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Explorations of Space-Charge Limits in Parallel-Plate Diodes and Associated Techniques for Automation


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.

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