ION BEAM EXTRACTION FROM ELECTRON CYCLOTRON RESONANCE ION
SOURCES AND THE SUBSEQUENT LOW ENERGY BEAM TRANSPORT
By
Daniel Winklehner

A DISSERTATION
Submitted to
Michigan State University
in partial fulﬁllment of the requirements
for the degree of
Physics - Doctor of Philosophy
2013

ABSTRACT
ION BEAM EXTRACTION FROM ELECTRON CYCLOTRON RESONANCE ION
SOURCES AND THE SUBSEQUENT LOW ENERGY BEAM TRANSPORT
By
Daniel Winklehner
Electron Cyclotron Resonance Ion Sources (ECRIS) are capable of delivering high currents
of Highly Charged Ions (HCIs) to heavy ion accelerators (e.g.: to the future FRIB). The
use of a sextupole magnet for conﬁnement of the plasma inside the source imposes a unique
triangular structure on the beam. This, together with the multitude of ion species that
are extracted at the same time and the high axial magnetic ﬁeld at the plasma aperture,
resulting from additional conﬁning solenoids, make the simulation and design of ECRIS
extraction systems particularly challenging. The ﬁrst objective of this thesis was to reﬁne
and test a semi-empirical simulation model of the formation and extraction of HCIs from ECR
ion sources as well as their transport through the subsequent Low Energy Beam Transport
(LEBT) system. To this end, a set of utility functions was written to simplify performing
the simulations.
In the LEBT system, another interesting, yet so far unanswered, question arises: The inﬂuence of space-charge eﬀects on the beam and the level of space-charge compensation in the
ECRIS beam line.
This interesting topic quickly became the second main objective of the thesis. A Retarding
Field Analyzer (RFA) was built and systematic measurements of the neutralization level in
ECRIS LEBT systems were done for the ﬁrst time as part of this thesis (this intensity and
pressure regime was previously not well explored). The measured neutralization levels for
typical ECRIS beams were found to be between 0 and 50 % and agreed reasonably well with

a simple formula developed by Gabovich et al. for highly neutralized proton and H − beams
after it was re-derived and extended in this thesis for low neutralization and multiple species.
Preliminary tests of the reﬁned and integrated simulation model for the ECR ion sources
VENUS and SuSI and their respective low energy beam transport systems include comparisons of measured beam currents, cross sections and emittances with the simulation results.
These tests suggest that the model is suited for the simulation of ion beam extraction and
transport for medium to high charge states of medium to heavy ions, but not for the lowest
charge states and lightest ions (He1+ , protons).
Finally, as an example application of the developed software, a variable-energy (300 kV - 3
MV) electrostatic accelerator was simulated and redesigned for the DIANA project, a new
proposed underground laboratory for nuclear astrophysics.

Dedicated to my grandmother, Dr. Sigilde K¨sner.
o

iv

ACKNOWLEDGMENTS

First and foremost I would like to express my gratitude towards my adviser Daniela Leitner.
She was always there to discuss ideas and answer my questions. It was a pleasure working
with her.
I would further like to thank my guidance committee for their support: Phillip Duxbury,
Felix Marti, Michael Syphers, and Kirsten Tollefson.
A lot of this thesis would not have been possible without the help of the many people doing
exceptional work here at the NSCL. These include, but are not limited to, the members
of the ECR group: Guillaume Machicoane, Dallas Cole, Larry Tobos, and Alain Lapierre,
people from the machine shop: Jay Pline and Susan LeCureux, and Ghulam Mujtaba of the
electronics department.
Furthermore, I would like to thank Dave Grote, and Damon Todd for their help with WARP.
I would also like to thank Janilee Benitez, Markus Strohmeier, and Claude Lyneis for their
support during my time in Berkeley.
There were many people participating in the DIANA collaboration, and it was a pleasure
working with all of them. I would especially like to thank Manoel Couder and Alberto
Lemut, with whom I worked closest, for the great teamwork and fun times.
My oﬃce mates Vincent Bader, Michael Bennett, and Michael Jones were a true source of
inspiration (work related and otherwise) during my time at MSU. I will miss our friendly
banter.
Last but not least, I would like to thank my family for their continuous support. Am´lie,
e
Birgit, Brigitte, Grete, Lucia, Robert, Sigilde, Stefanie, Valentin, I am looking forward to
spending more time with you soon!
v

TABLE OF CONTENTS

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

x

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xii

KEY TO SYMBOLS AND ABBREVIATIONS
Chapter 1 Introduction . . . . . .
1.1 General Introduction . . . . . .
1.2 Structure Of The Thesis . . . .
1.3 Conventions Used In This Work
1.3.1 Units . . . . . . . . . . .
1.3.2 Emittance . . . . . . . .
1.3.3 Brightness . . . . . . . .

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5
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Chapter 2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1 Charged Particle Beams . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Equations of motion . . . . . . . . . . . . . . . . . . . .
2.1.2 Particle Distributions . . . . . . . . . . . . . . . . . . . .
2.1.2.1 Liouville’s theorem . . . . . . . . . . . . . . . .
2.1.2.2 Transversal distribution . . . . . . . . . . . . .
2.1.3 Emittance . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.3.1 RMS Emittance . . . . . . . . . . . . . . . . .
2.1.3.2 Brightness . . . . . . . . . . . . . . . . . . . . .
2.1.3.3 Fractional Emittance . . . . . . . . . . . . . . .
2.1.3.4 Normalized Emittance . . . . . . . . . . . . . .
2.1.3.5 Normalized Brightness . . . . . . . . . . . . . .
2.1.4 Self Fields . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.5 Space Charge Compensation . . . . . . . . . . . . . . . .
2.1.5.1 Simple Generalization of the Result . . . . . . .
2.1.5.2 Multiple Ion Species . . . . . . . . . . . . . . .
2.1.5.3 Obtaining the Cross-Sections . . . . . . . . . .
2.1.5.4 An Example . . . . . . . . . . . . . . . . . . .
2.1.5.5 Some Comments and Limitations of this Model
2.2 Electron Cyclotron Resonance Ion Sources . . . . . . . . . . . .
2.2.1 General Overview . . . . . . . . . . . . . . . . . . . . . .
2.2.1.1 ECR Concept . . . . . . . . . . . . . . . . . . .
2.2.2 Magnetic Field Structure . . . . . . . . . . . . . . . . . .
2.2.3 Creation of Highly Charged Ions . . . . . . . . . . . . . .

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48
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Chapter 3 Software . . . . . . . . . . . . . . . . . . . . .
3.1 Simulation Codes . . . . . . . . . . . . . . . . . . . .
3.1.1 SIMION . . . . . . . . . . . . . . . . . . . . .
3.1.2 IGUN . . . . . . . . . . . . . . . . . . . . . .
3.1.3 WARP . . . . . . . . . . . . . . . . . . . . . .
3.1.3.1 Main Packages (python class name in
3.1.3.2 Particle Loading . . . . . . . . . . .
3.1.3.3 Fields and Lattice Elements . . . . .
3.1.3.4 Fieldsolvers . . . . . . . . . . . . . .
3.1.3.5 Structure of the Simulations . . . . .
3.2 Utility Programs . . . . . . . . . . . . . . . . . . . .
3.2.1 Electrode Designer . . . . . . . . . . . . . . .
3.2.1.1 IGUN export . . . . . . . . . . . . .
3.2.1.2 WARP export . . . . . . . . . . . .
3.2.2 WARP Postprocessor . . . . . . . . . . . . . .
3.2.3 NeMo Postprocessor . . . . . . . . . . . . . .

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Chapter 4 Hardware . . . . . . . . . . . . . . . . .
4.1 Electron Cyclotron Resonance Ion Sources . . .
4.1.1 LEDA injector ion source . . . . . . . .
4.1.2 Artemis A - Room Temperature ECRIS
4.1.3 SuSI - Superconducting ECRIS . . . . .
4.1.4 VENUS - Superconducting ECRIS . . .
4.2 Beam Diagnostic Instruments . . . . . . . . . .
4.2.1 Faraday Cup . . . . . . . . . . . . . . .
4.2.2 A/q Analyzer . . . . . . . . . . . . . . .
4.2.3 Beam Viewer . . . . . . . . . . . . . . .
4.2.4 Slit Proﬁle Scanner . . . . . . . . . . . .
4.2.5 Allison Emittance Scanner . . . . . . . .
4.2.6 Retarding Field Analyzer . . . . . . . . .

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2.2.6
2.2.7
2.2.8

ECR Heating . . . . . . . . . . . . . .
Plasma Sheath . . . . . . . . . . . . .
2.2.5.1 Electron Flux . . . . . . . . .
2.2.5.2 Ion ﬂux - The Bohm criterion
2.2.5.3 The Plasma Potential . . . .
Extraction Systems . . . . . . . . . . .
Extracted Current . . . . . . . . . . .
Theoretical Emittance of an ECRIS . .

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Chapter 5 Space-Charge Compensation Measurements
5.1 Preliminary Considerations . . . . . . . . . . . . . . . . .
5.1.1 Saturation Current . . . . . . . . . . . . . . . . .
5.1.2 Realistic Mesh Eﬀects . . . . . . . . . . . . . . .
5.1.2.1 Eﬀective Potential . . . . . . . . . . . .

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vii

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5.2

5.3

5.4

5.1.2.2 Voltage Dependent Transmission . . . . . . . . .
5.1.2.3 Reﬂections . . . . . . . . . . . . . . . . . . . . .
5.1.3 Magnetic Fields . . . . . . . . . . . . . . . . . . . . . . . .
Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.1 Calculating Saturation Currents . . . . . . . . . . . . . . .
5.2.2 Least Squares Fitting . . . . . . . . . . . . . . . . . . . . .
5.2.2.1 Secondary Ion Distribution Function . . . . . . .
5.2.2.2 Detector Transmission Function . . . . . . . . . .
5.2.3 High Energy Tail . . . . . . . . . . . . . . . . . . . . . . .
5.2.4 Error Estimation . . . . . . . . . . . . . . . . . . . . . . .
5.2.4.1 Statistical Errors in the Measurements . . . . . .
5.2.4.2 Systematic Errors in the Measurements . . . . .
5.2.4.3 Systematic Errors in the Theoretical Predictions
Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.1 LEDA injector ion source . . . . . . . . . . . . . . . . . .
5.3.1.1 Overview . . . . . . . . . . . . . . . . . . . . . .
5.3.1.2 Data Acquisition . . . . . . . . . . . . . . . . . .
5.3.1.3 Diagnostics . . . . . . . . . . . . . . . . . . . . .
5.3.1.4 Measurements . . . . . . . . . . . . . . . . . . . .
5.3.2 SuSI Superconducting ECRIS . . . . . . . . . . . . . . . .
5.3.2.1 Overview . . . . . . . . . . . . . . . . . . . . . .
5.3.2.2 Data Acquisition . . . . . . . . . . . . . . . . . .
5.3.2.3 Diagnostics . . . . . . . . . . . . . . . . . . . . .
5.3.2.4 Measurements . . . . . . . . . . . . . . . . . . . .
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 6 3D ECR Extraction Model . . . . . .
6.1 Overview . . . . . . . . . . . . . . . . . . . . . .
6.2 The Extraction Model . . . . . . . . . . . . . .
6.2.1 Initial Conditions with WARPxy . . . .
6.2.2 2D+ Extraction with WARP3d . . . . .
6.3 Beam Transport with WARPxy . . . . . . . . .
6.3.1 Electrostatic Elements . . . . . . . . . .
6.3.2 Automatic Space Charge Compensation
6.4 VENUS Simulations . . . . . . . . . . . . . . .
6.4.1 Heavy Ions . . . . . . . . . . . . . . . .
6.4.2 Light Ions . . . . . . . . . . . . . . . . .
6.5 SuSI Simulations . . . . . . . . . . . . . . . . .
6.6 Conclusion . . . . . . . . . . . . . . . . . . . . .

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160
160
161
161
165
169
169
170
172
173
174
178
186

Chapter 7 An Example Application: DIANA .
7.1 Introduction . . . . . . . . . . . . . . . . . . .
7.2 Layout . . . . . . . . . . . . . . . . . . . . . .
7.2.1 Front End . . . . . . . . . . . . . . . .
7.2.2 Ion Source . . . . . . . . . . . . . . . .
7.2.3 Accelerator Column . . . . . . . . . . .
7.2.4 Bending Section . . . . . . . . . . . . .
7.3 Custom Modiﬁcations . . . . . . . . . . . . .
7.3.1 Ion Source Extraction System . . . . .
7.3.2 Adjusted Voltage Gradient . . . . . . .
7.4 Simulations . . . . . . . . . . . . . . . . . . .
7.4.1 Ion Source Extraction . . . . . . . . .
7.4.2 Front End and Accelerator Tube . . .
7.5 Conclusion . . . . . . . . . . . . . . . . . . . .

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187
187
189
194
196
199
200
201
201
203
207
207
211
216

Chapter 8 Conclusion . . . . . . . . . . . . . . . . . . .
8.1 Summary . . . . . . . . . . . . . . . . . . . . . . . .
8.1.1 Space Charge Compensation Measurements .
8.1.2 ECRIS Extraction and Beam Line Simulations
8.1.3 The DIANA High Energy Accelerator . . . . .
8.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . .
8.2.1 Space Charge Compensation Measurements .
8.2.2 ECRIS Extraction and Beam Line Simulations
8.2.3 The DIANA High Energy Accelerator . . . . .

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217
217
217
218
219
220
220
222
222

APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
Appendix A Publications and Proceedings . . . . . . . . . . . . . . . . . . . . . 224
Appendix B DIANA - Resistor Values . . . . . . . . . . . . . . . . . . . . . . . 226
BIBLIOGRAPHY

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

ix

. . . . . . . . . . . . . . . . . 233

LIST OF TABLES

Table 2.1

M¨ller-Salzborn ﬁt parameters for charge-exchange cross-sections [1].
u

40

Table 4.1

LEDA - Ion Source Parameters . . . . . . . . . . . . . . . . . . . . .

74

Table 4.2

Artemis parameters. Magnetic ﬁeld values are nominal and subject
to optimization for each desired ion. . . . . . . . . . . . . . . . . . .

76

SuSI parameters. Magnetic ﬁeld values are nominal and subject to
optimization for each desired ion [2]. . . . . . . . . . . . . . . . . . .

78

VENUS parameters. Magnetic ﬁeld values are nominal and subject
to optimization for each desired ion [3]. . . . . . . . . . . . . . . . .

80

Table 4.3

Table 4.4

Table 5.1

NeMo - Copper mesh parameters. . . . . . . . . . . . . . . . . . . . 101

Table 5.2

Residual magnetic ﬁeld Bz (z) on axis at the diﬀerent neutralization
measurement locations. . . . . . . . . . . . . . . . . . . . . . . . . . 108

Table 5.3

Parameters for the neutralization measurement in the LEDA injector
ion source low energy beam transport line. . . . . . . . . . . . . . . 129

Table 5.4

Cross-sections for creation of slow secondary ions calculated for 70
keV proton beams in diﬀerent residual gases. Note: Only includes
single ionization and charge-exchange. . . . . . . . . . . . . . . . . . 132

Table 6.1

VENUS - Uranium simulation parameters . . . . . . . . . . . . . . . 174

Table 7.1

DIANA - High energy experiment performance criteria. . . . . . . . 192

Table 7.2

30◦ dipole speciﬁcations. For more information about Rogowski pole
tips see [4, 5] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

Table 7.3

DIANA - NANOGAN parameters. . . . . . . . . . . . . . . . . . . . 197

Table 7.4

Company guaranteed ion beam currents for NANOGAN. Vertical:
Ion species, Horizontal: Charge-state. Currents in eµA. From [6]. . . 198

x

Table 7.5

Simulations - Test case parameters. (case numbers kept consistent
with other documentation) The number of shorted rings corresponds
to the positions in Figure 7.9. . . . . . . . . . . . . . . . . . . . . . 212

Table 7.6

DIANA simulations - test case results. In case 16B it can be seen that
the beam current decreases by ∼ 7% du to the use of a collimator.
The beam diameters are <5 mm, and no case ﬁlls the 90◦ dipole more
than 75%. (case numbers kept consistent with other documentation) 215

Table 1

DIANA - Resistor values. . . . . . . . . . . . . . . . . . . . . . . . . 226

xi

LIST OF FIGURES

Figure 2.1

Figure 2.2

Figure 2.3

Figure 2.4

Figure 2.5

Figure 2.6

Figure 2.7

Figure 2.8

Figure 2.9

Beam cross-section of a K-V distribution. For interpretation of the
references to color in this and all other ﬁgures, the reader is referred
to the electronic version of this dissertation. . . . . . . . . . . . . . .

11

Two-dimensional transversal subspace (x-x’) of the phase space of a
K-V distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

Phase space evolution of a K-V beam without space charge in drift
space (no applied ﬁelds) for three diﬀerent instances along the beam
path (z). Note that the area of all three ellipses is constant, as is the
maximum divergence angle xb . x0 = xb (z = z0 ). . . . . . . . . . . .

13

Phase space distortion due to spherical aberrations in a periodic thin
lens focusing channel. Each period consists of a drift, a thin lens and
another drift. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

Radial beam potential and electrostatic self ﬁeld of a round 0.5 mA
16 O6+ beam. Beam radius 20 mm, grounded pipe radius 10 cm. . .

21

Solutions of the envelope equation for a 0.5 mA 16 O6+ beam in drift
space for diﬀerent initial angles a0 = rb (z = 0), initial radius rb (z =
0) = 10 mm, and zero emittance. Note that in the K-V model the
self-ﬁelds are linear in r and the emittance is constant. . . . . . . . .

22

Solutions of the envelope equation for a 0.5 mA 16 O6+ beam in drift
space for diﬀerent emittances, initial radius rb (z = 0) = 10 mm and
initial angle -0.1 rad. The emittance shown here is the normalized
full emittance of the beam ( n = 4 n−rms ). . . . . . . . . . . . . . .

23

Space charge compensation. Beam Cross section view with ions and
electrons created from residual gas ionization and charge exchange. .

26

Space charge compensation. Change in beam potential from uncompensated beam to 50% compensation. This is for a 2 mA oxygen
primary beam with round cross section (rb = 20 mm) and homogeneous charge distribution. The beam pipe (rp = 100 mm) is assumed
to be grounded. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

xii

Figure 2.10

Example of theoretical neutralization prediction for a 75 keV proton
beam with a 14 mm diameter in H2 residual gas. Experimental data
shown for comparison from [7]. The data was published without
errorbars. Left: pressure variation at a constant beam current of 100
mA, right: current variation at a constant gas density of 1.2 · 1018
m−3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

The magnet structure of the VENUS ECR ion source. Courtesy of
D. Leitner. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

Axial Field of the VENUS ECR ion source. It can be seen that the
mirror ratios are around 6 at injection (left peak ) and 4 at extraction
(right peak). Also indicated are the B values that correspond to the
two microwave frequencies 18 GHz and 28 GHz used in VENUS. The
two corresponding ecr surfaces then form nested shells. Image taken
from [8]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

Upper: Electron and Ion densities in collisionless plasma near wall.
Lower: Plasma potential of a collisionless plasma near wall. Image
by Lukas Derendinger under the GNU Free Documentation License.

49

Measured emittances of the Lawrence Berkeley National Laboratory
AECR-U compared to the theoretical prediction of magnetic emittances. Image taken from [9] . . . . . . . . . . . . . . . . . . . . . .

57

Figure 4.1

Schematic of the LEDA injector ion source. Image taken from [10]. .

73

Figure 4.2

Schematic of Artemis B (a duplicate of Artemis A). Image taken
from [11] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75

3D CAD model of the SuSI ECRIS. In the center the plasma chamber
can be seen, sorrounded by the six solenoid coils. Gas and microwaves
are fed from the left, the beam is extracted on the right. . . . . . . .

77

Schematic of the VENUS superconducting ion source. Image taken
from [3]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79

Faraday cup schematic. The positive ion beam deposits charge on the
cup. Electrons leaving the material due to the ion impact are pushed
back onto the cup by a suppressor on a negative potential. . . . . .

82

Cartoon picture of the A/q analyzer principle. . . . . . . . . . . . .

83

Figure 2.11

Figure 2.12

Figure 2.13

Figure 2.14

Figure 4.3

Figure 4.4

Figure 4.5

Figure 4.6

xiii

Figure 4.7

Example Charge State Distribution (CSD) of an oxygen beam from
the ECRIS SuSI at NSCL. Contamination by argon from a previous
experiment can also be seen as well as nitrogen from imperfect vacuum. 84

Figure 4.8

3D CAD model of a beam viewer. This beam viewer was designed
for the LEBT of the AECR-U ion source at LBNL. . . . . . . . . . .

86

Two images taken with a quartz crystal beam viewer. The beam
contained mostly argon and oxygen. Diﬀerent ion species (mass m,
charge state q) are focused diﬀerently in the extraction region of the
source and can be seen as ring-like structures on the viewer. The ﬁne
mesh covering the crystal can also be seen. . . . . . . . . . . . . . .

87

Figure 4.9

Figure 4.10

Cartoon of the slit proﬁle scanner operating principle. The two slits
can move independently. A computer program automatically sets the
slits and reads in the faraday cup current. By rastering the area of
the beam cross section a transversal beam intensity proﬁle is obtained. 88

Figure 4.11

Example beam proﬁle of an O3+ beam, ≈ 235 eµA. The beam exhibits a triangular structure, which is typical for many ECRIS beams. 89

Figure 4.12

Example beam proﬁle of an Ar8+ beam, ≈ 200 eµA, tuned for ’roundness’. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89

Schematic of an Allison-type electrostatic emittance scanner. Image
taken from [12]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

90

Allison emittance scan of a U36+ beam. Distortions can be seen
that make it hard to determine the ’full’ emittance of the beam (cf.
discussion in Section 2.1.3) the red ellipses denote the four-rms areas
of these two scans. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91

3D CAD model of the retarding ﬁeld analyzer. The measured ions
enter from the top and are repelled by grid 2 if their energy in eV
is lower than the grid voltage in V. The current measured with a
picoammeter connected to the collector is proportional to the number
of ions reaching the collector. . . . . . . . . . . . . . . . . . . . . . .

92

Cartoon demonstrating the working principle of a retarding ﬁeld analyzer. Upper image: Retarding grid (’Mesh 2’) at 0 V. All ions make
it to the collector. Lower image: Retarding grid at 15 V. Ions with
less energy (e.g. 10 eV) are repelled. In both cases electrons from the
outside are repelled by the electron suppressor (’Mesh 3’). . . . . . .

94

Figure 4.13

Figure 4.14

Figure 4.15

Figure 4.16

xiv

Figure 4.17

Figure 5.1

Example RFA spectrum. The directly obtained spectrum (solid line)
and the ﬁrst derivative -dI/dV (dashed line) are both normalized to
one for better visibility. The ﬁrst derivative of the obtained spectrum
represents the ion energy distribution. . . . . . . . . . . . . . . . . .

95

Cartoon of space-charge compensation through secondary electrons
and measurement of the energy distribution of the secondary ions
with the retarding ﬁeld analyzer ’NeMo’. Electrons and ions are
created inside the beam by ionization and charge-exchange of the
beam with the residual gas. Electrons are captured and lower the
beam space charge potential. ions are expelled and can be measured
with NeMo. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

97

Figure 5.2

Equipotential lines close to the mesh wires of Grid 2 calculated with
SIMION 8.1. Grids 1 and 3 are not shown as the distances from grid
to grid ( 6mm) are much larger than the wire spacing. Left: Grid
3 at -150 V. Right: Grid 3 at -500 V. Grid 1 and 2 at 0 V in both
cases. It can be seen that the holes between the wires have a lower
potential than the wires and that this depends on the setting of Grid 3.100

Figure 5.3

Two measured RFA curves exhibiting a shift towards higher potential
with increasing voltage on Grid 3 due to the formation of an eﬀective
potential lower than the applied voltage in the space between mesh
wires. Left curve with grid 3 at -150 V, right curve with grid 3 at
-500 V. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

Figure 5.4

Eﬀects of realistic mesh for three diﬀerent types of initial particles.
The rightmost curve is for 11 eV ions, the other two for 6 eV with
the one starting higher having smaller incident angles. All three are
with grid 3 on -150 V as during the measurements in the SuSI LEBT.
A bump is clearly noticeable in all three curves. This is due to the
fact that for grid 2 voltages close to the particle energy, incident ions
that would otherwise collide with mesh wires are now bent around the
wires. Changing the angle of incident changes the saturation current,
but not the horizontal shift and the ﬁnite width of the dropoﬀ (∆V ). 103

Figure 5.5

Comparison of slow ion reﬂections as seen for the lowest secondary ion
energies in the LEDA injector source measurements and simulations
under similar conditions (i.e. similar ion energies, same voltage of
-450 V on grid 3). This simulation did not include magnetic ﬁeld
eﬀects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

xv

Figure 5.6

+
Example trajectories of secondary H2 ions in the residual longitudinal magnetic ﬁeld of the two solenoid magnets of the LEDA injector
ion source. Trajectories were calculated with SIMION 8.1 for 5 eV
ions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

Figure 5.7

Collimator transmission curves for the case of the LEDA injector ion
+
source. The two secondary ion species H2 and H2 O+ are the two
biggest contributors to the residual gas pressure (as measured with a
residual gas analyzer (RGA) and have a ratio of ≈ 19:1. . . . . . . . 109

Figure 5.8

Mesh transmission curves for -450 V on grid 3. The ion energy increases from left to right. A drop in transmission as well as the formation of a peak can be seen towards the lowest energies. At higher
energies ( > 5 eV) the transmission curve only shifts further to the
right without changing its shape. . . . . . . . . . . . . . . . . . . . . 115

Figure 5.9

Mesh transmission curves for -150 V on grid 3. The ion energy increases from left to right. A drop in transmission as well as the formation of a peak can be seen towards the lowest energies. At higher
energies ( > 5 eV) the transmission curve only shifts further to the
right without changing its shape. . . . . . . . . . . . . . . . . . . . . 116

Figure 5.10

Example of a high energy tail. This spectrum was taken with a 180
µA Ar8+ beam at SuSI/Q014. The long tail can clearly be seen. The
tail contributes approximately 25 % to the saturation current. The
ﬁrst derivative corresponds to the energy distribution of the secondary
ions. Both, signal and derivative are normalized to a maximum of one
for better visibility. . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

Figure 5.11

Foto of the LEDA injector source at NSCL. The source is located on
the left, followed by diagnostic box 1 (the large box in the middle of
the image). NeMo was mounted where a transparent ﬂange can be
seen in the center of diagnostic box 1. Courtesy of Felix Marti. . . . 124

Figure 5.12

3D CAD drawing of diagnostic box 1 and the NeMo retarding ﬁeld
analyzer in the LEDA injector source beamline. The source itself is
to the left and not shown. . . . . . . . . . . . . . . . . . . . . . . . . 125

xvi

Figure 5.13

NeMo - LEDA injector source: Beam currents versus microwave
power. Bergoz 1 and the drain current are representative of the total beam current at the location of the neutralization measurements.
It can be seen that the extracted current behaves linearly with microwave power up to 600 W and then ﬂattens out. This could be
because not enough gas is supplied to sustain the higher currents or
because the source is outgassing, thereby contaminating the plasma.

126

Figure 5.14

A typical residual gas analyzer spectrum during a neutralization measurement. The data suggests a ratio of roughly 19:1 of molecular
hydrogen (H2 ) to heavier species. Note the logarithmic scale for the
pressure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

Figure 5.15

Transmission curves for H2 and H2 O as well as a mix of the two
with ratio 19:1 as observed in the RGA spectra. It is evident that
for heavier ions the magentic ﬁeld does not have a very big eﬀect,
thus H2 O is representative of all ion species heavier than 14 N . It can
also be seen that the mesh transmission is around 73% (as expected)
down to ion energies of about 2.5 eV where it rapidly drops towards
lower energies due to reﬂection. . . . . . . . . . . . . . . . . . . . . . 130

Figure 5.16

Corrected saturation currents. Note that the three data points at the
highest beam currents actually have lower secondary ion energies,
either due to a larger beam radius or higher space charge compensation. As discussed in Section 5.2.4 the systematic error bars on the
theoretical prediction are ≈ 82 − 85%. This is very large and makes
an absolute comparison mute. However, the relative change of saturation current with beam current can be examined and reproduced
by applying transmission factors (cf. text). . . . . . . . . . . . . . . 132

Figure 5.17

NeMo neutralization Spectra for 5 diﬀerent beam currents in the
LEDA injector ion source beam line diagnostic box 1. The voltages have not yet been corrected for the eﬀective potential inside the
meshes. The curves increase in current from left to right. . . . . . . 133

Figure 5.18

Results of the current variation with the LEDA injector ion source:
Neutralization and Beam sizes. It can clearly be seen that although
the overall beam radii are much too large, they reﬂect the observed
variations in neutralization. Note that changing the beam radius in
the analysis only changes the center potential of the beam (shifts the
spectrum left-right) and not the slope of the curve (i.e. the neutralization). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

xvii

Figure 5.19

Measured neutralization as a function of ion beam density. The solid
line is the theoretical prediction following the model described in
2.1.5, the shaded area includes the uncertainties in pressure and cross
sections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

Figure 5.20

Schematic of the SuSI LEBT system up to the second dipole magent.
Between the ﬁrst dipole and the location of NeMo (in the diagnostic
box Q013/14) there is a focusing and collimation channel consisting
of 3 solenoids, apertures, and slits. . . . . . . . . . . . . . . . . . . . 137

Figure 5.21

3D CAD schematic of diagnostic box Q014 in the SuSI LEBT. . . . 138

Figure 5.22

Beam proﬁle of O3+ , 1.1 · 10−6 Torr . . . . . . . . . . . . . . . . . . 141

Figure 5.23

Beam proﬁle of O3+ , 5.0 · 10−6 Torr . . . . . . . . . . . . . . . . . . 141

Figure 5.24

Beam proﬁle of O3+ , 8.0 · 10−6 Torr . . . . . . . . . . . . . . . . . . 142

Figure 5.25

Beam proﬁle of O3+ , 1.1 · 10−5 Torr . . . . . . . . . . . . . . . . . . 142

Figure 5.26

Beam proﬁle of Ar8+ , 1.0 · 10−6 Torr . . . . . . . . . . . . . . . . . 143

Figure 5.27

Beam proﬁle of Ar8+ , 2.5 · 10−6 Torr . . . . . . . . . . . . . . . . . 143

Figure 5.28

Beam proﬁle of Ar8+ , 5.0 · 10−6 Torr . . . . . . . . . . . . . . . . . 144

Figure 5.29

Beam proﬁle of Ar8+ , 7.5 · 10−6 Torr . . . . . . . . . . . . . . . . . 144

Figure 5.30

Beam proﬁle of Ar8+ , 1.0 · 10−5 Torr . . . . . . . . . . . . . . . . . 145

Figure 5.31

Saturation currents for three pressure variations in the SuSI LEBT
at Q014. The theoretical predictions are according to equation 5.3
with T = 0.65 and cross-section according to the M¨ller-Salzborn
u
empirical formula [1]. The transmission T depends slightly on the
ion energy and on the angle of incidence. . . . . . . . . . . . . . . . 146

Figure 5.32

Pressure variation of O3+ , neutralization measured and theory. It
can be seen that the theoretical prediction is signiﬁcantly lower than
the measured values. . . . . . . . . . . . . . . . . . . . . . . . . . . 147

Figure 5.33

Pressure variations of O6+ and Ar8+ , neutralization measured and
theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
xviii

Figure 5.34

Beam proﬁle of O3+ , 122 µA. . . . . . . . . . . . . . . . . . . . . . . 149

Figure 5.35

Beam proﬁle of O3+ , 203 µA. . . . . . . . . . . . . . . . . . . . . . . 150

Figure 5.36

Beam proﬁle of O3+ , 298 µA. . . . . . . . . . . . . . . . . . . . . . . 150

Figure 5.37

Beam proﬁle of O6+ , 195 µA. . . . . . . . . . . . . . . . . . . . . . . 151

Figure 5.38

Beam proﬁle of O6+ , 302 µA. . . . . . . . . . . . . . . . . . . . . . . 151

Figure 5.39

Beam proﬁle of O6+ , 404 µA. . . . . . . . . . . . . . . . . . . . . . . 152

Figure 5.40

Beam proﬁle of O6+ , 490 µA. . . . . . . . . . . . . . . . . . . . . . . 152

Figure 5.41

Beam proﬁle of O6+ , 604 µA. . . . . . . . . . . . . . . . . . . . . . . 153

Figure 5.42

Beam proﬁle of Ar8+ , 111 µA. . . . . . . . . . . . . . . . . . . . . . 153

Figure 5.43

Beam proﬁle of Ar8+ , 170 µA. . . . . . . . . . . . . . . . . . . . . . 154

Figure 5.44

Beam proﬁle of Ar8+ , 223 µA. . . . . . . . . . . . . . . . . . . . . . 154

Figure 5.45

Beam proﬁle of Ar8+ , 276 µA. . . . . . . . . . . . . . . . . . . . . . 155

Figure 5.46

Beam proﬁle of Ar8+ , 331 µA. . . . . . . . . . . . . . . . . . . . . . 155

Figure 5.47

Saturation currents for three beam current variations in the SuSI
LEBT at Q014. A transmission factor of T = 0.65 was used. The
lower measured saturation current for the Ar8+ series could be due
to the vertical asymmetry. . . . . . . . . . . . . . . . . . . . . . . . 156

Figure 5.48

Neutralization for three current variations. . . . . . . . . . . . . . . 157

Figure 6.1

Two helium beams extracted from VENUS. Left: without the radial
conﬁnement sextupole, right: under normal operating conditions with
the sextupole. The beam line layout can be seen in Figure 6.5. The
solenoid was set to 400 A in both cases. . . . . . . . . . . . . . . . . 161

xix

Figure 6.2

Plasma sputter marks inside VENUS. Left: The biased disk exhibits a
large star structure and a smaller clear triangle of about 5 mm height
etched into the plate. Right: The same (now rotated) discolorations
are seen, however the triangle would be inside the extraction aperture.162

Figure 6.3

Obtaining the initial conditions for a helium beam from the VENUS
ECRIS. Left: Particle distribution at the biased disk. Right: After
tracking through the sources magnetic ﬁeld. The circle represents the
extraction aperture. . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

Figure 6.4

Example of start and end distribution in an extraction simulation.
Shown is an argon beam with 4.6 mA extracted from SuSI. Left:
initial distribution. Note the slight vertical oﬀset due to a hole in the
source’s iron yoke. Right distribution after 15 cm. The rotation is
induced by the residual solenoid ﬁeld. . . . . . . . . . . . . . . . . . 168

Figure 6.5

VENUS - Source and LEBT. . . . . . . . . . . . . . . . . . . . . . . 172

Figure 6.6

Simulation of a 1.6 mA uranium beam extracted from VENUS. . . . 173

Figure 6.7

VENUS simulation of a helium beam without the sextupole for radial
conﬁnement. The simlations agree reasonably well with the measurements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

Figure 6.8

VENUS simulation of a helium beam with the sextupole for radial
conﬁnement energized, compared to measurements. Simulation and
measurement disagree by a factor of 2. . . . . . . . . . . . . . . . . . 176

Figure 6.9

Two images from the beam viewer. Both were taken with the same
settings: Solenoid lens current = 400 A, Puller position = 50 mm.
Left: Sextupole magnet on, right: sextupole magnet oﬀ. Due to the
conﬁning qualities of the sextupole, the extracted beam current in
the oﬀ case was lower (1.3 mA vs. 2 mA). . . . . . . . . . . . . . . . 177

Figure 6.10

Simulation results 825 mm after extraction (position of the beam
viewer). Both were simulated with the same settings: Solenoid lens
current = 400 A, Puller position = 50 mm. Left: Sextupole magnet
on, right: sextupole magnet oﬀ. The simulated currents were matched
to the measurements (left: 2.0 mA, right: 1.3 mA). . . . . . . . . . . 177

xx

Figure 6.11

Schematic of the SuSI LEBT system up to the second dipole magent.
Between the ﬁrst dipole and the location of NeMo (in the diagnostic
box Q013/14) there is a focusing and collimation channel consisting
of 3 solenoids, apertures, and slits. . . . . . . . . . . . . . . . . . . . 178

Figure 6.12

Envelopes of a O3+ beam in the SuSI beamline with a pressure of
1.1 · 10−5 Torr. The upper envelope is the horizontal one and the
lower envelope the vertical one. From left to right, the grey boxes
are: Source, einzel lens, analyzing magnet, and the three solenoids of
the collimation channel. . . . . . . . . . . . . . . . . . . . . . . . . . 181

Figure 6.13

Measured beam proﬁle of O3+ , 5.0 · 10−6 Torr . . . . . . . . . . . . 182

Figure 6.14

Simulated beam proﬁle of O3+ , 5.0 · 10−6 Torr . . . . . . . . . . . . 182

Figure 6.15

Measured beam proﬁle of O3+ , 8.0 · 10−6 Torr . . . . . . . . . . . . 183

Figure 6.16

Simulated beam proﬁle of O3+ , 8.0 · 10−6 Torr . . . . . . . . . . . . 183

Figure 6.17

Measured beam proﬁle of O3+ , 1.1 · 10−5 Torr . . . . . . . . . . . . 184

Figure 6.18

Simulated beam proﬁle of O3+ , 1.1 · 10−5 Torr . . . . . . . . . . . . 184

Figure 6.19

Neutralization along the beam line in the SuSI LEBT for a O3+ beam
in 1.1 · 10−5 Torr residual gas pressure. The center line represents
the simulation run with the charge exchange cross section obtained
by using the M¨ller-Salzborn formula. The three boxes correspond
u
to the solenoids in the collimation channel. The dashed lines are
element boundaries and apertures. Inside the einzel lens (0.4 - 0.8
m), the neutralization is zero, inside the dipole (1.1 - 1.9 m) the
beam contaminants are ﬁltered out. . . . . . . . . . . . . . . . . . . 185

Figure 7.1

DIANA - Facility layout. Courtesy of the DIANA collaboration. . . 189

Figure 7.2

High energy accelerator layout. From the top: Front end with ion
source and species separation. Accelerator column including an electrostatic quadrupole triplet (ESQ1). Bending section including an
electrostatic quadrupole triplet (ESQ2) and a 90◦ bending magnet.
Courtesy of NEC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

xxi

Figure 7.3

Front end layout. From the left: ECR ion source, extraction system,
collimator, einzel lens 30◦ , bending magnet, immersion lens. Courtesy
of NEC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

Figure 7.4

NANOGAN Sectional view. The beam is extracted from right to
left. This image shows the standard diode extraction system. ’Axial
FeNdB’ denotes the permanent magnets creating the solenoid ﬁeld,
’radial FeNdB’ the permanent magnet sextupole. Image adapted from
[13]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

Figure 7.5

Electrostatic accelerator column layout. The column consists of 4
units, each unit is composed of 2 segments ´ 23 rings (except unit 2
a
which holds the electrostatic quadrupole). Courtesy of NEC. . . . . 199

Figure 7.6

Cartoon picture of the NEC Pelletron accelerator tube including resistor chain. The resistor chain acts as a voltage divider. The HV
terminal voltage can be set from 300 kV to 3 MV. The Gap voltage
as well as the voltages on the respective rings is then determined by
the ratio of the resistors to each other. In the simplest case, all ring
resistors R are equal and thus the voltage diﬀerence between each
ring as well. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

Figure 7.7

Side-by-side comparison of the two diﬀerent extraction systems proposed for the Nano-PK ECR ion source. Diode system (top); Tetrode
system (bottom). Note that the ﬁnal beam energy (19 keV for singly
charged ions) is the same for both systems. The tetrode system gives
more tuning ﬂexibility by adjusting the voltage of the ’10000 V’ electrode (green). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

Figure 7.8

Comparison of linear voltage gradient and adjusted voltage gradient.
The voltage on each element of the column is shown versus the position along the accelerator axis. The horizontal axis corresponds to
the picture of the column below the ﬁgure. . . . . . . . . . . . . . . 204

Figure 7.9

Comparison of diﬀerent total voltages using the adjusted voltage gradient and the shorting rod technique. The voltage on each element
of the column is shown versus the position along the accelerator axis.
The horizontal axis corresponds to the picture of the column below
the ﬁgure, The vertical arrows denote the shorting positions. . . . . 205

xxii

Figure 7.10

Simpliﬁed layout of the simulated high energy accelerator and beamline. The ion source extraction simulation ends 88.25 mm after the
plasma aperture of the source. This is where the WARP simulation
starts, which covers the rest of the system in one continuous simulation.206

Figure 7.11

RZ-symmetric PANDIRA model of the permanent magnet structure
of NANOGAN. Dimensions are in cm, horizontal axis is Z, vertical is
R. Sextupole magnet not included. . . . . . . . . . . . . . . . . . . . 208

Figure 7.12

NANOGAN - Bz (z) on axis. As is typical for permanent magnets,
the ﬁeld (here in Gauss) becomes negative, before going to zero. The
plasma volume is located between the two peaks in the middle. . . . 208

Figure 7.13

IGUN simulation - Trajectory plot. The electrode voltages (from left
to right) are: Plasma electrode = 19 kV, puller electrode = 10 kV,
electron repeller = -2 kV, ground electrode = 0 kV. The axes are in
grid units (1 gu = 0.25 mm). The magnetic ﬁeld (secondary axis) is
shown as a green dashed line. . . . . . . . . . . . . . . . . . . . . . . 209

Figure 7.14

DIANA - WARP simulation results of a 300 keV 4 He1+ beam: Horizontal (upper half) and vertical (lower half) beam envelopes. . . . . 213

Figure 7.15

DIANA - WARP simulation results of a 300 keV 4 He1+ beam: Cross
section. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

Figure 7.16

DIANA - WARP simulation results of a 300 keV 4 He1+ beam: Horizontal phase space. . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

xxiii

KEY TO SYMBOLS AND ABBREVIATIONS

A . . . . . . . . . .Atomic mass number (sum of neutrons and protons in the nucleus of an atom)
amu . . . . . . . . . . . . . . . . . . . . . . . Atomic mass unit (1/12 of the mass of 12 C ≈ 1.661 · 10−28 kg)
ARTEMIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A roomtemperature ECRIS (at NSCL)
β . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relativistic factor beta (β = speed/c)
c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Speed of light in vacuum (≈ 3 · 108 m/s)
DIANA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dual Ion Accelerators for Nuclear Astrophysics
e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Elementary charge (≈ 1.602 · 10−19 C)
0

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vacuum permittivity (≈ 8.854 · 10−12 F/m)

ECR(IS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electron Cyclotron Resonance (Ion Source)
eµA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electrical micro-Amperes (10−6 C/s)
emA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electrical milli-Amperes (10−3 C/s)
γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relativistic factor gamma (γ = 1/ 1 − β 2 )
IGUN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A ion source extraction simulation code
K-V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Kapchinsky-Vladimirsky, a special particle distribution
LBNL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lawrence Berkeley National Laboratory
LEBT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Low Energy Beam Transport (System)
NEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . National Electrostatics Corporation
NeMo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Neutralization Monitor, the RFA used in this thesis
NSCL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . National Superconducting Cyclotron Laboratory
PDR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Preliminary Design Report
pµA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Particle micro-Amperes (10−6 particles/s)

xxiv

pmA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Particle milli-Amperes (10−3 particles/s)
PIC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Particle-In-Cell, a simulation method
q . . . . . . . . . . . . . . . . . . . . Charge-State (number of electrons missing for neutrality in an atom)
RGA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Residual Gas Analyzer
RFA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Retarding Field Analyzer
SOR . . . . . . . . . . . . . . . . . Successive Over-Relaxation, a ﬁnite diﬀerence method for ﬁeldsolves
SuSI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Superconducting Source for Ions (at NSCL)
VENUS . . . . . . . . . . . . . . . . . . . . . . . . . Versatile ECR ion source for NUclear Science (at LBNL)
WARP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A particle-in-cell simulation code
Z . . . . . . . . . . . . . . . . . . . . . . . . . . Atomic number (number of protons in the nucleus of an atom)

xxv

Chapter 1
Introduction

1.1

General Introduction

Since their inception in 1965 [14, pp. 313] electron cyclotron resonance ion sources (ECRIS)
have been a ﬁeld of rapid development. Their ability to produce very high charge state ions
of virtually any element make them the injector of choice for many heavy ion accelerator
facilities. A few examples are: the 88” Cyclotron at Lawrence Berkeley National Laboratory,
the Coupled Cyclotron Facility at Michigan State University or the Facility for Rare Isotope
Beams (FRIB) [15] under construction also at Michigan State University. Over time, the
demand for more beam current has grown and today’s third generation ECRIS are capable
of delivering many emA of total extracted beam current and several hundreds of eµA of
analyzed currents per species. However, facilities like FRIB already push the performance
limits of third generation ECRIS, and it stands to reason that in the future we will need to
continue the development of ECR ion sources. In his book ’Electron Cyclotron Resonance
Ion Sources and ECR plasmas’ Richard Geller presented simple scaling laws stating that
the charge state and beam currents extractable from ECR ion sources could be increased
by going to higher magnetic ﬁeld strengths and higher microwave frequencies [14, pp. 394].
A hypothesis that, over the last decades, proved to be reasonably valid, and today eﬀorts
are in place to develop a fourth generation ECRIS that will run at the highest ﬁelds and
frequencies yet, promising further increase in delivered beam current [16, 17]. Of course,
1

many other aspects of ECRIS physics inﬂuence the maximal achievable charge state, beam
intensities, and stability. Some of those are: Coupling of radio frequency microwaves into
the ECR plasma [18,19], the plasma dynamics inside the source in the presence of microwave
heating and strong magnetic ﬁelds [20, 21], the exact structure of the magnetic conﬁnement
ﬁeld [14], gas mixing [14, 22], and the position of the extraction aperture [22, 23]. In pulsed
mode, preglow [24, 25] and afterglow [26, 27] eﬀects were discovered, which can increase the
beam intensity for a brief period. In addition, many detailed studies have been performed
including, but not limited to: Precision measurements of the beam energy to determine the
plasma potential [28], X-ray measurements [29, 30], and measurements of the plasma energy
density with a diamagnetic loop [30]. Some of these results will be mentioned in Section 2.2
and chapter 6, but going into the details would go far beyond the scope of this thesis. Instead
I will concentrate on the subjects of ion beam extraction and low energy beam transport. At
the presently achievable beam intensities, space charge (the repulsion of neighboring beam
particles due to their own electric ﬁelds) can become an important factor for the low energy
beam transport (LEBT). In the case of ECR ion sources, the LEBT has two major functions:
• The transport of the beam with minimal losses and degradation of the beam quality.
• The selection of the desired ion species from the number of species provided by the
source.
A very important design tool both for the extraction systems and the LEBT are ion beam
transport simulations. However, the simulation of ion beam extraction from an ECRIS
plasma is not straight-forward and there are several challenges to overcome:
1. Modeling the extraction process from the plasma itself. This includes considering the
plasma electrons, the current density of the extracted ions, and the applied electrostatic
2

ﬁelds which, together, lead to the formation of a plasma meniscus.
2. Modeling the extraction and transport of a multitude of diﬀerent ion species with
diﬀerent masses and charge states.
3. Modeling the extraction from a region of high longitudinal magnetic ﬁeld, which introduces an increase in emittance.
4. Taking into account the non-rotationally symmetric beam due to the radial conﬁnement
sextupole ﬁeld in ECR ion sources.
And for the beam transport through the LEBT we have to consider:
5. The asymmetric beam from the extraction simulation.
6. Space charge and space charge compensation in the LEBT.
7. Realistic electrostatic and magnetostatic ﬁeld maps for the beam optics elements.
Item 1. has been studied extensively in the past and a one dimensional model for the plasma
sheath region in the source aperture was developed [31] which could be implemented in
computer codes like IGUN [31–33] to successfully simulate the extraction of rotationally
symmetric beams from plasma ion sources. Later, simulation codes like IGUN also included
items 3 and 4, multi-species extraction and magnetic ﬁelds. 3D codes like IBSimu [34],
KOBRA3 INP [35] and WARP [36] are now able to include 3D ﬁeld maps for the electrostatic
and magnetostatic ﬁelds and one can deﬁne arbitrary initial particle distributions. However,
the exact spatial and velocity distributions for those initial conditions are still a matter
of debate and while the aforementioned plasma simulations can provide some insight, no
satisfying solution has been found yet.
3

In this thesis, I will use a semi-empirical model for the 3D extraction of highly charged ions
from an ECRIS using the particle-in-cell code WARP. I will investigate the inﬂuence of the
extraction of up to 30 diﬀerent ion species at the same time on the beam dynamics in the
extraction system and the beginning of the LEBT.
I will furthermore investigate the level of space charge compensation (interchangeably also
called ’space charge neutralization’ or simply ’neutralization’) in the LEBT of an ECRIS
both theoretically and experimentally. The particle distributions obtained from the extraction simulations will then be used as initial conditions for the simulation of the beam
transport in the LEBT with neutralization determined by the results of the space charge investigation. Beam diagnostic devices like emittance scanners and beam viewers will be used
in experiments to compare the actual measured beam properties of the ECR ion sources
VENUS [3, 8, 37] and SuSI [38–40] to the simulations to investigate the applicability of the
two models for ECR extraction and space charge compensation for typical ECRIS ion beams.
At the end of the thesis, I will present a practical example of how the developed software
and the information from the LEBT studies can be put to use: The high energy electrostatic
accelerator of the DIANA project, which has recently completed the preliminary design
report (PDR) phase.

4

1.2

Structure Of The Thesis

A
Following L TEX convention, the thesis is structured in Chapters, Sections, and Subsections

and references to these will be labeled thus.
In accordance with the general introduction presented in the preceding section, this thesis
will focus on three main topics related to the understanding of the physics and engineering
challenges involved with ECRIS beam extraction and the subsequent LEBT:
• Ion extraction from an ECRIS and beam transport: Theory and simulations compared
to measurements.
• Space charge and space charge compensation: Theory and measurements.
• The DIANA project: A practical example of a small ECRIS used as the injector source
of an electrostatic accelerator.
These three topics are highly convoluted, because space charge has to be included in any
realistic simulation eﬀort of beams in the mA current regime. How strong its inﬂuence is
depends on beam current and space charge compensation, thus space charge compensation
becomes an important parameter in the simulations. At the same time, simulations have been
performed to conﬁrm the beam parameters (e.g. beam cross section) during the space-charge
compensation measurements. Consequently, no individual treatment is possible. Instead a
structure has been chosen in which the more general topics are treated ﬁrst (chapters 2Theory, 3-Software, 4-Hardware), followed by detailed chapters on the three topics described
above (chapters 5-Space-Charge Compensation Measurements, 6-3D ECR Extraction Model,
and 7-An Example Application: DIANA) with frequently given references between chapters.

5

1.3
1.3.1

Conventions Used In This Work
Units

Mostly, I will use the International System of units (SI - Le Syst`me international d’unit´s)
e
e
in this thesis. The most noticeable exception being the pressure which will generally be
given in Torr (1 Torr = 133.322 Pa). Also, sometimes it is more intuitive to use the metric
sub-unit of a base unit (e.g. mm instead of m for a beam diameter), which I will do freely.
Values cited from other works might not be in SI units, either. In all cases, it will be clearly
labeled which unit is used. If no units are given, SI units should be assumed (this is mostly
the case for presented formulae).

1.3.2

Emittance

Throughout the thesis, I will mostly use the normalized rms emittance n−rms and make
clear whenever I use one of the other deﬁnitions (e.g. for comparison with literature, other
experiments, etc.). See Section 2.1.3 for more information on emittance.

1.3.3

Brightness

In the rare cases I will use brightness (a ﬁgure of merit for the beam quality that includes
the total beam current) to characterize the beam, it will be a fractional brightness including
the fraction of the beam encompassed (e.g. B90% - including 90% of the beam current) See
Section 2.1.3 for more information on brightness.

6

Chapter 2
Theory

2.1

Charged Particle Beams

In this section I will introduce the basic principles of the physics of charged particle beams.
However, the fundamentals will only be reviewed brieﬂy and only where it is necessary
and helpful in understanding the more speciﬁc concepts related to ECR ion sources and
low energy, heavy ion beams. Very good treatments of ion beam physics and accelerator
physics can be found in textbooks on the topics [41–45] and the descriptions in the following
subsections (2.1.1 - 2.1.4) follow the discussion presented in those references.

2.1.1

Equations of motion

The motion of an individual charged particle (charge q · e, mass m) due to the Lorentz force
is governed by Newton’s equation of motion:

dP
= qe(E + v × B)
dt

(2.1)

with P the mechanical momentum, E and B the electric and magnetic ﬁelds, and v the
particle’s velocity vector. With
P = γmv

7

we can write the three components of equation 2.1 in Cartesian coordinates:

d
(γmx) = γmx + γm¨ = qe(Ex + yBz − zBy )
˙
˙ ˙
x
˙
˙
dt
d
(γmy) = γmy + γm¨ = qe(Ey + zBx − xBz )
˙
˙ ˙
y
˙
˙
dt
d
(γmz) = γmz + γm¨ = qe(Ez + xBy − yBx )
˙
˙ ˙
z
˙
˙
dt

(2.2a)
(2.2b)
(2.2c)

and in cylindrical coordinates r = (r, θ, z), v = (r, rθ, z):
˙ ˙ ˙

d
˙
˙
(γmr) − γmrθ2 = qe(Er + rθBz − zBθ )
˙
˙
dt
1d
˙
(γmr2 θ) = qe(Eθ + zBr − rBz )
˙
˙
r dt
d
˙
(γmz) = qe(Ez + rBθ − rθBr )
˙
˙
dt

(2.3a)
(2.3b)
(2.3c)

where the dot is equivalent to d/dt.

2.1.2

Particle Distributions

Let us consider the beam as an ensemble of particles, moving in largely the same direction.
Each individual particle can be described by the three spatial coordinates (x, y, z) and the
mechanical momentum coordinates (px , py , pz ). The beam can then be characterized by the
six-dimensional distribution function

f (x, y, z, px , py , pz , t)

(2.4)

which deﬁnes the beam in six-dimensional phase space. If z is the direction of propagation
of the beam, we can assume pz

px , py . This is called paraxial approximation. Simply

8

put, it means the velocity component in direction of propagation (z) is much larger than
the transversal velocity component (tightly connected to the transversal temperature of the
beam), which is usually the case. We can use z instead of t as the independent variable and
approximate the angles (x , y ) of the particle trajectories with the beam axis (sometimes
referred to as design trajectory) as: x = dx/dz ≈ px /pz = vx /vz and y ≈ vy /vz (note
that for non-relativistic beams px,y = mvx,y ). Thus we can express the four-dimensional
distribution function for the transversal phase space as

f (x, y, x , y )

2.1.2.1

(2.5)

Liouville’s theorem

Liouville’s theorem states that the volume of a given number of particles in phase space
is constant. If n(x, y, z, px , py , pz , t) is the six dimensional phase space density, this can be
expressed as
dn
=0
dt

(2.6)

or alternatively:
d3 qd3 p = const.

(2.7)

if only conservative forces are present. This is going to be important shortly, when emittance
is introduced. If there is no coupling between longitudinal and transversal motion, Liouville’s
theorem also holds for the four-dimensional transversal subspace and without coupling between the two transversal planes, it even holds for each of the two-dimensional horizontal

9

and vertical subspaces separately:

dxdydx dy = const.,

2.1.2.2

dxdx = const.,

dydy = const.

Transversal distribution

Restricting ourselves to paraxial, DC beams, we are mainly interested in the transversal
phase-space distribution f (x, y, x , y ). In [41] Reiser presents a self-consistent theory for
beams with ﬁnite velocity spread based on the Vlasov model [46] for systems for which the
Liouville theorem holds and collisions between particles can be neglected. The simplest and
probably best known solution to the self-consistent model is the Kapchinsky-Vladimirsky
(K-V) distribution [47]:

f (x, y, x , y ) = f0 · δ 

x2 x 2 +
b

2
x 2 x2 − 2 xx + x b x2
x
b b
2
x

+

2
yb y 2 +

2
2
y 2 yb − 2 yy + y b y 2
y
b
2
y


− 1 (2.8)

with xb , yb the maximum beam extent (b for ’beam’) in x and y directions, xb , yb the maximum angles, and x , y the (full) beam emittances, which will be discussed shortly. This
distribution uniformly ﬁlls the surface of the four-dimensional hyper-ellipsoid in the transversal phase-space (note the delta function). It has the interesting property that each projection
into one of the two-dimensional subspaces (x-y, x-x’, x-y’, y-y’, y-x’ x’-y’) is a uniformly ﬁlled
ellipse.

10

Figure 2.1: Beam cross-section of a K-V distribution. For interpretation of the references
to color in this and all other ﬁgures, the reader is referred to the electronic version of this
dissertation.

Figure 2.2: Two-dimensional transversal subspace (x-x’) of the phase space of a K-V distribution.

11

In Figures 2.1 and 2.2 the cross-section and x-x’ phase-space for the K-V distribution are
shown (arbitrary values for beam radius xb = yb , maximum divergence, and emittance were
chosen). Note that in x-y space the beam extents in x and y direction do not necessarily have
to be the same. An elliptical beam cross-section is the general form of the K-V distribution
and the circular beam was chosen in 2.1 for convenience. The other important property of
the K-V distribution is that it is the only solution to the self-consistent model that produces
linear self-ﬁelds (see Subsection 2.1.4). Of course, other solutions to the Vlasov theory exist
that are more realistic (e.g. waterbag distribution, Gaussian distribution) and if collisions
between beam ions are included, a Maxwell-Boltzmann distribution turns out to best represent realistic beams. But, the K-V distribution can be used to examine basic properties
of beams in transport systems with only horizontal and vertical focusing (electrostatic and
magnetic quadrupoles, dipoles) or axially symmetric systems.

2.1.3

Emittance

Emittance and brightness of ion beams are values used for quantiﬁcation of the beam quality.
Intuitively, emittance can be understood as the product of beam size and beam divergence,
thereby giving us an idea of the transportability of the beam. Brightness relates the beam
intensity to the emittance. The normalized emittance is particularly helpful when comparing
diﬀerent ion sources and accelerators that operate at diﬀerent beam energies. All of these
will now be discussed in more detail. Generally, the emittance is deﬁned as the phase space
area (A) of the two-dimensional projections (x-x’, y-y’, x-y’, y-x’) divided by π, e.g.:

x

=

Axx
π

[π-mm-mrad]

12

(2.9)

Figure 2.3: Phase space evolution of a K-V beam without space charge in drift space (no applied ﬁelds) for three diﬀerent
instances along the beam path (z). Note that the area of all three ellipses is constant, as is the maximum divergence angle xb .
x0 = xb (z = z0 ).

13

This emittance is given in units of π-m-rad or π-mm-mrad. The π in the units is not to be
taken as a factor π but as a reminder which deﬁnition of emittance is being used (there are
others that do not divide by π). This is also the emittance used in the deﬁnition of the K-V
distribution 2.8. Using Liouville’s theorem, we immediately see that if only conservative
forces are present the emittances x and y are conserved. If we allow coupling of the two
transverse phase spaces, only the product of the two ( x · y ) is conserved. The evolution of
the x-x’ phase space of a beam for three diﬀerent positions along the z-axis (three moments
in time) is shown in Figure 2.3. There are no applied ﬁelds and no self-ﬁelds and the area
of the ellipse (and thus the emittance) as well as the divergence are constant. The initial
upright ellipse represents a waist in the beam, i.e. the outermost angles (±xb ) - not to
be confused with the maximum angles xb - are zero. Unfortunately, real beams rarely have
perfect elliptical phase spaces. Distortions can be introduced into the beam by the ion source
itself, or by fringe ﬁelds and higher order aberrations in the focusing elements. Figure 2.4
shows phase space distortions of an originally K-V beam, due to spherical aberrations in a
periodic focusing channel. In accordance with Liouville’s theorem, the actual area covered
by the beam is still constant, but due to the ﬁlamentation, if we were to draw an ellipse
around the beam that contained all the particles (100% of the beam current) the emittance
would be artiﬁcially large. It is useful to introduce a new concept of emittance, that better
reﬂects the physical situation of distorted beams: the rms emittance.

2.1.3.1

RMS Emittance

If x2 is the second moment of our distribution f (x, y, x , y ); deﬁned as:

x2 =

x2 f (x, y, x , y )dxdydx dy
f (x, y, x , y )dxdydx dy
14

(2.10)

Figure 2.4: Phase space distortion due to spherical aberrations in a periodic thin lens focusing
channel. Each period consists of a drift, a thin lens and another drift.
and analogous for x 2 and xx , then the rms emittance can be deﬁned as:

rms

=

x2 x 2 − xx 2

15

[mm-mrad]

(2.11)

Because the rms emittance is a purely statistical value, we will use units of m-rad or mmmrad without the π for rms . It can be shown, that in the case of the K-V distribution
x-f ull

= 4 · x-rms , exactly. For other distributions, like the waterbag distributions and

the Gaussian distribution this factor varies from 6 to 16. It should also be noted that
combinations of distributions are possible, like Gaussian in transversal direction and uniform
in longitudinal direction. In practice, the beam density gets very low towards the outer
regions of the Gaussian distribution and it does not make much sense to calculate the ’full’
emittance. The same holds for otherwise distorted beams, like the ECRIS beams (as will be
seen in Chapter 6). Many people including Reiser [41] have adopted a concept of eﬀective
emittance which is just 4 · x-rms . On the other hand, what we are really interested in, is not
only the emittance, but to know how much beam current we can transport with a certain
emittance. This can be expressed as a brightness.

2.1.3.2

Brightness

The brightness is commonly deﬁned by

B=

J
dI
=
dΩ
dSdΩ

(2.12)

as current density per unit solid angle. Or in case of the K-V distribution (and waterbag
distribution as well)
2I
¯
B= 2
π x y

A
m2 -rad2

(2.13)

the emittances here are the full K-V emittances. As Reiser points out, sometimes the factor
2/π 2 is left out or the rms emittance is used. It is thus very important to be clear and
consistent in the deﬁnition and usage of emittance and brightness.
16

2.1.3.3

Fractional Emittance

From an experimentalist point of view, the practical thing to do is often to deﬁne an elliptical
emittance contour, that envelopes a certain fraction of the beam current, e.g. 90%, and
calculate the full emittance of the such deﬁned sub-ensemble of particles (e.g. x-90% ). This
contour should encompass ’most’ of the beam. The brightness can then be deﬁned as

2 · 0.9 · I
B90% = 2
π x-90% y-90%

(2.14)

If the beam has a very non-uniform distribution, the so obtained brightness will have a
maximum at some percent beam current value. Above the maximum, the emittances increase
faster than the current, thereby lowering the brightness. Below the maximum, the emittance
decreases slower than the current , thereby lowering the brightness. This can be used as a
ﬁgure of merit for the quality of the beam after ’cleaning it up’ (e.g. collimators).

2.1.3.4

Normalized Emittance

In order for the phase space area (volume) to be invariant also under acceleration, the
normalized emittance is deﬁned as:

n

=β·γ·

(2.15)

with β = v/c and γ = 1/ 1 − β 2 the relativistic factors. This deﬁnition is valid for rms
emittance as well as eﬀective or full emittance and lets us compare the quality of beams with
diﬀerent energies.

17

2.1.3.5

Normalized Brightness

Similarly we deﬁne the normalized brightness:

B
Bn = 2 2
β γ

(2.16)

The conventions for emittance and brightness used throughout this work are listed in the
introduction in Section 1.3.

18

2.1.4

Self Fields

In the discussion of self-ﬁelds, it is useful to start out with a simple beam. Hence we will
start with an axially symmetric K-V beam in cylindrical coordinates. This means that the
paraxial condition applies (vr

vb , vθ

vb , vz ≈ vb with vb the overall beam velocity).

At each longitudinal position z, the beam has the charge density ρ(r, θ, z) and velocity vz .
Associated with a distribution of charged particles is an electric ﬁeld, the self electric ﬁeld.
On the other hand, because the charged particles are in motion, a current can be associated
with the beam, which gives rise to a self magnetic ﬁeld. Because the ions are of same
sign charge, the self electric ﬁeld is defocusing and a coasting, perfectly parallel (i.e. zero
emittance) ion beam will start diverging given suﬃcient time. The self magnetic ﬁeld on the
other hand is focusing. It is useful to start the examination of the self ﬁelds’ eﬀects on the
beam with a cylindrical beam. Following Reiser [41, pp. 170], we use the simpliﬁcations:
1. The beam cross-section is circular with a sharp edge at radius rb inside a concentric
beam pipe with radius rp .
2. The beam radius changes slowly with z, so the paraxial assumption holds (vx
vy

vb ,

vb , vz ≈ vb ) and the radial magnetic ﬁeld component Br and axial electric ﬁeld

component Ez can be neglected.
3. The potential diﬀerence between beam center and drift tube wall is small compared to
the voltage equivalent of the particle kinetic energy (several 10 kV).
4. The ion beam density ρ is uniform inside the beam and zero outside. The velocity of all
particles is approximately the same (vb ) and therefore the current density is uniform
inside the beam and zero outside.

19

We can then write the charge density as

I
ρ= 2
rb πvb
with I the beam current, and get the electric ﬁeld (which has only a radial component) by
application of Gauss’s law:

Er =









I·r
2π 0 r2 vb
b
I
2π 0 rvb

for r ≤ rb
(2.17)
for r > rb .

By integrating, we obtain the electrostatic potential (assuming a grounded beam pipe,
φ(rp ) = 0):


 ∆φ 1 + 2 ln rp − r2

rb
r2
b
φ(r) =

rp

 ∆φ 2 ln
r

for r ≤ rb
(2.18)
for rb ≤ r ≤ rp

with
∆φ =

I
4π 0 vb

.

(2.19)

The magnetic ﬁeld (which has only an azimuthal component) is obtained by using Amp´re’s
e
law:



 µ I·r
 0 2
2πr
b
Bθ =


 µ0 I
2πr

for r ≤ rb
(2.20)
for r > rb .

As an example, the electrostatic ﬁeld and potential for a 0.5 mA oxygen beam of charge state
q = 5+ is shown in Figure 2.5. The self magnetic ﬁeld of such a low current, non-relativistic
beam is negligible.

20

Figure 2.5: Radial beam potential and electrostatic self ﬁeld of a round 0.5 mA 16 O6+ beam.
Beam radius 20 mm, grounded pipe radius 10 cm.
Using the electric and magnetic self ﬁelds in the radial force equation 2.3a, Reiser derives
the particle trajectory equation [41, p. 175] for a uniform, round beam in drift space:
d2 r
K
= 2r
dz 2
rb

(2.21)

2qeI
4π 0 mb β 3 γ 3 c3

(2.22)

with K the generalized perveance

K=

21

Figure 2.6: Solutions of the envelope equation for a 0.5 mA 16 O6+ beam in drift space for
diﬀerent initial angles a0 = rb (z = 0), initial radius rb (z = 0) = 10 mm, and zero emittance.
Note that in the K-V model the self-ﬁelds are linear in r and the emittance is constant.
which is connected to the generalized plasma frequency

2
ωp =

qeI
2
π 0 mb cβγ 3 rb

(2.23)

by
K=

2
ωp

2
rb
· 2 2.
2β c

(2.24)

The generalized perveance is a measure of the importance of space charge for the beam
transport. Without space charge compensation, it is always positive, as can easily be seen
22

Figure 2.7: Solutions of the envelope equation for a 0.5 mA 16 O6+ beam in drift space
for diﬀerent emittances, initial radius rb (z = 0) = 10 mm and initial angle -0.1 rad. The
emittance shown here is the normalized full emittance of the beam ( n = 4 n−rms ).
from equation 2.22 and thus the net eﬀect of defocusing electric ﬁeld and focusing magnetic
ﬁeld is a defocusing one. In the next subsection, we will see, that having stationary secondary
particles of the opposite charge inside the beam envelope can lead to a net focusing eﬀect.
In order to solve equation 2.21, we need an expression for the beam radius. This is given by
the beam envelope equation for a beam in drift space [41, p. 181]:
2
d2 r0
K
= 3+
2
rb
dz
rb

23

(2.25)

with

the beam emittance and rb (z) the envelope radius varying slowly in z. Depending on

the magnitudes of and K, we speak of emittance dominated versus space-charge dominated
beams. Equations 2.21 and 2.25 represent the equation of motion and envelope equation for
the axially symmetric K-V model a drift and examples of the beam envelopes for diﬀerent
initial angles and diﬀerent emittance values are shown in Figures 2.6 and 2.7, respectively.
For completeness, the equations of motion for the more general K-V model of separated
horizontal and vertical motion in linear external ﬁelds are:

x + κx (z)x −

2K
x=0
xb (xb + yb )

(2.26a)

y + κy (z)y −

2K
y=0
yb (xb + yb )

(2.26b)

with κx (z) and κy (z) the focusing functions representing the external ﬁelds, and xb and
yb the semi-axes of the beam cross-section ellipse. κx (z) and κy (z) are deﬁned solely by
the lattice of the accelerator (the magnets and electrostatic elements used for focusing and
bending). xb and yb have to be determined from the envelope equations:

xb + κx (z)xb −

2
2K
− x =0
(xb + yb ) x3
b

(2.27a)

yb + κy (z)yb −

2
2K
y
− 3 =0
(xb + yb ) yb

(2.27b)

This set of 2 × 2 coupled diﬀerential equations is usually solved numerically and is the basis
of many ion beam simulation codes.
We obviously over-simpliﬁed the beam in order to obtain these equations. For ECR ion
sources, the beam is not round, the ion distribution is not uniform and the beam is also not
concentric with the beam pipe (i.e. image ﬁelds might occur). One consequence of the non24

uniformity is that space-charge forces are no longer linear and thus Liouville’s theorem does
not hold anymore. Phase space volume and emittance are no longer preserved. Ultimately,
the problem of beam transport for many ECRIS beams gets so complex that the only way of
obtaining realistic results is by using simulation codes that can treat the three dimensional
problem of propagation of particle ensembles with applied and self ﬁelds. One of these codes
is WARP (a particle-in-cell code) and will be discussed in detail in Section 3.1. We have
seen, however, that already with a very simple beam concept, we can pinpoint some of the
problems that arise for ions beams with non-negligible self ﬁelds and we will continue the
analysis in the next subsection by including space charge compensation due to secondary
electrons created by the interaction of the beam with the residual gas in the beam line.

25

Figure 2.8: Space charge compensation. Beam Cross section view with ions and electrons
created from residual gas ionization and charge exchange.

2.1.5

Space Charge Compensation

The pressure in LEBT systems of ECR ion sources is typically between 10−5 to 10−8 Torr.
At these pressures, the interaction of the beam ions (primary ions) and the residual gas
molecules is not negligible. The positive beam ions interact with the gas molecules mainly
through two processes (assuming only one ion species in the beam) [44]:

Aq+ + X0 → A(q−1)+ + X+
Aq+ + X0 → Aq+ + X+ + e−

(charge exchange)
(ionization)

where A is the primary beam ion and X the gas molecule. The underline denotes the fast
particle. Both processes create slow secondary ions, which are expelled by the beam space
charge potential. Ionization also provides slow secondary electrons, which are captured inside

26

Figure 2.9: Space charge compensation. Change in beam potential from uncompensated
beam to 50% compensation. This is for a 2 mA oxygen primary beam with round cross
section (rb = 20 mm) and homogeneous charge distribution. The beam pipe (rp = 100 mm)
is assumed to be grounded.
the beam space charge potential well and eﬀectively lower said potential well. The process
is illustrated in Figures 2.8 and 2.9. In our simple model of a uniformly charged cylindrical
beam, the neutralization can be expressed in form of a neutralization-factor fe modifying
the total beam current in equation 2.19:

∆φ =

I · (1 − fe )
4π 0 vb

27

(2.28)

In a similar way, it can be incorporated into the linear beam model presented by M. Reiser [41]
by modifying the expression for the generalized perveance:

K=

2qeI
1 − γ 2 fe (τ )
4π 0 mb β 3 γ 3 c3

(2.29)

where β and γ are the relativistic factors, and τ is the time into the beam pulse. Since in ECR
LEBTs we are only concerned with non-relativistic beams, we can use γ = 1 and assume
that fe is a constant in time (steady-state of a DC beam). As an interesting aside, it should
be noted that in the case of fe > 1/γ 2 , K is negative and the beam is self-focusing due to the
self magnetic ﬁeld. This is called Budker condition of self-focusing. In the non-relativistic
case, the expression for K reduces to:

K=

2qeI
(1 − fe )
4π 0 mb β 3 c3

(2.30)

We will now see if we can ﬁnd an analytical expression for the magnitude of fe depending
on beam current, size and neutral gas density. The following derivation follows largely the
considerations in the papers written by I. A. Soloshenko et al. [48–50], and M. D. Gabovich
[51, 52]. I made an eﬀort, though, to write everything in SI units and not leave out any
essential steps in the derivation.
Let us consider the beam to be a plasma consisting of these three species:
• Primary beam ions (density nb , velocity vb , mass mb , charge state q, charge Qb =
q · e)
• Secondary ions (density ni , velocity vi , mass mi , charge Qi = e)
• Secondary electrons (density ne , velocity ve , mass me , charge Qe = −e)
28

Like in the previous chapter, we start with a uniform cylindrical charge distribution for the
primary beam ions. The trapped electrons are also assumed to uniformly ﬁll the beam envelope. As mentioned earlier, the secondary ions are expelled by the beam, but, of course,
spend some time inside the beam envelope. The only interaction between the beam and the
secondary electrons are Coulomb collisions (no collective processes are taken into consideration). The energy necessary for the secondary electrons to leave the potential created by
the beam is then given by:

dE
=L
dt out

rb

eϕ(r)

f (E) (eϕ(r) − E) dE

2πr dr
0

(2.31)

0

with L the length of the beam, rb the radius, ϕ(r) the potential at radius r, and f (E)
the secondary electron energy distribution. On the other hand, a fast beam (vb > ve ) will
transfer energy mostly through Coulomb collisions. A good treatment of Coulomb collisions
can be found here [53, 54]. In simple classical Coulomb collisions, the rate of transfer of
2
energy per unit length for one beam particle of energy Ekin = mb vb /2 to stationary target

particles can be written as

dE
= nt
d

Qb Qt 2 4π
bmax
· ln
2
4π 0
bmin
mt vb

(2.32)

which, for electrons as target species becomes
dE
q 2 e4 1
bmax
= ne
· ln
2
d
bmin
4π 2 me vb
0
where bmin and bmax are appropriate lower and upper limits of the impact parameter, and

29

q the charge state of the primary ion beam. We can now write the energy transfer to the
2
electrons per unit time for the sum of all beam particles Nb = nb rb πL with velocity vb as
2
nb ne q 2 e4 rb L
bmax
dE
· ln
=
2v
dt in
bmin
4me 0 b

(2.33)

Since the main contribution to the process comes from small angle scattering, a good choice
for bmin is b90 , the impact parameter, where the incident particle is scattered by 90◦ . b90 for
ion-electron collisions is given by

b90 =

−qe2 1
2
4π 0 mr vb

(2.34)

with mr the reduced mass, which for ion-electron collisions is mr ≈ me . The secondary
electrons are not bound to nuclei and essentially form a low density electron plasma. A good
value for bmax is then
bmax =

v
vb
= b·
ωpe
e

0 me

ne

(2.35)

with ωpe the electron plasma frequency. We expect the eﬀectiveness of the collisions to fall
oﬀ because of dielectric eﬀects for b > bmax = vb /ωpe . Substituting expressions 2.34 and
2.35 for bmin and bmax , respectively, we obtain

˜
L = ln 4π



3/2 3
3/2 me vb 
0
1/2
qe3 ne

(2.36)

˜
where we call L a Coulomb logarithm. Hence
2 ˜
nb ne q 2 e4 rb LL
dE
=
dt in
4me 2 vb
0

30

(2.37)

In steady-state, expressions 2.31 and 2.37 have to be equal in order to conserve energy:
rb

L

eϕ(r)

f (E) (eϕ(r) − E) dE =

2πr dr
0

or:

2 ˜
nb ne q 2 e4 rb LL

4me 2 vb
0

0

rb

eϕ(r)

f (E) (eϕ(r) − E) dE =

r dr
0

0

2˜
nb ne q 2 e4 rb L

8πme 2 vb
0

(2.38)

Let us now solve the integrals on the left hand side. At this point, Soloshenko et al. [49] and
Gabovich [52] make an approximation for f (E):

f (E) ∝

1
(eΦi + E)2

(2.39)

with eΦi the ionization energy of the gas molecules. They determine the proportionality
constant by requiring:

∞

f (E)dE =

∂ne
= nb vb n0 σe
∂t

0

with n0 the neutral gas density, and σe the electron creation cross -section. This expression
can be readily solved to yield
n v n σe · eΦi
f (E) = b b 0
(eΦi + E)2

31

(2.40)

And the energy integral in equation 2.38 can be solved to:
eϕ(r)

eϕ(r)

eϕ(r) − E
dE
(eΦi + E)2

f (E) (eϕ(r) − E) dE = η ·
0

0

ϕ(r)2 − Φ2
i + 1 − ln 1 + ϕ(r)
Φi (Φi + ϕ(r))
Φi

= η·

ϕ(r) − Φi
ϕ(r)
+ 1 − ln 1 +
Φi
Φi
ϕ(r)
ϕ(r)
− ln 1 +
= η·
Φi
Φi
= η·

(2.41)

where η was introduced for convenience as

η = nb vb n0 σe · eΦi

η is considered constant in r and E in our simpliﬁed view of the problem. The potential ϕ(r)
inside a homogeneously charged cylinder inside a beam pipe of radius R is given by 2.18:

ϕ(r) =

Ib
·
4π 0 vb

1 + 2 ln

rp r2
− 2
rb rb

with Ib the beam current and 0 the vacuum permittivity. Computing ∆ϕ = ϕ(rb ) − ϕ(0)
yields
∆ϕ =

Ib
4π 0 vb

Since, for our considerations, we are only interested in ∆ϕ, which is independent of the beam
pipe radius, we can set rp = rb , and

ϕ(r) = ∆ϕ ·

32

r2
1− 2
rb

.

(2.42)

Now we can carry out the radial integral in equation 2.38:
rb

r·
0

ϕ(r)
ϕ(r)
− ln 1 +
Φi
Φi
rb

r·

=
0

r2
1− 2
rb

2
2
rb
rb
−
2
4

∆ϕ
=
Φi
=

∆ϕ
Φi

dr =

− ln 1 +

∆ϕ
Φi

r2
1− 2
rb

r · ln 1 +

∆ϕ
Φi

r2
1− 2
rb

∆ϕ
∆ϕ
+ 1+
Φi
Φi

ln 1 +

rb

−
0

2
∆ϕ rb

−

2
Φi rb

−

Φi 4
∆ϕ 2
2
r ∆ϕ
Φ
+1− 1+ i
= b
2 2Φi
∆ϕ

ln 1 +

∆ϕ
Φi

dr

dr
∆ϕ
Φi
(2.43)

Expanding the logarithm for ∆ϕ/Φi < 1 yields
∆ϕ
ln 1 +
Φi

∆ϕ 1
−
=
Φi
2

∆ϕ 2 1
+
Φi
3

∆ϕ 3
+ O(4).
Φi

Substituting into the solution of equation 2.43, carrying out the multiplication and dropping
the terms of order 3 and higher gives
2
rb ∆ϕ
Φ
+1− 1+ i
2 2Φi
∆ϕ

·

∆ϕ 1
−
Φi
2

∆ϕ 2 1
+
Φi
3

33

∆ϕ 3
+ O(4)
Φi

2
rb (∆ϕ)2
≈
.
12 Φ2
i

And thus
rb

eϕ(r)

r dr
0

0

2
rb (∆ϕ)2
f (E) (eϕ(r) − E) dE = η ·
12 Φ2
i
2
rb (∆ϕ)2
= nb vb n0 σe eΦi ·
12 Φ2
i

r2 (∆ϕ)2
= nb vb n0 σe e · b
12 Φi

(2.44)

Then, 2.38 becomes
(∆ϕ)2 =

˜
3q 2 Φi ne e3 L
2
2πme vb n0 σe 2
0

(2.45)

The only unknown quantity now is the electron density. Here, Gabovich and Soloshenko use
quasi-neutrality:
ne = q · nb + ni

(2.46)

and the secondary ion balance equation (number of secondary ions created in the beam is
equal to the number of ions leaving the beam per unit time):

2
2rb πni vi = rb πnb vb n0 σi
¯

(2.47)

with vi the average secondary ion velocity. Thus the electron density is
¯
nvn σr
ne = q · nb + b b 0 i b
2¯i
v

34

(2.48)

We can now rewrite 2.45:
˜
nvn σr
3q 2 Φi e3 L
· q · nb + b b 0 i b
2n σ 2
2¯i
v
2πme vb 0 e 0
˜
q
3q 2 Φi nb e3 L
vσr
·
=
+ b i b .
2
n0 σ e
2¯i σe
v
2πme vb 2
0

(∆ϕ)2 =

(2.49)

Finally, we use the non-relativistic kinetic energy of the primary beam

qeU0 =

2
mb vb
,
2

2
with U0 the source voltage, to replace vb and obtain the formula presented in [49] (but in SI

units and for an arbitrary charge-state q of the primary ion beam):

(∆ϕneut )2 = 3L ·

mb Φi nb qe2
·
me U0 (4π 0 )2

q
vσr
+ b i b
n0 σ e
2¯i σe
v

.

(2.50)

˜
Note that we have absorbed a factor 4π into the deﬁnition of L (L = 4π L) to be consistent
with [49]. We also use equation 2.48 to replace the electron density in the Coulomb logarithm
(equation 2.36). We now have an explicit expression for ∆ϕ in terms of quantities either
experimentally accessible or calculable by theoretical models. The neutralization factor fe
can now be obtained by substituting the (partially) neutralized ∆ϕneut (equation 2.50) into
equation 2.28:
fe = 1 −

∆ϕneut
∆ϕf ull

where (with βc = vb the beam velocity):

∆ϕf ull =

35

I
4π 0 vb

(2.51)

2.1.5.1

Simple Generalization of the Result

One of the drawbacks of Gabovich’s result is the assumption of quasi-neutrality which naturally limits the usefulness of the formula to highly neutralized beams. If we modify equation
2.46 to reﬂect the fact that the electron density is only a fraction of the combined primary
and secondary ion densities (corresponding to the level of neutralization) we can write it as:

ne = f e · (q · nb + ni )
√

This only adds a factor

(2.52)

fe to the deﬁnition of ∆ϕneut and the new equation for calculation

of the compensation factor is:

fe = 1 −

1
∆ϕf ull

fe

3L ·

mb Φi nb qe2
·
me U0 (4π 0 )2

q
vσr
+ b i b
n0 σe
2¯i σe
v

(2.53)

Using the abbreviation

χ=

1
·
∆ϕf ull

3L ·

mb Φi nb qe2
·
me U0 (4π 0 )2

vσr
q
+ b i b
n0 σe
2¯i σe
v

(2.54)

we can easily solve for fe :
fe = 1 +

χ2 χ
−
2
2

χ2 + 4

(2.55)

Note: Technically, we cannot simply replace the electron density inside the Coulomb logarithm anymore, but a quick error propagation study shows that a relative error ∆(ne )rel
contributes little to ∆(∆ϕneut )rel :

∆(∆ϕneut )rel =

36

∆(ne )rel
˜
4·L

(2.56)

˜
which for typical values of 5-15 for L is a factor 20 - 60 smaller than ∆(ne )rel and thus
well below other sources of error due to our approximations. Still, a sort of crude relaxation
˜
process can be used, where fe is calculated using the full electron density inside L in a ﬁrst
˜
iteration. This preliminary fe can then be used inside L in the second iteration, making sure
that the resulting error really is negligible.

2.1.5.2

Multiple Ion Species

In the region between the ECR source and the ﬁrst dipole magnet, the beam is composed
of several ion species with diﬀerent mass-to-charge ratios m/q. Naturally, we are interested
to expand the theoretical model to include multiple ion species, to predict neutralization
in those parts of the beam line. Because of the linearity of the K-V model, we can easily
generalize equation 2.50 for multiple ion species in the beam. If we assume all species have
the same beam radius, this is achieved by summation over all k species in the beam:
k
j=1

k

dE
=
dt j, in
j=1

dE
.
dt j, out

(2.57)

With the full potential drop:
k

∆ϕf ull =
j=1

Ib,j
,
4π 0 vb,j

(2.58)

the individual Coulomb logarithms:


3/2

3
3/2 me vb,j

Lj = 4π ln 4π 0

37

1/2
q j e 3 ne


,

(2.59)

the individual ion beam densities

nb,j =

Ib,j

(2.60)

2
eqj rb πvb,j

and the sum expression for the electron density:
k

ne = fe ·

qj · nb,j +
j=1

nb,j vb,j n0 σi,j rb
2¯i
v

,

(2.61)

we obtain:

∆ϕ2
neut

3 Φi e 2
·
= fe ·
me U0 (4π 0 )2

k
j=1

qj nb,j rb nb,j vb,j σi,j
+
·
n0
2¯i
v

≡ f e · η2.

k
j=1 Ib,j mb,j Lj
Ib,j σe,j
k
j=1
qj

(2.62)

And with
η
∆ϕf ull

(2.63)

fe · χmulti

(2.64)

χmulti =
we can calculate fe from
fe = 1 −
to be (analogous to equation 2.55):

χ2
χ
fe = 1 + multi − multi
2
2

38

χ2
multi + 4.

(2.65)

2.1.5.3

Obtaining the Cross-Sections

Clearly, one of the sources of uncertainty in this model are the cross-sections σe and σi
for secondary electron and ion production. Remembering the two processes involved with
beam-residual gas-interaction mentioned at the beginning of this subsection, we can rewrite
σe and σi in terms of the actual reactions:

σe = σionization
σi = σcharge−exchange + σionization

In the past century, many measurements have been conducted and scaling models and semiempirical formulae have been developed to address the question of obtaining the ionization and charge-exchange cross-sections. For charge-exchange processes, A. M¨ller and E.
u
Salzborn presented an overview of previous measurements and scaling laws in 1977 [1]. Based
on the data they had collected, they proposed the following scaling law for the chargeexchange cross-section (in cm2 ):

β

σq,q−k = Ak · q αk · Φi k

k = 1, · · · , 4

(2.66)

with q the initial charge-state, k the number of exchanged electrons, Φi the ionization energy
in eV, and Ak , αk βk ﬁt parameters with values listed in Table 2.1.
For the ionization cross-section σionization , the situation is a bit more problematic. Data for
proton, helium and lithium beams in the desired range of 10 keV to 100 keV can be found
for gaseous hydrogen, helium and lithium targets [55–57], because the cross-sections are
interesting for fusion research. For other projectiles and targets, several measurements exist

39

at signiﬁcantly higher energies. Based on classical and quantum theories, several models have
been developed and compared to the existing measurements. A good summary of previous
eﬀorts was given by I.D.Kaganovich et al. in several papers between 2003 and 2005 [58–60].
He presented a scaling law that seems to ﬁt the data very well. Unfortunately, most cases we
are interested in for ECR ion source low energy beam transport are in energy regimes below
the applicability of this scaling law. If the velocity of the incident particle becomes too low,
the interaction time of projectile and target electron increases enough for tunneling eﬀects to
become an important factor in the ionization process and the Kaganovich ﬁt underestimates
the cross-section. But at the same time, the charge-exchange cross sections become much
larger than the ionization cross-sections at low velocities. To predict neutralization for
diﬀerent ECR beams, one has to ﬁnd measured ionization cross-sections and (very carefully)
scale known curves to the desired beams. The given references are very helpful in this.
Further information can be found in chapter 5 on space-charge compensation measurements
in ECR beam lines.

Table 2.1: M¨ller-Salzborn ﬁt parameters for charge-exchange cross-sections [1].
u
k

Ak

αk

βk

1

(1.43 ± 0.76) × 10−12

1.17 ± 0.09

−2.76 ± 0.19

2

(1.08 ± 0.95) × 10−12

0.71 ± 0.14

−2.80 ± 0.32

3

(5.50 ± 5.80) × 10−14

2.10 ± 0.24

−2.89 ± 0.39

4

(3.57 ± 8.90) × 10−16

4.20 ± 0.79

−3.03 ± 0.86

40

Figure 2.10: Example of theoretical neutralization prediction for a 75 keV proton beam with
a 14 mm diameter in H2 residual gas. Experimental data shown for comparison from [7].
The data was published without errorbars. Left: pressure variation at a constant beam
current of 100 mA, right: current variation at a constant gas density of 1.2 · 1018 m−3 .
2.1.5.4

An Example

Using cross-section data from [57] for a 75 keV proton beam going through a residual gas of
H2 , we ﬁnd σe ≈ 2.2 · 10−20 m2 (25% accuracy) and σi ≈ 3.2 · 10−20 m2 (30% accuracy).
Figure 2.10 shows the neutralization values calculated with the generalized model (equation
2.55). For comparison, data for the LEDA injector source presented by R. Ferdinand [7] is
shown in the graph. As in the measurements, the residual gas was assumed to be H2 , the
beam current was varied from 50 mA to 130 mA and the pressure from 2 · 10−5 to 3 · 10−4
Torr. The beam diameter was assumed to be 14 mm. The overall agreement is very good
and decreasing neutralization with decreasing beam current as well as decreasing residual
gas pressure can be observed in both experiment and theory.

41

2.1.5.5

Some Comments and Limitations of this Model

• The average secondary ion velocity vi obviously depends on the potential well depth
¯
of the compensated beam. The dependence goes as the square root of ∆ϕneut , though
making it a fourth root contribution to the calculated ∆ϕneut .Thus choosing a ion
temperature close to the expected ∆ϕneut gives good results and even a rather large
variation of it does not change the result signiﬁcantly.
• Gabovich and Soloshenko use a rather simplistic approximation for f(E) following
Thomson’s theory [61].
• Beam is approximated as a homogeneously charged cylinder. This can be a good
approximation for beams like the proton beams from, e.g., the LEDA source [7] (the
experimental data in Figure 2.10). For ECR sources the beam shape may have to be
taken into account. However, in general, the neutralization will exhibit the following
dependencies:
– Residual gas pressure (lower pressure →lower neutralization)
– Beam current (lower beam current →lower neutralization)
– Beam diameter (smaller beam diameter →lower neutralization)
• Charge-exchange and ionization are assumed to produce only ions with q = 0 →
1+. Ionization cross-sections for q = 0 → 2+ are generally an order of magnitude
lower; charge-exchange, however, may produce higher charge-states with non-negligible
probability.
• In the presented model, no collective processes are included (e.g. plasma waves, beam
instabilities, etc.) Further information can be found in [48] and Chapter 5 of this thesis.
42

• For the multiple species approach, in reality the diﬀerent radii will not be the same,
because individual beamlets experience diﬀerent focusing depending on their mass-tocharge ratio (m/q). This happens when they exit the source, as well as in solenoids or
einzel lenses in the LEBT system. Unfortunately, the densities nb , ni , and ne will no
longer be uniform in this case and the forces no longer linear so diﬀerent radii cannot
readily be incorporated into the linear theory.

43

2.2
2.2.1

Electron Cyclotron Resonance Ion Sources
General Overview

Very detailed treatment of ECR ion sources, the plasma physics applicable to those sources
and considerations regarding extraction of the beam can be found in the books by R. Geller
[14], and I.G.Brown [44, 45]. Furthermore, the development of ECR ion sources is a very
active ﬁelds and numerous publications are available. Here, I will review a few basic principles
of ECR ion sources, the related plasma physics and physics of ion beam extraction. I will give
references to related articles where corresponding topics are mentioned. The technological
and engineering aspects of the speciﬁc ECR ion sources used for experiments in this thesis
will be treated in chapter 4.1.

2.2.1.1

ECR Concept

The basic operating principle of the electron cyclotron resonance ion source is that of a
plasma ion source: A plasma consisting of ions and electrons is conﬁned by a magnetic ﬁeld.
Highly charged ions are created by successive (step-by-step) electron impact ionization. In
the ECR source, the electrons circle around the magnetic ﬁeld lines with their electron
cyclotron resonance frequency:
ωecr =

e·B
me

(2.67)

with B the magnetic ﬁeld, me the electron mass, and e the electron charge. Microwaves
of the same frequency (2πfrf = ωrf = ωecr ) are used to resonantly transfer energy to the
electrons (’heat’ them). Thus the magnetic ﬁeld does not only provide the ion conﬁnement,
but also the resonance condition necessary for the heating process. Typical frequencies frf

44

Figure 2.11: The magnet structure of the VENUS ECR ion source. Courtesy of D. Leitner.
go from 2.45 to 28 GHz with corresponding resonance ﬁelds from 0.0875 to 1 T.

2.2.2

Magnetic Field Structure

Typically, ECR ion sources consist of a longitudinal mirror ﬁeld produced by two or more
solenoids and a superimposed sextupole (also called a hexapole) magnet. Such a conﬁguration can be seen in Figure 2.11 for the case of the Lawrence Berkeley National Laboratory
ECR source VENUS. This type of conﬁguration is often called minimum-B, because from
the center of the source, the magnetic ﬁeld strength increases in all directions. The main
purpose of the sextupole ﬁeld is to suppress magneto-hydrodynamic (MHD) plasma instabilities [14]. There are ECR sources, with slightly diﬀerent magnetic ﬁeld conﬁgurations:
Some do use only solenoids (e.g. LEDA source), some use higher order multipoles for the
45

Figure 2.12: Axial Field of the VENUS ECR ion source. It can be seen that the mirror ratios
are around 6 at injection (left peak ) and 4 at extraction (right peak). Also indicated are
the B values that correspond to the two microwave frequencies 18 GHz and 28 GHz used in
VENUS. The two corresponding ecr surfaces then form nested shells. Image taken from [8].
radial conﬁnement (octupoles, dodecapoles). Solenoid-only operation is only used when high
currents of singly charged ions are desired. For a given rf-frequency, the area of magnetic ﬂux
B, where the ecr-condition (equation 2.67) is met forms a closed shell inside the ion source
chamber, the ecr surface. The longitudinal ﬁeld Bz (z) on axis of the VENUS source, as well
as the points where the ecr condition is met are shown in Figure 2.12. It was experimentally
found, that it can be beneﬁcial to use more than one frequency (18 GHz and 28 GHz in
case of VENUS) [62].

One important parameter of the magnetic ﬁeld is the mirror ratio

Rm = Bmax /Bmin . As can be seen in Figure 2.12, for VENUS this can be as high as 6 at
injection and 4 at extraction. Typical values for other ECR sources are between 2 and 4.

46

With the mirror ratio, a loss cone can be deﬁned and particles inside the loss cone escape the
magnetic mirror. Let us consider this for a simple mirror ﬁeld without radial conﬁnement,
collisions, or plasma potential. If α is the angle between v⊥ and v of an ion at the center of
1/Rm . A particle with

the source (where B = Bmin ), the loss cone is deﬁned by sin α0 =

α < α0 is not trapped and escapes the magnetic mirror. In reality many factors contribute
to the loss processes and the loss cones can be severely modiﬁed. More information can be
found in literature on ECRIS magnetic conﬁnement, e.g.: [14, 63–65].

2.2.3

Creation of Highly Charged Ions

As mentioned before, the high charge states of ions in an ECR ion source are achieved by
successive electron impact ionization. Two processes occur that work against the ionization
process: Charge-exchange with neutral atoms and losses to the walls (due to the loss cone).
The whole process can be described by a rate equation [66]:
q

dni
q−1
q
= ne σv ion
− ne σv ion
q−1→q ni
q→q+1 ni
dt
+ n0,i σv

q+1
cx
q+1→q ni

− n0,i σv

q
cx
q→q−1 ni

q

− q
τi

where the terms on the right hand side (in order of appearance) are:
• ionization from charge state q − 1 to q
• ionization from charge state q to q + 1
• charge-exchange from charge state q + 1 to q
• charge-exchange from charge state q to q − 1
• losses due to escaping from conﬁnement
47

ni

(2.68)

Here, σv denotes a rate coeﬃcient depending on cross section for either ionization or
charge-exchange and the electron velocity averaged over the electron energy distribution.
Cross-sections for electron-impact ionization are more available than those for ion-impact
ionization (cf. Section 2.1.5) and semi-empirical formulae have been proposed by M¨ller and
u
Salzborn [67], and Lotz [68, 69]. A scaling law for charge-exchange has been proposed by
M¨ller and Salzborn [1] (as mentioned in Section 2.1.5).
u

2.2.4

ECR Heating

In order to strip an electron from a target atom or ion, the ionization energy must be
provided by the projectile. The higher the charge state of the target ion, the more energy
is necessary to free the next electron. For instance, the 1st ionization potential of 40 Ar is
≈ 16 eV while the 18th (stripping the last electron) is ≈ 4.5 keV. Consequently, the typical
electron energies needed in ECR ion sources are 1 keV to 20 keV (depending on the element).
As mentioned earlier, the heating of electrons is achieved by coupling rf-microwaves into the
source volume. Resonant heating then occurs on the ecr surface. It is essential that the ecr
surface is closed and well within the source volume. In order for the microwaves to penetrate
the plasma and reach the ecr surface, the plasma frequency
ne e2
0 me

ωp =

should be lower than the microwave frequency: ωp < ωrf = 2πfrf . If this is not the case,
we speak of an overdense plasma. If the plasma is overdense, the highest charge states are
not reached and the plasma can experience turbulence (it is not quiescent) [14].

48

Figure 2.13: Upper: Electron and Ion densities in collisionless plasma near wall. Lower:
Plasma potential of a collisionless plasma near wall. Image by Lukas Derendinger under the
GNU Free Documentation License.

2.2.5

Plasma Sheath

Whenever a ﬂoating or negatively biased solid surface is introduced to the plasma (probe,
wall, extraction electrode, etc.), a so called plasma sheath forms between the plasma and the
solid surface (henceforth named ’wall’). This is necessary to shield the plasma from the negative potential (with respect to the plasma potential) created by the fact that substantially
more electrons than ions hit the wall (due to the diﬀerence in mass). Thus there must be
49

a potential drop from the (in general) positive plasma potential Φp to the less positive wall
potential Φw . Figure 2.13 shows the potential and ion/electron densities close to the wall.
It can also be seen that between plasma and sheath a pre-sheath with a less pronounced
potential drop forms, which accelerates the ions into the sheath. The plasma potential is
an important parameter in ECR extraction simulations. In the following we will derive an
expression for Φp following considerations by P.C. Stangeby et al. [70, 71] for a 1-D plasma
sheath. Assuming that the quasi-neutrality of the plasma

qi · ni = ne

(2.69)

i

(with qi the charge of a certain ion species, ni its particle density and ne the electrons’
particle density) is conserved, we can calculate the potential diﬀerence between wall and
plasma (Φp − Φw ) by setting the electron and ion currents to the wall equal (j + = j − ). The
sheath is very thin, so we can assume the following:
• No ionization or ion collision takes place within the sheath
• Consequently, all ions entering the sheath hit the solid electrode

2.2.5.1

Electron Flux

Inside the plasma we can assume the electrons to be in thermal equilibrium, thus having a
Maxwell-Boltzmann velocity distribution

f (ve )dve =

3/2
m v2
me
− e e
· e 2kTe · dve
2πkTe

50

(2.70)

The particle ﬂux in a given direction n is given by
ˆ

f (ve ) · ve · n · d3 ve
ˆ

(2.71)

V

Let’s now consider the random ﬂux through a surface element z dS located at the origin and
ˆ
only for particles coming from z < 0 (the electrons hit the wall only on the inside of the
chamber) of particles with velocities between v and v + dv and an angle of θ to the surface
normal. Thus v · z = v · cos θ. And in spherical coordinates we have d3 v = v 2 sin θ dθ dφdv.
ˆ
Thus the ﬂux is
∞

F =

π/2

f (v)v 3 dv

0

∞

2π

sin θ cos θ dθ
0

dφ = π
0

f (v)v 3 dv

(2.72)

0

with equation 2.70 as velocity distribution this yields
2/3 ∞
m v2
me
− e e 3
e 2kTe ve dve
Fe = π ne
2πkTe
0
kTe 1/2
= ne
2πme

Knowing that the mean velocity of electrons following a Maxwellian is given by

ve =
¯

8kTe
.
πme

(2.73)

we can rewrite the above result to

Fe =

1
· ne · ve
¯
4

51

(2.74)

In a repelling, conservative force-ﬁeld (as the electrons experience in the sheath), only the
number density is reduced and the distribution remains Maxwellian, thus we can write the
ﬂux of electrons to the wall as

1
¯
Fe = · ne · ve · e
4

2.2.5.2

e(Φw −Φp )
kTe

(2.75)

Ion ﬂux - The Bohm criterion

The shielding eﬀect of the sheath is not perfect and a small portion of the electric ﬁeld
extends into the plasma, causing a pre-sheath acceleration. A physically realistic, steadystate sheath solution requires that the ions enter the sheath with a speed of at least their
acoustic speed Cs , while from plasma analysis one ﬁnds that no such solution is possible
with an ion drift velocity exceeding Cs . Thus (in this simpliﬁed case) the ions must enter
the sheath with a net drift velocity equal to Cs . This is called the Bohm criterion [70, 71].
For isothermal (Ti = const.) ion ﬂow one ﬁnds

Cs =

2.2.5.3

k(Ti + Te )
mi

(2.76)

The Plasma Potential

We can now calculate the plasma potential with respect to the wall (on ion source high
voltage potential Φw ≈ 20 − 30 kV). For multiple ion species we cannot simply set the ﬂuxes
equal, but have to use the currents because we have to account for k diﬀerent charge states:
k

j+ = e ·

qj ni,j
j=1

52

k(Ti,j + Te )
mi,j

(2.77)

with e the elementary charge, k the number of diﬀerent species, qi the respective charge state,
ni,j the respective particle density, Ti,j the respective temperature and mi,j the respective
mass. Setting the currents equal and crossing out e on both sides gives

kTe
·e
2πme

ne ·

k

e(Φw −Φp )
kTe

=

qj ni,j
j=1

k(Ti,j + Te )
mi,j

(2.78)

thus

e
(Φw − Φp ) = ln 
kTe

k
j=1

qj ni,j
ne



2πme (Ti,j + Te )

Te mi,j

(2.79)

and ﬁnally, using equation (2.69) for ne ,

Φp = Φw +

kTe 
ln
e



k

qj ni,j − ln 
j=1



k

qj ni,j

2π

j=1

me
mi,j

1+

Ti,j

Te

(2.80)

The plasma potential diﬀerence to the wall (Φw −Φp ) is typically some tens of Volts. This has
been experimentally observed as well [28]. Typically, both the ion and electron temperatures
in the plasma sheath are assumed to be on the order of a few eV. In Section 3.1 I will describe
how the plasma density is used in two- and three dimensional self-consistent extraction
models for ion sources and ECR ion sources in particular.

2.2.6

Extraction Systems

The extraction system is a series of electrodes with the purpose of accelerating and guiding
the ions that come out of the source to the beam line. The minimum number of electrodes
is two (diode system). This includes the plasma electrode, which is in contact with the ion
source plasma and has a hole, the extraction aperture, through which the beam can exit the
source. The plasma electrode is at the same potential as the source itself. The distance
53

between plasma electrode and the next electrode is usually called extraction gap. Often a
puller electrode is added, which is biased negatively with respect to ground potential to keep
electrons created further along the beam line from back-streaming into the source. This
improves source stability. The simplest system including such a puller electrode consists
of three electrodes (plasma-, negative-, ground potential) and is called accel-decel triode
system. More sophisticated designs exist with even more electrodes on individually adjusted
potentials (four electrodes - tetrode system, etc.).

2.2.7

Extracted Current

For a planar diode geometry, the maximum current density J that can be extracted from a
plasma source is given by the Child-Langmuir law :

J=

1.67 · 10−3

3/2
Q 1/2 V0
[A/m2 ]
2
2
mc
d

(2.81)

with Q the charge, m the mass of the extracted particles, V0 the extraction voltage (diﬀerence
between the two electrodes of the diode system), c the speed of light, and d the distance of
the two electrodes. However, this is strictly an upper limit imposed on the system by the
space-charge of the beam. Two important factors limit this further:
1. The high voltage breakdown in vacuum limits the actual voltage that can be applied
to the electrodes.
2. The plasma must be able to provide the necessary current.
In reality, the maximum current from an ECR ion source is limited by the ﬂux of ions leaving
the plasma, the ion loss rate (cf. equation 2.77). In [66] Melin et al. also provide an alternate
54

expression depending on the conﬁnement time rather than the ion temperature:
q

q
Ii

1 ni qeVex
=
2 τiq

q

(2.82)

q

with ni the density of ion species i in charge state q, τi the conﬁnement time, and Vex the
plasma volume connected to the extraction aperture through magnetic ﬁeld lines. Both exq

pressions are somewhat academic, because τi is not really known and must be approximated
(which is done in [66]) and Vex is not static, but depends on collision and diﬀusion processes.
Similar problems exist with the ion density ni . But, let us assume we have a good idea of
what the current density at the extraction aperture will be for the ion source in question.
Then the extraction system (diode distance and shape) and voltage must be chosen such
that the beam is matched well into the subsequent LEBT. This means the focusing at the
extraction aperture due to the formation of a plasma meniscus should lead to a laminar ﬂow
through the extraction system. As shown by R. Geller [14, pp. 293], the shape of the meniscus can to ﬁrst order be approximated by the switching out J and d in the Child-Langmuir
1 + e(π):

law and modifying it with a geometry factor

2
d(π) = 0
3

3/4
2q 1/4 V0
m
J 1/2

1 + e(π)

(2.83)

where d is the distance from the puller electrode to a point π on the plasma surface. e(π)
takes into account the actual shape of the electrodes and is 0 for plane electrodes and a
homogeneous plasma [14]. The plasma meniscus and extraction of a beam from the plasma
surface will be discussed in detail in Chapter 6.

55

2.2.8

Theoretical Emittance of an ECRIS

Typically, the emittance of an ECR ion source will consist of two contributions [9, 14, 72]:
1. The thermal emittance: th (due to the transversal temperature the ions acquired while
they were inside the plasma)
2. The magnetic emittance: mag (due to the beam rotation induced by the decreasing
axial magnetic ﬁeld)
Ad 1. The thermal emittance can be estimated by assuming that the velocity distribution
of ions inside the plasma is Maxwellian [41, 73]. The normalized rms emittance is then:

th
n−rms

= 0.016 · r

kB Ti
A

(2.84)

in mm-mrad, with kB Ti the ion temperature in the plasma in eV, A the mass-number, and
r the radius of the extraction aperture in mm. This is for x-x’ as well as y-y’, assuming a
round beam.
Ad 2. Due to the high collisionality of the ions inside the plasma, we assume that they
have no ’memory’ of the transport processes before reaching the sheath at the extraction
aperture. They are thus said to be ’born’ close to the aperture, in a region of high magnetic
ﬁeld (up to 3 T for high performance ECRIS). That ﬁeld rapidly decreases to zero in axial
direction thus inducing rotation of the beam (conservation of canonical angular momentum
or Busch’s theorem [41, 74]). A short derivation of the emittance due to this can be found
in Geller’s book [14] and a paper by W. Krauss-Vogt et al. [75].

56

Figure 2.14: Measured emittances of the Lawrence Berkeley National Laboratory AECR-U
compared to the theoretical prediction of magnetic emittances. Image taken from [9]
The magnetic emittance can then be written as:

mag
n−rms

= 0.032 · r2 B0

1
A/q

(2.85)

in mm-mrad, with B0 the axial ﬁeld at the extraction aperture in T, r the radius of the
aperture in mm, and A/q the mass-number to charge-state ratio (dimensionless). The total
normalized rms emittance is then the sum of the two contributions. However, for typical
ECRIS parameters (Ti ≈ a few eV [66], B ≈ 1-3 T) the magnetic emittance clearly dominates
these theoretical values.

The theoretical emittance due to equations 2.84 and 2.85 is in

contradiction with measured emittances of high performance sources. For instance LBNL’s
57

AECR-U [9]. Figure 2.14 shows the measured emittance values and the prediction for the
magnetic emittance. It is assumed that the reason for this discrepancy is the fact that
higher charge states are conﬁned closer to the center of the source and the radius r used in
the theoretical approach has to be modiﬁed appropriately (see [76] for similar results with
another ECRIS). This behavior has been observed experimentally as well: (i) It was seen
that high charge state ion currents decrease less than lower charge states when the plasma
aperture diameter is decreased [77]. (ii) Measurements with a movable faraday cup close to
the extraction aperture showed bloated triangular distributions of ions with diﬀerent charge
states with sizes depending on the charge state [78]. From the theoretical emittance th
and the measured emittance act , an eﬀective plasma aperture, the virtual aperture, can be
deﬁned [9].
ref f = ract ·

act

(2.86)

th

Such an eﬀective aperture radius can be used in simulations, to better reproduce the measured emittances. Something similar has for example been done in [76].

58

Chapter 3
Software
In general, the electric and magnetic ﬁelds of the complex electrode and magnet geometries
in an ion source, beam transport system, or accelerator cannot be solved by hand anymore.
Similarly, calculation of the many possible diﬀerent trajectories of particles making up the
ion beam is a non-trivial problem. Especially when including the electrostatic self ﬁeld
interaction between individual beam particles. Consequently, simulations play an important
role in the design and study of ion sources, beam transport and accelerators. In this chapter,
I will describe the three simulation programs used in the course of this thesis as well as the
utility software I programmed.

3.1

Simulation Codes

In this thesis, I mainly used three ion beam simulation codes: SIMION, IGUN and WARP.
SIMION and IGUN are commercial software and will only be described brieﬂy in the following
Subsections 3.1.1 and 3.1.2. WARP is a particle in cell (PIC) code in ongoing development
by Lawrence Livermore National Laboratory (LLNL) and Lawrence Berkeley National Laboratory (LBNL) and was used extensively during this thesis work (simulations in Chapters
6 and 7). It will be described in greater detail in the following Subsection 3.1.3.

59

3.1.1

SIMION

SIMION 8 [79] is a widely used ion optics simulation program. It provides a graphical user
interface, the ion optics workbench, that allows the user to create and manipulate (position,
rotate, and scale) up to 200 instances of electrostatic and magnetostatic potential arrays
(PA). PA’s can be created either manually in the GUI, or through geometry (*.gem) ﬁles
written by the user, or through import from CAD data in the *.STL format. Mirror and
rotational symmetries can be exploited in the creation of the potential arrays to reduce
computation time and memory usage. Once a PA has been created, the potentials between
the electrodes are calculated by solving the Laplace equation on a square mesh (reﬁning in
SIMION jargon) through a ﬁnite diﬀerence method (over relaxation) [79, p 2-2]. Once the
solution has been calculated, SIMION can scale the potentials for diﬀerent voltages applied
to the respective electrodes (fast adjusting).
Ions can then be ﬂown inside the workbench. In SIMION, a fourth order Runge-Kutta
method is used for the calculation of the ion trajectories.
A very important feature that was heavily used in this work are workbench programs that
can be written by the user in the scripting language Lua. Through the workbench program, SIMION provides access to certain steps of the ion trajectory calculation letting the
user change, e.g., ion positions and velocities, electrode voltages and magnetic ﬁelds. Data
acquisition and output can be regulated through workbench programs as well.
A new feature of SIMION 8.1 over version 8.0 is the Poisson Solver that enables the user to
create an arbitrary charge distribution in a PA and solve for the electrostatic ﬁeld. This was
used in Chapter 5 to calculate the space charge potential for various asymmetric beams and
obtain secondary ion energy distributions of ions created by beam-residual gas interaction.

60

3.1.2

IGUN

IGUN [31–33] is a numerical simulation code for ion beam formation at a plasma boundary.
IGUN is a 2D code in RZ coordinates, exploiting the fact that most extraction systems are
rotationally symmetric and in many cases, the extracted beams are as well. The problem is
set up in IGUN by the user through an input ﬁle containing the basic parameters (INPUT1),
axial values of the magnetic ﬁeld (INPUT3), boundary points of the electrodes, and runtime
variables (INPUT5) [33].
IGUN then solves the problem by iterating multiple times over the following steps until a
stable solution has been reached [31]:
1. Solve Poisson’s equation for the potential map. This is done on a rectangular mesh
using a ﬁnite element method and a successive over-relaxation solver (SOR).
2. Solve the equations of motion for the macro-particles using ray-tracing with a fourth
order Runge-Kutta procedure.
3. Allocate space charge to the potential map. This is done in two steps: Inside the plasma
sheath, the space charge is calculated analytically using a 1D sheath model. Outside
the sheath the space charge density is generated from the particle trajectories obtained
in step 2 multiplied with a compensation term according to the thermal distribution
of electrons.
The magnetic ﬁeld data (if speciﬁed) is given as Bz (z) at r = 0 and the ﬁeld values for r > 0
are calculated through radial expansion up to order 6.

61

3.1.3

WARP

WARP [36, 80, 81] is a particle-in-cell (PIC) code originally developed for the simulation of
intense ion beams used in heavy-ion driven inertial conﬁnement fusion (HIF).
In PIC, macro-particles representing a number of real particles are advanced in time using
the Lorentz equation of motion. Electrostatic self-ﬁelds of the particles are calculated selfconsistently on a mesh. Particles are advanced through a combination of the Leapfrog and
isochronous Leapfrog methods [82].

3.1.3.1

Main Packages (python class name in italic)

The main package supervising the simulations is called top.
For advancing particles and solving for the electrostatic ﬁelds (external plus self-ﬁelds)
WARP has three diﬀerent geometry modes:
• WARP3d (particles: w3d, ﬁeld solver: f3d ). The full three dimensional model.
• WARPxy (particles: wxy, ﬁeld solver: fxy). A transverse slice model.
• WARPrz (particles: wrz, ﬁeld solver: frz ). A cylindrically symmetric model.
The separation of particle pusher and ﬁeld solver enables the user to combine 3D pusher and
RZ ﬁeld solver for cases where the electrodes are RZ symmetric, but a 3D treatment of the
beam is necessary.
In addition to the PIC model, there is also an envelope solver (package: env ). The envelope
solver has not been used in this thesis and will not be described further.

62

3.1.3.2

Particle Loading

Particles are loaded and injected separately. This is practical since the same initial distribution loaded can be used with diﬀerent injection methods or in diﬀerent geometry modes in
subsequent simulation runs. WARP provides a variety of built-in initial transversal particle
distributions: Gaussian, Semi-Gaussian or K-V (uniform). Longitudinal distributions can
be cigar shaped or uniform. WARP also lets the user specify arbitrary transversal and longitudinal particle distributions. The loaded beam can then be injected in one of two ways:
Constant current, or space charge limited.

3.1.3.3

Fields and Lattice Elements

WARP has four built in lattice elements: quadrupoles, dipoles, bends, and acceleration gaps.
Lattice segments can be periodically extended. Special attention should be given to the bend,
which is not a lattice element per se, but rather a region of coordinate transformation. In this
region, speciﬁed by a start point, end point and bending radius, Frenet-Serret coordinates
are used to transform the bend into a straight line. These elements are used in superposition
with all forms of dipole elements to keep the design trajectory straight.
In addition to the built-in lattice elements, WARP lets the user set electrostatic or magnetostatic external ﬁelds in a number of ways:
• Hard-edged multipole description
• Axially varying multipole description
• Gridded elements
For the ﬁrst two types of descriptions, WARP calculates the ﬁelds at the particle position

63

through multipole expansion. Gridded elements are given on a 3D grid and are interpolated
to the particle position as needed.
The user can also generate electrode geometries, specify their voltages, and have WARP
calculate the electrostatic ﬁelds directly.
In addition, there are supplemental python scripts for generating realistic models of common
beam optics elements like electrostatic quadrupoles or solenoids.

3.1.3.4

Fieldsolvers

In 3D and in transverse slice mode, the self ﬁeld of the beam is calculated on a Cartesian
mesh. WARP has a variety of ﬁeld solvers at its disposal (fast Fourier transforms - FFT,
capacity matrices, successive over relaxation - SOR, and the multigrid solver - an extension
of the SOR solver which utilizes additional coarser grids to speed up the calculations).
Because we need to resolve rather complex structures in some of the electrodes, the SOR
and multigrid solver are used for the simulations in this thesis. In WARP the variable fstype
sets the ﬁeldsolver with 3 being the SOR solver and 7 the multigrid solver.

3.1.3.5

Structure of the Simulations

Typically a script written to run a WARP simulation is called a deck. A WARP deck consists
of the following segments:
• Setting the basic run time parameters. This includes some descriptive strings, size, cell
size and boundary conditions.
• A call to setup(). This is mandatory and initializes some plotting routines.
• Setting up the lattice elements. This can be done in any of the ways described above.
64

• Loading the particles and specifying injection of the particles
• Selecting the package. This could e.g. be package(’wxy’) to select the WARPxy PIC
package.
• A call to generate(). This initializes the PIC simulation, allocates space, transforms
lattice arrays into the internal format for faster processing, etc. Note: In general the
simulation variables, particles and ﬁelds should be set up before the generate. However,
there are some convenience functions (e.g. addparticles() to load particles) that can
be called after the generate() because they can re-allocate the necessary memory.
• Finally, once the problem has been set up (whether in 3D, RZ or XY), the function
step() is called to run the simulation. Each call to step advances the simulation by
either one time step dt (3D, RZ) or one slice dz (XY). Step() can also be called with
a number of steps as an argument (e.g. step(100)).

In Chapter 6, I will go into more detail about the WARP packages used for the respective
simulations of ion beam extraction and low energy beam transport, how the particles were
initialized and loaded, and how the beam optics elements were set up.

65

3.2

Utility Programs

As mentioned in the previous section, WARP does not provide a graphical user interface
(GUI). For IGUN, there exists a suite of programs providing some GUI functionality, but
they were developed for Windows XP and are not well supported under Windows 7 and 8.
In IGUN, a user generated input ﬁle is read in upon start of the program and WARP is run
from inside a python script environment. In this section I will brieﬂy describe the utility
programs and postprocessors I wrote to simplify the use of IGUN and WARP.
All utility programs are written in python 2.7. The following libraries were used:
• Numpy [83]. Numerical library providing a number of array and matrix manipulation
functions.
• cPickle. A python library for fast and reliable saving and loading of python objects
(class instances, arrays, etc.).
• Matplotlib [84]. An extensive library for 2D plotting.
• PyGTK [85]. GUI development through GTK+2.0 API bindings for python.
In addition, cxFreeze [86] was occasionally used to create standalone executables from the
python scripts.

3.2.1

Electrode Designer

In both IGUN and WARP, the electrode geometry is provided by the user in form of arrays
of vertices and arcs. Without visual feedback, this can be cumbersome, especially if the
electrodes have complex shapes. It suggested itself to write a simple GUI to generate the

66

electrodes point-by-point and subsequently export them into IGUN or WARP. The electrode
designer provides an easy interface where the user can create and delete electrode objects
consisting of vertices and arcs. The state of the program can be saved and re-loaded at any
time. The image shown in the preview section can be saved as a *.png image ﬁle.
A 1D array containing magnetic ﬁeld data Bz (z) on axis can be loaded into the program
and displayed in the preview windows for WARP and IGUN.

3.2.1.1

IGUN export

In the IGUN preview window, the user can set a number of geometry constraints and specify
a set of ion species (m, q, current) to simulate. Exporting for IGUN creates a *.IN ﬁle which
can directly be run in IGUN. Additional parameters can be set by editing the exported text
ﬁle.

3.2.1.2

WARP export

Exporting for WARP does not include particles or setting up the whole simulation. This
is because WARP is a lot more complex than IGUN and many parameters have to be set
appropriately. When the electrode data is exported for WARP, three ﬁles are created: A ﬁle
containing the electrode geometry, a ﬁle containing meta data (voltages, id’s, and names of
the electrodes), and a simple WARP script that can be run immediately called ’main.py’. The
script reads in the electrode data, performs a single ﬁeld solve, and displays the calculated
electrostatic ﬁeld.

67

3.2.2

WARP Postprocessor

In order to evaluate the results from the WARP simulations, I developed a postprocessor
GUI. During the simulations, data is saved depending on the user settings and the type of
the simulation. In xy-slice mode, by default, the particle distributions at the start and the
end are saved as well as the envelopes (maximum extent and 2-sigma envelopes). Additional
locations to save the particle distributions can be speciﬁed. In 3D mode, only the particle
distributions at the start and the end are saved at this time. All ﬁles are written to disk
using cPickle. The so obtained data ﬁles can then be opened in the postprocessor. The main
area of the GUI is a notebook object with several pages:
• Pages 1-5: XY, XX’, YY’, XY’, YX’ distributions, respectively.
• Pages 6,7: Horizontal and vertical beam envelopes.
• Page 8: A combination plot of the horizontal and vertical envelopes.
• Page 9: A text window to display the ﬁle containing settings and results of the simulation.
• Page 10: A log of the operations performed in the postprocessor since starting the
program
Apart from standard graphics operations like changing labels and titles of the plots, zooming,
and setting the limits, etc., the postprocessor features the following functionalities:
• Loading multiple data sets for comparison.
• Switching on and oﬀ the viewing of individual data sets.
• Calculation of emittances and beam sizes from the particle distributions.
68

• Loading of lattice description ﬁles for display with the beam envelopes.
• Loading of IGUN *.TRJ ﬁles. These ﬁles contain the results of IGUN RZ symmetric
simulations. The postprocessor can calculate a XY particle distribution from the *.TRJ
ﬁle via a randomization process written by A. Lemut in C and ported to python. This
process includes skew angles due to the magnetic ﬁeld following a procedure described
by Chan et al. [87].
The postprocessor features the following export options:
• Saving the plots as *.png image ﬁles.
• Creating 2D histograms from the cross section and phase space plots to compare to
emittance scans and beam viewer images.
• Saving particle data as comma separated value (*.csv) ﬁles for import into Excel or
other programs.

69

3.2.3

NeMo Postprocessor

With the GUI foundation already in place, it suggested itself to also create a postprocessor
for the neutralization measurements discussed in chapter 5. As in the WARP postprocessor,
the main area is a gtk notebook with several tabs which correspond to the diﬀerent datasets
that can be loaded into the postprocessor and manipulated there:
• Pages 1,2: Neutralization spectra obtained with the retarding ﬁeld analyzer NeMo and
the simple xy plotter program ecrtune.
• Page 3: Charge state distributions (CSD) obtained with the analyzing magnet and
ecrtune.
• Page 4: Charts obtained by logging the EPICS-channels of the NSCL’s control system. These include faraday cup currents, ion gauge readings, and the ion source drain
current.
• Page 5: Slit scans of the ion beam cross section obtained with the four-jaw slits discussed in Subsection 4.2.4.
• Page 6: A log of messages from the program to the user.

70

In all viewports, limits, labels, linewidths, legends, titles, etc. can be set individually. Furthermore, the following options are available for data manipulation:
Ad pages 1,2, and 3: Analysis options for charge state distributions and NeMo spectra
include:
• Cutting the data sets individually
• Interpolating.
• Smoothing.
• Normalizing.
• Fitting peaks in the charge state distribution.
• Calculating ﬁrst and second derivatives of NeMo spectra.
• Calculating preliminary neutralization values from NeMo spectra.
Ad page 4 (the chart viewer): Channels can be scaled individually and deactivated from the
view screen. Average and standard deviation of all selected channels can be calculated for
the currently displayed time frame (horizontal axis).
Ad page 5 (slit scan viewer): Here, the maximum intensity can be set for the colormap
and the slit scan image can be exported to be used as a density map for creating charge
distributions and particle distributions in SIMION. The export options include: scaling of
the image size, rotating the image, and normalizing it to the correct total current.

71

Chapter 4
Hardware

4.1
4.1.1

Electron Cyclotron Resonance Ion Sources
LEDA injector ion source

The high current proton source was originally developed at Chalk River Laboratories [88]
and subsequently modiﬁed for operation as the ion source of the low energy demonstration
accelerator (LEDA) at Los Alamos National Laboratory [7, 89, 90]. Of the four ion sources
used in this thesis, the LEDA injector ion source is diﬀerent from the other sources in the
requirements and as a consequence in the design of the magentic ﬁeld. The LEDA injector
source was designed to provide very high currents (up to 130 emA) of protons. For single
ionization, the required energy of the electrons in the plasma is much lower than in highcharge-state ECRIS. Also, the necessary conﬁnement times for single ionization are short
compared to the step-by-step process necessary to achieve medium to high charge states. As
a consequence, there is no sextupole for radial conﬁnement. Instead, the two solenoids are
close enough together to form a single peak in the longitudinal magnetic ﬁeld structure. A
schematic of the source is shown in Figure 4.1 and the important parameters are listed in
Table 4.1. During peak operation under optimal conditions, the beam is composed of 85 +
90% protons and 10 - 15% H2 ions [90].

72

Figure 4.1: Schematic of the LEDA injector ion source. Image taken from [10].
73

The extraction system as of 1997 is a tetrode accel-decel extraction system. In 2012, the
LEDA source was transferred to NSCL for a short time. During this time, neutralization
measurements (described in Section 5.3.1) were conducted. It should be noted that during the time at NSCL, the extraction aperture diameter was decreased from 8.6 mm to 3
mm and, consequently, the extracted current was drastically reduced (during the presented
measurements it never exceeded 10 emA).
Table 4.1: LEDA - Ion Source Parameters
Parameter

Value

Extraction Voltage
Proton fraction

max. 75 kV
85 - 90%

Chamber ∅

90 mm

Chamber length

100 mm

Bmax

0.0875 T

fecr

2.45 GHz

MW power

max. 800 W

74

Figure 4.2: Schematic of Artemis B (a duplicate of Artemis A). Image taken from [11]

4.1.2

Artemis A - Room Temperature ECRIS

The Advanced Room TEMperature Ion Source Artemis [11,91,92] is located at the National
Superconducting Cyclotron Laboratory (NSCL) at Michigan State University. It is one of
the two injector ion sources of the coupled cyclotron facility (CCF). Its design is based on the
Advanced ECR (AECR) at LBNL. The longitudinal magnetic ﬁeld is provided by two roomtemperature solenoids and the radial ﬁeld by a permanent-magnet sextupole. It features
a triode accel-decel extraction system with movable puller electrode. The neutral material
for ionization is provided either in gaseous form through the gas lines or by evaporation in

75

an oven that is inserted radially between the two solenoid coils. A schematic view of the
source is shown in Figure 4.2 and the important operating parameters are listed in Table
4.2. Artemis A was designed to reliably deliver stable DC beams of medium charge state
ions over long periods of time (days to weeks).
Table 4.2: Artemis parameters. Magnetic ﬁeld values are nominal and subject to optimization for each desired ion.
Parameter

Value

Extraction Voltage

max. 24 kV

Chamber material

Aluminum

Chamber ∅

75 mm

Chamber length

290 mm

Binjection

1.8 T

Bmin

0.2 T

Bextraction

0.8 T

fecr

14.5 GHz

MW power

max. 2 kW

76

Figure 4.3: 3D CAD model of the SuSI ECRIS. In the center the plasma chamber can be
seen, sorrounded by the six solenoid coils. Gas and microwaves are fed from the left, the
beam is extracted on the right.

4.1.3

SuSI - Superconducting ECRIS

The SUperconducting Source for Ions (SuSI) [2, 38, 39, 93] is located at the National Superconducting Cyclotron Laboratory (NSCL) at Michigan State University. It is the second
of the two injector ion sources of the coupled cyclotron facility (CCF). A 3D CAD model
of SuSI can be seen in Figure 4.3. As can be seen in the ﬁgure, SuSI features six solenoid
coils, which can be adjusted individually, and a sextupole coil structure. All coils are superconducting and inside a cryostat which is kept at a temperature of about 4 K. Liquid

77

nitrogen and liquid helium for cooling are provided by the NSCL cryoplant via ﬁll lines. The
injection ﬂange is moveable, allowing the length of the plasma chamber to be adjusted from
400 to 500 mm. On the inside, facing the plasma, a biased disk is mounted on the injection
ﬂange. This biased disk is adjustable with respect to the injection ﬂange by 50 mm and can
be biased with voltages up to -300 V. The extraction system is a triode accel-decel system
with adjustable extraction gap width. The puller is usually biased with voltages between -1
kV and -3 kV. The use of two microwave frequencies can improve the stability of the source
and the production of high charge states. In the case of SuSI, the two frequencies are 14
and 18 GHz, but the magnet system is able to provide the necessary ﬁelds for 18 + 24 GHz
operation and a suitable Gyrotron providing the 24 GHz microwaves is being installed at
the moment.
Table 4.3: SuSI parameters. Magnetic ﬁeld values are nominal and subject to optimization
for each desired ion [2].
Parameter

Value

Extraction Voltage

max. 24 kV

Chamber material

Aluminum

Chamber ∅

100 mm

Chamber length

400-500 mm

Binjection

2.4 T

Bextraction

1.3 T

Brad,wall

1.3 T

fecr
MW power

14 + 18 GHz, (24 GHz)
max. 2 + 2 kW, (10 kW)

78

Figure 4.4: Schematic of the VENUS superconducting ion source. Image taken from [3].

4.1.4

VENUS - Superconducting ECRIS

The VENUS ECRIS [3, 8, 37, 94] is located at Lawrence Berkeley National Laboratory. It is
one of the injector ion sources of the 88” cyclotron. Unlike SuSI, the VENUS cryosystem
operates in a closed loop scheme, using four cryocoolers to keep the working temperature
of the liquid helium at 4.2 K. No transfer of helium is necessary during operation. The
longitudinal ﬁeld is provided by three superconducting solenoid coils and the sextupole ﬁeld
by six superconducting race-track coils. The important parameters (like peak magnetic
ﬁelds) are listed in 4.4.
The extraction system is a triode accel-decel system with adjustable extraction gap width.
The puller is usually biased with voltages between -1 kV and -3 kV. VENUS uses double
79

frequency heating (18 + 28 GHz), to improve the stability of the source and the production
of high charge states. The 18 GHz microwaves are provided by an oscillator and a klystron
ampliﬁer, while the 28 GHz microwaves are provided by a gyrotron system. Inside the
chamber, on the injection side, facing the plasma, a biased disk is located which can be
biased with a negative voltage up to -300 V.
X-ray production through bremsstrahlung is an important issue in ECR ion sources and
more so in superconducting sources, because the produced x-rays can heat the magnet coils
and thus lead to quenching, if the cryosystem cannot compensate fast enough. In VENUS,
a 2 mm tantalum shield was installed between plasma chamber and cryostat to act as x-ray
shielding [95].
Table 4.4: VENUS parameters. Magnetic ﬁeld values are nominal and subject to optimization for each desired ion [3].
Parameter

Value

Extraction Voltage

max. 25 kV

Chamber material

Aluminum

Chamber ∅

140 mm

Chamber length

500 mm

Binjection

4T

Bextraction

3T

Brad,wall

2.4 T

fecr

18 + 28 GHz

MW power

max. 2 + 10 kW

80

4.2

Beam Diagnostic Instruments

In this section, I will describe the diﬀerent diagnostic devices used during the measurements
for this thesis. They are:
• Faraday Cup invasive
• A/q Analyzer semi-invasive
• Beam Viewer invasive
• Slit-Scanner invasive
• Allison Emittance Scanner invasive
• Retarding Field Analyzer non-invasive
Devices labeled invasive block the beam from travelling on, the A/q analyzer is invasive
during the analysis, but is also used to single out the desired species for transport (hence
semi-invasive) Many of the diagnostic devices have been around for many years and are well
known. For these, only the basic principle will be recapitulated, and references will be given
to original material.

4.2.1

Faraday Cup

The faraday cup is a basic diagnostic device to measure the total beam current. A schematic
of its operating principle is shown in Figure 4.5. A conducting cup is used to collect the
positively charged beam ions and a Picoammeter is used to read out the charge per second
deposited on the cup. In the case of NSCL, specially designed beam current monitors
(BCM), which were designed and built in-house, are used. Because the impact of highly
charged ions on a surface leads to the escape of slow (typically 1-50 eV) electrons from the
81

Figure 4.5: Faraday cup schematic. The positive ion beam deposits charge on the cup.
Electrons leaving the material due to the ion impact are pushed back onto the cup by a
suppressor on a negative potential.
surface material, the current reading of the amperemeter would be artiﬁcially high (sum of
ions + sum of escaping electrons) for just the cup. To counteract this, an electron suppressor
is typically used, which is biased at a negative voltage (up to -300 V). This ring creates a
potential barrier that pushes the electrons back onto the cup (as indicated in the ﬁgure).
Regular faraday cups can withstand a beam power of up to a few hundred watt. Above that,
the heat dissipated in the material could destroy the cup. More elaborate designs exist, using
special shapes to better distribute the heat load on the cup surface, and cooling systems to
transport the heat away from the cup. The beams from the ECR ion sources with sextupole
(Artemis, SuSI, VENUS) are below 150 W, and regular faraday cups were used. The LEDA
ion source beams reached powers above 700 W. Here the currents were measured with Bergoz
current transformers [96] and on a cooled beam stop at the end of the short LEBT, as well
as by recording the drain current on the source high voltage power supply.

82

Figure 4.6: Cartoon picture of the A/q analyzer principle.

4.2.2

A/q Analyzer

Assuming the particles move in the horizontal plane, they are bent onto a curved trajectory
when transversing a vertical magnetic dipole ﬁeld. For the non-relativistic case and in
absence of an electric ﬁeld, the radius of curvature of the particle trajectory is given by:

r=

mv
Bqe

(4.1)

with B the ﬂux density of the vertical magentic ﬁeld, v the particle velocity, m ≈ A·amu the
particle mass, q the charge state, and e the elementary charge. In this context, the magnetic
rigidity [Bρ] is often used:
[Bρ] =

p
qe

(4.2)

with p the momentum of the beam particle and ρ = r the radius of curvature. This expression
is equivalent to equation 4.1. It follows directly from equation 4.1 that particles with diﬀerent
A/q ratios follow diﬀerent paths inside the dipole ﬁeld. This can be used to analyze the
distribution of diﬀerent ion species in the beam.
83

Figure 4.7: Example Charge State Distribution (CSD) of an oxygen beam from the ECRIS SuSI at NSCL. Contamination by
argon from a previous experiment can also be seen as well as nitrogen from imperfect vacuum.

84

In order to do this a dipole magnet is combined with slits at the magnet’s image point and a
faraday cup right after the slits (see ﬁgure 4.6). The magnetic ﬁeld is then slowly ramped up
while recording the ion current in the faraday cup. This leads to a plot commonly referred
to as a charge state distribution (CSD). An example of a typical CSD is shown in Figure 4.7
for an oxygen beam extracted from the SuSI ion source at NSCL.

4.2.3

Beam Viewer

The beam viewer or viewing screen (sometimes called a phosphor ) is a simple device that
allows us to obtain an image of the beam cross-section. It consists of a viewing plate, made of
a scintillator material (like SiO2 (Quartz), KBr, BaF2 , or YAG) that emits light upon being
hit by a beam particle, and a camera to record the picture. Usually, the viewing plate has an
angle of 45 degrees with the beam as well as with the camera, to get an undistorted picture
of the cross section. Figure 4.8 shows the 3D CAD model of such a beam viewer designed for
the AECR-U ion source at Lawrence Berkeley National Laboratory. An example of a beam
cross section for an argon beam taken with this beam viewer can be seen in ﬁgure 4.9. If the
beam particles deposit enough charge on the surface of the scintillator, slow beam ions can
be reﬂected locally by those charge-patches. In order to avoid this, a ﬁne, high-transparency
mesh is often used to cover the scintillator. Electrons extracted from the wires by passing
ions will then neutralize the deposited charge. The light yield of a scintillator depends on
many factors like: material, intensity of the beam, charge state of the beam ions, and the
duration of the exposure. One unfortunate property of scintillators is the degradation of light
yield with exposure time due to radiation damage [12,97,98]. Because this eﬀect depends on
the dose and energy, spots with higher beam intensity and lower charge states will degrade
faster, thereby creating a non-homogenous light yield distribution on the scintillator surface.
85

Figure 4.8: 3D CAD model of a beam viewer. This beam viewer was designed for the LEBT
of the AECR-U ion source at LBNL.
In order to get quantitative results about the beam intensity distribution, these eﬀects have
to be taken into consideration. This is further complicated by the fact that the crystals can
regenerate part of their scintillation strength on their own over time. In this thesis, beam
viewer images are only used in a qualitative way, to get information about the shape of the
beam cross section and compare them with each other and with slit proﬁle scans. In this
context the total light yield is not as important.

86

Figure 4.9: Two images taken with a quartz crystal beam viewer. The beam contained mostly
argon and oxygen. Diﬀerent ion species (mass m, charge state q) are focused diﬀerently in
the extraction region of the source and can be seen as ring-like structures on the viewer. The
ﬁne mesh covering the crystal can also be seen.

4.2.4

Slit Proﬁle Scanner

The slit proﬁle scanner is a combination of two slits (horizontal and vertical) and a faraday
cup. A schematic of the principle can be seen in Figure 4.10. The variant of a slit scanner
used at the NSCL has individually moveable left and right (upper and lower) plates, which
earned it the name 4-jaw. This way, not only the position of the slits can be changed, but
also the width. In a slit scan, the slits are moved stepwise to raster the cross sectional area
of the beam. At each slit setting, the partial current arriving at the faraday cup placed right
behind the slits is recorded. A beam intensity proﬁle is thus obtained. At the NSCL the
setting of the slits and readout of the faraday cup current is done with a MATLAB program.
In Figures 4.11 and 4.12 two such intensity proﬁles for beams extracted from SuSI at NSCL
are shown with slit widths of 2 mm and bicubic interpolation for a smoother image.

87

Figure 4.10: Cartoon of the slit proﬁle scanner operating principle. The two slits can move
independently. A computer program automatically sets the slits and reads in the faraday
cup current. By rastering the area of the beam cross section a transversal beam intensity
proﬁle is obtained.

88

Figure 4.11: Example beam proﬁle of an O3+ beam, ≈ 235 eµA. The beam exhibits a
triangular structure, which is typical for many ECRIS beams.

Figure 4.12: Example beam proﬁle of an Ar8+ beam, ≈ 200 eµA, tuned for ’roundness’.

89

Figure 4.13: Schematic of an Allison-type electrostatic emittance scanner. Image taken
from [12].

4.2.5

Allison Emittance Scanner

As discussed in Section 2.1.3, the emittance of the beam is a useful ﬁgure of merit to describe
its quality. Many methods exist to measure the emittance of a beam. Amongst them are:
Slit-wire/harp/slit [99–101] emittance scanners, (Allison) electrostatic emittance scanners
[102], and Pepperpot emittance scanners [103–105]. The only type of emittance scanner
used in this thesis was the Allison scanner. In order to determine the emittance of the
beam experimentally, one needs the spatial and angular distribution f (x, y, x , y ) of the
beam at a longitudinal position z. The Allison scanner can provide this information, albeit
only for one of the two transversal directions f (x, x ) or f (y, y ) while integrating over the
other. The scanning box is schematically depicted in ﬁgure 4.13. By using two such boxes
with perpendicular directions of movement and perpendicular slits, the horizontal (x − x )
and vertical (y − y ) phase space can be mapped separately. Let us now investigate the
example of the horizontal phase-space. The box is positioned with the slits vertical and
moved step-by-step in horizontal direction. For each horizontal position, a fraction of the

90

Figure 4.14: Allison emittance scan of a U36+ beam. Distortions can be seen that make it
hard to determine the ’full’ emittance of the beam (cf. discussion in Section 2.1.3) the red
ellipses denote the four-rms areas of these two scans.
beam is transmitted through the entrance slit of the scanning box. This is the spatial
information. The ions entering through the slit will have diﬀerent angles with respect to
the design trajectory. These angles are scanned by using deﬂection plates and a second slit
(as seen in Figure 4.13). For each horizontal position, the voltage on the plates is swept
from some negative value to the corresponding positive one and each time the current in the
faraday cup behind the second slit is recorded. Thus a 2 dimensional plot in x and x is
obtained where
x =

V (D − 2δ)
Ekin
4g

with V the plate voltage in V , Ekin the beam energy in eV, D the distance between entry
and exit slit, g the gap between the deﬂection plates, and δ the small gap between the slits
and the deﬂection plates [102]. By weighting each x/x pair with the current in the cup,
the rms emittance is then obtained. Because the x -voltage relation depends on the beam

91

Figure 4.15: 3D CAD model of the retarding ﬁeld analyzer. The measured ions enter from
the top and are repelled by grid 2 if their energy in eV is lower than the grid voltage in V.
The current measured with a picoammeter connected to the collector is proportional to the
number of ions reaching the collector.
energy, the maximum analyzable angle is a function of the scanner geometry (D, g, δ). The
resolution of the detector is a matter of entry- and exit-slit width. Both, geometry and
resolution have to be chosen for the respective application. As an example, a scan with the
LBNL Allison emittance scanner is shown in Figure 4.14 for a U36+ beam.

4.2.6

Retarding Field Analyzer

A retarding ﬁeld analyzer (RFA) is a device used to measure the energy distribution of
incoming ions by sweeping the voltage on a retarding ﬁeld grid while measuring the number

92

of ions still penetrating the grid. The RFA built in this thesis is a three-grid device with
two entry apertures (a 3D CAD drawing can be seen in Figure 4.15). The measured ions
enter through an aperture system, pass through the grounded grid 1 and are repelled by
grid 2 if their energy in eV is lower than the grid voltage in V. The current measured with
a picoammeter connected to the collector is proportional to the number of ions reaching the
collector.
The current measurement works exactly like in the case of a faraday cup and electron suppression is necessary to push back secondary electrons onto the collector plate. This is the
function of grid 3, which is negatively biased. Grid 3 also serves a second purpose: To
keep electrons from outside the detector from reaching the collector and contaminating the
measurement of the ion currents. A cartoon picture of the process is shown in ﬁgure 4.16.
In the presented case, the current readout was done with a Keithley 6485 Picoammeter or a
beam current monitor (BCM) built in-house at the NSCL and routinely used for measuring
very low beam currents with faraday cups. The voltage on the retarding ﬁeld grid was set
with a Kepco BOP-1000 bipolar power supply. Both BCM and Kepco were connected to
the laboratory’s EPICS system and a computer program was used to set the voltages and
read the corresponding collector current. In order to obtain a spectrum, the voltage on the
retarding grid is increased in small steps from 0 V to an upper limit depending on the ion
energies. At each step the collector current is measured. An example spectrum is shown in
Figure 4.17.
The measurement setup of the RFA at the diﬀerent locations in the beam lines of the ion
sources used for these experiments and the interpretation of the spectra and data analysis
will be treated in detail in Chapter 5.

93

Figure 4.16: Cartoon demonstrating the working principle of a retarding ﬁeld analyzer.
Upper image: Retarding grid (’Mesh 2’) at 0 V. All ions make it to the collector. Lower
image: Retarding grid at 15 V. Ions with less energy (e.g. 10 eV) are repelled. In both cases
electrons from the outside are repelled by the electron suppressor (’Mesh 3’).

94

Figure 4.17: Example RFA spectrum. The directly obtained spectrum (solid line) and the
ﬁrst derivative -dI/dV (dashed line) are both normalized to one for better visibility. The
ﬁrst derivative of the obtained spectrum represents the ion energy distribution.

95

Chapter 5
Space-Charge Compensation
Measurements
In ion source and accelerator design, one important topic are space-charge forces. These
usually become important for higher intensity beams, but can also already be an issue if
one wants to focus even a lower intensity beam into a very small beam spot. Especially
in the framework of ECR ion sources and the corresponding low energy beam transport
lines, with the ever growing demand for higher beam currents, space charge is a factor that
has to be included in any simulation eﬀort that claims to be accurate. The compensation
process reducing the space charge through ionization of residual gas molecules and subsequent trapping of slow free electrons inside the beam envelope has already been discussed
in theory in chapter 2.1.5. However, the model presented there is rather simpliﬁed and so
far only conﬁrmed for highly compensated beams. Very little data exists for the type of
beams produced in ECR ion sources and the ECRIS LEBTs where the neutralization can
be

100 %. In this chapter, I will report on measurements conducted at the NSCL in

four diﬀerent locations in the beam lines of various ECR ion sources using a retarding ﬁeld
analyzer (RFA). Several problems arise with using such a device in the current and beam
line pressure regimes associated with ECR ion sources which will be discussed in detail as
part of the data analysis.

96

Figure 5.1: Cartoon of space-charge compensation through secondary electrons and measurement of the energy distribution of the secondary ions with the retarding ﬁeld analyzer
’NeMo’. Electrons and ions are created inside the beam by ionization and charge-exchange
of the beam with the residual gas. Electrons are captured and lower the beam space charge
potential. ions are expelled and can be measured with NeMo.

97

5.1

Preliminary Considerations

As a reminder of the processes involved in space charge compensation and neutralization
measurement, Figure 5.1 shows a cartoon picture of an ECR ion source with beam and the
retarding ﬁeld analyzer ’NeMo’ (Neutralization Monitor). Before going into the details of the
measurements, I will discuss a few general topics and issues related to RFA measurements.

5.1.1

Saturation Current

The current arriving on the collector plate with the retarding ﬁeld grid (Grid 2) grounded
is called saturation current Isat . In theory, the saturation current are all the ions produced
by the beam and emitted into the solid angle of the detector aperture. Inside the detector,
positive ions are not repelled as long as the retarding ﬁeld grid is at 0 V. If we go back to
our simple view of a round uniform beam, we can estimate Isat . The total production of
ions per unit time inside a beam volume of length

dni
2
= rb πnb vb
dt

is:

ng σi,g

(5.1)

with ni the number of secondary ions, rb the beam radius, nb the primary ion beam density,
ng the partial densities of the residual gas species and σi,g the total secondary ion production
cross sections for those residual gas species interacting with the primary ion beam. Using
2
the beam current Ib = qerb πvb nb and the solid angle of emission into the detector aperture,

we can write down the saturation current as a function of beam current and residual gas
densities:
a
ra arctan rd Ib
Isat ≈
2q

98

ng σi,g

(5.2)

and for small angles (small ra , large d):

Isat ≈

2
ra Ib T

ng σi,g
2dq

(5.3)

with q the primary ion charge state, ra the radius of the detector’s entry aperture, and d
the distance of this aperture from the beam center. In addition, a transmission factor T was
added, to account for the losses on the three grids. In our case each grid has a (theoretical)
transmission of 90%. This leads to an estimation of T ≈ 73% for all grids combined. Because
of ion optical eﬀects due to the ﬁnite mesh sizes of the grids and the fact that we apply a
negative voltage on grid 3, the transmission is not constant but actually a function of grid
3 voltage, grid 2 voltage, and ion energy and angle of incidence. This will be explained
in more detail in the next subsection (5.1.2). The situation is further complicated in the
presence of (even weak) magnetic ﬁelds. The lower the beam potential (be it due to low
beam currents or high neutralization), the lower the secondary ion energy. For secondary
ions with only a few eV energy, even very low magnetic ﬁelds (< 10 Gauss) can change the
beam trajectories enough to keep the ions from entering through the aperture system (the
two aperture collimation system can be seen in Figure 4.15). The secondary ion energies
for our ECRIS beams are expected to be below 10 eV. As we will see, this eﬀect actually
acts as a high pass ﬁlter because higher energy ions are more likely to make it through the
collimator system. This problem will be discussed in Subsection 5.1.3. Both, mesh eﬀects
and magnetic ﬁelds have to be included in the data analysis.

99

Figure 5.2: Equipotential lines close to the mesh wires of Grid 2 calculated with SIMION
8.1. Grids 1 and 3 are not shown as the distances from grid to grid ( 6mm) are much larger
than the wire spacing. Left: Grid 3 at -150 V. Right: Grid 3 at -500 V. Grid 1 and 2 at 0 V
in both cases. It can be seen that the holes between the wires have a lower potential than
the wires and that this depends on the setting of Grid 3.

5.1.2

Realistic Mesh Eﬀects

As mentioned earlier, the retarding ﬁeld analyzer designed and used for the presented measurements is a three-grid device. Each grid is made of a copper wire mesh held by an
aluminum frame. The mesh was produced by Precision Eforming LLC. (Part #: MC-17)
and the details of the mesh are listed in Table 5.1. Each mesh has a maximum transmission
of 90 % due to the fact that wires block the path of the incoming ions. The theoretical
transmission of the three meshes combined would then be 0.9 · 0.9 · 0.9 ≈ 0.73 or 73 %. However, in reality the situation is more complicated. Due to the ﬁnite size of the wires and the
openings and the fact that there are neighbouring grids at diﬀerent potentials (Grid 1 = 0 V,
Grid 3 = -150 V to -500 V), the equipotential lines close to Grid 2 are not straight lines. An
example is shown in Figure 5.2. The implications for RFA measurements have already been

100

pointed out by, e.g., Hanson et al. [106, 107] and Sakai and Katsumata [108] and are:
1. Formation of an eﬀective potential between the wires of the mesh, which is lower than
the applied voltage and depends on the distances and voltages of the neighbouring
meshes [106].
2. A lens eﬀect from the potential depressions between the wires that changes the measured ions’ trajectories and thereby the transmission and introduces an energy spread
that leads to a ﬁnite detector resolution [108].
3. Reﬂected ions. Depending on the potentials on grid 2 and 3, the ion energies, and
whether or not a bias is applied to the collector (’faraday cup’) plate, ions can be
reﬂected back from the collector after making it past the retarding ﬁeld grid.
These three problems will now be discussed in more detail and will be included in the data
analysis 5.2 and the error estimation 5.2.4.

Table 5.1: NeMo - Copper mesh parameters.
Parameter

Value

Wires per mm

2.756

(Wires per inch)
Space

70
0.34417 mm

Wire ∅

0.018542 mm

Max. Transmission
Material

90 %
Copper

101

Figure 5.3: Two measured RFA curves exhibiting a shift towards higher potential with
increasing voltage on Grid 3 due to the formation of an eﬀective potential lower than the
applied voltage in the space between mesh wires. Left curve with grid 3 at -150 V, right
curve with grid 3 at -500 V.
5.1.2.1

Eﬀective Potential

The shift due to the eﬀective potential and its dependence on the voltage applied to Grid
3 (V3 ) was observed in all the measurements with the RFA. An example spectrum with
V3 = -150 V and -500 V is shown in Figure 5.3. In their paper, Hanson et al. present a
formula to calulate the shift in dependence on the applied voltages and the mesh parameters
(wire thickness, wire distance, inter-grid distances). Here, simulations of the mesh with very
high resolution (0.0037 mm/grid unit) in SIMION 8.1 were conducted in order to compare
the semi-empirical formulae of Hanson et al. and Sakai and Katsumata with simulations and
to better understand the ion optical eﬀects inside the detector. The observed shift between
102

Figure 5.4: Eﬀects of realistic mesh for three diﬀerent types of initial particles. The rightmost
curve is for 11 eV ions, the other two for 6 eV with the one starting higher having smaller
incident angles. All three are with grid 3 on -150 V as during the measurements in the SuSI
LEBT. A bump is clearly noticeable in all three curves. This is due to the fact that for grid
2 voltages close to the particle energy, incident ions that would otherwise collide with mesh
wires are now bent around the wires. Changing the angle of incident changes the saturation
current, but not the horizontal shift and the ﬁnite width of the dropoﬀ (∆V ).
V3 = -150 V and V3 = -500 V during the RFA measurements was ∆Vexp ≈ 3.1 V. With
the mesh parameters given in Table 5.1, the shift calculated with Hansons procedure [106] is
∆Vcalc = 2.78 V. The SIMION simulations suggest an even lower value of ∆Vsim = 2.25 V.

103

5.1.2.2

Voltage Dependent Transmission

Using the same SIMION simulation as in Subsection 5.1.2.1, it is possible to obtain simulated
detector signals for arbitrary secondary ion energies and angles of incidence in the detector.
In ﬁgure 5.4 three such curves are shown for 6 and 11 eV incident particles with angles
between 0 and 7 degrees. Characteristic of these curves is that, in accordance with the
prediction of Sakai et al. the curves exhibit a bump at grid 2 voltage close to the particles’
energy [108] and a falloﬀ that is not inﬁnitely steep as one would expect from a monoenergetic
beam, but has a ﬁnite slope that depends on the particle energy and adds to the RFA’s
uncertainty. That is to say the meshes act as a high pass ﬁlter (cf. Hanson et al. [106]). The
similarities of the curves for constant grid 3 (electron suppressor) voltage are such that we
can use the simulations to obtain a set of detector resolution curves for each voltage on grid 3
(i.e. -150 V for SuSI measurements and -450 V for LEDA measurements) which are scalable
with ion energy and can thus be used in the data analysis in a least squares ﬁt model (cf.
Section 5.2.2).

104

Figure 5.5: Comparison of slow ion reﬂections as seen for the lowest secondary ion energies
in the LEDA injector source measurements and simulations under similar conditions (i.e.
similar ion energies, same voltage of -450 V on grid 3). This simulation did not include
magnetic ﬁeld eﬀects.
5.1.2.3

Reﬂections

The third eﬀect of the meshes is the partial reﬂection or even oscillation of the slow ions
between the collector and the retarding ﬁeld grid 2 around the electron suppressor grid 3.
This only happens if the collector plate is grounded and can be avoided by biasing it at
a negative voltage (e.g. -50 V). This eﬀect leads to a signiﬁcantly lower collector current
halfway between 0 V on grid 2 and the secondary ion energy. An example is shown in ﬁgure
5.5. As can be seen, the dip eﬀect is well reproduced in the simulation which was performed
modelling a high resolution mesh in SIMION and applying similar voltages.

105

+
Figure 5.6: Example trajectories of secondary H2 ions in the residual longitudinal magnetic ﬁeld of the two solenoid magnets
of the LEDA injector ion source. Trajectories were calculated with SIMION 8.1 for 5 eV ions.

106

5.1.3

Magnetic Fields

As mentioned before, the presence of a magnetic ﬁeld at the location of the neutralization
measurement can have severe consequences for the outcome of said measurement. Unfortunately, when dealing with ECR ion sources, magnetic ﬁelds can not be avoided at all
locations. Of the four diﬀerent measurement locations, two were very close to the source
itself: SuSI/Q001 and the LEDA injector ion source (cf. Subsection 5.3.1). In both cases,
there are residual ﬁelds of the solenoid magnets of the sources’ conﬁnement ﬁelds, and the
inﬂuence of these ﬁelds can be observed in the measurements. At the second SuSI measurement location further downstream in the beam line (SuSI/Q014, cf. Subsection 5.3.2) a
focusing solenoid is close enough to the measurement location to potentially create similar
problems, but for the highest solenoid current setting of 170 A, the residual ﬁeld at the
+
measurement location is only ≈ 3 Gauss and the secondary ions are mostly N2 which has a
+
much higher magnetic rigidity than the H2 of the LEDA source and thus the magnet does

not pose a problem here. In the measurement in the beam line of Artemis B, no magnetic
ﬁeld was present distorting the particle trajectories. The measured or calculated residual
magnetic ﬁelds of the four measurement locations are listed in Table 5.2. In the two cases
with magnetic ﬁeld presence, the inﬂuence of the ﬁeld may be taken into account, but naturally imposes additional errors on the results. In the LEDA case, a POISSON Superﬁsh
calculation revealed a ﬁeld of only approximately 6 Gauss on axis, but as shown in Figure
5.6 this small magnetic ﬁeld is enough to bend ions onto curved trajectories. In this case,
the residual gas was mostly H2 and thus the secondary ions have a small magnetic rigidity.
In earlier experiments with a retarding ﬁeld analyzer and the same source (e.g. [7]), this
might not have been a problem, as the device had only one entrance aperture. In our case

107

we have a two-aperture collimation system and, as is illustrated in Figure 5.6, some of the
ions cannot pass through the second aperture.
This leads to a collimator transmission depending on:
1. Secondary ion mass
2. Secondary ion energy
3. Magnetic ﬁeld strength
4. Geometry (aperture size, detector distance from beam center)
With the help of SIMION, the collimator transmission of a given geometry, for a set of
secondary ion energies can be calculated. Such a curve is shown in Figure 5.7 for the case
+
of the LEDA injector source. As can be seen, H2 ions experience strong deﬂection, whereas

the heavier H2 O+ seems rather unperturbed. Similar curves have been calculated for the
two locations in the SuSI beamline. In Section 5.2 I will discuss how the transmission curves
are used to correct the saturation currents and ﬁt the neutralization curves.

Table 5.2: Residual magnetic ﬁeld Bz (z) on axis at the diﬀerent neutralization measurement
locations.
Location

Bz (z)

Method

LEDA Box 1

≈ 6 Gauss

Superﬁsh calculation

Artemis A

≈ 0 Gauss

No Magnet

SuSI Q001

≈ 60 Gauss

Measured

SuSI Q014

≈ 3 Gauss

Superﬁsh calculation

108

Figure 5.7: Collimator transmission curves for the case of the LEDA injector ion source. The
+
two secondary ion species H2 and H2 O+ are the two biggest contributors to the residual
gas pressure (as measured with a residual gas analyzer (RGA) and have a ratio of ≈ 19:1.

109

5.2

Data Analysis

In order to be consistent for the diﬀerent measurement locations and pressure and current
variations, the same procedure has been applied to all the data:
1. The saturation current for each series was plotted and checked against the theoretical
prediction with equation 5.3 with a transmission factor T modiﬁed for mesh eﬀects
and magnetic ﬁelds (cf. Subsection 5.2.1).
2. Neutralization spectra were ﬁtted with a least squares method including the various
transmission factors and mesh eﬀects (cf. Section 5.2.2).
3. The errors imposed on the results by the various approximations and uncertainties in
the data acquisition were estimated and added as errorbars to the results (cf. Section
5.2.4.

5.2.1

Calculating Saturation Currents

As discussed in Subsection 5.1.1, the saturation current Isat on the collector plate while grid
2 is at 0 V, should to ﬁrst order be directly proportional to the number of secondary ions
created by the interaction of the primary beam with the residual gas:

Isat ≈

2
ra Ib T

ng σi,g
2dq

(5.4)

It should grow linearly with the residual gas pressure and the primary beam current. Consequently the saturation current is a good ﬁrst check whether or not the measurements make
sense. Unfortunately, as discussed in Subsection 5.1.2, the transmission through the meshes

110

of the three grids, and as discussed in subsection 5.1.3, the transmission through the twoaperture system in the presence of a magnetic ﬁeld can greatly depend on the ion energy and
the setting of the negatively biased electron suppression grid and thus the saturation currents
do not necessarily exhibit the expected linear behaviour. However, as mentioned in Sections
5.1.2 and 5.1.3, it is possible to calculate transmission curves through the collimation system
and the meshes with grid 2 at 0 V. Thus the transmission factor T in equation 5.3 is no
longer constant, but instead a function of magnetic ﬁeld, secondary ion energy, mass, and
grid 3 voltage (m, B and V3 voltage are discrete parameters rather than free variables).

T = T (E, m, B, V3 ) = Tcoll (E, m, B) · Tmesh (E, V3 )

(5.5)

Here we have identiﬁed the two main contributions to T as the collimator transmission
depending on energy, mass and magnetic ﬁeld and the mesh transmission depending on
energy and grid 3 voltage.
In order to calculate Tcoll we ﬁrst have to determine the presence and strength of a magnetic
ﬁeld at the location of measurement. This was done either by directly measuring with a Gauss
meter or by calculating the magnetic ﬁeld with POISSON Superﬁsh. Secondly, a transmission
curve has to be calculated for each secondary ion species (this can be generalized into the
+
two categories H2 and heavier ions). The free parameter in these transmission curves is

the secondary ion energy. The transmissions are determined by starting a large number of
particles from the center of the primary ion beam and plotting the ratio of secondary ions
making it through the ﬁrst aperture to those reaching the collector plate.
The second part is to calculate Tmesh . This is done using SIMION as well by modeling the
meshes realistically in very high resolution and sending ions of diﬀerent initial energies and

111

angles through the meshes. This accounts for the three mesh eﬀects discussed in Section
5.1.2.
Examples of Tcoll and Tmesh for the LEDA injector source are shown in Figure 5.15. There
H2 is the light ion and H2 O is representative for all heavier ions.
From the RFA curves obtained with NeMo, we can now determine an estimate of the secondary ion energies by shifting the spectrum using the empirical formula for the eﬀective
potential presented in [106] and discussed in Section 5.1.2. Then we can use T = Tcoll ·Tmesh
in formula 5.3 to calculate the theoretical Isat and compare them to the measured ones. Examples are shown in Section 5 (e.g. Figure 5.16).

5.2.2

Least Squares Fitting

In general, all NeMo spectra were ﬁtted with a least squares method using python scripts.
The lmﬁt library provided by the University of Chicago [109] was used, which is a wrapper
around pythons scipy.minimize method, which in turn uses MINPACK. The least squares
method requires an objective function to minimize. In our case this is

fobj =

Icol,theo (V2 ) − Icol,meas (V2 )
σI (V2 )

(5.6)

with Icol,theo the theoretical collector current, Icol,meas the measured collector current and
σI the measured standard deviation of the collector current, all functions of the voltage V2 on
grid 2 (retarding ﬁeld). Note: The minimizer automatically squares the objective function.
Before the ﬁtting, a calibration curve is applied to the voltages of grid 2 (V2 ) corresponding
to the data points Icol,meas . This measured calibration curve shows that the KEPCO power
supply has an oﬀset of ≈ -0.75 V.
112

The theoretical collector current Icol,theo (V2 ) is calculated in a two step process:
1. Generation of the theoretical secondary ion energy distribution f(E)
2. Folding of this distribution with the transmission functions Tcoll and Tmesh .

5.2.2.1

Secondary Ion Distribution Function

We have two options for generating the secondary ion distribution function f(E).
The ﬁrst method (and the one used for the LEDA injection source) is to assume an axially
symmetric beam distribution (e.g. uniform or Gaussian) and calculate the theoretical beam
potential Φ(r). We can then generate an ensemble of ions with density according to the
distribution and beam radius rb (or σb ), and determine their kinetic energy from their place
of birth inside the beam envelope. For the case of the uniform cylindrical beam the energy
distribution can also be calculated and is simply:


 0




f (E) =
 const.




 0

for E < e · Φedge
for e · Φedge ≤ E ≤ e · Φcent

(5.7)

for E > e · Φcent

Generating particle ensembles is more versatile, though, and multiple beamlets with diﬀerent
radii and diﬀerent ion species can be used.
The second method is to measure the ion beam distribution at the location of the RFA
measurement (e.g. with a slit scanner or with a beam viewer) and use this measurement to
determine the energy distribution. This was done for the measurement in the SuSI beam
line at diagnostic box Q014. For each set of measurements (except one, because the image
acquisition was unavailable) a slit scan and a beam viewer image were obtained. The slit

113

scan using the 4-jaw slits gives a well resolved ion distribution, but is located ≈ 20 cm before
NeMo (Q013). There (Q014), a beam viewer with a quartz (and later a KBr) plate and
a CCD camera are available. The resolution of the viewing plates is not good enough to
determine the intensity distribution, but by comparing slit scans and beam imaging at the
NeMo location, it is possible to determine the rotation of the beam due to the solenoid, and
the change in beam size. The process of obtaining f(E) is then as follows:
• The slit scan and beam viewer image are compared and rotation and scaling of the
beam from Q013 to Q014 are determined.
• The density distribution from the slit scan is scaled and rotated accordingly and saved
into a ﬁle.
• This ﬁle is now loaded into a SIMION simulation of diagnostic box Q013/Q014 to
provide the charge density distribution of the primary beam.
• SIMION’s poisson solver is used to calculate the beam potential for an unneutralized
beam.
• Particles are initialized with a small random thermal energy (0.2 eV) at positions
according to the beam density proﬁle and experience the space charge ﬁeld of the
beam.
• Particles entering the aperture of NeMo are registered and their energy is saved to a
ﬁle, yielding f(E).
In both cases, the secondary ion energy distribution

f (E) = f (E, vb , rb , Ib , fe , rp )
114

Figure 5.8: Mesh transmission curves for -450 V on grid 3. The ion energy increases from
left to right. A drop in transmission as well as the formation of a peak can be seen towards
the lowest energies. At higher energies ( > 5 eV) the transmission curve only shifts further
to the right without changing its shape.
is a function of E, but depends also on the following parameters: beam velocity vb , beam
radius rb , beam current Ib , neutralization factor fe , and the pipe radius rp . For rp it should
be noted that in the case where f(E) is calculated with SIMION, the actual shape of the box
is taken into account, and in the analytical approach an eﬀective wall radius is approximated.

5.2.2.2

Detector Transmission Function

In order to include the mesh eﬀects and transmission into the data analysis, the high resolution SIMION model of the three grid system was used to to generate mesh transmission
curves for diﬀerent monoenergetic ions and linear interpolation was used to obtain the cor115

Figure 5.9: Mesh transmission curves for -150 V on grid 3. The ion energy increases from
left to right. A drop in transmission as well as the formation of a peak can be seen towards
the lowest energies. At higher energies ( > 5 eV) the transmission curve only shifts further
to the right without changing its shape.
responding curves for intermediate energies. Selected curves for the V3 = −450 V case
(LEDA source) are shown in Figure 5.8 and for the V3 = −150 V case (SuSI/Q014) in Figure 5.9. With f (E), Tcoll , and Tmesh , the theoretical collector current Icol,theo (V2 ) used in
the objective function 5.6 can be expressed as:
Emax

Icol,theo (V2 ) = A ·

f (Ei ) · Tcoll (Ei ) · Tmesh (Ei , V2 )

(5.8)

Ei =Emin

assuming a discrete set of energies, as is usually the case in numerical treatments. A is a
scaling factor that contains the normalization and Isat .
116

Figure 5.10: Example of a high energy tail. This spectrum was taken with a 180 µA Ar8+
beam at SuSI/Q014. The long tail can clearly be seen. The tail contributes approximately
25 % to the saturation current. The ﬁrst derivative corresponds to the energy distribution
of the secondary ions. Both, signal and derivative are normalized to a maximum of one for
better visibility.

5.2.3

High Energy Tail

Many of the RFA spectra exhibit features that diverge from the simple linear drop one would
expect from the energy distribution of a uniform round beam. Some of these features can
be explained by the transmission through the three grids of the RFA or reﬂection from the
collector plate (a dip in the low energy part, a bump before the steep dropoﬀ of the actual
ion distribution). These have been discussed in Section 5.1.2. In addition, non-uniform
distributions (e.g. Gaussian) or beam halo can lead to a signiﬁcant rounding of the lowenergy edge of the spectrum. However, the high energy edge of the spectrum should, in

117

theory, be relatively sharp. But most of the spectra taken with the RFA at the LEDA
source show a rounding also of the high energy side, and in all of the spectra taken with the
RFA in the SuSI beam line, a tail towards higher energies is observed. An example of such
a tail is shown in Figure 5.10 for the case of a 180 µA Ar8+ beam at SuSI/Q014. A long
tail as well as a pronounced bump can be seen, which, in the ﬁrst derivative, looks like an
additional separate ion population peaking at ≈ 12 V and linear falloﬀ to 0 between 35 and
40 V. Other authors have observed similar behaviour of their retarding ﬁeld analyzer spectra
(albeit not with such a pronounced tail) and have, depending on the respective situation,
found diﬀerent explanations:
1. Dissociative ionzation. A diatomic molecule can be excited by ion- or electron impact
and subsequently dissociate into ions with energies peaking around 8-10 eV [7, 110].
2. Beam halo. In [111] Sherman et al., found that depending on the aspect ratio of the
beam they sometimes saw mostly the beam halo consisting of H+ and possibly H+ , and
2
3
observed the actual beam contribution as the high energy tail. However, the energy
range of this tail was never as wide as the tail observed in this work.
3. Residual gas temperature. The gas molecules have a Boltzmann-distributed energy
with maximum around 0.2 eV. This is not much, but can contribute a little to the
rounding oﬀ of the edges.
4. Rapid oscillation of the beam between two space charge compensation levels due to
collective oscillations in the beam plasma [7, 48].
5. Ion beam oscillations. Recently, Toivanen et al. have published a paper on their
measurements of fast beam current oscillations in the ECR LEBT of the University
118

of Jyv¨skyl¨ [112]. These oscillations are too fast to be seen with a standard faraday
a
a
cup setup and might oﬀer an explanation for the long tail in the RFA spectra in the
SuSI diagnostic box Q013/Q014. The observed oscillations are in the hundreds of
Hertz to kHz regime. They depend on the ion source parameters like strength of the
solenoid ﬁelds and microwave power. 2σ oscillation amplitudes up to 80% relative to
the observed average beam current have been seen, although 10 - 20 % seem more
typical. If the beam intensity suddenly and for a short time increases, so does the
beam potential, and a few ions with higher energy might enter the detector (before the
beam goes back to steady state), leading to a tail.
6. Insulating layers building up on the grid surfaces. Sherman et al. [113] observed that it
was necessary to heat their RFA to ≈ 350◦ in order to prevent the buildup of material
on the grids and obtain consistent resolution and calibration. They cite a study by
Poole et al. [114] in which electron impingement on deposited layers of organic material
creates radicals and subsequently polymerizes them, thereby forming high resistance
ﬁlms. The main source for the initially deposited material was claimed to be vacuum
grease used in combination with rubber O-rings. In the SuSI set-ups presented here
only copper gaskets were used for sealing of vacuum components and thus heating was
considered, but deemed not feasible. However, with the appearance of the high energy
tails, heating might be considered as part of a future upgrade of the detector system.
7. Charge-exchange of two electrons. According to the M¨ller-Salzborn empirical scaling
u
law, the cross-sections for the exchange of two electrons is not negligible. If secondary
ions are in the q = 2+ state and pick up an electron on the way to the detector, their
energy could be up to twice eΦcenter .

119

5.2.4

Error Estimation

Because this measurement can not be considered a precision measurement, and because
of the rather large errors associated with the cross sections in the theoretical prediction I
will not attempt a detailed error analysis, but instead estimate in a broad way the largest
contributions to the errors associated with the presented measurements, thereby providing
only constraints to the maximum errors.

5.2.4.1

Statistical Errors in the Measurements

In the LEDA data sets, the collector current for each voltage step was calculated as the
average of a sample of 200 data points. The σ of each data point was used as individual
error in the least squares ﬁt. During the SuSI measurements, the signal on the collector
plate was monitored continuously and the σ of the data point with the lowest current in
each measurement was calculated from ≈ 100 samples and used as error for all data points
in the set (the largest error is associated with the lowest currents measured by the beam
current monitor). The absolute statistical errors on the grid voltage (V2 ) of ±0.05 V are
considered negligible (systematic errors of the grid voltage are corrected with a calibration
curve). The relative errors ∆fe of the neutralization factor resulting from including statistical
errors of the collector current are between 1% and 4%.

5.2.4.2

Systematic Errors in the Measurements

The main sources of systematic errors in the measurements are the uncertainty of the total
beam current, the beam line pressure, and the contribution of the high energy tail.
LEDA injector source
In the LEDA case, the beam current can conservatively be said to be known to ±0.75 mA
120

(this is the diﬀerence of drain current, and the higher Bergoz 1 reading, cf. Subsection 5.3.1).
This introduces a relative error from ∆fe of -1.4% to + 1.0% for the highest current (where
∆Ibeam = ±8%) to -13% to + 6% for the lowest beam current (where ∆Ibeam = ±26%).
The high energy tail can make up 5 - 20% of the saturation currents measured in the LEDA
injector source beam line. Fitting the tail with an underlying Gaussian energy distribution
gave good ﬁt results and only changed the neutralization factor by +1 to +2%. As a
conservative approximation, I will assume a systematic error of ±5% due to the tail. The
errors in fe due to tail and beam current are added in quadrature to the statistical error for
the errorbars in the plots.
SuSI/Q014
In the SuSI beam line at Q014, a faraday cup was used to measure the beam current ≈ 20
cm upstream of NeMo. In combination with the beam current monitors (BCM) produced
in-house at the NSCL, a conservative estimate of the accuracy of the beam current is ±10%.
The ion gauge used to measure the pressure of the residual gas was calibrated for N2 (which
is also the gas used in the experiments to raise the pressure in the beam line). Both, the
ion gauge and the leak valve for raising the pressure are located in the same diagnostic
box as NeMo. I am assuming an accuracy of the pressure values of better than ±10% as
well. The tail can account for up to 35% of the saturation current. Fitting with diﬀerent
background functions revealed an absolute error in neutralization factor fe of 5% to 10%.
As a conservative approximation, I will assume an absolute systematic error of ±10% due to
the tail. Compared to this error, the statistical error and the error due to the beam current
uncertainty are small. Added in quadrature, the total error is estimated to be ±11%. This
will be used as vertical error bars in all SuSI plots.

121

5.2.4.3

Systematic Errors in the Theoretical Predictions

Uncertainties in the theoretical calculations will be indicated by a shaded area in the plots.
LEDA saturation current. During the LEDA measurements, an ion gauge and a residual
gas analyzer were available. Both were uncalibrated and showed a large discrepancy. Using
the average of both readings as value for the pressure, and being very conservative, this
means the relative error on the pressure reading was ±75%! The ion creation cross sections
σi,g are only known to ±30% as well. As mentioned before, the ion beam currents are known
to ±0.75 mA (±8% to ±26% error in this current regime). Assuming that the transmission
factors calculated with SIMION are correct to ±10%, this leads to a total systematic error
for the theoretical saturation current of ±82 to ±85%.
LEDA neutralization. M¨ller and Salzborn claim that 2/3 of the measured charge exu
change cross sections are within ±35% of their empirical formula [1]. Ionization cross sections
are available for protons in the energy range of the LEDA source, and are cited with similar uncertainties. In the following I will assume that the cross sections for ionization and
charge exchange are known with ±35% accuracy for the LEDA source. Uncertainties in
beam current (±8% to ±26%) and beam line pressure (±75%) have to be considered as well.
SuSI saturation current and neutralization. Unfortunately, the typical energies of
ECRIS beams are below the validity of the Kaganovich ﬁt for ion-atom impact ionization [59].
However, at lower energies, charge exchange processes are dominating over ionization and
we can use the M¨ller-Salzborn ﬁt for σi . I assume that additional processes like single
u
ionization, dissociative ionization, etc. will not increase the cross section by more than 50%.
σe is approximated as a quarter of σi .

122

5.3

Measurements

Space charge compensation measurements have been attempted at four diﬀerent locations:
1. LEDA injector source, diagnostic box 1.
2. Artemis A, vertical beam line before the ﬁrst dipole.
3. SuSI, ﬁrst diagnostic box (Q001).
4. SuSI, diagnostic box after the dipole (Q013/Q014).
Detailed analysis has so far been carried out for the two locations LEDA injector source and
SuSI Q014 and will be presented in the following sections. Unfortunately, the beam current
measurements in the Artemis beam line are not reliable due to a faulty faraday cup. Since
knowledge of the total beam current is integral in the analysis of the RFA data, the Artemis
measurement will not be reported here. The analysis of the data in the SuSI diagnostic
box Q001 is more complex than LEDA and SuSI/Q014 because of the multitude of present
species in the ion beam (Q001 is before the ﬁrst analyzing magnet), which have diﬀerent
charge states and beam radii. In addition, the residual solenoid ﬁeld (≈ 60 Gauss) of SuSI
at Q001 further complicates the analysis. No results from this location will be reported in
this thesis, pending future thorough analysis of the data.

5.3.1

LEDA injector ion source

5.3.1.1

Overview

When the LEDA ion source was moved to the NSCL in 2012, an opportunity presented itself
to try and measure neutralization of high intensity proton beams and compare the results

123

Figure 5.11: Foto of the LEDA injector source at NSCL. The source is located on the left,
followed by diagnostic box 1 (the large box in the middle of the image). NeMo was mounted
where a transparent ﬂange can be seen in the center of diagnostic box 1. Courtesy of Felix
Marti.
to previous measurements on the same source, albeit under diﬀerent operating conditions.
Previous measurements were reported in 1998 by Ferdinand et al. [7] with a four-grid energy
analyzer developed in 1988 by Sherman et al. for H- beams [111, 113, 115]. The LEDA
injector ion source itself is described brieﬂy in section 4.1.1. Beside the design diﬀerences in
the retarding ﬁeld analyzer (three grids instead of four, not heated, two-aperture collimation
instead of one entry aperture), one major diﬀerence to the previous measurements is the
current regime. The measurements presented by Ferdinand et al. were obtained with beam
currents from 50 mA to 130 mA, whereas the results presented here are for much lower
currents from 5 mA to 10 mA; a result of the drastically reduced diameter of the extraction

124

Figure 5.12: 3D CAD drawing of diagnostic box 1 and the NeMo retarding ﬁeld analyzer in
the LEDA injector source beamline. The source itself is to the left and not shown.
aperture. This was done in order to decrease the beam emittance for another experiment.
The NeMo detector was mounted on the side of diagnostic box 1 which is shown in the
photograph in Figure 5.11 and as a CAD drawing in ﬁgure 5.12.

5.3.1.2

Data Acquisition

In contrast to the other measurements, and due to the location of the LEDA injector source
in a remote area of the laboratory, the data acquisition had to be a stand-alone system.
This was realized with LabView 2010 on a laptop computer communicating with a Keithley

125

Figure 5.13: NeMo - LEDA injector source: Beam currents versus microwave power. Bergoz
1 and the drain current are representative of the total beam current at the location of the
neutralization measurements. It can be seen that the extracted current behaves linearly
with microwave power up to 600 W and then ﬂattens out. This could be because not enough
gas is supplied to sustain the higher currents or because the source is outgassing, thereby
contaminating the plasma.
6485 Picoammeter for the collector current readout and a KEPCO BOP-500M bipolar power
supply for the retarding ﬁeld voltage via USB-to-serial port converters. The voltage on grid 2
(the retardign ﬁeld) provided by the KEPCO BOP-500M was changed manually. Each time
a data point was taken, the LabView program would read out the power supply’s voltage
and average over 200 picoammeter readings.

126

5.3.1.3

Diagnostics

The total beam current was measured at four diﬀerent locations in the beam line:
1. As drain current on the ion source high voltage power supply after subtracting the dark
current of ≈ 0.1 mA which was recorded each day at the beginning of the measurements
and frequently re-checked.
2. Shortly after the extraction system by means of a Bergoz DC current transformer
(Bergoz 1).
3. Shortly after the solenoid at the exit of diagnostic box 1 by means of another Bergoz
DC current transformer (Bergoz 2)
4. At the end of the short beamline on a copper beam stop.
Both Bergoz current transformers showed large ﬂuctuations during the measurements even
when drain current and beam stop current, as well as visual imaging of the beam would
suggest good beam stability. The most accurate account of the actual beam current in
diagnostic box 1 was thus given by the power supply drain current. A measurement of
beam current versus the microwave power is shown in Figure 5.13. For demonstration of
the high uncertainty of the current transformers, the errorbars on Bergoz 1 are shown. The
errobars on the drain current are smaller than the markers. It can be seen that the extracted
current behaves linearly with microwave power up to 600 W and then ﬂattens out. This
could be because not enough gas is supplied to sustain the higher currents or, more likely,
because the source is outgassing, thereby contaminating the plasma. Before and during the
measurements, the beam line had to be opened several times, leaving little time to condition
the source properly.
127

Figure 5.14: A typical residual gas analyzer spectrum during a neutralization measurement. The data suggests a ratio of roughly
19:1 of molecular hydrogen (H2 ) to heavier species. Note the logarithmic scale for the pressure.

128

The beam line pressure was measured with a hot cathode ion gauge close to the end of the
beam line and in parallel monitored with a residual gas analyzer (RGA). Both devices were
not calibrated and thus, unfortunately, can give only a ballpark idea of the correct pressure
in the beamline. However, the RGA can at least give us an idea of the composition of
the residual gas. A typical histogram plot of the RGA is shown in Figure 5.14. It can be
seen that the main contribution to the residual gas is hydrogen, which was to be expected
because hydrogen is the source gas and leaks into the beamline. The RGA data suggests a
total pressure of ≈ 5 · 10−7 Torr of H2 versus ≈ 2.6 · 10−8 Torr for the rest of the gas species
(mostly H2 O, N2 , N and O) which is a ratio of 19:1. The corresponding ion gauge readings
are in the regime of 4.25 · 10−6 Torr - almost an order of magnitude higher. This, of course
imposes a very large error on the entire measurement.

5.3.1.4

Measurements

Due to time constraints and diﬃculties setting up the data acquisition, only one dataset
was ﬁnally obtained with the LEDA injector source: A measurement where the total beam

Table 5.3: Parameters for the neutralization measurement in the LEDA injector ion source
low energy beam transport line.
Parameter

Value

Beam current

2.9 - 9.4 mA

Primary beam

protons

Total RGA pressure

5.26 · 10−7 Torr

Ion gauge pressure

4.25 · 10−6 Torr

Residual gas

95% hydrogen

NeMo grid 3

-450 V

129

Figure 5.15: Transmission curves for H2 and H2 O as well as a mix of the two with ratio
19:1 as observed in the RGA spectra. It is evident that for heavier ions the magentic ﬁeld
does not have a very big eﬀect, thus H2 O is representative of all ion species heavier than
14 N . It can also be seen that the mesh transmission is around 73% (as expected) down to
ion energies of about 2.5 eV where it rapidly drops towards lower energies due to reﬂection.
current was varied while the pressure was kept constant. The parameters of the series are
listed in Table 5.3.
As suggested in Section 5.2, we begin our analysis by plotting the saturation currents and
comparing them with the theoretical prediction. During this measurement a residual magnetic ﬁeld of ≈ 6 Gauss was present along the beam axis at the location of the measurement.
The residual gas molecules according to the RGA data were mostly H2 . As discussed earlier,
130

this warrants the use of a collimator transmission curve in order to correct the saturation
currents for losses in the collimation system and a grid transmission curve to account for
losses through reﬂections and focus eﬀects (the voltage on grid 3 was -450 V, hence the
reﬂection of ions is more pronounced in these measurements). The transmission curves as
calculated with SIMION are shown in ﬁgure 5.15. For lack of a better approximation, the
pressure was assumed to be halfway between the total RGA pressure and the ion gauge pressure reading, with a ratio of residual gases as given by the RGA. The necessary cross-sections
for ionization were calculated with the empirical formula provided by Kaganovich et al. [59],
the contribution from charge exchange was estimated by scaling a measured cross-section
curve at higher energies with the low-energy values obtained by using the M¨ller-Salzborn
u
scaling law [1]. The cross-sections are listed in Table 5.4.

131

Figure 5.16: Corrected saturation currents. Note that the three data points at the highest
beam currents actually have lower secondary ion energies, either due to a larger beam radius
or higher space charge compensation. As discussed in Section 5.2.4 the systematic error bars
on the theoretical prediction are ≈ 82 − 85%. This is very large and makes an absolute
comparison mute. However, the relative change of saturation current with beam current can
be examined and reproduced by applying transmission factors (cf. text).

Table 5.4: Cross-sections for creation of slow secondary ions calculated for 70 keV proton
beams in diﬀerent residual gases. Note: Only includes single ionization and charge-exchange.
Gas molecule

σi,g (m2 )

H2

2.98 · 10−20

N2

6.92 · 10−20

H2 O

1.28 · 10−19

Ar

7.29 · 10−20

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Figure 5.17: NeMo neutralization Spectra for 5 diﬀerent beam currents in the LEDA injector
ion source beam line diagnostic box 1. The voltages have not yet been corrected for the
eﬀective potential inside the meshes. The curves increase in current from left to right.
The corrected theoretical saturation currents for the given beam currents and pressures
alongside the uncorrected predictions and the measured currents are shown in Figure 5.16.
It can be seen that the corrected calculated saturation currents are aligning quite well with
the measured ones.
The next step in order to obtain the neutralization values is to analyze the NeMo spectra.
The ﬁrst 5 spectra of the current variation series (i.e. beam currents of 2.9 mA to 6.0 mA)
are plotted in ﬁgure 5.17. All of them have been normalized so they are easier to compare.
For the lowest beam current (2.9 mA) we observe a pronounced dip in the collector current
due to the reﬂection of ions discussed earlier (cf. 5.1.2). It is also seen for the next two

133

Figure 5.18: Results of the current variation with the LEDA injector ion source: Neutralization and Beam sizes. It can clearly be seen that although the overall beam radii are much too
large, they reﬂect the observed variations in neutralization. Note that changing the beam
radius in the analysis only changes the center potential of the beam (shifts the spectrum
left-right) and not the slope of the curve (i.e. the neutralization).
currents, albeit not as strong because the overall beam potential increases with current
and the secondary ions have slightly higher energies. Following the procedure described in
Section 5.2.2, we calculate the mesh transmission curves for the given setup (cf. ﬁgure 5.8)
and use the least squares ﬁtting method to calculate neutralization and beam radius for the
measured curves.
It should be mentioned that the obtained neutralization fe is independent of beam and beam
pipe radii (as we saw in chapter 2.1.5), the overall potential, however, is not. The obtained
value for the beam radius is highly dependent on the ion density distribution of the beam
and the sorrounding walls (in this case of diagnostic box 1). We have assumed a uniform
134

Figure 5.19: Measured neutralization as a function of ion beam density. The solid line is the
theoretical prediction following the model described in 2.1.5, the shaded area includes the
uncertainties in pressure and cross sections.
distribution, which is a good approximation for determining the neutralization, but may not
be ideal to give realistic beam radii just from neutralization spectra. In addition, SIMION
simulations revealed that the potential calculated from a round pipe corresponding to the
size of the square box is actually much too high and a smaller eﬀective beam pipe radius
rp of 175 mm should be used (as opposed to the 280 mm half-width) this is because the
half-length of the box is only 190 mm and beam pipes with diameters of only ≈ 100mm
lead away on both sides. That being said, the obtained beam radii are consistently an order
of magnitude too large (they are shown in Figure 5.18 alongside the neutralization results).
This could have several reasons:
135

• The eﬀective wall radius of 175 mm is still overestimated. Other objects in diagnostic
box 1 (e.g. cooling water pipes, cables) could further change the beam potential oﬀset.
• The Mesh transmission curves are overestimating the shift towards higher eﬀective
potentials.
• The beam distribution is not uniform.
Another possibility, though unlikely, should be pointed out: In case of a misalignment of the
detector and oﬀcentered beam, it is possible that mostly ions created in the beam halo enter
the detector and only at the highest beam currents, a contribution from the actual beam is
seen as tail in the spectrum (similar to item 2 in Subsection 5.2.3). The ions making up this
tail would have a higher energy, corresponding to a tighter beam. However, comparison with
previous work by Ferdinand et al. [7] shows that beam radii calculated from their presented
RFA data are also much too large (although this is not discussed in the paper).
Regardless of the overall mismatch of the beam radii which were expected to be between 10
and 20 mm, the obtained values can be used to calculate a beam density by scaling them all
with a factor 10 in order to compare the results with formula 2.50 derived in section 2.1.5.
The result can be seen in ﬁgure 5.19. The same cross sections for ion and electron production
as in the saturation current calculations were used as well as the same pressure of 2.4 · 10−6
Torr lying halfway between the ion gauge reading and the RGA total pressure. It can be
seen that the results match well to the theoretical prediction and that the neutralization
increases with beam density, as was expected.

136

Figure 5.20: Schematic of the SuSI LEBT system up to the second dipole magent. Between
the ﬁrst dipole and the location of NeMo (in the diagnostic box Q013/14) there is a focusing
and collimation channel consisting of 3 solenoids, apertures, and slits.

5.3.2

SuSI Superconducting ECRIS

5.3.2.1

Overview

The second location where space charge compensation measurements were performed is the
diagnostic box Q013/14 in the LEBT of the superconducting ECRIS SuSI. The location of
the diagnostic box in the SuSI LEBT system is shown in Figure 5.20. The beam optics
elements in the LEBT system between extraction and measurement location in order of
distance from the source are:
• Q002EL - Einzel lens directly after the extraction system
• Q003DH/DV - Horizontal and vertical steering magnets.
• Q005DS - Dipole magnet for selection of desired ion species by A/q.
137

Figure 5.21: 3D CAD schematic of diagnostic box Q014 in the SuSI LEBT.
• Q006DH/DV - Horizontal and vertical steering magnets.
• Q008SN - First collimation channel solenoid.
• Q010SN - Second collimation channel solenoid.
• Q012SN - Third collimation channel solenoid.
In between, there are various apertures and 4-jaw type slits (cf. section 4.2.4). One important
fact to notice is that this measurement was carried out after the analyzing magnet (Q005DS).
Hence the beam was composed mostly of one ion species and a small fraction of the charge
state below (q − 1) created by charge exchange processes.

138

5.3.2.2

Data Acquisition

The collector current of NeMo was measured with a beam current monitor (BCM) built
in-house at the NSCL. BCM and collector plate were connected by a double-shielded BNC
cable to decrease noise. Communication with the BCM was established via the laboratory’s
EPICS system. A simple x/y plotter program (ECR-tune) was used to set the voltage on
grid 2 (V2 ) with a remotely (EPICS) controlled KEPCO BOP-1000 power supply and read
the BCM current after a user deﬁned delay.

5.3.2.3

Diagnostics

The diagnostic box Q013/14 with NeMo and other diagnostic devices is shown in Figure
5.21. The box contains the following:
• Faraday cup (Q013).
• 4-jaw type slit (Q013).
• Hot cathode ion gauge (Q013).
• Leak valve connected to a N2 feed line (Q013).
• Viewing screen (exchangeable material: SiO2 or KBr coated metal screen) (Q014).
• NeMo retarding ﬁeld analyzer (Q014).

5.3.2.4

Measurements

The following six cases were measured at Q014 and will be discussed in detail:
1. O3+ Pressure 1.1 · 10−5 to 1.1 · 10−6 Torr, current approximately constant at 235 eµA.

139

2. O3+ Current 125 eµA to 298 eµA, pressure approximately constant at 5.0 · 10−6 Torr.
3. O6+ Pressure 1.0 · 10−5 to 1.0 · 10−6 Torr, current approximately constant at 700 eµA.
4. O6+ Current 194 eµA to 604 eµA, pressure approximately constant at 5.0 · 10−6 Torr.
5. Ar8+ Pressure 1.0 · 10−5 to 1.0 · 10−6 Torr, current approximately constant at 200 eµA.
6. Ar8+ Current 110 eµA to 331 eµA, pressure approximately constant at 5.0 · 10−6 Torr.
Pressure Series
Let us begin the examination by looking at the pressure variations. For each set of measurements (except for the O6+ pressure series number 3), a corresponding set of slit scans has
been made in order to approximate beam size and shape at the location of the RFA. For
the O3+ and for the Ar8+ series, these are shown in Figures 5.22 to 5.25 and ﬁgures 5.26
to 5.30, respectively. All slit scans were obtained using a 2 mm by 2 mm opening and the
images were interpolated afterward, using a bicubic algorithm. Unfortunately the MatLab
library for image acquisition was unavailable during the O6+ series. For O3+ we can observe
a triangular structure which is not uncommon in ECR ion sources. The beam seems to get a
bit more focussed with higher pressures. The Ar8+ appears to be better tuned for roundness.
The slight fading of the beam with higher pressure stems from a slight decrease in beam
current from 701 µA to 686 µA. The beam diameter increases slightly from lower to higher
pressure, hinting at a change in focussing of the beam, which in turn hints at a change in
space charge potential (i.e. higher neutralization).

140

Figure 5.22: Beam proﬁle of O3+ , 1.1 · 10−6 Torr

Figure 5.23: Beam proﬁle of O3+ , 5.0 · 10−6 Torr
141

Figure 5.24: Beam proﬁle of O3+ , 8.0 · 10−6 Torr

Figure 5.25: Beam proﬁle of O3+ , 1.1 · 10−5 Torr
142

Figure 5.26: Beam proﬁle of Ar8+ , 1.0 · 10−6 Torr

Figure 5.27: Beam proﬁle of Ar8+ , 2.5 · 10−6 Torr
143

Figure 5.28: Beam proﬁle of Ar8+ , 5.0 · 10−6 Torr

Figure 5.29: Beam proﬁle of Ar8+ , 7.5 · 10−6 Torr
144

Figure 5.30: Beam proﬁle of Ar8+ , 1.0 · 10−5 Torr
As with the LEDA measurement, the ﬁrst analysis step is to examine the saturation currents
Isat , which are plotted in ﬁgure 5.31. The last solenoid of the collimation channel which
is close to the diagnostic box, was at a low setting < 100A, leading to a residual magnetic
ﬁeld on axis at the measurement location of < 2 Gauss. This combined with the fact that
the background gas is mostly N2 , which is much heavier than H2 lets us expect a linear
dependence of Isat on the beam line pressure, which we indeed observe. The theoretical
values were calculated according to equation 5.3 with a transmission factor T = 0.65 and
cross-sections according to the M¨ller-Salzborn empirical formula for charge-exchange [1]. At
u
these energies, charge-exchange is the dominating contribution to σi . The mesh transmission
T depends slightly on the ion energy and on the angle of incidence, hence we expect it
to be less than the optimal value of 0.73 discussed earlier. Overall, considering the large

145

Figure 5.31: Saturation currents for three pressure variations in the SuSI LEBT at Q014.
The theoretical predictions are according to equation 5.3 with T = 0.65 and cross-section
according to the M¨ller-Salzborn empirical formula [1]. The transmission T depends slightly
u
on the ion energy and on the angle of incidence.
uncertainties on the cross-sections given by M¨ller and Salzborn, prediction and experiment
u
are in good agreement.

Fitting the RFA spectra with the least squares method described

in section 5.2.2 yields the neutralization values plotted in ﬁgures 5.32 and 5.33. It should be
mentioned that in some cases the contribution from the background in form of a Gaussian
energy distribution is considerable (up to 45 %). Because the reason for the high energy
tail (approximated by the Gaussian energy distribution) is unclear at the moment, this
consequently has to add signiﬁcantly to the estimated errors (cf. Section 5.2.4).

146

Figure 5.32: Pressure variation of O3+ , neutralization measured and theory. It can be seen
that the theoretical prediction is signiﬁcantly lower than the measured values.
In Figure 5.32 we can see that the measured values are signiﬁcantly higher than the theoretical prediction. The only explanation at this point is that the cross sections for ion and
electron production σi and σe are estimated falsly. Especially if the ratio of σe to σi is
increased, we quickly obtain much better agreement. On the other hand, the theoretical
prediction is not very sensitive to the absolute magnitude of the two cross sections if they
are changed by the same factor. In Figure 5.33, the measured and predicted values for Ar8+
and O6+ agree rather well.

147

Figure 5.33: Pressure variations of O6+ and Ar8+ , neutralization measured and theory.

148

Current Series
Let us now examine the current variations. Here we have a set of slit scans for each measurement. They are shown for O3+ , O6+ , and Ar8+ in Figures 5.34 to 5.36, Figures 5.37 to 5.41,
and ﬁgures 5.42 to 5.46, respectively. Immediately, we see that in the O3+ current series,
the beam changes its shape and size quite drastically. This is due to the fact that the biased
disk in the source shut down from the ﬁrst (Figure 5.36, 298 µA) to the second measurement
(Figure 5.35, 203 µA). Consequently, the plasma conditions changed, and with them the
beam shape and extracted current. I kept the series for completeness, but the results should
be viewed with caution. The O6+ series exhibits a slight asymmetry in the beginning, and
a round, almost uniform beam shape for higher currents. Ar8+ shows a triangular structure
and especially towards the higher currents a clear vertical asymmetry.

Figure 5.34: Beam proﬁle of O3+ , 122 µA.

149

Figure 5.35: Beam proﬁle of O3+ , 203 µA.

Figure 5.36: Beam proﬁle of O3+ , 298 µA.
150

Figure 5.37: Beam proﬁle of O6+ , 195 µA.

Figure 5.38: Beam proﬁle of O6+ , 302 µA.
151

Figure 5.39: Beam proﬁle of O6+ , 404 µA.

Figure 5.40: Beam proﬁle of O6+ , 490 µA.
152

Figure 5.41: Beam proﬁle of O6+ , 604 µA.

Figure 5.42: Beam proﬁle of Ar8+ , 111 µA.
153

Figure 5.43: Beam proﬁle of Ar8+ , 170 µA.

Figure 5.44: Beam proﬁle of Ar8+ , 223 µA.
154

Figure 5.45: Beam proﬁle of Ar8+ , 276 µA.

Figure 5.46: Beam proﬁle of Ar8+ , 331 µA.
155

Figure 5.47: Saturation currents for three beam current variations in the SuSI LEBT at
Q014. A transmission factor of T = 0.65 was used. The lower measured saturation current
for the Ar8+ series could be due to the vertical asymmetry.
In Figure 5.47 the saturation currents are plotted versus the beam currents. It can be seen
that for O6+ , the measured values agree well with the theoretical prediction. The measured
saturation currents for Ar8+ are consistently lower than the predicted ones, but more so for
higher beam currents. This could be attributed to the vertical beam asymmetry. The values
for O3+ are in the right regime, but the intercept of the measurement is not at 0 as would
be expected. As stated earlier, this measurement was under very unstable source conditions
and is only shown for completeness.

156

Figure 5.48: Neutralization for three current variations.
In Figure 5.48 the neutralization factor fe is plotted versus the ion beam currents for the
three series. It can be seen that O6+ and Ar8+ agree reasonably well with the theoretical
prediction, while the calculation for O3+ underestimates the neutralization factor. Considering a similar result from the pressure series, we must assume a systematic error that was
not taken into account.

157

5.4

Conclusion

5.4.1

Summary

In this chapter, I presented neutralization measurements at two diﬀerent locations: in the
beam line of the LEDA injector source and in the beam line of the SuSI superconducting
ECRIS.
The LEDA measurements agree with previous measurements at the same source with a similar detector and show neutralization values around 80-90 %, which depend on beam intensity.
It is possible to calculate the beam radius from the RFA spectra alongside the neutralization.
In case of the LEDA source this yielded very large beam radii which could not be explained.
A similar discrepancy showed up in previous measurements at the same source. Theoretical
calculations using reasonable beam radii compared well with the measured neutralization,
however.
The measurements in the beam line of SuSI suggest overall low neutralization factors (0% 50%) which compare reasonably well with the theoretical model introduced by Gabovich [51]
and discussed in detail in Chapter 2. They follow the prediction of higher neutralization with
higher residual gas pressure, larger beam radii and higher beam density. For typical ECRIS
LEBT settings, where the beam line pressure is kept as low as possible to prevent losses
due to charge exchange, this means very low neutralization factors and beam lines should
be designed accordingly. The reasonable agreement of the model with the measurements
suggests incorporating the simple theoretical treatment into the simulation of low energy
beam transport. This will be presented in chapter 6.

158

5.4.2

Outlook

The datasets taken at the two additional locations (Artemis beam line, SuSI diagnostic
box Q001) may still yield useful results upon further investigation. The analysis methods
presented in this chapter can be extended to accomodate multispecies ion beams and similar
collimator transmission curves as used in the LEDA injector source analysis can be obtained
for SuSI Q001. Preliminary analysis of data from the two locations suggests similar ﬁndings
as at the other locations (neutralization of 0% to 50% depending on beam size, beam current,
and residual gas pressure).
In order to settle questions of beam aspect ratio and to improve the statistics of the measurement by increasing the total measured current, a second identical detector put perpendicular
or under 45◦ to the existing detector might prove useful. The beam current monitors used
at the NSCL already provide two input channels and electron suppressor (V3 ) as well as
retarding ﬁeld (V2 ) could be provided in parallel by the same power supplies. Q014 in the
SuSI beam line with the perpendicular ports for the Allison emittance scanners would be an
ideal starting point.
The RFA setup as it is is certainly improvable as well. Possible changes that could beneﬁt
the setup are
• Applying a bias to the collector plate to suppress oscillations.
• Exploring the option of heating the detector in order to suppress insulating ﬁlm deposition on the grids.
• Trying diﬀerent combinations of apertures in the collimation system (e.g. taking out
the inner aperture, using smaller apertures).

159

Chapter 6
3D ECR Extraction Model

6.1

Overview

In this chapter I will describe in detail a semi-empirical model for extracting ion beams from
an ECRIS using the PIC code WARP. This model was developed in collaboration between the
88” Cyclotron’s Ion Source group at Lawrence Berkeley National Laboratory and the Heavy
Ion Fusion Science (HIF) Virtual National Laboratory (a collaboration of Lawrence Berkeley
National Laboratory and Lawrence Livermore National Laboratory) as an alternative to
existing extraction simulation models (IGUN [32], PBGUNS [116], KOBRA-INP [35]). The
success of WARP with intense heavy ion beams, its 3D capability and non- commercial
approach suggested expanding the code’s functionality with a 3D plasma extraction model
similar to IGUN’s 2D model. In addition, as discussed in Section 3.1.3 WARP has powerful
capabilities for multispecies beam transport simulations including space charge and so makes
end-to-end simulations of ion source and LEBT relatively easy. All simulations presented in
this chapter (from the source to the respective targets) were thus performed in WARP.

160

Figure 6.1: Two helium beams extracted from VENUS. Left: without the radial conﬁnement
sextupole, right: under normal operating conditions with the sextupole. The beam line
layout can be seen in Figure 6.5. The solenoid was set to 400 A in both cases.

6.2
6.2.1

The Extraction Model
Initial Conditions with WARPxy

One of the challenges with extracting ion beams from ECR ion sources is the triangular
beam structure coming from the sextupole conﬁnement ﬁeld. In Figure 6.1 two beam images
are shown for a helium beam extracted from VENUS. One without the use of the sextupole
magnet, which is clearly round, and one under normal operating conditions with the use
of the sextupole, which is clearly triangular (the position of the viewing screen can be seen
in Figure 6.5). In both cases, the focusing solenoid directly after extraction was set to
400 A to focus the beam onto the viewing plate. The question is now: How to create the
initial (triangular) distributions fi (x, y, z, vx, vy, vz) for each species to be extracted from the
ECRIS in the extraction simulation? Ideally, the plasma processes, rf-heating, and electron
impact ionization inside the source should be simulated self-consistently. And indeed, there
161

Figure 6.2: Plasma sputter marks inside VENUS. Left: The biased disk exhibits a large star
structure and a smaller clear triangle of about 5 mm height etched into the plate. Right:
The same (now rotated) discolorations are seen, however the triangle would be inside the
extraction aperture.
are several groups trying to model the plasma processes inside the source [21, 117, 118].
This theoretically yields the desired ion distributions and one of the most recent results for
an ECRIS at KVI Groningen looks very promising [20, 119]. However, these simulations
take a long time and are still highly specialized. In this work, I am using a simple semiempirical approach proposed by D. Todd and D. Leitner [120]. In this approach, the plasma
markings on the biased disk (shown e.g. for the VENUS source in Figure 6.2, left) are
used as reference to initialize the particles on the injection side of the source, give them a
Boltzmann distributed velocity and track them through the magnetic ﬁeld of the source to
the extraction aperture. In Figure 6.2 the discolorations and sputter marks on the biased
disk (left) and inside the plasma electrode with the extraction aperture (right) of the ECRIS
VENUS are shown. The 5 mm high triangle etched deeply into the bised disk is used as
reference for starting the particles.

162

In WARPxy the particle tracking to obtain the initial conditions is implemented in the
following way:
• The magnetic ﬁeld of the source is modeled in a program capable of calculating 3D
magnetic ﬁelds for coils and permanent magnets in the presence of other magnetic
materials like iron. In this thesis, the VENUS ﬁeld was calculated using VectorFields
Opera 3D [121] and the SuSI ﬁeld was calculated in a similar manner with LorentzEM [122].
• This ﬁeld is loaded into WARP as a gridded magnetic element (full 3D) via the convenience function:
addnewbgrd(zs, ze, xs, ys, sf, sc, dx, dy, bx, by, bz, nx, ny, nz)
with zs and ze the start and end coordinates in z direction in m, xs and ys the starting
coordinates in x and y in m. sf and sc are scaling factors and the ﬁeld will be scaled by
(sc + sf). dx, dy are the ﬁeld’s x and y grid cell sizes and nx, ny, nz are the number of
cells in the three respective directions. The ﬁeld data is stored in bx, by, bz which each
are three dimensional matrices with x the ﬁrst index, y the second, and z the third.
This function has to be called before the generate().
• Particles are initialized at the location of the biased disk inside the ECRIS with a
triangular spatial distribution and a Boltzmann velocity distribution of peak energy
between 2 and 5 eV. This is the energy regime of ions in an ECRIS and an adjustable
parameter in the simulations.
• Particles are then tracked through the source’s magnetic ﬁeld with a very small step
size (10−5 m) to resolve the spiraling motion.
163

Figure 6.3: Obtaining the initial conditions for a helium beam from the VENUS ECRIS.
Left: Particle distribution at the biased disk. Right: After tracking through the sources
magnetic ﬁeld. The circle represents the extraction aperture.
• The ﬁnal particle distributions fi (x, y, z, vx, vy, vz) are saved at a location a few mm
before the extraction aperture to leave some space for the subsequent extraction simulation to develop a plasma sheath.
An example of particle distributions for He1+ and He2+ at the VENUS biased disk and
the corresponding distribution at the extraction aperture after tracking through the sources
magnetic ﬁeld are shown in ﬁgure 6.3. Note the horizontal mirroring from injection to
extraction. The circle represents the extraction aperture of 4 mm radius.

164

6.2.2

2D+ Extraction with WARP3d

Recently, WARP was updated with a 2D (RZ) plasma extraction model similar to IGUN
and a 3D plasma extraction model using the same principles but on a 3D grid.
In these models, a plasma surface inside the source is deﬁned which is at plasma potential.
The plasma potential is usually 10 to 30 V higher than the wall potential (which, in the case
of ECR ion sources is typically between 20 and 30 kV).
From there, ions are tracked through the extraction aperture and the extraction system.
From the trajectories, the charge distribution of the beam is determined and accumulated
on the mesh. The poisson equation is solved for the combined external and beam space
charge potentials. In order to account for the plasma electrons, the so calculated potential
is updated with electrons distributed proportional to

∝ exp

e(V (r) − Vplasma )
kTe

with V (r) the potential determined from the external and space charge potentials and Te
the electron temperature.
The updated potential map is then used as an applied ﬁeld in the next run. This process is
repeated until a steady state solution has been reached. This is called a relaxation method.
It should be noted that in IGUN the potential distribution between the plasma surface and
the ion source wall (and extraction aperture) is calculated analytically using a one dimensional plasma sheath model [31], rather than through ray-tracing. Outside the aperture, the
process is the same as in WARP.
An (optional) magnetic ﬁeld can be added to the simulation in WARP as well as in IGUN.
This ﬁeld changes the trajectories of the extracted ions and thus the space charge map. It
165

is crucial to include the ﬁeld if it is present in the real ion source.
The two WARP models (RZ and 3D) were benchmarked against each other [123] and it was
found, that agreement between the 3D and the RZ model was only good if the mesh size in
the 3D simulation was much smaller than in RZ mode. This leads to excessive memory usage
and so Todd et al. proposed the 2D+ model which essentially runs in RZ mode until a steady
state solution has been reached, saves the potential array containing external potential and
space charge potential in a ﬁle and reloads that ﬁle for a subsequent single run in 3D mode.
This 2D+ method was benchmarked against the full 3D model, a pure 2D model, and IGUN
by Todd et al. and showed reasonably good agreement [123]. The 2D+ model has been used
in all the extraction simulations presented in this chapter.
The 2D+ model is implemented in the WARP deck in a way that the same script can be
used for the RZ run to obtain the relaxed solution and also for the subsequent 3D run to
track the actual triangular distribution by just setting an initial ﬂag. The RZ run consists
of these major steps:
• The particle data from the particle tracking to obtain the initial conditions is read in
and ion species data (masses, charge states) is set.
• RZ injection is switched on. The variables w3d.l inj rz and w3d.l inj regular are
set to True to ensure RZ injection and a higher particle density in the center to ensure
having particles there (the weights are adjusted accordingly, so the current distribution
is still uniform). The initial velocity in z direction is calculated from equation 2.76
(Bohm criterion).
• The plasma variables are set. The most important ones are:
– w3d.plasmapotential. The plasmapotential (in V) is calculated from equation
166

2.80.
– w3d.iondensity. The density of the extracted ions inside the plasma. Calculated
from the extracted current and the initial velocity.
– w3d.electrontemperature. The temperature of the Boltzmann electrons (eV).
– w3d.xbemin, w3d.xbemax, w3d.zbemin, w3d.zbemax. The extent over
which the Boltzmann distributed electrons are included (in m)
• The conductors geometry and the voltages are loaded from a ﬁle.
• Then the simulation is carried out in subsequent runs until a stable solution has been
reached (relaxation):
– gun() is called 4 times to obtain a preliminary solution.
– gun() is then called 6 more times with the current option. WARP will adjust
the particle weights in subsequent runs so that the total current at a location
speciﬁed by currentiz (usually near the end of the simulation) is the total current
desired by the user (e.g. the measured beam current).
– Finally, gun() is called 8 more times (still with the current option) with rhoparam
set to 0.5 which mixes in the last run’s potential array to ﬁnalize and smoothen
the solution.
• After this, the 3D potential map (external ﬁelds + space charge) is saved to a ﬁle for
the 3D run.
The number of calls to gun() and how much of the potential array of the last run should be
mixed in with the new one are, of course, parameters that the user can set. The ones listed
above are the ones that gave good results.
167

Figure 6.4: Example of start and end distribution in an extraction simulation. Shown is
an argon beam with 4.6 mA extracted from SuSI. Left: initial distribution. Note the slight
vertical oﬀset due to a hole in the source’s iron yoke. Right distribution after 15 cm. The
rotation is induced by the residual solenoid ﬁeld.
In the 3D run, no ﬁeld calculation is done, instead the ﬁeld from the RZ run is loaded from
the ﬁle. Most of the parameters are exactly the same, with the exeption, of course, of the
initial particle distributions fi (x, y, z, vx, vy, vz), which are loaded from a ﬁle. After all the
parameters have been set, gun() is called once to track the particles through the simulation
and obtain the ion distributions fe (x, y, z, vx, vy, vz) at the end of the extraction system
(0.25 m in the VENUS simulations, 0.15 m in the SuSI simulations). Examples of the initial
and ﬁnal ion distributions for the extraction of an argon beam from the SuSI ECRIS are
shown in Figure 6.4.

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6.3

Beam Transport with WARPxy

After successfully extracting ions from the ECRIS with the 2D+ extraction method in
WARP, it is necessary to simulate the beam transport to the locations of the beam diagnostic devices (for comparison of the simulations with measurements) or to the accelerator (for
design purposes). This can be done with a xy slice simulation in WARP as well, because
ECRIS beams are usually DC beams, thus the longitudinal space charge forces are negligible. At the same time, the beams are non-relativistic, so relativistic corrections of the ﬁelds
are not necessary and the self magnetic ﬁeld is negligible as well. As already discussed in
Subsection 3.1.3, lattice elements can be included in WARP simulations in a number of different ways (as hard-edged multipole elements, axially varying multipole elements, gridded
elements, electrodes which are solved with the SOR solver, and as one of the four built in
elements: quad, bend, dipo, and accl).

6.3.1

Electrostatic Elements

In order to include fringe ﬁeld eﬀects as accurate as possible in the beam transport simulations, but save computation time, the following approach was taken to include electrostatic
elements like einzel lenses or acceleration gaps:
• The element in question is given a unique ﬁlename including the applied voltages (e.g.:
”SuSI EL-21000V-v1.0 HR.wrp”).
• The main deck looks for a ﬁeld ﬁle with this name. If it exists, it loads the ﬁeld as a
gridded element from ﬁle.
• If it doesn’t exist, it calls another WARP script to load the geometry, apply the new

169

voltages and solve for the electrostatic ﬁeld. This is a fully separate run of WARP.
• The solved ﬁeld is now saved into the ﬁle with the ﬁlename above and the second
WARP run is terminated.
• The ﬁeld is reloaded into the main WARP deck as a gridded element.
This way, ﬁelds are only calculated when a new voltage is applied, but not if e.g. only the
particle distribution changes from run to run. Note that the geometry loaded in the second
WARP deck can be generated with the electrode designer utility presented in Section 3.2.

6.3.2

Automatic Space Charge Compensation

Because of the reasonably good agreement between measurements presented in Chapter 5
and the theoretical prediction with the extended formula originally developed by Gabovich et
al. (cf. Subsection 2.1.5), it suggested itself to include the option of automatically calculating
the neutralization factor from a set of cross sections σi and σe , and a beam line pressure
speciﬁed by the user and the beam currents, and species used in the simulation.
This is done in the following way:
• After each step, all the particles still alive are gathered in an array.
• An average 2σ beam radius is calculated for all species together. This is of course a
simpliﬁcation, but necessary for the multispecies extension of the Gabovich formula
to work. Anyway, multiple species are only present in the section between source and
analyzing magnet, which is usually very short.
• The beam velocities are calculated for each individual species.

170

• A function is called to calculate the neutralization according to the multispecies expression presented in equations 2.62 and 2.65. Only ion species with a number of particles
still alive > 0 are considered.
• The species currents and weights are then modiﬁed with the new neutralization factor
fe .
Note: This procedure was implemented after the measurement of the space charge compensation and has only been used with the latest simulations of the transport through the SuSI
LEBT.

Examples of extraction and beam transport simulations using the models described above
will be presented in the following sections.

171

Figure 6.5: VENUS - Source and LEBT.

6.4

VENUS Simulations

The layout of the VENUS LEBT is shown in Figure 6.5. The beam optics elements can be
seen in the ﬁgure and are:
• Solenoid lens (center 0.4 m after extraction).
• Dipole magnet (90◦ , 1m radius, 26.5◦ face angles).
The beam diagnostic elements are:
• Quartz viewing screen (0.8 m after extraction).

172

Figure 6.6: Simulation of a 1.6 mA uranium beam extracted from VENUS.
• Allison emittance scanner (3.3 m after extraction).
• Faraday cup (3.3 m after extraction).

6.4.1

Heavy Ions

The ﬁrst test of the simulation model was performed for uranium ions (mass 238 and charge
states from q = 20+ to q = 42+) extracted from VENUS in a measurement in 2005. Oxygen
was used as support gas, so oxygen ions were mixed in with the uranium (m = 16, q = 2+
to 6+). The simulation parameters are listed in Table 6.1 and the emittances are plotted
in Figure 6.6 for two diﬀerent ﬁxed neutralization values (70% and 80%). It can be seen,
that the vertical emittance agrees very well while the simulation overestimates the horizontal
emittance by a factor of 1.5. This could be due to the formation of a virtual aperture (cf.
Section 2.2.8) which was not taken into account at that time (initial triangles were the same
size for all species) in combination with an overestimated neutralization factor.

173

6.4.2

Light Ions

In order to test the simulation method on a more fundamental level, a simple beam was
chosen next: helium, high pressure, without support gas. Thus the extracted beam consisted mostly of He1+ . Here, the source was run in two diﬀerent ways: Without the radial
conﬁnement sextupole and in regular mode using the sextupole. Emittances measured with
the allison scanners compared to simulations with the model described above are shown in
Figure 6.7. It can be seen that the vertical emittances agree very well and the horizontal
emittances are still in the same regime.
In Figure 6.8 a variation of the puller distance with solenoid ﬁxed at 100 A under regular
operating conditions of the source (sextupole energized) is shown. It can be seen that the
measured and simulated emittances show similar trends with the puller variation, but the
simulated emittance is about a factor 2 smaller than the measured one. This behavior can
be explained by looking at the beam viewer right after the solenoid magnet (Figures 6.9
and 6.10). For the case with sextupole ON, the simulated beam is much smaller than the

Table 6.1: VENUS - Uranium simulation parameters
Parameter

Value

Total extracted current

1.6 emA

Source Voltage

20 kV

Puller Voltage

-2 kV

Bmax at extraction

2.1 T

Extraction Aperture ↔ Puller

31.5 mm

Te

5 eV

Ti

2 eV

174

measured one. It is possible that it is only transported through the good ﬁeld region of the
dipole magnet, thereby not undergoing signiﬁcant emittance growth.
It can be concluded that the semi empirical model does not work well for less conﬁned singly
charged ions, such as helium, which are created in a single ionization step.

175

Figure 6.7: VENUS simulation of a helium beam without the sextupole for radial conﬁnement. The simlations agree reasonably well with the measurements.

Figure 6.8: VENUS simulation of a helium beam with the sextupole for radial conﬁnement
energized, compared to measurements. Simulation and measurement disagree by a factor of
2.

176

Figure 6.9: Two images from the beam viewer. Both were taken with the same settings:
Solenoid lens current = 400 A, Puller position = 50 mm. Left: Sextupole magnet on, right:
sextupole magnet oﬀ. Due to the conﬁning qualities of the sextupole, the extracted beam
current in the oﬀ case was lower (1.3 mA vs. 2 mA).

Figure 6.10: Simulation results 825 mm after extraction (position of the beam viewer). Both
were simulated with the same settings: Solenoid lens current = 400 A, Puller position = 50
mm. Left: Sextupole magnet on, right: sextupole magnet oﬀ. The simulated currents were
matched to the measurements (left: 2.0 mA, right: 1.3 mA).

177

Figure 6.11: Schematic of the SuSI LEBT system up to the second dipole magent. Between
the ﬁrst dipole and the location of NeMo (in the diagnostic box Q013/14) there is a focusing
and collimation channel consisting of 3 solenoids, apertures, and slits.

6.5

SuSI Simulations

As a ﬁrst test of the automatic calculation of the neutralization factor in the LEBT beam
transport simulations, simulations were performed for the two pressure series O3+ and Ar8+
presented in chapter 5.3.2 to go along with the space charge compensation measurements.
This includes obtaining the initial conditions, extracting the beam from the ECRIS and
transporting it through the LEBT using the automatic calculation of neutralization presented
in Subsection 6.3.2. Unfortunately, no emittance scanners were available in this setup, so
I can compare the simulations only to the slit scans and beam images obtained during the
neutralization measurements. Figure 6.11 shows again the SuSI LEBT. Simulations were
terminated at Q014. An example of argon extracted from SuSI was shown in ﬁgure 6.4. The
cross sections for oxygen so close to the extraction aperture look very similar, so I will not

178

show more beam distributions directly after extraction, instead I will focus on the results at
Q014 and the beam envelopes through the LEBT. The following beam optics elements were
present in the LEBT and were included in the simulation as well:
• Q001EL - Einzel lens directly after the extraction system. Field calculated from
electrode data.
• Q003DH/DV - Horizontal and vertical steering magnets. In this model, a simple
thin kicker was used.
• Q005DS - Dipole magnet for selection of desired ion species by A/q. In this simulation,
the built-in hard edge model of WARP was used.
• Q006DH/DV - Horizontal and vertical steering magnets. In this model, a simple
thin kicker was used.
• Q007 - 4-jaw slits.
• Q008SN - First collimation channel solenoid. The ﬁeld was calculated in POISSON
superﬁsh and imported into WARP.
• Q009AP - 25 mm aperture.
• Q010SN - Second collimation channel solenoid. The ﬁeld was calculated in POISSON
superﬁsh and imported into WARP.
• Q011AP - 25 mm aperture.
• Q012SN - Third collimation channel solenoid. The ﬁeld was calculated in POISSON
superﬁsh and imported into WARP.

179

• Q013 - 4-jaw slits.
These are the two measurement series that will be presented:
1. O3+ Pressure 1.1 · 10−5 to 1.1 · 10−6 Torr, current approximately constant at 235 eµA.
2. Ar8+ Pressure 1.0 · 10−5 to 1.0 · 10−6 Torr, current approximately constant at 200 eµA.
As an example of the beam envelopes obtained from the beam transport simulations, the
envelopes for O3+ at 1.1 · 10−5 Torr are shown in Figure 6.12. It was necessary to use the
vertical steerers a little bit to compensate for an initial vertical angle due to the asymmetry
in the source’s iron housing where the magnet leads are entering the source through the
cryostat.
In Figures 6.13 to 6.18 the beam cross sections of the three highest pressures of the O3+
pressure series are compared to the measured slit scans. It can be seen, that shape and size
as well as a relative rotation with increasing pressure are reproduced. However, there is a
mismatch in absolute rotation.
Note that the only variable changed in this simulations was the pressure, which is only
used to calculate the neutralization factor. The argon pressure series (not shown) exhibits
similar tendencies the beam cross sections are round (compare to Figures 5.26 - 5.30) and
have the correct size. However, they appear more uniform in the simulations than in the
measurements.

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Figure 6.12: Envelopes of a O3+ beam in the SuSI beamline with a pressure of 1.1 · 10−5 Torr. The upper envelope is the
horizontal one and the lower envelope the vertical one. From left to right, the grey boxes are: Source, einzel lens, analyzing
magnet, and the three solenoids of the collimation channel.
181

Figure 6.13: Measured beam proﬁle of O3+ , 5.0 · 10−6 Torr

Figure 6.14: Simulated beam proﬁle of O3+ , 5.0 · 10−6 Torr

182

Figure 6.15: Measured beam proﬁle of O3+ , 8.0 · 10−6 Torr

Figure 6.16: Simulated beam proﬁle of O3+ , 8.0 · 10−6 Torr

183

Figure 6.17: Measured beam proﬁle of O3+ , 1.1 · 10−5 Torr

Figure 6.18: Simulated beam proﬁle of O3+ , 1.1 · 10−5 Torr

184

Figure 6.19: Neutralization along the beam line in the SuSI LEBT for a O3+ beam in
1.1 · 10−5 Torr residual gas pressure. The center line represents the simulation run with the
charge exchange cross section obtained by using the M¨ller-Salzborn formula. The three
u
boxes correspond to the solenoids in the collimation channel. The dashed lines are element
boundaries and apertures. Inside the einzel lens (0.4 - 0.8 m), the neutralization is zero,
inside the dipole (1.1 - 1.9 m) the beam contaminants are ﬁltered out.
In Figure 6.19 the neutralization factor according to the automatic calculation presented in
Section 6.3.2 along the SuSI beamline is shown. The part with zero neutralization in the
beginning is the einzel lens, where the presence of an electrode prohibits the accumulation
of electrons inside the beam envelope. It looks like neutralization can reach very high values
in the region before the bending magnet, where the beam current is still high (due to the
many species in the beam) and the beam radius is large.

185

6.6

Conclusion

In this chapter, I presented the three parts of the ECRIS extraction suite using the particle in
cell code WARP. The initial conditions are obtained semi-empirically from plasma markings
inside the source. The particles are then extracted from the plasma by means of a relaxation
process in which the steady state solution and the plasma meniscus form self-consistently.
The hybrid RZ-3D model called 2D+ saves computation time and space and compares reasonably well with full 3D simulations. The use of the same simulation code (WARP) for
the two parts of the ion source simulation and the beam transport together with a number
of python utility programs and GUI’s lets the user seamlessly go from one step to the next
thereby obtaining an end-to-end simulation of an ECRIS driven accelerator front-end. As
will be shown in Chapter 7, it is even possible to include the accelerator into the simulation.
First tests of the extraction simulations with uranium and helium beams extracted from
VENUS suggest the simulation model can be used for medium to heavy ions of medium to
high charge states, but the approach to obtain the initial conditions breaks down for very
light singly ionized ions. A more self-consistent model of the plasma as suggested by Mironov
et al. [20] might help obtain more realistic initial distributions for the subsequent extraction
simulation.
The implementation of the multispecies extension of the formula derived by Gabovich et
al. [51] as part of the LEBT beam transport simulation yielded promising ﬁrst results and
gave interesting insights in the distribution of space charge compensation along the beam
line. It might be possible that in the region before the analyzing magnet, space charge
compensation is as high as 90% due to the high beam current and large average beam
radius. This has to be conﬁrmed by future measurements, though.

186

Chapter 7
An Example Application: DIANA

7.1

Introduction

During the course of the presented thesis work, several simulation codes were used to simulate
extraction from Electron Cyclotron Resonance Ion Sources (ECRIS) and beam transport
through Low Energy Beam Transport (LEBT) systems. Utility programs were developed
in python to simplify the creation of the necessary input ﬁles for the simulations as well
as the post processing of the results. The Dual Ion Accelerators for Nuclear Astrophysics
(DIANA) [124–127] presented an interesting opportunity to apply some of the simulation
codes and utility programs, as well as know-how, to a new project. DIANA is a proposed
underground laboratory for nuclear astrophysics experiments currently being designed by a
collaboration of the University of Notre Dame, the University of North Carolina, Western
Michigan University, Michigan State University, Lawrence Berkeley National Laboratory,
and the Colorado School of Mines.
The scientiﬁc goal of DIANA is to measure charged particle nuclear cross- sections in low energy regimes, corresponding to stellar burning processes. These cross-sections are important
for determining the rates of nucleosynthesis processes from the Big Bang, through the early
development of the universe, up to the current stars, thereby shedding light on the abundance
of elements [128]. The charged particle nuclear cross- sections present a major uncertainty in
the stellar model simulations and, so far, for most of the reactions, low-energy data are only
187

available through extrapolation from higher energies. One of the challenges in measuring
these cross-sections is the fact that most of them are very small (pico- to femto-barn range)
at the desired energies. Background from cosmic rays in the detectors is therefore a major
problem. The two paths taken in DIANA to enable measuring these cross-sections are:
1. Minimize the background by going to an underground location and beneﬁt from the
natural shielding of the rock,
2. Design the accelerators for signiﬁcantly higher luminosity as compared to existing
facilities (e.g. LUNA [129]).
DIANA will consist of two accelerators: A low energy (50 - 400 keV) machine and a higher
energy (300 - 3000 keV) machine (the energies are quoted for q = 1+ charge-states and will
be higher for multiply charged ions). While the 400 kV accelerator certainly also presents
very interesting possibilities for the application of programs and results presented earlier in
this thesis, the focus of this chapter lies on the 3 MV machine. This accelerator includes
an ECRIS, a LEBT system, an electrostatic acceleration column, and aims to provide 1
mA of beam current (regime of non-negligible space-charge eﬀects) and is therefore an ideal
application of the software described in Chapter 3. Detailed simulations were carried out
for the base design of the accelerator and modiﬁcations were proposed and discussed with
the prospective vendor. In this chapter the base design will be described, the modiﬁcations
will be explained, and simulations will be presented to demonstrate the improvement and
feasibility of the ﬁnal design.

188

Figure 7.1: DIANA - Facility layout. Courtesy of the DIANA collaboration.

7.2

Layout

A general layout of the planned DIANA facility can be seen in ﬁgure 7.1. The most important
features are:
• Low energy accelerator. A ﬂat-ﬁeld (no multipole for radial conﬁnement) ECR ion
source placed on a high voltage platform, capable of delivering up to 100 emA of 4 He1+
with an energy range of 50 keV to 300 keV.
• High energy accelerator. A Pelletron electrostatic accelerator with a small permanent magnet ECR ion source inside the high voltage terminal, capable of delivering a
minimum of 850 eµA of 4 He1+ with an energy range of 300 keV to 3000 keV.

189

• Primary target stations. A solid target and a gas-jet target can both receive beam
from both accelerators to provide consistent data over a wide energy range.
• Secondary target stations. In a later phase of the project, a second set of targets
is envisioned to allow for individual operation of both accelerators at the same time.
As mentioned in the introduction to this chapter, the focus of the presented work lies on
the high energy accelerator. More precisely on the simulation of the ion extraction from
the small all-permanent-magnet ECR ion source and the transport through the electrostatic
acclerator. Therefore the other parts of the DIANA facility will not be discussed further.
The high energy accelerator is envisioned to be an oﬀ-the-shelf Pelletron by the National
Electrostatics Corporation (NEC) [130] with custom modiﬁcations to achieve the desired
beam intensity and quality over the full energy range. A Pelletron is similar to a Van de
Graaﬀ generator [131], but instead of a rubber band, charges are transferred to the high
voltage terminal by means of pellet chains. This particular system will have a set of 10
chains capable of delivering 150 eµA of current each, to provide the necessary charge to
extract a 1 emA beam. The whole system is typically enclosed in an SF6 vessel to minimize
high voltage breakdown, while the beam pipe in the center is evacuated.
The criteria necessary to reach DIANA performance level are listed in table 7.1. As can
be seen, the DIANA high energy accelerator features a wide energy range of 300 keV - 3
MeV throughout which a constant beam current with consistent beam quality is desired.
This was the motivation for doing extensive simulations and add custom modiﬁcations to
the standard design.
Figure 7.2 shows a more detailed schematic of the high energy accelerator system, loosely
divided into the three parts (cf. also ﬁgure 7.10):

190

• Front end. Ion production, species separation, and transport of the beam to the
accelerator. This section ends at the entrance of the accelerator.
• Accelerator column. Acceleration of the ions to the ﬁnal energy. This section
contains the electrostatic acceleration column and an electrostatic quadrupole triplet
(ESQ1) [41, pp. 98]. It ends at the exit of the column.
• Bending section. Bending of the beam from the vertical direction into the horizontal
plane. This part consists of an electrostatic quadrupole triplet ( ESQ2) after the exit
of the column, and a 90◦ bending magnet. This section ends at the image point of the
magnet.
These three parts will now be discussed in more detail, followed by a list of the custom
modiﬁcations (Section 7.3) and simulation results (7.4).

191

Table 7.1: DIANA - High energy experiment performance criteria.
Parameter

Performance Criteria

Comment

Output Energy (min)

300 keV

*ion charge state

Output Energy (max)

3000 keV

*ion charge state

Beam Currents:

(by species)

1H

∼ 1 mA

3 He,4 He

∼ 1 mA

Heavy Ions (A ≤ 22)
Maximum magnetic rigidity

> 50 pµA
1.44 Tm

Energy spread
Energy stability

0.05 %
0.01 %/h

192

Figure 7.2: High energy accelerator layout. From the top: Front end with ion source and
species separation. Accelerator column including an electrostatic quadrupole triplet (ESQ1).
Bending section including an electrostatic quadrupole triplet (ESQ2) and a 90◦ bending magnet. Courtesy of NEC.

193

Figure 7.3: Front end layout. From the left: ECR ion source, extraction system, collimator,
einzel lens 30◦ , bending magnet, immersion lens. Courtesy of NEC.

7.2.1

Front End

A detailed layout of the front end inside the high voltage terminal of the accelerator is shown
in Figure 7.3. The relevant elements are:
• ECR ion source. A small, all-permanent-magnet Electron Cyclotron Resonance Ion
Source (ECRIS). See Subsection 7.2.2.
• Extraction system. The standard extraction system of Nano-PK is a diode system (see Figure 7.7 top). In Subsection 7.3.1, an improved extraction system will be
introduced.
• Collimator. This collimator is part of the standard design and proves helpful in
restricting the beam size for low energies. It consists of two apertures with a center
distance of 50 mm and a diameter of 6-8 mm.

194

• Einzel lens. Unipotential electrostatic lens (see [41, pp. 78]). The maximum voltage
on the center element is +20 kV.
• 30◦ bending magnet. A double-focusing dipole magnet to separate ion species by
diﬀerent m (mass [amu] to charge-state [e] ratios). The magnet speciﬁcations are listed
q
in Table 7.2.
• Immersion lens. Bipotential electrostatic lens (see [41, pp. 78]). This lens accelerates
and focuses the beam into the entrance of the accelerator column. The maximum
voltage diﬀerence of the two elements as speciﬁed by NEC is 50 kV.

Table 7.2: 30◦ dipole speciﬁcations. For more information about Rogowski pole tips see [4,5]
Parameter
Type

Value
double focusing

Bending radius

300 mm

Bending angle

30◦

Face angles

8.8◦

Air gap

30 mm

Focal length

600 mm

Pole tip shape

Rogowski-type

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Figure 7.4: NANOGAN Sectional view. The beam is extracted from right to left. This
image shows the standard diode extraction system. ’Axial FeNdB’ denotes the permanent
magnets creating the solenoid ﬁeld, ’radial FeNdB’ the permanent magnet sextupole. Image
adapted from [13].

7.2.2

Ion Source

The Electron Cyclotron Resonance Ion Source is of the type Nano-PK (former NANOGAN
[13, 132]), produced by the company Pantechnik in France [6]. A section-view is shown in
Figure 7.4. The principles are the same as described in Sections 2.2 and 4.1 for larger ECRIS
with the exception that now all magnets are FeNdB permanent magnets.

196

Table 7.3: DIANA - NANOGAN parameters.
Parameter

Value

Chamber ∅

26 mm

Mirror length

100 mm

Bmax

0.80 T

Bmin

0.25 T

Bextraction

0.58 T

fecr

10 GHz

MW power

<100 W

197

Table 7.3 lists the important source parameters and table 7.4 the ion beam currents guaranteed for NANOGAN/NanoPK by the company on their webpage. This does not include
any information about beam size, diameter of the extraction aperture, and beam emittance.
A little more information can be found in several conference proceedings and review papers [13, 132, 133]. For DIANA the main focus lies on singly charged H and He. In [133]
A. Villari shows measurements of 99% emittances for a NANOGAN-II (which is similar to
NANOGAN) of 125 π-mm-mrad and 175 π-mm -mrad at 15 kV extraction voltage for He1+
and He2+ , respectively. Assuming 99% ∼ 4 × RM S this would correspond to normalized
RMS emittances of 0.089 mm-mrad and 0.175 mm-mrad, respectively. These values will
be important in Subsection 7.4.1 for comparison of simulations with actual measurements.
Improvements of the extraction system leading to better beam quality and matching into
the accelerator column will be discussed in subsections 7.3.1 and 7.4.1.

Table 7.4: Company guaranteed ion beam currents for NANOGAN. Vertical: Ion species,
Horizontal: Charge-state. Currents in eµA. From [6].
1+
H

1000

Ar

300

4+

6+

8+

9+

100

45

40

10+

10

11+

12+

13+

14+

8

15+

7

16+

1000

He

2+

100
1

Xe
Ta
Au

10
10

9

8

6

6

198

5

5
5

2

1

Figure 7.5: Electrostatic accelerator column layout. The column consists of 4 units, each unit
is composed of 2 segments ´ 23 rings (except unit 2 which holds the electrostatic quadrupole).
a
Courtesy of NEC.

7.2.3

Accelerator Column

The accelerator column consists of 4 units, each of which houses two electrostatic acceleration segments. Each segment consists of 23 concentric rings which are shaped to minimize
secondary electron emission. The layout of the column including the injector can be seen in
ﬁgure 7.5. The voltage grading of the immersion lens and the rings is realized by a resistor
chain parallel to the column (as depicted schematically in Figure 7.6). The standard gradient through the column is a linear voltage gradient. This corresponds to all the resistors
R having the same resistance (immersion lens excluded). Improvement of beam matching
into the column by using a custom, adjusted voltage gradient in unit one (by adjusting the
resistor values individually), and the NEC standard technique of using a shorting rod will be
discussed in Subsection 7.3.2. In unit two, the second segment was replaced by an electrostatic quadrupole triplet (ESQ1) for focusing the beam at lower energies, thereby providing
additional tuning ﬂexibility. The quadrupole triplet has an aperture radius of 19 mm, and
an electrode rod radius of 21.9 mm. The lengths of 1st , 2nd and 3rd element are 63.5 mm,
127.0 mm and 63.5 mm, respectively, with gaps of 19 mm in between the three elements.

199

Figure 7.6: Cartoon picture of the NEC Pelletron accelerator tube including resistor chain.
The resistor chain acts as a voltage divider. The HV terminal voltage can be set from 300 kV
to 3 MV. The Gap voltage as well as the voltages on the respective rings is then determined
by the ratio of the resistors to each other. In the simplest case, all ring resistors R are equal
and thus the voltage diﬀerence between each ring as well.

7.2.4

Bending Section

After reaching its ﬁnal energy at the exit of the column, the beam is transported through the
bending section where it is bent into the horizontal plane. The bending section consists of
a second electrostatic quadrupole triplet (ESQ2), and a double-focusing 90◦ dipole magnet.
ESQ2 is necessary to focus the beam into the object point of the dipole. The bending section
ends at the image point of the dipole, which is assumed at 2× the bending radius from the
exit of the dipole. At this point, the present design and simulation eﬀort is considered
concluded. A simpliﬁed layout including the bending section can be seen in Figure 7.10.
The resulting particle distributions are used in beam transport simulations to the two (four)
target stations by another group in the DIANA collaboration.

200

7.3

Custom Modiﬁcations

Preliminary simulations of the oﬀ-the-shelf design of the NEC Pelletron described in the
previous section showed that the beam quality and size could not be preserved through the
whole system mainly for two reasons:
• Strongly divergent beam out of the source. The diode system proposed for
NANOGAN/NanoPK by Pantechnik and NEC is insuﬃcient to achieve laminar ﬂow
of the beam through the extraction region which leads to a strongly divergent beam.
In the rest of the system, the inner diameter of most elements is too small to avoid
particle loss and emittance growth due to aberrations from fringe ﬁeld eﬀects (stronger
at larger radii).
• Strong focusing eﬀect at the entrance of the column. Due to the sharp change
in voltage gradient at the entrance of the column, the still rather low energy beam
(∼ 50 - 65 keV) experiences a strong focusing force which matches the beam poorly
into the column.

7.3.1

Ion Source Extraction System

There are several ways to remedy the divergent beam from the source (negative eﬀects
in parentheses): Smaller extraction aperture (less beam), collimate the beam (less beam),
increase the aperture sizes of the the rest of the system (more space required), add more
focusing elements (more space required). Instead, the extraction system was modiﬁed from
a simple diode system to a more complex tetrode system in order to improve the beam size
and beam quality from the source. The original diode system and the proposed tetrode
system are compared side-by-side in Figure 7.7, the results are discussed in Section 7.4.1.
201

Figure 7.7: Side-by-side comparison of the two diﬀerent extraction systems proposed for the
Nano-PK ECR ion source. Diode system (top); Tetrode system (bottom). Note that the
ﬁnal beam energy (19 keV for singly charged ions) is the same for both systems. The tetrode
system gives more tuning ﬂexibility by adjusting the voltage of the ’10000 V’ electrode
(green).

202

7.3.2

Adjusted Voltage Gradient

In order to improve the beam matching into the accelerator column, the grading of the ring
voltages in unit 1 (cf. Figure 7.5) was adjusted to be non-linear. The voltage distribution
in unit 1 now follows equation 7.1 with V0 = 2941 kV (3000 kV - 19 kV source - 40 kV
immersion lens), VV G = 300 kV, and L = 0.717 m.

V (z) = V0 − VV G

z 4
3
L

(7.1)

Here L is the optimal length of the adjusted part in order to achieve a voltage drop of VV G .
This would lead to a perfect matching of adjusted gradient and linear gradient thereafter (the
remaining 2641 kV). Unfortunately, this would also mean a much longer column. Instead,
the adjusted voltage gradient part is terminated after 0.52 m (physical length of unit 1) and
the linear voltage gradient part is shortened a bit as well (to ﬁt in units 2-4), leading to a
slightly higher gradient there. This procedure keeps the focusing at the entrance minimal,
thereby matching the beam well into the column. Although, due to this compromise, the
original focusing eﬀect is now shifted to the transition from unit 1 to unit 2, it is also
weakened a little and the beam has a higher energy and is thus less inﬂuenced by the eﬀect.
As a matter of fact, the focusing is actually helpful now, as will be seen in Section 7.4.2
(simulation results). In Figure 7.8 a comparison between a purely linear voltage gradient
and the adjusted voltage gradient is shown for the full 3 MV case, in order to illustrate the
points made above. In practice, it is not feasible to set all resistors of unit 1 individually.
Each one is custom made and it would be expensive to produce 41 diﬀerent resistors and
stock as many spares. Instead, the adjusted voltage gradient in unit 1 was approximated
piecewise linearly using only 5 diﬀerent resistor values. In appendix the ideal resistor values
203

Figure 7.8: Comparison of linear voltage gradient and adjusted voltage gradient. The voltage
on each element of the column is shown versus the position along the accelerator axis. The
horizontal axis corresponds to the picture of the column below the ﬁgure.
(normalized to a total of 1) are listed together with the practical values (in MΩ). This
includes also the resistor parallel to the immersion lens. Thus, the gap voltage scales with
the total acceleration voltage (HV terminal voltage).
The second technique that will be employed in the column is the well- established shorting
rod technique, which is part of many of NEC’s electrostatic accelerators. This is independent
of the grading of the rings and adds three connectors to the accelerator at which the column
can be grounded (see Figure 7.9, bottom) by inserting a shorting rod from the exit of the
column. This means that the rings coming afterwards are all at ground potential as well,

204

Figure 7.9: Comparison of diﬀerent total voltages using the adjusted voltage gradient and
the shorting rod technique. The voltage on each element of the column is shown versus the
position along the accelerator axis. The horizontal axis corresponds to the picture of the
column below the ﬁgure, The vertical arrows denote the shorting positions.
thereby eﬀectively shortening the column and keeping the gradients (and the immersion lens
gap-voltage) in the beginning of the column the same for diﬀerent HV terminal voltages. The
voltage distributions for a few HV terminal voltages are plotted in 7.9. Another beneﬁcial
eﬀect of grounding part of the accelerator for lower energies is that in the grounded section,
the beam will experience space-charge-compensation (which cannot occur in the presence of
strong electric ﬁelds).

205

Figure 7.10: Simpliﬁed layout of the simulated high energy accelerator and beamline. The ion source extraction simulation
ends 88.25 mm after the plasma aperture of the source. This is where the WARP simulation starts, which covers the rest of the
system in one continuous simulation.

206

7.4

Simulations

In this section, the simulations carried out for the DIANA high energy accelerator are described and results are presented. There are two diﬀerent parts: The ion source extraction
simulation and the subsequent simulation of front end, accelerator, and bending section (all
in one simulation). A layout overview can be found in Figure 7.10. The z = 0 point for all
the simulations is at the plasma aperture. Details to software (simulation codes and utility
programs) used in the two parts can be found in the respective subsections.

7.4.1

Ion Source Extraction

The electrode geometry of the tetrode extraction system was designed in the ElectrodeDesigner presented in Section 3.2.1, which includes an import function for simple magnetic ﬁelds and an option to export IGUN [31–33] input (*.IN) ﬁles. The magnetic ﬁeld
of NANOGAN was calculated in RZ geometry in PANDIRA, which is part of the Poisson
-Superﬁsh suit of software [134, 135]. This calculation did not include the sextupole magnet
(because it is not RZ symmetric), but the contribution of the sextupole to the longitudinal
B-ﬁeld on axis (Bz (z)) is very small. Figure 7.11 shows the ﬁeld calculated by PANDIRA and
Figure 7.12 the interpolated ﬁeld Bz (z) on the center axis using Poissons SF7 interpolator.
This ﬁeld was imported into the ElectrodeDesigner and an *.IN ﬁle was created which was
then used to simulate extraction from the source with IGUN.

207

Figure 7.11: RZ-symmetric PANDIRA model of the permanent magnet structure of
NANOGAN. Dimensions are in cm, horizontal axis is Z, vertical is R. Sextupole magnet
not included.

Figure 7.12: NANOGAN - Bz (z) on axis. As is typical for permanent magnets, the ﬁeld (here
in Gauss) becomes negative, before going to zero. The plasma volume is located between
the two peaks in the middle.
208

Figure 7.13: IGUN simulation - Trajectory plot. The electrode voltages (from left to right) are: Plasma electrode = 19 kV,
puller electrode = 10 kV, electron repeller = -2 kV, ground electrode = 0 kV. The axes are in grid units (1 gu = 0.25 mm).
The magnetic ﬁeld (secondary axis) is shown as a green dashed line.

209

As a ﬁrst step, the original diode system (see ﬁgure 7.7, top) was simulated for a plasma
aperture diameter of 7 mm and extracted current of 1.5 mA (He1+ + He2+ combined)
as found in literature to NANOGAN. The obtained normalized RMS emittance of 0.067
mm-mrad compares reasonably well with the 0.089 mm-mrad quoted for NANOGAN-II (cf.
Section 7.2.2) considering that the factor 4 used to transform from 99% to RMS emittance
is valid only for K-V beams and could easily be higher for realistic beams ( meaning the
RMS value of 0.089 could easily be lower). As it became clear that the diode system did
not produce suﬃcient beam quality, the tetrode system (see Figure 7.7, bottom) was used in
IGUN with a plasma aperture diameter of 6 mm. The resulting laminar beam ﬂow can be
seen in Figure 7.13. The ﬁnal beam diameter for 4 He1+ at z = 88.25 mm (where particles
are loaded into WARP) was 4.38 mm (∼34% smaller than with diode) with a normalized
RMS emittance of 0.0527 mm-mrad (comparable to diode system) and a beam current of
0.92 mA.

210

7.4.2

Front End and Accelerator Tube

This part of the simulations was carried out with WARP [36] in XY slice mode (cf. Subsection 3.1.3). This means space-charge is taken into account, but only in the two transversal
directions (x,y). Longitudinal space-charge eﬀects are assumed to be negligible, because the
beam is continuous. An overview of the system can be seen in Figure 7.10. Particle distributions (10000 macro-particles per species) are obtained from the previous IGUN simulations
and loaded into WARP at z = 88.25 mm after the plasma aperture of the source.
All the electrostatic ﬁelds (einzel lens, immersion lens, accelerator column, electrostatic
quadrupole triplets) were calculated by modeling the electrodes in WARP and calculating the
ﬁelds beforehand in 3D mode with 4-fold symmetry (only a quarter has to be calculated due
to symmetry of the elements). The magnetic ﬁeld of the 30◦ dipole was calculated beforehand
in 3D as well, using the ﬁnite element software Opera3d [121]. The magnetic ﬁeld was then
transformed into Frenet-Serret coordinates [42, pp. 22] (coordinate system that follows the
ideal particle trajectory through the magnet) and saved to a ﬁle. The transformation is
necessary because WARP also transforms the coordinates inside a bending element to follow
the design trajectory (cf. subsection 3.1.3). The 90◦ dipole was approximated by using
WARP’s internal hard-edge model.
The particle transport simulation was carried out by setting up the simulation space, loading
the respective ﬁelds and the particles and running the simulation step-by-step. The results
were analyzed with the PostProcessor (cf. Subsection 3.2). Field calculation and particle
transport during tuning of the elements were done in 2 mm/grid-cell resolution and then
conﬁrmed with a higher resolution of 1 mm/grid-cell.

211

Four test cases were picked to represent the wide range of desired beam energies:
• 14B: 4 He1+ , 3 MeV
• 15B: 4 He1+ , 1 MeV
• 16B: 4 He1+ , 300 keV
• 17B: 12 C2+ , 600 keV
Table 7.5 lists the relevant settings of source, einzel lens, and electrostatic quadrupole triplets
(ESQ1-A/B/C, ESQ2-A/B/C) for the four diﬀerent cases. Beam contaminants were taken
into account as well (4 He2+ in 14B, 15B, 16B; 4 He1+/2+ and 12 C1+/3+/4+/5+ in 17B).

Table 7.5: Simulations - Test case parameters. (case numbers kept consistent with other
documentation) The number of shorted rings corresponds to the positions in Figure 7.9.
Parameter

Case 14B

Case 15B

Case 16B

Case18B

2981

981

281

281

19

19

19

19

15.15

15.15

15.15

15.90

0

90

114

114

ESQ1-A (kV)

0.0

0.0

6.0

6.0

ESQ1-B (kV)

0.0

0.0

6.0

6.0

ESQ1-C (kV)

0.0

0.0

5.25

5.25

ESQ2-A (kV)

26.0

11.0

3.1

3.1

ESQ2-B (kV)

30.0

11.0

3.1

3.1

ESQ2-C (kV)

36.5

11.8

3.35

3.3

HV terminal (kV)
Source (kV)
Einzel Lens (kV)
Shorting (# of rings)

212

Figure 7.14: DIANA - WARP simulation results of a 300 keV 4 He1+ beam: Horizontal (upper half) and vertical (lower half)
beam envelopes.
213

Figure 7.15: DIANA - WARP simulation results of a 300 keV 4 He1+ beam: Cross section.

Figure 7.16: DIANA - WARP simulation results of a 300 keV 4 He1+ beam: Horizontal phase
space.
214

As an example, the resulting beam envelopes for case 16B (4 He1+ , 300 keV) are shown in
Figure 7.14 and the beam cross-section and horizontal phase space in Figures 7.15 and 7.16.
The results (beam size, emittance, and ﬁlling of the dipole) are listed in Table 7.6.

Conservatively, the simulations can be summarized to: The presented setup is able to create
and transport 4 He1+ beams of up to 850 µA beam current in the regime of 300 keV - 3 MeV
to the image point of the 90◦ dipole, with normalized RMS emittances < 0.06 mm-mrad and
diameters < 5 mm, ﬁlling the 90◦ dipole less than 75%.

Table 7.6: DIANA simulations - test case results. In case 16B it can be seen that the beam
current decreases by ∼ 7% du to the use of a collimator. The beam diameters are <5 mm,
and no case ﬁlls the 90◦ dipole more than 75%. (case numbers kept consistent with other
documentation)
Parameter Case 14B

Case 15B

Case 16B

Case18B

Energy (keV)

3000

1000

300

600

Initial Current (mA)

0.92

0.92

0.92

0.25

Final Current (mA)

0.92

0.92

0.86

0.24

εxx’-norm-rms (mm-mrad)

0.048

0.049

0.045

0.033

εyy’-norm-rms (mm-mrad)

0.055

0.054

0.051

0.036

X-diameter (mm)

3.0

3.3

4.7

3.8

Y-diameter (mm)

3.5

2.8

4.8

3.6

35 %

70 %

68 %

75 %

90◦ dipole ﬁll

215

7.5

Conclusion

The DIANA high energy accelerator presented an interesting opportunity to use the experience obtained during the earlier parts of this thesis, the same simulation software, and
the utility programs developed in this thesis, to model and improve a commercial 3 MV
electrostatic accelerator system. The requirements for the DIANA high energy accelerator
are to deliver H 1+ and He1+ beams of close to 1 emA beam current throughout the energy
range of 300 keV to 3 MeV, as well as selected heavier ions at lower currents. In order to
achieve this, extensive simulations using IGUN and WARP have been performed and custom
modiﬁcations were proposed to the vendor (NEC). Good communication with the vendor
was established and, after going through a few design iterations, a ﬁnal version was approved
for technical feasibility by the vendor. It is an oﬀ-the-shelf model of a 3 MV NEC Pelletron
with modiﬁed ion source extraction system, an adjusted voltage gradient in the ﬁrst unit of
the accelerator column, shorting rod technique, and an additional electrostatic quadrupole
triplet inside the column. Simulations of four test cases were carried out in order to test
the ﬁnal version of the modiﬁed accelerator. It could be shown that this design is able to
deliver beams close to the DIANA requirements. Only in the case of the lowest energy the
maximum obtainable beam current is reduced due to the use of a collimation system for preserving beam quality. The particle distributions created in this work were used by another
group in the DIANA collaboration to simulate the ﬁnal transport of the beam to the targets
with good results. DIANA has concluded the preliminary design report (PDR) stage and a
proposal for a 6-year plan to build and commission the facility is being put forward at the
moment.

216

Chapter 8
Conclusion
As discussed in the introduction, simulations are an important tool for designing and testing
extraction systems and LEBT systems for ECR ion sources. In order to be able to accurately
simulate these, the initial conditions inside the plasma need to be known. Furthermore,
a triangular structure of the beam due to the sextupole conﬁnement ﬁeld, multispecies
extraction and the presence of a strong solenoidal magnetic ﬁeld at the plane of extraction
increase the diﬃculty of such extraction simulations. To accurately transport the ion beam
through the LEBT, the level of space charge compensation needs to be known. I addressed
these concerns and challenges through novel simulation methods and measurements. The
following summary and outlook are essentially recapitulations of the respective conclusion
sections at the end of the major chapters of the thesis.

8.1
8.1.1

Summary
Space Charge Compensation Measurements

To answer the question about space charge compensation in typical ECRIS LEBT systems,
I presented neutralization measurements at two diﬀerent locations: in the beam line of the
LEDA injector source and in the beam line of the SuSI superconducting ECRIS.
The LEDA measurements agree with previous measurements at the same source with a

217

similar detector and show neutralization values around 80-90 %, which depend on beam
intensity. It is possible to calculate the beam radius from the RFA spectra alongside the
neutralization. In case of the LEDA source this yielded very large beam radii which could not
be explained. A similar discrepancy showed up in previous measurements at the same source
by Ferdinand et al. [7]. Theoretical calculations using reasonable beam radii compared well
with the measured neutralization, however.
The measurements in the beam line of SuSI suggest overall low neutralization factors (0 % 50 %) which compare reasonably well with the theoretical model introduced by Gabovich [51]
and discussed in detail in Chapter 2. They follow the prediction of higher neutralization with
higher residual gas pressure, larger beam radii and higher beam density. For typical ECRIS
LEBT settings, where the beam line pressure is kept as low as possible to prevent losses
due to charge exchange, this means very low neutralization factors and beam lines should
be designed accordingly. The reasonable agreement of the model with the measurements
suggested incorporating the simple theoretical treatment into the simulation of low energy
beam transport.

8.1.2

ECRIS Extraction and Beam Line Simulations

In this chapter, I presented the three parts of the ECRIS extraction suite using the particle in cell code WARP. The initial conditions are obtained semi-empirically from plasma
markings inside the source. The particles are then extracted from the plasma by means
of a relaxation process in which the steady state solution and the plasma meniscus form
self-consistently. The hybrid RZ-3D model called 2D+ saves computation time and space
and compares reasonably well with full 3D simulations. The use of the same simulation code
(WARP) for the two parts of the ion source simulation and the beam transport together with
218

a number of python utility programs and GUI’s lets the user seamlessly go from one step to
the next thereby obtaining a end-to-end simulation of an ECRIS driven accelerator frontend.
As will be shown in chapter 7 sometimes it is even possible to include the accelerator into
the simulation.
First tests of the extraction simulations with uranium and helium beams extracted from
VENUS suggest the simulation model can be used for medium to heavy ions of medium to
high charge states, but the approach to obtain the initial conditions breaks down for very
light singly ionized ions. A more self-consistent model of the plasma as e.g. suggested by
Mironov et al. [20] might help obtaining more realistic initial distributions for the subsequent
extraction simulation.
The implementation of the multispecies extension of the formula derived by Gabovich et
al. [51] as part of the LEBT beam transport simulation yielded promising ﬁrst results and
gave interesting insights in the distribution of space charge compensation along the beam
line. It might be possible that in the region before the analyzing magnet, space charge
compensation is as high as 90% due to the high beam current and large average beam
radius. This has to be conﬁrmed by future measurements, though.

8.1.3

The DIANA High Energy Accelerator

The DIANA high energy accelerator presented an interesting opportunity to use the experience obtained during the earlier parts of this thesis, the same simulation software, and
the utility programs developed in this thesis, to model and improve a commercial 3 MV
electrostatic accelerator system. The requirements for the DIANA high energy accelerator
are to deliver H 1+ and He1+ beams of close to 1 emA beam current throughout the energy
range of 300 keV to 3 MeV, as well as selected heavier ions at lower currents. In order to
219

achieve this, extensive simulations using IGUN and WARP have been performed and custom
modiﬁcations were proposed to the vendor (NEC). Good communication with the vendor
was established and, after going through a few design iterations, a ﬁnal version was approved
for technical feasibility by the vendor. It is an oﬀ-the-shelf model of a 3 MV NEC Pelletron
with modiﬁed ion source extraction system, an adjusted voltage gradient in the ﬁrst unit of
the accelerator column, shorting rod technique, and an additional electrostatic quadrupole
triplet inside the column. Simulations of four test cases were carried out in order to test
the ﬁnal version of the modiﬁed accelerator. It could be shown that this design is able to
deliver beams close to the DIANA requirements. Only in the case of the lowest energy the
maximum obtainable beam current is reduced due to the use of a collimation system for preserving beam quality. The particle distributions created in this work were used by another
group in the DIANA collaboration to simulate the ﬁnal transport of the beam to the targets
with good results. DIANA has concluded the preliminary design report (PDR) stage and a
proposal for a 6-year plan to build and commission the facility is being put forward at the
moment.

8.2

Outlook

Whenever a question is answered, at least two more arise and so it is not surprising that
there are a number of possibilities to continue this work.

8.2.1

Space Charge Compensation Measurements

As pointed out in Section 5.4, the data sets taken at the two additional locations (Artemis
beam line, SuSI diagnostic box Q001) may still yield useful results upon further investigation

220

and especially the SuSI/Q001 data might result in a validation of the 90% neutralization
predicted by the simple Gabovich formula. The analysis methods presented in this chapter
can be extended to accommodate multispecies ion beams and similar collimator transmission curves as used in the LEDA injector source analysis can be obtained for SuSI Q001.
Preliminary analysis of data from the two locations suggests similar ﬁndings as at the other
locations (neutralization of 0% to 50% depending on beam size, beam current, and residual
gas pressure).
In order to settle questions of beam aspect ratio and to improve the statistics of the measurement by increasing the total measured current, a second identical detector put perpendicular
or under 45◦ to the existing detector might prove useful. The beam current monitors used
at the NSCL already provide two input channels and electron suppressor (V3 ) as well as
retarding ﬁeld (V2 ) could be provided in parallel by the same power supplies. Q014 in the
SuSI beam line with the perpendicular ports for the Allison emittance scanners would be an
ideal starting point.
The RFA setup as it is is certainly improvable as well. Possible changes that could beneﬁt
the setup are
• Applying a bias to the collector plate to suppress oscillations.
• Exploring the option of heating the detector in order to suppress insulating ﬁlm deposition on the grids.
• Trying diﬀerent combinations of apertures in the collimation system (e.g. taking out
the inner aperture, using smaller apertures).
In addition, one might envision the use of NeMo as non-invasive beam diagnostic device. If
all processes are well understood and pressure and residual gas composition are known, a
221

RFA could be used to measure the beam current, beam radius, and beam stability.

8.2.2

ECRIS Extraction and Beam Line Simulations

While progress has been made in this thesis towards obtaining realistic initial particle distributions, they are still an approximation based on measured plasma markings. Here it
would be a great step forward to use the results of realistic ECR plasma simulations as
initial conditions which can self-consistently provide triangular distributions of varying size
for the diﬀerent ion species according to the plasma parameters. Since arbitrary particle
distributions can readily be loaded into the simulation, the extraction simulation and beam
transport could immediately be upgraded if a working plasma model was presented.
The theoretical model for the neutralization of multispecies ion beams would greatly beneﬁt
from a further generalization to beams with diﬀerent beam radii for each species. Unfortunately, this is not a trivial task and maybe even the scope of another PhD thesis.
Of course, the GUI programs developed for the ECRIS WARP simulations can be extended
in any number of ways from conveniently shifting electrodes in the electrode designer to a
full-blown GUI frontend for WARP, the possibilities are endless.

8.2.3

The DIANA High Energy Accelerator

Unfortunately, a few weeks ago, the decision was made to discontinue the DIANA project
for now. At this point I can only say: I really hope funding will be found at some point, this
is a great project!

222

APPENDICES

223

Publications and Proceedings
Papers and Refereed Proceedings
2012
Progress towards the development of a realistic electron cyclotron resonance ion source extraction model
D. Winklehner, D. Leitner, J. Y. Benitez, C. M. Lyneis, M. M. Strohmeier
Rev. Sci. Instrum. 83, 02B706 (2012)
DIANA - A Deep Underground Accelerator for Nuclear Astrophysics Experiments
D. Winklehner, A. Lemut, D. Leitner, M. Couder, A. Hodgkinson, M. Wiescher
AIP Proceedings CAARI (2012)

2011
Ion beam properties for ECR ion source injector systems
D. Leitner, D. Winklehner, M. Strohmeier
JINST 6 P07010 (2011) (http://jinst.sissa.it/jinst/)
Design of a 400 kV deep underground, high detector eﬃciency, high target density, high beam
intensity accelerator facility
A. Lemut, M. Couder, D. Winklehner, U. Greife, A. Hodgkinson, D. Leitner, M. Leitner,
J. S. Saba, P. A. Vetter, W. L. Waldron, M. Wiescher
Phys. Rev. ST Accel. Beams 14, 100101 (2011)

2010
Comparison of extraction and beam transport simulations with emittance measurements from
the ECR ion source VENUS

224

D. Winklehner, D. Todd, J. Benitez, M. Strohmeier, D. Grote and D. Leitner
JINST 5 P12001 (2010) (http://jinst.sissa.it/jinst/)

Conferences and Workshops
2012
Space Charge Compensation Measurements of Multi-charged Ion Beams Extracted from ECR
Ion Sources
D. Winklehner, D. Leitner, G. Machicoane, F. Marti, D. Cole, L. Tobos
Oral presentation: 20th Int. Workshop on ECR Ion Sources (ECRIS 2012) Sydney, Australia
(2012)
DIANA - A Deep Underground Ion Accelerator for Nuclear Astrophysics Experiments
D. Winklehner, A. Lemut, D. Leitner, M. Couder, A. Hodgkinson, M. Wiescher
Oral presentation: MSU Graduate Academic Conference, East Lansing, USA (2012)

2011
ECRIS extraction and LEBT simulations with the PIC code WARP
D. Winklehner, D. Leitner
Poster: 3rd DITANET School on beam diagnostics, Stockholm, Sweden (2011)

2010
Plasma-to-target WARP simulations of Uranium beams extracted from VENUS compared to
emittance measurements and beam images
D. Winklehner, D. Todd, D.Grote, D. Leitner, J. Benitez, M. Strohmeier
Oral Presentation: 19th International Workshop on ECR Ion Sources (ECRIS 2010) Grenoble, France (2010)
Comparison of extraction and beam transport simulations with emittance measurements from
the ECR ion source VENUS
D. Winklehner, D. Todd, D. Grote, D. Leitner, J. Benitez, M. Strohmeier, F. Aumayr
Poster: 14th Beam Instrumentation Workshop (BIW 2010), Santa Fe, NM, USA (2010)

225

DIANA - Resistor Values
The full set of resistor values, both for the idealized case where all values are unique and the
practical case that approximates the ideal case piecewise linearly by using only 5 diﬀerent
resistor values (not counting immersion lens), is shown here in Table 1.
Table 1: DIANA - Resistor values.
Element #

(m)
Total
Immersion lens
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17

Z

R
ideal
(normalized to 1)

R
practical
(MΩ)

0.00025
0.01320
0.02615
0.03910
0.05205
0.06500
0.07795
0.09090
0.10385
0.11680
0.12975
0.14270
0.15565
0.16860
0.18155
0.19450
0.20745
0.22040

1
0.013418
0
0.000498
0.000756
0.000899
0.001006
0.001095
0.001171
0.001238
0.001298
0.001354
0.001405
0.001452
0.001497
0.001539
0.001579
0.001618
0.001654

734
0
40
40
40
75
75
75
75
75
75
75
75
100
100
100
100
100

Continued on next page...
226

Table 1 (cont’d)
Element #

Z (m)

R (norm.)

R (MΩ)

18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50

0.23335
0.24630
0.25925
0.27220
0.28515
0.28515
0.29810
0.31105
0.32400
0.33695
0.34990
0.36285
0.37580
0.38875
0.40170
0.41465
0.42760
0.44055
0.45350
0.46645
0.47940
0.49235
0.50530
0.51825
0.53120
0.54415
0.55710
0.57005
0.57005
0.58300
0.59595
0.60890
0.62185

0.001689
0.001722
0.001754
0.001785
0
0
0
0.001816
0.001845
0.001873
0.001900
0.001927
0.001953
0.001978
0.002002
0.002026
0.002050
0.002073
0.002095
0.002117
0.002138
0.002160
0.002180
0.002200
0.002220
0.002240
0.002259
0
0
0
0.009185
0.009185
0.009185

100
100
100
100
0
0
0
124
124
124
124
124
124
124
124
124
124
124
124
124
124
124
124
124
124
124
124
0
0
0
555
555
555

Continued on next page...
227

Table 1 (cont’d)
Element #

Z (m)

R (norm.)

R (MΩ)

51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83

0.63480
0.64775
0.66070
0.67365
0.68660
0.69955
0.71250
0.72545
0.73840
0.75135
0.76430
0.77725
0.79020
0.80315
0.81610
0.82905
0.84200
0.85495
1.17205
1.18500
1.19795
1.21090
1.22385
1.23680
1.24975
1.26270
1.27565
1.28860
1.30155
1.31450
1.32745
1.34040
1.35335

0.009185
0.009185
0.009185
0.009185
0.009185
0.009185
0.009185
0.009185
0.009185
0.009185
0.009185
0.009185
0.009185
0.009185
0.009185
0.009185
0.009185
0
0
0
0.009185
0.009185
0.009185
0.009185
0.009185
0.009185
0.009185
0.009185
0.009185
0.009185
0.009185
0.009185
0.009185

555
555
555
555
555
555
555
555
555
555
555
555
555
555
555
555
555
0
0
0
555
555
555
555
555
555
555
555
555
555
555
555
555

Continued on next page...
228

Table 1 (cont’d)
Element #

Z (m)

R (norm.)

R (MΩ)

84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116

1.36630
1.37925
1.39220
1.40515
1.41810
1.43105
1.44400
1.45695
1.45695
1.46990
1.48285
1.49580
1.50875
1.52170
1.53465
1.54760
1.56055
1.57350
1.58645
1.59940
1.61235
1.62530
1.63825
1.65120
1.66415
1.67710
1.69005
1.70300
1.71595
1.72890
1.74185
1.74185
1.75480

0.009185
0.009185
0.009185
0.009185
0.009185
0.009185
0.009185
0
0
0
0.009185
0.009185
0.009185
0.009185
0.009185
0.009185
0.009185
0.009185
0.009185
0.009185
0.009185
0.009185
0.009185
0.009185
0.009185
0.009185
0.009185
0.009185
0.009185
0.009185
0
0
0

555
555
555
555
555
555
555
0
0
0
555
555
555
555
555
555
555
555
555
555
555
555
555
555
555
555
555
555
555
555
0
0
0

Continued on next page...
229

Table 1 (cont’d)
Element #

Z (m)

R (norm.)

R (MΩ)

117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149

1.76775
1.78070
1.79365
1.80660
1.81955
1.83250
1.84545
1.85840
1.87135
1.88430
1.89725
1.91020
1.92315
1.93610
1.94905
1.96200
1.97495
1.98790
2.00085
2.01380
2.02675
2.02675
2.03970
2.05265
2.06560
2.07855
2.09150
2.10445
2.11740
2.13035
2.14330
2.15625
2.16920

0.009185
0.009185
0.009185
0.009185
0.009185
0.009185
0.009185
0.009185
0.009185
0.009185
0.009185
0.009185
0.009185
0.009185
0.009185
0.009185
0.009185
0.009185
0.009185
0.009185
0
0
0
0.009185
0.009185
0.009185
0.009185
0.009185
0.009185
0.009185
0.009185
0.009185
0.009185

555
555
555
555
555
555
555
555
555
555
555
555
555
555
555
555
555
555
555
555
0
0
0
555
555
555
555
555
555
555
555
555
555

Continued on next page...
230

Table 1 (cont’d)
Element #

Z (m)

R (norm.)

R (MΩ)

150
151
152
153
154
155
156
157
158
159
160

2.18215
2.19510
2.20805
2.22100
2.23395
2.24690
2.25985
2.27280
2.28575
2.29870
2.31165

0.009185
0.009185
0.009185
0.009185
0.009185
0.009185
0.009185
0.009185
0.009185
0.009185
0

555
555
555
555
555
555
555
555
555
555
0

231

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232

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