CHARACTERIZATION OF ELECTRON CYCLOTRON RESONANCE ION SOURCE

INSTABILITIES BY CHARGED PARTICLE DIAGNOSTICS

By

Bryan Isherwood

A DISSERTATION

Michigan State University

in partial fulﬁllment of the requirements

Submitted to

for the degree of

Physics – Doctor of Philosophy

2020

ABSTRACT

CHARACTERIZATION OF ELECTRON CYCLOTRON RESONANCE ION SOURCE

INSTABILITIES BY CHARGED PARTICLE DIAGNOSTICS

By

Bryan Isherwood

Ion sources are invaluable tools for producing charge particles for scientiﬁc, industrial, and medical
applications. In particular, Electron Cyclotron Resonance (ECR) ion sources (ECRIS) are high
power sources capable of producing high intensity, high charge state beams of heavy ions. The
system uses microwaves to resonantly heat of electrons within an inhomogeneous magnetic trap.
However, the internal dynamics of the resulting plasma are complex and poorly understood. In
particular, the excitation of kinetic instabilities within the plasma can make operating these ion
sources diﬃcult and unpredictable. This thesis focuses on studying these instabilities to determine
ways to optimize ECRIS performance by minimizing their impact on the extracted beam current.
This study focuses on two measurements that look at the steady-state and time-resolved measure-
ments of charged particle currents escaping the ion source during stable and unstable operations.
The ﬁrst measurement is a novel diagnostic of electrons escaping conﬁnement from the plasma
chamber. The second was a measurement of high charge state ions (Ar8+) extracted from the plasma
chamber over a broad set of parameter spaces. These measurements provide insight into the ideal
operating conditions for an ECR ion source and the dynamics of the ion and electron populations
within its plasma.

Copyright by
BRYAN ISHERWOOD
2020

I produced a detailed tribute to my wrongness.
Nathan Pyle

iv

ACKNOWLEDGEMENTS

This thesis, and large amounts of the work presented, were made possible only through the time
and patience of Guillaume Machicoane, who helped see this projected through to its completion. I
would also like to acknowledge Eduard Pozdeyev and Yoshi Yamazaki for their support throughout
this study. This research was made possible by the National Science Foundation under NSF
Grant 1632761, the U.S. Department of Energy Award Number DE-SC0018362, the National
Superconducting Cyclotron Laboratory (NSCL), the Facility for Rare Isotope Beams (FRIB), and
Michigan State University.

I would also like to thank my committee Yoshishige Yamazaki, Guillaume Machicoane, Kirsten
Tollefson, Steven Lidia, and John Verboncoeur for their support and advice throughout this project.
Finally, I would like to thank all of those who have provided help, insight, and expertise, without
whom this project would not have been possible: Jeﬀry Stetson, Larry Tobos, Jesse Fogleman,
Shane Renteria, Derek Neben, Andreas Stoltz, Dirk Weisshaar, John Brandon, the entire FRIB and
NSCL Diagnostic, Electronics, and RF Groups, the NSCL operations group, Cody Norat, Steven
Lund, Olli Tarvainen, and Ivan Izotov.

v

TABLE OF CONTENTS

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Plasma Waves .

2.1 Plasma Physics

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INTRODUCTION .

LIST OF TABLES .
LIST OF FIGURES .
CHAPTER 1

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3
CHAPTER 2 ELECTRON CYCLOTRON RESONANCE ION SOURCES . . . . . . . . .
3
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4
2.1.1 Magnetic Conﬁnement
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
2.1.2 Oscillations and Collisions . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.3
8
Plasma Diﬀusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.4 Electronic Properties of the Plasma . . . . . . . . . . . . . . . . . . . . . . 10
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.1.5
2.2 Electron Cyclotron Resonance Ion Sources and their Plasmas . . . . . . . . . . . . 15
. . . . . . . . . . 15
2.2.1 Resonant Heating of Electrons Within ECR Ion Sources
2.2.2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2.3 Characteristics of the Electron Population . . . . . . . . . . . . . . . . . . 23
2.2.4 Characteristics of the Ion Population . . . . . . . . . . . . . . . . . . . . . 25
2.3 Operational Principles for ECR Ion Sources . . . . . . . . . . . . . . . . . . . . . 26
2.3.1 Electron Conﬁnement and the Minimum-B Magnetic Field Topology . . . . 27
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.3.2
2.3.3 Afterglow .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
Scaling Laws
2.3.4
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2.4 Kinetic Instabilities .
2.4.1 Diagnostic Observations
. . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.4.2 Role of Hot Electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
. . . . . . . 44
2.4.3 Theory: Quasi-linear Diﬀusion and Kinetic-Cyclotron Masers

Ion Conﬁnement
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Ionization Processes

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3.1 Superconducting Source for Ions (SuSI)
3.2 Diagnostics

CHAPTER 3 EXPERIMENTAL APPARATUS . . . . . . . . . . . . . . . . . . . . . . . 49
. . . . . . . . . . . . . . . . . . . . . . . 49
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3.2.1 Microwave Power Diode . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.2.2 Beam Spectrometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
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4-Jaw Collimator
3.2.3
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3.2.4 Beam Current .
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3.2.5 X-Ray Spectrometer
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CHAPTER 4 DISTRIBUTION OF ESCAPING ELECTRONS . . . . . . . . . . . . . . . 70
. 70
. 71
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4.1 Experimental Goals .
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4.2 Considerations for the Electron Current Measurement . . . . . . . . . . . . . . .
4.3 Parameter Space .
4.4 Results .

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. 85
4.4.1 Varying Power
4.4.2 Eﬀect of a Variable Pressure . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.4.3 Eﬀect of a Variable Hexapole . . . . . . . . . . . . . . . . . . . . . . . .
. 92
4.4.4 Eﬀect of Varying the Injection Side Magnetic Maximum . . . . . . . . . . 95
4.4.5 Eﬀect of Varying the Extraction Side Magnetic Maximum . . . . . . . . . 99
4.4.6 Eﬀect of Varying the Magnetic Minimum . . . . . . . . . . . . . . . . . . 102
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

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4.5 Concluding Thoughts .

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5.1 Goals
5.2 Parameter Space .
5.3 Results .

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CHAPTER 5 PARAMETER SWEEP . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
. 114
. 114
. 115
5.3.1 Beam Current Transient Proﬁles . . . . . . . . . . . . . . . . . . . . . . . 117
5.3.2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
5.3.3 Varying the Hexapolar Field Maximum . . . . . . . . . . . . . . . . . . . 137
5.3.4 Eﬀect of the Injection Side Magnetic Field upon Plasma Stability . . . . . . 142
5.3.5 Varying the Extraction Field Maximum . . . . . . . . . . . . . . . . . . . 146
Source stability for a varying magnetic minimum . . . . . . . . . . . . . . 153
5.3.6
5.3.7 Understanding the ’over-extraction’ transient proﬁles
. . . . . . . . . . . . 161
. 163

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5.4 Concluding Thoughts .

Stability Maps .

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CHAPTER 6 CONCLUSION .
APPENDICES .
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APPENDIX A MATLAB CODES . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
APPENDIX B LONGITUDINAL FIELD PROFILES FOR THE MEASUREMENT

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OF THE DISTRIBUTION OF ESCAPING ELECTRONS . . . . . . . 177

APPENDIX C EXTRA DATA FROM THE MEASUREMENT OF THE DISTRI-

APPENDIX D LONGITUDINAL FIELD PROFILES FOR THE PARAMETER

BUTION OF ESCAPING ELECTRONS . . . . . . . . . . . . . . . . 179

BIBLIOGRAPHY .

SWEEP MEASUREMENT OF ION STABILITY . . . . . . . . . . . 187
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
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vii

LIST OF TABLES

Table 2.1: A comparison of electron Debye lengths for the diﬀerent electron populations.
In accordance with the qualitative trends estimated in Mironov et al., it is
assumed that ne,cold ∼ 1 × 1011 cm-3 and ne,warm/hot ∼ 5 × 1011 cm-3 [58].
Table 2.2: A comparison of electron collision and gyration frequencies in a region where
B = 1 T, ne = 5 × 1011cm−3, ln Λ = 10 assuming the energy distribution fo
electrons follows a Maxwell-Boltzmann distribution. fce decreases due to the
γ−1 term in equation 2.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

. . 24

Table 2.3: The collision frequencies for Ar1+ and Ar8+ with ne = 5 × 1011cm−3, q = 8,
ln Λ = 10 assuming that the energy distribution of ions follows a Maxwell-
Boltzmann distribution. The right hand side of the plot compares the diﬀerent
collision frequencies against the gyration frequencies for both of these charge
states assuming B = 1 T.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Table 3.1: The measurable range could be set by the BCM controller. While there is
an upper limit of current that each range can measure, the actual value of the
range (ﬁrst column) does not determine the maximum value. That is to say,
the ﬁrst column is more of an order of magnitude reference. Increasing the
BCM measurable current range, generally, decreased the relative error of each
measurement. The variance increased along side the measurable range, but
did not have a signiﬁcant eﬀect upon each measurement.

. . . . . . . . . . . . . 59

Table 3.2:

Information of the calibration source used. A0 is the initial soruce activity,
measured on November 1st, 1988. A is the theoretical activity at the time of
the measurement. Calibration data was taken on February 15th, 2018.

. . . . . . 63

Table 4.1: Closing the vertical gap from 38 to 5 mm decreased the observable current by
about an order of magnitude. The relative current variance was calculated use
values similar to those in table 3.1. . . . . . . . . . . . . . . . . . . . . . . . . . 80

Table 4.2: Comparison between the longitudinal ﬁeld extrema of the NSCL standard op-

erating point for facility operations, and the control ﬁeld used for this measurement. 81

Table 4.3: The set of operating points used where chosen to measure the coupling be-
tween the magnetic minimum and maximum at injection and other measured
parameters. (cid:104)B(cid:105) is the average magnetic ﬁeld between the injection and ex-
traction maxima. The extraction side maxima was held constant at 1.100 T for
all operating points. . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

viii

Table 5.1: The ﬁeld maxima used were scaled to either be on average larger or smaller
than the scaling laws prescribe. The values above are there to demonstrate
the relative range of values used as the magnetic minimum was varied. As
calculated, the ﬁelds in each of the four ranges corresponds to Bmin/BRF =
0.725. The standard facility operating point uses Bmin/BRF = 0.711.

. . . . . . . 115

Table 5.2: Each of the diﬀerent transient proﬁles has been assigned a diﬀerent color. Gen-
erally speaking each of the diﬀerent proﬁles exists over a diﬀerent repetition
frequency range, with ion bust dominated events occurring over the smallest
time scale and over-extraction events occurring over the longest.

. . . . . . . . . 128

ix

LIST OF FIGURES

Figure 2.1: The series of solenoids generates a semi-uniform magnetic ﬁeld. The ﬁeld
strength reaches a minimum at some point between the centers of the coils,
represented by the less dense ﬁeld lines (top). The bottom left plot shows an
example ﬁeld proﬁle from the center of the above conﬁguration to the center
of the last coil. The bottom right plot shows an example loss cone.

5
Figure 2.2: The dispersion relation for an under dense plasma (ωce > ωpe). . . . . . . . . . 14
Figure 2.3: A 2D projection of the typical operational magnetic ﬁeld across the central
axis of the SuSI ECR ion source at the NSCL. This data was generated using
the program POISSON/Superﬁsh. . . . . . . . . . . . . . . . . . . . . . . . . . 17

. . . . . . .

Figure 2.4: A 3D projection of the magnetic ﬁeld magnitude within an ECR ion source.
This particular proﬁle was generated by using CST model of the SuSI ion
source at the NSCL. The black lines represent iso-B surfaces within the
volume of the magnetic trap.

. . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Figure 2.5: A contour projection of the minimum-B magnetic ﬁeld conﬁguration seen in
ﬁgure 2.4. The black rings indicate the position of iso-B surfaces within the
ﬁeld structure. The red ring represent the ECR surface over which the heating
of cold electrons occurs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Figure 2.6: A calculation of the Electron Impact Ionization cross section, given in [103]. . . 22

Figure 2.7: Golovanivsky’s plot which shows the criteria for producing high charge state
ions in an ECRIS plasma. The circles represent the atomic species which
is fully ionized for the given density and ion conﬁnement time. Those in
parenthesis to the right show the highest charge state achievable for other
atomic species. Reproduced from [17], with permission from AIP Publishing.

. 28

Figure 2.8: Comparison of the radial component of the magnetic ﬁeld generated by the

solenoids alone (left) and the hexapole alone (right). . . . . . . . . . . . . . . . 29

Figure 2.9: The three diﬀerent decay modes are dependent upon the degree of turbulence
of the plasma. The three curves show the typical decay forms of afterglow
pulses: a) Turbulent, b) Quiescent, c) Non-linear particle diﬀusion. Figure
originally printed in [18] and has been re-published with permission of Taylor
& Francis Group LLC.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

x

Figure 2.10: By pulsing RF power again after the initial afterglow burst, an instability can
be ignited which forces hot electrons out of the plasma. This results in a
second burst of ions. This method will work so long as the hot electrons have
not diﬀused out of the plasma by other means. Figure originally printed in
[18] and has been re-published with permission of Taylor & Francis Group LLC. 35

Figure 2.11: A demonstration of the magnetic scaling laws for intermediate to high charge
state argon ions. The measurement was taken using SuSI at the NSCL. Bmin
= 0.4 T, Pµ = 350 W, pAr = 214 nTorr.

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

. 37

Figure 2.12: Beam current losses for Ar8+, normalized by the largest measured value.

. . . . 40

Figure 2.13: Spectrogram of an unstable microwave emission in open space (no waveguide)
for a 14.056 GHz ECR ion source, 400 W microwave power, Bmin/BRF =
0.83. The solid line above 14.056 GHz results from the heating RF that
leaked out of the system. The microwave emission was so powerful that it
was possible to detect the emission in free space. Experimentally, this means
that frequencies above and below the waveguide cutoﬀs could be measured.
This plot has not been adjusted for transmission eﬃciency of the microwaves.
Reproduced from [25], with permission from AIP Publishing. . . . . . . . . .

. 41

Figure 2.14: Dispersion relationship of the extraordinary wave in an ECRIS plasma. fp =
6 GHz, fce = 14 GHz, θ = 5 °, fz = 2.22 GHz, fZ I = 14.1. The ’slow’ and
’fast’ qualiﬁers signify whether the phase velocity of the wave is less that or
greater than the speed of light in vacuum, respectively. The slow Z-mode
propagates between the plasma frequency and Z-Inﬁnity (Blue). The fast
Z-mode propagates in the orange region below the plasma frequency. The
X-mode propagates in the orange region above fx.

. . . . . . . . . . . . . . . . 42

Figure 2.15: Phase velocity of the extraordinary wave in an ECRIS plasma.

fp = 6.00
GHz, fce = 14.00 GHz, θ = 5°, fz = 2.22 GHz, fZ I = 14.01, and fx = 16.22.
The horizontal dashed line represents vθ = c. The ’slow’ and ’fast’ qualiﬁers
signify whether the phase velocity of the wave is less that or greater than the
speed of light in vacuum, respectively. The slow Z-mode propagates between
the plasma frequency and Z-Inﬁnity (Blue). The fast Z-mode propagates in
the orange region below the plasma frequency. The X-mode propagates in the
orange region above fx. The region above fZ I but below vθ = c is a forbidden
region with imaginary propagation. . . . . . . . . . . . . . . . . . . . . . . . . 43

Figure 3.1: A schematic of the source-beamline conﬁguration used for this research.

. . . . 50

Figure 3.2: A 3D model showing the cross-section of SuSI. Gas and microwave are
injected on the left and the ion beam is extracted to the right. Arrows on the
bottom point to the four active solenoid coils that surround the hexapole and
plasma chamber.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

.

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xi

Figure 3.3: An example of the on axis longitudinal ﬁeld proﬁle for SuSI. The three curves
were generated by POISSON/Superﬁsh. The vertical dashed line represents
the position of the ion source’s extraction aperture. The horizontal dashed
line represents the resonance ﬁeld for 18 GHz microwaves with cold electrons.

Figure 3.4: An example of SuSI’s radial ﬁeld as a function of distance from the center
axis, with 375 A going through each hexapole coil. The curve was generated
by a CST model of SuSI’s coil and yoke conﬁguration. The horizontal dashed
line represents the resonance ﬁeld for 18 GHz microwaves with cold electrons.

52

52

Figure 3.5:

(Top) The full geometry of the source and beamline model used in this study.
All models were taken from pre-rendered SolidWorks ﬁles created by NSCL
mechanical engineers. Those ﬁles were imported into CST studio suite, where
they were used to perform magnetic ﬁeld and beam trajectory calculations.
The model includes a yoke (in blue), plasma chamber, extraction geometry,
einzel lens, transport lines, dipole, and Faraday cup geometry. The cup
housing geometry has been simpliﬁed in order to improve simulation speed.
(Bottom) Coil geometry and current direction. On the left are the dipole
coils and on the right the source coils. The white arrows represent current
direction. In this picture the dipole is set to analyze electrons.

. . . . . . . . . . 53

Figure 3.6: The blue lines show the results of a trace simulation of Ar8+ ions traveling
through SuSI’s beamline up to the ﬁrst Faraday cup. Data from a charge
state distribution measurement was input into the program to calculate the
magnetic ﬁeld of the the ion source and analyzing dipole: Ebeam = 160 keV
(Vext = 20 kV), Veinzel = −20 kV, Vpuller = 0 V, Idip = 102.671 A. The beam
energy was predeﬁned, rather than relying upon the electrostatics simulator.
The simulation does slightly over approximate the ﬁeld in the dipole as it was
necessary to increase the beam’s energy by 4 keV (or extraction potential by
0.5 kV) in order to make the beam hit the center of the cup. The simulation
was performed using CST particle studio. . . . . . . . . . . . . . . . . . . . . . 54

Figure 3.7: Examples of the measured microwave signals coming from the ion source,
as recorded from an oscilloscope trace. Data was recorded in ’AC’ mode,
ignoring the steady state oﬀset. Each plot has been normalized such that
their largest value is 1. a) The instability shows a large burst of microwave
energy. The ’hump’ after the large peak is believed to be a result of a changing
reﬂection coeﬃcient. b) A moderate burst in microwave power which decays
back to a steady state value. c) A small burst in microwave power but possible
a much longer change in the reﬂection coeﬃcient of the plasma. . . . . . . . . . 56

Figure 3.8: The 4-jaw collimator (red and blue) in front of the ﬁrst Faraday cup (gray)
limited the current by closing a gap in the horizontal (dipole bending plane)
and vertical directions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

xii

Figure 3.9: The Faraday cup-to-oscilloscope transmission line consists of the cup, a
twinax cable, a twinax-to-coax converter box, and the oscilloscope (top). The
dashed lines on the attenuation curve show the -3 dB point of the transmission
line (bottom). The attenuation plot was generated using LTspice.

. . . . . . . . 59

Figure 3.10: Two lead blocks (grey) were used to collimate the x-rays coming from the
source. The tungsten slug (red) was only present in the block closest to the
dipole (see ﬁgure 3.1). Picture has been supplied by the courtesy of Derek Neben. 61

Figure 3.11: The acceptance area of the collimation system (red dashed line) falls on all

three electrodes of the extraction system.

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

Figure 3.12: An example Eu-152 emission spectrum as measured by the germanium de-

tector used in this study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

Figure 3.13: Channel-to-energy calibration for the germanium detector used in this study.

. . 63

Figure 3.14: The program known as TV, written by Andreas Fitzler from the university of
Köln, was used to integrate the observed peaks of the spectra. The program
uses the region surrounding the peak to subtract background counts. It then ﬁts
a Gaussian distribution to the background subtracted peak and then integrates
that Gaussian in order to determine the total number of photons observed.

. . . 65

Figure 3.15: Repeating the eﬃciency calibration three times helps conﬁrm the solid angle
calculation. Measurements were taken at 30, 60, and 100 cm distances away
from the radial edge of the detector (assuming cylindrical geometry).

. . . . . . 66

Figure 3.16: The two/three measurements were averaged together and a the resulting data
set was ﬁt with a 6th order polynomial. The ﬁt was generated by MATLAB’s
curve ﬁtting tool.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

.

.

.

Figure 3.17: Natural log of the bremsstrahlung intensity. The smaller plot show the ﬁt of the
linear region of the distribution. The domain was manually chosen to be 130
keV < Eγ < 320 keV. The line is described by ln Iγ = −0.01629× Eγ +6.261,
giving a spectral temperature of 61.39 keV ± 0.599 keV.

. . . . . . . . . . . . . 67

Figure 3.18: Choosing the ﬁtting region algorithmically, rather than blindly ﬁtting over a
region which worked for a diﬀerent operating point, dramatically improves
the overall ﬁt. Properly using this method can result in improved analytical
results when applied across multiple operating points.

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

xiii

Figure 4.1: A varying extraction voltage has little eﬀect on the hot-electron bremsstrahlung
distribution. The spectral temperature seems to ﬂuctuate at low voltages and
then decreases higher extraction voltages. This low order deviation is likely
an eﬀect of a changing background between measurements due to changing
facility operational conditions (compare to ﬁgure C.1). Plasma parameters
are Bmin = 0.397 T, P = 350 W, Tlive = 600 s, pressure = 212 nTorr.

. . . . . . . 72

Figure 4.2: The transient proﬁles of the extracted ion currents were measured for each
operating point used in Figure 4.1. While ions do escape conﬁnement at lower
voltages (< 2kV), it is diﬃcult to distinguish one charge state from another.
Bmin = 0.397 T, P = 350 W, pressure = 212 nTorr.

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

. 73

Figure 4.3: The residual magnetic ﬂux parallel and perpendicular to the beamline trans-

verse axis. Measurements were taken while all beamline elements were turned oﬀ. 74

Figure 4.4: Argon charge state distributions from SuSI for 20 kV, 10 kV, and 5 kV
extraction voltages measured at the ﬁrst Faraday cup after the ion source in
ﬁgure 3.1. Ion charge state increases from right to left, starting at Q = 1. Once
again, we see that the extracted current decreases with decreasing voltage.
Compared to ﬁgure 4.2, however, the purpose here is simply to increase the
domain over which the system’s magnetic rigidity has been measured.

. . . . . 75

Figure 4.5: Measurements of the integral form of the magnetic rigidity between the ion
source and the ﬁrst Faraday cup. The (red) dashed line shows the linear

best ﬁtÞ B(Idip) · dl = mIdip + b, where m = 6.98624 × 10−4 (6.98909 ×

, 7.01300×10−4) and b = 2.47924×10−4 (0.77833×10−4

10−4

, 5.73681×10−4). 76

Figure 4.6: The electron simulation under approximates the energy calibration created by
measuring the magnetic rigidity of the ions. This is likely due to the presence
of ambient and residual magnetic ﬁelds, as seen in ﬁgure 4.3.

. . . . . . . . . . 76

Figure 4.7: The largest diﬀerences between the ion rigidity and electron simulation energy
calibrations occur at lower energies. This is likely due to the presence of
ambient magnetic ﬁelds and interparticle interactions, which the simulation
did not take into account.

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

Figure 4.8: The yellow lines represent electron trajectories along the beamline within the
model ion source. Trajectories begin at a 5 mm radius extraction aperture
and end at the ﬁrst Faraday cup in ﬁgure 3.2. All electrons are emitted in a
direction perpendicular to the extraction aperture opening, with no angular
divergence. The color of the lines represent their energy. The sudden change
to blue at Faraday results from the -149 V potential on the cup’s electron
suppressor ring.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

.

.

.

.

xiv

Figure 4.9: The relative eﬃciency data (blue dots) approach a linear relationship with
energy as the energy of electrons increases. Each data point was calculated
by dividing the number of electrons emitted from the 6 mm extraction aperture
by the number of electrons which reached the Faraday cup. All electrons in
a given simulation/calculation were assumed to have escaped parallel to the
center axis of the ion source, with no divergence and at a single energy. As
such, these results best represent the eﬀect of the ion source’s and dipole’s
magnetic ﬁeld on the transmission of electrons of a given energy. The ﬁt was
performed with MATLAB’s CFit module and used a ’Robust’ Least Absolute
Residual ﬁtting option. The largest contributor to transmissions losses were
the fringe ﬁelds of the ion source. The charges are deﬂected away from the
beamline center by the locally diverging ﬁeld lines. This eﬀect can be seen
in ﬁgure 4.8.

. .

. .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

Figure 4.10: The magnetic ﬁeld maximum at injection and extraction were chosen to be 2.3
T and 1.1 T, respectively. These values roughly follow the scaling laws with
BInj/BRF ≈ 3.59 and BE xt/BRF ≈ 1.72. Throughout this measurement the
plasma volume and gradient at resonance are allowed to vary. The resonance
ﬁeld is given by the black dashed line. The extraction aperture is represented
by the vertical dashed line at x = 0 cm.

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

. 81

Figure 4.11: Values of the extraction side loss cone pitch angle used while varying the

magnetic minimum while holding the extraction maximum constant.

. . . . . . 82

Figure 4.12: Values of the injection side loss cone pitch angle used, here the magnetic

minimum is held constant. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

Figure 4.13: Values of the extraction side loss cone pitch angle used while varying the

extraction side maximum while holding the magnetic minimum constant. . . .

. 83

Figure 4.14: Values of the radial loss cone pitch angle used while varying the hexapole

with the longitudinal solenoid ﬁelds held constant. . . . . . . . . . . . . . . .

. 84

Figure 4.15: Measured electron distribution for the measurements ’control’ operating
point. The 50 keV reliability threshold is shown by the dashed line. Bmin =
0.4 T, BInj,max = 2.367 T, BExt,max = 1.100 T, Pµ = 350 W, Pressure = 212
nTorr, BRad,wall = 1.217 T. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

Figure 4.16: By applying the eﬃciency calibration described in the previous section, the
total current observed increases by several orders of magnitude, from 10−1−1
nA to 103 nA. The changing vertical gap distance had little eﬀect upon the
distribution once it was corrected for beamline and gap area transmission.
Bmin = 0.4 T, BInj,max = 2.367 T, BExt,max = 1.100 T, Pµ = 350 W, Pressure
= 212 nTorr, BRad,wall = 1.217 T.

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

xv

Figure 4.17: The total measured current was found to vary linearly with a varying square
acceptance area (equal horizontal and vertical gap widths). The red dashed
line demonstrates the linear ﬁt of the data points (blue).

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

Figure 4.18: Increasing the injected microwave power increased the measured electron
current uniformly across the entire energy range. The central energy of the
peak was largely unaﬀected by the changing microwave power.

. . . . . . . . . 87

Figure 4.19: The trend in the measured electron distribution is reﬂected in the bremsstrahlung
spectrum as the total number of photons increases while the energy spread
remains largely the same.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

Figure 4.20: The similarities between the measured electron and bremsstrahlung distribu-
tions is most easy seen in the comparison between the systems’ energy content
and measured particle numbers. Overall, the eﬀect of increasing power leads
to a more dense electron population, but not a more energetic one. All sets
shown have been normalized by the largest values in their sets: Epeak,max =
565 keV, Ts,max = 39.7 keV, XTs,max = 278 keV, Ipeak,max = -1383 nA, and
Nγ = 5.0 Mcounts. .

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

Figure 4.21: The changing electron distribution suggests that increased pressure leads to a

cooler, higher density plasma. . . . . . . . . . . . . . . . . . . . . . . . . . .

. 90

Figure 4.22: As the pressure increases, the measured bremsstrahlung distribution sees a

trend towards lower energy photons.

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

Figure 4.23: While the energy content of both measurements agree to an increased pressure
resulting in a cooler plasma, the trend total particles observed are inversely
proportional to one another. Both top and bottom plots are normalized by the
largest values in their sets: Epeak,max = 592 keV, Ts,max = 42.1 keV, XTs,max
= 297 keV, Ipeak,max = -1224 nA, and Nγ = 5.8 Mcounts.

. . . . . . . . . . . . 92

Figure 4.24: The increasing radial ﬁeld causes the overall current of the distribution to
increase, although this eﬀect is minimal compared to varying the microwave
power. The central energy of the peak is also, largely, unaﬀected by the
changing radial ﬁeld.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

Figure 4.25: The bremsstrahlung distribution was largely unaﬀected by the changing radial
ﬁeld, however, as the radial ﬁeld at the wall increases there is a preference for
the emission of lower energy photons.

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

xvi

Figure 4.26: The energy content of the distributions were largely unaﬀected by a changing
radial ﬁeld strength. The increasing radial ﬁeld causes an increase in the
both the height of the electron peak and total number of photons observed.
Epeak,max = 669 keV, Ts,max = 46.7 keV, XTs,max = 315 keV, Ipeak,max = -1061
nA, and Nγ = 6.3 Mcounts.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

Figure 4.27: The varying injection side magnetic maximum mostly aﬀected the lower

energy distribution of electrons, having minimum eﬀect on the high energy peak. 96

Figure 4.28: The bremsstrahlung distribution was largely unaﬀected by a varying injection

side magnetic maximum.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

Figure 4.29: All operating points were nearly identical to one another. The inverse relation-
ship between the electron peak amplitude and the total number of observed
photons can be seen, however, the largest eﬀect is likely on the lower energy
side of the distribution, as the electron measurement suggests. Epeak,max =
568 keV, Ts,max = 38.6 keV, XTs,max = 281 keV, Ipeak,max = -1357 nA, and
Nγ = 4.9 Mcounts. .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

Figure 4.30: The bias disk current decreased as the injection side magnetic ﬁeld increased,

|IBD,max| = 730 mA.

.

.

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

. 98

Figure 4.31: Increasing the extraction maximum slowly decreases the amplitude of the high
energy peak. Between 1.2 T and 1.23 T the change in peak amplitude increases
and results in a dramatic decrease in peaks height. These distributions have not
been corrected for changes in the transmission coeﬃcient due to a changing
extraction ﬁeld maximum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

Figure 4.32: The bremsstrahlung distribution remains largely unchanged as a result of a
changing extraction ﬁeld maximum, up to a point. Between 1.2 T and 1.23
T the number of intermediate to low energy photons observed decreases dra-
matically. These spectra have not been corrected for changes in the beamline
transmission coeﬃcient which result from a varying ﬁeld maximum at extraction.101

Figure 4.33: While the total number of photons observed decreases only slightly as the
extraction maximum is increased, the sudden sudden drop-oﬀ in extracted
beam current reduces the number of photons observed by more than 50%.
No appreciable change in the system’s energy content is observed. Epeak,max
= 585 keV, Ts,max = 38.6 keV, XTs,max = 270 keV, Ipeak,max = -1756 nA, and
Nγ = 4.9 Mcounts. Each data set has been normalized by the largest value
in their respective sets. The normalized electron peak amplitude has been
corrected for changes in the beamline transmission coeﬃcient which result
from a varying extraction maximum.

. . . . . . . . . . . . . . . . . . . . . . . 102

xvii

Figure 4.34: There is an obvious change in the energy content of the measured electron
distribution resulting from a changing magnetic minimum. While the central
energy of the peak increased, the height of the peak decreased.

. . . . . . . . . 103

Figure 4.35: Just as in ﬁgure 4.34, an increasing magnetic minimum pushes the photon
distribution to higher energies. The total number of counts, however, increases
where the electron peak’s height decreases. This suggests that overall electron
density increases, even if the measured electron current decreases.

. . . . . . . 104

Figure 4.36: Both the electron and bremsstrahlung distributions agree to a near linear
increased in the energy content of the plasma. The spectral temperature lags
behind due to the inﬂuence of lower energy electrons. Epeak,max = 742 keV,
Ts,max = 56.47 keV, XTs,max = 359 keV, Ipeak,max = -1916 nA, and Nγ =
6.7 Mcounts. Each data set has been normalized by the largest value in their
respective sets. .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

. .

Figure 4.37: Maximum and minimum energy limits for ﬁtting the spectral temperature as

the magnetic ﬁeld is varied.

. . . . . . . . . . . . . . . . . . . . . . . . . . . 105

Figure 4.38: There is a linear correlation between the decreasing trend of the height of the
high energy peak and the reciprocal square of the central energy of that peak.
Iγ/Iγ,max = 4.897 × γ−2 − 0.4523, R2 = 0.9938.

. . . . . . . . . . . . . . . . 106

Figure 4.39: The magnetic ﬁeld lines (blue arrows), within the acceptance area of the
extraction aperture (red dashed lines), are primarily parallel to the longitudinal
axis of the plasma chamber.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

Figure 4.40: There is a near-perfect inverse correlation between the trends of the decreasing
height of the high energy peak and the increasing number of photons observed
Ie/Ie,max = −0.9998 × Nγ/Nγ,max − 1.37, R2 = 0.9888 . . . . . . . . . . . . . 108

Figure 4.41: Bremsstrahlung distributions analyzed using the algorithm described in sec-
tion 3.2.5 show much better agreement with the electron distribution data than
those where an arbitrary energy domain was blindly applied to all operating
points. In this case, the disagreement is larger for operating points with lower
magnetic ﬁelds. Top plot: Bmin = 0.337 T. Bottom plot: Bmin = 0.468 T.

. . . . 112

Figure 5.1: Example Ar8+ current (orange) and microwave power (purple) diagnostics
while operating a stable ECR ion source plasma. The signals have been
normalized to their maximum value over a 400 ms period. The extracted
Ar8+ beam varies up to several percent even under stable conditions, in this
case δIAr8+ ∼ 7%. The 60 Hz noise, and its harmonics, in the microwave
signal results from the facility power source and is persistent throughout the lab. 117

xviii

Figure 5.2: Multiple measurements of the transient current of escaping electrons, at dif-
ferent energies, were taken using an oscilloscope. The energy of electrons
was selected by varying the energy in the dipole. The traces were then aver-
aged 128 using the oscilloscopes averaging function. Those traces were put
together, side-by-side, to plot the transient electron energy distribution. The
peak current is 8460 nA at an electron energy of 62.88 keV, almost twice the
central energy of the ’low energy peaks’ seen in chapter 4. Higher energy
electrons are minimally eﬀected by the instability and the high energy peak
is unaﬀected by the instability.

. . . . . . . . . . . . . . . . . . . . . . . . . . 118

Figure 5.3: The general semi-quantitative model of the transient ion current during the
unstable operation of an ECR ion source. A1, A2, and A3 represent the
amplitudes of the three predominant period of beam current variations that
result from an instability event. The time scales are meant to represent general
ranges over which the variation occurs. Not every instability event will result
in all three and, under most circumstances, there is one predominant feature
which best describes the beam. However, under rare circumstances, all three
can occur simultaneously.

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

. 119

Figure 5.4: An example of the observed time resolved beam current (orange) and mi-
crowave power (purple) diagnostics seen during a transient proﬁle dominate
by the ’ion burst’ eﬀect. Notice how there is a large but short-lived burst in
ions immediately after the burst of microwave energy. While present, the ’ion
loss’ proﬁle is not dominant. The majority of the microwave power is emitted
over a several milliseconds. Ar8+, Bmin = 0.39 T, BHex,wall = 1.2 T, Pµ = 150
W, pAr = 131 nTorr.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

.

Figure 5.5: An example of the observed time resolved beam current (orange) and mi-
crowave power (purple) diagnostics seen during a transient proﬁle dominate
by the ’ion loss’ eﬀect. Notice that almost 50% of the steady state current is
lost following the microwave burst. While present, the ’ion burst’ proﬁle is
not dominant, and can only be seen by focusing on the 100’s µs time scale
(bottom plot). The majority of the microwave power is emitted over 10’s µs
and very rapidly decays. The ’hump’ in the emitted power is believed to result
from a changing reﬂection coeﬃcient within the plasma chamber. Ar8+, Bmin
= 0.48 T, BHex,wall = 1.2 T, Pµ = 250 W, pAr = 131 nTorr.

. . . . . . . . . . . . 122

xix

Figure 5.6: An example of the observed time resolved beam current (orange) and mi-
crowave power (purple) diagnostics seen during a transient proﬁle dominate
by the ’over-extraction’ eﬀect. Extracted Notice the almost 30% increase in
extracted beam current following the exceptionally small burst of microwave
power from the system. While present, the ’ion loss’ proﬁle is not dominant.
Most importantly, this repetition frequency of this transient proﬁle is on the
order of 10’s Hz, compared to ﬁgures 5.4 and 5.5 which are on the order
of 100’s Hz (top plot). The emitted microwave power is exceptionally small
compared to the previous two cases (bottom plot). The ’hump’ in the emitted
power is believed to result from a changing reﬂection coeﬃcient within the
plasma chamber. Ar8+, Bmin = 0.48 T, BHex,wall = 1.32 T, Pµ = 350 W, pAr
= 131 nTorr.

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

Figure 5.7:

Figure 5.8:

’Ion burst’ dominated transient proﬁles are characterized by an increase in
the average extracted current at low charge states (Ar2+, blue, top trace) and a
decreased in the average extracted current for high charge states (Ar8+, orange,
bottom trace). Both current traces have been normalized to the largest value
in their respective scope traces. Bmin = 0.45 T, BHex,wall = 1.2 T, Pmu = 50
W, pAr = 141 nTorr, VBD = -17 V. . . . . . . . . . . . . . . . . . . . . . . . . . 124

’Ion loss’ dominated transient proﬁles are characterized by an increase in
the average extracted current at low charge states (Ar2+, Blue, bottom trace)
and a decreased in the average extracted current for high charge states (Ar8+,
orange, top trace). The plasma is also perturbed to a greater degree, compared
to the ’ion burst’ case. High charge state ion losses can be upwards of 80 -
100% of steady state beam currents, with much larger bursts in current for
low charge states. Both current traces have been normalized to the largest
value in their respective scope traces. Bmin = 0.52 T, BHex,wall = 1.2 T, Pmu
= 350 W, pAr = 123 nTorr, VBD = -17 V.

. . . . . . . . . . . . . . . . . . . . . 125

Figure 5.9: Unlike the other two cases, ’over-extraction’ dominated transient proﬁles are
characterized by a decrease in the average extracted current at low charge
states (Ar2+, blue, top trace) and an increase in the average extracted current
for high charge states (Ar8+, orange, top trace). Both current traces have been
normalized to the largest value in their respective scope traces. Bmin = 0.42
T, BHex,wall = 1.2 T, Pmu = 250 W, pAr = 131 nTorr, VBD = 20 V.

. . . . . . . . 126

Figure 5.10: Bmin = 0.42 T, BHex,wall = 1.2 T, Pµ = 350 W, pAr = 131 nTorr, VBD = +20 V.
Figure 5.11: Bmin = 0.42 T, BHex,wall = 1.2 T, Pµ = 350 W, pAr = 131 nTorr. VBD = +20 V . 127
Figure 5.12: Increasing the neutral gas pressure often changes the dominant transient mode
of the extracted beam current during unstable operating. Bmin = 0.515 T,
BHex,wall = 1.2 T, Pmu = 350 W. (Top) pAr = 96 nTorr, (Middle) pAr = 123
nTorr, (Bottom) pAr = 159 nTorr.

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

. 126

xx

Figure 5.13: Field maps for operating points where the average injection and extraction
maxima were above the limits described by the scaling laws: BInj,max/BRF ≈
4.09, BE xt,max/BRF ≈ 2.02.

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

Figure 5.14: Field maps for operating points where the average injection and extraction
maxima were above the limits described by the scaling laws: BInj,max/BRF ≈
3.71, BE xt,max/BRF ≈ 1.78.

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

Figure 5.15: Field maps for operating points where the average injection and extraction
maxima were above the limits described by the scaling laws: BInj,max/BRF ≈
3.43, BE xt,max/BRF ≈ 1.95.

. . . . . . . . . . . . . . . . . . . . . . . . . . . 130

Figure 5.16: Field maps for operating points where the average injection maximum was
larger than and extraction maxima small than the limits described by the
scaling laws: BInj,max/BRF ≈ 4.11, BE xt,max/BRF ≈ 1.61. . . . . . . . . . . . 130
Figure 5.17: Percentage of measured operating points with average currents greater than 60µA.132

Figure 5.18: Percentage of measured operating points with average currents greater than 80µA.132

Figure 5.19: Percentage of measured operating points with average currents greater than

.

.

.

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

. 133

100µA. .

.

.

.

.

Figure 5.20: Percentage of measured operating points where beam loss was greater than 30%. 133

Figure 5.21: Percentage of measured operating points where beam loss was greater than 60%. 134

Figure 5.22: Percentage of measured operating points where beam loss was greater than 80%. 134

Figure 5.23: Percentage of measured operating points with peak currents greater than 60µA. 135

Figure 5.24: Percentage of measured operating points with peak currents greater than 80µA. 136

Figure 5.25: Percentage of measured operating points with peak currents greater than 100µA. 136
Figure 5.26: Ar8+ beam current for a varying hexapole ﬁeld maxima: (Top left) BHex,wall
= 1.12 T, (Top right) BHex,wall = 1.18 T, (Bottom left) BHex,wall = 1.22 T,
(Bottom right) BHex,wall = 1.28 T. Bmin = 0.4 T, Pµ = 350 W, pAr = 213 nTorr.
Signals have been treated with a digital low pass ﬁlter using the Matlab signal
processing toolkit, with a cutoﬀ frequency of 333 kHz.

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

. 138

Figure 5.27: Average, maximum, and minimum beam current measurements of extracted
Ar8+ beam current for varying hexapole ﬁeld maxima: Bmin = 0.4 T, Pµ =
350 W, pAr = 213 nTorr. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

xxi

Figure 5.28: Microwave power signals for a varying hexapole ﬁeld: (Top left) BHex,wall
= 1.12 T, (Top right) BHex,wall = 1.18 T, (Bottom left) BHex,wall = 1.22 T,
(Bottom right) BHex,wall = 1.28 T. Bmin = 0.4 T, BExt,max = 1.1 T, Pµ = 350
W, pAr = 213 nTorr. Microwave power has been oﬀset by it’s minimum value,
over the entire 100 ms measurement period, to account for the AC coupling
of the oscilloscope. Signals have been treated with a digital low pass ﬁlter
using the Matlab signal processing toolkit, with a cutoﬀ frequency of 333 kHz. . 140

Figure 5.29: Fourier transform of the microwave power signals emitted from the ion source
for a varying hexapolar ﬁeld:
(Top left) BHex,wall = 1.12 T, (Top right)
BHex,wall = 1.18 T, (Bottom left) Bhex,max = 1.22 T, (Bottom right) Bhex,max
= 1.28 T. Bmin = 0.4 T, BExt,max = 1.1 T, Pµ = 350 W, pAr = 213 nTorr. . . . . . 141

Figure 5.30: Average and maximum peak microwave power for a varying hexapole ﬁeld

maxima: Bmin = 0.4 T, BExt,max = 1.1 T, Pµ = 350 W, pAr = 213 nTorr.

. . . . 141

Figure 5.31: Ar8+ beam current for a varying injection ﬁeld maxima: (Top Left) BInj,max
= 2.20 T, (Top Right) BInj,max = 2.33 T, (Bottom Left) BInj,max = 2.45 T,
(Bottom Right) BInj,max = 2.58 T. Bmin = 0.4 T, Pµ = 350 W, pAr = 213 nTorr.
Signals have been treated with a digital low pass ﬁlter using the Matlab signal
processing toolkit, with a cutoﬀ frequency of 333 kHz.

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

. 143

Figure 5.32: Average, maximum, and minimum Ar8+ beam currents, measured over a 100
ms period, for a varying injection ﬁeld maxima: Bmin = 0.4 T, Pµ = 350 W,
pAr = 213 nTorr.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

.

.

.

Figure 5.33: Microwave power signals for a varying injection ﬁeld maxima: (Top Left)
BInj,max = 2.20 T, (Top Right) BInj,max = 2.33 T, (Bottom Left) BInj,max =
2.45 T, (Bottom Right) BInj,max = 2.58 T. Bmin = 0.4 T, Pµ = 350 W, pAr
= 213 nTorr. Microwave power has been oﬀset by it’s minimum value, over
the entire 100 ms measurement period, to account for the AC coupling of the
oscilloscope. Signals have been treated with a digital low pass ﬁlter using the
Matlab signal processing toolkit, with a cutoﬀ frequency of 333 kHz.

. . . . . . 144

Figure 5.34: Fourier transform of the microwave power signals emitted from the ion source
for a varying injection ﬁeld maximum: (Top Left) BInj,max = 2.20 T, (Top
Right) BInj,max = 2.33 T, (Bottom Left) BInj,max = 2.45 T, (Bottom Right)
BInj,max = 2.58 T. Bmin = 0.4 T, Pµ = 350 W, pAr = 213 nTorr. . . . . . . . . . . 145

Figure 5.35: Average and maximum output microwave power, measured over a 100 ms
period, for a varying injection side ﬁeld maximum: Bmin = 0.4 T, Pµ = 350
W, pAr = 213 nTorr.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

.

xxii

Figure 5.36: Ar8+ beam current for a varying extraction ﬁeld maxima: (Top Left) BExt,max
= 0.99 T, (Top Right) BExt,max = 1.07 T, (Bottom Left) BExt,max = 1.15 T,
(Bottom Right) BInj,max = 1.23 T. Bmin = 0.4 T, Pµ = 350 W, pAr = 213 nTorr.
Signals have been treated with a digital low pass ﬁlter using the Matlab signal
processing toolkit, with a cutoﬀ frequency of 333 kHz.

. . . . . . . . . . . . . 147

Figure 5.37: Average, maximum, and minimum Ar8+ beam currents, measured over a 100
ms period, for a varying extraction ﬁeld maxima: Bmin = 0.4 T, Pµ = 350 W,
pAr = 213 nTorr. The dashed vertical line show where BExt,max/BRF = 1.8.

. .

. 148

Figure 5.38: Trends of the average and standard deviations of the average extracted Ar8+
beam current as the extraction ﬁeld maximum is varied, normalized to the
largest values in their respective sets: Bmin = 0.4 T, Pµ = 350 W, pAr = 213
nTorr. The vertical dashed line shows where BExt,max/BRF = 1.8.

. . . . . . . . 149

Figure 5.39: Microwave power signals for a varying extraction ﬁeld maxima: (Top Left)
BExt,max = 0.99 T, (Top Right) BExt,max = 1.07 T, (Bottom Left) BExt,max =
1.15 T, (Bottom Right) BInj,max = 1.23 T. Bmin = 0.4 T, Pµ = 350 W, pAr
= 213 nTorr. Microwave power has been oﬀset by it’s minimum value, over
the entire 100 ms measurement period, to account for the AC coupling of the
oscilloscope. Signals have been treated with a digital low pass ﬁlter using the
Matlab signal processing toolkit, with a cutoﬀ frequency of 333 kHz.

. . . . . . 150

Figure 5.40: Fourier transform of the microwave power signals emitted from the ion source
for a varying extraction ﬁeld maxima: (Top Left) BExt,max = 0.99 T, (Top
Right) BExt,max = 1.07 T, (Bottom Left) BExt,max = 1.15 T, (Bottom Right)
BInj,max = 1.23 T. Bmin = 0.4 T, Pµ = 350 W, pAr = 213 nTorr. . . . . . . . . . . 151

Figure 5.41: Maximum output microwave power for a varying extraction ﬁeld maxima:
Bmin = 0.4 T, Pµ = 350 W, pAr = 213 nTorr. The dashed vertical line shows
where BE xt/BRF = 1.8.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

Figure 5.42: Average and maximum output microwave power, measured over a 100 ms
period, for a varying extraction ﬁeld maxima: Bmin = 0.4 T, Pµ = 350 W, pAr
= 213 nTorr. The dashed vertical line shows where BE xt/BRF = 1.8.

. . . . . . 152

Figure 5.43: Ar8+ beam current for varying magnetic minimums (Top left) Bmin = 0.34
T, (Top right) Bmin = 0.35 T, (Bottom left) Bmin = 0.37 T, (Bottom Right)
Bmin = 0.4 T. Pµ = 350 W, pAr = 213 nTorr. Signals have been treated with a
digital low pass ﬁlter using the Matlab signal processing toolkit, with a cutoﬀ
frequency of 333 kHz.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

xxiii

Figure 5.44: Ar8+ beam current for varying magnetic minimums (Top left) Bmin = 0.43 T,
(Top right) Bmin = 0.44 T, (Bottom left) Bmin = 0.46 T, (Bottom Right) Bmin
= 0.47 T. Pµ = 350 W, pAr = 213 nTorr. Signals have been treated with a
digital low pass ﬁlter using the Matlab signal processing toolkit, with a cutoﬀ
frequency of 333 kHz.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

Figure 5.45: Average, maximum, and minimum Ar8+ beam currents, measured over a 100
ms period, for a varying magnetic minimum. (Top left) Bmin = 0.43 T, (Top
right) Bmin = 0.44 T, (Bottom left) Bmin = 0.46 T, (Bottom Right) Bmin =
0.47 T. Pµ = 350 W, pAr = 213 nTorr

. . . . . . . . . . . . . . . . . . . . . . . 156

Figure 5.46: Ar8+ beam current for varying magnetic minimums (Top left) Bmin = 0.34 T,
(Top right) Bmin = 0.35 T, (Bottom left) Bmin = 0.37 T, (Bottom Right) Bmin
= 0.4 T. Pµ = 350 W, pAr = 213 nTorr. Microwave power has been oﬀset by
it’s minimum value, over the entire 100 ms measurement period, to account
for the AC coupling of the oscilloscope. Signals have been treated with a
digital low pass ﬁlter using the Matlab signal processing toolkit, with a cutoﬀ
frequency of 333 kHz.

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

. 157

Figure 5.47: Ar8+ beam current for varying magnetic minimums (Top left) Bmin = 0.43 T,
(Top right) Bmin = 0.44 T, (Bottom left) Bmin = 0.46 T, (Bottom Right) Bmin
= 0.47 T. Pµ = 350 W, pAr = 213 nTorr. Microwave power has been oﬀset by
it’s minimum value, over the entire 100 ms measurement period, to account
for the AC coupling of the oscilloscope. Signals have been treated with a
digital low pass ﬁlter using the Matlab signal processing toolkit, with a cutoﬀ
frequency of 333 kHz.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

Figure 5.48: Fourier transform of the microwave power signals emitted from the ion source
for a varying magnetic minimums (Top left) Bmin = 0.43 T, (Top right) Bmin
= 0.44 T, (Bottom left) Bmin = 0.46 T, (Bottom Right) Bmin = 0.47 T. Pµ =
350 W, pAr = 213 nTorr . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 5.49: Fourier transform of the microwave power signals emitted from the ion source
for a varying magnetic minimums (Top left) Bmin = 0.43 T, (Top right) Bmin
= 0.44 T, (Bottom left) Bmin = 0.46 T, (Bottom Right) Bmin = 0.47 T. Pµ =
350 W, pAr = 213 nTorr . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. 159

. 159

Figure 5.50: Average and maximum output microwave power, measured over a 100 ms
period, for a varying magnetic minimum.
(Top left) Bmin = 0.43 T, (Top
right) Bmin = 0.44 T, (Bottom left) Bmin = 0.46 T, (Bottom Right) Bmin =
0.47 T. Pµ = 350 W, pAr = 213 nTorr

. . . . . . . . . . . . . . . . . . . . . . . 160

xxiv

Figure 5.51: The exponential decay time of the ’over-extracted’ current scales roughly as
q−2. The deviation can result from non-linear eﬀects in the plasma, such as
continuous heating, charge exchange, and ionizing collisions. BHex,wall = 1.2
T, pAr = 131 nTorr, For the red curve: Pµ = 350 W. For the green curve: Pµ
= 250 W.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

.

.

.

.

.

.

.

Figure B.1: Measured electron distribution for the measurements ’control’ operating
point. The 50 keV reliability threshold is shown by the dashed line. Bmin =
0.4 T, BInj,max = 2.367 T, BExt,max = 1.100 T.

. . . . . . . . . . . . . . . . . . 177

Figure B.2: Measured electron distribution for the measurements ’control’ operating
point. The 50 keV reliability threshold is shown by the dashed line. Bmin =
0.4 T, BInj,max = 2.367 T, BExt,max = 1.100 T.

. . . . . . . . . . . . . . . . . . 178

Figure B.3: Measured electron distribution for the measurements ’control’ operating
point. The 50 keV reliability threshold is shown by the dashed line. Bmin =
0.4 T, BInj,max = 2.367 T, BExt,max = 1.100 T.

. . . . . . . . . . . . . . . . . . 178

Figure C.1: A varying extraction voltage has little eﬀect on the hot-electron bremsstrahlung
distribution. The count rate and bremsstrahlung are much more stable than in
ﬁgure 4.1 due to the much more consistent background during the measure-
ment. The stability of the nuclear peak at 510 keV demonstrates this most
clearly. Plasma parameters are Bmin = 0.397 T, P = 350 W, Tlive = 600 s,
pressure = 212 nTorr.

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

Figure C.2: A varying magnetic ﬁeld does not appreciably change how the injected mi-

crowave power aﬀects the measured electron distributions. . . . . . . . . . . . . 180

Figure C.3: Similar to ﬁgure C.2, a varying magnetic ﬁeld does not appreciably change
how the injected microwave power aﬀects the measured bremsstrahlung dis-
tributions.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

.

.

.

.

.

.

.

Figure C.4: All ﬁeld distributions agree to a linear increase in the number of measured
particles with respect to the microwave power. Although a small change in
the spectral temperature is seen as the injection side magnetic ﬁeld is varied,
the overall trend indicates an insensitivity of electron energy to microwave power.181

Figure C.5: All ﬁeld distributions agree to a close to linear increase in the number of
measured particles with respect to the microwave power. Although a small
change in the spectral temperature is seen as the injection side magnetic ﬁeld
is varied, the overall trend indicates an insensitivity of electron energy to
microwave power.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

.

.

xxv

Figure C.6: The dependence of the high energy electron distribution on pressure was not
appreciably aﬀected by a varying longitudinal ﬁeld proﬁle. Although out of
the scope of this study, it should be recognized that low energy portion of
the distribution (Ekin < 70keV) is aﬀected by the both the pressure and the
. . . . . . . . . . . . . . .
average magnetic ﬁeld within the source chamber.

. 182

Figure C.7: The dependence of the bremsstrahlung distribution on pressure was not ap-

preciably aﬀected by a varying longitudinal ﬁeld proﬁle. . . . . . . . . . . . . . 183

Figure C.8: All measurements agreed to the same overall trend, regardless of longitudinal

ﬁeld proﬁle.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

.

.

.

.

.

.

.

.

.

.

.

.

Figure C.9: All measurements agreed to the same overall trend, regardless of longitudinal

ﬁeld proﬁle.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

Figure C.10: There was a greater eﬀect upon the measured electron distribution’s height as
a function of the radial magnetic ﬁeld maximum as the magnetic minimum
increased.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

.

.

.

.

.

.

.

Figure C.11: The longitudinal magnetic ﬁeld had no eﬀect on the measured bremsstrahlung

distribution for high energy photons.

. . . . . . . . . . . . . . . . . . . . . . . 185

Figure C.12: Comparison between the electron peak energy/intensity and bremsstrahlung
spectral temperature/total photon counts for across multiple ion source lon-
gitudinal magnetic ﬁeld proﬁles with varying radial ﬁeld maximum.

. . . . .

. 185

Figure C.13: Comparison between the electron peak energy/intensity and bremsstrahlung
spectral temperature/total photon counts for across multiple ion source lon-
gitudinal magnetic ﬁeld proﬁles with varying radial ﬁeld maximum.

. . . . .

. 186

Figure D.1: Measured electron distribution for the measurements ’control’ operating
point. The 50 keV reliability threshold is shown by the dashed line. Bmin =
0.4 T, BInj,max = 2.367 T, BExt,max = 1.100 T.

. . . . . . . . . . . . . . . . . . 187

Figure D.2: Measured electron distribution for the measurements ’control’ operating
point. The 50 keV reliability threshold is shown by the dashed line. Bmin =
0.4 T, BInj,max = 2.367 T, BExt,max = 1.100 T.

. . . . . . . . . . . . . . . . . . 187

Figure D.3: Measured electron distribution for the measurements ’control’ operating
point. The 50 keV reliability threshold is shown by the dashed line. Bmin =
0.4 T, BInj,max = 2.367 T, BExt,max = 1.100 T.

. . . . . . . . . . . . . . . . . . 188

Figure D.4: Measured electron distribution for the measurements ’control’ operating
point. The 50 keV reliability threshold is shown by the dashed line. Bmin =
0.4 T, BInj,max = 2.367 T, BExt,max = 1.100 T.

. . . . . . . . . . . . . . . . . . 188

xxvi

CHAPTER 1

INTRODUCTION

The ability to produce and accelerate charged particles has become an invaluable tool in many
industries worldwide. Nuclear and particle physics research facilities, medical treatment facilities
(such as proton and carbon therapy facilities), and semiconductor fabricators all use energetic
electrons and ions throughout their day to day operations [7]. As such, the research and development
of ion sources are essential for many scientiﬁc and engineering applications.

Many nuclear and particle physics research facilities rely upon producing high intensity, high
charge state ions for their operations. Accelerating highly charged ion beams is overall more
eﬃcient, as the electric force that a particle feels is directly proportional to its charge. Furthermore,
the production of large currents increases the probability of producing and meaningfully observing
rare isotopes. In many cases, scientists and engineers will turn to the Electron Cyclotron Resonance
Ion Source (ECRIS) for their facility’s needs.
Invented in 1974 by Richard Geller, the devices
quickly proved themselves to be powerful and reliable sources of highly charged ions [7]. The most
modern and advanced of these sources can produce 100’s eµA of uranium ion beams for use in
accelerator facilities. For example, the VENUS and SECRAL ion sources can produce up to 320
eµA of Xe20+ [42] and 360 eµA of Xe30+ [101], respectively.

The guiding principle of the ECRIS is rather straightforward:

ignite and heat a magnetized
plasma and then extract ions produced therein. Microwaves heat the plasma’s electrons through the
process of Electron Cyclotron Resonance (ECR). This process occurs as electromagnetic radiation
interacts with electrons gyrating about magnetic ﬁeld lines. The transferred energy is, principally,
stored in the electrons’ gyromotion. With stronger magnetic ﬁelds, the maximum stored energy
increases. The heated electrons can then inelastically collide with atoms and low charge state
ions to bring about electron impact ionization. Through this mechanism, and with large enough
magnetic ﬁelds, it is possible to strip even the heaviest of ions of most, if not all, of their orbital
electrons.

1

Nevertheless, behind this straight forward approach lie complications capable of crippling ion
source, and facility, operations. Under large enough magnetic ﬁelds, the electrons within the system
can suddenly release all the energy stored within their rotational motion. These kinetic instabilities,
caused by imbalances in the temperature distributions of electrons throughout the plasma, cause
sudden losses of extracted ion currents [81]. Overall, the instabilities limit the performance of ECR
ion sources.

As beam current stability is of vital importance for many ion source applications, these plasma
instabilities create limitations on ECR ion sources’ performance. This study seeks to better
understand these limiting plasma instabilities by studying the ion source plasma’s response to
various ranges of source parameters. Plasma and beam current diagnostics gathered data across a
series of measurements of both stable and unstable plasmas. This data provided insight into the
diﬀusion mechanisms of plasma electrons and ways to optimize the ion source’s performance.

This discussion in this thesis is broken down across ﬁve chapters.

It begins in chapter 2
with a discussion of ECR ion source plasmas, including the underlying plasma physics, operating
principles, and microinstabilites that cause losses of high energy electrons. Chapter 3 discusses
the experimental apparatuses and procedures used throughout this study. Chapters 4 and 5 discuss
the results of these measurements. Chapter 6 concludes this thesis by brieﬂy discussing the
ramiﬁcations of the results of these measurements on our understanding of ECRIS operations.

2

CHAPTER 2

ELECTRON CYCLOTRON RESONANCE ION SOURCES

2.1 Plasma Physics

Before entering the heart of this study (or even just its introduction), it is important to understand
some underlying concepts from plasma physics. Many authors have described and derived the
critical aspects of plasma physics on a fundamental level (see [5], [44]). Others have followed
in Geller’s [18] footsteps and have written speciﬁcally on aspects of plasma physics relevant to
ECR ion sources [88] or ECR heating in general (see [22, 29, 43, 59]) . As such, the good work
of others is referenced for in-depth discussions of plasma eﬀects and interactions in ECR heating
environments, and in general. The focus of this section is to reduce down their work to the material
most relevant for the discussion at hand.

A plasma is the so-called fourth state of matter. It is a state where, as most simply described
by Lieberman [44], is "a collection of free charged particles moving in random directions that is,
on average, electrically neutral." As a gaseous substance gains energy, it can eventually break apart
into a collection of electrons and ions. Although broken up into individual charges, the material as
a whole remains neutral. Magnetic ﬁelds can ensure that the system stays conﬁned to one region
of space. However, the individual charges remain unbounded within the plasma. As such, they
oscillate about ﬁeld lines and undergo multiple collisions. The collisions can lead to diﬀusion along
and across magnetic ﬁeld lines. The high mobility of the unbound charges gives the plasma unique
electronic characteristics; along its surface and through its volume. These unique characteristics
give rise to a variety of wave-plasma interactions, both globally and for individual particles.

The remainder of this section focused on the building blocks of plasma physics, which underline
ECR ion sources’ operation. We will begin by discussing magnetic conﬁnement, oscillations and
collisions, and then diﬀusion. The last two sections focus speciﬁcally on plasma ﬁeld interactions;
through discussions of plasma permittivity and then plasma wave propagation.

3

2.1.1 Magnetic Conﬁnement

Electrostatics do not provide stable conﬁnement of charged particles; proper conﬁnement requires
a magnetic ﬁeld.
In particular, the magnetic mirror is a simple and straightforward method of
conﬁning charged particles within a region of space. A charged particle acquires a magnetic
moment, | (cid:174)M| = I A, as it gyrates about a magnetic ﬁeld line. The magnetic moment is an invariant
of motion and is conserved [5]. As such:

(cid:174)M = −W⊥
B2

(cid:174)B = const,

(2.1)

where W⊥ = 1

2mv
magnetic ﬁled strength.

2⊥ is the energy of the perpendicular motion of the charge and B is the local

As the charged particle moves through space, towards increasing magnetic ﬁelds, it will see
its transverse energy increase to keep its magnetic moment and total energy constant. It does this
by decreasing its energy parallel to the magnetic ﬁeld lines, W(cid:107). A large enough magnetic ﬁeld
can convert all the particle’s longitudinal energy to rotational energy. At this point, the electron
will reverse direction and move along the external magnetic ﬁeld lines towards a region with lower
magnetic ﬂux. That ﬁeld maximum causes the electron to "reﬂect" backward into regions with
lower magnetic ﬁelds. We call that magnetic ﬁeld Bmirr. This, of course, does require there to be
a magnetic ﬁeld maximum capable of causing the mirroring eﬀect. If the longitudinal momentum
of the charge is too large, then the particle can penetrate through the magnetic maximum.

Now suppose we co-axially align two magnetic mirrors with some spatial separation, as seen
in ﬁgure 2.1. By combining the requirements of conservation of energy and magnetic moment, it
can be shown that a charged particle will remain conﬁned if:

Bmin
Bmax

=

1
R >

2⊥
2 = sin2

v

v

α0

(2.2)

where, Bmin is the minimum magnetic ﬁeld the particle interacts with, Bmax is the maximum
magnetic ﬁeld strength in the system (representing the last place where the particle can reverse
direction before escaping), R is the so-called mirror ratio, and α0 is the pitch angle deﬁning the

4

Figure 2.1: The series of solenoids generates a semi-uniform magnetic ﬁeld. The ﬁeld strength
reaches a minimum at some point between the centers of the coils, represented by the less dense
ﬁeld lines (top). The bottom left plot shows an example ﬁeld proﬁle from the center of the above
conﬁguration to the center of the last coil. The bottom right plot shows an example loss cone.

system’s loss cone. If the particle exists within the cone, then it is capable of escaping from the
plasma. However, any particles outside of the loss cone will remain conﬁned until they are scattered
into the loss cone [68]. This conﬁnement conﬁguration is known as the magnetic bottle.

The formalism for the magnetic bottle does a good job describing conﬁnement on a particle-by-
particle basis. Although it is sometimes useful to think about conﬁnement from the thermodynamic

5

perspective. To that end, we can describe the macroscopic stability of the plasma by [5]:

β =

nekBTe

B2
0
2µ0

(2.3)

where, kB is Boltzmann’s constant, Te is the electron distribution temperature, and B0 is the ﬁeld
strength at the plasma boundary. The parameter β is the ratio of plasma kinetic pressure to the
magnetic pressure at the boundary of the plasma. So long as β « 1 the plasma will be conﬁned.
At higher values, the kinetic pressure overtakes the magnetic pressure, and the plasma becomes
macroscopically unstable. This method of analyzing conﬁnement does not provide insight into the
particle-ﬁeld interaction. It is, however, very useful from a global perspective. Particularly when
it is diﬃcult to ascertain details about the individual particles in a system.

2.1.2 Oscillations and Collisions

Small electrostatic perturbations within the plasma bring about spatial oscillations of charges. The
charged particles oscillate with what is called the plasma frequency:

(cid:115)

ωp =

nq2
m0

(2.4)

where n is the charged particle density, q and m are the particle’s charge and mass, respectively,
and 0 is the permittivity of free space. In most cases, it is the lighter electrons which oscillate
around the heavier ions (mp/me ∼ 2000). As such, the electron-plasma frequency, ωpe is often
used as the plasma frequency.

As described by Lorentz’s force law, charged particles moving perpendicular to magnetic ﬁeld
lines will feel a force that causes oscillation around the ﬁeld lines. The frequency of oscillation
about the magnetic lines of force is known as the particle’s cyclotron frequency:

ωc = 2π fc =

qB
γm

=

Ωc
γ

(2.5)

where B is the local magnetic ﬁeld strength, γ is the Lorentz factor: (1 − v
2 , where c is the
speed of light in vacuum, and Ωc is the classical gyro frequency (sometimes called the ’cold’ gyro

2/c2)−1

6

frequency). It is handy to know that the classical electron gyro frequency is well approximated by
the relation:

fce(GHz) ≈ 28B.

where B is the magnetic ﬁeld in T. Likewise, the classical ion cyclotron frequency can be approxi-
mated by:

fci(MHz) ≈ 3.8qB
A

where q and A are the charge state and mass number, respectively.

The collision frequency describes how often, on average, a single particle will collide with

another particle. In general it is given by [64]:

νc = na(cid:104)σvb(cid:105) =

na
nb

Þ

vb fb(vb)σd3

v

(2.6)

where na is the target particle density, nb is the incoming particle density, σ is the cross-section of
interaction for a particular type of collision or interaction, fb is the velocity distribution function
of the incoming particles, and vb is the incoming particle velocity. The rate of collisions per unit
volume is then given by:

Rc = nbνc = nbna(cid:104)σvb(cid:105).

(2.7)

We can apply equation 2.6 to various diﬀerent types of collisions. These include electron-
electron (ee) collisions, electron-ion (ei) collisisons, ion-ion (ii), and neutral-electron (ne) or
neutral-ion (ni) collisions. For the moment, I will discuss electron-electron and ion-ion collisions.
A discussion on electron-ion and ion-neutral collisions is saved for section 2.2.2. Electron-electron
and ion-ion collisions are both elastic collisions dictated by the coulomb interaction.

For electrons, this interaction can be approximated by the frequency of collisions which cause

the electrons to deﬂect at a 90°angle [21]:

90

ee(cid:105) ≈ nee4 ln Λ
(cid:104)ν
3/2
1/2
e T
e

0m

(2.8)

where Te is the electron temperature in eV and ln Λ is the so-called coulomb logarithm. It can be
shown that Λ ∼ neλD [21]. It is usually assumed that ln Λ ≈ 10 − 12 in ECRIS plasmas. The

7

ion-ion collision rate can be similarly calculated by [53]:

(cid:104)νi j(cid:105) ≈ 6.8 × 10−8 ln Λi j

3/2
T
i

q2
i
Ai


j

(cid:113)

Aj


q

j q2
nq

j

where the ’ij’ indices indicate the collisions of the moving ion ’i’ on the target ion of species ’j’.
By assuming a single species, quasi-neutral plasma we can write the collision frequency as:

(cid:104)νii(cid:105) ≈ 6.8 × 10−8 ln Λi j ∗ q2
where the quasi-nutrality allows us to deﬁne neq =q nq

3/2
T
i

neq√
Ai

(2.9)

j q2

j , which is the mean-eﬀective charge of

the plasma.

The three quantities above, the plasma, cyclotron, and collision frequencies, dictate the move-
ment of an electron or ion within a plasma. The mechanism which dominates the time scales
of a plasma is dependent upon its density, degree of ionization, and temperature. In the case of
a magnetized plasma, the constraints of the externally applied magnetic ﬁeld or internal charge
currents will also help govern the kinematics of individual charges. The eﬀect which produces
the highest overall frequency will be the one which determines the equation of motion for a given
charge species. The dominant timescales for the electron and ion populations are estimated in
sections 2.2.3 and 2.2.4.

2.1.3 Plasma Diﬀusion

The previous section showed a selection of the various types of collisions that can aﬀect charged
particles. These collisions are responsible for determining how particles move throughout the
plasma. In general, collisions are largely responsible for the ways that electrons diﬀuse out of the
plasma.

Plasma pressure gradients and collisions can lead to diﬀusion of charges out of the plasma. For

example, the rate of change of the electron density can be described through Fick’s Law as:

∂ne
∂t

≈ De∇2ne

8

(2.10)

where

De =

kBTe
meνc

(2.11)

is the electron free-diﬀusion coeﬃcient [5]. The presence of a magnetic ﬁeld changes the situation,
of course, primarily by making the diﬀusion anisotropic.
If there is a uniform magnetic ﬁeld
(cid:174)B = B0 ˆz within the plasma, then the diﬀusion coeﬃcients perpendicular and parallel to the
magnetic ﬁeld lines go as:

D⊥ =

2
ν
c
2
2
c + Ω
ν
ce
D(cid:107) = De,

De

(2.12)

where νc is the eﬀective electron collision frequency and Ωce is the cold (non-relativistic) electron
cyclotron frequency. Using these diﬀusion coeﬃcients, it can be shown that:

(cid:32)

(cid:33)

∂ne
∂t

= D⊥

2ne
∂
∂x2 +

∂

2ne
2
∂ y

+ De

2ne
∂
∂z2 .

The diﬀusion coeﬃcients in equations 2.11 and 2.12 further demonstrate the conﬁning character-
istics of the magnetic ﬁelds. D⊥ decreases as B−2
0 , compared to De, which is not directly aﬀected
by the magnetic ﬁeld. As a result, systems with large magnetic ﬁeld strengths favor diﬀusion along
magnetic ﬁeld lines rather than across them.

Equation 2.11 is equally valid for ions as it is for electrons, exchanging the electron mass,
temperature, and collision frequency for those of ions. It is often the case that De > Di due to
the mass diﬀerence between electrons and ions. This, of course, means that electrons will diﬀuse
out of the plasma faster than ions. The electric imbalance that results from the species diﬀusion
imbalance causes ambipolar diﬀusion [5]. The electric ﬁeld, which is generated by the charge
imbalance, couples the ion and electron diﬀusion equations. The resulting diﬀusion coeﬃcient is:

Da =

kB(Te + Ti)
meνe + miνi

.

In the case of a fully ionized plasma, the diﬀusion across magnetic ﬁeld lines goes as [5]:

D⊥ =

nkB(Te + Ti)

e2B2
0

νc.

9

(2.13)

(2.14)

Once again, we predict that D ∼ B−2. However, experiments show that the diﬀusion coeﬃcient
will more often go as B−1. To compensate for this, Bohm determined the semiempirical formula
[5]:

D⊥ = DB =

kBTe
16eB .

(2.15)

This particular type of diﬀusion is known as Bohm diﬀusion.

Lastly, this leads to a discussion on the topic of mobility. Here, mobility is deﬁned as the ratio

of a particles drift velocity to the force which causes the drift:

|vd|
|F|

µd =

It is related to the diﬀusion coeﬃcient through Einsteins relation, which states [64]:

Through this, we can rewrite the mobility to take the form:

D = µd kBT

µd =

1

mνe f f

(2.16)

(2.17)

where νe f f is the eﬀective particle collision frequency. The mobility is valuable qualitatively as a
way to compare the movements of diﬀerent plasma species to one another. For example, electrons
will generally have a higher mobility than ions, due to the factor of 2000 diﬀerence in mass between
nucleons and electrons.

2.1.4 Electronic Properties of the Plasma

One of the more interesting properties of plasmas is their ability to act as either a conductor or
a dielectric medium. These are both due to the electrons’ ability to move throughout the plasma
volume freely. Generally speaking, plasma electrons are attracted to and attempt to surround the
positively charged ions.
If an external electric ﬁeld is present, the electrons will move around
the ion to compensate for the electric perturbation. This eﬀectively creates an electronic shield
around the ion, which will screen incoming electric ﬁeld lines. This well-known phenomenon is

10

known as Debye Shielding [5, 21, 96]. If we assume the area around the ion follows the Boltzmann
distribution, ne(r) = ne0 exp(eΦ(r)/kBTe), it can be shown that:

where Φ is the local electric potential, q is the ion charge, and λD is Debye’s length, given by:

Φ(r) =

−r
λD

q
r

e

(cid:115)

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

1 −2
1
2
0
0

0
0

3

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

ε = 0

1 = 1 +

pe(νc − iω)
2
ω
(νc − iω)2 + Ω
2
ce

i
ω

11

λD =

0kBTe
e2ne

.

(2.18)

Qualitatively, Debye’s length describes the distance over which electrostatic perturbations will
penetrate through the plasma [5].

In the absence of a magnetic ﬁeld a plasma will act like an isotropic conductor and dielectric

with their respective constants being [5, 80]:

nee2

σ0 =

me(νc − iω)

(cid:32)

and

 = 0 − σ0
iω

= 0

1 − i
ω

2
pe

ω

(νc − iω)

(cid:33)

where ω is the frequency of some propagating electromagnetic wave, should it exist. The presence of
magnetic ﬁelds changes this, principally by giving the plasma anisotropic conductive and dielectric
properties. As a result, we must generalize the conductive and dielectric constants in the form
of tensors [5, 80]. Here, we will focus on the dielectric tensor, as it is the overwhelmingly more
important quantity for this study. For simplicity assume a cold (non-relativistic) plasma in the
presence of a static magnetic ﬁeld (cid:174)B = B0 ˆz. Ignoring ion motion, it can be shown that:

(2.19)

where

2 =

i
ω

2
peωce

ω

(νc − iω)2 + Ω

2
ce

3 = 1 +

i
ω

2
pe

ω

(νc − iω) .

2.1.5 Plasma Waves

The wave equation for propagating electromagnetic waves can be derived from Maxwell’s equation,
assuming an isotropic dielectric constant [80]. However, the presence of a magnetic ﬁeld causes
the dielectric properties of the plasma to become anisotropic [5]. As a result, the wave propagation
will be directly aﬀected by its relationship to the external magnetic ﬁeld.

There are two ways that a wave’s relationship with the plasma can be determined. The ﬁrst
method involves solving Maxwell’s equations while considering the plasma to be either a dielectric
or conductive material. This method explicitly solves the wave equation for the plasma. The
second method is to solve Maxwell’s equations with a ﬂuid model of the plasma [5]. The latter
case does not provide an explicit solution to the wave equation. However, both methods provide
the dispersion relationship for the plasma, ω((cid:174)k).

The dispersion relationship is the more useful relationship for this study. The exact wave
interactions for ECRIS plasmas are unknown, as the medium has a complicated, inhomogeneous
magnetic ﬁeld. On the other hand, the range of possible electron cyclotron frequencies is well
known, and the system’s plasma frequency can be estimated [56]. Together these make the ﬂuid
description of the plasma more useful as the dispersion relationship determines the frequency and
polarization of propagating waves [39].

The dispersion of electromagnetic waves in a plasma is governed by the Altar-Appleton equation

which determines the index of refraction for a plasma [5, 39]:

N2± = k2c2/ω

2 = 1 −

2X(U − X)
2U(U − X) − Y2 sin2

θ ± ∆

(2.20)

where

∆ =

(cid:113)

Y4 sin4

θ + 4Y2(U − X)2 cos2

θ

12

and X = (ωpe/ω)2, Y = Ωce/ω, U = 1 + iνe/ω, ω is the angular frequency of the propagating
electromagnetic mode, νe is the eﬀective electron collision frequency, and θ is the angle at which
it propagates relative to the external magnetic ﬁeld. This equation is valid in the cold plasma
approximation, where charges are assumed to be stationary within a homogeneous plasma [5].
While the ECRIS plasma is neither cold nor homogeneous (due to the magnetic ﬁeld), the equation
can approximate the wave relationship in local areas within the plasma.

More precisely, assuming a plane wave, (cid:174)E ∼ exp(cid:16)

Individually, ω and (cid:174)k describe how a wave propagates through time and space, respectively.
√−1. However,
there is nothing that demands that either ω or (cid:174)k be real valued. In general form of either quantity
is omega = omegar + iωi and (cid:174)k = (cid:174)kr + i(cid:174)ki, where the subscripts r and i denote the real and
imaginary components of either quantity [8]. The imaginary components add a new dynamic to
the propagating wave by allowing it to grow or decay in time or space. Generally speaking, a
wave is unstable if it grows exponentially either temporally or spatially. However, since both ω
and (cid:174)k may be complex simultaneously, determining if a particular wave is unstable is not always a
straightforward process [80]. For simplicity, we will assume that (cid:174)ki = 0 within the ECR plasma.
Thus, all instabilities are time-like and occur only when ωi > 0.

(cid:16)(cid:174)k · (cid:174)x − ωt

(cid:17)(cid:17), where i =

i

The propagation direction of a wave determines many of its properties as well. In the case of a
longitudinally propagating wave (θ = 0), the plus and minus sign in equation 2.20 denote left-handed
(L) and right-handed (R) circularly polarized waves, respectively. For transversely propagating
waves (θ = π/2), the plus sign denotes ordinary waves (O) and the minus sign extraordinary waves
(X). The extraordinary wave has an upper and lower branch as well. Traditionally, the upper branch
of the extraordinary mode is called the X-mode with the lower branch called the Z-mode [37]. The
distinction is most common among space plasma physicists but will also be used here [80]. Figure
2.2 shows an example calculation of the dispersion relationship for O-, X-, and Z-mode waves.

Using equation 2.20 we ﬁnd various cutoﬀs (N = 0) and resonances (N = inf) that result from
the plasma dispersion. Ignoring ion motion, for propagation parallel to magnetic ﬁeld lines θ = 0
we ﬁnd that the R wave resonates at ω = Ωce and is cutoﬀ at ω = ωpe [5, 37]. This phenomena

13

Figure 2.2: The dispersion relation for an under dense plasma (ωce > ωpe).

is known as electron cyclotron resonance and will be discussed further in section 2.2.1. For
perpendicular propagation, resonance for the Z-mode occurs at the upper hybrid frequency, fuh.
The Z-mode has a lower cutoﬀ at fz. The X-mode propagates above it’s lower cutoﬀ frequency fx.
For propagation at angles 0 < θ < π2 , the Z-mode resonance changes to be the so-called Z-inﬁnity,
denoted by fZ I [37]. The Z-inﬁnity, in truth, is a generalization of the upper hybrid frequency for
oblique propagation. Those cutoﬀs and resonances are given in equation 2.21.

(2.21)

θ

fce

fz =

fuh =

(cid:118)(cid:116)

pe + f 2
f 2
ce

(cid:113)
2 (cid:169)(cid:173)(cid:171)−1 +
(cid:118)(cid:116)
2 (cid:169)(cid:173)(cid:171)1 +
1 + 4 f 2
(cid:114)
pe
(cid:113)
f 2
ce
uh − 4 f 2
f 4

1 + 4 f 2
pe
f 2
ce

(cid:170)(cid:174)(cid:172)
(cid:170)(cid:174)(cid:172) = fz + fce

ce f 2

pe cos2

f 2
uh

+

fx =

fZ I =

fce

1√
2

14

2.2 Electron Cyclotron Resonance Ion Sources and their Plasmas

The electron cyclotron resonance ion source is an open ended mirror machine designed for
the purpose of producing beams of high charge state ions. It does this by creating magnetic ﬁeld
structure that allows for cyclotron resonance heating of electrons within the plasma chamber [18].
The most powerful of these sources are capable of producing milliamperes of high charge state
currents for a range of ion mass species [79]. The most advanced sources rely upon cutting edge
advances in electrical, mechanical, microwave, and cryogenic engineering. The bare minimum
requirements to produce an ECRIS plasma, however, are rather minimal:

• Conﬁning Magnetic Field

• Microwave Power

• Adequate Neutral Gas or Vapor Pressure

The magnetic ﬁeld allows for resonant heating, which produces a wide, non-Maxwellian distribution
of electrons. A part of this distribution collides with low charge state ions and atomic species to
produce highly charged ions through electron impact ionization. Finally, an extraction system
accelerates ions away from the plasma and produces a beam. This next section will focus on how
electron heating and the ionization of atomic species produce the plasma properties of the ECRIS
plasma. Chapter 3 discusses speciﬁc conﬁnement and extraction details.

2.2.1 Resonant Heating of Electrons Within ECR Ion Sources

ECR ion sources rely upon the process of electron cyclotron resonance in order to stochastically
heat electrons. This process occurs as electromagnetic radiation propagating through the plasma
collides with electrons gyrating along magnetic ﬁeld lines. The electrons will resonate with and
gain energy from the EM radiation if [28, 97]:

ωRF = ωce(1 ± N(cid:107) β(cid:107))

(2.22)

15

where N(cid:107) is the longitudinal index of refraction and β(cid:107) = v(cid:107)/c is the electron’s velocity,
parallel to the external magnetic ﬁeld, normalized by the speed of light. The resonance interaction,
principally, heats the electrons transversely. This being due to the acceleration coming from
the right handed polarized component of the incoming wave. However, the electron will see an
increase in its longitudinal velocity due to the wave’s magnetic ﬁeld [20]. At very high energies
this interactions converts perpendicular energy into longitudinal energy; resulting in pitch angle
scattering.

The ECR ion source uses a combination of a series of co-linear solenoids and a hexapole magnet
in order to produce a magnetic ﬁeld. This superposition of ﬁelds serves two purposes: heating and
conﬁnement of electrons. Section 2.3.1 discusses the conﬁnement properties of this magnetic ﬁeld
geometry.

In general, the solenoids create a two peak magnetic ﬁeld distribution; a non-zero minimum
exists between the two peaks (Fig. 2.3). The three extrema must have values such that there is
a magnetic ﬁeld between them that will allow electrons to resonate with the incoming microwave
radiation. BRF denotes the ﬁeld strength where this interaction occurs.1 It is often the case that the
heating frequency used to heat the electrons acts as a constraint upon the conﬁning magnetic ﬁeld.
As a consequence of this, ECRIS plasmas are often compared to one another by the ratio Bmin/BRF
used during operation.2 The addition of the hexapole causes the magnetic ﬁeld to increase away
from the longitudinal axis of the plasma chamber. Together, these coils create what is known as the
Minimum-B ﬁeld structure (Min-B, Fig. 2.4). This 3D ﬁeld creates a closed surface for heating
the electrons (Fig. 2.5).3

1It is more common to use the symbol BECR to denote the magnetic ﬁeld where resonance
occurs. The resonance condition is velocity/energy-dependent, however, and what authors really
mean when they say BECR is ’the magnetic ﬁeld where non-relativistic resonance occurs’. As
such, this discussion uses BRF as it is more representative of what authors mean in the literature:
’the ﬁeld strength for classical electron resonance between the plasma and the microwave power I
injected into the system’.

2The relative height of each extremum to one another is also important but this discussion is

outside the scope of the current topic. See Sec. 2.3.4 for more details.

3For more details on the speciﬁc ion source which created these ﬁelds, see section 3.1.

16

Figure 2.3: A 2D projection of the typical operational magnetic ﬁeld across the central axis of the
SuSI ECR ion source at the NSCL. This data was generated using the program
POISSON/Superﬁsh.

The Min-B ﬁeld structure is a complicated, inhomogeneous ﬁeld topology that makes many
properties of the plasma diﬃcult to calculate, or even describe. Part of that complication manifests
in the stochastic heating of electrons. For our purposes, a stochastic system is a system that evolves
in time through random interactions [12]. The energy of electrons within the ECR plasma is a
good example of this.
Ideally, electrons will undergo numerous random collisions during their
’life-time’ in the plasma, the period over which the electron is ﬁrst conﬁned to when it escapes
conﬁnement. We want this to happen for two reasons. First, the electrons need to impact atomic
species to produce ions. Secondly, collisions make the global heating process more eﬃcient [59]
by randomizing the gyration phase of electrons with respect to the incoming RF. As a result, the
heating process occurs over multiple passes through the ECR region.

Mathematically, relativistic eﬀects make it is possible to heat electrons when Bmin > BRF. In
practice, however, the energy of these electrons is too high to sustain the plasma. While relativistic
heating is impractical, it is important to remember that the electron cyclotron frequency does change

17

Figure 2.4: A 3D projection of the magnetic ﬁeld magnitude within an ECR ion source. This
particular proﬁle was generated by using CST model of the SuSI ion source at the NSCL. The
black lines represent iso-B surfaces within the volume of the magnetic trap.

with large velocities, see 2.5. The relativistic factor qualitatively changes the resonant interaction
[97]. In particular, it changes the resonance from occurring along a line in momentum or velocity
phase space to being an ellipse. This can be shown by re-writing the resonance condition in terms
of velocity:

ω − Ωe ∗ (1 − v

2/c2)1/2 − ωNcosθ

v(cid:107)
c

= 0

and then further rewriting this to be in the form of an ellipse whose in velocity phase space:

2⊥
v
a2 +

(v(cid:107) − v0)2

b2

= 1

(2.23)

18

Figure 2.5: A contour projection of the minimum-B magnetic ﬁeld conﬁguration seen in ﬁgure
2.4. The black rings indicate the position of iso-B surfaces within the ﬁeld structure. The red ring
represent the ECR surface over which the heating of cold electrons occurs.

where

(cid:16)

(cid:32)

2
w
r
2
Ω
e

a2 = c2

v0 =

N2 cos2

cN cos θ
2
1 + w
r
2
Ω
e

N2 cos2

(cid:33)(cid:32)

θ − 1(cid:17)

+ 1

1 +

2
w
r
2
Ω
e

(cid:32)

(cid:33)−1

N2 cos2

θ

θ

2
w
r
2
Ω
e

(cid:33)−1

b2 = a2

1 +

N2 cos2

θ

2
w
r
2
Ω
e

There are several consequences to looking at resonance in an elliptical form. Even to the
lowest order approximation, the relativistic eﬀect results in qualitative changes to the resonance
condition [97]. Secondly, it allows the majority of electrons which satisfy the resonance condition
to exist in a region where ∂ fe/∂v⊥ > 0 [54]. Here, fe = fe((cid:174)v) is the electron velocity distribution
function. Under this condition, the resonant wave feels a net increase in energy at the expense of
the electrons [5, 10]. In other words, ωi > 0 for resonant ordinary and extraordinary waves. The

19

opposite case, ∂ fe/∂v⊥ < 0 where ωi < 0 increases the energy of electrons on average. This
eﬀect is commonly referred to as Landau damping, in the case of longitudinally propagating waves,
or Cyclotron damping, in the case of transversely propagating waves [5]. Further qualitative and
quantitative eﬀects reveal themselves as other non-linear eﬀects are introduced into the resonance
framework [54, 94, 97]. The ability for relativist electrons to produce unstable electromagnetic
modes is the most crucial point, at least for this study.
Instabilities that result from the highly
anisotropic electron distributions that the resonant heating mechanism creates are detrimental to
ion source operation [18, 82]. Of course, the appearance of such instabilities is a signiﬁcant focus
of this study, and they will be discussed further in section 2.4.

2.2.2

Ionization Processes

The production of highly charged ions within an ECR ion source takes place in progressive steps.
As the system consists of many mutually interacting particles (∼ 1011 cm−3), the exact growth rate
for a single charge state population is rather complicated. A single plasma species will undergo
many collisions of varying types before ﬁnally escaping the plasma. For simplicity this discussion
will focus on the dominant ionizing interactions: single electron-ion/neutral impact and charge
exchange [17, 88]. Electron Impact (EI) ionization is an inelastic scattering process where an
electron with a high enough energy removes an electron bound to an atomic shell. Charge exchange
(CX) occurs when an ion captures as an electron from a neutral species. The neutral is then ionized,
and the higher charge state ion’s ionization level decreases by one. Together, EI and CX interactions
increase and decrease the system’s average charge state, respectively. The sum of collision rates
describes the growth or decay of a given charge state population. The plasma only reaches a
steady-state once the EI ionization rate is equal to the sum of CX and conﬁnement loss rates.

The EI cross-section, and thereby collision rate, can be approximated by Lotz’s formula [45]

where:

20

σE I

q−1−→q ∝ ln Ee/Iqk
Ee/Iqk
q−1−→qve >

RE I = nenq−1 < σE I

(2.24)

q−1−→q

Here σE I

is the collision cross-section going from charge state q-1 to 1, RE I is the ionization
rate, Ee is the impacting electron’s energy, Iqk represents the binding energy of the electron to be
removed, ne is the electron density of the plasma, nq−1 is the density of ions in charge state q-1, and
ve is the velocity of electrons. While only an approximation, the formula has been shown to work
quite well for high charge states [4]. Most importantly, equation 2.24 is heavily dependent on the
impacting electron’s energy and bound electron’s binding energy. As a result, the increasing binding
energies associated with more tightly bound electrons cause the optimal energy for ionization to
increase. Simultaneously, the optimal cross-section associated with that energy decreases. As seen
in ﬁgure 2.6, the maximum cross-section of interaction for argon 40 decreases by ﬁve orders of
magnitude between initial ionization and the removal of the ﬁnal electron.

Alongside EI ionizing collisions are charge exchange collisions. As mentioned above, these
collisions act to reduce the average charge state of the plasma by the exchange of electrons between
neutral species and ions. The CX cross-section and collision rate are given by:

q+1−→q ∝ q( I0
σCX
I
q+1−→qvq+1 >
RCX = n0nq+1 < σCX

)3/2

(2.25)

q+1−→q

where σCX

is the cross section for an ion reducing it’s charge from q+1 to q, I0 is the Bohr
energy, I is the ionization potential of the neutral species, n0 is the density of neutral species, nq+1
is the density of ions of charge q+1, and vq+1 is their velocity. Importantly, the cross-section of
interaction is independent of either the ion or neutral atom’ s motion.

The characteristics of the steady-state population can be well approximated through single-

electron impact ionization, charge exchange, and diﬀusion:

∂ni
∂t

= RE I,i−1−>i + RCX,i+1−>i − RE I,i−>i+1 − RCX,i−>i−1 − ni
τi

(2.26)

21

Figure 2.6: A calculation of the Electron Impact Ionization cross section, given in [103].

where ni is the density of ions of charge state i and τi is the conﬁnement time of ions. The last term is
an extension of Fick’s law (equation 2.10) where we assume that D∇2ni ∼ ni/ti, where τi ∼ L2/D,
where L is some characteristic length of the plasma. The ﬁrst two terms on the right-hand side act
as source terms, increasing the population of the charge state of interest. The second two terms act
as sinks, taking ions from the population of interest to a diﬀerent one, reducing the population of
interest. The last term is a loss term that removes a particular species from the plasma altogether.
The second two terms do not have to be considered as losses, as the individual ions themselves
remain in the plasma, but exist in diﬀerent charge populations. An ion can be ionized from state
i− > i + 1 through EI and then reduced back, i + 1− > i, through charge exchange. Once the ion
has diﬀused out of the plasma, however, it is lost forever.

Comparing equations 2.24 and 2.25 shows an obvious diﬀerence between how the two reactions
respond to higher charge state ions. The EI reaction rapidly becomes less probable from one charge

22

state to the next. Charge exchange interactions, on the other hand, favor higher charge state ions.
The CX cross-section of interaction increases linearly with higher charge states ions. As a result,
having a higher neutral gas density becomes detrimental to the production of high charge states.

These two interactions alone deﬁne the core set of goals for producing highly charged ions of a

particular atomic species:

1. Increase Electron Energy to reach the EI cross section maximum.

2. Increase Electron Density in order to increase the probability of interaction.

3. Increase the ion conﬁnement time in order to increase the probability of colliding with an

electron.

4. Minimize the density of neutral atomic species within the plasma region.

2.2.3 Characteristics of the Electron Population
The typical ECRIS plasma is estimated to have electron densities in the range of 1011 − 1013 cm−3
[56, 58]. This corresponds to a GHz range plasma frequency (ωpe ≈ 6.34 GHz for ne = 5 ×
1011 cm−3). Experiments and simulations show that the electrons in an ECRIS plasma exist over
a large range of energies 10 − 106 eV [1, 2, 41, 56]. The distribution of these electrons is also
non-Maxwellian; the exact form of the distribution is unknown. Typically, the electron energy
distribution function (EEDF) is qualitatively approximated as a sum of three Maxwell-Boltzman
distributions. Each distribution is responsible for a diﬀerent property or mechanism within the
plasma. A cold population (Te ∼ 10− 100(cid:48)s eV) determines the dielectric properties of the plasma.
It is also believed to be responsible for conserving the plasma neutrality. A warm population
(Te ∼ 1 − 10(cid:48)s keV) is principally responsible for ionization of atomic species (see Fig. 2.6). A
hot population (Te ∼ 100− 1000(cid:48)s keV) is principally believed to aid in electron conﬁnement while
storing the majority of the free energy within the system [1]. The spread in temperature also comes
with a spread in the Debye length for the diﬀerent populations of electrons. Simulations of the

23

Te [eV]
λD [mm]

10
0.24

103
0.33

105
3.33

Table 2.1: A comparison of electron Debye lengths for the diﬀerent electron populations. In
accordance with the qualitative trends estimated in Mironov et al., it is assumed that
ne,cold ∼ 1 × 1011 cm-3 and ne,warm/hot ∼ 5 × 1011 cm-3 [58].

plasma, performed by Mironov et al., suggest that the density of the diﬀerent electron populations
are roughly on the same order of magnitude [58]. Following their estimates, we will assume
ne ∼ 1 × 1011 cm-3 for cold electrons and ne ∼ 5 × 1011 cm-3 for warm and hot electrons [58].
Table 2.1 shows a comparison between the Debye lengths for diﬀerent electron population (eq.
2.18).

e

It is worth pointing out the importance of the T−3/2

term in equation 2.8, which makes
the collision frequency highly dependent on the electron temperature. However, the magnetic
ﬁeld does the most to govern the motion of electrons within the plasma. Table 2.2 compares
the electron gyrofrequency to the electron-electron collision frequency using plasma parameters
typically associated with ECR ion sources. The physical mechanism (collisions, Lorentz force,
Coulomb force) acting over the smallest timescale determines the motion of electron, and ion,
within the plasma. For electrons, this is the static magnetic ﬁeld.
In practice, however, the
magnetic conﬁnement of electrons relies upon resonant heating. Section 2.1.1 explores the topic
of electron conﬁnement in greater detail. For now, it is suﬃcient to recognize that magnetic ﬁeld
eﬀects dominate the hierarchy of electron time scales.
It is important to note that the plasma
frequency for this model of the plasma is quite close to the gyro frequency. If the plasma becomes
overdense (ωpe > ωce), then the injected microwaves would be incapable of propagating through
the medium. This condition eﬀectively creates an upper limit on the density of electrons within the
plasma. See section 2.3.4 for more details.

24

Te [eV]
fce
(cid:104)νee(cid:105)

10

27.99 GHz
21.7 MHz

103

27.95 GHz
21.7 kHz

105

23.42 GHz
21.7 Hz

Table 2.2: A comparison of electron collision and gyration frequencies in a region where B = 1 T,
ne = 5 × 1011cm−3, ln Λ = 10 assuming the energy distribution fo electrons follows a
Maxwell-Boltzmann distribution. fce decreases due to the γ−1 term in equation 2.5.

2.2.4 Characteristics of the Ion Population

To preserve charge neutrality, the total density of ions needs to be on the order of the electron density.
However, since ions can be multiply charged, the density of a single ion species is approximately
an order of magnitude lower than the electron density (ni ∼ 1010) [56]. The energy distribution of
ions is also very diﬀerent to that of the electrons. ECRIS ions are believed to exist in a thermal
distribution at relatively low temperatures (1- 10’s eV). Traditionally, maximum ion temperatures
of 1 eV have been assumed. These values result from measurements of the extracted beam current,
such as those seen in [53]. However, several recently developed measurement techniques suggest
the ion temperature can be as high as 30 eV [36, 49]. Despite the small range of temperatures,
these results cannot be overlooked. An increase in temperature to even 5 eV dramatically changes
the collision characteristics of the plasma ions. Table 2.2.4 demonstrates this point by comparing
the eﬀects on an increasing ion temperature on Argon 1+ and 8+ by using equation 2.9. While the
collision frequency is exceptionally high for lower temperatures, easily breaking the gyromotion
of ions. As the temperature of ions increases, the ion gyro frequency can quickly overtake the
collision frequency. This could potentially indicate that the low charge state ions are magnetically
conﬁned. Section 2.3.2 discusses ion conﬁnement mechanisms in ECRIS plasmas in greater detail.
As a measure of completeness, equations 2.4 determines that the ion plasma frequency is on the
order of 10’s kHz.4

4This can be seen using eq. 2.4 for every charge state or by deﬁning an eﬀective ion plasma

frequency fpi,e f f =(cid:112)neqe2/4π

2mi0.

25

Te [eV]
ii (cid:105) [kHz]
(cid:104)ν
1+
(cid:104)ν
ii (cid:105) [MHz]
8+

1
430
2.75

5
38.5
2.46

30
2.62
0.168

fci
96.2
0.769

Table 2.3: The collision frequencies for Ar1+ and Ar8+ with ne = 5 × 1011cm−3, q = 8, ln Λ = 10
assuming that the energy distribution of ions follows a Maxwell-Boltzmann distribution. The
right hand side of the plot compares the diﬀerent collision frequencies against the gyration
frequencies for both of these charge states assuming B = 1 T.

2.3 Operational Principles for ECR Ion Sources

The previous section discussed the plasma parameters resulting from the heating and ionization
processes that naturally occur in an ECRIS plasma. Ion source scientists and engineers seek to
optimize these values to produce the most beam possible.5 A simple calculation can demonstrate
which parameters most appreciably aﬀect source performance. First, consider that, eventually,
each ion will leave the plasma. Each ion also must undergo multiple collisions in order to reach
a high charge state. We can maximize the number of electron-ion collisions by minimizing the
time between each collision. Let us consider the time it takes for a species to undergo q ionizing
collisions:

τq ≥

τi−1−>i =

1

=

νi−1−>i

1

ne(cid:104)veσi−1−>i(cid:105)

We can further simplify this by assuming a Maxwell-Boltzmann distribution to get:

q

i=1

q

i=1

q

i=1

neτq ≥ 5 × 104T

3/2
e,opt

(2.27)

where Te,opt is the optimal electron energy for ionization. Equation 2.27 deﬁnes the criteria for
high charge state production in ECR ion source:

• Large electron densities

• Long ion lifetimes

• Suﬃcient electron energies
5The emittance of the beam is also important, but a full discussion of extraction is outside the

scope of this study.

26

Golovanivsky’s plot, Fig. 2.7, visualizes these criteria [17]. This plot compares the product of
the electron density and ion lifetime necessary, ne · τi to achieve full ionization against the optimal
electron temperature, Te, for ionization. As the optimal electron density increases, so too must the
electron density and ion conﬁnement time.

Ultimately, the production of high charge state ions is a balancing act. It requires satisfying the
demands of atomic physics, electrodynamics, and thermal physics without overindulging in any
direction. As mentioned by Geller [17], many natural processes within the plasma can hamper
the production of higher charge state ions. For example, increasing the electron density promotes
high charge state production, while also increasing β (Sec. 2.1.1) and promoting a macroscopically
unstable plasma. This section focuses on how ECR ion sources meet Golvavanski’s criteria.
Furthermore, we will discuss operational techniques to enhance and optimize the ion source’s
performance by exploiting the physical properties of the plasma.

2.3.1 Electron Conﬁnement and the Minimum-B Magnetic Field Topology

The conﬁning magnetic ﬁeld is generated through the superposition of a series of solenoid coils
and a hexapole magnet. The solenoids provide two, lopsided magnetic mirrors, on either side of the
plasma chamber, that provide longitudinal conﬁnement (Fig. 2.3. The magnetic hexapole provides
a third mirror in the radial direction of the plasma chamber. Early ECRIS devices relied upon a set
of two-room temperature solenoids and a permanent magnet hexapole to fulﬁll these requirements.
More modern devices have built upon this model, adding more solenoids to gain more control over
the magnetic ﬁeld and variable magnet hexapole to control the radial ﬁeld. Sources such as SuSI
[99], VENUS [42], and SECRAL [101] use fully superconducting coil designs in order to achieve
the largest magnetic ﬁeld possible.

The superposition of all three mirrors is important as they provide both macroscopic stability
and a surface over which to heat the plasma. Together, these three ﬁelds create what is known
as the Min-B ﬁeld proﬁle. Beyond creating a closed resonance surface, the Min-B ﬁeld also
improves the conﬁnement and macroscopic stability characteristics of the plasma. The solenoids

27

Figure 2.7: Golovanivsky’s plot which shows the criteria for producing high charge state ions in
an ECRIS plasma. The circles represent the atomic species which is fully ionized for the given
density and ion conﬁnement time. Those in parenthesis to the right show the highest charge state
achievable for other atomic species. Reproduced from [17], with permission from AIP Publishing.

28

Figure 2.8: Comparison of the radial component of the magnetic ﬁeld generated by the solenoids
alone (left) and the hexapole alone (right).

alone have diﬃculty conﬁning the plasma radially. An electron or ion moving radially outward
from the center of the trap would see a decreasing ﬁeld magnitude, causing the ratio of kinetic to
magnetic pressures, β, to increase. An increasing value of β corresponds to lower conﬁnement and
increased diﬀusion of charged particles out of the plasma. The axially centered hexapole solves this
problem by introducing a radially increasing component of the magnetic ﬁeld. This conﬁguration
ensures that the system is macroscopically stable. Figure 2.8 compares the radial ﬁeld distributions
produced by a set of solenoid coils, alone, and a magnetic hexapole.

Simulations of the plasma provide a way to understand the plasma without the need for pertur-
bative diagnostics [84]. The NAM-ECRIS program does an excellent job of simulating the charge
state distributions of extracted ions [56]. Despite the advances of such programs, the spatial dis-
tribution of electrons is still mostly unknown. However, these calculations suggest that electrons’
energy distribution aﬀects their spatial distribution throughout the ECRIS volume. Low energy cold
electrons distribute uniformly throughout the plasma, while warm and hot electrons concentrate,

29

to varying degrees, at the center and turning points of the conﬁning magnetic ﬁeld [58]. They also
show that microwave heating is essential for electron conﬁnement, as it pushes the cold electrons
out of the loss cone. As a result, losses of electrons increase or decrease by turning the microwave
power oﬀ or on [57].

Perfect conﬁnement is not ideal for an ECRIS plasma, otherwise there would be no beam. It
is also exceptionally diﬃcult to perfectly conﬁne charged particles within a plasma. In one way or
another, the plasma species will ﬁnd a way to escape the plasma. Electrons escape the plasma by
entering the magnetic loss cone, such as collisional and pitch angle scattering trap[11]. Pitch angle
scattering plays an essential role in preventing the buildup of hot electrons within the plasma. At
high velocities, the resonant interaction causes transverse energy to convert to longitudinal energy
[20]. This mechanism becomes vital for limiting the density of hot, non-collisional electrons
within the plasma. However, experiments suggest new non-linear mechanisms for the diﬀusion of
electrons, as will be discussed in section 2.4.3.

2.3.2

Ion Conﬁnement

Conﬁnement in the case of ions is much diﬀerent. While electrons become better conﬁned over
time, ions will become more diﬀusive the longer they stay in the plasma. This is partially due to the
ion gyro-frequency increasing as q and ion collision frequency as q2 (see tab. 2.2.4). This creates
an interesting problem. Ion collisions are necessary in order to produce more highly charged ions.
Those collisions can also cause ions to diﬀuse out of the plasma before they are fully ionized.
However, if no ions diﬀuse out of the plasma, then there is no beam. While impressive in its own
right, a perfectly conﬁned ion plasma would be useless for accelerator operation. We then need to
understand how to balance the useful ion diﬀusive eﬀects with the useful non-diﬀusive collision
eﬀects.

While the exact diﬀusion constant is unknown, it has been shown to depend on the magnetic
ﬁeld strength as B−k, where 1 ≤ k ≤ 2 [86]. k = 1 would indicate a more turbulent Bohm diﬀusion
while k = 2 would indicate more traditional ambipolar diﬀusion, as discussed in section 2.1.3.

30

Similar to electrons, the ion diﬀusion constant generally decreases with larger magnetic ﬁelds.
However, their low mobility causes them to leave the plasma at a lower rate. The imbalance of
positive and negative charge ﬂow produces a small positive potential over the surface of the plasma.
This plasma potential pushes ions out of the system while retarding the diﬀusion of electrons. The
potential, however, is rather small with Φp ∼ 20 V.

The conﬁnement properties of ions may change as a function of their charge state. Some
evidence suggests that ions could be conﬁned magnetostatically at low charge states and electrostat-
ically at high charge states [36, 49]. High charge state conﬁnement, on the other hand, may require
electrostatic conﬁnement. These ions rely upon high density, hot electron populations to improve
their lifetime in the plasma. The well-conﬁned electrons exist in the center of plasma and create
an electric ﬁeld pointed into the center of the plasma [56]. This ﬁeld creates a potential barrier,
∆Φ, that ions must overcome before they can escape the plasma [49]. This type of phenomenon is
called a ’Potential Dip’. High charge state ions that have kinetic energies less than Q∆Φ are unable
to escape from the plasma. The diﬀusion time for an ion of charge q within this plasma model goes
as [36]:

√

q
τ
i

=

πRl
vT

exp |qe∆Φ|
T q
i

2 er2(cid:18)

π

(cid:19)(cid:32)

i

(2.28)

(2.29)

for a plasma of characteristic length l, mirror ratio R, and thermal velocity vT. The value of the dip
can be described as [18]:

(cid:33)

∆Φ =

1 + 2 ln l
r

ini − ne,hot

where the last term is the electrical balance of the local ion and electron density in a cylindrical
plasma of length l and radius r. What is important about equation 2.29 is that the potential is
directly proportional to the density of hot electrons. A large hot electron population creates a larger
potential dip, up to a point. Large densities of hot electrons can produce a wide variety of plasma
microinstabilities [9]. Particularly, once the total number of hot electrons, Nhot, is greater than the
number of cold electrons Ncold [92]. These instabilities are detrimental to the production of high
charge state ions, even as large hot-electron populations improve the conﬁnement of ions.

31

Unfortunately, the potential dip has never been measured. Simulations and estimates of the
ECRIS plasma suggest a small potential on the order of 0.01 - 1 V [13, 57]. These estimates
result in an overall lower ion temperature. However, measuring the Doppler broadening of spectral
emission lines from atomically excited ions suggests much higher ion temperatures than previously
suggested [36]. Ion temperatures up to 30 V were measured by analyzing the Doppler broadening
of atomic emissions lines coming from the source. If this is the case, then the potential dip must
also be on the order of 5 - 20 V; otherwise, it would be incapable of conﬁning those higher energy
ions. This increase in the depth of the potential also dramatically increases the lifetime of ions in
the plasma.

Simulations of the plasma demonstrate how ion conﬁnement is also aﬀected by the spatial
distribution of diﬀerent charge states [56]. Those calculations suggest that low-charge state ions
concentrate in the high ﬁeld region on the periphery of the plasma. The highest charge states
appear in the center of the plasma, where a large number of hot electrons are also present. The
described simulation does not include a potential dip, which would help draw high-charge state
ions into the central part of the plasma. The NAM-ECRIS simulation does a good job describing
ion dynamics within the plasma; however, it suggests that the depth of the potential dip is on the
order of 0.01-0.05 V [58]. This result is contradicted by the Doppler broadening measurements
described above, which suggest the existence of 30 V dips. However, it does not mean that the
simulation results are incorrect but instead limited by our present understanding of the physics of
the source plasma. As such, the simulation may give accurate results in the limit of collisions
dominating interparticle interactions.

2.3.3 Afterglow

While providing strong plasma conﬁnement is necessary for ECR ion source operations, manipu-
lating the conﬁnement of electrons and ions can enhance the system’s performance. One way of
doing so is by operating in the so-called afterglow mode. Afterglow, or plasma afterglow, is the
emission of radiation from a cooling plasma, once the driving power has been turned oﬀ. As the

32

emission of ions is a desirable eﬀect for ECR ion source, operating a plasma in an afterglow state
allows for the extraction of much higher beam currents than can be oﬀered in continuous wave
(CW) operation [86]. This is achieved in ECRIS devices by turning oﬀ the injected microwave
power after a period of CW operation [52, 90]. The technique was initially used at CERN and later
tested or adopted at other facilities, such as GSI [6, 40, 51, 76, 77].

Presently, there is no formal, analytical description of the phenomena in ECR ion sources.
Our current understanding suggests that the enhancement is a result of the loss of conﬁnement
characteristics immediately after the RF turnoﬀ [18, 86]. In the absence of power, electron losses
rapidly increase, as the heating mechanism was necessary for driving cold electrons out of the loss
cone [57]. Warm electrons will continue to undergo ionizing collisions, further increasing losses
by exchanging the energy, which keeps them conﬁned in the magnetic trap, to produce more poorly
conﬁned cold electrons. Consequently, the increased electron diﬀusion rate results in an increase
in the emission of x-rays from the plasma chamber [89]. The sudden loss of the cold electron
population breaks the plasma’s quasi-neutral state. As the plasma potential increases, the ions are
accelerated out of the ion source, creating a short term (∼ ms) burst of ions from the plasma. This
peak current decays exponentially, reﬂecting the exponential decrease in ion density resulting from
the diﬀusion process [40, 86].

Without a heating mechanism we can reduce equation 2.26 to:

∂ni
∂t

≈ −ni
τi

which can be solved as as ni(t) = n0 exp(−t/τi) [18]. This results in the characteristic exponential
decay seen in many studies of the afterglow eﬀect [40, 89]. The diﬀusion coeﬃcient for these ions
is largely dominated by collision eﬀects. Consequently, the decay time of ions in the afterglow
phase scales with charge states as (τ ∼ q−2), reﬂecting the charge dependence of the ion collision
frequency (eq. 2.9). Geller pointed out that there are a couple of diﬀerent afterglow decay modes
depending on the operation of the plasma (ﬁgure 2.9). Case a) demonstrates a turbulent Bohm-like
diﬀusion processes which scale as B−1 [87]. Case b) shows more quiescent decays that are heavily
dominated by collisional and RF diﬀusion eﬀects, resulting in more slowly decaying peaks. Geller

33

Figure 2.9: The three diﬀerent decay modes are dependent upon the degree of turbulence of the
plasma. The three curves show the typical decay forms of afterglow pulses: a) Turbulent, b)
Quiescent, c) Non-linear particle diﬀusion. Figure originally printed in [18] and has been
re-published with permission of Taylor & Francis Group LLC.

notes that hot electron plasmas would exhibit this kind of decay. Case c) involves the ignition of
non-linear processes, such as plasma instabilities, which cause rapid particle loss on time scales
much slower than the decay of the ion population. Such instabilities have been seen in afterglow
plasmas before [52, 76].

While many authors have written on and described this phenomenon, as of writing this disser-
tation, there is no formal deﬁnition for the afterglow mode of operation (at least none that is known
to this author). For our present purpose, let us use the following deﬁnition:

Deﬁnition 2.3.1. (Afterglow Mode) A mode of operation of ECR Ion Sources that temporary
enhances extracted beam current from the plasma chamber by means of electrostatic imbalance due
to loss of electrons from conﬁnement.

This deﬁnition is purposefully left broad to avoid specifying any singular underlying mechanism
for electron exodus. It is also purposefully vague in regard to the temperature of the electrons lost

34

Figure 2.10: By pulsing RF power again after the initial afterglow burst, an instability can be
ignited which forces hot electrons out of the plasma. This results in a second burst of ions. This
method will work so long as the hot electrons have not diﬀused out of the plasma by other means.
Figure originally printed in [18] and has been re-published with permission of Taylor & Francis
Group LLC.

from conﬁnement. Despite the literature’s focus on the eﬀect of cold electrons losses, a second hot
electron afterglow burst can occur by pulsing the RF power for a short period after the initial burst
[18]. The RF pulse perturbs the conﬁnement of hot electrons, causing the diﬀusion of high-charge
state ions, which were electrostatically trapped by the very well-conﬁned hot electron population.
Importantly, this eﬀect can occur up to 10’s of milliseconds after the microwave turn oﬀ, as the
hot electron population remains well conﬁned in the absence of heating. Time-resolved x-ray
measurements reinforce this perspective by demonstrating how the hot electron population’s decay
can occur over minutes [33]. Figure 2.10 shows an example of this type of burst. This second
method of producing afterglow has been successfully used in ion sources like MINIMAFIOS at
CERN in Switzerland and CAPRICE at GSI in Germany.

2.3.4 Scaling Laws

While there is no formal analytical description of the ECRIS plasma, many have developed empirical
and quasi-analytical scaling laws to guide source development. Equation 2.27, for example,
describes the scaling of conﬁnement time with electron density and energy. Geller et al. [15, 16]
and Gammino and Ciavola [14] determined laws which described how the optimal charge state and
current extracted from the ion source scaled with various ion source parameters. Hitz et al. [24] and

35

Lyneis [46] discovered that speciﬁc magnetic ﬁeld typologies also optimized ECRIs performance.
The nature of cyclotron heating puts an upper limit on the electron density within the plasma.
As mentioned in 2.1.5, right hand circularly polarized EM waves cannot propagate in regions where
ωpe = ωce. The cut oﬀ condition deﬁnes a critical density of the plasma above which incoming
RF waves will not propagate:

2

4π

0me f 2
RF
e2

0B2
ECR
me

.

=

ncrit =

(2.30)
For example, this limit occurs at 4.48×1011 cm-3 for 6 GHz and 97.5×1011 cm-3 for 28 GHz
microwaves. The plasma will become turbulent above ncrit leading to poor charged particle
conﬁnement [16].
Increasing the average magnetic ﬁeld, and thereby ωce, allows for higher
heating frequencies and subsequently larger electron densities. If we assume that ions diﬀuse in
an ambipolar manner, then the ion lifetime increases with the average magnetic ﬁeld, Bavg, as
τi ∝ B1.5

[15].

avg

The minimum energy necessary for producing a given charge state goes as the ionization
potential, which goes as q2 for a given ion. For low to moderate microwave power, the average
electron energy goes a PRF. At high energies, the power conversion becomes non-linear and goes
as P0.5
[16]. Thus, to fulﬁll the criteria for high charge state production, the ion source must be
RF
run with high power, magnetic ﬁelds, and microwave frequencies.

Gammino et al., Hitz et al., and Lyneis discovered adjusting the magnetic ﬁeld’s topology could
optimize the extracted beam current [14, 24, 46]. To begin with, we know from equation 2.3
that the magnetic pressure must be larger than the kinetic plasma pressure in order to have good
conﬁnement. Using equations 2.3 and 2.30:

nekBTe < ncrit kBTe =

which can be rewritten as:

ECR

B2
max
2µ0

= β

0kBTeB2
me

(cid:18) 2kBTe

(cid:19) 1

2

βmec2

Bmax
BRF

=

Bmax
BRF

≈ 2.

36

Let us assume that a ’good’ ECR plasma has β = 0.01 and kBTe ∼ 10keV to get [14]:

(2.31)

(2.32)

Figure 2.11: A demonstration of the magnetic scaling laws for intermediate to high charge state
argon ions. The measurement was taken using SuSI at the NSCL. Bmin = 0.4 T, Pµ = 350 W, pAr
= 214 nTorr.

Measurements from various ion sources found that the extracted beam is maximized when [24, 46]:

BRad,max = 2BRF
BInj,max = 3 − 4BRF

BE xt ≈ 0.9 − 1BRad = 1.8 − 2BRF

BMin ≈ 0.4BRad = 0.8BRF .

(2.33)

An example of these relations can be seen in ﬁgure 2.11.

While the magnetic ﬁeld scaling laws do a good job describing the source performance, they are
not well understood. Consequently, they act more like guidelines rather than rules. For example,
equation 2.32 sets a minimum lower limit on the last iso-B ﬁeld surface within the plasma chamber,
ensuring β = 0.01. A larger magnetic ﬁeld at the plasma chamber’s injection side and radial wall
maximizes the ion source’s performance by minimizing electron and ion diﬀusion rates towards
these boundaries. However, a lower extraction side ﬁeld maximum improves the performance of
the ion source. A higher value of β at extraction encourages the diﬀusion of electrons and ions
through the extraction aperture. It is diﬃcult to predict how all the ion source parameters will aﬀect

37

ECRIS performance, and the ideal operating points for one ion source may diﬀer from another.

Advancing our understanding of the scaling laws require more direct investigations of the
source’s performance. A better understanding of how the plasma parameters aﬀect the plasma will
allow scientists and engineers to improve the ion source’s overall capabilities. Not only that, but
further investigations can also reveal when and how those relations fail. To that end, this study
focuses on an experimental investigation of one of the possible mechanisms limiting the application
of the laws: kinetic plasma instabilities.

2.4 Kinetic Instabilities

Extracted high charge state currents typically reach a maximum when Bmin/BRF = 0.7−0.8, as
suggested by the scaling laws of equation 2.33 [83]. Directly observing the extracted current, on the
high ﬁeld side of the ion source’s operating parameter space, will often demonstrate quasi-periodic
µs range losses of extracted ion currents [81]. Experimental investigations determined that these
losses result from micro-instabilities, in the form of kinetic cyclotron maser instabilities, occurring
within the plasma. These instabilities result from the unstable ampliﬁcation of electromagnetic wave
modes within the plasma, which themselves result from large electron temperature anisotropies,
T⊥ >> T(cid:107) [54, 67, 71, 92].

The minimum-B ﬁeld conﬁguration prevents macroscopic instabilities from occurring. These
instabilities occur due to an imbalance between the outward thermal pressure of electrons and the
inward, conﬁning magnetic pressure. However, the ﬁeld structure cannot prevent micro-instabilities
from occurring. On the contrary, the large magnetic ﬁelds actively cause the instabilities to
occur [81, 83]. The resonant heating and loss cone conﬁnement scheme create large temperature
anisotropies, T⊥ >> T(cid:107), in the electron population.6 The hot, non-collisional electrons store large
amounts of free energy, which results in the excitation and ampliﬁcation of electromagnetic plasma
6This is a natural result of the loss cone and resonant heating combination. The energy in
the transverse motion of electrons, perpendicular to the magnetic ﬁeld lines, is increased through
the ECR process. Moreover, eq. 2.2 shows that electrons with large longitudinal velocities
(or energies) will enter the loss cone and escape conﬁnement. Together, these eﬀects create a
temperature anisotropy.

38

modes [47, 92]. Through this ampliﬁcation process, the electrons drive themselves into the loss
cone, breaking the conﬁnement of ions while also suppressing their production rate [26, 83]. If
the operating conditions are not changed, then the instabilities will repeat with a frequency range
of 10 Hz - 10 kHz, depending upon source parameters. This next section discusses the operational
ramiﬁcations of these instabilities in detail. Beginning with the diagnostic observations made
during unstable operation and leading into a discussion about the role of hot electrons and possible
mechanisms of suppressing the instability.

2.4.1 Diagnostic Observations

Three principal diagnostic signals characterize the kinetic instabilities within an ECRIS plasma:
emission of microwave power, bursts of bremsstrahlung radiation, and periodic variations of ex-
tracted beam currents. The beam current variations provided the initial observation of the insta-
bilities [38, 81]. Further investigation revealed bursts of microwave power and bremsstrahlung
radiation preceding the current variations [26, 81]. The microwave radiation signals the beginning
of the instability and lasts for several µs. A burst of bremsstrahlung radiation, lasting for 10’s -
100’s µs, immediately follows the microwave emission and result from the sudden loss of electrons
from conﬁnement. The absence of electrons in the system breaks quasi-neutrality and reduces the
production rate of ions. Eventually, the plasma will settle and the normal heating process will begin
again, leading to a recovery of the extracted ion currents, on the order of the electron heating time
(1-10’s ms). If none of the ion source parameters have changed, then the instability will ignite once
again and repeat in a quasi-periodic manner. Figure 2.12 shows an example of the beam current
losses for an extracted Ar8+ beam at the NSCL.

Figure 2.13 shows a spectrogram of the microwave emission from the JYFL 14 GHz source
during unstable operation [25]. Izotov et al. found that the emitted frequencies spanned from 6
GHz - 11 GHz (ﬁrst harmonic), with a second band in the 12 - 22 GHz range (second harmonic).
Increasing the ion sources average magnetic ﬁeld causes the plasma to emit electromagnetic radia-
tion, predominately, at lower microwave frequencies [25, 26]. However, the total range of emitted

39

Figure 2.12: Beam current losses for Ar8+, normalized by the largest measured value.

frequencies does not change. Despite this, the emission always appears to begin with higher
frequency microwaves and progress with the falling tones.

Mansfeld et al.

[47] thoroughly discussed why the Z-mode best ﬁts the observed dynamic
microwave spectrum. The mode most likely propagates quasi-longitudinally with respect to the
magnetic ﬁeld lines, that is to say, θ ≤ 1 Rad (see eq. 2.20). The excitation results from electron
temperature anisotropies, T⊥ >> T(cid:107), a natural consequence of the loss-cone conﬁnement and
resonant heating process [26, 47, 73]. The interested reader should take a moment to look at the
arguments put forth by Mansfeld et al. [47] as to why the Z-mode tentatively explains the observed
frequencies. For our purposes, however, we will only focus on the range of frequencies emitted
from the plasma chamber. Figures 2.14 and 2.15 shows the dispersion relation and phase velocity
for the extraordinary wave, respectively, for a plasma similar to the ECRIS plasma described in
Izotov et al. [26]. The domain labeled ’Slow Z-mode’ almost perfectly encapsulates the range
of observed microwave frequencies emitted from the plasma chamber during an instability event.
Furthermore, the subliminal (vφ < c) propagation of this mode makes the unstable resonant process

40

Figure 2.13: Spectrogram of an unstable microwave emission in open space (no waveguide) for a
14.056 GHz ECR ion source, 400 W microwave power, Bmin/BRF = 0.83. The solid line above
14.056 GHz results from the heating RF that leaked out of the system. The microwave emission
was so powerful that it was possible to detect the emission in free space. Experimentally, this
means that frequencies above and below the waveguide cutoﬀs could be measured. This plot has
not been adjusted for transmission eﬃciency of the microwaves. Reproduced from [25], with
permission from AIP Publishing.

more eﬃcient.

At this time, there are no reported measurements of the spectroscopic decomposition of the x-ray
burst that follows the microwave emissions. The brief timescale of the burst makes time-resolved
measurements diﬃcult, due to the small measurement window (∼ 100µs/event).7 However, mea-
surements of electrons escaping conﬁnement due to the instability demonstrate losses of electrons
over a wide range of energies [28]. The energy distribution of electrons escaping conﬁnement will
be discussed further in chapter 4.

7Multiple x-ray detectors could compensate for this, but doing so presents other practical issues,

such as space constraints, equipment costs, shielding, etc.

41

Figure 2.14: Dispersion relationship of the extraordinary wave in an ECRIS plasma. fp = 6 GHz,
fce = 14 GHz, θ = 5 °, fz = 2.22 GHz, fZ I = 14.1. The ’slow’ and ’fast’ qualiﬁers signify whether
the phase velocity of the wave is less that or greater than the speed of light in vacuum,
respectively. The slow Z-mode propagates between the plasma frequency and Z-Inﬁnity (Blue).
The fast Z-mode propagates in the orange region below the plasma frequency. The X-mode
propagates in the orange region above fx.

2.4.2 Role of Hot Electrons

The instability is the result of large temperature anisotropies, T⊥ >> T(cid:107), within a magnetized
In particular, slow Z-mode emission can occur when Nhot ≥ Ncold, that is when the
plasma.
number of hot electrons is greater than or on the order of cold electrons [92]. Measurements show
that instabilities only appear after several 10’s ms, corresponding to the time it takes to develop hot
electrons [65, 66, 81]. Instabilities in the afterglow regime demonstrate the role of hot electrons in
producing instability events. The lack of heating drives increases the diﬀusion of cold electrons,
increasing the ratio of hot to cold electrons [33, 57]. Even if the CW plasma is stable, the decaying
plasma will go through a series of instability events until the plasma fully decays [47].

The central role that hot electrons play can also explain one of the more interesting experimental
discoveries in the literature: the suppression of the instabilities by two frequency heating of the

42

Figure 2.15: Phase velocity of the extraordinary wave in an ECRIS plasma. fp = 6.00 GHz, fce =
14.00 GHz, θ = 5°, fz = 2.22 GHz, fZ I = 14.01, and fx = 16.22. The horizontal dashed line
represents vθ = c. The ’slow’ and ’fast’ qualiﬁers signify whether the phase velocity of the wave is
less that or greater than the speed of light in vacuum, respectively. The slow Z-mode propagates
between the plasma frequency and Z-Inﬁnity (Blue). The fast Z-mode propagates in the orange
region below the plasma frequency. The X-mode propagates in the orange region above fx. The
region above fZ I but below vθ = c is a forbidden region with imaginary propagation.

source plasma. Injecting a second, lower microwave frequency into the plasma can suppress the
instabilities, depending on the power of injected microwaves. The amount of power necessary to
suppress the instabilities increases with the system’s magnetic ﬁeld and, generally speaking, the
secondary microwave frequency [75]. In the most interesting case, the microwave frequency was
lower than the minimum cold electron gyrofrequency within the plasma. The only way that the
lower frequency microwaves could interact with the plasma is if they were interacting with very
high energy, relativistic, electrons. The stabilization may be a result of increased diﬀusion of hot
electrons that interact with the injected microwaves.

We can quantify the eﬀect of hot electrons by linearizing the excited microwave energy density

43

growth rate [47, 75, 92]:

dEµ
dt

≈ (γµ − δµ)Eµ

(2.34)

where Eµ is the energy density of unstable electromagnetic modes of growth rate γµ and decay
rate δµ. The growth rate, γµ, is proportional to the gradient of the electron distribution function,
but can be seen to go as Nhot/Ncold [75, 92]. The decay rate, δµ, is proportional to the electron
collision frequency. Once the growth rate overtakes the decay rate in equation 2.34, the excited
electromagnetic mode becomes unstable, and the microwave energy density increases exponentially.
The microwave energy begins resonantly interacting with electrons within the system and drives
them into the loss cone. More details on the theoretical description of this phenomenon are
discussed in the next section. What is important for the moment is that the onset of the instability
is highly dependent on the plasma’s ability to prevent the buildup of free energy within the hot
electron population of the system. One major goal in preventing the instabilities is to ﬁnd ways to
diﬀuse hot electrons out of the system quickly.

2.4.3 Theory: Quasi-linear Diﬀusion and Kinetic-Cyclotron Masers

A theoretical description of the instability phenomena is well outside the scope of this study. A
rigorous analysis would easily fulﬁll the requirements of a separate thesis. The framework is also
actively under construction, with minimal examples of its application to experimental systems.
However, it does exist, and Shalashov et al.
[73] recently presented a shallow description of
the phenomenon. This section covers an even more brief, qualitative description of Shalashov,
Gospodchikov, and Izotov’s heroic eﬀort.

The instability is formally known as the Kinetic-Cyclotron Maser (KCM) instability. This
unfortunate naming convention ﬁnds its roots in space plasma physics, where the phenomena
contribute to type I solar radio bursts, decametric radio emission from Jupiter, and auroral Z-mode
radiation from the Earth’s ionosphere. As Wu mentions, this phenomenon shares the same name
as the also-unfortunately named cyclotron maser instability, which is responsible for the emission
of coherent microwaves from gyrotron devices [97, 98]. The phenomenon that occurs with ECR

44

ion sources is NOT the same as the one which occurs within gyrotron devices. A comparison
between these instabilities has been discussed at length by Winglee [94]. Gyrotron ampliﬁcation
results from ﬂuid processes within the plasma, where KCM instabilities cannot are entirely kinetic
phenomena. Discussions of the theory of this instability, as it is applied to space plasmas, can be
found in these references [54, 55, 63, 91, 97].

There are distinct diﬀerences in how the instability manifests in space and laboratory plasmas,
however, requiring a new approach to the build-up of free energy within the system [73]. While
space plasmas gain their free energy through naturally occurring accelerating mechanisms, ion
source electrons gain their energy through the externally injected microwave radiation. As a result,
the ECRIS plasma is ignited and sustained by the energy source that actively drives the instability.
Before beginning the discussion of KCM interactions within ECR ion source plasmas, we must
ﬁrst discuss a second phenomenon known as quasilinear diﬀusion. This framework is a perturbative
approach to analyzing electron phase-space diﬀusion caused by wave-particle resonance events.
We can deﬁne a phase space for electrons, characterized by their transverse adiabatic invariant,
I⊥ = p2⊥/2meωce(z) and kinetic energy Ekin = (γ − 1)mc2. If we assume an electron distribution

governed by these values, (cid:101)f , we can discuss the Fokker-Planck interactions of the electrons with

monochromatic electromagnetic radiation, of frequency ω and harmonic number s. The theory is
simpliﬁed if we perform a coordinate rotation by deﬁning:

ξ = (sEkin + ωI⊥)/2
κ = (sEkin − ωI⊥)/2

which transforms our distribution: (cid:101)f(ξ, κ) = f(Ekin, I⊥). Under this transformation, the Fokker-

Planck equation takes the form:

(2.35)

(2.36)

(2.37)

(2.38)

∂ξ
where Dql is the quasilinear diﬀusion coeﬃcient, given by:

(cid:32)

(cid:33)
∂(cid:101)f

=

∂
∂ξ

Dql

∂(cid:101)f

∂t

Dql =

(cid:104)(∆ξ)2(cid:105)

2τ

45

where τ is the time between consecutive ’kicks’ with dispersion (cid:104)(∆ξ)2(cid:105). To a good approximation,
τ is given by the bounce time between mirror points τb. It is also shown that:

|Eω|2

zr

(cid:12)(cid:12)(cid:12)(cid:12)

Dql =

πe2
m

I⊥
2
τb
γ

(cid:12)(cid:12)(cid:12)(cid:12)zr

ωce

ω(cid:48)
ce (vz − kzI⊥/m)

,

(2.39)

where ω(cid:48)
ce = ∂ωce/∂z, Eω is the spectral intensity of the wave’s electric ﬁeld, and zr is the exact
location where the resonance condition is met. As electrons that interact with the monochromatic
radiation accelerate along lines of constant κ. Taking the time rate of change of the line integral
of the distribution along those curves allows us to determine that the electron will absorb power
equivalent to:

(cid:101)PRF =

Þ ξ2

ξ1

∂(cid:101)f

∂ξ

Dql

dξ.

(2.40)

The transfer of power from the electrons to the EM wave occurs by moving in the opposite direction

along the line of constant κ, such that:(cid:101)PRF,absorb = −(cid:101)PRF,emit .

(2.41)

This is the case during unstable operation. Of course, as the authors note, a full interaction require
the integration over all lines of κ. The resulting total power of the interaction is then:

PRF =

1
s2

ω

.

Þ

∂(cid:101)f

∂ξ

Dql

dξdκ.

We can now discuss resonant heating mechanism, as it pertains to an electron’s interaction
with a monochromatic electromagnetic wave. To simplify things, Shalashov describes the electron
energy in relation to the loss cone. In this case, the (I⊥, Ekin) phase space boundaries exist between
particles that have only transverse energy at the system’s magnetic minimum and those that have
only transverse energy at the system’s magnetic maximum. The upper bound is the maximum
kinetic energy that a particle can have while still existing in the loss cone. Analytically, these
boundaries take the form:

(cid:113)

Ekin,(min,max) =

m2c4 + 2mc2

ω(min,max)I⊥ − mc2

.

(2.42)

46

(cid:32)

(cid:33)

∂(cid:101)f

∂(cid:101)f

∂t

(cid:32)

(cid:33)

∂(cid:101)f

If an electron interacts with a particle anywhere between these phase-space boundaries, then it is
heated until the quasilinear diﬀusion process pushes it into the loss cone at an energy of:

E∗
kin

= 2mc2(ωce,max/ω − 1).

(2.43)

where ωce,max is the maximum angular gyration frequency along a magnetic ﬁeld line within the
trap, ωce(Bmax).

The ECR ion source KCM, as we know it, is not the result of monochromatic heating. As
described in section 2.4.1, the instability occurs as a result of a second-excited frequency within
the plasma, possibly more. To that end, we must discuss two-frequency heating. Suppose the
electron interacts resonantly with two frequencies ω1 and ω2. Extending the framework written
above, using equations 2.35, 2.36, and 2.37 we can write:

=

∂
∂ξ1

Dql,1

+

∂
∂ξ2

Dql,2

∂ξ1

κ1=const

∂ξ2

κ2=const

+ S,

(2.44)

where S is a source of electrons into the distribution. The lines of diﬀusion interact and interfere
with one another, allowing the electron to take multiple paths to the loss cone in (I⊥, Ekin) phase
space [73]. In principle, any number of diﬀerent frequencies, external or excited, can create a path
to diﬀusion for electrons. This mechanism may also explain how two frequency heating of the
ECRIS plasma improves the system’s overall stability [75].

The quasilinear diﬀusion framework sets up the kinetic instability theory known as the cyclotron
maser paradigm. This theory has been in development for quite some time as a model for stability
of mirror plasmas in laboratory settings [48, 47, 71, 72, 93, 92].
It has successfully described
some phenomena within the source plasma such as the so-called ’continuous wave mode’ of the
kinetic instability [70, 69]. Despite this, the theory is diﬃcult to apply due to the complexity of
the inhomogeneous nature of the plasmas in which they occur.8 Here, it will provide a qualitative
context to the underlying mechanism of the instability. As mentioned previously, Shalasov et al.
discussed the phenomenon more thoroughly [73].

8ECRIS plasmas are inhomogeneous in their energy, magnetic ﬁeld, constituent species, and

charge densities.

47

For this problem, our second frequency ω2 is unstable, that is to say, Im(ωs) > 0 [54, 97].
The second frequency is excited by the electrons being heating along quasilinear diﬀusion lines.
Consequently, the electrons can interact with the wave which they have excited and use it as a
second path into the loss cone. Thus, Dql,1 is determined by the externally injected microwaves
and Dql,2 by those excited internally (see eq. 2.39). The growth rate of the unstable mode goes as

(cid:12)(cid:12)(cid:12), where the plus or minus sign correspond to emission or absorption of the unstable wave

±(cid:12)(cid:12)(cid:12)∂(cid:101)f/∂κ

energy, respectively. This mechanism is also known as inverse Landau damping.

The development of the cyclotron maser paradigm for ECR ion sources has provided a much
needed analytical understanding of the observed kinetic instabilities within the source plasma. There
is still much work to do, however, as new diagnostic techniques are currently under development
[25, 27, 36, 49, 84, 85]. The remainder of this study focuses on direct measurements of the
electrons which diﬀuse out of the plasma and measurements of the ion beam under stable and
unstable operation. Through these measurements, we can ﬁnd trends that suggest how the plasma
responds to diﬀerent plasma parameters and how the instabilities may be suppressed or even utilized
to enhance source performance.

48

CHAPTER 3

EXPERIMENTAL APPARATUS

All measurements were taken using the Superconducting Source for Ions (SuSI) at the National
Superconducting Cyclotron Laboratory (NSCL)[99, 100]. Figure 3.1 shows the beamline conﬁg-
uration, which includes a series of corrector magnets, a dipole analyzing magnet, and a series
of focusing solenoids. Beam diagnostics consisted of two Faraday cups, which measured the
current coming from the ECR ion source. Microwave power detectors and a High Purity Germa-
nium Detector (HPGe) measured the electromagnetic radiation emitted from the ion source plasma
chamber.

This chapter will focus on these diﬀerent experimental apparatuses. The ﬁrst section of this
chapter will focus on the ion source itself, how to use the ion source to study plasma instabilities,
and the diﬀerent operational conﬁgurations used through this study. The latter section will focus
on the use and calibration of plasma and beamline diagnostics.

3.1 Superconducting Source for Ions (SuSI)

SuSI is an incredibly powerful ion source for both ion production as well as research.

Its
fully superconducting magnet design gives its magnetic ﬁeld an unparalleled degree of ﬂexibility,
making it the ideal ion source for studying source-based kinetic instabilities. Six coaxial solenoids
generate the system’s longitudinal magnetic ﬁeld, although only four coils are active at any one
time. This magnet design allows for the generation of longitudinal magnetic ﬁeld maxima in the
range of 2-3 T. A superconducting hexapole coil surrounds the plasma chamber to generate the
ﬁeld for radial conﬁnement [99]. Figure 3.2 shows a diagram of SuSI’s internal components.

The system’s principal advantage is the ﬂexibility of its magnetic ﬁeld. With four longitudinal
coils, it is possible to observe how diﬀerent aspects of the magnetic ﬁeld aﬀect the plasma. For
example, it is possible to vary the magnetic minimum while keeping the injection and extraction
side ﬁeld maxima constant (see ﬁg. 3.3). In general, it is possible to control the value of one of the

49

Figure 3.1: A schematic of the source-beamline conﬁguration used for this research.

ﬁeld extrema within the source chamber while leaving the other two extrema constant. However,
the system cannot control the gradient at resonance while also controlling two extrema. Thus, the
ﬁeld gradients were allowed to vary freely. Of course, the hexapole coils allow for a variable radial
conﬁnement scheme. Figure 3.4 shows an example calculation of the magnetic ﬁeld created by the
hexapole, in the absence of the solenoids.

A CST simulation model of SuSI performed magnetic ﬁeld and particle trajectory calculations
in the ion source and throughout its beamline (ﬁg. 3.5). The radial ﬁeld distribution in ﬁgure
3.4 is an example of one such magnetic ﬁeld calculation. Some simpliﬁcations were necessary to
minimize the computational time of each simulation. These include ignoring the steel frames that

50

Figure 3.2: A 3D model showing the cross-section of SuSI. Gas and microwave are injected on
the left and the ion beam is extracted to the right. Arrows on the bottom point to the four active
solenoid coils that surround the hexapole and plasma chamber.

support the ion source and analyzing dipole. Despite this, the simulation reliably reproduced the
trajectories of ions extracted from the ion source, within a reasonable error (see ﬁg. 3.6).

A klystron set to 18 GHz (BRF ≈ 0.64 T) propagates microwaves, via WR-62 waveguide, into
the plasma chamber in order to ignite and heat the plasma. A ULV 150 precision leak gas valve,
open and closed by a motor, controls the gas ﬂow rate into the ion source chamber. A bias disk, an
electrostatic probe aﬃxed to the injection baﬄe, acted as both a diagnostic and tuning probe in the
plasma chamber. An EPICS controller set the voltage on the disk while measuring the ion current
deposited onto it.

51

Figure 3.3: An example of the on axis longitudinal ﬁeld proﬁle for SuSI. The three curves were
generated by POISSON/Superﬁsh. The vertical dashed line represents the position of the ion
source’s extraction aperture. The horizontal dashed line represents the resonance ﬁeld for 18 GHz
microwaves with cold electrons.

Figure 3.4: An example of SuSI’s radial ﬁeld as a function of distance from the center axis, with
375 A going through each hexapole coil. The curve was generated by a CST model of SuSI’s coil
and yoke conﬁguration. The horizontal dashed line represents the resonance ﬁeld for 18 GHz
microwaves with cold electrons.

52

Figure 3.5: (Top) The full geometry of the source and beamline model used in this study. All
models were taken from pre-rendered SolidWorks ﬁles created by NSCL mechanical engineers.
Those ﬁles were imported into CST studio suite, where they were used to perform magnetic ﬁeld
and beam trajectory calculations. The model includes a yoke (in blue), plasma chamber,
extraction geometry, einzel lens, transport lines, dipole, and Faraday cup geometry. The cup
housing geometry has been simpliﬁed in order to improve simulation speed. (Bottom) Coil
geometry and current direction. On the left are the dipole coils and on the right the source coils.
The white arrows represent current direction. In this picture the dipole is set to analyze electrons.

53

Figure 3.6: The blue lines show the results of a trace simulation of Ar8+ ions traveling through
SuSI’s beamline up to the ﬁrst Faraday cup. Data from a charge state distribution measurement
was input into the program to calculate the magnetic ﬁeld of the the ion source and analyzing
dipole: Ebeam = 160 keV (Vext = 20 kV), Veinzel = −20 kV, Vpuller = 0 V, Idip = 102.671 A. The
beam energy was predeﬁned, rather than relying upon the electrostatics simulator. The simulation
does slightly over approximate the ﬁeld in the dipole as it was necessary to increase the beam’s
energy by 4 keV (or extraction potential by 0.5 kV) in order to make the beam hit the center of the
cup. The simulation was performed using CST particle studio.

3.2 Diagnostics

To understand how varying ion source parameters aﬀect diﬀerent plasma parameters, we must
ﬁrst have a way of measuring those parameters. This brings about one of the most challenging
aspects of ECRIS research:
the diagnostics. The plasma can only be reliably studied by non-
invasive diagnostics, such as extracted particle currents or emitted electromagnetic radiation [84].
Invasive methods, such as Langmuir probes, perturb the plasma and change the electron and ion
distributions [32, 57]. Any diagnostic used must perturb the plasma as little as possible.

The use of non-invasive diagnostics produces its other issues as well. Only observing particles
or energy that escapes the system provides a limited look into the internal interactions occurring
within the source plasma. They also tend to be coupled with non-plasma related eﬀects. For
example, ion source bremsstrahlung is a reliable method of determining the eﬀect of varying

54

plasma parameters. However, it is very diﬃcult to separate plasma produced x-rays from wall
produced x-rays. It is also eﬀectively impossible to deconvolute the multiple electron scattering
and photon scattering events from the measured spectra. At best, this method provides qualitative
information on the condition of the source plasma. Tarvainen et al. [84] and Girard et al. [19] have
provided detailed reviews of the diﬀerent diagnostics typically used to study the ECRIS plasma.

This study used three plasma diagnostics to understand the ion source plasma’s condition and
the instabilities aﬀecting its performance. Those are the beam current, ion source bremsstrahlung,
and a novel measurement of electrons that escape conﬁnement through the extraction aperture. This
section will discuss the strengths, weaknesses, and steps for analysis for each diagnostic method.

3.2.1 Microwave Power Diode

An HP 8473C Low Barrier Schottky Diode connected to a bi-directional coupler detected the
microwave power emitted from the plasma chamber. The diode has a frequency range of 0.01 -
26.5 GHz and has a maximum detectable rise time of 8 - 12 ns. The coupler attenuates power
coming from the plasma chamber by -40 dB and any power coming from the klystron by -65
dB. It is impossible to estimate the waveguide’s attenuation coeﬃcient, as we do not know the
frequency content of the emitted microwaves. The diode allows us to determine when an instability
event occurs by measuring variations in the microwave power directed out of the plasma chamber.
Those ﬂuctuations could result from a sudden burst of microwave power, emitted from the plasma
chamber, or a change in the reﬂected power due to changing ion and electron densities. Figure
3.7 shows several examples of microwave power signals, measured on an oscilloscope. Both the
microwave power and ion beam current diagnostics could be measured simultaneously on the same
oscilloscope, allowing for coincidence measurements.

3.2.2 Beam Spectrometer

SuSI’s injection beamline comes equipped with a 90° sector magnet that acts as a mass-to-charge
selector for injection into the CCF. However, when transmitting electrons through the dipole, the

55

Figure 3.7: Examples of the measured microwave signals coming from the ion source, as recorded
from an oscilloscope trace. Data was recorded in ’AC’ mode, ignoring the steady state oﬀset.
Each plot has been normalized such that their largest value is 1. a) The instability shows a large
burst of microwave energy. The ’hump’ after the large peak is believed to be a result of a changing
reﬂection coeﬃcient. b) A moderate burst in microwave power which decays back to a steady state
value. c) A small burst in microwave power but possible a much longer change in the reﬂection
coeﬃcient of the plasma.

system becomes an energy spectrometer. In general, we can say that the momentum of charged
particles capable of traversing the 90° bend is:

Þ

p =

2q
π

B · dl

(3.1)

where the integral of B is evaluated along the path the particle of charge q follows. When we can
assume a perfect "hard-edge" model for the dipole ﬁeld, causing the charge to move along a circle,
this reduces to:

[Bρ] =

p
q

(3.2)

where ρ is the radius of the circle that the particle will follow. Equations 3.1 and 3.2 are known as
the magnetic rigidity of the particle. This latter form of the magnetic rigidity is independent of the
particle’s angle of deﬂection.

56

Figure 3.8: The 4-jaw collimator (red and blue) in front of the ﬁrst Faraday cup (gray) limited the
current by closing a gap in the horizontal (dipole bending plane) and vertical directions.

For the case of ions, we can assume all charges coming out of the source are non-relativistic,

and equation 3.2 determines the mass-to-charge ratio of an ion by:

[Bρ] ≈ mpvi
e

A
Q

(3.3)

where mp is the proton mass, vi is the velocity of the ion, e is the electron charge, A is the mass
number, and Q is the ion’s charge state.

3.2.3

4-Jaw Collimator

A set of 4-jaw collimator slits resides within the beamline to control the amount of a single charge
state beam going into the solenoid lattice. The collimator consists of two sets of plates that can
move to change the width of a gap between them. One set closes down the beamline acceptance
in the horizontal plane or the bending plane of the analyzing dipole, and the other in the vertical
plane. At largest, each set of plates can open to a maximum and minimum gap width of 35 mm
and 5 mm, respectively. Figure 3.8 shows a cartoon of this system.

57

3.2.4 Beam Current

The extracted beam current is the most straight forward diagnostic of the plasma. We are gen-
erally safe to assume that Iq ∼ nq ∼ ne for a given change state. While it does not provide a
detailed description of the plasma electrons, it does give some understanding of the global electron
properties. Two Faraday cups reside in the beamline to perform current measurements, one before
the focusing solenoid array and one after it. The beam current impinged upon these cups can
be measured through a Tektronix MDO3054 oscilloscope or the laboratory beam current monitor
(BCM) system. A millisecond or faster range current measurements required an oscilloscope, with
the BCM used for slow precision measurements. Both cups come equipped with suppressor rings
set to -149 V; to prevent secondary electron emission from the cup. Without the ring, secondary
electrons would escape the surface and artiﬁcially increase/decrease the amount of current observed
or produce a large amount of noise.

The oscilloscope is ideal for taking measurements of the transient proﬁle of extracted particle
currents. The Faraday cup-to-oscilloscope transmission line must accommodate 1-100 µs time
scale beam current variations, considering the time scale of the emitted microwave and x-ray bursts.
Figure 3.9 shows a schematic of the circuit used to connect the Faraday cup to the oscilloscope. It
consists of a Twinax cable, a converter box, and the oscilloscope’s circuit components. The system
converts current to voltage signals at a rate of 40 µA/V. Calculations using the program LTspice
predict -3 dB attenuation just before 47 kHz (21 µs).

The faraday cup/BCM system can measure slowly varying currents on the order of 1 pA. Of
course, sampling the current multiple times before recording its value provides the best instrument
precision. The system is capable of recording up to 100 samples-per-second (10 ms/sample). The
system can also record measurements across several precision ranges. The 300 pA, 3 nA, and 30
nA ranges were the most important for this study. A Keithley 6220 DC Current Source was used
to test and calibrate this system across several of those current ranges. Table 3.1 shows selected of
the results of that measurement.

58

Figure 3.9: The Faraday cup-to-oscilloscope transmission line consists of the cup, a twinax cable,
a twinax-to-coax converter box, and the oscilloscope (top). The dashed lines on the attenuation
curve show the -3 dB point of the transmission line (bottom). The attenuation plot was generated
using LTspice.

Current Set [pA] BCM read [pA] (100 Samples/s) Variance [pA]

Range
300 pA
3 nA
30 nA

-50
-2,000
-50,000

-88.2
-2040
-49400

0.7
2
13.5

Table 3.1: The measurable range could be set by the BCM controller. While there is an upper
limit of current that each range can measure, the actual value of the range (ﬁrst column) does not
determine the maximum value. That is to say, the ﬁrst column is more of an order of magnitude
reference. Increasing the BCM measurable current range, generally, decreased the relative error of
each measurement. The variance increased along side the measurable range, but did not have a
signiﬁcant eﬀect upon each measurement.

59

3.2.5 X-Ray Spectrometer

An Ortec High Purity Germanium (HPGe) detector measured the energy-resolved ion source
bremsstrahlung distribution emitted axially from the plasma chamber. Two lead bricks and a
tungsten slug, with a 0.81 mm hole drilled through it, provided collimation for the x-ray beam (ﬁg.
3.10). The bricks (green squares in ﬁgure 3.1) look into the ion source’s longitudinal axis, through
a glass window ﬂange, on the analyzing dipole, 218 cm away from the ion source’s extraction
aperture. A distance of 45.3 cm separates the two collimators. The HPGe detector resides 30-40
cm away from the second brick, housed in a lead bunker. Only the brick closest to the dipole has
a tungsten slug in it. The tungsten collimator narrows the diagnostic’s acceptance angle down to
5 mrad, projecting a circular acceptance area of 413 mm2 (11.46 mm radius) onto the extraction
array. The extraction system uses an accel-decel triode, consisting of plasma, puller, and grounded
electrodes [95]. The plasma electrode has a radius of 6 mm, the puller a radius of 8 mm, and
the ground electrode a radius of 10 mm. The x-ray acceptance area is projected onto all three
electrodes, as seen in ﬁgure 3.11. In this geometry, 113 mm2 of the total acceptance area looks
directly into the plasma chamber, with the remaining 300 mm2 looking at the extraction array.
Interpretations of the bremsstrahlung distribution, which results from using this geometry, must
take the mixed origin of photons into account.

Leitner et al. performed simulations of the angular distribution of bremsstrahlung photons
moving through a similar collimation system for the VENUS ion source at Lawrence Berkely
National Laboratory [41, 62]. The simulation showed that most photons created traveled in the
forward direction, and as a result, electrons colliding with the extraction aperture produced the
majority of observed photons. As SuSI’s extraction system is similar to VENUS’s, it stands to
reason that, to a good approximation, the same conclusions apply here.

The detector includes a built-in pre-ampliﬁer that then sends pre-ampliﬁed, PUR, and busy
signals to a rack-mounted ampliﬁer and then into an Ortec Easy-MCA. The multi-channel analyzer
(MCA) creates a histogram with 8192 bins. It then ﬁlls each bin depending upon the height (voltage)
of the pulse signal sent from the ampliﬁer. Initially, the detector will deliver an output similar to

60

Figure 3.10: Two lead blocks (grey) were used to collimate the x-rays coming from the source.
The tungsten slug (red) was only present in the block closest to the dipole (see ﬁgure 3.1). Picture
has been supplied by the courtesy of Derek Neben.

Figure 3.11: The acceptance area of the collimation system (red dashed line) falls on all three
electrodes of the extraction system.

61

Figure 3.12: An example Eu-152 emission spectrum as measured by the germanium detector used
in this study.

ﬁgure 3.12. The x-axis is given in channels as the MCA can only record data relative to initial
input pulse height. The detector output must undergo channel-to-energy and count rate eﬃciency
calibrations before the results are considered meaningful.

The detector energy calibration relies on the MCA’s linearity. Comparing the well known hard
x-ray and soft γ-ray emission to the channel numbers associated with the measured emission peaks
creates a linear transformation between channel numbers and photon energies. We can compare
the relative heights of the emission spectra to that emission line’s relative decay probability to
determine the energy associated with each peak. Of course, this is easiest for isotopes such as
Co-60, which have two emission lines at 1.17 and 1.33 MeV. Using a source like Eu-152, while
more diﬃcult, does provide a more accurate calibration over a larger range of energies. As this
study focuses on continuous-spectrum bremsstrahlung measurements, it is essential to calibrate the
system for a wide range of energies. Data for the source used is given in table 3.2. Figure 3.13
shows an example energy calibration.

62

Source A0 [µCi]
152Eu
14.64

τ1/2 (Years) A [µCi]
3.26

13.522

Table 3.2: Information of the calibration source used. A0 is the initial soruce activity, measured
on November 1st, 1988. A is the theoretical activity at the time of the measurement. Calibration
data was taken on February 15th, 2018.

Figure 3.13: Channel-to-energy calibration for the germanium detector used in this study.

To measure the detector’s eﬃciency curve, the total number of photons observed must be
compared to the number of photons emitted from the test source during the calibration measurement.
The total number of observed photons is the integral area under each of the measured photon peaks,
assuming a Gaussian distribution. An example ﬁt can be seen in ﬁgure 3.14. The source’s activity at
the time of the measurement, A, determines the number of photons emitted from the radio-isotope
sample [34]:

where Pγ is the decay probability for a given energy level, ∆t is the ’live time’ (active measuring

Ntheo = APγ

Ω
4π

∆t

(3.4)

63

time) of the detector, and Ω is the theoretical solid angle given by [23]:

(cid:18)

(cid:19)

(cid:104)

ρ

Ω =

2H

H2 +(cid:0)ρ − π/4 − (5ρ)−1(cid:1)2(cid:105)1/2

1 + 1
2
6ρ

(3.5)

where H is the height of the germanium crystal and ρ is the distance of the source away from the
center of the crystal normalized by the crystal radius. The detector eﬃciency is then given by:

ε =

Nmeas
Ntheo

.

The measurement is repeated three times at three diﬀerent distances in order to conﬁrm the
calibration. Moving the source away from the detector makes the solid angle calculation less
sensitive to the measurement geometry but decreases the signal to background ratio.
If done
correctly, all three measurements should agree with one another (see ﬁgure 3.15). In some cases,
the measurement where the source was closest to the detector did not agree with the two subsequent
measurements. In these cases, the ﬁnal calibration curve included the latter two measurements, and
discarding the ﬁrst. The data sets that agreed with one another were averaged together and ﬁt with a
polynomial. Figure 3.16 shows an example of the ﬁt used. An estimate of the measured background
counts was subtracted from the measured spectra before applying the eﬃciency calibration. For this
study, the most signiﬁcant background sources were the other ion source, ARTEMIS, the cyclotrons
(during operation), and radiation from activated material within the experimental vault.

We can now take the corrected spectrum, using the measurement’s live-time to convert to
counts-per-second, and calculate what is known as the ’Spectral Temperature’, Ts. The spectral

temperature assumes an approximation of the plasma where Iγ ∝ exp(cid:16)− h fγ

(cid:17), where Iγ is the

Ts

bremsstrahlung count rate, h is Planck’s constant, fγ is the photon frequency. This relation comes
about by assuming the bremsstrahlung results from a Maxwell-Boltzmann distribution of electrons
[2, 30, 41]. The units of Ts are eV rather than K as the presence of Boltzmann’s constant is implicit,
as is the convention in much of the literature.

Usually, a linear region of the logarithm of the intensity vs. energy (ln Iγvs.Eγ) can be found.
Fitting this region with a line and taking the inverse reciprocal of the slope of that line will return the

64

Figure 3.14: The program known as TV, written by Andreas Fitzler from the university of Köln,
was used to integrate the observed peaks of the spectra. The program uses the region surrounding
the peak to subtract background counts. It then ﬁts a Gaussian distribution to the background
subtracted peak and then integrates that Gaussian in order to determine the total number of
photons observed.

spectral temperature. Figure 3.17 shows a plot of the log intensity of the measured bremsstrahlung
distribution and the ﬁtted region. Alternatively, ﬁtting an exponential, y(x) = a exp(−bx), will
return an equivalent value [61]. A weighted least-squares ﬁtting model was used to determine the
line given in ﬁgure 3.17 [74]. Although reported less often, the x-intercept of the ﬁtted line XTs
also provides valuable information about the distribution’s energy content. Appendix A shows the
equations for calculating the linear ﬁt, spectral temperature, and estimates of their respective errors.
While both Ts and XTs are measurements of the energy content of the ion source plasma,
any trends found using these measures must be carefully interpreted. For example, an increasing
XTs may correlate with an increasing maximum electron energy OR an increasing density of high
energy electrons. The calculation of the intercept is highly dependent upon the number of observed
photons at high energies. If the electron density increases across the entire energy domain, so that

65

Figure 3.15: Repeating the eﬃciency calibration three times helps conﬁrm the solid angle
calculation. Measurements were taken at 30, 60, and 100 cm distances away from the radial edge
of the detector (assuming cylindrical geometry).

Figure 3.16: The two/three measurements were averaged together and a the resulting data set was
ﬁt with a 6th order polynomial. The ﬁt was generated by MATLAB’s curve ﬁtting tool.

66

Figure 3.17: Natural log of the bremsstrahlung intensity. The smaller plot show the ﬁt of the
linear region of the distribution. The domain was manually chosen to be 130 keV < Eγ < 320
keV. The line is described by ln Iγ = −0.01629 × Eγ + 6.261, giving a spectral temperature of
61.39 keV ± 0.599 keV.

the observation of higher energy electrons becomes more probable, then it will appear as if the
plasma’s energy content has increased. In reality, the emission rate of those higher energy photons
has simply reached above the noise ﬂoor of the measurement. Measuring the x-ray distribution
for a longer period of time achieves a similar eﬀect, assuming a well-shielded detector or minimal
background.

It is essential to know where the log intensity is most linear before ﬁtting the distribution.
Unfortunately, many authors will use ﬁxed domains over which to calculate the spectral temperature
(see refs. [1, 2, 102].) While there are many practical reasons for doing so, it is best to avoid this
practice whenever possible. Other authors have made this same observation, recognizing that the
calculation of the spectral temperature changes with the ﬁtting domain [61, 62]. As the purpose
of the calculation is to determine a temperature like quantity for the plasma, it is obvious that the
energy domain associated with the temperature must change as the energy content of the system

67

If this were not the case, then it would imply that the model describing the energy

changes.
distribution itself is changing, making the deﬁnition of temperature meaningless.

To ﬁx this, the calculation of spectral temperatures in this study included an algorithm that
determined the high and low energy bounds of the ﬁtting domain. Based on work by Kroll et al.
[35], this algorithm uses non-linear regression analysis to determine the most linear region of the
distribution. An algorithmic approach provides a more objective way of determining a spectrum’s
linear region while also making the analysis less time-consuming.

The simplest way to determine the upper limit is to truncate the distribution at a ﬁxed count
rate. A survey of all measured spectra is necessary to determine the minimum count rate. For
this study, an intensity of 0.5 counts/s was suﬃcient to ensure the ﬁtting domain spanned the
spectrum’s linear region for all measurements. This procedure is arbitrary and biased towards less
energy-dense operating points but is more accurate than ﬁxing the upper limit. The lower energy
limit is chosen by progressively truncating the lower energy portion of the distribution until the ﬁt
meets a threshold given by the coeﬃcient of determination:

k(ln Iγ,k − yk)2
k(ln Iγ,k − mean(ln Iγ,k))2

R2 = 1 −

(3.6)

where yk is the evaluation of the ﬁt given by equations A.1 and A.3. If 0.99 < R2 < 1, then the ﬁt
would be used to calculate the spectral temperature. Otherwise, the code truncated the ﬁrst 5 points
from the data set, and the ﬁtting process began again. Occasionally, the lower limit on R2 needed
to be lowered to 0.985 for the algorithm to work as described. The 0.99 threshold is also arbitrary
but is also more accurate than ﬁxing the ﬁtting domain’s upper limit. Figure 3.18 shows compares
two ﬁts: one with blindly chosen ﬁtting domain and one where the described method chooses the
ﬁtting region. Applying the algorithm improves upon the ﬁt without sacriﬁcing too much of the
ﬁtting region in the process.

Rigorous analyses of the bremsstrahlung diagnostic and the ﬁtting algorithm are outside the
scope of this work. Rather than developing these tools further, section 4.5 discusses the advantages
and disadvantages of the presented procedure. While imperfect, this work presents a starting point

68

Figure 3.18: Choosing the ﬁtting region algorithmically, rather than blindly ﬁtting over a region
which worked for a diﬀerent operating point, dramatically improves the overall ﬁt. Properly using
this method can result in improved analytical results when applied across multiple operating
points.

for others to improve upon and reﬁne. The MATLAB code used for determining the spectral
temperature can be found in Appendix A.

69

CHAPTER 4

DISTRIBUTION OF ESCAPING ELECTRONS

4.1 Experimental Goals

Understanding the electron velocity distribution function (EVDF) and the electron energy
distribution function (EEDF) is imperative for understanding the internal processes of a plasma.
Currently, the energy distributions within ECRIS plasmas are poorly understood. As mentioned
in section 2.2.3, the distributions are often approximated as a combination of three Maxwell-
Boltzmann distributions [28, 52, 57]. However, this is only a phenomenological approximation.
Understanding how diﬀerent ion source parameters aﬀect the plasma requires direct measurements
of the plasma.

The number of diagnostics capable of probing the EEDF is limited. Langmuir probes are
too invasive and cause an unacceptable perturbation upon the plasma [31, 84]. The traditional
non-invasive method for probing the plasma is the characterization of longitudinal and radially
emitted bremsstrahlung [2, 50, 61]. However, x-rays are produced as electrons scatter through a
material; in this case, the plasma and plasma chamber [34]. It is diﬃcult to de-convolute the eﬀects
of electron scattering to varying plasma parameters. At best, we can use these results to make
qualitative arguments about how the plasma is changing [2].

A novel measurement method has recently been developed to overcome some of the limitations
of the Langmuir probe and bremsstrahlung diagnostics. This new technique looks at the energy
distribution of electrons that escape conﬁnement through the extraction aperture [28]. While not
perfect, this measurement provides new quantitative insight into the electron distribution and more
context to other well-established diagnostics.

70

4.2 Considerations for the Electron Current Measurement

To perform this measurement, most of the optical elements must be de-energized. Starting from
the source: The extraction electrode, puller, and einzel lens all need to be grounded. Doing so
prevents potential barriers for the escaping electrons. At the NSCL, this can be done by turning on
the electrodes’ power supplies and setting their voltage to zero.

It was possible that decreasing the extraction voltage could change the characteristics of the
plasma. Even relatively small voltages within the source chamber can aﬀect the charge state distri-
bution [31]. We can easily probe this eﬀect by observing the source bremsstrahlung distributions
and transient ion currents while decreasing the extraction voltage in progressive steps.1 The Einzel
lens voltage matched the extraction voltage for each measurement. Figures 4.1 and 4.2 show the
results of those measurements. A second measurement of the dependence of the bremsstrahlung
on the changing extracting voltage can be found appendix C (ﬁg. C.1).

Even though the extracted ion currents decreased dramatically as the extraction voltage de-
creases, the bremsstrahlung distribution is overwhelmingly unaﬀected. Figure 4.1 shows no visual
distinction across each distribution. While calculations of the spectral temperature suggest a small
increase in the system’s energy content for higher extraction voltages, each measurement is con-
sistent with every other measurement. The results suggest that the population of electrons with
energies above 80 keV were largely unaﬀected by a changing extraction voltage. This result makes
sense as hot-electrons are highly non-collisional and should be minimally aﬀected by changes to the
plasma parameters of ions or neutrals that may occur as a result of a changing extraction voltage.2
Most of the magnetic optics must also be de-energized during the measurement. Measuring
the electrons at the ﬁrst Faraday cup, immediately after the dipole, removes interference from the
solenoids (see Figure 3.2). To precisely measure the electron distribution, the analyzing dipole
required a smaller output current supply. Under its standard conﬁguration, the dipole’s power
1While ions do escape conﬁnement at lower voltages (< 2kV), it is diﬃcult to distinguish one

charge state from another.

in other ion sources[78].

2Although not known at the time of this measurement, this trend has been previously observed

71

Figure 4.1: A varying extraction voltage has little eﬀect on the hot-electron bremsstrahlung
distribution. The spectral temperature seems to ﬂuctuate at low voltages and then decreases higher
extraction voltages. This low order deviation is likely an eﬀect of a changing background between
measurements due to changing facility operational conditions (compare to ﬁgure C.1). Plasma
parameters are Bmin = 0.397 T, P = 350 W, Tlive = 600 s, pressure = 212 nTorr.

supply can output 0 to +500 A, to bend ∼ 10’s - 100’s keV ions. For this measurement, we used a
bipolar power supply designed to output less current (-12 A < I < 12 A) with more precise regulation
(δI = 10 mA).

The largest issue, however, is the presence of residual lines of magnetic ﬂux that exist in and
around the beamline. Decades of operation have saturated the iron in the surrounding area with
a magnetic ﬂux that impedes the electrons’ ability to traverse the beamline. The magnitude and
direction of the ﬂux were measured using a Lake Shore Model 410 Hall probe. Figure 4.3 shows
the results of those measurements. The residual ﬁelds act like optical elements which perturb the
trajectory of the electrons. The electrons feel a force that pushes them towards the wall of the
beam pipe, preventing low energy electrons from entering and passing through the dipole magnet.
This eﬀect must be compensated for when looking at the ﬁnal distribution of electrons. It was

72

Figure 4.2: The transient proﬁles of the extracted ion currents were measured for each operating
point used in Figure 4.1. While ions do escape conﬁnement at lower voltages (< 2kV), it is diﬃcult
to distinguish one charge state from another. Bmin = 0.397 T, P = 350 W, pressure = 212 nTorr.

determined that, at worst, electrons with energies less than 50 keV were unreliably measured. The
model in ﬁgure 3.5 it is not sophisticated enough to account for residual ﬁelds within all the iron
surrounding the beamline. For the remainder of this study, we will consider electrons beneath this
limit to be non-meaningful. Izotov et al. published measurements of the transmission-corrected
distribution [28].

High energy electrons which escape conﬁnement without being perturbed by the magnetic ﬁeld
will appear as a continuous spectrum. As a result, the dipole no longer acts as a mass-to-charge
analyzer but rather a spectrometer. We can determine the energy of electrons selected by the dipole
by using Equation 3.1. In this case, we want the integral form of the magnetic rigidity. It is not
wise to assume a hard edge model for the dipole, so the eﬀect of magnetic ﬁelds must be integrated
over their entire trajectory. Assuming relativistic electrons, equation 3.1 becomes:

(cid:113)

Þ

−2ec
π

B · dl =

tot − m2
E2
ec4

c

(4.1)

Where Etot is the total electron energy and me is the electron mass. We can then re-write this to

73

Figure 4.3: The residual magnetic ﬂux parallel and perpendicular to the beamline transverse axis.
Measurements were taken while all beamline elements were turned oﬀ.

ﬁnd the kinetic energy of electrons bent at 90°:

Þ

(cid:19)2

B · dl

+ m2

ec4 − mec2

(4.2)

Ekin =

(cid:115)(cid:18)−2ec
Þ

π

The integral expression in equation 4.2 can be determined experimentally by combining equa-

tions 3.1 and 3.3:

B(Idip) · dl ≈ πmpv
2e

A
Q

(4.3)

This calculation breaks down when the magnetic ﬁeld is large enough to saturate diople’s steel yoke.
The permeability of 1010 Steel becomes non-linear for ﬁelds larger than 1 T. CST simulations predict
coil currents of nearly 500 A are necessary to saturate the yoke, which is well above the maximum
current used. This approach also assumes that the electrons follow the same trajectory as the ions.
The program ECRtune sweeps the dipole current and records the charge state distribution of
the extracted ions. Equation 4.3 was then applied to the charge state distribution to determine
the magnetic rigidity of the system. The measurement was repeated multiple times over diﬀerent
extraction voltages (5 kV, 10 kV, 15 kV, 20 kV) to get the largest span of dipole currents possible.
A linear best ﬁt model was then applied to the data using the MATLAB CFit module. Figure 4.4

74

Figure 4.4: Argon charge state distributions from SuSI for 20 kV, 10 kV, and 5 kV extraction
voltages measured at the ﬁrst Faraday cup after the ion source in ﬁgure 3.1. Ion charge state
increases from right to left, starting at Q = 1. Once again, we see that the extracted current
decreases with decreasing voltage. Compared to ﬁgure 4.2, however, the purpose here is simply to
increase the domain over which the system’s magnetic rigidity has been measured.

shows examples of the ion distributions, and ﬁgure 4.5 shows the resulting peak data and ﬁt from
this measurement.

In practice, the ions will be perturbed far less than the electrons by the ambient ﬁeld. Not to
mention, the relation made by combining equations 4.2 and 4.3 may break down for low voltages
(as it appears to for extraction voltages lower than 2 kV). For example, ﬁgures 4.6 and 4.7 compare
the measured energy calibration to an ’idealized’ simulation of the beamline, using the model
described in ﬁgure 3.5. The two methods of calibrating the electron energy largely agree; however,
the idealized electron system cannot account for ambient ﬁeld or inter-particle interactions. As a
result, the simulation underapproximated the energy of transmitted electrons.

The simulation can also be used to determine the transmission eﬃciency of electrons moving
through the beamline (see ﬁgure 4.8). Figure 4.9 shows the results of the transmission calculation

75

Þ B(Idip) · dl = mIdip + b, where m = 6.98624 × 10−4 (6.98909 × 10−4

Figure 4.5: Measurements of the integral form of the magnetic rigidity between the ion source
and the ﬁrst Faraday cup. The (red) dashed line shows the linear best ﬁt
, 7.01300 × 10−4) and
b = 2.47924 × 10−4 (0.77833 × 10−4

, 5.73681 × 10−4).

Figure 4.6: The electron simulation under approximates the energy calibration created by
measuring the magnetic rigidity of the ions. This is likely due to the presence of ambient and
residual magnetic ﬁelds, as seen in ﬁgure 4.3.

76

Figure 4.7: The largest diﬀerences between the ion rigidity and electron simulation energy
calibrations occur at lower energies. This is likely due to the presence of ambient magnetic ﬁelds
and interparticle interactions, which the simulation did not take into account.

made using the CST model. The simulation model assumed an idealized beamline without residual
magnetic ﬁeld eﬀects, the presence of neutral gas in the beamline, and assumed a uniform and
non-divergent beam emittance. As such, the transmission of electrons is aﬀected only by the spatial
distribution and energy of electrons as well as the ion source and dipole magnetic ﬁelds. For our
purposes, the relative eﬃciency is εTrans = Nmeas/N0, where N0 is the initial number of electrons
emitted from an aperture of radius 6 mm, Nmeas is the total number of electrons collected in the
Faraday cup. The simulation does not include the 4-jaw collimator; thus, the total relative eﬃciency
goes as:

εTot =

εTrans A4−Jaw

Acup

(4.4)

where A4−Jaw is the opening area made by the 4-jaw collimator and Acup is the opening area of
the Faraday cup, given by a circle of radius 25 mm.

The coordinate (0,0) was artiﬁcially placed into the simulated data set. In reality, the residual
ﬁeld stored within the dipole’s iron yoke will cut the electron population oﬀ much earlier. The ﬁt
from equation 4.3 predicts a lower energy electron cutoﬀ of 6.828 keV, which should also relate

77

Figure 4.8: The yellow lines represent electron trajectories along the beamline within the model
ion source. Trajectories begin at a 5 mm radius extraction aperture and end at the ﬁrst Faraday cup
in ﬁgure 3.2. All electrons are emitted in a direction perpendicular to the extraction aperture
opening, with no angular divergence. The color of the lines represent their energy. The sudden
change to blue at Faraday results from the -149 V potential on the cup’s electron suppressor ring.

to a shifting of the entire energy spectrum. The transmission eﬃciency increases linearly at high
energies, but non-linearly at low energies. The data was ﬁt with a rational function to accommodate
both high and low energy trends. This way, the ﬁt is not biased by the behavior of lower energy
electrons. Figure 4.9 shows the ﬁtted model given by equation 4.5.

p1E2
kin

+ p2Ekin + p3
Ekin + q1

εTrans =
p1 = 2.289 × 10−5 (2.227 × 10−5
p2 = 0.002758 (0.002161, 0.003355)
p3 = −2.55 × 10−6 (−0.005826, 0.00582)
q1 = 8.844 (−1.204, 18.89)

, 2.35 × 10−5)

(4.5)

78

Figure 4.9: The relative eﬃciency data (blue dots) approach a linear relationship with energy as
the energy of electrons increases. Each data point was calculated by dividing the number of
electrons emitted from the 6 mm extraction aperture by the number of electrons which reached the
Faraday cup. All electrons in a given simulation/calculation were assumed to have escaped
parallel to the center axis of the ion source, with no divergence and at a single energy. As such,
these results best represent the eﬀect of the ion source’s and dipole’s magnetic ﬁeld on the
transmission of electrons of a given energy. The ﬁt was performed with MATLAB’s CFit module
and used a ’Robust’ Least Absolute Residual ﬁtting option. The largest contributor to
transmissions losses were the fringe ﬁelds of the ion source. The charges are deﬂected away from
the beamline center by the locally diverging ﬁeld lines. This eﬀect can be seen in ﬁgure 4.8.

The horizontal gap width of the 4-jaw collimator was set to its minimum of 5 mm for the duration
of the measurement. This was to improve the overall energy resolution of the measurement. The
vertical cap was allowed to vary, depending on the measurement, with a maximum gap width of
38 mm. Table 4.1 show performance examples of the current collecting system with 38 and 5 mm
vertical gaps.

79

Gap Area
5x38 mm2
5x5 mm2

Current Range Current Var.

∼1 nA
∼100 pA

0.2%
0.7%

Table 4.1: Closing the vertical gap from 38 to 5 mm decreased the observable current by about an
order of magnitude. The relative current variance was calculated use values similar to those in
table 3.1.

4.3 Parameter Space

A primary goal of this study was to determine how diﬀerent ion source parameters aﬀected
the distribution of escaping electrons. To that end, the electron distribution was measured across
several diﬀerent parameter spaces, including conﬁnement conﬁgurations, microwave power, and
gas pressure. Determining the set of magnetic ﬁeld parameters to include in the measurement
required a good deal of care. One signiﬁcant constraint when using SuSI is the possibility of
quenching the superconducting coils. If the magnetic coil currents change too quickly, the system
can lose its superconducting properties, and the ﬁeld will quickly collapse.

To minimize the risk of quenching, a new "control" operating point was deﬁned and used in
every measurement. The operating point was designed to have an extraction side maximum and
global minimum that were lower than the standard operating point for facility operations. Having
a lower extraction side maximum is the most important consideration as it improves the diﬀusion
of electrons out of the system.
If the ﬁeld were too large, it could prevent the observation of
electrons all together. Lowering the magnetic minimum helped prevent quenching that may have
occurred from changing the coil current densities for each measurement. This speciﬁc operating
point produced an unstable ion source plasma. However, it is highly likely to make an unstable
ECRIS plasma while varying the magnetic ﬁeld (see chapter 5 for more details), and attempting
to ﬁnd a set of stable operating points would limit scope of the measurement. Instead, making
the BCM take 100 samples before recording the current value helps combat the eﬀect that the
instabilities may be having on the observed distribution. A comparison between the control and
standard facility operating points are given in Table 4.2.

As described in section 3.1, SuSI’s magnetic ﬁeld is highly ﬂexible with the ability to vary the

80

Bmin/BRF BInj/BRF BE xt/BRF

Standard
Control

0.71
0.62

3.96
3.68

1.94
1.71

Table 4.2: Comparison between the longitudinal ﬁeld extrema of the NSCL standard operating
point for facility operations, and the control ﬁeld used for this measurement.

Figure 4.10: The magnetic ﬁeld maximum at injection and extraction were chosen to be 2.3 T and
1.1 T, respectively. These values roughly follow the scaling laws with BInj/BRF ≈ 3.59 and
BE xt/BRF ≈ 1.72. Throughout this measurement the plasma volume and gradient at resonance
are allowed to vary. The resonance ﬁeld is given by the black dashed line. The extraction aperture
is represented by the vertical dashed line at x = 0 cm.

magnitude of each of the ﬁeld extrema independently. This feature allowed the eﬀect of each of the
ﬁeld extrema on the escaping electron distribution to be observed independently on one another.
Figure 4.10 shows the longitudinal ﬁeld proﬁles used to measure the eﬀect of a varying magnetic
minimum. The variation in the loss cone pitch angles for each ﬁeld proﬁle can be seen in ﬁgures
4.11, 4.12, and 4.13. The full set of longitudinal ﬁeld proﬁles used can be seen in appendix B.

The eﬀect of varying the hexapole ﬁeld, microwave power, and neutral gas pressure were
performed after observing the eﬀect of varying longitudinal ﬁeld proﬁles. Figure 4.14 shows

81

Figure 4.11: Values of the extraction side loss cone pitch angle used while varying the magnetic
minimum while holding the extraction maximum constant.

Figure 4.12: Values of the injection side loss cone pitch angle used, here the magnetic minimum
is held constant.

82

Figure 4.13: Values of the extraction side loss cone pitch angle used while varying the extraction
side maximum while holding the magnetic minimum constant.

Bmin [T] BInj,max [T]
0.337
0.400
0.400
0.468

2.367
2.200
2.583
2.367

(cid:104)B(cid:105) [T]
1.080
1.065
1.114
1.151

Table 4.3: The set of operating points used where chosen to measure the coupling between the
magnetic minimum and maximum at injection and other measured parameters. (cid:104)B(cid:105) is the average
magnetic ﬁeld between the injection and extraction maxima. The extraction side maxima was held
constant at 1.100 T for all operating points.

how the pitch angle varied with the hexapole for each of these measurements. All three of these
parameters are known to aﬀect the distribution of ions and electrons. Their eﬀects upon the
escaping electrons were measured across multiple operating points with diﬀerent longitudinal
magnetic ﬁelds. Table 4.3 summarize the characteristics of theses operating points. For ease of
reading, the results from the four auxiliary ﬁeld measurements can be found in appendix C.

83

Figure 4.14: Values of the radial loss cone pitch angle used while varying the hexapole with the
longitudinal solenoid ﬁelds held constant.

4.4 Results

Figure 4.15 shows the measured distribution of escaping electrons while the ion source was in
the ’control’ operating point. The initial measurement shows a distribution consisting of two peaks,
one at high energies (200 - 1000 keV) range and one at more intermediate energies (10 - 70 keV
range). The ’low energy peak’ falls below the reliability threshold and, as a result, will be ignored for
the remainder of this study. However, the high energy peak is well above this threshold and provides
a large amount of qualitative and quantitative information on the distribution of escaping electrons.
As shown in table 4.1, closing down the acceptance area of the 4-Jaw collimator aﬀected the range
of currents measured. Correcting for transmission losses and area between the diﬀerent gap areas
shows a measurable but small diﬀerence at low to intermediate energies 4.16. Calculating the total
observed electron current measured as a function of the collimator area, such that the acceptance
area is a square, varies close to linearly with the acceptance area (see ﬁgure 4.17). However, the
spatial and momentum distributions of the electrons were not measured. At this time, no such
measurement exists in the literature either. The successful measurement of those distributions or

84

Figure 4.15: Measured electron distribution for the measurements ’control’ operating point. The
50 keV reliability threshold is shown by the dashed line. Bmin = 0.4 T, BInj,max = 2.367 T,
BExt,max = 1.100 T, Pµ = 350 W, Pressure = 212 nTorr, BRad,wall = 1.217 T.

the beam’s emittance would help reﬁne this type of measurement, particularly for understanding
the high energy peak.

Compared to previous measurements in the literature (see Izotov et al. [28]), the intermediate
energy region (50-150 keV) plateaus before reaching the high energy peak (ﬁgure 4.15). The
population of electrons with energies lower than 50 keV can vary signiﬁcantly from operating point
to operating point. However, the uncertainty in the transmission properties of these electrons makes
any arguments based on the number density of electrons diﬃcult to conﬁrm. Thus, the remainder
of this section focuses on variations in the high energy peak of the distribution.

4.4.1 Varying Power

The system’s energy content was largely insensitive to varying microwave power. This result agrees
with previous measurements that have suggested an increasing microwave power has little eﬀect
upon electron energies [2, 41, 46]. On the other hand, the increasing microwave power correlates
with increased electron currents, likely a result of an increasing plasma density [86]. Figure 4.18

85

Figure 4.16: By applying the eﬃciency calibration described in the previous section, the total
current observed increases by several orders of magnitude, from 10−1 − 1 nA to 103 nA. The
changing vertical gap distance had little eﬀect upon the distribution once it was corrected for
beamline and gap area transmission. Bmin = 0.4 T, BInj,max = 2.367 T, BExt,max = 1.100 T, Pµ =
350 W, Pressure = 212 nTorr, BRad,wall = 1.217 T.

shows how the energy and intensity of the high energy peak vary with microwave power; the
latter reﬂecting the increase in the system’s electron density. This growth is well reﬂected in the
bremsstrahlung distribution as the total number of photons observed also increases linearly (see
ﬁgure 4.19). The top plot in ﬁgure 4.20 shows how the high energy peak’s central energy and
bremsstrahlung distributions spectral temperature and ﬁt x-intercept scale with the power (see sec.
3.2.5 for details on bremsstrahlung analysis). The lower plot shows the eﬀect upon the electron
peak’s height and the total number of observed photons (without correcting for the detector’s
eﬃciency).

The linear ﬁt’s x-intercept increases with a downward curvature at higher microwave power
settings and likely results from the increased electron density at all energies. The higher population
density results in an increased count rate of very high energy photons but produces diminishing

86

Figure 4.17: The total measured current was found to vary linearly with a varying square
acceptance area (equal horizontal and vertical gap widths). The red dashed line demonstrates the
linear ﬁt of the data points (blue).

Figure 4.18: Increasing the injected microwave power increased the measured electron current
uniformly across the entire energy range. The central energy of the peak was largely unaﬀected by
the changing microwave power.

87

Figure 4.19: The trend in the measured electron distribution is reﬂected in the bremsstrahlung
spectrum as the total number of photons increases while the energy spread remains largely the
same.

returns as the power continually increases. On the other hand, the agreement of the spectral
temperature’s trend with that of the central energy of the electrons peak demonstrates how little
eﬀect the power has on the energy content of escaping electrons.

The results of the remaining operating points used for this part of the measurement are shown in
ﬁgures C.2, C.3, C.4, and C.5. The electron current and photon distributions trends were largely the
same regardless of the average magnetic ﬁeld within the system. While changing the magnetic ﬁeld
conﬁguration does seem to aﬀect the high energy peak amplitude, the trend of each data set shows a
near-linear relationship between microwave power and diﬀusing electron density. Simultaneously,
the central energy of the high energy peak is largely insensitive to changing microwave power. The
dependence of the spectral temperature on power seems to vary with the magnetic ﬁeld; however,
this may result from the ﬁtting process rather than the eﬀect of changing ﬁeld strength.

88

Figure 4.20: The similarities between the measured electron and bremsstrahlung distributions is
most easy seen in the comparison between the systems’ energy content and measured particle
numbers. Overall, the eﬀect of increasing power leads to a more dense electron population, but
not a more energetic one. All sets shown have been normalized by the largest values in their sets:
Epeak,max = 565 keV, Ts,max = 39.7 keV, XTs,max = 278 keV, Ipeak,max = -1383 nA, and Nγ = 5.0
Mcounts.

4.4.2 Eﬀect of a Variable Pressure

As ﬁgures 4.21 and 4.22 show, increasing the neutral gas pressure appears to have a small cooling
eﬀect upon the energy content of the diﬀusing electrons. Figure 4.23 shows that the electron
high energy peak’s central energy, spectral temperature, and bremsstrahlung ﬁt intercept decrease
slightly as a result of increasing gas pressure, suggesting that the increased pressure leads to a
cooler plasma. This result matches previous measurements that show the temperature content of
the plasma decreases with increasing pressure [1]. The increased density may lead to a more thermal
plasma as the collision frequency increases linearly with electron density (eq. 2.8). The eﬀect on
the measured distributions should be minimal as the increase in the neutral density is not large
enough to substantially aﬀect the collision frequency of hot electrons. Not to mention, we already
assume that changes in atomic or ion densities have little eﬀect on the hot electron component

89

Figure 4.21: The changing electron distribution suggests that increased pressure leads to a cooler,
higher density plasma.

of the plasma (ﬁg. 4.1). Oddly, there is also an inverse relationship in the number of measured
particles between the electron and bremsstrahlung measurements. This is contradictory to what
we would expect for an increased electron density, as the diﬀusion rate should increase across all
diﬀusion mechanisms. A further progression of this discussion requires a better understanding of
how diﬀerent ion source parameters may aﬀect the system’s energy content. For now, it is suﬃcient
to recognize how the pressure aﬀects each of the diﬀerent distributions.

As in the previous case, varying the magnetic ﬁeld strength does not appreciably alter how the
diﬀusing electrons respond to changing neutral gas pressure, at least over the range measured. In
all cases, an increasing pressure led to signiﬁcant but small decreases in measured electron energies
(Figure C.6). Once again, this is reﬂected in the bremsstrahlung spectra as the most signiﬁcant
changes across each operating point seem to result from the varying magnetic ﬁeld (Figure C.7).
Figures C.8 and C.9 summarize these results, showing consistent trends in the electron and photon
energy distributions for varying magnetic ﬁeld distributions. Most importantly, each data set shows

90

Figure 4.22: As the pressure increases, the measured bremsstrahlung distribution sees a trend
towards lower energy photons.

an inverse relationship between the height of the high energy peak and the number of photons
observed.

91

Figure 4.23: While the energy content of both measurements agree to an increased pressure
resulting in a cooler plasma, the trend total particles observed are inversely proportional to one
another. Both top and bottom plots are normalized by the largest values in their sets: Epeak,max =
592 keV, Ts,max = 42.1 keV, XTs,max = 297 keV, Ipeak,max = -1224 nA, and Nγ = 5.8 Mcounts.

4.4.3 Eﬀect of a Variable Hexapole

Varying the hexapole ﬁeld strength also showed signiﬁcant changes in the number of electrons
observed. The height of the high energy peak noticeably increases; however, the bremsstrahlung
distribution is mostly unaﬀected by the varying ﬁeld. This measurement agreed with previous
measurements, which showed an insensitivity of the plasma to a changing hexapole [3], at least
over the regime measured. Figures 4.24, 4.25, and 4.26 show the results of these measurements.
Unfortunately, a mistake was made during operation and the control operating point was not used
for this measurement.
Instead, the measurement used an operating point with Bmin = 0.43 T.
Despite this the other measurements were performed successfully.

A varying longitudinal magnetic ﬁeld proﬁle does appear to aﬀect how the hexapole aﬀected
the plasma. As ﬁgure C.10 shows, an increasing average magnetic ﬁeld causes the hexapole to
have a smaller eﬀect upon the diﬀusion of electrons out of the system. The magnetic minimum has

92

Figure 4.24: The increasing radial ﬁeld causes the overall current of the distribution to increase,
although this eﬀect is minimal compared to varying the microwave power. The central energy of
the peak is also, largely, unaﬀected by the changing radial ﬁeld.

the largest eﬀect over the scaling response of the electron distribution. Once again, this trend is
not reﬂected in the bremsstrahlung distribution. Figure C.11 shows how little the varying hexapole
ﬁeld aﬀected the measured bremsstrahlung distribution across multiple longitudinal ﬁeld proﬁles.
Figures C.12 and C.13 numerically summarize the previous results, providing more insight into the
issue. All operating points agree to virtually no change to the system’s energy content as a result
of varying the radial ﬁeld strength. The electron peak initially sees close to a factor of two increase
in its height across the radial ﬁelds used. This increase itself decreases to a factor of about 1.5 for
the system with the largest magnetic ﬂux. Oddly, this is also the only data set in which varying
the longitudinal maximum appears to have a unique eﬀect upon the measured photons. The lower
left plot of ﬁgure C.13 shows that a varying hexapole ﬁeld relates to a varying number of photons
counts, only when BInj,max = 2.58 T.

Together, these results suggest that the hexapole’s overall eﬀect, at least longitudinally, is to
change the diﬀusion properties of the plasma. Based upon simulations using the model described

93

Figure 4.25: The bremsstrahlung distribution was largely unaﬀected by the changing radial ﬁeld,
however, as the radial ﬁeld at the wall increases there is a preference for the emission of lower
energy photons.

in chapter 3, it is not an optical eﬀect from a lower ﬁeld at extraction.3 Most importantly, while
there is a signiﬁcant increase in the number of electrons that diﬀuse out of the plasma, the number
of photons observed changes very little per ﬁeld setting. Increasing the ﬁeld strength at injection
contradicts this trend. The almost 30% increase in photons can be accounted for by an increased
count rate of photons with energies less than 80 keV. Only counting the number of observed photons
with energies greater than 80 keV gives the trend in the lower-left plot of ﬁgure C.13 a less steep
slope, more closely matching the trends in the other three plots. The higher injection side maximum
may amplify the eﬀect of the hexapole.

By increasing the radial maximum, we are directly decreasing the pitch angle of their respective
loss cones (see ﬁgs. 4.12 and 4.14). It becomes easier for electrons to diﬀuse out of the plasma
through the loss cone on the extraction side of the ion source. This could explain why the increase
3Varying the hexapole had a negligible eﬀect upon the transmission of electrons through the

beamline, even at low energies.

94

Figure 4.26: The energy content of the distributions were largely unaﬀected by a changing radial
ﬁeld strength. The increasing radial ﬁeld causes an increase in the both the height of the electron
peak and total number of photons observed. Epeak,max = 669 keV, Ts,max = 46.7 keV, XTs,max =
315 keV, Ipeak,max = -1061 nA, and Nγ = 6.3 Mcounts.

in the current sees diminishing returns as the high energy peak shifts to higher energies. The data
also suggests that the plasma density remains relatively constant, as we would expect to see a much
larger increase in the number of observed photons across all operating points.

4.4.4 Eﬀect of Varying the Injection Side Magnetic Maximum

Varying the injection side magnetic ﬁeld also sees only small changes to either the measured
electron or bremsstrahlung distributions. Figures 4.27, 4.28, and 4.29 all indicate that the injection
side maximum has little to no eﬀect on the distribution of electrons escaping conﬁnement. The
increase in the injection side ﬁeld maximum minimally increases the energy content of the system.
The lack of change in the bremsstrahlung spectrum is expected as most of the electrons impacting
the plasma chamber’s injection side will direct their energy away from the HPGe detector. Despite
this, there is a change in the particle ﬂux to the biasdisk as the injection side ﬁeld is varied. Figure

95

Figure 4.27: The varying injection side magnetic maximum mostly aﬀected the lower energy
distribution of electrons, having minimum eﬀect on the high energy peak.

4.30 shows how the trend for the changing positive current deposited on the bias disk as the injection
side maximum varies. We once again observed an inverse relationship between the electron peak
amplitude and the total number of photon counts observed from the extraction side. This result is
unexpected, as the increased injection side magnetic ﬁeld should partially enhance the diﬀusion of
electrons through the extraction and radial loss cones, and increase both the measured electron and
photon intensities. Furthermore, focusing our attention on only x-ray photons with energies greater
than 80 keV does not change the slope of the trend in the photon count number. This means that
changing the electron or ion diﬀusion characteristics on the plasma chamber’s injection side is not
entirely responsible for the eﬀect seen in ﬁgure C.13.

The near insensitivity of the energy to the injection maximum is also expected. Varying the
injection side of the magnetic ﬁeld proﬁle should principally aﬀect the electron heating rate, not
necessarily their maximum energy. This result does not indicate that the injection side ﬁeld is not
important, as performing the same measurement over a much larger magnetic ﬁeld domain may

96

Figure 4.28: The bremsstrahlung distribution was largely unaﬀected by a varying injection side
magnetic maximum.

result in a vastly diﬀerent response. Particularly as the injection side maximum reaches the same
magnitude as the radial or extraction side ﬁeld maxima.

97

Figure 4.29: All operating points were nearly identical to one another. The inverse relationship
between the electron peak amplitude and the total number of observed photons can be seen,
however, the largest eﬀect is likely on the lower energy side of the distribution, as the electron
measurement suggests. Epeak,max = 568 keV, Ts,max = 38.6 keV, XTs,max = 281 keV, Ipeak,max =
-1357 nA, and Nγ = 4.9 Mcounts.

Figure 4.30: The bias disk current decreased as the injection side magnetic ﬁeld increased,
|IBD,max| = 730 mA.

98

4.4.5 Eﬀect of Varying the Extraction Side Magnetic Maximum

Varying the extraction side maximum changed the total number of electrons observed at the Faraday
cup. This result adds credibility to the hypothesis that the extraction side magnetic ﬁeld is mostly
controlling the diﬀusion rate of particles out of the system. In ﬁgure 4.31, increasing the extraction
side maximum above 1.2 T shows a sudden change in the conﬁnement characteristics of the
plasma. At this point, the total extracted beam current decreases drastically. These distributions
are not corrected for the changes in transmission due to a changing ﬁeld maximum at extraction.
The extraction ﬁeld aﬀects the transmission coeﬃcient of the system by perturbing the trajectory
of electrons. Figure 4.32 shows a similar trend in the total number of bremsstrahlung photons
observed, albeit less sensitive to the changing ﬁeld strength. The large drop in the current at 1.23 T
correlates with a sudden decrease in the number of photons observed for the same operating point.
The sudden change in both the electron and bremsstrahlung distributions suggests that the jump is
not a result of optical changes to the beamline. Figure 4.33 shows that the electron peak amplitude
and total number observed of photons see a 62% and 56% decrease in intensity, respectively. The
beam current results in ﬁgure 4.33 are corrected for beamline transmission, resulting in a slower
decline in the electron peak amplitude than in ﬁgure 4.31.

The decreasing current may result from changing the pitch angle of the loss cone at extraction
(see ﬁg. 4.13). As the loss cone pitch angle decreases, a smaller fraction of the electron velocity
distribution would exist inside it, decreasing the diﬀusion rate out of the system. However, without a
model for the plasma diﬀusion processes, or the electron velocity distribution function, it is diﬃcult
to compare the varying electron current and photon intensity to the changing loss cone.
If the
plasma had an isotropic electron velocity distribution, we could say that (cid:219)n ∼ α0; however, we know
that this is not the case. The resonant heating process preferentially increases the transverse energy
of electrons, creating a temperature anisotropy. Despite this, the electrons will have non-zero
longitudinal momentum at some point during their lifetime in the plasma. Thus, the decreasing
loss cone pitch angle will make it more diﬃcult for electrons to escape the system, which decreases
the observable current.

99

Figure 4.31: Increasing the extraction maximum slowly decreases the amplitude of the high
energy peak. Between 1.2 T and 1.23 T the change in peak amplitude increases and results in a
dramatic decrease in peaks height. These distributions have not been corrected for changes in the
transmission coeﬃcient due to a changing extraction ﬁeld maximum.

Benitez et al. also reported a change in the behavior of the x-ray distribution while varying
the extraction side maximum in the VENUS ion source [3]. They reported that increasing the
radial ﬁeld maximum above that of the extraction ﬁeld caused the measured spectral temperature
to change. This trend may indicate a change in the conﬁnement characteristics of the electron
population. While ﬁgure 4.31 shows a behavioral change, it is important to note that the results
reported here and by Benitez et al. contradict one another. Figure 4.33 shows no change in the
spectral temperature, but does see a change in the total number of electrons diﬀusing out of the
plasma chamber. However, a changing conﬁnement scheme for electrons may explain the rapid
decrease in the electron diﬀusion rate between 1.2 T and 1.23 T (the maximum ﬁeld strength
generated solely by the hexapole coils was calculated to be 1.22 T). Setting the extraction side
maximum above the radial ﬁeld maximum could have this eﬀect by encouraging radial diﬀusion of
electrons. However, the total ﬁeld at the plasma chamber wall, including the eﬀect of the solenoids,

100

Figure 4.32: The bremsstrahlung distribution remains largely unchanged as a result of a changing
extraction ﬁeld maximum, up to a point. Between 1.2 T and 1.23 T the number of intermediate to
low energy photons observed decreases dramatically. These spectra have not been corrected for
changes in the beamline transmission coeﬃcient which result from a varying ﬁeld maximum at
extraction.

is complicated and inhomogeneous, making it diﬃcult to conﬁrm this conﬁnement condition.
Measuring the bremsstrahlung emitted radially from the ion source as the extraction ﬁeld is varied
would be the best way to determine if such an eﬀect occurred. However, this kind of measurement
is diﬃcult to do with the SuSI ion source.

101

Figure 4.33: While the total number of photons observed decreases only slightly as the extraction
maximum is increased, the sudden sudden drop-oﬀ in extracted beam current reduces the number
of photons observed by more than 50%. No appreciable change in the system’s energy content is
observed. Epeak,max = 585 keV, Ts,max = 38.6 keV, XTs,max = 270 keV, Ipeak,max = -1756 nA, and
Nγ = 4.9 Mcounts. Each data set has been normalized by the largest value in their respective sets.
The normalized electron peak amplitude has been corrected for changes in the beamline
transmission coeﬃcient which result from a varying extraction maximum.

4.4.6 Eﬀect of Varying the Magnetic Minimum

The most striking results came when the magnetic minimum was varied. The energy associated with
the high energy peak increased as the minimum magnetic ﬁeld increased. The magnetic minimum
was the only feature of the magnetic ﬁeld that appreciably varied the peak’s energy. In conjunction
with its eﬀect upon the electron distribution, there is a sharp increase in the bremsstrahlung
distribution’s spectral temperature with an increasing magnetic minimum. Both distributions seem
to agree that there should be a factor of two increase in the system’s high energy content. Figures
4.34, 4.35, and 4.36 show how only changing the magnetic minimum can appreciably aﬀect the
energy distribution of electrons and photons escaping the system.

The top plot in ﬁgure 4.36 demonstrates how well the energy content of both measured distribu-

102

Figure 4.34: There is an obvious change in the energy content of the measured electron
distribution resulting from a changing magnetic minimum. While the central energy of the peak
increased, the height of the peak decreased.

tions scale with one another. Previous measurements of the ECRIS bremsstrahlung demonstrated
the relationship between the energy content of the plasma and the system’s magnetic minimum
[2, 3, 61]. Noland observed the same eﬀect in the spectral temperature of the radially emitted
bremsstrahlung [62]. The high degree of correlation between the linear ﬁt x-intercept and the high
energy peak’s central energy suggests that there is a relationship between the electron peak and the
high energy tail of the bremsstrahlung distribution. In particular, the higher central peak energies
correlate with increasing high energy photons in the tail of the bremsstrahlung distribution.

The lag in the spectral temperature may be an artifact of the mixed nature of the total electron
distribution. Here we hit upon a limit of the algorithm presented in chapter 3 (sec. 3.2.5). Figure
4.37 makes this failure obvious, as the maximum upper limit on the ﬁtting domain is 372 keV
for a ﬁeld minimum of 0.47 T. This is a bad ﬁt as both the escaping electron and bremsstrahlung
distributions have meaningful intensities at energies greater than 500 keV. This error results from
the simplicity of the ﬁtting algorithm. Figure 4.35 shows that the linear portion of the photon

103

Figure 4.35: Just as in ﬁgure 4.34, an increasing magnetic minimum pushes the photon
distribution to higher energies. The total number of counts, however, increases where the electron
peak’s height decreases. This suggests that overall electron density increases, even if the
measured electron current decreases.

distribution shifts to higher energies as the magnetic minimum increases, however, the entire log
scale distribution itself becomes more linear overall as the minimum increases as well. The careful
observer will also notice that, in particular, the log scale high energy tail of the bremsstrahlung
distribution also becomes more linear for higher magnetic minimums, especially for regions where
the count rate is less than 0.5 counts-per-second. Alongside this, although more linear on average,
the lower energy portion, particularly with count rates higher than 0.5 counts-per-second, has a
higher degree of curvature to it. This curvature biases the calculation of the spectral temperature
toward lower temperature values (large ﬁtted slopes).

The decrease in peak height of the electron distribution is unlikely to result from beamline
transport as the beamline transport coeﬃcient increases with increasing energy. Rather, it may
present evidence of a quasilinear diﬀusion process. Recall from chapter 2 that Dql ∼ γ−2
e , where
Increasing the
γe is the Lorentz factor of the quasilinearly diﬀusing electrons (see eq. 2.39).

104

Figure 4.36: Both the electron and bremsstrahlung distributions agree to a near linear increased in
the energy content of the plasma. The spectral temperature lags behind due to the inﬂuence of
lower energy electrons. Epeak,max = 742 keV, Ts,max = 56.47 keV, XTs,max = 359 keV, Ipeak,max =
-1916 nA, and Nγ = 6.7 Mcounts. Each data set has been normalized by the largest value in their
respective sets.

Figure 4.37: Maximum and minimum energy limits for ﬁtting the spectral temperature as the
magnetic ﬁeld is varied.

105

Figure 4.38: There is a linear correlation between the decreasing trend of the height of the high
energy peak and the reciprocal square of the central energy of that peak.
Iγ/Iγ,max = 4.897 × γ−2 − 0.4523, R2 = 0.9938.

resonant energy of electrons will results in a decreasing diﬀusion coeﬃcient [73]. We should then
expect that:

Ie =

dQ
dt

∼ −e

dne
dt

,

∼ Dqlne ∼ ne
2
e

γ

where Ie is the electron current diﬀused by quasilinear diﬀusion (high energy peak). Figure 4.38
shows that the peak amplitude increases linearly as the central energy of electrons in the high energy
peak decreases, by ﬁtting the current against γ−2
e . This does assume that the density of electrons
interacting with the electromagnetic waves is relatively constant.

An alternative explanation for this might be that the electron current decreases as the diﬀusive
interactions focus on higher energy electrons. The trend is then more of a demonstration of the
decreasing density in the high energy tail of the electron energy distribution. However, it is not
obvious what kind of mechanism would be responsible for the diﬀusion of electrons. As ﬁgure 4.39
shows, the direction of the ﬁeld lines a the source’s extraction aperture are essentially parallel to the
longitudinal axis of the plasma chamber. As a result, we can discount both fully ionized and Bohm
diﬀusion models (eqs. 2.14 and 2.15) as they require diﬀusion across magnetic ﬁeld lines. Classical

106

Figure 4.39: The magnetic ﬁeld lines (blue arrows), within the acceptance area of the extraction
aperture (red dashed lines), are primarily parallel to the longitudinal axis of the plasma chamber.

’free’ diﬀusion and ambipolar diﬀusion can act along magnetic ﬁeld lines, but are independent of
the magnetic ﬁeld and have diﬀusion coeﬃcients which scale with electron temperature (eqs. 2.11
and 2.13). Lastly, this may be the result of pitch angle scattering, but this hypothesis would require
numerical simulations to replicate the observed energy and amplitude trends.

The increased photon count rate does provide evidence for a classical electron diﬀusion model,
as the increasing loss cone would encourage electron diﬀusion (see ﬁg. 4.11). The wider angle
would encourage more electrons to escape from conﬁnement through all diﬀusion mechanisms and
across all energies. However, the height of the high energy peak and the total number of photons
observed by the HPGe detector appear inversely correlated. Figure 4.40 shows the relationship
between the total number of detected photons and the height of the high energy peak. Alongside
this, ﬁgure 4.34 brings into question whether a simple increase in the diﬀusion of electrons across
all energies and momenta makes sense. If that were the case, we might expect the observed electron
currents to increase across all energies, except those in the peak. This trend is not observed, as the

107

Figure 4.40: There is a near-perfect inverse correlation between the trends of the decreasing
height of the high energy peak and the increasing number of photons observed
Ie/Ie,max = −0.9998 × Nγ/Nγ,max − 1.37, R2 = 0.9888

measured distributions trends towards lower total electrons at the high energy side of the distribution.
This result implies that the number of electrons escaping from conﬁnement increases in regions
outside of the extraction aperture’s opening as the magnetic minimum increases. This eﬀect could
result from a spatially wider distribution of diﬀusing electrons, with a smaller concentration of
electrons escaping through the extraction aperture. Determining such a change will require rather
precise measurements of the emittance of escaping electrons and bremsstrahlung photons.

4.5 Concluding Thoughts

The results here agree with results seen in other measurements regarding the energy distribution
of the plasma with varying plasma parameters, both in regards to the bremsstrahlung [2, 3, 62] and
measured electron distributions [28, 73]. The magnetic minimum has the most control over the
energy content of the system. While injected microwave power, neutral pressure, and the radial
ﬁeld’s maximum can aﬀect the system’s energy content, their eﬀects are minimal compared to the

108

eﬀect of the magnetic minimum. The magnetic maxima had a minimal, if not negligible, eﬀect on
the diﬀused electrons’ energy content.

The intensities of both electron and photon distributions were found to be highly correlated,
particularly in through the electron high energy peak. The total rate of diﬀusion of electrons out
of system increased linearly with increasing microwave power, with little eﬀect upon the energy
content of the system (ﬁgs. 4.20, C.4, and C.5). The radial ﬁeld maximum was correlated with the
total observed electron current, although it had a minimal eﬀect upon bremsstrahlung distribution
(ﬁgs. 4.26, C.12, and C.13). While changing the magnetic maximum at injection was found to
appreciably change the rate of current deposition on the bias disk (ﬁg. 4.30), the varying ﬁeld
maximum had little eﬀect upon the diﬀusing electrons (ﬁg. 4.29). The extraction maxima had a
much larger eﬀect upon the distribution, particularly after the ﬁeld maximum surpassed the radial
ﬁeld maximum. In agreement with Benitez et al., increasing the extraction maximum’s magnitude
above that of the radial maximum caused a change in the plasma’s diﬀusion characteristics (ﬁg.
4.33) [3]. This suggests that changing the magnetic maximum relative to the radial maximum
at the wall, or vise versa, greatly aﬀects the plasma’s diﬀusion characteristics. However, without
corresponding radial measurements, it cannot be said whether this is a global or only a local eﬀect.
Finally, changing the magnetic minimum aﬀected both the energy content and intensity of electrons
diﬀusing out of the plasma.

The measurements revealed several correlations between the energy content of the ion source
bremsstrahlung and diﬀusing electrons. However, the decreasing electron current appears inversely
correlated with the intensity of x-rays emitted from the ion source. Moreover, the energy content
in both the bremsstrahlung and electron distributions increases as the minimum of the ion source’s
magnetic ﬁeld increases.
In section 4.4.6, it was suggested that this could be the result of a
changing spatial distribution of diﬀusing electrons. Although signiﬁcantly smaller, the same eﬀect
was observed in sections 4.4.2 and 4.4.4. This could indicate that the pressure and injection ﬁeld
maximum can measurably aﬀect the energy distribution of diﬀusing electrons (see ﬁgs. 4.23, C.8,
and 4.29). Although, if true, these parameters minimally aﬀect the energy of the diﬀusing electrons.

109

It also implies that, unlike in the case of power, the predominant eﬀect of varying the neutral gas
pressure is not to increase the density of hot electrons.

The high energy peak, or the ’hump’ described in Izotov et al.

[28], may result from the
quasilinear diﬀusion of electrons out of the magnetic trap, as described in section 2.4.3. This
may imply that that measured electron distribution is the combination of two distributions: one
where diﬀusion is caused by scattering4 and one which is quasilinearly diﬀused. However, proving
this hypothesis would be diﬃcult and would require more evidence than was provided by these
measurements.

The correlation between the high energy peak’s amplitude and its central energy does provide
some evidence for this hypothesis. The γ−2
trend is predicted by the quasilinear diﬀusion theory
e
(see eq. 2.39) [73]. On the other hand, the central energy of the high energy peak does not match the
predictions made by Shalashov et al., at least for single frequency diﬀusion. Let us use ﬁgure 4.15
as an example. Assuming BE xt,max/BRF ≈ 1.72, equation 2.43 determines that the central energy
of the high energy peak should occur at 727 keV, which is substantially greater than the measured
peak (550 keV). Furthermore, sections 4.4.5 and 4.4.6 demonstrated that the central energy is
dependent on the magnetic minimum, rather than the ﬁeld maximum in the direction of diﬀusion.
The introduction of at least one other microwave frequency could allow electrons to quasilinearly
diﬀuse out at lower energies [73]. However, no observations of excited electromagnetic modes
during the stable operation of the source have ever been reported. SuSI’s diagnostic array is also
not designed to make such a measurement. However, more traditional diﬀusion mechanisms also
do not explain the observed trends, as described in section 4.4.6. Pitch angle scattering may be
able to produce the observed peaks, but a simulation such as NAM-ECRIS would be necessary to
provide further evidence of the scattering process.

Comparing the electron and bremsstrahlung distributions further suggests that the measured
electron distributions are a combination of two diﬀerent distributions. This was most evident when
comparing the central energy of the high energy peak, spectral temperature, and the resulting ﬁt’s
4These can include either particle-on-particle Coulomb scattering or pitch angle scattering.

Recall that collisional scattering is very improbably at these energies [20].

110

x-intercept. The x-intercept near-perfectly followed the high energy diﬀusion peak as the peak
shifted towards higher energies. On the other hand, the spectral temperature varied slower due to
the convolution of the scattered electrons and those in the high energy peak. Moreover, as Bmin
increases, both the electron and photon distributions’ high energy sides decrease more smoothly.
The distribution of diﬀused electrons in the high energy peak becomes more distinct as its central
energy increases, correlating with an increasingly linear bremsstrahlung high energy tail. Together,
these suggest that the linear portion of the log-scale bremsstrahlung intensity is a measure of the high
energy electron peak. The non-linear region may result from some combination of pitch angle,
electron-electron, and electron-ion collisions. Further investigations, including computational
simulations, are necessary in order to further develop this hypothesis.

The importance of using a variable region for determining the spectral temperature cannot be
understated. Figure 4.41 shows the diﬀerence in calculated spectral temperatures as a function of
power across two diﬀerent operating points. The correction applied to the low ﬁeld measurement
is signiﬁcantly greater than the one with higher ﬁeld strength. Looking back at C.3, we see that
the increasing magnetic ﬁeld results in a qualitative change to the distribution. This result implies
several things: ﬁrst, that the initial guess for the calculation domain for spectral temperature
was more appropriate for higher energy systems. Secondly, a signiﬁcant enough change to the
distribution of photons in the system requires a re-evaluation of the spectral temperatures ﬁtting
region. This is a result of the correlation between the high energy peak and the linear region of
the log-scale bremsstrahlung distribution. The greatest improvement to the algorithm described in
section 3.2.5 would be the addition of a non-a prioir method of determining the maximum energy
of the electron ﬁt. In its current form, the algorithm requires the user to provide knowledge about
the minimum meaningful photon intensity within the distribution, i.e., where the distribution begins
to ﬂatten. Figure 4.35 shows an example of the linear region extending into beyond the 0.5 counts/s
limit. A more objective way to determine the upper limit of the ﬁtting domain would improve the
algorithm.

Lastly, it can be seen that the low energy portion of the electron distribution, the ’low energy

111

Figure 4.41: Bremsstrahlung distributions analyzed using the algorithm described in section 3.2.5
show much better agreement with the electron distribution data than those where an arbitrary
energy domain was blindly applied to all operating points. In this case, the disagreement is larger
for operating points with lower magnetic ﬁelds. Top plot: Bmin = 0.337 T. Bottom plot: Bmin =
0.468 T.

peak’, does vary with varying ion source parameters. In particular, its height and width seem to
vary, most notably, with the injected microwave power, average longitudinal magnetic ﬁeld, and
neutral gas pressure (see ﬁgs. 4.18, C.2, 4.21, C.6, 4.27, 4.31, and 4.34). As discussed earlier, the
residual and ambient magnetic ﬁelds present around and throughout the beamline make analyzing
this peak diﬃcult, and any trends or conclusions non-meaningful. Better understanding of the
escaping electrons in this energy range will require further reﬁnement of both the electron and
bremsstrahlung measurements. First, an electron spectrometer with a 10’s eV precision would
be necessary to measure it properly. SuSI’s beamline is presently incapable of this kind of
precision. Secondly, an x-ray detector that can accurately measure 100 eV - 100 keV photons would
be necessary to measure the bremsstrahlung distribution, which results from this peak.5 Most
importantly, like with the high energy peak, some simultaneous observation of an excited mode,

5This should be done concurrently with measurements in the 100 keV - 1 MeV range.

112

responsible for the diﬀusion of electrons, would be necessary to conﬁrm the hypothesis.

Altogether, the results presented here indicate that previous measurements of the bremsstrahlung
distribution may need to be re-interpreted. Previous measurements, such as those presented by
Noland [62], assume that a electrons obeyed a Maxwell-Boltzmann energy distribution. We have
seen that this is not the case. Future measurements will need to consider the apparent relationship
between the electron high energy peak and the bremsstrahlung distribution. Preferably, future
studies of the ECRIS axial bremsstrahlung will be performed concurrently with measurements of
the axially diﬀusing electrons and radially emitted bremsstrahlung.6 Further experimental studies
of the quasilinear diﬀusion mechanism will be necessary to determine the validity of the hypothesis,
as well as its eﬀect on ECR ion sources’ overall performance.

6The detector should be placed to look into the hexapole pole tip, as this will be the direction

which electrons diﬀuse outward from the center axis.

113

CHAPTER 5

PARAMETER SWEEP

5.1 Goals

It is well understood that the kinetic instabilities turn on and oﬀ depending upon the value of the
magnetic ﬁeld and injected microwave power [83, 81]. The goal of this measurement was to search
for similar thresholds while taking advantage of the ﬂexibility of SuSI’s magnetic ﬁeld. However,
the transient behavior of the extracted beam current is complex and almost immediately revealed
previously unknown qualitative characteristics. In exchange, the new complexity provided a new
perspective for understanding the kinetic cyclotron maser and its eﬀect upon the plasma. This
necessitated multiple ’stability maps’ of SuSI’s parameter space and highlighted the non-linear
relationship between the plasma, Min-B ﬁeld structure, and externally injected microwave power.
Despite this, many interesting and reproducible trends were discovered, pointing toward stabilizing
the plasma and maximizing extracted high charge state currents.

5.2 Parameter Space

A group for four sets of ﬁelds was calculated to probe SuSI’s parameter space, with the intent
of probing the magnetic scaling laws [46]. Contrary to the parameter spaces described in chapter
4, none of the extrema were ﬁxed for the duration of this measurement. This was done to observe
the largest possible range of magnetic minima while preventing the superconducting coils from
quenching. Rather, the ﬁeld maxima were set to either follow or deviate from the scaling laws
throughout each of the chosen parameter spaces. Each of the parameter spaces spanned 8 diﬀerent
magnetic ﬁeld operating points such that Bmin/BRF ∈ [0.55, 0.6,..., 0.9]. Table 5.1 shows the
average of the ratios between the magnetic maxima and cold resonance ﬁeld used in each set of
measurements. Appendix D shows a full list of operating points used. The exception to this

114

Binj/BRF
Bext/BRF

Standard Range 1 Range 2 Range 3 Range 4

3.43
1.95

4.11
1.61

3.96
1.94

4.09
2.02

3.71
1.78

Table 5.1: The ﬁeld maxima used were scaled to either be on average larger or smaller than the
scaling laws prescribe. The values above are there to demonstrate the relative range of values used
as the magnetic minimum was varied. As calculated, the ﬁelds in each of the four ranges
corresponds to Bmin/BRF = 0.725. The standard facility operating point uses Bmin/BRF = 0.711.

procedure was range 4, where operating the source at Bmin/BRF = 0.9 was unsafe for operation.1
In this case, the upper limit was set at Bmin/BRF = 0.85; however, since the intention was to go
higher, table 5.1 shows the of the average ﬁeld ratios including Bmin/BRF = 0.9.

Each of the longitudinal ﬁeld ranges were measured three times using each of the following
hexapole ﬁeld maxima, BHex,wall = 1.1 T, 1.2 T, and 1.322 T (BHex,wall/BRF = 1.71, 1.86, 2.05). For
reference, SuSI’s standard operational ﬁeld setting uses BHex,wall = 1.22 T. During the measurement
the microwave power was swept from 50 W to 550 W in 100 W steps for each magnetic ﬁeld setting.
The extracted beam current and reﬂected microwave power were observed and recorded to determine
if and when the plasma became unstable. A map of the source’s stability characteristics was created
by recording the transient characteristics of the extracted beam current and microwave power at
each operating point used.

5.3 Results

As most of this section will focus on unstable beam currents, it is essential to deﬁne what is
meant by ’unstable.’ This is not an easy task, however, as multiple authors have shown. Shalashov
et al. mathematically determined the unstable plasma to be a result of resonant interactions of
electrons with unstable electromagnetic modes along the quasilinear diﬀusion lines [72]. While
precise, this deﬁnition is impractical to use when the unstable modes cannot be measured. Similarly,
Naselli et al. created a parameter, Is, which can determine when the plasma is unstable [60]. In
essence, this parameter calculates the integral power of all microwave frequencies emitted from
1The plasma became too unstable with large variation in the neutral gas pressure and very large

bursts in x-ray energy.

115

the plasma, which are not reﬂections of the externally injected heating frequency. Once again, this
method requires us to have a way to discriminate the frequencies of the microwave power coming
from the source chamber.

In the absence of microwave spectrometry, we have to look at particle distributions emitted
from the source. Perhaps the best way is to have a scintillator and photomultiplier tube detector
nearby the source to detect bursts of bremsstrahlung radiation coming from the plasma chamber
[81]. The plasma is then unstable when quasi-periodic bursts of x-ray radiation are detected. This
creates other issues as the plasma does not produce battery-like steady states of charge diﬀusion,
there will always be some ﬂuctuation. Not to mention, the so-called ’CW mode’ of the instability
appears to be stable with minimal energy and current ﬂuctuations, but by the theory of Shalashov
et al., is still unstable [28, 73, 69]. So how do we know when the plasma is unstable? Is it when
the beam currents begin ﬂuctuation wildly, or steering is lost through the accelerator’s injection
line? Perhaps it is better to start with what we expect of a stable operating point. Figure 5.1
shows an example beam current and microwave power signal traces from a stable operating point,
as measured on an oscilloscope. An ’unstable’ operating point will then be deﬁned by the eﬀect of
the instability upon the observed microwave power signal and extracted beam current.

The complexity of the electron and ion dynamics makes the application of plasma physics
models to the extracted beam current exceptionally diﬃcult. As such, the results shown here
provide a general understanding of how the extracted beam current is aﬀected by the instabilities
to learn more about the concurrent state of the plasma. Beginning in section 5.3.1, a model for
describing the transient beam current extracted from an unstable plasma is deﬁned. Applying this
model to observed beam currents ’deﬁnes’ the stability of the diﬀerent operating points used in
the parameter sweeping measurements described in the previous section. Comparing the proposed
model to the measured beam current from each of the operating points makes it easier to determine
trends in the ion source plasma’s changing stability characteristics. These results create ’stability
maps’ of the diﬀerent measured parameter spaces. The maps and trends which result from them
will be discussed in section 5.3.2. Sections 5.3.3, 5.3.5, and 5.3.6 then look at the eﬀect that

116

Figure 5.1: Example Ar8+ current (orange) and microwave power (purple) diagnostics while
operating a stable ECR ion source plasma. The signals have been normalized to their maximum
value over a 400 ms period. The extracted Ar8+ beam varies up to several percent even under
stable conditions, in this case δIAr8+ ∼ 7%. The 60 Hz noise, and its harmonics, in the microwave
signal results from the facility power source and is persistent throughout the lab.

speciﬁcally varying the extraction and radial ﬁeld maxima and then the magnetic minimum has
upon the extracted current. In this case, the same operating points described in chapter 4 provide
a way to investigate the eﬀect of the varying ﬁeld extrema. This way, the trends found during the
electron measurement can be more easily applied to trends discovered while observing the extracted
ions. Finally, these results will be brought together in section 5.3.7 which will discuss what the
results in sections 5.3.2, 5.3.3, 5.3.5, and 5.3.6 imply about the model introduced in section 5.3.1
and produce hypotheses as to the nature of the underlying physics behind it.

5.3.1 Beam Current Transient Proﬁles

Let us begin our discussion by looking at how the extracted beam current is aﬀected in time by
the instability. At the beginning of the instability, a burst of microwave energy, lasting for 100’s
- 1000’s ns, is observed [81]. Almost immediately afterward, there is a rapid loss of electrons
from conﬁnement, most often seen as a burst of bremsstrahlung that can last for 100’s us. It is

117

Figure 5.2: Multiple measurements of the transient current of escaping electrons, at diﬀerent
energies, were taken using an oscilloscope. The energy of electrons was selected by varying the
energy in the dipole. The traces were then averaged 128 using the oscilloscopes averaging
function. Those traces were put together, side-by-side, to plot the transient electron energy
distribution. The peak current is 8460 nA at an electron energy of 62.88 keV, almost twice the
central energy of the ’low energy peaks’ seen in chapter 4. Higher energy electrons are minimally
eﬀected by the instability and the high energy peak is unaﬀected by the instability.

possible to observe this burst of electrons from conﬁnement using the procedure for measuring
the escaping electron distribution described in chapters 3 and 4. Figure 5.2 shows an example of
such a measurement on SuSI’s beamline. Following this loss of electrons from conﬁnement, the
extracted ion beam current will experience beam current variations on the order of 1 - 100’s ms.
The literature often focuses on operating points where a majority of the extracted high charge state
beam current is lost [75, 83, 81]. However, the beam current can be aﬀected by the instabilities
in multiple diﬀerent ways. Depending upon the ion source settings, high charge state currents can
either increase or decrease on average. Furthermore, the time scale over which the average current
changes depends upon the injected microwave power and magnetic ﬁeld settings. Figure 5.3 shows
a general transient proﬁle model of the extracted beam current while operating with an unstable
plasma.

118

Figure 5.3: The general semi-quantitative model of the transient ion current during the unstable
operation of an ECR ion source. A1, A2, and A3 represent the amplitudes of the three predominant
period of beam current variations that result from an instability event. The time scales are meant
to represent general ranges over which the variation occurs. Not every instability event will result
in all three and, under most circumstances, there is one predominant feature which best describes
the beam. However, under rare circumstances, all three can occur simultaneously.

A burst of ions, described by (t1, A1), follows the loss of electrons from conﬁnement. The
enhanced extraction of ions typically lasts for 100’s µs to milliseconds and ends with a rapid decrease
in current over 10’s - 100’s µs. This leads to (t2, A2) where the electron and ion populations have
reached a minimum [85]. Over the next 1-10’s ms, the hot electron and ion populations recover.
A2 tends to be the largest for intermediate repetition frequencies (10 - 100 Hz). Under some
circumstances, the extraction of high charge state currents increases above the steady-state level
and reaching a maximum at (t3, A3). After 1-100’s ms, the current will exponentially decay back to
the steady-state level until the next instability event. A3 tends to be the largest for lower frequencies
(0.1 - 10 Hz). The decreasing frequency range described above is no coincidence, as high instability
repetition frequencies prevent the accumulation of high charge state ions. Thus, operating points
with lower instability repetition rates tend to have higher extracted currents, for both average and
instantaneous currents.

The ’ion burst’ eﬀect is likely an instability driven afterglow event. Measurements have
demonstrated that up to 10% of the electrons escape from the conﬁnement following an instability
event [85]. Other measurements estimate that enough electrons are lost to increase the plasma

119

potential to over 1 kV [83]. The ’ion loss’ eﬀect results from a loss in the ion population due to this
burst, as well as a suppressed ionization rate. The physics of the ’over-extraction’ eﬀect is poorly
understood and is one of the major focuses of this study.

In practice, the extracted beam current will usually, but not always, have a single dominant
feature to it, either A1, A2, or A3. In many cases, one of the values is ∼ 0. The diﬀerent ’transient
modes’ can also be distinguished by the transient pattern of the microwaves emitted from the source
chamber. Operating points where A1 is dominant tend to have a tall, broad burst of microwave
power at the beginning of the instability. This burst decays roughly on the time scale of the
recovering beam current. If A2 is the dominant amplitude, then it is most typical for there to be
a massive burst of microwave power, which decays on an order of magnitude faster time scales
than the beam current recovers. A ’hump’ in the measured microwave power follows this burst and
recovers with the beam current. Lastly, when A3 is dominant, the microwave seldom demonstrates
a burst of microwave power. Rather, there is usually a ’hump’ which decays with the recovering
beam current. The ’hump’ seen in the latter two cases likely results from a changing reﬂection
coeﬃcient in the plasma chamber, rather than plasma emission. The fact that it decays alongside
the varying beam current, and therefore plasma density, supports this idea. Without a way of
measuring the frequency of the emitted microwaves, it is impossible to be certain. Figures 5.4, 5.5,
and 5.6 demonstrate all three ’transient modes’, respectively.

If the time between instability events is shorter than the production time of high charge state ions,
then the system’s ability to produce those high charge state ions is eﬀectively suppressed [83]. The
plasma never reaches a steady-state that balances the production of ions between diﬀerent atomic
interactions and so there is a greater emphasis on the production of lower charge state ions. Ion
burst and ion loss dominated transient modes are most often responsible for the observed decrease
in ECRIS performance. Although, the repetition frequency is, by far, the most detrimental quantity
to high charge state production, as it prevents the accumulation of ionizing electrons altogether.

Lastly, each of the modes acts diﬀerently across each of the extracted charge states. The ion
burst and ion loss dominated modes respond with an increased average beam at low charge states

120

Figure 5.4: An example of the observed time resolved beam current (orange) and microwave
power (purple) diagnostics seen during a transient proﬁle dominate by the ’ion burst’ eﬀect.
Notice how there is a large but short-lived burst in ions immediately after the burst of microwave
energy. While present, the ’ion loss’ proﬁle is not dominant. The majority of the microwave
power is emitted over a several milliseconds. Ar8+, Bmin = 0.39 T, BHex,wall = 1.2 T, Pµ = 150 W,
pAr = 131 nTorr.

and decreased beam at high charge states, as was reported in the literature [81]. The time scale of
the low charge state enhancement matches that of the ’ion loss’ at higher charge states. Interestingly
the over-extraction mode demonstrates the opposite eﬀect, decreasing the extracted beam current
for low charge states and increasing it for high charge states. Once again, this occurs on the order
of tens of milliseconds to seconds, being far longer than the observed losses of electrons from
conﬁnement. Figures 5.7, 5.8, and 5.9 show a comparison between the transient modes for Ar2+
and Ar8+.

At ﬁrst glance, the ’over-extraction’ eﬀect would appear to appreciable change the average
extracted current from the ion source plasma, although in many cases this is not true. Figure 5.10
shows the extracted beam currents of argon ions from the 1+ to 12+ charge states. Notice how,
despite the large increases in amplitude, the time over which that increase exists is relatively short.

121

Figure 5.5: An example of the observed time resolved beam current (orange) and microwave
power (purple) diagnostics seen during a transient proﬁle dominate by the ’ion loss’ eﬀect. Notice
that almost 50% of the steady state current is lost following the microwave burst. While present,
the ’ion burst’ proﬁle is not dominant, and can only be seen by focusing on the 100’s µs time scale
(bottom plot). The majority of the microwave power is emitted over 10’s µs and very rapidly
decays. The ’hump’ in the emitted power is believed to result from a changing reﬂection
coeﬃcient within the plasma chamber. Ar8+, Bmin = 0.48 T, BHex,wall = 1.2 T, Pµ = 250 W, pAr =
131 nTorr.

Figure 5.11 emphasizes this by comparing the peak, average, and steady-state beam currents to
one another for the same operating point. While the average extracted current is similar to the
steady-state current, the peak current is substantially larger. The overall eﬀect also scales with
increasing charge state. The percentage increase in the extracted beam current can be increased by
setting the bias disk voltage to positive values.

As a ﬁnal remark, under certain conditions, it is possible to convert one transient mode into
another, usually by varying the neutral gas pressure. Although common, the eﬀect depends on the
magnetic ﬁeld and injected power sustaining the plasma. Figure 5.12 shows an example of this
phenomenon by increasing the neutral gas pressure in the plasma chamber.

122

Figure 5.6: An example of the observed time resolved beam current (orange) and microwave
power (purple) diagnostics seen during a transient proﬁle dominate by the ’over-extraction’ eﬀect.
Extracted Notice the almost 30% increase in extracted beam current following the exceptionally
small burst of microwave power from the system. While present, the ’ion loss’ proﬁle is not
dominant. Most importantly, this repetition frequency of this transient proﬁle is on the order of
10’s Hz, compared to ﬁgures 5.4 and 5.5 which are on the order of 100’s Hz (top plot). The
emitted microwave power is exceptionally small compared to the previous two cases (bottom plot).
The ’hump’ in the emitted power is believed to result from a changing reﬂection coeﬃcient within
the plasma chamber. Ar8+, Bmin = 0.48 T, BHex,wall = 1.32 T, Pµ = 350 W, pAr = 131 nTorr.

123

Figure 5.7: ’Ion burst’ dominated transient proﬁles are characterized by an increase in the average
extracted current at low charge states (Ar2+, blue, top trace) and a decreased in the average
extracted current for high charge states (Ar8+, orange, bottom trace). Both current traces have
been normalized to the largest value in their respective scope traces. Bmin = 0.45 T, BHex,wall =
1.2 T, Pmu = 50 W, pAr = 141 nTorr, VBD = -17 V.

124

Figure 5.8: ’Ion loss’ dominated transient proﬁles are characterized by an increase in the average
extracted current at low charge states (Ar2+, Blue, bottom trace) and a decreased in the average
extracted current for high charge states (Ar8+, orange, top trace). The plasma is also perturbed to a
greater degree, compared to the ’ion burst’ case. High charge state ion losses can be upwards of
80 - 100% of steady state beam currents, with much larger bursts in current for low charge states.
Both current traces have been normalized to the largest value in their respective scope traces.
Bmin = 0.52 T, BHex,wall = 1.2 T, Pmu = 350 W, pAr = 123 nTorr, VBD = -17 V.

125

Figure 5.9: Unlike the other two cases, ’over-extraction’ dominated transient proﬁles are
characterized by a decrease in the average extracted current at low charge states (Ar2+, blue, top
trace) and an increase in the average extracted current for high charge states (Ar8+, orange, top
trace). Both current traces have been normalized to the largest value in their respective scope
traces. Bmin = 0.42 T, BHex,wall = 1.2 T, Pmu = 250 W, pAr = 131 nTorr, VBD = 20 V.

Figure 5.10: Bmin = 0.42 T, BHex,wall = 1.2 T, Pµ = 350 W, pAr = 131 nTorr, VBD = +20 V.

126

Figure 5.11: Bmin = 0.42 T, BHex,wall = 1.2 T, Pµ = 350 W, pAr = 131 nTorr. VBD = +20 V

Figure 5.12: Increasing the neutral gas pressure often changes the dominant transient mode of the
extracted beam current during unstable operating. Bmin = 0.515 T, BHex,wall = 1.2 T, Pmu = 350
W. (Top) pAr = 96 nTorr, (Middle) pAr = 123 nTorr, (Bottom) pAr = 159 nTorr.

127

5.3.2 Stability Maps

It was initially believed that a ’stability threshold’ existed in the magnetic ﬁeld vs. power parameter
space of the ion source [81]. A non-linear curve separated stable and unstable regions in this
parameter space, and even small variations in either quantity could change the stability of the
plasma. The neutral gas pressure, neutral species, and the source’s extraction potential controlled
the position of the threshold [83, 81]. Our ﬁrst measurement attempted to ﬁnd this threshold
in SuSI, albeit over a broader set of magnetic ﬁeld parameters. While the stability condition is
still sensitive to small parameter changes, the system lacks a singular division between stable and
unstable operating points.2 Instead, large and distinct unstable regions surround a smaller number
of stable operations points. A 2D color map replaced the stability threshold with the diﬀerent colors
representing diﬀerent predominant transient modes (see 5.3.2). Figures 5.13, 5.14, 5.15, and 5.16
show the colormaps generated by sweeping the parameter space in the SuSI ECR ion source. Each
ﬁgure shows the results from one set of pre-calculated magnetic ﬁeld parameters, as described in
section 5.2 and table 5.1, measured across three hexapolar ﬁeld maxima.

Designation

Stable
Ion Burst
Ion Loss

Over-Extraction

Description

Color Reference

No Oscillation or Microwave Burst

100’s-1000’s Hz
10’s-100’s Hz
0.1’s - 10’s Hz

Table 5.2: Each of the diﬀerent transient proﬁles has been assigned a diﬀerent color. Generally
speaking each of the diﬀerent proﬁles exists over a diﬀerent repetition frequency range, with ion
bust dominated events occurring over the smallest time scale and over-extraction events occurring
over the longest.

Immediately, we can see several trends that appear through the stability maps. The ’ion burst’
transient proﬁles (light blue) prefer lower energy density operating points, with low magnetic ﬁelds
or injected microwave powers.
’Ion loss’ (yellow) and ’over-extraction’ (red) transient proﬁles
appear most often in operating points with large energy densities, high microwave powers and

2Even small changes to the plasma parameters can make the plasma unstable.

In one case,
increasing the current by 1 A, or about 2%, through a single solenoid coil induced a periodic
instability.

128

Figure 5.13: Field maps for operating points where the average injection and extraction maxima
were above the limits described by the scaling laws: BInj,max/BRF ≈ 4.09,
BE xt,max/BRF ≈ 2.02.

Figure 5.14: Field maps for operating points where the average injection and extraction maxima
were above the limits described by the scaling laws: BInj,max/BRF ≈ 3.71,
BE xt,max/BRF ≈ 1.78.

magnetic ﬁelds. The presence and size of a stable region depend upon the topology of the magnetic
ﬁeld. The measurement with the most stable regions appeared in range 1 (ﬁg. 5.13), where the
systems used the largest magnetic ﬁelds.
In general, larger ﬁelds at extraction correlate with a
higher number of stable operating points. Lower extraction ﬁelds appear to make the ion burst
proﬁle occur more regularly, even at higher energy densities. The injection side ﬁeld appears not
to aﬀect the ion source plasma’s stability characteristics, at least over the measured range.

Stronger magnetic ﬁelds at the wall correlate with an increased probability of the plasma

129

Figure 5.15: Field maps for operating points where the average injection and extraction maxima
were above the limits described by the scaling laws: BInj,max/BRF ≈ 3.43,
BE xt,max/BRF ≈ 1.95.

Figure 5.16: Field maps for operating points where the average injection maximum was larger
than and extraction maxima small than the limits described by the scaling laws:
BInj,max/BRF ≈ 4.11, BE xt,max/BRF ≈ 1.61.

becoming unstable, as can be seen in ﬁgures 5.13, 5.14, and 5.15. The larger hexapole ﬁeld also
causes the two unstable regions to converge upon the stable region. In some cases, almost all the
stable operating points disappear at the highest hexapole ﬁeld strength used. The ﬁnal experimental
range reversed that trend, as the increasing radial ﬁeld correlated to an increase in the number of
stable operating points. The third map in ﬁgure 5.16 lacks the presence of over-extraction operating
points altogether. This case also happens to have the lowest ﬁeld extraction ﬁeld strength of all the
operating points.

130

We can look at the diﬀerence in overall performance between the diﬀerent experimental ranges
as well. Figures 5.17, 5.18, and 5.19 show the percentage of operating points measured with
average extracted Ar8+ currents larger than 60, 80, and 100 µA, respectively. These plots were
made by counting up the number of operating points with average currents and dividing them by
the total number of operating points in each range (N = 48). Recall that in range 4 all operating
points with Bmin/BRF = 0.9 were excluded so that NR4 = 42. Through this metric, we see that
the operating points with lower extraction maxima have, overall, larger extracted currents. The
injection side magnetic ﬁeld had little eﬀect on the ion source’s performance (ranges 1 and 4 have
large injection side maximums).
Increasing the radial ﬁeld maximum had little, or at most an
inconsistent eﬀect, on these statistics, with no visible pattern seen between the diﬀerent ranges.
The number of stable operating points does not seem to play a role in determining trends in the
average extracted current. Both operating ranges 1 and 3, which had the largest number of stable
operating points, are consistently outperformed by ranges 2 and 4. The trend in range 4 is the most
interesting by comparison as the lack of a large number of ’over-extraction’ operating points does
not aﬀect the average extracted Ar8+ current. By comparison, ranges 1 and 3 have an abundance of
’over-extraction’ operating points, but with very few operating points capable of boasting average
extracted currents up to 100 µA.

Conversely, ﬁgures 5.20, 5.21, and 5.22 show the percentage of operating points measured with
maximum losses of Ar8+ currents larger than 30%, 60%, and 80%, respectively. The losses measure
the decrease in current from the steady-state ion population in ﬁgure 5.3, making this a measure
of A2. Ranges 2 and 4, on average, lost the most current per instability event. Range 1, with the
largest injection and extraction ﬁeld maxima, performed best by this metric in large part due to the
number of stable operating points within its parameter space. Overall, the number of operating
points which lose more than 80% of their steady-state current is quite low, with a maximum of ∼
20% in range 4. Ranges 2,3, and 4 all perform similarly by this metric.

Lastly, we can look for similar trends in the peak current across each of the experimental ranges.
This is an indirect measure of either A1 or A3, as the peak is determined without reference to the

131

Figure 5.17: Percentage of measured operating points with average currents greater than 60µA.

Figure 5.18: Percentage of measured operating points with average currents greater than 80µA.

132

Figure 5.19: Percentage of measured operating points with average currents greater than 100µA.

Figure 5.20: Percentage of measured operating points where beam loss was greater than 30%.

133

Figure 5.21: Percentage of measured operating points where beam loss was greater than 60%.

Figure 5.22: Percentage of measured operating points where beam loss was greater than 80%.

134

Figure 5.23: Percentage of measured operating points with peak currents greater than 60µA.

steady-state current or the microwave burst (the distinction between ’ion burst’ and ’over-extraction’
is ignored). Figures 5.23, 5.24, and 5.25 show the percentage of operating points with peak extracted
Ar8+ currents larger than 60, 80, and 100 µA, respectively. Once again, ranges 2 and 4 perform the
best by this metric, with most of their peak currents being greater than 100 µA.

Together, these metrics seem to indicate that having a lower magnetic maximum at extraction
has a large eﬀect upon the stability of the plasma. While increasing the hexapole ﬁeld made
the plasma more unstable overall, its eﬀect upon the average ion source performance appears
negligible, at least across the measured parameter ranges. As mentioned previously, ranges 2 and
4 were designed with extraction ﬁelds close to or less than what is prescribed by the scaling laws
(sec. 2.3.4). There is no doubt that the extraction side ﬁeld aﬀected both the performance and
overall stability characteristics of the plasma. The magnetic minimum also plays a very large role
in determining how the plasma will become unstable, with there being a clear correlation between
the energy density of the plasma and how the instability aﬀects the extracted beam current. The
next section focuses on direct measurements of the changes that varying plasma parameters cause

135

Figure 5.24: Percentage of measured operating points with peak currents greater than 80µA.

Figure 5.25: Percentage of measured operating points with peak currents greater than 100µA.

136

by directly measuring the transient beam current proﬁle on an oscilloscope.

5.3.3 Varying the Hexapolar Field Maximum

We begin our discussion by looking at the radial ﬁeld’s eﬀect upon the measured ion transient proﬁle.
The same operating points used in chapter 4 were used here with two notable exceptions. First,
the extraction voltage and einzel lens voltages were ﬁxed at 20 kV; however, the puller remained
grounded. Secondly, the correct ’control’ operating point was used (Bmin = 0.4 T, BInj,max = 2.367
T, BExt,max = 1.100 T, Pµ = 350 W, Pressure = 213 nTorr, BHex,wall = 1.217 T).3

Figure 5.26 shows the beam current for diﬀerent hexapole ﬁeld strengths. The ’ion burst’ eﬀect
dominates the beam current’s transient proﬁle, with A1 > A2 > A3 = 0. Overall, changing the
hexapole had a negligible qualitative eﬀect over the measured range. Quantitatively, the increasing
hexapole ﬁeld saw the beam current decrease on average. Figure 5.27 demonstrates this more
clearly through the maximum, average, and minimum currents measured over a 200 ms period.
Across most of the domain, the maximum and minimum currents also begin to converge towards
an increasingly smaller average current.

The microwave signal also shows similar trends, with the repetition frequency and amplitude of
large microwave bursts (large area beneath peak in microwave power) increasing with the hexapolar
ﬁeld. This gives way to a gradual decline in the maximum burst amplitude, as seen in ﬁgure 5.28.
The microwave diagnostic shows an increase in the repetition frequency of the instabilities, with
relatively little change compared to the plasma’s response to a changing extraction maximum. The
Fourier transform of the microwave power signal, in ﬁgure 5.29, shows the shift in the frequency
content of the emitted microwave signals more clearly. Initially, the power across all frequencies
increases but then the signal appears to settle, but then begins to settle at higher ﬁeld strengths.
Figure 5.30 further clariﬁes this, showing that average microwave power is insensitive to the
hexapole ﬁeld.

3The wrong operating point was used for the ’control’ operating point during the electron

measurement.

137

Figure 5.26: Ar8+ beam current for a varying hexapole ﬁeld maxima: (Top left) BHex,wall = 1.12
T, (Top right) BHex,wall = 1.18 T, (Bottom left) BHex,wall = 1.22 T, (Bottom right) BHex,wall = 1.28
T. Bmin = 0.4 T, Pµ = 350 W, pAr = 213 nTorr. Signals have been treated with a digital low pass
ﬁlter using the Matlab signal processing toolkit, with a cutoﬀ frequency of 333 kHz.

Figure 5.27: Average, maximum, and minimum beam current measurements of extracted Ar8+
beam current for varying hexapole ﬁeld maxima: Bmin = 0.4 T, Pµ = 350 W, pAr = 213 nTorr.

138

In chapter 4, the electron measurement demonstrated that the increasing hexapole ﬁeld had little
eﬀect on the system’s energy content, but did seem to increase the diﬀusion of hot electrons out of
the system. Interestingly, the increased electron current coming from the plasma (see ﬁgure 4.24)
is not reﬂected with an increase in extracted ions, as we would expect if the ions were diﬀusing
ambipolarly. There is also little change to the plasma except for a slow decrease in the extracted
current at higher ﬁeld strengths.
Instability events do appear to occur more often, which may
indicate an increase in the density of electrons outside of the loss cone. This could explain the
increased current seen in chapter 4, as the increased current could result from more wave-particle
interactions between well-conﬁned electrons and excited waves. The better-conﬁned electrons
could also output more microwave power into the background plasma. Alternatively, an increased
axial diﬀusion rate, resulting from changing conﬁnement parameters, could globally aﬀect the
non-linear interactions of the electrons with the excited waves. Further study of this eﬀect may
require more sophisticated diagnostics to fully understand the eﬀect the hexapole ﬁeld plays on the
system.

139

Figure 5.28: Microwave power signals for a varying hexapole ﬁeld: (Top left) BHex,wall = 1.12 T,
(Top right) BHex,wall = 1.18 T, (Bottom left) BHex,wall = 1.22 T, (Bottom right) BHex,wall = 1.28 T.
Bmin = 0.4 T, BExt,max = 1.1 T, Pµ = 350 W, pAr = 213 nTorr. Microwave power has been oﬀset by
it’s minimum value, over the entire 100 ms measurement period, to account for the AC coupling
of the oscilloscope. Signals have been treated with a digital low pass ﬁlter using the Matlab signal
processing toolkit, with a cutoﬀ frequency of 333 kHz.

140

Figure 5.29: Fourier transform of the microwave power signals emitted from the ion source for a
varying hexapolar ﬁeld: (Top left) BHex,wall = 1.12 T, (Top right) BHex,wall = 1.18 T, (Bottom left)
Bhex,max = 1.22 T, (Bottom right) Bhex,max = 1.28 T. Bmin = 0.4 T, BExt,max = 1.1 T, Pµ = 350 W,
pAr = 213 nTorr.

Figure 5.30: Average and maximum peak microwave power for a varying hexapole ﬁeld maxima:
Bmin = 0.4 T, BExt,max = 1.1 T, Pµ = 350 W, pAr = 213 nTorr.

141

5.3.4 Eﬀect of the Injection Side Magnetic Field upon Plasma Stability

In chapter 4, it was shown how little of an eﬀect the injection side magnetic ﬁeld had upon the
diﬀusion of electrons out of the system. The stability maps reinforced this conclusion, demonstrating
that the injection side magnetic ﬁeld maximum appears largely uncorrelated to the plasma’s stability
or performance, at least over the measured domains. By looking speciﬁcally at the ion beam current,
we do see a degree of meaningful change for the ﬁrst time. Increasing BInj,max over the domain
described in chapter 4, causes the extracted beam current to increase very slowly. Initially, the ’ion
burst’ eﬀect dominates current transience. As the ﬁeld increases, both A1 and A2 begin to increase,
though the ’ion burst’ eﬀect remains the predominant feature of the transient proﬁle.Figures 5.31
and 5.32 show how the current varies across the magnetic ﬁeld domain. The increase in burst
amplitude correlates with a decrease in the repetition frequency of the instabilities. As the injection
side ﬁeld increases, the maximum and minimum measured currents begin diverging from the
average extracted current.

Once again, the eﬀect on the repetition frequency is easiest to see in ﬁgures 5.33 and 5.34, which
shows the microwave power signals and their respective Fourier transforms for selected operating
points. While the microwave power peak amplitude and signal frequency content generally decrease
over the measured domain, the average emitted power does appear constant. Figure 5.35 shows how
both the maximum and average microwave power are largely unaﬀected by an increasing injection
side maximum. This trend is a result of the width of individual microwave bursts increasing, despite
the decrease in their amplitude.

142

Figure 5.31: Ar8+ beam current for a varying injection ﬁeld maxima: (Top Left) BInj,max = 2.20
T, (Top Right) BInj,max = 2.33 T, (Bottom Left) BInj,max = 2.45 T, (Bottom Right) BInj,max = 2.58
T. Bmin = 0.4 T, Pµ = 350 W, pAr = 213 nTorr. Signals have been treated with a digital low pass
ﬁlter using the Matlab signal processing toolkit, with a cutoﬀ frequency of 333 kHz.

Figure 5.32: Average, maximum, and minimum Ar8+ beam currents, measured over a 100 ms
period, for a varying injection ﬁeld maxima: Bmin = 0.4 T, Pµ = 350 W, pAr = 213 nTorr.

143

Figure 5.33: Microwave power signals for a varying injection ﬁeld maxima: (Top Left) BInj,max =
2.20 T, (Top Right) BInj,max = 2.33 T, (Bottom Left) BInj,max = 2.45 T, (Bottom Right) BInj,max =
2.58 T. Bmin = 0.4 T, Pµ = 350 W, pAr = 213 nTorr. Microwave power has been oﬀset by it’s
minimum value, over the entire 100 ms measurement period, to account for the AC coupling of
the oscilloscope. Signals have been treated with a digital low pass ﬁlter using the Matlab signal
processing toolkit, with a cutoﬀ frequency of 333 kHz.

144

Figure 5.34: Fourier transform of the microwave power signals emitted from the ion source for a
varying injection ﬁeld maximum: (Top Left) BInj,max = 2.20 T, (Top Right) BInj,max = 2.33 T,
(Bottom Left) BInj,max = 2.45 T, (Bottom Right) BInj,max = 2.58 T. Bmin = 0.4 T, Pµ = 350 W, pAr
= 213 nTorr.

Figure 5.35: Average and maximum output microwave power, measured over a 100 ms period, for
a varying injection side ﬁeld maximum: Bmin = 0.4 T, Pµ = 350 W, pAr = 213 nTorr.

145

5.3.5 Varying the Extraction Field Maximum

Two rather interesting things happen as the maximum magnetic ﬁeld strength at extraction increases
while the plasma is unstable. First, the instability repetition rate increases while both A1 and A2
decrease. At the same time, the average current increases to a maximum at BExt,max = 1.15 T,
BExt,max/BRF = 1.8, and then decreases as the ﬁeld continues to increase. The standard deviation
of the current decreases on average until the average current reaches a maximum; the standard
deviation then reaches a local maximum at or around 1.2 T. Figures 5.36, 5.37, and 5.38 show
these trends through direct measurements and calculated values of scope traces of extracted Ar8+
current. As seen in the previous section, there appears to be an inverse correlation between the size
of the instability induced perturbation and its repetition frequency.

The microwave power emitted from the source varies quite dramatically over the measured
domain. As the ﬁeld strength increases, the amplitude of emitted microwaves ﬁrst increases then
decreases as the repetition rate of microwave bursts increases. Figure 5.39 shows examples of the
emitted microwave power, as seen on an oscilloscope, with their corresponding Fourier transforms
in ﬁgure 5.40.
Initially, the repetition frequency and emitted peak power increase. While the
frequency continues to increase, the power emitted per instability decreases. At the highest ﬁeld
setting, the emitted microwave power is almost indistinguishable from electronic noise, particularly
in ﬁgure 5.40. Figure 5.41 compares the average and maximum microwave power across all
operating points measured. The peak microwave power reaches a maximum at a ﬁeld strength of
1.07 T and continues to decrease for higher magnetic ﬁelds. However, this increase in the peak
amplitude correlates with a decrease in the pulse duration, which maximizes the average power at
1.02 T.

Notably, there is no dramatic change in the behavior of either the emitted ions or microwave
power at 1.2 T. The electron and bremsstrahlung distributions in chapter 4 saw a dramatic decrease
in their respective intensities after 1.2 T (see 4.4.5. However, concurrent measurements of the
bremsstrahlung distribution taken while measuring the ions show that the eﬀect still occurs. Figure
5.42 compares the trends in the total number of observed photons during both the ion and electron

146

Figure 5.36: Ar8+ beam current for a varying extraction ﬁeld maxima: (Top Left) BExt,max = 0.99
T, (Top Right) BExt,max = 1.07 T, (Bottom Left) BExt,max = 1.15 T, (Bottom Right) BInj,max =
1.23 T. Bmin = 0.4 T, Pµ = 350 W, pAr = 213 nTorr. Signals have been treated with a digital low
pass ﬁlter using the Matlab signal processing toolkit, with a cutoﬀ frequency of 333 kHz.

measurements. In both cases, the total number of observed photons decreases dramatically. There
is a 48% decrease while the extraction voltage is ﬁxed at 20 kV and 55% while the electrode is
grounded. While the extracted beam current and emitted microwave power sharply decrease beyond
1.6 T, we only observe the beginnings of the trend towards decreased electron diﬀusion (see ﬁgures
5.37 and 5.41). More measurements are necessary to determine how the ion distribution is aﬀected
by the sudden change in diﬀusing electrons’ behavior.

147

Figure 5.37: Average, maximum, and minimum Ar8+ beam currents, measured over a 100 ms
period, for a varying extraction ﬁeld maxima: Bmin = 0.4 T, Pµ = 350 W, pAr = 213 nTorr. The
dashed vertical line show where BExt,max/BRF = 1.8.

148

Figure 5.38: Trends of the average and standard deviations of the average extracted Ar8+ beam
current as the extraction ﬁeld maximum is varied, normalized to the largest values in their
respective sets: Bmin = 0.4 T, Pµ = 350 W, pAr = 213 nTorr. The vertical dashed line shows where
BExt,max/BRF = 1.8.

149

Figure 5.39: Microwave power signals for a varying extraction ﬁeld maxima: (Top Left) BExt,max
= 0.99 T, (Top Right) BExt,max = 1.07 T, (Bottom Left) BExt,max = 1.15 T, (Bottom Right)
BInj,max = 1.23 T. Bmin = 0.4 T, Pµ = 350 W, pAr = 213 nTorr. Microwave power has been oﬀset
by it’s minimum value, over the entire 100 ms measurement period, to account for the AC
coupling of the oscilloscope. Signals have been treated with a digital low pass ﬁlter using the
Matlab signal processing toolkit, with a cutoﬀ frequency of 333 kHz.

150

Figure 5.40: Fourier transform of the microwave power signals emitted from the ion source for a
varying extraction ﬁeld maxima: (Top Left) BExt,max = 0.99 T, (Top Right) BExt,max = 1.07 T,
(Bottom Left) BExt,max = 1.15 T, (Bottom Right) BInj,max = 1.23 T. Bmin = 0.4 T, Pµ = 350 W,
pAr = 213 nTorr.

Figure 5.41: Maximum output microwave power for a varying extraction ﬁeld maxima: Bmin =
0.4 T, Pµ = 350 W, pAr = 213 nTorr. The dashed vertical line shows where BE xt/BRF = 1.8.

151

Figure 5.42: Average and maximum output microwave power, measured over a 100 ms period, for
a varying extraction ﬁeld maxima: Bmin = 0.4 T, Pµ = 350 W, pAr = 213 nTorr. The dashed
vertical line shows where BE xt/BRF = 1.8.

152

5.3.6 Source stability for a varying magnetic minimum

In chapter 4, it was seen that the magnetic minimum controls the central energy of the high energy
peak. However, that trend tells us little about the plasma’s stability or how it is aﬀected by a varying
magnetic minimum. To that end, ﬁgures 5.43 and 5.44 show eight examples of the changing
extracted beam current as the magnetic minimum is varied, as seen by an oscilloscope. Initially,
the system extracts approximately 100 µ A of Ar8+ and displays a combined ion burst and ion
loss transient proﬁle.
Increasing the minimum magnetic ﬁeld strength causes both the average
extracted current and the ion bursts amplitude to increase. At the same time, A2 decreases as
the ’ion loss’ eﬀect becomes less prominent. At Bmin = 0.43, the repetition frequency begins to
decrease noticeably, and the beam current begins showing signs of ’over-extraction’. At this point,
A1 increases with the ﬁeld strength, A2 becomes visible once again, and the scope trace shows at
least one instance where A3 is non-zero. At a value of 0.47 T, Bmin/BRF = 0.73, the beam current
reaches its maximum while minimizing its upward and downward deviations. Figure 5.45 looks at
the average, maximum, and minimum currents observed over a 200 ms period of observing each
operating point. The maximum current diverges away from the average current between 0.4 <
Bmin < 0.44 and converges at higher magnetic ﬁelds. In this case, the increasing magnetic ﬁeld
appears to bring a degree of stability to the plasma, at least over the measured domain. The average,
maximum, and minimum currents show this by converging together as the average beam current
increase.

It is also important to look at the microwaves emitted from the system during this measurement.
The increasing magnetic ﬁeld leads to an increase in the burst repetition frequency and while
decreasing their amplitude. Figures 5.46 and 5.47 show scope traces of the measured microwave
power for various operating points, with the corresponding Fourier transforms in ﬁgures 5.48
and 5.49. The frequency content of emitted microwaves changes appreciably as the minimum
ﬁeld increases. Initially, the height, duration, and repetition frequency of the emitted microwaves
approximately uniform. At higher ﬁelds, the emitted power spreads out over a wider region of the
frequency domain. The higher ﬁeld strengths restore some of the uniformity of the signal, but this

153

Figure 5.43: Ar8+ beam current for varying magnetic minimums (Top left) Bmin = 0.34 T, (Top
right) Bmin = 0.35 T, (Bottom left) Bmin = 0.37 T, (Bottom Right) Bmin = 0.4 T. Pµ = 350 W, pAr
= 213 nTorr. Signals have been treated with a digital low pass ﬁlter using the Matlab signal
processing toolkit, with a cutoﬀ frequency of 333 kHz.

comes with a dramatic decrease in the repetition frequency of the emitted microwaves. Eventually,
the quasi-periodically emitted bursts of microwave power decrease beneath the system’s noise ﬂoor.
Figure 5.50 compares average emitted microwave power to the maximum measured power from
the plasma chamber. There is a local maximum in the maximum emitted microwave power at Bmin
∼ 0.4, which then decreases at either higher or lower ﬁeld strengths. At larger ﬁeld strengths, the
maximum emitted power decreases rapidly while the average power remains constant. This can
be seen in ﬁgure 5.47 where the microwave power signal becomes noisy, but otherwise appears
stable. This is also where the extracted current reaches a maximum (see ﬁgure 5.45). Once again,
there is a strong correlation between the lack of magnitude and frequency variations in the emitted
microwave power and large extracted currents. However, a vital measurement of the ion source is
absent from this discussion: the predominant frequencies emitted during the unstable event. While
both beam current and microwave signals reach a degree of stability, it is diﬃcult to make a strong
claim about the stability of the plasma in the absence of knowledge of the microwaves’ frequency

154

Figure 5.44: Ar8+ beam current for varying magnetic minimums (Top left) Bmin = 0.43 T, (Top
right) Bmin = 0.44 T, (Bottom left) Bmin = 0.46 T, (Bottom Right) Bmin = 0.47 T. Pµ = 350 W, pAr
= 213 nTorr. Signals have been treated with a digital low pass ﬁlter using the Matlab signal
processing toolkit, with a cutoﬀ frequency of 333 kHz.

composition.

The data acquired here reﬂects several of the trends seen in sections 5.3.3 and 5.3.5. Initially,
increasing the magnetic minimum increases the amplitude of ion burst instabilities events while
decreasing the overall instability repetition frequency. Once again, the extracted current correlates
less with the total emitted power per instability event and more about the number of instability
events that occur over time. To that end, a wider loss cone pitch angle correlates to less frequent
large bursts of microwave power and electron losses from the system. The qualitative change in
the behavior seen in ﬁgures 5.43 and 5.44 is unique to varying the magnetic minimum. As seen in
section 4.4.6, varying the magnetic minimum was also the only way to produce substantial changes
in the energy distribution of either the measured electrons or bremsstrahlung. As a result, the
qualitative change may result from a change in excited photons’ frequency or the energy of lost
electrons. Even if that were the case, this does not explain why the plasma seemingly becomes
stable as the magnetic minimum increases.

155

Figure 5.45: Average, maximum, and minimum Ar8+ beam currents, measured over a 100 ms
period, for a varying magnetic minimum. (Top left) Bmin = 0.43 T, (Top right) Bmin = 0.44 T,
(Bottom left) Bmin = 0.46 T, (Bottom Right) Bmin = 0.47 T. Pµ = 350 W, pAr = 213 nTorr

156

Figure 5.46: Ar8+ beam current for varying magnetic minimums (Top left) Bmin = 0.34 T, (Top
right) Bmin = 0.35 T, (Bottom left) Bmin = 0.37 T, (Bottom Right) Bmin = 0.4 T. Pµ = 350 W, pAr
= 213 nTorr. Microwave power has been oﬀset by it’s minimum value, over the entire 100 ms
measurement period, to account for the AC coupling of the oscilloscope. Signals have been
treated with a digital low pass ﬁlter using the Matlab signal processing toolkit, with a cutoﬀ
frequency of 333 kHz.

157

Figure 5.47: Ar8+ beam current for varying magnetic minimums (Top left) Bmin = 0.43 T, (Top
right) Bmin = 0.44 T, (Bottom left) Bmin = 0.46 T, (Bottom Right) Bmin = 0.47 T. Pµ = 350 W, pAr
= 213 nTorr. Microwave power has been oﬀset by it’s minimum value, over the entire 100 ms
measurement period, to account for the AC coupling of the oscilloscope. Signals have been
treated with a digital low pass ﬁlter using the Matlab signal processing toolkit, with a cutoﬀ
frequency of 333 kHz.

158

Figure 5.48: Fourier transform of the microwave power signals emitted from the ion source for a
varying magnetic minimums (Top left) Bmin = 0.43 T, (Top right) Bmin = 0.44 T, (Bottom left)
Bmin = 0.46 T, (Bottom Right) Bmin = 0.47 T. Pµ = 350 W, pAr = 213 nTorr

Figure 5.49: Fourier transform of the microwave power signals emitted from the ion source for a
varying magnetic minimums (Top left) Bmin = 0.43 T, (Top right) Bmin = 0.44 T, (Bottom left)
Bmin = 0.46 T, (Bottom Right) Bmin = 0.47 T. Pµ = 350 W, pAr = 213 nTorr

159

Figure 5.50: Average and maximum output microwave power, measured over a 100 ms period, for
a varying magnetic minimum. (Top left) Bmin = 0.43 T, (Top right) Bmin = 0.44 T, (Bottom left)
Bmin = 0.46 T, (Bottom Right) Bmin = 0.47 T. Pµ = 350 W, pAr = 213 nTorr

160

5.3.7 Understanding the ’over-extraction’ transient proﬁles

As discussed earlier, the ’ion burst’ eﬀect is most likely an afterglow event resulting from losses
of electrons due to an instability event. The afterglow phenomenon can also explain the ’over-
extraction’ mode. Using the evidence presented throughout this chapter, we can see that the ’over-
extraction’ eﬀect occurs in high energy density systems, particularly those with large magnetic
ﬁelds.
Increasing the magnetic ﬁeld may cause wave-electron interactions to occur at higher
energies or causes excitation of lower frequency unstable electromagnetic plasma modes [25, 26].
For suﬃciently strong magnetic ﬁelds, the aﬀected population would be the electrons responsible
for the potential dip, very high energy hot electrons. If this population were to escape conﬁnement
suddenly, then the potential dip would decrease, as ∆Φ ∼ (ni − ne). The sudden increase in the
potential would degrade the conﬁnement of highly charged ions within the system. This type of
afterglow is most similar to the so-called ’second afterglow’ described by Geller [18].

This latter hypothesis is rather diﬃcult to prove as it is nearly impossible to probe that deeply
into the plasma. However, it does make sense in the grand scheme of ECR operations and explains
many of the phenomena seen in the ’over-extraction’ mode. First, the low repetition frequency
and recovery time are some of the strongest evidence for an afterglow eﬀect on the high energy
electrons. Time-resolved bremsstrahlung measurements have shown that it can take 10’s-100’s ms
to produce steady-state hot-electron populations within ECRIS plasmas [33, 65, 66]. In fact, as
ﬁgures 5.10 and 5.51 show, the decay of the enhanced current goes as Iq ∼ Z−2, as we would
expect for collisionally diﬀused ions. Although there is not enough data to show it, the decay time
does appear to be inversely dependent upon the magnetic ﬁeld, as we would expect for either Bohm
or cross-ﬁeld diﬀusion [86].

This eﬀect may also explain the decrease in the extracted current of low charge state ions. As the
ion-ion collision frequency goes as νii ∼ ¯q, a temporary decrease in high charge state populations
may result in better conﬁnement of low charge states. As discussed in section 2.3.2, low charge
states may be magnetically conﬁned and their diﬀusion will not be appreciably aﬀected by losses
of hot electrons. Using equations 2.5 and 2.9 while assuming an ion temperature 5 eV, electron

161

density of 1011 cm-3, and average magnetic ﬁeld of 1 T shows that the ion-ion collision frequency
overtakes the ion cyclotron frequency between 2+ and 4+, with (cid:104)νii(cid:105)/ fci ∼ 1 for argon 3+. This
estimate agrees well with the trends seen in ﬁgure 5.10, which shows the percent change of extracted
ion currents for argon ions up to charge state 12+.

Lastly, the appearance of the ’over-extraction mode in high ﬁeld operating points suggests
increased interactions with higher energy electrons (see sec. 5.3.2). Izotov et al. demonstrated
that microwave power is predominantly emitted at lower microwave frequencies as the ion source
magnetic ﬁeld increases [26]. This could result from resonant interactions with increasingly higher
energy electrons, as fce ∼ B/γe.4 This relation presents two logical extremes that allow for
a decreasing resonance frequency.
If the magnetic ﬁeld where the resonant interaction occurs
remains constant, then the interaction must be occurring for a larger number of higher energy
electrons. Under the potential dip hypothesis, these electrons exist more towards the center of the
trap. If the energy of the interacting electrons remains constant, then the interaction must occur
deeper inside the plasma, due to the min-B ﬁeld structure (see sec. 2.3.1). Either way, the lower
value of the dominant emitted frequency suggests that electrons in the center of the trap are more
likely to diﬀuse during unstable operation with higher magnetic ﬁelds.

This hypothesis may also explain one of the stranger eﬀects observed with ’over-extraction’
operating. The absence of large (or sometimes any) microwave bursts immediately before any
beam current variations (see Figure 5.6). This eﬀect is notable, especially with the observations
of sizeable microwave power bursts during instability events that perturb the plasma to a lesser
degree. Electron losses focused at the center of the plasma may explain this observation. High
energy electrons would predominately resonate with microwaves at substantially lower frequencies
than in the other modes. If the majority of the plasma’s density appears around its extremities,
as simulations suggest, then the increasing density away from the center of the plasma would
make it diﬃcult for microwave radiation to penetrate out of the plasma [58, 80]. As a low order
approximation we can suppose a cold plasma that is divided into three spatially distinct regions: I)
4As an example of this trend, 100 keV electrons at the cold RF resonance position, fRF = 18GHz,
will resonant at 15.1 GHz and 1 MeV electrons will resonate at 6.1 GHz, excluding Doppler eﬀects.

162

ω > ωp(x) on − inf < x < 0 II) ωp(x) > ω on 0 < x < L III) ω > ωp(x) on L < x < inf. Here,
ω is the frequency of the excited and propagating wave, and ωp(x) is the spatially varying plasma
frequency. The transmission coeﬃcient is given by [80]:
1
4e−η

(cid:19)−2

(5.1)

T =

eη +

(cid:18)
Þ L

0

(cid:113)

where

η =

1
c

p(x) − ω
2

2dx.

(5.2)

ω

If we assume that electron losses occur at the plasma boundary, for ’ion burst’ and ’ion loss’ proﬁles,
then η ∼ ω
p ∼ ne(x), and therefore T, would decrease along the plasma boundary. However, as
2
the debye length is of the order of mm’s (in a system with a characteristic length of 50 - 100 cm),
even signiﬁcant losses at the plasma’s center could have little to no eﬀect upon the electromagnetic
boundaries within the system. Thus, it would be equally diﬃcult for the excited microwaves to
penetrate through the plasma boundary during an ’over-extraction’ type instability event. The waves
which can escape must then pass through a WR-62 waveguide, which has a lower cutoﬀ frequency
of 9.5 GHz, making the observation of the excited electromagnetic modes unlikely. To this author’s
knowledge, no one has observed the excitation of microwaves with frequencies lower than about 5
GHz. However, measurements show that the emitted spectral density of microwaves prefers lower
frequencies when the source is operated at higher magnetic minimums [27, 26].

The hypothesis assumes the existence of the potential dip or, at a minimum, a collection of hot
electrons in the center of the magnetic trap. Such a population of electrons itself is also a hypothesis
(see section 2.3.2), as measuring it without perturbing the plasma is nearly impossible. However,
in the absence of a better understanding of the source’s volumetric energy distribution, its presence
provides the best explanation for this measurement and many others [36, 49].

5.4 Concluding Thoughts

In this chapter, measurements of the stability of the SuSI ion source across a wide range of ion
source parameters were shown. The transient model for beam currents during unstable operation
provided a means to view and characterize trends across numerous operating points. These trends,

163

Figure 5.51: The exponential decay time of the ’over-extracted’ current scales roughly as q−2.
The deviation can result from non-linear eﬀects in the plasma, such as continuous heating, charge
exchange, and ionizing collisions. BHex,wall = 1.2 T, pAr = 131 nTorr, For the red curve: Pµ = 350
W. For the green curve: Pµ = 250 W.

seen through 2D color maps, demonstrated relationships between the system’s magnetic ﬁeld
topology and the plasma’s stability. Using the same operating points described in chapter 4,
brought further insight into the maps’ trends by individually varying important ﬁeld parameters.
Ultimately, as suggested by the maps, the magnetic minimum appears responsible for the transient
behavior of the extracted beam current.

The transient model provides a semi-quantitative method of characterizing the beam extracted
from an unstable plasma. Moreover, it revealed several patterns and trends across diﬀerent operating
parameter ranges (sec. 5.3). Those statistical trends across the stability maps clearly showed that
the best performing operating points were those with lower magnetic ﬁelds at extraction (sec.
5.3.2). Systematically varying the diﬀerent extrema of the magnetic ﬁeld topology demonstrate
correlations between high instability repetition frequencies and the suppression of extracted high
charge state currents. In general, longer-lasting microwave emission which causes large plasma
perturbations were correlated with higher extracted currents (secs. 5.3.4 and 5.3.5). The maximum
extracted current occurred when the size of the instability induced perturbations balanced against

164

their repetition frequency. However, it is the magnetic minimum that has the largest eﬀect on both
the transience and overall stability of the plasma (sec. 5.3.6).

The dependence of the beam current transience upon the magnetic ﬁeld indicates the presence
of an energy or frequency selection process occurring before or during an instability event. An af-
terglow hypothesis best explains the ’over-extraction’ transient mode seen in energy-dense plasmas.
As both the ’ion burst’ and ’over-extraction’ eﬀects rely on the same underlying phenomenon, we
can attempt to generalize the eﬀect using the knowledge that the magnetic minimum controls the
plasma’s energy content (sec. 4.4.6). As described in sections 2.4.1 and 5.3.7, varying the magnetic
minimum shifts the emitted microwave power to lower frequency microwaves [25, 26]. As such,
the microwaves will resonate with increasingly higher energy electrons during an instability event
[47]. At low ﬁeld strengths, the emitted microwave power shifts towards higher frequencies, which
would resonate with electrons near the extremity of the plasma. This causes the ’ion burst’ eﬀect,
an afterglow event similar to the externally driven afterglow phenomena [89]. At high enough ﬁeld
strengths, the electron losses predominantly focus on electrons with lower gyration frequencies in
the center of the plasma. In this case, the ’over-extraction’ afterglow is caused by a collapse of the
potential dip, leading to a loss of conﬁnement for high charge state ions. The ion loss predominant
proﬁles likely exist in some energy region in between these two extremes. This may also explain
the phenomena where increasing the neutral gas pressure changes the observed transient mode (ﬁg.
5.12).
Izotov et al. observed that changes to the neutral gas species or pressure can aﬀect the
spectral power density of emitted microwave radiation [26]. Although Izotov et al. changed both
the pressure and neutral species at the same time, and it is currently unknown if changing only one
of these variables will lead to the same eﬀect.

Under this hypothesis, the energy distribution of electrons driven out of the plasma could
vary signiﬁcantly across the diﬀerent transient modes. Unfortunately, time-resolved measurements
of the electron distribution are diﬃcult to perform. Figure 5.2 demonstrated losses of electrons
predominantly centered over a narrow band of energies in the 65-80 keV range (ﬁg. 5.2). This
operating point corresponded to an ’ion burst’ dominated transient proﬁle. Izotov et al. success-

165

fully performed multiple time-resolved electron measurements across various operating points and
heating schemes (CW and pulse mode) [28]. The measurements showed that the instability aﬀected
the entire warm and hot electron populations [28]. However, the authors do not establish whether
the warm electron population is always aﬀected. They did not report details of the corresponding
ion beams. Also, neither study included measurements of the frequency composition of emitted
microwaves. As a result, it cannot be concluded whether the resonance frequency or energies of
lost electrons determine the predominant ion transient proﬁle. Further understanding of the beam
current transient and its relation to the properties of the unstable plasma would require simul-
taneous measurements of the frequency composition of unstable microwaves and time-resolved
measurements of the escaping electron energy distribution.

Lastly, the results of these ion measurements revealed trends towards optimizing the ion source’s
overall performance by maximizing high charge state currents. The magnetic ﬁeld controls many
aspects of the plasma’s stability and performance. As demonstrated in sections 5.3.4, 5.3.5, and
5.3.6, even the individual ﬁeld extrema aﬀect the transient state of the extracted ion beam.

Overall, a lower extraction side magnetic ﬁeld correlates to higher extracted Ar8+ currents,
regardless of the injection side or hexapole ﬁeld maxima (sec. 5.3.2). Across all experimental
ranges, an extraction side maximum less than 1.16 T, BExt,max ≈ 1.8 BRF, produced the highest
extracted currents. The results even seem to disagree with previous work which suggested the
extraction ﬁeld for optimal performance scales as BExt,max ≈ 1.8 - 2 BRF [24, 46]. However, a
higher extraction side magnetic ﬁeld can produce a more stable plasma overall, suggesting that there
may be a balance between the stability and intensity of extracted currents for a given operating
point. The higher ﬁeld also correlates with less perturbative, but more frequent, instability events.
Bremsstrahlung measurements also demonstrated a change in the diﬀusion characteristics of the
plasma, which begins at ﬁeld strengths between 1.16 and 1.2 T (see ﬁg. 5.42). Further studies are
necessary to understand this eﬀect; however, it may provide insight into the physical mechanisms
which limit the extracted current at higher ﬁeld strengths. At this time, the eﬀect is not a good
diagnostic for optimizing performance. Observing the extracted beam current is more eﬃcient and

166

eﬀective, by a large margin (compare the slopes in ﬁgs. 5.37 and 5.42).

The role of the injection and radial ﬁeld maxima upon the stability of the plasma is unclear.
Varying injection side ﬁeld changes the repetition rate and amplitude of microwave bursts, with
an increasing amplitude at larger ﬁeld strengths (sec. 5.3.4). This contradicts the trend seen by
varying the extraction side ﬁeld, where larger amplitude instability events occurred at lower ﬁeld
strengths. It is possible that the trends shown in section 5.3.5 are a result of a changing electron
diﬀusion rate (see sec. 4.4.5). However, if that were the case, we would expect that varying the
injection side would produce similar trends. More measurements will be necessary to understand
these eﬀects.

Section 5.3.2 also demonstrated an odd relationship between the radial magnetic ﬁeld, by
varying the hexapole coils, and the stability of the plasma. While varying the hexapole aﬀected the
ion current extracted from the ion source, it did little to change the plasma’s transient characteristics
(sec. 5.3.3). However, it can signiﬁcantly aﬀect the global stability of the ion source’s magnetic
ﬁeld and power parameter space when coupled with a varying longitudinal magnetic ﬁeld topology
(sec 5.3.2). The larger radial ﬁelds make the plasma more unstable, overall, except when using
experimental range 4, where the increased ﬁeld improved the stability characteristics of the plasma.
The extraction side magnetic ﬁeld was lowest across this range (see ﬁg. B.2), and the change in the
characteristics of the plasma may result from the improved diﬀusion of electrons out of the system.
Sections 4.4.3 and 4.4.5 showed how an increasing hexapolar ﬁeld and a decreasing radial ﬁeld both
lead to improved diﬀusion of electrons out of the plasma. Combining these eﬀects could eﬀectively
suppress some instability events by increasing the diﬀusion of hot electrons out of the plasma.
Measurements that focus entirely upon the relationship between the hexapole and extraction ﬁelds
should be performed. Unfortunately, there are few ion sources capable of performing such a
measurement.

In total, these measurements revealed that the best-performing ion sources require as much
control over the conﬁning magnetic ﬁeld as possible. At minimum, the longitudinal conﬁguration
of future ECR ion sources should include three solenoids. This will provide a greater degree of

167

control over all three ﬁeld extrema, at least more so than the two coil conﬁgurations. In particular,
separating the value of the magnetic minimum from that of the extraction side maximum will
provide a much greater degree of control of both the stability and performance of the ion source.

168

CHAPTER 6

CONCLUSION

Electron cyclotron resonance ion sources are a powerful tool for producing high intensity, high
charge state ion beams. However, the dynamics of the resonantly heated plasma are poorly
understood. High energy electrons within the system can excite electromagnetic modes, inciting
kinetic instabilities that can enhance or suppress extracted high charge state currents. This study
attempted to better understand the dynamics of the plasma by measuring distributions of diﬀusing
electrons and ions extracted from the source during unstable operation. This work can provide
insight into improvements for the design of future ECR ion sources and directions for optimizing
the ion source’s performance during operation.

The quasilinear diﬀusion mechanism may provide a pathway for hot electrons to enter the
Min-B ﬁeld structure’s loss cone. However, a vital piece of the hypothesis, the excited plasma
modes, has never been observed during the stable ion source operations. Such a measurement
would be necessary to prove the hypothesis and, as such, further investigations of electron diﬀusion
mechanics are necessary.

Several diﬀerent parameters aﬀect the energy and density of the diﬀusing electrons. The
system’s magnetic minimum overwhelmingly controls the central energy of diﬀused electrons.
Most importantly, the distribution of diﬀusing electrons and the emission of hard x-rays are highly
correlated. This result has signiﬁcant implications for interpreting the bremsstrahlung diagnostic
going forward, particularly in regards to the spectral temperature.
If the underlying diﬀusion
mechanism includes an energy selection process that depends upon the magnetic minimum, it may
falsely imply an increase in the temperature of plasma electrons. At a minimum, it will make it
more challenging to determine the diﬀerence between an increase in the energy of trapped electrons
and escaping electrons. Narrowing the x-ray collimator’s acceptance angle, such that it does not
look onto the extraction aperture, may make the bremsstrahlung diagnostic a better reﬂection of the
energy distribution of plasma electrons.

169

The development of a semi-quantitative model of the beam current transience allowed for a
more detailed understanding of the plasma’s parameter space stability characteristics. Replacing
the idea of a stability threshold with a stability map enables us to characterize the plasma over a
wider domain of parameters. While the stability threshold may be valuable as a way to ﬁne-tune
ion source parameters along the border between stable and unstable regions, it lacks a global insight
into the relationship between the plasma’s stability and the Min-B ﬁeld structure.

The stability maps exposed correlations between the size of stable regions, the solenoid magnetic
ﬁeld proﬁle’s topology, and the hexapole ﬁeld strength. In general, systems with larger magnetic
ﬁelds are less stable. However, increasing the extraction side magnetic ﬁeld increases the stability of
the magnetic ﬁeld-power parameter space. Conversely, lower ﬁeld strengths at extraction maximize
the extracted current at the cost of a more unstable operating parameter space. These results also
provided insight into the semi-quantitative beam transience model.

The results from chapters 4 and 5 suggest that the three extrema of the transient proﬁle result
from an energy or microwave frequency selection process. Low energy, high-frequency events
cause afterglow events similar to the more traditional externally driven afterglow process, causing
the ’ion burst’ eﬀect. The data also suggests that high energy, low-frequency events result from
unstable interactions with hot electrons and result in an afterglow event most similar to the ’second
afterglow’. These ’over-extraction’ events may result from hot-electron losses, which perturb the
plasma’s potential dip. However, evidence of such a population is necessary to conﬁrm this
hypothesis. Understanding these diﬀerent eﬀects will help further the understanding of ion source
operation and performance during unstable operation. These results will provide insight into wave-
plasma interaction within the ion source and direct research and operators towards more reliable
and optimal operating points.

The work presented here, while revealing much about the ion source plasma, demonstrates that
both the escaping electron and bremsstrahlung diagnostics require further development. Expanding
the breadth and resolution of energy domain measurements to include 0.1 - 10 keV electrons and
photons is necessary to understand the ionization process within the source plasma. Focusing

170

solely on hot electrons and hard x-rays, while important, is complicated by the multitude of diﬀu-
sion mechanisms within the plasma. Furthermore, time-resolved versions of these measurements
may provide better insight into the plasma dynamics. These results also indicate the need for
ﬂexible magnetic ﬁeld conﬁgurations. Future ECR ion sources should include a minimum of three
longitudinal solenoid coils, to have greater independent control over the ﬁeld’s extrema. Further
studies of new heating mechanisms, including two frequency heating, are necessary to increase
these ion sources’ performance.

171

APPENDICES

172

APPENDIX A

MATLAB CODES

This appendix will cover the necessary steps for calculating the ﬁt of the log scale bremsstrahlung
distribution used to calculate the spectral temperature (see. 3.2.5). The slope of the log distribution
is given by:

with the standard deviation of the slope given by

where i indexes each energy bin with a central value of Ei ﬁlled with Iγ,i photon counts per second.
The y-intercept of this line is the given by:

(cid:32)i

slope =

δ(slope) =

(cid:32)i

Ei
e2
i

(cid:33)

1
e2
i

(cid:33)(cid:32)i
(cid:33)

(cid:33)2

Ei
e2
i

(cid:33)

ln Iγ,i
e2
i

ln Iγ,i
e2
i

(cid:33)2

−

Ei
e2
i

(cid:32)i

Ei
e2
i

(cid:33)(cid:32)i
(cid:32)i
(cid:118)(cid:117)(cid:117)(cid:117)(cid:117)(cid:117)(cid:117)(cid:117)(cid:117)(cid:117)(cid:117)(cid:117)(cid:117)(cid:117)(cid:116)
(cid:33)(cid:32)i
(cid:32)i
(cid:118)(cid:117)(cid:117)(cid:117)(cid:117)(cid:117)(cid:117)(cid:117)(cid:117)(cid:117)(cid:117)(cid:117)(cid:117)(cid:117)(cid:116)

(cid:32)i

Ei
e2
i

E2
i
e2
i

−

−

e2
i

(cid:33)

1
e2
i

1
e2
i

1
e2
i

(cid:33)

E2
i
e2
i

E2
i
e2
i

Ei ln Iγ,i

(cid:33)(cid:32)i
(cid:33)

(cid:32)i
(cid:32)i
(cid:32)i
(cid:33)(cid:32)i
(cid:32)i
(cid:32)i
(cid:33)
(cid:33)(cid:32)i
(cid:33)
(cid:33)(cid:32)i
(cid:32)i
(cid:33)
(cid:32)i
(cid:33)(cid:32)i

(cid:32)i

E2
i
e2
i

E2
i
e2
i

1
e2
i

−

−

Ei ln Iγ,i

(cid:33)2

e2
i

−

E2
i
e2
i

(cid:33)
(cid:113)

1
e2
i

(cid:33)2 .

Ei
e2
i

1(cid:112)Iγ,i

.

Iγ,i =

ei =

∂ ln Iγ,i
∂Iγ,i

δIγ,i =

1
Iγ,i

173

Y0 =

with a standard deviation of:

δY0 =

The per-bin error is given by:

(A.1)

(A.2)

(A.3)

(A.4)

The spectral temperature is then calculated as:

with a standard deviation of:

and the x-intercept is given by:

with a standard deviation of:

Ts = − 1
slope

δTs =

δ(slope)
slope2 .

XTs = − Y0
slope

δXTs =

= −Y0Ts

(cid:113)(TsδY0)2 + (Y0δTs)2

(A.5)

(A.6)

(A.7)

.

function output = linReg(Ene,xraycounts)

R2 = 0;

ll = 0;

Emin = 80;

Emax = 800;

while R2 < 0.99 %Define lower limit on regression

T1a = [];

T2a = [];

T3a = [];

T4a = [];

lc = [];

logerr = [];

[a,b] = min(abs(Ene - Emin)); %Find bin position of Minimum Energy

[c,d] = min(abs(Ene - Emax)); %Find bin position of Maximum Energy

en = Ene(b:d); %Use only the energy values between the Min and Max

m = 0;

n = 0;

Errsq = [];

G = [];

174

%Reshape the array which stores count rate data to be the same

%length as energy array.

k = 0;

for k = 1:(d-b+1)

if xraycounts(k+b) > 0.5 %Lower limit on photon intensity

counts = xraycounts(k+b);

lc = [lc ;log(counts)]; %create log(intensity) array

logerr = [logerr 1/sqrt(counts)]; %calculate error

m = m +1;

elseif ((xraycounts(k+b) == 0.5)||(xraycounts(k+b)<0.5))

Emax = en(k-1); %Set max energy based upon min intensity

en = en(1:k-1); %Set energy array to match intensity array

break;

end

end

%http://www.che.udel.edu/wp-content/uploads/2019/03/FittingData.pdf

%(citation)

%calculation of linear fit/fit error

for j = 1:length(lc)

errsqr = 1/(logerr(j)*logerr(j));

Errsq = [Errsq ; errsqr];

T1a = [T1a ; en(j)*errsqr];

T2a = [T2a ; lc(j)*errsqr];

T3a = [T3a ; en(j)*lc(j)*errsqr];

T4a = [T4a ; en(j)*en(j)*errsqr];

end

T1 = sum(T1a);

T2 = sum(T2a);

T3 = sum(T3a);

T4 = sum(T4a);

E = sum(Errsq);

175

slope = (T1*T2-T3*E)/(T1*T1-T4*E);

serror = sqrt(E/(T4*E-T1*T1));

intercept = (T1*T3-T2*T4)/(T1*T1-T4*E);

interror = sqrt(T4/(T4*E-T1*T1));

%test the fit against minimum regression limit

testY = slope*en+intercept;

numerator = (lc-testY).^2;

denominator = (lc-mean(lc)).^2;

R2 = 1-sum(numerator)/sum(denominator);

Emin = Emin+5;

ll = ll+1;

end

%output final results.

output = [slope serror intercept interror Emin Emax];

end

176

APPENDIX B

LONGITUDINAL FIELD PROFILES FOR THE MEASUREMENT OF THE

DISTRIBUTION OF ESCAPING ELECTRONS

Figure B.1: Measured electron distribution for the measurements ’control’ operating point. The
50 keV reliability threshold is shown by the dashed line. Bmin = 0.4 T, BInj,max = 2.367 T,
BExt,max = 1.100 T.

177

Figure B.2: Measured electron distribution for the measurements ’control’ operating point. The
50 keV reliability threshold is shown by the dashed line. Bmin = 0.4 T, BInj,max = 2.367 T,
BExt,max = 1.100 T.

Figure B.3: Measured electron distribution for the measurements ’control’ operating point. The
50 keV reliability threshold is shown by the dashed line. Bmin = 0.4 T, BInj,max = 2.367 T,
BExt,max = 1.100 T.

178

APPENDIX C

EXTRA DATA FROM THE MEASUREMENT OF THE DISTRIBUTION OF ESCAPING

ELECTRONS

C.1 Eﬀect of a varying extraction voltage on the energy-resolved ion source

bremsstrahlung distribution

Figure C.1: A varying extraction voltage has little eﬀect on the hot-electron bremsstrahlung
distribution. The count rate and bremsstrahlung are much more stable than in ﬁgure 4.1 due to the
much more consistent background during the measurement. The stability of the nuclear peak at
510 keV demonstrates this most clearly. Plasma parameters are Bmin = 0.397 T, P = 350 W, Tlive =
600 s, pressure = 212 nTorr.

C.2 Eﬀect of a Variable Power

179

Figure C.2: A varying magnetic ﬁeld does not appreciably change how the injected microwave
power aﬀects the measured electron distributions.

Figure C.3: Similar to ﬁgure C.2, a varying magnetic ﬁeld does not appreciably change how the
injected microwave power aﬀects the measured bremsstrahlung distributions.

180

Figure C.4: All ﬁeld distributions agree to a linear increase in the number of measured particles
with respect to the microwave power. Although a small change in the spectral temperature is seen
as the injection side magnetic ﬁeld is varied, the overall trend indicates an insensitivity of electron
energy to microwave power.

Figure C.5: All ﬁeld distributions agree to a close to linear increase in the number of measured
particles with respect to the microwave power. Although a small change in the spectral
temperature is seen as the injection side magnetic ﬁeld is varied, the overall trend indicates an
insensitivity of electron energy to microwave power.

181

C.3 Eﬀect of a Variable Pressure

Figure C.6: The dependence of the high energy electron distribution on pressure was not
appreciably aﬀected by a varying longitudinal ﬁeld proﬁle. Although out of the scope of this
study, it should be recognized that low energy portion of the distribution (Ekin < 70keV) is
aﬀected by the both the pressure and the average magnetic ﬁeld within the source chamber.

182

Figure C.7: The dependence of the bremsstrahlung distribution on pressure was not appreciably
aﬀected by a varying longitudinal ﬁeld proﬁle.

Figure C.8: All measurements agreed to the same overall trend, regardless of longitudinal ﬁeld
proﬁle.

183

Figure C.9: All measurements agreed to the same overall trend, regardless of longitudinal ﬁeld
proﬁle.

C.4 Eﬀect of a Variable Hexapole

Figure C.10: There was a greater eﬀect upon the measured electron distribution’s height as a
function of the radial magnetic ﬁeld maximum as the magnetic minimum increased.

184

Figure C.11: The longitudinal magnetic ﬁeld had no eﬀect on the measured bremsstrahlung
distribution for high energy photons.

Figure C.12: Comparison between the electron peak energy/intensity and bremsstrahlung spectral
temperature/total photon counts for across multiple ion source longitudinal magnetic ﬁeld proﬁles
with varying radial ﬁeld maximum.

185

Figure C.13: Comparison between the electron peak energy/intensity and bremsstrahlung spectral
temperature/total photon counts for across multiple ion source longitudinal magnetic ﬁeld proﬁles
with varying radial ﬁeld maximum.

186

APPENDIX D

LONGITUDINAL FIELD PROFILES FOR THE PARAMETER SWEEP

MEASUREMENT OF ION STABILITY

Figure D.1: Measured electron distribution for the measurements ’control’ operating point. The
50 keV reliability threshold is shown by the dashed line. Bmin = 0.4 T, BInj,max = 2.367 T,
BExt,max = 1.100 T.

Figure D.2: Measured electron distribution for the measurements ’control’ operating point. The
50 keV reliability threshold is shown by the dashed line. Bmin = 0.4 T, BInj,max = 2.367 T,
BExt,max = 1.100 T.

187

Figure D.3: Measured electron distribution for the measurements ’control’ operating point. The
50 keV reliability threshold is shown by the dashed line. Bmin = 0.4 T, BInj,max = 2.367 T,
BExt,max = 1.100 T.

Figure D.4: Measured electron distribution for the measurements ’control’ operating point. The
50 keV reliability threshold is shown by the dashed line. Bmin = 0.4 T, BInj,max = 2.367 T,
BExt,max = 1.100 T.

188

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