<p dir="ltr">Many phenomena in physics, engineering, and biology depend strongly on geometry; however, deriving analytic (and sometimes numerical or simulation) solutions to describe these phenomena for realistic geometries may be challenging or impossible. This thesis applies coordinate invariant mathematics to describe several key multidisciplinary problems.</p><p dir="ltr">The first phenomenon that we explore is space-charge-limited current (SCLC), which corresponds to the maximum steady-state current that can be injected into a diode. First derived by Child and Langmuir and described by the eponymous Child-Langmuir law for a one-dimensional, planar diode, SCLC is critical for numerous applications, including electric thrusters, Hall thrusters, directed energy, high-power microwaves, vacuum nanotransistors, and satellites. The SCLC is a critical limit to operation and many studies have sought ways to exceed it; however, this requires better understanding of the SCLC in more realistic geometries, motivating extensions to nonplanar and multidimensional geometries. However, many devices employ a crossed-field geometry in which a magnetic field is applied orthogonal to the electric field to enhance power output. This thesis applies variational calculus and capacitance to derive two sets of solutions for the SCLC in nonplanar crossed-field diodes.</p><p dir="ltr">The first set of solutions is found using scale factors and variational calculus. Variational calculus minimizes the gap energy to solve for the path of least resistance. The scale factors, which are the lengths of the local basis vectors, generalize the process. Models can be produced in variational calculus using the spatial domain alone, eliminating the need for the time domain transformation required by all other crossed-field approaches. This approach creates a powerful, numerically solvable solution for the SCLC in any orthogonal geometry, although it may be computationally expensive.</p><p dir="ltr">The second set of solutions is created by treating the diode as a capacitor and using the capacitance equations to find the SCLC. After finding a planar solution, the solution was generalized by combining conformal mapping and magnetic field mapping by leveraging the innately geometric definition of the Hull cutoff. The Hull cutoff, the magnetic field required to insulate the electron flow, is calculated across geometries to find a mapping factor for the magnetic field allowing the application of conformal mapping, a method of geometric translation that is normally unusable in crossed-field systems. This approach greatly reduces the computational expense and complexity present in other crossed-field approaches.</p><p dir="ltr">In Chapter 4, we apply Lie point symmetries to extend theories for spherical avascular tumor growth to spheroidal tumor growth. Lie point symmetries reduce the complexity of ordinary differential equations, providing a simpler, and sometimes the only, path to a solution. In this chapter, we apply Lie point symmetries to four types of tumors: prolate and oblate spheroids without a necrotic core, an area of dead cells often found at the center of larger tumors, and prolate and oblate spheroids with a necrotic core. Lie point symmetries simplify the differential equations in all four cases and make it possible to solve the prolate spheroid without a necrotic core.</p><p dir="ltr">The results from this thesis provide valuable insight to computational physicists benchmarking particle-in-cell simulations for determining SCLC for crossed-field diodes. Additionally, elucidating the physical phenomena in more realistic diodes can facilitate further optimization for many applications of crossed fields, such as magnetrons. The tumor growth models demonstrate the applicability of this approach to a dramatically different problem and could provide value to characterizing more realistic shapes.</p>

<p>  </p> <p>Electron emission plays a vital role in many modern technologies, from plasma medicine to heavy ion beams for fusion. An accurate theoretical model based upon the physics involved is critical to efficient operation of devices pushing the boundaries of complexity. The interactions between different electron emission mechanisms can severely alter device performance, especially when operating in extreme conditions. This dissertation studies electron emission from the perspectives of increasing geometric and physical mechanism complexities </p> <p>One half of this dissertation derives new relations for space-charge limited emission (SCLE) in non-planar geometries. SCLE is the maximum stable current that may be produced by electron emission before the electric field of the electrons themselves self-limits further emission. In planar devices, this is modeled by the well-established Child-Langmuir (CL) equation. The Langmuir-Blodgett (LB) equations remain the most commonly accepted theory for SCLE for cylindrical and spherical geometries after nearly a century; however, they suffer from being approximations based on a polynomial series expansion fit to a nonlinear differential equation. I derive exact, fully analytic equations for these geometries by using variational calculus to transform the differential equation into a new form that is fully and exactly solvable. This variational approach may be extended to any geometry and offers a full description of the electric field, velocity, and charge density profiles in the diode. </p> <p>SCLE is also an important mechanism for characterizing the operation of devices with an external magnetic field orthogonal to the electric field. This “crossed-field” problem decreases the limiting current as electrons travel longer, curved paths, effectively storing some charge in the gap (moving parallel to the emitter). At a critical magnetic field called the Hull cutoff, electron paths become so tightly curved that the circuit can no longer be completed, a condition called magnetic insulation. Crossed-field SCLE has been accurately modeled in planar devices by Lau and Christenson. Using the variational approach, I replicate their planar results and extend the calculation to cylindrical geometry, a common choice for magnetron devices. Further, I derive additional equations with simplified assumptions that, for the first time, provide an analytic description of experimental results below the Hull cutoff field. Following this I incorporate a series resistor: device resistance (or impedance) changes non-linearly with current and voltage, so I couple Ohm’s Law (OL) to all the models of crossed-field devices. For devices just below the Hull cutoff, I predict analytically and show in simulation novel bi-modal behavior, oscillating between magnetically insulated and non-insulated modes. With crossed-field device assessment, the variational calculus approach to space-charge may be used for numerous applications, including high power microwave sources, relativistic klystron devices, heavy ion beams, Hall thrusters, and plasma processing. </p> <p>The other half of this dissertation derives analytic theories to solve for emission current with three or more electron emission mechanisms simultaneously. In addition to the CL law, SCLE may also occur in neutral, non-vacuum diodes, modeled by the Mott-Gurney (MG) equation. These are the two limiting mechanisms I study; the other major modality of electron emission is direct electron production, the source of current in the device. Electrons are ejected when impelled by high temperature or electric field at the emission surface. These mechanisms are thermionic (or thermal) emission, modeled by the Richardson-Laue-Dushman (RLD) equation, and field emission, modeled by the Fowler-Nordheim (FN) equation, respectively. Additionally, just as I calculated the impedance of devices operating in a crossed-field configuration, all these models can be similarly coupled to OL. I derive models unifying FN, MG, and CL (with an extension linking OL, mentoring an undergraduate) and RLD, FN, and CL. These models are relevant for modern device design, especially as micro- and nano-scale devices seek to eliminate vacuum requirements and as space and military applications require higher temperature tolerances.</p> <p>While multi-physics models, like the ones described above, are important, the single-physics models (FN, RLD, MG, CL, OL) are still valid (and much easier to use) in their respective asymptotic limits. For example, a circuit behaves purely according to OL for very high resistances, according to MG for very high pressures, and so forth. Importantly, when devices operate in transition regions between these asymptotic limits, <em>none </em>of the asymptotic equations match the predictions of multi-physics models. Yet, intersections between the asymptotic equations are easily found, say for a certain set of voltage, gap distance, and pressure, CL=MG. Since both asymptotic equations give the same prediction, we may conclude that both must be inaccurate for those physical parameters! This gives rise to what I term “nexus theory:” solving two or more asymptotic equations simultaneously to rapidly and accurately predict sets of physical parameters at which multi-physics models (specifically, the physics leading to the “nexus point” parameters, points or curves at which nexus conditions are satisfied) are required for accurate device predictions. In fact, I show that multi-physics models are necessary within roughly one to two orders of magnitude from a nexus. In effect, nexus theory provides a simple, powerful tool to determine how complex a model is necessary for a particular device. Both nexus theory and multi-physics results in this dissertation have been successfully used to design devices to operate in specific transition regimes and identify the resulting device behavior.  </p>

Quel est le point commun entre des propulseurs à effet hall, des sources d'ions et les grandes machines de recherche sur la fusion magnétique ? Ils sont tous composés de plasmas interagissant avec des champs électrique et magnétique orthogonaux, et leurs tailles, sophistications et inaccessibilités rendent leur étude directe compliquée. Cette étude directe peut être menée à bien sur des machines plus simples, comme la colonne de plasma magnétisée Mistral utilisée dans ce travail, qui sont conçues pour l’étude de mécanismes fondamentaux. La tomographie, quant à elle, est couramment utilisée dans les tokamaks et stellarators, mais plus rarement sur des machines de laboratoire. Son intérêt majeur est de pouvoir reconstituer l’évolution temporelle de section 2D de plasma, et ce sans mesure intrusive. Dans le cadre de cette thèse un diagnostic de tomographie a été entièrement conçu, installé, calibré et testé. Les modèles existant de tomographie ont d'abord été adaptés à ce nouveau contexte, pour ensuite développer et valider le code complet d’inversion tomographique associé. Puis, une étude de faisabilité a été réalisée en mettant au point un diagnostic de tomographie utilisant un seul capteur avec un échantillonnage conditionnel sur des modes réguliers. L’attention est alors portée au développement, à la configuration et à l’application du diagnostic complet à 128 voies. Enfin, une étude paramétrique des modes réguliers a fait ressortir l'importance des paramètres de contrôle sur les modes (présence, fréquence et parité), et l'attention qui doit être portée à l'ensemble des paramètres expérimentaux, ainsi que l’évolution de leur forme et le comportement du plasma central. / What do satellites thrusters, ions sources, and fusion devices have in common? They all have plasmas with orthogonal electric and magnetic fields and their size, complexity and accessibility often make them hard to be directly studied. Simpler devices, like the linear magnetised plasma device Mistral used during this work, are conceived in order to understand, predict, and eventually control, some of their fundamental mechanisms. To this purpose, a tomography diagnostic is developed. Tomography is a well known diagnostic in tokamaks and stellarators, but remains seldom used in low temperature plasma studies. Its main advantages are to give access to the temporal evolution of a two-dimensional section of the plasma emissivity, and to be non-intrusive. In the frame of this thesis, a tomography diagnostic has been designed from scratch, implemented, calibrated and tested. The first step consists in the adaptation of existing tomography models in this context, and the full development and validation of the associated numerical code. Then, a proof of concept is conducted with a mono-sensor diagnostic using conditional sampling on coherent rotating modes. Following, the development, configuration, and application of the full 128 channels emission tomography diagnostics on Mistral are reported. New insights to characterise coherent rotating modes, such as the evolution of their shape and the behaviour of the core plasma, are given. Additionally, a parametric study of the rotating modes revealed the complex and intricated effect of control parameters on the modes (existence, frequency, and mode number), and the care that has to be put in monitoring many experimental parameters.