Ion Mass Spectrometry on the
Alcator C-Mod Tokamak
by
Robert Thomas Nachtrieb
B.S., Nuclear Engineering (1993)
University of Illinois, Urbana-Champaign
Submitted to the Department of Nuclear Engineering
in partial fulfillment of the requirements for the degree of
Doctor of Science in Applied Plasma Physics
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
March 2000
c 2000 Massachusetts Institute of Technology. All rights reserved.
Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Department of Nuclear Engineering
March 3, 2000
Certified by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Brian L. LaBombard
Research Scientist, Plasma Science and Fusion Center
Thesis Supervisor
Certified by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Ian H. Hutchinson
Professor, Department of Nuclear Engineering
Thesis Reader
Accepted by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Sow-Hsin Chen
Professor, Department of Nuclear Engineering
Chairman, Department Committee on Graduate Students
2
Ion Mass Spectrometry on the
Alcator C-Mod Tokamak
by
Robert Thomas Nachtrieb
Submitted to the Department of Nuclear Engineering
on March 3, 2000, in partial fulfillment of the
requirements for the degree of
Doctor of Science in Applied Plasma Physics
Abstract
A new ion mass spectrometry probe that operates at high magnetic field (∼ 8 tesla)
has been recently commissioned on Alcator C-Mod. The probe combines an omegatron E(t) × B ion mass spectrometer and a retarding field energy analyzer. The probe
samples the plasma in the far scrape-off layer (SOL), on flux surfaces between 25 and
50 millimeters from the separatrix.
Radio frequency (RF) power is used to collect ions with resonant cyclotron frequency on the side walls of an RF cavity. Scanning the frequency results in a spectrum
ordered by the ratio of ion mass to charge, M/Z. Resonances are resolved down to
signal levels as low as 5×10−4 times the bulk plasma species. Well-resolved resonances
have widths within a factor of two of theoretical values obtained from single-particle
orbit theory.
Impurity fluxes incident on the omegatron are quantified by varying the applied
RF power and recording the change of the amplitude of the resonant ion current.
Similar to that expected from theory, the resonant current I is observed to vary with
power P as I ≈ c0 (1 − e−P/c1 ). From the fitting parameters c0 and c1 it is possible to
extract absolute impurity flux and individual impurity temperature, respectively.
The ion spectra obtained by the omegatron probe always show the M/Z = 2
resonance dominant in deuterium plasmas. Most of the other persistant resonances
can be attributed to charge states of intrinsic impurities 10 B, 11B, and 12C with concentrations of a few percent. Resonances corresponding to charged states of 1 H, 3 He,
4
He, and 14N have been observed upon puffing those gases into tokamak discharges.
Impurity transport studies in the SOL are performed by puffing 3He gas into
tokamak plasmas. The ratio of charged state fluxes measured by the omegatron indicates that helium, which ionizes near the separatrix, is transported rapidly to the far
SOL plasma. Experimental measurements are matched by a one-dimensional radial
transport model with an outward convection velocity of 100 m/s and perpendicular
diffusion coefficient of 2 m2/s.
Results from the retarding field energy analyzer indicate that in ohmic L-Mode
plasmas the bulk ions have a two-temperature distribution, with 90% cold at the
Franck-Condon energy and the remainder hot at 20 electron volts, possibly the result
of charge exchange with fast neutrals. Significant secondary electron emission is
observed, which has important consequences for estimates of sputtering yields through
the influence on the sheath potential.
Thesis Supervisor: Brian L. LaBombard
Title: Research Scientist, Plasma Science and Fusion Center
4
Acknowledgments
I wish to acknowledge here just a few of the many people who helped bring this thesis
to conclusion.
Prof. Roy Axford set an early example for me of the highest mathematic and
scientific standards. Prof. Elias Gyftopoulos taught me to check premises all the way
back to the axioms, and cautioned me not to substitute familiarity for understanding.
I had fruitful discussions with Drs. P.C. Stangeby and G.M. McCracken regarding
sheath physics and mass spectrometer theory of the omegatron.
I thank the entire Alcator team, a dedicated and professional group, with whom I
thoroughly enjoyed working. Drs. Bruce Lipschultz and John Goetz brought me into
the group, and Prof. Ian Hutchinson as Alcator head renewed my funding semester
after semester while I fixed the omegatron. Prof. Hutchinson also served as thesis
reader and offered valuable advice at every stage of the thesis. Drs. Earl Marmar
and Jim Terry patiently answered my many questions. Ed Thomas Jr. (now Dr.)
with Dr. Brian LaBombard performed all the initial design work on the omegatron
hardware and electronics. Kathy Powers and Jason Thomas at the PSFC Library
were consistently helpful and friendly.
I profitted tremendously from discussions, debates, and derivations with fellow
graduate students and good friends, especially Chris Boswell, Sanjay Gangadhara,
Darren Garnier, Damien Hicks, Tom Hsu, Chris Kurz, Pete O’Shea, Jim Reardon,
Jeff Schachter, and Joe Sorci. Special thanks to Jeff, the continental version, who
introduced me to some great books, and to Darren and Suanne for their hospitality.
Profound thanks go to my advisor Brian LaBombard for being so generous with
his time, his electronics and physics insights, and his unflagging and inspirational
enthusiasm. Scores of times I have interrupted his own work for a “few minutes”
and we have ended up talking for hours about omegatron details. During my visits
the whiteboard usually gets covered with his colorful circuit diagrams, sketches of
hardware modifications, and graphical theoretical explanations. Although my name
appears alone on this thesis Brian surely deserves to be co-author.
Finally I thank my parents for their continuous support, and my wife Loretta for
her love and patience.
5
6
Contents
1 Introduction
27
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
1.1.1
Why Fusion? . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
1.1.2
Magnetic Confinement Fusion . . . . . . . . . . . . . . . . . .
28
1.1.3
Progress To Date . . . . . . . . . . . . . . . . . . . . . . . . .
33
1.2 Edge Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
1.2.1
Definition of Edge Plasma . . . . . . . . . . . . . . . . . . . .
37
1.2.2
Heat Loads to Wall . . . . . . . . . . . . . . . . . . . . . . . .
38
1.2.3
Impurities from Edge into Core Plasma . . . . . . . . . . . . .
40
1.2.4
Helium Ash Removal . . . . . . . . . . . . . . . . . . . . . . .
41
1.2.5
Influence of Edge Plasma on Core Plasma Properties . . . . .
42
1.3 Ion Mass Spectrometry . . . . . . . . . . . . . . . . . . . . . . . . . .
43
1.3.1
Omegatron History . . . . . . . . . . . . . . . . . . . . . . . .
43
1.3.2
Tokamak Ion Mass Spectrometry . . . . . . . . . . . . . . . .
45
1.3.3
Omegatron on a Tokamak . . . . . . . . . . . . . . . . . . . .
47
1.4 Goals of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
2 Diagnostic Description
51
2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
2.2 Probe Head . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
2.2.1
Internal Components . . . . . . . . . . . . . . . . . . . . . . .
7
53
2.2.2
External Components . . . . . . . . . . . . . . . . . . . . . . .
58
2.3 Linear Motion Subsystem . . . . . . . . . . . . . . . . . . . . . . . .
65
2.4 RF Amplifier Subsystem . . . . . . . . . . . . . . . . . . . . . . . . .
66
2.5 Grid Electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
68
2.6 Langmuir Probe Electronics . . . . . . . . . . . . . . . . . . . . . . .
70
2.7 Thermocouples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
3 Omegatron Probe Theory
73
3.1 Flux Tube Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
3.1.1
Simple Fluid Model . . . . . . . . . . . . . . . . . . . . . . . .
77
3.1.2
Sheath Drop with Secondary Electron Emission . . . . . . . .
80
3.1.3
Collisional Presheath . . . . . . . . . . . . . . . . . . . . . . .
83
3.1.4
Ion Distribution at the Sheath Edge
. . . . . . . . . . . . . .
86
3.2 Slit Transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
91
3.3 Retarding Field Energy Analyzer Model . . . . . . . . . . . . . . . .
96
3.3.1
Brillouin Flow . . . . . . . . . . . . . . . . . . . . . . . . . . .
96
3.3.2
3-D Space Charge . . . . . . . . . . . . . . . . . . . . . . . . .
97
3.3.3
RFEA 1-D Kinetic Model . . . . . . . . . . . . . . . . . . . .
99
3.4 Grid Transmission
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
3.4.1
Reflections from Space Charge . . . . . . . . . . . . . . . . . . 105
3.4.2
Space Charge Potentials . . . . . . . . . . . . . . . . . . . . . 109
3.5 Omegatron Ion Mass Spectrometer Model . . . . . . . . . . . . . . . 111
3.5.1
Single Particle Orbits . . . . . . . . . . . . . . . . . . . . . . . 112
3.5.2
Collection Frequency Range . . . . . . . . . . . . . . . . . . . 114
3.5.3
Dwell Time and Collection Energy Range
3.5.4
Dwell Time with Constant Potential . . . . . . . . . . . . . . 118
3.5.5
Dwell Time with Spatially Varying Potential . . . . . . . . . . 120
3.5.6
Determining Absolute Impurity Fluxes, Densities, and Temper-
. . . . . . . . . . . 117
atures using RF Power Scan . . . . . . . . . . . . . . . . . . . 120
8
3.5.7
Determining Impurity Temperature using RFEA Bias . . . . . 124
3.5.8
Broad Beam Modifications . . . . . . . . . . . . . . . . . . . . 124
3.5.9
Ion-ion Collisions . . . . . . . . . . . . . . . . . . . . . . . . . 127
4 Retarding Field Energy Analysis
129
4.1 Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
4.1.1
Current-Voltage Characteristic Features . . . . . . . . . . . . 129
4.1.2
Effect of ICRF . . . . . . . . . . . . . . . . . . . . . . . . . . 131
4.1.3
Flux Tube Boundaries . . . . . . . . . . . . . . . . . . . . . . 133
4.1.4
Effect of Magnetic Field Direction . . . . . . . . . . . . . . . . 133
4.2 Discussion of Characteristic Features . . . . . . . . . . . . . . . . . . 133
4.2.1
Comparison of IV Characteristic with Simple Theory . . . . . 136
4.2.2
Grid Transmission, Current Accounting . . . . . . . . . . . . . 140
4.2.3
Slit Transmission . . . . . . . . . . . . . . . . . . . . . . . . . 143
4.2.4
Slit Bias Scan . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
4.2.5
Space Charge . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
4.2.6
Secondary Electron Emission . . . . . . . . . . . . . . . . . . 148
4.2.7
Summary of Conclusions . . . . . . . . . . . . . . . . . . . . . 152
4.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
4.3.1
Time History of a Tokamak Discharge . . . . . . . . . . . . . 155
4.3.2
SOL Profiles: Ohmic Plasma . . . . . . . . . . . . . . . . . . . 155
4.3.3
SOL Profiles: ICRF Plasma . . . . . . . . . . . . . . . . . . . 158
4.3.4
Implications of Two-Temperature Ion Distribution . . . . . . . 160
4.3.5
Implications of Secondary Electron Emission . . . . . . . . . . 161
5 Omegatron Ion Mass Spectrometer
163
5.1 Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
5.1.1
Alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
5.1.2
Ambient Noise . . . . . . . . . . . . . . . . . . . . . . . . . . 166
9
5.1.3
Resonant Current . . . . . . . . . . . . . . . . . . . . . . . . . 169
5.1.4
Impurity Spectrum . . . . . . . . . . . . . . . . . . . . . . . . 172
5.1.5
Resonance Width Dependence on Non-resonant Current . . . 178
5.1.6
Resonance Width Dependence on Applied RF Power . . . . . 178
5.1.7
Resonance Amplitude Dependence on Applied RF Power . . . 181
5.1.8
Resonant Current Accounting . . . . . . . . . . . . . . . . . . 181
5.1.9
Summary of Conclusions . . . . . . . . . . . . . . . . . . . . . 186
5.2 Discussion of Spectrum Features . . . . . . . . . . . . . . . . . . . . . 187
5.2.1
Resolution and Broadening . . . . . . . . . . . . . . . . . . . . 187
5.2.2
Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
5.2.3
Oscillator Spectrum . . . . . . . . . . . . . . . . . . . . . . . . 188
5.2.4
Magnetic Fluctuations . . . . . . . . . . . . . . . . . . . . . . 191
5.2.5
Density profile . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
5.2.6
Degeneracies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
5.2.7
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
5.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
6
3
5.3.1
Impurity Densities, Temperatures from Applied RF Power Scan 194
5.3.2
Boronization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
5.3.3
H/D Scan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
5.3.4
Residual Gas Analysis . . . . . . . . . . . . . . . . . . . . . . 202
5.3.5
Neutral Pressure Measurement
He Transport
. . . . . . . . . . . . . . . . . 203
207
6.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
6.2 Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
6.3
3
He+ and 3 He++ Ionization in Local Flux Tube . . . . . . . . . . . . 213
6.4 Cross-Field Transport in Local SOL . . . . . . . . . . . . . . . . . . . 215
6.4.1
Deuterium Source in Local SOL . . . . . . . . . . . . . . . . . 219
6.5 Cross-Field 3He Transport Model . . . . . . . . . . . . . . . . . . . . 220
10
6.5.1
SOL Background Profiles . . . . . . . . . . . . . . . . . . . . . 223
6.5.2
Neutral Density Profile . . . . . . . . . . . . . . . . . . . . . . 223
6.6 Analytic Slab Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
6.7 Numerical Model with Experimental Profiles . . . . . . . . . . . . . . 226
6.8 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
6.8.1
Neglect of Recombination . . . . . . . . . . . . . . . . . . . . 231
6.8.2
Anomalous Cross-Field Transport . . . . . . . . . . . . . . . . 233
7 Summary
235
7.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
7.1.1
Hardware . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
7.1.2
Retarding Field Energy Analyzer . . . . . . . . . . . . . . . . 236
7.1.3
Ion Mass Spectrometer . . . . . . . . . . . . . . . . . . . . . . 238
7.1.4
3
He Transport in the Scrape-Off Layer . . . . . . . . . . . . . 238
7.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
7.2.1
Diagnostic Improvements . . . . . . . . . . . . . . . . . . . . . 239
7.2.2
Physics Experiments . . . . . . . . . . . . . . . . . . . . . . . 240
A Calculations
243
A.1 Kinetic Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
A.1.1 Single Particle . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
A.1.2 One-particle Distribution . . . . . . . . . . . . . . . . . . . . . 244
A.1.3 Moments of the Distribution . . . . . . . . . . . . . . . . . . . 246
A.2 Proof of Generalized Bohm Criterion . . . . . . . . . . . . . . . . . . 246
A.3 Hobbs and Wesson Fluid Sheath Model with Secondary Electron Emission
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
A.4 Electrostatic Potential due to a Block of Charge . . . . . . . . . . . . 252
A.5 Electrostatic Potential due to a Ribbon of Charge . . . . . . . . . . . 257
A.6 1-D Space Charge with Shifted Half-Maxwellian . . . . . . . . . . . . 262
11
A.6.1 General Development . . . . . . . . . . . . . . . . . . . . . . . 262
A.6.2 Space Charge Neglected . . . . . . . . . . . . . . . . . . . . . 269
A.6.3 Space Charge Included . . . . . . . . . . . . . . . . . . . . . . 269
A.7 Kinetic Sources and Collision Operators . . . . . . . . . . . . . . . . 274
B Electronics
279
B.1 Camac . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
B.2 Custom Electronics Schematics . . . . . . . . . . . . . . . . . . . . . 281
C Omegatron User’s Manual
287
C.1 Operation Widgets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
C.2 Analysis Widgets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
C.3 Generic Routines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
C.4 Control Routines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294
C.5 Control Routines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
12
List of Figures
1.1 Electrical energy consumption during 1996 of the OECD countries
versus their populations. The United States has the highest population and the highest total electrical energy consumption. Norway
has the highest electrical energy consumption per capita. Reference:
http://www.iea.org/stat.htm . . . . . . . . . . . . . . . . . . . . .
29
1.2 Nested surfaces with constant plasma pressure that result from ideal
MHD equilbrium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
1.3 Schematic of principle components of a tokamak. (Courtesy D. Garnier, 1996) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
1.4 Reaction rate parameter for nuclear fusion reactions with the highest
cross sections at keV temperatures. Reference: L.T. Cox, “Thermonuclear Cross Section and Reaction Rate Parameter Data Compilation,”
Phillips Laboratory, Edwards AFB CA 93523-5000, AL-TR–90-053
.
34
1.5 Lawson parameter nτ required as a function of temperature for different values of Q ≡ Pf /Ph in steady state (dW/dt = 0). “Breakeven” is
defined as Q = 1; “ignition” is defined as Q = ∞. Note that in 1998
the JT60-U tokamak team claimed to reach Q = 1.25 transiently with
“DT equivalent” conditions. . . . . . . . . . . . . . . . . . . . . . . .
36
1.6 Poloidal cross section of the Alcator C-Mod tokamak, with representative plasma last-closed flux surface. . . . . . . . . . . . . . . . . . .
13
39
2.1 Schematic of omegatron probe, showing slit, retarding field energy analyzer, and ion mass spectrometer portions mounted in a shielding box.
Figure courtesy B. LaBombard. . . . . . . . . . . . . . . . . . . . . .
52
2.2 Exploded view of internal components of omegatron probe retarding
field energy analyzer and ion mass spectrometer, showing: slit; grids;
RF plates; RF resistors; end collector; mica spacers and insulators; and
ceramic spacers and supports. Wires to the grids, RF plates, and RF
resistors omitted for clarity. . . . . . . . . . . . . . . . . . . . . . . .
54
2.3 Magnified image of the tungsten grid used on the omegatron probe.
Grid lines are 24.5 µm wide with 144 µm in between. Image courtesy
D. Hicks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
2.4 Exploded view of external components of omegatron probe, showing:
heat shield; shield box; coverplate; patch panel; mounting plate; lock
plate; and support plate. All wires and SMA connectors omitted for
clarity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
2.5 Poloidal cross section of Alcator C-Mod tokamak showing omegatron
(mirror image) inserted into upper divertor scrape-off layer plasma and
fast scanning Langmuir probe near midplane inserted to separatrix. .
62
2.6 Omegatron probe (mirror image) on Alcator C-Mod tokamak. Representative flux surfaces are shown, spaced two millimeters apart at the
midplane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
2.7 Block diagram of linear motion subsystem. . . . . . . . . . . . . . . .
64
2.8 Block diagram of RF amplifier subsystem. . . . . . . . . . . . . . . .
67
2.9 Block diagram of grid electronics board. Grids G1, G2, G3, RF plates,
and END collector each have a separate electronics board. . . . . . .
14
69
2.10 Block diagram of Langmuir probe electronics. Langmuir probes LP1,
LP2, LP3, and SLIT each have a separate electronics board. After E.E.
Thomas Jr., Technical Report PFC/RR-93-03, MIT Plasma Fusion
Center, 1993. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
2.11 Block diagram of thermocouples measuring bulk temperature of omegatron heat shield. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
3.1 Schematic of potential of a flux tube. Picture (a): Long flux tube,
L Lp . Picture (b): Short flux tube, L ≈ Lp . . . . . . . . . . . . .
78
3.2 Normalized electron current density to a surface as a function of normalized surface bias with different secondary electron emission coefficients γ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
82
3.3 Comparison of parallel transport time with characteristic slowing down
times and temperature equilibration times, for 20 eV ion minority (top)
or 3 eV ion minority (bottom) on 3 eV ion bulk. . . . . . . . . . . . .
85
3.4 Schematic of cross section of slit geometry, showing gap between 45
degree knife edges. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
91
3.5 Energy transmission function of deuterions through the slit for B = 5 T
and l = 25 µm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
3.6 Relative transmission through a slit with spacing d, edge thickness
t, angle θ, of a half-Maxwellian distribution with of temperature kT
shifted by energy qφ0 = w02 /2. Relative transmission decreases with
finite edge thickness. . . . . . . . . . . . . . . . . . . . . . . . . . . .
95
3.7 Schematic of the influence of space charge on the electrostatic potential
between two parallel surfaces of fixed potential. . . . . . . . . . . . .
97
3.8 Schematic cross section of omegatron and axial vacuum potential structure. Configuration with G2 as ion parallel energy selector is shown,
with SLIT grounded, V = 0 V. . . . . . . . . . . . . . . . . . . . . . . 101
15
3.9 Sketch of transmission of ions through grids if pitch angle is sufficiently
steep. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
3.10 Theoretical transmission of ions through the grid. Note the different
scales. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
3.11 Schematic of electrostatic potentials inside the omegatron. Grid bias
and/or potential due to space charge can reflect incoming ion flux. . . 106
3.12 Schematic of incident and reflected fluxes to all grids, normalized to
incident flux to grid G1. Each grid is assumed to attenuate the flux
passing through it in either direction by a factor ξ. A fraction gj of
the incident flux that passes through the jth grid arrives at the next
component downstream. . . . . . . . . . . . . . . . . . . . . . . . . . 107
3.13 Theoretical normalized current collected on RF plates as a function
of frequency for a typical cyclotron frequency for deuterium at the
omegatron location, ωc /(2π) ≈ 36 MHz, b = 2.6 and a = 1, 1/2, 1/8. . 126
4.1 Current-voltage characteristics from the omegatron in retarding field
energy analyzer mode. Dashed line is raw current to END collector,
solid line is current to END collector normalized by sum of currents to
grids G1, G2, G3, and to END collector and scaled to agree with the
raw saturation current. . . . . . . . . . . . . . . . . . . . . . . . . . . 130
4.2 IV characteristics before and during 2.5 MW of ion cyclotron resonance
auxiliary heating. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
4.3 Top: magnetic field lines tracing from omegatron probe face to molybdenum tiles on E-side tiles D-E limiter. Bottom: magnetic field line
connection lengths from omegatron probe for a typical plasma equilibrium and for different insertion depths. “Plunge” is insertion depth
from rest position. For equilibrium shown, insertion of 37 mm corresponds to poloidal flux surface ρ = 47 mm. Figures courtesy B. LaBombard. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
16
4.4 IV characteristics with normal field (B×∇B down) and abnormal field
(B × ∇B up). Current is always parallel to toroidal field to preserve
helicity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
4.5 Fractions of total measured current (G1+G2+G3+END) to G1, G2,
G3, and END as a function of voltage bias on G2, G3, and RF. Current
fraction to RF is always less than 10−3 . . . . . . . . . . . . . . . . . . 142
4.6 Current-voltage characteristics for ions and electrons for omegatron in
RFEA mode with different SLIT biases. Ion characterstics are largely
unaffected below 0 V, but shift above 0 V. Vertical lines correspond to
SLIT biases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
4.7 Current-voltage characteristics from the omegatron in retarding field
energy analyzer mode taken at different depths in the scrape-off layer
plasma. Current is obtained by normalizing END collector current by
sum of currents to grids G1, G2, G3, and to END collector and scaling
to agree with the average END collector current at reflector bias below
−40 V. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
4.8 Bias arrangement for secondary electron emission measurments. . . . 149
4.9 Effective coefficient of secondary electron emission versus acceleration
voltage. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
4.10 Processed IV characteristic, showing values of cold and hot ion temperatures and knee potential. Floating potential is obtained from Langmuir probes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
4.11 Ion and electron temperatures and sheath potential as a function of
time during a tokamak discharge. Electron temperature and floating
potential from Langmuir probe LP2 are also shown. . . . . . . . . . . 156
17
4.12 Cross-field profiles of electron and ion temperatures and sheath potential taken from omegatron RFEA and electron density, temperature,
and floating potential from Langmuir probe LP1, taken during ohmic
tokamak operation. ρ is the distance of the flux surface from the separatrix, measured at the midplane. . . . . . . . . . . . . . . . . . . . 157
4.13 Cross-field profiles of electron and ion temperatures and sheath potential taken from omegatron RFEA and electron density, temperature,
and floating potential from Langmuir probe LP1, taken during ICRFheated tokamak operation. ρ is the distance of the flux surface from
the separatrix, measured at the midplane. . . . . . . . . . . . . . . . 159
5.1 Electron current signal recorded on the RF plates as a function of
rotation of the omegatron about the vertical axis. . . . . . . . . . . . 165
5.2 Schematic of rotation of omegatron RF plates, viewed toroidally. Horizontal line between the plates represents the slit. Figure to left is
aligned, figure to right is rotated beyond cutoff. . . . . . . . . . . . . 165
5.3 Omegatron ambient noise spectrum without plasma (top), with plasma
but omegatron withdrawn (middle), and with plasma and omegatron
inserted (bottom).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
5.4 Top: applied RF frequency and resulting resonant frequency as functions of time. Bottom: resonant current vs applied RF frequency. Solid
line is current signal binned over regions 0.25 MHz wide, chosen to be
close to the theoretically expected resonance width. . . . . . . . . . . 170
5.5 Typical impurity spectrum: ratio of resonant current to non-resonant
current as a function of ratio species mass and charge. Annotations
near resonances identify possible isotopes. . . . . . . . . . . . . . . . 173
18
5.6 Top: intensity of spectroscopic line from helium versus time, looking
at the helium puff location. Middle: frequency of RF power applied
to omegatron versus time. Bottom: ratio of resonant ion current to
non-resonant ion current versus time. . . . . . . . . . . . . . . . . . . 177
5.7 Resonance widths of M/Z = 4 versus fluctuating non-resonant current,
showing contributions of Brillouin flow broadening, intrinsic broadening, and magnetic field variation. . . . . . . . . . . . . . . . . . . . . 179
5.8 Resonance widths of 3He+ and 3 He2+ versus applied RF power. Lower
solid lines represents single-particle prediction for homogenous magnetic field; upper solid line includes Brillouin flow broadening, assuming fluctuating beam current ∆I ≈ I, (ωc − ωr )/I = 0.007; dashed
lines include corrections for magnetic field variation. . . . . . . . . . . 180
5.9 Top: Normalized resonant ion current versus applied RF power. Solid
line is least squares fit of function y = c0 (1− e−x/c1 ); dotted lines represent one standard deviation change in each fitted parameter. Bottom:
Frequency full width at half maximum of resonance amplitude. Smooth
line is value predicted by theory including magnetic field variation,
Brillouin flow broadening with ∆I ≈ I, and intrinsic single particle
broadening. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
5.10 Current to grid G3, RF plates, and end collector for RF frequency fixed
at center frequency of bulk ion resonance (M/Z = 2) and RF power
switched between 0 watts and 8 watts. . . . . . . . . . . . . . . . . . 183
5.11 Influence of space charge on the magnitude of resonant current collected and on the fraction of the resonant current collected. Current
was decreased by withdrawing the omegatron further from the separatrix.185
19
5.12 Birdy circuit output and the calibrated frequency monitor (MHz) as
functions of time. The steps in the birdy signal are caused by the finite
resolution of the Bira frequency programming signal, corresponding to
approximately 25 kHz per bit. . . . . . . . . . . . . . . . . . . . . . . 189
5.13 Harmonics produced by the Wavetek model 1062 RF oscillator. Lines
connect the jth harmonic, j = 0 is the fundamental. . . . . . . . . . . 190
5.14 Fluctuation spectrum of poloidal magnetic field, recorded from poloidal
field coil BP09 JK near the omegatron. . . . . . . . . . . . . . . . . . 191
5.15 Impurity temperatures, flux fractions, and density fractions at sheath
edge, obtained from RF power scan technique for range 3 < M/Z < 12.
Labels identify assumed source of the resonances. . . . . . . . . . . . 195
5.16 Ion impurity spectrum before and after August 1999 boronization.
Note decrease in M/Z = 8 resonance. . . . . . . . . . . . . . . . . . . 198
5.17 Ion impurity spectrum before and after September 1999 boronization.
Note decrease in M/Z = 7 resonance. . . . . . . . . . . . . . . . . . . 199
5.18 Comparison of hydrogen to deuterium (H/D) density ratios from Balmer
spectroscopy and omegatron. Solid line is least-squares fit to data of
the form y = mx, where y represents the omegatron H/D and x represents the Balmer H/D. For comparison, dotted lines have slopes of 2m
and m/2. Omegatron H/D includes corrections for resonance broadening, collisional presheath, and finite applied RF power (assuming
kTH = 3 eV). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
5.19 Omegatron residual gas analyzer spectrum of M/Z of ion species formed
inside the omegatron by electron impact ionization. Note that M/Z =
4 resonance is dominant, probably corresponding to D+
2 . . . . . . . . 202
20
5.20 Neutral pressure in omegatron probe cavity as a function of time during
a tokamak discharge. Spikes represent resonant ion collection with
M/Z = 4 corresponding to D+
2 . Peak value of the spike corresponds to
the neutral pressure. Continuous signal is neutral pressure in E-Top
measured by an MKS baratron gauge.
. . . . . . . . . . . . . . . . . 204
6.1 Schematic of scrape-off layer geometry, showing directions parallel and
perpendicular to the magnetic field, and orientation of omegatron probe
face to separatrix and E-port ICRF limiter. . . . . . . . . . . . . . . 209
6.2 Poloidal cross section of Alcator C-Mod tokamak showing omegatron
(mirror image) inserted into upper divertor scrape-off layer plasma and
fast scanning Langmuir probe near midplane inserted to separatrix. . 212
6.3 Top: 3He impurity spectrum. Bottom: Asymptotic resonant current
fractions due to singly- and doubly-ionized helium, corrected for resonance broadening, assuming T = 3 eV for helium ions. . . . . . . . . . 214
6.4 Scale lengths for ion saturation current and electron density at omegatron face. Asterisks represent measurements from Langmuir probes;
squares represent possible corrections due to misalignment of the head
with local magnetic surfaces. . . . . . . . . . . . . . . . . . . . . . . . 216
6.5 Profiles of electron temperature, electron density, and rates of ionization and radiative recombination in scrape-off layer. Asterisks represent data points, smooth line is spline interpolation. . . . . . . . . . . 224
21
6.6 Comparison of calculated helium fluxes and densities in plasmas with
constant and ramped diffusion coefficient profiles. Solid, dotted, and
dashed lines represents neutral, singly-ionized, and doubly-ionized helium, respectively. Arrow heads indicated experimental data which the
model must match. The case of D⊥ = const, V = 0 yields fluxes which
do not match the observed values. Some form of ramped diffusion coefficient profile is necessary to reproduce experimental observations of
singly-ionized density and flux at the omegatron.
. . . . . . . . . . . 227
6.7 Calculated fluxes (g1 ) and densities (y1) of singly-ionized helium at
the omegatron in plasmas with constant diffusion coefficient profiles.
No constant diffusion coefficient profile reproduces both observed flux,
g1 (x1 ) ≈ 0.7 and observed density, y1 (x1) ≈ 2. . . . . . . . . . . . . . 228
6.8 Calculated density of singly-ionized helium at the omegatron for different ramped profiles of diffusion coefficient. Many different profiles
can reproduce the observed values of density and flux, but all of them
require an increase in diffusion coefficient across the scrape-off layer. . 230
6.9 Calculated density of singly-ionized helium at the omegatron for outward convection velocities with as a function of the amplitude of the
flat diffusion coefficient profile. Many flat profiles can reproduce the
observed values of density and flux, but all of them require an outward
convection velocity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
A.1 Sketch of the distribution of space charge between surfaces at x = ±a,
y = ±b, and z = ±c. Space charge is uniform inside rectangle of height
∆z = 2c , width ∆y = 2b and length ∆x = 2a = 2a, and zero elsewhere.254
22
A.2 Electrostatic potential profiles φ(x, y = 0, z = 0) in boxes of sides
|x| ≤ a, |y| ≤ b, |z| ≤ c. Ions pass through the boxes along x with
current I, velocity v, and cross sectional area 2b × 2c , giving charge
density ρ = I/(4vbc ). Top figure is volume between grids, where space
charge contributes negligibly to electrostatic potential. Bottom figure
is volume between RF plates, where space charge contributes noticibly
to electrostatic potential. . . . . . . . . . . . . . . . . . . . . . . . . . 258
A.3 Sketch of the distribution of space charge between surfaces at x = ±a.
Space charge is uniform inside ribbon of thickness ∆z = 2c and width
∆x = 2a = 2a, and zero elsewhere. . . . . . . . . . . . . . . . . . . . 259
B.1 Electrical schematic of omegatron grid ammeter circuit. . . . . . . . . 282
B.2 Electrical schematic of omegatron RF plate ammeter circuit. . . . . . 283
B.3 Electrical schematic of RF oscillator AM/FM control circuit. . . . . . 284
B.4 Electrical schematic of Langmuir probe ammeter circuit. . . . . . . . 285
C.1 Omegatron power supply and motion control widget. . . . . . . . . . 288
C.2 Omegatron bias and RF waveform widget . . . . . . . . . . . . . . . 289
C.3 Omegatron analysis widget for retarding field energy analyzer IV characteristics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
C.4 Omegatron analysis widget for ion mass spectrometer spectra. . . . . 293
23
24
List of Tables
1.1 Nuclear fusion reactions with the highest cross sections at keV temperatures. Notes: 1. easiest, 2. “advanced” (higher temperature), (3).
aneutronic with parasitic DD neutrons, (3). aneutronic . . . . . . . .
33
2.1 Comparison of slit and grid dimensions of selected tokamak retarding
field energy analyzer probes. All dimensions are in micrometers. . . .
56
3.1 Special cases of grid transmission and current accounting. Notes: (1)
full reflection from G2, (2) full reflection from G3, (3) full reflection
from RF, (4) no reflection. . . . . . . . . . . . . . . . . . . . . . . . . 108
4.1 Fraction of incoming current through slit that arrives at each component. Top number is calculated using attenuation factors, bottom
number is from measurements. . . . . . . . . . . . . . . . . . . . . . . 141
5.1 Frequently observed mass to charge ratios (M/Z) of resonances in spectra obtained with the omegatron, and charged states of isotopes with
nearby M/Z. Gas states of isotopes in parentheses have been puffed
into tokamak discharges; M/Z in parentheses can be attributed to no
other isotope. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
25
5.2 Typical cyclotron frequencies at omegatron location for stable isotopes
of molybdenum and argon within one megahertz of M/Z = 12. Isotopes are not resolved since resonance full width at half maximum is
∆f ≈ 0.5 MHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
A.1 Summary of dimensionless density for different conditions and in different regions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270
A.2 Summary of F (x) for different conditions and in different regions. . . 271
26
Chapter 1
Introduction
The sun is the principle power source for life on Earth, and it is a natural nuclear
fusion reactor. The ultimate objective of the magnetic fusion program is to recreate
on Earth many of the conditions in the sun to produce a new source of electrical
power. While this is a significant technical challenge, nuclear fusion promises to be
an abundant and clean source of power. Specifically, a principle component of the
fuel for fusion power is deuterium which is available in almost limitless quantity in
seawater. Electrical power produced by nuclear fusion would burn no fossil fuels and
would produce no greenhouse gases.
This chapter describes the need for nuclear fusion as a source of electrical power,
the importance of edge plasma physics in fusion research, and the importance of
ion mass spectrometry to edge plasma physics. The direct goal of this thesis is to
contribute to the fusion research effort, to be accomplished indirectly by describing
the construction, theory and operation of the omegatron probe on the Alcator C-Mod
tokamak, and by demonstrating the utility of the omegatron as a tool to study edge
plasma physics.
1.1
Motivation
27
1.1.1
Why Fusion?
The need for nuclear fusion as an abundant and clean power source is suggested
by present and projected energy consumption patterns. In 1996 the world consumed
14,000 TWh electricity, or approximately 2,500 kWh per capita. In the same year the
countries in the Organization for Economic Co-operation and Development (OECD)
consumed approximately 7,600 kWh per capita on average. Figure 1.1 shows the
electrical energy consumption of the OECD countries versus their populations. As
developing countries industrialize their electricity consumption will increase. We can
estimate a lower bound for the long term increase in electrical energy consumption
if we assume the OECD average per capita electricy consumption is typical for industrialized countries and then project industrialization of the entire world. If global
population remained at 1996 levels, we would expect electricity consumption to increase by at least 300%.
The International Energy Association has done a more careful near-term prediction of energy consumption including population changes and projected economic
growth of nations. They predict global energy consumption will increase by 65%
between 1995 and 2020, to be obtained mostly from coal, oil, and natural gas. The
above statistics suggest that fusion can become an important source of electricity in
the long-term, once it becomes too expensive to find or burn fossil fuels.
1.1.2
Magnetic Confinement Fusion
Gravity confines the plasma in the sun. The carbon-nitrogen-oxygen catalyst cycle
accelerates the process of proton-proton nuclear fusion [32, p.534], and the energy
released from the fusion reactions maintains the core temperature in the sun at 15
million Kelvin. To recreate the conditions necessary for nuclear fusion on earth it is
necessary to heat the fusion fuel to similar temperatures, but it is impractical to use
gravity confinement. A promising approach uses magnetic fields generated by electric
coils; it works by exploiting the behavior of charged particles in magnetic fields.
28
Figure 1.1: Electrical energy consumption during 1996 of the OECD countries versus
their populations. The United States has the highest population and the highest total
electrical energy consumption. Norway has the highest electrical energy consumption
per capita. Reference: http://www.iea.org/stat.htm
29
Reactor concepts based on this approach are referred to as “magnetic confinement
fusion” reactors.
A single particle of charge q and mass m moving with velocity v in a magnetic
field B experiences the Lorentz force and thus has the equation of motion
m
dv
= qv × B.
dt
By integrating the equation of motion it can be shown that the particle orbit describes
a helical motion around magnetic field lines with a radius that is inversely proportional
to the magnetic field. The magnetic field constrains the motion of the particle in
directions perpendicular to the magnetic field and has no effect on the motion of the
particle along the magnetic field. If the radius of orbit around the magnetic field line
is small compared to the radius of curvature of the magnetic field line, the particle is
practically “tied” to the magnetic field line. Furthermore, if the magnetic field line
can be made to close on itself in a relatively small region of space (of order meters for
a practical magnetic fusion reactor), the charged particle can be considered confined
to the same region of space.
A fusion plasma has many particles, typically greater than 1020 , and so the single
particle description is inadequate. It is often appropriate to describe the core plasma
with a fluid model known as ideal magnetohydrodynamics (MHD) [15]. In static
equilibrium with the fluid velocity v = 0 and ∂/∂t = 0, the equations which describe
the equilibrium configuration are
J × B = ∇p,
∇ × B = µ0 J,
∇ · B = 0,
where J represents the current density in the plasma, p represents the scalar plasma
pressure, and B represents the magnetic field as before but which now can include
fields generated by the plasma current density. From the equilibrium equations it
follows that B · ∇p = J · ∇p = 0, which means that the magnetic field and the
30
Figure 1.2: Nested surfaces with constant plasma pressure that result from ideal MHD
equilbrium.
plasma current density lie in surfaces of constant plasma pressure. The plasma can
be described as a series of nested surfaces similiar to those shown in Figure 1.2. Thus
a complicated core plasma geometry can be effectively described in one dimension
perpedicular to the surfaces.
For plasma confinement we will be interested in configurations for which the nested
surfaces close on themselves. For configurations described in cylindrical geometry and
that admit toroidal symmetry, the closed surfaces can be obtained through the GradShafranov equation using the poloidal flux coordinate ψ:
1
ψ≡
2π
Bp · dA,
∇ψ
R ∇·
R2
2
= −µ0R2
d
dp
+
(2πIp)2 ,
dψ dψ
where Bp represents the poloidal magnetic field, R represents the major radius from
the axis of symmetry, p represents the plasma pressure, Ip represents the plasma
current, and the functions p(ψ) and Ip(ψ) are assumed to be known. Surfaces of
constant ψ are known as “flux surfaces”. To a very good approximation quantities of
interest in the core can be considered constant on a flux surface in the core plasma;
this is not true for the edge plasma.
In a practical magnetic confinement reactor electromagnetic coils are used to create
31
Toroidal Magnetic Field Coil
Vacuum Vessel
ϕ
Z
R
R0
Equilibrium Field Coils
r θ
Ohmic Transformer Stack
Figure 1.3: Schematic of principle components of a tokamak. (Courtesy D. Garnier,
1996)
a magnetic topology such that field lines close on themselves without passing through
material surfaces. This is necessary to keep the core plasma from coming in contact
with the walls of the reactor. To date, one of the most promising configuration
of coils has been the tokamak, a schematic of which is given in Figure 1.3. It is
possible to show from ideal MHD that both toroidal and poloidal components of a
magnetic field B are necessary for equilibrium confinement of a plasma. External
toroidal magnetic field coils of the tokamak provide the toroidal component of the
magnetic field. Changing the current in a central coil called the ohmic transformer
stack changes the magnetic flux linking the plasma, thereby inducing a current to flow
in the plasma. The plasma current creates the poloidal magnetic field that, together
with the toroidal field, confines the core plasma.
32
reaction
T(d,n)α
d(d,n)3He
d(d,p)T
3
He(d,p)α
6
Li(p,α)3 He
11
B(p,α)2 α
Energy
(MeV)
17.6
3.3
4.0
18.3
4.0
8.7
note
1
2
2,(3)
2,(3)
2,3
2,3
Table 1.1: Nuclear fusion reactions with the highest cross sections at keV temperatures. Notes: 1. easiest, 2. “advanced” (higher temperature), (3). aneutronic with
parasitic DD neutrons, (3). aneutronic
1.1.3
Progress To Date
Many nuclear fusion reactions are possible, but the reaction with the highest cross
section at keV temperatures is a mix of two isotopes of hydrogen, deuterium (D) and
tritium (T), and produces a neutron (n) and a helium-4 nucleus (α):
D + T → n(14.1 MeV) + α(3.5 MeV)
Table 1.1 lists other nuclear fusion reactions with significant cross sections at keV
temperatures. Figure 1.4 plots the reaction rate parameters as functions of plasma
temperature for the reactions listed in Table 1.1. Note that a plasma with mean
energy of 10 keV corresponds to a temperature of 110 million Kelvin. Thus it is
crucial that any reactor concept effectively prevent the core plasma from coming in
contact with the walls.
Great progress has been made towards achieving net power production using the
tokamak concept. The Lawson model provides a simple but quantitative measure of
the progress. Consider a zero-dimensional plasma power balance
dW
= Pf + Ph − Pbr − Ptr ,
dt
33
Figure 1.4: Reaction rate parameter for nuclear fusion reactions with the highest cross
sections at keV temperatures. Reference: L.T. Cox, “Thermonuclear Cross Section
and Reaction Rate Parameter Data Compilation,” Phillips Laboratory, Edwards AFB
CA 93523-5000, AL-TR–90-053
34
where W represents the energy of the plasma, Pf represents the power produced by
fusion, Ph represents the external heating provided to the plasma, Pbr represents the
power lost from the plasma due to Bremsstrahlung radiation, and Ptr represents the
power lost from the plasma due to energy transport. For an equal mix of deuterium
and tritium fuel, and assuming quasineutrality such that nd + nT = ne , nd = nT ≡ n,
we can write forms for the terms in the power balance:
Pf = (n2 /4)σvEf ,
2
Pbr = Abr n2 Zeff
T 1/2,
Ptr = 3nT /τE ,
Pf /Ph ≡ Q.
We can rearrange the power balance equation to solve for nτE = f(T, Q, dW/dt).
The goal is for a reactor to operate in steady state (dW/dt = 0) and to ignite (Q =
∞). For a plasma temperature of T ≈ 10 keV, this would require nτE ≈ 3 × 1020
s/m3 . A significant milestone of the progress of fusion research is “breakeven” which
corresponds to Q = 1, such that the fusion power produced matches the external
heating. Although as of this writing no fusion reactor has yet reached breakeven,
the goal is within sight. Figure 1.5 shows curves of nτ required as a function of
temperature for different values of Q. It also shows world record values of nτ and T
achieved in actual tokamak experiments.
1.2
Edge Physics
The discussion of nuclear fusion in the previous section neglected any mention of the
contruction of the vessel which encloses the fusion plasma, and of the interaction
between the fusion plasma and the vessel. In fact the fusion plasma does interact
with the vessel and the effects of the interaction can not be neglected. The edge
plasma can be defined simply as the plasma region between core plasma and the
35
Figure 1.5: Lawson parameter nτ required as a function of temperature for different
values of Q ≡ Pf /Ph in steady state (dW/dt = 0). “Breakeven” is defined as Q = 1;
“ignition” is defined as Q = ∞. Note that in 1998 the JT60-U tokamak team claimed
to reach Q = 1.25 transiently with “DT equivalent” conditions.
36
vessel wall. The edge plasma is important because of its influence on: (1) core plasma
heat loads to the vessel wall, (2) core particle and energy confinement properties, (3)
introduction of fusion fuel into the core, (4) sources of impurities which penetrate into
the core plasma, and (5) removal of helium ash from edge. Readers interested in more
detail than presented in this section are referred to the thorough review of plasma
boundary phenomena in tokamaks by Stangeby and McCracken [64], who suggest
the importance of the edge plasma in fusion reactors when they note “for physical
systems in which the transport and other properties of the medium are fixed, central
conditions are entirely controlled by edge conditions.”
1.2.1
Definition of Edge Plasma
The previous section described toroidal MHD equilibria with nested flux surfaces.
Technically the boundary between the core plasma and the edge plasma is the last
closed flux surface (LCFS), also called the separatrix. Inside the separatrix flux
surfaces close on themselves without interruption, which provides good particle confinement. Outside the separatrix the flux surfaces penetrate through a solid surface
before closing on themselves, and thus from the perspective of particle confinement
the surfaces are considered open. Particles on open flux surfaces travel freely along
magnetic field lines until they interact with the wall: open flux surfaces have poor
confinement. By definition the edge plasma consists of open flux surfaces, so the edge
and core plasmas have very different properties.
Practically, the plasma temperature and density profiles decrease rapidly outside
the separatrix, over lengthscales of several millimeters. The wall acts as a strong
plasma sink since plasma travelling along open field lines recombines atthe wall. Particles in the edge plasma remain for times of order milliseconds, compared with particle confinement times in the core plasma several hundred times longer. In the Alcator
C-Mod tokamak, edge plasma temperatures at the edge are typically of order electron
volts (1 eV = 11600 K). In contrast with the core plasma, where plasma tempera37
tures are hot enough that almost all ions are fully stripped of electrons, edge plasma
temperatures are low enough that atomic processes can significantly influence particle
and energy balances. In addition, complex shapes of the vessel wall can give different
lengths and boundary conditions for each “flux tube” in the edge plasma, spoiling
the symmetry that in the core permits a one-dimensional description of the plasma.
Most tokamak reactors have one of two configurations to determine the separatrix: limiter or divertor. In limiter tokamaks a material surface intersects the core
plasma and therefore defines the boundary between open and closed flux surfaces.
In divertor tokamaks additional external magnetic coils with current running in the
same direction as the plasma current produce a null in the poloidal magnetic field;
the magnetic surface containing the poloidal field null is the separatrix. The Alcator
C-Mod tokamak has a coil set capable of producing a field null near a special region
of the vessel wall called a divertor, where the interaction of the edge plasma with
the wall is physically removed from the core plasma. C-Mod can also be run as a
limiter tokamak if desired. Figure 1.6 shows a poloidal cross section of Alcator C-Mod
vaccum vessel and coil set; also shown is the separatrix surface for a lower single-null
diverted plasma. Pitcher and Stangeby [53] review experimental results from divertor
tokamaks world-wide.
Most detailed work describing edge plasma is performed with multi-dimensional
computer codes which account for plasma interactions with neutral particles and the
wall. Despite the complexity of accurate modelling the edge plasma, some simple
models that make gross approximations admit analytic predictions which reproduce
many of the observed properties of edge plasmas and so can give considerable insight.
1.2.2
Heat Loads to Wall
A quick estimate can be made of the power loads to the wall in Alcator C-Mod.
Consider a plasma in steady state of major radius R = 0.67 m and minor radius
a = 0.2 m receiving input power Pin = 5 MW. If the plasma radiates half of the input
38
Separatrix
0.67 m
Figure 1.6: Poloidal cross section of the Alcator C-Mod tokamak, with representative
plasma last-closed flux surface.
39
power, the other half comes out as particles which follow the open flux surfaces to the
wall. We can estimate the wall surface area receiving this power by A = 2×2π(R+a)λ,
where λ represents the edge plasma width, λ ≈ 0.01 m. Thus the particle power
density is
PS ≈
Pin − Prad
,
4π(R + a)λ
which approaches 22 MW/m2 . Power loading on walls can reach 10 MW/m2 in
Alcator C-Mod, which is about as high as can be tolerated by present steady state heat
transfer technology; designs for burning plasma experiments such as FIRE project
heat loads up to 20 MW/m2 .[45] During pathological plasma operations local power
loading can be much higher, for example during disruptions (abrupt plasma current
termination), and therefore considerable wall damage can occur locally.
Note that the above formula suggests at least two ways to decrease the power load
to the wall. (1) Increase the fraction of power lost from the plasma due to radiation,
effectively spreading out the power. This is the idea behind the “dissipative divertor”
concept, in which impurities with high atomic number are introduced into the edge.
(2) Increase the edge plasma scale length λ. Both of these approaches involve the
edge plasma essentially. Thus for survival of the first wall in fusion reactor conditions
it is necessary to understand how the edge plasma affects power loading to the vessel
wall.
1.2.3
Impurities from Edge into Core Plasma
Survival of the vessel wall is a necessary but not sufficient criterion for successful
operation of a fusion reactor. High heat flux in the edge plasma can sputter and
ablate the wall material and unless special precautions are taken to prevent it, the
wall material can penetrate into the core plasma. Thus even if the wall survives the
heat flux, the core plasma fusion rate might not. Since most fusion reactor programs
are moving towards walls and plasma facing components made of metals with high
atomic number (Z), and since plasma impurities emit bremsstrahlung continuum
40
radiation with intensity proportional to Z 2 , it is important to reduce impurity flux
from the edge to the core.
An important component of edge plasma physics is understanding the generation
and transport of impurities in the edge plasma and their screening from the core
plasma. Since sputtering yields of energetic ions on surfaces generally increase with
ion energy, one approach to reduce impurity generation is to cool the edge plasma.
This can be accomplished by introduction of neutral gas to dilute the energy of the
plasma or by the intentional introduction of high-Z impurities in the edge plasma to
radiate away the edge plasma energy. These two approaches form the basis for the
“detached” and “radiative” divertor operation concepts.
Any impurities that are generated by or intentionally introduced into the edge
plasma must be kept from migrating to the core plasma. The original intent of
the divertor concept is to physically remove from the core the region where the edge
plasma and vessel wall interact, thus “screening” the core plasma from the impurities.
1.2.4
Helium Ash Removal
Operation of a fusion reactor in steady state will require removal of the fusion reaction
products. If the fuel employed is a mix of deuterium and tritium, the reaction products
will be neutrons and alpha particles (helium nuclei). The neutrons have no charge
and so leave the plasma without regard to the magnetic confinement. The alphas are
mostly confined by the magnetic field but eventually diffuse towards the separatrix
and the edge plasma. The alpha “ash” concentration in the core must be kept low,
otherwise it dilutes the heating power that is applied to the fuel and reduces the
fusion reaction rate.
Once past the separatrix the alphas flow to the wall with the edge plasma where
they recombine to form helium atoms. To prevent a buildup of helium gas the edge
plasma must be pumped. The efficiency of the pumping depends on the partial
pressure of the helium, which depends in turn on the configuration of the vessel wall
41
that interacts with the edge. A major objective of the divertor configuration is to
increase the pressure of the helium in the edge high as possible to reduce the size of
the pumps necessary to remove it.
An understanding of the edge plasma interaction with the wall is necessary to
predict the alpha transport to the wall and the neutral gas pressure at the wall, and
therefore the pumping efficiency of the helium ash.
1.2.5
Influence of Edge Plasma on Core Plasma Properties
The edge plasma has an important influence on the properties of the core plasma
beyond introduction of impurities. Edge conditions affect the shape of the temperature and density profiles, and appear to be related to core plasma energy confinement
times.
It is possible to modify the zero-dimensional core plasma power balance to include the effects of the “peakedness” of density and temperature profiles: fusion
performance improves with peaked profiles. Stangeby and McCracken [64, p.1271]
emphasize the importance of the particle and heat source functions on the shapes of
the density and temperature profiles, giving a simple example of a plasma with constant diffusion coefficients, fuelled at the edge and with a heat source in the center.
They show that the density profile is flat and that the particle “replacement time”
depends on conditions at the edge: τp ≈ aλiz /D⊥ , where a represents the plasma radius, λiz represents the ionization mean free path of neutrals, and D⊥ represents the
particle diffusion coefficient. Conversely, the temperature profile is peaked and the
energy confinement time depends only on core conditions: τE ≈ 3a2 /(2χ⊥ ), where χ⊥
represents the energy diffusivity. The effect of particle sources on the density profile
suggests that improved performance might be obtained using fuelling by pellets or
neutral beam injection.
42
1.3
Ion Mass Spectrometry
The previous section described the influence of the edge plasma on the core plasma
and the vessel wall and emphasized the need to understand the edge plasma in any
attempt to control it. A complete model of the edge plasma is difficult to realize
due to the many active processes in the edge, and validation of any type of model
relies heavily on experimental data. The modelling effort is hampered by a traditional
shortage of experimental measurements in the edge compared with the core plasma.
Langmuir probes and visible spectroscopy are the most common and reliable diagnostics of edge plasmas, giving density, temperature, and impurity measurements in
the edge. While more complicated to operate and less commonly found on tokamaks,
Thomson laser scattering has the potential to give detailed two-dimensional profiles
of electron density and temperature deep in the edge plasma [20]. In the divertor, a
residual gas analyzer gives composition of neutrals far from the plasma, and pressure
gauges give dynamic measurements of neutral gas pressure.[16]
Ion mass spectrometry complements the above suite of edge plasma diagnostics.
This section briefly reviews the history of ion mass spectrometry, particularly pertaining to tokamak research. The origins of the E(t)×B omegatron ion mass spectrometer
are elaborated, as well as the motivation to combine an ion mass spectrometer with
a retarding field energy analyzer.
1.3.1
Omegatron History
In early 1949 Thomas et al [68] used nuclear resonance in a magnetic field to measure
the proton moment. Later that year, Hipple et al [21] used the same magnet to
measure the cyclotron frequency of protons, from which they determined the mass
ratio of protons and electrons. Hipple et al confined protons axially with a dc electric
field and applied a variable frequency radio frequency electric field at right angles to
the magnetic field. The proton cyclotron frequency was determined by finding the
resonant frequency that caused the proton larmor radii to increase until they were
43
collected on side plates and measured with an amplifier. Hipple et al were able to
improve the frequency resolution by reducing the amplitude of the applied RF power.
Since their device measured frequency ω they suggested it be called an omegatron.
In 1954 Alpert and Buritz [1] used an omegatron in their studies of evacuated glass
systems to confirm that diffusion of atmospheric helium through the glass walls set
the lowest achievable pressure. They measured the spectrum of mass species present
in their system by observing the current collected as a function of the applied radio
frequency; their dominant masses were M/Z=4, 28, and 40, probably corresponding
to singly charged species of helium, diatomic nitrogen, and argon.
Wagener and Marth [76] performed similar work in 1957, but with the principle
objective to use the omegatron to analyze the partial pressures of component gases
at low pressures. They also measured a spectrum of mass species up to M/Z =
44 (carbon dioxide). They demonstrated the kind of sleuthing that is necessary to
identify degenerate resonances.
Operation of the omegatron as a routine residual gas analyzer for low pressure
systems was proposed by Averina [3] in 1961, who obtained rich mass spectra. Orientation of the omegatron in the magnetic field was obtained by noting when the
ionizing electron beam current to the collector plates was minimized. Averina noted
the principle loss of ions was along the direction of the magnetic field to the back of
the RF cavity and as a remedy he advocated a reflecting potential for the end plate;
he also noted the tradeoff between resolution (at low RF amplitude) and collection
efficiency (at high RF amplitude).
In 1962 Batrakov and Kobzev [5] described an omegatron that used metallic grids
for electrodes which enhanced evacuation of the region between the RF plates and
reduced the noise level. Turovtseva and Shevaleevskii [73] in 1963 used an omegatron
to study the influence of ionization sources on the equilibrium pressures of H2 and
CH4 over titatium plating. In 1964 Averina et al [4] described the operation of
a commerically available omegatron residual gas analyzer for high-vacuum systems
44
with mass range 2–150 amu. Widespread commercial use of the omegatron ceased
with the introduction of the radio frequency quadrupole residual gas analyzer.
All the above implementations of the omegatron had three features in common:
1. They analyzed ions formed by electron impact,
2. they employed permanent magnets, and
3. they were compact instruments on dedicated low pressure gas systems.
In 1990 Wang et al [79] used an omegatron to analyze the ions in the linear magnetized plasma device PISCES. Their experimental setup was essentially unchanged
from the Hipple and Sommer design, except that the magnetic field was provided by
external coils rather than permanent magnets and the ionizing electron beam was
omitted. In 1995 Mieno et al [46] described an omegatron configuration with plates
spaced 10 cm apart and 50 cm long which permited them to achieve mass spectrometry with exquisite resolution; however they performed their experiments on a
dedicated linear magnetized plasma device not much bigger than their omegatron.
1.3.2
Tokamak Ion Mass Spectrometry
Matthews was the first to employ in-situ ion mass spectrometry on a tokamak (DITE),
and in his original paper [38] he enumerated the particular requirements of a spectrometer probe:
1. The instrument had to exploit or be immune to the strong magnetic field,
2. the instrument had to accomodate a spread in ion velocities,
3. the geometry had to allow for ion motion parallel to the magnetic field, and
4. the geometry had to be simple to permit calculation of ion transmission.
Matthews also mentioned many of the particular challenges:
45
5. The probe had to be aligned to within a few degrees with the local magnetic
field,
6. the intensity of the ion source in the boundary plasma necessitated an attenuating slit to avoid space charge effects, and
7. the noisy electromagnetic environment of the tokamak set the noise and limited
the bandwidth of the electronics.
The plasma ion mass spectrometer (PIMS) probe Matthews developed exploited the
local magnetic field as did the omegatrons on the linear plasma devices previously, but
the perpendicular electric field was varied on timescales of tens milliseconds instead of
the inverse cyclotron frequency. The ion selectivity was based on the mass dependence
of the E × B drifts orbit radius, and a scan in electric field magnitude resulted in a
scan of M/Z. With Stangeby, Matthews [38, 43] compared the observed distribution
of ion species with a two-dimensional Monte Carlo neutral transport code lim and
found good agreement.
The original PIMS probe used a configuration in which all ions of a given M/Z
were collected on a wire a certain radius from the entrance slit, regardless of energy parallel to the magnetic field. In a subsequent modifcation to the PIMS probe,
Matthews and coworkers [41] divided the ion collection area into three separate regions to obtain a crude measure of the parallel energy distribution of ions with the
appropriate M/Z. Analyzing their experimental measurments along with results from
the Monte Carlo code Matthews and Stangeby concluded that the field structure near
the entrance slit effectively converted perpendicular ion motion into amplified parallel
energy dispersion, blurring the distinction between T and T⊥ .
Matthews et al [39] also obtained ion mass spectra from the scrape-off layer plasma
of the TEXTOR tokamak several days after the wall was boronized, using an UKAEA
PIMS V2.0 probe. Matthews et al used isotopic abundances to help resolve degeneracies in M/Z spectra for neon and boron. In the data analysis Matthews et al
46
used a multiparametric nonlinear least squares fit to spectra including instrumental
linewidth to help estimate ion species abundance; the abundance was used to calculate Zeff in SOL plasma. From energies and abundances of each ion species Matthews
et al estimated sputtering rates of the vessel wall.
1.3.3
Omegatron on a Tokamak
Matthews conclusively demonstrated the utility of ion mass spectrometry for helping
to diagnose the edge plasma conditions in tokamaks. In early 1992 Labombard and
Thomas [70] designed the first omegatron ion mass spectrometer probe for a tokamak.
This subsection describes the motivation for their design, as well as similarities and
differences from the PIMS probe by Matthews. Details of the omegatron probe design
are presented in the next chapter.
The UK Atomic Energy Agency developed the PIMS probe into a commercially
available product, but documentation noted that the probe could be used in a maximum field of approximately 3 tesla. This was insufficient for the Alcator C-Mod
tokamak (4-8 tesla), suggesting a different approach for ion mass spectrometry would
be required and leading LaBombard and Thomas [70, 69] to consider the omegatron
concept. Also, the PIMS probe allowed for no independent means of controlling the
mass resolution and the collector current signal; the radius of the cycloidal ion orbit
was set by the aperature spacing while the electric field amplitude was fixed for a given
M/Z. In constrast the omegatron electric field frequency selects M/Z; independent
control of the electric field amplitude permits trading improved mass resolution for
collection efficiency.
Matthews and Stangeby [41] noted that at low densities the ion temperature obtained by fitting data from a modified PIMS probe with Monte Carlo code output
exceeded the electron temperatures measured by Langmuir probes and that a retarding field energy analyzer (RFEA) would give a more direct measure of the sheath
potential drop. In a study using two separate probes, a PIMS probe and an RFEA
47
probe, Matthews et al [42] noted that “an elegant solution to the problem of impurity
effects [in determining the ion temperature] would be to incorporate retarding grids
into the mass spectrometer so that an analysis of individual charge state distributions
would be possible.”
1
A main uncertainty Matthews et al encountered in determining
the ion species concentrations was the assumption that all ion species had the same
temperature. Combining a gridded energy analyzer with an ion mass spectrometer
would have permitted them (in principle) to measure the temperature of each species
separately.
Combining a retarding field energy analyzer with an ion mass spectrometer was
one of the objectives in the design of the omegatron probe for Alcator C-Mod.
In comparison with the small ion currents collected by the omegatrons in lowpressure gas systems, of order 10−13 amperes, the ion mass spectrometer devices on
linear plasmas [79, 46] and tokamaks [38, 43, 41] measured much higher currents, of
order 10−9 –10−7 amperes. As Matthews noted [38], obtaining these measurements
in the noisy environment of the tokamak edge was challenging. The electronics for
the omegatron on C-Mod were designed to resolve resonant ion currents down to
sub-nanoampere levels with very good noise rejection. In addition to the measurement objectives, the design of the omegatron on Alcator C-Mod had to satisfy severe
engineering constraints: only a vertical diagnostic port was available so all vaccum
components had to fit within a cylinder 7.5 centimeters in diameter and two meters
from the plasma; and vacuum components had to be able to withstand the considerable heat loads that would result from possible plasma disruptions.
1.4
Goals of Thesis
The omegatron ion mass spectrometer designed by LaBombard and Thomas has been
completed, installed, debugged, operated and (mostly) optimized for use on the Alca1
Matthews’s thesis work involved retarding field energy analysis on DITE[37, 42]; Guo et al [17]
have applied RFEA to JET edge plasmas. The history of RFEAs will not be reviewed here.
48
tor C-Mod tokamak. The broad objective of this thesis is to demonstrate the utility
of the omegatron probe as an edge plasma diagnostic. The specific objectives of this
thesis are: to describe the construction of the omegatron probe and the electronics
(Chapter 2); to present background theory for modelling of the omegatron, tested
by tokamak discharge experiments (Chapter 3); to demonstrate data reduction techniques for the omegatron (Chapters 4 and 5); to apply the omegatron data analysis to
the specific topic of impurity transport in the edge plasma (Chapter 6); and to suggest
further improvements to the diagnostic and further experiments to perform (Chapter
7). Appendices contain useful but tedious calculations, electrical schematics, and a
user’s manual for researchers at MIT.
Communicating the knowledge gained about the operation of this diagnostic will
allow others to use the omegatron probe to diagnose the edge plasma content and
conditions, which will assist in impurity transport studies and contribute to the improved performance of the edge and core plasmas. It is hoped that this will contribute
to the broad objective of producing fusion power.
49
50
Chapter 2
Diagnostic Description
LaBombard and Thomas [69] designed the omegatron probe for Alcator C-Mod. They
constructed the first version of the probe and tested it on a benchtop linear plasma
device. Since then numerous additions and modifications were made to the hardware
and electronics to operate the omegatron probe on Alcator C-Mod. This chapter
describes the hardware and electronics. A description is also found in ref.[47].
2.1
Overview
Figure 2.1 shows a schematic of the key features of the omegatron probe. The probe
internal components are protected from plasma heat flux by a molybdenum heat
shield, which is connected electrically to the vacuum vessel. Inside the heatshield is
an electrically isolated shield box which contains the retarding field energy analyzer
and ion mass spectrometer. The axis of the probe is aligned along the local magnetic
field, which is predominantly in the toroidal direction. Plasma flows along field lines
through holes in the heat shield and shield box and is attenuated by a tungsten
slit before encountering the three grids that constitute the retarding field energy
analyzer. Ions and electrons that traverse the grids enter the RF cavity. Ions in the
cavity that have cyclotron frequency close to the frequency of applied RF power are
51
RetardingField Energy
Analyzer
Omegatron
RF Cavity
Heat Shield
(connected to
vacuum vessel)
Electrostatic Shield
(connected to slit)
Resonant Ions
Slit
100Ω
Load
Ion
Non-Resonant
Ions
Magnetic Field
Grid2
Grid1
Grid3
Balanced 50Ω End Collector
RF Coax Lines
(identical lengths)
1:1
to Langmuir
probe electronics
2:1
DC Break
RF Power
1-100 MHz,
< 30 watts
Resonant
Ion Current
RF Transformer
All Grid, Slit, and Collector Connections
use Coax with Isolated Shields
(shields are biased by electronics)
Figure 2.1: Schematic of omegatron probe, showing slit, retarding field energy analyzer, and ion mass spectrometer portions mounted in a shielding box. Figure courtesy
B. LaBombard.
52
called resonant ions; they absorb RF power and increase their perpendicular energy
until they collide with the RF plates. Electrons and non-resonant ions pass through
the RF cavity and are collected at the end plate.
An isolation transformer provides a DC break and applies half the RF power to
each RF plate with a 50 ohm coaxial feed, 180 degrees out of phase; a 100 ohm load
between the RF plates creates a virtual null in the RF electric field along the axis
between the RF plates. Resonant ion current collected on the RF plates is removed
through a center-tap of the transformer.
The slit, each of the grids, the RF plates, and the end plate each have an independent bias control. Current collected on each component is measured separately.
2.2
2.2.1
Probe Head
Internal Components
Figure 2.2 shows an exploded view of the retarding field energy analyzer and ion mass
spectrometer components.
The slit assembly consists of knife-edged pieces of tungsten coated with nickel
and spot-welded together. The knife-edge geometry is similar to that of Wan [77],
with one side flat and the other cut at forty-five degrees. The two flat sides face the
plasma. The gap between the knife edges presents an area 25 µm by 7 mm through
which the plasma may flow.
Behind the slit assembly are three grids, made from 150 lines-per-inch (nominal)
rectangular tungsten mesh, spot-welded to laser cut stainless steel window frames.
Each window frame has a tab for a wire, by which a voltage bias may be applied and
from which current may be collected.
Figure 2.3 shows a magnified picture of the grid material, taken with a CCD camera on a microscope. Calibrated pixel dimension and counting pixels in the digitized
image gives the grid line thickness, d = 24.5 µm, and space between grid lines, s = 144
53
ceramic dowel
mica sheet
RF plate
ceramic spacer
Grid Frame
End Collector
mica spacer
ceramic support
400 ohm RF
Resistor
ceramic
support
silver foil
SS washer
Slit Assembly
Figure 2.2: Exploded view of internal components of omegatron probe retarding
field energy analyzer and ion mass spectrometer, showing: slit; grids; RF plates;
RF resistors; end collector; mica spacers and insulators; and ceramic spacers and
supports. Wires to the grids, RF plates, and RF resistors omitted for clarity.
54
Figure 2.3: Magnified image of the tungsten grid used on the omegatron probe. Grid
lines are 24.5 µm wide with 144 µm in between. Image courtesy D. Hicks.
55
Probe
C-Mod omegatron
Wan[77, pp.71,75,76,82]
Matthews[37, pp.57,71,75,92]
Pitts[54, pp.97,99,100]
Guo[17]
typical
λD
20–200
25
40
7
20
Slit
width
25
30
30,70,100
5,25,100
30
Line
thick.
24.5
25
21
40
60
Line
spacing
168.4
101,170
250
500
400
Grid
spacing
700
2000
1000
1500,5000
2000
Table 2.1: Comparison of slit and grid dimensions of selected tokamak retarding field
energy analyzer probes. All dimensions are in micrometers.
µm. Optical transmission is estimated by s2 /(s + d)2 = 73%, which agrees quite well
with the the 71% optical transmission obtained by counting pixels in between the
lines.
The grids are isolated electrically from each other by laser-cut mica window frame
spacers, with approximately 0.7 mm spacing between grids (including the mica spacers and stainless frames). The slit and grids are packed together and are isolated
electrically from the side walls of the shield box using ground ceramic collar pieces.
The grids are isolated from the floor of the shield box by a laser-cut mica sheet.
Table 2.1 lists the dimensions of the slit opening, the grid line spaceing and the
grid line thickness for the omegatron probe retarding field energy analyzer. Similar dimensions are included for retarding field energy analyzers operated on other
tokamaks.
The slit and shield box are electrically connected together: a shimstock windowframe spring washer between the slit assembly and the shield box maintains electrical contact and mechanically compresses together the slit assembly, mica spacers,
and grids.
The probe head is designed to fit inside the circular portion of a Alcator vertical
diagnostic port, which has an inner diameter of 75 mm. This sets an upper bound on
the length and width of the RF plates of omegatron ion mass spectrometer. The choice
of plate spacing is a compromise between improving resonance resolution (improves as
56
spacing increases) and reducing required RF power (decreases as spacing decreases).
The RF plates are made from 0.5 mm thick stainless steel shim stock. Each RF
plate is approximately 30 mm wide and 40 mm long. The RF plates are supported
mechanically by cylindrical ceramic spacers underneath, between, and on top of the
plates. The orientation of the spacers is preserved by short ceramic dowels which pass
through the spacers and plates and into recessed holes in the floor of the shield box
and in the shield box coverplate. The coverplate and shield box effectively sandwich
the RF plates together and provide a uniform spacing of 5 mm between the RF plates.
Wires soldered to the RF plates deliver the RF power and remove the collected
resonant ions current.
Four RF resistors (400 ohms, ten watts each) are connected in parallel to the RF
plates to give a 100 ohm load for the RF amplifier. Stainless steel wires connect one
end of each resistor to the top RF plate and the other end of each resistor to the
bottom RF plate. The wires are connected to the plates with high temperature (200
C) silver solder.
Each resistor has an electrically isolated tinned copper base which acts as a heatsink. Thru holes in the resistor bases are tapped so that each resistor can be screwed
to the side wall of the shield box. Silver foil between the copper base and shield box
ensures that RF power dissipated in the resistor can be transferred to the shield box.
The power rating of the RF resistors approaches zero at 150 C, which sets the upper
limit of the operating temperature. The upper limit of the non-operation temperature
of approximately 250 C (bake-out) is set by the teflon insulation in SMA connectors.
The end collector must remain electrically isolated from the shield box since it
removes electrons and non-resonant ions from the cavity. A tab provides space for a
wire by which bias may be applied to the end collector and collected current removed.
Two ground ceramic collars fit around the side edges of the end collector to secure
it from moving in the plane of the cavity floor. Mica sheets which line the floor of
the cavity and the underside of the cavity cover complete the electrical isolation. A
57
band of annealed shim stock is spot-welded to the end collector and wrapped around
the middle of the two rear RF plate ceramic support posts. The band collects nonresonant current that would otherwise strike the posts and charge them to a floating
potential.
2.2.2
External Components
Figure 2.4 shows the external components of the omegatron probe head, including
the heat shield, the shield box and shield box coverplate, the patch panel, the lock
plate, the mounting plate and the angled adapter piece.
The internal components of the omegatron probe are protected from the scrape-off
layer plasma heat flux by a molybdenum heat shield. The face of the heatshield has
an elongated opening that lines up with the slit, and three recessions for Langmuir
probes. Langmuir probe LP1 is below the slit (closer to the plasma), LP2 is in line
with the slit, and LP3 is above the slit (further from the plasma). The heatshield
also has two vertical holes for thermocouples, each with a horizontal tapped hole for
a set screw.
The heatshield has a recession into which a copper cooling finger is inserted. The
cooling finger is at the end of a re-entrant tube. Compressed air is blown on the end
of the cooling finger through a stainless steel tube inserted into the re-entrant tube.
When the omegatron head is inserted into a tokamak discharge the bulk temperature
of the heat shield can increase from 30 C to 50–60 C, and up to 90 C after an
upward-going disruption. With the compressed-air cooling on, the temperature of
the heatshield drops by 20 C in ten minutes; with the compressed-air cooling off the
temperature of the heatshield drops only a few degrees in ten minutes.
We can approximate the heat transfer equation for the compressed air cooling of
the heat shield by mcpdT /dt = −hT , where T = TMo − Troom is the difference in
temperature between the heat shield and the compressed air, m = 0.231 kg is the
mass of the heat shield, and cp = 247 J/kg/K is the heat capacity of molydenum.
58
lock plate
adapter piece
patch panel
cover plate
RF assembly
shield box
mica sheet
heat shield
openings for
LP1
SLIT
LP2
LP3
Figure 2.4: Exploded view of external components of omegatron probe, showing: heat
shield; shield box; coverplate; patch panel; mounting plate; lock plate; and support
plate. All wires and SMA connectors omitted for clarity.
59
If the compressed air cools the heat shield from T1 = 30 K to T2 = 2 K in ∆t ≈
600 s, we can estimate the heat transfer coefficient of the compressed-air cooling by
h = mcp log(T1 /T2 )/∆t ≈ 0.3 W/K. This could be improved significantly by using a
liquid coolant and radiator.
The shield box fits into a recession in the heat shield and contains the assembled
components of the retarding field energy analyzer and ion mass spectrometer. The
shield box is made from stainless steel with a flame-sprayed aluminum-oxide coating
on the outside to isolate the shield box from the heat shield. The shield box has an
elongated opening that lines up with the slit (inside) and the heat shield elongated
opening (outside). The floor of the shield box has four recessions for the ends of the
ceramic dowels that maintain the alignment of the RF plates and ceramic spacers.
Inside the shield box near the front and back are lips which hold the grid assembly
and end collector ceramic collars, respectively.
The two sides of the shield box each have two pairs of thru holes, countersunk
and with clearance for socket head cap screws which secure the RF resistors. Two
vertical tapped thru holes at the sides and two vertical thru holes at the back are
used to align and secure the shield box coverplate and patch panel.
Laser-cut mica sheets between the shield box and the heat shield provide additional electrical isolation and prevent the shield box from rotating.
A copper cover-plate traps the grids and RF plates in the shield box. Openings
are provided for wires connected to the grids, RF plates, and end collector. The shield
box cover plate is in mechanical and electrical contact with the top edge of the shield
box, which is not flame-sprayed.
Wires from the slit, grids, rf plates, end collector, and Langmuir probes lead to
male SMA connectors, not shown in Figure 2.4, which are screwed into an aluminumoxide coated aluminum patch panel. The patch panel is screwed to the shield box
with two socket head cap screws; additional alignment is provided by pins near the
rear of the patch panel.
60
The patch panel allows the omegatron head to be assembled and checked out
before being mounted on the tube adapter piece. Semi-rigid coaxial cables terminated
with SMA connectors connect to the patch panel at one end and to SMA vacuum
feedthrus at the other end.
The subassembly consisting of the shield box, shield box coverplate, and SMA
patch panel is secured to the heat shield with a stainless steel mounting plate and
a lock plate. The lock plate has four thru holes with clearance for silver-plated
vented round-head cap screws. The screws secure the mounting plate and shield box
subassembly to the tapped thru holes in the heat shield.
Not shown in Figure 2.4 is a shallow (1 mm) recessed pattern on the underside
of the mounting plate into which the heat shield fits. The keyed fit mechanically
prevents rotation of the heat shield so the four screws do not have to perform this
function; the keyed fit also prevents plasma from entering the heat shield except
through the intended opening.
Four thru holes at the outer edges of the mounting plate are countersunk from
below. Flat-head silver-plated cap screws connect the mounting plate to the angled
adapter piece (not shown in Figure 2.4). The angled adapter piece is welded to the
end of a three meter stainless steel tube, also not shown in Figure 2.4.
Figure 2.5 and Figure 2.6 show scale drawings of the poloidal cross section of
Alcator C-Mod with the omegatron. To match the poloidal angle of typical plasmas
at the omegatron location, the head normal was rotated 45 degrees from horizontal.
Magnetic field reconstructions from plasmas at different toroidal field strengths and
different plasma currents were examined to determine approximate field pitch angles
at the omegatron location. The omegatron head was rotated 6 degrees in plane to
align the axis with the local magnetic field.
61
991028024 EFIT: 0.900
0.6
0.4
Z (m)
0.2
0.0
-0.2
-0.4
-0.6
0.3
0.4
0.5
0.6
0.7
R (m)
0.8
0.9
1.0
Figure 2.5: Poloidal cross section of Alcator C-Mod tokamak showing omegatron
(mirror image) inserted into upper divertor scrape-off layer plasma and fast scanning
Langmuir probe near midplane inserted to separatrix.
62
991028024 EFIT: 0.900
0.40
0.38
Z (m)
0.36
0.34
0.32
0.7
0.8
R (m)
0.8
Figure 2.6: Omegatron probe (mirror image) on Alcator C-Mod tokamak. Representative flux surfaces are shown, spaced two millimeters apart at the midplane.
63
Compressed Air
B3303
CAMAC
SMC24
CAMAC
Compressed
Air Piston
Power
Supply
B3303
CAMAC
Cooling Tube
Stepping
Motor
Limit Switches
Potentiometer
Circuit
Potentiometer
LG8252 A/D
CAMAC
Bellows
Rotation adjustment
Flange
Omegatron Head
Figure 2.7: Block diagram of linear motion subsystem.
64
2.3
Linear Motion Subsystem
Vertical position of the omegatron is controlled by a linear-motion vacuum bellows
and a stepping motor. Compressed air moves a cylinder to assist the stepping motor,
if necessary. A schematic of the linear motion subsystem is shown in Figure 2.7.
A custom 35 cm linear bellows with 15.25 cm inner diameter provides flexibility for
linear motion while preserving ultra high vacuum. The bellows assembly has two 25.4
cm flanges at the ends, the spacing between which is determined by bearing surface
sprockets on three threaded rods. The rotational orientation between the flanges is
preserved by three smooth stainless steel rods which pass through brass bushings in
the flanges.
The bellows are customized to permit ± 6 degrees of rotational freedom from the
equilibrium position; an external frame provides the rigid support for the smooth rods.
Alignment of the head with the toroidal magnetic field is accomplished by inserting
the probe head into an ECR plasma. The bellows are rotated until the measured
electron current to the RF ion collection plates is minimized (and the current to
the end collector is maximized) and then the bellows are clamped to the support
structure.
A stepping motor mounts directly to the bellows flange with a custom single-piece
alumium bracket. A chain drive connects the stepping motor to sprockets on each of
the threaded rods of the bellows.
The complete vacuum hardware weighs approximately 100 kilograms. Compressed
air at 280 kPa is used to extend two pistons that push the end plates of the vacuum
bellows apart and assist the stepping motor when withdrawing the omegatron. (The
compressed air is also used to cool the omegatron heat shield.) The compressed air
solenoid is energized by 120 VAC which is provided when a B3303 CAMAC module
energizes a relay.
Power for the stepping motor is provided through a relay energized by a B3303
CAMAC module, and the stepping motor is controlled through a CAMAC SMC24
65
module. A linear potentiometer moves with the bellows; the resistance is measured
with a custom potentiometer circuit, and the resulting voltage is sampled with a
LG8252 CAMAC module and converted into a position. Limit switches connect to
the stepping motor electrics to set bounds for the motion of the bellows.
The omegatron can be moved vertically at approximately 1 mm/s, which is sufficient to position the omegatron between plasma discharges.
2.4
RF Amplifier Subsystem
Figure 2.8 shows a block diagram of the RF oscillator and amplifier subsystem. A
Wavetek 1062 oscillator provides an RF signal with the appropriate frequency which
is amplified by an Amplifier Research 15 watt amplifier and sent to the isolation
transformer.
The Wavetek 1062 radiofrequency oscillator has a range from 1 to 400 MHz, of
which the range 1–100 MHz is used. It has custom analog electronics which accept
separate signals between −5 and +5 volts to control the power and frequency from
the oscillator. A channel of a BiRa 5910 D/A CAMAC module is used to control
the frequency; the 12 bit resolution of the BiRa divided amongst 100 MHz gives 25
kHz frequency resolution. The custom Wavetek analog electronics also offer signals
proportional to the power and frequency to monitor that the oscillator delivers what
is requested; output of these circuits is digitized with a TR16 CAMAC module.
The Wavetek 1062 features crystal oscillators at 5, 20, and 100 MHz which beat
with the oscillator; the “birdy” output is used to determine when the oscillator frequency passes over the crystal frequencies. A birdy circuit rectifies and low-pass filters
the beat signal of the RF oscillator with the 5, 20, and 100 MHz crystal oscillators;
the output of this circuit is digitized by a TR16 CAMAC module and is used to confirm when the oscillator frequency passes over a crystal frequency. The output of the
oscillator is sent to an Amplfier Research 15 watt wideband RF amplifier, which has a
variable gain between 22.9 and 47.4 dB selectable with a dial. Power to the amplifier
66
Bira 5910 D/A
CAMAC
B3303
CAMAC
RF Power
Program
Jorway J221
CAMAC
Omegatron
AND
RF Frequency
Program
power switch
Wavetek 1062
RF Oscillator
RF Amplifier
Birdy
Circuit
Isolation
Transformer
centertap
Frequency
Monitor
TR16 A/D
CAMAC
Power
Monitor
RF Signal
Electronics
Bira 5910 D/A
CAMAC
AM/FM Synchronous
Detection Electronics
Figure 2.8: Block diagram of RF amplifier subsystem.
67
is controlled to prevent accidental and/or continuous heating of the omegatron head
with RF power. Logical AND of output from a B3303 CAMAC module and from a
Jorway J221 gate pulse energizes a relay that provides 120 VAC to the amplifier.
The amplified RF power is applied evenly to both plates using a custom 1:2
isolation transformer; Figure 2.1 shows a schematic. The RF transformer consists of
two parts. The first section converts 50 ohm input to two balanced 50 ohm outputs
which are 180 degrees out of phase and mirror-symmetric with respect to the input
ground. The second section communicates with the first section through DC blocking
capacitors and allows the “RF null potential”, that is the DC potential at the center of
the RF plates, to be independently set. Low level currents resulting from ion collection
are extracted at the center tap of the output stage. This technique minimizes the RF
leakage into the low-level current detection electronics.
A BiRa B5910 D/A module programs the omegatron analog electronics to supply
the bias voltage of the RF plates, which is also applied to the isolation transformer.
The resonant ion current is collected through a center-tap of the transformer and is
sent to the omegatron RF analog electronics. The output of the RF electronics is
digitized with a TR16 CAMAC module.
2.5
Grid Electronics
LaBombard and Thomas [69] designed the original omegatron grid electronics; since
then significant modifications were made to the design. This section describes the
generic function of the omegatron grid electronics
The grid electronics measure very low currents (nanoamperes) with very high
common mode rejection (up to 200 volts). Each of the three grids, the centertap of
the RF transformer, and the end collecter have their own electronics board. Each
board has three stages of gain: ×1, ×10, and ×100, to improve dynamic range. All
boards except the RF board have input impedances as low as practical to reduce the
potential fluctuations that result with fluctuating current arriving at the component.
68
input
signal
U1
U3
filter
U5,U9
output signal
x1
virtual
signal
U2
U4
filter
U6,U10
output signal
x10
U7,U11
output signal
x100
U99
bias
request
U13
U13
U6,U10
bias monitor
V/40
Figure 2.9: Block diagram of grid electronics board. Grids G1, G2, G3, RF plates,
and END collector each have a separate electronics board.
This reduces the current induced by capacitive coupling with the RF plates. All
boards have inductors on signal inputs to filter RF noise.
A block diagram of the analog electronics is shown in Figure 2.9. The Appendix
contains a detailed electrical schematic. The current signal is converted to a voltage
by amplifier U1. The current signal from the analogous component of a “virtual
omegatron” is converted to a voltage by amplifier U2 and the difference between
these voltages is generated by amplifiers U3 and U4. The “virtual omegatron” is
a network of capacitors whose values are chosen to mimic the capacitive coupling
between the real omegatron components.
1
Potentiometers at the inputs of amplifiers U3 and U4 permit tuning the subtraction of the signals from the actual and “virtual” omegatrons. The gain of amplifier
U4 is set a factor of ten higher than the gain of amplifier U3. The ×1 and ×10 signals
are low-pass filtered and buffered. The filtered unbuffered ×10 signal is put through
a second buffer with another gain of ten. All output signals are digitized by a TR16
CAMAC module and are also available from front panel BNC connectors.
1
The “virtual omegatron” was not used while collecting the data for this thesis, and the input to
amplifier U2 was left open.
69
TTL Controls
Sweep A
Power Supply
Sweep B
Power Supply
External
Power Supply
To Probe
Current
Monitor
Current Signal
(V/Resistor)
Compensator
V/40 Signal
Figure 2.10: Block diagram of Langmuir probe electronics. Langmuir probes LP1,
LP2, LP3, and SLIT each have a separate electronics board. After E.E. Thomas Jr.,
Technical Report PFC/RR-93-03, MIT Plasma Fusion Center, 1993.
Amplifiers U1 and U2 have an isolated ground plane, the potential of which is set
to the component bias. This is to reduce DC leakage current and AC displacement
current between the center-conductor and the shield of the signal inputs. The isolated
ground plane can be driven from −100 volts to +200 volts. Each electronics board
has a circuit to drive the isolated ground plane voltage, a block diagram of which is
also shown in Figure 2.9. The desired voltage of the isolated ground plane divided
by forty is sent to the differential input of amplifier U13. The option exists to add a
unity-gain signal to the desired ground plane voltage using amplifier U14. The final
isolated ground plane voltage is divided by forty with a buffer and digitized by a
TR16 CAMAC module.
2.6
Langmuir Probe Electronics
Figure 2.10 shows a block diagram of the the Langmuir probe (LP) electronics designed and implemented by LaBombard [34] and used on the omegatron probe. (The
Appendix contains a detailed electrical schematic of the LP electronics.) The slit
and the three Langmuir probes on the face of the omegatron heatshield each have
their own LP electronics board. Each LP board applies a bias to the component and
70
LG8252 A/D
CAMAC
Omegatron
Heat Shield
Thermocouple
Electronics
Figure 2.11: Block diagram of thermocouples measuring bulk temperature of omegatron heat shield.
measures the resulting current; voltage signals proportional to the bias and current
are digitized by channels of a TR16 CAMAC module. Settings of a B3303 CAMAC
module energize relays on the LP boards which permit the user to measure either the
DC or AC bias voltage, to change the gain of the current-to-voltage measurement,
and to select from three separate power supplies.
2.7
Thermocouples
The omegatron heat shield has recessions for two chromel-alumel junction thermocouples. Thermocouple electronics produce voltages proportional to the junction
temperatures; each output is sent to a buffer, a block diagram of which is shown in
Figure 2.11. The buffer outputs are digitized by two channels of an LG8252 CAMAC
module.
71
72
Chapter 3
Omegatron Probe Theory
This chapter develops models describing the interaction of the edge plasma of Alcator
C-Mod with the omegatron probe and the behavior of plasma inside the omegatron
probe.
A fluid model is presented of the scrape-off layer plasma confined to the flux
tube bounded at one end by the omegatron probe face and at the other end by the
E-port ICRF limiter. A simplified model is presented first to show that when the
perturbation length of the probe is of order the length of the flux tube, the flux
tube plasma potential is set by the maximum of the wall boundary potentials and
the electron temperature in the plasma. The fluid model is extended to include
weak secondary electron emission at the boundaries and to show the influence on
the sheath potential drop. The theory from this section is used to explain results
of commissioning experiments described in the next chapter: dependence of sheath
potential on slit bias, and low values of sheath potential.
A single particle orbit model is presented of the transmission of ions and electrons
through the slit. (The model is also used to predict ion and electron transmission
through the grids.) It is found that ion and electron transmission through the slit
should approach optical transmission (the probability of transmission depends only
on the area of the opening). Secondary electron emission must be invoked to explain
73
experimental observations of the ratio of ion and electron collection through the slit,
described in Chapter 4.
The slit collimates the flux of ions and electrons entering the omegatron into a
beam. Since we can apply a bias to the slit, the flux of ions does not necessarily equal
the flux of electrons: the beam can be non-neutral. However the strong magnetic field
prevents the beam from expanding as long as the charge density is below the Brillouin
density. An estimate is obtained of the maximum beam current in the analyzer, above
which the beam will expand.
By collimating the beam, the slit reduces the contribution of the beam charge
density to the electrostatic potential. An estimate is obtained of the beam current
for which the contribution of space charge to the electrostatic potential is significant.
Below this current we can neglect space charge, and the electrostatic potential is
determined just by the applied grid potentials. This is important for the model of
the retarding field energy analyzer.
A kinetic model is presented of the interaction of the plasma with the potentials
of the grid. The model is simple, considering only the axial variations of the electrostatic potential and neglecting the effect of space charge in the Poisson equation.
The model shows that the temperature is extracted from inverse slope of semilog
plot of the current-voltage (IV) characteristic and the sheath potential occurs at the
“knee”. The model also predicts differences between ion and electron characteristics.
In the Chapter 4 the model is used to interpret the features of the experimental IV
characteristics presented.
Formal modifications to the kinetic RFEA model are made to include space charge
in the Poisson equation, and the expected influence on IV characteristic is shown.
The results suggest a mechanism with which to explain the rounding of the “knee”
in the IV characteristic at high current levels. The experimental observation that
the sheath potential is nearly equal to the floating potential, extracted from the IV
characteristics in the next chapter, is explained by secondary electron emission from
74
the slit.
A single particle orbit model is presented of the transmission of ions and electrons
through the grids. Ion current fractions to the grids are calculated based on the
assumption of optical ion transmission through the grids. This is tested against
experiment in the next chapter and the assumption of optical ion transmission is
found to be good. Electron transmission through the grids is predicted to be optical,
but the measured currents appear to yield electron transmission greater than optical.
This counter-intuitive result is resolved by including secondary electron emission from
electron impact.
A kinetic model is presented of the omegatron ion mass spectrometer. In the
model, ions pass through the RF cavity in a collimated beam, and an estimate is
presented of the frequency range over which resonant current is collected, including
the effect of magnetic field variation. An estimate is made of the influence of space
charge inside the RF cavity on the collection efficiency of resonant ions; reflection of
ions on space charge helps to explain the current accounting when the bulk species
is resonant, and helps to explain the trends observed when the density outside the
analyzer decreases. The dependence of resonant current on RF power is predicted
from the model. From the resonant as a function of applied RF power, it is shown
how to extract the temperature of the resonant species and the flux fraction.
Modifications to the single-particle omegatron model are made to include a noncollimated ion beam. The broadened beam complicates extraction of the temperature
using the RF power scan technique.
These theoretical results are used to explain the gross features of the omegatron
impurity spectra presented in Chapter 5. The model reproduces the center frequencies
of the resonances and can account for the frequency widths.
75
3.1
Flux Tube Model
A simple fluid model is presented of the plasma on a flux tube bounded at two ends.
Bulk plasma flow is neglected. Secondary electron emission at the surface is neglected
initially. It is shown that the potential of the plasma in the flux tube is practically set
by the higher of the two boundary potentials and the electron temperature, when the
probe perturbation length is of order the connection length. When the two bounding
surfaces have the same potential each of them receive no net current (floating condition). Neglecting secondary electron emission, the difference between the plasma
potential and the floating potential is predicted to be approximately three times the
electron temperature, in electron volts.
The simple fluid model is then modified to include secondary electron emission
at the surfaces. We consider the fluid sheath equations with cold ions and secondary
electrons similar to Hobbs and Wesson [22], but we do not restrict the net current
to the surface to be ambipolar (floating), which allows us to find an equation for
the net current to a surface in a plasma including secondary electron emission. This
model is appropriate for small coefficients of secondary electron emission or large
sheath potential drop; otherwise space charge becomes significant and a kinetic model
is needed. It is found that inclusion of secondary electron emission in the model
preserves two of the earlier conclusions (higher potential of the two surfaces determines
the plasma potential, floating condition when both surfaces at same potential) but
that it reduces the predicted difference between the sheath potential and the floating
potential.
Plasma flow is not considered in these models. Hutchinson[24, 26] has shown
that if there is subsonic ion flow in the presheath towards (away from) the surface,
the presheath potential drop required to accelerate ions to satisfy the Bohm sheath
criterion at the sheath is smaller (larger).
76
3.1.1
Simple Fluid Model
Consider a surface in contact with a plasma, as in part (a) of Figure 3.1. Standard
fluid sheath theory (see Hutchinson [25, pp.55–64], for example) provides the ion
and electrical current density received at a surface at potential φsurf inserted into a
plasma: J = e(Γi − Γe ). Electron density is assumed to satisfy a Boltzmann relation,
ne (x) = n∞ exp(e(φ(x) − φp )/kTe ), with a mean velocity ve =
8kTe /(πme ), where
n∞ is the electron density in the quasi-neutral plasma far from the surface, Te is the
temperature parameter describing the spread in the electron velocity distribution,
and φp is the electrostatic potential in the plasma far from the surface. Thus the
electron flux is
1
1
e(φsurf − φp)
Γe (xsurf ) = n(xsurf )ve = n∞ exp
4
4
kTe
8kTe
.
πme
Ions have no source in the sheath so the flux of ions to the surface equals the flux at
the sheath: Γi (xsurf ) = nis vis , and quasineutrality holds at the sheath edge so the ion
density equals the electron density: nis = n∞ exp(e(φs − φp )/kTe ), where φs is the
sheath potential. Ions are assumed to arrive at the sheath with kinetic energy only
such that their velocity is vis =
2e(φp − φs )/mi . In order that a sheath exist at
all, the ion fluid velocity at the sheath boundary equals the sound speed (the Bohm
sheath criterion), which fixes the difference between the sheath potential and the
plasma potential: e(φp − φs )/kTe = 1/2.
Combining all these relations, the current arriving at a surface can be written:
J
1
1
= exp −
−
2
2
en∞ Te /mi
2mi
e(φsurf − φp )
exp
.
πme
kTe
(3.1)
For a floating probe which draws no net current, J = 0, which occurs when the
77
PROBE POTENTIAL
PLASMA POTENTIAL
DISTANCE
IONS
NET ELECTRON
COLLECTION
AMBIPOLAR
NET ION
COLLECTION
PRESHEATH
SHEATH
PICTURE A. PLASMA POTENTIAL FIXED
WALL POTENTIAL
SHEATH
PROBE POTENTIAL
SHEATH
PLASMA
POTENTIAL
IONS
NET ELECTRON
COLLECTION
IONS
AMBIPOLAR
DISTANCE
PRESHEATH
PRESHEATH
NET ION
COLLECTION
PICTURE B. WALL POTENTIAL FIXED
Figure 3.1: Schematic of potential of a flux tube. Picture (a): Long flux tube,
L Lp . Picture (b): Short flux tube, L ≈ Lp .
78
difference between φsurf = φf and φp satisfies
e(φp − φf )
e1/2
= ln
kTe
2
2mi
πme
= 3.68 (deuterium).
Thus standard sheath theory predicts that the plasma potential is approximately
three-and-a-half times the electron temperature above the floating potential.
Consider now a flux tube bounded at two ends, as in part (b) of Figure 3.1. Bound
the flux tube at xa with a surface (the wall) at potential φa and at xb with a surface
(the probe) at potential φb , and in general φb = φa. If a plasma exists in between xa
and xb then a potential difference φa − φb = 0 will result in a net current flow at both
surfaces. In particular, if perpendicular flow into the flux tube is ambipolar, then
conservation of charge requires that the net current arriving at xa must have come
from xb such that Ja + Jb = 0. Using Equation (3.1) to find the relation between the
wall potentials φa and φb and the plasma potential φp :
1
2e−1/2 −
2
2mi
e(φa − φp )
1
exp
−
πme
kTe
2
2mi
e(φb − φp )
exp
πme
kTe
= 0,
or rearranging for φp :
eφp
exp
kTe
e1/2
=
4
2mi
eφa
exp
πme
kTe
eφb
+ exp
kTe
.
Note that when φa = φb the plasma potential is such that neither surface receives
net current, which is electrically equivalent to the floating condition. Also note that
the plasma potential always adjusts until it is at least 3 kTe /e above the maximum
of φa or φb .
The essential difference between the two pictures of the flux tube in Figure 3.1
is the ratio of the perturbation length to the connection length. Stangeby and McCracken [64, p.1233] estimate the perturbation length of a probe of area l1 × l2 by
equating the parallel flux to the probe, Γ = ncs l1 l2/2, with the perpendicular flux into
79
flux tube, Γ⊥ = 2(l1 + l2)Lp D⊥ n/λn . We take l1 ≈ 20 mm, l2 ≈ 10 cm, λn ≈ 20 mm.
Using D⊥ ≈ 0.7 m2 /s (as calculated from Langmuir probe measurements, see Section
6.4.1), Te ≈ 7 eV, and solving for the perturbation length gives Lp ≈ 0.7 m, which is
of the same order as the connection length, L = 0.3 m. Therefore for the bounded
flux tube model we use picture (b) as in Figure 3.1.
Experiments on DITE by Matthews and Stangeby [44] to measure the perturbaBohm
tion length of probe indicate that is is appropriate to use D⊥ ≈ D⊥
= 0.06Te [eV]/B[T] ≈
0.08 m2 /s. Using this value gives Lp ≈ 3 m, greater than the connection length.
3.1.2
Sheath Drop with Secondary Electron Emission
We can generalize the fluid model of the sheath edge to include the contribution of
secondary electron emission due to electron impact at the surface. Following the
simple model of Hobbs and Wesson [22] it is shown that secondary electron emission
can reduce the difference between the sheath potential and the floating potential to
e(φs −φf ) ≈ kTe , but preserves the other conclusions of the flux-tube potential model:
for a short flux tube the maximum surface potential determines the sheath potential,
and the floating condition is obtained when both surfaces have the same potential.
A simple fluid model is used to develop intuition; more sophisticated treatments
are available in the literature. For instance, full kinetic calculations by Schwager
[62] predict the presheath and sheath potential drops given the electron and ion
temperatures and coefficient of secondary electron emission. For the critical secondary
electron emission coefficient of γ = 0.9, which causes a local potential well to form
in front of the surface, Schwager predicts a deuterium-tritium plasma with Ti = Te
to have presheath and sheath potential drops of 0.4Te /e and 1.0Te /e, respectively.
Stephens [65] obtains a similar result.
In the presence of secondary electron emission from a surface, Hobbs and Wesson
80
find the density of electrons at the sheath edge ne (xs ) is given by

e(φsurf − φs )
ne1 (xs ) 
γ
= 1 + exp
Znis
2
kTe
E2 + e(φs − φsurf )
π
kTe
−1/2 −1
 ,
where φsurf is the potential of the surface, φs is the potential at the sheath, nis is the
ion density at the sheath edge, Z is the ion charge, γ is the ratio of secondary electron
flux at the surface to the incident electron flux due to all processes (e.g. including
low energy electron reflection), and E2 is the energy of secondary electrons leaving
surface. Evaluation of the Bohm sheath criterion gives

ne1 (xs ) 
πγ
e(φsurf − φs )
ZkTe
≤
1−
exp
2Ei
Znis
4
kTe
E2 + e(φs − φsurf )
π
kTe
−3/2 
.
These can be used to obtain the net current density J to the surface:
e(φsurf − φs )
ne1 (xs )
=1−
(1 − γ) exp
Znis
kTe
eZnis 2Ei /mi
J
mi kTe
.
2πme 2Ei
Full details of the calculations are presented in Section A.3.
Hobbs and Wesson consider the special case when J = 0, Z = 1, φs = 0, E2 e(φ(x) − φsurf ), Ei ≈ ZkTe /2, and me /mi 1 to find
1 (1 − γ)2 mi
e(φs − φf )
≈ ln
.
kTe
2
Z
2πme
The important feature of this result is that the difference between the sheath potential and the floating potential drops as the secondary electron emission coefficient γ
increases. When γ approaches a critical value near unity the fluid model breaks down
and a fully kinetic model must be used.
81
Figure 3.2: Normalized electron current density to a surface as a function of normalized surface bias with different secondary electron emission coefficients γ.
Langmuir Probe Characteristic
Figure 3.2 shows the normalized electron current density 1 − J/(eZnis 2Ei /mi ) to
the surface as a function of the normalized surface bias e(φs − φsurf )/kTe for different
values of secondary electron emission coefficient γ. It can be seen that increasing the
secondary electron emission depresses the absolute levels of electron current, but the
current still decreases exponentially for surface potentials below the sheath potential
by about kTe /e. Therefore we expect to extract a correct electron temperature using
IV characteristics from Langmuir probes even with significant secondary electron
emission. If secondary electron emission due to ion impact is negligible, the ion
saturation current is unmodified and the correct electron density can be extracted
from the Langmuir probe IV characteristic.
82
3.1.3
Collisional Presheath
In the following chapters we will employ a collisionless model for the evolution of the
distribution function inside the omegatron and the sheath outside the omegatron. In
this section we identify criteria to determine if the presheath is collisionless, which
in turn influences the initial conditions for the distribution functions of bulk and
impurity ions entering the omegatron. We find that in many cases the presheath
cannot be considered collisionless.
In fluid sheath theory similar to that employed in Section 3.1.1, the ions enter
the sheath at the sound speed, cs =
(kTe + kTi )/mi . The characteristic parallel
transport time in the presheath is the connection length divided by the ion sound
speed, τ = L /cs . We compare the parallel transport time with the characteristic
rate for an ion species α to acquire the flow velocity of the background plasma β, the
slowing-down rate[72]:
νsαβ
−14
= 6.8 × 10
nβ Zα2 Zβ2 λα,β
µβ /µα
µα + µβ
1/2
1
3/2
Tα
m3 /s,
where j = α, β represent the two species of ions, Tj represents the temperature in eV,
µj = mj /mp represents the mass in units of the proton mass, Zj represents the ion
charge, nj represent the density in m−3 , and λα,β is the Coulomb logarithm,

λα,β = 23 − log 
Zα Zβ (µα + µβ )
µα Tβ + µβ Tα
nα Zα2
Tα
nβ Zβ2
+
Tβ
1/2 

≈ 12.
If the parallel transport time is much longer than the inverse slowing-down rate
then we consider the test ions to have acquired the background plasma flow velocity.
In this case at the sheath edge the test ions have the bulk plasma sound speed,
vα,s = cs , and the test ion density at the sheath edge is related to the test ion flux at
the sheath edge by nα,s = Γα,s /cs .
On the other hand, if the parallel transport time is much shorter than the slowing
83
down time then we consider the presheath to be collisionless, and the test ions and
bulk ions can have different flow velocities at the sheath edge. The test ion density at
the sheath edge is obtained by integrating the distribution function over all velocities
at the sheath edge, and the fluid velocity and the flux are related by vα,s = Γα,s /nα,s .
Similarly we compare the parallel transport time with the temperature equilibration rate[72]:
ναβ
−13
= 1.4 × 10
nβ Zα2 Zβ2 λαβ
(µα µβ )1/2
(µα Tβ + µβ Tα)
3/2
m3/s.
(3.2)
If the parallel transport time is much longer than the temperature equilibration time
then we would expect the test ion species α to acquire the background plasma temperature, Tα = Tβ . Conversely, if the parallel transport time is long compared to the
temperature equilibration time then the test ion and background ion species can have
different temperatures, Tα = Tβ .
Figure 3.3 shows the characteristic times of slowing down and temperature equilibration for a background plasma with conditions similar to the flux tube at the
omegatron, density nβ ≈ 5 × 1017 m−3 and temperature Tβ ≈ 3 eV. The test ions
are hotter than the background, Tα ≈ 20 eV. It can be seen that all test ions more
than once ionized Zα ≥ 2 acquire both the temperature and flow velocity of the
background plasma while travelling along the presheath. Singly-ionized test species
Zα = 1 do not acquire the background ion temperature, but ions with mass µα ≤ 4
do acquire the background ion flow velocity. Since the temperature equilibration and
slowing down rates depend linearly on the background ion density, these conclusions
depend sensitively on the density of background plasma near the omegatron. Note
that temperature equilibration occurs more quickly at low ion temperatures.
84
Figure 3.3: Comparison of parallel transport time with characteristic slowing down
times and temperature equilibration times, for 20 eV ion minority (top) or 3 eV ion
minority (bottom) on 3 eV ion bulk.
85
3.1.4
Ion Distribution at the Sheath Edge
We use both the retarding field energy analyzer and the ion mass spectrometer of
the omegatron probe to perform measurements on portions of the ion distribution.
To relate the measurements inside the omegatron probe to the plasma outside the
omegatron we must adopt a kinetic model of the sheath and presheath, as opposed
to a fluid model. We briefly survey the literature of kinetic models of the presheath,
considering first collisionless models and then collisional models. For completeness
the sources and collision operators of the models described in this section are listed in
Section A.7. Most models in the literature with a realistic treatment of the presheath
assume a perfectly absorbing wall. On the other hand, most kinetic models in the
literature which consider emitting walls, for example refs.[63, 49, 62, 51, 50, 65] assume
that the ion distribution at the sheath edge is a shifted half-Maxwellian. Following
our literature survey we will conclude that it is acceptable to assume a shifted halfMaxwellian distribution for the ions at the sheath edge.
In 1929 Tonks and Langmuir[71] developed the first kinetic model of the sheath
and presheath, which included several now-common features: electrons are described
by a Boltzmann relation; the plasma is divided into sheath and presheath, where the
quasineutral presheath has spatial extent much greater than the collisionless, sourcefree sheath; and the wall is perfectly absorbing. Tonks and Langmuir considered an
ion source proportional to electron density raised to a power, S ∼ nγe , so the source
could be uniform (γ = 0), representative of volume ionization (γ = 1), multiple-stage
ionization (γ > 1), or even volume recombination (γ < 0). The ions were assumed to
be born with no thermal spread (cold ions), so the ion distribution could be obtained
directly from the ion source function; they also considered the case when ions were
born with a thermal spread, but Ti /Te 1.
In 1959 Harrison and Thompson [19] presented an analytic solution to the presheath
model of Tonks and Langmuir. They showed that quantities averaged over the ion
distribution at the sheath edge does not depend on the spatial form of the ion source
86
function in the presheath. The proof of Harrison and Thompson assumed a cold ion
distribution, but Emmert et al [14] later generalized the proof for a hot ion distribution. Harrison and Thompson also identified a criterion that any ion distribution at
the edge must satisfy in order that a sheath be formed, v −2 ≤ mi/(ZkTe ), which
generalized Bohm’s fluid model criterion for sheath formation. Their derivation is
reproduced in Section A.2.
In 1980 Emmert et al [14] considered a kinetic model of the collisionless plasma
presheath with a hot ion distribution. They considered a special source function
which, in the absence of potential gradients, gives a Maxwellian ion distribution.
With their special source they found an analytic solution in the presheath and solved
for the ion distribution at any location in the sheath and presheath. They also solved
the combined presheath and sheath problems numerically, finding good agreement
for the presheath in the limit of small Debeye length. They noted that an important
extension of the theory would be to include secondary electron emission from the
surface. In 1987 Bissel [7] verified that the ion distribution of the model of Emmert
et al satisfies the generalized Bohm criterion.
In 1987 Matthews et al [40] used an approximation of the presheath model of
Emmert et al to obtain the sheath potential from the IV characteristic of a retarding
field energy analyzer. Matthews et al noted that the model of Emmert et al resulted
in an ion distribution at the sheath edge which vanishes for ions with negative velocity,
decreases exponentially above a certain velocity, and in the transition region due to the
presheath is described with a complicated but analytic function. During the analysis
of their data Matthews et al neglected the presheath portion of the distribution
function, asserting that the systematic error would introduce at most 10% change in
their results (less than their experimental uncertainties). Thus the ion distribution
at the sheath edge that Matthews et al considered at the sheath edge was of the form



f(xs , v) = 

0
v < vmin ,
C(xs ) exp(−v 2/vt2 ) v ≥ vmin,
87
(3.3)
where x = xs is the sheath edge location, vt2 = 2kTi /mi , the normalization factor
C(xs ) depends on the flux to wall and the sheath potential, and vmin is determined
in part by the presheath potential drop.
Also in 1987 Bissel and Johnson [8] obtained an analytic solution to a collisionless
presheath model using a Maxwellian source function (rather than a source function
that results in a Maxwellian ion distribution in the absence of potential gradients,
as in the case of Emmert et al). In their analysis Bissel and Johnson imposed the
generalized Bohm criterion at the sheath edge; the presheath potential drop they
obtained was a factor of two larger than that of Emmert et al, and the potential
gradient was infinite at the sheath edge, whereas that of Emmert et al was finite.
In 1988 Scheuer and Emmert [61] reproduced the results of Bissel and Johnson
numerically, but without imposing the generalized Bohm sheath criterion as a boundary condition. Scheuer and Emmert obtained the ion distribution everywhere in the
presheath and found a finite electric field at the sheath edge. They noted that the
differences in sheath potentials between the models of Emmert et al and Bissel and
Johnson were in fact due to the different ion source functions in the presheaths, and
that while the source of Bissel and Johnson appropriately models ionization from a
population of neutrals with a given temperature, the resulting ion distribution at the
symmetry point is distinctly peaked and non-Maxwellian. They suggested that the
ion source proposed by Emmert et al is more applicable to other presheath models,
including a one-dimensional model with a “fictitous” cross-field transport source, in
which no ions are born with zero velocity.
Later in 1988 Chung and Hutchinson [11] considered a purely numerical model of
a collisionless presheath in a strongly magnetized plasma, where perpendicular transport into and out of the neighboring plasma was treated as a one-dimensional ion
source. The plasma next to the flux tube was assumed to be described by a shifted
Maxwellian distribution. Chung and Hutchinson obtained the ion distribution everywhere in the presheath, which far from the sheath was the same as the neighboring
88
plasma, and which at the sheath edge looked qualitatively similar to the distribution
obtained by Emmert et al. Chung and Hutchinson developed their model to interpret
flow measurements to probe surfaces, but as a special case they compared their zeroflow ion saturation currents with the models of Emmert et al and Bissel and Johnson,
with the Bissel and Johnson model giving the best agreement.
Chung and Hutchinson [12] generalized the ion source function in their model to
allow for a variable ratio of cross-field viscosity to diffusivity. The case of zero crossfield viscosity most nearly matched the ion source function of Bissel and Johnson,
and Chung and Hutchinson recovered the sharp ion distributions at the sheath edge.
Chung and Hutchinson showed that the influence of adding cross-field viscosity to the
ion source function is to round out the ion distribution at the sheath edge.
In 1991 Pitts [55] calculated the current he would have obtained with a retarding
field energy analyzer if the presheath had been governed by the analytic models of
Emmert et al and Bissel and Johnson, and then compared the results to experimental measurements with the retarding field energy analyzer on the DITE tokamak.
Despite the fact that the two models have very different source functions and, at
the symmetry point, very different ion distributions, Pitts showed that the predicted
IV characteristics are actually quite similar, and that it is not possible to distinguish
between them using his experimental data. Based on the plausibility of the ion distribution at the symmetry point, Pitts indicated a preference for the model of Emmert
et al. Pitts also noted that neglect of presheath portion of the ion distribution at low
temperature or high secondary electron emission might not be justified.
In 1988 Scheuer and Emmert [60] considered a presheath model with low to moderately collisionality using a Bhatnagar-Gross-Krook (BGK) collision operator in the
Boltzmann equation. They modeled ion-ion collisions by requiring that the BGK
collision operator conserve particles, momentum and energy. Over the range of collisionality for which their numerical scheme would converge, λmfp /L ≥ 1, they found
that ion-ion collisions had almost no noticible effect on the ion distribution at the
89
sheath edge. Since they used the same ion source term as Emmert et al [14], the ion
distributions of Scheuer and Emmert looked the same as those of Emmert et al, that
is, exhibited rounding at the sheath edge. In 1989 Koch and Hitchon [30] extended
the numerical work of Scheuer and Emmert [60], showing that at high collisionality
λmfp /L ≈ 0.1 the ion distributions at the sheath edge resulting from the ion sources
of Emmert et al [14] and Bissel and Johnson [8] are indistinguishable.
Several authors have treated kinetic presheath models including ion-neutral collisions, mostly in the context of weakly-ionized plasmas. See refs.[59, 35, 6, 60] for
example. It will be shown in Chapter 6 that the ionization source near the omegatron is much less than the parallel flux. We use that result here to justify neglect of
ion-neutral collisions.
Therefore in the remainder of this thesis we adopt the assumption that ions at the
sheath edge have a shifted half-Maxwellian distribution of the form Equation (3.3).
We determine the lower velocity such that the distribution satisfies the generalized
Bohm criterion [19],
ns =
∞
vmin
fs dv,
1
ns
∞
vmin
mi
fs
dv
≤
.
v2
ZkTe
The integrals can be evaluated using
∞
x
1
exp(−t2)tadt = Γ( a+1
, x2 ),
2
2
where Γ(a, x) is the complementary incomplete gamma function [2, pp.566–569],
which has the recursion relation Γ(a + 1, x) = aΓ(a, x) + xa exp(−x). The generalized Bohm criterion is satisfied if
u−1 exp(−u2)
kTi
,
≤ 1+
1
2
Γ( 2 , u )
ZkTe
where u ≡ vmin /vt. For kTe = 7 eV, kTi = 3 eV, Z = 1, we require vmin/vt ≥ 0.81.
90
3.2
Slit Transmission
In this section the transmission of ions through the slit is estimated. It is found
that the 45 degree knife-edge of the slit admits transmission of a half-Maxwellian
distribution of ion through the slit equivalent one quarter of optical transmission.
For the omegatron slit geometry transmission increases by a factor of two if the halfMaxwellian distribution is shifted forward such that the minimum energy is twice the
thermal energy.
t
z
θ=45˚
d = 25 um
α'
x
α''
Figure 3.4: Schematic of cross section of slit geometry, showing gap between 45 degree
knife edges.
Figure 3.4 shows a schematic of the cross section of the slit geometry. The x
direction is taken parallel to the magnetic field, and the slit is along the y direction.
The tungsten pieces which define the slit present area A1 = 30 mm2 to the plasma,
and the gap between the pieces (the slit) presents area A2 = 0.2 mm2 to the plasma.
We can define “optical transmission” through the slit as the ratio of the gap area to
the total area: ξopt,slit ≡ A2/A1 = 0.0067.
The class of ions which can pass through the slit must meet at least one of two
criteria:
1. The ions have Larmor radius smaller than the slit spacing. This restricts the
maximum possible perpendicular energy of the ion, but places no restrictions
91
on the parallel energy. For this case the energy transmission function is
@(vx , vy , vz ) =



1 v⊥ ≤ v⊥,max,


0 v⊥ > v⊥,max,
2
= vx2 + vy2.
where v⊥
2. The ions have a forward pitch angle greater than the slit knife-edge angle. This
restricts the ratio of parallel to perpendicular energy to be greater than 45
degrees. For this case the energy transmission function is
@(vx , vy , vz ) =



1 vx /vz ≥ tan α,


0 vx /vz < tan α,
where α is the angle formed by the nearest material surface.
The energy transmission function is sketched in Figure 3.5 for deuterons in a magnetic
field of 5 T. By inspection the pitch-angle criterion permits a larger range of total
energies than the Larmor radius criterion.
For ions which meet one of the two criteria the tranmission is considered optical,
that is the fraction of ions which pass through is proportional to the ratio of the gap
area to the total tungsten area. The class of ions which do not meet either criteria
cannot pass through the slit; the distribution of ions on the other side of the slit is
depleted of this class ions (unless it is filled in by collisions).
For the Larmor radius criterion the effective transmission is approximately equal to
the product of the optical transmission and the fraction of ions which have perpendicu2
lar energy less than the maximum perpendicular energy: ξeff /ξopt ≈ erf(mv⊥,max
/2kT ),
where v⊥,max depends on the magnetic field strength and the ion mass. For deuterions
2
/2 = 0.1 eV. Characterat B = 5 T and slit dimension l = 25 µm we have mv⊥,max
istic temperatures are typically much higher than this, so we need consider only the
pitch-angle criterion.
92
Figure 3.5: Energy transmission function of deuterions through the slit for B = 5 T
and l = 25 µm
93
We can estimate the fraction of a shifted half-Maxwellian distribution of ions that
passes through an aperature formed by pieces with angle θ, spacing d, and edges
of thickness t, as in Figure 3.4. The algebra is tractable only if we assume that
the ion orbits are straight lines, that is the ion larmor radius is much less than the
slit dimensions. We assume that transmission is optical for ions with pitch angle
vx /vz ≥ tan α, otherwise it is zero. For the ion shown in Figure 3.4, if vz > 0 then
tan α = tan θ, and if vz < 0 then tan α = (d/2 − z)/t. Consider flux incident on the
slit from within the horizontal strip of height dz at z:
Γx (z) =
∞
0
vx dvx
∞
−∞
dvy
∞
−∞
dvz f(vx , vy , vz ).
The transmitted flux can be written
Γx (z) =
=
∞
0
∞
0
vx dvx
vx dvx
∞
−∞
∞
−∞
dvy
dvy
∞
dvz f(vx , vy , vz )@(vx, vy , vz ),
−∞
vx tan α
−vx tan α
dvz f(vx , vy , vz ),
where tan α = min[tan θ, (d/2 + z)/t], tan α = min[tan θ, (d/2 − z)/t], and −d/2 ≤
z ≤ d/2. The relative transmission at z is defined as ξr (z) = Γx /Γx . We find the
total relative transmission by finding the average of ξr (z) over the slit area,
1
ξslit
=
ξops
d
d/2
−d/2
ξr (z)dz.
For a half-Maxwellian shifted in the x direction the distribution can be written
f(vx , vy , vz ) = C exp
m (vx − v0)2 + vy2 + vz2 ,
2kT
for vx ≥ v0 and for all vy , vz . The transmitted flux is
Γx (z) = C
∞
−∞
dwy e−wy
2
∞
w0
wx dwx e−(wx −w0 )
94
2
wx tan α
−wx tan α
dwz e−wz ,
2
Figure 3.6: Relative transmission through a slit with spacing d, edge thickness t,
angle θ, of a half-Maxwellian distribution with of temperature kT shifted by energy
qφ0 = w02 /2. Relative transmission decreases with finite edge thickness.
and the relative transmission becomes
∞
ξr (z) = 1 −
w0
wxdwx e−(wx −w0 ) Γ( 12 , (wx tan α )2 ) + Γ( 12 , (wx tan α )2 )
2
1 + Γ( 12 )w0 Γ( 12 )
,
where wj = vj / 2kT /m, vj = vx, v0. Figure 3.6 shows the average relative transmission for ions through a slit of with tan θ = 1 (45 degrees) and various values of l/d for
a shifted maxwellian as a function of w02 /2 = qφ0/kT , where φ0 represents the sheath
potential drop. Note that the relative transmission for 1 ≤ t/d ≤ 2 almost doubles
as the sheath potential drop increases from zero to two times the ion distribution
temperature. Also note the decrease in average transmission as the edges of the slit
become more blunt. Visual inspection of the knife-edges for the omegatron probe slit
gives 1 < t/d < 2.
95
3.3
Retarding Field Energy Analyzer Model
The shape and bias of the slit have two important influences on the plasma which
passes through the gap: (1) the transmitted plasma forms a collimated ribbon-shaped
beam, and (2) the beam can be non-neutral. First we obtain an estimate of the current
limit for equilibrium configurations of the non-neutral beam. Then we estimate the
current limit for which beam space charge has a significant effect on the electrostatic
potential. These two current limits establish the operating range of the omegatron
where single-particle theory applies. The single-particle model of the retarding field
energy analzer is presented, with which it is shown how to extract the distribution
temperature and sheath potential. Formal modifications to the theory are presented
to include the effect of space charge.
3.3.1
Brillouin Flow
Brillouin [10] first pointed out in 1945 that a non-neutral column of charged particles
will remain confined by a magnetic field up to a space-charge limit set by the column
charge density and the magnetic field strength. The maximum charge density that
2
, or n = nB ≡
can be obtained in an equilibrium configuration occurs when ωc2 = ωpi
@0B 2 /mi , the Brillouin density. See Krall and Trivelpiece[31] or Davidson[13]. For
deuterons in a magnetic field of approximately 5 T, the Brillouin density is nB ≈
7 × 1016 m−3 .
Consider a beam of ions moving with velocity v along a magnetic field of strength
B. If the beam has cross sectional area A then the current carried by the beam is
I = qnvA, where n is the density of the ions in the beam and q is the charge per
ion. Note that the beam cross sectional area at the slit is Aslit and is unlikely to
decrease further downstream, so we can take A ≥ Aslit. If the density of the ions
in the beam exceeds the Brillouin density then the non-neutral plasma column will
no longer be in equilibrium. The beam will expand until the ion density is at or
below the Brillouin density. We can estimate a current limit for ions which move
96
GRID A
POTENTIAL
MAXIMUM
POTENTIAL WITH
SPACE CHARGE
GRID B
POTENTIAL
DIFFERENCE
DUE TO
SPACE CHARGE
MAXIMUM
POTENTIAL,
VACUUM ONLY
DISTANCE
Figure 3.7: Schematic of the influence of space charge on the electrostatic potential
between two parallel surfaces of fixed potential.
with the thermal speed v =
kTi /mi by requiring the density to be at or below
the Brillouin density: I ≤ Imax,B ≡ qnB Aslit kTi /mi . For deuterons at Ti = 3 eV,
Aslit = (7 mm)(25 µm), and B = 5 T gives Imax,B = 27 µA. For I Imax,B we neglect
broadening of the beam and consider the beam to be confined.
3.3.2
3-D Space Charge
Consider a collimated and confined non-neutral beam of ions. We want to estimate
the current at which the space charge from the ions contributes significantly to the
electrostatic potential. The motivation for this exercise is suggested in Figure 3.7,
which sketches the potential structure between two parallel grids in vacuum (straight
line) and in the presence of significant space charge (curved line). In the retarding
field energy analyzer grids are used to reflect charged particles which have kinetic
energy lower than the grid potential. If significant space charge is present then it is
possible for the plasma to determine the maximum potential rather than the grids.
In this case the mapping between the maximum potential and the grid potential is no
longer trivial. To determine the potential structure inside the grids requires solving
97
the Poisson equation consistently with the equation of motion for the ions.
Here we are interested only to find the current limit below which we can neglect
the influence of space charge on the electrostatic potential structure. Therefore we
can solve a simpler problem by calculating the electrostatic potential due to the beam,
neglecting the influence of the potential on the ion equation of motion. If the total
electrostatic potential is much less than the average energy of ions in the beam then
our neglect is justified. We can estimate the current limit by solving for the ion beam
current where the total electrostatic potential is of order the average ion energy.
We consider just the contribution of the space charge. We can always add a
homogeneous solution to the Poisson equation (i.e. the vacuum electrostatic potential)
if the applied grid potentials are different from ground. Consider a point charge
inside a rectangular cavity with grounded walls. The electrostatic potential due to
the point charge, a Green function, can be found by solving the Poisson equation with
a delta function source. The electrostatic potential due to a beam of ions is found by
integrating the product of the Green function and the beam charge density over the
volume of the cavity. The details are given in the appendix, both for a rectangular
volume of charge density in a grounded box (Section A.4) and for an infinite ribbon
of charge density between two grounded surfaces (Section A.5).
Let the rectangular cavity be defined by two grids at x = ±a, the side walls at
y = ±b, and the top and bottom covers at z = ±c. The region of non-zero space
charge extends along the field lines between the grids in the region −a ≤ x ≤ a,
where a = a. The width of the space charge is defined by the slit width −b ≤ y ≤ b,
and the slit height −c ≤ z ≤ c . We are particularly interested in the potential along
y = z = 0 and 0 ≤ x ≤ a:
∞
φ(x, 0, 0)
8 cos kl x (−1)l sin km b sin kn c
,
=
2
ρ/@0
klmn
kl a km b
kn c
l,m,n=0
2
2
where kl = (l + 12 )π/a, km = (m + 12 )π/b, kn = (n + 12 )π/c, and klmn
= kl2 + km
+ kn2 .
The charge density ρ is found from the beam current through I = ρv(2b)(2c ).
98
The dependence of the maximum electrostatic potential on the beam thickness can
be seen easily for the case when the beam width is much bigger than both the beam
thickness and the beam length: φ(x = 0, z = 0) ≈ (a2ρ/(2@0 ))[1 − exp(−πc/(2a))].
In the limit that the beam thickness is much bigger than the spacing between the
grids c a, we recover the familiar slab result.
Consider a beam of ions of mass mi with an average ion velocity v =
kTi/mi .
We estimate the current limit by solving for the case when
e a2 Imax,φ
eφ(0, 0, 0)
=
kTi
kTi 2@0 4bc
mi
F (a, b, c, a, b, c ) ≈ 1,
kTi
where F (a, b, c, a, b, c ) is the summation term divided by a2/2. It can be shown that
in the limit that b = b = c = c → ∞ the factor F (a, b, c, a, b, c) → 1, leaving the
familiar infinite slab result.
Dimensions typical of a ribbon-shaped beam of ions passing between two RFEA
grids, a = a = 0.35 mm, b = 7.5 mm, b = 3.5 mm, c = 5 mm, and c = 15 µm, give
F (a, b, c, a, b , c) = 0.052. Deuterons with kTi = 3 eV give a space-charge limiting
current of Imax,φ ≈ 21 µA. (For an infinite slab model with the same grid spacing
the space-charge limiting current would be ≈ 1 µA.) Thus we expect if the ion beam
current I Imax,φ we can neglect space charge in ion equation of motion. This will
allow us to consider the influence of just the vacuum potential structure on the ion
distribution function in the retarding field energy analyzer.
3.3.3
RFEA 1-D Kinetic Model
If we operate below the current limits for beam confinement and space charge effects
then a one-dimensional, single-particle description of the ion dynamics is appropriate.
The retarding field energy analyzer (RFEA) consists of three grids parallel to each
other and normal to a beam of ions. To measure the ion distribution function: the
bias of the first grid is set very negative to reject electrons; the bias of the second grid
99
is swept (reflector grid) and only ions with sufficient parallel kinetic energy can pass it;
the bias of the end collector is set below the bias of the second grid, and the bias of the
third grid is set below the bias of the the end collector to repel secondary electrons
emitted by the end collector back to the end collector. Plotting the end collector
current as a function of the reflector grid bias gives the so-called IV characteristic.
From the derivative of the IV characteristic with respect to voltage it is theoretically
possible to extract the distribution function of the ions or electrons. In practice the
distribution function is often Maxwellian, so one usually fits an exponential to the IV
characteristic and reports a temperature.
Figure 3.8 shows a schematic of the omegatron cross section and the voltage bias
of each component during typical RFEA operation measuring the ion distribution
function.
A one-dimensional model of the retarding field energy analyzer is presented for
two reasons: (1) it is simple, so it gives predictions of performance quickly which
can be used to build intuition, and (2) it accurately represents the behavior of the
diagnostic in low density plasmas.
A one-dimensional kinetic model is used for the plasma flowing between two grids.
The plasma is assumed to be composed of either ions or electrons but not both, so it is
not quasineutral. The plasma density is assumed to be low enough that the influence
of space charge on the electrostatic potential can be neglected; space charge effects
are considered later. The complete model of the retarding field energy analyzer is
generated by considering a series of grids, with appropriate attenuation of plasma as
it passes through each grid.
Hutchinson [25, p.79] gives gives a brief description of RFEA operation; the appendix reviews kinetics fundamentals. It is found that for a flux Γs of particles with
a half-Maxwellian distribution of velocities incident on the RFEA, the fraction that
100
PLASMA
SLIT
G1
G2
G3
RF
END
+70V
ION MODE
φsheath
0V
-40 V
-70 V
+70V
ELECTRON MODE
+40V
φsheath
0V
-70 V
Figure 3.8: Schematic cross section of omegatron and axial vacuum potential structure. Configuration with G2 as ion parallel energy selector is shown, with SLIT
grounded, V = 0 V.
101
is collected downstream is equal to




1
φmax ≤ φs ,
Γ(x)
=
 exp q(φs − φmax )
Γs

φmax ≥ φs ,

kT
(3.4)
where φs is the sheath potential and φmax is the maximum potential in the analyzer,
set by reflector grid bias when space charge is negligible. Thus by sweeping out the
reflector grid bias and measuring the current downstream it is possible to extract the
distribution temperature.
Details in the appendix show that for this one-dimensional model, space charge
is negligible in the omegatron probe RFEA when the total current to the analyzer
is below Imax ≈ 20 µA. Above this current space charge determines the maximum
potential in the analyzer φmax rather than the reflector grid bias, and plotting the
downstream current as a function of the reflector grid bias gives a current-voltage
characteristic with a “soft knee”. Thus interpreting the knee in the characteristic as
the sheath potential when space charge cannot be neglected gives a sheath potential
lower than expected.
3.4
Grid Transmission
Transmission of individual ions through the grids depends on the ion kinetic energy
perpendicular and parallel to the magnetic field. It is possible to estimate which ions
pass through the grids, based on the grid geometry and the parallel and perpendicular energy of the ions, and it is found that most ions have the same probability
of transmission (optical). This permits a significant theoretical simplification: the
distribution of ions that has passed through a grid is equal to the distribution before
the grid but attenuated by a scalar factor. A technique is developed by which the
transmission coefficients for each grid can be determined using experimental measurements.
102
s
d
Larmor radius
pitch angle
Figure 3.9: Sketch of transmission of ions through grids if pitch angle is sufficiently
steep.
We can estimate grid transmission geometrically. As with transmission through
the slit, there are two criteria:
1. For the class of ions with Larmor radius less than the space between grids, rL ≤
s, the transmission approaches optical transmission. This restricts v⊥ ≤ ωc s,
which is a horizontal line on the E⊥ vs E graph. Here s represents the space
between grid lines.
2. A class of ions with pitch angle steep enough can penetrate the grid even if the
Larmor radius exceeds the grid spacing: v/v⊥ ≥ d/s, where d represents the
grid line thickness. This class of ions passes through the grid in a time much
shorter than its cyclotron period. The situation is sketched in Figure 3.9.
Figure 3.10 shows the theoretical transmission through grids as a function of the
ion energy parallel and perpendicular to the magnetic field (perpendicular and parallel
to the grid, respectively); note the different axis scales. The class of ions with low
parallel energy and high perpendicular energy are attenuated completely, but from
Figure 3.10 it is clear that for deuterium ions this constitutes a small fraction of
the solid angle. The pitch angle transmission almost completely relaxes the Larmor
103
Figure 3.10: Theoretical transmission of ions through the grid. Note the different
scales.
104
radius restriction. To a first approximation we can take the grid transmission to equal
optical transmission, independent of energy. Okubo et al [48] perform more detailed
calculations of ion orbits passing through grids and arrive at essentially the same
result.
As was done for the slit, we can calculate the effective transmission through the
grid for a shifted half-Maxwellian distribution. Here the pitch angle restriction is
v/v⊥ ≥ d/s = 0.17, giving a transmission within one percent of optical transmission
for shift energy qφ0/kT > 0.01.
A more sophisticated treatment would consider the detailed truncation of the
perpendicular distribution function with respect to the parallel distribution function.
A fully realistic treatment would include the mixing of parallel and perpendicular
distributions that might occur at grids due to effective scattering of ions off grids.
These treatments are beyond the scope of this thesis.
3.4.1
Reflections from Space Charge
Consider the case when a fraction of the incoming ion flux is reflected, either due
to grid bias or due to space charge. Figure 3.11 shows an schematic of a possible
potential structure.1
The flux of ions which arrives at grid G1 from the slit is attenuated by a factor
ξ as it passes through. If there is space charge between grids G1 and G2, then only
the fraction g1 of the flux arrives at grid G2, composed of ions with sufficient kinetic
energy to pass the maximum electrostatic potential; the remaining fraction 1 − g1 is
reflected back to grid G1. The flux that arrives at grid G2 is attenuated by another
factor of ξ as it passes through the grid, and then only the sufficiently energetic
fraction g2 arrives at grid G3, etc. The incident and reflected fluxes, normalized to
the original flux incident on grid G1, are shown in Figure 3.12. Since ion parallel
energy is assumed to be conserved, ions lose no energy by passing through a grid.
1
Theoretical estimates of the potential structure between the grids and in the RF cavity are given
in Section A.4 and Section A.5.
105
PLASMA
SLIT
G1
G2
G3
RF
END
φ_3
φ_2
potentials
φsheath
φ_1
0V
Figure 3.11: Schematic of electrostatic potentials inside the omegatron. Grid bias
and/or potential due to space charge can reflect incoming ion flux.
106
G1
1
G2
ξ
g1ξ
G3
g1ξ^ 2
g2g1ξ^ 2
(1-g1 )ξ^2
(1-g1 )ξ
(1-g2 )g1ξ^ 4
(1-g2 )g1ξ^ 3
(1-g2 )g1ξ^ 2
(1-g3 )g2g1ξ^ 6
(1-g3 )g2g1ξ^ 5
(1-g3 )g2g1ξ^ 4
END
g2g1ξ^ 3 g3g2g1 ξ^3
(1-g3 )g2g1ξ^ 3
Figure 3.12: Schematic of incident and reflected fluxes to all grids, normalized to
incident flux to grid G1. Each grid is assumed to attenuate the flux passing through
it in either direction by a factor ξ. A fraction gj of the incident flux that passes
through the jth grid arrives at the next component downstream.
There are assumed to be no ions trapped in electrostatic potential wells.
The current measured on each grid is just the captured fraction 1 − ξ times the
sum of the fluxes arriving at the grid; the end collector is assumed to capture all
of the flux that arrives there. Using Figure 3.12 we can write down the normalized
current to each component:
IG1 = (1 − ξ)[1 + (1 − g1 )ξ + (1 − g2 )g1 ξ 3 + (1 − g3 )g2 g1 ξ 5 ],
IG2 = (1 − ξ)[g1 ξ + (1 − g2 )g1 ξ 2 + (1 − g3 )g2 g1 ξ 4 ],
IG3 = (1 − ξ)[g2 g1 ξ 2 + (1 − g3 )g2 g1 ξ 3 ],
IEND = g3 g2 g1 ξ 3 .
In general the flux fractions gj which pass between the grids can vary between
zero and unity. By applying extreme biases to the grids it is possible to ensure that
all of the ions have sufficient energy to pass to the end collector, or to ensure that
(practially) none of them do. In such cases the fractions gj are either zero or unity,
and the expressions simplify for the current collected on each component, depending
107
note
(1)
(2)
(3)
(4)
g1
0
1
1
1
g2
g3
0
1
1
0
1
IG1
(1 − ξ)[1 + ξ]
(1 − ξ)[1 + ξ 3 ]
(1 − ξ)[1 + ξ 5 ]
1−ξ
IG2
0
(1 − ξ)[ξ + ξ 2 ]
(1 − ξ)[ξ + ξ 4 ]
(1 − ξ)ξ
IG3
0
0
(1 − ξ)[ξ 2 + ξ 3 ]
(1 − ξ)ξ 2
IEND
0
0
0
ξ3
Table 3.1: Special cases of grid transmission and current accounting. Notes: (1) full
reflection from G2, (2) full reflection from G3, (3) full reflection from RF, (4) no
reflection.
only on the attenuation factor ξ. Table 3.1 shows the expected currents on each
component (normalized to incident flux on grid G1) when different components are
used to reflect all ions.
It is possible experimentally to enforce gj = 1 or gj = 0. By comparing the
currents measured on each component with the predictions of Table 3.1 we can check
(1) whether the attenuation factor ξ is indeed the same for each grid, and (2) whether
the attenuation factor is close to the optical value. It is found in Chapter 4 that the
attenuation factor ξ ≈ 0.66 for all grids, close to the optical value.
Once the attenuation factor ξ has been determined, the transmission factors gj
can be determined for arbitrary grid biases. Using measured currents, obtain the
following ratios:
IG3
(1 − ξ)[g3 (−ξ) + (1 + ξ)]
=
IEND
g3 ξ
g2 (−ξ + (1 − g3 )ξ 3 ) + (1 + ξ)
IG2
=
IG3
g2 (ξ + (1 − g3 )ξ 2 )
IG1
g1 (−ξ + (1 − g2 )ξ 3 + (1 − g3 )g2 ξ 5 ) + (1 + ξ)
.
=
IG2
g1 (ξ + (1 − g2 )ξ 2 + (1 − g3 )g2 ξ 4 )
Then solve sequentially for transmitted fractions g3 , g2 , g1 :
g3
IG3 ξ
= (1 + ξ)
+ξ
IEND 1 − ξ
−1
108
,
g2
g1
IG2
= (1 + ξ)
(ξ + (1 − g3 )ξ 2 ) − (−ξ + (1 − g3 )ξ 3 )
IG3
IG1
= (1 + ξ)
(ξ + (1 − g2 )ξ 2 + (1 − g3 )g2 ξ 4 )−
IG2
(−ξ + (1 − g2 )ξ 3 + (1 − g3 )g2 ξ 5 )
−1
−1
,
.
Values of transmission fractions gj < 1 imply reflected ions. If the maximum bias of
the grids is less than the sheath potential then space charge must be present. Once
the grid attenuation factor ξ is known one need only to measure the grid currents to
determine if space charge is present.
Electron transmission through grids is predicted to be optical because of the
smaller Larmor radii, but observations are not consistent with current ratios predicted
for optical transmission. The problem is with the simple model here, which neglects
secondary electron emission due to electron impact. Henceforth electron transmission
is assumed to be optical and it is understood that there can be significant secondary
electron emission from electron impact.
3.4.2
Space Charge Potentials
If we assume that the distribution of bulk ions in the omegatron is described by a
half Maxwellian, f ∼ exp(−µ(x, v)/kT ), µ = mv 2/2 + qφ(x), then we can calculate
the maximum potentials due to space charge between the grids that would reproduce
the observed transmission fractions g1 , g2 , g3 .
Assume that the ions incident on grid G1 have minimum parallel energy µs , which
2
we evaluate at the sheath edge: µs = mvmin
/2 + qφs . Ions in the distribution have
forward speed greater than vmin such that the distribution satisfies the generalized
Bohm criterion, see Section 3.1.4.
Let ions that have passed through grid G1 on their way to grid G2 have flux Γ1 , of
which only the fraction g1 ≡ Γ2 /Γ1 composed of ions with minimum energy µ1 = qφ1
109
arrives at G2:
Γ1 =
∞
µs
dµ
f(µ) ,
m
Γ2 =
∞
f(µ)
µ1
dµ
.
m
Evaluating the integrals and taking the ratio gives g1 = exp[(µs − µ1 )/kT ]. Solve
for µ1 to find the minimum energy of the bulk ions arriving at grid G1: µ1 =
µs + kT log(1/g1 ), where kT is the bulk ion temperature. The potential is found
by dividing the parallel energies by the charge of the bulk ion species, q = e for
deuterium. Similar calculations can be performed for the maximum electrostatic
potentials between G2/G3 and G3/END:
µs kT
+
log(1/g1 ),
e
e
kT
= φ1 +
log(1/g2 ),
e
kT
= φ2 +
log(1/g3 ).
e
φ1 =
φ2
φ3
Now consider a trace impurity with distribution f ∼ exp(−µ/kT ), with minimum parallel energy µs . Since the impurity quantity is trace we can neglect the
contribution of its space charge to the electrostatic potential. Therefore we can use
the electrostatic potential maxima established by the bulk ion species φ1, φ2 , φ3, to
calculate the fractions of impurity distribution that arrive at each grid.
Let the impurity ions which have passed through grid G1 and are moving towards
grid G2 have a flux given by Γ1 , of which only the fraction Γ2 /Γ1 composed of ions
with minimum energy µ1 = q φ1 arrives at G2:
Γ1
=
∞
µs
dµ
f (µ) ,
m
Γ2
=
∞
q φ1
f (µ)
dµ
.
m
The impurity transmission factor is given by Γ2 /Γ1 = exp[(µs − qφ1)/kT ]. If we
assume that the trace impurities have the same temperature as the bulk ions kT ≈
kT , then Γ2 /Γ1 = g1Z , where g1 is the transmission factor found for the bulk ion
species.
110
3.5
Omegatron Ion Mass Spectrometer Model
First we consider a model of single particle orbits in the omegatron ion mass spectrometer. This allows us to predict the range of RF frequencies over which which
resonant ions are collected, the so-called resonance width. The derivation of the resonance width is performed with a homogeneous magnetic field. A magnetic field which
varies over the width of the omegatron broadens the resonance, and this broadening
effect is calculated.
In this simple model ions are assumed to enter the RF cavity midway between the
RF plates, in a collimated beam. The ions are collected on the plates if they spend
enough time in the RF cavity to acquire enough perpendicular energy from the RF
field that their Larmor radius increases to half the plate spacing. Otherwise the ions
exit the RF cavity without begin collected.
If the resonant ions entering the RF cavity have a shifted half-Maxwellian distribution of parallel energies, and if the ion beam is well collimated, then it is possible to
calculate the total resonant ion current to the RF plates as a function of the applied
RF frequency and applied RF power. From the variation of the resonant current with
the applied RF power, holding all other quantities fixed, it is shown how to extract
the temperature of the resonant ion species. From the ratio of the resonant current
to the non-resonant (bulk) current it is shown how to extract the resonant ion flux
fraction.
If the ions do not enter the cavity on the axis then the assumption of a collimated
beam is violated. Modifications to the single-particle model are presented showing the
influence of a broad beam on the resonance width and the current collected. A broad
beam significantly complicates the relationship between the resonant ion current and
the applied RF power, making it difficult to extract the resonant ion temperature.
Evidence is presented in Chapter 5 that an electron beam passing through the cavity
is well collimated, but we have no direct information yet about the collimation of an
ion beam.
111
Finally we justify the omission of ion-ion collisions from our model of ion orbits
inside the RF cavity. Neglect of ion-neutral collisions is justified from experimental
measurements of the neutral pressure inside the RF cavity, see Section 5.3.5.
3.5.1
Single Particle Orbits
First we solve the single-particle equation of motion in a magnetic field, with a perpendicular oscillating electric field. Then we estmate the influence of space charge on
ion orbit.
The equation of motion for a single particle of charge q and mass m in an electric
field E and a magnetic field B = Bez is
q
dr
d dr
= ωc × ez + E.
dt dt
dt
m
For the case when E = 0 the equation of motion describes rotation about the z axis
with an angular frequency equal to the cyclotron frequency, ωc = qB/m. Transform
to a rotating coordinate system,
 



x











 

y
=










cos(ω t)
 




sin(ω t)
x













.



 

− sin(ω t) cos(ω t)
y
We use the short-hand notation r = U(ω t)r, r = U † (ω t)r = U(−ω t)r , where
U represents the unitary rotation transformation. Note that UU † = U † U = 1,
d2 U/dt2 = −(ω )2U, and UdU † /dt = −ω ez ×. In the transformed coordinates the
equation of motion becomes
d dr
dr
= (ωc − 2ω )
× ez + (ω )2 − ω ωc r + U(ω t)E.
dt dt
dt
For special choice of the coordinate system rotation frequency ω the radial term or
the velocity term can be made to vanish. Note that for the case when E = 0 and
(ω )2 − ω ωc = 0 the equation of motion again describes rotation about the z axis, but
112
this time in the rotating coordinate system and with an effective frequency ωc − 2ω .
We consider the special case when the electric field E = Ej cos(ωj t)ex is applied
externally. In a coordinate system for which the rotation frequency satisfies ωc −2ω =
0, the equation of motion becomes
d2 x
ωc 2 qEj
+
x =
[cos(ω+ t) + cos(ω− t)] ,
2
dt
2
2m
2
qEj
ωc
d2 y +
y = −
[sin(ω+ t) + sin(ω− t)] ,
2
dt
2
2m
where we have used the double-angle formulae and ω± ≡ ωc /2 ± ωj . The rotated
coordinates x and y have the solutions
x = c0 cos(ωc t/2) + c1 sin(ωc t/2) + c+ cos(ω+ t) + c− cos(ω− t),
(3.5)
y = c4 cos(ωc t/2) + c5 sin(ωc t/2) − c+ sin(ω+ t) − c− sin(ω− t),
(3.6)
c± =
qEj
2m
ωc
2
2
−1
2
− ω±
=
qEj
[∓ωj (ωc ± ωj )]−1 .
2m
(3.7)
Transforming back to the lab coordinate system and neglecting terms proportional to
(ωc + ωj )−1 , it can be shown that the Larmor radius for ions entering on axis increases
according to
rL (t) =
Ej t
qEj 2 sin((ωc − ωj )t/2)
≈
,
2m
ωj (ωc − ωj )
2B
(3.8)
where (ωc − ωj )t 1 is the condition for resonant ions.
This is the same result as in Thomas’s thesis[69]. The more general form summing
over a spectrum of frequencies ωj in Equations (3.5)–(3.7) must be used if the source
of applied RF power has more than one significant Fourier component. However we
shall see in the next chapter that while the RF oscillator does produce harmonics in
addition to the desired (fundamental) frequency, the amplitudes of the harmonics are
low enough to be negligible.
Now consider the case when the total electric field has an additional radial compo113
nent due to a cylinder along the z axis with uniform space charge density qn0.2 The
electric field inside the cylinder due to the space charge only is E = (qn0/2@0 )r. Inserting this into the ion equation of motion and transforming to a rotating coordinate
system gives
ωp2 qEj
d dr
dr
2
= (ωc − 2ω )
× ez + (ω ) − ω ωc +
cos(ωj t)U(ω t)ex .
r +
dt dt
dt
2
m
where ωp2 = q 2 n0/(@0 m) represents the plasma frequency. If we choose ω = ωc /2
as before then the ion motion in the rotating coordinate system is described as in
Equations (3.5)–(3.7) except with ωc /2 → ωc /2, where
2ωp2
ωc
ωc
=
1− 2
2
2
ωc
1/2
.
Therefore the Brillouin flow that results from space charge inside the analyzer has the
effect of reducing the resonance frequency. In the lab frame the resonance frequency
is down-shifted. For ωp2 /ωc2 1,
ωres − ωc ≈ −
3.5.2
ωp2
.
2ωc
(3.9)
Collection Frequency Range
Next we calculate the range of frequencies for which a resonant ion can be collected.
Ions are assumed to enter the cavity a distance d = D/2 to the closest RF plate.
To be collected on the RF plates, resonant or near-resonant ions must increase their
Larmor radii by d before they exit, otherwise they will pass to the end collector. The
Larmor radius of a near-resonant ion, Equation (3.8), takes on a maximum value of
rL (t) ≤
2
1
E
,
B |ω − ωc |
Thanks to I. Hutchinson for suggesting this mechanism.
114
which sets the collection criterion for the range of frequencies:
|ω − ωc | ≤
E
.
Bd
If the ions enter on axis, then d = D/2 and the above formula sets the ideal, singleparticle resonance frequency half-width.
Recall the calculation of the frequency range for resonant ion collection assumed
a homogeneous magnetic field. If the magnetic field is not homogeneous then the
frequency range increases: the resonance is broadened. The effect of magnetic field
variation is estimated next.
Collection Frequency Range with Magnetic Field Variation
The frequency range calculated previously assumes a homogenous magnetic field, or
equivalently a beam of ions so narrow that the magnetic field does not vary significantly over the beam radius. In fact the toroidal magnetic field does change over the
omegatron entrance slit, and the ion beam is in the shape of a ribbon which occupies
a finite major radial extent, approximately 5 mm.
Heuristically one can think of the ion beam ribbon as consisting of multiple pencil
beams. Each pencil beam is at a different major radius so each beam has a different
magnetic field strength and the ions in each beam have a different cyclotron frequency
ωc . For two adjacent beams the magnetic field is different by ∆B, and so the cyclotron
frequencies of the two beams are different by ∆ωc . If the magnetic field is dominated
by the toroidal field then the difference in magnetic field ∆B of the two pencil beams
is related to the difference in major radius ∆Rmaj of the two beams. The differences
in magnetic field, cyclotron frequency, and major radius of the two pencil beams are
related by (absolute values)
∆ωc
∆Rmaj
∆B
=
=
.
ωc
B
Rmaj
115
With the cyclotron frequency for a particular resonant ion species ωc , the slit center
major radius Rmaj = 735 mm, and the major radial extent of the slit ∆Rmaj = 5 mm,
one can calculate the broadening ∆ωc due to variation in the magnetic field.
Collection Frequency Range with Brillouin Flow
It was shown earlier that the Brillouin flow of a magnetically confined non-neutral
plasma influences down-shifts the resonance frequency, see Equation (3.9). Here we
obtain a crude estimate of the contribution to resonance broadening.
If the current is quiescent, the Brillouin flow frequency shift by itself does not
broaden the frequency range over which ions are resonant. But in fact the nonresonant current does fluctuate, with relative amplitude near 100%. In this case the
frequency range becomes
∆ωres =
1
∆(ωp2 ).
2ωc
We do not directly measure the fluctuations in the charge density inside the RF cavity,
but we can measure the fluctuations in the end collector current, and we can measure
the shift in the resonance center. If we suppose that
∆(ωp2)
∆I
,
≈
2
ωp
I
then we can obtain an estimate of the Brillouing flow broadening directly:
∆ωres ≈ (ωc − ωres )
∆I
.
I
Recall each pencil beam has its own inherent broadening. To get the resonance
width of the ribbon beam the effects of inherant broadening, magnetic broadening,
and Brillouin flow broadening must be convolved. If we assume the three broadening mechanisms produce Gaussian response functions then the square of the total
resonance width is the sum of the squares of the individual mechanism resonance
widths.
116
3.5.3
Dwell Time and Collection Energy Range
We want to calculate the total resonant ion current collected on the RF plates, given a
flux of ions entering the cavity on the axis. The first step is to consider the collection
criteria for individual ions as they traverse the cavity. Resonant ions which do not
spend enough time in the RF cavity, either because they pass through the cavity
too quickly or because they are reflected right at the entrance, are not collected. In
general the resonant current collected depends on the electrostatic potential structure
inside the RF cavity.
It was shown in the derivation of Equation (3.8) that the Larmor radius of a
resonant ion in the RF cavity increases linearly in time. If we assume the resonant
ion enters the RF cavity midway between the plates, then the condition rL (τ ) = D/2
sets the spin-up time required for the ion to be collected on the plates, τ = BD/E.
The time an ion spends in the RF cavity, the dwell time, must exceed the spin-up
time for the ion to be collected. Note that the spin-up time depends only on the RF
plate spacing D, the ambient magnetic field magnitude B, and the amplitude of the
applied RF electric field E, but not on the ion mass or charge.
The electric field amplitude E can be found from the potential amplitude V
√
through E = V/(D/2) = P R/(D/2), where P is the total RMS power delivered
to the omegatron.
3
Using D = 5mm, d = D/2, P = 15W, and B = 5T at the
omegatron location gives E = 11 V/mm and τ = 2.3 µs.
The dwell time td depends on the time-history of the parallel velocity of the ion
3
We can relate the electric field amplitude to the RMS power delivered by the RF amplifier
as follows. In the cavity, we have Ẽ(t) = Ṽ (t)/(D/2), where Ẽ(t) = E sin ωt. The potential
Ṽ (t) = V sin ωt is applied across a resistor of value R = 50 ohms between the RF plate and the
virtual ground node midway between the RF plates. Thus the instantaneous current that flows in
˜
the resistor is I(t),
and the instantaneous power that is deposited in the resistor is P̃ (t) = Ṽ 2 (t)/R.
We measure the RMS power sent to each leg, from which we can calculate the potential amplitude
of the sinusoidally varying potential: Prms ≡ Ṽ 2 (t)/R √
= V 2 /(2R). The RMS power sent to each
leg is half of the total RMS power Ptot, so we have V = RPtot. Henceforth we drop the subscript
“tot” from the total RMS power.
117
while it is inside the RF cavity:
L
td =
0
dx
v(x)
or
td = 2
x∗
0
dx
,
v(x)
where the first form holds for ions which pass through the cavity, and the second form
holds for ions which are reflected (at x = x∗) inside the cavity.
Suppose that the parallel energy of the ion stays constant as it moves in the RF
cavity: µ = mv 2/2 + qφ(x) = const. Then the parallel velocity of an ion with parallel
energy µ is known as a function of position:
v(x) =
2
(µ − qφ(x))
m
1/2
.
In general to determine the dwell time requires a knowledge of the longitudinal potential structure, φ(x).
Once the dwell time is known for an ion with parallel energy µ, the range of
parallel energies for which resonant ions are collected µmin ≤ µ ≤ µmax is determined
by the criterion td ≥ τ . Then the resonant current collected on the RF plates is
determined by
µmax
IRF = qA
µmin
fRF (µ)
dµ
,
m
(3.10)
where fRF is the distribution of ions entering the RF cavity, m and q are the ion mass
and charge respectively, and A is the cross section area of the ion beam.
3.5.4
Dwell Time with Constant Potential
We can extract the necessary physical insight of resonant ion collection by considering
a constant electrostatic potential in the RF cavity. Calculations of the electrostatic
potential due to a ribbon of contant space charge density (albeit inconsistent with the
ion equation of motion) indicate that the potential profile along the axis of the RF
cavity is nearly flat, changing sharply only near grid G3 and the end collector. The
118
profile is flat inside the RF cavity because of the nearby RF plates. While a more
thorough calculation would solve both the Poisson equation and the ion equation of
motion consistently, the vertical boundary conditions imposed by the RF plates would
not change, which means the resulting electrostatic potential on the axis would still
be flat far from the entrance.
Therefore let us consider the special case when φ(x) = φ0 for 0 < x < L, where
x = 0 at the entrance to the RF cavity and L is the length of the cavity. Then we
can evaluate the dwell time:
td =




0



L
2
(µ − qφ0)
m
−1/2
µ ≤ qφ0,
µ ≥ qφ0.
We find the range of ion parallel energy for which the resonant ion is collected by the
criterion td ≥ τ :
µmin = max (qφ0, µs ) ,
µmax = qφ0 +
m L
2 τ
2
.
(3.11)
A deuterium ion in a cavity of length L = 40 mm with the dwell time calculated
earlier gives m(L/τ )2/2 = 3 eV.
Note that the fraction of the distribution function collected depends on the distance d, through the upper limit of the parallel energy µmax . RF plates closer together
means d = D/2 is smaller and a larger fraction of the distribution function is collected. Also note that the frequency range |ω − ωc | for which resonant ion collection
occurs depends on d: smaller d = D/2 gives a larger range of frequencies ω over which
resonant ions are collected.
Note that in the limit φ0 = φ(x = 0) = φG3 the grid potential, this reduces to
case with no space charge in the RF cavity. For φ0 > φG3 , the electrostatic potential
has the effect of shifting the range of collection to higher energies, with the shift
increasing with the charge of the ion.
119
In this simple model, ions which reflect on the electrostatic potential in the RF
cavity have zero dwell time (prompt reflection), and thus have no chance to be collected on the RF plates.
3.5.5
Dwell Time with Spatially Varying Potential
If instead of the abrupt change of electrostatic potential in the RF cavity we have
φ(x) =



φG + (φ0 − φG )(x/∆x), 0 ≤ x ≤ ∆x,


φ0 ,
∆x ≤ x < L,
then the dwell time for reflected ions becomes
2
td =
q
∆x
2m(µ − qφG ).
φ0 − φG
For ∆x = 0 we recover the prompt reflection of the constant potential model. Two
reflected ions with different masses m = m but the same charge and the same parallel
energy have dwell times related by td/td =
m/m : the lighter reflected ion has a
shorter dwell time. Therefore the collection efficiency is lower for the lighter reflected
ion than it is for the heavier reflected ion.
We use the results of this section only to show that a (more) realistic potential
profile gives finite dwell time. In practice, if ∆x L, or if space charge is negligible,
then the main contribution to the dwell time will come from the transit time through
the cavity, and the constant potential model holds. We use the constant potential
model in the next section to determine the impurity temperatures and fluxes.
3.5.6
Determining Absolute Impurity Fluxes, Densities, and
Temperatures using RF Power Scan
It is clear that reducing the applied RF power will reduce the resonant ion current
collected on the RF plates, since it increases the spin-up time τ . With the simple
120
model developed so far it is possible to predict the relationship between the RF power
and the resonant ion current. Given the assumptions of the model, it is possible to
extract the temperature of the resonant ion species from the current-power curve.
Say the resonant ions at the sheath edge have a half-Maxwellian distribution of
the form fs (µ) = Γs (m/kT ) exp((µs − µ)/kT ) when µ ≥ µs and fs = 0 elsewhere,
where Γs is the flux of resonant ions incident on the slit, and µs is the minimum
parallel energy of the ion distribution at the sheath edge. By the time the ions reach
the RF cavity the distribution has been attenuated by three grids and the slit, and
all ions with energy µ ≤ qφ2 have been reflected, where φ2 is the maximum potential
due to space charge before the RF cavity. The distribution of resonant ions entering
the rf cavity is
fRF (µ) =



ξ 3 ξs Γs (m/kT ) exp


0
µs − µ
kT
µ ≥ qφ2,
µ ≤ qφ2,
where ξs is the slit transmission and ξ is the grid transmission. Evaluating Equation
(3.10) with the energy limits given by the flat potential model, Equation (3.11), gives
the resonant ion current collected:
IRF
µs − qφ0
= qAξ ξs Γs exp
kT
3
−m(L/τ )2
1 − exp
2kT
.
(3.12)
The simple dependence of the collected resonant current in Equation (3.12) on the
applied RF power suggests a technique to measure the temperature of the impurity
species.
For fixed grid biases and for a steady distribution function the electrostatic potential in the RF cavity does not change. The upper bound of the parallel kinetic energy
of resonant ion collected on the plates depends on the applied RF power through the
dwell time, τ −2 ∼ P . If the ion parallel distribution function is Maxwellian then the
121
formula for the collected resonant current Equation (3.12) holds, in which case
IRF = c0 1 − e−P/c1 .
We can measure the resonant ion current as we vary the RF power P and fit the data
to a function of the form given above. If the ion mass is known, the temperature kT
is extracted from the fitting parameter c1 by
2L2 R
kT
= c1 2 4 ,
m
B D
where L is the length of the RF cavity, R = 50 ohm is the apparent impedance for
each leg of the omegatron, B is the magnetic field magnitude at the omegatron, and
D is the spacing between the RF plates.
The fitting parameter c0 gives the current that would have been collected if infinite
RF power had been applied. Equation (3.12) predicts that even with infinite applied
RF power only the fraction e(µs −qφ0 )/kT of the flux entering the RF cavity is collected,
because according to the model ions with parallel energy less than qφ0 have identically
zero dwell time. A real potential structure would change gradually just inside the RF
cavity, so all ions would have a finite dwell time, and applying infinite RF power (in
principle) would result in all resonant ions being collected.
We can estimate the uncertainty introduced by the flat potential model in the
calculation of the impurity flux. In one extreme, none of the reflected ions are spun
up, so we take φ0 = φ3 , the maximum potential in the RF cavity calculated in
Section 3.4. In the other extreme, reflected resonant ions in the RF cavity have the
same chance to be spun up as the transmitted resonant ions, so we take φ0 = φ2, the
maximum potential before the RF cavity. The actual resonant current collected is
likely to be somewhere in between:
µs − qφ3
exp
kT
c0
µs − qφ2
≤
≤ exp
.
3
qAξ ξs Γs
kT
122
In either case the fraction of the non-resonant ions that arrive at the end collector
must have minimum parallel energy greater than qφ3.
Often we are interested in the ratio of the resonant impurity flux to the bulk
nonresonant flux at the sheath edge. Assuming we have scanned the power to obtain
the asymptotic value of the resonant current IRF = c0,
(IRF /Z) exp[(µs − qbulkφ3)/kTi ]
Γs
=
,
Γbulk
(IEND /Zbulk ) exp[(µs − qφ0)/kT ]
where kTi is the bulk ion temperature, q = Zbulke is the bulk ion charge, φ3 is the
maximum electrostatic potential before the end collector, we assume φ2 ≤ φ0 ≤ φ3,
and the bulk ions and the impurity ions are assumed to have the same attenuation
coefficients through the slit and grids. In the case when the impurities have approximately the same temperature as the bulk ions kT ≈ kTi, and assuming resonant ions
that reflected in the RF cavity have a finite dwell time and so can be collected, then
IRF Zbulk
1
Γs
=
,
Γbulk
IEND Z (g1 g2 g3 )Z−1
(3.13)
where g1 , g2 , g3 are the transmission fractions for the bulk species obtained in Section
3.4.
The density of the resonant ions at the sheath edge is found by relating the
resonant ion flux at the sheath edge to the fluid velocity at the sheath edge. If the
presheath is highly collisional then the ion impurities acquire the bulk ion flow velocity
at the sheath edge, the fluid sound speed:
vs =
kTe + kTi
.
mi
In this case relating the impurity density to the impurity flux at the sheath edge gives
ns = Γs /vs . Since both impurities and bulk ions have the same flow velocity at the
123
sheath edge:
ns
Γs
=
.
nbulk
Γbulk
3.5.7
Determining Impurity Temperature using RFEA Bias
The retarding field energy analyzer modifies the parallel distribution function of all ion
species in the analyzer, including impurities; the omegatron ion mass spectrometer
selects the ions with a specific M/Z from the stream of ions passing through the
RF cavity. Combining the two diagnostics gives an obvious way to measure the
temperature of individual ion impurities, provided there exists sufficient impurity ion
signal compared to the noise floor.
A potential drawback of this approach, in contrast with the technique of determining the ion impurity temperature by scanning the applied RF power, is that the
non-resonant current also decreases as the reflector bias is increased. This can change
the space charge electrostatic potential in the RF cavity, which alters the efficiency
with which resonant ions are collected.
3.5.8
Broad Beam Modifications
A beam of ions entering the RF cavity with a broadened density profile results in
resonant current collection over a wider range of frequencies would be observed for a
narrow beam of ions. The maximum amplitude of ion Larmor radii is smaller for RF
frequencies farther from the cyclotron frequency. But if the beam density profile is
broad then there are ions close to the plates that will collected further off resonance.
It is shown that an ion beam with a Gaussian profile with 1/e width less than D/16
produces negligible frequency broadening, and the conclusions of the previous section
hold.
For the slice of the ion beam density profile close to the plates, a larger fraction of
the parallel distribution function is collected than for the slice of the profile midway
between the plates. It is shown that this significantly complicates the simple relation
124
previously established between the applied RF power, the collected resonant current,
and the resonant ion temperature. Thus it is desirable to operate the omegatron in
a regime where the assumption of a collimated ion beam is valid.
Consider a flux of ions entering the cavity distributed over the distance between
the RF plates according to a symmetric function p(z), where z represents the distance
from the axis between the plates and p(z) is normalized according to 2
D/2
0
dz p(z) =
1. Ions entering the RF cavity at distance z from the axis must be spun up to Larmor
radius d = D/2 − z before they can be collected.
The total current collected on the RF plates is sum of the currents collected from
each distance z, weighted by p(z). Thus
D/2
IRF = 2qA
µmax (E,z,ω)
dz p(z)
zmin
µmin
fRF
dµ
,
m
where now the dependence of the upper bound of the parallel energy µmax on the
electric field E, spin-up distance z, and applied RF frequency ω is shown explicitly.
Only those distances d which satisfy the resonance condition will contribute any
current. Invert the resonance width criterion to see that for a fixed electric field E
and a fixed frequency ω, current will be collected only from those regions in the cavity
that satisfy
E
D
D
−
≤z≤ .
2
B|ω − ωc |
2
When E/(B|ω − ωc |) ≥ D/2 then the entire distance between the RF plates meets
the resonance condition, and the lower bound of z is 0. Thus we have


 D
zmin =
2


0
−
E
B|ω−ωc |
when E/(B|ω − ωc |) ≤ D/2,
when E/(B|ω − ωc |) ≥ D/2.
For the case when the distribution of parallel velocities is described by a Maxwellian
125
Figure 3.13: Theoretical normalized current collected on RF plates as a function of
frequency for a typical cyclotron frequency for deuterium at the omegatron location,
ωc /(2π) ≈ 36 MHz, b = 2.6 and a = 1, 1/2, 1/8.
it is possible to perform the velocity integral analytically:
D/2
IRF = 2qA
zmin
µs − qφ0
dz p(z) ΓRF exp
kT
−m(L/τ (E, z))2
1 − exp
2kT
,
For a gaussian beam density profile, p(z) = p0 exp(−z 2/σ 2 ), where p0 is the appropriate normalization constant. Neglecting all constants, and defining χ ≡ d/(D/2)
we have
√
IRF ∼
πa
erf(1/a)
2
−1 where
σ
a≡
,
D/2
0
b≡
χ0
−(1 − χ)2
dχ exp
a2
m 2EL
,
2kT BD
−b2
1 − exp
χ2
,
(3.14)
2E
χ0 = min
,1 .
BD|ω − ωc |
Note that in the limit that a → 0 the on-resonant current for a Maxwellian distribution reduces to IRF ∼ 1 − exp(−µmax /kT ), as before.
For P = 15 W we have E = 11 V/mm. Inserting other appropriate values for
the cavity length, the plate spacing, and kT = 3 eV for deuterium gives b ≈ 2.6.
126
Figure 3.13 shows the normalized current as a function of frequency for b = 2.6
and a = 1, 1/2, 1/8, where the wider beam density profiles collect more off-resonant
current.
The relationship between applied RF power and the on-resonant current collected
from a broad beam, Equation (3.14), is much more complicated than that for onresonant current collected from a collimated beam, Equation (3.12). In this case it
is not possible to extract analytically the temperature from the current-power curve.
As noted previously, if the ion beam has a Gaussian density profile (say) which is
sufficiently narrow (say an eighth of the plate spacing) then we recover the collimated
beam approximation.
3.5.9
Ion-ion Collisions
We neglect the effect of ion-ion collisions on the grounds that typical ions suffer much
less than one collsion with another ion as they traverse the RF cavity. (Neglect
of ion-neutral collisions is justified experimentally in the Section 5.3.5 by using the
omegatron as a residual gas analyzer to measure neutral pressure in the RF cavity.) A
key assumption in the single-particle model of the omegatron ion mass spectrometer
is that collisions can be neglected, so establishing the validity of the collisionless
approximation is important.
Consider an ion with parallel energy e∆φ and mass mi . Then it has velocity
vi =
e∆φ/mi and it traverses the cavity of length L in time L/vi . Ions with charge
Ze and mass µmp with density ni and in thermal equilibrium with temperature Ti
collide with each other at frequency[23]
νi = 4.80 × 10−14 ni Z 4 λα,β
1
3/2
µ1/2Ti
,
where ni is in units of m−3 , Ti is units of eV, and νi is in units of s−1 , and λα,β ≈ 12
is the Coulomb logarithm.
127
Taking e∆φ ≈ Ti, we can find the temperature for which νi = vi/L such that
particles suffer one collision on average as they traverse the RF cavity. Note that for
a magnetically confined non-neutral plasma beam in equilbrium we require the ion
density to be less than or equal to the Brillouin density, ni ≤ nB ≡ @0 B 2/mi , and so
νi (ni ) ≤ νi (nB ). Setting ni = nB , B = 5 T, λα,β = 12, L = 5 cm, and using deterium
mass and charge, it can be shown that for temperatures above Ti ≥ 0.5 eV ions suffer
less than one collison. Since ion tempertures measured in the edge are typically much
higher than this we are justified to neglect ion-ion collisions.
128
Chapter 4
Retarding Field Energy Analysis
This chapter describes the interpretation of data from the omegatron when operated
as a retarding field energy analyzer. The first section presents typical data and
calls attention to specific features. The next section discusses the features, along
with experiments performed to understand the features. The final sections describe
applications of the omegatron retarding field energy analyzer and implications of the
results.
4.1
Observations
In this section observations are presented from typical operation of the omegatron
retarding field energy analyzer, particularly ion and electron current-voltage (IV)
characteristics. Important features of the IV characteristics are noted, but any attempt to explain the data is postponed until the next section.
4.1.1
Current-Voltage Characteristic Features
Figure 4.1 shows a typical current-voltage characteristic for the ions. Biases of components were set as in Figure 3.8, with grid G2 as reflector. Several features of the
data can be noted:
129
Figure 4.1: Current-voltage characteristics from the omegatron in retarding field
energy analyzer mode. Dashed line is raw current to END collector, solid line is
current to END collector normalized by sum of currents to grids G1, G2, G3, and to
END collector and scaled to agree with the raw saturation current.
130
• The raw current fluctuates with large amplitude.
• The characteristic exhibits slight hysteresis.
• At a very negative reflector bias there exists a saturation current level.
• At a certain reflector bias the current starts to decrease (“voltage knee”).
• The reflector bias at the knee is below 0 V.
• For reflector biases higher than the knee voltage, the ion current decreases
exponentially.
• The exponential decrease in ion current has a break in slope, which occurs at a
current level approximately 2–8% the saturation current level.
In addition, it should be noted that the floating potential φf , measured separately
with Langmuir probes, is near 0 volts.
4.1.2
Effect of ICRF
Figure 4.2 shows the influence of ion cyclotron radio frequency auxiliary heating on
the IV characteristics of ions and electrons. Several features of the data can be noted:
• The break in ion IV characteristic is observed in ohmic discharges only; the
break disappears during ICRF discharges.
• Knee of ion IV characteristic becomes much more rounded during ICRF discharges.
• During ICRF discharges the ion IV characteristic has a slope similar to the
shallower portion of the slope during ohmic discharges.
• The electron IV characteristic has single slope in ohmic and ICRF discharges.
• The slope of the electron IV characteristic becomes more shallow during ICRF
discharges.
131
Figure 4.2: IV characteristics before and during 2.5 MW of ion cyclotron resonance
auxiliary heating.
132
4.1.3
Flux Tube Boundaries
Figure 4.3 shows the typical mapping of magnetic field lines intersecting the face of
the omegatron probe. The “plunge” depth is the distance the omegatron is inserted
into the plasma from its retracted rest position. For the indicated equilibrium the
omegatron was at a plunge distance of 37 mm, corresponding to a the poloidal flux
surface ρ = 47 mm from the separatrix at the midplane. It can be noted that:
• The flux tubes that connect to the omegatron face almost always end at E-port
limiter tiles.
• The connection length depends weakly on insertion depth for typical plasma
equilibria.
4.1.4
Effect of Magnetic Field Direction
Figure 4.4 shows the influence on the ion and electron IV characteristics of changing
the direction of the toroidal field (and the plasma current). From the data it can be
noted that:
• Changing the direction of the magnetic field makes no significant difference in
the ion or electron IV characterstics.
It is also true that the Langmuir probe data obtained at the slit location (LP2)
does not change with changing the magnetic field. While it is assumed that the
above results are general, it should be noted that only limited abnormal-field data
were obtained. In addition no data were obtained at substantially different field
magnitudes.
4.2
Discussion of Characteristic Features
The important features of the retarding field energy analyzer IV characteristics are
discussed in this section. First the features are compared against simple theory; if all
133
Connection Length: 1.83 meters
Toroidal Turns: -0.35 F_Start: 2.75 G_Start:
1.60
Ended field line on SIDE of E-Side Tiles of D-E Antenna
Shot#990820008 0.6000 sec
0.6
F
1.0
E
G
0.4
0.5
D
0.2
H
0.0
0.0
-0.2
C
J
-0.5
-0.4
-0.6
0.4
0.6
0.8
1.0
B
1.2 -1.0
-1.0
K
A
0.0
-0.5
0.5
1.0
CCW Connection Length (m)
CW Connection Length (m)
Omegatron Connection Lengths
20
Non-Divertor Surface
15
10
5
0
5
20
4
40
60
80
100
120
80
100
120
Plunge (mm)
E-Side Tiles of D-E Antenna
G-H Full Limiter
Outer Divertor
3
2
1
0
20
40
60
Plunge (mm)
Figure 4.3: Top: magnetic field lines tracing from omegatron probe face to molybdenum tiles on E-side tiles D-E limiter. Bottom: magnetic field line connection lengths
from omegatron probe for a typical plasma equilibrium and for different insertion
depths. “Plunge” is insertion depth from rest position. For equilibrium shown, insertion of 37 mm corresponds to poloidal flux surface ρ = 47 mm. Figures courtesy
B. LaBombard.
134
Figure 4.4: IV characteristics with normal field (B × ∇B down) and abnormal field
(B × ∇B up). Current is always parallel to toroidal field to preserve helicity.
135
the features agreed with the textbook theory then we could proceed immediately to
extract temperature information from the IV characteristics. But some non-standard
features are not described by simple theory, notably the voltage of the knee and the
break in the exponential portion. A series of experiments is described to determine the
influence of different omegatron probe components on the ion distribution function.
It is found that the transmission of ions through the slit and grids is nearly optical,
and that the voltage of the knee can be explained with secondary electron emission
from the slit and space charge inside the analyzer. Since the non-standard features
can mostly be explained with collisionless mechanisms, which do not significantly
rearrange the distribution function, we proceed to interpret the inverse slope of the
exponential portions of the IV characteristic as temperatures, and the break in slope
as evidence of a two-temperature ion distribution.
4.2.1
Comparison of IV Characteristic with Simple Theory
In this subsection the features of the IV characteristic are compared with the simple
theory of retarding field energy analyzer operation. It is found that the voltage of the
knee and the break in slope cannot be explained by the simple theory. If we could
invoke the textbook theory directly, we might interpret the break in slope as evidence
that two ion populations exist with different temperatures. But unless we understand
the other non-standard feature, the voltage of the knee, we cannot be sure that the
distribution function inside the analyzer is representative of (or that we can map it
to) the distribution function outside the analzer.
Fluctuations
It can be seen from Figure 4.1 that the raw current arriving at the END collector
(dashed line) fluctuates with modulation amplitude almost one hundred percent. The
fluctuations also appear on the currents received by the other grids (not shown).
Since the electronics for each grid has identical bandwidth it is possible to remove
136
the fluctuations from the characteristic. The solid line in Figure 4.1 shows the END
current fraction (the END current divided at each instant in time by the sum of
the currents arriving on G1, G2, G3 and the END) as a function of G2 bias. The
dimensionless current fraction is scaled by the average current arriving at the END
during the saturation portion of the characteristic (when the G2 bias is between −70
volts and −40 volts).
Hysteresis
The IV characteristic in Figure 4.1 exhibits a slight hysteresis. While a physical
mechanism like space charge could explain the hysteresis, in this case the hysteresis
is more likely due to an electronics artifact. The TTE 5 kHz passive filters used in
the grid electronics have a variable phase delay as a function of frequency such that
the filtered signal is delayed in time from the original signal by about a millisecond.
Since time appears as a parameter when plotting the current-voltage characterstic, a
time delay in the current introduces hysteresis.
The hysteresis could be reduced experimentally by sweeping the selector bias more
slowly, but this reduces the number of characteristics that can be collected during
a plasma discharge. The hysteresis could be eliminated during analysis by carefully
shifting the timebase of the current signal, but in practice it is easier and more
objective to bin the signal with respect to the signal of the independent variable, so
long as the quantity being binned changes linearly in time and has equal upward-going
and downward-going ranges.
Saturation Current, Presence of Knee
The existence of a saturation current in the IV characteristic is expected from the
simple theory. At low current levels we expect that as the reflector bias drops below
the sheath potential full ion transmission obtains. This is similar to the ion saturation
portion of a Langmuir probe IV characteristic.
137
The presence of a knee in the IV characteristic is also expected from the simple
theory. When the reflector bias approaches the sheath potential, ions with low parallel
energy are reflected, decreasing the current downstream.
Voltage of Knee
The simple theory predicts the voltage of the knee to occur near the sheath potential,
which for the case of negligible secondary electron emission is approximately 3kTe
above the floating potential. It was noted earlier that the floating potential, measured
with Langmuir probes on the face of the heat shield, is near machine ground, 0 V.
But the voltage of the knee is clearly below the floating potential. Also the simple
theory predicts the current to decrease abruptly as the reflector bias exceeds the
sheath potential: the knee should be sharp. But the knee in Figure 4.1 is distinctly
rounded. Therefore the voltage of the knee is not explained by the simple theory.
Exponential Decrease
The ion current decreases exponentially as the reflector bias is increases above the
voltage of the knee. The simple theory predicts that above the sheath potential an
increase of the reflector bias of dV results in an decrease of the downstream current
by dI proportional to the distribution function. If the distribution function of ions
incident on the analyzer is a shifted half-Maxwellian, the simple theory predicts an
exponential decrease in current. Therefore if we assume that the distribution function
is a shifted half-Maxwellian then the exponential decrease is consistent with the simple
theory.
Break in Slope
If the incident distribution function is a shifted half-Maxwellian then the simple theory
predicts the exponential decrease in ion current will continue indefinitely and not have
a break in slope. We can attribute the break in slope to two populations of ions inside
138
the analyzer with different temperatures: the slope of the IV characteristic changes
at the parallel energy when the population of hot minority species dominates the cold
majority species.
If the low voltage of the knee is the result of some collisional process, say with
the slit or the grids, then the distribution function inside the analyzer cannot be
mapped to the distribution function outside the analyzer. Before we can assert that
the ions outside the analzyer are comprised of two populations, a cold bulk and a
hot minority, we need to be sure the low voltage of the knee is not the result of a
collisional mechanism. However we shall conclude that since transmission through the
slit and grids appears optical at high ion energy, and since collisions would tend to
isotropize the ion distribution, the break in slope is in fact due to two ion populations
with different temperatures.
ICRF Effects
The slope of the IV characteristic increases during ICRF. We might be tempted to
attribute the change to more energetic ions arriving at the omegatron. As with
the break in slope, once we understand the interaction of the omegatron with the ion
distribution function, we are able to conclude that the bulk ion temperature increases
during ICRF heating.
Magnetic Field Direction
The theory of the short flux tube predicts no difference in geometry for abormal field
direction, and fluid presheath theory by Hutchinson [24, 26] predicts only a modest
change in presheath potential with plasma flow. The direction of the plasma current
on Alcator C-Mod is always parallel to the direction of the toroidal magnetic field to
preserve the field helicity, since some plasma-facing tiles are sloped. Therefore when
the field is reversed the omegatron flux tube still terminates on E-port limiter tiles,
the connection length stays the same, and the relationship between the connection
139
length and the perturbation length is not modified.
Pitts [57] and Wan [78] report changes in toroidal plasma flow with reversal of
toroidal field of limited machines DITE and Alcator C, respectively. Mach probes
in the lower divertor on Alcator C-Mod indicate similar reversal of flow[27] with
abnormal field direction, although plume emissions from the inner wall show that the
flow reversal does not extend all the around the scrape-off layer. If any asymmetry
effects are present near the omegatron they are not yet resolvable.
We now describe a series of experiments intended to determine the influence of grid
transmission, slit transmission, slit bias, space charge, secondary electron emission on
the distribution function inside the analyzer.
4.2.2
Grid Transmission, Current Accounting
We looked for the effects of the grids on the ion distribution function. The distribution
function would be truncated if the grids severely restricted the class of ion energies
that could pass, or reshaped if passing ions suffered Coulomb collisions with the grid
wires. We might expect that if passing through one grid truncated the distribution
function then passing through multiple grids should multiply the effect.
The experiment was set up as follows: the omegatron was run as a retarding
field energy analyzer with different components acting as reflector. The current to
all components was monitored. The grid currents were compared with the currents
predicted from optical transmission. Figure 4.5 shows the results of the experiment.
In Chapter 3 it was predicted that the transmission through the grids should
optical for ions with most energies. Here the predictions are tested and are found to
agree with experiment within ten percent. When grid biases are set to ensure that
ions pass through each grid only once, calculations using the current fraction to each
grid give the single-pass transmission for each grid: ξ1 = 0.67, ξ2 = 0.62, ξ3 = 0.66.
By comparison, calculations from the Chapter 3 predicted optical transmission for a
half-Maxwellian distribution through each grid to be ξopt = 0.71.
140
reflector
G2
G3
RF
none
SLIT
0.445
0.4 ± 0.4
0.173
0.2 ± 0.5
0.075
−0.1 ± 0.5
0
G1
0.555
0.583 ± 0.008
0.419
0.449 ± 0.004
0.370
0.391 ± 0.004
0.333
G2
0
−0.016 ± 0.004
0.408
0.398 ± 0.004
0.320
0.322 ± 0.004
0.252
G3
0
−0.001 ± 0.001
0
−0.011 ± 0.001
0.235
0.226 ± 0.006
0.142
END
0
0
0
0.274
Table 4.1: Fraction of incoming current through slit that arrives at each component. Top number is calculated using attenuation factors, bottom number is from
measurements.
With the attenuation factor for each grid it is possible to predict the current
fraction that appears on each component as of G2, G3 or RF are used to reflect the
ions. Good agreement of these estimates with measured current fractions is shown in
Table 4.1.
Several conclusions can be drawn from the data:
• The observations agree with predictions of the model of ion transmission from
Section 3.4.
• Any of grid G2, grid G3, or the RF plates can be used as reflector component.
This means that the biases can be set so that space charge is not reflecting the
ions.
• Ion currents measured on components are consistent with “optical” transmission
through the grids. There is no excessive attenuation or truncation of distribution function even after passing through multiple grids.
An additional experiment was peformed in ion RFEA mode with the slit bias was
held very negative and grid G1 used to reflect the ions. The IV characteristics obtained were compared with IV characteristics obtained during a similar discharge but
using grid G3 as ion reflector. The IV characteristics gave the same knee potentials
141
Figure 4.5: Fractions of total measured current (G1+G2+G3+END) to G1, G2, G3,
and END as a function of voltage bias on G2, G3, and RF. Current fraction to RF is
always less than 10−3 .
142
and similar temperatures. We interpret these observations as evidence against ion
pitch-angle scattering on grid G1.
The grid transmission experiment suggests that there is no modification of distribution function (aside from attenuation due to current collected on the grids). The
influence of the grids does not explain shifted knee of ion characteristic.
4.2.3
Slit Transmission
An experiment was performed to determine the transmission of ions through the slit.
The motivation was to see if the slit significantly truncated the distribution function,
even if the grids did not, through selective transmission.
The experiment was set up as follows: the slit was allowed to float electrically so
that it received equal fluxes of ions and electrons. The total ion current inside the
analyzer was compared with the total electron current. The ratio of the currents gave
the relative transmission of ions to electrons. The experiment was performed off-line
in an electron cyclotron discharge cleaning (ECDC) plasma.
It was observed that the ion transmission is 25% the electron transmission when
the slit floats.
From the theory of the previous chapter, transmission through the slit of a shifted
half-Maxwellian distribution of ions is expected to be optical for a modest shift energy, e∆φ ≈ 2kTe and drop to 20%–30% of optical for zero shift and finite slit edge
thickness, see Figure 3.6. If we assume that electron transmission through the slit
is optical (marginally satisfied for an ECDC plasma, in which the electron Larmor
radius is of order the slit width), then the ion transmission agrees with the predicted
transmission for an unshifted half-Maxwellian distribution.
4.2.4
Slit Bias Scan
An experiment was performed to determine if the ions entering the slit were suffering Coulomb scattering, transferring parallel energy to perpendicular energy and
143
distorting the parallel distribution function. If Coulomb scattering of ions off the slit
was producing the shift in the knee to lower voltage, then increasing the ion velocity
through the slit should have reduced the interaction time, thus reducing the effect of
Coulomb scattering, thus increasing the knee voltage.
The experiment was set up as follows: the omegatron was configured as a retarding
field energy analyzer with the biases of all components except the slit set as in Figure
3.8. For each sweep of the reflector component (G2) a different bias was applied to
the slit: −10, −4, 0.8, or 20 volts. We looked for the knee in the IV characteristic
to increase for slit biases below the floating potential. Figure 4.6 shows the results of
the experiment.
Several observations can be made of the ion IV characteristics:
• Decreasing the slit bias below 0 V has little influence on the ion IV characteristic.
The knee in the IV characteristic remains near 0 V.
• Increasing the slit bias above 0 V shifts the ion IV characteristic to more positive
voltages. The knee in the ion IV characteristic follows the slit voltage.
• The ion saturation current changes by approximately a factor of two over the
range of slit voltages applied (−10 V to +20 V).
• The slit bias has no influence on the break in slope (ohmic discharges only)
• The slit bias has no influence on roundedness of knee.
In a separate experiment the slit bias was set to −70 V. The ion IV characteristic
was indistinguishable from the ion IV characteristic shown in Figure 4.6 with slit bias
of −10 V.
Observations can also be made of the electron IV characteristics:
• The saturation current level increases exponentially with slit bias.
• The knee in the electron IV characteristic always occurs near the slit bias,
regardless of whether the bias is above or below 0 V.
144
Figure 4.6: Current-voltage characteristics for ions and electrons for omegatron in
RFEA mode with different SLIT biases. Ion characterstics are largely unaffected
below 0 V, but shift above 0 V. Vertical lines correspond to SLIT biases.
145
From the results of the experiment we can draw several conclusions:
• The knee in ion IV characteristic is still at lower voltage than we expect. Since
it does not depend on the energy ions have as they pass through the slit the
effect is not due to Coulomb scattering at the slit.
• If there was a sheath potential drop of ≈ 3Te in front of the floating slit, then
the distribution of ions would be accelerated and the transmission should be
nearly optical. In this case decreasing the slit bias below the floating potential
should have no effect on the ion transmission through the slit. In fact the ion
transmission through the slit improves by a factor of two as the slit bias is
decreased from the floating potential. This suggests that the sheath potential
drop is much less that 3Te when the slit floats.
• The trends observed are consistent with short flux tube theory:
– The floating potential (obtained with Langmuir probes) is near ground
because the E-port limiter tiles are at machine ground (by definition).
– The voltage of the knee follows the higher bias of the two biases: E-port
limiter tiles or the slit; the knee potential is related to sheath potential,
but apparently not equal to it.
– Below the floating potential, the slit bias has little effect on the shape of
the IV characteristic: the adjacent plasma potential is not set by the slit.
Despite a change of slit bias above and below the floating potential, the voltage
of the knee is still below the floating potential.
4.2.5
Space Charge
An experiment was performed to determine the relationship between the voltage of
the knee and the current in the analyzer. If significant space charge was present in
146
Figure 4.7: Current-voltage characteristics from the omegatron in retarding field
energy analyzer mode taken at different depths in the scrape-off layer plasma. Current
is obtained by normalizing END collector current by sum of currents to grids G1, G2,
G3, and to END collector and scaling to agree with the average END collector current
at reflector bias below −40 V.
the analyzer then when the reflector bias was near the sheath potential the maximum
potential between the grids would be set by the plasma. Attenuation of the ion
current would start at a reflector bias below the sheath potential, as we do observe.
If the current in the analyzer was decreased, then the space charge would decrease,
and the voltage of the knee would become more positive.
The experiment was set up as follows: the omegatron was configured as a retarding
field energy analyzer with all components biases set as in Figure 3.8. Over a series of
identical discharges the vertical position of the omegatron was changed to different
depths in the scrape off layer plasma. Shots with the omegatron deeply inserted
resulted in high currents; shallow insertion resulted in lower currents.
Figure 4.7 shows the results of the experiment. We can make several observations:
147
• The ion saturation current decreases as plasma density decreases (that is, decreases further from the separatrix).
• The knee becomes sharper as current decreases.
• The voltage of the knee approaches 0 V as current decreases.
• The break in slope persists (in ohmic discharges).
It should also be noted that the floating potential, obtained separately from Langmuir probes on the face of the omegatron heat shield, was persistently near machine
ground during the density scan.
From these observations we can make several conclusions:
• The shift of the knee behaves as we would expect if it was due to space charge.
• The ion saturation currents observed are near the simple estimate of the space
charge current limit calculated in the previous chapter, Imax,φ ≈ 20 µA, which
results in e∆φ ≈ kTi . Below this limit the influence of space charge should
decrease.
• As the current decreases (and presumably the space charge goes to zero) we
still have φs − φf ≈ 0. So space charge doesn’t explain why we do not see the
expected sheath drop φs − φf ≈ 3kTe .
4.2.6
Secondary Electron Emission
We performed an experiment to determine if secondary electron emission was present
in the omegatron. As Hobbs and Wesson [22] point out, effective1 secondary electron
emission from electron impact reduces the sheath potential required to obtain ambipolar flow to a surface. For coefficient of secondary electron emission high enough,
the slit would effectively float within ≈ kTe of the plasma potential.
1
“Effective” secondary electron emission includes reflection of primary electrons, emission from
ion impact, and thermionic emission as well as secondary emission from primary electrons.
148
PLASMA
SLIT
G1
G2
G3
RF
END
S.E.E. SUPPRESSED
+70V
+40V
φsheath
0V
S.E.E. RELEASED
+70V
+40V
φsheath
0V
Figure 4.8: Bias arrangement for secondary electron emission measurments.
To measure the secondary electron emission from the slit directly would have required a separate probe outside the omegatron in front of the slit, which was not
available. Instead a simpler (but less definitive) test was performed to measure the
effective coefficient of secondary electron emission inside the omegatron, using a technique pioneered by Pitts[54, 56] and similar to that of and Böhm and Perrin [9].
The experiment was set up as follows: the omegatron was configured as in Figure
4.8 to operate as a retarding field energy analyzer, with components biased for electron
IV characteristics. For alternating sweeps of the reflector bias the end collector bias
was set above and below (≈ 30 V) the bias of the neighboring component (RF plates)
149
to alternately suppress and admit the emission of secondary electrons. Electron IV
characteristics from the end collector with and without secondary electron emission
were thereby obtained. The ratio of the end collector currents obtained with admitted
and suppressed secondary electrons should have been unity if there was no secondary
electron emission, and it should have been less than unity if there was secondary
electron emission: the difference from unity gave the effective coefficient of secondary
electron emission, γeff .
Figure 4.9 shows the results of the experiment, from which we observe:
• At low reflector bias the coefficient of effective secondary electron emission approaches unity, γeff ≈ 1.
• At positive reflector bias (as the mean electron energy increases) the coefficient
decreases.
From these observations we can conclude:
• There is significant emission of secondary electrons inside the omegatron from
electron impact on the (stainless steel) end collector.
We observe from Figure 4.9 that at low electron energies the coefficient of secondary electron emission approaches γeff = 1. Similar results at low electron energy
were obtained by Pitts, who found 1 ≤ γeff ≤ 1.8 for a molybdenum end collector and
γeff ≈ 1 for a carbon end collector.
E.W. Thomas [67] provides analytic fits to secondary electron emission from different surface materials as a function of electron beam energy: γe (E) ≈ γ(Emax ) ∗
√
(2.72)2 y exp(−2 y), y ≡ E/Emax , where Emax is the electron energy where the coefficient takes its maximum value. Matthews [40] has averaged the coefficient over
a Maxwellian distribution of energies graphite; Ordonez [51, 50] has done the same
for all of the materials tabulated by Thomas, and calculated the temperature of the
electron distribution required to give a coefficient of secondary electron emission that
150
f e (v)
fe (v)
bias near +60 V
bias near 0 V
v
v
Figure 4.9: Effective coefficient of secondary electron emission versus acceleration
voltage.
151
results in space charge saturation, γ = 0.9. For a surface coated with boron the
critical temperature is Te,c = 15 eV; for iron and tungsten the critical temperatures
are Te,c = 35 eV and Te,c = 53 eV, respectively. The end collector is made from stainless steel, but the coefficient for boron would be appropriate if the end collector was
coated with diboronane, say from the boronization procedure.
Like Pitts, we observe a high level of secondary electron emission or an effect
like it, but we do not have a satisfactory quantitative explanation. The predicted
coefficient of secondary electron emission from electron impact on a boron surface is
still lower than we observe by a factor of two. Reflection of low energy electrons and
secondary electron emission from ion impact are too low to make up the difference.
If we hypothesize that there is similar secondary electron emission from the slit
as there is from the end collector, then the shift of the voltage of the knee in the ion
IV characterstic is explained: Secondary electron emission depresses the difference
between the sheath potential and the floating potential, and the voltage of the knee
in the characteristic appears below the floating potential because of space charge.
Note that depression of the sheath potential, say by secondary electron emission, is
consistent with the improvement of transmission of ions through the slit by dropping
the slit bias below the floating potential. If there is secondary electron emission from
the slit, then when the slit floats there is little sheath drop, so the ion distribution
is not shifted much beyond its shift at the sheath, and so transmission is less than
optical. When the slit is biased below the floating potential the sheath drop increases,
the distribution is shifted, and transmission increases by a factor of two. If there was
no secondary electron emission from the slit then there would always be sufficient
sheath drop (≈ 3kTe ) to shift the distribution so that transmission was optical, and
dropping bias of the slit would have no effect on transmission.
4.2.7
Summary of Conclusions
We summarize the results of the experiments as follows:
152
• Transmission of ions through slit and grids is nearly optical.
• Effective secondary electron emission can be significant at low electron energies
typical of the SOL plasma, γeff ≈ 1.
• The rounded, shifted knee in the in IV characteristic is explained with a combination of space charge and effective secondary electron emission.
• Neither space charge nor secondary electron emission significantly rearrange the
ion distribution function. Only the slit truncates the (unshifted) distribution
to a 45 degree velocity cone. If there is a modest sheath potential then the
distribution shifts and the slit transmission approaches optical.
• Therefore we interpret the exponential portion of IV characteristic as the temperature of the distribution outside analyzer. We associate the voltage of the
knee of the ion IV characteristic (at low currents) with the sheath potential.
• We interpret the break in slope evidence of two ion populations with different
temperatures (to be precise, different kT /q).
The bulk ion temperature is extracted from the IV characteristic as shown in
Figure 4.10. The potential of the knee, φk , is extracted graphically; in the limit that
space charge effects become negligible we would expect the voltage of the knee to
approach the sheath potential. We estimate the fraction of the ion distribution that
is hot by dividing the current at the break by the saturation current.
Since secondary electron emission does not significantly change the interpretation
of Langmuir probes, as shown in the previous chapter, we are able to reliably extract
electron temperature and density from analysis of Langmuir probe IV characteristics.
4.3
Applications
With the basic features of the IV characteristics explained, we discuss simple applications of the omegatron retarding field energy analyzer.
153
Figure 4.10: Processed IV characteristic, showing values of cold and hot ion temperatures and knee potential. Floating potential is obtained from Langmuir probes.
154
4.3.1
Time History of a Tokamak Discharge
As an application of the omegatron RFEA, time histories of electron and ion temperatures are presented for a typical tokamak discharge. The results are compared with
data obtained from Langmuir probes on the face of the omegatron heat shield. The
influence of ICRF auxiliary heating is observed on the electron and ion distribution
functions.
Figure 4.11 shows the results. There are several features to note:
• The RFEA electron temperature is consistently a few eV below the Langmuir
probe electron temperature, but the error bars on the RFEA electron temperature are very small during the ohmic portions of the discharge.
• Both RFEA and LP electron temperatures are relatively constant except for an
increase during ICRF.
• During ohmic portions, the fractions of the ion distribution that are hot and
cold, are relatively constant.
• During ICRF portions all ions become hot.
• The knee potential is consistently lower than the floating potential, indicating
significant space charge.
4.3.2
SOL Profiles: Ohmic Plasma
As a further application of the omegatron RFEA, cross-field profiles of electron and
ion temperatures are obtained from a sequence of similar tokamak discharges. The
results are compared with data obtained from Langmuir probes on the face of the
omegatron heat shield. Profiles are presented at two times, with and without auxiliary
ICRF heating.
Figure 4.12 gives the results during an ohmic portion of the discharge. There are
several features to note:
155
Figure 4.11: Ion and electron temperatures and sheath potential as a function of
time during a tokamak discharge. Electron temperature and floating potential from
Langmuir probe LP2 are also shown.
156
Figure 4.12: Cross-field profiles of electron and ion temperatures and sheath potential
taken from omegatron RFEA and electron density, temperature, and floating potential
from Langmuir probe LP1, taken during ohmic tokamak operation. ρ is the distance
of the flux surface from the separatrix, measured at the midplane.
157
• The plasma density (and thus the current) decreases by a factor of 10 over
∆ρ ≈ 10 mm
• RFEA and LP electron temperature profiles are flat over the profile. LP electron
temperatures is a few eV higher than RFEA electron temperature.
• The cold and hot portions of ion distribution are constant over the profile.
• The temperature of the hot ion portion appears to increase over the profile, but
the low signal further out increases the errorbars.
• As the current decreases the floating potential remains constant.
• As the current decreases the sheath potential approaches the floating potential
(from below).
4.3.3
SOL Profiles: ICRF Plasma
Figure 4.13 gives the results during an ICRF auxiliary heated portion of the discharge.
There are several features to note:
• The density increases by a factor of ten over ohmic and does not decay as quickly
with radius.
• The RFEA and LP electron temperature profiles are approximately flat and
approximately equal, but the error bars increase over ohmic shots.
• The (single) ion temperature profile is flat, and equal to the electron temperature.
• The floating potential profile is flat but is noisier over ohmic.
• The sheath potential profile increases with radius.
158
Figure 4.13: Cross-field profiles of electron and ion temperatures and sheath potential
taken from omegatron RFEA and electron density, temperature, and floating potential
from Langmuir probe LP1, taken during ICRF-heated tokamak operation. ρ is the
distance of the flux surface from the separatrix, measured at the midplane.
159
4.3.4
Implications of Two-Temperature Ion Distribution
The omegatron observes two ion populations in ohmic plasmas, the bulk with kTi /Z ≈
3 eV and the minority (2–8%) with kTi /Z ≈ 20 eV. The implications are discussed in
this section.
Umansky [75] observes that on Alcator C-Mod the main chamber recycling fluxes
often greatly exceed the divertor fluxes, leading him to conclude that plasma flow in
the scrape-off layer is dominated by radial transport to the main chamber walls rather
than by parallel transport to the divertor. The cold bulk is consistent with the picture
of main chamber recycling proposed by Umansky. Most of the ions (deuterons) have
approximately the Franck-Condon energy from dissociation of molecular deuterium,
the minimum energy expected. If instead of being ionized in the far scrape-off layer,
as Umansky suggests, the ions were ionized inside or near the separatrix, then they
would have a temperature representative of the separatrix.
Note that 20 eV is a lower bound for the temperature of the hot species. With
the retarding field energy analyzer on DITE, Pitts [54] saw a similar break in slope
in the ion IV characteristics which he attributed to impurities. If the break in slope
is indeed due to impurities, then the temperature must be greater than 20 eV for
impurities with Z > 1. (Recall from Equation (3.4) that the change of current with
grid bias depends only on the ratio of temperature and charge).
Consider first the possibility that the hot ion population is transported to the
omegatron from a hotter region of the plasma. We assume that the two ion populations are both deuterium, with a cold bulk kTi,c = 3 eV and a hot minority,
kTi,h = 20 eV. We calculate the temperature equilibration rate given in Section
3.1.3 time for two species. For µα = µβ = 2, Zα = Zβ = 1, Tβ = 3 eV, Tα = 20 eV,
bulk ion density near the omegatron nβ = 5 × 1017 m−3 , we have (να,β )−1 = 200 µs.
If the break in slope of the ion IV characteristic is to be interpreted as ion species
with different temperatures, the time for the hot ions to be transported to the
omegatron must be less than the temperature equilibration time. Recall that the
160
equilibration rate increases approximately linearly with the density but decreases
approximately with T −3/2. The cross-field profiles of ne and Te , measured by a
scanning Langmuir probe, give ne Te−3/2 ∼ να,β approximately constant across the
scrape-off layer. Therefore the equilibration rate profile is approximately constant,
so we take (να,β )−1 as an estimate of the time it takes for the the hot species to
be transported to the omegatron from a region of hotter plasma, say the separatrix. If the flux is mostly radial the hot ions have an approximate transport velocity
(∆ρ)να,β ≈ (40 mm)/(200 µs) ≈ 200 m/s. As we shall see in Chapter 6, outward
transport velocity of this magnitude is consistent with diffusive and convective transport of 3He.
4.3.5
Implications of Secondary Electron Emission
Secondary electron emission is important since it can influence impurity sources due to
sputtering. Sputtering yields depend sensitively on ion energy. Therefore an accurate
estimate of the sputtering yield requires an accurate estimate of the ion energy, most
of which is acquired from the sheath potential drop. As Stangeby and McCracken
[64, p.1298] point out, uncertainties in the coefficient of secondary electron emission
can lead to uncertainties in sheath drop, which leads to uncertainties in evaluating
sputtering rates.
The effect of secondary electron emission might be important only for surfaces at
angles nearly normal to the magnetic field. Low energy secondary electrons emitted
from surfaces at acute angle to the magnetic field will execute a fraction of a gyro-orbit
and be recollected on the surface.
Although significant secondary electron emission has been measured from the
omegatron end collector and inferred from the omegatron slit, it is not clear whether
similar secondary electron emission occurs from other surfaces. In the literature describing measurements of secondary electron emission from ion and electron beam
impact, the target surfaces are carefully cleaned to obtain reproducible surfaces. The
161
surface conditions of the omegatron slit and end collector are essentially uncharacterized. At present other surfaces of plasma facing components on Alcator C-Mod can
be characterized neither in-situ nor in real-time.
162
Chapter 5
Omegatron Ion Mass Spectrometer
This chapter presents typical data obtained from the ion mass spectrometer portion
of the omegatron and gives an interpretation of the features. Simple applications are
described showing the utility of the omegatron.
5.1
Observations
This section presents characteristic features of ion mass spectrometer data. Discussion
of additional resonance broadening mechanisms is postponed until the next section.
5.1.1
Alignment
For proper operation of the omegatron ion mass spectrometer, non-resonant ions
must traverse the RF cavity along magnetic field lines without being collected on
the RF plates. If the omegatron is misaligned with the magnetic field non-resonant
current collection will dominate over resonant current collection and the ion mass
spectrometer is useless.
Precise alignment of the omegatron probe axis with the toroidal magnetic field is
achieved using an electron cyclotron resonance (ECR) cleaning plasma. Current is
passed through the toroidal field coils which produces a field of 0.0875 T inside the
163
vacuum chamber. The ECR plasma is formed by a magnetron operating at 2.45 GHz
and 3 kW. The resonance location is swept radially by changing the current in the
toroidal field coils. For the omegatron alignment the resonance location is fixed on
the inner wall. Voltage biases of the grids and end collector are set to reject ions and
to accelerate electrons through the RF cavity to the end collector. The omegatron is
rotated about its vertical axis and the current to the RF plates is measured, shown in
Figure 5.1. The omegatron is misaligned if there is significant electron current to the
RF plates; alignment is achieved by minimizing the current. The alignment procedure
also gives the effective electron beam full width in an ECR plasma, indicated by the
range of rotational angle required over which current to the RF plates increases from
minimum to maximum.
Figure 5.1 shows the signal from the RF plates due to non-resonant current as
the omegatron is rotated about the vertical axis in an electron cyclotron resonance
(ECR) plasma. Two notable features of the data are:
• The non-resonant current to the RF plates is small if the rotational alignment
is within two degrees from center.
• If the omegatron is rotated beyond two degrees from center by one degree the
non-resonant current increases by orders of magnitude.
Figure 5.2 shows a sketch of the rotation of the omegatron RF plates. When
the omegatron axis is aligned with the toroidal magnetic field, particles travelling
toroidally through the slit pass parallel to the RF plates to the end collector. If
the omegatron is rotated beyond a critical angle θ then particles travelling toroidally
through the slit intercept an RF plate instead of the end collecter. The optical cutoff
criterion for the slit is given by
L sin θ =
D/2
,
sin α
where L = 50 mm represent the distance from the slit to the end collector, D = 5 mm
164
Figure 5.1: Electron current signal recorded on the RF plates as a function of rotation
of the omegatron about the vertical axis.
e
Z
e
Z
θ
e
n
e
n
D
α
L
Figure 5.2: Schematic of rotation of omegatron RF plates, viewed toroidally. Horizontal line between the plates represents the slit. Figure to left is aligned, figure to
right is rotated beyond cutoff.
165
is the plate spacing, and α = 45 deg. is the angle of the surface normal en of the RF
plates from vertical axis ez . Optical cutoff of the slit is predicted at θ = 4 degrees, in
good agreement with Figure 5.1.
Figure 5.1 also shows that for a rotation of about one degree near the critical
angle the electron current through the slit to the RF plates goes from its maximum value to its minimum, giving an estimate of the beam profile full width of
2σ/(D/2) ≤ (1 degree)/(4 degrees); for the plate spacing given above the beam halfwidth is predicted to be σ ≤ 0.3 mm.
We conclude:
• Proper alignment of the omegatron is critical to obtain geometric separation of
resonant and non-resonant ions; if it is not aligned, the ion mass spectrometer
is useless.
• The best alignment with the toroidal field has been achieved in situ by using the
actual toroidal magnetic field, e.g. with an ECR plasma and rotatable bellows.
• The geometry of the RF plates and the rotation angle over which the electron
beam current to the RF plates increases gives an electron beam width less than
0.6 mm; if this is comparable to the ion beam width in tokamak discharges then
the beam can be considered collimated.
5.1.2
Ambient Noise
Without processing the signal after it has been collected, the minimum resonant current that can be measured is limited by the noise signal level on the RF plates. During
a plasma discharge, bipolar noise is observed on the RF plates with amplitude a few
tens of nanoamperes equivalent. For maximum resonance amplitudes of deuterium
less than one microamp this gives a signal to noise S/N < 100, which is not acceptable if we want to resolve a factor of ten change in amplitude of impurities present
at concentrations of less than one percent of the deuterium bulk.
166
Figure 5.3 shows the ambient noise spectrum recorded on the omegatron rf plates
with and without plasma. Several features of data can be noted:
• In absence of plasma the noise floor of the RF ammeter electronics is at level of
a few bits, equivalent to a current of IRF,noise ≈ 0.02 nA RMS. Harmonics can
be seen, probably acoustic coupling to vacuum pump vibrations.
• During a plasma dicharge with the omegatron withdrawn, the noise floor increases by two to three orders of magnitude, with large harmonics visible above
1 kHz.
• With the omegatron inserted into a plasma discharge the noise increases by
another order of magnitude above 1 kHz, to equivalent current IRF,noise ≈ 20 nA
RMS.
The present design of the omegatron has the slit electrically connected to the
box surrounding the RF cavity; an unintentional consequence is around 30 picofarads
capacitive coupling between the RF plates and the slit. Therefore fluctuating voltage
on the slit induces a current on the RF plates. The slit typically receives plasma
current that fluctuates with large amplitude and a broad power spectrum. If the
impedance of the slit to ground is high the fluctuating current produces a fluctuating
voltage. To reduce this effect the input impedance of the slit ammeter electronics is
set as low as possible, around half an ohm.
A more effective solution would be to break the capacitive coupling between the
RF cavity altogether, which requires electrically isolating the slit from the shield box.
This could potentially reduce the noise on the RF plates by three orders of magnitude.
Amplitude modulation (AM) and frequency modulation (FM) synchronous detection electronics for the RF plates ammeter have been designed, built, installed, and
tested. The electronics modulates the RF power or frequency at 10 kHz and demodulates the RF signal back down below 1 kHz; components of the signal to the RF
plates which do not correlate with the modulated RF power are effectively removed.
167
Figure 5.3: Omegatron ambient noise spectrum without plasma (top), with plasma
but omegatron withdrawn (middle), and with plasma and omegatron inserted (bottom).
168
During discharges in which the omegatron is fully withdrawn and is in contact with
no plasma the synchronous detection reduces the noise on the RF plates by at least
a factor of fifty, reducing the noise of acoustic coupling with pumps and other low
frequency vibrations shown in Figure 5.3.
However synchronous detection was not used for this thesis. The synchronous
detection technique works only if the spectral power of the noise is lower at the
modulation frequency than at the orignal frequency. Figure 5.3 shows that when the
omegatron is in contact with plasma, the noise signal on the RF plates at 10 kHz is
as high as the noise signal at 1 kHz. Again, the proposed fix is to reduce the source
of noise by breaking the capacitive coupling between the slit and the RF plates.
We conclude:
• The signal to noise of the raw signal is not good, S/N ≈ 100.
• The primary source of noise is coupling between RF plates and slit; we propose
a fix.
• We have implemented additional analog electronics techniques to reduce noise
by orders of magnitude, 10 kHz AM/FM synchronous detection.
• In absence of hardware or electronic techniques, improvement of signal to noise
must rely on digital signal processing.
5.1.3
Resonant Current
Figure 5.4 shows an example of the resonant current collected on the RF plates IRF as
a function of the frequency of the applied RF power. Several features can be noted:
• Bipolar noise with RMS amplitude 5–10 nA can be seen in the RF signal, with
a frequency near one kilohertz.
• There is a large feature (a resonance) centered at t = 0.688 s.
169
Figure 5.4: Top: applied RF frequency and resulting resonant frequency as functions
of time. Bottom: resonant current vs applied RF frequency. Solid line is current
signal binned over regions 0.25 MHz wide, chosen to be close to the theoretically
expected resonance width.
170
• The center of the resonance occurs when the frequency of the applied RF power
is f = 19 MHz; since the magnetic field magnitude near the omegatron is B =
4.6 T, this corresponds to M/Z = 3.7 ≈ 11/3.
• The smooth line is the digitally filtered signal; the filtering procedure is intended
to leave intact features with widths near expected resonance width.
• The full frequency width at half maximum (fwhm) of the resonance is approximately 0.5 MHz.
Simple digital signal filtering (binning) reduces the noise in Figure 5.4 by a factor
of five or more, which improves the signal to noise to S/N > 500. We exploit the
theoretical predictions for the frequency width to remove much narrower features. For
instance, the width of the resonance shown in Figure 5.4 is expected to be 0.3 MHz,
considering just intrinsic single-particle resonance convolved with magnetic field variation, so the binning width is chosen to be 0.25 MHz. For monotonically varying
frequencies the binning procedure is essentially a boxcar average. Filtering by convolution with a Savitzky-Golay kernel [58, p.650] gives similar results. The binning
technique can also be used with periodic signals which change linearly in time, say
multiple sweeps over the same frequency range, to cancel portions of the signal from
different periods which do not correlate; this amounts to a histogram.
Resonances in the RF current signal are identified as increases in the signal that are
correlated with the RF frequency only. Without synchronous detection the correlation
with frequency is done mostly by eye; fluctuations in the bulk plasma arriving at the
omegatron can also cause changes in signal level. In practice the quantity we are
interested in is the fraction of plasma arriving at the omegatron corresponding to each
impurity species. A simple way to remove the gross effects of plasma fluctuations is
to divide the resonant RF current by the non-resonant end current.
Figure 5.5 shows a typical resonant current ratio spectrum, binned, plotted as a
function of the ratio of species mass and charge (assuming the frequency of resonance
171
is the cyclotron frequency, and that the magnetic field at the omegatron is dominated
by the toroidal field). It shows that even with the capacitive coupling between the
slit and RF cavity, and even without synchronous detection, digital binning permits
resolution of resonances down to levels of 0.1% of the non-resonant current.
Conclusions:
• Simple binning improves the signal to noise S/N > 500.
• Digital processing is necessary to see features with amplitudes 10−3 the nonresonant current amplitude.
5.1.4
Impurity Spectrum
Figure 5.5 shows a typical impurity spectrum, with the digitally filtered current to
the RF plates normalized by the non-resonant current to the end collector. Several
features can be noted:
• The noise floor for the digitally filtered, normalized signal is IRF /IEND ≈ 5 ×
10−4 .
• In deuterium plasma, the dominant resonance is at M/Z = 2, and the M/Z = 4
resonance is often present.
• Resonances are observed at charge to mass ratios M/Z which correspond to
the charged states of the following isotopes:
10
B3+ ,
12
C+ ,
12
C2+ ,
16
O3+ ,
14
N2+ ,
14
11
B+ ,
11
B2+ ,
11
B3+ ,
10
B+ ,
10
B2+ ,
N3+ .
• Other resonances, some of which are not shown in Figure 5.5, have been observed
to increase when impurity gases have been puffed. For instance, puffing H2
results in an increase in the resonance corresponding to 1H+ ; puffing 3 He results
in increases in the resonances corresponding to 3 He+ and 3 He2+ ; puffing 4 He
results in increases in the resonances corresponding to 4 He+ ; and puffing N2
results in an increase in the resonances corresponding to
172
14
N2+ and
14
N3+ .
Figure 5.5: Typical impurity spectrum: ratio of resonant current to non-resonant
current as a function of ratio species mass and charge. Annotations near resonances
identify possible isotopes.
173
M/Z
(1)
(3/2)
2.0
3.0
3.5
3.7
4.0
4.7
5.0
Possible Isotopes
(1 H+ )
(3 He2+ )
(D+ ), (4He2+ )
(3 He+ ),12 C4+
(14N4+ )
11 3+
B
+ 12 3+
4
(D+
2 ), ( He ), C
(14N3+ ),19 F4+
10 2+
B
M/Z
5.5
6.0
6.3
7.0
8.0
10.
11.
12.
Possible Isotopes
11 2+
B
12 2+
C
19 3+
F
(14 N2+ )
16 2+
O
10 + 40
B , ( Ar4+ )
11 +
B
12 + 98,96,95
C ,
Mo8+
Table 5.1: Frequently observed mass to charge ratios (M/Z) of resonances in spectra
obtained with the omegatron, and charged states of isotopes with nearby M/Z. Gas
states of isotopes in parentheses have been puffed into tokamak discharges; M/Z in
parentheses can be attributed to no other isotope.
• The resonances corresponding to the charged states of boron appear in approximately the same proportions as the isotopic abundances (19.9%
11
10
B, 80.1%
B).
• For M/Z > 12 the resonances are not well resolved.
Figure 5.5 shows a typical, rich spectrum of ion impurities obtained by the omegatron. Table 5.1 lists resonances frequently observed in ion mass spectra, along with
charged states of isotopes with nearby ratios of mass to charge. Note that M/Z = 1
and M/Z = 3/2 are resolvable resonances which can be attributed uniquely to singly
ionized hydrogen and doubly ionized 3 He, respectively. In fact both of those resonances have been observed upon puffing the appropriate isotopes.
Most other resonances can be attributed to more than one ion. The list of candidate ions need not be restricted to those with ionization energies of order the electron
temperature near the omegatron (around 10 eV). Experiments with 3 He gas puffs
show that ions with ionization energies up to 54 eV are observable by the omegatron.
With some simple estimates it is often possible to narrow the list of isotope candiates.
174
For example, a resonance is commonly observed at M/Z = 11 with current of
order one percent the bulk ion flux. A common isotope of iron has a charge state,
56
Fe5+ , with M/Z = 11.187 and an ionization energy of 75 eV, to which we might
be tempted to attribute the resonance near M/Z = 11. However additional evidence
allows us to disqualify iron as the sole cause.
1. Appropriate resonances corresponding to other charged states of iron do not
appear in the omegatron spectrum. A resonance near M/Z = 8 sometimes
appears (but not always, see the section on boronization in this chapter) which
could be attributed to
to
56
56
Fe7+ , but a resonance near M/Z = 9.3 corresponding
Fe6+ is not observed.
2. Other diagnostics do not see the concentrations of iron implied by attributing
all of the M/Z = 11 resonance to iron. Iron is observed with other spectroscopic diagnostics[66], but core plasma Zeff measurements do not indicate a
concentration of one percent iron.
It is more likely that the resonance at M/Z = 11 corresponds to the singly ionized isotope of boron-11. The walls in Alcator C-Mod are conditioned with a coating of B2 D6
(diborane) to reduce impurity fluxes into the core plasma, so observation of boron
ions in the scrape off layer plasma is expected. Resonances corresponding to M/Z of
other charged states of
11
B do appear in the omegatron spectrum. Resonances also
appear corresponding to the M/Z of the charged states of the other stable isotope 10B,
and the intensities of the resonances for each charged state appear in approximately
the same ratio as the natural isotopic abundance. In addition, boron is observed on
other spectroscopic diagnostics[66] and the total concentration is estimated to be of
order one percent the bulk ion (deuterium) density.
The resonance sometimes observed at M/Z = 8 is probably due to
16
O2+ . An
increase in the M/Z = 8 resonance is observed after a vacuum break, which correlates
with an increase in the mass 18 resonance (H2 O) on a radio frequency quadrupole
175
residual gas analyzer. The RGA results confirm that water vapor enters the vessel
during a vacuum break. Oxygen is also observed spectroscopically.
Other resonances can be partially resolved by active experiment. For instance the
resonant at M/Z = 4 is a persistant feature of the impurity spectrum in deuterium
plasmas. Spectroscopic diagnostics with bandpass filters for line radiation of helium
see little signal when helium is not puffed, but see a dramatic increase in signal when
helium is puffed. See Figure 5.6. During a helium puff the intensity of the M/Z = 4
resonance, which is proportional to the helium concentration, increases as the integral
of the spectroscopic signal, which is proportional to the helium source since it looks at
the puff location. With no helium puff the intensity of the M/Z = 4 resonance remains
mostly constant. Therefore we attribute the increase in the M/Z = 4 resonance to
4
He+ . The remainder could be due either to an intrisic impurity like
12
C+3 , or to a
molecular form of the bulk plasma, D+
2.
Molybdenum is observed spectroscopically in the core plasma and at the edge.
However the omegatron cannot resolve ions of isotopes of molybdenum that have
charge less than +8 since the frequency range for each resonance (resonance width)
is larger than the difference in cyclotron frequencies for the isotopes.
We conclude:
• Ion mass spectrum is dominated by deuterium in deuterium plasmas, as expected.
• Isotopes of boron, carbon, and oxygen are likely present as intrinsic impurities
in the scrape off layer plasma.
• Charged states of hydrogen, helium, and nitrogen are observed unambiguously
when those gases are puffed.
• With the present status of the hardware and with digital signal processing,
resonances are identifiable with amplitudes as low as 5 × 10−4 the non-resonant
current.
176
Figure 5.6: Top: intensity of spectroscopic line from helium versus time, looking at
the helium puff location. Middle: frequency of RF power applied to omegatron versus
time. Bottom: ratio of resonant ion current to non-resonant ion current versus time.
177
5.1.5
Resonance Width Dependence on Non-resonant Current
Figure 5.7 shows the down-shift of the M/Z = 4 resonance as a function of the
non-resonant current arriving at the end collector (top) and the contribution of the
Brillouin flow to the resonance width as a function of the fluctuating current. Several
features of the data can be noted:
• The center frequency of the resonance has the expected dependence on the
non-resonant current, approaching the cyclotron frequency at zero current and
shifting below the cyclotron frequency at high non-resonant current.
• The frequency widths of the resonances are largely independent of the current
fluctation level. Brillouin flow does contribute to the resonance broadening, but
it appears to be dominated by intrinsic broadening and magnetic field variation.
At high currents the measured widths are in a range that can be explained by
theory within 25%. At low currents the resonance widths apparently increase, but
the resonant current at these values is near the noise floor.
Several other mechanisms of resonance broadening will be discussed further in
Section 5.2.
5.1.6
Resonance Width Dependence on Applied RF Power
Figure 5.8 shows the frequency widths of resonances of the charged states of 3 He, as
a function of the applied RF power.
3
He was chosen for this example because the
resonances are unique: no other isotope has M/Z = 3/2; the M/Z = 3 resonance,
while not unique, still is observed only when 3 He is puffed. Several features of the
data can be noted:
• The frequency widths of the resonances extrapolate to a non-zero value at zero
power.
178
Figure 5.7: Resonance widths of M/Z = 4 versus fluctuating non-resonant current,
showing contributions of Brillouin flow broadening, intrinsic broadening, and magnetic field variation.
179
Figure 5.8: Resonance widths of 3 He+ and 3He2+ versus applied RF power. Lower
solid lines represents single-particle prediction for homogenous magnetic field; upper
solid line includes Brillouin flow broadening, assuming fluctuating beam current ∆I ≈
I, (ωc − ωr )/I = 0.007; dashed lines include corrections for magnetic field variation.
180
• The widths depend weakly on applied RF power.
• The widths for resolved and unique resonances (for example charged states of
helium with isotopic mass three) can be reproduced by simple theory to within
a factor of two.
5.1.7
Resonance Amplitude Dependence on Applied RF Power
Figure 5.9 shows the change of the resonance amplitude with an increase in applied
RF power, for the M/Z = 4 resonance. Several features of the data can be noted:
• The amplitude of the resonance goes to zero at zero applied RF power and
increases with applied RF power.
• The amplitude reaches a saturation value.
• The widths of unresolved, degenerate resonances (for example M/Z = 4, which
+
4
in principle could be due to D+
2 , He ,
12
C3+ ,
16
O4+ ,
20
Ne5+ , etc.) can be
reproduced by simple theory to within a factor of two.
5.1.8
Resonant Current Accounting
Figure 5.10 shows current collected on the RF plates when the bulk species M/Z = 2
is resonant and the applied RF power is switched on and off; also shown are the
non-resonant current collected downstream on the end collector and upstream at grid
G3. Several features of the data can be noted:
• The resonant current to RF plates effectively vanishes when the RF power turns
off. Turning the RF power on increases the current to the RF plates.
• Collection of bulk ion current to RF plates results in a decrease in current to
end collector.
• The current to grid G3 also changes when bulk ions are collected on RF plates.
181
Figure 5.9: Top: Normalized resonant ion current versus applied RF power. Solid line
is least squares fit of function y = c0 (1 − e−x/c1 ); dotted lines represent one standard
deviation change in each fitted parameter. Bottom: Frequency full width at half
maximum of resonance amplitude. Smooth line is value predicted by theory including
magnetic field variation, Brillouin flow broadening with ∆I ≈ I, and intrinsic single
particle broadening.
182
Figure 5.10: Current to grid G3, RF plates, and end collector for RF frequency fixed
at center frequency of bulk ion resonance (M/Z = 2) and RF power switched between
0 watts and 8 watts.
183
• The current to grid G3 exceeds the sum of currents to RF plates and end
collector.
• Fluctuations observed in the currents to G3, RF, and end collectors are visibly
correlated.
Figure 5.10 shows how resonant collection of the bulk species affects current up
stream on grid G3 and downstream at the end collector. That the current collected
to the RF plates has any effect at all on the current to grid G3 is evidence of space
charge in the RF cavity. Further evidence is provided by the ratio of currents to
G3 and the end collector. Recall from Section 3.4 that IG3/IEND > (1 − ξ)/ξ ≈ 0.5
implies reflected current, where ξ is the grid attenuation factor; Figure 5.10 clearly
shows IG3/IEND > 1.
Figure 5.11 shows that reflection of resonant ions on space charge in the RF cavity
has an important influence on the resonant ion current collected to the RF plates.
We can make several observations:
• The transmission coefficient for current in the RF cavity g3 decreases from unity
(100% transmission) to approximately 40% over range of currents in RF cavity.
• The absolute level of resonant current to RF plates increases as amount of
current in RF cavity increases.
• The fraction of resonant RF current decreases as current in RF cavity increases.
Significant space charge in the RF cavity complicates the modelling of resonant
current collection. The middle panel of Figure 5.11 shows the motivation for operating
at high current levels: as the current in the analyzer increases, the absolute level of
resonant current also increases. For a fixed noise floor, this monotonically improves
the signal to noise ratio. But the improved signal to noise comes at a price, shown
in the bottom panel: a smaller fraction of the total distribution is collected. This
supports the prompt-reflection model of space charge presented in Section 3.5.4.
184
Figure 5.11: Influence of space charge on the magnitude of resonant current collected
and on the fraction of the resonant current collected. Current was decreased by
withdrawing the omegatron further from the separatrix.
185
We conclude:
• Current ratio data support a model that includes space charge inside RF cavity.
• Resonant ions reflected on space charge inside RF cavity are not collected.
• When total current in RF cavity (and space charge) decrease, the fraction of
resonant ions collected increases.
• To observe resonant current without spacecharge requires reduction in current
to the RF cavity, which in turn requires a reduction in the noise signal level.
5.1.9
Summary of Conclusions
We summarize the discussion of the typical ion mass spectrometer features as follows:
• With proper alignment of the omegatron with the magnetic field, and with
digital processing of data, it is possible to obtain impurity resonance spectra
with noise levels a factor 5 × 10−4 lower than the non-resonant current.
• The intrinsic impurity spectrum is dominated by charged states of isotope of
boron, and perhaps carbon, at collected current levels less than 2% the bulk
deuterium current. Charged states of puffed impurities have been observed as
well.
• All resonance amplitudes increase with applied RF power up to a saturation
value.
• The frequency widths can be reproduced by simple theory, but only if Brillouin
flow broadening is included. Magnetic field variation and intrinsic single-particle
broadening also contribute to the resonance widths.
• Evidence exists for space charge in the RF cavity which reflects resonant and
non-resonant ions. Any model of resonant ion collection must include this.
186
5.2
Discussion of Spectrum Features
This section relates the characteristic features of the ion mass spectrometer data
to the theory of Chapter 3, specifically the resolution of the ion mass spectrometer
and resonance broadening mechanisms. Several additional broadening mechanisms
are examined and rejected. The broadening mechanisms identified to contribute all
preserve resonant current.
5.2.1
Resolution and Broadening
The frequency range over which significant resonant current is collected is referred
to as the resonance width. There are several possible mechanisms, instrumental and
physical, which would cause the measured resonance widths to be wider than the
theoretical minimum width.
Understanding the impurity resonance widths is important to verify the physics
model of resonant impurity collection; this helps to relate the measured resonance
amplitudes to the theoretical quanties, and helps to suggest regimes to operate the
diagnostic to obtain optimum resolution.
Several possible resonance broadening mechanisms are discussed, and estimates
are given of each contribution to the overall resonance width. All of the mechanisms
identified thus far to contribute to the resonance width also preserve the integral of
resonant current over the frequency range. Thus, if a broadening mechanism causes
the resonant frequency range to increase, the current amplitude decreases appropriately. The distinction is important since we can recover what the ideal resonance
current amplitude would be in the absence of any broadening mechanisms.
5.2.2
Filtering
If the RF oscillator frequency is swept too quickly over the cyclotron frequency then
the electronics responds on the bandwidth timescale, delaying and extending the time
187
the signal has significant amplitude. A naive mapping of current to frequency with
time as a parameter gives a resonance which appears broadened in frequency. This
effect has been measured by applying external current pulses to the RF electronics
and observing the output widths. Off-line tests of the electronics show that for input
pulses with gaussian shapes and fwhm between 1 ms and 10 ms the electronics can
add 10% to the fwhm, but that the broadening preserves the product of the resonance
amplitude and the resonance width, ∆fmeas ≈ 1.1∆f.
5.2.3
Oscillator Spectrum
There are two ways in which the RF oscillator could conceivably contribute to resonance broadening. (1) If the oscillator produces a spectrum of frequencies near the
desired frequency, or if the center frequency is modulated about the desired frequency,
then requesting any frequency within the modulation range of the cyclotron frequency
will result in resonant current collection. (2) If the oscillator generates multiple frequencies simultaneously, for instance harmonics of a fundamental frequency, then
each frequency can result in resonant current collection.
We shall see that neither of these mechanisms play a significant role in resonance
broadening.
Figure 5.12 shows the output of the birdy circuit as the oscillator frequency scans
over the 5 MHz crystal. Several features can be noted:
• The birdy signal increases when the RF oscillator frequency approaches the
crystal frequency, and drops sharply when the RF oscillator frequency passes
over the crystal frequency.
• Discrete changes in the birdy signal correspond to approximately 25 kHz change
in RF frequency.
• The birdy can be used as a diagnostic to check if the frequency request voltage
is noisy.
188
Figure 5.12: Birdy circuit output and the calibrated frequency monitor (MHz) as
functions of time. The steps in the birdy signal are caused by the finite resolution of
the Bira frequency programming signal, corresponding to approximately 25 kHz per
bit.
189
Figure 5.13: Harmonics produced by the Wavetek model 1062 RF oscillator. Lines
connect the jth harmonic, j = 0 is the fundamental.
Figure 5.13 shows the peak power of harmonics produced by the RF oscillator as
a function of the fundamental frequency. Several features can be noted:
• The power of each harmonic decreases as the harmonic order increases: the first
harmonic has a lower amplitude than the fundamental frequency, the second
harmonic has a lower amplitude than the first harmonic, etc.
• The power of each harmonic decrease as the fundamental frequency increases.
• Fundamental frequencies above 4 MHz (corresponding to M/Z > 18.5 for B =
5.4T on axis) have all harmonics with at least 30 dB lower power than the
fundamental frequency.
We conclude from the birdy circuit in Figure 5.12 that we are able to obtain the
requested frequency within 25 kHz, which is much less than typical resonance widths.
190
Figure 5.14: Fluctuation spectrum of poloidal magnetic field, recorded from poloidal
field coil BP09 JK near the omegatron.
Therefore the oscillator center frequency is suffiently clean that it does not contribute
to resonance broadening. From Figure 5.13 we conclude that oscillator harmonics
contribute negligibly to resonances with M/Z < 18 (for typical magnetic fields on
Alcator). Since resonances above M/Z = 12 are poorly resolved anyway, we conclude
that the oscillator is not a source of resonance broadening for M/Z < 12.
5.2.4
Magnetic Fluctuations
The influence of magnetic field variation on the resonance width was mentioned in
Section 3.5.2. The same mechanism could contribute to resonance broadening if the
magnetic field fluctuates with a significant amplitude on the same timescale as the
resonance sweep. Figure 5.14 shows the power spectrum of poloidal magnetic field
fluctuations taken from a poloidal field coil near the omegatron.
191
Over timescales of order 10 milliseconds, the poloidal magnetic field near the
omegatron fluctuates with an amplitude of approximately ∆B ≈ 1.0 mT. For deuterium in a typical magnetic field, fc = 36 MHz,
∆fc = fc
∆B
= fc 2 × 10−4 ≈ 0.007 MHz fwhm,
B
Since this is much smaller than the intrinsic resonance width we neglect magnetic
field fluctuations as a mechanism to broaden resonances.
5.2.5
Density profile
Ions enter the RF cavity in a beam with finite thickness. Ions near the edge of the
beam are closer to the RF plates than ions at the center of the beam, so it takes
less perpendicular energy to collect them, so they are collected over a wider range of
frequencies.
Recall the estimate from Section 3.3.1 of the Brillouin current limit for 3 eV
deuterons: Imax,B ≡ qnB Aslit kTi /mi ≈ 27 µA. Note this level of current is observed
routinely in the omegatron, particularly during ion mass spectrometer operation.
Therefore it is likely that some beam spreading does occur at these high currents.
However it was shown in Section 3.5.8 that for beam thickness less than the RF
plate spacing by d ≤ D/16 the beam can be considered collimated. The current
required to give d = D/16 with n = nB and the same ion mass and temperature as
above is I ≈ 300 µA, which exceeds the non-resonant current routinely observed in
the omegatron RF cavity. Therefore we consider the beam to be collimated for the
typical range of currents observed.
Thus far we have no measure of the width of the ion beam in a tokamak plasma
discharge, which is an important component of the single-particle theory presented in
Section 3.5.8. Whatever the shape of beam density profile at the center, it must be
small near the RF plates since we typically reject non-resonant current from the RF
192
plates by a factor of 10−3 or 10−4 . If we assume that the ion beam width in a tokamak
plasma discharge is equal to the electron beam width in an ECR plasma then we have
σ/(D/2) ≈ 1/8. It was shown in Chapter 3 that ion beams with this thickness or less
can be considered collimated for the purposes of ion mass spectrometry. A simple
estimate here confirms the result: ∆d = σ = (D/2)/8 = 0.31 mm, which gives a
frequency spread of
∆f =
5.2.6
E
σ
1
≈ 0.035 MHz fwhm
2π B(D/2) D/2
Degeneracies
Since the typical resonance widths are of order 0.5 MHz, resonances above about
M/Z = 12 are difficult to resolve. For example we might wish to resolve isotopes
of molybdenum, an intrinsic impurity on Alcator C-Mod, and argon, an impurity
injected to measure plasma rotation at the core. Consider the charge states of stable
isotopes of molybdenum and argon with cyclotron frequencies within one megahertz
around M/Z = 12, listed in Table 5.2. It can be seen that the isotopes all have
resonances within 0.5 MHz of each other so that they will all overlap into a continuum.
But we also observe broadening of the resolved and unique 3He resonances a factor
of two beyond theoretical predictions, which resonance degeneracy cannot explain.
5.2.7
Summary
In summary, we observe resonance widths which can be reproduced by simple theory including only intrisic broadening, magnetic field variation, and Brillouin flow
broadening. All of the broadening mechanisms identified thus far to contribute to the
resonance width preserve the integral of the resonance amplitude over the width. Additional mechanisms which could conceivably contribute to the resonance broadening
have been considered and rejected.
193
Isotope
94
Mo7+
40
Ar3+
92
Mo7+
100
Mo8+
98
Mo8+
97
Mo8+
96
Mo8+
95
Mo8+
94
Mo8+
92
Mo8+
M/Z
13.415
13.321
13.130
12.488
12.238
12.113
11.988
11.863
11.738
11.488
fc ( MHz)
5.49
5.53
5.61
5.90
6.02
6.08
6.15
6.21
6.28
6.42
isotopic
abundance (%)
9.25
99.60
14.84
9.63
24.13
9.55
16.68
15.92
9.25
14.84
Table 5.2: Typical cyclotron frequencies at omegatron location for stable isotopes of
molybdenum and argon within one megahertz of M/Z = 12. Isotopes are not resolved
since resonance full width at half maximum is ∆f ≈ 0.5 MHz.
5.3
Applications
Having discussed the features of typical ion mass spectrometer data, we turn our
attention to simple applications
5.3.1
Impurity Densities, Temperatures from Applied RF
Power Scan
A mosaic of the ion impurity spectrum from 3 < M/Z < 12 was obtained at several
different applied RF powers during a sequence of plasma discharges. For each identifiable resonance, the amplitude versus applied RF power was fit by least squares to
a function of the form IRF = c0 [1 − exp(−P/c1 )]. The ratio of temperature to mass
for each resonance was obtained from the fitting coefficients c1 , and the asymptotic
impurity flux fractions were found from the fitting coefficients c0, using the theory
of Section 3.5.6. The densities are obtained from the fluxes assuming the impurities
have the same flow velocity at the sheath edge as the bulk ions (collisional presheath).
The results are shown in Figure 5.15. Several features of the results can be noted:
194
Figure 5.15: Impurity temperatures, flux fractions, and density fractions at sheath
edge, obtained from RF power scan technique for range 3 < M/Z < 12. Labels
identify assumed source of the resonances.
195
• Impurity temperatures are all between 2–3 eV, within the uncertainties of the
fits and with the shown impurities assigned to the resonances.
• The qualitative features of the flux fractions (and density fractions) can be
recognized in the raw current to the RF plates, normalized by the non-resonant
current to the end collector, IRF/IEND vs M/Z.
• The calculated impurity flux fractions (and density fractions) are all below ten
11 +
percent, with M/Z = 4(2 H+
2 ) and M/Z = 11( B ) the largest.
• The flux (and density) fractions of the charged states of isotopes of boron appear
in approximately the same ratio as the isotopic abundance, within the error bars.
We can compare the temperature equilibration time with the parallel transport
time. If the equilibration time is short compared to the parallel transport time then
even if ion impurities arrived at the omegatron flux tube with different temperatures,
we would expect them to equilibrate with the background ions in the flux tube as
they are accelerated in the presheath.
The temperature equilibration time Equation (3.2) in the plasma near the omegatron with density ne ≈ 5 × 1017 m−3 and bulk ion temperature Ti ≈ 3 eV of a species
with similar mass and charge as the bulk ions, is shorter than the parallel transport
time for all species with Z > 1. For impurities with Z = 1 we have να,β τ ≈ 0.2. See
Figure 3.3. Note that at higher densities the temperatures equilibrate faster. Thus
we expect all species with Z > 1 to equilibrate to the bulk ion temperature.
Pappas et al [52] have inferred the neutral molybdenum particle influx from spectroscopic measurements from neutral molybenum, and they have matched the molybenum influx using a sputtering model assuming a flux of B3+ between 2–5 % the
deuterium flux to the outer divertor. This level of boron is consistent with the flux
fractions shown in Figure 5.15. However, it is not clear if the two estimates can
be compared directly. The spectroscopic estimates come the 1995-1996 experimental
196
campaign when spectroscopic intensity (and therefore the inferred molybdenum influx) was about a factor of ten higher than observed for the 1999 campaign; therefore
to match the molybdenum influx with the sputtering model the boron flux would have
to change by a similar amount. Also the spectroscopic measurements were performed
near the strike point in lower divertor; the omegatron measurements were performed
in the far scrape-off layer plasma of the upper divertor.
A lower bound for the effective atomic number in the edge plasma and the uncertainty can be calculated using the impurity densities and the charges of the isotopes
that have been assigned to the resonances:
Zeff
nj Zj2
≡ ,
j nj Zj
j
∆Zeff
Zeff
2
=
k
Zk2
Zk
−
2
j nj Zj
j nj Zj
2
(∆nk )2 .
Using the impurity density fractions from Figure 5.15 gives Zeff = 1.3 ± 0.2. This
represents a lower bound because unresolved impurities with M/Z > 12 have not
been included in the calculation.
5.3.2
Boronization
Figure 5.16 shows the impurity spectra recorded before and after the August, 1999
boronization. Figure 5.17 shows the impurity spectra recorded before and after the
September, 1999 boronization. From the impurity spectra before and after boronization several features can be noted:
• Many resonances persist with similar amplitudes after the boronization: M/Z =
2, 4, 6, 10, 11. Specifically, the resonances attributable to boron do not change
significantly.
• The M/Z = 8 resonance disappears after the August 1999 boronization.
• The M/Z = 7 resonance decreases after the September 1999 boronization.
197
Figure 5.16: Ion impurity spectrum before and after August 1999 boronization. Note
decrease in M/Z = 8 resonance.
198
Figure 5.17: Ion impurity spectrum before and after September 1999 boronization.
Note decrease in M/Z = 7 resonance.
199
5.3.3
H/D Scan
Some ICRF auxiliary heating schemes heat a hydrogen minority species. The efficiency of the heating depends on the hydrogen concentration, so knowledge and
control of the hydrogen concentration is important. In an experiment designed to
find the optimum hydrogen concentration for ICRF heating[36], the hydrogen concentration was varied from 2.5% up to 20% and back down. The ratio of hydrogen to
deuterium concentration was determined spectroscopically from Balmer emission[74]
in the edge plasma and scrape-off layer.
The omegatron also measured hydrogen concentration during the experiment.
Since the M/Z = 1 resonance was identified uniquely the two measurements could
be compared. Figure 5.18 compares the omegatron and Balmer spectroscopic measurements of relative concentration of hydrogen. Several features of the results can
be noted:
• The omegatron measurements have positive correlation with the Balmer measurement.
• The mean omegatron H/D measurement is lower than the corresponding Balmer
H/D measurement by a factor of 0.6.
• Over a sequence of tokamak discharges the hydrogen concentration was scanned
up and then down. No hysteresis was apparent in the relationship between
Balmer H/D and Omegatron H/D.
The H/D fraction calculated from the omegatron data includes corrections for
resonance broadening, and assumes a collisional presheath such that the hydrogen
and deuterium fluid velocities are the same at the sheath edge. A scan of applied
RF power in the omegatron was not performed, so the asymptotic values of the
resonant hydrogen current are not available. Instead peak measured currents are
used in calculations for Figure 5.18 and a temperature correction is applied assuming
200
Figure 5.18: Comparison of hydrogen to deuterium (H/D) density ratios from Balmer
spectroscopy and omegatron. Solid line is least-squares fit to data of the form y =
mx, where y represents the omegatron H/D and x represents the Balmer H/D. For
comparison, dotted lines have slopes of 2m and m/2. Omegatron H/D includes
corrections for resonance broadening, collisional presheath, and finite applied RF
power (assuming kTH = 3 eV).
201
Figure 5.19: Omegatron residual gas analyzer spectrum of M/Z of ion species formed
inside the omegatron by electron impact ionization. Note that M/Z = 4 resonance is
dominant, probably corresponding to D+
2.
the the hydrogen has equilibrated with the bulk ions near the omegatron, TH = 3 eV.
Doppler broadening of the Dα line gives neutral deuterium temperatures of order
kT ≈ 2 eV (near where the neutral deuterium line radiation is measured).
5.3.4
Residual Gas Analysis
By a simple change of bias to the grids, the omegatron can be operated as a residual
gas analyzer. The grid biases are set to reject plasma ions and to accelerate plasma
electrons into the RF cavity; ions formed in the RF cavity by electron impact are collected with the ion mass spectrometer. Figure 5.19 shows the ion spectrum obtained
from operating the omegatron as a residual gas analyzer. Several features of the data
can be noted:
202
• The resonance at M/Z = 4 is dominant.
• The resonances with the next highest amplitudes are lower by a factor of five
with M/Z = 3, 2.
• Several other resonances are observed.
• The resonance spectrum is completely different from the plasma ion spectra.
We could attribute some of the resonances to ionized molecules, for example D+
2
+
at M/Z = 4, HD+ at M/Z = 3, H+
2 at M/Z = 2, D3 at M/Z = 6, and so forth.
Futher attempt to identify the resonances in the RGA spectrum is beyond the scope
of this thesis.
5.3.5
Neutral Pressure Measurement
The neutral gas density inside the omegatron cavity can be determined by operating
the omegatron as a residual gas analyzer. Figure 5.20 compares the neutral pressure
inside the omegatron with the upper divertor pressure measured by a baratron gauge.
Several features of the data can be noted:
• The neutral pressure calculated in the omegatron is of the same order as the
neutral pressure measured by the baratron gauge in the upper divertor.
• The neutral pressure in the omegatron does not change on the same timescale
as the divertor neutral pressure.
We can estimate the neutral density as follows. The ionization of molecular hydrogen by electron impact has a maximum cross section of 10−16 cm2 at 60 eV [29],
approximately the energy of the electrons in the omegatron during RGA mode. Let
the rate of ion production by a beam of electrons with energy E on stationary hydrogen molecules be given by
RR = ne n0σi (E)ve ,
203
Figure 5.20: Neutral pressure in omegatron probe cavity as a function of time during a tokamak discharge. Spikes represent resonant ion collection with M/Z = 4
corresponding to D+
2 . Peak value of the spike corresponds to the neutral pressure.
Continuous signal is neutral pressure in E-Top measured by an MKS baratron gauge.
204
where
ne ve =
Ie
,
Ae
RR =
IRF
,
Axe@
where Ie represents the electron current through the cavity (measured at the end
collector), IRF represents the resonant current of D+
2 (with M/Z = 4, measured on
the RF plates), @ represents the collection efficiency of resonant current, x represents
the length of the volume through which the electrons ionize neutrals, and A represents
the area of the electron beam. The above equation can be rearranged to solve for the
neutral density,
n0 =
IRF
1
,
Ie @σi(E)x
and a lower bound for n0 can be obtained by noting that @ ≤ 1, σ(E) ≤ σmax , and
x ≤ L, which gives
n0 ≥
IRF 1
.
Ie σmaxL
Using IRF = 3 × 10−6 A, Ie = 1 × 10−3 A, L = 5 cm, and σmax = 1 × 10−16 cm−2
gives n0 ≥ 6 × 1018 m−3 . If the neutral gas is assumed to be at room temperture,
T0 ≈ 0.025 eV, then the neutral gas has a pressure of approximately 0.024 Pa.
We can estimate the probability that an ion will undergo a collision with a neutral
during its transit through the RF cavity as follows. The uncollided flux of particles
passing through a medium with collision cross section σs and density n0 is attenuated
as
Γ(x) = Γ0 exp(−n0σs x),
and thus the probability that a particle will suffer a collision in the cavity is given by
P = 1 − exp(−n0σs L),
≈ n0 σs L for n0σs L 1.
If we approximate the scattering cross section σs by the proton-hydrogen excitation
cross section given by Janev [29], σs ≈ 10−15 cm2 , and use the cavity length and
205
neutral density obtained above we get P ≈ 0.03. Since ions suffer much less than
one collision with a neutral while inside the omegatron we are justified to neglect
ion-neutral collisions in the ion equation of motion.
206
Chapter 6
3He Transport
6.1
Overview
Experiments were performed on Alcator C-Mod to characterize the transport of helium ions in the scrape-off layer plasma. Helium gas with atomic number three was
puffed from the wall into tokamak discharges, and the omegatron ion mass spectrometer was used to record the absolute concentrations and fluxes of singly- and
doubly-charged helium ions.
Helium is a convenient impurity for transport experiments: it has only two charged
states, it forms no molecules, and excited states can be neglected, so it is simple
to model; it is a recycling impurity, so steady transport behavior is independent
of the gas puff location; in deuterium majority plasmas we safely neglect helium
charge exchange; if we also neglect backscattering of helium ions as neutrals from
wall surfaces then neutral helium atoms have the wall temperature; trace amounts
of helium are benign for machine operation, so experiments can often proceed in
“piggy-back” mode; finally, the charge to mass ratios are either unique (M/Z = 3/2)
or uncommon (M/Z = 3), so the helium resonances can be identified unambiguously
with the omegatron.
The ratios of doubly-charged to singly-charged helium ions flux and density mea207
sured by the omegatron provide information about impurity transport in the scrapeoff layer. Qualitatively, if the impurity transport out of the hot plasma is rapid,
there is insufficient time to form doubly-ionized helium, and the inward flux of neutral helium is balanced by an outward flux of singly-ionized helium. If instead the
impurity transport out of the hot plasma is slow, then most of the singly ionized
helium becomes doubly-ionized, and the inward flux of neutral helium is balanced by
an outward flux of doubly-ionized helium.
It is found that the ratio of doubly-charged to singly-charged helium ion flux
measured by the omegatron is near unity. The electron density and temperature near
the omegatron are too low for the helium ions to have been produced locally, thus
the helium must have been transported from a hotter region of the plasma. A simple
one-dimensional diffusive model reproduces the observed values of density and flux,
but only if the cross-field transport is rapid and increases with distance from the
separatrix.
Umansky[75] observed that in Alcator C-Mod the neutral flux from the wall,
inferred from visible radiation, far exceeded the parallel ion flow to the divertor,
measured by Langmuir and Mach probes. This implied large cross-field transport, and
to model observed profiles of electron density Umansky required an effective diffusion
coefficient profile which increased with distance outward from the separatrix.
Following Umansky, we consider a one-dimensional, perpendicular (cross-field)
transport model, including particle diffusion and convection. Although we use a
diffusive model for transport, the diffusion process is considered to be anomalous,
that is, the transport process could actually be due to mixing of turbulence eddies.
Only volume ionization from the ground states is included. We neglect radiative
recombination; this assumption is justified once the flux profiles have been obtained
by showing that the effect of radiative recombination on the flux at the boundary is
small.
Figure 6.1 shows a schematic of the two-dimensional cross section of the scrape208
Core Plasma
x=x0
Separatrix
Scrape-off Layer (SOL)
Γ He0
Γ He+
∆ρ=40 mm
Γ He++
x=x1
Γ|| He++
Γ|| He+
Omegatron
Slit
x
Local SOL
E-Port ICRF
Antenna
Figure 6.1: Schematic of scrape-off layer geometry, showing directions parallel and
perpendicular to the magnetic field, and orientation of omegatron probe face to separatrix and E-port ICRF limiter.
209
off layer. Helium, if it is ionized near the separatrix, must transport across magnetic
field lines through the scrape-off layer (SOL) to arrive at the Local SOL shared by
the omegatron and the E-Port ICRF antenna. Using the notation as shown in Figure
6.1, the outline of the remainder of this chapter is as follows:
1. The parallel fluxes of singly- and doubly-ionized 3 He are measured in the omegatron RF cavity. The densities of singly- and doubly-ionized 3 He are inferred at
the sheath edge (Section 6.2).
2. Helium ionization rates near the omegatron are compared with parallel transport times. It is shown that to assume that the helium ions are generated in
the Local SOL is inconsistent with the observed fluxes of helium ions, and that
the helium ions must be formed outside the Local SOL and transported into it
(Section 6.3). This is the motivation to consider a model of cross-field helium
transport.
3. Measurements of the characteristic decay lengths of parallel deuterium flux in
the Local SOL are used to relate parallel flux to the omegatron and perpendicular flux into the Local SOL. Assuming that the helium and deuterium ions are
subject to the same transport mechanisms, and knowing that the presheath is
highly collisional, we relate the helium ion measurements at the omegatron to
helium fluxes into the Local SOL (Section 6.4).
4. A cross-field 3 He transport model is developed to relate the fluxes and densities
of neutral, singly-ionized, and doubly ionized 3 He in the SOL (Section 6.5).
5. The background plasma electron temperature and density profiles in the SOL
are obtained from scanning Langmuir probe measurements and are used to
calculate helium ionization rates in the SOL (Section 6.5.1).
6. Since atomic helium enters the SOL plasma at the wall temperature, the profiles
of neutral helium density and the singly-ionized helium source can be calculated
210
directly (Section 6.5.2).
7. An analytic model of transport in a homogeneous slab SOL is used to estimate
the magnitude of effective perpendicular diffusion required to reproduce the
observed values of helium ion fluxes at the boundary between the SOL and the
Local SOL (Section 6.6).
8. A numeric model of transport in a slab SOL with the measured temperature
and density profiles is considered. Cross-field transport is adjusted (via D⊥ and
V ) to yield measured values of density and perpendicular flux of helium ions
arriving at the boundary between the SOL and Local SOL (Section 6.7).
9. The results are discussed and compared with other estimates of cross-field transport in the SOL (Section 6.8).
6.2
3
Observations
He gas was puffed into a standard Alcator C-Mod tokamak plasma, with toroidal
field B = 5.4T on axis, plasma current Ip = 0.8 MA, line-averaged electron density
ne = 1020 m−3, and ohmic heating only. Both charged states of 3 He were observed
unambiguously with the omegatron ion mass spectrometer. The geometry of the flux
surfaces near the omegatron for a typical plasma is shown in Figure 6.2. Figure 6.3
shows the 3 He impurity spectrum. The resonant currents collected due to singly- and
doubly-ionized 3He are approximately equal.
As the RF power applied to the omegatron was increased the amplitude of resonant
current collected also increased. For each power level a calculation was performed as
in Section 3.5.6 to determine the asymptotic current that would have been collected if
infinite power had been applied. The helium ions were assumed to have temperature
3 eV, since the RFEA bulk ions typically show this temperature in ohmic L-mode
discharges and the presheath is highly collisional. The asymptotic currents for singly211
991028024 EFIT: 0.900
0.70
0.60
0.50
Z (m)
0.40
0.30
0.20
0.10
0.00
0.6
0.7
0.8
R (m)
0.9
1.0
Figure 6.2: Poloidal cross section of Alcator C-Mod tokamak showing omegatron
(mirror image) inserted into upper divertor scrape-off layer plasma and fast scanning
Langmuir probe near midplane inserted to separatrix.
212
++
+
and doubly-ionized 3He are shown in Figure 6.3, IHe
/IHe
= 0.8 ± 0.1. That the
asymptotic currents are independent of applied RF power indicate that the helium
ion temperature is approximately 3 eV. Analysis of the data using the procedure
+
++
+
outlined in Section 3.5.6 gives n+
He /nD = 3.9% ± 0.3% and nHe /nD = 2.5% ± 0.5%.
6.3
3
He+ and 3He++ Ionization in Local Flux Tube
First we consider the possibility that all the helium ions detected by the omegatron
are formed by ionization in the local flux tube connecting the face of the omegatron
++
and the E-port ICRF antenna. Let n0He , n+
He , nHe represent the density of neutral,
singly-ionized, and doubly-ionized helium, respectively. The continuity equation for
the jth charge state of helium can be written
∂njHe
+ ∇ · ΓjHe = S j − njHe Aj , for j = 0, +, ++
∂t
where ΓjHe is the flux, S j is the ionization source of nj and nj Aj is the sink due to
ionization to the next charge state.
Assume that local ionization sources and sinks are large compared to the divergence of perpendicular flux near the omegatron. In this case, the charge states of
helium are produced locally due to ionization in the plasma with electron density ne
and temperature Te , and losses are due only to transport parallel to field lines. We
neglect volume recombination and we consider ionization from the ground state only.
++
In steady state the continuity equations for n+
He and nHe are given by
n+
He
+
++
= n0He ne σv+
He − nHe ne σvHe ,
τ
n++
He
++
= n+
He ne σvHe ,
τ
where τ = L / kTe /mi is the parallel transport time, L ≈ 1 m is half the connection
++
length for the omegatron, and σv+
He and σvHe represent the ionization reaction
213
Figure 6.3: Top: 3 He impurity spectrum. Bottom: Asymptotic resonant current fractions due to singly- and doubly-ionized helium, corrected for resonance broadening,
assuming T = 3 eV for helium ions.
214
rate parameters for electron impact on neutral helium and singly-charged helium,
respectively. Solve for the density ratios:
n+
ne σv+
He
He
=
,
n0He
(1/τ ) + ne σv++
He
n++
He
= τ ne σv++
He .
n+
He
++
Polynomial fits to the reaction rate parameters σv+
He , σvHe are found in Janev
[29, pp.263,264]. The electron density ne ≈ 2 × 1017 m−3 and electron temperature
kTe ≈ 7 eV are available from Langmuir probes at the omegatron, giving ne σv+
He ≈
+
++
+
−1
0
−3
−5
43 s−1 , ne σv++
He ≈ 0.2 s , τ = 67 µs, nHe /nHe ≈ 3 × 10 , and nHe /nHe ≈ 10 .
+
The results of this simple model contradict observation, which gives n++
He /nHe ≈ 1.
Also, the helium neutral pressure implied by the simple model is larger than expected,
equal to the total neutral pressure observed in the upper divertor. Therefore it is a bad
assumption to neglect perpendicular transport, so we consider the opposite extreme,
where perpendicular transport to the omegatron dominates over local ionization.
6.4
Cross-Field Transport in Local SOL
From Langmuir probe measurements along the face of the heat shield and from resonant helium current measurments with the ion mass spectrometer, helium ion densities and perpendicular fluxes at the boundary of the Local SOL are obtained. We
have no direct measurement of the helium transport in the Local SOL, but we postulate that the helium and the deuterium ions are subject to the same transport
mechanisms. It is shown that both helium and deuterium sources are negligible in
the Local SOL, and thus the perpendicular fluxes into the Local SOL are related to
the parallel fluxes to the omegatron. Since the presheath is highly collisional all ion
species have the same fluid velocity at the sheath edge.
In the Local SOL the deuterium flux is divergence free (to be shown in Section
6.4.1). Then
− Γ+
D,⊥ (x1 )L +
∞
x1
215
Γ+
D, (x)dx = 0,
(6.1)
Figure 6.4: Scale lengths for ion saturation current and electron density at omegatron
face. Asterisks represent measurements from Langmuir probes; squares represent
possible corrections due to misalignment of the head with local magnetic surfaces.
216
where Γ+
D,⊥ is the average perpendicular flux along the Local SOL boundary. Figure
6.4 shows the exponential decay of ion saturation current and electron density with
distance from the edge of the heat shield in the Local SOL,
+
Γ+
D, (x)/ΓD, (x1 ) = exp[−(x − x1 )/λ ],
(6.2)
so we can perform the integral in Equation (6.1) easily. Then we have a simple
relationship between the parallel and perpendicular flux in the Local SOL:
+
Γ+
D,⊥ (x1 )/ΓD, (x) = (λ /L ) exp[(x1 − x)/λ ].
(6.3)
In the Local SOL the helium flux is divergence free, as shown in Section 6.3. Then
again by the divergence theorem
− ΓjHe,⊥ (x1 )L +
∞
x1
ΓjHe, (x)dx = 0,
(6.4)
We assume that the parallel flux of helium in the Local SOL is also related by an
exponential factor:
ΓjHe, (x)/ΓjHe, (x1 ) = exp[−(x − x1)/λ ],
(6.5)
Then Equation (6.3) applies for helium:
ΓjHe,⊥ (x1)/ΓjHe, (x) = (λ /L ) exp[(x1 − x)/λ ],
(6.6)
Using omegatron theory as given by Equation (3.12) and Equation (3.13) we relate
measured current to flux:
+
++
+
Γ++
He, (x2 )/ΓHe, (x2 ) = (IHe /IHe )/(2gtrans ) ≡ α.
217
(6.7)
Then the perpendicular helium fluxes at x1 are related to the currents measured:
+
Γ++
He,⊥ (x1 )/ΓHe,⊥ (x1 ) = α.
(6.8)
Relate perpendicular helium fluxes using conservation of mass flux,
++
Γ0He,⊥ (x1 ) + Γ+
He,⊥ (x1 ) + ΓHe,⊥ (x1 ) = 0.
(6.9)
Solve for the helium ion fluxes normalized by the neutral flux:
0
Γ+
He,⊥ (x1 )/ΓHe,⊥ (x1 ) = 1/(1 + α),
0
Γ++
He,⊥ (x1 )/ΓHe,⊥ (x1 ) = α/(1 + α).
(6.10)
The presheath is highly collisional, see Section 3.1.3. Then
+
+
+
++
++
Γ+
D, (x)/nD (x) = ΓHe, (x)/nHe (x) = ΓHe, (x)/nHe (x) = cs (x)/2.
(6.11)
It then follows that
+
n++
He (x)/nHe (x) = α.
(6.12)
Relate the helium ion density to the perpendicular helium flux:
+
n+
He (x1 ) = ΓHe, (x1 )/(cs (x1 )/2),
(6.13)
= Γ+
He,⊥ (x1 )(L /λ )/(cs (x1 )/2),
(6.14)
= (1 + α)−1 Γ0He,⊥ (x1 )(L/λ )/(cs (x1)/2).
(6.15)
Since neutral helium flux has the wall temperature,
Γ0He,⊥ (x1) = n0He (x1)vt /4.
218
(6.16)
The helium ion densities normalized by the neutral density are
0
−1
n+
He (x1 )/nHe (x1 ) = (1 + α) (L /λ )(vt /(2cs (x1 ))),
(6.17)
0
−1
n++
He (x1 )/nHe (x1 ) = α(1 + α) (L /λ )(vt /(2cs (x1 ))).
(6.18)
++
+
/IHe
= 0.8 ± 0.1 and gtrans = 0.65 ± 0.07 gives α = 0.6 ± 0.1. For
Measuring IHe
λ = 8.3 ± 1.3 mm, L ≈ 0.9 m, and kTe = 7.4 ± 1.9 eV and vt0 ≈ 1400 m/s we have
Γ+
He,⊥ (x1 )
= 0.62 ± 0.05,
n0He (x1)vt0
n+
He (x1 )
= 2.1 ± 0.9,
n0He (x1 )
which are used as boundary conditions for a cross-field transport model for helium
ions in the SOL.
6.4.1
Deuterium Source in Local SOL
It is shown that ionization of deuterium in the flux tube can be neglected in the conti+
0
nuity equation. The deuterium ion density satisfies ∇ · Γ+
D = nD ne σvD . Integrating
this over the volume of x ≥ x1 and using the divergence theorem,
−Γ+
D,⊥ (x1 )L
+
Γ+
D, (x1 )λ
= L
∞
x1
dx n0D ne σv+
D.
We obtain an estimate of the neutral deuterium density n0D by equating the deuterium
ion perpendicular flux out of the flux tube with the neutral deuterium perpendicular
flux into the flux tube: Γ0D,⊥ + Γ+
D,⊥ = 0, where the neutral deuterium flux is given by
Γ0D,⊥ = n0D (x1 )vFC , and vFC is the speed of deuterium atoms with the Franck-Condon
energy. Inserting this into the deuterium ion continuity equation gives
+
+
−Γ+
D,⊥ (x1 )L + ΓD, (x1 )λ ≤ L |ΓD,⊥ (x1 )|
219
ne (x1)λn max(σv+
D)
.
vFC
For ionization of atomic deuterium by electron impact we have max(σv+
D) = 3 ×
10−14 m3/s. Taking ne (x1) ≤ 1018 m−3, λ ≈ 8 mm, and vFC ≈ 2 × 104 m/s, we
have [ne (x1)λ max(σv+
D )]/vFC ≈ 0.01. Therefore we safely neglect ionization of
deuterium in the flux tube volume and take ∇ · Γ+
D = 0, giving
Γ+
D,⊥ (x1 )
=
Γ+
D, (x1 )
L
λ .
If we assume the perpendicular flux into the Local SOL is the result of a diffusive
transport mechanism, we can calculate an effective diffusion coefficient:
Γ+
D,⊥ (x1 ) = −D⊥ (x1 )
dn+
D
dx
= D⊥ (x1 )
x1
n+
D (x1 )
.
λn
The parallel flux and the density at the sheath edge are related by
Γ+
D, (x1 )
n+ (x1)
= D
2
kTe + kTD
.
mD
Solving for D⊥ (x1) gives
λ λn
D⊥ (x1) =
2L
kTe + kTD
.
mD
Figure 6.4 shows the exponential decrease of the ion saturation current and the
electron density across the Langmuir probes on the omegatron face. For λn =
6.4 ± 1.6 mm, λ = 8.3 ± 1.3 mm, L ≈ 0.9 m, and kTe = 7.4 ± 1.9 eV we have
D⊥ (x1) = 0.6 ± 0.2 m2 /s.
6.5
Cross-Field 3 He Transport Model
We shall take as the steady state continuity equations
d
vt0
−n0He
dx
4
= −n0He ne σv+
He ,
220
(6.19)
d +
+
++
ΓHe,⊥ (x) = n0He ne σv+
He − nHe ne σvHe ,
dx
d ++
++
(x) = n+
Γ
He ne σvHe ,
dx He,⊥
dnjHe
j
ΓHe,⊥ = −D⊥ (x)
+ V (x)njHe ,
dx
(6.20)
(6.21)
(6.22)
where x is the direction perpendicular to the magnetic field, x = x0 is near the
separatrix, x = x1 is near the edge of the omegatron probe, vt0 =
8kTwall /(πm) is
the thermal speed of neutrals into the plasma, D⊥ (x) is the (anomalous) diffusion
coefficient and V (x) is the convection velocity, positive away from the separatrix.
It is assumed that the cross-field transport of singly- and doubly-ionized helium is
described by a diffusive and convective process for which both ion species have the
same diffusion coefficient and convection velocity. We require D⊥ (x) > 0. Once the
electron density and temperature profiles ne , Te are known, the ionization reaction
++
rates ne σv+
He and ne σvHe are also known.
Equation (6.19) is a first order ordinary differential equation so it requires one
boundary condition. Since Equation (6.19) is homogeneous we can determine n0He
up to a scale factor. Note that Equations (6.19)–(6.21) are linear in the densities
njHe . For convenience we normalize all helium densities profiles by the neutral helium
density at the Local SOL boundary, yj (x) ≡ njHe (x)/n0He (x1), and normalize all fluxes
by the neutral helium flux at the boundary. The transport equations become
dg0
= −y0(x)A0(x),
dx
dg1
= y0(x)A0(x) − y1(x)A1(x),
dx
dg2
= +y1(x)A1(x), ,
dx
−D⊥ (x) dyj V (x)
+
yj (x)
gj =
vt0/4 dx
vt0 /4
(6.23)
(6.24)
(6.25)
(mod j = +, ++),
(6.26)
++
where A0 = (ne σv+
He )/(vt0 /4) and A1 = (ne σvHe )/(vt0 /4) represent the absorption
coefficients to to ionization.
221
Before solving the continuity equations, some general observations can be made.
The continuity equation for the neutral helium has no volume source term, only a sink
term. Therefore the flux of neutrals entering the plasma is monotonically attenuated.
Since the sink term in the continuity equation depends on the neutral density itself,
deep into the plasma the neutral density decreases exponentially. The flux of neutrals
is always towards the separatrix.
The continuity equation for the doubly ionized helium has no volume sink term,
only a source term: the doubly ionized helium is lost only by recombination at the
wall. Since the flux of doubly ionized helium vanishes deep in the plasma and increases
further out, the flux of doubly-charged helium ions is positive (out of the plasma)
everywhere.
Note by summing up Equations (6.23)–(6.25) that the helium mass flux is constant
in space. Deep into the plasma where the ionization rates are high we expect the fluxes
of neutral and singly ionized helium to vanish. To maintain steady state the flux of
doubly ionized helium must also vanish deep in the plasma. Since the fluxes all vanish
deep in the plasma and since the sum of the fluxes is constant, the sum of the fluxes
is zero everywhere.
With the normalizations g0 (x) = −y0(x) and g0 (x) + g1 (x) + g2 (x) = 0. The
0
boundary conditions are y0 (x1) = 1, y1(x0 ) = 0, y1 (x1) = n+
He (x1 )/nHe (x1 ) = 2.1 ±
0
0.9, g1 (x1) = Γ+
He,⊥ (x1 )/(nHe (x1 )vt0 /4) = 0.62 ± 0.05. Note that we mathematically
overconstrain the problem to supply boundary conditions for the singly-ionized helium
density at the center and the edge and a boundary condition for the singly-ionized
helium flux at the edge. We shall see that to find solutions which match all the
boundary conditions we must restrict the possible forms of the diffusion coefficient
and convection velocity.
222
6.5.1
SOL Background Profiles
Figure 6.5 shows profiles of electron temperature, electron density, ionization and
radiative recombination rates across the scrape-off layer, mapped along magnetic flux
surfaces to the midplane. The coordinate ρ corresponds to the distance of a flux
surface from the separatrix measured at the midplane. The electron temperature and
density are obtained from the A-port fast scanning Langmuir probe and the Langmuir
probes on the omegatron heat shield.
Over most of the scrape-off layer plasma the electron temperature is high enough to
neglect radiative recombination in the ion continuity equations. We neglect excitationionization compound reactions of the form He+ (1s) → He+∗ (n ≥ 2) → He++ . Since
the ionization rates are much slower than the de-excitation rate (Einstein coefficient)
of excited singly helium, A ≈ Z 4 6 × 108 s−1 , we expect the density of excited singlyionized helium to be negligible.
6.5.2
Neutral Density Profile
The normalized neutral helium density is found directly:
ne σv+
1 dy0
He
=
,
y0 dx
vt0/4
6.6
y0 (x) = exp −
x1
x
ne σv+
He
dx .
vt0/4
Analytic Slab Model
First we consider a simple slab model with constant temperature, density, and diffusion coefficient. We use the results of this model to to develop physical intuition, to
estimate the magnitude of diffusion coefficient profiles to be used with more realistic
profiles, and to provide analytic results with which to check a numeric code that
solves non-constant profiles.
For convenience we choose a coordinate system with the origin at the omegatron
and increasing towards the core plasma, and take the perpendicular coordinate vari223
Figure 6.5: Profiles of electron temperature, electron density, and rates of ionization and radiative recombination in scrape-off layer. Asterisks represent data points,
smooth line is spline interpolation.
224
able s ≡ A0(x1 − x)/(vt0/4), where Aj = ne σvjHe. Then the neutral, singly-ionized,
and total ionized helium densities are given by
y0(s) = e−s ,
y1 (s) + y2(s) = y1(0) + y2(0) −

2
vt0
(1 − e−s ),
DA0
DA0 A1
−
y1(s) = y1(0) +
2
vt0
A0
k2 ≡
−1 
 e−ks
DA0 A1
−
−
2
vt0
A0
−1
e−s ,
2
A1vt0
.
DA20
The perpendicular flux of singly-ionized helium (positive towards the omegatron)
at the omegatron is related to the density of singly-ionized helium at the omegatron
by
g1 (0) = −

vt0
DA1
y1(0) + 1 +
2
vt0
A0
−1
A1 
D
.
A necessary but not sufficient criterion for the flux of singly-ionized helium to be
++
towards the omegatron is that y1 (0)A1/A0 ≤ 1, or n0 (0)σv+
He ≥ n1 (0)σvHe : the
source of singly ionized helium at the edge must exceed the sink. The magnitude of
the singly ionized flux towards the omegatron has a permitted range 0 ≤ g1 (0) ≤
(1 − y1 (0)A1/A0 )2 . The extremum in the flux is obtained when
1
DA0
=
−
2
vt0
y1 (0)
A1
.
A0
For example, if Te = 10 eV, ne = 1019 m−3 , vt0 = 1400 m/s, and y1 (0) = 2.2 then
A0 = 7.8 × 103 s−1 , A1 = 110 s−1 , and the extremum g1 (0) = 0.68 is obtained when
D = 78 m2 /s. Note that this value of diffusion coefficient is much larger than typical
effective diffusion coefficient in the SOL, D⊥ ≈ 1 m2 /s.
225
6.7
Numerical Model with Experimental Profiles
The profiles of electron temperature and density, obtained by the A-port scanning
Langmuir probe shown in Figure 6.5, are used to obtain profiles of ionization rates.
Equation (6.24) is integrated numerically. The flux of doubly ionized helium is found
from the sum of the neutral and singly ionized helium fluxes, and the profile of doubly
ionized helium density is obtained by integration. As with the analytic slab model,
solutions need not exist which simultaneously match the flux and density conditions
near the separatrix and at the edge.
Electron temperatures and densities are assumed to be constant on poloidal magnetic flux surfaces, which permits data obtained from the scanning Langmuir probe to
be mapped to the omegatron probe. Helium ion flux near the omegatron balances the
neutral helium flux, which ignores the magnetic field. Therefore the temperature and
density gradients in physical space (not magnetic coordinates) near the omegatron are
relevant. In the numeric calculation the physical coordinate is obtained by taking the
distance between magnetic surfaces near the omegatron, not near the midplane as is
commonly done. The flux surfaces expand in the upper divertor near the omegatron,
doubling the distance between flux surfaces compared to the midplane.
Figure 6.6 shows three representative cases of flat and ramped diffusion coefficient
profiles, with zero or non-zero outward convection velocity. The cases were chosen
to have the same helium ion densities at the edge, close to experimentally observed
values (represented by the square symbols in Figures 6.7–6.9.) The first case (constant
diffusion) yields 3 He fluxes at the boundary which do not match the observed values.
Note that while the second and third cases (ramped diffusion coefficient with zero
convection velocity, and flat diffusion coefficent with non-zero convection velocity,
respectively) both match the flux and density and the boundary, their predictions for
the doubly-ionized densities in the core are different by a factor of ten. Thus it might
be possible to distinguish between the two cases if further data of core helium density
is available.
226
Figure 6.6: Comparison of calculated helium fluxes and densities in plasmas with
constant and ramped diffusion coefficient profiles. Solid, dotted, and dashed lines represents neutral, singly-ionized, and doubly-ionized helium, respectively. Arrow heads
indicated experimental data which the model must match. The case of D⊥ = const,
V = 0 yields fluxes which do not match the observed values. Some form of ramped
diffusion coefficient profile is necessary to reproduce experimental observations of
singly-ionized density and flux at the omegatron.
227
Figure 6.7: Calculated fluxes (g1 ) and densities (y1 ) of singly-ionized helium at the
omegatron in plasmas with constant diffusion coefficient profiles. No constant diffusion coefficient profile reproduces both observed flux, g1 (x1) ≈ 0.7 and observed
density, y1(x1 ) ≈ 2.
228
For the experimental temperature and density profiles it is found that no constant
profile of diffusion coefficient satisfies the (overdetermined) boundary conditions at
the edge and separatrix. Figure 6.7 shows the results of the numeric integration. The
initial conditions of the singly-ionized density at x = x0 are adjusted until the flux
at the edge matches a specified flux, g1 (x1) = 0.1, 0.2, or 0.7. The density at the
edge y1(x1 ) is obtained and plotted as a function of the (constant) amplitude of the
diffusion coefficient profile.
At very high values of diffusion coefficient the density of singly-ionized helium
becomes small for all values of flux. The flux of ions leaving the plasma still balances
the flux of neutrals entering the plasma, but most of the helium ions are doublycharged. The high diffusion coefficient permits the singly-ionized helium ions to
penetrate further into plasma regions of high ionization rate.
At lower values of diffusion coefficient physical solutions do exist, but only for
smaller flux, say g1 (x1) ≈ 0.1, when most of the flux of ions is from doubly-charged
helium. If we artificially impose a higher value of singly-ionized helium flux at the edge
with lower values of diffusion coefficient, say g1 (x1) ≈ 0.7, non-physical values of the
density result, y1 (x1) < 0. Note from Figure 6.7 that there is no value of the diffusion
coefficient amplitude which gives results within the uncertainties of experimentally
observed values of the flux g1 (x1 ) ≈ 0.7 and (inferred) values of the density y1(x1 ) ≈ 2.
If instead the diffusion coefficient profile is allowed to increase with increasing
x then solutions matching the overdetermined inner and outer boundary conditions
can be obtained. Note that these solutions are not necessarily unique. Figure 6.8
shows the density y1(x1) for cases with g1 (x1) = 0.7 and diffusion coefficient profiles
that increase from D⊥ (x) = 0.1 m2 /s at some location (“foot”) to a higher value over
∆x = 15 mm. The location of the “foot” and the maximum value of D⊥ are varied.
The results of Figure 6.8 can be interpreted as follows: in order for the observed
flux of singly-ionized helium to reach the omegatron, a region of low transport is
required at locations where the rate of ionization to doubly-charged helium is suffi229
D
15 mm
Dmin
x
x foot
Figure 6.8: Calculated density of singly-ionized helium at the omegatron for different
ramped profiles of diffusion coefficient. Many different profiles can reproduce the
observed values of density and flux, but all of them require an increase in diffusion
coefficient across the scrape-off layer.
230
ciently high. If instead a region of high transport exists at those locations, then singly
ionized helium can diffuse there and be ionized, reducing the flux of singly-ionized
helium observed at the omegatron.
It is also possible to match the observed values of density and flux at the boundary
if the transport is due to a mix of outward convection and diffusion. The convection
velocity profile is assumed to be constant over must of the region outside the separatrix; inside the separatrix the the convection velocity is assumed to be zero, and there
is a continuous transition between the two regions. The diffusion coefficient profile
is constant. Figure 6.9 shows the normalized values of singly-ionized helium density
at the edge that result from different magnitudes of convection velocity and diffusion
coefficient. Once again, note that the profiles are not unique.
6.8
6.8.1
Discussion
Neglect of Recombination
With the profiles of helium ion density shown in Figure 6.6 we now justify the neglect
of radiative recombination. The continuity equations for the normalized fluxes and
densities, including recombination, are written,
dg0
= −y0A0 + y1 R1 ,
dx
dg1
= y0A0 − y1 A1 + y2 R2 − y1 R1 ,
dx
dg2
= +y1A1 − y2 R2 ,
dx
where Aj = ne σvi,j /vt and Rj = ne σvr,j /vt represent the ionization and recombination rates. Integration of the left hand sides from the center to the edge gives
the fluxes at the edge. Since the right hand sides are now known we can compare
the fluxes at the edge including and neglecting recombination. In order to neglect
231
V
10 mm
Vc
x
x=0
Figure 6.9: Calculated density of singly-ionized helium at the omegatron for outward
convection velocities with as a function of the amplitude of the flat diffusion coefficient
profile. Many flat profiles can reproduce the observed values of density and flux, but
all of them require an outward convection velocity.
232
recombination it is sufficient to show that
x1
x0
y1R1 dx x1
x0
x1
y0A0 dx
and
x0
y2R2 dx x1
x0
y1A1dx.
By numeric integration of the ionization and recombination rates A0, . . . , R2 with the
density profiles y0 , y1 obtained numerically, it can be shown that the above inequalities
are satisfied by a margin of 103 . Therefore we are justified to neglect recombination
in the original analysis.
Inclusion of recombination in the helium analysis would increase the helium neutral velocity above the wall temperature thermal speed. Therefore helium neutrals
would penetrate further into the plasma before being ionized, and the outward transport would have to increase even further to produce the observed ratios of doublyand singly-ionized helium. Therefore the transport estimates neglecting recombination represent a lower bound.
6.8.2
Anomalous Cross-Field Transport
The results obtained with the 3 He transport are consistent with the picture of rapid,
radially outward transport proposed by Umansky. The observed values of helium density and flux at the edge can be reproduced in the model only by including an effective
diffusion coefficient that increases further from the separatrix. Note that the Bohm
Bohm
= 0.06Te [eV]/B[T] ≈
diffusion coefficient for these SOL plasmas is of order D⊥
0.1 m2 /s. This value, which is often taken to be approximately the maximum value
for transport due to microturbulence, is orders of magnitude smaller than the D⊥
values obtained for the V = 0 case, see Figure 6.8. This suggests that the outward
convection model, Figure 6.9 is more likely to simulate the actual transport mechanism. Note that convection velocities on the order of 100 m/s imply poloidal electric
fields of ≈ 4 V/cm, a reasonable value on open field lines that have 10 eV temperature
variations in 1 cm, Figure 6.5.
233
This physical conclusion should be an important constraint for understanding the
mechanisms of impurity transport in the edge plasma. That the result could be
obtained from such a simple measurement demonstrates the utility of the omegatron
probe.
234
Chapter 7
Summary
This chapter summarizes the results from the retarding field energy analyzer and
ion mass spectometer portions of the omegatron, as well as the impurity transport
experiments in the scrape-off layer plasma. Suggestions are given for improvements
to the omegatron hardware and electronics and further applications of the omegatron
are mentioned.
7.1
7.1.1
Results
Hardware
An omegatron ion mass spectrometer was operated on a tokamak for the first time,
extending ion mass spectrometry to high-field regimes with B > 4 T. A robust design
of the vacuum components was found. Critical issues of diagnostic alignment and
noise signal reduction were identified and resolved.
Ion resonances are resolved observed for for M/Z < 12 at signal levels as low as
5 × 10−4 times the bulk plasma species. Well-resolved resonances have widths within
a factor of two of theoretical values obtained from single particle orbit theory. A
new technique to measure the temperature of individual ion impurities was found
by varying the applied power to the omegatron ion mass and analyzing the resuling
235
resonant current.
7.1.2
Retarding Field Energy Analyzer
The flux tube bounded on one side by the face of the omegatron heatshield is almost
always bounded on the other side by the E-Port ICRF antenna. The connection
length of the flux tube is approximately the same as the plasma perturbation length,
which means that the plasma potential of the flux tube is essentially determined by
the boundary potentials. This fact is important for the interpretation of the portion
of the IV characteristic that gives the sheath potential, especially the variation with
slit bias.
Experiments were performed to determine how the slit and grids modify the distribution function of ions that pass through. Results indicate that the probability of
transmission through the grids is the same for the large majority of ions, and that
the probability depends only on the area of the holes of the grid. In this case the
grid modifies the ion distribution by a scalar attenuation factor slightly less than the
grid optical transparency. The same results apply for the slit, provided the ions are
accelerated through the slit by several volts. This is convenient for calculations since
it means the shape of the ion distribution function outside the analyzer is preserved as
ions pass through the slit and grids, and therefore the retarding field energy analyzer
can be used to determine the bulk ion temperature outside the analyzer.
The ratio of currents collected on the grids indicates that space charge is present in
the analyzer during routine operation and can significantly modify the electrostatic
potential structure. Therefore a complete analysis of the IV characteristics of the
retarding field energy analyzer must accomodate the effects of space charge. A simple,
three-dimensional calculation of the electrostatic potential from a ribbon of uniform
space charge is used to estimate the current limit below which space charge can be
ignored. Using the ratios of currents to the grids and end collector, and assuming
the forward portion of the ion distribution in the retarding field energy analyzer is
236
a shifted half-Maxwellian, the maximum electrostatic potential between each grid is
calculated. This is useful for determining the portion of the distributions of trace
impurities which reach the RF cavity.
The repeller bias at which the knee of the IV characteristic occurs suggests that
significant secondary electron emission occurs from the slit. Supporting evidence of
secondary electron emission from the slit is provided by the increase in ion transmission through the slit when the slit bias is below the floating potential; secondary
electron emission depresses the sheath potential, which reduces the shift of the ion
distribution through the slit, so fewer ions meet the pitch angle transmission criterion.
Since secondary electron emission depresses the sheath potential, the results have important implications for ion sputtering from surfaces which meet the magnetic field
at near-normal angles. In a related experiment, secondary electron emission from the
end collector was measured with coefficient of order unity.
Across the far scrape-off layer plasma, in ohmic L-mode discharges, the bulk ion
population near the omegatron is observed to have two temperatures: the majority
is cold at approximately three electron volts, the minority hotter at approximately
twenty electron volts. During ICRF auxiliary heating the entire ion population becomes hot. No data were obtained during H-mode: the ion density decreases at the
omegatron location below the noise floor. The bulk cold ion population supports the
picture of main-chamber recycling proposed by Umansky et al: deuterium is ionized
near the wall, so the minimum energy the ions have is the dissociation energy of the
deuterium molecule, the Franck-Condon energy.
The ion temperatures recorded by the omegatron retarding field energy analyzer,
and the electron densities recorded by the Langmuir probes in the omegatron heatshield, indicate that the presheath is collisional during much of the operation range
of the omegatron. Therefore the trace impurity ions are expected to acquire the flow
velocity of the bulk plasma at the sheath edge. Since the omegatron ion mass spectrometer measures impurity flux, it is necessary to know the relationship between the
237
impurity ion density and flow velocity at the sheath edge to calculate the impurity
ion density.
7.1.3
Ion Mass Spectrometer
In deuterium plasmas the ion resonance with mass to charge ratio M/Z = 2 always
dominates, and M/Z = 4 is often observed, probably corresponding to singly ionized
molecular deuterium. Ion mass spectra have been obtained with good resolution
for 1 ≤ M/Z < 12, down to signal levels 5 × 10−4 the non-resonant current. The
resonance spectra of intrinsic impurities are dominated by charged states of boron
and carbon at concentrations less than two percent. The impurity spectrum does not
change dramatically before and after boronization, except that oxygen appears to be
reduced.
Resonances corresponding to charged states of hydrogen, 3He, helium-four, and
nitrogen have been observed following gas puffs of those isotopes.
A simple kinetic model has been developed for the resonant current collected in
the omegatron as a function of the resonant ion temperature as well as the physical
dimensions and parameters of the hardware. The model correctly predicts the range
of frequencies for which ions species are collected within a factor of two, for resolved
and unique resonances.
The simple kinetic model also predicts the variation of the resonant current amplitude with applied RF power. From the observed variation of resonant current with
power, and accounting for resonant ions reflected on space charge in the RF cavity, it
is possible to determine the temperature of individual ion impurity species. For the
data obtained it is found that the impurity ions acquire the bulk ion temperature.
7.1.4
3
3
He Transport in the Scrape-Off Layer
He gas was puffed from the wall into ohmic L-mode discharges discharges and the
charged states were measured with the omegatron ion mass spectrometer. It is found
238
that in the concentrations of singly- and doubly-ionized helium at the edge are approximately equal. The electron temperature and density at the omegatron are too
low to account for ionization of helium in the local flux tube, therefore the helium
must have been ionized in a hotter region of the edge plasma and been transported
to the omegatron.
The measurements by the omegatron of the charge states gives information about
the transport in the edge plasma. If ion transport is slow, then ion dwell time in
the hot plasma is sufficiently long to be ionized to the second charge state, and the
omegatron should see mostly doubly ionized helium. Conversesly, if ion transport is
rapid, ions do not have sufficient time to be ionized again and the omegatron should
see mostly singly-ionized helium.
A simple one-dimensional radial transport model reproduces the observed values of
charge state flux and density only if rapid outward transport is included, increasing
with distance from the separatrix, with diffusion coefficients of order 2 m2/s and
outward convection velocity of order 100 m/s. These conclusions are similar to those
of Umansky, but are based on completely independent measurements.
7.2
Future Work
Much of the work of this thesis is devoted to commissioning of the omegatron on
Alcator C-Mod, and establishing a theoretical framework within which to interpret
the measurements. Several possible improvements to the omegatron hardware and
electronics are suggested. But even in its present form the omegatron has impressive
potential as a physics tool, and several experiments are proposed.
7.2.1
Diagnostic Improvements
• Isolate the slit from the shield box, and electrically connect the slit to grid G1.
Apply the same DC bias to the shield box as the RF plates. This breaks the
239
capacitive coupling between the RF plates and the shield box and potentially
reduces the noise on the RF plates by three orders of magnitude. This requires
in-vaccum hardware modifications to the shield box.
• If noise at 10 kHz has dropped, implement the synchronous detection electronics.
This should result in further improvement in signal-to-noise by a factor of ten.
• Improve residual gas analyzer operation of the omegatron by improving communication of neutrals inside and outside the probe.
• Improve the measurement of applied RF power that is actually delivered to
the omegatron head. This is important for determining the temperature of
impurities. It might be possible to use a directional coupler at high power, as
the ICRF system does, to monitor forward and reflected power. This would
required only air-side electronics modifications.
• Find replacements for teflon SMA connectors and teflon wire insulation. Find
RF resistors that maintain their power rating at higher temperatures. Then the
omegatron head can operated at higher temperatures, which means the probe
can be inserted closer to the separatrix.
• Cool the heat shield with a liquid coolant, perhaps closed loop with radiator
and fan. This should dramatically decrease the cooling time between shots,
which with the relaxation of the absolute temperature restrictions above should
permit operation closer to the separatrix. This would also reduce the burden
on the cell compressed air system.
7.2.2
Physics Experiments
• Determine the dependence of the hot portion of IV characteristic on the magnetic field magnitude and the plasma current.
240
• Characterize the surface of the slit, end collector, and determine the precise
source of the significant emission of secondary electrons.
• Further calibrate of the omegatron, as with the H/D scan, but with reduced
scatter in the data and with other isotopes.
• With the required hardware modifications, run the omegatron at current levels
where spacecharge in the RF cavity is small, so that the assumptions of collimated beam and negligible reflections are guaranteed. Then the uncertainty of
the collection efficiency of reflection ions is removed and the temperatures and
fluxes can be measured with higher precision.
• Perform a systematic exploration of the resonances, and attempt to quantify
the components of degenerate resonances. Look for trace impurities known or
believed to be present: argon, molybdenum, fluorine.
• Perform further transport studies with non-recycling impurities, for example
with nitrogen.
• Reperform the impurity transport experiments in H-Mode plasmas, once the
signal-to-noise has been improved. Compare the results of L-Mode, H-Mode,
enhanced Dα H-Mode.
• Investigate further the resonance spectrum obtained during omegatron residual
gas analyzer operation.
241
An appropriate quote concludes this thesis.
“No one devotes years to, say, the development of a better spectrometer
or the production of an improved solution to the problem of vibrating
strings simply because of the importance of the information that will be
obtained. . . Though its outcome can be anticipated, often in detail so great
that what remains to be known is itself uninteresting, the way to achieve
that outcome remains very much in doubt. Bringing a normal research
problem to a conclusion is achieving the anticipated in a new way, and it
requires the solution of all sorts of complex instrumental, conceptual, and
mathematical puzzles. The man who succeeds proves himself an expert
puzzle-solver, and the challenge of the puzzle is an important part of
what usually drives him on.” T.S. Kuhn [33, p.36], on Normal Science
as Puzzle-solving
242
Appendix A
Calculations
A.1
A.1.1
Kinetic Fundamentals
Single Particle
Consider a particle of charge q and mass m in a magnetic field of magnitude B(x)
and in an electrostatic potential field φ(x). Let
1 2
1
+ qφ(x)
H = mv2 + mv⊥
2
2
represent the total energy of the particle, and let
µB =
2
1 mv⊥
2 B(x)
represent the diamagnetic moment of the particle. In many situations both the total
energy and diagmagnetic moment of the particle are constant. Further, if B(x) is
approximately constant, then the parallel kinetic energy of the particle depends only
on the electrostatic potential at the location of the particle:
1 2
mv + qφ(x) = H − BµB ≡ µ(x, v) = constant.
2 243
Let x represent the position of the guiding center of a particle along a magnetic field
line, and w represent the velocity of the guiding center (parallel to the magnetic field).
Then we refer to µ(x, w) = mw2/2 + qφ(x) as the “parallel energy.”
A particle at x with velocity between w and w + dw has parallel energy between
µ(x, w) and µ(x, w + dw) = µ(x, w) + dµ(x, w), where dµ(x, w) = mw dw to first order
in dw.
Consider a situation for which the parallel energy of a particle is constant as it
moves. If at position x1 a particle has velocity w1 , then at position x2 the particle
will have velocity w2 , where x1 , w1, x2 , w2 satisfy the relation µ(x1 , w1) = µ(x2 , w2).
Similarly, if at x1 a particle has velocity w1 + dw1 , at then position x2 it must have
velocity w2 + dw2 such that µ(x1 , w1 + dw1) = µ(x2, w2 + dw2 ). Expanding both sides
gives dµ(x1 , w1) = dµ(x2 , w2), or to first order mw1dw1 = mw2dw2 .
A.1.2
One-particle Distribution
Consider a one-particle velocity distribution, such that f(x, w) dw dx represents the
number of particles in the one-dimensional volume dx at x with velocities between w
and w + dw. Then f(x, w)dw represents the number density of particles at x with
velocities between w and w + dw. The number of particles per unit time arriving at
x with velocities between w and w + dw is given by the particle flux f(x, w)w dw.
The particles at x with velocities between w and w + dw have parallel energy
between µ(x, w) and µ(x, w) + dµ(x, w). Therefore f(x, w)w dw also represents the
number of particles per unit time arriving at x with parallel energy between µ(x, w)
and µ(x, w) + dµ(x, w).
If the number of particles per unit time that arrive at x1 with parallel energy
between µ(x1 , w1 ) and µ(x1 , w1) + dµ(x1 , w1) is the same as the number of particles
per unit time that arrive at x2 with parallel energy between µ(x2, w2 ) and µ(x2 , w2) +
dµ(x2 , w2 ), then
f(x1 , w1)w1dw1 = f(x2 , w2)w2 dw2 .
244
In addition if the parallel energy of each particle is constant between x1 and x2 then
f(x1, w1 ) = f(x2 , w2 ), with µ(x1 , w1) = µ(x2, w2 ).
(A.1)
The most general form of the distribution function which satisfies Equation (A.1) is
f(x, w) = f(µ(x, w)), an arbitrary function of the parallel energy (constant of the motion). If we know the distribution function at one location f(µ(x1 , w1 )), then Equation
(A.1) allows us to map it to another location x2 by f(µ(x2 , w2)) = f(µ(x1 , w1)).
Aside: If such a constant of the motion µ(x, w) exists, then any function of the
constant of the motion f(µ(x, w)) solves the steady Vlasov equation,
w
∂f
∂f
+a
= 0,
∂x
∂w
a=−
q dφ
.
m dx
However, if the number of particles per unit time that arrive at x1 with parallel
energy between µ(x1 , w1) and µ(x1 , w1) + dµ(x1 , w1 ) is not the same as the number of
particles per unit time that arrive at x2 with parallel energy between µ(x2 , w2) and
µ(x2 , w2) + dµ(x2 , w2), then f(x1 , w1)w1 dw1 = f(x2 , w2 )w2dw2 . Even if the parallel
energy of a particle travelling from x1 to x2 is constant such that w1dw1 = w2 dw2,
we still have f(x1 , w1 ) = f(x2 , w2 ). Even if at one location the distribution function
can be written as a function of the parallel energy f(x1 , w1) = f(µ(x1 , w1)), it does
not follow that we can map the distribution function to another location as before.
Recall that f(x, w) = f(µ(x, w)) obtains from a steady state, source-free, collisionless collection of particles. But it is also true that a collection of particles in a
stable equilibrium with temperature T has a one-particle distribution of the above
form, specifically, f(x, w) = C exp(−µ(x, w)/kT ), even if particles do suffer collisions.
Collisions imply µ(x1, w1 ) = µ(x2 , w2 ). Particle sources imply f(x1, w1 )w1dw1 =
f(x2 , w2)w2 dw2 .
The presence of either collisions or particle sources breaks the
mapping of the distribution function from one location x1 to another location x2:
f(µ(x1 , w1 )) = f(µ(x2 , w2 )).
245
A.1.3
Moments of the Distribution
The total number density of particles at x and the total forward particle flux at x are
found by integrating over the relevant velocities:
n(x) =
∞
−∞
dwf(x, w),
Γ(x) =
∞
f(x, w)w dw.
0
For distribution functions which depend only on the particle energy, integrals to determine the particle flux may be evaluated easily by changing the variable of integration:
µmax
f(µ(x, w))
Γ(x) =
µmin
1
dµ.
m
If we use a half-Maxwellian distribution we can find the normalization constant by
evaluating the flux at a location where the potential and the minimum ion velocity
are known. Consider the sheath edge, where the minimum parallel energy of the ions
is µs = mvs2/2 + qφs :
Γs =
A.2
C
m
∞
e−µ/kT dµ =
µs
CkT −µs /kT
Γs m µs /kT
→C=
.
e
e
m
kT
Proof of Generalized Bohm Criterion
Here we reproduce the proof of the Generalized Bohm Criterion as presented by
Harrison and Thompson [19]. Let the ion distribution be given by f(x, v), and the
electrons be described by a Boltzmann relation, ne = n0 exp(eφ/kTe ). Then the
Poisson equation for a one-dimensional configuration is written
∞
d2 φ
e
− 2 =
Z
f(x, v)dv − n0 exp(eφ/kTe) .
dx
@0
−∞
Let the sheath edge x1 be defined as the last position where quasineutrality holds, and
let x2 be just inside the sheath. If there are no sources inside the sheath then the flux
246
of ions within a given energy range is constant: f(x1 , v1)v1dv1 = f(x2, v2)v2 dv2, where
x1, v1 , x2, v2 are related through µ(x1, v1 ) = µ(x2 , v2) and µ(x, v) = mv 2/2 + qφ(x) is
the parallel energy.
Evaluate the Poisson equation just inside the sheath, and expand the potential in
the ion distribution and the electron density near the value at x1, φ(x2) = φ(x1) +
∆φ + . . . :
f(x2 , v2)dv2 = f(x1 , v1)
v1
1/2
+ 2q(φ(x1) − φ(x2 ))/m]
q
= f(x1 , v1) 1 +
∆φ + . . . dv1,
mv12
e(φ(x1) + ∆φ + . . .)
,
ne (x2) = n0 exp
kTe
e∆φ
eφ(x1)
1+
+ ... .
= n0 exp
kTe
kTe
[v12
dv1 ,
Insert these into the Poisson equation to get, to first order,
e2
Z
d2
− 2 ∆φ = ne (x1)
dx
@0
mne (x1)
∞
−∞
1
f(x1, v1 )
dv1 −
∆φ,
2
v1
kTe
where we have used quasi-neutrality at the sheath edge. In order that the solution
not oscillate we require (Z/m)v1−2 ≤ 1/kTe , the generalized Bohm sheath criterion.
Using the Schwarz inequality, A2 B 2 ≥ AB2, we can show that v12v1−2 ≥ 1,
in which case we obtain : v12 ≥ ZkTe /m.
A.3
Hobbs and Wesson Fluid Sheath Model with
Secondary Electron Emission
Hobbs and Wesson [22] treat the influence of secondary electron emission on the
floating potential of a surface, using a cold fluid model of the plasma sheath. Here
the fluid model is generalized slightly to allow for net current to flow to the surface.
247
Consider a one-dimensional coordinate system normal to a wall immersed in a
plasma. Let x = 0 denote the location of the wall and let x = xs denote the sheath
edge. We consider the region 0 ≤ x ≤ xs . At the sheath edge and beyond quasineutrality is satisfied:
Znis = ne1 (xs ) + ne2 (xs ),
(A.2)
where nis ≡ ni (xs ) represents the ion density at the sheath edge, ne1 represents the
density of primary electrons, and ne2 represents the density of secondary electrons.
We assume the wall is biased to collect ions and that ions arrive at the sheath edge
2
with energy Ei = (mi/2)vis
. We neglect ion sources in the sheath (ionization, recom-
bination), so the one-dimensional continuity equation for ions gives
ni (x)vi(x) = nis
2Ei
.
mi
(A.3)
Ions are assumed to travel through the sheath without collisions, thus the total energy
for each ion is conserved:
mi
vi (x)2 + eZφ(x) = Ei + eZφs,
2
(A.4)
where φs represents the electrostatic potential at the sheath edge. If the wall potential
φ0 is sufficiently negative with respect to the sheath potential φs then the plasma
electron distribution function is approximately Maxwellian and the density is given
by the Boltzmann relation,
e(φ(x) − φs )
ne1 (x) = ne1 (xs ) exp
,
kTe
(A.5)
where kTe represents the electron temperature. Secondary electrons are assumed to
be released from the wall with an energy E2 = (me /2)ve2 (0)2 . The secondary electrons
are assumed to be accelerated through the sheath without collisions, thus the total
248
energy for each secondary electron is conserved:
me
ve2 (x)2 − eφ(x) = E2 − eφ0.
2
(A.6)
Since there are no sources or sinks of secondary electrons in the sheath the flux of
secondary electrons in the sheath is constant. The flux of secondary electrons released
from the wall is proportional to the flux of primary electrons arriving at the wall, with
proportionality constant γ. Combining these two relations,
1
e(φ0 − φs )
ne2 (x)ve2 (x) = γ ne1 (xs ) exp
4
kTe
8kTe
,
πme
(A.7)
where the flux of primary electrons is given by the random thermal flux of a Maxwellian
distribution. Inside the sheath the electrostatic potential is determined from the Poisson equation,
−
e
d2 φ
= (Zni (x) − ne1 (x) − ne2 (x)) ,
2
dx
@0
(A.8)
which has the boundary conditions φ(xs ) = φs and dφ/dx = 0 at x = xs .
We will combine the above equations and substitute into the Poisson equation
forms for ion and electron densities that depend only on constants and φ(x). We
will then multiply both sides by an integrating factor (dφ/dx) and integrate to get
an equation for (dφ/dx)2 . We will require this quantity to be positive to obtain
monotonic solutions for φ(x); imposing this condition near x = xs will restrict the
possible values of Ei and constitute the Bohm sheath criterion. We will then find the
form for the net current arriving at the wall as a function of the wall potential φ0,
the secondary electron emission coefficient γ, the electron temperature kTe and the
energy of released secondary electrons E2 .
Solve the ion energy equation Equation (A.4) for vi(x) and insert into ion continuity equation Equation (A.3) to get
ni (x) = nis [1 + eZ(φs − φ(x))/Ei ]−1/2
249
Solve secondary electron energy equation Equation (A.6) for ve2(x) and insert into
the secondary electron continuity equation Equation (A.7) to get
γ
e(φ0 − φs )
ne2 (x) = ne1 (xs ) exp
2
kTe
E2 + e(φ(x) − φ0 )
π
kTe
−1/2
.
Combine the above equation and the primary electron density equation Equation
(A.5) and factor out the primary electron density at the sheath edge.
e(φ(x) − φs )
ne2 (x) + ne1 (x)
+
= exp
ne1 (xs )
kTe
γ
e(φ0 − φs )
exp
2
kTe
E2 + e(φ(x) − φ0 )
π
kTe
−1/2
.
Apply quasineutrality at the sheath edge Equation (A.2) to solve for ne1 (xs ) in terms
of nis :

γ
e(φ0 − φs )
ne1 (xs ) 
= 1 + exp
Znis
2
kTe
E2 + e(φs − φ0)
π
kTe
−1/2 −1
 .
(A.9)
Note that ne1 (xs )/(Znis ) does not depend on Ei . Insert these forms for the ion and
electron densities into the Poisson equation Equation (A.8):
−@0 d2 φ
= [1 + eZ(φs − φ(x))/Ei ]−1/2 −
eZnis dx2

γ
e(φ(x) − φs )
e(φ0 − φs )
ne1 (xs ) 
exp
+ exp
Znis
kTe
2
kTe
(A.10)
E2 + e(φ(x) − φ0)
π
kTe
−1/2 
.
Multiply the form of the Poisson equation above by dφ/dx and integrate from x = xs
to x = x, using the boundary conditions at x = xs to get
2
2Ei @0
1 dφ
1/2
=
[1 + eZ(φs − φ(x))/Ei ] − 1 +
Znis kTe 2 dx
ZkTe
nei (xs )
e(φ(x) − φs )
exp
− 1+
+
Znis
kTe
250

1/2
γ
e(φ0 − φs )  E2 + e(φ(x) − φ0 )
exp
2
kTe
kTe /π
E2 + e(φs − φ0)
−
kTe /π
1/2 

 .

The net current density to the wall J is found by
ne1 (xs )
e(φ0 − φs )
=1−
(1 − γ) exp
Znis
kTe
eZnis 2Ei /mi
J
mi kTe
.
2πme 2Ei
(A.11)
Bohm Sheath Criterion
In order to have non-oscillatory solutions to the Poisson equation we require
d d2 φ
dφ dx2
≥ 0,
φs
the Bohm sheath criterion. Evaluate the Bohm sheath criterion for the form of the
Poisson equation in Equation (A.10) to get

ZkTe
e(φ0 − φs )
ne1 (xs ) 
πγ
exp
≤
1−
2Ei
Znis
4
kTe
E2 + e(φs − φ0 )
π
kTe
−3/2 
.
(A.12)
Note that the right hand side above does not depend on Ei , which makes it easy to
substitute for Ei in the Poisson equation and in the formula for the net current to
the surface.
Ions arrive at the sheath edge with energy Ei , which they are assumed to have acquired by dropping to the sheath potential from the plasma potential in the presheath:
Ei = eZ(φp − φs ).
Comparison with Hobbs and Wesson
Hobbs and Wesson consider J = 0, which gives
1
e(φs − φ0 )
ne1 (xs )
=
exp
Znis
1−γ
kTe
251
2πme 2Ei
.
mi kTe
Equating the two forms for ne1 (xs )/(Znis ) gives
1
1−γ
2πme 2Ei
e(φs − φ0 )
exp
mi kTe
kTe
γ
=1−
1−γ
me
Ei
.
mi E2 + e(φs − φ0)
Inserting this into the equations for the secondary and primary electron densities
gives
γ
ne2 (x)
=
Znis
1−γ
ne1 (x)
=
Znis
me
Ei
,
mi E2 + e(φ(x) − φ0 )
γ
1−
1−γ
me
Ei
mi E2 + e(φs − φ0 )
e(φ(x) − φs )
exp
,
kTe
which are the same as the formulae for Hobbs and Wesson when Z = 1, φs = 0, and
E2 e(φ(x) − φ0 ). The singularities at γ = 1 are not of concern since the fluid
approximation breaks down before γ = 1.
Note for γ = 0 the Bohm sheath criterion reduces to Ei ≥ ZkTe /2. Using Ei ≈
ZkTe /2 and assuming E2 e(φs − φ0 ) we can find the approximate form for the
floating potential φf = φ0 when J = 0:
e(φs − φf )
exp
kTe
&
'
'm
1 − γ 1 + ( e

≈

 ZkTe /2 
mi e(φs − φf )
mi
.
2πme Z
If we use me /mi 1 then as Hobbs and Wesson find,
1 (1 − γ)2 mi
e(φs − φf )
≈ ln
.
kTe
2
Z
2πme
A.4
Electrostatic Potential due to a Block of Charge
Consider a domain of volume V bounded by grounded surfaces at x = ±a, y = ±b,
and z = ±c. Inside the volume there is space charge within the volume bounded by
surfaces at x = ±a, y = ±b and z = ±c; the charge density is zero elsewhere. We
want to find the potential φ at any point within the volume V . In particular we are
252
interested in the potential along x for y = z = 0. The geometry is shown in Figure
A.1.
The potential solves the Poisson equation, ∇2φ = −ρ/@0 with the homogeneous
Dirichlet boundary conditions on the volume surfaces. The problem has symmetry
about the origin so we need only consider the problem over one eighth of the volume:
the boundary of the domain we will consider is defined by the planes at x = 0, y = 0,
z = 0, x = a, y = b, and z = c.



for 0 ≤ x ≤ a, 0 ≤ y ≤ b , 0 ≤ z ≤ c,
−ρ/@0
∂ φ ∂ φ ∂ φ
+
+
=
∂x2 ∂y 2 ∂z 2 
 0
2
2
2
φ(x = a, y, z) = 0,
∂φ
∂x
elsewhere,
φ(x, y = b, z) = 0,
= 0,
x=0
∂φ
∂y
= 0,
y=0
φ(x, y, z = c) = 0,
∂φ
∂z
= 0.
z=0
We will solve the above second order linear partial differential and associated equation
boundary value problem using Green functions. See Jackson[28, p.121], for example.
We want to find a function G(x, x) that satisfies the Poisson equation for a pointcharge source, ∇2G(x, x) = −δ(x − x ). Once we have G(x, x) then we find a
particular solution to the full Poisson equation by
φ(x) =
V
ρ(x)
G(x, x)d3 x +
@0
)
S
∂φ
∂
G(x, x ) − φ(x) G(x, x) dA,
∂n
∂n
where the domain of volume V is bounded by surface A. If the Green function
G(x, x ) is chosen so that it satisfies the same boundary conditions as the potential
φ(x) then the surface integral above vanishes.
We will find a Green function that satisfies the given boundary conditions by an expansion in a complete set of eigenfunctions which themselves satisfy the given boundary conditions. We find the complete set of eigenfunctions by solving the Helmholtz
2
ψlmn = 0. The normalized eigenfunctions that
equation in the volume: ∇2ψlmn + klmn
253
y
b
b'
-a,-a'
a,a'
x
-b'
-b
z
c
c'
-a,-a'
a,a'
x
-c'
-c
phi(x,y=0,z=0)
-a
a
x
Figure A.1: Sketch of the distribution of space charge between surfaces at x = ±a,
y = ±b, and z = ±c. Space charge is uniform inside rectangle of height ∆z = 2c ,
width ∆y = 2b and length ∆x = 2a = 2a, and zero elsewhere.
254
solve the Helmholtz equation and satisfy the given boundary conditions are
cos kl x cos km y cos kn z
,
ψklm (x, y, z) = a/2
b/2
c/2
2
2
= kl2 + km
+ kn2 .
where kl = (l + 12 )π/a, km = (m + 12 )π/b, kn = (n + 12 )π/c, and klmn
We will look for a Green function of the form
G(x, x ) =
∞
Clmn (x, y , z )ψlmn (x, y, z).
l,m,n=0
Substitute this form into the Poisson equation for the Green function (point source)
to find
∞
2
Clmn (x , y , z )klmn
ψlmn (x, y, z) = δ(x − x)δ(y − y )δ(z − z ).
l,m,n=0
Multiply both sides by an eigenfunction ψl m n (x, y, z) and integrate over the whole
domain. Since the eigenfunctions are orthonormal all terms in the sum vanish except
2
l = l, m = m, n = n, giving Clmn (x , y , z )klmn
= ψlmn (x , y , z ). Insert the
coefficients Clmn (x , y , z ) into the Green function to get
G(x, x ) =
∞
ψlmn (x, y, z)ψlmn(x, y , z )
,
2
klmn
l,m,n=0
which exhibits the symmetry property we expect of the Green function, G(x, x) =
G(x , x).
Integrating the Green function over the volume of non-zero charge density gives
the potential:
φ(x, y, z)
=
ρ/@0
∞
8 cos kl x cos km y cos kn z
2
abc klmn
l,m,n=0
a
0
cos kl x dx
b
0
cos km y dy 255
c
0
cos kn z dz ,
=
∞
8 cos kl x cos km y cos kn z sin kl a sin km b sin kn c
.
2
abc klmn
kl
km
kn
l,m,n=0
We are particularly interested in the potential along y = z = 0 and 0 ≤ x ≤ a. Then
for a = a we have
∞
φ(x, 0, 0)
8 cos kl x (−1)l sin km b sin kn c
=
.
2
ρ/@0
klmn
kl a km b
kn c
l,m,n=0
It can be shown that for c = c, b = b, and in the limit that c/a → ∞ and
b/a → ∞ this solution reduces to the one dimensional parabolic solution, φ(x, y =
0, z = 0) = (a2ρ/(2@0 ))(1 − (x/a)2 ).
Consider the volume in between two omegatron grids. Define the coordinate
system such that x axis is along the magnetic field. Choose the origin so that grids
are located at x = ±a, where 2a = 0.7 mm. In the retarding field energy analyzer
portion of the shield box the side walls are at b = ±7.5 mm and the shield box top and
bottom are at c = ±5 mm. Ions pass between the grids in a ribbon shaped beam with
cross sectional dimensions 2b = 7 mm and 2c = 0.03 mm. Since the current passes
between the grids we have a = a. The charge density is found from the ion current
I and velocity v by ρ = I/(4vbc ). We take as a simplifying assumption v = const.,
which gives a uniform charge density.
Figure A.2 shows the potential half-profile between the grids for I = 10 µA and v =
20 m/s. For this current the space charge contribution to the electrostatic potential is
negligible, approximately one volt. Compare this result with the potential we would
have expected from a slab approximation, φ0 = a2ρ/(2@0 ) ≈ 16 V.
Now consider the volume in between the omegatron RF plates. Choose the origin
so that Grid 3 and the end collector are located at x = ±a, where 2a = 40 mm.
The side walls are at b = ±15 mm and the RF plates are at c = ±2.5 mm. The
ion beam has the same cross sectional area except now it is longer, a = a. Figure
A.2 shows the potential half-profile in the RF cavity for the same ion current that
256
passed through the grids. For this current the space charge contribution to the
electrostatic potential has a noticible contribution, approximately ten volts. Compare
this result with the potential we would have expected from a slab approximation,
φ0 = a2 ρ/(2@0 ) ≈ 54 × 103 V (clearly non-physical).
If the ion current and velocity remain constant then the charge density changes
as the beam cross sectional area changes: I = 4ρvbc . In the limit that km b 1 and
kn c 1 and for constant ion current and velocity the maximum potential φ(0, 0, 0)
does not depend on the beam cross section dimension, b, c .
A.5
Electrostatic Potential due to a Ribbon of Charge
Consider two infinite grounded surfaces at x = ±a. The region between the surfaces
has an infinite ribbon of uniform charge density ρ in the y direction and for −c ≤
z ≤ c ; the charge density is zero elsewhere. We want to find the potential φ at any
point between the grids. In particular we are interested in the potential along x for
z = 0. The geometry is shown in Figure A.3.
The potential solves the Poisson equation, ∇2 φ = −ρ/@0 with the boundary conditions φ(x = ±a, y, z) = 0 , ∂φ/∂y = 0 at y = ±∞, and φ(x, y, z = ∞) = 0. The
charge density and the boundary conditions are symmetric in the y direction so we
can consider just the two dimensional problem in the x, z plane. The problem also
has symmetry about the origin so we need only consider the quarter plane problem:
the boundary of the domain we will consider is defined by the planes at x = 0, x = a,
z = 0 and z = ∞.



−ρ/@0
∂ φ(x, z) ∂ φ(x, z)
+
=
2
2

∂x
∂z

0
2
2
φ(x = a, z) = 0,
φ(x, z = ∞) = 0,
∂φ
= 0,
∂x x=0
257
for 0 ≤ z ≤ c, 0 ≤ x ≤ a = a
elsewhere,
Figure A.2: Electrostatic potential profiles φ(x, y = 0, z = 0) in boxes of sides |x| ≤ a,
|y| ≤ b, |z| ≤ c. Ions pass through the boxes along x with current I, velocity v, and
cross sectional area 2b ×2c , giving charge density ρ = I/(4vbc ). Top figure is volume
between grids, where space charge contributes negligibly to electrostatic potential.
Bottom figure is volume between RF plates, where space charge contributes noticibly
to electrostatic potential.
258
c'
-a,-a'
a,a'
x
-c'
phi(x,z=0)
-a
a
x
Figure A.3: Sketch of the distribution of space charge between surfaces at x = ±a.
Space charge is uniform inside ribbon of thickness ∆z = 2c and width ∆x = 2a = 2a,
and zero elsewhere.
259
∂φ
∂z
= 0.
z=0
We will solve the above second order linear partial differential and associated equation
boundary value problem using Green functions. See Haberman[18], for example. We
want to find a function G(x, x ) that satisfies the Poisson equation for a point-charge
source, ∇2G(x, x ) = −δ(x − x). Once we have G(x, x) then we find a particular
solution to the full Poisson equation by
φ(x) =
V
ρ(x)
G(x, x)d3 x +
@0
)
∂φ
∂
G(x, x ) − φ(x) G(x, x) dA,
∂n
∂n
S
where the domain of volume V is bounded by surface A. If the Green function
G(x, x ) is chosen so that it satisfies the same boundary conditions as the potential
φ(x) then the surface integral above vanishes.
Let us try for a Green function of the form
G(x, x ) =
∞
Cn (z) cos(kn x).
n=0
This form automatically satisfies the boundary condition at x = 0 for all z. We
choose the kn such that cos(kn a) = 0 to match the boundary condition at x = a,
that is kn = (n + 12 )π/a. Substitute this form into the Poisson equation for the Green
function (point source) to find
∞
n=0
d2 Cn (z)
− kn2 Cn (z) cos(kn x) = −δ(x − x )δ(z − z ).
2
dz
Multiply both sides by cos(km x), where cos(km a) = 0, and integrate from x = 0 to
x = a. All terms in the sum except one vanish, giving
d2 Cn (z)
− cos(kn x)
2
−
k
C
(z)
=
δ(z − z ).
n n
dz 2
a/2
We will solve this second order ordinary differential equation for Cn (z) as follows. The
260
equation is homogenous for two regions: in the region with z > z we find a solution
which satisfies the boundary condition at z = ∞; in the region for z < z we will
find a solution which satisfies the boundary condition at z = 0. We will match the
solutions at z = z , and the derivatives of the solutions at z = z will be discontinuous
due to the delta function:
dCn
dz
z+
z−
− cos(kn x )
=
.
a/2
We have
Cn (z) =



γn1 exp(kn z) + γn2 exp(−kn z)
for z > z ,


γn3 exp(kn z) + γn4 exp(−kn z)
for z < z ,
and



kn [γn1 exp(kn z) − γn2 exp(−kn z)]
dCn (z)
=

dz
 k [γ exp(k z) − γ exp(−k z)]
n n3
n
n4
n
for z > z ,
for z < z .
The boundary condition at z = ∞ requires γn1 = 0. The boundary condition at z = 0
requires γn3 = γn4 . Matching the solutions at z = z requires
γn2 exp(−kn z ) − γn3 [exp(kn z ) + exp(−kn z )] = 0.
The jump condition in the derivative at z = z requires
γn2 exp(−kn z ) + γn3 [exp(kn z ) − exp(−kn z )] =
cos(kn x)
.
kn a/2
Simultaneous solution of the these equations is given by
γn2 =
cos(kn x ) cosh(kn z )
,
kn a/2
γn3 =
261
exp(−kn z ) cos(kn x)
.
kn a
Insertion of γn2 and γn3 back into the coefficients Cn (z) gives
G(x, x ) =

∞


n=0
cos(kn x) cos(kn x) cosh(kn z ) exp(−kn z)/(kn a/2)
for z ≥ z ,

 ∞
n=0
cos(kn x) cos(kn x) cosh(kn z) exp(−kn z )/(kn a/2)
for z ≤ z ,
which exhibits the symmetry property we expect of the Green function, G(x, x) =
G(x , x).
Integrating the Green function over the volume of non-zero charge density gives
the potential:
ρ
φ(x, z) =
@0
a
dx
0
c
dz G(x, z, x, z ).
0
We are particularly interested in the potential along z = 0 and 0 ≤ x ≤ a. Then we
have
a
c
∞
ρ exp(−kn z ) φ(x, 0) =
cos(kn x)
cos(kn x )dx
dz ,
@0 n=0
kn a/2
0
0
∞
ρ (−1)n [1 − exp(−kn c)]
.
=
cos(kn x)
@0 n=0
kn
kn2 a/2
In the limit that c /a → ∞ this solution reduces to the one dimensional parabolic
solution, φ(x, 0) = (a2 ρ/(2@0 ))(1 − (x/a)2 ). To a very good approximation φ(0, 0) ≈
(a2ρ/(2@0 ))[1 − exp(−πc/(2a))].
A.6
A.6.1
1-D Space Charge with Shifted Half-Maxwellian
General Development
Consider the following setup: a cavity of length xb − xa has potentials at the entrance
φ(xa) = φa and the end φ(xb) = φb fixed. Particles enter the cavity at x = xa where
the distribution f(x = xa , v ≥ 0) is known. The distribution function evolves in the
cavity collisionlessly and without sources. The density n(x) at any location is determined by integrating the distribution function over all velocities, and the potential is
262
determined self-consistently through the Poisson equation, −d2 φ(x)/dx2 = qn(x)/@0.
At any location x the distribution function f(x, v) depends on the potential structure
“downstream” which may reflect particles. Thus the problem of solving the Poisson
equation is highly nonlinear.
We will show that when space charge has neglible influence on the electrostatic
potential, n(x) ≈ 0, which decouples the solution of the Poisson equation from the
plasma distribution function and considerably simplifies the algebra. We will obtain
the solution of the Poisson equation when space charge is included, for the case of
Maxwellian distribution function.
Mapping the Distribution Function
Consider a semi-infinite distribution function at one location x1 , for v1 ≥ 0 given by



f(x1 , v1) = 

0
v1 < va1,
f(µ(x1 , v1)) va1 ≤ v1 ≤ ∞.
where the velocity limit va1 is given by µ(x1 , va1) = qφ0. We want to demonstrate how
this distribution maps to other locations x = x2. The subscript on v1 is a reminder
that the velocity portion of phase space is at x = x1 . From x1 to x2 the form of the
distribution function is invariant, as well as the form of the velocity bounds, which
map as follows (postive root): µ(x2, va2) = µ(x1 , va1).
No Reflection, φc ≤ φ0
For the case of no reflections, at a different location x2 the distribution function
becomes



f(x2 , v2) = 

0
v2 < va2,
f(µ(x2 , v2)) va2 ≤ v2 < ∞,
The case for full transmission obtains if the slowest particles in the distribution
still have finite forward velocity at x = xc . Specifically, full transmission obtains if
263
µ(x2 , va2) ≥ qφc . Thus it is sufficient but not necessary that the potential decrease
monotonically from x = x1 to x = x2 to obtain full transmission.
Partial Reflection, φc ≥ φ0
Now consider the case when a portion of the distribution of particles is reflected.
The class of particles which are transmitted have finite forward velocity at x = xc ;
the rest are reflected before or at x = xc . The velocity bound vc1 that divides the
distribution into transmitted and reflected portions satisifes the condition (postive
root): µ(x1 , vc1) = qφc .
The distribution function at x = x1 is then identitical to the unreflected distribution function for forward moving particles, but it also includes particles reflected
from the space charge potential:










f(x1 , v1) =









−∞ < v1 < −vc1 ,
0
f(µ(x1 , v1)) −vc1 ≤ v1 ≤ −va1,
−va1 < v1 < va1,
0
f(µ(x1 , v1)) va1 ≤ v1 < ∞.
Thus at x2 ≤ xc and for φ(x2 ) ≤ φ0 (equivalent to va2 ≥ 0) the distribution function
maps as follows:










f(x2 , v2) =









−∞ < v2 < −vc2 ,
0
f(µ(x2 , v2)) −vc2 ≤ v2 ≤ −va2,
−va2 < v2 < va2,
0
f(µ(x2 , v2)) va2 ≤ v2 < ∞,
where the velocity bounds map as follows (postive roots): µ(x2 , va2) = qφ0, µ(x2 , vc2) =
264
qφc . At x2 ≤ xc and for φ0 ≤ φ(x) ≤ φc the distribution function maps as follows:







−∞ < v2 < −vc2,
0
f(x2 , v2) =  f(µ(x2 , v2 )) −vc2 ≤ v2 ≤ 0,





f(µ(x2 , v2 )) 0 ≤ v2 < ∞,
with the same condition for the velocity bound. For x2 ≥ xc the distribution function
maps as follows:
f(x2, v2 ) =



0


f(µ(x2 , v2)) vc2 ≤ v2 < ∞.
−∞ < v2 < vc2,
Maxwellian Distribution
Consider a distribution function of the form f(µ(x, v)) = C exp(−µ(x, v)/kT ), and
let particles which enter at x = 0 have a minimum velocity due to a potential drop:
µ(x0 , 0) = µ(0, va0) = qφ0, where φ(x0) = φ0 represents a sheath potential drop.
Then we immediately have at the entrance x = 0
f(0, v0 ) =



0


C exp(−µ(0, v0 )/kT ) va0 ≤ v0 < ∞.
−va0 < v0 < va0,
The distribution function for the other velocities depends on whether the potential
structure “downstream” reflects particles back “upstream”.
We can find the normalization constant C from a known forward flux:
Γ0 =
=
=
∞
0∞
va0
∞
qφ0
v0f(0, v0 )dv0 ,
v0f(µ(0, v0 ))dv0 ,
f(µ)
dµ
,
m
C∞
=
exp(−µ/kT )dµ,
m qφ0
265
=
CkT
exp(−qφ0/kT ),
m
or C = Γ0 (m/kT ) exp(qφ0/kT ). At any other location in the cavity we can find the
forward flux in a similar fashion. In particular we will be interested in the flux at xb,
which we obtain by
Γ(x) =
∞
vj
=
C
m
vf(x, v)dv,
∞
qφj
exp(−µ/kT )dµ,
q(φ0 − φj )
,
= Γ0 exp
kT
where φj = φ0 if φc ≤ φ0 and φj = φc if φc ≥ φ0 . Thus the normalized current can
be written



1
φc ≤ φ0 ,
Γ(x)
=

Γ0

exp (q(φ0 − φc )/kT ) φc ≥ φ0 .
(A.13)
We can integrate the distribution function at any location to find the density. Say
we have f(x, v) = 0 for v1 ≤ v ≤ v2. Then
v2
C exp(−µ(x, v)/kT )dv,
n(x) =
v1
Γ0 exp(qφ0/kT ) v2
−1 1 2
exp
mv + qφ(x) dv,
(kT /m)
kT 2
v1
v2
Γ0
q(φ0 − φ(x))
−mv 2
exp
=
exp
dv.
(kT /m)
kT
2kT
v1
=
Define vt =
2kT /m and change variables t = v/vt to get
v2 /vt
Γ0
q(φ0 − φ(x))
exp
2
n(x) =
vt
kT
exp(−t2)dt.
v1 /vt
Recall the definition of the normalized complimentary incomplete gamma function,
266
Q(a, z), defined by:
Q(a, z) = 1 −
1
Γ(a)
z
e−t ta−1 dt,
0
(a) > 0,
which satisifies the recursion relation[2, p.569]
dm z
[e Q(a, z)] = ez Q(a − m, z),
dz m
where Γ(a) (not to be confused with the particle flux) is the familiar (complete)
√
gamma function that satisfies Γ(a + 1) = aΓ(a) and Γ(1/2) = π. Note that
limz→∞ Q(a, z) = 0 and Q(a, 0) = 1. By a change of variable of integration it is
possible to show that
z
2
exp(−t2)dt =
0
√
π(1 − Q(1/2, z 2 )).
Thus the density becomes
√
q(φ0 − φ(x))
1 v12
1 v22
Γ0 π
,
,
exp
Q
−Q
,
n(x) =
vt
kT
2 vt2
2 vt2
for f(x, v) = 0 for v1 ≤ v ≤ v2 and vt =
(A.14)
2kT /m.
No Reflection, φc ≤ φ0
For the case of no reflections, it is possible to find the density as a function of location
from Equation (A.14). Recall that the lower limit of the distribution function satisfies
µ(x0 , 0) = µ(x, vax) such that
1 2
mv + qφ(x) = qφ0.
2 ax
267
2
Then vax
/vt2 = q(φ0 − φ(x))/kT and we have
√
Γ0 π y0 (x)
e
Q(1/2, y0 (x)),
n(x) =
vt
where y0(x) ≡ q(φ0 − φ(x))/kT .
Partial Reflection, φc ≥ φ0
For the case of partial reflections it is also possible to find the density as a function
of location from Equation (A.14), but now the reflected portion of the distribution
function must be included in the density. Recall that the lower limit of the distribution
function satisfies µ(x0, 0) = µ(0, va0 ) such that
1 2
mv = qφ0.
2 a0
2
/vt2 = qφ0/kT . Then we have for x ≤ xc and φ(x) ≤ φ0
Then va0
√
Γ0 π y0 (x)
n(x) =
e
[Q(1/2, y0 (x)) + Q(1/2, y0 (x)) − Q(1/2, yc (x))] ,
vt
where yc (x) ≡ q(φc − φ(x))/kT , and µ(x, vcx) = µ(xc , 0) means
1 2
mv + qφ(x) = qφc .
2 cx
For x ≤ xc and φ0 ≤ φ(x) ≤ φc we have
√
Γ0 π y0 (x)
e
[1 + 1 − Q(1/2, yc (x))] ,
n(x) =
vt
and for x ≥ xc we have
√
Γ0 π y0 (x)
e
Q(1/2, yc (x)).
n(x) =
vt
268
It can be shown that the above forms for the density all agree when φc = φ0 at
the appropriate locations of x, and that when x = xc the forms of the density to the
left and right of xc match.
A.6.2
Space Charge Neglected
If we assume that space charge can be neglected then n(x) ≈ 0 and we obtain the
vacuum solution to the Poisson equation, which in slab geometry is linear:
φ(x) =
φa x b − φb x a
xb − xa
φb − φa
+
x,
xb − xa
The mapping between the boundary potentials and the maximum potential between
the boundaries is trivial: the maximum potential occurs at one of the boundaries,
φc = max(φa, φb ). As one sweeps out the potential φb and measures the resulting flux
at xb , the “knee” in the curve occurs when φb = φ0, see Equation (A.13). Thus one
can easily obtain the value of φ0 . The electrostatic potential φ(x) can be inserted
into the definition of y0 (x) and the density is obtained everywhere.
A.6.3
Space Charge Included
When space charge cannot be neglected we must retain n(x) in the Poisson equation. The resulting second order ordinary differential equation is highly nonlinear. In
this case the mapping between the boundary potentials and the maximum potential
between the boundaries is non-trivial, and in general the location of the maximum
potential is inside the boundaries. For solutions of the Poisson equation that have
the maximum potential at one of the boundaries, space charge is effectively negligible
and all of the conclusions of the previous subsection apply.
We will consider a subset of the solutions to the Poisson equation, those for which
the gradient of the potential vanishes between the boundaries. Let xc represent the
position at which the potential gradient vanishes, and let φ(xc ) = φc represent the
269
x
x ≤ xc
x ≤ xc
x ≥ xc
φc
φc ≤ φ0
φ(x) ≤ φ0 ≤ φc
φ0 ≤ φ(x) ≤ φc
φ0 ≤ φc
n(x)
ey0 (x)Q(1/2, y0 (x))
ey0 (x) [2Q(1/2, y0 (x)) − Q(1/2, yc (x))]
ey0 (x) [2 − Q(1/2, yc (x))]
ey0 (x)Q(1/2, yc (x))
Table A.1: Summary of dimensionless density for different conditions and in different
regions.
potential maximum. Particles enter the cavity with a minimum kinetic energy equal
to qφ0. Therefore if φc ≤ φ0 all particles will traverse the cavity; if φc ≥ φ0 then some
of the particles will be reflected.
We can rewrite the Poisson equation in dimensionless form using
x = x/λD ,
y0 = q(φ0 − φ(x))/kT,
√
n = n/(Γ0 π/vt),
√
λ−2
= q 2(Γ0 π/vt )/(@0 kT ),
D
and then dropping the primes to get
d2 y0
= n(x),
dx2
(A.15)
where n(x) is the dimensionless density. Table A.1 summarizes the density conditions.
First Integral of Poisson Equation
Multiply both sides of Equation (A.15) by dy0 /dx. Both sides can be written as a
perfect differentials:
d2 y0 dy0
1 d
=
2
dx dx
2 dx
dy0
dx
270
2
=
dF (x)
.
dx
x
x ≤ xc
x ≤ xc
x ≥ xc
φc
φc ≤ φ0
φ(x) ≤ φ0 ≤ φc
φ0 ≤ φ(x) ≤ φc
φ0 ≤ φc
F (x)
ey0 (x)Q(1/2, y0 (x))
ey0 (x) [2Q(3/2, y0 (x)) − Q(3/2, yc (x))]
ey0 (x) [2 − Q(3/2, yc (x))]
ey0 (x)Q(3/2, yc (x))
Table A.2: Summary of F (x) for different conditions and in different regions.
Use the recursion relation for the incomplete gamma function,
d y0 (x)
dy0
e
γ(a + 1, y0 (x)) = ey0 (x) Q(a, y0(x))
,
dx
dx
Note that since all yj are all equal to within a constant we have dy0 /dx = dyc /dx,
and so
e
y0 (x)
dy0
dyc
q(φ0 − φc ) yc (x)
Q(a, yc (x))
e
Q(a, yc (x))
= exp
,
dx
kT
dx
q(φ0 − φc ) d yc (x)
e
Q(a + 1, yc (x)) ,
= exp
kT
dx
d y0 (x)
=
e
Q(a + 1, yc (x)) .
dx
Table A.2 lists the first integrals F (x) of the right hand side of Equation (A.15) for
the different cases and locations.
Integrate the Poisson equation once formally to get
dy0
dx
2
−
x2
dy0
dx
2
= 2(F (x2 ) − F (x1)),
x1
Integrate from x1 = xc where dφ/dx = dy0 /dx = 0 to get
dy0
= ± 2(F (x) − F (xc),
dx
(A.16)
where the appropriate root is chosen based on whether the gradient is evaluated to
271
the left (positive) of the potential with zero gradient or to the right (negative).
Critical Solution
The critical solution occurs when the maximum potential due to space charge, φc ,
just equals the plasma potential, φ0. No particles are reflected as the slowest particles
just have enough energy to pass the maximum space charge potential. Thus Poisson’s
equation for this case is:
d2 y0
= ey0 (x)Q(1/2, y0 (x)),
dx2
with the boundary conditions φ(xc ) = φ0 (or y0(xc ) = 0) and dy0 /dx = 0 at x = xc .
Multiplying both sides by dy0/dx and integrating from x1 to x2 gives
dy0
dx
2
−
x2
dy0
dx
2
= 2 ey0 (x2 ) Q(3/2, y0 (x2 )) − ey0 (x1 )Q(3/2, y0 (x1)) .
x1
Use the boundary conditions to reduce this equation to a first order ordinary differential equation for y0(x), from which the potential φ(x) may be extracted:
dy0 y (x)
= 2 (e 0 Q(3/2, y0 (x)) − 1).
dx
This separable first order ordinary differential equation can be solved by integration. The integrand can be evaluated near y0(xc ) = 0 using
lim ey Q(3/2, y) − 1 = y.
y→0
Numeric integration may proceed more easily for large values of y using the series
form:
∞
Γ(n + 1 − a)
y a−1
(−1)n
,
e Q(a, y) =
Γ(a)Γ(1 − a) n=0
yn
y
which grows as
ey Q(a, y) ≈
y a−1
for large y.
Γ(a)
272
We can find an analytic approximation to the critical grid spacing required for
space charge effects. Use



√
2y
dy
≈ 2√y
dx 

Γ(3/2)
0 ≤ y ≤ y ∗,
y ∗ ≤ y ≤ ∞,
where y ∗ = 4/π is the value where the two limiting functions are equal. With this approximation the separable first order ordinary differential equation can be integrated
by hand:



√
2y
x(y) ≈ √

 2y ∗ +
y ≤ y ∗,
2π1/4 3/4
[y
3
− (y ∗)3/4] y ≥ y ∗.
For given boundary values ya and yb the total solution is approximately xab ≈ x(ya) +
x(yb ).
We can use this approximate formula to determine how xab λD varies with large
and small kT . Note that since y ∼ kT −1, large (small) T implies small (large) y.
First consider the limit of small T , that is y y ∗ .
lim x(y)λD
T →0
2π 1/4
=
3
q(φ0 − φ(x))
kT
3/4 


@0
2
(q(φ0 − φ(x)))3/4  2
=
3
q Γ0
@0kT
π 1/2q 2Γ0
1/2
2kT 
m
,
1/2
2
m
,
which depends only on Γ0 , which is proportional to the current. Next consider the
limit of large T , that is y y ∗ .
lim x(y)λD =
T →∞
=
q(φ0 − φ(x))
kT
&
'
'
(
1/2 

2q(φ0 − φ(x))@0
@0kT
π 1/2q 2 Γ0
2
mπ
1/2
2kT 
m
,
kT
,
q 2 Γ0
which is proportional to the square root of the ratio of temperature and flux.
273
In typical experimental situations we have y y ∗, which gives a critical current.
For currents below the critical current space charge cannot play a significant role. For
instance, let kTe = 10 eV such that φ0 ≈ 30 V (assuming secondary electron emission
is negligible), φa = −70 V, φb = 0 V, λD (xa + xb ) = 1 mm, m = 2mp , and q = e.
Using these numbers gives Γ0 = 5 × 1020 m−2 s−1 ; with A = (10 mm)(25 µm) and
q = e gives I = 20 µA, a level of current routinely collected in the omegatron. For the
same numbers above but kTe = 0 eV (such that the difference between the floating
potential and the plasma potential vanishes) the critical current drops to I = 5 µA,
still large compared to the smallest currents observed.
The conclusion from this one-dimensional model is space charge can be neglected
for current levels below the microamp level, and for which currents the selector component bias at the knee in the IV characteristic reflects the actual maximum potential
between the grids.
A.7
Kinetic Sources and Collision Operators
Harrison and Thompson [19] use the source function S(φ),
S(φ) =
λnγe ,
eφ
ne = n0 exp
,
kTe
where λ, γ are constants. Ions are born cold, which means the ion distribution function
can be obtained directly from the source function.
Emmert et al [14] use a collisionless model, holding E = mvz2/2 + qφ(z) constant
for ions. Obtaining the ion velocity vz from E, they use phase space coordinates
(z, E) rather than (z, vz ), and denote direction of ion travel by σ = ±1. They solve
the kinetic equation
∂(gvz )
= S(z, E, σ),
∂z
and pick the source function such that for no potential gradient the ion distribution
274
is Maxwellian,
−(E − qφ(z))
1
.
exp
g∼
mvz (z, E)
kTi
The ion source function S(z, E, σ) with this property is
qφ(z)
−E
S0 h(z)
exp
exp
.
S(z, E) =
2kTi
kTi
kTi
Bissel [8] note that the source function of Emmert et al is equivalent to
−mi vz2
mi vz
exp
dvz .
S(z, vz )dvz = S0 h(z)
kTi
2kTi
Bissel and Johnson [7] use a collisionless model, such that for ions mvz2/2 + qφ(z)
is constant. They solve for the ion distribution f(z, vz ) using a Maxwellian ion source
function S(z, vz )
mi
S(z, vz )dvz = Rnn ne (z)
2πkTi
1/2
−mivz2
exp
dvz .
2kTi
Chung and Hutchinson [11] use a collisionless model, E = mvz2/2+qφ(z) constant,
and solve the Boltzmann equation numerically,
q dφ ∂
∂
−
vz
f(z, vz ) = Sf ,
∂z m dz ∂vz
with the ion source Sf
Sf = W (z)[f∞ (vz ) − f(z, vz )].
They assume the ion distribution outside fluxtube f∞ is shifted Maxwellian. Chung
and Hutchinson [12] extended the model of their ion source term to include ionization
and variable ratio of viscocity to diffusivity, Sf = σtSt + (1 − σt)Si , where
St = W (z) {α[f∞ (vz ) − f(z, vz )] + (1 − α)[1 − n(z)/n∞ ]f∞ } ,
Si = σvionne (z)fn (z, vz ).
275
If the neutral distribution fn (z, vz ) is a Maxwellian, and if ne (z)σvion ∼ |vz | then
Si reduces to the ion source of Emmert et al; if σvion is constant then Si reduces to
the ion source of Bissel and Johnson.
Riemann [59], using the dimensionless coordinates x = z/L, y = mvz2/(2kTe ), χ =
−eφ/kTe , q(y) = L/λ(vz ), solves for the ion distribution function f(χ, y), normalized
by
ni =
∞
0
y −1/2f(χ, y)dy or ji =
∞
f(χ, y)dy.
0
He considers Boltzmann’s equation for charge exchange with cold neutrals
dχ
dx
∂f
∂f
+
∂χ ∂y
where
C(χ) =
= −q(y)f(χ, y) + C(χ)δ(y),
∞
q(y)f(χ, y)dy + σ(χ)
0
designates total rate of charge exchange and ionization.
Main [35] obtains asymptotic aproximations to the ion distribution from the Boltzmann equation,
dU
dz
1 ∂f
∂f
+
vz
∂U
m ∂vz
=
∂f
∂t
,
c
using the notation U = qφ, where q is electron charge, and the potential sign convention such that increasing potential repels electrons. He considers charge exchange
collisions with constant collision frequency:
∂fi
∂t
=
c
1
fn (vz )
τ nn
∞
−∞
fi (u)du − fi (vz )
∞
−∞
fn (u)du ,
and a quasi-constant cross-section collision term,
∂f
∂t
=σ
c
∞
−∞
fn (vz )f(u)|vz − u|du −
∞
−∞
f(vz )fn (u)|vz − u|du .
Scheuer and Emmert [60] consider collisions with constant mean free path using a
276
BGK collision operator and obtain results for low to moderate collisionality. Starting
with a form of the ion distribution f(z, vz ), they write the Boltzmann equation,
vz
where
q ∂φ ∂f
∂f
−
= −νf + νΦ(z, vz ) + S,
∂z mi ∂z ∂vz
mi
Φ(z, vz ) = nb (z)
2πkTb
1/2
−m(vz − qb (z))2
exp
.
2kTb
With a change coordinates, they define g(z, E, σ) = f(z, vz )/(mi v), where E =
mv 2/2 + qφ(z), v = |vz |, σ = ±1 is direction of travel. The Boltzmann equation
becomes
d(gσv)
−gv
Φ
=
+
+ S(z, E).
dz
λ
mi λ
They use the ion source function of Emmert et al,
qφ(z) − E
1
exp
.
S(z, E) = S0h(z)
2kTs
kTs
277
278
Appendix B
Electronics
B.1
Camac
The omegatron electronics cabinet has its own Camac crate. The diagnostician running the omegatron communicates with it through the modules in the Camac crate
via a serial fiber optic highway. The functions of the camac modules can be divided
roughly according to data input and data output functions.
Camac Inputs
The camac inputs digitizes voltage signals from the omegatron analog electronics and
stores them until they are retrieved. The omegatron Camac crate has two Joerger
TR16S/H sample-and-hold 12-bit digitizers with 16 channels each and 16 kB per
channel. Digitizing at 10 kHz gives approximately 1.6 seconds of data, sufficient
for typical C-Mod plasmas. This digitization rate forms a fundamental limit on
the electronics bandwidth, and the rest of the analog electronics employ filtering
so that the TR16s are not aliased. In addition the omegatron Camac crate has an
Aurora 12 digitizer to provide supplemental digitization of signals at faster rates (and
subsequently shorter durations).
A LeCroy LG8252 slow digitizer provides 16 channels of scalars on demand. The
279
LG8252 is used to record the position, head temperature, and power supply voltages.
Camac Outputs
The camac outputs provides voltages to control the postion and operation of the
omegatron. The omegatron Camac crate has its own MPB decoder which looks for
events on the fiber optic timing highway and generates clock, trigger and gate TTL
pulses. The omegatron crate also has a Jorway J221 module which accepts a clock
from the MPB decoder and produced trigger and gate TTL pulses.
The omegatron Camac crate has three BiRa 5910 12 bit digital-to-analog waveform
generators with four channels each. They can be programmed to draw arbitrary
waveforms, typically within -5 to +5 volts, and typically at 10 kHz. The frequency
can be changed with software; the voltage range can be changed only with jumpers
(5 volt or 10 volt, unipolar or bipolar). The outputs of the 5910s are used to program
the analog electronics to apply biases to components, and to program the RF power
and frequency. The BiRa 5910 uses 2-pin Lemo connectors for the outputs; each
channel of the output is driven independently of ground, so caution must be used
when interfacing to a BNC connector. The safest interface to the 5910 uses only
isolated BNC inputs with differential amplifiers.
The BiRa 3303 relay board accepts 24 volts from a power supply and applies it to
16 channels upon command from the Camac highway. The 3303 channel outputs are
used to close other relays which turn on the compressed air, the RF amplifier, and
the motor power supply.
The 3224 board has 24 output registers; it is used to control the gain and power
supply relays of the Langmuir probe analog electronics cards.
The Joerger SMC24 stepping motor controller card provides the timing and channels the current to drive the stepping motor. It also records the status of the stepping
motor, the stepping motor power supply, and the limit switches.
280
B.2
Custom Electronics Schematics
Enlarged paper or electronic versions of these schematics may be obtained by contacting Dr. Brian LaBombard at labombard@psfc.mit.edu.
281
High bandwidth grid current monitor
+15V
(isolated)
0.1uf
282
Figure B.1: Electrical schematic of omegatron grid ammeter circuit.
7 +
4.7uf
2
OPA627BP
3
-15V
(isolated)
Omegatron Grid/Collector Arrangement
Heat Shield
(connected to
vacuum vessel)
RF Cavity
Non-Resonant
Z/M
100Ω
Load
Current Sig X 10
BNC3
Current Sig X 100
BNC4
330uH
1:1
to Slit input on each
channel and to Langmuir
probe electronics
2:1
+
4.7uf C59
R7
C1
Grid
(2) 10Ω
RN55D
R6
3
R8
2
(2) 10Ω
RN55D
8
1kΩ RN55C
Match with R17
R18
Vishay
S106K-20k-.01%
R1
0.1uf
Arco No. C404
trim capacitor
R24
0.033uf Mallory
CK05BX333K
C23
C28
0.1uf
4
1kΩ RN55C
6
In
1500pf C6
C29
C18
33pf Mallory
CK05BX330K
+
4.7uf C63
-15V
1kΩ RN55C
R48
10 kΩ (match with R1)
+15V
100Ω RN65C
Out
R21
OPA627BP
R23
F1
R33
20kΩ RN55C
Match with R32
+
7 4.7uf
C66
2
U5
3
LT7 - 5kHz - 1kΩ - 1245
4
8
0.1uf C34
R39
+15V
+
7 4.7uf
C62
U3
1kΩ RN55C
C32 0.1uf
49.9Ω
RN55C
6
R13
0.1uf C33
1 +
4.7uf C68
X1 Output
BNC #1 - Front Panel
4 OPA633KP 8
U9
0.1uf C35
5
2,3,6,7
Lowpass Filter
Common
10 kΩ (match with R3)
BNC5
R32
+15V
OPA627BP
10Ω Trim
Bourns 3006P
C9
+
4.7uf C67
-15V
Case
100Ω RN65C
-15V
R40
+
4.7uf C69
DIN connector
Row C, pin#8
150pf
C7
R3
BNC6
Virtual Grid
BNC7
Grid V Out
10pf
RF Power
RF Transformer
BNC9
Slit
Virtual
Grid/Collector
BNC #6 - Front Panel
330uH
with High Bandwidth Grid Current Monitor
C13
Vishay
S106K-20k-.01%
B. LaBombard
Modifications to original board
R. Nachtrieb
5/1/99
are indicated in BLUE and RED
Tel.:(617) 253-6942 Fax: (617) 253-0627
R4
OPA627BP
R56
10Ω Trim
Bourns 3006P
6
-15V
(isolated)
R20
Vishay
S106K-20k-.01%
+
4.7uf C61
R36
+15V
(2) 10Ω
RN55D
R10
3
R12
2
(2) 10Ω
RN55D
8
0.1uf
R28
0.015uf Mallory
CK05BX153K
C30
C36
1kΩ RN55C
1kΩ RN55C
Match with R19
U4
4
C31
0.1uf
In
C8
C15
10pf Mallory
150pf
CK05BX100K
+
4.7uf C65
-15V
2kΩ RN55C
F2
R22
Out
2
U6
4
8
0.1uf C38
+15V
100Ω RN65C
OPA627BP
R27
LT7 - 5kHz - 1kΩ - 1245
2kΩ RN55C
R41
+
7 4.7uf
C70
3
R37
200kΩ RN55C
Match with R36
C22
0.1uf
+15V
+
7 4.7uf
C64
6
OPA627BP
R11
3 U2
0.1uf C27
4
8
C12
PC402
4-20pf
Arco PC402 4-20pf
variable capacitor
R9
+
7 4.7uf
C60
2
100Ω RN55C
Match with R2
1kΩ RN55
Match with R20
R19
0.1uf C26
Arco No. PC402
trim capacitor
4 - 20pf
200kΩ RN55C Match with R37
R49
+15V
(isolated)
330uH
Modified Omegatron Grid Electronics
C3
C11
Arco PC402 4-20pf
variable capacitor
Camac Waveform
Program
BNC8
M.I.T. Plasma Fusion Center
175 Albany St.
Cambridge, MA 02139
R5
R55
Arco PC402 4-20pf
variable capacitor
High bandwidth Grid Current Monitor
Z/M
Collector
All Grid, Slit, and Collector Connections
use Coax with Isolated Shields
(shields are biased by electronics)
1kΩ RN55C
Match with R18
R17
10pf
End Collector
Balanced 50Ω
RF Coax Lines
(identical lengths)
DC Break
12 - 65pf
20kΩ RN55C Match with R33
R47
-15V
(isolated)
Handle
Grid1 Grid2 Grid3
6
8
C10
PC402
4-20pf
Grid V/40 Sig
1500pf
C14
Vishay
S106K-20k-.01%
2 U1
0.1uf C25
4
100Ω RN55C
Match with R4
330uH
Slit
Ion
OPA627BP
Current Sig X 1
BNC2
C5
+
7 4.7uf
C58
3
R2
BNC1
Electrostatic Shield
(connected to slit)
Resonant Z/M
C24
+15V
(isolated)
Omegatron
Grid/Collector
BNC #5 - Front Panel
Magnetic Field
Retarding Field
Energy Analyzer
+
4.7uf
0.1uf
Front Panel Layout
High Bandwidth Monitor
BNC on Front Panel
6
0.1uf
4
49.9Ω
RN55C
6
1 +
4.7uf C72
4
R14
0.1uf C37
OPA633KP
U10
5
2,3,6,7
X10 Output
BNC #2 - Front Panel
8
0.1uf C39
Lowpass Filter
Common
+
4.7uf C71
-15V
Case
R50
+
100Ω RN65C
-15V
4.7uf C73
R42
DIN connector
Row C, pin#12
Gray Area Denotes Region with Separate Ground
and +/- 15V Power Planes (isolated)
0.0022 uf 200WVDC
C20
R34
200kΩ RN65C
Camac Waveform
BNC #8 - Front Panel
Cut ground
connection
DIN connector
Row A, pin#1
Cut Connection
+100V
R31
4.99kΩ RN55C
R38
0.1uf 200VDC
6
4.99kΩ
- 3583
+ U13
7
Cut
ground
0.0022 uf
200KΩ
330uH
R53
C54
1
+15V
(isolated)
0.022 uf 200WVDC
C16
20KΩ S106K
1
0.1uf 200 VDC
20KΩ S106K
R54
20KΩ
S106K
6
C17
7
0.022 uf
0.1uf
C56
- 3583
+ U14
C48
Isolated Ground
Plane
2
5
R52
Slit Potential Input
BNC #9 - Front Panel
0.1uf 200VDC
+200V
Case
Grounded
-200V
L1
0.1uf
R51
C55
0.0015uf Mallory
CK05BX152K
C21
20KΩ S106K
2
5
0.1uf
2
C1
C50
3
-V1
10
+V2
0.1uf
100Ω 1/4 W
C49
1
-15V
(isolated)
0.1uf 200 VDC
L2
+V1
C51
9
C2
8
-V2
Burr-Brown
722
U15
0.1uf
C40
330uH
P+
11
V+
20
E
18
V-
16
R26 1.3kΩ
RN55C
+ 4.7uf
C82
+15V
L3
0.1uf
C52
0.1uf
L4
330uH
2
OPA627BP
Required equipment for board tuning:
Extender card
Function generator
Oscilloscope
3 BNC cables
BNC T
Banana plugs to put 100 kOhm in series with center conductor
R25
100Ω RN65C
-15V
Board tuning procedure:
0.1uf C46
+15V
R45
7 +
4.7uf C78
2
6
OPA627
3 U8
0.1uf C45
4
8
196KΩ
RN65C
R35
R30
-15V
+
4.7uf C79
33pf
1 +
4.7uf C80
C19 R16
4
8
OPA633
49.9Ω RN55C U12
0.1uf C47
5
2,3,6,7
R46
-15V
100Ω RN65C
+
4.7uf C81
Grid Potential/40 Output
BNC #4 - Front Panel
Connections to 32 pin DIN connector:
Signal Connections
Notes
Symbol refers to large ground plane,
common to external power supplies
DIN connector
Row C, pin#3
6
3 U7
0.1uf C41
4
8
1KΩ RN55C
0.1uf C44
+15V
Grid Potential Output
BNC #7 - Front Panel
4.99KΩ
RN55C
R29
+
7 4.7uf
C74
49.9Ω
RN55C
0.1uf C42
1 +
4.7uf C76
4
R15
2,3,6,7
OPA633KP
U11
5
X100 Output
BNC #3 - Front Panel
8
0.1uf C43
330uH
C57
-100V
Case
Grounded
0.47 uf
200WVDC
2kΩ RN55C
C53
20kΩ RN55C
R43
+15V
100Ω RN65C
+15V
Symbol refers to a local isolated ground plane,
connected to common (pins 2 & 9) of
U15 (722 DC/DC convertor) and
connected to output (pin 1) of U14 (3583)
A
C
C
C
C
1
3
8
12
24
CAMAC Waveform Input +
Potential Monitor Signal Output +
Current Monitor x1 Signal Output +
Current Monitor x10 Signal Output +
Current Monitor x100 S
Power Supply Connections
Row#
Pin#
Power Supply
A
18
+200VDC
A
19
+100VDC
A
20
-100VDC
A
21
-200VDC
A
31
-15VDC
C
31
+15VDC
All other pins are connected to ground
Grid board being tuned should be on extender card with component-side cover
removed.
(1) DC offset tuning:
Start with nothing plugged in to front panel. Apply 10 Vpp 0 V offset square wave
at 3 Hz to V/40 pin on front panel. Turn on high voltage power supply. CAUTION:
HIGH VOLTAGE +/- 200 VOLTS ON FRONT PANEL! Observe square wave signal from Ix1
(Ix10) on oscilloscope and tune R55 (R56) to minimize amplitude. Turn OFF high
voltage power supply when finished.
(2) Common mode adjustment:
High voltage power supply is OFF. Start with nothing plugged into front panel.
Drive ground plane voltage externally by applying 20 Vpp 0 V offset sine wave at
3.2 kHz to Vout on front panel. Observe sine wave signal from Ix1 (Ix10) on
oscilloscope and adjust C14 (C15) to minimize amplitude.
(3) Virtual Omegatron balance:
High voltage power supply is OFF. Apply 20 Vpp 0 V offset sine wave at 3.2 kHz
through 100 kOhm on center conductor to both Grid (aka BNC 5) and Virtual Grid
(aka BNC 6), using BNC T. Observe sine wave signal from Ix1 or Ix10 on
oscilloscope and balance C9 and C11 to minimize the amplitude.
+
4.7uf C75
+
100Ω RN65C
4.7uf C77
-15V
R44
DIN connector
Row C, pin#24
283
Figure B.2: Electrical schematic of omegatron RF plate ammeter circuit.
C24
0.1uf
330 pf
C5
+15V
(isolated)
Omegatron
RF Plate Collector
7 +
4.7uf C58
3
100mH
R2
OPA627BP
22uH
6
C2
330 pf
12 - 65pf
1kΩ RN55C
Match with R18
R17
+
4.7uf C59
R5
-15V
(isolated)
R1
R24
0.15uf Mallory
CK05BX333K
(2) 10Ω
RN55D
0.1uf
4
8
C32 0.1uf
GRIDS:
LT7 - 5kHz - 1kΩ - 1245
1kΩ RN55C
6
-15V
1kΩ RN55C
C29
C18
33pf Mallory
CK05BX330K
R21
F1
R33
20kΩ RN55C
Match with R32
OPA627BP
R23
U5
3
RF: ALT7-1kHz filter unit
0.1uf C34
1 +
4.7uf C68
4 OPA633KP 8
R13
U9
0.1uf C35
5
2,3,6,7
49.9Ω
RN55C
6
0.1uf C33
4
8
1kΩ RN55C
R39
+15V
100Ω RN65C
7 +
4.7uf C66
2
Out
In
330 pf C6
+
4.7uf C63
X1 Output
Lowpass Filter
Common
R48
1 MegΩ RN60D(match with R1)
C23
+15V
U3
2
1kΩ RN55C
Match with R17
C28
7 +
4.7uf C62
3
R8
R18
Vishay
S106K-20k-.01%
Remove (4) variable
1 MegΩ RN60D (match with R3)
capacitors
0.1uf
+15V
(2) 10Ω
RN55D
OPA627BP
6.6 pf
C1
R32
R6
R55
10Ω Trim
Bourns 3006P
R7
CK05BX Style
Arco No. C404
trim capacitor
20kΩ RN55C Match with R33
R47
2 U1
0.1uf C25
4
8
10kΩ RN55C
Match with R4
22uH
C14
Vishay
S106K-20k-.01%
+
4.7uf C67
-15V
Case
100Ω RN65C
-15V
R40
+
4.7uf C69
33 pf
C7
R3
C3
CK05BX Style
C13
Vishay
S106K-20k-.01%
6.6 pf
+15V
(isolated)
Virtual
RF Plate Collector
100mH
R4
OPA627BP
10kΩ RN55C
Match with R2
22uH
C4
Vishay
R20
S106K-20k-.01%
+
4.7uf C61
R28
0.068uf Mallory
CK05BX153K
+15V
(2) 10Ω
RN55D
R10
3
R12
2
0.1uf
C30
C36
(2) 10Ω
RN55D
8
1kΩ RN55C
Match with R19
0.1uf
7 +
4.7uf C64
6
U4
1kΩ RN55C
4
C15
10pf Mallory
CK05BX100K
+
4.7uf C65
-15V
2kΩ RN55C
In
C8 33 pf
C31
0.1uf
7 +
4.7uf C70
2
Out
F2
R22
OPA627BP
R27
U6
3
R37
200kΩ
RN55C
Match
with R36
C22
RF: ALT7-1kHz filter unit
49.9Ω
RN55C
6
0.1uf C38
1 +
4.7uf C72
4
R14
0.1uf C37
4
8
2kΩ RN55C
R41
+15V
100Ω RN65C
+15V
GRIDS:
LT7 - 5kHz - 1kΩ - 1245
OPA627BP
R11
C27
0.1uf
330 pf
-15V
(isolated)
R56
10Ω Trim
Bourns 3006P
6
5
U2 1
3
4
8
22uH
R9
7 +
4.7uf C60
2
R36
1kΩ RN55
Match with R20
R19
0.1uf C26
Arco No. PC402
trim capacitor
4 - 20pf
200kΩ RN55C Match with R37
R49
OPA633KP
U10
5
2,3,6,7
X10 Output
8
0.1uf C39
Lowpass Filter
Common
+
4.7uf C71
-15V
Case
R50
+
100Ω RN65C
4.7uf C73
-15V
R42
+15V
(isolated)
Gray Area Denotes Region with Separate Ground
and +/- 15V Power Planes (isolated)
R34
0.0068uf Mallory
CK05BX152K
C21
330uH
+15V
(isolated)
L1
0.1uf
Isolated Ground
Plane
C49
-15V
(isolated)
0.1uf
C48
C50
0.1uf
0.1uf
L2
C51
1
+V1
2
C1
3
-V1
10
+V2
9
C2
8
-V2
Burr-Brown
722
U15
C40
330uH
P+
11
V+
20
E
18
V-
R26 1.3kΩ
RN55C
+ 4.7uf
C82
C52
0.1uf
0.1uf
L4
0.1uf
R43
+15V
100Ω RN65C
+15V
+15V
L3
2kΩ RN55C
C53
2
20kΩ RN55C
7 +
4.7uf C74
6
49.9Ω
RN55C
OPA627BP
R29
3 U7
0.1uf C41
4
8
16
330uH
0.1uf C42
1 +
4.7uf C76
4
R15
2,3,6,7
OPA633KP
U11
5
0.1uf C43
330uH
-15V
+
4.7uf C75
DIN connector
Row C, pin#24
+
100Ω RN65C
4.7uf C77
-15V
R44
x10 High bandwidth monitor
0.1uf
+15V
2
7 +
4.7uf
OPA627BP
3
4
-15V
0.1uf
+
4.7uf
X10 High Bandwidth Monitor
6
X100 Output
284
Figure B.3: Electrical schematic of RF oscillator AM/FM control circuit.
INSIDE WAVETEK 1062
SIDE VIEW OF MY9-1 OSCILLATOR
U3
AD633
W = (X1-X2)(Y1-Y2)/10 + Z
X1
+VS
1
2-PIN LEMO
8
+
1
+
0.1 uF
9
+18 V
4.7 uF
-18V
4
1
+18V
LEVEL
1.1 K
2
-
2
RF POWER REQUEST
0V TO +10V INPUT YIELDS
FULL RANGE OF AMPLITUDE
CONTROL
+
1/10 V
Y1
+18 V
∏
∑
6
W
2
7
FREQ
REQUEST
CW
ENABLE
-10V TO 0V
7
X2
8
5
MONITOR
Z
3
6
100 K
+
178 K
-
-VS
4
5
Y2
ISOLATED BNC
-18 V
0.1 uF
4.7 uF
+
232 K
AMPLITUDE MODULATION INPUT
0V TO +10 V INPUT YIELDS FULL
AMPLITUDE MODULATION.
INPUT DEFAULTS TO +10V
200 K
10 PF
0.1 uF
3
10 K
-
+4.7 uF
TAP # 2
POWER
MONITOR
TAP # 1
LEVEL
6
1N4148
4
2
SHIELDED COAX
SHIELDED COAX
10 K
ISOLATED BNC
RF POWER MONITOR
OUTPUT
7
-
+
U5
OPA627
3
10 K
-18 V
0.1 uF
+18 V
+4.7 uF
100 K
8
100 PF
0.1 uF
330 K
500
U4
OPA627
2
+18V
RF OSCILLATOR MY9-1
EFFECTIVE GAIN
V_MON/V_LEVEL = -3.5
7
+
IN WAVETEK
RF OUTPUT
1-100 MHZ
+18 V
4.7 uF
+
TAP # 6
FREQUENCY
REQUEST
6
4
8
TAP # 4
+18 V
SHIELDED COAX
100 K
+18V
0.1 uF
TAP # 9
-18 V
SHIELDED TWISTED PAIR
-18V
-18 V
4.7 uF
+
330 PF
CASE
-6 TO -3.5 V YIELDS
1 TO 100 MHZ
3.01 K
10 K
33 pF
10 pF
2-PIN LEMO
1
0.1 uF
2
+18 V
0.1 uF
10 K
2
1K
+18 V
+4.7 uF
7
-
10 K
+
U1
OPA627
10 K
6
2
+
8
36.5 K
-18 V
RF FREQUENCY
REQUEST
-5V TO +5V
YIELDS 1-100 MHZ
U2
OPA627
0.1 uF
4.7 uF
+
+
U6
OPA627
ISOLATED BNC
6
4
3
6
8
4
8
-18 V
+18V
0.1 uF
0.1 uF
7
-
51
3
3.01K
2
7
-
4
3
33 pF
+18 V
+4.7 uF
-18 V
4.7 uF
+
PLASMA FUSION CENTER
MASSACHVSETTS INSTITVTE
OF TECHNOLOGY
OMEGATRON RF AMPLITUDE
CONTROL CIRCUIT VERSION 2
DRN
CHKD
APPD
SHEET
R. NACHTRIEB
1 OF 1
DRAWING #
RTNAC-990128
19990421
285
Figure B.4: Electrical schematic of Langmuir probe ammeter circuit.
Vacuum Flange
RG58/U coax, 21ft long
(5 coax cables per AIR/FLEX Assembly)
3/4" I.D. AIR/FLEX Cable Shield,
13ft long
Vacuum Feedthru
(10-pin Instrumentation type)
Isolated BNC
on Front Panel
SMA connector
patch-panel
Triax Shield Driver
5 of these circuits are on one card, in one cable assembly
Power is supplied by a separate twisted pair cable
Compensation Network
1
0.1uf 50VDC
Mallory M20R104M5 (typ.10 places)
C7
0.1uf CSD3
BNC1
2
C51
+15V
R3
1 +
4.7uf
20kΩ RN55C
4
+15V
C8
4.7uf 50VDC
0.1 uf 200VDC
Mallory TDC475M050NLF Mallory M40R104K2
(typ. 10 Places)
C13
U1
R4
50kΩ Trim
Bourns 3299W
8
OPA633KP
5
6
0.1uf
C10
180pf 200VDC
Mallory CK05BX181K
182Ω RN55C
-15V
C3
C11
C4
S1 - AROMAT HD1E-M-DC24V
(NEWARK Stock No. 46F5747)
0.1 uf 200VDC
Mallory M40R104K2
R26
1MegΩ Trim
Bourns 3299W
R7
+24V 2
C2
R25
D1
1kΩ 1%
5W GN-5C
D4
S9
1kΩ 1%
5W GN-5C
5
2
2
+24V
S3
4
+24V
8
7
5
D7
+24V
8
4
3
+5VDC
C29
D2
4
6
4 1
+24V
8
8
10
3 Amp Traces
on PC Board
1
3
5
12
7
+24V
8
S6
4
R18
7
5
R19
R21
R32
Re
(2) 1.0Ω 1%
5W GN-5C
Rg
R33
14
1
2
3
D10
+24V
8
Calibrate
2
1
Isolated BNC
on Front
DIN Row: A
Panel
Pin #: 2
BNC4
13
+24V
8
Current
R37
2 U4
2
(2) 1Ω 5%
1/2W
(carbon)
R40
Rf
R42
R44
Rh
(2) 10Ω
RN55D
4
U3
Y3
C27
10pf Mallory
CK05BX100K
+
4.7uf C21
-15V
7
9
Y6
Y7
10
Y8
6
5
100Ω RN65C
-15V
1) 1N4001 or 1N4004 diodes are used on each relay coil
0.1uf C24
2) Relays S2 thru S11 are Aromat JR2-DC24V
(Newark Stock No. 46F5686)
3) The following DIN pins are to be tied to ground plane:
Row A: 5, 9, 10, 12, 13, 14, 17, 18, 19, 20,
21, 22, 24, 25, 26, 27, 28, 29, 30
+
4.7ufC25
Row C: 5, 10, 12, 14, 17, 18, 19, 20, 21, 22,
2
20kΩ RN55C
R55
11
Y9
16
1
A
15
2
B
14
8
D
4
C
13
5) Thick lines indicate wires which can carry 10+ amps for a 3 second pulse
12
6) Relays are shown in power-off position
0.1ufC28
7) Resistors R1 and R2 will be RN55 style. The PC Board should have
.060" diameter holes with .085" dia. pads at locations R1 and R2 for plug-in sockets
8) Capacitors C1 thru C6 will be CK05BX style. The PC Board should have
.060" diameter holes with .085" dia. pads at locations C1 thru C6 for plug-in sockets
Float
P2
C
V
F
C
2
C
4
A
4
A
7
C
11
Layout of Front Panel
Fabrication Notes:
SN7445 BCD-to-Decimal Decoder/Driver
8
7
6
Y5
DIN Row C, Pin #6
DIN Row A, Pin #8
8
U5
R54
1kΩ RN55C
Match with Rg
R38 R39
5
Y4
OPA633KP
R53
DC Supply
DIN Row A, Pin #6
1 +
4.7ufC23
4
BNC4
Probe Current Out
Isolated BNC - Front Panel
DIN Row C, Pin #8
R52
49.9Ω
RN55C
6
C20
0.1uf
0.1uf C22
100Ω RN65C
+15V
OPA627BP
R49
4
2
1
2Ω 5%
1/2W
(carbon)
0.1uf C18
7 +
4.7ufC19
3
R50
3
S8
4
1
4
Y2
+5V
P1
Isolated BNCs - Amphenol Type 31-10
(NEWARK Stock No. 38F1322)
BNC3
Shunt Resistor Select
P0
DIN
Row: C A C A C A C C
Pin#: 13 15 15 16 16 23 23 25
BNC2
Voltage
2
+15V
(2) 10Ω
RN55D
R48
R46
10Ω Trim
Bourns 3006P
6
3
D11
S7
2
R36
3
Y1
Voltage
Monitor
Gain
+DC
SB
R51
1kΩ RN55C
Match with Rh
7
5
6
10Ω
RN55D
R35
BCD
11
9
+
4.7uf C61
1
R41 (4) 20kΩ RN65C R43
Match group with Re
Y0
Decimal
Power Suply Select
-DC
3
D9
S5
SN7406 Hex Inverter Buffer/Driver
SA
2.0Ω 1%
5W GN-5C
6
2
4
2
1
1
49.9Ω
200Ω
Rd RN55C
RN55C
Match Rc
Match Ra
R34
0.1uf
U2
Ground
Switch
Power Supply Connections
R17
(2) 20Ω
RN65D
A
2
2
3
D8
S4
1
B
S11
7
R16
5
6
4
1
Rb
+24V
8
7
5
6
D3
2
Probe
DIN Row C, Pin #1
DIN Row A, Pin #3
C17
1
D5
+24V
8
3
BNC1
20kΩ RN55C
R47
S10
5
R15
49.9Ω
RN65C
Match Rd
200Ω R14
RN65C Rc
Match Rb
R13
Ra
3
"Float" switch
D6
6
Overcurrent
3 - 17pf
R31
1
4
100Ω RN65C
-15V
Probe Voltage Out
Isolated BNC - Front Panel
DIN Row A, Pin #1
DIN Row C, Pin #3
0.1ufC60
Arco No. PC402 trimmer capacitor
(4) 20kΩ RN65C
Match group with Rf
R20
6
8
R12
2
5
5
2
BNC3
7
4
7
U7
1
8
R30
S2
1
R11
BNC2
1 +
4.7uf C59
49.9Ω RN55C
R27
+
4.7uf C56
-15V
red LED - Dialight# 521-9246
Mounting clip - Dialight# 515-0004
0.1ufC58
OPA633KP
6
A
40:1 or 4:1 AC couple
voltage divide selection
499Ω RN55C
7
3
+15V
R29
47pf Mallory
CK05BX470K
C57
4
6
0.1uf C55
4
5
5
+24V
8
3
5
3 U8
Current Monitor
6
6
OPA637BP
1,6
6
3A Circuit Breaker
(external to PCB)
calibration resistor
7
S1
4
R23
5 Ω 50W
Power Resistor
(external to PCB)
100Ω RN65C
7 +
4.7uf C54
2
3
R1, R2, C3, C4, C5, C6 - mounted on DIP sockets, values to be determined after hookup to probe
Power Supply Switching
R28
1KΩ RN55C
0.1uf C53
4.99kΩ RN55C
402kΩ RN60C
R8 plug-in
sockets
0.1uf 50VDC Mallory
M20R104M5 (2 places)
4.7uf 50VDC Mallory
TDC475M050NLF (2 Places)
+
4.7uf CSD2
R24
R22
C16
402kΩ RN60C
75KΩ RN55C
0.1uf
Potential Monitor
392kΩ RN60C
0.1 uf 200VDC
Mallory M40R104K2
C6
R6
402kΩ RN60C
R5
C5
5
+15V
C52
Arco No. PC402
Trim Cap.
3-17pf
R2
6
52.3Ω RN55C
C1
plug-in
sockets
8
OPA633KP
USD1
CSD4
-15V
C1 - approx .2 - .5 pf (to be determined
C2 - approx 180 pf
during final tests)
R10
C12
180pf 200VDC
Mallory CK05BX181K
75KΩ RN55C
Thick lines indicate wires
which can carry 10+ amps
for a 3 second pulse
C15
1MegΩ Trim
Bourns 3299W
4
RSD2
RSD1
0.1 uf 200VDC
Mallory M40R104K2
1 +
4.7uf CSD1
56.2kΩ RN55C
R45
Arco No. C404
Trim Cap.
8-60pf
R1
CSD5
CSD6
680pf 200VDC
Mallory CK05BX681K
182Ω RN55C
R9
C14
680pf 200VDC
Mallory CK05BX681K
C9
+
4.7uf
Probe
In-Vacuum Triax Cable
Probe Cable
R0
R1
R2
C
7
C
9
A
11
A. Adjusting Current Monitor Null (zero current with no plasma)
DIN Row:
Pin #:
Overcurrent Protection
Probe Current Power Supplies
Isolated BNC on
120VDC Power Supply
+DC
Sweep A
R57
120 VDC Supply
(1 for each group
of 3 probes)
2
1
Audio Amp #1
(common to
all probes)
D13
Audio Amp #2
(common to
all probes)
Waveform Input #1
from CAMAC
Waveform Input #2
from CAMAC
3) Adjust trim capacitor (C17) to null out the part of the
current signal which is proportional to the derivative of the input
signal. ~50 kHz Triangle-wave input is best.
7
Vs
D14
MCT2E
OPA627BP
5
D15
C26
50VDC
+ 4.7uf
Mallory
TDC475M050NLF
2
Sweep B
+15V
10MegΩ RN60D
(4) 1N4004 or 1N4001 Diodes
D12
-DC
1) Apply a voltage waveform to sweep "A" or sweep "B" inputs. Note:
probe cables must be connected.
2) Select 0.5 ohm shunt resistor. Adjust 10 ohm trim (R46) to balance resistor
bridge and null out the part of the current signal which is directly
proportional to the input signal. ~100 Hz Square-wave input is best.
R56
20KΩ RN55C
U6
3
6
U9
4
-15V
4
6-pin DIP
Design Parameters:
Trigger on: Vs = -6V
Tirgger off: Vs = +12V
Discharge time (+12V to -6V) = 3 sec
Charge time (-6V to 12V) = 91 sec
-15V
15kΩ
RN55C
+15V R58
4.99kΩ RN55C
15kΩ
R60
RN55C
R59
10kΩ RN55C
B
R62
R61
1kΩ RN65C
4) Select 200 ohm shunt resistor. Try various R1, R2, and adjust 1MegΩ
potentiometers (R10 ,R8) to compensate for leakage resistance of triax cables.
Adjust ~100Hz A.C. leakage first using R1 & R10. Adjust D.C. leakage
(~1Hz) with R2 & R8. Squ
Q1
2N2222A
LED1
LED on
Front Panel
(Dialight 521-9246)
TO-18
Metal Can
Package
6) Adjust 50kΩ (R3) potentiometer to optimize high frequency null. A ~ 500kHz
sine-wave input is best.
B. Adjusting Voltage Monitor
21V x 4.7uf / 3sec => 33 uA
MCT2E Current Transfer Ratio = .2
=> .165 ma through LED => 30kΩ resistor for OPA633 output of 5V
1) Apply a voltage waveform to sweep "A" or sweep "B" inputs.
2) Try various C1 and C2 to optimize flatness of voltage monitor response
(in 4:1 AC mode) at high frequencies. (These capacitors compensate
for stray capacitance in S1.)
M.I.T. Plasma Fusion Center Flush-Mount Probe Electronics
Designed by B. LaBombard, revised 3/21/92.
175 Albany St.
power resistor and circuit breaker added 9/21/98
Cambridge, MA 02139
Tel.:(617) 253-6942 Fax: (617) 253-0627
286
Appendix C
Omegatron User’s Manual
The omegatron probe is a custom diagnostic. The diagnostician who operates the
omegatron will need to be familiar with its many components and their interaction.
Dozens of modular programs have been written in IDL to configure and control the
omegatron, the most important of which are described here; all the routines reside in
the directories listed below.
Most of the analysis of the data for the omegatron was performed to make specific
figures for this thesis. At the upper right hand corner of each figure is the full path
of the IDL routine that generates the plot. Most often the same routine that plots
the data also performs the analysis of the data.
To run the omegatron and analyze data from it, the following directories should
be in the diagnostician’s IDL path:
USER10:[NACHTRIEB.IDL]
USER10:[NACHTRIEB.OMEGATRON]
USER10:[NACHTRIEB.OMEGATRON.WIDGET]
USER10:[NACHTRIEB.OMEGATRON.WAVE]
USER10:[NACHTRIEB.OMEGATRON.ANALYSIS]
USER10:[NACHTRIEB.OMEGATRON.TITE]
287
Figure C.1: Omegatron power supply and motion control widget.
C.1
Operation Widgets
Figure C.1 shows the widget used to move the omegatron via CAMAC control, to
toggle the compressed air cooling, the RF amplifier, and the synchronous detection,
and to monitor the omegatron power supplies. Test shots may be taken by typing in
the shot number and pressing the “take shot” button; once the test shot cycle has
been completed the shot number is incremented. The “5900” button erases all test
shots in the range 5900–5999 and resets the shot number to 5900. The control widget
is run by typing
$ ccl remcam
$ idl
IDL> .r otron
IDL> otron
Figure C.2 shows the widget used to program the waveforms for the grid biases,
the RF power, and the RF frequency. Note that by supplying the magnetic field on
288
Figure C.2: Omegatron bias and RF waveform widget
289
axis the user can specify the range of M/Z to scan instead of the cyclotron frequency.
Several common waveform shapes can be requested with exclusive list buttons. The
user can request the waveform in units appropriate to the component (e.g. volts for
grid G1, watts for RF power). The resulting waveforms are drawn in a window
below; abscissa dimensions can be displayed in volts (output from Bira B5910) or in
desired units (volts, megahertz, watts, etc.). Custom waveforms can be drawn, saved,
restored, and edited with the field at the top. The waveform widget is run by typing
$ idl
IDL> .r waveform
IDL> .r otron_wave1
IDL> otron_wave1
Several custom waveform files already exist. To save or restore a file, type the
name without the .wvs extension.
Directory USER10:[NACHTRIEB.OMEGATRON.WAVE]
BORON.WVS;1
H_TO_D.WVS;1
IMS2.WVS;1
IMS_SLITGND.WVS;3
OPT_BIAS_H2D.WVS;1
OTRON_IMSRGA2.WVS;1
OTRON_RGA.WVS;3
TEST.WVS;1
TITE_IMS.WVS;2
TITE_SEE_G2.WVS;2
TI_G2.WVS;1
HELIUM.WVS;1
HELIUM_SLITGND.WVS;1
H_TO_D_ICRF.WVS;2 IMS.WVS;2
IMS_HE3.WVS;1
IMS_ICRF.WVS;2
IMS_T2.WVS;5
OPT_BIAS.WVS;4
OPT_BIAS_SLIT.WVS;1 OTRON_IMSRGA.WVS;4
OTRON_IMSRGA_4.WVS;1
OTRON_RGA2.WVS;1
RAMP_SLIT.WVS;2
TE_SEE.WVS;6
TITE.WVS;7
TITE_SEE.WVS;1
TITE_SEE_G1.WVS;1
TITE_SEE_G3.WVS;1 TI_G1.WVS;2
TI_G3.WVS;2
TI_SLIT.WVS;4
The IDL widget routines for omegatron control and waveform editing are listed
in the directory below. The raw widget files are listed as well; these may be edited
with IDL’s wided widget builder.
Directory USER10:[NACHTRIEB.OMEGATRON.WIDGET]
OTRON.PRO;81
OTRON_WAVE1.WID;7
OTRON.WID;9
OTRON_WAVE1.PRO;101
290
C.2
Analysis Widgets
Figure C.3 shows a widget for displaying IV characteristics of the retarding field
energy analyzer. Figure C.4 shows a widget for displaying impurity spectra collected
from the omegatron ion mass spectrometer. IDL source files for both analysis widgets
and the associated widget files may be found in:
Directory USER10:[NACHTRIEB.OMEGATRON.WIDGET]
OTRON_IVCHAR.PRO;66
OTRON_IVCHAR.WID;8 OTRON_RESONANCES.PRO;130
OTRON_RESONANCES.WID;9
The analysis widgets rely on other IDL routines found in
Directory USER10:[NACHTRIEB.OMEGATRON.ANALYSIS]
Directory USER10:[NACHTRIEB.OMEGATRON.TITE]
C.3
Generic Routines
Several generic routines written for data processing and analysis are listed below.
Note that some of these routines appear in almost every omegatron control or analysis
routine.
Directory USER10:[NACHTRIEB.IDL]
APPEND.PRO;14
ARRSTR.PRO;5
GAUSS.PRO;1
GET_NUM.PRO;4
NOTE.PRO;2
OPEN_SHOTS.PRO;5
OPLOTE.PRO;36
PLOTBOX.PRO;10
PLOTPSIGRID_ROB.PRO;7
PLOT_FFT_SPECTRUM.PRO;7
PRESS_RETURN.PRO;7 PUT_NUM.PRO;1
RA.PRO;5
SAVGOL.PRO;1
SPLI6.PRO;15
STR.PRO;1
SVDDN.PRO;9
SVDDP.PRO;7
SVDT.PRO;6
SVI_HE0.PRO;5
291
FFT_DERIV.PRO;9
GET_SIG.PRO;16
OPEN_TREE.PRO;7
PLOTE.PRO;4
PUT_SIG.PRO;2
SPB11.PRO;12
SVDD.PRO;2
SVDHE3.PRO;9
SVI_HE1.PRO;3
Figure C.3: Omegatron analysis widget for retarding field energy analyzer IV characteristics.
292
Figure C.4: Omegatron analysis widget for ion mass spectrometer spectra.
293
SVPB11.PRO;7
SVR_HE2.PRO;3
WLSF.PRO;9
C.4
SVPLI6.PRO;6
SVX_HEI.PRO;6
SVR_HE1.PRO;2
SVX_HEII.PRO;2
Control Routines
The control and waveform widgets enable the user to perform the most common
operations for the omegatron. For more detailed control dozens of routines are listed
below to be run from the IDL prompt.
Position control of the omegatron is possible from an IDL prompt. A testshot
procedure also exists which initializes all the omegatron hardware, triggers it, and
records all the data.
Directory USER10:[NACHTRIEB.OMEGATRON]
OMEGATRON_MOVE.PRO;44
OMEGATRON_TEST.PRO;34
The following routines take the desired parameter (e.g. a frequency in MHz, a
power in watts) and return the output Bira 5910 voltage necessary to obtain the
parameter. The inverse can be obtained as well (e.g. supply voltage, get frequency).
Dozens of small sub-routines are listed below, grouped approximately according to
function, which draw portions of waveforms, modify expressions in the tree, etc.
Directory USER10:[NACHTRIEB.OMEGATRON.WAVE]
F_RF.PRO;14
V_LP.PRO;8
P_RF.PRO;34
V_G.PRO;5
WV_EXP.PRO;5
WV_FTEST.PRO;2
WV_MW.PRO;8
WV_RAMP.PRO;3
WV_STEPS.PRO;7
WV_EXP2.PRO;5
WV_GATE.PRO;10
WV_MW2.PRO;4
WV_RAMPS.PRO;5
WAVEFORM.PRO;225
WAVE_1.PRO;10
WV_DOUBLEV.PRO;4
WV_EXP_T.PRO;5
WV_HILL.PRO;3
WV_MZ.PRO;17
WV_STEP.PRO;2
294
WAVE_1FREQ.PRO;5
WAVE_1RAMP.PRO;1
WAVE_1RAMP_PRF.PRO;1
WAVE_2FREQ.PRO;29
WAVE_5.PRO;4
WAVE_COPY.PRO;1
WAVE_FLAT_ALL.DAT;3
WAVE_GRIDRAMP.PRO;5 WAVE_GRIDRAMP2.PRO;2
WAVE_IMS.PRO;8
WAVE_IMSRAMP.PRO;16 WAVE_IMSRAMP_2.PRO;10
WAVE_ISAT.PRO;3
WAVE_LP.PRO;1
WAVE_MULTIGAUSS.DAT;7
WAVE_MULTIGAUSS.PRO;3
WAVE_NOCURRENT.PRO;1
WAVE_ONE.PRO;3
WAVE_OTRON.PRO;47 WAVE_OTRON_2.PRO;6
WAVE_PRFSCAN.PRO;27 WAVE_PRFVAR.PRO;3 WAVE_PULSE.PRO;19
WAVE_SIGNAL_TO_NOISE.PRO;3
WAVE_SLOW.PRO;7
WAVE_TI.PRO;9
WAVE_TI2.PRO;3
WAVE_TITE.PRO;41
WAVE_TITE2.PRO;4
WAVE_TITE3.PRO;1
WAVE_TRANSMISSION.PRO;4
WAVE_TRANSMISSION2.PRO;2
APPEND_SIG.PRO;5
FILL_SIG.PRO;4
C.5
CALIB_FREQ_3PT.PRO;12
MAKE_SIG.PRO;4
Control Routines
Finally, several useful scopes can be found in the following directory.
Directory USER10:[NACHTRIEB.SCOPES]
OMEGATRON_ALL.DAT;63
BIRDY_CHECK.DAT;5 BMOD.DAT;3
GLOBAL_PARAM.DAT;59
OMEGATRON_LP.DAT;13
DISRUPTION.DAT;3
295
296
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