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mm mm.“

EXPERIMENTAL STUDY OF AN
ION CYCLOTRON RESONANCE ACCELERATOR

by

Christopher T. Ramsell

A DISSERTATION
Submitted to:
Michigan State University

Department of Physics and Astronomy
National Superconducting Cyclotron Laboratory

in partial fulﬁllment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

2000

ABSTRACT

EXPERIMENTAL STUDY OF AN
ION CYCLOTRON RESONANCE ACCELERATOR

by

Christopher T. Ramsell

The Ion Cyclotron Resonance Accelerator (ICRA) uses the operating principles of
cyclotrons and gyrotrons. The novel geometry of the ICRA allows an ion beam to drift
axially while being accelerated in the azimuthal direction. Previous work on electron
cyclotron resonance acceleration used waveguide modes to accelerate an electron beam
[5]. This research extends cyclotron resonance acceleration to ions by using a high ﬁeld
superconducting magnet and an rf driven magnetron operating at a harmonic of the
cyclotron frequency. The superconducting solenoid provides an axial magnetic ﬁeld for
radial conﬁnement and an if driven magnetron provides azimuthal electric fields for
acceleration. The intent of the ICRA concept is to create an ion accelerator which is
simple, compact, lightweight, and inexpensive. Furthermore, injection and extraction are
inherently simple since the beam drifts through the acceleration region. However, use of
this convenient geometry leads to an accelerated beam with a large energy spread.
Therefore, the ICRA will be most useful for applications which do not require a mono-
energetic beam. An ICRA designed to accelerate protons to 10 MeV would be useful for
the production of radioisotopes, or neutron beams, as well as for materials science

applications.

As a ﬁrst step toward producing an ICRA at useful energies, a low energy ICRA
has been designed, built, and tested as a demonstration of the concept. Analytical theory
and a full computer model have been developed for the ICRA. Beam measurements
taken on the ICRA experiment have been compared with theory.

The ICRA computer model uses realistic ﬁelds of the solenoid, magnetron, and
electrostatic bend. This code tracks single particle trajectories from the ion source
through the entire system to a target face. A full emittance injected beam can be
modeled by tracking many single particle trajectories.

The ICRA experiment is designed to accelerate a proton beam from 5 keV to 50
keV in 5 turns. A superconducting solenoid provides a 2.5 Tesla axial magnetic field.
The accelerating structure built for the experiment operates at 152 MHz (4th harmonic)
and provides 3 kV across 8 gaps. Measurements of the accelerated beam current vs.
beam orbit radius indicate an energy distribution ranging from near zero to near the full
design energy, with 7% of the beam current above 24 keV and 1% above 42 keV.

Energy distributions generated using the ICRA computer model show reasonable
agreement with the experimental data. After a small correction of the bend voltage, the
computer model shows good agreement with the magnitude and shape of the
experimental data for a wide range of turn number.

Finally, a scheme for optimization of the basic ICRA design is given. Design
parameters are identiﬁed which minimize cost and which maximize the accelerated beam
current. Three 10 MeV proton designs are given which offer a compromise between low

cost and a'high quality beam.

DEDICATION

To my beautiful wife, Katrina.

She gives me strength and makes life worth living.

iv

ACKNOWLEDGMENTS

I want to thank Terry Grimm. The ICRA concept was his idea, and I am
thankful to Terry for giving me the opportunity to design and build an entire particle
accelerator and gain the all the experience that comes with such a project. Terry is a
very smart guy, and he has taught me most of what I know about accelerators.

I want to thank my advisor, Richard York, for being very fair and easy to talk to
and for providing funding for this project. York is relentlessly practical and matter of
fact. What I learned from him is best stated in his own words: “Don’t worry about the
mysteries of the universe. Just get the damn thing working !”

A special thanks goes to Felix Marti. Felix was not originally involved in my
project, however he had the knowledge and expertise that we needed, so he ended up
getting sucked into it anyway.

Also thanks to Chris Compton, a good friend who has been here through most of
the project. Over the years Chris helped on many things, including design of the support
structure for the ICRA and taking data, just to mention a few.

I also want to acknowledge the contribution of each student who worked on this
project for me. In chronological order they are:

Scott Rice built components of the rf cavity, and toiled with me through the magnetic
mapping process. Thanks Scott !

Erica Blobaume and I built the ion source and Einzel lens together as an REU project.
Thanks Erica !

Nathan Pung was very helpful with the experimental set up, building the support
structure, alignment of components, and running the ICRA. His help enabled me to
move ahead at a time when I really needed it. Thanks Nate !

Joel Fields worked on the setup and alignment as well as computer modeling of the

50 keV ICRA. Thanks Joel !

TABLE OF CONTENTS

LIST OF SYMBOLS AND ABBREVIATIONS .......................... ix
LIST OF TABLES .................................................. xii
LIST OF FIGURES ................................................. xiv
1. INTRODUCTION ................................................ 1
1.1 Cyclotrons ................................................... 2

1.2 Gyrotrons .................................................... 4

1.3 The Magnetron ............................................... 6

1.4 Ion Cyclotron Resonance Acceleration ............................ 7

1.5 Applications ................................................. 10

2. THEORY ...................................................... 12
2.1 Magnetic Field ............................................... 12

2.2 Vacuum .................................................... 15

2.3 Ion Source .................................................. 16

2.4 RF Cavity ....... ‘ ............................................ 17
2.4.1 Waveguides ............................................ 18

2.4.2 Coaxial Quarter Wave Cavity .............................. 24

2.4.3 RF Driven Magnetron .................................... 34

2.4.4 Equivalent Circuit Representation ........................... 39

2.4.5 The Hybrid Quarter Wave — Magnetron Cavity ................. 42

2.5 Analytical Model of the Beam Trajectory .......................... 46
2.5.1 Electrostatic Bend ....................................... 51

2.5.2 Conservation of Magnetic Moment ......................... 53

2.5.3 Mirror Condition ........................................ 58

2.5.4 Mass Spectrum of the Injected Beam ......................... 61

2.5.5 Number of Turns Through the Acceleration Region ............. 64

2.5.6 Acceleration Region ..................................... 65

2.5.7 RF Phase and Magnetic Field Flatness Criterion ............... 70

2.5.8 Axial Turn Length in the Extraction Region .................. 75

vi

2.6 Computer Model of the Beam Trajectory .......................... 77

2.6.1 The Computer Code ..................................... 79
2.6.2 Phase Dependence and Acceleration ........................ 81
2.6.3 Cavity End Field Effects ................................. 92
2.6.4 Magnetic Field and Phase ................................ 95
2.6.5 Complete Single Particle Trajectories ....................... 97
2.6.6 Acceptance Phase Space of the RF Driven Magnetron ........... 112
2.6.7 Injected Beam Emittance .................................. 118
2.6.8 Energy Distribution of the Accelerated Beam .................. 124

3. DESIGN ........................................................ 127
3.1 Basic System Parameters ....................................... 128
3.2 Components of the System ..................................... 139
3.2.1 Superconducting Magnet ................................. 148
3.2.2 Vacuum System ........................................ 148
3.2.3 Ion Source, Einzel lens, and Electrostatic Bend ............... 151
3.2.4 RFCavity ............................................. 156
3.2.5 Beam Diagnostics ....................................... 162
3.2.6 Mounting Structure ...................................... 166

4. EXPERIMENTAL RESULTS ..................................... 168
4.1 Magnet Mapping ............................................. 169
4.2 Ion Source .................................................. 172
4.3 RF Cavity .................................................. 174
4.3.1 Experimental Methods ................................... 174
4.3.2 Summary of RF Measurements ............................ 189

4.4 Alignment .................................................. 190
4.5 Measurements of the Injected Beam ............................. 193
4.5.1 Ion Spectrum ........................................... 194
4.5.2 Radius of Beam Orbit .................................... 200
4.5.3 Mirror Curve ........................................... 201
4.5.4 Number of Turns Through the Acceleration Region ............ 203
4.5.5 Injected Beam Emittance ................................. 211

vii

4.6 Measurements of the Accelerated Beam ........................... 214

4.6.1 Energy Distributions ..................................... 215

4.6.2 Resonance Width ....................................... 226

5. DISCUSSION ................................................... 228
5.1 Discussion of Results ......................................... 228
5.2 The Next Step for the ICRA .................................... 232

6. CONCLUSION ................................................. 252
APPENDIX ...................................................... 254
REFERENCES .................................................. 263

viii

CTN

mnpnno

I>Loss

LIST OF SYMBOLS AND ABBREVIATIONS

radius of inner conductor of coaxial quarter wave cavity
radius of outer conductor of coaxial quarter wave cavity
speed of light

conductance for vacuum pumping [Liters/sec]
capacitance

capacitance of coaxial section of length Z

capacitance of magnetron section

beam energy (also use K)

electric ﬁeld

Faraday cup

resonant frequency

cyclotron frequency

rf frequency

gap size between bending plates
ion cyclotron resonance accelerator
inner diameter

kinetic energy (also use E)

inductance
axial length
axial length of the rf driven magnetron

effective length of the bending plates in the direction of the beam.

Niobium-Titanium, a material commonly used to make superconducting wire

National Superconducting Cyclotron Laboratory

number of orbits the beam goes through while traversing the cavity length

network analyzer

outer diameter

pressure [Torr]

positron emission tomography

power lost due to resistance in the rf cavity

ix

Pﬁ rf power

p" component of momentum parallel to the local magnetic ﬁeld

p 1 component of momentum perpendicular to the local magnetic ﬁeld
pz axial component of momentum

Q Quality factor of a resonant circuit or resonant cavity

Q throughput [Torr Liter/sec]

rf radio frequency

R inner radius of magnetron cavity

Rs shunt resistance

s arc length

S pumping speed of turbo pump [Liter/sec]

Tc critical temperature of a superconducting material

TE transverse electric

TEM transverse electric and magnetic

TM transverse magnetic

wan initial voltage the beam is accelerated thru when extracted from the ion source
med voltage difference between the upper and lower electrostatic bending plates
x horizontal position in the accelerator centered coordinate system
xs horizontal position in the coordinate system of the ion source

x’ slope dx/dz

y vertical position in the accelerator centered coordinate system

ys vertical position in the coordinate system of the ion source

y’ slope dy/dz

z axial position in the accelerator centered coordinate system

Zc characteristic impedance of a coaxial quarter wave cavity

2., impedance of free space Z0 = 110 /80 = 37752

25 axial position in the coordinate system of the ion source

Greek symbols

or shorthand notation for (p l/pu)

(10 p ijn at the entrance to the acceleration region

8

skin depth

Az.um axial turn length (section 2.5.8)

8

ebend

d'D

£53?

beam emittance
permittivity of free space so = 8.8500“) CNm

relativistic mass factor 7 =1+$ (assume "yr—1 unless stated otherwise).

permeability of free space 110 = 41t(10'7) Tin/A

rf phase of the particle (¢ = (mt - n0)

azimuthal position of particle in the accelerator centered coordinate system
angle the beam is bent by the electrostatic bending plates (see Figure 25)
radius of curvature through the electrostatic bend (see Figure 25)

phase of the rf (1: = to“ t)

cyclotron angular frequency [rad/s]

resonant frequency

rf frequency

xi

LIST OF TABLES

Table 1. Gap voltage in the rf cavity required for different turn numbers using equation
2.80 with injection energy of 5 keV and ﬁnal energy of 50 keV. pg.130

Table 2. Compare the cavity voltage (V0) calculated from the analytical model w/ the
computer model. Notice that in the computer model, the reduced axial momentum
causes an increased number of turns. The lower p2 and accurate electric ﬁeld cause a
signiﬁcantly lower cavity voltage required. pg.132

Table 3. Total phase drift over the acceleration region. Compares what we would expect
analytically with the actual computer result (for optimum acceleration case). pg.135

Table 4. Final system parameters for a 5 turn trajectory through the 50 keV ICRA.
pg.138

Table 5. A summary of the basic characteristics of each of the main 3 beam diagnostics.
pg. 163

Table 6. Two points on the magnetic axis (relative to the axis of the magnet bore).
Azimuthal variation for all 2 planes is less than 0.15% of the central ﬁeld value

(Bo = 251‘). pg. 170
Table 7. Approximate beam currents of each constituent in a typical beam used in the

50 keV ICRA. pg. 172
Table 8. Results of method 1: Parameters for the hybrid rf cavity. pg.18l

Table 9. Three voltage ratios measured by the low power calibration method. pg.186

Table 10. Comparison between low power measurements made by method 1 and
method 2. pg. 187

Table 11. Summary of rf cavity measurements. pg. 189

Table 12. Comparison between calculation and experimental data of the H+ beam orbit
radius in the high B-ﬁeld region. In each case the beam is just below mirroring

(P1 = pm). pg-ZOO

Table 13. p2 and Apz as well as NW and AN...ms of the injected beam calculated from
experimental mirror data of Figure 91. pg.207

xii

Table 14. Arc lengths observed visually on the extraction Faraday cup (Vginzel = 2.8 kV).
' pg.209

Table 15. Rate of change of total number of turns (N) through the 50 keV ICRA.
Visual observations compared with computer result (using Vmim, = 745 V for trailing

edge of image). pg.210

Table 16. Three different 10 MeV ICRA designs with or = 2,4,8. pg.249

xiii

LIST OF FIGURES

Chapter 1.
Figure 1. An example of a two dee cyclotron and the coordinate system used. pg.2
Figure 2. Side view of a gyrotron. pg.4

Figure 3. Cross section of the annular electron beam in a gyrotron. Copied from
reference [4] pg.5

Figure 4. Electric a) and magnetic b) ﬁelds in a magnetron structure with 8 oscillators.
Copied from [8]. pg.6

Figure 5. Hole and slot conﬁguration of a single oscillator and its equivalent LC circuit.

Copied from [9]. pg.7
Figure 6. Basic components of an ICRA pg.9
Chapter 2.

Figure 7. An ion moving in a helical path in a constant axial B-ﬁeld. pg.13

Figure 8. Cross sections of a Helmholtz coil (left) and a solenoid (right) with equal coil
dimensions pg. 14

Figure 9. Axial ﬁeld proﬁle for a Helmholtz coil (left) and a solenoid (right) with equal
dimensions. pg.14

Figure 10. Examples of waveguides. pg.20

Figure 11. a) A TE ﬁeld pattern in which E is transverse only, but B has transverse and
axial components. b) A TM ﬁeld pattern in which B is transverse only, but E has

transverse and axial components. pg.2]
Figure 12. Cross section of a coaxial transmission line. pg.24
Figure 13. Geometry of a quarter wave cavity. pg.27

xiv

Figure 14. Voltage and current proﬁles for the quarter wave cavity. pg.27

Figure 15. A four gapmagnetron structure showing the n-mode at two different times

The 2 axis points into the page. pg.34
Figure 16. Geometry of a single oscillator and its equivalent circuit. pg.35
Figure 17. Equivalent circuit for a four gap magnetron in the 1: mode. pg.36
Figure 18. A four gap magnetron made of hyperbolic vanes. pg.38

Figure 19. The CII(RO+L) circuit which represents the frequency response of a resonant
cavity. pg.39

Figure 20. The parallel RLC circuit representation of a resonant cavity uses a ﬁctitious
shunt resestance (Rs). pg.40

Figure 21. An extra capacitance added across the end of a foreshortened quarter wave
cavity. pg.42

Figure 22. Assumed voltage proﬁle and lengths of the hybrid cavity for the 50 keV
ICRA. pg.43

Figure 23. Top and side view of the trajectory through an ICRA with trajectory shown in
red, and magnetic ﬁeld lines shown in blue. The axial magnetic ﬁeld proﬁle is plotted

below. pg.47
Figure 24. Beam orbit as seen looking in the axial direction as bend voltage is increased

pg.48
Figure 25. Geometry of the electrostatic bend. pg.51

Figure 26. Geometry of conservation of magnetic moment (copied from reference 19).
pg.54

Figure 27. A comparison of the beam orbit radius through the injection region calculated
using the analytical model and using ICRAcyclone. The magnetic ﬁeld proﬁle is shown
in blue. pg.55

Figure 28. Mirroring the beam is analogous to when a ball does not have enough kinetic
energy to make it to the top of a hill. pg.60

XV

Figure 29. Axial view of a beam containing three ions species.The bend voltage is ﬁxed
at the point for which the Argon orbit is centered around the z axis. pg.62

Figure 30. An rf driven magnetron structure with n=4 oscillating in the n—mode. pg.66
Figure 31. Azimuthal component of the E-ﬁeld vs azimuthal position. pg.66

Figure 32. Lowest order term for the radial dependence of cavity voltage in an 8 gap
magnetron. pg.68

Figure 33. Beam orbit radius from spread sheet calculation plotted next to magnetron
inner radius. pg.69

Figure 34. Magnetic ﬁeld proﬁle for the 50 keV ICRA with ﬂatness of 8B/Bo < 0.5%
over the 5 cm acceleration region. pg.73

Figure 35. Axial turn length depends on beam radius (or energy) in the extraction region
pg.76

Figure 36. Equipotential lines in an 8 gap magnetron structure at entrance (left) and exit
planes (right). Solution computed using RELAX3D. pg.80

Figure 37. Final proton energy vs initial phase for single particle trajectories through the
computer model of the 50 keV ICRA. The curve is repeated for three different rf cavity
voltages. pg.82

Figure 38. Cavity voltage required for “Optimal acceleration” vs. number of turns
through the acceleration region of the 50 keV ICRA. pg.83

Figure 39. Final proton energy vs initial phase for ﬁxed V0 in the 50 keV ICRA. As Otis
increased, the beam gains more energy and eventually strikes the wall. Here oro indicates
p lip" of the central ray. pg.85

Figure 40. Beam orbit radius vs 2 position for the three trajectories in Figure 39 with
o=68° and Aer/0t = (+2%, 0, and -2%). pg.86

Figure 41. A snapshot in time of an ion (red dot) at the moment of peak electric ﬁeld
across the gap. Two cases are shown. On the left the ion lags the rf and on the right the
ion leads the rf. pg.88

Figure 42. Axial momentum through the acceleration region (rf on) for three different
initial phases. pg.89

xvi

Figure 43. Three ions are started with different initial phases. Tracking their phase
through the acceleration region reveals phase pulling toward 90°. Notice the red line
goes through the greatest number of turns and the green line goes through the fewest
turns. pg.90

Figure 44. The red line shows ﬁnal vs. initial phase of protons accelerated through the
magnetron. (50 keV ICRA design). The solid black line represents ﬁnal phase equal to
the initial phase. Dotted lines have been placed at ¢=90° and ¢=(360°+90°) for
comparison. pg.9]

Figure 45. Final proton energy vs. initial phase for a ﬁxed V0 in the 50 keV ICRA.
The beam which passes through the end ﬁelds before entering the cavity shows a higher
energy gain and a shift in optimum initial phase. pg.93

Figure 46. The ion with initial ¢>90° gets a decrease in pz through end ﬁelds, but the ion
with initial ¢<90° gets an increase in p,. The particle moves left to right. The
acceleration region is between the dashed lines. The entrance end ﬁelds are just to the
left of the acceleration region. pg.94

Figure 47. Phase vs. axial position over the acceleration region. pg.96

Figure 48. Top and side view of a proton trajectory through the entire computer model for
the 50 keV ICRA. Here Vszend = 722 Volts. The beam mirrors before reaching the
accelerating cavity. pg.98

Figure 49. T0p and side view of a proton trajectory through the entire computer model for
the 50 keV ICRA. Here l/2Vbe,,d = 718 Volts. The beam is just below the mirror voltage.
RF is off. pg. 100

Figure 50. Axial view of the same trajectory as shown in Figure 49 except that the
trajectory is only shown through the injection region. Vszcnd = 718 Volts. The beam is
just below the mirror voltage. RF is off. pg. 102

Figure 51. Top and side view of a proton trajectory through the entire computer model for
the 50 keV ICRA. Here l/2Vb¢,.d = 710 Volts, which gives Nmms = 11 through the cavity
length. RFis off. pg.104

Figure 52. T 0p and side view of a proton trajectory through the entire computer model
for the 50 keV ICRA. Here l/2V.,¢.,.d = 680 Volts, which gives Nmms = 5 through the
cavity length. RF is off. pg.106

Figure 53. Top and side view of a proton trajectory through the entire computer model
for the 50 keV ICRA. Again ‘/2me = 680 Volts, which gives Nmms = 5 through the
cavity length, but now the rf is on. pg. 108

xvii

Figure 54. Axial view of the same proton trajectory that was shown in Figure 53 except
that here the trajectory is shown only through the acceleration region so that the increase
in radius can be seen. Again l/2V.,._.,,d = 680 Volts, Nmms = 5 through the cavity length, and
the rf is on. pg.110

Figure 55. Initial particle coordinates at the ion source are expressed in a Cartesian
coordinate system centered on the central ray, but the acceptance phase space at the
entrance plane of the magnetron must be expressed in an accelerator-centered cylindrical
coordinate system. _ pg.55

Figure 56. Two planes through the acceptance phase space at the entrance plane of the
magnetron for a 1 MeV proton ICRA. Each data point represents a summation over all
rf phases. pg.116

Figure 57. Final positions of protons shot forward from the ion source through the
injection fringe ﬁelds and stopped at the entrance plane of the magnetron. The
acceptance phase space from Figure 56 is outlined with dotted lines for comparison.

pg.117

Figure 58. The emittance used at the ion source is a coupled phase space of
3x3x3><3=81 particles pg.118

Figure 59. Final x, y positions of protons on the extraction Faraday Cup after being shot
from the ion source through the entire system. Here the bending voltage is high enough
to give an arc length of more than 360°. pg.120

Figure 60. Final 9 position of protons on extraction Faraday Cup vs their axial
momentum at 2:0. pg.121

Figure 61. A mirror curve generated using ICRAcyclone. The fraction of particles,

which reach the extraction Faraday cup without being mirrored, is plotted vs. bend
voltage. pg.6]

Figure 62. Mirror voltage of 81 protons vs. their axial momentum at z=0 at Vbcnd =
690V. The solid line is included for visual comparison with linear. pg.123

Figure 63. An integrated Energy distribution for the 50 keV ICRA generated using the
computer model. Emittance of the injected beam was 2.21: mm mrad. pg.125

Figure 64. Scanning over cavity voltage shows a peak in the beam current accelerated
above a given energy. Emittance of the injected beam was 2.21: mm mrad. Data
generated using the computer model for the 50 keV ICRA design. pg. 126

xviii

9AM

Figure 65. Beam radius vs. magnetic ﬁeld for several beam energies. pg.129

Figure 66. Phase vs. 2 for trajectories started with different initial phases in the 50 keV
ICRA. The red, blue, and green curves use the computer model with a solenoidal B-ﬁeld,
cavity end ﬁelds, and acceleration through the magnetron. The oscillation (once per
turn) is caused by motion of the orbit center. The blue oscillating curve gives optimum
acceleration. The smooth blue curve is the analytical calculation of phase based on
equation 2.83 which assumes a centered beam and no phase pulling. Vertical dotted
lines represent the ends of the acceleration region. Both blue curves have the same
initial phase at the cavity entrance so the total phase change over the acceleration region
can be compared. - pg. 134

Figure 67. Parallel and perpendicular momentum and beam radius calculated
analytically through the injection region. pg. 136

Figure 68. Side view of the 50 keV ICRA with the author added to give a sense of scale.
The large blue dewar contains the superconducting solenoid. The ion source is on the left
and the extraction end is on the right. The vertical tube leads up to the vacuum turbo
pump which is just off the top of the photo. pg. 140

Figure 69. A view looking down on the ion source. Notice the angle between the

accelerator z axis and the source axis is because the source is aligned with a B-ﬁeld line.
The motion feed through mounted on the left side of the junction piece (red tape around
it) is used for moving the injection Faraday cup into the beam. pg. 141

Figure 70. A side view of the ion source assembly, which shows the four main
components. From left to right, these are: the water cooling jacket, the insulating glass
break, the electrical feed through, and the bellows which connects the source to the
vacuum junction. The actual ion source is inside the water cooling jacket, and the Einzel
lens can be seen through the glass break. The electrostatic bend is inside the vacuum
junction. Compare this photo with the mechanical drawing of Figure 72. The source is
mounted on two V-blocks which are rigidly connected by an aluminum plate. Pusher
blocks on the sides of the V-blocks are used to adjust the position and angle of the source
horizontally, and vertical bolts through the aluminum plate are used to adjust the position
and angle of the source vertically. pg. 142

Figure 71. The extraction end of the 50 keV ICRA. The rf accelerating cavity
(aluminum tube) is mounted in the bore of the superconducting magnet (blue). The V-
block below the cavity can be adjusted in order to align the rf cavity with the magnetic
axis. The extraction port covers the extreme downstream end of the system and contains
the extraction Faraday cup and vieWports for observing the beam. The mechanical
feedthrough on t0p of the extraction port is used to move the radial probe, in the radial
direction (see Figure 73). pg. 143

xix

 

Figure 72. Top view of a cross section through the 50 keV ICRA. All major
components are labeled, however the radial probe is not shown in this view (see Figure
73). B-ﬁeld lines are shown in blue and a trajectory is shown in red. The magnetron
section only is cut through two different planes in order to show the inner and outer vanes
both. Figure 76 shows this more clearly. pg.144

Figure 73. Side view of a cross section through the 50 keV ICRA. All major
components are labeled. B-ﬁeld lines are shown in blue and a trajectory is shown in red.
The magnetron section only is cut through two different planes in order to show the inner
and outer vanes both. Figure 76 shows this more clearly. pg. 146

Figure 74. Simple representation of the vacuum system showing typical pressures during
high gas load into ion source. Conductances are calculated using formulas in [17].

pg. 1 50

Figure 75. Schematic of the electronics for the 50 keV ICRA. g.154

Figure 76. A cutaway view of the rf cavity. Notice that in the magnetron section only,
the cross section has been taken through two different planes in order to show a cross
section through an inner vane and an outer vane both. pg. 159

Figure 77. Components of the magnetron section before assembly. pg.161

Figure 78. Components of the hybrid rf cavity. Here the magnetron is assembled. The
magnetron, inner conductor, and shorting plate (all made of copper) are about to be
inserted into the aluminum outer conductor. pg.161

Figure 79. A view of the fully assembled rf cavity looking into the extraction end. The 8
tapered magnetron vanes of the acceleration section can be seen inside the ID of the exit

mounting ring. pg. 162
Figure 80. Pivot points of the mounting structure. pg. 167
Chapter 4.

Figure 8]. Measured axial magnetic ﬁeld at r = 0. Data has been centered axially and
normalized. pg. 171

Figure 82. Measured axial magnetic ﬁeld in the acceleration region. The smooth line is

a parabolic curve ﬁt to the data The ﬂat ﬁeld length for AB/Bo < 0.5% is 2.1 inches =
5.3 cm. pg. 171

XX

Figure 83. Schematic drawing of the experimental setup for measurement of Q. The
matched state is the condition used when running the cavity at full power and
accelerating beam. However, measured Q approaches the true unloaded Q only in the
unloaded state. pg. 179

Figure 84. Transmission frequency response curves used for measurement of the Q.

The matched state (top) gives fo/Af = 802, and the unloaded state (bottom) gives fo/Af =
1550 pg.180

Figure 85. Experimental setup for the calibration of method 2. V is a voltage at a probe,
and V' is a voltage at the end of a cable. pg.183

Figure 86. Geometry of the ion beam as one constituent clips on the entrance to the
magntron. Here the bending voltage is turned high enough that the H; beam (larger
radius) has clipped on the entrance to the magntron and the beam with the H" (smaller
radius) is centered on the z axis. pg.196

Figure 87. Ion spectrum taken on the extraction FC. The solid black line shows
experimental data. Theoretical values where each ion should clip have been calculated
using equation 2.73 and are shown as vertical red dotted lines. From left to right: Ar“,
Ni, 02*, Cf, Ar”, H20+, N“, 0*, Ar”, C", Hf, Hf, and H“. The experimental value at
which the H+ beam mirrors is shown as a blue dotted line. Using these calculated
values, the expected theoretical curve is plotted as a solid red line. pg. 197

Figure 88. Drawing of the beam image observed on the extraction Faraday cup. Here
the bending voltage is high enough that both ion beams produce a full 360° ring. At
lower bend voltages the image observed is less than 360° as will be described in section
4.5.4. pg. 198

Figure 89. Experimental data. Black diamonds represent beam current measured on the
radial probe which is shielded from electrons showers. Dotted lines mark the range of
bend voltage where the Hf beam was observed to clip visually, and where the W beam
mirrors. pg. 199

Figure 90. Experimental mirror curve. Proton beam current on the radial probe vs. bend
voltage for several different Einzel lens settings. pg.20]

Figure 91. Experimental data. Proton beam current on the radial probe vs. bend
voltage. The Einzel Lens setting is 3.0 kV. pg.202

Figure 92. Analytical calculation of pz vs bend voltage compared with the result from the
full computer model. The analytical calculation depends on the mirror voltage, therefore
the two curves meet at the mirror point a-priori. pg.205

xxi

 

Figure 93. Analytical calculation of the spread in p; vs bend voltage. pg.206

Figure 94. Drawing of the proton beam image observed on the extraction Faraday cup
The arc length can only be measured for low bending voltages, where A0 < 360°.pg.208

Figure 95. Comparison between computer generated mirror curves and experimental
data. Experimental data (black diamonds) is normalized to its peak value (45 nA). The
5 solid lines represent computer generated mirror curves for 5 different injected
emittances. pg.212

Figure 96. Arc length increases with bend voltage. The experimental data from Table
14 is plotted. The computer model is used to generate arc lengths for three different
injected beam emittances: 0.11:, 0351:, and 0.81: mm mrad (emittances are un-
norrnalized). pg.213

Figure 97. Measured radial distribution with rf on and off (raw data). Beam current on
the radial probe is plotted vs. radial position of the radial probe. Vertical dotted lines
mark the calculated radius of the injected beam and maximum radius corresponding to
the exit diameter of the magnetron. pg.215

Figure 98. Top graph: Raw data and corrected data are shown for rf on and off. The
solid line shows a theoretical curve for the rf off case only. Bottom graph: The
corrected data for rf on and off is shown on a log scale in order to show the
accelerated data more clearly. pg.217

Figure 99.‘ Drawing of visual observations on the extraction Faraday cup. This axial
view shows the extraction Faraday cup which is located z 7” further down stream (into
the page) than the radial probe. The radial probe casts a shadow at 90° rotation from the
probe. The edge of the shadow moves inward as the radial probe is moved inward.

pg.219

Figure 100. Measured energy distribution with rf on and off. Data is taken from the
corrected radial distribution of Figure 98. Dotted lines mark the injection energy and
peak design energy. pg.220

Figure 101. Comparison between corrected experimental data and the computer
generated energy distribution. In each case: Vo = 1.7 kV, l/2Vbend = 740V, Vain“. =
2.5 kV. The computer model uses a 2.2 1: mm mrad injected beam emittance. pg.221

Figure 102. Measured energy distributions for several bending voltages. Cavity voltage
is 1.4 kV, VEinzel = 2.7 kV, Mirror voltage ranges from 690 - 745 V. pg.222

xxii

Figure 103. Experimental data from Figure 102 is compared with the computer result.
Here, bending voltages of 690 V and 710 V are used in the computer and experiment
both. pg.223

Figure 104. Experimental data from Figure 102 is compared with the computer result.
In the tOp graph the same bend voltage is used for computer and experiment. In the
bottom graph, bending voltages in the computer model have been shifted lower to obtain
matching energy distributions. pg.224

Figure 105. Experimental data showing the accelerated beam current vs. magnetic ﬁeld.
The position of the radial probe is 19.3 m (E > 38 keV), and Vszend = 683 V

(Nu.ms = 5). pg.226

Figure 106. Comparison between experimental data and the computer generated
resonance width. The computer model uses: Vszcnd = 683 V, E > 38 keV (for R-probe

at 19.3 mm) and injected beam emittance of 2.2 1: mm mrad. pg.227

Chapter 5.

Figure 107. Measured energy distribution of the proton beam with if on and with rf off.
pg.229

Figure 108. Comparison between measured energy distributions and computer generated

energy distributions for two different bend voltages. pg.230

Figure 110. Cross section of a magnetron with two linear tapers used to approximate the
square root shape of the acceleration trajectory. pg.233

Figure 111. The inﬁnitesimal energy spread caused by an inﬁnitesimal spread
momentum spread is plotted vs. 0: and for several values of n. pg.238

Figure 112. A graphical map of the basic design equations for the acceleration trajectory
of the ICRA. Input parameters are E0 or, B0, 11, F, Egan, and the average radius fraction
(r/R). pg.241

Figure 113. Design equations for the resonant structure of the rf driven magnetron.

Input parameters are: 2,; , n , and Q. Parameters taken from Figure 112 are: g , Zea. , frf ,
and V0 . pg.242

Figure 114. The dotted line shows the diameter required for resonators of the magnetron
(nﬂ) as a function of magnetic field. The solid line shows total diameter D = 2(r.,¢am +
ngmm) (for a 10 MeV beam) needed for the warm bore of the superconducting
magnet. pg.247

xxiii

1. INTRODUCTION

The Ion Cyclotron Resonance Accelerator (ICRA) is an ion accelerator which
uses novel geometry. It combines the principles of cyclotrons and gyrotrons and uses an
rf driven magnetron as the accelerating structure. The intent of the ICRA concept is to
create an accelerator which is simple, compact, and lightweight. An ICRA designed to
accelerate protons to 10 MeV would be useful for the production of radioisotopes, or
neutrons, and may also have applications in materials science. The ICRA concept and

10 MeV design were ﬁrst published in 1997 [1].

As a demonstration of the concept, a computer model has been developed to study
particle trajectories in the ICRA, and a 50 keV proton ICRA has been built and tested.
The ﬁrst experimental results were published in April 1999 [2]. This thesis describes the
theory that has been developed for the ICRA, then presents the design of the 50 keV
prototype, and compares the experimental results with the theory; Finally it gives

recommendations for future research, andpresents an improved 10 MeV design.

This chapter begins by introducing the reader to cyclotrons, gyrotrons, and
magnetrons. In section 1.4 an overview of the ion cyclotron resonance accelerator and its
basic components are given. Section 1.5 discusses the applications for which the ICRA is

the most well Suited.

Chapter 2 covers the theory which has been deveIOped for the ICRA, including
both analytical and, computer model. Chapter 3 then presents the speciﬁc design of the
50 keV experiment’. Chapter 4 shows the experimental results, and compares with the
theory. Chapter 5 then gives recommendations for future research, presents the design of

a 10 MeV ICRA, and gives a conclusion.

1.1 Cyclotrons

In 1931 Ernest Lawrence demonstrated the cyclotron, by accelerating protons to
an energy of 80 keV. This new tool for probing the nucleus fueled our understanding of
the atom and lead to the rise of nuclear and high energy physics. Today, synchrotrons
achieve much higher energies than cyclotrons, however cyclotrons are still the leading
choice in ﬁelds such as the production of radioisotopes for medical applications, proton

and neutron beam therapy for treatment of cancer, as well as nuclear physics.

Cyclotrons use time varying electric ﬁelds to accelerate charged particles in the

azimuthal (0) direction, and an axial (2) dc magnetic ﬁeld to bend the particle beam into a

closed orbit so that it will pass through the same accelerating structure many times. The
geometry and coordinate system for a simple cyclotron are shown in Figure 1. The
accelerating structure is a set of hollow conductors called “dees”, which serve the same
function as drift tubes in a linear accelerator. As a particle passes through a dee, the

voltage potential is changing with time, but is constant with respect to position so the

electric ﬁeld inside the dee is zero. However, since each adjacent dee has a different

voltage, particles are accelerated by electric ﬁelds across the gaps between the dees.

 

Figure 1. An example of a two dee cyclotron and the coordinate system used.

The voltage on the dees alternates at a constant rf frequency (frf). In order to be
continuously accelerated, the beam must cross each gap only at times when the E-ﬁeld is
pointing. forward (azimuthally). Thus, the rf frequency must be a harmonic of the

cyclotron orbital frequency (fc). This is the resonance condition required for acceleration.

Because of the “drift tube” nature of the dees, the beam will strike the inner
surface of a dee if it is not kept in the median plane (2:0). For this reason, weak axial
focusing was used to conﬁne ions. At higher energies relativistic effects prevented the
use of weak focusing. Therefore, strong axial focusing was developed using steel pole

tips to create an azimuthal variation in the magnetic ﬁeld.

As the beam gains kinetic energy, its orbit radius in the magnetic ﬁeld increases.
Extraction is usually achieved by allowing the beam that reaches full radius to pass
behind a thin septum into a region where a strong dc electric ﬁeld pulls the beam in the

radial direction. At large radius, the radial spacing between turns may become extremely
small (r cc JE) , in which case, some beam current will be lost by scraping on the

septum. Beam current hitting the septum creates thermal and radiation issues which
contribute to limiting the beam current in the cyclotron. An alternative extraction
method is to accelerate a negative ion beam, then use a foil to strip ions to positive charge
state which changes the radius of curvature of the beam. In either case, only the beam
that reaches full radius is extracted, so the cyclotron has a relatively narrow energy

spread in comparison with the ICRA which has no such constraint.

1.2 Gyrotrons

Gyrotrons are a source of high power coherent microwaves. Theoretical gyrotron
research began in the late 1950’s by Twiss in Austrailia, J. Schneider in the US. and
A.Gaponov in the USSR, though experimental veriﬁcation was not obtained until the
mid 1960’s [3]. Today gyrotrons are available from industry with average rf power levels

of a megawatt and efﬁciencies greater than 50%.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Magnet
C°"5 B—Field
.[ . Line
Electron Gun Y /
} ' / W
- - .22.:-_—_.'._.'.'/._._.v._._. ___________________________________
“Hum? ................... W
[vaqu \{ """" Microwaves
e- Beam Resononi COVHY

Figure 2. Side view of a gyrotron.

Gyrotrons convert the kinetic energy of a dc electron beam into high frequency
electromagnetic ﬁelds by exciting waveguide modes in a cylindrical resonant cavity. A
magnetic ﬁeld in the direction of the cavity 2 axis, conﬁnes an annular electron beam to
spiral around B-ﬁeld lines. The e' beam is hollow in the center and contains many tiny

beamlets around the circumference as shown in Figure 3.

The source of the electron beam is an electron gun located outside the resonant

cavity in the fringe ﬁeld region of the magnet. Here the velocity component parallel to

the B-ﬁeld is set so that the beam will drift along ﬁeld lines into the high ﬁeld region of
the resonant cavity where the interaction occurs. Inside the waveguide structure, the

relativistic electron beam interacts with azimuthal electric ﬁelds which causes bunching

within each beamlet. [4]

 

Figure3. Crdsssecﬁonoftheannularelectronheaminagyrotron
Copied from reference [4]

Bunching and the transfer of energy from the electrons to the cavity ﬁelds both
depend on a resonance between the cyclotron frequency of the electron orbits and the

frequency of the cavity ﬁelds.

Even at the time that gyrotrons were being investigated as a source of
microwaves, it was recognized that the inverse should also be possible, i.e. to accelerate
electrons by driving a gyrotron structure with microwave power. This was demonstrated
by Jory and Trivelpiece in 1968. Using electric ﬁelds from the TB” mode of a circular

waveguide, they accelerated a 10 mA electron beam and measured an energy gain of

460 keV [5].

1.3 The Magetron

Magnetrons, like gyrotrons, generate microwave radiation by converting the
kinetic energy of an electron beam into electromagnetic ﬁelds in a resonant cavity which
is immersed in an axial magnetic ﬁeld [6]. Development of the magnetron in the late
1930’s and early 1940’s was instrumental in the successful use of radar during World
War II [7]. Today magnetrons generate microwaves at 2.45 GHz in microwave ovens all

over the world, and are so common that you can purchase a replacement tube for about

$50.

 

Figure 4. Electric a) and mgnetic b) ﬁelds in a mgnetron structure with 8 oscillators.
Copied from [8]

The magnetron structure itself is a resonant cavity, but it is much different than
the open waveguide structure of the gyrotron. The cavity is comprised of several coupled
oscillators as shown in Figure 4. The hole and slot conﬁguration of each oscillator means

that the magnetron structure behaves like a lumped circuit with isolated inductance and

capacitance (Figure 5). The additional inductance and capacitance lowers the resonant
frequency and allows the magnetron to be much smaller than a wavelength. Our interest
in the magnetron is as an accelerating structure because an rf driven magnetron can be

used to generate electric ﬁelds in the azimuthal direction.

 

Figure 5. Hole and slot conﬁguration of a single oscillator and its equivalent LC circuit.
‘ Copied from [9].

1.4 Ion Cyclotron Resonance Acceleration

The ion cyclotron resonance accelerator (ICRA) combines the principles of
cyclotrons and gyrotrons. Like the cyclotron, ions are conﬁned radially (r) while being
accelerated in the azimuthal (9) direction. However, in the axial (z) direction, the beam
is not conﬁned. Instead, the beam is allowed to drift through the accelerating structure
just as the electron beam drifts through the waveguide of a gyrotron. For this reason, an

appropriate name for the ICRA is an “axial drift cyclotron”.

As mentioned previously, Jory and Trivelpiece demonstrated cyclotron resonance

acceleration by accelerating an axially drifting electron beam in a waveguide. The

 

research documented here extends cyclotron resonance acceleration to m for the ﬁrst
time. The waveguide structures used in gyrotrons are on the order of a wavelength and
would be too large at the low frequencies required to accelerate ions. However, by using
an rf driven magnetron operating at a harmonic of the cyclotron frequency, together with
a high ﬁeld superconducting magnet, the accelerating structure becomes small enough to
ﬁt into the bore of a common superconducting magnet. This means that a magnet which

is available from industry can be used.

The main components of an ICRA are shown in Figure 6. These are the
superconducting magnet, ion source, electrostatic bend, accelerating structure, and the
target. A dc ion beam is extracted from the ion source directly along a B-ﬁeld line so that
the VXB force on the beam is zero. The electrostatic bend deﬂects the beam so that it
has a component of momentum perpendicular to the B-ﬁeld which. causes it to orbit
around ﬁeld lines. The remaining momentum parallel to the B-ﬁeld causes the beam to
spiral axially into the high ﬁeld region. At the acceleration region, the B-ﬁeld is
relatively ﬂat and the beam drifts axially through the magnetron structure. While inside
the magnetron, rf electric ﬁelds accelerate the beam in the azimuthal direction so that the
radius of the beam orbit increases. Upon exit from the magnetron, the beam spirals into
the lower ﬁeld of the extraction region, until striking a target downstream. A detailed

description of the beam trajectory is given in section 2.5.

 

 

 

 

 

 

 

 

 

 

 

 

NV

 

 

 

 

 

 

 

 

ET

 

 

 

 

B-ﬁelll lino

 

a-w Iii

 

 

 

 

 

 

 

Figure 6. Basic components of an ICRA

This acceleration scheme is inherently simple, compact, and inexpensive.

In principle, the rf driven magnetron can be cut out of a single piece of c0pper.
Superconducting magnets of the proper size are available from industry. The high
magnetic ﬁeld means the machine is very compact. Furthermore, because of the lack of
axial focusing, no steel is needed to shape magnetic ﬁelds so the entire machine is
lightweight. Since anlICRA would be compact and lightweight, it might be designed to
be portable which could open up some ﬁeld applications for accelerators. Beam
extraction is inherently simple because nearly all of the beam drifts through the
accelerating cavity to the target region where it can be isolated from the accelerator

mitigating maintenance and radiation shielding issues.

As we will see, the penalty for all of these advantages is that the beam accelerated
by the ICRA contains a large energy spread. In fact the extracted beam will contain

energies ranging from near zero to the full design energy. This is caused by three

vwr'
1 all.

its.

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a“!

1
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factors. 1) For simplicity the injected beam is dc, therefore the part of the beam which is
in phase with the rf is accelerated, while the rest of the beam is decelerated. 2) Any
spread in the axial momentum through the acceleration can cause a difference in the
number of kicks an ion receives. 3) A radial dependence of the accelerating ﬁelds in the
rf driven magnetron causes radial defocusing. The large energy spread simply means that
the ICRA will be most useful for applications in which an energy spread does not matter,

such as the production of radioisotopes, or neutron beams.

1.5 Applications

Today proton andiheavier ion beams at energies below a few MeV are typically
produced by electrostatic accelerators or radio frequency quadrupoles. Beams above a
few MeV are typically produced by cyclotrons, or linacs, with the highest energies being
attained by synchrotrons. In principle, an ICRA could be designed to accelerate any ion
to any energy range. However, for the purpose of limiting the scope of this discussion,
we will concentrate on a particular energy regime from 3 to 12 MeV. In particular, an
ICRA designed to accelerate protons or deuterons to 10 MeV would be useful for many
applications. Three areas for which the ICRA is the most well suited are 1) production of
short lived radioisotopes, 2) as an accelerator based neutron source, and 3) for materials
science applications.

The use of radioisotopes in medicine and industry has experienced steady growth
over the past two decades. Medium energy accelerators (10 < E < 30 MeV) are used in

commercial production of radioisotopes with half lives long enough for shipment. Lower

10

energy machines (E < 10 MeV) are used in hospitals for on site production of relatively
short lived isotopes. The most common of these are the positron emitters: 11C, 13N, ‘50,
and l8F which are used for positron emission tomography (PET) [10,11]. A 10 or 12
MeV ICRA delivering 10 to 50 11A of protons to the upper half of the energy range would
be useful for production of these PET isotopes. If that ICRA were designed to be
portable, it could be shared by several institutions.

Applications for neutron beams include thermal neutron radiography, fast neutron
radiography, fast neutron analysis, and neutron activation analysis. Of these, fast neutron
analysis has recently gained attention as a method for rapidly identifying materials in
applications such as bomb/drug/weapon detection for airport security. A more common
application of neutron beams, thermal neutron radiography, is used to produce an image
of the internal cOmponents of an object by passing neutrons through the object and
imaging the neutrons on ﬁlm. The majority of neutron radiographs are made at nuclear
reactors where high neutron ﬂuxes are available. The disadvantage here is that equipment
to be radiographed must be brought to the reactor site. Accelerator based neutron sources
produce somewhat lower neutron ﬂuxes than reactors, but offer the possibility of being
portable and therefore would be more useful for ﬁeld applications. A 10 MeV ICRA
producing only lOuA of protons (upper half of AE) would produce a neutron rate useful
for neutron radiography or as a research tool for fast neutron analysis.

Finally there are numerous materials science applications for which a 10 MeV
ICRA would also be useful including: ion implantation, charged particle activation

analysis, and radiation damage studies.

11

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at, 4‘

2. THEORY

This chapter covers the theory needed to understand and design an ion cyclotron
resonance accelerator (ICRA). Sections 2.1, 2.2, and 2.3 explain basic considerations for
the magnetic ﬁeld, vacuum, and ion source. Section 2.4 covers the theory of resonant rf
cavities. The last two sections comprise the majority of the chapter. Section 2.5 covers
all aspects of the beam trajectory that can be calculated analytically. Section 2.6 covers
those aspects of the beam trajectory which can only be calculated using a full computer

model. Several ﬁgures in this chapter are presented in color.

2.1 Mametic Field

Consider a positive ion with charge (q) and mass (m) in a region of constant

magnetic ﬁeld, B. If the ion has some momentum in a direction perpendicular to the B-
ﬁeld (p i), then the V x B term of the Lorentz force equation provides a centripetal force

which causes the ion to move in a circular path. The radius (r) of this orbit is given by:
pi = qBr (2.1)
The angular frequency, called the cyclotron frequency, is given by:

B
doc = — (2.2)
'an
where y is the relativistic mass factor. Notice that for 7 =1 the cyclotron frequency

depends on the B-ﬁeld and on r—g- of the ion, but does not depend on the ion velocity.

In the direction parallel to magnetic field lines, force on the ion is zero and the ion

will simply drift (p, = constant). Therefore in a region of constant magnetic ﬁeld, an ion

' with momentum p = (p: + p") will move in a helical path.

 

Figure 7. Anion moving in a helical path in a constant axial B-field.

As in the cyclotron, acceleration in the ICRA requires a resonance between the rf
accelerating ﬁelds and a harmonic of the cyclotron frequency, (section 2.5.6). However,
since ions in the ICRA will also drift along B-ﬁeld lines (2 direction), the magnetic ﬁeld
must be nearly constant over the axial (2) length of the acceleration region (for ya 1). At
higher energies, the cyclotron frequency will decrease as 7 becomes greater than 1. In
this case it will be necessary to add a slope (dB/dz) to the B-ﬁeld, which matches the
increase in y, in order to maintain resonance. Since the focus of this work is a low energy
50 keV proton accelerator, the reader should assume that y = l, for the remainder of this
document unless stated otherwise.

A Helmholtz coil pair provides a longer axial ﬂat ﬁeld length than a solenoid of
equal coil radius (RC) and axial length (2,). However, either coil geometry will work.

The B-ﬁeld on the z axis (r=0), of a single current loop is given by:

R3
Bz(z)=B0 _Tc_——2_l' (2.3)
(Rc + z )2

13

where the ﬁeld has been normalized'to the central ﬁeld value Bo. This equation can be
used to calculate the ﬁelds on the axis of a Helmholtz coil or a solenoid by simply
placing coils at appropriate locations and then superimposing the B-ﬁeld of each coil.
Solving for the magnetic ﬁeld off axis (r it 0) is more difﬁcult (see section 2.6.1).

Figure 8 shows the geometry of a Helmholtz coil and a solenoid coil. Figure 9 shows the
axial proﬁle of each B—ﬁeld. Notice the Helmholtz coil provides a signiﬁcantly longer

ﬂat ﬁeld length.

Helmholtz Coil Solenoid

A" Mr

 

 

ii
i

E"
C)
O—h
C)

 

 

 

V

 

t..—
i‘

 

 

 

 

 

 

 

 

inf -.....

Figure 8. Cross sections of a Helmholtz coil (left) and a solenoid (right) with equal coil dimensions

 

 

1.1 1.1
1.0

0.9
0.3
0.7

5 0.6

a 0.5
0.4
0.3
02
0.1

00 liinliililllillimil lLJllllllIllJLlllll

I I U l I I I I l

   

 

 

 

 

 

 

 

0.1 1111111111111111111 llllllllllllllllllL
-2.0 -1.5 -1.0 0.5 0.0 0.5 1.0 1.5 2.0 ~20 -1-5 ~10 ~05 0.0 0-5 1-0 1-5 2-0
2(Arbilrarytongmuas) “Arbitrary Length Units»)

 

Figure 9. Axial field proﬁle for a Helmholtz coil (left) and a solenoid (right) with equal dimensions.

14

For the ICRA, the choice of B-ﬁeld profile is an important consideration because:
1) the ﬂat ﬁeld length limits the axial length available for acceleration, and 2) the ion
source must be able to operate in the chosen fringe ﬁeld region. Furthermore we will see
in section 2.5.2 that the ratio of ﬁeld at the acceleration region to ﬁeld at the source is

involved in determing the increase in the transverse momentum spread of the beam.

lama!!!

Preferably the vacuum system for any accelerator should maintain a complete
absence of atoms. In reality, the pressure in the ICRA will be dominated by the mass
ﬂow of gas fed into the ion source, and to a smaller extent by outgassing from materials
inside the vacuum chamber. The mean free path (7:) is the average distance that a particle
travels before colliding with another particle. For this experiment, it is sufﬁcient to have
a mean free path longer than the path of the beam. Details of the vacuum system are

given in section 3.2.2.

15

2.3 Ion Source

A wide variety of ion sources are adequate to supply the beam for an ICRA. The
only requirement is that the source must operate in the fringe ﬁeld of the magnet at the
chosen location. However, a high brightness source is preferred because the acceptance
phase space of the rf driven magnetron together with the brightness of the injected beam

are crucial parameters which determine the ﬁnal accelerated beam current.

In most ion sources, the energy spread in the extracted beam is small, because all
ions are accelerated through nearly the same potential from the extraction aperture to the
puller electrode. However, the beam will have some ﬁnite spread in transverse and
longitudinal position and momentum. These beam dimensions occupy a 6 dimensional
volume in phase space. The transverse spread in position and momentum is normally
described in terms of two dimensional areas called the beam emittance. The computer

model of the injected beam emittance is discussed in section 2.6.7 of this chapter.

In chapter 3, we will see that the ion source chosen for the 50 keV ICRA is a
simple electron impact ion source. Electrons emitted from a hot filament are accelerated
through approximately 100 volts toward an anode. The electrons impact and ionize H2
gas creating H” and Hf. An ion beam is extracted from the source at 5 - 10 keV. The
energy spread in this type of source is due to both the temperature of the ions and

variations in the potential at which the ion was created. The maximum energy spread is

still small. The computer model of section 2.6 uses the assumption that AB = 0.

The source chosen for the 50 keV ICRA also includes an Einzel lens for focusing
the beam before injection into the ICRA. This electrostatic lens provides azimuthally

symmetric electric ﬁelds, which are effective for focusing the low velocity beam, and

also has an adjustable focal length. For the purposes of this chapter on theory, the Einzel
lens should be thought of as part of the ion source apparatus. Details of the ion source

and the Einzel lens are given in section 3.2.3 of the chapter on design.

W

Initial designs for the ICRA used an rf driven magnetron in an 8 Tesla axial dc
magnetic ﬁeld [1]. In chapter 3 on the design of the 50 keV ICRA we will see that the
availability of a 2.5 Tesla superconducting magnet created the opportunity to build a
proof of principle device at very low cost. The disadvantage of this lower magnetic ﬁeld
is that a pure magnetron would not ﬁt into the bore of the available magnet. The solution

was to build a hybrid coaxial 2- magnetron cavity.

This section covers the electromagnetic theory needed to design the rf
accelerating structure of the ICRA. The theory is well known and is given here as
background. Section 2.4.1 begins with plane waves, then reviews waveguide TE and
TM modes. Section 2.4.2 covers TEM waves in a coaxial cavity, then the ﬁelds in a
coaxial quarter wave cavity are derived. Section 2.4.3 gives the ﬁelds in the central
region of the magnetron structure. Section 2.4.4 simpliﬁes the theory by representing a
resonant cavity as an equivalent lumped circuit. Finally, the method used to describe the

hybrid cavity is given in section 2.4.5.

17

2.4.1 Wavegpides

We begin with the time harmonic form of Maxwell’s equations in a source free

region of empty space.

VXE = -i0)B (2-4)

VXB zi—(f—E (2.5)
c

VOE =0 (2-6)

VOB =0 (2-7)

In this form, oscillatory time dependence is assumed. Therefore it is only necessary to
solve for the Spatial dependence, E(‘r’) , then the full time dependent solution is recovered
by multiplying by an oscillatory factor: E(f, t) = E(‘r’)e"'°" . This approach is correct in
general because any non-oscillatory time dependence can be constructed with a Fourier

series. However, for this work we are only interested in oscillatory solutions.

Taking the curl 0f equation 2.4 and substituting equations 2.5 and 2.6 gives:

VxVsz—ioXVxB)

V(V-E)-V2E=-iw(-ié%’E)
VZE =—9—’—l‘=.

C2

Where the total wave vector is defined as:

18

This gives the vector Helmholtz wave equation:

 

 

(V2 +k2)E=O (2.8)

 

 

 

 

 

Similarly for the magnetic field: (V2 + k2 )B = O (2.9)

 

Where each vector equation actually represents three sealer wave, equations. An
important solution of the wave equation in open space (no boundary conditions) is the

plane wave. The electric ﬁeld for a plane wave can be expressed in the form:

E(r,i) = E0e1(k'f“”‘) (2. 10)
The direction that the wave propagates is given by the wave vector, k , but the direction

of the actual electric ﬁeld is given by E. The velocity of propagation is v = 51? = c .

Notice that imposing V . E = 0 on equation 2.10 gives k . E0 =0. This means that the

E-ﬁeld cannot point in the direction of propagation [12, 13]. Furthermore, Using

equation 2.4 to solve for the magnetic ﬁeld leads to:

B('r',t) = ["xE(i,i)] (2.11)

8|—

Evidently the magnetic part of the wave has the same form, but points in a direction
perpendicular to the direction of propagation and to the electric ﬁeld. For this reason

plane waves are referred to as transverse electric and magletic or TEM waves. For

 

example, if the wave propagates in the z direction, then R . ‘r’=kz and E0 = XE, + 9Ey

so the plane wave would be written as:

E(f,t) = (213,, +yEy)e‘(kZ“”‘)

19

and the B-ﬁeld would be:

B('r’,t) = “—5305,, -XEy)ei(kZ-‘°t)

Wavegpides

Waveguides are hollow conductors with cross section of any shape that remain
constant along their axial (2) length. Examples are shown in Figure 10. Of course
electrostatic ﬁelds cannot exist inside a waveguide because the metal walls all have the
same potential. However, if plane waves are introduced into a waveguide the waves will
reﬂect off of the conducting walls. The incident and reﬂected waves superimpose to
create a standing wave pattern along the transverse dimensions, and a travelling wave

along the axial dimension as shown in Figure 11.

ewe/0  

Figure 10. Examples of waveguides

 

 

 

 

 

Since the wave pattern must satisfy the boundary conditions at the conducting walls,
either E or B must have a component in the direction of propagation. Thus ﬁelds in a
waveguide are either TE (transverse electric) or TM (transverse magnetic), but
waveguides do not support TEM waves. Notice that whether the waves are TE or TM

depends on the initial polarization of the wave (or the orientation of the driving probe).

20

 

 

W

 

 

 

V

 

 

Figure 11

a) A TE ﬁeld pattern in which E is transverse only, but B has transverse and axial components.
b) A TM field pattern in which B is transverse only, but E has transverse and axial components.

In the usual treatment of waveguides [12, 13] the ﬁelds are separated into transverse and
axial dependence using separation of variables.
Assume that any of the six components Ex, E,, E2, Bx, B,, or B2 can be written as:

W?) = f (x.y)g(z) (2.12)

The wave equation can be broken into:

2

(Vi+ki)f(x,y)=0 and ._2
dz

+k:)g(z)=0

where k2 = k2 —ki (2.13)

21

If k2 is real, the solution for g(z) is oscillatory, so the electric and magnetic ﬁelds are both
of the form:

E(f.t)=E(x,y)ei(kZz-wt) (2.14)
Where E(x,y) could have components in both the transverse (l) and longitudinal ( 2)
directions. When solving for the ﬁelds in a speciﬁc waveguide geometry, forcing
E(x, y) to satisfy the boundary conditions leads to an expression for k i in terms of the
transverse waveguide dimensions.

Obviously, propagation down the waveguide depends on kz being real, but notice
the implication of equation 2.13. The wave vectors, k, k i, and kz, are related by a triangle
equality (see Figure 11). Therefore, if k depends on frequency of the wave (k=a)’c), and
k i is ﬁxed by the cross sectional dimensions of the waveguide, then kz, is simply the
remaining side of the triangle. The result is that, if the frequency (or k) becomes small
enough, kz will become imaginary and the wave will not propagate through the
waveguide.

For this reason, waveguides have a cut off frequency, below which waves
introduced into the waveguide will not propagate along the length. The cut off frequency
occurs when k = k i, and depends exclusively on the transverse dimensions of the
waveguide being used.

ck
f = t
C 21:

 

Since the z dependence is only in the exponential factor, we can write

V=(V, —ik12) (2.15)

where: V1. = x 53; + y for Cartesian coordinates.

A
3y
This allows ﬁeld components to be separated into transverse and longitudinal directions.
Substituting equation 2.14 into equation 2.4 and then using 2.15 to separate terms in the

transverse and longitudinal directions gives :

Longitudinal: (i7i x EL) 2 —i(1)Bz 2 (2.16)

Transverse: (leziaz) —ikz(2in) =—imB,l (2.17)
Where two analogous equations come from substituting B(r,t) into (2.5). Equations 2.16
and 2.17 relate the transverse and longitudinal components of the ﬁelds in a waveguide.
Notice that equation 2.16 says that the longitudinal magnetic ﬂux through the cross
sectional area of the waveguide causes a transverse electric ﬁeld.

Now consider the special case of a TEM wave by letting Ez and B2 both go to
zero. Equations 2.16 and 2.17 reduce to:

(v,xe,)=o (2.18)
B. = £(2x1‘3,) (2.19)
(1)

Where we have let k2 = k since k1 will be zero. From equation 2.18 we see that without
any magnetic ﬂux through the cross sectional area of the waveguide, the transverse ﬁeld
(E i) can only be caused by an electrostatic potential (E; =— V<I> ). In other words TEM
waves cannot exist inside a waveguide, unless there is a voltage difference between the
walls to support a transverse electric ﬁeld. Notice that equation 2.19 is the same result as
for TEM plane waves in equation 2.11.

Since the conducting walls of a waveguide are all at the same potential, TEM

waves cannot exist in waveguides. However, a transmission line with a voltage

23

difference between two separate conductors can support TEM waves. In the next section
we begin by studying one example of a two conductor transmission line, called the

coaxial transmission line.

2.4.2 Coaxial Quarter nge Cavity

TEM waves can exist in transmission lines which have two conductors. An
example is the coaxial transmission line. The cross section of a coaxial line is shown in
Figure 12. Where the radius of the inner conductor is “a” and the radius of the outer

conductor is “b”.

 

Figure 12. Cross section of a coaxial transmission line

Recall that the solution of the wave equation is separable into transverse and
longitudinal dependence (equation 2.12), and that the ﬁelds of a TEM wave can be
derived from an electrostatic potential (equation 2.18). This means that we can solve the
Laplace equation for the electrostatic potential, and the transverse electric ﬁeld, then
iikz —itot

multiply by the z dependence (e

) and the time dependence (e ) to obtain a

complete solution.

24

Solving the Laplace equation for the transverse dependence only gives:

 

V2<I> = O
_ ln(r/ b)
(M) "' ° ln(a/b)
Differentiating gives the electric ﬁeld:
E = — V,<I>(r)
_. _ VO f
E(r) — ln(a/b) r (2.20)

and the magnetic ﬁeld can be found using equation 2.19.

.. _ V0 9-
B“) _ cln(a/b) r (221)

The surface current (J S) can be f0und using the boundary condition which comes from

 

Ampere’s law: I = “1 (ﬁxB) (2.22)

S
0

where n is a unit vector normal to the surface. For the inner conductor (11 =f‘ ) the

surface current is:

J — —1—-———V° 2
s — Z0 aln(b/a)

Where Zo = 1.10/80 = 3779 is called the “impedance of free space”.

Integrating around the circumference of the inner conductor gives the total current in
terms of the voltage.

I=27taJs

_ _1_ 21:Vo
° ‘ zo ln(b/a)

 

25

Therefore, we can solve for the ratio of the voltage to the current.

V0 20
ZC — Io — Elnw/a) (2.23)

 

L is known as the characteristic impedance of a coaxial transmission line. Zc relates the
peak voltage to the peak current (or the E-ﬁeld to the B-ﬁeld) and depends only on

geometry of the transmission line.

The Quarter nge Cavity

Now consider a coaxial quarter wave cavity, i.e. a section of a coaxial
transmission line whose length (5) is one quarter of a wavelength (M1)»). A shorting plate

at one end connects the inner and outer conductors while the other end is left open, as
shown in Figure 13. The short forces the voltage difference between the inner and outer '
conductors to be zero at all times, but the open end can oscillate between iVo. Therefore,
we expect the voltage proﬁle to look like a quarter of a wave as shown in Figure 14.

As with any electromagnetic oscillator, stored energy is transferred back and forth
between the electric and magnetic ﬁelds. It is useful to think of charge as bouncing back
and forth between the inner and the outer conductor, where the shorting plate provides a
path for current to ﬂow between the two. At the moment of maximum charge
separation, the voltage difference is maximum and the electric field points in the radial
direction. One quarter of a cycle later, charge ﬂowing along the conductors and across
the shorting plate causes a magnetic ﬁeld in the azimuthal direction. The expected
current proﬁle is also shown in Figure 14. Obviously the magnetic ﬁeld is largest at the

short, and the electric ﬁeld is largest at the Open end.

26

Shorting Inner Outer

 

 

 

 

 

Plate Conductor Conductor
é////////////////////////////////////////// X
.LaV///////ﬂ////////////////////////17///// Z 7
o a Open
b 5 End

 

 

 

 

 

Figure 13. Geometry of a quarter wave cavity.

_e .5
O N
L‘ J

Voltage I Vo
.0 0 .0
A a: 00

.0
re

 

 

.0
o

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

.0
o

1.2 1
1.0 .
0.8 ~

0.6 «

Current / 10

0.4 «

oai

 

 

0.0 i I I I . I l
0.0 0.1 0.2 0.3 0.4 0.6 0.7 0.8 0.9 1.0

N9 __
‘01

Figure 14. Voltage and current proﬁles for the quarter wave cavity

27

Since fields in the quarter wave cavity are TEM, they satisfy an electrostatic
solution. Thus, we can solve for the transverse electrostatic ﬁelds, then add on the z
dependence and time dependence last. The axial dependence can be found by

superimposing the left and right moving waves.

V(z) = Ae‘“z + Be”ikz

I(z) = Ce’kz — De‘ikz
Where voltages add, but currents moving in Opposite directions subtract. Applying
boundary conditions (V = 0 at z = 0) at the shorted end, and (V = V0 at z = f) at the open
end gives: V I ‘ V(z) = V0 sing?) (2.24)
Similarly, maximum current (10) ﬂows on the short, and I = O at the open end.

1(2) = Io cos(-72%) (2.25)

Combining the radial, and the axial dependence, together with the time dependence, the

electric ﬁeld becomes.

 

“ _ V0 1 - TCZ ~10):
E.(r.2)—ln(a/b) r sm(2€)e 1“ (2.26)

 

 

 

Notice that this electric ﬁeld is entirely in the radial direction, thus it does not account for

fringe ﬁelds at the open end of the quarter wave cavity. Similarly the magnetic ﬁeld is:

 

i it; ~1011" ,
cos( 2€)e 0 (2.27)

3“”) = c ln(ii/b) r

 

 

 

When calculating the capacitance (C), we must account for the axial proﬁle. It can be

found from the total energy stored in the electric ﬁeld (U5).

UE =%CV§

28

Where the stored energy is:
_ l - 2
UE —-§EOJ|E| d3r
Substituting equation 2.26 and integrating gives:
1 11:8 8 2
U = — ° V
E 2 (ln(b/a)] 0

Therefore the total capacitance of a quarter wave coaxial cavity is:

 

 

_ 1:806
— ln(b/a)

 

(2.28)

 

 

 

The inductance (L) is more difﬁcult to calculate. Although equation 2.25 correctly
accounts for the axial current distribution on the inner and outer conductors, the current
which ﬂows on the shorting plate (at z = 0) also makes a signiﬁcant contribution to the
magnetic ﬁeld near-z = O. A much simpler method is to use the relationship between

wavelength and frequency 7.0 = c/ f o and the fact that the resonant frequency of the cavity

(f0) is related to the inductance and capacitance by:

1
f :—
° 2n./Lc

Thus the total inductance is:

 

L = ﬂit—rig ln(b/ a) (2.29)

 

 

 

The Quality factor, or the “Q” is deﬁned as:

(1)o x (energy stored in cavity ﬁelds)

 

 

 

Q = time averaged power loss
(0 U
or Q = ° (2.30)
Ploss

 

 

 

29

The full analytical expression for the Q of a coaxial quarter wave cavity can obtained by

ﬁrst calculating the time averaged power loss caused by surface currents ﬂowing on the

conductors.

The surface current (1,) on the inner conductor is obtained from equation (2.22):

 

 

1‘, = 1(ﬁme)
Substituting the B—field of equation 2.27 gives:
'° — ____°_ EZ 4“"
J,— Zoaln(b/a) cos(2€)e 2 (2.31)

The general expression for power loss due to current ﬂowing through a resistive material

is: 13105.: II-E d3r

vol
where I is the current density and d3 r is a differential element of volume. Ohm’s law can

be used to express the electric ﬁeld in terms of I .

JzoE

2
Pr... = H; d3r (2.32)

vol
where 0' is the conductivity of the metal. The power loss on the inner conductor is

obtained by assuming that the current ﬂows with uniform density over a depth of one
skin depth (8).

5 = _1_.. (2.33)

Using 3:3. /5 for case of the inner conductor, equation 2.32 reduces to:

30

o In.

Pinner = 2%1 3: dz

Substituting equation'2.3 1, then integrating, and taking the time average gives:

2 E T
21: V. 2 1:2 1 - 2
Fume, = — ., , cos ——. dz — srn at (it
< > a 50(lnb/a)'Z; i (22) Ti ( )

 

V02 1
(Pinner> - 475602: [Ea-J (2.34)

where Zc is the characteristic impedance from equation 2.23.

Similarly the power loss due to current on the outer conductor is:

V2 P
< om) 41:65ZE (2b) (2 35)

Since maximum current ﬂows at the 2:0 end of the cavity, we must also account
for power loss on the shorting plate. Again, using equation 2.22 for 2:0:

I =10“le
ll

 

 

)

5 2:0

Substituting the magnetic ﬁeld from equation (2.27) gives:

V i’ -
J 2 O - -I(DI .
’ Zo ln(b/a) r e (2 36)

As before, the power loss is calculated assuming the current has uniform density over one

skin depth (8 ). For the shorting plate the general expression for power loss reduces to:
21: b
P5,,“ = 8—6 I J:- rdr

Substituting equation 2.36 for the surface current, then integrating, and taking the time

average gives:

31

2

V0
(1)910”) = W ln(b/a) (2.37)

C

The total power loss is then:
<Ploss> = (Filmer) + (Power) + (Pshon>

Substituting equations 2.34, 2.35, and 2.37 yields:

V02 1 1 1 2
(P1055) — W '5 [a + B- + 7ln(b/a)] (2.38)

Finally we can calculate an analytical expression for the Q using

(no UE
P

loss

 

Q:

Where UE = é—CVOZ , and the capacitance was given in equation 2.28. After signiﬁcant

rearrangement we arrive at:

2 ln(b/a)

l i Z
5[a + b + Zln(b/a)]

 

Q = (2.39)

When using equations 2.38 and 2.39 one should be aware that in real resonant
cavities, the conductivity of most conductors is signiﬁcantly reduced from that of the
pure metal [14]. Machining, cutting, and bending metal create crystalline defects that
reduce the conductivity of metal [15]. Furthermore the conductivity of metal decreases
as the operating temperature of the cavity increases.

Finally, it is useful to consider the ratio of R,/Q, where R5 is the shunt resistance.
In section 2.4.4 the shunt resistance is deﬁned in terms of the power loss and the peak

cavity voltage by the expression:

Pm = 21g (2.40)

 

Thus, the (Rs/Q) becomes:

R V02

S

Q = 2<QP....)

 

Substituting equations 2.38 and 2.39 and rearranging, we ﬁnd that the conductivity

cancels and Rle depends only on the geometrical factors a and b.

R
_. = 1,20 ln(b/a) (2.41)
Q TE”
Using equations 2.28 and 2.29 it is easy to show that —E— gives the same result.
‘ R
Therefore: . . 6‘— = —% (2.42)

Although equation 2.41 is speciﬁc for the geometry of the quarter wave cavity, in
general Rle depends only on geometry. This is useful because although the effective

conductivity is difﬁcult to determine, the power loss can still be calculated very simply

by measuring the Q, and either measuring or calculating Jig , to determine Rs.

RS=Q

Ole:

Once R5 is known, equation 2.40 can be used to calculate the power loss for any cavity

voltage.

33

2.4.3 RF Driven Mametron

As explained in section 1.3, the magnetron structure is normally used to generate
microwave power, by converting the kinetic energy of an electron beam into oscillating
electromagnetic ﬁelds in the resonant cavity. However, the inverse is also possible. RF
power can be used to drive oscillating electric and magnetic fields in a magnetron
structure. We wish to use those ﬁelds to accelerate an ion beam.

Figure 15 shows an example of a four gap magnetron structure. The ﬁelds in a
magnetron are most easily understood if one thinks of the charge which bounces back and
forth from one side of an oscillator to the other. At the moment of maximum charge
difference across a slot (+/-), the electric ﬁeld in the slot is maximum. However, when
charge is ﬂowing from one side to the other, the current around the hole creates a
magnetic ﬁeld which points into the page. Thus the slot acts like a capacitor and the hole
acts like an inductor.

Half an rf cycle later
Capacitors

nductors

.0» —+

 

. I.
.i' a 5

Figure 15. A four gap magnetron structure showing the n-mode at two diﬂ'erent times

The 2 axis points into the page.

34

 

 

., I ~11—

. l 3......
L

 

 

 

 

 

Figure 16. Geometry of a single oscillator and its equivalent circuit

In this example, the magnetron structure is a set of four coupled harmonic
oscillators. There are four modes of oscillation, however only the 1t-mode is of interest
here. Strapping can be used to eliminate unwanted modes [6]. Figure 15 illustrates the
1:-mode in which the charge on each adjacent electrode is opposite. In other words each
oscillator is 180° out of phase with the one next to it, hence the name “1:-mode”.

Each oscillator behaves like an LC circuit. The geometry and equivalent circuit
for a single oscillator are shown in Figure 16. A good estimate for the inductance and

capacitance of a single oscillator can be calculated from the geometry alone.

C=e ° (2.43)

 

Where (can, is the cavity length into the page.

2
mL
° I

C8V

(2.44)

 

L=u

The resonant frequency of a single oscillator is evidently:

1 C gc
: __.._ _-_-_ __ 2.45
(9° a/LC rL Vnic ( )

 

35

where little c is the speed of light and all other geometrical parameters are deﬁned in

Figure 16.

 

, c e
LEZC £31. :> JT-i':

Figure 17. Equivalent circuit for a four gap magnetron in the 1: mode.

 

 

Figure 17 shows the equivalent circuit for the entire magnetron operating in the

1:-mode. The total inductance, L’, is:

and the total capacitance, C' , is:

C'=4C

(t)

0

(D, : —l—— = —l—— :
° JL’C’ JLC
Thus, in the 1:-mode, the resonant frequency of the entire magnetron is the same as the

resonant frequency of a single oscillator.

We are interested in the electric ﬁelds in the central region because they will be

usedto acceleration ions. The magnetic ﬁelds in the central region are negligible
therefore Vx E = 0 so the electric ﬁelds can be solved in terms of an electrostatic

potential. The voltage potential is a solution to the Laplace equation in cylindrical

36

coordinates with radial and azimuthal dependence only. Matchin g to the boundary

conditions, leads to a Fourier series in 0.

V(r,0,t) = vo [An (1%)“ sin(n0) + A2,, (#2“ sin(2n0) + A3,, (ﬁfn sin(3n0) +.... ]eiwrt

 

Where: Amn = § [-—%—]sin(ﬂ) for: m = 1,2,3,
.. m- 2

1: f
Here n is the harmonic number and f is a fraction which deﬁnes the gap width in terms of
the angle subtended by a gap (0g) and the angle subtended by the remaining wall (0“,) at

r = R (see Figure 15).

0, 0
f 2 ° - g
9,, + 6g 21:/Ngaps
Or written more compactly
V(r 0 t) - V -—8— i {—1— sin(1n—75f—) (L)m" sin(mn0)] ei‘”rft (2 46)
9 s _ 0 2 2 2 R .
' It f m=l m

A good estimate of the voltage can be obtained by using only the lowest order

term. For m = 1 only:
V(r,0, t) = V0 (71%)" sin(n0) eimrft (2.47)

This expression represents the voltage due to hyperbolic vanes as shown in Figure 18
The azimuthal electric ﬁelds and energy gain of an ion accelerating through the electric

ﬁeld will be derived in section 2.5.6.

37

-i-VO

 

Figure 18. A four gap magnetron made of hyperbolic vanes

38

2.4.4 Eguivalent Circuit Representation

Generally resonant cavities have electric and magnetic ﬁelds which exist in the
same volume, therefore their inductance and capacitance are distributed. On the other
hand, a lumped LC circuit has inductance and capacitance which are separated spatially.
In spite of this difference, the frequency response of any resonant cavity can be
accurately represented by an equivalent lumped LRC circuit, at least in the region near
resonance.

The equivalent circuit representation is important because it presents the theory in
terms of gross quantities such as L, R, C, and Q which can be measured. This section
draws these quantities together and presents a simpler scheme for the theory of resonant
cavities.

The circuit which most accurately represents the frequency response of a
resonant cavity is the parallel-series combination shown in Figure 19. However, this
circuit is complicated mathematically, and the series resistance (R) is difﬁcult to
measure directly. The common practice among rf engineers is to use the parallel circuit of
Figure 20. This circuit is much simpler to solve, and although the shunt resistance (R,) is
a ﬁctitious quantity, R, can be measured directly. The two circuits are equivalent at

resonance if we choose R5 = QZRo .

 

At Resonance

 

L
V00 ::C R § @Vo QZRO

 

 

 

Figure 19. The CII(R°+L) circuit which represents the frequency response of a resonant cavity

39

At Resonance

3:
\/o® C L RS f7> @Vo RS

 

 

 

 

 

Figure 20. The parallel RLC circuit representation of a resonant cavity uses a ﬁctitious shunt
resestance (R,)

First consider the impedance (ZLC) of only the parallel LC combination from the circuit

 

of Figure 20.
i011.
ZLC = ———-—l 1 = 2
imC + ,——— — 92,—
1601. (06

Where the resonant frequency is:

l

0):—

° JLC

The impedance of the parallel LC combination becomes inﬁnite when (to—>010).
Therefore, at resonance the parallel circuit reduces to Rs in series with V0, as shown in

Figure 20. This means that the power loss at resonance is due to the current through the

shunt resistance only.

k)

V
P - ° (2.48)

loss _ 2

 

W

S

In the full parallel R,LC combination, the voltage across the capacitor is the same as the

voltage across the resistor. Therefore, the Q can be found using U; = 1/2 CV,,2 and

equation 2.48 for the power loss.

 

Substituting these into: Q = ‘ I:

gives three useful expressions for the Q

 

 

 

 

 

 

 

 

Q = mORSC (2.49)
- 1 R.
usm 000 =———: Q = (2.50)
g JLC (0.1-
or eliminating coo : Q = R5 %- (2.51)

 

 

 

Recall that this same relationship was found for the quarter wave cavity (equation 2.42).

Thus the power loss can be calculated using two simple equations:

 

 

 

 

P = where RS=Q {-3- (2.52)

 

 

 

 

 

When designing a resonant cavity to be used as an accelerating structure, the
bottom line is usually “how much power does it take to generate a voltage Vo ?
Therefore the shunt resistance is an important quantity to know. For experimental

measurements it is useful to think of the shunt resistance as the Q times ,/L/ C .

The Q depends on the resistivity of the metal and on cavity geometry both [16], but can

be measured directly from the frequency response of the cavity. ,/L/ C depends only on

the cavity geometry and usually can be calculated (equations 2.28, 2.29 or 2.43, 2.44).
If expressed in terms of the resonant frequency, then fo can be measured and only one of

L or C must be calculated.

 

Olt“
I

21tf°C

41

2.4.5 The Hybrid Coaxial - Maggetron Cavity

Section 2.4 mentioned that the if cavity used for the 50 keV ICRA is a hybrid
between a coaxial quarter wave cavity and a magnetron structure. The chosen rf
frequency of 152 MHz meant that the inductors of a pure magnetron would extend too far
in the radial direction to ﬁt into the warm bore of the available 2.5 Tesla superconducting
magnet. The solution was to attach the vanes of a magnetron to the open end of a quarter
wave cavity. In this way the vanes of the magnetron supply the electric ﬁelds needed for
acceleration and the quarter wave cavity acts as the resonant structure needed for rf
oscillation.

The speciﬁc geometry of this hybrid cavity will be shown in section 3.2.4. For
the rf theory it is only necessary to know that adding the vanes of the magnetron adds a
large capacitance to the end of the coaxial cavity as shown in Figure 21. The additional

capacitance means that the coaxial cavity must be shortened to less than N4.

 

 

 

7//////////////////////// _________ ”I \2

 

 

 

 

\\\\\\\\\\§\\ \\\\\\\\

§

s

\

s

\

s

s

P

\

 

IIIr/IIIIIJJIIIJTI’IIIII/111111 [Ir/[Illfllllrn IIIIIIIII

 

 

 

Figure 21. An extra capacitance added across the end of a foreshortened quarter wave cavity

42

 

Va

 

 

_._._.__.__._._.___.__._.__._,._._...__—___.
_______..-_________J__________

\
/
Coaxial Section Magnetron Z
9.7 cm Vanes
5cm

Figure 22. Assumed voltage profile and lengths of the hybrid cavity for the 50 keV ICRA

The capacitance of the magnetron section (Cm) is large compared to the remaining
capacitance of the coaxial Section (C,). The approach used here is to assume that the

inductance of the magnetron vanes is negligible since they are in the low magnetic ﬁeld

region. Under this assumption the voltage proﬁle would be‘as shown in Figure 22.

The new length of the coaxial section can be found from the desired resonant frequency.

or LC = 1 = ﬁxed

 

where both L and C are proportional to the length of the coaxial section, but Cm is a

constant. Thus the length (I) can be found by solving the quadratic equation:

(L’£)(C’€+Cm) = = ﬁxed (2.53)

_1_
00%

43

In order to calculate the power needed to generate the voltage V0, we need to

know the shunt resistance R5 = QE . Then the power can be calculated using

2

V .
equation (2.48) Ploss = 31%. Thus we need to know the change inf-(IE: and in the Q

caused by the additional capacitance (Cm -> C E + Cm).

 

The change in R5 is dominated by the new J—E- . Luckily this is simple to calculate.
L = _1__ 1
\E mocm " 21tf0(Cg +cm) (2'54)

The new theoretical Q can be calculated by repeating the procedure of section 2.4.2 with
two changes. First, the integrals over currents on the inner and outer conductors will be
taken over the new shorter length of the coaxial section and second, the current must be
scaled up due to the additional capacitance. Since the charge on a capacitor at any time

B:q=CV,

differentiating gives: I = 3:15:- = Cid‘tl = (nCV

Thus the current is proportional to the capacitance, so the new current is simply:

I0 —9 Io[——Cé:im]

Although the theoretical expression for the Q is straight forward to derive, it is of
little use because the Q depends on the conductivity (0’). In practice the conductivity of
the cavity depends on the resistance of joints, connections, and surface oxidation on the

conductors. The conductivity of the pure conductor is almost never achieved.

Furthermore, the Q can be measured from the frequency response curve of the actual

cavity.

The method used to determine the power loss and V0 in the hybrid cavity is to
measure f0, measure the Q, and measure Cm. then calculate the remaining capacitance of
the coaxial section. With these quantities known, the shunt resistance can be calculated
so that the power required to generate V0 is known (equation 2.52). These measurements

are given in chapter 4.

45

2.5 Analm'cal Model of the Beam Traiectogy

This section covers all aspects of the beam trajectory that can be calculated
a“alytically. The analytical model describes only the central ray through the system, i.e.
abeam of zero emittance. Multi-paiticle trajectories are dealt with using the computer

QCiel in section 2.6. Limitations of the analytical model are discussed in the beginning

aftﬁcCtion 2.6.

We begin with a detailed overview of the beam trajectory which includes some

moti Vation for each sub-section below. Figure 23 shows atop and side view of the entire
beam trajectory through the ICRA. The basic components of the ICRA are the ion
8011 re :3, electrostatic bend, superconducting magnet, rf driven magnetron, and a target.
Notice that the trajectory is divided into three distinct regions: the injection region, the
accel eration region, and the extraction region.
A superconducting magnet provides a B—ﬁeld which is constant in time. The ion
beam is extracted from the source directly along a B-ﬁeld line so that the v XE force on
the beam is zero. We will see in chapter 4, that this beam actually contains multiple ion
Spec i es. The beam passes between a pair of electrostatic bending plates which are
located at some radius away from the z axis. All ibn species in the beam are deﬂected to
the Same angle Shem (section 2.5.1). After the beam is deflected, it will have momentum

c:C)111I><)nents perpendicular (p i) and parallel (pH) to the local magnetic ﬁeld.
pJ. = ptotal Sin(ebend) pll : ptotal cos(ebend)
'1‘
he perpendicular component causes the beam to orbit at radius r = p l / qB , and the

pat-
al 1e] component will cause the beam to move in the axial direction toward the high

‘F
1‘ e1(1 region as shown in Figure 23.

A

46

 

 

TOp View Superconducting
Solenoid

Magnetron

 

Electrostatic

'°n Bend
°Urce

Injection Region Extraction Ragion

SESIF'<::I teas:

Figure 23. Top and side view of the trajectory through an ICRA with
trajectory shown in red, and magnetic ﬁeld lines shown in blue.

The axial magnetic field profile is plotted below.

47

 

Imagine for a moment what happens if we begin with zero voltage on the bending
Plates and then gradually turn up the bend voltage. Obviously the bend angle will
i"Crease, which increases p i and decreases p". Since p .L increases, the radius of the beam
orbit will grow. Figure 24 shows that if we look in the axial direction, we see that the
Ismut on the orbit where the beam initially passed through the bending plates is a ﬁxed
mint- As the orbit radius increases, the center of the orbit shifts over in a direction
perpendicular to the direction of the kick given by the electrostatic bend. We want the
bearn to be centered around the z axis so that it will be centered in the magnetron when it
an-i ves at the acceleration region, and there is only one bending voltage (or bend angle)
for which the beam is centered around the z axis. Therefore, the initial ﬁeld line with
which the ion source is aligned must be at the proper radial distance away from the z axis

SO mat the beam will be centered when the desired ratio of (p l/p") is reached.

 
   
 

 

 

 

Bending
Plates
WW Fixed
/ .— o
\ -l I Pomt
X mm

Figure 24. Beam orbit as seen looking in the axial direction as bend voltage is increased

48

 

As the beam spirals intothe high B-ﬁeld region, the orbit radius follows magnetic
ﬁeld lines. This means that r and p” decrease, but p i increases. Section 2.5.2 shows that
assuming conservation of magnetic moment, these three parameters can be easily
calculated at any point in the injection trajectory if the magnetic ﬁeld proﬁle is known.

If the peak B-ﬁeld is high enough to drive the axial momentum (p2) to zero, the beam will
mirror (section 2.5.3). This condition is important because it gives an experimental
reference point from which we can determine the axial momentum. As mentioned above,
the beam actually contains multiple ion species, however we are only interested in
accelerating the W (proton) portion of the beam. Section 2.5.4 shows how the injection
region of the ICRA can be used as a mass spectrometer to measure the constituents in the
beam and to eliminate all but the W for injection into the acceleration region.

After traveling through the injection region, the ratio of p Jpn has been
transformed so that the proton beam arrives at the entrance to the acceleration region with
the desired radius and axial momentum. The B-ﬁeld is relatively constant through the
acceleration region, so pz can be assumed to be constant. The axial velocity determines
the number of turns the beam goes through while traversing the cavity length (section
2.5.5). Inside the magnetron rf electric ﬁelds accelerate the beam in the azimuthal (9)
direction. If the cyclotron frequency remains in resonance with the rf frequency, the
beam is accelerated across every gap in the magnetron. This causes the orbit radius to
increase while the beam continues to drift axially. Because of the radial dependence of
the E-field, the inner diameter of the magnetron should be tapered (dR/dz) for maximum
energy gain (section 2.5.6). The acceleration trajectory is ideal when the ion skims along

the inner diameter of the cavity.

49

When the beam exits the magnetron it continues to drift axially into the
extraction region. Since the extracted beam has a large energy spread from nearly zero
to the full design energy, the beam in the extraction region should be thought of as a solid
rotating cylinder of ions with high kinetic energies at large radius and the lowest energy

at the center (E ~ r2 ).

The equations of section 2.5.2 that were used for the injection region, can now be
applied to each energy component of the beam in the extraction region. As the B-ﬁeld
drops off, the beam follows ﬁeld lines. The perpendicular momentum decreases while
the parallel component and the orbit radius grow. The axial length between turns is
calculated in section 2.5 .8. This is important to consider when designing a target or beam

diagnostics in the extraction region.

50

2.5.1 Electrostatic Bend

Let us derive the bending equation for an electrostatic bend in a region with no
magnetic ﬁeld. The electrostatic bend is essentially two ﬂat plates which are separated
by a gap, g, and with voltage difference Vbcnd. Using the bending geometry illustrated in
Figure 25, we assume that the beam will follow a curved path with radius of curvature p,

and that the electric ﬁeld always points radially toward the center of this curvature.

 

 

Figure 25. Geometry of the electrostatic bend

Equating the centripetal force with the electric force supplied by the E-ﬁeld

between the bending plates gives:

where q, m and v are the charge, mass and velocity of ions in the beam passing between

the bend plates. The electric ﬁeld between the bend plates is approximately

 

 

_. V . . 0
IBM“, = :“d and the beam from the ion source has kinetic energy K = 1/2 mv“ = qum

q Vbend = zqvbeam

therefore, we can write: g p

so the bending radius is:

V

 

 

p = 2g ”‘3’“
Vbend
We can solve for the bend angle in radians:
.LV £2
6 n = i = iﬂ 4“— [radians] (2.55)
be d p Vbeam ( g '

 

 

 

where a change in notation has been introduced from the arc length (5) to the effective
length of the bending plates (Keg). There are two important points to notice about equation
2.55. First, the bend angle does not depend on the mass or charge of the ion. Therefore,
different ion species are all bent to the same angle. Second, the bend angle (Ohm) is
proportional to the bend voltage Vbcndg This will be useful when calculating the
trajectory through the injection region. In practice we do not need to know (cf; because as
we will see in section 2.5.3, the mirror condition can be used to solve for this constant
(see equation 2.69).

Of course a real electrostatic bend has fringe ﬁelds and edge effects that focus
the beam. Since the analytical model deals only with the central ray, these higher order

effects are not dealt with analytically, but will be accounted for in the computer model.

52

2.5.2 Conservation of Magnetic Moment

After the beam leaves the electrostatic bend, it has momentum perpendicular (p i)

and parallel (pH) to the magnetic ﬁeld. The perpendicular component causes the beam to
orbit around the B—ﬁeld lines at radius r = 2% , and the component parallel to the B-ﬁeld

causes the beam to move axially into the high ﬁeld region. In order to calculate the
trajectory of a single ion through the fringe ﬁeld of the solenoid, we use the fact that the
magnetic moment (umag) is conserved in slowly varying magnetic ﬁelds [18, 19].
That is:
ttmg = constant (2.56)
The magnetic moment is the current ( I) times the area (A) encircled 'by an orbit. '
it... = IA” (257)

Using I = qfc , and p L=qBr, this can be expressed as

[1 = P1 ___ El.
mag 2mB B

 

Or substituting directly into equation 2.57.

' ' _ _ _ (1B 2
”mag '- IA — qch - q(-2?m—]nr
This means that the magnetic ﬂux ((Dmag) is also conserved.

(Dmag = BA = constant (2.58)

Thus, the beam orbit follows magnetic ﬁeld lines as shown in Figure 26.

 

Therefore: Br,2 2 B 2r:
£2. _ 5
r, Bz (2.59)

 

 

 

53

 

 

 

 

 

 

Figure 26. Geometry of conservation of mgnetic moment (copied from reference 19).

This means that if we know the radius of the orbit (n) in the low ﬁeld region (3;), then
the radius (r2) in the high ﬁeld region (B2) will be reduced by the square root of the ratio

of the B-ﬁelds. Similarly we,can calculate the perpendicular momentum by substituting:

pJ-z B2
= /— 2.60
PI, Bl ( )

Notice that p 1 increases as the square root of the Béﬁeld ratio. Finally we can calculate pl.

pi = qBr for r.

 

 

 

 

 

using the fact that kinetic energy of the beam is conserved in magnetic ﬁelds.

Therefore: pi = p30“, —p:2 (2.61)

B
PI2 = thzotal ‘13:.[32'] (2.62)
1

Since p i increases while spiraling into the high ﬁeld, and the total momentum remains

 

 

 

 

 

constant, then p" must decrease.

54

As a check of the accuracy of this approximation, Figure 27 shows the
analytical calculation of the orbit radius through the injection trajectory. The result is
compared with data from the computer model of section 2.6. The computer model
contains no approximations of this kind and should be considered completely accurate for
the purposes of this comparison. The two calculations were started at the same radius in
the high B-ﬁeld region and then calculated backwards though the injection region so that
the error can be seen in the region of large radius. The error in the analytical
approximation (relative to computer model) grows to 1.5 % for the magnetic ﬁeld proﬁle
shown. The error will be greater for a ﬁeld which drops off faster or for a trajectory

which is relatively more parallel (fewer turns).

 

 

 

 

 

 

 

 

 

0.50 1.2

0.45 1

0.40 -
’5? I
2 -
8 0.35 : O
.3 _ m

'- [\4

f5 0 30 : —Analytical in
'O ' ..
E _ — Corrputer

0.20 E

0 15 h 1 l l 1 l 1 l l l 1 1 l l 0 O

14 12 10 -8 -6 -4 2 0

Z (inches)

Figure 27. A comparison of the beam orbit radius through the injection region calculated using the
analytical model and using ICRAcyclone. The magnetic ﬁeld proﬁle is shown in blue.

55

At the beginning of section 2.5 we said that the analytical model of the beam
trajectory deals only with the central ray through the system, i.e. a beam of zero
emittance. However, conservation of magnetic moment has some implications on how
the spreads in the beam radius and momentum change. Therefore we will diverge brieﬂy
to discuss these here.

Imagine that a real beam entering the solenoidal fringe ﬁelds contains a spread

in beam radius, Ar. From the geometry of the ﬁeld lines alone, it is easy to see that the

 

radial spread in the beam will decrease as the beam spirals into the high ﬁeld (Figure 23).

This can also be seen by differentiating equation 2.59.

B
Ar, = Ar, l—B—l (2.63)
2

So the radial spread (Al'g) in the high ﬁeld region (Bg) is less than the radial spread (Ari) v

 

 

 

 

in the low ﬁeld region (Bl) by a factor of J—BBL .
2

The spread in the perpendicular momentum is found by differentiating equation 2.60.

 

 

 

 

 

 

B, .
APLZ : Apil B- (264)
1
. . B.
Hence, the spread in p _L increases by a factor of B” .
l

There is no simple form for Ap... However, if we use 2.60 to express 2.61 in terms of

parallel momentums only:

 

56

then differentiating gives:

 

 

 

Apll2 = ' Apll] [_:_2)£§[1_] (2-65)
1 nz

Although the actual parallel momentum decreases while spiraling into the high ﬁeld

 

region, both (B2/B1) and (pm/pug) are greater than 1, so the spread in p" increases.

57

2.5.3 Mirror Condition

As discussed in section 2.5.2, when the beam goes from a region of low
magnetic ﬁeld to a region of high ﬁeld adiabatically, the perpendicular momentum
increases by the root of the B-ﬁeld, and the parallel momentum decreases such that the
kinetic energy of the beam is conserved. If the magnetic ﬁeld becomes high enough,
eventually the parallel momentum goes to zero and all of the momentum will be
perpendicular to the B-ﬁeld. At this point, the axial motion of the beam has come to a
stop, but the axial force on the beam (F = — qveB,2) is still in the negative 2 direction
(see Figure 26), so the beam will be reflected. This is referred to as “mirroring”.

In the caSe of the injection region of the ICRA, we want to determine what
conditions, back in the low ﬁeld region at the electrostatic bend, will lead to a beam
which mirrors when it reaches the high ﬁeld region. Here, we take B1 to be the low ﬁeld
and B2 to be the high ﬁeld. Since we know the conditions at B2 for mirroring, we simply
need to impose the mirror condition (pl,2 ——>'O ), and then solve for the conditions at B1 in

terms of those at B2. Begin by conserving kinetic energy between region (1) and (2).
. Ialz = (2,12
pi+pi = pé+pL

Now impose the mirror condition (p,,2 —> 0 ).

pi+pL = 0+pL

Use (2.60) to express the p 2 in the high ﬁeld region in terms of p 2 back at the bend.

B
pi. = pi. is?“ j

58

We are interested in the bend angle which causes mirroring, so solve for the ratio of

perpendicular to parallel momentum, then use: tan(ebcnd) = & -

lll

Pu - B1
‘—| = J (2.66)
pm minor Bz —BI

so that: amino, = tan ’1 [ ——B-'—] (2.67)

 

 

 

 

 

BZ-Bl

Notice that the momentum ratio at the electrostatic bend, which leads to
mirroring at the high ﬁeld region, is known from the magnetic ﬁelds alone. This is the
form that is most useful for the ICRA. However, this result can be understood intuitively

if we invert and square equation (2.66) to obtain a kinetic energy ratio.

Km B2 — B,
——- = —-— (2.68)
KJJ ‘ Bi

mirror
The ratio of parallel to perpendicular energy needed to reach the high B-ﬁeld region (B2)
is simply the fractional increase in the B-ﬁeld. In fact this expression is exactly
analogous to what one would derive for a golf ball rolling up a hill (zero rotational
energy), in which the ratio of initial kinetic energy (K) to potential energy (U) is simply

the fractional increase in height (h).

 

Kl +UI = O+U2
Kl _ hz'Fhl
U1 — hl

This analogy occurs because in the process of rolling up the hill, K is converted into U

(just as K" is converted into K 2) while the total energy is conserved, and because U

59

changes linearly with h (just as K 2 changes linearly with B (eqn 2.53) ). Therefore, if one
is willing to think of K i as a potential energy that is stored in the radius of the beam, and
of K” as the kinetic part of the energy, then equation 2.68 says that the kinetic energy
needed to reach the top of the hill is simply the potential energy times the fractional
increase in the B-ﬁeld.

The mirror condition provides a simple way to solve for the constants in
bending equation 2.55. The mirror voltage (meor) will be measured experimentally
(section 4.5.3), and the mirror angle is easily calculated from the B-ﬁeld. Therefore,

since the bend angle (Ghend) scales linearly with bend voltage (Vbend) equation 2.55 can be

expressed as:

 

V
9 = .
bend 9 mirror ( V

 

 

 

“"d J (2.69)

mirror

 

   

 
  

“A‘AIAINMIMMW
v-V-v-uwu-rmmn .

 

 

N
NV

 

 

Figure 28. Mirroring the beam is analogous to when a ball does not have enough kinetic energy to

make it to the top of a hill.

60

2.5.4 Mass Spectrum of the Injected Beam

In the ICRA, beam is extracted from the ion source along a B-ﬁeld line so that
the V x‘B force on the beam is zero. All ions are accelerated through the same voltage '
(Vbeam) from the ion source aperture to the puller. However, because different ions may
have different charge or different masses, different ions will have different total

momentum.

PM = VZmE = (/2mqV.,.,m (2.70)

When entering the electrostatic bend, the beam is parallel to the B-ﬁeld, p 2 = 0, and p" =
pm“. In section 2.5.1 we showed that the electrostatic bend will bend all ions to the
same angle (9mm) regardless of their mass or charge (equation 2.55). Refer to Figure 25
to see that:

p. = pm... sin(9.,...) (2.71)
As soon as the beam obtains a component of momentum perpendicular to the B-ﬁeld,

ions will experience a ‘v x B force tending to make them orbit around a gyro center with a

p .

radius of: r = q—B

This can be expressed in terms of the bend angle, and beam energy using equations (2.70)

 

and (2.71)
1' ___ protalsin(ebend) : _—Mam sin(9 )
(1B (1B bend
Therefore:
2 V
r = Mame“) (2.72)
qB

61

Equation 2.72 gives the radius of the orbit immediately after the beam exits the

electrostatic bend. Next, let the beam spiral from the bend (B1) to the high B-ﬁeld region

(B2). Using equation (2.59) gives:

[Bl ’2mec .
r, = r — = ——l“— srn(9 n ) (2.73)
b 1 B2 quB2 be d

The important thing to notice here is that the radius of each different ion beam is
proportional to the root of its mass to charge ratio, but the radius of all ions will increase
as we increase the bend angle. Figure 29 shows the relative sizes of each constituent in

an ion beam containing H+, H2+ and Ar+ ions for a ﬁxed bend angle.

 

Figure 29. Axial view of a beam containing three ions species.
The bend voltage is ﬁxed at the point for which the Argon orbit is centered around the z axis.

In the 50 keV ICRA, the narrowest aperture for the orbiting beam in the entire
system is at the entrance to the magnetron. Experimentally we can turn up Vbcnd while
measuring the beam current on a Faraday Cup at the extraction end. As each ion beam

clips on the entrance to the magnetron, the current will dr0p at discrete bend voltages.

62

The resulting graph of beam current vs. bend voltage gives a spectrum of each ion species
in the beam.
The voltages where each constituent in the beam clips can be calculated from

equation 2.73. First we simplify by lumping all the known constants into a constant C.

r2 = CJ—rqisinwbmd)

Setting r2 equal to the radius at which the beam will clip and solving for the bend angle

we obtain:

 

 

. -1 rl'
ebend =sm . [ﬁg—9 {13] (2.74)

 

 

Finally, the bend angle can be converted to the bend voltage using equation (2.55 or
2.69). Section 4.5.1 shows an ion Spectrum measured experimentally and compares

with the theoretical results using equation 2.7.4.

2.5.5 Number of Turns Through the Acceleration Region

After spiraling through the injection region, the beam arrives at the entrance to
the acceleration region with r, p 2, and p” which have been transformed according to the

equations in section 2.5.2. We now wish to calculate the number of turns (Nmms) that the
beam goes through as it traverses the axial length of the cavity (6%.).

The magnetic ﬁeld must be relatively constant over the length of the accelerating
region in order to maintain resonance. Therefore, it is a very good approximation to
assume the ﬁeld is constant in the analytical model. The number of turns through the
acceleration region is simply the ratio of the total time spent traversing the cavity length
to the period for one revolution.

N : tcav

turns T
' C

. . . f -
The total time spent in the cavrty depends on the axral velocrty (v2 = i) and the

cav

 

 

 

period for one revolution is related to the cyclotron frequency by: -—1— = fc = iB—u
Tc 21cm
So the number of turns becomes:
3 qu’
N _ f C8V _ cav .
turns C Vz 216132 (2 75)

 

 

 

When using this formula, one should be aware that equation 2.75 gives the
number of turns through the length of the acceleration region only when the rf is off (no
acceleration). When the rf is turned an, axial components of the electric ﬁeld (caused by
the tapered inner diameter of the magnetron) lead to an axial momentum change which

depends on the phase of the beam. This effect is explained in section 2.6.2.

2.5.6 Acceleration Region

Section 2.4.3 showed that the electric fields in the central region of the rf driven
magnetron can be written as a Fourier series in the azimuthal dependence. The lowest
order contribution is for the case of hyperbolic vanes (refer to equation 2.47 and Figure
18). Using this ﬁrst order term gives a good approximation of the cavity voltage and is

useful for illustrating the important aspects of acceleration through the magnetron.

The lowest order term in the voltage, as a function of radius (r) azimuthal

position (9) and time (t), is given by:
V(r,6, t) = vo (ﬁr—r sin( n6) e‘mrrt (2.76)

Here R is the inner radius of the cavity, V0 is the voltage on the cavity wall and n is the
azimuthal mode number of the cavity, which comes from the number of gaps (Ngaps)
N gaps = Zn

The azimuthal component of electric ﬁeld is found by differentiating equation (2.76).

E = - Vv
E9 = -31-::°— (31%)"-l cos(nO) e‘wrr' (2.77)

Consider an 8 gap magnetron structure as shown in Figure 30. For this case

n = 4. The azimuthal electric ﬁeld (E9) is plotted as a function of 6 in Figure 31. The dc

magnetic ﬁeld points out of the page, so ions orbit in a counter-clockwise direction at

65

 

 

 

 

 

Figure 30. An rf driven magnetron structure with n=4 oscillating in the n-mode.

Ti
<:>
<+
II
8|
3,

E9 Ion

p>
"-9
>
---<.-------
2P.--
CDV

R)
:l

-------.>___-
A---

CDV

\

--<

4:; :
<

 

 

C)

R)

:l

O
.9...

Figure 31. Azimuthal component of the E-ﬁeld vs azimuthal position.

the cyclotron frequency (Dc = am]: (for 7 = 1). The phase of the rf signal is deﬁned as:

t = ant
and assuming the orbit is perfectly centered, the azimuthal position is:

0:034.

If an ion crosses the ﬁrst gap (9 = 0) at time t = 0, the electric field is maximum and the

ion will be accelerated in the forward (6) direction. If the cyclotron frequency is such

that the ion arrives at the second gap (8 = 45°) at the time when the rf phase has changed
by half a cycle (1: = 180°) then the electric ﬁeld points forward and the ion is accelerated

——Zl——) equal to the time for one

sapsc

again. Setting the time to travel between gaps (t =

half of an rf cycle (I = 515— ), leads to the resonance condition required for acceleration.
rf

 

 

(2.78)

 

mﬁznm

C

 

The kinetic energy gained in crossing a single gap can be found by integrating

along a path at constant radius.

Km =JF-ds
.lL .
Kgap = --an0 (1%)" I; cos(nG) em)rft d6

Keep = lzt'qu (1%)"

' B

If we deﬁne the total voltage difference across a gap as Vgap = 2Vo , the energy gained

across a single gap becomes

 

 

 

Kgap = %qvgap(i-)“ (2.79)

 

where 1t/4 = 0.79 is the transit time factor.

Notice that the multipole nature of the magnetron, leads to an rn radial

dependence in the energy gain. This radial dependence is plotted in Figure 32 for n=4.

67

The beam will gain maximum energy if it remains as close as possible to the cavity wall
(r = R) throughout the acceleration trajectory. For this reason the inner diameter of
magnetron should be tapered with 2 so that the beam will remain close to the cavity wall
as its orbit radius increases. For this research, a linear taper was chosen (dR/dz =

constant), so that the actual magnetron would be simple to manufacture.

A i

 

 

 

>
r.

m.~--------------..----..-.'..-

Figure 32. Lowest order term for the radial dependence of cavity voltage in an 8 gap magnetron.

Equation 2.79 can be used to calculate the orbit radius and energy gain through
the acceleration region. One method is to choose the cavity slope (dR/dz), initial radius
fraction (r/R), and axial momentum (p2), then let the particle gain energy Kgap across the
ﬁrst gap, then recalculate: r, z, and R, so that Kgap can be calculated for the second gap.
Continue this iteration until the cavity length has been traversed. If this is done as a
simple spread sheet calculation, then the cavity voltage (V0) can be adjusted until an ideal

trajectory is obtained. Figure 33 shows a calculation of orbit radius using this method.

A quicker estimate of the total energy gain can be made by estimating the

average radius fraction throughout the acceleration trajectory <-I;—> and assuming that the

68

 

>

 

-—t-——-———

 

5cm Z

Figure 33. Beam orbit radius from spread sheet calculation plotted next to magnetron inner radius

energy gained across each gap remains constant throughout the trajectory. In this case
using the number of gaps per turn (Ngaps = 2n) and the total number of turns, (Nmms), the

total energy gain is simply:

 

 

 

Kgain = _-2£qvg,p<—I%> 2nNturns (2.80)

 

Finally, note that in this discussion the radial component of the E-ﬁeld has been
completely neglected (see Figure 15). This is a valid approximation because when
averaged overtime, 1-3r does not cause any net acceleration. Furthermore, the tapered
inner diameter of the magnetron will cause a component of electric ﬁeld in the z
direction. Similarly the end ﬁelds, resulting from the ﬁnite axial length of the magnetron,
will certainly have components in the z direction. For the most part these effects have
been found to be small, however they will be discussed in sections 2.6.2 and 2.6.3. The
full 3 dimensional solution of the electric ﬁeld in the magnetron is accounted for in the

computer model (see section 2.6).

69

2.5.7 RF Phase and Magnetic Field Flatness Criterion

The analytical model of the beam trajectory does not account for any variation in
rf phase. In fact, the discussion and the timing arguments of section 2.5.6 describe the

case in which the phase of the ion is constant relative to the rf ( a),f = n (Dc ). Furthermore

the derivation of the energy gain across a gap (equation 2.79) uses the assumption that the
phase relative to the rf is 6 = 90° so that the ion crosses a gap when the cavity voltage is
at its maximum. However, in section 2.6.2 we will see that the computer model shows
that the ICRA exhibits signiﬁcant phase bunching. Also the experimental results show a
resonance width that is much wider than expected (section 4.6.2). Therefore, it is

necessary to derive analytical expressions for phase changes, as a basis for comparison.

This section ﬁrst gives an introduction to rf phase and then covers phase changes
caused by detuning of the magnetic ﬁeld. Next the magnetic ﬁeld ﬂatness criterion is
deﬁned and an equation is found for the phase change caused by the magnetic ﬁeld

proﬁle over the acceleration region .

The general equation which describes the phase of a particle during acceleration
is:
¢(t) = ((1)0 + (orft —— n9)
or simply
A¢(t) = (coﬂt — n6) (2.81)
If the beam orbit is centered, then the azimuthal position is:

Ozwct.

7O

so that: A¢(t) = (ti)rf — no)c )t (2.82)
In this form it is easy to see that the phase equation is simply a comparison of two rates.
At resonance (wrf = nwc), the phase will remain constant at its initial value ((1)0).
However, if (m 2: no)C then the phase will drift overtime.

The time dependence of the rf signal is taken to be sin((n,ft), therefore we expect
optimum acceleration across any gap to occur when (1) = 90°. We wish to determine how

far the phase will drift through the acceleration region if the magnetic ﬁeld is not at

resonance.

Eh_ase Change Caused by Magnetic Field Detuning
Consider the case where the rf frequency is ﬁxed at (or, =na)co , but the B-ﬁeld

(constant with respect to z) is detuned away from B0 by an amount AB. In this case
equation 2.82 becomes:

A¢(t) = [nonco — n((oco + Amc)]t

qB.

 

Where wco = , and the zero subscript indicates the B-ﬁeld required for resonance

with the rf. In terms of the fractional change in the B-ﬁeld this becomes:
A¢(t) = - nd)co ﬁ-B—t (2.83)
Here A4) is the total change in phase over the time, t, caused by AB. This equation is

valid for any time (t). However, if we want the phase change over the time the ion

71

spends in the accelerating cavity, it is useful to express A4) in terms of the number of

turns in the cavity length. Using:

 

 

 

 

 

 

t
Nturns : ﬂ : fCOtCEIV
TCO
we obtain: Adm = — 27m 1%}; Nturns in radians (2.84)
or: Aqu = — 360° n 33 Nturns in degrees (2.85)
O

 

 

 

These equations describe the phase change that occurs through the cavity length if the
B-ﬁeld is constant (no 2 dependence), but is detuned away from resonance with the rf
frequency. These equations are valid only if acceleration does not cause a phase change.
However, in section 2.6.2 we will see that eleCIric fields in the rf driven the magnetron do

cause a phase shift:

Phase Change Caused by Magnetic Field Profile

The characteristics of Helmholtz and solenoidal ﬁelds were shown in section 2.1.
The ideal magnetic ﬁeld for the ICRA would be perfectly ﬂat over the acceleration region
in order to maintain resonance. However, this is not possible for any real coil. Therefore,
it is useful to characterize the magnetic ﬁeld ﬂatness in terms of the percent drop off

(dB/Bo) at the ends of the acceleration region. The geometry is shown in Figure 34.

72

 

1.005

   

1 .000 .............

--‘-———q

I

I

I

I

I

I

I

I

I
.-----
I

I

I

I

I

I
IE
I

I

I

I
L

 
   

--------------------------------------------------

0.995

0.990

Bz/Bo

 

0.985

  

t

0.980 -

0.975

I

 

 

t------------

 

--—------

 

0.970 1 1 . 1

lb r
y.-
b
in-
D

2 (cm)

Figure 34. Magnetic ﬁeld proﬁle for the 50 keV ICRA with

ﬂatness of SB/Bo < 0.5 % over the 5 cm acceleration region

Assume that the central value of the B-ﬁeld (B0) is at resonance with the rf.
Therefore, the subscript ‘zero’ will now indicate both the central ﬁeld value and
resonance. In this case, AB is a function of 2. Therefore, the total phase change requires

integration. Using equation 2.83.

LB“) dt

A¢(t) = — Ind)“, B

0

The equation can be converted to an integration over 2, by assuming that the axial

velocity (v2) is constant.

73

Kt

[CS

A¢(t) = 7319'?" j AB(z) dz

The B-ﬁeld proﬁle of any coil over this central region can be closely approximated as a

parabola. B(z) = Bo —5B 1 Z” ,
(ﬂea-Y

 

Now integrating over this B-ﬁeld proﬁle from -‘/2 [cav to +‘/2 (my and simplifying gives:

1 5B
A¢mv = "-3- “COCO 'B—o' tcav

Once again, expressing this in terms of N rnrns. gives the same result as equation 2.85 but

reduced by a factor of 1/3.

 

 

 

 

 

A¢cav = ”:14: 21mg}; mm in radians (2.86)
Aocav = —-;— 360°n—g—3Nmms ' in degrees (2.87)
O

 

 

 

Keep in mind that these equations are for when the central ﬁeld value (B0) is matched for
resonance with the rf. Certainly A¢ over the acceleration region can be further reduced
by increasing the central ﬁeld (Bo) so that resonance occurs at some compromise between
Bo and (Bo - 8B). Again, equations 2.86 and 2.87 are valid only if acceleration does not
cause the phase to change, however in section 2.6.2 we will see that electric ﬁelds in the

rf driven the magnetron do cause a phase shift.

74

2.5.8 Axial Turn Length in the Extraction Region

Once the beam exits the acceleration region it spirals into the lower B-ﬁeld
region of the target. This is the extraction trajectory (refer to Figure 23). An important
quantity to know in this region is the axial distance between turns, or the axial turn

length. The axial turn length (Azmm) can be calculated by multiplying the axial velocity

(v2) times the time for a single turn.

Using the cyclotron frequency (f c = ~1—1—) and expressing V2 in terms of the axial

C

momentum. we obtain a convenient expression:

 

= .p_z.
Azmm 21c (18 (2.88)

 

 

 

Where, B = B2(z) and p2 are both local variables, [and p2 can be calculated using the

adiabatic approximation discussed in section 2.5.2. Recall that as the B-ﬁeld drops off
through the extraction region, p 2 is converted into p“, and therefore pz increases.
Immediately after acceleration, nearly all of the energy is in p 2, (p 2/ p" is large so p" is
negligible). Therefore, in a region far enough downstream that the B-ﬂeld has dropped
off signiﬁcantly, the value of pz depends almost entirely on what has been converted from
p ,L to p" . Thus pz will depend strongly on the total energy of the beam. The result is that
in the extraction region, the beam at large radius has a larger axial turn length than the
beam near the center. In fact after pz has grown signiﬁcantly, Azwm goes roughly as the
root of the beam energy and therefore is roughly linear with radius. The analytical
calculation plotted in Figure 35 shows the axial turn length vs. beam orbit radius for

NBC = 0.34 in the 50 keV ICRA.

75

 

 

 

 

 

 

 

 

5 - "
g .-
o 4 -
.5 ,2..-
5- i' f/
a ,
3 F
E L
3
(U
“i -Nurns=5
< :—

_Murns=6
1 —
0 1 1 4 I r 1 L L 1 1 L 1 l L
0 2 4 6 8 10 12 14 16 18

Radius of the Beam (mm) at z = 9"

Figure 35. Axial turn length depends on beam radius (or energy) in the extraction region

It is important to understand the axial turn length when designing a target or a
beam diagnostic for measurement of the extracted beam. In fact this result will be used
for the corrections to the radial probe data in sections 3.2.5 and 4.6.1.

One should also be aware that equation 2.88 gives only the instantaneous value
of the axial turn length. The actual length of a turn in the extraction region depends on
the value of the B-ﬁeld at every point throughout the orbit. This means that if there is a
large change in B-ﬁeld over a single turn, then the axial turn length will grow

signiﬁcantly and equation 2.88 will give a poor estimate of the actual length of a turn.

76

2.6 Computer Model of the Beam Trajectory
This section covers the computer model used to study the beam trajectory

through the ICRA. A complete computer model is necessary for two reasons:

1) There are several effects which are not accounted for in the analytical model (listed
below) and 2) the computer model allows the capability to simulate a beam of ﬁnite
emittance by running thousands of single particle trajectories, each with slightly different
initial conditions. Modeling a full emittance beam is the ultimate goal of the computer

model because this allows the accelerated beam current to be estimated.

The analytical model discussed in section 2.5 gives a good estimate of the
trajectory, but there are several approximations made which limit the accuracy, especially
in the acceleration region. In the analytical model, the beam is assumed to remain
centered on the z axis and it would be difﬁcult to accurately account for an off centered
beam. During acceleration, ions are assumed to remain in phase with the rf (at 4) = 90°).
In reality, acceleration at each gap causes motion of the orbit center, and whenever the
beam is off center there is a corresponding oscillation in the phase (section 2.6.4). Most
importantly, the analytical model cannot accurately predict (r/R) at every step through the
acceleration trajectory and this ratio has major impact on the cavity voltage needed for
optimum acceleration because the slope of voltage vs. position is so steep (Figure 32).
For this reason, the average r/R in equation 2.80 is useful only as a ﬁrst approximation.
The computer model makes no assumptions of this sort and all of the effects mentioned
above are accounted for. Accuracy of the computer model is limited mainly by the

accuracy of the electric and magnetic ﬁelds that are input into the code.

77

Section 2.6.1 below describes the computer model. This section then begins
with the simple case of acceleration through the magnetron in a ﬂat magnetic ﬁeld. Each
section thereafter adds realistic effects, until the entire system is accounted for. Section
2.6.2 shows the dependence of ﬁnal energy on the initial phase. End ﬁelds of the
magnetron are added in section 2.6.3. A realistic magnetic field is added in section 2.6.4.
At this point, the beam can be launched from the location of the ion source and pass
through the entire system using realistic ﬁelds. Section 2.6.5 shows several views of

single particle trajectories through the entire system.

Multi-particle trajectories begin in section 2.6.6. Here, initial conditions of each
particle are adjusted in order to map out the acceptance phase space of the rf driven
magnetron. Next a full emittance beam is launched from the ion source and
characteristics of the injected beam (rf off) are discussed insection 2.6.7. Finally, the full

emittance beam is accelerated and the energy distribution is obtained in section 2.6.8.

78

2.6.1 The Computer Code

A complete computer model has been developed for the ICRA. This model uses
three computer codes: ICRAcyclone, RELAX3D, and BRZcoil.
ICRAcyclone is a modiﬁed version of Z3CYCLONE which was developed here

at the N SCL and has been used for many years to model particle trajectories in cyclotrons

at this lab [20, 21]. ICRAcyclone uses 4th order Runge Kutta integration to step an ion
through its complete trajectory in three dimensions. An ion can be started from the ion
source, then passed through the electrostatic bend, through the injection, acceleration, and
extraction regions, until it stops at the target face. This code handles only single particle
trajectories. However, a beam of ﬁnite emittance can be simulated using many single
particle trajectories. Space charge forces are not accounted for.

The complete off axis analytical solution of the magnetic ﬁeld [12 pg.177] is
calculated using a code named BRZcoil [22, 23]. These Helmholtz, or solenoidal
magnetic ﬁelds are two dimensional (azimuthal symmetry). BRZcoil calculates the
magnetic ﬁeld arrays for B, and B2 over theentire volume of interest, including injection,
acceleration, and extraction regions. These arrays are then imported by ICRAcyclone.

The full three dimensional solution for voltage potentials in the magnetron and
the electrostatic bend are computed using RELAX3D, a code developed at TRIUMF in
Vancouver, Canada [24]. RELAX3D uses successive overrelaxation to solve the Laplace
equation for electrostatic potentials. The voltage arrays are imported to ICRAcyclone
where electric ﬁelds are obtained by numerical differentiation.

The electric ﬁeld of the electrostatic bend is constant in time. However, when

modeling the electric ﬁelds in the magnetron, the electrostatic voltages are scaled by

79

sin((n,f t). Using an electrostatic solution to model the ﬁelds in the magnetron is valid
because. only the central region is modeled where electric ﬁelds are dominant. The time
varying magnetic ﬁelds of the magnetron are negligible in the central region which
means that V x F. = 0. Therefore, the ﬁelds can be derived from an electrostatic potential.
Furthermore, the lumped circuit nature of the magnetron allows the cavity to be much

smaller than a wavelength. Figure 36 below shows the RELAX3D solution of voltage

potentials for a tapered magnetron with 8 gaps.

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 36. Equipotential lines in an 8 gap magnetron structure at

entrance (left) and exit planes (right). Solution computed using RELAX3D.

80

2.6.2 Phase Dependence and Acceleration

This section deals only with acceleration through the rf driven magnetron. All
trajectories through the magnetron use a ﬂat B-ﬁeld so that the effects discussed here can
be isolated. In this section, graphs of ﬁnal energy vs. initial phase are used to illustrate
two topics which are very central to understanding acceleration in the ICRA. The ﬁrst is
the dependence on two operating parameters: cavity voltage and number of turns. The
second, is a look ahead at the why the spread in (p l/p") of the injected beam has
important implications on the amount of beam current that can be accelerated. This
section then covers two unique effects caused by acceleration through a magnetron
structure. These are: l) a change in axial momentum which is phase dependent, and

2) a change in phase which tends to bunch the beam.

Recall from section 2.5.7, that the general equation which describes the phase of
a particle during acceleration is given by:
am 2 (ch + (orft — n0)
or simply
A¢(t) = (wrft — n0) (2.89)
Since the time dependence of the rf signal is sin((0,rt), optimum acceleration across any
gap occurs for (I) = 90°. If 0 > 90° the ion lags the rf (arrives at the gap late), but if 0 < 90°

the ion leads the rf (arrives at the gap early).
Figure 37 shows a graph which is very important for understanding the

characteristics of the ICRA. Ions are started from the cavity entrance plane and

accelerated through the magnetron. The initial phase ((1)0) of each ion is varied, and the

81

ﬁnal energies are plotted. As expected, ions that enter the magnetron with initial phase
near 90° are accelerated across every gap and continue to gain energy until they leave the
acceleration region. Ions with initial phase near 270° are decelerated. N are that
deceleration is less effective than acceleration because if ions loose energy, their radius

decreases and they fall toward the central region where cavity voltages go to zero.

Now consider the three different curves in Figure 37. Obviously as the cavity
voltage increases, the beam gains more energy until it eventually strikes the wall.
Remember that at every point throughout the acceleration region, the maximum orbit
radius of the beam is limited by the inner radius of the magnetron. Any beam that goes

beyond this boundary is lost.

50

 

45 -1
V0 = 1.280 kV

V0 = 1.230 kV

 

V0 = 1.180 kV

N (A) w 8
01 O 01
1 1 1 1

 

Final Energy (keV)
to
o

..11
0"

Initial Energy

10 \ ° .

0 45 90 1 35 1 80 225 270 31 5 360
Initial Phase (deg.)

 

 

Figure 37 . Final proton energy vs initial phase for single particle trajectories through the computer

model of the 50 keV ICRA. The curve is repeated for three different rf cavity voltages.

82

Next consider what would happen if the axial momentum (p2) is increased. The
voltage required for the beam to strike the wall would be higher because the beam goes
through fewer turns in the cavity. This means that there is a one to one correspondence
between V0 and Ntums (or between V0 and p2). In order to proceed, it is necessary to have
a reference point which deﬁnes how to set the cavity voltage. Therefore, let us deﬁne
“optimum acceleration” as the single particle trajectory at the peak of the E-cb curve
which reaches maximum ﬁnal energy without striking the cavity wall. Figure 38 shows
the cavity voltage required for Optimum acceleration, vs. Ntums (or l/pz). This is just the
same inverse relationship between V0 and Ntums that is evident in equation 2.80 for a

ﬁxed ﬁnal energy.

 

L4

L2 -

   
 
  

Some beam hits
the wall

Vo (W)

53
a:
1

Beam not fully
Accelerated

04 -

02 -

 

 

 

0 2 4 6 8 10 12 14 16 18 20 22

Ntums

Figure 38. Cavity voltage required for “optimal acceleration” vs. number of turns through the

acceleration region of the 50 keV ICRA

83

Keep in mind that since Ntums cc l/pz, the information in Figure 38 could also be
presented as Vo vs. p2. Ntums was chosen here, only because the graph seems more
intuitive. Figure 38 is a map of the operating parameters, V0 and p2. Notice that for any
operating point which is below thecurve, the beam will not be fully accelerated. For any

operating point above the curve, some portion of the beam will strike the wall.

The width of the peak of the Ed) curve in Figure 37 has direct impact on how
much beam current is accelerated by the ICRA. For an actual device using a dc injected
beam, ions will enter the acceleration region at all phases. In the limit that the beam
emittance goes to zero (central ray only), the fraction of the beam which is accelerated is
simply the width of the E-¢ curve (above some chosen cutoff energy) divided by the full
360° range. For this reason, m3 is an important parameter in the beam phase space for

determining what fraction of the beam is accelerated (see section 2.6.6).

At this point we must focus on a point that is more subtle, but also very central
for understanding the ICRA. While the‘E-¢ curve does contain information about gross
operating parameters, such as the cavity voltage (Vo)'and the Nmms setting, it also

contains important implications on the full emittance beam which can be accelerated.
Speciﬁcally, the range of (p j/pu) must be small in order for signiﬁcant current to be
accelerated. To see why, consider the Ed) curve shown in Figure 39 below. This time
the cavity voltage (V0) is held constant and the initial ratio of (p i/pi.) is varied. Here we

use the notation:

 

a E — 2.75
P" ( )

 

 

 

84

 

50

 

 

 

 

45 4 Aer/arc: +2°/o
40 “ ' Act/a0 = 0
g 35 4 Aa/ao = -2%
o
5 30 l \
>
a:
5 25 .
r:
“‘ J
i 20
.E .
IL. 15 . Initial Energy \ /
10 .
5 _ ....................................................
O [1'11 trumrTui . r T 1T Te Iri Vi IT .rrr
0 45 90 1 35 1 80 225 270 31 5 360

Initial Phase (deg.)

Figure 39. Final proton energy vs initial phase for ﬁxed V0 in the 50 keV ICRA. As at is increased,
the beam gains more energy and eventually strikes the wall. Here (1. indicates p Jpn of the central

ray.

Figure 39 clearly shows that as the initial ratio of at is increased, the beam is
accelerated to higher energies until it eventually strikes the wall. Recall that in the

computer model all ions are injected at the same initial energy.
- 2 _ 2 2 2 _ .
IPI - P. + 139 + pz — constant for all ions

Therefore, increasing p9 decreases p2, and vise versa. Here we see that acceleration in an
rf driven magnetron causes radial defocusing. Any initial spread in (1 leads to a radial
spread and therefore a corresponding energy spread. To understand the source of the
spread, consider two effects which determine how much energy the beam gains.
Obviously, the beam will gain more energy if Ntums is high (pz low) and less if Ntums is

low (pz high). This effect is then compounded by the radial dependence of the cavity

85

voltage. Throughout the trajectory, the ion whose radius fraction mm is high (large p 2),
sees a higher cavity voltage and moves to ahigher (r/R). On the other hand the ion with a
lower radius fraction (smaller p1) sees a relatively lower cavity voltage, therefore it
moves to a smaller (r/R) relative to the ﬁrst ion. These two coupled effects work together
to cause the beam to gain more energy if (p llpu) is high, and less energy if (p i/pu) is low.
The result is that, the ICRA will accelerate maximum beam current to full energy when

the beam entering the acceleration region has the minimum spread in (p (lpu).

The radial defocusing is seen more clearly in Figure 40 below. Here the beam
radius vs. axial position is plotted for the same three trajectories at the peak of the E41)
curve in Figure 39 above. The ion with initial A0110!o = +2% strikes the wall, while the

ion with initial Aer/ato = -2% is not fully accelerated.

0.8

 

l

0.7

l

0.6

 

lrll

x

0.5

0.4 -

 

 

 

 

 

0.3 -

,.
0.2 i-

)—
0.1 t-

i.
0 1 l 1 l 1 l 1 l 1 l 1 l J_1 l 1 l 1 l 1 l g l 1 l

-1.2 -1.0 -O.8 -0.6 -O.4 43.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

2 (inch)

Figure 40. Beam orbit radius vs 2 position for the three trajectories in Figure 39

with (1),, ° and Aer/0t = (+2%, 0, and -2%) .

86

~ Blame Dependent Axial Momentum Change

In sections 2.5.5 and 2.5.6 of the analytical model, the axial momentum (p2) was
assumed to remain constant through the acceleration region. In the absence of any axial
forces, this would be the case. However, notice that in Figure 39 the maximum energy
gain actually occurs at 00 = 68°, rather than ()0 = 90°. This ocCurs because the taper of
the cavity inner radius causes a change in the axial velocity which depends on phase.
The beam which slows down axially, goes through more turns and therefore gains more

energy.

It is not critically important for the reader to understand this effect because in
practice we do not care which phase of the beam gets accelerated. However, since the
magnitude of this effect does depend on the cavity slope, it is conceivable that this could

place a limit on the maximum slope used. Therefore the effect is explained here.

Consider an ion orbiting in the counter clockwise (+0) direction as shown in

Figure 41 below. For acceleration in the + 0 direction the ion will always pass through
an E-ﬁeld just before each gap which has a component in the + 2 direction, and an
E-ﬁeld just after each gap which has a component in the — 2 direction. If the ion is in
phase with the rf (q) = 90°), then it receives an axial kick forward just before each gap,
and backwards just after each gap, and the net change in the axial momentum is zero.
However, if the ion lags the rf (d) > 90°), then it will experience the maximum electric
ﬁeld just before passing the gap. Hence, its net change in axial momentum will be
positive. Likewise, if an ion leads the rf (0 < 90°), it will experience maximum electric
ﬁeld just after passing the gap, so it will have a net decrease in axial momentum. Figure

41 shows a schematic drawing which aids in visualizing the geometry. The decreased

87

axial momentum causes the ion which leads the rf to go through more turns and gain
more energy. Obviously if the ion leads the rf by too much (for example ¢—)O°), then the
energy gain across each gap is reduced. The phase for peak energy gain (¢=68° in this

case) occurs for an optimum gain in Ntums, yet an insigniﬁcant degradation in Kgap.

Logging the rf Leading the rf
(¢>90°) (¢<90°)

 

 

 

 

 

 

 

 

Figure 41. A snapshot in time of an ion (red dot) at the moment of peak electric field across the gap.
Two cases are shown. On the left the ion lags the rf and on the right the ion leads the 11'.

88

Figure 42 shows the evolution of the axial momentum through the acceleration
region for three ions with initial phases (1)0: 60°, 90°, and 120°. The ion which leads the
rf shows reduced p2 and the ion which lags the rf shows an increase in p2. The magnetron
cavity used here had 8 gaps. The small oscillations within each curve occur because the

ion gets pushed forward then backward 8 times per turn.

 

 

 

 

 

 

 

 

 

 

I I

1— ' I

I I

1.24 — 1 .

I I

r ; '

1.22 _ ' WW

.— ' 1

' 1

1.20 — 1 -

I I

‘3’ ’ ‘ '

0 1.16 ; E i

I
g 1.14 — i
N _ .
I

n' 1.12 — 1 i

I I

1- I I

1.10 - i i

n : —H'IIO‘—‘60deg. '

I

1-08 " : —Phi0=90deg. :

1— I I

1.06 _ I —Fhi0=120deg. :

l. : .

1.04 11 4 1 '1

1 5 -1.0 0.5 0.0 0.5 1.0 1.5

2 (inch)

Figure 42. Axial momentum through the acceleration region (rf on) for three different initial phases.

89

Phase Bunching

Using the computer model to track the phase through the acceleration region
reveals a phase change which tends to bunch the beam near (it = 90°. The source of the
phase shift is the radial component of electric ﬁeld in the magnetron which causes a
rotation in the motion of the orbit center. Figure 43 shows an example in which three
ions started at three different phases each migrate towards ¢ = 90°. The oscillation in the
phase occurs whenever the beam is off center from the accelerator center (acceleration
always shifts the orbit center). Notice that the general trend is to bunch the beam near
90°. An additional view of the phase bunching with all computer modeling effects

present can be seen in Figure 66 of chapter 3

140

 

130

120

110

100

llIlllIlﬁ

   
   

80

Phi (deg)

70

 

60
— Fhi0=90deg.
— Fhi0= 120 deg.

50

 

 

4O

 

llﬁrlllurir’

 

 

 

r—----— ------ -----—-- -—--
--------—---- - -—--—-------—4

30

 

d
0"
U
.5
o

-0.5 0.0 0.5
2 (inch)
Figure 43. Three ions are started with different initial phases. Tracking their phase

.0.
o

1.5

through the acceleration region reveals phase pulling toward 90°. Notice the red line

goes through the greatest number of turns and the green line goes through the fewest turns.

90

A more global view of this effect can be seen by plotting the ﬁnal phase after
acceleration vs. the initial phase when entering the cavity for the full range of initial
phases. Figure 44 shows the result. If no phase bunching occurred, then the ﬁnal phase
would be equal to the initial phase. A black line with slope = 1 has been plotted to show
this case. However, results using the computer model (red line) show that a wide range

of initial phases migrate toward (1) = 90°.

495

 

450 --------------------------------------

405

360

315

270

225

180

Final Phase (deg.)

135

90 ----------------------------

45

 

O 45 90 135 180 225 270 315 360

hitial Phase (deg)

Figure 44. The red line shows final vs. initial phase of protons accelerated through the magnetron.
(50 keV ICRA design). The solid black line represents final phase equal to the initial phase.

Dotted lines have been placed at ¢=90° and ¢=(360°+90°) for comparison.

91

2.6.3 Cavity ‘End Field Effects

Just before entering the acceleration region, the injected beam will pass through
the electric end ﬁelds of the magnetron. These end ﬁelds are accounted for in the
computer model because in the 3D solution of the electric ﬁelds (using RELAX3D) the
magnetron has ﬁnite axial length, and ground plates are placed one cm away from each
end. Allowing an ion to travel through the end ﬁelds before entering the acceleration
region causes two effects. Both effects are relatively insigniﬁcant. They are mentioned
here for completeness. All trajectories in this section use a constant magnetic ﬁeld so

that the effects of the end ﬁelds can be isolated.

The ﬁrst effect is a small additional energy gain through the end ﬁeld region.
This causes the beam to enter the magnetron at a slightly higher energy. Therefore, since
the peak energy is limited by the inner diameter of the accelerating cavity, the voltage

must be decreased. so that the ion does not hit the wall. Figure 45 shows an E-q) curve

which compares a beam that has passed through the end ﬁelds, with the beam that did not
pass through the end ﬁelds. The cavity Voltage was held constant in each case, so the
small additional energy gain through the end ﬁelds sets the beam on a different trajectory

towards a much higher energy gain.

The second effect is a shift in the optimum initial phase. This occurs because
the end ﬁelds, just before entering the magnetron, have large 2 components. If an ion
lags the rf (0 > 90°), then the ion will experience the peak electric ﬁeld just before it
passes the gap. On the other hand if the ion leads the rf (q) < 90°) then it will experience

the peak electric ﬁeld just after passing the gap (review Figure 41). In the end field

region, the electric ﬁeld just before the gap has an axial component in the — 2 direction,

92

but the E-ﬁeld just after the gap has an axial component in the + 2 direction. Therefore,
the ion which lags the rf (q) > 90°) will have a net decrease in axial momentum (p,) and

therefore gains more energy than the ion which leads the rf.

 

 

 

 

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45 E
40 5 I e e
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o IIIIIIIIIIIIIAIJIIn“ILIIIIILJIIIIIIII“:11111111111111:IIIIIIIIJIIIII
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hitial Phase (deg)

Figure 45. Final proton energy vs. initial phase for a ﬁxed V. in the 50 keV ICRA.
The beam which passes through the end ﬁelds before entering the cavity

shows a higher energy gain and a shift in optimum initial phase.

Figure 46 shows the change in axial momentum through the end ﬁelds and
acceleration region. The acceleration region is between the dotted lines. The particle
moves from left to right so the entrance end ﬁelds are in the region to the left of the

acceleration region. Although the ion with 4)., = 60° has decreasing pz in the acceleration

93

region, it is the ion with (1)0 = 120° which has decreased axial momentum through the

entrance end ﬁelds.

0.068

0.067

0.066

0.065

0.064

0.063

0.062

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0.061

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Figure 46. The ion with initial ¢>90° gets a decrease in p, through end fields, but the ion with initial

¢<90° gets an increase in p, The particle moves left to right. The acceleration region is between the

dashed lines. The entrance end fields are just to the left of the acceleration region.

End ﬁelds at the exit side of the acceleration region cause a similar additional

energy gain, however the effect is less signiﬁcant since the beam is now at high energy.

Notice the large oscillation in p; caused by the exit ﬁelds in Figure 46. The net axial

momentum change (Apz) caused by the exit end ﬁelds is not signiﬁcant because phase

bunching (section 2.6.2) through the acceleration region causes ions to have phase near

90°.

94

2.6.4 Magnetic Field and Phase

From this point forward in this chapter, a realistic magnetic ﬁeld is used in the
computer model (either solenoidal or Helmholtz). This section discusses phase changes
caused by the B-ﬁeld proﬁle over the acceleration region. The characteristics of
Helmholtz and solenoidal ﬁelds were discussed in section 2.1. The full off axis solution

of magnetic ﬁelds was discussed in section 2.6.

Recall that the magnetic ﬁeld flatness is characterized in terms of the drop at the
ends of the acceleration region (5B). B-ﬁeld ﬂatness criterion was deﬁned in section
2.5 .7, and the ﬁeld proﬁle over the acceleration region of the 50 keV ICRA was shown in

Figure 34.

Using a realistic ﬁeld introduces a phase drift anywhere where no)c at (on. In the
ICRA it is useful to plot phase vs. axial position. Figure 47 shows 0 vs. 2 as the ion
accelerates through the magnetron. Notice that the oscillation in phase has one cycle per
turn. This occurs because acceleration at each gap pushes the center of the beam around,
and whenever the beam is off center there is a corresponding oscillation in the phase.
This effect is well known in cyclotrons. The phase oscillation is of no great consequence,
although it should be minimized by centering the beam.

More important is the overall phase drift caused by regions where the B-ﬁeld is
not in resonance with the rf frequency. Whenever the B-ﬁeld is low (no)C < (an), equation
2.81 (section 2.5.7) predicts that the phase will increase in time. The reader should
compare the phase diagram in Figure 47 with the B-ﬁeld proﬁle shown in Figure 34 to

see that the phase drift past the ends of the cavity makes sense.

95

 

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Phase graphs like the one in Figure 47 are useful for checking that the beam
remains in resonance and will be particularly important when designing an ICRA with a
sloped B-ﬁeld for acceleration to relativistic energies.

The addition of the realistic magnetic ﬁeld over the entire volume of the
accelerator allows particles to be shot through the entire system from ion source to target.

A set of complete single particle trajectories is shown in the following section.

2.6.5 Complete Single Particle Trajectories

The previous three sub-sections (2.6.2 — 2.6.4) covered the complete computer
model, starting from acceleration through the magnetron in a flat magnetic ﬁeld, and then
adding end ﬁelds of the magnetron and a realistic solenoid. Section 2.6.6 will begin
discussion of multi-particle trajectories used to model a beam of ﬁnite emittance.
However, before leaving single particle trajectories, we wish to show a series of views of
the beam. In this section the electrostatic bending plates have been added, so that all
realistic ﬁelds are now in place. The ﬁgures which follow show several examples of
single particle trajectories shot through the entire system using the ICRA computer

model.

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2.6.6 Acceptance Phase Space of the RF Driven Magnetron

Any serious analysis of the maximum amount of current that could be accelerated
by an ICRA, must be based on an understanding of the acceptance phase space of the
acceleration region used. The acceptance phase space can be mapped out by using
ICRAcyclone to start particles at the entrance plane of the magnetron, then accelerating
each particle, and tabulating their ﬁnal energies. The initial conditions of each trajectory
are varied over some range in order to map out the boundaries of the phase space of the
beam which is accelerated to the desired energy. Since the ICRA accelerates beam to
some large energy spread, a cutoff energy must be chosen as a criterion for an acceptably
accelerated beam. Obviously if you choose EC.“ = 0.9 13,.m the acceptance phase space

will be smaller than if you choose EC," = 0.5 Em .

It may be tempting to describe this phase space in terms of Cartesian variables
(x, y, z, px, py, pz, t), but attempts at this have proven to be futile. The natural coordinate
system for the magnetron cavity in an axial B-ﬁeld is cylindrical. Therefore, the
acceptance phase space must be mapped in (r, 6, z, p,, p9, pz, t,) using a coordinate system
whose origin is at the center of the magnetron. This “accelerator-centered coordinate
system” is shown in Figure 55. To see why this is true, imagine that you have just run
the trajectory for some chosen central ray, then you change the initial position of the
particle by Ay to see the effect on the ﬁnal energy. In doing this, the initial conditions of
the particle have actually changed, by r and 6 both. As we will see, the acceptance phase
space does depend on r but it does not depend on 9, therefore in changing y, you have

missed the azimuthal symmetry.

112

 

 

 

 

 

 

 

 

 

 

 

 

Accelerator Centered

Beam Centered Coordinate System
Coordinate System
WY W
YS r
e
. z x

Figure 55. Initial particle coordinates at the ion source are expressed in a Cartesian coordinate
system centered on the central ray, but the acceptance phase space at the entrance plane of the
magnetron must be expressed in an accelerator-centered cylindrical coordinate system.

Before proceeding, four cautionary statements are given here:

1) Once the initial acceptance phase space of the magnetron has been mapped out, that
does not necessarily mean that a beam can be produced to ﬁll that space. In fact a large
source of optimization for the ICRA is in tailoring a beam to ﬁll the desired phase space
as much as possible.

2) The method used here to relate the injected beam to the acceptance at the entrance
plane of the magnetron, was to choose an emittance back at the ion source, then shoot

each particle from the ion source, forward through the fringe ﬁelds, and stop the beam at

the entrance plane of the magnetron. When doing this, the initial coordinates of each

113

particle at the ion source should be expressed in a Cartesian coordinate system with the
origin at the central ray of the beam as shown in Figure 55. The coordinates of each

particle can then be expressed in the cylindrical accelerator—centered system so that the

beam can be shot forward, then stopped at the entrance plane of the magnetron, and
compared with the acceptance plots to determine the fraction of particles that landed

inside the acceptance phase space.
3) And ﬁnally, notice that it is not necessary to map out the acceptance phase space.

It is only a tool for comparing what beam the injection region produces with what beam
the magnetron could accelerate if that beam could be produced. The accelerated fraction
of the beam can also be determined by completely bypassing this pit stop and simply

running particle trajectories through the entire system.

Results of the analysis of the acceptance ghase space.

In order to map out a 7 dimensional coupled phase space (r, 0, 2, pr, p9, pz, t,) we
ﬁrst scan over one variable at a time to see its effect. Those parameters whose effects are
negligible are then drOpped. Three variables can be eliminated. The effect of 0 is

negligible, and 2 was chosen to be constant at the entrance plane of the magnetron.

One momentum can be eliminated because all ions in the computer model begin with the
same energy, therefore pz, p9 and pr are not independent. Thus mapping all three would
be redundant. The arbitrary choice was made to drop pz, therefore the relevant phase
space parameters become: L435, Pg, and time (or (be). We already know that the

accelerated beam depends on rf phase (review section 2.6.2) and we must sum over the

114

full 0° —- 360° phase range because the injected dc beam will contain all phases. Scanning
each of the remaining three parameters independently shows that the acceptance phase

space is very sensitive to p9, and only moderately sensitive to r and p,. Recall the

discussion of the spread in (p Jpn) in section 2.6.2

Figure 56 shows the results of scans over two planes through the remaining three
dimensional phase space volume. These are Ape, vs. Ar, and Ape, vs. Apr. Here A
indicates a change relative to the central ray. Each data point on the graph includes a
summation over all rf phases. In other words each point represents the width of the E-tl)
curve of Figure 39 for a cutoff energy of E > 0.9 Edcsign. Thus if a beam of zero emittance

was placed at one of the red data points, then the fraction accelerated would be between

5% - 10 % (see the legend of Figure 56). ,

Figure 57 shows the ﬁnal positions of particles tracked from the ion source,
forward through the injection region, and stopped at theentrance plane of the acceleration
region. For comparison, the baundary of the acceptance phase space is outlined with a
dotted line in Figure S7. The beam outside this boundary is not accelerated above the
cutoff energy. Notice that. the injected beam has a wider p9 spread, and a narrower radial
spread, than what the magnetron will accept. This tendency occurs because the injection
trajectory causes a decrease in the radial spread of the beam, and an increase in the

momentum spreads (Ape, and Apz). This is explained at the end of section 2.5.2.

I This particular result was for an 8 Tesla, 1 MeV proton design. The injected
beam had an emittance of 271: mm mrad (unnormalized) and the fraction accelerated

above 0.9 MeV was 1.9 %.

115

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Figure 56. Two planes through the acceptance phase space at the entrance plane of the magnetron

for a 1 MeV proton ICRA. Each data point represents a summation over all rf phases.

116

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Figure 57. Final positions of protons shot forward from the ion source through the injection

fringe ﬁelds and stopped at the entrance plane of the magnetron.

The acceptance phase space from Figure 56 is outlined with dotted lines for comparison.

117

2.6.7 Injected Beam Emittance

This section deals with characteristics of the injected beam only. Although the
beam will pass through the injection, acceleration, and extraction regions, the rf power in
the accelerating cavity is o_ff for all cases. All trajectories in this section are tracked from

the ion source, through the electrostatic bend, and through the realistic magnetic ﬁeld.

A beam of some ﬁnite emittance can be modeled by using ICRAcyclone to run
many single particle trajectories, one after another. Recall that ICRAcyclone does not
account for space charge effects (section 2.6.1). Initial conditions for each particle back
at the ion source are expressed in a Cartesian coordinate system with the origin at the
central ray of the beam (see Figure 55 section 2.6.6). The 4 transverse phase space
dimensions are xs , x; , ys, y: , where x; = p,s /p,°m' is simply the slope of a particular ray
relative to the central ray. In the plots which follow, the injected beam emittance was

modeled by a fully coupled set of 3x3x3x3 = 81 particles. The (xs , x2) and (ys, yg)

planes are shown in Figure 58.

 

 

 

 

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118

In section 4.5.3 we will see that the axial momentum spread of the beam (Apz)
can be measured experimentally in two ways; the arc length at the extraction end, and the
mirror curve (both discussed below). These measurements can be reproduced using the
computer code and used to determine whether the beam emittance has been chosen

correctly.
Arc Length at the Extraction FaradaxLCup

The extraction Faraday Cup (EXFC) is a phosphor screen that is used for
viewing the beam at the far downstream extraction end of the 50 keV experiment. The
EXFC is oriented perpendicular to the z axis so the image seen is in an x-y plane. The
reader may want to look ahead to Figure 72 in Section 3.2. If the beam were a single
particle trajectory, one would expect to see the beam hit at a single point on the extraction
Faraday cup. However, for a real beam, any spread in the axial momentum causes the
image on the extraction Faraday cup to spread in the 9 direction. This spread (A6) is

referred to here as the “arc length”.

In order to check the arc length in the computer model, the full emittance of 81
particles was tracked from the ion source, through the entire system, to the 1 plane of the
extraction Faraday cup. The x-y positions are then plotted. An example is shown in

Figure 59.

It is simple to see how a spread in axial velocity causes the image of the beam to
smear out in 9 if the magnetic ﬁeld were constant. However, the solenoidal fringe ﬁelds
in the extraction region of the ICRA are certainly not constant. Therefore, it is necessary
to check whether the arc length is a good measure of the spread in axial momentum at

2:0. This was done by using the computer code to track the full emittance beam from the

119

ion source, through the entire system, to the extraction Faraday cup. The axial
momentum of each ion was obtained at 2:0 as the ion passed through the center of the
system. Then the ﬁnal angle of each ion on the extraction Faraday cup is plotted vs. l/pz .

The results in Figure 60 below show that the relation between 1/pz and 6 is extremely

linear even with the solenoidal ﬁeld. Therefore, the arc length is a direct measure of Apz.

 

 

 

 

 

 

1.0
0.8 -
0.6 _ o .L o
6 e
L a
O.
0-4 _ e. ' o
.. O
O .9
O
0.2 - .
A ' f 0.
E e
.E 0-0 g I;
- O
>- O
-0.2 - ’ .9
’ e
- O
. O
-o.4 — g . 0
- O
O .O
-06 ... ~Q .
’8 o b 09 °
-O.8 -
4.0 ¥ 1 1 1 1 I 1 l 4 . 1 l 1 1 1 l 1 I 4
-1 -O.8 -0.6 -0.4 -O.2 O 0.2 0.4 0.6 0.8 1

x (inch)

Figure 59. Final x, y positions of protons on the extraction Faraday Cup
after being shot from the ion source through the entire system.

Here the bending voltage is high enough to give an arc length of more than 360°.

120

 

 

 

 

E i
315 E; 0’
270 '5'- :
E e
5 3
¢ 225 E .0
.1: s . .
.9 180 E .3
: o
g : ’ I
135 g I
E i
90 i e. /e
45 E- 4”
E 9 e
O E 1 1 J A’l l l
0.45 0.46 0.47 0.48 0.49 0.50 0.51 0.52
1/Pz (c/MeV)

Figure 60. Final 9 position of protons on extraction Faraday Cup vs their axial momentum at 2:0.

Mirror Curve

The second useful experimental measurement of Apz is the mirror curve.
Experimentally, beam current on the extraction Faraday cup is measured while the
bending voltage is increased. If the spread in pZ were zero, then the axial momentum
would continue to decrease until the mirror angle is reached at which point the current
reaching the extraction end would instantly drop to zero (section 2.5.3). On the other
hand, if the beam contains a spread in pz, then different components of the beam will
mirror at different bend voltages. This causes the beam current to drop over some ﬁnite

range of bending voltage. The plot of current vs. bend voltage is called a “mirror curve”

The mirror curve can be reproduced with the computer model by ﬁrst selecting
an injection emittance, then tracking all particles through the entire system. The number
of ions which arrive at the extraction Faraday cup are plotted as a function of bend

voltage. An example is shown in Figure 61.

 

1.1

1.0

 

0.9 -

0.7 r-

0.6 -

N/No

0.5 -

0.4 -

O.3 -

0.2 ~—

0.1 '-

 

 

 

 

0.0 1 J .
670 680 690 700 710 720 730 740 750 750 770

Bending Voltage (V)
Figure 61. A mirror curve generated using ICRAcyclone. The fraction of particles,

which reach the extraction Faraday cup without being mirrored, is plotted vs. bend voltage.

In order to check whether the mirror voltage is a good measure of axial
momentum at 2:0, an entire emittance was tracked through the system for one constant
bend voltage, and the value of pz for each particle is obtained as it passes z=O. Then the
mirror voltage of each particle is obtained and plotted vs. pZ at 2:0. Figure 62 shows
that the result is fairly linear, indicting that mirror voltage also is a good measure of the

axial momentum at z = O.

122

 

750

740 —

‘ §
1

720 F

Mirror Voltage (V)

710 -

7001"

 

 

l g L 1 I

 

 

690
0.7

1.0 1.1 ' 1.2

Pz at z=0 (MeV/c)

1.3

1.4

1.5

Figure 62. Mirror voltage of 81 protons vs. their axial momentum at 2:0 at V5... = 690 V.

The solid line is included for visual comparison with linear.

123

2.6.8 Energy Distribution of the Accelerated Beam

All trajectories in this section are tracked from the ion source through the entire

system with rf on. The injected beam emittance used is the one shown in section_2.6 7. .

A fully coupled emittance of 3x3x3x3=81 particles is shot through the system for every 2
degrees in phase. After the full emittance beam is accelerated, the number of particles at
each energy are counted and the percent of particles is plotted vs. energy. Figure 63

shows the result. Here Vbend = 700V which corresponds to Ntums s 11.

The large energy spread is a basic characteristic of the ICRA because nearly all
of the beam (except what strikes the wall) drifts through the accelerating cavity whether it
is accelerated or not. Speciﬁcally two factors cause the spread. First, since the dc
injected beam contains ions over the full range of rf phase, even part of the beam near
¢=270° which is decelerated still contributes to the energy distribution curve. Second,
the radial defocusing discussed in section 2.6.2 causes an energy spread in the accelerated

portion of the beam. A

124

 

100

 

— V0 = 0.0 kV
— V0 = 0.65 kV

 
  
  
 
 

—VO=1.0kV
—VO=1.2kV

.e
o

o 1 1111111] 1ﬁﬁ1r11rﬁ 1 1
m1-

Vo=1.4 kV

 

 

 

Percent of Partcles Above Energy

 

 

 

I I I I I I I

10 15 20 25 30 35 40 45 50

Final Energy (keV)

 

.°
cub

Figure 63. An integrated Energy distribution for the 50 keV ICRA generated

using the computer model. Emittance of the injected beam was 2.21: mm mrad.

Figure 63 is an integrated distribution, meaning that any point on the curve
represents the percentage of particles accelerated above a given energy. Thus, if zero
particles hit the wall, then the number of particles above E = 0 would be 100%. Notice
where the curves intersect the percent particles axis (E = 0). Figure 63 shows that for
V0 = 1.0 kV, the current above E=0 has dropped to 50%, meaning that 50% of the
particles were intercepted by the cavity wall. As the cavity voltage is increased, the
beam current at high energy increases (right hand side increases), but also more of the

beam strikes the wall (left hand side of the curve drops). Thus, for any cutoff energy

125

chosen, the beam current will pass though a maximum. The curve in Figure 64 shows the

peak in the current for ions accelerated above 42 keV.

0.7

 

0.6 -

0.5 -

0.4 —

0.3 -

Percent of Particles Above 42 keV

0.1 #-

 

 

 

0.0 1 m 1 1 1 1 1 1
0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5

Cavity Voltage (kV)
Figure 64. Scanning over cavity voltage shows a peak in the beam current accelerated above a given

energy. Emittance of the injected beam was 2.21: mm mrad. Data generated using the computer
model for the 50 keV ICRA design .

126

3. DESIGN

From the outset of this research, the intent was to build an operating device as a
demonstration of the ICRA concept. Initial designs for a 1 MeV demonstration ICRA
used a 488 MHz magnetron operating in an 8 Tesla Helmholtz coil with a 10 cm flat ﬁeld
length. Although this Helmholtz coil is available from industry, it would have been the
single most expensive component in the system (~ $100K). However, a superconducting
solenoid available at the NSCL made it possible to perform this research at very low cost.
This solenoid has an operating magnetic ﬁeld of 2.5 Tesla and a ﬂat ﬁeld length of 5 cm.
In order to use this solenoid for the ICRA experiment, the beam energy was scaled down
to 50 keV to accommodate the lower B-field and the shorter acceleration region, and the

rf cavity was redesigned to operate at 152 MHz.

This chapter details the design of the 50 keV proton ICRA. Section 3.1 walks
the reader through the choice of all basic system parameters. Section 3.2 begins with an
overview of the actual components and then discusses the main components in greater
detail in sub-sections. Section 3.2.1 covers the superconducting magnet. Section 3.2.2
covers the vacuum system. Section 3.2.3 covers the ion source, the Einzel lens, and the
electrostatic bend. Design of the rf cavity is given in section 3.2.4. Section 3.2.5
discusses the beam diagnostics used to determine the location of the beam and make
beam measurements. Finally, section 3.2.6 covers the mounting structure used to

position and align all components.

Several ﬁgures in this chapter are presented in color.

127

3.1 Basic System Parameters

This section gives a brief account of how the basic system parameters were
chosen for the 50 keV ICRA. The discussion is primarily meant to be a guide to how one
would design any ICRA. Unfortunately, during the building process some experimental
realities (chapter 4) forced changes in the ﬁnal system parameters. In these cases, the
explanation has been given in footnotes so as not to confuse the continuity of the main
discussion. If the reader is interested only in how to design an ICRA, these footnotes
should be ignored. Table 4 lists the ﬁnal system parameters which may differ slightly

from those initial values in the discussion.

The design process begins at the exit‘plane of the acceleration region and works
backwards to the ion source. Figure 65 shows a useful graph of proton orbit radius vs.
magnetic ﬁeld for a range of energies. The choice of 50 keV at 2.5 Tesla1 means that the
ﬁnal r = 1.3 cm, easily ﬁts inside the 8”_bore of the solenoid (section 3.2.1). The ICRA
is capable of operating over a wide range of Ntums (review Figure 38). For example
5 — 15 turns is a reasonable choice for the 50 keV ICRA. Since the fewest Ntums requires
the highest V0, we use Ntums = 5 to check the E-ﬁeld in the cavity for vacuum sparking
across a chosen gap width.

The choice of injection energy limits the cavity gap width. For example, if we

choose E10 = 5 keV, the orbit radius at the cavity entrance is: r0 = 0.41 cm. Using the

computer model, an initial radius fraction of r(,/Ro z 0.75 works well and allows the

1) The original design used an 8 gap magnetron operating at 154 MHz in B0 = 2.53 Tesla, however the
ﬁnal resonant frequency of the rf cavity is 152 MHz, thus B0 = 2.49 Tesla. Since the rf cavity was already
built with exit radius R = 1.27cm, this small drop in B-ﬁeld restricts the ﬁnal energy to 48 keV.

128

 

 

Beam Radius (cm)

 

 

 

 

 

 

0 1 1 1 I 1 J 1 I L l 1 1 rﬁ‘ 1 i x i L
o 1 2 _3 4, 5 6 7 3 9 10
Magnetic Field (Tesla)

Figure 65. Beam radius vs. magnetic ﬁeld for several beam energies.

curved acceleration trajectory to come close to the linear taper of the magnetron yet exit

near r/R = 0.98 for full energy gain; So the cavity radius at the entrance plane is chosen

ro

tobe: R°=075

 

= 0.55 cm. At the entrance plane, assume that the gap between

magnetron vanes is the same as the width of the vane. Thus, the gap width for an 8 gap

magnetron follows from the circumferencez.

_ 21tRo
g’zN

83135

= 0.216 cm.

 

2) Note that since g can increase with R0 , this means that the maximum cavity voltage (limited by vacuum
sparking) increases with the injection energy.

129

Now check the electric ﬁeld for this choice of N mm, and gap width. Using
equation 2.75, the parallel injection energy for 5 turns is Euo = 0.8 keV, so the total
injection energy3 is E0 = 5.8 keV. The total energy gain is: Egan, = (50 - 5.8 keV) = 44.2
keV and the energy gain across a single gap, is about 1.1 keV. The average radius

fraction throughout the acceleration region can be estimated by:

0.75 + 0.98
__I' = :
<R> 2 0.87

 

Using equation 2.80 for the energy gain gives Vo = 2.5 kV across a gap4. Therefore, the

E-ﬁeld is IEI z Vo/g = 12 kV/cm. Even after ﬁeld enhancement (due to sharp comers)

this is far below the Kilpatrick criterion for vacuum sparking which is IEI = 100 kV/cm

for dc, and several times higher for rf E—ﬁelds [25] . For higher tum numbers the cavity

voltage will be less of course. Table 1 lists V0 for several Ntums.

Table l. Gap voltage in the rf cavity required for different turn numbers using
equation 2.80 with injection energy of 5 keV and ﬁnal energy of 50 keV.

N V

 

3) While running the experiment the total injection energy was increased to 6.4 keV in order to clip the H;
beam (section 4.5.1). However the cavity was initially designed based on E E 5 keV (Em, = 5.8 keV).

4) Notice that when using equation 2.80 we set Vgap = V0 , rather than V8,“, = 2V0. This is because the outer
conductor (and the outer vanes) of the hybrid rf cavity remain at ground so the total gap voltage is V0 , as
opposed to a pure magnetron, where all vanes would go to iVo , giving a total of 2Vo across a gap.

130

The procedure above uses the analytical model to calculate the basic design
parameters. One parameter that requires more attention is the rf cavity voltage (V0).
Table 2 below shows a comparison between Vo calculated by four different methods with
increasing accuracy from left to right. In each case the injection energy is E0 = 6.4 keV,
and the initial axial momentum is set for 5 turns.

The ﬁrst column in Table 2 uses the analytical expression for the energy gain
(equation 2.80) which assumes a constant radius fraction (r/R) throughout the trajectory.
The second column uses the spread sheet method discussed in section 2.5.6. Here the
orbit radius and energy gain (equation 2.79) are recalculated at every gap so that r/R
varies naturally throughout the trajectory. Both of these two analytical methods use only
the ﬁrst term in the Fourier series to approximate the radial dependence of the cavity
voltage. On the other hand the computer model uses the complete electric ﬁeld and
allows r/R to vary naturally. Furthermore, the computer model accounts for the change in
pz caused by the cavity taper (section 2.6.2). The third column gives Vo using the
computer model with a ﬂat magnetic ﬁeld. Notice that the accurate electric ﬁeld and
variable pz cause signiﬁcant reduction in V0. The fourth column uses the computer
model, but this time a realistic magnetic field proﬁle over the acceleration region is used
and the beam passes through the end ﬁelds at entrance and exit. On the surface one
might think that the cavity voltage should be higher in this case to account for phase drift
where the B-ﬁeld is not perfectly in resonance with the rf. However, the small additional
energy gain, caused by the end ﬁelds at the entrance, means that the cavity voltage must
be decreased. Furthermore, the decreased B-ﬁeld at the end of the cavity causes a slight

increase in the axial momentum and leads to a curvature in the trajectory so that the beam

131

cannot reach full radius at exit without striking the wall at an earlier point in the cavity.

The net effect of adding the realistic B-ﬁeld over the acceleration region is to degrade the

ﬁnal energy by 6% to 45 keV. This small loss in ﬁnal energy could be regained by

simply increasing the cavity inner radius. The choice of magnetic ﬁeld proﬁle over the

acceleration region is discussed on the following page.

Table 2. Comparison of the cavity voltage (V 0) calculated from the analytical model with that from

the computer model. Notice that in the computer model, the reduced an'al momentum causes an

increased number of tum. The lower p1 and accurate electric ﬁeld cause a significantly lower cavity

 

 

 

 

 

 

 

 

 

 

 

 

voltage required
Analytical Analytical Computer Computer w/
W] constan w/ variable w/ ﬂat 8B/Bo < 0.5%
r/R ‘r/R B-Field and end ﬁelds
E, 6.4 keV 6.4 keV 6.4 keV 6.4 keV
Initial p, 1.19 MeV/c 1.19 MeV/c 1.19 MeV/c 1.21 MeV/c
Final pz 1.19 MeV/c 1.19 MeV/c 1.11 MeV/c 1.28 MeV/c
Ntums 5.0 5.0 5.23 5.0
V. 2.16 kV 1.94 kV 1.23 kV 1.08 keV
Erm 48 keV 1 48 keV 48 keV 45 keV

 

 

Magnetic Field Flatness

At this point it is necessary to choose the magnetic ﬁeld flatness over the
acceleration region (deﬁned in Figure 34). Of course the smaller the ﬁeld variation (8B),
the more expensive the superconducting magnet will be. Thus the largest B-ﬁeld
variation should be chosen which does not signiﬁcantly degrade acceleration. Magnetic
ﬁeld ﬂatness can be chosen by considering the phase drift caused by the B—ﬁeld proﬁle
over the acceleration region. The analytical expression was derived in section 2.5.7.
Since the phase drift depends on the number of turns (equation 2.87), trajectories with
Nmm, E 15 are used here as a worst case situation.

Comparing the phase drift through the acceleration region with what one would
calculate analytically is not straight forward because, since acceleration causes phase
bunching, the total phase change of any trajectory depends on the initial phase. However,
we can choose the initial phase which gives optimum acceleration (peak of the E-oo
curve) for comparison with analytical.

Figure 66 shows phase vs. 2 for particles started at three different initial phases.
The red, blue, and green curves (oscillating once per turn) have been calculated using the
computer model. Each trajectory uses the full solenoidal magnetic ﬁeld (Figure 34), and
travels through end ﬁelds, and through the acceleration region. Vertical dotted lines
mark the ends of the acceleration region. Even with all effects present, acceleration does

cause phase bunching (review Figure 43).

133

 

180
170
160
150
140
130
120
110

————1

111l1rr111111r1|1|111

 

 

 

 

 

 

 

 

8’ 100 l
B
8 w — —| ——_ ___r _________
.2 8° I |
‘1 70 | |
60 I
so Z : | — Analytical
40 I : — 01110 = 0 deg.
so _ : | — phiO = 30 deg.
20 : I — phiO = 60 deg.
|
10 | |
O 1 1. l 1 I 1 1
-2.0 -1.5 -1.0 0.5 0.0 0.5 1.0 1.5 2.0 2.5

2 (inch)

Figure 66. Phase vs. 2 for trajectories started with different initial phases in the 50 keV ICRA. The
red, blue, and green curves use the computer model with a solenoidal B-ﬁeld, cavity end ﬁelds, and
acceleration through the magnetron. The oscillation (once per turn) '5 caused by motion of the orbit
center. The blue oscillating curve gives optimum acceleration. The smooth blue curve is the
analytical calculation of phase based on equation 2.83 which assumes a centered beam and no phase
pulling. Vertical dotted lines represent the ends of the acceleration region. Both blue curves have
the same initial phase at the cavity entrance so the total phase change over the acceleration region

can be compared.

Now consider only the two blue curves shown in Figure 66. These two curves
provide a comparison between the analytical calculation and the computer model (for the
B-ﬁeld proﬁle of Figure 34). The smooth blue curve was calculated using equation 2.83
which assumes a centered bean and no phase pulling. The blue curve which oscillates
once per turn is the computer result. These two curves start with the same phase at the
entrance to the acceleration region so that the total phase change over the acceleration

region can be compared. Table 3 below summarizes the result.

134

Table 3. Total phase drift over the acceleration region. Compares what we would expect

analytically with the actual computer result (for optimum acceleration case).

 

A0 V/Vo = sin( 90°+A¢)
Analytical Calculation + 30° 0.87
Computer Model - 15° 0.97

 

 

 

 

 

 

 

Based on the B-ﬁeld proﬁle only (analytically) we would expect the phase to drift
forward, yet the computer model shows that phase bunching dominates and actually
causes the phase to decrease. Since the computer model gives a phase change which is
much less than expected, the analytical result might be taken as the worst case prediction.
However, an accurate determination of the total phase change certainly requires the full
computer model.

For the 50 keV the B—ﬁeld ﬂatness was chosen to be 83 < 0.5% based on the
analytical calculation. This choice does cause a 6% degradation in ﬁnal energy (of a
single particle trajectory) as shown in Table 2. Yet, based on phase change alone, Figure
66 and Table 3 indicate that this ﬁeld ﬂatness may be more stringent than was necessary.
In fact the degradation in ﬁnal energy is caused by an increase in axial momentum near
the cavity exit. This causes a slight increase in the axial momentum and leads to a
curvature in the trajectory so that the beam cannot reach full radius at exit without
striking the wall at an earlier point in the cavity. The ﬁnal design energy could be
regained by simply increasing the inner radius of the cavity or changing the cavity slope

to allow for the additional curvature in the trajectory.

Complete magnetic ﬁeld mapping results are given in section 4.1.

135

 

inject
1,111. i

131111.11

N011"

inject

F jg.

Injection Region
Once the magnetic ﬁeld proﬁle is known, (See Figure 81 in section 4.1) the

trajectory through the injection region can be completely speciﬁed in terms of Ntums and
ro/R0 at the cavity entrance. First let’s review what we know about the conditions at the
entrance plane

0 Perpendicular: For r0 = 0.41 cm —) P10 = 3.1 MeV/c (equation 2.1)

0 Parallel : For Ntums = 5 —> Pllo = 1.2 MeV/c (equation 2.75)
Now (r, p _L, p", and Azmm) can be found at any axial position in the lower B—ﬁeld of the

injection region using equations 2.59, 2.60, 2.62, and 2.88. Figure 67 shows the result.

 

 

 

 

 

 

 

 

 

4 4
\ .
A 3 1- -1 3
8 ’ d A
v _. ... g
.3 - - c
a .. 1 g
a: 2 L - 2 E
3: 3
_ ‘ ﬂ
8 _ -Fiadius - 5
g * —Pperp 1 g
m 1 _ —Pparal|e| _ 1
0 I I I A I I I I I I I I I I L L I I L I I I I I I o
-26 -24 -22 -20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0

Z (inches)

Figure 67. Parallel and perpendicular momentum and beam radius calculated analytically through

the injection region.

136

Location of the Electrostatic Bend
We want the electrostatic bend to approach an instantaneous kick so that the

beam turns through only a small fraction of an orbit while traversing the bend plates.
(Zeff << Azmm). However, a short axial length of the bend plates requires a high Vbend.
Moving back to a lower B-ﬁeld causes a higher induced momentum spread (section
2.5.2), but ebend is lower so Vbend will also be lower. Furthermore, the bend plates should

be at a convenient location (inside the magnet bore would be difﬁcult). At 2 = -12”,

B = 0.35 Tesla , so the bend angle is only 0bend = tan ‘1 (P 1 / pl.) = 20° Using equation

2.55, with [eff = 2 cm and g = 0.8 cm, gives Vszend below ilkV. This allows a

convenient power supply to be used.

Location of the Ion Source

The source should be as close as possible to the high ﬁeld region in order to
minimize the induced momentum spread. However the ion source assembly was made of
standard components to keep the cost low. The minimum length for the basic
components needed (seetion 3.2.3) pushed the source aperture back to 12.9” behind the
E-static bend. Hence, the source location is z = -”6.2” where the B-ﬁeld is 0.030 Tesla.
Equation 2.59 can be used to follow the ﬁeld line back further and place the source
aperture at a radius of r = 3.66”.

The proceeding discussion gives a procedure for designing an ICRA. As
mentioned in the beginning of this section, certain experimental realities force small
changes in the design (see footnotes). The ﬁnal parameters for the 50 keV ICRA are

listed in Table 4 below.

137

Table 4. Final system parameters for a 5 turn trajectory through the 50 keV ICRA. The values
given here differ slightly from those in the text to account for changes made during the experiment.

See footnotes on the previous three pages for eXplanation.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Ion Source Aperture
. , Ion: . protons
z = -26.2”
' B/Bo = 0.013
radial position: r = 3.88 cm
extraction voltage Vbeﬂn = 6.4 keV
Electrostatic Bend
2 = -12.25”
B/Bo = 0.141
orbit radius: r = 1.16 cm
131/Du = 0-37
0b. = 20°
= 2 CITI
g = 0.8 cm
Vbend = ~ i700 V
Cavity Entrance
ILL/p” = 2.74
orbit radius: r = 0.436 cm
cavity radius: R = 0.55 cm
p2 = 1.19 MeV/c
Acceleration Region
B0 = 2.493 Tesla
ﬁeld ﬂattness: 8B/Bo < 0.5 %
n = 4
frf = 152 MHZ
rf off: Nturns = 5
rf on: N,,,_ms = 5.23
V0 = 1.08 kV
cavity length [m = 5 cm
g = 0.216 cm
peak cavity voltage: VO = 3 kV @ 100W input
Cavity Exit
cavity radius: R = 1.27 cm
orbit radius: r = 1.22 cm
Elﬁn—a] = 45 keV

 

138

The design presented here is based purely on single particle trajectories.
Knowledge of the accelerated beam current requires tracking many single particle
trajectories as discussed in section 2.6.8, and also involves knowledge of the emittance of
the beam extracted from the ion source. Theoretical and experimental injected beam

emittance and accelerated beam currents are presented in sections 4.5 and 4.6.

MW

We begin with an overview of the system, then the sub-sections below will cover
each of the major components in greater detail. Figure 68 shows a photo of the 50 keV
ICRA. The large blue_dewar is the superconducting solenoid. The ion source is on the
left and the extraction end on the right. The source is at high voltage, so it is covered
with a clear box made of polycarbonate. The entire system is approximately 5 feet long

and 6 feet tall. The vacuum turbo pump is just off the top of the photo on t0p of the

139

vertical tube. The ion source can be seen more clearly in Figure 69. In this photo the
source is aligned with a B-ﬁeld line, so the angle between the accelerator z axis and the
source axis can be clearly seen. Figure 70 shows a close up of the ion source and the
vacuum junction which connects source to the rf cavity and the turbo pump. Figure 71
shows the extraction end of the system. This view shows the rf cavity mounted inside the
magnet bore. The extraction port, which covers the far downstream end, has view ports
for observing the accelerated beam, and a mechanical feed through on top for moving the

radial probe (section. 3.2.5).

 

Figure 68. Side view of the 50 keV ICRA with the author added to give a sense of scale. The large
blue dewar contains the superconducting solenoid. The ion source is on the left and the extraction
end is on the right. The vertical tube leads up to the vacuum turbo pump which is just off the top of
the photo.

140

The mechanical drawing in Figure 72 shows a top view of a cross section through the
50 keV ICRA. The major components are labeled. The proton beam is shown in red and

B—ﬁeld lines are shown in blue. Figure 73 shows a side view cross section.

3

ti};

 

 

Figure 69. A view looking down on the ion source. Notice the angle between the accelerator z axis
and the source axis is because the source is aligned with a B-ﬁeld line. The motion feed through
mounted on the left side of the junction piece (red tape around it) is used for moving the injection
Faraday cup into the beam.

141

 

Figure 70. A side view of the ion source assembly, which shows the four main components. From

left to right, these are: the water cooling jacket, the insulating glass break, the electrical feed

through, and the bellows which connects the source to the vacuum junction. The actual ion source is
inside the water cooling jacket, and the Einzel lens can be seen through the glass break. The
electrostatic bend is inside the vacuum junction. Compare this photo with the mechanical drawing of
Figure 72. The source is mounted on two V-blocks which are rigidly connected by an aluminum
plate. Pusher blocks on the sides of the V-blocks are used to adjust the position and angle of the
source horizontally, and vertical bolts through the aluminum plate are used to adjust the position

and angle of the source vertically.

142

 

Figure 71. The extraction end of the 50 keV ICRA. The rf accelerating cavity (aluminum tube) is
mounted in the bore of the superconducting magnet (blue). The V-block below the cavity can be
adjusted in order to align the rf cavity with the magnetic axis. The extraction port covers the
extreme downstream end of the system and contains the extraction Faraday cup and viewports for
observing the beam. The mechanical feedthrough on top of the extraction port is used to move the

radial probe, in the radial direction (see Figure 73).

143

Figure 72. Top view of a cross section through the 50 keV. ICRA. All major components are labeled,
however the radial probe is not shown in this view (see Figure 73). B-ﬁeld lines are shown in blue
and a trajectory is shown in red. The magnetron section only is cut through two different planes in

order to show the inner and outer vanes both. Figure 76 shows this more clearly.

144

 

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labeled. B-ﬁeld lines are shown in blue and a trajectory is shown in red. The magnetron section
only is cut through two different planes in order to show the inner and outer vanes both. Figure 76

shows this more clearly.

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3.2.1 Superconducting Magnet

The magnet used is a superconducting solenoid made in the 1970’s by
Intermagnetics General Corporation. This solenoid has an 8” diameter warm bore and a
flat ﬁeld length of 5 cm for a ﬁeld variation of less than 0.5%. The NbTi coil is rated at
3.5 Tesla for 125 Amps. However, due to the age of the solenoid, it was decided to run
the ﬁeld at 2.5 Tesla (89 Amps). The superconducting solenoid is inside the large blue
dewar which can be seen in Figure 68. The 8” warm bore can be seen in Figure 71.
Magnetic ﬁeld mapping results are given in section 4.1. The axial ﬁeld proﬁle is shown

in Figure 81.

3.2.2 Vacuum System

The vacuum system consists of a turbo pump, a rough pump, and the vacuum
walls of 5 main components: the ion source, vacuum junction, vertical extension tube, the
outer conductor of the rf cavity, and the extraction port. The system was simpliﬁed by
using the aluminum outer conductor of the rf cavity as a vacuum wall. All vacuum seals
have conflat knife edges and use OFE copper gaskets, except the two seals to the
aluminum outer conductor of the rf cavity which use viton O-rings. The turbo pump is
mounted on top of the vertical extension tube (~ one meter long) so that the fan blades

are in a magnetic ﬁeld below 50 Gauss.

148

A simple representation of the vacuum system is shown in Figure 74. The
system was initially designed for a throughput of Q = 104 Torr Liter/sec, however higher
pressure was required in the ion source to ignite the hydrogen plasma. Therefore,

Q = 10'3 Torr Liter/sec is representative of actual operating conditions. Pressure in the
system is dominated by the gas flow into the ion source. The poor conductance of the
long vertical tube reduces the pumping speed from 500 Us at the turbo pump to 85 Us at
the vacuum junction. The conductance of the injection components further reduces the
pumping speed to 30 Us near the ion source aperture. For conductance formulas see

reference [ 17].

Typical Operating pressure at the junction is P; = 1(10'5) Torr and a typical
pressure near the ion source is P3 = 3(10’5) Torr. A calculation of the mean free path (7L),
using the cross section (0') for electron capture by an incident proton [26], can be

expressed in terms of the pressure (P).

_ 3 mTorr meter
n 0' P

 

Thus, for an average pressure along the beam path of about 2(10'5) Torr, the mean free

path is: 7. = 150 meters

The accelerated beam in the 50 keV ICRA has a total path length (2) of 3 meters, so the

beam lost by charge exchange with gas in the system can be estimated using:

f

This gives: Mo = 98 %, therefore about 2 % of the beam is lost to recombination.

149

This calculation shows that the vacuum system is marginal. However, it is adequate for
the purposes of the ICRA experiment. While running the experiment, care was taken not

to raise pressures higher than those stated here.

Turbo Pump

 

 

 

 

 

 

 

 

5:500 L/s
. G (3:10-3 Torr L/s
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Ion 'r’ \QD RF C "(y
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c,,=so L/s

Figure 74. Simple representation of the vacuum system showing typical pressures during

high gas load into ion source. Conductances are calculated using formulas in [17].

150

3.2.3 Ion Source, Einzel Lens, and Electrostatic Bend

This section covers all components of the injection region. The major external
components of the ion source assembly can be seen in Figure 70. These are the water
cooling jacket, the glass break, the electrical feed through, and the bellows, which
connects the entire source assembly to the vacuum junction. The major internal
components can be seen in Figure 72. The ion source itself is mounted inside the water
cooling jacket, the Einzel lens is mounted inside the glass break, and the electrostatic
bend is mounted inside the vacuum junction. These three components are discussed

below.

Ion Source

The actual ion source is a simple electron ﬁlament source purchased from
Colutron Research Corp. This component includes the ﬁlament, anode, and electrical
and gas feedthroughs. A dc current heats the ﬁlament-so that electrons are emitted.
Electrons are then accelerated through ~ 100 volts toward the anode. Gas is leaked in
through the rear of the source and ionized by the electron beam. Ions are extracted
through a 0.020” diameter aperture in the anode. The gas pressure inside the source
needed to light a plasma ranges from 30 mTorr for Argon to over 1000 mTorr for pure
Hydrogen. The 0.020” aperture is small enough to keep high pressure in the source, yet
allow low pressure in the beam line. However, gas ﬂow through the aperture is the major

gas load to the vacuum system.

An electrical schematic is shown in Figure 75. The power supplies for the

ﬁlament and anode are isolated by an isolation transformer and floated at high voltage

151

(5 — lOkV) so that the ion source can be maintained at high voltage. The source aperture
(at high voltage) and the puller (at ground) are separated by a 6 mm gap. The strong
electric ﬁeld across this gap extracts the ion beam from the source and accelerates it to
the injection energy (quwn). The glass break electrically isolates the water cooling
jacket from the rest of the beam line which is grounded. A clear box made of
polycarbonate covers the water cooling jacket for safety, and a plywood and acrylic box
covers the power supplies. Dials on the power supplies have been replaced by long

plastic rods so that the source can be adjusted while the beam is on.

Einzel Lens

The Einzel Lens consists of three coaxial drift tubes called the “puller”, the “mid
tube”, and the “exit tube”. The tubes are held on the same axis by being trapped between
4 ceramic rods. The 4 ceramic rods are held in place by mounting rings at the entrance
and the exit. Set screws in the mounting rings allow alignment of the puller aperture with
the source aperture. The puller aperture is D = 0.070”, after which the inner diameter of
all three tubes opens to D = 1/4”. All three drift tubes are electrically isolated so that the
beam current hitting each tube can be monitored. The mid tube is normally placed at
high positive voltage (2-3kV) while the other two tubes are kept at ground. Fringe ﬁelds
in the gaps between the drift tubes focus the beam. The focal length can be adjusted by
adjusting the voltage of the mid tube (VEinzel)- Einzel lenses are well understood,

therefore minimal detail is given here [27].

152

Electrostatic Bend

The electrostatic bend is made of two ﬂat copper plates separated by an 8 mm
gap. The bend angle for the injected beam is about 20°, therefore the bend plates are
tilted at 10°. See the geometry of Figure 25. The bend plates are also rotated about the y

(vertical) axis by 4.7 degrees so that the upstream face of the plates is perpendicular to

the oncoming beam. Voltage used on the bend plates is less than 800 volts each.

153

Figure 75. Schematic of the electronics for the 50 keV ICRA.

154

 

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3.2.4 . RF Cavity

At an operating magnetic ﬁeld of 2.5 Tesla, the cyclotron frequency for protons
is 38 MHz. Therefore, a magnetron operating at the 4th harmonic (8 gaps) would have an
rf frequency of 152 MHz. At this frequency a pure magnetron will not ﬁt into the 8”
warm bore of the superconducting solenoid because the inductors extend too far in the
radial direction. The solution was to replace the inductors of the pure magnetron with
some inductance which extends in the axial direction where space is not limited. A
coaxial quarter wave cavity has the geometry needed to move the inductance axially.
Therefore the vanes of the magnetron which are mostly capacitive, have been mounted
across the open end of a coaxial section which provides the inductance needed. The

result is the “hybrid coaxial - magnetron cavity” shown in Figure 76.

The structure can be difﬁcult to visualize. Start by considering Figure 15 which
shows an example of a 4 gap magnetron Operating in the 1: mode. Notice that in the 1:
mode (frozen in time), the charge difference between each vane alternates, + — + — .
Thus, every other vane should be connected to the inner conductor of the coaxial section,
while the remaining half of the vanes should be connected to the outer conductor. The
cutaway view of Figure 76 is taken in two different planes (magnetron section only) in
order to show a cross section through an inner vane which is electrically connected only
to the inner conductor, and an outer vane which is electrically connected only to the outer
conductor. Ceramic insulators allow all vanes to be mechanically connected. This
conﬁguration has one additional beneﬁt. While a pure magnetron is a set of coupled

oscillators with 2n—l possible modes of oscillation, the hybrid cavity is naturally strapped

to allow oscillation only in the 7: mode.

156

The vanes Of the rf cavity were made by electron discharge machining (EDM).
Figure 77 shows the individual components Of the magnetron section. Figure 78 shows
the magnetron and the inner conductor and the shorting plate assembled together just
before being inserted into the aluminum outer conductor. Figure 79 shows a view

looking into the fully assembled rf cavity from the extraction end.

The rf cavity has two coupling loops. The large loop is for coupling power into
the cavity, and the small loop is a pickup loop which gives a voltage signal that is
proportional tO the cavity voltage (V0). Inductive coupling was chosen because high B-
ﬁelds at the shorting plate are accessible but the high rf voltage end at the magnetron is at
the center of the superconducting solenoid and so wound be difﬁcult to access. During
construction Of the cavity, the area Of the drive loop was adjusted in order to obtain an
impedance match with the 500 cables used. The area Of the pickup 100p was adjusted tO
obtain an output signal in a convenient voltage range (VSlvolt). Both ﬁnal loops are
rigid (cannot be adjusted). Furthermore the rf cavity has no tuning mechanism. The
resonant frequency is ﬁxed at 152 MHz, therefore the resonance required for acceleration

is hit by adjusting the magnet current.

The rf cavity is designed tO generate 3 kV across the gaps Of the magnetron for
an input power Of less than 100 Watts (review section 2.4.5). The experimental results
are given in section 4.3. The input rf signal is generated by a Hewlett Packard 8660B
signal generator and then ampliﬁed by a TB Systems 1546RA rf amp. This narrow band
rf amplifier has an Operating window from 150 - 160 MHz, and a gain Of 40 dB (100W
output for lOmW input). Power output Of the rf amp is measured with a Bird 43 watt

meter before going tO the drive loop Of the rf cavity. The signal from the cavity pickup

157

loop is read on an oscilloscope. This pickup voltage is then used to calculate the voltage

across the magnetron vanes (V0).

158

Figure 76. A cutaway view of the rf cavity. Notice that in the magnetron section only, the cross
section has been taken through two different planes in order to show a cross section through an inner

vane and an outer vane both.

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Figure 78. Components of the hybrid rf cavity. Here the magnetron is assembled. The magnetron,
inner conductor, and shorting plate (all made of copper) are about to be inserted into the aluminum

outer conductor.

161

 

 

 

Figure 79. A view of the fully assembled rf cavity looking into the extraction end. The 8 tapered

magnetron vanes of the acceleration section can be seen inside the ID of the exit mounting ring.

3.2.5 Beam Diagnostics

Beam diagnostics are the tools used to determine the location of the beam and
make measurements of the beam. This section will discuss the three main beam
diagnostics used. These are: the injection Faraday cup, the radial probe, and the
extraction Faraday cup (below).

Other diagnostics include the bend plates and an aperture on the downstream side
of the bend plates. The top and bottom bend plates and the aperture are all 3 electrically
isolated so that the beam current hitting them can be monitored. Furthermore, the top
and bottom bend plates and the aperture all have phosphor coating (KBr). This is used to

determine visually if the beam is too high, or too low, or too far left, or too far right.

162

What follows is a description of the three main diagnostic tools mentioned

above. The three most important characteristics Of each tOOl are summarized in Table 5.

Table 5. A summary of the basic characteristics of each of the 3 main beam diagnostics.

 

 

 

 

 

Electrically Phosphor
Isolated Coating Moveable
Injection F.C. yes yes yes
Radial Probe yes - yes
Extraction F.C. yes yes -

 

 

 

 

 

Injection FaradaLCQ and Viewer

The injection Faraday cup is an aluminum plate that is mounted on a motion feed
through. It is electrically isolated so that the beam current hitting it can be measured, and
it has a phosphor coating (KBr) which allows an image Of the beam tO be seen. The
injection Faraday cup can be moved into the beam at a point immediately upstream Of the
electrostatic bend. It is shown in two positions on Figure 72. This tOOl is useful for
aligning the ion source with a B-ﬁeld line. Since the beam actually contains 3 different
ion Species, a misalignment with the B-ﬁeld causes the beam to split, so that three
separate beam spots are seen on the injection Faraday cup. By adjusting the angle Of the
ion source, the three beam spots can be brought together at one point. (In section 4.5 .5

we will see that the computer model shows that the point at which the three beams come

163

together is actually about 05° away from a B-ﬁeld line. However, this condition is

reproducible, and therefore it is useful.)

Radial Probe

j The radial probe is a Faraday cup which can be moved radially into the extracted
beam. Geometry Of the radial probe is shown in Figure 73. The-mechanical feedthrough
which moves the probe can be seen in the photo Of Figure 71. The radial probe is used
to measure the radial distribution Of the accelerated beam (beam current vs. radius).
Recall that the extracted beam is a rotating cylinder Of ions all centered around the z axis
but orbiting at different radii because they have different energies (E cc r2 ). The energy
distribution Obtained from the computer model was shown in Figure 63. The radial probe
is designed for experimental measurement Of this same curve so that a comparison

between theory and experiment can be made (see section 4.6.1).

The radial probe has an electron blocker mounted along the upstream edge which
prevents electrons from reaching the Faraday cup portion Of the probe. The electron
blocker works on the fact that electrons orbit around B-ﬁeld lines at such a’ small radius
that r can be neglected (~ 0.1 mm). Hence, electrons effectively move along magnetic
ﬁeld lines and therefore the electron blocker casts a shadow that electrons cannot reach.
The Faraday cup resides in this electron shadow. However, protons orbit at much larger
radius (r S 2cm), and therefore can pass behind the e‘ blocker and strike the Faraday cup.
A small portion Of the proton beam does strike the electron blocker. This effect on radial

probe data will be accounted for with a mathematical correction.

164

Figure 73 shows that the inner edge Of the radial probe is leped (dy/dz). Since
the proton beam in the extraction region follows B-ﬁeld lines (section 2.5.2), and since
the radial probe has some axial length (Al), the slope Of the inner edge of the probe must
match the local ﬁeld line in order for the current at any single energy to be measured
correctly. However, ﬁeld lines at large radius have a steep slope, while the slope goes tO
zero at the z axis. Therefore, the radial probe has the correct slope only for one energy.
This was chosen to be the full accelerated beam energy (50 keV). Measurements at any

lower energies will require a mathematical correction.

Finally, for some conditions, the radial probe is not long enough in the axial
direction to catch all Of the beam (see section 2.5.8). This will require a third
mathematical correction. These three mathematical corrections to the radial probe data

are discussed in section 4.6.1 and details are given in Appendix A.

Extraction FaradaLCup and Viewer

The extraction Faraday cup is a round aluminum plate that is mounted at the far
downstream end Of the 50 keV ICRA. See Figure 72. It is electrically isolated for
measurement of beam current, but more importantly it has a phosphor coating (KBr)
which allows an image Of the beam to be seen through a viewport. The information
Obtained from visualizing the beam on the extraction Faraday cup has been instrumental
in understanding much Of the behavior of the beam in the ICRA. The extraction Faraday
cup allows measurement Of the beam arc length (section 2.6.7 and 4.5.4), it is useful for
understanding the ion spectrum (section 2.5 .4 and 4.5. 1) and it provided proof that the

accelerated beam consists Of ions orbiting around the z axis at large radius (section 4.6.1).

3.2.6 Mounting Structure

The purpose of the mounting structure is to hold all components in place and
allow adjustments so that the components can be aligned relative to each other.
Alignment consists of the following four criteria.

0 The axis of the rf cavity must coincide with the magnetic axis.

0 The magnetron must be in the centerlof the magnet axially so that the
acceleration region is in the region Of ﬂattest B-ﬁeld.

0 The ion source must be at the correct location relative to the rf cavity so that the
spiraling beam will be centered (x,y) when entering the rf cavity.

0 The axis of the ion source should be aligned with the local B-ﬁeld line.

Four pivot points were chosen which satisfy these criteria and yet can be
implemented in a simple structure. The rf cavity and the vacuum junction (contains the
bend plates) are connected together as one rigid piece. The entire ion source assembly is
also one rigid piece. However, a bellows between the source and the vacuum junction
allows the ion source to move relative to the rf cavity and the electrostatic bend. The rf
cavity/junction assembly has a pivot point at each end. Similarly, the ion source
assembly has a pivot point at each end. All four pivot points allow vertical and
horizontal motion. The location of each pivot point is shown in Figure 80. Notice that
one pivot point is located at the exit Of the Einzel lens and another pivot point is located
at the electrostatic bend. This allows these two points to be held stationary while the
other end Of a component is moved. Pusher blocks are used for horizontal motion and

threaded rod is used for vertical motion. Examples can be seen in Figure 70 and 71.

166

 
 
    

Side View

Bellows

 

 

 

 

 

 

 

 

Pivot ' Pivot ’ . Pivot
Point Point Point

Pivot
Point

Figure 80. Pivot points of the mounting structure

167

4. EXPERIMENTAL RESULTS

The major components of the 50 keV ICRA were assembled and aligned during
December of 1998. Accelerated beam was ﬁrst Obtained in January of 1999. This
chapter covers the experimental results and discusses comparisons with the theory.
Section 4.1 summarizes the results of mapping the magnetic ﬁeld Of the superconducting
solenoid. Section 4.2 covers the ion source. Section 4.3 covers the rf cavity
experimental methods (4.3.1) and has a separate summary Of results section (4.3.2).
Finally, the beam measurements are divided into two sections. Section 4.4 covers
measurements of the injected beam (rf off), and section 4.5 covers measurements of the

accelerated beam (rf on).

Several ﬁgures in this chapter are presented in color.

168

4.1 Magget Mapping
It was necessary to map the magnetic ﬁeld Of the superconducting magnet before
the design of the 50 keV ICRA could be ﬁnalized. This section gives a summary Of the

magnetic ﬁeld mapping results.

The purpose of mapping the magnetic ﬁeld was to accomplish four main goals.

1) Locate the magnetic axis of the magnet.

The magnet coil is mounted inside the dewar, but is not necessarily coaxial with the

dewar bore.
2) Locate the axial position of the magnetic center.

The magnetron should be placed at the correct axial location which provides the

ﬂattest magnetic ﬁeld over the length of the acceleration region.

3) Obtain the full axial magnetic ﬁeld proﬁle.

The ﬁeld profile is needed to calculate the trajectory, and choose locations for the
ion source and electrostatic bend. The axial ﬁeld proﬁle also provides all

information needed to calculate B-ﬁeld arrays for the computer model.
4) Determine the ﬂat field length for AB/Bo = -0.5 %.

The resonance condition requires that the axial length of the acceleration region be

‘ limited to this flat ﬁeld length.

The magnetic ﬁeld was mapped using a two-axis Hall probe (B2, B,). The Hall

probe is mounted inside an acrylic tube, and two mounting rings hold the acrylic tube

169

inside the magnet bore. The Hall probe can be mounted at several discrete radial
positions inside the acrylic tube, and the mounting rings allow the acrylic tube to move
azimuthally and axially relative to the dewar. Therefore, the mapping apparatus provides
motion of the Hall probe in all three dimensions Of the cylindrical coordinate system Of
the acrylic tube (r, 0, 2). Position errors are estimated to be less than 1 mm (0.040”).
Furthermore, the two mounting rings which hold the acrylic tube inside the magnet bore
are adjustable. This allows the position of the entire mapping apparatus to be adjusted

relative to the magnet bore until the mapper axis is aligned with the magnetic axis.

The magnetic axis was located by scanning B, and B, vs. 0 for several different 2
planes, and then adjusting the position of the mapper until a location was found which
gives minimal azimuthal variation in the ﬁeld. Table 6 below gives the location Of the

magnetic axis relative to the magnet bore.

Table 6. Two points on the magnetic axis (relative to the axis of the magnet bore). Azimuthal
variation for all 2 planes is less than 0.15% of the central field value (B0 = 2.5T).

 

at z = +8.25” 7 at z = -8.25”
x = +0.064” x = +00%”
y = -0.186” y = -0.119”

 

 

 

 

 

 

The location Of the axial center was calculated from three different ﬁeld proﬁles.

The average is 2mm = + 0.35”, with a standard deviation Of 0.02”. Figure 81 shows the
full magnetic ﬁeld proﬁle which has been centered in z and normalized to the central

ﬁeld value. Figure 82 below shows the measured B-ﬁeld over the central ﬁeld region.

170

A parabola has been ﬁt to the data so that the ﬂat ﬁeld length can be calculated. The ﬂat

ﬁeld length is 5.3 cm for AB/Bo <- 0.5%.

1.1

 

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Figure 82. Measured axial magnetic field in the acceleration region. The smooth line is a parabolic

curve fit to the data. The ﬂat ﬁeld length for ABIBO < 0.5% is 2.1 inches = 5.3 cm

171

4.2 lon Source

The design of the ion source was discussed in section 3.2.3. The ﬁlament
supplies electrons which ionize the gas inside the source, however signiﬁcant ion beam
current is extracted only when the source pressure becomes high enough to ignite a
plasma. The pressure at which a plasma will ignite depends on the properties of the
particular gas used. Pure Argon will ignite with gas pressure inside the source as low as
40 mTorr. However, pure H2 gas requires about 2000 mTorr. In order to keep pressure
in the rest of the vacuum system below 10'5 Torr it was necessary to use Argon as a
support gas. This combination of H2+Ar allows a reasonable proton beam with a source

pressure of 500 mTorr.

With pure Argon in the source, a beam of up to 10 11A can be extracted.
However, with pure H2 gas in the source, the majority of the beam is Hf, and only a
small fraction is H". This small fraction of H+ produced is exacerbated by the pressure
limitations explained above. Between these two effects, a typical beam used for
measurements in the 50 keV ICRA contains only about 5 — 50 nA of protons, rather than
the l — 10 M originally expected. Table 7 shows typical beam currents for the ion
source running in this Hg-t-Ar mode. Section 4.5.1 explains the ion spectrum in greater
detail.

Table 7.

Approximate beam currents of each constituent in typical beam used in the 50 keV IC RA.

 

 

 

 

Ion Beam Current
Ar+ 1 — 10 uA
H5“ 50 —500 nA
H+ 5 -— 50 nA

 

 

 

 

172

Operating the ion source in a magnetic ﬁeld creates many concerns such as: the
ability to ignite a plasma, and to extract a beam from the source, the performance of the
Einzel lens, etc. The magnetic ﬁeld at the source is about 300 Gauss. However,
electrons in the source are pulled along the z axis of the source, and the source is aligned
with the local magnetic ﬁeld. Experimental results show that Operating the source in this
magnetic ﬁeld is not a problem. In fact the beam appeared to be more stable with the B-
ﬁeld on, than with the B-ﬁeld Off. Furthermore, the Einzel lens provided a wide range of
focusing capability and whether the B-field was on or Off caused no discemable change
in performance.

A major disadvantage Of this particular source design is that each ﬁlament only
lasts for about 50 - 100 hours, and replacing a ﬁlament requires complete disassembly of
the source. After replacing the ﬁlament, re-assembly of the source requires re-alignment ’

of the Einzel Lens and connection to the ICRA vacuum system.

173

4.3 RF Cavity

The purpose of the rf cavity measurements is to determine the gap voltage (V0)
for a given input power (Pin). The rf cavity is a hybrid magnetron Icoaxial cavity. The
design was discussed in section 3.2.4 and a drawing of a cross section through the cavity
was shown in Figure 76. The theory was explained in section 2.4.5. Section 4.3.1 covers
the if measurement techniques used, and presents the experimental results. A summary

Of the results is given in section 4.3.2.

4.3.1 Experimental Methods

The rf cavity gap voltage (V) cannot be measured by attaching a probe directly
across the Open end Of the cavity because any measurement, which couples signiﬁcant
power out of the cavity, changes the power loss and results in an incorrect measurement
Of V0. Even when using probes with impedance as high as 1 MI) and capacitance as low
as 1.5 pF, the additional capacitance increases the losses in the cavity and gives an

erroneous measurement.

* A pure magnetron would have Vgap = 2V0. However the hybrid cavity has Vgap = V0.

174

The method used here is to determine Vo indirectly by two independent low
power methods. Recall from section 2.4.5 that the, power loss (P) can be expressed in

terms Of the shunt resistance (Rs):

 

V2
p —_- o 4.
ms < 1)
Where: R5 = Q J:— = Q (4.2)

 

C 21tfo(C€+Cm)

If R8 is known, then the cavity voltage is known for a given input power.

Method 1 is to measure Q, f0, and the capacitance of the magnetron section (Cm),
then calculate the capacitance of the coaxial section (C,). From these, Rs can be
calculated so that V0 is known for a given input power. The results Of method 1 are

summarized in Table 8 below.

Method 2 is a calibration of the pickup and drive loops. Rather than driving the
cavity from the drive loop and attempting to measure VO across the open end. The cavity
is driven from the Open end so that V0 is known, and the voltage at the drive loop is

measured. The results of method 2 are summarized in Table 9.

175

Method 1 (Measurement of Cavity Parameters)

If the shunt resistance (R5) is known, equation 4.1 can be used to calculate the
cavity voltage (V0) for a given input power (P). The shunt resistance can be calculated
from Q, L, and C. However, the inductance (L) is difﬁcult to measure, while the resonant
frequency (f0) is easy to measure. Therefore, it is convenient to express Rs in terms of Q,

f0 and the capacitance.

R5 = Q(_S] = Q\l(:= 21th (C +C m)

The inner and outer vanes Of the magnetron are electrically isolated, therefore the

 

capacitance of the magnetron section (Cm) can be measured with a simple capacitance
meter. On the other hand, the capacitance of the coaxial section (Ct) must be calculated
using equation 2.28, which accounts for a sin(z) voltage proﬁle along the length Of the
coaxial section (2). The resonant frequency is easily found as the peak in the
transmission curve on a network analyzer. However, accurate measurement Of the Q

takes more care.

Mea_surement 0f the Unloaded 0

Measurements of the Q were made using a Hewlett Packard 8714C network
analyzer (NWA). A brief account Of the method is given here. The signal generator
inside the NWA scans over frequency and the output signal is transmitted through the rf

cavity in order to Obtain a frequency response curve for the cavity. The Q is then

176

calculated from the peak in the transmission curve using Q = fo/Af, where fo is the center

frequency of the peak and Af is the full width at half of the maximum power.

The Q must be calculated from a peak in the transmitted signal, not from .the dip
in the reﬂected signal. The reason for this becomes obvious if one considers the meaning
of each curve. A dip in the reflection curve means that power was sent toward the cavity,
but did not return to the NWA for whatever reason. On the other hand the peak in a
transmission curve means that power was successfully coupled into the cavity which
excited rf ﬁelds in the cavity, then a small portion of that signal was coupled out by the
pickup probe and transmitted back to the NWA. Obviously a transmission measurement
is representative of the actual frequency response of the rf cavity, but a reflection curve is

not necessarily.

Obtaining a true transmission curve also depends on placement of the drive and
pickup probes inside the cavity. The purpose of the pickup probe is to Obtain a voltage
which is proportional to the cavity ﬁelds. Therefore it is important that the pickup probe
be located far enough away from the drive probe that it couples only to the cavity ﬁelds

but does not cOuple any power from evanescent waves emitted directly from the drive

probe.

The actual Q of the resonant cavity (deﬁned in equation 2.30) is called the
“unloaded Q” or Q0. If any power is coupled into Or out Of the cavity then the cavity
becomes loaded and the measured Q will be less than Q0. In other words the loaded Q is
always less than the unloaded Q (Q, < Qo). Since measurement of the Q requires
transmitting rf power through the cavity, it is not possible to measure the true Q0.

However, the measured Q (Qm) does approach Qo asymptotically as coupling to the

177

cavity becomes very weak. For the speciﬁc case that the drive loop is matched to the
input line (no reﬂection, so all power flows into the cavity) the half power theorem says
that 1/2 of the power will be dissipated in the cavity and 1/2 of the power will be dissipated

in the external circuit. In this case the measured Q is 1/2 of the unloaded Q [16].

 

(2...:2

In order to conﬁrm that the Q is being measured correctly, transmission curves should be
taken for both a matched condition and for a weakly coupled condition. If the ratio of

these two Q measurements approaches 2, then the correct Qo has been found.

Figure 83 shows a schematic drawing of two experimental conditions used for
measurement of Q. In the matched state the drive loop is matched to the 50 Q
characteristic impedance Of the input coaxial line so that all power from the NW A flows
into the cavity. In this state, the measured Q is 1/2 Q0. In the unlogcﬁi state power is
weakly coupled into the cavity through an additional capacitive probe near the open end
of the cavity. This probe is adjusted for a poor match (majority Of incident power is
reﬂected back to the NW A). “Weakly coupled” means that a negligible amount of rf
power is coupled into and out of the cavity. A factor of 1000 at both the input and output
was found to be sufﬁcient. In this case —40 dB at input and -40 dB at output gave a
transmitted signal below -80 dB down from the input. The frequency response curve for

the hybrid rf cavity is shown in Figure 84.

178

Matched State

 

E pickup
Transmitted l

g °""

PH

in

 

 

 

 

Unloaded State

 

E Pickup

PT

ronsmiﬂod
g Drive

 

 

 

Figure 83. Schematic drawing of the experimental setup for measurement of Q.
The matched state is the condition used when running the cavity at full power and accelerating beam.

However, measured Q approaches the true unloaded Q only in the unloaded state.

179

 

-41—

-42 -

Transmitted Power (dB)

 

 

 

.45 L l— 1 _1 I l 1

151.8 151.9 152.0 ‘ 152.1 152.2
Frequency (MHz)

 

Transmitted Power (dB)

 

 

 

'100 i J 1 l 1 l L
151.8 151.9 152.0 152.1 152.2

Frequency (MHz)

 

Figure 84. Transmission frequency response curves used for measurement of the Q.

The matched state (top) gives fo/Af = 802, and the unloaded state (bottom) gives fJAf = 1550

180

The measured Q of the hybrid rf cavity is Q E 1600. Therefore, calculation of R,
gives 41,400 k9, and the cavity voltage for 100 input power is Vo = 2880 V. The results

of method 1 are given in Table 8 below.

Table 8. Results of method 1: Parameters for the hybrid rf cavity.

 

Inner conductor radius: a = 1/2 (0.75”) = 0.95 cm
Outer conductor radius: b = 1/2 (4.5”) = 5.72 cm

 

 

Length Of coaxial section: £7 = 9-7 cm

Capacitance of coaxial section: C, = 1-5 PF
Capacitance of magnetron section: Cm = 39 pF

Resonant frequency: fo = 152.02 MHz
Rs/Q = 25.85 a
Quality factor: Q 1600
Shunt resistance: R8 = 41 ,400 Q
Gap voltage (for Pin = 100W): Vo = 2,880 V

 

 

 

 

 

 

 

 

 

 

 

181

Method 2 (Calibration of Drive and Pickup Loops)

As in method 1, the purpose here is to determine the gap voltage V0 for a given
input power. The calibration technique of method 2 provides two important
measurements. First, the ratio of gap voltage to pickup voltage (VG/V2) is measured.
This calibration factor allows V0 to be determined from the pickup voltage while the rf
cavity is running at full power. Second, the ratio of gap voltage to drive voltage (VG/V1)
is measured. This allows the shunt resistance (R,) to be calculated and therefore is useful

for conﬁrming Vo that was calculated by method 1.

The three experimental conditions used for the calibration method are shown in
Figure 85 . The matched state is the same conﬁguration that the cavity will run in when
accelerating the beam. In this state, the drive loop is matched to the input line so that the
impedance of the cavity looks like 50 Q and all power flows into the cavity without being
reflected. The pickup loop couples to the cavity ﬁelds and provides a small voltage
signal which is proportional to V0. In the forced state, a probe is attached directly across
the Open end of the cavity so that V0 can be forced to a known value. In this state either
the drive voltage (V1) or the pickup voltage (V2) can be measured to Obtain ratios (Vo/V1)
and (Vo/V2)-
The three conﬁgurations in Figure 85 are used to measure three voltage ratios:

(VI/V2) in the matched state

(VG/V2) in the forced state (for the pickup voltage)

(VON 1) in the forced state (for the drive voltage)

182

V2, . Matched State

Fl’ransmit’red V2
1 V.

Ea“
Fit/‘9

 

 

 

 

Forced State (pickup)

 

V.
‘%V.
Piransmitiad V0

E3 W 500%

 

 

 

 

Forced State (drive)

 

Er
Kg)“ 509%

Re“— SP

ransmittod in

 

V,

 

 

 

 

Figure 85. Experimental setup for the calibration of method 2.

V is a voltage at a probe, and V' is a voltage at the end of a cable.

Voltage ratios are determined from power ratios measured with the network
analyzer, therefore care must be taken to insure that the power can be related to the

voltage on the line (i.e. no standing waves). For example, in the matched state, the

183

voltage at the input loop is known only if the drive loop is matched to the 50 Q

characteristic impedance of the input cable. In that case the voltage is related to the input

v3

2(509) '

 

ower b : P = Now consider the forced state for measurino Vo/V2 as shown
p y l —— a

in Figure 85. The 50 Q resistor across the open end of the cavity insures that no
reﬂections occur and allows V0 to be related to the input power (Po) by:

2
P0 = V°
2(500)

 

Since power coupled out by the pickup loop (P2) is sent to the transmission port of the

NWA, P2 and V2 are related in the same way.

P = V22 -
3 2(500)

 

Under these conditions the voltage ratio can be calculated directly from the power ratio.

where the power ratio is measured with the NWA in dB. Similarly for the drive ratio
(VD/V1). The three measured voltage ratios (after correcting for cable losses) are given

in Table 9 at the end of this section.

When running the accelerator, the cavity will be run in the matched state,

 

 

° measured in the forced state with the actual ° in

h f ldl'k t t
t ere are we wou l e oequae V2 V2

the matched state. However, before making this assumption we must check that the

measurements in the forced state did not alter the ﬁeld proﬁle in the cavity. If the ﬁeld

184

state should agree:

'4)

proﬁle is unchanged, then the drive to pickup voltage ratio (V.N 2) measured in each
V1

V V
_1 _°_ _ (4. 3 )
V0 forced V2 forced V2 matched
state state State

For actual measurements made on the hybrid rf cavity:

. . V1 V 1810
the left hand srde ives: — —-9— = — = 58.0
g [V0 Jforced£V2 ]forced 312
state state
. . . V1
and the right hand srde gives: — = 60.4
V2 matched
state

The error between these two ratios is 4%, which indicates that measurements in the

forced state give good agreement with the actual voltage ratios in the matched state.

 

 

V V
Thus we can equate: ° - = —°
V9 matched V2 forced
° state state
. vo _ v0
and. — —
VI malCth V] forced
state state

This allows two important conclusions.

1) The ﬁrst equation allows us to calculate the gap voltage from the pickup voltage when

 

V
the cavity is running in the matched state. The measured value of V0 f a is 1810,
9 OICC

state

 

therefore: Vo =1810 V2 (4.4)

 

 

 

185

2) The second equation allows us to calculate R, and V0 and therefore make a
comparison with the results of method 1. If the drive loop is matched to 50 9 so that no
reﬂections occur, then the power on the input cable is equal to the power which ﬂows
into the cavity.

Pline z Pcavity

Relating power to voltage in the matched state gives:

 

 

V,2 _ V02
2(509) ’ 2R,
V 2
sothat: R, = 5052 [7] (4.5)
, l

 

 

 

V
The measured value of —°—| is 31.2 therefore:
I forced

R = 48,7000

8
The shunt resistance allows us to calculate the cavity voltage for a given input power

using equation 4.1. Results of calibration measurements using method 2 are summarized

in Table 9.

Table 9. Three voltage ratios measured by the low power calibration method.

 

 

 

Matched State: VIN 2 = 60.4
Forced State: VON] = 31.2
Forced State: VON 2 = 1810

 

Shunt resistance: R, = 48,700 Q
Gap voltage (P,,, = 100W): Vo = 3,120 V

 

 

 

 

 

186

Comparison Between Methods 1 and 2

The rf cavity voltage has been indirectly measured by two independent low power
methods. The error in V, between the two methods is 8%. Table 10 below gives a
comparison of the results. The cavity voltage will be calculated using the average

between these two methods.

Table 10. Comparison between low power measurements made by method 1 and method 2.

 

 

 

 

 

 

 

 

Low Power RF Measurements
Method 1 Method 2 Average
Vo/V2= 1670 1810 1740
VON, = 31.2
Rs/Q = 25.85 (2
= 1600
R, = 41 ,400 Q 48,700 0 45,050 (2
(for Pin=100W) Vo = 2,880 V 3,120 V 3000 V

 

 

 

 

 

 

187

Higp. Power Measurements

Thetwo measurement techniques above both use low power signals from a
network analyzer. When running the cavity at high power, conditions may change due
to rf heating. High power measurements also include errors due to the calibration of the
Bird Watt Meter used to measure P1, and the oscilloscope used to measure V2. When
using high power, the cavity is run only in the matched state, therefore the ratio V1N2 is

used to make a comparison between high power and low power measurements.

V
High power measurements in the matched state give: [‘71-] . = 62.2
2 £231..
. . , V1
The low power matched state results were found prevrously: ? = 60.4
Low

Power
The error between, low power and high power is 3 70. Since the uncertainty in the low
power measurement is greater than 3%, this is considered good agreement. Therefore,
the low power measurements summarized in Table 10 will be used to calculate Vo from
the measured pickup voltage (V2).

Finally, it is important to note that while running the cavity at full power, only the
pickup voltage should be used to determine V0. In cases such as multipactoring, or if the
cavity develops an arc, or even if the input signal is off resonance, then losses in the
cavity will increase causing a mismatch at the drive loop, therefore some input power
will be reﬂected. In such cases, the gap voltage is not proportional to the drive voltage.
However since the pickup voltage couples power only from the actual cavity ﬁelds, V2 is

always a reliable indicator of the cavity gap voltage

188

4.3.2 Summary of RF Measurements

Section 4.3.1 covered the experimental methods used to determine the cavity gap
voltage (V,) by two independent methods. In method 1 measurements were made of the

quality factor (Q), the resonant frequency (f0), and the capacitance of the magnetron
section (Cm), then the capacitance of the coaxial section (C,) was calculated. From these,

R, was calculated using equation 4.2, so that V0 is known for a given input power
(equation 4.1). Method 2 is a calibration of the pickup and drive loops. The cavity is
driven from the Open end so that V0 is known, then the voltages at the drive loop (V1) and
the pickup 100p (V2) are measured. Measurement of (VG/V 1) allows us to calculate R, for
comparison with method 1 (equation 4.5). Measurement Of (V C,IV 2) allows us to
calculate the gap voltage from the pickup voltage when operating the accelerator. The

error in V0 between these two low power measurements is 8 ‘70.

High power measurements include additional errors due to rf heating and
calibration of the Bird Watt meter and the oscilloscope. The error between high power
measurements and low power measurements is 3%.. This is good agreement, therefore
the average between the two low power results will be used to calculate the gap voltage
(V0) from the pickup voltage (V2) during the 50 keV ICRA experiment. Final rf results

are summarized in Table 11 below.

Table 11. Summary of rf cavity measurements.

 

 

 

 

 

 

fo = 152.02 MHZ
VJV 2 = 1740
Rs = 45,050 9
(for Pin=100W) Vo = 3000 V

 

 

189

4.4 Aligment

Alignment of the 50 keV ICRA was done in two phases.

1) During the initial assembly, all major components of the ICRA were aligned

according to design speciﬁcations.

2) Later, the alignment was adjusted based on measurements of the injected beam.

A brief description of each process is given below.

Aligpment to Designa Sgciﬁcations

The mounting structure discussed in section 3.2.6 has four pivot points which
allow vertical and horizontal motion for positioning the major components (review
Figure 80). The rf cavity was aligned with the magnetic axis and centered axially based
on the magnetic ﬁeld mapping results. A surveying scope was then aligned along the
axis of the rf cavity by viewing center marks at each end. The scope provided a reference
so that each component could be located relative to the magnetic axis. The surface Of the
lower ion source table (Figure 70) was shimmed to be level with the magnetic axis.
A dial indicator standing on this table was used to level the electrostatic bending plates.
Cross hairs were attached to the bend plates and the bend was shimmed to meet the z axis
by sighting through the scope. Next, a square standing up on the table surface was
viewed through the scope and a line was marked on the table directly below the z axis.
From this reference line, the angled position of the ion source was plotted on the surface
of the lower table, and the upper table of the ion source was aligned with the lines below.

Position errors for this alignment process are estimated to be less than 1 mm.

190

Aligpment Based on Beam Measurements

The design alignment was sufﬁcient for the injected beam to pass through the
entire system, however the alignment was improved by making small position _
adjustments based on beam measurements. The improved alignment is based on two

criterion:
0 The ion source should be aligned with the local magnetic ﬁeld line.

0 The proton beam orbit should be centered on the axis of the accelerating cavity.

The injected beam contains three different ion species and each ion has a different

radius of curvature in the B-ﬁeld. If the beam extracted from the ion source is parallel

with the B-ﬁeld, then the V 313 force is zero and all ions move together in a straight line.
However, a misalignment. with the B-ﬁeld causes the beam to split so that three separate
beam spots are visible on the injection Faraday cup. By adjusting the angle of the ion
source, these three beam spots were brought together into a single beam spot with a

diameter of about 1 mm.

Later, computer modeling revealed that even when the beam is extracted directly
along a ﬁeld line, the curvature of the ﬁeld line causes separation before the beam
reaches the injection Faraday cup. The alignment which brings all 3 beam spots together
on the injection Faraday cup is actually 0.5° off from parallel with the B-ﬁeld line.

Since all experimental data was taken using this condition, the same alignment has been

used in the computer code.

191

The rf cavity was centered around the H+ beam by moving the extraction end of
the entire rf cavity vertically and horizontally until the beam clips at the entrance plane of
the magnetron (review pivot points Figure 80). The bend voltage is increased until the
proton beam is just below mirroring so that there is zero axial spacing between turns.

The entrance to the magnetron is the narrowest aperture in the system. When the beam
clips there, the image observed on the extraction Faraday cup disappears. The process
can be described as: up, down, move to center, then left, right, move to center. The result
is to center the entrance of the magnetron around the proton orbit, as shown in Figure 86.
The smallest resolution in the motion of the magnetron is 0.004”, which is less than 1%
of the entrance diameter (0.433”). Alignment data also provides a measurement of the

beam orbit radius (see section 4.5.3)

The disadvantage of this method is that since the extraction end of the cavity is
moved but the injection end is not, the ﬁnal position of the cavity may be at some small
angle to the magnetic axis. For the 50 keV ICRA the magnetron entrance was displaced
by 0.063” which creates an angle of 5 mrad or 0.3°. This angle effectively causes a 2%
degradation in the exit radius Of the magnetron which corresponds to a 4% degradation in
the ﬁnal beam energy. A more complicated procedure can be imagined in which the
injection end and the extraction end are both moved iteratively in order to center the
cavity vertically and horizontally, while simultaneously keeping the cavity aligned with
the magnetic axis. This procedure was not attempted because the degradation in ﬁnal

accelerated beam energy was considered insigniﬁcant.

192

4.5 Measurements of the Injected Beam

The remainder Ofthis chapter covers beam measurements made on the 50 keV
ICRA experiment. In this section the goal is to understand the injected beam therefore
the accelerating rf voltage is _o_f_t_‘ in all cases. Beam measurements with rf pp are covered

in section 4.6.

The ion spectrum of the injected beam is studied in section 4.5.1. This is
important so that only the proton beam is injected into the acceleration region because
other ions can obscure measurements of the accelerated beam in section 4.6. Section
4.5.2 shows several measurements of the proton beam orbit radius which verify that the
correct beam has been injected. Section 4.5.3 shows measured mirror curves which can
be used to determine the spread in the axial momentum of the beam. 4.5.4 covers the
methods used to determine the axial momentum and number of turns through the
acceleration region. In section 4.5.5 the mirror curve and are lengths are each used to
estimate the, emittance of the injected beam. These two estimates are used to place
bounds on the emittance of the injected beam that will be used for computer simulation of

the accelerated beam in section 4.6.

193

 

4.5.1 Ion Spectrum

The purpose of studying the ion spectrum of the injected beam is so that we can
block all ions from entrance into the acceleration region except the proton (HT) beam.
This is necessary so that extraneous beams do not obscure measurements of the
accelerated proton beam made later in section 4.6. This section begins with a” summary

of the results, then each Figure is discussed in greater detail below.

Summag

The full ion spectrum (Figure 87) gives only rough agreement with the theoretical
calculations because it is plagued with many complexities that are not accounted for by
the theory. Results indicate that ions produced by the source are probably Ar‘“, Hf, and
H“, however we cannot draw any deﬁnite conclusions. These three ions are also
observed visually as three distinct rings on the phOsphor screen at the extraction end of
the system (Figure 88). Visual observations of the beam are more reliable because they
allow the user to see exactly when each ion beam clips. The technique used was to
increase the bending voltage until all ion beams have clipped except the innermost proton
beam (see Figure 86). Later in section 4.5.2, measurements of the beam radius are used
to conﬁrm that this is the proton beam. Figure 89 shows a comparison between the visual
observations and an ion spectrum taken with the radial probe. This Figure is important
because it allows us to identify at what bending voltage the Hf clips and where the

mirror curve begins. The mirror curve is covered in section 4.5.3

194

Current vs. Bending Voltage

Recall that both Argon and H2 gas are leaked into the ion source (section 4.2).
Therefore, the most likely constituents in the ion beam are: Ar", Ar”, Ar”. .. and the
ionized molecules: H“, Hf, and Hf. However, other common ions such as C", N+, 0+,

CO“, C02”: H20+ and others are also possible.

As explained in section 2.5.4, ions with different charge or mass will have a
different momentum, and therefore will orbit with different radius in the magnetic ﬁeld.
According to equation 2.72 the radius of each ion orbit is proportional to the root of its

mass to charge ratio:

Protons have the highest q/m available and therefore will have the smallest orbit radius of
all constituents in the beam. As the bending voltage is increased, the radius of all ion
beams increases until the outer beams clip on the entrance aperture of the magnetron.

Figure 86 shows the geometry.

195

 

 

 

~®Z

 

x\ 7

¢//

////,  /

Figure 86. Geometry of the ion beam as one constituent clips on the entrance to the magntron. Here
the bending voltage is turned high enough that the 112* beam (larger radius) has clipped on the

entrance to the magntron and the beam with the H” (smaller radius) is centered on the z axis.

Figure 87 shows the full ion spectrum taken by scanning over bending voltage and
plotting the current on the extraction FC. We would expect the beam current to stairstep
downward each time an ion beam clips'on the entrance of the magnetron. Vertical dotted

lines show the theoretical calculation for where each ion‘ should clip (equation 2.73)

Unfortunately the spectrum is plagued with additional complexities which are not
described by the theory such as secondary electrons, beam scraping on the bend plate at
low bend voltage, and an entrance aperture that is not perfectly round. Furthermore, the
accuracy is limited by the fact that the beam is a spiral, so there is some variation in the
radius at which the beam actually clips. This spectrum shows only rough agreement
with theoretical values. The major constituents of the ion beam appear to be Ar“, Hf and

H“. However, we cannot draw any deﬁnite conclusions from this graph.

196

 

 

 

 

H,"
H;

-----—--------—---d

I

I H+

I
MIRBORS

 

 

 

Current on Extraction F.C. (nA)

éaoaaeeééé
l

 

 

-----—-----------——-----------d

   

 

 

:\ -

----p--—-———__- -—4p-—---___-_----_-----_----.

0.".
I'll.
thlllllllll

 

 

p---p---- -———-

I
lllllllllljjgllllllll

01002003004005006007008009001000
BendingVoltage

 

Figure 87. Ion spectrum taken on the extraction FC. The solid black line shows experimental data.
Theoretical values where each ion should clip have been calculated using equation 2.73 and are
shown as vertical red dotted lines. From left to right: Ar’, Ni, 0;, Cf, Ar“, 1120*, N‘, 0*, Ar", C*,
Hf, Hf, and H’. The experimental value at which the H+ beam mirrors is shown as a blue dotted
line. Using those calculated values, the expected theoretical curve is plotted as a solid red line.

In theory this technique for analyzing the beam injected into an ICRA should be
very useful. However in practice, the method must be improved in order to obtain more
convincing results. Since agreement between the ion spectrum and theoretical
calculations is so crude, it is necessary to rely on visual observations and on

measurements of the beam orbit radius to identify the proton beam.

197

Visual observations

Visual observations were important for understanding the ion spectrum. Phosphor
coating on the extraction Faraday cup (described in section 3.2.5) produced an image of
the beam at the extreme downstream end of the system. Figure 88 shows a drawing of
the beam image that was observed. The Argon ring is not shown here because it clips at
such a low bending voltage when the image of Figure 88 is still very small. As the bend
voltage is increased, the H; beam eventually clips on the cavity entrance and the outer
ring in Figure 88 disappears. If the bend voltage is increased still further, the image of
the inner ring eventually fades away as the H” beam mirrors. In order to clip the Hf
beam before the H+ mirrors, it was necessary to increase the injection energy from SkeV
to 6.4keV. Even at this beam energy, the H+ beam mirrors before its radius becomes

large enough to clip on the cavity entrance.

H+

Figure 88. Drawing of the beam image observed on the extraction Faraday cup. Here the bending
voltage is high enough that both ion beams produce a full 360° ring. At lower bend voltages the
' image observed is less than 360° as will be described in section 4.5.4.

The are lengths shown in Figure 88 are greater than 360° because the bend voltage is high

(near mirroring), however at lower bending voltages, are lengths are less than 360°. Arc

198

 

length data is used in section 4.5.4 as a measure of the spread in the axial momentum.

These visual observations can be used together with the radial probe (which has an
electron blocker) to obtain a reliable ion spectrum. Figure 89 shows excellent
correlation between the visual observations and drops in the beam current measured on
the radial probe. This technique shows when the Hf beam has clipped, and therefore
allows us to obtain the range of bending voltage for which only the proton beam is
injected into the acceleration region. Furthermore, we can identify the region where the
proton beam mirrors. The mirror curve is important because it provides information
about the axial momentum spread. Section 4.5 .3 will concentrate on the “mirror curve”

in greater detail.

 

      
 

 

 

 

 

2° ‘ I I
I I
A Z | I
:5 16 a I l
7;; 14 L I I
.o F | I
g 12 - I I
a 10 L | I |
'6 - : : visually :
to
8 I—
“: - I I W I
o 6 _ | | mirrors |
E - | I I
g 4 I I
(:3 2 ” I I l
7 I I I
P I I I
O
_ I I I I ’
_2 1 1 1 1 l l 1 l I 1 l 1 1 1 l _L 1 J 1 1 1 1 1 J 4
600 650 700 750 800 850
Bending Voltage (V)

Figure 89. Experimental data. Black diamonds represent beam current measured on the radial
probe which is shielded from electrons showers. Dotted lines mark the range of bend voltage where

the H; beam was observed to clip visually, and where the H+ beam mirrors

199

4.5.2 Radius of Beam Orbit

The radius of the proton beam was measured using three different techniques.

These measurements provide conﬁrmation that the innermost ring in the beam image is

the proton beam (review Figure 88). The results are summarized in Table 12 below.

Table 12. Comparison between calculation and experimental data of theH+ beam orbit radius

in the high B-field region. In each case the beam is just below mirroring (p j 2 pm)

 

 

 

 

 

 

 

 

H+ orbit Error relative
Method radius ‘0 calculated
Calculated from known Vbcam and B—ﬁeld 4.65 mm -
Measured using radial probe. ‘ 4.5 mm - 3 %
Measured from alignment data: 4.2 mm - 10 %
Estimated from size of image on extraction FC 4.3 mm - 8 %

 

For comparison, the radius of the H; beam is calculated to be 6.58 mm. The most

accurate measurement of the H+ orbit radius is done using the radial probe. An example

of a radial probe scan is shown in Figures 97 and 98 in section 4.6.1.

Spread in beam radius

The radial spread in the injected proton beam (Ar) was measured using alignment

data (distance rf cavity is moved to cut across the beam). The result is.

Ar=0.3i0.1 mm (atz=0).

200

 

4.5.3 Mirror Curve

As explained in section 2.6.7, if an injected beam with zero emittance mirrors,
then the current will drop to zero at one particular bending voltage. However the real
injected beam has some ﬁnite emittance which leads to a spread in axial velocity.
Therefore, when the beam mirrors, current will drop over some range of Vbcnd. Section
2.6.7 showed that the range of Vbend is proportional to the spread in the axial velocity of
the beam (Figure 62). Therefore, the mirror curve provides information about the
emittance of the injected beam. In section 4.5.4 the mirror curve will be used to
determine the axial momentum spread, and in section 4.5.5 the mirror curve will be used

as a measure of the emittance of the injected beam.

 

 
 
 
 
   
  
 
     

Einzel Lens Setting

 

 

 

 

g 4‘ +2.36kV
g _ eaaakv
a. 3,, +2.40kv
a
g .. +2.44kv
a: 2— +2.50kv
C
O
8
e
3
o

 

 

 

 

700 720 740 760 780 800 820 340
Bending Voltage (V)

Figure 90. Experimental mirror curve. Proton beam current on the radial probe vs. bend voltage

for several different Einzel lens settings.

201

Figure 90 shows a measured mirror curve. Proton beam current vs. bending
voltage is plotted for several different Einzel lens settings. The Einzel lens affects the
axial velocity spread, because it changes how parallel the beam is to B-ﬁeld lines in the

region before entering the electrostatic bend.

Small changes in the alignment of the ion source cause a signiﬁcant shift in the
mirror voltage. Figure 91 below shows the mirror curve which is relevant to the
accelerated beam shown in section 4.6. 1. This mirror data will ﬁrst be used to determine

the spread in p2 and the emittance injected beam in the following two sections.

50

 

N (a) #
O O O
l l l

Current on Radial Probe (nA)
l

d
O
T

 

 

 

 

680 690 700 710 720 730 740 750 760
Bending Voltage (V)

Figure 91. Experimental data. Proton beam current on the radial probe vs. bend voltage. The
Einzel Lens setting is 3.0 kV.

202

4.5.4 Number of Turns Through the Acceleration Region

In this section beam measurements are used to determine how many turns the
beam goes through in the acceleration region (Ntums). Recall that if pz is known then
Ntums is also known (equation 2.75). Again, our purpose is to quantify only the injected
beam (i.e. rf is off). Beam measurements have been made which allow two methods of

determining Ntums and two methods of determining the spread in Ntums.

1) Nmms determined from the mirror curve
2) ANtums determined from the mirror curve
3) ANmm, determined from the arc len gth on the extraction Faraday cup

4) Ntums determined from the change in the totgl # of turns thru the system

These four methods are discussed below.

mums from the Mirror Curve

 

The number of turns through the acceleration region (Ntums) is obviously a

function of the bend voltage (Vbcnd). The calculation can be represented conceptually by:
Vbend —) ebend '_) p2 -) Ntums

The ﬁrst step requires knowledge of the actual mirror voltage. The measured mirror
curve of Figure 91 shows that the beam mirrors over a range of Vbend from 690V to 740V
with a center mirror voltage of 715V. The mirror voltage allows the bend angle (Band) to
be scaled linearly with Vbcnd (equation 2.69). The parallel momentum at the exit of the

electrostatic bend is simply pll = pma, COS(6bend)- The beam then spirals into the high

ﬁeld region where pZ is given by equation 2.62. Once pz is known at z = 0, equation 2.75

gives Ntums.

The last step (p2 —> Ntums) is not in question because the magnetic ﬁeld in the
acceleration region is quite ﬂat, and no other assumptions are involved. However, the
analytical expressions for the ﬁrst three steps involve several assumptions, therefore we

must question the accuracy in calculating: (Vbend —) Obend ——> p2).

Figure 92 shows a comparison between pz vs. Vbcnd calculated analytically and
with the computer model. The maximum difference is less than 4% at Vbcnd = 700V.
Therefore, pz can be calculated analytically from experimental bend voltage. Keep in
mind that Figure 92 is an extrapolation based on the experimental mirror point to obtain

pZ at lower bend voltages.

204

2.5

 

 

 

 

 

__ I
_ I
. l
r —Computer |
2.0 : --Analytical :
8 I I
II _ I
5 1.5 — I
m _
v _ I
’e‘ _ I
3 - I
N r- l
o. _ I
0.5 — . I
- I
” |
00 - 1 I J J 1 I 1

 

 

 

 

550 570 590 610 630 650 670 690 710 730 750
' 1/2 Bend Voltage (V)

Figure 92. Analytical calculation of pz vs bend voltage compared with the result from the full
computer model. The analytical calculation depends on the mirror voltage, therefore the two curves

meet at the mirror point a-priori.

9mm, from the Mirror Curve

Figure 91 shows that the injected beam mirrors over a range of bend voltages
from 690V to 740V. Figure 92 shows that Van” can be used to calculate pz analytically
for lower bend voltages. Similarly, the spread in mer, allows us to determine Apz for
lower bend voltages. Figure 93 shows how simply changing the mirror voltage in the

analytical calculation causes a spread in pl.

205

 

2.5

 

2.0

:0)

1.5

1.0

Pz (MeV/c) (at z

0.5

llIIlIlllllllIflIlllll

 

 

0.0 L L 1 I 1 I 1 I 1 I L I 4 4 I 1 I 1 A I 1
550 570 590 610 630 650 670 690 710 730 750 770 790
1/2 Bend Voltage (V)

 

Figure 93. Analytical calculation of the spread in p, vs bend voltage

This approach involves an approximation. Changing Vmim,r in equation 2.69:

_ Vbcnd
ebend - emirror {m}
effectively changes the length of the bending plates (4,“) in equation 2.55. Obviously

changing the effectiveness of the bending plates does cause a spread in pz. However, the

source of the spread will be mostly dy/dz while the spread in a real beam is due to the full

beam emittance which contains spreads in (X5 , x2, ys, y; ). Since section 2.6.7 showed

that the mirror voltage is nearly linear with pz, regardless of the source of the spread, this

approximation is thought to be good.

206

Figure 93 is useful for visualizing how the spread in p; changes with bend
voltage. The spreads in p2 and Ntums both increase with Voend- In the limit that the beam
mirrors, the spread in Ntums becomes inﬁnite, however the spread in pz ranges from 0 to

p2. Table 13 summarizes the calculation of p2 and Apz and gives ANmms and Ntums.

Table 13. p, and Apz as well as Nmm and AN“ of the injected beam calculated from experimental

 

 

 

 

 

 

 

 

mirror data of Figure 91.
1/2 VW. pz ApJpz Ntums ANturns

(V) (MeV/c) (MeV/c)

650 1.49 38 % 4.0 1.7
660 1.38 45 % ' 4.3 2.3
670 1.26 57 % 4.7 3.5
680 1.13 76 % 5.3 6.2
690 0.98 100 % 6.1 co
700 0.79 100 % 7.5 ca
710 0.53 100 % 11.1 00

 

 

 

 

 

 

 

207

Apz from the Arc Length on the Extrafction Faraday cup

The image of the beam shown in Figure 88 has an arc length (A9) greater than
360° because the bending voltage is high. However at lower bending voltages, the arc
length observed is less than 360° as shown in Figure 94. It was shown that the arc length
is proportional to the spread in pz (section 2.6.7), therefore are lengths less than 360° can

be measured and used to determine Apz.

 

 

Figure 94. Drawing of the proton beam image observed on the extraction Faraday cup

The arc length can only be measured for low bending voltages, where A9 < 360°.

For the 50 keV ICRA, the beam mirrors at 690 — 740 V. Operating bend voltages
are in the range of 650 to 710 (4 - 11 turns, Table 13), however above 610 V the arc

length becomes greater than 360° and therefore cannot be measured.

208

Table 14. Arc lengths observed visually on the extraction Faraday cup (Van... = 2.8 kV)

 

 

 

 

1/2 Vbend A9
540 V 120°
600 V 1800
610 V 360°

 

 

 

 

Since A9 is greater than 3600 at the operating bend voltages, we must measure A9
at lower bend voltages and then extrapolate forward to higher Vbend- The method used
here is to choose an emittance which produces an arc length that is consistent with the
measured data, then use the computer code to track this emittance at higher bend

voltages. This method is used in section 4.5.5.

Measurement of the arc length should be useful for future ICRA designs,
particularly if the emittance of the injected beam is small enough (or bend voltage low

enough) that are lengths are less than 360° at the operating bend voltage. This would

provide a direct measure of Apz without extrapolating.

209

p2 from Total Number of Turns Through the System

As the bending voltage increases, the total number of turns through the entire
system increases. This means that the image of the beam arc length on the extraction PC

will rotate in the same direction that the beam turns (clockwise as viewed along + 2)

Angular positions of the trailing edge of the image were recorded for several bending

voltages. This data provides the rate of change of the ﬁnal angle vs. bend voltage

d6 ﬁnal

(1 , which can be veriﬁed using the computer model. The results are shown in
bend

Table 15.

Table 15. Rate of change of total number of turns (N) through the 50 keV ICRA.

Visual observations compared with computer result (using Vnim, = 745 V for trailing edge of image).

 

 

 

 

 

 

Experimental Computer
1/2 Vbend Observation Result
500 V N 17.2
530 V N + 0.5 17.2 + 0.70
560V N+1.0 17.2-+1.43
580V N+l.5 17.2+ 1.92
Average AN = 0.5 0.64

 

 

 

 

 

The disagreement between observation (AN = 0.5) and the computer model

(AN = 0.64) is small and could easily be attributed to errors in the visual observation.
Therefore, this data is taken as conﬁrmation that the computer results of Figure 92 are

COITCCI.

210

4.5.5 Injected Beam Emittance

Section 4.6 will present measurements of the accelerated beam current. In order
to compare with theory, an injected beam of ﬁnite emittance must be simulated by using
the full computer code to track many single particle trajectories. Therefore it is ﬁrst

necessary to determine the emittance of the injected beam.

In section 2.6.6 we saw that the acceptance phase space of the rf driven
magnetron is most sensitive to the spread in pa (or equivalently, the spread in pz).
Section 2.6.7 showed that the mirror curve and the arc length are both good measures of
the spread in p2. Therefore, the method used here is to use the computer model to choose
an emittance which is consistent with beam measurements of the mirror curve, and the
arc length. In this way, the emittance chosen will be consistent with the Apz and
therefore will produCe a similar accelerated beam current in section 4.6. The model used

for the emittance of the injected beam was shown in section 2.6.7.

Mirror Curve

Figure 95 shows mirror curves generated for several different injected beam
emittances using the computer model. The experimental mirror curve (relevant to the
accelerated beam in section 4.6) has been normalized to its peak value and is shown in
black. The beam with 0.351: mm mrad (red) has a much narrower range of mirror voltage
than the experimental data. However, the beam with 3.71: mm mrad (orange) clearly has
a much wider range of mirror voltage than the experimental data. The range from 0.81t to

2.211: mm mrad gives close agreement with the experimental data.

211

1.1

 

 

 

 

 

 

 

1 j A 0.351: mm mrad
0.801: mm mrad
0'9 T ‘ 1.41: mm mrad
0.8 -
3.71: mm mrad
0.7 I-
0 6 _ 2.21: mm mrad
o .
Z
2 0.5 —
0.4 __ Measured Data
0.3 —
0.2 -
0.1 —
O 1 l 1 l 1 L 1 l 1 .
670 680 690 700 710 720 730 740 750 760 770

Bending Voltage (V)

Figure 95. Comparison between computer generated mirror curves and experimental data.
Experimental data (black diamonds) is normalized to its peak value (45 nA). The 5 solid lines

represent computer generated mirror curves for 5 different injected emittances.

212

Arc Leth

Section 2.6.7 described the how the computer model can be used to reproduce the
arclength on the extraction Faraday cup. Experimental measurements of the beam arc
lengths were given in Table 14. Figure 96 shows a comparison between experiment and
the computer model for three different injected beam emittances. Arc lengths indicate

that the injected beam emittance is roughly 0.351: mm mrad.

Comparing the results from the mirror curve and the arc length indicates that the
emittance of the injected beam is in the range from 0.351: to 2.21: mm mrad

(unnormalized).

405

 

IUIIIY

360 ------------------------------------------ - - - -
o lVeasured

a Corrp. Model

 

 

315

 

 

 

270

225

180

Arc Length (degrees)

135

90

 

 

IIIIIIIIIIIIIIIIUIIIIIIIIIIIIIIIIll[IIIIFYIIIIIIIIIIIIUIIIIIII

 

45 4 l I l l I l l
530 540 550 560 570 580 590 600 610 620
1/2 Bend Voltage (V)

Figure 96. Arc length increases with bend voltage. The experimental data from Table 14 is plotted.

The computer model is used to generate arc lengths for three different injected beam emittances:

0.11:, 0.351:, and 0.81: mm mrad (emittances are un-normalized) .

213

4.6 Measurements of the Accelerated Beam

In the previous section, measurements of the mirror voltage and the arc length
were used to place bounds on the injected beam emittance. Section 4.6.1 below shows
the measured radial distribution of the beam with rf on and with rf off. The radial
distribution is then converted to an energy distribution. An injected beam emittance is
selected within the bounded range and used in the computer model to generate energy
distribution curves. Computer results for several energy distribution curves are

compared with measured data.

Finally, section 4.6.2 shows the measurement of the accelerated beam current
while sweeping over the B-ﬁeld. This “resonance width” is then compared with the

theoretical resonance width using the computer model.

214

 

 

4.6.1 Energy Distributions

This section covers the measured radial distribution and energy distributions for
the accelerated beam. First, the radial probe data is corrected for errors in order to
recover the full beam Current at each radial position. Visual data is discussed. Then the
energy distribution is calculated from the radial distributions and the result is compared

with the results from the computer model.

Figure 97 shows raw experimental data taken on the radial probe with the rf on
and off. The beam current which strikes the radial probe is measured as the probe is
moved from large radius inward toward r = 0. When the rf is on, beam current is clearly

accelerated to a much larger radius than with rf off.

 

 

8
7 — +RFofI
6" Q'RFOH

 

 

 

Current on Radial Probe (nA)
A
I

 

 

 

 

Radius (mm)

Figure 97. Measured radial distribution with rf on and off (raw data). Beam current on the radial
probe is plotted vs. radial position of the radial probe. Vertical dotted lines mark the calculated
radius of the injected beam and maximum radius corresponding to the exit diameter of the

magnetron.

215

 

 

The reader may want to review the design of the radial probe as explained in
section 3.2.5 on beam diagnostics. Since the radial probe is designed to capture 100% of
the beam when it is moved all the way in to r = 0, the data is an integrated distribution.

In other words, every point on the graph represents the sum of all beam current with

radius equal to or greater than the radius read off the graph.

Consider the unaccelerated beam data shown in Figure 97. According to
‘calculations we expect the 6.4 keV proton beam at the radial probe to have an orbit radius
of r = 7.9 mm and a radial spread of about Ar = 0.4 mm. Thus, we would expect the
measured beam current to be zero for any radius above roughly 8 mm, but then rise to
100% and remain constant for all lower radii. This theoretical curve is shown as a solid
line in Figure 98 below. The apparent energy spread in the raw data of Figure 97 is
caused by the sloped inner edge of the radial probe cutting across the beam orbit radius
(refer to Figure 7.3). This is the ﬁrst of three effects on the radial probe data which must
be mathematically corrected. Corrections to the radial probe data were explained brieﬂy

in section 3.2.5. They are:
1) Slope on the inner edge of the probe
2) Beam lost on the electron blocker
3) Axial turn spacing longer than the probe length

The effects of l) and 2) are largest for beams with small radius. The effect of 3) is largest
for beams with large radius. The m data has been corrected for all three effects. The
$91; data has been corrected for 2) and 3) only. The effect of 1) on the accelerated data
has not been corrected for. The distortion caused by this effect is largest at low radius

and goes to zero at 22mm (50 keV ﬁeld line). Furthermore, the correction would shift

216

 

 

16
3? 14
.5
g 12
9
q. 10
a:

c 8

O

E

9 6

:3

O 4
2
o
100

—L
C)

I TTIIIIII

Current on Radial Probe (nA)
p

0.01

Figure 98. Top graph: Raw data and corrected data are shown for rf on and off. The solid line

shows a theoretical curve for the rf off case only.

 

 

 

 

 

_

—

b

i.—

b

_

h

.. <

b

b

t. -.

.—

r- C

I—- “

l— l: ‘

I— . o

" - t : “ -
44 l 1 1.4, I A

 

 

4- Raw Data
+ Corrected Data
— Theory (RF off)

 

 

RFoﬁ

RF on

10 12 14 16
Radius(mm)

§'-s;—:

 

 

l FITrW

I I ljlllll

I I ITITIII

 

 

 

+ Corrected Data RF off
+ Corrected Data RF on

 

 

 

 

10 12 14

Radius (mm)

16 18

Bottom graph: The corrected data for If on

and off is shown on a log scale in order to show the accelerated data more clearly.

217

 

beam current toward higher radius, therefore neglecting this correction leads to a
conservative energy distribution. Details of the radial probe corrections are given in the

Appendix.

In Figure 98 above, the top graph shows raw data, and corrected data from the
radial probe (rf on and off). The theoretical curve is shown as a solid line (for rf off
only). The lower graph in Figure 98 shows the corrected experimental data on a log
scale. The log plot provides a better view of the accelerated beam current and allows
one to see at what beam radius the current drops off. The accelerated proton beam

current is 0.5 nA (3 %) at 19.3 mm and drops to 0.2 nA (1 %) by 20.3 mm.

The purpose of the radial probe is to obtain an energy distribution (current vs.
energy). The radial distribution can be converted to an energy distribution using E oc r2.
However before doing this, it is ﬁrst necessary to consider visual observations which

show that this relation is valid.

Visual observations

 

We wish to convert radial probe data to beam energy in order to obtain a current

vs. energy distribution. However, relating radius to energy by (E oc r2) assumes that each

orbit in the accelerated beam is centered on the z axis. Computer modeling indicates that

the extracted beam is centered around the z axis, however we require experimental

verification. Visual observation provides this veriﬁcation.

The ability to observe an image of the beam on the extraction Faraday cup is as

useful for understanding the accelerated beam as it was for understanding the injected

218

 

p

 

beam. When the rf power is turned on, the image of the injected beam with narrow Ar
(see Figure 88) is observed to bloat up into a large diffuse cloud which can cover nearly
the entire extraction Faraday cup. When the radial probe is moved into the accelerated
beam, it casts a shadow into this image as shown in Figure 99. The shadow observed is
at approximately 90° azimuthal rotation from the orientation of the radial probe.
Furthermore, the shadow moves inward as the probe moves in, leaving a circular image
in the center undisturbed. These observations are consistent with a spiral trajectory and
are taken as evidence that the beam does consist of large orbits centered around the z

axis.

Radial Probe

Viewport

 

 

Extraction
Faraday Cup

Outer Edge
of Image

 

Figure 99. Drawing of visual observations on the extraction Faraday cup. This axial view shows the
extraction Faraday cup which is located 2 7” further down stream (into the page) than the radial
probe. The radial probe casts a shadow at 90° rotation from the probe. The edge of the shadow

moves inward as the radial probe is moved inward.

219

EnergvDistribution

When converting the beam orbit radius to energy, the radius must ﬁrst be
transformed back to the exit of the magnetron (equation 2.59) where the momentum is
nearly all in the 0 direction. Then using the known magnetic ﬁeld, p9 can be calculated
(equation 2.1) and converted to the beam energy. This method neglects the axial beam
energy at the exit of the magnetron (about 1 keV in this case). Figure 100 below,
shows the data from the corrected radial distribution of Figure 98 which has been

converted to an energy distribution.

100

 

   

 

 

  

 

 

 

 

' i
2 10 I I
5 ; .
(D I :
.Q ._ : :
9 - .
0.. ' :
(I 15 ' .
C E I :
C E ' :
set F :
g 01% ' i

0.01 111111111111111J1111I1111I1111111111111111111111IL
O 5 10 15 20 25 30 35 4O 45 50

Energy (keV)

Figure 100. Measured energy distribution with rf on and oﬁ‘. Data is taken from the corrected
radial distribution of Figure 98. Dotted lines mark the injection energy and peak design energy.

220

The injected beam energy (6.4 keV), and the energy corresponding to the exit radius of
the magnetron (48 keV) are both marked with vertical dotted lines. The accelerated
proton beam current is 7 % above 24 keV (0.5 Edesign), then drops to l % above 42 keV.

(z 0.9 Edesign). A smaller fraction may be accelerated as high as 46 keV.

Figure 101 shows a comparison between experimental data and the computer
model. The experimental data has been normaliZed using the peak value of the
unaccelerated beam. In section 4.5.5 measurements of the arclength and mirror curves
were used to place bounds on the emittance of the injeCted beam. An emittance was

selected within this range (0.351: - 2.21:) which gives reasonable agreement with the

experimental energy distribution. The emittance used in Figure 101 is 2.21: mm rrrrad.

 

 

 

 

 

 

 

 

 

100*
—ComputerModel
+ Measured Corrected
10:
:\°‘ :
o ..
Z
\ .-
Z
1:
o11111111411111111111I1111111111111111111111111111
0 5 10 15 20 25 30 35 40 45 50

Final Energy (keV)

Figure 101. Comparison between corrected experimental data and the computer generated energy
distribution. In each case: Vo = 1.7 kV, V2Vbend = 740V, V5.” = 2.5 kV. The computer model

uses a 2.2 1: mm mrad injected beam emittance.

221

Figure 102 below shows experimental energy distributions taken for several
different bending voltages while V0 is held constant. This data illustrates two important
characteristics. First, at low bend voltages (axial momentum is high so Nmms is low) the
beam is not accelerated to full energy. As the bend voltage is increased (axial momentum
decreases so Ntums increases) the accelerated beam current moves to higher energies. The
experimental data of Figure 102 shows the expected trend. The second trend to notice is
that when the beam is accelerated to higher energies, (high Vbend) the total beam current
(above E = 0) decreases. This indicates that a signiﬁcant portion of the beam strikes the

wall inside accelerating cavity. This trend was observed in Figure 63.

 

 

 

 

10.000 _ .
E Q} Ignore 5
' Hgbeam :
2‘ . :
5 - \ 670V i
‘D :
g 1.000. / .
6: : :
.72 I :
P - :
g .
_ 690V 1
8 / :
E 0.100 : :
g) E 650V :
:3; I 710v :
I 660V \
0.010 llllIlllLLLLlLllll l1 111111111111LL111|1 1111111
0 5 10 15 20 25 30 35 40 45 50
Energy(keV)

Figure 102. Measured energy distributions for several bending voltages.

Cavity voltage is 1.4 kV, V5,". = 2.7 kV, Mirror voltage ranges from 690 - 745 V.

222

 

Finally, notice that for Vbend = 650, 660, and 670 V the Hf beam is present and shows up
as a large increase in beam current at about 12 — 13 keV. The conversion from radial
distribution to energy distribution is valid only for the proton beam, therefore this data

should be ignored.

Comparison between this experimental data and the computer generated energy
distributions shows good correlation at high bend voltages but increasingly worse
agreement at lower bending voltages. Figure 103 compares computer with experimental
data for bending voltages of 690 V and 710 V. Figure 104 shows the comparison for

650, 660, and 670 V. The emittance used in the computer code is 2.2 1: mm mrad.

 

 

 

 

100
Measured
- ‘. Vbend=690V
\\
ComputerModel
10 : Vbend=690V
E: E "
o _ \
g _
Measured
1 : Vbend=710V
I Computer Model
O1111111111I111Jl11L111L41I111411111111111111111111
0 5 10 15 20 25 30 35 40 45 50

Final Energy (keV)

Figure 103. Experimental data from Figure 102 is compared with the computer result.

Here, bending voltages of 690 V and 710 V are used in the computer and experiment both.

223

 

 

 

 

 

  

 

 

 

100
I ‘x\
_ ‘\
\ Computer Model
10 : ~ 650 V
§ : \ 660 v
0 * 670 v
g * Measured \
1 650 V
E 660 v
: 670 v
0.1111111141I11L111111l1111I1111I1111I1111I11 1l111
0 5 10 15 20 25 30 35 40 45 50
Final Energy (keV)
100
F \\ Measured 670 V
f \t Computer Model 640 V
10 :
A : ‘\ Measured 660 V
o\° C Computer Model 630 V
o - ‘
E -
2 Measured
1 : 650V
: Computer Model
I 605 V
0.1llllIIlJllll_LllllllIllllIllllIl lllllll lllllllli

o 5 1o 15 20 25 30 35 4o 45 50
Final Energy (keV)

Figure 104. Experimental data from Figure 102 is compared with the computer result. In the top
graph the same bend voltage is used for computer and experiment. In the bottom graph, bending
voltages in the computer model have been shifted lower to obtain matching energy distributions.

224

Figure 104 shows that the energy distributions can be made to match the
experimental data, if the bend voltage in the computer code is (shifted lower. The
maximum discrepancy in bend voltage is 45 volts or 7% of 650V. The reason for this
discrepancy is unknown at this time. The most likely cause is an error in the position or

the alignment of the ion source.

225

 

 

4.6.2 Resonance Width

Section 2.5.7 explained the phase drift caused by magnetic ﬁeld detunin g only.
Using equation 2.85, a beam with Nturns = 5 would experience a phase drift of At) = 45°
for a magnetic ﬁeld error of AB/Bo = 0.6 %. Therefore, we would expect a total
resonance width of about 1.2% if acceleration through the magnetron does not contribute
to the phase change. Experimental measurements of the accelerated beam show a
resonance width of approximately 3%. Figure 105 shows the measured data. If the
magnetic ﬁeld is detuned by 1.5% from center, equation 2.85 gives Act) = 108° which is

clearly not compatible with a beam accelerated over 5 turns.

005

 

004 -

003 —

002 —

001 —

000

 

Current on Radial Probe (nA)

-O01 -

 

 

0.02 1 I 1 I 1 I l I 1
86 87 88 89 90 91

Magnet Current (Amps)

 

Figure 105. Experimental data showing the accelerated beam current vs. magnetic ﬁeld.

The position of the radial probe is 19.3 mm (E > 38 keV), and 1AV”... = 683 V (NW = 5).

226

The computer model has been used to simulate acceleration of a full emittance
beam. Varying the rf frequency to simulate magnetic ﬁeld detuning indicates a
resonance width of over 4%. Figure 106 shows the comparison between the computer
result and the experimental data. This is taken as further evidence that acceleration
thrOUgh the rf driven magnetron does cause phase bunching as discussed in sections 2.62
and 3.1. The ‘wider than expected’ resonance width means a less stringent accuracy in

setting the magnetic ﬁeld, and is therefore a desirable characteristic of the ICRA.

 

120

 

1 10 -
+ Measured

— Corrputer

100 I-

 

 

 

N/No (%)
s
l

  

 

 

 

 

 

O .
.20 mm 1 1 l 1 1 1 1 I 1 1 L1 I 1 1 1 1 1 1 1 1 I 1 1 1 1

-4 -3 -2 -1 0 1 2 3
dB/Bo (%)

 

Figure 106. Comparison between experimental data and the computer generated resonance width.
The computer model uses: l/2V.,..,.. = 683 V, E > 38 keV (for R-probe at 19.3 mm) and injected beam

emittance of 2.2 1: mm mrad.

227

5. DISCUSSION

 

5.1 Discussion of Results

The Ion Cyclotron Resonance Accelerator uses novel geometry which allows the
beam to drift axially while being accelerated in the azimuthal direction. A super-
conducting magnet provides an axial B-ﬁeld and an rf driven magnetron provides

azimuthal electric ﬁelds. This accelerator design is simple, compact and lightweight.

A 10 MeV ICRA would be useful for production of radioisotopes or neutron beams.

The objective of this research has been to provide experimental veriﬁcation of the

ICRA acceleration method. This task has two steps:

1) Develop analytical theory and a computer model which can be used to predict the

fraction of beam accelerated by an ICRA

2) Design, build, and test a 50 keV proton ICRA. Then compare the experimental results

with theory.

These objectives have been successfully completed. The analytical theory shown
here can be used to calculate single trajectories that are centered on the z axis. This
analysis is accurate for most design work. The computer model developed for the ICRA
can be used to track particle trajectories from ion source to target. Many single particle
trajectories have been used to model a full emittance injected beam and predict the

fraction of beam current accelerated.

An ICRA has been designed to accelerate protons to 50 keV. The design uses an

existing superconducting solenoid. The ion source, vacuum system, accelerating rf

228

 

cavity, and beam diagnostics have been built for this experiment. Beam measurements

have been made in order to understand both the injected and the accelerated beam.

Figure 107 shows a measured energy distribution. A proton beam was injected at
6.4 keV (rf off). When the rf is on, the beam is accelerated to a distribution ranging from
near zero up to near the design energy. 7% of the beam current is accelerated above 24

keV (0.5 Edam.) and 1 % is accelerated above 42 keV (z 0.9 Edesign). A smaller fraction

may be accelerated as high as 46 keV.

1000.0

 

    

 

 

 

 

     

 

 

 

 

I I
I I
100.0 I
I I
Z\°‘ : 5
O 1005 l :
3 E : :
Z _ . :
" I .
1.0g :
’ I I
01 1111111~ IIIIILLLLLLLILIIILIIlIJJLlIIIIIILLIII 1I1
0 5 1O 15 20 25 30 35 4O 45 50

Energy (keV)
Figure 107. Measured energy distribution of the proton beam with rf on and with rf off.

All beam parameters needed for input into the ICRA computer code have been
determined from beam measurements, rf cavity measurements, magnetic ﬁeld mapping,

and alignment measurements. These include: cavity gap voltage (V0), bend voltage

229

(Vbcnd), injection energy (E0), B-ﬁeld proﬁle, and the location of all components.
Beam measurements have been used to estimate a range of emittance of the injected
beam. An emittance was selected in this range which gives the best match with the

measured energy distributions.

 

 

 

 

100 '
\‘ Measured
\\ vm=690v
10 3\ Computer Model
’0‘ : Vbend=690V
o\ _. ,_
o _ \ \
Z _
Z
1 5 Measured
: Vbend=710V
: Computer Model
_ Vbend=710V
O1111111111L1111I111111111144111111111111J_1111L1111
0 5 10 15 20 25 30 35 40 45 50

Final Energy (keV)

Figure 108. Comparison between measured energy distributions and computer generated energy

distributions for two different bend voltages.

Using these input parameters, the computer code has been used to reproduce
energy distributions for the accelerated beam. Figure 108 shows a comparison between
the computer results and measured energy distributions for two different bend voltages.
The computer data shows good agreement with the magnitude and shape of the measured
energy distributions, and even correctly predicts the drop in beam current caused by the

beam striking the inner diameter of the magnetron. With a small correction to Vbcnd, the

230

computer model yields good agreement over a wider range of bend voltages. This
agreement indicates that the ICRA computer code can be used to correctly predict the

energy distribution of the beam accelerated in an ICRA.

Finally, having benchmarked the ICRA computer model, the code should be a
useful tool for designing the next higher energy ICRA. This topic is addressed in section

5.2 below.

231

5.2 The Next Step for the ICRA

The 50 keV ICRA experiment successfully demonstrated the ICRA. The
analytical model is adequate for most of the design work. The computer model has been
used to estimate the faction of the accelerated beam current. Clearly, the next step should
be to design and build an ICRA which will accelerate beam to useful energies. The ﬁnal
energy was chosen at 10 MeV. Beam currents of 10 —- 50 11A in an energy range of
5 S E S 10 MeV would be useful for the applications discussed in section 1.5. The goal
here is to optimize the design parameters to obtain a compromise between maximum
accelerated beam current and minimum cost of the components. This section presents a
scheme for optimization of basic design equations. After which, three 10 MeV proton

ICRA designs are presented.

M32netic Field Proﬁle

A proton at 10 MeV has a relativistic mass factor of ‘y = 1.01 . Therefore, relativity
begins to signiﬁcantly affect the cyclotron frequency. Since the energy gain in the ICRA
trajectory is roughly linear with axial (2) position, simply sloping the B-ﬁeld at a rate of
==1% over the cavity length will match 7 so that the beam remains in phase with the rf.
The B-ﬁeld slope can be obtained by using a Helmholtz coil pair with independent
current control, or by simply adding a trim coil to a ﬂat ﬁeld conﬁguration. The designer
should be aware that this slope in the B-ﬁeld will cause the axial momentum to decrease
throughout the acceleration region, which creates the possibility of mirroring the beam
inside the acceleration region. This effect should be compensated for by increasing the

axial momentum at the cavity entrance.

232

Magpetron Taper

Since the energy gain in the ICRA trajectory is roughly linear with 2 position, the
orbit radius of the beam is roughly proportional to the square root of 2. Thus, if the inner

radius of the magnetron increases linearly with z (dR/dz = constant), the beam will move

away from the wall to lower voltage VO cc (1%)" during the latter part of the acceleration

region. This effect is negligible in the 50 keV ICRA; therefore only one lineartaper was
used. However, the 10 MeV should have at least two different linear tapers as shown in
Figure 110. The slopes are chosen to approximate the natural shape of the acceleration
trajectory so that the beam remains close to the inner diameter of the accelerating

SII'LlCIUl' e.

p—————————

 

 

\
/

Z

Figure 110. Cross section of a magnetron with two linear tapers used to approximate the square root

shape of the acceleration trajectory.

233

 

Two Cost Saving Technologies

Ideally, the 10 MeV ICRA should utilize two technological developments which

will lead to signiﬁcant cost savings.

1) Cryogen free superconducting magnets utilize high Tc current leads in order to
minimize heat transfer into the liquid helium. This allows a compact refrigeratibn system
mounted directly on the side of the magnet to produce a closed helium system. Cryogen
free magnets such as this would be particularly beneﬁcial in the design of a portable
ICRA. Superconducting magnets which use NbTi coils produce magnetic fields up to
about 8 Tesla and are relatively inexpensive. The cost scales roughly with the volume of
the warm bore. Magnetic ﬁelds higher than 8 Tesla, require more advanced

superconducting materials (i.e. Nngn), therefore the cost rises signiﬁcantly.

2) Industrial magnetrons at 460 MHz and 950 MHz and output powers in the range of

50 to 100 kW would provide an inexpensive rf power source for the ICRA.

234

 

Energy Spread Caused by Radial Dependence of the Cgvity Voltage

One challenge in designing a 10 MeV ICRA is that the radial defocusing
discussed in section 2.6.2 will have a larger effect because the induced energy spread
depends on the energy gain. This means that a smaller fraction of the beam will be
accelerated above the desired cutoff energy for the same injected beam emittance. In
order to design an ICRA which will accelerate signiﬁcant current to the desired energy
range, it is important to understand the source of the energy spread, so that we can

determine which design parameters affect the spread.

In order to determine the absolute energy spread which will be induced over the
entire acceleration trajectory, it is necessary to choose a particular cavity geometry and
run computer simulations. However, it is passible to gain some insight into the relative
energy spread by considering only the spread induced by an inﬁnitesimal energy gain at

the entrance of the cavity.

As the beam spirals through the injection region (toward higher B-ﬁeld), the ratio
of o: = p i/p" increases. Assuming that all ions in the injected beam have the same energy

but a slightly different angle relative to the local B-ﬁeld, we have:
pita = pi + pi
Differentiating gives: 0 = p i 5pi + p" 5p”

Rearranging yields a convenient expression.

 

(5.1)
PI PI.

 

 

 

235

Equation 5.1 is valid through the injection region, up to the cavity entrance,
however it is not valid in the acceleration region (where 0: increases, yet 5pu/pn remains
roughly constant). We deﬁne 01., as the ratio of p i/p” at the entrance to the acceleration
region. For an ICRA of any reasonable length, 010 will be greater than 1. Therefore, at
the entrance to the acceleration region, the fractional spread in p“ is greater than the

fractional spread in p i by a factor of 0102.

Now consider equation 2.79, which gives an analytical expression for the energy

I!
gained across a single gap. E gap = %qu (é)

Two particles with different orbit radii (r) will see two different gap voltages. This leads

to a spread in the energy gained. Differentiating with respect to r gives:
LE” = “BI
Egap i r

as ,
Since pr 2 qB,r, we can write: [ E g p ] = nFS—pi—J (5.2)
gap J_

 

Similarly, a spread in the number of kicks (N = 2nNmms) leads to a spread in the

energy gain. This can be found using equation 2.80.
1: r "
Egain : 2'qu <E> N

Differentiating with respect to N gives:

6E gain
raga,n I N

 

236

 

If we take N to be continuous rather than discrete, then this equation is also true for the

energy gain across a single gap.

 

SEgap = _5_N_
E I N

or: 5 V
we can write: {—521 = - ﬂ (5.3)
I

gap

 

since pll cc
kicks

Since r/R is constant in equation 2.80, equation 5.3 neglects any variation in r/R due to
variation in p... If we neglect this coupling term, the inﬁnitesimal energy spreads caused

by spreads in parallel and perpendicular momentum can be added directly.

8E sap : 6E sap + 8E :39
E 139 E sap J_ E sap

Substituting equations 5.2 and 5.3 leads to:

8B
= “[514] ,. [£11]
Esap pl pm

This equation is valid for an inﬁnitesimal energy spread at any point in the acceleration

 

 

region. At the entrance to the cavity 5p“ and 5p; are related by equation 5.1. It is

convenient to express the energy spread in terms of the perpendicular momentum spread

. 5 .
rather than the parallel momentum spread srnce & remains constant throughout the

Pl

8
injection region, while —I-)-"- does not (section 2.5.2).

 

Pu
513,, ., a
——:" = (map—pi (5.4)
Egap cavity pl
entrance

 

 

 

237

 

Equation 5.4 shows that for a particular choice of harmonic number (n), this initial energy

E /E
gap gap for several

spread will never go below 11. Figure 11 1 shows the ratio of

5pt/pi
values of n and for a range of 010. For 010 greater than about 2, the energy spread is
dominated by 010 (or 8p"). While for 010 less than 2, the spread is dominated by n (or 01);).
Obviously, for high accelerated beam current, n and 010 should both be as low as possible.

However, it is more important to have low 010 because the second term in equation 5.4

varies as 0102.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

40
C
35:
30"
5'25“
e? E
:20
ET I:
D15”
3 —
10:
5: i
0" 1
o 1

Alphao

Figure 111. The inﬁnitesimal energy spread caused by an infinitesimal spread momentum spread is
plotted vs. 0., and for several values of n.

238

 

Optimization of ICRA Design Parameters

Optimization of ICRA design parameters requires two criterion.
1) Minimize cost
2) Maximize the accelerated beam current

The cost can be minimized by: small magnet volume, and low rf power, and by using the
cryogen free superconducting magnet as well as an industrual magnetron for an rf power
source. High accelerated beam current requires low harmonic number (n) and low 0: as

discussed above.

In the optimization scheme below, the ICRA design parameters have been divided

into natural input parameters, and output parameters.

 

Input parameters: E0, or, B0, 11, F, Ega-m, (r/R), 8p“- Q, [C

1
l

Olltpllt Parameters: frf , Va 9 gcav, I'beam 9 Rmagnetron, PIT,

Where E0 is the injection energy, 0: is the ratio of p l to Pu at the cavity entrance, B0 is

the central magnetic ﬁeld, 11 is the harmonic number, F is the ratio of the rf cavity

voltage (V0) to the maximum cavity voltage set by the Kilpatrick criterion for sparking,

Egan is the total energy gained through the acceleration region (Egain = Eﬁnal — E0). (r/R)

 

rs the average radius fraction throughout the acceleration trajectory, 1:); rs the spread 1n
.L

the perpendicular momentum of the injected beam, Q is the quality factor of the If driven

239

 

 

magnetron, and EC is the radial length of the capacitor gap in the magnetron (deﬁned in
Figure 16, pg 35).

Two input parameters should be ignored or taken as constant because we do not
have much choice in their values. For an average trajectory through the acceleration

region (r/ R) 5 0.85, and in order to obtain the lowest possible capacitance (and a high

RJQ), EC should be no more than 5 0.5 cm.

The output parameters are the rf frequency (f,f), and the cavity voltage (V0), the
length of the acceleration region (Km), the orbit radius of the ﬁnal accelerated beam
(rheam), the radial distance needed for the resonators of the magnetron (Rmagncmn), and the

rf power needed to run the magnetron (Prf). Notice that 6cm, , rm , and ngncmn

together deﬁne the volume of the superconducting magnet, because [Gav is also the
magnet ﬂat ﬁeld length, and the diameter of the magnet warm bore is given by D =
2(r1mm + Rmmmn).

In an effort to be brief, all design equations have been summarized in Figure 112,
and Figure 113. All equations have been presented previously in this dissertation, except
the Kilpatrick criterion (below). The equations in Figure 112 deal only with the trajectory
through the acceleration region. The equations in Figure 113 deal only with the resonant
structure of the magnetron and the rf power required to run the magnetron. The equations
represent the simplest analytical approach to relate all design parameters. Only lowest
order effects are used. For example, the trajectory equations assume a centered proton
orbit, the electric ﬁeld in the magnetron is represented by ﬁrst harmonic only, and

relativistic effects are neglected.

240

 

 

 

Inputs

 

 

 

 

 

 

{——
E0 9‘ ptotal=V2mBo
a
P =poal
i H 1+0:2
a an
1
P =P tal—

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Egain g

III
52
E

 

 

 

 

 

 

 

 

 

 

 

 

_ a ,
5- e... - Z'IqBoINm’ f—J

Figure 112. A graphical map of the basic design equations for the acceleration trajectory of the

 

IC RA. Input parameters are E. a, 8., n, F, E“, and the average radius fraction (r/R).

241

 

 

 

From
Figure 112

 

 

 

 

 

 

 

 

 

 

__1_ c g
(C 1 [._§ rL"2I21t14,,] nr,
frf '

[cav ____>, C = 8°[fﬁl]

 

 

 

 

 

 

 

 

 

 

= (c +2:L

 

 

 

 

 

 

 

 

 

 

 

 

 

 

" I
._1.. Q
Q > R5 2n£21rfﬁC
v:
V0 %‘ Prf —2—Rs-

 

 

 

Figure 113. Design equations for the resonant structure of the rf driven magnetron. Input

parameters are: [c , n , and Q. Parameters taken from Figure 112 are: g , [no , f" , and V0 .

 

Using the equations of Figure 112, we see that choosing E0 and 0: deﬁnes p _L and

Pu at the cavity entrance. Choosing B0 sets the beam orbit radius at the cavity entrance

(r0). The beam should enter the magnetron at an initial radius fraction of about rO/Ro =
0.75, therefore choosing Bo also sets the initial radius of the magnetron (R0). The width
of a gap is limited by the circumference of the magnetron at its entrance. A good
assumption is to divide the circumference equally among 2n gaps and 2n vanes.
Choosing the harmonic number deﬁnes the gap width (g) and the rf frequency (frf).
Using the Kilpatrick criterion for vacuum sparking, the maximum cavity voltage (V0)
depends on frf but also on the gap width [Kilpatrick, Humphries]. The actual cavity
voltage should be set to some fraction (F) of the Kilpatrick voltage (Vo = FVK) to allow
for ﬁeld enhancement and a safety margin. Once V0 is known, choosing Egan, sets the

number of turns through the acceleration region (Nmms). Finally, the flat ﬁeld length

(ﬂaw) is known from Nmms, pll , and B0.

In Figure 113, the capacitance of the magnetron (C) is calculated from geometry
(review Figure 16). Once C, fry, n, and the Q of the cavity are known, we can calculate
the shunt resistance (R,). The rf power required to run the magnetron is then calculated
from V0 and Rs. Since the capacitance and f,; are both known, the radius of the required
inductor (n) can be calculated. This simple geometry assumes a round hole for the
inductor (Figure 16), However, the inductor can always be deformed to reduce its radial

extent, therefore the equation for rL assumes a reduction by a factor of 1/2. Finally, the
radial distance needed for the magnetron oscillators (Rmagncmn ) can be calculated from (C

and rL. As mentioned above, the magnet diameter required is simply
D = 2(rbeam '1' Rmagnetron)-

243

 

 

Using the equations in Figure 112 and 113, it is straightforward to derive

expressions for each output parameter in terms of only input parameters and constants.

This allows us to look at the effect of each input parameter on the output parameters.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

R = I + mc‘jz 1 (2mEo)”4 J;
magnetron c 3 ‘ﬂ: (enBo)3/2 (Ii-(12)!”

 

 

 

 

 

 

 

16 8o n2 [c o:
P = — — FE E ainB —— —
”f 3 m ° g ° (r/R)" Q 1+0:2

 

 

 

 

5; = (n+012)-6p—i
E PI.

 

 

 

We want [my and Rmagne, both small to reduce the cost of the superconducting

magnet. Similarly, the rf power should be low to reduce the cost of the rf power system.

And the energy spread should be low in order to maximize the accelerated beam current.

244

 

 

 

From these equations we can make several useful observations about the effect of
each input parameter. For this example, Egan. 5 10 MeV has been chosen because it is a
useful beam energy for the applications discussed in chapter 1. As stated previously,

(r/ R) E 0.85, and in order to obtain a high RS/Q without causing an unreasonably low

capacitance we require EC 5 0.5 cm. No claims about the Q of the magnetron structure

will be made until rf after measurements are done. If an estimate of the rf power is

needed, then a conservative value for the Q would be 2000. Furthermore, -8—pi— is the

1

result of the injected beam emittance and the electrostatic bend. Again, if an estimate is

needed, then the value from the 50 keV ICRA could be used % E 8.7 %.
.1.

This leaves only 5 input parameters: E9, B9, FA nggand 0:. The effect of these 5 input

parameters is summarized below.

Effect of Iniection Energy (E9)
1) High E0 causes shorter [m (cheaper magnet)

2) High Eo causes smaller injected beam emittance (higher accelerated beam current)

3) High Eo allows larger gaps and higher Vo (potentially higher rf power). However we

can still choose F low to counteract this effect.

Effect of Magnetic Field L139)

4) High Bo causes shorter [m (cheaper magnet).

245

5) High Bo causes smaller rm (cheaper magnet).
6) High B0 causes higher rf frequency / smaller Rmagnetron (cheaper magnet).
7) High Bo causes higher rf power.

8) High B0 may require smaller injected beam size.

Effect of Cavity Voltage Fraction (F)
9) High F causes high V0 (high rf power).

10) High F causes shorter [my (cheaper magnet).

Effect of Harmonic Number (11)
11) High It requires higher rf power (Prf ~ 112 because RS ~ n'z).
12) High 11 causes a larger energy spread (lower accelerated beam current).

13) High 11 causes higher rf frequency and a smaller ngnwon (cheaper magnet).

14) High 11 causes shorter [Gav (cheaper magnet).

Effect of Alpha (0:)
15) High 0: causes a larger energy spread (lower accelerated beam current)
16) High 0: causes a shorter Em (cheaper magnet)

17) High 0: lowers the rf power.

246

Using the observations listed above we are now in a position to choose some input
parameters. It is obvious from points 1-3 that Eo should be as high as is convenient. We
choose E0 = 100 keV. Similarly, B0 should be as high as possible without inducing
excessive cost. Current NbTi technology allows Bo =18 Tesla. Above this level the cost
rises steeply. Therefore, we choose B0 = 8 TeslaThe harmonic number 11 should be low
for a high beam quality, however we want n high so that the magnetron ﬁts inside the
magnet bore. The choice of n = 4 gives frf = 488 MHz which allows the use of an
industrial magnetron as the rf power source. Furthermore, fr; = 488 MHz makes the
radius of the magnetron resonators less than 2 inches. Together with the ﬁnal beam
diameter, this leads to a warm bore of roughly 8 inches which is a convenient size. Figure

114 shows the magnet bore diameter as a function of magnetic ﬁeld for n = 4.

25

 

    
 

 

 

 

20 -—
E I—
8 15 - Resonators+Beam
f I
2’. _\
a . \
.9 10 "" \\
D ._ \
5 — 7 ~~~~~~~~
0 .- 1 l 1 l 1 I 1 L 1 l 1 1 1
2 3 4 5 6 7 8 9 10
Ba (Tesla)

Figure 114. The dotted line shows the diameter required for resonators of the magnetron (n=4)
as a function of magnetic field. The solid line shows total diameter D = 2(r.,.g.m + Rum”)

( for a 10 MeV beam) needed for the warm bore of the superconducting magnet.

247

The two remaining input parameters are F and 0:. Since the beam quality has
already been pushed low by using n = 4, it is important to have 01 as low as possible.
However, low 0: causes a long cavity length. Since high F will lower (my, and F is the
only parameter left to adjust, we should choose F as high as possible so that we can
choose a as low as possible and yet still obtain a reasonable cavity length. Choosing F
high requires higher rf power, therefore F controls the trade off between rf power and

cavity length. The highest possible voltage fraction is F = 0.25 (0.5 for ﬁeld

enhancement, and 0.5 for a safety factor). Table 16 below shows the resulting three

designs with o: = 2, 4, and 8.

248

 

 

 

Table 16. Three different 10 MeV ICRA designs with o: = 2,4,8.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

a 2 4 8
ion proton proton proton _
Eﬁnal 10 MeV 10 MeV 10 MeV
E0 100 keV 100 keV 100 keV
B0 8 Tesla 8 Tesla 8 Tesla
n 4 4 ' 4
Ngaps 8 8 8
fc 122 MHz 122 MHz 122 MHz
fr; 488 MHz 488 MHz 488 MHz
g 2.7 mm 2.9 mm 3.0 mm
F 0.25 0.25 0.25
V() 56 kV 66 kV 69 kV
Ntums 54 46 44
ro 0.51 cm 0.55 cm 0.57 cm
R0 0.68 cm 0.74 cm 0.76 cm
rem 5.7 cm 5.7 cm 5.7 cm
Rem 6.0 cm 6.0 cm 6.0 cm
(W 34 inches 16 inches 8 inches
C 14 pF 6 pF 3pF
L 7x10'9 Henry 2><10'8 Henry 4x10'8 Henry
Rle 3 a 7 a 14 Q
Q 2000 2000 2000
Rs 5.7 kg 13.3 kg 28.0 kQ
Pg 270 kW 160 kW 84 kW
[c 0.5 cm 0.5 cm 0.5 cm
rL 2.02 cm 2.10 cm 2.13 cm
Rmagnwon 4.5 cm 4.7 cm 4.8 cm
Magnet ID 8.3 inches 8.4 inches 8.5 inches
n +08 8 20 68

 

 

 

249

 

4th harmonic, and an injection energy of 100 keV. Varying 0: = 2, 4, 8 controls the trade
off between cavity length and quality of the accelerated beam. Each design has a magnet

Discussion
The three ICRA designs presented in Table 16 all use an 8 Tesla central ﬁeld,
bore of reasonable diameter (ID = 8”). However, the o: = 2 design requires a flat ﬁeld

 

length of 34” while the o: = 8 design ﬁts into a very reasonable 8” length. The 0: = 2

design will certainly accelerate higher beam current because (n + 012) = 8, while the 0: = 8

'_m a

design will accelerate much lower beam current since (n + 012) = 68. The longer cavity
design also requires signiﬁcantly more rf power because of the increased capacitance of

the accelerating cavity.

It is useful to compare these results with a previous 10 MeV design which was
computer modeled, but not put through this optimization process [1]. This design also
used an 8 Tesla B-ﬁeld, n = 4, for a 488 MHz magnetron, and had at = 4.1. However the
injection energy was 54 keV, and the fraction of Kilpatrick voltage was F = 0.5 which

caused a cavity length of only 10 inches. Computer modeling results showed 1311A

 

(0.67 %) of the beam accelerated to the range from 5 to 10 MeV, assuming an injected

proton beam of 2 mA in an emittance of 21: mm mrad (unnonnalized) [1, 26].

The 0: = 4 design in Table 16 should do better than the previous 10 MeV results
for several reasons. The factors which control the energy spread (n, 0:, and Egain) are the
same in both designs. However, the optimized design in Table 16 has a higher injection
energy, which causes a lower beam emittance at injection. Also the previous 10 MeV

results were obtained for a condition where the beam is not scraping the wall. Experience

250

from the 50 keV ICRA experiment shows that peak accelerated beam current is obtained
for a condition such that the beam current is scraping the wall (section 2.6.8).
Furthermore, the analysis of the acceptance phase space in section 2.6.6 showed that the
acceptance is independent of azumuthal (0) position. This leaves open the option of
using multiple ion sources at different azimuthal positions, or a single source with

multiple extraction apertures.

 

T1;—

As a conservative goal, if the o: = 4 design in Table 16 accelerated only 2% of the

beam current to an energy distribution between 5 - 10 MeV, this would produce a 40 11A

Ir- ._ .-..

beam which would be useful for applications. The 0: = 2 design would accelerate higher

current and the 0: = 8 design would accelerate less.

All three designs in Table 16 have the distinct advantage of being able to utilize
an existing industrial magnetron for the rf power source. If a cryogen free magnet is also
used, this combination would lead to an inexpensive accelerator. At this point, the main
question is how much current each design will accelerate. In order to answer this
question, the ICRA computer code (section 2.6) should be used to model a full emittance

beam through each of the three designs. In this way a proper analysis can be made

 

between cost vs. accelerated beam current. This task is recommended as future research.

251

6. CONCLUSION

This experimental study of a 50 keV Ion Cyclotron Resonance Accelerator
(ICRA) has successfully demonstrated ion acceleration using the same axial drift
geometry that is characteristic of gyrotrons. Although, cyclotron resonance acceleration
has been previously demonstrated using electrons [5], this experiment marks the ﬁrst
time that this geometry has been used to accelerate an ion beam.

An rf driven magnetron operating at 152 MHz was mounted in 2.5 Tesla axial
magnetic ﬁeld supplied by a superconducting solenoid. A 6.4 keV proton beam was
injected into the high magnetic ﬁeld region such that the beam spirals around magnetic
ﬁeld lines while continuing to drift axially through the acceleration region. RF electric
ﬁelds of the magnetron accelerated the beam in the azimuthal direction. Measurements .
of the accelerated beam show an energy distribution with 7% of the beam current above
24 keV and 1% above 42 keV.

Measurements of the injected beam were used to determine the radius of the
proton orbit (r) and the number of turns through the acceleration region (Nmms) as well as
the spreads in r and Ntums. Beam measurements have also been used to estimate the
emittance of the injected beam between 0.351: and 2.21: mm mrad (unnormalized).

Analytical theory and a complete computer model have been deve10ped for the
ICRA. A full emittance injected beam has been simulated using the computer model to
track many single particle trajectories. The computer model was used to obtain energy
distributions of the accelerated beam.

Computer generated energy distributions show reasonable agreement with the

experimental energy distributions. After a small correction of the bend voltage the

252

computer model gives good agreement with the magnitude and shape of the experimental
data over a range of turn number.

The agreement between the computer model and experiment is considered a
benchmark of the ICRA computer code. Therefore the code will be a useful tool for
designing a higher energy ICRA.

A scheme for optimizing an ICRA design has been given. Design parameters
which minimize cost and maximize the accelerated beam current have been identiﬁed.
Three different 10 MeV designs have been proposed which offer a range of the trade off
between cost vs. accelerated beam current. A full cost analysis and prediction of the
accelerated beam current using the ICRA computer code for each design is suggested as

future research.

253

APPENDIX

Radial Probe Corrections

Ideally the radial probe should block all of the beam current above a certain radius
and read an integrated current vs radius. Errors in the radial probe data were explained in
section 3.2.5. Three mathematical corrections are needed to correct the radial probe data:

1) Account for the slope on the inner edge of the probe
2) Account for beam lost on the electron blocker
3) Account for axial turn lengths which are longer than the probe length

The slope on the inner edge of the radial probe matches the slope of the B-ﬁeld for the

50 keV orbit radius. At any smaller radius, the slope on the probe induces an apparent
energy spread in the measured data. Corrections for the electron blocker and for the axial
turn Spacing both depend on the beam energy. Therefore, correction 1) must be made

before 2) or 3). The order of 2) and 3) does not matter.

Slog on the Inner Edge

The slope on the inner edge of the radial probe matches the slope of the B-ﬁeld
for the 50 keV orbit radius only. At any radius smaller than this, the slope of the ﬁeld
line is less than the slope of the probe. Thus the probe induces an energy spread on the
measured data. Unaccelerated beam data can be corrected using the simple thin shell

model. Accelerated beam data can only be corrected using the thick shell model.

254

Thin Shell Model

The thin shell model is only valid for the unaccelerated beam. We begin with a
simple example for constant magnetic ﬁeld. Assume the beam is a uniform rotating
cylindrical shell with radius ro and zero thickness (Ar = 0). According to Figure Al, the

beam current measured on the R-probe (1",) is simply the total beam current (10) times the

fraction intercepted by the probe.

_ dz
g_n[€] (40

Using the triangle at the tip of the probe, Az can be expressed in terms of the beam radius

(r0), the measured radius (rm) and the probe slope (mp).

(ro —rm)=mpAz

 

 

II

 

<
7

 

e—AZ >,,,.///I B-Field Line
//”,,,
I: 7‘?

m \Z
/

 

 

 

 

 

Figure Al. Geometry of the radial probe showing how the sloped inner edge cuts across an

un-accelerated beam of radius r0. In this example the magnetic field is constant.

255

 

"—l

I; i
‘t‘

 

 

Therefore, the measured beam current as a function of rm becomes:

I'o -rm
1,, _ 1, IT, i I

Solving for I0 gives a formula to recover the actual beam current from the measured data.

 

— rm

mf .
10:14 P ] for A23: (A2)

If the B-ﬁeld is not constant, but the ﬁeld line has slope m3, then the length intercepted

by the probe (At’) is more complicated.

 

 

 

A€ = \/Az2 +(mB Az)2 where A2 = [Q]
I (m —m )4
Therefore: I0 = m p B AzSE (A.3)

 

 

for
(Il+mf3 (ro—rm)

This formula can be applied directly to the measured data (Im vs. rm).

 

256

 

distortion caused by the slope on the inner edge of the radial probe. The distortion is
largest in the lower energy portion of the distribution and goes to zero at 50 keV.
Correcting for this effect would shift current toward higher energies, therefore neglecting

Thick Shell Model
Radial distributions of the accelerated beam have po_t been corrected for the
this correction leads to conservative energy distributions. Furthermore, some of the

 

beam is never picked up at the low end of the energy spectrum (0 — 5 keV), therefore not

enough information is known to correctly perform the correction. For completeness, the

1-

effect is explained below.

The accelerated beam is not concentrated at one radius, but rather is spread out in
a radial distribution which ranges from near r = 0 to the radius of the maximum energy.
Therefore, correcting accelerated data for the effect of the probe slope requires a thick

shell model.

As an example, assume that the B-ﬁeld is constant (horizontal ﬁeld lines) and the
beam current is distributed over a range from r = (0 — 20) mm. The radial probe is moved
inward in 2mm steps. As the probe moves from rm = 6 mm to rm = 4 mm, a 2 mm wide

strip along the sloped inner edge of the probe picks up some additional beam current

(A14 = 14 - Io)- Although this measured current has been recorded at rm = 4 mm, the slope
on the inner edge of the radial probe has cut across shells with radius 6,8,10,12,and
14mm. Therefore the proper correction must take a fraction of the current measured at

low radius, and redistribute it to higher radii.

257

 

 
  

 

-IZEDCD--. ___________ ______ m—Il/lllll/l”_'_ ______

 

 

Figure A2. The slope on the inner edge of the radial probe cuts across shells at several radius.

The probe is shown here as it moves from r... = 6m to 4mm.

For each radial step, the additional current measured (Ali) can be decomposed into
the contribution from each shell. The contribution from each shell is the area times the

current density of that shell (A, 1,).
A12 = A313 + A515 + A717 + A919 +
A14: 0 + A515 +A7J7+A919+
A16: 0 + 0 + A7J7+A9J9+...

Thus, the transformation is a matrix equation.

If the total number of radial steps taken is “”,n then Aij is an n x n matrix with zeros
below the diagonal. J J- is the actual current density in each radial shell and AI, is the

measured increase in current in going from r = i+2 to i.

258

 

The actual beam current can be recovered by inverting this matrix equation:
_ —l

The distortion in the measured radial distribution is less for the actual B-ﬁeld than
for the constant magnetic ﬁeld case illustrated in Figure A2. In reality the slope of the
radial probe matches the 50 keV ﬁeld line, therefore the distortion goes to zero at full

energy. Figure A3 shows the range of energies that are mixed.

 

55
50 --

45—

35—

30: ————————————

25—

20_

Actual Energy (keV)

15-

10—

 

 

I
I
|
l
l
I
I
|
|
I
I
I

I I I I

p

 

0 I l I I
0 5 1 0 1 5 20 25 30 35 40

Measured Energy (keV)

.h
(11

50 55

Figure A3. The measured beam current at a particular energy actually contains a spread in beam
energies bounded by the two lines shown here. For example, the beam current measured at 24 keV
actually contains energies ranging from 24 keV to 31 keV. The spread goes to zero at 50 keV.
Vertical dotted lines mark the injected beam, half maximum, and the full the beam energy (48 keV)

259

thaw-thrush:
‘

I‘.

"i. —

 

 

Beam Lost on the Electron Blocker

The electron blocker is mounted along the upstream edge of the radial probe as
shown in Figure A4. Since electrons are trapped on magnetic ﬁeld lines, the electron
blocker casts a shadow over the Faraday cup portion of the probe that electrons cannot
reach. On the other hand, ions orbit around the z axis with large radius therefore they can

pass behind the electron blocker and strike the Faraday cup.

Mounﬁng ‘ Mounﬁng

Rod Rod
Electron
Blocker
\_ , -
Electron
X Blocker

Faraday
Cup

SIDE VIEW AXIAL VIEW

 

 

 

 

 

Figure A4. Geometry of the radial probe and the electron blocker.

Since the ion beam moves forward in 2 as it rotates azimuthally, a small portion of
the ion beam will be intercepted by the electron blocker. Figure A5 shows the geometry.
The fraction (f) of the beam that is lost is just the arc length across the width of the
blocker divided by the circumference of the entire orbit (f = s/21:r = 0/21: ). Since the
width of the blocker is constant, error induced will be largest for beams with small

diameter. If the e— blocker has width (w) and the beam has radius (r) then:

1 __ - 0
2w — rsrn(2)

260

Beam
Orbit

Electron
Blocker

 

 

 

 

Figure A5. Geometry showing the fraction of an ion orbit
circumference which is intercepted by the electron blocker.

So the fraction (f) of the orbit that is blocked is

 

_ i - —I 1".
f — “srn ( 21F) (A.4)

 

 

 

The measured current (Im) is reduced from the actual current (10) by:

I = 10(1—f)

111

Therefore the corrected beam current is:

 

I
10 = (1 ff) (A.5)

 

 

 

 

Notice that this correction depends on knowing the radius of the beam correctly.
Therefore, the energy spread caused by the slope on the leading edge of the probe must
be corrected for before the electron blocker correction is applied.

The electron blocker correction must be applied only to the current at a particular
radius. Therefore, since data taken with the radial probe is an integrated distribution, the
currents must ﬁrst be unstacked to recover the actual beam current at each radius

(AI, = I, —Ii+1) then correction factor is applied, then data is resummmed (I, = Ii+1+AIi).

26]

 

 

 

Axial Turn length

In the extraction region, the axial momentum of the beam (pz) increases as the
magnetic ﬁeld drops off . The axial length between turns (Azmm) is proportional to p2.
This relation was derived in section 2.5.8. Furthermore, the increase in Azturn depends on
the energy of the beam upon exit from the acceleration region. Section 2.5.8 showed

that Az.urn is approximately linear with the beam orbit radius in the extraction region.
If the axial turn length is longer than the axial length of the radial probe (6), then

the Faraday cup on the radial probe will intercept only a fraction of the total beam current

(10). The measured beam current (1",) is simply:

 

 

(A.6)

where Azturn must be calculated for each beam energy in the distribution using equations
2.62 and 2.88. Since Az,um increases with beam energy, this correction effects mostly the
upper energy portion of the radial distribution.

This correction depends on knowing the radius of the beam correctly. Therefore,
the energy spread caused by the slope on the leading edge of the probe must be corrected
for before correcting for the axial turn spacing. Correction for the axial turn spacing must
be applied only to the current at a particular radius. Therefore, since data taken with the
radial probe is an integrated distribution, the currents must ﬁrst be unstacked to recover

the actual beam current at each radius (AL = I, —Ii+1) then correction factor is applied, then

data is resummmed (I, = I,+1+AI,).

262

10.

ll.

12.

13.

14.

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