Le savant n'étudie pas la nature parce que cela est utile ; il l'étudie parce qu'il y prend plaisir et il y prend plaisir parce qu'elle est belle.

-Henri Poincaré

The scientist does not study nature because it is useful to do so. He studies it because he takes pleasure in it, and he takes pleasure in it because it is beautiful. -Henri Poincaré

1

Abstract (English)

Neutral Beam Injectors (NBI) are one of the external heating systems used to control and sustain the fusionplasma in thermonuclear fusion devices such as tokamaks. They rely on the acceleration and neutralization of several beamlets extracted from a negative ion source of hydrogen or deuterium. For the design of the large-scale fusion reactor ITER (International Thermonuclear Experimental Reactor), two NBI are foreseen, each of them providing a power of 16.5 MW to the plasma. This PhD relates the work I did with Consorzio RFX in the field of negative ion accelerators for NBI. The scope of this thesis is to improve the knowledge and understanding of the physics of negative ion acceleration through simulations and analysis of experiments. The first chapter of this thesis addresses the technology of nuclear fusion reactors. First, a brief historical introduction is presented, for purposes of contextualization of the challenges and expectations of fusion reactors. In particular, the ITER project is shortly described. The rest of the chapter details the basic science of nuclear fusion reactors, and more particularly the magnetic confinement devices relying on the Deuterium-Tritium reaction. The next chapter gives more details on the physics and technology of NBI, and defines the terminology used in the rest of the thesis. The main components constituting a NBI for fusion are presented with typical orders of value foreseen for the case of ITER and details on the chemical-physical processes in the plasma source. The third chapter is a state of the art of the experimental facilities and simulation tools on which I worked during this PhD. The fourth chapter presents some analytical results useful for the understanding of negative ion acceleration physics. The chapter is divided in two sections: the first one recalls classical results found in the scientific literature about the extraction and acceleration of a negative ion beam (approximations of the potential profile in the accelerator, thin-lens approximation and Davisson-Calbick formula, potential generated by a charged beam, space charge limited extraction). In the second part, we propose an analytic formalism to describe with a great precision and a relative simplicity the electric potential and field inside a multigrid multiaperture accelerator, based on an approximation found in the literature. The results obtained and their simplification are discussed in detail and confronted with simulations. The equations obtained are useful for analytic developements in the following chapters. In the fifth chapter, we describe an analysis of experiments led during a collaboration between Consorzio RFX and QST Japan on the NITS facility (Naka, Japan). In the first section, we develop an analytic study of the beamlet deflection under the effect of a magnetic field, which provides a physical understanding of the dependence of the beamlet deflection on operating and design parameters. The second part tests those conclusions with numerical tools. It is shown in this part that the magnetic deflection measured during the experimental campaign is significantly underestimated by numerical tools. Besides, the magnetic deflection appears to depend on properties of the plasma source, which is not assumed by simple beam acceleration codes. In the sixth chapter, we study the hypothesis that the discrepancy between measurements and simulations is caused by the wrong hypothesis that the beamlet current density is uniform at the surface where it is extracted from the ion source (referred to as meniscus). First of all, I detail the assumptions made in order to improve the numerical tool used to predict the beamlet deflection, by including the effect of an asymmetric current distribution at the meniscus. In a second part, the effect of this asymmetry on the beamlet deflection is investigated through analytic calculations. I show how several physical effects (the beamlets self-electric force, the grid electric forces exerted on an asymmetric charge distribution and the lens effect of the grids) combine to generate an additional deflection linear to the magnitude of the asymmetry at the meniscus. This analysis is confirmed in a third part by a large number of simulation, allowing to compare the beamlet deflection with the asymmetry. The relation obtained between both quantities is used in a fourth part to heuristically deduce for each measurement on NITS the asymmetry that must be included in simulations to correctly reproduce the measured deflection. This approach allows to propose an empirical law to quantify the current asymmetry at the meniscus. The seventh chapter focuses on the work I performed during an exchange period with IPP Garching (Germany). The objective of this collaboration was to explain the differences in terms of beamlet divergence between the NITS and the BATMAN Upgrade (IPP Garching) facilities. First, details are given on the definition and measurement of the divergence in the frame of NBI accelerators. Through theory and simulations, we show what are the physical parameters with a large influence on the divergence. I then present a large number of simulations with various conditions for both experimental devices, and I provide a probable answer to the problematic, relying on the fact that a substantial part of the beamlet impinges on the grids with BATMAN Upgrade, making it impossible to reach small values of divergence. We provide arguments for this explanation based on experimental measurements, simulations and analytic calculations. In the rest of this section, other possible explanations are presented. In particular, the effect of the radio-frenquency used in the ion source of BATMAN Upgrade on the meniscus is shortly investigated through a simple toy model. The eigth chapter focuses on an experimental campaign led at QST (Japan) on the experimental facility MTF in 2019. The measurements made during this campaign allow to propose a benchmark of the numerical tools used for calculations of beam optics and thermal loads on the accelerator components. The empirical law on the meniscus asymmetry presented in the sixth chapter is confirmed by measurements. A final chapter summerizes the results obtained in this PhD. Finally, a first appendix presents ongoing work with Particle In Cells simulations, aiming at better describing the current density asymmetry at the meniscus that forms in negative ion sources. The study presented in this appendix is an ongoing work that constitutes the successive step to the semi-empirical description of the meniscus asymmetry presented in this PhD. A second appendix shows how the findings of this PhD may imply the need to improve the design of MITICA (the real-size prototype of ITER NBI) if the same phenomenon related to the current asymmetry at the meniscus causes an additional deflection not foreseen by the numerical codes used for its sizing.

Sommario (italiano)

Gli iniettori a fascio neutro (NBI) sono uno dei sistemi di riscaldamento esterno utilizzati per controllare e sostenere un plasma di fusione nei dispositivi di fusione termonucleare come i tokamak. Essi si basano sull'accelerazione e la neutralizzazione di parecchi beamlets estratti da una sorgente di ioni negativi di idrogeno o deuterio. Per la progettazione del reattore a fusione su larga scala ITER (International Thermonuclear Experimental Reactor), sono previsti due NBI, ognuno dei quali fornisce una potenza di 16,5 MW al plasma. Questo dottorato di ricerca descrive il lavoro che ho svolto con il Consorzio RFX nel campo degli acceleratori di ioni negativi per NBI. Lo scopo di questa tesi è quello di migliorare la conoscenza e la comprensione della fisica dell'accelerazione di ioni negativi attraverso simulazioni e analisi di esperimenti. Il primo capitolo di questa tesi riguarda la tecnologia dei reattori a fusione nucleare. In primo luogo, viene presentata una breve introduzione storica, ai fini della contestualizzazione delle sfide e delle aspettative relativi ai reattori a fusione. In particolare, viene brevemente descritto il progetto ITER. Il resto del capitolo descrive in dettaglio la scienza di base dei reattori a fusione nucleare, e più in particolare i dispositivi di confinamento magnetico che si basano sulla reazione deuterio-trizio. Il capitolo successivo fornisce maggiori dettagli sulla fisica e la tecnologia degli NBI, e definisce la terminologia utilizzata nel resto della tesi. I componenti principali che costituiscono un NBI per la fusione sono presentati con i tipici ordini di valore previsti per il caso di ITER e i dettagli sui processi chimico-fisici nella sorgente di plasma. Il terzo capitolo è uno stato dell'arte degli impianti sperimentali e degli strumenti di simulazione su cui ho lavorato durante questo dottorato di ricerca. Il quarto capitolo presenta alcuni risultati analitici utili per la comprensione della fisica dell'accelerazione di ioni negativi. Il capitolo è diviso in due sezioni: la prima ricorda i risultati classici che si trovano nella letteratura scientifica sull'estrazione e l'accelerazione di un fascio di ioni negativi (approssimazioni del profilo del potenziale nell'acceleratore, approssimazione della lente sottile e formula di Davisson-Calbick, potenziale generato da un fascio carico, estrazione limitata dalla carica spaziale). Nella seconda parte si propone un formalismo analitico per descrivere con grande precisione e relativa semplicità il potenziale e il campo elettrici all'interno di un acceleratore multigriglia multiapertura, basato su un'approssimazione trovata in letteratura. I risultati ottenuti e la loro semplificazione sono discussi in dettaglio e confrontati con le simulazioni. Le equazioni ottenute sono utili per gli sviluppi analitici dei capitoli seguenti. Nel quinto capitolo si descrive un'analisi degli esperimenti condotti durante una collaborazione tra il Consorzio RFX e QST Japan sull'acceleratore NITS (Naka, Giappone). Nel primo capitolo, sviluppiamo uno studio analitico della deflessione del fascio sotto l'effetto di un campo magnetico, che fornisce una comprensione fisica della dipendenza della deflessione del fascio con parametri di operazione e di design. La seconda parte verifica queste conclusioni con strumenti numerici. In questa parte si dimostra che la deflessione magnetica misurata durante la campagna sperimentale è significativamente sottovalutata dagli strumenti numerici. Inoltre, la deflessione magnetica sembra dipendere dalle proprietà della sorgente di plasma, cosa che non è assunta dai semplici codici di accelerazione del fascio. Nel sesto capitolo, si studia l'ipotesi che la discrepanza tra le misure e le simulazioni sia causata dall'ipotesi errata che la densità di corrente del beamlet sia uniforme sulla superficie dove gli ioni vengono estratti dalla sorgente ionica (denominata menisco). Prima di tutto, dettaglio le ipotesi fatte per migliorare lo strumento numerico utilizzato per prevedere la deflessione del beamlet, includendo l'effetto di una distribuzione di corrente asimmetrica al menisco. In una seconda parte, l'effetto di questa asimmetria del menisco sulla deflessione del beamlet è studiato attraverso calcoli analitici. Mostro come diversi effetti fisici (la forza autoelettrica del beamlet, le forze elettriche delle griglie esercitate su una distribuzione asimmetrica di carica e l'effetto lente delle griglie) si combinano per generare una deflessione aggiuntiva lineare alla grandezza dell'asimmetria al menisco. Questa analisi è confermata in una terza parte da un gran numero di simulazioni, permettendo di confrontare la deflessione del beamlet con l'asimmetria. La relazione ottenuta tra le due quantità è utilizzata in una quarta parte per dedurre euristicamente per ogni misura su NITS l'asimmetria che deve essere inclusa nelle simulazioni per riprodurre correttamente la deflessione misurata. Questo approccio permette di proporre una legge empirica per quantificare l'asimmetria della corrente al menisco. Il settimo capitolo descrive un lavoro che ho svolto durante un periodo di scambio con IPP Garching (Germania). L'obiettivo di questa collaborazione era quello di spiegare le differenze in termini di divergenza di fascio tra NITS e l'acceleratore BATMAN Upgrade (IPP Garching). In primo luogo, vengono forniti dettagli sulla definizione e la misura della divergenza nel quadro degli acceleratori per NBI. Attraverso la teoria e le simulazioni, mostriamo quali sono i parametri fisici con una grande influenza sulla divergenza. Poi presento un gran numero di simulazioni con varie condizioni per entrambi i dispositivi sperimentali, e fornisco una risposta probabile, basandomi sul fatto che una parte sostanziale del beamlet impatta sulle griglie con BATMAN Upgrade, rendendo impossibile raggiungere piccoli valori di divergenza. Fornisco argomenti per questa spiegazione basati su misure sperimentali, simulazioni e calcoli analitici. Nel resto di questa sezione, vengono presentate altre possibili spiegazioni. In particolare, l'effetto della radiofrequenza usata nella sorgente ionica di BATMAN Upgrade sul menisco è brevemente studiato attraverso un semplice modello giocattolo. L'ottavo capitolo si concentra su una campagna sperimentale condotta a QST (Giappone) sull'impianto MTF nel 2019. Le misure effettuate durante questa campagna permettono di proporre un benchmark degli strumenti numerici utilizzati per il calcolo dell'ottica del fascio e dei carichi termici sui componenti dell'acceleratore. La legge empirica sull'asimmetria del menisco presentata nel sesto capitolo è confermata dalle misurazioni. Un capitolo finale riassume i risultati ottenuti in questo dottorato di ricerca. Infine, una prima appendice presenta il lavoro in corso con le simulazioni di Particle In Cells, con l'obiettivo di descrivere meglio l'asimmetria della densità di corrente al menisco che si forma nelle sorgenti di ioni negativi. Lo studio presentato in questa appendice è un lavoro in corso che costituisce il passo successivo alla descrizione semi-empirica dell'asimmetria del menisco presentata in questo dottorato. Una seconda appendice mostra come i risultati di questo dottorato di ricerca possano implicare l'obbligo di migliorare il design di MITICA (il prototipo a grandezza reale dello NBI di ITER) se lo stesso fenomeno legato all'asimmetria della corrente al menisco provoca una deflessione supplementare non prevista dai codici numerici utilizzati per il suo dimensionamento.

Abstract (Nederlands)

Injectoren met bundels van neutrale deeltjes, Neutral Beam Injectors (NBI), zijn één van de externe verwarmingssystemen die worden gebruikt om het fusieplasma in thermonucleaire fusiemachines met magnetische opsluiting, zoals tokamaks en stellarators, te controleren en te onderhouden. Het verhitten van het plasma tot extreem hoge temperaturen, van enkele miljoenen graden, is noodzakelijk om de fusiereacties te laten doorgaan. Neutrale bundels zien het omringende magneetveld niet en kunnen doordringen in het plasma. De NBI injectie systemen maken gebruik van de versnelling van ionen-bundels tot hoge energieën, die dan worden geneutraliseerd en in het plasma gestuurd. Hiervoor worden typisch ionen van waterstof of deuterium gebruikt. Voor het ontwerp van ITER, de International Thermonuclear Experimental Reactor, in aanbouw in Cadarache (Frankrijk), zijn twee NBI systemen voorzien. Elk injector systeem zal een vermogen van 16,5 MW aan het plasma leveren. De meeste bestaande fusiemachines beschikken over NBI injectoren die vertrekken van bundels met positieve ionen. De energie die de bundels moeten hebben voor de huidige machines, is van de orde van 100 keV. Echter voor grotere machines, zoals ITER, moet de energie van de ionen bundels groter zijn, van de orde van MeV. Dit is nodig opdat de neutrale bundel zou kunnen doordringen tot het centrum van het plasma vooraleer hij wordt geïoniseerd en zijn energie afgeeft aan de omringende plasma ionen. Bij te lage energie zou de bundel enkel het randplasma bereiken. Echter, bij dergelijke hoge energieën is de neutralisatie-efficiëntie van de positieve bundels quasi nul, en dus niet bruikbaar. Om die reden moet worden omgeschakeld naar bundels met negatieve ionen. Dit doctoraat heeft betrekking op het werk dat ik aan het Consorzio RFX heb gedaan in verband met versnellers van negatieve ionen voor deze neutrale beam injectoren. Het doel van dit proefschrift is het verbeteren van de kennis en het begrip van de fysica van deze versnellers door middel van simulaties en analyses van experimenten. Het eerste hoofdstuk van het proefschrift gaat over de technologie van kernfusie reactoren. Eerst wordt een korte historische inleiding gegeven, om de uitdagingen en verwachtingen van fusiereactoren in de juiste context te plaatsen. In het bijzonder wordt het ITER-project kort beschreven. De rest van het hoofdstuk gaat over de basiswetenschap van kernfusie reactoren, en meer in het bijzonder over de configuraties die gebruik maken van magnetische opsluiting van plasmas. De machines zijn gebaseerd op de reactie tussen deuterium en tritium, twee isotopen van waterstof. Het volgende hoofdstuk geeft meer details over de fysica en technologie van NBI systemen, en definieert de terminologie die in de rest van het proefschrift zal worden gebruikt. Het derde hoofdstuk geeft een overzicht van de 'state of the art' van de experimentele faciliteiten en simulatieinstrumenten waaraan ik heb gewerkt tijdens dit doctoraat. Het vierde hoofdstuk laat een aantal analytische resultaten zien, die nuttig zijn voor het begrijpen van de fysica van de versnellers van negatieve ionen. Het hoofdstuk is opgedeeld in twee: het eerste deel verwijst naar de klassieke resultaten die in de wetenschappelijke literatuur over de extractie en versnelling van een bundels met negatieve ionen zijn gevonden. In het tweede deel stellen we een analytisch formalisme voor om met een grote precisie en een relatieve eenvoud de elektrische potentiaal en het veld in een versneller met multiple grids en multiple apertures te beschrijven. De verkregen resultaten en de gebruikte vereenvoudigingen en aannames worden in detail besproken en geconfronteerd met simulaties. De verkregen vergelijkingen zijn nuttig voor de analytische ontwikkelingen die in de volgende hoofdstukken aan bod komen. In het vijfde hoofdstuk beschrijven we een analyse van experimenten die geleid hebben tot een samenwerking tussen Consorzio RFX en QST Japan op de NITS-faciliteit in Naka, Japan. In het eerste hoofdstuk ontwikkelen we een analytische studie van de beamlet deflectie onder invloed van een magnetisch veld. Dit geeft een goed fysisch inzicht in de afhankelijkheid van de beamlet deflectie van operationele en ontwerp parameters. In het tweede deel worden deze conclusies getest met behulp van numerieke tools. Hierin wordt aangetoond dat de magnetische doorbuiging, gemeten tijdens de experimentele fase, significant wordt onderschat door de numerieke tools.

In het zesde hoofdstuk bestuderen we de hypothese dat de discrepantie tussen de metingen en de simulaties wordt veroorzaakt door de verkeerde veronderstelling dat de stroomdichtheid van de bundel uniform is aan het oppervlak waar deze wordt geëxtraheerd uit de ionen-bron (aangeduid als meniscus). Het effect van de meniscusasymmetrie op de afbuiging van de beamlet wordt onderzocht door middel van analytische berekeningen en simulaties. De verkregen resultaten worden toegepast op de analyse van de in het vorige hoofdstuk gepresenteerde experimenten. Een empirische wet op de grootte van de meniscus-asymmetrie wordt verkregen. Het zevende hoofdstuk beschrijft een werk dat ik heb uitgevoerd tijdens een uitwisselingsperiode met het Institut für Plasmaphysik (IPP) in Garching, nabij München, Duitsland. Het doel van deze samenwerking was het verklaren van de verschillen in beamlet divergentie tussen de faciliteiten van NITS (van het instituut in Naka, Japan) en BATMAN Upgrade (van het IPP Garching). We presenteren een mogelijk antwoord, gebaseerd op het feit dat een substantieel deel van de beamlet de rasters beïnvloedt in de BATMAN Upgrade opstelling, waardoor het onmogelijk is om kleine waarden voor divergentie te bereiken. We geven argumenten voor deze verklaring op basis van experimentele metingen, simulaties en analytische berekeningen. Het achtste hoofdstuk richt zich op een experimentele fase onder leiding van de groep van QST (in Japan) op hun experimentele faciliteit, de Megavolt Test Facility (MTF), in 2019. De metingen die tijdens deze experimenten zijn verricht, maken het mogelijk om een benchmark voor te stellen van de numerieke instrumenten die worden gebruikt voor berekeningen van de optiek van de bundel, en de thermische belasting van de versneller componenten. De empirische wet van de meniscus-asymmetrie die we in het zesde hoofdstuk hebben gepresenteerd, kan inderdaad worden bevestigd door deze metingen. In het laatste hoofdstuk worden alle resultaten die tijdens het doctoraatsonderzoek zijn verkregen, samengevat. Tot slot wordt in een eerste bijlage het werk dat nog aan de gang is i.v.m. Particle In Cells (PIC) simulaties, gepresenteerd. Dit onderzoek heeft als doel de asymmetrie van de stroomdichtheid bij de meniscus, die zich in bronnen met negatieve ionen vormt, beter te beschrijven. Een tweede bijlage laat zien hoe de bevindingen van dit doctoraat de noodzaak kunnen impliceren om het ontwerp van MITICA (Megavolt ITER Injector and Concept Advancement, het real-size prototype van de NBI voor ITER) verder te verbeteren. Nuclear fusion is one of the countless natural phenomena brought to light by the joint revolutions of physics and chemistry across the XX th century in fields such as atomic theory, special relativity, quantum mechanics or astronomy.

List of Abbreviations

The physical understanding of matter profoundly changed in 1896 with the discovery of radioactivity [START_REF] Bécquerel | Sur les radiations invisibles émises par les corps phosphorescents[END_REF][START_REF] Sklodowska | Recherches sur les substances radioactives[END_REF].

As the existence of the atom was already widely accepted by scientists, the study of radioactive materials and emissions revealed that atoms were themselves constituted of more primary elements [START_REF] Rutherford | Lxxix. the scattering of α and β particles by matter and the structure of the atom[END_REF] and that atomic transmutation was possible. In this same period, A. Einstein developed the theory of special relativity, from which he derived the equivalency between mass and energy. He immediately perceived the connection between this result and the radioactivity experiments [START_REF] Einstein | Ist die trägheit eines körpers von seinem energieinhalt abhängig?[END_REF]. In nuclear transformations such as radioactive decay, atoms can spontaneously turn into more stable atomic species, keeping the total content of mass and energy involved equal throughout the transformation. The binding energy of the initial atom, which contributed to its total mass before the transformation, is then partially converted into kinetic energy of the products. In 1913, Paul Langevin understood that the small deviations of atom mass measurements from the Prout's law (stating that the mass of any atom is a whole multiple of the hydrogen atom mass, or in other words that each element is a combination of hydrogen atoms) come from the binding energy released when light atoms combine to form heavier ones with a more stable configuration [START_REF] Langevin | L'inertie de l'énergie et ses conséquences[END_REF]. His intuition was confirmed, first by the precise measurement of He and H masses by F. W. Aston in 1920 [START_REF] Fw Aston | The constitution of the elements[END_REF] and later by the achievement of the D-D fusion reaction in laboratory by M. Oliphant and E. Rutherford [START_REF] Laurence | Transmutation effects observed with heavy hydrogen[END_REF] in 1934.

In 1919, J. Perrin briefly suggested in a treaty of chemical mechanics that nuclear fusion might be the source of energy of the Sun [8]. At this time, the origin of the power radiated by the Sun was still an unsolved issue. The hypotheses of gravitational and chemical sources predicted an age of some fifteen million years for the Sun at most, in flagrant contradiction with the billions of years estimated by geologists. In 1920, A. S. Eddington got the same intuition and thoroughly developed it, thus founding the modern theory of the constitution of the stars [START_REF] Eddington | The internal constitution of the stars[END_REF]. He immediately realized that the energy released by the nuclear fusion of hydrogen could be exploited for the needs of humanity: "If, indeed, the sub-atomic energy in the stars is being freely used to maintain their great furnaces, it seems to bring a little nearer to fulfilment our dream of controlling this latent power for the well-being of the human race -or for its suicide."

The birth of the fusion reactor concept

If the first hydrogen bomb test (Ivy Mike) in 1952 confirmed the last part of Eddington's prediction, it is also in the years 1950s that the first concrete researches on peaceful applications of nuclear fusion began.

The first record of a design of constructible reactor is a proposal of O. A. Lavrent'ev made in the years 1950s [START_REF] Bondarenko | Role played by o a lavrent'ev in the formulation of the problem and the initiation of research into controlled nuclear fusion in the USSR[END_REF]. Lavrent'ev was a soldier of the USSR who started his education in nuclear physics in 1941. As he served as a radiotelegraph operator on the island of Sakhalin, Russia, he completed his scientific education and delivered lectures on nuclear technology to the army officers. In 1949 he exposed his idea of using thermonuclear fusion for industrial energy production in a letter addressed to the Central Committee of the Communist Party of the Soviet Union and proposed a design of reactor, that could be described in modern terms as a fusor (a fusion device based on inertial electrostatic confinement). Fig. 1.1 shows an interpretation of this first reactor design based on the description made in [START_REF] Bondarenko | Role played by o a lavrent'ev in the formulation of the problem and the initiation of research into controlled nuclear fusion in the USSR[END_REF]. The reactor is made of two spherical concentric electrodes. The inner electrode is a transparent grid with a strong negative potential, emitting electrons. The outer electrode is connected to an ion source that emits D + ions. Both species form a plasma in the space between the electrodes. A small fraction of the accelerated D + passes through the grid resulting in a very high density of D + at the center of the device, which generates D-D fusion reactions.

Lavrent'ev's idea got the interest of some the most influent USSR nuclear physicists, A. Sakharov and I. E. Tamm, who developed the concept of torus shaped magnetic fusion reactors in the 1950s [START_REF] Ie Tamm | Teoriya magnitnogo termoyadernogo reaktora (the theory of magnetic fusion reactor) pt. i, iii[END_REF][START_REF] Sakharov | Teoriya magnitnogo termoyadernogo reaktora (the theory of magnetic fusion reactor) pt[END_REF]. This concept later evolved to become the tokamak, from "toroidal'naya kamera s magnitnymi katushkami" ("toroidal chamber with magnetic coils"). In the same period, the first working tokamak, the T-1, was conceived by N. A. Yavlinsky and built in 1958. In 1951 the American scientist L. S. Spitzer, inspired by the (false) news of the achievement of an operating fusion reactor in Argentina, also approached the problem of nuclear fusion for energy production. He elaborated a 8-shaped reactor, called Stellarator [START_REF] Spitzer | A proposed stellarator[END_REF], in the Princeton Plasma Physics Laboratory. In the next years the United States funded intensive endeavors to develop a fusion reactor, in the frame of the Sherwood Project [START_REF] Amasa | Project sherwood: the us program in controlled fusion[END_REF]. On the belief that plasma equilibrium was impossible to reach with a tokamak configuration, the project focused on alternative reactor designs from 1951 to 1958, such as the Spitzer's stellarator and its further enhanced versions, or reactor concepts based on plasma pinch. The researches on fusion reactors became an international matter when I.V. Kurchatov shared the advancements of the USSR on fusion with the rest of the world in 1956 [START_REF] Kurchatov | The possibility of producing thermonuclear reactions in a gaseous discharge[END_REF]. In 1953, the United States launched the "Atoms for Peace" program as an attempt to pool the efforts of all nations towards peaceful nuclear applications. Consequently to the creation of this program, international conferences to share the knowledge on atomic reactors (International Conferences on the Peaceful Uses of Atomic Energy) were regularly organized. The second conference, held in 1958, was centered on the use of nuclear fusion to produce energy at an industrial scale and affirmed the necessity of an international collaboration to reach this goal [START_REF]Proceedings of the second united nations international conference on the peaceful uses of atomic energy[END_REF], while scientists started to share their efforts for this purpose [START_REF]Harnessing nuclear fusion[END_REF]. In a context of relaxation of the strained relations between the USA and the USSR in the years 1980s, the idea of an international collaborative megaproject appeared as the best way to achieve the control of nuclear fusion energy for civil applications. In a joint statement of the Geneva Summit of 1985, Ronald Reagan and Mikhail Gorbachev "emphasized the potential importance of the work aimed at utilizing controlled thermonuclear fusion for peaceful purposes" [START_REF]Union of soviet socialist republics -united states: documents from the geneva summit[END_REF]. Two years later the Soviet Union, the European Atomic Energy Community, the United States, and Japan launched an international project named ITER (International Thermonuclear Experimental Reactor) aiming at building a nuclear fusion reactor based on the tokamak concept. A first conceptual design was agreed upon in 1990 [START_REF] Tomabechi | ITER conceptual design[END_REF], followed by two phases of Engineering Design Activity, from 1992 to 1998 and from 1998 to 2001. The United State briefly left the project after the first phase but joined it again, along with China and South Korea in 2003, and India in 2005. In 2006 the construction of the experiment at Cadarache (France) was decided and the ITER organization was created in 2007, officializing the beginning of the project. At the time of this PhD, the first plasma of ITER is announced for 2025. The assembly phase of the machine is beginning and the completion to first plasma is amounted to 71.1% [18]. In parallel to the studies on the design and engineering of ITER, an intensive research work on tokamaks has been led all around the world, significantly broadening the physical and technological knowledge on fusion reactors. A non-exhaustive list of milestones, taken from the more complete list of [21], is displayed in table 1.1. Along with technological achievements, the better understanding of tokamak plasma behavior revealed a high difficulty to maintain a good confinement because of plasma instabilities. Rather than a fusion reactor, ITER is now conceived as a full-scale experiment devoted to the understanding of hot plasma physics in a large tokamak and to the demonstration that nuclear fusion can be used for industrial purpose. The achievement of production of electricity on the grid is foreseen for a successive step, defined in 2004 under the name of DEMO (Demonstration Power Plant) [START_REF] Llewellyn | The fast track to fusion power[END_REF], designating the full-scale reactor(s) built with the knowledge acquired on ITER.

Joint international researches on fusion and the ITER project

Role of nuclear fusion in the future

Throughout the XX th century, the control of thermonuclear fusion for energy production was mainly viewed as a scientific challenge and a step toward the amelioration of human condition. During the Cold War it became a symbol of appeasement, pooling the efforts of all nations towards a peaceful application of nuclear physics. More recently, the growing awareness of resource scarcity and of the climate change, caused mostly by the greenhouse gas emissions, have made of alternative clean energies a major social issue. The anomalous increase in temperature of the atmosphere observed in the last century appears to be directly related to the massive use of fossil fuel since the beginning of the industrial area. As the population and the global development of mankind grow, the use of fossil fuel at the global scale keeps increasing (Fig. 1.3). The unpredictable consequences of the climate change are a major society problem, which has configured the researches on energy sources throughout the XXI st century. The use and development of available clean energies, such as nuclear and renewable energies, are essential to mitigate climate change. Among all the energy sources currently available, nuclear fission is the only one to combine: a high capacity factor (70-100%) [START_REF] Van | The spatial extent of renewable and non-renewable power generation: A review and meta-analysis of power densities and their application in the us[END_REF] a large surface power density, around 0.2-0.3 kW m -2 [START_REF] Van | The spatial extent of renewable and non-renewable power generation: A review and meta-analysis of power densities and their application in the us[END_REF] low carbon emissions throughout the life cycle (10-30 g/kWh of CO 2 ) [START_REF] Victor Nian | Life cycle analysis on carbon emissions from power generation-the nuclear energy example[END_REF] a high degree of safety (0.07 deaths resulting from pollution and accidents per TW h) [START_REF] Markandya | Electricity generation and health[END_REF] For this reason, nuclear energy is expected to play a substantial role in the future energy mix. Indeed, all the scenarios presented in 2018 by the IPCC (Intergovernmental Panel on Climate Change) where global warming is limited to 1.5°C involve an increase of nuclear share in the energy mix during the next thirty years [START_REF] Rogelj | Mitigation pathways compatible with 1.5 c in the context of sustainable development[END_REF]. The main drawbacks of nuclear fission are the following: uranium is a limited resource and its extraction is a complex technical and geopolitical issue, a nuclear disaster can have worldwide consequences, and nuclear use by geopolitical players who would lack the required expertise in safety can constitute a worldwide threat, nuclear plants produce long-lived radioactive waste, whose treatment is a technical and ethical challenge.

If future nuclear fusion power plants become able to reach the performances of current fission plants in terms of capacity factor and surface power density, which is expected by current theories, they could play the same role in abundant clean energy production while avoiding the three above-mentioned problems. Economically speaking, nuclear fusion reactors appear especially relevant as baseload power plants in an energetic mix that includes renewable energy, in an economy where a CO 2 policy penalizes the most polluting energy sources [START_REF] Lako | The long-term potential of fusion power in western europe: Markal scenarios until[END_REF]. 1.2 Scientific basis of nuclear fusion for industrial applications

Nuclear binding energy

The configuration of an atom nucleus is governed by two interactions:

the electric interaction, carried by the photons. This force repels the particles of same charge (Coulomb repulsion), has an infinite range and an intensity that increases at shorter range.

the strong interaction, carried by mesons in the case of interactions between nucleons. If the two nucleons are very close (that is, closer than a threshold value around 0.8 fm) the strong interaction is repulsive and indefinitely increases as the distance between the nucleons gets smaller, as a result of Pauli exclusion force. However, beyond this threshold the strong force is attractive with a large intensity. This intensity vanishes to a negligible value in a range of a few fm. At a short range the strong force overcomes the Coulomb repulsion, which allows the binding of the protons and ensures the cohesive integrity of atom nuclei. The structure of an atom nucleus is determined by a balance between those two forces. For an isolated proton of a nucleus, the interactions with its close neighbors are dominated by the attractive strong interaction while the interactions with the distant ones are dominated by Coulomb repulsion (Fig. 1.5). As a consequence, light nuclei become more stable as their number of nucleons increases because their protons remain close to each other and the number of bounds governed by strong interaction increases. As the nuclei get bigger, the distant neighbors of an isolated proton become more numerous, which reduces the stability of the nucleus as Coulomb repulsion starts to overcome the strong interaction at the scale of the whole nucleus. The stability of an atom nucleus is quantified by its binding energy per nucleon. For a stable nucleus this energy is negative as the dominating interaction between the nuclei is attractive, and the nucleus is more stable when this energy gets higher in absolute value. The stability of atomic species tends to increase with the number of nucleus, until the Iron-56 ( 56 Fe), which is the most stable atomic configuration. Beyond this point, increasing the mass of the nucleus decreases its stability, and there are no stable elements heavier than the Lead-208 ( 208 Pb). This behavior is summarized by the Aston curve (Fig. 1.6). Although the binding energy is presented here as a negative quantity, it is sometimes defined as the energy that must be brought to separate the nucleus components (in which case it is a positive quantity).

Fusion reactions in a plasma

A fusion reaction occurs when two atomic nuclei assemble to generate a heavier one. If the final atomic configuration is more stable than the initial one (that is, if the total binding energy per nucleon gets lower in the new configuration), the reaction can spontaneously occur once the atoms are close enough for the strong interaction to overcome the Coulomb barrier. A part of the binding energy of the initial configuration is then converted into kinetic energy of the products. With few exceptions, nuclear fusion reactions of elements lighter than the Iron are exoenergetic.

As explained above (Fig. 1.5), a fusion reaction can occur only if the distance between the nuclei is of the order of some fm. Before reaching such a small distance, the nuclei are repelled by an increasingly strong Coulomb force. To overcome the Coulombian barrier, two atoms must head toward each other with an extremely high velocity. In the case of a fusion reactor, the fusion reactions must occur in a closed and fixed area, usually in a confined gas. In such case, the velocity of particles is mostly thermal velocity, which means that fusion reactions can happen only at extremely high gas temperatures. At those temperatures, the atoms of the gas lose their electrons and the particles form a plasma (a gas of atoms, ions and free electrons). More precisely, the transition from the state of neutral gas to the state of plasma is characterized by the degree of ionization of the gas, that is, the proportion of ions in the gas. For a gas characterized by a density n i of ions and n n of neutrals, the ionization degree is:

α = n i n i + n n
While in a non-ionized gas the main interaction between particles is collisions, the motion of a particle in plasma affects other particles through electromagnetic interactions, even at long ranges. One can consider that the plasma state is reached when the ionization degree α is high enough so that collective behavior driven by electromagnetic interactions develop at a large scale. In fusion plasma, the temperature is so high that α ≈ 1 (plasma fully ionized). In colder plasmas such as ion source plasmas, the gas is only partially ionized. Another property of plasma is its quasi-neutrality: while the ionized gas contains ions and free electrons, its global electric charge is almost zero. This is a consequence of the electric screening occurring in the plasma due to the high mobility of the charge carriers. An electric potential generated locally by a concentration of charges inside the plasma is screened with a characteristic length called Debye length, much smaller than any typical dimension of the extension of the plasma. When the ion temperature is negligible with respect to the electron temperature, the Debye length is:

λ D = ε 0 k B T e n e e 2
where T e and n e are the electronic temperature and density. Finally, the energy of the plasma relates to the thermal energy of its species. For a given ion temperature, the thermal velocity of different species of ions decreases with their mass. Therefore, for a same power injected inside a plasma, its particles collide with higher velocity if they are lighter species, resulting in a higher fusion reaction rate. For this reason, fusion reactions with light nuclei should be used for industrial purpose, in order to reach a correct reaction rate with achievable plasma temperature. A list of possible fusion reactions for industrial energy production is found in [START_REF] Kikuchi | Fusion physics[END_REF]: Because of technological limits, it is impossible to confine a large volume of plasma for a reasonable time with a temperature much higher than a few tens of keV. At such temperatures, the most convenient reaction is the reaction 1 (Fig. 1.7). In this case, the energy extracted by the reactor is carried by the neutrons. If future technologies make it possible to confine plasma at much higher temperature for a long time, the aneutronic fusion reactions (such as reactions 3, 5, 7, 8, 9, 10) would become possible. This would result in much more efficient reactors, but also eliminate some critical problems linked to the recuperation of the neutron kinetic energy (neutron radiation, ionizing damage, material activation). In present researches, the scope of an industrial fusion reactor is to produce and sustain a plasma of D and T hot enough to reach a good reaction rate, for a time sufficient to produce a large amount of energy (comparable to the production of a fission reactor over the long term) so that the reactor efficiency becomes high enough for industrial purpose. For the D-T reaction with a plasma temperature in the order of 10 -100 keV, these conditions are described by the simple Lawson criterion:

1. D + T --→ 4

D-T fusion reaction

nT τ > (nT τ ) threshold
where T is the plasma temperature, n the plasma density and τ the confinement time (the rate at which the plasma energy is dissipated). When the triple product reaches the threshold value, the power produced inside the plasma by the fusion reactions equals the power lost by the plasma and the fusion reactions are in theory self-sustained. The reactor efficiency is determined by the ratio Q = P f usion /P heat , where P heat is the power brought to the plasma by external heating systems. In steady-state operation P heat = 0.

In the current researches on D-T fusion, two important approaches are considered to fulfill the Lawson criterion:

the magnetic confinement: the plasma is confined by magnetic fields so as to obtain long confinement times (some seconds). In this case the plasma dispersion is countered by the magnetic pressure, but this is achievable only with a small plasma density and magnetic fields of large magnitude. For a magnitude of the field around 1-10 T inside the reactor, which is the present technological limit, the plasma density should not exceed 10 20 -10 21 m -3 . Beyond this point the magnetic pressure is not enough to counter the expansion of the plasma.

the inertial confinement: the plasma is first compressed at extremely high density and then let free to disperse. Only the particle inertia prevents the confinement time to be zero, but during this short time the plasma density and reaction rate get very high (for current experiments the goal is a density around 10 3 kg m -3 ). This solution requires to inject a large and focused power on a small target to compress it at high density. For example, in the case of the hydrogen bomb, a target of D and T is compressed at very high density by the power transmitted by the explosion of a fission bomb in a first stage. For industrial applications, the target should be compressed in a controlled and reproducible way and the energy brought to it as low as possible. Currently, the most promising solution for this goal is to use a large number of high power lasers that deliver a large energy in a short pulse to a target of some millimeters.

Both technologies rely on different ways to satisfy the Lawson criterion and face different kind of challenges specific to their approaches. In the case of inertial fusion, the major problem encountered is the anisotropy of the target compression caused by plasma instabilities. If the target does not compress with a very good isotropy, the fusion reaction becomes much less efficient. The physics of a target compression at very high density is not fully understood and the numerical simulations are very complex. Currently, the largest experiments of inertial fusion are the Laser Megajoule (Bordeaux, France) and the National Ignition Facility (Livermore, California).

In the case of magnetic fusion, several concepts have been elaborated (tokamak, stellarator, spheromak, RFP...) among which the tokamak is the most studied at the present time. The major problems encountered in such configurations come from the difficulty to confine and sustain the unstable plasma for a long duration (in particular, to prevent and mitigate disruptions), and to recover the energy of the neutrons without damaging the reactor components. This PhD is dedicated to the reactors based on magnetic confinement, and the challenges and achievements of this technology will be more detailed in the following. There exist other concepts aiming at controlling the D-T fusion or other reactions, especially the aneutronic ones. Those concepts include for example the Z-machine, exploiting the zeta pinch [START_REF] Haines | Fusion research at imperial college[END_REF][START_REF] Vp Smirnov | Fast liners for inertial fusion[END_REF], the Dense Plasma Focus concept [START_REF] Krishnan | The dense plasma focus: A versatile dense pinch for diverse applications[END_REF] or the colliding beam fusion concept [START_REF] Maglich | The migma principle of controlled fusion[END_REF]. At the present time, such alternative approaches have not given results as promising as the tokamak configuration, but constitute an active part of researches on fusion reactors.

Magnetic confinement

The approach of the magnetic confinement to satisfy the Lawson criterion is to maximize the confinement time of the plasma by forcing its particles to follow magnetic lines. In the case of the tokamak configuration, the charged particles of the plasma are trapped inside a torus-shaped vessel. The direction tangent to the torus major circumference is called "toroidal" while the direction tangent to the minor circumference is called "poloidal". A first set of coils circling the torus minor section (toroidal field coils) generate a magnetic field along the toroidal direction. However, following those magnetic lines, the particles are deflected along the poloidal direction by a curvature drift, with a direction depending on their charges. If there were only this first set of coils, the particles of opposite charges present in the plasma would be deflected in opposite directions by the curvature drift. This would result in a charge separation and therefore a polarization drift directing the particles towards the reactor walls. The solution used in tokamaks to prevent this effect is to add a second magnetic field along the poloidal direction by circulating an electric current inside the plasma. The current is induced by coils wound along the toroidal direction, inside the torus (central solenoid) and outside (poloidal field coils). Current can also be driven in the plasma by other means than magnetic induction, as will be detailed further. The current circulating through the plasma and the toroidal coils generate helicoidal magnetic lines inside the torus. Charged particles that follow those lines are trapped inside the vessel as the curvature drift direction is constantly changing. The plasma current is induced by the central solenoid and the poloidal coils. Picture from [START_REF] Kikuchi | Fusion physics[END_REF].

The principle of magnetic confinement is therefore to prevent the plasma from heading toward the walls under the effect of its pressure by forcing the particles to follow the lines of magnetic field. For this reason, a high magnetic field and a low plasma temperature and pressure should be preferable. However a high temperature and a sufficient density are also necessary to achieve a correct fusion reaction rate (Fig. 1.7), which means that the plasma pressure cannot be arbitrary low. Indeed, the plasma pressure for ITER is foreseen to be up to a few bars. To confine a plasma at such pressure, the magnetic field inside the reactor must have a magnitude of several T, and for those high values the coils must be made of superconductor materials (niobium-tin Nb 3 Sn and niobium-titanium Nb-Ti for ITER). Theoretically, a magnetic fusion device is able to reach a steady state and continuously produce energy as long as it is fueled. The 3.5 MeV 4 He + ions produced by the fusion reactions are confined by the magnetic field and deposit their excess energy in the plasma, while the neutrons escape the magnetic barrier and deposit their energy outside of the vessel. The fusion reactions become self-sustaining when the power brought by the 4 He + products equals the power lost by the plasma. In this case it is no more necessary to heat the plasma, P heat is zero and the factor Q becomes infinite. This limit is called the ignition. For reactors foreseen in the short and medium term the ignition will not be reached, the goal of ITER is to reach a value Q=5 while successive reactors should go up to a range of 10-100. Before the fusion reactions happen, external heating systems are necessary to increase the plasma temperature. Initially, the plasma is heated by Joule effect by the current flowing through it, but the plasma resistivity decreases with the electron temperature [START_REF] Robert S Cohen | The electrical conductivity of an ionized gas[END_REF] and the ohmic heating becomes irrelevant once high temperatures are reached. To fill the gap between the temperature where Joule effect becomes too weak and the 10-30 keV required to reach the ignition, two main technologies are employed: bring energy to the plasma through electromagnetic waves and through high momentum neutral particles. The first solution relies on antennas that emit high-energy waves inside the plasma. The electromagnetic energy of the wave is transmitted to the particles by cyclotron heating: the frequency of the wave is in resonance with the angular frequency of the charged particles in their circular motion around the magnetic lines. For ITER, two kinds of device are used to inject power inside the plasma through waves:

Ion Cyclotron Resonance Heating, injecting waves of frequency 40 to 55 MHz, heating the plasma ions. Heated ions can acquire an energy comparable to the alpha particles produced by D-T fusion reactions (some MeV).

Electron Cyclotron Resonance Heating, injecting waves of frequency 100 to 200 GHz, heating the plasma electrons. Besides the heating function, ECRH also has applications for the control of the plasma density and temperature profile. Indeed, the resonance frequency of the electrons depends on the magnetic field magnitude, which varies inside the vessel. By selecting a precise frequency, it is therefore possible to locally heat the plasma electrons.

The second solution consists in injecting a beam of high momentum neutral particles inside the plasma, so that they deposit their power by collisional processes. This heating system is called Neutral Beam Injector (NBI). A detailed description of this technology is given in next chapter. A heating power of 50 MW is foreseen for ITER, among which 33 MW will be brought by two NBIs. As understanding of tokamak plasma behavior evolves, it appears that external heating systems are not only useful to reach the ignition, but also to control the plasma instabilities. Moreover, it has been discovered that, beyond a threshold heat power level, the confinement time of the plasma abruptly increases (typically by a factor of 2, or even more), passing in what is called the H-mode. The H-mode confinement allows a large reduction of the tokamak size, and has practically become a requirement for large size reactors.

NBI integration and function

Because of the magnetic barrier inside the tokamak, the only entities that can be sent directly inside the plasma are waves and neutral particles. NBI aims at focalizing and accelerating neutral atoms to inject them inside the plasma. To avoid the introduction of impurities inside the tokamak plasma, the NBI must inject a beam of neutral H or D atoms. The neutral beam brings heat to the plasma, through the kinetic energy of the injected particles dissipated by collisional processes, and momentum to the plasma particle, thus affecting the rotation speed and current of the plasma. For the operation of a fusion reactor, a NBI can serve the following purposes:

Plasma heating: the neutral beam penetrates the plasma and its atoms are ionized by collision with the plasma species or by charge exchange process. The mean free path of the neutral atoms in the vessel is proportional to their energy E:

λ[m] ≈ 5.6 × 10 17 E n[m -3 ]A[u]
with n the plasma density and A the species atomic mass. For ITER nominal parameters, the mean free path of injected neutral D is around 2 m. For large tokamaks, a high beam energy of the order of 1 MeV is necessary for the neutral particle to reach the central part of the plasma. Once the injected atoms become fast ions, a part of them release their energy to the ions and electrons of the tokamak plasma.

Current drive: the ionization of the neutral beam results in an ion current in the direction of injection. The background plasma electrons, dragged by the fast ions, tend to reduce the ion current but do not cancel it totally. Thus, if the velocity of the neutral beam has a toroidal component (co or counterinjection), a current is driven in the toroidal direction. If the plasma current is magnetically induced by the poloidal and central coils, the reactor is necessarily pulsed and the duration of each cycle is limited by the maximal increase of the primary winding current. Driving a part of the current plasma by other means than magnetic induction allows to extend the duration of a cycle. For ITER, the current driven by the two NBI is estimated around 3 MA (20% of the total plasma current). Non-magnetic current drive is also necessary for devices that do not include poloidal field coils (such as the stellerator). Some scenarios for future tokamaks to be conceived after ITER rely on a steady-state operation of the machine. In this case the plasma current must be driven by other means that magnetic induction. In those configurations, NBI plays a major role as a current driving device.

Shape the plasma current density profile: the stability of the plasma is very dependent on the plasma current density profile. Similarly to ECRH, NBIs allow to act on this profile during operations by driving current locally in the plasma. Indeed, the penetration of the beam inside the plasma directly depends on its energy, and consequently, it is possible to heat a local region of the plasma by choosing a correct neutral beam energy. For example, it has been shown that a NBI allows the control of sawtooth oscillations in the tokamak JET [START_REF] It Chapman | Sawtooth control using off-axis nbi[END_REF]. Neutral Beam Injection was also used on DIII-D to access a H-mode without Edge Localized Modes (Quiescent H-mode) [START_REF] Burrell | Diii-d quiescent h-mode experiments with co plus counter neutral beam injection[END_REF], by using at the same time co and counter-injection. NBI are a powerful tool for fusion reactors, but a precise control of the injected energy is needed, and, for large scale tokamaks such as ITER, the injected energy must be very high, of the order of one MeV.

Chapter 2

Physical and technological description of NBI for fusion

Components of a NBI

A Neutral Beam Injector for fusion is a complex system whose function is to produce a well-focused high energy neutral beam of H or D. Since directly accelerating and focusing neutral particle is not technologically practicable, it is necessary to first produce, accelerate and focus a beam of H and D ions, and then neutralize it with a good efficiency. Most of the existing tokamaks with NBI use positive ions, because this choice prevents free electrons to be extracted and accelerated together with the ions. However, for large tokamaks such as ITER where a high energy is needed, the neutralization cross-sections of H + and D + ions fall to unacceptably low values (Fig. As the ion beam energy in the neutralizer gets higher, it is more and more difficult to neutralize the positive ions. At 500 keV (order of magnitude for ITER) the neutralization efficiency is considerably higher with negative ions. Curve from [START_REF] Koch | Fast particle heating[END_REF].

For this reason, NBI for ITER must rely on the extraction and acceleration of Hand Dions, despite the issues caused by the electrons co-extraction and co-acceleration. A negative ion based NBI can be globally described in four parts: a plasma source that produces specific species of ions in a confined plasma, keeping them available in an extraction area for extraction and acceleration.

an acceleration stage that extracts the ions from the source to form a beam (or a group of smaller beams, refered to as beamlets, that later combine to form a single larger beam). The beam or beamlets extracted are then accelerated by an electrostatic field towards the exit of the accelerator.

Chapter 2. Physical and technological description of NBI for fusion a neutralizer where a fraction of the ion beam particles is neutralized while keeping a high kinetic energy and a good focus. The neutralizer is associated with an Electron Dump to eliminate the electrons extracted from the source or produced inside the accelerator stage by stripping reactions. At the exit of the neutralizer a well-focused beam of highly energetic neutral particles propagates towards the tokamak plasma, going through the NBI duct.

a Residual Ion Dump system located downstream of the neutralizer, to eliminate the non-neutralized ions. Although this technology is optional for small or medium NBI, it becomes necessary for large NBI where a high power is carried by the residual ions.

Plasma source

Plasma sources are used for various applications including particle accelerators, ion thrusters for spacecraft propulsion, sputter deposition, mass spectrometry, ion etching, isotope separation, etc. Given this wide range of applications, a large diversity of models of ion sources exists with many specificities such as the ion species, the value of current extracted from the source, the interface with the acceleration stage (slit, circular aperture, multi-apertures...). In this PhD, we detail the technological characteristics of ion sources for NBI used in fusion experiments, and especially for ITER. The numerical orders of value of the source parameters are given for an ITER-like NBI. The design of the ITER NBI ion source is based on a single plasma chamber with 8 RF plasma drivers. Large machines for fusion experiments require the productions of hydrogen or deuterium negative ions. For such large experiments the current extracted from the source is around 10-50 A, which is usually achieved with an extracted current density around 100-300 A m -2 over an extraction area around 0.1-0.2 m 2 . [START_REF] Hemsworth | Overview of the design of the ITER heating neutral beam injectors[END_REF][START_REF] Takeiri | Negative ion source development for fusion application[END_REF]. The ion source for ITER NBI has the structure of a tandem source, meaning that it consists in two adjacent regions: a driver region, located at the back of the source, where the plasma is generated, and an expansion region where the ions circulate from the driver to the extraction area (Fig. 2.3). At the extraction area the negative ions (along with an unwanted fraction of co-extracted electrons) are extracted through the apertures of a Plasma Grid (PG) by an electrostatic field (the extraction field).

Various concepts exist to generate the plasma in the driver region, but for current NBI experimental devices two of them are commonly used: in arc sources electrons are thermo-emitted by hot cathodes (tungsten filaments heated to 2000-3000 K). The charged particles produced and already present in the source are accelerated by the electric field between the cathodes and the source vessel, and produce a plasma by electronic avalanche. Both the thermionic emission rate and the intensity of the electron avalanche are determined by the filament voltage and current, which allows an active control of the plasma density inside the driver. The plasma density is directly related to the electrical power flowing through the filaments, noted P arc .

in RF sources a coil surrounding the driver induces an oscillating electric field (the field frequency is commonly around 1 MHz), that accelerates the electrons until their kinetic energy is high enough to strip them from the atoms and generate a plasma (referred to as inductively coupled plasma). The rate of electron stripping -and therefore the plasma density -relates on the RF power delivered to the plasma inside the driver, noted P RF . Also in this case, an active control of the plasma density is possible.

In both cases, the power transmitted to the plasma is around 10-100 kW for nominal operations. Unlike RF sources, the arc sources have the disadvantage that the filaments in contact with the plasma erode and can break, which imposes frequent and unpredictable maintenance of the source. Minimizing and facilitating the maintenance of all components is a critical specification for the design of ITER, and RF sources have therefore been chosen for ITER HNB. After its generation in the driver, the plasma circulates in the expansion chamber, so called because the space available for the plasma is larger than in the driver. Consequently, the plasma density decreases (typically by a factor of 10). The plasma is confined by permanent magnets placed outside the plasma chamber walls, which generate a cusp magnetic field on the walls of the chamber with an intensity of some tenth of Tesla at some millimeters from the edges. The electrons enter the expansion region with energy around 10 eV and density around 10 18 m -3 , which is about ten times higher than the ion density. It is necessary to reduce their energy while they propagate towards the extraction area because the negative ions have a low binding energy and are easily lost by collision with the background electrons. Besides, if the electrons are not slowed down, a large electronic flux is directed toward the PG, causing the extraction of a high current electron beam alongside with the negative ion beam.

To solve these issues, a transverse magnetic field called filter field is generated inside the source to eliminate the electrons of highest velocity. Indeed, the Lorentz force exerted on the electrons that propagate towards the extraction area is stronger for the most rapid ones, hence the filter effect. The flux of electrons towards the extraction zone is reduced and the electronic temperature drops to about 1 eV. In small sources, the filter field is usually generated by permanent magnets placed around the source vessel. For larger sources, however, permanent magnets located on the edges of the vessel would be too far from its center, and filter field intensity here would be too low. In this case, a better solution (also adopted for ITER HNB design) is to generate a current flowing through the Plasma Grid, which induces a transverse magnetic field inside the source. The intensity of this field is around some tenths of Tesla in the center of the vessel.

A complementary strategy to reduce the co-extraction of electrons is to positively bias the Plasma Grid so that more electrons are directed toward it by electrostatic attraction. This solution is efficient to reduce the co-extraction of electrons, but slightly decreases the negative ion extraction current. The bias voltage is usually of the order of 1-10 V. The neutral gas contained in the source is a gas of H 2 or D 2 (the next paragraph focuses on hydrogen). The negative ions inside the accelerator are produced in two different processes:

Volume production: the production of negative ions happens inside the plasma by physical-chemical reactions. For a hydrogen plasma (ionized H 2 gas) the negative ions mostly form by electron collisions processes. The dominant processes are the dissociative electron attachment to H 2 molecules (1), the dissociative electron attachment to vibrationally excited H 2 (v > 5) molecules (2) the polar dissociation of the H 2 molecule (3) or the dissociative recombination of H + 2 and H + 3 ions with an electron (4). [START_REF] Hamilton | H-and d-production in plasmas[END_REF] (1) H 2 + e --→ H -2 --→ H -+ H The H -2 molecular ion formed by capture of an electron is unstable and has a large probability to dissociate after some fs in a H atom and a Hion.

(2) H 2 (v > 5) + e --→ H -+ H +

The dissociative electron attachment to H 2 has a much larger cross-section when the H 2 molecule is vibrationally excited (for a vibrational level v beyond 5). [START_REF] Allan | Effect of vibrational and rotational exictation on dissociative attachment in hydrogen[END_REF][START_REF] Cacciatore | Coupled solution of boltzmann equation and non equilibrium vibrational kinetics in h2 volume discharges: an analysis of the input data[END_REF] The H 2 molecule gets vibrationally excited by electronic collisions, or the H 2 (v > 5) are a product of interactions of H, H + 2 and H + 3 on the walls of the vessel or with the electrons. Surface production: the negative ions are a byproduct of collisions of neutral atoms and positive ions on the surface of the PG. The PG is normally made of copper coated by molybdenum but, in order to enhance the Hproduction, vapor of caesium is injected inside the source. The caesium deposits on the grid surface and lowers its work function. The main processes leading to the formation of Hions are backscattering of H atoms and H + ions on the grid and desorption of hydrogen negative ions caused by caesium and hydrogen impacts on the grid [START_REF] Jr Hiskes | Mechanism for negative-ion production in the surface-plasma negative-hydrogen-ion source[END_REF]. Experiments usually show that a large part of the negative ions are produced by surface interaction processes. Indeed, beam extraction experiments dominated by surface production processes are characterized by a low fraction of co-extracted electrons relatively to the ions (1 or less).

Although the negative ion creation processes were presented here for a hydrogen plasma, similar mechanisms apply for a deuterium plasma.

In the extraction area, the extraction electric field repels the positive ions towards the source and extracts the negative species from the plasma. This forms a negative ion dominated zone close to the PG apertures. The penetration of the extraction field inside the plasma is limited by the opposite longitudinal electric field generated by the negative ions space charge in this area. Upstream of the PG aperture there is a curved surface beyond which the extraction electric field vanishes and the plasma remains quasi-neutral. Because of its shape, this surface is referred to as meniscus.

The meniscus separates the quasi-neutral plasma of the source from the beam region, where the negative species dominate and their motion is governed by the applied extraction voltage. The position and shape of the meniscus are defined by a local space-charge limited condition: the meniscus recedes inside the source until the available thermal flux on its surface coming from the plasma bulk equals the space-charge limited ion current density [START_REF] Humphries | Charged particle beams[END_REF]. Further details on the physics of the meniscus will be given in the following parts, as it determines the optics of the extracted ion beam.

Since the extraction field repels the positive ions and attracts the negative one, a virtual cathode forms just upstream of the meniscus. Different definitions of the meniscus are possible depending on whether the accelerated beamlet or the source plasma is studied. For example, it can be described as an equipotential of the Plasma Grid in the first case, and as the surface where the positive ion density drops or where the negative ion flux jumps in the second case.

Acceleration stage

An electrostatic negative ion accelerator for NBI is composed of a series of parallel grids of increasing voltages with a large number of aligned small apertures with diameter generally around 1 cm (this dimension must remain small to reduce the number of co-extracted electrons). Through each series of aligned aperture, a beamlet is formed and accelerated from the meniscus to the exit of the acceleration stage. The accelerator comprises different kinds of grids:

the Plasma Grid (PG) separates the plasma source from the acceleration stage. The meniscus is located inside the apertures of the PG. An unwanted beamlet of co-extracted electrons forms together with the negative ions. This beamlet has to be suppressed close to the entrance of the accelerator, otherwise the electrons gain a large energy in the accelerator and can seriously damage the beamline components.

the Extraction Grid (EG) (sometime refered to as extractor ) is the grid located just after the PG. Its difference of electric potential with the PG defines the extraction electric field. This parameter has a large influence on the meniscus shape and therefore on the beam optics, and on the particle trajectories. Inside the EG, permanents magnets are embedded in order to generate a magnetic field perpendicular to the beamlet axis, called suppression field. This field deflects the beam of co-extracted electrons on the grid surface, to prevent it from propagating beyond the EG. Since the hydrogen ions are much heavier than the electrons, the ion beam is not very affected by this magnetic field. More precisely, if θ s is the angular deflection of the species s caused by a magnetic field perpendicular to the beam propagation direction:

θ ions θ electrons = m electrons m ions θ H - θ electrons = 2.3% θ D - θ electrons = 1.7%
Even though the effect of the suppression field on the ion beam deflection is small, it is not negligible.

On the contrary, it constitutes a key issue for the accelerator system design, as will be seen later in this thesis. The magnetic deflection, alongside with the initial deflection of the beam, must be corrected, and several strategies are developed for this purpose, and will be described later.

a Steering Grid (SG) refers to a grid whose aperture axes are shifted with respect to the other grids. This generates a beam deflection due to the electrostatic lens effect [START_REF] Clinton | Electron lenses[END_REF]. This kind of grid is installed on some accelerators to correct the ion beam deflection.

the Grounded Grid (GG) is the last grid of the accelerator, biased at the ground potential. The beamlets exiting the Grounded Grid must be well-focused and should have no deflection.

Accelerating Grids (AG) refer to the grids located between the EG and the GG. The AGs have gradually increasing electric potential so that the negative particles are constantly accelerated. There is a lower risk of breakdown in the acceleration stage if the potential drop between the GG and the EG is distributed over multiple AG.

a Repeller Grid (or Repeller Electrode) is normally located just before or after the GG. It generates an electric field directed towards the exit of the accelerator (in the opposite direction with respect to the rest of the accelerator) in the last gap of the accelerator, which tends to slow down the negative ion beamlets but also eliminates the co-accelerated electrons.

The grids are usually made of copper, and the PG is covered by a layer of molybdenum around 5 µm thick. This choice is motivated by the relatively small yield for D + and D + 2 sputtering on a Mo surface. The heat load on the grids is a key point for the system design, and a maximal temperature of 200 °C is foreseen for ITER HNB [START_REF] Hemsworth | Overview of the design of the ITER heating neutral beam injectors[END_REF]. The acceleration stage of ITER NBI should perform a 1 MV acceleration of the negative ion beam. Two designs have been proposed to reach this goal: the MAMuG (Multi-Aperture Multi-Grid) configuration [47] featuring a gradual acceleration of five steps of 200kV and the SINGAP (Single-Aperture Single-Gap) configuration [START_REF] Hpl De Esch | Updated physics design iter-singap accelerator[END_REF] with only two gaps of 40kV and 960kV (Fig. 2.6). It appears that, without intermediate grids, the secondary electrons produced in the acceleration gap reach the accelerator exit with a very large energy and may damage the beamline components. For this reason the MAMuG configuration has been retained for ITER design.

Neutralizer and Residual Ion Dump systems

After exiting the accelerator stage the negative ion beam is no more focused by the grids and the electrostatic self-electric force exerted by the beam particles on each other should defocalize it. However, the interactions between the beam and the background gas in the drift region beyond the GG generate positive ions, whose space charge counters the self-electric force and prevents the beam divergence to increase. Those interactions are mostly ionization of the H 2 molecules of the background gas by collisions with the hydrogen ions (1) and single (resp. double) detachment of hydrogen atoms (resp. negative ions) (2) [START_REF] Ratko K Janev | Collision processes in low-temperature hydrogen plasmas[END_REF].

(1) H + + H 2 --→ H + + e + H + 2 H -+ H 2 --→ H -+ e + H + 2 (2) H + H 2 --→ H + + H 2 + e H -+ H 2 --→ H + + H 2 + 2 e
It is estimated [START_REF] Veltri | Spatial characterization of the space charge compensation of negative ion beams[END_REF][START_REF] Heinemann | Upgrade of the batman test facility for h-source development[END_REF][START_REF] Cristhian | Negative ion beam space charge compensation by residual gas[END_REF] that the beam divergence stops increasing some centimeters downstream of the GG. Thus, the beam stays focalized as it heads towards the neutralizer. A neutralizer for high power negative ion NBI is a cell where the additional electrons of the beam negative ions are stripped. The fraction of neutralized ion must be high enough to get a correct neutralization efficiency (the ratio between the power of the neutral beam downstream of the neutralizer and of the ion beam upstream). The solution retained for ITER is a gas neutralizer: the ion beam passes through a cell where a neutral gas circulates. By collisional interactions, a fraction of the fast ions of the beam lose their excess electron thus generating fast neutral atoms (single electron detachment). Some fast negative ions can also lose two electrons (double electron detachment) resulting in fast positive ions. A fraction of those fast positive ions can retrieve an electron from the background gas atoms (charge exchange), forming fast neutral ions, although this reaction happens with a low probability at high beam energy. Finally, a part of the fast neutral atoms produced by those reactions can be ionized again by collision with the background gas, resulting in fast positive ions (reionization). Subscripts indicate the fast particles. [START_REF] Serianni | Neutralisation and transport of negative ion beams: physics and diagnostics[END_REF] In current designs, the gas neutralizer has an efficiency around 55%. For some preliminary models of DEMO, it is a requirement to enhance significantly the NBI system performances (aiming at 70% of efficiency) [START_REF] Sonato | Conceptual design of the DEMO neutral beam injectors: main developments and r&d achievements[END_REF]. R&D on alternative neutralizer concepts is being carried out to reach this goal. If, instead of a neutral gas, a deuterium plasma circulates in the Neutralizer cell, the efficiency could theoretically reach 80% [START_REF] Kh Berkner | Plasma neutralizers for h-or d-beams[END_REF], and experiments have indeed displayed a significant improve of the efficiency by using a plasma cell instead of a neutral gas cell [START_REF] Hanada | Experimental comparison between plasma and gas neutralization of high-energy negative ion beams[END_REF]. Recent studies have explored the possibility of forming such plasma inside the neutralizer chamber by the passage of the beam itself through a neutral gas background [START_REF] Surrey | The beam driven plasma neutralizer[END_REF][START_REF] Sartori | Preliminary studies for a beam-generated plasma neutralizer test in nio1[END_REF]. An important challenge in such device is the necessity to produce, sustain and confine a secondary plasma. Another alternative solution consists to intercept the beam with a large number of lasers, relying on photodetachment (D -+ hν --→ D + e) rather than collisional processes. This concept of photoneutralization [START_REF] Chaibi | Photo-neutralization of negative ion beam for future fusion reactor[END_REF][START_REF] Fiorucci | Thermal effects in high power cavities for photoneutralization of d-beams in future neutral beam injectors[END_REF][START_REF] Simonin | R&d around a photoneutralizer-based nbi system (siphore) in view of a demo tokamak steady state fusion reactor[END_REF] could achieve a neutralization efficiency around 70% and is planned for some DEMO concepts [START_REF] Sonato | Conceptual design of the DEMO neutral beam injectors: main developments and r&d achievements[END_REF]. The main challenge of this concept is the complexity and brittleness of the optical system to be deployed in the neutralizer cell.

Even with the best solutions, a large part of the negative ion beam is not neutralized, and a significant fraction of the acceleration power does not reach the tokamak region. The residual ions must be eliminated, otherwise they would damage the duct that joins the NBI to the tokamak and induce a large reionization of the neutral beam exiting the neutralizer. For high power NBI, trying to recover the energy of the non-neutralized ions is worthwhile to increase the global efficiency of the system. The Electrostatic Residual Ion Dump (ERID) of ITER extracts the ions from the neutral beam with an electrostatic field. The neutral beam passes through vertical biased panels, where the ions are intercepted.

The power density on these panels is estimated at some MW m -2 at most. The panels are actively watercooled.

Chapter 3

State of the art The extraction voltage is around 9 kV and the total acceleration voltage around 1 MV. The filter field is generated by a current flowing through the PG. The suppression field is generated by a specific combination of SmCo permanent magnets: the Co-extracted Electrons Suppression Magnets (CESM) are horizontal rows of magnets embedded in the EG. They are polarized along the beam direction, with alternate directions of polarization. This generates around each aperture a cusp vertical magnetic field.

the Asymmetric Deflection Compensation Magnets are vertical magnets embedded inside the EG. They are polarized along the vertical direction. All the magnets of a horizontal row have the same direction of polarization, and this direction alternates from one row to the next. This configuration is inspired by the Halbach array, a magnet configuration that generates a stronger magnetic field on one side of the array than on the other [START_REF] Halbach | Application of permanent magnets in accelerators and electron storage rings[END_REF][START_REF] Chitarin | Cancellation of the ion deflection due to electron-suppression magnetic field in a negative-ion accelerator[END_REF]. This asymmetric magnetic field corrects the ion deflection generated by the CESM.

the Stripped Electrons Suppression Magnets (SESM) have a configuration similar to the CESM and are embedded in the acceleration grids. They deflect the secondary electrons produced in the source. The polarization is such that the magnetic deflection it generates on the negative ion beam is compensated at the exit of the acceleration stage. The accelerator of MITICA includes other specific features dedicated to the optics of the beam:

thick metallic plates, referred to as "kerbs" are attached on the sides of each groups of 5 beamlets. The kerbs are designed to compensate the repulsive electric force between the neighbouring beamlets (see section 4 of [START_REF] Hemsworth | Overview of the design of the ITER heating neutral beam injectors[END_REF]), a tilting of the grid segments so that the groups of beamlets are accelerated towards a convergent point, which reduces the divergence of the total beam.

The operation of MITICA is foreseen for 2021. SPIDER [START_REF] Serianni | First operation in spider and the path to complete mitica[END_REF] is the real size prototype ion source for MITICA. It is an experiment focused on the validation and improvement of the ITER ion source design for specifications such as the extracted current, the pulse length, the electron co-extracted ratio or the uniformity of the plasma across the source. SPIDER is composed of eight 1 MHz RF drivers located at the back of the source, each of them providing 100 kW of power, a parallelepipedal plasma source and a grid system similar to MITICA (1280 apertures, extraction surface of 0.2 m 2 , see Fig. 3.1). SPIDER must fulfill the requirements of MITICA (Table 3.1) in terms of extracted current, co-extracted electron ratio, source pressure and pulse duration. The filter field is generated inside the source by a current flowing through the PG. The exact configuration of the busbar returning the current to the PG has a large impact on the source plasma uniformity. Indeed, the transverse filter field tends to create a top-bottom asymmetry inside the whole source. This asymmetry must be reduced at 10% or less for the whole beam current density. Besides, the magnetic configuration must be such that the magnetic field is weak inside the RF drivers, inside the accelerator and just downstream of the PG [START_REF] Agostinetti | Optimisation of the magnetic field configuration for the negative ion source of iter neutral beam injectors[END_REF][START_REF] Chitarin | Improvements of the magnetic field design for spider and mitica negative ion beam sources[END_REF]. The grid geometry of SPIDER is inspired by the optimum geometry computed for MITICA, but it features only a PG, an EG and a GG with a total acceleration voltage around 100 kV. SPIDER suppression field includes only CESM. Different sections of the grids include different ways of compensating the beamlet deflection: the apertures of the GG are shifted (SG concept), a magnetic field is generated by permanent magnets embedded inside the GG, in a section of the grid the magnetic deflection is left uncompensated in order to evaluate the efficiency of the compensation solutions applied to the other sections. [START_REF] Pimazzoni | Assessment of the spider beam features by diagnostic calorimetry and thermography[END_REF].

SPIDER operations of plasma generation and beam extraction have already begun with reduced acceleration and extraction voltage, and reduced number of diagnostics

NITS (Negative Ion Test Stand) at QST (Japan)

NITS is a Haccelerator, with a multi-cusp plasma source characterized by a special shape of truncated cylinder, known as "kamaboko" (from a Japanese fish paste of similar shape). The plasma is produced by hot tungsten filaments positioned around the cylinder and is confined by a multi-cusp shaped magnetic field produced by permanent magnets located all around the cylinder. The extraction area is located at the planar part. The Filter Field is generated by permanent magnets also located around the cylinder. During the experiments reported in this thesis [START_REF] Chitarin | Experimental validation of an innovative deflection compensation method in a multi-beamlet negative-ion accelerator[END_REF], the suppression field was generated by a special combination of magnets (Fig. 3.6): one group of apertures (on the top) included CESM and ADCM while the bottom group included only CESM to evaluate the corrective effect of the ADCM on the beam trajectory. MTF is a testbed accelerator for the negative ion NBI of the tokamak JT-60 SA [START_REF] Ishida | Overview of the jt-60sa project[END_REF] and for ITER HNB. It normally includes eight grids (PG, EG, SG, four AGs and GG) although in the experiments described in this thesis only two AG were included [START_REF] Hiratsuka | Achievement of high power and long pulse negative ion beam acceleration for jt-60sa nbi[END_REF]. The grid system has been optimized to increase the voltage holding of the accelerator, which is a critical issue for accelerators with the MAMuG configuration [START_REF] Kojima | Development of design technique for vacuum insulation in large size multi-aperture multi-grid accelerator for nuclear fusion[END_REF][START_REF] Hiratsuka | Achievement of high power and long pulse negative ion beam acceleration for jt-60sa nbi[END_REF]. The objective of MTF for JT-60 SA is to accelerate a Hbeam of current density 130 A m -2 with a voltage of 500 kV for 100 s. Such target was achieved in the experiments described in this thesis. The extraction voltage is around 5-10 kV The ion source of MTF is a kamaboko source similar to the one described above (Fig. 3.5), operated with an arc power around 30 kW and a plasma pressure around 0.3 Pa. As in the case of NITS, the filter field is generated by permanent magnets located around the source vessel. The suppression field is generated only by CESM magnets, the deflection compensation is purely electrostatic (by the SG). For JT-60 SA, five sections of 9 × 5 apertures (1080) are foreseen. In the experiments described, the number of beamlets was limited to 3x3.

BATMAN (BAvarian Test Machine for Negative ions) Upgrade at IPP Garching (Germany)

BATMAN Upgrade (BUG) [START_REF] Heinemann | Upgrade of the batman test facility for h-source development[END_REF] is a Haccelerator with a 1 MHz RF driven source (1 MHz, 150 kW). During the experiments presented here, the accelerator included a PG, an EG, a RE (Repeller Electrode) and a GG.

The filter field can be generated by permanent magnets displayed around the source vessel or by a current flowing through the PG [START_REF] Schiesko | Langmuir probe investigations of different magnetic filter field configurations at batman upgrade[END_REF]. The suppression field is generated only by CESM and the magnetic deflection is not compensated. The typical values of operational parameters are very close to NITS in terms of extraction and acceleration voltage and current density.

ELISE (Extraction From a Large Ion Source Experiment) at IPP Garching (Germany)

ELISE [START_REF] Heinemann | The negative ion source test facility elise[END_REF] is a RF driven Hand Dsource and accelerator, that approximately corresponds to a half size prototype of SPIDER ion source (Fig. 3.9), as it includes four 1 MHz RF drivers of ≈100 kW each and a similar configuration as SPIDER, with a 0.1 m 2 extraction surface. The accelerator is a three grid system (PG, EG, GG), with an extraction voltage usually around 5-7 kV. The filter field is generated by a current flowing through the PG and the suppression field is generated by CESM. The magnetic deflection is not compensated. ELISE is an important testbed for SPIDER and ITER HNB in terms of Hand Dextraction, Cs dynamics, electron co-extraction ratio and source plasma uniformity. It achieved extraction of around 250-300 A m -2 of Hand Dfor pulse durations around 10 s with a co-extraction ratio below 1 for hydrogen operations. In the case of Dextraction, the electron co-extraction ratio is still smaller than 1 but rapidly increases during the pulse [START_REF] Wünderlich | Formation of large negative deuterium ion beams at elise[END_REF]. A beam homogeneity around 90% was achieved [START_REF] Wünderlich | Achievement of iter-relevant accelerated negative hydrogen ion current densities over 1000 s at the elise test facility[END_REF], close to the specifications of SPIDER. The experiments on ELISE focus rather on the source properties and beam extraction than on the beam optical properties across the accelerator. The module SCALA is conceived to simulate the trajectories of charged particles in electrostatic and magnetostatic fields in 3D. The calculation is iterative to include the effects of space-charge, self-magnetic fields and relativistic corrections.

Simulation tools and numerical models

In a given iteration, the electrostatic field is calculated by solving the Laplace equation at the first iteration and the Poisson equation in the next ones. The complete trajectory is calculated from the field distribution and from the model of emitter in the simulation, solving the equation of motion of each macroparticle (a macroparticle is a cluster of particles whose masses and charges are summed and that is treated as a single particle). The space charge is assigned to the mesh along each trajectory.

In the next iteration the calculated potential is applied with a relaxating factor set by the user. The fields are recalculated including the space charge and -optionally -the beam self-magnetic field. The trajectories are updated. The error in the trajectory is estimated by comparing the results of solving the model with a 4th order and 5th order Runge-Kutta method ("Dormand-Prince 5 method"). SCALA provides several models of emitters, and one of them is specifically dedicated to the extraction of ions from a plasma (the Free Plasma Emitter type 103 ). In this model the volumetric properties of the plasma are contained in the surface properties of the meniscus, summarized by three parameters: the meniscus voltage, the extracted current density and the ion temperatures in the plasma. The emitter is defined on a plane located upstream of the extraction aperture and the macroparticles drift from this plane to the meniscus without influence of the electric and magnetic fields. The shape and position of the meniscus are calculated in an iterative way.

Advanced model for the definition of the meniscus

In another model of emitter (the User Defined Emitter type 9 ), the property of the emission surface is mainly defined by the user. It allows to reproduce other types of emitters while modifying the properties of the meniscus and the hypotheses that determine its shape and position. This emitter is also defined on a plane upstream of the extraction area, but a drift condition can be imposed: the macro-particles drift in the direction normal to the plane with no influence of the external field until they reach a threshold potential defined by the user. The current density of the macroparticles at the emitter follow a spatial profile defined by the user. This emitter can be set to reproduce the model of the type 103 emitter with the following conditions:

the macro-particles start by drifting until a threshold:

V -V start < V meniscus j emitter = j meniscus * cos ((E init , n))
where E init is the electric field at the meniscus and n the vector normal to the meniscus surface

v init = 8k B T ions πm ions
This more general model has the advantage that the simplifying assumptions of uniform current density j meniscus over the meniscus surface can be replaced by a spatially dependent parameter j meniscus (x, y, z), which will be exploited in this thesis. An example of simulations with this kind of emitter is shown in Fig. 2.4 a).

Additional physical features of Opera

Opera also includes a module (TOSCA magnetostatics) for the computation of the magnetostatic field generated by permanent magnets and current flows. TOSCA allows to generate tables of magnetic map that can be read by SCALA simulations or by other softwares.

In SCALA, the secondary particles can be included by defining volumetric secondary emitters (for secondaries generated by interaction with the background gas) and surface emitters (for backscattered particles, sputtering, erosion...).

Hypothesis of the models and post-treatment

A realistic model of negative ions acceleration includes the following elements:

the grid system geometry and the voltage applied to each grid. Most of the time accurate results are obtained by simulating a single aperture of each grid of an accelerator. This does not take account of the mutual electrostatic repulsion of the beamlets passing through neighboring apertures, but this effect can often be neglected in first approximations, the Filter and Suppression magnetic fields in the simulation domain, although sometimes only the suppression field is taken into account as the filter field is weak downstream of the meniscus and its effect can be neglected, the co-extracted and secondary electrons in the simulation domain, but they are often neglected because of their very weak mass compared to the ions, which implies that their contribution to the overall space charge is very weak, a physically accurate model of the meniscus since, as explained above, the definition of the meniscus shape and position can be precisely defined in the model, using the type 9 emitter.

Several hypotheses are usually made in Opera. The most common one are the following:

the beam magnetic field is neglected. Indeed, the beam magnetic field is a relativistic effect and even with an accelerator of 1 MV the particle velocities are much smaller than the velocity of light, the fields are considered to be static, this approximation is good when the length of the experiments is much larger than the time of formation of the meniscus, which is always the case in the experiments considered in this thesis.

Two main results are expected from a simulation of an accelerator with Opera: the data of all particle trajectories: the number of macro-particles intersecting any plane inside the simulation domain and, for any macro-particle, the current carried, the Time of Flight, the velocity vector and the position in space, the total electro-magnetic field (including space-charge effects) in the simulation domain.

From the first set of data, the optical properties of the beamlets can be defined and computed, as explained in the following sections.

COMSOL Multiphysics ®

Beam trajectory calculation

COMSOL (COMputer SOLutions) Multiphysics ® [START_REF]Comsol multiphysics[END_REF] is a FEA multiphysics software, applicable to a broad range of problems (mechanical, thermal, fluid and flow simulations, electromagnetics...). COMSOL includes a particle tracing module suited for simulations of electron and ion beam propagation. The beam trajectory and electromagnetic map inside the accelerator are computed in an iterative way (using a Dormand-Prince 5 method) by mutual interactions between the particle tracing module and the electrostatic module ("bidirectional coupling"). The magnetic field can be included with an external table (for example made with Opera) or using the magnetostatic module of COMSOL. It is possible to conceive a self-consistent simulation that determines the meniscus shape and position, however there is at present no simple model to simulate the extraction of a beam from a plasma. For this reason, it is common to import the geometry of the meniscus from other codes (usually Opera).

As Opera, COMSOL provides the coordinates of the beam macroparticles in the space phase, as well as the electromagnetic map inside the accelerator. It has been shown (see section 5.4 of [START_REF] Baltador | Numerical and experimental characterization of beams of negative ions and investigation on strategies for beam efficiency improvements[END_REF] and [START_REF] Chitarin | Benchmark of 3d multi-beamlet numerical models for the optics design of negative ion accelerators[END_REF]) that Opera and COMSOL are well consistent in the calculation of an ion beam trajectory and space charges when the meniscus geometry is imported in COMSOL from Opera.

Secondary emission and heat load simulations

COMSOL is particularly suitable to calculate the heat loads generated by the particle collisions on the accelerator elements. The density of the background gas leaking from the source inside the accelerator is determined by a free molecular flow simulation with the molecular flow module of COMSOL. Indeed, for NBI, the molecular mean free path within the background gas inside the accelerator is usually of some tenth of centimeters (pressure and temperature around 0.1 Pa and 100°C for H 2 or D 2 ) while the typical distance between the accelerator elements (e.g. the gap between the grids) is of a few tens of millimeters. The approximation of molecular flow is, therefore, reasonable as the Knudsen number is at least around 10. The cross-section of the volume and surface processes generating secondary particles inside the source are then included in the particle tracing simulation.

Collisions can be modeled as a continuous force or, as it is the case in the simulations discussed in this thesis, using a Monte Carlo model in which particles can undergo elastic or inelastic collisions with the background gas. Surface process induced by particle collisions on the grids can also be included.

On each node of the grids, the power deposited by the impinging particles is computed as E * ν with E = mv 2 2 the particle energy and ν = I q the emission frequency (current by charge). The heat loads caused by the particle impacts are then exported to the Heat Transfer Module of COMSOL that computes the heat distribution on the grids.

SLACCAD

SLACCAD (Stanford Linear Accelerator Center -Computer Aided Design) is an adaptation of an electron tracing code developed at the Stanford Laboratory [START_REF] Wb Hermannsfeld | Electron trajectory program[END_REF][START_REF] Pamela | A model for negative ion extraction and comparison of negative ion optics calculations to experimental results[END_REF] dedicated to the simulation of electron and ion extraction from a plasma and acceleration. SLACCAD applies the same solving pattern as common particle tracing codes (Fig. 3.10) but simulates the trajectory of the particles in a 2D plane with an axisymmetric boundary that greatly reduces the solving time (a few tens of second whereas Opera and COMSOL require a several tens of minutes), but neglects some 3D phenomena, such as: the mutual electrostatic repulsion between the neighboring beamlets (beamlet-beamlet repulsion), the external magnetic field inside the accelerator, and consequently the co-extracted electrons, the non-radially symmetric profile of the current density at the meniscus.

SLACCAD includes complementary modules dedicated to the inclusion of secondary particles in the simulation. Several improvements of those modules and of the simulation post-treatment have been made (see for example section 5.1 of [START_REF] Veltri | Physics of negative ion extraction and acceleration in neutral beam injectors[END_REF]). SLACCAD is not suitable to compute the beam trajectory, as it neglects the 3D effects and therefore assume no beamlet deflection, but is commonly used to quickly estimate the beam divergence for a scan of input parameters (grid potential, extracted current density, background gas density, grid geometry...).

EAMCC

EAMCC (Electrostatic Accelerator Monte Carlo Code) is a 3D particle tracing code developed at the CEA (Commissariat à l´Énergie Atomique et aux Énergies Alternative) by G. Fubiani [START_REF] Fubiani | Modelling of secondary emission processes in ion based electrostatic accelerator of the international thermonuclear experimental reactor[END_REF]. In EAMCC simulations, the electromagnetic field is not computed in a self-consistent way but imported from other codes such as SLACCAD or Opera. EAMCC is particularly suited to the computation of secondary particle production and trajectories, as it includes an elaborate physical description of the collisions with the grids and the background gas (using a Monte Carlo method), of the secondary emission yields (using a semi-analytical approach and databases such as the ORNL Controlled Fusion Atomic Data Center "redbooks" [START_REF] Barnett | Ornl's controlled fusion atomic data center[END_REF]) and of the secondary particle trajectories. The heat loads on the grids are calculated from the kinetic energy of the particle hitting the accelerator grids.

NBImag

NBImag [START_REF] Aprile | Nbimag: A useful tool in the design of magnetic systems for the iter neutral beam injectors[END_REF] is a code conceived at RFX for the simulation of complex magnetic field configurations. It is fitted to the study of NBI accelerators, that often feature complex arrays of permanent magnets (Fig. 3.2) and current flow induced magnetic fields. NBImag uses an integral formulation which allows a more efficient calculation than FEA codes. It also computes magnetic forces and coil inductance. It is consistent with FEA computations, and in particular with the magnetostatic module of Opera (see section 5.2 of [START_REF] Baltador | Numerical and experimental characterization of beams of negative ions and investigation on strategies for beam efficiency improvements[END_REF]).

Chapter 4

Theoretical elements on the beam propagation in a NBI accelerator

Even though no analytic model is precise enough to replace particle tracing simulations with Finite Element Analysis, a correct understanding of the physics of the acceleration of the beam is important to interpret the results of the numerical tools.

In the first section of this chapter, we present a brief summary of common elements of theory used in the frame of negative ion acceleration.

In the second section, we develop a formalism to describe the electric potential and field inside a NBI accelerator, which will be applied to different calculations in the following chapters.

Common simple approaches to the physics of beam acceleration 4.1.1 Built functions approximating the electric potential inside the accelerator

In negative ion accelerators made of successive grids, the electric potential of each grid increases along the accelerator, so that the particles are constantly accelerated by the electric field inside the gaps (except when a Repeller Grid is installed).

We use the following conventions to describe the electric potential inside the accelerator:

the accelerator is made of n grids numbered from 1 to n (1 being the PG and n the GG).

the ith grid has an inlet aperture of radius r in,i located at a position z i , an outlet aperture of radius r out,i and a thickness t i , the gap lengths (the shortest distance between two successive grids) is

d i = z i+1 -(z i + t i )
the ith grid has a potential V i with V 1 = 0 by convention. Although the GG is grounded in the real system, setting the PG at the potential 0 in the calculations allows to use only positive voltages and to simplify the equations.

the norm of the electric field between the ith and the i + 1th grids is noted E i , and is given by:

E i = V i+1 -V i d i
ẑ is the axial direction (the beam accelerates from -z towards +z), x and ŷ are respectively the horizontal and vertical directions.

The simplest way to approximate the potential generated by the grids is to use a continuous piecewise function V with the property that there is a at least one point inside the ith grid where V (z) = V i . A simple example of such function is the piecewise linear function defined so that the potential is V i at the middle point of the grid

thickness z M,i = z i + t i 2 (Fig. 4.1) ∀i ∈ [[0, n -1]], z ∈ [z M,i , z M,i+1 [ =⇒ V (z) = V i + z -z M,i z M,i+1 -z M,i (V i+1 -V i )
A slightly more elaborate function considers that the potential stays constant inside the grid (Fig. 4.2).

∀i

∈ [[0, n -1]]    z ∈ [z i , z i + t i [ =⇒ V (z) = V i z ∈ [z i + t i , z i+1 [ =⇒ V (z) = V i + z -(z i + t i ) z i+1 -(z i + t i ) (V i+1 -V i )
Approximating the electric potential by a piecewise linear function amounts to approximating the axial component of the electric field by a step function. The non-continuity of the electric field can raise problems in some applications. From then on, it can be useful to approximate the potential by a piecewise quadratic function, that preserves the continuity of the electric field.

A simple and fairly accurate approximation (Fig. 4.3) is the function that ensures the continuity of the electric potential and electric field, with the two following conditions:

V (z M,i ) = V i
the inflection points of the potential profile (that is, the extrema of the axial component of the electric field) are located at the middle points of the gaps:

z G,i = z i + t i + d i 2 ∀i ∈ [[0, n -1]]            z ∈ [z M,i , z G,i [ =⇒ V (z) = (z -z M,i ) 2 (V i+1 -V i ) (z G,i -z M,i )(z M,i+1 -z M,i ) + V i z ∈ [z G,i , z M,i+1 [ =⇒ V (z) = (z -z M,i+1 ) 2 (V i+1 -V i ) (z G,i -z M,i+1 )(z M,i+1 -z M,i ) + V i+1
Finally, approximations of even higher polynomial order are sometimes used, referred to as "spline" [START_REF] Szilagyi | Electron and ion optics[END_REF]. Such approximations are rather employed for numerical solving than for analytic calculations. The errors made by built functions are substantial (of the same order as the extraction voltage) but these approximations are still useful in some ranges of application.

Electrostatic lens

In a simplified model, a charged particle passing through a circular aperture in a charged planar surface is deflected by an angle proportional to its distance from the central axis, like a light ray passing through a spherical diopter in geometrical optics. As will be shown later, the resulting electrostatic deflection is independent on the charge-to-mass ratio of the particle. Consequently, the "refractive" power of the circular aperture is an intrinsic property that does not depend on the particle species. For this reason, such circular apertures are designated by the name of electrostatic lenses. Similarly to usual lenses, electrostatic lenses are characterized by a focal length:

f = 4V (r particle , z aperture ) E d -E u
where E d and E u are the asymptotic values of the electric field magnitude downstream and upstream of the plane, and V (r particle , z aperture ) is the potential with the convention: V = 0 where the particles have no velocity (here at the PG) [START_REF] Clinton | Electron lenses[END_REF]. This approximation is correct for thin lenses, as long as the refractive power is weak, that is, for focal length much larger than the aperture radius. This representation is useful for its simplicity, to estimate the effect of the electrostatic deflection caused by the grids. However, this often leads to large errors for NBI accelerators, as the grids are often thick and the focal lengths comparable to the aperture radii. Even a small error on the deflection angle leads to a large error in the beam trajectory.

To illustrate this, we simulate with Opera the trajectory of an electron in the NITS accelerator, starting in the extraction gap and passing through the EG. We compare the previsions of the electrostatic lens formula (using the values of V , E d and E u calculated by FEA) with the focal lengths directly estimated from the electron trajectory (Fig. 4.4). From the lens formula, using the potential and electric fields calculated with Opera:

f in,EG = 2966.5V 6.5 × 10 5 V m -1 = 18.3mm f out,EG = 6761.6V -1.3 × 10 6 V m -1 = -20.8mm
From the trajectory of a an electron passing through the EG (Fig. 4.4), starting at x = 4 mm:

f in,EG = 3.4mm 0.0335 + 0.1426 = 19.3mm f out,EG = 3.7mm -0.0961 -0.0335 = -28.6mm
From the trajectory of an electron starting at x = 2 mm:

f in,EG = 1.7mm 0.0201 + 0.0628 = 20.5mm f out,EG = 1.9mm -0.0458 -0.0201 = -28.8mm
In the case of NITS, whose EG is quite thick, the lens formula gives a correct order of magnitude (especially for the inlet surface) but the error on the outlet surface focal length is around 25%, which generates a large error on the beam trajectory. The lens formula is a consequence of assuming a piecewise linear potential profile and a thin lens. More precise models exist to approximate the effect of a thick lens on the beam deflection, but such models are not used in this thesis, as the electrostatic lens is sufficient to obtain useful qualitative results. We retain here that, in the case of NITS, this formula underestimates the focal length of the EG outlet aperture.

Beam space charge

Potential generated by a cylindrical beam

The electromagnetic field of the beam has a significant influence on its trajectory and shape. The beam selfmagnetic field is usually neglected because it is a relativistic effect and, in the case of negative ion accelerators for fusion, the particles travel at non-relativistic velocities.

The electric potential of the beam is generated by its charge density ρ, with:

ρ(x, y, z) = j(x, y, z) v(x, y, z)
j is the current density of the beam and v its velocity. Since the beam current density is approximately conserved along its path, the effect of the space charge decreases as the beam accelerates.

The potential generated by the beam is obtained by assuming the shape of the beam (usually cylindrical) and the charge density profile, applying Poisson's equation to the space inside the beam and Laplace's equation the space outside.

A general model for the space charges generated by the beam is the following one:

we assume that the beam propagates through a cylindrical tube of radius R 0 of constant grounded potential (this is not the case in general for NBI accelerators except when the beam passes through the grids), the beam has a circular section of radius r 0 and its center is located at a distance s from the tube axis, the charge density profile ρ(r) is radially symmetric. The general solution for this case is found in [START_REF] Regenstreif | Sur le potentiel de charge d'espace produit par un faisceau décentré non uniforme[END_REF]: outside of the beam:

Φ O = r 0 2ε 0 f (r 0 ) ln (R 2 0 -s 2 ) 2 + s 2 r 2 -2sr(R 2 0 -s 2 ) cos θ R 2 0 r 2
inside the beam:

Φ I = r 0 2ε 0 f (r 0 ) ln (R 2 0 -s 2 ) 2 + s 2 r 2 -2sr(R 2 0 -s 2 ) cos θ R 2 0 r 2 0 + f (r 0 ) -f (r) ε 0 with: f (r) = v=r v=0 1 v u=v u=0 uρ(u)du dv f (r) = 1 r u=r u=0 uρ(u)du
This corresponds to the following electric field:

outside of the beam:

E r,O = r 0 ε 0 r f (r 0 ) (R 2 0 -s 2 )(R 2 0 -s 2 -sr cos θ) (R 2 0 -s 2 ) 2 + s 2 r 2 -2sr(R 2 0 -s 2 ) cos θ E θ,O = r 0 s ε 0 f (r 0 ) -(R 2 0 -s 2 ) sin θ (R 2 0 -s 2 ) 2 + s 2 r 2 -2sr(R 2 0 -s 2 ) cos θ
inside the beam:

E r,I = r 0 s ε 0 f (r 0 ) (R 2 0 -s 2 ) cos θ -sr (R 2 0 -s 2 ) 2 + s 2 r 2 -2sr(R 2 0 -s 2 ) cos θ + f (r) ε 0 E θ,O = E θ,I
We illustrate the validity of this model by considering the potential distribution generated by a Hbeamlet propagating in the NITS accelerator. We consider a case with V ext = 4.4kV and V ext = 21.1kV, and we study a section of the beam in the middle of the GG. The following model is studied: we consider that the beam section is circular with a radius r 0 = 3.9mm, off-centered by a distance s = 0.87mm, propagating through a grounded tube of radius R 0 = 8mm corresponding to the GG, we assume a Gaussian profile cut-off at the beam edge r = r 0 :

r ≤ r 0 =⇒ ρ(r) = A exp -r 2 w 2 r > r 0 =⇒ ρ(r) = 0 being I the beam current, the beam linear charge density is λ(z) = I v z . λ(z) = θ=2π θ=0 r=+∞ r=0 ρ(r)rdrdθ = A2π w 2 2 1 -exp - r 2 0 w 2
Hence:

A = I πw 2 1 -exp - r 2 0 w 2 v z
with v z = 2.13 × 10 6 m s -1 and I = -0.0166A

from the simulation we find that the Gaussian profile is a good approximation with w = r 0 ln 4 (the charge density at the edge of the beam is one fourth of the charge density at its center)

With those assumptions the model is in good agreements with the Opera simulation: The maximum current density in a space charge limited regime in the vacuum is given by the Child Langmuir formula, obtained from the equation of motion of the ions, the conservation of the current, and Poisson's equation:

j SCL = 4ε 0 9 2e m i V 3/2 ext d 2
This formula allows to define two quantities: the beam perveance: being I the beamlet current, considered positive, the perveance is defined by Π = IV -3/2 ext , the normalized perveance: being j the beamlet current density, we can assume that at the PG:

j = I πr 2 P G . The normalized perveance is defined by Π norm = Π Π 0 with Π 0 = 4ε 0 9 2e m i πr 2 P G d 2
The Child Langmuir law can therefore be rewritten as:

j j SCL = Π norm
The normalized perveance quantifies how close the beamlet current density is from the space charge limitation. The meniscus surface is defined by a local space charge limited condition: the distance d of its surface from the extractor is such that j SCL equals the thermal flux of particles coming from the plasma [START_REF] Humphries | Charged particle beams[END_REF]. When the perveance decreases the meniscus recedes inside the plasma source, getting more concave, and on the contrary it gets flatter (and even convex) when the perveance increases (Fig. 4.7). Finding an analytic expression for the electric potential inside the accelerator is very complex and the exact expression would depend on the precise geometry specific to each system. A way to approximate with great accuracy the electric potential inside an accelerator was found by E. Regenstreif [START_REF] Regenstreif | Théorie de la lentille électrostatique a trois électrodes[END_REF]. It consists in approximating a multigrid accelerator as follows (Fig. 4.8):

Only the inlet and outlet surfaces of the grids are taken into account.

Those surfaces are approximated by infinite planes with a circular aperture.

Between the inlet and outlet surfaces of a grid the electric field is zero. Fig. 4.8 illustrates how a two stage accelerator is approximated by a superposition of circular apertures in infinite planes. It should be noted that the inlet surface of the first grid and outlet surface of the last grid are not included, as the electric field does not change from one side of those apertures to the other. This representation can be applied to an arbitrary number of grids. The main point of this approximation is that there exists a simple exact analytic solution for the electric potential created by an electric field located on one side of an infinite plane with a circular aperture (see [START_REF] Thornton | Two problems in potential theory[END_REF], eq. ( 23) for the demonstration and [START_REF] Scott | Calculation of the ion extraction boundary of a plasma ion source[END_REF] eq. ( 2) and ( 3) for the form of the equation used in this paper).

Analytic formula of the electric potential and field generated by the grids

The electric potential φ(x, z) generated by a circular aperture of radius a centered on (r = 0, z = z 0 ) in an infinite plane with an external electric field of norm E 0 directed along ẑ on the part z > z 0 is given by:

φ(E 0 , a, r, z 0 , z) = E 0 (z -z 0 ) + ψ if z ≥ z 0 ψ if z < z 0 (4.1)
With:

                   ψ = aE 0 π R -λ 2 - |z -z 0 | a arctan 2 R + λ λ = (z -z 0 ) 2 + r 2 -a 2 a 2 R = λ 2 + 4(z -z 0 ) 2 a 2 (4.2)
We keep the same notation as previously. The ith grid has an inlet aperture of radius r in,i located at a position z i , an outlet aperture of radius r out,i , a thickness t i and an electric potential V i (with V 1 = 0 by convention). The norm of the electric field between the i th and the i + 1 th , E i is defined as previously. Since the Maxwell equations are linear, the superposition principle gives the total electric potential as the sum of the potentials of all the apertures. The electric potential generated inside the accelerator (taking into account a single aperture of each grid) is then given by:

φ 0 (r, z) = n-1 i=1 φ (E i , r out,i , r, z i + t i , z) + φ (-E i , r in,i+1 , r, z i+1 , z) (4.3)

Space charges and formula for the total potential inside the accelerator

We also need to take account of the beam space charge. For z increasing, the space charge effect becomes weaker as the beam particles gain velocity. Besides, the potential generated by the grids increases rapidly and becomes much higher than the beam electric potential. For this reason, we do not develop here a physical model to take account of the space charge but simply define an effective beam potential that ensures the two following conditions:

the beam potential vanishes for z -→ ∞ the electric potential is zero at the meniscus.

An example of function that meets these criteria without making calculations overly complex is the following one:

φ SC (r, z) = -φ 0 (r, z men ) exp - z -z men z n -z men (4.4)
This space charge model is accurate almost everywhere inside the accelerator. To calculate the potential near the meniscus or at the exit of the accelerator, a more realistic and physical model would be needed.

The potential in the accelerator stage in the presence of the beam is:

V (r, z) = φ 0 (r, z) + φ SC (r, z) 4.2.

Electric field inside the accelerator

Since the electric potential is analytically and completely defined inside the accelerator, we can also derive the electric field, using the parameters defined in equation 4.2.

We define ε, K 1 and K 2 as:

ε = sign(z -z 0 ) K 1 = R -λ K 2 = R + λ
The partial derivatives of R and λ are:

     ∂λ ∂r = 2r a 2 ∂λ ∂z = 2(z -z 0 ) a 2      ∂R ∂r = 2λ R ∂λ ∂r ∂R ∂z = λ + 2 R ∂λ ∂z
We notice that:

2|z -z 0 | a = R 2 -λ 2 (4.5)
Hence:

∂λ ∂z = ε √ R 2 -λ 2 a (4.6)        ∂K 1 ∂r = 2λ -R R 2r a 2 ∂K 1 ∂z = λ -R + 2 R ε √ R 2 -λ 2 a        ∂K 2 ∂r = 2λ + R R 2r a 2 ∂K 2 ∂z = λ + R + 2 R ε √ R 2 -λ 2 a ∂ψ ∂r = aE 0 π 1 2 1 √ 2K 1 ∂K 1 ∂r + |z -z 0 | a 1 √ 2K 2 (K 2 + 2) ∂K 2 ∂r ∂ψ ∂r = rE 0 √ 2πa |z -z 0 | a 2 √ R + λ R 1 2 + R + λ - √ R -λ R
Therefore, using the equation 4.5, we obtain after simplification:

∂φ(E 0 , a, r, z 0 , z) ∂r = - √ 2r a E 0 π √ R -λ R(R + λ + 2) ∂ψ ∂z = aE 0 π 1 2 1 √ 2K 1 ∂K 1 ∂z + |z -z 0 | a 1 √ 2K 2 (K 2 + 2) ∂K 2 ∂z - ε a arctan 2 K 2 ∂ψ ∂z = E 0 a π 1 2 √ 2 λ -R + 2 R √ R -λ + 2|z -z 0 | a 1 R √ R + λ ∂λ ∂z - ε a arctan 2 R + λ
Using equation 4.5 and simplifying:

∂ψ ∂z = E 0 a π 1 √ 2R √ R -λ ∂λ ∂z - ε a arctan 2 R + λ
Using equation 4.6 and simplifying:

∂ψ ∂z = ε E 0 π 1 R R + λ 2 -arctan 2 R + λ ∂φ(E 0 , a, r, z 0 , z) ∂z =        E 0 + E 0 π 1 R R+λ 2 -arctan 2 R+λ if z ≥ z 0 - E 0 π 1 R R+λ 2 -arctan 2 R+λ if z < z 0
Therefore, the electric field inside the accelerator is:

E r (E 0 , a, r, z 0 , z) = √ 2r a E 0 π √ R -λ R(R + λ + 2) E z (E 0 , a, r, z 0 , z) =        -E 0 - E 0 π 1 R R+λ 2 -arctan 2 R+λ if z ≥ z 0 E 0 π 1 R R+λ 2 -arctan 2 R+λ if z < z 0
The space charge electric field is obtained by differentiation of equation 4.4:

E r,SC (r, z) = - ∂φ 0 ∂r (r, z men ) exp - z -z men z n -z men E z,SC (r, z) = φ 0 (r, z men ) z n -z men exp - z -z men z n -z men
The total electric field inside the accelerator is obtained by applying the superposition principle as above:

E r,tot (r, z) = E r,SC (r, z) + n-1 i=1 E r (E i , r out,i , r, z i + t i , z) + E r (-E i , r in,i+1 , r, z i+1 , z) E z,tot (r, z) = E z,SC (r, z) + n-1 i=1 E z (E i , r out,i , r, z i + t i , z) + E z (-E i , r in,i+1 , r, z i+1 , z)

Comparison with Finite Element Analysis

Electric potential for various accelerator geometries

This approximation gives satisfying results for a large range of geometrical parameters. To illustrate it, we compare the potential calculated by this method with the results of Opera for four different accelerator geometries. Besides, some physical properties, such as the transverse direction of the electric field or the location of the inflection points are well reproduced. This means that this approach includes physical properties of the potential profile for any geometry, which is not possible with a built function. Such physical properties are useful to describe the optics of the beam (for example to determine the position where the beam envelope changes from convergent to divergent).

Quantification of the error on the electric field for MITICA accelerator

It is helpful to quantify precisely the accuracy of this formula. A difficulty comes from the fact that the electric potential is defined as zero on the PG by convention. This convention impacts the relative error, which increases near the point where the zero potential is set. To avoid this difficulty, we compare the norm of the electric field estimated by the formula with the result of a FEA simulation. Besides, we perform this comparison with the MITICA geometry, as it is the largest accelerator. The comparison is performed along a section of the accelerator going from the meniscus location (center of the PG) to the outlet surface of the GG (that is, a length of 550 mm), with r from 0 to 5mm, discretized by steps of 0.25mm (46221 points).

The average relative error is :

E formula (r, z) -E FEA (r, z) E FEA (r, z) = -0.02
The absolute value of the relative error is distributed as follows: The normalized maximal error is:

max E formula (r, z) -E FEA (r, z) max E FEA (r, z) -min E FEA (r, z) = 0.15
The formula computes with a great accuracy the average electric field inside the accelerator, but also makes substantial errors in small regions where the approximation of Fig. 4.8 fails, that is, close to the grids inlet and outlet surfaces and near the meniscus.

Further simplifications

The formula for the electric potential as previously expressed is already simple enough for a rapid numerical integration but it displays some interesting features that allow to further simplify it, aiming at a result almost as simple as other common approximations (piecewise linear, piecewise quadratic, spline).

Close to the beamline axis the potential varies little with r:

V (δr, z) = V (0, z) + o (δr) 2
For this reason, the potential at a position r not too distant from the central axis can be approximated by its value along the central axis.

The parameter ψ (equation 4.2) has a simple expression along the central axis (r = 0):

ψ(E 0 , a, z 0 , z) = aE 0 π 1 - z -z 0 a arctan a z -z 0
ψ quickly converges to 0 with z. For example, with ψ 0 = ψ(z = z0) = aE0 π :

|z -z0|> 2.5a =⇒ ψ < 0.05ψ 0 Far from the aperture, the function φ can therefore be simplified as follows:

φ(E 0 , a, z 0 , z) ≈ E 0 (z -z 0 ) if z z 0 0 if z z 0
For most high extraction voltage accelerators (that is, with an extraction voltage of some kV) the space charge effect quickly becomes negligible, already before the EG inlet aperture.

Therefore we can simplify the formula as follows:

For z ∈ [z i + t i , z i+1 ]: φ 0 (z) = i-1 j=1 (φ (E j , r out,j , z j + t j , z) + φ (-E j , r in,j+1 , z j+1 , z)) + φ (E i , r out,i , z i + t i , z) + φ (-E i , r in,i+1 , z i+1 , z) + n-1 j=i+1 (φ (E j , r out,j , z j + t j , z) + φ (-E j , r in,j+1 , z j+1 , z)) φ 0 (0, z) ≈ i-1 j=1 E j (z j+1 -(z j + t j )) + E i (z -z i -t i ) + ψ(E i , r out,i , z, z i + t i ) + ψ(-E i , r in,i+1 , z, z i+1 ) + 0 With: i-1 j=1 E j (z j+1 -(z j + t j )) = i-1 j=1 V j+1 -V j z j+1 -(z j + t j ) (z j+1 -(z j + t j )) = V i And: φ(E i , r out,i , z i + t i , z) + φ(-E i , r in,i+1 , z i+1 , z) = E i (z -(z i + t i )) + ψ(E i , r out,i , z i + t i , z) -ψ(E i , r in,i+1 , z i+1 , z) = E i (z -(z i + t i )) + r out,i E i π 1 - z -(z i + t i ) r out,i arctan r out,i z -(z i + t i ) - r in,i+1 E i π 1 - z -z i+1 r in,i+1 arctan r in,i+1 z -z i+1 = E i π r out,i -r in,i+1 + (z -(z i + t i )) π -arctan r out,i z -(z i + t i ) + (z -z i+1 ) arctan r in,i+1 z -z i+1
Chapter 4. Theoretical elements on the beam propagation in a NBI accelerator 50 And, since the space charge effect is neglected, for z ∈ [z i + t i , z i+1 ]:

V (z) ≈ V i + E i π r out,i -r in,i+1 + (z -(z i + t i )) π -arctan r out,i z -(z i + t i ) + (z -z i+1 ) arctan r in,i+1 z -z i+1 (4.7) For z < z 1 + t 1 : V (z) ≈ φ 0 (z) + φ SC (z)
With:

φ 0 (z) ≈ E 1 π (r out,1 -r in,2 + (z -(z 1 + t 1 )) π -arctan r out,1 z -(z 1 + t 1 ) + (z -z 2 ) arctan r in,2 z -z 2
With the same approximation we assume:

φ 0 (z men ) ≈ φ(E 1 , r out,1 , z 1 + t 1 , z men ) φ 0 (z men ) = r out,1 E 1 π 1 - z 1 + t 1 -z men r out,1 arctan r out,1 z 1 + t 1 -z men And: φ SC (z) = -φ 0 (z men ) exp - z -z men z n -z men
We define L = z n -z 1 , the total length of the accelerator, and δ = z 1 + t 1 -z men , the distance between the outlet surface of the PG and the meniscus location. This parameter δ is not a design parameter but a physical property depending mostly on the extraction voltage and the current density. In most cases, it can be considered that the meniscus is located near the middle of the grid, without causing a large error on the electric potential profile.

φ SC (z) = - E 1 π r out,1 -δ arctan r out,1 δ exp - z -z i -t i + δ δ + L -t 1

Application to the evaluation of the grid electrostatic lens focal lengths

The piecewise approximation of section 4.2.3 gives a correct approximation of the potential inside the gaps between the grids. The potentials do not recombine inside the grids, but the approximation remains accurate at the bounds of the interval z = z i + t i and z = z i+1 . This is especially useful to give a general analytic evaluation of the trajectory of the particles between two grids (the accelerated beam or the secondary particles).

To illustrate such application to a concrete case, we compute the focal lengths of the electrostatic lenses, using the lens formula previously described.

Here we employ the same decomposition in sum of circular apertures than we used to determine the potential (Fig. 4.8). We also apply the simplification described in section 4.2.3.

The focal lengths at the inlet and outlet apertures are given by:

f out,i = 4V (0, z i + t i ) -E i f in,i+1 = 4V (0, z i+1 ) E i
Chapter 4. Theoretical elements on the beam propagation in a NBI accelerator 51

We determine the potential at the grid inlet (z = z i ) and outlet (z = z i + t i ) apertures.

We first consider the case z > z 1 + t 1 where space charges are neglected. The equation 4.7 yields:

V (z i + t i ) ≈ V i + E i π r out,i -r in,i+1 + d i arctan r in,i+1 d i V (z i+1 ) ≈ V i + E i π r out,i -r in,i+1 + d i π -arctan r out,i d i
It is interesting to notice that the potentials at the grid depends in fact only on the aspect ratios (aperture radius divided by the gap length). We define the angles:

α i = π -arctan r out,i d i β i = arctan r in,i+1 d i
The potential at the apertures is then rewritten as:

V (z i + t i ) = V i + V i+1 -V i π (β i -tan α i -tan β i ) V (z i+1 ) = V i + V i+1 -V i π (α i -tan α i -tan β i )
The geometric definition of α i and β i is shown in Fig. 4.16 To express the focal lengths in the most simple way, we define the voltage ratio R V i as:

R V i = V i+1 V i R V 1 → +∞
Chapter 4. Theoretical elements on the beam propagation in a NBI accelerator

52

The focal length at the inlet and outlet apertures of the grids are then given by:

f out,i = - 4d i R V i -1 - 4d i π (β i -tan α i -tan β i ) f in,i+1 = 4d i R V i -1 + 4d i π (α i -tan α i -tan β i )
The gain in accuracy with respect to a built function is not the point of this approach, because the imprecision of the lens formula itself is anyway more important than the errors on the potential value at the grids. Rather, the advantage is to qualitatively evaluate the dependence of the focal lengths on the accelerator geometric parameters, and more precisely on the angles α i and β i . Indeed, using the following identities:

∂α i ∂d i = sin 2α i 2d i ∂ tan α i ∂d i = -tan α i d i ∂β i ∂d i = -sin 2β i 2d i ∂ tan β i ∂d i = -tan β i d i ∂ tan α i ∂r out,i = - 1 d i ∂α i ∂r out,i = - cos 2 α i d i ∂ tan β i ∂r in,i+1 = 1 d i ∂β i ∂r in,i+1 = cos 2 β i d i
we obtain the partial derivative of the focal length with respect to the geometric parameters of the accelerator.

∂f out,i ∂d i = - 4β i -2 sin (2β i ) π - 4 R V i -1 < 0 ∂f out,i ∂r in,i+1 = -4 sin 2 β i π < 0 ∂f out,i ∂r out,i = 4 π > 0
Therefore, the focal length at the ith outlet aperture depends on the geometry as follows:

df out,i = 4 π dr out,i -sin 2 β i dr in,i+1 -β i - sin 2β i 2 + π R V i -1 dd i
In the same way, we can show:

df in,i+1 = 4 π sin 2 α i dr out,i -dr in,i+1 + α i + sin 2α i 2 + π R V i -1 dd i
For i = 1: We already showed:

φ SC (z) = - E 1 π r out,1 -δ arctan r out,1 δ exp - z -z i -t i + δ δ + L -t 1
Besides:

φ 0 (z 1 + t 1 ) = E 1 π r out,1 -r in,2 + d 1 arctan r in,2 d 1
And:

V (z 1 + t 1 ) = φ 0 (z 1 + t 1 ) + φ SC (z 1 + t 1 )
Therefore:

f out,1 = - 4 π d 1 (β 1 -tan α 1 -tan β 1 ) -r out,1 -δ arctan r out,1 δ exp - δ δ + L -t 1
Chapter 5

Data analysis of the Joint Experiments on NITS

The National Institutes for Quantum and Radiological Science and Technology (QST) is a Japanese organization formed from the reunion in 2016 of the National Institute of Radiological Sciences (NIRS) and a part of the activities of the Japan Atomic Energy Agency (JAEA). The Fusion Energy Directorate is composed of two institutes located in Naka (Ibaraki prefecture) and Rokkasho (Aomori prefecture). Both centers are conducting researches on NBI for fusion. In particular, the center of Naka hosts the tokamak JT-60 SA, which will operate at conditions very close to ITER (high density plasma, H mode conditions, superconducting magnets). In 2016 and 2017, QST and RFX performed Joint Experiments at Naka with the negative ion accelerator testbed NITS (Negative Ion Test Stand), described in section 3.1.3. These experiments aimed at measuring by optical calorimetry the properties of accelerated Hbeamlets with a particular magnet configuration (CESM+ADCM, see Fig. 3.6). The analysis of measurements reveals two facts that are not predicted by the particle tracing simulations: in contradiction with beam acceleration simulations and simplified analytic models, the trajectory of the beam depends on properties of the ion source, the magnetic deflection of the beamlet is always higher than foreseen by the simulations.

Analytic model of the beam trajectory

Magnetic field generated by the CESM array

We recall that the same convention, used all through this thesis, defines the ẑ-axis as the beamline axis, oriented so that the beam propagates from -z to +z. The ŷ-axis (resp. x-axis) is along the vertical (resp. horizontal) direction. The magnetic moments of the magnets are directed along ẑ and in opposite directions from one row to the next. Between two rows of magnets a magnetic field with a dominant vertical component is generated. This vertical component also changes sign from one row to the next. The horizontal dimension of the permanent magnets is much larger than the two other dimensions, and the hypothesis of an infinitely long magnet can be used as a good approximation. This implies that the magnetic field generated has only two components: along ŷ and ẑ. Let us consider a unique permanent magnet of the CESM array. We assume an infinite dimension along x, the thicknesses of the magnet along ŷ and ẑ are 2h y and 2h z , its magnetic remanence is noted B r,C and its magnetic moment follows +ẑ. At first, we define the origin point as the middle point of the magnet cuboid. The two components of the magnetic field generated by this single magnet are given by ( [START_REF] Edward | Permanent magnet and electromechanical devices: materials, analysis, and applications[END_REF]):

       B Cy (y, z) = B r,C 4π ln (y + h y ) 2 + (z -h z ) 2 (y + h y ) 2 + (z + h z ) 2 -ln (y -h y ) 2 + (z -h z ) 2 (y -h y ) 2 + (z + h z ) 2 B Cz (y, z) = B r,C 2π arctan 2h z (y + h y ) (y + h y ) 2 + z 2 -h 2 z -arctan 2h z (y -h y ) (y -h y ) 2 + z 2 -h 2 z
Chapter 5. Data analysis of the Joint Experiments on NITS

54

Let us now consider the vertical component of the magnetic field generated by two adjacent rows of CESM (Fig. 5.1). The magnet centers are separated by a distance 2L y along ŷ. We calculate their magnetic field in the vector basis of the whole accelerator: the basis vector x, ŷ, and ẑ stay identical and the centers of both magnets are located at positions (0, +L y , z EG ) and (0, -L y , z EG ). The bottom (resp. top) magnet has a magnetic moment oriented along +ẑ (resp. -ẑ).

B Cy (y, z) = B r,C 4π ln (y + L y + h y ) 2 + (z -z EG -h z ) 2 (y + L y + h y ) 2 + (z -z EG + h z ) 2 -ln (y + L y -h y ) 2 + (z -z EG -h z ) 2 (y + L y -h y ) 2 + (z -z EG + h z ) 2 + -B r,C 4π ln (y -L y + h y ) 2 + (z -z EG -h z ) 2 (y -L y + h y ) 2 + (z -z EG + h z ) 2 -ln (y -L y -h y ) 2 + (z -z EG -h z ) 2 (y -L y -h y ) 2 + (z -z EG + h z ) 2
On the horizontal plane passing by the origin, the ŷ component of the magnetic field B Cy (0, z) is:

B Cy (0, z) = B r,C 4π ln (L y + h y ) 2 + (z -z EG -h z ) 2 (L y + h y ) 2 + (z -z EG + h z ) 2 -ln (L y -h y ) 2 + (z -z EG -h z ) 2 (L y -h y ) 2 + (z -z EG + h z ) 2 + -Br 4π ln (L y -h y ) 2 + (z -z EG -h z ) 2 (L y -h y ) 2 + (z -z EG + h z ) 2 -ln (L y + h y ) 2 + (z -z EG -h z ) 2 (L y + h y ) 2 + (z -z EG + h z ) 2 B Cy (0, z) = B r,C 2π ln (L y + h y ) 2 + (z -z EG -h z ) 2 (L y + h y ) 2 + (z -z EG + h z ) 2 (L y -h y ) 2 + (z -z EG + h z ) 2 (L y -h y ) 2 + (z -z EG -h z ) 2
An advantage of this configuration is that the vertical magnetic field at the level of the aperture does not vary much in the vicinity of the beamline axis. Indeed, at a small distance δy from this axis:

B Cy (δy, z) = B Cy (0, z) + o δy L y 2
Another advantage is that the ẑ-component of the magnetic fields generated by the two magnets cancel each other at their mid-plan: B Cz (0, z) = 0, thus reducing the second order transverse deflection caused by the transverse velocity of the beam.

Magnetic field generated by the ADCM array

For the ADCM array, the hypothesis of infinitely long magnet does not hold. We consider two ADCM magnet of thicknesses 2H x , 2H y and 2H z along x, ŷ and ẑ. Their remanence is noted B r,A and both their magnetic moments follow +ŷ. For the sake of brevity we derive only the ŷ component of the field. The magnet centers Chapter 5. Data analysis of the Joint Experiments on NITS 55 are located in (-L x , 0, z EG ) and (+L x , 0, z EG ). B Ay is given by ( [START_REF] Edward | Permanent magnet and electromechanical devices: materials, analysis, and applications[END_REF]):

B Ay (x, y, z) = B r,A 4π (j,k,m,n)∈{-1,1} 4 kmn arctan         (x + jL x + kH x )(z -z EG + mH z ) -(y + nH y )   x + jL x + kH x y + nH y z -z EG + mH z          
The magnetic field generated by the ADCM does not vary much in the vicinity of the beamline axis. At a small distance δy from this axis the difference of magnetic field is:

           B Ay (δx,

Model used to describe the trajectory of the beamlet

The propagation of a group of charged particles in a electromagnetic field is a difficult problem because of the large number of individual particles and the complex mutual interactions between their trajectories and the electromagnetic field surrounding them. To get a physical picture of the beamlet behavior in the accelerator with a reasonable complexity, several hypotheses have to be made. While most of them are widely accepted and corroborated by experiments, others can be at the origin of mismatches between models and measurements.

the following assumption are not questioned in this thesis:

-The ions of the beam travel at a nonrelativistic speed in the accelerator and relativistic effects can consequently be neglected, although they are usually taken into account in simulations. -The fields are static, and two particles starting from the same position in the phase-space at different times will follow the same trajectories (continuous beam). -The angles of the particle trajectories with the beamline axis are small (paraxial approximation).

-The space charges of the co-extracted electrons in the acceleration stage can be neglected. Indeed, the space charge forces generated by a beam of particle is proportional to its charge density. For a particle species s of charge q s and mass m s with a density n s in a beam of current density j s and velocity v s , this charge density is given by:

ρ s = q s n s = j s v s
The requirement for a NBI based on negative ions NIis a co-extracted ratio about 1 in the worst case, and thus j e ≈ j NI -. Therefore, the ratio of charge densities of electrons and negative ions is:

ρ e ρ NI - ≈ v NI - v e
The beams of electrons and negative ions are accelerated by the same potential ∆V so they velocity can be estimated by:

         v e ≈ 2q e ∆V m e v NI -≈ 2q NI -∆V m NI - Hence: ρ e ρ NI - ≈ m e m NI -
This ratio is only 2.3% for Hions and 1.7% for Dions.

the following hypotheses are used for a first step, for the sake of simplicity:

-The effect of the interaction of the ion beam with the background gas in the acceleration stage is neglected, the beam is supposed to propagate in vacuum. -The mutual electrostatic repulsion between the several accelerated beamlets is neglected.

-The initial transverse velocity of the particles is neglected and it is assumed that the ions are extracted with a velocity perpendicular to the meniscus surface. -The current distribution of particles inside the beam is considered radially-symmetric.

Rather than solving the trajectory of all the beam particles, we consider the trajectory of a single particle inside the beam, located on a point where the beam self-electromagnetic forces are weak. The trajectory of this reference particle (or central orbit) depends only on the external electromagnetic forces generated by the accelerator beamline components. Therefore, it behaves as a material point. The beam is always spread around this reference particle, therefore the beam trajectory can be defined as the reference particle trajectory. The position of the beam center is defined as the average position of the beam particles. For a radially-symmetric beam this position is exactly the point where the beam self-electromagnetic forces vanish and it is close to the beamline axis.

Chapter 5. Data analysis of the Joint Experiments on NITS 57

Magnetic deflection of the beamlet

A reference particle of mass m and charge q propagating along the beamline axis with a velocity v undergoes a magnetic force:

F M (x, y, z, t) = qv × B
whose projection on x is:

F M,x (x, y, z, t) = q (B z (y, z)v y (x, y, z, t) -B y (y, z)v z (x, y, z, t)) (5.1)
The following assumptions are made:

The particle is close to the beamline axis, therefore B z (y, z) ≈ 0 and B y (y, z) ≈ B y (0, z)

At t = 0 the particle is located at the meniscus, at a short distance upstream of the PG knife-edge, at the position (0, 0, z men )

At a short distance downstream of the meniscus:

v z (x, y, z, t) v z (0, 0, z men , t = 0) v z (x, y, z, t) v x (x, y, z, t) v z (x, y, z, t) v y (x, y, z, t)
With those hypotheses, the horizontal velocity of the particle is obtained by integration of eq. 5.1:

v x (x, y, z, t) = -q m t t=0 B y (0, z)v z (x, y, z, t)dt + v x (0, 0, z men , 0) v x (x, y, z, t) = -q m z zmen B y (0, z)dz + v x (0, 0, z men , 0)
The particle is accelerated by an electrostatic potential V (x, y, z), set to 0 at the PG. The meniscus is an equipotential of the PG, hence V (0, 0, z men ) = 0. The conservation of energy states:

qV (x, y, z) = 1 2 m v 2 -v(t = 0) 2 qV (x, y, z) ≈ 1 2 m (v z (x, y, z, t)) 2 v z (x, y, z, t) ≈ 2q m V (x, y, z)
The horizontal angular magnetic deflection of the particle is defined as:

θ M (x, y, z) = dx dz = v x (x, y, z, t) v z (x, y, z, t) θ M (x, y, z) = 2m q v x (0, 0, z men , 0) - 2q m z zmen B y (0, z)dz V (x, y, z) -1/2
Chapter 5. Data analysis of the Joint Experiments on NITS 58

Electrostatic deflection of the beamlet

The electrostatic deflection of the beam is described by the lens approximation.

The meniscus is close to the PG inlet surface (z = z P G ), therefore the electrostatic deflection of the reference particle is negligible at this point.

To make the notations more explicit, since we study the case of NITS, we replace the index "1", "2" and "3" used in section 4 by "PG", "EG", "GG" and R V refers to V acc V ext .

At the PG outlet surface (z = z P G,out ) the focal length is:

f P G,out = 4V (x, y, z P G,out ) -E ext
At the EG inlet surface (z = z EG,in ):

f EG,in = 4V (x, y, z EG,in ) E ext
At the EG outlet surface (z = z P G,out ):

f EG,out = 4V (x, y, z EG,out ) -E acc
At the PG inlet surface (z = z GG,in ):

f GG,in = 4V (x, y, z GG,in ) E acc
Along its trajectory, the particle always moves in the +ẑ direction. As a consequence, there is a bijection between the time t and the ẑ-coordinate of the particle along its trajectory. For this reason, the particle position can be equivalently described by the coordinates (x(z), y(z), z) instead of (x(t), y(t), z(t), t).

Besides, the particle starts its trajectory close to the plane defined by y = 0 and we consider that it stays close to this plane all along the trajectory. We can set y = 0 and use a 2D model in the plane (x, z).

To include in a same formula the lens effect of the three grids, we use a Heaviside step function:

∀(x, a) ∈ R 2 , H(x, a) = 0 if x ≤ a 1 if x > a.
The total angular horizontal deflection of the reference particle is then:

θ(x, z) =θ M (x, z) + H(z, z P G,out ) x(z P G,out ) f P G,out + H(z, z EG,in ) x(z EG,in ) f EG,in + H(z, z EG,out ) x(z EG,out ) f EG,out + H(z, z GG,in ) x(z GG,in ) f GG,in (5.2)

Estimation of beamlet deflection in the case of NITS

The horizontal position of the reference particle at a point z 1 is:

x(z 1 ) = x(z 0 ) +

                           x P G,out = x(z men ) + z P G,out zmen θ M (x(z), z)dz x EG,in = x(z men ) + d 1 f P G,out x P G,out + z EG,in zmen θ M (x(z), z)dz x EG,out = x(z men ) + d 1 + t EG f P G,out x P G,out + t EG f EG,in x EG,in + z EG,out zmen θ M (x(z), z)dz x GG,in = x(z men ) + d 1 + t EG + d 2 f P G,out x P G,out + t EG + d 2 f EG,in x EG,in + d 2 f EG,out x EG,out + z GG,in zmen θ M (x(z), z)dz
To simplify the calculation, it is more convenient to use a matrix formulation. We define:

X =     x P G,out x EG,in x EG,out x GG,in     I 4,1 =     1 1 1 1     I θ =           z P G,out zmen θ M (x(z), z)dz z EG,in zmen θ M (x(z), z)dz z EG,out zmen θ M (x(z), z)dz z GG,in zmen θ M (x(z), z)dz           A =             0 0 0 0 d 1 f P G,out 0 0 0 d 1 + t EG f P G,out t EG f EG,in 0 0 d 1 + t EG + d 2 f P G,out t EG + d 2 f EG,in d 2 f EG,out 0            
The previous system is them rewritten as follows:

X = I 4,1 x(z men ) + I θ + A X
Being Id the identity matrix: X = (Id -A) -1 I 4,1 x(z men ) + I θ

We define F = (Id -A) -1

F =     1 0 0 0 F 2,1 1 0 0 F 3,1 F 3,2 1 0 F 4,1 F 4,2 F 4,3 1                                                F 2,1 = d 1 f P G,out F 3,1 = d 1 + t EG f P G,out + d 1 f P G,out t EG f EG,in F 3,2 = t EG f EG,in F 4,1 = d 1 + t EG + d 2 f P G,out + d 1 f P G,out t EG + d 2 f EG,in + d 2 f EG,out d 1 + t EG f P G,out + d 1 f P G,out t EG f EG,in d 2 f EG,out F 4,2 = t EG + d 2 f EG,in + t EG f EG,in d 2 f EG,out F 4,3 = d 2 f EG,out
The equation 5.2 for the angular horizontal deflection can be rewritten with X.

θ(x, z) = θ M (x, z) + H(z, z P G,out ) f P G,out H(z, z EG,in ) f EG,in H(z, z EG,out ) f EG,out H(z, z GG,in ) f GG,in X
Chapter 5. Data analysis of the Joint Experiments on NITS 60 Hence:

θ(x, z) = θ M (x, z) + H(z, z P G,out ) f P G,out H(z, z EG,in ) f EG,in H(z, z EG,out ) f EG,out H(z, z GG,in ) f GG,in F I 4,1 x(z men ) + I θ
After the inlet aperture of the GG there are no more electrostatic lens and the magnetic field is negligible. The angular deflection tends toward a limite referred to as residual deflection. This residual deflection must be as low as possible to ensure a good focusing of the beam that results of the merging of all the beamlets. The residual deflection is given by:

θ r = lim z→∞ θ M (x, z) + 1 f P G,out 1 f EG,in 1 f EG,out 1 f GG,in F I 4,1 x(z men ) + I θ
The trajectory of the beam is thus calculated by numerical integration, using the expression previously found for θ:

x(z) = x(z men ) + z zmen θ(x(z), z)dz

Numerical evaluation of the beamlet trajectory and residual deflection

The software Mathematica was used to solve the equation defined in the previous section, with the formula for the electric potential along the beamline axis obtained in section 4.2.3. The indefinite integral of the vertical component of the magnetic field along the beamline axis has an explicit expression:

B Cy (0, u)du = B r,C 2π (j,k)∈{-1,1} 2 j -2z -2(kh y + L y ) arctan jkh z -z + z EG kh y + L y -(jkh z -z + z EG ) ln (L y + kh y ) 2 + (z -z EG -jkh z ) 2 B Ay (0, u)du = B r,A 2π (j,k,m)∈{-1,1} 3 km (z -z EG + mH z ) arctan         (jL x + kH x )(z -z EG + mH z ) H y   jL x + kH x H y z -z EG + mH z           + H y arctanh           jL x + kH x H y z -z EG + mH z   jL x + kH x        
Therefore, θ M (x, y, z) has an explicit analytic expression, but I θ must be numerically integrated. In Fig. 5.4, the numerical solving of the beam trajectory x(z) is compared with an Opera simulation with V ext = 4.4kV, V acc = 21.1kV and B r,C = 1.1T. We do not include yet the ADCM in the calculation. We notice a significant error caused by an excessive deflection of the reference particle at the EG outlet surface. As explained in section 4.1.2, the focal length at the outlet surface of the EG of NITS is underestimated by the lens formula. Even though the error is quite small, it generates a large deviation in the final trajectory. Consequently, the analytic model has been corrected by multiplying by a factor 1.5 the focal length at the outlet aperture of the EG, as suggested by the comparison shown in Fig. 4.4. This factor may depend on the cases and the correction brought here likely is not generally valid, but it was found to give correct results for the cases tested, as shown on Fig. 5.5). In this figure, a case with ADCM (B r,A = 1.1T) is also plotted, and a good agreement is still observed. The trajectory and deflection along the axis are computed numerically as there is no explicit integral of θ M , but the residual deflection can be estimated by evaluating θ at the limit z z GG,out .

Being K 1 , K 2 and K 3 parameters depending only on the design parameters of the accelerator and on the mass per charge ratio of the particles:

lim z→∞ - 2q m z zmen B y (0, z)dz = K 1 B r,C + K 2 B r,A lim z→∞ -V (x, y, z) = K 3 (V ext + V acc ) -1/2 = K 3 (R V + 1) -1/2 V -1/2 ext
The focal lengths are determined by the geometric parameters of the accelerator, as explained in section 4.2.4.

                         f P G,out = -3.03mm f EG,in = 18.85mm f EG,out = -9.62mm - 48mm R V f EG,out,corrected = -14.43mm - 72mm R V f GG,in = 39.22mm + 48mm R V
We find that:

1 f P G,out 1 f EG,in 1 f EG,out 1 f GG,in F = Π 2 Π 1 Π 3 Π 1 Π 4 Π 1 Π 5 Π 1
where Π i (i ∈ [ [START_REF] Bécquerel | Sur les radiations invisibles émises par les corps phosphorescents[END_REF][START_REF] Langevin | L'inertie de l'énergie et ses conséquences[END_REF]]) are five polynomials of second order of R V whose coefficients depend only on the design parameters of the accelerator

The dependence of the vector I θ on R V is negligible. Therefore, being K 4 , K 5 and K 6 three 4 × 1 vectors whose components depend only on the accelerator geometric parameters:

I θ = K 4 B r,C + K 5 B r,A + K 6 v x (0, 0, z men , 0)
Therefore, by putting together all the components, and calling x(z men ) = x men and v x (0, 0, z men , 0) = v x0 we obtain the following formula for the residual deflection:

θ r = 1 √ V ext b 0 + b 1 R V + b 2 R 2 V a 0 + a 1 R V + R 2 V - C √ R V + 1 B r,C + d 0 + d 1 R V + d 2 R 2 V a 0 + a 1 R V + R 2 V - E √ R V + 1 B r,A + f 0 + f 1 R 2 V a 0 + a 1 R V + R 2 V - G √ R V + 1 v x0 + h 0 + h 1 R V + h 2 R 2 V a 0 + a 1 R V + R 2 V x men
For the NITS accelerator, the value of the constants in SI units are: This analytic study provides a better physical understanding of the operation parameters influence on the beamlet deflection. To qualitatively compare the impact of each parameter (magnetic field, voltage ratio... here generically denoted by p) on the residual deflection, we first need to define typical values of p during the experiments, noted p 0 , and typical ranges of variation for p, noted ∆p. Then, the residual deflection θ r is plotted against p = p -p 0 ∆p in Fig. 5.6.

During the Joint Experiments several magnetic configuration have been tried, with magnetic remanences around B 0 r = 1T and typical variations around ∆B r = 0.3T.

As will be explained later, a variation of the beamlet average position at the meniscus can be expected, with an order of value around ∆x men = 0.1mm. It is initially supposed that the reference particle starts on the beamline axis: x 0 men = 0.

The initial transverse velocity is also assumed to be zero (v 0 x0 = 0), but, as will be explained later, an initial transverse velocity can be expected with variation of the order of ∆v x0 ≈ 10 3 m s -1 .

During the Joint Experiments a typical value of

V ext is V 0 ext = 3kV with variations of ∆V ext = 1kV.
The voltage ratio is set at R 0 V = 5 during most of the measurements, with variations around ∆R V = 1.

First of all, we can see that the deflection caused by the ADCM is comparable to the deflection generated by the other parameters. This indicates that the addition of ADCM constitutes an effective solution to cancel the beamlet horizontal deflection. We also observe that the initial horizontal position and velocity of the beam, set to be 0 in most numerical models, actually strongly modify the horizontal deflection. The effect of an initial horizontal velocity and position on the reference particle trajectory is better seen in Fig. 5.7.

Finally, with a voltage ratio of R V = 5, the residual deflection of the beamlet is given by: Therefore, the trajectories of the beamlets of one horizontal row should be exactly mirror-symmetric to the trajectories of the next row with respect to the vertical plan (ŷ, ẑ).

θ r [mrad] = 25.1B r,A [T] -41.4B r,C [T] -5.4 × 10 -3 v x0 [m s -1 ] V ext [kV] -51.75x men [mm] (5.3)
For this reason it is sufficient to simulate a single horizontal row of beamlets to determine the horizontal deflection of all of them. The resulting pattern formed by all the beamlets deflected with alternated directions is referred to as "criss-cross" deflection. In Fig. 5.9 and 5.13, the criss-cross deflection caused by the noncompensated magnetic deflection is clearly visible.

After the acceleration stage exit, all the beamlets merge together to form the total ion beam. If all the beamlets were deflected in the same direction, this would generate a deflection of the total beam, which could be quite easily corrected (for example by changing the orientation of the beamline axis, as done in MITICA accelerator [START_REF] Agostinetti | Detailed design optimization of the mitica negative ion accelerator in view of the iter nbi[END_REF]). This is the case for the vertical deflection of the beamlets, mostly caused by the filter field and in the same direction for all of the beamlets. Instead, the criss-cross deflection results in an increase of divergence of the total beam. Correcting this global divergence after the acceleration stage is much more challenging.

In Opera simulations, the position, velocity and current of each track are calculated. We call n the number of tracks, I k and (x k , y k , z k , v x,k , v y,k , v z,k ) the current and coordinates in the phase-space of the k th track, and I tot the sum of the currents of all the tracks. Considering the tracks of a single beamlet, for a coordinates χ k of the phase-space, the average coordinate χ is: In the same way, the deflection of the beam is defined by the mean value of the deflection of each track weighted by the current. Thus, the horizontal and vertical deflections of the beam are:

χ = n k=1 I k I tot χ k
           θ x = n k=1 arctan v x,k v z,k I k I tot θ y = n k=1 arctan v y,k v z,k I k I tot
As will be explained later, the deflection was measured on the Joint Experiments as a distance corresponding to the displacement of the beamlets on a calorimeter located 937mm upstream of the GG outlet surface. Thus, in order to directly compare the deflection calculated by Opera with the measurements, the horizontal deflection in Opera is defined as the horizontal displacement of the beamlet 937 mm after the GG exit. To apply this definition with the Opera model, we calculate the angular horizontal deflection θ x and average position x of each beamlet 50 mm upstream of the GG exit and use this value to linearly prolong the beamlet trajectories. Indeed, a few tens of mm after the GG, the magnetic field magnitude and the transverse electric field are very close to zero and it is realistic to assume that the beamlets continue their trajectories in straight lines. The horizontal deflection of a beamlet is then described by the parameter: ∆x = x z=50mm + (887mm) tan θ x,z=50mm -x z=0

(5.4) ∆x is the horizontal displacement of the beamlet with respect to the center of the PG aperture from which it is extracted. During the Joint Experiments, 15 beamlets of Hwere accelerated together, forming a 5 × 3 array. We call ∆x(i, j) the horizontal displacement of the beamlet of the i th row and j th column.

To achieve a simple systematic study the criss-cross deflection, it is useful to quantify it by a single number. First of all, we define the average displacement of the five beamlets of a horizontal row:

∆x(i) = 1 5 5 j=1
∆x(i, j)

The trajectories followed by the beamlets of two successive horizontal rows are mirror-symmetric (Fig. 5.10), which formally implies:

∀{i, k} ∈ [[1, 3]] 2 ∀j ∈ [[1, 5]] |i -k|= 1 =⇒ ∆x(i, j) = -∆x(k, 6 -j) Therefore: ∀{i, k} ∈ [[1, 3]] 2 |i -k|= 1 =⇒ ∆x(i) = -∆x(k)
This allows to define the average deflection of all the fifteen beamlets:

∆x = 1 2 ∆x(1) + ∆x(3) 2 -∆x(2) (5.5)
With this definition, the CESM magnetic deflection increases ∆x and the ADCM magnetic deflection reduces it. Therefore, a positive (resp. negative) value of ∆x means that the magnetic deflection is undercompensated (resp. overcompensated).

We assume that the interactions between the neigbouring beamlets cancel each other in the average deflection of eq. 5.5. Therefore, we consider that applying the definition of eq. 5.5 to fifteen beamlets gives the same result as applying the definition of eq. 5.4 to a single beamlet in a model where the beamlet-beamlet interactions are not included. This means that, for a sake of simplicity, it is sufficient to simulate the trajectory of a single beamlet, with a single aperture for each grid. The first Opera model simulates the extraction of the ions from the source with the Free Plasma Emitter type 103 (section 3.2.1). In this section, we use a series of simulations to evaluate the dependence of ∆x on the operation parameters B r,A , B r,C and R V , since those parameters strongly influence the deflection (Fig. 5.

Results of a preliminary Opera model

6).

As already shown in [START_REF] Baltador | Numerical and experimental characterization of beams of negative ions and investigation on strategies for beam efficiency improvements[END_REF], the deflection does not vary much with the current density at the meniscus and this parameter is left unchanged, with j 0 [A m -2 ] = 15.2(V ext [kV]) 3/2 (the optimum value for the beam optics). Several Opera simulations were performed with the following input parameters: 0.5 T, 1.1 T, 1.6 T B r,A 0, 0.5 T, 1.1 T It is known by the analytic approach presented in the previous section that the deflection is described by the following formula:

∆x[mm] = A 1 (R V )B r,C [T] + A 2 (R V )B r,A [T] V ext [kV] (5.6)
Besides, it can be assumed with a good approximation that the deflection evolves linearly with R V for small variations around the optimal value of R V = 5 (this is visible on Fig. 5.6). For this reason, A 1 and A 2 are approximated by affine functions of R V . The fit of eq. 5.6 is in very good agreement with Opera with the following coefficients:

A 1 (R V ) = 5.16R V + 0.29 A 2 (R V ) = 4.20R V -46.09 (5.7)
The maximum error with respects to the simulations is 0.71mm and the average absolute error is 0.18mm (all the values are found in tables 5.4 and 5.5). The remanence of the ADCM and CESM should be chosen to cancel the horizontal deflection: we call design point the magnetic configuration for which ∆x = 0. Using equations 5.6 and 5.7, this condition is equivalent to:

red: V ext = 2kV blue: V ext = 3kV black: V ext = 4.4kV R V = 4 R V = 4.5 R V = 5 R V = 5.5 R V = 6 B r,C = 0.5T B r
red: V ext = 2kV blue: V ext = 3kV black: V ext = 4.4kV R V = 4 R V = 4.5 R V = 5 R V = 5.5 R V = 6 B r,C = 0.5T B r,A = 0 -0.
B r,A = 5.16R V + 0.29 46.09 -4.2R V B r,C
For R V = 5, the optimum ratio of ADCM to CESM remanence is close to 1 (Fig. 5.12), the deflection is expected to cancel out when the CESM and ADCM have roughly the same remanence.

Description of the Joint Experiments and measurements of deflection

The magnetic configuration used during the Joint Experiments is described in section 3.1.3. Only the bottom part includes ADCM, which allows a direct comparison with the case where only CESM are mounted. Throughout the two campaigns, three different sets of values were used for the remanences of the CESM and ADCM: After the exit of the accelerator, the beamlets continue their trajectory and deposit their power on a Carbon Fiber Composite (CFC) tile located 937mm after the GG exit. An IR camera facing the tile records the temperature map. The thermal images are then treated with Matlab: the temperature map just before the impact of the beamlets is recorded and defines the background temperature, the increase in temperature caused by the beamlets impact on the tile is obtained by substracting the background temperature to the temperature map obtained just after the impact of the beamlets on the tile, the perspective distortion of the temperature map, caused by the angle of the line of sight of the IR camera with respect to the tile, is corrected.

The thermal image obtained after treatment shows exactly where the beamlets deposit their power and with what magnitude. On the maps obtained, the criss-cross deflection and the corrective effect of the ADCM are clearly visible (Fig 5 .13). If the divergence of the beamlets is small enough to distinguish each single footprint on the tile, the total map of temperature increase can be fitted by a sum of 30 Gaussians corresponding to the two groups of beamlets. The horizontal displacement of the beamlets on the tile is then defined using equation 5.5: being x i,j the x coordinates center of each Gaussian on the tile and x0 i,j the x coordinates of the center of the PG apertures:

2∆x = 1 5   5 j=1 (x 1,j -x0 1,j ) + (x 3,j -x0 3,j ) 2   - 1 5   5 j=1 (x 2,j -x0 2,j )   And ∀j ∈ [[1, 5]] x0 1,j = x0 2,j = x0 3,j , therefore: 2∆x = 5 j=1
x 1,j + x 3,j -2x 2,j 10 -

5 j=1 x0 1,j + x0 3,j -2x0 2,j 10 ∆x = 5 j=1 x 1,j + x 3,j -2x 2,j 20
A translation of the temperature map (x i,j → x i,j + x of f set ) leaves unchanged the value of ∆x. Therefore, it appears that this definition is valid for any origin of the frame (x,ŷ,ẑ), and does not involve the position of the centers of the PG apertures. This is convenient because the exact position of the CFC tile with respect to the accelerator grids is not always precisely known during the experiments.

The measurements displayed in this study were made after the conditioning of the source and the inclusion of caesium. The co-extraction ratio is then around 1 and for a majority of the pulses recorded by the IR camera, the divergence of the beamlets is small (less than 10 mrad). For such cases, every single beamlet is distinguishable, thus allowing a precise measurement of their horizontal displacements on the CFC tile.

Evolution of the measured horizontal deflection with the magnetic field, extraction voltage and voltage ratio

The analytic model and the Opera simulation predict a specific evolution of the horizontal deflection with the operating parameters:

with the convention chosen for ∆x, the horizontal displacement linearly increases with B r,C and linearly decreases with B r,A , the horizontal deflection is inversely proportional to the square root of V ext , if the voltage ratio stays close enough to 5 (which is the case for all measurements with good optics), ∆x linearly increase with R V .

The large number of measurements made during the two joint experiments allow to test those predictions. First of all, the horizontal deflection is compared to the extraction voltage for 21 measurements with B r,C = The horizontal deflection is then compared with the remanence of the CESM and ADCM. Five magnetic configurations were used throughout the experimental campaigns (Table 5.6). Besides, the condition ∆x = 0 for B r,A = B r,C = 0 is included among the points. To compare all points with different extraction voltage, the deflection is scaled by the square root of V ext (∆x √ V ext ). Although the effect of B r,C is indeed linear, we observe a saturation when increasing the ADCM remanence, and the scaled deflection does not decrease when B r,A goes from 0.88 to 1.1 T. It should be noted that the arc power was significantly higher in average during the second part of the first Joint Experiments, which, as will be explained later, increases the horizontal deflection. This explains the non-linear dependence of the deflection on B r,A and the quite large dispersion of the measured scaled divergence (see the black points of Fig. 5.15). Finally, several voltage ratio scans were made in the experiments with R V close to 5, and the deflection indeed increases linearly with the voltage ratio. The slope slightly decreases when ADCM are present, which is well consistent with the predictions of Opera (eq. 5.7). General behaviors foreseen by the analytic analysis and the simulations do indeed appear in the measurements, but with some limitations: ∆x √ V ext does not decrease linearly with B r,A , the evolution of ∆x with V ext would be much better fitted by a function aV

-1/2 ext
+ b with b = 0. This is in contradiction with the simple Opera simulations (eq. 5.6), but is actually consistent with the analytic model if the initial average position of the beamlet at the meniscus is not 0 (eq. 5.3).

Discrepancy of the measurements with the predictions of Opera

While Opera predicts that, for a voltage ratio around 5, the deflection should be cancelled with the second configuration tested (B r,A /B r,C = 1, see Fig. 5.12), a significant deflection of ∆x ≈ 5 -10mm was still measured. An example of thermal footprint with the second configuration is visible in Fig. 5.17, the large non-compensated deflection is clearly visible, even in the bottom part where ADCM are included. For this reason, a configuration with a larger ADCM to CESM remanence ratio was used in the second Joint Experiments (third configuration), which effectively allowed to cancel out the deflection.

More details about the underestimation by Opera and other codes of the horizontal deflection measured during the Joint Experiments are detailed in [START_REF] Baltador | Numerical and experimental characterization of beams of negative ions and investigation on strategies for beam efficiency improvements[END_REF][START_REF] Chitarin | Experimental validation of an innovative deflection compensation method in a multi-beamlet negative-ion accelerator[END_REF][START_REF] Chitarin | Improving a negative ion accelerator for next generation of neutral beam injectors: Results of qst-consorzio rfx collaborative experiments[END_REF][START_REF] Denizeau | Experimental and numerical investigation on the asymmetry of the current density extracted through a plasma meniscus in negative ion accelerator[END_REF]. Throughout the three phases of the Joint Experiments, more than 400 calorimetry measurements were made, among which more than 200 with a very good optics (every single beamlet is discernable, like on Fig. 5.17). The deflection was calculated from the thermal maps with the Matlab post-treatment routine previously described, with an automatic fitting process that allowed to get measurements for this large number of IR images. It was impossible to simulate each single case with Opera, but, as explained previously, the fit of equation 5.6 with the coefficients of 5.7 is a very good estimation of the results of Opera for a voltage ratio around R V = 5, and can be used to replace Opera simulations. This allows a systematic comparison of Opera with the results of the Joint Experiments. In the second Joint Experiments, the extraction voltage had to be kept low due to technical difficulties. As a consequence, large errors on the actual voltage ratio were made, resulting in large fluctuations on the deflection predicted by Opera for similar pulses. For this reason, we made the hypothesis only for the pulses of the second experiments that the voltage ratio is identical for all the pulses and equal to the average value. This hypothesis is consistent with the experiments, where no large variation of measured deflection was observed, meaning that R V stayed rather constant. Fig. 5.18 compares the results of all the pulses measured with at least a good optics (that is, at least discernible rows of beamlets). It can be seen that a large majority (more than 85%) of the Opera simulations underestimate the real deflection with more than 2 mm of error. Actually, another fact observed by analysis of all the measurements demonstrates that the Opera model is lacking physical elements: the deflection scaled by the square root of the extraction voltage is correlated with the power injected inside the plasma through the tungsten filaments. In the Opera simulation (and in most particle tracing codes) the characteristics of the source are contained only in the description of the meniscus, and no plasma property at the scale of the whole source is included in the calculation of the beam trajectory. This is also the case in the analytic approach described in section 5.1.3. In Fig. 5.19, the scaled deflection ∆x √ V ext is compared to the measurements for the pulse with the following properties:

a good optics and a small divergence (all beamlet distinguishable), B r,C = 1.1T and B r,A = 1.1T or 0,

R V ∈ [4.

95, 5.05]

A roughly linear evolution of the scaled deflection is observed in both cases (with and without ADCM) with a same slope around 0.6mm kV 1/2 kW -1 . These observations lead to the conclusion that a phenomenon linked to the behavior of the plasma inside the ion source significantly increases the horizontal deflection of the beamlets. In the next section, it is hypothesized that the origin of this additional horizontal deflection is the incorrect assumption, made in most particle tracing codes (including Opera), that the current density profile at the meniscus is uniform, or at least radially symmetric.

The underestimation of the criss-cross deflection is a serious issue for ITER. If the criss-cross deflection is left uncompensated, it significantly increases the divergence of the whole beam formed by the merging of all the beamlets, which propagates towards the accelerator. The consequence would be a severe reduction of the neutralization efficiency [START_REF] Agostinetti | Detailed design optimization of the mitica negative ion accelerator in view of the iter nbi[END_REF]. The design of components that eliminate this criss-cross deflection (be it by electrostatic or magnetic compensation) is made by simulation tools, and an error in the design of the large accelerator MITICA would be notably challenging to rectify. Asymmetry of the current density profile at the meniscus and influence on the deflection

In this chapter, the acronym CDPM is used for Current Density Profile at the Meniscus.

The discrepancy between the measurements of the Joint Experiments and the simulation is not a novelty: it has been reported before [START_REF] Umeda | Long-pulse beam acceleration of mev-class h-ion beams for iter nb accelerator[END_REF], but also later [START_REF] Denizeau | Benchmark of beam acceleration codes on a high voltage negative ion accelerator for fusion with a new hypothesis on the beam meniscus[END_REF], that the particle tracing simulations tend to underestimate the criss-cross deflection for negative ions. Parallelly, it has been shown by Particle In Cells (PIC) simulations that the CDPM and the meniscus shape are not radially symmetric but are distorted by the magnetic field present in the acceleration stage that penetrates the source plasma [START_REF] Fubiani | Plasma asymmetry due to the magnetic filter in fusion-type negative ion sources: Comparisons between two and three-dimensional particle-in-cell simulations[END_REF][START_REF] Gutser | Transport of negative hydrogen and deuterium ions in rf-driven ion sources[END_REF][START_REF] Miyamoto | Study of negative hydrogen ion beam optics using the 3d3v pic model[END_REF][START_REF] Taccogna | Pic modeling of negative ion sources for fusion[END_REF][START_REF] Abe | Effects of the extraction voltage applied by the puller-electrode on the h-extraction in the linac4 negative ion source[END_REF]. Very recently, measurements of beamlet phase space structure by Y. Haba et al. directly proved the existence of an asymmetry of accelerated negative ion beamlet current density profile [START_REF] Haba | Characterisation of negative ion beam focusing based on phase space structure[END_REF].

It has been demonstrated with Opera and COMSOL that the asymmetry of the CDPM results in an increase of deflection, such that a beam with higher current on one side will be more deflected towards the opposite side [START_REF] Veltri | Effects of negative ion source characteristics on beam optics: the case of spider[END_REF][START_REF] Agostinetti | Proposal of additional experiments with mitica-like extraction grid at nits[END_REF]. An interpretation of the underestimation of the horizontal deflection measured during the Joint Experiments is that it results from the wrong assumption of symmetric CDPM [START_REF] Agostinetti | Proposal of additional experiments with mitica-like extraction grid at nits[END_REF][START_REF] Baltador | Numerical and experimental characterization of beams of negative ions and investigation on strategies for beam efficiency improvements[END_REF]. In this chapter, we present a study of this interpretation, whose main steps are already published in [START_REF] Denizeau | Experimental and numerical investigation on the asymmetry of the current density extracted through a plasma meniscus in negative ion accelerator[END_REF]. First, the effect of an asymmetry of the CDPM is studied through the evolution of the deflection with a figure of merit describing the magnitude of the asymmetry. This provides a generalization of the fit of eq. 5.6. The new fit obtained then allows to perform a systematic analysis of the error on the measured deflection to heuristically deduce the magnitude of the asymmetry for each pulse and to compare it with the operating parameters.

6.1 Improvement of the Opera model to include the CDPM asymmetry

Description and quantification of the CDPM asymmetry

The hypothesis of uniform current density profile is a default assumption of the meniscus model used in the previously presented Opera simulation (Free Plasma Emitter type 103 ). To improve it, another meniscus model (User Defined Emitter type 9 ) is used in order to define a spatial dependent CDPM, as described in section 3.2.1.

It is supposed that the asymmetry at the meniscus is caused by the penetration of the vertical magnetic field inside the source, resulting in an asymmetric current distribution along the horizontal x direction. The filter field may also generate a current asymmetry along ŷ but we ignore it for now, as it does not impact the horizontal criss-cross deflection. Consequently, the asymmetry is set between the left (x < 0) and right (x > 0) parts of the meniscus in the new Opera model. Besides, we suppose that the direction of the asymmetry is driven by the Lorentz force: a larger current density is extracted from the direction -eẑ × B y (z men ). Let us consider a case with only CESM: the direction of the magnetic deflection of the beamlet is also driven by the Lorentz force, but with the vertical magnetic field downstream of the EG, B y,d . Since the vertical magnetic field changes direction before and after the EG (Fig. 5.2), B y (z men ) and B y,d have opposite directions. Therefore, if we consider for example the case where the CESM field deflects the beam towards the right (∆x > 0), it must be assumed that the current density is higher on the left part of the meniscus (Fig. 6.1). The CDPM is noted j(x, y, z). The surface of the meniscus is calculated in a self-consistent way, so that z depends on x and y and therefore: j(x, y, z(x, y)) = j(x, y). In order to characterize the asymmetry by a single figure of merit, we lay down the following definitions: the meniscus is divided in two halves by the vertical plane (ŷ,ẑ) that contains the center of the PG aperture, being x > 0 and x < 0 the "right" and "left" sides of the meniscus, we define j right and j lef t as the average current densities over the right and left surfaces of the meniscus, the total average current density j tot = j right + j lef t 2 corresponds to the current density measured during the experiments, we define the unbalancing factor α unb as: j lef t -j right j lef t + j right . This quantity is positive when j lef t > j right , which corresponds to the case where the CESM generate a positive horizontal deflection (Fig. 6.1).

the CDPM is "centered" if the absolute deviation of the current density from its average value (|j(x, y)j tot |) is symmetric with respect to the vertical plane that divides the meniscus in two halves. Being x P G the x coordinate of the PG center, this means:

∀(x P G + x) ∈ meniscus : |j(x P G + x, y) -j tot |= |j(x P G -x, y) -j tot |
We assume that the CDPM is centered.

From this last condition we deduce:

j lef t -j tot = j tot -j right
Therefore:

j right = (1 -α unb )j tot j lef t = (1 + α unb )j tot
There are several current density profiles that could follow those criteria, however it appears that similar beamlet deflection and divergence are predicted notwithstanding the profile shape, as shown in next section.

Description of the improved Opera model

The Opera model was improved by using the User Defined Emitter type 9 as explained in section 3.2.1, with a CDPM j(x) that follows the conditions defined above. r P G is the radius of the PG, and we assume that it also is the radius of the meniscus. At first, j(x) is defined with the arctan function to obtain a smooth step along x. When α unb = 0, j(x) is constant and equal to j tot . The following function is used:

j(x) = (1 -α unb a arctan (bx + c))
We use the following conditions to determine the constants:

1 2r P G r P G -r P G j(x)dx = j tot =⇒ c = 0 1 r P G r P G 0 j(x)dx = j right = (1 -α unb )j tot =⇒ a = 1 arctan (br P G ) - ln (b 2 r 2 P G +1) 2br P G
b is a free parameter that determines the smoothness of the profile, going from a linear function to a step function:

       lim b→0 j(x) = 1 -α unb 2x r P G j tot lim b→∞ j(x) = 1 -α unb |x| x j tot
Finally, to simplify the result and to get a smoothness parameter that goes from 0 (linear) to 1 (step) we define:

β = 2 π arctan (br P G )
The current density profile is then: For each of those profiles, an Opera simulation was made without magnetic field, with j tot = 140A m -2 , α unb = 0.2, V ext = 4.4kV, R V = 5.68. The trajectories for each case are very similar, and the error on ∆x is less than 1 mm. The divergence is also not affected by the choice of CDPM. For this reason, the asymmetry at the meniscus is simply modeled by a step function:

j(x) =   1 -α unb tan β π 2 arctan x r P G tan β π 2 β π 2 tan β π 2 + ln cos β π 2   j tot
j(x) = 1 -α unb |x| x j tot
As in the previous chapter, the Opera simulations presented here are made at the optimal perveance:

j tot [A m -2 ] = 15.2(V ext [kV]) 3/2

Effect of the asymmetry on the beamlet trajectory

As seen in Fig. 6.3 and described in [START_REF] Veltri | Effects of negative ion source characteristics on beam optics: the case of spider[END_REF][START_REF] Denizeau | Experimental and numerical investigation on the asymmetry of the current density extracted through a plasma meniscus in negative ion accelerator[END_REF], a larger current density on one side of the meniscus generates a deflection toward the opposite side. Since the current density is higher on the left side when the CESM deflection is towards the right direction (and vice versa), the CDPM asymmetry systematically amplifies the CESM deflection. This means that the asymmetry can indeed explain the systematic underestimation of the horizontal deflection by simulations with a uniform CDPM. Several physical phenomena explain this modification of the beam trajectory.

Initial horizontal position of the beamlet at the meniscus

The average horizontal position of the beam at the meniscus is not 0, as it would be with a radially symmetric distribution, because the part of the beamlet with higher current density has a higher weight in the average position. We assume that the charged particles form a continuous current distribution across a cross-section of the beamlet, and we assume a flat circular meniscus of radius r P G . The average horizontal position of the beamlet at the meniscus is:

x = 1 j tot πr 2 P G x=r P G x=-r P G y= √ r 2 P G -x 2 y=- √ r 2 P G -x 2 xj(x)dxdy x = 1 j tot πr 2 P G x=0 x=-r P G y= √ r 2 P G -x 2 y=- √ r 2 P G -x 2 xj(x)dxdy + x=r P G x=0 y= √ r 2 P G -x 2 y=- √ r 2 P G -x 2 xj(x)dxdy x = 1 j tot πr 2 P G x=0 x=-r P G y= √ r 2 P G -x 2 y=- √ r 2 P G -x 2 x(1 + α unb )j tot dxdy + x=r P G x=0 y= √ r 2 P G -x 2 y=- √ r 2 P G -x 2
x(1 -α unb )j tot dxdy Using the change of variable χ = -x in the second integral:

x = 1 πr 2 P G x=0 x=-r P G y= √ r 2 P G -x 2 y=- √ r 2 P G -x 2 x(1 + α unb )dxdy + χ=-r P G χ=0 y= √ r 2 P G -χ 2 y=- √ r 2 P G -χ 2 χ(1 -α unb )dχdy x = 1 πr 2 P G x=0 x=-r P G y= √ r 2 P G -x 2 y=- √ r 2 P G -x 2 x(1 + α unb )dxdy - χ=0 χ=-r P G y= √ r 2 P G -χ 2 y=- √ r 2 P G -χ 2 χ(1 -α unb )dχdy x = 2α unb πr 2 P G x=0 x=-r P G y= √ r 2 P G -x 2 y=- √ r 2 P G -x 2 xdxdy
And finally:

x = 2α unb πr 2 P G -2r 3 P G 3 = - 4r P G 3π α unb
For example, for the NITS accelerator (r P G = 7mm):

x [mm] ≈ -3α unb
This is consistent with the case presented in Fig. 6.3 and the trajectories of Fig. 6.10 (the small errors come from the assumption of flat circular meniscus). As explained in section 5.1.7, because of the electrostatic lenses of the accelerator the average horizontal position of the beamlet at the meniscus has a very large influence on the beamlet deflection. A shift of a tenth of mm generates several mrad of residual deflection (see Fig. 5.6). This additional deflection was estimated numerically as:

θ asym [mrad] = -2.79 × 10 2 -3.63 × 10 2 R V -45R 2 V 6.10 + 6.21R V + R 2 V x men
This allows a first estimation of the influence of the asymmetry on ∆x. We consider the case R V = 5 and we approximate ∆x by ∆x[mm] ≈ 0.937 * θ asym [mrad] (the calorimeter is located 937 mm downstream of the GG). As it turns out, the influence of the effect of α unb on the trajectory is overestimated by one order of magnitude by this quick estimation. The reason is that the first analytic study cannot be directly applied here, because the reference particle hypothesis is not valid anymore. Indeed, this assumption can be made only if the space charge forces exerted on a particle located at the position x is close to zero. In the case of an asymmetric beam, the total self electric field of the beamlet is not 0 and this implies the existence of another transverse force besides the one generated by the accelerator grids. In the next section, we give a brief analytic description of the electric forces acting on an asymmetric beamlet. We consider the cross-section S of a cylindrical beam of radius R, characterized by an asymmetric charge surface density ρ(x, y) = (1 -|x| x α unb )ρ 0 and we calculate the electric field on this section. We assume that the field on S is mostly generated by the particle inside S. The current density is connected to ρ by j(x, y) = νρ(x, y), where ν is the emission frequency of the particles, supposed to be constant over S (or at least radially symmetric). The electric field at the coordinates x 0 , y 0 generated by the charge located in x, y is:

dE(x 0 , y 0 , x, y) = ρ(x, y)dxdy 4πε 0 ((x -x 0 ) 2 + (y -y 0 ) 2 ) 3/2   x -x 0 y -y 0 0   x,ŷ,ẑ
And its projection along x:

dE x (x 0 , y 0 , x, y) = ρ(x, y)(x -x 0 )dxdy 4πε 0 ((x -x 0 ) 2 + (y -y 0 ) 2 ) 3/2
The total electric field exerted by the beam particles in x 0 , y 0 is:

E x (x 0 , y 0 ) = R -R √ R 2 -y 2 - √ R 2 -y 2 dE x (x 0 , y 0 , x, y) E x (x 0 , y 0 ) = - ρ 0 4πε 0 R -R K(y, x 0 , y 0 )dy with: K(y, x 0 , y 0 ) = √ R 2 -y 2 - √ R 2 -y 2 -1 -|x| x α unb (x -x 0 )dx ((x -x 0 ) 2 + (y -y 0 ) 2 ) 3/2
We separate the integration on both halves of S and then apply the change of variable x → -x to the first integral:

K(y, x 0 , y 0 ) = 0 - √ R 2 -y 2 -(1 + α unb )(x -x 0 )dx ((x -x 0 ) 2 + (y -y 0 ) 2 ) 3/2 + √ R 2 -y 2 0 -(1 -α unb )(x -x 0 )dx ((x -x 0 ) 2 + (y -y 0 ) 2 ) 3/2 K(y, x 0 , y 0 ) = √ R 2 -y 2 0 (1 + α unb )(x + x 0 )dx ((x + x 0 ) 2 + (y -y 0 ) 2 ) 3/2 - √ R 2 -y 2 0 (1 -α unb )(x -x 0 )dx ((x -x 0 ) 2 + (y -y 0 ) 2 ) 3/2
K(y, x 0 , y 0 ) can be rewritten as a sum of two terms K 1 (y, x 0 , y 0 ) and α unb K 2 (y, x 0 , y 0 ):

           K 1 (y, x 0 , y 0 ) = √ R 2 -y 2 0 x + x 0 ((x + x 0 ) 2 + (y -y 0 ) 2 ) 3/2 - x -x 0 ((x -x 0 ) 2 + (y -y 0 ) 2 ) 3/2 K 2 (y, x 0 , y 0 ) = √ R 2 -y 2 0 x + x 0 ((x + x 0 ) 2 + (y -y 0 ) 2 ) 3/2 + x -x 0 ((x -x 0 ) 2 + (y -y 0 ) 2 ) 3/2              K 1 (y, x 0 , y 0 ) = 1 ( R 2 -y 2 -x 0 ) 2 + (y -y 0 ) 2 - 1 (R 2 -y 2 + x 0 ) 2 + (y -y 0 ) 2 K 2 (y, x 0 , y 0 ) = 2 x 2 0 + (y -y 0 ) 2 - 1 ( R 2 -y 2 -x 0 ) 2 + (y -y 0 ) 2 - 1 (R 2 -y 2 + x 0 ) 2 + (y -y 0 ) 2
The horizontal component of the self electric field over S is a sum of a constant term and a term proportional to the unbalancing factor α unb .

E x (x 0 , y 0 ) = E x,sym (x0, y0) + E x,asym (x0, y0) with:          E x,sym (x 0 , y 0 ) = - ρ 0 4πε 0 R -R K 1 (y, x 0 , y 0 )dy E x,asym (x 0 , y 0 ) = -α unb ρ 0 4πε 0 R -R
K 2 (y, x 0 , y 0 )dy (6.1) E x (x 0 , y 0 ) must be numerically integrated. The result is displayed in Fig. 6.5.

Self eletric force on the beamlet

We can calculate the total horizontal force generated by the self electric field.

F self,x = (x0,y0)∈S ρ(x 0 , y 0 ) -ρ 0 4πε 0 R -R
(K 1 (y, x 0 , y 0 ) + α unb K 2 (y, x 0 , y 0 ))dy dx 0 dy 0 (6.2)

We define:

K 3 (y, x 0 , y 0 , k) = ρ(x 0 , y 0 ) (k R 2 -y 2 + x 0 ) 2 + (y -y 0 ) 2 with k ∈ {-1, 0, 1}.
We first solve the case k ∈ {-1, 1}: Using polar coordinates (Fig. 6.4): ρ(α)

                     y = R sin θ with θ ∈ - π 2 , π 2
x 0 = r cos α with r ∈ [0, R] and α ∈ - π 2 , 3π 2
y 0 = r sin α ρ(α) = ρ 0 1 - |cos α| cos α α unb IK 3 (k) = (x0,y0)∈S R -R K 3 (y, x 0 , y 0 , k)dydx 0 dy 0 = (r cos α,r sin α)∈S θ=π/2 θ=-π/2 ρ(α)R cos (θ)dθrdrdα R 2 + r 2 + 2krR(cos θ cos α -k sin θ sin α) = (r cos α,r sin α)∈S θ=π/2 θ=-π/2 ρ(α)R cos (θ)dθrdrdα R 2 + r 2 + 2krR cos (kθ + α) = α=3π/2 α=-π/2 ρ(α) θ=π/2 θ=-π/2 r=R r=0 R cos (θ)dθrdrdα R 2 + r 2 + 2krR cos (kθ + α)
θ=π/2 θ=-π/2 R cos θ R 2 + r 2 + 2krR cos (kθ + α) -kR cos (kθ + α) arctanh r + kR cos (kθ + α) R 2 + r 2 + 2krR cos (kθ + α) r=R r=0 dθdα = α=3π/2 α=-π/2 ρ(α) θ=π/2 θ=-π/2 R 2 cos (θ) 2 1 + k cos (kθ + α) 2 -1 + cos (kθ + α) arctanh (cos (kθ + α)) -k arctanh 1 + k cos (kθ + α) 2 dθdα
We separate the integral on both halves of S:

Chapter 6. Asymmetry of the current density profile at the meniscus and influence on the deflection

84 = α=π/2 α=-π/2 ρ 0 (1 -α unb ) θ=π/2 θ=-π/2 R 2 cos (θ) 2 1 + k cos (kθ + α) 2 -1 + cos (kθ + α) arctanh (cos (kθ + α)) -k arctanh 1 + k cos (kθ + α) 2 dθdα + α=3π/2 α=π/2 ρ 0 (1 + α unb ) θ=π/2 θ=-π/2 R 2 cos (θ) 2 1 + k cos (kθ + α) 2 -1 + cos (kθ + α) arctanh (cos (kθ + α)) -k arctanh 1 + k cos (kθ + α) 2 dθdα
Using the change of variable β = π -α in the second integral:

IK 3 (k) = α=π/2 α=-π/2 ρ 0 (1 -α unb ) θ=π/2 θ=-π/2 R 2 cos (θ) 2 1 + k cos (kθ + α) 2 -1 + cos (kθ + α) arctanh (cos (kθ + α)) -k arctanh 1 + k cos (kθ + α) 2 dθdα + β=π/2 β=-π/2 ρ 0 (1 + α unb ) θ=π/2 θ=-π/2 R 2 cos (θ) 2 1 -k cos (-kθ + β) 2 -1 + cos (-kθ + β) arctanh (cos (-kθ + β)) + k arctanh 1 -k cos (-kθ + β) 2 dθdβ
Let f be the function:

f (k, α, θ) = 2 1 + k cos (kθ + α) 2 -1+cos (kθ + α) arctanh (cos (kθ + α)) -k arctanh 1 + k cos (kθ + α) 2
Then:

IK 3 (k) = ρ 0 R 2 α=π/2 α=-π/2 θ=π/2 θ=-π/2 cos (θ) ((1 -α unb )f (k, α, θ) + (1 + α unb )f (-k, α, θ)) dθdα
We now solve the case k = 0:

2IK 3 (0) = (x0,y0)∈S R -R
2ρ(x 0 , y 0 )

x 2 0 + (y -y 0 ) 2 dydx 0 dy 0 = (x0,y0)∈S ρ(x 0 , y 0 ) j∈{-1,1} arctanh R + jy 0 x 2 0 + (R + jy 0 ) 2 dx 0 dy 0 = α=3π/2 α=-π/2 ρ(α) j∈{-1,1} r=R r=0 arctanh R + jr sin (α) R 2 + r 2 + 2jr sin (α) rdrdα = α=3π/2 α=-π/2 ρ(α) j∈{-1,1} R R 2 + r 2 + 2jrR sin (α) -jrR sin (α) ln R 2 + r 2 + 2jrR sin (α) + jR sin (α) + r + r 2 arctanh R + jr sin (α) R 2 + r 2 + 2jrR sin (α) r=R r=0
dα Chapter 6. Asymmetry of the current density profile at the meniscus and influence on the deflection

85 = R 2 α=3π/2 α=-π/2 ρ(α) j∈{-1,1} 2 1 + j sin (α) 2 -1 -j sin (α) ln 2 1 + j sin (α) + 1 + arctanh 1 + j sin (α) 2 dα = R 2 α=π/2 α=-π/2 ρ 0 (1 -α unb ) j∈{-1,1} 2 1 + j sin (α) 2 -1 -j sin (α) ln 2 1 + j sin (α) + 1 + arctanh 1 + j sin (α) 2 dα +R 2 α=3π/2 α=π/2 ρ 0 (1 + α unb ) j∈{-1,1} 2 1 + j sin (α) 2 -1 -j sin (α) ln 2 1 + j sin (α) + 1 + arctanh 1 + j sin (α) 2 dα
Using the change of variable β = α -π in the second integral:

= R 2 α=π/2 α=-π/2 ρ 0 (1 -α unb ) j∈{-1,1} 2 1 + j sin (α) 2 -1 -j sin (α) ln 2 1 + j sin (α) + 1 + arctanh 1 + j sin (α) 2 dα +R 2 β=π/2 β=-π/2 ρ 0 (1 + α unb ) j∈{-1,1} 2
1 -j sin (β) 2 -1 + j sin (β) ln 2 1 -j sin (β) + 1 + arctanh 1 -j sin (β) 2 dβ
Therefore, let g be the function:

g(j, α) = 2 1 + j sin (α) 2 -1 -j sin (α) ln 2 1 + j sin (α) + 1 + arctanh 1 + j sin (α) 2
Then:

2IK 3 (0) = ρ 0 R 2 α=π/2 α=-π/2 j∈{-1,1} ((1 -α unb )g(j, α) + (1 + α unb )g(-j, α)) = ρ 0 R 2 α=π/2 α=-π/2 (1 -α unb )   j∈{-1,1} g(j, α)   + (1 + α unb )   j∈{-1,1} g(-j, α)   = 2ρ 0 R 2 α=π/2 α=-π/2 j∈{-1,1}
g(j, α)dα

We now use the identities:

     1 + sin α 2 = cos 2 x 2 - π 4 1 -sin α 2 = sin 2 x 2 - π For α ∈ -π 2 , π 2 :      cos α 2 - π 4 > 0 sin α 2 - π 4 < 0 hence:        1 + sin α 2 = cos α 2 - π 4 1 -sin α 2 = -sin α 2 - π 4
And therefore:

           g(1, α) = 2 cos α 2 - π 4 -1 -sin (α) ln 1 + 1 cos α 2 -π 4 + arctanh cos α 2 - π 4
g(-1, α) = -2 sin α 2 - π 4 -1 + sin (α) ln 1 - 1 sin α 2 -π 4 + arctanh sin α 2 - π 4
With the change of variable γ = π 4 -α 2 we can rewrite 2IK 3 (0) as:

4ρ 0 R 2 γ= π 2 γ=0
2(cos (γ) + sin (γ) -1) + cos (2γ) ln 1 + csc (γ) 1 + sec (γ) + arctanh (cos γ) -arctanh (sin γ) dγ Besides:

∀γ ∈ 0, π 4 : arctanh cos π 4 -γ -arctanh sin π 4 -γ = -arctanh cos π 4 + γ -arctanh sin π 4 + γ
Therefore:

γ= π 2 γ=0
(arctanh (cos γ) -arctanh (sin γ)) dγ = 0 And finally:

2IK 3 (0) = 4ρ 0 R 2 sin γ -cos γ + sin γ cos γ ln 1 + csc γ 1 + sec γ α=π/2 0 = 8ρ 0 R 2
We can now calculate the horizontal component of the self electric force F self,x . Rewriting eq. 6.2 with IK 3 (k):

F self,x = -ρ 0 4πε 0 (IK 3 (-1) -IK 3 (1) + α unb (2IK 3 (0) -IK 3 (-1) -IK 3 (1)) With:          IK 3 (-1) -IK 3 (1) = α unb 2ρ 0 R 2 α=π/2 α=-π/2 θ=π/2 θ=-π/2 cos (θ) (f (1, α, θ) -f (-1, α, θ)) dθdα -IK 3 (-1) -IK 3 (1) = -2ρ 0 R 2 α=π/2 α=-π/2 θ=π/2 θ=-π/2 cos (θ) (f (1, α, θ) + f (-1, α, θ)) dθdα
Therefore:

F self,x = α unb ρ 2 0 R 2 πε 0 α=π/2 α=-π/2 θ=π/2 θ=-π/2 cos (θ)f (-1, α, θ)dθdα -2 F self,x = α unb 3.66ρ 2 0 R 2 πε 0 6.2.

Effect of the external electric field generated by the grids

We consider the same cylindrical beam model and we assume that the beamlet is centered on the beamline axis. We calculate the transverse electric force exerted by the grid systems on the beamlet section S, considering that S is located near the center of the acceleration gap (between the PG and the EG). The transverse electric field generated by the grid is given by the formula obtained in section 4.2.1.2. To simply estimate the electrostatic field, we use the following simplifications: the space charge effects are neglected, the contribution of the GG to the electric field is neglected, we approximate E r,tot (r, z) by its first order series expansion at r = 0 and z =

z P G,out + z EG,in 2
The external transverse electric field is then:

E r,tot (r, z) = 16 π V ext 1 d 1 r 3 P G,out (4r 2 P G,out + d 2 1 ) 2 - r 3 EG,in (4r 2 EG,in + d 2 1 ) 2 -8 r 3 P G,out (4r 2 P G,out + d 2 1 ) 2 + r 3 EG,in (4r 2 EG,in + d 2 1 ) 2 z - z P G,out + z EG,in 2 r =(A 0 + A 1 z)V ext r with A 0 ≈ 9.5 × 10 3 m -2 and A 1 ≈ -2.0 × 10 6 m -3 if we set z men = 0.
Using the same notations as in the previous section, we calculate the total force exerted on the particles in the section S with polar coordinates:

F ext = r=R r=0 α=3π/2 α=-π/2 ρ(α)E r,tot (r, z) êr rdrdα With: êr = cos α sin α x,ŷ F ext = (A 0 + A 1 z)V ext R 3 3 α=π/2 α=-π/2
(1 -α unb )ρ 0 êr drdα + α=3π/2 α=π/2

(1 + α unb )ρ 0 êr drdα With the change of variable β = α -π:

F ext = (A 0 + A 1 z)V ext R 3 3 α=π/2 α=-π/2 (1 -α unb )ρ 0 êr dα + β=π/2 β=-π/2 (1 + α unb )ρ 0 (-êr )dβ = -(A 0 + A 1 z)V ext R 3 3 2α unb ρ 0 α=π/2 α=-π/2 êr dα = -(A 0 + A 1 z)V ext R 3 3 2α unb ρ 0 α=π/2 α=-π/2 cos α sin α x,ŷ dα = -(A 0 + A 1 z)V ext R 3 3 2α unb ρ 0 2 0 x,ŷ F ext = -(A 0 + A 1 z)V ext R 3 3 4α unb ρ 0 x
In the case of negative ions, a negative radial field (A 0 +A 1 z < 0) is defocalizing since it expands the beam radius while a positive radial field (A 0 + A 1 z > 0) is focalizing. The direction of the force caused by the asymmetry is different in the first and second case. When the external field is focalizing (that is, close to the outlet surface of a grid, here the PG), the resulting force has the same direction as the force caused by the beam self electric field, and an opposite direction when the external field is defocalizing (close to the inlet surface of a grid, here the EG). This is illustrated by Fig. 6.6. We can use the two formulae obtained in this section to compare the effects of both forces: We observe that the self-electric field of the beamlet is predominant, however the external electric field effect was slightly underestimated by the assumption that the beamlet is centered on the beamline axis. Besides, as could be expected, the self electric force becomes less dominant when the charge density over the beamlet section decreases and when the extraction voltage increases. The self electric force is predominant. Close to the PG both force deflect the beamlet towards the same direction (+x if the charge density is higher on the side x < 0).

F ext,x F self,x = - 4π 3 * 3.66 ε 0 V ext R ρ 0 (A 0 + A 1 z)

Evolution of α unb along the beamlet trajectory

We found in the previous sections that the intensity of the forces generated by the asymmetry is proportional to the unbalancing factor α unb of the section S of the beamlet. However the value of α unb changes along the beamlet path. Since the asymmetry is quantified by the CDPM asymmetry α unb (z men ), it is useful to study the dependence of α unb (z) on α unb (z men ).

The self-electric force exerted on the beamlet particles is stronger on the half with higher charge density and thus tends to cancel the asymmetry. Another way to evidence this phenomenon is to calculate the beamlet electrostatic potential energy:

E p = 1 4πε 0 r1=R r1=0 r2=R r2=0 α1=3π/2 α1=-π/2 α2=3π/2 α2=-π/2 ρ(α 1 )ρ(α 2 )dα 1 dα 2 r 2 1 + r 2 2 -2r 1 r 2 cos (α 1 -α 2 ) r 1 dr 1 r 2 dr 2 = ρ 2 0 4πε 0 r1=R r1=0 r2=R r2=0 α1=π/2 α1=-π/2 α2=π/2 α2=-π/2 (1 -α unb ) 2 dα 1 dα 2 r 2 1 + r 2 2 -2r 1 r 2 cos (α 1 -α 2 ) + α1=π/2 α1=-π/2 α2=3π/2 α2=π/2 (1 -α 2 unb )dα 1 dα 2 r 2 1 + r 2 2 -2r 1 r 2 cos (α 1 -α 2 ) + α1=3π/2 α1=π/2 α2=π/2 α2=-π/2 (1 -α 2 unb )dα 1 dα 2 r 2 1 + r 2 2 -2r 1 r 2 cos (α 1 -α 2 ) + α1=3π/2 α1=π/2 α2=3π/2 α2=π/2 (1 + α unb ) 2 dα 1 dα 2 r 2 1 + r 2 2 -2r 1 r 2 cos (α 1 -α 2 ) r 1 r 2 dr 1 dr 2
Using the changes of variable

β 1 = α 1 -π and β 2 = α 2 -π: = ρ 2 0 4πε 0 r1=R r1=0 r2=R r2=0 α1=3π/2 α1=-π/2 α2=3π/2 α2=-π/2 dα 1 dα 2 r 2 1 + r 2 2 -2r 1 r 2 cos (α 1 -α 2 ) + α1=π/2 α1=-π/2 α2=π/2 α2=-π/2 (-2α unb + α 2 unb )dα 1 dα 2 r 2 1 + r 2 2 -2r 1 r 2 cos (α 1 -α 2 ) + α1=π/2 α1=-π/2 β2=π/2 β2=-π/2 -α 2 unb dα 1 dβ 2 r 2 1 + r 2 2 + 2r 1 r 2 cos (α 1 -β 2 ) + β1=π/2 β1=-π/2 α2=π/2 α2=-π/2 -α 2 unb dβ 1 dα 2 r 2 1 + r 2 2 + 2r 1 r 2 cos (β 1 -α 2 ) + β1=π/2 β1=-π/2 β2=π/2 β2=-π/2 (2α unb + α 2 unb )dβ 1 dβ 2 r 2 1 + r 2 2 -2r 1 r 2 cos (β 1 -β 2 ) r 1 r 2 dr 1 dr 2 E p = E p (α unb = 0) + 2α 2 unb ρ 2 0 4πε 0 α1=π/2 α1=-π/2 α2=π/2 α2=-π/2 r1=R r1=0 r2=R r2=0 1 r 2 1 + r 2 2 -2r 1 r 2 cos (α 1 -α 2 ) - 1 r 2 1 + r 2 2 + 2r 1 r 2 cos (α 1 -α 2 ) r 1 r 2 dr 1 dr 2 dα 1 dα 2
With the change of variable r 1 → -r 1 :

E p = E p (α unb = 0) + α 2 unb ρ 2 0 πε 0 α1=π/2 α1=-π/2 α2=π/2 α2=-π/2 r1=R r1=-R r2=R r2=0 r 1 r 2 dr 1 dr 2 dα 1 dα 2 r 2 1 + r 2 2 -2r 1 r 2 cos (α 1 -α 2 ) E p ≈ E p (α unb = 0) + α 2 unb 3.1ρ 2 0 R 3 πε 0
The particles of the beamlet tend to reorganize into a configuration that dissipate the potential energy, and therefore that cancels α unb . The tendency of the beam to reorganize into a symmetric configuration is stronger for higher potential energy and therefore higher values of α unb .

We do not further study the dynamics of the particles, and for the sake of simplicity we consider that the reduction of the asymmetry along the beamlet trajectory is direction proportional to α unb :

dα unb dz (z) = - 1 ζ α unb (z)
where ζ is a characteristic distance. This implies:

α unb (z) = α unb (z men ) exp - z ζ
In Opera simulations, we observe that the beamlet asymmetry tends to vanish shortly after the EG outlet surface, therefore, being d 1 the extraction gap length, we can assume ζ ≈ d 1 . This approximation is correct at least inside the extraction gap (Fig. 6.9). The converging electrostatic lens at the EG deflects the right part of the beam towards the left and vice versa. For this reason, the asymmetry tends to reverse, resulting in small negative values near the exit of the acceleration stage. Anyhow, the asymmetry gets so small after the exit of the EG that its effect can be neglected.

Summary of the effects of the CDPM asymmetry on the beamlet trajectory

We still assume that, at the meniscus, the beamlet current density is higher on the side x < 0 (left side). α unb (z) refers to the unbalancing factor at a transverse section of the beamlet at a position z while the initial asymmetry at the meniscus is simply noted α unb (= α unb (z men )). The main processes by which the current density asymmetry modifies the beamlet deflection are as follows:

1. At the meniscus, the beamlet average position is shifted toward the left by a distance proportional to α unb : x (z men ) < 0 and x (z men ) ∝ α unb .

2. The beamlet current density asymmetry generates two electrostatic forces. One is caused by the beam self electric field and always pointed in the direction (+α unb x) (that is, toward the right for α unb > 0). This force is proportional to α unb (z). The other force is caused by the external transverse electric field generated by the grids. It is directed toward +α unb x when this electric field focalizes the beamlet and toward -α unb x when it defocalizes it. This force is also proportional to α unb (z) and is usually much smaller than the self electric force. The consequence of the CDPM asymmetry on the beamlet trajectory is an additional deflection towards +x, directly proportional to α unb , since all the processes described above are proportional to α unb . The beamlet asymmetry rapidly vanishes after the extraction gap. Fig. 6.10 and 6.11 give a summary of the effect of the asymmetry on the trajectory in four steps: 1 O the beamlet initial position is shifted towards -x and a horizontal deflection towards +x immediately builds up, 2 O as the unbalancing factor exponentially decreases, the horizontal deflection decreases and almost vanishes inside the EG, 3 O the beamlet is still shifted towards -x as it enters the EG and the EG electrostatic lens increases the deflection towards +x 4 O the beamlet exits the GG with a residual deflection towards +x proportional to α unb .

Systematic study of the effect of the asymmetry on the horizontal deflection with Opera

The asymmetry of the CDPM generates a significant additional horizontal deflection (in the sense that it adds up to the CESM deflection), which can explain the underestimation of the horizontal deflection by the first Opera model (Fig. 5.18). Once again, we need to compare the results of the second Opera model with hundreds of measurements. For this reason, we derive in this section an improvement of the fits of equations 5.6 and 5.7 when the parameter α unb is included in the simulations, to compare Opera predictions with the measurements without simulating each single case. First of all, the parameter ∆x was defined in the previous section under the following condition: the horizontal deflection of the beamlets was assumed to be mirror symmetric from one horizontal row to the next. This allowed to quantify the deflection with a single number ∆x, and to calculate it with a single-aperture singlebeamlet model. Since the direction of the asymmetry is directly related to the orientation of the vertical magnetic field (Fig. 6.1), the sign of α unb is also alternated row by row. As seen previously, all the effects of the asymmetry on the beamlet trajectory are linear with α unb . Therefore the deflection is also an odd function of (B r,C , B r,A , α unb ): θ(x, B r,C , B r,A , α unb ) = -θ(x, -B r,C , -B r,A , -α unb ). The illustration of Fig. 5.10, the ∆x definition of equation 5.5 and the single-beamlet single-aperture approximation remain valid and are applied in this section. More details on the beamlet mutual repulsion effect further are set out below.

Effect of the magnetic field, the extraction voltage and the current density profile on the trajectory with an asymmetric CDPM

In the previous section we did not consider any effect of the magnetic field on the deflection when an asymmetry of the CDPM is considered. Indeed, we consider that the axial velocity of the particles (depending mostly on the grid voltages) remains uniformly distributed, and consequently that the asymmetry has no effect on the global magnetic force on the beamlet. Actually, the magnetic field could generate second order effect by modifying the distance of the beam from the axis, and consequently the magnitude of the transverse force generated by the external electric field F ext , but we showed that this force has a negligible effect on the trajectory with respect to the beam self electric force (Fig. 6.7).

To confirm it, we simulate the evolution of ∆x with four different magnetic configuration (Fig. 6.12), with V ext = 4.4kV and R V = 5. In simulations we observe that ∆x evolves linearly with α unb , as expected. The slope of this evolution does not vary much with the magnetic configuration, and stays around 33mm. 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 19, 20, 21, 25, 26, 30, 33, 35} All those simulations were made with R V = 5 and j[A m -2 ] = 15.2(V ext [kV]) 3/2 . As expected, no large influence of V ext on the evolution of ∆x is distinguishable, and all the points follow a linear function of α unb with a slope around 33 -34mm (Fig. 6.13). We also observe that the current density j has no effect on the beam trajectory, at least at constant perveance.

α unb [%] ∈ {1, 2,
We also verify that, as assumed in section 6.1.2, the choice of representing the asymmetry by a step function (among other possible profiles such as those plotted in Fig. 6.2) does not significantly impact the effect of the asymmetry on the beam trajectory. To do so, we plot in Fig. 6.14 the evolution of ∆x with α unb , using a linear CDPM. The deflection obtained in this case is still a linear function of α unb , with a slope of 34mm.

Effect of the voltage ratio on the trajectory with an asymmetric CDPM

The additional horizontal deflection of the beamlet when an asymmetric CDPM is assumed is largely caused by the electrostatic lens at the EG (Fig. 6.11). For this reason, the voltage ratio is expected to significantly influence the beam trajectory: the unbalancing should generate a larger additional deflection when R V is higher.

To verify it, we plot in Fig. 6.15 the evolution of ∆x(α unb ) -∆x(α unb = 0) with α unb for R V = 4.5, 5.5 and 6. We observe that the deflection evolves linearly with α unb , with a slope that increases with R V . Once again we can assume that the deviations from the optimal value R V = 5 are small, which allows to linearize the evolution of ∂∆x ∂α unb with the voltage ratio. By doing so we find:

∆x(α unb ) -∆x(α unb = 0) ≈ (4.8R V + 8.9)α unb
This allows to complete the fit of eq. 5.6 as follows:

∆x[mm] = (5.16R V + 0.29)B r,C [T] + (4.20R V -46.09)B r,A [T] V ext [kV] + (4.8R V + 8.9)α unb (6.3)
This approximate fit is expected to be less precise than the one with a uniform CDPM because, as visible for example on Fig. 6.13, errors around ±1mm are made by this approximation. At this point, we did not provide yet a physical way to quantify α unb for each pulse. Consequently, the next step of this study is to calculate for each pulse of the Joint Experiments the value of α unb that should be implemented in the Opera model in order to reproduce the measured value of ∆x. Being ∆x exp the deflection measured during the experiments and ∆x sym the deflection predicted by a model Opera with symmetric CDPM, α unb is given by: ∆x exp [mm] -∆x sym [mm] = (4.8R V + 8.9)α unb And therefore:

α unb = 1 4.8R V + 8.9 ∆x exp [mm] - (5.16R V + 0.29)B r,C [T] + (4.20R V -46.09)B r,A [T] V ext [kV] (6.4)
This allows to treat the unbalancing factor α unb as an output parameter of the measurements and to study its evolution with the operating parameters.

Effect of the beamlet electrostatic mutual interaction

In all the Opera simulations we neglected the effects of beamlets mutual electrostatic interaction, because we made the assumption that this effect cancels out when averaging the deflection over the five beamlets of a horizontal row. Based on this hypothesis we supposed that a single beamlet single aperture model would give the same deflection as an average over fifteen beamlets.

In this section, we shortly study the effects of the beamlet beamlet interaction with an asymmetric current density at the meniscus. We compare two simulations: one with a single beamlet and a single aperture, and the other one of a full horizontal row of the accelerator (Fig. 6.16). The simulation parameters are the following:

j beamlet 140 A m -2 V ext 4.4 kV R V 5.68 B r,C 1.1 T B r,A 1.1 T α unb 0, 0.05, 0.1, 0.2, 0.3, 0.5
First of all we study the evolution of the deflection of each beamlet of the horizontal row, sequentially numbered from left to right. The deflection of a beamlet is defined as its horizontal displacement on the calorimeter with respect to its initial position at the PG (this corresponds to the parameter ∆x(1, i) of Fig. 5.10). In Fig. 6.17, we can make the following observations: The deflection is substantially lower (resp. higher) for the beamlets on the left (resp. right), with a difference around 10 mm between the two most distant beamlets. This shows that the beamlet beamlet interaction is an important effect that cannot be neglected if we study each beamlet individually.

The average deflection of the five beamlet is very close to the deflection of the central beamlet: this reflects the fact that the mutual electrostatic forces that the external beamlets exert on the central beamlet compensate each other. This justifies the single beamlet single aperture approximation: indeed, the deflection of the central beamlet in a row, the average deflection over the row and the deflection of the beamlet in a single aperture model are all very close (Fig. 6.18), even with an asymmetric CDPM.

The deflection of each single beamlet linearly evolve with α unb with a similar slope. The deflection of a beamlet in a single aperture model linearly increases with α unb with the same slope.

This example justifies the approximation of a single beamlet single aperture model, even to study the effect of the asymmetry. First of all, we assume for physical reasons that the magnitude of the asymmetry depends on parameters that affect the plasma behavior upstream of the meniscus, inside the plasma source. For this reason, the accelerator dimensions and the grid voltages are expected not to significantly affect α unb . The physical explanation of the local asymmetry at the scale of the PG radius (which will be detailed later) involves the magnetic field that penetrates the source near the meniscus and the ion flux towards the PG inside the plasma [START_REF] Fubiani | Plasma asymmetry due to the magnetic filter in fusion-type negative ion sources: Comparisons between two and three-dimensional particle-in-cell simulations[END_REF][START_REF] Taccogna | Pic modeling of negative ion sources for fusion[END_REF]. For this reason, it is expected that the ion and electron current (directly correlated to the arc power of the source P arc ) and the magnetic field penetrating the source downstream of the meniscus have a significant influence on the asymmetry magnitude. Because of the particular magnetic configuration of NITS during the experiments (Fig. 5.13), two measurements of ∆x were simultaneously made for each pulse with almost the same source conditions, over fifteen beamlets with ADCM and fifteen beamlets without. Differences of local source conditions between the two parts of the accelerator can be expected because, due to the global plasma asymmetry at the scale of the whole source, the plasma conditions may change from the top of the source to the bottom. The evolution of α unb (calculated by eq. 6.4) along the two experimental campaigns is plotted in Fig. 6.19. The magnitude of the asymmetry follows an approximately similar evolution on the top and bottom parts of the source, but is larger on the part with ADCM. This is consistent with the hypothesis that the asymmetry increases with the ŷ component of the magnetic field in the extraction area. Indeed, in the case of NITS, the absolute value of the magnetic field ŷ near the meniscus is approximately:

B y,men [mT] = 17B r,C [T] + 2.9B r,A [T] (6.5)
In Fig. 6.20 the average value of α unb for the pulses with best optics (each beamlet distinguishable) with similar arc power P arc is plotted against B y,men . The lack of points and the large dispersion around the average values of α unb make it difficult to determine a precise relation between both quantities, but there is a clear correlation, and we assume a linear evolution of α unb with B y,men for lack of better. Consequently, we can define a reduced unbalancing factor as αunb = α unb B y,men . The assumption of a linear relation between α unb and B y,men implies that αunb is independant on the magnetic field and should be roughly equal on the top and bottom parts of the grid. Actually there are still some differences between the reduced unbalancing factor with and without ADCM for a same pulse. Therefore, we take for each pulse the average value αunb = 1 2 ( αunb (with ADCM)+ αunb (w/o ADCM)). If we now compare αunb with the arc power of the source for the pulses with best optics, we observe a correlation, Chapter 6. Asymmetry of the current density profile at the meniscus and influence on the deflection 98 especially during the first Joint Experiments (in the second Joint Experiments the arc power was not much varied), as shown in Fig. 6.21. This correlation explains the unexpected dependence of ∆x on P arc shown in Fig. 5.19. It is also consistent with the increase of the slope ∂α unb ∂B y,men with P arc , visible on Fig. 6.20.

New definition of the unbalancing factor and application to the Joint Experiments

In Fig. 6.22, αunb is plotted against the arc power of the source for the 377 measurements of the Joint Experiments corresponding to the pulses with good or at least correct optics (at least discernible rows of beamlets). We calculate a Pearson correlation coefficient between both quantities of:

r Parc, αunb = Cov[P arc , αunb ] σ Parc σ αunb = 0.38
with a slope of 0.4T -1 kW -1 . To interpret the Pearson coefficient, it is necessary to take account of the sample size (377 elements) and to make sure that the value of 0.38 does not come from a statistical fluctuation. If we consider two uncorrelated random variables X, Y and a sample of X and Y of N elements, the distribution of the sample Pearson correlation coefficient is well known and can be approximated by a Gaussian of standard deviation N -1/2 centered on 0 [START_REF] Hotelling | New light on the correlation coefficient and its transforms[END_REF][START_REF] Ruben | Some new results on the distribution of the sample correlation coefficient[END_REF] (red curve of 6.23). In the case of the measurements of αunb with P arc the sample size is N = 377, which yields a standard deviation of 0.052. Therefore, the correlation between both parameters exists with a 7σ confidence, which excludes the hypothesis of a simple statistical fluctuation.

Chapter 6. Asymmetry of the current density profile at the meniscus and influence on the deflection 100

To confirm this analytic prediction, we compute the sample distribution of Pearson's correlation between the 377 values of P arc and 10 6 random data sets of 377 element, uniformly distributed in [0, 1] (this interval does not impact the result as the correlation coefficient is invariant to location and scale transformations). The result (Fig. 6.23) confirms that the value of 0.38 could not be reached with a random set of values for α unb . Therefore, the sample number is high enough to unequivocally interpret this value of r Parc, αunb as a weakmoderate correlation between the arc power of the source and the unbalancing factor [START_REF] Taylor | Interpretation of the correlation coefficient: a basic review[END_REF]. This results has important consequences on the way we look at the CDPM asymmetry: up to this point, this hypothesis was an ad hoc addition to the Opera model, consisting in rejecting the simplifying hypothesis of radially symmetric CDPM, even though this assumption gives satisfying results for the simulation of positive ion beams. The value of α unb was deduced ex post from the measurements, with a value determined by the sole requirement of matching the Opera simulation with the experiments. The assumption of asymmetric current density could explain the underestimation of the horizontal deflection but had no predictive ability, because the value of α unb changes for each pulse. The fact that α unb does not take random values means that the underestimation of the measured deflection is not a simple experimental or numerical uncertainty, but depends on physical phenomena not included on the preliminary simulations. Besides, the correlation between α unb and P arc is meaningful because, as will be detailed later, a relation of causality is expected between these parameters. Indeed, P arc directly influences the ion and electron fluxes toward the extraction area and thus the magnitude of the asymmetry. The same is true for the dependence of α unb on the vertical component of the magnetic field near the meniscus. The results of the last section allow to now define the unbalancing coefficient independently from the simulations, as follows:

1. we suppose that a beamlet extracted from the plasma source has an approximately circular shape, with a center of coordinates (x B , y B ), 2. we suppose that the current density j of the beamlet is asymmetric and we call unbalancing factor the

quantity α unb = j x<x B -j x>x B j ,
3. in the case of the NITS accelerator with a kamaboko ion source, near the optimum values of source pressure, bias voltage, perveance and voltage ratio:

α unb ≈ -0.4B y,men [T]P arc [kW] (6.6)
where B y,men is the algebric value of the magnetic field vertical component at the meniscus (α unb > 0 if the magnetic field is directed towards -ŷ), 4. we assume that the beam trajectory is correctly simulated by a numerical model where the CDPM is approximated by a profile similar to the ones displayed in Fig. 6.2.

Using this definition, we can now calculate the deflection predicted by the improved Opera model for each pulse of the Joint Experiments. To do so, we use equations 6.3, 6.5 and 6.6. The result, plotted in Fig. 6.24, shows a much better agreement with the measurements than the first model (Fig. 5.18): in this case, 75% of the measurements are correctly reproduced within an error bound of ±2mm. To conclude this section, we give a concrete example of applicability of the improved Opera model by simulating the pulse 12436, shown in Fig. 5.17. We consider only the part with ADCM. For this case, a model with uniform CDPM at the meniscus predicts a deflection of 0.9 mm while a deflection of 7.9 mm was actually measured. For this pulse P arc = 25.30kW and B y,men = -0.0219T, hence α unb = 0.22. According to the fit of eq. 6.3, Opera should give a deflection of ∆x = 8.2mm. When solving the actual Opera model, we obtain a horizontal deflection ∆x = 7.8mm very close to the value given by eq. 6.3, but more importantly in excellent agreement with the measured value. However, the fact that an almost identical value was obtained should not be over-interpreted: uncertainties are expected in the simulations because of the intrinsic imprecision of Opera, the definition of ∆x, the choice of a step profile to describe the asymmetry, the approximate value of B y,men and other approximations. The correction brought by the inclusion of the asymmetry avoids large errors of 5-10 mm observed with the preliminary model, but does not predict the deflection with an extreme accuracy, and an error bar around ±2mm should be considered in the case of NITS. Fig. 6.25 compares the trajectories of the beam from the meniscus to the CFC tile with and without asymmetry. The residual deflection caused by the asymmetry accounts for the large discrepancy between the measurements and the model with uniform CDPM.

Summary of the main results

We now present a short summary of all the useful results obtained in this chapter. The study presented here is a complement and an improvement of the paper [START_REF] Denizeau | Experimental and numerical investigation on the asymmetry of the current density extracted through a plasma meniscus in negative ion accelerator[END_REF] (the main improvement being the inclusion of the influence of the parameter R V , previously neglected). From the analysis of the Joint Experiments we draw the following conclusions:

in the case of negative ion acceleration, it is not correct to assume a radially symmetric current density profile at the meniscus, and doing so leads to a large underestimation of the deflection in simulations, the deflection of a beamlet (whether defined as a displacement of the beamlet or as an angle) with a CESM (and CESM+ADCM) configuration is very well approximated by the following numerical fit:

(a 0 R V + a 1 )B r,C + (a 3 R V + a 4 )B r,A √ V ext + (a 5 R V + a 6 )α unb
where the coefficients {a i } depend only on the accelerator geometry. Once the coefficients {a i } are known, this formula can replace simulations, at least for pulses close to the optimum operation parameters, in order to study a large number of cases.

the magnitude of the current density asymmetry at the meniscus is correlated to the power injected inside the plasma source and to the magnetic field penetrating the source near the meniscus.

In this chapter, the CDPM asymmetry was heuristically studied by an indirect measurement of its effects on the deflection of the beamlets on the NITS accelerator. However the reduced asymmetry coefficient αunb was shown to have a physically meaningful correlation with the arc power P arc . Appendix A presents a short study of the emergence of the current density asymmetry in the extraction area under the effect of the magnetic field, using PIC simulations. The results obtained in this chapter are used and tested in chapter 8.

Chapter 7

Explanation of the difference of beamlet divergence between NITS and BUG Max Planck Institut für Plasmaphysik (IPP) is a German physics institute whose activities include experimental and theoretical researches in fusion. It comprises two centers located in Garching bei München (Bavaria) and Greifswald in northeast Germany. The Garching center was founded in 1960. It notably hosts the ASDEX tokamak, which has been holding an important role in fusion researches since its construction (discovery of H mode in 1982).

A part of the ITED divison of IPP Garching is in charge of researches on NBI towards the conditions of ITER HNB, with the BUG and ELISE experiments (sections 3.1.5 and 3.1.6), two graded steps towards SPIDER.

In 2019, the divergence measured by spectroscopy on BUG and ELISE beams was around 15-20 mrad, significantly higher than measured on similar devices, such as NITS (section 3.1.3). I was hosted four months inside the ITED division to study this problem, in collaboration with researchers in charge of BUG and ELISE operations. The idea of this collaboration was to use the numerical models applied to NITS and the experimental results of the Joint Experiments in order to quantify and explain the difference of measured divergence between BUG and NITS, despite similar dimensions and operating conditions. Several explanations have been investigated and, in particular, it was important to understand if the mode of production of the plasma inside the source (RF or filament) had an impact on the optics. The main finding of this exchange period is that, due to the conception of the grid geometry of BUG, a part of the beamlets impinges on the EG when their current density exceed a threshold. Because of this limitation, it is impossible to reach the optimum conditions of beam current and extraction voltage to minimize divergence. This conclusion, presented at the end of my working period, was confirmed shortly later by a joint campaign of experiments between IPP Garching and RFX [START_REF] Wimmer | Beamlet scraping and its influence on the beam divergence at the batman upgrade test facility[END_REF].

Motivation of this study 7.1.1 Definition of the beamlet divergence, calculation with particle tracing simulations

The divergence of a beamlet quantifies the dispersion of the transverse velocities of the particles that constitute it. It is minimal when all the particles follow parallel trajectories and increases with the deviation of the particles from the beamlet average trajectory. The divergence is measured in a transverse plane (x,ŷ) intercepting the beam at a position z. There are at least two common ways of quantifying the divergence of a beamlet of particles:

Being (θ i ) with i ∈ [[1, n particles ]
] the distribution of the angles of the particle trajectories with respect to the beamline axis, the divergence can be defined as the variance of the angular distribution. It is then referred to as the RMS divergence:

ε RM S = (θ i -θ ) 2
With this definition, it is clear that the RMS divergence is 0 when all the particles move along parallel trajectories (∀i θ i = θ ) and increases with the dispersion of the particle transverse velocities around θ .

The angular distribution of the particles on a vertical cross-section of the beamlet is usually represented by a Gaussian distribution:

f (θ) = f 0 exp - θ 2 2σ 2
with f 0 the amplitude of the Gaussian and σ its standard deviation. The e-folding divergence of the distribution is the value ε 1/e such that the particle fraction at θ = ε 1/e is f 0 e . This corresponds to the width of the Gaussian:

ε 1/e = √ 2σ.
The RMS divergence is useful for particle tracing simulations, where the angle of each particle (or macroparticle) is known. For example, in Opera or SLACCAD the divergence of a beamlet is defined as follows: we call n the number of tracks, I k and (x k , y k , z k , v x,k , v y,k , v z,k ) the current and coordinates in the phase-space of the k th track, and I tot the sum of the currents of all the tracks.

ε RM S = n k=1 I k I tot   arctan   v 2 x,k + v 2 y,k v z,k   -   n j=1 I j I tot arctan   v 2 x,j + v 2 y,j v z,j       2 (7.1)
In a similar way, we can also define the RMS divergence of the beam only on the x and y directions:

                   ε x,RM S = n k=1 I k I tot   arctan v x,k v z,k -   n j=1 I j I tot arctan v x,j v z,j     2 ε y,RM S = n k=1 I k I tot   arctan v y,k v z,k -   n j=1 I j I tot arctan v y,j v z,j     2 (7.2)
The paraxial approximation states that the angles of the particles with respect to the beamline axis are small, therefore:

arctan   v 2 x,k + v 2 y,k v z,k   = dr dz = r
where r is the radial position of the particles (r 2 = x 2 + y 2 ) and the z-derivative is indicated by .

Similarly: arctan v x,k v z,k = x and arctan v y,k v z,k = y .
The distribution of the particles in the space (x , y ) is commonly represented by a 2D Gaussian: being f 1 (x , y ) the fraction of particles with angles (x , y ), and f 2 (r , α) the fraction of particles with an angle (r cos α, r sin α):

           f 1 (x , y ) = 1 2πσ 2 exp - x 2 2σ 2 - y 2 2σ 2 f 2 (r , α) = 1 2πσ 2 exp - r 2 2σ 2
The fraction of beam particle with the angles x , y or r is:

                       g 1 (r ) = α=2π α=0 1 2πσ 2 exp - r 2 2σ 2 rdα g 2 (x ) = y =+∞ y =-∞ 1 2πσ 2 exp - x 2 2σ 2 - y 2 2σ 2 dy g 2 (y ) = x =+∞ x =-∞ 1 2πσ 2 exp - x 2 2σ 2 - y 2 2σ 2 dx
Chapter 7. Explanation of the difference of beamlet divergence between NITS and BUG 105

           g 1 (r ) = r σ 2 exp - r 2 2σ 2 g 2 (x ) = 1 √ 2πσ exp - x 2 2σ 2
The difference between the distribution of the particles on x or y and on r is illustrated by Fig. 7.1. The RMS divergence varies by a factor √ 2 whether it is calculated along a specific direction or for any transverse direction.

               ε RM S = r =+∞ r =0 g 1 (r ) r - χ =+∞ χ =0 g 1 (χ )χ dχ 2 ε x,RM S = x =+∞ x =-∞ g 2 (x ) x - χ =+∞ χ =-∞ g 2 (χ )χ dχ 2 ε RM S = √ 2σ ε x,RM S = ε y,RM S = σ
Therefore, if we assume that the particle distribution on (x , y ) is Gaussian, the RMS and e-folding divergences are the same parameter. It should be noted, however, that the divergence along one specific direction (e.g. x) is given by: ε x,RM S = ε 1/e / √ 2. If we refer to the e-folding divergence along the x direction, this corresponds to the quantity ε x,1/e = √ 2ε x,RM S . In the rest of this study we assume that the beam is indeed distributed in angles according to a 2D Gaussian and the divergence is referred to as ε = ε RM S = ε 1/e . The reasoning presented here is valid only when the beam is circular. The more general form for the beam distribution would be:

g(x , y ) = 1 2πσ 2 exp - x 2 2σ 2 x - y 2 2σ 2
y with σ x = σ y , ε x,RM S = σ x and ε y,RM S = σ y . In this study, we consider that both parameters are close enough to assume a circular shape for the beam. Under asymmetric conditions (external magnetic field, asymmetry of the beam current density) some mrad of difference between the two directions can be calculated.

To test the accuracy of the 2D Gaussian representation for the angular distribution of the particles, we calculate this distribution for an illustrative simulation. Opera is used to simulate the propagation of a Hbeam in the NITS accelerator, with a symmetric beamlet of current density j = 145A m -2 , V ext = 4.4kV and R V = 5.68, with a CESM+ADCM configuration (B r,C = B r,A = 1.1T). We consider a transverse cross-section located 50mm downstream of the GG outlet surface. To calculate the distribution function of the particles, we discretize x and y in samples of size 2 mrad and calculate the cumulative current fraction of the particles for each sample. The distribution simulated with Opera is well approximated by a Gaussian, with σ x = 9.5mrad and σ y = 10.5mrad. This small difference of 1 mrad is likely caused by the magnetic field but can be neglected. The e-folding divergence of the beam is around 10 mrad. The measured increase in temperature is supposed to be directly proportional to the power density of the beamlets intercepting the calorimeter, and therefore to the current density (all particles are accelerated by the same potential difference). As previously explained (Fig. 2.7), the outward radial self electric force caused by the beam space charge is compensated by the background gas ionization some cm downstream of the GG. For this reason, we can suppose that the beam is laminar with a fixed angular distribution as it propagates from the exit of the accelerator to the CFC tile. This angular distribution is a Gaussian of standard deviation σ r noted f (r ). For a given particle, the hypothesis of laminar beam and straight trajectories yield r = r 0 + r L where r is the particle radial position on the calorimeter and r its angle with the beamline axis, constant between the GG and the calorimeter. r 0 is the radial position of a particle as it exits the GG. Therefore, the particles hitting the calorimeter are radially distributed as f r L + r 0 , which is a Gaussian of r of standard deviation Lσ r . It is common to neglect the value of r 0 , as it is generally assumed to be significantly smaller than the beam radius at the calorimeter. This approximation gives an upper bound of the divergence, corresponding to the assumption that the beam has a radius 0 as it exits the GG. The current density profile of the beam is then a Gaussian at the calorimeter: j(x, y) = A exp -

(x -x 0 ) 2 + (y -y 0 ) 2 2(Lσ r ) 2 .
The temperature map obtained in calorimetry measurements is fitted by a function:

∆T = N k=1 A k exp - (x -x 0 ) 2 + (y -y 0 ) 2 2σ 2 k
Since we assumed that the increase in temperature is proportional to the current density, the e-folding divergence of each beamlet is directly given by:

ε = √ 2σ r = √ 2σ k L
Measurements by calorimetry are most precise when the beamlets have a very small divergence, ideally when each single beamlet is distinguishable, as in the case of NITS or MTF. At larger values of divergence, the fitting procedure is more delicate and require the use of simulation tools to get a correct value.

Spectroscopy

Measurements by Beam Emission Spectroscopy (BES) rely on analysis of the light spectrum produced by the interactions between the beam and the background gas. The beam and background particles are excited by collisions, resulting in radiations associated to the atom de-excitation. The main radiation observable is Hα (Dα for deuterium beam), which corresponds to a spectrum line around 653 nm. Since the beam propagates at high velocity, the Doppler effect shifts the beam Hα line, with a magnitude that depends on the beam velocity (determined by the grid voltages) and on the angle of the detector with respect to the beam direction. Two peaks corresponding to the radiations of the beam particles and the background gas are normally visible. Furthermore, the particles neutralized by stripping reactions in the PG-EG gap propagate inside the accelerator, with the velocity that the extraction voltage provided them before their neutralization. For this reason, when the measurement is made downstream of the EG, a third peak corresponding to the stripped neutral particles is distinguishable. A typical spectrum measured by BES is shown in Fig. 7.4. The BES provides a plurality of information on the beam characteristics (stripping losses, velocity, deflection). This diagnostic also allows to measure the beam divergence: the dispersion of the velocity distribution of the beam particles generates a broadening of the Doppler peak. This broadening can be directly related to the e-folding divergence of the beam using numerical models or empirical fits [START_REF] Barbisan | Beam characterization by means of emission spectroscopy in the elise test facility[END_REF]. Limitations of the divergence measurement by spectroscopy are the necessity to average the divergence over the Line of Sight of the BES in a multi-beamlet accelerator, and the errors caused by the criss-cross deflection, leading to an overestimation of the divergence. Differences of some mrad can be found between BES and calorimetry measurements, with a deviation that decreases as the divergence gets smaller (see for example [START_REF] Ott | Beam divergence and ion current in multiaperture ion sources[END_REF][START_REF] Wimmer | Beamlet scraping and its influence on the beam divergence at the batman upgrade test facility[END_REF]).

Difference of divergence for RF-driven and filament-driven facilities

In a multibeamlet accelerator, the beamlet divergence is a key parameter to define the quality of the beam optics: the divergence of the total beam resulting from the merging of all the beamlets is at least equal to the divergence of the beamlets. A requirement for ITER HNB is to generate a beam with at most 7 mrad of e-folding divergence. While most of the filament driven negative beam accelerators have reached this criterion (NITS [START_REF] Chitarin | Experimental validation of an innovative deflection compensation method in a multi-beamlet negative-ion accelerator[END_REF], MTF [START_REF] Denizeau | Benchmark of beam acceleration codes on a high voltage negative ion accelerator for fusion with a new hypothesis on the beam meniscus[END_REF], NIFS [START_REF] Veltri | Optics of the nifs negative ion source test stand by infrared calorimetry and numerical modelling[END_REF], RNIS [START_REF] Haba | Characterisation of negative ion beam focusing based on phase space structure[END_REF]) with beamlet divergences of ≈ 4-7 mrad, significantly larger values were measured on BATMAN Upgrade (BUG) and ELISE, both RF-driven sources, with divergences in the range of 14-20 mrad. This chapter focuses on a comparison between the performances of NITS and BUG in terms of divergence, because both accelerators have a comparable grid geometry and operating parameter ranges (see Fig. 7.5 and table 7.1). The data presented here was collected from discussions with IPP and QST researchers and are valid for the versions of NITS and BUG operated at the time of this PhD (2019). Actually, the RMS and e-folding definitions of the divergence are always defined as positive quantities, whether the particles are converging towards the beamlet center or diverging away from it. In reality, one branch of the smiling curve corresponds to a globally converging beam and the other one to a globally diverging beam, with a superposition of both converging and diverging trajectories at the minimum. The voltage ratio determines the refractive power of the accelerator and therefore strongly influences the beamlet divergence. As a preliminary study, we use a simplified model of geometric optics to evaluate the effect of R V on ε. We apply to NITS and BUG the approximation presented in section 4.2: the accelerator is represented by a series of electrostatic lenses, whose focal lengths depend only on the voltage ratio, the extraction gap length and the aspect ratii of the accelerator. We use the same notations for the geometric parameters (the angles α and β are defined in Fig. 4. [START_REF]Proceedings of the second united nations international conference on the peaceful uses of atomic energy[END_REF]). In this model, the RE (Repeller Electrode) of BUG is assumed to be grounded. Consequently the GG does not modify the divergence of the beam. The RE is considered here as the third grid of the model.

f EG,out [mm] -9.6 - 48 R V -1 -8.2 - 60 R V -1 f GG,in [mm] 39.2 - 48 R V -1 51.0 - 60 R V -1
Here we do not apply the empirical correction of f EG,out used in section 5.1.7 since we are merely looking for a qualitative description. The particles start their trajectory with an initial angle that depends on the meniscus shape. Here the meniscus is approximated by a spherical surface of radius of curvature ρ men (positive for concave meniscus). The particles located on the menicus at a radial distance r start their trajectory with a deflection r = -r/ρ men : the meniscus behaves like a convergent lens of focal length ρ men . We use the matrix formulation of geometric optics to calculate the transfer matrix of the whole accelerator L. To do so, we define the following functions, corresponding to a lens of focal length f and to a propagation along a distance d:

F (f ) = 1 0 -1 f 1 D(d) = 1 d 0 1
With the notations of section 4.2.4, the transfer matrix of the three grid accelerator is:

L = F (f GG,in ).D(d 2 )
.F (f EG,out ).D(t EG ).F (f EG,in ).D(d 1 ).F (f P G,out ).D(δ).F (ρ men ) The refractive power of the whole accelerator is given by: 1 f acc = L 2,1 . Fig. 7.8 shows the evolution of this parameter with the voltage ratio for NITS and BUG. For low (resp. high) R V the accelerator acts as a convergent (resp. divergent) lens, and there is an optimal value of R V for which the accelerator does not deflect the particles that propagate through it. This optimum value is higher for BUG than for NITS, as a result of the difference of geometry between both accelerators. In this simple model, the radius of curvature of the meniscus has little impact on the accelerator refractive power.

The influence of the perveance on the divergence is partially explained by the change of curvature of the meniscus, getting less concave as the perveance increases (see section 4.1.3.2), but the main effect comes from the radial forces caused by the beam space charge on itself. The total charge carried by the beam increases with the perveance, causing a larger radial outward force on the beam. A more physical approach to evaluate the beamlet divergence is to calculate its envelope, that is, the smallest closed surface that contains all of its particles. This envelope is supposed to be radially symmetric, with a varying radius r(z). With paraxial approximation, the envelope equation is given as a function of the voltage along the beamline axis V (z) as:

r (z) + V (z) 2V (z) r (z) + V (z) 4V (z) r(z) = m i 2e 1 4πε 0 r(z) I V (z) 3/2 + T EM I
where e and m i are the charge and mass of the ions and I the beam current (see [START_REF] Reiser | Theory and design of charged particle beams[END_REF] for the demonstration and [START_REF] Veltri | Design of the new extraction grid for the nio1 negative ion source[END_REF] for the form of the equation used here). The term T EM I can usually be neglected for NBI accelerators [START_REF] Serianni | Neutralisation and transport of negative ion beams: physics and diagnostics[END_REF]. This equation can be rewritten with the following scaled parameters:

r = r r P G z = z 3d 1 Ṽ = V V ext
The symbol now represents the derivation with respect to z when applied to a scaled parameter.

r P G 9d 2 1 r (z) + Ṽ (z) 2 Ṽ (z) r (z) + Ṽ (z) 4 Ṽ (z) r(z) = m i 2e 1 4πε 0 r P G r(z) I V ext Ṽ (z) 3/2 Being Π 0 = 4ε 0 9 2e m i πr 2 P G d 2 : r (z) + Ṽ (z) 2 Ṽ (z) r (z) + Ṽ (z) 4 Ṽ (z) r(z) = Π Π 0
The potential along the axis of a triode grid set can be written in the form V (z) = φ 1 (z)V ext + φ 2 (z)V acc , where the functions φ 1 and φ 2 only depend on the grid geometry (see section 4.2.1.1). With scaled parameters: Ṽ (z) = φ1 (z) + φ2 (z)R V . Therefore:

r (z) + φ1 (z) + φ2 (z)R V 2 φ1 (z) + φ2 (z)R V r (z) + φ1 (z) + φ2 (z)R V 4 φ1 (z) + φ2 (z)R V r(z) = Π norm (7.3)
If we consider a circular beam with uniform current density and no magnetic field and apply the paraxial approximation, the envelope formula implies that the beam envelope (and therefore its divergence) only depends on the voltage ratio, the normalized perveance and the geometric parameters of the accelerator.

7.2.2 Effects of the magnetic field, the beamlet asymmetry and other secondary parameters the particles are deflected with the same force, the RMS of the velocity distribution does not change: only the deflection of the beam increases, and not its divergence.

In accelerators for NBI, the distribution of axial velocity of the particles is mainly driven by the electric force created by the grid differences of potential. For this reason, all the beam particles acquire similar axial velocities, resulting in a very low dispersion of the axial velocity profile (Fig. 7.9). In any case, the transverse magnetic force on a particle is much smaller than the transverse electric force near the apertures of the grids. Therefore, it is likely that the Lorentz force does not significantly affect the beamlet divergence. Another effect of the magnetic field could be to shift the beam from the beamline axis, thus generating an asymmetric effect of the space-charge (the model presented in 4.1.3.1 shows that an asymmetric electric field is exerted on a beamline off-centered from the beamline axis). However, it is assumed that this effect does not significantly affect the beamlet divergence. At most it generates differences of the order of 1 mrad between the divergence along x and ŷ (Fig. 7.2). This is confirmed, for example, by the Opera simulations of the NITS accelerator presented in section 5.2.2. In Fig. 7.10, the evolution of the beamlet divergence with the voltage ratio is plotted for different magnetic configurations. No significant difference is induced by the presence and amplitude of the magnetic field. For this reason, the simulations presented in this section do not include the magnetic field. The asymmetry of the current density profile of the beamlet described in chapter 6 modifies the space charge distribution inside the beamlet and therefore has an impact on the beam divergence. Besides, if we consider the RMS definition of the divergence, the difference of current modifies the calculation of the angular distribution standard deviation because the part of the beam with higher current has a higher weight in the root mean square. However, the asymmetry vanishes before the exit of the accelerator (Fig. 6.9) and the beam can be considered as symmetric where its divergence is usually measured (some cm or dm downstream of the GG). Effects on the divergence may come from the different dynamics of the particles along the horizontal and vertical direction, which modify the shape of the beamlet. For this reason, significant differences between ε x and ε y can be expected. Such differences have been demonstrated experimentally by Y. Haba et al. [START_REF] Haba | Characterisation of negative ion beam focusing based on phase space structure[END_REF]. This is also visible in Opera simulations (Fig. 7.11): the e-folding divergence along the x direction (defined as ε x,1/e = √ 2ε x,RM S , as explained in section 7.1.1) is higher than on the ŷ direction, and this difference increases with the asymmetry. The total divergence of the beamlet also increases with α unb but, for the values of α unb estimated for NITS (0.1-0.3) the additional divergence is only of 2-3mrad. The effect of the asymmetry on the shape and divergence of the beam is substantial and is an open issue. However, in the simulations presented here, the current density profile of the beam was assumed to be symmetric, which could lead to a certain underestimation of the real divergence by the numerical tools. Moreoever, we did not take into account the halo effect [START_REF] Hpl | Negative ion beam halo mitigation at the 1 mv testbed at irfm[END_REF]: in negative ion accelerators, it is assumed that, besides the focused accelerated beamlet, a secondary co-axial beam originates from particles generated on the chamfer of the PG. This beam, referred to as halo, can be very divergent and does not follow the Gaussian distribution of the main beamlet. As a consequence, the beam halo is assumed to increase the measured divergence of some mrad. Finally, we neglected the effect of the background gas, the co-extracted electrons and the secondaries generated in the accelerator. The contributions of the electrons and secondaries to the space charge can be neglected: the electrons are much lighter than the negative ions, and the density of secondaries is usually much smaller than the ion density in the beam. An effect of secondary particles is to slow down the beam by reducing its energy through collisions, and to intercept a part of the accelerated ions, thus reducing the beam current density between the meniscus and the GG. The effect of secondaries on the divergence was not studied, and only negative ions were included in the simulations presented here. With the approximations described in the previous section, the divergence of the beamlet for a given geometry depends only on Π norm and R V . It is difficult to study separately the effect of each parameter through a series of "smiling curves", because the optimum value of R V depends on Π norm and vice versa. For this reason, a better representation of the evolution of the divergence is obtained if we describe the divergence as a map in the plane (R V , Π norm ). For both geometries, 187 simulations were made with Opera and SLACCAD, with the following parameters: First of all we can see that, for the NITS geometry, both Opera and SLACCAD predict a similar structure for the map of perveance: this is indicated by the shape and orientation of the divergence isolines. However, a large difference between both codes is that the Opera map seems to cover only a fraction of the SLACCAD map: it is as if the Opera map was strechted along the Π norm direction (this is particularly visible by observing the deformations of the isolines). Consequently, large differences are obtained in terms of optimal normalized perveance. We cannot provide an explanation of this discrepancy on the effect of Π norm , but it appears, at least in the case of NITS, that Opera is the most precise code. Indeed, during the Joint Experiments, the optimal normalized perveance was found around 0.32-0.35, in good agreement with Opera prediction (Fig. 7.13).

Concerning the voltage ratio, both codes slightly overestimate the optimal value for NITS (5.5 instead of 5 in experiments).

While SLACCAD and Opera are quite consistent with the measurements for NITS (with a minimal divergence measured around 9 mrad), they fail to reproduce the large divergence measured with BUG. Opera is, however, not so far from the ≈ 14 -15 mrad measured on BUG, with an error that can be considered as acceptable.

The issue to solve when I started my exchange period was to understand why, while simulations predict very close values of minimal divergence for both NITS and BUG geometries (less than 1 mrad of difference), the measurements reveal a significantly higher divergence with BUG. Thus, the issue to solve in this chapter is not a discrepancy in numerical tool predictions, but rather a difference of measurements between both devices, seemingly not predictable by simulations. The radial space charge force acting on the beamlet and increasing its envelope becomes stronger when the perveance of the beamlet increases. Beyond a threshold, the beamlet radius becomes so large that a significant part of it impinges on the grids. In simulation, this phenomenon of scraping causes anomalies in the calculation of the beam trajectory and divergence. In real operations, the beam extraction and acceleration cannot be maintained when this effect becomes too important, because of the large heat loads generated on the grids. The scraping of the beam is characterized by a decorrelation of the beam current measured downstream the accelerator exit (accelerated current density) and the current density extracted at the PG (injected current density), as shown in Fig. 7.14. At low current density, both parameters are almost equals (possible differences come from the interaction of the beamlet with the background gas). As the injected current density keeps increasing, a saturation of the accelerated current density appears, corresponding to a threshold point. Beyond this point, more and more current get lost by scraping on the grids, and the accelerated current density drops. In simulations, different modes of scraping are identified:

If the beam reaches the EG with a radius larger than its inlet aperture, a part of the grid is lost on the vertical face of the EG. This kind of scraping is observed for BUG (Fig. 7.15). It happens because the clearance of the beam at the EG (the minimal distance between the beam envelope and the grid surface) is too small.

If the beam becomes too large inside the EG a part of the external particles impinges inside the grid aperture, against the chamfer. This kind of scrapping happens with the NITS accelerator (Fig. 7.14).

Finally, the beamlet deflection generated by magnetic field and beam asymmetry worsen both modes of scraping, and scraping occurs also with the NITS accelerator when the magnetic deflection is not compensated.

Opera predicts that, because of scraping, it is impossible to reach the optimal perveance with BUG. This appears clearly on the divergence map of Fig. 7.12: the minmum of divergence is located on the line of threshold perveance. On this map, there are no closed isoline of divergence, which means that no curve of ε against Π norm display an overperveant part (a part where the divergence increases with Π norm after reaching its minimum). The optimum perveance for BUG predicted by Opera is greater than or equal to the threshold of beamlet scraping. Unlike Opera, SLACCAD actually predicts that the optimal perveance is below the scraping threshold with a substantial margin, and that the curve of divergence against perveance should include an overperveant part (this is identified by the existence of close divergence isoline on the map of Fig. 7.12). This is another consequence of the discrepancy between Opera and SLACCAD, appearing as a "stretching" along Π norm of the Opera divergence map. The predictions of Opera are in better agreement with the experiments: the measurements on BUG usually display no overperveant part in the curves ε against Π norm (see for example Fig. 7.16).

Experimental evidence of beam scraping with BUG

In the experiments, there are several ways to identify a phenomenon of scraping of the beam:

as explained above, the impossibility to obtain an overperveant part in the curve ε against Π norm is an indication that the perveance threshold for scraping is lower than the optimal perveance. This behavior is observed in ELISE and BUG. when the ions of the beam start impinging on the grids, they deposit a large thermal power. For this reason, a jump in the measured heat loads on the EG indicates that the beamlet is scraping on the grid.

in the experiments, the diagnostics are usually located downstream of the GG and measure the accelerated current density. The injected current density is correlated in a roughly linear relation with the power injected inside the plasma source. Therefore, unless a large fraction of the beam current is lost in the accelerator, the accelerated current density should evolve linearly with the power injected inside the plasma in nominal operating space. For this reason, a decorrelation between the measured current density and the injected power indicates a scraping of the beam (although it may also be caused by space charge limit of the current extraction).

Opera predicts that scraping should occur close to the nominal operating parameters for BUG but not for NITS. One way to test this prediction is to examine the evolution of the accelerated current density with the power injected inside the plasma for NITS and BUG. Fig. 7.17 evidences an approximately linear relation between the arc power and the accelerated ion current measured throughout both Joint Experiments with NITS. No saturation of the current density is observed, with P arc in the range 15-30 kW. Concerning BUG, the legend of Fig. 7.18 must be interpreted as follows: the ion current jH -is measured downstream of the GG and corresponds to the ion beam current density. However, the electron current density je -is estimated by the heat loads on the grids: it is assumed that all the co-extracted electrons and only Chapter 7. Explanation of the difference of beamlet divergence between NITS and BUG 120 electrons impinge on the accelerator grids. The slope change of the estimated co-extraction ratio (red curve) near P = 25kW actually corresponds to a large increase of heat loads on the grids caused by the beamlet scraping (ions hitting the EG generate large heat loads). At the same threshold injected power, we observe a saturation of the accelerated ion current (black curve) and a decorrelation with the injected power. This joint behavior of the measured heat loads and acceleration current proves that beam scraping occurs during the operation. Here, the threshold current density is around 65-70 A m -2 with V ext = 3kV, which corresponds to a normalized perveance of 0.26-0.28. This is well consistent with other perveance scans on BUG, where the maximum normalized perveance is measured around 0.22-0.23 (see for example Fig. 7.16). Therefore, the difference of divergence between both accelerators is at least partially explained by the low perveance threshold for scraping of BUG geometry. Even though Opera predicts that, at the limit of scraping, a low divergence similar to NITS (11 mrad) could be reached with BUG (Fig. 7.12), it is likely that the real scraping threshold is even lower than predicted, due for example to the magnetic field, the beamlet current density asymmetry or other effects not included in the simulations. If this threshold is slightly lower (for example Π max ≈ 0.25, as suggested by measurements), the minimal achievable divergence becomes significantly higher, and substantial differences between NITS and BUG in terms of minimal divergence are then expectable. With a limit of 0.25 in normalized perveance, Opera predicts for BUG a minimal reachable divergence of 13.9 mrad, which corresponds to the limit reached in experiments. The question raised by this conclusion is why the scraping limit is much lower for BUG than for NITS, despite both devices having similar dimensions. To provide a first qualitative answer, it is useful to compare the shape of the beam for both accelerators with typical parameters. This comparison is shown in Fig. 7.19, with the following parameters: Despite the similarities between both facilities, the beam profile with NITS and BUG is quite different inside the EG: the shorter EG radius and thickness of BUG result in a much lower clearance (around 0.5 mm in Fig. 7. [START_REF]Union of soviet socialist republics -united states: documents from the geneva summit[END_REF]). This short clearance indicates that, at Π norm = 0.33, the beam is close to the threshold perveance.

parameter R V Π norm V tot [kV] ε[mrad] NITS 5

Analytic model of the beam envelope in the extraction gap

To confirm this observation, we calculate the beam envelope between the PG and the EG, in order to determine the threshold perveance for scraping with both geometries.

To do so, we use the equation 7.3 with the approximation of section 4.2.3, which gives the electric potential between the PG and the EG as: And

φ(z) ≈ V ext πd 1 r P G,out -r EG,in + (z -z P G,out ) π -arctan r P G,out z -z P G,out + (z -z EG,in ) arctan r EG,in z -z EG,in
V (z) = φ(z) -κ V ext d 1 (z -z P G,out
) with κ = 0 if z > z P G,out and 1 otherwise (eq. 4.1).

Using the angles α and β defined in Fig. 4.16:

φ(z) V ext = 1 π -tan α 1 -tan β 1 + z -z P G,out d 1 π + arctan d 1 tan α 1 z -z P G,out + z -z EG,in d 1 arctan d 1 tan β 1 z -z EG,in
We now use the scaled parameters of eq. 7.3. Besides, we define z0 = z P G,out 3d 1 . This implies:

z -z P G,out d 1 = 3(z -z0 ) z -z EG,in d 1 = z -(z P G,out + d 1 ) d 1 = 3(z -z0 ) - 1
We set z0 = 0. Therefore:

Ṽ (z) = 1 π -tan α 1 -tan β 1 + 3z π + arctan tan α 1 3z + (3z -1) arctan tan β 1 3z -1 - 3zκ
Using this expression in the envelope equation allows to calculate the scaled envelope r(z). We consider that scraping of the beam occurs when the beam radius exceeds the EG radius at the EG inlet surface (z = d 1 and z = 1/3), which corresponds to:

r 1 3 > r EG,in r P G = - tan β 1 d 1 r P G
Finally, for NITS and BUG (d 1 = 6mm, r P G = 7mm) we can define the threshold perveance for scraping Π max by:

r (z) + Ṽ (z) 2 Ṽ (z) r (z) + Ṽ (z) 4 Ṽ (z) r(z) = Π max =⇒ r 1 3 = 6 7 tan β 1
We note that this definition depends only on the angles α 1 and β 1 and on the initial radius and angle of the beam envelope, at the knife edge (z = z ke ). The initial beam radius is assumed to be r P G . The initial angle actually depends on the beam perveance: at higher perveance it is close to r = 0, because the meniscus gets flatter as the perveance increases. A complete model would include the evolution of the initial angle with Π norm .

Here, in order to provide a simple qualitative result, we consider initial angles of the beam envelope of r init =0°, 5°and 10°. The results are shown in Fig. 7.20, for three different values of normalized perveance. For all of those values, the threshold of scraping is never reached for NITS. For BUG, scraping is observed with an initial angle of 5°, with a threshold of Π max = 0.24. With an initial angle of 0 °the beam is always scraping at the EG at any value of Π norm . We observe in simulations that the meniscus of BUG is rather flat at low divergence conditions (see Fig. 7.15 for example), and the initial value of r is expected to be low. On the NITS accelerator, we observe that the meniscus is usually more curved near optimal conditions. This is a consequence of the design of the PG of NITS, aimed at generating a deeper meniscus to increase the current extracted from the source [START_REF] Kashiwagi | Beam optics study in qst[END_REF].

In conclusion, we observe in the experiments, the simulations and in a simple analytic model that the risk of scraping is higher for BUG than for NITS, for at least three reasons: because of the different voltage profile along the beamline axis, the radius of the beam is larger with BUG, because of the smaller EG radius, the scraping limit is significantly more stringent on BUG, one of the objectives of the design of NITS was to obtain a deep meniscus, which increases the initial convergent angle of the beam envelope and reduces the risk of scraping.

Additional explanations

Other hypothesis are being explored to account for the difference in reachable divergence between both accelerators. We shortly detail two of them:

Initial transverse velocity of the particles

In SLACCAD and Opera, it is assumed that all particles are emitted with a velocity normal to the meniscus surface. This simplification does not take account of the thermal transverse velocities of the particles at the meniscus. As shown in section 5.1.7 (Fig. 5.7), the transverse velocity of a particle at the meniscus has a substantial effect on its trajectory. In simulations with IBISimu3D, adding a transverse thermal velocity of 0.1 and 1 eV increases the divergence of 10-20 mrad with respect to SLACCAD, and gives results consistent with Chapter 7. Explanation of the difference of beamlet divergence between NITS and BUG 123 experiments [START_REF] Heinemann | Beam optics and divergence investigations at ipp test facilities[END_REF]. It is also observed in experiments an effect of the source pressure on the divergence. A problem of this explanation is that the hypothesis of initial transverse velocity is not necessary to reproduce the correct order of magnitude for the divergence measured on NITS, and on other facilities such as MTF.

Besides, experiments led at QST in the years 2012-2013 allowed to compare the beamlet thermal footprints on a calorimeter with a flat and chamfered PG. Even though the initial transverse velocity is significantly different in both cases, no large differences in beamlet size were observed on the thermal images [START_REF] Kashiwagi | Beam optics study in qst[END_REF]. This difference may be justified by the fact that the ion energy seems to be significantly higher in RF sources than in filament sources. This topic is an open question at the time of this PhD.

Potential oscillations caused by the RF driver

A hypothesis formulated at the beginning of my exchange period was that the radiofrequency might degrade the beamlet optics, with respect to a filament source. It has been demonstrated that, in a RF driven plasma source, oscillations of the plasma potentials are observed, following the RF frequency [START_REF] Shinto | Observation of beam current fluctuation extracted from an rf-driven h-ion source[END_REF]. During BUG operations, such oscillations were seen, with a magnitude around 1 V.

A first hypothesis testable with particle tracing simulations is that the plasma potential oscillations cause current density oscillations, which means that, even at optimal conditions, the normalized perveance oscillates around its optimal value, thus increasing the beamlet divergence. However, it has been shown that such effect would not cause more than 1 mrad of divergence increase [START_REF] Veltri | Introduction to beam optics and divergence[END_REF].

Another phenomenon that could be more relevant is the oscillations of the electrons in the extraction area. Indeed, unlike in positive ion accelerators, the electrons in negative ion accelerators have a large influence on the meniscus formation. Because of their much smaller inertia, electrons are much more influenced by the potential oscillations than the negative ions. Electron oscillations could modify the meniscus shape, and consequently the beamlet divergence.

With the beam simulation tools used in this PhD, it is impossible to get a good understanding of the RF effect on the beam optics, because SLACCAD and Opera do not take account of the electron behavior in the plasma source and their influence on the meniscus formation: the meniscus in both codes is calculated in the same way for positive and negative ions. For this reason, it was impossible to go further on this topic in the frame of this PhD, but studies are currently investigating the RF effect on the beam focusing [START_REF] Haba | Response of beam focusing to plasma fluctuation in a filament-arc-type negative ion source[END_REF].

We propose here rapid considerations to show how the RF could affect the meniscus formation. It has been shown by S. Kovaleski [START_REF] Scott | Calculation of the ion extraction boundary of a plasma ion source[END_REF] that the meniscus surface calculated by 2D PIC simulations (XOOPIC [START_REF] John P Verboncoeur | An object-oriented electromagnetic pic code[END_REF]) with positive ions is well approximated by an analytic model using eq. 4.1, when the meniscus is assumed to be an isoline of electric field with: E = k B T e qλ D (T e is the electron temperature, and λ D the Debye length). This hypothesis can be kept for negative ions, and interpreted as follows: the extraction electric field is screened by the electrons and penetrates the plasma only on a small volume located between two spherical cups, with a depth λ D (typical screening length). We call V screening and S screening this volume and the surface of the spherical cups that delimit it. Inside this volume, the positive ions are repelled by the extraction field coming from the accelerator, and pushed by the plasma pressure from the source. Consequently, the meniscus is the surface where both forces compensate: it is the last surface where positive ions are present, and therefore the frontier of the source plasma. Hence: P S screening = QE eq E eq is the external electric field in the volume. P is the plasma pressure, approximately equal to n e k B T e (neglecting the ion temperature). Q is the total positive charge in the volume, equal to n + V screening q, with q the positive ion charge and n + their density. The plasma is quasi-neutral and we assume that the negative ion density is negligible with respect to the electron density. This yields n + = n e = n. Therefore, the electric field in the volume is:

E eq = S screening V screening k B T e q = k B T e λ D q
With this model, the meniscus intercepts the beamline axis at a coordinate z men such that E z (r = 0, z men ) = E eq . We now use the expression obtained for the axial electric field in section 4.2.1.2, considering only the knife edge of the PG and setting z ke = 0: The variation of potential in the extraction area affects the plasma parameters (density and electronic temperature), and consequently the equilibrium electric field at the meniscus and the meniscus position. For example, if we consider a thermalized plasma with a constant electronic temperature:

E eq = E z (E ext , r P G , r = 0, z ke = 0, z) = E ext π 1 R R + λ 2 -arctan 2 R + λ
n e = n 0 exp - k B T e eφ plasma
If we consider a small variation δφ of the plasma potential:

n e [φ plasma + δφ] = n e 1 + k B T e eφ 2 plasma δφ
This modifies the Debye length as follows:

1 λ D [φ plasma + δφ] = e 2 n e ε 0 k B T e 1 + k B T e eφ 2 plasma δφ 1/2 = 1 λ D 1 + k B T e 2eφ 2
plasma δφ And therefore:

E eq [φ plasma + δφ] = E eq 1 + k B T e 2eφ 2 plasma δφ = E eq + δE eq
With:

δE eq = E eq k B T e 2eφ 2 plasma δφ
Besides, by differentiation of E eq :

δE eq = 2E ext π r P G z 2 men (r 2 P G + z 2 men ) 2 δz men
Hence: In this model, the oscillations of the meniscus position are significant, and may largely impact the beam optics. It should be noted, however, that the meniscus formation was treated here with a similar approach than for positive ions. The different behavior of electrons in both cases makes this problem more complex. For this reason, the toy model presented here should be seen as a simple illustration of the processes by which plasma potential oscillations could affect the beam optics. PIC simulations and experimental measurements where filament and RF sources are used with a same grid system may allow to better understand the effect of RF on the electron motion in the extraction area and consequently on the meniscus shape and beam optics.

δz men = k B T e 4eφ 2 plasma πE eq E ext (r 2 P G + z 2 men ) 2 r P G z 2 men δφ = k B T e 4eφ 2 plasma

Summary of the main results

In this chapter, we showed that, despite the similarities between the size and operating parameters of NITS and BUG (table 7.1), differences in grid conceptions (especially of aspect ratii) generate quite large differences in beamlet shapes (Fig. 7.19 and 7.20). Consequently, scraping of the beam at the EG is much more likely to occur near optimal parameters with BUG than with NITS. This is confirmed by Opera, but not SLACCAD (Fig. 7.12). Opera predicts a maximal normalized perveance of Π norm ≈ 0.35 and no overperveant part in the curves ε vs Π norm . In the experiments, scraping of the beam is indeed observed near nominal parameters (7.18) with a limit perveance lower than predicted by Opera (around 0.25 [START_REF] Heinemann | Beam optics and divergence investigations at ipp test facilities[END_REF]). This difference can be explained by the simplifications of the Opera model, which neglects the magnetic field, the beamlet asymmetry, the secondaries, the halo and the transverse thermal velocity of the particles at the meniscus. No overperveant part is observed in the experimental perveance scan, and consequently the minimum of divergence cannot be reached with BUG geometry. An experimental campaign subsequent to my exchange period confirmed the limiting effect of scraping on the beam divergence [START_REF] Wimmer | Beamlet scraping and its influence on the beam divergence at the batman upgrade test facility[END_REF]. A possible explanation to this intrinsic problem of BUG geometry might be the fact that its grid system was designed with SLACCAD, where the effect of scraping is not limiting. The difference of predictions between Opera and SLACCAD is an open question. Besides, the RF driver might have an impact on the beamlet optics with the following process:

the radiofrequency generates oscillation of the plasma potential in the extraction area, the electrons in the extraction area follow the potential oscillations, the Debye length and the screening effect of the electrons vary with the plasma potential, the meniscus shape and position depend on the screening of the extraction voltage by the plasma. Therefore, oscillation of the meniscus are expected, following the plasma potential oscillations.

A toy model was used to give an order of magnitude of this effect, and we showed that, with typical values for BUG, the displacement of the meniscus is not negligible. This topic is an open question at the time of this PhD [START_REF] Veltri | Introduction to beam optics and divergence[END_REF][START_REF] Haba | Response of beam focusing to plasma fluctuation in a filament-arc-type negative ion source[END_REF].

Chapter 8

2019 experimental campaign on MTF (QST), benchmark of numerical tools

In 2019, experiments were conducted on the MTF accelerator (section 3.1.4) at QST (Naka, Ibaraki, Japan), aiming at accelerating a Hbeam with high power density and good optics for a long duration. The goal of those experiments was to reach the operation parameters foreseen for the NBI of JT-60SA. For these experiments, the accelerator design has been optimized with the goal to increase the voltage holding of the grid system through an intensive numerical and experimental work [START_REF] Kojima | Development of design technique for vacuum insulation in large size multi-aperture multi-grid accelerator for nuclear fusion[END_REF]. This allowed to reach a beam acceleration of 500 keV with a current density of 154 A m -2 for a duration of 100 s, unprecedented for such high power density [START_REF] Hiratsuka | Achievement of high power and long pulse negative ion beam acceleration for jt-60sa nbi[END_REF]. This experimental campaign provided measurements of heat loads and beamlet deflection and divergence in conditions close to ITER HNB. The measurements obtained during this campaign were compared to the simulation tools usually applied at Consorzio RFX to negative ion accelerators and presented in section 3.2 (Opera, COMSOL, SLACCAD, EAMCC). In particular, in the frame of this PhD, the measurements of deflection with MTF allowed to test the semi-empirical fit determined in chapter 6 from the Joint Experiments with NITS, with a similar Kamaboko source (Fig. 3.5) and a lower power in inside the plasma. The experimental set-up for the 2019 campaign is described with more details in section 3.1.4. In this chapter, we present a comparison of the predictions of numerical tools with the measurements of MTF in terms of beam optics (deflection and divergence) and heat loads on the accelerator grids. A discussion of the discrepancies observed is proposed. To adapt to the alternated vertical magnetic field, the horizontal offset of the EG is inverted from one row to the next. Consequently, it can still be assumed that the beamlet trajectories are mirror-symmetric from one horizontal row to the next, and the definition of ∆x of equation 5.5 remains applicable to MTF. In the measurements presented here, 9 beamlets were accelerated, in a 3 × 3 array. We define again the x, ŷ and ẑ as the horizontal, vertical and beam acceleration directions. The CESM field is directed towards +ŷ on the first and third row, and towards -ŷ on the second one. For this reason, the CESM deflect the beamlet toward the right (x > 0) on the first and third row and toward the left (x < 0) on the second row. As in the case of NITS, we use the convention that the deflection caused by the CESM increases ∆x.

Chapter 8. 2019 experimental campaign on MTF (QST), benchmark of numerical tools 127

In order to correct the ion beamlet deflection caused by the CESM, the SG axis is shifted towards +x on the first and third row and towards -x on the second row. Therefore, a positive value of ∆x means that the magnetic deflection is not fully corrected by the SG, while a negative value means that the deflection is overcompensated. The magnetic field (mostly the filter field) also generates a vertical deflection of the beamlets, compensated by a shift of the SG in the ŷ direction. In this section, we consider only the horizontal deflection. Being θ x (i, j) the deflection of the beamlet of the i th row and j th column, the deflection is given by:

θ x = 1 2 3 j=1 1 3 θ x (1, j) + θ x (3, j) 2 -θ x (2, j)
During the experiments, the beam optical properties were measured by optic calorimetry, with a CFC calorimeter located 2.68 m upstream of the GG outlet surface.

To benchmark the different simulation tools with identical conditions, we chose a reference pulse, corresponding to a measurement where the beam optics is particularly good (small divergence, all beamlets distinguishable on the CFC tile), numbered as 8374 during the experimental campaign. The operating parameters for this pulse are: Four different models have been used to determine the beam deflection:

for the reference, a first Opera model was used with a uniform current density profile at the meniscus, with a single beamlet and a single aperture, an advanced Opera model was built, including the 9 beamlets and an asymmetric current density profile at each meniscus, using the definition and conventions of chapter 6. The magnitude and sign of α unb were determined with the semi-empirical equation 6.6 (Fig. 8.4).

A 3D model of EAMCC (EAMCC3D) was used to simulate a single beamlet and single aperture with periodic boundary conditions on the lateral edges of the simulation domain. The asymmetry of the current density at the meniscus is not included in this model. In the EAMCC3D model used here, the particle trajectories are calculated in an electromagnetic field imported from the first Opera simulation. The gas density profile was calculated with COMSOL (Fig. 8.10). The beam halo is included with a halo fraction of 8%, following the suggestions of similar works [START_REF] Agostinetti | Benchmark of the slaccad code against data from the manitu testbed at ipp[END_REF].

COMSOL was used to simulate all nine beamlet together with the inclusion of the background gas and the collisional processes. The meniscus is imported from EAMCC3D and the background gas density profile is calculated with the molecular flow module (Fig. 8.10). For the simulation of the reference pulse, the beam halo was not included in the model.

As explained in section 3.2, it is known that Opera and COMSOL are in good agreement in the calculation of the beam trajectory (see section 5.4 of [START_REF] Baltador | Numerical and experimental characterization of beams of negative ions and investigation on strategies for beam efficiency improvements[END_REF] and [START_REF] Chitarin | Benchmark of 3d multi-beamlet numerical models for the optics design of negative ion accelerators[END_REF]). A consistency check was also performed to compare the predictions of the first model of Opera and EAMCC3D on MTF, with V acc = 165kV -V ext and j ≈ 150A m -2 , with V ext varying from ≈ 2.5 to ≈ 7 kV. Both codes predict the same deflection for the beam, with at most ≈ 1mrad of error. [START_REF] Aprile | Report on joint experimental activities and simulations at qst laboratories 13 may -19 july 2019[END_REF] The deflection calculated by the different models for the reference pulse is displayed in table 8.2. The results confirm the observation made on NITS and on other experimental device, in the sense that the horizontal deflection is underestimated by the particle-tracing codes with uniform or radially symmetric current density profile at the meniscus. Indeed, all models with symmetric meniscus are consistent to predict a low horizontal deflection, slightly overcompensated by the SG. This result is expectable, since the SG horizontal shift was calculated with similar tools. Instead, the deflection measured by calorimetry is significantly higher, and undercompensated. In fact, the deflection pattern visible for example on Fig. 8.2, 8.6 and 8.9 (right-left-right) visually proves that the deflection is actually undercompensated. The advanced Opera model with an asymmetric current density at the meniscus given by equation 6.6 is consistent with this measurement: it predicts an undercompensation of the magnetic deflection, with a value of ∆x and θx close to the measurements. COMSOL and EAMCC predict similar low values of deflection for other pulses tested (table 8.6). In the case of Opera, a systematic study of the beamlet deflection was performed with the advanced model in order to test the validity of equation 6.6. The evolution of the deflection with α unb was simulated for five pulses with good optics, numbered as 8304, 8349, 8374 (reference pulse), 8381 and 8394. For each pulse, the deflection evolves linearly with α unb (Fig. 8.3), and the line equations allow to determine the value of α unb that should be chosen to reproduce the measurements.

The comparison of this value with the predictions of equation 6.6 is shown in table 8.3. The magnetic field at the meniscus has a value around 16.5 mT. We can see that the semi-empirical equation gives a satisfying order of magnitude for α unb , although it slightly overestimates the values obtained from measurements. The reduced unbalancing factor αunb = α unb /B y,men follows the trend observed in the Joint Experiments on NITS (see Fig. 8.5), with values of 10-15 T -1 for a power injected in the plasma around 34-37 kW. It should be noted that NITS and MTF use the same kamaboko source but are significantly different, both in terms of geometry (PG shape and size, extraction gap length, number of grids, SG for MTF) and operating parameters (much higher voltage, longer pulses and no ADCM for MTF). This is a strong indication that the underestimation of the horizontal deflection measured with both facilities is linked to phenomena occuring inside the plasma source.

Divergence of the beam

The divergence on MTF was measured by calorimetry (section 7.1.2.1). In the case of MTF, the calorimeter is located almost three times further than in the case of NITS (2.68 m against 0.94 m). If we assume that the beam is laminar with a fixed angular distribution as it propagates from the GG to the calorimeter, the size of the beamlets on the calorimeter should be around three times larger than on NITS, making the measurement less precise, as the beamlet footprints are more likely to overlapse. However, the divergence in MTF is also smaller and on the pulses with best optics each beamlet is distinguishable on the CFC, allowing a quite precise determination of ε. In the measurements and multibeamlet simulations, the divergence was measured for each separate beamlet footprint and ε is defined as the average divergence over the nine beamlets. In addition to the numerical models described in the previous section (Opera with and without asymmetry, EAMCC3D and COMSOL), the divergence on MTF was studied with SLACCAD, in an axisymmetric model neglecting the magnetic field and the interaction between the beam and the background particles.

The consistency check made with Opera and EAMCC3D shows that EAMCC3D predicts a higher divergence than Opera, with a difference around 30%. [START_REF] Aprile | Report on joint experimental activities and simulations at qst laboratories 13 may -19 july 2019[END_REF] All five models were tested on the reference pulse 8374 (Fig. 8.2). The comparison between the predictions of each model and the measurement on the reference pulse is displayed in table 8.4. All the models tested correctly predict the beamlet divergence, with less than 1.5 mrad of error. The effect of Π norm and R V on the divergence was studied with Opera and SLACCAD. The minimum divergence is found around 7 mrad for R V ≈ 25 and Π norm ≈ 0.15 (Fig. 8.8). The scan of V ext made with EAMCC3D confirms that the divergence is minimal around R V ≈ 27. A similar order of value was observed during the experiments: for 11 pulses with good optics, the divergence was found in a range [6. In Fig. 8.8, the scans of ε with R V and Π norm is compared to the operating space during the experiments. Both models correctly predict the optimal operating parameters and the minimal divergence. If we compare SLACCAD and Opera we observe that: both codes predict similar shape for the "smiling curve" of ε vs R V , with a slightly different optimal value, as in the case of NITS and BUG, Opera predicts a curve of ε vs Π norm less concave, with a lower slope for the overperveant part than SLACCAD.

Both Opera and SLACCAD predict that the acceleration grids intercept the beam (scraping) for Π norm > 0.2 and Π norm < 0.08 for Opera and 0.05 for SLACCAD.

Heat loads 8.2.1 Comparison between the COMSOL and EAMCC3D models and the measurements

During the experiments, the heat loads on the grids were monitored using water calorimetry technique on the cooling system. This diagnostic was not reliable when the CFC tile was installed, due to the short pulse length (≈ 0.2 s). Consequently, it is not possible to measure at the same time the optics of the beam and the thermal loads on the grids. This is an issue because the heat loads depend on the beamlet deflection and divergence. When a simulation of the beam is compared to the heat loads measurements, it is assumed that the simulated beam has the same optical properties as the real one. To simultaneously compare the beam optics and the heat loads in simulations and measurements, we used two pulses with similar operation parameters, but different pulse lengths (a short pulse with the calorimeter mounted and a long one without calorimeter), assuming that both have similar optics. For optics measurements, we used the pulse indexed as 8318 (Fig. 8.9) and, for heat loads measurement, the pulse indexed as 7817. The density profile of the background gas inside the accelerator, calculated with COMSOL, is shown in Fig. 8.10. A halo fraction of 8 % was assumed in both codes. The predictions of COMSOL and EAMCC3D for the pulse 8318 are compared in table 8.6 with the optics measurement of the pulse 8318. First of all, we can see that COMSOL and EAMCC3D are in good agreement in their predictions of beamlet divergence and deflection when the halo effect is included. The divergence measured by calorimetry is slightly lower, and in good agreement with the calculation of COMSOL if the beam halo is ignored. It is possible that, because of the large distance between the GG and the calorimeter, the halo part of the beam is not distinguishable from the background temperature on the CFC tile. This would explain the relatively low value of measured divergence, corresponding to the beamlet core. As expected, the horizontal deflection is underestimated in the simulations. The inclusion of the meniscus asymmetry in the EAMCC3D model is an ongoing activity. During the experiments, the heat loads were measured separately on the AG1, AG2 and GG. In table 8.7, the thermal loads on each of those three grids simulated by COMSOL and EAMCC3D are compared with the measurements. Although both codes give similar order of value for the total power, the repartitions of the heat loads on the grids are quite different. We can see it more clearly in 8.11. COMSOL predicts much more electrons on the AG1 than EAMCC3D (Fig. 8.13 and 8.12) and therefore higher heat loads, which is more consistent with the measurements. On COMSOL, this large number of electrons results from the interaction of the halo part of the beam with the background gas. The additional heat loads caused by the halo effect on the heat loads is weaker on EAMCC3D. On the AG2, COMSOL predicts that a majority of the heat loads is caused by the electrons, and EAMCC3D by the ions. Both codes predict similar heat loads repartition on the GG.

Possible explanations to the thermal loads underestimation by simulations

Even though the order of value of the total deposited power predicted by simulations is correct, the heat loads on the AG1 and AG2 are substantially underestimated. This issue is important, because the thermal loads on the grids are a critical point for the design of MITICA and ITER HNB.

In the simulations, the heat load is expected to be higher on the last grids, because of the larger number of secondary electrons and ions of higher energy impinging on the grids. On the other hand, the measurements indicate that the heat loads on the AG1 and AG2 have values comparable to the load on the GG. These values happen to be much higher in the experiments than in the simulations, with a difference of the order of some kW.

Three hypotheses are presented here to explain this discrepancy:

the current density profile asymmetry at the meniscus was neglected in the simulations. Consequently, the deflection of the beam is largely underestimated. The inclusion of the meniscus asymmetry in the COMSOL and EAMCC3D models of MTF is an ongoing work, but preliminary simulations have shown that the differences with the radially symmetric case is not very large (less than 2 kW on each grid and in total).

on the thermal footprints of most of the pulses recorded, an additional row of beamlets is visible, which follows the criss-cross deflection and the disposition of the grid apertures (Fig. 8.14). The footprint on the CFC tile is most likely caused by ions, because electrons would be at least 40 more deflected than the Hbeamlets. If those additional beamlets are continuously accelerated during the pulse, the heat loads caused by those ill-focused particles can be significant. This would be the case, for example, if part of the source gas leaks inside the accelerator through the mask set on the PG. If, however, the additional beamlet footprint are caused by residual gas inside the vessel, the additional beamlets would last only for a very short time and not influence the total heat loads. in the simulations, each beamlet is assumed to carry the same optimum current density value, which minimizes the divergence. Although the average current density is indeed close to the optimum value, the ion density and velocity distribution can be non-homogeneous at the scale of the whole negative ion source (mainly due to the presence of the transverse magnetic filter field). For this reason, some of the nine beamlets have a weaker or stronger current and are consequently more divergent than simulated (this is visible for example on the pulse studied here, on Fig. 8.9). For low current density (underperveant beamlet, Fig. 8.15 a)), two factors impact the beamlets heat loads with opposite effects: the beamlet divergence increases, generating more heat loads on the grids, but the beamlet power density is also lower, which reduces the overall deposited power. Instead, when the current density is higher (overperveant beamlet, Fig. 8.15 b)), both effects combine to increase the thermal loads: the beam power density and divergence are higher than in the optimum case. Preliminary EAMCC3D simulations with realistic values for each beamlet showed that the total heat loads are indeed higher when variations of current density are considered. The plot of Fig. 8. [START_REF]Proceedings of the second united nations international conference on the peaceful uses of atomic energy[END_REF] shows the evolution of the thermal loads of a single beamlet with its current density. The model studied here is an EAMCC3D model with an extraction voltage V ext = 5.5kV, corresponding to an optimal current density of j = 150A m -2 . As explained previously, the heat loads does not vary much for underperveant beamlets (the high divergence is compensated by the low power density), but a large increase is observed for overperveant beamlets. However, the repartition of the heat loads over the grids starts to significantly change only for large deviation from the optimal current density. For example, table 8.8 shows how the repartition of the thermal loads changes in two extreme cases (j = 3j opt and j = 0.26j opt , see Fig. 8.15). Even with a highly overperveant beam, the high fraction of power loads on the AG1 cannot be reproduced by EAMCC3D. More work needs to be done to explain the difference between EAMCC3D and COMSOL, and to improve the models so as to reproduce the real heat load repartition. Nonetheless, it should be noted that both codes manage to correctly estimate the heat loads on the GG, which is the most stringent design parameter, and to give a satisfying order of magnitude for the total heat loads.

Summary of the main results

The experiments on MTF allowed to confront the simulation tools commonly used at Consorzio RFX with measurements on a Haccelerator close to ITER HNB performances in term of pulse duration, beam power density and acceleration voltage. Concerning the beam optics, the following results were obtained:

it is confirmed that it is not accurate to assume a radially symmetric current density profile at the meniscus, and doing so causes a significant underestimation of the magnetic deflection.

the improved Opera model described in chapter 6 with the semi-empirical formula 6.6 gives a correct evaluation of the deflection. The unbalancing factor α unb deduced from measurements follows the trend observed with NITS at lower arc power (Fig. 8.5).

the four codes applied to MTF (Opera, SLACCAD, EAMCC and COMSOL) give a correct evaluation of the beamlet divergence.

SLACCAD and Opera correctly estimate the optimum voltage ratio and normalized perveance (Fig. 8.6 and 8.8).

Concerning the heat loads on the accelerator grids:

COMSOL and EAMCC3D correctly predict the heat loads on the GG and estimate a correct order of magnitude for the total heat loads, COMSOL and EAMCC3D underestimate the heat loads on the AG1 and AG2. The error is particularly large for EAMCC3D on the AG1.

COMSOL and EAMCC3D do not predict the same distribution of particles impinging the grids for the AG1 and AG2, but are in good agreement for the GG (Fig. 8.11).

Some simplifying assumptions of both models (symmetric meniscus, same current density for all the beamlets) partially explain this discrepancy, but no clear explanation is given yet.

The study presented here can be considered as a benchmark for the beam optics simulations, and to a lesser extent for the thermal load simulations, in terms of approximate estimation of the total heat loads.

issue, based on the scraping of the beam in the BUG accelerator. I then provided experimental, numerical and analytic evidences in line with this conclusion. An open and critical issue to address in the future is to understand why SLACCAD and Opera predict different evolutions of the beamlet divergence with the normalized perveance: SLACCAD calculates an optimal normalized perveance below the threshold limit for scraping in the case of BUG while Opera predicts an optimal point beyond or at the limit for scraping.

In Chapter 8, we provide a benchmark of the numerical tools commonly applied to negative ion accelerators at RFX, based on the experimental campaign with MTF led in 2019 at QST. A discusion of the results obtained is presented, and some points are still open issues (mainly, the discrepancy between the thermal loads simulated and measured). I showed that the unbalancing factor deduced from the measurements follows the trend observed in Chapter 6.

The work presented in this thesis opens up prospects for future researches on the physics of NBI.

It is becoming increasingly clear that a good physical understanding of the plasma dynamics inside the ion source is crucial to correctly predict the beam optics, be it to determine the current density asymmetry at the meniscus or to investigate the effect of Radio Frequency on the divergence of the beamlets.

In particular, this PhD presents a first step to implement such physical effect in beam optics simulations. Successive activities should continue the work detailed in Appendix A, aiming at a physical description of the meniscus asymmetry (its dependence on the source parameters and size, on the ion species...) to correctly design the deflection compensation systems of ITER HNB. In this regard, experiments of negative ion beam acceleration with a systematic measurement of the criss-cross deflection under various source conditions could provide very valuable results. field in the (ŷ, ẑ) plane, but also takes account of the particle loss in the third direction. For this reason this kind of simulation is referred to as 2.5D. More detailed explanations on the assumptions of the 2.5D model and a comparison with the results of the 3D model are found in [START_REF] Fubiani | Plasma asymmetry due to the magnetic filter in fusion-type negative ion sources: Comparisons between two and three-dimensional particle-in-cell simulations[END_REF].

The advantage of this model is that it is solved much faster than the 3D models, allowing to perform simulations in a relatively short time. However, the 2.5D model is not suited to simulate the plane perpendicular to the suppression field. For this reason, the asymmetry studied is this section is the asymmetry generated by the filter field at the meniscus. The plasma is scaled with ε0 = 64ε 0 . The accelerator dimensions are almost identical to the previous case, but the width of the ADCM: θ x (w ADCM = 2.5mm, α unb = 0.1) ≈ θ x (w ADCM = 1.95mm, α unb = 0)

For higher values of α unb , for example in a range of 0.15-0.25, the horizontal deflection becomes significant, with values around 0.8-1.4 mrad.

In such case, it would become necessary to enlarge even more the groove in the EG, which would require a modification of the grid geometry in order to satisfy the mechanical and optical criteria at the same time.

For this reason, it would be an important result to correctly estimate the magnitude of the asymmetry of the current density profile at the meniscus for MITICA.

Figure 1 . 1 :

 11 Figure 1.1: The reactor is made of two spherical concentric electrodes. The inner electrode is a transparent grid with a strong negative potential, emitting electrons. The outer electrode is connected to an ion source that emits D + ions. Both species form a plasma in the space between the electrodes. A small fraction of the accelerated D + passes through the grid resulting in a very high density of D + at the center of the device, which generates D-D fusion reactions.

Figure 1 . 2 :

 12 Figure 1.2: a) ITER construction site, the large buildings in the center of the picture are the tokamak building and the assembly hall b) location of ITER site c) ITER tokamak illustration and comparison with human size Photos from [18]

Figure 1 . 3 :

 13 Figure 1.3: Although fossil fuel consumption tends to locally decrease in developed countries, it remains a necessary tool for developing countries and the worldwide consumption keeps increasing at a fast rate.

Fig. 1 .

 1 [START_REF] Einstein | Ist die trägheit eines körpers von seinem energieinhalt abhängig?[END_REF] shows an example of energetic scenario for Western Europe in 2100 where fusion is involved.

Figure 1 . 4 :

 14 Figure 1.4: Example of energetic mix for Western Europe in 2100 in the "rational perspective" scenario of [27] (an ecologically driven and collaborative scenario). The share of fusion increases as the objective on CO 2 reduction becomes more stringent. (The energy sources appear on the same vertical order on the plot bars than on the legend.)

Figure 1 . 5 :

 15 Figure 1.5: The ratio of the magnitudes of the strong and Coulomb forces that a proton exerts on another one is plotted against their distance. The total force is repulsive beyond a distance of about 5-6 fm. The strong force is derived from the Reid potential.

Figure 1 . 6 :

 16 Figure 1.6: As the mass number increases, the binding energy per nucleon decreases until the more stable elements (Fe). It then increases again as the nuclei become less stable due to the Coulomb repulsion.

Figure 1 . 7 :

 17 Figure 1.7: Reactivity of the reactions 1, 2 and 3 against the ion temperature. The dashed area is the operation range expected for ITER while the dashed line is the highest ion temperature ever recorded in a tokamak (45 keV on JT-60U [21]). At those temperatures the reactivity of the reaction 1 is considerably higher.

Figure 1 . 8 :

 18 Figure 1.8: Illustration of the D-T reaction, in fusion reactors the neutron energy is recovered while the He energy heats the plasma.

Figure 1 . 9 :

 19 Figure1.9: The current circulating through the plasma and the toroidal coils generate helicoidal magnetic lines inside the torus. Charged particles that follow those lines are trapped inside the vessel as the curvature drift direction is constantly changing. The plasma current is induced by the central solenoid and the poloidal coils. Picture from[START_REF] Kikuchi | Fusion physics[END_REF].

Figure 1 .

 1 Figure 1.10: a) Integration of the three NBI foreseen for ITER b) Three modes of injection of the neutral particles are possible, characterized by injection direction relatively to the plasma current direction.

Figure 2 . 1 :

 21 Figure 2.1:As the ion beam energy in the neutralizer gets higher, it is more and more difficult to neutralize the positive ions. At 500 keV (order of magnitude for ITER) the neutralization efficiency is considerably higher with negative ions. Curve from[START_REF] Koch | Fast particle heating[END_REF].

Figure 2 . 2 :

 22 Figure 2.2: Components of ITER HNB (Heating Neutral Beam). Picture from [37].

Figure 2 . 3 :

 23 Figure 2.3: Schematic representation of a tandem ion source for NBI with RF (top) or filament (bottom) driver. Typical parameter of the source plasma are indicated.

( 3 )

 3 H 2 + e --→ H -+ H + + e (4) H + 2 + e --→ H -+ H + H + 3 + e --→ H -+ H + 2

Figure 2 . 4 :

 24 Figure 2.4: a) Beamlet acceleration simulation: here the meniscus is an input parameter calculated as an equipotential of the PG b) Particle In Cells simulation of the source plasma, the meniscus is distinguishable, as the surface where the negative ion flux abruptly jumps

Figure 2 . 5 :

 25 Figure 2.5: Section of the grids for ITER HNB, the beamlets are accelerated through each series of aligned apertures. Photo from [45].

Figure 2 . 6 :

 26 Figure 2.6: Comparison between the MAMug and SINGAP geometries (the beamlets propagate from left to right). Figure from [48].

Figure 2 . 7 :

 27 Figure2.7: At first the beam quickly expands after the GG due to its self-electric force, but after a short time the positive ions generated by interactions between the beamlet and the background gas compensate the space charge forces and the beam divergence stops increasing.

  electron fraction < 0.5 < 1 MITICA source is based on the design of SPIDER, which will be detailed next. The accelerator of MITICA is composed of seven successive grids: a PG, an EG, four AGs and a GG. Each grid includes 4 × 4 × 5 × 16 (1280) apertures (Fig. 3.1).

Figure 3 . 1 :

 31 Figure 3.1: MITICA grids, a total of 1280 apertures is foreseen. Picture from [63].

Figure 3 . 2 :Figure 3 . 3 :

 3233 Figure 3.2: Configuration of the permanent magnets that generate the suppression field of MITICA, each array of magnets is embedded inside one of the grid (picture generated with NBImag [66])

3. 1 . 2 Figure 3 . 4 :

 1234 Figure 3.4: top-left: illustration of SPIDER components and size bottom-left: photo of SPIDER installed right: extraction of Hbeamlets with SPIDER

Figure 3 . 5 :

 35 Figure 3.5: a) sketch of NITS kamaboko-type plasma source b) photograph of the kamaboko mounted on the accelerator

Figure 3 . 6 :Figure 3 . 7 :

 3637 Figure 3.6: Permanent magnets embedded in NITS PG (picture generated by NBImag)

Figure 3 . 8 :

 38 Figure 3.8: Exploded view of the BUG facility. Picture form [51].

Figure 3 . 9 :

 39 Figure 3.9: Evolution from BUG (1 driver) to ELISE (4 drivers) to SPIDER and ITER HNB (8 drivers). Picture form [77].

Figure 3 . 10 :

 310 Figure 3.10: Schematic representation of the typical solving process of particle tracing simulations

Figure 3 . 11 :

 311 Figure 3.11: Example of simulation of the NITS accelerator with a single aperture and a type 103 emitter. The particles start from a plane upstream of the PG. The red beam is the ion beam and the blue particles are the electrons, deflected by the suppression field.

Figure 3 . 12 :

 312 Figure 3.12: Calculation of the trajectory of copper eroded from the grids by the impacts of the ions on MTF.

Figure 3 . 13 :

 313 Figure 3.13: Example of Hbeam (left) and secondary electrons (right) trajectory calculation in MTF with COMSOL.

Figure 3 . 14 :

 314 Figure 3.14: Example of simulation with SLACCAD (post-treatment with Matlab).

Figure 3 . 15 :

 315 Figure 3.15: Beam (blue rays) and secondaries (green, red and black rays) trajectories on MTF calculated with EAMCC3D.

Figure 4 . 1 :Figure 4 . 2 :Figure 4 . 3 :

 414243 Figure 4.1: First piecewise linear profile compared to an Opera simulation (NITS accelerator without beamlet)

Figure 4 . 4 :

 44 Figure 4.4: Trajectory of two electrons passing through the EG, launched just upstream of the PG with x=2mm and x=4mm. The dotted lines represent the inlet and outlet surfaces of the EG.

Figure 4 . 5 :

 45 Figure 4.5: Transverse section of the beam propagating in a circular tube

Figure 4 . 6 :

 46 Figure 4.6: a) beamlet potential inside the GG predicted by Opera (in V) b) beamlet potential inside the GG predicted by the analytic model described above, plotted with Mathematica (in V)

Figure 4 . 7 :

 47 Figure 4.7: Comparison between the meniscus shape with Πnorm = 0.08 (left) and Πnorm = 0.2 (right) with the MTF geometry.

4. 2

 2 Analytic expression for the electric potential and electric field inside a NBI accelerator

Figure 4 . 8 :

 48 Figure 4.8: Typical geometry of a two stage accelerator (Plasma Grid (PG), Extraction Grid (EG) and Grounded Grid (GG)), approximated by a superposition of circular apertures

Figure 4 . 9 :

 49 Figure 4.9: Circular aperture in an infinite plane with an external electric field on one side

Figure 4 . 10 :Figure 4 . 12 :

 410412 Figure 4.10: electric potential along ẑ with r=0 and r=6 for two different acceleration voltages in NITS

Figure 4 . 14 :

 414 Figure 4.14: MTF geometry: in the top part the error bars are scaled with the extraction voltage

Figure 4 . 16 :

 416 Figure 4.16: Geometric definition of αi and βi for a three grid configuration

Figure 5 . 1 :

 51 Figure 5.1: Schematic representation of two adjacent rows of CESM

x 2 Bx 2 Figure 5 . 2 :Figure 5 . 3 :

 225253 Figure 5.2: Magnetic field generated along the beamlet axis between two rows of CESM of remanence 1.1 T (here zEG = 0)

  (z), z)dz Chapter 5. Data analysis of the Joint Experiments on NITS 59 Hence:

Figure 5 . 4 :

 54 Figure 5.4: Comparison between the numerical solving of the trajectory equation and the result of an Opera simulation for a case without ADCM

Figure 5 . 5 :

 55 Figure 5.5: Comparison between the numerical solving of the trajectory equation with a corrected value for fEG,out and the result of an Opera simulation for the cases with and without ADCM

Figure 5 . 6 :

 56 Figure 5.6: Influence of the different parameters included in the model on the horizontal deflection of the beamlet. The resultant deflection is plotted against p. On this plot, each parameter takes values from p0 -2∆p to p0 + 2∆p.

Figure 5 . 7 :Figure 5 . 8 :

 5758 Figure 5.7: Trajectory of the beamlet with initial horizontal position and velocity, the initial velocity has a small but not negligible impact on the deflection and the initial position of the reference particle strongly modifies the beam trajectory.

Figure 5 . 9 :

 59 Figure 5.9: Image obtained from OPERA simulation of NITS accelerator, showing the expected position (in terms of power density distribution) of two beamlet groups (3 × 5 + 3 × 5 beamlets) at a distance of about 900 mm downstream of the GG. The position of each GG aperture and of the corresponding axis is also shown in black. The beamlets in the lower beamlet group exhibit the typical pattern of criss-cross deflection due to the CESM magnetic field. The beamlets in the upper group are subjected to a much reduced deflection thanks to the combined effect of CESM+ADCM. Illustration taken from Fig. 5.17 of [82].

Figure 5 . 10 :

 510 Figure 5.10: Illustration of the calculation of ∆x with two horizontal rows: the crosses represent the centers of the GG apertures, the first (resp. second) row is deflected toward the right (resp. left). The deflection pattern is mirror-symmetric from one horizontal row to the next: ∆x(1, 1) = -∆x(2, 5), ∆x(1, 2) = -∆x(2, 4), ∆x(1, 3) = -∆x(2, 3)...

Figure 5 . 11 :

 511 Figure 5.11: Preliminary model of NITS and calculation of the beam trajectory. The beam average position and horizontal angular deflection are calculated on a plane represented by the dashed line (50 mm upstream of the GG). The trajectory is then prolonged as a straight line.

Figure 5 . 12 :

 512 Figure 5.12: Evolution of the ADCM to CESM remanence that cancels the magnetic deflection with the voltage ratio

Figure 5 . 13 :

 513 Figure 5.13: (a) magnetic configuration used in the Joint Experiments (b) example of thermal image measured

1 .

 1 1T and B r,A = 1.1T or 0. The voltage ratio and arc power are approximately constant (R V ∈ [4.9, 5.1], P arc ∈ [22kW, 25kW]). The measurements with similar extraction voltage are averaged. Qualitatively, the result observed in Fig. 5.14 is in agreement with the analytic results, which predict a linear evolution of the deflection with the inverse square root of the extraction voltage (∆x = aV -1/2 ext + b), but the fit is not so accurate when assuming b = 0. The value of the coefficient a decreases when ADCM are present, as predicted by the model.

Figure 5 . 14 :

 514 Figure 5.14: Evolution of the measured horizontal deflection with the extraction voltage, RV ≈ 5, Parc ≈ 25kW

Figure 5 . 15 :

 515 Figure 5.15: Evolution of the measured horizontal deflection with the CESM and ADCM remanences, RV ≈ 5

Figure 5 . 16 :

 516 Figure 5.16: Evolution of the measured horizontal deflection with the voltage ratio, Vext ≈ 5.0kV, Parc ≈ 25kW

Figure 5 . 17 :

 517 Figure 5.17: Pulse 12436: Br,A = Br,C = 1.1T, RV = 5.07, Vext = 4kV, j = 134A m -2 , Opera predicts an almost perfect compensation on the part with ADCM ∆x = 0.9mm, but a large deflection ∆x = 7.9mm is measured. The distance between the average position of the left beamlets of the horizontal rows is shown for illustrative purpose.

Figure 5 . 18 :

 518 Figure 5.18: Comparison of ∆x predicted by Opera with 752 measurements on the NITS accelerator (326 with ADCM and 326 without). The systematic underestimation of the deflection appears clearly, especially in the first Joint Experiments.

Figure 5 . 19 :

 519 Figure 5.19: Comparison of ∆x √ Vext at constant voltage ratio and magnetic field with the power injected inside the source plasma. A clear correlation is visible, in contradiction the simple Opera model where the parameter Parc is not included.

Figure 6 . 1 :

 61 Figure 6.1: If only CESM are included, the plasma drift inside the source and the beam horizontal deflection have opposite directions: if ∆x > 0 the current density is higher on the left part x < 0 of the meniscus.

Figure 6 . 2 :

 62 Figure 6.2: Exemple of different profiles with varying smoothness, from a linear profile to a step profile. jtot = 140A m -2 α unb = 0.2

  x [mm] at z = 100mm 0.45 0.45 0.43 0.41 0.41 e-folding divergence [mrad] at z = 100mm 15.5 15.4 15.4 15.3

Figure 6 . 3 :

 63 Figure 6.3: Beamlet trajectory for the different profiles testesd, with jtot = 140A m -2 α unb = 0.2

  ∆x[mm] = 0.937 * (-52.79) * (-3α unb ) ≈ 150α unb

Figure 6 . 4 :

 64 Figure 6.4: Beamlet section with an asymmetric charge density

Figure 6 . 5 :

 65 Figure 6.5: left column: horizontal component of the electric field generated by a radially symmetric beamlet at the outlet surface of the PG, right column: horizontal component of the additional electric field generated by an asymmetry of the beamlet charge density, top row: map calculated with Opera, bottom row: map obtained by numerical integration of eq. 6.1. The dashed line represents the beam circonference. For this example α unb =0.3, Vext = 4kV, j = 140A m -2 , ρ0 = -1.5 × 10 -5 C m -2 .

Figure 6 . 6 :

 66 Figure 6.6: The radial electric field generates stronger radial forces on the beamlet half with more charges. The total force over the whole beamlet is directed along +x for an outward radial electric field (as shown in the figure) and along -x for an inward field.

Fig. 6 .

 6 Fig. 6.7 shows the ratio between the two forces along the beamline axis inside the extraction gap with the following typical values: V ext = 3kV, R = 6mm, ρ 0 = -1.5 × 10 -5 C m -2 .We observe that the self-electric field of the beamlet is predominant, however the external electric field effect was slightly underestimated by the assumption that the beamlet is centered on the beamline axis. Besides, as could be expected, the self electric force becomes less dominant when the charge density over the beamlet section decreases and when the extraction voltage increases.

Figure 6 . 7 :

 67 Figure 6.7: Ratio between the force generated by the external radial electric field and the beam self electric force. The self electric force is predominant. Close to the PG both force deflect the beamlet towards the same direction (+x if the charge density is higher on the side x < 0).

Figure 6 . 8 :

 68 Figure 6.8: Current density over a beam section at different positions along the beamline. The current density asymmetry rapidly vanishes and the beamlet is almost symmetric after exiting the EG. The color scale goes from blue to red from lower to higher current density in absolute value.

Figure 6 . 9 : 4 . 5 .

 6945 Figure 6.9: Evolution of the unbalancing factor along the beamline axis. α unb (z) approximately follows an exponential decay inside the extraction gap. The dashed lines represent the fitting functions z → α unb (zmen) exp (-z/d1)

Figure 6 . 10 :

 610 Figure 6.10: Trajectory of the beamlet for different values of α unb .

Figure 6 . 11 :

 611 Figure 6.11: Horizontal deflection of the beamlet for different values of α unb .

Figure 6 . 12 :

 612 Figure 6.12: Evolution of ∆x calculated with Opera against α unb for four different magnetic configurations.

Figure 6 . 13 :

 613 Figure 6.13: Evolution of ∆x(α unb ) -∆x(α unb = 0) calculated with Opera against α unb for four different magnetic configurations and extraction voltages. Here the increase of deflectionwith respect to a symmetric case is plotted, in order to compare all the cases together.

Figure 6 . 14 :

 614 Figure 6.14: Evolution of ∆x against α unb with a linear CDPM.

Figure 6 .

 6 Figure 6.15: a) Evolution of the deflection with α unb for different values of the voltage ratio. b) The slope ∂(∆x)/∂(α unb ) can be linearized for RV close to 5.

Figure 6 . 16 :

 616 Figure 6.16: Simulation of an entire horizontal row of beamlets with asymmetric CDPM.

Figure 6 . 17 :

 617 Figure 6.17: Evolution of the deflection with α unb for each beamlet, sequentially numbered from left to right. The dashed plot is the average deflection over the five beamlet.

Figure 6 .

 6 Figure 6.18: Comparison between the average deflection of the five beamlets in a multibeamlet configuration with the deflection of a single beamlet in a single aperture model.

6. 4

 4 Application to the Joint Experiments 6.4.1 Comparison of the unbalancing factor deduced from the measurements with the operating parameters

Figure 6 . 19 :

 619 Figure 6.19: Evolution of α unb throughout the Joint Experiment campaigns on the part with (red) and without (blue) ADCM. The moving average is defined by the average of the last and next three points, weighted by the quality of the optics (the pulses where each single beamletis distinguishable are given a weight twice higher than the others).

Figure 6 . 20 :

 620 Figure 6.20: Evolution of the unbalancing factor with the absolute value of the ŷ component of the magnetic field near the meniscus: each points represents the average of α unb over pulses with same arc power at ±0.5kW. The arc power indicated is the average over all those points.

Figure 6 .

 6 Figure 6.21: Comparison between the arc power and the reduced unbalancing factor for the pulses of the first Joint Experiments with the best optics. A similar evolution is observed between the two parameters.

Figure 6 . 22 :

 622 Figure 6.22: Evolution of the reduced unbalancing factor with the arc power: a weak moderate correlation is visible.

Figure 6 . 23 :

 623 Figure 6.23: The coefficient of correlation between Parc and a random data is calculated for 10 6 points. As predicted by the theory, it follows an approximately Gaussian distribution centered on 0, with a standard deviation of 0.052.

Figure 6 . 24 :

 624 Figure 6.24: Comparison of ∆x predicted by Opera with 752 measurements on the NITS accelerator, the simulation results are much more correct than in the first model (Fig. 5.18).

Figure 6 . 25 :

 625 Figure 6.25: Comparison of the beamlet trajectory from the meniscus to the calorimeter for the pulse 12436 with uniform CDPM and with α unb =0.22 (linearly prolonged from z = 92.5mm). The residual deflection caused by the asymmetry accounts for the large discrepancy between the measurements and the model with uniform CDPM.

Figure 7 . 1 :

 71 Figure 7.1: a) distribution of the particles along x (specific direction): the particles are sampled in infinite rectangles {x ∈ [x, x + dx[, y ∈] -∞, +∞[} b) distribution of the particles along r (any transverse direction): the particles are sampled in circles {r ∈ [r, r+dr[, α ∈ [0, 2π[}

Figure 7 . 2 :

 72 Figure 7.2: Angular distribution of a negative ion beamlet simulated with Opera, some cm after exiting the accelerator.

7. 1 . 2

 12 Calorimetric and spectroscopic measurements of the divergence7.1.2.1 CalorimetryIn the frame of NBI for fusions, two main kinds of diagnostics are used to measure the divergence of the beam and beamlets, relying on calorimetry and spectroscopy. First of all, measurements with calorimetry were already detailed with the example of NITS in section 7.1.2.1, and examples of exploitable temperature map are visible in Fig.5.13 b) and Fig.5.17. The temperature map obtained from thermography, after Matlab post-treatment, can be approximated by a sum of N Gaussians, with N the number of beamlets.

Figure 7 . 3 :

 73 Figure 7.3: Gaussian fit of a single beamlet from a temperature map obtained on the MTF accelerator (the smooth Gaussian fit overlaps the measured temperature 3D map, it has been slightly shifted on the picture for visibility).

Figure 7 . 4 :

 74 Figure 7.4: Typical spectrum measured by BES for a D beam. The three peaks corresponding to the background gas, the beam particles and the stripped neutrals are visible. The Doppler peak is broadened with respect to the background gas peak, thus providing a measurement of the divergence. Figure from [111].

Figure 7 . 5 :

 75 Figure 7.5: Comparison between the grid systems of NITS and BUG: both geometries are quite similars (same PG radius, approximately similar aspect ratii).

Figure 7 . 6 :

 76 Figure 7.6: Example of "smiling curve" obtained with Opera for the NITS accelerator (the continuous lines are 4 th order polynomial fits). It can be noted here that the optimum value of RV depends on Πnorm and vice versa: ε reaches its minimum at a certain position in the space (RV , Πnorm).

Figure 7 . 7 :

 77 Figure 7.7: Representation of the accelerator as as series of lenses to calculate its transfer matrix.

Figure 7 . 8 :

 78 Figure 7.8: Evolution of the accelerator refractive power with the voltage ratio for NITS and BUG geometries.

Figure 7 . 9 :

 79 Figure 7.9: Distribution of the axial velocity (relative to the maximum value) of the particles intercepting a vertical plane (z constant). The inhomogeneity, defined as the amplitude of the relative velocity distribution, is very low and decreases along the accelerator beamline. a) z = z knif e edge + 5 mm, inhomogeneity of 5% b) z = z knif e edge + 30 mm, inhomogeneity of 2% c) z = z knif e edge + 75 mm, inhomogeneity of 0.5%

Figure 7 . 10 :

 710 Figure 7.10: Evolution of the divergence with the voltage ratio for the NITS accelerator various magnetic configurations (CESM+ADCM), with Πnorm = 0.317 (simulated with Opera). The magnetic field induces at most 2-3 mrad of difference.

Figure 7 . 11 :

 711 Figure 7.11: Evolution of the e-folding divergence of the beamlet with the unbalancing factor for the NITS accelerator (simulated with Opera). The significant difference between the divergence on the x and ŷ directions indicates the deformation of the beam shape. For the value of α unb estimated for NITS, the asymmetry increases the divergence of some mrad.

7. 3

 3 Explanations of the difference of beamlet divergence between NITS and BUG 7.3.1 Simulation with Opera and SLACCAD, comparison with experiments

Figure 7 . 12 :

 712 Figure 7.12: Maps of divergence in the space (RV , Πnorm) for BUG and NITS, with SLACCAD and Opera. The white point is the minimal divergence εmin and the white lines are isolines of divergence at 1.1εmin, 1.5εmin and 3.5εmin. The blue stripe on the top part of the maps of BUG corresponds to the region where a substantial part of the beam impinges on the grids (scraping).

Figure 7 . 13 :

 713 Figure 7.13: Evolution of the divergence with Πnorm at the optimal RV and vice versa with SLACCAD and Opera for NITS, comparison with the experimental values. The evolution of ε with Pnorm is better reproduced by Opera.

7. 3 . 2

 32 Scraping of the beamlet on the grids 7.3.2.1 Description of the scraping phenomenon

Figure 7 . 14 :

 714 Figure 7.14: Beamlet scraping at high injected current density on the NITS accelerator with Vext = 4kV and RV = 5.1. The blue rays in SLACCAD simulations represent the current lost on the grids.

Figure 7 . 15 :

 715 Figure 7.15: Evolution of the divergence with the normalized perveance for NITS and BUG with RV = 5.1. Before reaching the minimum of divergence, scraping occurs on the EG of BUG.Unlike the case of NITS, there is no overperveant part of the curve with BUG.

Figure 7 . 16 :

 716 Figure 7.16: Example of perveance scan with BES measurement on BATMAN Upgrade [119]. The "smiling curve" is interrupted (no overperveant part can be observed). The scraping threshold is measured around Πnorm = 0.20 -0.25. Data courtesy of BUG team.

Figure 7 . 17 :

 717 Figure 7.17: Evolution of the accelerated current with the arc power of the source for NITS (experimental data).

Figure 7 .

 7 Figure 7.18: Evolution of the accelerated ion current density (A m -2 ) and co-extraction electron current density (evaluated by the heat loads on the grids) with the RF power injected inside the plasma source (kW). Data courtesy of BUG team.

Figure 7 . 19 :

 719 Figure 7.19: Comparison between the beam profiles calculated with Opera for NITS and BUG geometries with typical operating parameters (2D slice).

Figure 7 . 20 :

 720 Figure 7.20: Beam envelope from the PG knife edge to the EG inlet aperture with NITS and BUG, solved with Mathematica for different values of normalized perveance and initial angle of the beam envelope (r init ). The scraping is observed for BUG but not for NITS. The beam envelope is larger with BUG.

Figure 7 .

 7 Figure 7.21: Physical model used to approximate the meniscus surface. The forces represented apply on the positive ions.

r 2 P G + z 2 men z men r 2 P G + z 2

 222 men r P G z men arctan z men r P G -1 δφ With typical values: z men = -1mm, r P G = 7mm, T e = 1eV, φ plasma = 10V and δφ = 0.5V we find: δz men = 0.13mm = 13 %z men Chapter 7. Explanation of the difference of beamlet divergence between NITS and BUG 125

Figure 8 . 1 :

 81 Figure 8.1: Section of the MTF accelerator (made with Opera).

8. 1

 1 Beam optics 8.1.1 Deflection of the beamIn MTF, the co-extracted electrons are suppressed by a set of CESM magnets with a remanence B r,C = 1.1T and the magnetic deflection of the beamlets is corrected by a Steering Grid located downstream of the EG.

Figure 8 . 2 :

 82 Figure 8.2: Thermal image of the reference pulse.

Figure 8 . 5 :

 85 Figure 8.5: Addition of the values of αunb for five pulses of MTF (blue diamonds) to the trend of Fig. 6.22.

  5mrad 7.7mrad] with R V ∈ [19.8 27.7] and Π norm ∈ [0.10 0.19]. The measurements of divergence for 11 pulses near the optimum values of R V and Π norm are shown in Fig. 8.6. Examples of thermal images for those pulses are visible in Fig. 8.7.

Figure 8 . 6 :

 86 Figure 8.6: Measurements of divergence for the pulses near the optimum parameters. The size of the circles is related to the measured divergence by a linear function. The optimum parameters are around RV = 26 and Πnorm = 0.12.

Figure 8 . 7 :

 87 Figure 8.7: Example of thermal images of the pulses used for Fig. 8.6.

Figure 8 . 8 :

 88 Figure 8.8: Scans of divergence with Πnorm and RV with SLACCAD and Opera: the green rectangle represents the operating space during the experiments with MTF.

Figure 8 . 9 :

 89 Figure 8.9: Thermal image of the pulse 8318.

Figure 8 . 10 :

 810 Figure 8.10: Density profile accross MTF accelerator (calculation by T. Patton).The pressure is 230 mPa inside the source and 6.1 mPa at the exit of the accelerator, according to measurements. The source temperature was set to 1000 K and the accelerator components temperature to 300 K.[START_REF] Aprile | Report on joint experimental activities and simulations at qst laboratories 13 may -19 july 2019[END_REF]

Figure 8 . 11 :

 811 Figure 8.11: Repartition between the species causing the major heat loads (electrons, negative and neutral hydrogen) ; the green lines represent the total heat load measured by water calorimetry on each grid.

Figure 8 . 12 :

 812 Figure 8.12: EAMCC3D: electron trajectory. The number of electrons on the AG1 is lower than on the AG2 and GG.

Figure 8 . 13 :

 813 Figure 8.13: COMSOL: electron trajectory and heat loads on the AG1, AG2 and GG. The electrons deposit a high power on the AG1. The color bars represent the number of particles and the power deposited on the grids.

Figure 8 . 14 :

 814 Figure 8.14: Example of additionnal rows of beamlets observed on some thermal images (the most visible beamlet is indicated by the white arrows). On some thermal images, the intensity of the first beamlet of this additionnal row is comparable to the intensity of the other beamlets.

Figure 8 .

 8 Figure 8.15: a) Trajectory of an underperveant beamlet (j = 0.26opt) b) Trajectory of an overperveant beamlet (j = 3jopt).

Figure 8 . 16 :

 816 Figure 8.16: Evolution of the total power load deposited on the grids by a single beamlet, calculated with EAMCC3D (Vext = 5.5kV).

Figure A. 4 :

 4 Figure A.4: Cross-section of the beam at the EG. It is discernable by eye that there is a higher density of particles on the side x < 0, which is the direction toward which the beamlet is deflected in the extraction gap.

Figure A. 5 :

 5 Figure A.5: Skewness calculated for 10 6 normally distributed samples of 300000 elements. The skewness observed in the experiments is way outside the range of statistical fluctuations.

Figure B. 4 :

 4 Figure B.4: Groove inside the EG of MITICA where the ADCM are inserted, visible on the horizontal section of the accelerator (taken from [62]).

Figure B. 5 :

 5 Figure B.5: Evolution of the horizontal deflection with the unbalancing factor for MITICA geometry, with j = -293A m -2 , Vext = 8.4kV and Vacc = 191.6kV -Vext for two magnetic configurations.

Table 1 . 1 :

 11 Milestones in tokamak achievements

	achievement	tokamak	year
	First tokamak to demonstrate plasma disruptions	TM-3, USSR	1963
	First machine to produce "significant" neutrons	TM-3	1971
	First use of poloidal divertor	JFT-2a, Japan	∼1975
	First demonstration of H-mode ASDEX, Germany 1982
	First D-T plasma	JET, EU	1991
	Greatest triple product n e τ E T = 0.51(n e τ E T ) breakeven	JT-60, Japan	1996
	Greatest fusion power output (16.1 MW, Q=0.67)	JET, EU	1997
	First fully superconducting tokamak	EAST, China	2006

Table 3 . 1 :

 31 Specifications of MITICA

Table 4 . 1 :

 41 Geometries tested

	Accelerator	Number of grids	Acceleration range	Specific features
	NITS [94]	3	Some tens of kV
	BATMAN Upgrade (BUG) [51] 4	Some tens of kV Repellor Electrode
	MTF 3 stages [72]	6	500 kV	Steering Grid
	MITICA [62]	7	1000 kV

Table 4 . 2 :

 42 Distribution of absolute relative error

	≤ 0.05	70%
	> 0.05 and ≤ 0.2 20%
	> 0.2	10%

Table 5 .

 5

	1: NITS geometric parameters
	Geometric parameter Value for NITS
	α 1	2.21 rad
	β 1	0.83 rad
	α 2	2.53 rad
	β 2	0.59 rad
	L	41 mm
	δ	3 mm
	This corresponds to the following values:

Table 5 .

 5

		2: Coefficients for NITS
	a 0	6.10	d 2	-1.17
	a 1	6.21	E	2.15
	b 0	4.28	f 0 -1.60 × 10 -4
	b 1	8.36	f 1	-2.72 × 10 4
	b 2	-3.86	G -1.44 × 10 -4
	C	1.22	h 0	-2.79 × 10 2
	d 0	0.440	h 1	-3.63 × 10 2
	d 1	4.66	h 2	-45.0

Table 5 . 3 :

 53 Parameters used in the set of Opera simulations

	Parameter Values tested on Opera
	R V	4, 4.5, 5, 5.5, 6
	B r,C

Table 5 .

 5 4: ∆x in mm predicted by Opera for 50 simulations

Table 5 .

 5

		,A = 0	7.20 4.59	5.43	9.45 6.25 7.44	7.02	7.65
	B r,C = 0.5T B r,A = 0.5T	-2.36	-1.01	0.34 0.27	1.47	2.51
	B r,C = 0.5T B r,A = 1.1T	-10.28	-8.79	-8.34 -6.96	-5.22	-3.68
					20.31
	B r,C = 1.1T	B r,A = 0	10.26	12.01	16.30	15.30	16.67
					13.66
	B r,C = 1.1T B r,A = 0.5T	3.50	5.65	9.64 7.69	9.65	11.49
					0.58
	B r,C = 1.1T B r,A = 1.1T	-4.42	-1.81	0.70	3.11	5.29
					0.71
	B r,C = 1.6T	B r,A = 0	15.79	17.78	19.96	22.16	24.17
	B r,C = 1.6T B r,A = 0.5T	9.17	11.45	13.65	16.12	18.23

5: Error in mm: ∆xOpera -∆x f it

Table 5 . 6 :

 56 Magnetic configurations during the Joint Experiments

	Magnetic configurations	B r,C	B r,A
	1 st Joint Experiments -phase 1	1.1 T 0.88 T
	1 st Joint Experiments -phase 2	1.1 T	1.1 T
	2 nd Joint Experiments	0.77 T 1.1 T

Table 6 . 1 :

 61 Optical properties of the beamlet for different CDPM

Table 7 . 1 :

 71 Comparison between characteristics of BUG and NITS Effect of the voltage ratio and normalized perveance on the beamlet divergenceDuring operations of ion beam extraction and acceleration, the divergence is mostly dependent on the voltage ratio R V and the normalized perveance of the beamlet Π norm . The evolution of the divergence with both parameters can be described by a concave curve with two branches and a global minimum. The optimum voltage ratio and perveance refer to the values of R V and Π norm that minimize ε. Examples of such curves, sometimes called smiling curve, are given in Fig.7.6.

		Parameter	BUG	NITS
	Source	Source volume	31cm × 58cm × 23cm 47 L	880cm 2 × 34cm 30 L
		Extraction area	108 cm 2	46 cm 2
		Power injection	RF 150 kW	8 filaments ≈ 25 kW
	Grid system	Number of apertures	70: 5 × 14	30: 5 × 6
		PG aperture diameter	14mm	14mm
		EG aperture diameter	12mm	13mm
		PG thickness	9mm	6mm
		Extraction gap	6mm	6mm
		Distance between	∆x = 20mm	∆x = 19mm
		aperture centers	∆y = 20mm	∆y = 21mm
	Operating parameters	PG temperature	≈200°C	≈210-230°C
		Pulse duration	Plasma 6-7s Extraction 4-5s	Plasma 1.5s Extraction 0.5s
		Space charge compensation	Background particles + RG to enhance SCC	Background particles
	Measure of divergence ε	Diagnostic	BES	CFC calorimeter
		Abscissa where ε is measured	260mm from GG (BES1) 1290mm from GG (BES2)	937mm from GG
		Typical values	15-20 mrad	5-10 mrad
	7.2 Parameters affecting the beamlet divergence
	7.2.1

  The geometric parameters of NITS and BUG are:

	Focal length	NITS		BUG
	f P G,out [mm]	-3.0		-3.7
	f EG,in [mm]	18.8		20.9
	Geometric parameter	NITS	BUG
	α 1		2.21 rad 2.12 rad
	β 1		0.83 rad 0.79 rad
	α 2		2.53 rad 2.70 rad
	β 2		0.59 rad 0.46 rad
	L		41 mm	38.7 mm
	δ		3 mm	3 mm
	This corresponds to the following focal lengths:

Table 7 . 2 :

 72 Parameters used for Opera and SLACCAD simulations

		R V	6	Π norm 0.2270 0.3247
		4	6.4	0.1022 0.2433 0.3410
		4.4 6.8	0.1294 0.2596 0.3573
		4.8 7.2	0.1565 0.2759 0.3735
		5.2 7.6	0.1836 0.2922 0.4007
		5.6	8	0.2108 0.3084 0.4278
	The extraction voltage is kept at 3.9kV, and the acceleration voltage and current density are varied. The
	curves obtained are fitted by 4 th order polynomials (see Fig. 7.6 for an example on four curves) and extrapolated
	over a grid with R V from 4 to 8 by steps of 0.1 and Π norm from 0.1 to 0.4 by steps of 0.01. The results are
	displayed in Fig. 7.12.
	The minimal divergence and the optimum parameters are:
	parameter NITS (Opera) NITS (SLACCAD) NITS (exp) BUG (Opera) BUG (SLACCAD)
	ε min [mrad]	11.52	8.34	≈ 8.9	11.47	7.99
	R V	5.5	5.5	≈ 4.9 -5	6.2	6.1
	Π norm	0.36	0.28	≈ 0.35	0.34	0.26

Table 8 . 1 :

 81 Parameters of the reference pulse (Vacc is the voltage between the EG and the AG1, the AG2 and GG have voltages 2(Vext + Vacc) and 3(Vext + Vacc)).

	141.5	6.2	162.3	36.8	16.5	0.12	25.9

j[A m -2 ] V ext [kV] V acc [kV] P arc [kV] B y,men [mT] Π norm R V

Table 8 . 2 :

 82 Comparison between simulations and measurements for the reference pulse

		∆x[mm] θx[mrad]
	Measurement	6.5	2.5
	COMSOL	-1.3	-0.5
	EAMCC3D	-0.8	-0.3
	Opera (α unb =0)	-0.5	-0.2
	Opera (α unb =0.24)	7.8	2.9

Table 8 . 4 :

 84 Comparison between simulations and measurements for the reference pulse

		ε[mrad]
	Measurement	6.5
	COMSOL (without halo)	7.3
	EAMCC3D	7.9
	Opera (α unb =0)	6.5
	Opera (α unb =0.24)	6.8
	SLACCAD	7.0

Table 8 . 5 :

 85 Comparison between the properties of two similar pulses used to compare separately the heat loads and the optics with the simulationsIndex Pulse duration [s] Current density [A m -2 ] V ext [kV] V acc [kV] P arc [kW]The heat loads on the accelerator components have been studied with COMSOL and EAMCC3D. Both models include the following reactions:

	7817	51	153.3	5.5	162.9	34.4
	8318	0.2	149.4	5.2	163.6	36.4
	Single stripping: H -+ H 2 --→ H + H 2 + e -
	Double stripping: H -+ H 2 --→ H + + H 2 + 2 e -
	Background gas ionization:	H -+ H 2 --→ H -+ H + 2 + e -H + H 2 --→ H + H + 2 + e -

Table 8 . 6 :

 86 Optical properties of the pulse 8318 simulated with EAMCC3D and COMSOL

		θ x [mrad]	ε[mrad]
	Measurement (pulse 8318)	2.7	6.7
	EAMCC3D	-0.3	7.5 with halo
	COMSOL	-0.5	6.8 without halo 7.7 with halo

Table 8 . 7 :

 87 Thermal loads on the accelerator grids[kW]

	Grid	AG1 AG2 GG Total
	Measurements	3.22 4.83 5.15 13.21
	EAMCC3D	0.47 3.36 5.60	9.52
	COMSOL	2.37 2.00 5.86 10.24

Table 8 . 8 :

 88 Repartition of the thermal loads over the grids for strongly underperveant and overperveant beamlets simulated with EAMCC3D. The number in parentheses is rapported to the total heat loads (the sum over a row is not necessarily 100% because the thermal loads on the PG, EG and SG are not shown here, since they are negligible).

		j[A m -2 ]	AG1	AG2	GG	Total
	Measurements	149.7	3.22 (24%) 4.83 (37%) 5.15 (39%) 13.21
	Near optimal	150	0.47 (5%) 3.36 (35%) 5.60 (59%)	9.52
	Underperveant	40	0.25 (6%) 2.49 (55%) 1.77 (39%)	4.51
	Overperveant	450	2.30 (14%) 4.37 (26%) 9.97 (60%) 16.64

Figure 4.11: electric potential along r with a total voltage of 19.5kV in NITS

The magnetic field inside the accelerator influences the divergence of the beam through the Lorentz force, which modifies the transverse acceleration of the beam particles in proportion to their axial velocities. However, if all

Acknowledgements

Before drawing up a list, hopefully as complete as possible, of all the people that I would like to thank, I should shortly remind what my gratitude is for. First of all, my drive for starting this PhD is my passion for physics and technology, and my objective was above all to do science. I feel like I could accomplish this goal and for this I am really grateful to my supervisor, Giuseppe Chitarin, for his precious guidance and responsiveness. I am also very thankful to Daniele Aprile for all the help and support he gave me. Working with them has been a pleasure on the scientific and human levels. More generally, I had a very good experience collaborating with the scientists of Consorzio RFX and I thank all the people with whom I worked during my thesis. I am thankful to Piero Agostinetti, Antonio Pimazzoni, Pierluigi Veltri, Gianluigi Serianni, Emanuele Sartori, Tommaso Patton, Mattia Dan, Fabio Veronesi, Carlo Baltador, and Matteo Brombin, who answered some of my questions and helped me improve my work. During these three years, I also had the good fortune to collaborate with several scientists of other institutes. I am thankful to Gwenael Fubiani, Francesco Taccogna and Pierpaolo Minelli for the generosity with which they shared their work and their time. Not being able to visit the LAPLACE laboratory because of the coronavirus situation is the only regret of my PhD. I am grateful to the people of QST for their very kind hospitality during my working period at Naka. More generally, working with them during these three years has been a great and continuous pleasure and my thesis relies mainly on the scientific data resulting from this collaboration. Among others, I would like to thank Junichi Hiratsuka, Mieko Kashiwagi and Glynnis Mae Saquilayan. I also thank the ITED division of IPP Garching, and particularly Dirk Wünderlich, Andrew Hurlbatt, Federica Bonomo and Loïc Schiesko. My stay at IPP really gave me another perspective on the research world and its functioning. I thank the director and the professors of the National PhD School Ferdinando Gasparini for the instructive week I spent at the university of Napoli. Finally, I am very grateful to the University of Padova, which funded this thesis, the Ghent University and Consorzio RFX. I thank Paolo Bettini, Piergiorgio Sonato, Jean-Marie Noterdaeme and Kristel Crombé for everything they did for my PhD.

Oltre all'aspetto scientifico, la mia motivazione per fare il dottorato in Italia era di acquisire nuove esperienze fuori dalla mia "zona comfort", e scoprire un'altra cultura e un altro modo di lavorare. Devo dire che la mia esperienza è andata ancora meglio di quanto mi aspettassi, grazie alla gentile ospitalità delle persone che mi hanno accolto a RFX. Ho avuto la fortuna di trovare amici sul posto di lavoro, grazie ai quali posso dire di essermi integrato nella cultura italiana. Ringrazio tutte le persone che ho conosciuto a RFX e tutti i miei colleghi dottorandi. Guardando indietro, non vedo altro che bei ricordi durante il mio soggiorno in Italia. In particolare, vorrei ringraziare Daniele, Carlo, Luca, Margherita e Francesco che mi hanno aiutato molto dopo un infortunio sugli sci, non dimenticherò questo atto di gentilezza.

Ma passion pour la physique est née et a grandi principalement grâce à des professeurs capables d'animer et d'entretenir une vocation scientifique. J'exprime ma gratitude à chacun d'entre eux, mais je suis particulièrement reconnaissant à Mme Kilic et M. Barnier, qui ont le plus marqué mes études. Enfin, tout ce travail serait vain en l'absence de ce qui compte vraiment, à savoir mes proches. Je remercie Deborah pour son appui constant et précieux, ainsi que sa patience durant mes longs séjours à l'étranger, mon frère et ma soeur pour leur soutien, et mes parents, qui m'ont toujours laissé libre de mes choix et m'ont toujours apporté un support indispensable dans tout ce que j'ai entrepris.

Chapter 9

Summary

In this final chapter, we synthetize the findings of this PhD that could require further investigations and may be useful for the researches on negative ion accelerators for NBI.

In the second section of Chapter 4, I presented a formalism that provides very general and precise equations for the electric potential and field inside a multigrid accelerator for NBI. The theory used to get this result is well-known, but I showed how the results compare with simulations for different accelerators. I applied the equations obtained for three different important calculations: the trajectory of a negative ion beamlet with an external magnetic field, the electric forces exerted on a beamlet (in particular when an asymmetric current density profile is assumed) and the beamlet envelope.

In the first section of Chapter 5 I calculated the trajectory of a negative ion beamlet in an accelerator with an external magnetic field. I proposed a rather simple matrix formulation that makes the numerical integration of the trajectory simple. I showed how the thin lens formula applied to the EG of NITS causes a large error, which necessitates to add an ad hoc corrective factor. For this reason, the results obtained with the analytic model have no general value but indicate how the beamlet deflection is expected to depend on the operating and design parmeters of the accelerator.

In the second section of Chapter 5, I continuated an ongoing work of analysis and interpretation of the Joint Experiments on NITS. I gave details on the discrepancies observed in terms of horizontal deflection. In Chapter 6, I investigated further the explanation of this discrepancy, based on the current density profile asymmetry at the meniscus, with a systematic simulation work. First, an analytic study explains different mechanism causing the additional deflection in an asymmetric beamlet and compares their effect. This part allows to physically explain the dependence of the deflection on α unb and other parameters, observed with Opera. A useful conclusion of this chapter is that the deflection predicted by Opera (and other particle tracing codes) for accelerators with a CESM+ADCM configuration can be very well approximated by the following numerical fit:

where the coefficients {a i } depend only on the accelerator geometry. Once the coefficients {a i } are known, this formula can be used instead of simulations. This is relevant when a very large number of simulations is required. The correlation found between the parameter α unb heuristically deduced from experiments and the plasma source parameters (magnetic field magnitude, power absorbed by the plasma) is a good evidence of the existence of an asymmetry at the meniscus. An empirical law was deduced for the kamaboko source used during the Joint Experiments: α unb = -0.4B y,men P arc (B y,men is the algebraic magnetic magnetic field at the meniscus.)

A collaboration aimed at better understanding the emergence and magnitude of the asymmetry in a negative ion source with PIC simulations is ongoing. This field of research is quite important, as the value of α unb on the large MITICA accelerator may play a crucial role in its design.

In Chapter 7, I presented the problem of the difference of minimum divergence reached with the NITS and BUG facilities. Based on an intensive simulation work, I was able to propose an explanation to this Chapter 9. Summary Appendix A

Study of the emergence of current density asymmetry in a plasma source

The conclusion of the chapter 6 is that, in the extraction area of negative ion sources for NBI accelerator, a substantial asymmetry of the current density profile develops along the direction perpendicular to the suppression magnetic field. The magnitude of this asymmetry is correlated to the strength of the magnetic field and to the power injected inside the plasma. Beam simulations usually calculate the meniscus shape and position with a very simplified model, treating in the same way negative and positive ions, which cannot describe the appearance of this asymmetry. To investigate the behavior of the plasma inside the extraction area, PIC (Particle In Cells) simulations are more suited. The emergence of an asymmetry of the current density profile at the meniscus is a known fact in the field of PIC simulations for NBI ion sources [START_REF] Fubiani | Plasma asymmetry due to the magnetic filter in fusion-type negative ion sources: Comparisons between two and three-dimensional particle-in-cell simulations[END_REF][START_REF] Gutser | Transport of negative hydrogen and deuterium ions in rf-driven ion sources[END_REF][START_REF] Miyamoto | Study of negative hydrogen ion beam optics using the 3d3v pic model[END_REF][START_REF] Taccogna | Pic modeling of negative ion sources for fusion[END_REF][START_REF] Abe | Effects of the extraction voltage applied by the puller-electrode on the h-extraction in the linac4 negative ion source[END_REF]. In this appendix, we detail preliminary results of an ongoing activity on this topic. The ultimate goal of this approach would be to get a physical representation of the current density profile at the meniscus, correct enough to accurately predict the criss-cross deflection, and general enough to be easily included in beam simulations with softwares like Opera. The results presented in this appendix have been partially published in [START_REF] Denizeau | Experimental and numerical investigation on the asymmetry of the current density extracted through a plasma meniscus in negative ion accelerator[END_REF]. They have been obtained with the code of Gwenael Fubiani, a detailed description of which can be found in [START_REF] Fubiani | Modeling of negative ion extraction from a magnetized plasma source: Derivation of scaling laws and description of the origins of aberrations in the ion beam[END_REF].

A.1 Description of the 3D PIC-MCC code

In this part, we present the results of a simulation of a plasma source and extractor with a 3D Particle-In-Cell Monte Carlo collisions (PIC-MCCs) explicit algorithm. The simulation domain in the vicinity of the plasma grid aperture is shown in Fig. A.1. The code is parallelized in a hybrid manner using OpenMP [127] and MPI libraries. From the extractor geometry, plasma parameters, initial conditions and boundary conditions, the code computes the trajectories and densities of the plasma species and the electrostatic field in the simulation domain.

The magnetic filter field B x is approximated by a Gaussian profile along ẑ, and the electron suppression field is defined as a cusp field surrounding the PG aperture (the exact equations are found in [START_REF] Fubiani | Modeling of negative ion extraction from a magnetized plasma source: Derivation of scaling laws and description of the origins of aberrations in the ion beam[END_REF], equations ( 1) and ( 2)), with B y ≈ 20mT directed along -ŷ, which is of the same order as the value applied in NITS or MTF, and foreseen for MITICA.

To reduce the computation time, the plasma is scaled down in density by increasing the vacuum permittivity ε0 = 64ε 0 . The behavior of the plasma with a permittivity ε0 is approximately equivalent to a plasma 64 times denser with a permittivity ε 0 , and an extraction voltage 16 times higher (the validity of this method is discussed in detail in [START_REF] Fubiani | Plasma asymmetry due to the magnetic filter in fusion-type negative ion sources: Comparisons between two and three-dimensional particle-in-cell simulations[END_REF][START_REF] Fubiani | Modeling of negative ion extraction from a magnetized plasma source: Derivation of scaling laws and description of the origins of aberrations in the ion beam[END_REF]). The accelerator dimensions are reduced: the PG has a radius of 4 mm, a thickness of 2 mm and the extraction gap is 3.5 mm long. A single circular aperture is modelled with periodic boundary conditions on the edges, and the geometry of the grid is approximated by ignoring the chamfers. The cells of the mesh are cubic with an edge of size ∆ cell = 125µm, smaller than the scaled Debye length (∆ cell /λ D = 0.814). The extraction voltage is 400 V. A detailed description of the case studied here is found in [START_REF] Fubiani | Modeling of negative ion extraction from a magnetized plasma source: Derivation of scaling laws and description of the origins of aberrations in the ion beam[END_REF].

The negative ions are magnetized in the extraction area, and therefore they are deflected by a E × B drift, as shown in current distribution inside the PG. This is why the direction of the Hasymmetry at the meniscus follows an E × B drift. This is visible for example in Fig.

A.1 a), where the flux of negative ions is higher on the bottom part (x < 0) of the PG aperture. A more advanced description of the plasma parameters asymmetry caused by the plasma drifts is found in [START_REF] Fubiani | Plasma asymmetry due to the magnetic filter in fusion-type negative ion sources: Comparisons between two and three-dimensional particle-in-cell simulations[END_REF][START_REF] Taccogna | Pic modeling of negative ion sources for fusion[END_REF][START_REF] Fubiani | Modeling of negative ion extraction from a magnetized plasma source: Derivation of scaling laws and description of the origins of aberrations in the ion beam[END_REF].

The simulation outputs include current density maps of the negative ions in the planes (x,ẑ) and (ŷ,ẑ). Consequently, the unbalancing factor α unb can be explicitly computed as an output parameter of the simulations (in Chapter 6 it was heuristically deduced from measurements).

Appendix A. Study of the emergence of current density asymmetry in a plasma source 143

A.2 Evidence and quantification of the current density profile asymmetry at the meniscus

The first step of this study is to demonstrate that the PIC simulation indeed predicts an asymmetry of the current density profile near the meniscus. We start by redefining the unbalancing factor in a way that is consistent with its definition in Opera: we first consider the current density profile along x at a given z with y = 0 (see Fig. We determine the edges of the beam along the x axis. They correspond to the points where the current density vanishes. More precisely, if j max , the maximum current density along the profile, is at a position x 0 , the beam edge positions x 1 and x 2 (x 1 < x 2 ) are the first points below and beyond x 0 where the current density falls to 10 -4 j max .

Those edges being defined, the coordinate of the middle point of the beam is given by

. The asymmetry factor is then defined by:

The average value of α unb inside the PG aperture is 9.1 %, which is the same order of magnitude as assumed for the NITS accelerator. The value obtained in PIC simulations is lower, probably because of the scaled down plasma parameters and accelerator size. However, we can see on Fig. A.3 that the shape of the current density profile is rather different that assumed in beam simulations: the profile is not centered (as defined in Chapter 6) and the maximum current density is reached in the part where the average current density is lower. The difference of current density between the two halves of the beamlet comes from the steeper increase of the profile near the edge of the beamlet in the left part. Because of the filter field B x , a vertical asymmetry is also present in the beamlet, but much weaker than the horizontal one, as visible in Fig. A.1 b). The average value of the vertical unbalancing factor at the PG is 1.3%.

A.3 Use of the skewness (third moment) to quantify the asymmetry

To apply the definition of α unb given in Chapter 6, it is necessary to define the "right half" and "left half" of the beamlet. This distinction is actually delicate in the frame of PIC simulations, because the vertical centerline that separates the beam envelope in two semicircles of same area is not defined as clearly as it is with ray-tracing 144 codes, and a small error on the centerline position results in a significant error on α unb . Furthermore, the current density profiles (for example Fig. A.2) are not centered profiles like those used in Opera (plotted in Fig. 6.2), making the definition of α unb less natural. In future steps, the current density profile in beam simulations should be approximated by a more realistic profile where the α unb quantity could be delicate to compute.

To avoid the difficulties encountered to set the value of α unb , a quantity that can be studied is the skewness (3 rd moment) of the particle horizontal position distribution. Being x the distribution of horizontal position of the particle along a transverse section, the skewness µ 3 is defined by:

In the PIC simulation studied here, the position and velocity of the particles are recorded on the plane (x,ŷ) at the entrance of the EG. We calculate the skewness of the distribution in this plane.

The distribution of horizontal position of the particles has a variance µ 2 = 0.52mm 2 and a skewness µ 3 = 0.21. To interpret this number, it is necessary to compare it with the sample skewness of a symmetric normal distribution with similar variance (0.52mm 2 ) and sample size (300000 elements), to ensure that it does not come from a statistical fluctuation. To do so, Matlab was used to generated 10 6 sets of 300000 elements normally distributed with a variance 0.52 mm 2 . We obtain that the sample skewness of a symmetric distribution follows a Gaussian (Fig. A.5), with a standard deviation σ = 0.0071 for a sample size of 300000, almost 30 times lower than the skewness observed in the simulation.

Therefore, the skewness computed from the PIC simulation is not a simple statistical fluctuation but a clear evidence and characterization of the beamlet asymmetry. Thus, the third moment allows to quantify the asymmetry without assumptions about the current density profile and the beam envelope shape.

A.4 Effect of the magnetic field on the asymmetry

In this part, a 2D version of the code was used to investigate the effect of the magnetic field on the magnitude of the asymmetry. The 2D model presented here solves the plasma densities, particle trajectory and electric 146 here a chamfer was added on the PG. The PG has a radius and a thickness of 4 mm, and the extraction gap is 3.5 mm long. A single circular aperture is modelled with periodic boundary conditions on the edges. The cells of the mesh are squared, and two different mesh sizes have been used in the simulations: ∆ cell,1 = 62.5µm and ∆ cell,2 = 125µm, both smaller than the scaled Debye length (∆ cell,1 /λ D = 0.407, ∆ cell,2 /λ D = 0.814). The extraction voltage is 150 V. The magnetic filter field B x is approximated by a Gaussian profile along ẑ, and the electron suppression field is defined as a cusp field. The equations for both fields correspond to equations ( 1) and ( 2) of [START_REF] Fubiani | Modeling of negative ion extraction from a magnetized plasma source: Derivation of scaling laws and description of the origins of aberrations in the ion beam[END_REF] with B 0,y = 7.5mT for the cusp field. The value of B 0,x for the filter field has been tested at: 6 mT and 20 mT with ∆ cell,1 = 62.5µm, 0, 4 mT and 10 mT with ∆ cell,2 = 125µm

In this model, the Hions are produced inside the volume and on the surface of the chamfer. Here, the asymmetry was calculated for all the ions together. It is an open question to determine if the ions produced in volume are more sensible to the magnetic field and therefore generate more asymmetry. The flux maps are displayed in Fig. A.6. For each case, the coefficient α unb is calculated as in the previous section, taking the average value in the space between the knife-edge of the PG and its outlet surface. We can see on Fig. A.6 the transverse Hflux becoming larger as the filter field magnitude increases. The growth of the asymmetry appears clearly if we compare the current density profiles along a horizontal diameter of the PG (along the x axis at the outlet surface of the grid) for different magnitudes of filter field. For example, in Fig. A.7, we compare the current density profiles for the cases with 0, 4 and 10 mT. When the filter field increases, the current density profile becomes wider, but its size increases more on one side, which causes the asymmetry. This can be shown by plotting the difference between the current density (rapported to its maximum value) for the cases with 10 and 0 mT (see Fig. A.8).

The evolution of α unb with B 0 is plotted in Fig. A.9. As observed in the measurements with NITS, there is a correlation between the intensity of the magnetic field in a plane and the asymmetry in the perpendicular plane. The evolution is not linear: the curve is slightly concave and we can expect a saturation of α unb for stronger magnetic fields.

A.5 Conclusion and limitations

The scan presented here have some limitations: first of all we calculated the asymmetry produced by the filter field. Its dependence on the field magnitude may change with respect to the cusp field generated by the EG magnets, and its magnitude is lower than the asymmetry in the perpendicular plane, causing much larger relative errors. Besides, the model is a zoom on a single aperture, the eflux in the whole source may not be precisely reproduced, and it strongly impacts the transverse Hflux. The increase of α unb with the filter field magnitude in a simplified model is an indication that a similar phenomenon may occur in the real source with the suppression field. Activities with PIC simulations are ongoing to better investigate the evolution of the asymmetry at the meniscus with source parameters. The preliminary design of the accelerator of MITICA (section 3.1.1) has been calculated with particle tracing codes, assuming a uniform or symmetric current density profile at the meniscus [START_REF] Agostinetti | Detailed design optimization of the mitica negative ion accelerator in view of the iter nbi[END_REF]. As demonstrated with the Joint Experiments with NITS (chapter 6) and MTF (chapter 8), this hypothesis does not hold, at least for a certain kind of plasma source (filament driven with H -). Given that the current density asymmetry grows under the effect of the suppression field in the extraction area, there is no indication that it should not exist for a RF source with Dlike MITICA. In this appendix, we investigate with Opera the effect of an asymmetric current density profile at the meniscus on the expected deflection on MITICA. The resulting modifications of the grid system design are discussed.

B.1 Empirical considerations on the order of value of α unb for MITICA

The empirical equation 6.6 applied in Opera for NITS and MTF may be specific to the kamaboko source used on both experiments, but, as a first step, we make the hypotheses that the unbalancing factor α unb is correlated to the vertical component of the magnetic field at the meniscus and to the power injected inside the source by the RF drivers also in the case of MITICA. We suppose, based on the results of PIC simulations (appendix A), that the asymmetry is a consequence of a polarization drift of the negative ions in the extraction area. The Dions have twice the mass of the H -, and consequently half their mobility in the plasma. This indicates that the magnitude of the asymmetry of a Dsource could be expected to be lower than the asymmetry of a Hsource. This hypothesis is to be investigated by PIC simulations. If we compare MITICA with NITS and MTF, we obtain that: 150 the magnetic field at the meniscus for MITICA is quite close to NITS and MTF, with a magnitude around 15-20 mT [START_REF] Chitarin | Improvements of the magnetic field design for spider and mitica negative ion beam sources[END_REF].

the RF power foreseen for MITICA is in the range of 100-1000 kW, much higher than for NITS and MTF. the current density extracted in MITICA is around 300 A m -2 , around twice the current density extracted in NITS or MTF (100-160 A m -2 ).

Of course, the linear evolution of α unb with the power injected (Fig. 8.5) has been observed only over a short range of ≈ 30 kW and is probably not valid for much higher power as in the case of MITICA. Rather, a saturation of α unb at high power would be more physical. Besides, the size of the source is probably an important factor: since the asymmetry at the meniscus is a local phenomenon, it is more likely related to the plasma species densities in the extraction area and more precisely in the small volume in front of each aperture of the PG. Consequently, it may be more physical to scale the asymmetry by the current density extracted at each meniscus (we recall here that this quantity is also roughly proportionnal to the power inside the source, see Fig. 7.17 for example). For MITICA operations, current densities around 2 or 3 times higher than on MTF and NITS are expected. Therefore, two main factors with opposite effects must be considered: the lower mobility of the Dreduces α unb , and the larger extracted current density increases it. Future experimental and numerical work should investigate these conclusions, and study the effect of the source size and ion species on the asymmetry at the meniscus. As a first step for MITICA, we assume that both effects approximately compensate, and that it is reasonible to expect a magnitude of α unb in the same range as observed on NITS on MTF: 10-30%.

B.2 Opera simulation with an asymmetric meniscus

We now study a single beamlet single aperture model of MITICA, with an asymmetric current density profile at the meniscus, described by a step profile. First of all, a consistency check was performed to ensure that the divergence and deflection in a symmetric case are consistent with the values obtained with the more detailed models used for the design of the grids. Scraping of the beam is observed with MITICA, beyond the optimal point of minimal divergence. A 2 nd order polynomial fit of the scan (excluding the points with scraping) allows to estimate a minimal divergence of 151 5.4 mrad for V ext = 8.4kV, in good agreement with previous simulations (see section 2.1 of [START_REF] Agostinetti | Detailed design optimization of the mitica negative ion accelerator in view of the iter nbi[END_REF]), as expected. Opera foresees a larger divergence than SLACCAD, but still lower than the 7 mrad required for ITER operations (section 3.1.1). With uniform current density at the meniscus, a residual horizontal deflection of 0.43 mrad is observed, which means that the ADCM and SESM of MITICA correctly compensate the deflection induced by the CESM. We now investigate the effect of α unb , fixing V ext at its optimal value 8.4 kV. A scan of horizontal deflection θ x was made against the unbalancing factor α unb . As in the case of NITS, we observe a substantial effect of the asymmetry, causing an additional deflection around 7.25α unb in mrad. For α unb in a range of 0.1-0.25, the deflection is found in a range 1.2-2.2 mrad. Such high values of deflection would be a critical issue for MITICA. At higher values of α unb , scraping of the beam occurs due to its large deflection.

B.3 Possible correction of the grid system design

The permanent magnets for MITICA accelerator were designed based on numerical models with a uniform current density profile at the meniscus. Consequently, the beamlet deflection in the present design may be underestimated, in which case stronger ADCM are needed to fully cancel the horizontal deflection. In MITICA, the magnets are at the maximum remanence for the material (samarium-cobalt). Therefore, the most convenient way for increasing the ADCM strength is increasing their size. In the case of NITS, perfect compensation was reached by reducing the CESM remanence [START_REF] Chitarin | Experimental validation of an innovative deflection compensation method in a multi-beamlet negative-ion accelerator[END_REF]. It is difficult to apply this solution to MITICA, because the excessive co-extracted electrons, accelerated at high energy, could damage the accelerator components.

For this reason, it may be necessary to improve the design of MITICA by increasing the size of the groove machined in the EG where the ADCM are inserted (shown in Fig. B.4), so as to use larger magnets to generate a stronger field. However, great care must be taken of the modifications of the mechanical properties caused by an enlargement of this groove and the reduction of the copper thickness around it. The mechanical resistance of the accelerator components to the thermal loads is a critical point for MITICA.

In the current design of MITICA, the ADCM width is 1.95 mm. In a second set of simulations, a scan of θ x with α unb is made with a new magnetic configuration in which the ADCM width is set to 2.5 mm. This value is chosen as a first step, as it seems compatible with the mechanical requirements of the grid system (however no mechanical simulation have been made yet with such design). The evolution of the deflection with α unb is plotted in Fig. B.5. With wider ADCM, the horizontal deflection is smaller, as expected, with a value of -0.1 mrad when the meniscus is symmetric. The deflection increases with α unb , with a slope 6.00mrad. For α unb in the range 0.1-0.15, the horizontal deflection is around 0.5-0.8 mrad, which is very close to the design point: being w ADCM