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Investigation of Sub-Nanosecond Breakdown through Experimental and
Computational Methods
by
Jordan Elliott Chaparro, B.S.E.E, M.S.E.E.
A Dissertation
In
ELECTRICAL ENGINEERING
Submitted to the Graduate Faculty
of Texas Tech University in
Partial Fulfillment of
the Requirements for
the Degree of
DOCTOR OF PHILOSOPHY
Approved
Hermann Krompholz
Lynn Hatfield
Andreas Neuber
Thomas Gibson
Fred Hartmeister
Dean of the Graduate School
August, 2008
© 2008
JORDAN CHAPARRO
All Rights Reserved
Texas Tech University, Jordan Chaparro, August 2008
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ACKNOWLEDGMENTS
I would like to thank my committee for their advice and guidance on the work
conducted over last four years. I would like to especially thank Dr. Krompholz for
serving as my committee chairman, for always being available for discussion, and for
keeping the project on track. Dr. Hatfield has also never failed to provide sound
advice, which has helped immensely in the development of my work. I would like to
recognize the Texas Tech Physics Department and specifically, Dr. Gibson, Dr.
Volobouev, and Mr. Burnside, for providing access and support for development on
the Gamera cluster and helpful advice for the development of the models created
under this project. I would also like to thank Dr. John Krile for his help and patience
in adapting his previous modeling efforts to this project.
I owe special thanks to the staff of the P3E group at Texas Tech, specifically,
Danny Garcia, Shannon Gray, Dino Castro, and Marie Byrd. I also give my gratitude
to the physics department’s machine shop including Kim Zinsmeyer and Phil Cruzan
who have done an outstanding job in fabricating the components needed to conduct
this research. I also thank my fellow undergraduate colleagues for their assistance and
support. To the colleagues who I directly collaborated with on this work including,
Kevin Kohl, Dr. Han-Yong Ryu, and Willie Justis, I thank you for the many
contributions that made this project possible. Finally I would like to thank the Air
Force Office of Scientific Research for funding the research presented here.
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TABLE OF CONTENTS
ACKNOWLEDGMENTS ............................................................................................ ii
ABSTRACT .............................................................................................................. v
LIST OF TABLES ...................................................................................................... vi
I. INTRODUCTION ................................................................................................... 1
II. BACKGROUND THEORY AND PRIOR RESEARCH ................................................... 3
DISCHARGE REGIMES IN PULSED BREAKDOWN ...................................................................... 3
Townsend Regime................................................................................................... 3
Streamer Regime .................................................................................................... 5
Post Streamer Regime ............................................................................................ 7
PHYSICAL PROCESSES IN PICOSECOND BREAKDOWN ............................................................... 9
Field Emission ......................................................................................................... 9
Electron – Neutral Collisions ................................................................................. 13
Runaway Electrons ............................................................................................... 15
Explosive Electron Emission .................................................................................. 17
RECENT RESEARCH EFFORTS ............................................................................................ 19
III. NUMERICAL MODEL ........................................................................................ 26
DEVELOPMENT AND OPERATING ENVIRONMENT .................................................................. 26
PARTICLE-IN-CELL IMPLEMENTATION ................................................................................. 27
Meshing ................................................................................................................ 28
Multigrid Poisson Solver ....................................................................................... 29
Particle Trajectories .............................................................................................. 34
MONTE-CARLO COLLISIONS ............................................................................................. 35
FOWLER-NORDHEIM EMISSION ........................................................................................ 39
IV. EXPERIMENTAL SETUP ..................................................................................... 40
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PULSE FORMING SYSTEM ................................................................................................ 40
TRANSMISSION LINE SYSTEM ............................................................................................ 41
EXPERIMENTAL CHAMBER AND VACUUM SYSTEM ................................................................ 50
CAPACITIVE VOLTAGE DIVIDERS AND DIGITIZERS .................................................................. 51
X-RAY DETECTION AND LUMINOSITY MEASUREMENTS .......................................................... 52
V. RESULTS ........................................................................................................... 56
RADIAL VOLUME BREAKDOWN AT HIGH OVERVOLTAGE ........................................................ 56
Equivalent circuit model ....................................................................................... 57
Breakdown Characteristics ................................................................................... 60
Modeling formative delay .................................................................................... 62
Monte-Carlo estimate of formative delay ............................................................ 64
Scaling Law ........................................................................................................... 67
GEOMETRIC BREAKDOWN STRUCTURE ............................................................................... 71
RUNAWAY ELECTRON ENERGY DISTRIBUTIONS .................................................................... 77
STATISTICAL DELAY ........................................................................................................ 86
VI. CONCLUSIONS ................................................................................................. 91
REFERENCES ......................................................................................................... 94
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ABSTRACT
Sub-nanosecond breakdown, at sub-atmospheric pressures, is governed by
significantly different physics when compared to standard breakdown processes. Applied
field risetimes of 100s of ps combined with high peak amplitudes and short gap spacing
allows for overvoltage to develop in the gap greatly exceeding static breakdown
conditions. These conditions lead to a significant portion of electrons in the runaway
mode and highly inhomogeneous charge distributions that greatly affect the scaling
relationships for the discharge. The continued progression of pulsed power applications
to shorter time scales makes a full understanding of such discharges necessary for the
future development of devices relying on ultrafast, high voltage pulses. Insights into the
physical background of sub-nanosecond breakdown are provided in this dissertation
through both empirical analysis and numerical modeling. The modeling of the discharge
is implemented through a customized particle-in-cell code combined with Monte-Carlo
methods for simulating particle collisions. The results of the model show reasonable
agreement to experimental results across the full range of test parameters. Additional
insights into physical mechanisms that are not easily empirically measured are provided.
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LIST OF TABLES
Table 4-1 - Relative photon sensitivity for 3 absorber foils. ....................................... 55
Table 5-1 Intensity ratios between foils for a variety of maximum
electron energies ................................................................................... 80
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LIST OF FIGURES
Figure 2.1 - Boundary between Townsend and Streamer processes as a
function of overvoltage percentage and p*d [4]. .................................... 5
Figure 2.2 – Schematic drawing of avalanche with developing fast
electron filament [11]. ............................................................................ 6
Figure 2.3 – Breakdown formative time as a function of electric field
and pressure as measured by Felsenthal and Proud [14]. ....................... 7
Figure 2.4 - The image force potential (-e2/4x), the external applied
potential (-eEx), and the modified barrier potential U(x)
as a function of distance from the metal surface. Up is
the total potential well depth for the metal and φ is the
metal’s work function. All potentials are given in terms
of eV [23]. ............................................................................................ 11
Figure 2.5 – Current from 10-8
cm2 brass micro-point with an
enhancement factor of 200 as a function of applied
macro-field. .......................................................................................... 12
Figure 2.6 – Cross sections for electron-neutral processes in argon. The
excitation curve is the sum of 37 individual electronic
excitation cross-sections [27]. .............................................................. 13
Figure 2.7 – First 11 electronic collision cross sections for argon along
with the excitation potential lost by the incident electron
in the collision [27]. .............................................................................. 14
Figure 2.8 – Approximation of frictional force as a function of electron
energy. I represents the average inelastic energy loss. ......................... 16
Figure 2.9 - SEM image of exploded micro-tip protrusion [34]. ................................. 18
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Figure 2.10 - Electron escape curves as derived by Yakovlenko [42].
The region above the upper portion of the curve
corresponds to the runaway regime while the area
between the upper and lower branch represents the
amplification region. the area below the bottom branch
corresponds to the drift regime. ............................................................ 21
Figure 2.11 - Streak camera imaging of discharge with the slit aperture
parallel to the axis of the discharge. Results show
intense regions of luminosity near the cathode indicating
highly localized ionization processes. .................................................. 23
Figure 2.12 - Streak imaging with the slit aperture perpendicular to the
discharge axis. The channel width expansion and
development can be seen clearly and increased
multichannel probability is observed with increasing
pressure. ................................................................................................ 24
Figure 2.13 - (Left) Calculated electron arrival energies as a function of
pressure and pulse amplitude. (Right) Average number
of ionizing collisions in the transit of a 1 mm gap as a
function of pressure and applied voltage as determined
by simple force model [43]................................................................... 25
Figure 3.1 – Operational flow of time step in PIC model. ........................................... 28
Figure 3.2- Restriction from a 12 element 1D grid to 6 element grid
resulting in a relative shift in error mode frequency [47]. .................... 30
Figure 3.3 – Semi-Coarsening method for progressing from fine to
coarse grids with descending aspect ratios. .......................................... 33
Figure 3.4 – Result of spherical charge test with boundaries at ± 5 mm
either set to free space conditions or grounded at zero
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potential. The red curve shows the analytical solution
which agrees well with the free-space results from the
Multigrid solver. ................................................................................... 34
Figure 3.5 – Collisional frequencies for three fictional processes
summed to form the maximum null frequency [54]. ............................ 36
Figure 3.6 – Scattering angle as a function of a random element R for a
variety of incident electron energies..................................................... 37
Figure 3.7 - Normalized secondary electron energy as a function of a
random variable for several incident energies. ..................................... 38
Figure 3.8 - Results from FN emission model. Shown is the FN curve
(red) the Child-Langmuir relationship for space charge
limited current (blue) and points (magenta) representing
steady state current magnitudes from the numerical
model as a function of field amplitude. ................................................ 39
Figure 4.1 - Oil-filled coaxial setup (not to scale). ...................................................... 40
Figure 4.2 - Schemata of pulse slicer assembly. Adjustable peaking
and chopping gaps are used in high pressure nitrogen to
reduce the risetime and FWHM of the generated pulses
[59]. ...................................................................................................... 41
Figure 4.3 - Hyperboloidal Rexolite lens transitioning from planar
wavefront in coaxial geometries to spherical fronts for
the biconical section. ............................................................................ 43
Figure 4.4 - Determination of the curvature of the lens by comparison
of electrical path lengths. ...................................................................... 44
Figure 4.5 - Output of Spire Pulser with 1kV amplitude ............................................. 45
Figure 4.6 - Lens fitting on tapered inner conductor section with two
sealing o-rings. ..................................................................................... 46
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Figure 4.7 - Axial and radial electrode geometries for biconical gap
assembly. Gaps may be varied from 1 - 4 mm for both
configurations. ...................................................................................... 47
Figure 4.8 - Gap assembly for testing statistical breakdown delays near
the FN threshold. Gap spacing of up to 11 mm can be
accommodated yielding fields an order of magnitude
lower in amplitude than those with the biconical
assembly. .............................................................................................. 48
Figure 4.9 - 3D view of gap assembly and optical viewports for
imaging and x-ray analysis. .................................................................. 49
Figure 4.10 - Tri-branch resistive load for termination of the
transmitted side of the coaxial line. Nominal impedance
is 46 ohms and reflections from the load are less than
10%. ...................................................................................................... 50
Figure 4.11 - Images of constructed incident (left) and transmitted
(right) oil-filled coaxial transmission lines coupled to
the experimental test chamber. ............................................................. 50
Figure 4.12 - Experimental test chamber. The front optical window is
in the center of the image. .................................................................... 51
Figure 4.13 - Schematic view of capacitive voltage divider. Total area
of copper shim along with the dimensions of the line
determine the capacitances shown which yield sub
100ps risetimes and > 10 ns fall. The divider ratio is
250:1. .................................................................................................... 52
Figure 4.14 - 3D cutaway view of PMT assembly for measuring x-ray
emission from the anode. The assembly sits in a test
chamber view port angled at 24 degrees from the
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vertical axis and corresponding viewports have been
drilled into the gap assembly. ............................................................... 53
Figure 5.1 - Measured voltage output for numerous shots from the
RADAN pulser. Amplitude and risetime variation are
less than 10%. ....................................................................................... 57
Figure 5.2 - Lumped element model for gap. ............................................................... 58
Figure 5.3 - Measured transmitted voltages resulting from breakdown
from four different pulse amplitudes and across the full
range of tested pressures. ...................................................................... 59
Figure 5.4 - Sample current pulses obtained from lumped element
model for 4 pulse amplitudes and a number of pressures. ................... 60
Figure 5.5 - Measured breakdown voltages for radial discharges with
pulse amplitudes from 50 - 150 kV. ..................................................... 61
Figure 5.6 - Formative delay times from experimental results and
lumped element modeling for pulse amplitudes between
50 - 150 kV. .......................................................................................... 62
Figure 5.7 - Calculated formative delays as a function of pressure for
several pulse amplitudes (given in kV/mm) using simple
force modeling. ..................................................................................... 63
Figure 5.8 - Formative delays from streamer derived model proposed
by Yakovlenko resulting from a 200 ps risetime ramped
step [64]. ............................................................................................... 64
Figure 5.9 – Simulated effect of the field enhancement factor on the
formative delay times with a pulsed field amplitude of
50 kV/mm. ............................................................................................ 66
Figure 5.10 - Simulated formative delay times for field amplitudes
between 25 and 150 kV/mm over the full pressure range. ................... 67
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Figure 5.11 - Simulated (green) and experimental (blue) E/p vs pt plot.
Simulated results show reasonable agreement to
experimental efforts. Red points simulate the influence
of UV illumination of the gap prior to voltage
application and shift the relationship towards the black
line representing the accepted streamer regime scaling
law determined by Felsenthal and Proud [14]. ..................................... 68
Figure 5.12 - Simulated pressure normalized ionization frequencies
plotted against E/p. The red curve is the curve fit argon
ionization frequencies from Yakovlenko [42] which is
claimed to be accurate for E/p up to 104 V/(cm torr).
Black circles represent empirically derived frequencies
determined from the product of measured Townsend
coefficients [28] and electron drift velocities. Dashed
lines represent curve fits for simulated data resulting
from specific pressures. ........................................................................ 70
Figure 5.13 - Simulated electron energy distributions resulting from
application of E/p of 104 V/(cm torr).. ................................................. 73
Figure 5.14 - Simulation of ionization processes per 15 µm x 15 µm
pixel on logarithmic scale, showing the strong
concentration of ionizations in front of the cathode, and
channel constriction increasing with pressure. From top
to bottom, the pressures for the images are 100, 200,
300, and 600 torr. ................................................................................. 73
Figure 5.15 - XY slice of the gap at the center of Z showing the time
development of space-charge fields for a pressure of 200
torr. Each successive picture represents a one quarter
step of the formative time. .................................................................... 74
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Figure 5.16 - 1D plot of space charge field development for quarter
steps of the formative time at 600 torr. .............................................. 767
Figure 5.17 - 1D plot of space charge field development for quarter
steps of the formative time at 100 torr. ................................................ 77
Figure 5.18 - Metallic absorber foils sensitivity for energies up to 150
keV. ...................................................................................................... 78
Figure 5.19 - Product of approximated Bremsstrahlung spectrum
resulting from monoenergetic 100 keV electron beam
with the metallic foil sensitivity curve (dashed lines).
The ratio of the areas of the dashed regions relates to the
maximum electron energy in the runaway distribution. ..................... 801
Figure 5.20 - Maximum PMT intensities measured from 150 kV pulsed
discharges in argon for the four absorber foils. .................................. 812
Figure 5.21 - j2td metric measuring the delay time to explosive electron
emission as a result of applied fields (dashed) both for a
case where space-charge is tracked (green) and one
where it is neglected (blue). The green curve in this
example is for a pressure of 50 torr.. .................................................... 83
Figure 5.22 - Simulated EEE delays for 150 kV/mm pulsed fields. The
local minimum near 75 torr may explain the minimum
in silver and lead PMT data around the same pressure.. ...................... 84
Figure 5.23 - Simulated electron energy distributions at the anode over
the duration of the EEE delay. .............................................................. 59
Figure 5.24 - Electron energy distribution of particles in the test gap at
the time of the breakdown condition for a pulsed field of
50 kV/mm. ............................................................................................ 86
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Figure 5.25 - Incident (red) and transmitted (black) traces used to
measure statistical delay. The pulse with on the incident
pulse is around 1.25 ns.. ....................................................................... 87
Figure 5.26- Plotted delays for 20 shots at each pressure. The red
marks indicate the average and standard deviation of the
incident pulses for the pressure. Plotted delays for 20
shots at each pressure. The red marks indicate the
average and standard deviation of the incident pulses for
the pressure. .......................................................................................... 88
Figure 5.27 - Laue plots for 75 (top), 115 (middle) and 175 (bottom)
kV pulses. ............................................................................................. 89
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CHAPTER 1
INTRODUCTION
Electrical discharges, initiated by field emission in conditions with very high E/p
ratios, greater than 100s of V/(cm torr), exhibit markedly different scaling properties
when compared to those predicted by standard Townsend and streamer models. Such
conditions are brought about when high voltage pulses, with risetimes no greater than
a few hundred picoseconds, are applied over short gaps at pressures up to atmospheric
conditions. The field amplitudes attained during such events may be more than an
order of magnitude greater than the standard volume breakdown threshold, observed
with DC conditions. This leads to a breakdown process that is initiated by short
duration beams of highly energetic runaway electrons (REs) which are accelerated
continuously across the gap to the maximum energy allowed by the applied field.
Collisional carrier amplification is primarily limited to a narrow region of the gap near
the cathode resulting in the development of a highly inhomogeneous charge
distribution, which leads to space-charge fields that have an effect on the further
development of the discharge. To this point, a complete understanding of the physics
behind such discharges is far from complete. Continuing advancements in the
development of high-voltage generators, capable of producing sub-nanosecond pulses
with high-repetition rates, has led to several emerging applications, such as ultra-
wideband radar systems [1], and ultra-fast switches [2], where conditions for such
discharges can arise. Therefore, a deeper understanding of the physics behind ultra-
fast breakdown is desirable for the continued development of these and future
applications.
Research into discharge mechanism of any type is complicated by the sheer
number of variables at play. It is for this reason that scaling relationships are sought
out which simplify breakdown description by grouping attributes in certain
combinations which describe discharge properties over a wide range of conditions.
For instance, one of the most well known and commonly observed scaling
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relationships is expressed by the Paschen law which states that for the constant
product of pressure and gap distance, representing the number of neutral particles
between the electrodes, the static breakdown voltage remains constant as well. It can
be further shown, that this relationship holds true over a certain range which diverges
with strong fields among several other conditions [3]. Likewise, most of the
commonly cited scaling laws do not extend as predicted to the range where FE
initiated picosecond discharges occur. Over the last few years, there have been
extensive experimental efforts to resolve the relationships for this regime.
Numerical models, on the other hand, are much less limited in the number of
variables they can consider. Using statistical sampling techniques and fundamental
physical laws, numerical models can provide reasonable estimates for breakdown
metrics over an expansive range of conditions. In addition, such models can be used
to investigate and expand scaling laws and the boundaries where they breakdown.
The continuing advancement of computer architecture and parallel computing has
made the implantation of complex models feasible and highly attractive.
This dissertation aims to expand scaling laws to the picosecond regime through
experimental and numerical efforts. In addition, efforts are made to increase the
physical understanding of such phenomena and to explain the observed divergence
from existing scaling relationships. This document is organized as follows: in chapter
2, an overview of the present physical theory behind picosecond discharge is given.
Chapters 3 and 4 detail the implementation of a particle-in-cell based numerical model
intended to directly simulate experimental conditions and detail the experimental
apparatus respectively. In Chapter 5, empirical and numerical results are compared
and commented on. Conclusions and summary can be found in Chapter 6.
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CHAPTER 2
BACKGROUND THEORY AND PRIOR RESEARCH
Before an investigation of picosecond breakdown can take place, it is necessary to
examine the existing background theory for the physical process governing pulsed
discharge. As pulser and digitizer capabilities have advanced, pulsed discharge has
been researched in several distinct operating regimes where different scaling and
physical relationships have been established. Progression has been towards faster
risetimes and subsequently higher overvoltages, where the current picosecond time
scales are found. This section provides an overview of what the previous research into
pulsed discharge has revealed to this point and then discusses important physical
processes that govern the picosecond regime along with summarizing current research
efforts.
Discharge Regimes in Pulsed Breakdown
Pulsed discharge regimes can be in part distinguished by the degree that the static
breakdown voltage is exceeded during the development time of the breakdown event.
The risetime of the pulse, in comparison with the formative delay time for the
discharge, governs the degree of overvoltage that can be attained by the time when
breakdown occurs. In addition to degree of overvoltage, recent research has shown
that the nature of the source of initiatory electrons can have significant effects on
scaling relationships for high E/p regimes.
Townsend Regime
For small overvoltages, not exceeding 10%, the standard Townsend model is
considered an accurate description for most pressures [4]. The Townsend process,
described in detail in [5], is an event governed by growth through collisional
avalanches, with exponential growth rates. With this exponential relationship, the
current of the discharge through the gap can be described as:
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)exp(0 dII ⋅⋅= α (2.1)
where α is the collisional ionization coefficient representing the number of new
electrons produced over a 1 cm path, I0 is the initiatory current, and d is the gap
distance. Secondary avalanches, initiated by photoelectric and positive ion effects at
the cathode, are also included in more detailed versions of the model with the
introduction of the secondary ionization coefficient γ. The model generally maintains
its validity to the point where the space-charge fields of an avalanche begin to distort
the applied field. This condition can be related to a critical electron density, which
when attained, indicates a transition from the Townsend regime to the streamer one.
The transition from Townsend to streamer conditions was first investigated by
Fischer and Bederson [6] who found that the formative times scaled as a function of
the percentage of overvoltage. Later, work by Allen and Phillips [4] experimentally
measured transitionary boundary between Townsend and streamer processes for
several gases as a function of overvoltage. Figure 2.1 shows the measured boundary
between the two regimes for an air filled gap as a function of overvoltage and pd. As
seen in the figure, low values of pd (< 200 torr*cm) require much higher overvoltages
to diverge from Townsend theory.
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Figure 2.1 - Boundary between Townsend and Streamer processes as a function of
overvoltage percentage and p*d [4].
Many of the scaling relationships for the Townsend regime are expressed as a
function of the pressure and gap distance product. Additionally, it has been
established that both α/p and γ can be expressed as functions of E/p. Expanding on
these relationships, Schade [7] established a scaling law for predicting formative times
(τd) for Townsend processes that depends on both E/p and p*d.
Streamer Regime
The observation of time delays too short to be explained by Townsend theory, led
Loeb [8], Meeks [9], and Raether [10], to independently propose the fundamental
streamer model. The essential basis of the theory was based on secondary avalanches
initiated, through photoionziation, joining with the primary avalanche to form
conducting plasma channels that could bridge the gap at velocities orders of
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magnitude faster than drift velocities. It was later demonstrated that in addition to
photoionziation, secondary avalanches could be initiated by fast electrons escaping
from the front of the avalanche head due to Coulomb forces [11], [12]. Figure 2.2
illustrates the development of a filamentary channel of fast electrons with some
electrons escaping the avalanche head to reach the runaway state.
Figure 2.2 – Schematic drawing of avalanche with developing fast electron filament
[11].
According to Raether [10], the formative time necessary for the avalanche to
reach streamer conditions can be expressed as:
v
NCR
⋅=
ατ
)ln(
(2.2)
where NCR is the critical electron density, and v is the electron drift velocity. The
propagation time of the streamer is much smaller than τ and is neglected. Because α/p
= f(E/p), equation 2.2 dictates that streamer discharges follow the similarity law pτ =
f(E/p). Work by Fletcher [13] attempted to better define the mathematical framework
for the streamer regime by making comparisons to experimental discharges resulting
from nanosecond pulses. It was concluded from this investigation that a critical
electron number of Nc = 108 agreed with results for fields below 10
5 V/cm.
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The scaling relationship between pτ and E/p was further expanded upon by
Felsenthal and Proud [14], who experimentally measured delay times for several
gasses using 4 – 30 kV nanosecond voltage pulses. Application of these pulses with
gap pressures of 1 – 760 torr and electrode spacing of 0.5 – 5 cm led to measured
formative times from 0.3 – 30 ns. Ultraviolet illumination of the gap prior to
breakdown was used to limit statistical lag and seeded the cathode with approximately
104 initiatory electrons. Figure 2.3 shows the measured scaling relationship for the
nine tested gases.
Figure 2.3 – Breakdown formative time as a function of electric field and pressure as
measured by Felsenthal and Proud [14].
Post Streamer Regime
When overvoltage percentages reach several times the volume breakdown
threshold, deviations from streamer behavior may be noted. Delay times can reach
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durations too short for photoionziation processes to occur (typical de-excitation times
~10-9
s). This points to the influence of runaway electrons as the source of the leading
avalanches which was first explained by Babich and Stankevich [15]. The number of
initiatory electrons has also been shown to have a significant effect on the
development times observed in this regime [16], [17].
Stankevich and Kalinin [18] first observed runaway electrons, in atmospheric
conditions, by measuring x-ray emissions from the anode as a result of
bremsstrahlung. The voltage source for the experiment produced 46 - 58 kV pulses
with 2 ns risetime that were applied across a gap pressurized with dry air. The
resulting x-ray pulses were measured using an organic scintillator and a
photomultiplier tube with absorbing metallic foils between the anode and the
scintillator. With known absorption profiles, the metallic foils were used to quantify
the energy of the radiation resulting from the discharge. Results were used to estimate
the average photon energy of 6 keV which suggested that the electrons gained energy
comparable to one tenth of the applied voltage. Similar experimental demonstrations,
showing x-ray radiation in the keV range, were conducted by Kremnev and Karbatov
[19] and Tarasova et al. [20].
The physical behavior of the regime was further distinguished by Mesyats and
collaborators [3], [16], who investigated the role of initiatory electrons and explosive
emission processes after measuring delay times that did not conform to streamer
regime similarity laws. He described two different breakdown types; one where some
mechanism seeded the cathode with a large number of initiatory electrons and one
where the only source of initial electrons was through field emission from the cathode.
It was recognized that the early work of Felsenthal and Proud [14] and Fletcher [13],
adhered to the similarity law in high E/p regimes with short gaps due to the large
number of initiatory electrons that were produced with the intention of preventing
statistical delays. Mesyats indicated that these initial electrons were necessary to
maintain the scaling relationship because individual avalanches under these conditions
could not reach critical densities within the short gap spacing. This was attributed, in
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part, to electron avalanche self-braking which slows the growth rate of the avalanche
significantly from exponential rates when fields exceed 105 V/cm.
As a result of this self-braking of the avalanche growth, field emission initiated
discharges in the same E/p range tend to have delay times one to two orders of
magnitude slower then multielectron initiated discharges. This has been demonstrated
by experiments reported in [16] where applied fields of E = 1.4 x 106 V/cm were
applied across highly polished electrodes resulting in delay times of τ ≈ 1 ns. By
scratching the surface of the cathode intentionally, the delay times were reduced to
below 100 ps due a large number of field emission sites and eventual explosive
emission along the edges of the scratches. Even with good cathode production
techniques, there will inevitably be a limited number of micro-point protrusions on the
surface which lead to localized single point emission sites [21]. Limiting the number
and degree of these protrusions on the cathode surface lengthens the formative delay.
The localized field emission initiated, high E/p regime most closely relates to the
picosecond time scale discharges discussed in this thesis.
Physical Processes in Picosecond Breakdown
The discharges studied in this paper are field emission initiated and take place
with maximum applied fields ranging from 4 x 105 to 2 x 10
6 V/cm. In argon, these
fields correspond to overvoltage factors ranging from 8 – 800. As such the breakdown
process operates in the nebulously defined post streamer operating regime and these
conditions lead to particular dominant physical characteristics that will be explored in
greater detail in this section.
Field Emission
The field emission (FE) process is the only statistically significant source of
initiatory electrons for the discharges discussed in this thesis and as such plays a major
role in determining the measureable properties of the breakdown. Field emission is a
quantum mechanical effect where electrons tunnel through potential barriers, caused
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by image forces, between a metal and vacuum interface. Strong electric fields reduce
the width of the potential barrier at the interface and increase the probability of an
electron tunneling from the solid into the vacuum. In order to achieve any significant
emission current, very high field amplitudes, greater than 107 V/cm, are required.
Typically, field amplitudes of this degree can only be achieved in localized region
around micro-protrusions or surface imperfections that have high enhancement
factors. Localized enhancement from such protrusions on otherwise planar electrodes
can reach magnitudes of several hundred with total areas around 10-8
cm2
[21].
The field emission process is described by the Fowler-Nordheim (FN) theory
which quantifies the current density as a function of electric field amplitude [22]. The
theory is based upon the probability for an electron to tunnel through the potential
barrier which is calculated by integrating the barrier transparency, D, and the flow of
electrons incident on the barrier, N, over the electron energy on which both D and N
are dependent. The potential barrier, when altered by an applied electric field E, is
described by the following function:
,4
)(2
xEex
exU ⋅⋅−
⋅−=
(2.3)
where e is the electron charge and x is the distance from the metal surface. A
schematic diagram of the image force potential before and after being altered by an
applied field can be seen in Figure 2.4 [23].
For such a potential barrier the transparency is given by:
),(3
28exp),(
2/3
yE
Ex
eh
mFExD υ
π⋅
⋅⋅⋅⋅⋅
−= (2.4)
where Ex is the electron energy and υ(y) is the Nordheim function. Integrating the
function over all energies for electrons incident on 1 cm2 of the barrier surface from
within the metal leads to the classic FN formula:
,)(3
28exp
)(8
2/3
2
23
⋅⋅
⋅⋅−
⋅⋅⋅
⋅= y
Eeh
m
yth
Eej
e
FN υφπ
φπ (2.5)
Texas Tech University, Jordan Chaparro, August 2008
11
where φ is the work function for the cathode material in eV, E is in V/cm, and jFN is in
A/cm2. The functions t(y) and υ(y) are recent inclusions to the FN function due to the
effect of image charges [24].
Figure 2.4 - The image force potential (-e
2/4x), the external applied potential (-eEx),
and the modified barrier potential U(x) as a function of distance from the metal
surface. Up is the total potential well depth for the metal and φ is the metal’s work
function. All potentials are given in terms of eV [23].
The local field enhancement factor, β, can be simply included into the FN model
as a multiplier on the field amplitude. Figure 2.5 shows FN predicted current from a
micro-point site with an area of 10-8
cm2 and β = 200 as a function of electric field
values in the range of the experiments conducted in this paper. Brass is used as the
electrode material for this example. The work function of brass (4.55 eV) is estimated
as a weighted average of copper (4.65 eV) and zinc (4.33 eV) which has been
experimentally shown to be reasonable [25].
Texas Tech University, Jordan Chaparro, August 2008
12
Figure 2.5 – Current from 10-8
cm2 brass micro-point with an enhancement factor of
200 as a function of applied macro-field.
Behavior at the threshold where FE leads to the initial electron production is
important when considering statistical time lag. Recent research efforts [26] have
focused on counting the number of electrons emitted in FE and thermal emission
events. Experimental emission events were controlled from a modified electron
microscope using tungsten point filaments which could be heated to 2800 K for
thermal processes or positioned in the high field acceleration region of the microscope
for FE. An electron counting system using a linear, energy dispersive detector was
employed. Results indicated that thermal emission emitted random single electrons as
expected while FE events tended to emit multiple electrons in bursts. As many as 11
electrons were detected from isolated random FE events. For discharges operating
with fields near the threshold for FE, the statistics of multiple electron emission could
significantly impact delay times.
Texas Tech University, Jordan Chaparro, August 2008
13
Electron – Neutral Collisions
Electron collisions with background gas particles dictate many of the
characteristics of discharge physics. The frequency of which these collisions occur
can be simply quantified by the ratio between electron velocity and mean free path,
which is derived from the inverse of the gas number density and collisional cross-
section area product. Figure 2.6 shows the collisional cross sections for electron-
neutral processes in argon.
Figure 2.6 – Cross sections for electron-neutral processes in argon. The excitation
curve is the sum of 37 individual electronic excitation cross-sections [27].
For low energy electrons (< 10eV), elastic collisions are most common. In these
collisions, a fractional quantity of the electron’s kinetic energy is transferred to the gas
particle as:
Texas Tech University, Jordan Chaparro, August 2008
14
i
e EM
mE )cos1(
2θ−=∆ (2.6)
where me is the mass of the electron, M is the mass of the gas particle, θ angle of
deflection, and Ei is the incident energy. The scattering angle is nearly uniformly
distributed for low energies and with increasing energy become more skewed towards
low angles. Because the ratio between the mass of the electron and the neutral particle
is approximately 2 x 10-5
, momentum transfer from electron-neutral elastic collisions
is negligible.
Inelastic processes include excitation and ionization events, in which part of the
kinetic energy of the impacting electron is converted into potential energy to either
excite or liberate an electron in the neutral particle. The amount of energy converted
is equal to the energy required to change the state of the electron. The cross-sections
for the first 11 electronic excitation states for argon can be seen in Figure 2.7.
Figure 2.7 – First 11 electronic collision cross sections for argon along with the
excitation potential lost by the incident electron in the collision [27].
Texas Tech University, Jordan Chaparro, August 2008
15
For ionizing collisions, the energy balance must account for the energy of the
newly liberated electron in addition to the ionization potential. Ionization rate
coefficients are often used to describe the growth rate for an avalanche and are
dependent on the total electron energy distribution which is intern, dependent on the
ratio of E/p. For low to moderate E/p (<103 V/(cm torr) ), reasonable approximations
for ionization rates have been determined that are comparable to experimental
measurements [28]. For increasing E/p ratios, the high energy tail of the electron
distribution becomes more significant and a number of electrons may exceed the
energy maximum of the ionization frequency leading to total ionization rates that
begin to level off and even decrease at very high E/p ratios. Currently, there is very
little experimental data for ionization rates with E/p > 105 V/(cm torr). Numerical
calculations from Monte-Carlo methods are useful for determining rate coefficients for
unexplored regimes and have been used to extend the known range of these metrics
for some gases [29].
Runaway Electrons
Runaway electrons play a large role in discharges with extremely high
overvoltages and arise in high E/p conditions where the energy gained by an electron
over the unit path is greater than the energy lost to inelastic collisions. The local
electron runaway criterion can be derived from the following rough energy balance
equation:
)(εε
FeEdx
d−= (2.7)
where x is the distance from the cathode and F(ε) is the frictional force from collisions
with the background gas. If elastic scattering is disregarded, the frictional force
resulting from inelastic losses is based on the Bethe approximation as [30]:
I
zneF
εε
πε
⋅⋅⋅⋅⋅=
2ln
2)( 0
4
, (2.8)
Texas Tech University, Jordan Chaparro, August 2008
16
where n0 is the number density of the neutral gas, z is the number of electrons in the
molecule, ε is the kinetic energy of the electron, and I is the average inelastic energy
loss. A generalized plot of the frictional force is given in Figure 2.8.
Figure 2.8 – Approximation of frictional force as a function of electron energy. I
represents the average inelastic energy loss.
Solving for the maximum of equation 2.8, it is shown that F(ε) has a peak at ε =
2.72 I/2. It follows that there exists some critical field, Ec = F(2.72 I/2)/e, where
electrons with energy ≥ I/2 are continuously accelerated. This field is given by:
I
zneEc
⋅
⋅⋅⋅⋅=
72.2
4 0
3π. (2.9)
Replacing the number density with pressure at 300 K and substituting for the
numerical constants gives the runaway criterion in terms of E/p:
I
z
p
Ec ⋅×= 31038.3 . (2.10)
For example, for N2 Z = 14, I = 80 eV, and Ec/p = 590 V/(cm torr).
The above formulation is rough and neglects some notable processes that
invalidate the E > Ec runaway criteria. The most important point not taken into
consideration is the multiplication of electrons. More detailed research of runaway
criteria in fully ionized plasmas (i.e. ionization is absent) has led to critical fields for
Texas Tech University, Jordan Chaparro, August 2008
17
runaway conditions similar to the one expressed in equation 2.9 [31]. Adjusting for
ionization processes, equation 2.7 can be adjusted to account for ionization processes
as [32]:
,)( ***
εαεε
⋅−−= iFeEdx
d (2.11)
where ε* is the mean electron energy, and αi is the Townsend electron multiplication
coefficient. This new term from equation 2.9 accounts for the energy that is lost to the
secondary electrons produced through ionization. In reality, there are many statistical
considerations that lead to some small percentage of electrons being accelerated
continuously for fields E > Ec that is not easily quantified. However, examining
statistical processes is much more readily accomplished through numerical simulation
and accurate examination of runaway distributions is possible through Monte-Carlo
models [33].
With the definition of runaway electrons established, the role they play in pulsed
discharge can be reviewed. Mesyats [3] has shown that for multielectron initiated
discharges, runaway electrons have little or no effect on formative times. For single
electron FE initiated discharges, the role of the runaway electron is much more
substantial. The majority of the emitted electrons form a slow moving ionization front
of which runaway electrons stream off of at velocities far greater than the front’s drift
propagation velocity. While collisions for electrons in the runaway state are
uncommon, they are not impossible and statistically do occur creating weakly ionized
channels across the rest of the gap space in the path of the beams. At the point of
explosive electron emission of the micro-point, the jets become discharge channels
closing the gap [3], [21].
Explosive Electron Emission
In FE initiated discharges with high applied fields, exceeding 106 V/cm, explosive
electron emission plays a major role. Explosive electron emission (EEE) occurs when
joule heating of the micro-protrusion tip from FE current brings the temperature of the
Texas Tech University, Jordan Chaparro, August 2008
18
metal to its boiling point. At this point the molten metal transitions to a dense plasma
which is ejected into the gap space leaving cratered regions where the tip was. This
ejection of material leads to new micro-protrusions being created along the edges of
the crater. Figure 2.9 shows a SEM image of a cratered EEE site taken by Bergmann
et al. [34]. From the image it can be seen that new micro-tip protrusions can form not
just along the crater’s perimeter, but also further out as ejected molten metal comes
back into contact with the surface.
Figure 2.9 - SEM image of exploded micro-tip protrusion [34].
The heating of the micro-protrusion tip can be modeled as a point emitter which
allows for the temperature to be expressed as a function of the FE current density.
Mesyats, who covers the derivation of this model in detail in [21], comes to an
expression quantifying the conditions for explosive emission that is determined only
by the physical properties of the cathode material and it is given as:
Texas Tech University, Jordan Chaparro, August 2008
19
00
2 lnT
Tctj crd κ
ρ ⋅≅ . (2.12)
Here, ρ, c, and κ0 are, respectively, the density, the specific heat, and resistivity of the
cathode material. Tcr is the critical temperature of the material and T0 is the initial
temperature. Thus, the product of the current density squared and the time of applied
current pulse correlating to explosive emission are determined by the physical
properties of the material. They physical properties of many metals lead to j2td values
of ~ 109 A
2 s/cm
2 [21].
Recent Research Efforts
In the last few years, there has been extensive research into highly overvoltaged
discharges mainly originating out of Russia. Many of these investigations have been
focused on resolving the duration of runaway electron beams through novel
experimental measurement techniques [35] , [36] , [37], [38], [39], [40], and numerical
methods [40].
Experimental setups for each of these studies were based around some type of
RADAN pulser [41], producing pulses with risetimes < 0.5 ns and with amplitudes of
100s of kV. Beam development was investigated in mostly sub-atmospheric
conditions. The method for measuring the duration of picosecond runaway beams in
each of the experiments is similar. A thin foil anode was used followed by a beam
collimator, with narrow radial slits intended to attenuate the output of the current
collector so that no coaxial attenuators would have to be used. The collector was
connected directly by wideband cable to a fast digitizer (6 GHz for [34-37], 15 GHz
for [38-39]). Results showed that the runaway beam currents are extremely short with
Mesyats measuring beams as short as 45 ps in air. Overall, beams varied in duration
from 45-500 ps depending on pulse amplitude, risetime, and background gas. It was
concluded that the beams could be injected from field emitters or off the ionization
front. Mesyats proposes that the short duration of the beams is a result of screening of
the micro-emitters by ionized gas brought about in the transition from FE to EEE.
Texas Tech University, Jordan Chaparro, August 2008
20
Both Mesyats and Tarasenko proposed potential electron accelerator applications
utilizing such discharges to produce picosecond duration electron beams with energies
ranging from 150-500 keV.
Other work from Yakovlenko [42] has focused on mathematically modeling
picosecond discharge by accounting for runaway effects and expanding similarity
relationships. The work is based on an assumption that runaway electrons begin to
dominate when the distance d between the electrodes becomes comparable to the
characteristic multiplication length αi-1
. Accordingly, the criterion for the critical
electric field strength Ec is given as:
1),( =⋅ dpEciα . (2.13)
The Townsend coefficient is then described as a function of pressure and E/p as
αi(E,p) = p*ξ(E/p). For flat electrodes, E = U/d, and Ec = Uc/d and the critical
condition for runaway electron in the gap space can be expressed as:
11 =
⋅=
⋅
pd
Ucpdor
p
Epd c ξξ . (2.14)
This expression presents the dependence of the critical voltage Uc(pd) on the pressure
and gap distance product pd. The curve Uc(pd) is termed the “electron escape curve”
as it separates the regions of electron multiplication from the region where electrons
accelerate out of the gap without multiplication. Electron escape curves for a number
of gases are shown in Figure 2.10 [42].
Texas Tech University, Jordan Chaparro, August 2008
21
Figure 2.10 - Electron escape curves as derived by Yakovlenko [42]. The region
above the upper portion of the curve corresponds to the runaway regime while the area
between the upper and lower branch represents the amplification region. the area
below the bottom branch corresponds to the drift regime.
’
The presence of a maximum in the ξ(E/p) function leads to the horseshoe shape of
the function. The shape can be thought of as having two branches in the upper and
lower part of the horseshoe shape with the boundary point between them being the pd
minimum. This minimum value of pd is also the position of the maximum of the
pressure normalized Townsend coefficient and ξ(E/p). The upper branch is thus due
to the drop of the Townsend coefficient with increasing E/p and represents the border
between multiplication (below) and runaway regions (above). While it has been
shown through simulation that the upper branch reasonable predicts the runaway
threshold, it does not predict the percentage of the total population that are in the
Texas Tech University, Jordan Chaparro, August 2008
22
runaway state. The lower branch corresponds to reduction of the Townsend
coefficient and represents the boundary between electron multiplication (above) and
electron drift with insufficient energy for multiplication (below). A third branch may
be considered if relativistic effects are considered (see Helium curve in figure 2.10).
This branch is due to the increase in ionization cross sections at high energies due to
relativistic effects.
Recent efforts out of Texas Tech University have also led to findings about the
ultra-fast breakdown process. In the research reported in [43] and [44], pulses from a
RADAN pulse system are tested across various volume gap conditions in sub-
atmospheric argon and dry air. Voltage pulses with 150 ps risetimes, 300 ps FWHM,
and up to 180 kV amplitudes are applied to the gap leading to various measurements
of breakdown metrics. In addition, streak photography of the process is used to reveal
discharge structure for the event. In Figure 2.11, streak images for various pressures
are given with the slit aperture of the camera in the plane of the discharge. Results are
shown with spatial dimensions in the vertical direction and time in the horizontal with
a streak speed of 50 ps/mm. It can be seen that a thin layer, less than 200 µm in width,
develops along the surface of the cathode with high levels of luminosity representing
the primary ionization front. Diffuse, low level luminosity (two orders of magnitude
less intense), covers the remainder of the gap space as a result of the secondary
processes emerging from runaway electron beams. Optical estimates of the formative
time can be taken from the slope of the luminosity which shows average gap transit
velocities of 5 x 108 to more than 10
9 cm/s which correlate to formative times on the
order of hundreds of ps. After the build up phase, longer duration streak imaging
showed that the structure remained stationary for 100s of ns with decreasing
luminosity over time.
Texas Tech University, Jordan Chaparro, August 2008
23
Figure 2.11 - Streak camera imaging of discharge with the slit aperture parallel to the
axis of the discharge. Results show intense regions of luminosity near the cathode
indicating highly localized ionization processes. Gap width is 1 mm.
With the slit aperture in the opposite orientation, perpendicular to the axis of the
discharge, the nature of channel expansion and tendency for multiple channel
formation was investigated. Figure 2.12 shows results from four pressures. The
channel expansion was found to last about 200 ps with expansion velocities
determined to be around 2.5 x 108 cm/s. The tendency towards the development of
multiple discharge channels was found to increase with pressure.
Texas Tech University, Jordan Chaparro, August 2008
24
Figure 2.12 - Streak imaging with the slit aperture perpendicular to the discharge axis.
The channel width expansion and development can be seen clearly and increased
multichannel probability is observed with increasing pressure. Gap width is 1 mm.
Also developed, was a simple force based model intended to roughly describe
electron kinematics under the influence of collisions. From the model, rough
approximations of electron arrival energies and multiplication rates versus pressure
were attained. Electron motion was modeled simply as:
ce dt
dv
m
eE
dt
dv
−= , (2.15)
where the subtracted acceleration term is from collisional losses. This average
collisional loss term is described in terms of both inelastic and elastic components.
The elastic component is given by:
Texas Tech University, Jordan Chaparro, August 2008
25
2vndt
dx
dx
dv
dt
dvel
elastic
⋅⋅==
σ , (2.16)
and inelastic loss is:
in
i
inelastic
nm
E
dt
dvσ⋅=
, (2.17)
where n is gas density, σel and σin are elastic and inelastic momentum transfer cross
sections respectively, and Ei is the excitation or ionization energy. The model
predicted runaway conditions for reduced electric fields in argon of Ec/p = 2.23 x 103
V/(cm torr), which is nearly four times higher than the condition as predicted by
equation 2.10 which neglects elastic scattering all together. The essential results of the
model calculation are presented in Figure 2.13.
Figure 2.13 - (Left) Calculated electron arrival energies as a function of pressure and
pulse amplitude. (Right) Average number of ionizing collisions in the transit of a 1
mm gap as a function of pressure and applied voltage as determined by simple force
model [43].
The left-hand plot in Figure 2.13 shows the arrival energy of electrons at the anode as
a function of pressure for different applied amplitudes. The curves show slight
pressure dependence in the runaway regime and quickly drop off when transitioning to
the multiplication region. The right-hand plot represents primary ionizations from
electrons starting at the cathode with zero velocity.
Texas Tech University, Jordan Chaparro, August 2008
26
CHAPTER 3
NUMERICAL MODEL
Numerical methods for charged particle simulation have been employed since the
1960s. Such models employ statistical representation of particle interactions with
fundamental equations, allowing for simulations to retain much of the underlying
physics of the phenomena. The model discussed in this paper utilizes particle-in-cell
(PIC) techniques with Monte Carlo (MC) sampling to model the picosecond
discharge. In this chapter, the fundamental techniques used to implement the model
will be reviewed.
Development and Operating Environment
The model, from the beginning has been developed in a UNIX environment with
early prototyping and testing of individual program modules done in Python, which
was chosen for its high level nature and numerous visualization tools. Due to speed
limitations of the language, the complete build of the program was done in C++. The
code is designed with object-oriented concepts and a modular focus with custom
classes built around STL data types. In addition, the NIST designed, Template
Numerical Toolkit (TNT) package has been used for some linear algebra algorithms.
There have been numerous implementation of the code progressing from model
progressing from an early 1D-1V implementation to the current 3D-3V relativistic
version. Eventually, the code was adapted for operation in a parallel environment
using Message Passing Interface (MPI) protocol.
The parallel environment is a 16 node Beowulf cluster located in the Physics
department at Texas Tech. The cluster has 64 gigabytes of aggregate RAM and 640
gigabytes of total disk space. Each node uses Intel Core 2 Quad Q6600 2.4 GHz
Texas Tech University, Jordan Chaparro, August 2008
27
processors and the nodes are connected through a Gigabit Ethernet. The Open-MPI
environment, using MPI-2 standards, is the installed MPI interface.
The limitation on particle trajectories that can be tracked is based on memory
constraints and is reached well before the densities for the breakdown condition are
reached. The particle limit, for both electron and ion species, has been set to 106.
When this limit is reached, half of the particles are discarded randomly and the charge
of the remaining particles is doubled. The simulation is then able to continue running
with impaired resolution affecting primarily the low population regions of the
distribution.
Particle-in-Cell Implementation
The PIC method is a particularly well established technique for modeling plasma
structures and is described in detail in publications by both Birdsall and Langdon [45]
and Hockney and Eastwood [46]. At the most basic level, PIC methods distribute the
charge of each particle in the simulation to grid points through some defined
weighting scheme, and then the electric field is calculated, usually through the
solution of Poisson’s equation, where the charge density on the grid serves as the
source term. Particle motion can then be integrated over the time step using the
derived field values and the process repeats. Additional emission, absorption, and
collisional mechanism can be simply integrated into the cycle to expand on the
complexity of the model. The general flow of the model, over a single timestep, is
given in Figure 3.1.
Texas Tech University, Jordan Chaparro, August 2008
28
Figure 3.1 – Operational flow of time step in PIC model.
Meshing
One of the key requirements to ensure accuracy in PIC simulations is maintaining
sufficient mesh resolution. In particular, a plasma with a Debye length less than the
mesh spacing cannot be accurately simulated. If this is the case, charge separation
effects within the cell are not reflected in the calculated field leading to accumulation
error in the model. From the streak camera images, presented in Figures 2.11 and
2.12, it is obvious that the charge distributions in the gap are highly non-uniform. In
the fraction of the gap space where the charge concentration is high, the Debye length
is much shorter than in diffuse space. Obtaining sufficient mesh resolution using an
equispaced grid requires a large number of grid points which results in wasted
computational effort in regions where densities are low. In a 3D simulation, where the
number of grid points can be very high, using an equidistant scheme to model a non-
uniform plasma is not feasible. One way of resolving this issue is to employ a grid
utilizing non-equispaced points, which leads to a mesh where the resolution is
adaptively set to be sufficiently high in regions with dense charge formations and low
in volumes with low charge populations. The decision to use an adaptive mesh leads
to complications in the implementation of the Poisson solver which will be discussed
in the next section.
Texas Tech University, Jordan Chaparro, August 2008
29
Multigrid Poisson Solver
The solution of the space-charge field is the basis of the PIC formulation. The
earliest PIC versions used basic electro-static relationships to solve the fields on the
grid which required O(N2) operations, where N is the total number of grid points.
With progression from 1D models, this quickly became inadequate and numerical
methods for solving Poisson’s equations were developed. Most commonly, fast
Fourier transformation (FFT) of a finite difference discretization of Poisson’s equation
is employed which improves runtime to O(N*log(N)). While meshless, non-
equispaced Fourier methods have been developed recently [47], the standard
formulation is invalid on adaptive grids.
Multigrid methods are iterative schemes for solving differential equations by
progressing through a series of progressively coarser discretizations. They are well
suited to handling adaptive non-equispaced meshes and have runtimes on the same
order as Fourier methods. A detailed description of Multigrid techniques can be found
in [48]. The essential idea behind the technique is similar to standard relaxation
methods. In such a problem, if the boundary elements are known, a guess can be
made in the rest of the grid space and the discretization of the PDE can be iteratively
solved many times until the solution is converged upon. The principle issue with
simple relaxation is that once the high frequency error modes are eliminated, future
convergence is very slow. This is addressed in Multigrid methods by transferring the
residual error of a solution to a coarser discretization with fewer grid points. As seen
in the very simple 1D case portrayed in Figure 3.2, transferring to a coarse grid by
linear interpolation can lead to a relative shift in the frequency of the error mode
allowing for relaxation iterations to be effective again. Once the error has been solved
for on the coarser grid, it can be interpolated back as a correction factor to the finer
grid. Repetition of these steps, relaxation, restriction to coarse grids, and interpolation
to fine grids, is the basis of all Multigrid schemes.
The non-equispaced Multigrid solver for the model developed here is similar to
those developed by Pöplau [49], [50], [51].
Texas Tech University, Jordan Chaparro, August 2008
30
Figure 3.2- Restriction from a 12 element 1D grid to 6 element grid resulting in a
relative shift in error mode frequency [47].
Poisson’s equation for this problem is given as:
,3
0
R⊂Γ=∆− inερ
ϕ (3.1)
with boundary conditions on perfectly conducting walls resulting from a known
potential g:
,1Γ∂= ongϕ (3.2)
and on free space boundaries:
,0 2Γ∂=+∂∂
onrn
ϕϕ (3.3)
where ρ is the charge density, ε0 is the dielectric constant, φ is the potential, and r is
the radius from the center of the simulation space to the boundary point [50]. The
Texas Tech University, Jordan Chaparro, August 2008
31
domain Г is discretized along the x, y, and z axis in Nx, Ny, and Nz subintervals. The
length of each subinterval, for example on x, is given by hx,0, hx,1, … , hx,Nx-1.
Furthermore, average spacing between two neighboring intervals is introduced as:
=
−=+
=
−
x
ix
x
ixix
ix
Nih
Nihh
h
,0,2
1,...,1,2~
,
,1,
, , (3.4)
with analogous spacing defined for y, and z.
In the general case, the discretization of the second order derivative with second
order finite difference is given by:
)~
(~),,(),,(2
~),,(),,( 2
,
,,
1
1,,1,,
1
2
2
ix
ixix
kji
ixix
kji
ixix
kjikjihO
hh
zyx
hh
zyx
hh
zyx
x
zyx++−≈
∂
∂ +
−−
− ϕϕϕϕ. (3.5)
With non-equidistant grid spacing, and in three dimensions, the discretization of
Poisson’s equation is:
kjikzjyix
kji
kz
kji
kzkz
kji
kz
jyix
kji
jy
kji
jyjy
kji
jy
kzix
kji
ix
kji
ixix
kji
ix
kzjy
fhhh
hhhhhh
hhhhhh
hhhhhh
,,,,,
1,,
,
,,
,1,
1,,
1,
,,
,1,
,
,,
,1,
,1,
1,
,,
,,1
,
,,
,1,
,,1
1,
,,
~~~
1111~~
1111~~
1111~~
=
−
++−+
−
++−+
−
++−
+−
−−
+−
−−
+−
−−
ϕϕϕ
ϕϕϕ
ϕϕϕ
, (3.6)
for i = 1,… , Nx-1, j = 1,…,Ny-1, k = 1,…,Nz-1. This is sufficient for Dirichlet boundary
conditions where the boundaries are not operated on. For the free space boundary
condition, the evaluation of the derivative term requires evaluation with the boundary
Texas Tech University, Jordan Chaparro, August 2008
32
point. This is enforced by adding additional equations to the linear system for points
on the boundary given by:
−
+ + kji
x
kji
x
kzjyhrh
hh ,,1
0,
,,
0,
,,
111~~ϕϕ , (3.7)
for left hand boundary points and:
−
+ −
−kji
Nxx
kji
Nxx
kzjyhrh
hh ,,1
1,
,,
,
,,
111~~ϕϕ , (3.8)
for right hand points.
The relaxation scheme used by the solver is Gauss-Seidel with SOR (Successive
Over Relaxation). The optimal SOR parameter for this implementation has been
determined through trial and error.
Another important consideration for non-equispaced Multigrid is the method of
restriction to coarse grids. In equispaced grids, a typical means of restriction is simply
removing alternating lines from the grid. This does not work well with non-
equispaced meshes because the convergence speed of the solver decreases
dramatically when mesh elements have high aspect ratios. Thus, the goal of restriction
on a non-equispaced grid is to move to coarser grids with descending aspect ratios.
This is accomplished through a method called semi-coarsening. Typically, a
minimum mesh spacing hmin is determined for each particular axis. If removing a line
from the grid would make the new cell more than a certain factor of the minimum
spacing, it is not removed. The results of such an algorithm can be seen in the 2D
grids in Figure 3.3.
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Figure 3.3 – Semi-Coarsening method for progressing from fine to coarse grids with
descending aspect ratios.
The 3D version of the implemented solver was tested with the following routine.
30,000 macro-particles, with a total charge of -1 nC, were randomly positioned in a
sphere with a 1 mm radius at the center of the simulation space (i.e. x=0, y=0, z=0).
The simulation space was defined to be 10 mm long in each axis and centered around
zero. The non-equidistant grid spacing was defined with a hyperbolic sine function
producing grid spacing like those seen in Figure 3.2. The results of the test are
directly comparable to the analytical solution for the potential profile of a uniformly
charged sphere, which is well known and is given by:
−⋅
⋅⋅+
⋅⋅
≥⋅⋅
=Φ
otherwisera
a
Q
a
Q
arifr
Q
r
20404
04)(
22
3επεπ
επ, (3.9)
where Q is the charge of the sphere and a is the radius. The solver was used to solve
for the potential with both free-space boundary conditions and with the boundaries
grounded as perfectly conducting planes. A 2D slice of the results of the test are
plotted in Figure 3.4. With the mixed boundary conditions, the solution converges to
the expected solution. With potential on the boundaries fixed to zero, the potential is
underestimated, illustrating the need for free space boundary conditions.
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Figure 3.4 – Result of spherical charge test with boundaries at ± 5 mm either set to
free space conditions or grounded at zero potential. The red curve shows the
analytical solution which agrees well with the free-space results from the Multigrid
solver.
Particle Trajectories
Once the field amplitudes are solved on the grid, they are linearly interpolated to
each particle and the trajectory change is determined. This is done through a standard
4th
order Runge-Kutta scheme [52] which has been determined to have machine level
accuracy for the scale of time steps used in the simulation. The force on the particle is
determined from the interpolated field as:
)( BvEqFrrrr
×+= , (3.10)
where q is the charge of the particle, E is the electric field, v is the particle velocity,
and B is the magnetic field. In this particular implementation, no external magnetic
fields are applied and self fields resulting from charge current is neglected leaving
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35
only the influence of the electric field. The relativistic equation of motion follows
from:
dt
vdmv
dt
md
dt
dF
rr
rr
γγρ
0
0 )(+== , (3.11)
which leads to:
γ0m
c
vF
c
vF
dt
vd
⋅−=
rr
rr
r
, (3.12)
where γ is the Lorentz factor, m0 is the base mass of the particle, and c is the speed of
light. The model tracks both ion and electron trajectories, but because of the relatively
immobility of the ions when compared to the lighter electrons, their trajectories are
updated far less frequently.
Monte-Carlo Collisions
For simplicity, the model takes into account only electron-neutral collisions for
argon gas. A compilation of 38 individual and lumped inelastic cross-sections ranging
from excitation to the 4s[3/2]2 state at 11.55 eV to single ionization at 15.76 eV are
used in addition with the elastic collision cross-section [27]. The model uses linear
interpolation, between the given data points from the individual cross-section files, for
collisional energies below 500 eV. The high energy tails of the distributions are
implemented as a curve fit of the of the modified Bethe relationship defined in [53] as:
+
−++= −
Dx
xCxBxAxx
)1()ln()ln()( 1σ , (3.13)
where x is the ratio of energy to threshold energy for the process with x ≥ 1.
The Monte-Carlo sampling scheme used to model collisions is based around the
null-collision method. Because computing collisional probabilities for each particle is
computationally expensive, a maximum collisional frequency, independent of energy
and position, is defined as [54]:
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36
))((max))((maxmax vExn TE
gx
σν = , (3.14)
with ng(x) representing the spatially varying target density, σT(E) the total summed
cross-sections, and v the electron velocity. Figure 3.5 shows the fictional collisional
frequencies summed to yield the null frequency.
Figure 3.5 – Collisional frequencies for three fictional processes summed to form the
maximum null frequency [54].
With the definition of maximum collisional frequency, the timestep can be chosen
for the model with the constraint requiring νmax∆t << 1 for accuracy. For the selected
time step, the total probability of a collision can be expressed as:
),exp(1 max tPT ∆−−= ν (3.15)
and the number of collisions to sample out of the total particle list is Nc = PTN. Thus a
time step selected as 10% of the mean free path would require 10% of the total
particles to be sampled for collisions on every cycle.
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When a particle is selected for a collision, a random number (R) is mapped for
collisional frequencies 0 ≤ R ≤ νmax. This random number is then compared against the
summation of individual collisional frequencies until it is exceeded. At this point the
last contributing frequency is defined as the collision type. If the random number is
never exceeded a null-collision is assumed and there is no modification to the particles
trajectory.
For all collisions, either elastic or inelastic differential cross section data [55] for
electron-argon collisions up to 300 keV is used to determine the scattering angle and
final velocity for the electron. To do this, the differential cross sections are
integrated, normalized, and compared against a randomly generated parameter R =
[0,1). The scattering angle for several such cross sections is plotted against R in
Figure 3.6. From the plot, the forward scattering tendency of high energies can be
observed. The change in energy for a particle with respect to a stochastically sampled
scattering angle θ was given in Equation 2.6.
Figure 3.6 – Scattering angle as a function of a random element R for a variety of
incident electron energies.
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For excitation processes the final energy of the electron can be estimated by
additionally subtracting the threshold energy for the particular process as Ef = Ei - ∆E -
Eth. For ionizations the energy of the ionized electron must also be subtracted from
the initial energy and is determined from the following expression proposed by Green
and Sawada [56]:
−⋅=
B
EERBE izi
2arctantan2 , (3.16)
where B is an empirically derived factor that is known for many gasses ( Bargon =
10 eV). The relationship, normalized to the maximum energy E2 can attain, is plotted
against the random factor in Figure 3.7. From the plot it can be seen that for
increasing energies, the impacting electron tends to retain a greater percentage of the
initial energy.
Figure 3.7 - Normalized secondary electron energy as a function of a random variable
for several incident energies.
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Fowler-Nordheim Emission
Current emission is implemented using equation 2.5 and the calculated normal
electric field on the cathode surface given from the Poisson solver. The equation
when used in conjunction with space charge field calculations near the cathode in
vacuum conditions has shown reasonable agreement to the Child-Langmuir (CL) law
for space charge limited current density. Figure 3.8 shows a plot with the FN curve,
the CL curve, and the steady state results from the PIC model for current emission
under vacuum conditions. FE processes have been similarly modeled by Feng and
Verboncoeur to investigate the transition to space-charge limited currents [57].
Figure 3.8 - Results from FN emission model. Shown is the FN curve (red) the Child-
Langmuir relationship for space charge limited current (blue) and points (magenta)
representing steady state current magnitudes from the numerical model as a function
of field amplitude.
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CHAPTER 4
EXPERIMENTAL SETUP
The experimental setup used for testing picosecond discharge is based primarily
on an oil-filled coaxial transmission line system with field shaping lenses. The line is
connected to a pulse generator joining a test gap in a controllable atmosphere and is
terminated by a second transmission line of similar length with a matched resistive
load. The schematic in Figures 4.1 illustrates the assembly of the setup. In this
chapter, the specific components of the system will be reviewed.
Figure 4.1 - Oil-filled coaxial setup (not to scale).
Pulse Forming System
A RADAN 303A high voltage pulser along with an SN4 pulse slicer, served as
the source for the system. The pulser-slicer combination is capable of providing 20-
150kV pulse amplitudes with rise times as low as 150 ps and pulse durations from 250
to 1500 ps. The nominal impedance of the pulser is 45 Ω. The 303A pulsed source is
based on an oil-insulated Blumlein line charged by a Tesla pulse transformer [58].
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The Tesla transformer is supplied from energy stored in two low inductance capacitors
(40µF, 750V) and is switched through a fast thyristor (current rate up to 5kA/µS). The
load of the Blumlein line is connected to a high-voltage pressurized switch which
produces a ~5 ns pulse with a 1 ns rise time that is coupled into the SN4 slicer which
features two adjustable high-pressure spark gaps intended to reduce the risetime and
width of the pulse [59]. The highly pressurized axial gap, known as the peaking gap,
controls the risetime of the pulse while a radial gap, termed the chopping gap, limits
the width of the pulse. The gap is designed to operate at 4 MPa with nitrogen gas.
Figure 4.2 illustrates the orientation of the SN4 pulse slicer spark gaps.
Figure 4.2 - Schemata of pulse slicer assembly. Adjustable peaking and chopping
gaps are used in high pressure nitrogen to reduce the risetime and FWHM of the
generated pulses [59].
Transmission Line System
An oil-filled coaxial transmission line system has been designed for the system in
order to accommodate geometric matching from the geometry of the transmission line
to the biconical shape of the test gap housing and to limit undesired reflections. The
previously used apparatus utilized a 1.2 m or 8 m section of RG-19 coaxial cable (52
Ω) between the pulser and the test chamber. A 15 m section of RG-19 cable
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terminated with a 50 Ω resistive load served as a matched load on the opposite side of
the chamber. At the point of connection between the RG-19 cable and the test
chamber, there existed a tapered section to transition from the inner and outer
conductor dimensions of the cable to those needed for the biconical test gap. While at
the end of the transition section the impedance was constant with respect to the RG-19
cable, the tapered stages leading up to the larger dimensions led to many undesirable
reflections in the transmission line which clouded reflected data from the test gap.
This was confirmed through a finite element simulation of the geometry of the line.
Additionally, the geometry inside the test chamber abruptly transitioned from a
coaxial to biconical design which distorted the pulse, slowing the risetime at the gap.
These issues presented by the RG-19 based system made the creation of a new
system desirable. The introduction of a refined setup also presented the opportunity to
attempt to improve rise times and pulse distortion in the gap. The geometric mismatch
at the coaxial to biconical transition along with the transition from oil to gas causes
distortion of the wave and degradation of the risetime. To prevent this type of wave
distortion, the newly created setup was designed to feature a hyperbolic shaping lens
at the coaxial to biconical transition with the intention of converting the planar waves
in the coaxial section to a spherical wavefront matched to the dimensions of the
biconical section as seen in Figure 4.3.
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Figure 4.3 - Hyperboloidal Rexolite lens transitioning from planar wavefront in
coaxial geometries to spherical fronts for the biconical section. The curved surface at
the oil, Rexolite boundary is intended to lengthen the surface path to suppress possible
flashover events.
The curvature of the hyperboloidal lens face was determined using a simple
relationship between the electrical path-length through the lens and the length from the
face of the lens to the center of the gap [60]. In order to match the lens from the
coaxial dimensions to those of the biconical section properly, all rays must have the
same electrical length. Figure 4.4 illustrates the condition that governs the shape of
the lens.
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Figure 4.4 - Determination of the curvature of the lens by comparison of electrical
path lengths.
The condition stated above for the curvature of the lens face was combined with
equations governing the impedance of a biconical section to obtain the dimensions of
the biconical gap [60].
In order to facilitate the desired improvements in both wave dispersion and
distortion in the transmission line setup, the RG-19 coaxial cable was replaced with
oil-filled copper piping with copper rod inner conductors suspended by spacers within
the pipe. The dimensions of the pipe’s inner diameter (D = 37.59 mm), the inner
conducting rod’s diameter (d = 11.67 mm), and the permittivity of transformer oil (εr =
2.30) were all chosen such that the line impedance would closely approach that of the
pulser (45 Ω).
Ω=
= 23.46log*138
d
DZ
rε, (4.1)
The spacers used to center the inner conductor and the hyperbolic lenses were
made out of Rexolite (εr = 2.50). Tests of the completed transmission line were
carried out using a Spire Pulser capable of 500 ps rise times and amplitudes up to 1
kV. Figure 4.5 shows an example pulse from the Spire pulser.
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Figure 4.5 - Output of Spire Pulser with 1kV amplitude
The transformer oil was cycled through the line through a vacuum system to
remove any air dissolved in the oil. The line was connected to the pulser through a 50
Ω coaxial cable. Results, measured from reflected amplitudes from the cable to oil-
filled line transition, showed an approximate characteristic impedance of 46.8 Ω
which closely matched the predicted impedance.
The tapering section used to transition to a larger dimensions in the RG-19 based
system was removed and the copper piping was directly coupled to the test chamber
with a sealing system consisting of a flange with a beveled inner edge that clamped
down on an o-ring around the pipe. This allowed for a uniform geometry from the
pulser output to the biconical gap section while maintaining the controllable
atmosphere within the test chamber. The pipes that were used to couple into the test
chamber are both open ended and are sealed by the lenses which have o-rings that seal
against both the inner surface of the pipe and the surface of the inner conductor. The
lens is held in place on the inner conductor by the beginning of the inner conductor’s
biconical taper as seen in Figure 4.6.
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Figure 4.6 - Lens fitting on tapered inner conductor section with two sealing o-rings.
Between the open ended pipes inside the test chamber fits the biconical gap
section. There are two different biconical sections that fit between the two open ended
pipes inside the test chamber, one designed for an axial gap and another for a radial
gap. The design of both biconical sections is similar. The inner conductor taper
begins at the lens face and tips of different lengths can be attached to vary the gap
distance. To preserve the angle of the biconical section, all electrodes feature the
same angle of taper to a point where they become straight until the desired electrode
length is reached. The axial electrodes have a 3 mm diameter hemispherical tip. In
the case of the radial gap, a copper tube section with a shaped electrode is slid over the
straight portion of the axial electrodes and is held in place against the beginning of the
taper on the electrodes. A quarter-inch hemispherical electrode serves as the anode.
The gap distances for both orientations are adjustable and range from 1 to 3 mm. The
orientation of both electrode configurations can be seen in Figure 4.7.
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Figure 4.7 - Axial and radial electrode geometries for biconical gap assembly. Gaps
may be varied from 1 - 4 mm for both configurations.
For tests conducted to determine the statistical delays, resulting from picosecond
risetime fields with maximum amplitudes near the threshold for field emission, larger
gap distances were required. In order to accommodate this, a third gap assembly was
created that remained coaxial throughout. This setup can be seen in Figure 4.8, and
allowed for gap spacing, with a radial configuration, between 1 and 11 mm. New
“lenses” were designed to serve primarily as end-caps for the oil section and were
designed with flat faces.
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Figure 4.8 - Gap assembly for testing statistical breakdown delays near the FN
threshold. Gap spacing of up to 11 mm can be accommodated yielding fields an order
of magnitude lower in amplitude than those with the biconical assembly.
The outer conductors of each of the designed gap assembly sections are made of
aluminum. They all feature front and rear quarter-inch viewports and a quarter inch x-
ray port drilled at 24˚ off the vertical axis pointing at the anode. In the case of the
radial gap, the front and rear viewports are drilled at a 13.24˚ angle from the horizontal
axis and the entire body of the outer conductor is tilted by this angle to align the
angled ports with the front and rear viewport windows. The reasoning behind this
design was that when the radial gap was perfectly aligned on the horizontal axis, the x-
ray viewport’s view of the anode was blocked by the cathode. By rotating the
orientation of the gap it was possible to see the surface from the 24˚ angle off the
vertical axis needed to align the viewport with the correct ports on the test chamber.
Figure 4.9 below illustrates the assembled radial biconical section with the various
viewports.
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Figure 4.9 - 3D view of gap assembly and optical viewports for imaging and x-ray
analysis.
The load for the system is an oil-insulated three branched resistor tree within the
final section of pipe on the opposite side of the test chamber from the pulser. Each of
the three branches in the load consists of nine 3-watt ceramic resistors connected in
series over 21 cm. The end of each branch is soldered to a brass disc that makes
contact against the inside of the pipe and is connected to a copper rod that extends
through a swagelok on the sealing back-plate of the terminating pipe section. The rod
allows for adjustments to the position of the inner conductor on the side of the load.
Figures 4.10 and 4.11 show the resistor tree and fully constructed images of both the
load and pulser side of the oil-filled setup. Details on the characterization of the
transmission line system through finite element simulation can be found in previous
works by Chaparro [61].
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Figure 4.10 - Tri-branch resistive load for termination of the transmitted side of the
coaxial line. Nominal impedance is 46 ohms and reflections from the load are less
than 10%.
Figure 4.11 - Images of constructed incident (left) and transmitted (right) oil-filled
coaxial transmission lines coupled to the experimental test chamber.
Experimental Chamber and Vacuum System
A large bell jar is used to control the atmosphere in the test gap. The chamber
features twelve 3 inch ports at its base which are utilized for a mechanical vacuum
pump, a turbo-molecular pump, three different pressure gauges for varying ranges
from rough vacuum to atmosphere, and an ion pressure gauge tube which is used for
high vacuum measurements. The mechanical vacuum pump can bring the chamber to
a rough vacuum of several mtorr and the turbo-molecular pump is used to bring the
pressures down to high vacuum (10-6
torr). Evenly spaced around the center of the
bell jar, there are two 8 inch ports and two 6 inch ports. Two of the ports are used to
couple the copper pipe transmission line into the bell jar while the other two are used
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as viewports for optical diagnostics. At the top of the bell jar are four 6 inch ports
each at an angle of 24 degrees from the vertical axis pointing at the center of the bell
jar. These ports are used for the photo-multiplier tube based x-ray diagnostic system.
The specific port used varies depending on the configuration of the gap. Figure 4.12
shows an image taken of the bell jar and illustrates the port orientation.
Figure 4.12 - Experimental test chamber. The front optical window is in the center of
the image.
Capacitive Voltage Dividers and Digitizers
Capacitive voltage dividers with rise times of <100 ps and divider ratios on the
order of 103 were used to capture signals on both the incident and transmitted sides of
the test gap. The distance from the dividers to the gap was sufficient to separate
incident signals from reflected ones. In the oil-filled system, the dividers were built
into short, 10 cm pipe sections that were fit between the longer sections going to the
test gap and the pulser or load. Devcon 2 ton epoxy adhesive, has been tested over 12
months in transformer oil to determine its resistance to the oil and has shown no signs
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of degradation. It was used to adhere the insulation and metallic shim to the inside of
the pipe. These dividers were positioned 1.1 m from the gap and calibration showed a
250:1 divider ratio. A schematic representation of the divider sections can be seen in
Figure 4.13.
Figure 4.13 - Schematic view of capacitive voltage divider. Total area of copper shim
along with the dimensions of the line determine the capacitances shown which yield
sub 100ps risetimes and > 10 ns fall. The divider ratio is 250:1.
The capacitive voltage divider signals are recorded with a Tektronix TDS 6604
digitizer (50 ps sampling period, 6 GHz bandwidth). An HP Infinium 5825A digitizer
(500 ps sampling period, 500 MHz bandwidth) was used to record the photo-multiplier
output for x-ray detection.
X-ray Detection and Luminosity Measurements
X-ray emission from the anode is measured by means of a photo-
multiplier/scintillator combination positioned to have line of sight of the anode
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through viewports drilled in the outer conductor assembly. The photo-multiplier used
is a Hamamatsu R1828-01. The R1828-01 features a spectral sensitivity that peaks at
420 nm and extends across the visible range. It has a 1.3 ns risetime and a gain up to
3.0 x 107 when supplied with a 2.5 kV source. The PMT is suspended approximately
5 inches above the test gap. Between the PMT and the gap sits a metallic foil holder, a
NE102A equivalent scintillator, a Lexan window that serves as a vacuum seal, and a
neutral density filter that attenuates the visible light coming from the scintillator.
Figure 4.14 shows the schematic view of the x-ray detection system. The front face
surrounding the view port of the system is shielded with a ring of 5 mm thick lead to
prevent undesired radiation from bypassing the test foil.
Figure 4.14 - 3D cutaway view of PMT assembly for measuring x-ray emission from
the anode. The assembly sits in a test chamber view port angled at 24 degrees from
the vertical axis and corresponding viewports have been drilled into the gap assembly.
Using the conversion and transmission efficiencies of the system, it is possible to
derive an approximate ratio of PMT output to a single x-ray photon of known energy.
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Using this method with the measured x-ray intensities it is possible to estimate the
number of electrons hitting the anode in a known energy range as well as the radiation
intensity output from the anode [62].
A solid angle Ωc-sr of 0.083 sr was determined from the physical dimensions of
the x-ray viewport. By dividing by 4π, the radiation percentage seen by the scintillator
can be determined. The Bicron BC-400 plastic that was used for scintillation has a
fluorescence efficiency, ηfe, of 3%. This efficiency allows the following conversion to
be made between the numbers of visible photons produced from a single x-ray photon:
visible
rayx
feE
EN
−=η
where N is the number of visible photons produced, Ex-ray is the energy of the x-ray
photon, and Evisible is the energy of the visible photon. The refractive index of the
scintillator is 1.58 which yields an output efficiency by internal reflection of ηoe = 0.11
[19]. The energy conversion efficiency, δ, for Bremsstrahlung radiation can be used to
estimate the percentage of gap electrons at a specific energy that yield x-rays by:
ZE910−=δ
where Z is the atomic number of the metallic anode and E is the kinetic energy of the
electron in eV. The previously presented transmission coefficient, ηaf, for the absorber
foil must also be taken into account. The final constants that must be known are
related to the operation of the PMT tube and are the quantum efficiency of the
photocathode (ηpmt = 0.1), the gain of the PMT (α = 3 X 107), and the ratio of
scintillator visible light output that is coupled to the PMT (Ωsc-pmt = 0.1). Thus the
ratio, η, of an x-ray producing electron to electrons at the PMT-output can roughly be
equated as:
pmtpmtscoeafscx N αηηδηη −− ΩΩ= .
In Table 4.1, the relative sensitivity per photon at a given energy is given for
each absorber foil.
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Table 4-1 - Relative photon sensitivity for 3 absorber foils.
Photon Energy [keV] Aluminum
Copper
Silver 30 558.72 48.70 0.00
60 2245.17 1573.73 108.95
80 3993.05 3369.39 993.89
100 6240.20 5636.43 2886.90
150 14042.91 13372.51 10569.76
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CHAPTER 5
RESULTS
The results of both experimental and numerical efforts will be presented in this
chapter. From experimental efforts, measured data is used with an equivalent circuit
model to obtain current waveforms which are used to produce formative delay
estimates for a variety of pressures and pulse amplitudes. These are directly compared
with numerical results for the same metrics and scaling comparisons are made against
results from previous research. In addition, the model is used to explore the
dependence of the formative time on a number of gap parameters.
The spatial development and energy distribution of electrons in picosecond
discharge is investigated through the numerical model and results are compared to
previous streak camera data and experimental x-ray analysis of the breakdown event.
The model is used to examine the development of space-charge structures and the role
they play in the discharge mechanisms and the transition from FE to EEE.
Finally, experimental results from picosecond discharges with overvoltage ratios
an order of magnitude lower than those previously tested are investigated to examine
the statistical variation of fast discharge at the field emission threshold.
Radial Volume Breakdown at High Overvoltage
Picosecond discharge, across 1 mm radial gaps, depicted in Figure 4.7, is the
primary focus of this research. Overvoltage ratios of 80-800 were obtained from the
application of voltage waveforms with 150 ps risetimes and amplitudes from 40 – 150
kV. The FWHM of the pulses is no more than 300 ps and discharges were studied in
pressures from 10-5
– 600 torr. Waveforms from the RADAN pulser, seen in Figure
5.1, have shot to shot variation in amplitude and risetime of no more than 10%. For
analysis of the electrical characteristics of the discharge, the incident and transmitted
side capacitive voltage dividers were the only sensors employed. Subsequent analysis
from this raw data through lumped-element modeling techniques was used to discern
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other electrical properties of the discharge. Risetime degradation in the transmission
line system, due to geometrical matching imperfections slows the front of the pulse by
no more than 10-20 ps at the gap, as shown by COMSOL simulations [61].
Figure 5.1 - Measured voltage output for numerous shots from the RADAN pulser.
Amplitude and risetime variation are less than 10%.
Equivalent circuit model
In order to discern the characteristic properties of the discharge, current
waveforms are required in addition to the experimentally measured voltages. This is
accomplished by applying Kirchhoff’s voltage law to a lumped element model of the
gap with the known waveforms from the incident and transmitted sensors. With the
radial orientation of the electrodes, the equivalent circuit model of the gap, seen in
Figure 5.2, can be simply implemented as a capacitor bridging the outer and inner
electrodes with a current source representing the discharge current in parallel.
Kirchhoff’s voltage law defines V
side, Vc is the voltage across the ca
Similarly, currents can be defined for the model as:
where I1 is the current on the incident side, I
I is the current conduction current through the gap.
Figure 5.2 - Lumped element model for gap.
Transmission line equations relating the incident (V
to the input voltage (V
and
where Z is the gap impedance.
system, which have been investigated in detail in [
signal, it is more convenient to express these relationships in terms of
transmitted waveforms
where the 0 superscript represents the “no breakdown” case at high vacuum where
very little current is lost through the gap. The gap capacitance can be found from the
equation:
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Kirchhoff’s voltage law defines V1 = Vc = V2, where V1 is the voltage on the input
is the voltage across the capacitor, and V2 is the transmitted voltage.
Similarly, currents can be defined for the model as:
21 IIdt
dVCI ++⋅= , (5.1)
is the current on the incident side, I2 is the current on the transmitted
I is the current conduction current through the gap.
Lumped element model for gap.
Transmission line equations relating the incident (V→) and reflected (V
(V1) and current (I1) and can be written as:
←→ += VVV1 , (5.2)
Z
VVI
)(1
←→ −= ,
where Z is the gap impedance. With geometric reflections in the transmission line
system, which have been investigated in detail in [61], convoluted with the reflected
it is more convenient to express these relationships in terms of
transmitted waveforms. Because Vc = V2, the discharge current can be written as:
)()(2
2
0
22
0
2 VVdt
dCVV
ZI −+−= , (5.4)
where the 0 superscript represents the “no breakdown” case at high vacuum where
rrent is lost through the gap. The gap capacitance can be found from the
Texas Tech University, Jordan Chaparro, August 2008
is the voltage on the input
pacitor, and V2 is the transmitted voltage.
, (5.1)
is the current on the transmitted side, and
reflected (V←) signals
, (5.2)
, (5.3)
transmission line
with the reflected
it is more convenient to express these relationships in terms of incident and
, the discharge current can be written as:
, (5.4)
where the 0 superscript represents the “no breakdown” case at high vacuum where
rrent is lost through the gap. The gap capacitance can be found from the
Thus, approximated current waveforms can be obtained from just transmitted and
incident waveforms and ref
Figure 5.3 shows sample transmitted waveforms for the full pressure range and a
number of pulse amplitudes.
raw transmitted data, but these relationships can be better e
breakdown voltage and formative delay time analysis
waveforms resulting from the lumped model can be seen in Figure 5.4.
noteworthy, that for nearly all pressures, the current pulse
edge of the waveform, limiting the maximum attained field amplitude in the gap.
Figure 5.3 - Measured transmitted voltages resulting from breakdown from four
different pulse amplitudes and across the full rang
Texas Tech University, Jordan Chaparro, August 2008
59
)(2')]'()'([ 0
2
0
0
2tZCVdttVtV
t
=−→∫ , (5.5)
Thus, approximated current waveforms can be obtained from just transmitted and
incident waveforms and reflected data can be neglected.
Figure 5.3 shows sample transmitted waveforms for the full pressure range and a
number of pulse amplitudes. An obvious pressure dependence can be observed
raw transmitted data, but these relationships can be better explained through
breakdown voltage and formative delay time analysis. Several plots of sample current
waveforms resulting from the lumped model can be seen in Figure 5.4.
noteworthy, that for nearly all pressures, the current pulse initiates duri
edge of the waveform, limiting the maximum attained field amplitude in the gap.
Measured transmitted voltages resulting from breakdown from four
different pulse amplitudes and across the full range of tested pressures.
Texas Tech University, Jordan Chaparro, August 2008
, (5.5)
Thus, approximated current waveforms can be obtained from just transmitted and
Figure 5.3 shows sample transmitted waveforms for the full pressure range and a
pressure dependence can be observed in the
xplained through
. Several plots of sample current
waveforms resulting from the lumped model can be seen in Figure 5.4. It is
during the rising
edge of the waveform, limiting the maximum attained field amplitude in the gap.
Measured transmitted voltages resulting from breakdown from four
e of tested pressures.
Texas Tech University, Jordan Chaparro, August 2008
60
Figure 5.4 - Sample current pulses obtained from lumped element model for 4 pulse
amplitudes and a number of pressures.
Breakdown Characteristics
Breakdown voltages, given in Figure 5.5 for a variety of pulse amplitudes, can be
simply measured as the maximum amplitude of the transmitted voltage signal.
Vertical bars, representing standard deviation of a 10 sample average, are plotted for
each data point. From the plot, it appears that the increase in breakdown voltage, at
pressures approaching one atmosphere, decreases with increasing pulse amplitude.
Formative times are defined in this case as the point from the time from the start
of the voltage pulse to 10% of the impedance limited current and are plotted as a
function of pressure for several pulse amplitudes in Figure 5.6. It should be noted that
due to the fairly long sampling time of 50 ps, interpolation techniques have been used
on the raw data in the model to artificially increase resolution in the lumped model.
This does not influence the general validity of the results but in light of this along with
the inevitably inconsistent shot
of gap spacing, the limited accuracy of the resu
represents an average of 5 samples per data point with standard deviations around
15%.
Figure 5.5 - Measured breakdown voltages for radial discharges with pulse amplitudes
from 50 - 150 kV.
Texas Tech University, Jordan Chaparro, August 2008
61
the inevitably inconsistent shot-to-shot electrode conditions, and rough
the limited accuracy of the results should be kept in mind
represents an average of 5 samples per data point with standard deviations around
Measured breakdown voltages for radial discharges with pulse amplitudes
Texas Tech University, Jordan Chaparro, August 2008
rough reproducibility
lts should be kept in mind. Figure 5.6
represents an average of 5 samples per data point with standard deviations around
Measured breakdown voltages for radial discharges with pulse amplitudes
Figure 5.6 - Formative delay times from experimental results and lumped element modeling for pulse
amplitudes between 50 - 150 kV.
Modeling formative delay
Applying basic physical models to predict formative time is instructive to
understanding discharge physic
experimental efforts, investigations of simplifications and neglected
light on the processes of the p
based on existing theory of discharge will be considered along with results from the
PIC / Monte-Carlo simulation.
First, the simple force model discussed in Chapter 2 (Eq. 2.15
[63]. Results show a fairly reasonable agreement for pressures beyond the minimum
for all applied amplitudes.
in figure 5.6. The voltage ramp was modeled as a simple linear rise
amplitude over 200 ps. Breakdown condition in the model is defined as 30 collisions
per primary electron which equates to a final electron
Texas Tech University, Jordan Chaparro, August 2008
62
Formative delay times from experimental results and lumped element modeling for pulse
150 kV. Error is roughly 15% for each data point.
formative delay
physical models to predict formative time is instructive to
understanding discharge physics. Even if a model does not perfectly describe
experimental efforts, investigations of simplifications and neglected physics
light on the processes of the phenomena. Here, two simple mathematical models
based on existing theory of discharge will be considered along with results from the
Carlo simulation.
First, the simple force model discussed in Chapter 2 (Eq. 2.15 – 2.
. Results show a fairly reasonable agreement for pressures beyond the minimum
for all applied amplitudes. Deviation is less than 20% from experimental values seen
The voltage ramp was modeled as a simple linear rise
amplitude over 200 ps. Breakdown condition in the model is defined as 30 collisions
per primary electron which equates to a final electron number of about 10
Texas Tech University, Jordan Chaparro, August 2008
Formative delay times from experimental results and lumped element modeling for pulse
physical models to predict formative time is instructive to
s. Even if a model does not perfectly describe
physics can shed
henomena. Here, two simple mathematical models
based on existing theory of discharge will be considered along with results from the
2.17) is employed
. Results show a fairly reasonable agreement for pressures beyond the minimum
erimental values seen
The voltage ramp was modeled as a simple linear rise to the applied
amplitude over 200 ps. Breakdown condition in the model is defined as 30 collisions
of about 109.
Texas Tech University, Jordan Chaparro, August 2008
63
Figure 5.7 - Calculated formative delays as a function of pressure for several pulse
amplitudes (given in kV/mm) using simple force modeling.
A second model was introduced in the previously discussed work by Yakovlenko
[42], and is based, in part, on the streamer definition for formative time as the time to
reach a charge carrier amplification of 108. An empirically derived function for
ionization rate, which was reported as being accurate to E/p ratios up to 104 V/(cm
torr), was used to describe the growth rate of the avalanche from the formula:
dtp
tEp
n
n⋅
= ∫
τ
ψ00
)()ln( , (5.6)
where n is the electron count, n0 is the initial electron number, p is the pressure in torr,
E is the electric field in V/cm, and t is time in ns. ψ represents the ionization rate
reduced to pressure with units of (ns torr)-1
. For argon, ψ is given as:
( ) ),109.128exp(102 42/18.02 xxxxAr
−−− ×−−⋅⋅×=ψ (5.7)
where x is the pressure reduced electric field in V/(cm torr). The electric field for the
model is a ramped step with a risetime of 200 ps. Results agree fairly well for small
amplitudes but diverge at high E/p ratios due to inaccuracy in the estimated ionization
rate.
Figure 5.8 - Formative delays from streamer derived m
resulting from a 200 ps risetime
Monte-Carlo estimate of
Monte-Carlo methods have
sampling techniques to ac
those where runaway elect
in chapter 4 was configured to match the test conditions as closely as possible
mm gap spacing, argon background gas, and Gaussian applied voltage pulses with
risetime and FWHM similar to the experiment
times at high E/p ratios has been shown to be highly dependent on the source of
initiatory electrons. As such, two primary parameters were adjusted to modulate the
number of free electrons produced through field emission. The first is the localized
electric field enhancement factor β, whose effect on delay times
for an applied field of 50 kV/mm, as an example.
Texas Tech University, Jordan Chaparro, August 2008
64
amplitudes but diverge at high E/p ratios due to inaccuracy in the estimated ionization
Formative delays from streamer derived model proposed by Yakovlenko
resulting from a 200 ps risetime ramped step [64].
Carlo estimate of formative delay
Carlo methods have the advantage of being able to utilize raw statistical
sampling techniques to achieve accuracy over a wide range of E/p ratios, including
those where runaway electrons play a significant role [29]. The PIC model discussed
was configured to match the test conditions as closely as possible
mm gap spacing, argon background gas, and Gaussian applied voltage pulses with
similar to the experiment. As stated in chapter 2, the formative
times at high E/p ratios has been shown to be highly dependent on the source of
electrons. As such, two primary parameters were adjusted to modulate the
number of free electrons produced through field emission. The first is the localized
electric field enhancement factor β, whose effect on delay times is shown in
ld of 50 kV/mm, as an example. Increasing β tends to shift the
Texas Tech University, Jordan Chaparro, August 2008
amplitudes but diverge at high E/p ratios due to inaccuracy in the estimated ionization
odel proposed by Yakovlenko
the advantage of being able to utilize raw statistical
hieve accuracy over a wide range of E/p ratios, including
. The PIC model discussed
was configured to match the test conditions as closely as possible with 1
mm gap spacing, argon background gas, and Gaussian applied voltage pulses with
. As stated in chapter 2, the formative
times at high E/p ratios has been shown to be highly dependent on the source of
electrons. As such, two primary parameters were adjusted to modulate the
number of free electrons produced through field emission. The first is the localized
is shown in Figure 5.9
Increasing β tends to shift the
Texas Tech University, Jordan Chaparro, August 2008
65
minimum delay time to lower pressures and the rise of delay time with pressure is
increased. The degree of the response to changing β varies with applied field
amplitudes, with lower field amplitudes showing increased sensitivity to the
parameter. While β affects the current from a single micro-point emission site, also
important is the density of such sites. This is analogous to the experimental results
reported by Mesyats [16] cf. chapter 2 where scratching a smooth electrode enough
led to a multi-electron like discharge due to the many field emission sites. Even
without obvious enhancement features like scratches, different degrees of polishing
effort would lead to different density of suitable emitters. In the absence of more
intricate techniques like modeling multiple emitters from a surface geometry profile,
as in [65], simply randomly selecting a number of micro-emission sites in a defined
area according to a prescribed density should be sufficient for adequately accounting
for different degrees of surface roughness.
Texas Tech University, Jordan Chaparro, August 2008
66
Figure 5.9 – Simulated effect of the field enhancement factor on the formative delay
times with a pulsed field amplitude of 50 kV/mm.
The best comparisons to experimental results were obtained using 250 micro-
emission sites with areas of 10-8
cm2
randomly distributed over a total area of 10-3
cm2.
Enhancement factors at the micro-sites were set at β = 300 which is not unreasonable
[21]. The resulting delay times can be seen in figure 5.10 for pulse amplitudes
between 25 kV/mm and 150 kV/mm. Results are in fairly good agreement with
experimental data in terms of delay minimum and overall magnitude with less than
20% deviation across the entire tested range. The largest departures from
experimental values are found in the pressures approaching atmosphere where the
delay time did not slow as much as indicated by empirical results. The general
characteristic of wider minimums at higher applied voltage, and increased slowing of
the delay at high pressures with decreasing amplitude are reproduced by the model.
Texas Tech University, Jordan Chaparro, August 2008
67
Figure 5.10 - Simulated formative delay times for field amplitudes between 25 and
150 kV/mm over the full pressure range.
Scaling Law
Below, in Figure 5.11, the results from experimental and simulated discharges are
plotted according to the scaling relationship pτ = f(E/p). Also plotted, is the similarity
law results for argon taken by Felsenthal and Proud [14] and simulated results from
the model where the surface of the cathode was initially seeded with 104 randomly
distributed electrons with the intention of simulating conditions brought about by UV
illumination. The general agreement, between the simulated and experimental data is
again evident. With the additionally seeded electrons, pτ values shift nearly to the
curve determined by Felsenthal and Proud as predicted by Mesyats [3]. The change in
slope at high E/p values, seen in the experimental and simulated data, can be explained
by considering the increasing number of runaway electrons. As E/p rises above 5 x
103 V/(cm torr), increasing populations of electrons reach runaway mode
effectively reduces the population of particles which can take part in amplification
processes. This leads to a
energy distribution, which i
relationship.
Figure 5.11 - Simulated (green) and experimental (blue) E/p vs pt plot. Simulated
results show reasonable agreement to experimental efforts. Red points simulate the
influence of UV illumination of
relationship towards the black line representing the accepted streamer
law determined by Felsenthal and
The increase in slope
Felsenthal and Proud.
potentially explained by simulated results that resolve
103 – 10
5 V/(cm torr). Because achieving such
pulsed fields, there is very little experimental data
Texas Tech University, Jordan Chaparro, August 2008
68
increasing populations of electrons reach runaway mode
effectively reduces the population of particles which can take part in amplification
leads to a decrease in ionization rate, due to shifts in the electron
which increases delay times and in turn the slope of the
Simulated (green) and experimental (blue) E/p vs pt plot. Simulated
results show reasonable agreement to experimental efforts. Red points simulate the
influence of UV illumination of the gap prior to voltage application and shift the
relationship towards the black line representing the accepted streamer
law determined by Felsenthal and Proud [14].
The increase in slope for high E/p is not as pronounced in the data reported by
Felsenthal and Proud. Aside from differences in the measurement technique, t
explained by simulated results that resolve ionization rates at high E/
V/(cm torr). Because achieving such E/p magnitudes requires very fast
ere is very little experimental data, even bordering on the runaway
Texas Tech University, Jordan Chaparro, August 2008
increasing populations of electrons reach runaway mode which
effectively reduces the population of particles which can take part in amplification
due to shifts in the electron
the slope of the scaling
Simulated (green) and experimental (blue) E/p vs pt plot. Simulated
results show reasonable agreement to experimental efforts. Red points simulate the
the gap prior to voltage application and shift the
relationship towards the black line representing the accepted streamer regime scaling
in the data reported by
Aside from differences in the measurement technique, this is
ionization rates at high E/p from
E/p magnitudes requires very fast
even bordering on the runaway
Texas Tech University, Jordan Chaparro, August 2008
69
regime, which gives ionization rates. Some previous efforts with Monte-Carlo codes,
have shown that the Townsend coefficient levels off and eventually decreases as E/p
extends into the runaway regime [29]. Simulated results, from the model presented in
this dissertation, seem to indicate that once beyond the threshold for runaway
production, the ionization frequency no longer scales with E/p and instead depends on
the pressure and field independently (see Figure 5.12). The method of determining
ionization rates for this test is simple. A population of N electrons is initially seeded
at the cathode. Field emission is disabled, and the number of electrons is such that no
space-charge effects arise. The electrons are accelerated in DC fields and the ionizing
collisions resulting from the initial electrons are counted over some time period
(usually ~50 ps). The averaged number of ionizing collisions over the known time
frame yields estimates of ionization frequency.
Texas Tech University, Jordan Chaparro, August 2008
70
Figure 5.12 - Simulated pressure normalized ionization frequencies plotted against
E/p. The red curve is the function for argon ionization frequencies from Yakovlenko
[42] which is claimed to be accurate for E/p up to 104 V/(cm torr). Black circles
represent empirically derived frequencies determined from the product of measured
Townsend coefficients [28] and electron drift velocities. Dashed lines represent curve
fits for simulated data resulting from specific pressures.
The effect in which increasing pressure causes reductions in ionization frequency
in the runaway regime can be explained from the electron energy distributions as seen
in Figure 5.13. For test cases with the same E/p, the higher pressure / higher voltage
case has a distribution that has a greater percentage of the electrons beyond the
ionization peak stretching out to the maximum energy allowed by the applied field.
The result is a stretching of the distribution leaving the intermediate region, where
ionization frequencies are the highest, under populated. This does not occur below the
runaway threshold because the peak of the ionization maximum is never overcome.
As a result, high values of E/p that rely on very low pressures see less of a reduction in
the ionization rate and less of an increase in slope of the scaling relationship.
number of initiating electrons also slightly reduces the increase in slope in the scaling
law at high E/p.
Figure 5.13 – Simulated electron energy distributions resulting from application of
of 104 V/(cm torr).
Geometric Breakdown
The breakdown structure has been previously experimentally investigated by
streak camera imaging and results can be seen in Figures 2.11 and 2.12. Numerical
investigation of the structure allows, not only for confirmation of the experimental
imaging, but also the opportunity to investigate the time development of space charge
fields which are not easily determined
Texas Tech University, Jordan Chaparro, August 2008
71
the ionization rate and less of an increase in slope of the scaling relationship.
number of initiating electrons also slightly reduces the increase in slope in the scaling
Simulated electron energy distributions resulting from application of
Breakdown Structure
reakdown structure has been previously experimentally investigated by
streak camera imaging and results can be seen in Figures 2.11 and 2.12. Numerical
investigation of the structure allows, not only for confirmation of the experimental
but also the opportunity to investigate the time development of space charge
fields which are not easily determined through experimental methods. The
Texas Tech University, Jordan Chaparro, August 2008
the ionization rate and less of an increase in slope of the scaling relationship. A high
number of initiating electrons also slightly reduces the increase in slope in the scaling
Simulated electron energy distributions resulting from application of E/p
reakdown structure has been previously experimentally investigated by
streak camera imaging and results can be seen in Figures 2.11 and 2.12. Numerical
investigation of the structure allows, not only for confirmation of the experimental
but also the opportunity to investigate the time development of space charge
experimental methods. The
Texas Tech University, Jordan Chaparro, August 2008
72
development of these fields creates a non-uniform environment that plays a large role
in the development of picosecond discharge.
In Figure 5.14, ionizations are tracked on a per pixel basis until breakdown
conditions are reached. The general structure is in good agreement with the
previously obtained streak imaging with a region of intense ionization taking place in
a thin layer (30 – 120 µm) near the surface of the cathode. This layer gets wider and
brighter, in contrast to the rest of the gap, with increasing pressure. Also, Coulomb
forces in this densely populated region near the cathode lead to channel expansion
rates like those seen in the streak images. The inconsistent and spotty regions in the
latter portions of the gap, which are especially evident with high pressures, are a result
of rapid renormalization. The particle renormalization is necessary to keep the
number of particles tracked by the computer contained in the available memory space
and to limit collisional computation times. The end result is lower resolution in
sparsely populated areas which leads to spotty regions of high intensity instead of a
smooth low intensity distribution. In general, the bright cathode layer is 2-3 orders of
magnitude greater in intensity than the remainder of the gap space which agrees well
with the streak camera results where the luminosity is roughly correlated to excitation
and ionization processes.
Figure 5.14 - Simulation of ionization processes per 15 µm x 15 µm pixel on
logarithmic scale, showing the strong concentration of ionizations in front of the
cathode, and channel constriction increasing with pressure. From top to bottom, the
pressures for the images are 100, 200, 300, and 600 torr.
Examination of the space
picosecond discharge because of the highly non
accumulation of separated charge regions near the cathode has a la
emitted current which drives the development of the discharge. As a result of having a
large effect on FE, the space
Texas Tech University, Jordan Chaparro, August 2008
73
Simulation of ionization processes per 15 µm x 15 µm pixel on
logarithmic scale, showing the strong concentration of ionizations in front of the
cathode, and channel constriction increasing with pressure. From top to bottom, the
es are 100, 200, 300, and 600 torr.
Examination of the space-charge development is especially important in
picosecond discharge because of the highly non-uniform charge distributions. The
accumulation of separated charge regions near the cathode has a large effect on field
emitted current which drives the development of the discharge. As a result of having a
large effect on FE, the space-charge also plays a significant role in the time delay to
Texas Tech University, Jordan Chaparro, August 2008
Simulation of ionization processes per 15 µm x 15 µm pixel on
logarithmic scale, showing the strong concentration of ionizations in front of the
cathode, and channel constriction increasing with pressure. From top to bottom, the
charge development is especially important in
uniform charge distributions. The
rge effect on field
emitted current which drives the development of the discharge. As a result of having a
charge also plays a significant role in the time delay to
EEE. Figure 5.15, shows the
discharge event with a
torr. Each successive image in the series represents one quarter steps of the total
formative time. Since negative polarity pulses are applied, positive spa
fields should be viewed as retarding the applied field.
amplitude has been subtracted from the total field to leave just the contribution from
the space-charge.
Figure 5.13 - XY slice of th
space-charge fields for a pressure of 200 torr. Each successive picture represents a
one quarter step of the formative time.
Texas Tech University, Jordan Chaparro, August 2008
74
, shows the time development of the space-charge fields for a
a pulse amplitude of 150 kV and background pressure of 200
Each successive image in the series represents one quarter steps of the total
Since negative polarity pulses are applied, positive spa
fields should be viewed as retarding the applied field. The applied macro
amplitude has been subtracted from the total field to leave just the contribution from
XY slice of the gap at the center of Z showing the time
charge fields for a pressure of 200 torr. Each successive picture represents a
one quarter step of the formative time.
Texas Tech University, Jordan Chaparro, August 2008
arge fields for a
pulse amplitude of 150 kV and background pressure of 200
Each successive image in the series represents one quarter steps of the total
Since negative polarity pulses are applied, positive space-charge
The applied macro-field
amplitude has been subtracted from the total field to leave just the contribution from
e gap at the center of Z showing the time development of
charge fields for a pressure of 200 torr. Each successive picture represents a
Texas Tech University, Jordan Chaparro, August 2008
75
Initially (top image), electron emission during the rising edge of the pulse slightly
diminishes the field on the surface of the cathode due to a net negative charge from the
emitted current. In a vacuum environment, this is the effect that leads to space-charge
limited current described by the Child-Langmuir law. In the second image from the
top, ionizing collisions begin to build up a region of positive charge about 50 µm off
the surface of the cathode. The electrons resulting from these ionizations begin to
form a net negatively charged region following the ionization front. In the third
picture, the net retarding field at the cathode surface has nearly vanished due to the
ionizing front moving closer to the cathode which leads to a more neutral plasma. In
the bottom picture, the numerous ionizations that have built up since the beginning of
the discharge near the cathode surface have finally created a strong net positive charge
region which enhances the cathode surface field and drives field emission higher
which leads to eventual explosive electron emission. The negatively charged region
following the ionization front significantly reduces the local field causing
accumulation of more electrons. Beyond this region, a slightly enhanced region
extends across the rest of the gap space.
Figures 5.16 and 5.17 show space-charge development along the center axis of the
discharge for the same pulse amplitude and pressures of 600 and 100 torr. Results
from the 600 torr case (5.15) demonstrate an overall less intense space-charge
contribution when compared to results from lower pressures down to 50 torr (5.16).
This is due to the higher pressure not allowing for as much charge separation between
the negatively and positively charged regions as a result of increased collisional
friction. This leads to electron distributions that on the whole, are more heavily
skewed towards lower energies and have lower runaway populations. Results across
all pressures show that the highest amplitude space-charge fields occur at around, or
just below, the pressures where the delay times are the lowest. This is due to a
combination of charge amplification rates being relatively high for these pressures,
and also that these pressures allow for more charge separation leading to dense regions
where one charged particle dominates over the other. The higher magnitude space-
charge fields, in the lower to
higher field emission current which drives shorter simulated formative delay times
real discharges, there
cathode drive field emission to the poin
closing time of the gap.
Figure 5.16 - 1D plot of space charge field development for quarter steps of the
formative time at 600 torr.
Texas Tech University, Jordan Chaparro, August 2008
76
in the lower to intermediate sections of the pressure range, lead to much
higher field emission current which drives shorter simulated formative delay times
is an analogous effect where dense ion concentrations near the
cathode drive field emission to the point of explosive emission which shortens the
gap.
1D plot of space charge field development for quarter steps of the
formative time at 600 torr.
Texas Tech University, Jordan Chaparro, August 2008
intermediate sections of the pressure range, lead to much
higher field emission current which drives shorter simulated formative delay times. In
ion concentrations near the
t of explosive emission which shortens the
1D plot of space charge field development for quarter steps of the
Figure 5.17 - 1D plot of space charge field development for quarter steps of the
formative time at 100 torr.
Runaway Electron Energy Distributions
As mentioned in chapter 2,
energies are often done through
employed here to provide a rough qualitative description of the runaway electron
energies. Using a PMT, organic scintillator, and metallic absorber foils, it is possible
to obtain rough energy spectrums by
signal and the metallic
multiplying the absorption curve of the scintillator with the transmission curves for the
metals. The foils act as filters fo
180 µm thick copper foil
keV. In total, four different foils were used to obtain a rough spectrum of the arrival
energies of runaway electr
Texas Tech University, Jordan Chaparro, August 2008
77
1D plot of space charge field development for quarter steps of the
formative time at 100 torr.
Runaway Electron Energy Distributions
As mentioned in chapter 2, experimental investigations of runaway electron
often done through x-ray analysis. This diagnostic technique was also
here to provide a rough qualitative description of the runaway electron
Using a PMT, organic scintillator, and metallic absorber foils, it is possible
to obtain rough energy spectrums by considering the amplitude of the measured
metallic sensitive curve. The sensitivity curves are determined by
multiplying the absorption curve of the scintillator with the transmission curves for the
metals. The foils act as filters for x-ray radiation of different energies. For instance, a
180 µm thick copper foil typically blocks x-ray photons with energies less than 20
different foils were used to obtain a rough spectrum of the arrival
energies of runaway electrons: Aluminum (20 µm), Copper (180 µm), Silver (500
Texas Tech University, Jordan Chaparro, August 2008
1D plot of space charge field development for quarter steps of the
experimental investigations of runaway electron
This diagnostic technique was also
here to provide a rough qualitative description of the runaway electron
Using a PMT, organic scintillator, and metallic absorber foils, it is possible
considering the amplitude of the measured PMT
sensitive curve. The sensitivity curves are determined by
multiplying the absorption curve of the scintillator with the transmission curves for the
radiation of different energies. For instance, a
photons with energies less than 20
different foils were used to obtain a rough spectrum of the arrival
µm), Copper (180 µm), Silver (500
Texas Tech University, Jordan Chaparro, August 2008
78
µm), and Lead( 1000 µm). The sensitivity curves for these foil thicknesses can be
seen in Figure 5.18.
Figure 5.148 - Metallic absorber foils sensitivity for energies up to 150 keV.
The x-ray production from runaway electron impact at the anode is a result of the
Bremsstrahlung effect which produces a spectrum from mono-energetic electrons
approximated by Kramer’s formula [66]:
),()( εε −= ukZP (5.8)
where u is the electron energy, ε is the energy of x-ray radiation, Z is the atomic
number of anode material, and k is a proportionality factor. In general, the spectrum
resulting from mono-energetic electron beams is wide making it difficult to work back
to exact electron energy distributions from measured intensities. Still, obtaining rough
approximations of the maximum energy and the relative number of electrons striking
the anode is possible by comparing the ratio of measured PMT intensities when using
different foils. The relationship between measured intensity ratio and energy can be
Texas Tech University, Jordan Chaparro, August 2008
79
formulated as follows. The spectrum of emitted photons resulting from a beam of
monoenergetic electrons impacting the anode is approximated as a line beginning at
the impact energy of the electrons and extending back at some arbitrary slope to zero.
The product of this approximated spectrum and the metallic sensitivity curve when
integrated yields a relative intensity that should correlate to those measured by the
PMT. Figure 5.l9 plots the foil sensitivity, an approximated bremsstrahlung spectrum
resulting from 100 keV electrons, and the product of the two quantities. The ratio of
the integrated products (the area below the dashed lines) should relate roughly to the
ratios between the measured PMT intensities for the different foils. As shown in
Table 4.1, higher energy photons correspond to much higher measured PMT signals
and as such, the ratios should roughly correspond to the highest electron energies in
the runaway beam. Table 5.1 lists the relative integrated ratios from these products for
several different monoenergetic electron energies.
Texas Tech University, Jordan Chaparro, August 2008
80
Figure 5.19 - Product of approximated Bremsstrahlung spectrum resulting from
monoenergetic 100 keV electron beam with the metallic foil sensitivity curve (dashed
lines). The ratio of the areas of the dashed regions relates to the maximum electron
energy in the runaway distribution.
Table 5-1 Intensity ratios between foils for a variety of maximum electron energies
Energy [keV] Al
(20 micron)
Cu
(180 micron)
Ag
(500 micron)
Pb
(1 mm)
25 1 0.0019 0.0 0.0
50 1 0.0676 0.0006 0.0
75 1 0.1738 0.0053 0.0007
100 1 0.2684 0.0254 0.0047
125 1 0.3452 0.0586 0.0074
150 1 0.4070 0.0972 0.0104
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Examining measured intensities from the PMT, which are displayed in Figure
5.20 resulting from breakdown with 150 kV pulse voltage, reveals general
relationships of the energy distribution of the runaway electrons. First, the fact the
lead foil has measurable intensities for pressures up to atmosphere confirms that
across the entire pressure range, there is at least some component of high energy
electrons > 50 keV striking the anode.
The local minimum in the silver and lead curves at around 50 torr when compared
with the derived ratios from the table above, indicates maximum electron energies
around 50 keV. At very low pressures near vacuum, maximum energies are > 75 keV
and, from 200 to 600 torr, around 60 keV.
Figure 5.15 - Maximum PMT intensities measured from 150 kV pulsed discharges in
argon for the four absorber foils.
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The minimum at 50 torr is interesting, as the formative delay time at that pressure
is fairly long which allows for the applied field to reach its maximum amplitude
before breakdown. A possible explanation of the reason behind this minimum is given
by Mesyats [40] and Tarasenko [35], who have experimentally investigated the nature
of picosecond runaway electron beams, in sub-atmospheric gas diodes, generated with
pulses similar to those in this study. They have found that the beams have very short,
typically 50 – 200 ps, durations that are often much shorter than the breakdown delay
time for the applied pulse. Mesyats has proposed that the duration of these short
beams is governed by the time for Joule heating to cause explosive electron emission.
When a field emitter explodes, a dense plasma is formed which changes the charge
distribution substantially within 100 nm of the cathode. The large number of electrons
in the plasma begins to ionize near the cathode leading to a quick space-charge
transition that effectively screens the emitters on the surface of the cathode. Since the
majority of runaway electrons are a result of the very high fields that develop near
field emitters, the beam is effectively quenched.
Using the developed numerical model, the time delay to EEE can be investigated.
Figure 5.21 shows an example of the EEE delay for a pressure of 50 torr compared to
a case where the space-charge is neglected. In dashed blue, the voltage ramp of the
pulser is plotted without the influence of space-charge. The dashed green line
represents the average field amplitude normal to the surface of the cathode when
space-charge is tracked.
The field under the influence of the space charge shows the same general trend as
in Figures 5.15-5.17 where the field is initially retarded by electron emission and
eventually transitions to very high values due to the rapid positive charge build up
near the cathode.
The solid lines represent the time integrated j2td metric from equation 2.12 and the
dashed red line is the EEE threshold for copper. As seen from the plot, space-charge
contributions near the cathode drives field emission higher, resulting in a shorter EEE
delay when compared to the space-charge neglected case.
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Figure 5.21 - j2td metric measuring the delay time to explosive electron emission as a
result of applied fields (dashed) both for a case where space-charge is tracked (green)
and one where it is neglected (blue). The green curve in this example is for a pressure
of 50 torr.
The delays across all pressures have been plotted in Figure 5.21 for pulse
amplitudes of 150 kV. It is evident that a local minimum exists, in this case near 75
torr. The shorter the EEE delay, the lower the corresponding field amplitude is when
the beam is stopped, which results in less energetic runaway electrons. Electrons at
the leading edge of the ionization front may still runaway, have less distance to
acquire energy from the field and are subject to reduced amplitude fields in front of
the ionization region. The electron energy distributions at the anode from the
numerical model up to the point of indicated EEE are plotted in Figure 5.23 and agree
fairly well with the observations made from Figure 5.22.
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Figure 5.22 - Simulated EEE delays for 150 kV/mm pulsed fields. The local
minimum near 75 torr may explain the minimum in silver and lead PMT data around
the same pressure.
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85
Figure 5.23 - Simulated electron energy distributions at the anode over the duration of
the EEE delay.
While the model does not currently have any implementation to directly model
the EEE process, it is interesting to note that the rapid increase in current brought
about by the constant build up of ions near the surface of the cathode has an analogous
effect of rapidly advancing the electron count towards the breakdown condition. This
may be why reasonable estimates for formative delays are possible when neglecting
such a critical mechanism.
Finally, the electron energies in the gap space at the time of breakdown are
examined in Figure 5.24. Results show that distributions extend to higher electron
energies for lower pressures despite the trend for slightly higher breakdown voltages
for pressures beyond the delay time minimum.
Figure 5.24 - Electron
breakdown condition f
Statistical Delay
In order to investigate statistical delay, some modifications to the experimental
setup were necessary. The biconical gap section was replaced with a coaxial one
where gap distances of up to 11 mm
investigate the transition from the threshold region
leads to statistical variance of the breakdown delay
emission current is sufficient to le
Figure 5.25 shows sample voltage pulses that were used to gauge statistical delay
times. The red trace corresponds to the incident waveform from the pulser
to generate the pulse, which has a FWHM of around 1.25 ns, th
pulse slicer was opened entirely. The peaking gap was used to maintain sub 200 ps
Texas Tech University, Jordan Chaparro, August 2008
86
energy distribution of particles in the test gap at the time of the
ition for a pulsed field of 50 kV/mm.
In order to investigate statistical delay, some modifications to the experimental
setup were necessary. The biconical gap section was replaced with a coaxial one
where gap distances of up to 11 mm were attainable. The goal of the research was to
investigate the transition from the threshold region, where low level field emission
leads to statistical variance of the breakdown delay, to the region where the field
emission current is sufficient to leave only formative times.
shows sample voltage pulses that were used to gauge statistical delay
trace corresponds to the incident waveform from the pulser
generate the pulse, which has a FWHM of around 1.25 ns, the chopping gap of the
pulse slicer was opened entirely. The peaking gap was used to maintain sub 200 ps
Texas Tech University, Jordan Chaparro, August 2008
energy distribution of particles in the test gap at the time of the
In order to investigate statistical delay, some modifications to the experimental
setup were necessary. The biconical gap section was replaced with a coaxial one
. The goal of the research was to
low level field emission
to the region where the field
shows sample voltage pulses that were used to gauge statistical delay
trace corresponds to the incident waveform from the pulser. In order
e chopping gap of the
pulse slicer was opened entirely. The peaking gap was used to maintain sub 200 ps
Texas Tech University, Jordan Chaparro, August 2008
87
10-90% risetimes. The maximum gap spacing of 11 mm was used and pulse
amplitude was varied yielding tested field amplitudes from 50 – 160 kV/cm.
Figure 5.25 - Incident (red) and transmitted (black) traces used to measure statistical
delay. The pulse width on the incident pulse is around 1.25 ns.
The FWHM of the black trace in Figure 5.24 represents the breakdown delay for
the trial. Figure 5.26 shows the breakdown delay for 20 shots per pressure for a
number of different pulsed field amplitudes. Red marks indicate the averaged FWHM
for the incident pulse.
Texas Tech University, Jordan Chaparro, August 2008
88
Figure 5.26 - Plotted delays for 20 shots at each pressure. The red marks indicate the
average and standard deviation of the incident pulse FWHM.
Often it is more instructive to plot statistical processes as the dependence
|log(nt/n0)| = f(t) where nt is the number of breakdowns with a delay of t or longer and
n0 is the total number of trials. This type of plot is often referred to as a Laue plot [67].
The time at the beginning of the curve in such a distribution represents the formative
time and the slope of the line indicates the degree of statistical variation. Figure 5.27
shows the delay times plotted in such a manner for voltage amplitudes of 75, 115, and
175 kV.
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89
Figure 5.27 - Laue plots for 75 (top), 115 (middle) and 175 (bottom) kV pulses.
It is important to realize that, especially on the picosecond timescale, statistical
and formative delay times cannot be completely separated and it is expected that
statistical fluctuations in the amplification mechanism exist. This is especially
relevant at low pressures below 50 torr. Examination of the Laue plot for the 175 kV
pulse amplitude illustrate the reasoning behind this statement. An obvious transition
in processes occurs between the 115 and 175 kV amplitudes as minimum formative
Texas Tech University, Jordan Chaparro, August 2008
90
times shift nearly 300 ps and random variance, seen at lower field amplitudes, nearly
completely vanishes for all pressures except 50 torr. At this magnitude of pulse
amplitude, it is unlikely that any shot to shot variation in the initial electron current is
significant as the time spent in threshold region of the FN relationship is minimal.
This indicates possible amplification variations that are not masked by excessive
currents from the cathode. Pressures below 50 torr do not show this variation because
their delay times are generally longer than the applied pulse.
As reported in chapter 2, ongoing research indicates that the field emission
process, per se, seems to prefer the emission of multiple electrons per emission act
[26], so that the observed distribution of breakdown times in Figure 5.26 is assumed to
be mainly determined by the statistics of field enhancement factors, which is of
technical importance only in the vicinity of emission thresholds, i.e. at fields on the
order of less than 50 kV/cm.
Analysis of statistical delay through numerical modeling showed little delay
variance resulting from stochastic sampling of the field emission threshold region and
from emitting random numbers of electrons per emission event. There was a much
greater variation seen with small changes, below a factor of 10, made in the field
enhancement factor. Most likely, a combination of a number of factors such as shot to
shot differences in the quantity and sharpness of micro-emitters and statistical effects
with the field emission process govern the statistical breakdown delay at threshold
field amplitudes.
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91
CHAPTER 6
CONCLUSIONS
Picosecond discharges, with high overvoltages, have been investigated with an
emphasis on exploring and expanding upon existing pulsed discharge similarity laws
and revealing the underlying physical processes behind the phenomena. To facilitate
this goal, a customized particle-in-cell model, combined with Monte-Carlo collisional
sampling, has been designed, tested, and implemented to both predict picosecond
discharge metrics and to explore underlying physics.
In the investigated E/p range, it was shown that fundamental shifts in the behavior
of the scaling relationships are due principally to the effects of runaway electrons and
the number of initiatory electrons. As macroscopic and localized fields greatly exceed
the runaway threshold at fields > 7 x 103
V/(cm torr), the amplification rate of the
discharge decreases significantly, limiting the growth rate of the field and increasing
delay times. This is seen in similarity plots as a change in the slope of the curve that
extends to the highest tested E/p ranges. The original experimental observation of this
effect was confirmed by the PIC model. Attempts to achieve the same effect with
analytical models based on either simple forces or streamer mechanism failed to
reproduce reasonable estimates across the entire tested voltage range. Additionally,
the PIC model was used to extend measurements of ionization frequencies and
indicated a divergence from the scaling law α = f(E/p) for conditions well above the
runaway threshold.
The role of the initiatory electron source was also investigated. Through
parametric adjustments the estimated density of micro-point field emitters was
estimated and the role of the field enhancement factor on formative delay times was
investigated. It was found that higher field enhancement factors led to shifts in the
formative delay time pressure minimum and in the slowing of the delay times at
higher pressures. The model also confirmed assertions by other authors [3] indicating
that a large number of initial electrons shifts delay times towards those predicted by
Texas Tech University, Jordan Chaparro, August 2008
92
streamer theory as the sheer number of initial avalanches are able to overcome the
effects of self braking due to runaway phenomena.
The PIC model was also used to investigate the structural development of the
discharge as it relates to charge distributions and space charge development. The
model confirmed previously obtained streak imaging results which indicated that the
majority of ionization processes took place in a narrow region near the surface of the
cathode. It was shown that significant pressure dependence for the development of
space charge fields exists and that it has a significant role on breakdown development.
One of the key roles played by the space-charge is in the driving of field emission
currents that lead to explosive electron emission effects. It was shown that the
pressure dependence of the delay to explosive electron emission is primarily an effect
of the space-charge development and that this delay can account for previously
unexplained features in experimental x-ray diagnostics. Specifically, the EEE delay is
believed to limit the duration of runaway electron beams which strike the anode
causing x-ray radiation. The shortest simulated EEE delays correspond fairly well to
minimums in the measured x-ray amplitudes at around 50 torr.
Finally experimental measurements were used to investigate the statistical delay
at field amplitudes near the field emission threshold. A sudden transition in the
process was measured for fields between 105 and 160 kV/cm where statistical and
formative delays were drastically reduced. Below 105 kV/cm the formative times are
fairly constant and statistical delays decrease steadily with increasing field amplitudes.
It was concluded through numerical analysis that small variations brought about by
sampling the initial stages of field emission are insufficient to explain the statistical
delays and that they most likely depend on the statistics of field emission sites which
can frequently change on a shot to shot basis.
Improving PIC based modeling to include more intricacies of the phenomena
holds a great amount of promise for future advancements in the understanding of
picosecond discharge. Such simulations have proven to be capable of spanning the
different operating regimes and are capable of investigating processes not easily
Texas Tech University, Jordan Chaparro, August 2008
93
observable by experimental means. Investigation of the transition from field to
explosive emission with the inclusion of EEE mechanisms and the inclusion of a
greater degree of collisional processes could reveal even more about ultrafast
breakdown. In addition, common breakdown parameters such as ionization
coefficients and drift velocities could be expanded upon in unexplored regimes by
numerical simulation. It is expected that increasingly sophisticated numerical
techniques will play a major role in the future advancement of pulsed discharge
theory.
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94
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