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This is to certify that the
dissertation entitled

MINIATURE MICROWAVE PLASMAS OF HYDROGEN AND
ARGON INVESTIGATED USING OPTICAL EMISSION
SPECTROSCOPY

presented by

DAVID STORY

has been accepted towards fulfillment
of the requirements for the

PHD degree in ELECTRICAL ENGINEERING

 

W

’IVlajor Professor’s Signature

Mdy 26. 2006
/ I

Date

MSU is an Alfinnative Action/Equal Opportunity Institution

LIBRARY
Michigan State
University

 

 

 

 

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PLACE IN RETURN Box to remove this checkout from your record.
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MINIATURE MICROWAVE PLASMAS OF HYDROGEN AND ARGON
INVESTIGATED USING OPTICAL EMISSION SPECTROSCOPY

By

David Story

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements

for the degree of

DOCTOR OF PHILOSOPHY

Department of Electrical Engineering

2006

Abstract

MINIATURE MICROWAVE PLASMAS OF HYDROGEN AND ARGON
INVESTIGATED USING OPTICAL EMISSION SPECTROSCOPY

By

David Story

Research on miniature microwave plasmas is motivated in part by the interest in
generating on-chip plasma sources for applications such as miniature spectroscopy,
sterilization of on-chip laboratories, and local area plasma-assisted etching and chemical
vapor deposition.

The goal of this work is to determine the properties of miniature plasma
discharges generated by microwave energy. Specifically, small discharges of argon and
hydrogen with volumes of less than 1 cubic centimeter are investigated. Various
properties of the plasma discharges are measured including plasma gas temperature,
electron density, and internal plasma electromagnetic field strength.

The discharges are measured across a wide pressure range from 0.1 Torr to over
100 Torr using non-invasive optical emission spectroscopy techniques. Specific optical
emission diagnostic techniques utilized includes Stark broadening of atomic hydrogen
emissions to determine electron density, molecular hydrogen rotational temperature,
Zeeman splitting in molecular hydrogen emissions to determine both the microwave
magnetic field strength and the plasma temperature.

Modeling of the plasma discharges is also done using particle and energy balance

equations.

ACKNOWLEDGMENTS

I would like to thank my advisor, Dr. Timothy Grotjohn, and collaborators Dr. I es
Asmussen and Dr. Donnie Reinhard for the opportunity to work in the plasma laboratory
in the Department of Engineering Research at Michigan State University, and Mr. Terry

Casey for generously supplying unlimited access to the machine shop.

iii

Table of Contents

List of Tables .................................................................................................................... vii
List of Figures .................................................................................................................. viii
Chapter 1 Introduction ..................................................................................................... 1
Chapter 2 Background ..................................................................................................... 5
2. 1 Miniature Plasma Sources ................................................................................... 5
2.1.1 Micro-Cell Plasma Display Panels ............................................................. 5
2.1.2 Micro-Strip Line Sources ............................................................................ 8
2.1.3 Capacitive Sources .................................................................................... 10
2.1.4 Inductive Sources ...................................................................................... 12
2.1.5 Microwave Torch and Arc Discharges ..................................................... 13
2.1.6 Micro-Hollow Cathode Tubes .................................................................. 16

2. 2 Microwave Plasma Sources .............................................................................. 20
2.2.1 2.45 GHz Microwave Plasma Cavity Resonator ...................................... 21
2.2.2 Surface Wave Plasma Reactor .................................................................. 23
2.2.3 Electron Cyclotron Resonance (ECR) Reactor ......................................... 23
Chapter 3 Experimental Setup ....................................................................................... 27
3. 1 Miniature Microwave Plasma System .............................................................. 27

3. 2 Plasma Diagnositics .......................................................................................... 33
3.2.1 Optical Emission Spectroscopy ................................................................ 33
3.2.2 Optical Emission Spectroscopy Design .................................................... 38
3.2.3 Optical Emission Spectroscopy Test ........................................................ 43

3. 3 Preliminary Findings ......................................................................................... 43
3.3.1 Preliminary Experiments .......................................................................... 45
3.3.2 Preliminary Diagnostic Results ................................................................. 45
Chapter 4 Global Model Theory .................................................................................... 49
4. 1 Global Model .................................................................................................... 49
4.1.1 Low Pressure, Steady-State Approximations ........................................... 53
4.1.2 Intermediate Pressure, Steady-State Approximations ............................... 55
4.1.3 High Pressure, Steady-State Approximations ........................................... 58
Chapter 5 Spectroscopy Theory: Zeeman Effect ........................................................... 66
5. 1 Introduction: Quantum Theory ......................................................................... 67
5. 2 Eigenvectors ...................................................................................................... 69
5.2.1 Harmonic Oscillator .................................................................................. 70
5.2.2 Central Potential ........................................................................................ 74
5.2.2.1 Angular Momentum Operators ............................................................. 74

5.2.2.2 Central Potential Hamiltonian ............................................................... 82

5. 3 Electron Spin ..................................................................................................... 84

iv

5. 4 Angular Momentum Addition: C lebsch-Gordan Coefficients .......................... 88

5.4.1 Diatomic Hydrogen: Clebsch-Gorden Coefficients .................................. 91
5.4.2 Atomic Hydrogen: Clebsch-Gorden Coefficients ..................................... 96

5. 5 Perturbation Theory .......................................................................................... 96
5.5.1 Spin-Orbit Interaction ............................................................................. 100
5.5.2 Relativistic Mass Correction ................................................................... 104
5.5.3 Anomalous Zeeman Effect ..................................................................... 105

5. 6 Rotational Spectrum for Diatomic Hydrogen ................................................. 108
5. 7 Fine Structure of Atomic Hydrogen ............................................................... 110
5. 8 Nominal Fine Structure Transition Intensity .................................................. l 10
Chapter 6 Atomic Hydrogen: Stark Effect .................................................................. 122
6. l Stark Splitting: Spherical Coordinates ............................................................ 123
6.1.1 Perturbation Matrix ................................................................................. 123
6.1.2 Stark Energy Spectrum: Spherical Coordinates ...................................... 131

6. 2 Parabolic Coordinates ..................................................................................... 131
6.2.1 Parabolic Transform ................................................................................ 134
6.2.2 Runge-Lenz Vector ................................................................................. 139
6.2.3 Parabolic Energy Levels and Wave Functions ....................................... I43

6. 3 Stark Effect Perturbation ................................................................................. 146
6.3.1 Stark Effect: Parabolic Wave Functions ................................................. 147
6.3.2 Stark Effect: Fine Structure .................................................................... 151

6. 4 Coordinate Transforms ................................................................................... 157
6.4.1 Parabolic Ladder Operators .................................................................... 157
6.4.2 Clebsch-Gordan Coefficients .................................................................. 161
6.4.3 Semi-Parabolic Coordinates .................................................................... 163
Chapter 7 Results ......................................................................................................... 168
7. l Spectrometer Set-Up ....................................................................................... 168
7. 2 Argon Results .................................................................................................. 171
7.2.1 Global Model Results ............................................................................. 172
7.2.2 Argon Spectroscopy Measurements ....................................................... 172
7.2.3 Argon 4300.1 A Line Shape ................................................................... 178

7. 3 Hydrogen Results: Diatomic Hydrogen .......................................................... 178
7.3.1 Diatomic Hydrogen: Rotational Spectrum .............................................. 184
7.3.1.1 Diatomic Hydrogen Temperature: Interband Transitions ................... 184
7.3.1.2 Diatomic Hydrogen Temperature: Intraband Transitions ................... 191
7.3.2 Diatomic Hydrogen: Zeeman Shift ......................................................... 205

7. 4 Hydrogen Results: Atomic Hydrogen ............................................................. 206
7.4.1 Atomic Hydrogen: Stark Shift ................................................................ 210
7.4.1.] Stark Shift: Ha ..................................................................................... 211
7.4.1.2 Stark Shift: Hg ..................................................................................... 218
7.4.1.3 Stark Shifi: Hy ..................................................................................... 226
7.4.2 Electron Density ...................................................................................... 229
Chapter 8 Conclusion .................................................................................................. 233
8. 1 Experimental Results ...................................................................................... 233

V

 

8.1.1 Results: Electric Field Polarization ......................................................... 233

8.1.2 Results: Electric Field Magnitude ........................................................... 234
8.1.3 Results: Atomic Hydrogen Spectral Resolution ..................................... 238

8. 2 Discussion ....................................................................................................... 242
Appendix A Plasma System and Components ............................................................ 244
Appendix B Fiber Optic Feed-Through ...................................................................... 250
References ....................................................................................................................... 252

vi

List of Tables
Table 1 Global Model Predictions for Argon Plasmas. .................................................. 173
Table 2 Global Model Predictions for Argon Plasmas. .................................................. 174
Table 3 Global Model Predictions for Argon Plasmas. .................................................. 175
Table 4 Hydrogen Rotational Transitions. ...................................................................... 187
Table 5 Hydrogen Rotational Energy Levels .................................................................. 197

vii

List of Figures

Figure l Micro-Cell Plasma Display Panels. ...................................................................... 6
Figure 2 Micro-Stripe Line Sources. .................................................................................. 9
Figure 3 Miniature Mass Spectrometer (Capacitive). ....................................................... l I
Figure 4 Inductive Sources. .............................................................................................. 14
Figure 5 Torch and Arc Discharges. ................................................................................. 17
Figure 6 Micro-Hollow Cathode Tubes. ........................................................................... 18
Figure 7 2.45 GHz Microwave Plasma Assisted CVD Reactor. ...................................... 22
Figure 8 Surface-Wave Microwave Plasma Assisted CVD Reactor. ............................... 24
Figure 9 Electron Cyclotron Resonance CVD Reactor. ................................................... 26
Figure 10 Experimental System. ....................................................................................... 29
Figure 11 Plasma Source. ................................................................................................. 30
Figure 12 Optical Emission Diagnostics ........................................................................... 35
Figure 13 Observed Electronic States (H2). ...................................................................... 36
Figure 14 Rotation Temperature (Gas). ............................................................................ 37
Figure 15 Optical Emission Preliminary Design. ............................................................. 39
Figure 16 Optical Emission Spectroscopy Design ............................................................ 40
Figure 17 Optical Emission F eed-through. ....................................................................... 41
Figure 18 Spherical Lens Design. ..................................................................................... 42
Figure 19 H3 (100 mT) Optic Test. ................................................................................... 44
Figure 20 Power Density in Argon Plasma ....................................................................... 47
Figure 21 Experimental Argon Ignition Power. ............................................................... 48
Figure 22 ngden~ vs. Te for Maxwell Electrons in Argon. .................................................. 59
Figure 23 Collisional Energy Loss vs. Te in Argon. ....................................................... 65

viii

Figure 24 Angular Momentum Operator: Spherical Coordinates ..................................... 77

Figure 25 Angular Momentum Diatomic Hydrogen. ....................................................... 90
Figure 26 Precession of Vibrating Diatomic. ................................................................... 94
Figure 27 Zeeman Energy Levels. .................................................................................. 109
Figure 28 Ha Fine Structure Transitions. ........................................................................ 1 l 1
Figure 29 Hp Fine Structure Transitions. ........................................................................ l 12
Figure 30 Ha Fine Structure Relative Transition Intensity. ............................................ l 13
Figure 31 Ha Fine Structure Peaks Near Band Center .................................................... 120
Figure 32 Ha Fine Stucture: Absorption Spectroscopy, Pulsed Dye Laser. ................... 12]
Figure 33 Ha Stark Energy Spectrum: Spherical Coordinates ........................................ 132
Figure 34 H3 Stark Energy Spectrum: Spherical Coordinates. ....................................... 133
Figure 35 Hyperbolic Transform of Constant z Surfaces. .............................................. 135
Figure 36 Parabolic Transform in the Complex Plane .................................................... 136
Figure 37 Classical Relationships for Runge-Lenz Vector ............................................. 140
Figure 38 Ha Stark Effect Transitions: Parabolic Coordinates. ...................................... 149
Figure 39 H5 Stark Effect Transitions: Parabolic Coordinates. ...................................... 150
Figure 40 Ha Stark Effect Fine Structure Splitting. ........................................................ 156
Figure 41 Semi-Parabolic Coordinate Representation of Stark, Zeeman Effect. ........... 167
Figure 42 PMT Noise Response. .................................................................................... 170
Figure 43 Global Model Predictions for Argon Plasma Electron Temperature. ............ 176
Figure 44 Global Model Predictions for Argon Plasma Electron Density. .................... 177
Figure 45 H5 Line, P2100 T, 60 W., FWHM=2.025 A. ................................................. 179
Figure 46 Argon Electron Density (40 W). .................................................................... 180
Figure 47 Argon Line Shape, 4300.] A., 100 Torr. ........................................................ 181
Figure 48 Argon Line Shape Ll. .................................................................................... 182

ix

Figure 49 Argon Line Shape L2. .................................................................................... 183

Figure 50 Hydrogen Rotational Spectrum. ..................................................................... 186
Figure 51 Fortrat Plot. ..................................................................................................... 189
Figure 52 Rotational Energy Transition. ........................................................................ 190
Figure S3 Rotational Temperature vs. Pressure. ............................................................. 192
Figure 54 H2 Zeeman Splitting: Tight BPF, 0.5 Torr. .................................................... 194
Figure 55 H2 Zeeman Splitting: Relaxed BPF, 0.5 Torr. ................................................ 195
Figure 56 B Field (B=35 mT). ........................................................................................ 199
Figure 57 B Field vs. Pressure. ....................................................................................... 200
Figure 58 Intraband Energy Distribution. ....................................................................... 202
Figure 59 Intraband Temperature vs. Pressure. .............................................................. 204
Figure 60 H2 Zeeman Splitting: Cross-Correlation Filter, 0.5 Torr. ............................... 207
Figure 61 H2 Zeeman Splitting: 5.0 Torr, 60 W. ............................................................ 208
Figure 62 H2 Zeeman Splitting: 50 Torr, 60 W. ............................................................. 209
Figure 63 Ha Parabolic Transition Intensities ................................................................. 212
Figure 64 Ha Spectral Response: Gross and Fine Structure, 50 Torr. ............................ 213
Figure 65 Ha Experimental Spectrum, 50 Torr ............................................................... 214
Figure 66 Ha Spectral Response: 0-4000V. Continuum. ................................................ 215
Figure 67 Ha Experimental Spectrum, 5.0 Torr .............................................................. 216
Figure 68 Ha Experimental Spectrum, 0.5 Torr .............................................................. 217
Figure 69 H3 Parabolic Transition Intensities. ................................................................ 220
Figure 70 H3 Spectral Response: Gross and Fine Structure, 50 Torr. ............................ 221
Figure 71 H5 Experimental Spectrum, 50 Torr. .............................................................. 222
Figure 72 Hp Spectral Response: 0-4000V. Continuum. ................................................ 223
Figure 73 H3 Experimental Spectrum: 5.0 Torr .............................................................. 224

X

Figure 74 HB Experimental Spectrum, 0.5 Torr. ............................................................. 225

Figure 75 Hr Parabolic Transition Intensities. ................................................................ 227
Figure 76 Hy Experimental Spectrum (Gross Structure), 50 Torr. ................................. 228
Figure 77 Electric Field vs. Pressure. ............................................................................. 230
Figure 78 HB Spectrum, 100 Torr, 60 W., FWHM=1.241 A. ........................................ 231
Figure 79 Hydrogen Electron Density vs. Pressure (60 W) ............................................ 232
Figure 80 Electric Field Polarization for Molecular Hydrogen. ..................................... 235
Figure 81 Electric Field in Resonant Reactor Chamber. ................................................ 239
Figure 82 Miniature Microwave Plasma System. ........................................................... 244
Figure 83 Gas Flow Meter Bank (4 Channel) ................................................................. 245
Figure 84 Electronics Control Board. ............................................................................. 245
Figure 85 Plasma Reactor Chamber. .............................................................................. 246
Figure 86 Fiber Optic Feed-Through (13 Channels). ..................................................... 247
Figure 87 Optical Fiber Micro-Positioner (OES). .......................................................... 247
Figure 88 Hydrogen Plasma; 0.5 Torr, 60 W .................................................................. 248
Figure 89 Hydrogen Plasma; 5.0 Torr, 60 W .................................................................. 248
Figure 90 Hydrogen Plasma; 10.0 Torr, 60 W ................................................................ 249
Figure 91 Hydrogen Plasma; 50 Torr, 60 W ................................................................... 249
Figure 92 Reactor Chamber with Fiber Optic F eed-Through ......................................... 250
Figure 93 Fiber Optic Feed-Trough. ............................................................................... 250
Figure 94 Feed-Trough Micro-Lens System (13 Channels). .......................................... 251
Figure 95 Feed-Through Construction Tool Set. ............................................................ 251

xi

 

Chapter 1 Introduction

The creation and characterization of miniature microwave plasma sources is a
relatively new and under-investigated field. Some miniature plasma sources have been
developed for the pixel cells in flat panel displays, as well as to investigate the possibility
of bringing mass spectrometry and optical emission spectroscopy functions to the
computer processor unit (CPU). However, none of the previously mentioned sources are
created with microwave power, which allows for more flexible geometries and a wider
range of pressure variations.

The first objective of this investigation is to establish the operating conditions
for a microwave plasma source that allows the creation of miniature discharges, and then
to measure the properties of the resulting plasma discharges. An additional objective is
to develop a predictive understanding of miniature microwave plasma behavior by using
plasma global models, and by comparing model results to the measured plasma
properties. The overall goal is to add to the scientific understanding and engineering
design principles for miniature microwave discharges.

To this end, investigations are performed in both noble and molecular gases
(argon and hydrogen) across a range of pressure and microwave powers. The
investigation includes the implementation of instrumentation for non-invasive optical
emission spectroscopy. The plasma discharge properties focused on in this investigation
include discharge shape and size, plasma power density, plasma electron density, plasma

gas temperature, and electric and magnetic field strength in the plasma.

In this investigation, hydrogen and argon plasmas are formed at pressures
ranging from 0.1-100 Torr, and at powers from 5-60 W. To obtain both high optical
emission sensitivity and spectral resolution a special optical system is designed to bring
lenses to within 5 mm of the plasma center. The optical system permits non-invasive
measurements of the intense plasma discharges.

Argon discharges are analyzed experimentally to determine plasma density and
plasma discharge power density. Two techniques are compared to determine the electron
density from argon discharges.

Analysis of hydrogen data was extensive, including plasma discharge size and
shape, plasma power density, plasma electron density, plasma gas temperature, and
electric and magnetic field strength in the plasma. Optical spectrum measurements reveal
peaks in the diatomic hydrogen rotation spectrum used to estimate rotational temperature.
Higher resolution measurements of the sub-band structure of diatomic hydrogen were
used to determine resident magnetic fields consistent with Zeeman splitting. This
suggests hydrogen plasmas have a partially discrete or constant magnitude magnetic field
component, which varies with pressure.

Atomic hydrogen spectroscopic readings demonstrated sub-band structure as
well. Peaks within the hydrogen alpha, beta, and gamma bands were consistent with
energy level splitting seen in the Stark effect. As a result, the magnitude of the resident
electric field was estimated across the pressure regime.

Chapter 2 provides a background for the study of miniature microwave plasma

sources by presenting the current state of miniature plasma sources.

Chapter 3 presents the experimental set-up. The experimental set-up includes
designs and builds for both the plasma system and the diagnostic system, a multi-channel
fiber optic feed-through. The diagnostic set-up required the build of a new optics system
to penetrate the reactor and focus on the center of the plasma discharge. Chapter 3
concludes with test results for both the plasma reactor system and fiber optic feed-
through.

Chapter 4 presents the global model, a theory that describes the plasma physics
of monotonic gases such as argon. Low to medium pressure plasmas can be described
accurately with the global model. The global model is found ineffective at higher
pressures; this was substantiated on preliminary test sets made during initial system
testing.

Chapter 5 presents the spectroscopic theory for diatomic molecules and for
single electron atoms. Both sets of theory are directly applicable to hydrogen plasmas.

Chapter 6 applies spectroscopic theory from Chapter 5 to predict the peak
amplitude and splitting in atomic and rotational spectra associated with hydrogen.
Chapter 6 introduces spectral theory specific to hydrogen-like (Rydberg) atoms, without
which determination of the Stark spectrum would be impossible.

Chapter 7 accumulates the experimental results, and makes direct comparisons
between the experimental results and predictions made by the global model in Chapter 4
and the spectroscopy theory developed in Chapter 5 and Chapter 6. Chapter 7 records
experimental results for diatomic and atomic hydrogen spectra, and matches these results

to the Zeeman and Stark effects developed in Chapter 5 and Chapter 6.

Chapter 8 summarizes project results, and lays the groundwork for future
experiments aimed to get at the root of plasma behavior. Chapter 8 also suggests future
experimental techniques to provide more insight into the nature of the hydrogen plasma

behavior, specifically hi gh-pressure contraction.

 

Chapter 2 Background

The current research activity in miniature plasma sources and microwave plasma
sources is presented in the following two subchapters. The miniature microwave plasma
source designed for this project is detailed in Chapter 3. The miniature microwave
plasma source design is similar to larger microwave plasma sources, but requires

fundamental knowledge of miniature source operation to be successful.

2. 1 Miniature Plasma Sources

This brief overview presents the current state of miniature plasma sources. The
following plasma sources will be discussed in the proceeding paragraphs: Micro-cell
plasma display panels, micro-strip line sources, capacitive sources, inductive sources,

torch and are discharges, and micro-hollow cathode tubes.

2.1.1 Micro-Cell Plasma Display Panels

Micro-cell plasma display pixel cells consist of two parallel glass plates fitted
with electrodes on their surfaces, as shown in Figure 1 [1]. Each electrode is covered by
thin dielectric layer and coated with MgO. The cell is filled with various combinations of
Xenon, Neon, Helium, and trace amounts of Argon. The cell is sealed; the cell pressure
can vary from 100 torr to 500 torr, depending on other cell parameters including gas
mixture and excitation frequency. The cell is approximately 1 mm cubed in dimension.

The MgO layer produces secondary electrons on impact by electrons, greatly

multiplying the number of electrons in the plasma and the number of collisions that

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generate excited radicals. In fact, secondary electron emission is by far the main source
of electrons in micro-cell plasma displays. The MgO layer provides a high secondary
electron emission rate, and hence increases the cell efficiency rate. The cell efficiency
rate is defined as the ratio of the power absorbed per unit volume that produces excited
states of Xenon to the total power absorbed per unit volume.

The breakdown voltage, or the voltage necessary to ignite the plasma, and the
self-sustaining voltage are a function of the ionization energy of the fill gas, the
frequency, the cell capacitance, lifetimes for each of the gas species, and the secondary
electron emission rate of the MgO layer. The typical breakdown voltage for a cell is
approximately 300 volts. During self-sustaining operation, at a frequency of 50 kHz, the
plasma electron density is approximately 10'1-1012 electrons per cubic centimeter [2].

Application of a high voltage pulse across the electrodes initiates the plasma
discharge. The energetic free electrons excite Xenon atoms through atomic collision.
Excited Xenon atoms release photons as electrons fall from higher energy states
(resonant, excimer, and metastable states) to the ground state. The photons are emitted in
the ultraviolet range. The ultraviolet radiation reacts with the phosphor coating on the
cell walls, which converts the ultraviolet light into visible light- red, green, or blue,
depending on the type of phosphor coating.

Investigation shows that the mixtures relatively lean in Xenon produce the lowest
breakdown voltages while still delivering high ionization rates. Neon-Xenon and
Helium-Xenon ratios of 95%-5% reduce the breakdown voltage from 300 volts,
necessary for 100% Xenon cells, to approximately 125 volts. Xenon efficiency rates

peak at 90% for 100% Xenon cells, and drop-off moderately to approximately 70% as the

Xenon concentration is reduced to 5%. Cell efficiency rates are higher for Neon-Xenon
mixtures than for Helium-Xenon mixtures in all concentrations. The effect of Argon as
an additive is negligible [3].

Researchers have also studied the optimum shape and operating frequency of
these plasmas. Two-dimensional modeling of the plasma cell predicts that much higher
cell efficiency rates and electron densities can be achieved with a cylindrically shaped
cell operating at radio frequencies (13.56 MHz) [4]. The cylindrical shape geometry
allows for greater plasma volume for a given surface area. As a result, the cell can be
made smaller, and the necessary breakdown voltage and self-sustaining voltage reduced.

The advantages of smaller size cannot be realized if the frequency is not increased
as well. Although much less mobile than the electrons, ionized Helium still has enough
time to pass through the sheath to the walls at relatively low frequencies. Applying radio
frequency voltage helps trap the Helium ions in the reduced plasma volume. As a result,
plasma electron densities can be increased by a factor of five to ten, reaching 1.0 X 10'3

electrons per cubic centimeter.

2.1.2 Micro-Strip Line Sources

Miniature microwave frequency plasma sources are targeted for on-chip
applications, including micro-strip line technology. Micro-strip line sources, as shown in
Figure 2 [5], consist of approximately one-millimeter square channels in fused silica
dielectrics, or simply 0.3-1 mm silica tubes, and the corresponding ground plane and
microwave matching elements formed on the top and bottom of the channel. Argon is
flowed through the channel; the plasma is ignited with a piezoelectric sparking device

and sustained with approximately 15 Watts of microwave power at 2.45 GHz.

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The resulting plasma is very bright as viewed looking into the open-ended
channel. The micro-strip line plasma has been demonstrated at one atmosphere, allowing
contaminants to be introduced from the environment, and hypothetically, detected by
matching the contaminant to its atomic emission spectra. The benefit of a source of this

type is an on-chip optical emission or atomic emission spectrometer.

2.1.3 Capacitive Sources

A simple plasma source geometry is that of the capacitive source. In general, a
high DC, rf (13.56 MHz) or microwave (2.45 GHz) voltage is set up across parallel
plates. The resulting electric field ionizes neutrals, producing ions and free electrons.
The free electrons accelerate under the influence of the electric field, and collide with
neutrals and ions. If the free electrons are given sufficient energy, these collisions
generate more free electrons, and the plasma becomes self-sustaining.

In DC discharges, electron acceleration is strictly a function of the applied electric
field and the mean free path of the electron, which is a function of pressure. In an RF or
microwave power discharge, the effective mean free paths can be made shorter if
collisions reverse the electron momentum at a frequency roughly equal to the frequency
of the applied electric field. Optimal coupling occurs when the frequency of the applied
power matches the electron collision frequency, which occurs at a pressure of
approximately 5 torr for an applied RF power at 13.56 MHz.

One specific capacitive source application is the miniature mass spectrometer, as
shown in Figure 3 [6]. The plasma is coupled to the incoming gas by accelerating plasma
electrons through a two-grid electrode system. The plasma electrons are focused into a

narrow beam as they enter the sample gas ionization chamber to keep the ion

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energy distribution as narrow as possible. The electrons ionize the gas sample; the ions
are then accelerated and deflected as they travel along the mass spectrometer channel by
a series of alternating voltage pulses synchronized to periodically spaced terminals.

Microwave power is the preferred source for two reasons. Firstly, sputter damage
to the plasma cell walls is reduced as the plasma ions are trapped by the high frequency
electric fields. Secondly, and more importantly, the high frequency electric fields used to
generate the plasma discharge have a negligible effect on the heavy ions in the mass
spectrometer channel. Obviously, since the theory of operation for the mass spectrometer
is the ionization, acceleration, deflection, and accurate deflection detection of the gas
species, spurious electric fields must be avoided or the entire system will be
compromised.

Technically, the micro-cell plasma display discussed in section 2.1.1 is a
capacitive plasma source. Also, the micro-strip line plasma source presented in the

previous section can be generated as a capacitve discharge or a surface wave discharge.

2.1.4 Inductive Sources

Large-scale inductive sources dominate the microchip fabrication landscape.
Miniature inductive plasma sources could be used as part of a microprocessor based
emission spectrometer or mass spectrometer, or could be the basis for thrust generation in
ion beam drives for space propulsion. Recent work has demonstrated the ability to create
5 mm, 10 mm, and 15 mm diameter planar inductively coupled plasmas (ICPs) at
pressures below 10 torr, powered by 1-20 Watts RF power between 13.56 MHz-500

MHz.

12

Miniature planar ICPs, as shown in Figure 4 [7], are constructed by masking off a
20-turn spiral pattern, 15 mm in diameter for the largest of the three sources. The planar
spiral is fixed directly above a 1.8-mm glass window, which contains the plasma. Two
high-Q capacitors are placed in series with the helix to adjust the tuning; the tuning is
effected by the inter-winding capacitance. The plasma containment vessel is filled with
Argon or air, and operated at pressures between 0.01 torr and 10 torr.

The miniature ICP sources accurately follows the same trends for plasma
potential, electron temperature (when the plasma sheath is correctly removed from the
calculation), and ignition frequency (electron elastic collision frequency equals rf source
frequency) as do large-scale ICPs. But, both experimental Langmuir probe and
interferometer measurements (35 GHz) yield electron densities (approximately 1.0 x 10'0
to 1.0 x 10H electrons per cm3) which are an order of magnitude lower than that predicted
by global plasma models. This discrepancy is thought to be a function of wall
recombination, resulting from the relatively low volume to surface area ratio. Similar

effects were mentioned previously in the low frequency micro-cell plasma display cell.

2.1.5 Microwave Torch and Arc Discharges

Torch and are discharges have been investigated for over four decades. In present
torch and are configurations, gas is forced through a small (~lmm) diameter nozzle
supersonically, and ignited by microwave power. The resulting plasma can take two
forms in general, corona and torch. The plasma forms as the high electric field at the
electrode or nozzle tip accelerates electrons into neutrals at a high enough velocity to
ionize the neutrals. This form is known as the corona, and is concentrated at the very tip

of the nozzle.

13

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As the plasma slowly begins to absorb more power, the vibration energy and
translation energy in the gas increases, as well as the ionization. This reduces the
effective electric field near the nozzle, gradually extinguishing the corona form of the
plasma near the nozzle, but exciting the working gas that is farther from the nozzle. The
plasma appears as a flame, with a hollowed center where the corona discharge is
extinguished. This form of the plasma is called the torch. The electric field present when
the corona discharge forms is approximately 14,000 volts per centimeter; the electric
field in the torch discharge is approximately 300 volts per centimeter [8]. Electron
temperatures in the range of 5000-5200K have been recorded for similar experimental
sets [9].

As the gas flow rate is increased, the gas will flow around the torch, and the
vibration temperature and translation temperature of the gas will be reduced due to gas
cooling. The torch effectively runs out of fuel, the microwave energy again begins to
accelerate electrons near the tip, and the corona form of the plasma returns as the torch
appears to be blown out.

When the microwave power dissipated in the torch discharge increases above a
critical point, the sharp electrode edge is heated to produce thermionic electron emission.
The resulting plasma looks more like a controlled arc than a torch, and is referred to as an
arc torch discharge. The thermionic emission provides enough electrons to prohibit the
return of the corona plasma form, stabilizing the discharge.

Another attempt to stabilize the torch discharge is the introduction of a conical
nozzle that contains, or redirects the working gas such that the plasma consumes nearly

all of the flowed reactant. Typical electron densities, as measured by the resonant

15

frequency and bandwidth shift, registered approximately 1.0 x 109 to 1.0 x 10” cm’3 for
this form of the plasma torch, operating at one atmosphere. This variation of the torch
discharge is shown in Figure 5 [10].

Torch or are torch discharges have been developed for pressures ranging from 0.5
torr up to one atmosphere. Torch discharges have been used for surface treatment and
cleaning, and for thin film depositions on internal cavity walls, holes, vias, and on
substrates of complex shape.

A variation of the torch or are torch is the microwave powered plasma pencil
[11]-[13]. The experimental set-up is similar to that given for the microwave torch and
are discharge. The difference is that the plasma pencil utilizes the gas delivery tube as a
hollow cathode to supply the microwave power. Research in this area includes attempts
to focus the plasma beam with a high-current magnetic lens system. This is similar to the
focusing achieved in modern microscopy, such as the electron microscope. Plasma
diagnostics of the plasma pencil yielded electron temperatures from 5200K to 5800K,

with gas temperatures on the order of 700K to 950K, operating at one atmosphere [13].

2.1.6 Micro-Hollow Cathode Tubes

The micro-hollow cathode tube refers to a structure, as shown in Figure 6 [14], in
which the plasma form-s between a hollow cathode and an arbitrarily shaped anode. The
micro-hollow cathode tube is characterized by an initial pre-discharge. The initial pre-
discharge plasma is shaped by the electric field. As the applied DC voltage and current
are increased, the pre-discharge forms a column extending from the hollow cathode to the

anode.

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The pre-discharge potential is pinned to the anode. As a result, the electrons
follow electric field lines and accelerate radially inward. When the pressure is such that
the mean free path of the electrons closely matches the diameter of the hollow tube, the
electrons (fast electrons) gain enough energy to ionize the gas species and form a
negative potential discharge within the hollow cathode tube. The electrons (fast
electrons) oscillate between the negative discharge and negative cathode [15].

The electrons (fast electrons) generate ions and electrons on collision with the gas
species. Ions and electrons follow field lines axially along the hollow cathode tube. As a
result of these interactions, the plasma potential drops as the current through the plasma
increases. This regime, where the effective resistance of the plasma is negative, is the
normal operating regime and referred to as the ‘hollow cathode discharge’.

The hollow cathode discharge often has a spherical shape, confined by the hollow
cathode and the anode. With increasing current, the voltage begins to increase, and the
plasma breaks into filaments, as commonly seen in high-voltage discharges between
small, sharp-edged gaps.

The critical discharge figure of merit for the hollow cathode discharge is pD; the
plasma pressure (p) multiplied by the diameter (D) of the hollow cathode. The hollow
cathode discharge forms for pD values from a fraction of a torr-cm to 10-20 torr-cm.
Electron energies, determined by spectroscopy, are greater than 10 eV [14].

The anode and cathode are made from molybdenum [l4], and separated by a 250-
micron mica layer. Argon gas is flowed through the hollow cathode tube. Hollow
cathode discharges have been formed with hole diameters as small as 200 microns, and at

pressures approaching 900 torr (17.9 torr-cm).

l9

2. 2 Microwave Plasma Sources

This study focuses on microwave plasma sources. Microwave plasma sources
offer several advantages over plasma sources driven at lower frequencies. First, when
microwave energy is focused in a resonator cavity, the electric field strength, which is a
function of potential and wavelength, is strong enough to excite a discharge. Second, the
microwave energy can propagate through dielectric media; hence, the microwave probe
does not need to come in contact with the plasma itself, making the discharge
electrodeless. This is not true with low frequency discharges, which require putting the
electrodes in direct contact with the plasma. Potential damage to or contamination from
metal electrodes by collisions with high-energy plasma species is eliminated.

A second advantage to higher frequencies is seen in miniature plasmas. The fast
electric field reversal maintains the electrons in the center of the discharge, reducing the
number of collisions with the container wall. By trapping the electrons, fewer electrons
are lost to the walls and more energy is absorbed by the electrons, resulting in greater
ionization. This effect was discussed in section 2.1.1 when examining micro-cell plasma
displays, which were dominated by secondary electron emission, in contrast to direct
ionization within the plasma itself.

In general, microwave plasmas operate with smaller plasma potentials, thus
reducing the plasma sheath potential, which affects the energy at which the gas species
exit the plasma. Such reduced gas species energy is necessary for the success of many
surface reactions involved in plasma-assisted chemical vapor deposition (PACVD).

The literature covering microwave plasma sources is extensive. The following

subsections examine three common designs: the 2.45 GHz microwave plasma cavity

20

resonator, the surface wave plasma reactor, and the electron cyclotron resonance (ECR)

reactor.

2.2.] 2.45 GHz Microwave Plasma Cavity Resonator

A common microwave plasma source design, developed at Michigan State
University, is the 2.45 GHz microwave plasma cavity resonator, shown in Figure 7 [l6].
Microwave power is introduced to a cylindrical cavity through a coaxial probe,
penetrating the cavity axially from the top or the side. The height of the cavity and the
probe depth are adjusted for cavity microwave field resonance with the applied
microwave frequency. In one design, the cavity diameter is 17.8 cm, and the height is
adjusted to 21 cm. The resulting resonant mode is TM 013. Microwave power levels
range from 500W-5kW. Operating pressures run from 5 torr to 180 torr. Such systems
have been developed for PACVD of diamond.

The plasma discharge forms within a sealed quartz dome, mounted at the base of
the cylindrical reactor. The discharge is initiated by the electric field focused in the
quartz dome. The reactant gases are injected from the base plate of the reactor with high
velocity, mixing in the quartz dome before ignition. Premixing the gases improves
deposition uniformity in plasma assisted chemical vapor deposition (PACVD) reactions.
Uniform deposition can be maintained on wafers up to four inches in diameter.

The substrate holder is interchangeable and adjustable in height, to better interact
with the plasma formed above it. In high-pressure experiments, the substrate holder has
been water-cooled to better facilitate deposition. The system has been scaled up to

accommodate 915 MHz power supplies. The 915 MHz reactor is 45 cm in diameter; the

21

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Figure 7 2.45 GHz Microwave Plasma Assisted CVD Reactor.

22

largest possible substrate size is 33 cm. The 915 MHz reactor power requirement is

8kW-18kW.

2.2.2 Surface Wave Plasma Reactor

The surface wave plasma reactor’s geometry is completely different from that of
the 2.45 GHz-microwave plasma cavity. In the surface wave reactor, the microwave
power is transmitted from the waveguide through a sealed 2.5-cm diameter quartz tube,
into a waveguide surfatron, which functions as a double-stub tuner. The quartz tube is
filled with reactant gases; the pressure can be adjusted from 1-60 torr. The surface wave
reactor uses 1 kW microwave power at 2.45 GHz. The surface wave plasma reactor
schematic is given in Figure 8 [17].

The plasma fills the quartz tube, and distends several centimeters below the
waveguide structure at low pressures. The plasma excitation along the plasma column is
facilitated by the propagation of microwave energy along the column via surface waves
that travel along the boundary of the plasma. Below the waveguide structure, the quartz
tube diameter can be increased to accommodate substrates up to 8 cm in diameter. The
plasma expands to fill the quartz tube below the waveguide, allowing for complete

coverage of the substrate during deposition.

2.2.3 Electron Cyclotron Resonance (ECR) Reactor

The electron cyclotron resonance (ECR) reactor is similar in geometry to the 2.45
GHz-microwave plasma cavity, however the nature of the plasma is quite different. In

the ECR reactor, electron heating -motion and collision- is a result of the electron

23

 

 

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Figure 8 Surface-Wave Microwave Plasma Assisted CVD Reactor.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Heater

 

 

 

 

 

 

 

 

 

 

 

24

 

 

cyclotron effect; the force imposed on charged particles that results from an oscillating
electric field in the presence of a permanent magnetic field (875 gauss).

In the reactors described previously, the most efficient heating occurs at pressures
where the mean free path of the electrons give rise to a collision frequency that matches
the microwave frequency. At the point of collision with an atom, the electron momentum
is randomized. At the same instant, the electric field reverses to accelerate the electron,
increasing its average velocity with each field reversal and collision, until the electron has
enough energy to ionize the atom or molecule.

In ECR reactors operating at resonance frequency, the electron revolves around
the magnetic field lines with an angular rate equal to the frequency of the applied
microwave power. Each field reversal accelerates the electron for one-half revolution
before the next reversal. The electron will ionize an atom upon collision if it has been
given enough time to build up sufficient energy.

A specific example of an ECR source is the compact ion and free radical model
#610 plasma source developed at Michigan State University, shown in Figure 9 [18].
The reactor is a stainless steel cylinder with 5.8-cm outer diameter. The front half of the
cylinder is the coaxial microwave power feed, terminated with a loop antenna. The back
half is filled with a 3.6-cm x 3.0-cm quartz reaction vessel. The operating pressure is
kept between 0.1 mtorr and 3.0 mtorr, much lower than the operating pressure for the
2.45-GHz microwave plasma cavity resonator described in section 2.2.1. Microwave

power levels range from 50W-200W.

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26

Chapter 3 Experimental Setup

The primary research objective is to quantify the operating characteristics of
miniature microwave plasmas with sizes ranging from 0.25-10mm. To this end, a new
microwave plasma system must be built that can create miniature plasmas in the specified
range at controlled pressures. Additionally, it should allow for multiple gas feeds at
controllable flow rates. It should be safe, affordable, run at low power, and ideally,
portable.

Plasma diagnostics must be investigated and developed. Diagnostics must
provide the following plasma characteristics: electron density, gas temperature, and
plasma power density. Diagnostics should also be portable, requiring only standard
laptop computer interface.

The following section describes the design, construction, operation, test, and
function of the miniature microwave plasma reactor and system designed specifically for
this investigation. The next section describes the plasma diagnostic set, and the extra
design work that was necessary to extract the required plasma characteristics from such a
small, low-power source.

Section 3.3 provides valuable initial test data from the plasma system, giving
insight into plasma ignition and plasma operating conditions that drive diagnostic and

theoretical development decisions.

3. 1 Miniature Microwave Plasma System

A miniature microwave plasma source and experimental system was designed,

built, and tested at Michigan State University. The experimental system, as shown in

27

Figure 10, consists of the plasma source, vacuum chamber, microwave power system,
pressure control system, and gas delivery system. The microwave plasma source, shown
in Figure l 1, is a 6.5-cm outer diameter coaxial waveguide, with lO-mm diameter center
probe. The waveguide is terminated with an adjustable short. The center probe can be
adjusted to vary the center conductor gap, where the plasma is formed. The distance
from the short to the center conductor gap is adjusted to approximately one-half the
wavelength of the applied microwave power (2.45 GHz). A quartz tube slips over the
center probe, surrounding the gap and enclosing the plasma.

The plasma source is connected to a 100 W microwave power supply (2.45 GHz)
through a circulator and a series of directional couplers and terminators or loads. The
circulator is fixed to the microwave source output to protect its magnetron from reflected
power. Thermistors convert transmitted and reflected microwave power into current,
which drives the associated power meters.

The pressure control system functions to stabilize the system pressure. It consists
of two Baratron pressure sensors (20 torr and 1000 torr), a 2-atmosphere pressure gauge,
manual pressure sensor selector, three independent pressure control setting channels, two
digital pressure display units, and automatic pressure control feedback circuitry to fix
pressures from 1 mtorr to 1000 torr. The pressure control feedback drives a throttle
valve, which determines the rate the reactant gas is evacuated from the system. An
impeller pump (Alcatel, 40 liter/min) develops the vacuum.

The automatic pressure control circuitry receives signals from the manually
selected Baratron pressure sensors. The 20 torr head measures pressure accurate to l

mtorr, for pressures less than 10 torr. The 1000 torr head measures pressure accurate to

28

 

 

 

 

   
   
  

 

   
  
  
  

 

 

 

 

 

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29

 

  

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30

0.1 torr, up to pressures of 1000 torr. The pressure controller compares the Baratron
input to that of the selected pressure setting, and drives the throttle valve to converge to
the control setting. The pressure and target pressures are registered on the digital
displays.

The gas delivery system includes a 4-channel bank of gas flow meters (Hastings:
model #CPR-4A, MKS: Type 247). Three flow meters are rated for flows up to 1000-
standard cubic centimeters per minute (seem); the fourth flow meter is limited to 10-
sccm, and as a result, provides the highest resolution. The 4-channel flow control unit
actuates all four flow meters. The controller drives the flow meters with the difference
between the selected flow rate and the flow rate feedback from the flow meters. The
flow rate through each of the four flow meters is registered on controller digital displays.
Each gas channel is connected to 2500-psi gas cylinders, regulated to 15 psi. The gas
cylinders are secured to the side of the plasma source system. The gas channels and gas
canisters are completely interchangeable. This allows for experiments using any
combination of up to four gases.

The experimental system is sealed by metal-to-metal fittings (VCR seals, 64 total
seals). The base pressure is less than 1 mtorr under normal operation (impeller pump
only); the base pressure drops to less than 1.0 x 10'7 torr during leak tests, which requires
the addition of an auxiliary turbo pump (Alcatel, 100 liter/min). Leak tests consistently
register leaks less than 1 mtorr for 16-hour intervals. The system volume is 78 liters.

To reduce contamination and water vapor accumulation, the system is closed
during system purge. Argon, regulated to just under one atmosphere, brings the system

back up to pressure when the experiment is complete.

31

The following chart summarizes the current state of the miniature microwave

plasma source and experimental system. The system specifications include the plasma

source, vacuum chamber, microwave power system, pressure control system, and gas

delivery system.

Base pressure (roughing pump): < lmtorr
Base pressure (turbo pump): < 10'7 torr
Leak rate (w/o reactor): < lmtorr/16 hrs
Plasma ignition power: 10W

Power meters:

1 forward power meter following 50/50 splitter
l reflected power meter following circulator

Gas channels:

3 1000-seem channels
1 lO-sccm channel

Pressure heads:
1 1000-tort Baratron transducer
1 20-torr Baratron transducer
1 2-atmosphere head

Pressure display for each pressure head:

Digital display: Baratron heads
Analog display: 2-atmosphere head

Automatic pressure control select between 1000-torr and 20-torr Baratron heads
Accurate automatic pressure control from 1 mtorr to 1000 torr

Three pressure control setting channels

Automatic Argon system purge to 1 atmosphere with adjustable pressure regulator

Additional air valve isolation from roughing pump

32

Manual equalization valve to bring roughing pump to 1 atmosphere
Nitrogen vent to roof:

Adjustable Nitrogen flow rate
Shut-off valve to prevent backflow from neighboring DLC system

All seals metal-to-metal (VCR) fittings

In summary, the following input parameters can be controlled and monitored:

. Pressure: 0.5 torr-2 atmospheres

. Power: 0.5-100 W

. Probe diameter (plasma diameter): 0.2-10mm
. Plasma height: 0.2mm-20mm
. Gas flow: 1.25-10003ccm (velocity function of nozzle size)

. Gas species: Argon, Nitrogen, Hydrogen, Air, Hydrogen/Methane mixture

The flexibility in design allows for plasma investigation at a wide range of

pressures, at different discharge aspect ratios, at power levels from 0.5 W to 100 W, and

with reconfiguration capability on all four-gas channels.

3. 2 Plasma Diagnositics

The plasma diagnostics proposed to investigate miniature microwave plasmas

created by the plasma source built for this investigation are limited to spectroscopy due to

the configuration of the source. The following sections describe the diagnostic set up for

the optical emission spectrometer.

3.2.1 Optical Emission Spectroscopy

Optical plasma diagnostic techniques include plasma-induced emission and laser-

induced fluorescence [19]-[20]. Other radiation based non-intrusive techniques include

33

optogalvanic, infrared, spontaneous and stimulated Raman, and multi-photon
spectroscopy [21]. Optical diagnostic techniques, specifically plasma-induced emission,
will be used to estimate electron density, electron temperature, and gas temperature in
this investigation. The experimental set up for plasma-induced emission, or optical
emission spectroscopy (OES), is given in Figure 12.

Line broadening is seen in high-density plasmas where high local electric fields
are present, which result from localized charge imbalances. This effect is called Stark
broadening, or electric field broadening. Estimates can be made from Stark broadening
for translation temperature and electron density.

Stark broadening of the Hydrogen Balmer series (Ha, Hg) as a function of electron
temperature has been computed by Griem [22]. Electron density and temperature
determine the broadening for purely Stark broadened Ha lines. Deconvolving the Stark
shape from the total spectrum line leaves a Doppler broadened HOI curve, and gives an
estimate for Hydrogen translation energy and electron density (assuming a Maxwell
distribution) [23]-[24].

Gas temperature is measured using the optical emission lines corresponding to H2
and N2 rotational temperature; molecular Hydrogen electronic configurations and
rotational energy levels and transitions are shown in Figures 13 -14 [25].

Rotational temperature transitions within the same electronic configuration and

vibration energy produce line intensities in accordance with the Boltzmann distribution.

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S J' J" E Hoal-London factor

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3.2.2 Optical Emission Spectroscopy Design

The intensity of the light that was gathered by the optical emission spectrometer
from plasma emission was found to be so weak in preliminary testing that virtually no
signal could be detected by the optical emission spectrometer. The plasma light intensity
itself was well above any detectable threshold, very visible to the naked eye in all cases.
However, the simple lens and fiber optic system used to focus the light into the
McPherson model 216.5 optical emission spectrometer was insufficient. This preliminary
design is shown in Figure 15.

In an attempt to increase the emission intensity, the lens system was plunged into
the plasma reactor, focusing the plasma emission on an array of optical fibers inside the
reaction chamber. The vacuum was sealed with a double O-ring feed-through, similar to
seals used in electron microscopy. Light was focused into the fibers, and collimated at
the end of the fibers, by specially designed and cut spherical lenses. The collimated light
at the end of the fibers was refocused into the McPherson 216.5 optical emission
spectrometer. Figures 16-18 detail the diagnostic setup, fiber feed-through design, and

spherical lens specifications, respectively.

38

 

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39

 

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42

3.2.3 Optical Emission Spectroscopy Test

Initial test results for the optical emission spectroscopy design are given in Figure
19. Photomultiplier tube currents in excess of 200 nA were recorded for the Hp line with
an accelerating voltage of ~900V. Vacuum pressures were unaffected by the new feed-
through; there was no discemable difference in leak rate after the feed-through
installation.

The plasma formation was unaffected by the new feed-through, and there was no
detectable microwave energy leak around the feed-through mount. To compensate the
light blocked at the reactor window by the new feed-through, its unused fibers were used

to channel light into the cavity to adjust the probe in the absence of the plasma.

3. 3 Preliminary Findings

Preliminary findings are restricted to a set of experiments conducted immediately
following the miniature microwave plasma system build (June-August, 200]). The first
set of experiments tested the miniature microwave system functions, such as leak rate,
base pressure, pressure control, flow control, and microwave power measurement. The
second set of experiments was concerned with plasma formation and stability. In the
second set of experiments, Argon plasmas were formed at pressures ranging from 1
mtorr-760 torr (1 atmosphere). These experiments were conducted to verify that plasmas
could be formed, controlled, and operated safely over the required pressure range.

The miniature microwave system leak rate registered less than 1 mtorr over a
period of 16 hours. Base system pressure measured less than 9.0 x 10'8 torr while

pumping with an auxiliary turbo pump. System pressure was monitored to 0.1 mtorr.

43

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44

System pressure could be stabilized with no gas flow at pressures as low as
lmtorr. System pressure could be stabilized with gas flow at pressures approaching 10
mtorr. The pressure at which the miniature microwave system pressure can be stabilized
is limited by the resolution of the flow meters, not the throttle valve feedback control
loop.

Argon plasmas were ignited at pressures between 5 torr-IO torr. The microwave
power (2.45 GHz) necessary for ignition was approximately 30W-40W. The microwave
power necessary for a self-sustaining plasma was as small as 0.2 W for pressures less

than 100 torr.

3.3.1 Preliminary Experiments

Preliminary experiments concentrated on Argon plasmas and their characteristics.
Argon plasmas are easily formed, as monotonic gases ionize more readily. Plasmas were
ignited at pressures between 10 ton-15 torr. Pressure settings were adjusted such that
stable plasmas were formed at pressures from 1 mtorr-1000 torr.

Argon plasmas formed at pressures below 1 torr diffused through the gaps in the
quartz tube, filling the entire reactor. Plasmas formed at pressures greater than 400 torr
began to collapse, pulling away from the quartz tube. Plasmas greater than 800 torr were
spherical. In general, higher pressure Argon plasmas formed discharge filaments when

the plasma impedance was not matched to the impedance of the microwave power circuit.

3.3.2 Preliminary Diagnostic Results

Preliminary diagnostics were restricted to plasma size, shape, and power density,

as recorded by digital imaging. Measurements for plasma size and shape were taken

45

directly from the digital image. Microwave power meters recorded transmitted and
reflected microwave power. The resulting data is summarized in the series of plots given
in Figures 20-21, first published June 15, 2001 [26].

Specifically, power density is recorded for pressures from 100 torr-760 torr for
the Argon plasma, and plotted in Figure 20. The power density, calculated from the
diagnostic data, is used in section 7.2.1 to calculate electron density and temperature
using the global model. Ignition power was recorded for pressures from 5 mtorr-760 torr,

and is plotted in figure 21.

46

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48

Chapter 4 Global Model Theory

To complete the characterization of the miniature microwave plasma, it is
necessary to model the plasma mathematically. Several models have been proposed for
low-pressure plasmas [27]—[29], moderate pressure plasmas [30], and high-pressure
plasmas (~1 atmosphere) [31]-[32]. Matching these models to the diagnostic estimates is
necessary to prove the validity of these models, such that these models can be used in the
future for miniature plasma source design.

Global models for non-equilibrium plasmas calculate electron density (n) and
electron temperature (Tc) as a function of input power (Pabs), pressure (P), gas
concentrations and plasma reactor geometry.

Briefly, global models require species balance, momentum balance, and energy
balance in the Boltzmann transport equations. Conservation of these three quantities are
commonly referred to as the zero, first, and second moment Boltzmann equations. The
global models balance these equations macroscopically, as opposed to other finite
difference analysis techniques [33] that balance these equations for each small volume
element included in the microwave reactor system. Global models can incorporate
chemical reactions and reaction rates for specific species.

Global models do not consider convective flow dominated conditions, as found at

higher pressures (> 100 torr).

4. 1 Global Model

The global model development begins with the general Boltzmann transport

equations. This set of equations can be simplified by limiting the plasma behavior.

49

‘ .n-« “a.“ wv

Pinning the plasma boundary conditions to the edge of a collisionless sheath reduces the
equation set further. The resulting set of equations require pressure dependent
relationships, valid over limited pressure regimes. The global model, in its final form,
combines Boltmann transport particle, momentum, and energy balance equations,
matched at the edge of a collisionless sheath. Solved iteratively, the global model
predicts electron and ion densities, electron temperatures, and electron and ion flux. The
mathematical development proceeds directly from texts by Bittencourt [34], Lieberman
[35], Goldston [36], Bird [37], and Chen [38].

The global model requires balancing zero (mass/species), first (momentum), and
second (energy) moments of the Boltzmann transport equations. More complicated
mathematical models require balancing higher order Boltzmann transport equations; for
example, heat transfer through convective flows requires balancing the third moment
Boltzmann transport equation. These equations are critical in developing mathematical

models. The zero moment Boltzmann equation is given as follows:

9%"t—a—+V ' (pmafia) = Sa

(4.1)

3
so = —pa—’;’Q- = m€(Kine _ krnez _ kane)

collision

pm, 2 a density
17a, 5 a average velocity
Sa, 5 a ionization rate
me E electron mass
ne 5 electron number density

K ,- , k, , ka 5 ionization, recombination, attachment rates

50

Equation 4.1 is called the continuity equation, and represents the conservation of

mass. Physically, the difference between the rate at which particles or flow from a

differential volume (dV) and the rate the particles are generated (Sq) is equal to the time

rate of change of the particles or within the differential volume. The first moment

Boltzmann equation is given as,

”

D _ 2 , -
A = naqa (E + iiaxB) + pmag - VPa + Au

pma Dt
(4.2)
1.4.0, = 8(pmal70)
a:

col l ixion

a acceleration due to gravity

é
Pa, 5 a partial pressure
Aa, E a momentum collision rate

Equation 4.2 is referred to as the equation of motion, and represents the
conservation of momentum. Physically, as expected, the mass density times the time
derivative of the average velocity is equal to the sum of the forces. In Equation 4.2, the

forces are composed of the Lorentz force and forces resulting from gravity and pressure.

a

The additional term, A 0, represents the mean momentum change with respect to time of

the a particles as a result of collisions within the plasma.

51

The second moment Boltzmann equation is given as,

 

 

p a _ .. _. g g n .
fill-3%) + (121](V o ua) + (VPO, 0 V) o ua + qa, = M0, — ua, 0 Ag, + (ua2/2)Sa
8(pma, < v2 >a, /2)I
Ma 2
at , ,
col/Ismn
Ma, 5 a energr collision rate (4.3)

Equation 4.3 is called the energy transport equation, and represents the
conservation of energy. The first term represents the total thermal energy rate of change
of a differential volume moving with average velocity u. The second term represents the
thermal energy entering and leaving the differential volume. The third term represents
the work performed on the species within the unit volume by the forces (pressure) on the
surface. The fourth term represents the heat flux through the differential volume. The
terms on the right side of the equation represent the energy change as a result of particle
collisions.

The global model follows directly from the first three Boltzmann moment
equations. Approximations to the Boltzmann moment equations can be made, given the
plasma pressure regime. Sections 4.1.1-4.1.3 examines approximations made for the low,
moderate, and high-pressure regimes, respectively. In each pressure regime, the plasma

is assumed to be in steady state operation.

52

4.1.] Low Pressure, Steady-State Approximations

The Boltzmann transport equations can be simplified dramatically by assuming no
change in state in the plasma over time; that is, the plasma density function is constant in
phase space, both distance and velocity, at every point in the plasma. At low-pressure,
electron diffusion immediately counteracts the effects of internal forces, such as electric
field. As a result, there is no net electron acceleration in the plasma, and the total
derivative with respect to time is set equal to zero in Equation 4.2, when considering
electrons. Ion diffusion is much slower; drift due to the electric field dominates
diffusion. For ions, given a constant state, the partial derivative with respect to time is set
equal to zero in Equation 4.2. Thus,

—.

mne—Dlltfie = eneE-I-Vl’e = —eneV¢+VPe = 0; where: E = —V¢

VPe = kTeV 0e isothermal plasma (4.4)

:> fie = nOeWTe Boltsmann distribution formula

I
DEMui2+e¢i+(Pi/7li):0 (4'5)

:>-;-Mu,-2+e¢,-=0; for: H50

53

 

Solving Equations 4.4 and 4.5 for u,, and substituting into Equation 4.1 gives:

 

1..

1/2
V'(t777)=Y0 [_2€Teln_77_] 77 =v-Jl

M I70
’7 = ”e = ”i
mo 0.425
(4.6)
,1,- < (R,L)

xi,- 5 mean free path
Us 5 density at edge of collisionless sheath
770 E bulkdensity

The solution to Equation 4.6 can be found in closed-form. The ratio of the
density at the edge of the plasma sheath to the bulk density is a constant. Combining

Equations 4.4 and 4.5 with Poisson’s equation, that is,

u B a Bohm velocity (velocity at sheath edge) (4 7)
M 5 ion mass °

e

W = —(n. — n.)
«‘30

Poisson's equation

54

6‘0
(4.8)

1 2

e8s =3Mus

1/2

eTe
:>u = — 4.9
B (M) ( )

The Bohm velocity (uB) is defined as the velocity on the edge of the plasma
sheath, when the sheath is collisionless. In the global model development, the plasma
sheath is always considered collisionless; the Bohm velocity development is valid for

each of the pressure regions considered in this study.

4.1.2 Intermediate Pressure, Steady-State Approximations

Intermediate pressures are defined as pressures in which ion motion is still
dominated by drift. However, the mean free path is less than the plasma reactor
dimensions. Therefore, the collision term in Equation 4.2 must be included at

intermediate pressures. Thus,

Vm =ui//I.,-
2e -
l
(4.10)
”i =flrE

Vm E momentum reversal rate, xi, 5 ion mean free path

M 5 ion mobility

55

Equations 4.10, taken with the Boltzmann distribution function and the time
invariant continuity equation, Equation 4.1, gives the following non-linear differential

equation:

1/2 _ 1/2
”[321] iI_,,_42I =v,-..rz (4.11)

The analytic solution to Equation 4.11 does not converge to the low-pressure
analytic solution in section 4.1.1, as the mean free path goes to infinity. Godyak found an
approximate solution that does converge to the low-pressure solution. According to
Godyak, the following ratios are to be used to relate plasma density at the sheath edge to

plasma density in the bulk, given a cylindrical discharge with radius R and length L:

—I/2
112 = E a 0.86I3 +—€-I (4.12)
770 241
fl —I/2
hR = J; a 0.80I4 + ——I (4.13)
770 ’I'i

And the ionization rate is given as:

-I/2
vi, = 1’4 5 2.2i’iI4-15I (4.14)
770 R 47'

The density ratios are used in the global model to find the ratio of the plasma

volume to the effective plasma area; that is,

56

d ._l_5£__
‘7” 2RhL+LhR

”SA = 770/1th7
(4. I 5)

volume/ effective area

deff
77s
Aeff 5 effective area

density at sheath edge

In both low and intermediate pressure regimes, the plasma density is constant, or
nearly constant, through the bulk of the plasma, and sharply driven to zero in the sheath
between the bulk plasma and reactor walls. The flat distribution is due to the uneven
diffusion rates of the two charged species, electrons and ions. At higher pressures, the
ion diffusion rate is not negligible, and the bulk plasma density is no longer constant.

Returning to Equation 4.], given constant densities:

(IFOdS = IKizngndvol; where: K [277g 5 Viz, 77g 5 neutral gas densities
S

vol

 

u 3770 (11,. 2m + h 22:18): KiZUgIIOIIzRZL) (4.16)
K), _ l
“B deffr;g

Equation 4.16 is solved iteratively for Te, as both KI and uB are functions of T...
The ratio den is used in the global model as part of the global model power balance
equation. A description of the power balance equation can be found at the end of section

4.1.3. The relationship between ngden(Te) and Te for Argon in the low to moderate

57

pressure regime is shown in Figure 22 [19]. Figure 22 also gives ngdcn(Te) as a function

of TC in the high pressure regime, which is addressed in section 4.1.3.

4.1.3 High Pressure, Steady-State Approximations

High-pressures are defined as pressures in which the ion diffusion rate is not
dominated by ion drift. That is, ion diffusion and electron diffusion, and the resulting
drift due to internal electric fields, must balance such that ion density and ion flux is
equal to electron density and flux at every point in the plasma. Accordingly, in steady-

state, Equation 4.2 and the isothermal assumption gives:

 

 

e kT
1‘0. = naE - 0‘ V’lar = #aUaE - DaV 77a

 

mana ma ma
”1' = ”e = 77
(4.17)
I“i :re :1“: ”iui :neuezllu
at 5 ions, electrons
Dar 5 a diffusion
F = ”u = _.uiDe +fleDi V77 5 —DVr7
tut + .ue
(4.18)

D E ambipolar diffusion coejf

Substituting Equation 4.18 into the continuity equation, Equation 4.1, gives the

following second—order differential equation:

58

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59

Solving Equation 4.19 in cylindrical coordinates gives the following solution set:

 

 

I] = ”0J0[X2Ir]COS(%J I X0] = 2.405(15! zero 0f .10)
87] Id) X r
(4.20)

Fir Er—fluxatz

 

F- =—D§l= X0113

212
0’ Br R TloJ1IX01 )COS(—l.—l

J,(x0,)=0.519
Returning to the steady-state continuity equation, integrating with respect to

volume, and applying Green’s theorem gives:

«If-as = Iv,, 7](r,z)dvol (4.21)

5 vol

Integrals on the right and left side of Equation 4.21 can be found in closed-form
with the relationships given in Equations 4.20. Setting the right and left side of Equation

4.21 equal gives the following:

2 2
e1 14
viz -511: R L 1

2
D778 D ”g defl ”g
(4.22)
Kizngne E Viz7le

60

Equation 4.22 is of the same form as Equation 4.16, with D(Te) replacing uB(T,.).
Equation 4.22 is solved iteratively for Tc, with the aid of the Figure 22 [39], which gives
ngden(Te) as a function of TC in the high pressure regime.

The energy conservation equation, Equation 4.3, is simplified by assuming
relatively constant differential volumes, and by neglecting convection. These
assumptions eliminate the third and fourth terms in equation 4.3. The assumption that the
plasma is steady-state requires the partial derivative of thermal energy with respect to

time to be zero. Applying the chain rule to the total derivative and gradient:

2I&I_§_I&I,2,.m
Dt 2 at 2 2
(4.23)
~ 3 _. 3 - - 3- —
VOIEPauJ=3PaVOu+EuOVPa
D3130, 3--~3- a3]~(3-I
— — +—PVou=Vo —Pu +— —P 2V0 —Pu 4.24
01(2l2a Izalaz(2“ 2" I)

The derivative with respect to time on the left side of Equation 4.24 is equal to the
total power absorbed in the plasma volume, defined as Sabs, less the power lost in
electron-neutral collisions that ionize neutrals. The gradient on the right side of Equation

4.24 is equal to the thermal energy flux to the reactor walls. Specifically:

61

Sub], = e(ee + e,- )<II“ - dS + eec. I K izngne dvol
5

vol

2
4f’dS : 47113770Jl (XOI)I:£XOI +£R—:I
S It L X0]

(4.25)

2

L R
I Kizngnedvol = 47d<izngno ——Ji (X01)
7! X0]

vol

ee , 8,- , 8C are loss terms

Where the integrals in Equation 4.25 are exactly the same integrals found in
Equations 4.20-4.22. The loss terms represent thermal energy lost in the electrons and
ions as they diffuse to the reactor walls, and ionization collisions in the plasma bulk,

respectively. Combining Equations 4.25 with Equation 4.21 gives:

—__SGA_
770 — ~
eDAefl 87*
. L II R2
Aeff = 4flJI(XOI -fl_—XOI +252}; (4.26)

67" =€e+€i+8c

Equation 4.26 is also valid for low and intermediate pressure regimes, with us
replacing D, and the effective area given in Equations 4.15. Note the effective area in
Equation 4.26 has units of distance.

The collection of equations in section 4.1.1, section 4.1.2, and section 4.1.3
provide the global model equation set. Solving the continuity equation gives the electron
temperature; solving the power balance, or energy conservation equation, yields electron

62

and ion density. To complete the equation set, it is necessary to determine the power loss
terms, Sc, 8,, and 8c. The electron density distribution, f, is critical in finding these terms.
The global model places only one restriction on the electron energy distribution; that is,

electron-ion collisions are elastic. As a result:

I 2 3
—mv dv+ dv =—kT,
I2 f IM 2 .

(4.27)

for monotonic gases (normalized)

Therefore, the maximum entropy of the distribution function f, constrained by
Equation 4.27, sets the electron energy distribution function equal to the Maxwell
distribution. The average energy flux for the electron, given a Maxwell distribution, is

2eTe; the average velocity is given by:

 

8T 1/2
<Ve >=[ 9] (4.28)

Assuming only elastic collisions, the ions pass from the plasma bulk to the reactor
wall with no change in energy. The difference between the bulk plasma potential and the
reactor wall potential is equal to the energy flux per ion. The potential difference is
found in two parts. Firstly, from the plasma bulk to the sheath edge; secondly, from the
sheath edge to the reactor wall. The former potential (a) is found by invoking energy
conservation from the center of the discharge to the sheath edge; the latter potential (b) is

found by balancing electron and ion flux to the reactor wall. Thus,

63

2
I 2 I T
a e =—Mu =—M —' : ,=—e
() ¢e 2 B 2 £ J we 2

(b) Ifie : ri => gillsew /Te < Ve >= ”SUB

(4.29)
3 "IIUSBIAW ”e (EST/2 = USIEXEJI I2 2 —¢W = %ln(3A;n—]
(c122,- = .1, — a. = IZe—Ii +1.16%nd
And,
er = ee+e,- +56 2 2Te+-ge—Il+lnI3A7:—n-)I+gc(fe) (4.30)

The rate energy is lost per unit volume per ionization collision is a function of
electron temperature, and is given by the curve presented in Figure 23. Calculations of

electron density and temperature for experimental data are given in chapter 7.

64

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65

Chapter 5 Spectroscopy Theory: Zeeman Effect

Extracting information from spectroscopy results requires an understanding of
Quantum Theory. Bohr-Sommerfeld Theory adequately explains simple atomic spectra
classically, with given ad hoc quantization rules. For example, the Balmer formula, a
direct result of Bohr-Sommerfeld, accurately accounts for the principle peaks in the
visible atomic Hydrogen spectrum. The theory also accounts for the quantization of
angular momentum, and applies to the vibration and rotation spectrum of simple
molecules, and the normal Zeeman effect.

However, Quantum Theory is necessary to explain complex atomic spectra, the
anomalous Zeeman effect, and fine structure. Quantum Theory is necessary to formulate
angular momentum coupling (spin-orbit) and coupling to magnetic moments. Quantum
theory is necessary to address relativistic effects (Thomas Precession, Darwin Shift) and
multi-body effects (Lamb Shift); effects that are pronounced in atomic Hydrogen spectra.
Additionally, a systematic analysis of spectral data is not possible without the constructs
of Quantum Theory.

The first two sections introduce Quantum Theory fundamentals, followed by
sections that describe the quantum effects of fields on particles. In a plasma, these
quantum interactions effect changes in spectral lines. Specifically, the effects that
contribute to spectral peak splitting found in atomic and diatomic hydrogen are discussed

in the final sections.

66

5. 1 Introduction: Quantum Theory

One starting point for Quantum Theory is the Schrodinger equation, proposed by
E. Schrodinger in 1925. The Schrodinger equation defines the wave function state; that
is, the wave function position and momentum. Classically, the equations of motion are
found by following the stationary path defined by the action integral (Maupertuis,
Hamilton). In Quantum Theory, the equations of motion —the Schrodinger equation- is
found by following all possible paths. In 1948, R. Feynman developed the Schrodinger
equation formally by summing all possible paths constrained by the action integral and
uncertainty in conjagate state variables position and momentum. The Schrodinger

equation is given as follows:

2
--r5—V2y/(x,t)+ Viz/(Lt): flirt/(Lt) (51)
2m 3!

Setting the potential energy term (V) to the energy stored in the near parabolic
energy well of an atomic bond, the Schrodinger equation yields Hermite polynomials as
the wave function (‘1') solution for the harmonic oscillator. Setting the potential energy
term to the potential that results from a central potential, the Schrodinger equation yields
spherical harmonic functions (associated Legendre polynomials) as the wave function
solution for a single electron orbiting the nucleus.

It should be noted that the wave function solutions for the harmonic oscillator and

the central charge can both be constructed without the use of the Schrodinger equation.

67

Dirac constructed eigenvectors and developed solutions for the harmonic oscillator based
strictly on the constructs of Hilbert space and conjugate relations. Born, Heisenberg, and
Jordan did the same with angular momentum operators to solve for the angle dependent
solutions to a central potential.

For the purpose of studying spectroscopy peaks, the Schrodinger equation will be
temporarily set aside. First, the eigenvector equations and operator functions for the
harmonic oscillator and central charge will be briefly illustrated. This tact will
demonstrate the powerful nature of the eigenvector technique, particularly for
spectroscopy, where the only results needed are the corresponding eigenvalues, which set
the energy levels of the system.

The eigenvector approach will introduce the angular momentum operators that are
used to determine degenerate energy levels in central charge potentials. These operators
will then be used to find the energy levels and degeneracies in coupled angular
momentum problems. Perturbation theory will show how these energy levels split —the
degeneracies are removed- with the effect of applied magnetic fields (Zeeman effect).
Results will be applied to the hydrogen rotational spectrum.

The Schrodinger equation will be used to address effects caused by changes made
to the potential energy of the system. Perturbation theory is needed to calculate energy
shifts that result from magnetic fields (Zeeman) and electric fields (Stark). The
Schrodinger equation must be modified to account for the interaction of the electron spin
with the orbital angular momentum of the electron. Also, the Schrodinger equation must
be adjusted to account for the relativistic mass of the electron. Results will be applied to

the diatomic and atomic hydrogen spectrums; the complete energy spectrum for diatomic

68

hydrogen is given in Chapter 6; the complete energy spectrum for atomic hydrogen (Ha,

Hp, H7) is given in Chapter 7.

5. 2 Eigenvectors

Mathematically, the eigenvector equation is given by the following:

A | x) 2 11 | x) (5-2)

Where A is a vector operator, x is a set of eigenvectors or eigenfunctions, and A is
the eigenvalue diagonal matrix. Physically, the linear algebra terms observability and
projection space mean the physical quantity or operator (A) can be observed and
measured (7t) if the object (x) can be projected without distortion.

An example is a microscope. The microscope objective lens operates (A) on the
light reflected from the object (x -LHS) to create an image projected onto the focal plane
(x —RHS) of the eyepiece. In this case, the operation of the objective lens is observable if
the image is clear; that is, x-RHS = x-LHS. The eigenvalue for the microscope is simply
its magnification.

Operators that commute can be observed by the same set of eigenfunctions. This

can be seen for operators A and B in the following:

69

A l X) = 1 | x)
A | x') = II x')
(x'l AB | x) = /l'(x' | B | x)
(5.3)
(x'lBAlx)=/i(x'|B|x)
<x’ I [A.Bll x> = (X— lX-r’l B l x>
[A,B]=0:>(x'|B|x)=0:> B|x) =,t,,x
For the microscope example, a compound objective lens commutes; it does not

matter whether the higher magnification occurs first or second. The next two sections

describe the operators for the harmonic oscillator and central potential.

5.2.1 Harmonic Oscillator

The infinitesimal translator operator changes the wave function position argument

asfollows:
T(ax)=1—ax’-i=1—j.-jhi.31y=1—j.p.ajy (5.4)
a ax 7’ ’1

Where p is the momentum operator, and is Hermitian. The Hamiltonian —the

energy operator- is given by:

70

——-H =—(P2+Q2) (5.5)

p = w/mth (5.6)

And mm2 is the spring constant (k) of the system. Q and P represent derivative
position and momentum operators, which like true momentum operators p and q, do not

COITIITIUIC.

[Q.Pl = 1 (5.7)

The one-dimensional harmonic oscillator potential energy is a fiinction of
compression- or translation -which can be discretized; allowed transitions increase or
decrease compression by one unit. Operators that change the energy of the wave function
are commonly called “ladder operators”.

Creation and annihilation operators [40] for the one-dimensional harmonic
oscillator are given in the following. These operators are unitless, and represent the
infinitesimal energy change that results from the infinitesimal translation, given in

Equation 5.4.

71

(5.8)
+ l
a = — - P
JE(Q J )
And,
[a,a+] =1 (5.9)
Now,
fiH2%(P2+Q2)=%(aa++a+a)=N+§ (5.10)
Where,
N = a+a (5-11)

Returns the original wave function as the eigenvector, with eigenvalue equal to

the number of units of energy stored in compression (n). In eigenvector notation:

N|n)=n|n) (5.12)

And with,

[N,a] = [a+a,a] = a+[a,a]+ [a+,a]a = -a

(5.13)

[N,a+] = [a+a,a+] = a+[a,a+]+[a+,a+]a = a+

72

It is clear that,

Naln) =([N,a]+aN)| n) =(n—l)a | n)
(5.14)

Na+ ln) = ([N,a+]+a+N)| n) = (n+1)a+ In)

Which implies a+|n> and a|n> are also eigenvectors of N, with eigenvalues n+1

and n-1, respectively. Relating Equations 5.13 and 5.14, it follows that:

aln>=JZIn—1>
(5.15)
61+ | n) =Jn+l |n+1)
Returning to the Hamiltonian,
H|n)=h¢{N+-;:)|n)=hw(n+l/2)|n) (5.16)

Therefore, the energy levels for the one-dimensional harmonic oscillator are given

by the eigenvalues:

En=hw(n+1/2) (5.17)

73

5.2.2 Central Potential

Understanding the central charge potential requires a thorough understanding of
the angular momentum component, which contains the most interesting spectral
information -—that of degenerate peaks that split in the presence of an applied electric or
magnetic field. The angular momentum component and angular momentum operators
will be covered in the next section. The central potential Hamiltonian will be presented

in the following section.

5.2.2.1 Angular Momentum Operators

The infinitesimal rotation operator changes the wave function position argument

as follows [41]:

(5.18)

This is exactly analogous to the infinitesimal translation operator presented in the
previous section. The second order expansion of the infinitesimal rotation operator leads

directly to the angular momentum commutation relations:

74

R: (84>) = R2 (aw 2)R; (897 2)

:(l-j'Jz 'a‘%h)(l_j'Jz 'a%h]
=1—j-Jz'a%'J§(a%h)2

And,

R.(a¢')Ry(a¢')— R,,(a¢')ze.<a¢') = R. (aw )—1

Which implies,

[1-14,-a‘%-J§[a‘%h)2j[l’j'1y'a‘IZ‘JfP‘IZth
_[l_j.Jy.a%_J§(a¢%h)2:”:l—j.Jx.a%—J3(a%h)2]

= (Jny — Jny(a%)= — j . J, -(a¢’2)/h
:> [Janylz thz

And, in general,

[Ji’Jj]= jhg'ijk

75

(5.19)

(5.20)

(5.21)

(5.22)

The angular momentum components Jx, Jy, and JZ do not commute. Likewise,
orbital angular momentum operators Lx, LV, and L2 do not commute. Recasting the

components of L in spherical coordinates, as shown in Figure 24:

I I I . I a . a
L}, 2L1: =Lx ijL}, :[i£+jCOt65—6j

(5.23)

Where the primed coordinates are body-axis coordinates. Now, raising the
operator dimension by one gives the horizontal component of the angular momentum

magnitude, in both body-axis and inertial frame:

I

2
_1 —a—sin B—a— + cot2 H-a——
Slng 86 86 a¢2

Li 4%, = —;-(L+ L_ +L_ L+ )2 —

 

(5.24)
[141’ L2] = 0

Here it is clear that L),2 and LI commute. Therefore, L2 and L2 commute and have
the same eigenfunctions. L2 is identical to the Laplacian operator in spherical

coordinates, and is given by:

 

2
1.2 l a ' a 1 _a_ (5.25)

- ——sr

n —+
81118 86 86 sin2 9 a¢2

76

 

 

5/ j 8(1)

 

 

 

 

 

 

 

 

—8/89

 

 

Figure 24 Angular Momentum Operator: Spherical Coordinates.

 

 

 

77

 

 

 

 

—cot6 5654)

 

 

L2 is periodic about the z axis. Its eigenvector equation must be the following:

12117,"): m 1 Ylm>
(5.26)
2 WW) = F(6)e-"""’

The eigenvector equation for L2 is different for each m. For m = l, the eigenvector

equation is:

2
—a—sin6—a—+ [2 |)’/)

L2|Y1’>=- . .
$111936 86 sm 6

 

 

(5.27)
L2 | sin, 9) = 1(1+ 1) | sin] 9)

Where the corresponding eigenfunction and eigenvalue are sin'B and 1(l+l). The
angular momentum ladder operators are the infinitesimal angular momentum operators

from Figure 24 and Equation 5.23. More succinctly, they are given as:

L+ = L,C 3‘.ij = [igag—mcotfi] (5.28)

Where m is the eigenvalue for L2. Physically, m is the dimension of the

divergence operator (V 0), and depends on the dimension —or eigenvalue- of the wave

function.

78

Commutator relations are as follows:

[2.2. 1: A
[L2,L_]= —L_

[L+,L—]= 2L2
With Equation 5.24:
L2 =L}, + L3. = %(L_L+ + L+L_)+ L‘Z:

L_L+ = L2 -L,(L,. +1)
L+L_ = L2 -L,.(L,_. —1)

And,
[L2,L,-]= o :5 [L2,L+]= [L2,L_]= 0
So,

1.221115"): 1:27- 11,m>=1(1+1>L_+_ Il,m>
LzLi |l,m) = Li(Lz il) l I,m) = (mi1)L_t | l,m)

:Lillam>:cillamil>

79

(5.29)

(5.30)

(5.31)

(5.32)

Equation 5.32 implies Lt |1,m) are both eigenvectors of L2 and L2, with
eigenvalues 1 (1+1) and mil, respectively. Further, L: are ladder operators, analogous

to the ladder operators for the harmonic oscillator found in Equation 5.15. Using

Equation 5.30, and the fact the ladder operators are Hermitian:

c3 = |L+ |1,m)]2 = (1,m I L_L+ |1,m) = [1(1 + 1) - m(m +1)](/,m |1,m)

c3 = |L_ |1,m)|2 = (I,m I L+L__ |1,m) = [1(1+ l)—m(m —1)](1,m|1,m)
(5.33)

 

:>ci =,/1(1+1)-m(m:1)

 

12+. |1,m)=\/l(l+l)—m(mi1)|1,mi1):>—ISmS+l

Before leaving this introduction, it is useful to see that the results given in
Equation 5.33 can be arrived at be restricting oneself to the physical interpretation of the
operators. The operators can be written as successive gradient/divergence operators, each
changing the dimension of the waveform by one. As seen in Equation 5.30, the anti-
commutation of L». and L- yields the Laplacian for the horizontal-plane component of

angular momentum. Using the identity in Equation 5.28 for a given m:

fi=%LQ+AL)

 

l a . +(m+1) 1 a . _,,,
L_L =- —sm -——s1n 6 5.34
+ sin+zm+lj 6 39 sin-m 6 39 ( )
1 a —(m-l). 1 +m 6

Ladder operators applied to the RHS of Li increase (decrease) the dimension by
one; that is, the exponent on the sin function (m) increases (decreases) to reflect the
dimension of the wavefunction. Starting with m = l, L: | Y,’) =1 I Y,’) , and applying the
L. operator 3 times, m -> m —s, and the new wavefunction that satisfies Equation 5.34 is

given by:

 

LiL, 119’): LzL-i | Y,’) = (l—s) |( 151nm (9)175 (5.35)

sin+m 6 39

The spherical harmonic function, Ylm(9,¢), is multiplied by the radial wave
function component to complete the wave function. The complete central potential
Hamiltonian and radial wave function component are covered briefly in the next section.

Finally, the angular momentum operator can be connected to the harmonic
oscillator (Schwinger) [42] by mixing the fields of uncoupled harmonic oscillators, each
with independent commutation relations. One operator (L+) creates one unit of +h/ 2
angular momentum (L2) and annihilates one unit of —h/ 2 angular momentum.
Likewise, its conjugate (L-) annihilates one unit of + h/ 2 angular momentum and creates
one unit of —h/ 2 angular momentum. This connection reinforces the results from the

Clebsch-Gordan calculation in section 5.4.].

81

5.2.2.2 Central Potential Hamiltonian

Returning to the Schrodinger equation, the central potential Hamiltonian

(hydrogen atom) is given in spherical coordinates as:

 

 

2 2
H _ 31+ L + V(r)
2m 2mr2
h 1 8
r : 'T——r
j r 8r
(5.36)
2
L2 = -h2 . isinfli+_l___2_
81116 86 86 sin2 6 a¢2
2
V(r) = —5-
r
And,
H l W) = E I u!) (5.37)

Substituting as follows:

82

 

: ’ : 2
h
2 2
= 5"- m (5.38)
hc — 2E
_l,
rtfl = lee 2 v,
Equation 5.37 can be rewritten:
d2 a’
x—+(21+2—x)——-—(l+l—v) v,=0 (5.39)
(1x7- dx

Solving by Taylor series expansion gives [43]-[44]:

v
F(I + l + p — v) (214-1) x_ x-l-l—vex
r(1+1—v) (21+1+p)1p1 x-—>°°

 

v, = F(l+l—v,2/+2;x)= Z
p=0
l
—x
:> ruI————>x"’e7-
x—-)oo
(5.40)

Which does not converge for large r. However, “I! in Equation 5.38 does

converge if the polynomial is finite, that is:

1+ 1 — v = O,—l,—2,—3,... :> v = n : n — (I + 1) = O,1,2,3,...

(5.41)
n=1,2,3,... H Iw>=nlw>
:5 I=O,1,2...n-1=> L2 |y1)=1(l+1)|1//)
-ISms+l Lz|W>=mW>

83

Where (n, l(l+l), m) are the eigenvalues —or quantum numbers- for a central
potential. The principle quantum number, n, defines the energy level, l(l+l) the
rotational energy (angular momentum), and m, bounded in Equation 5.33, the magnetic

moment.

5. 3 Electron Spin

Electron spin follows the eigenfunction precepts detailed for orbital momentum.
However, spin is a more elusive concept. In 1922, 0. Stern and W. Gerlach carried out a
series of experiments in Frankfurt (Stem-Gerlach Experiments) that illustrated just how
illusive a concept spin is [42]. Randomly oriented electrons were ejected from a
collimating slit, passed through a gradient magnetic field, and recorded on a screen. Two
peaks were observed, corresponding to spin up and spin down orientations. These peaks
were identified as 8; and 82'.

Then, Sz+ is passed through a second gradient magnetic field, perpendicular to the
first. Again, two peaks result, identified as 8: and Sx'. The SS beam of electrons was
then passed through a third gradient magnetic field, oriented identically to the first. The
result: both spin up and spin down peaks were observed (82+, 82'), even though 82' had
been removed in the first step of the experiment.

The conclusion is that the SC measurement —or filtering- restores the missing 8;

spin. Mathematically, this can be seen with the following operator set:

84

h
S. = 310 +><— 1)+ (I —><+ 1)]

S). =§1—j<1 +><- 1)+j(1 —><+ 1)] (5.42)

h
S. = 51(1+><+ 1)— (1 —><— 1)]

Physically, an exact measurement of SC means that there is no certainty to the
measurement of S2 -that 8; and S; are equally likely- as Sx and Sz do not commute. This
relationship is analogous to the relationship between position and momentum.

Spin commutator relationships are identical to those of the orbital angular

momentum commutators, given in Equation 5.22.

[S,-, S )- ]= jive/15;. (5.43)

In addition, spin has the following anti-commutator relationships:
{S;,S-} =lh26, :5 52 =(—+—+—]h2 =31)2 (5.44)

And spin ladder operators are given by:

h
15.-’54.]: 55+

[3,,s_]= —%s_ (5.45)

71
[54-25;]: 2552

85

Where:

 

Si |s,m) = Js(s+1)—m(mi 1) |s,mi l) :> -213 S m S +% (5.46)

For a system of two spins,

s=s,+s2

5'2 12> = (51 +5212 11> =s<s+1>rz2 12>
(5.47)
512 11> = "1171120

52 Iz>=(Slz+Szz)lz>=(m1+m2)lz>=mlz>

The full eigenfunction —or wavefunction- solution for the central potential

problem is simply the product of the spatial and spin wavefunctions:

11/(r1t)= ¢(x,t)l(l,2)
(5.48)

1713 = (¢(x1,x2)-¢(x2,x1))23(1,2)

1V1 = (¢(X1,x2)+¢(x2,x1))2’1(1,2)

The Pauli Exclusion Principle excludes two identical particles from the same state
(position, momentum) [45]; therefore, the wavefunction must be anti-symmetric. The

first wavefunction (W3) is anti-symmetric in space, symmetric in spin; there are three

86

configurations of spin (triplets) that satisfy the symmetry condition. The second
wavefunction (W1) is symmetric in space, anti-symmetric in spin; there is only one
configuration of spin (singlet) that satisfies the anti-symmetry condition.

In diatomic molecules, the triplet wavefunction density is lower between atoms
than that of the singlet; that is, the inner product term in the square (exchange density) is
smaller. The triplets represent anti-bonding orbitals, set at a higher energy than the
singlet bonding orbitals.

Triplet and singlet states can be built from individual spin states, using ladder

operators and the orthogonality principle, as shown in the following.

(|s=1,m=1)=|++)

|s=1,m=0)=S_|s=1,m=1)=(Sl_+S2_)|s=l,m=1)

 

 

1 1 1 1
12> J1(1+1)—1(1—0) ls _ 1,m -0) _‘/3(-2-+1)—3(-2--1)(|—+)+1+—))

<=>|s=l,m=0)=-‘Tl_2—(|—1-)+|—I-))

|s=1,m=—1)=S_|s=1,m=0)=(Sl_+Sz_)|s=l,m=O)

 

 

|:> fil+l)—O(0+1) Is =1,m =0) = ‘/%(%+l)-%(%-l)x/2(I —)+|—))

(5.49)

 

12>| s = 1,m = —1)=|—)

And,

87

1s=0,m=0>=~/§(1—+>—1+—>)
(5.50)

:(s=o,m=0|s=1,m=0)=(72(|—+)—1+—))1J2(1—+)+1+—)))=0

Equations 5.49 and 5.50 convert individual spins to total spin. The total angular
momentum is energy degenerate, but not so under the influence of a magnetic field.
Total spin and total magnetic moment are needed to calculate this interaction. The next

section shows how to combine angular momentum terms, spin and otherwise.

5. 4 Angular Momentum Addition: Clebsch-Gordan Coefficients

The addition of angular momentum requires the conversion from the |11,m1;12,m2>
(L12, L12; L22, L22) representation to the U.,j3;j,m> (J 12, .122; 12, J2) representation. The
elements in the square matrix that perform this transformation are called the Clebsch-
Gordan coefficients.

The representation transformation is important in spectral analysis. All four
elements in both representations commute for spherically symmetric groups. For groups
that are cylindrically symmetric, but not spherically symmetric, only the latter

representation commutes. Specifically, for diatomic molecules such as hydrogen, L o L:

does not commute with L12 or L22. In spin-orbit coupling found in atomic hydrogen,
L o S does not commute with L2 or 82. But in both cases, they do commute with all the
elements in the latter representation; therefore, that representation is observable and
complete.

Equations 5.49 and 5.50 are an example of Clebsch-Gordan coefficients; in this

case, transforming from (512, 8.2; 822, 822) to (812, 822; 82, SI) representations. For

88

diatomic hydrogen in the ground state (2g, 2,) —that is, no orbital angular momentum-
two electron spins couple with the molecular angular momentum (R) to give the total
angular momentum, as shown in Figure 25. For atomic hydrogen, one electron spin
couples with its orbital angular momentum (spin-orbit coupling). These two cases are

examined in the next two sections.

89

 

 

 

 

 

 

 

 

 

 
 

 

 

 

 

 

 

 

 

Spin
1/2

 

 

 

 

 

 

 

Magnetic Angular
Moment Momentum
(m)
A) """""""""""
Rotational Angular
Momentum
Electron
Orbital
Spin
l/2

 

 

 

Figure 25 Angular Momentum Diatomic Hydrogen.

90

5.4.1 Diatomic Hydrogen: Clebsch-Gorden Coefficients

Special attention will be given to the portion of the rotational spectrum of
diatomic hydrogen where electrons fall from the charge-transfer excited state (2”)
molecular orbital to the ground (2g) molecular orbital. Each peak represents a transition
from an angular momentum state one unit higher, lower, or equal to the final state [46].

Both 2” and 28 are degenerate in orbital angular momentum. Although they are
both s-orbitals, and spherically symmetric, the angular momentum eigenvalue is unity
(1:1); this is a result of the rotation about the axis perpendicular to the internuclear axis
(m = +1, 0, -l), as shown in Figure 25 [47]. The spin degeneracy for two electrons, as
described in section 5.3, accounts for an additional degeneracy in each of the orbital
momentum states. Represented in triplet and singlet form, the spin degeneracy is unity
(8:1). The total degeneracy in each ground state is equal to (21+1)(28+1) = 9.

Equations 5.49 and 5.50 are a simple example of the Clebsch-Gordan coefficients;
the angular momentum (s) and magnetic moment (52) of individual electrons are added to
give the total angular momentum (S) and magnetic moment (82). The Clebsch-Gordan
coefficients summing electron spins were found by applying operators S+ and S'. In
general, Clebsch-Gordan coefficients summing angular momentum are found with the

following recursion relationships, from application of the ladder operators J+ and J'.

 

J(j:m)(j:m+1)(j1,j2;m1,m2 IjiJzulmJ—tl)

 

= J01 f m1Xj1im1+11<j1aj2W1f1,m211151517") (5-51)

 

+\/(jz ’7 "72sz imz +1)<j1,j2;m1,m2 :1 | j1ifz;j,m>

91

Adding the diatomic hydrogen orbital angular momentum eigenvector to one

electron spin eigenvector gives:

(5.52)

 

(5.53)

_ l _
'=l+—,m
I} 2 >

 

= (5.54)

 

 

 

, 1
IJ:I_Eam>

 

 

 

92

Addition of the second electron spin gives the following 3x4 matrix as the sum of

orbital angular momentum and two electron spins:

 

 

r - Im—l;+—,+-1->
|l+1,m) 2 2
MI+l,++ Ml+l,+- MI+l,-+ Ml+l,-— Ina-1:1)
|1,m) = 114,,++ M,,+_ M,,_+ M,,__ f 3- (5.55)
M1-1,++ Ml-l,+— M1—1,—+ MI—l ——j lm’"§’+'§>
-11” 1”")3 |m+1-_l -1
|- 3 9 2 J

 

 

An identical 3x4 Clebsch-Gordan coefficient matrix exists with m replaced by
negative m. These two matrices are coupled for diatomic molecules. P. Zeeman
discovered and explained the coupling physically in 1902. Simultaneous forward (m)
and reverse rotations (-m) rotations sum to a single vibration, which precesses in the
presence of a magnetic field [48]-[49], as shown in Figure 26. As a result, nine distinct
degeneracies are present for each total angular momentum >0; three for rotation (+, 0, -),

and three spin (+, 0, -) for each rotation.

93

 

 

+801)

 

 

 

 

 

 

+50)

 

+6)

 

 

 

 
    
  
 

Lorentz Force (—(1))

A
V

Lorentz Force (-(u)

 

 

 

 

 

 

AAAA
VVYVV

 

 

 

 

 

 

/ +801)

 

 

 

AA
VVVV

 

 

Figure 26 Precession of Vibrating Diatomic.

94

The following lists the matrix elements for the rotation matrix in Equation 5.55.

 

 

1
Ml+l,++ =(m1-l,+-;—,+-2-Il+l,m) :Jl+m\[l+m+l

 

 

 

 

21+] 21+2
l l l+m+l l-m+1

M _= m,+—,-— 1+l,m =
”1"“ <’ 2 2I > \/21+1 \/21+2
] ] 1+m+11-m+l

M _ = m,——,+-— l+l,m =
”1” <’ 2 2I > \/121+1\[21+2

 

l 1 m+l
M __= m +1,--—, —— I
”l < ’ 2 2| "12> ’\/21+1\[21+2

l l I m+l l+m
M = —1,+—,+— 1, —
”H ("1’ 2 2l m) [21+] ($21M [21+2 ]

l ] 1+m+l 1+m+1
M/’+_ = (m,,+—,—— I 1,177) = -J121 "1le +\/ J
2 2 21+] 21+] 21+2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(5.56)
1+_]_|1+lm l+m_ l— m+l l—m+1
M,,_+=(m],-—
2 +2' 21+] 21+l 21+2
M,___=(m,+1,——,--l—|l,m)=Jll+m+1\/l-2[m l— m
’ 2 2 21+1+121+2
1 l l—m I—m+1
M_ = —1,+—,+— 1-1, =
I l,++ (ml 2 2' m) J21 J 21+]
1 l l+m
M_ _= ,+—,-— 1—1,m
"’+ (m’ 2 2I >:\/21\/21+1
1 l l+m
M _ ,——,+— 1-1, —
" + (m’ 2 2l m) \/21\/21+1
] l+m 1+m+l
M __= m +1,—,——— I-l,
1-,1 <1 2 | '71:) \[211/21+1

Entries in the matrix for Equations 5.55-5.56 are the Clebsch-Gordan coefficients
for the sum of orbital angular momentum and two electron spins for a diatomic molecule,
such as hydrogen. These coefficients are combined to find rotation and spin degeneracies.

95

5.4.2 Atomic Hydrogen: Clebsch-Gorden Coefficients

The total angular momentum and magnetic moment for atomic hydrogen can be
found by the addition of one unit of orbital angular moment and one electron spin, as
given in Equation 5.54 in the previous section. These coefficients are necessary to find
the energy change as a result of spin-orbit coupling, a result that follows from

perturbation theory.

5. 5 Perturbation Theory

Perturbation theory allows additional operators to be included in the Hamiltonian
to account for small changes in energy. Energy changes result from applied fields, and
energy corrections can be made for the relativistic mass of the electron and spin-orbit
coupling. Energy changes caused by electric (Stark) and magnetic (Zeeman) fields
remove orbital and spin degeneracies.

In general, the Hamiltonian can be appended with additional energy operators.
The set of equations on the following page summarize the perturbation mathematics for
non-degenerate energy levels, such as those found in the fine structure of atomic
hydrogen. Further development later in this section allows for the perturbation of

degenerate energy levels, found in the mixing of atomic wave functions.

96

Holno>=Eolno>
(H0+/lV)| n0) 2 501(n0+%)
n=no+723,<no 1% =0
(Eo—H0)1n> =(4V—An)1n>

<n01(Eo—Ho)1n>=<no1(4V—A,,)1n>=0

(AV — An )1 108%

An|n>=/1(n0|V|n)|n> (557)

That is, the change in energy along the nth eigenvector is the projection of the
potential operating on |n0>. Finding |n> should be as easy as applying (Bo-Ho)I to both
sides of Equation 5.57, and it is. But, (Eu-H0) maps |n0> to 0, so (Ea-Ho)"l is ill-defined

for n. However, (Bo-Ho)" is not ill-defined for HO , which is orthogonal to n. Defining (1),,

orthogonal to n results in the following:

9,, 1 n>=(1—1no><no 1)1 198%
(5.58)

1n> 21n>+1123>=1no>+—&—(4V-An)1n>

(ED—Ho)

An=ll(n0|V|n)

 

1

Equation 5.58 can be solved iteratively for eigenvectors |n>; eigenvectors |ni> will

be combinations of eigenvectors orthogonal to n0>, the set of eigenvectors for the

97

unperturbed Hamiltonian. Given the notation, the first and second order perturbations for

energy level and eigenfunctions are given in the following two sets of equations [42]:

Vkl=<nk|Vlnl>

 

732:0
(5.59)
A1,, =(nOI/1V M8): ZVnn
A2,,=(n0|V|n(l))=(n0|/1V ‘1’" 4V1n8>= Maw/122M”—
(Bo-Ho) “"05" E1)
Vk1=<nk WW)
"8) =1 "0)
n5)=1n8)+—i’—2V1n8)=1n0)+12—_—~—11)
(50'- H0) ””1133: £1)
2 0 (Dr: 1
n0)=|n0)+ lVan)
(50' 0)
<0 0 <1> <1>,,
=|n8)+———"——/1V|n0)+—"—/1V|——"——/lV|n8)
(150—110) (150-170) (150—110 )
-| "OH/1 1k)
glE—nV—L “Ek)
(5.60)
V V V V
+112 [(1 1n )1 k)_ nn kn )lk>
E15,“): “EkXEn—El) k;(—_EnEk)

98

Intuitively, Equations 5.59-5.60 state that the first order perturbation is the
projection back to the zero order eigenvectors of XV, which operates on the zero order
eigenvector set. The second order perturbation operates and projects a second time. On
convergence, XV|n> projects back onto |n>, returning exactly the eigenvector equation.

Equations 5.59-5.60 are predicated on the fact that E" at E 1- fork 1: n; that is, the
energy levels are non-degenerate. For degenerate energy levels, Equations 5.59-5.60 fail
to produce perturbed energy levels and wave functions.

However, degenerate energy levels allow the freedom to mix eigenfunctions
within a given level. The new eigenfunction representation can be composed such that

the inner product terms an go to zero for each E" =Ekwherek ¢n. Returning to

Equations 557-560:

I
Pm:21mi><mil
i=0

1
11}- >=1D,,,|1j >=Z|m,. xm, 11,- >
i=0
(5.61)

o=<m21(Eo—Ho)11§~’>=<m91(4V—Am)1l?>

I l
=52<m312V|mP><mP|1§h=ij <m911j?)=A,,, <m911j?)
i=0 i=0

Equation 5.6] transforms the wave function representation into one in which the

new inner product term matrix (an) is diagonal; that is, an=0 for allk ¢n. The

99

eigenvalues for Equation 5.61 are the perturbed energy levels; the eigenvectors for
Equation 5.6] are the linear transform coefficients mapping |m> to |l>. Equations 5.59-
5.60 are now valid for degenerate energy levels; summation is over all remaining non-
degenerate states, all states with unique, non-zero eigenvalues.

The next three sections look at spin-orbit coupling in atomic hydrogen, the
relativistic mass correction for atomic hydrogen, and the anomalous Zeeman effect. Each

of these effects can be accurately approximated by perturbation theory.

5.5.1 Spin-Orbit Interaction

A magnetic field will interact with the orbital angular momentum and electron
spin of an atom, splitting the energy lines in the visible spectrum. This effect is called the
anomalous Zeeman effect. But first, it is important to look at just the interaction of the
orbital angular momentum with the electron spin —the spin-orbit interaction.

The central potential in the Schrodinger equation is not strictly a central potential
due to the shape of the electron cloud surrounding the nucleus. A moving electron
accelerates radially in response to a field gradiant just as it would to an applied magnetic

field, namely:

V = e<1>(r)
(5.62)

B =-XxE=leV
c ec

The electron spin couples with B; the energy correction operator represents the

work done to rotate the electron spin away from the magnetic field.

100

Thus,

eS

mec

 

(5.63)

 

 

Harm- 212[S.(,-,x_;3.d_V)]-[1) 2121.416.”
mec r dr 2 mec r dr
Where the extra multiplicative constant 1/2 is due to Thomas precession [50].

(L 0 S )does not commute with L2 or S2, but does commute with total angular momentum

J2 and .12. There are two total angular momentum terms, from section 5.4.1:
L-S =—l—(J2 -L2 —52)
2
2
': ': h . . 3 ': ':

1

l 5712; j=l+l/2
'=/i1/2 '=/il/2

(Y/ ”"|1‘111..91Y1j ’m> =1

_[_+_1h2;j=1—l/2
l 2

 

ldV

l '=1:1/ 2, °=11r1 / 2
Anlm :: __fi<__> (YIJ "7 l HLS 1Y1] ,m)
Zmec r dr ”1

1
l —h2; j=1+l/2

A _ 1 <1a1V>< 2
nlm" 2 2 d
2mec r r n] _l_+_1h2;j=1_1/2

l 2

 

 

(5.64)

101

Equation 5.64 is Lande’s interval rule [42]. Referencing Equations 5.36-5.41, the

potential gradient term can be found in steps:

 

 

2
l e
(Wm/erVl‘PnOZZEn :-—2‘
a0
1 l
smhrhwn=7_—
n a0
2
2 e
(Wnll'a— 81W) ‘3—
n ‘10
4 meez 4 1
20nn% 1%»: 3 2 = 37-
r2(21+1)n h ()0 (21+l)n (10

21(1 +1»?2 + 33

 

 

(‘anlé— 9:11 | ‘i’n/>= (‘Pn/ l— 3 2 l‘l’nn =0
mer r
=><‘III‘I‘¥>—””"2 <w I} fll >— 2 1
I — l “ I I - '—
" H " n22m+ +1) " ” IwHw+m££ ma)
1 a 2 e2

 

=>(‘Pn l-—V|‘Pn >= —
I rra 1(1+1)(1+2)n3a3

102

The ratio of the energy shift for atomic hydrogen to the Balmer intervals is on the

order of 08:1/1372 [42], that is:

 

 

 

 

 

 

hz
00 -
mee2
2
a=9_=_1_ (5.66)
hc 137
772 (1:11) ’22 ._2 2
AIM," __ 2mgc2 r d" n] 2m§c2 (18 :2. _L2
82 32 62 hc 137
200 200 200
Which means A'm can be written as:
i I. ° 1 1/2
_, = +
1 62 2 1 2 ]
A"hf-“217“” 11 2 3i
0 (+)(+ )n _l+l;j=1—1/2
2
(5.67)
i "I. ' I 1/2
_, = +
2 a4 l 2 J

 

<
2,,4 1(1 + l)(l + 2)

 

L 2

_fl;j=1—1/2

This expression will be combined with the relativistic mass correction in the next

section.

103

5.5.2 Relativistic Mass Correction

The perturbation operator for relativistic mass comes directly from the relativistic

energy term:

 

 

 

 

 

 

 

E3, = mgc4 + pzc2
(5.68)
p2 p2 p2 1 p2 2
->En,—m962=mec2 1+ 2 2—1 = - 2
2m? me C Zme Zmec Zme
Solving for Alnlz
2
2 2 2
l l e e
Alnl : —_—f<\ynll P PPM) : _'_2<En +_ I En + >
2mec 2mg 2mec r r
= — 1 [E3 + ZEne2<l> + e2<-1—>]
2 2
2mec r r
(5.69)

 

_ng_( 2 -1]
"n 21+] 4n

2 a4( 2n 3]
=mec --—— -—
2n4 21+1 4

 

Now, adding the spin-orbit interaction to the relativistic mass correction, with

total angular momentum j substituted for orbital angular momentum 1:

Al =_meCZL " -3 (5.70)

 

104

And the total energy, including rest mass, is [50]:

 

— —— +... (5.71)

Briefly, an additional correction term —the Darwin term [51]- allows for s orbital
(1:0) corrections in Equation 5.69. Both the Darwin term and Lamb Shift evolve from
the relativistic quantum field equation —the Dirac equation. The Lamb shift makes a very
small correction to remove the degeneracies in orbital angular momentum [42]. Its effect
is nearly negligible for this set of experiments, and will not be pursued here.

Equation 5.71 completely describes the energy levels associated with the fine
structure of atomic hydrogen. The fine structure of atomic hydrogen, and nominal

transition intensities, are addressed further in sections 5.7 and 5.8.

5.5.3 Anomalous Zeeman Effect

Degeneracies in the spin-orbit interaction are lifted with an applied magnetic
field. This effect is the anomalous Zeeman effect. The perturbation term enters as the
electron momentum interacts with the field momentum, the field that produces the

magnetic moment.

105

Thus, for a relatively weak magnetic field,

A = jgum) z. A = —%(.§By —}7'Bx)

 

 

 

 

 

 

2 2 2
)A ) A
p—>p—‘ :>H= p +V— ‘ (p-A+A-p)+ e
0 2m? 2mec Zmec2
(5.72)
l
pOA = AOp—thOA = A0p+0= AOp =%B(—ypx+xpy)=§BLz
A2 = AOA =%Bz(x2 +y2)
Ignoring the smaller quadratic term:
p2 eB
H = + V - (L, + 25,.) (5.73)
2me mec

Where the factor of two on the spin term is due to the g-factor of the electron [50].

To summarize:

 

 

 

2
H0: P +V
2m.
1 ldV
HLS _ 2 2 (Los) (5.74)
Zmec r dr
88 e8
HB — 2 (Lz+25z)_ 2 (Jz+Sz)
2mec 2mec

106

The transformation to total angular momentum and total magnetic moment is

given in Equation 5.54 (C lebsch-Gordan coefficients), namely:

 

 

 

 

 

l = _
I] = Him) = 2) (5.75)
l¥m+—
= m+—,m ——
21+1 I ’ ‘ 2>
Therefore, the first order energy perturbation is:
eB , 1 . 1
AIB=- (j=li—,m|(Jz+Sz)|j=li—,m)
2mec 2 2
l _ l l _ l
B lim+— 1+m+— hlim+— 1+m+—
A18 = _ e 2 2 2

m + +- — (5.76)
ZnQC 21+1 21+1 2 21+1 21+1

 

 

=— e3 mh[li 1 ]
Zmec 21+1

For a stron er field, JZ no lon er commutes; on] L2, 82, L2 and SI remain as
g g Y

commuting operators.

107

As a result,

 

 

A'Bz- (1:1: ,m|(Lz+2Sz)|j=I+—,m)
Zch
8
=— e h(m,+2m,.) (5.77)
mec

2
m m h l a .
Zmecz r 3r
Not all degeneracies are removed in a strong magnetic field; in], ms combinations

yield the same first order energy correction. Line splitting where the applied magnetic

field effect exceeds that of the spin-orbit interaction is called the Paschen-Back limit.

5. 6 Rotational Spectrum for Diatomic Hydrogen

The intraband rotational spectrum, the series of peaks for a constant vibration
eigenvalue and single angular momentum transition, becomes evident as the applied
magnetic field removes the degeneracies on orbital and spin angular momentum. Figure
27 shows the energy level diagram for the Zu—Zg transition [52]. Each of the fifteen
transitions is associated with a unique energy difference; the reason: the energy split in
orbital angular momentum is approximately 50 percent larger (28.4/20) for Eu than 2g
[53] due to the higher rotational inertia of the charge-transfer orbitals.

Zeeman splitting for a free electron (gEBB) in an applied magnetic field of 5 T is
approximately 4.7 cm-l (0.2 A/T at 4627.66 A) [54]. Accordingly, the magnetic field in

the plasma can be calculated by tracking the intraband peak separation in the rotational

108

 

J=3

 

 

 

 

 

 

M=3

 

 

 

 

 

 

 

 

 

 

 

Energy

 

 

 

 

J=2

 

 

 

 

 

 

MZ-l

 

 

 

M=-2

 

 

 

 

 

M=-3

 

 

Transitions: AM=O,+/-l

 

 

 

 

M=2

 

 

 

 

 

Figure 27 Zeeman Energy Levels.

 

 

 

 

 

 

109

 

 

 

hydrogen spectrum. Peak intensities are a function of the CIebsch-Gordan coefficients,
given in Equation 5.54, and energy state population, which is approximately a Boltzmann
distribution. Over a narrow energy band, the Boltzmann distribution is linear, with slope
l/kT. Plasma magnetic field calculations and temperature estimates from experimental

rotational spectra are found in Chapter 7.

5. 7 Fine Structure of Atomic Hydrogen

The fine structure of the atomic hydrogen spectrum is composed of the assembly
of corrections formulated in section 5.5, and specified for H, and H3 in Figures 28 [50]-
29. Degeneracies are removed by the energy corrections. However, the fine structure of
atomic hydrogen is further mixed upon application of an electric field. It is this
additional separation in peaks that complicate the atomic hydrogen spectrum in low to
medium electric fields (~1000 V/cm-5000 V/cm). Calculations for the electric field

directly from the Stark shifted spectrum are summarized in the following chapter.

5. 8 Nominal Fine Structure Transition Intensity

Ha nominal atomic hydrogen fine structure line intensity ratios are given in Figure
30. Fine structure line intensity ratios are a product of the Clebsch-Gordan coefficients
that construct the fine structure energy levels, and the overlap integrals that connect these
energy levels. The inner product of the angular waveforms relies on the following

identity [55]:

 

 

 

 

COSQ'YIm=\/(l+m+])(1_m+l)Y/+1m+\[(l+m)(l_m)YI-Im (5.78)
’ (21+l)(21+3) ’ (21+l)(21-1) ’

110

 

 

0.108

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.222

 

 

 

 

0.036

 

 

 

 

 

 

 

 

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

 

“3/2
3/2

 

M=5/2
M=l/2

M
M

 

 

 

 

 

 

 

3

 

 

 

 

 

 

 

3

 

N—

 

 

 

 

lll

 

Figure 28 Ha Fine Structure Transitions [50].

AE=15,232.951 cm-l

 

 

 

N:

 

 

 

 

 

0.046

 

 

 

 

 

0.305

 

 

 

 

 

 

0.015

 

 

 

 

M=5/2

 

 

M=3/2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

N=4

 

M=l/2

 

 

 

 

 

 

 

 

M=3/2

 

 

 

3

 

 

 

AE=20,570.502 cm-l

Figure 29 H3 Fine Structure Transitions.

 

 

N:

 

 

 

112

 

l/2>3/2

 

 

 

 

5/2>3/2

 

 

 

 

 

3/2>3/2

 

 

 

 

 

 

l/2>l/2

 

 

 

 

 

3/2>l/2

 

 

 

t!—

 

0.108

 

 

 

D---—-u-

 

 

 

AE=15,232.951 cm-l

 

Figure 30 Ha Fine Structure Relative Transition Intensity.

 

0.036

 

 

 

 

0.222

 

 

 

113

 

 

0.108

 

 

 

And the orthogonality of the waveforms. As a result,

 

 

1\/(j:m)(j:m+1) (5.79)

*
<Y'—1/2m:1/2|0089|Y'+1/2mi~1/2>=— . .
J j 2 J(J+1)

Using the identity given in Equation 5.78, the transitional wave functions can be

summarized by the expressions,

l j—m+1

+ Rn.j+l/2 _2(j+1) Yj+l/2,m—l/2 =>(5.81)
j+m+l

_ Rn.j+l/2‘/ij+1 / 2,m+1/2 => (5-82)

unJ-Jn =< (C — G) (5.80)
j + m
21'

 

+ Rn,j—l/2 Y'—1/2,m—l/2 3 (5-83)

j—m

2]

 

+ Rn,j——1/2 Y'—l/2,m+l/2 => (584)

 

l

 

 

Where:
i (j+m+1)(j—m+2)Y.
4(j+1)(j+2) 1+3/2,m-1/2
Yj+1/2,m_1/2C086=< + (5.81)

 

 

j—l / 2,m—l / 2

 

Jmmw—ml)
1 4J'(j+1)

ll4

 

 

j+3/2,m+l/2

lJ(j—m+1)(j+m+2)
4(j+1)(j+2)

Yj+1/2,m+]/2C056=< + (5.82)

 

 

j—1/2.m+1/2

 

th—mwmu)
1 4j(j +1)

 

 

.
(j+m)(j-m+1)
4].”. +1) j+l/2,m—1/2

Yj_1/2,m_l/2cost9=< + (5.83)

 

 

 

Yj—3 / 2,m—l / 2

Jo—mwm—I)
1 4117—1)

 

 

lJ(j—m)(j+m+l) .
4j(j+l) 1+1/2.m+1/2

Yj_1 /2,m+1/2 €086 =4 + (5.84)

 

 

 

Y-_
4].”. _1) ) 3/2,m+l/2

\lomxj—m—l)
The notation um;m represents both the initial and final wave functions, with fine
structure quantum numbers n,j,m as given in section 5.5. The relative amplitude of

transitions between energy levels is given by:

2 2
21¢), 1.. >1 = 210%,, |R|un.,j.,m >| (5.85)
m;j.j'

 

n0, '0 _-
211,]! —1< ¢n,j

115

Where Equation 5.85 specifies the inner products of radial wave functions in
Equations 5.81-84. These inner products are only a subset of all possible transitions,
limited by the orthogonality property of spherical harmonics.

Specifically, the H0! fine structure transition amplitudes are given by the

following, with allowable transitions limited to AJ 2 0, +/-l.

3,P1/2 _ 2
22’51/2 — 2K R3,] 1R1 R2,0 >1 *
m=1/2

 

 

 

[J(l/2—m+l)\/(l/2+m)(l/2-—m+l) (l/2+m)

 

 

 

 

2(1/2+1) 4.1/2(1/2+1) 2-1/2
(5.86)
2
+ (l/2—m+l) (l/2+m)(1/2—m+1) (l/2+m)
2(1/2+1) 4-1/2(1/2+1) 2-1/2
=l.O4
2395.18: ZI< R3,61R182,1>12*
m=1/2
(l/2+m) (l/2—m+1) (l/2+m)(l/2-m+1)
2.1/2 2(1/2+1) 4-1/2(1/2+1)
(5.87)

 

 

 

 

2
+ (1/2—m) (1/2+m+1) (l/2—m)(1/2+m+1)
2-1/2 2(1/2+l) 4-1/2(l/2+l)

=0.10

116

3,P3/2 _ 2
22,51/2 — Zl< R31 1R1 R20 >1 *
m=l/2

 

 

(3/2+m) (3/2—m)(3/2+m—1) (l/2+m)
2-3/2 4-3/2(3/2—1) 2-1/2
(5.88)

 

 

2
+ (3/2—m) (3/2+m)(3/2-m—l) (l/2—m)
2-3/2 4-3/2(3/2—1) 2-1/2

= 2.08

3.03/2 _ 2
22,Pl/2 " ZI<R3.21R1R2.1>1 *
m=l/2

(3/2—m+l) (l/2-m+l) (1/2+m+l)(l/2—m+2)
2(3/2+1)o 2(1/2+1) 4(1/2+1)(1/2+2)

+ (3/2+m+l) (1/2+m+l) (1/2—m+1)(1/2+m+2) 2
2(3/2+1) 2(1/2+l) 4(1/2+1)(1/2+2)

 

 

(5.89)

 

 

 

 

= 5.01

3.51/2 _ 2
22,103/2 " Zl< R3.0 1 R l R2.1 >1 *
m=l/2

 

 

 

J(l/2+m) (3/2+m) (3/2—m)(3/2+m—1)
2-1/2 2-3/2 4-3/2(3/2—1)

(5.90)

 

 

+\[(1/2-m) (3/2—m) (3/2+m)(3/2—m—1) 2
2-1/2 2-3/2 4-3/2(3/2—1)

= 0.20

117

3,D3/2 __ 2
22.P3/2 — 2K R32 1R1 R21 >| *
m=1/2,3/2

 

 

[J(3/2—m+l)‘/(3/2+m)(3/2—m+l) (3/2+m)

 

 

 

 

2(3/2+1) 4-3/2(3/2+1) 2-3/2
(5.91)
2
_ (3/2+m+l) (3/2—m)(3/2+m+1) (3/2—m)
2(3/2+l) 4-3/2(3/2+1) 2-3/2
=1.00
23:’B§/§= ZI<R3,21R1R2,1>12*
m=1/2,3/2
(5/2+m) (5/2—m)(5/2+m—l) (3/2+m)
2-5/2 4-5/2(5/2—l) 2-3/2
(5.92)

 

 

2
+ (5/2—m) (5/2+m)(5/2—m—1) (3/2—m)
2-5/2 4-5/2(5/2-—l) 2-3/2

= 9.02

118

Where the pertinent normalized radial wave functions are given as [56]:

l 5 (27 — 18R + 2R2)e_R/ 3 (5.93)

R =———
3'0 74920.7

1 640m?“3

R =-——(
3" J2460.375

l _
R26 R/3

R z
3’2 J12,301.875

In a plasma discharge, fine structure line intensities vary dramatically as a
function of electron density [57]-[58]. For example, peak l/2>1/2 in Figure 30, barely
detectable in the nominal case, becomes as strong or stronger than peaks 5/2>3/2 and
3/2>1/2 at electron densities >10l4 cm'3 [42]. Figures 31 [59]—32 [60] demonstrate fine
structure peak ratios in experimental conditions closer to experiments run for this study.
Both are taken from low-pressure gas discharges. However, electron densities are not
recorded for either experiment. As evident, the characteristic shape of the fine structure
peaks can be used as a signature for identifying Stark effect splitting in hydrogen

plasmas.

119

 

 

 

5/2(3/2)>1/2

 

 

 

l/2>l/2

 

 

 

 

 

 

5/2(3/2)> 1/2

 

 

 

 

l/2>l/2

 

 

 

 

 

 

Figure 3] Ha Fine Structure Peaks Near Band Center [59].

I l 1 l
I f I l
meal :2... m.

120

 

 

 

 

 

 

 

5/2>3/2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 32 Ha Fine Stucture: Absorption Spectroscopy, Pulsed Dye Laser [60].

 

 

 

 

121

 

 

 

 

 

3/2>1/2
LambShift 1H
1/2>1/2
LambShift
I J L I I I l I
I I I r I l I I
OGH lOGH

Chapter 6 Atomic Hydrogen: Stark Effect

Chapter 5 covers Zeeman splitting in the rotation spectrum of molecular
hydrogen, and the fine structure of atomic hydrogen. Each of these effects are
completely described in spherical coordinates. Chapter 6 addresses Stark splitting in
atomic hydrogen. Stark splitting can be expressed in spherical coordinates as well, but
spherical coordinates limit spectral analysis of atomic hydrogen when considering the
gross structure splitting in combination with fine structure splitting. The gross structure
waveforms mix so thoroughly that initiating fine structure points is intractable.

Chapter 6 addresses this shortcoming. The following sections first solve for the
Stark effect splitting in spherical coordinates, then the problem is moved to parabolic and
semi-parabolic coordinates to better match the symmetry of atomic hydrogen. Wave
functions that result from a coulombic central force, such as that found in atomic
hydrogen, have an additional degree of freedom when expressed in parabolic coordinates.
The additional freedom represents an additional symmetry that was hidden —and not
necessary- in spherical coordinates.

As a result, the Stark effect does not mix the resulting waveforms in parabolic
coordinates, and the gross structure is predictable in the presence of fine structure
splitting.

Sections on Stark fine structure splitting immediately follow the treatment of
Stark splitting in parabolic coordinates. For both gross and fine structure splitting,

transition amplitudes are included with experimental spectral data in Chapter 7.

122

6. 1 Stark Splitting: Spherical Coordinates

The wave functions for atomic hydrogen are developed in spherical coordinates in
section 5.2.2.2. The next two sections address the Stark effect for atomic hydrogen in the
spherical coordinate system. The first section solves the Schroedinger equation by direct
application of perturbation theory, developing the perturbation —or overlap— matrices
associated with the given potential operator. Then, solves for the eigenvalues of the
perturbation matrices, which immediately give the Stark shifted atomic hydrogen energy

levels. The second section details the Stark shifted spectrum for atomic hydrogen.

6.1.1 Perturbation Matrix

The perturbation matrix is composed of all possible wave function overlap
integrals; that is, the integrals of each pair of degenerate wave functions and the
applicable potential energy operator. The potential energy operator for the Stark effect is

related to the applied electric field as follows:

852 ; LinearPo/arization

VStark = (6'1)
eE (x i jy) ; C ircularPo/arization

Due to reactor geometry, the polarization of the electric field for this set of
experiments is strictly linear. F irst-order approximations to energy level shifts are the
eigenvalues of the perturbation matrix.

Spherical wave functions of atomic hydrogen associated with electronic energy

levels two through four are given on the following pages [56],[6l].

123

3

1 Z _ 72 . -°

_ ==-—————-’-——pe p. sn16 e j¢
1112.1,1 ,——2”.32 03

Z3

1
W2,1,0- F—Zrt-l6 03

3
1 Z _ '
=._______ 1.. P/Zshjg.e+1¢
WQJAJ EZFTSE agpk'

pe’p/zcosd

 

 

1r 1 Z:(2- ple p”
3”:°° J27: 9841.5

1 Z3 .. /3 . _'¢
._ = -——-6-— ) p su16 8 J
11’3"" 727:.1640.25\lag( p p8
W310: 1 2—316-131W-p/30059
” fim164025 03

3
1 Z — /3 — /2 - +°¢
= -—— 6- )pe p e p snag e 1
V3” f2n-1640.25\lag( p

 

 

 

124

(6.2)

(6.3)

4 . g

Ze _p/3sin26.e—j2¢

Wn’ 2=—— 27: 6561 6;"
Z3
W352, 1: l gfepz p/3sin6cos6l-e_j¢
72;: 1640.25 00
W _ 123p2e—p/3(3c0526—1)
3’2’0 J27: 19683

Z3
= sin6cos6-e J
V33“ (27: 1640. 25 \1 _ep

Ze —p/3

 

 

 

sin2 (9 - e+j2¢

lV3,,2+2=————— 2” 6561 a—:pe

 

3
W400 = l 12701192—144/2'1‘24PZ __-p3Lp/4

72;: 1179648

 

52: 1310720

3
'#410 = 1 1123 (80 201022021 M40086

«[27: 655360

’_ 3
W4,1,—1 = 1 Z (3)—(80 20p+p2}- p/4sin6. e+j¢

727: 1310720

3
11’4,1,-1=1 12735 (80— 20p+p2¥ p/4sinl9- e M (6.4)

 

 

125

 

l//4,2,—2 = Jz

1 ‘23 2 /4 2 _.2
—(12 -e,0)p p sin 8.9 J ¢

 

3 o
l _3'1‘2(Z“1€’P)P2 p/4sin6cosa.e‘J¢
Ir-786432 a3

 

3
I Z 2 /4 2
2 —p 3cos 6—1
47185921103—0 plpe 1 )

 

W4,2,—1 = J2
V4.20 = J2”
V4,2,+l = J

Zn-

3 .
7186432 11002 "—0 1"?- P10213—p/4sintélcost9-e‘Vi’>

 

W4.2,+2 = J2

3 e
l -Z_—(12—p)p2e—p/451n268+12¢
n~3145728 03

 

'Z—333P3 p/4sin36.e‘f3¢
18874368 03

 

Z:3P3 p/4sin2 610030 6 J24)
3145728 00

 

Eiep3 —p/4 Sin‘(65cos 2.6—1) e J¢
31457280 a3

 

3e_p/4( (5/3cos36— 0086)

oh:

-262 1440 1100

 

—3'3P3 p/4sin6(25cos 0—1)e+J¢
31457280 0(3)

P36 —p/4sin2 6cos6l e+J2¢

 

£313
3

.23

3145728 gap

P33 p/4sin3 0 e+J3¢

 

W4,3,—3 = 42”
W4,3,—2 = J27;
W4,3,-1 = 42”
W430 = J27;
V4,3,+1 = J27:
V4,3,+2 = «[22-
V4,3,+3 = ‘5”.

18874368]’::M

126

The non-zero elements of the perturbation matrices for each energy level can be

summarized in the following:

n = 2:
02 =< W2,0,0 | Z l W4,1,0 >= 30000
n = 3:

a3 =< W3,0,0 | Z l W3,1,0 >= 7-3500
1’3 =< W3,1,—1 lz | W3,2,—1 >= 45000
C3 =< W3,1,0 | Z l W3,2,0 >= 53000
d3 =< W3,1,+1 IZ | W3,2,+1 >= 45000

n=4:

04 =< W4,0,0 I Z | W4,1,0 >=13-4200
b4 =< W4,1,-1 |2 | W4,2,—1 >= 93000
C4 =< W4,1,0 I Z | W4,2,0 >=10-73ao
d4 =< W4,1,+1 I Z | W4,2,+1 >= 93000
64 =< W4,2,-2 lz l W4,3,—2 >= 60000
f4 =< W4,2,-1 | Z | W4,3,-1 >= 7-5900
84 =< W4,2,0 |2 | W4,3,0 >= 8«0500
’14 =< W4,2,+1 I Z | W4,3,+1 >= 7-5900
i4 =< W4,2,+2 | Z | W4,3,+2 >= 63000

(6.5)

Where the perturbation -or overlap- matrices are expressed by the following for

each of the electronic energy levels two through four:

127

 

 

 

 

0

0

0

0

0

0
000
012400

0
0
0
0
0
0
0
0

0
0
0
0
0
0

0
O
0
0
0
0

00000

0

0
6’4

00000

b4
0 O
0

00000

0d4OOOO

0

a4

0
0
0
0

0 0

0

00064

0
0
0
0
0
O
0
0

0

0000/4
00000g4

00000

0

C4
000d4

0 0

i4

00000

0
0

0
0
0
0
0
0
0

0
0
0
0
0
0
0

00000

00000

6’4
0 0f4

0

00000

0 0
0

0

00000

0084

0000

’14

0
0
0

0
0
0

14000

O

0

0

0000

 

 

)

6.

6

(
.000000000
000%300000
0003000000
0.030000000
000000000
0000000..w30
@OOOOOQOO
0000063000
00 @000000
—

__

I3

H

128

The characteristic equation and solution for each of the electronic energy levels is

given by the following:

i 0,0,0

 

 

 

4502:0th a2+g2+c2)-4a2g3

L

(6.7)

129

Substituting from Equation 6.5 gives the following Stark energy level shifts in

electronic energy levels two through four for linearly polarized electric fields:

n = 2 :
_ 0
— i 36(10E
n = 3
0
(SE = i4.5eaOE
It 9.0600E
n = 4 :
' 0 (6.8)
+
g = < ._ 6.060013
i 12.08001?
1:18.08ea05

 

Which can be summarized easily by the following equation:

E = 60015 ' -:—m'

(6.9)
OSi<n

Where n is the electronic quantum number.

130

6.1.2 Stark Energy Spectrum: Spherical Coordinates

Figures 33-34 are diagrams of the Stark shifted energy levels for atomic hydrogen
transitions Ha and H13, respectively. Each figure is specific to an electric field that is
linearly polarized. Transition lines are left out of Figures 33-34, and transition
amplitudes are not addressed.

In spherical coordinates, allowable transitions in a linearly polarized electric field
are limited to those in which quantum numbers Al=+/-1 and Am=0. However, Stark
shified wave functions assigned to each energy level can no longer be expressed as
spherical harmonics; rather, they are linear combinations of spherical harmonics, and the
mixing that results from the applied electric field is not trivial. As a result, the allowed
transitions and transition amplitudes are extremely difficult to generate in spherical
coordinates.

Following mixing, each eigenfunction is generated from the eigenvalues of the
perturbation matrices, and the perturbation matrices themselves. The presence of
degenerate eigenvalues (e.g. multiple zeroes) requires very complicated operators to
generate the eigenfunctions, or wave functions. These operators turn out to be the ladder
operators associated with a new, cylindrically symmetric coordinate system. Instead of
working through the ladder operators, it is far simpler to solve the Schroedinger equation

for atomic hydrogen in the new coordinate system, that of parabolic coordinates.

6. 2 Parabolic Coordinates

The application of an electric field destroys the symmetry in the radial component

of the generalized central force problem; it is now cylindrically symmetric. As a result,

131

 

|0,0>+|1,0>+|2,0>

 

 

 

 

/

5E=4.5a0eE (typ)

 

 

 

 

 

 

/ |1,+/-1>+|2,+/-1>

 

 

 

 

 

N=3 12,-2>, |2,2>

 

 

 

 

 

 

 

 

|1,+/-1>+|2,+/-1>

 

 

 

 

 

10,0>+11,0>+|2,0>

 

 

 

 

 

l0.0>+| 1 .0>

 

 

 

 

/

5E=3.aneE (typ)

 

 

 

 

 

 

 

 

 

 

 

 

N=2 / |1,1>, |1,1>

 

 

[0,0>+|1,0>

 

 

 

Figure 33 Ha Stark Energy Spectrum: Spherical Coordinates.

132

 

|0,0>+| l ,0>+|2,0>+|3,0>

 

 

 

 

 

|1,+/-l>+|2,+/- l>+|3,+/-l>

 

 

 

 

f

5E=6.0aneE (Wu)

 

 

 

 

 

/ l0,0>+| l ,0>+|2,0>+|3,0>

 

 

 

 

|2,+/-2>+|3,+/-2>

 

 

 

 

 

 

 

N=4 I33-3>9 I393)

 

 

 

 

 

 

|0,0>+| l ,0>+|2,0>+|3,0>

 

 

 

|2,+/-2>+|3,+/-2>

 

 

 

 

I l ,+/-l>+|2,+/— l>+l3,+/- l >

 

 

 

 

 

|0,0>+|1,0>+|2.0>+|3.0>

 

 

 

 

 

|0,0>+| l ,0>

 

 

 

 

/

5E=3.0aneE (tvnl

 

 

 

 

 

 

 

 

 

:2 / lls'l>all,l>

 

 

 

 

 

|0,0>+| l ,0>

 

 

 

Figure 34 H5 Stark Energy Spectrum: Spherical Coordinates.

133

the orbital wave functions are no longer separable in spherical coordinates. More
importantly, the mixing of the wave functions that results from perturbation analysis
makes ferreting out the spherical harmonic components associated with each energy level
very difficult. Both issues are resolved by moving to a new coordinate system that

matches the symmetries of the problem.

6.2.1 Parabolic Transform

Cylindrical symmetry is retained when folding up R3 space in such a way that the
x-y plane forms a right circular cone about the z axis, and each additional plane with
constant 2 folds into hyperboloid sheets, as shown in Figure 35. Every plane intersects
the infinite set of hyperboloid sheets to form circles, ellipses, parabolas, and hyperbolas;
conic sections that define the dynamics associated with a central force proportional to
l/R. (Each conic section is actually a geodesic with respect to rotated SO(2,1) space, or
Lorentzian measure [62]). The inverse map of this three-dimensional folding is the
parabolic transformation.

The parabolic transform map is shown in Figure 36; this map generates the

parabolic coordinate system [63]. Using complex variables:

 

q =§w2 =-;-(€+M)2 =l(¢'2 -772)+j§n

Fag—.772) (6.10)
2

R=J[§(62-n2)] +8612 =§(:2+n2)

 

constant

 

 

 

 

 

+x

 

+z

 

 

 

 

 

 

\
s
s
s
\ \
s
‘ s
\ s
\ \
‘
‘\ ‘s I
~ ’
--- ....... ‘ ’-":---
\
\
s
\
\
s‘ ’

I I
I I
I
I
I
I
I I
I I
I I
I I
I I
I I
I I
I, ’
I
I I
I, ’
i I
I I
, I
, I
I I
I I
I I
I’ I
I
1’ ’
I
I
I
I
---’ - ........................
I
I
I

 

 

 

 

 

 

+Y

 

 

 

Figure 35 Hyperbolic Transform of Constant z Surfaces.

135

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

+jv

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

u Z
+ + W.)
_ _ +
_ am - 2
o . n u
. — . .- «M
g
. u .. .
. .
_ u a .. :II - .
. . o . \\ I — ~
. ._ 111111 . . 4 / . .
IIIIII _IIIII.II|I£IIIIIIIIIII — a \ I . s
- - a — \ I. u
. . u .. . ss I. .. \nl
“ " e0 .. x a. 11111 u: . V
. . . .s or .I’ H u
. . as \’ I . a Q (
..... T--"q"'If"""-"---'-'-' O. . \ 0” - I . w
s s .
u u . ... . . . . ...
. . .. . ~ . . .
. . .s. . .s . C
. . n
. . .oa . la. 6
u . a. . P 3
. — l
. . . . Z k
— _ s o u. p
. . . . .. m
. . s .. . O
. . x s as ob
- P OJ 1 “ C
u . + x .3 .— C
o
. u s .o a cam
. . s - o — n
. . . .
. . . . . . . a Q m
. . . . . . . . l O
. o a
u n s o O Q Q O N“ n
. . ~ . — . . . 1+. mm
. . . . . . . ( C
. . av a; ~. ~ u “I-
u " ls. .. 6 Im
. - s r
. . a
. u up. .. P
. - s 6
H n W— A \s 3
g
m

 

 

 

Now, R2 is separable as the difference of fourth-order terms. Further, the

Laplacian can be found directly from the differentials generated by the map [64].

l a a a
V2=———8§A +85A -— 8634 --—
(SVOI{ [ ’79 BS§J+ [ :6 BSU]+ { 6’7 850 J}

2- 1 _. .3. .9. .2. __ 1 __j’:
V —§n(§2+nz){3§[&736] 3 “[6 Gem} 527720612

(6.11)

With a change of variables,

6:6

(6.12)

77:5

The Laplacian can be expressed as,

 

2
V2: 4 1(u3—j4u3—(vi) 4.1-3— (6.13)
u+v Bu Bu 8v 3v uvagz

And,

1
2:50-»

(6.14)

1
R=-— +
26 v)

137

Consequently, the Schroedinger equation in parabolic coordinates, with a

coulombic central force and applied electric field, is written as,

hz
——V2+V (Iszw

2m

2: 4 1(,2)._a_(,2).1_2: 6,,
2m u+v Bu Bu 8v av uvag2 W (' )

Z62

 

 

+ 1

+eE-l—(u—v) t/I=Wi//
5(U'1'V) 2

Where Z is the number of protons of the single electron atom (Rydberg atom),
which is equal to one in the case of hydrogen. The energy term (W) must match the
electronic energy found in spherical coordinates. Equation 6.15 is separable into the

following independent equations:

2 2
h— i[ui)—l 31— —-]—Wu+lZue?’+1195}:2 U=0
2m au Bu 4 u 4 2 8

2 2
h_ i(,,.§.]-l '"_ _le+lzve2_leEv2 V=O (6.16)
2m 0v 3v 4 v 4 2 8

Z=Zu+Zv=1

138

The separation of equations in u and v requires the existence of an additional
invariant of motion, covered in the next section. The inability to make this further
separation in spherical coordinates is the reason that spherical harmonics remain mixed
after solving for the eigenfunctions of the Stark shifted energy levels. The Stark effect
perturbation matrices for both equations in u and v, on the other hand, are diagonal. This

will be demonstrated in section 6.2.3.

6.2.2 Runge-Lenz Vector

Figure 37 illustrates an additional constant of motion, the Runge-Lenz vector,
particular to dynamics where the central potential is proportional to UK A classical

development for the procedure follows:

mR2 =Z
mR2(I1X(3xR)= I:

xié=Zx—iié=ix,3 (6.17)

-mk(i€xg§xl§ 2
R

\_/

139

 

 

+Y

 

 

 

 

 

‘/2(pr)

 

   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

A0~focal length

+x

 

 

 

 

Figure 37 Classical Relationships for Runge-Lenz Vector.

140

 

The quantum mechanics form of the Runge-Lenz vector is [65]:

)

(“xp—px”)+me2R (6.18)

Where,

— ARcosq) = —Z 0(Rxfi) + mezR = -L2 +mezR

 

 

(6.19)
2 2
1 me me
—= 1+ cos¢]=—[l+acos¢]
R L2 [ me2 1.2
a z i
me2

Therefore, the Runge-Lenz invariant fixes the eccentricity of the orbital trajectory.
Section 6.3.1 and 6.3.3 use Equation 6.19 to develop operators that connect elliptical
paths of constant energy, but differing eccentricity.

The Runge-Lenz vector is not independent of the other two invariants, angular

momentum and energy. The operator relationships are given as follows:

(6.20)
A2 = 2W(L2 +1)+1

141

With units of electron mass and charge, and c=l. The Runge-Lenz vector does

not commute with angular momentum L, but upon rescaling:

 

[aha/1 = jgijkLk

(6.21)
[Lil/'1: jgijkLk
[Liaaj] = Jggjkak
Now, let:
1
112 = 5(1. i a) (6.22)
Equation 6.22 gives,
[JaJ’JflJ] = j§afl£UkJa,k (6-23)

Linear combinations of the invariants angular momentum and the scaled Runge-

Lenz vector yield two uncoupled, commuting angular momenta (J 1, 12). As a result,

142

1
JEZ=Z(L2iL0AiA0L+a2)

 

=l(L2+az)='l-[L2+ ‘ A2] (6.24)
4 4 -2W

=l(_l__1)=l(n2_1)

4 2W 4

Where W is discrete energy, with principle quantum number 11. Now, each
angular momentum satisfies the previously derived central potential relationships. That
is,

2 . . . . 1 ,
J1 llimi >= Ji(Ji +1)l Jim! >= 3(n-1)(n+1)|11mi>
Jz,i | jimi >= ”’1' l Jimi > (6-25)
" Ji ‘5 mi 5 11

6.2.3 Parabolic Energy Levels and Wave Functions

Removing the quadratic Stark effect, solutions to Equations 6.16 are identical.
Each equation is equivalent to the Schroedinger equation in spherical coordinates, with

m/2 replacing angular momentum 1. With the following substitutions:

U=e 2 uzfu(u),x=

 

(6.26)

1 m
——nv — 1

V=e2 v2 v, = v
fv()y W

 

143

The two differential equations can be written as functions of x and y as:

2
[xi-5+ (m+l—x)§;+[—_LZWZ,, —%(m+ 1)]qu = 0

(6.27)

a2 a 1 1
—— 1—4— ——Z —— 1 =0
[yay2+(m+ })ay+( ,—__2W v 2(m+ )va

Where Equations 6.27 are of exactly the same form as the reduced differential
equation for the radial component of the wave function in spherical coordinates, given in

Equation 5.38. As a result, the solutions are:

r(" nu +P) r0") ii ix-nu—m—lex
F(- nu) l"(m+19) p! H“

 

fl, =F(— nu,m+l,x)= i

p=0

F(— nv + p) l"(m) xp

 

 

 

__ _ -n -m—l x
fv-F(—nv,m+1,X)—p§0 r(_nv) r(m+p)?—__)x-9°° x V e
(6.28)
nu: l Zu—-1-(m+l)=nZu-l(m+l)
-2W 2 2
nv= I ——]—(m+1)=nZ ——1-(m+l)
J—2W 2 “ 2

144

Where the last equality for nu and 11V holds for the zero perturbation case. Now,

f.” converges if the polynomial is finite, that is:

— nu‘v = 0,—l,—2,—3,... :> nu», = O,1,2,3,...
(6.29)
:> n(Zu +Zv)=n = nu +nv+m+l

The energy levels in parabolic coordinates are discrete and degenerate, defined by
two electric quantum numbers, nu and n,, which replace the angular momentum quantum
number 1 found in spherical coordinates. Wave functions in parabolic coordinates are of
the same form as the radial component of the wave function in spherical coordinates,
both generated by the same differential equation form. The radial component in spherical
coordinates and the parabolic wave function are given in the following. The radial
component in spherical coordinates is taken directly from Equations 5.37 and 5.39; the

parabolic wave function is taken directly from Equations 6.26 and 6.28.

1

p
Rm, =crple 2 F(—(n—I—l),21+2,p)

Wnu,nv,m = W
(6.30)

m 1
—-—u

U=c;',u2e 2 F(—nu,m+l,u)

V =c;v2e 2 F(-nv,m+1,v)

145

Now, the orthogonality of the radial wave function,

(6.31)

2
Jdrr Rnl'Jl' an‘Jj = 6!]

lmplies the following inner product relationship for the parabolic wave functions:

[ 2U . U . = ..

id“ nL,m "5,... 5'!

< (6.32)
2 — .-

LJ‘dVV VnL,mVn‘-{,m — 6’]

 

Therefore, the perturbation matrix is diagonal in parabolic coordinates, and the
eigenfunctions, or wave functions, do not mix; the parabolic manifold is unchanged [66].

As a result, the allowed transitions and transition amplitudes are tractable.

6. 3 Stark Effect Perturbation

The linear Stark effect perturbation removes degeneracy from the parabolic wave
functions, and further mixes the fine structure of atomic hydrogen in response to the
application of a constant value electric field. The next two sections address the Stark
effect with respect to both parabolic wave functions (gross structure) and the fine
structure of atomic hydrogen, and develop the transition intensities for both gross and
fine structure that govern Ha and H13 bands of the atomic hydrogen spectrum. For
comparison, transition intensity bar charts are included along side experimental spectra

results in Chapter 7.

146

6.3.] Stark Effect: Parabolic Wave Functions

From the previous sections, specifically Equations 6.16, 6.28, 6.30, and 6.32, the

perturbation matrix elements can be calculated as follows:

 

 

 

 

_ 1 1 —u[ m I2
5211“,»: — ZeE —2W (nu :“m)!3 Iduuz-u me Lnu+m(u)
_ 2 2
—+—eE (621,, +6num+m +6nu+3m+2)
(2 —l E l d V[L’" 12 633
"Wm—4e —2W(n,, + n)m!3 I v v2 vm 8 WWW ( ' )
--lE (62+6 + 2+6 +3 +2)

14,,“ m+m(u)= (- 1) m(nu+m)!(n " +m]F(—nu,m+l,u)
m

Where the last equation gives the relationship between the Lagurre polynomials

and the hypergeometric function [67]. Summing Z terms from Equations 6.28 and 6.33,

1: J— 2W(nu +%(m+ 1))+ J- 2W(nv +%(m+ 1)]

+leE-
4

 

1 6n2+6n m+m2+6n +3m+2 (6.34)
W ll u ll

(6m? + 6nvm + m2 + 6nv + 3m + 2)

 

147

Which, for small 5W, gives [68],

1: \l—2Wn+-:—eEn2 -n(nu -n,.)

(6.35)
[W = —-—l—+3eE-n(nu -n,,)
2112 2

”14,12 2 O,1,2,3,...

 

[ n=nu+nv+m+l

The Stark energy levels in parabolic coordinates exactly match the eigenvalues
for spherical coordinates, found in Equation 6.9. Figures 38-39 illustrate the Stark shifted
energy levels for atomic hydrogen bands Ha and Hg, and transitions resulting from linear
and circular polarization [69].

Wave functions associated with the shifted energy levels are not mixed in
parabolic coordinates. Transition amplitudes can be found from direct integration of the
unperturbed eigenfunctions. Gordon [70] determines atomic hydrogen transition
amplitudes resulting from linear polarization, as represented in parabolic coordinates, in

the following:

148

 

 

=3 8E=4.5aer (typ)

 

 

 

 

 

n1-n2=+2

 

 

 

 

 

 

11 1:2, n2=0, m=0

 

 

 

n l -n2=+ l

 

 

 

 

Am:

 

 

 

 

 

n1=l,n2=0,m=1

 

 

n1-n2=0

 

 

 

 

 

 

|Am|=l

 

 

 

 

nl=l, n2=l, m=0
nl=0, n2=0, m=2

 

 

 

n1-n2=-1

 

 

 

 

 

n1=0, n2=l, m=1

 

 

n 1 -n2=-2

 

 

 

 

 

 

 

 

 

 

nl=0, n2=2, m=0

 

 

nl-n2=+1

 

 

 

 

 

 

 

 

 

 

n1=1, n2=0,m=0

 

 

 

n1-n2=0

 

 

 

 

 

 

 

V

 

 

 

 

 

nl=0, n2=0, m=1

 

 

nl-n2=-l

 

 

 

 

 

 

 

=2 8E=3.0aer (typ)

 

 

 

 

Figure 38 Ha Stark Effect Transitions: Parabolic Coordinates.

149

 

 

nl=0, n2=1, m=0

 

 

 

 

 

 

 

 

 

 

 

 

I N=4 II 8E=6.aneE(typ) I
11 1:3. n2=0, m=0
m |Am| 1 nl-n2=+2

 

nl=2, n2=0, m=1

n1—n2=+1

ill

 

 

I n1=2(l),n2=1(0), m=0(2) I

 

n l -n2=0

 

nl=l, n2=l, m=1

n l -n2=-1

 

I n1=l(0),n2=2(1),m=0(2) I

nl-n2=-2 I

.—

 

n l=0, n2=2, m=0

n l -n2=-3

 

E.
11
.o
E;
11
E"
B
5

nl-n2=l

 

 

 

 

 

 

 

 

nl=l, n2=0, m=0

n l -n2=0

 

 

 

 

 

 

 

 

5
II
o
:1
N
O
B
u

a s

n 1-n2=-l

 

 

 

 

 

 

 

V V

 

 

 

o
:1
N
3
<9:

:3

u a

I N=2 II 8E=3.aneE(typ)

 

Figure 39 H5 Stark Effect Transitions: Parabolic Coordinates.

150

 

 

I I
znf”"1”m = (_1)”il +"i; 2a0 (”u ‘1' m)! ("v + m)! (nu + m)! (nv + m)!
flu aann . 4m!2 nu! "V! "I"! "I"!

I "1+2 I 11+",
4nn In—n)
X —-—,2 I
(n—n) n+n

2 I2 ’
, , n +n 2nn , ,
xII(n1 — n2 )-——7-2- — (n1 — "2 )*—, 2 IWm("1"1)Wm("1"1)
(n + n) (n + n)

 

(6.36)

" ["1Vm(nln1 _1)Wm(n2n’2)_ n2V/m("ln1)'//m(n2n’2 "' 1)] I

Where,

 

1
, , —4nn' (—n-)(—n1) 1 -—4nn'
. . = F __ ,’_ ,’ +1, :l+_—L_—l__ _ +... 6.37
Wm(",",) n, n, m ("_nr)2I (m+1) I!I(n_nr)2I ( )

Figure 63 and Figure 69 in Chapter 8 give the HCl and Hg transition intensities.

6.3.2 Stark Effect: Fine Structure

The Stark effect on the fine structure of atomic hydrogen, which is negligible with
respect to Stark splitting in high external fields, is important under conditions of a
relatively low applied electric fields (<1000 V/cm) [60]. Energy level shifts are a
function of the atomic hydrogen non-degenerate fine structure (Equation 5.71), the
associated Clebsch-Gordan coefficients (Equation 5.53), and results from overlap
integrals similar to those calculated in section 6.1.1 (Equation 6.5). Calculating the

energy perturbation elements for states defined by total angular momentum (njm):

151

aE =< V’n,j—l/2.m 1 Z l V’n,j+l/2,m >

 

 

 

=< Rn.j—l/2 IR 1 Rn,j+1/2 > (6-38)
V 1IJ+m* ‘Ij— m+l
< Y;1—/,—2m 1/ 2|0059|——Yj+1/2.m—1/2>
‘I2j 1I2j+2
-<
j—m . —Jj+m+l
< . Yj—1/2.m+1/2|C039| . Yj+1/2,m+1/2>
I ‘IZJ J2j+2

 

The inner product of the radial waveforms can be found in closed form with the

use of the generating function for Laguerre polynomials, namely:

 

1) r 1’— "I'
(TE—[FTw- :24 (30’: (6.39)

Where the radial waveforms are given by,

R ,(R): (n—l-l)! (3)3/2e-RMI2R) [42,7185]
n, (n+1)!3(2n) n n "+ n

R I 1(R)= (n-I)! (ET/ze—R/"IZ—RI L21111I2_RI
"’_ (n+1—1)13(2n) n n "+ n

 

 

(6.40)

 

 

152

Substituting,

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

_- a -
21+ (n +1)! dn-I-l e—pl—l;
Ln [11,013 1 1 2! 2
(n—I—l)!da"" (l-a) +
_ aa=0
(6.41)
' ,2.“
+1—1 ' d"“’ ""3
L21-111(p )_ (n ) 8
02—0! d6“ (1-13)”
L _ [3:0
Which gives,
1 I "2 _12
< RnJ-l IR 1 RnJ >:_ .
4(n+l)!(n-1)1
_ a 1 ' )3 ‘
_p_ ‘P—
d 21+2e —p 61"”-1 e "a . dn-l 3 Hg
1 pp da""“ (1— a)”+2 dfl’” (l-my
p:
_ —a=0 — ~P=0
2, .-p[1+.—“-+.—”2l
—a —
[dpp +
_1 nz—l2 (inn-[:1 dn—jP
2 2 -I-1 n-l
:1 ,In _, (21+2)! d" d -a)(1-fl)3 (6.42)
4(n+1)!(n-I)! da”"“ 46"" l(l-amm a=fl=0

153

Now, using differentials of the geometric series,

1 (IN 1

__ : (N+i)!
N! aim/3)” (1 —afl)"’+'

N!i!

 

 

 

(afl)’

=i

i=0

Equation 6.42 is reduced to:

l \lnz—l2

Z(n+1)1(n -1)1'

 

< RnJ—l I R I RnJ >=

00

- 2

i=0

(21+ 2 +1)! d"”"

I!

 

[WI-00] -dn—[
0...

dan-I-I 0 dfln—I

3

= -§n\/n2 —I2

And,

 

(6.43)

[M - 2613] H (6.44)

 

3 .
< Rn.j—1/2 | R 1 R2221” >= 74'? -(1+1/2)2

The inner product of the angular waveforms

Equation 5.78, and repeated here:

 

(6.45)

relies on the identity given in

 

 

 

0086.),1'" =‘/(1+m+1)(/-m+l)
’ (21+1)(2I+ 3)

154

(I + m)(1 — m)
Y’H’m + [(21 + 1)(2/ — 1) Y’"”"

(6.46)

And the orthogonality of the waveforms. As a result,

 

 

l\[(j-T~m)(jim+l) (6.47)

*
<Y. |cost9lY- ‘1 >=—
j—l / 2,mi'l / 2 H / .Jnil / 2 - -
1 2 1(1 +1)

Using Equations 6.38, 6.45, and 6.47, the fine structure perturbation matrix
elements connecting each orbital pair (njm) can be expressed explicitly by the following

[70]:

aEnjm :< ij—l/ZJn I Z I Wn,j+l/2,m >

 

3 2 - 2
En‘fiz —(J+l/2) [(j+m)(j_m+])—(j—m)(j+m+l)]

 

 

 

 

BEN-m =eEa - (6.48)
J 0 2Jj(j+ I) ZJJ'U' +1)
2_ . 2
=eEa0--3-‘/n .(y+1/2) nm
4 J(J+1)

Thus, each fine structure energy level is split into 2j+1 equally spaced energy
levels, identified by magnetic quantum number m, where— j S m S + j. The uppermost
energy level in each fine structure element (j=n-1/2) remains degenerate. The Stark effect
energy level shifts in the atomic hydrogen Ha fine structure line are summarized in
Figure 40. The fine structure Stark effect is critically important to the spectral analysis of

data in Chapter 7.

155

-A ..............

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 
 

 

 

156

 

 

 

 

 

 

 

J """"""""""""""""""""""""""""""""""""""""" q .............
. . .
. .
. 2 . m
. / . /
. I... . V
. . z .
. . . m
. . m .
. . . w
. . .
. . . __
2 2 . 2 2 . . E
/ / c / / c p
3 II .I. 3
. . . . . .
__ m u __ __ u u
m . m m . 2 . «III.
. . V. . .I..
. . __ .
. . . __
. . m . m
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. .
. .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . . m
. . . k
. . .
. . . V
. . . 0
. . . __
. . E
. .
. .
2 2 2
y y H m a
__ __ __ __ __
J J J J J

 

 

 

 

 

 

 

 

 

 

 

 

 

=3
=2

N
N

 

 

 

 

 

 

Figure 40 Ha Stark Effect Fine Structure Splitting.

6. 4 Coordinate Transforms

Connecting spherical and parabolic coordinates requires the development in
section 6.2.2, which generates two uncoupled, commuting angular momenta that are
operators for the coulombic central force problem in any coordinate system in which R2
is separable, given stationary L2. Consequently, spherical and parabolic coordinates are
connected by ladder operators that follow in form the ladder operators developed for
angular momentum in section 5.2.1.1.

In the next sections, the ladder operators in parabolic coordinates are developed;
then, the ladder operators are shown as the connection between parabolic and spherical
coordinates, generating the associated Clebsch-Gordan coefficients [71]. Finally, semi-
parabolic coordinates are used to represent the angular momentum operators found in
parabolic coordinates as two coupled, two dimensional harmonic oscillators (Schwinger)
acting in three dimensional Lorentz space SO(2,1), for constant LZ [60]; that is,
oscillators composing angular momenta on the hyperboloid surfaces described in section

6.2.1.

6.4.1 Parabolic Ladder Operators

The angular momentum operators in parabolic coordinates, and their associated
properties, are developed in section 6.2.2, and given in Equation 6.25. With the

additional constraint of Equation 6.35,

157

. . . . 1 .
J12 111’"; >= Ji(./i +1)| Jimi >= 3(n-1)(n+1)| Jimi >

Jz,1' ljimi >= ”71 l 11% >
(6.49)

‘jismigji

n=nl+n2+m+1

Where u is replaced by l, and v is replaced by 2 in the last equation to match the
angular momenta notation. Now, the angular momentum operators in parabolic
coordinates exactly mirror the angular momentum operator developed in section 5.2.2.].
Consequently, ladder operators in parabolic coordinates can be composed in an identical
way, and the ladder operators themselves, with exception given to the additional

constraint of Equation 6.38, yield identical results. That is,
Jf-fz = J,’f2 i le-fz (6.50)

And,

 

1
Jli |n1,n2,ml,m2 >= ‘lzmz -1)-m1(m1i1) lan-l,n2,ml i1,m2 >

(6.51)

 

1
J2i |n1,n2,m],m2 >= \lzmz —1)—m2(m2 il) l n],n2 T- 1,m],m2 i1 >

158

Where,

n=nl+n2+m+l

(6.52)

m = m1 + ”12
But, ml and m; are arbitrary. Setting them equal, and using Equation 6.28,

1
n1=§(n—l)-m1

(6.53)

l
nl=§(n—1)—ml

Which results in the following ladder operators in parabolic coordinates [72],

J? | n,n],n2,m >= 1("l("‘"1) I n,n1 —1,n2,m+l >

 

J1- | n,n],n2,m >= \/("1 +l)[n—(nl +1)] | n,n1 +1,n2,m—1 >
(6.54)
J; I n,n1,n2,m >= ‘1n2(n—n2) |n,n],n2 -1,m+1>

 

J2— | n,n1,n2,m >= J(n2 + l)[n-(n2 +1)] I n,n],n2 +1,m-l >

By symmetry, the operators H, L2, and AZ commute in parabolic coordinates.

Consequently, the Stark perturbation potential can be expressed in terms of the rescaled

159

Runge-Lenz vector a such that it commutes with H and L2, and its eigenvalues are equal

to the Stark energy level shifts found in Equation 6.35.

3
V5mrk = eEz,r —-> -2-na
3

:> VStark [n,nl,n2,m >= -2—eEnaz |n,n1,n2,m >

3
= EeEnULZ —J2,z) | n,nl,n2,m > (6.55)

3
= EeEn(m1—m2)|n,n1,n2,m >

3
= EeEnOrz “’71) | n,n1,n2,m >

Mixing J); in 80(3) x 80(3) space generates a single three dimensional angular

momentum operator 3. in 80(4), defined as follows [60]:

,1 =(1x24y242) = (11,): - 124,11)» — j2,y’jl,z + 12,2)

[473/111 = fey/(’11:

22 1112131212 >= AM +1) 1 1121221222 > (6.56)
22 1112162222 >= m 1 13131222 >

wig/12st

160

Motivation for this transform is given in section 6.2.1. The rotation operator
ROFe‘0le connects ellipses with constant energy (W) but different eccentricity (or), where

‘3 . . .
0t =A/me", as g1ven in section 6.2.2.

6.4.2 Clebsch-Gordan Coefficients

Wave functions in spherical and parabolic coordinates each have exactly one state

in which the shared quantum number m is a maximum. That is,

| 11,] = n — 1,m = n —l >sphericalEl mm = O,n2 = 0,m = n -1 >lmmh0h-C (6.57)

As a result, these states are identical functions in both coordinate systems.

Operating on each side of Equation 6.57 with its prescribed ladder operator L’ yields:

L_ |n,l = n — 1,m = n—l >5th~mlz ‘/2(n— l) | 11,]: n —l,m = n— 2 >spherical
17 1’79"] : 0,)?2 : 0”" = "—1 >parabolic

= (J1- +J2-) I "’"l = 0372 : 0,m : "—1 >parabolic

= MI mm 21,112 = 0,m = n—2 >pamb0h-c

+Vn—l |n,n] 2 0,112 = 1,m = "—2 >parab0/ic

(6.58)

161

Setting the spherical and parabolic results equal generates the expected Clebsch-

Gordan coefficientsi 1/J2 , matching results from eigenvalue/eigenvector calculations,
and mechanizing the process.

The following tables give Clebsch-Gordan coefficients, transforming from
spherical to parabolic coordinates, for constant values of n-m== 2, 3, and 4, respectively
[73]-[74]. The rows are defined by parabolic quantum number m, the columns by the

spherical quantum number t=l-m.

61~61~o
61:61-—

 

 

0 1 2
m+2 l 1 m+1

112(2m + 3) 72 212(2),: + 3)
1 m+l 0 m+2
2m+3 2m+3
m+2 -l m+l

2(2m + 3) 72 2(2m + 3)

162

0 1 2 3

0 1 m+3 _1_ ’3m+9 1’3m+3 _l_ ’m+1
2 2m+3 2 2m+5 2 2m+3 2 2m+5

1 l ’m+1 -1 [m+3 —1 ’3m+9

- - 2m+5 7 2m+3 -2_ 2m+5
2 113m+3 __1 m+l :1,m+3 l ’3m-1-9
2 2m+3 2 2m+5 2 2m+3 2 2m+5

3 l,m+3 —_1 3m+9 _l_/3m+3 :_1 m+1 (6.59)
2 2m+3 2 2m+5 2 2m+3 2 2m+5

 

 

 

 

 

 

 

 

 

 

 

 

 

Such that:

n1+n2

_ 1,m
¢n1.n2.m - IZISI C 1.1m, 1.2m W221)" (6.60)
=m

Where (1) and ‘1’ are the parabolic and spherical wave functions, respectively.

6.4.3 Semi-Parabolic Coordinates

Returning to the original parabolic transform given in Equations 6.10 and 6.11,

the Schroedinger equation for a central coulombic potential can be written:

 

l 1 {i[fni]+—a-[§n-a—J}+—l—-i w
2 :0“? +712) Bf af 3?] 377 52,72 362
(6.61)

2
+ W+—+—— w=0
[ (31:02)]

163

And is easily separable, for constant LZIm, into the following:

2 2
-a—2+i—a——i”—+2W.§2 +42l F(,‘)=0
a; 535 :2
2 2
_B__ +l—a———m7+2Wrz2 +4Z2 Gm): 0 (6-62)
817277377 77
214-22:1

1n units of electron charge and mass, c=1. Each differential equation in Equation
6.62 represents a two dimensional oscillator in polar coordinates. The angular
momentum (m2), potential energy (W), and charge fractions (Z13) connect the two
equations. For a harmonic oscillator with unit frequency, operators for the first of the

two equations can be written:

=— D

19:55: 5
V562 (6-63)
22-2 .2 2-232 _2D,,,_m_:

E53 84‘ :2 :2 6

164

Where D is shorthand for the first derivative with respect to Q. As a result:

[pi1,V(H—V)] =0
[Pi l2V](H -V) = -V[p2H - V]

(6.64)
[PiLV] =1§D2Vl = €[D2V]+[§2V]D = 2V

:>[pil,H—V]=-2(H—V)

And,
1(H-V>,V1=;:,-D¢D:-§—:-D§Déé= (p+1)2 —<p—1)2 =4p (6.65)

Combining Equations 6.64 and 6.65 [75],

[p,H] = (—H +2V) a S;

[H,(—H + 2V)] = 4p 5 5,7 (6.66)

[(-H + 2V),p] = —H E ‘Sw

And, identical operators (T,) can be developed for the second equation in
Equation 6.62. Physically, 8 and T are each angular momentum in 80(2,1) space, fixed

to the common hyperboloid LZZm, precessing independently about the common axis n=n,

165

perpendicular to the plane of motion. Figure 41 presents this physical interpretation [60];
that is, the orbital mechanics in semi-parabolic coordinates.

Mixing operators 8 and T in 80(2,1) x 80(2,l) generates a new set of three
dimensional operators (W) in 80(3) that commute with L), and with 12, from Equation

6.56.

W 5(S§_T§’SU_TI7’SCU+T(0) (6.67)

Where (82, T2, W2, W2) all commute with LZ and 3.2. Consequently, the rotation
operator RaZelaS“, analogous to the rotation operator defined in section 6.4.1, connects
ellipses with constant energy but different eccentricity. Physically, mixing operators 8
and T rotates the normal vector 11 to n’ in Figure 41 [60]; the orbital plane intersecting the
hyperboloid. As a result, the operator W represents all elliptical orbital paths in semi-
parabolic coordinates, whereas 8 and T each represented strictly circular paths about the
'1] axis.

Stark and Zeeman splitting can be combined by way of the 8 and T vectors in
semi-parabolic coordinates, and this is necessary for analysis of the fine structure of
atomic hydrogen for large applied magnetic fields. For this experiment, Zeeman splitting
in the fine structure is negligible [60], due to the relatively low level of the magnetic

fields, as determined from the rotational spectrum of molecular hydrogen.

166

 

+
N

 

 

 

 

 

 

 

 

-q----.’

 

l

 

 

    

‘
O
o
o
n
'-
—
"

 

 

 

 

+u

 

 

 

 

Figure 41 Semi-Parabolic Coordinate Representation of Stark, Zeeman Effect [60].

167

 

 

 

 

Chapter 7 Results

Results concentrate on matching the spectroscopy readings with theory presented
in the previous three chapters. The first section gives spectrometer set-up background,
critical in assembling high-resolution scans with as low signal-to-noise ratio as possible.
The next section ties the spectrometer readings from Argon to the electron density for
Argon, and determines how accurately the measurements for Argon fit the global model
presented in Chapter 4.

The final sections examine both diatomic and atomic hydrogen spectroscopy
results. The spectroscopy readings are interpreted according to theory presented in
Chapter 5 and Chapter 6; calculations are made as to the temperature, electron density,
and electric and magnetic field strengths of the hydrogen plasma. From these
calculations, conclusions are drawn in the final chapter, Chapter 8, as to the nature of the

hydrogen plasma contraction, which begins to occur at pressures as low as 5 Torr.

7. l Spectrometer Set-Up

Initially, thirteen 1 mm diameter fiber optic channels were focused on the plasma
center and available to project the plasma emission through the spectrometer slit.
Focusing multiple channels to the spectrometer slit increased the signal to the
spectrometer, but drastically reduced the resolution.

Next, one fiber optic channel was set less than 0.25 mm from the slit. Steadily
increasing the distance from the slit had the predictable result of reducing signal-to—noise

and increasing resolution. Maximum spectrometer resolution of 0.3 A (FWHM) was

168

reached at a distance of 8 mm, nearly matching the manufacturers specification for a light
cone of ratio 9:1.

Accelerating voltage was adjusted to increase signal-to-noise. Figure 42 shows
the spectrometer step response to accelerating voltage supplied to the photomultiplier
tube (PMT); each step represents a 100 Volt increase in accelerating voltage, from 0 V to
900 V. Evident is a definite nonlinear response, beginning at approximately 500 V. The
noise response of the spectrometer is consistent with an older PMT [76].

The response is extremely sensitive to both accelerating voltage and the time
derivative (dI/dt) of the excitation signal. Reducing the accelerating voltage to the point
that maximized the PMT nonlinear response magnified dI/dt, and allowed the PMT to
effectively operate as a detector.

At low accelerating voltages, signal-to-noise was decreased by the accuracy of the
pico-ammeter. The pico-ammeter resolution was specified at 10''4 A, but the accuracy —
the random signal error- was an order of magnitude worse. As a result, several
experiments needed to be run for each test case to determine the best accelerating voltage
setting to maximize signal-to-noise.

Even afier determining the accelerating voltage, the signal still suffered from
undershoot. Consistently, sharp drops (0.2 pA) in signal occurred after consecutive
readings with high slope (dl/dt). To determine the fidelity of these responses, it was
necessary to slow the spectrometer to the slowest accurate scan rate. Scan rates lower
than 1 A/minute produced data with less resolution and more noise than slightly higher
scan rates; that is, the motor control for the spectrometer mirror was not as accurate for

the lowest scan rates.

169

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oo+wood

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(v) wanna ma

07mom...

 

OTMOON

o..-mom.N

 

or-mco.m

170

All of the aforementioned adjustments allowed for resolution fine enough to
correlate fine structure peaks present in atomic hydrogen spectra, providing a signature
for each spectrum. Peaks could be identified that were separated by as little as 0.08 cm",
approximately 0.04 A. at the HO, line (6562.85 A.) —7.5 times better than the FWHM
resolution of the spectrometer. Evidence for resolution of this order is demonstrated in
the atomic hydrogen Ha peaks, presented in Section 7.4.1.1.

The line shape of the spectral responses appeared to have first-order decay at the
trailing edge; although not universal, often enough for concern. Reducing the accelerating
voltage did effect a change, but did not eliminate the decay from the hydrogen rotational
band. To determine whether the decay was real or measurement error, all measurements
were run forward and backward. In each case, with the reduced accelerating voltages, the
forward and backward curves matched exactly; the data was real, and the fine structure

peaks were further confirmed.

7. 2 Argon Results

The electron density is measured experimentally and compared to the global
model predictions for Argon. Gas temperatures for the Argon plasmas in this study are
taken from Rogers [77]. The theoretical and experimental data can be expected to diverge
at higher pressures (>100 torr), as convective flows begin to dominate diffusion as the
major transport mechanism [77].

The first section presents the results from the global model, based on
experimental inputs. The next section summarizes the set of experiments run to

determine the electron density for the Argon plasma and compares the experimental

171

results for electron density to the global model. The final section suggests an alternative

model for electron density based on the line shape of the Argon spectrum at 4300.1 A.

7.2.1 Global Model Results

The global model predicts electron density and electron temperature as a function
of input power, pressure, gas concentrations, and plasma geometry. In this case, the
global model prediction for electron density will be compared with experimental values.
The tables in Tables 1-3 and the plots given in Figures 43-44 summarize the global model

results.

7.2.2 Argon Spectroscopy Measurements

Hydrogen was added to the argon plasma at a ratio of 1:25. Stark broadening in
the atomic Hydrogen beta (Hg) spectrum was used to determine the electron density for
the argon plasma, a result first calculated by Griem [22],[78], and parameterized by

Nikolic, et a] [79]. The following summarizes the electron density estimate:

3

Ne = [3.99x108 0 (MN 2 cm—3
a

 

2 2 2

A"l'Stark = \/A’1 — A2175 - Ajllrzstrument

a = 0.0762 (7.1)
Alps = 0.07724

AAlrlstrument = 0'30

172

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Power=40W,(R,L)=(5mm,5mm)

 

 

Electron Temperature

Pressure (log torr)

 

 

 

 

Power=40W,(R,L)=(5mm,1 0mm)

(eV)

 

Electron Temperature

.4 -2 0 2 4

Pressure (log torr)

 

 

 

 

Power=40W,(R,L)=(5mm,15mm)

 

Electron Temperature
(0V)

-4 -2 0 2 4

Pressure (log torr)

 

 

 

Figure 43 Global Model Predictions for Argon Plasma Electron Temperature.

176

 

Power=40W,(R,L)=(5mm,5mm)

 

 

Electron Temperature
(9V)

Pressure (log torr)

 

 

 

 

Power=40W,(R,L)=(5mm,10mm)

(eV)

 

Electron Temperature

-4 -2 0 2 4
Pressure (log torr)

 

 

 

 

Power-40W,(R,L)=(5mm,1 5mm)

 

Electron Temperature
(9V)

-4 -2 O 2 4
Pressure (log torr)

 

 

 

Figure 44 Global Model Predictions for Argon Plasma Electron Density.

177

Equation 7.1 is specified for the Hydrogen beta (Hg) line in plasmas with
approximate electron temperatures of 5000K, and electron densities on the order of 10”-

10'5 cm’3; ALA/1,3,A/l,w,-m,mu, are full-width half-maximum (FWHM) line widths for

the spectrum, fine structure, and spectrometer resolution, respectively. The H5 spectrum
is shown in Figure 45 for a pressure of 100 Torr. Figure 46 plots both the Stark
broadened electron density from experiment and the global model prediction for pressure

ranging from 100 mtorr to 100 Torr.

7.2.3 Argon 4300.1 A Line Shape

Argon spectroscopy concentrated on the Ar*->Ar transition at 4300.1 A. In all
readings, accelerating voltage for the photomultiplier tube was set to 300 V. Figure 47
demonstrates the Argon line shape at 4300.1 A, with a pressure of 100 Torr. The
wavelength shift from center of the lower (L1) and upper (L2) sidelobes are plotted
against pressure in Figures 48-49, as suggested by Milosavljevic, et al [80]. The L1
curve is similar to the theoretical electron density plotted in Figure 46; the L2 curve is

similar to the electron density plotted in Figure 46 for experimental data.

7. 3 Hydrogen Results: Diatomic Hydrogen

Hydrogen results are divided into two categories: diatomic and atomic hydrogen.
First, spectrographic data is used to find the rotational temperature of molecular
hydrogen; calculations are made for data both within a single vibration band and within a
single rotation band. Then, Zeeman splitting applied to the fine structure of the rotational

spectrum is used to estimate the internal magnetic field of the hydrogen plasma.

178

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183

7.3.1 Diatomic Hydrogen: Rotational Spectrum

Diatomic hydrogen spectroscopy concentrates on the rotational transitions in the
first vibration band of the electronic transition Zu-Zg; that is, the excited ground state -or
ls state- of H2 to the ground state of H3. Transitions in this region emit photons in the
visible spectrum, from 4540-4600 A. In this region, the fine structure is simplified, as
there is no orbital angular momentum (1:0) intrinsic to the molecule.

First, rotational temperature calculations are made based on the intensity of peaks
across the band of rotational transitions (interband transitions). Next, temperature is
calculated based on populations in fine structure peaks within a single rotation transition
(intraband transitions). Energy differences for fine structure transitions are small, and
provide linear temperature curves. Finally, the curves will be compared across the

pressure regime.

7.3.1.1 Diatomic Hydrogen Temperature: Interband Transitions

Allowed diatomic hydrogen transitions are prescribed in Figure 27, section 5.7.
Relative transition line strengths are a fimction of the rotational level of the upper
vibration band [53]. Relative populations in the upper band (J ’) are a function of the
rotational energy term and rotational inertia of H2. As a result, the relative line intensity

for an approximate Boltzmann distribution is given in the following formula:

 

B,,'J’(J’+I)hc]

kTr

1 oc (J’ + l)e-[ (7.2)

184

The H2 rotational spectrum is very difficult to analyze, as can be seen in Figure
50; it is difficult to identify transition peaks. Table 4 gives a hypothesis set of transition
peaks. Transitions that lose one unit of angular momentum (R branch) emit photons of
shorter wavelength (higher energy) than the band center; transitions that gain one unit of
angular momentum (P branch) emit photons of longer wavelength (lower energy) than
the band center.

The lowest energy transition (highest emitted wavelength) in H2 is not the ground
transition. This is due to the rotational inertia increase in H2 from 20.0 cm'l to 28.4 cm'1
in transitions from 2,, to 23 [53]; that is, the excited state electrons, concentrated at the
center of the molecule, effectively pull in the protons, reducing the rotational inertia.

In H2, a nonlinear centrifugal stretching term reduces the rotational energy at low
frequencies; that is, the rotational inertia of the protons increases proportional to J(J+l),
the rotational energy of the electrons decreases proportional to .IZ(J+1)2 [81].

The Fortrat diagram, which plots quantum number as a function of transition
energy, is one available test to confirm transition peak identification. The Fortrat

diagram is defined by the set of equations given in the following [82]:

185

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The Fortrat diagram was generated using coefficients from an OriginR curve fit.

Coefficients matched rotational inertia values to within 10%; that is:

(B; + 3;) = 54.24
(B; — 3;) = —12.09 (7.4)
a4(aB;2 — 3332): —0.046

The match should confirm the peak transition identification.
Figure 52 plots from Table 4 the log of the scaled peak intensity vs. the energy of
the upper transition band, or rotational energy number, following Equation 7.2. Assuming

a Boltzmann energy distribution, the slope of the curve is given as:

188

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=> 7, = —28.8 . 1 = 508K = 235C

s/opeu'm-l )

 

Repeating rotational temperature calculations generates the pressure vs. rotational

temperature plot of interband rotation transitions given in Figure 53.

7.3.1.2 Diatomic Hydrogen Temperature: Intraband Transitions

As can be seen in Figure 52, the plot of the log of the scaled peak intensity vs. the
energy of the upper transition band is not entirely linear. This indicates that either the
energy distribution is not a Boltzmann distribution, or the rotation transition peaks are too
difficult to identify accurately in H2. A solution to both of these difficulties is to
calculate temperature within one rotation band.

The theory from section 5.6 provides background. Each rotation transition peak
in Figure 50 is degenerate; angular momentum is constant, but Jz is not. Application of
an electric or magnetic field destroys the spherical symmetry, and removes the
degeneracies.

The electric field effect (Stark) is second order —and negligible- for spherically
symmetric wavefunctions such as Eu and 28; the magnetic field effect (Zeeman) is first
order, and causes splitting in Eu and 28 energy levels, and corresponding peak splitting in

transitions.

191

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192

Figures 54-55 represent a pair of high-resolution images of the peak identified as
m=2 (J: 2->2) in the R-branch of the diatomic hydrogen rotation spectrum, taken directly
from Figure 50.

Fine structure due to Zeeman splitting is identified. There are fifteen main peaks,
corresponding to fifteen transitions within the band. Figure 27 in section 5.7 indicates
this peak must represent m=3 (J: 3->2). The bands in Figure 50 were incorrectly
assigned; that is, the side lobe at 4581.3 A in Figure 50 is the new m=1 peak, and each
assigned m must be incremented by one (parenthetical values). The resulting Fortrat
diagram is shifted, but unperturbed.

The magnetic field strength is not strong enough to decouple the orbital angular
momentum and spin. Consequently, Clebsch-Gordan coefficients from section 5.4
determine energy levels for both upper and lower bands. Calculations are simplified by
concentrating on the orbital angular momentum shifts, the first term in equation 5.73.
This term alone is the Larmor precession, or normal Zeeman effect.

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195

Table 5 fits energy levels from Equation 7.5 onto Figures 54-55, for a given
magnetic field strength B. In Table 5, the mean-squared error is minimized for B=35
mT; experimental vs. theoretical magnetic field magnitude is plotted in Figure 56. Figure
57 plots magnetic field strength for pressures from 0.5-50 Torr.

Transition amplitudes come directly from application of the spherical tensor
Tq(k). Briefly, the spherical tensor is related to the angular momentum eigenvectors in

the following way:

k k
[12.71; ’I=th.§ ’
(7.6)

 

[1.73“] = Mk .—. q)(k :q +1)Tq(k)

And, the spherical tensor is transformed from representation or to or’ by a multi-

dimensional Clebsch-Gordan rotation, that is:

(a’;j’,M'| Tq(k) |a;j,m> =
(7.7)
«221’ II T)“ ll an”)
JZj +1

 

<j.k;m,q I j,k;j’.m’><a’;j’ l 72;") I an)
Equation 7.7 is the Wigner-Eckart theorem [43]. For Tq(k), an eigenfunction:

jam/f" (12¢)an (mu/2'9 (22¢)
(7.8)

 

 

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Equations 7.10 and 7.11 by dipoles rotating about the z-axis.

Finally, the following three special cases account for dipole transitions:

 

 

_§_\/(L+M+l)(L—M+l)

dQYM“ , Y0 ’ YM ’ :‘/
I L (19¢)1(19¢)L+I(’9¢) 4,, (2L+1)(2L+3)

 

(L+M+l)(L—M+l)
(2L+ l)(2L+3)

 

=> JdQYLM (29,¢>2YL“1.(29,¢)=\/

 

 

YM“ y. M+l :J: (L+M+1)(L+M+2)
Ida L (09¢) l(§9¢)YL+l (09¢) SIZ'J (2L+1)(2L+3)

 

 

M* .,YM :_\/I (L+M+1)(L+M+2)
:> deYL (29,¢)(x+1)) “101W 2 (2L+l)(2L+3)

 

 

(mix ’ Y4 YM+I , :JE (L-M)(L-M-l)
I L (19¢)1 (19415)“: (29¢) ml (2L—1)(2L+1)

 

 

M“ _. M+l _1 (L-M)(L-M-1)
: [my]. (19,¢)(X Jy)YL—l (12¢) _J;\/ (2L—1)(2L+1)

(7.9)

(7.10)

(7.11)

Equation 7.9 accounts for transitions effected by dipoles in the z direction;

Table 5. Transition rates are proportional to the z dipole strength.

Transition rates from

Equation 7.9 are included with peak amplitudes found in Figures 54-55, and charted in

Deconvolved transition amplitudes divided by transition probabilities (rates)

distribution [54].

201

given in Equation 7.9 are plotted against ml energy level differences in Figure 58; the

slope, calculated in the following, is proportional to rotational temperature, regardless of

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parabolic deflection of Figure 58. The accuracy of Equation 7.12 is independent of
rotational level distribution. Figure 59 plots intraband rotational temperature vs.

pressure.

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7.3.2 Diatomic Hydrogen: Zeeman Shift

Peaks for the Zeeman shifted rotation bands were found by deconvolution of a
collection of forward and reverse spectral scans. Scans in both directions were necessary
to verify peak location, and to avoid increased photomultiplier tube signal-to-noise level
evident on the falling edge of the band, as discussed in section 7.] and demonstrated in
Figure 42.

To verify Zeeman peaks, both forward and reverse scans were band-passed
filtered to remove modulation at the sampling frequency. The band-pass filter results are
given at the baseline of Figures 54-55. The reverse scans are flipped to align with the
forward scans. The forward scan amplitudes in Figure 54 are artificially reduced. All
spectral readings were taken with the accelerating voltage set to 550 V. The plasma
conditions were identical for each reading; the pressure was 0.5 Torr, the gap size 5 mm,
the power set to 60 W, and the hydrogen flow rate was 100 sccm. The wavelength of the
spectrum served as the time element for the band-pass filter; that is, l A = 1 second.

Figures 54-55 attempt to demonstrate the consistency of the spectral data; peaks
lined up vertically —as demonstrated by the band-pass filter results- confirm peak
location. Peaks are identified by the OriginR cross-correlation software package.

Band-pass filter results suffer from time shifts, as is expected from broadband
finite impulse response (FIR) filters. This is seen in Figure 54; the cut-off frequency on
the forward scan is set to 15 Hz. Figure 54 identifies peaks with amplitudes greater than
0.2 E-ll A. Time shifis from the FIR filter are evident. Sofiware peak identification is
unreliable for time shifts of this magnitude; as an example, peaks 5-6 should register for

both forward and reverse scans, and do not.

205

Figure 55 examines the same two spectral scans with relaxed band-pass filters
(fcut-off=25 Hz) on both forward and reverse scans. Additionally, peak triggers are
raised to 0.5 E-ll A. on both forward and reverse scans to counteract the raised noise
floor associated with the wider filters. Peaks identified in Figure 55 are cataloged in
Table 5 and compared to peaks expected from normal Zeeman splitting for the J: 3->2
band of the Zu->2g transition in molecular hydrogen.

A second filtering method combined both inputs in a moving average windowed
cross-correlation filter. Results from this filter are shown graphically in Figure 60, with
results from Figure 55 given along the baseline for comparison.

Results for both filtering methods (Figure 55, Figure 60) identify peaks consistent
with normal Zeeman splitting for hydrogen in the given rotational band, further
strengthening the evidence for Zeeman energy shifts in the diatomic hydrogen rotational
spectrum.

Figures 61-62 are plots of the H2 rotational spectrum for pressures of 5.0 Torr and
50 Torr. The five transition peaks identified represent transitions effected by the
circumferential electric field. Magnetic field strengths are calculated for the given
pressures and included in Table 5 and plotted in Figure 57 in section 7.3.1.2. The

magnetic field strengths at 5.0 Torr and 50 Torr are 28 mT and 25 mT, respectively.

7. 4 Hydrogen Results: Atomic Hydrogen

J. Balmer described atomic Hydrogen lines, in the visible spectrum, in 1885. N.
Bohr first explained the Balmer Series in 1913. The energy differences (13,.) found in the
Schrodinger equation with central potential energy match the Balmer series. Three of

these lines, Ha, H3, and H7, are investigated in this study.
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209

Gross and fine structure spectral results from atomic hydrogen lines Ha, Hg, and
HY are studied in the next section to determine the resident electric field in the hydrogen
plasma. In the final section, the Stark broadened H5 line is used to determine the electron

density of the plasma.

7.4.1 Atomic Hydrogen: Stark Shift

Section 6.3.1 develops the concepts that govern Stark effect energy shifts and
transitions for the atomic hydrogen gross structure. Included are direct calculations for
the resulting H; and Hg energy spectrum shifts (Figures 38-39), and Equations 6.36-6.37
defining relative transition amplitudes. Gross structure splitting is dominant with applied
fields on the order of 5000 V/cm [60] and larger.

Section 6.3.2 develops the concepts that govern the Stark effect for the fine
structure of atomic hydrogen. Fine structure analysis is in general limited to fields on the
order of 1000 V/cm [60]; larger fields mix the fine structure wave functions,
complicating the analysis.

Analysis of intermediate fields requires coupling both gross and fine structure
effects. For exact solutions, all perturbation potentials must be reformulated in the
Schroedinger equation. Accurate approximate solutions have been found up to the field
limits, where gross and fine structures mix.

Electric fields in this investigation are anticipated to be on the order of 2000-5000
V/cm (see section 8.1.2) —a consequence of the small electrode gap that was necessary to
confine miniature plasmas. Fields at this level effect a uniform distortion in the parabolic
orbital wave functions [60]; the outer shell of the wave function retains parabolic

symmetry, the wave function core approximates spherical symmetry. The anticipated

210

spectral response is an interleaving of the gross structure splitting and fine structure
splitting. It is anticipated that gross structure spectral shifts and relative amplitudes
should follow closely those outlined in section 6.3.1.

Further, it is anticipated that the fine structure splitting should be convolved with
the gross structure. The fine structure amplitudes will deviate from the nominal values
found in section 5.8 (Figure 30), a result of the relatively high electron density (N- ~ 10”-
10IS cm'3) of the plasma [60]. The fine structure signature should more closely follow
that of experiments at approximately the same electron densities. Figures 31-32 in
section 5.8 demonstrate fine structure spectrums from experiments conducted in low-
pressure gas discharges under similar conditions.

Even at much higher applied electric fields, the parabolic wave functions do not
interact with each other. Instead, each series of components {(n.,n2,m), (n1+1,n2,m)...}

are superimposed, with spacing proportional to E3”4 [60].

7.4.1.1 Stark Shift: Ha

Sections 6.3.1-6.3.2 develop the gross structure and fine structure for Stark effect
atomic hydrogen transitions from principle quantum numbers n=3 to n=2, the Ha energy
spectrum. Figures 38 and 40 give a graphical depiction of the gross and fine structure
Stark effect for the H(x spectrum.

Figure 63 [83] gives the Ha transition amplitudes for the parabolic eigenvectors,
calculated using Equations 6.36-6.37 in section 6.3.]. Figure 65 and Figures 67-68 are
the Ha spectrums from discharge experiments where the pressure was controlled at 50

Torr, 5.0 Torr, and 0.5 Torr, respectively. In each case, discharge conditions were set to

211

 

 

1:: linear polarization

 

 

 

 

 

 

 

 

 

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Figure 63 Ha Parabolic Transition Intensities.

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maximize power density; the electric probe gap fixed at 5 mm, the input power at the
maximum power level of the microwave supply, 60 W. Hydrogen flow meters were set
uniformly to 100 sccm. Critical peaks are identified on each of the three figures.

Figure 64 shows graphically the theoretical mixing of gross and fine structures at
50 Torr. The theoretical Ha spectrum given in Figure 64 and the experimental spectrum
given in Figure 65 are consistent with an electric field estimate of 3858 V/cm. Critical
peaks in Figures 64 and 65 are labeled.

The gross spectrum peaks are separated uniformly by 6 units (0.1 A) in Figure 64,
approximately equal to the expected gross structure splitting (~6.40 units=0.1067 A) for
the given electric field. The sharp nature of the peaks is unexpected; that is, Doppler
broadening is not evident in the fine structure transitions. This point is addressed in
Chapter 8 (section 8.1.3).

Figures 67-68, representing pressures of 5 Torr and 0.5 Torr, do not give as much
detail. However, peak identification is still possible. Figure 66 gives the theoretical Ha
spectrum response for applied electric fields that vary continuously through 4000 V/cm.
Figure 66 connects the fine structure with no applied field to the mixing at ~4000 V/cm
(3858 V/cm at 50 Torr). The peaks identified in Figure 66 map onto Figures 67-68; the

associated electric fields are 2083 V/cm and 1875 V/cm for 5 and 0.5 Torr.

7.4.1.2 Stark Shift: Hg

Sections 5.8 and 6.3.1 develop the fine structure and gross structure for atomic
hydrogen transitions from principle quantum numbers n=4 to n=2, the Hg energy

spectrum. Figure 29 and Figure 39 give a graphical depiction of the fine and gross Hg

218

spectum. Note that the k=0 transition is allowed for Hg (Figure 39), but the composite
parabolic functions (1,1,1) and (0,0,1) are orthogonal.

Figure 69 [83] gives the Hg transition amplitudes for the parabolic eigenvectors,
calculated using Equations 6.36-6.37 in section 6.3.1. Figure 71 and Figures 73-74 are
the Hg spectrums from discharge experiments where the pressure was controlled at 50
Torr, 5.0 Torr, and 0.5 Torr, respectively, and all other operating conditions matched
those given in the previous section for Ha experiments. Critical peaks are identified on
each of the three figures.

The Hg peaks do not present the striking character of the Ha peaks; the peaks are
neither sharp nor is the fine structure as easily unraveled. Gross structure peaks are
Doppler broadened. However, it is still very possible to repeat the theoretical spectrum
analysis of the previous chapter with fewer points.

Figure 70 shows graphically the theoretical mixing of gross and fine structures at
50 Torr for Hg. Figure 72 connects the fine structure with no applied field to the response
at «4000 V/cm (3890 V/cm at 50 Torr). The peaks identified in Figure 70 and Figure 72
map onto Figures 71 and Figures 73-74, respectively. The associated electric fields are
given in Figure 72. As can be seen, the electric field strength of the hydrogen plasma
taken from Hu and Hg spectral data, under identical operating conditions, are nearly
idenficaL

In the theoretical Hg spectrum given in Figure 72, the gross spectrum begins to

emerge and dominate at fields >4000 V/cm, and it becomes possible to estimate the electric

field from gross structure splitting alone. Parabolic wave function energy levels

219

 

 

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Figure 69 Hg Parabolic Transition Intensities.

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are separated uniformly by 3.3 units, or 0.236 cm'l (0.056 A). The corresponding electric

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for the H.- band.

7.4.1.3 Stark Shift: HY

Figure 75 [83] gives the parabolic transition intensities for both linear and circular
electric dipoles. Figure 76 gives the H7 spectral data for operating conditions identical to
those for H0l and Hg, at 50 Torr. Although noisier, it is still relatively easy to see the
Doppler broadened gross structure peaks, separated by 2.7 units, or 0.237 cm’] (0.045 A).

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Figure 75 H‘y Parabolic Transition Intensities.

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Therefore, the electric field strength of the hydrogen plasma at 50 Torr, taken
independently from Ha, Hg, and H7 spectral data, agrees to within less than 3%. The
electric field is plotted vs. pressure in Figure 77 on the following page. The resulting
electric field strength is considerably greater in this set of experiments than that found in

previous spectroscopic studies using higher principle quantum numbers (n=l4-20) [84].

7.4.2 Electron Density

Stark broadening measurements of Hg lines were carried out in section 7.2.2 to
ascertain electron densities for Argon. Stark broadening measurements for hydrogen
follow this procedure; a sample Hg line is given at 100 Torr, in Figure 78. Figure 79
plots hydrogen electron density estimates of the hydrogen plasma for pressures from 0.1-
100 Torr. For each experiment, the probe separation was set to 5 mm, the flow rate 100

sccm, the power 60 W.

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232

Chapter 8 Conclusion

The purpose of this investigation was to design and build a miniature microwave
plasma system and associated diagnostic optics to collect spectroscopic information that
would possibly reveal reasons for the plasma behavior as pressures are increased.

To this end, a new plasma system was created as a flexible test bed for
experiments. A new optical system was designed and built to bring the collection lens
system to within 5 mm of the plasma center. A sophisticated measurement technique was
discovered to zero in on some of the fine structure associated with hydrogen and argon
lines; taking advantage of nonlinearities in the response of the photomultiplier tube
(PMT), the monochromagraph resolution was pushed beyond performance specifications.

The following sections target specific areas where the experimental results
appeared contradictory or at odds with what might be expected from the experimental

parameters.

8. 1 Experimental Results

Results from molecular and atomic hydrogen studies must be consistent with each
other, and with each element of the theory that predicts these results. The next few
sections examine the consistency of the results with respect to the electric field

polarization, electric field magnitude, and atomic hydrogen spectral resolution.

8.1.1 Results: Electric Field Polarization

Assumptions made about the polarization of the electric field resident in the
plasma were corroborated by experiments with atomic hydrogen: the absence of a center

peak in both Ho- and H7 spectrums eliminates the possibility of a rotating electric field.

233

Further, the solid match of the atomic hydrogen experimental data to the
theoretical spectrum verifies the presence of a linearly polarized electric field, a field that
changes strength with changing pressure.

The experimental data for molecular hydrogen appears to contradict the absence
of a rotating electric field. Zeeman splitting generates fifteen peaks; five peaks result
from transitions involving linearly polarized fields, and five each from right and left hand
circularly polarized fields.

This apparent contradiction is resolved graphically in Figure 80. The linearly
polarized electric field is fixed along the z-axis, connecting the electric probes of a nearly
capacitive discharge. In the reference frame of molecular hydrogen spinning about the
circumferential axis, the electric field appears to be rotating; molecular orbitals
experience a circularly polarized field. In the reference frame of molecular hydrogen
spinning about the z-axis, the electric field appears stationary, and linearly polarized.
The combination of hydrogen rotations —the molecular hydrogen domains aligned
helically around the cylindrical discharge- explain the transitions associated with both

linearly polarized and circularly polarized fields.

8.1.2 Results: Electric Field Magnitude

The electric field, calculated from the Stark shift spectral data, approximates an
independent electric field calculation that follows from resonance principles. The quality
factor can be expressed in two ways; the ratio of the resonant frequency to the full-width

half-maximum (FWHM) frequency band, and the ratio of the stored energy to the applied

234

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 80 Electric Field Polarization for Molecular Hydrogen.

235

power. The former can be used to find the quality factor while the experiment is running;
the latter allows for an approximation of the resident electric field.

The quality factor for the resonant cavity was found for experimental conditions
identical to those of the experimental set that was used to find the Stark shifi in the
hydrogen plasma, with pressure set to 50 Torr. At low power, the input power dropped to

half of its original value when the cavity length was increased by approximately 0.5 cm.

 

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AFWHM
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FWHM

236

The transverse electric field in the resonant cavity (E,) can be approximated by,

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P P

 

Q= 5V0]

2
_ 2%.).95053 3 R0

(8.2)
2 2 2
= (271') R0805, iclni
P 2

 

= 1.063x10‘3 Erz(v/cm)

Where R/Ro is the ratio of the outer to inner coaxial diameters. For gaps (Ad)
much less than one-quarter wavelength, the resonant (transverse) field E, is related to the

field between the two probes Ep by,

_AV_ 1 WEde R0_ E_C__R01n
P: Ad Ad IM “R0
2 er _lresl
(8.3)
= 0.266Er(v/cm)

Now, the ratio of the electric field at the plasma sheath (Es) to the electric field
between the probes is approximately equal to the ratio of their surface areas. Combining

with Equations 8.1-8.3:

237

 

 

2 2 2
: Asheath : flsheath : [Bx/teat}: ] : [1.0mm ] 2 0.01

 

7mm: - Aprohe — fig 00 10mm
ES 2 —]—Ep(v/cm) = 0266 1 Q 3 (8.4)
7 area 7 area 1.063xl 0—

E 2725(v/cm)

The electric field at the sheath is of the same order of magnitude as that found in
sections 7.4.1-7.4.3 (3858 V/cm) using the Stark effect shift. Figure 81 illustrates the

electric field structure in the resonant cavity.

8.1.3 Results: Atomic Hydrogen Spectral Resolution

The spectral resolution is sharper and the magnitude greater in the set of peaks
corresponding to H0- fine structure transitions (n,j)=(3,3/2)>(n,j)=(2,1/2), and to a lesser
extent, the other peaks as well. Normally, these peaks are not this sharp, the resolution
reduced by instrument broadening and Doppler broadening.

The effects of Instrument broadening were reduced by adjusting the accelerating
voltage to push the photo-multiplier tube (PMT) to operate in its nonlinear range (section
7.1, Figure 42). As a result, the slope of each dI/dt response from the PMT that was
above the threshold was magnified. The PMT effectively took on characteristics of a
detector.

The narrowing of Doppler broadened spectra is more involved. Figure 64, in

Section 7.4.], shows the theoretical interweaving of the Stark effect for both gross and

238

 

 

 

 

 

Resonant Field

 

 

--------------1--------------

  

.--------------.E.-------------

L
r

 

 

Er

 

 

Ep

 

 

 

 

 

 

 

S

E

 

 

 

 

 

plasma

 

 

 

 

 

 

 

Er

 

 

 

 

 

 

 

Figure 81 Electric Field in Resonant Reactor Chamber.

239

fine structure in the HCl line. Figure 64 allows for the location of 25 independent
transitions, shown graphically on Figure 65. Figure 64 also helps explain why the H;
peaks are much sharper than expected, and why H5 and H7 peaks are not. The explanation
follows.

With an applied field of exactly 3858 V/cm, levels j=3/2 and j=l/2 in n=2 are
coupled by the microwave power source at 2.45 GHz. Electrons in the lower level are
raised to the upper level by the strong overlap between (n,j)=(3,5/2) and (n,j)=(2,3/2)
waveforms.

Now, transitions from (n,j)=(3,5/2) to (n,j)=(2,3/2) and (n,j)=(2,l/2) are recorded
by the spectrometer. Additionally, the photon released in these transitions interacts with
atomic hydrogen orbitals that are immersed in a strong microwave standing field. This
modulates the interaction energy of the photon by +/-2.45 GHz. The new modulated
energy of the photon is exactly the amount necessary to raise electrons from the
(n,j)=(2,l/2) and (n,j)=(2,3/2) levels into several of the (n,j)=(3,1/2) gross structures. The
energy released in the transition of these energy levels to a lower state is modulated, and
continues a chain reaction in which each of the upper states are tied to each other through
a series of transitions in the optically thick plasma.

The important point is that this chain reaction is initiated by microwave energy
absorbed in the lower band [85], and kept going by the same strong microwave source,
frequency modulating the photon energy released in upper-to-lower band transitions. To
absorb the exact amount of energy, the hydrogen atoms must be relatively stationary with
respect to the standing electromagnetic field [86]. Thus, the emission in the visible

spectrum will come from atoms with very small velocities, defeating Doppler

240

 

 

 

broadening. This effect is similar to that found in laser spectroscopy [87], and was only
observed in this set of experiments where the electric field was found to be ~3800 V/cm.

Another explanation for the relative absence in Doppler broadening is that, as the
plasma is optically thick, the only emissions escaping to the PMT are emissions from
atoms along the perimeter of the plasma. Whether confined in motion by the quartz tube,
or confined by the same forces that constrain the plasma, atoms at the perimeter of the
plasma have very low velocity; the edge of the plasma is the turning point for atoms with
velocities less than escape velocity. As a result, the Doppler broadened line width is
narrowed, reflecting the nearly static hydrogen atoms at the plasma edge, in the direction
that the light is emitted.

This condition is not related to the constant velocity, or Bohm velocity (uB), of
hydrogen ions at the plasma sheath (section 4.1.1). Obviously, H+ ions have no electrons,
and therefore no electronic emission.

One additional point. The gross structure Stark splitting at 3858 V/cm is predicted
to be 0.1067 A, or 6.40 units, from theory. The gross structure splitting on either side of
the centerline is exactly 6.0 units. The gross structure splitting between the lefi and right
side —that is, the energy difference between the k=+2 and k=—2 parabolic energy levels-
does appear to be exactly 6.40 units.

A rigorous explanation for this effect would be very difficult. The gross structure
energy levels are locked to the fine structure transitions by the nonlinear nature of the
interaction between photon and the atomic orbital and spin-orbit coupling, and in this

case the microwave field. This effect, called mode locking, is common place in physics;

241

 

 

from optical scattering in multi-mode fiber, to clock pendulums mounted on a common

wall.

8. 2 Discussion

Results from this investigation indicate that a constant magnitude magnetic field
and a constant magnitude electric field are both sustained by the plasma, evident in the
hydrogen spectrum by strong Zeeman and Stark splitting. These fields are not the
applied microwave electromagnetic components; the applied microwave components are
sinusoidal, and would imprint a continuous spectrum about the center wavelength, which
is also evident.

Furthermore, the impressed magnetic and electric field strengths are not related;
the magnetic field decreases with pressure, the electric field increases, as shown in
Figures 57 (section 7.3.1.2) and 77 (section 7.4.2).

One possible explanation for the constant magnitude magnetic field is the
following: the hydrogen molecules spin in the same direction and align in concentric
rings around the plasma center under the strong influence of the microwave H field.
Hydrogen is diamagnetic, with very small k (~-0.2x10'2) [53], but the collection of
hydrogen molecules forms domains that are circumferential to the plasma.

If this were the case, one would expect very high rotation temperatures at low
pressures. Rotation temperatures should first decrease, due to collisions, then increase as
the pressure is increased. This is seen in experimental data.

As with domains in ferrous material, the hydrogen magnetic field does not flip

until the microwave field has reversed itself hard enough. At that point, the molecular

242

 

 

spin reverses to its maximum value nearly at once. As a result, the H vs. B plot looks like
a standard hysterisis curve.

Furthermore, contractions at higher pressures can be seen as a direct interaction of
the spinning hydrogen molecules and the magnetic field gradient set up at the edge of the
plasma by internal collisions. The Lorentz force imbalance exerted on the current loop
defined by the protons rotating about their center axis pushes the plasma to the center of
the reactor. At lower pressures, the magnetic field gradient is not present.

The constant electric field is more difficult to understand, and there will be no
proposal for its mechanism at this time.

Improvements to the experiment mostly involve equipment. Hydrogen bonding
and spin effects could be monitored by infrared and microwave spectroscopy,
respectively. In the optical spectrum, higher pressures could be monitored by CCD
spectroscopy; the higher frame rates would eliminate the concern for noise jitter due to
instabilities in contraction. Further, spatial resolution —multiple optics channels- would
provide interesting comparisons between the plasma center and edge, where magnetic
field gradients are suspected.

It is uncertain whether higher resolution optical spectroscopy is the answer. The
Lummer-Gehrcke plate [88] requires no slit, increasing signal intensity, and provides
spectral data accurate to 10‘4 cm". But, the sinusoidal microwave fields may overwhelm
the finer structure, and blur out any advantage.

At this point, there are more questions than answers, and almost limitless avenues
to pursue in the understanding of the miniature plasma formed by microwave plasma

sources.

243

Appendix A Plasma System and Components

 

Figure 82 Miniature Microwave Plasma System.

244

 

.- “.11!“

1x a ‘u

.v. ““1111th
I- 1" » -

 

Figure 83 Gas Flow Meter Bank (4 Channel).

 

Figure 84 Electronics Control Board.

245

 

Figure 85 Plasma Reactor Chamber-

246

 

 

-._ - . .
4 .

Figure 87 Optical Fiber Micro-Positioner (CBS).

247

 

Figure 88 Hydrogen Plasma: 0.5 Torr, 60 W.

 

Figure 89 Hydrogen Plasma; 5.0 Torr, 60 W-

248

 

Figure 90 Hydrogen Plasma; 10.0 Torr, 60 W.

 

Figure 91 Hydrogen Plasma; 50 Torr, 60 W.

249

Appendix B Fiber Optic Feed-Through

 

Figure 92 Reactor Chamber with Fiber Optic Feed-Through.

 

Figure 93 Fiber Optic Feed—Trough.

250

 

\{L‘
, «'4

   

hannels).

Figure 94 Feed-Trough Micro-Lens System (13 C

 

Figure 95 Feed-Through Construction Tool Set.

251

[3]

[8]

[9]

[10]

[11]

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