Space-charge limited flow is a topic of much interest and varied application.

We extend existing understanding of space-charge limits by simulations,

and develop new tools and techniques for doing these simulations along the way.

The Child-Langmuir limit is a simple analytic solution for space-charge limited

current density in a one-dimensional diode. It has been previously extended to two dimensions

by numerical calculation in planar geometries. By considering an axisymmetric cylindrical system with axial emission from a circular cathode of finite radius $r$ and outer drift tube $R > r$

and gap length $L$, we further examine the space charge limit in two dimensions.

We simulate a two-dimensional axisymmetric parallel plate diode of various aspect ratios ($r/L$),

and develop a scaling law for the measured two-dimensional space-charge limit (2DSCL)

relative to the Child-Langmuir limit as a function of the aspect ratio of the diode.

These simulations are done with a large ($100 T$) longitudinal magnetic field to restrict electron

motion to 1D, with the two-dimensional particle-in-cell simulation code OOPIC.

We find a scaling law that is a monotonically decreasing function of this aspect ratio,

and the one-dimensional result is recovered in the limit as $r >> L$.

The result is in good agreement with prior results in planar geometry,

where the emission area is proportional to the cathode width.

We find a weak contribution from the effects of the drift tube for current at the beam edge,

and a strong contribution of high current-density ``wings'' at the outer-edge of the beam,

with a very large relative contribution when the beam is narrow.

Mechanisms for enhancing current beyond the Child-Langmuir limit remain a matter of great importance.

We analyze the enhancement effects of upstream ion injection on the transmitted current in a one-dimensional parallel plate diode.

Electrons are field-emitted at the cathode,

and ions are injected at a controlled current from the anode.

An analytic solution is derived for maximizing the electron current throughput in terms of the ion current.

This analysis accounts for various energy regimes, from classical to fully relativistic.

The analytical result is then confirmed by simulation of the diode in each energy regime.

The simulation approach involved iteratively testing injected ion current,

and treating the measured transmitted electron current as a feedback mechanism.

The feedback loop was automated, allowing for a single simulation to locate the optimized current.

By tuning the injected ion current, we are able to optimize the transmitted electron current.

This tuning of the ion current is automated by the integration of a high-level Python interface,

wrapping the C++ particle-in-cell simulation code OOPD1.

In this particular system, analysis showed that simulation runtime would be a function of transit time, and thus ion mass.

By experimenting with reduced ion mass, we were able to significantly reduce simulation times,

while recovering the same physical results.

Field-limited emission is an approach for using Gauss's law to to satisfy the space charge limit for emitting current in particle-in-cell simulations.

We find that simple field-limited emission models make several assumptions,

which introduce small, systematic errors in the system.

We make a thorough analysis of each assumption,

and ultimately develop and test a new emission scheme that accounts for each.

The first correction we make is to allow for a non-zero surface field at the boundary.

Since traditional field-emission schemes only aim to balance Gauss's law at the surface,

a zero surface field is an assumed condition.

But for many systems, this is not appropriate,

so the addition of a \emph{target} surface field is made.

The next correction is to account for nonzero initial velocity,

which, if neglected, results in a systematic underestimation of the current,

due to assuming that all emitted charge will be weighted to the boundary,

when in fact it will be weighted as a fraction strictly less than unity,

depending on the distance across the initial cell the particle travels in its initial fractional timestep.

A correction is made to the scheme, to use the actual particle weight to adjust the target emission.

The final analyses involve geometric terms,

analyzing the effects of cylindrical coordinates,

and taking particular care to analyze the center of a cylindrical beam,

as well as the outer edge of the beam, in Cartesian coordinates.

We find that balancing Gauss's law at the edge of the beam is not the correct behavior,

and that it is important to resolve the profile of the emitted current,

in order to avoid systematic errors.

A thorough analysis is done of the assumptions made in prior implementations,

and corrections are introduced for cylindrical geometry,

non-zero injection velocity, and non-zero surface field.

Particular care is taken to determine special conditions for the outermost node,

where we find that forcing a balance of Gauss's law would be incorrect.

The new emission scheme is tested in a two-dimensional periodic simulation,

to demonstrate that the Jaffe limit for a one-dimensional diode with finite initial velocity is recovered.

We also extend the iterative scheme developed earlier,

and apply it to determine a scaling law for the Child-Langmuir limit

in an axisymmetric planar diode, with finite initial velocity.

We find that the new scheme reproduces prior results,

and in significantly less computation time due to no longer needing to overinject,

and leads to rapid convergence of the surface field,

using our new algorithmic optimization wrapper to seek the local limiting current along an emitter.

Capacitively driven radio frequency (rf) discharges are commonly used for plasma-assisted material processing. Because of the significant difference in the mobility of electrons and ions, a thin layer of sheath is always established at the boundary, which separates the discharge into two regions: quasi-neutral bulk plasma and positively charged sheath. Within the sheath, ions are accelerated by electric fields and therefore, bombard the electrode with significant energies. The ion energy distribution (IED) on the substrate is essentially important in the optimization of discharge operations. For plasmas sources typically operated at higher densities and lower pressures, the ion motion in the rf sheath is mainly collisionless since the ion mean free path is much larger than the sheath width. At high operating pressures and large sheath voltage drops, the sheaths are typically collisional.

A fast and simple model consisting of a series of computational steps is of great value to predict the plasma parameters and IEDs, given the control parameters of the discharge. Respective models for IEDs in collisionless and collisional capacitive rf sheaths are developed in this dissertation, based on the sheath models developed in late 1980s. Both models do not rely on any intermediate parameters from simulation or experimental results and only take a few seconds (collisionless) or minutes (collisional) to get the final IEDs. Ion-neutral charge exchange reactions are considered for collisional rf sheaths. Energy dependent ion mean free path is taken into account. Particle-in-cell (PIC) simulations are used to verify the previous sheath models and the IED models.

The PIC code OOPD1, being developed with a powerful capability as the development of the collisional IED model goes on, is introduced in this dissertation. Comparisons with XPDP1 are presented, which confirms OOPD1 as a trustable, friendly, and extensible simulation tool to observe the rf discharges.

The goal of this dissertation was to gain an understanding on the relative

importance of microwave power, neutral pressure, and magnetic field

configuration on the behavior of the hot electrons within an Electron Cyclotron

Resonance Ion Source (ECRIS) plasma. This was carried out through measurement of

plasma bremsstrahlung with both NaI(Tl) (hv > 30 keV) and CdTe (2 keV < hv <

70 keV) x-ray detectors, and through measurement of the plasma energy density

with a diamagnetic loop placed around the plasma chamber. We also examined the

anisotropy in x-ray power by simultaneously measuring the x-ray spectra in two

orthogonal directions: radially and axially, using NaI(Tl) detectors.

We have seen that for a 6.4 GHz ECRIS, both the x-ray power produced by confined

electrons and the plasma energy density behave logarithmically with microwave

power. The x-ray flux created by electrons lost from the plasma, however, does

not saturate. Thus, the small increase in plasma density that occurred at high

microwave powers (> 150 W on a 6.4 GHz ECRIS) was accompanied by a large

increase in total x-ray power. We suggest that the saturation of x-ray power and

plasma energy density was due to rf-induced pitch-angle scattering of the

electrons. X-ray power and plasma energy density were also shown to saturate

with neutral pressure, and to increase nearly linearly as the gradient of the

magnetic field in the resonance zone was decreased. All of these findings were

in agreement with the theoretical models describing ECRIS plasmas.

We have discussed the use of a diamagnetic loop as a means of exploring various

plasma time scales on a relative basis. Specifically, we focused much of our

attention on studying how changing ion source parameters, such as microwave

power and neutral pressure, would effect the rise and decay of the integrated

diamagnetic signal, which can be related to plasma energy density. We showed

that increasing microwave power lowers the e-fold times at both the leading edge

and the trailing edge of the microwave pulse. Microwave power, however, had

almost no impact on the ignition times of the plasma.

The plasma energy density e-fold times were insensitive to both neutral pressure

and magnetic field setting. Neutral pressure, however, had a dramatic effect on

the time of first appearance of the diamagnetic signal ("plasma ignition time").

In addition to neutral pressure, ignition times were also a function the

relative abundance of electrons in the plasma chamber at the beginning of a

microwave pulse. In all instances, the rise time of the integrated diamagnetic

signal was seen to be faster than the decay time.

By comparing the unintegrated diamagnetic signal to the ratio of reflected to

forward microwave power we theorized that the initial, exponential rise in the

diamagnetic signal at the leading edge of a microwave pulse was due to rapid

changes in both the average electron energy and density. During the slowly

decaying portion of the diamagnetic loop signal, only the hot tail of the

electron population was increasing. This theory was supported by time resolved,

low energy x-ray measurements that showed that the period of rapid change of the

ratio of reflected to forward microwave power coincided with a rapid change in

average photon energy.

We have also showed that x-rays production in an ECRIS plasma was highly

anisotropic, with radial x-ray counts being much greater than axial x-ray

counts. This was shown to be true for both the "ECR" (operating at 6.4 GHz) and

the higher performance "AECR-U" (operating at 14 GHz). Based on this, we can

make the qualitative statement that the electron energy was also highly

anisotropic, with a much larger perpendicular energy than parallel energy. The

degree of anisotropy was shown to increase with the operating frequency of the

ion source. This increase was most likely attributable to the higher power

density and greater confinement associated with higher performance machines, and

implies that superconducting ECRIS operating at very high frequencies will have

extremely anisotropic x-ray power deposition.

The radial spectral temperature on the "AECR-U" was over twice as large as the

axial spectral temperature. However, in the "ECR" the radial and axial spectral

temperatures were similar. Hence, the anisotropy in spectral temperature also

showed dependence on the magnetic field strength and operating frequency of the

ECRIS. The combination of higher energies, and intensity of high energy x-rays

in the radial direction has important implications in the x-ray heat load

estimates for superconducting ECR ion source cryostats.

Improved modeling of angled-dielectric insulation in high-voltage

systems is described for use in particle-in-cell (PIC) simulations.

Treatment of non-orthogonal boundaries is a significant challenge in

modeling angled-dielectric flashover, and conditions on boundaries are

developed to maintain uniform truncation error in discretized space

across the dielectric angles studied. Extensive effort was expended in

isolating particular operating regimes to illustrate fundamental

phenomenological surface effects that drive the discharges studied

herein; consequently, this document focuses on the phenomenology of

two specific dielectric angles at 6.12° for multiplicative breakdown

(the so-called single-surface multipactor) and 22.9° for a

non-multiplicative discharge that evolves into a dark current at

steady state.

Phenomenological results for simulations in vacuum through "ultra-low

pressures" on the order of a few hundred mTorr are presented. A

multipactor front forms via net emission of electrons from impact on

the dielectric surface, where emission leads to saturated field

conditions in the wake of the front, producing a well-defined

forward-peaked wave. A treatment of the gain and saturation

characteristics is presented, isolating the surface electric fields as

the driving contributor to both metrics. Physical models include

oftenneglected effects such as space-charge, dielectric-surface

charging, and particle distributions in energy and space. For the

discharges treated in this study, breakdown voltages of the typical

Paschen form are not applicable, since multiplicative conditions are

driven primarily by surface effects.

Phenomenological results are also presented for simulations at low

pressure (~ 1Torr), which is shown to be a transitional limit where

volume effects become appreciable compared to surface effects. A

coupling between space charge and surface charge is shown to lead to

oscillatory effects in otherwise DC discharges. Surface multipactor

leads to increased ionization and space charge, and the ensuing

space-charge momentum alters what would have been a steady-state

saturation as in the case of vacuum-like discharges. Models for

diffusive outgassed species are developed and implemented, extending

the capabilities of the PIC suite.

The overarching theme of this study is to communicate the dependence

of multiplicative discharges dominated by surface effects on

near-surface electric field conditions. It is shown through various

examples from vacuum through low pressures, and in diffusive gases,

that single-surface multipactor conditions can be expressed solely in

terms of the dielectric surface field angles. This treatment lays the

foundation for a novel extension of RF breakdown susceptibility theory

[1] to the DC regime, grounding breakdown susceptibility to the

well-established fundamentals on secondary emission [2, 3]. This

theory shows that breakdown characteristics can be modeled in an

a-priori framework, hence the lack of a Paschen-type curve.

Finally, the effect of the seed source on discharge characteristics is

also studied. A comparison between a constant-waveform source, a

Fowler-Nordheim source, and an application of a modified source based

on theoretical treatment from [4] are presented, showing that the seed

is a necessary but insufficient condition for surface flashover, where

the dominant contributor is the configuration of the surface fields

downstream of the seed source. While the seed can influence upstream

conditions to alter the injected current, the gain characteristics of

the downstream region are still well described by the framework

developed in the remainder of this document.