High deposition rates on large areas are industrial needs for mass production of microcrystalline silicon (μc-Si:H) solar cells. This doctoral work aims at exploring the usefulness of Matrix Distributed Electron Cyclotron Resonance (MDECR) plasmas to process the intrinsic layer of μc-Si:H p-i-n solar cells at high rates. With the high dissociation of silane achieved in MDECR plasmas, deposition rates as high as 6nm/s and 2.8nm/s have been demonstrated in our lab for amorphous and microcrystalline silicon respectively, without hydrogen dilution. This technique is also promising because it can be easily scaled up on large areas, just by extending the matrix of elementary microwave applicators. This subject was a unique opportunity to cover the whole chain of this field of research : A new MDECR reactor has been specially designed and assembled during this project. Its maintenance and its improvement have been important technical challenges : for example, the addition of a load-lock enabled us to lower the oxygen concentration in our films by a factor of 10. The impact of the deposition parameters (e.g. the ion energy, the substrate temperature, different gas mixtures, the microwave power) has been explored in extensive parametric studies in order to optimize the material quality. Great efforts have been invested in the characterization of the films. Our strategy has been to develop a wide range of diagnostics (ellipsometry, Raman spectroscopy, SIMS, FTIR, XRD, electrical characterizations etc.). Finally, p-i-n cells have been processed with the selected interesting materials. The successive successful improvements in the material quality (e.g. diffusion lengths of holes parallel to the substrate as high as 250 nm) did unfortunately not result in high efficiency solar cells. Their limited performance is in particular due to a very poor response in the red part of the spectrum resulting in low current densities. Consequently, the potential sources of limitation of the reactor, the material and the device have been studied : e.g. the presence of “cracks” prone to post-oxidation in the highly crystallized materials and the risk of deterioration of the ZnO substrate or of the p-doped layer by a too high process temperature or by hydrogen diffusing from the plasma.

Les microplasmas correspondent à des décharges confinées sur de faibles échelles spatiales (typiquement quelques 100 µm) offrant ainsi une grande stabilité pour des densités de puissance de l'ordre de 100 kW/cm3. L'étude de ces décharges s'est fortement développée à partir du début des années 2000 tant sur le plan théorique qu'expérimental. Leur intérêt réside dans la possibilité de générer des plasmas à moyenne et haute pression avec de faibles tensions et dans les nombreuses applications potentielles telle que le traitement de surface, les sources de lumière ou bien encore la micro-propulsion spatiale. Les microdécharges à cathode creuse (MHCD) constituent une configuration particulière de microplasmas. La décharge est générée dans l'argon en appliquant une différence de potentiels entre les deux électrodes qui sont assemblées sous forme d'un sandwich électrode-diélectrique-électrode percé sur toute la longueur par un trou de quelques centaines de micromètres. D'un point de vue expérimental, une étude par imagerie et spectroscopie d'émission nous a permis de déterminer les mécanismes d'émission de la lumière, la structure de la décharge, la température du gaz ainsi que la densité électronique. D'un point de vue théorique, nous avons développé des modèles semi-analytiques permettant d'obtenir des lois d'échelle et des comportements macroscopiques, et de vérifier ainsi les interprétations expérimentales. Malgré le nombre important d'hypothèses et d'approximations sur lesquelles reposent les différents modèles développés, la comparaison entre résultats théoriques et expérimentaux nous a permis de capter une partie de la physique impliquée au sein des MHCD.

<p>The overall goal of the research presented in this thesis is to extend the physical understanding of the unsteady external aerodynamics associated with highly maneuverable delta-wing aircraft by using and developing novel, more efficient computational fluid dynamics (CFD) tools. More specific, the main purpose is to simulate and better understand the basic fluid phenomena, such as vortex breakdown, that limit the performance of delta-wing aircraft. The problem is approached by going from the most simple aircraft configuration - a pure delta wing - to more complex configurations. As the flow computations of delta wings at high angle of attack have a variety of unusual aspects that make accurate predictions challenging, best practices for the CFD codes used are developed and documented so as to raise their technology readiness level when applied to this class of flows.</p><p>Initially, emphasis is put on subsonic steady-state CFD simulations of stand-alone delta wings to keep the phenomenon of vortex breakdown as clean as possible. For half-span models it is established that the essential characteristics of vortex breakdown are captured by a structured CFD code. The influence of viscosity on vortex breakdown is studied and numerical results for the aerodynamic coefficients, the surface pressure distribution and breakdown locations are compared to experimental data where possible.</p><p>In a second step, structured grid generation issues, numerical aspects of the simulation of this nonlinear type of flow and the interaction of a forebody with a delta wing are explored.</p><p>Then, on an increasing level of complexity, time-accurate numerical studies are performed to resolve the unsteady flow field over half and full-span, stationary delta wings at high angle of attack. Both Euler and Detached Eddy Simulations (DES) are performed to predict the streamwise oscillations of the vortex breakdown location about some mean position, asymmetry in the breakdown location due to the interaction between the left and right vortices, as well as the rotation of the spiral structure downstream of breakdown in a time-accurate manner. The computed flow-field solutions are visualized and analyzed in a virtual-reality environment.</p><p>Ultimately, steady-state and time-dependent simulations of a full-scale fighter-type aircraft configuration in steady flight are performed using the advanced turbulence models and the detached-eddy simulation capability of an edge-based, unstructured flow solver. The computed results are compared to flight-test data.</p><p>The thesis also addresses algorithmic efficiency and presents a novel implicit-explicit algorithm, the Recursive Projection Method (RPM), for computations of both steady and unsteady flows. It is demonstrated that RPM can accelerate such computations by up to 2.5 times.</p>

Les agrégats de gaz rare constituent un état de la matière intermédiaire entre les cibles solides et les atomes en phase gazeuse. Il a été démontré que les agrégats irradiés sont sources d'ions, d'électrons, de neutrons énergétiques ainsi que de rayonnement allant du visible aux X durs. Cette source peut-être produite avec un taux de répétition élevé et a l'avantage de ne pas produire de débris, et de présenter une très forte conversion de l'énergie laser incidente. Nous nous intéressons au rayonnement X particulièrement, en le caractérisant en intensité, spectre et durée, comme préalable à toute application. En collaboration avec l'INRS-Energie (Varenne, Canada), nous avons mis en œuvre une caméra à balayage de fente dont la résolution temporelle est de 800 fs rms. En focalisant des impulsions laser courtes (30 fs - 5 ps) et intenses (jusqu'à 1e17 W/cm2) sur des agrégats d'argon (15 - 30 nm), nous avons démontré que l'émission X dont l'énergie est supérieure à 2 keV est plus courte que la résolution temporelle. En couplant la caméra à un cristal tronconique, nous nous sommes intéressés au rayonnement de couche K dans la gamme 2,9 - 3,2 keV. Nous avons démontré que ce rayonnement a une durée inférieure à la résolution temporelle, et que les raies étaient émises avec un écart temporel relatif inférieur à 1 ps. Une simulation basée sur un modèle nano-plasma et sur un code collisionnel-radiatif a été développée au CELIA. Les spectres X résolus en temps calculés reproduisent à la fois la brièveté d'émission du rayonnement X et les états de charge élevés observés.

Pour concevoir les futurs réacteurs à fusion nucléaire, une bonne compréhension des mécanismes régissant l'intéraction plasma-paroi est requise. En particulier, il est nécessaire d'estimer quantitativement les flux de chaleurs impactant les matériaux et la contamination du coeur par les impuretés provenant du mur. Dans ce contexte, le code fluide SolEdge2D a été développé pour simuler le transport dans le plasma de bord. L'interaction plasma-paroi est prise en compte grâce à une méthode de pénalisation innovante et originale. Cette méthode permet en particulier de modéliser la géométrie complexe des éléments face au plasma avec une grande flexibilité. En parallèle, une étude plus théorique sur les propriétés du transport dans les milieux faiblement collisionels a été conduite avec les physiciens du groupe CSDC de l'université de Florence. Une généralisation de la loi de Fourier prenant en compte les corrélation spatio-temporelle à longue distance à été obtenue par l'analyse de modèles stochastiques 1D. Cette loi retrouve en particulier la transition entre un régime diffusif à forte collisionalté et un régime balistique à faible collisionalité. / In the scope of designing future nuclear fusion reactors, a clear understanding of the plasma-wall interaction is mandatory. Indeed, a predictive estimation of heat flux impacting the surface and the subsequent emission of impurities from the wall is necessary to ensure material integrity and energy confinement performances. In that perspective, the fluid code SolEdge2D has been developed to simulate plasma transport in the tokamak edge plasma. The plasma-wall interaction is modeled using an innovative penalization technique. This method enables in particular to take complex plasma facing components geometry into account. In parallel to this numerical effort, a theoretical work has been achieved to find appropriate corrections to fluid closures when collisionality drops. The study of stochastic 1D models has been realized in collaboration with physicists from the CSDC group in Florence. A generalized Fourier law taking long range spatio-temporal correlations has been found to properly account for ballistic transport in the low collisional regime. This formulation is expected to be used to model parallel heat flux or turbulent cross-field transport in tokamak plasmas.

Le plasma est un gaz ionisé qui possède des caractéristiques physiques intéressantes dans le domaine des hyperfréquences. En simplifiant, on peut le caractériser comme un milieu diélectrique dispersif dont la permittivité est fonction de deux paramètres : la pulsation plasma (wp) et la fréquence de collision électron-neutre (Vp). En pratique, ces paramètres dépendent principalement de la densité électronique du gaz et de sa pression. Ainsi, en contrôlant les caractéristiques du plasma, on contrôle sa permittivité diélectrique, ce qui permet d’envisager son application dans le domaine de la reconfigurabilité en hyperfréquence.Parmi les topologies pouvant générer une décharge plasma, nous nous sommes focalisés sur l’utilisation de topologies récentes, à savoir les microdécharges plasma. Ces microdécharges sont intéressantes de par leur facilité d’intégration dans un dispositif RF : petite taille, stabilité, température proche de la température ambiante et perspectives d’utilisation à plus haute pression, voire à la pression atmosphérique.Devant la difficulté de modéliser précisément l’effet du plasma sur une onde guidée, une approche expérimentale a été privilégiée. Deux dispositifs de mesure ont ainsi été conçus pour caractériser cette interaction : une ligne de transmission microruban classique et une inversée intégrant une microdécharge en leurs centres. Grâce au protocole expérimental mis en œuvre, les paramètres S de la ligne de transmission sont obtenus et comparés à ceux des lignes sans plasma dans une large gamme paramétrique, qu’il s’agisse de la pression du gaz, de la fréquence ou encore du courant injecté à la décharge. Les résultats obtenus montrent deux phénomènes particulièrement intéressants: un déphasage de l’onde électromagnétique en présence de la décharge plasma et / ou une absorption importante de la puissance par la décharge.Deux dispositifs antennaires ont finalement été conçus en exploitant ces résultats. Le premier est une antenne imprimée accordable en fréquence dans une plage de l’ordre du pourcent, grâce à une décharge plasma contrôlée.Le plasma modifie alors la constante diélectrique entre les deux conducteurs constitutifs de l’antenne. Le second dispositif est une antenne anneau imprimée qui peut protéger son récepteur d’une attaque microondes de forte puissance. Ainsi, lorsqu’un champ incident dépasse un seuil prédéfini, réglable dans une certaine mesure par une tension continue externe, une décharge plasma apparaît au sein de l’élément rayonnant. Elle crée alors de la désadaptation et de l’absorption qui limitent de façon non linéaire la puissance restituée à l’accès. / Plasma is an ionized gas with physical characteristics that are of interest to the microwave domain. To simplify, we can characterize it as a dispersive medium whose dielectric permittivity depends on two parameters :the plasma pulsation wp and the electron-neutral collision frequency Vp. These two parameters depend mainly on the electron density of the gas and its pressure. If we can control the characteristics of the plasma, we can also control its dielectric permittivity, which allows us to consider the plasma for applications in the field of microwave reconfigurability.Among the structures that can generate a plasma discharge, we have focused on the use of recent topologies,known as plasma microdischarges. These microdischarges are of interest because of the possibility of easily integrating them into a RF device : small size, stability, temperature near room temperature and potential use at high pressures, including at atmospheric pressure.Given the difficulties in accurately modelling the effects of the plasma on a guided wave, an experimental approachwas preferred. Two measuring devices have been designed to characterize this interaction : a conventional microstrip transmission line and an inverted microstrip transmisison line, both including a microdischarge in their centers. With this experimental characterization, the S-parameters of the transmission line with the plasma are obtained and compared to those without plasma as a function of a wide range of parameters, such as gas pressure,frequency and current injected into the discharge.The results show two particularly interesting phenomena : a phase shift of the electromagnetic wave in presence of the plasma discharge and/or an important absorption of the incident power by the discharge. Two devices have been designed to exploit these results. The first is a frequency tunable microstrip patch antenna over a range of the order of one percent. In that case, the plasma changes the dielectric constant between the two conductors of the antenna. The second is a microstrip ring patch antenna that can protect the receiver from a high-power microwave (HPM) attack. When an incident electric field exceeds an adjustable preset threshold (tuned by an external DCvoltage source), a plasma discharge appears in the radiating element. The plasma then creates a mismatch and an absorption effect that limits, in a non-linear way, the received power at its input.

Um dos problemas enfrentados pelos cientistas para o confinamento de plasma em Tokamaks, para se obter fusão termonuclear controlada, é o transporte radial de partículas pela borda do plasma. Nessa dissertação, estudamos o transporte através de um modelo que relaciona as flutuações eletrostáticas na borda do plasma às ondas de deriva. Essas ondas criam no plasma regiões de fluxo convectivo, formando ilhas que são, eventualmente, separadas por barreiras. Para apenas uma onda, o sistema é integrável e todas as trajetórias do plano de fase são curvas invariantes que, se não existirem barreiras, estão em ilhas divididas por separatrizes. Foi verificado que, quando uma segunda onda com velocidade de fase diferente da primeira é utilizada, o sistema não é mais integrável e a região anteriormente ocupada pelas separatrizes torna-se caótica. Com a quebra de separatrizes ocorre o transporte caótico de partículas. Quando uma separatriz é quebrada, surge em seu lugar uma estrutura que ainda preserva algumas características da separatriz, mas se modifica no espaço de fases ao longo do tempo. Essa estrutura é conhecida como Estrutura Lagrangiana Coerente (ELC). Nessa dissertação verificamos que as ELCs, por um lado, funcionam como barreiras de transporte, pois nenhuma trajetória consegue atravessa-la e, por outro lado, criam regiões no espaço de fases onde o transporte é alto, pois trajetórias próximas a elas tendem a ser aceleradas. Uma das principais contribuições obtidas ao se estudar ELCs no problema de duas ondas de deriva, aplicado ao confinamento de plasmas em Tokamaks, é a possibilidade de se prever a existência de ilhas, que funcionem como barreiras de transporte, no plano de fases que, por sua vez, são um importante mecanismo de aprisionamento de partículas. / One of the problems facing scientists in the confinement of plasma in tokamaks, to obtain controlled thermonuclear fusion, is the radial transport of particles at the plasma edge. In this dissertation, we study particle transport through a model that relates the electrostatic fluctuations at the edge of the plasma with drift waves. These waves create regions inside the plasma with convective flow, forming islands that are eventually separated by barriers. For one wave, the system is integrable and all the trajectories of phase space are invariant curves that are divided by separatrices. It was found that when a second wave with phase velocity different from the first is used, the system is no longer integrable and the region previously occupied by the separatrix becomes chaotic. With the destruction of the separatrix the transport of particles is chaotic. When a separatrix is broken, appears in its place a structure that preserves some features of the separatrix, but it is changing in phase space over time. This structure is known as Lagrangian Coherent Structure (LCS). In this dissertation we found that the LCSs, on the one hand, act as transport barriers, since no trajectory can cross it and, moreover, creates regions in phase space where particle transport is high, because trajectories close to them tend to be accelerated. One of the main contributions obtained by studying LCSs in the problem of two drift waves, applied to the confinement of plasma in tokamaks, is the ability to predict the existence of islands, which act as transport barriers, which are an important mechanism of trapping particles.

Propõe-se analisar teoricamente a instabilidade disruptora que ocorre em plasmas confinados em Tokamaks, investigando-se a influencia de campos helicoidais ressonantes em seu equilíbrio. Para isso, considerou-se a superposição do campo magnético do plasma em equilíbrio MHD estático com o campo associado às ressonâncias, levando-se em conta a geometria toroidal do Tokamak. Devido a falta de simetria, as linhas do campo magnético total resultante desta superposição devem formar superfícies magnéticas apenas em algumas regiões do plasma. Usando o método da media, foram obtidas (analiticamente) funções de superfícies aproximadas que contem, em torno das regiões de ressonância do plasma, as linhas deste campo magnético total. Verificou-se que estas superfícies aproximadas têm estruturas de ilhas magnéticas. / It is proposed to analyse theoretically the disruptive instability that occurs in confined plasmas in Tokamaks, through investigation of the influence of resonant helical fields on their equilibrium. With this aim, a superposition of the magnetic field of the plasma in static MHD equilibrium with the field associated with the resonances is considered, taking into account the toroidal geometry of the tokamak. Due to the lack of symmetry, the lines of the total magnetic field resulting from this superposition must form magnetic surfaces only around some regions of the plasma. Using the averaging method, functions of approximate magnetic surfaces are obtained (analitically) around the regions of resonances of the plasma (they contain the lines of this total magnetic field). It was verified that these approximate surfaces have structures of magnetic islands.

Este trabalho descreve o estudo das instabilidades magneto-hidro-dinâmicas (MHD) comumente observadas nas descargas elétricas de plasma no tokamak TCABR, do Instituto de Física da USP. Dois diagnósticos principais foram empregados para observar essas instabilidades: um conjunto poloidal de 24 bobinas magnéticas (bobinas de Mirnov) colocadas próximas à borda do plasma e um medidor de emissões na faixa do Ultra Violeta e de raios X moles com 20 canais (sistema SXR), cujo circuito de condicionamento de sinais foi aprimorado como parte deste trabalho. Esses diagnósticos foram escolhidos porque fornecem informações complementares, uma vez que o sistema SXR observa a parte central da coluna de plasma, enquanto as bobinas de Mirnov detectam as instabilidades MHD na região mais externa da coluna. As informações coletadas por esses diagnósticos foram submetidas à análise espectral com resolução temporal e espacial, possibilitando determinar a evolução das características espectrais e espaciais das instabilidades MHD observadas. Essas análises revelaram que durante a etapa inicial da formação do plasma (quando a corrente de plasma ainda está aumentando) ilhas magnéticas com números de onda decrescente, identificadas como sendo modos kink de borda, são detectadas nas bobinas de Mirnov. Após a formação do plasma, quando os parâmetros de equilíbrio estão relativamente estáveis (platô), oscilações são detectadas tanto nas bobinas de Mirnov quanto no sistema de SXR, indicando a presença de instabilidades MHD em toda a coluna de plasma. Em geral as oscilações medidas nas bobinas de Mirnov tem baixa amplitude e correspondem a pequenas ilhas magnéticas que foram identificadas como sendo modos de ruptura (modos tearing). Por outro lado, as instabilidades na região central foram identificadas como dentes de serra, que correspondem a relaxações periódicas da região interna à superfície magnética com fator de segurança q=1 e que são acompanhadas de oscilações precursoras, cuja amplitude depende da fase do ciclo de relaxação. Devido à essa modulação de amplitude, aparecem picos de frequência satélite nos espectrogramas dos sinais do SXR. Além disso, devido ao fato dos ciclos de relaxação não serem sinusoidais, os harmônicos da frequência de relaxação também aparecem nesses espectrogramas. No entanto, em muitas descargas do TCABR, a intensidade das oscilações medidas nas bobinas de Mirnov aumentam significativamente durante o platô, com efeitos sobre a frequência de todas as instabilidades MHD, até mesmo sobre os dentes de serra localizados na região central da coluna. Em todos os casos, observou-se que durante o platô a frequência das ilhas magnéticas coincide com a frequência das oscilações precursoras do dente de serra, apesar de serem duas instabilidades distintas, localizadas em posições radiais muito diferentes. Essa coincidência de frequências possibilitou descrever a evolução em frequência de todas as oscilações detectadas em diversos diagnósticos com base em apenas duas frequências básicas: a dos ciclos de relaxação dente de serra e a das ilhas magnéticas. / This work describes the study of magneto-hydro-dynamic instabilities (MHD) commonly observed in plasma discharges in tokamak TCABR (at Instituto de Física da USP). Two main diagnostics were employed to observe these instabilities: a poloidal set of 24 magnetic coils (Mirnov coils) placed near the plasma border and a detector of emissions in the Ultra Violet and soft X-ray range with 20 channels (SXR system) which improvement of the signal conditioning circuit was done as part of this work. These diagnostics were chosen because they provide complementary information, since the SXR system measures the central part of the plasma column, while the Mirnov coils detect the MHD instabilities in the outer part of the column. The information collected by these diagnoses was submitted to spectral analysis with temporal and spatial resolution, making it possible to determine the evolution of the spectral and spatial characteristics of the observed MHD instabilities. These analyzes revealed that during the initial stage of the plasma formation (when the plasma current is still increasing) magnetic islands with decreasing wave numbers, identified as edge kink modes, are detected in the Mirnov coils. After the plasma formation, when the equilibrium parameters are relatively flat (plateau), oscillations are detected in both Mirnov coils and SXR system, indicating the presence of MHD instability in the whole plasma column. In general, the fluctuations measured by the Mirnov coils have low amplitude corresponding to small magnetic islands, which were identified as tearing modes. On the other hand, the instabilities at the central region were identified as sawteeth oscillations that correspond to periodic relaxations in the internal region of the magnetic surface with safety factor q = 1 and that are accompanied by precursor oscillations which amplitude depends on the phase of the relaxation cycles. Due to this amplitude modulation, frequency satellite peaks appear in the spectrograms of the SXR signals. Furthermore, due to the fact that relaxation cycles are not sinusoidal, harmonics of the relaxation frequency also appear in the spectrograms. However, in many TCABR discharges, the intensity of the oscillations measured by the Mirnov coils increase significantly during the plateau, with affects the frequency of all MHD instabilities, even over the sawteeth in the central region of the column. In all cases, it was observed that during the plateau the frequency of the magnetic islands coincides with the frequency of the sawtooth precursors, although they are two different instabilities located in separated radial positions. This coincidence of frequencies allowed describing the frequency evolution of all measured oscillations by considering only two basic frequencies: the cycles of sawtooth relaxation and the magnetic islands.

Mapas simpléticos têm sido amplamente utilizados para modelar o transporte caótico em plasmas e fluidos. Neste trabalho, propomos três tipos de mapas simpléticos que descrevem o movimento de deriva elétrica em plasmas magnetizados. Efeitos de raio de Larmor finito são incluídos em cada um dos mapas. No limite do raio de Larmor tendendo a zero, o mapa com frequência monotônica se reduz ao mapa de Chirikov-Taylor, e, nos casos com frequência não-monotônica, os mapas se reduzem ao mapa padrão não-twist. Mostramos como o raio de Larmor finito pode levar à supressão de caos, modificar a topologia do espaço de fases e a robustez de barreiras de transporte. Um método baseado na contagem dos tempos de recorrência é proposto para analisar a influência do raio de Larmor sobre os parâmetros críticos que definem a quebra de barreiras de transporte. Também estudamos um modelo para um sistema de partículas onde a deriva elétrica é descrita pelo mapa de frequência monotônica, e o raio de Larmor é uma variável aleatória que assume valores específicos para cada partícula do sistema. A função densidade de probabilidade para o raio de Larmor é obtida a partir da distribuição de Maxwell-Boltzmann, que caracteriza plasmas na condição de equilíbrio térmico. Um importante parâmetro neste modelo é a variável aleatória gama, definida pelo valor da função de Bessel de ordem zero avaliada no raio de Larmor da partícula. Resultados analíticos e numéricos descrevendo as principais propriedades estatísticas do parâmetro gama são apresentados. Tais resultados são então aplicados no estudo de duas medidas de transporte: a taxa de escape e a taxa de aprisionamento por ilhas de período um. / Area-preserving maps have been extensively used to model chaotic transport in plasmas and fluids. In this work we propose three types of maps describing electric drift motion in magnetized plasmas. Finite Larmor radius effects are included in all maps. In the limit of zero Larmor radius, the monotonic frequency map reduces to the Chirikov-Taylor map, and, in cases with non-monotonic frequencies, the maps reduce to the standard nontwist map. We show how the finite Larmor radius can lead to chaos suppression, modify the phase space topology and the robustness of transport barriers. A method based on counting the number of recurrence times is used to quantify the dependence on the Larmor radius of the threshold for the breakup of transport barriers. We also study a model for a system of particles where the electric drift is described by the monotonic frequency map, and the Larmor radius is a random variable that takes a specific value for each particle of the system. The Larmor radius\' probability density function is obtained from the Maxwell-Boltzmann distribution, which characterizes plasmas in thermal equilibrium. An important parameter in this model is the random variable gamma, defined by the zero-order Bessel function evaluated at the Larmor radius\'particle. We show analytical and numerical computations related to the statistics of gamma. The set of analytical results obtained here is then applied to the study of two numerical transport measures: the escape rate and the rate of trapping by period-one islands.