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Explorations of SpaceCharge Limits in ParallelPlate Diodes and Associated Techniques for Automation
 Author(s): RaganKelley, Benjamin
 Advisor(s): Lieberman, Michael;
 Verboncoeur, John
 et al.
Abstract
Spacecharge limited flow is a topic of much interest and varied application.
We extend existing understanding of spacecharge limits by simulations,
and develop new tools and techniques for doing these simulations along the way.
The ChildLangmuir limit is a simple analytic solution for spacecharge limited
current density in a onedimensional 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 twodimensional axisymmetric parallel plate diode of various aspect ratios ($r/L$),
and develop a scaling law for the measured twodimensional spacecharge limit (2DSCL)
relative to the ChildLangmuir 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 twodimensional particleincell simulation code OOPIC.
We find a scaling law that is a monotonically decreasing function of this aspect ratio,
and the onedimensional 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 currentdensity ``wings'' at the outeredge of the beam,
with a very large relative contribution when the beam is narrow.
Mechanisms for enhancing current beyond the ChildLangmuir limit remain a matter of great importance.
We analyze the enhancement effects of upstream ion injection on the transmitted current in a onedimensional parallel plate diode.
Electrons are fieldemitted 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 highlevel Python interface,
wrapping the C++ particleincell 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.
Fieldlimited emission is an approach for using Gauss's law to to satisfy the space charge limit for emitting current in particleincell simulations.
We find that simple fieldlimited 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 nonzero surface field at the boundary.
Since traditional fieldemission 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,
nonzero injection velocity, and nonzero 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 twodimensional periodic simulation,
to demonstrate that the Jaffe limit for a onedimensional 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 ChildLangmuir 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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