Keywords: Experimental nuclear physics, Direct and indirect methods, Globular clusters distribution . . . . .

Globular clusters are essential objects for studying stellar evolutionary patterns and the early phases of galaxy formation. The abundance anomalies observed in the globular cluster NGC 2419, such as potassium overabundance and magnesium depletion, can be explained if an earlier generation of stars contaminates the currently observed stars. However, the nature and properties of this first generation of stars are not clearly identified. The temperature and density range of this one depends on a number of reaction rates. The 30 Si(p,γ) 31 P reaction is one of the few reactions identified as having sufficient influence to constrain the nature of the stellar site(s) that contaminated NGC 2419. The objective of this thesis is to reduce the nuclear uncertainties associated with the 30 Si(p,γ) 31 P reaction by determining the resonance strengths in the energy range of astrophysical interest. The study of this reaction was done via two distinct experiments. For low energy resonances, up to 500 keV above the threshold of La thèse est un périple long et parfois difficile, mais j'ai eu la chance d'embarquer avec un équipage formidable. Comme tout marin qui part en mer, j'ai fait face à des défis et des incertitudes, mais je pouvais toujours compter sur mes collègues, mes amis, ma famille et mes directeurs de thèse pour me soutenir et me guider. Ensemble, nous avons navigué dans les eaux parfois tumultueuses de la recherche et chaque vague a été surmontée grâce au soutien de chaque personne présente à mes côtés.

Je tiens tout d'abord à remercier chaleureusement les membres de mon jury, pour leur écoute attentive et leurs commentaires constructifs. C'est un honneur pour moi d'avoir Elias, Alison, François et Freddy comme jury, d'autant plus que j'ai grandement apprécié vous côtoyer au laboratoire, ou pendant des manips au Ganil ou à Vancouver. Leurs conseils avisés ont été précieux tout au long de ma thèse. Un grand merci en particulier à Alison et François d'avoir accepté d'être les rapporteurs de mon travail.

Je suis également reconnaissante envers mes directeurs de thèse, Faïrouz et Nicolas, qui ont su me guider et m'encourager tout au long de ce parcours. Leur complémentarité, leur rigueur et leur confiance, m'ont permis de réaliser un travail de qualité. Je leur suis particulièrement reconnaissante pour leur soutien sans faille lors de la rédaction de ma thèse. Au-delà de la Physique Nucléaire, je suis également reconnaissante pour les recommandations cinéma de Fairouz, et les discussions autour de la musique électronique avec Nicolas, et bien d'autres sujets qui continueront, j'en suis sûre, d'alimenter nos échanges. Cette thèse repose grandement sur l'aide précieuse de Richard Longland B Integral of the exponentially modified gaussian . . . . . . . . .

Titre : Étude expérimentale de la réaction 30 Si(p,γ) 31 P pour comprendre les anomalies nucléosynthétiques dans les amas globulaires Mots clés : Physique nucléaire expérimentale, mesures directes et indirectes, Amas globulaires Résumé : Les amas globulaires sont des objets essentiels pour étudier les modèles d'évolution stellaire et les premières phases de la formation des galaxies. Les anomalies d'abondance observées dans l'amas globulaire NGC 2419, telles que la surabondance de potassium et la déplétion de magnésium, peuvent être expliquées si une génération antérieure d'étoiles contamine les étoiles actuellement observées. Cependant, la nature et les propriétés de cette première génération d'étoiles ne sont pas clairement identifiées. La plage de températures et de densités de celle-ci dépend d'un certain nombre de taux de réaction. La réaction 30 Si(p,γ) 31 P est l'une des rares réactions identifiées comme ayant une influence suffisante permettant de contraindre la nature du ou des sites stellaires ayant contaminé NGC 2419. L'objectif de cette thèse est de réduire les incertitudes nucléaires associées à la réaction 30 Si(p,γ) 31 P en déterminant les forces de résonances dans la gamme d'énergie d'intérêt astrophysique. L'étude de cette réaction s'est faite via deux expériences distinctes. Pour les résonances à basse énergie, jusqu'à 500 keV au-dessus du seuil de l'émission proton, la réaction de transfert 30 Si( 3 He,d) 31 P a été réalisée au Tandem du laboratoire MLL à Munich. Les particules légères produites au cours de la réaction ont été analysées en moment par le spectrographe magnétique Q3D de très haute résolution et leurs distributions angulaires ont été interprétées dans le cadre de la DWBA (Distorded Wave Born Approximation) pour obtenir les facteurs spectroscopiques proton. Ces derniers sont essentiels pour le calcul des largeurs proton servant à calculer les forces des résonances. Pour les résonances à plus haute énergie, une mesure directe des forces de résonances a été effectuée à l'aide du séparateur de recul DRAGON installé à TRIUMF à Vancouver au Canada. Les résultats de ces expériences ont été utilisés pour calculer le nouveau taux de la réaction 30 Si(p,γ) 31 P et son impact astrophysique a été étudié afin de mieux contraindre les conditions stellaires pouvant expliquer les abondances observées dans les étoiles de l'amas globulaire NGC 2419.

proton emission, the 30 Si( 3 He,d) 31 P transfer reaction was performed at the Tandem of the MLL laboratory in Munich. The light particles produced during the reaction were momentum analyzed by the very high resolution Q3D magnetic spectrometer and their angular distributions were interpreted in the DWBA (Distorded Wave Born Approximation) framework to obtain the proton spectroscopic factors. These are essential for the calculation of the proton widths used to compute the strengths of the resonances. For higher energy resonances, a direct measurement of the resonance strengths was performed using the DRAGON recoil sperctrometer installed at TRIUMF in Vancouver, Canada. The results of these experiments were used to calculate the new rate of the 30 Si(p,γ) 31 P reaction and its astrophysical impact was studied in order to better constrain the stellar conditions that can explain the abundances observed in the stars of the globular cluster NGC 2419.

et Phil Adsley. Phil a commencé le projet de recherche sur lequel je me suis appuyée, et a été d'une grande disponibilité pour m'orienter dans les premiers temps de ma thèse. Richard, quant à lui, a mis à ma disposition l'accès aux codes d'inférence bayésienne et de calcul de nucléosynthèse, ce qui m'a permis d'approfondir mes résultats et de les interpréter avec plus de précision. Je suis particulièrement reconnaissante envers eux pour leur générosité et leur soutien. J'en profite également pour remercier l'équipe DRAGON, plus particulièrement Chris Ruiz, Matt Williams et Teja Upadhyayula, qui m'ont accueillie à TRIUMF, géré l'expérience pendant qu'on était en Europe à cause d'une pandémie mondiale, mais aussi accompagnée durant l'analyse des données. Grâce à vous tous, j'ai pu vivre une expérience de recherche enrichissante malgré la distance i géographique, cela témoigne de la solidité des liens scientifiques et humains, et je suis fière d'avoir pu travailler avec vous.

Je ne saurai oublier de remercier mes collègues doctorants, avec qui j'ai partagé de nombreux moments marquants pendant ces années de recherche. Les sessions musicales avec Massyl, les heures passées à débuguer ROOT avec Alessandro, les discussions sans fin avec Malia, la visite de Giverny avec Wenling et Anahi, les quiz avec Damien, mais aussi l'accueil qu'Anne m'a réservé à mon arrivée, sans oublier Guillem, Louis, Rémy, et bien sur Clément, le coeur de la ruche. Tous ces moments resteront gravés dans ma mémoire et ont rendu cette expérience inoubliable. Je tiens également à remercier les membres de l'ex-groupe Nester pour leur bienveillance, leurs conseils et leur soutien. Les pauses-café à l'ombre du cerisier en fleur et sandwich dégustées au bord de l'Yvette pendant les courses Coincoin me manqueront. Je tiens également à remercier Thomas, qui m'a accompagnée depuis le M1, et qui a su me soutenir dans les moments difficiles et partager d'autres moments plus festifs, mais également Sullivan, Florian et Yann pour les pauses thé agrémentées de débats et discussions passionnantes.

Je tiens à exprimer ma profonde gratitude envers Céline, qui a été bien plus qu'une collègue. Elle a été ma confidente, ma conseillère, celle qui a su me réconforter dans les moments difficiles et partager mes joies. Sa présence à mes côtés a été une immense source de réconfort tout au long de cette aventure. Je remercie toutes ces personnes pour m'avoir permis de vivre une expérience de recherche exceptionnelle, où je me suis sentie comme dans une famille. Leurs encouragements, leurs conseils et leur amitié resteront pour toujours dans mon coeur.

Enfin, je voudrais exprimer ma gratitude envers mes parents, ma famille et mes amis qui m'ont soutenu tout au long de cette aventure. Vos encouragements et vos attentions ont été d'une grande aide pour moi. Je sais que j'ai pu compter sur vous à chaque étape de ce processus, que ce soit pour m'écouter parler de ma thèse pendant des heures, pour m'aider à démêler des problèmes ou pour me rappeler de prendre soin de moi. Votre présence et votre soutien ont été un cadeau inestimable pour moi, et je suis infiniment reconnaissante de vous avoir dans ma vie.

Sarah

In the first chapter, we introduce the key notions of stellar evolution for low and intermediate-mass stars and more particularly their evolution within globular clusters. Observed abundances will be presented, as well as sensitivity studies to identify the key temperature and density conditions and nuclear reaction rates needed to reproduce the observations. The thermonuclear reaction rate will be introduced and experimental approaches to measure it will be presented. An overview will be given of the state of the art of the 30 Si(p,γ) 31 P reaction in which this thesis is embedded.

In the second chapter, we present the experimental study of the one-proton transfer reaction 30 Si( 3 He,d) 31 P using the high energy resolution Q3D magnetic spectrometer. The experimental setup as well as the data analysis that followed the experiment are presented. The obtained excitation energies and angular distributions are presented and discussed. The interpretation of these results has to be placed in the framework of the Distorted Wave Born approximation (DWBA). In chapter 3 we present the key concepts of this theory and the ingredients involved in the calculations, the proton spectroscopic factor and proton widths of 31 P states are obtained from the comparison between these calculations and the experimental differential cross-sections. We compare the different existing codes and approaches, before concluding the chapter with a Bayesian study to estimate the uncertainties on the proton widths.

The second experiment consists of an inverse kinematic study of the 30 Si(p,γ) 31 P radiative capture reaction. This experiment was carried out using the DRAGON spectrometer, which components are detailed in chapter 4. The direct measurements of the resonance strengths is carried out by detecting the 31 P recoils, in coincidence with the γ-decays. The corresponding data analysis is presented and followed by a discussion of the results obtained on the energies and strengths of the 30 Si+ p resonances.

Finally, the properties of the resonances will be summarized at the beginning of chapter 5, where the calculation of the reaction rates by a Monte Carlo method will be presented and the subsequent results discussed. We conclude this thesis by studying the impact of our new 30 Si(p,γ) 31 P reaction rate may have on the chemical elements produced in the globular clusters, and their influence on the temperature and density conditions of the polluters in the NGC 2419 globular cluster. As far as humanity recalls, we enjoyed looking at stars, and classifying them, whether in constellations, or in groups according to their brightness and colours. While science did not make sense of the meaning behind the shapes of the constellations, the brightness and the colours of the stars did catch the attention of the astronomers. The brightness of the stars is measured by the apparent magnitude m, usually in a given spectral band. The difference between magnitudes measured in different bands gives a measure for the colour of the star. A famous photometric system is the UBV one, for which the mean wavelengths of response functions are 364 nm for U, 442 nm for B, 540 nm for V. The spectral intensity for stars can be described by a black-body distribution, the effective temperature T ef f can thus be deduced using Wien's displacement law [START_REF] Böhm-Vitense | Introduction to Stellar Astrophysics[END_REF].

The apparent magnitude of the star does not inform about the intrinsic brightness of the star, for example, some star may appear fainter than another star closer to Earth with similar intrinsic brightness. This bias is taken into account in the definition of the absolute magnitude M , which expresses the brightness that a star would have if it was 10 pc1 away from Earth. The absolute magnitude is related to the apparent one as [START_REF] Böhm-Vitense | Introduction to Stellar Astrophysics[END_REF]:

M = m -5(log 10 d pc -1), (1.1)
where d pc is the distance of the stars in parcsec. The absolute magnitude is also linked to the luminosity of the star L, which is the total amount of energy emitted by the star per unit time, in all wavelengths.

Plotting the luminosity of a group of stars as a function of their effective temperatures gives the Hertzsprung-Russell (HR) diagram. The HR diagram for a multitude of star clusters and galaxies shows that the stars are not randomly distributed, but are rather found in well-defined paths and branches. This distribution depicts the different phases of the life of a star, and its location in the HR diagram depends mainly on its age, initial mass and in a lesser extent on its chemical composition. Figure 1.1 shows a schematic HR diagram with the Sun and some famous stars, it also shows the main branches which are the Main Sequence (MS) the Giant and Supergiant branches, and the White Dwarfs (WD) locus. The figure also shows the evolutionary paths for a star with 1 solar mass and for a 10 M ⊙ massive star. The stellar evolution of low to intermediate mass stars will be described in the following subsection.

Stellar evolution

The evolution of a star is governed by the interplay between the pressure from fusion reactions, and the gravitational forces. The balance between these forces is called the hydrostatic equilibrium. We usually distinguish between low and intermediate mass stars, (0.7 M ⊙ < M < 8 M ⊙ ) and massive stars M ≥ 8 M ⊙ .

Low and intermediate mass stars

The location on the HR diagram is linked to the evolution phases of the star during its lifetime. The stars begin their journey in the Main Sequence (MS) phase, where they burn hydrogen in their core. The stars spend the majority of their lifetime in the MS, that is why it is the most populated part of the HR diagram (90%). When the hydrogen in the core is exhausted and has been almost completely converted to helium, the star reaches the turn-off point. If the star has a mass < 0.5 M ⊙ the nuclear reactions cease and the star ends its life as a White Dwarf with a helium core. For stars with higher masses, the remaining hydrogen continues to burn in a shell surrounding the helium core. During this phase, the star is in the sub-giant branch (SGB), the core contracts and its temperature increases. The outer layers expand and cause the surface temperature to drop, the luminosity remains approximately constant since the energy is more effectively radiated through the convective outer layers of the star. The stars spend only a few million years in the SBG phase, and that creates a gap in the HR diagram because very few stars are observed in there. Once the envelope cools enough, the first dredge-up happens as the outer convection zone extends deeper into the star, and the H-burning happens more effectively in larger sections of the star. This causes the luminosity of the star to increase, and becomes part of the Red Giant Branch (RGB). The H-burning continues in the shell of the star and the formed He is added to the core that grows steadily in mass and temperature due to gravity and contraction. When the core temperature reaches 100 MK, helium burning is initiated, which causes the temperature to further increase, and the He-burning rate to increase too, this runaway condition is called the helium flash. The core luminosity increases tremendously and the released energy increases the pressure which causes the core to expand, until the inward force of gravity balances, and the star starts fusing helium in a hydrostatic equilibrium state.

At this point, the star leaves the RGB and as the core expands and cools, the outer hydrogen-rich envelope reacts by contracting and thus the effective temperature increases. Metal poor and low mass stars have higher temperatures for the same luminosity, and thus will be distributed horizontally on the HR diagram, forming the so-called horizontal branch (HB).

As for hydrogen burning, when helium is exhausted in the core, its burning becomes located in a shell surrounding a carbon-oxygen core, the star witnesses an expansion of its envelope and the convective zone reaches deeper layers, the luminosity increases while the effective temperature decreases, and the star is in the Asymptotic Giant Branch (AGB). The energy released in the first part of the AGB phase, called early-AGB, is dominated by the He-burning in the shell. Later, the H-burning shell dominates and the freshly formed He causes instabilities in the He-shell. This stage is called the thermally pulsing AGB. Most of the hydrogen envelope is lost due to the effects of winds. At the end of the AGB phase, stars that had initial masses between 3 and 8 M ⊙ end-up with M ≤ 1.4 M ⊙ . The lost mass composes the so-called Planetary Nebula. The remaining star is a Carbon-Oxygen core (CO) surrounded by hot helium, but the nuclear reactions cease, and the star becomes a compact dense star composed of degenerate matter, and is called a White Dwarf. The maximum mass a WD can have corresponds to the limit mass supported by electron degeneracy pressure, and is called the Chandrasekhar mass.

The White Dwarf may be part of a binary system, either with another WD, or with any other type of star. If the WD accretes matter from its companion with a sufficiently high rate, the WD reaches the critical Chandrasekhar mass, the nuclear reactions in the carbon-oxygen core of the WD are reignited, and a runaway happens, releasing enough energy causing a violent explosion. The WD that precedes the explosion is completely destroyed. These spectacular astrophysical phenomena are called Type Ia Supernovae (SN Ia), or thermonuclear supernovae. Due to their characteristic light curve, they are standard candles [START_REF] Branch | Type Ia Supernovae as Standard Candles[END_REF] and help to constrain the expansion history of the Universe and the dark energy it contains [START_REF] Riess | Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant[END_REF]. Despite their importance, the models describing the SN Ia explosions are still debated. Recent observations have unravelled a variety among SN Ia, with different types of possible progenitors, such as the single and double degenerated scenarios [START_REF] Nomoto | Type Ia Supernova Models and Progenitor Scenarios[END_REF].

Massive stars

For stars with initial masses higher than 8M ⊙ , the evolution phases may continue with carbon, neon, oxygen and silicon burning, at the end of which heavy elements up to iron are formed. The star ends up with a Fe core surrounded with successive layers of lighter elements up to a final hydrogen shell. The star is said to have an onion-skin structure. The density of the core increases until it is comparable to the density in a nucleus (around 3 × 10 17 kg m -3 ). The repulsive nuclear forces are at play and resist the attractive gravitational force, but the mass is too high, and the star ends up by collapsing gravitationally. The core bounces back, which creates a shock wave that propagates outwards, and the outer layers of the star explode. This phenomenon is a core-collapse supernova, also called Supernova of type II. A remnant is left behind, usually a neutron star or a black-hole.

Astronomical spectroscopy

Starlight decomposition and the study of the corresponding spectrum go back to Isaac Newton, when he looked at a sun beam through a glass prism. Nowadays, the techniques are much more elaborated and the light of distant stars can be studied in different wavelengths using a diffraction grating, the resulting spectrum is studied by looking for absorption bands and lines. These features correspond to the chemical species present in the atmosphere of the star, where each absorbed wavelength corresponds to a particular element. The abundances of the observed elements can be deduced from the strength of the spectral lines. It is more common to deal with relative abundances, where the abundance of element X is expressed with respect to the abundance of element Y such as:

[X/Y] ⋆ = log 10 (N X /N Y ) ⋆ -log 10 (N X /N Y ) ⊙ , (1.2)
where N X(Y ) is the number of atoms of element X (Y) per unit volume. This bracket notation is a dimensionless expression of the abundances, and the subscript ⋆ indicates that the latter are measured for a given star while ⊙ is for the solar abundances. The abundance of element X is usually expressed as [X/H] or [X/Fe]. With this definition, the metallicity is denoted as [Fe/H]. Astrophysicists call metals all the elements heavier than helium. The metallicity is thus also given by:

Z = 1 -X -Y, (1.3)
where X and Y are the mass fractions for hydrogen and helium, respectively. Note that the metallicity is increasing since the beginning of the Universe. It is thus an indicator for the age of the star, or more precisely, for the time at which the star formed.

Globular clusters 1.2.1 General properties

Globular Clusters (GCs) are unique objects orbiting in the halo of galaxies at large distances of several kpc from the galactic centre. They consist of ten thousand to millions (10 4 -10 6 ) of low mass stars, strongly bound together by the gravitational force. These fascinating objects are to astrophysics and cosmology what fossils are to archaeologists.

The oldest observed globular cluster orbiting the Milky Way is t GS = 13.3 ± 0.3 Gyr old, setting a lower limit for the age of the Universe at t U = 13.5 ± 0.3 Gyr; this value is in agreement with the recent λ-CMB model estimation of t U = 13.8 ± 0.02 Gyr [5]. Globular clusters are also a unique testing ground for galactic models: they are the first baryonic structures to form after the reionization era that have witnessed the formation, evolution, and dynamics of the galaxies [START_REF] Duncan A Forbes | Globular cluster formation and evolution in the context of cosmological galaxy assembly: open questions[END_REF]. Globular clusters are also privileged laboratories for understanding and constraining stellar models: due to their old age, only low mass stars remain (≲ 1 M ⊙ ). These stars are distributed in the HR diagram along the main sequence, Subgiant and Red Giant, Horizontal, and Asymptotic Giant branches, where they mainly burn hydrogen. Furthermore, the typical metallicity of GC is between -2.3 < [Fe/H] < 0, and exhibits homogeneous abundances for heavy elements (iron-peak, neutron-capture and α-elements) within the clusters and between clusters. These properties helped to understand the role of the metallicity in the stellar evolution, as well as to better constrain models describing convection, rotation-induced mixing, mass loss and nuclear reactions in the stellar interior [START_REF] Vandenberg | Models of Metalpoor Stars with Gravitational Settling and Radiative Accelerations. II. The Age of the Oldest Stars[END_REF].

Stellar population within globular clusters

The understanding of globular clusters is closely linked to the resolving power of the instruments they were observed with. They were first thought to be single stars, or even comets, since they appeared as fuzzy blobs through the lenses of Charles Messier in the 18 th century. Years later, astronomers had access to larger telescopes and could recognize the cluster nature of these objects. The photometric observation during the 20 th century resulted in several Hertzsprung-Russell diagrams for globular clusters. Figure 1.2 shows the colour magnitude diagram2 for globular cluster NGC 1851 that exhibits a single evolutionary curve, called isochrone. It represents a path in the diagram, along which stars with the same age and different initial masses evolve. The estimated age of NGC 1851 is 9.9 Gyr [START_REF] Harris | A New Catalog of Globular Clusters in the Milky Way[END_REF]. These observations indicated that all stars within such clusters were formed from a unique proto-cluster 3 gas, with the same initial composition. The paradigm of the single stellar population of globular clusters was thus postulated [START_REF] Harris | Supergiant Molecular Clouds and the Formation of Globular Cluster Systems[END_REF].

At the turn of the 21 st century, the advance of very large telescopes pushed further the resolution limits, and improved photometric and spectroscopic observations called into question the paradigm of the unique stellar population. Figure 1.3 shows again colour-magnitude diagrams for Globular cluster NGC 1851, but with measurements taken 15 years after the ones of figure 1.2. It exhibits two distinct loci, evidence that [START_REF] Rosenberg | Photometric catalog of nearby globular clusters* -I. A large homogeneous (V, I) color-magnitude diagram data-base[END_REF] not all stars within this cluster have formed with the same initial composition [START_REF] Milone | The Hubble Space Telescope UV Legacy Survey of Galactic globular clusters -IX. The Atlas of multiple stellar populations[END_REF].

Chemical signature of multiple stellar populations

High-resolution spectroscopy of bright red giant stars in globular clusters revealed a star-to-star variation in the abundances of some light chemical species, such as carbon, nitrogen, oxygen, sodium, aluminium and sometimes magnesium, all produced during hydrogen burning [START_REF] Piotto | A Triple Main Sequence in the Globular Cluster NGC 2808[END_REF]. This difference in the abundances can be more than an order of magnitude from one star to another. Furthermore, these variations are coupled to a characteristic anti-correlation between pairs of elements, typically between C-N, O-Na, Mg-Al [START_REF] Gratton | Mixing along the red giant branch in metal-poor field stars[END_REF]. All globular clusters within the Milky Way exhibit an O-Na anticorrelation [START_REF] Carretta | Na-O anticorrelation and HB*** -VIII. Proton-capture elements and metallicities in 17 globular clusters from UVES spectra[END_REF], and figure 1.4 shows the case of NGC 2808, along with the abundances for metal-poor stars in the halo field, that are believed to be representative of the original abundances of the proto-cluster. The comparison with the latter suggests a distinction between two types of stars: oxygen-rich sodium-poor that we shall call first population, and oxygen-depleted sodium-enriched stars, called second population. This distinction is made because the temperature of the sodium-enriched stars is too low to explain the observed overabundance. The observed red giant stars must thus have inherited their O-Na abundances [START_REF] Carretta | Na-O anticorrelation and HB*** -VIII. Proton-capture elements and metallicities in 17 globular clusters from UVES spectra[END_REF]. Red points are RGB stars with a higher colour-index, closer to the red wavelengths. The inset shows a zoomed-in view around the SGB. Figure taken from [START_REF] Milone | The Hubble Space Telescope UV Legacy Survey of Galactic globular clusters -IX. The Atlas of multiple stellar populations[END_REF].

Under the light of these observations, the evolutionary model for globular clusters have been updated and includes several star formation episodes. Figure 1.5 summarizes schematically these episodes. In a first episode zero-metallicity stars have formed from the primordial elements, these stars are believed to have had very large masses (100 M ⊙ to 1000 M ⊙ ) and ended up very quickly in Supernovae explosions [START_REF] Heger | The Nucleosynthetic Signature of Population III[END_REF] enriching the proto-cluster gas with metals up to iron. In a second episode, the first generation of stars of the cluster forms, a fraction of these stars undergoes hydrogen burning at temperatures such that the oxygen is depleted, and the sodium overproduced, in addition to the other characteristic anti-correlations being created within these first generation stars often called the polluters. This processed material is ejected in the intracluster medium and mixed up with pristine gas, which is metal poor and corresponds to the composition of the proto-cluster. A small fraction of the first generation stars (30% of the observed stars) remains, and they are differentiated by their pristine composition. The rest of the stars of the cluster are second generation and are believed to have formed from a mixture of the pristine proto-cluster gas and processed material that polluter stars have ejected.

Halo Field Figure 1.4: Sodium abundances with respect to oxygen abundance, as observed in globular cluster NGC 2808 and in the stars of the halo-field of the Galaxy. The uncertainties on the observations are indicated by the black cross, upper limits on observed oxygen abundances are indicated in blue arrows, while finite observations are indicated with red circles for the cluster. The halo-field stars have a metallicity -2 ≤ [Fe/H]≤ -1 [START_REF] Gratton | Mixing along the red giant branch in metal-poor field stars[END_REF]. The figure is adapted from [START_REF] Carretta | Na-O anticorrelation and HB*** -VIII. Proton-capture elements and metallicities in 17 globular clusters from UVES spectra[END_REF]. The type and nature of these polluter stars are still unknown and are actively debated in the community. Prantzos et al. [START_REF] Prantzos | Light nuclei in galactic globular clusters: constraints on the self-enrichment scenario from nucleosynthesis[END_REF] used the proto-cluster abundances as a starting point for nucleosynthesis calculations performed at constant temperature and density, and managed to reproduce the observed abundances in NGC 6752. They found that for producing the sodium enhancement, polluters must have burned hydrogen at temperatures around T = 75 MK. The ashes of this burning process were then mixed with pristine matter to produce the observed anti-correlations. The results put constrains on the polluters' nature, and it was suggested that AGB stars [START_REF] Prantzos | On the self-enrichment scenario of galactic globular clusters: constraints on the IMF[END_REF], massive and super-massive stars (up to 10 5 M ⊙ ) [START_REF] Denissenkov | Supermassive stars as a source of abundance anomalies of proton-capture elements in globular clusters[END_REF] could be good polluters candidates. Further scenarii have been suggested later (fast rotating massive stars [START_REF] Decressin | Fast rotating massive stars and the origin of the abundance patterns in galactic globular clusters[END_REF], classical novae (CN) [START_REF] Prantzos | Revisiting nucleosynthesis in globular clusters -The case of NGC 2808 and the role of He and K[END_REF] etc.), but all fail to explain all the observed features of multiple populations in globular clusters. The polluters' nature is still an open question.

Case of NGC 2419

While all globular clusters exhibit the O-Na anti-correlation, few clusters show an additional anti-correlation: magnesium depletion and potassium enhancement. This is observed in NGC 2808 and NGC 2419, the latter being one of the oldest and biggest globular cluster of the Milky Way (see figure 1.7. The chemical abundances observed in this cluster are shown in figure 1.8 for 10 elements displayed as a function of potassium abundance for red giant stars [START_REF] Mucciarelli | News from the Galactic suburbia: the chemical composition of the remote globular cluster NGC 2419[END_REF]. These stars have an average metallicity [Fe/H]= -2.09±0.02, with no variation from star to another. This eludes the Supernova IIa as a polluter candidate, as it would enhance both iron and potassium abundances in second generation stars. Furthermore, the abundances of Si, Ca, and Ti do not vary from star-to-star, which eliminates candidates undergoing helium burning. NGC 2419 must then have witnessed a self-pollution episode that caused the Mg-K anti-correlation, and this raises many interesting questions: what kind of polluter stars caused this anti-correlation? How is the Mg-K anti-correlation linked to the more common O-Na anti-correlation that is also observed in the NGC 2419 cluster?

Insight from Nuclear Physics

So far, little is known about the polluter stars in NGC 2419. Studies have suggested potential candidates such as AGB and super-AGB stars undergoing hydrogen hot bottom burning4 [START_REF] Ventura | SUPER-AGB-AGB EVOLUTION AND THE CHEMICAL INVENTORY IN NGC 2419[END_REF], fast rotating massive stars [START_REF] Decressin | Fast rotating massive stars and the origin of the abundance patterns in galactic globular clusters[END_REF], or even hypothetical stars such as supermassive starts with M ≥ 10 4 M ⊙ [START_REF] Denissenkov | Supermassive stars as a source of abundance anomalies of proton-capture elements in globular clusters[END_REF]. Iliadis et al. [START_REF] Iliadis | On Potassium and Other Abundance Anomalies of Red Giants in NGC 2419[END_REF] tackled the question in a more general way, using a single-stage, one-zone hydrogen-burning model, as it was done in the work of Prantzos et al. [START_REF] Prantzos | Light nuclei in galactic globular clusters: constraints on the self-enrichment scenario from nucleosynthesis[END_REF]. A Monte-Carlo procedure was used to calculate the processed abundances using a nucleosynthesis network where the temperature, density, fraction of hydrogen consumed, reaction rates and some initial abundances were varied. This procedure is presented in more details in chapter 5. Results from Iliadis et al. [START_REF] Iliadis | On Potassium and Other Abundance Anomalies of Red Giants in NGC 2419[END_REF] NGC 2419

Sensitivity study and key reactions

A sensitivity study was performed by Dermigny et al. [START_REF] Dermigny | Sensitivity to Thermonuclear Reaction Rates in Modeling the Abundance Anomalies of NGC 2419[END_REF] in order to identify the critical reactions that play a role in the over-abundance of potassium in NGC 2419. The 2373 reaction rates from STARLIB [START_REF] Sallaska | STARLIB: A Next-generation Reaction-rate Library for Nuclear Astrophysics[END_REF] were used in a Monte Carlo nucleosynthesis network calculations, where their uncertainties were allowed to vary. The study found that the spread of the temperature range of hydrogen burning strongly depends on the uncertainties on the reaction rates. Figure 1.10 (from Dermigny et al. [START_REF] Dermigny | Sensitivity to Thermonuclear Reaction Rates in Modeling the Abundance Anomalies of NGC 2419[END_REF]) shows the temperature distributions for fixed rates (in red) and varied rate (in blue), with density fixed at ρ = 100 g/cm 3 . The broadening δ is defined as the difference in temperature between the 16 th and 84 th percentiles of the distribution. In a second step, the rates were varied individually: each reaction was allowed to vary within its associated statistical uncertainties, while other reaction rates were fixed at their recommended values.

The procedure was repeated by varying each rate individually, and the width of the resulting temperature distribution δ, is compared to δ mr corresponding to all rates fixed. This was done for all the reaction rates, 2373 broadening values have been obtained, and are displayed in figure 1.11 where the x-axis of the histogram represents the fraction δ/δ mr . The uncertainty of few proton capture reaction rates have been found to contribute significantly to the broadening of the temperature range. 30 Si(p,γ) 31 P, 37 Ar(p,γ) 38 K and 38 Ar(p,γ) 39 K increase the temperature spread by 40%, 12% and 15%, respectively. The following dissertation will focus on the 30 Si(p,γ) 31 P reaction rate. 30 Si(p,γ) 31

Importance of the

P reaction

In order to understand the link between the magnesium and potassium abundances, the silicon nucleosynthesis at high temperatures (T > 100 MK) needs to be eluded first. Figure 1.12 shows the nuclear reaction network for hydrogen burning conditions for T = 160 MK. The 28 Si isotope is the most abundant one for T < 140 MK, its abundance increases smoothly via the proton capture 27 Al(p,γ) 28 Si reaction. At higher temperatures, the 28 Si destruction rate overcomes the production one, and successive proton captures followed by β + decays converts the 28 Si into 30 Si isotope. The amount of 30 Si formed is efficiently consumed via the 30 Si(p,γ) 31 P rate for T > 160 MK. Note that for the temperature range considered, 31 P is a bottleneck isotope for the nucleosyn- thesis network converting Mg into K. This highlights the importance of the 30 Si(p,γ) 31 P reaction rate on which this thesis is focused.

1.3.3 30 Si(p,γ) 31 P reaction rates for SN Ia

The 30 Si(p,γ) 31 P reaction rate is also found to be of importance for Type Ia Supernovae (SN Ia) nucleosynthesis. A sensitivity study, performed by Parikh et al. [START_REF] Parikh | The effects of variations in nuclear interactions on nucleosynthesis in thermonuclear supernovae[END_REF], identified the reaction rates to which different explosion models were sensitive. To do so, the authors varied the reaction rates by an arbitrary factor 10 (up and down) and recalculated the final chemical abundances with a nucleosynthesis network in explosive conditions. Figure 1.13 shows the results on the nucleosynthesis yields for species with masses between 10 ≤ A ≤ 50. The impact of the 30 Si(p,γ) 31 P reaction rate can be seen for elements with 30 ≤ A ≤ 40 where the abundances vary with a factor 2 to 8 difference. While the variation of the reaction rates with an arbitrary factor 10 allows the identification of the sensitive reactions, it is needed to investigate the existing uncertainties. The 30 Si(p,γ) 31 P reaction rates have already been measured experimentally as its uncertainties will be detailed in chapter 1.5, however, a precise determination of these rates around T ≈ 1 GK would still be appreciated.

Figure 1.13: Effects on the yield of some nuclei when varying the reaction rates by a factor of ten up and down in SN Ia calculations. The reaction plotted have effect on at least three species. The effect of the 30 Si(p,γ) 31 P reaction is highlighted in red. Figure adapted from [START_REF] Parikh | The effects of variations in nuclear interactions on nucleosynthesis in thermonuclear supernovae[END_REF].

Nuclear physics provides key elements for understanding astrophysical observations. It is used, for example, to estimate the energy generated during the fusion reactions, which drives the stellar evolution. Nuclear astrophysics also studies the nucleosynthesis processes that are responsible for the element production and their abundances observed in stars. In this section, I will present the physical concepts and quantities needed to explain the astrophysical observations. Those concepts are namely the nucleosynthesis networks, reaction cross-sections and reaction rates.

Nuclear reaction networks

The elements synthesized in stars are linked together through a nuclear network. The abundance of each isotope depends on the availability of some other isotopes, and on the temperature conditions that allow the transmutation of one nucleus to another. This entanglement can be described through a set of couples differential equations that have the form:

dN i dt = jk N j N k ⟨σv⟩ jk→i creates N i - m N i N m ⟨σv⟩ im destroys N i , (1.4)
where N i is the number of nuclei i per unit volume, and ⟨σv⟩ jk→i is the reaction rate of the interactions between particles j and k that produce the nucleus i, while i is also destroyed by the reaction between nuclei i and m with rate ⟨σv⟩ im . This equation can be generalized to any interaction: beta decay, photo-disintegration etc. But we will focus in the following on the thermonuclear reaction rate.

Thermonuclear reaction rates

Nuclear cross-sections describe the interaction between two interacting nuclei, it is a function, among many other parameters, of the relative energy between the nuclei. In stellar interiors, particles are not mono energetic, their energies are rather described by a statistical distribution. During quiescent burning, the star is in a state of thermodynamic equilibrium, and the kinetic energy of the particles in the stellar plasma is distributed according to the Maxwell-Boltzmann distribution. In such condition, it is convenient to use the thermonuclear reaction rate, which represents the convolution of the energy distribution and the nuclear cross-section. The reaction rate between nuclei 1.4 From Stars to Nuclei a and A is given by [START_REF] Iliadis | Nuclear Physics of Stars[END_REF]:

⟨σv⟩ aA = 8 πµ 1/2 1 (kT ) 3/2 ∞ 0 E σ(E)e -E/kT dE, (1.5)
where µ is the reduced mass of particles a and A, and E is their energy in the centre of mass system. k is the Boltzmann's constant, T the temperature of the stellar medium and σ(E) is the nuclear cross-section. The integrand of this equation shows the competition between two contributions: on one hand nuclear cross-section σ(E) which is very small at low energies, due to the centrifugal barrier and the Coulomb repulsion in the case of positively charged particles a and A, and on the other hand, the Maxwell-Bolzmann distribution which is maximal for E = kT . For a stellar temperature of T ≈ 100 MK, the kinetic energy is E = 9.5 keV. This energy is way below any Coulomb barrier, the smallest being the one between two protons with ∼ 600 keV.

Astrophysical S-factor and Gamow peak

Despite the height of the Coulomb barrier, quantum mechanics gives a finite probability for the reactions to occur in the stellar temperatures through the tunnel effect. The probability for a particle with ℓ = 0 to go through a Coulomb barrier, also called the Gamow factor, is given as [START_REF] Gamow | The Quantum Theory of the Atomic Nucleus[END_REF]:

P = e -2πη , (1.6)
where η is the Sommerfeld parameter defined as:

η = α µ 2E Z a Z A , (1.7)
with Z a and Z A the charge numbers of the interacting particles a and A, E their energy in the centre of mass system, and α the fine structure constant.

In order to disentangle between the Coulomb and nuclear forces, the cross-section can be written as [START_REF] Iliadis | Nuclear Physics of Stars[END_REF]:

σ(E) = S(E) E e -2πη , (1.8)
where S(E) is called the astrophysical S-factor and encapsulates the nuclear contributions to the cross-section, and the 1 E term describes the energy dependency of the geometrical cross-section. The S-factor span fewer order of magnitude compared to the cross-section, which makes the numerical computations more manageable. The reaction rate in Eq. (1.5) can be rewritten as [START_REF] Iliadis | Nuclear Physics of Stars[END_REF]:

⟨σv⟩ = 8 πµ 1/2 1 (kT ) 3/2 ∞ 0 S(E)e -2πη e -E/kT dE.
(1.9) e -E/kT The probability e -2πη e -E/kT , i.e. the product of the Gamow and Boltzmann factors is the so-called Gamow peak. It defines the energy region where the reaction rate is maximal. Figure 1.14 shows in purple the Gamow peaks at T = 100 MK and T = 200 MK for the radiative proton capture by 30 Si, while the dashed blue and red curves show the Boltzmann and the Gamow factors, respectively. We observe that the energy at the maximum of the Gamow peak and its width increase when the temperature increases. The maximum of the Gamow peak is found at the energy [START_REF] Iliadis | Nuclear Physics of Stars[END_REF]:

E 0 = π ℏ 2 (Z a Z A e 2 ) 2 µ 2 (kT ) 2 1/3 , (1.10)
and the corresponding width is [START_REF] Iliadis | Nuclear Physics of Stars[END_REF]:

∆ = 4 √ 3 E 0 kT . (1.11)
Thermonuclear reactions occur mainly within the Gamow window defined between E 0 -∆/2 and E 0 + ∆/2.

Resonant and non-resonant reaction rates

There are two main energy dependencies for the S-factor: smooth variations for nonresonant processes, as opposed to resonant ones where the S-factor varies dramatically around given energies called resonance energies. The expressions for the reaction rate for each case is now detailed.

Non-resonant reaction rate

For non-resonant reactions between charges particles, the S-factor varies smoothly with the energy. In most cases, it can be expanded into a Taylor series around E = 0 as [START_REF] Iliadis | Nuclear Physics of Stars[END_REF]: .12) this form facilitates the integration of the S-factor for reaction rates calculations.

S(E) ≈ S(0) + S ′ (0)E + 1 2 S ′′ (0)E 2 , ( 1
A common example of non-resonant process is the direct capture, which is an electromagnetic process where the capture occurs on bound states of the final nucleus in a one-step process without forming of a compound nucleus in an intermediate state. Experimentally, the cross-section of the direct capture is obtained by measuring the excitation function of the reaction, i.e., measuring the reaction yield as a function of the incident beam energy. When such measurements are not available, the direct capture cross-section can be calculated by summing over all final bound states of the residual nucleus, such as [START_REF] Iliadis | Spectroscopic factors from direct proton capture[END_REF]:

σ DC total = ℓ i ,ℓ j C 2 S(ℓ f )σ DC theory (ℓ i , ℓ j ), (1.13)
where C 2 S is the spectroscopic factor of the final bound state, and it shall be explained in great details in chapter 3, and ℓ the relative orbital angular momentum. The theoretical cross-section σ DC theory is calculated with a single-particle potential model. The non-resonant S-factor is then calculated as:

S(E) = σ DC total (E) E e 2πη . (1.14)
These formulae are used to calculate the S-factor for different bombarding energies and then fit the values with the expression (1.12).

To the zero order, one can assume that the S-factor is constant over the Gamow window (S(E) ≈ S(E 0 )), the corresponding reaction rate is:

⟨σv⟩ DC = 8 µπ 1/2 1 (kT ) 3/2 S(E 0 ) π/2 (1.15)

Resonant reaction rate

The reaction cross-section may present sharp maxima at certain energies, these resonance energies E r correspond to unbound levels in the formed compound nucleus with excited energies:

E x = E r + Q, (1.16)
where Q is the reaction Q-value. Resonances are parametrized by their width Γ, which is related to the half life of the resonance as Γ = ℏ/τ . The cross-section for an isolated5 resonance is described by the Breit-Wigner formula [START_REF] Iliadis | Nuclear Physics of Stars[END_REF]:

σ BW (E) = λ 2 4π (2J + 1) (2j a + 1)(2J A + 1) Γ i Γ j ((E r -E) 2 + Γ 2 /4 , (1.17)
where λ is the de Broglie wavelength, J the spin of the resonance state, j a and J A are the spins of the interacting nuclei, the partial widths Γ i and Γ j are the decay widths for the incoming and outgoing channels, and the sum over partial widths gives the total width Γ. For a (p,γ) reaction, the partial widths are simply Γ p and Γ γ . When the width is sufficiently small (Γ ≪ E r ) the resonance is said to be narrow and the partial widths are nearly constant in the region E r ± Γ/2. By replacing the cross-section from Eq. (1.17) into the expression for the reaction rate in Eq. (1.5), and with Γ i and Γ j constant with respect to the energy, one obtains:

⟨σv⟩ narrow res = √ 2πℏ 2 (µkT ) 3/2 e -Er/kT (2J + 1) (2j a + 1)(2J A + 1) Γ i Γ j Γ ∞ 0 Γ/2 (E r -E) 2 + Γ 2 /4 dE.
(1.18) The integral in Eq. (1.18) can be resolved analytically and the reaction rate for a narrow and isolated resonance is given as [START_REF] Iliadis | Nuclear Physics of Stars[END_REF]:

⟨σv⟩ narrow res = 2π µkT 3/2
ℏ 2 e -Er/kT ωγ, (1.19) where ωγ is the resonance strength defined as:

ωγ = (2J + 1) (2j a + 1)(2J A + 1) Γ i Γ j Γ .
(1.20)

1.5 Current knowledge of the 30 Si(p,γ) 31

P reaction

The resonant properties of the 30 Si+p system are needed to calculate the 30 Si(p,γ) 31 P reaction rate. For the temperature regions of interest for Globular Cluster's polluters, the corresponding Gamow windows are E 0 = 150 ± 83 keV for T 9 = 0.16 and E 0 = 240 ± 148 keV for T 9 = 0.2. For Type Ia Supernovae, the relevant temperature is T 9 = 1, corresponding to Gamow window of E 0 = 700 ± 570 keV. The interesting resonance energy range is thus E c.m. r = 60 -1200 keV above the proton separation threshold at S p = 7296.55 keV, corresponding to excited states in the 31 P between E x = 7350 and 8450 keV. The structural information of these excited states are helpful for the evaluation of the 30 Si(p,γ) 31 P reaction rate. For these excitation energies, only two decay modes are possible, proton and γ-ray emission.

Figure 1.15 shows the level scheme for 31 P for the region of interest. The excitation energies, J π and ℓ values are adopted from the NNDC compilation [31]. The reaction rate has been estimated with data prior to the present work [START_REF] Dermigny | Thermonuclear reaction rate of 30 Si(p ,γ )[END_REF], and the contribution to the total reaction rate is shown in figure 1. [START_REF] Prantzos | Light nuclei in galactic globular clusters: constraints on the self-enrichment scenario from nucleosynthesis[END_REF]. This figure highlights the main resonances contributing to the 30 Si(p,γ) 31 P rate, and we shall focus on these resonances in the following discussion.

Direct measurements for resonances below E c.m. r = 400 keV are tedious due to the high Coulomb barrier. The proton channel is therefore suppressed and the γ-decay mode dominates. The resonance strength given in Eq. (1.20) becomes:

ωγ = ωΓ p , assuming Γ p ≪ Γ γ .
(1.21)

The proton widths were previously calculated using the spectroscopic factors determined by Vernotte et al. [START_REF] Vernotte | Si( 3 He,d) 31 P reaction at 25[END_REF] through the transfer reaction 30 Si( 3 He,d) 31 P (see section 1.6.1). However, the energy resolution achieved did not allow the separation of doublet states that lie within the region of interest, mainly the doublets at E x = 7719-7737 keV and E x = 7898 -7911 keV. Furthermore, contamination peaks were hiding low populated states at E x = 7346, 7446 and 7465 keV that correspond to resonance energies of E c.m. r = 51, 146 and 171 keV, respectively. For these resonances, an upper limit for the spectroscopic factor was taken, as C 2 S ≤ 1 for E r = 51 and 146 and as C 2 S ≤ 0.003 for E r = 171 keV. Details on this procedure can be found in chapter 5. These upper limits can be seen as the hashed areas on the figure 1 In that work, the measurements are relative to the weighted average of the previous measurements of Lyons et al. [START_REF] Lyons | Total yield measurement in 27 Al(p,γ) 28 Si[END_REF] and those of Pain and Sargood [START_REF] Paine | p,γ) resonance strengths in the s-d shell[END_REF]; this value is represented with a green point in figure 1.17. Note that the strength of the resonance at E c.m. r = 422 keV was measured once and ωγ 422 = (1.14 ± 0.25) × 10 -4 eV, while the one at E c.m. r = 555 keV has never been measured directly. Resonances at higher energies are relevant for the nucleosynthesis processes within In this work, we propose a remeasurement of the resonances indicated in figures 1.16 and 1.17, including the reference resonance at E c.m. r = 602 keV, in order to improve the precision of the 30 Si(p,γ) 31 P reaction rate. The experimental strategies for doing so are presented in the following subsection.

wg (eV)

Figure 1.17: Direct measurements of the 30 Si +p resonance strengths (ωγ). In the work of Pain and Sargood [START_REF] Paine | p,γ) resonance strengths in the s-d shell[END_REF], the strength of the E c.m. r = 602 keV resonance was measured in three different ways: absolute measurement using thick and thin targets and relative measurement with respect to the E p = 632 keV in 27 Al(p,γ) 28 Si. The green data point is the strength adopted by Dermigny et al. for the E c.m. r = 602 keV resonance, and is the weighted average of values from Pain and Sargood [START_REF] Paine | p,γ) resonance strengths in the s-d shell[END_REF] and from Lyons et al. [START_REF] Lyons | Total yield measurement in 27 Al(p,γ) 28 Si[END_REF]. Values from Wolff et al. [START_REF] Wolff | Spin determinations of 31P levels from the 30Si(p, ) reaction[END_REF] are relative measurement normalized with respect to the value of the ωγ 602 from Smith and Endt [START_REF] Smith | Resonant Absorption of Gamma Radiation from the Si 30 (p, γ)P 31 Reaction[END_REF], renormalization with respect to the adopted value of Dermigny et al. is shown in blue points.

Experimental approach

There are different approaches to determine a resonance strength, depending on the centre-of mass resonance energy. In the present work, the 30 Si(p,γ) 31 P resonant strengths were determined using two different approaches, the 30 Si( 3 He,d) 31 P transfer reaction, and the radiative direct capture with a thick target.

Transfer reactions

For low energy resonances with E c.m. r ≤ 400 keV, the resonance strength needs to be estimated through an indirect method. We take advantage of the situation described by equation (1.21) which shows that the strength is proportional to the proton partial width. The proton width can be expressed as [START_REF] Iliadis | Nuclear Physics of Stars[END_REF]:

Γ p = C 2 S p Γ s.p , (1.22)
where C 2 S p is the proton spectroscopic factor, and Γ s.p is the single particle proton width. The spectroscopic factor expresses to which extent the state can be described by a single particle configuration, i.e. in the case of the levels in 31 P how close the configuration is to a 30 Si+p system. C 2 S = 1 being the case of a purely single particle configuration. This spectroscopic factor can be determined with a proton transfer reaction. The 30 Si( 3 He,d) 31 P reaction has been studied for this purpose at the Meier Leibniz Laboratory (MLL) with the Q3D spectrometer [START_REF] Enge | Proc. 3rd Int. Conf. On Magnet Technology[END_REF]. The spectroscopic information for 31 P levels between E x = 6820 and 8105 keV have been extracted. The experiment, data analysis and the subsequent results are detailed in chapter 2. The theory underlying the transfer reaction and the interpretation of the measurements to obtain the proton widths are given in great details in chapter 3.

Direct measurements

Direct measurement of the resonance strengths were also performed in the scope of this thesis for resonances between E c.m. r = 485 keV and E c.m. r = 950 keV. Measuring the reaction cross-section in the laboratory is equivalent to the determination of the reaction yield, which is the number of reactions N R that occur for a given number of incident beam particles N b :

Y = N R N b . (1.23)
The yield is related to the cross-section σ and the areal density of the target as [START_REF] Iliadis | Nuclear Physics of Stars[END_REF]:

Y = σn∆x, (1.24)
where n and ∆x are the density and the thickness of the target, respectively.

In such experiments, the beam loses energy in a thick target material. The crosssection therefore depends on the depth where the interaction occurs. The total yield is given by the integral over the target thickness L:

Y = L 0 σ(x)n(x)dx.
(1.25)

The previous equation can be given as a function of the energy:

Y = E b -∆E E b σ(x)n(x)dx dx dE(x) dE(x) dx , (1.26)
where ∆E is the total energy loss through the thickness L, and E b is the initial beam energy.

The stopping power ε(E) is defined as:

ε(E) = - 1 n(x) dE dx , (1.27)
then the yield becomes:

Y = E b E-∆E σ(E) ε(E) dE. (1.28)
In the case of a thick target, i.e. Γ ≪ ∆E, and using the Breit-Wigner parametrization for the cross-section, the yield can be expressed as:

Y = λ 2 r 2 m a + m A m A ωγ ε r , (1.29)
where m a and m A are the masses of the beam and the target, respectively; they appear in the equation because it is expressed in the centre-of-mass system. λ r and ε r are the de Broglie wavelength and the stopping power, respectively calculated at the resonance energy E r . This expression provides the link between the resonance strength ωγ and the measured yield. The strengths of the 30 Si +p resonances have been studied using the DRAGON recoil spectrometer [START_REF] Hutcheon | Measurement of radiative capture resonance energies with an extended gas target[END_REF] at TRIUMF. The experiment, data analysis and the subsequent results are detailed in chapter 4. Proton spectroscopic information in the 31 P nucleus can be extracted from the study of one-proton transfer reaction. 30 Si( 3 He,d) 31 P is a more suitable reaction for this purpose than 30 Si(d,n) 31 P, for example, since it consists in detecting light charged particles, which is experimentally more accessible than detecting neutrons. The 30 Si( 3 He,d) 31 P transfer reaction has already been performed at Orsay using the Split-Pole spectrometer by Vernotte et al. [START_REF] Vernotte | Si( 3 He,d) 31 P reaction at 25[END_REF], but the study failed to observe some excited levels that are of astrophysical interest, mainly the low populated levels at E x = 7446 and 7470 keV, along with the unresolved doublet at E x = 7719 -7737 keV.

The 30 Si( 3 He,d) 31 P reaction was performed in May 2019 at the Maier-Leibnitz Laboratorium in Munich, Germany. A 3 He beam was accelerated to an energy of 25 MeV with a 14 MV Tandem, and the reaction products were momentum analysed with the high resolution Q3D magnetic spectrometer.

In this chapter, the experimental set-up used to study the 30 Si( 3 He,d) 31 P reaction is presented and the methods for calibration and normalization are detailed. Then the excitation energies and angular distributions obtained are presented and discussed.

Experimental set-up 2.1.1 Beam preparation

The 3 He beam was produced at the MLL tandem facility. First, an atomic gas is injected through the Electron Cyclotron Resonance (ECR) ionizer, where atoms become positively charged by electron impact with an efficiency of few percent [START_REF] Hertenberger | The Stern-Gerlach polarized ion source for the Munich MP-tandem laboratory, a bright source for unpolarized hydrogen and helium ion beams as well[END_REF]. The positive ions are then transported by an electrostatic field to a charge exchanger that consists of a Caesium vapour jet. The ions successively pickup two electrons and become negatively charged. The MLL ion source can produce up to 10 µA of unpolarized 3,4 He -ions [START_REF] Hertenberger | The Stern-Gerlach polarized ion source for the Munich MP-tandem laboratory, a bright source for unpolarized hydrogen and helium ion beams as well[END_REF], but for safety reasons related to radiation build-up, the intensity is restricted to 1 µA.

The ions are extracted from the source with a tension V pre-acceleration and injected into the Tandem. The latter is a Van de Graaf electrostatic accelerator with a terminal voltage at its centre that can reach 14 MV [START_REF] Van De Graaff | Tandem Electrostatic Accelerators[END_REF]. Figure 2.1 shows its working principle: the negatively monocharged ions are accelerated toward the centre of the Tandem, where they go through a thin carbon foil that fully strips the ions from their electrons. The ions become positively charged and continue to be accelerated under the effect of the same potential difference between the terminal voltage in the centre and the ground voltage downstream the accelerator. The ion beams are accelerated to an energy of:

E = eV pre-acceleration + e(1 + q)V T andem (2.1)
where e is the elementary charge and q the final charge state of the accelerated ions. The beam is deflected by a 90°analyzing magnet, which is used to select the desired beam energy. A feedback loop measures the beam current after the analysing magnet, if the currents are different between the right and left slits, feedback is sent to the tandem, and its terminal voltage is automatically adjusted, and the energy resolution achieved is ∆E/E ≤ 10 -4 . The beam is then directed to the different experimental areas with a switching magnet followed by a set of magnetic dipoles and quadrupoles. In the experiment presented here, the fully stripped 3 He 2+ ions were accelerated to an energy of 25 MeV with a terminal voltage of 8.24 MV. The intensity delivered to the target chamber of the Q3D was between 150 and 200 enA.

Reaction chamber and targets

A target holder is placed inside the vacuum sealed reaction chamber. In this experiment, we used a SiO 2 target enriched by 95% in 30 Si with areal thickness of 16 µg/cm 2 on a 40 µg/cm 2 carbon backing. In addition, natural silicon oxide and natural carbon targets, with areal thicknesses of 0.8 and 20.5 µg/cm 2 , respectively, were used to identify any contaminant reaction and to characterize the background.

The target thickness was determined using the Rutherford Back Scattering (RBS) technique at the JANNuS/SCALP facility [START_REF] Bacri | SCALP, a platform dedicated to material modifications and characterization under ion beam[END_REF] (IJCLab, France) and at the Triangle Universities Nuclear Laboratory (Durham, North Carolina, USA), where the targets were manufactured. The analysis of target specifications will be presented in Section 2.2.1.

The accumulated incident beam charge, needed for data normalization, is measured with a Faraday cup located within the reaction chamber at 0°downstream of the target (figure 2.2). The Q3D is a high-resolution spectrometer that consists of one magnetic quadrupole and three dipoles magnets [START_REF] Enge | Proc. 3rd Int. Conf. On Magnet Technology[END_REF], as shown in figure 2.2. The quadrupole focuses the reaction products within the range of the focal plane, while the dipoles separates the ejected particles according to their magnetic rigidity. The core principle for a spectrometer is to measure the momenta of particles with charge q based on their magnetic rigidity Bρ, where B is the magnetic field and ρ is the curvature radius. The relation between magnetic rigidity and momentum is:

Q3D magnetic spectrometer

Bρ = p q (2.2)
The curvature radius ρ is related to the position of the incident particle in the focal plane detection system. For the Q3D spectrometer, the curvature radius spans between ρ min = 97 and ρ max = 102 cm. Since the momentum bite pmax p min = ρmax ρ min = 1.05 of the Q3D is rather small, the excitation energy range covered by a single magnetic field value is limited. Several magnetic field exposures are then necessary in order to investigate broad excitation energy ranges. Various field gradients such as hexapole, octupole and decapole fields can be generated by a corrective element placed between the first and second dipoles. These fields are perpendicular to the beam direction and are used to correct the kinematic effects induced by the large angular acceptance of the Q3D. The corrective elements provide the Q3D a high energy resolution of ∆E/E ≤ 1/5000 at full solid angle of 13 msr.

Detection system at the focal plane

A sectional schematic view of the focal plane detection system is shown in Figure 2.3. The latter is placed at an angle between 40°and 50°with respect to the incident particles. The entrance is a 25 µm Kapton entrance foil [START_REF] Hertenberger | A light ion focal plane detector for the Q3D magnetic spectrograph with periodic cathode readout[END_REF][START_REF] Wirth | New q3d focal plane detector with cathode-strip readout became operational[END_REF], which is followed by two multi-wire proportional counters filled with 500 mbar of isobutane gas. The gas is ionized by incident charged particles, and free electrons are generated along the particle's path. The resulting avalanche generates in the anode wire a signal proportional to the energy loss of the particle in the first counter ∆E 1 . The avalanche generates as well a positive charge distribution near the cathode foil. The second proportional counter has two anode wires and the sum of both their signals is ∆E, but they can also be read separately to check the correct position of the detector. Furthermore, in the second proportional counter, the cathode foil consists in 255 strips of 3 mm width, electrically isolated with 0.5 mm spacing between each strip. The charge distribution generated by the avalanche has a Gaussian shape, and its mean corresponds to the position where the incident particle crossed the anode wire. This allows the determination of the position of the incident particle on the focal plane with a resolution better than 0.5 mm (FWHM) [START_REF] Wirth | New q3d focal plane detector with cathode-strip readout became operational[END_REF]. Figure 2.4 shows schematically the working principle of the proportional counters with the cathode strips foil.

The two proportional counters are followed by a 7 mm thick plastic scintillator where the particles are stopped. The energy of the particles is converted to scintillation light, which is collected and amplified by four photomultiplier tubes (PMT). The resulting signal is proportional to the residual energy E res . Comparing the energy losses of the two proportional counters, ∆E 1 and ∆E, along with the residual energy E res , provides a powerful mean for particle identification.

Data acquisition

The data acquisition system is triggered by signals in both multi-wire proportional counters and in the scintillator. The charge collected in these three detectors are The charge distribution can be approximated by a Gaussian description (dashed line) [START_REF] Wirth | New q3d focal plane detector with cathode-strip readout became operational[END_REF]. recorded, along with the charge collected on each individual cathode strip of the second proportional counter. For each run, two scalers are recorded, named scaler1 and scaler3. Scaler1 is connected to the Faraday cup downstream the target and counts the incident beam charge, a pulse is generated after a certain amount of charge is accumulated. Scaler3 is incremented whenever the DAQ is busy, which gives a measure for the dead time of the acquisition. These quantities are used for the normalization of the data (see section 2.4.1).

When the DAQ is triggered, the cathode wire is read out. If a cathode signal is above the threshold, the readout module scan adjacent strips. The number of neighbouring strips with signal above threshold has to be between 3 and 7 so that the signal is considered as a valid one. The position information is then deduced from the charge distribution, as shown in the bottom part of figure 2.4. Each cathode strip has its own preamplifier, pulse shaper and Peak-Hold/Discriminator, and the signals are preprocessed within the Q3D hall for a fast treatment of the coincidence signals. Only the valid events are sent to the acquisition system in the control room.

Data Analysis

Target thickness

The targets used in the present experiment were analysed in order to determine their thickness and chemical composition. These data are crucial for the normalization of the differential cross-section discussed in section 2.4.1.

a/ Method

The technique used for target characterization is the Rutherford Backscattering Spectrometry (RBS) [START_REF] Rubin | Chemical Analysis of Surfaces by Nuclear Methods[END_REF], it is a technique based on the accuracy with which Rutherford cross-sections are known and described analytically. The technique consists in bombarding the target with an ion beam, such as protons or α-particles, and measuring the scattered ions at backward angles, where the mass separation is maximum. The incident energy is chosen to be below the Coulomb barrier so that the cross-sections are dominated by the Rutherford scattering.

b/ Measurements and data analysis

The RBS measurements to characterize the nat SiO 2 , 30 SiO 2 and nat C targets were performed with a 1.5 MeV 4 He + beam, delivered by the electrostatic accelerator ARAMIS at IJCLab [START_REF] Cottereau | ARAMIS: an accelerator for research on astrophysics, microanalysis and implantation in solids[END_REF]. The beam is normal to the target and the scattered α-particles are detected at θ lab = 165°in a circular silicon detector with a 25 mm 2 active area situated at 8 cm from the target, thus covering a solid angle of 3.9 msr. The cumulated charge is measured with a Faraday-cup at 0°. Data were analysed with the simnra software [START_REF] Mayer | SIMNRA, a simulation program for the analysis of NRA, RBS and ERDA[END_REF] which calculates the expected number and energies of detected α-particles according to elemental thicknesses and stoichiometries. The program fits the calculated spectra to the experimental data points in order to deduce the thickness of the layers of the target. Figure 2.5 shows the experimental 4 He + spectrum in red points and its best fit, obtained for the enriched oxide silicone target. The peaks, from lowest to highest energies, correspond to increasing mass elements, namely nat C, 16 O and 30 Si. The shape and width of the peaks are related to the thickness of the layer where the element is present. For thin layers (< 20 µg/cm²) the peak shape is close to a gaussian response of the silicon detector, but for thicker layers, the peak starts to have a flat-top shape, as we can see for the 12 C peak, used for the backing of the target. The isotopes fractions contribute to the shape of the peak: some shouldering at higher (respectively lower) indicates the presence of a heavier (respectively lighter) isotope than the most abundant one in the sample, for example 13 C represents less than 2% of the nat C, and it can be seen around channels 250 -300 on the spectrum. The fit of the Si peak corresponds to an enriched sample with (95±2)% of 30 Si, the 28 Si is observed as a small peak between channels 650 and 700.

c/ Results

Table 2.1 summarizes the thickness and stoichiometries for each target used. The obtained thickness for the 30 SiO 2 target is 16.5 µg/cm 2 , and it corresponds to the thickness estimated with an independent RBS analysis using α-particles at 2 MeV performed by the target manufacturing laboratory at the Triangle Universities Nuclear Laboratory (Durham, North Carolina, USA). The thicknesses of the carbon backing and the nat C target also correspond to the values provided by the manufacturing laboratory. The natural silicon oxide target contains a very small number of SiO 2 molecules. A visual inspection shows that the target was degraded during transportation. These results confirm the online spectra obtained during the experiment, where the counting rates were extremely low when we used this target.

Target N (10

Particle Identification

Several exit channels are energetically available for the same reacting particles, and hence different reaction products having similar magnetic rigidity can be detected in the focal plane detectors. Figure 2.6 shows the expected positions on the focal plane for different reaction products corresponding to allowed reaction channels with positive Qvalues (see appendix A for kinematic calculations). The reactions can occur on different nuclei present in the target. We see that the tritons from the channel 30 Si( 3 He, 3 H) 30 P are expected to overlap the deuterons corresponding to populated levels of interest in 31 P. Hence, a particle identification is needed and is achieved by combining the physical information measured in the different detectors of the focal plane presented in section 2.1.4. Note that the elastic scattering channel ( 3 He, 3 He) and the one proton pick-up channel ( 3 He, α) have negative Q-values for the energy region of interest. Figure 2.6 shows that only deuterons and tritons are expected to be detected. 30 Si along with other nuclei present in the target.

a/ Method

A charged particle of mass A and charge Z, with energy E loses ∆E in a given material according to the Beth-Bloch formula [START_REF] Segre | Experimental nuclear physics[END_REF]:

∆E ∝ AZ 2 ln(E) E . (2.3)
This formula can be expressed as a function of the magnetic rigidity: considering the equation (2.2) and the non-relativistic expression for the energy : E = 1/2mv 2 , the energy loss can be expressed as:

∆E ∝ A 2 (Bρ) 2 ln( Z 2 A (Bρ) 2 ), (2.4)
according to this expression, two incident particles with the same magnetic rigidity but different masses, are better distinguished according to their energy loss, since it scales with A 2 . In other words, the energy loss is more sensitive to the mass of the incident particle than to its charge. It will lose more energy in the proportional counter and less in the scintillator (see Section 2.1.4). The 2-dimensional (2D) spectra (E, ∆E) exhibits distinct loci for particles with different masses and charges.

b/ Results

Figure 2.7 shows the 2D spectra used for particle identification in our experiment. In addition to the (E res , ∆E) spectrum, the 2D spectrum (∆E, ∆E 1 ) provides an additional mean of separation between particles, and disambiguate between any particles with different energies and charges that would lose the same energy in one proportional counter. In the latter, we clearly identify two loci, the highest energy loss corresponds to tritons, as expected from Eq. (2.3), both groups have magnetic rigidities distributed along the whole focal plane. The (E res , ∆E) spectrum shows as well a separation between the deuterons and tritons, but we can also see that some particles with the same energy loss in the proportional counter, deposit a smaller energy in the plastic scintillator (zone C of figure 2.7, bottom panel). These events correspond to a bad collection of the scintillation photons and represent less than 0.5% of the detected deuterons. They are thus excluded from selected events. Event selection is checked for each run at all the measured angles. Statistical error due to events' selection is less than 1%.

Deuteron magnetic rigidity spectra

Figure 2.8 shows the magnetic rigidity deuteron spectra in analogue-to-digital converter (ADC) units, for the events detected in the focal plane position detection system, and selected according to the previous identification cuts. Measurements have been done at θ Q3D = 6°, 10°, 12°, 16°, 20°, 23°and 32°. The narrow peaks are associated to 31 P states and their corresponding excitation energies are labelled vertically in black.

The spectra contain broad peaks corresponding to levels produced by proton transfer to isotopes present in the target, apart from the 30 Si nuclei, such as levels of 13 N induced by reaction on 12 C, and 17 F induced by reaction on 16 O. These peaks, labelled in green in the figure, are much broader than deuteron peaks associated to 31 P states, due to the fact that the multipole fields are tuned for a A = 31 mass.

Indeed, due to the angular acceptance of the Q3D, particles entering the spectrometer with extreme angles θ + ∆θ, have different trajectories than particles with central incident angles. The trajectories converge on a different focal plane than the expected position, and the difference in the position of the focal plane is called the kinematic displacement. For the Q3D spectrometer, the focal plane detection system is not movable, the kinematic displacement is corrected with the multipole fields. Furthermore, the trajectories of the particles within the spectrometer depend on their kinematics, thus reaction products with masses different from A = 31 would have different values of the kinematic displacement that are not compensated.

Reaction kinematics can also be investigated by comparing the magnetic rigidity of the peaks for different detection angles. The relation between the particle's momentum and its emission angle is given in appendix A. The magnetic field is usually chosen in order to cover the same excitation energies in 31 P for all angles. Consequently, the deuteron peaks associated to 31 P levels will be at the same position (in ADC channels) at all angles, while the peaks associated to other reactions will have different positions on the focal plane from an angle to another. Figure 2.9 shows the expected peak positions, as a function of the detection angle for different levels populated by proton transfer reaction to nuclei present in the target, along with the measured positions on the spectra. Levels of 13 N, 17 F and 29 P are induced by reactions on 12 C, 16 O and 28 Si, respectively. The expected positions are calculated based on the magnetic rigidity calibration of the focal plane.

a/ Lineshape analysis and spectral resolution

A peak description is needed for the calibration of the focal plane position detector, and for extracting the number of counts for cross-section determination. The peaks are hence fitted with an analytical function. For deuteron peaks associated with 31 P Other peaks, with green labels, correspond to levels populated from contaminating nuclei present in the target, mainly 12 C, 14 N, 16 O and 28 Si. Levels with asterisk on the energies are used for the focal plane position detectors calibration.

states, the lineshape function used is a skewed Gaussian obtained from the convolution of a Gaussian with an exponential tail at low energies. The skewed shape of the peak results from optical aberrations induced by magnetic multipole fields. The analytical Figure 2.9: Expected (solid lines) and measured (black dots) peak positions on the focal plane detection system for states apart from those in 31 P. Levels of 13 N, 17 F and 29 P are induced by reactions on 12 C, 16 O and 28 Si, respectively. expression of the function used is [START_REF] Benamara | Étude des réactions 26 Al(n,p) 26 Mg et 26 Al(n,α) 23 Na par diffusion inélastique pour application à la nucléosynthèse de l' 26 Al dans les étoiles massives[END_REF]:

f (x) = A exp x -x 0 λ erfc x -x 0 σ + σ 2λ , (2.5)
where A, x 0 , σ and λ are the peak amplitude, the mean, the width of the Gaussian, and the exponential tail parameter, respectively. The function erfc(x) is the complementary error function. The integral of f (x) is given by:

I = 2Aλ exp -σ 2 4λ 2 , (2.6)
and the analytical calculation is detailed in appendix B. Figure 2.10 shows the fit function applied to a single 31 P peak.

Note that this lineshape function can be used for deuteron peaks associated to states in nuclei with masses close to A = 31 such as 29 P states induced from ( 3 He,d ) reaction on 28 Si contamination, since the kinematic displacement are close in value. The σ and λ parameters for deuteron peaks corresponding to 29 P states are similar (less than 30% difference) to the parameters for peaks associated to 31 P states. Gaussian as described by Eq. (2.5), x 0 , σ and λ are in ADC units.

The width σ and the exponential tail parameter λ are related to the experimental resolution and the optical aberrations. They are thus fixed to be common to all peaks from the same nucleus, given that the levels do not have a natural width. The contribution of the broad contamination peaks is modelled by an exponential tail whenever they are close to a peak associated to 31 P state.

The contributions from each deuteron peak are summed into a unique function, and the whole spectrum is fitted with the resulting function. The fit results for the whole spectrum are shown in figure 2.11 for the detection angle θ Q3D = 12°.

b/ Focal plane position detector calibration

The centroids (in ADC channels) of all populated levels are determined from the fit of the spectra over the whole focal plane. Several populated levels of 31 P in the region of astrophysical interest have energies known with a good precision, better than 4 keV, thanks to previous studies of the nucleus [31], and particularly through the same 30 Si( 3 He,d) 31 P reaction measurements of Vernotte et al. [START_REF] Vernotte | Si( 3 He,d) 31 P reaction at 25[END_REF]. Furthermore, several peaks are well populated and isolated, which makes them good candidates for a calibration of the focal plane. The levels used for calibration are marked with an asterisk, in table 2.2 and in figure 2.11 as well. This auto-calibration has the advantage of not requiring another reaction for calibration. The deuteron magnetic rigidities of the reference levels are calculated from their excitation energies and for each measured angle with use of two-body kinematics (see appendix A). The corresponding curvature radii, ρ, are obtained by dividing the magnetic rigidities by the value of the magnetic field B, set for each measurement.

The focal plane is calibrated with a second order polynomial. Indeed, the curvature of the focal plane induces a non-linearity between the rigidities and positions in the focal plane. The relation between the curvature radius ρ and the positions on the focal plane detection system in ADC channels is:

ρ = p 2 ch 2 + p 1 ch + p 0 (2.7)
with ch the channel in ADC units.

In the magnetic rigidity calculations for the levels used for calibration, the energies of the deuterons are corrected with the energy losses in the target. We make the assumption that the interaction occurs at the middle of the target, which is a good approximation since layers of the target are very thin (≤ 50 µg/cm 2 ). First, 3 He beam is slowed down to the interaction point, the reaction occurs at the energy E( 3 He)-E loss1 . For the 16.5 µg/cm 2 thick 30 SiO 2 target tilted at -13°, E loss1 = 1.4 keV. The deuterons produced in the reaction are then slowed down in the other half of the target and in the carbon backing before entering the Q3D with the energy E( 2 H)-E loss2 . The measured magnetic rigidity corresponds to a smaller deuteron energy than the one corresponding to the excitation energy of the level populated (1.97 to 2.13 keV smaller, the smallest difference being at 12°since the target is tilted at -13°). The energy losses of the 3 He and 2 H ions are calculated using the Stopping and Range of Ions in Matter (srim) program [START_REF] Ziegler | SRIM -The stopping and range of ions in matter[END_REF]. The target densities used in the calculations have been estimated in Section 2.2.1.

Figure 2.12 shows the curvature radii as a function of the ADC channels, and the corresponding fit, along with the residuals centred around zero, as a way to represent the degree of misfit, which is less than 4 keV for all the measured angles. This value is less than the experimental energy resolution which is between 4 and 8 keV, for detection angles of 6°and 32°respectively. Curvature radius ρ for detected deuterons as a function of the ADC channel and its fit with a second order polynomial (eq. 2.7). Bottom panel: Calibration residuals expressed in excitation energy difference between expected energy and value calculated with the calibration fit.

Excitation energies

The magnetic rigidity (Bρ) values obtained from the calibration of the focal plane position detector are converted to excitation energies. Equation (2.2), kinematic relations (appendix A) and energy losses in the targets are once more used to extract the excitation energies E x of the levels populated in the 31 P nucleus.

Excitation energies uncertainties

The uncertainty associated to the determined excitation energies includes errors arising from the various calculations steps. For a function F that depends on the correlated variables x i , the variance of F is given by:

σ 2 F = i j ∂F ∂x i ∂F ∂x j σ x i ,x j , (2.8)
where σ x i ,x j are the covariance matrix elements of the fit. Two effects should be considered for the error propagation: (i) the calibration function (Eq. 2.7), and (ii) the peak fitting (Eq. 2.5). The uncertainty on the magnetic rigidity is hence given by:

∆(Bρ) = B σ 2 p 0 + ch 2 σ 2 p 1 + ch 4 σ 2 p 2 + chσ p 0 ,p 1 + ch 2 σ p 0 ,p 2 + ch 3 σ p 1 ,p 2
error from calibration function

+ (p 1 + 2p 2 ch) 2 σ 2 ch
error from the spectrum fit (2.9) where σ p i ,p j are the elements from the covariance matrix for the calibration parameters p i . σ ch is the elements from the covariance matrix of the spectrum fit, and corresponds to the error on parameter x 0 in Eq. (2.5).

Error from the calibration on the focal plane position detector dominates by more than 95% the uncertainties on the magnetic rigidity. These calibration errors have maximal values (0.05% of Bρ) at the extremities of the focal plane, and at the centre of the latter. This reflects the lack of well populated and isolated levels suitable for calibration in these regions.

The uncertainty on the excitation energy E x depends on the error on magnetic rigidity ∆(Bρ) (eq. 2.9) and on the error on the angle of the measurement θ Q3D . The errors on these quantities are measured in an uncorrelated way, and thus they can be summed quadratically:

(∆E x ) 2 = ∂E x ∂Bρ 2 (∆Bρ) 2 + ∂E x ∂θ Q3D 2 (∆θ Q3D ) 2 .
(2.10)

Note that the masses of all interacting nuclei are known with great precision, and the beam energy is assumed to have negligible uncertainty; thus these quantities do not contribute to the total uncertainty. The derivations and explicit formula for the uncertainty on excitation energy can be found in appendix A. The uncertainty on the spectrometer angle is set to be the measurement reading error: ∆θ Q3D = 0.5°. The final uncertainty on E x is completely dominated by the magnetic rigidity term.

Table 2.2 summarizes the energies and their associated errors obtained for all observed levels in 31 P, and for all measured angles.

Adopted excitation energies

Since the calibration procedures are independent for each angle, a mean excitation energy can be calculated as a weighted average of all the obtained energies. 2.4 Angular distributions excitation energy and its associated error are expressed as:

E x = E x i /(∆E x i ) 2 1/(∆E x i ) 2
(2.11)

∆E x = 1 1/(∆E x i ) 2 ,
(2.12)

where E x i are the excitation energies at each angle i, and ∆E x i their associated uncertainties. Table 2.2 shows the adopted energies and their associated uncertainties for all the observed levels in 31 P. The table also displays the available information from literature, namely from ENSDF [31] compilation and from the work of Vernotte et al. [START_REF] Vernotte | Si( 3 He,d) 31 P reaction at 25[END_REF]. For the well populated levels in 31 P, the excitation energies obtained in the present work are in excellent agreement with literature. The uncertainties have been reduced thanks to the better energy resolution of the Q3D, its good sensitivity, and the different independent measurements performed at different angles. These performances also allowed the separation of doublets of interest, such as E x = 7719 -7737 keV and E x = 7897 -7911 keV. Low populated states have also been observed, and a detailed discussion is dedicated to these states in the Section 2.5

Angular distributions

In this section, the method used for extracting the angular distributions for levels observed at different angles will be described. The differential cross-section at each angle is given by:

dσ dΩ (θ c.m.) = N c (θ lab ) N b (θ lab )N t Ω lab LT DAQ J(θ lab ) (2.13)
where N c is the number of counts for each deuteron peak and at each laboratory angle, N b the number of incident beam particles, N t the areal density of the target (see Section 2.2.1), Ω the solid angle of the Q3D in laboratory frame, LT DAQ the live time of the data acquisition system, and J(θ lab ) is the Jacobian determinant of the transformation from the laboratory frame to the centre of mass one (appendix A). The number of counts for each peak N c is simply the integral of the peak function obtained with Eq. (2.6). The other quantities are detailed in the following subsections.

Beam particle normalization

The total number of beam particles N b that are incident on the target is measured with a Faraday cup connected to a current integrator, and placed at 0°downstream the target in the reaction chamber. Most of the beam remains unreacted and thus is stopped in the Faraday cup. The current is converted into digital pulses and integrated, for each run, in a scaler module named scaler1. The integrated number of beam particles is given by:

N b = scaler1 full scale 1000 q e , (2.14)
where q is the charge state of the incident ions, in our experiment the 3 He incident ions were fully stripped so q = 2, 1000 is the digitizing rate of the integrator (1 kHz), the full scales used in the experiment are 2 × 10 -7 and 6 × 10 -7 C, depending on the beam conditions.

Solid angle estimation

The reaction products enter the Q3D through a diamond shaped aperture, with a width 2A = 73.5 mm and a height 2B = 63 mm, as shown in figure 2.13. The solid angle is controlled by moving the horizontal and vertical slits. The distance between the target and the slits is fixed at D = 354.2 mm. The solid angle Ω for this aperture shape is given by [START_REF] Rebeiro | Nuclear structure studies in the A=136 region using transfer reactions[END_REF]:

Ω = 1 D 2 4∆x∆y -2(∆x -A + A∆y B )(∆y -B + B∆x A ) , (2.15)
where ∆x and ∆y are the horizontal and vertical openings. The same slits opening was used during the whole experiment, ∆x = +13 mm for the x-slits and ∆y = +24.5 mm for y-slits. The error on the slit opening is set to 0.05 mm. The solid angle during the whole experiment was Ω = 4.18 ± 0.04 msr.

Dead time corrections

When the DAQ system is busy while processing an event, any incoming new event received during this time is not accounted for. In such situation, the current integrator increments a quantity called scaler3. The dead time is thus related to the ratio between scaler1 and scaler3, and the live time is simply:

LT DAQ = 1 - scaler3 scaler1 .
(2.16)

The live time varies, for the present experiment between 99.94 and 87.4%.

Uncertainties

The relative uncertainty of the extracted differential cross-sections is the quadratic sum over the non-correlated relative uncertainties of the different physical quantities upon x-slits represents the horizontal opening ∆x while the distance between y-slits is the vertical opening ∆y [START_REF] Rebeiro | Nuclear structure studies in the A=136 region using transfer reactions[END_REF].

which rely the differential cross-sections' calculation:

∆ dσ dΩ dσ dΩ 2 = ∆N c N c 2 + ∆N b N b 2 + ∆N t N t 2 + ∆Ω Ω 2 + ∆LT LT 2 + ∆J J 2 .
(2.17) The N b and LT DAQ depend on the scalers, which statistical uncertainties are given by: ∆(scaler) = √ scaler. For the Jacobian determinant, the uncertainty is related to the Q3D angle, and we take: ∆J = ∂J ∂θ ∆θ. Overall, the relative error on the differential cross-section varies between 12% and 35% for the least populated states. Figures 2.15 shows the angular distributions for all observed level, listed in table 2.2.

Results and discussion

In the present experiment, 26 31 P levels have been populated through the 30 Si( 3 He,d) 31 P reaction. These levels have excitation energies between E x = 6.8 and 8.1 MeV. Several weakly populated states have also been observed. Figure 2.14 shows a close up view of the deuteron magnetic rigidity spectra, associated to levels that have not been observed in the previous study of the 30 Si( 3 He,d) 31 P reaction [START_REF] Vernotte | Si( 3 He,d) 31 P reaction at 25[END_REF]. In the present work, levels observed at excitation energies of E x = 7159, 7347, 7446, 7470, 7691 and 7863 keV are not populated in the work of Vernotte et al. because of the weak sensitivity and energy resolution. The latter was about FWHM = 25 keV instead of 7 keV in our case, this is the reason why some doublets, well resolved in the present work, are not separated in the previous 30 Si( 3 He,d) 31 P study, such as the E x = 7719 -7737 keV doublet. Note that all aforementioned levels have been populated through different reactions. We now give a review for these states:

• E x = 7159.7 ± 1.6 keV: This level was previously reported in the 32 S(d, 3 He) 31 P reaction [START_REF] Vernotte | One-nucleon pickup reactions on 32s: Experimental results and shell-model calculations[END_REF] at an excitation energy of E x = 7158 keV. It was not observed in the previous 30 Si( 3 He,d) 31 P reaction study [START_REF] Vernotte | Si( 3 He,d) 31 P reaction at 25[END_REF] due to its proximity with the strongly populated level at E x = 7141 keV, which has an angular distribution strongly peaked at forward angles. This made the extraction of differential cross-sections for the E x = 7159 keV possible at angles θ Q3D ≥ 20°only.

• E x = 7347.0 ± 1.2 keV: This state has a reported energy E x = 7346 ± 6 keV based on the study of the 29 Si( 3 He,p) 31 P reaction [START_REF] Al-Jadir | States of high E x in 31 P[END_REF].

• E x = 7445.7±2.8 keV: This state is assumed to be associated to the E x = 7441.4 keV level observed using the 27 Al(α,γ) 31 P reaction [START_REF] De | Levels of 31p from -particle capture in 27al[END_REF]. This state is of interest in the temperature region of stars in Globular Clusters. But unfortunately, the differential cross-section was only extracted at θ Q3D ≥ 20°, due to the strong contaminating peak induced by proton transfer on oxygen. Its angular distribution will be further discussed in the following chapter.

• E x = 7470.5 ± 2.3 keV: This level is associated to the E x = 7466 ± 2 keV state reported in the literature. It was first observed through the 27 Al(α,γ) 31 P reaction [START_REF] De | Levels of 31p from -particle capture in 27al[END_REF], and later with the 28 Si(α,pγ) 31 P [56] and 29 Si( 3 He,p) 31 P [54] reaction. The level was not observed at forward angle, only at θ Q3D = 20°, 23°and 32°. The angular distribution will be further discussed in the following chapter.

• E x = 7691.1 ± 1.0 keV: This level is associated with the E x = 7687.2 ± 2.0 keV state which has been observed in a single experiment using the 30 Si(p,γ) 31 P reaction [START_REF] De Neijs | Levels of 31 P from proton capture in 30 Si[END_REF]. It has been observed unambiguously at forward angles: θ Q3D = 6°, 10°a nd 12°. At higher angles, its associated peak overlaps with the one corresponding to E x = 3105 keV populated by the 28 Si( 3 He,d) 29 P. The differential cross-section at large detection angles was obtained by subtracting the contribution of the contaminating 29 P. The latter was deduced from the angular distribution obtained in the 28 Si( 3 He,d) 29 P study [START_REF] Dykoski | Single proton transfer to 29,30 P states[END_REF], obtained at the same beam energy as the present experiment. tion [START_REF] Ionescu-Bujor | High spin structure and intruder configurations in 31 P[END_REF]. The state has not been observed in other experiments, probably due to its proximity with the E x = 7851.4 ± 0.8 keV level. The limited resolution would not allow the separation between these two levels.

• E x = 7863.4 ± 1.
Despite the good energy resolution and sensitivity achieved in the experiment described in this chapter, there are some levels listed in the literature that have not been populated, this is the case of the levels at E x = 7117, 7356, 7572, 7825, 7994 and 8031 keV reported on ENSDF compilation [31]. This is mainly due to the selectivity of the transfer reaction 30 Si( 3 He,d) 31 P reaction which populates mainly levels with angular orbital momenta ℓ ≤ 3, as it will be shown and discussed in the following chapter. Thus, levels with higher angular orbital momenta, such as (E x = 7117.7 keV, J π = 9/2 + , ℓ = 4) and (E x = 7442.3.7 keV, J π = 11/2 -, ℓ = 5) are not expected to be strongly populated by this transfer reaction. Some other levels are not observed simply due to contaminating peaks, especially the ones induced by reactions on nuclei with masses close to A = 31. For example, peaks corresponding to 29 P are present in the same region of the focal plane at all angles. This was the case for levels at E x = 7994 and 8031 keV, which were hidden by the 13 N(2365 keV) at forward angles and then by the 29 P(3448 keV) at all other measured angles. Noteworthy that levels at E x = 7356 and 7572 keV have only been observed in the experimental study of the 33 S(d,α) 31 P reaction [START_REF] Teterin | Study of excited states of 31 p with the aid of the reaction 33 s(d, α) 31 p[END_REF], with no spin-parity assignments. Same for the E x = 7825 ± 9 keV level, observed only once using the 29 Si( 3 He,p) 31 P reaction [START_REF] Moss | Excitation energies of levels in 31 P[END_REF], in contradiction with the latter study of the same transfer reaction at different incident energy [START_REF] Al-Jadir | States of high E x in 31 P[END_REF]. These levels are discarded from the rest of the present study.

High-resolution spectra of 31 P excited states were obtained with the Q3D magnetic spectrometer at the Maier-Leibnitz Laboratorium in Munich. With the one-proton 30 Si( 3 He,d) 31 P transfer reaction, 26 31 P excited states with excitation energies between 6.8 and 8.1 MeV were populated. The calibration of the focal plane detection system and the fit of the spectra with a multi-peak function allowed the determination of excitation energies of the observed states with great precision (less than 3 keV uncertainty). After normalization with respect to the number of incident beam ions, the target thickness and solid angle, differential cross-sections were calculated for all measured angles, between 6°and 32°. The obtained angular distributions will be analysed in the DWBA framework in the following chapter. The angular distributions that have been extracted in the previous chapter need to be compared to some theoretical calculation, in order to extract the spectroscopic information needed for proton width determination, and subsequently resonance strength calculation. The theoretical framework that is adopted is the Distorted Wave Born Approximation (DWBA). This model is suited for the description of direct transfer reactions. It relies on the assumption that the cross-section is dominated by the elastic scattering, while the one-step transfer of the nucleon is weak enough to be treated as a first order theory.

Elements of reaction theory are presented in the first part of this chapter, followed by the description of the ingredients needed to perform DWBA calculation. We also present a quantitative comparison between the different methods and prescriptions used for the calculation of the proton widths for unbound levels. The subsequent results are presented and analysed. Finally, the impact of the uncertainties attached to the potentials used in the determination of the spectroscopic factor and proton width are investigated in a statistically meaningful way, using a Bayesian framework.

Reaction theory

The study of nuclear reactions allows the determination of spectroscopic properties of nuclei needed to calculate the astrophysical reaction rates. These nuclear reactions can be categorized according to the timescales during which they occur: nuclei may fuse for long enough to "forget" how the compound nucleus was formed, and it will decay into a channel independent of the way it was formed. Direct reactions, on the contrary, are designated as surface processes where the interaction between two reacting nuclei occurs in such a short timescale that only few nucleons are involved. Direct reaction amplitude depends on the overlap between the initial and final states, and thus properties such as the spectroscopic factors and orbital angular momenta can be extracted from the reaction observables.

Elements of scattering theory

Scattering theory provides the framework within which a definition of the physical model for nuclear reaction can be introduced. The mathematical structures of the theory are first introduced without any consideration for the intrinsic spins of the interacting particles. The spins and angular momenta will be introduced later in subsection 3.1.4. We consider here the cases involving only pairs of particles, that will be called partitions.

In the general case, a nuclear reaction can be written as A(a, b)B, and thus two channels α = a + A and β = B + b are considered:

A + a channel α → B + b channel β (3.1)
The internal state of a partition is described by the wave function :

ψ α = ψ a ψ A with corresponding internal energy ϵ α = ϵ a + ϵ A and Hamiltonian H α , such as H α |ψ α ⟩ = ϵ α |ψ α ⟩. The total energy of the channel is E = E α + ϵ α where E α is the relative kinetic energy, defined as E α = ℏ 2 k α 2 /2µ α ,
where ⃗ k α is the wave vector corresponding to the relative motion of the two nuclei, that can be described by a plane-wave, and µ α is the reduced mass of the partition :

µ α = m a m A /(m a + m A ). The corresponding Hamiltonian is K α .
The total interaction potential is the sum of the interactions between nucleons in a with nucleons in A, and is written as: V α = a i=1 a+A j=1+a v ij where i refers to nucleons in a and j to nucleons in A. Only the two-body interactions are considered here. The total Hamiltonian for the system is:

H = H α + K α + V α . (3.2)
The total wave function |Ψ⟩ of the system is the solution of the Schrödinger equation :

(E -H) |Ψ⟩ = |0⟩ . (3.3)
For nuclear reactions in the framework of scattering theory, the α channel usually describes a beam particle a incident on a target nucleus A. At large distances from the target the total wave function of the system also includes outgoing scattered waves in all energetically available channels. The wave function is thus denoted by

Ψ (+) α ( ⃗ k α )
, reflecting the fact that the outgoing (+) waves arise from the incident wave in α channel. Note that the time-reversed solution corresponds to incoming scattered waves noted Ψ (-)

β ( ⃗ k β )
. At large distances from the target (r β → ∞), the outgoing scattered wave functions become spherical, and the system can be described by the wave function written as [START_REF] Messiah | Quantum Mechanics[END_REF]:

Ψ (+) α ( ⃗ k α ) → r β →∞ e i ⃗ kα• ⃗ rα |ψ α ⟩ + β f α→β ( ⃗ k β , ⃗ k α ) e ik β •r β r β |ψ β ⟩ . (3.4) The coefficient f α→β ( ⃗ k β , ⃗ k α )
is the scattered amplitude of an outgoing spherical wave in channel β induced by a plane-wave of unit amplitude in α channel. The angle defined by the vectors ⃗ k β and ⃗ k α is the scattering angle θ at which the outgoing particles are detected. The differential cross-section for a reaction A(a, b)B corresponding to the channel transition α → β is given by [START_REF] Satchler | Direct nuclear reactions[END_REF]:

dσ α→β dΩ = v β v α |f α→β ( ⃗ k β , ⃗ k α )| 2 , (3.5)
where v i are the channel velocities in the centre-of-mass of each channel. It is more convenient to use the transition amplitude, defined as [START_REF] Satchler | Direct nuclear reactions[END_REF]:

T α→β = - 2πℏ 2 µ β f α→β . (3.6)
The differential cross-section becomes:

dσ α→β dΩ = µ α µ β (2πℏ 2 ) 2 k β k α |T α→β ( ⃗ k β , ⃗ k α )| 2 . (3.7)
Considering a single channel β, the asymptotic form of the total wavefunction become:

Ψ (+) α ( ⃗ k α ) → r β →∞ e i ⃗ kα• ⃗ rα |ψ α ⟩ - µ β 2πℏ 2 e ik β •r β r β T α→β ( ⃗ k β , ⃗ k α ) |ψ β ⟩ . (3.8)
For a given interaction potential V β describing the exit channel, the Schrödinger equation (3.3) can be expressed as:

(E -H β -K β ) Ψ (+) α ( ⃗ k α ) = V β Ψ (+) α ( ⃗ k α ) . (3.9)
This representation is called post form because the interaction potentials of the exit channel β appear in the expression. In the prior representation, the roles of the entrance and exit channels are interchanged. The mechanisms governing the nuclear reactions are believed to be invariant under time-reversal, the post and prior representations are thus equivalent. The post representation will be used in the rest of this work.

Using the Green function technique [START_REF] Messiah | Quantum Mechanics[END_REF], the Schrödinger equation (3.9) can be written as:

Ψ (+) α ( ⃗ k α ) = e i ⃗ kα• ⃗ rα |ψ α ⟩ - µ β 2πℏ 2 e ik β | ⃗ r β -⃗ r ′ β | | ⃗ r β -⃗ r ′ β | V β Ψ (+) α ( ⃗ k α ) d ⃗ r ′ β .
(3.10)

When taking the asymptotic limit of this solution, r β ≫ r ′ β , and comparing with the asymptotic form (3.8), one deduces the transition amplitude for a plane-wave scattered by the potential V as:

T α→β ( ⃗ k β , ⃗ k α ) = J e i ⃗ k β • ⃗ r β ψ β V β Ψ (+) α ( ⃗ k α ) , (3.11)
where J is the Jacobian for the transformation to coordinates of the channel β.

Plane and Distorted Waves Approximations

The expression (3.10) does not provide a solution for the Schrödinger equation ( 3.3) because it still involves the unknown total wave function

Ψ (+) α ( ⃗ k α ) .
Approximations are needed at this point in order to extract any observable quantity.

Born approximation [START_REF] Born | Quantenmechanik der Stoßvorgänge[END_REF]: If the interaction potential is much smaller than the relative kinetic energy of the system, the scattering part of equation (3.10) is negligible when compared to the plane-wave part. The scattering wave function can thus be approximated by a plane-wave:

Ψ (+) α ( ⃗ k α ) ≈ e i ⃗ kα• ⃗ rα |ψ α ⟩.
The Plane Wave Born approximation of the transition amplitude is given by:

T P W BA α→β = J e i ⃗ k β • ⃗ r β ψ β V β e i ⃗ kα• ⃗ rα ψ α (3.12)
This approximation is poor since the differential cross-section deduced from the scattering amplitude does not contain contributions from channels with flux absorption or emission, i.e., the transfer, absorption, or emission of particles.

Distorted waves [START_REF] Mott | The Theory of Atomic Collisions[END_REF]: The potential V can be separated into two distinct and arbitrary parts, such as:

V β ( ⃗ r β , x β ) = U β (r β ) + W β ( ⃗ r β , x β ) with W β ≪ U β . (3.13)
This separation can be chosen so that the majority of the interaction is encapsulated in U β which depends solely on the relative distance between b and B, and thus W β can be treated as a perturbation. U β is usually taken as the potential describing the elastic scattering B(b, b)B, that we shall call optical potential, and elastic scattering of the entrance channel A(a, a)A can be described in an analogue way. These potentials are adjusted to experimental data of elastic scattering. The Schrödinger equation (3.3) can be reformulated as:

(E -H β -K β -U β ) Ψ (+) α = (V β -U β ) Ψ (+) α (3.14) (E β -K β -U β )ξ β = ⟨ψ β | W β Ψ (+) α (3.15)
The homogeneous equation corresponding to (3.15) admits the solution:

(E β -K β -U β )χ (+) β ( ⃗ k β , ⃗ r β ) = 0, (3.16)
where The exact solution of (3.15) leads to the following transition amplitude [START_REF] Messiah | Quantum Mechanics[END_REF]:

χ (+) β ( ⃗ k β , ⃗ r
T α→β ( ⃗ k β , ⃗ k α ) = J χ (-) β ( ⃗ k β )ψ β U β e i ⃗ kα• ⃗ rα ψ α + χ (-) β ( ⃗ k β )ψ β W β Ψ (+) α ( ⃗ k α ) , (3.17) with χ (-) β ( ⃗ k β ) = χ (+) β * (-⃗ k β ).
The * superscript stands for the complex conjugate function, and the (-) superscript indicates that the spherical waves are incoming. It is the time-reverse of χ

(+) β ( ⃗ k β ).
Since U β ≫ W β , the right-hand side of (3.15) is negligible, and the distorted wave is expected to be identical to the total wave function, at least outside the nucleus. The wave function can be thus approximated by: Ψ

(+) α ( ⃗ k α ) ≈ χ (+) α ( ⃗ k α , ⃗ r α ) |ψ α ⟩.
The transition amplitude (3.17) becomes, in the Distorted Wave Born approximation (DWBA):

T DW BA α→β ( ⃗ k β , ⃗ k α ) = J χ (-) β ( ⃗ k β )ψ β U β e i ⃗ kα• ⃗ rα ψ α + χ (-) β ( ⃗ k β )ψ β W β χ (+) α ( ⃗ k α )ψ α . (3.18)
The first part of Eq. (3.18) equals to zero except when α = β because by definition, the potential U β cannot induce transition between different channels in elastic scattering. The second part of Eq. (3.18) describes the transition amplitude, its integral form is:

T DW BA α̸ =β ( ⃗ k β , ⃗ k α ) = J d⃗ r β d⃗ r α χ (-) * β ( ⃗ k β , ⃗ r β ) ⟨ψ β | W β |ψ α ⟩ χ (+) α ( ⃗ k α , ⃗ r α ). (3.19)
The overlap term ⟨ψ β | W β |ψ α ⟩ is called the form factor. It contains the information about the nuclear structure and the selection rules of the transition. The form factor will be further detailed for the case of a transfer reaction in the following subsections.

Case of transfer reactions

For transfer reactions, the general case A(a, b)B consists in the following:

a(= b + x) + A → b + B(= A + x). (3.20)
In the present case, x is a single nucleon, transferred in one step to nucleus A, forming the composite nucleus B = A + x. The residual interaction in the post form is: W bB = V bB -U bB where U bB describes the elastic scattering between b and B. Since B = A+x, the interacting potential can be divided as:

V bB = V bA + V bx .
The residual interaction becomes:

W bB = V bx + V bA -U bB . (3.21)
For medium-mass/heavy targets, the approximation V bA ≈ U bB can be made. The residual interaction is reduced to:

W bB = V bx . (3.22)
Furthermore, the form factor can be factorized into [START_REF] Satchler | Direct nuclear reactions[END_REF]:

⟨ψ b ψ B | V bx |ψ a ψ A ⟩ = ⟨ψ b | V bx |ψ a ⟩ projectile-like ⟨ψ B |ψ A ⟩ target-like . (3.23)
Projectile-like and target-like overlaps will be detailed in sections 3.1.5 and 3.2.1 respectively.

Selection rules

So far, the formalism was introduced for spinless particles only, the spins and angular momenta are now considered. Suppose that the nucleon x has an intrinsic spin s and has an orbital angular momentum l 1 relative to b within nucleus a:

⃗ j 1 = ⃗ l 1 + ⃗ s, ⃗ J a = ⃗ J b + ⃗ j 1 . (3.24)
Similarly, when x is within the nucleus B, it has an orbital angular momentum

l 2 relative to A ⃗ j 2 = ⃗ l 2 + ⃗ s, ⃗ J B = ⃗ J A + ⃗ j 2 . (3.25)
The transferred angular momentum is:

⃗ l = ⃗ j 2 -⃗ j 1 = ⃗ l 2 -⃗ l 1 . (3.26)
This result can be formulated in terms of total angular momenta:

⃗ l = ⃗ J B + ⃗ J b -⃗ J A -⃗ J a . (3.27)
The parity conservation gives:

π a = π b π x (-1) l 1 , π B = π A π x (-1) l 2 , (3.28)
so the total change in parity induced by the transfer reaction must satisfy:

∆π = π A π a π B π b = (-1) l 1 +l 2 (3.29)

Finite Range (FR) calculations

The parentage decomposition of the projectile a = x + b implies that the projectile-like overlap in relation (3.23) can be written as [START_REF] Satchler | Direct nuclear reactions[END_REF]:

⟨ψ b | V bx |ψ a ⟩ = d⃗ r bx dξ A xb ψ b (ξ) * V bx (⃗ r xb )ψ b (ξ)φ xb (⃗ r xb ) (3.30) = d⃗ r bx A xb V bx (⃗ r xb )φ xb (⃗ r xb ), (3.31)
where ξ are the internal variables of b, and φ xb (⃗ r xb ) is the wave-function of nucleon x relative to b and A xb is the probability amplitude of the configuration x + b inside a. This overlap is usually denoted as:

d⃗ r bx D(⃗ r bx ) = d⃗ r bx A xb V bx (⃗ r xb )φ xb (⃗ r xb ) (3.32)
This overlap function describes how the nucleon x interacts with the core b within the projectile a. Historically, several approximations were made on this projectile-like overlap, and the Zero-Range (ZR) approximation was the most used due to the limited computational capacities in the past. It consists in assuming : D(r) ≈ D 0 δ(⃗ r) where D 0 is the volume integral of D(r) and δ the Dirac function. This approximation relies on the assumption that the form factor has a short range, either because the interaction has a short range or the internal wave function is confined in a small volume, which is the case for s-waves. This reduces the number of integrals in relation (3.19) from six to three. The zero-range approximation is a good assumption for one-nucleon transfer reactions such as (d,p) and ( 3 He,d ) because the nuclei have small-size and the projectiles are in an s-wave internal state. The modern DWBA codes and average computer can perform the calculation in the Finite range frame, without any further approximation made on the overlap function D(⃗ r). However, ZR-calculations can still be performed in order to compare with results obtained in previous studies.

Summary

The DWBA is a suitable framework for the study of direct reactions with one-step mechanism. It is based on the assumption that the entrance and exit channels of the process are dominated by the elastic scattering, while other channels are treated as perturbations. It is noteworthy that the DWBA framework is not a simple first order theory. The use of optical potentials, that are fitted to experimental data of elastic scattering, includes higher order effects. The DWBA framework is a powerful tool, with a strong element of phenomenology that is needed for a good description of direct reaction mechanisms. That's why the optical and binding potentials used should be chosen with great care.

Spectroscopic information extraction 3.2.1 Spectroscopic factor

Given the parentage decomposition B = A + x, the internal state wave-function of the formed nucleus can be decomposed into [START_REF] Satchler | Direct nuclear reactions[END_REF]:

|ψ B ⟩ = jℓmM b A ℓj xA ⟨J A j; M A m|J B M B ⟩ ⟨T A t; N A n|T B N B ⟩ Y lj m ( ⃗ r xA )φ nlj xA (r xA ) |ψ A ⟩ , (3.33
) where J i and M i are the total angular momentum and its projection for the nucleus i, while T i and N i are the isospin and its 3 rd component, j and t are the total angular momentum and isospin, of the transferred nucleon x, while

m = M B + M b -M A - M a and n = N B -N A , Y lj m ( ⃗ r xA
) are the spherical harmonics and ⃗ r xA is the unitary vector defining the angles between x and A, while φ nlj xA (r xA ) is the radial part of the wavefunction. The coefficients A ℓj xA are the probability amplitude of the configuration A + x with relative orbital angular momentum ℓ and projection m. In the shellmodel description, it represents the probability amplitude that the x nucleon occupies the nℓj orbital. The isospin Clebsh-Gordan coefficient is often abbreviated as C = ⟨T A t; N A n|T B N B ⟩. Note that both |ψ B ⟩ and |ψ A ⟩ are fully antisymmetric.

Radial wave function

φ nlj xA (r xA )
is the radial part of the single-particle wave-function, it represents the normalized solution of the two-body equation describing the motion of x with respect to A:

ℏ 2 k 2 xA 2µ xA + V xA (r xA ) + V Coul (r xA ) + ε φ nlj xA (r xA ) = 0, (3.34)
where ε is the binding energy of x within B. The depth of the radial part of the potential V xA (commonly chosen with a Wood-Saxon shape) is adjusted to reproduce the measured binding energy ε. Beyond the range of nuclear forces, the radial wavefunction of the charged particle x has the asymptotic form [START_REF] Mukhamedzhanov | Connection between asymptotic normalization coefficients, subthreshold bound states, and resonances[END_REF]:

φ nlj xA (r xA ) → r→∞ b nlj W -η,ℓ+1/2 (2κr Ax ) r A x , (3.35)
where b nlj are the single-particle asymptotic normalization coefficients (ANC), W is the Whittaker function, κ = √ 2µ xA ε is the wave-number of the state with binding energy ε, and η = Z A z x µ xA /κ is its associated Sommerfeld parameter.

Radial form factor

Given the wave-function decomposition of (3.33), the target-like overlap of (3.23) becomes:

⟨ψ B |ψ A ⟩ = j CA ℓj xA ⟨J A j; M A m|J B M B ⟩ Y lj m ( ⃗ r ′ xA )φ nlj xA (r xA ), (3.36)
where C is the isospin Clebsh-Gordan coefficient defined previously. The radial part of this overlap function is

I B xA nℓj = A ℓj xA φ nlj xA (r xA )
and is called the radial form factor. The spectroscopic factor is defined as [START_REF] Iliadis | Nuclear Physics of Stars[END_REF]:

S B ℓj = ∞ 0 dr r 2 |I B xA nℓj | 2 , (3.37)
and since the radial wave-functions are normalized to unity, one obtains :

S B ℓj = A ℓj xA 2 .
The spectroscopic factor is often referring to the product:

C 2 S B ℓj = ⟨T A t; N A n|T B N B ⟩ 2 (A ℓj xA ) 2 . (3.38)
The transition amplitude in Eq. (3.18) is also proportional to the spectroscopic factor of the projectile, which can be defined, according to (3.32), in the same fashion as: S a ℓj = (A xb ) 2 . Its knowledge is crucial for calculating the C 2 S B ℓj . The measured cross-section for the A(a, b)B transfer reaction is then expressed as:

dσ dΩ b exp = C 2 B S B × C 2 a S a × dσ dΩ DW BA (3.39)

Transfer to unbound levels

When the residual state |ψ B ⟩ is unbound against the emission of the nucleon x, the state corresponds to a scattering resonance. If the width of the resonance is sufficiently narrow, (i.e., the state sufficiently long-lived), the total reaction A(a, b)B → x+A * can be considered sequential, and the transfer part treated in the same way as for bound states.

When final states of the transfer reaction are unbound, it is more suitable to express the double differential cross-section [START_REF] Vincent | New Method for Distorted-Wave Analysis of Stripping to Unbound States[END_REF]:

d 2 σ(a, bx) dΩ b dE x = dσ B dΩ b × 2µ xA πℏ 2 κ xA (E x )|T xA | 2 scattering A + x , (3.40)
which is expressed as the product of the cross-section for the resonant process, times the one for the particle emission. E x is the excitation energy of the system B = A + x, which is related kinematically to the energy of the detected particle b, and to the wavenumber

κ xA = √ 2µ xA E x .
The term T xA is the transition amplitude for the (x + A) scattering system. The cross-section dσ B dΩ b has the same shape as the one corresponding to the transfer to a bound state. However, the radial wave function in the case of an unbound level has a different asymptotic behaviour. Indeed, since the binding energy between A and x becomes negative, the radial wave function oscillates at large distances, and do not decay exponentially as in Eq. (3.35). The radial wave function is not normalizable anymore, and has an infinite distribution in energy.

Radial integration of form factor

As for the bound states, the radial wave function φxA is the solution of equation (3.34), but with ε < 0. In this case, the radial wave function is asymptotically proportional to the irregular Coulomb function G ℓ (κr)/r [START_REF] Mukhamedzhanov | Connection between asymptotic normalization coefficients, subthreshold bound states, and resonances[END_REF].

The integration of the corresponding form factor is numerically difficult since the radial wave function do not decay exponentially at large radii, as it is the case for the Whittaker function. Vincent and Fortune [START_REF] Vincent | New Method for Distorted-Wave Analysis of Stripping to Unbound States[END_REF] made a numerical ansatz for computing the integral: the range of integration is first divided into an inner part: 0 < r < R and an outer part R < r < ∞, where R is chosen so that the nuclear forces become negligible in the outer part, i.e., the radial wave function becomes asymptotic with respect to the Coulomb functions. The inner part of the integral is evaluated in the same way as for the bound states, whereas for the outer part, the contour of the integral in the complex plane is deformed in such a way that the integrand undergoes under a few oscillations and then tends to zero exponentially at large values.

Case of isolated resonances

If the system A + x has an isolated resonance with energy E R = E x -Q and partial width Γ x , the transition amplitude can be approximated by the Breit-Wigner form [START_REF] Vincent | New Method for Distorted-Wave Analysis of Stripping to Unbound States[END_REF]:

T xA ≈ 1/2 Γ x E x -E R -1/2iΓ x . (3.41)
Furthermore, the shape of the cross-section dσ B dΩ b is independent of E x , thus when replacing (3.41) in (3.40) and integrating over E x , the transfer cross-section is [START_REF] Vincent | New Method for Distorted-Wave Analysis of Stripping to Unbound States[END_REF]:

dσ(a, b) dΩ b = Γ x µ xA κ xA (E R ) ℏ 2 dσ B (E R ) dΩ (3.42)
The single-particle width, defined as Γ x = C 2 S ×Γ s.p x , has the general expression [68] :

Γ s.p x = 2P ℓ (E, R)γ 2 s.p , γ 2 s.p = ℏ 2 R 2µ xA φ 2 (R) (3.43)
where P ℓ is the penetrability of the Coulomb and centrifugal barrier, γ 2 s.p the reduced single-particle width and contains the nuclear structure information. φ(R) is the radial part of the wave function estimated at the interaction radius R, and it is assumed to be normalized to unity inside the interaction radius; i.e. R 0 dr r 2 φ 2 (r) = 1. This expression is independent of how the state B = A + x has been formed.

Weakly bound approximation

Given the difficulties to make the radial integration of the form factor, several approximations have been proposed. The method using quasi-bound states for stripping to unbound levels was first proposed by Cole et al. [START_REF] Cole | Method of pseudo-bound states for stripping to unbound levels[END_REF]. It consists in cutting-off the oscillating tail of the radial part of the wave function and thus considering the levels as bound states. The weakly bound approximation is an equivalent method, it consists in calculating the form factor for a small binding energy, few tens of keV. The form factor is then used to calculate the single particle partial width. For the spectroscopic factor, the prescription proposed by Becchetti et al. [START_REF] Becchetti | 12C(7Li, t)16O and stellar helium fusion[END_REF] is usually followed: the spectroscopic factor is estimated from the normalization of the experimental data with respect to cross-section calculated at different small binding energies, typically between 50 and 200 keV. The values of the spectroscopic factor are then extrapolated to the correct binding energy, which is negative. The validity of this procedure has been tested in the case of the 30 Si( 3 He,d) 31 P reaction, with ZR calculations (see section 3.3.2.b). This approximation is applied for levels with orbital angular momenta ℓ < 2. Levels with higher orbital momenta are considered as quasi-bound, given the large centrifugal barrier. 30 Si( 3 He,d) 31

Case of the

P reaction

The theoretical framework developed in the previous section will be now applied to the 30 Si( 3 He,d) 31 P transfer reaction that has been presented in Chapter 3.

DWBA calculations inputs a/ Optical potential for entrance and exit channels

The optical potentials, introduced in Eq. (3.13) describe the elastic scattering for the entrance and exit channels. The potentials have an imaginary part in order to describe the lost flux due to the transfer reaction. The general form of the optical potentials is:

U = V c -V f (r, r v , a v ) + h m π c 2 V s.o 1 r d dr f (r, r s.o , a s.o )⃗ σ. ⃗ ℓ -i W f (r, r W , a W ) -4W D d dr f (r, r D , a D ) , (3.44)
where the function f (r, r i , a i ) is the Wood-Saxon function:

f (r, r i , a i ) = 1 1 + exp r-r i A 1/3 a i , (3.45)
with r i and a i the radius and the diffuseness, respectively, V c the Coulomb potential for a charged sphere of radius R c = r c A 1/3 , V, V s.o , W and W D the depths of the volume, spin-orbit, volume and surface absorption potentials, respectively. For the entrance channel, the parameters were fitted to data from elastic scattering measurements of 3 He on 30 Si at E beam = 25 MeV. These measurements could not be performed during the MLL experiment due to reduced beam time, thus the data of Vernotte et al. [START_REF] Vernotte | Optical model analysis of 3 He elastic scattering from s-d shell nuclei at 25 MeV[END_REF] were used, since the measurements were performed at the same incident energy as the present work. Three sets of parameters were obtained, but the study of the transfer reaction 30 Si( 3 He,d) 31 P in the work of Vernotte et al. [START_REF] Vernotte | Si( 3 He,d) 31 P reaction at 25[END_REF] showed that the set with the deepest volume potential reproduces better the data of the most populated levels.

For the exit channel, no experimental data for the scattering of d on 31 P are available. As in the study of Vernotte et al. [START_REF] Vernotte | Si( 3 He,d) 31 P reaction at 25[END_REF], the global parametrization of Dahenick et al. [START_REF] Daehnick | Global optical model potential for elastic deuteron scattering from 12 to 90 MeV[END_REF] was used. The parametrization was obtained from fitting more than 100 angular distributions for elastic scattering of deuterons on nuclei ranging from 27 Al to 238 Th, with incident energies between 12 and 90 MeV.

The optical potentials parameters used for both entrance and exit channels are listed in Table 3.1. 30 Si + 3 He 189. [START_REF] Harris | A New Catalog of Globular Clusters in the Milky Way[END_REF] The interaction between the proton and the 30 Si core is described with a Wood-Saxon potential with volume and spin-orbit terms. The depth of the volume part is adjusted for each level in order to reproduce the binding energy E b of the corresponding 31 P state, which is related to the excitation energy E x via the Q p value of the reaction:

V W S r r a r W V W D r i a i V s.o. r s.o. a s.o. r C (MeV) (fm) (fm) (MeV) (MeV) (fm) (fm) (MeV) (fm) (fm) (fm)
E b = Q p -E x with Q p = 7296.55(2) keV [73]. (3.46)
The spin-orbit term of the binding potential is calculated considering the relative angular orbital momentum ℓ, which is deduced from the selection rules (section 3.1.4).

Assuming that the 30 Si core is in its ground state ⃗ J π 30 Si = ⃗ 0 + one obtain: c/ Binding potential for 2 H + p

ℓ = J31 P ± 1/
The overlap ⟨ 3 He 2 H+p has been calculated by Brida et al. [START_REF] Brida | Quantum Monte Carlo calculations of spectroscopic overlaps in A⩽7 nuclei[END_REF] with a Green's function Monte Carlo (GFMC) technique, based on realistic two-and three-nucleon interactions that reproduced well the low-lying spectra of light nuclei. The overlaps are extended to regions beyond the nuclear surface by means of Wood-Saxon fits. The results for an s-wave are parametrized as:

V (r) = V c -V W S f (r, r v , a v ) -β exp -(r/ρ) 2 + h m π c 2 V s.o 1 r d dr f (r, r s.o , a s.o )⃗ σ. ⃗ ℓ -i W f (r, r W , a W ) -4W D d dr f (r, r D , a D ) , (3.48)
with V c the Coulomb potential for a charged sphere of radius R c = 1.25 × 2 1/3 = 1.575 fm, V W S = 179.94 MeV, r v = 0.54 fm, a v = 0.68 fm. The potential includes a Gaussian "wine-bottle" term to provide an additional flexibility at short range, and its parameters are β = 1.13 and ρ = 0.64. The corresponding binding energy is: Q = 5.49 MeV. The calculated spectroscopic factor corresponding to this overlap is C 2 S( 3 He) = 1.33 [START_REF] Brida | Quantum Monte Carlo calculations of spectroscopic overlaps in A⩽7 nuclei[END_REF].

In order to compare consistently the results obtained in the present work with those of Vernotte et al. [START_REF] Vernotte | Si( 3 He,d) 31 P reaction at 25[END_REF], Zero-Range DWBA calculations have also been performed. The latter are also useful for the study of the levels with unbound energies. In this case, the volume integral is taken as : D 0 = -172.8 MeV fm 3/2 [START_REF] Bassel | Normalization of and Finite-Range Effects in ( 3 He, d) and (t, d) Reactions[END_REF]. The corresponding spectroscopic factor is C 2 S( 3 He) = 1.48 [START_REF] Bassel | Normalization of and Finite-Range Effects in ( 3 He, d) and (t, d) Reactions[END_REF].

Methods and codes used for DWBA calculations

Two different codes were used for the DWBA calculations. The spectroscopic factors were obtained from the normalization of angular distribution curves calculated with fresco [START_REF] Thompson | Coupled reaction channels calculations in nuclear physics[END_REF] and dwuck4 [77] codes, with respect to experimental data. fresco code was developed to perform coupled-reaction channels calculations in nuclear physics, and perform Zero and Finite Range calculations. However, the published version of the code is not suited to perform DWBA calculations for unbound states.

In another hand, the code dwuck4 [77] was used to perform calculations for unbound energies. The BIND subroutine of the code, implemented by Comfort [START_REF] Comfort | Extended version of DWUCK4[END_REF], calculates single-particle radial wave functions ϕ(r) corresponding to the solution of the Schrödinger equation with a real potential and for a single set of quantum numbers, with the procedure of Vincent and Fortune [START_REF] Vincent | New Method for Distorted-Wave Analysis of Stripping to Unbound States[END_REF] mentioned in subsection 3.2.2. The code performs calculations in ZR approximation only, so it was used to calculate the single-particle proton widths Γ s.p .

The results obtained from both codes and for different approximations have been compared, and the details are presented in the following subsections.

a/ Comparison between ZR and FR calculations

A set of strongly populated 31 P levels has been chosen for the comparison between Zero-Range and Finite-Range DWBA calculations, with dwuck4 and fresco codes, respectively. Note that the equivalence between both codes has been checked by comparing the results obtained by each code in ZR calculation. The results were identical.

For the study, we considered the levels at E x = 7141, 7898, 7946 and 7911 keV, which have transferred orbital angular momenta ℓ = 0, 1, 2 and 3, respectively. The first level at E x = 7141 keV is bound by 154 keV, for the three other levels, an arbitrary binding energy of 1 MeV was considered, since the FR-DWBA calculation need to be performed at bound energies. Figure 3.2 shows the measured angular distributions of the levels and the corresponding DWBA curves, blue ones are finite range calculations while green curves are the zero range calculations. The FR-curves reproduce slightly better the experimental angular distribution for ℓ = 0, while it is equivalent to ZR-curves for higher transferred angular momenta.

The extracted proton spectroscopic factors C 2 S and the calculated single particle reduced widths γ 2 s.p calculated from radial wave functions are listed in table 3.2. The spectroscopic factor obtained from the normalization of ZR-calculations is systematically higher by 33 to 38%. This was expected from the theoretical calculations investigating the finite range and non-locality corrections, where a factor of 0.77 reduction in the spectroscopic factor was proposed [START_REF] Bassel | Normalization of and Finite-Range Effects in ( 3 He, d) and (t, d) Reactions[END_REF]. However, the single particle reduced width is compatible for both ZR and FR calculations, with less than 2% difference. This is expected since equation (3.34) shows that the radial wave function depends on the interaction between the core and the transferred particle only, and the internal structure of the projectile does not play any role. tion 3.2.3. Some precautions have to be made first: the radial form factor has to be calculated in the region where the nuclear forces become negligible, this translates for the bound form factors as an asymptotic behaviour with respect to the Whittaker function. However, the weakly bound approximation is based on the assumption that the radial part of the wave function is similar for both bound and unbound levels. This is almost true in the inner region of the nucleus (see figure 3.3). Therefore, when taking into account the different conditions aforementioned, the radial part of the wave function should be calculated at the surface of the nucleus, where the wave function for the unbound state is equal to both the bound wave function, and the irregular Coulomb wave function. Figure 3.3 shows the normalized radial part of the wave functions for both bound and unbound levels, where we can see that they are equal in the inner part (r < R), and then have different asymptotic behaviours (r > R), corresponding to Whittaker and Coulomb functions, respectively.

For consistency, the DWBA calculations are performed in zero range approximation with the same code dwuck4. The same levels as in subsection 3.3.2.a are considered. This time, the form factor is calculated for different binding energies. The sign convention of dwuck4 is used here: bound states have negative energies, while those of unbound ones are positive. The calculations are performed from E b = -800 keV up to E b = +1 MeV. Such high unbound energy was chosen to test the limits of the approximation studied here. For each binding energy, the form factor was calculated and both the reduced width γ 2 s.p and differential cross-sections were extracted. The latter was normalized with respect to the measured data points in order to obtain the corresponding spectroscopic factor. The results are summarized in figure 3.4 where the evolution of the spectroscopic factor as a function of the binding energy is shown. Each point is a result from the dwuck4 calculations. The prescription of Becchetti et al. [START_REF] Becchetti | 12C(7Li, t)16O and stellar helium fusion[END_REF] is applied: the spectroscopic factors obtained for bound energies (negative) are fitted linearly and extrapolated up to unbound energies (positive). The dashed lines show this extrapolation. The difference between the calculated and the extrapolated spectroscopic factor at E b = 1 MeV is less than 10% and is within the error bars. The slope of the linear extrapolation is larger for high ℓ-values. Note that the spectroscopic factor tendency starts to fluctuate for ℓ = 0 at unbound energy of +800 keV, this fluctuation appears at higher energies for higher ℓ-values. Noteworthy that Becchetti et al. made their prescription based on an α-particle transfer, which may be more suitable for these kinds of approximations than for a proton transfer.

The weakly bound approximation is then compared to exact calculations at unbound energies. Note that for the reduced single-particle width γ 2 s.p , the experimental data are not needed. The calculations are performed at two different channel radii, R = 5 and 7 fm. Figure 3.5 shows the calculated reduced single-particle width in solid lines according to the relation (3.43), while the dashed lines indicate the results of the weakly bound approximation, i.e., the reduced width calculated for a bound energy of E b = 50 keV and assumed the same for all unbound energies. For the channel radius R = 5 fm, the evolution of the reduced single particle width is not linear for ℓ = 0 and ℓ = 1: the wave function did not reach its asymptotic behaviour yet for low momenta, while it is more confined due to the centrifugal barrier corresponding to higher orbital angular orbital momenta. For the radius channel R = 7 fm, the wave-functions have reached the asymptotic behaviour, as shown in figure 3.3, and the evolution of the reduced single-particle width is linear. However, as for the spectroscopic factor, the smooth evolution of the reduced single-particle width is lost at E r = 800 keV for the ℓ = 0 transferred orbital angular momentum.

Finally, the proton widths Γ p = C 2 S × Γ s.p are computed using the different prescriptions presented. Figure 3.6 shows the evolution of the proton width as a function of the binding energy, with channel radius set at R = 7 fm. From this comparison, it appears that the weakly bound approximation, with both spectroscopic and form factors calculated at a small binding energy, induces between 2% to ≈30% error on the width for levels that are unbound with 1 MeV. The accuracy of this approximation depends widely on the transferred angular momentum, without any explicit trend for the dependence.

Summary

From the comparison between different calculation approaches, the strategy for extracting the partial proton widths is as follows:

-The spectroscopic factor C 2 S is deduced from the normalization, with respect to data, of differential cross-sections calculated in Finite Range approach. For unbound levels, the spectroscopic factor is calculated for different binding energies and then extrapolated to the correct unbound energy. The code fresco will be used for this purpose.

-The reduced single particle width γ 2 s.p is deduced from the radial form factor calculated at the exact binding energy, whether the level is bound or not, since the γ 2 s.p does not depend on the ZR nor FR calculations. The code dwuck4 will be used for this purpose.

For both codes used, the same optical and binding potentials are used.

Results and discussion

The angular distributions obtained in Chapter 3 fitted with the calculated FR-DWBA curves are displayed in figures 3.7 and 3.8, showing the angular distributions for bound and unbound levels, respectively. For unbound levels, the spectroscopic factors are obtained from the extrapolation technique to the binding energy of the populated state (see section 3.3.2.b). The single particle widths are calculated at the correct The results are summarized in table 3.3, along with results obtained in the previous study of Vernotte et al. [START_REF] Vernotte | Si( 3 He,d) 31 P reaction at 25[END_REF]. Since two different J values are possible for each transferred ℓ value, J = ℓ ± 1/2, the spectroscopic factor is given as the product (2J + 1)C 2 S. The excitation energies and spin-parity assignments of the literature [31] are also reported. The spectroscopic factors obtained in the present work are 20% to 40% smaller than those obtained in the work of Vernotte et al. which were obtained using ZR-DWBA calculations. For bound levels, the reduced single particle proton γ 2 s.p width is given, while the proton width is given for unbound levels.

All observed levels in the previous 30 Si( 3 He,d) 31 P study have been observed as well in the present work. For the strongly populated states, the extracted angular distributions and their corresponding DWBA fits are in a good agreement with the previous 30 Si( 3 He,d) 31 P study. However, some angular distributions are better described with FR-DWBA curves corresponding to different ℓ values. For instance, the angular distribution of the level at E x = 6933 keV is better fitted with a ℓ = 3 curve, even though it disagrees with the J π = 5/2 + spin-parity assignment, rather corresponding to ℓ = 2, for which the results are also displayed. For the level at E x = 7976 keV, the angular distribution is better described by an ℓ = 2 angular momentum transfer, where the previous work has suggested both ℓ = 2 or 3.

For levels at excitation energies E x = 6826, 7082, 7851 and 8078 keV, angular distributions have been extracted but were not well described with DWBA calculations in the work of Vernotte et al. [START_REF] Vernotte | Si( 3 He,d) 31 P reaction at 25[END_REF]. In the present study, the angular distributions are well described by FR-DWBA curves corresponding to ℓ = 5, 1, 1 or 2 and 1 respectively, and agree with the existing spin-parity assignments, except for the E x = 8078 keV state which has a J π = 11/2 -spin-parity assignment deduced from a 24 Mg( 16 O,2α p γ) 31 P [START_REF] Ionescu-Bujor | High spin structure and intruder configurations in 31 P[END_REF]. Note that the angular distribution of this state lacks data at forward angles, so no strong assumption can be made from our work.

The improved resolution of the present study made possible the separation of several doublets and thus the extraction of the angular distributions for each component. That is the case for the doublets E x = 7719 -7737 keV and E x = 7851 -7863 keV for which the angular distributions for all components have been extracted and fitted with DWBA curves. For the E x = 7719 keV state, the orbital angular momentum transfer ℓ = 3 is obtained, leading to J π = 5/2 -or 7/2 -spin assignment. A recent 30 Si(p,γ) 31 P direct measurement [START_REF] Dermigny | Thermonuclear reaction rate of 30 Si(p ,γ )[END_REF] suggests J π = (3/2, 5/2) -based on the combination of constrains coming from the γ-ray transitions measured in the 30 Si(p,γ) 31 P study, and the 29 Si( 3 He,p) 31 P reaction [START_REF] Al-Jadir | States of high E x in 31 P[END_REF]. From the combination of all existing spinparity assignments, we suggest that the E x = 7719.5 keV state has J π = 5/2 -. The angular distribution of the E x = 7863 keV state can be described by either ℓ = 0 or ℓ ≥ 3 transferred orbital angular momenta. However, the very weak population of this state may suggest a bad matching condition for the transfer reaction, which would favour large transferred angular momenta. In addition, this resonance has not been reported in the low-energy study of the 30 Si(p,γ) 31 P reaction [START_REF] Kuperus | Energy measurements of proton resonances in light nuclei[END_REF] although two ℓ = 1 resonances were observed in its close vicinity corresponding to E x = 7781 keV and E x = 7898 keV. This could suggest that the contribution of the present resonance is suppressed by the Coulomb and centrifugal barriers, which would indicate a rather high spin. Therefore, ℓ = 3 value is considered for this state.

Weakly populated levels at E x = 7159, 7347, 7446, 7470 and 7691 keV have been observed for the first time through the 30 Si( 3 He,d) 31 P reaction. The angular distribution associated to the E x = 7347 keV state can be described by ℓ = 1 transferred angular momentum, in line with the spin parity assignment J π = 3/2 -, 5/2 -deduced from a 29 Si( 3 He,p) 31 P reaction [START_REF] Al-Jadir | States of high E x in 31 P[END_REF]. However, the ℓ = 2 curve gives similar reduced chi-square values for the fit of the angular distribution at forward angles.

The states at E x = 7159, 7446 and 7470 keV have been observed at angles θ Q3D ≥ 20°, which added an ambiguity to ℓ value assignment. Level at E x = 7159 keV could be fitted with either ℓ = 1 or 3 orbital angular momentum. The level at E x = 7446 keV could correspond to the low energy component of the doublet reported in the literature at E x = 7441 -7442 keV with spin assignment J π = (3/2 + -9/2) and J π = 11/2 + respectively. This would correspond to either ℓ = 2, 3 or 4 values for the transferred orbital angular momentum. The limited angular range of the measured data does not allow discriminating between these possible values, ℓ = 2 and 3 were considered in the present work. Higher ℓ values were not considered since for ℓ = 3, the resonance would significantly not contribute to the total reaction rate, this will be discussed in chapter 5. The E x = 7470 keV state is associated to the E x = 7466 keV level observed in 27 Al(α,γ) 31 P [55], 28 Si(α,pγ) 31 P [56] and 29 Si( 3 He,p) 31 P [START_REF] Al-Jadir | States of high E x in 31 P[END_REF] reactions. Based on these experimental studies, the spin-parity of this state is restricted to J π = (7/2, 9/2) - [START_REF] Endt | Energy levels of A = 21-44 nuclei (VII)[END_REF]. Such spin-parity assignment corresponds to an ℓ = 3 or ℓ = 5 angular momentum transfer for the 30 Si( 3 He,d) 31 P reaction. The spectroscopic factor was extracted for the lower angular momentum transfer, having in mind the contribution of the resonance to the total reaction rate.

The level at E x = 7691 keV has been unambiguously observed in the present experiment at θ Q3D = 6°, 10°and 12°. At higher angles, its associated peak overlaps with the 29 P state at E x = 3105 keV populated by the 28 Si( 3 He,d) 29 P reaction. For large Q3D detection angles, the angular distribution of the 31 P state at E x = 7691 keV was thus obtained after subtracting the contribution from the 29 P level. Such contribution was estimated using the angular distribution obtained in the 28 Si( 3 He,d) 29 P study [START_REF] Dykoski | Single proton transfer to 29,30 P states[END_REF] performed at the same incident energy as the present experiment. The angular distribution of the E x = 7691 keV state of 31 P is best described by an ℓ = 3 orbital angular momentum transfer, giving J π = 5/2 -or 7/2 -. This level was not observed in the work of Ref. [START_REF] Vernotte | Si( 3 He,d) 31 P reaction at 25[END_REF] because of the presence of the 29 P contamination at all angles.

Uncertainty estimation

The uncertainty propagation and estimation is a crucial part in the determination of the proton width. This uncertainty has multiple contributions: (i) the optical potential parameters for the entrance and the exit channels, (ii) the binding potential for the relative 30 Si+p wave function, and (iii) the experimental uncertainties associated with the cross-section normalization. The uncertainties corresponding to the potential used in DWBA calculations are the ones dominating the final results. They are investigated in this section using various Monte Carlo methods. The results for the spectroscopic factor and partial width will be presented.

Bayesian approach

Quantifying the uncertainties related to optical potential and reaction models have been a long-standing issue despite the widespread use of the DWBA method in nuclear reaction study. Even though the DWBA calculations have been used for more than 4 decades, the propagation of the associated uncertainties in a statistical meaningful way has been tackled only a few years ago. This was made possible thanks to the decreasing computational time. The Bayesian approach has been introduced to low energy nuclear theory and nuclear reactions by a series of recent papers [START_REF] King | Direct Comparison between Bayesian and Frequentist Uncertainty Quantification for Nuclear Reactions[END_REF][START_REF] King | Uncertainty quantification due to optical potentials in models for (d, p) reactions[END_REF][START_REF] Lovell | Constraining transfer cross sections using Bayes' theorem[END_REF] where both frequentist and Bayesian approaches have been applied and compared.

The Bayesian inference aims to establish the credibility of a given model built around a set of parameters θ, to describe some experimental data D. Models and parameters are not chosen ex nihilo, but are rather selected based on some "belief", which can be the physicist's intuition, or experience gained from previous experiments performed in different conditions. Bayesian statistics gives the mean to rationally update these beliefs based on the newly observed data. Note that this inductive reasoning is very much how scientific thinking should be.

For DWBA calculations, the parameters θ would be the optical and binding potential parameters, and data D would correspond to the differential cross-sections for elastic scattering and transfer. The Bayesian approach is based on Bayes theorem, that is expressed as [START_REF] Joyce | Bayes' Theorem[END_REF]:

P (θ|D) = P (D|θ)P (θ) P (D) , (3.49)
where the notation P (X|Y ) denotes the probability of X given that Y is true. P (θ) are called the prior probability distributions of the model parameters and represents our "beliefs" about whether the parameters are suitable for describing the data. P (θ|D) is the posterior probability, P (D|θ) is the likelihood function and represents the probability of observing the data given the chosen distribution for the model's parameters. The likelihood is deduced from fitting the model to the data. Finally, P (D) is called the evidence1 and represents the probability to observe the data given any possible model, in other words, evidence can be expanded as:

P (D) = θ P (D|θ)P (θ)dθ. (3.50)
Computing the evidence numerically is highly demanding in computational resources, and with high dimensional problems such as the optical potential fits, estimating the integral (3.50) becomes analytically impossible. Furthermore, the probability of the whole model θ is not the most interesting part of a statistical analysis. We rather want to extract the individual posterior distribution of each parameter θ i , which can be obtained such as:

P (θ i |D) = j̸ =i P (θ|D)dθ j , (3.51)
this operation is called the marginalization of parameters other than θ i and is also a tedious-to-calculated multidimensional integral.

Markov Chain Monte Carlo

Monte Carlo sampling is a common method to overcome the complexity of analytical expressions with random variables. It is based on the random generation of numerous samples and the study of the subsequent ensembles using a probabilistic model to approximate numerical solutions [START_REF] Metropolis | The Monte Carlo Method[END_REF]. However, for the Bayesian approach, a classical Monte-Carlo sampling is not convenient because it needs to evaluate the convolution of two probabilities that need to be normalized by the evidence, which is as mentioned above, an integral over the whole parameters space. The Markov Chain Monte Carlo (MCMC) algorithms give the solution for sampling the relevant phase space and reduce the dimensionality of the problem. A Markov Chain is a random process where the probability of pulling the sample x t+1 at step t + 1 depends solely on the outcome obtained at step t, i.e. P (x t+1 |x t , x t-1 ...x 0 ) = P (x t+1 |x t ). Monte-Carlo sampling is used to generate the transition probability from step t to t + 1. Furthermore, the properties of such systems allow the sampling of a desired probability distribution π(x) of the random variable x, provided that we know a function f (x) proportional to π(x).

The most famous MCMC method is the Metropolis-Hastings algorithm [START_REF] Hastings | Monte Carlo sampling methods using Markov chains and their applications[END_REF]. It is suitable for sampling multiple parameters. The procedure is as follows:

• Initializing at t = 0 with x t = x 0 .

• A sample x ′ is drawn from a chosen probability distribution.

• The acceptance rate α = f (x ′ ) f (xt) = π(x ′ ) π(xt) is calculated. ⋄ If α ≥ 1 the sample x ′ is accepted and the next step is x t+1 = x ′ .
⋄ Otherwise, a random number u sampled uniformly as u ∈ [0, 1] * If α ≥ u the sample x ′ is accepted and the next step is x t+1 = x ′ . * Otherwise, the sample x ′ is rejected.

It is common to express the MCMC sampling as "walkers" exploring the phase space of θ, and the probabilities of each parameter is updated at each step of these walkers. MCMC algorithms are very useful for the Bayesian statistics, where the posterior distribution P (θ|D) can be sampled from P (D|θ)P (θ), without calculating the evidence. Marginal posteriors are also easily calculated with MCMC methods.

In order to sample a posterior distribution that is the closest to the real one, the first samples of the chain are discarded in order to get rid of the influence of the initialization. This procedure is called burn-in and is, in other words, the process of "forgetting the history" of the system. It is also important to select samples each N jump steps, so correlations between samples are not introduced. A good practice for MCMC methods is to consider multiple Markov Chains simultaneously, with different initial parametrization. In the following subsection the Bayesian framework will be used to vary the optical potential parameters and explore their effect on the spectroscopic factor and the fit of the angular distributions for the elastic scattering and the transfer reaction.

Influence of the optical potential parameters

The influence of the optical parameters on the spectroscopic factor was investigated with a Bayesian MCMC method. The calculations have been done with a MCMC code based on ZR-fresco calculation. Zero Range was used to keep the computational time reasonable. The Python Fresco UNcertainty Funkutation (PFUNK) [START_REF] Marshall | Bayesian analysis of the 70 Zn(d, 3 He) 69 Cu transfer reaction[END_REF] varies the desired potential parameters by a MCMC sampling, where the prior probabilities are updated at each step by fitting the results to angular distributions' data. For the optical potential parameters, the DWBA calculations are fitted to both elastic scattering and proton transfer data. As a first step, only optical potential parameters for the 30 Si+ 3 He entrance channel are varied. The prior distributions of the parameters have normal probability distributions, with mean values taken from table 3.1, and with Scatter plot of the real potential depth V r and the radius r r for the entrance channel, each point reproduces the 30 Si + 3 He elastic scattering and 30 Si( 3 He,d) 31 P transfer data. The red curve is the fit of the data points with a power law.

width of 20% of the mean values: σ θ = 0.20µ θ with θ the index running over optical potential parameters for the entrance channel. Note that the radius of the real part of the potential r r has been kept fixed, as it was done for the original fitting procedure in ref [START_REF] Vernotte | Optical model analysis of 3 He elastic scattering from s-d shell nuclei at 25 MeV[END_REF]. Indeed, they observed a continuous degeneracy for the parameters, especially between the depth V r and the radius r r , that can be expressed as V r r n r = c te . This relation is reproduced with the procedure presented in this work, when fitting elastic and transfer data. Figure 3.9 shows the scatter plot of both parameters corresponding to the posterior distributions obtained with MCMC.

The spectroscopic factor is considered as a parameter of the model, and hence its prior distribution is defined. The code PFUNK assumes a half-normal2 distribution with a mean µ = 0.01 and a large width σ = 0.1. The influence of the parametrization of the C 2 S prior distribution has been tested (µ = 0.01 -0.5 and σ = 20% -500%) and found to have no influence on its posterior distribution when the sampling is sufficiently large.

The sampling is performed with 500 walkers and 1000 steps. The first N burn-in = 500 steps are discarded in order to keep the most converged samples, and data are picked every N jump = 20 steps. The final samples consist of 12500 points for each sampled parameter. Figure 3.10 shows the pairwise correlation plots obtained for the E x = 7898 keV state.

The posterior distribution for the spectroscopic factor is shown in the insert included in figure 3.10. The results are best fitted with a log-normal distribution. The median value and variance of this probability distribution can also be easily deduced from the cumulative distribution by considering the 50 th percentile for the median, and the 16th and 84th percentiles to extract the variance (see appendix C). When parameters on the entrance potential are varied only, the 1σ uncertainty is estimated to be less than 7%. The pair-wise correlation plots of figure 3.10 show some expected correlations such as the ones between W v and r i similar to the one for the real part of the potential, but also between r i and a i . The correlation between real and imaginary diffusenesses a r and a i is however not expected.

A set of 1000 sampled parameters of the entrance potential has been randomly selected, it was used to calculate the angular distribution of the elastic scattering. Figure 3.11 shows the results compared to the elastic scattering data points. The coloured bands represent the 1 and 2 σ confidence intervals.

The influence of the exit channel optical potential parameters is investigated in a second step. Since there is no available experimental data for elastic scattering of deuterons on 31 P, the prior distributions for the exit potentials are chosen to be more informative: the width of the normal distribution is taken as 10% of the median values. The calculations are done in the same way as described earlier. The correlations between the parameters are shown in figure 3.12 and the spectroscopic factor distribu- tion is shown in the insert, the corresponding uncertainty is around 25%. The pairwise correlation plots for the parameters of the entrance channel are similar to figure 3.10. These results show that the spectroscopic factor is sensitive to the uncertainty on the optical potentials, and the uncertainty is dominated by the exit channel, due to the lack of experimental data for constraining the elastic scattering. The impact of the optical potentials, both for the entrance and exit channels, is represented in Figure 3. [START_REF] Gratton | Mixing along the red giant branch in metal-poor field stars[END_REF] showing the transfer angular distribution and the corresponding confidence intervals for the calculated distribution using 1000 sampled sets of potentials. The transfer data are better reproduced by the Bayesian fit, especially between θ = 20°and 30°.

The subsequent spectroscopic factor is 7% larger than the one obtained with a simple DWBA fit. The improvement of the fit is mainly due to the adjustment of the potential parameters of the exit channel.

The optical potentials have an impact only on the spectroscopic factor, since the single-particle width depends on the binding potential exclusively. The impact of the latter is investigated in the next subsection.

Influence of the binding potential parameters

The influence of the overlap functions on the spectroscopic factors for (d, 3 He) transfer reactions was highlighted in the work of Flavigny et al. [START_REF] Flavigny | Single-particle strength from nucleon transfer in oxygen isotopes: Sensitivity to model parameters[END_REF]. In the present subsec- The red curve is obtained with simple ZR-DWBA calculations using parameters listed in table 3.1, and with binding energy of E b = 100 keV. tion, we investigate the impact of the binding potential parameters using a Direct MC method because it requires sampling two parameters only. Indeed, since the depth of the volume part of the potential is adjusted to reproduce the observed binding energy, only the radius and the diffuseness of the potential well remain to be sampled. The depth of the spin-orbit is fixed to its value listed in table 3.1. For this part, the calculations are performed using dwuck4 because it allows calculations for unbound levels, and the analysis is done in simple frequentist framework. The radius r and the diffuseness a are sampled according to a normal distribution with mean values of r = 1.25 fm and a = 0.65 fm, and with widths of 15% and 10%, respectively. The spectroscopic factor C 2 S and the single particle width Γ s.p are extracted for each sample. The probability distributions obtained for the E x = 7898 keV state, are shown in figure 3.14, along with the correlation plot and the probability distribution of the product Γ p = C 2 S × Γ s.p . A strong correlation is observed between the spectroscopic factor and the single-particle width. This gives rise to a strongly peaked probability distribution for the proton width, with very low uncertainties. This distribution cannot be described analytically, and thus the cumulative distribution function is used to extract the 1σ uncertainty interval.

The strong correlation between the C 2 S and Γ s.p results from the fact that both quantities are obtained using the same radial wave function of the system. The proton width is most sensitive to the region of the total radial wave function that is probed by the direct reaction used, i.e. ( 3 He,d), which takes place close to the nuclear surface [START_REF] Keeley | Private Communication[END_REF].

Uncertainty of proton widths

The investigation of the impact of the parameters of the potentials used in DWBA calculations showed that the final uncertainties on the proton width are dominated by the parameters of the optical potential, which induce about 25% uncertainty on the spectroscopic factor. These uncertainties are dominated by the exit channel, since the potential parameters are not constrained with experimental elastic data points. The impact of the binding potential on the final uncertainties of proton width is negligible, and this is due to the high correlation between the spectroscopic factor and the singleparticle width. The DWBA model was used to describe the angular distributions obtained through the one proton transfer reaction 30 Si( 3 He,d) 31 P. The spectroscopic factors were determined from the normalization of the theoretical differential crosssection with respect to those measured experimentally. The shape of the angular distributions was used to deduce the transferred angular orbital momentum. These spectroscopic quantities were used to calculate the proton width of 31 P unbound states. The codes fresco and dwuck4 were compared and the assets and liabilities of each one were discussed. We showed that the main source of uncertainties arises from the optical potential used in the DWBA calculations. The impact of these uncertainties were estimated using a Bayesian approach and a typical 25% uncertainty for the proton widths was found.

4

Direct measurements of the 30 Si(p,γ) 31 30 Si+p resonances using the 30 Si( 3 He,d) 31 P transfer reaction has shown its limits for resonance energies E c.m. r ≳ 600 keV. At these energies, the proton width starts to be non-negligible compared to the total width of the resonances. Other ingredients, such as the total γ-width, become necessary to compute the resonance strength. Luckily, at those higher energies, direct measurements of resonance strengths become possible, without the need for a model to describe the reaction mechanism.

The strength determination of

In this chapter, we will present the direct strength measurements for 6 resonance energies through the 30 Si(p,γ) 31 P radiative proton capture reaction in inverse kinematics. It allows the detection of both reaction products: the compound nucleus will have the same momentum as the beam in the laboratory frame, and will thus exit the target, and be measurable as single events, or in coincidence with the emitted γ-rays. The experiment was performed in August 2021 using the DRAGON spectrometer at TRIUMF facility. The beam production and experimental setup are presented in the two first section, followed by the description of the data analysis procedures and the attached challenges. Finally, the results are presented and discussed.

ISAC facility at TRIUMF

The direct measurement of the 30 Si(p,γ) 31 P reaction in inverse kinematics was performed in August 2021 at TRIUMF laboratory, Canada's particle accelerator centre. The stable and radioactive beams used at TRIUMF are produced at the Isotope Separator and ACcelerator (ISAC) facility. The facility consists of two experimental areas: ISAC-I where isotopes are accelerated up to 1.8 MeV/u, and ISAC-II, where isotopes with masses up to A = 150 can be accelerated to an energy of 6 MeV/u, and up to 16 MeV/u for lighter isotopes with A < 30. The Advanced Rare Isotope Laboratory (ARIEL) is a new facility under construction that will soon deliver high intensity exotic beams. The ISOL (Isotopic Separation Online) technic is used for the production of radioactive beams: high intensity (50-100 µA) protons accelerated by the cyclotron collide with a thick target made of a heavy isotope, this creates radioactive species through spallation, fragmentation and/or fission reactions. The target is heated to temperature up to 2300°C, for a quick release of short-lived isotopes. Target materials are chosen according to the desired beam.

The stable isotopes are produced with the Off-Line Ion Source (OLIS) [START_REF] Jayamanna | Off line ion source terminal[END_REF]. It has been used to produce the 30 Si beam, and it is described in the following.

OLIS ion source

The Off-Line Ion Source facility consists of a high voltage terminal containing three different setups: a Microwave Ion source (MWIS) that can provide up to 40 different singly or doubly charged isotopes, a surface ion source which can operate in a hybrid surface-arc discharge mode, which is mainly used for the production of alkali and semi-alkali isotopes, and a multi-charge ion source (MCIS). The OLIS facility can operate simultaneously with the online radioactive ion source terminal, delivering either ISAC-I or ISAC-II experimental halls.

For the 30 Si(p,γ) 31 P experiment, the 30 Si beam was produced within the multicharge ion source (MCIS). It is a mobile and self-contained station, based on the Supernanogan electron cyclotron resonance (ECR) commercialized by Pantechnik, in which the magnetic circuit is entirely made with permanent magnets both for the radial and longitudinal fields, so the total electrical power is extremely low. The maximum extracting voltage is 30 kV. The source can provide highly charged ions, but it is restricted by the ISAC-I accelerators that accept beams with A/q < 7 for energies above 150 keV/u.

Acceleration and beam transport

An overview of the beam lines from OLIS to DRAGON is shown in figure 4.2. In the ISAC-I experimental area, the beam is accelerated in two stages. First, the beam is extracted from the ion source with an energy of 2 keV/u and goes through a prebuncher that separates beam bunches by 85 ns. This step improves the beam quality and transmission. The beam is transported through the Low Energy Beam Transport (LEBT) line up to the Radio Frequency Quadrupole (RFQ) accelerator [START_REF] Poirier | CW performance of the TRIUMF 8 meter long RFQ for exotic ions[END_REF]. The latter consists in nineteen ring resonator with a frequency of 35 MHz, which create an oscillating electric field that accelerates the beam ions up to 150 keV/u. The transmission through the RFQ accelerator is around 80% [START_REF] Poirier | CW performance of the TRIUMF 8 meter long RFQ for exotic ions[END_REF].

The beam is then transported to the second stage of the acceleration through the Medium Energy Beam Transport (MEBT) line. The Drift Tube Linac (DLT) [START_REF]Acceleration of radioactive ions[END_REF] accelerates the beam to its final energy ranging from 117 keV/u to 1.8 MeV/u with more than 95% transmission efficiency. The DLT accepts beams with 2 ≤ A/q ≤ 7 which is well below the A/q ≤ 30 that the RFQ can handle. If this condition is not fulfilled, a carbon stripping foil is inserted at the MEBT section and the desired charge state is selected with two magnetic dipoles.

Finally, the accelerated beam is transported through the High Energy Beam Transport (HEBT) line up to the Prague magnet where the beam energy can be measured. A subsequent buncher can be enabled to reduce the energy spread down to ∆E/E ≈ 0.1%, as well as an additional carbon stripping foil to eliminate isobaric contaminating elements. The beam is then transported to the experimental stations at ISAC-I: DRAGON, TUDA, etc. Note that the beam can be deflected downstream the magnet Prague toward the experimental hall ISAC-II, where the beam can be accelerated up to 6 MeV/u for masses A < 150 and up to 16 MeV/u for masses A < 30.

DRAGON spectrometer

The Detector of Recoils And γ-rays Of Nuclear reactions (DRAGON) is an electromagnetic separator that was specifically designed to measure the cross-sections of proton and alpha radiative capture reactions at energies relevant for astrophysical processes. The beam suppression factor, that can reach 10 -13 when coincidences with γ-rays are considered, is a key feature for measuring cross-sections of the processes occurring at sub-Coulomb barrier energies, corresponding to stellar temperatures. DRAGON has measured over 2 decades reactions with various beams from A = 3 up to A = 76, the curvature radius of its first magnetic dipole being the main limitation for measurements at higher masses. DRAGON is equipped with a windowless gas target surrounded by an array of BGO γ-ray detectors, an electromagnetic separator and heavy ion detectors at the focal plane, a sketch of the overview is shown in figure 4.3. Each of these components is detailed in the following subsections.

Windowless gas target

The gaseous targets used for DRAGON experiments are contained in a trapezoidal cell that can contain up to 10 Torr of hydrogen or helium gas, to study proton (p, γ) or alpha (α, γ) radiative capture reactions, respectively. The gaseous targets have the advantage to be chemically pure, and the low background is also achieved by the use of a windowless cell, thus getting rid of the background signal that can be induced from the interaction of the beam with the window's material, as well as achieving a better beam transmission and less energy loss and straggling. Figure 4.4 shows the sectional view of the windowless gas target. The gas is contained by multiple stages of differential pumping, figure 4.5 shows a schematic of the pumping system. In the first stage, the gas leaking from the beam apertures of the gas cell are recirculated by 5 large Roots blowers that raise the outflow gas pressure to more than 40 Torr [START_REF] Hutcheon | The DRAGON facility for nuclear astrophysics at TRIUMF-ISAC[END_REF]. The recirculation of the gas can induce impurities from pump-oil vapour for example. The impurities are reduced by liquid-hydrogen-cooled zeolite adsorption trap. The trapezoidal shape of the target cell deflects the gas jets induced by the recirculation process. The second pumping stage uses 6 turbo-molecular pumps to reduce the pressure outside the cell to ≈ 10 -6 Torr. This strong pressure differential keeps the beamline under vacuum for optimal operating conditions. All the pumping systems are enclosed in a thin aluminium box in order to reduce the γ-ray absorptive material between the target and the BGO detectors. The differential pumping induces a distortion in the target density profile. The latter was measured using a BGO detector that was moved along the beam axis along the length of the target, while measuring the yield of the 12 C( 3 He,pγ) 14 N [94]. The gas cell is equipped with two Surface Silicon Barrier (SSB) detectors mounted at 30°and 57°with respect to the beam axis (see figure 4.4). These detectors are used to measure the number of elastically scattered target particles, which is proportional to the number of incoming beam ions. Unfortunately, only the detector at 57°was available during the 30 Si beam run. The pressure and the temperature are recorded along the duration of the run. The determination of the incident beam particles will be discussed in the analysis section 4.3.1.

BGO γ-ray detector array

The target cell is surrounded by an array of 30 Bismuth Germanate Oxide (BGO) crystals, with hexagonal shape suitable for a compact stacking, covering ≈ 90% of 4π solid angle. Figure 4.7 shows the design of a single BGO detector, and the whole array in the DRAGON configuration surrounding the gas cell. BGO scintillators were chosen for their good timing capability to reject background induced by radioactive beam particles, and their high density which leads to good detection efficiency. Each crystal is coupled to a Photo-Multiplier Tube (PMT) which converts the scintillation photons into an electric signal which is fed to the Data acquisition system (DAQ) of DRAGON. The BGO detector array is used to detect prompt γ-rays in order to identify in coincidence the recoils that are detected in the focal plane. A good energy resolution is thus not crucial and the typical 10% (FWHM) is sufficient for most experiments conducted at DRAGON. The efficiency of the BGO array is crucial for extracting the experimental yields in coincidence with a good statistics. The detection efficiency, which varies between 45 and 80%, depends on the decay scheme of the resonance, and it is determined using geant3 simulations (see section 4.3.5) [START_REF] Gigliotti | Efficiency calibration measurement and simulation of the DRAGON BGO gamma ray array at TRIUMF[END_REF].

Electromagnetic Mass Separator

The DRAGON electromagnetic elements were optimized for measuring radiative capture in inverse kinematics at energies of astrophysical interest with high beam suppression factor. For proton and α capture reactions, the momenta of the recoils and the beam particles are very close, but with different energies. The DRAGON electromagnetic mass separator consists of two magnetic (M) and two electric dipoles (E), arranged in an M-E-M-E configuration (see figure 4.3). The characteristics of each dipole is listed in table 4.1. Additional quadrupole and sextupole lenses are used for focusing and trajectory corrections. The particles exiting the target are separated in two steps: according to the charge then to the mass. Recoils and beam particles exit the gas target with a distribution of charge states, the charge exchange occurs from the interaction of the particles with the target molecules. The charge state distribution can be calculated theoretically or measured experimentally, this part will be detailed in section 4.3.8. The first magnetic dipole MD1 is used to select a single charge state. From the Lorentz law and the expression of the centripetal force, one obtains:

qvB = mv 2 ρ M =⇒ ρ M = mv qB , (4.1)
the magnetic field B is set so that the particles with charge state q are bent with a fixed radius ρ M . Downstream MD1, a pair of slits are displaced in order to let through the particles with the desired curvature radius. Particles with different charge states are deflected and stopped by the slits. A Faraday cup (FCCH) is also located downstream MD1 and is used to calculate the charge state distributions of ions exiting the target. The subsequent electric dipole ED1 separates the particles according to their masses. Particles with different masses than the recoils mass are deflected and stopped by slits located downstream ED1. The electric rigidity is:

|E |ρ E = mv 2 q =⇒ ρ E = mv 2 E , (4.2)
where E is the electric field set by the dipole ED1. The charge and mass separation are repeated a second time with the help of MD2 and ED2 in order to achieve a suppression factor ranging between 10 -10 and 10 -15 . The DRAGON BGO array surrounding the gas target. Both figures adapted from [START_REF] Hutcheon | The DRAGON facility for nuclear astrophysics at TRIUMF-ISAC[END_REF]. The electromagnetic element (MD1) can be used to measure the energy of the incident beam, according to the relation [START_REF] Hutcheon | Measurement of radiative capture resonance energies with an extended gas target[END_REF]:

Dipoles

E A = c M D1 qB A - 1 2uc 2 E A 2 , (4.3)
where c is the speed of light, q the charge state of the beam, A the mass number, u the atomic mass unit (in amu), B the magnetic field (in Tesla) set in MD1 that has the constant c M D1 = 48.15 [START_REF] Vandenberg | Models of Metalpoor Stars with Gravitational Settling and Radiative Accelerations. II. The Age of the Oldest Stars[END_REF] MeV T -2 . The mass-to-charge ratio of the beam can also be derived from the electromagnetic elements of DRAGON using the relation [START_REF] Hutcheon | The DRAGON facility for nuclear astrophysics at TRIUMF-ISAC[END_REF]:

A q = 2468 B 2 V , (4.4)
where B is the magnetic field in MD1 in Tesla and V is the voltage of ED1 in kV. The angular acceptance of DRAGON is θ max = ±20 mrad, which is large enough for most (p, γ) reactions in inverse kinematics. Indeed, for the 30 Si(p,γ) 31 P reaction, the recoil cone ranges between ±8.5 mrad and ±6.5 mrad, for beam energies between 500 keV/u and 1 MeV/u, respectively. However, some (α, γ) reactions with low beam mass have larger cone angles. This reduces significantly the transmission efficiency ε t of the separator, which is estimated by the mean of geant3 simulations of the gas cell and the electromagnetic elements of the separator (see section 4.3.6).

Focal plane detection system

Once the recoils have been selected by the electromagnetic mass separator, they are detected by a set of heavy ion detectors located at the focal plane of the separator. The system consists in a pair of Micro-Channel Plate (MCP) detectors for local timeof-flight measurements, followed by a hybrid chamber that consists in a Double Sided Silicon Strip Detector (DSSSD) enclosed in an Ionization Chamber (IC).

The working principle for MCP detectors is as follows [START_REF] Vockenhuber | A transmission time-of-flight system for particle identification at a recoil mass separator at low energies[END_REF]: a thin diamond-like carbon (DLC) foil (≈ 5 µg/cm 2 ) is placed perpendicularly to the trajectory of the recoils. When an ion goes through the DLC, electrons are produced, and then reflected by a 45°electrostatic mirror. The electrons are detected by the MCP. The latter is made from a highly resistive material to which a bias is applied, with thousands of tubes with small diameter (15-50 µm) over the surface of the plate. These tubes have a similar function as a PMT: each electron entering a tube produces up to 10 6 secondary electrons. The gain is not controlled, but the timing properties are well above any available PMT. The MCP are used at DRAGON for their excellent timing resolution of 300-400 ps: the two MCP detectors are separated by 59 cm, with the downstream detector MCP1 being twice as large as MPC0, in order to have larger acceptance for diverging recoils [START_REF] Vockenhuber | A transmission time-of-flight system for particle identification at a recoil mass separator at low energies[END_REF]. Figure 4.8a shows the local time-of-flight detection system at DRAGON along with a scheme for the MCP working principle. Unfortunately, during the 30 Si run, only MCP0 detector was functional, the local-time-of flight measurements could not be performed.

The second detection stage is the hybrid chamber that works as an ∆E -E detector (figure 4.8b). First the recoils lose part of their energy in the IC chamber [START_REF] Vockenhuber | Improvements of the DRAGON recoil separator at ISAC[END_REF]. The latter is 25 cm long equipped with 3 anode strips, and can contain between 10 and 20 Torr of isobutane gas, separated from the MCP detection chamber by a thin Mylar window. The passage of charged particles creates pairs of electrons and ions that are accelerated by a 50 V/cm electric field. A Frisch grid on ground potential placed between the anode and the cathode corrects for any position dependence on the pulse signal and improves the timing resolution. The mass separation capacity of the IC detector is optimum for A ≥ 20 and E ≤ 500 keV/u. The recoils are fully stopped in the DSSSD located at the end of the hybrid chamber. It is a Micron W1 double-silicon strip detector, with 16 strips of dimension 3 × 50 mm 2 on each side. The charge collection is performed by a 200 nm thick aluminium grid. Figure 4.9 shows the design of the DSSSD and the scheme of the strip. The DSSSD is used for its good energy resolution in heavy ion detection, achieved thanks to the 3.6 eV electron-hole gap and thus a large number of charge carriers. The DSSSD is also used for its timing resolution, around 10 ns, which is used to obtain accurate separator time-of-flight measurements.

The separator time of flight is used to obtain additional 2-3 orders of magnitude to the suppression factor of the DRAGON electromagnetic spectrometer. It corresponds to the time spent by the recoils to travel from the gas target to the DSSSD, which represents 20.7 m length. The triggers used for the separator time of flight will be detailed in the following, along with the data acquisition systems of the different parts of DRAGON.

Data acquisition system

The DRAGON's data acquisition system consists of two independent DAQs, each with independent triggering and readout; the head DAQ for the BGO array and tail DAQ for the rest of the detectors. This configuration reduces the DAQ dead time, and coincidences are reconstructed with the help of digital timestamps, without the fastidious use of hardware gating.

The trigger and timestamps are controlled by FPGA (Field-Programmable Gate Array) integrated to a general purpose VME board called IO32, designed and manufactured in TRIUMF. The timing is set with reference to a 20 MHz quartz oscillator crystal. The trigger logic diagram is displayed in figure 4.10.a. The head DAQ contains the anode signal from the 30 BGO array detectors. Each signal is split into logic and analogue parts. The analogue branch is delayed then sent to a charge to digital converter (QDC), while the logic branch is sent through a Constant Fraction Discriminators (CFD) module. Individual logic signals are feeding a Time to Digital Converter (TDC) while the total OR output is used for the head trigger. The signals of the focal plane detector system, along with the SSB detectors, are also split and converted to ADC and TDC signals. The RFQ timing signal is also fed to both DRAGON DAQs to provide additional timing reference.

The data from both DAQ systems are transferred from the frontend VME processor to the backend computer every second, for efficiency reasons. The events are buffered in the VME which orders the events in a First In, First Out (FIFO) structure. The time difference between the earliest and the latest event t diff is calculated each time a new event is placed in the buffer. If the difference is greater than a user specified value (4 s for example), the coincidence selection is performed: events with time stamps difference lower than 10 µs are sent to the coincidences and singles processors. Other events are sent to the singles' processor only. This matching algorithm is summarized in the flow diagram in figure 4.10.b.

The data acquisition is based on the Maximum Integrated Data Acquisition System (MIDAS) software [START_REF]MIDAS Documentation[END_REF]. TRIUMF also uses the Experimental Physics and Industrial Control Software (EPICS) to control beamline elements and apparatuses, such as gas pump and valves, and to record variables such as pressure and temperature within the gas, dipole fields, and Faraday cup currents.

DAQ live time

Measuring the live time of the acquisition systems is very important for an accurate estimation of the experimental yields. The live time is the fraction of the run time over which the DAQ is available to accept new triggers. It can be estimated by comparing the number of acquired and accepted triggers for single events. But for coincidence ones, the live time calculation becomes more complex as it must take into account the triggers of two independent DAQs. The busy time is calculated instead. For each event that is being recorded, there are a number of lost events, that can be estimated with Poisson statistics as [START_REF] Christian | Design and commissioning of a timestamp-based data acquisition system for the DRAGON recoil mass separator[END_REF]:

n lost = λ n i=0 τ i = λτ, (4.5)
where n is the number of recorded events, λ the rate of generated events (λ = n/T ), τ i is the busy time for the i th event, τ the total busy time and T the total measurement time. The total number of events N collected by the detectors is:

N = n + n lost = n + λτ = n + (N/T )τ N = n 1 -τ /T
The live time, η live = n/N is thus:

η live = 1 -τ /T. (4.6)

Data analysis

The reaction yield is calculated experimentally as the number of recoil ions formed from the proton capture reaction over the number of incident beam ions. The total number of recoils have to be inferred from the number of recoils detected, which depends on the different detection and transmission efficiencies. The yield is given by:

Y = N det r N beam ε BGO τ DRAGON τ M CP ε DSSSD η live f q , (4.7)
where N beam is the number of incident beam particles, ε BGO and ε DSSSD are the detection efficiencies of the BGO array and the DSSSD detectors, τ DRAGON and τ M CP are the transmission efficiencies through the separator and through the MCP grids respectively. f q is the fraction of the selected recoils with charge state q, and η live is the live time fraction of the data acquisition system, for both head and tail DAQs and is given by Eq. (4.6).

If the detected particles at the focal plane are clearly identified as the recoils, with few contaminating leaky beam particles, the yield can be measured with DRAGON in the "singles" configuration. This means that the detection of the γ-rays in coincidence is not needed. In this case, the detection efficiency ε BGO is not considered, and only the live time fraction of the tail DAQ is taken into account.

The procedure for beam normalization, particle identification and efficiency estimation is now presented in the following subsections.

Beam particle normalization

Elastic scattering is used to determine the number of incident beam ions. The scattered protons are detected in the Silicon Surface Barrier (SSB) (see section 4.2.1), and their number is given by:

N SSB p = dσ Ruth dΩ N beam N target ∆Ω, (4.8)
where dσ Ruth dΩ is the differential Rutherford cross-section for elastic scattering, N beam is the number of incident beam ions, N target is the areal density of the target, and ∆Ω is the solid angle covered by the SSB detector. For sufficiently small pressures, the gas in the target cell can be considered as an ideal gas and thus N target is simply:

N target = N A P L ef f R T , (4.9)
where N A is the Avogadro number, R the ideal gas constant, P the pressure, T the temperature, and L ef f the effective length of the gas cell. It is convenient to encapsulate all the constant quantities in a constant that we shall call the normalization coefficient R. From Eq. (4.8) and (4.9) this constant is equal to:

R = R T N A L ef f dσ Ruth dΩ ∆Ω . (4.10)
The number of incident ion beams is then given by:

N beam = R N SSB P (4.11)
Experimentally, the number of beam particles is determined directly from the measurements of the beam current. Indeed, the incident beam particles are stripped from q electrons, so an incident current Ī can be measured, and we have:

N beam = Ī eq , (4.12)
where Ī is the mean value of the beam current measured in the Faraday cup located upstream the target, e is the elementary charge and q the charge state of the beam which is equal to 7 + during the whole experiment. The R coefficient is thus determined as:

R = Ī eq ∆t P N SSB p , (4.13)
where N SSB p is the proton number counted during the time window ∆t, corresponding to the first (and last) 5 minutes of the run. The current is measured at the beginning (and end) of the run, and the beam conditions are assumed to be stable during ∆t. The measurement of each of these physical quantities is detailed in the following.

Current

The current is measured hourly by the Faraday cup FC4, which is located upstream the target. Figure 4.11a shows the current measured over a time-lapse at the beginning of a run. The runs are started and stopped automatically each hour. We compute the mean current value and the associated dispersion σ(I) in a 20 s time interval chosen in order to avoid the peak at the beginning of each measurement corresponding to the insertion of FC4.

Pressure

The pressure in the gas target is measured over the whole duration of the run. Figure 4.11b shows the pressure variations over the run 10140. For the calculations of the R coefficient, the mean pressure over the first 5 minutes is considered. The mean pressure is also estimated for the last 5 minutes in order to calculate the R end coefficient, which is simply the coefficient encapsulating the experimental conditions at the end of the run. This is used to check whether these conditions are stable during the run.

SSB counts

The silicon surface barrier (SSB) detector located inside the gas target cell is used to detect the elastically scattered protons during the whole run. As for the pressure, the number of scattered protons is calculated for the first and last 5 minutes of the run. Figure 4.11c shows the energy spectrum of the SSB1 for run 10140, only for the first 5 minutes. The red dashed lines shows the energy interval for which the number of counts is integrated. Those energy limits were chosen for each beam energy.

Transmission efficiency

The transmission of the beam through the target is never perfect, and it needs to be accounted for in order to have a good estimation of the number of incident beam ions that actually go through the target so that the R coefficient can be corrected. The transmission is computed as the fraction of the beam measured downstream, in Faraday cup FC1 over the beam measured upstream, in FC4: ϵ t = F C1 F C4 . Ideally, these measurements should be performed with an empty gas target since the charge state of the beam is the same in both FC1 and FC4 cups. However, these measurements are sparse. When the target is filled, the charge state distribution (CSD) of silicon ions should be considered for the measurements in FC1 (see Sec. 4.3.8), where the beam is stopped after crossing the gas target. Figure 4.12 shows the results for transmission measurements for both filled and empty gas target. The error bars on the transmission efficiency are bigger when the target is filled with gas due to the additional contribution of the uncertainty on the charge state distribution. These error bars are much larger for the first runs where the beam current was instable. But the values are still compatible with empty target measurements that were done immediately before or after the filling of the cell, except for E beam = 778 keV. The trend in transmission values in squares informs on any change in beam conditions. This can be seen in figure 4. [START_REF] Gratton | Mixing along the red giant branch in metal-poor field stars[END_REF] where the green points are the R coefficient values corrected with the transmission efficiency as R × ϵ t . For example, for beam energy 628 keV/u (E c.m. r = 602 keV), between runs 10170 and 10178, the increasing trend for R value (black points) is compensated by the decreasing transmission seen in figure 4.12. Indeed, a change in the alignment of the beam would change the transmission value and the number of scattered particles detected in SSB1, since the scattering cross-section is sensitive to the incident and emission angles. The second SSB detector would have given more insight and control over this kind of situation since it is positioned at a different angle than SSB1.

Beam ions number

We recall the expression for the number of beam ions as:

N beam = R N SSB P (4.14)
This time, N SSB and P are estimated over the whole duration of the run. The 30 Si beam intensity was high enough to treat each run separately and the number of incident beam ions can be determined for each run. We will also compute the average R coefficient and use it to determine the total number of incident ions for each beam energy.

Independent runs

The level of statistics achieved in the present experiment allows considering each run individually. The total number of incident ion beams can be calculated as the sum of ions of each run using R i , the coefficient computed for each run:

N T ot beam = i R i N SSB i P i , (4.15)
and the associated variance is simply the sum of the variances of each run:

σ 2 (N T ot ) = i R i N SSB i P i × σ 2 (R i ) R 2 i + σ 2 (P i ) P 2 i + 1 N SSB i . ( 4

.16)

Average R One can compute for each beam energy the average R coefficient, by fitting linearly the R i , (as displayed in figure 4.13) with a constant function. The associated reduced χ 2 is :

χ 2 = 1 ν Σ ν i (R i -R) 2 σ 2 i , (4.17)
where σ 2 i is the error on R i for each run. For the experiment presented here, all the considered runs have similar experimental condition, and same duration. For measurement at all beam energies, the fits give χ 2 ≤ 1 (see table 4.2). However, if χ 2 ≥ 1, this would mean that there is an unknown systematic uncertainty σ u that needs to be considered. The uncertainty on R i on each run would be inflated as: σ

′ 2 i = σ 2 i + σ 2 u
, and the value of σ u is found by requiring that χ 2 = 1.

Once the R coefficient determined, the total number of incident beam ions is calculated as:

N T ot beam = R i N SSB i P i , ( 4
σ 2 (N T ot ) = N T ot 2 beam ×     σ 2 ( R) R2 + i N SSB i P i 2 × σ 2 (P i ) P 2 i + 1 N SSB i ( i N SSB i P i ) 2     . (4.19)
Table 4.2 shows the obtained number of ions for both approaches described above. The total number for both procedures is very similar, this justifies the approach using R, but one must make sure that the experiment conditions are stable. The larger errors for the number of ion beam for E c.m. r = 950 keV is due to the beam instability during the corresponding runs, which are the first one to be taken in this experiment.

Separator Time of Flight

The Separator Time of Flight (ToF) is the time difference between the head trigger and the tail trigger (BGO and DSSSD triggers precisely). For beam energies between E lab =507 keV/u and 981 keV/u, the calculated ToF for the recoils ranges from 2.18 to 1.56 µs, respectively. A good agreement is found with the measured time of flight, as shown in figure 4.14 for the lowest beam energy at E lab =507 keV/u. The peak is fitted with a gaussian distribution, and a time window of 3 × τ RF = 260 ns is defined around the mean of the distribution. τ RF = 86 ns is the Radio Frequency period of the pulsed beam. The time window is chosen to include all events that could be beam-correlated.

DSSSD energy spectrum

The 31 P recoils are identified at the focal plane of the DRAGON spectrometer mainly with the mean of their DSSSD energy spectra. A careful energy calibration of the DSSSD is first performed before analysing the resulting spectra.

Energy calibration

The energy calibration is performed for each strip of the DSSSD detector independently, since each strip has its own electronic readout. First, the energy calibration is performed with the use of a triple alpha source, containing 239 Pu, 241 Am and 244 Cm nuclei that emits alpha particles when disintegrating. The main alpha transition for each isotope of the source is fitted with a gaussian. The region containing the satellite peaks corresponding to lower intensity transitions, is excluded from the fit in order to have a good description of the main transition. The same gaussian width is used for all transition. The top panel of figure 4.15 shows the energy spectrum of the triple alpha source for strip number 1 taken as an example. The positions of the peaks, in ADC channels, are then fitted linearly to alpha particles energies listed in table 4.3, according to the relation:

E α = c 0 + ch ADC × c 1 .
The calibration procedure is focused on a good energy alignment between the strips in order to have a resolution sufficient to distinguish between the recoils peak and any leaking beam peak. The bottom panel of figure 4.15(b) shows the energy calibration fit for the strip number 1. Figure 4.16 shows the stacked energy spectra of all strips before (left panel) and after (right panel) the calibration. The peaks for α transitions do not appear at the same ADC in the left panel, the misalignment is corrected after applying the calibration where we can see that the α peaks appear at the same position for all strips. In this calibration procedure, the dead layer of the DSSSD is ignored since the absolute energy of the recoils is not needed for their identification, especially since their final energy highly depends on the isobutane gas pressure in the ionization chamber.

Energy spectra analysis

In our analysis, only the front strips are used for the particle identification since the energy resolution of the back strips is worse than the resolution of the front ones. The multiplicity of the events for the front strips was also investigated, and events with multiplicity > 1, i.e. detected in two or more adjacent strips, are found to be less than 0.6%. Figure 4.17 shows a DSSSD pulse height spectrum obtained with an attenuated 30 Si beam at E lab = 507 keV/u. The DRAGON spectrometer parameters were set for a charge state of q = 8 + . Three main features can be distinguished:

• Full-energy peak: is associated to good events which have deposited all their energy in the bulk of the DSSSD with a good charge collection.

• Low-energy tail: originates from extra energy loss of the incident particles crossing the aluminium grid and the SiO 2 insulator of the DSSSD (see figure 4.9). This represents ∼ 7% of the events.

• Inter-strip events: when incident particles hits the DSSSD in the gap between two adjacent strips, the charge is shared between the two strips resulting in an incomplete energy collection, and the energy is at most half the incident energy. These events represent 3% of the total number of the events, which is in line with the geometric efficiency of the DSSSD which is (96.15 ± 0.15)% [START_REF] Wrede | A double sided silicon strip detector as a DRAGON end detector[END_REF].

Once the features of the DSSSD pulse height spectra obtained with the attenuated 30 Si beam are studied, one can analyse the spectra corresponding to a run with recoils. .18 shows the DSSSD energy spectra of the recoils for both the highest and lowest energies measured, both for single (black histogram) and coincidence events (blue histogram). A peak at a higher energy than the main one is observed at all beam energies measured in the present experiment. However, this peak is better separated for the highest beam energies. This highest energy peak is associated to unreacted beam particles that were not suppressed by the spectrometer. This leaky beam is completely suppressed when considering the coincidence events, i.e., events that are correlated with a γ-ray event detected in the BGO array. The spectra measured during attenuated beam runs are superimposed to confirm the peak position of the leaky beam. For the measurements at beam energy E lab = 506 keV/u, a fit with a double Gaussian was performed on the recoils' spectrum, in single events. The highest energy component reproduces well the shape of the peak obtained with the attenuated beam.

An attempt can be made to extract the number of recoils from the singles spectra by subtracting the fitted events under the leaky beam peak. However, the centroids of the recoils peaks for singles and coincidences do not match: for the resonances where the DSSSD spectrum were fitted with a double gaussian (E c.m. r = 485 keV and E c.m. r = 555 keV) the difference between the singles and the coincidence peak position is up to 250 keV. This indicates that the procedure for extracting the number of recoils in singles is not very reliable for low energy resonances. Surprisingly, for higher energy resonances, the position of the singles peaks do not coincide perfectly with the position of the coincidences either, and the difference is up to 160 keV in the case of the E c.m. r = 752 keV resonance. The difference varies from a resonance energy to another, and this may be a hint for the presence of a leaky beam with different charge states under the recoils peak. This point will be discussed in section 4.4.1. Figure 4.19 shows the DSSSD spectra for all measured resonances. The difference between the number of counts under the singles and coincidences peak is related to the γ-ray detection efficiency, which is treated in the following subsection.

Singles Coincidence Singles' fit

Leaky beam

Figure 4.19: DSSSD spectra obtained for both singles, and events measured in coincidence with a γ-ray event detected in the BGO. The latter is displayed as a light blue, plain histogram. For resonances at low energy, the peak associated to the leaky beam is fitted with a gaussian, in addition to the recoils gaussian fit.

Focal plane Particle Identification

The MCP and IC detectors of the focal plane can be used to better identify the particles incident on the focal plane of DRAGON. Figure 4.20 shows bi-dimensional spectra for MPC energy, local ToF between MCP and DSSSD, and the energy loss in the IC chamber, as a function of the DSSSD energy. These spectra were obtained for the resonance at E c.m. r = 752 keV energy. The singles' data in blue dots are superimposed with coincidence events in red dots. Similar results are obtained for all measured resonances. The leaky beam events are visible and separated from the rest of the events thanks to the difference in the energy deposited in the DSSSD. No further separation is observed between singles and coincidence events using the energy measured by the MCP, which has a worse energy resolution than the DSSSD. The local time of flight between the MCP and the DSSSD detectors is also the same for the recoils and leaky beam events. Unfortunately, the difference in the energy loss of 30 Si and 31 P ions in the MCP and in the IC chamber is negligible since the masses are very close, and the charge states should be similar after the EMS separation. The MCP detector and the Ionization Chamber won't be used in the rest of the analysis.

BGO energy spectra

The detection of the γ-rays plays a crucial role in the determination of the coincidence events and subsequently the determination of the separator time of flight. For the sake of keeping the BGO counting rate reasonable, the BGO array DAQ is triggered only if a γ-ray event has an energy above a predefined hardware threshold. If such an event is detected, all the BGOs of the array are read and the measured energies are stored. The detection efficiency of the BGO array is determined via simulations using geant3 code. In order to keep the analysis manageable when comparing the measured spectra to the simulated ones, we consider for each event the highest deposited energy in a BGO crystal only, this energy is labelled E γ 0 , the corresponding spectrum is specific to the BGO array as a whole.

BGO calibration and alignment

The BGO detectors have been calibrated only once, at the beginning of the experiment. The calibration is performed by placing a 244 Cm/ 13 C source inside the BGO array. The alpha particle produced from the disintegration of 244 Cm initiate the 13 C(α,n) 16 O* reaction. The 16 O is populated in an excited state and de-excite by emitting a 6.13 MeV γ-ray. The corresponding peak is manually selected on the spectra and a script adjusts the bias of the photomultiplier of each BGO crystal in order to shift the peak This alignment is found insufficient since it can drift during the experiment that spans several days. Furthermore, a misalignment is observed in the low energy region of the spectrum since the procedure is based upon a single calibration point at the relatively high energy of 6.13 MeV. For all these reasons, an additional calibration procedure has been performed and was inspired from the previous work of Karpesky's thesis [101]. The procedure consists in the following:

• First, the runs for each measured beam energy are summed together, and the energy spectra of each BGO are extracted, both in singles and coincidences.

• In singles spectra, the peaks associated to the natural background radiations of • For coincidences spectra, the peak associated to the transition from the resonant level to the ground state in 31 P, i.e., the highest expected γ-ray energy, is fitted along with its corresponding first-escape peak. The peaks are thus fitted with a double gaussian (see figure 4.22, right panel). The position of the escape peak is not used for the calibration, but it is considered in the fit of the spectrum to better constrain the full energy peak.

• The detected energy corresponding to the highest γ transition is deduced from simulations output, where a single transition is considered with 100% probability of decaying to the ground state. The detected energy E BGO differs from one crystal to another, since they have different positions and thus different angles θ with respect to the beam axis. This difference is due to the Doppler effect which is taken into account in the following steps.

• The correspondence of the BGO numbers between the simulation and the experiment is ensured by comparing the angle θ of each detector. For the simulation, the angles are deduced using the Doppler shift of the simulated peaks. The angles of the physical BGO are deduced from the (x, y, z) coordinates. Figure 4.21a shows the correspondence between the BGO numbers and their corresponding angles θ for both physical and simulated BGO detectors. The difference in angle is due to solid angle effects, which depend on the BGO layout around the gas cell. (figure 4.21).

BGO angles

q (deg)

Physical position

Simulated position (a) The angles of the BGO detectors as a function of their numbers. The angles of physical detectors correspond to the centre of each crystal, while the angles for the simulated detectors are deduced from the Doppler shifts of the measured γ-rays for the transition at E γ = 7.898 MeV. • A first order calibration is then performed with the 3 data points: peaks at 1.461 MeV and 2.615 MeV, and the main transition. • The calibration coefficients are applied to all the runs. Figure 4.23a shows the residues of the calibration procedure for each run. A disparity is observed from a run to another for the same measured resonance. This is due to the drift in the bias that may be correlated to temperature changes for example.

• A realignment procedure is performed for each run this time. The peaks are fitted in the same fashion as previous steps, but the fits are performed on E γ 0 spectra. The subsequent 3 data points are then fitted with a second order polynomial. Figure 4.23b shows the residues of the calibrations for each run after the realignment procedure. The errors on the fitted positions of the peaks induced by the background radiation are less than 20 keV for all the runs, while the errors on the γ-ray transitions in 31 P vary according to the statistics of the runs. For the strong resonance at E c.m. r = 602 keV the deviations are less than 50 keV, while it can deviate by 250 keV for weaker resonances. The energy resolution for γ-rays at 8 MeV is around 500 keV FWHM in E γ 0 spectra. This is due to the fact that Doppler effect is not taken into account in order to remain coherent with the simulated spectra. This calibration is satisfying for the rest of the analysis because the BGO array is used as a calorimeter to deduce the number of detected γ-rays, rather than performing a spectroscopy study of the resonance.

Case of the E c.m. r = 950 keV resonance: This resonance was the first one to be measured. Unfortunately, the hardware threshold for the BGO DAQ was kept at 2500 keV. This high threshold value is used for the BGO calibration runs. For this resonance energy, the software threshold is applied at 2800 keV. The runs at this resonance energy were calibrated and aligned differently than the other runs since the background radiation from 40 K at 1.461 MeV was not measured and the one from 208 Tl at 2.615 MeV was completely dominated by the near-threshold induced noise.

The calibration was performed by considering the γ-ray transitions at 2.233 keV, 3.134 keV, 6.013 keV and the transition to the ground state at 8247 keV, with its corresponding first escape peak. The Doppler effect was considered to deduce the expected detection energy. In order to estimate the γ-ray detection efficiencies for each measured resonance, the BGO spectra must be compared to simulated spectra. The DRAGON simulation toolkit is based on geant (GEometry ANd Tracing) program which uses Monte Carlo techniques to simulate the interaction of particles with matter through electromagnetic and hadronic processes. It is capable of tracing the particles through complex geometries, enabling a complete simulation of the detection setup. DRAGON simulation toolkit is based on a previous version: geant3 which is written in fortran programming language. It simulates the windowless gas target and the 30 surrounding BGO detectors, along with the beam lines and the electromagnetic elements of the separator.

The input file for each measured resonance includes the decay scheme of the resonant state through the 6 strongest primary transitions. The excitation energies and lifetimes of the considered levels are specified along with the subsequent γ-ray transitions, fed by the primary transitions. For each resonance energy, 10000 events have been simulated.

The simulated E γ 0 spectra have to be convoluted in order to match with the experimental resolution of the energy spectra. The resolution that has been coded in the DRAGON geant3 simulation corresponds to the BGO photomultiplier resolutions as measured 20 years ago, with optimum alignment and calibration. Unfortunately, the energy resolution is degraded with time and this has to be taken into account. Each simulated spectrum is convoluted with a Gaussian with a width given by σ = 2.5%E. This relation was obtained by trial and error until the experimental spectra were well reproduced.

Spectra comparison

The γ-ray branching ratios listed in the NNDS compilation [31] are taken as a starting point for simulating the BGO spectra. The highest energy transition is taken as a reference for the normalization, so that relative intensities of the other transitions can be qualitatively compared, and the branching ratios modified if needed.

If the simulated spectra do not reproduce well the data, the TFractionFitter class from ROOT package [103] is used to obtain the optimum branching ratios for each primary decay. The procedure followed is:

• Each primary decay is simulated separately: 100% probability is assumed for the transition from the resonant state to the state fed by the primary branch. The subsequent cascades are included with their original branching ratios.

• For some resonances, different or additional decay branches can be considered.

• TFractionFitter is called and takes for argument the experimental spectrum and the different simulated spectra for each decay branch. The routine fits, with a likelihood method using Poisson statistics, the fraction of the simulated spectra for each decay branch, so that their sum reproduces the best the experimental spectrum.

• The fit results are, straightforwardly, the new branching ratios for each primary decay. A new simulation is performed with these new ratios and the results are once more compared to the data.

A hardware threshold for the BGOs trigger was set at 500 keV. In order to be insensitive to near-threshold background, the γ-ray detection efficiencies are calculated for γ-ray energies above 800 keV. The simulated and experimental spectra are thus compared considering this energy cut. The efficiency is given as:

ϵ γ = N simu γ N simu reactions . (4.20)
The corresponding statistic uncertainty is related to the simulated number of detected γ-rays and is given by N simu γ . In addition, a 10% systematic uncertainty is considered for all γ-ray efficiencies, coming from the geometry and the geant3 simulations [START_REF] Gigliotti | Efficiency calibration measurement and simulation of the DRAGON BGO gamma ray array at TRIUMF[END_REF]. The detailed analysis for each measured resonance energy is now presented. E c.m. r = 485 keV resonance This resonance was measured in two different sets of runs, with different recoils charge states for each set: q = 7 + and q = 8 + . The two measurements are analysed separately, but the extracted γ-ray detection efficiency will be the same since it depends only on the outcome of the simulations. Figure 4.25a shows the measured BGO spectrum, in blue, and the simulated spectrum in orange, using the decay branching ratios listed in NNDC in a first step. The results reproduce well the overall shape of the measured spectrum, but the intensity of some transitions are underestimated in the simulated spectrum, namely, E γ 0 at 3.13 and 4.65 MeV, which corresponds to the cascade from the resonance state at E x = 7779 keV to E x = 3.134 keV then to the ground state. The big peak at around 511 keV in the simulated spectrum corresponds to γ-rays resulting from pair annihilation. The TFractionFitter method was used, and the results are shown in figure 4.25b. The new branching ratios are listed along with the ones from NNDC in table 4.4. We can see that the fitted branching ratio for the transition R→ 3.134 MeV is significantly higher than the NNDC value. For both sets of decay branching ratios, the detection efficiency is ϵ γ = 0.72 ± 1.7%(stat.). 4.4: Decay schemes and branching ratios used for geant3 simulations, first using data from NNDC [31], and after using TFractionFitter when needed. The deduced γray detection efficiencies are also given.

Decays to the first excited states in 31 P were added, and the spectrum was fitted according to the procedure detailed above. The results are shown in figure 4.26b. The overall structure of the spectrum is better reproduced, even though some transition intensities are not well-matched between the data and the simulation. The low level of statistics does not allow a further improvement of the situation. The results are summarized in table 4.4, and the simulations with the new decay scheme give a detection efficiency of ϵ γ = 0.76 ± 1.7%(stat.), instead of the 0.67 ± 1.7%(stat.) for the single decay branch.

E c.m. r = 602 keV resonance
This resonance is the strongest one that has been measured in the present experiment, and it is usually taken as a reference resonance in relative strength measurements. The branching ratios for the decays of the corresponding excited state at E x = 7898 keV are thus well known. The branching ratios given in the NNDC compilation were considered for the simulation and the results are shown in figure 4.27. The simulation reproduces very well the data. The detection efficiency for this resonance is ϵ γ = 0.65 ± 0.07 .28a shows the comparison between the measured BGO spectrum and the simulated one obtained using the branching ratios listed in NNDC. The overall shape is reproduced, but the intensities of the transitions do not concur between data and simulations. The experimental spectrum is thus fitted using the TFractionFitter method, and the new decay branching ratios are presented in table 4.4. The new simulated spectrum is shown in figure 4.28b. The correspondence between the data and the simulation is well improved, and all transitions are matching. The γ-ray detection efficiency deduced from the simulations is ϵ γ = 0.72 ± 0.3%(stat.). This resonance corresponds to an excitation energy of E x = 8207 keV. The simulated BGO spectrum based on NNDC reproduces very well the measured spectrum (see figure 4.29), the overall shape and important transitions are matching. All attempts to have a better fit of the branching ratios didn't give better results. The γ-ray detection efficiency deduced from the simulations is ϵ γ = 0.69 ± 1.7%(stat.).

E c.m. r = 950 keV resonance
This resonance corresponds to an excitation energy of E x = 8247 keV. The experimental spectrum is compared to the simulation results obtained with NNDC decay branching ratios. Both spectra are shown in figure 4.30a. The transitions between 3 and 6 MeV do not match between the simulated and experimental BGO spectra. The fitting of the spectrum is then performed, and the new decay branching ratios presented in table 4.4. The new simulated spectrum is shown in figure 4.30b. The correspondence between the data and the simulation is improved, and some transitions are better described, however, it is not perfect and any attempt to further improve the situation failed. The γ-ray detection efficiency deduced from the simulations is ϵ γ = 0.59 ± 1.9%(stat.). This value is the lowest among the calculated BGO efficiency, and this is expected since the threshold is set at a higher value.

Recoils transmission

The transmission efficiency of the recoils through the DRAGON spectrometer is also calculated using the same geant3 simulations. It is deduced from the ratio of simulated recoils that hit the detectors on the focal plane, over the number of simulated reactions.

The angular cone emission of the recoils for the measured reactions spans between 6.5 mrad and 8.5 mrad, for the resonances at E c.m. r = 950 keV and E c.m. r = 485 keV, respectively. This is well below the angular acceptance of the DRAGON spectrometer (± 20 mrad), and the transmission efficiency is found to be more than 98% for all the measured energy resonances. The associated systematic uncertainty is 2% [START_REF] Gigliotti | Efficiency calibration measurement and simulation of the DRAGON BGO gamma ray array at TRIUMF[END_REF], and a statistical uncertainty is also calculated from the number of recoils and reactions in the simulation. The results are summarized in table 4 Furthermore, the recoils have to go through the MCP devices before getting detected in the DSSSD detector at the focal plane. The thin grid of the MCP detector induces some lost in the recoils passing through it. During the present experiment, only one MCP was inserted. The transmission efficiency through the MCP was calculated several times with different beam energies and the results are displayed in figure 4. 31. The value at τ M CP = 0.65 (red point) is quite deviant from the rest of the measured values. Note that this measurement was performed at the same beam energy than the second point at τ M CP = 0.79 which is compatible with other measurements.

The weighted average is calculated considering the black data points of figure 4.31, and is found:

τ M CP = 0.82 ± 0.07. (4.21)
The dispersion of the MCP efficiencies is high compared to the usual values obtained in previous DRAGON experiments, even though the obtained value is compatible with In the rest of the analysis, we consider the measured MCP efficiencies.

Incident beam energy and stopping power

The incident beam energy and the beam stopping power in the gas target can be measured directly using the DRAGON spectrometer. This avoids the use of semiempirical formulae and their attached systematic uncertainties for the determination of the beam stopping power. The incoming beam energy can be measured by DRAGON operators using MD1 dipole (Eq. 4.3) with an empty gas target. If the incoming beam energy is too high to be bent by MD1, several measurements are performed at different value of the gas pressure. The data points are then linearly extrapolated down to P = 0 Torr. The slope of the linear fit of the outgoing beam energy as a function of the gas density is the stopping power ϵ DRAGON . The results are compared to the stopping power calculated with srim program [START_REF] Ziegler | SRIM -The stopping and range of ions in matter[END_REF]. Figure 4.32 and table 4.6 show the measured data points fitted and the extracted values for the stopping power, respectively.

The measured stopping powers are systematically smaller than those calculated with srim. This is partially due to the fact the ion beams are partially stripped (q = 7 + ), while the srim code calculates the energy losses of fully stripped ions. A 30 Si beam in H 2 gas. The measured stopping powers are compared to those obtained from srim program. Beam energies with an asterisk are deduced from the linear extrapolation of the measured energies with filled target and estimated for P = 0 Torr.

E beam ϵ DRAGON ϵ srim Difference (keV/u) eV /(10
rough estimation of the difference can be done with the screening coefficient calculated with Slater's formula [104], a reduction factor of ≈ 0.73 is expected in the stopping power. However, this correction does not allow to explain the observed difference.

Resonance energy

The previous determination of the incident beam energy and of the stopping power is needed to determine the resonance energy through the BGO array hit pattern method [START_REF] Hutcheon | Measurement of radiative capture resonance energies with an extended gas target[END_REF]. As the beam goes through the target, it loses energy as a function of the position. The direct measurement of narrow resonances is based on this property (see section 1.6.2 for more details). Most of the reactions will occur in a small section in the centre of the target, given that the pressure and incident beam energy were carefully chosen. The counts observed in the BGO detectors can thus be used to reconstruct the reaction vertices along the beam axis, defined as the z-axis. = 602 keV resonance. The spectra are obtained after selecting coincidence events and applying the BGO energy threshold.

We can use the geant3 simulations to deduce z true the position of the resonance in the target. The pressure of the gas cell is varied, and the corresponding positions z BGO = f (z true ) are fitted linearly:

z BGO = 0.84 z true + 0.47cm. 1 , (4.23)
The left panel of figure 4.34 shows the fit of this relation obtained for the 30 Si(p,γ) 31 P reaction. The results are compared to the original work of Hutcheon et al. [START_REF] Hutcheon | Measurement of radiative capture resonance energies with an extended gas target[END_REF] where the authors used the 24 Mg(p,γ) 25 Al reaction to benchmark the setup and the simulations. The coefficients are compatible within the error bars, and this is expected since 24 Mg and 30 Si nuclei have close masses. Additional caution may be needed for beams with masses significantly different from A = 24 when using the relation (4.23).

1 the constant correction is added if there is an additionnal shielding around the beam entrance tube, which is the case for radioactive beams. Figure 4.34: z BGO as a function of z true for the 30 Si(p,γ) 31 P and the 24 Mg(p,γ) 25 Al reactions, and their corresponding linear fits.

The z true position can be expressed as the fraction of the gas target crossed by the beam before reacting, and it is defined as:

f = 0.5 + z true /L ef f , (4.24)
where L ef f is the effective length of the gas target, and is equal to 12.3 ± 0.5 cm. Using the quantities defined above, the beam energy at the fraction f of the target, corresponding to the resonance energy in the laboratory frame E r lab , is given by [START_REF] Hutcheon | Measurement of radiative capture resonance energies with an extended gas target[END_REF]:

E r lab = (1 -f )E i + f E o -f (1 -f ) (E i -E o ) 2 E i + E o R, (4.25)
where E i and E o are the incoming and outgoing beam energies in the laboratory frame, respectively, and R = (E/S)/(∆E/∆S) describes the energy dependence of the stopping power S, and is deduced from srim code. For the 30 Si(p,γ) 31 P reaction, R = 3.02, note that the second degree term of Eq. (4.25) is a very small correction (≈ 0.2 -0.6 keV).

Since the measurements of the 30 Si(p,γ) 31 P capture reaction have a high level of statistics, it allows performing a run by run analysis. The resonance energy for each run has been calculated, and the mean value extracted for each group of run. Figure 4. [START_REF] Lyons | Total yield measurement in 27 Al(p,γ) 28 Si[END_REF] shows the difference between the resonance energy estimated using DRAGON, and the values listed in the literature. Red lines are the weighted mean energies for each group of runs. A discussion of these results is done in section 4. [START_REF] Nomoto | Type Ia Supernova Models and Progenitor Scenarios[END_REF] [31]. Grey boxes correspond to spread in the literature energy while pink ones correspond to 1σ standard deviation of the measurements.

Charge State Distribution

Both beam and recoil particles exit the gas target with a charge state distribution.

As explained in the description of the DRAGON separator, the MD1 dipole selects a single charge state. The number of recoils transmitted to the focal plane detection system has to be corrected with the charge state fraction in order to estimate the real number of recoils that have been formed. The charge state distribution (CSD) can be measured experimentally by selecting different charge states with different MD1 settings, and measuring the corresponding fraction F q = FCCH/FC1. FC1 is the Faraday cup located downstream the target, while FCCH is the Faraday cup located after the first magnetic dipole (MD1). The fractions F q are then fitted with a normal distribution, suitable for light targets. The mean charge state q after equilibrium is simply the mean value of the distribution, defined as :

q = q qF q . (4.26)
The equilibrium condition is considered to be achieved if the target thickness is greater than a critical thickness of the order of 10 17 atoms/cm 2 , which is the case for DRAGON targets. The charge state distributions for the phosphorus were measured during a different beam time at TRIUMF. The 31 P beam was produced by OLIS and the charge state distributions were measured for 6 different incident beam energies. Figure 4.37 shows all the measurements and their corresponding gaussian fits.

The beam energies at which those charge states were measured do not correspond exactly to the energies of the 31 P recoils for the resonances studied in the present work. However, the measurements can be used to fit the semi-empirical formula for the mean charge state q, and the width of charge state distributions d. The mean charge state q is given by [START_REF] Liu Wenjie | Charge state studies of heavy ions passing through gas[END_REF]:

E beam (keV/u) E recoils (keV/u) q q recoils F q 507.
q = Z p 1 -exp - A Z γ p E E ′ + B , (4.27)
and the distribution width is:

d = d 1 Z ω p , (4.28)
where Z p is the charge of the incident particle, E is the laboratory energy in MeV/u, E ′ is the reduced expression for the electron velocity and is given as E ′ = 0.067635 MeV/u. Figure 4.38 shows the fit of the two previous equations using the 6 measured data sets. The obtained parameters are: A = 1.742, B = 0.826, γ = 0.427, d 1 = 0.69 and ω = 0.13.

Noteworthy that these semi-empirical formulae are usually used when there is no experimental measurements of the CSD, and the adopted values are A = 1.55563, B = 0.48386, γ = 0.47299, d 1 = 0.668262 and ω = 0.1116143. These values are not from the original work on CSD in DRAGON [START_REF] Liu Wenjie | Charge state studies of heavy ions passing through gas[END_REF] but comes from an updated result in 2020 including new measurements results for 6 ≤ Z p ≤ 19 and 162keV /u ≤ E ≤ 1173keV /u [START_REF] Lovely | Proton Capture on 34 S in the Astrophysical Energy Regime[END_REF]. The corresponding curve is displayed in blue dashed lines in the top panel of figure 4.38.

Once the mean charge state q and the width d are calculated for the right incident energy, the charge state fraction can be calculated as:

F q = 1 √ 2πd exp - (q -q) 2 2d . (4.29)
The uncertainties arising from the various fits that have been performed (gaussian fit, q = f (E)) are propagated using the corresponding covariance matrices. The final uncertainty on the charge state fractions ranges between 0.5% when the selected fraction is the closest to the mean charge state, to 5% when F q is the most different from q. Table 4.7 summarizes the results for the charge states fractions.

Results and discussion

The different physical quantities extracted in the previous sections are now combined. The resonance strengths are first presented and discussed, followed by the results for the resonances energies.

Resonance strength

The resonance strength as shown in Chapter 1, is given by the equation:

ωγ = 2Y λ 2 r m t m t + m b ϵ lab , (4.30)
where Y is the thick target yield, λ r is the de Broglie wavelength, ϵ lab the laboratory stopping power, and m t and m b are the target and beam masses, respectively. The experimental yield, as given in Eq. (4.7) is reminded here:

Y = N det r N b ε DRAGON , (4.31)
where N det r is the number of detected recoil particles, N b is the number of incident beam particles, and ε DRAGON is the total detection efficiency of the DRAGON recoils separator. This detection efficiency for singles is given as:

ε singles DRAGON = F q • τ M CP • ε DSSSD • τ DRAGON • η tail live , (4.32)
whereas the total efficiency for the coincidence events is given as:

ε coinc DRAGON = F q • τ M CP • ε DSSSD • τ DRAGON • ε γ • η coin live . (4.33)
All values required to calculate the resonance strengths for each measured energy are summarized in table 4.8. The values of the resonance strengths are displayed in figure 4.39 where the DRAGON singles and coincidence measurements, displayed in red and blue respectively, are compared to previous strength measurements. Black data points are absolute direct measurements, except for Wolff et al. [START_REF] Wolff | Spin determinations of 31P levels from the 30Si(p, ) reaction[END_REF] and Dermigny et al. [START_REF] Dermigny | Thermonuclear reaction rate of 30 Si(p ,γ )[END_REF], where the strengths were measured relatively to the E c.m. r = 602 keV resonance. Purple data points are data normalized to the value ωγ 602 = 1.95 ± 0.10 eV which is a weighted average of strengths measured by Lyons et al. [START_REF] Lyons | Total yield measurement in 27 Al(p,γ) 28 Si[END_REF] and Pain & Sargood [START_REF] Paine | p,γ) resonance strengths in the s-d shell[END_REF]. For the low energy resonances at E c.m. r = 485 keV and E c.m. r = 555 keV, the strengths are also compared to those obtained from the transfer reaction measurements performed at the Q3D [START_REF] Harrouz | Experimental study of the 30 Si( 3 He, d) 31 P reaction and thermonuclear reaction rate of 30 Si(p, γ) 31 P[END_REF], in green colour. Weighted average of all previous measurements is calculated and displayed with the associated uncertainty, in gray area. The energies measured using the DRAGON spectrometer are displayed in red and blue, corresponding to singles and coincidence measurements respectively. The statistical uncertainties are smaller than the data points, and the coloured boxes represent the systematic uncertainties.

For the resonance of reference at E c.m. r = 602 keV, the mean strength value adopted by Dermigny et al. [START_REF] Dermigny | Thermonuclear reaction rate of 30 Si(p ,γ )[END_REF] is in agreement with the DRAGON's measurements in coincidence, and not with the singles' one. The weighted average of all previous measurements except the one from Riihonen et al. is shown with the dashed line and the coloured band is the associated error. It corresponds to a resonance strength of ωγ = 1.9 ± 0.08 eV, which is in good agreement with the strength measured using coincidence events. The discrepancy between the strength values measured in singles and in coincidence is observed for all the measured resonances with DRAGON. Several comments can be made: = 555 keV resonances, the coincidence values are compatible with the strength calculated using the proton partial widths determined with the Q3D transfer measurements, while the singles are not. Noteworthy that for the E c.m. r = 555 keV resonance, the direct strength determination in coincidence is in agreement with the ℓ = 2 case. This result discards the ℓ = 1 transferred orbital momentum which the corresponding strength would be an order of magnitude higher compared to the coincidences.

• For the E c.m.
Each element of the aforementioned discussion is insufficient if taken individually, but the whole picture favours the coincidence measurements, and we discard the strength values determined with single events.

The discrepancy between the singles and coincidences measurements does not exhibit a dependency with respect to the incident beam energy. It is also interesting to remind the reader about the energy difference observed between the singles and coincidences peaks on some DSSSD spectra (section 4.3.3). This could be explained by some beam particles that lose a fraction of their energy due to scattering and straggling on the optical elements of the separator. The energy loss induces a change in the charge state and such ions may have an (energy -charge) configuration such that they follow the same path as the recoils through the separator. Such leaky beam would be indistinguishable from the recoils without requiring a detected γ events in coincidence. 4.8: Parameters used to calculated the resonance strengths and their associated statistical (second line) and systematic uncertainties (third line) for each resonance energy. The resonance strengths are calculated using both singles and coincidence events.

The adopted values for the resonance strengths will be the one measured using events in coincidence with γ-ray emission.

Resonance energies

The resonance energies are determined by following the procedure described in subsection 4.3.7. The weighted average of the energies determined for each run is calculated for the different incident beam energies. The results are summarized in figure 4. [START_REF] Lyons | Total yield measurement in 27 Al(p,γ) 28 Si[END_REF] where the energies measured at DRAGON are compared to the previous measurements found in the literature.

Measurements using γ-rays detections are displayed in black dots. In the work of Kuperus et al. [START_REF] Kuperus | Energy measurements of proton resonances in light nuclei[END_REF], Wolff et al. [START_REF] Wolff | A study of the excited states of 31 P with the 30 Si(p,γ) 31 P reaction[END_REF] and Dermignyet al. [START_REF] Dermigny | Thermonuclear reaction rate of 30 Si(p ,γ )[END_REF] the resonances were studied through the 30 Si(p,γ) 31 P reaction. Shikazono & Kawarasaki [START_REF] Shikazono | Study of odd-a nuclei in the 2s-1d shell by means of the (γ, γ ′ )reaction[END_REF] studied 31 P through the 31 P(γ,γ ′ ) 31 P reaction. Green data points correspond to resonance energies deduced from the excitation energies2 measured through the 30 The procedure used to determine the resonance energies might be tarnished by systematic errors that are not yet identified for the 30 Si(p,γ) 31 P measurements. These energies are sufficiently high so that uncertainties less than few keV have little to no influence on the reaction rate. This effect will be nonetheless investigated in the next chapter. The adopted resonance energies for the rest of the present work are the ones determined using the Q3D spectrometer for E c.m. r < 800 keV, since it provides a consistent way of determining the energies, and the values measured by Wolff et al. for higher energies.

In this chapter, we presented the direct strength measurements of 30 Si(p,γ) 31 P resonances using the DRAGON recoil spectrometer. The strength of 6 high energy resonances were determined through absolute measurements with improved precision. The strength of the E c.m. r = 555 keV was measured directly for the first time. The results of these measurements are complementary to the strengths of resonances at lower energies that have been determined using the proton partial widths, calculated in the DWBA framework. We also presented the determination of the resonance energies. All these results will be used to calculate the updated value of the 30 Si(p,γ) 31 P reaction rate at energies relevant for astrophysical temperatures.

In the course of previous chapters, the nuclear properties of 30 Si + p resonances have been deduced from measurements using two different experimental approaches. The resonance strengths have been either measured directly, or calculated through the determination of the proton widths using the 30 Si( 3 He,d) 31 P transfer reaction. In this chapter, we present a compilation of all known 30 Si + p resonances. The reaction rates at the stellar temperatures are computed, and used to investigate the temperature and density conditions for polluter candidates in globular clusters.

5.1 30 Si + p resonance properties 30 Si(p,γ) 31 P reaction rate in the temperature range between 10 MK and 10 GK is dominated by the contribution of the narrow resonances that we shall detail in the following.

Resonances at E r ≲ 400 keV have been studied exclusively through indirect reactions. For these resonances, the strength is approximated as (see Chapter 3):

ωγ = 0.5(2J + 1)Γ p (5.1)
• E c.m. r = 17.2 ± 1.6 keV: This resonance corresponds to the excitation energy E x = 7313.7 keV, the level was observed through the 29 Si( 3 He,p) 31 P transfer reaction [START_REF] Al-Jadir | States of high E x in 31 P[END_REF], with the associated spin-parity J π = 1/2 + , 3/2 + . There is no indication of this level in our experimental data.

• E c.m. r = 19.6 ± 0.9 keV: This resonance corresponds to the first unbound 31 P state observed in the present work, with an excitation energy of E x = 7316.2 keV. The analysis of the angular distribution from the 30 Si( 3 He,d) 31 P data is best described by a ℓ = 3 transferred orbital angular momentum, thus corresponding to a 5/2 -or 7/2 -assignment. This is in line with the results obtained from a previous study of the 30 Si( 3 He,d) 31 P transfer reaction [START_REF] Vernotte | Si( 3 He,d) 31 P reaction at 25[END_REF]. Note that the latter work observed this state at E x = 7314 ± 4 keV, which is in agreement with our value within error bars. The strength of this resonance is estimated in the present work as ωγ = 1.2× 10 -38 eV.

• E c.m. r = 50.5 ± 1.2 keV: This resonance corresponds to an excited energy E x = 7347.0 keV. This state is weakly populated through the 30 Si( 3 He,d) 31 P experiment, and its associated angular distribution is best described by ℓ = 1 or ℓ = 2 momentum transfer, giving similar reduced chi-square when only the three most forward angles are considered. Regarding the previous study of this level through the 29 Si( 3 He,p) 31 P reaction [START_REF] Al-Jadir | States of high E x in 31 P[END_REF] that reported spin-parity J π = 3/2 -, 5/2 -, and considering the parity conservation in the present transfer reaction, this leads to ℓ = 1 and J π = 3/2 -. This orbital angular momentum implies a resonance strength of ωγ = 1.04×10 -20 eV.

• E r = 59.5 keV:

This resonance corresponds to the excited state with energy E x = 7356 keV. No spin-parity assignment has been reported in the literature. This state has been populated by the 33 S(d,α) 31 P reaction [START_REF] Teterin | Study of excited states of 31 p with the aid of the reaction 33 s(d, α) 31 p[END_REF], however the α-particle peak associated to this state appears as a weak shoulder in a contamination peak. This prevented an accurate determination of the excitation energy, and indeed no associated uncertainty is given in that work. Although not cited in the ENSDF compilation [31], the study of the 29 Si( 3 He,p) 31 P reaction [START_REF] Moss | Excitation energies of levels in 31 P[END_REF] reports a state at E x = 7356 ± 9 keV. However, this should be considered as tentative since the author gives the energy between brackets (see his Table 1). Interestingly, in a subsequent 29 Si( 3 He,p) 31 P reaction study [START_REF] Al-Jadir | States of high E x in 31 P[END_REF] there is no indication of the E x = 7356 keV state, and it was instead associated to the E x = 7347 ± 6 keV state. This level has not been populated in the Q3D experiment, in line with its non-observation in all other ( 3 He,d) studies. We conclude that the existence of the E x = 7356 keV state is not firmly established, and we thus do not further consider it in this work.

•

E r = 145.8 ± 0.9 keV and E r = 149.2 ± 2.9 keV:

These resonances correspond to a doublet reported in Ref. [31] at E x = 7441.4 keV and E x = 7442.3 keV. The low energy component of the doublet was first observed using the 27 Al(α,γ) 31 P reaction [START_REF] De | Levels of 31p from -particle capture in 27al[END_REF] and a spin assignment J = (3/2 -9/2) was derived [31]. This state mainly decays through two 7/2 + levels, thus adding another constraint for its parity, which is most likely positive in case of a J = 3/2 assignment. The other component of the doublet has been observed in several experimental studies [START_REF] Twin | Gamma-ray spectroscopy in 31 P: levels below 7.5 MeV (populated in 28 Si(α,pγ) reaction)[END_REF][START_REF] Ionescu-Bujor | High spin structure and intruder configurations in 31 P[END_REF][START_REF] Jenkins | Mirror energy differences in the A = 31 mirror nuclei, 31 S and 31 P, and their significance in electromagnetic spin-orbit splitting[END_REF] pointing to a J π = 11/2 + assignment. Note that the two levels of this doublet have never been observed in the same experiment.

In the 30 Si( 3 He,d) 31 P experiment presented in this work, we observe a weakly populated state at the three angles θ Q3D ≥ 20°, with a weighted average excitation energy of E x = 7445.7 ± 2.8 keV. Since the present transfer reaction is better matched for low transferred angular momenta, we assume that the level observed corresponds to the low spin component of the doublet. The spin-parity assignment discussed previously corresponds to either ℓ = 2, 3 or 4. However, the limited angular range of the angular distribution and the lack of data at forward angles does not allow discriminating between the different possible transferred relative angular momenta. For the reaction rate calculations, three cases were considered: ℓ = 2, ℓ = 3, and no resonance at E c.m. r = 149 keV.

• E c.m. r = 174.0 ± 2.3 keV: This resonance corresponds to level at E x = 7470.5 keV. A deuteron peak is unambiguously associated to this state at θ Q3D = 20°, 23°and 32°. We associate this level to the one reported at E x = 7466 ± 2 keV in the literature. The E x = 7466 keV state was first observed using the 27 Al(α,γ) 31 P reaction [START_REF] De | Levels of 31p from -particle capture in 27al[END_REF], and later with the 28 Si(α,pγ) 31 P [56] and 29 Si( 3 He,p) 31 P [START_REF] Al-Jadir | States of high E x in 31 P[END_REF] reactions. Based on these experimental studies, the spin-parity of this state is restricted to J π = (7/2, 9/2) - [START_REF] Endt | Energy levels of A = 21-44 nuclei (VII)[END_REF]. Such spin-parity assignment corresponds to an ℓ = 3 or ℓ = 5 angular momentum transfer for the 30 Si( 3 He,d) 31 P reaction. The reaction rate has been calculated assuming the two possible ℓ values.

•

E r = 276 keV:

This resonance corresponds to the excited state observed at E x = 7572 keV in the experimental study of the 33 S(d,α) 31 P reaction [START_REF] Teterin | Study of excited states of 31 p with the aid of the reaction 33 s(d, α) 31 p[END_REF], no spin-parity is given. There is no evidence of this state in the present study, in agreement with the results from other 30 Si( 3 He,d) 31 P study [START_REF] Vernotte | Si( 3 He,d) 31 P reaction at 25[END_REF]. This resonance is thus not considered for the reaction rate calculation.

• E c.m. r = 394.6 ± 1.0 keV: This resonance corresponds to an excited energy of E x = 7691.1 keV. This state has been unambiguously observed in the present 30 Si( 3 He,d) 31 P experiment at θ Q3D = 6°, 10°and 12°. At higher angles, the state at E x = 7691 keV overlaps with the 29 P state at E x = 3105 keV populated by the 28 Si( 3 He,d) 29 P reaction. For large Q3D detection angles, the angular distribution of the 31 P state at E x = 7691 keV was thus obtained after subtracting the contribution from the 29 P level. Such contribution was estimated using the angular distribution obtained in the 28 Si( 3 He,d) 29 P study [START_REF] Dykoski | Single proton transfer to 29,30 P states[END_REF] performed at the same incident energy as the present experiment. The angular distribution of the E x = 7691 keV state of 31 P is best described by an ℓ = 3 orbital angular momentum transfer, giving J π = 5/2 -or 7/2 -. This level was not observed in the work of Ref. [START_REF] Vernotte | Si( 3 He,d) 31 P reaction at 25[END_REF] because of the presence of the 29 P contamination at all angles.

We associate the present level with the E x = 7687.2±2.0 keV state, which has been observed in a single experiment using the 30 Si(p,γ) 31 P reaction [START_REF] De Neijs | Levels of 31 P from proton capture in 30 Si[END_REF]. Based on the γ-ray feeding and decay properties, J ≤ 9/2 is suggested [31], which is in agreement with the present spin-parity assignment. The corresponding strength is ωγ = 4.4×10 -6 eV.

• E c.m. r = 423.0 ± 0.8 keV: This resonance corresponds to the 31 P exited level at E x = 7719.5 keV, which has been observed using the 29 Si( 3 He,p) 31 P [54], 30 Si(d,n) 31 P [START_REF] Uzureau | Spectroscopic study of 31p and 32s by the (d, n) reaction at ed = 7 mev[END_REF] and 30 Si( 3 He,d) 31 P [START_REF] Vernotte | Si( 3 He,d) 31 P reaction at 25[END_REF] transfer reactions. No spin-parity assignment could be established previously because of the proximity of the E x = 7737 keV state forming an unresolved doublet in the aforementioned experiments. The resolution obtained in the present 30 Si( 3 He,d) 31 P experiment allowed to extract the angular distribution of this state at all Q3D angles. The analysis of this distribution shows an ℓ = 3 orbital angular momentum transfer pattern, leading to J π = 5/2 -or 7/2 -. A recent 30 Si(p,γ) 31 P direct measurement [START_REF] Dermigny | Thermonuclear reaction rate of 30 Si(p ,γ )[END_REF] suggests J π = (3/2, 5/2) -based on the combination of constrains coming from the γ-ray transitions and the 29 Si( 3 He,p) 31 P reaction [START_REF] Al-Jadir | States of high E x in 31 P[END_REF]. From the combination of all existing spin-parity assignments, we suggest that the E x = 7719.5 keV state has J π = 5/2 -.

The resonance in the 30 Si(p,γ) 31 P reaction was observed at E c.m. r = 421.9±0.3 keV with a strength ωγ = (1.14 ± 0.25) × 10 -4 eV [START_REF] Dermigny | Thermonuclear reaction rate of 30 Si(p ,γ )[END_REF]. Using the value of the proton width deduced from the spectroscopic factor (2J + 1)C 2 S = 0.045, and under the reasonable approximation Γ p ≪ Γ γ at this energy, we find the strength of this resonance ωγ = (7.43±2.23)×10 -5 eV which is 1σ compatible with the strength measured directly. The latter was favoured for the reaction rate calculations.

• E c.m. r = 440.8 ± 0.7 keV: This resonance corresponds to the excited level at E x = 7737.3 keV. It has only been observed through the 30 Si( 3 He,d) 31 P reactions, both in the Q3D one presented in chapter 2, and in Vernotte et al. [START_REF] Vernotte | Si( 3 He,d) 31 P reaction at 25[END_REF] at E x = 7736±4 keV. This level forms a doublet with the E x = 7719 keV state. Both studies suggest an ℓ = 3 transferred angular momentum for this level based on its angular distribution. This corresponds to a strength value of ωγ = 3.92×10 -4 eV.

• E c.m. r = 484.6 ± 0.8 keV: It corresponds to an excitation energy of E x = 7779.7 keV, in line with the results of the 30 Si( 3 He,d) 31 P experiment at E x = 7791.1±0.8 keV. It has been observed by several transfer reactions [START_REF] Vernotte | Si( 3 He,d) 31 P reaction at 25[END_REF][START_REF] Al-Jadir | States of high E x in 31 P[END_REF][START_REF] Uzureau | Spectroscopic study of 31p and 32s by the (d, n) reaction at ed = 7 mev[END_REF] and in the direct measurement of the 30 Si(p,γ) 31 P reaction [START_REF] Dermigny | Thermonuclear reaction rate of 30 Si(p ,γ )[END_REF][START_REF] De Neijs | Levels of 31 P from proton capture in 30 Si[END_REF]. These direct measurement experiments established a J π = 3/2 -spin-parity assignment [31]. The angular distribution obtained in the present work at all angles corresponds to a transferred angular orbital momentum of ℓ = 1 in line with the known spin-parity assignment. The resonance in the 30 Si(p,γ) 31 P reaction is observed at E c.m. r = 486.2 ± 0.2 keV with a strength ωγ = (0.188 ± 0.014) eV [START_REF] Dermigny | Thermonuclear reaction rate of 30 Si(p ,γ )[END_REF]. Using the value of the proton width deduced from the spectroscopic factor (2J + 1)C 2 S = 0.016, and under the reasonable approximation at this energy that ωγ = 0.5(2J +1)Γ p , we find a strength of ωγ = 0.12±0.04 eV for this resonance.

The direct measurements using DRAGON spectrometer give a resonance strength of ωγ = 0.150 ± 0.012 eV, which is compatible with the value calculated using the spectroscopic factor determined with the Q3D. However, this value is ∼ 20% smaller than the value measured by Dermigny et al. [START_REF] Dermigny | Thermonuclear reaction rate of 30 Si(p ,γ )[END_REF]. For the reaction rate calculation we adopt the DRAGON's value of the strength.

• E c.m. r = 528.5 ± 9 keV: This resonance corresponds to the excited state at E x = 7825 keV. It has only been observed in the study of the 29 Si( 3 He,p) 31 P reaction [START_REF] Moss | Excitation energies of levels in 31 P[END_REF] even though it has not been confirmed by the study of the same transfer reaction at a different beam energy [START_REF] Al-Jadir | States of high E x in 31 P[END_REF]. In our Q3D experiment, there is an indication of a deuteron peak at the expected energy at a detection angle of 10°. However, this indication at a single angle is not enough to conclude for a positive observation of this state. We therefore discard this level in the rest of the present study.

• E c.m. r = 554.9 ± 0.3 keV: This resonance was measured using both DRAGON and Q3D spectrometers. It corresponds to an excited energy of E x = 7851.3 keV. It has been observed in the 29 Si( 3 He,p) 31 P experiment [START_REF] Al-Jadir | States of high E x in 31 P[END_REF] at E x = 7856 ± 6 keV with J = 1/2 -5/2, and with the 32 S(d, 3 He) 31 P reaction [START_REF] Vernotte | One-nucleon pickup reactions on 32s: Experimental results and shell-model calculations[END_REF] at E x = 7851 ± 5 keV. This state was also observed at E x = 7855 ± 4 keV in a previous 30 Si( 3 He,d) 31 P experiment [START_REF] Vernotte | Si( 3 He,d) 31 P reaction at 25[END_REF], but the corresponding angular distribution could not be reproduced by DWBA calculations. This is probably due to the presence of the tail of the strongly populated state at E x = 7898 keV, together with the existence of an unresolved state at E x = 7863 keV. Indeed, the deuteron spectrum obtained with the 30 Si( 3 He,d) 31 P experiment is better described by two close components, E x = 7851.4 ± 0.8 keV and E x = 7863.4 ± 1.4 keV (χ 2 /ndf = 0.95 against χ 2 /ndf =1.52 for one-component fit). The angular distribution of the E x = 7851.4 ± 0.8 keV state could be extracted and is well described by a DWBA calculation corresponding to either ℓ = 1 or ℓ = 2.

The strength of the resonance when measured directly is ωγ = 0.024 ± 0.003 eV, which favours the ℓ = 2 assignment that gives ωγ = 0.017 eV rather than ℓ = 1 corresponding to ωγ = 0.181 eV. Note that the gamma width of this resonance has been deduced from a (γ,γ') experimental study [START_REF] Shikazono | Study of odd-a nuclei in the 2s-1d shell by means of the (γ, γ ′ )reaction[END_REF] and estimated as (2J +1)Γ 2 γ = (1.05± 0.17)Γ. For a spin parity assignment of J = 5/2 + , this corresponds to Γ γ = 0.18 eV. The strength measured directly is used for reaction rate calculation.

• E c.m. r = 566.9 ± 1.4 keV: This resonance corresponds to the level at E x = 7863.4 keV. Despite its weak population, it has been observed at 5 Q3D angles. The only state that could be associated to our observation would be the state at E x = 7859.8 ± 0.4 keV reported in the 24 Mg( 16 O,2αpγ) 31 P reaction [START_REF] Ionescu-Bujor | High spin structure and intruder configurations in 31 P[END_REF] with a spin parity assignment of sured directly, however we expect from the high spin parity assignment J π = 5/2 -, 7/2 - that this resonance does not contribute significantly to the total reaction rate, especially since it has an energy close to the resonance of reference at E c.m. r = 601 keV which dominates the rate.

• E c.m. r = 649.9 ± 0.8 keV: This resonance corresponds to an excited level at E x = 7946 keV, that has been observed in the 30 Si( 3 He,d) 31 P experiment performed at the Q3D, as well as in the study of Vernotte et al. [START_REF] Vernotte | Si( 3 He,d) 31 P reaction at 25[END_REF]. The angular distribution analysis indicates ℓ = 2 in both works. The strength of this resonance has been measured directly by Kuperus et al. [START_REF] Kuperus | Energy measurements of proton resonances in light nuclei[END_REF] and normalized with respect to the strength of the resonance of reference at E c.m. r = 601 keV. This results in ωγ = 0.071 eV.

• E c.m. r = 679.9 ± 0.8 keV: This resonance corresponds to the 31 P excited level at E x = 7976.4 keV that has been populated in the 30 Si( 3 He,d) 31 P experiment. DWBA analysis presented in chapter 3 suggests a first spin parity assignment of J π = 3/2 + , 5/2 + , corresponding to a transferred angular momentum of ℓ = 2. The resonance has not been measured directly, and based on the conclusions for the resonance at E c.m. r = 648 keV, with ℓ = 2 too, this resonance is not expected to contribute to the 30 Si(p,γ) 31 P total reaction rate.

• E c.m. r = 697.5 ± 6 keV: This corresponds to an excited state at E x = 7994 keV, which has only been observed using the 29 Si( 3 He,p) 31 P reaction [START_REF] Al-Jadir | States of high E x in 31 P[END_REF] where the associated proton peak is strongly contaminated. This level was not observed in the Q3D experiment nor in the work of Vernotte et al. [START_REF] Vernotte | Si( 3 He,d) 31 P reaction at 25[END_REF]. The existence of this state is questionable, and it is not considered in the calculation of the reaction rate.

• E c.m. r = 734.7 ± 0.9 keV: This resonance corresponds to the excited state at E x = 8031.2 keV, it was not observed in the 30 Si( 3 He,d) 31 P transfer reaction, but the resonance was observed through direct measurements of Kuperus et al. [START_REF] Kuperus | Energy measurements of proton resonances in light nuclei[END_REF]. The strength, after renormalization is ωγ = 0.081 eV. .

• E c.m. r = 751.8 ± 1.1 keV: This resonance corresponds to an excited level at E x = 8047.7 keV, which has been populated through the 30 Si( 3 He,d) 31 P reaction studied at the Q3D, as well as in the study with the same reaction done by Vernotte et al. [START_REF] Vernotte | Si( 3 He,d) 31 P reaction at 25[END_REF], both DWBA analysis concluding with an ℓ = 1 transferred orbital momentum. The same conclusion has been drawn from the study of the 29 Si( 3 He,p) 31 P [START_REF] Al-Jadir | States of high E x in 31 P[END_REF], and the 30 Si(d,n) 31 P [START_REF] Uzureau | Spectroscopic study of 31p and 32s by the (d, n) reaction at ed = 7 mev[END_REF] reactions. The strength of this resonance has been measured directly through the 30 Si(p,γ) 31 P reaction by Rihoonen et al. [START_REF] Riihonen | Hydrogen burning of 29,30 Si in explosive carbon burning[END_REF], and the resulting strength, after renormalization with respect to the reference resonance at E c.m. r = 601 keV, was ωγ = 0.44 eV, with no experimental uncertainty associated to it. This resonance has 10% contribution to the total reaction rate around T=3 GK, as it can be seen in figure 5.3b, thus its strength has been remeasured with DRAGON spectrometer, giving ωγ = 0.57 ± 0.06 eV, this value is adopted for the new reaction rate calculations.

• E c.m. r = 781.5 ± 1.7 keV: This resonance corresponds to an excitation energy of E x = 8078.0 keV. It has been populated by the 30 Si( 3 He,d) 31 P reaction, but the fit of the associated differential cross-section, suggesting ℓ = 1, disagrees with the J π = 11/2 -spin-parity assignment deduced from a 24 Mg( 16 O,2αpγ) 31 P reaction [START_REF] Ionescu-Bujor | High spin structure and intruder configurations in 31 P[END_REF]. No direct measurements have been performed for this resonance. But if one considers the case that maximizes the resonance strength, i.e. ℓ = 1, the corresponding obtained proton spectroscopic factor (see chapter 3) would imply a strength 6 times smaller than for the resonance at E c.m. r = 751 keV. The strength would be even much smaller for J π = 11/2 -. This shows that the resonance would not significantly contribute to the total reaction rate.

• E c.m. r = 808.3 ± 1.5 keV: This resonance corresponds to the E x = 8103.6 keV excited state, observed through the 30 Si( 3 He,d) 31 P reaction, which suggests an ℓ = 2 transferred orbital momentum, both in the work presented in chapter 3 and in Vernotte et al. [START_REF] Vernotte | Si( 3 He,d) 31 P reaction at 25[END_REF]. This is in agreement with the J π = 5/2 + spin-parity assignment deduced from the 30 Si(p,γ) 31 P measurements of the resonance [START_REF] Broude | The Energy Levels ofsup31/supP II: Angular Distributions and Correlations[END_REF]. The strength measured by Kuperus and renormalized to the present E c.m. r = 601 keV resonance strength, is ωγ = 0.18 eV with no associated uncertainty.

• E c.m. r = 912.5 ± 1.0 keV: This resonance corresponds to an excitation energy of E x = 8207.9 keV. The corresponding level has been populated through the 31 P(γ,γ ′ ) 31 P reaction [START_REF] Hough | Level widths in 31 P[END_REF], which suggested a J π = 3/2 + spin parity assignment, in agreement with the work from [START_REF] Broude | The Energy Levels ofsup31/supP II: Angular Distributions and Correlations[END_REF]. The strength was first measured by Wolff et al. [START_REF] Wolff | A study of the excited states of 31 P with the 30 Si(p,γ) 31 P reaction[END_REF], and after renormalization with respect to the E c.m. r = 601 keV resonance, the strength was ωγ = 0.886 eV, without associated uncertainty. Note that the resonance strength from Wolff et al. [START_REF] Wolff | A study of the excited states of 31 P with the 30 Si(p,γ) 31 P reaction[END_REF] are measured relatively to the 27 Al(p,γ) 28 Si. The strength of this resonance has been remeasured using DRAGON spectrometer, the new value which is adopted for the rate calculations is ωγ = 0.73 ± 0.08 eV.

• E c.m. r = 927.9 ± 1.0 keV: This resonance corresponds to an excitation energy of E x = 8224.4 keV. The spin parity of this level was determined through the study of the angular distributions of primary and secondary γ-ray transitions in the 30 Si(p,γ) 31 P reaction [START_REF] Wolff | Spin determinations of 31P levels from the 30Si(p, ) reaction[END_REF] and corresponds to J π = 7/2 -. The strength of the resonance has been measured by Wolff et al. [START_REF] Wolff | A study of the excited states of 31 P with the 30 Si(p,γ) 31 P reaction[END_REF], and the value after renormalization is ωγ = 0.13 eV.

• E c.m. r = 946.2 ± 1.0 keV: The associated excited state to this resonance has an energy of E x = 8242.7 keV, and represents with the level at E x = 8247.6 keV a doublet, that could not have been resolved in the work of Vernotte et al. [START_REF] Vernotte | Si( 3 He,d) 31 P reaction at 25[END_REF]. The spinparity of this state has been measured as J π = 5/2 -by Wolff et al. [START_REF] Wolff | Spin determinations of 31P levels from the 30Si(p, ) reaction[END_REF], and the strength of the resonance has been estimated as ωγ = 0.63 eV [START_REF] Wolff | A study of the excited states of 31 P with the 30 Si(p,γ) 31 P reaction[END_REF], after renormalization. The strength has not been remeasured in the present work, but is renormalized with respect to our value of the strength of the resonance of reference.

• E c.m. r = 951.1 ± 1.0 keV: This resonance corresponds to the excited state at E x = 8247.6 keV, which has been populated through several transfer reactions such as 29 Si( 3 He,p) 31 P [54], 30 Si( 3 He,d) 31 P [START_REF] Vernotte | Si( 3 He,d) 31 P reaction at 25[END_REF], and 30 Si(d,n) 31 P [START_REF] Uzureau | Spectroscopic study of 31p and 32s by the (d, n) reaction at ed = 7 mev[END_REF] which suggests a spin parity of J π = 3/2 -. The resonance has been measured by Wolff [START_REF] Wolff | A study of the excited states of 31 P with the 30 Si(p,γ) 31 P reaction[END_REF] and the strength is found to be ωγ = 0.79 eV. This resonance represents the strongest component of the doublet with the E c.m. r = 946 keV, and has the advantage to strongly decay to the ground state, which is not the case for the other resonance. The detection of the high energy γ-ray is less sensitive to background signal, and the detection of the coincidence events are unequivocal. The adopted value for the resonance strength measured with DRAGON is ωγ = 1.54 ± 0.18 eV.

• E r ≥ 1 MeV

Higher resonances correspond to excitation energies where the density of levels becomes more important. The 30 Si + p resonances have been measured for E r > 1 MeV in several works [START_REF] Van Rinsvelt | The 30 Si (p, γ) 31 P reaction for bombarding energies between 1.00 and 1.53[END_REF][START_REF] Harris | Properties of Excited States of P 31 . II. Gamma-Ray Angular Distributions and Correlations[END_REF]. The strengths have been renormalized with respect to the resonance of reference at E c.m. r = 601 keV. Resonances at such high energies contribute to the total reaction rate for temperature above 5 GK, which is extremely high for stellar hydrogen burning.

All the new determined resonance strengths are summarized in table 5 30 Si+p threshold adopted in present work. For resonances with E c.m. r < 400 keV, a 30% uncertainty is considered for the adopted resonance strength.

5.2 30 Si(p,γ) 31

P reaction rates

We recall here the expression of the thermonuclear reaction rate in the general case [START_REF] Iliadis | Nuclear Physics of Stars[END_REF]:

⟨σv⟩ = 8 πµ 1/2 1 (kT ) 3/2 ∞ 0 E σ(E)e -E/kT dE.
(5.2)

Each of the parameters contributing to the calculations has an associated uncertainty, and a statistically meaningful estimation of the reaction rates has to take into account all these uncertainties simultaneously. The Monte Carlo Method for Reaction rates (RatesMC) [START_REF] Longland | Charged-particle thermonuclear reaction rates: I. Monte Carlo method and statistical distributions[END_REF] has been developed for this purpose and is detailed in the following.

Monte Carlo approach

The RatesMC code [START_REF] Longland | Charged-particle thermonuclear reaction rates: I. Monte Carlo method and statistical distributions[END_REF] is a powerful tool to calculate the reaction rates and their associated uncertainties. It is based on Monte Carlo techniques, where each parameter is sampled according to a distribution that takes into account the nature and the magnitude of the uncertainties attached to the parameters. It has the advantage to take into account the correlations that may exist between the different parameters. For example, in the case of the reaction rate estimation for narrow resonances, the same random value of the resonance energy has to be considered in the calculation of both the reaction rate formula 5.2 and the penetration factor to which is multiplied the reduced particle width.

a/ Probability distributions of nuclear inputs

Each of the nuclear physics parameters contributing to the reaction rates is sampled according to a probability distribution P (x) with an associated expected value, or mean E[x] , and a corresponding variance V [x], given by:

E[x] = +∞ -∞ xP (x)dx, V [x] = +∞ -∞ (x -E[x]) 2 P (x)dx, (5.3)
where P (x) is normalized as +∞ -∞ P (x)dx = 1. The form of the probability distributions depends on how the variable x has been determined, and on how it is related to the physical quantities that have been measured experimentally. The measuring principle is nuclear physics experiments is based on event counting, which leads, given enough statistics, to normal distributions for measured quantities. This is given by the central limit theorem that states when independent random variables are summed, the distribution approaches a normal distribution, no matter how these variables are distributed. The distributions of each nuclear input data is discussed in the following.

Resonance energies In the present work, as well as in most experiments, the resonance energies have been determined either through the measurement of excitation energy of the corresponding level in the compound nucleus, through the relation: E r = E x -Q, or through energy loss of the incident beam in the target. In both cases, the resonance energy determination is affected by a sum of several contributions: the beam energy calibration, the target density profile and the beam straggling in the target, the detection of either the deuterons in the gas detectors (for the transfer reaction) or the γ-rays in the BGO crystal (for the direct measurements), and the noise in the electronic data acquisition system. For each experiment, all these contributions to the total uncertainty are summed and by the virtue of the central limit theorem, the resulting resonance energy can be described by a normal distribution:

P (x) = 1 σ √ 2π e (x-µ) 2 /2σ 2 , (5.4)
where µ and σ are the mean and the standard deviation of the distribution, respectively. The sampled energies in the negative part of the normal distribution are treated as subthreshold resonances and the contributions of their tail are treated in a same way as in the case of broad resonances.

Resonance strengths Resonance strengths are either measured directly, through yield measurements and stopping power determination (see Chapter4), or through the determination of the partial widths, which is also linked to the spectroscopic factor determination from cross-section normalization (see chapters 2 and 3). Note that in both cases, the strength is given by the product and the quotient of positive quantities, such as the number of measured counts for reaction products, number of incident beam particles, detection efficiencies, solid angles, etc. The logarithm of the resonance strength can be seen as the sum of independent random variables, and thus is normally distributed. Consequently, the strength is said to be distributed according to the lognormal distribution, which is given by:

P (x) = 1 σ √ 2π 1 x e (ln x-µ) 2 /2σ 2 , (5.5)
where µ and σ are the mean and the standard deviation of the corresponding normal distribution of ln(x), respectively. The statistical properties of the log-normal distribution are detailed in appendix A. Based on similar reasons, the partial particle and γ-ray widths are also distributed according to log-normal distributions. One practical property of this distribution is to predict only strictly positive values, which is suitable for resonance strengths and partial widths.

Upper limits The experimental resonance strength determination is sometimes impossible. It is the case for resonances close to the threshold, where direct measurements of the cross-sections are difficult. Similarly, the corresponding excited state may be expected yet not observed through the transfer reactions, due to the presence of contamination, low statistics etc. In such cases, an upper limit is set for the resonance strength or the spectroscopic factor. The challenge consists in incorporating these upper limits in the reaction rate calculation with a meaningful statistical distribution. To do so, the measured upper limits on the resonance strength ωγ ul , or the partial widths Γ ul p , are expressed in terms of the dimensionless reduced width θ 2 p ul , which is defined as [START_REF] Iliadis | Nuclear Physics of Stars[END_REF]:

θ 2 p = C 2 S p × |φ 2 (R)| 2 R , (5.6)
where φ(R) is the radial part of the wave function calculated at the radius R (see section 3.2.2). The corresponding proton partial width and resonance strength would be:

Γ p = 2ℏ 2 µR 2 P ℓ θ 2 p , (5.7
)

ωγ ≈ 0.5(2J + 1)Γ p , (5.8)
(5.9)

with µ the reduced mass of the system, and P ℓ the penetrability. The probability density function of θ 2 is given by a Porter-Thomas distribution [START_REF] Guhr | Randommatrix theories in quantum physics: common concepts[END_REF], defined as:

f (θ 2 ) = c √ θ 2 e -θ 2 /2⟨θ 2 ⟩ , (5.10)
where c is a normalization constant, and ⟨θ 2 ⟩ is the local mean value for a given single channel, i.e. for a specific nucleus, angular momentum and channel spin for unbound levels. Note that ⟨θ 2 ⟩ is usually deduced from incomplete systematic studies based on few measurements. The measured upper limit information is incorporated in the sampling process by truncating the Porter-Thomas distributions at θ 2 ul :

f (θ 2 ) =    c √ θ 2 e -θ 2 /2⟨θ 2 ⟩ if θ 2 < θ 2 ul 0 if θ 2 > θ 2 ul (5.11)
The mean value of dimensionless reduced width that has been used in the present work is ⟨θ 2 ⟩ = 0.0003 taken from [START_REF] Pogrebnyak | Mean proton and α-particle reduced widths of the Porter-Thomas distribution and astrophysical applications[END_REF].

Uncertainty on spin-parity For some resonances, the orbital angular momentum ℓ cannot be determined without ambiguity. For instance, the differential cross-section for some populated excited states in 31 P lack data at forward angles and the DWBA fit is not well constrained. In such cases, different ℓ-values can be considered. In previous works [START_REF] Dermigny | Thermonuclear reaction rate of 30 Si(p ,γ )[END_REF], the different orbital angular momenta were considered simultaneously while calculating the reaction rate, assuming a Monte Carlo sampling of 100% N chance of contribution from each of the N possible values for ℓ. In the present work, the different possibilities for the ℓ of a given resonance were considered independently, considering that the resonance cannot be a mix of different parities, and the uncertainty is rather discrete regarding the possible quantum numbers for a level. This approach is discussed and justified in section 5.1 for the E c.m. Table 5.2: Parameters of the S-factor for the direct proton capture on 30 Si. Values come from a fit performed in Ref [119].

Direct radiative capture The non-resonant processes contributing to the total reaction rate are conveniently expressed via the astrophysical S-factor (see Chapter 1). The calculations are simplified by considering the Taylor expansion of the S-factor around E = 0 MeV:

S(E) ≈ S(0) + S ′ (0)E + 1 2 S ′′ (0)E 2 .
(5.12)

For Monte Carlo reaction rate estimations, the expansion parameters of the S-factor are sampled according to a log-normal distribution, where the mean values corresponding to the 30 Si(p,γ) 31 P reaction are given in table 5.2, and the square root of the variance is taken as 40%.

b/ Reaction rate output distribution

The total reaction rate, in the case of narrow resonances, is, as it appears in equation 5.2, a sum of the resonant contributions. At low temperatures, few resonances contribute to the rate since the Gamow peak is narrow. The reaction rate is then best described by a log-normal distribution, which is coherent with several considerations: (i) the rate is dominated by the non-resonant direct capture, and thus is proportional to the S-factor, which is log-normally distributed. Otherwise, the rate is dominated by a single or a few resonances, with the main source of uncertainty being: (ii) the resonance energy (normal distribution), thus the reaction rate, proportional to e Er/kT is log-normally distributed, and/or (iii) the uncertainty is dominated by the resonance strength, log-normally distributed too. At higher temperatures, the Gamow peak becomes larger, and the density of levels in the compound nucleus increases. The sum in the reaction rate formula includes much more resonances, and the central limit theorem becomes valid as the reaction rate distribution tends toward a Gaussian distribution. Note that the Gaussian distribution can be described by a log-normal parametrization. Figure 5.1 shows the 30 Si(p,γ) 31 P reaction rate distributions for different stellar temperatures. As explained above, the rate distribution is skewed and distributed with a log-normal density function, in red. The distribution becomes more symmetric as the temperature increases, the skewness factor1 is indicated for each temperature, and one can see that it tends towards zero when the temperature increases. Note that the log-normal density function (red curve) is not fitted on the reaction rate distribution (blue histogram), but is rather calculated from its parametrization: µ = E[ln(x)] and σ 2 = V [ln(x)]. These parameters are deduced from the cumulative distribution of the reaction rate calculated with the Monte Carlo technique. The low, median, and high rates correspond to the 0.16, 0.50 and 0.84 quantiles, respectively. The log-normal parameters are related to these quantities as follows [START_REF] Longland | Charged-particle thermonuclear reaction rates: I. Monte Carlo method and statistical distributions[END_REF]:

µ = ln(x med ) = ln(x low .x high ), σ = ln x high x med = ln x high x low . ( 5

.13)

From all the discussion above, one can mistakenly deduce that the reaction rate is always distributed according to a log-normal density function. However, none of the aforementioned arguments prove this. In the work of Iliadis et al. [START_REF] Iliadis | Charged-particle thermonuclear reaction rates: III. Nuclear physics input[END_REF], several reaction rates have been studied, and their distributions are often far from being lognormally distributed. For example, if an upper limit is considered for a partial width of a given resonance, the resulting rate won't necessarily be log-normally distributed. However, a statistical test can be made to check the validity of such a description. The Anderson-Darling test [START_REF] Stephens | EDF Statistics for Goodness of Fit and Some Comparisons[END_REF], noted A-D, indicates that the log normal distribution is a good approximation if A-D≲ 1. This condition is relaxed by Iliadis et al. [START_REF] Iliadis | Charged-particle thermonuclear reaction rates: III. Nuclear physics input[END_REF] where values of A-D ≈ 1 -30 are accepted, based on a visual inspection of the distribution and the corresponding log-normal distribution. For the 30 Si(p,γ) 31 P reaction rate, we find 0.2 ≲A-D≲ 20.

All the input parameters used in the RatesMC input file are given in appendix D

Results and discussions

The Monte Carlo calculations used for the determination of the 30 Si(p,γ) 31 P reaction rates were performed with RatesMC using 20000 samples for each parameter. The results are presented in Tables 5.3 and 5.4, where the low, median, and high rates represent the 16 th , 50 th , and 84 th percentiles of the rates' distribution. The reaction rate as a function of the temperature, and its associated 68% coverage probability are presented in figure 5.2. The experimental study of the 30 Si(p,γ) 31 P reaction rate through the 30 Si( 3 He,d) 31 P transfer reaction has reevaluated the recommended rate and reduced the uncertainty for low temperature T ≲ 50 MK. This can be seen in figure 5.4 where the present rate is displayed in purple, along with the reaction rates from the most recent 30 Si(p,γ) 31 P evaluation [START_REF] Dermigny | Thermonuclear reaction rate of 30 Si(p ,γ )[END_REF] in gray colour. All the rates are normalized to the recommended rate calculated in the present work. The uncertainty is now 50 times smaller, with a median Table 5.3: Thermonuclear reaction rates for the 30 Si(p,γ) 31 P reaction as a function of the temperature in units of cm 3 mol -1 s -1 . The E c.m. r = 149 keV resonance was assumed to have ℓ p = 2 in this calculation. Table 5.4: Thermonuclear reaction rates for the 30 Si(p,γ) 31 P reaction as a function of the temperature in units of cm 3 mol -1 s -1 . The E c.m. r = 149 keV resonance was assumed to have ℓ p = 3 in this calculation and the temperature range is restricted to where the rates are different with respect to those displayed in Table 5 rate twice as high as in the previous work of Dermigny et al. [START_REF] Dermigny | Thermonuclear reaction rate of 30 Si(p ,γ )[END_REF] for T ≲ 50 MK. This dramatic reduction of uncertainty is due to the experimental determination of the proton partial widths for 31 P levels close to the emission threshold, while the previous rates were based on upper limits of the spectroscopic factors.

In the temperature range 50 MK ≤ T ≤ 200 MK, the new recommended rate is up to seven times higher. This is a consequence of the experimental observation of the E c.m. This is shown in figure 5.2 where three sets of reaction rates have been calculated assuming ℓ = 2, ℓ = 3 and no contribution at all; the three cases are normalized to the ℓ = 2 recommended reaction rate. We chose to present the different rates separately in Table 5.3 for ℓ = 2 and Table 5.4 for ℓ = 3 for 50 MK ≤ T ≤ 200 MK where the rates are different.

Note that the previous study of Dermigny et al. [START_REF] Dermigny | Thermonuclear reaction rate of 30 Si(p ,γ )[END_REF] has sampled the contribution of this resonance with equal probabilities between ℓ = 2, ℓ = 3 and ℓ = 4. Given that the recommended reaction rate is defined as the median of the cumulative distribution function (50 th percentiles) the Dermigny et al. [START_REF] Dermigny | Thermonuclear reaction rate of 30 Si(p ,γ )[END_REF] recommended rate naturally corresponds to the ℓ = 3 case, which is a misleading conclusion. Furthermore, the 1σ probability coverage includes continuous rates values that lie between contributions of ℓ = 2 and ℓ = 3, for example, this is physically not possible due to the quantified nature of the orbital momentum.

For temperatures above 200 MK, the present rate is 25% smaller than the previous rate. This is due to the remeasurement of the resonance of reference at E c.m. r = 602 keV. Our new 30 Si(p,γ) 31 P reaction rates are also compared in figure 5.5 to the rates from Iliadis et al. [START_REF] Iliadis | Charged-particle thermonuclear reaction rates: III. Nuclear physics input[END_REF] since it was used in the nucleosynthesis calculations presented in chapter 1. For low temperatures T ≲ 50 MK, the rate is smaller compared to the previous evaluation. This is due to the experimental determination of the spectroscopic factor of the E c.m. r = 50 keV resonance which dominates the rate in this temperature range. In the temperature range between 50 MK and 150 MK, the rate published by Iliadis et al. is smaller than our rate with ℓ = 2, and higher than with ℓ = 3. This is due to the fact that the E c.m. r = 149 keV resonance was not considered in the previous evaluation; thus the rate is dominated by the direct capture, which was since reevaluated to a smaller contribution [119]. For the range 150 MK ≤ T ≤ 1 GK, the previous rate is one order of magnitude higher. This is due to the lack of experimental assignment of spin-parity fot the resonance at E c.m. r = 418 keV. Iliadis et al. considered an ℓ = 0 for this state, which was later found to have an ℓ = 3 instead. This made the contribution of the resonance negligible. The difference induced by the updated 30 Si(p,γ) 31 P reaction rate will be investigated in the following section. 30 Si(p,γ) 31

Impact of the

P reaction rates

In order to estimate the impact of our new 30 Si(p,γ) 31 P reaction rates on the temperature and density conditions of polluter stars reproducing the astrophysical observations, nucleosynthesis network calculations have been performed following the same framework as in the work of Iliadis et al. [START_REF] Iliadis | On Potassium and Other Abundance Anomalies of Red Giants in NGC 2419[END_REF] and Dermigny et al. [START_REF] Dermigny | Sensitivity to Thermonuclear Reaction Rates in Modeling the Abundance Anomalies of NGC 2419[END_REF]. The calculations are detailed in the following subsections.

Nucleosynthesis network calculations

In order to reproduce the observed abundances in globular cluster NGC 2419, a nucleosynthesis network calculation is performed for hydrogen burning conditions. The network consists of 213 nuclides from proton up to 55 Cr. Thermonuclear reactions rates are adopted from STARLIB library [START_REF] Sallaska | STARLIB: A Next-generation Reaction-rate Library for Nuclear Astrophysics[END_REF] version 67a 0 22619. This library contains the rates for charged particles induced reactions, neutron induced reactions, and also weak interactions such as β decays of the nuclides. This library has the advantage to include the uncertainties associated to the reaction rates. They are either deduced from the experimental uncertainties whenever available, or corresponds to an arbitrary factor of 10 uncertainty when the rates are calculated with a statistical model.

The initial abundances that were considered correspond to the observed abundances in halo field stars2 with the same metallicity as NGC 2419, namely [Fe/H] = -2.1. The initial mass fractions are listed in table 5.5, and are derived from the results of a one-zone chemical evolution model of the Milky Way halo [START_REF] Goswami | Abundance evolution of intermediate mass elements (C to Zn) in the Milky Way halo and disk[END_REF]. The adopted initial abundances are also guided by the red giant stars observed in NGC 2419 that do not exhibit potassium enhancement ([K/Fe] ≈ 0), since their composition is believed to be unaffected by the polluters' material. The initial mass fractions of 23 Na, 24 Mg, 27 Al, 28 Si, 39 K, 40 Ca and 51 V are adjusted to match the observed abundances of red giants in NGC 2419.

The hydrogen burning process is assumed to occur in a one-zone model with constant temperature and density, which is a reasonable approximation for the hydrostatic burning in the core of massive and supermassive stars, and in the hydrogen-shell of AGB stars.

Firstly, the evolution of the abundances during H-burning is explored. The calculations are performed at temperature T = 160 MK, and density ρ = 100 g/cm 3 and initial hydrogen abundance X i H = 0.75. The calculations are stopped when the remaining hydrogen mass fraction is X f H = 0.7. The obtained mass fractions for the magnesium and the potassium are displayed in figure 5.6. The abundances obtained for a temperature of T = 160 MK are compared to the results of the calculations performed at T = 100 MK (dashed lines). The abundance of Mg is not altered at T = 100 MK, and needs higher temperatures to be converted into K. We clearly see that the amount of Mg decreases as the K abundance increases for T = 160 MK. This indicates that the final abundances of Mg and K strongly depend on the temperature, and that the observed Mg-K anticorrelation may result from very specific (Tρ) conditions.

Processed matter and dilution

The abundances resulting from the nucleosynthesis network calculations need to be compared to the abundances observed in NGC 2419. First, we see from figure 5.6 that for T = 160 MK and ρ = 100 g.cm -3 , the magnesium is completely destroyed since its mass fraction falls to 10 -10 , that would correspond to an abundance of [Mg/Fe]≲ -7, which is too small to be detected. The abundances observed in the most magnesium poor stars are around [Mg/Fe]∼ -1.5, and would then correspond to a fraction of the initial Mg abundance X pris that has been mixed with the processed matter X proc .

All the range of abundances can be reproduced by considering this dilution process, which is given as:

X mix = X proc + f X pris 1 + f , (5.14)
where f is the dilution factor. We vary this dilution factor as :f = 0, 0.02, 0.05, 0. The observed abundances that must be reproduced by the nucleosynthesis calculations are indicated by the black boxes in figure 5.7, and given by:

1.3 < [K/Fe] < 2.0 -1.5 < [Mg/Fe] < -0.8 0.1 < [Ca/Fe] < 0.7 -0.2 < [Ti/Fe] < 0.7 0.4 < [Si/Fe] < 1.1 0.4 < [Sc/Fe] < 1.3 -0.2 < [V/Fe] < 0.6 (5.15)
These abundances correspond to the highest amount of potassium observed. The dilution factor must be between f = 0.01 and f = 0.04 to reproduce the smallest magnesium abundances observed.

Monte Carlo sampling

Monte Carlo sampling is used to explore the space parameters that can reproduce the observed abundances. The temperature T and density ρ are sampled according to a uniform logarithmic distribution, and the final hydrogen mass fraction is sampled according to a uniform distribution:

log 10 (T ) ∼ log 10 (T min ) + (log 10 (T max ) -log 10 (T min )) × p T , log 10 (ρ) ∼ log 10 (ρ min ) + (log 10 (ρ max ) -log 10 (ρ min )) × p ρ , X f H ∼ X min + (X max -X min ) × p X , (5.16)
with T min = 50 MK, T max = 300 MK for the temperature range, ρ min = 10 -4 g.cm -3 , ρ max = 10 11 g.cm -3 for the density range, and X min = 0.10, X max = 0.75 for the final hydrogen mass fraction. The parameters p T , p ρ and p X are uniform random numbers between 0 and 1, used to sample the temperature, density and final hydrogen mass fraction, respectively. The reaction rates and their associated uncertainties are a function of the temperature, the rate ⟨σv⟩(T ) is sampled as [START_REF] Longland | Recommendations for Monte Carlo nucleosynthesis sampling[END_REF]:

⟨σv⟩(T ) ∼ e µ(T ) × e σ(T )px(T ) , (5.17)
where µ is the recommended median rate, and σ is the associated uncertainty, as defined in the equations (5.13). The parameter p x (T ) is sampled following the prescription of [START_REF] Mucciarelli | News from the Galactic suburbia: the chemical composition of the remote globular cluster NGC 2419[END_REF][START_REF] Cohen | The Bizarre Chemical Inventory of NGC 2419, An Extreme Outer Halo Globular Cluster[END_REF]. The black boxes indicate the selected abundances that need to be reproduced by the nucleosynthesis network calculations. The black curve with crosses indicates the results of the mixing between the processed matter and the pristine matter (f between 0 (rightmost cross) and 1000 (leftmost cross)). The calculations were performed for T = 160 MK, ρ = 100 g/cm 3 and X f H = 0.7.

Longland [START_REF] Longland | Recommendations for Monte Carlo nucleosynthesis sampling[END_REF] with a hyperbolic tangent function, in order to sample both low and high rates depending on the temperature region. The experimentally deduced uncertainties for the reaction rates are considered whenever available. Otherwise, the factor uncertainty is set to f.u = e σ(T ) = 10 for reaction rates calculated with a statistical model. The initial abundances of elements that are not constrained by the observations in NGC 2419 (indicated by an asterisk in table 5.5) are also varied during the Monte Carlo calculations, and are sampled according to a log normal distribution (Eq. 5.5).

The factor of uncertainty for the initial abundances is taken as f.u = 2.5 following the prescription of Iliadis et al. [START_REF] Iliadis | On Potassium and Other Abundance Anomalies of Red Giants in NGC 2419[END_REF]. The samples that reproduce the observed abundances given by Eq. (5.15) with a dilution factor 0.01 ≤ f ≤ 0.04 are selected, and the corresponding T, ρ and X f H are stored. In order to compare consistently the impact of the 30 Si(p,γ) 31 P reaction rate calculated in the present work with the one considered previously during the original works of Iliadis et al. [START_REF] Iliadis | On Potassium and Other Abundance Anomalies of Red Giants in NGC 2419[END_REF], and Dermigny et al. [START_REF] Dermigny | Sensitivity to Thermonuclear Reaction Rates in Modeling the Abundance Anomalies of NGC 2419[END_REF], we first reproduced their results within the framework presented above. In the following, we briefly present the benchmark of the method, before presenting the results for our new 30 Si(p,γ) 31 P reaction rates. a/ Benchmark using the STARLIB 30 Si(p,γ) 31

P rate

The rates used in both works of Iliadis et al. [START_REF] Iliadis | On Potassium and Other Abundance Anomalies of Red Giants in NGC 2419[END_REF], and Dermigny et al. [START_REF] Dermigny | Sensitivity to Thermonuclear Reaction Rates in Modeling the Abundance Anomalies of NGC 2419[END_REF] are first used in the nucleosynthesis network calculations. Figure 5.8 shows the stellar density as a function of the temperature for two sets of network calculation: the red circles are obtained for calculations performed with fixed recommended rates, while the blue dots are obtained by sampling the reaction rates within their uncertainties. The broadening of the (T, ρ) locus is a clear effect of the reaction rates uncertainties. In order to have a quantitative estimation of this effect, we select a density between 80 g.cm -3 and 120 g.cm -3 , and we project the realizations within this range into the temperature axis.

Figure 5.9a shows the corresponding temperature histograms, in red for fixed rates and in blue for all rates varying. The broadening δ of the temperature distribution is defined as the region between the 16 th and the 84 th percentiles. The broadening corresponding to calculations with the recommended median rates is δ mr = 9 MK, while the one obtained while varying all the rates is δ all = 18.5 MK. The ratio is δ all δmr = 2.1, slightly higher than the δ all δmr = 19 11 = 1.7 ratio obtained by Dermigny et al. [START_REF] Dermigny | Sensitivity to Thermonuclear Reaction Rates in Modeling the Abundance Anomalies of NGC 2419[END_REF].

One has to bear in mind that the Monte Carlo nucleosynthesis code used in the present work is not the same as in the original works of Iliadis et al. The new recommended 30 Si(p,γ) 31 P rate induces a shift of the median value for the temperature locus when the abundances are reproduced. This median temperature shifts from T = 158.8 MK to T = 166.3 MK. Furthermore, the spread of the temperature locus increases, from δ mr = 9 MK to δ mr = 16.4 MK. Interestingly, the difference in the ℓ values for the E c.m. r = 149 keV resonance has no significant impact on the temperature locus.

The main difference is that the 30 Si(p,γ) 31 P reaction rate from Iliadis et al. has a recommended value 10 times higher than the present rate around T ≈ 170 MK (see discussion 5.2.2). This causes 30 Si to be much more destroyed at T ≳ 170 MK when compared to our new reaction rate. Note that for this temperature range, 28 Si is converted to 30 Si which becomes the dominant silicon isotope, and the elemental abundances are closely linked to its mass fraction.

Figure 5.11 shows the variation of the silicon abundance in nucleosynthesis networks with both previous and updated 30 Si(p,γ) 31 P rates, for T = 160 and 175 MK. Interestingly, when the temperatures exceed ≈ 170 MK, the silicon abundance obtained with the STARLIB rates falls drastically and do not match the observations anymore (gray bands). This explains why the temperature locus is truncated when using STARLIB's high rate, when comparing to the temperature distribution obtained with our new reaction rate.

Impact of uncertainties

The impact of the uncertainties associated to the 30 Si(p,γ) 31 P rate is investigated in a second step. Figure 5.12 shows the comparison between the temperature distributions obtained for the different 30 Si(p,γ) 31 P rates. These rates were now allowed to vary within their uncertainties, represented by the coloured areas in figure 5.5. All other reaction rates were kept fixed at their recommended values.

Despite the larger uncertainty bands of the rates from Iliadis et al., the corresponding temperature locus is narrower compared to the case with our new rates. This indicates that for the lowest rate from Iliadis et al., silicon is still efficiently destroyed at T ≈ 180 MK. Now we get to see the impact of the updated rate uncertainties with respect to the recommended rate. Figure 5.13 shows the temperature distributions obtained with calculations with fixed rates (red histogram) and only varying the 30 Si(p,γ) 31 P rate (green histogram), for both cases ℓ = 2 and ℓ = 3 for the E c.m. r = 149 keV resonance. The fraction δ/δ mr is found ≈ 1 in both cases, indicating that the 30 Si(p,γ) 31 P rates with the present uncertainties are constrained enough for the stellar temperature and density conditions in the polluters of NGC 2419. The same procedure is repeated while varying all the rates, and is shown as the blue histogram in figure 5.13. The broadening ratio is δ all δmr = 20.7/16.5 = 1.25, which means that the uncertainties corresponding to the rest of the STARLIB reaction rates induce a 25% broadening of the temperature range when using the updated 30 Si(p,γ) 31 P recommended rates presented in this work. For this particular model and with our new 30 Si(p,γ) 31 P reaction rates, there is little room for improving the constraints on (Tρ) conditions from a nuclear perspective. However, the present study showed that the update of a single median rate could induce a significant change in the final abundances, having a strong impact on the temperature locus. In that case, a sensitivity study similar to the one of Dermigny et al. should be performed to evaluate the impact of such variation for the existing recommended rate.

In this chapter, we presented our updated 30 Si(p,γ) 31 P reaction rates based on the 30 Si + p resonance strengths obtained in our work. The rates were calculated using the RatesMC Monte Carlo code, and statistically meaningful uncertainties were obtained. The new rates are compared to previous results, and their impact was explored in the context of observed abundances in globular cluster NGC 2419. With the present reaction rates, the observations can be reproduced at higher temperatures compared to the previous 30 Si(p,γ) 31 P rate. Our new reaction rates are found to be sufficiently constrained for the elemental observations in NGC 2419, and any improvement of the temperature broadening should be investigated regarding other key reaction rates.

Conclusion

Globular clusters are complex objects that continue to arouse the interest of physicists and astronomers. The improvement of photometric and spectroscopic observation techniques has made it possible to highlight the multiplicity of star populations within globular clusters. The imprint of past generations of stars, called polluters, can be seen in the observation of anti-correlations between pairs of light chemical elements. Some of these anti-correlations, such as Na-O, are observed in almost all globular clusters. The cluster NGC 2419, however, harbours an even more curious anti-correlation, between magnesium and potassium. Sensitivity studies have indicated that the 30 Si(p,γ) 31 P reaction and its associated uncertainties have the greatest impact on the plausible temperature and density ranges for polluter stars in globular cluster NGC 2419. In this thesis, we focus on the improvement of our understanding of the 30 Si(p,γ) 31 P reaction with different experimental approaches aiming at the determination of the 30 Si+p resonance strengths.

In a first experimental approach, the 31 P excited states were populated using the one-proton 30 Si( 3 He,d) 31 P transfer reaction. This allowed the determination of the spectroscopic properties of states close to the proton separation threshold, where direct measurements would be difficult due to the high Coulomb barrier. The reaction products were measured at different angles using the high energy resolution Q3D magnetic spectrometer. The excitation energies of 31 P levels were extracted, as well as their corresponding angular distributions. Comparison of the latter with the DWBA model allowed us to deduce the transferred orbital angular momenta, and the associated spectroscopic factors, for 26 states in 31 P. Calculations of the proton widths using the same theoretical model allowed us to deduce the resonance strengths associated to unbound states above the proton threshold. The analysis was accompanied by a statistical study using Bayesian methods where all the ingredients of the calculations, i.e. optical and binding potentials, elastic scattering and transfer data, were considered to estimate the uncertainties on the proton widths.

In a second experimental approach, the strengths of higher energy resonances have been measured directly with the DRAGON recoil spectrometer. Measurements have been made for 6 resonances, between E c.m. r = 485 keV and 950 keV. These resonances lay within the Gamow window corresponding to polluter stars in globular clusters, and also for type Ia Supernovae for the higher energy ones. The resonance strengths were determined from the number of 31 P recoils detected in coincidence with γ-ray decays. This analysis was accompanied by a series of detailed Monte Carlo simulations to determine the various efficiencies, including the γ-ray detection efficiencies and the recoil transmission through the DRAGON spectrometer. Resonance energies were also determined using measurements of beam energies and stopping power in the gas target.

A compilation of the properties of 30 Si+p resonances at energies up to 1 MeV was presented in the last chapter. The strength of these resonances were used for the calculation of the 30 Si(p,γ) 31 P reaction rates using Monte Carlo techniques, allowing the determination of the median recommended rate, as well as the associated high and low rates. These new rates have been integrated into the STARLIB library in order to study the impact of the nucleosynthesis network calculations on the abundances observed in the globular cluster NGC 2419. We conclude that the new rates allow reproducing the observations for a more extended temperature range than previously, namely between 150 and 200 MK for a density of 100 g.cm -3 .

The main conclusion of this work is that the uncertainties associated with the new 30 Si(p,γ) 31 P rate no longer contribute to the ill-definition of the conditions on the temperatures and densities of the polluter sites. The remaining uncertainties associated to other reaction rates expand the temperature locus by 25%. However, these results are not immune to modification if any of the key reaction rates are changed upon future new evaluations. Finally, these results should be seen in the context of more advanced models for hydrogen burning, where temperature and density profiles for explosive processes should be considered. The nature of polluter stars in globular cluster NGC 2419 are still a mystery to be solved, and we expect, under the light of the present work, that the unravelling of this enigma would come from fine-tuned astrophysical scenarios and improved astronomical observations. the latter can be related to the energy in the entrance channel through the Q-value as T ′ f = T ′ i + Q, and we have:

T ′ f = Q + T a m A m a + m A (5.36)
Furthermore, the velocity of the emitted particle after the collision is given by:

⃗ v ′ b = ⃗ v b -⃗ v c.m. , (5.37)
when projecting on the beam axis in the laboratory frame, we obtain:

v ′ b cos(θ ′ ) = v b cos(θ) -v c (5.38)
One can define the parameter γ as the ratio of velocities of the centre-of-mass frame and of particle b in the said frame:

γ = v c v ′ b = m a m b T a m B (m b + m B )Q + m B (m B + m b -m a )T a (5.39)
The relation between the emission angle in the laboratory frame and in the centreof-mass frame is given as: cos(θ) = γ + cos θ ′ 1 + γ 2 + 2γ cos θ ′ (5.40)

The differential cross-section is defined for a given solid angle element dω in the direction defined by θ, it is thus impacted by the frame transformation. We have:

dσ dΩ θ dΩ = dσ dΩ ′ θ ′ dΩ ′ (5.41)
The Jacobian of the transformation is defined as: Finally, the integral of the exponentially modified gaussian (5.44) is given by: Very often, the exponentially modified gaussian (5.44) is not used as it is, and we add an amplitude factor A. The corresponding amplitude is simply: ! J p r o j 0 .

I = δ exp - δ 2 2ν 2 (a
I = 2Aν
! J t a r g e t 0 .

! J e x i t p a r t i c l e (=0 when o n l y 2 c h a n n e l s open ) 7 2 9 6 .

Résumé en Français Introduction

Il existe une grande variété d'étoiles, en termes de taille, de masse, d'âge, de couleur et de luminosité. Elles sont toutes regroupées dans des galaxies, qui ont, elles aussi, des morphologies différentes. Mais à la périphérie de ces galaxies, il existe des groupes d'étoiles en orbite, étroitement liés par la gravité. Ces amas globulaires, isolés du reste des étoiles, comptent parmi les objets les plus anciens jamais observés. Les plus grands abritent quelques millions d'étoiles, à peine plus lourdes que notre soleil.

Les amas globulaires sont des objets complexes qui continuent à susciter l'intérêt des physiciens et des astronomes. L'amélioration des techniques d'observation photométrique et spectroscopique a permis de mettre en évidence la multiplicité des populations d'étoiles au sein des amas globulaires. L'empreinte des générations antérieures d'étoiles, appelées polluantes, est visible dans l'observation des anti-corrélations entre paires d'éléments chimiques légers. Certaines de ces anti-corrélations, comme Na-O, sont observées dans presque tous les amas globulaires. L'amas NGC 2419 abrite une anti-corrélation encore plus curieuse, celle entre le magnésium et le potassium.

Des études de sensibilité ont indiqué que la réaction 30 Si(p,γ) 31 P et les incertitudes qui y sont associées ont le plus grand impact sur les plages de température et de densité plausibles pour les étoiles polluantes de l'amas globulaire NGC 2419.

Cette thèse vise à apporter des réponses du point de vue de la physique nucléaire en contraignant le taux de la réaction 30 Si(p,γ) 31 P à l'aide de deux approches expérimentales différentes : les réactions de transfert et les mesures directes.

Éléments d'astrophysique stellaire

Dans le premier chapitre, nous introduisons les notions clés de l'évolution stellaire pour les étoiles de faible masse et de masse intermédiaire et plus particulièrement leur évolution au sein des amas globulaires. Le diagramme Hertzprung-Russel est introduit à cet effet, et les différentes phases de combustion au sein des étoiles de masse faible à intermédiaire sont exposées. Des notions de spectroscopie en astronomie sont introduites, afin de faire le lien avec les abondances élémentaires observées sur l'atmosphère des étoiles.

Dans une seconde partie, les propriétés des amas globulaires sont présentées, notamment les masses typiques des étoiles qui les peuples et les métallicités moyennes. Les observations indiquant la présence de plusieurs générations d'étoiles dans les amas globulaires sont présentées, notamment la présence d'anti-corrélation entre les abondances de paires d'éléments chimiques légers, et les observations photométriques qui montrent un dédoublement des branches de géantes rouges sur les diagrammes Hertzprung-Russel correspondants. L'amas globulaire NGC 2419 est discuté en particulier. Les abondances observées sont confrontées aux calculs de nucléosynthèses, effectuées par Iliadis et al. [START_REF] Iliadis | Charged-particle thermonuclear reaction rates: III. Nuclear physics input[END_REF] et Dermigny et al. [START_REF] Dermigny | Sensitivity to Thermonuclear Reaction Rates in Modeling the Abundance Anomalies of NGC 2419[END_REF], ainsi, ces auteurs ont réalisé des études de sensibilité pour identifier les conditions clés de température et de densité et les taux de réaction nucléaire nécessaires pour reproduire les observations. Le taux de réaction thermonucléaire est introduit ainsi que les notions clés de force de résonance, largeurs proton, la fenêtre de Gamow, et les équations correspondantes pour le cas de réactions résonantes et non résonantes. Une vue d'ensemble est donnée de l'état de l'art de la réaction énergies d'excitation et les distributions angulaires de 26 états excités du 31 P sont présentées et discutées. Ces états ont des énergies d'excitation entre 6,8 et 8,1 MeV. L'étalonnage du système de détection du plan focal et l'ajustement des spectres à l'aide d'un système multifonctionnel ont été réalisés. L'ajustement des spectres à l'aide d'une fonction multi-pics a permis de déterminer les énergies d'excitation des états observés avec une grande précision (incertitude inférieure à 3 keV). Après normalisation par rapport au nombre d'ions du faisceau incident, l'épaisseur de la cible et l'angle solide, les sections efficaces différentielles ont été calculées pour tous les angles mesurés, entre 6°et 32°.

Calculs théoriques en DWBA

Les distributions angulaires obtenues doivent être comparées à des calculs théoriques afin d'extraire l'information spectroscopique nécessaires à la détermination de la largeur du proton, puis de la force de résonance. Le cadre théorique adopté est l'approximation de Born de l'onde distordue (DWBA). Distorted Wave Born Approximation (DWBA). Ce modèle est adapté à la description des réactions de transfert directes. Il repose sur l'hypothèse que la section efficace est dominée par la diffusion élastique, tandis que le transfert en une étape du nucléon est suffisamment faible pour être traité comme une théorie du premier ordre.

Les éléments de la théorie des réactions sont présentés dans le chapitre 3, suivis de la description des ingrédients nécessaires pour effectuer le calcul DWBA. Nous présentons également une comparaison quantitative entre les différentes méthodes et prescriptions utilisées pour le calcul de la largeur proton pour les niveaux non liés.

Les facteurs spectroscopiques sont déterminés à partir de la normalisation de la section efficace différentielle théorique par rapport à celle mesurée expérimentalement. La forme des distributions angulaires a été utilisée pour déduire le moment orbital angulaire transféré. Ces quantités spectroscopiques ont été utilisées pour calculer la largeur proton des états non liés du 31 P Les codes fresco et dwuck4 sont comparés, et les avantages et inconvénients de chacun sont discutés. Nous avons montré que la principale source d'incertitudes provient du potentiel optique utilisé dans les calculs DWBA. L'impact de ces incertitudes a été estimé à l'aide d'une approche bayésienne et une incertitude typique de 25% pour les largeurs de protons a été trouvée.

Mesure directe des forces de résonance

La détermination de la force des résonances 30 Si+p à l'aide de la réaction de transfert 30 Si( 3 He,d) 31 P a montré ses limites pour les énergies de résonance E c.m. r 600 keV. À ces énergies, la largeur proton commence à être non négligeable par rapport à la largeur totale des résonances. D'autres éléments, tels que les largeurs γ, deviennent nécessaires pour calculer la force de résonance. Heureusement, à ces énergies plus élevées, des mesures directes des forces de résonance sont possibles, sans qu'il soit nécessaire de décrire le mécanisme de réaction.

Dans le chapitre 4, nous présentons les mesures directes de la force de résonance pour 6 énergies de résonance, par la réaction de capture radiative de protons 30 Si(p,γ) 31 P en cinématique inverse. La configuration utilisée permet de détecter les deux produits de réaction : le noyau composé aura le même impulsion du moment que le faisceau dans le référentiel du laboratoire, et sortira donc de la cible. Il sera mesurable en tant qu'événement unique ou en coïncidence avec les rayons γ émis. L'expérience a été réalisée en août 2021 à l'aide du spectromètre DRAGON dans l'installation TRIUMF. La production du faisceau et le dispositif expérimental sont présentés dans les deux premières sections du chapitre 4. S'ensuit la description des procédures d'analyse des données et des défis qui y sont associés. Les forces de résonance à 6 énergies, entre E c.m. r = 485 keV et E c.m. r = 950keV ont été ainsi mesurées, dont la force de la résonance à E c.m. r = 555 keV mesurée directement pour la première fois. Les forces de résonance ont été déterminées à partir du nombre de reculs du 31 P détectés en coïncidence avec des désintégrations γ. Cette analyse a été accompagnée d'une série de simulations Monte Carlo détaillées afin de déterminer les différentes efficacités, y compris les efficacités de détection des rayons γ et la transmission des noyaux de recul à travers le spectromètre DRAGON. Les énergies de résonance ont également été déterminées à l'aide de mesures de l'énergie des faisceaux et du pouvoir d'arrêt de la cible gazeuse.

Les résultats de ces mesures sont complémentaires aux forces des résonances à des énergies plus basses qui ont été déterminées à l'aide des largeurs partielles calculées dans le cadre du DWBA. Nous avons également présenté la détermination des énergies de résonance. Tous ces résultats sont utilisés pour calculer la valeur actualisée du taux de réaction 30 Si(p,γ) 31 P aux énergies pertinentes pour les températures astrophysiques. Les résonances du système 30 Si+p et taux de réaction 30 Si(p,γ) 31 P Pour finir, une compilation des propriétés des résonances 30 Sip à des énergies allant jusqu'à 1 MeV a été présentée dans le dernier chapitre. La force de ces résonances a été utilisée pour le calcul du taux de réaction 30 Si(p,γ) 31 P à l'aide du code de Monte Carlo RatesMC, permettant de déterminer le taux médian recommandé, ainsi que les limites hautes et basses associées au taux. Les résultats des taux sont présentés sous forme tableau permettant de les utiliser dans le calcul de réseau de nucléosynthèse, et la contribution au taux total de chaque résonance est également calculée. La figure 5.14 montre la contribution fractionnaire de chaque résonance en fonction de la température stellaire

Calculs de nucléosynthèse

Ces nouveaux taux ont été intégrés à la bibliothèque STARLIB afin d'étudier l'impact du réseau de nucléosynthèse sur les abondances observées dans l'amas globulaire NGC 2419. Nous concluons que les nouveaux taux permettent de reproduire les observations pour une gamme de température plus étendue que précédemment, à savoir entre 150 et 200 MK pour une densité de 100 g.cm -3.

Conclusion

La principale conclusion de ce travail est que les incertitudes associées au nouveau taux de la réaction 30 Si(p,γ) 31 Pne contribuent plus à la mauvaise définition des conditions sur les températures et les densités des sites pollueurs. Les incertitudes restantes associées aux autres taux de réaction élargissent le domaine en température de 25%. Cependant, ces résultats ne sont pas à l'abri d'une modification si l'un des taux de réaction clés est changé lors de nouvelles évaluations.

Enfin, ces résultats doivent être considérés dans le contexte de modèles plus avancés pour la combustion de l'hydrogène, où les profils de température et de densité pour les processus explosifs devraient être pris en compte. La nature des étoiles polluantes de l'amas globulaire NGC 2419 reste un mystère à résoudre, et nous pensons, à la lumière du présent travail, que la résolution de cette énigme proviendra de scénarios astrophysiques finement ajustés et d'observations astronomiques améliorées.

1

 1 Stellar evolution . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Hertzsprung-Russell diagram . . . . . . . . . . . . . . . . . . 1.1.2 Stellar evolution . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 Astronomical spectroscopy . . . . . . . . . . . . . . . . . . . 1.2 Globular clusters . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 General properties . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Stellar population within globular clusters . . . . . . . . . . . 1.2.3 Chemical signature of multiple stellar populations . . . . . . 1.2.4 Case of NGC 2419 . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Insight from Nuclear Physics . . . . . . . . . . . . . . . . . 1.3.1 Sensitivity study and key reactions . . . . . . . . . . . . . . . 1.3.2 Importance of the 30 Si(p,γ) 31 P reaction . . . . . . . . . . . . 1.3.3 30 Si(p,γ) 31 P reaction rates for SN Ia . . . . . . . . . . . . . . 1.4 From Stars to Nuclei . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Nuclear reaction networks . . . . . . . . . . . . . . . . . . . . 1.4.2 Thermonuclear reaction rates . . . . . . . . . . . . . . . . . . 1.4.3 Astrophysical S-factor and Gamow peak . . . . . . . . . . . . 1.4.4 Resonant and non-resonant reaction rates . . . . . . . . . . . 1.5 Current knowledge of the 30 Si(p,γ) 31 P reaction . . . . . . . 1.6 Experimental approach . . . . . . . . . . . . . . . . . . . . . 1.6.1 Transfer reactions . . . . . . . . . . . . . . . . . . . . . . . . 1.6.2 Direct measurements . . . . . . . . . . . . . . . . . . . . . . .

Figure 1 . 1 :

 11 Figure 1.1: Schematic Hertzsprung-Russel diagram showing the Absolute magnitude (on the left) as a function of the colour index (B-V). On the right, the luminosity in solar units is also shown. The main stellar classes are indicated, and the evolutionary paths for a star with 1 M ⊙ (in blue) and 10 M ⊙ (in red) are shown. Locations of 8 famous stars in the HR diagram are shown.

Figure 1 . 2 :

 12 Figure 1.2: (Left) Colour magnitude diagram in V vs (V-B) bands for globular cluster NGC 1851. The main evolution stages are indicated. (Right) Covered field of the cluster, corresponding to right ascension (RA) 05h 14m 06.76s and declination (Dec) -40°02' 47.6. Figure adapted from Ref. [10]

Figure 1 . 3 :

 13 Figure 1.3: Colour magnitude diagram of NGC 1851 obtained with high resolution optical (λ = 336 nm) and infrared (λ = 814 nm) photometry of the Hubble Space Telescope. Red points are RGB stars with a higher colour-index, closer to the red wavelengths. The inset shows a zoomed-in view around the SGB. Figure taken from [11].

Figure 1 . 5 :

 15 Figure 1.5: Schematic history of globular clusters. Main episodes are indicated with arrows, and elements enriching the intra-cluster medium from those episodes are listed.

Figure 1 . 6 :Figure 1 . 7 :

 1617 Figure 1.6: Hubble Space Telescope image of NGC 2419

Figure 1 . 8 :

 18 Figure 1.8: Elemental abundances as a function of potassium abundances observed in globular cluster NGC 2419. Each dot is the observed abundances in a red giant star.Orange data points are from Mucciarelli et al.[START_REF] Mucciarelli | News from the Galactic suburbia: the chemical composition of the remote globular cluster NGC 2419[END_REF] and purple data points are from Cohen & Kirby[START_REF] Cohen | The Bizarre Chemical Inventory of NGC 2419, An Extreme Outer Halo Globular Cluster[END_REF]. Errors are sometimes smaller than data point. Figure adapted from[START_REF] Iliadis | On Potassium and Other Abundance Anomalies of Red Giants in NGC 2419[END_REF].

Figure 1 . 9 :

 19 Figure 1.9: Stellar density vs. temperature. Blue dots are conditions that reproduce the observed abundances in NGC 2419. The results are obtained from random sampling of the temperature, density, final hydrogen mass fraction, reaction rates and initial abundances. Black tracks are hydrogen burning conditions for some polluter candidates (see text). Figure taken from [23].

Figure 1 . 10 :

 110 Figure 1.10: Distributions of temperature conditions that reproduce the observed abundances in NGC 25419 for a density of ρ = 100g/cm 3 . The red histogram is obtained with reaction rates fixed at their recommended values, while the blue histogram corresponds to sampling with all reaction rates varying within their statistical uncertainties.Figure taken from [25].

Figure 1 . 11 :

 111 Figure 1.11: Broadening of the temperature conditions due to the uncertainty of each reaction in the STARLIB network. Figure taken from [25].

Figure 1 . 12 :

 112 Figure 1.12: Nucleosynthesis network for hydrogen burning at T = 160 MK and ρ = 900g/cm 3 . The relevant part for the Mg-K anti-correlation observed in globular clusters is represented. The thickness of the arrows is a scale for the reaction flux.

Figure 1 . 14 :

 114 Figure 1.14: Gamow peak (purple line) for the 30 Si(p,γ) 31 Preaction, at T = 100 MK and T = 200 MK. The red dashed line is the Gamow factor and the blue dashed line is the Boltzmann factor.

  .16 between T = 50 MK and T = 200 MK, covering a large portion of the Gamow window for polluters in Globular Clusters. The strongest resonances with energies above 400 keV have been measured directly in multiple works. The compilation of direct measurements for the main resonances with E r < 1 MeV is shown in figure 1.17. The strength of several of these resonances is estimated relatively to the strength of the resonance of reference at E c.m. r = 601 keV. That was the case for resonances at E c.m. r = 422 keV and E c.m. r = 486 keV that have been measured through γγ coincidences in the work of Dermigny et al. 2020.

Figure 1 . 15 :

 115 Figure 1.15: Level scheme for31 P nucleus with excitation energies and spin parities indicated. Only ground and unbound states are shown. The proton energy threshold is given along with energy in centre of mass of the system30 Si + p. Blue arrows show the energy region defined by Gamow windows for astrophysical temperatures of T 9 = 0.1, 0.2 and 1. All values are from NNDC compilation[31] in its 2021 version.

Figure 1 . 16 :

 116 Figure 1.16: The fractional contributions of resonances in the 30 Si(p,γ) 31 P reaction, along with the contribution of the direct capture (labelled DC), to the total reaction rate. The thickness of each band represents the 68% coverage probability, hashed areas are used for resonances for which an upper limit on the strength was considered. The black dotted lines shows the contribution of resonances with energies greater than 648 keV. Figure from Dermigny et al. [32].

Figure 2 . 1 :

 21 Figure 2.1: Schematic representation of the tandem accelerator of the MLL.

Figure 2 . 2 :

 22 Figure 2.2: Schematic representation Q3D spectrometer at MLL

Figure 2 . 3 :

 23 Figure 2.3: Cross-sectional schematic view of the detectors at the focal plane of the Q3D. Beam enters from the left.

Figure 2 . 4 :

 24 Figure 2.4: Working principle of the focal plane detectors with stripped cathode. Bottom part: The incoming particles generate ionization electrons along its track. The electrons drift toward the anode wire and induce positive charges on the cathode strip foil. Top part: histogram of the amount of induced charges collected by each strip (strip step a = 3.5 mm). The charge distribution can be approximated by a Gaussian description (dashed line) [45].

Figure 2 . 5 :

 25 Figure 2.5: Energy spectrum of the scattered 4 He + from the RBS measurement on the enriched 30 Si target. Red points are the experimental data while the curves are the best fit.

Figure 2 . 7 :

 27 Figure 2.7: Particle identification spectra. The top panel is the (∆E, ∆E 1 ) spectrum, while the bottom panel is (E, ∆E). The dashed red lines enclose deuteron and tritons blobs, while the green dashed correspond to the rejected events.

Figure 2 . 8 :

 28 Figure 2.8: Deuteron magnetic rigidity spectra at Q3D spectrometer angles of 10°(top) and 20°(bottom). The excitation energies corresponding to each peak are deduced, preliminary, from the expected rigidities of known levels in 31 P, as shown in figure 2.6.Other peaks, with green labels, correspond to levels populated from contaminating nuclei present in the target, mainly 12 C, 14 N,16 O and28 Si. Levels with asterisk on the energies are used for the focal plane position detectors calibration.

Figure 2 . 10 :

 210 Figure 2.10: Fit of a 31 P peak with a skewed

Figure 2 . 11 :

 211 Figure 2.11: Deuteron position spectrum at spectrometer angle θ Q3D = 12°. The best fit is shown in red solid line, while individual contributions are in red dashed lines for 31 P states and in green dashed lines for contamination contributions.

Figure 2 . 12 :

 212 Figure2.12: Calibration fit of the positions at focal plane detector system. Top panel: Curvature radius ρ for detected deuterons as a function of the ADC channel and its fit with a second order polynomial (eq. 2.7). Bottom panel: Calibration residuals expressed in excitation energy difference between expected energy and value calculated with the calibration fit.

Figure 2 . 13 :

 213 Figure 2.13: Schematic representation of the Q3D opening. The distance betweenx-slits represents the horizontal opening ∆x while the distance between y-slits is the vertical opening ∆y[START_REF] Rebeiro | Nuclear structure studies in the A=136 region using transfer reactions[END_REF].

Figure 2 . 14 :

 214 Figure 2.14: Deuteron magnetic rigidity spectra showing close-up on weakly populated levels in 31 P. Spectrum showing the channel region 1020-1220 (middle left panel) have been rebinned with a factor of 4.

Figure 2 . 15 :

 215 Figure 2.15: Experimental angular distribution for populated stats through the 30 Si( 3 He,d) 31 P transfer reaction. Each data point, in red, is the differential crosssection measured at a detection angle, expressed in the centre of mass frame.

Figure 2 . 15 :

 215 Figure 2.15: Experimental angular distribution for populated stats through the 30 Si( 3 He,d) 31 P transfer reaction. Each data point, in red, is the differential crosssection measured at a detection angle, expressed in the centre of mass frame.

3 . 1

 31 Reaction theory . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Elements of scattering theory . . . . . . . . . . . . . . . . . . 3.1.2 Plane and Distorted Waves Approximations . . . . . . . . . . 3.1.3 Case of transfer reactions . . . . . . . . . . . . . . . . . . . . 3.1.4 Selection rules . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.5 Finite Range (FR) calculations . . . . . . . . . . . . . . . . . 3.2 Spectroscopic information extraction . . . . . . . . . . . . 3.2.1 Spectroscopic factor . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Transfer to unbound levels . . . . . . . . . . . . . . . . . . . 3.2.3 Weakly bound approximation . . . . . . . . . . . . . . . . . . 3.3 Case of the 30 Si( 3 He,d) 31 P reaction . . . . . . . . . . . . . . 3.3.1 DWBA calculations inputs . . . . . . . . . . . . . . . . . . . 3.3.2 Methods and codes used for DWBA calculations . . . . . . . 3.3.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . 3.4 Uncertainty estimation . . . . . . . . . . . . . . . . . . . . .3.4.1 Bayesian approach . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Markov Chain Monte Carlo . . . . . . . . . . . . . . . . . . . 3.4.3 Influence of the optical potential parameters . . . . . . . . . 3.4.4 Influence of the binding potential parameters . . . . . . . . . 3.4.5 Uncertainty of proton widths . . . . . . . . . . . . . . . . . .

Figure 3 .Figure 3 . 1 :

 331 Figure 3.1: Schematic representation of the diffusion of a plane wave with wave vector k α and the diffused spherical wave with wave vector k β . V is the diffusion potential.

  β ) are called the distorted waves, and are the eigenfunctions of U β describing the scattering of b on B due to the potential U β .

2

 2 and π31 P = (-1) ℓ . (3.47) The values of the potential parameters are: r c = r v = r s.o = 1.25 fm, a r = a s.o = 0.65 fm and V s.o = 6.25 MeV [71].

Figure 3 . 2 :

 32 Figure 3.2: Measured angular distributions corresponding to different transferred orbital angular momenta, in red dots, fitted with DWBA curves calculated for a binding energy of 1 MeV, both in zero range (green) and finite range (blue).

Figure 3 . 3 :

 33 Figure 3.3: Normalized radial part of wave functions for bound and unbound states, in plain lines. The dashed lines correspond to Whittaker and irregular Coulomb wave function, that governs the asymptotic behaviour of the wave function. The vertical dashed line shows the radius at which the form factors are calculated and is R = 7 fm.

Figure 3 . 4 :

 34 Figure 3.4: Evolution of the spectroscopic factor as a function of the binding energy. Negative energies correspond to bound states. The values of the spectroscopic factor are normalized with respect to C 2 S at E=-100 keV. Solid lines are the exact calculations with unbound energies with dwuck4, while the dashed lines are the linear extrapolation from bound energy region, up to an unbound energy of 1MeV.

Figure 3 . 5 :

 35 Figure 3.5: Evolution of the reduced single particle width γ 2 s.p as a function of binding energy, for two channel radii: R = 5 and 7 fm. The values are normalized with respect to γ 2 s.p at E b = 50 keV. Solid lines are the results of the exact calculations, while the dashed lines are the value of the chosen reduced width for all unbound energies accordingly to the weakly bound approximation.

Figure 3 . 6 :

 36 Figure 3.6: Evolution of the proton width (in relative units) as a function of the binding energy. Negative energies correspond to bound states. The values of the widths are normalized with respect to Γ p at E=-100 keV. Solid lines are the exact calculations with unbound energies, while the dashed lines are the value of the calculated width accordingly to the weakly bound approximation.

Figure 3 . 7 :

 37 Figure 3.7: Angular distributions of 31 P bound states populated in the 30 Si( 3 He,d) 31 P reaction. Curves represent finite-range DWBA calculations normalized to the data. The error bars are typically smaller than the data points.

Figure 3 . 8 :

 38 Figure 3.8: Angular distributions of 31 P proton-unbound states populated in the 30 Si( 3 He,d) 31 P reaction. Curves represent finite-range DWBA calculations normalized to the data. The error bars are typically smaller than the data points.

Figure 3 . 9 :

 39 Figure 3.9: Scatter plot of the real potential depth V r and the radius r r for the entrance channel, each point reproduces the30 Si + 3 He elastic scattering and30 Si( 3 He,d)31 P transfer data. The red curve is the fit of the data points with a power law.

Figure 3 .

 3 Figure 3.10: Pair-wise correlation plots for posterior distributions of entrance optical potential parameters. The spectroscopic factor corresponds to E x = 7898 keV state, and its distribution is fitted with a log-normal probability distribution function (pdf). median value and corresponding 1σ uncertainty are given.

Figure 3 . 11 :

 311 Figure 3.11: Bayesian fit of the elastic data. Purple and blue bands represent the 68% and 95% credibility intervals. The red curve represents the initial parameters of the fit, obtained by Vernotte et al. [71]

Figure 3 .Figure 3 . 13 :

 3313 Figure 3.12: Pair-wise correlation plots for posterior distributions of exit optical potential parameters. The spectroscopic factor corresponds to E x = 7898 keV state, and its distribution is fitted with a log-normal probability distribution function (pdf). median value and corresponding 1σ uncertainty are given.

Figure 3 . 14 :

 314 Figure 3.14: Statistical distributions for (top left) the spectroscopic factor, (top right) the single particle partial width, and (bottom left) the proton partial width. The red curves are the log-normal distributions fitted to the data. The bottom right panel shows the correlation between the spectroscopic factor and the single particle widths, in log scales.

  facility at TRIUMF . . . . . . . . . . . . . . . . . . . 4.1.1 OLIS ion source . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Acceleration and beam transport . . . . . . . . . . . . . . . . 4.2 DRAGON spectrometer . . . . . . . . . . . . . . . . . . . . 4.2.1 Windowless gas target . . . . . . . . . . . . . . . . . . . . . . 4.2.2 BGO γ-ray detector array . . . . . . . . . . . . . . . . . . . . 4.2.3 Electromagnetic Mass Separator . . . . . . . . . . . . . . . . 4.2.4 Focal plane detection system . . . . . . . . . . . . . . . . . . 4.2.5 Data acquisition system . . . . . . . . . . . . . . . . . . . . . 4.3 Data analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Beam particle normalization . . . . . . . . . . . . . . . . . . . 4.3.2 Separator Time of Flight . . . . . . . . . . . . . . . . . . . . 4.3.3 DSSSD energy spectrum . . . . . . . . . . . . . . . . . . . . . 4.3.4 Focal plane Particle Identification . . . . . . . . . . . . . . . 4.3.5 BGO energy spectra . . . . . . . . . . . . . . . . . . . . . . . 4.3.6 Recoils transmission . . . . . . . . . . . . . . . . . . . . . . . 4.3.7 Incident beam energy and stopping power . . . . . . . . . . . 4.3.8 Charge State Distribution . . . . . . . . . . . . . . . . . . . . 4.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . 4.4.1 Resonance strength . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Resonance energies . . . . . . . . . . . . . . . . . . . . . . . .

Figure 4 .

 4 [START_REF] Böhm-Vitense | Introduction to Stellar Astrophysics[END_REF] shows the overview of the different facilities at TRIUMF laboratory.

Figure 4 . 1 :

 41 Figure 4.1: Schematic view of the TRIUMF laboratory with the cyclotron and different experimental areas. DRAGON is located in ISAC-I hall.

Figure 4 . 2 :

 42 Figure 4.2: Schematic representation of the beam lines in ISAC-I. OLIS is connected to DRAGON through three transport beam lines: LEBT, MEBT and HEBT. Image adapted from [92].

Figure 4 . 3 :

 43 Figure 4.3: Diagram of DRAGON mass spectrometer showing the gas cell and the BGO array in the head part, the electromagnetic elements of the separator, and the focal plane detection system in the tail part[START_REF] Hutcheon | The DRAGON facility for nuclear astrophysics at TRIUMF-ISAC[END_REF]

Figure 4 . 4 :

 44 Figure 4.4: Sectional view of the DRAGON windowless gas target. The trapezoidal shaped gas cell is contained in an aluminium box. The target is equipped with Silicon Surface Barrier detectors for monitoring elastic scattering. Figure adapted from [93].

Figure 4 . 5 :

 45 Figure 4.5: Schematic of the DRAGON pumping system. Figure adapted from [93].

Figure 4 . 6 :

 46 Figure 4.6: DRAGON target density profile as a function of the position, along beam axis. Data taken from [94].

Figure 4 . 7 :

 47 Figure 4.7: (a) Schematic of a single γ-ray detector, consisting of a BGO scintillating crystal and its photomultiplier tube. (b)The DRAGON BGO array surrounding the gas target. Both figures adapted from[START_REF] Hutcheon | The DRAGON facility for nuclear astrophysics at TRIUMF-ISAC[END_REF].

( a )

 a Schematic of DRAGON's local time-of-flight detection system, composed by two MCP detectors. Components of the detectors are shown in the zoom in the upper right part. Lower right part shows the working principle of the Micro Channel Plate. FCF is the final Faraday cup. (b) Design of DRAGON's ionization chamber (IC) used for the hybrid chamber at the focal plane. The DSSSD is placed at the right edge of the chamber (red box).

Figure 4 . 8 :

 48 Figure 4.8: DRAGON's focal plane detection system.

Figure 4 . 9 :

 49 Figure 4.9: Design of DSSSD W1 (Micron Semiconductor Ltd). (a) Illustration of the DSSSD. (b) Sketch of the profile of a strip of the detector. Strips are implanted using an Al contact grid covering 2% of the active area.

( a )

 a FC4 current measurement, the current Ī is the mean value over the 20s time slice between red dashed lines. (b) Pressure readings. Red dashed lines show 5 first and 5 last minutes of the run. SSB energy spectrum of the 5 first minutes of the run, the red dashed lines show the energy cut considered.

Figure 4 . 11 :

 411 Figure 4.11: Plots of various parameters used for computing the R coefficients for the run 10140.

Figure 4 . 12 :

 412 Figure 4.12: Transmission efficiency measured for all runs, blue squares are measurements performed with filled gas target, and thus the current values were corrected regarding the charge state distributions of the beam (CSD), red circles are transmission measurements with empty gas cell.

Figure 4 .

 4 Figure 4.13: R factors for each run, at the beginning of the run, and at the end of it displayed as run+0.5. Green points are R factors corrected with transmission values. The dispersion σ(R) is given for each case.

Figure 4 . 14 :

 414 Figure 4.14: Separator Time of Flight spectrum for E lab =507 keV/u. Red dashed lines define the 260 ns time window for selecting coincidence events (see text).

Figure 4 . 15 :

 415 Figure 4.15: Energy calibration of one strip of the DSSSD detector. The calibration was performed with a triple alpha source. (a) Energy spectrum and corresponding fit. (b) Energy calibration with a first degree polynomial (red line).

( a )

 a Stacked energy spectra (in arbitrary units) for the front strips in ADC channels, before the energy calibration (b) Stacked energy spectra for the front strips after calibration. The peaks associated to the three α disintegrations are aligned.

Figure 4 . 16 :

 416 Figure 4.16: Effect of energy calibration for the front strips of the DSSSD detector obtained with a triple α calibration source.

Figure 4 . 17 :

 417 Figure 4.17: DSSSD energy spectrum for an attenuated beam run at E lab = 506 keV/u.

Figure 4

 4 Figure 4.18 shows the DSSSD energy spectra of the recoils for both the highest and lowest energies measured, both for single (black histogram) and coincidence events (blue histogram). A peak at a higher energy than the main one is observed at all beam energies measured in the present experiment. However, this peak is better separated for the highest beam energies. This highest energy peak is associated to unreacted beam particles that were not suppressed by the spectrometer. This leaky beam is completely suppressed when considering the coincidence events, i.e., events that are correlated with a γ-ray event detected in the BGO array. The spectra measured during attenuated beam runs are superimposed to confirm the peak position of the leaky beam. For the measurements at beam energy E lab = 506 keV/u, a fit with a double Gaussian was performed on the recoils' spectrum, in single events. The highest energy component reproduces well the shape of the peak obtained with the attenuated beam. An attempt can be made to extract the number of recoils from the singles spectra by subtracting the fitted events under the leaky beam peak. However, the centroids of the recoils peaks for singles and coincidences do not match: for the resonances where the DSSSD spectrum were fitted with a double gaussian (E c.m.

Figure 4 .

 4 Figure 4.18: DSSSD energy spectra for recoils run in singles (black histogram) and coincidence events (blue histogram). The spectrum obtained with an attenuated beam is superimposed after scaling to the data (red histogram). The singles data are fitted with a double gaussian for the low energy resonance in the top panel (E lab = 507 keV/u) (green+ red dashed lines) or a simple gaussian for the highest energy resonance in the bottom panel (E lab = 981 keV/u) (black solid line).

  IC energy vs DSSSD energy

Figure 4 . 20 :

 420 Figure 4.20: Bi-dimensional spectra of various variable corresponding to detectors at the focal plane for the resonance at E c.m.r

30 ������������������������������������������������������������������������������������������������������������������������������������������������������������Figure 4 . 21 :

 30421 Figure 4.21: Layout of the BGO detectors in the simulation along with their number. The length of the PMT in yellow has been reduced for clarity. Figure taken from [102].

Figure 4 .

 4 [START_REF] Cohen | The Bizarre Chemical Inventory of NGC 2419, An Extreme Outer Halo Globular Cluster[END_REF], bottom panel, shows the calibration adjustment for BGO n3 taken as an example, for the resonance at E c.m. r = 602 keV.

Figure 4 . 22 :

 422 Figure 4.22: BGO energy calibration. Top panels, left and middle panels show the fit of the spectrum in singles for the 40 K and 208 Tl respectively. The top right panel shows the BGO spectrum in coincidence with events in the focal plane detector. The peak is fitted with a double gaussian corresponding to the E x = 7898 keV level and its first escape peak. The background is modelled with a constant function. The bottom panel shows the fit calibration with a first order polynomial.

Figure 4 .

 4 [START_REF] Ventura | SUPER-AGB-AGB EVOLUTION AND THE CHEMICAL INVENTORY IN NGC 2419[END_REF] shows the fit of the spectrum for one BGO crystal, along with the calibration fit in the bottom panel of the figure.

( a )

 a After the first step of calibration.

  After the second step of calibration.

Figure 4 . 23 :

 423 Figure 4.23: Residues of energy calibration of BGO array after each step of calibration. The positions are fitted on the E γ 0 spectrum for each run.

Figure 4 . 24 :

 424 Figure 4.24: BGO spectrum energy calibration for E c.m. r = 950 keV. Top panel shows the coincidence spectrum and the gaussian fit of several γ-ray transition at 2.233 keV, 3.134 keV, 6.013 keV and the transition to the ground state at 8247 keV. Bottom panel shows the energy calibration of the BGO detector with a linear fit.

( a )

 a Simulated spectrum using NNDC branching ratios.(b) Simulated spectrum using branching ratios obtained with TFractionFitter.

Figure 4 . 25 :

 425 Figure 4.25: Comparison between E γ 0 ray energy spectrum measured experimentally, and the spectrum obtained from geant3 simulations.

( a )

 a Simulated spectrum using NNDC branching ratios.(b) Simulated spectrum using branching ratios obtained with TFractionFitter.

Figure 4 . 26 :

 426 Figure 4.26: Comparison between E γ 0 energy spectrum measured experimentally, and the spectrum obtained from geant3 simulations.

Figure 4 . 27 :

 427 Figure 4.27: Comparison between E γ 0 ray energy spectrum measured experimentally, and the spectrum obtained from geant3 simulations using NNDC branching ratios.

Figure 4

 4 Figure 4.28a shows the comparison between the measured BGO spectrum and the simulated one obtained using the branching ratios listed in NNDC. The overall shape is

( a )

 a Simulated spectrum using NNDC branching ratios.(b) Simulated spectrum using branching ratios obtained with TFractionFitter.

Figure 4 . 28 :

 428 Figure 4.28: Comparison between E γ 0 ray energy spectrum measured experimentally, and the spectrum obtained from geant3 simulations.

Figure 4 . 29 :

 429 Figure 4.29: Comparison between E γ 0 ray energy spectrum measured experimentally, and the spectrum obtained from geant3 simulations using NNDC branching ratios.

  spectrum using NNDC branching ratios.

  Simulated spectrum using branching ratios obtained with TFractionFitter.

Figure 4 . 30 :

 430 Figure 4.30: Comparison between E γ 0 ray energy spectrum measured experimentally, and the spectrum obtained from geant3 simulations.

Figure 4 .

 4 Figure 4.31: MCP transmission efficiency measured with different beam energies. The red point is discarded from the calculation of the weighted mean. The latter is displayed as a solid line and the associated uncertainties are in dashed lines

( a )Figure 4 . 32 :

 a432 Figure 4.32: Beam energy as a function of gas target density, the stopping power is calculated from the slope of the linear fit.

Figure 4 .Figure 4 . 33 :

 4433 Figure 4.33: Hit pattern of the BGO array for the E c.m.r

Figure 4 . 35 :

 435 Figure 4.35: Difference between resonance energies measured with DRAGON and mean energy from the literature[31]. Grey boxes correspond to spread in the literature energy while pink ones correspond to 1σ standard deviation of the measurements.

Figure 4 . 36 :

 436 Figure 4.36: Charge State Distribution of 30 Si ions in hydrogen target, for different beam energies.

Figure 4 .

 4 [START_REF] Wolff | Spin determinations of 31P levels from the 30Si(p, ) reaction[END_REF] shows the fit of the corresponding CSD. These values are used to correctly estimate the beam transmission efficiency through the target (see section 4.3.1).

Figure 4 . 37 :

 437 Figure 4.37: Charge State Distribution of 31 P ions in hydrogen target, for different beam energies.

Figure 4 . 39 :

 439 Figure 4.39: Compilation of resonances strengths found in the literature. Black data points are absolute 30 Si(p,γ) 31 P measurements and purple points are relative values renormalized to ωγ 602 = 1.95 eV. Green data points are strengths deduced from the 30 Si( 3 He,d) 31 P transfer reaction. Weighted average of all previous measurements is

r= 485 •

 485 keV and the E c.m. r = 752 keV resonances, the strengths measured in coincidence are in agreement with previous direct measurements while singles measurements are not. Note however that for the E c.m. r = 911 keV resonance, both DRAGON strengths values are compatible with the literature, mainly due to the large uncertainties of the previous measurements. In the case of the E c.m. r = 950 keV resonance, both strength values are higher than previous measurements, with the coincidences' value being closer to previous measurements. For the E c.m. r = 485 keV and E c.m.

  r

Si( 3

 3 He,d) 31 P transfer reaction. The resonance energies measured at DRAGON are compatible with the ones measured by Kuperus et al. and Wolff et al., as well as the energies proposed by Shikazono & Kawarasaki, which have very large uncertainties. The three highest energies measured at DRAGON are also in agreement with the energies measured through 30 Si( 3 He,d) 31 P transfer reaction. However, DRAGON's measurements are incompatible with the energies measured by Dermigny et al., for the E c.m. r = 485 keV and E c.m. r = 602 keV resonances. They are also incompatible and systematically smaller than the energies measured with the Q3D spectrometer.

Figure 5 . 1 :

 51 Figure 5.1: Monte Carlo samples of the30 Si(p,γ)31 P reaction rate for different temperatures. The rates are normalized by the median (0.5 quantile) of the distribution. The corresponding log-normal density functions, in red, are calculated using Eq.5.13. The skewness factors are displayed in bold italic figures at the right of each distribution. The rate approaches the normal distribution as the temperature increases.

Figure 5 .

 5 [START_REF] Riess | Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant[END_REF] shows the fractional contribution of30 Si(p,γ)31 P resonances to the total reaction rate, and we see that the E c.m. r = 50 keV resonance dominates the rate for T ≲ 50 MK. For the temperature range relevant for polluter candidates in globular clusters between 50 and 200 MK, the E c.m. r = 149 keV resonance dominates the rate when considering an ℓ = 2 transferred angular momentum. Otherwise, the rate is rather dominated by the direct capture (DC). This induces a factor of 10 difference in the reaction rate for cases ℓ = 2 or ℓ ≤ 3 as shown in figure 5.2. Figure 5.3 also shows that the rate is dominated by the E c.m. r = 485 keV resonance for temperatures between 200 and 500 MK, and by the resonance of reference at E c.m. r = 601 keV between 500 MK and 3 GK. Resonances at E c.m. r = 752, 912 and 950 keV have a non-negligible contribution to the total reaction rate at T ≳ 2 GK as shown in figure 5.3b.

Figure 5 . 2 :

 52 Figure 5.2: Thermonuclear rates of the 30 Si(p,γ) 31 P reaction calculated for different transferred orbital angular momenta for the resonance at E c.m. r = 149 keV. All rates are normalized to the recommended value.

31 P( a )

 31a Fractional contribution of30 Si(p,γ)31 P resonances, along with direct capture contribution (labelled "DC") to the total reaction rate. The thickness of each band represents the uncertainty of the contribution.The dotted black line represents the contribution to the total rate of resonances with energies above E c.m. r = 648 keV. The E c.m. r = 149 keV resonance was assumed to have ℓ p = 2 in this calculation (see text for details). (b) Zoom-in on the fractional contribution of 30 Si(p,γ) 31 P resonances to the total reaction rate, for high temperature domain. The thickness of each band represents the uncertainty of the contribution. The dotted black line represents the contribution to the total rate of resonances with energies above E c.m. r = 950 keV.

Figure 5 . 3 :

 53 Figure 5.3: Fractional contribution of the 30 Si(p,γ) 31 P resonances.

r= 149

 149 keV and E c.m. r = 174 keV resonances in the present work through the30 Si( 3 He,d)31 P reaction. These resonances were treated as upper limits in previous works. A special care has to be given to the rate in this temperature region, where it is dominated by the contribution of the E c.m. r = 149 keV resonance, but only if an ℓ = 2 is considered for its associated orbital momentum. The other possibility, corresponding to ℓ > 2, implies that the rate is rather dominated by the contribution of the direct capture. The median rate would be 10 times smaller. This emphasizes the importance of the E c.m. r = 149 keV resonance in the temperature range 50 MK ≤ T ≤ 200 MK.

Figure 5 . 4 :

 54 Figure 5.4: Reaction rates from the present work (purple) and the evaluation of Dermigny et al. [32] (gray) normalized to the present recommended rate. The shaded areas represent 68% coverage probabilities. The E c.m. r = 149 keV resonance was assumed to have ℓ p = 2 in this calculation. The uncertainty of this ℓ assignment leads to an additional uncertainty around 0.1 GK (see figure 5.2 and text for details).

Figure 5 . 5 :

 55 Figure 5.5: Thermonuclear reaction rates from the present work with both cases ℓ = 2 and ℓ = 3, in purple and orange respectively. The rate from Iliadis et al. [120] is represented in gray. The coloured areas represent 68% coverage probability. All the rates are normalized to the ℓ = 2 case.

Figure 5 . 6 :

 56 Figure 5.6: Evolution of the mass fractions for Mg and K during H-burning at constant temperatures T = 100 MK (dashed lines) and T = 160 MK (solid lined) and density ρ = 100 g/cm 3 .

Figure 5 . 7 :

 57 Figure 5.7: Elemental abundances observed in NGC 2419[START_REF] Mucciarelli | News from the Galactic suburbia: the chemical composition of the remote globular cluster NGC 2419[END_REF][START_REF] Cohen | The Bizarre Chemical Inventory of NGC 2419, An Extreme Outer Halo Globular Cluster[END_REF]. The black boxes indicate the selected abundances that need to be reproduced by the nucleosynthesis network calculations. The black curve with crosses indicates the results of the mixing between the processed matter and the pristine matter (f between 0 (rightmost cross) and 1000 (leftmost cross)). The calculations were performed for T = 160 MK, ρ = 100 g/cm 3 and X f H = 0.7.

  and Dermigny et al., where the difference in the sampling subroutines may induce such differences. These All rates fixed All rates varied (a) Temperature distribution for fixed rates (red) and all rates varying (blue).

  Temperature distribution for fixed rates (red) and just30 Si(p,γ) 31 P rate varying (green).

Figure 5 . 9 : 3 (

 593 Figure 5.9: Temperature distributions reproducing the observed abundances in NGC 2419, with densities between 80 g.cm -3 and 120 g.cm -3

Figure 5 . 10 :

 510 Figure 5.10: Stellar conditions that reproduce the observed elemental abundances in NGC 2419. The gray dots and black histogram are obtained by considering only the rates of the STARLIB library, the purple and orange colours correspond to the rates measured in the present work, while considering ℓ = 2 or ℓ = 3 for the E c.m. r = 149 keV, respectively. All the calculations are made with rates fixed at their median recommended values.

Figure 5 . 11 :

 511 Figure 5.11: Variation of silicon abundances at T = 160 MK (dashed lines) and T = 175 MK (solid lines). The calculations are performed at ρ = 100 g.cm -3 and stopped when X f H = 0.7. A dilution factor of f = 0.04 was applied. The horizontal shaded band indicates the range of acceptable abundances to match with the observations in NGC 2419.

1 +

 1 γ cos θ ′ (1 + γ 2 + 2γ cos θ ′ ) 3/2 = 1 -γ 2 sin 2 θ (γ cos θ + 1 -γ 2 sin 2 θ) 2(5.43)To evaluate this integral between -∞ and +∞, we shall use the following limits:

  30 S i ( p , g ) 3 1P * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 1 ! Z p r o j 14 ! Z t a r g e t 0 ! Z e x i t p a r t i c l e (=0 when o n l y 2 c h a n n e l s open ) 1 . 0 7 8 ! Aproj 2 9 . 9 7 3 7 7 ! A t a r g e t 0 ! A e x i t p a r t i c l e (=0 when o n l y 2 c h a n n e l s open ) 0 .

5 5 !

 5 p r o j e c t i l e s e p a r a t i o n e n e r g y ( keV ) 0 . ! e x i t p a r t i c l e s e p a r a t i o n e n e r g y (=0 when o n l y 2 c h a n n e l s open ) 1 . 0 ! Radius parameter R0 ( fm ) 2 ! Gamma-ray c h a n n e l number (=2 i f e j e c t i l e i s a g-ray ; =3 o t h e r w i s e ) * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 1 . ! Minimum e n e r g y f o r n u m e r i c a l i n t e g r a t i o n ( keV ) 20000 ! Number o f random samp l e s ( >5000 f o r b e t t e r s t a t i s t i c s ) 0 ! =0 f o r r a t e output a t a l l t e m p e r a t u r e s ; =NT f o r r a t e output a t s e l e c t e d t e m p e r a t u r e s 0 1 ! ( r e s , E) =0 f o r no c o r r e l a t i o n . =1 f o r c o r r e l a t i o n * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * Non-r e s o n a n t c o n t r i b u t i o n S ( keVb ) S ' ( b ) S ' ' ( b/keV ) f r a c E r r C u t o f f Energy ( keV ) 2 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *

31 PFigure 5 . 14 :

 31514 Figure 5.14: Contribution fractionnelle des résonances au taux de réaction30 Si(p,γ)31 P ainsi que la contribution de la capture directe au taux. La largeur de chaque bande représente l'incertitude associée à la contribution. La courbe en trait pointillé noir représente la contribution des résonances à des énergies supérieures à E c.m. 650 keV.

  Study of the30 Si( 3 He,d)31 P transfer reaction

	Chapter 2 -
	2
	Study of the 30 Si( 3 He,d) 31 P
	transfer reaction
	Contents
	2.1 Experimental set-up . . . . . . . . . . . . . . . . . . . . . . .
	2.1.1 Beam preparation . . . . . . . . . . . . . . . . . . . . . . . .
	2.1.2 Reaction chamber and targets . . . . . . . . . . . . . . . . . .
	2.1.3 Q3D magnetic spectrometer . . . . . . . . . . . . . . . . . . .
	2.1.4 Detection system at the focal plane . . . . . . . . . . . . . . .
	2.1.5 Data acquisition . . . . . . . . . . . . . . . . . . . . . . . . .
	2.2 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . .
	2.2.1 Target thickness . . . . . . . . . . . . . . . . . . . . . . . . .
	2.2.2 Particle Identification . . . . . . . . . . . . . . . . . . . . . .
	2.2.3 Deuteron magnetic rigidity spectra . . . . . . . . . . . . . . .
	2.3 Excitation energies . . . . . . . . . . . . . . . . . . . . . . .
	2.3.1 Excitation energies uncertainties . . . . . . . . . . . . . . . .
	2.3.2 Adopted excitation energies . . . . . . . . . . . . . . . . . . .
	2.4 Angular distributions . . . . . . . . . . . . . . . . . . . . . .
	2.4.1 Beam particle normalization . . . . . . . . . . . . . . . . . . .
	2.4.2 Solid angle estimation . . . . . . . . . . . . . . . . . . . . . .
	2.4.3 Dead time corrections . . . . . . . . . . . . . . . . . . . . . .
	2.4.4 Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . .
	2.5 Results and discussion . . . . . . . . . . . . . . . . . . . . .

B = 0.924 T °E = 25.000 MeV Angle = 10.0 30Si(3He,2H)31P 6.668 6.796 6.910 7.073 7.158 7.314 7.441 7.572 7.687 7.779 7.860 7.945 8.031 28Si(3He,2H)29P 2.423 3.106 3.448 12C(3He,2H)13N 2.365 14N(3He,2H)15O 6.793 7.276 7.556 16O(3He,2H)17F 0.000 0.495 30Si(3He,3H)30P 7.178 7.282 7.347 7.472 7.561 7.636 7.688 7.742 7.803 7.874 7.932 7.997 8.053 8.107 8.166 95.0 cm 97.0 cm 99.0 cm

	Figure 2.6: Expected peak positions in the focal plane detection system for different
	reactions products induced by reactions on

Table 2 .

 2 The mean Chapter 2 -Study of the 30 Si( 3 He,d) 31 P transfer reaction Present work 16 * 7141.1 17 * 7140.8 19 * 7140.2 24 * 7140.5 20 * 7140.2 24 * 7139.4 34 * 7140.16 * 7214.5 17 * 7214.2 18 * 7214.8 26 7214.3 21 * 7213.6 24 * 7213.3 34 * 7214.4 8 * 7214.3 20 7316.2 17 * 7316.9 19 * 7315.6 20 * 7315.8 25 * 7315.3 23 16 * 7719.8 17 * 7719.6 18 * 7719.2 24 * 7718.6 21 * 7719.2 24 * 7718.8 35 * 7719.4 8 * 7718 4 7737.3 15 * 7737.6 16 * 7737.5 17 * 7737.0 23 * 7737.3 20 * 7737.0 23 * 7736.5 33 * 7737.3 8 * 7736 4 7781.1 15 * 7781.3 17 * 7781.0 19 * 7780.9 25 * 7780.9 21 * 7781.16 * 7945.8 17 * 7946.7 18 * 7947.0 24 * 7946.5 21 * 7946.0 24 * 7946.9 35 * 7946.2 8 * 2: Excitation energies in keV of 31 P levels populated by the 30 Si(

	6°10°12°16°20°23°32°Mean	Vernotte et al. NNDC
	6826.5 18	6825.6 22	6826.2 21	6827.2 29	6826.8 22	6827.1 26	6826.4 36	6826.5 9	6824.2 20
	6844.3 17	6843.2 18 * 6844.2 19 * 6844.1 25 * 6844.0 21 * 6844.3 25 * 6844.6 37 * 6844.0 8 *	6841.9 15
	6911.4 16 *			6910.9 24 * 6911.3 21 * 6911.4 24 * 6911.2 33 * 6911.3 10 *	6909.6 16
	6933.4 16 * 6934.3 17 *		6931.0 24	6933.0 21 * 6932.7 24 * 6933.6 34 * 6933.2 9 *	6931.9 15
	7069.2 16	7068.7 17	7068.2 19	7067.8 25			7067.4 35	7068.5 9	7073 4
		7082.7 19	7080.5 30	7082.6 30			7083.6 38	7082.4 14	7079.9 19
	7141.0 7 8 *	7141.1 18
					7159.4 20	7158.4 31	7165.8 55	7159.7 16	7158 5
	7215.0 7316.3 28		7316.1 9 *	7314 4
	7346.5 19	7349.9 29	7346.8 36	7344.8 33	7347.2 28			7347.1 12	7346 6
					7446.4 50	7447.7 45	7441.5 56	7445.7 29	7441.4 10
					7471.2 31	7469.1 33	7472.6 69	7470.4 22	7466 2
	7695.0 19	7694.5 21	7692.2 23		7686.9 26	7677.4 29		7690.9 10	7687.2 20
	7719.6 3 24 *		7781.1 8 *	7779 1
	7852.2 16	7851.4 18	7852.3 19	7850.6 26	7849.3 24	7848.5 29		7851.2 8	7852 4
	7870.8 44	7864.0 23	7867.6 31	7863.3 34	7860.7 27	7861.2 31		7863.8 14	7859.8 4
	7897.6 15 * 7897.5 16	7897.9 17 * 7897.7 23 * 7898.4 20 * 7898.8 23 * 7899.7 32 * 7898.0 8 *	7896 1
	7911.6 15	7911.2 16	7911.7 17	7911.6 23	7911.6 20	7911.8 23	7911.1 33	7911.5 8 *	7913 4
	7945.6 7945 1
	7974.1 17	7975.9 17	7977.3 18	7977.3 24	7977.4 21	7977.3 24	7977.8 34	7976.4 8	7980
			8048.2 18 * 8048.4 24 * 8048.2 20 * 8048.6 24 * 8048.8 33	8048.4 11 *	8048 1
					8078.8 24	8077.8 28	8076.2 38	8078.0 17 *	8077.0 4
					8105.4 21 * 8105.0 25 * 8103.2 35 * 8104.9 15 *	8104 1

3

He,d)

31

P reaction at different measured angles. Italic numbers correspond to the excitation energy uncertainties. Levels labeled with an asterisk have been used for the calibration.

Table 3 .

 3 1: Potential parameters used in DWBA calculations

	Channel

Table 3 .

 3 2: Proton spectroscopic factors and reduced single-particle widths values obtained via Zero-Range and Finite-Range calculations, and the difference between the two approaches.

	b/ Determination of the spectroscopic factor and particle widths for
	unbound states

For unbound states, integrating the radial form factor is tedious. We propose in this study a test of the validity of the weakly bound approximation presented in sec-

Table 3 .

 3 3: Excitation energy, angular momentum, proton spectroscopic factor and single-particle width of 31 P levels populated by the30 Si( 3 He,d) 31 P reaction are reported. Italic numbers correspond to excitation energy uncertainties. Comparison with available information from the literature is provided. S Γ sp or γ 2 sp (eV) E x (keV) 1 ℓ p (2J+1)C 2 S 2 E x (keV)

		Present work		Vernotte et al. [33]	ENSDF [31]
	E x (keV) ℓ p	(2J+1)C 2 J π
	6826.5 9	0.061	1.16 × 10 4	6826		6824.2 20	11/2 -
						6828 3
	6844.0 8	0.034	2.90 × 10 4	6843 3	0.04 6841.9 15	5/2 -
	6911.3 10	0.052	1.58 × 10 5	6910 1	0.08 6909.6 16	3/2 -
	6933.2 9	0.085	2.96 × 10 4	6932 2	0.08 6931.9 15	5/2 +
	or 2	0.040	6.13 × 10 4
	7068.5 9	0.041	3.06 × 10 4	7068 3	0.04	7073 4	5/2 -, 7/2 -
						7073 6	1/2 + , 3/2 +
	7082.4 14	0.004	1.65 × 10 5	7081		7079.9 19	3/2 -, 5/2 +
						7084.0 17	3/2 + , 5/2, 7/2 +
						7117.7 10	9/2 +
	7140.7 8	0.099	1.61 × 10 5	7139 0	0.22 7141.1 18	1/2 +
	7159.7 16	0.002	1.60 × 10 5			7158 5
	or 3	0.003	3.12 × 10 4
	7214.4 8	0.020	1.61 × 10 5	7214 1	0.032 7214.3 20	1/2 -, 3/2 -
						7313.7 16	1/2 + , 3/2 +
	7316.1 9	0.0075	2.44 × 10 -36	7314 3	0.016	7314 4	5/2 -, 7/2 -
	7347.1 12	0.0007	2.78 × 10 -17			7346 6	3/2 -, 5/2 -
	or 2	0.0012	6.60 × 10 -19
						7356 16
	7445.7 29	0.0007	6.71 × 10 -8			7441.4 10	(3/2 to 9/2)
	or 3	0.0006	1.19 × 10 -9
						7442.3 3	11/2 +
	7470.4 22	0.0008	1.60 × 10 -8			7466 2 5/2 -, 7/2 -, 9/2 -
						7572
	7690.9 10	0.006	1.53 × 10 -3			7687.2 20
	7719.4 8	0.045	3.34 × 10 -3	7718		7718 4
	7737.3 8	0.114	6.89 × 10 -3	7736 3	0.16	7736 4	5/2 -, 7/2 -
	7781.1 8 *	0.015	16.25	7780 1	0.02	7779 1	3/2 -
						7825 9
	7851.2 8	0.009	55.75	7855		7852 4	1/2,3/2,5/2 +
	or 2	0.0114	2.56
	7863.8 14	0.004	0.074			7859.8 4	11/2 -
	7898.0 8	0.115	112.6	7900 1	0.16	7896 1	1/2 -
	7911.5 8	0.190	0.215	7913 3	0.12	7913 4	5/2 -, 7/2 -
	7946.2 8	0.033	8.711	7949 2	0.04	7945 1	3/2 + , 5/2 +
	7976.4 8	0.023	16.122	7980 2	0.032	7980
						7994 6	5/2 -
						8031 1	5/2 +
	8048.4 11	0.034	733.5	8051 1	0.04	8048 1	3/2 -
	8078.0 17	0.004	975.3	8080		8077.0 4	11/2 -
	8104.9 15	0.023	68.11	8107 2	0.018	8104 1	5/2 +

  .18)

	E c.m r	q tune		Approach with R	Independent runs	Difference
	(keV/u)		R (×10 9 )	χ 2 /ndf N T ot beam (×10 13 )	N T ot beam (×10 13 )
	485	7+	0.74 ± 2.08% 4.81/11 2.41 ± 3.15%	2.42 ± 2.21%	-0.14%
	485	8+	0.78 ± 1.33% 0.32/ 5 7.18 ± 1.47%	7.18 ± 1.45%	-0.02%
	555	7+	1.08 ± 2.04% 0.42/ 5 7.78 ± 2.17%	7.82 ± 3.57%	-0.47%
	602	8+	1.25 ± 0.79% 2.62/13 17.6 ± 1.81%	17.6 ± 1.03%	0.08%
	752	9+	1.93 ± 2.34% 0.15/ 3 3.10 ± 3.01%	3.10 ± 3.04%	-0.26%
	911	9+	2.72 ± 3.45% 0.21/ 3	3.61± 4.15%	3.59 ± 4.12%	0.38%
	950	10+ 2.82 ± 5.18% 1.76/ 7 2.34 ± 7.92%	2.33 ± 5.85%	0.26%

Table 4 . 2

 42

: Number of beam ions for each group of runs, computed with two different approaches with the associated uncertainty:

Table 4

 4

	Isotope	E α (MeV)	I α (%)
		5.15659(14) 70.77(14)
	239 Pu	5.1443(8) 17.11(14)
		5.1055(8)	11.94(7)
		5.48536(12) 84.8(5)
	241 Am	5.44280(13) 13.1(3)
		5.388	1.66(2)
	244 Cm	5.80477(5) 76.90(10) 5.76264(3) 23.10(10)

.3 summarizes the energies of the emitted alpha particles and their associated relative intensity. A pulse generator is used to check the linearity of each strip. The procedure is performed online, and no data have been saved for this step.

Table 4 .

 4 3: Energies and relative intensities of alpha particles emitted by isotopes present in the source used for the calibration of the DSSSD detector[31].

  40 K and 208 Tl at 1.461 MeV and 2.615 MeV, respectively, are identified and fitted with gaussian distributions summed to the exponentially decaying background created at low energies. The fits are displayed in the left and middle top panels of figure4.22.

Table 4 . 5

 45 .5.

	E c.m. (keV) Transmission efficiency (%)
	485	99.26 ± 4.96
	555	99.35 ± 4.96
	602	98.53 ± 2.43
	752	99.45 ± 4.96
	911	99.32 ± 4.96
	950	99.38 ± 4.96

: Transmission efficiencies of the recoils through DRAGON spectrometer. The results are extracted from geant3 simulations of the electromagnetic mass spectrometer elements.

Table 4 .

 4 15 atoms/cm 2 ) eV /(10 15 atoms/cm 2 ) 6: Measured incident energies and stopping powers for

	%

  .2.

					Literature
					DRAGON
	951.8	483.0	554.8	600.2	752.3	483.2	914.0	E res (keV)
			Run number

Table 4 .

 4

	1	462.9	6.82	7	0.392 ±0.004
				8	0.199 ±0.01
	577.7	529.9	7.60	7	0.333 ±0.007
	628.5	575.9	8.07	8	0.397 ±0.002
	778.8	720.6	9.26	9	0.384 ±0.002
	944.0	827.2	9.95	9	0.254 ±0.004
	981.2	910.1	10.40	10	0.367 ±0.003

7: Charge state fractions obtained with the semi-empirical formula given in equation (4.27), fitted to the experimentally measured data for 31 P recoils. The width of charge state distribution is found to be d = 1.005 according to Eq. (4.28).

  Chapter 4 -Direct measurements of the 30 Si(p,γ) 31 P reaction with DRAGON E c.m.

	r (keV)	N beam ×10 13	N det recoils (singles) (coinc) N det recoils	F q	τ DRAGON ε γ	η tail live	η coin live	ϵ lab (eV)	ωγ (eV) ωγ (eV) (singles) (coinc)
	485±1	2.42	3605	1463	0.392	0.993	0.722	0.945	0.781	122	0.219	0.148
			1.66%	2.61%		1.45%	1.7%				2.21%	3.12%
		2.21%			0.9%	2%	10%	0.005%	0.005%	0.6%	3.75%	10.49%
	485±1	7.18	6912	2290	0.199	0.993	0.722	0.912	0.754	117.5	0.278	0.153
			1.20%	2.09%		1.45%	1.7%				1.88%	2.69%
		1.45%			5.2%	2%	10%	0.005%	0.005%	0.6%	6.14%	11.56%
	555 ±1	7.82	3079	628	0.333	0.993	0.763	0.903	0.756	119.7	0.076	0.024
			1.80%	3.99%		1.45%	1.7%				2.32%	4.32%
		3.57%			1.9%	2%	10%	0.005%	0.005%	0.5%	4.69%	10.86%
	599±1	17.6	202690	89585	0.397	0.989	0.650	0.910	0.751	123.6	2.201	1.787
			0.22%	0.33%		0.23%	0.3%				0.32%	0.44%
		1.03%			0.5%	2%	10%	0.005%	0.005%	0.4%	3.14%	10.29%
	752 ±1	3.10	13691	4473	0.384	0.995	0.71	0.950	0.785	125.6	1.047	0.574
			0.85%	1.5%		1.45%	1.7%				1.68%	2.27%
		3.04%			0.6%	2%	10%	0.005%	0.005%	0.4%	4.13%	10.63%
	913 ±1	3.59	8850	3814	0.254	0.993	0.69	0.954	0.789	114.3	0.940	0.729
			1.08%	1.62%		1.45%	1.7%				1.81%	2.18%
		4.12%			1.6%	2%	10%	0.005%	0.005%	2.2%	4.87%	10.94%
	950 ±1	2.33	15879	7162	0.367	0.993	0.59	0.976	0.855	107.4	1.783	1.54
			0.79%	1.18%		1.45%	1.9%				1.65%	2.22%
		4.12%			1.6%	2%	10%	0.005%	0.005%	2.2%	6.54%	11.77%
	Table

  .1. From Dermigny et al.[START_REF] Dermigny | Thermonuclear reaction rate of 30 Si(p ,γ )[END_REF]. d The complete resonance strength formula has been used (see text). e From Kuperus et al.[START_REF] Kuperus | Energy measurements of proton resonances in light nuclei[END_REF].

	E c.m. r	(keV) E x (keV)	ℓ	(2J+1)C 2 S	Q3D Γ p (eV)	ωγ (eV)	DRAGON ωγ (eV)	Adopted ωγ (eV)
	18.6 16	7313.7 16			1.45 × 10 -35 a 1.45 × 10 -35		1.45 × 10 -35
	19.6 9	7316.2 9	3	0.0075	2.94 × 10 -39	1.18 × 10 -38		1.18 × 10 -38
	50.5 12	7347.0 12	1	0.0007	5.20 × 10 -21	1.04 × 10 -20		1.04 × 10 -20
	59.4 16	7356 16
	145.8 3	7442.3 3			4.33 × 10 -17 a 2.60 × 10 -16		2.60 × 10 -16
	149.2 29	7445.7 27 b	2	0.0007	1.16 × 10 -11	2.33 × 10 -11		2.33 × 10 -11
	174.0 23	7470.5 23	3	0.001	1.59 × 10 -12	6.38 × 10 -12		6.38 × 10 -12
	276		7572
	394.6 10	7691.1 10	3	0.006	1.47 × 10 -6	4.40 × 10 -6		4.40 × 10 -6
	423.0 8	7719.5 8	3	0.044	2.48 × 10 -5	7.42 × 10 -5		(1.14 ± 0.25) × 10 -4c
	440.8 7	7737.3 7	3	0.114	9.79 × 10 -5	3.92 × 10 -4		3.92 × 10 -4
	484.6 8	7781.1 8	1	0.015	0.061	0.123	0.150 ± 0.013	0.150 ± 0.013
	528.5 9	7825 9
	554.9 8	7851.4 8	2	0.0114	0.028	0.017 d	0.024 ± 0.003	0.024 ± 0.003
	566.9 16	7863.4 14	3	0.004	5.55 × 10 -5	1.67 × 10 -4		1.67 × 10 -4
	601.3 7	7897.8 7	1	0.115	6.49		1.787 ± 0.18	1.787 ± 0.18
	614.9 8	7911.5 8	3	0.190	0.04
	649.9 8	7946.2 8	2	0.033	0.29			0.07 ± 0.02 e
	679.9 8	7976.4 8	2	0.023	16.122
	697 6		7994 6
	734.7 9	8031.2 9						0.08 ± 0.03 e
	751.8 11	8048.8 11	1	0.034	24.9		0.57 ± 0.06	0.57 ± 0.06
	781.5 17	8078.0 17	(1)	0.004	3.9
	808.3 15	8103.6 15	2	0.023	1.57			0.18 ± 0.54 e
	912.5 10	8207.9 10					0.73 ± 0.08	0.73 ± 0.08
	927.9 10	8224.4 10						0.13 ± 0.04 f
	946.2 10	8242.7 10						0.63 ± 0.2 f
	951.1 10	8247.6 10					1.54 ± 0.18	1.54 ± 0.18
	a A dimensionless reduced width ⟨θ 2 p ⟩ = 0.0045 is assumed.
	b 7441.4 keV in literature.

c f From Wolff et al.

[START_REF] Wolff | A study of the excited states of 31 P with the 30 Si(p,γ) 31 P reaction[END_REF]

.

Table 5 .

 5 1: Properties of resonances above the

  .3.

				30 Si(p, γ)
		1.2	486 keV 50 keV 149 keV 19 keV	601 keV
		1.0	DC
	Contribution	0.4 0.6 0.8
		0.2
		0.0 0.01	0.1	1	10
				Temperature (GK)
				T (GK)	Low	Recommended	High
				0.050	1.17×10 -18	1.59×10 -18	2.22×10 -18
				0.060	1.91×10 -17	2.55×10 -17	3.44×10 -17
				0.070	3.15×10 -16	4.36×10 -16	6.08×10 -16
				0.080	4.04×10 -15	5.54×10 -15	7.69×10 -15
				0.090	3.58×10 -14	4.81×10 -14	6.60×10 -14
				0.100	2.29×10 -13	3.07×10 -13	4.16×10 -13
				0.110	1.13×10 -12	1.50×10 -12	2.02×10 -12
				0.120	4.49×10 -12	6.02×10 -12	8.14×10 -12
				0.130	1.53×10 -11	2.04×10 -11	2.80×10 -11
				0.140	4.70×10 -11	6.31×10 -11	8.54×10 -11
				0.150	1.47×10 -10	1.90×10 -10	2.53×10 -10
				0.160	5.48×10 -10	6.58×10 -10	8.16×10 -10
				0.180	1.16×10 -08	1.26×10 -08	1.36×10 -08
				0.200	1.98×10 -07	2.12×10 -07	2.27×10 -07

Table 5 .

 5 5: Initial mass fractions adopted from a Galactic chemical evolution model, and adjusted for the23 Na,24 Mg, 27 Al,28 Si, 39 K and40 Ca, in order to reproduce the observed abundances in NGC 2419. Table taken from Ref[START_REF] Iliadis | On Potassium and Other Abundance Anomalies of Red Giants in NGC 2419[END_REF]. figure 5.7 by the black crosses. Note that a horizontal dilution curve indicates that the initial abundances are not altered and remain constant, this is the case for Ca, Ti and V.

parsec, 1 pc = 1/tan(1 arcsec) a.u ≈ 3.26 light-years, where 1 a.u is the astronomical unit defined as the mean Earth-Sun distance.

Observational equivalent of the HR diagram.

from ancient Greek protos, means first, prior.

happening in the lower part of the outer convective zone.

When resonances do not overlap.

T 9 ≡ T in GK

-6 -4 -2 -

Jargonized synonym of data.

normal distribution truncated to only positive values

Using the relation E c.m. r = E x -Q

skewness = (e σ

+ 2) × (e σ 2 -1

which are believed to have formed from pristine matter.

https://nvlpubs.nist.gov/nistpubs/jres/73b/jresv73bn1p1_a1b.pdf

Si(p,γ)

P dans laquelle s'inscrit cette thèse. et les approches expérimentales pour le mesurer sont présentées.

Remerciements

J π = 11/2 -. The angular distribution for this state can be described by either a ℓ = 0 or ℓ ≥ 3 transferred orbital momentum with similar statistical likelihoods. However, the very weak population of this state may suggest a bad matching condition for the transfer reaction, which would favour large transferred angular momenta.

In addition, this resonance has not been reported in the low-energy study of the 30 Si(p,γ) 31 P reaction [START_REF] Kuperus | Energy measurements of proton resonances in light nuclei[END_REF] although two ℓ = 1 resonances were observed in its close vicinity at E c.m. r = 484. [START_REF] Duncan A Forbes | Globular cluster formation and evolution in the context of cosmological galaxy assembly: open questions[END_REF] keV and E c.m. r = 601.5 keV. This could suggest that the contribution of the present resonance is suppressed by the Coulomb and centrifugal barriers, which would indicate a rather high spin. We therefore consider for this state ℓ = 3, and this must be considered as maximizing its contribution to the reaction rate. The corresponding resonance strength is ωγ = 1.67×10 -4 eV.

• E c.m. r = 601.3 ± 0.7 keV: This resonance is considered as the reference one for the 30 Si(p,γ) 31 P reaction rate. It corresponds to an excitation energy of E x = 7898.8 keV. It has been observed in several transfer reactions [START_REF] Vernotte | Si( 3 He,d) 31 P reaction at 25[END_REF][START_REF] Al-Jadir | States of high E x in 31 P[END_REF][START_REF] Uzureau | Spectroscopic study of 31p and 32s by the (d, n) reaction at ed = 7 mev[END_REF], and through γ-ray studies of 31 P decays [START_REF] De Neijs | Levels of 31 P from proton capture in 30 Si[END_REF][START_REF] Kuperus | Energy measurements of proton resonances in light nuclei[END_REF][START_REF] Shikazono | Study of odd-a nuclei in the 2s-1d shell by means of the (γ, γ ′ )reaction[END_REF]. The direct 30 Si(p,γ) 31 P measurements established a J π = 1/2 -spin-parity assignment and total width of Γ = 68 ± 9 eV [START_REF] Kuperus | Energy measurements of proton resonances in light nuclei[END_REF]. This state has been well populated at all angles in the 30 Si( 3 He,d) 31 P experimental work and the analysis of its angular distribution is well described by a ℓ = 1 transferred angular momentum, in agreement with the accepted spin-parity assignment.

This resonance was also measured with the DRAGON experiment, with extensive statistics, as shown in chapter 4. The resonance strength that was adopted from these measurements is ωγ = 1.79 ± 0.18 eV, this value is in agreement with the one adopted by Dermigny et al. [START_REF] Dermigny | Thermonuclear reaction rate of 30 Si(p ,γ )[END_REF] ωγ = 1.95 ± 0.10 eV which is obtained from a weighted average of the results from Paine and Sargood [START_REF] Paine | p,γ) resonance strengths in the s-d shell[END_REF] and that of Lyons et al. [START_REF] Lyons | Total yield measurement in 27 Al(p,γ) 28 Si[END_REF]. Noteworthy, the strength obtained using the proton width through the equation ωγ ≈ 0.5(2J +1)Γ p gives a very different value (6.5 eV) indicating that for resonances with energy E r ≥ 600 keV, the approximation Γ ≈ Γ γ is no longer justified.

The strengths of the resonances at higher energies that have been determined in previous works relatively to the strength of the E c.m. r = 601 keV resonance are renormalized to our present value ωγ = 1.79 ± 0.18 eV.

• E c.m. r = 614 ± 0.8 keV: This resonance corresponds to the 31 P excited level at E x = 7911.5 keV that has been populated in the 30 Si( 3 He,d) 31 P experiment. DWBA analysis presented in chapter 3 indicates an ℓ = 3 transferred angular momentum, in agreement with Vernotte et al. [START_REF] Vernotte | Si( 3 He,d) 31 P reaction at 25[END_REF]. The strength of this resonance has never been mea- difference may also come from the width of the density range, where the authors of previous works chose a finite density at ρ = 100 g/cm 3 . All in all, the same conclusions are derived from both results, and most importantly, these results are used to set a comparison point with the new 30 Si(p,γ) 31 P reaction rates, that are investigated in the following.

The network calculations are also performed with all rates fixed except for the 30 Si(p,γ) 31 P rate which is allowed to vary within its uncertainties. The results are shown as the green histogram in figure 5.9b where the red histogram is the same as before. The ratio of the broadening is 1.6, again slightly higher than the 1.4 obtained by Dermigny et al. [START_REF] Dermigny | Sensitivity to Thermonuclear Reaction Rates in Modeling the Abundance Anomalies of NGC 2419[END_REF]. In these conditions, we find as pointed out by Dermigny et al., that the 30 Si(p,γ) 31 P reaction rate and its associated uncertainties are predominantly responsible for the broadening of the temperature locus.

b/ Impact of the present 30 Si(p,γ) 31

P rates

We distinguish two sets of rates for the 30 Si(p,γ) 31 P reaction: either considering ℓ = 2 for the E c.m. r = 149 keV resonance, or an ℓ = 3 orbital angular momentum. We will present each case separately, then compare them.

We repeat the same calculation as in the previous subsection, but we replace the STARLIB 30 Si(p,γ) 31 P reaction rate by the one determined in the present work. are reproduced, with densities 80 g.cm -3 ≤ ρ ≤ 120 g.cm -3 , red histograms are obtained with fix rates, green ones correspond to the results when varying the 30 Si(p,γ) 31 P rate only, while all rates are varied within their uncertainties for the blue one.

Appendix

A Two body kinematics

A Two body kinematics

The excitation energy of a nucleus formed through a given reaction can be linked to the kinematic properties of the second particle that have been formed in the said reaction.

We first consider the reaction of the type:

where A is the target nucleus (and has no velocity in the laboratory frame), a is the beam nucleus, b is the particle that can be detected while B is the residual nucleus which spectroscopic properties are being studied.

In the following, the total energy of each particle is noted E, while its kinetic energy is T , its mass is m and its momentum is ⃗ p, we have:

A reaction channel is said to be available if the reaction Q-value is positive, the latter s defined as:

The total energy available E tot is defined as:

(5.21)

The momentum-energy P of each nucleus can be written as:

The momentum conservation

where θ b is the angle at which the particle b is detected, and is defined with respect to the beam axis. Furthermore, the energy conservation

Finally the excitation energy of nucleus B, E x is given as:

(5.25)

Experimentally speaking, T a is the beam energy, and p b is measured using the magnetic rigidity, which is defined as:

where q is the charge of the particle b.

A.1 Excitation energy uncertainty

The uncertainty on the excitation energy determined through the magnetic rigidity measurement is defined as:

we consider here that the masses of the interacting nuclei are known with great precision. By combining all the given relations, one obtain:

and

A.2 Transformation from the laboratory frame to the centreof-mass frame

We consider here non-relativistic formulae, since the velocities of the involved particles are sufficiently small compared to c. The centre-of-mass of the system before the collision has a velocity:

the velocities of particles a and A expressed in the centre-of-mass frame are:

(5.31) thus, the corresponding kinetic energies are:

(5.32)

The available kinetic energy in the entrance channel is then:

After the collision, conservation of the total momentum in the centre-of-mass frame gives:

and the total kinetic energy in the exit channel is:

B Integral of the exponentially modified gaussian

The exponentially modified gaussian used in the Q3D analysis to fit the peaks from the reaction of interest is:

Where:

• x 0 is the mean of the peak,

• ν is the exponential decay coefficient,

• δ is the width of the peak, Considering the following variable changes:

the differential element becomes:

The integral of the function (5.44) is then:

While the (a) part is constant with respect to u, the integral (b) is of the type :

This integral is given in "A table of Integrals of the Error functions" by E. W. Ng and M. Geller 1968 3 . (3) is found in paragraph 4.2 , 2nd integral:

Equation (5.51) is then :

If the covariance matrix is given, the variance of the integral is:

C Statistical distributions and their properties

We will summarize here the expressions for different statistical distributions used in the present thesis, along with their properties.

A parameter x can be distributed according to the probability density function P (x). The associated expected value, or mean E[x] , and the variance V [x], are given by:

where P (x) is normalized as

The cumulative distribution is a useful function, and it is defined as:

using this function, one can define x q , the quantile of order q, as F (x q ) = q. The most frequently used quantile is the median x 1/2 . The 68% probability coverage can be defined between the quantiles of order q = 16 and q = 84, that shall be called lower and upper bounds, respectively.

C.1 Gaussian distribution

The Gaussian, also called the normal distribution, is defined as:

where µ and σ are the mean and the standard deviation, respectively, and are related to the expectation value and variance as:

Normal distribution is defined between -∞ and +∞, it is symmetrical and thus its mean and median values coincide. the lower and upper bounds are given by µσ and µ + σ, respectively.

C.2 Log-Normal distribution

If a continuous random variable y is Normally distributed, and it is related to the random variable x such as y = ln(x), then the variable x is said to be log-normally distributed, such as:

where µ and σ are the mean and standard deviation for the gaussian distribution of ln x, the corresponding parameters for the log-normal distribution are:

The median value of the log-normal distribution is e µ , while the upper and lower bounds are given by e µ-σ and e µ+σ . For the log-normal distribution, it is convenient to define the uncertainty factor f.u., defined as: f.u. = e σ . The log-normal distribution is asymmetric and defined for 0x + ∞, which is convenient for describing physical quantities that take only positive values.

C.3 Porter-Thomas distribution

This distribution is also called the chi-squared distribution, it corresponds to the sum of the squares of k normally distributed variables, and k is called the number of degree of freedom.

For k = 1, the distribution is given as:

(5.60) the distribution has no adjustable parameters and is defined for 0x+∞. More generally, the mean and variance are given by: