the permittivity and complex permeability, of the ferrite material to be loaded into the RF resonant structure will be presented. The final cavity design guided by the extended transmission line theory, numerical electromagnetic field simulations, and the physics of electron storage rings will be presented for the ALPHA storage ring. The summary of cavity construction will be given. Microwave instrument characterization measurements as well as measurements with electron beam will presented to verify the operation of the 15 MHz RF ferrite-loaded quarterwave cavity installed in the ALPHA storage ring. The design concepts presented here can be applied to future ferrite-loaded cavities used in other compact circular accelerators such as storage rings and synchrotrons. The ferrite-loaded resonant cavity that has been constructed constitute the most compact structure in current literature that is designed to operate at low RF frequencies [51].

6.1

Ferrite-Loaded Transmission Line Theory

For a given quarter-wave structure with an accelerating gap with line capacitance Cgap and line inductance Lgap , the input impedance can be modeled as a shorted lossless transmission line given by the expression

Zin = iωLgap = Z` tan βè ,

(6.1)

72


where Z` is the characteristic impedance of the line and è is the effective quarterwave cavity length. In fulfillment of the resonant condition, the resonant frequency of the transmission line satisfies the expression ωo2 =

1 , Lgap Cgap

(6.2)

leading to the important transcendental expression for quarter-wave transmission lines 1 = Z` tan βè , ωo Cgap

(6.3)

where β = ωo /v is the propagation wave number. A typical geometry and ferrite ring configuration in a quarter-wave coaxial resonant structure is given in Fig. 6.1. As an approximation, the effective permittivity e and permeability µe of the line is calculated by averaging over all ferrite rings of thickness d1 and ferrite separation gaps of d2 to yield the expressions

e =

 d1     k + (1 − k) d1 + d2 , when d2 are air filled,    

1 , (1 − k)

(6.4)

when d2 are metallic cooling plates.

and µe = (1 + k(µ − 1))

d1 , d1 + d2

(6.5)

for either case where the d2 gaps are filled with air or filled with metallic cooling plates. The constant that encapsulates all the radial geometries of the resonant structure is given as k = ln

r3 r4 / ln , r2 r1

(6.6)

where r1 , r2 , r3 , r4 is the radial distance from the center of the beam pipe to the inner conductor, the radial distance from the center of the beam pipe to the inner ferrite

6.1 Ferrite-Loaded Transmission Line Theory

(a)

73

d2

(b) d2 d1

d2 d1

d2 d1

d1

d1

d1 r3 r4

r2 r1

,µ `

Figure 6.1: Cross-sectional schematic diagram of a quarter-wave resonant structure loaded with ferrite rings with permittivity and permeability = e o and µ = µe µo . (a) Side-view diagram depicting longitudinal dimensions where d1 is the thickness of the ferrite rings (shown in gray), d2 is the spacing between ferrite rings that could be composed of metallic cooling plates or air, and ` is the overall length of the structure. The ceramic gap (shown black) will allow for the interaction of the particles and the fields generated by the structure. (b) Perspective-view diagram depicting the radial geometry of the ferrite-loaded quarter-wave structure where the radial distance from the origin to the inner conductor is r1 , to inner radius of the ferrite ring is r2 , to the outer radius of the ferrite ring is r3 , and to the outer conductor is r4 .

74


ring, the radial distance from the center of the beam pipe to the outer ferrite ring, and the radial distance from the center of the beam pipe outer conductor, respectively. Now with the ability to calculate the effective permittivity and permeability of the line, the characteristic properties of the transmission line can be calculated. The capacitance per unit length for such a transmission line is given by the expression C` =

2πe o , r4 ln r1

(6.7)

and the inductance per unit length by, 1 L` = µe µo ln 2π

r4 r1

.

(6.8)

Derivations for all characteristic quantities for ferrite-loaded resonator cavities are given in Appendix B. The wave velocity in the given transmission line is c 1 =√ v=√ , e µe L` C`

(6.9)

where c is the speed of light. And finally, the characteristic impedance of the transmission line can be expressed as r r r L` 1 µe µo r4 Zo µe r4 Z` = = ln = ln , C` 2π e o r1 2π e r1 p where Zo = µo /o is the impedance of free space.

6.2

(6.10)

Design of Ferrite-loaded Cavity

The initial direction of the ALPHA project was to re-purpose the ferrite-loaded RF cavity, originally used on the Cooler Injector Synchrotron (CIS), for operation on

6.2 Design of Ferrite-loaded Cavity

75

the ALPHA storage ring. However, preliminary studies [54] carried out indicated that the performance limitations at 15–30 MHz for the ferrite materials used in the CIS cavity and space constraints on the ALPHA storage ring necessitate the design of a compact RF quarter-wave resonant cavity. The design will fulfill the space constraints and utilizes an alternative ferrite material with improved core loss at the intended operation frequencies. This section will describe the design of the compact quarter-wave cavity motivated by electromagnetic field simulations and transmission line theory. The section will conclude with the final design specifications of the ALPHA 15 MHz ferrite-loaded quarter-wave coaxial resonator cavity.

6.2.1

Physical Design Constraints

The geometry of the cavity was heavily constraint by the limited space that is available on the storage ring between third dipole magnet and the second damping wiggler magnet as shown in Fig. 6.2. Other beamline accoutrement and instrumentation such vacuum pumps, ion pumps, residual gas monitors, vacuum pipe bellows, and even thickness of the ConFlat (CF) flanges that mate with the beam pipe used to mount the RF cavity to the rest of vacuum system adds to further restrict the longitudinal dimensions of the accelerating structure. The radial size of the beam pipe, and the inner and outer radii of ferrite rings will constraint the radial dimensions of the RF cavity. A section will be devoted to fully describing the ferrite rings that will include geometries, electrical properties, and characterization measurements. Table 6.1 summarizes the final geometric constraints on the ALPHA storage ring for the initial design process. Fig. 6.3 shows the beam pipe section of the RF cavity that includes the ceramic vacuum break, designed by C. Romel and T. Luo, that will further restrict the overall length of the resonant cavity.

76


Damping Wiggler #2

Dipole #3

Figure 6.2: Diagram of the ALPHA storage ring populated by magnetic elements, vacuum pumps, and vacuum diagnostics. The blue hatched region depicts the planned location for the RF resonant cavity, between Dipole #3 and the Damping Wiggler #2.


Figure 6.3: (top) Vacuum pipe portion of the RF cavity with dimensions labeled in inches. (bottom) Prospective view of the vacuum pipe showing the mounting brackets and ceramic (acceleration) gap.

77

78

6. Ferrite-loaded Quarter-wave Coaxial Resonator Cavity Table 6.1: Summary of the geometric constraints in the initial design process of the 15 MHz quarter-wave resonant cavity.

6.2.2

Geometric Parameters

Maximum Value

Minimum Value

Overall Length

37.32 cm

13.26 cm

Outer Radius

–

10.16 cm

Inner Radius

6.35 cm

2.54 cm

Storage Ring Cavity Requirements

The specifications of the RF system must be established with knowledge of the ALPHA storage ring to ensure that requirements and objectives of the project are met. Parameters of the RF resonant structure such as the cavity harmonic number, required gap voltage, and tunability range of the operating frequency must be suitably selected. This section summarizes the physics that will set limits on these parameters. The harmonic number h of a resonant cavity is related to the operational RF frequency ωRF and revolution frequency ωo of the stored particle by ωRF = hωo .

(6.11)

The harmonic number will determine the bunch spacing as well the maximum number of particles contained in each bunch. Since the ALPHA storage ring will be used as a debuncher and bunch broadener, a harmonic number of h = 1 was chosen to provide the longest RF bucket for the task. This would mean that operational RF frequency of the cavity is the equal to the revolution frequency of the stored particle bunch and


79

that only a single macro-particle bunch can be stored and manipulated within the ring. Furthermore, this choice of harmonic number will provide the largest number of particles per bunch for the given machine design. The revolution frequency of stored particles are, of course, determined by the velocity of the particle bunch centroid and the circumference of the ring. This makes the storage ring operating frequency of the RF cavity fixed and do not require complex frequency ramping schemes as is used for proton and ion accelerator rings. In a typical electron storage ring, the RF cavity serves two primary purposes: replenishing the energy loss by the electrons via synchrotron radiation and provide a RF bucket for longitudinal focusing of the electron bunch. Since the energy loss via synchrotron radiation is relatively small for 50–100 MeV electrons (on the order of a few eV), providing a viable RF bucket for particle confinement is the essential objective of the RF resonant cavity. The exact path length of particles traversing the ring cannot be determined by the geometries alone, thus variability in the operating frequency of the RF structure about the design frequency is required. This implies the need for shifting the resonance frequency of the RF cavity to efficiently couple in RF power to accommodate frequency shifting of the RF system. The absolute frequency tuning range of the RF system will determined by the geometry of the resonant structure and other hardware considerations. This discussion will be deferred until later a section. The required peak voltage at the accelerating gap of a RF structure in a low energy electron storage ring is determined by two primary factors: the momentum spread of the injected beam and large-angle Coulomb beam-beam interaction that results in particle loss. The ability for a given ring to efficiently capture and store beams of non-vanishing momentum spread is referred to as the momentum acceptance of the ring. This important parameter is related to the peak RF voltage Vp , among other

80


beam parameters, and is expressed as 1/2 2eVp Y (φs ) δacc = πβ 2 Eo h |η| where the RF bucket height factor is given as 1/2 π − 2φ s Y (φs ) = cos φs − sin φs , 2

(6.12)

(6.13)

Eo is the beam energy, h is the cavity harmonic number, β = v/c is related to the particle speed, φs is the synchronous phase, and η is the phase slip factor that is related to the momentum compaction factor of the ring. The specified peak RF voltage must be large enough to provide a momentum acceptance that will accommodate the momentum spread of the injected beam. Beam-beam Coulomb interaction can be divided into two main categories: singlecollision interactions and multiple-collision interactions. In multiple-collision interactions, the momentum exchange between orthogonal planes of motion are not large enough to result in particle loss, but will ultimately limit beam emittance. These multiple-collision Coulomb interactions are also commonly referred to as intrabeam scattering. In a single-collision interaction, two particles collide where their momentum transfer is large enough for both particles to be lost due to the lattice acceptance limitations. Since particles executes both betatron motion and synchrotron motion as it revolves around the ring, transverse momentum can transfer into longitudinal momentum and vice versa via coulomb interaction. However, the process into which longitudinal momentum is transferred into transverse momentum resulting in particle loss is insignificant in particle accelerators. Therefore only large-angle Coulomb scattering where transverse momentum exchange resulting in particles escaping the longitudinal limits of the RF bucket and are lost will be considered. This effect was first discovered Bruno Touschek, et al. [36] at the Italian Anello d’Accumulazione (AdA) [37, 38], the first electron storage ring ever constructed, in 1961.


81

The Touschek lifetime τT of an electron beam is the time required for the beam to decay to half of its initial intensity given as [39, 40, 41] I 1 re2 cq 1 F (ξ) = ds 2 (s) τT 8πeγ 3 σs C σx (s)σz (s)σx0 (s)δacc where the function F (ξ) is defined to be Z 1 1 1 1 F (ξ) = − ln − 1 e−ξ/u du, u 2 u 0

(6.14)

(6.15)

ξ = (δacc (s)/γσx0 (s))2 , and γ is the Lorentz factor not to be confused with the γx Twiss parameter. Here σx , σz , σs , and δacc are defined as the rms transverse beam widths, rms bunch length, and momentum acceptance, respectively. The σx0 term does not represent the full rms horizontal beam divergence, but the rms divergence when x ∼ 0 as the two interacting electron are assumed to share the same spatial positions. This dispersion correcting term is given as s x H(s)σδ2 σx0 (s) = 1+ , σx (s) x

(6.16)

where σδ is the rms momentum spread, and the H-function is given as H(s) = γx (s)D2 (s) + 2αx (s)D(s)D0 (s) + βx (s)D02 (s).

(6.17)

Here D(s) and D0 (s) are the dispersion function and its derivative, and αx (s), βx (s), and γx (s) are the Twiss parameters in the horizontal direction where γx (s) =

1 + αx2 (s) . βx (s)

(6.18)

For an electron beam of 50 MeV with a rms momentum spread of 0.5%, the Touschek lifetime of the beam is shown in Fig. 6.4 with varying peak gap voltage of the RF cavity. Since the properties of the ALPHA lattice is augmented with changing gradient damping wiggler bending radius ρw , the numerical integration was performed

82


for each lattice configuration. Lattice parameters such as the dispersion function and Twiss parameters were obtained through MAD-X simulations and was crossreferenced with SimTrack simulations. Since the Touschek lifetime is inversely proportional to the total charge within a bunch, the lifetime is lower for the case where Qtotal = 1000 nC compared to the case where Qtotal = 500 nC. The Touschek lifetime also scales with the cube of the energy, therefore a drastic enhancement of the lifetime is expected with higher beam energies. Fig. 6.4 shows that a peak RF gap voltage of 1 kV will fulfill the design specification and leave enough headroom for wide machine operating ranges.

6.2.3

Ferrite Material Characterization

The ferrite rings that will be loaded into the cavity will need to minimize inductive core losses up to the operating frequency of 30–40 MHz. The Toshiba M4 C21A ferrite rings were chosen for their desired core loss properties and availability. The physical geometries of the M4 C21A ferrite rings are summarized in Table 6.2 in terms of dimensional variables previously defined in Fig. 6.1. This section will introduce several dispersion models that are guided by measurements of permeability and core loss of the chosen ferrite material. This information will be vital in the simulation and design of the RF cavity.

The Toshiba M4 C21A ferrite rings were utilized at the Los Alamos Proton Storage Ring (PSR) in 1998 in an upgrade to mitigate the space-charge forces of the higher intensity proton beams [42, 43, 44]. The ferrite rings enclosed inside three pill-box cavities were installed into the evacuated beamline. The inductive forces from the


83

8 Qtotal = 1000 nC Qtotal = 500 nC Eo = 50 MeV

7

Touschek Lifetime τT (hr)

6

5

4

3

2

1

0

1 Wiggler Bending Radius ρw (m)

2

Figure 6.4: Touschek lifetime τT as a function of the damping wiggler bending radius ρw where the peak RF voltage is Vp = 0.1 kV (blue), Vp = 0.5 kV (red), Vp = 1 kV (black), and Vp = 2 kV (green). The lifetime calculations were carried out for electrons with beam energy of Eo = 50 MeV. The peak in the lifetimes are for ρw in the vicinity of the isochronous condition.

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Table 6.2: Summary of the geometric parameters of the M4 C21A ferrite rings to be utilized in the 15 MHz quarter-wave resonant cavity.


Corresponding Values

Inner Radius

r2 = 6.35 cm = 2.5 in

Outer Radius

r3 = 10.16 cm = 4.0 in

Thickness

d1 = 2.54 cm = 1.0 in

ferrite/beam interaction was used to cancel the space-charge repelling forces. Experimental results indicated that the additional inductive forces did indeed cancel a large proportion of the space-charge force contribution from the intense proton beam. This is evident in the shortened bunch lengths attained in the presence of the installed ferrite inserts with no bias current in the solenoidal wire windings. Furthermore, when a bias current is applied, the bunch is lengthened due to the diminishing inductive compensation that counteracted the space-charge force. However, longitudinal microwave instabilities were observed as the result of the upgrade. To closely study the instability, the Toshiba M4 C21A ferrite rings were modeled and characterized with varying frequencies and temperatures at FermiLab in 2000 by K Y. Ng and D. Wildman [45]. A description of the relevant models and material characterization measurements of the Toshiba M4 C21A ferrite rings will be presented. The relative permeability of a single ferrite ring can expressed in a complex form to incorporate core loss given as µ = µ0 + iµ00 ,

(6.19)


85

Lp

Rp

Figure 6.5: A simple two-element model of the ferrite ring where Lp is the ideal inductance and Rp is the ideal resistance in parallel.

where µ0 and µ00 are real numbers. The impedance across a single ferrite ring of thickness d1 , outer radius r3 , and inner radius r2 can be expressed as r3 f errite 0 00 0 00 Zo , = −i (µ + iµ ) ωLo = −i (µ + iµ ) ωµo d1 ln r2

(6.20)

where Lo = µo d1 ln (r3 /r2 ) is the inductance of a ferrite ring due solely to geometric contributions when the relative permeability is µ = 1. The simplest electrical model for a given ferrite material consist of an ideal inductance Lp and an ideal resistor Rp in parallel as depicted in Fig. 6.5. In this simple model, the impedance of the ferrite core that satisfy the dispersion relation is given as Zof errite (ω) = −iωLp

1 + ω/ωr , 1 + ω 2 /ωr2

(6.21)

where the resonance frequency is ωr =

Rp , Lp

and the real and imaginary parts of the permeability can be found to be Lp Lp 1 ω/ωr 0 00 µ = , µ = . 1 + ω 2 /ωr2 Lo 1 + ω 2 /ωr2 Lo

(6.22)

(6.23)

The permeability measurement of the ferrite rings were carried out by heating

86


the ferrite material to high temperatures, then letting it slowly cool in a controlled manner, while measuring the impedance across the ferrite. To do this, a test fixture that consisted of a 9 in diameter and 1.25 in thick aluminum disk was machined with the inner portion of the disk milled out to accommodate a single ferrite ring with an outer diameter of 8 in (r3 = 4 in) and thickness of d1 = 1 in. This inner portion of the aluminum disk was milled 5 × 10−3 in undersized so that it could be heated in an oven to thermally expand and comfortably seat the ferrite ring. Good electrical and thermal contact was ensured once the aluminum fixture cools and contracts to bind in the ferrite ring. Now that only one face and the inner diameter of the ferrite ring is left exposed, another 9 in diameter aluminum disk with a small diameter cut out from the center was placed on the remaining exposing face leaving only the inner diameter of the ferrite ring exposed for measurements. The two aluminum disk were fastened together and conductive cooper tape was used to maintain good electrical connection. The test fixture was placed on a hot plate and covered with two fire bricks for heat retention and temperature stability. The entire test fixture was heated to 175 ◦ C and allowed to slowly cool during measurements. A probe from a Hewlett Packard 4193A vector impedance meter was placed directly across the inner edge of the ferrite ring. Impedance measurements were taken in the frequency range of 10 MHz to 110 MHz in incremental steps of 10 MHz and in the temperature range of 150◦ C to 25◦ C in incremental steps of 25◦ C. A small hole was drilled and tapped into the aluminum test fixture where a Fluke 80T-150U universal temperature probe can monitor the temperature of the ferrite core as it slowly cools. The measurement of the input impedance as a function of frequency were used to calculate the inductance Lp using Eq. 6.21 and from the resonance condition in Eq. 6.22 the resistance Rp . From Eq. 6.23, the relative permeability µ0 and ferrite core loss µ00 can be calculated for the given electrical model. This is shown is Fig. 6.6


87

and Fig. 6.7 where the diamonds indicate data that was inferred from measurement of the complex input impedance. Another experimental setup, similar to the one built by FermiLab scientists, that uses a vector network analyzer instead of a vector impedance meter was constructed to characterize ferrite rings at Los Alamos National Laboratory in 2001 by A. Browman. The two-port scattering matrix measurements were use by C. Beltran [46, 47] in conjunction with the MAFIA electromagnetic simulation software package to model the electrical properties of the Toshiba M4 C21A ferrite ring. The microwave test fixture used at Los Alamos is shown in Fig. 6.8 where RF power is coupled through the port at the center of the fixture. Summary of the study will be presented. The capacitance measurement of the test fixture with and without loading of the ferrite ring were in good agreement with MAFIA simulation results given the fixture geometries and properties of the material within. Confirmation of the relative permittivity specified by the ferrite ring manufacture to be ∼ 13 was accomplished by measuring the capacitance of the ferrite-loaded test fixture and comparing it to measurements. The relative permittivity was found to be effectively constant for all frequencies and in the temperature ranges of interest from 25–125◦ C. The measurement of the complex permeability is a bit more involved. First the reference plane for the measurement must be established by calibrating the network analyzer (with open, short, and 50 Ω load calibration standards) at the input port of the test fixture. To properly model the complex permeability behavior of the M4 C21A ferrite rings, the simulation reference plane must match that of the measurement reference plane. This is accomplished by adjusting the input port distance in the simulation so that the difference between the phase from the S11 measurement and the phase obtained by MAFIA agree within a defined tolerance.

88


100

150◦ C 125◦ C 100◦ C 75◦ C 50◦ C 25◦ C

Permeability µ0

80

60

40

20

0

0

20

40

60 80 Frequency (MHz)

100

120

Figure 6.6: The relative permeability µ0 of a single Toshiba M4 C21A ferrite ring enclosed in the heated aluminum test fixture as a function of frequency from 10–110 MHz at temperatures from 25◦ C up to 150◦ C. (Data courtesy of K. Y. Ng and D. Wildman).


30

150◦ C 125◦ C 100◦ C 75◦ C 50◦ C 25◦ C

25

Ferrite Core Loss µ00

89

20

15

10

5

0 0

20

40


100

120

Figure 6.7: The ferrite core loss µ00 of a single Toshiba M4 C21A ferrite ring enclosed in the heated aluminum test fixture as a function of frequency from 10–110 MHz at temperatures from 25◦ C up to 150◦ C. (Data courtesy of K. Y. Ng and D. Wildman).

90


Figure 6.8: A similar test fixture constructed at Los Alamos National Laboratory designed to measure the impedance of a ferrite ring using a vector network analyzer. (Courtesy of C. Beltran)

To analyze the permeability, the dispersion model was chosen to be the Debye model with the complex permeability expressed as µ(ω) = µo µ∞ +

iωµo ∆µ , iω − ω 2 τ

(6.24)

where µ∞ = 1 and ∆µ = µo − µ∞ . The real and imaginary part of Eq. 6.24 corresponding to the permeability and ferrite core loss can be written as µ0 (ω) = µ∞ +

∆µ , 1 + ω2τ

µ00 (ω) =

∆µ . ωτ + 1/ωτ

(6.25)

By varying the µ0 and µ00 parameters within the simulation, the corresponding frequency dependent S11 scattering parameter can be obtained and compared with actual measurement data. Once the difference in the amplitude and phase (that form the S11 scattering parameter) between simulation and measurement are within defined tolerances, a functioning model for the permeability and ferrite core loss has been constructed. This information will be used in the model of the ferrite-loaded RF


91

cavity for ALPHA. The real and imaginary component of the complex permeability obtained from the efforts is summarized in Fig. 6.9 The real component of the magnetic permeability µ0 of ferrite core at room temperature is relatively constant in the frequency range from 20–100 MHz with variation ∆µ0 ≈ 10.4 and the peak µ0peak ≈ 50.5 located at about 70 MHz. When the ferrite core temperature is increased, the location of µ0peak is shifted to lower frequencies. For ferrite core temperature of 125◦ C, the real permeability was found to be µ0peak ≈ 73 located at about 20 MHz. The imaginary component of the magnetic permeability µ00 also exhibits similar behavior. The rising edge of µ00 shifts to lower frequencies as the ferrite core temperature is increased. The M4 C21A ferrite models presented by K. Y. Ng, et al. differs slightly from the model presented by C. Beltran, et al. Most notably, a lack of a peak versus a very pronounced peak in the real component of the magnetic permeability. These details of ferrite characteristics is important when frequency ramping is required, i.e., in the acceleration of protons or charged ions. Since the operation of an electron storage ring only requires a fixed operating RF frequency, and small deviation tuning about this fixed RF frequency during commissioning, we are only concerned with the properties of the ferrite around 15 MHz where the two models converge. The relative magnetic permeability value of µ0 ≈ 43 and relative permittivity ≈ 13 will be used to simulate the ferrite-loaded resonant structure at room temperature.

6.2.4

Resonant Cavity Simulations

The intrinsic characteristics of a resonant cavity is determined by the geometry and materials used in the construction of the RF structure. In the previous section, the

92


80 µ0 @ 125◦ C

Permeability µ0 and Ferrite Core Loss µ00

70 60 50

µ0 @ 25◦ C

40 30 µ00 @ 125◦ C

20

µ00 @ 25◦ C

10 0 -10

0

20

40


100

120

Figure 6.9: Characterization summary of the M4 C21A ferrite. The solid line corresponds to the real component of the magnetic permeability and the dot-dashed line corresponds to the imaginary component of the magnetic permeability as a function of frequency. The lines in blue and red represents the complex permeability (real and imaginary) with ferrite core at 25◦ C and 125◦ C, respectively.


93

proposed M4 C21A ferrite material was characterized so that it may be incorporated into the modeling of the ferrite-loaded cavity. A collection of programs, under the title Poisson-Superfish, will be used to calculate the static electric and magnetic fields, and other properties of a user-defined resonant structure geometry. The software package can also handle problems involving non-linear and anisotropic materials. The PoissonSuperfish package is currently maintained by the Los Alamos Accelerator Code Group and has been thoroughly tested in the 40 years since it was first introduced [48, 49]. Electromagnetic fields can be solved in either a 2-dimensional Cartesian coordinate or an axially symmetric cylindrical coordinate by generating a triangular mesh that is optimized and fitted to user defined materials with geometries. In this section, Poisson-Superfish will be used to determine and optimize resonant cavity characteristics, while considering all constraints, varying with geometry and material property. Since the software package is a static field solver, more complicated phenomena such as skin depth and material losses are not considered. Consider the following quarter-wave resonant structure to be studied using PoissonSuperfish. Fig. 6.10 shows the graphical summary of the Poisson-Superfish simulation, plotted using the subroutine WSFPLOT, where the inner conductor radius is r1 = 2.84 cm, outer radius of r4 = 25 cm, and overall cavity length of ` = 35 cm. The concentration of electric field lines and magnetic field lines are precisely where one would expect of a quarter-wave structure, i.e., regions of maximal electric field and magnetic field are on opposing ends of the structure. Furthermore, the concentration of electric field lines exhibit variation in the longitudinal direction whereas the concentration of magnetic field lines exhibits variation in the axial direction. The electric field can be seen most highly concentrated in the region of the ceramic accelerating gap (with known material properties that were applied in the simulation). This will provide a “window” for traversing electrons to be influenced by the fields generated at

94


−5

0

5

10

15

20

25

30

35

30

30

25

25

20

20

15

15

10

10

5

5

0

0 −5

0

5

10

15

20

25

30

35

Figure 6.10: An example output of a Poisson-Superfish simulation run where a slice of the a quarter-wave resonant structure is shown axially symmetric about the horizontal axis. The geometry of this structure had an inner conductor radius of r1 = 2.84 cm, outer radius of r4 = 25 cm, and overall cavity length of ` = 35 cm yielding a resonant frequency of 160.28 MHz. The electric field contours are depicted with magenta lines where the density of lines represent the regional electric fields strengths and the magnetic fields are depicted with circles where the size of the circles represents the regional magnetic field strengths. The scales in the horizontal and vertical axes are in units of cm.


95

the accelerating gap. The resultant structure in Fig. 6.10 yielded a resonant frequency of 160.28 MHz. Suppose a geometry is defined where the inner conductor radius is comparable to the size of the vacuum beam pipe and fixed at r1 = 2.84 cm while allowing the outer conductor radius r4 and the overall cavity length ` to be varied. The result of this simulation will give us an intuitive sense of how the resonant frequency of the structure scales with geometry. The results of the parameter mapping are summarized in Fig. 6.11. The conductor material was defined by default to be perfect electrical conductors (PEC), the empty spaces between material regions to be vacuum, and other relevant geometric parameter such as the inner conductor radius was held fixed at r1 = 2.84 cm in the simulation. Refer to Fig. 6.1 to find the definitions for each relevant cavity parameter. The resonant frequency of the quarter-wave structure decreases as the overall length ` of the cavity was increased. This is no surprise as the resonant frequency of the structure scales as with the wavelength of electromagnetic fields that is confined within the structure. As ` approaches the length constraint of 37.32 cm in the storage ring, the resonant frequency can only be reduced to a minimum of ∼ 185.22 MHz in the case where r4 = 10.79 cm. As the outer conductor radius is increased, further reduction in the resonant frequency is observed. In principle, r4 can increase without bound and thus eventually enabling a structure with resonant frequency of 15 MHz. In practice however, having a resonant structure with extreme large r4 will negate the feature of compactness and drive up construction cost of the cavity. There is also constraints on r4 imposed by the distance from the axial center of the cavity to the ground which makes attaining a resonant frequency of 15 MHz through enlarging r4 impossible. The simulation for a structure with ` = 37.32 cm and r4 = 35 cm resulted

96


425 (a) 400 375 (b) Resonant Frequency (MHz)

350 325

(c)

300

(d)

275

(e)

250

(f)

225 200 175 150 125

10

15

20 25 30 Overall Cavity Length ` (cm)

35

40

Figure 6.11: Resonant frequency of the quarter-wave RF structure where no ferrite rings were loaded as a function of the overall cavity length ` and outer conductor radius r4 where (a) r4 = 10.79 cm, (b) 15 cm, (c) 20 cm, (d) 25 cm, (e) 30 cm, and (f) 35 cm. The inner conductor radius of the structure was held fixed at r1 = 2.84 cm in the Poisson-Superfish simulation.


97

in a resonant frequency of 140.48 MHz. This is nowhere near the specified h = 1 operating frequency of 15 MHz, thus other geometric parameters must be explored to be able to achieve the storage ring specifications. The inner conductor radius is heavily constrained by physical factors such as the radius of the vacuum beam pipe and the inner radius of the M4 C21A ferrite rings. In Fig. 6.11, r1 = 2.84 cm was studied as it was the case where r1 was comparable to the radius of the vacuum beam pipe. In this following simulation, r1 = 5.72 cm is used so that the inner conductor radius is comparable to the inner radius of the ferrite ring. An example of this new geometry is given in Fig. 6.12 where ` = 35 cm, r1 = 5.72 cm, r4 = 25 cm, and with no ferrite loaded. This choice of r1 was motivated by the need for an air gap between the conductor and the ferrite ring for temperature moderation in high power operation. A more detailed discussion will be given in the sections ahead on the construction and fabrication considerations of the resonant structure. When ` and r4 are varied with the inner conductor radius fixed at r1 = 5.72 cm, the results of this simulation is summarized in Fig. 6.13. When comparing Fig. 6.11 to Fig. 6.13, the dependency of the resonant frequency as a function of ` exhibits a more linear trend and less of the inverse-square bowing in the case where r1 = 5.72 cm. The simulation for the resonant structure, shown in Fig. 6.12, with ` = 37.32 cm, r1 = 5.72 cm, and r4 = 35 cm resulted in a resonant frequency of 172.8 MHz. This higher than in the case where r1 = 2.84 cm with resonant frequency of 160.28 MHz. This is due to the reduction in the resonating volume of the structure with larger inner conductor radius. With all conventional geometric constraints adjusted and optimized, still the specified operational resonant frequency of 15 MHz have yet to be achieved. This forces the design direction to consider the possibility of introducing resonant frequency altering materials into the

98


−5

0

5

10

15

20

25

30

35

30

30

25

25

20

20

15

15

10

10

5

5

0

0 −5

0

5

10

15

20

25

30

35

Figure 6.12: An example output of a Poisson-Superfish simulation run where a slice of the a quarter-wave resonant structure is shown axially symmetric about the horizontal axis. The geometry of this structure had an inner conductor radius of r1 = 5.72 cm, outer radius of r4 = 25 cm, and overall cavity length of ` = 35 cm yielding a resonant frequency of 172.8 MHz. The electric field contours are depicted with magenta lines where the density of lines represent the regional electric fields strengths and the magnetic fields are depicted with circles where the size of the circles represents the regional magnetic field strengths. The scales in the horizontal and vertical axises are in units of cm.


99

425 (a) 400 375 (b) Resonant Frequency (MHz)

350 325

(c)

300

(d)

275

(e)

250

(f)

225 200 175 150 125

10

15

20 25 30 Overall Cavity Length ` (cm)

35

40

Figure 6.13: Resonant frequency of the quarter-wave RF structure where no ferrite rings were loaded as a function of the overall cavity length ` and outer conductor radius r4 where (a) r4 = 10.79 cm, (b) 15 cm, (c) 20 cm, (d) 25 cm, (e) 30 cm, and (f) 35 cm. The inner conductor radius of the structure was held fixed at r1 = 5.72 cm in the Poisson-Superfish simulation.

100


#1

#2

#3

#4

#5

#6

#7

Figure 6.14: Cross-sectional diagram of a quarter-wave resonating structure with numbers corresponding to the relative position of a single ferrite ring introduced within the simulation.

resonating structure. Ferrite materials with properties of high µe could be used to impede the electromagnetic wave velocity within the resonant cavity and thereby further enhance the reduction of the resonant frequency of the structure. Having the M4 C21A ferrite material characterized in the previous section, it can be introduced into the Poisson-Superfish simulations by creating material boundary regions that matches the physical geometry of the ferrite rings and having material properties of µe = 43 and e = 13 at room temperature. Since for a quarter-wave resonant structure the region of maximum magnetic field and electric field occur at opposing ends of the structure, it is reasonable to assume that the resonant frequency would be dependent of the placement of the ferrite material within the resonant cavity. Fig. 6.14 depicts a cross-sectional diagram of the resonant cavity that shows label numbers corresponding to relative positions of single ferrite rings within the


101

resonant structure with respect to the location of the accelerating gap (i.e., position #1 corresponds to the ferrite ring that is positioned closest to the accelerating gap and position #7 corresponds to the ferrite ring that is positioned furthest from the accelerating gap). Given the length limitation of the resonant structure and the thickness of the ferrite rings, a maximum of only seven M4 C21A ferrite rings can be loaded into the cavity. In the following simulations, a single ferrite ring was placed at various positions within a resonant cavity with the geometry held fixed to an overall length ` = 37.32 cm, and inner conductor radius r1 = 5.72 cm with outer conductor radius allowed to vary between r4 = 10.79 − 35.0 cm. Fig. 6.15 summarizes the resonant frequency dependency on the position of the ferrite ring relative to the accelerating gap. In the absence of the ferrite material loaded into the cavity, the starting point resonant frequency of the structure was simulated to be 218.54 MHz. When a single M4 C21A ferrite ring was introduced into the cavity placed in position #1, the resonant frequency was greatly reduced to 115.37 MHz. As the ferrite ring was positioned further and further away from the accelerating gap, the resonant frequency continues to decrease. Fig. 6.16 shows the case where a single M4 C21A ferrite ring is placed in position #5 yielding a resonant frequency of 85.43 MHz. Both electric and magnetic fields are seen be concentrated in the region of the ferrite material. When the ferrite ring was finally placed in the position furthest from the accelerating gap, position #7, the resonant frequency was minimally reduced to 77.96 MHz for the given geometries. This makes intuitive sense as the ferrite material now resides in the region of maximum magnetic field and thus the augmentation to the resonant frequency is maximized. This of course also agrees with the assertion that at the opposing end of the cavity, the electric field is indeed a maximum where the accelerating gap is located. There is also an important but subtle point to be understood by these results. Current

102


220 r4 r4 r4 r4 r4 r4

200

Resonant Frequency (MHz)

180

= 10.79 cm = 15.00 cm = 20.00 cm = 25.00 cm = 30.00 cm = 35.00 cm

160

140

120

100

80

60

0

1

2 3 4 5 Single Ferrite Ring Position in Cavity

6

Figure 6.15: Resonant frequency of the quarter-wave RF structure as a function of the location of a single ferrite ring with respect to the accelerating gap calculated using Poisson-Superfish. Geometric parameters such as the overall length and inner radius were held fixed at ` = 37.32 cm and r1 = 5.72 cm as the outer conductor radius r4 was allowed to vary. The material properties of the ferrite ring was set to µe = 43 and e = 13 for nominal room temperature operation.

7


103

biasing by coiling a current-carrying wire around one or many ferrite rings will have the greatest effect in the rings furthest away from the accelerating gap. This current biasing scheme can be used to tune the resonant frequency and maximize the coupling of RF power into the cavity. Suppose if more than one ferrite ring were to be loaded into the resonant cavity of given geometry. Intuition dictates that the resonant frequency of the cavity should continue to decrease with increasing number of ferrite rings loaded into the structure. In simulations, a single ferrite ring is placed in position #1 (ferrite ring position closest to the accelerating gap), and then subsequent ferrite rings are added one by one to the resonant structure. Each ferrite ring is longitudinally spaced apart by d2 ≈ 1 cm. This separating space between each ferrite ring will be used for forcedair cooling to stabilize the temperature dependent electromagnetic properties of the ferrite materials in high power cavity operations. The cooling of the ferrite rings will be optimized for peak cavity performance and stability. Fig. 6.17 summarizes the resonant frequency dependency on the number of ferrite rings loaded into the resonant structure where the overall length and inner radius were held fixed at ` = 37.32 cm and r1 = 5.72 cm as the outer conductor radius r4 was allowed to vary. In the absence of the ferrite material loaded into the cavity, the starting point resonant frequency of the structure is again 218.54 MHz for ` = 37.32 cm, r1 = 5.72 cm, and r4 = 10.79 cm. When the first ferrite ring is placed in position #1, the resonant frequency of the structure decreases abruptly to 115.37 MHz. As subsequent ferrite rings are loaded, spaced d2 ≈ 1 cm apart, the resonant frequency reduction was further enhanced. Fig. 6.18 shows the electromagnetic fields within the resonant structure when three M4 C21A ferrite rings at room temperature are loaded. The resulting in a resonant frequency of this structure was found to be 48.01 MHz. When

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Figure 6.16:

10

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30

35

Poisson-Superfish simulation result with a single

M4 C21A ferrite ring placed at position #5 where the inner conductor radius is r1 = 5.72 cm, outer radius of r4 = 10.79 cm, and overall cavity length of ` = 37.29 cm yielding a resonant frequency of 85.43 MHz. The electric field contours are depicted with magenta lines where the density of lines represent the regional electric fields strengths and the magnetic fields are depicted with circles where the size of the circles represents the regional magnetic field strengths. The scales in the horizontal and vertical axes are in units of cm.


105

220 r4 r4 r4 r4 r4 r4

200


180

= 10.79 cm = 15.00 cm = 20.00 cm = 25.00 cm = 30.00 cm = 35.00 cm

160 140 120 100 80 60 40 20

0

1

2 3 4 5 Number of Ferrite Rings Loaded

6

Figure 6.17: Resonant frequency of the quarter-wave RF structure as a function of number of ferrite rings loaded into the cavity using Poisson-Superfish where the first ferrite ring was loaded into the position closest to the accelerating gap. Geometric parameters such as the overall length and inner radius were held fixed at ` = 37.32 cm and r1 = 5.72 cm as the outer conductor radius r4 was allowed to vary. The material properties of the ferrite ring was set to µe = 43 and e = 13 for nominal room temperature operation.

7

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Figure 6.18: Poisson-Superfish simulation result where three M4 C21A ferrite rings were placed in positions furthest from the accelerating gap ∼1 cm apart. The inner conductor radius was r1 = 5.72 cm, outer radius r4 = 10.79 cm, and overall cavity length ` = 37.29 cm yielding a resonant frequency of 48.01 MHz. The electric field contours are depicted with magenta lines where the density of lines represent the regional electric fields strengths and the magnetic fields are depicted with circles where the size of the circles represents the regional magnetic field strengths. The scales in the horizontal and vertical axises are in units of cm.


107

the cavity was completely loaded with seven M4 C21A ferrite rings, the resonant frequency of 30.29 MHz was obtained. This is the lowest resonant frequency that can be obtained when the resonant cavity is fully loaded with seven M4 C21A ferrite rings at room temperature while still maintaining geometric compactness of the structure in the radial direction (r4 = 10.79). Aside from varying the geometry of the resonant structure, changing the number of the M4 C21A ferrite rings loaded and their placement within the structure, where the material properties of the ferrite rings was held fixed for normal operating conditions at room temperature, was simulated with Poisson-Superfish. As seen in the discussion on the characterization of the M4 C21A ferrite rings in Section 6.2.3, the magnetic permeability µ0 of the ferrite material (as well as the ferrite core loss µ00 ) will change with varying ferrite core temperatures. The µ0 and µ00 is also dependent on the degree of field saturation within the ferrite core. This can be brought on and manipulated by introducing a biasing current through a wire coiled around the ferrite rings. The amount of core saturation will be proportional to the current within the wire and number of wire coils around the ferrite ring. Suppose a resonant structure configuration is given such that all seven ferrite rings are loaded with separation d2 ≈ 1 cm, and geometry is fixed at ` = 37.32 cm, and r1 = 5.72 cm, as shown in Fig. 6.21. The relative magnetic permeability and outer conductor radius can then be varied. Fig. 6.19 depicts the dependency of the resonant frequency as a function of the relative magnetic permeability µ0r . The control of the ferrite core temperature for the purpose of tuning the resonant frequency of the structure can accomplished by placing the entire ferrite-loaded RF cavity into an oven configured to maintain a constant operating temperature. While this is yet another method to tune the cavity resonant frequency, in practice

108


175


150

r4 r4 r4 r4 r4 r4

= 10.79 cm = 15.00 cm = 20.00 cm = 25.00 cm = 30.00 cm = 35.00 cm

125

100

75

50

25

0

1 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 Relative Magnetic Permeability µ0r

Figure 6.19: Resonant frequency of the quarter-wave RF structure as a function of the magnetic permeability µ0 of the ferrite material. Geometric parameters such as the overall length, inner radius, and outer radius of the structure were held fixed at ` = 37.32 cm, r1 = 5.72 cm, and r4 = 10.79 cm, respectively and the ficed material properties were set to e = 13, in the Poisson-Superfish simulation.


109

the tuning range is narrow and may be costly in implementation. As a stand-alone method of resonant frequency tuning, the response time of frequency tuning will be very slow as the entire structure and the ferrite materials enclosed within must reach thermal equilibrium for stable operation. Even more critically, for most ferrite materials, including the M4 C21A ferrites, high temperature ranges that are required for higher permeability µ0 values would also increase the core loss µ00 rendering the ferrite material much too lossy for feasible applications in particle accelerators. As an advantage however, enclosing the ferrite-loaded resonant structure within a temperature controlled environment would increase the overall stability of the system in short and long term operation when used in conjunction with other methods of resonant frequency tuning. The resonant frequency calculated in the Poisson-Superfish simulations summarized in Fig. 6.11 and Fig. 6.13 when various resonant structure geometries were considered seems at first glance to contradict the results in which ferrite materials where introduced into the resonating medium as shown in Fig. 6.15, Fig. 6.17, and Fig. 6.19. The results in Fig. 6.11 and Fig. 6.13 indicates that for larger outer conductor radius r4 , a reduction in resonant frequency should be expected. Yet the results in Fig. 6.15, Fig. 6.17, and Fig. 6.19 shows that when ferrite materials are introduced into the resonant structure, larger r4 did not enhance the reduction of the resonant frequency. Structures with r4 > 15 cm seem to have approached some parameter regions of diminishing return as resonant frequency reduction are greater for r4 < 15 cm and most notably in the case where r4 = 10.79 cm. This can be explained by understanding that by having more electromagnetic fields contained within the resonant structure interact with the ferrite material, the further the wave velocity of the fields will be impeded and slowed. This causes the overall resonant frequency of the structure to be further enhanced as a result.

110


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Figure 6.20: Poisson-Superfish simulation result where seven M4 C21A ferrite rings are loaded and thus filling the cavity. The inner conductor radius was r1 = 5.72 cm, outer conductor radius r4 = 30 cm, and overall cavity length ` = 37.29 cm yielding a resonant frequency of 44.96 MHz. The electric field contours are depicted with magenta lines where the density of lines represent the regional electric fields strengths and the magnetic fields are depicted with circles where the size of the circles represents the regional magnetic field strengths. The scales in the horizontal and vertical axises are in units of cm.


111

The resonant structure shown in Fig. 6.20 depicts the electromagnetic field within a cavity that is fully loaded with seven M4 C21A ferrite rings. Here the geometric parameters were chosen to be ` = 37.29 cm, r1 = 5.72 cm, and r4 = 30 cm which resulted in a resonant frequency of 44.96 MHz. The electromagnetic fields are spread out over the volume of the structure. The interaction with the ferrite material is limited to a fraction of the cavity volume. The structure shown is Fig. 6.21 also depicts a fully loaded cavity with similar geometries with the exception that the outer conductor radius is now r4 = 10.79 cm instead of r4 = 30 cm as was the case in Fig. 6.20. The electromagnetic field confined to a smaller volume is concentrated in the region surrounding the M4 C21A ferrite rings. The concentration of field lines in a confined region increases interaction with the ferrite material and thereby enhances the reduction of the resonant frequency to 30.29 MHz. This optimal geometry resonant structure, shown in Fig. 6.21, will be adopted for further studies. Table 6.3 summarizes the cavity geometries that were obtained from simulation. The specified operating frequency of 15 MHz has not been achieved through geometry alone. However, by imparting capacitance at accelerating gap the RF cavity, transmission line theory will be used show that attaining the specified resonance frequency will be possible.

6.2.5

Application of Transmission Line Theory

For the given optimized quarter-wave structure in Fig. 6.21, transmission line theory extended for ferrite-loaded quarter-wave structures developed in Section 6.1 will be used to predict the feasibility in augmentation of the gap capacitance for the tuning of cavity resonant frequency. The total gap capacitance can be written as Cgap = Ccav + Cext where Ccav is the intrinsic gap capacitance of the cavity and Cext is the

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Figure 6.21: Poisson-Superfish simulation result where seven M4 C21A ferrite rings are loaded and thus filling the cavity. The inner conductor radius was r1 = 5.72 cm, outer conductor radius r4 = 10.79 cm, and overall cavity length ` = 37.29 cm yielding a resonant frequency of 30.29 MHz. The electric field contours are depicted with magenta lines where the density of lines represent the regional electric fields strengths and the magnetic fields are depicted with circles where the size of the circles represents the regional magnetic field strengths. The scales in the horizontal and vertical axises are in units of cm.


113

Table 6.3: Summary of the final geometric parameters of the ferriteloaded quarter-wave cavity optimized using Poisson-Superfish.



Overall Cavity Length

` = 37.29 cm = 14.68 in

Inner Conductor Radius

r1 = 5.72 cm = 2.25 in

Outer Conductor Radius

r4 = 10.79 cm = 4.25 in

Ferrite Ring Separation

d2 = 0.96 cm = 0.38 in

external tuning capacitance. A variable capacitor configured in parallel with Ccav will be used to impart more capacitance to the accelerating gap. For a shorted lossless transmission line at resonate frequency, the important expression will be summarized here given as 1 = Z` tan βè = Ze (ωo ), ωo Cgap

(6.26)

where ωo = 2πfo is the angular resonant frequency, Z` is the characteristic impedance of the transmission line, β = ωo /v and v the propagation wave number and wave velocity within the resonant structure, and è is the effective length of the quarterwave cavity measured from the center of the accelerating gap to the far end of the structure. Given the resonant structure geometries, the effective permeability µe and the effective permittivity e was calculated for the case where the ferrite rings are separated by air gaps spaced d2 = 0.96 cm apart. Using Eq. (6.7) and Eq. (6.8), the line ca-

114


Table 6.4: Summary of transmission line parameters relevant to the resonant frequency tuning calculations for the 15 MHz ferrite-loaded quarter-wave RF cavity.

Transmission Line Parameters


Effective Permeability

µe = 23.3

Effective Permittivity

e = 2.29

Geometric Factor

k = 0.74

Line Capacitance

C` = 200.16 pF/m

Line Inductance

L` = 2.97 µH/m

Characteristic Impedance

Zc = 121.69 Ω

Resonant Wave Velocity

v = 4.11 × 107 m/s

6.3 Construction of Ferrite-loaded Cavity

115

pacitance C` and line inductance L` of the quarter-wave structure can be calculated. The characteristic impedance of the line Z` and propagation wave velocity v within the resonant structure follows. Table 6.4 summarizes the relevant transmission line parameters used to predict the gap capacitance necessary to reduce the operating resonant frequency down to 15 MHz. Using the parameters summarized on Table 6.4 in conjunction with the quarterwave transmission line equation given in Eq. (6.26), the required gap capacitance Cgap was calculated. The result of the calculation has been graphically represented in Fig. 6.22 where the intersection points of the 1/ωo Cgap curve and the Ze (ωo ) curve gives the fundamental resonant frequency as well as the 3rd and 5th resonant harmonics of the ferrite-loaded quarter-wave structure. When the gap capacitance is Cgap = 102.74 pF, the fundamental resonant frequency of 15 MHz is obtained. For structures with quarter-wave symmetry, only odd harmonics exist. Therefore Fig. 6.22 also shows the 3rd harmonics at 70.76 MHz, and 5th harmonic at 135.91 MHz that were calculated. These higher order harmonics might be useful in beam applications that utilizes augmented RF potentials by exciting the cavity with multiple resonant frequencies. An industrial high voltage variable capacitor in the tuning range of 8-650 pF is currently available in the market and will be used to tune the resonant frequency of the cavity. More details on the variable capacitor will be given in the sections ahead.

6.3

Construction of Ferrite-loaded Cavity

The 15 MHz quarter-wave resonant cavity was constructed at the Indiana University Center for Exploration and Matter (IU CEEM), formerly known as the Indiana University Cyclotron Facility (IUCF). All machining, metal work, engineering

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400

Effective Characteristic Impedance Ze (fo ) (Ω)

350

300

250 (d) 200 (c)

150

100

(b)

50

0

(a)

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20

40 60 80 100 Resonant Frequency fo (MHz)

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140

Figure 6.22: The fundamental resonant frequency, 3rd , and 5th harmonic solutions to the transcendental quarter-wave transmission line equation given in Eq. (6.26). Four curves represents 1/ωo Cgap where (a) Cgap = 600 pF, (b) 102.74 pF, (c) 50 pF, and (d) 25 pF. The intersection points of the fundamental resonant frequency at 15 MHz, 3rd harmonic at 70.76 MHz, and 5th harmonic at 135.91 MHz was calculated for the gap capacitance of Cgap = 102.74 pF.


117

Figure 6.23: Engineering drawing showing the assembled vacuum ceramic break and specific welding requirements for each joint in the construction of the 15 MHz quarter-wave resonant cavity.

drawings, and cavity assembly were accomplished on-site at IU CEEM by Larry Boot, Jak Doskow, and me. The beam pipe assembly shown in Fig. 6.3 was constructed by the Kurt J. Lesker Company, who specializes in the construction of highvacuum chambers and pipes of varying materials. A weldable vacuum ceramic break (Part#CFT08V2371), made up of a high purity alumina ceramic insulation cylinder (to provide the 0.75 in accelerating gap) and low-expansion Kovar mount fitting, was carefully welded on to the 304 stainless steel vacuum beam pipe sections with diameter of 2 in. These welds must be smooth on the inner portion of the system to minimize beam pipe impedance effects that may cause beam instabilities. The engineering drawing in Fig. 6.23 specified all welding requirements in the construction of the ceramic vacuum break. For the remaining portions of the resonant structure, Oxygen Free High Conductivity Copper (OFHC) was selected for its exceptional performance and ease of

118


machining. Sheets of 50 mil (0.05 in = 18 Gauge) thick OFHC were rolled to form the cylindrical inner and outer conductor of the resonant cavity. Sets of small holes were intentionally drilled into optimal locations on the OFHC sheets to allow for forced air cooling. The end plates of the resonant structure were machined from thicker 250 mil (0.25 in = 3 Gauge) OFHC to allow it to be drilled into and tapped for the purpose of fastening the rolled inner and outer conductors in place. Brass screws, with lower resistivity, were used instead of stainless steel screws to prevent localized hot-spots due to the alternating currents. These end plates are fastened to stainless steel mounting flanges stitch welded on to the vacuum ceramic break assembly. All components made from stainless stain were designed to exist outside of the resonant structure and will not contribute to the electrical properties of the cavity. The seven M4 C21A ferrite rings required in the design must be placed into the resonant structure prior to enclosing it with the rolled OFHC outer conductor. To ensure that these ferrite rings remain stationary and maintain the separation distance from one another recommended by the design, thin guides machined from G-10 epoxy laminate materials were implemented. These guide were fastened to the rolled OFHC outer conductor through the use of nylon screws. Fig. 6.25 shows the seven M4 C21A ferrite rings loaded around the vacuum break assembly and into a partially assembled resonant cavity. The green G-10 guides are placed at four locations around the edge of the ferrite rings to support and separate the rings from the inner and outer conductor. A three-point-of-contact stand, machined from aluminum, will be used in combination with the G-10 guide to shift the weight of the ferrite rings away from the vacuum pipe. The stand has also been built to work with the industrial extruded aluminum erector set from 80/20, Inc. where solid aluminum plates can be mounted to form a box and completely enclose the resonant structure. This will shield the resonant structure from electromagnetic inferences, maintain the cavity’s geometric rigidity,


Figure 6.24: Engineering drawing showing an example of an end piece machined for use on the 15 MHz quarter-wave resonant cavity. This particular OFHC part fastens the rolled inner conductor to the vacuum ceramic break mounting flange.

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120


and act as a thermal buffer from temperature change.

6.3.1

Resonant Frequency Tuning Capacitor

For resonant frequency tuning of the cavity, a high voltage vacuum variable capacitor (Part#CV05C-650N/5) built by Comet was selected. The capacitor has a large variability range from 8–650 pF in about 7.9 rotations of the tuning shank, and have a maximum voltage rating of 5 kV. Being compact in size and pre-equipped with a mounting bracket, the capacitor adequately fit into the accelerating gap region of the resonant structure. Since there is no other easy method of configuring this variable capacitor across the accelerating gap, the introduction of the capacitor into the resonating medium will slightly perturbed the electromagnetic field enclosed. The resonant frequency will shift as a result. Simulations have been carried out to show that this perturbation will be small and that reaching a resonant frequency of 15 MHz can still be easily obtained. The tuning capacitor can be seen mounted in the resonant cavity in Fig. 6.26.

6.3.2

RF Power Coupling

The excitation of the RF cavity is achieved by the method of coaxial power coupling. RF power from a linear RF power amplifier is inductively coupled into the cavity in the region of maximum magnetic fields to obtain optimal cavity excitation. This coupling point was chosen to be between ferrite ring #6 and #7. This is shown in Fig. 6.26. The inductive loop is created by establishing a current path from the source toward the inner conductor through a female N-Type receptacle soldered to machined


Figure 6.25: Cut-away engineering drawing showing an example of an OFHC end piece machined and mounted for use on the 15 MHz quarter-wave resonant cavity. These end pieces fastens the rolled OFCHC inner and outer conductor to the mounting flange welded on the vacuum ceramic break pipe. The green G-10 guides ensure the correct placement of the ferrite rings. The 3-point stand supports the entire cavity so that the weight of the ferrite rings are not on the vacuum beam pipe.

121

122


circular OFHC bus bar that is connected to the inner conductor. The current path is closed through the shortest return path to the source along the skin of the inner conductor, end plate, and outer conductor. When an alternating current is applied, azimuthal magnetic field are generated by the inductive current loop thereby exciting the RF cavity and driving the alternating electric field at the ceramic accelerating gap. The precise length of this inductive loop can be simulated to obtain a 50 Ω input impedance to ensure an electrical impedance match with the RF power source, but due to the limitation on placement of the drive point within the cavity, the implementation of an impedance matching network between the power source and the RF cavity is to be expected.

6.4

Cavity Characterization Measurements

This section will summarize all low-power cavity characterization measurements. These measurements will be used (along with other measurements shown ahead) to quantify the performance of the resonant structure and provide vital calibration details in aiding the design and implementation of low-level RF electronics required to properly operate the RF system. The RF cavity was characterized using available RF equipment from the Indiana University Center for Exploration of Energy and Matter (IU CEEM), Crane Naval Surface Warfare Center (NSWC Crane), Indiana University Medical Cyclotron Operations Group (IU Medical CycOps), Department of Physics, and Department of Chemistry. The RF test equipment utilized includes the Hewlett-Packard 8753D 6 GHz Vector Network Analyzer, the Hewlett-Packard 8561B 50 Hz–6.5 GHz Spectrum Analyzer, the Agilent 8648B 9 kHz–2 GHz Signal Generator, Hewlett-Packard 4815A Vector Impedance Meter, and Tektronix DPO7254 Digital

6.4 Cavity Characterization Measurements

123

(e) (c)

(a)

(d) (b)

Figure 6.26: Cut-away engineering drawing of the 15 MHz quarterwave ferrite-loaded RF cavity with important components denoted. (a) Inductively coupled RF drive points between ferrite #6 and #7, (b) seven M4 C21A ferrite rings in placement, (c) high-voltage variable capacitor, (d) accelerating gap made of a high-purity alumina ceramic welded in place with low-expansion Kovar fittings (e) capacitor tuning motor and fixture for remote cavity tuning.

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Phosphor Oscilloscope.

6.4.1

Cavity Impedance Matching Network

The placement of the M4 C21A ferrite rings within the fully loaded cavity only allow inductive axial coupling access to the inner conductor through the separations between the ferrite rings of width d2 = 0.96 cm. For the ferrite rings configuration in the design, the coupling point between ferrite ring #6 and #7 will yield a cavity input impedance closest to 50 Ω. The Smith Chart in Fig. 6.27 shows the complex reflection coefficient Γ, superimposed on to circles of normalized impedance and admittance, as a function of frequency measured with a calibrated vector network analyzer (VNA). The measurement was conducted with the measurement reference plane calibrated to be at the input port of the RF cavity without the implementation of a matching network. The data collected spans a frequency range from 6 MHz to 29 MHz with a maximum of 1601 data points across the frequency range. The complex reflection coefficient at 14.999 MHz was measured to be Γunmatched = −0.227 − i0.396. cavity This corresponds to an input complex impedance of the ferrite-loaded RF cavity of unmatched Zcavity = (23.825 − i23.829) Ω referenced to a 50 Ω load at 14.999 MHz. For this

impedance measurement, the resonant frequency of the cavity was tuned to 15 MHz by adjusting the variable capacitor and tracking the log |S11 | absorption curve with the VNA. The data point corresponding to the complex reflection coefficient at 14.999 MHz, shown in Fig. 6.27, is highlighted with a red circle. The data points to the left corresponds to Γunmatched = −0.304 − i0.346 at 15.013 MHz, and the data point to the lef t right corresponds to Γunmatched = −0.143−i0.435 at 14.984 MHz are highlighted in the right

6.4 Cavity Characterization Measurements

70

(+ jX /Z

45

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IND UCT IVE

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5.0

10 0.25 0.26 0.24 0.27 0.23 0.25 0.24 0.26 0.23 ECTION COEFFICIEN 0.27 L T F E I N R D EGRE LE OF ES ANG ISSION COEFFICIENT IN TRANSM DEGR LE OF EES ANG

8

0.

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RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo)

50

0.2

20

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10

-10 1.0

E IV CT DU IN

2.0

5 -4

0.15

0.14 -80

0.48

0.

06

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-1

40

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0.4 1

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1

15

TOWARD LOAD —> 10 7 5

1 1 30

0

0.1

0 0

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0 1

0.99

0.9

CENTER 1

1.1

4

1.1

1.2

1.3 1.4

0.2

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0.6

1.2

1.3 0.95

1.4

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0.9

WAVELE 0.49 NGTH S TOW ARD 0.0 D AV W 160 10 7 5

1 1 30

0

0.1

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0.99

0.9

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WAVELE 0.49 NGTH S TOW ARD 0.0 D AV W 160 85◦ . Now the right-most of the three islands of the 3 : 1 resonance have successfully cleared the phase space origin and particles are no longer confined to the strong resonance structure. The bunch now initially residing at the edge the chaotic region is once again allowed to diffuse outward to fill the thick chaotic layers of overlapping higher-order resonances surrounding the 3 : 1 and 1 : 1 stable fixed-points. When the modulation amplitude was increased further, the phase space structure continued to shift leftward as the synchrotron tune continued to further detune. When the modulation amplitude approaches am > 99◦ , the initial position of the bunch moves out of the chaotic region and is now confined within the tori about the stable fixed-points of a higher-order resonance. No diffusion was observed in this region of the am parameter space. The non-diffusive phase space structure when am = 104◦ , shown in Fig. 7.13, depict a structure riddled with many higher-order resonance islands that has yet to collapse. The initial bunch can be trapped in one

7.2 Simulation Results

201

1.5

Normalized Momentum δ

1

0.5

0

-0.5

-1

-1.5 -2.5

-2

-1.5

-1 -0.5 Phase φ (rad)

0

0.5

Figure 7.13: Poincaré surface of section depicting the non-diffusive longitudinal phase space structure on the verge of particle loss where am = 104◦ and ∆φo = 30◦ . Note that instead of the well-behaved tori, the structure is bounded by layers of overlapping resonances that will drive particle loss.

202

7. Phase Space Dilution of Beams

or many sets of these resonance islands. The edge of the phase space structure, that was previously occupied by the well-behaved bounding tori, is now replaced by the chaotic layers of overlapping and disrupted higher-order resonances with increasing modulation amplitude. At this point, any large region of chaos generated by the phase modulation that lacks the well-behaved bounding tori will result in beam loss. Finally, when the modulation amplitude was increased past am ≈ 115◦ , rapid particle loss was observed. Here the particle bunch continues to diffuse beyond the disrupted chains of higher-order resonances that once formed the well-behaved tori bounding the region of chaotic particle motion. These unbounded chaotic phase space structures are unsuitable for the controlled diffusion of the particle bunch.

7.2.2

Choice of Modulation Tune and Amplitude

Simulation results show that when the modulation amplitude am is varied, resonance islands within the phase space structure are disrupted and may collapse and shrink. The shape and arrangement of the resonance structures are altered as a result. These disruptions of resonances in proximity of the initial bunch location can be used to drive particles from confinement near the origin of phase space. Furthermore, the multitude of disrupted resonances can overlap to form thick layers of localized regions of chaos in which particles can diffuse into and occupy. This behavior was illustrated in the previous section where the modulation tune was held fixed at νs = 2νs for ∆φo = 30◦ and the modulation amplitude am was varied. In this section, the equilibrium properties of the particle bunch in the presence of phase modulation for the given phase space structure will be calculated when both the modulation amplitude and tune are varied. The limitations on the νm and am parameters in which the dilution


203

process will be useful in beam physics applications will be discussed. Longitudinal particle-tracking simulations were employed to exhaustively map the νm /νs and am parameter space to find the parameter regions exhibiting optimal particle bunch diffusion and equilibrium bunch stability. Rapid particle diffusion typically equilibrates after about 1 × 105 revolutions when appropriate parameters are chosen. The rms bunch length and momentum spreads were calculated for 1 × 104 macroparticles after 1 × 106 revolutions to insure that only bounded diffusion processes remained in the analysis. Thresholds for maximum rms bunch length and rms momentum spread were selected to consider physical constrains of the system and maintain feasibility of the technique. This includes considerations such as excluding bunch lengths greater than 75% of a given ring to allow for beam extraction and momentum spreads that are beyond the momentum acceptance of a typical storage ring. The parameter maps of the rms bunch lengths and momentum spreads in the modulation amplitude and modulation fraction parameter space are given in Fig. 7.14 and Fig. 7.15 for the phase difference parameter ∆φo = 30◦ . When the modulation fractions were selected to be νm /νs ∼ 1 and the modulation amplitude small, the particles were unable to be driven away from the origin of phase space due the weak influence of the applied phase modulation. As soon as the modulation amplitude approaches am ∼ 20◦ , particle loss sharply occurs and will persist with increase am . This result was expected as the particle bunch was predicted to be overdriven by the dominant driving resonant strengths of the lower harmonic parametric resonances. When modulation fraction were chosen to be νm /νs ∼ 3, the particle bunch remains confined to resonance islands near the origin of phase space unless considerably large modulation amplitude is applied (am > 120◦ ). This result is also consistent with the

204


1.4

3

1.2

1

0.8 2 0.6

1.5

0.4

0.2 1 0

20

40 60 80 100 Modulation Amplitude am (deg)

120

0

Figure 7.14: (color) Parameter map of the rms bunch length as a function of the modulation amplitude and modulation fraction where ∆φo = 30◦ . The dark blue regions represent parameters where the particles are confined and no diffusion occurs. The dark red regions represent parameters where the diffusion is unbounded and particles were lost in the process.

RMS Bunch Length σφ (rad)

Modulation Fraction νm /νs

2.5


205

predictions of a particle bunch severely under-driven by the weak driving resonance strengths of higher harmonic parametric resonances. In Fig. 7.14 and Fig. 7.15, the dark blue regions in the parameter space represent bunch sizes that remained small after 5 × 105 revolutions, i.e., bunches that have not fully undergone diffusion due to being confined to stronger resonances near the initial location of the bunch. The dark red regions represent bunch sizes that surpassed an upper operable threshold in which particle loss has been defined rendering the arrangement of resonant islands not useful for controlled longitudinal emittance blowup. It is in between these two extreme parameters limits in which bounded diffusion can be attained. In the modulation fraction range 1.6 . νm /νs . 2.0, a sizable region of parameters in which bounded particle diffusion was observed has been found for ∆φo = 30◦ . The rms bunch lengths within this parameter region are in the range of 0.5 rad . σφ . 0.9 rad and rms momentum spread in the range of 0.4 . σδ . 0.6. More importantly, this region of parameters was found to be relatively large with respect to current operating tolerances of phase modulated RF systems used in particle accelerators. The parameter insensitivity of the applied phase modulation in the beam dilution technique indicates that experimental results can be readily obtained without impractical and high cost of tightly constrained control electronics for the RF system.

7.2.3

Phase Difference Parameter ∆φo = 45◦

In the previous case where ∆φo = 30◦ , Fig. 7.5 and Fig. 7.6 show that a small bunch initially at the phase space origin driven by the 5 : 2 parametric resonance, and eventually by the dominant 3 : 1 resonance resulting in the surface of section shown

206


1.4

3

1.2

1

0.8 2 0.6

1.5

0.4

0.2 1 0

20


120

0

Figure 7.15: (color) Parameter map of the rms normalized momentum spread as a function of the modulation amplitude and modulation fraction where ∆φo = 30◦ . The dark blue regions represent parameters where the particles are confined and no diffusion occurs. The dark red regions represent parameters where the diffusion is unbounded and particles were lost in the process.

RMS Momentum Spread σδ


2.5


207

in Fig. 7.10 where diffusion has equilibrated. The approximate intercepts of the 3 × Qs /νs curve with the νm /νs = 2 horizontal dotted-line is far from the phase space origin shown in Fig. 7.6. This implies that the initial bunch location is far away from the stable fixed points of the 3 : 1 resonance, therefore particles are restricted from falling into the chaotic layers surrounding the 3 : 1 resonance islands unless sufficiently large modulation amplitude is applied. Applying phase modulation with large am can disrupt the well-behaved bounding tori required for successful bunch broadening. Since the resonance strength of the 2 : 1 resonance is much larger than that of the 3 : 1 resonance as shown in Fig. 7.7. This motivates our search for the double RF parameters that will better utilize the driving strengths of the dominant 2 : 1 resonance. The location of the potential well minimum is shifted by changing the phase difference parameter to ∆φo = 45◦ resulting in the new potential well minimum location at φo = 29.12◦ . This shows that moderate changes to the phase difference parameter from the maximum offset resulted in only slight shifts of the potential well minimum location. This insensitivity to phase difference parameter shifts that are about ∆φo = 30◦ , shown in Fig. 7.4, can be an advantage in beam dilution experiments when the phase difference parameters are being explored. The synchrotron tune Qs /νs as a function of the action J for ∆φo = 45◦ is shown in Fig. 7.16 where the 2 × Qs /νs curve had shifted upwards to meet the νm /νs = 2 horizontal dotted-line. With the boxed interception points occurring at approximately J = 0.085, the 2 : 1 resonance islands, and more importantly, the unstable fixed-points of the 2 : 1 resonance are now very close the initial location of the bunch at J = 0.106. Now relatively smaller modulation amplitudes are expected for particle diffusion to occur. The resonance strengths |gn (J)| for ∆φo = 45◦ , shown in Fig. 7.18, were found to deviate only slightly from the case where ∆φo = 30◦ with no other unique features

208

7. Phase Space Dilution of Beams 2.5 3×

Normalized Synchrotron Tune Qs /νs

8/3× 2 5/2× 2×

1.5

3/2× 1 1×

0.5

0

0.5

1

1.5 Action J

2

2.5

Figure 7.16: Normalized synchrotron tune Qs /νs and its multiples as a function of the action J. The νm /νs = 2 horizontal dotted-line interception illustrates the parametric resonances driven by phase modulation where the double RF parameters are r = 1/2, h = 2, and ∆φo = 45◦ . The vertical dotted-line where J = 0.106 depict the initial location of the bunch at the origin of phase space and the circular intercept points depict the two relevant 8 : 3 resonances that are required to be disrupted to enable a larger chaotic region to be filled.

3


209

1.5


1

0.5

0

-0.5

-1

-1.5 -2

-1.5

-1

-0.5 0 0.5 Phase φ (rad)

1

1.5

Figure 7.17: Poincaré surface of section depicting the non-diffusive longitudinal phase space structure with parameters am = 7◦ , νm /νs = 2, and ∆φo = 45◦ .

2

210


to be noted. The non-diffusive longitudinal phase space structure when am = 7◦ and ∆φo = 45◦ is illustrated in Fig. 7.17. Immediately, the 2 : 1 resonance structure can be identified near the origin of phase space, as predicted. The small bunch, however, can only spread out inside the thin chaotic layers surrounding the 2 : 1 resonance islands and cannot be driven to the larger chaotic region of 7 : 3, 8 : 3 and higher-order resonances beyond. In order for the bunch to diffuse into these aforementioned regions, the thin higher-order chains of islands enclosing the 2 : 1 resonance structure must be disrupted and collapse. The sharp increase in the rms spreads σφ and σδ , shown in Fig. 7.18, when the modulation amplitude is increased to about am ≈ 9◦ marks the condition where the collapse of the thin bounding higher-order chains of resonance islands allows for the diffusion of the particle bunch. With the modulation amplitude increased just beyond am ≈ 9◦ , the bunch sizes briefly level off and then increases again for the second time at am ≈ 23◦ . This second sudden increase in σφ and σδ could be explained by the existence of multiple 8 : 3 resonances each with their own sets of resonance islands. Their individual positions in action-space are depicted as circles in Fig. 7.16. The first set of resonance islands associated with the 8 : 3 resonance nearest to J = 0 collapses resulting in the sharp increase in the bunch sizes when am ≈ 9◦ . As the modulation amplitude increases to am ≈ 23◦ , the second set of 8 : 3 resonance islands also collapses resulting in the sharp increase in the bunch sizes for the second time. The Poincaré surface of section in Fig. 7.19 depict the condition just before the second set of 8 : 3 resonance islands collapses and shrink. The chain of eight islands associated with the second 8 : 3 resonance can be seen surrounding the entire phase space structure. The remnants of both sets of the 8 : 3 resonance islands, after the modulation amplitude has been


211

RMS Bunch Length σφ (rad) and Momentum Spread σδ

1

σφ

0.8

σδ

0.6

0.4

0.2 0

10 20 30 Modulation Amplitude am (deg)

40

Figure 7.18: The rms bunch spreads σφ and σδ as a function of modulation amplitude am where ∆φo = 45◦ . The hatched region depict the approximate regions where no diffusion was observed. The solid band depict the approximate region where beam loss was encountered. Data points where the error is much less than 1% have negligible error bars.

212


1.5


1

0.5

0

-0.5

-1

-1.5 -2

-1.5

-1

-0.5 0 Phase φ (rad)

0.5

1

1.5

Figure 7.19: Poincaré surface of section depicting the diffusion of a small bunch where am = 23◦ and ∆φo = 45◦ just as the bunch sizes sharply increase for the second time. Note the islands of the 8 : 3 resonance at the edge of the diffused structure just before collapse and subsequent shrinking.


213

increased beyond am = 23◦ , can be readily identified in Fig. 7.20 as small empty spaces in an otherwise uniform structure. The Poincaré surface of section where am = 28◦ and ∆φo = 45◦ is shown in Fig. 7.20. As compared to Fig. 7.10, the large empty regions of the 3 : 1 resonance stable fixed-points are now replaced by smaller empty regions, pertaining to the stable fixed-points of the 2 : 1 and 8 : 3 resonance. These resonance islands are now completely enveloped by a larger chaotic region. The overall phase space structure can be seen to be broader and more rectangular as compared to the case where ∆φo = 30◦ . The rms bunch length and momentum spreads, calculated after equilibration, were σφ = 0.93 ± 0.02 rad and σδ = 0.689 ± 0.004. The choice of these parameters provides a greater degree of uniformity of the longitudinal particle distribution as shown in particle distribution histogram in Fig. 7.21. Furthermore, the diffusion process converges and equilibrates about twice as quickly, after 6.5×104 revolutions shown in Fig. 7.22, as opposed to 1.5×105 revolutions. These improvements were attained at a modulation amplitude of am = 28◦ , which is about half the modulation amplitude as compared to am = 58◦ in the case where the potential well minimum offset was greatest from the phase space origin when ∆φo = 30◦ . The parameter maps of the rms bunch lengths and momentum spreads in the am and νm /νs parameter space for the phase difference parameter ∆φo = 45◦ , shown in Fig 7.23 and Fig 7.24, depict a slightly smaller parameter region where bounded diffusion occurs when compared to ∆φo = 30◦ . Yet a sizable region of parameter space still exists, ensuring that particle diffusion will be observed in experiments without unnecessarily strict constrains on operating parameters.

214


1.5


1

0.5

0

-0.5

-1

-1.5 -2

-1.5

-1


0.5

1

1.5

Figure 7.20: (Color) Black: Poincaré surface of section of a small Gaussian bunch of rms length σφ = 0.001 rad initially at the phase space center at the last modulation periods of 1.5 million revolutions where am = 28◦ and ∆φo = 45◦ . The red points depict single-particle tracking of test particles placed initially near the edge of the phase space structure.


215

1500 am = 58◦ ∆φo = 30◦ 1000

Particle Number

500

0 -2

-1.5

-1


0.5

1

1.5

1500 am = 28◦ ∆φo = 45◦ 1000

500

0 -2

-1.5

-1


0.5

1

1.5

Figure 7.21: Histogram of 30 bins spanning the entire length comparing the linear particle distribution of the two different double RF settings after diffusion. The black line depict the turn-by-turn averaged of the individual bins 1 × 106 revolutions after diffusion, confirming the stability of the structure.

216


1.2

σφ2

RMS Bunch Spreads σφ2 (rad2 ) and σδ2

1

0.8

0.6

σδ2

0.4

a = 28◦

0.2

0

0

∆φo = 45◦

0.5 1 Revolution Number (1 × 105 )

1.5

Figure 7.22: Longitudinal bunch sizes as a function of revolution number for am = 28◦ and ∆φo = 45◦ as the diffusion process evolves and equilibrate after about 6.5 × 104 revolutions.


217

1.4

3

1.2

1

0.8 2 0.6

1.5

0.4

0.2 1 0

20


120

0

Figure 7.23: (color) Parameter map of the rms bunch length as a function of the modulation amplitude and modulation fraction where ∆φo = 45◦ . The dark blue regions represents parameters where the particles confined and no diffusion occurs. The dark red regions represents parameters where the diffusion is unbounded and particles are loss.

RMS Bunch Length σφ (rad)


2.5

218


1.4

3

1.2

1

0.8 2 0.6

1.5

0.4

0.2 1 0

20


120

0

Figure 7.24: (color) Parameter map of the rms normalized momentum spread as a function of the modulation amplitude and modulation fraction where ∆φo = 45◦ . The dark blue regions represents parameters where the particles confined and no diffusion occurs. The dark red regions represents parameters where the diffusion is unbounded and particles are loss.

RMS Momentum Spread σδ


2.5


7.2.4

219

Modulation Phase ηm

The modulation phase is related to the synchronization of the applied phase modulation with the combined double rf waveform, referenced to the revolution of the particle bunch. Though seemingly innocuous, the modulation phase ηm when varied, yielded some interesting results that merits further examination. For the case where ∆φo = 30◦ , am = 28◦ , and νm = 1.6νs , the entire phase space structure rotates about the potential well minimum when η is varied. This is shown in Fig. 7.25 where 1 × 104 particles were tracked for 1 × 106 revolutions, with ηm configured from ηm = 0◦ to ηm = 150◦ in increments of 30◦ . It is interesting to note that by varying the modulation phase, the resulting phase space structure redistributes particles in a way such that both the bunch length and linear particle density are altered. This will be important in synchronization consideration for the extraction of the particle bunch. In the special circumstances where large and strong resonance islands happen to exist near the origin of phase space where the particle bunch initially resides, the rotation of the phase space structure may cause the small bunch to be trapped in a nearby resonance island and inhibit particle diffusion. For the case where ∆φo = 30◦ , am = 58◦ , and νm = 2νs , the rotation of the phase space structure was observed as η was varied. However, when ηm = 60◦ the edge of the large central 1 : 1 resonance coincides with the initial location of the particle bunch. The particles are confined to the large central resonance island and no diffusion is observed. This is shown in Fig 7.26, depicting the rotating phase space structure with varying η where diffusion has been inhibited when ηm = 60◦ .

220


2 1.5 1 0.5 0 −0.5 −1 −1.5 −2 −3


−2

ηm = 0◦

ηm = 30◦

ηm = 60◦

ηm = 90◦

ηm = 120◦

ηm = 150◦

−1 0 1 Phase φ (rad)

2

3

Figure 7.25: Poincaré surface of sections depicting the diffusive longitudinal phase space structures for varying modulation phase ηm where am = 28◦ , ∆φo = 30◦ , and νm = 1.6νs . The phase space structure is seen to rotate about the potential well minimum.


221

ηm = 0◦

ηm = 30◦

ηm = 60◦

ηm = 90◦


1.5 1 0.5 0

−0.5 −1

−1.5 −2

−1 0 1 Phase φ (rad)

2

Figure 7.26: Poincaré surface of sections depicting the phase space structures for varying modulation phase η where am = 58◦ , ∆φo = 30◦ , and νm = 2.0νs . When η = 60◦ , the initial particle bunch at the origin of phase space is confined within the large 1 : 1 resonance island and diffusion was inhibited.

222

7.3


Implementation of the Dilution Method

The ALPHA project, introduced in Section 5.2.4, was designed to manipulate electron beams for extreme environment radiation effects experiments. The electron bunch must be broadened before accumulation in order to obtain the large total bunch charge of up to 600 nC while circumventing adverse space-charge effects. It has been proposed that particle bunch broadening can be achieved by applying phase modulation to a specially configured double RF system to produce a bunch of length up to 40 ns. A schematic outline of how the bunch broadening will be incorporated into the charge accumulation will be given as follow. An electron bunch is initially injected into the ring at full energy from a 50–100 MeV electron linac. The bunch is allowed to undergo radiation damping to reach an equilibrium bunch length of ∼100 ps in the RF bucket generated by the double RF cavity system where the phase difference is set as ∆φo = 0◦ . To rapidly broaden the small radiation damped bunch, the phase difference parameter immediately switches to ∆φo = 45◦ and appropriate phase modulation is applied to both the RF cavities. After the process of phase space dilution has equilibrated, after about 4 ms, charge accumulation can begin. The pair of magnetic bumpers will introduce a localized closed-orbit distortion in the vicinity of the Lambertson septum and the bumper fields are allowed to decay in synchronization with subsequent injection of electrons from the linac. This process, known as the 4-turn closed-orbit distortion injection scheme, will continue until the desired total bunch charge has been attained. A 100 kV traveling-wave fast-extraction kicker will be used to displace the electron trajectory so that it may coincide with the field region of the Lambertson septum for extraction. The electron bunches that are extracted from the ring after bunch broadening and accumulation will have variable bunch lengths up to 40 ns, variable total bunch charge up to 600 nC, be free of S-band

7.3 Implementation of the Dilution Method

223

micro-structures, and have properties of longitudinal particle distribution uniformity. The field-programmable gate array (FPGA) based low-level RF controls for the double RF system are currently being designed by M. Ng. The design will include a low-noise master oscillator for timing reference, output RF signals of harmonic number h1 = 1 and h2 = 2 to drive the RF power amplifiers, and input cavity samples for feedback controls. The control electronics will be able to interface with LINUX and the Experimental Physics and Industrial Control System (EPICS) software for remote phase and amplitude controls integration with existing accelerator control systems. As it turns out, an implementation solution using AC-dipoles exist that can greatly simplify RF hardware and operating consideration required for beam dilution.

7.3.1

Alternative Implementation with an AC Dipole

An alternative implementation of the phase space beam dilution method can be realized by using an AC dipole in conjunction with a double RF system with optimally flattened potential well. The AC dipole deflecting particles in the horizontal plane in a non-zero dispersive region will effectively replace the phase modulation applied to the both RF cavities. Some of the advantages of choosing the AC dipole mode of implementation are as follow: AC dipoles are easy to manufacture, they are relatively inexpensive, easy to operate, and will greatly simplify the RF control electronics required to drive rapid particle diffusion in the process of phase space dilution. In this section, the equations of motion in the presence of an AC dipole will be derived and the Hamiltonian constructed to show that the implementation of AC dipoles can indeed be used as a vehicle to carry out the phase space beam dilution. A discussion on the optimal placement of an AC-dipole in a dispersive region, and required peak

224


magnetic fields and frequency of oscillation will be presented. Consider the path length of a particle circulating in a storage ring given by the expression C=

I q

2

(1 + x/ρ) +

x02

+

z 02

I ds ≈ Co +

x ds + . . . , ρ

(7.21)

where x0 and z 0 terms are path length deviations due to the horizontal and vertical betatron motion, ρ is the dipole bending radius keeping the particle in circulation, and Co is the path length of an ideal reference particle on the design trajectory. Since the oscillation of the betatron motion is much faster (on the order of 10’s of MHz for ALPHA) compared to the alternating frequency of the AC dipole, proportional to the synchrotron frequency (10’s of kHz for ALPHA), the path length difference due to betatron motion averages to zero. Only the x/ρ dipole term is considered. For dipole errors distributed about a given circular accelerator, the closed-orbit (CO) displacement can be obtained by a linear superposition of each dipole kick distributed around the ring expressed as Z t+To ∆B(t) Gx (s, t) xCO (s) = dt, Bρ t where the horizontal Green’s function of Hill’s equation is given as p βx (s)βx (t) Gx (s, t) = cos (πνx + ψx (s) − ψ(t)) , 2 sin (πνx )

(7.22)

(7.23)

βx (t) betatron function over the region of distributed dipole errors, βx (s) betatron function at the point of interest s, νx is the horizontal betatron tune, ψx (s) is the horizontal phase advance function, ψ(t) is a phase determined by initial conditions, and the distributed dipole error function can be defined as ∆Θ(t) =

∆B(t) . Bρ

(7.24)


225

For a single localized dipole kick Θ(so ) at location so , the difference in the resulting closed-orbit path length as compared to the design trajectory can be expressed as I Gx (s, so ) ds = Dx (so )Θ(so ), (7.25) ∆C = C − Co = Θo ρ where the horizontal dispersion function at location so is defined as p I I p βx (so ) βx (s) Gx (s, so ) ds = cos (πνx − |ψx (s) − ψ(so )|) ds. Dx (so ) = ρ 2 sin πνx ρ (7.26) An angular deflection of Θ(so ) in the dispersive region where the dispersion is Dx (so ) at the location so give rise to differences in path length deviating from the ideal reference particle. In terms of synchronous phase φ, referenced to the primary RF cavity of harmonic number h1 = 1, Eq. (7.25) can be re-expressed as 2π 2π ∆φ = −h1 ∆θ = −h1 ∆C = −h1 Dx (so )Θo (so ) Co Co

(7.27)

For an AC dipole with harmonic time-dependency that is referenced to the primary RF cavity, the change in synchronous phase can now be expressed as ! âc ` B 2π ∆φ = −h1 Dx (so ) sin (νac θ + ηac ) , Co Bρ

(7.28)

âc ` is the integrated peak AC dipole field amplitude, ` is the length of the where B AC dipole, Bρ is the momentum rigidity, νac is the AC dipole tune, and ηac is the AC dipole phase. The time-dependent variable θ is the angular displacement of the circulating particle where θ = 2π is one complete revolution around the ring. The equation of motion for φ with respect to the time-dependent variable θ can be expressed as dφ = νs δ − h1 dθ

2π Co

Dx (so )

âc ` B Bρ

! sin (νac θ + ηac ) .

(7.29)

226


Now the perturbation to the Hamiltonian of the double RF system, shown in Eq. (7.1), due to the harmonic time-dependent kick of the AC dipole is given as ! âc ` B 2π Dx (so )δ sin (νac θ + ηac ) . H1ac = −h1 Co Bρ

(7.30)

The original perturbation to the double RF Hamiltonian where phase modulation was applied to both RF cavities is given as H1 = am δνs sin (νm θ + ηm ) .

(7.31)

When comparing Eq. (7.30) to Eq. (7.31), a simple transformation relating the two different implementation can be obtained am νs −h1

2π Co

Dx (so )

âc ` B Bρ

! ,

νm νac ,

ηm ηac .

(7.32)

The set of transformations presented in Eq. (7.32) must be satisfied for the two different implementation of phase space beam dilution to be equivalent and the optimal bunch broadening parameters found to be still applicable through the use of an AC dipole. Furthermore, this method makes a few assumptions that need to be addressed. The dispersion was assumed to be only in the horizontal direction and no dispersion in the vertical direction as the bending magnets bend in the horizontal plane to maintain particle circulation. The synchro-beta coupling is assumed to be negligibly small so that more complicated coupling dynamics and resonances are not encountered. Both of these assumptions are true for the ALPHA storage ring. Suppose an AC dipole of length ` = 0.1 m was to be build for the ALPHA storage ring with the primary (h1 = 1) RF cavity providing a peak voltage of V1 = 1 kV at the accelerating gap and the secondary (h2 = 2) RF cavity V2 = 0.5 kV. Fig. 7.27 shows âc product for the two cases phase space dilution parameters studies in the Dx (so )B


227

140 50 MeV 75 MeV 100 MeV

120

âc (Gauss-m) Dx (so )B

100 ∆φo = 30◦ am = 58◦ 80

60 ∆φo = 45◦ am = 28◦ 40

20

0 0.4

0.6

0.8 1 1.2 1.4 1.6 1.8 Wiggler Bending Radius ρw (m)

2

2.2

Figure 7.27: The product the dispersion function and peak magnetic âc as a function of the gradient damping wiggler bendfield Dx (so )B ing radius ρw and injection beam energy. Two different phase space beam dilution parameters, ∆φo = 30◦ am = 58◦ and ∆φo = 45◦ am = 28◦ , are depicted.

228


âc the previous section: ∆φo = 30◦ am = 58◦ and ∆φo = 45◦ am = 28◦ . The Dx (so )B product is shown as a function of the gradient damping wiggler bending radius ρw and for various beam energies to be injected into the ring from the electron linac.

Chapter 8 Conclusion

The development of compact storage rings represent the natural progression in the evolution of circular particle accelerator technology. These compact particle accelerators have played critical roles in enabling smaller institutions, university-based laboratories, and medical facilities unprecedented access to modern tools and capabilities that were once limited to large-scale national and international scientific collaborations. These devices have been utilized for x-ray production and higher-order manipulation of beams for experiments that span and encompass a multitude of disciplines. The wide-spread dissemination of particle accelerator technology would also imply that new concepts and techniques pertaining to beam physics can be readily tested, developed, and further refined to guide the design of future machines critical to new and exciting scientific discoveries that are to come. Compact circular accelerators have unique developmental hurdles to overcome that will require careful design considerations. Operating in the lower energy regime, space-charge effects begin to complicate the beam dynamics. In some cases it may 229

230

8. Conclusion

lead to lattice instabilities and uncontrolled emittance blowup. Practical constraints such as space limitation for instruments and beam diagnostics on compact rings will also need to be considered. Critical particle accelerator components such as bending magnets, focusing magnets, steering magnets, and RF resonant structures must be designed with strict dimensional constrains to allow for proper implementation on the compact ring. A compact ferrite-loaded quarter-wave RF resonant cavity has been designed, constructed, and tested to meet the requirements of the ALPHA storage ring and the radiation effects experiments at Indiana University. The design of the RF resonant cavity was guided by the Poisson-SUPERFISH field simulator and ferrite-loaded cavity theory. Since the natural length of a quarter-wave structure scales inversely with operating RF frequency, the ALPHA storage ring operating at 15 MHz would yield a RF structure that will be about 5 m in length. Thus the use of a conventional resonant structure will occupy too much space in the 20 m circumference storage ring. The M4 C21A ferrite rings were chosen for their low-loss performance for RF frequencies 60 MHz and under, low cost, availability, and previously used in particle accelerator projects at Los Alamos and FermiLab. The loading of ferrite material into the resonant medium will impede the wave velocities of the electromagnetic fields contained within the resonant cavity. This will effectively reduce the resonant frequency of the compact accelerating structure. A variable capacitor (8–650 pF) across the accelerating voltage gap was used to enabling resonant frequency tuning of the RF cavity in the range 6.9–27.6 MHz. The optimal geometries for the ferrite-loaded resonant structure operating at 15 MHz has been found with overall cavity length of ` = 37.3 cm, inner conductor radius of r1 = 5.7 cm, and outer conductor radius r4 = 10.8 cm. The performance of the ferrite-loaded quarter-wave RF cavity has been bench-

Conclusion

231

marked at the Indiana University Center for Exploration of Energy and Matter. Direct gap voltage measurements and bead-pull measurements were used to characterize the response of the RF cavity to the low input RF power. The shunt impedance of the RF cavity including the matching network was measured to be Rs = 5.565 kΩ. A calibrated vector network analyzer was used to measure the loaded Q-factor to be Qloaded = 65.7. The [R/Q] figure of merit was calculated to be [R/Q] = 40.8 which is consistent with ferrite-loaded cavity designs used at other accelerator facilities. From the Touschek scattering calculations, the requirement of the at least 1 kV peak gap voltage has been established to provide a deep RF bucket for adequate beam lifetime in the radiation effects experiments. For the shunt impedance measured, the RF power amplifier requirement of 1 kW has been specified to drive the quarter-wave RF cavity. Several beam-based measurements were used to measure the frequency of the synchrotron oscillations and ascertain the influence of the cavity gap voltage on the electron bunch. The gap voltage measurement obtained from the beam response agrees well with the gap voltage prediction from the low-power RF cavity characterization experiments (within 5%). The resulting ferrite-loaded quarter-wave structure built is currently the most compact structure of its kind in literature. This RF cavity design can potentially be applicable in the development and operation of future compact storage rings. The radiation effects experiments at Indiana University requires the delivery of an electron beam that is broadened up to 40 ns in length, with large total bunch charge up to 600 nC, low peak current, high average current, and enhanced duty factor. Due to physical scale of the compact electron storage ring, conventional methods of bunch broadening cannot be applied to a stored electron bunch that may fulfill the radiation effects experimental requirements. These method include the enhancement of the equilibrium bunch length due to a double RF system with flattened potential well,

232

8. Conclusion

RF barrier bucket manipulations, and band-limited phase noise excitation. Motivated by several experiments that explored phase modulation driven bunch diffusion at the Indiana University Cooler Storage Ring in 1997, a phase space beam dilution method has been developed. The method will combine both the flattened double RF potential and the applied phase modulation bring about the controlled longitudinal emittance blowup of a stored radiation damped electron bunch. The applied phase modulation will excite parametric resonances in the bunch. The more strongly the phase modulation is applied, i.e., for applied phase modulation with larger am , the more numerous distinct parametric resonances are driven in the bunch. Resonant islands corresponding to the driven parametric resonances will eventually be disrupted and overlap to form large regions of chaotic particle motion in the synchrotron phase space. These regions will allow particles to undergo diffusion for bunch broadening. Even when large regions of chaos have been generated, the particles may still be confined to strong resonant islands in the vicinity of the initial location of the bunch. These confined particles are unable to escape and will not traverse the region of chaotic particle flow to undergo diffusion. A solution has been found that allows for the shifting of these chaotic regions of overlapping resonances. By changing the relative phase between the two RF cavities, the large regions of chaos are configured to coincide with the initial location of the radiation damped particle bunch. When optimal parameters are configured, the small particle bunch can be rapidly broadened in a manner that is useful for the radiation effects experiments. A simulation study exploring the dependence of driven parametric resonances on the applied phase modulation parameters am , νm , and ηm has been presented for the double RF system with variability of phase difference parameter ∆φo . Analysis has

Conclusion

233

been carried out on the continuous-time Hamiltonian of particle motion to predict specific parametric resonances that will be driven by the applied phase modulation and how strongly each m = 1 order parametric resonance are driven. The results were used to interpret the particle tracking simulations. The framework for describing distinct parametric resonances generated have been presented. Longitudinal symplectic multi-particle tracking have been used to uncover the resulting structure and interaction of many parametric resonances driven by the applied phase modulation. Poincaré surfaces of section were introduced to analyze the particle tracking data when beam dilution parameters such as the phase difference ∆φo , modulation amplitude am , modulation tune νm , and modulation phase ηm are varied. The phase space beam dilution parameters that will optimally broadening the small electron bunch for the ALPHA storage ring have been reported. Furthermore, the resulting bunch after the dilution process will exhibit longitudinal particle distribution uniformity that is critical for the radiation experiments. Using the particle tracking data in conjunction with the analytical double RF theory, various features and anomalous bunch diffusion behavior have been explained. An exhaustive mapping of the modulation fraction νm /νs versus modulation amplitude am parameter space have been carried out. The results shows that large regions of parameter space exist where phase space beam dilution occurs and that the process is not overtly parameter sensitive. This will be important in the implementation of the dilution method in that the specification of the RF control electronics will not have to be cost prohibitive. The precise phase modulation of both the RF cavities may be difficult to achieve. An alternative implementation of the beam dilution method has been presented that will utilize an AC dipole together with a double RF system that will greatly simplify RF control electronics required. The alternating dipole field placed in a non-zero dispersive region of the storage ring will yield harmonic time-

234

8. Conclusion

dependent path length change that was shown to be equivalent to phase modulation applied to both the RF cavities. The product of the horizontal dispersion function at âc , have been found the location of the AC dipole and peak magnetic field, Dx (so )B as a function of storage ring parameter such as the damping wiggler bending radius ρw and beam energy Eo relevant to the ALPHA storage ring that will being about phase space beam dilution. The phase space beam dilution method will be important for the handling of high intensity beams in future particle accelerators. By decreasing the peak current of the beam through bunch broadening, space-charge effects can be mitigated. The method will also have application in industrial radiation effects experiments, and many other applications that require bean uniformization and direct beam impingement.

Appendix A Synchrotron Radiation

In this appendix, we revisit the mathematical derivation and formalism in classical electrodynamics that forms the basis of our understanding on the topic of synchrotron radiation in circular particle accelerators. We will start with a discussion of the Liénard–Wiechert Potentials to describe the electromagnetic fields of a moving charged particle, then use the results to derive Larmor’s formula.

A.1

Li´ enard–Wiechert Potentials

The scalar potential field ϕ and the vector potential field A equation pair for a source point charge q at position rs traveling at velocity β is known as the Liénard–Wiechert potentials and are given as 235

236

A. Synchrotron Radiation

q , (1 − n · β) |r − rs | tr µo c qβ A(r, t) = , 4π (1 − n · β) |r − rs | tr

1 ϕ(r, t) = 4πo

(A.1a) (A.1b)

where β(t) = vs (t)/c, tr = t − Rs (tr )/c denotes the quantities evaluated at the retarded time, and Rs (tr ) is the distance from the source at the retarded time. The electric and magnetic fields can be derived directly from the potentials using the definitions, E = −∇ϕ −

∂A , ∂t

(A.2a)

B = ∇ × A.

(A.2b)

The resulting electric and magnetic fields when combining Eq. (A.2a) and Eq. (A.2b) with the potentials Eq. (A.1a) and Eq. (A.1b) are given as   ˙ qn × (n − β) × β q(n − β) 1   , E(r, t) = + 4πo γ 2 (1 − n · β)3 |r − rs |2 c(1 − n · β)3 |r − rs |

(A.3a)

tr

B(r, t) =

qn × n × (n − β) × β˙ µo  qc(β × n) + 4π γ 2 (1 − n · β)3 |r − rs |2 (1 − n · β)3 |r − rs | 

  ,

(A.3b)

tr

where n(t) =

A.2

r − rs , |r − rs |

1 γ(t) = q . 1 − |β(t)|2

Larmor’s Formula

Larmor’s equation, first derived by physicist J. J. Larmor in 1897, describes the total power radiated by a non-relativistic accelerating point charge. The relativistic extension to Larmor’s formula was developed by Alfred-Marie Liénard in 1898 and

A.2 Larmor’s Formula

237

independently by Emil Wiechert in 1900 which will form the foundation of our understanding of synchrotron radiation. The Poynting vector in SI units is given as S=

1 E × B. µo

Substituting Eq. (A.3a) and Eq. (A.3b) into Eq. (A.4) yields the expression ˙ 2 q 2 n × (n × v) S= . 16π 2 o c cr

(A.4)

(A.5)

When the angle between the acceleration vector and the observation vector is defined to be θ, Eq. (A.5) becomes S=

q2 v˙ 2 ˆ. sin2 θ n 16π 2 o c3 r2

(A.6)

Eq. (A.6) gives the power radiated per unit solid angle by an accelerating charge. To get the total power radiated, Eq. (A.6) is integrated over the entire solid angle to arrive at the well-known expression Pγ =

1 2q 2 v˙ 2 . 4πo 3c3

(A.7)

Appendix B Derivation of Ferrite-loaded Cavity Quantities

B.1

Effective Permittivity and Permeability

B.1.1

Line Capacitance

The line capacitance can be obtained by appealing to the electric potential in the quasi-electrostatic regime. With geometries define on Fig. 6.1, the electric potential per unit length within the resonant structure can be expressed as Z r4 Z r2 Z r3 Z r4 V = E(r) · dr = Eair (r) · dr + Ef errite (r) · dr + Eair (r) · dr, (B.1) r1

r1

r2

r3

where the electric field per unit length is defined as E(r) =

q rˆ, 2πo r

238

(B.2)

B.1 Effective Permittivity and Permeability

239

and the relative permittivity is =

  1 in air  

(B.3)

.

in ferrite

Solving through the integrals and rearranging terms, the line capacitance is written as 2πo . (B.4) r2 1 r3 r4 ln + ln + ln r1 r2 r3 The effective permittivity for a single ferrite ring will be defined such that it is related C` =

q = V

to the inner and outer conductor radius of the cylindrical cavity given as 1 e = . r4 r2 r3 1 r4 ln ln + ln + ln r1 r1 r2 r3

(B.5)

A geometric factor encapsulating all of the radial geometry of the cylindrical can be defined as k = ln

r3 r4 / ln , r2 r1

(B.6)

such that the effective permittivity for a single ferrite ring can be expressed as e =

. k + (1 − k)

(B.7)

When averaging over the air gap between each ferrite ring within the cavity, the effective permittivity can be written as e = k + (1 − k)

d1 d1 + d2

,

(B.8)

and thus arriving at the final expression for the line capacitance of a ferrite-loaded cylindrical cavity shown in Eq. 6.7

d1 2πo 2πe o d1 + d2 . C` = = r4 r4 ln [k + (1 − k)] ln r1 r1

(B.9)

240

B.1.2

B. Derivation of Ferrite-loaded Cavity Quantities

Line Inductance

For a current I flowing through a coaxial line, the line inductance is given as Z r2 Z r3 Z r4 Z r4 µo I µµo I µo I B(r) · dr = dr + dr + dr (B.10) L` I = 2πr r1 2πr r2 r3 2πr r1 Solving through the integrals, the line inductance is written as r3 r4 µo r2 + µ ln + ln . L` = ln 2π r1 r2 r3

(B.11)

The effective permeability for a single ferrite ring will be defined such that it is related to the inner and outer conductor radius of the cylindrical cavity given as r4 r2 r3 r4 µe ln = ln + µ ln + ln . r1 r1 r2 r3

(B.12)

Using the same geometric factor previously defined in Eq. B.6, the effective permeability for a single ferrite ring can be expressed as µe = 1 + k (µ − 1) .

(B.13)

When averaging over the air gap between each ferrite ring within the cavity, the effective permeability can be written as µe = 1 + k (µ − 1)

d1 d1 + d2

,

(B.14)

and thus arriving at the final expression for the line inductance of a ferrite-loaded cylindrical cavity shown in Eq. 6.8 µe µo r4 µµo r4 d1 L` = ln = [1 + k (µ − 1)] ln . 2π r1 2π r1 d1 + d2

(B.15)

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