Investigation of Sub-Nanosecond Breakdown through ...

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

ii

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.


iii

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


iv

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


v

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.


vi

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


vii

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


viii

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


ix

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


x

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


xi

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


xii

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


xiii

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


xiv

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


1

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


2

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.


3

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:


4

)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.


5

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


6

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.


7

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


8

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


9

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


10

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)


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].


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.


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:


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].


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)


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


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


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:


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.


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].


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


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.


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.


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:


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.


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


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.


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.


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].


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


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


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.


33

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.


34

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


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]:


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.


37

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.


38

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.


39

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.


40

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].


41

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


42

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.


43

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.


44

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.


45

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.


46

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.


47

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.


48

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.


49

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].


50

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


51

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


52

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


53

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.


54

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.


55

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


56

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


57

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:


58

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)


rrent is lost through the gap. The gap capacitance can be found from the


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)


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


59

)(2')]'()'([ 0

2

0

0

2tZCVdttVtV

t

=−→∫ , (5.5)


incident waveforms and reflected data can be neglected.


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


Measured transmitted voltages resulting from breakdown from four

different pulse amplitudes and across the full range of tested pressures.


, (5.5)



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


Measured transmitted voltages resulting from breakdown from four

e of tested pressures.


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.


61

the inevitably inconsistent shot-to-shot electrode conditions, and rough

the limited accuracy of the results should be kept in mind


Measured breakdown voltages for radial discharges with pulse amplitudes


rough reproducibility

lts should be kept in mind. Figure 5.6


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


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


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


per primary electron which equates to a final electron number of about 10


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


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


of about 109.


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.


64


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


similar to the experiment. As stated in chapter 2, the formative


electrons. As such, two primary parameters were adjusted to modulate the


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



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


. As stated in chapter 2, the formative


electrons. As such, two primary parameters were adjusted to modulate the


is shown in Figure 5.9

Increasing β tends to shift the


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.


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.


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


68

increasing populations of electrons reach runaway mode


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


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


increasing populations of electrons reach runaway mode which


due to shifts in the electron

the slope of the scaling

Simulated (green) and experimental (blue) E/p vs pt plot. Simulated


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


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.


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


71

the ionization rate and less of an increase in slope of the scaling relationship.


Simulated electron energy distributions resulting from application of

Breakdown Structure

reakdown structure has been previously experimentally investigated by



but also the opportunity to investigate the time development of space charge

fields which are not easily determined through experimental methods. The


the ionization rate and less of an increase in slope of the scaling relationship. A high


Simulated electron energy distributions resulting from application of E/p

reakdown structure has been previously experimentally investigated by



but also the opportunity to investigate the time development of space charge

experimental methods. The


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


73

Simulation of ionization processes per 15 µm x 15 µm pixel on



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


large effect on FE, the space-charge also plays a significant role in the time delay to


Simulation of ionization processes per 15 µm x 15 µm pixel on



charge development is especially important in

uniform charge distributions. The

rge effect on field


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.


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


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.


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


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


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.


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



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


Figure 5.17 - 1D plot of space charge field development for quarter steps of the


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


77



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


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



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


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


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


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.


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


81

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.


82

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.


83

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.


84

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.


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


86

energy distribution of particles in the test gap at the time of the

ition for a pulsed field of 50 kV/mm.



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



energy distribution of particles in the test gap at the time of the



. 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



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.


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.


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


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.


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


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


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.


94

REFERENCES

[1] W. D. Prather, C. E. Baum, J. M. Lehr, et. al., "Ultra-wideband source research".

IEEE 12th International Pulsed Power Conference. Montery, CA. pp. 185-189.

1999.

[2] A. R. Dick, S. J. MacGregor, M. T. Buttram, et. al., “Breakdown Phenomena in

Ultra-Fast Plasma Closing Switches”. Proc. of 12th IEEE Int. Pulsed Power

Conference. Monterey, CA. pp. 1149-1152. 1999.

[3] G. A. Mesyats, “Similarity laws for pulsed gas discharges”. Uspekhi Phys., Vol.

49, pp. 1045-1065. 2006.

[4] K. R. Allen, L. Phillips., “Mechanism of Spark Breakdown”. Electrical Rev.,

Vol. 173, pp. 779-783. 1963.

[5] E. Nasser, Fundamentals of Gaseous Ionization and Plasma Electronics. New

York : John Wiley & Sons, 1966.

[6] L. H. Fisher, B. Bederson., “Search for the Transition of Streamer to Townsend

Form of Spark in Air”. Phys. Rev., Vol. 75, p. 1615. 1949.

[7] R. Schade, Z. Phys., Vol. 108, pp. 353-375. 1938.

[8] L. B. Loeb, Fundamental Processes of Electrical Discharge in Gases. New

York : Wiley, 1939.

[9] J. M. Meek, "A Theory of Spark Discharge”.Phys. Rev., Vol. 57, pp. 722-730.

1940.

[10] H. Raether, Electron Avalanches and Breakdown in Gases. London :

Butterworth, 1964.

[11] E. E. Kunhardt, W. W. Byszewski, “Development of Overvoltage Breakdown at

High Gas Pressure”. Phys. Rev. A.,, Vol. 21, pp. 2069-2077. 1980.

[12] L. P Babich, T. V. Loiko, V. A., Tsukerman, "High-Voltage Nanosecond

Discharge in a Dense Gas at a High Overvoltage with Runaway Electrons".

Phys. Uspekhi,, Vol. 33, pp. 521-540. 1990.

[13] R. C. Fletcher, “Impulse Breakdown in the 10^-9-Sec. Range of Air at

Atmospheric Pressure”. Phys. Rev. 76, Vol. 76, pp. 1501-1511. 1949.


95

[14] P. Felsenthal, J.M. Proud, “Nanosecond-Pulse Breakdown in Gases". Phys. Rev.

A., Vol. 139, pp. 2069-2077. 1965.

[15] L. P. Babich, Y. L. Stankevich, “Transition from Streamers to Continuous

Electron Acceleration”. Sov. Phys. Dokl., Vol. 17, pp. 1333-1336. 1973.

[16] G. A. Mesyats, Y.I.Bychkov, A.I. Iskol'dskir, “Nanosecond Formation Time of

Discharges in Short Air Gaps”. Sov. Phys. Tech., Vol. 13, pp. 1051-1055. 1969.

[17] V. V. Krmnev, G. A. Mesyats, “Growth of a Nanosecond Pulsed Discharge in a

Gas with One-Electron Initiation”. Zh. Prikl. Mekh. Tekh. Fiz., Vol. 1, pp. 40-

45. 1971.

[18] Y. L. Stankevich, V. G. Kalinin, “Fast Electrons and X-Ray Radiation During

the INitial Stage of Growth of a Pulsed Spark Discharge in Air”. Sov. Phys.

Dokl., Vol. 12, pp. 1042-1043. 1968.

[19] V. V. Kremnev, Y. A. Kurbatov, “Study of X-ray from Gas Discharge in High

Electric Fields”. Sov. Phys. Tech., Vol. 17, p. 626. 1972.

[20] L. V. Tarasova, L. N. Khudyakova, T. V. Loiko, V. A. Tsukerman, “Fast

Electrons and X-rays from Nanosecond Gas Discharges at 0.1-760 Torr”. Sov.

Phys. Tech., Vol. 19, pp. 351-353. 1974.

[21] G. A. Mesyats, D. I. Proskurovsky. Pulsed Electrical Discharge in Vacuum.

Berlin : Springer-Verlag, 1989.

[22] R. H. Fowler, L. Nordheim, “Electron Emission in Intense Electric Fields”.

Proc. R. Soc., Vol. 119, pp. 173-181. 1928.

[23] G. Fursey, Field Emission in Vacuum Microelectronics. New York : Springer.

2005.

[24] C. A. Spindt, I. Brodie, L. Humphrey, E. R. Westerberg, “Physical Properties of

Thin-Film Field Emission Cathodes with Molybdenum Cones”. J. Appl. Phys.,

Vol. 57, p. 5248. 1976.

[25] O, Yamashita, “Effect of Metal Electrode on Seebeck Coefficient of p- and n-

type Si Thermoelectrics”. J. Appl. Phys., Vol. 95, pp. 178-183. 2004.

[26] M. A. Piestrup, H.R. Puthoff, P. J. Ebert, “Measurement of Multiple Electron

Emission in Single Field-Emission Events”. J. Appl. Phys., Vol. 82, p. 5864.

1997.


96

[27] A. Yanguas-Gil, J. Cotrino, L. L. Alves, “An Update of Argon Inelastic Cross-

Sections for Plasma Discharges”. J. Appl. Phys., Vol. 38, pp. 1588-1598. 2005.

[28] M Suzuki, T Taniguchi, H Tagashira, “Momentum Transfer Cross Section of

Argon Deduced from Electron Drift Velocity Data”. J. Appl. Phys., Vol. 23, pp.

842-850. 1990.

[29] G. V. Kolbychev, “Gas Ionization by Runaway Electrons”. Rus. Phys. J., Vol.

45, pp. 1209-1214. 2005.

[30] A. V. Gurevich, “On the Theory of Runaway Electrons”. Zh. Eksp. Teor. Fiz.,

Vol. 39, pp. 1296-1307. 1960.

[31] R. M. Kulsrud, Y. C. Sun, N. K. Windsor, H. A. Fallon, “Runaway Electrons in

Plasma”. Phys. Rev. Lett., Vol. 31, pp. 690-693. 1973.

[32] V. F. Tarasenko, S. I. Yakovlenko, “The Electron Runaway Mechanism in

Dense Gases and the Production of High-Power Subnanosecond Electron

Beams”. Phys. Uspekhi, Vol. 47, pp. 887-905. 2004.

[33] L. P. Babich, I. M. Kutsyk, E. N. Donskoy, A. U. Kudryavtsev, “New Data on

Space and Time Scales of Relativistic Runaway Electron Avalanche for

Thunderstorm Environment: Monte Carlo Calculations”. Phys. Lett. A., Vol.

245, pp. 460-470. 1998.

[34] K. Bergmann, J. Kiefer, C. Gavrilescu, W. Neff, R. Lebert, “Electrode

Phenomena and Current Distribution in a Radial Pseudospark Switch”. J. Appl.

Phys., Vol. 84, pp. 4180-4184. 1998.

[35] V. F. Tarasenko, “Effect of the Amplitude and Rise Time of a Voltage Pulse on

the Formation of an Ultrashort Avalanche Electron Beam in a Gas Diode”. Tech.

PHys., Vol. 52, pp. 534-536. 2007.

[36] E. K. Baksht, M. I. Lomaev, D. V. Rybka, V. F. Tarasenko, “High-Current-

Density Subnanosecond Electron Beams Formed in a Gas-Filled Diode at Low

Pressures”. Tech. Phys. Lett., Vol. 32, pp. 948-950. 2006.

[37] E. K. Baksht, E. V. Balzovskii, A. I. Kilmov, I. K. Kurkan, M. I. Lomaev, D. V.

Rybka, V. F. Tarasenko, “A Collector Assembly for Measuring a

Subnanosecond-Duration Electron Beam Current”. Inst. Exp. Tech., Vol. 50, pp.

811-814. 2007.


97

[38] V. F. Tarasenko, D. V. Rybka, E. K. Baksht, I. D. Kostryrya, M. I. Lomaev,

“Generation and Measurement of Subnanosecond Electron Beams in Gas-Filled

Diodes”. Inst. Exp. Tech, Vol. 51, pp. 213-219. 2008.

[39] G. A. Mesyats, S. Korovin, K. Sharypov, V. Shpak, S. Shunailov, M. Yalandin,

“Dynamics of Subnanosecond Electron Beam Formation in Gas-Filled and

Vacuum Diodes”. Tech. Phys. Lett., Vol. 32, pp. 18-22. 2006.

[40] G. A. Mesyats, M. I. Yalandin, K. A. Sharypov, V. G. Shpak, S. A. Shunailov,

“Generation of a Picosecond Runaway Electron Beam in a Gas Gap With a

Nonuniform Field”. Acepted for publication in IEEE Trans. Plas. Sci. 2008.

[41] G. A. Mesyats, V. G. Shpak, M. I. Yalandin, S. A. Shunailov, “RADAN-

EXPERT Portable High-Current Accelerator”. Albuquerque, NM : Proc. of 10th

IEEE Int. Pulsed Power Conf. pp. 539-543. 1995.

[42] S. I. Yakovlenko, “Similarity Law for a Homogeneous Discharge in the Presence

of a Strong Field and Analogues of the Stoletov Constant for the Pulsed

Regime.” Laser Phys., Vol. 17, pp. 1-14. 2007.

[43] H. G. Krompholz, L. L. Hatfield, A. A. Neuber, K. P. Kohl, J. E. Chaparro, H.

Y. Ryu, “Phenomenology of Subnanosecond Gas Discharges at Pressures Below

One Atmosphere”. IEEE. Trans. Plas. Sci., Vol. 34, pp. 927-937. 2006.

[44] J. E. Chaparro, W. J. Justis, H. G. Krompholz, L. L. Hatfield, A. A. Neuber,

“Breakdown Delay Times for Subnanosecond Gas Discharges at Pressures

below One Atmosphere”. Accepted for publication in IEEE. Trans. Plas. Sci.,

2008.

[45] C. K. Birdsall, A. B. Langdon. Plasma Physics Via Computer Simulation. New

York : Taylor & Francis Group, 2005.

[46] R. W. Hockney, J. W. Eastwood. Computer Simulation Using Particles. New

York : Taylor & Francsis Group, 1988.

[47] D. Potts, G. Steidl, M. Tasche. “Fast Fourier Transforms for Nonequispaced

Data: A Tutorial”. in P. J. Ferreira, J. J. Benedetto. Modern Sampling Theory:

Mathmatics and Applications. Boston : Birkhauser, 2001.

[48] W. L. Briggs, A Multigrid Tutorial. Philadelphia : Society for Industrial and

Applied Mathmatics, 1987.


98

[49] G. Pöplau, “Multigrid Solvers for Poisson's Equation in Computational

Electromagnetics”. 3rd International Workshop in Scientific Computing in

Electrical Engineering. 2000.

[50] G. Pöplau, U. van Rienen, B. van der Geer, M. de Loos, “Multigrid Algorithms

for the Fast Calculation of Space-Charge Effects in Accelerator Design”. IEEE

Trans. Mag., Vol. 40, pp. 714-718. 2004.

[51] G. Pöplau, U. van Rienen, “A Self-Adaptive Multigrid Technique for 3-D Space

Charge Calculations”. IEEE. Trans. Mag., Vol. 44, pp. 1242-1246. 2008.

[52] N. J. Giordano, H. Nakanishi. Computational Physics. New York : Pearson

Prentice Hall, 2006.

[53] Y. K. Kim, M. Inokuti, “Total Cross Sections for Inelastic Scattering of Charged

Particles by Atoms and Molecules. V. Evaluation to the Next Order beyond the

Bethe Asymptote”. Phys. Rev. A., Vol. 3, pp. 665-678. 1971.

[54] J. P. Verboncoeur, “Particle Simulation of Plasmas: Review and Advances”.

Plas. Phys. Ctrl. Fusion, Vol. 47, pp. 231-260. 2005.

[55] NIST Standard Reference Database64, NIST Electron Elastic-Scattering Cross-

Section Database, August 2003.

[56] A. E. Green, T. Sawada, “Ionization Cross Sections and Secondary Electron

Distributions”. J. Atmo. Terr. Phys., Vol. 34, p. 1719. 1972

[57] Y. Feng, J.P. Verboncoeur, “Transition from Fowler-Nordheim Field Emission

to Space Charge Limited Current Density”. Phys. Plas., Vol. 13, pp. 13-21. 2006.

[58] RADAN 303B Technical Manual.

[59] Slicer SN4 Technical Manual.

[60] E. G. Farr, D. E. Ellibee, J. M. Elizondo, C. E. Baum, J. M. Lehr, “A Test

Chamber for a Gas Switch Using a Hyperboloidal Lens, Switching notes”.

AFRL, Kirtland AFB. Albuquerque, NM, Note 30. 2000.

[61] J. E. Chaparro, “Ultrafast Gaseous Breakdown for Pressures below One

Atmosphere”. Electrical and Computer Engineering, Texas Tech University.

Lubbock, TX. Masters Thesis. 2006.


99

[62] F. Hegeler, “Fast Electrical and Optical Diagnosis of the Early Phase of

Dielectric Surface Flashover”. Electrical and Computer Engineering, Texas Tech

University. Lubbock, TX. Masters Thesis. 1995.

[63] J. E. Chaparro, W. Justis, H. Krompholz, L. L. Hatfield, A. A. Neuber, “Scaling

Laws for Sub-Nanosecond Breakdown in Gases with Pressures below one

Atmosphere”. Albuquerque, NM : Proc. 16th International IEEE Pulsed Power

Conference. 2007.

[64] J. E. Chaparro, H. Krompholz, L. L. Hatfield, A. A. Neuber, “Statistical and

Formative Delay Times for Sub-Nanosecond Breakdown at Sub-Atmospheric

Pressure”. Las Vegas, NV : Proc. IEEE. Power Modulator Conference. 2008.

[65] Y. Feng, J. P. Verboncoeur, “A Model for Effective Field Enhancement for

Fowler-Nordheim Field Emission”. Phys. Plas., Vol. 12, pp. 12-18. 2005.

[66] H. W. Koch, J. W. Motz, “Bremsstrahlung Cross-Section Formulas and Related

Data”. Rev. Mod. Phys., Vol. 31, pp. 920-957. 1959.

[67] M. von Laue, Annalen der Physik, Vol. 76, pp. 261-265. 1925.

Investigation of Sub-Nanosecond Breakdown through ...

Documents

Transcript of Investigation of Sub-Nanosecond Breakdown through ...