Improved Nuclear Predictions of Relevance to the R-Process of Nucleosynthesis

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Improved Nuclear Predictions of Relevance to

the R-Process of Nucleosynthesis

Mathieu Samyn

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Improved nuclear predictions

of relevance to the r-process of nucleosynthesis

Amélioration des prédictions nucléaires

nécessaires aux calculs de nucléosynthèse du processus r

Th` ese de doctorat

présentée en vue de l’obtention du diplôme de

Docteur en sciences de l’Universit´e Libre de Bruxelles

par

Mathieu Samyn

le 22 janvier 2004

et r´ ealis´ ee sous la direction de

M. Marcel Arnould Directeur de th`ese

M. St´ephane Goriely Co-directeur de th`ese

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Remerciements

Je souhaiterais remercier en premier lieu mes parents, Gilles et France, pour leur constant soutien, leur enthousiasme, leur intérêt, le respect qu’ils ont toujours su exprimer dans la plupart de mes activités, et particulièrement au cours de cette thèse de doctorat. Je leur dédie ce travail.

Je voudrais ensuite exprimer ma profonde et sincère gratitude envers Marcel Arnould, professeur et directeur de l’Institut d’Astronomie et d’Astrophysique (IAA) qui, en grande partie grâce à lui, est un endroit vivant, ouvert, multi-disciplinaire et accueillant nombres de personnes de qualités. Après m’avoir guidé, avec Stéphane, dans mon travail de fin détudes, il m’a offert la possibilité de commencer cette thèse. Il n’a ensuite cessé de m’encourager et de me soutenir sous de multiples facettes pour que je mène à bien ce travail.

Mais c’est grâce à la persévérance de Stéphane que ce travail a vraiment pris forme. Il n’a pas seulement été un (co)promotteur de thèse disponible, enthousiaste et infatiguable:

c’est un collège de travail obstiné, exigeant mais compréhensif, et dont les qualités sont utilisées à bon escient. Une grande partie des calculs liés aux tables de masses présentées dans ce travail a été réalisée en collaboration avec lui, et n’auraient jamais abouti sans lui.

Il reste trois personnes que j’aimerais remercier tout particuli`erement: Mike Pearson, Paul-Henri Heenen et Michael Bender.

Riche de sa longue expérience, Mike fut d’une aide précieuse pour analyser, ”assaison- ner”, critiquer et commenter nombres des résultats présentés ici. Je n’oublierai pas mon séjour à Montréal et les quelques bons moments passés en sa compagnie.

Je suis reconnaissant envers Paul-Henri pour ses nombreux conseils et explications, ainsi que pour son aide dans la programmation de la résolution des équations Hartree- Fock-Bogoliubov, de Lipkin-Nogami et dans les conseils sous-jacents aux méthodes de projection, notamment pour la projection de la fonction d’onde sur la bonne parité lorsque celle-ci est brisée.

L’aide de Michael Bender fut également précieuse. Je lui suis reconnaissant pour les nombreuses discussions que nous avons eues à Maubuisson, à Bruxelles ou à Trento, et ses conseils dans la programmation de l’énergie du centre de masse et de la projection sur le nombre exact de particules. Ses compétences et son expérience nous ont souvent

éclairés sur des difficultés liées à la théorie de champ moyen.

Je souhaiterais également remercier Elias Khan et Shinya Wanajo. La collaboration scientifique avec Elias m’a permis de découvrir l’Institut de Physique Nucléaire d’Orsay et ses membres, dont Elias, doué et toujours aimable, ainsi que d’ajouter à notre code HFB son code QRPA pour l’étude des résonances géantes (dipolaires) et de l’influence de nos recherches sur celles-ci. Shinya m’a donné l’opportunité d’étudier l’influence des améliorations de deux formules de masses sur la nucléosynthèse du processus r.

Enfin, à Fran¸cois Tondeur, je dois la version originale des “codes sphérique et déformé”

Hartree-Fock plus BCS, c’est à dire le point de départ de ma thèse.

Au-delà de ce “cercle restreint” de personnes qui ont joué un rôle déterminant dans la

poursuite des mes recherches, il y a les nombreuses personnes de l’IAA, que j’aimerais citer

en guise de remerciement pour les aides techniques, discussions, fous rires, . . . J’ai d’abord

partag´e un bureau avec Maya (je n’oublierai pas le duo clarinette-flˆ ute du repas de No¨el),

Stéphania et Marie, dont la générosité fut très appréciée (thé, mandarines, chocolats,

. . . ). J’ai ensuite partag´e mon bureau avec d’autres occupants: Gabriel, Laurent, Viviane

(qui s’est attaquée au problème du calcul des taux de fission spontannée), et Gwaënaelle.

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informatique fut souvent appréciée, ainsi que celle de Marc, l’administrateur système en charge des clusters d’ordinateurs et dont l’expérience sur les problèmes de fission fut aussi très utile, Ivan, avec qui malheuresement je n’ai pas eu le temps de travailler sur les désintégrations β de noyaux déformés (. . . ), Sophie, Alain, Fernando, Lionel, Claire, Ana, Sylvie, Benoˆıt, Abdallah, Abdel, autant de personnes dont j’ai toujours apprécié la compagnie.

Pour conclure par le commencement, c’est Katharine qui, un dimanche soir en compag- nie de tous nos compagnons de Polytech et comprenant mes hésitations sur la direction à prendre, m’a clairement déconseillé de m’orienter vers le privé . . . et c’est un ami de bien, Tenzin Nyima, qui m’a encouragé et soutenu pour que je mène à bien cette thèse.

Avant de tourner cette page reconnaissante à l’histoire, je remercie mon épouse, Soyang, pour les joies et le soutien qu’elle m’a apportés, et notre fille, Kalzang, pour ses sourires et ses pleurs qui forcent l’apprentissage de la vie.

Je remercie ´egalement le Fonds National de la Recherche Scientifique pour son soutien

financier indispensable.

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Form is empty; emptiness is form.

Emptiness is no other than form;

form is no other than emptiness.

Buddha

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The rapid neutron-capture process, known as the r-process, is responsible for the origin of about half the stable nuclei heavier than iron observed in nature. Though the r-process is believed to take place in explosive stellar environments and to involve a large number (few thousands) of exotic nuclei, this nucleosynthesis process remains poorly understood from the astrophysics as well as nuclear physics points of view. On the nuclear physics side, the nuclei are too exotic to be studied in the laboratory, even though great efforts are constantly made to extend the experimental limits away from the β − stability re- gion. Therefore, theoretical models are indispensable to estimate the nuclear properties of interest in the r-process nucleosynthesis modelling. So far, models used to predict the properties of the exotic nuclei were based on parametrized macroscopic-type approaches the reliability of which is questionable when extrapolating far away from the experimen- tally known region.

This work is devoted to the improvement of nuclear predictions, such as the nuclear ground- and excited-state properties, needed as input data to model the r-process. In order to give the predictions a reliable character, we rely on the microscopic mean-field Hartree-Fock theory based on the Skyrme-type interaction. Pairing correlations play an important role in the description of nuclei, and become essential for nuclei located near the drip lines, since the scattering of pairs of quasi-particles into the continuum increases significantly. In this work, we brought to the Hartree-Fock model the self-consistent treatment of the pairing correlations within the Hartree-Fock-Bogoliubov (HFB) theory.

Further improvements are made in the restoration of symmetries broken by correlations added in the form of additional degrees of freedom in the wave function. These include the translational invariance restored by calculating the recoil energy, the particle-number symmetry by an exact projection after variation, the rotational symmetry by an approx- imate cranking correction and the parity symmetry for reflection asymmetric shapes. In addition, the renormalization of the HFB equations has been studied as well and al- lows to eliminate the dependence of the total energy with respect to the cutoff energy.

The effective nucleon-nucleon interaction is determined by adjusting its parameters on all available experimental masses, with some constraints derived from fundamental nuclear matter properties. A systematic study of the influence on mass predictions for each of the above cited improvements as well as of some uncertainties affecting the particle-hole and particle-particle interactions has been conducted. In spite of quite important differ- ences in the input physics, we find a great stability in the mass predictions for exotic neutron-rich nuclei, though local mass differences can be significant.

Each of the Skyrme force derived in the present work has been tested on the predictions of basic ground-state properties (including charge radii, quadrupole moments, single- particle levels), fission barriers and electric dipole γ − ray strengths. The HFB predictions globally reproduce experimental data with a level of accuracy comparable with the widely- used droplet-like models. The microscopic character of the approach followed in the present work makes however the predictions for exotic neutron-rich nuclei involved in the r-process more reliable.

The influence of such improved nuclear mass predictions on the r-process abundance

distribution is studied in the specific scenario of the prompt supernova explosion mecha-

nism.

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Le processus de capture rapide de neutrons, appelé processus r, est responsable de la production de la moitié des éléments stables plus lourds que le Fer observés dans la nature.

Bien qu’il soit généralement admis que le processus r se déroule dans des environnements stellaires explosifs et est associé à la production d’un grand nombre (quelques milliers) de nucléides exotiques, celui-ci demeure mal compris du point de vue astrophysique ainsi que du point de vue de la physique nucléaire. En ce qui concerne la physique nucléaire, les noyaux sont trop exotiques pour être étudiés dans le laboratoire, même si d’importants efforts sont constamment consentis pour étendre les limites expérimentales loin de la région de stabilité nucléaire. Ainsi, les modèles théoriques sont un outil indispensable pour estimer toutes les propriétés nucléaires nécessaires à la modélisation du processus r.

Jusqu’à récemment, les modèles utilisés pour prédire les propriétés des noyaux exotiques

étaient basés sur des approches paramétrisées de type “goutte liquide” dont la fiabilité des extrapolations loin de la région connue expérimentalement peut être mise en doute.

Ce travail est dédié à l’amélioration des prédictions nucléaires, telles que les propriétés de l’état fondamental et des états excités des noyaux, utilisées comme bases de données pour modéliser le processus r. Afin de conférer aux prédictions la meilleure fiabilité possi- ble, nous utilisons la théorie microscopique de champ moyen Hartree-Fock (HF) basée sur l’interaction effective de Skyrme. Les corrélations d’appariement jouent un rôle impor- tant dans la description des noyaux et deviennent essentielles dans les noyaux situés près des zones limites d’émission spontanée de nucléons. Dans ce travail, nous avons ajouté au modèle HF le traitement des corrélations d’appariement de manière cohérente dans le cadre de la théorie Hartree-Fock-Bogoliubov (HFB). Des améliorations supplémentaires sont également apportées et concernent essentiellement le rétablissement de symétries brisées par les corrélations ajoutées à la fonction d’onde sous forme de degré de lib- erté supplémentaire. Celles-ci incluent l’invariance par translation restaurée par le calcul de l’énergie du centre de masse, la symétrie liée au bon nombre de particules par une projection exacte après variation, l’invariance par rotation par une correction approchée de type cranking et la symétrie de parité lorsque la symétrie gauche-droite est brisée.

En outre, la renormalisation des équations HFB, qui permet d’éliminer la dépendance de l’énergie totale par rapport à l’énergie de cutoff, a été étudiée. L’interaction effective nucléon-nucléon est déterminée en ajustant ses paramètres sur toutes les masses nucléaires disponibles, en tenant compte de certaines contraintes liées aux propriétés de la matière nucléaire. Une étude systématique de l’influence sur les prédictions de masses de cha- cune des améliorations citées ci-dessus ainsi que de certaines incertitudes affectant les interactions dans les voies particule-trou et particule-particule a été menée. Bien que d’importantes modifications aient été apportées au modèle physique, nous trouvons une grande stabilité des prédictions de masses. Cependant, des différences de masse peuvent être significatives localement.

Chacune des forces de Skyrme déterminées dans ce travail a été testée sur les prédictions des propriétés de l’état fondamental des noyaux (comprenant les rayons de charge, les moments quadrupolaires, les niveaux à une particule), ainsi que sur les barrières de fission et les distributions de force de l’émission dipolaire électrique de photons. Les prédictions HFB sont en accord global avec les données expérimentales avec un niveau de précision comparable à celui des modèles de type goutte liquide largement utilisés. Le caractère microscopique de l’approche suivie dans ce travail confère cependant un degré de fiabilité plus grand aux prédictions des propriétés des noyaux exotiques riches en neutron.

L’influence des prédictions de masses nucléaires ainsi améliorées est étudiée sur la distri-

bution d’abondance du processus r de nucléosynthèse calculée dans le scénario particulier

de l’explosion prompte d’une ´etoile en supernova.

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Introduction

The understanding of nucleosynthesis processes, taking mainly place during a particular stage of the life of a star, is an essential part of nuclear astrophysics. The rapid neutron capture process (the r-process), rapid with respect to β − decays, is known to be responsible for about half of nuclei heavier than iron, but remains a mystery for two (fundamental) reasons.

On the one hand, the astrophysical site(s) of this process is (are) still unknown: the deepest layers ejected during the explosion of massive stars in type II supernovae and the rapidly expanding matter escaping from the collision of two neutron stars or one neutron star and a black hole are potential sites that are under study, as they seem to present the adequate conditions for the development of the r-process. In particular, some hydrodynamical models of type II supernova explosion suggest that the matter of the region of the star located at the basis of the ejecta of a supernova could be subject to a strong neutron irradiation after α particles have recombined in nuclei of atomic mass A ≤ 110. This neutrino-driven wind revives the shock wave stalled in the stellar core due to iron photodesintegrations. In stars with M ≈ 10M , the prompt explosion could be successful, i.e. essentially driven by the shock wave. However, to date, none of the realistic hydrodynamical models is able to predict an explosion, and therefore to provide the conditions to drive a successful r-process in these environments.

On the other hand, nuclei synthesized during the r-process are extremely neutron rich, most of them inaccessible to experiments. The only way to investigate the r-process of nucleosynthesis is to use theoretical models to predict, for thousands of unknown nuclei, all the nuclear properties of astrophysical interest, among others the ground-state properties, neutron and neutrino capture cross sections, β − decay rates, fission probabilities, . . . . Models used to calculate such properties require a detailed description of the nuclear structure in its ground state.

When estimating these different nuclear inputs, two major features of the nuclear theory must be considered, namely its microscopic and universal aspect. A microscopic descrip- tion by a physically sound model based on first principles ensures a reliable extrapolation away from experimentally known region. On the other hand, a universal description of all nuclear properties within one unique framework for all nuclei involved ensures a coherent prediction of all unknown data.

Different classes of nuclear models can be contemplated according to their reliability,

i.e. their ability to accurately predict properties that have not been included in the ad-

justment of the parameters of the model, starting from local macroscopic approaches up

to global microscopic approaches. We find in between these two extremes, approaches like

the classical (e.g liquid drop, droplet), semi-classical (e.g Thomas-Fermi), macroscopic-

microscopic (e.g classical with microscopic corrections), semi-microscopic (e.g microscopic

with phenomenological corrections) and fully microscopic (e.g mean field, shell model,

QRPA) approaches. In a very schematic way, the higher the degree of reliability, the

less accurate the model reproduces the bulk set of experimental data. The classical or

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phenomenological approaches are highly parametrized and therefore often successful in reproducing experimental data, or at least much more accurate than microscopic calcu- lations. The low accuracy usually obtained with microscopic models mainly originates from computational complications making the determination of free parameters by fits to experimental data time-consuming. These difficulties become gradually less critical with the rapid increase of computer facilities.

With the concern to improve the reliability of extrapolations of nuclear properties of interest for the r-process nucleosynthesis, and to describe all of these within one unique framework, we have brought, in this thesis, some important developments to a microscopic model. Moreover, this model is shown to compete with more phenomenological highly- parametrized models in the reproduction of many experimental data, such as static nuclear properties, fission barriers and giant dipole resonances.

The present study is organized as follows.

Chapter 1 is an introduction to the r-process of nucleosynthesis problem. After pre- senting the abundance distribution of nuclei produced by the r-process, we mention the various astrophysical sites that are believed to give adequate hydrodynamical conditions for the r-process to take place, and describe briefly the prompt explosion mechanism of a 10 solar mass star at the end of its evolution, that will be referred to in Chap. 8 to study the impact of our new nuclear mass data on nucleosynthesis. We then mention the various nuclear reactions to include in the nuclear reaction network, and highlight the various nuclear properties of importance to calculate the different reaction rates. We end this first chapter by emphasizing the importance of using microscopic models to reliably predict unknown nuclear properties.

Chapter 2 covers our present knowledge of experimental observables used as a refer- ence or a validity test for microscopic nuclear properties calculations, mentioning briefly the techniques to measure them. These are essentially the nuclear masses (and mass dif- ferences), single-particle energies, the charge radii, the quadrupole moments, the electric dipole γ − ray strengths and the fission barriers.

Chapter 3 introduces in detail our microscopic framework, namely the Skyrme-Hartree- Fock approach, and presents, together with Apps. B & C, the developments that we have brought to that model: the generalization of the BCS approximation by solving the Hartree-Fock-Bogoliubov equations (Sect. 3.3.5), the restoration of broken translational (App. C.2), particle number (App. C.1) and parity (App. C.4) symmetries by projection techniques; the restoration of the rotational invariance was part of the existing framework, but since we have slightly modified the prescription, it is discussed as well (App. C.3). We have also studied the recently suggested renormalization scheme of the HFB equations (Sect. 3.3.11). In order to consistently describe the giant dipole resonance of impor- tance for neutron capture cross section calculations, we have coupled our spherical HFB calculation with the Quasi-Random Phase Approximation (QRPA) developed in Orsay [KSGN02], that we briefly present. Finally we end Chap. 3 by mentioning the method defined in [Rei00] for calculating the mass parameters.

In Chap. 4, we give our procedure to adjust the phenomenological effective interaction on nuclear masses and infinite nuclear matter properties, and present a series of new sets of Skyrme force parameters as well as the corresponding nuclear mass tables, using various prescriptions for the pairing interaction, the isoscalar effective mass and for variances in the mean-field framework as described in Chap. 3 and App. C. This chapter is ended by an overview of all Skyrme forces adjusted.

The quality of the various mass formulas to predict ground-state properties other than

the binding energy is depicted in Chap. 5, analyzing successively the agreement of theo-

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INTRODUCTION 3

retical predictions with experimental single-particle spectra, radii, quadrupole moments, neutron separation energies and β − decay Q-values, as well as shell and pairing gaps, all being defined in Chap. 2.

The ability of the newly-derived Skyrme forces with their underlying model to predict fission barriers is studied in detail in Chap. 6. We first present our method for calculating and analyzing the three-dimensional potential energy surface (PES) and apply it for a particular case. We then critically analyze the influence, on reflection symmetric fission barriers, of both the HF framework and the Skyrme interaction (in both the particle-hole (ph) and the particle-particle (pp) channels). We calculate with one Skyrme force and compare with experimental data the reflection asymmetric outer fission barrier of all nuclei known to present a left-right asymmetric outer saddle-point, and study the sensitivity of potential energy surface properties with the rotational correction prescription. We finally study the barrier height of some super-heavy elements (and its critical dependence on the pp channel of the interaction) as well as very neutron-rich nuclei. We briefly go beyond the static path calculation and present one result of fission half-life.

The electric dipole γ − ray strength is estimated in Chap. 7 within the HFB+QRPA framework on the basis of the newly-derived Skyrme forces. The location of the giant dipole resonance (GDR) is compared with experimental data. One of the Skyrme force leading to an accurate estimate of the experimental GDR position and width is used to calculate the electric dipole γ − ray strength and radiative neutron-capture rates for all nuclei of relevance for the r-process nucleosynthesis.

The impact of nuclear masses on r-process nucleosynthesis calculations is studied in Chap. 8 in the specific prompt explosion scenario. For this, the predictions based on two of our HFB masses are compared with the FRDM results.

Our summary, conclusions and outlook are finally given.

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Chapter 1 R-process: the problem

1.1 Introduction

1.1.1 Universe and Solar System nuclidic composition

Our knowledge of the composition of the Universe, and in particular, of the solar system, is the result of

• the analysis of electromagnetic radiation of different sources: our galaxy (non- exploding and exploding stars of all kinds, and the interstellar medium (ISM)), and external galaxies.

• the chemical analysis of accessible matter, composed mainly of solar-system con- stituents (earth, moon, meteorites), a small fraction being in the form of (extra- )galactic cosmic rays; solar and non-solar neutrinos bring essential information as well.

More information on the many observational data can be found in [AT99] and in the references therein.

1.1.2 Solar System: elemental and isotopic composition

The analyses of a special class of rare meteorites, the CI1 carbonaceous chondrites ¹ , have allowed the determination of the elemental abundance distribution of the solar system at the time of its formation some 4.6 Gy ago, displayed in the left panel of Fig. 1.1. Solar spectroscopy is also well suited to provide information on volatile element abundances like H, He, C, N, O and Ne. For some elements (Ar, Kr, Xe, Hg), the abundance determination rely entirely on interpolations based on nucleosynthetic considerations. It is possible to explain the present solar system elemental composition if secondary physico-chemical and geological processes that have been taking place since the formation of the solar system are taken into account. For example, H and He have evaporated from the earth atmosphere, like the O from Mars; the analysis of the solar spectrum indicates the presence of all elements (the Sun is a star of population I ² ), but the elemental abundances of rare gases and Mercury differ from the one observed in meteorites. Generally speaking, the

1 Meteorites made of a carbon matrix and chondrules, inclusions of refractory materials of various origins, considered as the least-altered samples of primitive solar matter available [AG89]

2 Relatively rich in heavy elements, population I stars are concentrated towards the disk of the Galaxy,

and their motion is dominated by the common rotation of the Galaxy.

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Figure 1.1: Bulk elemental (left panel) and nuclear (right panel) compositions of the galactic material from which the solar system formed about 4.6 Gy ago [AG89]. The abundances are normalized to 10 ⁶ Si atoms. The elemental abundances are largely based on the analysis of meteorites of the CI1 carbonaceous chondrite type. The nuclear composition is derived from those elemental abundances with the use of the terrestrial isotopic composition of the elements, except for H and the noble gases. For a given A, the most abundant isobar is shown by a filled circle, and the others, if any, by open circles

elemental composition varies from one constituent to another. This is not the case of the isotopic (nuclear) composition which is relatively constant throughout the different constituents, i.e. the isotopic ratios for a given element (with the exception of H and the noble gases) are the same in the solar system and are not altered by the physico-chemical and geological processes. For this reason, terrestrial materials have been adopted as the primary standard for the isotopic composition characteristic of the primitive solar nebula. The nuclear abundance distribution deduced from the elemental abundances and the terrestrial isotopic compositions is shown in the right panel of Fig. 1.1.

From the two figures 1.1, it appears that

• H and He are the most abundant elements,

• the abundance of Li, Be and B is much lower than their neighbors,

• Characteristic peaks are superposed to a uniform decrease with the atomic Z or mass A number: in particular, in the right panel, peaks are located at a multiple of A = 4, until the Fe peak at A = 56 is reached; a broad peak is located in the 80 < A < 90 region, and double peaks are observed at A = 130, 138 and A = 195, 208.

For practical reasons and to bring to the fore the link between observations and nu- cleosynthesis models that are aimed to explain the isotopic compositions, nuclei with A > ∼ 60 are separated in three categories, according to their location in the (N, Z ) plane.

We distinguish the s-nuclei, located at the bottom of the β − stability valley (’s’ for ’slow’

neutron-capture process, neutron-capture rates being small compared to β − decay rates),

the r-nuclei of the neutron-rich side of the bottom of the valley (’r’ for ’rapid’ neutron-

capture process, neutron-capture rates being high compared to β − decay rates), and the

p-nuclei populating the proton-rich side of the same valley (’p’ for ’protons’ enrichment

process, including (γ, n) reactions, a combination of (p, γ) and β − decays, or a combination

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1.1. INTRODUCTION 7

63 64 65 66 68 69 70 71 72 73 74

Cd In Sn Sb

67 62

12.5% 12.8% 24.1% 12.2% 28.7% 7.5%

95.7%

14.7% 7.7% 24.3% 4.6%

42.7%

57.3%

129d

71.9s

110 111 112 113 114 116

115 113

112 114 115 116 117 118 119 120 122 124

121 123

4.3% 14s

0.4%

0.7%

115d 1.0%

2.7d

27h 32.4%

8.6% 5.6%

s−process path

r−process p

p

r

r r

r

53.4h p

N Z

p s

s sr sr sr sr

sr sr sr

sr

Figure 1.2: The s-process path through the Cd, In, Sn and Sb isotopes. The r-process takes place in the neutron-rich part of the nuclide chart, via rapid neutron captures, and the β − decay cascades are represented by large arrows. Thick dashed lines stand for the production of pure r-nuclei. Some stable nuclei are synthesized by both the s- and the r-processes, and are usually labelled sr. The relative abundance is given for all stable isotopes; the experimental β − decay half-life is indicated for unstable isotopes. Adapted from [Cla68]

of (γ, n), (γ, p) or (γ, α) and β − decays; cf. [AG03]). The s- and r-processes are illustrated in Fig. 1.2 for a particular area of the (N,Z) nuclide chart, around the tin region. The s-process path (thick line) lies at the bottom of the β − stability valley. The r-process path is not shown since it enters deeply the neutron-rich side away from the β − stability valley. Instead, the β − decay cascades occurring at the end of the r-process (cf. § 1.2.2) are represented by double lines. As seen in Fig. 1.2, five of the stable Sn isotopes are produced by both the s- and r-processes; these isotopes are therefore labelled ’sr’. Dotted lines are β − decay cascade originating from the r-process but producing r-only nuclei: if an unstable isotope lies between two stable neutron-rich isotopes, the s-process cannot access the heaviest stable isotope(s) of that element. These heavy isotopes can therefore be produced by the r-process only. The Cd element has one such isotope, Sn two and Sb one. Also indicated are the p-nuclei, resulting from the p-process. Although some nuclear and astrophysics uncertainties still affect the s-process modelling, parametrized models can explain the abundance distribution of the s-only nuclei with a relative high accuracy (e.g. [KBW89, Gor99]) and can be used to separate the s- and r- process contributions to the abundance of the sr-nuclei. The abundance distributions of s-, r- and p-nuclei are shown in Fig. 1.3 for A > ∼ 70. The origin of the various peaks in the right panel of Fig. 1.1 appears therefore to be associated with particular nucleosynthesis processes. The peaks located at A = 138 ( ¹³⁸ ₅₆ Ba 82 ) and A = 208 ( ²⁰⁸ ₈₂ Pb 126 ) are associated with the s-process, while the r-process is responsible for the A = 130 ( ¹³⁰ ₅₂ Te 78 ) and A = 195 ( ¹⁹⁵ ₇₈ Pt 117 ) peaks.

They are a signature of a specific nuclear property, the nuclear shell structure leading to an increase of the binding for specific “magic” nuclei with respect to their neighbours.

The s-nuclei abundance peaks are located at the N = 82 and N = 126 magic numbers,

since the s-process flow does not deviate from the bottom of the β − stability valley, while

the position of the r-abundance peaks, shifted to lower N with respect to these magic

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10 ^-6 10 ^-5 10 ^-4 10 ^-3 10 ^-2 10 ^-1 10 ⁰ 10 ¹ 10 ²

80 100 120 140 160 180 200

A b u n d a n c e s [ S i= 1 0 6 ]

A

s s

r

p

Figure 1.3: s-, r- and p- nuclear abundance distributions

numbers, will be understood in § 1.2.2 within a schematic view of the r-process. The p-nuclei are much less abundant (of a factor 10 to 1000); for a review on the p-process, see [AG03].

The nuclear composition of the solar-system is not due to a unique event, but to a large number of nucleosynthetic events that have taken place along the life of the Galaxy in very different astrophysical sites. In particular, the matter synthesized by a star is, sooner or later, ejected in the interstellar medium that serve as a base material for future generations of stars.

The main objective of the theory of nucleosynthesis is to explain, by various processes, the solar-system nuclear composition and to identify the astrophysical sites that provide the necessary conditions for these processes to occur. The nuclear reactions invoked are thermonuclear and non-thermonuclear (spallation) reactions. The former class of reactions plays a major role in the Big-Bang primordial nucleosynthesis and during stellar evolution.

The latter one is connected with the reactions taking place in diluted cold environments:

the ISM (that interacts with the galactic cosmic rays), and circumstellar environments (that interact with the energetic stellar particles). The Big-Bang nucleosynthesis produced the major part of H, D, and ⁴ He, and a part of ³ He and ⁷ Li. The other nuclei present in the Universe have mainly been (are) produced by thermonuclear processes during the evolution of a star (non-explosive and explosive stellar nucleosynthesis, [AT99, AS01]).

Generally speaking, charged-particle (protons, α-particles, and heavier elements like ¹² C

and ¹⁶ O) and neutron-induced reactions, as well as photodisintegrations take place in

stellar plasmas. Charge-induced reactions mainly deal with nuclei of mass number A ≤

60 − 70, the limit being fixed by the Coulomb barrier increasing with the nuclear charge

and consequently increasing the timescales for the reactions to take place; they produce

the energy compensating the energy losses at the surface of the star, therefore preventing

the star to contract, and modify the composition of the star. Neutron captures can involve

heavier nuclei but usually do not contribute to the energy production.

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1.2. ASTROPHYSICAL ASPECTS OF THE R-PROCESS 9

Among the three nucleosynthesis processes that lead to nuclei heavier than the iron peak, the r-process, occurring in explosive stellar events and generating thousands of neutron-rich nuclei, is by far the most complex and remains poorly understood from the astrophysics as well as nuclear physics point of views (see Sect. 1.2). The present work is devoted to improve the nuclear physics aspects of the r-process.

1.2 Astrophysical aspects of the r-process

1.2.1 Astrophysical sites

The astrophysical sites of the r-process are still unknown. Various sites have been sug- gested as candidates able to provide the adequate conditions to produce r-nuclei. However, none fulfills all the requirements. The r process requires (cf. e.g [BBFH57]) high neutron densities (N n ≥ 10 ²⁰ cm ⁻³ ), high temperatures (T ≥ 10 ⁹ K) and small neutron-irradiation timescales (τ ≈ 1 s). While the need for a large neutron number per seed nucleus is easy to understand if nuclei as heavy as Pb or the actinides are to be synthesized, the high temperature and the small irradiation timescale are essential to understand the shape (i.e.

the position and width of the abundance peaks) of the r-nuclei abundance distribution shown in Fig. 1.3.

The explosion of massive stars (M ≥ 12M , where M is the solar mass) in type II supernovae has since long been proposed as a promising site. In these explosions, two types of scenarii are favoured:

• the first class of models considers the explosive nucleosynthesis in various outer shells of the massive star: the shock wave, produced by the collapse of the star, generate the explosion of the He- or C-rich shells, in which the high-temperature increase could lead to the production of neutrons by the reactions ²² Ne(α, n) ²⁵ Mg and ²⁵ Mg(α, n) ²⁸ Si. Important difficulties remain; it appears notably that in re- alistic astrophysical conditions, this process does not release enough neutrons per seed nuclei to produce elements heavier than A ' 90. The ¹³ C(α, n) ¹⁶ O reaction has also been suggested to be a potential neutron source, but conclusions stay un- changed, and the existence of this reaction itself in realistic environments has not been demonstrated by stellar models.

• The second class of models considers the nucleosynthesis in the region located be- tween the forming supernova remnant (a neutron star or a black hole) and the base of the ejecta of the supernova; this region is commonly called the “neutrino-driven wind”, because the shock wave, stalled in the outer part of the iron core due to the Fe photodisintegration, is revived by the strong neutrino winds. One and two dimensional simulations do however not confirm the explosion, the heating by neu- trinos winds being too weak [JBK ⁺ 03, BRJK03]. Even if the explosion is forced, the conditions do not lead to a successful r-process.

The explosion of stars with M ≈ 8 − 12M has been less studied. At the end of

their evolution, these stars may either die in a the form a C-O or O-Ne white dwarf or

collapse before ending in a “prompt supernova explosion” [HNW84, Nom87, STM ⁺ 01,

WTI ⁺ 03]. The evolution, the collapse and the explosion of a M ≈ 9M , together with

the thermodynamical conditions for the r-process to occur in the ejected material are

discussed in Sect. 1.2.3. We however point out that no realistic calculation leads to a

significant explosion, hence to a r-process.

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“Neutron star mergers” [RJ98, FRT99, RJ01] also show some promise, possibly leading to an r-process during the ejection and the decompression of very neutron-rich matter.

The more exotic site of a neutron star–black hole coalescence is also under study [RJ01].

1.2.2 The canonical model

To understand the various nuclear mechanisms taking place during the r-process, as well as to explain the r-abundance distribution in the solar system, a simple site-independent model can be invoked. In this so-called canonical model [GA96], the temperature and the neutron density are constant during the neutron irradiation time τ . The neutron concentration and the temperature are assumed to be high enough for the (n,γ) and (γ,n) reactions to be always faster than the β − decays, until the neutron flux is switched off at t = τ . It starts from the seed nucleus ⁵⁶ Fe formed earlier. When nuclei approach the neutron-drip line ³ the binding energy of a neutron decreases strongly. The (n,γ) reactions slow down and the (γ,n) reactions speed up. If the temperature and the neutron density are high enough, a (n,γ)-(γ,n) equilibrium takes place for each isotopic chain ⁴ before any β − decay occurs. This process is represented in Fig. 1.4. Based on the nuclear Saha

Production neutron rich

of very nuclei

slow

neutron capture

fast

slow fast

neutron emission β− desintegration Z+1

Z

N−2 N+1 N N+1 N+2

Figure 1.4: Schematic representation of the (n,γ)-(γ,n) equilibrium in an isotopic chain, and of the waiting-point nucleus (N, Z)

equation, the abundances can be derived as log N (Z, A + 1)

N(Z, A) = log G(Z, A + 1)

G(Z, A) − 34.075 + log N n (1.1)

− 3

2 log( A

A + 1 T 9 ) + 5.04 T 9

S n (Z, A + 1)

where N (Z, A) and N _n are, respectively, the number densities [cm ⁻³ ] of the nucleus (Z, A) and of the neutrons, T 9 = T /(10 ⁹ K ) is the temperature, S n [MeV] is the neutron separa- tion energy, and G(Z, A) is the partition function of the (Z, A) isotope, defined as

G(Z, A) = 1 2J ₀ + 1

X

i

(2J i + 1) exp( − E i /kT ) (1.2)

where J i and E i are respectively the spin and the energy of the i ^th excited level (0 stands for the ground state), and k the Boltzmann constant. Inside the isotopic chain, only the few nuclei characterized by a neutron separation energy

S n (Z, A) ' S _a ⁰ [M eV ] = (34.075 − log N n + 3

2 log T 9 ) T 9

5.04 (1.3)

3 The drip line is defined as the limit where nuclei become unstable with respect to particle emission, i.e. the nucleon separation energy vanishes S q (Z, A) = 0, q = n, p

4 The limits of validity of the hypothesis of an (n,γ)-(γ,n) equilibrium (i.e. the waiting-point approxi-

mation) in the (T 9 , N n ) plane is studied in detail in [GA96]

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1.2. ASTROPHYSICAL ASPECTS OF THE R-PROCESS 11

are produced significantly. This equation therefore defines the r-process path, i.e. the set of nuclei through which the nuclear flow travels, the transition rate between each (n,γ)-(γ,n) equilibrium (i.e. element) being dictated by the β − decay rates: if N (Z ) = P

A N (Z, A) is the abundance of an isotopic chain, and if λ ^Z _β = P

A λ ^Z,A _β N (Z, A)/N (Z ) is the effective β − decay rate from one isotopic chain to the other, we can write

dN (Z)

dt = N (Z − 1)λ ^Z−1 _β − N (Z)λ ^Z _β . (1.4)

Explanation of the peaks in the abundance distribution The radiative neutron capture by a neutron closed-shell nucleus (a neutron magic nucleus) leads to a particularly rapid neutron emission through photodisintegration, due to the pronounced decrease of the neutron separation energy right across the filling of the shell ⁵ . Therefore, when the r-process path encounters magic nuclei with N =50, 82, or 126, the nuclear flow stops and waits for the β − decays to act. The daughter nucleus, after having captured a neutron, becomes in turn magic in N . Through this process, the path deviates along the isotone N =50, 82, or 126, until the (γ, n) reactions become weak enough and S n high enough for the path to proceed towards heavier isotopes. In other words, nuclei accumulate around the shell closure, in an “accumulation point”, since β − decay rates decrease when heavier isotones are produced.

When neutrons are no more available (t ≥ τ), all unstable nuclei transform into stable nuclei through β − decays. The matter gathered at the accumulation points generates the peaks.

1.2.3 A 9 M star model

The canonical model is very useful, in particular to understand the abundance peak distribution, and the importance and impact of nuclear predictions on the r-abundance distribution. It can also be used to study the sensitivity of the r-process to any change in the modelling of the nuclear properties (cf. § 1.3) or of the astrophysical input. However, it does not refer to any known stellar environment. We have therefore chosen to present the result of the modelling of one of the possible r-process sites mentioned in § 1.2.1, that provides (after some artificial tuning) the thermodynamical conditions for a successful r-process nucleosynthesis (Chap. 8).

The purpose of this paragraph is to briefly present the evolution of a 9 M star for which some models predict a collapse followed by a possible explosion [HNW84, Nom87, WTI ⁺ 03].

Evolution The hydrostatic evolution of a star is made of an alternate succession of quasi-static gravitational contractions and much slower thermonuclear burning stages.

The gravitational contraction increases the temperature in the central region of the star, leading to the ignition of the thermonuclear reactions that stabilize the structure and bring the star into a thermal equilibrium ⁶ . This burning stage ends when the nuclear fuel is no more available in the central region of the star, and is followed by a contraction:

the stellar core heats up until the ashes of the previous nuclear activity start burning. A new nuclear stage takes place. Fig. 1.5 gives the chemical evolution of the helium core

5 This shell structure may actually disappear when the drip line is approached (cf. Chap. 5), but this consideration does not change the general conclusions drawn here

6 An equilibrium is said to be thermal when the amount of heat received by a mass element equals the

amount of heat emitted by this mass element

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of initial mass 2.2 M (corresponding to a stellar mass of approximately 9–11 M ), from the core helium burning to the formation of a O+Ne+Mg core [Nom87]. The H burning stage as well as the first quasi-static gravitational contraction that ignite He burning have been neglected in this calculation that starts from an “He star”.

Figure 1.5: Chemical evolution of the helium core of initial mass 2.2 M from core helium burning to the formation of O-Ne-Mg core. Convective regions due to helium and carbon burning are curled, and the surface convection zone is shaded. From [Nom87].

Collapse The late evolution of a 9 M star is briefly summarized, according to the cal- culations of [HNW84, Nom87, WTI ⁺ 03]. When the O-Ne-Mg core mass reaches 1.375M , electron captures on ²⁴ Mg, ²⁴ Na, ²⁰ Ne, and ²⁰ F take place ⁷ . The electron concentration Y e 8 decreases (so does the electron pressure), which induces the rapid contraction of the core on the timescale of electron captures. The rapid contraction ignites Oxygen (ρ c > 10 ¹⁰ g/cm ³ ) and a deflagration front heats up the material into a nuclear statistical equilibrium (NSE), the matter composition of which is given by the nuclear Saha equa- tion. The collapse of the core is accelerated owing to the rapid electron capture onto NSE matter. The oxygen deflagration front advances in mass to increase the mass of the NSE core. At this stage, the central density is ρ c = 4.4 10 ¹⁰ g/cm ³ , the temperature 1.3 10 ¹⁰ K, the mass of the core in NSE M _c ^{N SE} = 0.1M , and the central electron concentration rather low Y e = 0.37, due to electron captures.

This late evolution can conveniently be described as a track in the central density log(ρ _c ) and temperature log(T _c ) diagram, as shown in Fig. 1.6.

Bounce The last values of ρ c , T c , Y e and M c given above are the initial (t = 0) ther- modynamical conditions of the pre-supernova model of [WTI ⁺ 03]. What follows in the numerical calculations of [WTI ⁺ 03] is a core bounce after ≈ 90 ms; at this time, the central density is ρ c = 2.2 10 ¹⁴ g/cm ³ (the nuclear density being about 2.6 10 ¹⁴ g/cm ³ ),

7 For lower mass stars, these electron captures will not occur; instead, the star will continue to cool down and will evolve into a White Dwarf

8 Y e is defined by the relation Y e = ⁽ⁿ ^e− _n ⁻ⁿ ^e ⁺ ⁾

b = ¹

1+ ^Nn _Np , where n _e − , n _e ⁺ and n b represent the number

densities of electrons, positrons and baryons, respectively; N n and N p are the total neutron and proton

densities (free and bound), n b = N n + N p and N n /N p is the net neutron to proton ratio. For Y e < 0, 5,

the neutron excess η = 1 − 2Y e is positive.

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1.2. ASTROPHYSICAL ASPECTS OF THE R-PROCESS 13

Figure 1.6: Evolution of the O-Ne-Mg core in the central density and temperature diagram. From [Nom87].

the temperature 2.1 10 ¹⁰ K, the mass of the core in NSE M _c ^{N SE} = 1.0M , and the central electron concentration Y e = 0.34. The equation of state (EOS) plays an important role in the evolution of the collapse, the core bounce and the explosion mechanism, but will not be discussed here. It is found in [WTI ⁺ 03] that under such conditions, only a very weak explosion takes place: not more than 0.008M is ejected and the explosion energy is 2 10 ⁴⁹ ergs. These results disagree with the calculations of [HNW84] where a very energetic supernova explosion with a total energy of about 2 10 ⁵¹ ergs was obtained. The lowest Y e

value in the outgoing ejecta is 0.45 (lack of free neutrons). No r-process ensues due to a too low entropy.

Since we are mainly interested in analyzing the consequence of the nuclear physics inputs like the nuclear masses on r-process abundances, it is necessary to influence artifi- cially the explosion. It is shown in [WTI ⁺ 03] that to end the explosion with an important production of r-nuclei, one must scale the shock-heating term ⁹ by a factor of 1.6, i.e.

largely above the usual values prescribed in hydrodynamics. We note that the agreement of r-nuclei synthesis with the solar abundances is not a necessary condition to be ful- filled by the astrophysical site, since the solar abundances may arise from different events.

However, the observed composition of r-nuclei in very metal poor stars [SLI ⁺ 03] resembles the solar distribution, which suggests that one supernova event could be able to provide already the main part of the solar distribution.

The r-process will be significantly strong in mass trajectories characterized by small electron concentrations Y e ≈ 0.14 − 0.18. The mass-trajectories time evolution used in the present r-process nucleosynthesis calculation is described in Fig. 1.7. The radius, the temperature and the density of mass elements from 90 ms before the bounce until 0.91 s after the bounce is shown as a function of time. We have indicated by thick lines the limits of the region characterized by 0.14 ≤ Y e ≤ 0.45 used in the r-process calculations (Chap. 8).

9 In order to avoid the excessively high resolution needed to see and follow a shock wave in hydrody-

namical simulations, a “shock heating” term is introduced; it consists in the “capture” of a shock wave

in an hydrodynamical resolution scheme based on the grid method, by adding a parametrized artificial

viscosity in the flow equations.

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2 3 4 5

9 10 11

0 0.5 1

5 10

t (s)

Figure 1.7: Time variations of radius (upper panel), temperature (central panel), and density (lower panel) for selected mass trajectories (with roughly an equal mass interval) in the energetic prompt explo- sion of a 9 M star, in which the shock-heating energy is enhanced artificially by a factor of 1.6. From [WTI ⁺ 03]. Dotted and dashed trajectories are discussed in Chap. 8

1.2.4 Nucleosynthesis in the supernova explosion

Behind the shock wave, a region is expanding rapidly. We explain hereafter the variety of nuclear processes occurring in this environment.

α-process

The temperature of the matter after the core bounce is such that matter is in NSE, as

explained above. The medium is mainly composed of free nucleons, helium (α − particles)

and electrons. Neutrons are overabundant, due to the core neutronization during the

collapse (electron captures by protons and nuclei). When the temperature decreases, the

NSE favours the formation of heavier nuclei by the recombination of α particles (with

each other) and by free nucleon captures. The reactions that lead to heavy nuclei are

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1.2. ASTROPHYSICAL ASPECTS OF THE R-PROCESS 15

mainly the following three-body reactions:

α (αα, γ) ¹² C , (1.5)

α (αα, p) ¹¹ B (α, p) ¹⁴ C , (1.6)

α (αn, γ) ⁹ Be(α, n) ¹² C and (1.7)

α (αd, p) ⁹ Be . (1.8)

The lower the density (the higher the entropy), the weaker these reactions. The α − and n − capture rates on newly formed heavy nuclei are high: from the very beginning of α − recombinations, nuclei of relatively high mass (A ≈ 100, Z ≈ 40) can be created, starting from the reactions

12 C(n, γ) ¹³ C(α, n) ¹⁶ O(α, γ) ²⁰ N e . . . (1.9)

14 C(α, n) ¹⁷ O(α, n) ²⁰ N e . . . (1.10)

For 60 ≤ A ≤ 100 nuclei, photodisintegrations are fast enough to hinder further α − particle capture. When the temperature decreases, photodisintegrations as well as α-captures slow down. As a consequence, no heavier nuclei will be synthesized. The (n,p) and (n,α) reac- tions also contribute to this nucleosynthesis limitation. For these reasons, the so-defined α − process does not lead to nuclei of mass number A > 100. When the temperature reaches T 9 = 4, the α − process freezes, i.e. the rates of the previously defined reactions become negligible with respect to the other rates and to the expansion timescale of the environment. The α − process typically develops in an environment characterized by tem- peratures of the order of T 9 ' 7 − 4, and densities ρ ' 10 ⁶ − 10 ⁵ g/cm ³ ).

r-process

The expanding medium above the nascent neutron star is, at the beginning, neutron-rich.

When the α − process freezes out, the neutron captures by heavy nuclei can contribute to the realization of an r-process, at lower temperatures. It actually depends on the strength of the α − process: If it is important, it leads to a large number of heavy nuclei, and the number of neutrons per seed nucleus available is reduced ¹⁰ . In fact, the importance of the α − particle recombinations in heavier nuclei is regulated by the relation existing between the three-body reactions ααα and ααn, and the expansion and cooling timescale. For a high enough entropy and/or a rapid expansion, only few α − particles are recombined, so that the number of seed nucleus of the r-process is low and the number of neutrons available per seed nucleus high enough to lead to a successful r-process, that will be described in detail in Chap. 8.

10 If < A ^f _h > is the average mass number of seed nuclei at the end of the α − process and if (Y n /Y h ) ^f is

the number of neutrons available per seed nucleus, the maximum mass number reached at the end of the

r-process is < A ^f _h > +(Y n /Y h ) ^f

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1.3 Nuclear aspects of the r-process

Another difficulty that hinders the understanding of the synthesis of r-nuclei is due to the uncertainties in the theoretical predictions of nuclear data far from the β-stability region, for which essentially no experimental data exist. In particular, mass predictions for neutron-rich nuclei play a key role since they affect all the nuclear quantities of relevance in the r-process, namely the neutron capture, photodisintegration and β-decay rates, as well as the fission probabilities (cf. § 1.3.5).

The next paragraph briefly comments on the various nuclear reactions that enter the r-process calculation. A short overview of the different models that are (have been) used to calculate nuclear masses, radiative neutron-capture cross sections, β − decay rates and fission properties for the r-process application will be successively given to better introduce the next chapters.

1.3.1 Nuclear reaction network

The canonical model of the r process studied in subsect. 1.2.2, as already mentioned, does not refer to any realistic environment: it cannot be used in the context of the nucle- osynthesis in the neutrino-driven wind or the prompt explosion, where the temperature and the neutron density are not constant and where the initial composition is essentially made of free nucleons and leptons in NSE, and not of Fe. In addition, the assumption of the (n,γ)-(γ,n) equilibrium, that allows not to calculate the corresponding reaction rates, might not be valid in realistic environments especially in the neutron freeze-out phase.

To model the r-process, we have to know, for each nucleus, the rate of all the relevant reactions shown in Fig. 1.8.

p

(α, ) p

β ⁺ β ⁻

N Z

Fission (sf,(n,f),( ,f)) _β ( ,γ) p

(α, ) n

( ,γ) n (γ, ) n

(γ, ) ( ,α)

p

( ,α) n

β2 β1

β3

(α,γ)

(n,p) n n n (p,n)

(γ,α)

Figure 1.8: The various reactions a nucleus can undergo during a nucleosynthesis process, represented

in the (N, Z) chart

Improved Nuclear Predictions of Relevance to the R-Process of Nucleosynthesis

Improved Nuclear Predictions of Relevance to

the R-Process of Nucleosynthesis

Mathieu Samyn

Improved nuclear predictions

of relevance to the r-process of nucleosynthesis

Amélioration des prédictions nucléaires

nécessaires aux calculs de nucléosynthèse du processus r

Th` ese de doctorat

présentée en vue de l’obtention du diplôme de

Docteur en sciences de l’Universit´e Libre de Bruxelles

par

Mathieu Samyn

le 22 janvier 2004

et r´ ealis´ ee sous la direction de

M. Marcel Arnould Directeur de th`ese

M. St´ephane Goriely Co-directeur de th`ese

Remerciements

Mais c’est grâce à la persévérance de Stéphane que ce travail a vraiment pris forme. Il n’a pas seulement été un (co)promotteur de thèse disponible, enthousiaste et infatiguable:

Il reste trois personnes que j’aimerais remercier tout particuli`erement: Mike Pearson, Paul-Henri Heenen et Michael Bender.

Riche de sa longue expérience, Mike fut d’une aide précieuse pour analyser, ”assaison- ner”, critiquer et commenter nombres des résultats présentés ici. Je n’oublierai pas mon séjour à Montréal et les quelques bons moments passés en sa compagnie.

éclairés sur des difficultés liées à la théorie de champ moyen.

Enfin, à Fran¸cois Tondeur, je dois la version originale des “codes sphérique et déformé”

Hartree-Fock plus BCS, c’est à dire le point de départ de ma thèse.

Au-delà de ce “cercle restreint” de personnes qui ont joué un rôle déterminant dans la

poursuite des mes recherches, il y a les nombreuses personnes de l’IAA, que j’aimerais citer

en guise de remerciement pour les aides techniques, discussions, fous rires, . . . J’ai d’abord

partag´e un bureau avec Maya (je n’oublierai pas le duo clarinette-flˆ ute du repas de No¨el),

Stéphania et Marie, dont la générosité fut très appréciée (thé, mandarines, chocolats,

. . . ). J’ai ensuite partag´e mon bureau avec d’autres occupants: Gabriel, Laurent, Viviane

(qui s’est attaquée au problème du calcul des taux de fission spontannée), et Gwaënaelle.

Avant de tourner cette page reconnaissante à l’histoire, je remercie mon épouse, Soyang, pour les joies et le soutien qu’elle m’a apportés, et notre fille, Kalzang, pour ses sourires et ses pleurs qui forcent l’apprentissage de la vie.

Je remercie ´egalement le Fonds National de la Recherche Scientifique pour son soutien

financier indispensable.

Form is empty; emptiness is form.

Emptiness is no other than form;

form is no other than emptiness.

Buddha

Contents

Abstract vii

R´ esum´ e viii

Introduction 1

1 R-process: the problem 5

1.1 Introduction . . . . 5

1.1.1 Universe and Solar System nuclidic composition . . . . 5

1.1.2 Solar System: elemental and isotopic composition . . . . 5

1.2 Astrophysical aspects of the r-process . . . . 9

1.2.1 Astrophysical sites . . . . 9

1.2.2 The canonical model . . . 10

Explanation of the peaks in the abundance distribution . . . 11

1.2.3 A 9 M star model . . . 11

Evolution . . . 11

Collapse . . . 12

Bounce . . . 12

1.2.4 Nucleosynthesis in the supernova explosion . . . 14

α-process . . . 14

r-process . . . 15

1.3 Nuclear aspects of the r-process . . . 16

1.3.1 Nuclear reaction network . . . 16

1.3.2 Nuclear masses . . . 18

1.3.3 Radiative neutron-capture cross sections . . . 20

1.3.4 β-decay rates . . . 21

1.3.5 Fission properties . . . 22

Fission barriers . . . 23

Spontaneous and neutron-induced fission . . . 23

Mass distribution of fission fragments . . . 23

1.3.6 Impact of neutrino-capture reactions . . . 24

1.4 Microscopic models and present studies . . . 24

2 Observables 27 2.1 Introduction . . . 27

2.2 The atomic mass . . . 27

2.2.1 Mass measurements . . . 27

Production of exotic nuclides . . . 27

Measurements . . . 28

Q-value measurements (reactions and decays) . . . 29

direct measurements . . . 29

2.2.2 Overview of data . . . 30

2.2.3 Finite-point mass-difference formulae . . . 30

Pairing gaps . . . 31

Shell gaps . . . 32

2.3 Single-particle energies . . . 33